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Lecture 7: Photonic Crystal Waveguides
And Photonic Circuits
5 nm
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Super-lens and Super-prism effects
H. Kosaka et al, Phys. Rev. B. 58, 10096, 1998
H. Kosaka et al, Appl. Phys. Lett. 74, 1370, 1999
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Snell’s law in terms of a constant frequency circle
Example: using constant frequency diagram to derive Snell’s law and
the condition for total internal reflection.
Snell’s law
n1 = 1
n2 = 1.5
θ 1
θ 2
n1sin"
1= n
2sin"
2
x
yn2 "
2
c2= k x
2+ k y
2
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Constant frequency contour in a 2D crystal
Luo et al, Phys. Rev. B 65, 201104, 2002; M. Notomi, Phys. Rev. B 62, 10692, 2000
• At low frequencies, the constant frequency diagram approaches a circle, the photonic crystal behaves as a uniform dielectric as far as diffraction is concerned
• With increasing frequencies, the constant frequency contour becomes more complicated,
leading to effects including superprism, superlens, negative refraction, and self-collimation.
Constant frequency diagram for the first band
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Super-lens and constant frequency
ω=0.165 2πc/a
Air
P. C.
Γ X
M
Vg = ∂ k ω(k) group velocity
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Vg = ∂k ω(k) group velocity
Photonic Band Engineering
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All-angle negative refraction
Luo et al, Phys. Rev. B 65, 201104, 2002
Source Image
Photonic crystal
• An all-convex constant frequency contour.
• All incoming wave are included within the
constant frequency contour of the photonic
crystal.
• The frequency is below 0.5 2 π c/a.
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3D Photonic crystals and PC slabs
• 3D photonic crystals.
• Photonic crystal slab structures.
• Omnidirectional reflectors.
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3D photonic crystal with complete band gap
Ho, Chan, Soukoulis, PRL, 65, 3152 (1990)
Complete band gap observed in both air spheres and dielectric spheres
Refractive index ratio needs to exceed 2 in order for band gap to open
Optimal structure consists of connected dielectric and air networks.
34 % when spheres are touching each other
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Variants of diamond structure, practical 3d structures
Chan et al, Solid State Communication, 89, 413-6 (1994)
S. Lin et al, Nature, vol. 394, p. 251-3, (1998)
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Self-assembled 3D photonic crystal structures
Y. Vaslov et al, Nature, vol. 414, p. 289, (2001)
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Photonic crystal slab structures
Low-index materials
High-index materials
In plane 2D photonic band gap provides complete in plane confinement.Out of plane confinement provided by high index guiding
Ease of fabrication
In complete confinement in the third dimension
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Photonic band diagram for photonic crystal slabs
Radiation modes above the light line.Losslessly guided modes below the light line.
Incomplete band gap in the guided mode spectrum
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Light line and radiation modes in a uniform slab
Light line separates the radiation modes from the guided modes.
air
dielectric
air
ω
kk
Light-lineω=ck/n
Below the light line, i.e. k > ω/c, electromagnetic field decays
exponentially away from the dielectric slab.
Radiation
modes
Guided
modes
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Uniform slab, repeated zone scheme with period a
a
ω
k-2 π /a 0 2 π /a-π /a π /a
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Effect of in-plane index contrast
a
ω
k
Photonic band gap
Guided resonance
π /a-π /a
Upper frequency
cutoff
ω = 0.5*2πc/a
!
c
"#$
%&'
2
=
d r ( 1
) r( )* + H(r)
2
d r ( H(r)2
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Photonic crystal LED
Uniform region
Photonic Crystal
e n h a n c e m e n t
wavelength (nm)700 800 900 1000
0
10
20
0.0
1.0
Pump wavelength
r e
f l e c t i v i t y
0.5
S. Fan et al, Physical Review Letters, 78, 3294 (1997);
A. A. Erchak, D. J. Ripin, S. Fan et al, Applied Physics Letters, 78, 563 (2001)
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Strong reflection from a single dielectric film
r=0.4a
Frequency (c/a)
O. Kinic, W. Suh, S. Kim, A. Sudbo, M. F. Yanik, S. Fan and O. Solgaard, Optics Letters, (in press)
W. Suh, M. F. Yanik, O. Solgaard, S. Fan, Applied Physics Letters, 82, 1999 (2003).
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Photonic crystal LED
Uniform region
Photonic Crystal
e n h a n c e m e n t
wavelength (nm)700 800 900 1000
0
10
20
0.0
1.0
Pump wavelength
r e
f l e c t i v i t y
0.5
S. Fan et al, Physical Review Letters, 78, 3294 (1997);
A. A. Erchak, D. J. Ripin, S. Fan et al, Applied Physics Letters, 78, 563 (2001)
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Wavevector (2π/a)0.30 0.40 0.50
0.15
0.20
0.25
0.30
0.35
0.40
F r e q
u e n c y ( c / a )
Slab modes
Radiation modes
gap
Oxide
Si
-1.0 1.00.0
Magnetic field
Waveguides in dielectric slabs
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Omni-directional reflection from multi-layer film
TE modes (s)
E
H
TM modes (p)
H
E
parallel wavevector
Fink, Winn, Fan, et al, Science. 282, 1679, 1998.
n1 = 4.6, n2 = 1.6