§9.6 High-Frequency Modulation Considerations

Lecture 16

In practice, the modulation signal is often at very high frequencies and may occupy a large bandwidth, such that the wide frequency spectrum of lasers can be efficiently used.

In this section, we consider some basic factors limiting the highest usable modulation frequencies in some basic experimental situations.

(A) Maximum Modulation Bandwidth

sR

LRLCV

sR: Internal resistance of source

C: Capacitance of EO crystal

tieV 0~ Assume:

ticc eVtV 00)(

Then voltage drop:

EO crystal:

Internal Resistance:

§9.6 High-Frequency Modulation Considerations

)()(

)()( 0 tVCiRdt

tdVCRRtItV cs

csss

10 )( CRs when

We have )()( tVtV cs Most of voltage is waste !

Capacitor impedance:

This can also be obtained from the impedance

Resistance impedance:sR

C0

1

Voltage drop is proportional to its impedance.

§9.6 High-Frequency Modulation Considerations

The solution:

Parallel connect a inductance (L) and a resistance (RL), and

make the circuits resonant to obtain the maximum impedance

120 )( LCi.e., 1

00 )/(1 CL or

Then the RLC circuits impedance: LRZ

As long as: sL RR Most of voltage will drip across crystal

However, above results is subject to the bandwidth limitation.

CRL

2

1

2

The maximum modulation bandwidth

§9.6 High-Frequency Modulation Considerations

CCRL 2

11Electric Power:

Consume Power & Peak Retardation

L

m

R

VP

2

2

c

Vrn mo 633

EO retardation:

Capacitance:l

AC

c2

lrn

AP m

263

60

22

4

§9.6 High-Frequency Modulation Considerations

(B) Transit-Time Limitation

Transit-Time: the time that light pass through the crystalc

nld

What happen if Electric field change is comparable with transit-time?

aEl

t

t

l

d

dtten

cadzzeat

')'()()(

0

')'( tim

meEte ti

dm

im

dm

ei

et

1)( 0

mmd alEEnca )/(0Peak retardationat

2/

)2/sin(1||

dm

dm

dm

i

i

er

dm

1dm

Reduction Factor

§9.6 High-Frequency Modulation Considerations

nl

cdmm 42

2/

2

9.0|| r2/ dm

For example: KDP crystal

5.1n

cm1lGHz5m

Modulation frequency

§9.6 High-Frequency Modulation Considerations

(C) Traveling-Wave Modulators

There is a way to eliminate the transit-time limitation, which makes the optical and modulation filed have a same phase velocity such that a portion of an optical wavefront will exercise the same instantaneous E field through the crystal.

§9.6 High-Frequency Modulation Considerations

)/1(

1)/1(

mdm

ncci

ncci

er

mdm

Reduction Factor

The same as previous one, exceptd

)'()'( ttn

ctz Optical wavefront position at time t’

dt

tdttzte

n

act

')]'(,'[)(Retardation:

)]')(/('[)'(),'( ttncktim

zktim

mmmm eEeEzte Modulation field:

ti

mdm

nccim

mdm

encci

et

)/1(

1)(

)/1(

0

mmm ck /

)/1( md ncc

mcPhase velocity of modulation field

§9.6 High-Frequency Modulation Considerations

mcnc / 1r Transit-time limitation eliminate

)/1(4 mm nccnl

c

Modulation frequency

Lecture 17

Chapter X Interaction of Light and Sound

Highlights

1. Scattering of Light by Sound

3. Bragg Diffraction of Light by Acoustic Waves - Analysis

2. Raman-Nath and Bragg Diffraction

Controlling the frequency, intensity and direction of an optical beam

Propagation of laser beams in crystals with acoustic waves

4. Deflection of Light by Sound

Partially Reflecting Mirror Model

Particle Picture

§10.1 Scattering of Light by Sound

A sound wave consists of sinusoidal perturbation of the density of the material, or strain, that travels at the sound velocity

Index of refraction

s

sv AverageIndex

Distance

0

z

sv

)sin(),( zktntzn ss

sss vk /

Diffraction of light by sound waves was predicted by Brillouin in 1922 and demonstrated experimentally some ten years later.

