§9.6 high-frequency modulation considerations lecture 16 in practice, the modulation signal is...
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§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: