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Maxwell Equations & Propagation in Anisotropic Media (39

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Solo Hermelin
Solo Hermelin

Describes the propagation of electromagnetic waves in anisotropic electrical media. Please send comments to solo.hermelin@gmail.com. For more presentations on different subjects visit my website at http://www.solohermelin.com. The presentation is not properly downloaded. Please go to my website and open it in the Optics Folder.

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1 Maxwell Equations & Propagation in Anisotropic Electric Media SOLO HERMELIN Updated 12.06.06 23.12.12 http://www.solohermelin.com
2 Maxwell Equations & Propagation in Anisotropic MediaSOLO TABLE OF CONTENT Maxwell’s Equations History of Maxwell’s Equations Symmetric Maxwell’s Equations Constitutive Relation Boundary Conditions Energy and Momentum Monochromatic Planar Wave Equations Planar Waves in an Source-less Anisotropic Electric Media Lorentz’s Lemma Planar Wave Group Velocity Phase Velocity, Energy (Ray) Velocity for Planar Waves Energy Flux and Poynting Vector Energy Flux and Poynting Vector for a Bianisotropic Medium Wave equation Wave-Vector Surface Double Refraction at a Uniaxial Crystal Boundary Refractive Index of Some Typical Crystals Polarization History
3 Maxwell Equations & Propagation in Anisotropic MediaSOLO TABLE OF CONTENT Planar Waves in an Source-less Anisotropic Electric Media Orthogonality Properties of the Eigenmodes Phase Velocity, Energy (Ray) Velocity for Planar Waves Ray-Velocity Surface Index Ellipsoid Conical Refraction on the Optical Axis Equation of Wave Normal References
4 POLARIZATION Erasmus Bartholinus,doctor of medicine and professor of mathematics at the University of Copenhagen, showed in 1669 that crystals of “Iceland spar” (which we now call calcite, CaCO3) produced two refracted rays from a single incident beam. One ray, the “ordinary ray”, followed Snell’s law, while the other, the “extraordinary ray”, was not always even in the plan of incidence. SOLO History Erasmus Bartholinus 1625-1698 http://www.polarization.com/history/history.html
5 POLARIZATIONSOLO History Étienne Louis Malus 1775-1812 Etienne Louis Malus, military engineer and captain in the army of Napoleon, published in 1809 the Malus Law of irradiance through a Linear polarizer: I(θ)=I(0) cos2 θ. In 1810 he won the French Academy Prize with the discovery that reflected and scattered light also possessed “sidedness” which he called “polarization”.
6 POLARIZATION Arago and Fresnel investigated the interference of polarized rays of light and found in 1816 that two rays polarized at right angles to each other never interface. SOLO History (continue) Dominique François Jean Arago 1786-1853 Augustin Jean Fresnel 1788-1827 Arago relayed to Thomas Young in London the results of the experiment he had performed with Fresnel. This stimulate Young to propose in 1817 that the oscillations in the optical wave where transverse, or perpendicular to the direction of propagation, and not longitudinal as every proponent of wave theory believed. Thomas Young 1773-1829
7 POLARIZATIONSOLO http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html Methods of Achieving Polarization Polarization is based on one of four fundamental physical mechanisms: 1. Dichroism or selective absorbtion 3. Reflection 2. Scattering 4. Birefrigerence To obtain Polarization we must have some asymmetry in the optical process.
8 POLARIZATIONSOLO http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html Methods of Achieving Polarization Polarization by Birefrigerence (continue – 1) Polarization can be achieved with crystalline materials which have a different index of refraction in different planes. Such materials are said to be birefringent or doubly refracting. Nicol Prism The Nicol Prism (1828) is made up from two prisms of calcite cemented with Canada balsam. The ordinary ray can be made to totally reflect off the prism boundary, leving only the extraordinary ray. William Nicol(1768 ?– 1851) Scottish physicist
9 POLARIZATIONSOLO http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html Methods of Achieving Polarization Polarization by Birefrigerence (continue – 2) Polarization can be achieved with crystalline materials which have a different index of refraction in different planes. Such materials are said to be birefringent or doubly refracting. Wollaston Prism William Hyde Wollaston 1766-1828
10 POLARIZATIONSOLO http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html Methods of Achieving Polarization Polarization by Birefrigerence (continue – 3) Polarization can be achieved with crystalline materials which have a different index of refraction in different planes. Such materials are said to be birefringent or doubly refracting. Glan-Foucault Polarizer
11 POLARIZATION SOLO Polarizations Prisms Overview http://www.unitedcrystals.com/POverview.html
12 POLARIZATION SOLO Polarizations Prisms Overview http://www.unitedcrystals.com/POverview.html Return to Table of Content
13 MAXWELL’s EQUATIONSSOLO Magnetic Field IntensityH  [ ]1− ⋅mA Electric DisplacementD  [ ]2− ⋅⋅ msA Electric Field IntensityE  [ ]1− ⋅mV Magnetic InductionB  [ ]2− ⋅⋅ msV Current DensityJ  [ ]2− ⋅mA Free Charge Distributionρ [ ]3− ⋅⋅ msA James Clerk Maxwell (1831-1879) A Dynamic Theory of Electromagnetic Field 1864 Treatise on Electricity and Magnetism 1874 Return to Table of Content
14 SOLO Electrostatics Charles-Augustin de Coulomb 1736 - 1806 In 1785 Coulomb presented his three reports on Electricity and Magnetism: -Premier Mémoire sur l’Electricité et le Magnétisme [2]. In this publication Coulomb describes “How to construct and use an electric balance (torsion balance) based on the property of the metal wires of having a reaction torsion force proportional to the torsion angle”. Coulomb also experimentally determined the law that explains how “two bodies electrified of the same kind of Electricity exert on each other”. -Sécond Mémoire sur l’Electricité et le Magnétisme [3]. In this publication Coulomb carries out the “determination according to which laws both the Magnetic and the Electric fluids act, either by repulsion or by attraction”. -Troisième Mémoire sur l’Electricité et le Magnétisme [4]. “On the quantity of Electricity that an isolated body loses in a certain time period , either by contact with less humid air, or in the supports more or less idio-electric”.
15 SOLO Electrostatics Charles-Augustin de Coulomb 1736 - 1806 1 2 12 123 0 12 1 4 q q F r rπ ε =   Coulomb’s Law q1 – electric charge located at 1r  q2 – electric charge located at 2r  12 1 2r r r= −    The electric force that the charge q2 exerts on q1 is given by: If the two electrical charges have the same sign the force is repulsive, if they have opposite signs is attractive. 12 0 8.854187817 10 /Farad mε − = ×Permittivity of vacuum Accord to the Third Newton Law of mechanics: 21 12F F= −   12 1 12F q E=   Define 2 12 123 0 12 1 4 q E r rπ ε =   where is the Electric Field Intensity [N/C]
16 SOLO During an evening lecture in April 1820, Ørsted discovered experimental evidence of the relationship between electricity and magnetism. While he was preparing an experiment for one of his classes, he discovered something that surprised him. In Oersted's time, scientists had tried to find some link between electricity and magnets, but had failed. It was believed that electricity and magnetism were not related. As Oersted was setting up his materials, he brought a compass close to a live electrical wire and the needle on the compass jumped and pointed to the wire. Oersted was surprised so he repeated the experiemnt several times. Each time the needle jumped toward the wire. This phenomenon had been first discovered by the Italian jurist Gian Domenico Romagnosi in 1802, but his announcement was ignored. 1820Electromagnetism Hans Christian Ørsted 1777- 1851
17 SOLO 1820Electromagnetism André-Marie Ampère 1775 - 1836 Danish physicist Hans Christian Ørsted's discovered in 1820 that a magnetic needle is deflected when the current in a nearby wire varies - a phenomenon establishing a relationship between electricity and magnetism. Ørsted's work was reported the Academy in Paris on 4 September 1820 by Arago and a week later Arago repeated Ørsted's experiment at an Academy meeting. Ampère demonstrated various magnetic / electrical effects to the Academy over the next weeks and he had discovered electrodynamical forces between linear wires before the end of September. He spoke on his law of addition of electrodynamical forces at the Academy on 6 November 1820 and on the symmetry principle in the following month. Ampère wrote up the work he had described to the Academy with remarkable speed and it was published in the Annales de Chimie et de Physique. Ampère and Arago investigate magnetism dl uu an infinitesimal element of the contour C J  curent density A/m2 dS uu a differential vector area of the surface S enclosed by contour C Ampère’s Law Magnetic Field IntensityH  [ ]1− ⋅mA
18 SOLO 1820 Electromagnetism Biot-Savart Law 0 2 1 4 rI dL dB r µ π × = uu u uu Magnetic Field of a current element Jean-Baptiste Biot 1774-1862 ( )( ) 1 1 2 21 1 1 21 1 2 1 20 1 2 3 1 2 4 c c c F I dl B dl dl r r I I r r µ π = × × × − = − ∫ ∫∫ uu  uu uu     Ampère was not the only one to react quickly to Arago's report of Orsted's experiment. Biot, with his assistant Savart, also quickly conducted experiments and reported to the Academy in October 1820. This led to the Biot-Savart Law. Félix Savart 1791 - 1841
19 SOLO 1820 Electromagnetism Biot-Savart Law Jean-Baptiste Biot 1774-1862 ( ) ( )2 30 0 ' ' 4 ' J r A J A r d r r r µ µ π ∇ = − ⇒ = −∫       0 0 H J B H B B A µ∇× = = ∇× = → = ∇×       ( ) ( ) 2 0B A A A Jµ∇× = ∇× ∇× = ∇ ∇× −∇ =     choose 0A∇× =  ( ) ( ) ( ) ( ) ( )3 30 0 3 ' ' ' ' ' 4 ' 4 ' r r J r J r r r B r A r d r d r r r r r µ µ π π   × − = ∇ × = ∇ × = ÷ ÷− −  ∫ ∫            Where we used ( ) ( ) 0 ' : 1/ 'r r rJ J r J r rφ φ φ φ∇ × = ∇ × + ∇ × = −       14243 Derivation of Biot-Savart Law from Ampère’s Law Poison’s Equation Solution for an unbounded volume Ampère Law Félix Savart 1791 - 1841
20 SOLO 1831Electromagnetism On 29th August 1831, using his "induction ring", Faraday made one of his greatest discoveries - electromagnetic induction: the "induction" or generation of electricity in a wire by means of the electromagnetic effect of a current in another wire. The induction ring was the first electric transformer. In a second series of experiments in September he discovered magneto-electric induction: the production of a steady electric current. To do this, Faraday attached two wires through a sliding contact to a copper disc. By rotating the disc between the poles of a horseshoe magnet he obtained a continuous direct current. This was the first generator. Michael Faraday 1791- 1867 Magnetic Field IntensityH  [ ]1− ⋅mA Electric DisplacementD  [ ]2− ⋅⋅ msA Electric Field IntensityE  [ ]1− ⋅mV Magnetic InductionB  [ ]2− ⋅⋅ msV →→ ⋅ ∂ ∂ −=⋅ ∫∫∫ dS t B dlE SC   t B E ∂ ∂ −=×∇  → dl → dS C B  E  The voltage induced in a coil moving through a non-uniform magnetic field was demonstrated by this apparatus. As the coil is removed from the field of the bar magnets, the coil circuit is broken and a spark is observed at the gap. The first transformer: Two coils wound on an iron toroid. http://www.ece.umd.edu/~taylor/frame1.htm
21 MAXWELL’s EQUATIONS 1. AMPÈRE’s CIRCUIT LW (A) (1820) SOLO 2. FARADAY’s INDUCTION LAW (F) (1831) Electric DisplacementD  [ ]2− ⋅⋅ msA Magnetic Field IntensityH  [ ]1− ⋅mA Current DensityJ  [ ]2− ⋅mA André-Marie Ampère 1775-1836 →→ ⋅ ∂ ∂ −=⋅ ∫∫∫ dS t B dlE SC   t B E ∂ ∂ −=×∇  → dl → dS C B  E  Electric Field IntensityE  [ ]1− ⋅mV Magnetic InductionB  [ ]2− ⋅⋅ msV Michael Faraday 1791-1867 The following four equations describe the Electromagnetic Field and where first given by Maxwell in 1864 (in a different notation) and are known as MAXWELL’s EQUATIONS
22 MAXWELL’s EQUATIONSSOLO 4. GAUSS’ LAW – MAGNETIC (GM) dV → dS V 0=⋅∫∫ → S dSB  B  0=⋅∇ B  Magnetic InductionB  [ ]2− ⋅⋅ msV 3. GAUSS’ LAW – ELECTRIC (GE) dV → dS V ∫∫∫∫∫ =⋅ → VS dVdSD ρ  D  ρ=⋅∇ D  ρ Electric DisplacementD  [ ]2− ⋅⋅ msA Free Charge Distributionρ [ ]3− ⋅⋅ msA GAUSS’ ELECTRIC (GE) & MAGNETIC (GM) LAWS developed by Gauss in 1835, but published in 1867. Karl Friederich Gauss 1777-1855 Return to Table of Content
23 James C. Maxwell (1831-1879) ELECTROMAGNETICSSOLO Magnetic Field IntensityH  [ ]1− ⋅mA Electric DisplacementD  [ ]2− ⋅⋅ msA Electric Field IntensityE  [ ]1− ⋅mV Magnetic InductionB  [ ]2− ⋅⋅ msV Electric Current DensityeJ  [ ]2− ⋅mA Free Electric Charge Distributioneρ [ ]3− ⋅⋅ msA Fictious Magnetic Current DensitymJ  [ ]2− ⋅mV Fictious Free Magnetic Charge Distributionmρ [ ]3− ⋅⋅ msV MAXWELL’S SYMMETRIC EQUATIONS FOR THE ELECTROMAGNETIC FIELD e S e C J t D HSd t D JdlH      + ∂ ∂ =×∇⇒•        ∂ ∂ +=• ∫∫∫ → )1( AMPÈRE’S LAW t B JESd t B JdlE m S m C ∂ ∂ −−=×∇⇒•        ∂ ∂ +−=• ∫∫∫ →     )2( FARADAY’S LAW e V e S DdvSdD ρρ =•∇⇒=• ∫∫∫∫∫  )3( GAUSS’ ELECTRIC LAW m V m S BdvSdB ρρ =•∇⇒=• ∫∫∫∫∫  )4( GAUSS’ MAGNETIC LAW
24 ELECTROMAGNETICSSOLO SYMMETRIC MAXWELL’s EQUATIONS Magnetic Field IntensityH  [ ]1− ⋅mA Electric DisplacementD  [ ]2− ⋅⋅ msA Electric Field IntensityE  [ ]1− ⋅mV Magnetic InductionB  [ ]2− ⋅⋅ msV Electric Current DensityeJ  [ ]2− ⋅mA Free Electric Charge Distributioneρ [ ]3− ⋅⋅ msA Fictious Magnetic Current DensitymJ  [ ]2− ⋅mV Fictious Free Magnetic Charge Distributionmρ [ ]3− ⋅⋅ msV 1. AMPÈRE’S CIRCUIT LW (A) eJ t D H    + ∂ ∂ =×∇ 2. FARADAY’S INDUCTION LAW (F) mJ t B E    − ∂ ∂ −=×∇ 3. GAUSS’ LAW – ELECTRIC (GE) eD ρ=⋅∇  4. GAUSS’ LAW – MAGNETIC (GM) mB ρ=⋅∇  Although magnetic sources are not physical they are often introduced as electrical equivalents to facilitate solutions of physical boundary-value problems. André-Marie Ampère 1775-1836 Michael Faraday 1791-1867 Karl Friederich Gauss 1777-1855
25 ELECTROMAGNETICSSOLO SYMMETRIC MAXWELL’s EQUATIONS (continue – 1) eJ t D H    + ∂ ∂ =×∇mJ t B E    − ∂ ∂ −=×∇ eD ρ=⋅∇  mB ρ=⋅∇  From the Symmetric Maxwell’s Equations we can see that we obtain the same equations by performing the following operations. DUALITY           ⇓           ⇓           − ⇓           ⇓             − ⇓             ⇓             − ⇓             ⇓             ⇓             − ⇓ µ ε ε µ ρ ρ ρ ρ e m m e e m m e J J J J E H H E B D D B             The Maxwell’s Equations are symmetric and dual. eJ t D H    + ∂ ∂ =×∇ mJ t B E    − ∂ ∂ −=×∇ mB ρ=⋅∇  eD ρ=⋅∇ 
26 ELECTROMAGNETICSSOLO SYMMETRIC MAXWELL’s EQUATIONS (continue – 2) From the Symmetric Maxwell’s Equations ( ) 00 = ∂ ∂ +⋅∇⇒⋅∇+⋅∇ ∂ ∂ =×∇⋅∇=⇒           =⋅∇ + ∂ ∂ =×∇ t JJD t H D J t D H e ee e e ρ ρ      ( ) 00 = ∂ ∂ +⋅∇⇒⋅∇−⋅∇ ∂ ∂ −=×∇⋅∇=⇒           =⋅∇ − ∂ ∂ −=×∇ t JJB t E B J t B E m mm m m ρ ρ      CONSERVATION OF ELECTRICAL AND MAGNETIC CHARGES 0= ∂ ∂ +⋅∇ t J e e ρ 0= ∂ ∂ +⋅∇ t J m m ρ Therefore Return to Table of Content
27 ELECTROMAGNETICSSOLO CONSTITUTIVE RELATIONS Homogeneous Medium – Medium properties do not vary from point to point and are the same for all points.ε µ Isotropic Medium – Medium properties are the same in all directions and are scalars.ε µ Linear Medium – The effects of all different fields can be added linearly (Superposition of different fields). For Linear and Isotropic Medium we have: ED  ε= HB  µ= where: 0εε eK= 0µµ mK= - Dielectric Constant (or Relative Permitivity)eK - Relative PermeabilitymK The simpler case:
28 ELECTROMAGNETICSSOLO CONSTITUTIVE RELATIONS (continue - 1) The most general form of Linear Constitutive Relations is: Classification of Media dyadicsxwhere H E B D 33,,, =                 =         µζξε µζ ξε        Classification according to the functional dependence of the 6x6 matrix         µζ ξε   1. Inhomogeneous: function of space coordinates 2. Nonstationary: function of time 3. Time-dispersive: function of time derivatives 4. Space-dispersive: function of space derivatives 5. Nonlinear: function of the electromagnetic field
29 ELECTROMAGNETICSSOLO CONSTITUTIVE RELATIONS (continue - 3) Classification of Media (continue - 2) In general 0,0,0,0  ≠==≠ µζξε Anisotropic Electromagnetic - medium described by both 0,0  ≠≠ µε Anisotropic Electric - medium described by 0  ≠ε Anisotropic Magnetic - medium described by 0  ≠µ If is symmetric, it can be diagonalized0  ≠ε           = z y x ε ε ε ε 00 00 00  Uniaxial zyx εεε ≠= (thetragonal, hexagonal, rombohedral crystals) Biaxial zyx εεε ≠≠ (orthohombic, monoclinic, triclinic crystals) Isotropic zyx εεε == This is called an Anisotropic Medium. If or is Hermitian; i.e.0  ≠ε 0  ≠µ ( ) Transposeconjugate conjugateTranspose a aj ja T z -T,-* ,-H UUUU *H ==           −= 00 0 0 α α Is gyroelectric or gyromagnetic depending on whether stands for or . If both tensors are of this form the medium is gyroelectromagnetic. ε  µ  U         µζ ξε  
30 ELECTROMAGNETICSSOLO CONSTITUTIVE RELATIONS (continue - 4) Classification of Media (continue - 3) In general 0,0,0,0  ≠≠≠≠ µζξε This is called an Bianisotropic Medium. Such properties have been observed in antiferromagnatic chromium oxide ( antiferromagnetic materials are ones in which it is enerically favorable for neighboring dipoles to take an antiparallel orientation; see Ramo, Whinery, and Van Duzer (1965), pg. 145)                 −− −− −= ** ** 4 µµζζ ξξεεω   j G mediumlosslessundefinite mediumpassivedefinitenegative mediumactivedefinitepositive ⇒ ⇒ ⇒ G G G In this case, we have the following classification (see development later) Return to Table of Content
31 SOLO ELECTROMAGNETICS BOUNDARY CONDITIONS Boundary Conditions ( ) ( ) ldtHtHhldtHldtHldH h C 2211 0 2211 ˆˆˆˆ ⋅+⋅=Θ+⋅+⋅=⋅ →→ ∫  where are unit vectors along C in region (1) and (2), respectively, and21 ˆ,ˆ tt 2121 ˆˆˆˆ −×=−= nbtt - a unit vector normal to the boundary between region (1) and (2)21 ˆ −n - a unit vector on the boundary and normal to the plane of curve Cbˆ Using we obtainbaccba  ⋅×≡×⋅ ( ) ( ) ( )[ ] ldbkldbHHnldnbHHldtHH e ˆˆˆˆˆˆ 21212121121 ⋅=⋅−×=×⋅−=⋅− −−  Since this must be true for any vector that lies on the boundary between regions (1) and (2) we must have: bˆ ( ) ekHHn  =−×− 2121 ˆ ∫∫∫ ⋅        ∂ ∂ +=⋅ → S e C Sd t D JdlH    ( ) dlbkbdlh t D JSd t D J e h e S e ˆˆ 0 ⋅=⋅        ∂ ∂ +=⋅        ∂ ∂ + → ∫∫      AMPÈRE’S LAW [ ]1 0 lim: − → ⋅        ∂ ∂ += mAh t D Jk e h e  
32 SOLO ELECTROMAGNETICS BOUNDARY CONDITIONS Boundary Conditions (continue – 1) ( ) ( ) ldtEtEhldtEldtEldE h C 2211 0 2211 ˆˆˆˆ ⋅+⋅=Θ+⋅+⋅=⋅ →→ ∫  where are unit vectors along C in region (1) and (2), respectively, and21 ˆ,ˆ tt 2121 ˆˆˆˆ −×=−= nbtt - a unit vector normal to the boundary between region (1) and (2)21 ˆ −n - a unit vector on the boundary and normal to the plane of curve Cbˆ Using we obtainbaccba  ⋅×≡×⋅ ( ) ( ) ( )[ ] ldbkldbEEnldnbEEldtEE m ˆˆˆˆˆˆ 21212121121 ⋅−=⋅−×=×⋅−=⋅− −−  Since this must be true for any vector that lies on the boundary between regions (1) and (2) we must have: bˆ ( ) mkEEn  −=−×− 2121 ˆ ∫∫∫ ⋅        ∂ ∂ +−=⋅ → S m C Sd t B JdlE    ( ) dlbkbdlh t B JSd t B J m h m S m ˆˆ 0 ⋅=⋅        ∂ ∂ +=⋅        ∂ ∂ + → ∫∫      FARADAY’S LAW [ ]1 0 lim: − → ⋅        ∂ ∂ += mVh t B Jk m h m  
33 SOLO ELECTROMAGNETICS BOUNDARY CONDITIONS Boundary Conditions (continue – 2) ( ) ( ) SdnDnDhSdnDSdnDSdD h S 2211 0 2211 ˆˆˆˆ ⋅+⋅=Θ+⋅+⋅=⋅ → ∫∫  where are unit vectors normal to boundary pointing in region (1) and (2), respectively, and 21 ˆ,ˆ nn 2121 ˆˆˆ −=−= nnn - a unit vector normal to the boundary between region (1) and (2)21 ˆ −n ( ) ( ) SdSdnDDSdnDD eσ=⋅−=⋅− −2121121 ˆˆ  Since this must be true for any dS on the boundary between regions (1) and (2) we must have: ( ) eDDn σ=−⋅− 2121 ˆ  ( ) dSdShdv e h e V e σρρ 0→ ==∫∫∫ GAUSS’ LAW - ELECTRIC [ ]1 0 lim: − → ⋅⋅= msAhe h e ρσ ∫∫∫∫∫ =• V e S dvSdD ρ 
34 SOLO ELECTROMAGNETICS BOUNDARY CONDITIONS Boundary Conditions (continue – 3) ( ) ( ) SdnBnBhSdnBSdnBSdB h S 2211 0 2211 ˆˆˆˆ ⋅+⋅=Θ+⋅+⋅=⋅ → ∫∫  where are unit vectors normal to boundary pointing in region (1) and (2), respectively, and 21 ˆ,ˆ nn 2121 ˆˆˆ −=−= nnn - a unit vector normal to the boundary between region (1) and (2)21 ˆ −n ( ) ( ) SdSdnBBSdnBB mσ=⋅−=⋅− −2121121 ˆˆ  Since this must be true for any dS on the boundary between regions (1) and (2) we must have: ( ) mBBn σ=−⋅− 2121 ˆ  ( ) dSdShdv m h m V m σρρ 0→ ==∫∫∫ GAUSS’ LAW – MAGNETIC [ ]1 0 lim: − → ⋅⋅= msVhm h m ρσ ∫∫∫∫∫ =• V m S dvSdB ρ 
35 SOLO ELECTROMAGNETICS BOUNDARY CONDITIONS Boundary Conditions (summary) ( ) mkEEn  −=−×− 2121 ˆ FARADAY’S LAW ( ) ekHHn  =−×− 2121 ˆ AMPÈRE’S LAW [ ]1 0 lim: − → ⋅        ∂ ∂ += mAh t D Jk e h e   [ ]1 0 lim: − → ⋅        ∂ ∂ += mVh t B Jk m h m   ( ) eDDn σ=−⋅− 2121 ˆ  GAUSS’ LAW ELECTRIC [ ]1 0 lim: − → ⋅⋅= msAhe h e ρσ ( ) mBBn σ=−⋅− 2121 ˆ  GAUSS’ LAW MAGNETIC [ ]1 0 lim: − → ⋅⋅= msVhm h m ρσ
36 SOLO ELECTROMAGNETICS BOUNDARY CONDITIONS Boundary Conditions for Perfect Electric Conductor (PEC) ( ) 0ˆ 2121  =−×− EEn FARADAY’S LAW ( ) ekHHn  =−×− 2121 ˆ AMPÈRE’S LAW ( ) eDDn σ=−⋅− 2121 ˆ  GAUSS’ LAW ELECTRIC ( ) 0ˆ 2121 =−⋅− BBn  GAUSS’ LAW MAGNETIC ekHn  =×− 121 ˆ 0ˆ 121  =×− En eDn σ=⋅− 121 ˆ  0ˆ 121 =⋅− Bn  02  =H 02  =B 02  =E 02  =D
37 SOLO ELECTROMAGNETICS BOUNDARY CONDITIONS Boundary Conditions for Perfect Magnetic Conductor (PMC) ( ) mkEEn  −=−×− 2121 ˆ FARADAY’S LAW ( ) 0ˆ 2121  =−×− HHn AMPÈRE’S LAW ( ) 0ˆ 2121 =−⋅− DDn  GAUSS’ LAW ELECTRIC ( ) mBBn σ=−⋅− 2121 ˆ  GAUSS’ LAW MAGNETIC 0ˆ 121  =×− Hn mkEn  −=×− 121 ˆ 0ˆ 121 =⋅− Dn  mBn σ=⋅− 121 ˆ  02  =H 02  =B 02  =E 02  =D Return to Table of Content
38 SOLO ELECTROMAGNETICS BOUNDARY CONDITIONS Energy and Momentum Let start from Ampère and Faraday Laws               ∂ ∂ −=×∇⋅ + ∂ ∂ =×∇⋅− t B EH J t D HE e      EJ t D E t B HHEEH e      ⋅− ∂ ∂ ⋅− ∂ ∂ ⋅−=×∇⋅−×∇⋅ ( )HEHEEH  ×⋅∇=×∇⋅−×∇⋅But Therefore we obtain ( ) EJ t D E t B HHE e      ⋅− ∂ ∂ ⋅− ∂ ∂ ⋅−=×⋅∇ First way This theorem was discovered by Poynting in 1884 and later in the same year by Heaviside.
