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Chapter 9. Electrooptic Modulation of Laser Beams

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(1)

Nonlinear Optics Lab . Hanyang Univ.

Chapter 9. Electrooptic Modulation of Laser Beams

9.0 Introduction

Electrooptic effect :

# Effect of change in the index of refraction of medium (crystal) by an external (DC) electric field

# Nonlinear polarization :

effect) (Pockels

effect EO

linear

n E

jk

k j

ijk

i

E E

P (  ) 2 

(2)

(   0 ) (  ) ( 0 )

#

(2)

Nonlinear Optics Lab . Hanyang Univ.

9.1 Electrooptic Effect

Index Ellipsoid

j

j ij

i E

D

Displacement current :

Energy density :

ij

j i ij

E E

U

2 1 2

1 D E

: for the principal axes





  

zz z yy

y xx

x D D

U D

2 2 2

2 1

Put, x y Dz

z U U D

y U D

x

2 / 1 2

/ 1 2

/ 1

2 , 1

2 , 1

2

1

2

1

2 2

2 2

2

  

z y

x

n

z n

y n

x

: Index ellipsoid

(3)

Nonlinear Optics Lab . Hanyang Univ.

General expression of index ellipsoid

 ,

j

j ij

i

D

E

where, ij (1)ij : impermeability tensor element

Energy density :

ij

j i ij

D D

U

2 1 2

1 D E

Put, x y Dz

z U U D

y U D

x

2 / 1 2

/ 1 2

/ 1

2 , 1

2 , 1

2

1

2 / 1 2

/ 1 2

/ 1 2

/ 1 2

/ 1 2

/ 1

) 2 (

, 1 ) 2 (

, 1 ) 2 (

1

x z z

y y

x D D

zx U D

U D yz D

U D

xy

1 2

2

2 12 23 13

2 33 2

22 2

11      

x

y

z

xy

yz

xz

3 33 2

2 22 2

1 11 2

, 1 , 1

1

n n

n

21 12 6 31 2

13 5 32 2

23 4

2

, 1 , 1

1

n n

n

Put,

1 1 1 2

1 2 1 2

1 1

6 2 5

2 4

2 2

3 2 2

2 2 2

1

2

xy

xz n yz n

z n y n

x n n

(4)

Nonlinear Optics Lab . Hanyang Univ.

Impermeability :

  

l

kl

k ijkl k

k ijk ij

ij

  E S E E

(0)

Linear EO Quadratic EO

Kleinman symmetric medium :

ijk

hk

h = 1 2 3 4 5 6

ij = 11 22 33 23,32 13,31 12,21

) , , ,

1 3 ( (1) (2) (3)

2 E i j x y z

nij ij j

 



where,

ij : Electrooptic tensor (element)

(9.1-3)

(5)

Nonlinear Optics Lab . Hanyang Univ.

3 2 1

63 62

61

53 52

51

43 42

41

33 32

31

23 22

21

13 12

11

6 2

5 2

4 2

3 2

2 2

1 2

1 1 1 1 1 1

E E E

n n n n n n



 





 





 





 





 





 



(9.1-3) 

ex) E=0,

1 0 1

1

1 , 1

1 , 1

1 1

6 0 2 5 0

2 4 0

2

2 3 0

2 2

2 0 2 2

1 0 2

E E

E

E z E y

E x

n n

n

n n

n n

n n

n1 i 0 for all i

2  

 



(6)

Nonlinear Optics Lab . Hanyang Univ.

(7)

Nonlinear Optics Lab . Hanyang Univ.

Example) EO effect in KH2PO4 (KDP ; negative uniaxial crystal, symmetry group)

When the electric filed is applied along the z-axis, the equation of the index ellipsoid is given by m

2 4

1 2 63

2 2 2

2 2

2    E xy

n z n

y n

x

z e

o o

The Sij matrix is

2 63 2

2 63

0 1 0

1 0 1 0

e o

z

z o

ij

n E n

n E

S

Report : Summary (pp. 333-339)

i j

ij

S

S   

For the principal axis,

i

Condition for nontrivial solution (eigenvalue equation) :

0 0 1

0

1 0 1 0

2 63 2

2 63

n S n S

E

E n S

e o

z

z o

(8)

Nonlinear Optics Lab . Hanyang Univ.

