wave optics

1. 1
Wave Optics

2. Content
2
 Introduction
 What are waves?
 Wave equation
 Wave optics
 Gaussian Beam
 Diffraction
 Interference
 Coherence

3. 3
Introduction
Light is part of the family of electromagnetics
radiation
“Electromagnetics is the study of electric and
magnetic phenomena and their engineering
applications”
Light, i.e. a family of Electromagnetic radiation or
wave, constitutes a time varying electric and
magnetic fields propagating in certain direction

4. 4
Electromagnetic radiation propagating in z direction

5. 5
 Electromagnetic (EM) waves includes many types of
signal: visible light, radio waves, infrared waves,
gamma rays, x rays, …
 All EM waves share the following properties
– Phase velocity in vacuum is c = 3 x 108
m/s
– In vacuum, for any EM wave λ = c/ f
Each EM waves is distinguished by its own
wavelength, or equivalently by its own oscillation
frequency f.
The Electromagnetic Spectrum

6. 6

7. 7
What are waves?
 Waves are a natural consequence of many physical
processes:
- waves and ripples on oceans and lakes,
- mechanical waves on stretched strings,
- sound wave that travel trough air,
- electromagnetic waves (light,..)
- earthquake waves,…

8. Common properties of EM waves
8
• Moving waves carry energy
• Waves have velocity
It takes time for a wave to travel from one point
to another
Light wave travel at 3 x 108
m/s
Sound wave travel at 330 m/s
• Some waves exhibit a property called linearity
Waves that do not affect the passage of other
waves are called linear.
The total of two linear waves is the sum of the
two

9. 9
Wave equation
The basis for understanding the wave theory of
transmission is through 4 sets of equations known
as Maxwell's equations.
Maxwell’s Equations
Modern electromagnetism is based on these set of
four fundamental relations given by:
∇ •D = ρv,
∇ ×E = -dB/dt
∇ •B = 0,
∇ ×H = J + dD/dt

10. 10
Symbols
E and D are electric field quantities interrelated by D
= εE, with ε being the electrical permittivity of the
material;
B and H are magnetic field quantities interrelated by
B = µH, with µ being the magnetic permeability of
the material;
ρv is the electric charge density per unit volume; and
J is the current density per unit area.

11. 11
Wave equation
The wave equation can be derived from the Maxwell’s
Equations by assuming the conductivity σ and volume charge
density ρν are both zero.
2
2
2
2
2
tc
n
∂
∂
=∇
ψ
ψ
The wave equation
Where ψ may represent a component of the E or H field and c is
the velocity given by c=1/(εoµo)0.5
in the dielectric medium.
From here the general solution is given by
( ){ }ztj βωψψ −= exp0
ψ0 – amplitude
ω - angular frequency
t – time
β - propagation constantComplex representation

12. Wave equation
Wave in a lossless Medium
12
A medium is said to be lossless if it does not
attenuate the amplitude of the wave traveling within it
or on its surface.
y(x,t) = A cos (ωt-kx)
where
A amplitude
f (Hz) Frequency
T (s) Period
ω(rad/s) = 2π/T = 2πf Angular frequency
λ (m) Wavelength
k (rad/m) = 2π/λ Phase constant / wavenumber
up = λf = ω/β phase velocity

13. Phase & Phase Velocity
 Phase was defined as the argument of the sine function.
j = kx ± wt
 At t = x = 0,
which is a special case.
 The sine function can be rewritten as
where ε is the initial phase.
( ) ( ) 00,0,
0
0 ==
=
= ψψ
t
xtx
( ) )sin(, εωψ +−= tkxAtx
13

14. 14
 The initial phase angle is just the constant contribution to the
phase arising from the generator.
 It is independent of how far or how long the wave has
travelled.
 The phase (kx – ωt) and (ωt – kx) were used to describe the
sine wave functions earlier.
 Both describe waves moving in the positive x–direction that are otherwise
identical except for a relative phase difference of p.
 The initial phase is of no particular significance; thus literatures abound with
both expressions.