39 SOLO ELECTROMAGNETICS BOUNDARY CONDITIONS Energy and Momentum (continue -1) We identify the following quantities -Power density of the current densityEJe  ⋅ ( )HEDE t BH t EJe  ×⋅∇−      ⋅ ∂ ∂ −      ⋅ ∂ ∂ −=⋅ 2 1 2 1       ⋅ ∂ ∂ =⋅= BH t pBHw mm  2 1 , 2 1       ⋅ ∂ ∂ =⋅= DE t pDEw ee  2 1 , 2 1 ( )HEpR  ×⋅∇= eJ  -Magnetic energy and power densities, respectively -Electric energy and power densities, respectively -Radiation power density For linear, isotropic electro-magnetic materials we can write( )HBED  00 , µε == ( )DE tt D E ED     ⋅ ∂ ∂ = ∂ ∂ ⋅ = 2 10ε ( )BH tt B H HB     ⋅ ∂ ∂ = ∂ ∂ ⋅ = 2 10µ
40 SOLO ELECTROMAGNETICS BOUNDARY CONDITIONS Energy and Momentum (continue – 3) Let start from the Lorentz Force Equation (1892) on the free charge ( )BvEF e  ×+= ρ Free Electric Chargeeρ [ ]3− ⋅⋅ msA Velocity of the chargev  [ ]1− ⋅sm Electric Field IntensityE  [ ]1− ⋅mV Magnetic InductionB  [ ]2− ⋅⋅ msV Hendrik Antoon Lorentz 1853-1928 eρ Force on the free chargeF  [ ]Neρ Second way
41 SOLO ELECTROMAGNETICS BOUNDARY CONDITIONS Energy and Momentum (continue – 4) The power density of the Lorentz Force the charge ( ) ( ) EJBvEvp e Bvv Jv e ee    ⋅=×+⋅= =×⋅ = 0 ρ ρ or ( ) ( ) ( ) ( ) ( )[ ] ( )HE t B HE t D E t D HEEH E t D HEJp t B E HEHEEH J t D H e e                 ×⋅∇−        ∂ ∂ ⋅+⋅ ∂ ∂ −= ⋅ ∂ ∂ −×⋅∇−×∇⋅= ⋅        ∂ ∂ −×∇=⋅= ∂ ∂ −=∇× ×⋅∇=∇×⋅−∇×⋅ + ∂ ∂ =∇× eρ
42 SOLO ELECTROMAGNETICS BOUNDARY CONDITIONS Energy and Momentum (continue – 5) ( )HEDE t BH t EJe  ×⋅∇−      ⋅ ∂ ∂ −      ⋅ ∂ ∂ −=⋅ 2 1 2 1 Let integrate this equation over a constant volume V ∫∫∫∫∫∫∫∫∫∫∫∫ ×∇−      ⋅−      ⋅−=⋅ VVVV e dvSdvDE td d dvBH td d dvEJ  2 1 2 1 If we have sources in V then instead of we must use E  source EE  + Use Ohm Law (1826) ( )source ee EEJ  += γ         = ∂ ∂ ∫∫∫∫∫∫ VV td d t Georg Simon Ohm 1789-1854 source e e EJE  −= γ 1 For linear, isotropic electro-magnetic materials ( )HBED  00 , µε ==
43 SOLO ELECTROMAGNETICS BOUNDARY CONDITIONS Energy and Momentum (continue – 6) ∫∫∫∫∫∫∫∫∫∫∫∫∫ ⋅∇+      ⋅+      ⋅+=⋅ VVVR n V source e dvSdvDE td d dvBH td d dRIdvEJ  2 1 2 12 ∫∫∫       ⋅= V FieldMagnetic dvBH td d P  2 1 ∫∫∫       ⋅= V FieldElectric dvDE td d P  2 1 ∫∫∫∫∫ ⋅=⋅∇= SV Radiation SdSdvSP  ∫∫∫ ⋅= V source eSource dvEJP  ( ) ( ) ∫∫∫∫ ∫∫∫∫∫∫∫∫∫∫∫∫∫∫∫ ⋅−= ⋅−⋅=⋅−⋅⋅=⋅ V source e R n V source e L S e ee V source e L S e ee V e dvEJdRI dvEJ dS dl dSJdSJdvEJldSdJJdvEJ   2 11 γγ ∫= R nJoule dRIP 2 RadiationFieldMagneticFieldElectricJouleSource PPPPP +++= For linear, isotropic electro-magnetic materials( )HBED  00 , µε == R – Electric Resistance Define the Umov-Poynting vector: [ ]2 / mwattHES  ×= The Umov-Poynting vector was discovered by Umov in 1873, and rediscovered by Poynting in 1884 and later in the same year by Heaviside.
44 ELECTROMAGNETICSSOLO EM People John Henry Poynting 1852-1914 Oliver Heaviside 1850-1925 Nikolay Umov 1846-1915 1873 “Theory of interaction on final distances and its exhibit to conclusion of electrostatic and electrodynamic laws” 1884 1884 Umov-Poynting vector HES  ×= Return to Table of Content
45 ELECTROMAGNETICSSOLO Monochromatic Planar Wave Equations Let assume that can be written as:( ) ( )trHtrE ,,,  ( ) ( ) ( ) ( ) ( ) ( )tjrHtrHtjrEtrE 00 exp,,exp, ωω  == where are phasor (complex) vectors. ( ) ( ){ } ( ){ } ( ) ( ){ } ( ){ }rHjrHrHrEjrErE  ImRe,ImRe +=+= We have ( ) ( ) ( ) ( ) ( )tjrEjtj t rEtrE t 00 expexp, ωωω  = ∂ ∂ = ∂ ∂ Hence ( ) ( ) ( )             =⋅∇ =⋅∇ −−=×∇ +=×∇ ⇒           =⋅∇ =⋅∇ − ∂ ∂ −=×∇ + ∂ ∂ =×∇ = ∂ ∂ m e m e j t m e m e B D JBjE JDjH BGM DGE J t B EF J t D HA ρ ρ ω ω ρ ρ ω         )(
46 ELECTROMAGNETICSSOLO Note The assumption that can be written as:( ) ( )trHtrE ,,,  ( ) ( ) ( ) ( ) ( ) ( )tjrHtrHtjrEtrE 00 exp,,exp, ωω  == is equivalent to saying that has a Fourier Transform; i.e.:( ) ( )trHtrE ,,,  ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∫∫ ∫∫ ∞ ∞− ∞ ∞− ∞ ∞− ∞ ∞− =−= =−= ωωω π ωω ωωω π ωω dtjrHtrHdttjtrHrH dtjrEtrEdttjtrErE exp, 2 1 ,&exp,, exp, 2 1 ,&exp,,   A Sufficient Condition for the Existence of the Fourier Transform is: ( ) ( ) ( ) ( ) ∞<= ∞<= ∫∫ ∫∫ ∞ ∞− ∞ ∞− ∞ ∞− ∞ ∞− ωω π ωω π drHdttrH drEdttrE 22 22 , 2 1 , , 2 1 ,   ( ) ( ) ( ) ( ) ( )[ ] ( ) ( ) ( )[ ] ( ) ( )ωωδωω ωωωω −=−= −=−= ∫ ∫∫ ∞ ∞− ∞ ∞− ∞ ∞− 00 0 exp expexpexp,, rEdttjrE dttjtjrEdttjtrErE   End Note
47 ELECTROMAGNETICSSOLO Fourier Transform The Fourier transform of can be written as:( ) ( )trHtrE ,,,  ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∫∫ ∫∫ ∞ ∞− ∞ ∞− ∞ ∞− ∞ ∞− −== −== dttjtrHrHdtjrHtrH dttjtrErEdtjrEtrE ωωωωω π ωωωωω π exp,,&exp, 2 1 , exp,,&exp, 2 1 ,   ( ) ( ) ( ) ( ) ∞<= ∞<= ∫∫ ∫∫ ∞ ∞− ∞ ∞− ∞ ∞− ∞ ∞− ωω π ωω π drHdttrH drEdttrE 22 22 , 2 1 , , 2 1 ,   JEAN FOURIER 1768-1830 A Sufficient Condition for the Existence of the Fourier Transform is:
48 ELECTROMAGNETICSSOLO ( ) ( )       ×∇−×∇−=×∇×∇ ×∇+×∇=×∇×∇ −−=×∇ +=×∇ ⇒     −−=×∇×∇ +=×∇×∇ = = m e m e ED HB m e JHjE JEjH JHjE JEjH JBjE JDjH ωµ ωε ωµ ωε ω ω ε µ ( ) me JJjEkE  ×∇−−=−×∇×∇ ωµ2 ( ) em JJjHkH  ×∇+−=−×∇×∇ ωε2 λ ππ µεω λ 22 f c c f k = ∆ === Using the vector identity ( ) ( ) ( ) AAA  ∇⋅∇−⋅∇∇=×∇×∇ For a Homogeneous, Linear and Isotropic Media:       =⋅∇ =⋅∇ ⇒    =⋅∇ =⋅∇ = = µ ρ ε ρ ρ ρ ε µ m e ED HB m e H E B D ε ρ ωµ e me JJjEkE ∇ +×∇+=+∇ 22 µ ρ ωε m em JJjHkH ∇ +×∇−=+∇ 22 and we obtain Monochromatic Planar Wave Equations (continue - 1)
49 ELECTROMAGNETICSSOLO Assume no sources: we have Monochromatic Planar Wave Equations (continue - 2) 0,0,0,0 ==== meme JJ ρρ  022 =+∇ EkE 022 =+∇ HkH nkk n k 0 00 00 0 =         == ∆   εµ µε εµωµεω ( ) ( ) ( ) ( ) ( ) ( )    == == ⋅− ⋅− rktjtj rktjtj eHerHtrH eEerEtrE     ωω ωω ω ω 0 0 ,, ,, ( ) 022 =+∇⇒ ⋅−=∇⋅∇⇒−=∇ ⋅− ⋅−⋅−⋅−⋅− rkj rkjrkjrkjrkj ek ekkeekje    Helmholtz Wave Equations satisfy the Helmholtz wave equations( ) ( )ωω ,,, rHrE  ( ) ( )    = = ⋅− ⋅− rkj rkj eHrH eErE     0 0 , , ω ω Assume a progressive wave of phase ( )rkt  ⋅−ω (a regressive wave has the phase) ( )rkt  ⋅+ω For a Homogeneous, Linear and Isotropic Media k  0E 0H r  t k  Planes for which constrkt =⋅−  ω
50 ELECTROMAGNETICSSOLO To satisfy the Maxwell equations for a source free media we must have: Monochromatic Planar Wave Equations (continue - 3) we haveUsing: 1ˆˆ&ˆˆ =⋅== sss c n sk ωεµω         =⋅∇ =⋅∇ −=×∇ =×∇ 0 0 H E HjE EjH ωµ ωε           =⋅ =⋅ =× −=× 0ˆ 0ˆ ˆ ˆ 0 0 00 00 Hs Es HEs EHs ε µ µ ε sˆ Planar Wave 0E 0H r         =⋅− =⋅− −=×− =×− ⇒ ⋅− ⋅− ⋅−⋅− ⋅−⋅− −=∇ ⋅−⋅− 0 0 0 0 00 00 rkj rkj rkjrkj rkjrkj ekje eHkj eEkj eHjeEkj eEjeHkj rkjrkj           ωµ ωε        =⋅ =⋅ =× −=× 0 0 0 0 00 00 Hk Ek HEk EHk     µω εω For a Homogeneous, Linear and Isotropic Media: Return to Table of Content
51 SOLO ELECTROMAGNETICS Planar Waves in an Source-less Anisotropic Electric Media     0 0 0 0 m e m e B D J t B E J t D H ρ ρ =⋅∇ =⋅∇ − ∂ ∂ −=×∇ + ∂ ∂ =×∇           ( ) ( ) ( ) ∫ +∞ ∞− ⋅− = dkekEtrE rktj   ω 0, ωω ωω j t eje t rs c n tjrs c n tj → ∂ ∂ ⇒= ∂ ∂       ⋅−      ⋅−  ˆˆ kjs c n jes c n je rs c n tjrs c n tj  −=−→∇⇒−=∇       ⋅−      ⋅− ˆˆ ˆˆ ωω ωω 0ˆ 0ˆ ˆ ˆ =⋅ =⋅ =× −=× Bs Ds BEs c n DHs c n ( ) ( )DEBH tt D E t B H HEEHHES HB ED         ⋅+⋅ ∂ ∂ −= ∂ ∂ ⋅− ∂ ∂ ⋅−= ×∇⋅−×∇⋅=×⋅∇=⋅∇ ⋅= ⋅= 2 1µ ε ( ) emUDEBHS k Ss c n =⋅+⋅=⋅=⋅ 2 1 ˆ ω  Maxwell’s Symmetric Equations Planar Wave 0 0 =⋅ =⋅ =× −=× Bk Dk BEk DHk     ω ω ωj t → ∂ ∂ s c n j ˆω−→∇ ωj t → ∂ ∂ kj  −→∇ vector phasor vector phasor s c n k ˆω= 
52 SOLO ELECTROMAGNETICS Planar Wave Group Velocity Consider a Planar Wave composed of frequencies centered around ω0. ( ) ( ) ( )( ) ∫ +∞ ∞− ⋅− = dkekEtrE rktkj   ω 0, ( ) ( ) +−        += 0 0 0 kk kd d k ω ωω Expand ω(k), using Taylor series around k0 ( ) ( ) ( ) ( ) ∫ ∞+ ∞− −         − ⋅− −      ⋅− = dkekEetrE kk kk rkk t kd d j rktj 0 0 0 0 00 0,      ω ω The Phase of the Group of Waves is ( ) 0 0 0 0 kk kk rkk t kd d g    −           − ⋅− −      = ω ϕ To find the velocity of phase of the group of waves consider the points for which φg = const ( ) 0 0 0 0 = − ⋅− −      ⇒ kk rdkk td kd d g d   ω ϕ The Group Velocity of the Planar Wave is ( ) 0 0 0 kk kk kd d krd g td rd v      − −       = = = ω δ Return to Table of Content
53 SOLO ELECTROMAGNETICS Phase Velocity, Energy (Ray) Velocity for Planar Waves The phase of the field is given by       ⋅−=⋅−= rs c n trkt  ˆωωφ For a constant phase front we have       ⋅−=⋅−=⋅−== rds c n dtrdskdtrdkdtd  ˆˆ0 ωωωφ s n c s kdt rd v srr const p ˆˆ: ˆ === = = ω φ   Phase Velocity B  D  pv  E  H  HES  ×=α α k  ev  PlanarWaves n c ktd rd s ==⋅ ω  ˆ n c S S vs e =⋅ :ˆ ( ) α α cosˆ ˆ cos p S Ss e v Ss S n c v ⋅ = = ⋅ = Define the phase velocity as Define the energy flow velocity in the Poynting vector direction as S S vSv ee = → 1 ( ) emUDEBHS k Ss c n =⋅+⋅=⋅=⋅ 2 1 ˆ ω  ( ) em ee U S S Ssn c S S vv = ⋅ == ˆ 1 :  Energy (Ray) Velocity The electro-magnetic energy propagates along the Poynting vector . For anisotropic media the Poynting vector is not collinear with .k  HES ×= Return to Table of Content Electromagnetic Energy Density
54 SOLO ELECTROMAGNETICS Planar Waves in an Source-less Anisotropic Electric Media A wave packet can be viewed as a superposition of monochromatic waves, with different frequencies ω and wave vector that must satisfy the Maxwell’s equationsk  EHk HEk ⋅−=× ⋅=× εω µω   Suppose that the wave vector changes by an infinitesimal amount , that determines changes k  k  δ HE δδωδ ,, HHHEkEk ⋅⋅+⋅=×+× δµωµωδδδ  ( )EEEHkHk −⋅⋅−⋅−=×+× δεωεωδδδ  ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )EEEEEHkHEk HHHHHEkHEk δεωεωδδδ δµωµωδδδ ⋅⋅+⋅⋅=×⋅−×⋅ ⋅⋅+⋅⋅=×⋅+×⋅   ( ) ( ) ( ) ( ) 0 2 00 =⋅+×⋅+⋅−×⋅−= ⋅⋅+⋅⋅−×⋅          EHkEHEkH HHEEHEk εωδµωδ µεωδδ ( ) ( ) ( )BACACBCBA  ×⋅≡×⋅≡×⋅
55 SOLO ELECTROMAGNETICS Planar Waves in an Source-less Anisotropic Electric Media ( ) ( ) e em vk U S k HHEE HEk      ⋅=⋅= ⋅⋅+⋅⋅ ×⋅ = δδ µε δ ωδ 2 1 ( ) gk vkk  ⋅=∇⋅= δωδωδ - Energy (Ray) Velocityev  From the definition of Group Velocity: gv  since those relations are true for all: k  δ eg vv  = For a monochromatic wave δω=0, therefore ( ) 0=×⋅=⋅ HEkSk  δδ ( ) ( ) 02 =⋅⋅+⋅⋅−×⋅ HHEEHEk µεωδδ 
56 ELECTROMAGNETICSSOLO Consider an Electric Anisotropic Media General Symmetric        =⋅∇ =⋅∇ −=×∇ ⋅=×∇ 0 0 B D HjE EjH µω εω  For a Homogeneous, Linear and Anisotropic Media:           = 333231 232221 131211 εεε εεε εεε ε            = z y x D ε ε ε ε 00 00 00  Diagonalized Principal Axis TTD  εε 1− =     −−=×∇×∇ +⋅=×∇×∇ m e JHjE JEjH µω εω  ( ) ( )   ×∇−×∇−=×∇×∇ ×∇+⋅×∇=×∇×∇ m e JHjE JEjH ωµ εω  ( ) me JJjEE ×∇−×∇−=×∇×∇ µω ε ε µεω 0 0 2  Planar Waves in an Source-less Anisotropic Electric Media Return to Table of Content
57 ELECTROMAGNETICSSOLO According to the principle of superposition these two fields can exist separately or be superimposed without disturbing each other. ( )1221 HEHE  ×−×⋅∇ Consider two distinct fields and. 11, HE  22 , HE  Assumptions: 1.The two fields are harmonic functions of time and of the same frequency. 2.The medium is linear. 3.The point considered is not inside a source and Ohm Law applies.         ∂ ∂ +⋅+ ∂ ∂ ⋅+        ∂ ∂ +⋅− ∂ ∂ ⋅= t D JE t B H t D JE t B H ee 1 12 2 1 2 21 1 2         12212112 HEEHHEEH  ×∇⋅+×∇⋅−×∇⋅−×∇⋅= ( ) ( )1122122112 DjJEBHjDjJEBHj ee j t ωωωω ω +⋅+⋅++⋅−⋅−= = ∂ ∂ ( ) ( ) 0 0 1122122112 21 ==⋅= ⋅= =⋅+⋅+⋅⋅+⋅+⋅−⋅⋅−= ee JJ ee ED HB EjJEHHjEjJEHHj εωµωεωµω ε µ    Lorentz’s Lemma
58 ELECTROMAGNETICSSOLO Lorentz’s Lemma (continue) ( ) ( )1221 HEHE  ×⋅∇=×⋅∇ For the two distinct fields and. 11, HE  22 , HE  Hendrik Antoon Lorentz 1853-1928 For a Monochromatic Planar Wave ( ) ( ) ( ) ( ) ( ) ( )    == == ⋅− ⋅− rktjtj rktjtj eHerHtrH eEerEtrE     ωω ωω ω ω 0 0 ,, ,, ( ) ( ) ( )rktjrktjrktj eskjekje   ⋅−⋅−⋅− −=−=∇ ωωω ˆ Lorentz’s Lemma can be written ( ) ( )1221 HEkHEk  ×⋅=×⋅
59 ELECTROMAGNETICSSOLO Lorentz’s Reciprocity Theorem ( ) ( )222 111 se se EEJ EEJ   += += σ σ This is the more general case that does not exclude the sources. ( ) 12211221 ee JEJEHEHE  ⋅−⋅=×−×⋅∇Ohm’s Law: are the applied electric field intensities within the sources21, ss EE  ( )1221 HEHE  ×−×⋅∇         ∂ ∂ +⋅+ ∂ ∂ ⋅+        ∂ ∂ +⋅− ∂ ∂ ⋅= t D JE t B H t D JE t B H ee 1 12 2 1 2 21 1 2         12212112 HEEHHEEH  ×∇⋅+×∇⋅−×∇⋅−×∇⋅= ( ) ( )1122122112 DjJEBHjDjJEBHj ee j t ωωωω ω +⋅+⋅++⋅−⋅−= = ∂ ∂ ( ) ( ) 122112211212 21122211122112 esesssssss ssssee JEJEEEEEEEEE EEEEEEEEEEJEJE   ⋅−⋅=⋅−⋅−⋅+⋅= ⋅−⋅=+⋅−+⋅=⋅−⋅= σσσσ σσσσ ( ) ( )1122122112 EjJEHHjEjJEHHj ee ED HB ⋅+⋅+⋅⋅+⋅+⋅−⋅⋅−= ⋅= ⋅= εωµωεωµω ε µ   
60 ELECTROMAGNETICSSOLO Lorentz’s Reciprocity Theorem (continue - 1) Integrate over a very large volume: ( ) 12211221 ee JEJEHEHE  ⋅−⋅=×−×⋅∇ ( ) ( )∫∫ ×−×⋅∇=⋅−⋅ VV eses dvHEHEdvJEJE 12211221  ( ) 11 22 21 21 0 0 0 0 1221 0 VoutsideE VoutsideE EE HH S s s S S sdHEHE      = = == == ⇐=⋅×−×= →∞ →∞ ∫ We obtained ( ) ( )∫∫ ⋅=⋅ V es V es dvJEdvJE 1221  ( ) 212121 bs V es V es IVdsJdlEdvJE =⋅=⋅ ∫∫  ( ) 121212 bs V es V es IVdsJdlEdvJE =⋅=⋅ ∫∫ 
61 ELECTROMAGNETICSSOLO Lorentz’s Reciprocity Theorem (continue - 2) The Reciprocity Theorem 1221 bsbs IVIV = The current induced in 2 when 1 is energized, divided by the voltage applied on 1, is the same as the current induced in 1 when 2 is energized, divided by the applied voltage on 2, as long as the frequency and the impedances remain unchanged. 1221 bbss IIVV =⇔= Return to Table of Content
62 SOLO ( ) ( ) ( ) ( ) ( ) ( )[ ]{ } ( ) ( ) ( ) ( )[ ] ( ) ( ) ( ) ( )[ ] ( ) ( ) ( ) ( ) ( ) ( )[ ]∫ ∫ ∫ ∫∫ −+⋅+= −+⋅−+= = ⋅=⋅= = T T T TEDT e dttjrErErEtjrE T dttjrEtjrEtjrEtjrE T dttjrEal T dttrEtrE T dttrDtrE T w 0 2**2 0 ** 0 2 00 2exp,,,22exp, 4 1 exp,exp,exp,exp, 4 1 exp,Re 1 ,, 1 ,, 1 ωωωωωω ε ωωωωωωωω ε ωωε ε ε      But ( ) ( )[ ] ( ) ( ) ( )[ ] ( ) 0 2 2exp 2exp 2 1 2exp 1 0 2 2exp 2exp 2 1 2exp 1 0 0 0 0 ∞→ ∞→ → − =−=− →== ∫ ∫ T T T T T T Tj Tj tj Tj dttj T Tj Tj tj Tj dttj T ω ω ω ω ω ω ω ω ω ω Therefore ( ) ( ) * 00 * 00 0 * 22 1 ,, 2 EEeEeEdt T rErEw rkjrkj T e εε ωω ε === ⋅−⋅ ∫   Let compute the time averages of the electric and magnetic energy densities ELECTROMAGNETICS Energy Flux and Poynting Vector For a Homogeneous, Linear and Isotropic Media
63 SOLO In the same way ( ) ( ) ( ) ( ) * 00 00 2 ,, 1 ,, 1 HHdttrHtrH T dttrBtrH T w TT m µ µ === ∫∫  Using the relations ( ) 00 ˆ HsEA ×−= ε µ ( ) 00 ˆ EsHF ×= µ ε since and are real values , where * is the complex conjugate, we obtain S∇ )**,( SS ∇=∇= ( ) ( ) ( ) ( ) ( ) ( )[ ] ( ) e m e wHkEHkEHkE EkHEkHHHw HkEHkEEEw =×⋅=×⋅=×⋅= ×⋅=×⋅=⋅= ×⋅=×⋅=⋅= * 00 * 0 * 00 ** 0 * 00 * 00 * 00 * 00 * 00 * 00 ˆ 2 ˆ 2 ˆ 2 ˆ 2 ˆ 22 ˆ 2 ˆ 22 µεµεµε µε µ εµµ µε ε µεε ELECTROMAGNETICS ( ) ( ) * 00 * 00 0 * 22 1 ,, 2 EEeEeEdt T rErEw rkjrkj T e εε ωω ε === ⋅−⋅ ∫   Energy Flux and Poynting Vector (continue – 1) For a Homogeneous, Linear and Isotropic Media: Time-averaged Electric Energy Density Time-averaged Magnetic Energy Density
64 SOLO Therefore ( ) ( )( )* 00 ˆ 2 rHkrEww me  ×⋅== µε Within the accuracy of Geometrical Optics, the Time-averaged Electric and Magnetic Energy Densities are equal. ( ) ( ) ( ) ( ) ( ) ( )( )* 0000 * 00 ˆ 22 rHkrErHrHrErEwww me  ×⋅=⋅+⋅=+= ∗ µε µε The total energy will be: The Poynting vector is defined as: ( ) ( ) ( )trHtrEtrS ,,:,  ×= ( ) ( ) ( ) ( ) ( ) ( )[ ] ( ) ( )[ ] ( ) ( )[ ]∫ ∫∫ −− +×+= ×=×=×= T tjtjtjtj T tjtj T dterHerHerEerE T dterHerEal T dttrHtrE T trHtrES 0 ** 00 ,, 2 1 ,, 2 11 ,,Re 1 ,, 1 ,, ωωωω ωω ϖϖϖϖ ϖϖ  ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ] ( ) ( ) ( ) ( )[ ]ωωωω ωωωωωωωω ωω ,,,, 4 1 ,,,,,,,, 4 11 ** 0 2****2 rHrErHrE dterHrErHrErHrEerHrE T T tjtj ×+×= ∫ ×+×+×+×= − [ ] [ ]0 * 0 * 00 0 * 0 * 00 4 1 4 1 HEHE eHeEeHeE rkjrkjrkjrkj ×+×= ×+×= ⋅−⋅⋅⋅−  The time average of the Poynting vector is: John Henry Poynting 1852-1914 ELECTROMAGNETICS Energy Flux and Poynting Vector (continue – 2) For a Homogeneous, Linear and Isotropic Media: Return to Table of Content
65 SOLO Assume the linear general constitutive relations ELECTROMAGNETICS ( ) ( ) ( )        =⋅∇ =⋅∇ −−=×∇ +=×∇ m e m e BGM DGE JBjEF JDjHA ρ ρ ω ω )( dyadicsxwhere H E B D 33,,, =                 =         µζξε µζ ξε        ( ) ( ) ( ) ( )    −⋅+⋅−=×∇ +⋅+⋅=×∇ m e JHEjEF JHEjHA µζω ξεω   Energy Flux and Poynting Vector for a Bianisotropic Medium
66 SOLO ELECTROMAGNETICS Let compute the following ( ) ( )[ ] ( )[ ] ( ) me HH me HH JHJEHHEHHEEEj JHEjHEJHEj EHHEHE ⋅−⋅−⋅⋅−⋅⋅−⋅⋅+⋅⋅= −⋅+⋅−⋅+⋅+⋅+⋅−−= ×∇⋅+×∇⋅−=×⋅∇ ****** **** *** µζξεω µζωξεω   ( ) ( )[ ] ( ) ( )[ ] ( ) ****** **** *** me HH m HH e JHJEHHHEHEEEj HJHEjJHEjE EHHEHE ⋅−⋅−⋅⋅−⋅⋅−⋅⋅+⋅⋅−= ⋅−⋅+⋅−−++⋅+⋅⋅−= ×∇⋅+×∇⋅−=×⋅∇ µζξεω µζωξεω   ( ) ( ) ( ) ( ) ****** ****** ** me HH me HH JHJEHHHEHEEEj JHJEHHEHEHEEj HEHE ⋅−⋅−⋅⋅−⋅⋅−⋅⋅+⋅⋅− ⋅−⋅−⋅⋅−⋅⋅−⋅⋅+⋅⋅= ×⋅∇+×⋅∇ µζξεω µζξεω   ( ) ( ) ( ) ( )    −⋅+⋅−=×∇ +⋅+⋅=×∇ m e JHEjEF JHEjHA µζω ξεω   Energy Flux and Poynting Vector for a Bianisotropic Medium (continue – 1)
67 SOLO ELECTROMAGNETICS ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ] ( ) ( ) [ ] ( ) ( )mmeeHH HH mmee HHHH me HH me HH JHJHJEJE H E jHE JHJHJEJE HHEHHEEEj JHJEHHHEHEEEj JHJEHHEHEHEEj HEHE ⋅+⋅−⋅+⋅−        ⋅                 −− −− −⋅= ⋅+⋅−⋅+⋅+ ⋅−⋅+⋅−⋅+⋅−⋅+⋅−⋅−= ⋅−⋅−⋅⋅−⋅⋅−⋅⋅+⋅⋅− ⋅−⋅−⋅⋅−⋅⋅−⋅⋅+⋅⋅= ×⋅∇+×⋅∇ ****** **** **** ****** ****** ** µµζζ ξξεε ω µµζζξξεεω µζξεω µζξεω      ( ) ( ) ( ) ( )[ ] ( ) ( ) ( ) ( )[ ]ωωωωωω ,,,, 4 1 ,,Re 2 1 ,, ** rHrErHrErHrEaltrHtrES ×+×=×=×=  We found the time average of the Poynting vector We see that ( ) ( )[ ] [ ] ( ) ( )mmee HH HH JHJHJEJE H Ej HEHEHES ⋅+⋅−⋅+⋅−         ⋅                 −− −− −⋅=×⋅∇+×⋅∇=⋅∇ **** **** 4 1 4 1 44 1 µµζζ ξξεεω    Energy Flux and Poynting Vector for a Bianisotropic Medium (continue – 2)
68 SOLO ELECTROMAGNETICS Let integrate the mean value of the Poynting vector over the volume V [ ] ( ) ( ) ( ) ( )[ ] ( ) ∫∫ ∫ ∫∫ ∫∫ →→ ⋅=⋅×+×= ×⋅∇+×⋅∇= ⋅+⋅−⋅+⋅−         ⋅                 −− −− −⋅=⋅∇ SS Gauss V V mm V ee V HH HH V dSnSdSnHEHE dVHEHE dVJHJHdVJEJE dV H Ej HEdVS 11 4 1 4 1 4 1 4 1 4 ** 1 ** **** **     µµζζ ξξεεω ( ) ( )[ ] [ ] ( ) ( )mmee HH HH JHJHJEJE H Ej HEHEHES ⋅+⋅−⋅+⋅−         ⋅                 −− −− −⋅=×⋅∇+×⋅∇=⋅∇ **** **** 4 1 4 1 44 1 µµζζ ξξεεω    Energy Flux and Poynting Vector for a Bianisotropic Medium (continue – 3)
69 SOLO ELECTROMAGNETICS We recognize the following                 −− −− −= HH HH j µµζζ ξξεεω   4 G mediumlosslessundefinite mediumpassivedefinitenegative mediumactivedefinitepositive ⇒ ⇒ ⇒ G G G [ ] ( ) ( )∫∫ ∫∫ ⋅+⋅−⋅+⋅−         ⋅                 −− −− −⋅=⋅ → V mm V ee V HH HH S dVJHJHdVJEJE dV H Ej HEdSnS **** ** 4 1 4 1 4 1 µµζζ ξξεεω    [ ] dV H Ej HE V ∫         ⋅                 −− −− −⋅ ** ** ** 4 µµζζ ξξεεω   ( )∫ ⋅+⋅ V ee dVJEJE ** 4 1 ( )∫ ⋅+⋅ V mm dVJHJH ** 4 1 ∫ → ⋅ S dSnS 1  Time average of the Radiated Energy through S (Irradiance) Time average of Electromagnetic Energy in V Time average of Joule Energy in V Time average of Fictious Joule Energy in V Energy Flux and Poynting Vector for a Bianisotropic Medium (continue – 4) Return to Table of Content
70 SOLO ELECTROMAGNETICS Planar Waves in an Source-less Anisotropic Electric Media     0 0 0 0 m e m e B D J t B E J t D H ρ ρ =⋅∇ =⋅∇ − ∂ ∂ −=×∇ + ∂ ∂ =×∇           ( )       ⋅− = ⋅− == rs c n tj s c n k rktj eEeEE    ˆ 0 ˆ 0 ω ω ω ωω ωω j t eje t rs c n tjrs c n tj → ∂ ∂ ⇒= ∂ ∂       ⋅−      ⋅−  ˆˆ  sjks c n jes c n je k rs c n tjrs c n tj ˆˆˆ ˆˆ −=−→∇⇒−=∇       ⋅−      ⋅− ωω ωω  0ˆ 0ˆ ˆ ˆ =⋅ =⋅ =× −=× Bs Ds BEs c n DHs c n ( ) ( )DEBH tt D E t B H HEEHHES HB ED         ⋅+⋅ ∂ ∂ −= ∂ ∂ ⋅− ∂ ∂ ⋅−= ×∇⋅−×∇⋅=×⋅∇=⋅∇ = ⋅= 2 1µ ε ( ) emUDEBHS k Ss c n =⋅+⋅=⋅=⋅ 2 1 ˆ ω  Maxwell’s Symmetric Equations Planar Monochromatic Wave 0 0 =⋅ =⋅ =× −=× Bk Dk BEk DHk     ω ω ωj t → ∂ ∂ s c n j ˆω−→∇ ωj t → ∂ ∂ kj  −→∇ vector phasor vector phasor c n k ω=:
71 SOLO ELECTROMAGNETICS Planar Waves in an Source-less Anisotropic Electric Media 0ˆ 0ˆ ˆ ˆ =⋅ =⋅ =× −=× Bs Ds BEs c n DHs c n dyadicsxwhere H E B D 33,,, µζξε µζ ξε                        =         HHIB ED   µµ ε =⋅= ⋅= For Anisotropic Electric Media the Linear Constitutive Relations are Planar Waves in an Source-less Media The most general form of Linear Constitutive Relations is: HES ×= Poynting Vector B  D  pv  E  H  HES  ×=α α k  ev  PlanarWaves ( ) DHs c n Bs c n Ess c n µµ −=×=×=××      ˆˆˆˆ 2 ( ) ( )[ ]EssE c n Ess c n D ⋅−      =××      −= ˆˆ 1 ˆˆ 1 22 µµ SDEs ,,,ˆ are coplanar and normal to H I  µµζξε === ,0,
72 SOLO ELECTROMAGNETICS Planar Waves in an Source-less Anisotropic Electric Media B  D  pv  E  H  HES  ×=α α k  ev  PlanarWaves( ) ( )[ ]EssE c n Ess c n D ⋅−      =××      −= ˆˆ 1 ˆˆ 1 22 µµ SDEs ,,,ˆ are coplanar and normal to H  ( ) DEnEsDsDE c n DD c ⋅=      ⋅⋅−⋅      =⋅ = 2 0 1 0 2 02 ˆˆ 1 ε µ µε DE DD n ⋅ ⋅ = 0 2 1 ε ( ) ( ) ( ) ( )( )2222 2 2 2 2 2 ˆ DEDED DDEED D DE E D DDE E D D D D EE D D D D EE s ⋅− ⋅− = ⋅ − ⋅ − =       ⋅−       ⋅− = ( ) ( ) ( ) ( )( )2222 2 2 2 2 2 : DEDEE EDEDE E DE D E DDE D E E E E DD E E E E DD S S t ⋅− ⋅− −= ⋅ − ⋅ − −=       ⋅−       ⋅− −==  If and are not collinear:DE
73 SOLO ELECTROMAGNETICS Planar Waves in an Source-less Anisotropic Electric Media Phase Velocity, Energy (Ray) Velocity The phase of the field is given by       ⋅−=⋅−= rs c n trkt  ˆωωφ For a constant phase front we have       ⋅−=⋅−=⋅−== rds c n dtrdskdtrdkdtd  ˆˆ0 ωωωφ s n c s kdt rd v srr const p ˆˆ ˆ === = = ω φ   Phase Velocity B  D  pv  E  H  HES  ×=α α k  ev  PlanarWaves n c ktd rd s ==⋅ ω  ˆ n c S S vs e =⋅ˆ ( ) α α cosˆ ˆ cos p S Ss e v Ss S n c v ⋅ = = ⋅ = Define the phase velocity as Define the energy flow velocity in the Poynting vector direction as S S vSv ee = → 1 ( ) emUDEBHS k Ss c n =⋅+⋅=⋅=⋅ 2 1 ˆ ω  ( ) em ee U S S Ssn c S S vv = ⋅ == ˆ 1 Energy (Ray) Velocity The electro-magnetic energy propagates along the Poynting vector . For anisotropic media the Poynting vector is not collinear with .k  HES ×= Return to Table of Content s c n k ˆω=  Electromagnetic Energy Density
74 SOLO ELECTROMAGNETICS Planar Waves in an Source-less Anisotropic Electric Media BEk DHk ω ω =× −=×   HHIB ED   µµ ε =⋅= ⋅= Wave Equation ( ) EDHkBkEkk EDDHkHBBEk ⋅−=−=×=×=×× ⋅=−=×==× εµωµωµωω εωµω   22 2222 0 0 0 ˆ kkkkkk kk kk kk k k k k s s s kskk zyx xy xz yz z y x z y x =++=⋅           − − − =×⇒           =           ==  ( ) ( ) ( ) ( )            +− +− +− =           − − −           − − − =×× 22 22 22 0 0 0 0 0 0 yxzyzx zyzxyx zxyxzy xy xz yz xy xz yz kkkkkk kkkkkk kkkkkk kk kk kk kk kk kk kk            = z y x ε ε ε ε 00 00 00                                        =                   = 2 2 2 0 0 0 0 0 0 2 2 00 00 00 00 00 00 c n c n c n z x x z y x ω ω ω εµ εµ εµ εµ εµ εµ εµ εµ εµω εωµ  In principal dielectric axes: 0 2 0 2 0 2 ,, ε ε ε ε ε ε z z y y x x nnn === ( ) 02 =⋅+×× EEkk εµω  ( ) 021 =+×× − DDkk µωε  or
75 SOLO ELECTROMAGNETICS Planar Waves in an Source-less Anisotropic Electric Media Wave Equation (continue – 1) ( ) 02 =⋅+×× EEkk εµω  ( ) ( ) ( ) 0 22 2 22 2 22 2 =                                     +−      +−      +−      z y x yx z zyzx zyzx y yx zxyxzy x E E E kk c n kkkk kkkk c n kk kkkkkk c n ω ω ω A nonzero solution exists only when ( ) ( ) ( ) 0det 22 2 22 2 22 2 =                       +−      +−      +−      yx z zyzx zyzx y yx zxyxzy x kk c n kkkk kkkk c n kk kkkkkk c n ω ω ω
76 SOLO ELECTROMAGNETICS Planar Waves in an Source-less Anisotropic Electric Media Wave Equation (continue – 2) ( ) ( ) ( )                       +−      +−      +−      =                       +−      +−      +−      ++= 22 2 22 2 22 2 22 2 22 2 22 2 2222 det z z zyzx zyy y yx zxyxx x kkkk yx z zyzx zyzx y yx zxyxzy x kk c n kkkk kkkk c n kk kkkkkk c n kk c n kkkk kkkk c n kk kkkkkk c n zyx ω ω ω ω ω ω We obtain         −−        −−                 −+        −+        −         −        +−= 2 2 22 222 2 22 222 2 22 22 2 22 22 2 22 2 2 22 22 2 22 k c n kkk c n kkk c n kk c n kk c n k c n kk c n y zx z yx y z z y zy x x ωωωωωωω 02 2 22 2 2 22 22 2 22 2 2 22 22 2 22 2 2 22 22 2 22 2 2 22 2 2 22 =         −        −+        −        −+        −         −+        −         −        −= k c n k c n kk c n k c n kk c n k c n kk c n k c n k c n yx z zx y zy x zyx ωωωωωωωωω Divide by (assumed nonzero!!)         −         −        −− 2 2 22 2 2 22 2 2 22 k c n k c n k c n zyx ωωω 1 2 22 2 2 2 22 2 2 2 22 2 2 = − + − + − c n k k c n k k c n k k z z y y x x ωωω This is a quadratic equation of k2 (k6 terms drop – next slide). For each set sx, sy, sz it yields two solutions for k2 : k1 2 and k2 2           =           z y x z y x s s s k k k k 2 2 22 2 2 2 22 2 2 2 22 2 2 1 k c n k s c n k s c n k s z z y y x x = − + − + − ωωω ( )skkk ˆ,, 21 ε  =
77 SOLO ELECTROMAGNETICS Planar Waves in an Source-less Anisotropic Electric Media 02 2 22 2 2 22 222 2 22 2 2 22 222 2 22 2 2 22 222 2 22 2 2 22 2 2 22 =         −        −+        −        −+         −        −+        −         −        − k c n k c n skk c n k c n skk c n k c n skk c n k c n k c n yx z zx y yz x zyx ωωωωωωωωω ( ) ( ) ( ) ( ) ( ) 02222 4 4 4222 2 2 622222 4 4 4222 2 2 622222 4 4 4222 2 2 62 222 6 6 2222222 4 4 4222 2 2 6 =++−+++−+++−+ +++−+++− ksnn c ksnn c ksksnn c ksnn c ksksnn c ksnn c ks nnn c knnnnnn c knnn c k zyxzyxzyzxyzxyxzyxzyx zyxzyzxyxzyx ωωωωωω ωωω ( ) ( ) ( ) ( )[ ] 0111 222 6 6 2222222222 4 4 4222222 2 2 =+−+−+−−++ zyxxzyyzxzyxzzyyxx nnn c ksnnsnnsnn c ksnsnsn c ωωω This is a quadratic equation of k2 . For each set sx, sy, sz it yields two solutions for k2 : k1 2 and k2 2 ( )skkk ˆ,, 21 ε  = 2 2 22 2 2 2 22 2 2 2 22 2 2 1 k c n k s c n k s c n k s z z y y x x = − + − + − ωωω 02 2 22 2 2 22 22 2 22 2 2 22 22 2 22 2 2 22 22 2 22 2 2 22 2 2 22 =         −        −+        −        −+        −         −+        −         −        − k c n k c n kk c n k c n kk c n k c n kk c n k c n k c n yx z zx y zy x zyx ωωωωωωωωω           =           z y x z y x s s s k k k k Wave Equation (continue – 2a) 1 222 =++ zyx sss
78 SOLO ELECTROMAGNETICS Planar Waves in an Source-less Anisotropic Electric Media Wave Equation (continue – 3) ( ) ( ) ( ) 0 22 2 22 2 22 2 =                                     +−      +−      +−      z y x yx z zyzx zyzx y yx zxyxzy x E E E kk c n kkkk kkkk c n kk kkkkkk c n ω ω ω A solution (self-mode) is: ( ) ( ) ( ) ( ) ( ) ( ) 0 0 0 222 2 222 2 222 2 =+++         ++−      =+++         ++−      =+++         ++−      ⋅ ⋅ ⋅             Ek zzyyxxzzzyx z Ek zzyyxxyyzyx y Ek zzyyxxxxzyx x EkEkEkkEkkk c n EkEkEkkEkkk c n EkEkEkkEkkk c n ω ω ω                                   −      −      −      2 2 2 2 2 2 k c n k k c n k k c n k z z y y x x ω ω ω ( )Ek ⋅−  ( ) ( ) ( )                                   −      ⋅ −      ⋅ −      ⋅ −=                         2 2 2 2 2 2 k c n Ekk k c n Ekk k c n Ekk E E E z z y y x x z y x ω ω ω    In principal dielectric axes Return to Table of Content
79 SOLO ELECTROMAGNETICS Planar Waves in an Source-less Anisotropic Electric Media Wave-Vector Surface The above equation can be represented by a three-dimensional surface in k space. ( ) ( ) ( ) 0det 22 2 22 2 22 2 =                     +−      +−      +−      yx z zyzx zyzx y yx zxyxzy x kk c n kkkk kkkk c n kk kkkkkk c n ω ω ω The understand how this surface look let find the intersection of the surface with kz = 0. ( ) ( ) 0 00 0 0 det 222 2 2 2 22 2 22 2 2 2 2 2 =         −         −              −              +−      =                     +−      −      −      yxx y y x yx z yx z x y yx yxy x kkk c n k c n kk c n kk c n k c n kk kkk c n ωωω ω ω ω
80 SOLO ELECTROMAGNETICS Planar Waves in an Source-less Anisotropic Electric Media Wave-Vector Surface (continue – 1) The intersection of the surface with kz = 0 is given by two equations, derived from: ( ) 0 22 2 222 2 2 2 =         +−              −         −              −      yx z yxx y y x kk c n kkk c n k c n ωωω 2 22       =+ c n kk z yx ω ( ) ( ) 1 // 22 =+ cn k cn k x y y x ωω ( ) 0 00 0 0 22 2 2 2 2 2 =                                     +−      −      −      z y x yx z x y yx yxy x E E E kk c n k c n kk kkk c n ω ω ω                 =                 = zz y x EE E E E 0 0 2                     −      − =                 = 0 2 2 1 y x yx z y x k c n kk E E E E ω 021 =⋅ EE Ellipse The electric fields corresponding to those two solutions are: on Ellipse on Circle Since the coefficients of the electric fields are real the two solutions represents two linear polarized planar waves. Circle
81 SOLO ELECTROMAGNETICS Planar Waves in an Source-less Anisotropic Electric Media Wave-Vector Surface (continue -2) Intersection of the surface with kz = 0, are a circle and an ellipse in kx, ky plane. 2 22       =+ c n kk z yx ω ( ) ( ) 1 // 2 2 2 2 =+ cn k cn k x y y x ωω Intersection of the surface with ky = 0, are a circle and an ellipse in kx, kz plane. 2 22       =+ c n kk y zx ω ( ) ( ) 1 // 2 2 2 2 =+ cn k cn k x z z x ωω Intersection of the surface with kx = 0, are a circle and an ellipse in ky, kz plane. 2 22       =+ c n kk x zy ω ( ) ( ) 1 // 2 2 2 2 =+ cn k cn k y z z y ωω Suppose nx < ny < nz. ( ) 0 222 2 2 2 22 2 =         −         −              −              +−      yxx y y x yx z kkk c n k c n kk c n ωωω 0 2 0 2 0 2 ,, ε ε ε ε ε ε z z y y x x nnn ===
82 SOLO ELECTROMAGNETICS Planar Waves in an Source-less Anisotropic Electric Media Wave-Vector Surface (continue – 3) The complete k surface is double; that is, it consists of an inner shell and an outer shell. For any given direction of the vector k, there are two possible values for the wave-number k (eigenmodes). Therefore are also two values for the phase velocity. From the figure we can see that the inner and outer shells of the k surface intersect at four points in each quadrant of the kx, kz plane (for nx<ny<nz). Those points define two directions for which the two values of k are equal. Those two directions are called Optical Axes. The structure of an anisotropic medium permits two monochromatic plane waves that are polarized orthogonally with respect to each other. If nx,ny and nz are different the crystal has two optical axes and is said to be biaxial. 0 2 0 2 0 2 ,, ε ε ε ε ε ε z z y y x x nnn === nx < ny < nz.
83 SOLO ELECTROMAGNETICS Planar Waves in an Source-less Anisotropic Electric Media Wave-Vector Surface (continue -4) Let compute the Optical Axis direction. 0 2 0 2 0 2 ,, ε ε ε ε ε ε z z y y x x nnn === We can see that the Optical Axis is in x, z plane, therefore           =           =           = δ δ ω cos 0 sin c n s s s k k k k k OA OAz y x OA OAz y x OA  2 22       =+ c n kk y zx ω ( ) ( ) 1 // 2 2 2 2 =+ cn k cn k x z z x ωω 22       =      c n c n yOA ωω 1 cossin 2 22 2 22 =+ x OA z OA n n n n δδ must be at the intersection ofOAk  and yOA nn = ( ) 1 coscos1 2 22 2 22 =+ − x y z y n n n n δδ 22 22 2 cos −− −− − − = zx zy OA nn nn δ 22 22 2 sin −− −− − − = zx yx OA nn nn δ 22 22 2 tan −− −− − − = zy yx OA nn nn δ If δ is less than 45º, the crystal is said to be a positive biaxial crystal. If δ is greater than 45º, the crystal is said to be a negative biaxial crystal. nx < ny < nz.         − − ±= −− −− − 22 22 1 2,1 sin zx yx OA nn nn δ
84 SOLO ELECTROMAGNETICS Planar Waves in an Source-less Anisotropic Electric Media Wave-Vector Surface (continue -5) 0 2 0 2 0 2 ,, ε ε ε ε ε ε z z y y x x nnn ===nx < ny < nz.
