EE 232: Lightwave Devices

Lecture #2 – Optical gain and laser

cavities

Instructor: Seth A. Fortuna

Dept. of Electrical Engineering and Computer Sciences

University of California, Berkeley

Gain Medium

Gain medium(𝐺)𝐼0 𝐼0𝑒

𝐺𝐿

𝐿

(

)( (

) ( ) (

)

)

)(

z

I I L I

I

L

z z

z G L

G

I

I

z

z

I

+

+

= −

= −

=

1G

I

LI=

Gain is the fractional increasein light intensity per unit length(Units are 𝑐𝑚−1)

Gain Medium (with internal loss)

Gain medium(𝐺, 𝛼𝑖)

𝐼0 𝐼0𝑒𝐺𝐿

𝐿

)( (

( ) ( )(

)

)

)

)

(

(

( )

i

i

I I L I

I L I

I G L

z z

z I z G z

z

= −

= −

=

+

+ −

−

1( )i

I

LG

I

−

=

Internal loss (𝛼𝑖) is the fractional decreasein light intensity per unit length (unrelated to fundamental absorption)(Units are 𝑐𝑚−1)

Gain Medium (with internal loss)

Gain medium(𝐺, 𝛼𝑖)

𝐼0 𝐼0𝑒𝐺𝐿

𝐿

1 1( )i

dIG

I

I

d IL z− →

=

(

0

)( ) iG z

I z I e−

=

Gain with cavityGain medium (𝐺, 𝛼𝑖)

𝐿

((o))Mirror 1 Mirror 2

2 1

)2(( 01

2 )e

( 0 )th iG LL

RI

I zR

z

−

+

+== =

+

=

Round-trip gain

(Threshold conditionfor self-sustainingoscillation)

𝑧

Gain with cavityGain medium (𝐺, 𝛼𝑖)

𝐿

((o))Mirror 1 Mirror 2

(Threshold conditionfor self-sustainingoscillation)

𝑧

2 1

1 1ln

2th i

m i

GRL R

=

= +

+

Photon lifetime (τ𝐩)Gain medium (𝐺, 𝛼𝑖)

𝐿

Mirror 1 Mirror 2

𝑧

𝑁𝑝

After one round trip: 1 2(1 ) pR R N− photons are lost from the cavity(ignoring 𝛼𝑖)

1 2( ) ( ) (1 )RT

R

p p P p

RTT

N t N t RN N

t

R

+ −=

−= −

/

0pt

p pN N e−

= where 1 2

2 /

1p

nL c

R R =

−(photon lifetime)

Quality factor (Q)

Energy stored in cavity

Energy lost per cycle2Q =

0

N2

2p

pp

p

p

QN T

Q

T

=

=

=

Gain medium (𝐺, 𝛼𝑖)

𝐿

Mirror 1 Mirror 2

𝑧

𝑁𝑝

Alternative expression for 𝑮𝒕𝒉Gain medium (𝐺, 𝛼𝑖)

𝐿

((o))Mirror 1 Mirror 2

𝑧

cG

nis the fractional increase in photons per second

01p

p

th p

p

th th

NcG N

n

c nG G

n Q c

=

= → =

Semiconductor laser

𝐿

Cleavedfacet

𝑧

21

~ 30% for 3.51

nR n

n

− = =

+

𝑛2

𝑛2

𝑛1Cleavedfacet

𝑛1(gain)

Semiconductor laser

𝐿

Cleavedfacet

𝑧

𝑛2

𝑛2

Cleavedfacet

Power of optical mode in gain regionConfinement factor:

Total power of optical mode =

𝑛1(gain)

01th

nG

Q c

=

Fabry-Perot cavity modes

𝐿

Mirror 1 (𝑟1, 𝑡1) Mirror 2 (𝑟2, 𝑡2)

𝑧

𝐸𝑖𝑛𝑐𝐸𝑡

Field just to the right of Mirror 1 and propagating in +z direction:

2 4 2 3

1 1 2 1 2 1 2 1 2(1 ( ) ...)jk L jk L jk L

incE E t r r e r r r r e r r e+ − − −= + + + +

1k

k aar

r

=−

1 2

1 2

1 where

1nc jiE LE t

r r ek

+

−=

−=

Evaluatefield here

𝐸+ is the field traveling to the right (+z direction)

Fabry-Perot cavity modes2

2

2

0

2

0

2

2

1 22

1 2

2

2 2

1 2

22

1 2

2 1 2 1

1 0

22

1 2

2 2

1

22

2 0

2

1 2

2Transmission

2

1

1

1

(1 ) (1 )

1

(1 )(1 )

1

v

tt

inc inc inc

j

inc

j

cavi

inc

ca ityty

j

j

E t

SIT

I S E

E t tr r e

E

t t

r r e

r r

r r e

r r

r r e

+

−

−

− −

−

−

= = =

−=

=−

− −=

−

− −=

−

0

1

1

2

: Impedance of medium outsid

o

e cavity

: Impedance of m

: Time-average Poynting vector (intensity)

