1. Quantum Dots in Photonic Structures
Lecture 4: Photonic crystals
Jan Suffczyński
Wednesdays, 17.00, SDT
Projekt Fizyka Plus nr POKL /11 współfinansowany przez Unię Europejską ze środków Europejskiego
Funduszu Społecznego w ramach Programu Operacyjnego Kapitał Ludzki

2. Plan for today 2. Two-dimensional photonic crystals – band structure
DBR mirrors
3. Two-dimensional photonic crystal – fabrication methods

3. Reminder Refractive index: 𝑫= 𝜀 0 𝜀 𝑟 (𝜔)𝑬 Polarization by EM wave
𝑫= 𝜀 0 𝜀 𝑟 (𝜔)𝑬
Polarization by EM wave
Complex dielectric function

4. After: András Szilágyi
Reminder
Refractive index:
𝑫= 𝜀 0 𝜀 𝑟 (𝜔)𝑬
Polarization by EM wave
Complex dielectric function
Simultaneous description of refraction and absorption
Speed light in a medium:
𝑣= 𝑐 𝑛
𝜀 𝑟 (𝜔) =𝑛 𝜔 +𝑖∙𝜅(𝜔)
Dispersion
After: András Szilágyi

5. Reminder
Photonic crystal: periodic arrangement of dielectric (or metallic…) objects
periodic refractive index contrast
the period comparable to the wavelength of light in the material.
1D photonic crystal:
Distributed Bragg Reflector (DBR)
Example calculation:
Transfer Matrix Method

6. Bragg mirror from the University of Warsaw
MgTe
High n: Cd0.86Zn0.14Te
Cd0.86Zn0.14Te buffer
GaAs substrate
20 stack
DBR
Cd0.86Zn0.14Te
Low n: 20 x Superlattice 1:1
1 mm
SEM: T. Jakubczyk
J.-G. Rousset

7. Refractive index engineering
For a good DBR we need a pair of materials that have:
large refractive index contrast Δn = nhigh-nlow
lattice paramters as close as possible

8. Bragg mirror lattice matched to CdTe
The structure
15 par
ZnTe 53nm
superlattice
18 periods
ZnTe 53 nm
ZnTe 0.7 nm
superlattice
MgTe 0.9 nm
ZnTe buffer
1000 nm
ZnTe 0.7 nm
MgSe 1.3 nm
ZnSe 62 nm
GaAs
1 μm
W. Pacuski, UW

9. Bragg mirror lattice matched to CdTe
The structure
15 par
ZnTe 53nm
supersieć
18 powtórzeń
ZnTe 53 nm
ZnTe 0.7 nm
supersieć
MgTe 0.9 nm
ZnTe buffer
1000 nm
ZnTe 0.7 nm
MgSe 1.3 nm
ZnSe 62 nm
GaAs
W. Pacuski, UW

10. Bragg mirror lattice matched to CdTe
The structure
15 par
ZnTe 53nm
supersieć
18 powtórzeń
ZnTe 53 nm
MgSe 1.3 nm
ZnTe 0.7 nm
MgTe 0.9 nm
1 nm
ZnTe 0.7 nm
supersieć
MgTe 0.9 nm
ZnTe buffer
1000 nm
ZnTe 0.7 nm
MgSe 1.3 nm
ZnSe 62 nm
GaAs
W. Pacuski, UW

11. DBR mirror and DBR cavity reflectivity
Microcavity
W. Pacuski, UW

12. DBR mirror and DBR cavity reflectivity
Q = λ/Δλ = 3600
Microcavity

13. CdTe based microcavity – 60 pairs
But sometimes technology makes jokes…

14. Planar cavity with DBR mirrors
Stop band Δλ=(n1-n2)/π(n1+n2)
Δθ ~ 20o typically in the case og GaAs/AlAs DBR
Reflectivity
Antinode of the field in the center of the cavity
lBr
λ-cavity
Electric field distribution
Reflection of a planar DBR microcavity consisting of a top and bottom mirror with 15 and 21
repeats of GaAs/AlAs. In (a) a 240 nm-thick bulk GaAs cavity is incorporated.
Field distributions corresponding to (a) at wavelength of (c) 841 nm
Cavity mode
Exponential decay of the stationary field from the center of the cavity

15. Towards 2D and 3D photonic crystals
Low index of refraction
High index of refraction
3D photonic crystal

16. Photonic crystals – how it works?
a>>l incoherent scattering a
a~l coherent scattering a
a<<l
averaging a
Photonic crystals

17. 2D, 3D photonic bandgap?
J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade,
Photonic Crystals: Molding the Flow of Light