15.  The phase of a disturbance is a function of x and t.
 The partial derivative of ϕ wrt t or the rate-of-change of phase
with time,
 It is given as the angular frequency of the wave at any fixed location.
 It indicates the rate at which a point oscillates up and down.
 For each cycle, ϕ changes by 2π.
( ) ( )εωϕ +−= tkxtx,
ω
ϕ
=





∂
∂
xt
15

16.  The partial derivative of ϕ wrt x or the rate-of-change of
phase with distance,
 These two expressions combined yields
 The term on the LHS represents the speed of propagation of the
condition of constant phase.
 It gives the speed at which the profile moves and is known
commonly as the phase velocity of the wave.
 It is positive when the wave moves in positive x direction; negative
when in direction of decreasing x.
( )
( )
v
kx
t
t
x
t
x
±=±=
∂∂
∂∂−
=





∂
∂ ω
ϕ
ϕ
ϕ
k
x t
=





∂
ϕ∂
16

17. 17
 In a lossy medium, the amplitude of the wave will
decrease as an exponential decaying factor e-αx
 Thus
y(,x, t)= A e-αx
cos (ωt-βx+φ0)
 e-αx
: attenuation factor
 and α: attenuation constant of the medium (Neper
per meter, Np/m)
 Notice the wave amplitude is now A e-αx
.
Wave in a Lossy Medium

18. 18

19. Wave Optics
19
In describing the propagation of light as
a wave we need to understand:
wavefronts: The surfaces joining all
points of equal phase are known as
wavefronts or a surface passing
through points of a wave that have
the same phase and amplitude.
rays: a ray describes the direction of
wave propagation. A ray is a vector
perpendicular to the wavefront.

20. Wavefronts
20
 We can chose to associate
the wavefronts with the
instantaneous surfaces where
the wave is at its maximum.
 Wavefronts travel outward
from the source at the speed
of light: c.
 Wavefronts propagate
perpendicular to the local
wavefront surface.

21. Light Rays
21
 The propagation of the
wavefronts can be
described by light rays.
 In free space, the light
rays travel in straight
lines, perpendicular to
the wavefronts.

22. Huygens’ Principle:
22
 Huygens’ principle (Christaan Huygens, 1629-1695, published
about 1690) describes how a wavefront moves in space.
 According to this principle, we imagine that each point on the
wavefront acts as a point source that emits spherical wavelets.
 These wavelets travel with the velocity of light in the medium.
 At any later time, the total wavefront is the envelope that
encloses all of these wavelets.
 That is, the tangent line that joins the front surface of each
one of them.

23. Huygens’ Principle:
23
All the points on a
wavefront can be
considered as point source
for the production of
spherical secondary
wavelets. At the later time,
the new position of the
wavefront will be the
surface of tangency to
these secondary wavelets.

24. Superposition Principle of Waves
 The 1-D differential wave equation reveals an intriguing
property of waves; its solutions are according to the
Superposition Principle.
 This can be easily proven to be true. Consider two
different wave functions y1 and y2; both are separate
solutions to the differential wave equation.Thus,
2
2
2
22
2
2
2
1
2
22
1
2
1
and
1
tvxtvx ∂
∂
=
∂
∂
∂
∂
=
∂
∂ ψψψψ
24

25.  Adding those two equations yield
 It is established that (ψ1 + ψ2) is indeed a solution itself.
 Physically it translate that when two separate waves arrive
at the same place in space wherein they overlap, these
waves simply add (or subtract from) one another without
permanently destroying of disrupting either wave.
( ) ( )212
2
2212
2
2
2
2
22
1
2
22
2
2
2
1
2
1
11
ψ+ψ
∂
∂
=ψ+ψ
∂
∂
∂
ψ∂
+
∂
ψ∂
=
∂
ψ∂
+
∂
ψ∂
tvx
tvtvxx
25