0 ) 1 (

1 2

63 2

2

2









z

o e

E n S

n S

z o

z o

e

n E S n E

n S

S 2 2 63 12 63

1 ,

1 ,  

1) For S’

1 0 1

1 0 1 1 0 1

2 3 3 2

3 33 2

32 1 31

2 2 1 2

63 2

3 23 2

22 1

21

2 63 2 1

1 2 3

13 2

12 1 11













e o

e o z

z e

o

n S n

S S

S

n E n

S S

S S

n E S n

S S

S

arbitrary ,

0 3

2

1  

  

(0,0,1)

χ

(9)

Nonlinear Optics Lab . Hanyang Univ.

2) For S’’

3 2 63

2 3

2 63 1 63

2 63 1 63

1 1

0 0

 













z o

e

z z

z z

n E n

E E

E E

0 0

3 2 1







) 2 0 1 2 ( 1 (1,1,0)

,

,



χ

3) For S’’’

0 0

3 2 1







) 2 0 1 2 ( 1

(1,-1,0) , ,



χ Similarly,

z Z y

x Y

y x

X ( ) ,

2 , 1

) (

2 1

1 1 1

2 2 2

2 63 2

2 63    

 

 

 



 

 

e z

o z

o n

y z n E

x

n

E

New principal axes :

The equation of the index ellipsoid in the new principal coordinate system :

(10)

Nonlinear Optics Lab . Hanyang Univ.

9.2 Electrooptic Retardation

For a wave propagating along the z-direction, the equation of the index ellipsoid is

1 1

1 2

2 63 2

2 63   

 

 

 



 

  E y

x n

no

Ez o

z

2 63



o

z

n

E

Assuming,

z o

o y

z o

o x

n E n n

n E n n

63 3

63 3

2 2

 

 

Field components polarized along x’ and y’ propagate as

 

 

 

t c n n E z

i y

z E n

n c t

i z

n c t

i x

z o

o

z o

o x

Ae e

Ae Ae

e

6 3 3

6 3 3

) 2 / ( ) / (

) 2 / ( ) / ( ]

) / ( [

(11)

Nonlinear Optics Lab . Hanyang Univ.

Phase difference at the output plane z=l

between the two components (Retardation) :

63 ,

3

c V n

o

y x

 

  

c l l n

E

V

z ,

x

 

x

where,

The retardation can also be written as

V

V V

l Ez

63

2 3

where,

n

o

V

(Halfwave retardation voltage)

(12)

Nonlinear Optics Lab . Hanyang Univ.

9.3 Electrooptic Amplitude Modulation

t e

t e

y x

cos A

cos A

A ) 0 (

A ) 0 (

y x

E E

or, using the complex amplitude notation

2 2

* 2

A 2 ) 0 ( )

0

(

x y

i E E

I E E

A ) (

A ) (

l E

e l

E

y

-iΓ

x ( 1)

2 ) A

(Ey o e-iΓ

)]

1 )(

1 2 [(

) A ( ) (

2

*

y o y o -iΓ

o E E e e

I

sin 2 A

2 2 2Γ

(13)

Nonlinear Optics Lab . Hanyang Univ.

The ratio of the output intensity to the input :

 

 

 

 

V Γ V

I I

i o

sin 2 sin2 2

: amplitude modulation

(14)

Nonlinear Optics Lab . Hanyang Univ.

Sinusoidal modulation)

mt

m

 sin

2 

 

t

I t I

m m

m m

i

 

sin sin

2 1 1

2 sin sin2 4

0

 

 

 

1

m

sin

m

t

2

1  

1

m

(15)

Nonlinear Optics Lab . Hanyang Univ.

9.4 Phase Modulation of Light

Electric field does not change the state of polarization, but merely changes the output phase by

l c E

r n n

c l

z x

x 2

63 2 0 '

      

If the bias field is sinusoidal ; EzEm sin

mt

] sin

cos[ t t

A

e

out

    

m

  n r E l

c l E r

n03 63 m 03 63 m 2

 

 

where,

: Phase modulation index )

exp(

) ( )

sin

exp(i t J in mt

n n

m  



t n i n

n out

e

m

J A

e

(  )

( )



: side band (harmonics)

(16)

Nonlinear Optics Lab . Hanyang Univ.

9.5 Transverse Electrooptic Modulators

Longitudinal mode of modulation : E field is applied along the direction of light propagation Transverse mode of modulation : E field is applied normal to the direction of light propagation

# Transverse mode is more desirable :

1) Electrodes do not interfere with the optical beam

2) Retardation (being proportional to the crystal length) can be increased by use of longer crystal 3) Can make the crystal have the function of /4 plate

 

 

 

 

 

d

r V n n

c n l

e z

x 63

3 0 0

'

2

 

(17)

Nonlinear Optics Lab . Hanyang Univ.

9.6 High-Frequency Modulation Considerations

If Rs > (oC)-1, most of the modulation voltage drop is across Rs

 wasted !

Solution : LC resonance circuit + Shunting resistance, RL >>Rs

LC parallel circuit,

2 2

2

1

LC

R L

Z

 

2

2

2 2 2

1 1 1

1 1

1 1 1

so,

 

L CL R

L CL iR

R Z

c L R i

Z

L L L L

Total impedance :

 

 

 

2 / 2 1

2 2 2

2 2 2

2 2 2

1 1

1

1

1

L CL R

L CL R

L CL R R R

Z

L L

L L s

At the resonance [=0=1/(LC)1/2], Z RL !

(18)

Nonlinear Optics Lab . Hanyang Univ.

Maximum bandwidth :

C R

L

2 1 2 

Required power for the peak retardation m :

L m

R P V

2

2

where,

A

: cross-sectional area of the crystal normal to l

2 , )

(

63 3 0

r l n

E

V

m z m m

   c  

R

L 2

1

2 63 6 0 2 2

4 l n r P

m

A

 

 

(19)

Nonlinear Optics Lab . Hanyang Univ.

Transit-Time Limitation to High-Frequency Electrooptic Modulation

(9.2-4)  aEl (a

n03r63/c)

But, if the field E changes appreciably during the transit time through the crystal,

t

t l

d

dt t n e

a c dz z e a t

' ) ' ( )

( )

(

0

Taking e(t) as a sinusoid ; e(t')Emeimt'

Phase change during the transit-time,d=nl/c

where, 0a(c/n)

dEmalEm : Peak retardation Reduction Factor, r (Fig. 11-17)

Practically, in order to obtain |r|=0.9,

t i d

m i t

t

t i m

m d

m d

m

i e e

dt e

n E a c t

 

  

1

' )

(

0

'

c

d nl

m

/2, and

d /

nl

m) c/4

( max

Ex) KDP, n=1.5, l=1cm,

GHz 5

) ( max

m

(20)

Nonlinear Optics Lab . Hanyang Univ.

Traveling wave Modulators

: matching the phase velocities of the optical and modulation fields by applying the modulation signal in the form of a traveling wave

Consider an element of the optical wavefront that enters the crystal at z=0 at time t )

' ( )

'

( t t

n t c

z  

The retardation exercised by this element is given by

t

d

t

dt t z t n e

t ac

' )]

' ( ,' [ )

(

(21)

Nonlinear Optics Lab . Hanyang Univ.



 

 

) /

1 ( ) 1

(

) / 1 ( 0

m d

m

nc c i

t i

nc c i

e e t

m d

m

m

 

)]

' )(

/ ( ' [ ]

'

)

[

'.

( t z E

m

e

i mt kmz

E

m

e

i mt km c n t t

e

The traveling modulation field :

where, 0a(c/n)

dEmalEm : Peak retardation

Reduction factor :

) /

1 (

) /

1

(

1

m d

m

m d

m

nc c i

nc c

e

i

r

# c/n=cm  r = 1

# Maximum modualtion frequency (|r|=0.9) :

) / 1 ( ) 4

( max

m

m nl c nc

c

case) field

(static

4 / )

( m maxc nl

(22)

Nonlinear Optics Lab . Hanyang Univ.

9.7 Electrooptic Beam Deflection

Deflection angle inside the crystal ( dx dn n

l Dn

n l D

y 

'

0

n )

External deflection angle (By Snell’s law)

dx l dn D

l n

n     

 

'

Δn) c(n

TA l

: A ray upper of

me Transit ti

c n TB l : B ray lower of

me Transit ti

n l n T

n T c

B A

y ( )

: B respect to A with

ray of Lag

(23)

Nonlinear Optics Lab . Hanyang Univ.

z

A n r E

n

n 63

3 0 0 2

z

B n r E

n

n 63

3 0 0 2

n r E

z

D l

63 3

0

 

Double-prism KDP beam deflector

Number of resolvable spots, N (for a Gaussian beam) :

z beam

E l r

N n

63

3 0

2

 

# (9.2-7)

63 3

2 0

)

(

n lr

E

z

     1

4 2

| 2

63 3 0 63 3

0

 

 

n lr

lr

N

V V

n

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