EE 232: Lightwave Devices

Lecture #19 – Semiconductor Laser

modulation rate - Small signal analysis

Instructor: Seth A. Fortuna

Dept. of Electrical Engineering and Computer Sciences

University of California, Berkeley

4/11/2019

2Fortuna – E3S Seminar

Small signal analysis

Power

Current

time

time

0( ) ( )P t P p t= +

0( ) ( )I t I i t= +

0P

0ISmall time-varying current withDC offset is applied to the device:

0( ) ( )I t I i t= +

thus producing a time dependentoutput power:

Time-varying current is small suchthat device characteristic can bedescribed by a linear extrapolationaway from the bias point.

0( ) ( )P t P p t= +

3Fortuna – E3S Seminar

Small signal analysis - Laser( ) ( )

( ) ( )( )i g

dv

n t J tg t S t

dR

t qt

d= − −

( ) ( )( ) ( ) ( )sp sp g

p

dS t S tR t g t S t

dtv

= − +

0

0

0

( )

( )

( )

( )

( )

( )

n n

t

t n

J t J

S t

J

SS

t

t

= +

=

+ =

+

Let

( ) ( )) ( )( sp AugeSR rHR tt R R t Rt + +=

00 0 0 0 ][(

( ) ( )[ ( )] ( ))( ( ))i g

J t nd tn t R

Jn g g

dt qdtv n t S S

= − −

++ − + +

0 0

( ) ( ) ( )( ) ( ) ][i g

d n t J

t

t ntg g

d qd

tv S n t S

= − − +

(Ignoring 2nd

harmonic term)

Similarly,

Then,

( )g n

n0n

0g

0 0

( )( ) ( )( ) ( ) ][sp g

p

S t S tv S t n t

d n tg Sg

dt

= + + −

0

0( ) ~ ( ) ( )n n

gg n g n n t

n =

+

4Fortuna – E3S Seminar

Sinusoidal excitation – Laser

We assume the excitation is sinusoidal

( ) Re[ ( )

( ) Re[ ( )

( ) R ]e[

]

]

( )

i

i

t

i

t

t

n t n e

J t J e

S t S e

−

−

−

=

=

=

0 0

( )( )( [) ]g

p

S tt g gS i v S nS

− + −

Recall,0

1g g th

p

vgv g

= =

Then,0g

in S

v Sg

− =

0 0

( )( ) ][i g

J n tg g

qdn i v S nS

− = − − +

0( ) i g

Jn i v Sg

qd

− + = −

1

0

02

g

g

r

p

g

v g

S

S

v

−=

+

=

Note:

And,

2 2

0

( )r i

g

S Ji

qSg dv

− =

−

0

2 2

/i g

r

S gv S

J i

qd

−

=

− (relaxation oscillation frequency)

(damping)

5Fortuna – E3S Seminar

Sinusoidal excitation – Laser (cont’d)

0

2 2

12

1

2 2

12

2 2

12

1

2

/ //

1

1

1 (

( )( )

)

g m act i g

g m act

r

i g m p

r r

mi

m i r r

mi r p r

m i r r

S h v v SPh v

J J i

Ph i

I

hi

P hi

V g qdV

v q

I

q

q

−

−

−

−

−

−

−

= = −

= −

= −

= −

−+

− +

+

Let’s write the transfer function in terms of power

6Fortuna – E3S Seminar

3dB-frequency

Electrical 3dB-frequency is given by

12

13 3

2

11

2( )dB dB

r p r

r r

i

−

− − + =−

It is often the case that and 1p p 1r . Then,

12

3

22

11 dB

r

−

−3 1 2dB rf f= +

Writing in terms of output power,

0

3 0 0

1.55 1.55 1.55 1

2 2 2

g g g m idB

p g p m cav cav m

Sf

h v

v g v g v

h

gP P

V V

+=

= =

3

1.55

2

g idB th

cav

v gf I I

q V

= −

Writing in terms of drive current,For high speed:(1) Maximize differential gain(2) Minimize cavity volume (mode volume)(3) Maximize drive current relative to

threshold current

7Fortuna – E3S Seminar

Gain saturation

0 0( ( ) )( , )

( )

( )

1

n t ng

g n

S

gn S

t

+

+

−=

When the photon density is high, gain may decrease with further increase in photon density. This is called nonlinear gain saturation or gain compression.This can be accounted for with the following model.

: gain compression factor

0

0

0

( )

( )

( )

( )

( )

( )

n n

t

t n

J t J

S t

J

SS

t

t

= +

=

+ =

+

Let

0 0

0

0

0 0

2

0 0 0

( , ) ( ) ( )

( )

1

1 1 (1) (

)

n n n n

g n S n t SS

g g g

dn dS

g gg

t

n t S tS S S

= =

+ +

= +

+−

+

+ +

( ) ( ) ( ) /

( ) ( ) 0

iA Dd nn t t J t qd

S t S tC Bdt

+ =

−

Then,

( )0

2

0

1

1

g

p

Bv

S

g

+

−=0

0

1

1

gv gA

S

S+=

+

0

01

gv S

S

gC

=

+

0

2

0(1 )

g gD

v

S=

+

8Fortuna – E3S Seminar

Sinusoidal excitation with gain saturation

( ) Re[ ( )

( ) Re[ ( )

( ) R ]e[

]

]

( )

i

i

t

i

t

t

n t n e

J t J e

S t S e

−

−

−

=

=

=

Let

( ) ( ) /

( 0)

in J qd

S

i A D

C i B

− =

+

− − +

2( )

/

( )

iC

AB CD i A

J qd

BS

+ −− +

=

2

0 0

2

0 0

02

0

1 1 1

1 (1 )

1

1

r

g g

p p

g

r

p

B

S v

A CD

v g g

g

S

S

v

S

S

= +

=

− −

+

+ + ( )0

0 0 0

1 2

1

1 11

g sp

g p

r

A

v

f

g R

v

S

S

B

gS S

K

−

=+

= +

+ + + +

+24 p

g

Kv g

= +

9Fortuna – E3S Seminar

Sinusoidal excitation with gain saturation

( )

( )

( )

10 1 2 2

0

10 1 2 2

0

10 1 2 2

0

12

1

2 2

2

2

(1

/ (1

)

1

1

1

)

( )

g

i r

g

g m act i g m act r

g

i g m r

i g p m

r r

mi

m i r

v SSqd

J S

v SP Sh v h v qd

J J S

v SPh v q

I S

h v q

g

i

hi

gi

gV V i

i

q

−−

−−

−−

−

−

−

+

−+

= −

= = −

= −

= −

−

−=

−

−

+

+

1

2

12

2 1

21 ((2 ) )

r

mi r r

m i r r

P hi K

I q

−

−

− −

+

−

= − +

Then,

10Fortuna – E3S Seminar

K-factor

Comparison with our previous result where gain saturation was not considered

1 2

24r

K

−= +1 2

p r −= +

With gain saturation Without gain saturation

Typical K-factor is about 100 times larger than typical photon lifetime (Coldren pg. 260). Therefore, as drive current is increased and the relaxation oscillation frequency is increased, the damping factor becomes non-negligible.Previously, we ignored the damping to calculate the 3dB frequency; however, thisis not accurate at high photon densities (i.e. high drive current) in the presenceof gain saturation.

:K Referred to as the “K factor”. Units are seconds. As we will see, the K factor sets the upper limit of the intrinsic laser modulation speed.

11Fortuna – E3S Seminar

3dB-frequency1

22 1

2

12

3 3

2 2

12 2

2

3 3

2 2

1 (

11

11

2

(2 ) )

2

mi r r

m i r r

dB dB

r r

dB dB

r r

P hi K

I

i

q

−

− −

−

−

−

−

=

= − ++

− =

−

+

Let’s try to find the maximum 3dB-frequency that is possible

We can see that the 3dB frequency is maximized whenThen,

2

3

201 dB

r

− =

12

3 ,max

2

1

2

dB

r

−

=

2 1

3 ,max 3 ,

2 22dB r dB max

Kf

− →=

K-factor is an intrinsic parameter that sets the upper limit of the modulation speed

12Fortuna – E3S Seminar

3dB-frequency

3 (GHz)dBf

10

20

log

(d

B,

no

rm.)

P I

0P

010P

0100 P

01000P

At low power (low photon density): Damping is small. 3dB frequency is increasedby increasing the relaxation oscillation frequency through increased current injection.

At high power (high photon density): Damping is large. Relaxation oscillationfrequency saturates due to gain compression. Maximum 3dB frequency islimited by K-factor.

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