26. The resulting disturbance at each point in the region of overlap is the
algebraic sum of the individual constituent waves at that location.
26
Superposition of two equal-wavelength sinusoids
-2
-1
0
1
2
-1 0 1 2 3 4
kx (rad)
ψ (x, 0)
ψ1
ψ2
ψ
)(1 oxψ
)(2 oxψ
)(1 oxψ
)()( 21 oo xx ψψ +
At every point, the resultant wave is the summation of the
individual waves
( )rad0.1sin9.0
sin0.1
2
1
+=ψ
=ψ
kx
kx

27.  The two constituent waves are in-phase (phase-angle
difference is zero).
 The composite wave of substantial amplitude is sinusoidal with
the same frequency and wavelength as the component waves.
27
-2
-1
0
1
2
-1 0 1 2 3 4 5 6 7
kx
ψ
ψ
ψ1
ψ
2

28.  The resultant amplitudes
diminishes as the phase-angle
difference increases.
 It is expected to vanish when the
phase difference equals π.At that
point, the waves are said to be
out-of-phase.
 The fact that at this point the
resultant wave amplitude is zero
gives rise to the name
interference.
-2
-1
0
1
2
-1 0 1 2 3 4 5 6 7
kx
ψ
28
-2
-1
0
1
2
-1 0 1 2 3 4 5 6 7
kx
ψ
-2
-1
0
1
2
-1 0 1 2 3 4 5 6 7
kx
ψ

29. 29
 Before proceeding with other wave phenomena, it
should be obvious that analysis using sine and cosine
functions involving trigonometric manipulations are
rather involved and somewhat tedious.
 Consider the following solution for the algebraic
addition of two (or more) overlapping waves that
have the same frequency and wavelength.

30. Algebraic Method
 A solution of the differential wave equation can be written
in the form
in which Eo is the amplitude of the harmonic disturbance
propagating along the positive x-axis.
 To separate the space and time part of the phase, let
so that
[ ])(sin),( ε+−ω= kxtEtxE o
[ ]),(sin),( εα+ω= xtEtxE o
30
( ) ( )ε+−=εα kxx,

31.  Suppose then that there are two such waves
and
each with the same frequency and speed, coexisting in space.
 The resultant disturbance is the linear superposition of these
waves:
E = E1 + E2
[ ]111 sin),( α+ω= tEtxE o
[ ]222 sin),( α+ω= tEtxE o
31
( )222
111
sincoscossin
sincoscossin
αω+αω+





 αω+αω=
ttE
ttEE
o
o

32.  Separate the time-dependent terms,
 Let
 Take the summation of the square of the two equations
above:
which the sought-after expression for the amplitude (Eo) of
the resultant wave.
2211
2211
sinsinsin
coscoscos
α+α=α
α+α=α
ooo
ooo
EEE
EEE
32
( )
( ) tEE
tEEE
oo
oo
ωα+α+
ωα+α=
cossinsin
sincoscos
2211
2211
( )1221
2
2
2
1
2
cos2 α−α++= ooooo EEEEE

33.  The phase of the resultant wave is obtained from
 The total disturbance then becomes
or
 The composite wave is harmonic and of the same frequency
as the constituents, although its amplitude and phase are
different.
tEtEE oo ωα+ωα= cossinsincos
( )α+ω= tEE o sin
33
2211
2211
coscos
sinsin
tan
α+α
α+α
=α
oo
oo
EE
EE

34.  When Eo1 >> Eo2, α ≅ α1 and when Eo2 >> Eo1 , α ≅ α2 .The
resultant wave is in-phase with the dominant component
wave.
 The flux density of a light wave is proportional to its
amplitude squared.
 The resultant flux density is not simply the sum of the
component flux densities; there is an additional
contribution, known as the interference term.
2211
2211
coscos
sinsin
tan
α+α
α+α
=α
oo
oo
EE
EE
( )1221
2
2
2
1
2
cos2 α−α++= ooooo EEEEE
34