18. Dispersion relation and origin of the band gap in 1D
Consider medium of refractive index n1
and light of wavelenght of l = 2a
Light line:
free space
Light line:
medium n1
frequency ω
Rob Engelen
standing wave in n1
π/a
wave vector k

19. Dispersion relation and origin of the band gap in 1D
Consider the stack of layers of refractive indeces n1 and n2
and light of wavelenght of l = 2a n1 n2 n1 n2 n1 n2 n1
frequency ω a
Rob Engelen
π/a
n1: high index material
n2: low index material
wave vector k

20. Dispersion relation and origin of the band gap in 1D
Consider the stack of layers of refractive indeces n1 and n2
and light of wavelenght l = 2a
standing wave in n2 n1 n2 n1 n2 n1 n2 n1
frequency ω
Rob Engelen
standing wave in n1
π/a
n1: high index material
n2: low index material
wave vector k

21. Dispersion relation and origin of the band gap in 1D
Consider the stack of layers of refractive indeces n1 and n2
and light of wavelenght l = 2a
standing wave in n2 n1 n2 n1 n2 n1 n2 n1
frequency ω
bandgap
Rob Engelen
standing wave in n1
π/a
n1: high index material
n2: low index material
wave vector k

22. frequency ω
wave vector k
π/a
-π/a
Bloch wave with wave vector k is equal to Bloch wave with wave vector k+m2p/a:
modified slide from Rob Engelen

23. k + 2π/a is equivalent to k
Band diagram
k is periodic:
k + 2π/a is equivalent to k

24. Band diagram frequency ω wave vector k π/a -π/a frequency ω
π/a
-π/a
frequency ω
wave vector k
π/a
-π/a
-2π/a
2π/a
-2π/a
2π/a
This is the first Brillouin zone
modified slide from Rob Engelen

25. Band diagram – one more view
Anticrossing of modes leading to formation of the band gap

26. Another look- Bragg scattering conditions
When a wave impinges on a crystal it will be reflected at a particular set of lattice planes characterized by its reciprocal lattice vector g only if the so-called Bragg condition is met
If the Bragg condition is not met, the incoming wave just moves through the lattice and emerges on the other side of the crystal (when neglecting absorption)

27. Photonic crystals - introductory example from the prevous lecture
1. Bragg scattering
Regardless of how small the reflectivity r is from an individual scatterer, the total reflection R from a semi infinite structure:
Complete reflection when:
Propagation of the light in crystal inhibited when Bragg condition satisfied
Origin of the photonic bang gap

28. Reciprocal lattice

29. Dispersion relation for 2D photonic crystal
Dispersion relation for a 1D photonic crystal (solid lines). The boundary
of the first Brillouin zone is denoted by two vertical lines. The dispersion lines in
the uniform material are denoted by dashed lines. They are folded into the first
Brillouin zone taking into account the identity of the wave numbers which differ
from each other by a multiple of 2π/a. When two dispersion lines cross, they repel
each other and a photonic bandgap appears.
2D square lattice

30. Dispersion relation for 2D photonic crystal
2D hexagonal lattice
Top view of the geometry of the 2D crystal for the numerical calculation
of the transmission and the Bragg reflection spectra by means of the plane-wave
expansion method
Transmittance (right-hand side) and the dispersion relation (left-hand
side) of a hexagonal lattice for (a) the E polarization and (b) the H polarization in
the Γ-X direction. The ordinate is the normalized frequency. On the left-hand side
of these figures, solid lines represent symmetric modes and dashed lines represent
antisymmetric modes. The latter cannot be excited by the incident plane wave
because of the mismatching of their spatial symmetry, and so they do not contribute
to the light transmission. Opaque frequency regions are clearly observed in the
transmission spectrum where no symmetric mode exists. (After [49])
Band gap:
no propagation possible at that frequency
density of optical states (DOS) is 0

31. Dispersion relation for 2D photonic crystal vs transmission

32. Photonic crystals in Nature
Sea mouse
Opal
McPhedran et al.

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

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

35. Making Rods & Holes Simultaneously
Steven G. Johnson, MIT
side view s u b s t r a t e Si
top view
Steven G. Johnson, MIT

36. Making Rods & Holes Simultaneously
expose/etch
holes A A A A s u b s t r a t e A A A A A A A A A A A A A A A A A A A A A
Steven G. Johnson, MIT

37. Making Rods & Holes Simultaneously
backfill with
silica (SiO2)
& polish A A A A s u b s t r a t e A A A A A A A A A A A A A A A A A A A A A
Steven G. Johnson, MIT

38. Making Rods & Holes Simultaneously
deposit another
Si layer l a y e r 1 A A A A s u b s t r a t e A A A A A A A A A A A A A A A A A A A A A
Steven G. Johnson, MIT