85 SOLO ELECTROMAGNETICS Planar Waves in an Source-less Anisotropic Electric Media Double Refraction at a Birefringent Crystal Boundary ( ) 0ˆ  =−× ti kkn At the Crystal Boundary defined by the normal the following condition must be satisfied nˆ The refracted wave is, in general, a mixture of two eigenmodes ( ) ( )222111 ,,, sn c ksn c k  ε ω ε ω == We can write ( ) ( ) 222111 sin,sin,sin θεθεθ skskk ii  == This looks like Snell’s Law but k1 and k2 are not constant, and we must add the quadratic equation in k2 : ( ) ( ) ( ) ( )[ ] 0111 222 6 6 2222222222 4 4 4222222 2 2 =+−+−+−−++ zyxxzyyzxzyxzzyyxx nnn c ksnnsnnsnn c ksnsnsn c ωωω 0 2 0 2 0 2 ,, ε ε ε ε ε ε z z y y x x nnn ===
86 SOLO ELECTROMAGNETICS Planar Waves in an Source-less Anisotropic Electric Media Wave-Vector Surface (continue – 6) If two of the refraction indices are equal the crystal has one optical axis and is said to be uniaxial. zyx εεε ≠=nx = ny =no≠ nz=ne For each set sx, sy, sz it yields two solutions for k2 : k1 and k2 ( )skkk ˆ,, 21 ε  = The quadratic equation of k2 is: 2 2 22 2 2 2 22 2 2 11 k c n k s c n k s e z o z = − + − − ωω ( )[ ] ( ) ( )[ ] 0111 24 6 6 222222 4 4 42222 2 2 =+++−−+− eozezoozezo nn c ksnsnn c ksnsn c ωωω ( ) ( ) ( ) ( )[ ] 0111 222 6 6 2222222222 4 4 4222222 2 2 =+−+−+−−++ zyxxzyyzxzyxzzyyxx nnn c ksnnsnnsnn c ksnsnsn c ωωω nx = ny =no≠ nz=ne Uniaxial Crystals One other way is to substitute in:
87 SOLO ELECTROMAGNETICS Planar Waves in an Source-less Anisotropic Electric Media Wave-Vector Surface (continue – 7) zyx εεε ≠=nx = ny =no≠ nz=ne ( )[ ] ( ) ( )[ ] 0111 24 6 6 222222 4 4 42222 2 2 =+++−−+− eozezoozezo nn c ksnsnn c ksnsn c ωωω ( )( ) 01 2 2 2 22222 2 2 2 2 22 =      −−−+      − oeozoe n c kknnsn c kn ωω ( )[ ] 01 22 2 2 222222 2 2 2 =       −+−      − eoezozo nn c knsnsn c k ωω The Wave-Vector surface decomposes into two separate shells 0 2 2 2 2 =− on c k ω ( )[ ] 0 22 2 2 222222 =−++ eoezoyx nn c knsnss ω 0 2 2 2 222 =−++ ozyx n c kkk ω Spherical surface 01 2 2 2 2 2 2 2 22 =−+ + o z e yx n c k n c kk ωω Ellipsoid of Revolution surface These two surfaces are tangent along the z axis. Uniaxial Crystals (continue – 1) Return to Table of Content
88 SOLO ELECTROMAGNETICS Planar Waves in an Source-less Anisotropic Electric Media Double Refraction at a Uniaxial Crystal Boundary For the Uniaxial Crystal is easy to solve the Boundary condition graphically: ( ) eeooii skkk θεθθ sin,sinsin 2  == The Wave-Vector surface decomposes into two separate shells: • the sphere of the ordinary wave with • the ellipsoid of extraordinary wave with .constko = ( )2, ske  ε is found graphically using the equality: Snell’s Lawooii kk θθ sinsin =o k  The intersection of the normal to the boundary from the end with the ellipse defines the direction of ek  o k  The intersection of the plane defined by and passing through the common centers of the sphere and the ellipsoid will give a circle and an ellipse, respectively. nki  , ii k θsin ik ok i θ oθ Intersection of normal surface with plane of incidence in ii k θsin Uniaxial Crystal Incident ray n  Optical Axis oE  eo nn >
89 SOLO ELECTROMAGNETICS Planar Waves in an Source-less Anisotropic Electric Media Double Refraction at a Uniaxial Crystal Boundary
90 SOLO ELECTROMAGNETICS Planar Waves in an Source-less Anisotropic Electric Media Wave-Vector Surface (continue – 8) nx = ny = nz=n If all three indices are equal, then the Wave-vector surface degenerates to a single sphere, and the crystal is optically isotropic. εεεε === zyx 2 22 2 c n k ω = 0 2 2 2 =                       z y x zzyzx zyyyx zxyxx E E E kkkkk kkkkk kkkkk 2222 zyx kkkk ++= ( ) 02 =⋅+×× EEkk εµω  We can see that: { } { } { } 0 ,, ,, ,, det 2 2 2 ≡           =             zyxz zyxy zyxx zzyzx zyyyx zxyxx kkkk kkkk kkkk kkkkk kkkkk kkkkk and: 0=⋅=++ EkEkEkEk zzyyxx  0/εε=n Isotropic Crystal Return to Table of Content
91 SOLO ELECTROMAGNETICS Optically Isotropic (Cubic) Crystals n Sodium Chloride (NaCl) Diamond Fluorite CdTe GaAs 1.544 2.417 1.392 2.69 3.40 Uniaxial Positive Crystals nO nE Ice Quartz Zircon Rutile ( TiO2 ) BeO ZnS 1.309 1.310 1.544 1.553 1.923 1.968 2.616 2.903 1.717 1.732 2.354 2.358 Uniaxial Negative Crystals nO nE Beryl Sodium Nitrate Calcite Tourmaline 1.598 1.590 1.587 1.336 1.658 1.486 1.669 1.638 Refractive Index of Some Typical Crystals
92 SOLO ELECTROMAGNETICS Biaxial Crystals n1 n2 n3 Gypsum Feldspar Mica Topaz NaNO2 SbSI YAlO3 1.520 1.523 1.530 1.522 1.526 1.530 1.552 1.582 1.588 1.619 1.620 1.627 1.344 1.411 1.651 2.7 3.2 3.8 1.923 1.938 1.947 Refractive Index of Some Typical Biaxial Crystals
93 SOLO ELECTROMAGNETICS Planar Waves in an Source-less Anisotropic Electric Media Crystal Types
94 SOLO ELECTROMAGNETICS Planar Waves in an Source-less Anisotropic Electric Media Crystal Types (continue -1)
95 SOLO ELECTROMAGNETICS Planar Waves in an Source-less Anisotropic Electric Media Crystal Types (continue – 2)
96 SOLO ELECTROMAGNETICS Planar Waves in an Source-less Anisotropic Electric Media Crystal Types (continue – 3) A.Mermin, “Solid State Physics”, Holt, Reinhart and Winston, 1976, p.122 Return to Table of Content
97 SOLO ELECTROMAGNETICS Planar Waves in an Source-less Anisotropic Electric Media Orthogonality Properties of the Eigenmodes Assume the two monochromatic waves corresponding to a given direction are defined by and sˆ 1111 ,,, nDEH  2222 ,,, nDEH  Start with Lorentz Reciprocity Theorem Use HB BEs c n µ= =×  ˆ Es c n H ×= ˆ µ ( )[ ] ( )[ ]12 2 21 1 ˆˆˆˆ EsEs c n EsEs c n ××⋅=××⋅ µµ ( ) ( ) ( ) ( )12 2 21 1 ˆˆˆˆ EsEs c n EsEs c n ×⋅×=×⋅× µµ ( ) ( )1221 ˆˆ HEsHEs ×⋅=×⋅ Since we have21 nn ≠ ( ) ( ) 0ˆˆ 21 =×⋅× EsEs ( ) ( ) 0ˆˆ 1221 =×⋅=×⋅ HEsHEs ( ) ( )1221 HEHE  ×⋅∇=×⋅∇ s c n j ˆω−→∇ 0=⋅=⋅ sHEH 
98 SOLO ELECTROMAGNETICS Planar Waves in an Source-less Anisotropic Electric Media Orthogonality Properties of the Eigenmodes (continue – 1) We obtain 0ˆˆˆˆ 221121212121212121 =⋅=⋅=⋅=⋅=⋅=⋅=⋅=⋅=⋅=⋅=⋅ HEHEHsHsDsDsEEEDDEDDHH ( ) ( ) 0ˆˆ 1221 =×⋅=×⋅ HEsHEs ( ) ( ) ( ) ( ) 0ˆˆ 0ˆˆ 21 1 2 21 2 ˆ 2121 21 1 ˆ 2121 2 2 2 1 1 1 11 =⋅=⋅=×⋅−=×⋅ =⋅=⋅×=×⋅ − −=× =× = DD n c DE n c HsEHEs HH n c HEsHEs D n c Hs B n c Es HB ε µ µ  ( ) ( ) ( ) ( ) 0ˆˆ 0ˆˆ 12 1 1 12 1 ˆ 1212 12 2 ˆ 1212 1 1 1 2 2 2 22 =⋅=⋅=×⋅−=×⋅ =⋅=⋅×=×⋅ − −=× =× = DD n c DE n c HsEHEs HH n c HEsHEs D n c Hs B n c Es HB ε µ µ 
99 SOLO ELECTROMAGNETICS Planar Waves in an Source-less Anisotropic Electric Media Orthogonality Properties of the Eigenmodes (continue – 2) We obtained The two solutions correspond to two different possible linear polarizations of a wave propagating in direction and these two solutions have mutually orthogonal polarizations. sˆ 0ˆˆˆˆ 221121212121212121 =⋅=⋅=⋅=⋅=⋅=⋅=⋅=⋅=⋅=⋅=⋅ HEHEHsHsDsDsEEEDDEDDHH Return to Table of Content
100 SOLO ELECTROMAGNETICS Phase Velocity, Energy (Ray) Velocity for Planar Waves The phase of the field is given by       ⋅−=⋅−= rs c n trkt  ˆωωφ For a constant phase front we have       ⋅−=⋅−=⋅−== rds c n dtrdskdtrdkdtd  ˆˆ0 ωωωφ s n c s kdt rd v srr const p ˆˆ: ˆ === = = ω φ   Phase Velocity B  D  pv  E  H  HES  ×=α α k  ev  PlanarWaves n c ktd rd s ==⋅ ω  ˆ n c S S vs e =⋅ :ˆ ( ) α α cosˆ cos ˆ p S Ss e v Ss S n c v = ⋅ = ⋅ = Define the phase velocity as Define the energy flow velocity in the Poynting vector direction as S S vSv ee = → 1 ( ) emUDEBHS k Ss c n =⋅+⋅=⋅=⋅ 2 1 ˆ ω  ( ) em ee U S S Ssn c S S vv = ⋅ == ˆ 1 :  Energy (Ray) Velocity The electro-magnetic energy propagates along the Poynting vector . For anisotropic media the Poynting vector is not collinear with .k  HES ×= Return to Table of Content
101 SOLO ELECTROMAGNETICS Planar Waves in an Source-less Anisotropic Electric Media Ray-Velocity Surface We defined the Energy-Ray Velocity as Let compute ( ) HDHEDEH U DHE U Dv emUemem e =               ⋅−         ⋅=××=×   0 11 ( ) ( ) DEHHEHEH U HHE U HvDvv emUemem eee 1 /0 1111 − −=−=                 ⋅−      ⋅=××=×=×× ε µµµ    In principal dielectric axes:                     =                     =− 2 2 2 2 2 2 1 00 00 00 1 00 0 1 0 00 1 1 z y x z y x n c n c n c εµ εµ εµ ε µ  ( ) em ee U S S Ssn c S S vv = ⋅ == ˆ 1 : 
102 SOLO ELECTROMAGNETICS Planar Waves in an Source-less Anisotropic Electric Media Ray-Velocity Surface (continue – 1) We can write ( ) 0 1 1 =+×× − DDvv ee ε µ  as ( ) ( ) ( ) 0 22 2 22 2 22 2 =                                       +−      +−         +−      z y x eyex z ezeyezex ezeyezex y eyex ezexeyexezey x D D D vv n c vvvv vvvv n c vv vvvvvv n c We obtain non-zero solutions only when ( ) ( ) ( ) 0det 22 2 22 2 22 2 =                       +−      +−         +−      eyex z ezeyezex ezeyezex y eyex ezexeyexezey x vv n c vvvv vvvv n c vv vvvvvv n c
103 SOLO ELECTROMAGNETICS Planar Waves in an Source-less Anisotropic Electric Media Intersection of the surface with vez = 0, are a circle and an ellipse in vex, vey plane. 2 22       =+ z eyex n c vv ( ) ( ) 1 // 2 2 2 2 =+ x ey y ex nc v nc v Intersection of the surface with vey = 0, are a circle and an ellipse in vex, vez plane. 2 22         =+ y ezex n c vv ( ) ( ) 1 // 2 2 2 2 =+ x z z ex nc k nc v Intersection of the surface with vex = 0, are a circle and an ellipse in vey, vez plane. 2 22         =+ x ezey n c vv ( ) ( ) 1 // 2 2 2 2 =+ y ez z ey nc v nc v Suppose nx < ny < nz. Ray-Velocity Surface (continue – 2) The previous equation has the same structure as the Wave-Vector Surface, therefore can be represented by a three-dimensional surface in ve space. vex, vey and vez are the components of ve in the principal axes of the dielectric. 0 2 0 2 0 2 ,, ε ε ε ε ε ε z z y y x x nnn ===
104 SOLO ELECTROMAGNETICS Planar Waves in an Source-less Anisotropic Electric Media nx < ny < nz. The complete ve surface is double; that is, it consists of an inner shell and an outer shell. For any given direction of the vector ve, there are two possible values for the energy-ray velocity ve (eigenmodes): ve1 and ve2. From the figure we can see that the inner and outer shells of the ve surface intersect at four points in each quadrant of the vex, vez plane (for nx<ny<nz). Those points define two directions for which the two values of ve are equal. Those directions are called Ray Axes and are distinct from the Optic Axes of the dielectric. Ray-Velocity Surface (continue – 3)
105 SOLO ELECTROMAGNETICS Planar Waves in an Source-less Anisotropic Electric Media Let compute the Ray Axis direction. We can see that the Ray Axis is in x, z plane, therefore           ==           = → RA RA RA RARAe RAez ey ex RAe n c Sv v v v v δ δ cos 0 sin 1  2 22         =+ y ezex n c vv ( ) ( ) 1 // 2 2 2 2 =+ x ez z ex nc v nc v 22         =      yRA n c n c 1 cossin 2 22 2 22 =+ RA RAx RA RAz n n n n δδ must be at the intersection ofRAev  and yRA nn = ( ) 1 coscos1 2 22 2 22 =+ − y RAx y RAz n n n n δδ 22 22 2 cos zx zy RA nn nn − − =δ 22 22 2 sin zx yx RA nn nn − − =δ 22 22 2 tan zy yx RA nn nn − − =δ         − − ±= − 22 22 1 2,1 sin zx yx RA nn nn δ 0 2 0 2 0 2 ,, ε ε ε ε ε ε z z y y x x nnn === Ray-Velocity Surface (continue – 4) RA x z yz xy x z OA n n nn nn n n δδ 2 2 2 22 22 2 2 2 tantan = − − = Ray Axes are distinct from Optical Axes.
106 SOLO ELECTROMAGNETICS Planar Waves in an Source-less Anisotropic Electric Media ( ) ( ) ( )                       +−        +−         +−        =                       +−        +−         +−        ++= 22 2 22 2 22 2 22 2 22 2 22 2 2222 det eze z ezeyezex ezeyeye y eyex ezexeyexexe x vvvv eyex z ezeyezex ezeyezex y eyex ezexeyexezey x vv n c vvvv vvvv n c vv vvvvvv n c vv n c vvvv vvvv n c vv vvvvvv n c ezeyexe We obtain         −−        −−                 −+        −+        −         −        +−= 2 2 2 222 2 2 222 2 2 22 2 2 22 2 2 2 2 2 22 2 2 e y ezexe z eyexe y eze z eye z e y exe x v n c vvv n c vvv n c vv n c vv n c v n c vv n c 0 2 2 2 2 2 2 22 2 2 2 2 2 22 2 2 2 2 2 22 2 2 2 2 2 2 2 2 =         −        −+        −        −+        −         −+        −         −        −= e y e x eze z e x eye z e y exe z e y e x v n c v n c vv n c v n c vv n c v n c vv n c v n c v n c Divide by (assumed nonzero!!(         −         −        −− 2 2 2 2 2 2 2 2 2 e z e y e x v n c v n c v n c 1 2 2 2 2 2 2 2 2 2 2 2 2 = − + − + − z e ez y e ey x e ex n c v v n c v v n c v v Ray-Velocity Surface (continue – 5( Return to Table of Content
107 SOLO ELECTROMAGNETICS Planar Waves in an Source-less Anisotropic Electric Media Index Ellipsoid The electric energy density is given by: ∗∗−∗ ⋅⋅=⋅⋅=⋅= EEDDDEUe εε  2 1 2 1 2 1 1 Let find the Surfaces of Equal Energy expressed in the principal dielectric axes: z z y y x x e DDD DDU εεε ε 222 1 2 ++=⋅⋅= ∗− We have: ( ) ( ) zyxin iii ,,// 00 2 === εεεµεµ Define: ezeyex UEzUEyUEx 2/:2/:2/: 000 εεε === The surfaces of equal energy expressed in the principal dielectric axes are: 1 /1/1/1 2 2 2 2 2 2 =++ zyx n z n y n x Fresnel’s Ellipsoid 0ˆˆˆˆ 221121212121212121 =⋅=⋅=⋅=⋅=⋅=⋅=⋅=⋅=⋅=⋅=⋅ HEHEHsHsDsDsEEEDDEDDHH We found that the two eigenmodes, of each direction , must satisfy: s  222 2 zzyyxxe EEEEEU εεεε ++=⋅⋅= ∗
108 SOLO ELECTROMAGNETICS Planar Waves in an Source-less Anisotropic Electric Media Index Ellipsoid (continue – 1( The electric energy density is given by: ∗∗−∗ ⋅⋅=⋅⋅=⋅= EEDDDEUe εε  2 1 2 1 2 1 1 z z y y x x e DDD DDU εεε ε 222 1 2 ++=⋅⋅= ∗− We have: ( ) ( ) zyxin iii ,,// 00 2 === εεεµεµ Define: 000 2/:2/:2/: εεε ezeyex UDzUDyUDx === The surfaces of equal energy expressed in the principal dielectric axes are: 12 2 2 2 2 2 =++ zyx n z n y n x Index Ellipsoid, Optical Indicatrix, Reciprocal Ellipsoid 0ˆˆˆˆ 221121212121212121 =⋅=⋅=⋅=⋅=⋅=⋅=⋅=⋅=⋅=⋅=⋅ HEHEHsHsDsDsEEEDDEDDHH We found that the two eigenmodes, of each direction , must satisfy: s  222 2 zzyyxxe EEEEEU εεεε ++=⋅⋅= ∗ Let find the Surfaces of Equal Energy expressed in the principal dielectric axes:
109 SOLO ELECTROMAGNETICS Planar Waves in an Source-less Anisotropic Electric Media Index Ellipsoid (continue – 2( Since we found that the index ellipsoid is used to find the two indexes of refraction and the two corresponding directions that are in the plane normal to that passes through ellipsoid center O and defines the ellipse. 0ˆˆ 21 =⋅=⋅ DsDs 21 DandD sˆ Sπ Assume that is one of the expected solutions. 0 2/ εe UDOP = → The normal at P to the ellipsoid PV is: E U z D y D x D U z z y y x x ez z y y x x e zyx PV  2 111 2 111 00 0 ε εεε ε εεε ε =         ++=         ++= ∧∧∧ ∧∧∧→ Since are coplanar PV is in the plane defined by and sˆDπ D sED  ,, The plane tangent to ellipsoid at P is normal to PV ( (, intersects plane along PT that will be normal to and , therefore it gives direction. Tπ E E H Sπ sˆ 0ˆˆ 221121 =⋅=⋅=⋅=⋅ HEHEHsHs
110 SOLO ELECTROMAGNETICS Planar Waves in an Source-less Anisotropic Electric Media Index Ellipsoid (continue – 3( PT is at the intersection of the tangent plane to the ellipsoid at P to the plane normal to and containing PO. Therefore PT is tangent to the ellipse containing PO. Tπ Sπ sˆ - Since PT is in , it is normal toSπ sˆ E - Since PT is in the tangent plane to the ellipsoid, it is normal to PV ( ( Tπ sˆ H EH- Since is also normal to and PT defines the direction of is also normal toH D The tangent to an ellipse is only normal to the radius vector if it is one of the axes of the ellipse. We conclude that the directions of are the directions of the axes of the ellipse obtained by the intersection of the ellipsoid with the plane normal to . 21 DandD sˆ
111 SOLO ELECTROMAGNETICS Planar Waves in an Source-less Anisotropic Electric Media Index Ellipsoid (continue – 4( To find for the propagation direction we find the ellipse obtained by the intersection of the plane normal to , at the origin O, with the Index Ellipsoid. 21 DandD sˆ sˆSπ 21 DandD are in the direction of ellipsoid axes, and therefore perpendicular to each other is on the normal to ellipsoid at OP = and in the plane of and . In the same plane and normal to is E sˆ E Dπ D Dπ HE HE S × × = → 1
112 SOLO ELECTROMAGNETICS Planar Waves in an Source-less Anisotropic Electric Media Index Ellipsoid (continue – 5( 12 2 2 2 2 2 =++ zyx n z n y n xnx < ny < nz. For the propagation direction of the Optical Axis the intersection of the plane normal to , at the origin O, with the Index Ellipsoid is a Circle. OAsˆSπ OAsˆ 1 D 21 DandD are in any direction on this circle, and therefore for any we can find the perpendicular to it.2 D Conical Refraction on the Optical Axis We want to find the direction of vectors. E We have: [ ] ET E E E EsDs z y x zxOAOA ⋅=           ⋅=⋅⋅=⋅=  δεδεε cos0sinˆˆ0 where: [ ]T zxT δεδε cos0sin:=            = δ δ cos 0 sin ˆOAsFor nx < ny < nz we found that in principal dielectric axes         − − ±= −− −− − 22 22 1 2,1 sin zx yx nn nn δ SDEs ,,,ˆ are coplanar and normal toH We also found: DE DD n ⋅ ⋅ = 0 2 1 ε const n D DE y ==⋅ 2 0 2 ε We found: y nn =
113 SOLO ELECTROMAGNETICS Planar Waves in an Source-less Anisotropic Electric Media Index Ellipsoid (continue – 6( 12 2 2 2 2 2 =++ zyx n z n y n xnx < ny < nz. Conical Refraction on the Optical Axis (continue – 1( OAsˆ Let draw in principal dielectric axes coordinate O,x,y,z, starting from O, the vectors and . They define the plane Bπ T  0=⋅ ET  Since we can find in vectors such thatEBπ constDE =⋅ and , , and are in the same plane. E OAsˆD HE HE S × × = → 1 Tπ Therefore the locus of is the projection of the circle of from plane to an ellipse in plane having the Oy axis in common. E D Sπ Since are in the same plane with and will be on a conical surface with O as a vertex having and on its surfacesT  OAsˆDHE HE S × × = → 1 → S1 OAsˆ
114 SOLO ELECTROMAGNETICS Planar Waves in an Source-less Anisotropic Electric Media Index Ellipsoid (continue – 7( Conical Refraction on the Optical Axis (continue – 2( Take a biaxial crystal and cut it so that two parallel faces are perpendicular to the Optical Axis. If a monochromatic unpolarized light is normal to one of the crystal faces, the energy will spread out in the plate in a hollow cone, the cone of internal conical refraction. When the light exits the crystal the energy and wave directions coincide, and the light will form a hollow cylinder. This phenomenon was predicted by William Rowan Hamilton in 1832 and confirmed experimentally by Lloyd, a year later (Born & Wolf(. Because it is no easy to obtain an accurate parallel beam of monochromatic light on obtained two bright circles (Born & Wolf(. William Rowan Hamilton (1805-1855(
115 SOLO ELECTROMAGNETICS Planar Waves in an Source-less Anisotropic Electric Media Index Ellipsoid (continue – 8( Conical Refraction on the Optical Axis (continue – 3( Return to Table of Content
116 SOLO ELECTROMAGNETICS Planar Waves in an Source-less Anisotropic Electric Media BEs c n DHs c n   =× −=× ˆ ˆ HHIB ED   µµ ε =⋅= ⋅= ( ) E n c D n c Hs n c Bs n c Ess ⋅−=−=×=×=×× εµµµ  2 2 2 2 ˆˆˆˆ 1ˆˆ 0 0 0 ˆˆ 222 =++=⋅           − − − =×⇒           = zyx xy xz yz z y x sssss ss ss ss s s s s s           = z y x ε ε ε ε 00 00 00                    == 0 2 0 2 0 2 22 2 00 00 00 1 ε ε ε ε ε ε ε µε µεµ n n n nn c z y x  ( ) ( ) ( ) ( )             +− +− +− =             +− +− +− =           − − −           − − − =×× 2 2 2 22 22 22 1 1 1 0 0 0 0 0 0 ˆˆ zzyzx zyyyx zxyxx yxzyzx zyzxyx zxyxzy xy xz yz xy xz yz sssss sssss sssss ssssss ssssss ssssss ss ss ss ss ss ss ss Equation of Wave Normal ( ) 0ˆˆ 2 2 =⋅+×× E n c Ess εµ  ( ) 0ˆˆ 2 2 1 =+×× − D n c Dss µε  or
117 SOLO ELECTROMAGNETICS Planar Waves in an Source-less Anisotropic Electric Media ( ) 0ˆˆ 2 2 =⋅+×× E n c Ess εµ  0 1 1 1 2 0 2 2 0 2 2 0 2 =                             +− +− +− z y x z z zyzx zyy y yx zxyxx x E E E s n ssss sss n ss sssss n ε ε ε ε ε ε The solution is obtained when       −      −+      −      −+      −      −+      −      −      −= 111111111 0 2 0 2 2 0 2 0 2 2 0 2 0 2 2 0 2 0 2 0 2 ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε nn s nn s nn s nnn yx z zx y yz x zyx                   +− +− +− = 2 0 2 2 0 2 2 0 2 1 1 1 det0 z z zyzx zyy y yx zxyxx x s n ssss sss n ss sssss n ε ε ε ε ε ε       −−      −−            −+      −+      −      −      +−= 1111111 0 2 22 0 2 22 0 2 2 0 2 2 0 2 0 2 2 0 2 ε ε ε ε ε ε ε ε ε ε ε ε ε ε n ss n ss n s n s nn s n y zx z yx y z z y zy x x Equation of Wave Normal (continue – 1(
118 SOLO ELECTROMAGNETICS Planar Waves in an Source-less Anisotropic Electric Media 0111111111 1 1 1 det 0 2 0 2 2 0 2 0 2 2 0 2 0 2 2 0 2 0 2 0 2 2 0 2 2 0 2 2 0 2 =        −        −+        −        −+        −        −+        −        −        −= =                     +− +− +− ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε nn s nn s nn s nnn s n ssss sss n ss sssss n yx z zx y yz x zyx z z zyzx zyy y yx zxyxx x Divide by (assumed nonzero!!(       −      −      −− 111 0 2 0 2 0 2 2 ε ε ε ε ε ε nnn n zyx 2 0 2 2 0 2 2 0 2 2 1 nn s n s n s z z y y x x = − + − + − ε ε ε ε ε ε 222 2 22 2 22 2 1 nnn s nn s nn s z z y y x x = − + − + − 0 2 0 2 0 2 ,, ε ε ε ε ε ε z z y y x x nnn === Equation of Wave Normal (continue – 2( Fresnel equation of wave normal Fresnel's wave surface, found by Augustin-Jean Fresnel in 1821, is a quartic surface describing the propagation of light in an optically biaxial crystal Augustin-Jean Fresnel (1788 – 1827(
119 SOLO ELECTROMAGNETICS Planar Waves in an Source-less Anisotropic Electric Media Equation of Wave Normal (continue – 3( Fresnel equation of wave normal 02 0 2 0 222 0 2 0 222 0 2 0 222 0 2 0 2 0 =        −        −+        −        −+        −        −+        −        −        − nnsnnnsnnnsnnnn yx z zx y yz x zyx ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε 022 00 42 00 6222 00 42 00 6222 00 42 00 62 000 2 000000 4 000 6 =+        +−++        +−++        +−+ +        ++−        +++− nsnsnsnsnsnsnsnsns nnn z yx z yx zy zx y zx yx zy x zy x zyxzyzxyxzyx ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ( ) ( ) ( ) 0111 000 22 00 2 00 2 00 42 0 2 0 2 0 =+      −+−+−−        ++ ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε zyx x zy y zx z yx z z y y x x nsssnsss 0111111111 0 2 0 2 2 0 2 0 2 2 0 2 0 2 2 0 2 0 2 0 2 =        −        −+        −        −+        −        −+        −        −        − ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε nn s nn s nn s nnn yx z zx y yz x zyx 6 n This is a quadratic equation of n2 . For each set sx, sy, sz it yields two solutions for n2 : n1 2 and n2 2 ( )snnn ˆ,, 21 ε  = 222 2 22 2 22 2 1 nnn s nn s nn s z z y y x x = − + − + − 1 222 =++ zyx sss Return to Table of Content
Maxwell Equations & Propagation in Anisotropic Media (39
Maxwell Equations & Propagation in Anisotropic Media (39
Maxwell Equations & Propagation in Anisotropic Media (39
Maxwell Equations & Propagation in Anisotropic Media (39
Maxwell Equations & Propagation in Anisotropic Media (39
Maxwell Equations & Propagation in Anisotropic Media (39
Maxwell Equations & Propagation in Anisotropic Media (39
Maxwell Equations & Propagation in Anisotropic Media (39
Maxwell Equations & Propagation in Anisotropic Media (39
Maxwell Equations & Propagation in Anisotropic Media (39
Maxwell Equations & Propagation in Anisotropic Media (39
Maxwell Equations & Propagation in Anisotropic Media (39
Maxwell Equations & Propagation in Anisotropic Media (39
Maxwell Equations & Propagation in Anisotropic Media (39
Maxwell Equations & Propagation in Anisotropic Media (39
Maxwell Equations & Propagation in Anisotropic Media (39
Maxwell Equations & Propagation in Anisotropic Media (39
Maxwell Equations & Propagation in Anisotropic Media (39
Maxwell Equations & Propagation in Anisotropic Media (39
Maxwell Equations & Propagation in Anisotropic Media (39
Maxwell Equations & Propagation in Anisotropic Media (39

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Maxwell Equations & Propagation in Anisotropic Media (39

  • 1. 1 Maxwell Equations & Propagation in Anisotropic Electric Media SOLO HERMELIN Updated 12.06.06 23.12.12 http://www.solohermelin.com
  • 2. 2 Maxwell Equations & Propagation in Anisotropic MediaSOLO TABLE OF CONTENT Maxwell’s Equations History of Maxwell’s Equations Symmetric Maxwell’s Equations Constitutive Relation Boundary Conditions Energy and Momentum Monochromatic Planar Wave Equations Planar Waves in an Source-less Anisotropic Electric Media Lorentz’s Lemma Planar Wave Group Velocity Phase Velocity, Energy (Ray) Velocity for Planar Waves Energy Flux and Poynting Vector Energy Flux and Poynting Vector for a Bianisotropic Medium Wave equation Wave-Vector Surface Double Refraction at a Uniaxial Crystal Boundary Refractive Index of Some Typical Crystals Polarization History
  • 3. 3 Maxwell Equations & Propagation in Anisotropic MediaSOLO TABLE OF CONTENT Planar Waves in an Source-less Anisotropic Electric Media Orthogonality Properties of the Eigenmodes Phase Velocity, Energy (Ray) Velocity for Planar Waves Ray-Velocity Surface Index Ellipsoid Conical Refraction on the Optical Axis Equation of Wave Normal References
  • 4. 4 POLARIZATION Erasmus Bartholinus,doctor of medicine and professor of mathematics at the University of Copenhagen, showed in 1669 that crystals of “Iceland spar” (which we now call calcite, CaCO3) produced two refracted rays from a single incident beam. One ray, the “ordinary ray”, followed Snell’s law, while the other, the “extraordinary ray”, was not always even in the plan of incidence. SOLO History Erasmus Bartholinus 1625-1698 http://www.polarization.com/history/history.html
  • 5. 5 POLARIZATIONSOLO History Étienne Louis Malus 1775-1812 Etienne Louis Malus, military engineer and captain in the army of Napoleon, published in 1809 the Malus Law of irradiance through a Linear polarizer: I(θ)=I(0) cos2 θ. In 1810 he won the French Academy Prize with the discovery that reflected and scattered light also possessed “sidedness” which he called “polarization”.
  • 6. 6 POLARIZATION Arago and Fresnel investigated the interference of polarized rays of light and found in 1816 that two rays polarized at right angles to each other never interface. SOLO History (continue) Dominique François Jean Arago 1786-1853 Augustin Jean Fresnel 1788-1827 Arago relayed to Thomas Young in London the results of the experiment he had performed with Fresnel. This stimulate Young to propose in 1817 that the oscillations in the optical wave where transverse, or perpendicular to the direction of propagation, and not longitudinal as every proponent of wave theory believed. Thomas Young 1773-1829
  • 7. 7 POLARIZATIONSOLO http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html Methods of Achieving Polarization Polarization is based on one of four fundamental physical mechanisms: 1. Dichroism or selective absorbtion 3. Reflection 2. Scattering 4. Birefrigerence To obtain Polarization we must have some asymmetry in the optical process.
