Quantum electronics

Introduction to laser theory

lightmatter

M. FoxQuantum Optics – An IntroductionChapter 4Ed. Oxford University Press (2006)

P. Meystre and M. Sargent IIIElements of Quantum OpticsChapter 6Ed. Springer Verlag (1990)

Laser

active medium

(R1=1) (R2<1)

output

LLc

power supplyhigh reflector output coupler

Arrangement of mirrors to form an optical cavity

Gain medium inserted into the cavity to compensate for the lossesthrough stimulated emission of radiation

Gain balancing losses yields oscillation/laser action

Light Amplification by Stimulated Emission of Radiation

Properties of laser light

Directionality

Monochromaticity ω0ω

( )I ω

Brilliance

k

Coherence

dIdΩ

high

narrow

( )I ω

well-defined

k

Low phase drift( )tδφ

sin ( )kz t tω δφ− +⎡ ⎤⎣ ⎦

Gain medium (1/2)

Two-level transition of gain medium interacting with

the (laser) light beam

Total rate of stimulated emission:

Total rate of absorption:

Net rate of stimulated emission:

Net rate of photon emission:

→= −Γ&a a b aSE

N N

→ →= Γ = −Γ&a b a b a b bABS

N N N

( )→= −Γ −&a a b a bN N N

( )p a a b a bN N N N→= − = Γ −& &

Light amplification requires POPULATION INVERSION ( )∆ = − > 0a bN N N

a

b

0ωh

ωh

Gain medium (2/2)

stimulated-emission rate constant

2

0

( )a b

π ω ω ηε→℘

Γ = ⋅ ⋅ ⋅h

l

a b tN gη η→= Γ ⋅ ∆ = ⋅&2

0

( , ) ( )tg N Nπω ω ω

ε℘

∆ = ⋅ ∆ ⋅ ⋅h

l

gain rate constantη η ⋅=( ) (0) tg tt e

20 0

12

U Eε ω η= = ⋅h e.m. energy density (J·cm-3)

η = photon density

( )222(1) 02

sin /2( ) ( )

4 / 4a b b

tP t C t

δδ→

Ω= = ⋅

(0) 1aC =

0 0( ) ( ) ( )a b a bP t P t dω ω→ →→ ⋅∫ l

20 ( )

2a b

a b

dPdt

π ω→→Γ = = Ω l

spectral lineshape( )0 0( ) 1dω ω =∫ l

a

b

a

b

0ωh

ωh

Pumping population inversion

Two-level system with pumping to and decay from both levels

( )

( )

a a a a a b a b

b b b b a b a b

N N N N

N N N N

ω

ω

κ γκ γ

→

→

= + − − Γ −⎡ ⎤⎣ ⎦= + − + Γ −⎡ ⎤⎣ ⎦

&

&

Equations of motion in rate-equation approximation

(very short coherence time γ −1):

20 2 2

0

1( ) ( ), ( )

2 ( )a b

π γω ω ωπ ω ω γ→Γ = Ω =

− +l l where

In steady state :( 0)a bN N= =& &

ω∆

∆ =+ ⋅ l

0

1 ( )

NN

I unsaturated population difference

κ γ κ γ− −∆ = ⋅ − ⋅1 10 a a b bN

( )γ γ− −= +1 11

12 a bT2

0 1I Tπ= Ω

Lorentzian lineshape

a

b

aγ

bγ

κa

κb

0ωh

ωh

Pumping of n-level lasers

three-level laser four-level laser

Pumping:- optical- electrical

fast fast

fast

b

a

a’

0a bN N− >

a

b

a’

b’

Photons emitted (spontaneously) into a laser cavity mode initiate

light amplification through stimulated emission

ground state ground state

Gain spectrum

0( , ) ( ) ( )g N ω ω ω ω∆ ∝ ⋅ l l

Unsaturated gain spectrum

hom 0 2 20

1( )

( )γω ω

π ω ω γ− =

− +l

Inhomogenous (e.g., Gaussian) lineshape

( ) ( )2

0inh 2

1exp

22 ωω

ω ωω

σσ π

⎡ ⎤−⎢ ⎥= −⎢ ⎥⎣ ⎦

l

Homogeneous (Lorentzian) lineshape

( ) ( ) ( )gain inh hom dω ω ω ω ω′ ′ ′= −∫l l l

Gain lineshape given by spectral convolution (Voigt profile):

ω0ω

0( , )g N ω∆

homl

inhl

Laser cavity

Linear laser configuration

Active medium between the

reflectors

Brewster windows help enforce

conditions that only one field

polarization exists in the cavity

Ring laser configuration

Many other configurations available

z

Cavity modes (1/2)

( , ) ( ) )n n nn

E z t E z tω φ= +∑ cos(

LONGITUDINAL modes

0( )( ) sin( )n n nE z E k z=

The total phase acquired in a cavity round-trip must be equal to an

integer multiple of 2π:

Electromagnatic field in the laser cavity written as a superposition of

plane waves:

nC

k nLπ

= 2nC

k nLπ

=

two-mirror laser ring laser

n

rk ncω

=

rn = refractive index

dispersion

relation

Field dependence on the tranverse coordinates (x,y) --- Gaussian beams

2 2

2( , , ) ( ) ( ) ( ) exp( )njk n j k

x yE x y z E z H x H y

w z⎛ ⎞+

⋅ ⋅ ⋅ −⎜ ⎟⎝ ⎠

πλ

= =2

00

22

wb z

Cavity modes (2/2)TRANSVERSE modes

= +2

0 20

( ) 1z

w z wz

beam waist

Rayleigh length

x

y

Cavity loss: Q-factor

Electromagnetic energy stored in the cavity decays in time due to

internal losses and mirror losses

quality factor/n

n nn

UQ

dU dtω≡ ⋅

n nn

n

dUU

dt Qω⎛ ⎞

= − ⋅⎜ ⎟⎝ ⎠

p,/( ) (0) n

n n

tU t U e

τ−=

p,n

nn

Qτω

= photon decay time constant

The electric field decays in time according to the equation:

p,/2 )(( ) (0) n n nn n

tt iE t E e eτ φω +− −=

2

2 2

1( )

( ) ( /2 )nn n n

EQ

ωω ω ω

∝− +

%(FWHM)

nn

n

Qω

δω=

Laser threshold (time domain)

Intensity builds up exponentially before gain gets saturated:

0( )t tgain lossg Nη η η α η= + = ∆ −⎡ ⎤⎣ ⎦& & &

( )0( ) exp ( )t tt g N tη α⎡ ⎤∝ ∆ −⎣ ⎦

net gain [s-1]

1pt Q

ωα τ −= =

loss [s-1]

( )t tg N αΓ ⋅ ∆ =0gain lossgain lossV Vη η⋅ + ⋅ =& &

gain

loss

V

VΓ = optical confinement factor

modal gain [s-1]

Steady-state laser action is reached when the (saturated) gain in the

active medium equals the losses:

Laser threshold (space domain)

Round-trip amplification

Threshold condition:

1k kI A I+ = ⋅ 1 2 intexp 2 exp 2 cA R R gL Lα= ⋅ ⋅ −⎡ ⎤ ⎡ ⎤⎣ ⎦ ⎣ ⎦where

1A = mod int mirr int1 2

1 1ln

2c c

Lg g

L L R Rα α α

⎛ ⎞= ⋅ = + = +⎜ ⎟

⎝ ⎠

mirror loss[cm-1]mirr

1 2

1 1ln

2 cL R Rα

⎛ ⎞= ⎜ ⎟

⎝ ⎠

intα internal loss [cm-1]

modg g= Γ ⋅ modal gain[cm-1]

/ cL LΓ = confinement factor

Lc

R1 R2

L

Connection between time and space domains

RR 1 2 intexp[ ] 1 exp 2 1t cT R R L

η α αη

∆= − − = − −⎡ ⎤⎣ ⎦

Losses

2 r cR

n LT

c= round-trip time

1 11 1 1[ ] [cm ]t

r r

d d dz c d cg s g

dt dz dt n dz nη η η

η η η− −= ⋅ = ⋅ ⋅ = ⋅ ⋅ = ⋅

Gain

fractional round-trip loss

1 1[cm ] [s ]rt

ng g

c− −= ⋅

-1 1mirr int[cm ] [ ]r

t

ns

cα α α α −= + =

Laser characteristics

Pout

PinPth

Slope efficiency (SL)

out

in

PSL

P∆

=∆

int extSL η η= ⋅ mirrext

mirr int

αηα α

=+

internal efficiencyintη =

extη = extraction efficiency

Large increase in slope efficiency (SL) above lasing threshold

Population inversion ∆N (and thus gain) gets clamped to the value

at threshold

Laser types and emission wavelengths

Gas lasers (He-Ne, CO2, ...)

Dye lasers (Coumarin, ...)

Chemical lasers (HF, DF, ...)

Solid-state (Ti:Sapphire, Nd:YAG, ...)

Semiconductor lasers (AlGaAs, GaN, ...)

Other lasers (metal-vapour, free-electron, ...)

Types of laser operation (1/2)

CONTINUOUS-WAVE (CW)

nI

ωnω

CW SINGLE-MODE

jI

ω

j jr r c

c ck j

n n Lπω = =Frequencies of the

longitudinal modes

( )g ωα

CW MULTI-MODE

( ) cos( )n n nE t E tω φ= + ( ) cos( )j j jj

E t E tω φ= +∑2

n nI E∝ 2j

j

I E∝ ∑ random phases)( jφ

Types of laser operation (2/2)

PULSED OPERATION

Q-switching

Switching the cavity Q-factor from low to high value results in fast

depletion of the energy stored in the active medium with consequent

giant pulse emission (duration down to 1 ns)

Mode-locking

( ) cos( )j j jj

E t E tω φ= +∑

Oscillation modes are phased locked to one another to yield a train of

short pulses at time intervals equal to the cavity round-trip time

1 ( )j jfφ φ+ =with

t t+TRt

R 2 /r cT n L c=

outp

ut

2 0.44( )

( )t FWHM

FWHMπ

ω⋅

∆ =∆

Time-bandwidth limit for

Gaussian pulses:

Example of laser device

threshold

Longitudinal modes with

free spectral range (FSR):

1c

n nk k kLπδ −= − =

2

2

22

r

r cn

nk

n Lk

π λδλ δ =

TEM image

cavi

ty

GaN-based monolithic laser

S. Park et al., Appl. Phys. Lett. 83, 2121 (2003)