35.  The crucial factor is the difference in phase between the two
interfering waves E1 and E2,
δ ≅ α2 − α1
 When δ = 0, ±2π, ±4π,… the resultant amplitude is maximum,
whereas
δ = ±π, ±3π,… the resultant amplitude is minimum at any point in
space.
 In the former case, the waves are said to be in-phase; while in the
latter case, the waves are 180o
out-of-phase.
-2
-1
0
1
2
-1 0 1 2 3 4 5 6 7
kx
ψ
35
-2
-1
0
1
2
-1 0 1 2 3 4 5 6 7
ψ
kx
ψ
ψ1
ψ2

36. Gaussian Beams
36
 Light can take the form of beams that come as close as
possible to spatially localised & non-diverging waves –
paraxial approximation of the wave equation.
 The beam power is concentrated within a small cylinder
surrounding the beam axis and intensity distribution in the
transverse plane follows Gaussian distribution.
 The wavefronts are approximately planar near the beam
waist but they gradually curve and become approximately
spherical far from the waist.

37. 37

38. Gaussian Beam Math
38
The expression for a light
beam's electric field can then be written as:
where:
w(z) is the spot size (1/e2
width) vs. distance from the waist,
R(z) is the beam radius of curvature, and
ψ(z) is a phase shift.
This equation is the solution to the wave equation when we require that
the beam be well localized at some point (i.e., its waist).
Ex,y,z( )∝
exp−ikz−iψz()[ ]
wz()
exp
x2
+y2
w2
z()
−i
π
λ
x2
+y2
Rz()








The paraxial approximation ensures that
the plane wave is modulated by a
complex amplitude that is a slowly
varying function of position

39. Gaussian Beam Spot, Radius, and Phase
39
The expressions for the spot size,
radius of curvature, and phase shift:
where zR is the Rayleigh Range (the distance over which the beam remains
about the same diameter), and it's given by:
w z( )=w0
1+ z/zR( )
2
R z( )=z+zR
2
/z
ψ z( )=arctan z/zR( )
zR
=πw0
2
/λ

40. Gaussian Beam Collimation
40
Twice the Rayleigh range is the
distance over which the beam
remains about the same size,
that is, remains “collimated.”
_____________________________________________
.225 cm 0.003 km 0.045 km
2.25 cm 0.3 km 5 km
22.5 cm 30 km 500 km
_____________________________________________
 Tightly focused laser beams expand quickly.
 Weakly focused beams expand less quickly, but still expand.
Collimation Collimation
Waist spot Distance Distance
size w0 λ = 10.6 µm λ = 0.633 µm
Longer wavelengths
expand faster than
shorter ones.
2
02 2 /Rz wπ λ=

41. Gaussian Beam
Divergence
41
 Far away from the waist, the
 spot size of a Gaussian beam will be:
 The beam 1/e2
divergence half angle is then w(z) / z as z → ∞ :
 or…
 The Rayleigh range zR is the ratio of the beam waist radius to the half
angle divergence.
 If we substitute the expression: into the one above, we
obtain:
 So the product of the waist and the divergence depends only on the
wavelength. For a specific laser wavelength,the smaller the waist and the
larger the divergence angle.
wz()=w0
1+z/zR( )
2
≈w0
z/zR( )
2
=w0
z/zR
tanθ1/e2 =
wz()
z
=
w0
z
zR
z
=
w0
zR
zR
=
w0
θ
zR=πw0
2
/λ
w0
θ=λ/π

42. Focusing a Gaussian Beam
42
 A lens will focus a collimated Gaussian beam to a new spot size.
 It turns out that there is a relationship (derived later) between the input beam
size and the new beam waist:
w0 ≈ λ f / πw,
where w is the beam radius at the lens.
 So the smaller the desired focus, the BIGGER the input beam must be!
It should be noted that the beam radius must be less than 2/3 of the
lens diameter D. Beyond this the beam will lose its Gaussian character
and diffraction rings will begin to emerge.