Electro-Optic Effects: definition

• Electro-optic effects refer to changes in the refractive index of a material induced by the application of an external electric field– Which therefore modulates the optical properties– The applied field is not the electric field of any light wave but a

separate external field• We can apply an external field by placing electrodes on

opposite faces of a crystal and connecting these electrodes to a battery– The presence of the field distorts the electron motions in the

atoms/molecules of the substance– Distorts the crystal structure resulting in changes in optical

properties

Electro-optic effect

• An applied external field can cause an optically isotropic crystal such as GaAs to become birefringent– The field induces principal axes and an optic axis– Typically changes in the refractive index are small– The frequency of the applied field has to be such that the

field appears static over the time scale it takes for the medium to change its properties, as well as for any light to cross the substance

• The electro-optic effect are classified according to first or second order effects

Field induced refractive index

• Take the refractive index n to be a function of the applied E-field, that is n=n(E), we can expand this as a Taylor series in E. The new refractive index n’ would be:n’= n + a1E + a2E 2+.…– where the coefficients are called the linear electro-optic effect and second

order electro-optic effect coefficients.• The change in n due to the first E term is called the Pockels effect

n = a1E • The change in n due to the second E 2 term is called the Kerr

effect and a2=K where K is called the Kerr coefficient

n = a2 E 2 = (K) E 2

Pockels Effect

• Suppose x, y and z are principal axes of a crystal with refractive indices n1, n2 and n3 along these directions– For an optically isotropic crystal, these would be the same– For a uniaxial crystal n1= n2 n3

• Apply a voltage across a crystal and thereby apply an external dc field Ea along z-axis– In Pockels effect, the field will modify the optical indicatrix.– The exact effect depends on the crystal structure– GaAs (isotropic) with a spherical indicatrix becomes birefringent– KDP (potassium dihydrogen phosphate) that is uniaxial becomes

biaxial

Pockels Effect: KDP (KH2PO4)

• The field Ea along z rotates the principal axes by 45 about z

• Changes the principal indices as shown in Fig.10(b) – The new principal indices are now n1’ & n2’, which means the

cross section is now an ellipse– Propagation along the z-axis under an applied field now

occurs with different refractive indices n1’ & n2’– The applied field induces new principal axes x’ & y’ for this

crystal

xz Ea

n1 = no

y

(a)

xn2 = no n1

n2

z

(b)

x

45

(c)

xz

KDP, LiNbO 3 KDP LiNbO 3

n1

n2

y Eay

(a) Cross section of the optical indicatrix with no applied field, n1 = n2 = no (b) Theapplied external field modifies the optical indicatrix. In a KDP crystal, it rotates theprincipal axes by 45 to x and y and n1 and n2 change to n1 and n2 . (c) Appliedfield along y in LiNbO2 modifies the indicatrix and changes n1 and n2 change to n1and n2 .

© 1999 S.O. Kasap, Optoelectronics (Prentice Hall)

Fig 10: Pockels Effect

Pockels Effect: LiNbO3

• In the case lithium niobate (uniaxial crystal), a field Ea is applied along the y-direction– It does not significantly rotate the principal axes – changes the principal refractive indices n1 & n2 (both equal to

no) to n1’ & n2’ as shown in Fig 10(c) • Consider a wave propagating along the z-direction

(optic axis) in the crystal– Before a field Ea is applied, this wave experience n1=n2=no

whatever in the polarization as Fig 10(a)– In the presence of an applied field Ea, the light propagates as

two orthogonally polarized waves (parallel to x and y) experiencing different refractive indices n1’ & n2’

Pockels Effect: LiNbO3, cont

• The applied field thus induces a birefringence for light traveling along the z-axis.

• The field induced rotation of principal axes is neglected.

• The Pockels effect gives the new refractive indices n1’ & n2’ in the presence of Ea as

n1 ’ n1 + ½ n13

r22 Ea & n2 ’ n2 – ½ n23

r22 Ea

where r22 is a constant, called a Pockels coeffient that depends on the crystal structure and the material.

Phase modulator

• It is clear that the control of the refractive index by an external applied field is a distinct advantage that enables the phase change through a Pockels crystal to be controlled or modulated– Such a phase modulator is called a Pockels cell

• In the longitudinal Pockels cell phase modulator, the applied field is in the direction of light propagation -> Fig 10(b)

• In the transverse phase modulator, the applied field is transverse to the direction of light propagation -> Fig 10(c)

Transverse Phase Modulator

• The applied field Ea = V/d is applied parallel to the y-direction (normal to the direction of light propagation along z)

• The incident beam is linearly polarized at 45 to the y-axis.– It is represented in terms of polarizations (Ex & Ey components) along the x and y

axes– Ex & Ey experience refractive indices nx & ny.

