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Lecture 7: Photonic Crystal Waveguide s And Photonic Circuits 5 nm

Photonic Crystal Waveguides And Photonic Circuits

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Page 1: Photonic Crystal Waveguides And Photonic Circuits

8/8/2019 Photonic Crystal Waveguides And Photonic Circuits

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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