39. Making Rods & Holes Simultaneously
dig more holes
offset
& overlapping l a y e r 1 B B B B A A A A s u b s t r a t e B B B B A A A A B B B A A A B B B B A A A A A B A B A B A B A B A B A B B B B A A A
Steven G. Johnson, MIT

40. Making Rods & Holes Simultaneously
backfill l a y e r 1 B B B B A A A A s u b s t r a t e B B B B A A A A B B B A A A B B B B A A A A A B A B A B A B A B A B A B B B B A A A
Steven G. Johnson, MIT

41. Making Rods & Holes Simultaneously
etcetera
(dissolve
silica
when
done) l a y e r 3 A A A A
one
period l a y e r 2 C C C C l a y e r 1 B B B B A A A A s u b s t r a t e C B C B C B C B A A A A C B C B C B C A A A C B C B C B C B A A A A C A B C A B C A B C C A B C A B C A B C A B C B C B C B C A A A
Steven G. Johnson, MIT

42. Making Rods & Holes Simultaneously
etcetera l a y e r 3 A A A A
one
period l a y e r 2 C C C C l a y e r 1 B B B B
hole layers A A A A s u b s t r a t e C B C B C B C B A A A A C B C B C B C A A A C B C B C B C B A A A A C A B C A B C A B C C A B C A B C A B C A B C B C B C B C A A A
Steven G. Johnson, MIT

43. Making Rods & Holes Simultaneously
etcetera l a y e r 3 A A A A
one
period l a y e r 2 C C C C l a y e r 1 B B B B
rod layers A A A A s u b s t r a t e C B C B C B C B A A A A C B C B C B C A A A C B C B C B C B A A A A C A B C A B C A B C C A B C A B C A B C A B C B C B C B C A A A
Steven G. Johnson, MIT

44. 7-layer E-Beam Fabrication
Point out 60nm veins
30nm inter-layer alignment
[ M. Qi, et al., Nature 429, 538 (2004) ]

45. Three-dimensional Si photonic crystal
Y. A. Vlasov et al., Nature 414, 289 (2001)
S.-Y. Lin et al., Nature 394, 251 (1998)

46. Two-Photon Lithography
2-photon probability ~ (light intensity)2
lens
some chemistry
(polymerization)
3d Lithography
…dissolve unchanged stuff
(or vice versa)

47. Lithography – the best friend of a man
λ = 780nm
resolution = 150nm
7µm
S. Kawata et al., Nature 412, 697 (2001).
780 nm with a 150-femtosecond pulse width was used as an exposure source. Bull sculpture produced by raster scanning; the process took 180 min.
To depict the tiny features of our microbulls, a fabrication accuracy of about 150 nm was needed, which we achieved by using nonlinear processes of the photochemicalreactions involved. Polymerization occurred only in the vicinity of the focal spot and the size of solidified voxels (three-dimensional volume elements) was reduced because of the quadratic dependence of TPA probability on the photon fluence density. Furthermore, photogenerated radicals in this volume were subject to scavenging by dissolved oxygen molecules, so polymerization reactions were not initiated and propagated if the exposure energy was less than a critical value. This property defined a TPA threshold and excluded the low-intensity lobe (the zero-order edge and the subsidiary maxima of the Airy pattern) from polymerization and thus further reduced the voxel size (Fig. 1g). The diffraction limit imposed by Rayleigh criteria was simply a measure of focal-spot size, and did not put any restraint on the actual size of solidified voxels.
(3 hours to make)
2µm
S. Kawata et al., Nature(2001)

48. Holographic Lithography
Four beams make 3d-periodic interference pattern
k-vector differences give reciprocal lattice vectors (i.e. periodicity)
absorbing material
Photonic crystals for the visible spectrum by holographic lithography
AUTHOR: Sharp,-D.-N.; Campbell,-M.; Dedman,-E.-R.; Harrison,-M.-T.; Denning,-R.-G.; Turberfield,-A.-J.
SOURCE: Optical-and-Quantum-Electronics. Jan.-March 2002; 34(1-3): 3-12
University of Oxford
[ D. N. Sharp et al., Opt. Quant. Elec. 34, 3 (2002) ]
(1.4µm)
beam polarizations + amplitudes (8 parameters) give unit cell
[ D. N. Sharp et al., Opt. Quant. Elec. 34, 3 (2002) ]