  • 8. 8 POLARIZATIONSOLO http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html Methods of Achieving Polarization Polarization by Birefrigerence (continue – 1) Polarization can be achieved with crystalline materials which have a different index of refraction in different planes. Such materials are said to be birefringent or doubly refracting. Nicol Prism The Nicol Prism (1828) is made up from two prisms of calcite cemented with Canada balsam. The ordinary ray can be made to totally reflect off the prism boundary, leving only the extraordinary ray. William Nicol(1768 ?– 1851) Scottish physicist
  • 9. 9 POLARIZATIONSOLO http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html Methods of Achieving Polarization Polarization by Birefrigerence (continue – 2) Polarization can be achieved with crystalline materials which have a different index of refraction in different planes. Such materials are said to be birefringent or doubly refracting. Wollaston Prism William Hyde Wollaston 1766-1828
  • 10. 10 POLARIZATIONSOLO http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html Methods of Achieving Polarization Polarization by Birefrigerence (continue – 3) Polarization can be achieved with crystalline materials which have a different index of refraction in different planes. Such materials are said to be birefringent or doubly refracting. Glan-Foucault Polarizer
  • 13. 13 MAXWELL’s EQUATIONSSOLO Magnetic Field IntensityH  [ ]1− ⋅mA Electric DisplacementD  [ ]2− ⋅⋅ msA Electric Field IntensityE  [ ]1− ⋅mV Magnetic InductionB  [ ]2− ⋅⋅ msV Current DensityJ  [ ]2− ⋅mA Free Charge Distributionρ [ ]3− ⋅⋅ msA James Clerk Maxwell (1831-1879) A Dynamic Theory of Electromagnetic Field 1864 Treatise on Electricity and Magnetism 1874 Return to Table of Content
  • 14. 14 SOLO Electrostatics Charles-Augustin de Coulomb 1736 - 1806 In 1785 Coulomb presented his three reports on Electricity and Magnetism: -Premier Mémoire sur l’Electricité et le Magnétisme [2]. In this publication Coulomb describes “How to construct and use an electric balance (torsion balance) based on the property of the metal wires of having a reaction torsion force proportional to the torsion angle”. Coulomb also experimentally determined the law that explains how “two bodies electrified of the same kind of Electricity exert on each other”. -Sécond Mémoire sur l’Electricité et le Magnétisme [3]. In this publication Coulomb carries out the “determination according to which laws both the Magnetic and the Electric fluids act, either by repulsion or by attraction”. -Troisième Mémoire sur l’Electricité et le Magnétisme [4]. “On the quantity of Electricity that an isolated body loses in a certain time period , either by contact with less humid air, or in the supports more or less idio-electric”.
  • 15. 15 SOLO Electrostatics Charles-Augustin de Coulomb 1736 - 1806 1 2 12 123 0 12 1 4 q q F r rπ ε =   Coulomb’s Law q1 – electric charge located at 1r  q2 – electric charge located at 2r  12 1 2r r r= −    The electric force that the charge q2 exerts on q1 is given by: If the two electrical charges have the same sign the force is repulsive, if they have opposite signs is attractive. 12 0 8.854187817 10 /Farad mε − = ×Permittivity of vacuum Accord to the Third Newton Law of mechanics: 21 12F F= −   12 1 12F q E=   Define 2 12 123 0 12 1 4 q E r rπ ε =   where is the Electric Field Intensity [N/C]
  • 16. 16 SOLO During an evening lecture in April 1820, Ørsted discovered experimental evidence of the relationship between electricity and magnetism. While he was preparing an experiment for one of his classes, he discovered something that surprised him. In Oersted's time, scientists had tried to find some link between electricity and magnets, but had failed. It was believed that electricity and magnetism were not related. As Oersted was setting up his materials, he brought a compass close to a live electrical wire and the needle on the compass jumped and pointed to the wire. Oersted was surprised so he repeated the experiemnt several times. Each time the needle jumped toward the wire. This phenomenon had been first discovered by the Italian jurist Gian Domenico Romagnosi in 1802, but his announcement was ignored. 1820Electromagnetism Hans Christian Ørsted 1777- 1851
  • 17. 17 SOLO 1820Electromagnetism André-Marie Ampère 1775 - 1836 Danish physicist Hans Christian Ørsted's discovered in 1820 that a magnetic needle is deflected when the current in a nearby wire varies - a phenomenon establishing a relationship between electricity and magnetism. Ørsted's work was reported the Academy in Paris on 4 September 1820 by Arago and a week later Arago repeated Ørsted's experiment at an Academy meeting. Ampère demonstrated various magnetic / electrical effects to the Academy over the next weeks and he had discovered electrodynamical forces between linear wires before the end of September. He spoke on his law of addition of electrodynamical forces at the Academy on 6 November 1820 and on the symmetry principle in the following month. Ampère wrote up the work he had described to the Academy with remarkable speed and it was published in the Annales de Chimie et de Physique. Ampère and Arago investigate magnetism dl uu an infinitesimal element of the contour C J  curent density A/m2 dS uu a differential vector area of the surface S enclosed by contour C Ampère’s Law Magnetic Field IntensityH  [ ]1− ⋅mA
  • 18. 18 SOLO 1820 Electromagnetism Biot-Savart Law 0 2 1 4 rI dL dB r µ π × = uu u uu Magnetic Field of a current element Jean-Baptiste Biot 1774-1862 ( )( ) 1 1 2 21 1 1 21 1 2 1 20 1 2 3 1 2 4 c c c F I dl B dl dl r r I I r r µ π = × × × − = − ∫ ∫∫ uu  uu uu     Ampère was not the only one to react quickly to Arago's report of Orsted's experiment. Biot, with his assistant Savart, also quickly conducted experiments and reported to the Academy in October 1820. This led to the Biot-Savart Law. Félix Savart 1791 - 1841
  • 19. 19 SOLO 1820 Electromagnetism Biot-Savart Law Jean-Baptiste Biot 1774-1862 ( ) ( )2 30 0 ' ' 4 ' J r A J A r d r r r µ µ π ∇ = − ⇒ = −∫       0 0 H J B H B B A µ∇× = = ∇× = → = ∇×       ( ) ( ) 2 0B A A A Jµ∇× = ∇× ∇× = ∇ ∇× −∇ =     choose 0A∇× =  ( ) ( ) ( ) ( ) ( )3 30 0 3 ' ' ' ' ' 4 ' 4 ' r r J r J r r r B r A r d r d r r r r r µ µ π π   × − = ∇ × = ∇ × = ÷ ÷− −  ∫ ∫            Where we used ( ) ( ) 0 ' : 1/ 'r r rJ J r J r rφ φ φ φ∇ × = ∇ × + ∇ × = −       14243 Derivation of Biot-Savart Law from Ampère’s Law Poison’s Equation Solution for an unbounded volume Ampère Law Félix Savart 1791 - 1841
  • 20. 20 SOLO 1831Electromagnetism On 29th August 1831, using his "induction ring", Faraday made one of his greatest discoveries - electromagnetic induction: the "induction" or generation of electricity in a wire by means of the electromagnetic effect of a current in another wire. The induction ring was the first electric transformer. In a second series of experiments in September he discovered magneto-electric induction: the production of a steady electric current. To do this, Faraday attached two wires through a sliding contact to a copper disc. By rotating the disc between the poles of a horseshoe magnet he obtained a continuous direct current. This was the first generator. Michael Faraday 1791- 1867 Magnetic Field IntensityH  [ ]1− ⋅mA Electric DisplacementD  [ ]2− ⋅⋅ msA Electric Field IntensityE  [ ]1− ⋅mV Magnetic InductionB  [ ]2− ⋅⋅ msV →→ ⋅ ∂ ∂ −=⋅ ∫∫∫ dS t B dlE SC   t B E ∂ ∂ −=×∇  → dl → dS C B  E  The voltage induced in a coil moving through a non-uniform magnetic field was demonstrated by this apparatus. As the coil is removed from the field of the bar magnets, the coil circuit is broken and a spark is observed at the gap. The first transformer: Two coils wound on an iron toroid. http://www.ece.umd.edu/~taylor/frame1.htm
  • 21. 21 MAXWELL’s EQUATIONS 1. AMPÈRE’s CIRCUIT LW (A) (1820) SOLO 2. FARADAY’s INDUCTION LAW (F) (1831) Electric DisplacementD  [ ]2− ⋅⋅ msA Magnetic Field IntensityH  [ ]1− ⋅mA Current DensityJ  [ ]2− ⋅mA André-Marie Ampère 1775-1836 →→ ⋅ ∂ ∂ −=⋅ ∫∫∫ dS t B dlE SC   t B E ∂ ∂ −=×∇  → dl → dS C B  E  Electric Field IntensityE  [ ]1− ⋅mV Magnetic InductionB  [ ]2− ⋅⋅ msV Michael Faraday 1791-1867 The following four equations describe the Electromagnetic Field and where first given by Maxwell in 1864 (in a different notation) and are known as MAXWELL’s EQUATIONS
  • 22. 22 MAXWELL’s EQUATIONSSOLO 4. GAUSS’ LAW – MAGNETIC (GM) dV → dS V 0=⋅∫∫ → S dSB  B  0=⋅∇ B  Magnetic InductionB  [ ]2− ⋅⋅ msV 3. GAUSS’ LAW – ELECTRIC (GE) dV → dS V ∫∫∫∫∫ =⋅ → VS dVdSD ρ  D  ρ=⋅∇ D  ρ Electric DisplacementD  [ ]2− ⋅⋅ msA Free Charge Distributionρ [ ]3− ⋅⋅ msA GAUSS’ ELECTRIC (GE) & MAGNETIC (GM) LAWS developed by Gauss in 1835, but published in 1867. Karl Friederich Gauss 1777-1855 Return to Table of Content
  • 23. 23 James C. Maxwell (1831-1879) ELECTROMAGNETICSSOLO Magnetic Field IntensityH  [ ]1− ⋅mA Electric DisplacementD  [ ]2− ⋅⋅ msA Electric Field IntensityE  [ ]1− ⋅mV Magnetic InductionB  [ ]2− ⋅⋅ msV Electric Current DensityeJ  [ ]2− ⋅mA Free Electric Charge Distributioneρ [ ]3− ⋅⋅ msA Fictious Magnetic Current DensitymJ  [ ]2− ⋅mV Fictious Free Magnetic Charge Distributionmρ [ ]3− ⋅⋅ msV MAXWELL’S SYMMETRIC EQUATIONS FOR THE ELECTROMAGNETIC FIELD e S e C J t D HSd t D JdlH      + ∂ ∂ =×∇⇒•        ∂ ∂ +=• ∫∫∫ → )1( AMPÈRE’S LAW t B JESd t B JdlE m S m C ∂ ∂ −−=×∇⇒•        ∂ ∂ +−=• ∫∫∫ →     )2( FARADAY’S LAW e V e S DdvSdD ρρ =•∇⇒=• ∫∫∫∫∫  )3( GAUSS’ ELECTRIC LAW m V m S BdvSdB ρρ =•∇⇒=• ∫∫∫∫∫  )4( GAUSS’ MAGNETIC LAW
  • 24. 24 ELECTROMAGNETICSSOLO SYMMETRIC MAXWELL’s EQUATIONS Magnetic Field IntensityH  [ ]1− ⋅mA Electric DisplacementD  [ ]2− ⋅⋅ msA Electric Field IntensityE  [ ]1− ⋅mV Magnetic InductionB  [ ]2− ⋅⋅ msV Electric Current DensityeJ  [ ]2− ⋅mA Free Electric Charge Distributioneρ [ ]3− ⋅⋅ msA Fictious Magnetic Current DensitymJ  [ ]2− ⋅mV Fictious Free Magnetic Charge Distributionmρ [ ]3− ⋅⋅ msV 1. AMPÈRE’S CIRCUIT LW (A) eJ t D H    + ∂ ∂ =×∇ 2. FARADAY’S INDUCTION LAW (F) mJ t B E    − ∂ ∂ −=×∇ 3. GAUSS’ LAW – ELECTRIC (GE) eD ρ=⋅∇  4. GAUSS’ LAW – MAGNETIC (GM) mB ρ=⋅∇  Although magnetic sources are not physical they are often introduced as electrical equivalents to facilitate solutions of physical boundary-value problems. André-Marie Ampère 1775-1836 Michael Faraday 1791-1867 Karl Friederich Gauss 1777-1855
  • 25. 25 ELECTROMAGNETICSSOLO SYMMETRIC MAXWELL’s EQUATIONS (continue – 1) eJ t D H    + ∂ ∂ =×∇mJ t B E    − ∂ ∂ −=×∇ eD ρ=⋅∇  mB ρ=⋅∇  From the Symmetric Maxwell’s Equations we can see that we obtain the same equations by performing the following operations. DUALITY           ⇓           ⇓           − ⇓           ⇓             − ⇓             ⇓             − ⇓             ⇓             ⇓             − ⇓ µ ε ε µ ρ ρ ρ ρ e m m e e m m e J J J J E H H E B D D B             The Maxwell’s Equations are symmetric and dual. eJ t D H    + ∂ ∂ =×∇ mJ t B E    − ∂ ∂ −=×∇ mB ρ=⋅∇  eD ρ=⋅∇ 
  • 26. 26 ELECTROMAGNETICSSOLO SYMMETRIC MAXWELL’s EQUATIONS (continue – 2) From the Symmetric Maxwell’s Equations ( ) 00 = ∂ ∂ +⋅∇⇒⋅∇+⋅∇ ∂ ∂ =×∇⋅∇=⇒           =⋅∇ + ∂ ∂ =×∇ t JJD t H D J t D H e ee e e ρ ρ      ( ) 00 = ∂ ∂ +⋅∇⇒⋅∇−⋅∇ ∂ ∂ −=×∇⋅∇=⇒           =⋅∇ − ∂ ∂ −=×∇ t JJB t E B J t B E m mm m m ρ ρ      CONSERVATION OF ELECTRICAL AND MAGNETIC CHARGES 0= ∂ ∂ +⋅∇ t J e e ρ 0= ∂ ∂ +⋅∇ t J m m ρ Therefore Return to Table of Content
  • 27. 27 ELECTROMAGNETICSSOLO CONSTITUTIVE RELATIONS Homogeneous Medium – Medium properties do not vary from point to point and are the same for all points.ε µ Isotropic Medium – Medium properties are the same in all directions and are scalars.ε µ Linear Medium – The effects of all different fields can be added linearly (Superposition of different fields). For Linear and Isotropic Medium we have: ED  ε= HB  µ= where: 0εε eK= 0µµ mK= - Dielectric Constant (or Relative Permitivity)eK - Relative PermeabilitymK The simpler case:
  • 28. 28 ELECTROMAGNETICSSOLO CONSTITUTIVE RELATIONS (continue - 1) The most general form of Linear Constitutive Relations is: Classification of Media dyadicsxwhere H E B D 33,,, =                 =         µζξε µζ ξε        Classification according to the functional dependence of the 6x6 matrix         µζ ξε   1. Inhomogeneous: function of space coordinates 2. Nonstationary: function of time 3. Time-dispersive: function of time derivatives 4. Space-dispersive: function of space derivatives 5. Nonlinear: function of the electromagnetic field
  • 29. 29 ELECTROMAGNETICSSOLO CONSTITUTIVE RELATIONS (continue - 3) Classification of Media (continue - 2) In general 0,0,0,0  ≠==≠ µζξε Anisotropic Electromagnetic - medium described by both 0,0  ≠≠ µε Anisotropic Electric - medium described by 0  ≠ε Anisotropic Magnetic - medium described by 0  ≠µ If is symmetric, it can be diagonalized0  ≠ε           = z y x ε ε ε ε 00 00 00  Uniaxial zyx εεε ≠= (thetragonal, hexagonal, rombohedral crystals) Biaxial zyx εεε ≠≠ (orthohombic, monoclinic, triclinic crystals) Isotropic zyx εεε == This is called an Anisotropic Medium. If or is Hermitian; i.e.0  ≠ε 0  ≠µ ( ) Transposeconjugate conjugateTranspose a aj ja T z -T,-* ,-H UUUU *H ==           −= 00 0 0 α α Is gyroelectric or gyromagnetic depending on whether stands for or . If both tensors are of this form the medium is gyroelectromagnetic. ε  µ  U         µζ ξε  
  • 30. 30 ELECTROMAGNETICSSOLO CONSTITUTIVE RELATIONS (continue - 4) Classification of Media (continue - 3) In general 0,0,0,0  ≠≠≠≠ µζξε This is called an Bianisotropic Medium. Such properties have been observed in antiferromagnatic chromium oxide ( antiferromagnetic materials are ones in which it is enerically favorable for neighboring dipoles to take an antiparallel orientation; see Ramo, Whinery, and Van Duzer (1965), pg. 145)                 −− −− −= ** ** 4 µµζζ ξξεεω   j G mediumlosslessundefinite mediumpassivedefinitenegative mediumactivedefinitepositive ⇒ ⇒ ⇒ G G G In this case, we have the following classification (see development later) Return to Table of Content
  • 31. 31 SOLO ELECTROMAGNETICS BOUNDARY CONDITIONS Boundary Conditions ( ) ( ) ldtHtHhldtHldtHldH h C 2211 0 2211 ˆˆˆˆ ⋅+⋅=Θ+⋅+⋅=⋅ →→ ∫  where are unit vectors along C in region (1) and (2), respectively, and21 ˆ,ˆ tt 2121 ˆˆˆˆ −×=−= nbtt - a unit vector normal to the boundary between region (1) and (2)21 ˆ −n - a unit vector on the boundary and normal to the plane of curve Cbˆ Using we obtainbaccba  ⋅×≡×⋅ ( ) ( ) ( )[ ] ldbkldbHHnldnbHHldtHH e ˆˆˆˆˆˆ 21212121121 ⋅=⋅−×=×⋅−=⋅− −−  Since this must be true for any vector that lies on the boundary between regions (1) and (2) we must have: bˆ ( ) ekHHn  =−×− 2121 ˆ ∫∫∫ ⋅        ∂ ∂ +=⋅ → S e C Sd t D JdlH    ( ) dlbkbdlh t D JSd t D J e h e S e ˆˆ 0 ⋅=⋅        ∂ ∂ +=⋅        ∂ ∂ + → ∫∫      AMPÈRE’S LAW [ ]1 0 lim: − → ⋅        ∂ ∂ += mAh t D Jk e h e  
  • 32. 32 SOLO ELECTROMAGNETICS BOUNDARY CONDITIONS Boundary Conditions (continue – 1) ( ) ( ) ldtEtEhldtEldtEldE h C 2211 0 2211 ˆˆˆˆ ⋅+⋅=Θ+⋅+⋅=⋅ →→ ∫  where are unit vectors along C in region (1) and (2), respectively, and21 ˆ,ˆ tt 2121 ˆˆˆˆ −×=−= nbtt - a unit vector normal to the boundary between region (1) and (2)21 ˆ −n - a unit vector on the boundary and normal to the plane of curve Cbˆ Using we obtainbaccba  ⋅×≡×⋅ ( ) ( ) ( )[ ] ldbkldbEEnldnbEEldtEE m ˆˆˆˆˆˆ 21212121121 ⋅−=⋅−×=×⋅−=⋅− −−  Since this must be true for any vector that lies on the boundary between regions (1) and (2) we must have: bˆ ( ) mkEEn  −=−×− 2121 ˆ ∫∫∫ ⋅        ∂ ∂ +−=⋅ → S m C Sd t B JdlE    ( ) dlbkbdlh t B JSd t B J m h m S m ˆˆ 0 ⋅=⋅        ∂ ∂ +=⋅        ∂ ∂ + → ∫∫      FARADAY’S LAW [ ]1 0 lim: − → ⋅        ∂ ∂ += mVh t B Jk m h m  
  • 33. 33 SOLO ELECTROMAGNETICS BOUNDARY CONDITIONS Boundary Conditions (continue – 2) ( ) ( ) SdnDnDhSdnDSdnDSdD h S 2211 0 2211 ˆˆˆˆ ⋅+⋅=Θ+⋅+⋅=⋅ → ∫∫  where are unit vectors normal to boundary pointing in region (1) and (2), respectively, and 21 ˆ,ˆ nn 2121 ˆˆˆ −=−= nnn - a unit vector normal to the boundary between region (1) and (2)21 ˆ −n ( ) ( ) SdSdnDDSdnDD eσ=⋅−=⋅− −2121121 ˆˆ  Since this must be true for any dS on the boundary between regions (1) and (2) we must have: ( ) eDDn σ=−⋅− 2121 ˆ  ( ) dSdShdv e h e V e σρρ 0→ ==∫∫∫ GAUSS’ LAW - ELECTRIC [ ]1 0 lim: − → ⋅⋅= msAhe h e ρσ ∫∫∫∫∫ =• V e S dvSdD ρ 
  • 34. 34 SOLO ELECTROMAGNETICS BOUNDARY CONDITIONS Boundary Conditions (continue – 3) ( ) ( ) SdnBnBhSdnBSdnBSdB h S 2211 0 2211 ˆˆˆˆ ⋅+⋅=Θ+⋅+⋅=⋅ → ∫∫  where are unit vectors normal to boundary pointing in region (1) and (2), respectively, and 21 ˆ,ˆ nn 2121 ˆˆˆ −=−= nnn - a unit vector normal to the boundary between region (1) and (2)21 ˆ −n ( ) ( ) SdSdnBBSdnBB mσ=⋅−=⋅− −2121121 ˆˆ  Since this must be true for any dS on the boundary between regions (1) and (2) we must have: ( ) mBBn σ=−⋅− 2121 ˆ  ( ) dSdShdv m h m V m σρρ 0→ ==∫∫∫ GAUSS’ LAW – MAGNETIC [ ]1 0 lim: − → ⋅⋅= msVhm h m ρσ ∫∫∫∫∫ =• V m S dvSdB ρ 
  • 35. 35 SOLO ELECTROMAGNETICS BOUNDARY CONDITIONS Boundary Conditions (summary) ( ) mkEEn  −=−×− 2121 ˆ FARADAY’S LAW ( ) ekHHn  =−×− 2121 ˆ AMPÈRE’S LAW [ ]1 0 lim: − → ⋅        ∂ ∂ += mAh t D Jk e h e   [ ]1 0 lim: − → ⋅        ∂ ∂ += mVh t B Jk m h m   ( ) eDDn σ=−⋅− 2121 ˆ  GAUSS’ LAW ELECTRIC [ ]1 0 lim: − → ⋅⋅= msAhe h e ρσ ( ) mBBn σ=−⋅− 2121 ˆ  GAUSS’ LAW MAGNETIC [ ]1 0 lim: − → ⋅⋅= msVhm h m ρσ
  • 36. 36 SOLO ELECTROMAGNETICS BOUNDARY CONDITIONS Boundary Conditions for Perfect Electric Conductor (PEC) ( ) 0ˆ 2121  =−×− EEn FARADAY’S LAW ( ) ekHHn  =−×− 2121 ˆ AMPÈRE’S LAW ( ) eDDn σ=−⋅− 2121 ˆ  GAUSS’ LAW ELECTRIC ( ) 0ˆ 2121 =−⋅− BBn  GAUSS’ LAW MAGNETIC ekHn  =×− 121 ˆ 0ˆ 121  =×− En eDn σ=⋅− 121 ˆ  0ˆ 121 =⋅− Bn  02  =H 02  =B 02  =E 02  =D
  • 37. 37 SOLO ELECTROMAGNETICS BOUNDARY CONDITIONS Boundary Conditions for Perfect Magnetic Conductor (PMC) ( ) mkEEn  −=−×− 2121 ˆ FARADAY’S LAW ( ) 0ˆ 2121  =−×− HHn AMPÈRE’S LAW ( ) 0ˆ 2121 =−⋅− DDn  GAUSS’ LAW ELECTRIC ( ) mBBn σ=−⋅− 2121 ˆ  GAUSS’ LAW MAGNETIC 0ˆ 121  =×− Hn mkEn  −=×− 121 ˆ 0ˆ 121 =⋅− Dn  mBn σ=⋅− 121 ˆ  02  =H 02  =B 02  =E 02  =D Return to Table of Content
  • 38. 38 SOLO ELECTROMAGNETICS BOUNDARY CONDITIONS Energy and Momentum Let start from Ampère and Faraday Laws               ∂ ∂ −=×∇⋅ + ∂ ∂ =×∇⋅− t B EH J t D HE e      EJ t D E t B HHEEH e      ⋅− ∂ ∂ ⋅− ∂ ∂ ⋅−=×∇⋅−×∇⋅ ( )HEHEEH  ×⋅∇=×∇⋅−×∇⋅But Therefore we obtain ( ) EJ t D E t B HHE e      ⋅− ∂ ∂ ⋅− ∂ ∂ ⋅−=×⋅∇ First way This theorem was discovered by Poynting in 1884 and later in the same year by Heaviside.
  • 39. 39 SOLO ELECTROMAGNETICS BOUNDARY CONDITIONS Energy and Momentum (continue -1) We identify the following quantities -Power density of the current densityEJe  ⋅ ( )HEDE t BH t EJe  ×⋅∇−      ⋅ ∂ ∂ −      ⋅ ∂ ∂ −=⋅ 2 1 2 1       ⋅ ∂ ∂ =⋅= BH t pBHw mm  2 1 , 2 1       ⋅ ∂ ∂ =⋅= DE t pDEw ee  2 1 , 2 1 ( )HEpR  ×⋅∇= eJ  -Magnetic energy and power densities, respectively -Electric energy and power densities, respectively -Radiation power density For linear, isotropic electro-magnetic materials we can write( )HBED  00 , µε == ( )DE tt D E ED     ⋅ ∂ ∂ = ∂ ∂ ⋅ = 2 10ε ( )BH tt B H HB     ⋅ ∂ ∂ = ∂ ∂ ⋅ = 2 10µ
  • 40. 40 SOLO ELECTROMAGNETICS BOUNDARY CONDITIONS Energy and Momentum (continue – 3) Let start from the Lorentz Force Equation (1892) on the free charge ( )BvEF e  ×+= ρ Free Electric Chargeeρ [ ]3− ⋅⋅ msA Velocity of the chargev  [ ]1− ⋅sm Electric Field IntensityE  [ ]1− ⋅mV Magnetic InductionB  [ ]2− ⋅⋅ msV Hendrik Antoon Lorentz 1853-1928 eρ Force on the free chargeF  [ ]Neρ Second way
  • 41. 41 SOLO ELECTROMAGNETICS BOUNDARY CONDITIONS Energy and Momentum (continue – 4) The power density of the Lorentz Force the charge ( ) ( ) EJBvEvp e Bvv Jv e ee    ⋅=×+⋅= =×⋅ = 0 ρ ρ or ( ) ( ) ( ) ( ) ( )[ ] ( )HE t B HE t D E t D HEEH E t D HEJp t B E HEHEEH J t D H e e                 ×⋅∇−        ∂ ∂ ⋅+⋅ ∂ ∂ −= ⋅ ∂ ∂ −×⋅∇−×∇⋅= ⋅        ∂ ∂ −×∇=⋅= ∂ ∂ −=∇× ×⋅∇=∇×⋅−∇×⋅ + ∂ ∂ =∇× eρ
  • 42. 42 SOLO ELECTROMAGNETICS BOUNDARY CONDITIONS Energy and Momentum (continue – 5) ( )HEDE t BH t EJe  ×⋅∇−      ⋅ ∂ ∂ −      ⋅ ∂ ∂ −=⋅ 2 1 2 1 Let integrate this equation over a constant volume V ∫∫∫∫∫∫∫∫∫∫∫∫ ×∇−      ⋅−      ⋅−=⋅ VVVV e dvSdvDE td d dvBH td d dvEJ  2 1 2 1 If we have sources in V then instead of we must use E  source EE  + Use Ohm Law (1826) ( )source ee EEJ  += γ         = ∂ ∂ ∫∫∫∫∫∫ VV td d t Georg Simon Ohm 1789-1854 source e e EJE  −= γ 1 For linear, isotropic electro-magnetic materials ( )HBED  00 , µε ==
  • 43. 43 SOLO ELECTROMAGNETICS BOUNDARY CONDITIONS Energy and Momentum (continue – 6) ∫∫∫∫∫∫∫∫∫∫∫∫∫ ⋅∇+      ⋅+      ⋅+=⋅ VVVR n V source e dvSdvDE td d dvBH td d dRIdvEJ  2 1 2 12 ∫∫∫       ⋅= V FieldMagnetic dvBH td d P  2 1 ∫∫∫       ⋅= V FieldElectric dvDE td d P  2 1 ∫∫∫∫∫ ⋅=⋅∇= SV Radiation SdSdvSP  ∫∫∫ ⋅= V source eSource dvEJP  ( ) ( ) ∫∫∫∫ ∫∫∫∫∫∫∫∫∫∫∫∫∫∫∫ ⋅−= ⋅−⋅=⋅−⋅⋅=⋅ V source e R n V source e L S e ee V source e L S e ee V e dvEJdRI dvEJ dS dl dSJdSJdvEJldSdJJdvEJ   2 11 γγ ∫= R nJoule dRIP 2 RadiationFieldMagneticFieldElectricJouleSource PPPPP +++= For linear, isotropic electro-magnetic materials( )HBED  00 , µε == R – Electric Resistance Define the Umov-Poynting vector: [ ]2 / mwattHES  ×= The Umov-Poynting vector was discovered by Umov in 1873, and rediscovered by Poynting in 1884 and later in the same year by Heaviside.
