lecture on photonic crystal

1. Photonic Crystals (Tinh thể quang tử):
Principles , Techniques, and Applications

2. Outline
waves in periodic media
Photonic crystals in theory and practice
Intentional defects and devices
Index-guiding and incomplete gaps

3. Outline
waves in periodic media
Photonic crystals in theory and practice
Bulk crystal properties
Intentional defects and devices
Index-guiding and incomplete gaps
Perturbations, tuning, and disorder

4. planewave
( )
, ~ i k x t
E H e 
 
2
/
k c



 

5. for most , beam(s) propagate
through crystal without scattering
(scattering cancels coherently)
...but for some  (~ 2a), no light can propagate: a photonic band gap
planewave
( )
, ~ i k x t
E H e 
 
2
/
k c



 

6. Photonic Crystals
periodic electromagnetic media
with photonic band gaps: “optical insulators”
(need a more
complex
topology)

7. Photonic Crystals in Nature
wing scale:
Morpho butterfly
6.21µm
[ L. P. Biró et al.,
PRE 67, 021907
(2003) ]
Peacock feather
[J. Zi et al, Proc. Nat. Acad. Sci. USA,
100, 12576 (2003) ]
[figs: Blau, Physics Today 57, 18 (2004)]

8. Photonic Crystals
periodic electromagnetic media
with photonic band gaps:
“optical insulators”
for holding and controlling light
can trap light in cavities and waveguides (“wires”)

9. Photonic Crystals
periodic electromagnetic media
But how can we understand such complex systems?
Add up the infinite sum of scattering?

10. A mystery from the 19th century

11. A mystery from the 19th century

12. A mystery solved…
electrons are waves (quantum mechanics)
1
waves in a periodic medium can
propagate without scatterings)
2
The foundations do not depend on the specific wave equation.

13. Electronic and Photonic Crystals
atoms in diamond structure
wavevector
electron
energy
Periodic
Medium
Bloch
waves:
Band
Diagram
dielectric spheres, diamond lattice
wavevector
photon
frequency
interacting: hard problem non-interacting: “easy” problem

14. Fun with Math
1
1
E H i H
c t c
H E J i E
c t c
 

 
 

   
   
0
dielectric function (x) = n2(x)
First task:
get rid of this mess
eigen-operator eigen-value eigen-state
+ constraint
2
1
H H
c


 
    
 
0
H
 

15. Hermitian Eigenproblems
Hermitian for real (lossless) 
well-known properties from linear algebra:
 are real (lossless)
eigen-states are orthogonal
eigen-states are complete (give all solutions)
eigen-operator eigen-value eigen-state
+ constraint
2
1
H H
c


 
    
 
0
H
 

16. Periodic Hermitian Eigenproblems
[ G. Floquet, “Sur les équations différentielles linéaries à coefficients périodiques,” Ann. École Norm. Sup. 12, 47–88 (1883). ]
[ F. Bloch, “Über die quantenmechanik der electronen in kristallgittern,” Z. Physik 52, 555–600 (1928). ]
if eigen-operator is periodic, then Bloch-Floquet theorem applies:
can choose:
periodic “envelope”
planewave
Corollary 1: k is conserved, i.e. no scattering of Bloch wave
Corollary 2: given by finite unit cell,
so  are discrete n(k)
 
( , ) ( )
i k x t
k
H x t e H x

 

k
H

17. Periodic Hermitian Eigenproblems
1
2
3

k
band diagram (dispersion relation)
map of
what states
exist &
can interact
?
range of k?
Corollary 2: given by finite unit cell,
so  are discrete n(k)
k
H

18. Periodic Hermitian Eigenproblems in 1d
1 2 1 2 1 2 1 2 1 2 1 2
(x) = (x+a)
a
Consider k+2π/a:
periodic!
satisfies same equation as Hk
= Hk
k is periodic:
k + 2π/a equivalent to k
“quasi-phase-matching”
( ) ( )
ikx
k
H x e H x

2 2
( )
2 2
( ) ( )
i k x i x
ikx
a a
k k
a a
e H x e e H x
 
 

 
 
  
 

19. band gap
Periodic Hermitian Eigenproblems in 1d
1 2 1 2 1 2 1 2 1 2 1 2
(x) = (x+a)
a
k is periodic:
k + 2π/a equivalent to k
“quasi-phase-matching”
k

