K8 Optical Modulators - ETH Zpeople.ee.ethz.ch/~fyuriy/oe/oe_optcom_chapters/Optoelectronics_2… · 8-9 K8 In chap.4 codirectional couplers realized by adjacent optical wave guides

K8 8-1

8 Optical Modulators (korr. 24.05/2008) 40 GHz Lithium-Niobat Mach-Zehnder-Modulator (package)

1 2L

constantlight input

controlvoltage Vc

opticaldata pulses

ΔΦ=π phase shift

electro-opticwaveguide

transmission

extinction

VC

Optical Input

Power Splitter

Elektode 2

Elektode 1

Waveguide 2

Waveguide 1

Monolithic integrated GaAs-Mach-Zehnder-Modulator (cut-out)

Δn(VC) --> ΔΦ=2πΔnL/λ

____________________________________________________________________________________________________________________________________________________________________________j Electronics Laboratory: Optoelectronics and Optical Communication 24.05.2008

K8 8-2

Goals of the chapter:

• Optical carrier waves can be modulated in Amplitude, Phase and Frequency in order to carry information

• Modulators modulated the carrier wave by changing the material properties of attenuation α and refractive index n

• in suitable materials α and n can be modulated at high frequencies by time dependent electrical fields (EO), magnetic fields (MO) or acoustic fields (AO)

• The external modulators produce much less chirp (dispersion) than eg. current-modulated SC-diode lasers

Methods for the Solution:

• similar to electronic modulators a nonlinearity of the WG material is required. For EO-modulators the nonlinearity of the polarization in strong EM-fields is used

• Analysis of the Electro-Optic effect in anisotropic crystals answers the questions: How do optical waves propagate in anisotropic crystals? How do strong EM-fields E change the refractive index n

• Analysis of basic modulator configurations for optical AM, FM and PM at high frequencies


K8 8-3

8 Optical Modulators Introduction:

Laser diodes are broadband (-40 GHz) current-modulated light oscillators with the following drawbacks for long distance communication:

• To operate diode lasers with a single optical, resp. wavelength under dynamic current modulation (large signal operation) complicated DFB- or DBR-diode structures with internal gratings are required

• Even under single-frequency operation there is a frequency chirping caused by the modulation current leading to Chirping and Pulse Dispersion in fibers

• Diode lasers have today a limited small signal modulation bandwidth of about 20-40 GHz

In order to avoid chirping diode lasers for long distance communication are used a as single frequency DC-sources for the optical carrier and their output wave is modulated by an electrically driven external modulator. External optical modulator have small λ-chirp, but are relative complex and expensive devices, with substantial insertion losses and limited potential for monolithic integration. Goals of the chapter:

• Qualitative concept for optical modulators • Analysis of wave propagation in materials with anisotropic index of refraction → Method of Fresnel-Equation or Method of the Index-Ellipsoid • Description of the Electro-Optic Effect for the modification of the refractive index by strong electrical RF-

field • Implementation of the EO-Effect in practical realizations of optical AM-, FM- and PM-modulators • Outlook on alternative SC-absorption-modulators


K8 8-4

Coherent optical waves can be considered as quasi-sinusoidal carrier waves, which are characterized by their - Amplitude E - Polarization direction - Frequency ω and - Phase φ In complete analogy to RF-waves information can be imprinted on optical waves by four modulation techniques: - Optical Amplitude-Modulation AM (and Intensity-Modulation) - Optical Frequency-Modulation FM - Optical Phase-Modulation PM - Optical Polarization-Modulation

Bild

Electrical information signal

modulates the Optical Amplitude: Amplitude Shift Keying Optical Phase: Phase Shift Keying

Optical Frequency: Frequency Shift Keying


K8 8-5

All following modulator concepts make use in one or other form of the electrical modulation of α und n. Advantages of external modulators are:

- Small optical frequency-chirp - Very high modulation bandwidth ( → 100 GHz) Drawbacks of external modulators are:

- relative high signal voltages (~2-7V) - relative high insertion losses by fiber coupling - relative large size (mainly for EO-modulators) - necessity of a separate, narrowband DC-light source - relative high component cost (modulator and electrical broadband driver

8.1 Concepts of Optical Modulation

8.1.1 Absorptions (α)- / Gain (g)-Modulation 1) Gain- / Attenuation-Modulation in forward polarized pn-Diodes

Operation principle:

Considering the spectral gain g(I,ω) of an SOA at a specific signal wavelength λs, resp. signal frequency ωopt and pump current I, we observe either

1) Transparency for ωopt gE<2) Gain 3) Absorption

λs


K8 8-6

As the optical gain g(ω,n) is a function of carrier density n resp. pump-current I of the SOA the gain g(I,ω) can be modulated, resp. the optical intensity of the input signal Pin can be switches between attenuation and gain:

( ) ( )( )Dout in

Lg , I tP t P e

ω= Pin = constant

This kind of modulation is limited in the achievable bandwidth B, due to the relative long spontaneous lifetime τspont (ns), to about B≈1/τspont and is not suitable for high speed modulation. However the SOA can be used as quasi-static optical switch.

2) Attenuation α Modulation in reverse polarized pn-Dioden (qualitative)

In reverse biased pn-diodes a strong external field E=VD/w can change the effective bandgap Eg(VD(t)) by the Frank-Keldysh-Effect (w= depletion layer width). Due to the strong band bending carriers can tunnel into the band gap leading to a reduction of the effective bandgap for band-to-band transitions. This results in a shift of the absorption spectrum α(ωopt,VD) and an increased in absorption with increasing VD, which can be used to modulate the DC-input-power Pin.

Due to the small depletion layer capacitance Cj and the absence of carrier injection this absorption modulator can operate up to high frequencies (→ 40-60 GHz).

Franz-Keldysh- and Quantum-Well Stark-Effect-Modulator and their modulation characteristic α(VD):

• The strong band bending allows the penetration (tunnelling) of the wavefunctions of electrons and holes into the gap.

• Transitions between these “gap-level” see a smaller

energy difference than EG, leading to absorption ar wavelengths λ below the gap energy.


Eg

E = 0 E<0 (reverse bias, band bending)

K8 8-7


Modulation of the effective bandgap Eg,eff(VD), shift of the absorption edge towards lower wavelengths. Attenuation spectra and modulation characteristics α(VD): Absorption spectra α(ω, VD) at different reverse bias voltages VD.

Width increasing revers bias VD the absorption α increases by about 3 orders of magnitude.

λS

~30 dB


K8 8-8

8.1.2 Modulation of Refractive Index (n)

Principle of Operation: Electro-Optic Effect

So far we have assumed that the polarization P(E) is a linear function of the electric field E, according to the relation EP χ= . In this case the relative dielectric constant r 1ε χ= + and refractive index n 1 χ= + do not depend on the magnitude of E (linear medium).

However in the presence of a strong electric field ERF (here a low frequency RF-field) the polarization becomes nonlinear such that the ε(ERF), resp. n(ERF) at optical frequencies depends on ERF and can be modulated by ERF. The modulation of ε(ERF) and n(ERF) by an electric field ERF is called the Electro-Optic Effect.