  • 44. 44 ELECTROMAGNETICSSOLO EM People John Henry Poynting 1852-1914 Oliver Heaviside 1850-1925 Nikolay Umov 1846-1915 1873 “Theory of interaction on final distances and its exhibit to conclusion of electrostatic and electrodynamic laws” 1884 1884 Umov-Poynting vector HES  ×= Return to Table of Content
  • 45. 45 ELECTROMAGNETICSSOLO Monochromatic Planar Wave Equations Let assume that can be written as:( ) ( )trHtrE ,,,  ( ) ( ) ( ) ( ) ( ) ( )tjrHtrHtjrEtrE 00 exp,,exp, ωω  == where are phasor (complex) vectors. ( ) ( ){ } ( ){ } ( ) ( ){ } ( ){ }rHjrHrHrEjrErE  ImRe,ImRe +=+= We have ( ) ( ) ( ) ( ) ( )tjrEjtj t rEtrE t 00 expexp, ωωω  = ∂ ∂ = ∂ ∂ Hence ( ) ( ) ( )             =⋅∇ =⋅∇ −−=×∇ +=×∇ ⇒           =⋅∇ =⋅∇ − ∂ ∂ −=×∇ + ∂ ∂ =×∇ = ∂ ∂ m e m e j t m e m e B D JBjE JDjH BGM DGE J t B EF J t D HA ρ ρ ω ω ρ ρ ω         )(
  • 46. 46 ELECTROMAGNETICSSOLO Note The assumption that can be written as:( ) ( )trHtrE ,,,  ( ) ( ) ( ) ( ) ( ) ( )tjrHtrHtjrEtrE 00 exp,,exp, ωω  == is equivalent to saying that has a Fourier Transform; i.e.:( ) ( )trHtrE ,,,  ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∫∫ ∫∫ ∞ ∞− ∞ ∞− ∞ ∞− ∞ ∞− =−= =−= ωωω π ωω ωωω π ωω dtjrHtrHdttjtrHrH dtjrEtrEdttjtrErE exp, 2 1 ,&exp,, exp, 2 1 ,&exp,,   A Sufficient Condition for the Existence of the Fourier Transform is: ( ) ( ) ( ) ( ) ∞<= ∞<= ∫∫ ∫∫ ∞ ∞− ∞ ∞− ∞ ∞− ∞ ∞− ωω π ωω π drHdttrH drEdttrE 22 22 , 2 1 , , 2 1 ,   ( ) ( ) ( ) ( ) ( )[ ] ( ) ( ) ( )[ ] ( ) ( )ωωδωω ωωωω −=−= −=−= ∫ ∫∫ ∞ ∞− ∞ ∞− ∞ ∞− 00 0 exp expexpexp,, rEdttjrE dttjtjrEdttjtrErE   End Note
  • 47. 47 ELECTROMAGNETICSSOLO Fourier Transform The Fourier transform of can be written as:( ) ( )trHtrE ,,,  ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∫∫ ∫∫ ∞ ∞− ∞ ∞− ∞ ∞− ∞ ∞− −== −== dttjtrHrHdtjrHtrH dttjtrErEdtjrEtrE ωωωωω π ωωωωω π exp,,&exp, 2 1 , exp,,&exp, 2 1 ,   ( ) ( ) ( ) ( ) ∞<= ∞<= ∫∫ ∫∫ ∞ ∞− ∞ ∞− ∞ ∞− ∞ ∞− ωω π ωω π drHdttrH drEdttrE 22 22 , 2 1 , , 2 1 ,   JEAN FOURIER 1768-1830 A Sufficient Condition for the Existence of the Fourier Transform is:
  • 48. 48 ELECTROMAGNETICSSOLO ( ) ( )       ×∇−×∇−=×∇×∇ ×∇+×∇=×∇×∇ −−=×∇ +=×∇ ⇒     −−=×∇×∇ +=×∇×∇ = = m e m e ED HB m e JHjE JEjH JHjE JEjH JBjE JDjH ωµ ωε ωµ ωε ω ω ε µ ( ) me JJjEkE  ×∇−−=−×∇×∇ ωµ2 ( ) em JJjHkH  ×∇+−=−×∇×∇ ωε2 λ ππ µεω λ 22 f c c f k = ∆ === Using the vector identity ( ) ( ) ( ) AAA  ∇⋅∇−⋅∇∇=×∇×∇ For a Homogeneous, Linear and Isotropic Media:       =⋅∇ =⋅∇ ⇒    =⋅∇ =⋅∇ = = µ ρ ε ρ ρ ρ ε µ m e ED HB m e H E B D ε ρ ωµ e me JJjEkE ∇ +×∇+=+∇ 22 µ ρ ωε m em JJjHkH ∇ +×∇−=+∇ 22 and we obtain Monochromatic Planar Wave Equations (continue - 1)
  • 49. 49 ELECTROMAGNETICSSOLO Assume no sources: we have Monochromatic Planar Wave Equations (continue - 2) 0,0,0,0 ==== meme JJ ρρ  022 =+∇ EkE 022 =+∇ HkH nkk n k 0 00 00 0 =         == ∆   εµ µε εµωµεω ( ) ( ) ( ) ( ) ( ) ( )    == == ⋅− ⋅− rktjtj rktjtj eHerHtrH eEerEtrE     ωω ωω ω ω 0 0 ,, ,, ( ) 022 =+∇⇒ ⋅−=∇⋅∇⇒−=∇ ⋅− ⋅−⋅−⋅−⋅− rkj rkjrkjrkjrkj ek ekkeekje    Helmholtz Wave Equations satisfy the Helmholtz wave equations( ) ( )ωω ,,, rHrE  ( ) ( )    = = ⋅− ⋅− rkj rkj eHrH eErE     0 0 , , ω ω Assume a progressive wave of phase ( )rkt  ⋅−ω (a regressive wave has the phase) ( )rkt  ⋅+ω For a Homogeneous, Linear and Isotropic Media k  0E 0H r  t k  Planes for which constrkt =⋅−  ω
  • 50. 50 ELECTROMAGNETICSSOLO To satisfy the Maxwell equations for a source free media we must have: Monochromatic Planar Wave Equations (continue - 3) we haveUsing: 1ˆˆ&ˆˆ =⋅== sss c n sk ωεµω         =⋅∇ =⋅∇ −=×∇ =×∇ 0 0 H E HjE EjH ωµ ωε           =⋅ =⋅ =× −=× 0ˆ 0ˆ ˆ ˆ 0 0 00 00 Hs Es HEs EHs ε µ µ ε sˆ Planar Wave 0E 0H r         =⋅− =⋅− −=×− =×− ⇒ ⋅− ⋅− ⋅−⋅− ⋅−⋅− −=∇ ⋅−⋅− 0 0 0 0 00 00 rkj rkj rkjrkj rkjrkj ekje eHkj eEkj eHjeEkj eEjeHkj rkjrkj           ωµ ωε        =⋅ =⋅ =× −=× 0 0 0 0 00 00 Hk Ek HEk EHk     µω εω For a Homogeneous, Linear and Isotropic Media: Return to Table of Content
  • 51. 51 SOLO ELECTROMAGNETICS Planar Waves in an Source-less Anisotropic Electric Media     0 0 0 0 m e m e B D J t B E J t D H ρ ρ =⋅∇ =⋅∇ − ∂ ∂ −=×∇ + ∂ ∂ =×∇           ( ) ( ) ( ) ∫ +∞ ∞− ⋅− = dkekEtrE rktj   ω 0, ωω ωω j t eje t rs c n tjrs c n tj → ∂ ∂ ⇒= ∂ ∂       ⋅−      ⋅−  ˆˆ kjs c n jes c n je rs c n tjrs c n tj  −=−→∇⇒−=∇       ⋅−      ⋅− ˆˆ ˆˆ ωω ωω 0ˆ 0ˆ ˆ ˆ =⋅ =⋅ =× −=× Bs Ds BEs c n DHs c n ( ) ( )DEBH tt D E t B H HEEHHES HB ED         ⋅+⋅ ∂ ∂ −= ∂ ∂ ⋅− ∂ ∂ ⋅−= ×∇⋅−×∇⋅=×⋅∇=⋅∇ ⋅= ⋅= 2 1µ ε ( ) emUDEBHS k Ss c n =⋅+⋅=⋅=⋅ 2 1 ˆ ω  Maxwell’s Symmetric Equations Planar Wave 0 0 =⋅ =⋅ =× −=× Bk Dk BEk DHk     ω ω ωj t → ∂ ∂ s c n j ˆω−→∇ ωj t → ∂ ∂ kj  −→∇ vector phasor vector phasor s c n k ˆω= 
  • 52. 52 SOLO ELECTROMAGNETICS Planar Wave Group Velocity Consider a Planar Wave composed of frequencies centered around ω0. ( ) ( ) ( )( ) ∫ +∞ ∞− ⋅− = dkekEtrE rktkj   ω 0, ( ) ( ) +−        += 0 0 0 kk kd d k ω ωω Expand ω(k), using Taylor series around k0 ( ) ( ) ( ) ( ) ∫ ∞+ ∞− −         − ⋅− −      ⋅− = dkekEetrE kk kk rkk t kd d j rktj 0 0 0 0 00 0,      ω ω The Phase of the Group of Waves is ( ) 0 0 0 0 kk kk rkk t kd d g    −           − ⋅− −      = ω ϕ To find the velocity of phase of the group of waves consider the points for which φg = const ( ) 0 0 0 0 = − ⋅− −      ⇒ kk rdkk td kd d g d   ω ϕ The Group Velocity of the Planar Wave is ( ) 0 0 0 kk kk kd d krd g td rd v      − −       = = = ω δ Return to Table of Content
  • 53. 53 SOLO ELECTROMAGNETICS Phase Velocity, Energy (Ray) Velocity for Planar Waves The phase of the field is given by       ⋅−=⋅−= rs c n trkt  ˆωωφ For a constant phase front we have       ⋅−=⋅−=⋅−== rds c n dtrdskdtrdkdtd  ˆˆ0 ωωωφ s n c s kdt rd v srr const p ˆˆ: ˆ === = = ω φ   Phase Velocity B  D  pv  E  H  HES  ×=α α k  ev  PlanarWaves n c ktd rd s ==⋅ ω  ˆ n c S S vs e =⋅ :ˆ ( ) α α cosˆ ˆ cos p S Ss e v Ss S n c v ⋅ = = ⋅ = Define the phase velocity as Define the energy flow velocity in the Poynting vector direction as S S vSv ee = → 1 ( ) emUDEBHS k Ss c n =⋅+⋅=⋅=⋅ 2 1 ˆ ω  ( ) em ee U S S Ssn c S S vv = ⋅ == ˆ 1 :  Energy (Ray) Velocity The electro-magnetic energy propagates along the Poynting vector . For anisotropic media the Poynting vector is not collinear with .k  HES ×= Return to Table of Content Electromagnetic Energy Density
  • 54. 54 SOLO ELECTROMAGNETICS Planar Waves in an Source-less Anisotropic Electric Media A wave packet can be viewed as a superposition of monochromatic waves, with different frequencies ω and wave vector that must satisfy the Maxwell’s equationsk  EHk HEk ⋅−=× ⋅=× εω µω   Suppose that the wave vector changes by an infinitesimal amount , that determines changes k  k  δ HE δδωδ ,, HHHEkEk ⋅⋅+⋅=×+× δµωµωδδδ  ( )EEEHkHk −⋅⋅−⋅−=×+× δεωεωδδδ  ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )EEEEEHkHEk HHHHHEkHEk δεωεωδδδ δµωµωδδδ ⋅⋅+⋅⋅=×⋅−×⋅ ⋅⋅+⋅⋅=×⋅+×⋅   ( ) ( ) ( ) ( ) 0 2 00 =⋅+×⋅+⋅−×⋅−= ⋅⋅+⋅⋅−×⋅          EHkEHEkH HHEEHEk εωδµωδ µεωδδ ( ) ( ) ( )BACACBCBA  ×⋅≡×⋅≡×⋅
  • 55. 55 SOLO ELECTROMAGNETICS Planar Waves in an Source-less Anisotropic Electric Media ( ) ( ) e em vk U S k HHEE HEk      ⋅=⋅= ⋅⋅+⋅⋅ ×⋅ = δδ µε δ ωδ 2 1 ( ) gk vkk  ⋅=∇⋅= δωδωδ - Energy (Ray) Velocityev  From the definition of Group Velocity: gv  since those relations are true for all: k  δ eg vv  = For a monochromatic wave δω=0, therefore ( ) 0=×⋅=⋅ HEkSk  δδ ( ) ( ) 02 =⋅⋅+⋅⋅−×⋅ HHEEHEk µεωδδ 
  • 56. 56 ELECTROMAGNETICSSOLO Consider an Electric Anisotropic Media General Symmetric        =⋅∇ =⋅∇ −=×∇ ⋅=×∇ 0 0 B D HjE EjH µω εω  For a Homogeneous, Linear and Anisotropic Media:           = 333231 232221 131211 εεε εεε εεε ε            = z y x D ε ε ε ε 00 00 00  Diagonalized Principal Axis TTD  εε 1− =     −−=×∇×∇ +⋅=×∇×∇ m e JHjE JEjH µω εω  ( ) ( )   ×∇−×∇−=×∇×∇ ×∇+⋅×∇=×∇×∇ m e JHjE JEjH ωµ εω  ( ) me JJjEE ×∇−×∇−=×∇×∇ µω ε ε µεω 0 0 2  Planar Waves in an Source-less Anisotropic Electric Media Return to Table of Content
  • 57. 57 ELECTROMAGNETICSSOLO According to the principle of superposition these two fields can exist separately or be superimposed without disturbing each other. ( )1221 HEHE  ×−×⋅∇ Consider two distinct fields and. 11, HE  22 , HE  Assumptions: 1.The two fields are harmonic functions of time and of the same frequency. 2.The medium is linear. 3.The point considered is not inside a source and Ohm Law applies.         ∂ ∂ +⋅+ ∂ ∂ ⋅+        ∂ ∂ +⋅− ∂ ∂ ⋅= t D JE t B H t D JE t B H ee 1 12 2 1 2 21 1 2         12212112 HEEHHEEH  ×∇⋅+×∇⋅−×∇⋅−×∇⋅= ( ) ( )1122122112 DjJEBHjDjJEBHj ee j t ωωωω ω +⋅+⋅++⋅−⋅−= = ∂ ∂ ( ) ( ) 0 0 1122122112 21 ==⋅= ⋅= =⋅+⋅+⋅⋅+⋅+⋅−⋅⋅−= ee JJ ee ED HB EjJEHHjEjJEHHj εωµωεωµω ε µ    Lorentz’s Lemma
  • 58. 58 ELECTROMAGNETICSSOLO Lorentz’s Lemma (continue) ( ) ( )1221 HEHE  ×⋅∇=×⋅∇ For the two distinct fields and. 11, HE  22 , HE  Hendrik Antoon Lorentz 1853-1928 For a Monochromatic Planar Wave ( ) ( ) ( ) ( ) ( ) ( )    == == ⋅− ⋅− rktjtj rktjtj eHerHtrH eEerEtrE     ωω ωω ω ω 0 0 ,, ,, ( ) ( ) ( )rktjrktjrktj eskjekje   ⋅−⋅−⋅− −=−=∇ ωωω ˆ Lorentz’s Lemma can be written ( ) ( )1221 HEkHEk  ×⋅=×⋅
  • 59. 59 ELECTROMAGNETICSSOLO Lorentz’s Reciprocity Theorem ( ) ( )222 111 se se EEJ EEJ   += += σ σ This is the more general case that does not exclude the sources. ( ) 12211221 ee JEJEHEHE  ⋅−⋅=×−×⋅∇Ohm’s Law: are the applied electric field intensities within the sources21, ss EE  ( )1221 HEHE  ×−×⋅∇         ∂ ∂ +⋅+ ∂ ∂ ⋅+        ∂ ∂ +⋅− ∂ ∂ ⋅= t D JE t B H t D JE t B H ee 1 12 2 1 2 21 1 2         12212112 HEEHHEEH  ×∇⋅+×∇⋅−×∇⋅−×∇⋅= ( ) ( )1122122112 DjJEBHjDjJEBHj ee j t ωωωω ω +⋅+⋅++⋅−⋅−= = ∂ ∂ ( ) ( ) 122112211212 21122211122112 esesssssss ssssee JEJEEEEEEEEE EEEEEEEEEEJEJE   ⋅−⋅=⋅−⋅−⋅+⋅= ⋅−⋅=+⋅−+⋅=⋅−⋅= σσσσ σσσσ ( ) ( )1122122112 EjJEHHjEjJEHHj ee ED HB ⋅+⋅+⋅⋅+⋅+⋅−⋅⋅−= ⋅= ⋅= εωµωεωµω ε µ   
  • 60. 60 ELECTROMAGNETICSSOLO Lorentz’s Reciprocity Theorem (continue - 1) Integrate over a very large volume: ( ) 12211221 ee JEJEHEHE  ⋅−⋅=×−×⋅∇ ( ) ( )∫∫ ×−×⋅∇=⋅−⋅ VV eses dvHEHEdvJEJE 12211221  ( ) 11 22 21 21 0 0 0 0 1221 0 VoutsideE VoutsideE EE HH S s s S S sdHEHE      = = == == ⇐=⋅×−×= →∞ →∞ ∫ We obtained ( ) ( )∫∫ ⋅=⋅ V es V es dvJEdvJE 1221  ( ) 212121 bs V es V es IVdsJdlEdvJE =⋅=⋅ ∫∫  ( ) 121212 bs V es V es IVdsJdlEdvJE =⋅=⋅ ∫∫ 
  • 61. 61 ELECTROMAGNETICSSOLO Lorentz’s Reciprocity Theorem (continue - 2) The Reciprocity Theorem 1221 bsbs IVIV = The current induced in 2 when 1 is energized, divided by the voltage applied on 1, is the same as the current induced in 1 when 2 is energized, divided by the applied voltage on 2, as long as the frequency and the impedances remain unchanged. 1221 bbss IIVV =⇔= Return to Table of Content
  • 62. 62 SOLO ( ) ( ) ( ) ( ) ( ) ( )[ ]{ } ( ) ( ) ( ) ( )[ ] ( ) ( ) ( ) ( )[ ] ( ) ( ) ( ) ( ) ( ) ( )[ ]∫ ∫ ∫ ∫∫ −+⋅+= −+⋅−+= = ⋅=⋅= = T T T TEDT e dttjrErErEtjrE T dttjrEtjrEtjrEtjrE T dttjrEal T dttrEtrE T dttrDtrE T w 0 2**2 0 ** 0 2 00 2exp,,,22exp, 4 1 exp,exp,exp,exp, 4 1 exp,Re 1 ,, 1 ,, 1 ωωωωωω ε ωωωωωωωω ε ωωε ε ε      But ( ) ( )[ ] ( ) ( ) ( )[ ] ( ) 0 2 2exp 2exp 2 1 2exp 1 0 2 2exp 2exp 2 1 2exp 1 0 0 0 0 ∞→ ∞→ → − =−=− →== ∫ ∫ T T T T T T Tj Tj tj Tj dttj T Tj Tj tj Tj dttj T ω ω ω ω ω ω ω ω ω ω Therefore ( ) ( ) * 00 * 00 0 * 22 1 ,, 2 EEeEeEdt T rErEw rkjrkj T e εε ωω ε === ⋅−⋅ ∫   Let compute the time averages of the electric and magnetic energy densities ELECTROMAGNETICS Energy Flux and Poynting Vector For a Homogeneous, Linear and Isotropic Media
  • 63. 63 SOLO In the same way ( ) ( ) ( ) ( ) * 00 00 2 ,, 1 ,, 1 HHdttrHtrH T dttrBtrH T w TT m µ µ === ∫∫  Using the relations ( ) 00 ˆ HsEA ×−= ε µ ( ) 00 ˆ EsHF ×= µ ε since and are real values , where * is the complex conjugate, we obtain S∇ )**,( SS ∇=∇= ( ) ( ) ( ) ( ) ( ) ( )[ ] ( ) e m e wHkEHkEHkE EkHEkHHHw HkEHkEEEw =×⋅=×⋅=×⋅= ×⋅=×⋅=⋅= ×⋅=×⋅=⋅= * 00 * 0 * 00 ** 0 * 00 * 00 * 00 * 00 * 00 * 00 ˆ 2 ˆ 2 ˆ 2 ˆ 2 ˆ 22 ˆ 2 ˆ 22 µεµεµε µε µ εµµ µε ε µεε ELECTROMAGNETICS ( ) ( ) * 00 * 00 0 * 22 1 ,, 2 EEeEeEdt T rErEw rkjrkj T e εε ωω ε === ⋅−⋅ ∫   Energy Flux and Poynting Vector (continue – 1) For a Homogeneous, Linear and Isotropic Media: Time-averaged Electric Energy Density Time-averaged Magnetic Energy Density
  • 64. 64 SOLO Therefore ( ) ( )( )* 00 ˆ 2 rHkrEww me  ×⋅== µε Within the accuracy of Geometrical Optics, the Time-averaged Electric and Magnetic Energy Densities are equal. ( ) ( ) ( ) ( ) ( ) ( )( )* 0000 * 00 ˆ 22 rHkrErHrHrErEwww me  ×⋅=⋅+⋅=+= ∗ µε µε The total energy will be: The Poynting vector is defined as: ( ) ( ) ( )trHtrEtrS ,,:,  ×= ( ) ( ) ( ) ( ) ( ) ( )[ ] ( ) ( )[ ] ( ) ( )[ ]∫ ∫∫ −− +×+= ×=×=×= T tjtjtjtj T tjtj T dterHerHerEerE T dterHerEal T dttrHtrE T trHtrES 0 ** 00 ,, 2 1 ,, 2 11 ,,Re 1 ,, 1 ,, ωωωω ωω ϖϖϖϖ ϖϖ  ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ] ( ) ( ) ( ) ( )[ ]ωωωω ωωωωωωωω ωω ,,,, 4 1 ,,,,,,,, 4 11 ** 0 2****2 rHrErHrE dterHrErHrErHrEerHrE T T tjtj ×+×= ∫ ×+×+×+×= − [ ] [ ]0 * 0 * 00 0 * 0 * 00 4 1 4 1 HEHE eHeEeHeE rkjrkjrkjrkj ×+×= ×+×= ⋅−⋅⋅⋅−  The time average of the Poynting vector is: John Henry Poynting 1852-1914 ELECTROMAGNETICS Energy Flux and Poynting Vector (continue – 2) For a Homogeneous, Linear and Isotropic Media: Return to Table of Content
  • 65. 65 SOLO Assume the linear general constitutive relations ELECTROMAGNETICS ( ) ( ) ( )        =⋅∇ =⋅∇ −−=×∇ +=×∇ m e m e BGM DGE JBjEF JDjHA ρ ρ ω ω )( dyadicsxwhere H E B D 33,,, =                 =         µζξε µζ ξε        ( ) ( ) ( ) ( )    −⋅+⋅−=×∇ +⋅+⋅=×∇ m e JHEjEF JHEjHA µζω ξεω   Energy Flux and Poynting Vector for a Bianisotropic Medium
  • 66. 66 SOLO ELECTROMAGNETICS Let compute the following ( ) ( )[ ] ( )[ ] ( ) me HH me HH JHJEHHEHHEEEj JHEjHEJHEj EHHEHE ⋅−⋅−⋅⋅−⋅⋅−⋅⋅+⋅⋅= −⋅+⋅−⋅+⋅+⋅+⋅−−= ×∇⋅+×∇⋅−=×⋅∇ ****** **** *** µζξεω µζωξεω   ( ) ( )[ ] ( ) ( )[ ] ( ) ****** **** *** me HH m HH e JHJEHHHEHEEEj HJHEjJHEjE EHHEHE ⋅−⋅−⋅⋅−⋅⋅−⋅⋅+⋅⋅−= ⋅−⋅+⋅−−++⋅+⋅⋅−= ×∇⋅+×∇⋅−=×⋅∇ µζξεω µζωξεω   ( ) ( ) ( ) ( ) ****** ****** ** me HH me HH JHJEHHHEHEEEj JHJEHHEHEHEEj HEHE ⋅−⋅−⋅⋅−⋅⋅−⋅⋅+⋅⋅− ⋅−⋅−⋅⋅−⋅⋅−⋅⋅+⋅⋅= ×⋅∇+×⋅∇ µζξεω µζξεω   ( ) ( ) ( ) ( )    −⋅+⋅−=×∇ +⋅+⋅=×∇ m e JHEjEF JHEjHA µζω ξεω   Energy Flux and Poynting Vector for a Bianisotropic Medium (continue – 1)
  • 67. 67 SOLO ELECTROMAGNETICS ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ] ( ) ( ) [ ] ( ) ( )mmeeHH HH mmee HHHH me HH me HH JHJHJEJE H E jHE JHJHJEJE HHEHHEEEj JHJEHHHEHEEEj JHJEHHEHEHEEj HEHE ⋅+⋅−⋅+⋅−        ⋅                 −− −− −⋅= ⋅+⋅−⋅+⋅+ ⋅−⋅+⋅−⋅+⋅−⋅+⋅−⋅−= ⋅−⋅−⋅⋅−⋅⋅−⋅⋅+⋅⋅− ⋅−⋅−⋅⋅−⋅⋅−⋅⋅+⋅⋅= ×⋅∇+×⋅∇ ****** **** **** ****** ****** ** µµζζ ξξεε ω µµζζξξεεω µζξεω µζξεω      ( ) ( ) ( ) ( )[ ] ( ) ( ) ( ) ( )[ ]ωωωωωω ,,,, 4 1 ,,Re 2 1 ,, ** rHrErHrErHrEaltrHtrES ×+×=×=×=  We found the time average of the Poynting vector We see that ( ) ( )[ ] [ ] ( ) ( )mmee HH HH JHJHJEJE H Ej HEHEHES ⋅+⋅−⋅+⋅−         ⋅                 −− −− −⋅=×⋅∇+×⋅∇=⋅∇ **** **** 4 1 4 1 44 1 µµζζ ξξεεω    Energy Flux and Poynting Vector for a Bianisotropic Medium (continue – 2)
  • 68. 68 SOLO ELECTROMAGNETICS Let integrate the mean value of the Poynting vector over the volume V [ ] ( ) ( ) ( ) ( )[ ] ( ) ∫∫ ∫ ∫∫ ∫∫ →→ ⋅=⋅×+×= ×⋅∇+×⋅∇= ⋅+⋅−⋅+⋅−         ⋅                 −− −− −⋅=⋅∇ SS Gauss V V mm V ee V HH HH V dSnSdSnHEHE dVHEHE dVJHJHdVJEJE dV H Ej HEdVS 11 4 1 4 1 4 1 4 1 4 ** 1 ** **** **     µµζζ ξξεεω ( ) ( )[ ] [ ] ( ) ( )mmee HH HH JHJHJEJE H Ej HEHEHES ⋅+⋅−⋅+⋅−         ⋅                 −− −− −⋅=×⋅∇+×⋅∇=⋅∇ **** **** 4 1 4 1 44 1 µµζζ ξξεεω    Energy Flux and Poynting Vector for a Bianisotropic Medium (continue – 3)
  • 69. 69 SOLO ELECTROMAGNETICS We recognize the following                 −− −− −= HH HH j µµζζ ξξεεω   4 G mediumlosslessundefinite mediumpassivedefinitenegative mediumactivedefinitepositive ⇒ ⇒ ⇒ G G G [ ] ( ) ( )∫∫ ∫∫ ⋅+⋅−⋅+⋅−         ⋅                 −− −− −⋅=⋅ → V mm V ee V HH HH S dVJHJHdVJEJE dV H Ej HEdSnS **** ** 4 1 4 1 4 1 µµζζ ξξεεω    [ ] dV H Ej HE V ∫         ⋅                 −− −− −⋅ ** ** ** 4 µµζζ ξξεεω   ( )∫ ⋅+⋅ V ee dVJEJE ** 4 1 ( )∫ ⋅+⋅ V mm dVJHJH ** 4 1 ∫ → ⋅ S dSnS 1  Time average of the Radiated Energy through S (Irradiance) Time average of Electromagnetic Energy in V Time average of Joule Energy in V Time average of Fictious Joule Energy in V Energy Flux and Poynting Vector for a Bianisotropic Medium (continue – 4) Return to Table of Content
  • 70. 70 SOLO ELECTROMAGNETICS Planar Waves in an Source-less Anisotropic Electric Media     0 0 0 0 m e m e B D J t B E J t D H ρ ρ =⋅∇ =⋅∇ − ∂ ∂ −=×∇ + ∂ ∂ =×∇           ( )       ⋅− = ⋅− == rs c n tj s c n k rktj eEeEE    ˆ 0 ˆ 0 ω ω ω ωω ωω j t eje t rs c n tjrs c n tj → ∂ ∂ ⇒= ∂ ∂       ⋅−      ⋅−  ˆˆ  sjks c n jes c n je k rs c n tjrs c n tj ˆˆˆ ˆˆ −=−→∇⇒−=∇       ⋅−      ⋅− ωω ωω  0ˆ 0ˆ ˆ ˆ =⋅ =⋅ =× −=× Bs Ds BEs c n DHs c n ( ) ( )DEBH tt D E t B H HEEHHES HB ED         ⋅+⋅ ∂ ∂ −= ∂ ∂ ⋅− ∂ ∂ ⋅−= ×∇⋅−×∇⋅=×⋅∇=⋅∇ = ⋅= 2 1µ ε ( ) emUDEBHS k Ss c n =⋅+⋅=⋅=⋅ 2 1 ˆ ω  Maxwell’s Symmetric Equations Planar Monochromatic Wave 0 0 =⋅ =⋅ =× −=× Bk Dk BEk DHk     ω ω ωj t → ∂ ∂ s c n j ˆω−→∇ ωj t → ∂ ∂ kj  −→∇ vector phasor vector phasor c n k ω=:
  • 71. 71 SOLO ELECTROMAGNETICS Planar Waves in an Source-less Anisotropic Electric Media 0ˆ 0ˆ ˆ ˆ =⋅ =⋅ =× −=× Bs Ds BEs c n DHs c n dyadicsxwhere H E B D 33,,, µζξε µζ ξε                        =         HHIB ED   µµ ε =⋅= ⋅= For Anisotropic Electric Media the Linear Constitutive Relations are Planar Waves in an Source-less Media The most general form of Linear Constitutive Relations is: HES ×= Poynting Vector B  D  pv  E  H  HES  ×=α α k  ev  PlanarWaves ( ) DHs c n Bs c n Ess c n µµ −=×=×=××      ˆˆˆˆ 2 ( ) ( )[ ]EssE c n Ess c n D ⋅−      =××      −= ˆˆ 1 ˆˆ 1 22 µµ SDEs ,,,ˆ are coplanar and normal to H I  µµζξε === ,0,
  • 72. 72 SOLO ELECTROMAGNETICS Planar Waves in an Source-less Anisotropic Electric Media B  D  pv  E  H  HES  ×=α α k  ev  PlanarWaves( ) ( )[ ]EssE c n Ess c n D ⋅−      =××      −= ˆˆ 1 ˆˆ 1 22 µµ SDEs ,,,ˆ are coplanar and normal to H  ( ) DEnEsDsDE c n DD c ⋅=      ⋅⋅−⋅      =⋅ = 2 0 1 0 2 02 ˆˆ 1 ε µ µε DE DD n ⋅ ⋅ = 0 2 1 ε ( ) ( ) ( ) ( )( )2222 2 2 2 2 2 ˆ DEDED DDEED D DE E D DDE E D D D D EE D D D D EE s ⋅− ⋅− = ⋅ − ⋅ − =       ⋅−       ⋅− = ( ) ( ) ( ) ( )( )2222 2 2 2 2 2 : DEDEE EDEDE E DE D E DDE D E E E E DD E E E E DD S S t ⋅− ⋅− −= ⋅ − ⋅ − −=       ⋅−       ⋅− −==  If and are not collinear:DE
  • 73. 73 SOLO ELECTROMAGNETICS Planar Waves in an Source-less Anisotropic Electric Media Phase Velocity, Energy (Ray) Velocity The phase of the field is given by       ⋅−=⋅−= rs c n trkt  ˆωωφ For a constant phase front we have       ⋅−=⋅−=⋅−== rds c n dtrdskdtrdkdtd  ˆˆ0 ωωωφ s n c s kdt rd v srr const p ˆˆ ˆ === = = ω φ   Phase Velocity B  D  pv  E  H  HES  ×=α α k  ev  PlanarWaves n c ktd rd s ==⋅ ω  ˆ n c S S vs e =⋅ˆ ( ) α α cosˆ ˆ cos p S Ss e v Ss S n c v ⋅ = = ⋅ = Define the phase velocity as Define the energy flow velocity in the Poynting vector direction as S S vSv ee = → 1 ( ) emUDEBHS k Ss c n =⋅+⋅=⋅=⋅ 2 1 ˆ ω  ( ) em ee U S S Ssn c S S vv = ⋅ == ˆ 1 Energy (Ray) Velocity The electro-magnetic energy propagates along the Poynting vector . For anisotropic media the Poynting vector is not collinear with .k  HES ×= Return to Table of Content s c n k ˆω=  Electromagnetic Energy Density
  • 74. 74 SOLO ELECTROMAGNETICS Planar Waves in an Source-less Anisotropic Electric Media BEk DHk ω ω =× −=×   HHIB ED   µµ ε =⋅= ⋅= Wave Equation ( ) EDHkBkEkk EDDHkHBBEk ⋅−=−=×=×=×× ⋅=−=×==× εµωµωµωω εωµω   22 2222 0 0 0 ˆ kkkkkk kk kk kk k k k k s s s kskk zyx xy xz yz z y x z y x =++=⋅           − − − =×⇒           =           ==  ( ) ( ) ( ) ( )            +− +− +− =           − − −           − − − =×× 22 22 22 0 0 0 0 0 0 yxzyzx zyzxyx zxyxzy xy xz yz xy xz yz kkkkkk kkkkkk kkkkkk kk kk kk kk kk kk kk            = z y x ε ε ε ε 00 00 00                                        =                   = 2 2 2 0 0 0 0 0 0 2 2 00 00 00 00 00 00 c n c n c n z x x z y x ω ω ω εµ εµ εµ εµ εµ εµ εµ εµ εµω εωµ  In principal dielectric axes: 0 2 0 2 0 2 ,, ε ε ε ε ε ε z z y y x x nnn === ( ) 02 =⋅+×× EEkk εµω  ( ) 021 =+×× − DDkk µωε  or
  • 75. 75 SOLO ELECTROMAGNETICS Planar Waves in an Source-less Anisotropic Electric Media Wave Equation (continue – 1) ( ) 02 =⋅+×× EEkk εµω  ( ) ( ) ( ) 0 22 2 22 2 22 2 =                                     +−      +−      +−      z y x yx z zyzx zyzx y yx zxyxzy x E E E kk c n kkkk kkkk c n kk kkkkkk c n ω ω ω A nonzero solution exists only when ( ) ( ) ( ) 0det 22 2 22 2 22 2 =                       +−      +−      +−      yx z zyzx zyzx y yx zxyxzy x kk c n kkkk kkkk c n kk kkkkkk c n ω ω ω
  • 76. 76 SOLO ELECTROMAGNETICS Planar Waves in an Source-less Anisotropic Electric Media Wave Equation (continue – 2) ( ) ( ) ( )                       +−      +−      +−      =                       +−      +−      +−      ++= 22 2 22 2 22 2 22 2 22 2 22 2 2222 det z z zyzx zyy y yx zxyxx x kkkk yx z zyzx zyzx y yx zxyxzy x kk c n kkkk kkkk c n kk kkkkkk c n kk c n kkkk kkkk c n kk kkkkkk c n zyx ω ω ω ω ω ω We obtain         −−        −−                 −+        −+        −         −        +−= 2 2 22 222 2 22 222 2 22 22 2 22 22 2 22 2 2 22 22 2 22 k c n kkk c n kkk c n kk c n kk c n k c n kk c n y zx z yx y z z y zy x x ωωωωωωω 02 2 22 2 2 22 22 2 22 2 2 22 22 2 22 2 2 22 22 2 22 2 2 22 2 2 22 =         −        −+        −        −+        −         −+        −         −        −= k c n k c n kk c n k c n kk c n k c n kk c n k c n k c n yx z zx y zy x zyx ωωωωωωωωω Divide by (assumed nonzero!!)         −         −        −− 2 2 22 2 2 22 2 2 22 k c n k c n k c n zyx ωωω 1 2 22 2 2 2 22 2 2 2 22 2 2 = − + − + − c n k k c n k k c n k k z z y y x x ωωω This is a quadratic equation of k2 (k6 terms drop – next slide). For each set sx, sy, sz it yields two solutions for k2 : k1 2 and k2 2           =           z y x z y x s s s k k k k 2 2 22 2 2 2 22 2 2 2 22 2 2 1 k c n k s c n k s c n k s z z y y x x = − + − + − ωωω ( )skkk ˆ,, 21 ε  =
  • 77. 77 SOLO ELECTROMAGNETICS Planar Waves in an Source-less Anisotropic Electric Media 02 2 22 2 2 22 222 2 22 2 2 22 222 2 22 2 2 22 222 2 22 2 2 22 2 2 22 =         −        −+        −        −+         −        −+        −         −        − k c n k c n skk c n k c n skk c n k c n skk c n k c n k c n yx z zx y yz x zyx ωωωωωωωωω ( ) ( ) ( ) ( ) ( ) 02222 4 4 4222 2 2 622222 4 4 4222 2 2 622222 4 4 4222 2 2 62 222 6 6 2222222 4 4 4222 2 2 6 =++−+++−+++−+ +++−+++− ksnn c ksnn c ksksnn c ksnn c ksksnn c ksnn c ks nnn c knnnnnn c knnn c k zyxzyxzyzxyzxyxzyxzyx zyxzyzxyxzyx ωωωωωω ωωω ( ) ( ) ( ) ( )[ ] 0111 222 6 6 2222222222 4 4 4222222 2 2 =+−+−+−−++ zyxxzyyzxzyxzzyyxx nnn c ksnnsnnsnn c ksnsnsn c ωωω This is a quadratic equation of k2 . For each set sx, sy, sz it yields two solutions for k2 : k1 2 and k2 2 ( )skkk ˆ,, 21 ε  = 2 2 22 2 2 2 22 2 2 2 22 2 2 1 k c n k s c n k s c n k s z z y y x x = − + − + − ωωω 02 2 22 2 2 22 22 2 22 2 2 22 22 2 22 2 2 22 22 2 22 2 2 22 2 2 22 =         −        −+        −        −+        −         −+        −         −        − k c n k c n kk c n k c n kk c n k c n kk c n k c n k c n yx z zx y zy x zyx ωωωωωωωωω           =           z y x z y x s s s k k k k Wave Equation (continue – 2a) 1 222 =++ zyx sss
  • 78. 78 SOLO ELECTROMAGNETICS Planar Waves in an Source-less Anisotropic Electric Media Wave Equation (continue – 3) ( ) ( ) ( ) 0 22 2 22 2 22 2 =                                     +−      +−      +−      z y x yx z zyzx zyzx y yx zxyxzy x E E E kk c n kkkk kkkk c n kk kkkkkk c n ω ω ω A solution (self-mode) is: ( ) ( ) ( ) ( ) ( ) ( ) 0 0 0 222 2 222 2 222 2 =+++         ++−      =+++         ++−      =+++         ++−      ⋅ ⋅ ⋅             Ek zzyyxxzzzyx z Ek zzyyxxyyzyx y Ek zzyyxxxxzyx x EkEkEkkEkkk c n EkEkEkkEkkk c n EkEkEkkEkkk c n ω ω ω                                   −      −      −      2 2 2 2 2 2 k c n k k c n k k c n k z z y y x x ω ω ω ( )Ek ⋅−  ( ) ( ) ( )                                   −      ⋅ −      ⋅ −      ⋅ −=                         2 2 2 2 2 2 k c n Ekk k c n Ekk k c n Ekk E E E z z y y x x z y x ω ω ω    In principal dielectric axes Return to Table of Content
  • 79. 79 SOLO ELECTROMAGNETICS Planar Waves in an Source-less Anisotropic Electric Media Wave-Vector Surface The above equation can be represented by a three-dimensional surface in k space. ( ) ( ) ( ) 0det 22 2 22 2 22 2 =                     +−      +−      +−      yx z zyzx zyzx y yx zxyxzy x kk c n kkkk kkkk c n kk kkkkkk c n ω ω ω The understand how this surface look let find the intersection of the surface with kz = 0. ( ) ( ) 0 00 0 0 det 222 2 2 2 22 2 22 2 2 2 2 2 =         −         −              −              +−      =                     +−      −      −      yxx y y x yx z yx z x y yx yxy x kkk c n k c n kk c n kk c n k c n kk kkk c n ωωω ω ω ω
  • 80. 80 SOLO ELECTROMAGNETICS Planar Waves in an Source-less Anisotropic Electric Media Wave-Vector Surface (continue – 1) The intersection of the surface with kz = 0 is given by two equations, derived from: ( ) 0 22 2 222 2 2 2 =         +−              −         −              −      yx z yxx y y x kk c n kkk c n k c n ωωω 2 22       =+ c n kk z yx ω ( ) ( ) 1 // 22 =+ cn k cn k x y y x ωω ( ) 0 00 0 0 22 2 2 2 2 2 =                                     +−      −      −      z y x yx z x y yx yxy x E E E kk c n k c n kk kkk c n ω ω ω                 =                 = zz y x EE E E E 0 0 2                     −      − =                 = 0 2 2 1 y x yx z y x k c n kk E E E E ω 021 =⋅ EE Ellipse The electric fields corresponding to those two solutions are: on Ellipse on Circle Since the coefficients of the electric fields are real the two solutions represents two linear polarized planar waves. Circle
  • 81. 81 SOLO ELECTROMAGNETICS Planar Waves in an Source-less Anisotropic Electric Media Wave-Vector Surface (continue -2) Intersection of the surface with kz = 0, are a circle and an ellipse in kx, ky plane. 2 22       =+ c n kk z yx ω ( ) ( ) 1 // 2 2 2 2 =+ cn k cn k x y y x ωω Intersection of the surface with ky = 0, are a circle and an ellipse in kx, kz plane. 2 22       =+ c n kk y zx ω ( ) ( ) 1 // 2 2 2 2 =+ cn k cn k x z z x ωω Intersection of the surface with kx = 0, are a circle and an ellipse in ky, kz plane. 2 22       =+ c n kk x zy ω ( ) ( ) 1 // 2 2 2 2 =+ cn k cn k y z z y ωω Suppose nx < ny < nz. ( ) 0 222 2 2 2 22 2 =         −         −              −              +−      yxx y y x yx z kkk c n k c n kk c n ωωω 0 2 0 2 0 2 ,, ε ε ε ε ε ε z z y y x x nnn ===
  • 82. 82 SOLO ELECTROMAGNETICS Planar Waves in an Source-less Anisotropic Electric Media Wave-Vector Surface (continue – 3) The complete k surface is double; that is, it consists of an inner shell and an outer shell. For any given direction of the vector k, there are two possible values for the wave-number k (eigenmodes). Therefore are also two values for the phase velocity. From the figure we can see that the inner and outer shells of the k surface intersect at four points in each quadrant of the kx, kz plane (for nx<ny<nz). Those points define two directions for which the two values of k are equal. Those two directions are called Optical Axes. The structure of an anisotropic medium permits two monochromatic plane waves that are polarized orthogonally with respect to each other. If nx,ny and nz are different the crystal has two optical axes and is said to be biaxial. 0 2 0 2 0 2 ,, ε ε ε ε ε ε z z y y x x nnn === nx < ny < nz.
  • 83. 83 SOLO ELECTROMAGNETICS Planar Waves in an Source-less Anisotropic Electric Media Wave-Vector Surface (continue -4) Let compute the Optical Axis direction. 0 2 0 2 0 2 ,, ε ε ε ε ε ε z z y y x x nnn === We can see that the Optical Axis is in x, z plane, therefore           =           =           = δ δ ω cos 0 sin c n s s s k k k k k OA OAz y x OA OAz y x OA  2 22       =+ c n kk y zx ω ( ) ( ) 1 // 2 2 2 2 =+ cn k cn k x z z x ωω 22       =      c n c n yOA ωω 1 cossin 2 22 2 22 =+ x OA z OA n n n n δδ must be at the intersection ofOAk  and yOA nn = ( ) 1 coscos1 2 22 2 22 =+ − x y z y n n n n δδ 22 22 2 cos −− −− − − = zx zy OA nn nn δ 22 22 2 sin −− −− − − = zx yx OA nn nn δ 22 22 2 tan −− −− − − = zy yx OA nn nn δ If δ is less than 45º, the crystal is said to be a positive biaxial crystal. If δ is greater than 45º, the crystal is said to be a negative biaxial crystal. nx < ny < nz.         − − ±= −− −− − 22 22 1 2,1 sin zx yx OA nn nn δ
  • 84. 84 SOLO ELECTROMAGNETICS Planar Waves in an Source-less Anisotropic Electric Media Wave-Vector Surface (continue -5) 0 2 0 2 0 2 ,, ε ε ε ε ε ε z z y y x x nnn ===nx < ny < nz.
  • 85. 85 SOLO ELECTROMAGNETICS Planar Waves in an Source-less Anisotropic Electric Media Double Refraction at a Birefringent Crystal Boundary ( ) 0ˆ  =−× ti kkn At the Crystal Boundary defined by the normal the following condition must be satisfied nˆ The refracted wave is, in general, a mixture of two eigenmodes ( ) ( )222111 ,,, sn c ksn c k  ε ω ε ω == We can write ( ) ( ) 222111 sin,sin,sin θεθεθ skskk ii  == This looks like Snell’s Law but k1 and k2 are not constant, and we must add the quadratic equation in k2 : ( ) ( ) ( ) ( )[ ] 0111 222 6 6 2222222222 4 4 4222222 2 2 =+−+−+−−++ zyxxzyyzxzyxzzyyxx nnn c ksnnsnnsnn c ksnsnsn c ωωω 0 2 0 2 0 2 ,, ε ε ε ε ε ε z z y y x x nnn ===
  • 86. 86 SOLO ELECTROMAGNETICS Planar Waves in an Source-less Anisotropic Electric Media Wave-Vector Surface (continue – 6) If two of the refraction indices are equal the crystal has one optical axis and is said to be uniaxial. zyx εεε ≠=nx = ny =no≠ nz=ne For each set sx, sy, sz it yields two solutions for k2 : k1 and k2 ( )skkk ˆ,, 21 ε  = The quadratic equation of k2 is: 2 2 22 2 2 2 22 2 2 11 k c n k s c n k s e z o z = − + − − ωω ( )[ ] ( ) ( )[ ] 0111 24 6 6 222222 4 4 42222 2 2 =+++−−+− eozezoozezo nn c ksnsnn c ksnsn c ωωω ( ) ( ) ( ) ( )[ ] 0111 222 6 6 2222222222 4 4 4222222 2 2 =+−+−+−−++ zyxxzyyzxzyxzzyyxx nnn c ksnnsnnsnn c ksnsnsn c ωωω nx = ny =no≠ nz=ne Uniaxial Crystals One other way is to substitute in:
  • 87. 87 SOLO ELECTROMAGNETICS Planar Waves in an Source-less Anisotropic Electric Media Wave-Vector Surface (continue – 7) zyx εεε ≠=nx = ny =no≠ nz=ne ( )[ ] ( ) ( )[ ] 0111 24 6 6 222222 4 4 42222 2 2 =+++−−+− eozezoozezo nn c ksnsnn c ksnsn c ωωω ( )( ) 01 2 2 2 22222 2 2 2 2 22 =      −−−+      − oeozoe n c kknnsn c kn ωω ( )[ ] 01 22 2 2 222222 2 2 2 =       −+−      − eoezozo nn c knsnsn c k ωω The Wave-Vector surface decomposes into two separate shells 0 2 2 2 2 =− on c k ω ( )[ ] 0 22 2 2 222222 =−++ eoezoyx nn c knsnss ω 0 2 2 2 222 =−++ ozyx n c kkk ω Spherical surface 01 2 2 2 2 2 2 2 22 =−+ + o z e yx n c k n c kk ωω Ellipsoid of Revolution surface These two surfaces are tangent along the z axis. Uniaxial Crystals (continue – 1) Return to Table of Content
  • 88. 88 SOLO ELECTROMAGNETICS Planar Waves in an Source-less Anisotropic Electric Media Double Refraction at a Uniaxial Crystal Boundary For the Uniaxial Crystal is easy to solve the Boundary condition graphically: ( ) eeooii skkk θεθθ sin,sinsin 2  == The Wave-Vector surface decomposes into two separate shells: • the sphere of the ordinary wave with • the ellipsoid of extraordinary wave with .constko = ( )2, ske  ε is found graphically using the equality: Snell’s Lawooii kk θθ sinsin =o k  The intersection of the normal to the boundary from the end with the ellipse defines the direction of ek  o k  The intersection of the plane defined by and passing through the common centers of the sphere and the ellipsoid will give a circle and an ellipse, respectively. nki  , ii k θsin ik ok i θ oθ Intersection of normal surface with plane of incidence in ii k θsin Uniaxial Crystal Incident ray n  Optical Axis oE  eo nn >
  • 89. 89 SOLO ELECTROMAGNETICS Planar Waves in an Source-less Anisotropic Electric Media Double Refraction at a Uniaxial Crystal Boundary
  • 90. 90 SOLO ELECTROMAGNETICS Planar Waves in an Source-less Anisotropic Electric Media Wave-Vector Surface (continue – 8) nx = ny = nz=n If all three indices are equal, then the Wave-vector surface degenerates to a single sphere, and the crystal is optically isotropic. εεεε === zyx 2 22 2 c n k ω = 0 2 2 2 =                       z y x zzyzx zyyyx zxyxx E E E kkkkk kkkkk kkkkk 2222 zyx kkkk ++= ( ) 02 =⋅+×× EEkk εµω  We can see that: { } { } { } 0 ,, ,, ,, det 2 2 2 ≡           =             zyxz zyxy zyxx zzyzx zyyyx zxyxx kkkk kkkk kkkk kkkkk kkkkk kkkkk and: 0=⋅=++ EkEkEkEk zzyyxx  0/εε=n Isotropic Crystal Return to Table of Content
  • 91. 91 SOLO ELECTROMAGNETICS Optically Isotropic (Cubic) Crystals n Sodium Chloride (NaCl) Diamond Fluorite CdTe GaAs 1.544 2.417 1.392 2.69 3.40 Uniaxial Positive Crystals nO nE Ice Quartz Zircon Rutile ( TiO2 ) BeO ZnS 1.309 1.310 1.544 1.553 1.923 1.968 2.616 2.903 1.717 1.732 2.354 2.358 Uniaxial Negative Crystals nO nE Beryl Sodium Nitrate Calcite Tourmaline 1.598 1.590 1.587 1.336 1.658 1.486 1.669 1.638 Refractive Index of Some Typical Crystals
  • 92. 92 SOLO ELECTROMAGNETICS Biaxial Crystals n1 n2 n3 Gypsum Feldspar Mica Topaz NaNO2 SbSI YAlO3 1.520 1.523 1.530 1.522 1.526 1.530 1.552 1.582 1.588 1.619 1.620 1.627 1.344 1.411 1.651 2.7 3.2 3.8 1.923 1.938 1.947 Refractive Index of Some Typical Biaxial Crystals
  • 93. 93 SOLO ELECTROMAGNETICS Planar Waves in an Source-less Anisotropic Electric Media Crystal Types
  • 94. 94 SOLO ELECTROMAGNETICS Planar Waves in an Source-less Anisotropic Electric Media Crystal Types (continue -1)
  • 95. 95 SOLO ELECTROMAGNETICS Planar Waves in an Source-less Anisotropic Electric Media Crystal Types (continue – 2)
  • 96. 96 SOLO ELECTROMAGNETICS Planar Waves in an Source-less Anisotropic Electric Media Crystal Types (continue – 3) A.Mermin, “Solid State Physics”, Holt, Reinhart and Winston, 1976, p.122 Return to Table of Content
  • 97. 97 SOLO ELECTROMAGNETICS Planar Waves in an Source-less Anisotropic Electric Media Orthogonality Properties of the Eigenmodes Assume the two monochromatic waves corresponding to a given direction are defined by and sˆ 1111 ,,, nDEH  2222 ,,, nDEH  Start with Lorentz Reciprocity Theorem Use HB BEs c n µ= =×  ˆ Es c n H ×= ˆ µ ( )[ ] ( )[ ]12 2 21 1 ˆˆˆˆ EsEs c n EsEs c n ××⋅=××⋅ µµ ( ) ( ) ( ) ( )12 2 21 1 ˆˆˆˆ EsEs c n EsEs c n ×⋅×=×⋅× µµ ( ) ( )1221 ˆˆ HEsHEs ×⋅=×⋅ Since we have21 nn ≠ ( ) ( ) 0ˆˆ 21 =×⋅× EsEs ( ) ( ) 0ˆˆ 1221 =×⋅=×⋅ HEsHEs ( ) ( )1221 HEHE  ×⋅∇=×⋅∇ s c n j ˆω−→∇ 0=⋅=⋅ sHEH 
  • 98. 98 SOLO ELECTROMAGNETICS Planar Waves in an Source-less Anisotropic Electric Media Orthogonality Properties of the Eigenmodes (continue – 1) We obtain 0ˆˆˆˆ 221121212121212121 =⋅=⋅=⋅=⋅=⋅=⋅=⋅=⋅=⋅=⋅=⋅ HEHEHsHsDsDsEEEDDEDDHH ( ) ( ) 0ˆˆ 1221 =×⋅=×⋅ HEsHEs ( ) ( ) ( ) ( ) 0ˆˆ 0ˆˆ 21 1 2 21 2 ˆ 2121 21 1 ˆ 2121 2 2 2 1 1 1 11 =⋅=⋅=×⋅−=×⋅ =⋅=⋅×=×⋅ − −=× =× = DD n c DE n c HsEHEs HH n c HEsHEs D n c Hs B n c Es HB ε µ µ  ( ) ( ) ( ) ( ) 0ˆˆ 0ˆˆ 12 1 1 12 1 ˆ 1212 12 2 ˆ 1212 1 1 1 2 2 2 22 =⋅=⋅=×⋅−=×⋅ =⋅=⋅×=×⋅ − −=× =× = DD n c DE n c HsEHEs HH n c HEsHEs D n c Hs B n c Es HB ε µ µ 
  • 99. 99 SOLO ELECTROMAGNETICS Planar Waves in an Source-less Anisotropic Electric Media Orthogonality Properties of the Eigenmodes (continue – 2) We obtained The two solutions correspond to two different possible linear polarizations of a wave propagating in direction and these two solutions have mutually orthogonal polarizations. sˆ 0ˆˆˆˆ 221121212121212121 =⋅=⋅=⋅=⋅=⋅=⋅=⋅=⋅=⋅=⋅=⋅ HEHEHsHsDsDsEEEDDEDDHH Return to Table of Content
  • 100. 100 SOLO ELECTROMAGNETICS Phase Velocity, Energy (Ray) Velocity for Planar Waves The phase of the field is given by       ⋅−=⋅−= rs c n trkt  ˆωωφ For a constant phase front we have       ⋅−=⋅−=⋅−== rds c n dtrdskdtrdkdtd  ˆˆ0 ωωωφ s n c s kdt rd v srr const p ˆˆ: ˆ === = = ω φ   Phase Velocity B  D  pv  E  H  HES  ×=α α k  ev  PlanarWaves n c ktd rd s ==⋅ ω  ˆ n c S S vs e =⋅ :ˆ ( ) α α cosˆ cos ˆ p S Ss e v Ss S n c v = ⋅ = ⋅ = Define the phase velocity as Define the energy flow velocity in the Poynting vector direction as S S vSv ee = → 1 ( ) emUDEBHS k Ss c n =⋅+⋅=⋅=⋅ 2 1 ˆ ω  ( ) em ee U S S Ssn c S S vv = ⋅ == ˆ 1 :  Energy (Ray) Velocity The electro-magnetic energy propagates along the Poynting vector . For anisotropic media the Poynting vector is not collinear with .k  HES ×= Return to Table of Content
  • 101. 101 SOLO ELECTROMAGNETICS Planar Waves in an Source-less Anisotropic Electric Media Ray-Velocity Surface We defined the Energy-Ray Velocity as Let compute ( ) HDHEDEH U DHE U Dv emUemem e =               ⋅−         ⋅=××=×   0 11 ( ) ( ) DEHHEHEH U HHE U HvDvv emUemem eee 1 /0 1111 − −=−=                 ⋅−      ⋅=××=×=×× ε µµµ    In principal dielectric axes:                     =                     =− 2 2 2 2 2 2 1 00 00 00 1 00 0 1 0 00 1 1 z y x z y x n c n c n c εµ εµ εµ ε µ  ( ) em ee U S S Ssn c S S vv = ⋅ == ˆ 1 : 
  • 102. 102 SOLO ELECTROMAGNETICS Planar Waves in an Source-less Anisotropic Electric Media Ray-Velocity Surface (continue – 1) We can write ( ) 0 1 1 =+×× − DDvv ee ε µ  as ( ) ( ) ( ) 0 22 2 22 2 22 2 =                                       +−      +−         +−      z y x eyex z ezeyezex ezeyezex y eyex ezexeyexezey x D D D vv n c vvvv vvvv n c vv vvvvvv n c We obtain non-zero solutions only when ( ) ( ) ( ) 0det 22 2 22 2 22 2 =                       +−      +−         +−      eyex z ezeyezex ezeyezex y eyex ezexeyexezey x vv n c vvvv vvvv n c vv vvvvvv n c
  • 103. 103 SOLO ELECTROMAGNETICS Planar Waves in an Source-less Anisotropic Electric Media Intersection of the surface with vez = 0, are a circle and an ellipse in vex, vey plane. 2 22       =+ z eyex n c vv ( ) ( ) 1 // 2 2 2 2 =+ x ey y ex nc v nc v Intersection of the surface with vey = 0, are a circle and an ellipse in vex, vez plane. 2 22         =+ y ezex n c vv ( ) ( ) 1 // 2 2 2 2 =+ x z z ex nc k nc v Intersection of the surface with vex = 0, are a circle and an ellipse in vey, vez plane. 2 22         =+ x ezey n c vv ( ) ( ) 1 // 2 2 2 2 =+ y ez z ey nc v nc v Suppose nx < ny < nz. Ray-Velocity Surface (continue – 2) The previous equation has the same structure as the Wave-Vector Surface, therefore can be represented by a three-dimensional surface in ve space. vex, vey and vez are the components of ve in the principal axes of the dielectric. 0 2 0 2 0 2 ,, ε ε ε ε ε ε z z y y x x nnn ===
  • 104. 104 SOLO ELECTROMAGNETICS Planar Waves in an Source-less Anisotropic Electric Media nx < ny < nz. The complete ve surface is double; that is, it consists of an inner shell and an outer shell. For any given direction of the vector ve, there are two possible values for the energy-ray velocity ve (eigenmodes): ve1 and ve2. From the figure we can see that the inner and outer shells of the ve surface intersect at four points in each quadrant of the vex, vez plane (for nx<ny<nz). Those points define two directions for which the two values of ve are equal. Those directions are called Ray Axes and are distinct from the Optic Axes of the dielectric. Ray-Velocity Surface (continue – 3)
  • 105. 105 SOLO ELECTROMAGNETICS Planar Waves in an Source-less Anisotropic Electric Media Let compute the Ray Axis direction. We can see that the Ray Axis is in x, z plane, therefore           ==           = → RA RA RA RARAe RAez ey ex RAe n c Sv v v v v δ δ cos 0 sin 1  2 22         =+ y ezex n c vv ( ) ( ) 1 // 2 2 2 2 =+ x ez z ex nc v nc v 22         =      yRA n c n c 1 cossin 2 22 2 22 =+ RA RAx RA RAz n n n n δδ must be at the intersection ofRAev  and yRA nn = ( ) 1 coscos1 2 22 2 22 =+ − y RAx y RAz n n n n δδ 22 22 2 cos zx zy RA nn nn − − =δ 22 22 2 sin zx yx RA nn nn − − =δ 22 22 2 tan zy yx RA nn nn − − =δ         − − ±= − 22 22 1 2,1 sin zx yx RA nn nn δ 0 2 0 2 0 2 ,, ε ε ε ε ε ε z z y y x x nnn === Ray-Velocity Surface (continue – 4) RA x z yz xy x z OA n n nn nn n n δδ 2 2 2 22 22 2 2 2 tantan = − − = Ray Axes are distinct from Optical Axes.
  • 106. 106 SOLO ELECTROMAGNETICS Planar Waves in an Source-less Anisotropic Electric Media ( ) ( ) ( )                       +−        +−         +−        =                       +−        +−         +−        ++= 22 2 22 2 22 2 22 2 22 2 22 2 2222 det eze z ezeyezex ezeyeye y eyex ezexeyexexe x vvvv eyex z ezeyezex ezeyezex y eyex ezexeyexezey x vv n c vvvv vvvv n c vv vvvvvv n c vv n c vvvv vvvv n c vv vvvvvv n c ezeyexe We obtain         −−        −−                 −+        −+        −         −        +−= 2 2 2 222 2 2 222 2 2 22 2 2 22 2 2 2 2 2 22 2 2 e y ezexe z eyexe y eze z eye z e y exe x v n c vvv n c vvv n c vv n c vv n c v n c vv n c 0 2 2 2 2 2 2 22 2 2 2 2 2 22 2 2 2 2 2 22 2 2 2 2 2 2 2 2 =         −        −+        −        −+        −         −+        −         −        −= e y e x eze z e x eye z e y exe z e y e x v n c v n c vv n c v n c vv n c v n c vv n c v n c v n c Divide by (assumed nonzero!!(         −         −        −− 2 2 2 2 2 2 2 2 2 e z e y e x v n c v n c v n c 1 2 2 2 2 2 2 2 2 2 2 2 2 = − + − + − z e ez y e ey x e ex n c v v n c v v n c v v Ray-Velocity Surface (continue – 5( Return to Table of Content
  • 107. 107 SOLO ELECTROMAGNETICS Planar Waves in an Source-less Anisotropic Electric Media Index Ellipsoid The electric energy density is given by: ∗∗−∗ ⋅⋅=⋅⋅=⋅= EEDDDEUe εε  2 1 2 1 2 1 1 Let find the Surfaces of Equal Energy expressed in the principal dielectric axes: z z y y x x e DDD DDU εεε ε 222 1 2 ++=⋅⋅= ∗− We have: ( ) ( ) zyxin iii ,,// 00 2 === εεεµεµ Define: ezeyex UEzUEyUEx 2/:2/:2/: 000 εεε === The surfaces of equal energy expressed in the principal dielectric axes are: 1 /1/1/1 2 2 2 2 2 2 =++ zyx n z n y n x Fresnel’s Ellipsoid 0ˆˆˆˆ 221121212121212121 =⋅=⋅=⋅=⋅=⋅=⋅=⋅=⋅=⋅=⋅=⋅ HEHEHsHsDsDsEEEDDEDDHH We found that the two eigenmodes, of each direction , must satisfy: s  222 2 zzyyxxe EEEEEU εεεε ++=⋅⋅= ∗
  • 108. 108 SOLO ELECTROMAGNETICS Planar Waves in an Source-less Anisotropic Electric Media Index Ellipsoid (continue – 1( The electric energy density is given by: ∗∗−∗ ⋅⋅=⋅⋅=⋅= EEDDDEUe εε  2 1 2 1 2 1 1 z z y y x x e DDD DDU εεε ε 222 1 2 ++=⋅⋅= ∗− We have: ( ) ( ) zyxin iii ,,// 00 2 === εεεµεµ Define: 000 2/:2/:2/: εεε ezeyex UDzUDyUDx === The surfaces of equal energy expressed in the principal dielectric axes are: 12 2 2 2 2 2 =++ zyx n z n y n x Index Ellipsoid, Optical Indicatrix, Reciprocal Ellipsoid 0ˆˆˆˆ 221121212121212121 =⋅=⋅=⋅=⋅=⋅=⋅=⋅=⋅=⋅=⋅=⋅ HEHEHsHsDsDsEEEDDEDDHH We found that the two eigenmodes, of each direction , must satisfy: s  222 2 zzyyxxe EEEEEU εεεε ++=⋅⋅= ∗ Let find the Surfaces of Equal Energy expressed in the principal dielectric axes:
  • 109. 109 SOLO ELECTROMAGNETICS Planar Waves in an Source-less Anisotropic Electric Media Index Ellipsoid (continue – 2( Since we found that the index ellipsoid is used to find the two indexes of refraction and the two corresponding directions that are in the plane normal to that passes through ellipsoid center O and defines the ellipse. 0ˆˆ 21 =⋅=⋅ DsDs 21 DandD sˆ Sπ Assume that is one of the expected solutions. 0 2/ εe UDOP = → The normal at P to the ellipsoid PV is: E U z D y D x D U z z y y x x ez z y y x x e zyx PV  2 111 2 111 00 0 ε εεε ε εεε ε =         ++=         ++= ∧∧∧ ∧∧∧→ Since are coplanar PV is in the plane defined by and sˆDπ D sED  ,, The plane tangent to ellipsoid at P is normal to PV ( (, intersects plane along PT that will be normal to and , therefore it gives direction. Tπ E E H Sπ sˆ 0ˆˆ 221121 =⋅=⋅=⋅=⋅ HEHEHsHs
  • 110. 110 SOLO ELECTROMAGNETICS Planar Waves in an Source-less Anisotropic Electric Media Index Ellipsoid (continue – 3( PT is at the intersection of the tangent plane to the ellipsoid at P to the plane normal to and containing PO. Therefore PT is tangent to the ellipse containing PO. Tπ Sπ sˆ - Since PT is in , it is normal toSπ sˆ E - Since PT is in the tangent plane to the ellipsoid, it is normal to PV ( ( Tπ sˆ H EH- Since is also normal to and PT defines the direction of is also normal toH D The tangent to an ellipse is only normal to the radius vector if it is one of the axes of the ellipse. We conclude that the directions of are the directions of the axes of the ellipse obtained by the intersection of the ellipsoid with the plane normal to . 21 DandD sˆ
  • 111. 111 SOLO ELECTROMAGNETICS Planar Waves in an Source-less Anisotropic Electric Media Index Ellipsoid (continue – 4( To find for the propagation direction we find the ellipse obtained by the intersection of the plane normal to , at the origin O, with the Index Ellipsoid. 21 DandD sˆ sˆSπ 21 DandD are in the direction of ellipsoid axes, and therefore perpendicular to each other is on the normal to ellipsoid at OP = and in the plane of and . In the same plane and normal to is E sˆ E Dπ D Dπ HE HE S × × = → 1
  • 112. 112 SOLO ELECTROMAGNETICS Planar Waves in an Source-less Anisotropic Electric Media Index Ellipsoid (continue – 5( 12 2 2 2 2 2 =++ zyx n z n y n xnx < ny < nz. For the propagation direction of the Optical Axis the intersection of the plane normal to , at the origin O, with the Index Ellipsoid is a Circle. OAsˆSπ OAsˆ 1 D 21 DandD are in any direction on this circle, and therefore for any we can find the perpendicular to it.2 D Conical Refraction on the Optical Axis We want to find the direction of vectors. E We have: [ ] ET E E E EsDs z y x zxOAOA ⋅=           ⋅=⋅⋅=⋅=  δεδεε cos0sinˆˆ0 where: [ ]T zxT δεδε cos0sin:=            = δ δ cos 0 sin ˆOAsFor nx < ny < nz we found that in principal dielectric axes         − − ±= −− −− − 22 22 1 2,1 sin zx yx nn nn δ SDEs ,,,ˆ are coplanar and normal toH We also found: DE DD n ⋅ ⋅ = 0 2 1 ε const n D DE y ==⋅ 2 0 2 ε We found: y nn =
  • 113. 113 SOLO ELECTROMAGNETICS Planar Waves in an Source-less Anisotropic Electric Media Index Ellipsoid (continue – 6( 12 2 2 2 2 2 =++ zyx n z n y n xnx < ny < nz. Conical Refraction on the Optical Axis (continue – 1( OAsˆ Let draw in principal dielectric axes coordinate O,x,y,z, starting from O, the vectors and . They define the plane Bπ T  0=⋅ ET  Since we can find in vectors such thatEBπ constDE =⋅ and , , and are in the same plane. E OAsˆD HE HE S × × = → 1 Tπ Therefore the locus of is the projection of the circle of from plane to an ellipse in plane having the Oy axis in common. E D Sπ Since are in the same plane with and will be on a conical surface with O as a vertex having and on its surfacesT  OAsˆDHE HE S × × = → 1 → S1 OAsˆ
  • 114. 114 SOLO ELECTROMAGNETICS Planar Waves in an Source-less Anisotropic Electric Media Index Ellipsoid (continue – 7( Conical Refraction on the Optical Axis (continue – 2( Take a biaxial crystal and cut it so that two parallel faces are perpendicular to the Optical Axis. If a monochromatic unpolarized light is normal to one of the crystal faces, the energy will spread out in the plate in a hollow cone, the cone of internal conical refraction. When the light exits the crystal the energy and wave directions coincide, and the light will form a hollow cylinder. This phenomenon was predicted by William Rowan Hamilton in 1832 and confirmed experimentally by Lloyd, a year later (Born & Wolf(. Because it is no easy to obtain an accurate parallel beam of monochromatic light on obtained two bright circles (Born & Wolf(. William Rowan Hamilton (1805-1855(
  • 115. 115 SOLO ELECTROMAGNETICS Planar Waves in an Source-less Anisotropic Electric Media Index Ellipsoid (continue – 8( Conical Refraction on the Optical Axis (continue – 3( Return to Table of Content
  • 116. 116 SOLO ELECTROMAGNETICS Planar Waves in an Source-less Anisotropic Electric Media BEs c n DHs c n   =× −=× ˆ ˆ HHIB ED   µµ ε =⋅= ⋅= ( ) E n c D n c Hs n c Bs n c Ess ⋅−=−=×=×=×× εµµµ  2 2 2 2 ˆˆˆˆ 1ˆˆ 0 0 0 ˆˆ 222 =++=⋅           − − − =×⇒           = zyx xy xz yz z y x sssss ss ss ss s s s s s           = z y x ε ε ε ε 00 00 00                    == 0 2 0 2 0 2 22 2 00 00 00 1 ε ε ε ε ε ε ε µε µεµ n n n nn c z y x  ( ) ( ) ( ) ( )             +− +− +− =             +− +− +− =           − − −           − − − =×× 2 2 2 22 22 22 1 1 1 0 0 0 0 0 0 ˆˆ zzyzx zyyyx zxyxx yxzyzx zyzxyx zxyxzy xy xz yz xy xz yz sssss sssss sssss ssssss ssssss ssssss ss ss ss ss ss ss ss Equation of Wave Normal ( ) 0ˆˆ 2 2 =⋅+×× E n c Ess εµ  ( ) 0ˆˆ 2 2 1 =+×× − D n c Dss µε  or
  • 117. 117 SOLO ELECTROMAGNETICS Planar Waves in an Source-less Anisotropic Electric Media ( ) 0ˆˆ 2 2 =⋅+×× E n c Ess εµ  0 1 1 1 2 0 2 2 0 2 2 0 2 =                             +− +− +− z y x z z zyzx zyy y yx zxyxx x E E E s n ssss sss n ss sssss n ε ε ε ε ε ε The solution is obtained when       −      −+      −      −+      −      −+      −      −      −= 111111111 0 2 0 2 2 0 2 0 2 2 0 2 0 2 2 0 2 0 2 0 2 ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε nn s nn s nn s nnn yx z zx y yz x zyx                   +− +− +− = 2 0 2 2 0 2 2 0 2 1 1 1 det0 z z zyzx zyy y yx zxyxx x s n ssss sss n ss sssss n ε ε ε ε ε ε       −−      −−            −+      −+      −      −      +−= 1111111 0 2 22 0 2 22 0 2 2 0 2 2 0 2 0 2 2 0 2 ε ε ε ε ε ε ε ε ε ε ε ε ε ε n ss n ss n s n s nn s n y zx z yx y z z y zy x x Equation of Wave Normal (continue – 1(
  • 118. 118 SOLO ELECTROMAGNETICS Planar Waves in an Source-less Anisotropic Electric Media 0111111111 1 1 1 det 0 2 0 2 2 0 2 0 2 2 0 2 0 2 2 0 2 0 2 0 2 2 0 2 2 0 2 2 0 2 =        −        −+        −        −+        −        −+        −        −        −= =                     +− +− +− ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε nn s nn s nn s nnn s n ssss sss n ss sssss n yx z zx y yz x zyx z z zyzx zyy y yx zxyxx x Divide by (assumed nonzero!!(       −      −      −− 111 0 2 0 2 0 2 2 ε ε ε ε ε ε nnn n zyx 2 0 2 2 0 2 2 0 2 2 1 nn s n s n s z z y y x x = − + − + − ε ε ε ε ε ε 222 2 22 2 22 2 1 nnn s nn s nn s z z y y x x = − + − + − 0 2 0 2 0 2 ,, ε ε ε ε ε ε z z y y x x nnn === Equation of Wave Normal (continue – 2( Fresnel equation of wave normal Fresnel's wave surface, found by Augustin-Jean Fresnel in 1821, is a quartic surface describing the propagation of light in an optically biaxial crystal Augustin-Jean Fresnel (1788 – 1827(
  • 119. 119 SOLO ELECTROMAGNETICS Planar Waves in an Source-less Anisotropic Electric Media Equation of Wave Normal (continue – 3( Fresnel equation of wave normal 02 0 2 0 222 0 2 0 222 0 2 0 222 0 2 0 2 0 =        −        −+        −        −+        −        −+        −        −        − nnsnnnsnnnsnnnn yx z zx y yz x zyx ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε 022 00 42 00 6222 00 42 00 6222 00 42 00 62 000 2 000000 4 000 6 =+        +−++        +−++        +−+ +        ++−        +++− nsnsnsnsnsnsnsnsns nnn z yx z yx zy zx y zx yx zy x zy x zyxzyzxyxzyx ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ( ) ( ) ( ) 0111 000 22 00 2 00 2 00 42 0 2 0 2 0 =+      −+−+−−        ++ ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε zyx x zy y zx z yx z z y y x x nsssnsss 0111111111 0 2 0 2 2 0 2 0 2 2 0 2 0 2 2 0 2 0 2 0 2 =        −        −+        −        −+        −        −+        −        −        − ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε nn s nn s nn s nnn yx z zx y yz x zyx 6 n This is a quadratic equation of n2 . For each set sx, sy, sz it yields two solutions for n2 : n1 2 and n2 2 ( )snnn ˆ,, 21 ε  = 222 2 22 2 22 2 1 nnn s nn s nn s z z y y x x = − + − + − 1 222 =++ zyx sss Return to Table of Content

Notas del editor

  1. M.V.Klein, T.Furtak,&amp;quot;Optics&amp;quot;, pp.34-35 E. Hecht, A. Zajac,&amp;quot;Optics&amp;quot;,pp.8, pp.225-226 W.Swindell,Ed.,&amp;quot;Polrized Light&amp;quot;, BenchmarkPapers in Optics, V.1, pg.10 and 25
  2. M.V.Klein, T.Furtak,&amp;quot;Optics&amp;quot;, pp.34-35 E. Hecht, A. Zajac,&amp;quot;Optics&amp;quot;,pp.8, pp.225-226
  3. M. Born, E. Wolf,&amp;quot;Principles of Optics&amp;quot;, pp. xxiii M.V. Klein, T. Furtak,&amp;quot;Optics&amp;quot;, pp.35
  4. http://en.wikipedia.org/wiki/Charles-Augustin_de_Coulomb
  5. http://en.wikipedia.org/wiki/Charles-Augustin_de_Coulomb
  6. http://chem.ch.huji.ac.il/history/oersted.htm
  7. http://en.wikipedia.org/wiki/Amp%C3%A8re%27s_Law http://www-history.mcs.st-andrews.ac.uk/Biographies/Ampere.html http://www.ampere.cnrs.fr/activites/act_ampere_recherche.php?qt=titre&amp;titre=Rapport&amp;page=7&amp;total=205&amp;lang=fr
  8. http://www-history.mcs.st-andrews.ac.uk/history/Biographies/Biot.html http://www-history.mcs.st-andrews.ac.uk/Biographies/Ampere.html http://www.cec.uchile.cl/~cutreras/apuntes/node82.html#SECTION00710000000000000000
  9. http://www-history.mcs.st-andrews.ac.uk/history/Biographies/Biot.html http://www.cec.uchile.cl/~cutreras/apuntes/node82.html#SECTION00710000000000000000
  10. http://chem.ch.huji.ac.il/history/faraday.htm http://www.ece.umd.edu/~taylor/frame1.htm
  11. Maffett, &amp;quot;Topics for a Statistical Description of Radar Cross Section&amp;quot;, pp.47
  12. Maffett, &amp;quot;Topics of Statistical Description of Radar Cross Section&amp;quot;, 1988, pp.25-27 and 287-289
  13. Maffett, &amp;quot;Topics of Statistical Description of Radar Cross Section&amp;quot;, 1988, pp.25-27 and 287-289
  14. Maffett, &amp;quot;Topics of Statistical Description of Radar Cross Section&amp;quot;, 1988, pp.25-27 and 287-289
  15. A. Yariv, P. Yeh,&amp;quot;Optical Waves in Crystals&amp;quot;, John Wiley &amp; Sons, 1984, pp.79-82 A. Maffett,&amp;quot;Topics of Statistical Description of Radar Cross Sections&amp;quot;, John Wiley &amp; Sons, 1988, pp.314
  16. G.R. Fowles,&amp;quot;Introduction to Modern Optics&amp;quot;, 2nd Ed., Dover, p. 176 Yariv &amp; Yeh,&amp;quot;Optical Waves in Crystals&amp;quot;, John Wiley &amp; Sons, 1984, p.85
  17. Yariv &amp; Yeh, &amp;quot;Optical Waves in Crystals&amp;quot;,John Willey &amp; Sons, 1984, p.85 G.R. Fowles, &amp;quot;Introduction to Modern Optics&amp;quot;, 2nd Ed., Dover, 1975, p.176
  18. A. Yariv, P. Yeh &amp;quot;Optical Waves in Crystals&amp;quot;,1984, John Wiley &amp; Sons, p.83 A. Mermin, &amp;quot;Solid State Physics&amp;quot;,Holt, Reinhart and Winston, 1976, p.116 http://www.tulane.edu/~sanelson/eens211/32crystalclass.htm
  19. A. Yariv, P. Yeh &amp;quot;Optical Waves in Crystals&amp;quot;,1984, John Wiley &amp; Sons, p.83 A. Mermin, &amp;quot;Solid State Physics&amp;quot;,Holt, Reinhart and Winston, 1976, p.116 http://www.tulane.edu/~sanelson/eens211/32crystalclass.htm
  20. A. Yariv, P. Yeh &amp;quot;Optical Waves in Crystals&amp;quot;,1984, John Wiley &amp; Sons, p.83 A. Mermin, &amp;quot;Solid State Physics&amp;quot;,Holt, Reinhart and Winston, 1976, p.116 http://www.tulane.edu/~sanelson/eens211/32crystalclass.htm
  21. A. Yariv, P. Yeh,&amp;quot;Optical Waves in Crystals&amp;quot;, John Wiley &amp; Sons, 1984, pp.79-82 A. Maffett,&amp;quot;Topics of Statistical Description of Radar Cross Sections&amp;quot;, John Wiley &amp; Sons, 1988, pp.314
  22. G.R. Fowles,&amp;quot;Introduction to Modern Optics&amp;quot;, Dover, 1975, pp.178-180
  23. Born &amp; Wolf, &amp;quot;Principle of Optics&amp;quot;, 6th Ed., pp.686-690 M.V. Klein,&amp;quot;Optics&amp;quot;, 1970, pp.609
  24. Born &amp; Wolf, &amp;quot;Principle of Optics&amp;quot;, 6th Ed., pp.686-690 M.V. Klein,&amp;quot;Optics&amp;quot;, 1970, pp.609
  25. http://en.wikipedia.org/wiki/Wave_surface