Kerr Effect
Dn = lKEa
Kerr coefficient
2
Kerr effect
term
Applied field
An applied electric field, via the Kerr
effect, induces birefringences in an
otherwise optically isotropic material
Kerr Effect
(a) An applied electric field, via the Kerr effect, induces birefringences in an
otherwise optically isotropic material. (b) A Kerr cell phase modulator.
Electro-Optic Properties
Pockels (r) and Kerr (K) coefficients in a few selected materials Values in parentheses
for r values are at very high frequencies
Material
LiNbO3
Crystal
Uniaxial
Indices
Pockels Coefficients
K
m/V2
no = 2.286
´ 10-12 m/V
r13 = 9.6 (8.6); r33 = 309
(30.8)
ne = 2.200
Comment
l  633 nm
Uniaxial
no = 1.512
r22 = 6.8 (3.4); r51 =32.6
(28)
r41 = 8.8; r63 = 10.3
(KH2PO4)
KD*P
(KD2PO4)
Uniaxial
ne = 1.470
no = 1.508
r41 = 8.8; r63 = 26.8
l  546 nm
GaAs
Isotropic
ne = 1.468
no = 3.6
r41 = 1.43
l  1.15 mm
Glass
Nitrobenzene
Isotropic
Isotropic
no  1.5
no  1.5
0
0
KDP
3×10-15
3×10-12
l  546 nm
Kerr Effect Example
Example: Kerr Effect Modulator
Suppose that we have a glass rectangular block of thickness (d) 100 mm and length (L) 20
mm and we wish to use the Kerr effect to implement a phase modulator in a fashion
depicted in Figure 6.26. The input light has been polarized parallel to the applied field Ea
direction, along the z-axis. What is the applied voltage that induces a phase change of p
(half-wavelength)?
Solution
The phase change Df for the optical field Ez is
Df 
2pDn
l
L
2p (lKEa2 )
l
2pLKV 2
L
d2
For Df = p, V = Vl/2,
Vl / 2
d
(100  106 )


 9.1 kV
3
15
2 LK
2(20  10 )( 3  10 )
Although the Kerr effect is fast, it comes at a costly price. Notice that K depends on the
wavelength and so does V1/2.
Integrated Optical Modulators
The electro-optic effect takes place over the spatial overlap region between the
applied field and the optical fields.
The spatial overlap efficiency is represented by a coefficient G
The phase shift is Df and depends on the voltage V through the Pockels effect
2p
L
Df  G
nr
V
l
d
3
o 22
Integrated Optical Modulators
Induced phase change
Length of electrodes
2p
L
Df  G
nr
V
l
d
Spatial overlap efficiency = 0.5 – 0.7
3
o 22
Applied voltage
Electrode separation
Pockels coefficient
Different for different
crystal orientations
Integrated Optical Modulators: An Example
2p
L
Df  G
nr
V
l
d
3
o 22
Df depends on the product V×L
When Df = p, then V×L = Vl/2L
Consider an x-cut LiNbO3 modulator with d  10 mm,
operating at l = 1.5 mm
This will have Vl/2L  35 Vcm
A modulator with L = 2 cm has Vl/2 = 17.5 V
By comparison, for a z-cut LiNbO3 plate, that is for light propagation along the y-direction
and Ea along z, the relevant Pockels coefficients (r13 and r33) are much greater than r22 so
that
V l/2L  5 Vcm
A LiNbO3 Phase Modulator
A LiNbO3 based phase modulator for use from the visible spectrum to
telecom wavelngths, with modulation speeds up to 5 GHz. This particular
model has Vl/2 = 10 V at 1550 nm. (© JENOPTIK Optical System GmbH.)
A LiNbO3 Mach-Zehnder
Modulator
A LiNbO3 based Mach-Zehnder amplitude modulator for use from the
visible spectrum to telecom wavelengths, with modulation frequencies up
to 5 GHz. This particular model has Vl/2 = 5 V at 1550 nm. (© JENOPTIK
Optical System GmbH.)
Integrated Mach-Zehnder Modulators
An integrated Mach-Zehnder optical intensity modulator. The input light is split
into two coherent waves A and B, which are phase shifted by the applied voltage,
and then the two are combined again at the output.
Integrated Mach-Zehnder Modulators
Approximate analysis
Input C breaks into A and B
A and B experience opposite phase
changes arising from the Pockels
effect
A and B interfere at D. Assume they have the same amplitude A
But, they have opposite phases
Eout  Acos(wt + f) + Acos(wt  f) = 2Acosf cos(wt)
Output power Pout  Eout2
Pout (f )
 cos 2 f
Pout (0)
Amplitude
Mach-Zehnder Modulator
Courtesy of Thorlabs
Coupled Waveguide Modulators
(a) Cross section of two closely spaced waveguides A and B (separated by d) embedded in a substrate.
The evanescent field from A extends into B and vice versa. Note: nA and nB > ns (= substrate index).
(b) Top view of the two guides A and B that are coupled along the z-direction. Light is fed into A at z =
0, and it is gradually transferred to B along z. At z = Lo, all the light has been transferred to B. Beyond
this point, light begins to be transferred back to A in the same way.
Coupled Waveguide Modulators
Lo = Transfer distance
If A and B are identical,
full transfer of power from
A to B occur over a
coupling distance Lo,
called the transfer
distance
Coupling Efficiency
D b = bA  bB
= Mismatch between propagation constants
bA
When the mismatch
Db = p3/Lo
then, power transfer is prevented
We can induce this mismatch by
applying a voltage (Pockels effect)
bB
Coupled Waveguide Modulator
3ld
Vo 
3
2Gn rLo
Applied voltage
 2p
Db  Dn AB 
 l
Voltage induced
mismatch

 1 3 V  2p 
  2 n rG  
d  l 

2
Pockels
coefficient
Coupled Waveguide Modulator
Voltage needed to switch
the light off in B
3ld
Vo 
2Gn 3rLo
Modulated Directional Coupler
An integrated directional coupler. The applied field Ea alters the
refractive indices of the two guides (A and B) and therefore changes
the strength of coupling.
Modulated Directional Coupler: Example
Example: Modulated Directional Coupler
Suppose that two optical guides embedded in a substrate such as LiNbO3 are coupled as
in Figure 6.31 to form a directional coupler, and the transmission length Lo = 10 mm. The
coupling separation d is ~10 mm, G  0.7, the operating wavelength is 1.3 mm where
Pockels coefficient r  10´1012 m/V and n  2.20. What is the switching voltage for this
directional coupler?
Solution
3ld
3 (1.3  106 )(10  106 )
Vo 

 15.1 V
3
3
12
3
2Gn rLo 2(0.7)( 2.2) (10  10 )(10  10 )
Acousto-Optic Modulator
Fiber-coupled acousto-optic modulator
(Courtesy of Gooch & Housego)