0 π/a
–π/a
irreducible Brillouin zone

20. Any 1d Periodic System has a Gap
1
k

0
[ Lord Rayleigh, “On the maintenance of vibrations by forces of double frequency, and on the propagation of
waves through a medium endowed with a periodic structure,” Philosophical Magazine 24, 145–159 (1887). ]
Start with
a uniform (1d) medium:
1
ck




21. Any 1d Periodic System has a Gap
1
(x) = (x+a)
a
k

0 π/a
–π/a
[ Lord Rayleigh, “On the maintenance of vibrations by forces of double frequency, and on the propagation of
waves through a medium endowed with a periodic structure,” Philosophical Magazine 24, 145–159 (1887). ]
Treat it as
“artificially” periodic
bands are “folded”
by 2π/a equivalence
,
cos , sin
i x i x
a a
e e
x x
a a
 
 
 
   
    
   

22. (x) = (x+a)
a
1
Any 1d Periodic System has a Gap

0 π/a
[ Lord Rayleigh, “On the maintenance of vibrations by forces of double frequency, and on the propagation of
waves through a medium endowed with a periodic structure,” Philosophical Magazine 24, 145–159 (1887). ]
x = 0
Treat it as
“artificially” periodic
sin
cos
x
a
x
a


 
 
 
 
 
 

23. (x) = (x+a)
a
1 2 1 2 1 2 1 2 1 2 1 2
Any 1d Periodic System has a Gap

0 π/a
[ Lord Rayleigh, “On the maintenance of vibrations by forces of double frequency, and on the propagation of
waves through a medium endowed with a periodic structure,” Philosophical Magazine 24, 145–159 (1887). ]
Add a small “real” periodicity 2 = 1 + D
x = 0
sin
cos
x
a
x
a


 
 
 
 
 
 

24. band gap
Any 1d Periodic System has a Gap

0 π/a
[ Lord Rayleigh, “On the maintenance of vibrations by forces of double frequency, and on the propagation of
waves through a medium endowed with a periodic structure,” Philosophical Magazine 24, 145–159 (1887). ]
Add a small
“real” periodicity
2 = 1 + D
(x) = (x+a)
a
1 2 1 2 1 2 1 2 1 2 1 2
x = 0
Splitting of degeneracy:
state concentrated in higher index (2)
has lower frequency
sin
cos
x
a
x
a


 
 
 
 
 
 

25. Some 2d and 3d systems have gaps
• In general, eigen-frequencies satisfy Variational Theorem:
Inverse “potential”
“kinetic”
bands “want” to be in high-
…but are forced out by orthogonality
–> band gap (maybe)
 
1
1
2
1
2 2
1 2
1
0
( ) min
E
E
ik E
k c
E



 
  



2
2
*
1 2
2
2
0
0
( ) min " "
E
E
E E
k



 
 



26. A 2d Model System
a
Square lattice of dielectric rods ( = 12 ~ Si) in air ( = 1)

27. Solving the Maxwell Eigenproblem
H(x,y) ei(kx – t)
where:
constraint:
1
Want to solve for n(k),
& plot vs. “all” k for “all” n,
Finite cell  discrete eigenvalues n
Limit range of k: irreducible Brillouin zone
2 Limit degrees of freedom: expand H in finite basis
3 Efficiently solve eigenproblem: iterative methods
   
2
2
1 n
n n
i i
c


      
k k H H
  0
i
   
k H

28. Outline
Preliminaries: waves in periodic media
Photonic crystals in theory and practice
Intentional defects and devices

29. 2d periodicity, =12:1
a
irreducible Brillouin zone
G
X
M
k

30. 2d periodicity, =12:1
Ez
– +
Ez
(+ 90° rotated version)
gap for n > ~1.75:1

31. 2d periodicity, =12:1
a
irreducible Brillouin zone
G
X
M
k

32. 2d photonic crystal: TE gap, =12:1
gap for n > ~1.4:1

33. 3d photonic crystal: complete gap , =12:1
I.
II.
[ S. G. Johnson et al., Appl. Phys. Lett. 77, 3490 (2000) ]
gap for n > ~4:1

34. The First 3d Bandgap Structure
K. M. Ho, C. T. Chan, and C. M. Soukoulis, Phys. Rev. Lett. 65, 3152 (1990).
overlapping Si spheres
MPB tutorial, http://ab-initio.mit.edu/mpb
for gap at  = 1.55µm,
sphere diameter ~ 330nm