This field induced change in n can be used in EO-modulators to modify the dielectric properties of the medium. The strong, modulating RF-field can be applied to the dielectric medium by suitable metal electrode structure. 1) Directional coupler as a modulator

In this EO-modulator the RF-field is used to change the coupling constant κ(ERF) between two parallel waveguides 1 and 2, by small changes Δn in the refractive index nS(ERF) of the medium between the waveguides. Schematic Structure of an EO-β-Coupler (codirectional coupler): (see chap.4)

Structure: Coupling Characteristic (IKIz)

Electrode

ns

ns

ns

ng

ERF

Pin I1(L, V)V

Δns(V

L

ng

Iκ(V)I z

I2(L, V)

Pout2 V=0

V≠0

Pout1


K8 8-9

In chap.4 codirectional couplers realized by adjacent optical wave guides (refractive index ng) in a medium (nS) have been considered. The coupling constant κ is a function of the waveguide geometry, the coupling length L and the refractive indices ng und nS(V):

( ) ( ) ( )2

20t m

t

kE f n E f function E2

κ δβ

= =

The modulating field ERF, resp. voltage V is applied by the two electrodes and changes eg. nS(ERF), resp κ(ERF):

Δn(ERF) ~ r ERF

Depending on the voltage V the optical input power Pin in WG1 stays in WG1 (Pout1) or is transferred to WG2 (Pout2), thus the light Pout at a particular WG is modulated by V. 2) Mach-Zehnder (MZ)-Interferometer-Modulator (Modulation of the phase difference between arm 1 and arm 2)

Waveguide cross section:

n1

Wellenleiter: n2

V

E

Elektroden

An interferometer where the refractive index in the 2 arms (1,2) is modulated by V, Δn1(V), Δn2(V),and where the 2 phase-shifted waves from arm 1 and arm 2 interfere at the output Y-coupler, can be used as a EO-modulator.

Bild waves from arm 1 and 2 total wave at output

Relative phase difference ( )12 1 2Vφ φ φΔ = Δ − Δ Δφ1 ~ LΔn1(V) EO-phase change Arm 1 WG1

WG2 Δφ2 ~ LΔn2(-V) EO-phase change Arm 2

L

V

-V


K8 8-10


The MZ-modulator has two arms 1 and 2 of length L. The E-field applied by the 2 electrode-pairs changes the refractive index n1(E) and (n2(-E)) in the arms by the electro-optic effect. Because n2(-E) ≠ n1(E) the two waves have a relative optical phase difference Δφ12(V)= Δφ1(V)- Δφ2(V) at the exit port, which modifies the interference of the two waves.

Depending on the phase difference Δφ12(V) the two waves interfere constructively (transmission) or distracively (attenuation), resulting in an amplitude modulation of Pout(V) (see chap.8.4.3).

8.2 Propagation of optical wave in anisotropic media Many materials, which show an EO-effect are highly anisotropic crystals, where the refractive index n and the relative dielectric constant ε depend on the direction of the E-field vector E relative to the crystal axis. Many modulators are based on the following effects:

a) Anisotropy of the refractive index (Doppelbrechung)

b) Electro-Optic Effect (refractive index n(ωopt, E) is a function of E)

Question ?:

How does an optical field propagate in an isotropic medium, where n depends 1) on the propagation direction S and 2) on the polarization of the wave ? What are the solutions for the propagating modes ?

• Maxwells-Equations are valid also in an isotropic medium

• In a isotropic medium the relative dielectric constant εr and the susceptibility χ are scalars, meaning that the E- and D-vectors have the same direction ( ) rD E Eε= .resp. ( )P E Eχ= .

• In an anisotropic medium the E- and D-vectors do not have the same direction. The function relation between the D and E is described by the symmetric (3x3) tensor of the dielectric constant , resp. jkε ( )D E E= jkε .


K8 8-11

As a consequence the refractive index n experienced by a plane wave with the wavevector β depends on the propagation direction and the polarization of the wave in a anisotropic medium. S

Plane waves in an anisotropic medium (schematic)

sDa

Db

Eb

Ea

na , λa

nb , λb

anisotropes Material

1b

6b

6a

1a

Β=Δφ The analysis would show (without rigorous proof, because of very lengthy mathematics), that in an anisotropic medium for an excited plane wave in propagation direction ( )X Y ZS S ,S ,S ; S 1= = and with a frequency ω:

a) there are 2 polarized plane waves-solutions (modes) (a: Da, Ea, na, vph,a , b: Db, Eb, nb, vph,b) for a propagation direction s

b) both modes a and b propagate with different velocities according to 2 different refractive indices na and nb

c) the D-vectors baD ; D are linear polarized and orthogonal to each other and also orthogonal to the propagation direction s

d) die E-Vectors baE ; E

D a E E E= + = +

of the two modes are linear polarized but not necessarily orthogonal to each other

e) because the two modes propagate at different velocities there is a variable phase difference Δφ between the modes leading to a changing polarization of the resulting wave vectors bD D nda b a

≡E

H S


K8 8-12

8.2.1 Dielectric properties of anisotropic media

8.2.1.1 The Dielectric Tensor εij

Up to know we assumed for an isotropic media that ε is a scalar and linear:

1) the polarization ( )EP and ( )D E are parallel to the electrical field E and

2) the polarization ( )EP is linear Because many crystalline EO-materials are strongly anisotropic the directions of D and E are not the same. Therefore the dielectric constant becomes a tensor, that is represented by a 3x3 matrix.

The material equation of the anisotropic medium relating the D and E -Vector is represented by a linear dielectric tensor εij relation.

Bild Tensor

x xx xy xz x

y yx yy yz y ij

z zx zy zz z

D ED E D ED E

ε ε εε ε εε ε ε

⎛ ⎞⎛ ⎞ ⎛ ⎞⎜ ⎟⎜ ⎟ ⎜ ⎟= =⎜ ⎟⎜ ⎟ ⎜ ⎟

⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠

ε

( )ijε = Dielectric Tensor (9 elements εij) Width the formal inverse relation (definition):

ijE Dη= ; 1ij ijη ε −=

( )ijη = Impermeability-Tensor


K8 8-13

The permeability tensor( )ijε has the following properties:

1) Tensor ijε is symmetric jk kjε ε= (without proof)

2) Any symmetric tensor in a particular coordinate system x,y,z can be transformed by the choice of a special coordinate system x‘,y‘,z‘ into a purely diagonal tensor ijε , diagonalization-transformation (without proof).

x‘,y‘,z‘ is called principle axis system (Hauptachsensystem).

EDEEE

000000

DDD

ij

'z

'y

'x

'z'z

'y'y

'x'x

'z

'y

'x

ε=⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

εε

ε principle axis representation

It follows for the inverse relation:

x' x' x' x'

y' y' y' y' ij

z' z' z' z'

E 1 / 0 0 DE 0 1/ 0 D E DE 0 0 1/ D

εε

ε

⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟ ⎜ ⎟= =⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠

η principle axis representation

2x' x'x' x'

1 2y' 0 y' y' y'

2z' z'z' z'

1 / n 0 0E DE 0 1/ n 0 DE D0 0 1/ n

ε −

⎛ ⎞⎛ ⎞ ⎛ ⎞⎜ ⎟⎜ ⎟ ⎜ ⎟= ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠

The refractive indices nij of an anisotropic medium are defined in analogy the the isotropic case n2= εr:

2jj jj 0n /ε ε= ε0 is the permeability constant of vacuum

Definition: njj is the refractive index for the principle axis j relating the component of the j-axis components of Ej and Dj. Remark: In the following we will assume that we are always operating in the principle axis system x‘,y‘,z‘ , that is we assume that the x,y,z is the principle axis system omitting the “ ‘ “.


