Quantum Electronics

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Quantum Electronics Lecture 5

Lecturer:

Bozena Jaskorzynska

Royal Institute of Technology (KTH) Sweden, bj@kth.se

Electro-optical

modulation of light

Lectures co-financed by the European Union in scope of the European Social Fund

(2)

Contents

♦ Introduction to light modulation

♦ Linear Electro-optic effect, phase retardation

♦ Electro-optic modulation of amplitude or phase

♦ Traveling wave modulators

Saleh Ch. 20

(3)

Optical beam modulation

Modulation formats:

Amplitude Modulation (AM), Phase Modulation (PM), Frequency Modulation (FM)

Optical carrier beam Amplitude Modulation Frequency Modulation

Optical field - very high frequency carrier (e.g. 200 THz for λ=1.5 μm)

large modulation frequency possible

large amounts of information can be coded

Applications: data encoding in optical communication,

active mode locking of lasers, short pulse generation, beam deflectors, etc

(4)

Optical beam modulation for data encoding

Communication system: a physical variable (light intensity, field

amplitude, frequency, phase, or even polarization) is modulated at one point and detected at another point

Modulator

CW light in

Modulated light

Laser

Transmitter

Receiver

signal Signal V(t)

(5)

Amplitude modulation

Most popular for optical fiber communication systems, primarily due to the simplicity of envelope photo-detection

Non-Return-to-Zero

Return-to-Zero

For modulation 2.5 Gb/s and above external modulators preferred to avoid chirp 40 Gb/s commercially available (Lithium Niobate), Target: Terabit (1000 Gb/s) speed

(6)

High speed modulator: Beyond 40 Gb/s

1980 1985 1990 1995 2000 2005 2010

10-2 10-1 100 101 102

Year

Single channel datarate [Gb/s]

10MbE (Coaxial)

100MbE (Twisted Pair) 1GbE

(Twisted Pair/Fiber) 10GbE (Fiber)

40Mb/s (First fiberoptical system) STM-16

STM-64

STM-256

Potential 100Gb/s Ethernet standard 2006

Motivation:

Investigate transmitter technologies suitable for 100Gb/s

40 Gb/s and above:

External modulators necessary due to speed and chirp.

2.5 Gb/s and above:

External modulators preferred due to chirp.

Synchronous Transport Mode

(7)

Electro-Optic (EO) effects

 1

 2

(2)

 3

(3) 2

 3

(3)

 ...

  E   E E   E E   E E E

P

o o o o o o

Combine a DC (or low frequency field) E

o

with a wave E

ω cos(ω t)

at optical frequency ω:

For E

o

<< E

ω

 

(2)

 

2 (3)

 

3

) 1

(

     cos(  t)     cos(  t)

E E

E E

E E

P

o o o o o o

DC Kerr effect > AC Kerr effect Pockels effect >

Friedrich Pockels (1865 - 1913)

(8)

Linear EO (Pockels) effect

Linear electro-optic effect discovered by Pockels in 1883

Relation between the 2nd order susceptibility and the Pockels tensor

External variation of the DC field provides phase modulation of light

optical switching, wavelength tuning

 

1

 2

(2)

 ...

  E   E E

P

o o o

E n E

P E

D  

0

    

0 2

0 ) 2 ( 0 2

0

n  2  E

   

ijk ji ii ijk(2)

n

2

n

2

r

2

 1

 

k

k ijk jj

ii

ij

n n r E

n

2 2 2

Refractive indices along the principal axes

(9)

Impermeability tensor

Convenient to describe the induced changes in terms of impermeability tensor η

2

1

ij ij

n

E r E n n r

dn n

d

  

 

  

 

 

  

 

 

 

3 3

2 1

 2

E r n n

n dn n

n dn     

 



2 2 2 4

n

 

n3r

E

2 1

k

k ijk ij

ij

(0)

r E

k k

ijk ij

ij

r E

n

 

 

  1

2

ij

j i

ij

( E ) x x 1

Index ellipsoid

Useful scalar relations to estimate order of magnitudes:

(10)

2

1

ij ij

n

Impermeability tensor – contracted form

In lossless and optically inactive media is symmetric:

ji

ij

 

Hence 3x3 matrix can be reduced (contracted) to a column of 6 independent elements:

2 3

3 2 2

2 2 2

2 1

1 2

1 , 1

1 , 1

1 1

z y

x n n n n

n

n

  

 

 

 

 

 

 

 

   

2 6

6 2 2

5 5 2

2 4

4 2

1 , 1

1 , 1

1 1

xy xz

yz n n n n

n

n

  

 

 

 

 

 

 

 

   

From diagonal elements

From off-diagonal elements

(11)

Pockels tensor – symmetries, contracted notation

To be further reduced by spatial (group) symmetry

In lossless and optically inactive media: Permutation symmetry

Contracted notation:

r

ijk

r

lk

6 5

4

3

2

1 :

12,21

31,13

23,32

33 22 11 :

l ij

3 , 2 ,

 1 k

In a centrosymmetric crystal :

r  0

From the symmetry should not change under lattice inversion:

From physics (linear charge displacement under DC field):

r r

inv

r

r

r

inv

  r  0

jik

ijk

r

r

(12)

Impact of linear EO effect in contracted notation

k k

ijk ij

ij

r E

n

 

 

  1

2

2 1

2

2 11 12 13

21 22 23

2

3 31 32 33

41 42 43

2

4 51 52 53

61 62 63

2 5

2 6

1

1

1

1

1

1

x y z

n

r r r

n

r r r

n r r r E

r r r E

n r r r E

r r r

n

n

   

     

 

    

       

   

      

         

      

       

           

   

       

   

   

 

 

   

 

   

 

The change induced by the DC electric field E=(Ex, Ey, Ez):

can now be expressed in the contracted form:

k lk l

l

r E

n   

 

 

  1

2

summation over repeated indices!

(13)

Linear EO effect – impact on Index ellipsoid

Unperturbed (E=0) Index ellipsoid

External electric field E distorts the Index ellipsoid. Possible impact:

2

1

2 2

2 2

2

  

z y

x

n

z n

y n

x

off-diagonal (mixed) terms appear in

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2

3 2 2

2

2 2 2

2

1 2 2

1 1

1 1

1

1

z

n y n

n x n

n

nx y z

rotation

change of the axes length (diagonal elements of )

k lk l

E n    r

 

  1

2

1 1 1 2

1 2 2

6 2 5

2 4

2

  

 

 

 

 

 

 

 

 

xy

xz n yz n

n

(14)

Examples of electro-optic tensors

41

41

63

0 0 0

0 0 0

0 0 0

0 0

0 0

0 0 r

r

r

 

 

 

 

 

 

 

 

 

 

22 13

22 13

33 51

51 22

0 0

0 0

0 0

0 0

0 0

r r

r r

r r

r r

  

 

 

 

 

 

 

 

  

 

KDP: KH2PO4 Potassium Dihydrogen

Phosphate

LiNbO3:

Lithium Niobate

m 2 4

GaAs

Gallium Arsenide

m 3 4

Isotropic Anisotropic

m

3

(15)

Linear EO effect in KH 2 PO 4 (KDP)

41 41

63

0 0 0

0 0 0

0 0 0

0 0

0 0

0 0 r

r r

 

 

 

 

 

 

 

 

 

 

2 2 2

41 41 63

2 2 2

0 0

2

x

2

y

2

z

1

e

x y z

r E yz r E xz r E xy

nnn    

Obtain the equation for the index ellipsoid

Diagonalize the equation. Here, by rotating the reference system by 45o. For field polarized along z one obtains:

2

2 2

63 63

2 2 2

0 0

1 1

z z

1

e

r E x r E y z

n n n

   

 

    

   

   

External field Ez induces the difference between

n

x'and

n

y'

z

x n n r E

n ' 0 03 63

2

 1

ny' n0 n03r63Ez

2

 1

Electrically tunable birefirngence ! Consider DC field along the optic axis z:

E  ( 0 , 0 , E

z

)

(16)

Ellectrically induced birefringence

GaAs GaAs

y

a

E

E

z

a

E

E

LiNbO3

 0 E

a

Isotropic GaAs became uniaxial

Uniaxial KDP and LiNbO3 became biaxial

(17)

Pockels cell – phase retardation

Consider a light beam passing through a ”cell” made of an electro-optic crystal, with its Index of ellipsoid modified by Ea field.

Assume the input light linearly polarized at some angle to the crystal axis (e.g. 450 as in the figure).

Decompose the input field into two fields Ex and Ey polarized along the crystal axes, propagate them separately, and add at the cell output.

The acquired phase retardation Г between Ex and Ey will be determined by:

L E n E

n

k

x a y a

y

x

 

0

[ ( )  ( )]

  

EO crystal

L

Polarization state can thus be tuned by Ea, and if desired converted to amplitude or frequency modulation

(18)

Electro-optic retardation – longitudinal geometry

External electric field along the direction of light propagation

For KDP:

Retardation depends on V but not on length

Γ

=

Ez x

y

Half voltage Vπ - voltage for which the pase shift

Γ

=

π

63 3

2 r n

0

V

Here:

V

V



V r n k z E r n k z

E r n n

k z

E r n n

k

z z z

y

x 63

3 0 0 63

3 0 0 63

3 0 0

0 63

3 0 0

0

)

2 ( 1

2 )

( 1  

 

  

 

 

 

 

(19)

Electro-optic retardation – transverse geometry

1. Phase retardation depends on voltage V, length l and thickness d

2. Γ has a term not depending on the applied voltage:

birefringence effect

For KDP:

External electric field normal to the direction of light propagation

At a given voltage V one can increase retardation by increasing modulator length l and/or decreasing its thickness d

l Ez

 

0 633

x z o e

2

n r V

l n n

c d

 

       

 

63 3 0 0 63

3 0

) (

2

r n

d n n

l r n

V   d  

e

(20)

Malus’s Law

Phase modulation converted to intensity modulation

(21)

Converting phase shift to transmitted intensity

After the polarizer P:

1 1

1 1 2 1

2 0 / 2

/

) 2 / 1 sin(

1 2 1 0

0 1

1

1 1

2

1 i E

e e

i

i

 

 

 

 

 

 

Relative transmitted intensity:

) 2 / ( sin

2

in out

I I

With a quarter wave plate (QWP):

1 sin( )

2 ) 1

cos( 2 2 )

sin(

2 2 1

) 1 2 / 4

/ (

sin

2

  

 

   

 

in out

I

I

(22)

Amplitude modulation – longitudinal geometry

z x

´

y

´ l

x

y V

t V V

V msinm 2

V

V

 

t

I t I

m m

m m

in out

 sin

2 1 1

) sin sin(

2 1 1

2

2 0 63

V n r

(for KDP) For Гm<<1 - linear replica of the modulating voltage:

(23)

Phase (frequency) modulation

If an optical wave is incident normally on the x’-y’ plane with its E vector along the x’ direction, the electro-optic effect will simply change the output phase, without change of the polarization:

3 0 63

x

2

z

n r E l c

  

 

0 633

exp sin

out m

2

m

e A i t t n r E l

c

    

       

Phase modulation index

the output becomes (disregarding the constant phase factor):

t) (i exp A

e

in

 

For an input beam and the external field E

z

 E

msin

(i 

m

t)

(24)

Transit time limitation

3 0 63

x y z

n r V

V E l c

    

' ) ' ( )

0

( 

 

t

t l

d

dt t n E

dz c z

E

) exp( i t E

E

m

m

Phase retardation for DC (or very slowly varying) field:

For E changing appreciably during the transit time τ

d

=nl/c of the light through the crystal:

r

-

decrease in peaking retardation resulting from the finite transit time.

Modulation gets “averaged out”

For abs(r)=0.9 as a threshold, the maximum modulation frequency:

 

max

m

4 v c

nl

Where the wave enters the crystal at time t- τ

d

, and leaves the crystal at time t

For

:

0 1 2 3 4 5 6

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

m

d/

abs(r)

2 1

0

1

i m d i mt

m d

e e

i

  

 

  

    

 

El

0

(25)

Traveling wave modulators

m m

i t ik z

EE e

m

For perfect phase-velocity matching (cm=c/n) r=1 –no frequency limitations

With r=0.9 the maximum operating frequency:

c

m is the phase velocity of the modulation field

 

m max

41 /

m

v c

nl c nc

 

The modulation signal - a traveling wave phase matched with the optical wave To overcome the transit –time

limitation

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

abs(r)

1.0 0.8 0.6 0.3 0.1

1

c nc

/

m

 

2

 

 

1 / 0

1

1 /

m d m

m

i c nc

i t

m d m

e e

i c nc

  

 

  

       

r

Increased by factor:

1/ !!

Strip transmission line

(26)

Integrated transverse Pockels cell

d V r L n

o3 23

2

   

Constant Γ =0.5-0.7

(27)

EO tunable Mach-Zehnder Interferometer (MZI)

y V

c axis (Z)

Ez optical Ez optical

LiNbO3 d

V

c axis (z)

Ez optical Ez optical

LiNbO3 d

(28)

Electrooptic Mach-Zehnder modulator

up to several cm

Pin

V

n(E)

optical waveguides

• LiNbO

3

• GaAs-AlGaAs

• InP-InGaAsP

• (polymers)

Mach-Zehnder Interferometer (MZI)

The input beam is split at the Y-junction into two beams that propagate in each of identical arms.

With no voltage applied they constructively interfere and recombine in the output Y-junction – MZI has no effect.

When the voltage is applied the refractive index of the arms becomes different.

The output is the sum of the two beams and the output intensity depends of their relative phase:

 

inp

i i

out

Ee Ee I

I

 1  cos 

2 1 2

1 2

1

1 2 2

(29)

Lithium niobate Mach-Zender modulator

(30)

Electrooptical coupler modulator

Applied voltage alters the refractive indices and the induced phase mismatch decouples the guides

(31)

THz all-optical modulation in a Si–polymer hybrid system

Michael Hochberg, Caltec 2006

The gate signal has its intensity modulation pattern transferred to the source via Cross Phase Modulation due to NL Kerr effect in the polymer cladding

Nonlinear polymer cladding

Si substrate

No phase –matching needed between gate and source Can be used to convert modulation to another wavelength

(32)

EO modulators: pros and cons

Pros:

Very low optical loss

High power handling capability Broad bandwidth

Zero or tunable chirp

Temperature insensitivity

Cons:

Large size

Bias-drifting issue Polarization sensitive

Difficult to integrated with other components

High costs for large volume production

(33)

40 Gb/s Si laser modulator

Interesting blog:http://blogs.intel.com/research/2007/07/40g_modulator.html

Micrometre-scale 40 Gb/s laser modulator in Si developed by Intel (2007)

Based on free-carrier plasma dispersion effect – silicon’s refractive index is changed when

the density of free carriers (electrons/holes) is varied

Future terabit per second optical chip – Intel vision

Obraz

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