49. Holographic Lithography
[ D. N. Sharp et al., Opt. Quant. Elec. 34, 3 (2002) ]
Photonic crystals for the visible spectrum by holographic lithography
AUTHOR: Sharp,-D.-N.; Campbell,-M.; Dedman,-E.-R.; Harrison,-M.-T.; Denning,-R.-G.; Turberfield,-A.-J.
SOURCE: Optical-and-Quantum-Electronics. Jan.-March 2002; 34(1-3): 3-12
University of Oxford
[ D. N. Sharp et al., Opt. Quant. Elec. 34, 3 (2002) ]
In this article we describe a new holographic process (Campbell et al. 2000) that is intrinsically well adapted to the production of microstructured photonic materials. A 3D laser interference pattern exposes a photosensitive polymer precursor (photoresist) rendering exposed areas insoluble; unexposed areas are dissolved away leaving a 3D photonic crystal formed of cross-linked polymer with air-ҐҐled voids. This process allows ҐҐxible design of the structure of a unit cell and thus of the optical properties of the microstructured material.
We create an interference pattern at the intersection of four non-coplanarbeams from a frequency-tripled, injection-seeded Nd:YAG laser (wavelengthk 1/4 355 nm) (Campbell et al. 2000). Such a pattern has 3D translationalsymmetry; its primitive reciprocal lattice vectors are equal to the diҐҐrencesbetween the wavevectors of the beams. Patterns with any Bravais lattice maybe generated in this way, and interference patterns have been used to createperiodic optical traps for laser-cooled atoms (Grynberg et al. 1993) (we notethat Berger et al. (1997) have independently suggested lithography using 3Dinterference, and that Shoji and Kawata (2000) have used a combination oftwo-dimensional (2D) and one-dimensional patterns to create 3D microstructure).We require a pattern with high intensity contrast that can betranslated into high solubility contrast in the photoresist; we also require theinsoluble polymer to form a connected but open structure. For a particularset of wavevectors, eight parameters describing the intensities and polarizationvectors of the four beams are required to deҐҐe the intensity distributionwithin a unit cell (i.e. the basis of the interference pattern). The eight freebeam parameters give considerable freedom to engineer the content of theunit cell, which in turn determines the photonic band structure (Ho et al.1990; Yablonovitch et al. 1991b).
Fig. 3. (a) SEM image of a fcc polymeric photonic crystal generated by exposure of a 10 µm film ofphotoresist to the interference pattern shown in Fig. 2(a). The top surface is a (111) plane; the film hasbeen fractured along (111) cleavage planes of the photonic crystal structure. Scale bar, 10 µm. (b) Close-upof (111) surface and {111} cleavage planes. Scale bar, 5 µm. (c) Simulation of (b) formed by cutting aconstant-intensity surface along narrow bonds. (d) Close-up of a (111) surface. Scale bar, 1 µm. (e) Titaniaphotonic crystal with higher refractive index contrast produced by using the polymeric structure as atemplate. The surface is slightly tilted from the (111) plane. Scale bar, 1 µm.
10µm
huge volumes, long-range periodic, fcc lattice…backfill for high contrast

50. Colloidal photonic crystals
Colvin, MRS Bulletin 26, (2001)

51. 3D Bandgap Structure frequency (c/a) for gap at λ = 1.55µm,
K. M. Ho, C. T. Chan, and C. M. Soukoulis, Phys. Rev. Lett. 65, 3152 (1990).
frequency (c/a)
11% gap
MPB tutorial,
K. M. Ho, C. T. Chan, and C. M. Soukoulis, "Existence of a photonic gap in periodic dielectric structures," Phys. Rev. Lett. 65, 3152 (1990).
for gap at λ = 1.55µm,
sphere diameter ~ 330nm
overlapping Si spheres
MPB tutorial,

52. Experimental Air-guiding Photonic Crystal Fiber
[ R. F. Cregan et al., Science (1999) ]
5µm

53. How to create a cavity within a photonic crystal?
Reminder: Bragg mirror cavity
Two equivalent views:
Cavity = mirror + resonator+ mirror
Cavity = mirror with a „good” defect

54. Single-Mode Cavity Bulk Crystal Band Diagram A point defect
frequency (c/a)
A point defect
can push up
a single mode
from the band edge G X M G Γ X M

55. …here, doubly-degenerate
“Single”-Mode Cavity
Bulk Crystal Band Diagram
frequency (c/a)
A point defect
can pull down
a “single” mode X
…here, doubly-degenerate G X M G M Γ X

56. Tunable Cavity Modes
frequency (c/a)
Ez:
monopole
dipole

57. “1d” Waveguides + Cavities = Devices
resonant filters
waveguide splitters
high-transmission
sharp bends
channel-drop filters

58. Outlook: Optical Microcavities
Vahala, Nature 424, 839 (2003)
Microcavity characteristics: Quality factor Q, mode volume V