• When Ex traverses the length distance L, its phase changes by 1.

• When Ey traverses the distance L, its phase changes by 2, given by a similar expression except that r22 changes sign. Thus, the phase changes between two components is

d

Vrnn

LL

noo 223

211

1

22

Vd

Lrno 22

321

2

Polarization modulator

• The applied voltage thus inserts an adjustable phase different Df between the two field components– The polarization state of the output wave can be controlled

by the applied voltage and the Pockels cell is a polarization modulator.

• The medium can be changed from a quarter-wave to a half-wave plate by simply adjusting V.– The voltage V = Vl/2 , the half-wave voltage and generate a

half-wave plate (D f =p)

Outputlight

z

x

Ex

d

EyV

z

Ex

Eyy

Inputlight Ea

Tranverse Pockels cell phase modulator. A linearly polarized input lightinto an electro-optic crystal emerges as a circularly polarized light.

© 1999 S.O. Kasap, Optoelectronics (Prentice Hall)

Fig 11: Transverse Phase modulator

Transverse Intensity Modulator

• From the polarization modulator in Fig.11, an intensity modulator can be built as shown in Fig.12 – by inserting a polarizer P and an analyzer A before and after the phase

modulator– P and A have their transmission axes at 90 to each other

• The transmission axis of P is at 45 to the y-axis – The light entering the crystal has equal Ex and Ey components

• In the absence of applied voltage, two components travel with the same refractive index and polarization output is the same as its input– There is no light detected at the detector as A and P are at the right angle

Fig 12: Transverse Intensity Modulator

Left: A tranverse Pockels cell intensity modulator. The polarizer P and analyzer A havetheir transmission axis at right angles and P polarizes at an angle 45 °to y-axis. Right:Transmission intensity vs. applied voltage characteristics. If a quarter-wave plate

is inserted after P , the characteristic is shifted to the dashed curve.

© 1999 S.O. Kasap, Optoelectronics (Prentice Hall)

Transmission intensity

V

I o

Q

0 V l /2

V

°

Inputlight

P ADetector

Crystal

zx

y

QWP

Transverse Intensity Modulator, 2

• An applied voltage inserts a phase difference Df between the two E-field components– The light leaving the crystal now has an elliptical polarization and hence

a field components along the transmission axis of A– A portion of this light will therefore pass through A to the detector– The transmitted intensity now depends on the applied voltage V.

• Suppose that Eo is the amplitude of the wave incident on the crystal face– The amplitude along x- and y-axis will be Eo/2 each.– Ex is along the –x direction

• The total field E at the analyzer is

tE

ytE

x oo cos2

ˆcos2

ˆE

Transverse Intensity Modulator, 3

• A factor cos(45) of each component passes through A.– We can resolve Ex and Ey along A’s transmission axis– Then add these components and use trigonometric identity to obtain

the field emerging from A– The final result is

E = Eo sin( ½Df ) sin(wt + ½Df )• The intensity I of the detected beam is

I = Io sin2( ½Df ) or

I = Io sin2(p/2 V/V /2l )where Io is the light intensity under full transmission

and V /2l is an applied voltage needed to allow full transmission

Example: Pockel Cell Modulator

• What should be the aspect ratio d/L for the transverse LiNiO3 phase modulator in Fig.11 that will operate at a free-space wavelength of 1.3mm and will provide a phase shift Df of p (half wavelength) between the two field components propagating through the crystal for an applied voltage of 24V? ( no = 2.2, r22 = 3.410–12 m/V)

Solution

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103.124104.32.2103.1

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Letting

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2-3-

2-

3123

62/223

2/223

2/

d/L

d

d/L

d/L

VrnL

d

Vd

Lrn

VV

EE

o

o

yx

Integrated optical modulators

Phase and polarization modulation

• Integrated optics refers to the integration of various optical devices and components on a single substrate such as lithium niobate.– In integrated electronics, all necessary devices are integrated

in the same semiconductor crystal substrate• There is a distinct advantage to implementing various

optical communicated devices on the same substrate– E.g. laser diodes, waveguides, splitters, modulators,

photodetectors etc in a miniature device

Polarization modulator

• Polarization modulator is shown Fig.13– An embedded waveguide has been fabricated by implanting a