K8 8-14

Remark:

In general the directions of the principal axis (x’, y’, z’) has a specific relation to the crystal structure of the medium, resp. to the lattice vectors (Kristallgittervektoren) ( )1 2 3a a , a , a= (for details see Lit. [6, 3]).

This relation has to be considered when designing anisotropic devices in a particular material system.

Example: lattice vectors, crystal planes and the principle axis of GaAs, InP

Principal Axis:

ˆx' aˆy' bˆz' c

a1, a2, a3 = lattice vectors

x’, y’, z’ = principle axis Characterization of the refractive index properties in the principle axis system: Isotropic medium: xx yy zzn n n index sphere= = → Uniaxial medium: xx yy zzn n n n n rotational-symmetric Index-Ellipsoid= = ≠ → Biaxial medium: xx yy zzn n n general Index-Ellipsoid≠ ≠ →


K8 8-15

8.2.1.2 Fresnel-Equation for plane waves in anisotropic media

Question: Under the assumption that a plane wave, propagating in an anisotropic medium in direction s , can be decomposed into 2 linear polarized modes a and b, - what are the 2 associated refractive indices na, nb and the corresponding field vectors

ba a bandD , E D , E of the resulting plane wave propagating in the direction ( )x x zs s ,s ,s , s 1= = :

( ) 2 nE r,t E exp j t s r E exp j t s rc

n index of refraction of the waver space positionvector

π ωω ωλ

⎛ ⎞ ⎛ ⎞⎡ ⎤ ⎡ ⎤= − = −⎜ ⎟ ⎜ ⎟⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦⎝ ⎠ ⎝ ⎠==

Planwave

a b

a b

D D D

E E E

= +

= +

( )snn = ?

Possible solutions for wave with a propagation direction s have to fulfill:

1) Maxwells equation and

2) the relationship of the dielectric tensor as a material equation

3) we assume that the material are not magnetic μr=1

To solve this basic problem two methods have been developed: 1) Fresnel-Equation and 2) the Index-Ellipsoid

1) Method of Fresnel-Equation:

Assumptions: xyz is the principle axis system of the ijε -tensor and the ijε -tensor is known. ____________________________________________________________________________________________________________________________________________________________________________j Electronics Laboratory: Optoelectronics and Optical Communication 24.05.2008

K8 8-16

Procedure:

1) Basic relations between s , E, D, H and the Pointing vector S in anisotropic media

From Maxwell-equations for plane waves is can be derived by use of several vector identies and the basic properties of the rotation-operation rot rot x x= ∇ × :

nj r sc

Maxwell : H D and E H with , ,t t x y z

applying the rotation operation on e and j we get :t

n n nj s H j D D s H and H s Hc c cD s ; D H ; H E ; H s

ω

μ

ω

ω ωμ

−

⎛ ⎞∂ ∂ ∂ ∂ ∂∇ × = ∇ × = − ∇ = ⎜ ⎟∂ ∂ ∂ ∂ ∂⎝ ⎠

∂− ∇ × =

∂

− × = → = − × = − ×

⇒ ⊥ ⊥ ⊥ ⊥

Observe: the flow of energy is not identical with the propagation direction S s . 2) Calculation of the refractive index of a plane wave in a anisotropic medium (Lit.[6, 2])

We assume a direction of propagation s and the principle axis indices of refraction n11=nx, n22=ny, n33=nz

From Maxwells equation above we get:

( ) ( )2 2

2 2vectoridentity

1 1 1 0 1 1 2 2 2 0 2 2 3 3 3 0 3 3

n nD s H and H s Ec c

n n nD s H s s E E s s E *c c c

in the principle axis system we can write:D = E ' E ; D = E ' E ; D = E ' E

μ

μ μ

ε ε ε ε ε ε ε ε ε

= − × = − × ⇒

⎡ ⎤= − × = − × × = −⎣ ⎦

= = =

n= assumed index of refraction n for a planar wave


K8 8-17

If we insert the last equation by components in equation (*):

( )

( ) ( )

( ) ( ) ( ) ( ) ( )

22

k 0 k k 0 k k k k2

2k

k 2 2k

k

2 2 2 2 2 21 2 3

1 1 2 2 3 3 2 2 2 2 2 21 2 3

nD E n E E s s E ; k 1,2,3c

n sE s En n

multiplying the last equation with s and summing up for k 1,2,3

n s n s n ss E s E s E s E s En n n n n n

ε ε εμ

⎡ ⎤= = = − = →⎣ ⎦

=−

=

⎧ ⎫⎪ ⎪+ + = = + +⎨ ⎬− − −⎪ ⎪⎩ ⎭

The unknown refractive index n is a solution of the quadratic Fresnel-equation which has two solutions na and nb for a given direction of propagation ( )x x zs s ,s ,s , s 1= = :

22 2yx z

2 2 2 2 2 2 2x y z

n , n for the 2 mode of propagationa bss s 1 s permits the calculation of

n n n n n n n→+ + =

− − −

Fresnel-equation

As a consequence for each propagation direction there exist two wave solutions a and b with the two index of refraction na and nb. The Frensnel-equation with known nx, ny, nz gives na and nb, but we still have to determine the direction and magnitudes of the Ea,b and Da,b-fields.


K8 8-18

Maxwells-equations gave the following properties of the field vectors:

1) Orthogonality a bD D s⊥ ⊥ 2) aE and bE are not orthogonal

3) Vector direction for a bE and E

It can be shown that the direction of the E-field baE and E (polarization) of the two solutions is given by:

x2 2

x

y2 2

y

z2 2

z

sn n

sE E e

n n

sn n

⎡ ⎤⎢ ⎥−⎢ ⎥⎢ ⎥

= ≈ ⎢ ⎥ with n = n−⎢ ⎥

⎢ ⎥⎢ ⎥

−⎢ ⎥⎣ ⎦

e and ea or nb we get the two possible polarization direction ba of the E-fields

Proof:

In the derivation of section 2) we got the relation

( ) ( ) ( )2 2

k kk k2 2 2 2

k k

n s n sE s E k 1,2,3 E qed.n n n n

= = →− −

∼

From that we obtain the directions of aE and bE as well as the two propagation vectors aa n2sλπβ = and bb n2s

λπβ = .


K8 8-19

4) The boundary conditions of the exciting field determines the values of aE and bE .

5) Finally using the inverse dielectric tensor 1ij ijη− =ε we determine the dielectric flux densities a ij a b ij bD E and D Eη η= =

Observe: if the propagation direction s is chosen in such a way that na or nb is identical to one of the index of refraction of the principle axis n1, n2, n3, then the corresponding component of the polarization direction vector e diverges, meaning that the E-field is polarized in the direction of the corresponding principal axis. In this case also the D-field has the same direction.

Summary:

In an anisotropic medium the plane wave solution can be represented as the sum of two plane waves Da, Db, having 1) the same propagation direction s, 2) orthogonal polarizations but different refractive indices na, nb (different phase velocities) → giving rise to birefringence (Doppelbrechung) refraction

( )x y z

a b a b a b a bij

DielektischerFresnel Rand TensorGleichung bedingung

Given : s s ,s ,s

s n , n e ,e E , E D , Dε− −

=

⎯⎯⎯⎯⎯→ ⎯⎯→ ⎯⎯⎯⎯⎯→ ⎯⎯⎯⎯⎯⎯→

2) Concept of the method of the refractive Index-Ellipsoid

To simplify the calculation of the Fresnel-equation, using the method of the refractive index ellipsoid provides a graphical method for the determination of na, nb and the directions of bD ,a D for a given propagation direction s of a plan-wave.