: transmission of mirror 1

: reflectivity of mirror 1

: transmission of mi

edium inside cavity

rr

cavity

S

t

r

t

2

2 1 2

1 0 1

2 1 2

2 0 2

r 2

: reflectivity of mirror 2

1

1

cavity

cavity

r

t r

t r

−

−

+ =

+ =

Assume that𝑟1,2 and 𝑡1,2are real forsimplicity

Fabry-Perot cavity modes

2

2

2

2

2

2

2

2 2

1

22

1 2

2 2

1

2 2

1 2 1 2

2 2

1

2 2 2

1 2 1 2

2 2

1

2

1 2 1 2

2 2

1

2

1 2 1 2

2

1

(1 )(1 )

1

(1 )(1 )

(1 )(1 )

(1 )(1 )

1 (

(1 )(1 )

1 2 cos 2 (

(1 )(1 )

1 2 1 2sin (

(1 )(

)

)

)

1

j

j j

j j

r r

r r e

r r

r e r

r r

r r e e r

r r

r r r

r

r

r r e

r

r

r

r

r r

r r

−

−

−

− −=

−

− −=

− −

− −=

− + +

− −=

− +

− −=

− − +

− −=

2

2 2

1 2 1 2 1 2

2 2

1

2 2

1 2 1 2

2

)

1 2 4 sin (

(1 )(1 )

(1 ) 4 sin

)r r

r kL

r r r r

r rT

r r r

− + +

− −=

− +

0

1

1

2

: Impedance of medium outsid

o

e cavity

: Impedance of m

: Time-average Poynting vector (intensity)

: transmission of mirror 1

: reflectivity of mirror 1

: transmission of mi

edium inside cavity

rr

cavity

S

t

r

t

2

2 1 2

1 0 1

2 1 2

2 0 2

r 2

: reflectivity of mirror 2

1

1

cavity

cavity

r

t r

t r

−

−

+ =

+ =

Fabry-Perot cavity modes

𝐿

Mirror 1 Mirror 2

𝑧

𝐸𝑖𝑛𝑐

𝐸𝑡

Evaluatefield here

Resonance condition: 2

0 r

s 0

m = intege2

in kL m

mL

n

= → =

=

Fabry-Perot cavity modes

𝜆𝑚 𝜆𝑚+1 𝜆𝑚+2 𝜆𝑚+3 𝜆𝑚+4

Tra

nsm

issi

on

𝑟 = 0.5𝑟 = 0.7

𝑟 = 0.99

2

2nL

Fabry-Perot cavity modes

𝜆𝑚 𝜆𝑚+1 𝜆𝑚+2 𝜆𝑚+3 𝜆𝑚+4

Loss / G

ain(𝒄𝒎

−𝟏)

αm + α𝑖

Gain

e.g. with simple gain curveThreshold is achieved for one Fabry-Perot mode.

Tra

nsm

issi

on

Population inversion

2E

1E

Spontaneous emission Absorption Stimulated emission

2 21

2 1

21 1B WNWN B

N N

21

21

1

2

Spontaneous emission rate

Stimulated emission / absorption rate

Electromagnetic energy density

Number of electrons in ground state

N Number of electrons in excited state

N Number of photonsp

A

B W

W

N

=

=

=

=

=

=

Gain will occur if stimulated emission > absorptionwhich implies that number of electrons in the excited state is higher than the ground state (i.e. population inversion)

2-level system

2E

1E

21

2121 1 21 2

dNA N B WN B

dtWN= − + −

Not physical,population inversion not possiblein a 2-level system (at equilibrium)!

2 1

21 21 21

21 21 21 21

21 21 21

For population inversion

2

2

B W B W

B B

W A B W

N N

AN N

A W A W

B

+

+

+ +

2 211 2 1 21 2

At equilibrium,

0 A N B WN B WN= − + −

1 2

21 211

21 21

21

21 2

2

1

Let

2

2

N N N

AN

A

B W

WB

B W

N

N NA WB

+

+

+

=

=

+

=

3-level system

2E

1E

0E

pu

mp

τ small

Lasing transition

Population inversion (𝑁2 > 𝑁1) is possible if a state with energy smaller than 𝐸1 is added; providedelectrons quickly decay from state 1 into state 0.In other words, 𝐴10 must be larger than 𝐴21.

Gain in a semiconductor

𝐹𝑐

𝐹𝑣

𝑘𝑐

𝑘𝑣

) ) / ]

() ) / ]x

1( )

1 exp[( (

1)

1 e p[( ( v v

c c

c c

v v

kE k

f

fF kT

kF kTE k

−

−

=+

=+

(1 ) (1 )c v p v c p

pdNB f f N B f f N

dt = − − −

Stimulated emission Absorption

transition rate coeffic

sdensity of stat

ient

e

B

=

=

)0 for ( )1 (1p

c v v c

dNf f f f

dt − −

c vF F − Bernard-DuraffourgCondition

More on this as we progress in the course