LiNbO3 substrate with Ti atoms which increase the refractive index

– Two coplanar strip electrodes run along the waveguide and enable the application of a transverse field Ea to light propagation direction z

• The external modulating voltage V(t) is applied between the coplanar to drive electrodes– By virtue of the Pockels effect, induces a change Dn in the

refractive index and hence a voltage dependent phase shift through the device

V(t)

Ea

Cross-section

LiNbO3

d

Thin buffer layerCoplanar strip electrodes

EO Substratez

y

x

Polarizedinputlight

WaveguideLiNbO 3

L

Integrated tranverse Pockels cell phase modulator in which a waveguide is diffusedinto an electro-optic (EO) substrate. Coplanar strip electrodes apply a transversefield Ea through the waveguide. The substrate is an x-cut LiNbO3 and typically thereis a thin dielectric buffer layer (e.g. ~200 nm thick SiO2) between the surfaceelectrodes and the substrate to separate the electrodes away from the waveguide.

© 1999 S.O. Kasap, Optoelectronics (Prentice Hall)

Fig 13: Polarization modulator

Polarization modulator, 2

• The light propagation along the guide can be represented in terms of two orthogonal modes, Ex along x and Ey along y

– These two modes experience symmetrically opposite phase changes– The phase shift Df between the Ex and Ey polarized waves would normally

be given by Pockels effect• In the case of the applied field is not uniform between the

electrodes and further not all applied field lines lie inside the waveguide– The electro-optic effect takes place over the spatial overlap region

between the applied field & the optical fields– This spatial overlap efficiency is lumped into a coefficient G and the

phase shift Df is written as = (2 / ) Df G p l (no

3r22)(L/d )Vwhere G 0.5-0.7 for various integrated polarization modulator of this type

Optical Switching:Mach-Zehnder Modulator

• One potential application of self-phase modulation is in optical switching to switch the output from low to high intensity in fs time scale.

• In optical switching, induced phase shift by applied voltage can be converted to an amplitude variation by using an interferometer– Interferometer is a device that interferes two waves of the

same frequency but different phase

V(t)

LiNbO3 EO Substrate

A

BIn

OutC

DA

B

Waveguide

Electrode

An integrated Mach-Zender optical intensity modulator. The input light issplit into two coherent waves A and B, which are phase shifted by theapplied voltage, and then the two are combined again at the output.

© 1999 S.O. Kasap, Optoelectronics (Prentice Hall)

Fig 14: Optical Switching:Mach-Zehnder Modulator

Mach-Zehnder Modulator, 2

• Consider the structure shown in Fig.14, which has implanted single mode waveguide in a LiNbO3 substrate in the geometry.– The waveguide at the input braches out at C to two arms A and B – These arms are later combined at D to constitute the output– The splitting at C and combining at D involve a simple Y-junction

waveguides• In the ideal case, the power is equally split at C so that the field

is scaled by a factor 2 going into each arm– The structure acts as an interferometer because the two waves traveling

through the arm A and B interfere at the output port D– The output amplitude depends on the phase difference (optical path

difference) between A and B branches

Mach-Zehnder Modulator, 3

• Two back-to-back identical phase modulators enable the phase changes in A and B to be modulated.– The applied field in branch A is in opposite direction to that

in branch B– The refractive index changes are opposite and phase

changes in arm A and B are also opposite• If applied voltage induces a phase change of p/2 in

arm A, this will be –p/2 in arm B so that A & B would be out of phase by p.– These two waves will interfere destructively and cancel each

other at D.– The output intensity would be zero

Mach-Zehnder Modulator, 4

• Since the applied voltage controls the phase difference between the two interfering waves A and B at the output– This voltage also control the output light intensity (the

relationship is not linear)• The relative phase difference between the two waves A

and B is doubled with respect to a phase change f in a single arm– The switching intensity can be predicted by adding waves A

and B at D with A as amplitude of wave A & B:

Eoutput A cos(wt+f) + A cos(wt–f) = 2A cosf coswt

Mach-Zehnder Modulator, 5

• The output power is proportional to E2output which is maximum

when f = 0. Thus,

• The derivation represents approximately the right relationship between the power transfer and the induced phase change per modulating arm.

• The power transfer is zero when = f p/2. • In practice, the Y-junction losses and uneven splitting results in

less than ideal performance– A and B do not totally cancel out when = f p/2

2cos0out

out

P

P