35. Layer-by-Layer Lithography
• Fabrication of 2d patterns in Si or GaAs is very advanced
(think: Pentium IV, 50 million transistors)
So, make 3d structure one layer at a time
…inter-layer alignment techniques are only slightly more exotic
Need a 3d crystal with constant cross-section layers

36. A Schematic
[ M. Qi, H. Smith, MIT ]

37. 7-layer E-Beam Fabrication
[ M. Qi, et al., Nature 429, 538 (2004) ]

38. The Woodpile Crystal
[ S. Y. Lin et al., Nature 394, 251 (1998) ]
gap
(4 “log” layers = 1 period)
http://www.sandia.gov/media/photonic.htm
Si
[ K. Ho et al., Solid State Comm. 89, 413 (1994) ] [ H. S. Sözüer et al., J. Mod. Opt. 41, 231 (1994) ]

39. Two-Photon Lithography
Atom
e E0
hn = ∆E
hn
photon
hn
photon
2 2-photon probability ~ (light intensity)2
lens
some chemistry
(polymerization)
3d Lithography
…dissolve unchanged stuff
(or vice versa)

40. 2µm
Lithography is a Beast [ S. Kawata et al., Nature 412, 697 (2001) ]
 = 780nm
resolution = 150nm
7µm
(3 hours to make)

41. Holographic Lithography
[ D. N. Sharp et al., Opt. Quant. Elec. 34, 3 (2002) ]
absorbing material
Four beams make 3d-periodic interference pattern
(1.4µm)
k-vector differences give reciprocal lattice vectors (i.e. periodicity)
beam polarizations + amplitudes (8 parameters) give unit cell

42. One-Photon
Holographic Lithography
[ D. N. Sharp et al., Opt. Quant. Elec. 34, 3 (2002) ]
huge volumes, long-range periodic, fcc lattice…backfill for high contrast
10µm

43. Mass-production II: Colloids
microspheres (diameter < 1µm)
silica (SiO2)
sediment by gravity into
close-packed fcc lattice!
(evaporate)

44. Inverse Opals
fcc solid spheres do not have a gap…
…but fcc spherical holes in Si do have a gap
Infiltration
sub-micron colloidal spheres
Template
(synthetic opal)
3D
Remove
Template
“Inverted Opal”
complete band gap
~ 10% gap between 8th & 9th bands
small gap, upper bands: sensitive to disorder
[ H. S. Sözüer, PRB 45, 13962 (1992) ]

45. In Order To Form
a More Perfect Crystal…
meniscus
silica
250nm
Convective Assembly
[ Nagayama, Velev, et al., Nature (1993)
Colvin et al., Chem. Mater. (1999) ]
• Capillary forces during drying cause assembly in the meniscus
• Extremely flat, large-area opals of controllable thickness
Heat Source
80C
65C
1 micron
silica spheres
in ethanol
evaporate
solvent

46. A Better Opal

47. Inverse-Opal Photonic Crystal
[ Y. A. Vlasov et al., Nature 414, 289 (2001). ]

48. Outline
Preliminaries: waves in periodic media
Photonic crystals in theory and practice
Intentional defects and devices

49. Intentional “defects” are good

50. Cavity Modes
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51. Cavity Modes
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finite region –> discrete 

52. Cavity Modes: Smaller Change
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53. Cavity Modes: Smaller Change
L
Bulk Crystal Band Diagram

54. Cavity Modes: Smaller Change
L
Defect bands are shifted
up (less )
with discrete k
# ~ ; ( ~ 2 / )
2
L k


 

55. Single-Mode Cavity
A point defect
can push up
a single mode
from the band edge
0
field decay ~
curvature
 


56. Projected Band Diagrams
conserved k!
1d periodicity
G X
M
conserved
not
conserved
So, plot  vs. kx only…project Brillouin zone onto G–X:
gives continuum of bulk states + discrete guided band(s)

57. Air-waveguide Band Diagram

58. (Waveguides don’t really need
a complete gap)
Fabry-Perot waveguide:
This is exploited e.g. for photonic-crystal fibers…

59. Guiding Light in Air!
mechanism is gap only vs. standard optical fiber:
“total internal reflection”
— requires higher-index core
no hollow core!
hollow = lower absorption, lower nonlinearities, higher power