D- and E-field are related finally by the tensor equation EDij,ij

⎯⎯⎯ →← ηε .


K8 8-20

Starting with the D-vector of the desired solution for propagation in the direction s we write for D :

( ) ( ) ( )x y z x y z x y zD D , D , D D d mit d D , D , D / D d , d , d

d unity polarization direction vector of D

= = = =

=

Calculating the energy density we of the EM-field in the principal axis system (x‘,y‘,z‘) using the η-tensor:

2 2 22 2 2 2y' y'T T T x' z' x' z'

e ij 2 2 2 200 xx yy zz xx yy zzusing the using the

definition of the definition of Drefractive index n

D D D1 1 1 D D 1 D Dw E D D D2 2 2 22 n n n n

ηεε ε ε ε

⎡ ⎤ ⎡ ⎤= = = = + + = + + →⎢ ⎥ ⎢ ⎥

⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

22 2y'x' z'

2 2 2 2xx yy zz

dd1 dn n n n

= + + n is the solution for a given direction d of the D-vector

Ellipsoid-equation for na, nb for a direction d of the dielectric flux D

From Maxwells-equation we know that d and s are orthogonal, meaning can be replaced by d s . Determination of na, nb and b from the refractive index ellipsoid for a given propagation direction ad , d s :

A more common situation is that the propagation direction s is given and we are searching for bad , d and na , nb.


K8 8-21

Definition of Refractive index ellipsoid:

Considering the index ellipsoid ( )1 2 3n ,n ,n

22 2y'x' z'

2 2 2 2xx yy zz

22 2y'x' z'

2 2 2xx yy zz

d1 d d definingn n n n

dd d1 index ellipsoidn n n

= + + →

= + +

with the principle axis length of this ellipsoid nxx, nyy, nzz.

and using the shorter notation: nxx = n1 , nyy = n2 , nzz = n3. The following geometric procedure (without proof) makes use of the index-ellipsoid in order to determine for a given propagation direction s 1) the polarization directions of the two Dd a and Db-fields and 2) The two associated indices of refraction na and nb

1) because D and s are orthogonal, s determines a normal plane through the center of the ellipsoid. This normal plane creates an intersection with the ellipsoid, which is an ellipse on the surface of the ellipsoid.

2) the principal axis of the intersection-ellipse determine the two possible polarization direction ba ndd a d of the two solutions of aD and bD

3) the two principal axis of the intersection ellipse are the values of the refractive indices na and nb.

ad bd

Normal plane of s

intersection-ellipse

The detailed, lengthy proof can be found in (Lit.3,2). The proof calculates the intersection ellipse and shows directly that the values of the principle principal axis of the ellipse are the two solutions of the Fresnel-equation.


K8 8-22

The E-fields baE and E ndare related to baD a D by the inverse principal-axis dielectric tensor ηij :

( )( )

( )

20 x' x'

x' x'2 1

y' 0 y' y' y' ij ij

2z' z'0 z' z'

1 / n 0 0E DE 0 1/ n 0 D E D DE D0 0 1/ n

ε

ε η

ε

−

⎛ ⎞⎛ ⎞ ⎛ ⎞⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟= = =⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠⎜ ⎟

⎝ ⎠

ε

The absolute values of and ba E,E ba D,D are determined by the boundary conditions (excitation of the field).

( )x y z

a b a b a b a b1 2 3 ij

inverse dielectricrefractive index ellipsoid boundary tensorn , n , n conditions

given : s s ,s ,s

s n ,n ,d ,d D , D E , Eη−

=

⎯⎯⎯⎯⎯⎯⎯⎯⎯→ ⎯⎯⎯⎯⎯→ ⎯⎯⎯⎯⎯⎯⎯→

Observe: wave excitation boundary condition:

Depending on excitation of the wave both or only one of the polarization a and b can excist.

If the exciting field has only a D-vector in the direction of a principle axis, then only this polarization (eg. a) propagates and the corresponding E-vector (eg. Ea) has the same direction as D in the principal axis direction (diagonal η-tensor).

In the special case that the propagation vector s has the direction of

s

Da

DbEb

Ea

anisotropes Material

sD E

Externe Anregungin Hauptachsen-richtung

z'

x'y'

a principal axis, the two orthogonal polarized waves Da and Db can be excited, with the polarization of (Ea, Da) and (Eb, Db) of these two waves in the direction of the other two principal axis.

These two waves propagate in general with different phase velocities

gr ,a gr ,a

gr ,b gr ,b

v c / n

v c / n

=

=.


K8 8-23

Example: Index-Ellipsoid for uniaxial crystals

Because of the practical relevance for modulator applications we consider the uniaxial index ellipsoid in the principle axis system using the definitions:

( )( )

xx yy 0

zz e

n n n ordinary ordentlichen index of refraction,

n n extraordinary ausserordentlichen index of refraction

= =

=

The uniaxial index ellipsoid has a rotational symmetry around the z-principal axis. Representation of an uniaxial Index-Ellipsoid: s is the unity-vector of the wave propagation direction in polar coordinates ( )φθ ,,s . Due to the rotational symmetry φ is not relevant.

Bil

( )( ) ( )

( )

( )

( ) ( ) ( )

ordinary 0

extraordinary e

2 2

2 2 2e 0 e

2 2

a e b 02 20 e

n n f ,

n n f

with s 0, sin , cos

1 cos sinn n n

cos sinn n 1 / , n nn n

θ φ

θ φ

φ θ θ

θ θθ

θ θθ θ φ

= ≠

= ≠

= =

= + →

= = + =

For a propagation direction s with the angle θ relative to the z-axis the index-ellipsoid gives us the 2 refractive indices na and nb for the two wave solutions:

( ) ( )( )( ) ( )

0

e

s 0,sin ,cos in x,y,z -coordinates

a : 1,0,0 n

b : 0,cos ,sin n ?

= Θ Θ

→

Θ Θ → Θ =


K8 8-24

8.2.1.3 Polarisations-Propagation and Birefringence:

Because the two waves Ea(x,t) and Eb(x,t) propagate with different phase velocities vph,a=c/na and vph,b=c/nb, a phase difference Δφ(x) between these 2 wave develops with increasing propagation distance, resulting in a modification of the polarization of the resulting field (eg. polarization rotation). This properties will be used in EO-modulators.

( ) ( ) ( ) xc

nnxcn

cnx22xx

0ba

0

b

0

a

baba

ωωωλπ

λπββφ −=⎟⎟

⎠

⎞⎜⎜⎝

⎛−=⎟⎟

⎠

⎞⎜⎜⎝

⎛−=−=Δ Polarization: z=const t=const for vpha=vphb but Δφ=const

Graphical representation of the polarization modification: linear → elliptic → circular → orthogonal linear

βa

βb

Ea

Wave a

linear elliptic circular elliptic linear polarization (Δφ=π , x=πc/ω/(na-nb)

Wave b local polarization

Eb

x

x z’

x

y’

z z’ y’

y

Δφ=0 π/2 π


K8 8-25

We consider the situation of two orthogonal (unity vectors bas , s ) polarized waves a and b with identical amplitudes Eo propagating in a anisotropic material in the direction s ( ba s,s and s are mutually orthogonal). sa and sb are the in the direction of the principal axis z’ and y’:

Bild

We calculate the polarization at position x of the resulting field ( )t,xE in a coordinate system (x,y,z) , which is rotated by 450:

z’

z

y y’

( )( ) ( ) ( )

( ) ( ){ } ( ) ( ){ }( ) ( ) ( )

( ){ }

( ) ( ){ }

a b b a b

b

b

a b

a 0 a b 0 b

j t x j t x j t x j x0 a b 0 a b

b a b a

j t x j0 a b

j t xj0 a b

s 1,0,0 , E E

E t,x s E cos t x s E cos t x

E Re s e s e E Re e s e s

mit x x x n n xc

E Re e s e s

E t,x Re E s e s e

ω β ω β ω β β β

ω β φ

ω βφ

ω β ω β

ωφ β β β

− − − − +

− Δ

−Δ

= =

= − + −

⎡ ⎤= + = + =⎣ ⎦

Δ = − = Δ = − →

⎡ ⎤= +⎣ ⎦→

⎡ ⎤= +⎣ ⎦ ( ) ( ){ }( )

bj t xt

jt a b

Re E x e

with E x s e s

ω β

φ

−

Δ

=

= +


K8 8-26

Planar vector wave with position dependen polarization → POLARIZATION-ROTATION z

Bild Ming p694

y’ z’

x y

Transforming the field components in to the rotated orthogonal coordinate system z , y,x :

( ) ( )

( ) ( )

a b a b

a b

1 1z s s , y s s2 21 1s z y , s z y2 2

= + = − +

= − = +

z and y are orthogonal

gives the vector wave in the (x, y, z)-coordinate system:

( ) ( )( ) ( )( ) ( )( ) ( )( )0 b 0 bE t ,x 2E cos x / 2 cos t x x / 2 x 2E sin x / 2 sin t x x / 2 yφ ω β φ φ ω β φ= Δ − + Δ + Δ − + Δ

At a fixed position x, the E-field vector rotates on a ellipse as a function of time

At a fixed time t the E-field vector propagates on a spiral in x direction at Δφ(x)=nπ n=0, 1, 2 one of the two components of the polarization vanishes → locations of linear polarization (z- or y-direction).

Conclusions:

the birefringence Δφ in anisotropic crystals determines the spatial evolution of the polarization from the linearer, elliptical and circular polarization state.

if the birefringens Δφ(x) can be modulated, then the polarization can be modulated opening the possibility to convert the polarization modulation into an amplitude-modulation using a polarizing element.


K8 8-27

Concept of an EO-intensity modulator by polarization rotation: Basic idea and necessity of the anisotropy

Bild: Yariv p.338

Absorption

Transmission

Δφ(V) Transmission für Δφ(V)=π x,y,z = principal axis of the EO-crystal, modulating electrical field E=V/L in the x-direction, propagation in z-direction.

8.3 Electro-Optic (EO) Effect

The concept of the EO-modulator by controlling the birefringence of a medium of a fixed length L requires an effect to change the index-ellipsoid of a birefringent medium, by an external modulating electrical field E=V/L.

• Electro-Optic Effect: a strong external field Eex can modify the microscopic lattice potential (electronic polarization) of a EO-crystal → therefore the dielectric tensor ( )exEε becomes a function of the external field Eex.

• Because the dielectric tensor ijε as well as the interaction with the external field Eex depend strongly on the crystalline structure and on the orientation between light field, electric control field and crystal structure the functional relations can be complicated tensor relations.


K8 8-28

8.3.1 Linear and quadratic Electro-Optic Effect

Because electro-optic modulators are of considerable importance we will concentrate on the modification ot the index-ellipsoid by an external electrical field. Similar effects resulting from magnetic field are not considered.

Details related to the microscopic crystalline structure are phenomenologically simplified. In view of practical applications we consider only a few very common crystalline structures, eg. for III-V-compound semiconductors.

Modification of the index-ellipsoid by an external field E:

Concept:

The effect of a “slow” external electrical control field Eex results in a field dependent deformation of the index-ellipsoid, that can be considered as a generalized elongation and rotation of its principal axis (has to be proven).

Because the 3x3 dielectric tensor ( )exij Eε becomes a function of the E-field Eex, therefore the same is true for the inverse tensor ( )exij Eη :

( )

( )

ij ex

ij ex

D E E

with the inverse relationship:

E E D

=

=η

ε ( )exij Eη = Impermeability-tensor

( ) ( ) ( )

( )

ij ,krlineare d e of E

−Pockels Effekt ( )

ex

ij ex ij ex ij

ependenc quadratic dependence of von E

the change Δη of η caused by E be approximated by:

E E 0

with

E r E s E E

η η

η= = =

−

= Δ +

Δ = +∑ ∑ ∑

η

Kerr Effekt ij ,kls( )

3 3 3

ij ex ij ,k k ij ,kl k lk 1 k 1 l 1

Δηij = modification of the ij-element of the inverse dielectric tensor

Ek components of the external field vector (E1, E2, E3) in the principal axis system x, y, z.


K8 8-29

For the derivation of the index-ellipsoid we made use of the impermeability-tensor ηij , so it is obvious that the index-ellipsoid is going to be changed by the EO-effect. Repeating the derivation leads after a lengthy, but simple derivation to the modified index-ellipsoid (Lit. [2]):

2 2 2xx x' yy y' zz z'2 2 2 2

xx yy zz

yz y' z' xz x' z' xy x' y'

1 1 1 1d d dn n n n

2 d d 2 d d 2 d d

η η η

η η η

⎛ ⎞⎛ ⎞ ⎛ ⎞= + Δ + + Δ + + Δ +⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠ ⎝ ⎠

Δ + Δ + Δ

Modified Index-Ellipsoid To simplify this general expression a particular material system has to be considered, where due to the crystal structure and symmetry many elements rij,k and sij,kl of the EO-tensors become small or zero. Linear E/O-Effect (Pockel-Effect)

Adopting the international convention we rewrite the last equation with:

nxx = n1 , nyy = n2 , nzz = n3 , .... etc. (6 coefficients)

dx’ = x, dy’ = y, dz’ = z

2 2 21 2 32 2 2 2

1 2 3

23 13 12

1 1 1 1x y zn n n n

2 y z 2 x z 2 x y

η η η

η η η

⎛ ⎞⎛ ⎞ ⎛ ⎞= + Δ + + Δ + + Δ +⎜ ⎟⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠ ⎝ ⎠Δ + Δ + Δ


K8 8-30

For the linear Pockels-Effect the 6 coefficient Δη are linear functions of the field vector E:

1 11 12 1311

2 21 22221

3 33332

4 233

5 13

6 61 6312

r r rr r .

E. . r

E. . .

E. . .

r . r

η ηη ηη ηη ηη ηη η

Δ ΔΔ ΔΔ Δ

= =Δ ΔΔ ΔΔ Δ

Linear Pockel-Effect (Electro-Optical Tensor, 6x3 matrix with 18 elements)

8.3.2 Material Parameters

Electro-Optic Materials, III-V-Semiconductors, Niobates

There are a large number of EO-crystals, characterized by the principal axis refractive indices n1, n2, n3 and the matrix ijr - all these

parameters are basically a function of wave length λ. As a practical important EO-material we consider the III-V-compound SC GaAs, InP.

m34 at λ=0.9μm: (similar to InP) Example: GaAs, cubic crystal

Werte Yariv, Ming

• rij ~0 • rij >0 r41=r52=r63>0

GaAs is for E=0 isotropic (n1=n2=n3=n) and becomes biaxial in an external field E>0.


K8 8-31

Assuming an E-field in the principal axis direction z ( )zE,0,0E = , the by inserting the field in the equation for the Pockels-effect we get an index-ellipsoid:

( )2 2 2 2 2 2 2 2 212 63 z2 2 2 2 2 2 2 2

1 241 z

3

1 1 1 1 1 1 1 1x y z 2 x y x y z 2r E x y x y zn n n n n n

2r En

x yn

η⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞

= + + + Δ = + + + = + + +⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎝ ⎠

Transforming the principle axis system x,y,z (E =0) in the rotated (450 rotation around the z-axis) system → x‘,y‘,z‘ , new principal axis system for E ≠ 0):

( ) ( )

( ) ( )'x'y2

1y,'y'x2

1x

.respyx2

1'y,yx2

1'x,'zz

−=+=

+=−==

by insertion we obtain:

( ) ( ) ( )( )

( ) ( )

2 2 241 z2 2

22 2 2 2 2 2 2

41 4 21 z 41 2z2 2r E r Ey'n nn n

+ + =⎟ ⎜ ⎟⎠ ⎝ ⎠

− +z2

1 12 2n n1 2

1 1 1x' y' y' x' z' r E x' y' y' x'2 2n n

1 11 z 1x' y' z r E y' x' x'n

⎛ ⎞= + + − + + + − =⎜ ⎟ ⎜ ⎟⎝ ⎠⎝ ⎠

⎛ ⎞ ⎛ ⎞ ⎛ ⎞+ + + − =⎜ ⎟ ⎜

⎝ ⎠ ⎝

x

y

z, z’

y’ x’

EZ

1⎛ ⎞

( ) ( ) 21 41 z

22 41 z

1n n 1 r E n2

1n n 1 r E n2

⎛ ⎞≅ +⎜ ⎟⎝ ⎠

⎛ ⎞≅ −⎜ ⎟⎝ ⎠

fix 2 2 2 2 2

41 z 1 41 z 41 z2 21

41 z2 22

21 r E nz41

1 1 r E , n n / 1 r E n n 1 r E nn n

1 1 r En n

>>

= − = − ≅ + →

= + →


K8 8-32

In the new, field induced principal axis coordinate system x‘,y‘,z=z’ we obtain for the

biaxial index-Ellipsoid-Equation:

( ) ( ) 2

2

z22

2

z21

2

2 nz

En'y

En'x

n1

++=

For GaAs the index-sphere transforms (rotation, dilatation or compression) into a biaxial ellipsoid, where the principal axis in the field direction is not modified. Remark: the deformation of the index-ellipsoid in the principal axis is extremely small (Δn/n ~10-4 – 10-6 ), this fact has been used for the previous simplifications. The birefringence for propagation in the z-direction becomes

( ) ( ) ( )3z 1 2 41 z

2 2B E / x n n r n Eπ πφλ λ

= Δ Δ = − = −

B is the phase change per unit length. Because B is very small (~10-4 – 10-6) the propagation distance must be 104 – 106 λ long to get a phase delay of ~π! Summary:

• GaAs is without E-field isotrop, its refractive index is independent of the propagation direction (index sphere) • By the EO-effect GaAs becomes biaxial, the index-ellipsoid is “deformed” with increasing field and 2

orthogonally polarized wave can exist. Each wave “sees” its own refractive index according the index ellipsoid • The EO-Effect allows the propagation velocity and the polarization state to be modulated by an

external electrical RF-field All consideration would have been identical if we had hosen the x‘- oder y‘-direction for the E-field.


K8 8-33

Examples of EO-Data of different materials:

InP (λ=1.55μm) r41= -1.63 n=3.17


K8 8-34

Electro-Optic Tensors rij of different material classes:

° r = -r

• rij ~0

• rij >0


K8 8-35

8.4 Electro-Optic Modulators Electro-optic modulators serve as components for optical communication technology to modulate the phase, the frequency, the amplitude or the polarization of optical carrier waves.

The basic mechanism for all these functions is the modulation of the index-ellipsoid (n1(E), n2(E), n3(E)) by an external electrical control field applied to an EO-material.

Depending if one or two polarizations are excited the EO-modulator introduces a pure phase modulation Δφ(Ε) without polarization change or a modulation of the polarization direction. Basically one has to determine how to align 1) the principal axis (n1,n2, n3) of the index ellipsoid des Index-Ellipsoids,

( )z,y,xs = and 2) the optical propagation direction ( ) ( )321 E,E,EtE = relative to each other. 3) the direction of the modulating external E-field

There are to basic configuration for the orientation of the E-field and the propagation direction s:

a) Longitudinal modulators:

The direction of the E-field and the propagation s of the optical field are equal, making transparent electrodes almost an necessity.

Qualitatively the relation between birefringence Δφ and applied voltage V becomes:

( )exL E LV / L V ` f LφΔ ≈ = ≈ ≠ L = length of the E0-Mediums

In longitudinal modulators the phase change is independent of L. Relative large control voltages are a consequence. b) Transverse modulators:

The direction of the E-field and the propagation s of the optical field are orthogonal.


K8 8-36

exL E LV / dφΔ ≈ ≈ L = length of the EO-medium, d = electrode separation

Qualitatively the relation between birefringence Δφ and applied voltage V becomes:

In transverse modulators the phase change increases with L and decreases with d, leading to relative low control voltages.

8.4.1 E/O-Phase Modulators

The simplest E/0-modulator is is a phase modulator consisting of an optional input polarizer, transmitting just one polarization, an EO-crystal with two electrodes for the applied external control voltage V in the propagation direction.

The optical propagation is 1) in the direction of the z-principal axis and 2) with the polarization in the x’-principal axis direction.

Bild

x’,y’,z’ = principal axis only the x’-polarized wave exists

( ) ( ) ( )( )

z

1 z1

L,E L 0

2 L n E2 Ln

φ φ φ

ππλ λ

Δ = − =

Δ+

Eex

Δφ(v)

Eopt

x’-principal axis excitation


K8 8-37

Transverse Phase EO-Modulator:

We assume an isotropic EO-crystal that becomes biaxial if a voltage is applied (eg. GaAs, InP).

The n3-principale axis is in the propagation direction, the n2-principal axis is vertical to the planar electrodes (n2 is invariant).

Bild

nz=n3+Δn3

n1+Δn1

n2=konst.

d

The polarization at the input z=0 is oriented in the direction of the n2-principle axis, such that only one polarization

propagates in the EO-crystal in the direction of the n1-principle axis → no polarization-rotation.

( ) ( ) 31 1 14

1 Vn ' V / d n n V / d n r n2 d

= + Δ = + → ( ) VnrdLnL2V 3

14λπ

λπφ =

Δ=Δ

Phase Modulation For an efficient transversal phase modulator we require a high EO-coefficient (Pockels) r, a long modulator length L and a small electrode separation d. This leads to a compromise between low operation voltage Vπ and electrical bandwidth B.


K8 8-38

8.4.2 E/O-Frequency Modulators

Similar to electrical RF-modulators, we can realize an optical frequency modulator for the optical instantaneous frequency ω0pt(t) by an phase modulator be cause the instantaneous frequency is the just derivative of the optical phase.

( ) ( ) ( )tVt

nrdLt

tt 3

41opt ∂∂

=Δ∂∂

=Δλπφω

Frequenzmodulation

The optical frequency modulation is proportional to the time derivation of the electrical control voltage.

8.4.3 E/O-Amplitude Modulator As the EO-effect does not introduce an attenuation modulation, the natural phase modulation has to be converted to an amplitude modulation by:

AM EO-modulator principles

Polarization-modulators, where two polarized waves are excited, make use of the rotation of the polarization direction between in- and output of the EO-crystal. The polarization rotation is converted into amplitude modulation with the help of an polarizing element.

Interference-modulators, modulate the phase-difference between two beams of originally of the same phase, without changing the polarization state. Conversion to amplitude modulation is achieved by interference of the two beams with a relative phase difference Δφ at the output of the modulator by a y-combiner.

Waveguide-Coupling-modulator,

Spatial modulation of light beams, see chap. 8.1.2.


K8 8-39

Concepts and building-blocks a) Polarization modulator: transversal E-field

We assume, that the cubic EO-crystal (with the m34 -crystal structure) is isotropic and becomes biaxial if the E-field is applied, eg. for GaAs. The n2-principle axis is the propagation direction y‘, the n3-principal axis (z‘-direction) be vertical to the electrode planes. This situation is realized if the crystal axis are arranged as shown in the picture below (without proof):

induced principle axis

Bild: Yariv 347

E1

E3

n1 , (x’) Δn ≠ 0

n2 , (y’) Δn ≠ 0

n3 , (z’=z) Δn =0

z’

x’ y’

001

x’- and z’-principal axis excitation

E

Input- and output polarizers are rotated by 90o relative each other The (001)-crystal plane is parallel to the electrodes, resp. vertical to z‘ and the (100)-plane is rotated by 450 relative to the x‘-propagation direction (not shown). The polarization of the optical wave at the input y=0 is arranged in such a way, that both polarization in x‘- and z‘-direction are excited with equal amplitudes in the EO-crystal. The polarization direction is rotated by 450 with respect to the normal of the electrode. The external electrical field strength is E:

( ) ( )d/V,0,0E,0,0E z ==


K8 8-40

Under our assumptions for the crystal-orientation we can use the simple example from chap.8.3.2 directly and we obtain for the changes of the refractive indices of the principle axis x‘, y‘ and z‘ caused by the Pockel-effect:

( )

( )

( )

21 41 z

22 41 z

3

1n n 1 r E n x' Richtung , angeregte Polarisation, n 021n n 1 r E n y' Richtung , Ausbreitungsrichtung , n 02

n n z' Richtung , angeregte Polarisation, n 0

⎛ ⎞≅ − − Δ ≠⎜ ⎟⎝ ⎠⎛ ⎞≅ + − Δ ≠⎜ ⎟⎝ ⎠

≅ − Δ =

Using the above equation we find the following two waves with polarization directions x‘ and z‘ result at y‘=L:

( ) ( ) ( )( )

( )( )

( )( )

( ) ( )

( ) ( )

3411

0 001

341

00 0

L r n V / dL nL n VL j tj tj t 2 c 2cc1

L r n V / dL n L njj t j t2cc c j V

341

x Polarisation : n 0

1 1 1E t,L E t,0 e E t,0 e E t,0 e2 2 2

1 1E t,0 e e E t ,0 e e2 2L r n V / d

V

ωω ωωω πλ

ωω ωω ωφ

πφ

λ

⎛ ⎞⎛ ⎞⎛ ⎞ − +⎜ ⎟−− ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠

⎛ ⎞⎛ ⎞ ⎛ ⎞⎜ ⎟− −⎜ ⎟ ⎜ ⎟⎜ ⎟ Δ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

− Δ ≠

= = =

= =

Δ =

( ) ( ) ( )( )

( )3

3 0 0

L n VL L nj t 2 j t j tc c

3

z Polarisation : n 0

1 1 1E t,L E t,0 e E t ,0 e E t,0 e2 2 2

ω ωω π ω ωλ

⎛ ⎞ ⎛ ⎞ ⎛ ⎞− − −⎜ ⎟ ⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠ ⎝ ⎠

− Δ =

= = =

At the output polarizer at x’=L with its rotated in its transmission axis rotated by 900 relative to the input-polarizer, we obtain for the transmitted components of the two polarized waves (projection of the x-, z-polarized partial waves onto the transmission direction of the polarizer):


K8 8-41

( ) ( ) ( ) ( )

( ) ( )( )

0 0

0

L n L nj t j tc cj V

out

L nj tc j V

1 1 1 1E L,t E t ,0 e e E t,0 e2 2 2 2

1 E t,0 e 1 e2

ω ωω ωφ

ωωφ

⎛ ⎞ ⎛ ⎞− −⎜ ⎟ ⎜ ⎟Δ⎝ ⎠ ⎝ ⎠

⎛ ⎞−⎜ ⎟ Δ⎝ ⎠

= − =

= −

( )

( ) ( ) ( ) ( ) ( )( ) ( )( ) ( )( )

( ) ( ) ( ) ( ) ( )

2out

j V j V2 2out out

2

For the intensity I L,t E one gets:1 1I L,t ~ E L,t E L,t E t ,0 1 e 1 e E t,0 2 2cos4 4

1I L,t ~ I 0,t 1 cos I 0,t sin / 22

Transmission-Characteristic of the EO-Polarizations-Modulator:

φ φ φ

φ φ

∗ Δ − Δ= − − = − Δ

− Δ = Δ⎡ ⎤⎣ ⎦

( ) ( )( )

32 241

341

I L,t L r n V V dT V sin sin mit VI 0,t 2 d 2 V Lr nπ

π

π π λλ

⎛ ⎞ ⎛ ⎞= = = =⎜ ⎟ ⎜ ⎟

⎝ ⎠⎝ ⎠

Vπ is the voltage required to change the transmission from its minimum to the maximum. This type of voltage dependence of T(V) is typical for many polarization modulators based on cos2-relations. For large beam diameters d the voltage Vπ can reach values in the order of 100-1000V. For integrated WG Vπ can be as low as Vπ=3-7V.

Transfercharacteristic of EO-Modulators:


K8 8-42

Example for Vπ: of a GaAs-Modulator:

GaAs @ λ=1.55μm, L=1mm, d=1μm, r41=1.43 10-12m/V, n=3.43 → Vπ=29.1V b) Mach-Zehnder-Modulator: interference modulator

Principle of operation:

In order to avoid the use of polarizers, which are difficult to integrate, one makes use of interferometer-type, eg. the Mach-Zehnder-Interferometer(MZI). Here the conversion of the modulated phase difference change ( )VφΔ into amplitude, resp. intensity change occurs by interference in the output coupler (constructive / destructive ) of the two waves from arms 1 a d 2 of the interferometer. The interferometer arms containing the EO-medium can be realized: free-space, fibers, planer waveguides etc.

Hybrid MZI-Modulator: Integrated Waveguide-MZI-Modulator:

Bil L=L’ + L’’ L

We assume an isotropic E0-crystal becoming biaxial with an applied field, eg. GaAs. The n2-principle axis is in the propagation direction , the n3-principle axis is vertical to the electrode plane.

The polarization of the two waves at the input z=0 (after the input beamsplitter) is such that only one polarization in the n1-direction of the EO-crystal of each arm is excited. In arm 1 the phase of the wave gets modulated by the EO-effect by Δφ1(V).

Δφ(v)

L’

L’’

Δn ≠ 0, → Δφ(v)

Δn = 0

E

E/√2

E/√2

Z

Arm 1 Output y-coupler Arm1

V(t)


K8 8-43

At the output both waves are recombined, resp. interfered. If the interferometer is completely symmetric (both arms have the same optical length nL and the same attenuation α) then both waves cancel each other for Δφ1(Vπ)=π , meaning that the transmission T(Vπ)=0. For Δφ=0, that is for V=0 the two waves interfere constructively and the transmission T(0) reaches its maximum T=1.

( ) ( )( )

32 241

341

Transmission Characteristic of the Mach Zehnder Interferenz Modulator :

I L,t L r n V V dT V cos cos with VI 0,t 2 d 2 V Lr nπ

π

π π λλ

− − − −

⎛ ⎞ ⎛ ⎞= = = =⎜ ⎟ ⎜ ⎟

⎝ ⎠⎝ ⎠

Proof:

( ) ( )

( ) ( )

out 1 22

out out 1 2 1 2

1 1 2 2 1 2 1 2 in

2 2 2

E E E

I E E E E E *

E E E E 2 E E cos mit E E E / 2

E 1 cos E cos / 2

φ

φ φ

∗ ∗

= +

≅ = + + =

= + + Δ = =

= + Δ = Δ


K8 8-44

Schematic Representation of the operation principle of the MZI-modulator: View of a AlGaAs-based MZ-EO-Modulator

1 2L

constantlight input

controlvoltage Vc

opticaldata pulses

ΔΦ=π phase shift

electro-opticwaveguide

transmission

extinction

splitter ground plane electrode

Δn(VC) --> ΔΦ=2πΔnL/λ

VC


K8 8-45

Frequency response and bandwidth of EO-modulators

As the calculation of the frequency response T(ω) is part of an exercise we just summarize the results.

Usually the frequency response of the dipoles of the EO-medium is not limiting because the time constant of the polarization is in the fs-range.

A basic frequency limitation of the modulator bandwidth is the transit time T of the optical wave through the EO-interaction area with the length L. As the light wave transits through L it integrates (LP) over the spatial phase modulation due to the external field Eex(t)=V(t)/d thar the wave “sees” during its trajectory. Here we assume that the electrical field is distributed across the electrodes instantaneously.

phv/LT = Transit time of the optical wave

x

d

0 L

β

vphV(t)

vph =phase velocity of the optical wave n

Transit-time Limitation:

The frequency response can be determined as the Fourier-transform of the impulse response of Δφ(t) to a voltage impulse V(t)=V0δ(t) or as the response to the harmonic excitation V(t)=V0cos(ωt). Because during the transit time T=L/vgr the wavefronts see a time depending local refractive index of the form Δn(t,x)=r n3 V0 cos(ωt +φ(x)). φ(x) represents r^the phase shift due to the time of travel x/vgr. The accumulated phase shift Δφ(ω) is just the integral over the local phase changes during the transit x=0 → L, resp. t=0 → T.

( ) ( ) ( )( ) ( ) ( )

( ) ( )d

VLnr0/

/sin02/T

2/Tsin0 03

0N

N

λπφ

ωωωωφ

ωωφωφ =ΔΔ=Δ=Δ

As a measure of the bandwidth one often uses the first zero of ( )ωT ; T/2N =ω

Short modulators have high bandwidth, but require relative high operation voltages , eg. Vπ. ____________________________________________________________________________________________________________________________________________________________________________j Electronics Laboratory: Optoelectronics and Optical Communication 24.05.2008

K8 8-46

Traveling wave modulator:

In order to avoid the transit time limitation T, the electrodes can be designed as distributed transmission lines, such that the modulating electrical voltage wave V(x,t) propagates with the same speed as the optical wave. This is not trivial to achieve. In the ideal case when vph,e=vph,o , the wave front „sees“ always the same electrical modulation voltage V(t) travelling „along“. Otherwise the waves drift apart with the velocity difference Ivph,e-vph,o I .

Bild

Electrical Wave Guide with output termination Because the condition vph,e=vph,o is difficult to fulfil and because the transmission lines have losses (damping) there is still a finite bandwidth of the travelling wave E-modulator:

( ) ( )

( ) ( ) ( ) ( )

S

S

c / nj T 1c

c / nj T 1c

2

e 10

t 1 1T t sin T t t ....2 2 2

ω

ωφ ω φ

φφ

⎛ ⎞−⎜ ⎟

⎝ ⎠

⎛ ⎞−⎜ ⎟

⎝ ⎠

−Δ = Δ

Δ⎛ ⎞= → = + Δ +⎜ ⎟

⎝ ⎠


K8 8-47

( ) ( ) ( )S

S

π

c / nj T 1c

Sc / nj T 1c

S

Linearized for the small signal modulation at the bias voltage V V /2:

c / nsin T 1ce 1T / 0

c / nT 1c

ω

ω

ωω φ ω φ

ω

⎛ ⎞−⎜ ⎟

⎝ ⎠

⎛ ⎞−⎜ ⎟

⎝ ⎠

=

⎛ ⎞⎛ ⎞−⎜ ⎟⎜ ⎟

− ⎝ ⎠⎝ ⎠= Δ Δ = =⎛ ⎞

−⎜ ⎟⎝ ⎠

⎟⎟⎠

⎞⎜⎜⎝

⎛−

=

S

N

cn/c1T

2ω

c/n = vph,0 = Phase velocity of the optical wave cS = vph,e =Phase velocity of the electrical wave Τ = Transit time of the optical wave ω = Signal frequency Remind:

• The principle of reducing the frequency limitation due to transit time effects in extended structures by travelling wave structures is very general and finds applications in distributed photodiodes, travelling wave electronic amplifiers, laser diodes etc..


K8 8-48

8.5 Electro-Absorptions-Modulators (attachment)

In last few years semiconductor absorption modulators have become important because of their high bandwidth, low operation voltage ~ 2-3V, small volume and the possibility of integration. For details see Appendix 8A.

8.6 Examples of commercial modulators Commercial MZ-Interferometer-Modulators based on SC or LiNbO3 realize bandwidth upto 40 → 80 GHz. High Speed E/O-Mach-Zehnder-Interferometer (semiconductor, Niobates)


K8 8-49

Conclusions / Summary:

• The linear and nonliner E/O-Effect provides an efficient electrical modulation of the refractive index n and the electrical control of optical wave propagation

• The relation between the vectors of the polarisation and the polarizing field are described by linear tensors • The wave propagation in anisotropic materials can be described as a superposition of two normal modes of

different propagation velocities and different polarization (refractive index-ellipsoid construction) • The technique of the refractive index ellipsoid in combination with the EO-effect, allows a simple

determination of the two polarisation directions and the related index of refractions • The bandwidth of electronic EO-modulators is mostly limited by transit-time effects of the optical or electrical

wave in the modulator structure • EO-modulators are sensitive to the polarization of the input signal → polarization control in the fiber channel

can be mandatory • Electrical field dependent control of absorption in semiconductors (Bandgap modifications in bulk- and

quantum well material) makes fast and compact EA-modulators possible


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K8 Optical Modulators - ETH Zpeople.ee.ethz.ch/~fyuriy/oe/oe_optcom_chapters/Optoelectronics_2… · 8-9 K8 In chap.4 codirectional couplers realized by adjacent optical wave guides