§10.1 Scattering of Light by Sound

I. Partially Reflecting Mirrors Model

s

sv

x

Incident

optical beam

i

A

r i r

Diffra

cte

d beam

B CD

0x

n

mx ri

)cos(cos

,...2,1,0 m

(A) All the points on a given mirror contribute in phase to the diffraction direction…

Optical path difference: AC-BD

ri 0m

For example:

Interfere constructively condition -----

§10.1 Scattering of Light by Sound

(B) Diffraction from any two acoustic phase fronts add up in phase in the reflected direction…

s

svIncident

optical beam

A

Diffra

cted

beam s

B

0

Moving sound wavefunction s

For example:

Optical path difference: AO+OB

ns /sin2

Bragg diffraction

example mn 5.0/ MHz500s m/s300sv

cm106/ 4 sss v 5.3rad104 2

§10.1 Scattering of Light by Sound

II. Particle Picture of Bragg Diffraction

Dual particle-wave nature of light

incidentk

diffractedk

sk

Conservation of momentum

sid kkk

i

s

dConservation of energy

sid

i

s

d

sid

ns /sin2

§10.1 Scattering of Light by Sound

III. Doppler Derivation of the Frequency Shift

nc

v

/2

nc

vs/

sin2

ns /sin2 s

s

sv 2

/2 c

sid

The Doppler shift changes sign when the sound wave direction is reversed, so

sid

v

nc

vi /

nc

vd /

§10.2 Raman-Nath and Bragg Diffraction

I. Raman-Nath Diffraction

Low sound wave frequency, short interaction length, and si kk

Phase grating

II. Bragg Diffraction

Higher sound wave frequency, longer interaction length, and

ns /sin2

Raman-Nath

+1

+2

III. criterion

22

s

LQ

1Q

1Q

Raman-Nath Diffraction

Bragg Diffraction

§10.3 Bragg Diffraction of Light by Acoustic Waves

I. Coupled Wave Function

)sin(),( rkr sstntn Index of refraction modulation

Additional electric polarization ),(),(2),( 0 ttnt rerrp

Wave equation ),(),( ,2

2

2,

2

,2 t

ttt di

didi rp

ere

Total field

..)(2

1),( )( ccerEte iiti

iii rkr

..)(2

1),( )( ccerEte dd ti

ddd rkr

Slow amplitude variation assumptioniiii drdEkE /2

§10.3 Bragg Diffraction of Light by Acoustic Waves

di

i Eidr

dE At bragg condition

id

d Eidr

dE c

ndidi 2

,

Coordinate transform

cosd

i Eid

dE

cosid Ei

d

dE

Initial conditions0)0( dE

222 |)0(||)(||)(| iiddii ErrErE

§10.3 Bragg Diffraction of Light by Acoustic Waves

II. Diffraction Efficiency

nc

l

E

E

I

I

i

diffracted

incident

diffracted

2sin

)0(2

2

2

spn

n2

3

3

2

s

acoustic

v

Is

acousticacoustic

sincident

diffracted MIl

Iv

pnl

I

I

2sin

2sin 2

3

262

3

26

sv

pnM

Diffraction figure of merit

§10.4 Deflection of Light by Sound

Incident light

Diffracted beam at vs

Diffracted beam at vs+vs

incidentk

diffractedk

sk

Initially Then sss

k

k

s

ss vk

2

Deflection of optical beam can be achieved by changing the sound frequency near the Bragg-diffraction condition

A

B

O

§10.4 Deflection of Light by Sound

sss vk /)(2 s

s

s

nvk

k

Number of resolvable spots

/s

sdiffracted s s

DN

v D v

example

120MHzMHz80 s

cm/s101.3 5svN

MHz40 s

cm1D

Calculate: