Si Photonics (2018/2) W.-Y. Choi

Lect. 10: Diffraction Gratings

Si Photonics (2018/2) W.-Y. Choi

Lect. 10: Diffraction Gratings

sin2

2

y

y

ak

ak

slit shape (y) far-field (ky)

Fourier Transform

Si Photonics (2018/2) W.-Y. Choi

Lect. 10: Diffraction Gratings

a

d

● ● ● ● ● ● *

×

FourierTransform

● ● ●● ● ●

2d

yk

a

yd

● ● ●

ya

d

Diffracted light from periodic slits

(Assume infinitely many for simplicity)

==> Far-field only for discrete ky's

yk yk

(Diffraction Grating)

2sinyk k md

Si Photonics (2018/2) W.-Y. Choi

Lect. 10: Diffraction Gratings

a

d

2yk m

d

Grating equation

2 2sin md

sind m

yk

Bragg Condition

William Lawrence Bragg(1890-1971)

William Henry Bragg(1862-1942)

Nobel Prize in Physics (1915)

W. L. Bragg is the youngest Nobel Physics winner

The only father-son joint Nobel winner

Si Photonics (2018/2) W.-Y. Choi

Lect. 10: Diffraction Gratings

sind m

dz

y

Incidentlight wave

Diffraction grating

One possiblediffracted beam

a

Intensity

y

m = 0m = 1

m = -1

m = 2

m = -2

Zero-orderFirst-order

First-order

Second-order

Second-order

Single slitdiffractionenvelope

dsin

Width for each diffracted beam?

Si Photonics (2018/2) W.-Y. Choi

Lect. 10: Diffraction Gratings

i

Input with tilted angle

(sin sin )ijkytotal

AE e dyR

(2)(1)y

R

siny

sin iy (sin sin )ijkyA e

R

sinjkyA eR

sinjkytotal

AE e dyR

Si Photonics (2018/2) W.-Y. Choi

Lect. 10: Diffraction Gratings

Tilted incidence on grating

msinsind i

sind m

(a) Transmission grating

Incidentlight wave

Zero-orderFirst-order

First-order

(sin sin )ijkytotal

AE e dyR

Si Photonics (2018/2) W.-Y. Choi

Lect. 10: Diffraction Gratings

Reflection-type grating

R=1

R=0

Grating is often realized with periodic shaping of reflection surface

msinsind i

Same diffraction equation applies

Incidentlight wave

m = 0

m = -1

m = 1Zero-order

First-orde

First-order

(b) Reflection grating

i

Si Photonics (2018/2) W.-Y. Choi

Lect. 10: Diffraction Gratings

sin sinout ind m

2 2sin sinout in md

,x outk

Grating shifts kx by the integer multiples of 2/d

2 2 2sin sinout in md

K-vector perspective?

,x ink 2 md

Si Photonics (2018/2) W.-Y. Choi

Lect. 10: Diffraction Gratings

Grating imposes BC on the incident wave

( , 0)x ( )inE x y f x( , 0)rE x y

Far-Field Diffraction:

( )f x 2 m

md

F.T. of ( , 0)x ( )inE x y f x

( )x outE k ( ) x inE k

,x outk ,x ink2 md

Spatial modulation Sidebands formation

● ● ●

kx2d

● ● ●

*

Si Photonics (2018/2) W.-Y. Choi

Lect. 10: Diffraction Gratings

2-D Diffraction Grating

Take x and y separately

Si Photonics (2018/2) W.-Y. Choi

Lect. 10: Diffraction Gratings

How do we get light in and out of PIC?

Good coupling possible with spot-size conversion

Requires - Precise alignment- Polishing fiber facet- AR coating - Dicing of a chip

No dicing required

Wafer-level optical testing possible

Chap. 5 in C&H

Si Photonic Integrated Circuit

Light source (laser) cannot be integrated

Externally supplied

Si Photonics (2018/2) W.-Y. Choi

Lect. 10: Diffraction Gratings

Condition for vertical coupling out?

i 90 0

d With m=-1

In

Reflection type grating with period d

Out

m=-2, 90 msinsind i

But with d=

Also satisfied Not desired

Si Photonics (2018/2) W.-Y. Choi

Lect. 10: Diffraction Gratings

i 90

In

Reflection type grating with period d

Out

msinsind i

sin 1d m

( 1)1 sin

d m

No other diffracted output

If

Si Photonics (2018/2) W.-Y. Choi

Lect. 10: Diffraction Gratings

sineff c c

dn n

, ,2

x out x ink k md

2 2 2sinc c effn n md

For m=-1

sineff airn

Snell’s Law

Diffraction at 180°- c is also allowed

Si Photonics (2018/2) W.-Y. Choi

Lect. 10: Diffraction Gratings

Design Exercise: Grating coupler