Chapter 1.

Laser: Theory and Applications

Reading: Siegman, Chapter 6 and 7

Laser Basics

Stimulated emission

1E

0EPopulation inversion (N1 > N0) laser, maser

Light Amplification by Stimulated Emission of Radiation

Gain medium High reflector Out coupler

Optical cavity

Pumping (optical, electrical, etc.) for population inversion

Transition rate

for stimulated emission )()(

)(01

2

01012

01

01

0

2

2

2

014

4

IM

Ie

cmW

Incident light intensity

ℏ𝜔

ℏ𝜔01 = 𝐸1 − 𝐸0

Ruby Laser

http://web.phys.ksu.edu/vqm/laserweb/Ch-6/C6s2t1p2.htm Energy level diagram of Ruby laser

• This system is a three level laser with

lasing transitions between E2 and E1.

• The excitation of the Chromium ions is

done by light pulses from flash lamps

(usually Xenon).

• The Chromium ions absorb light at

wavelengths around 545 nm (500-600 nm).

As a result the ions are transferred to the

excited energy level E3.

• From this level the ions are going down to

the metastable energy level E2 in a non-

radiative transition. The energy released

in this non-radiative transition is transferred

to the crystal vibrations and changed into

heat that must be removed away from the

system.

• The lifetime of the metastable level (E2) is

about 5 msec.

He-Ne Laser

http://en.wikipedia.org/wiki/Helium-neon_laser Schematics

Energy level diagram

Applications

• Marking

• Micromachining

• Laser Ablation

• Laser Annealing

• Surface Structuring

• Laser Vision Correction

• Optical Testing and Inspection

• Pulsed Laser Deposition

• Fiber Bragg Gratings

Excimer Lasers

http://tftlcd.khu.ac.kr/research/poly-Si/chapter4.html

• Gain medium: inert gas (Ar, Kr, Xe etc.)

+ halide (Cl, F etc.)

• Excited state is induced by an electrical

discharge or high-energy electron beams.

• Laser action in an excimer molecule

occurs because it has a bound

(associative) excited state, but a repulsive

(disassociative) ground state.

Excimer Wavelenth

Fe 157 nm

ArF 193 nm

KrF 248 nm

XeCl 308 nm

Semiconductor Lasers – laser diodes

en.wikipedia.org/wiki/Laser_diode, www.mtmi.vu.lt/pfk/funkc_dariniai/diod/led.htm

Diode laser structure Vertical cavity surface emitting lasers

(VCSEL) structure

Band structure near a semiconductor p-n junction

e h e h

forward bias off forward bias on

Specific Laser Systems Laser media:

gas, dye, chemical, excimer, solid-state, fiber, semiconductor, free-electron

http://en.wikipedia.org/wiki/Image:Laser_spectral_lines.png

Longitudinal Modes in an Optical Cavity

Boundary condition:

:

:

L

EM wave in a cavity

mc

mc

mL

22

mL

cm

L

c

m

L ,,

2

2

m = 1, 2, 3, …

Round-trip time of flight: mc

LT

2

Typical laser cavity: L = 1.5 m, = 0.75 mm

nsec secsec/m

m1010

103

32 8

8

c

LT

!! milion m.

m4104

10750

32 6

6

Lm

MHz Hz 100101 8 T

R

𝜆 = 2𝐿

𝜆 = 𝐿

𝜆 =2𝐿

3

Single mode

22

0 tttI coscos)(

-100 -50 0 50 100 Time

Inte

nsity

Frequency 1

Inte

nsi

ty

Spectrum

Cavity Quality Factors, Qc

End mirror

R 1

Out coupler

T = 1 ~ 5 %

L

Energy loss by reflection, transmission, etc.

t

tiTteeE c 02/

0

Ti

E

dteeeEE

c

titiTtc

2/)(

)(

0

0

0

2/

00

222

0

2

02

4/)()(

T

EE

c

Emission spectrum

0 c

c

QT

~

2)(E Lorentzian

Energy of circulating EM wave, Icirc(t)

,exp)()(

T

tItI ccirccirc 0

Number of round trips in t

: round-trip time of flight c

LT

2

t

QItI

c

circcirc

exp)()( 0

where cc

c

LTQ

14

Q-factor of a RLC circuit

R

LQ

Typical laser cavity: L = 1 m, = 0.8 mm, c = 0.01 (~1% loss/RT)

9

61061

010

1

1080

4

.

..

cQ Hz610

)()()(

)()()(

)()(

2212112

22121112

21

tNtNtNW

tNWtNW

dt

tdN

dt

tdN

Two Level Rate Equation

Total number of atoms: constant)()( 21 tNtNN

Population difference: )()()( 21 tNtNtN

)()()()()()(2

)(2)()(2)(

2121212112

2212112

tNtNtNtNtNtNW

tNtNtNWtNdt

d

Two Level Rate Equations and Saturation

1E

2E

221NW 221N

2N

1N

112NW

: non-radiative decay rate 1

21

1

T

1

12

)()(2)(

T

NtNtNWtN

dt

d

Steady-State Atomic Response: Saturation

1

12

)()(20)(

T

NtNtNWtN

dt

d

11221 TW

NNN ss

0.01 0.1 1 10 0.0

0.2

0.4

0.6

0.8

1.0

1.2

N

N ss

TW122Signal strength,

Saturation

Intensity, I=Isat

112 2

1,

/1

1

TW

WWN

Nsat

sat

ss

Gain coefficient

in laser materials

Nm

sat

mm

III

/1)( 0

Transient Two-Level Solutions

11

12

1

12 )(1

2)(

)(2)(T

NtN

TW

T

NtNtNWtN

dt

d

Initial population difference at t = 0: ANN ss )0(

1

112

1212

21exp21

)0(21

)(T

tTW

TW

NN

TW

NtN

,1

2exp)(1

12

t

TWANtN ss

TW

NNss

1221

0 1 2 3 0.0

0.5

1.0

N

tN )(

1/Tt

Transient saturation behavior

following sudden turn-on of an

applied signal

02 112 TW

5.0

1

2

4

Effective signal-stimulated transition probability: 21122

1WWWeff

1

)()(2)(

T

NtNtNWtN

dt

deff

Rate equation:

Atomic time constants: T1 and T2

T1 : longitudinal (on-diagonal) relaxation time

population recovery or energy decay time

T2 : dephasing time, transverse (off-diagonal) relaxation time

time constant for dephasing of coherent macroscopic polarization

Two-Level Systems with Degeneracy

1E

2E 2g

1N

1g

2N

)()()( 21

1

2 tNtNg

gtN

Population

difference:

Stimulated transition rates: 212121 WgWg

)()()()()(

2212111221 tNWtNW

dt

tdN

dt

tdN

Rate equation:

Steady state population:

,1

141

4

4

4 NWNW

WN p

p

p

if 14 pW

Normalized pumping rate

Rate equation for level 4

pWWW 4114

,/)(

)()(

4441

4414243414

NNNW

NNNWdt

dN

p

p

4142434

4

1

where

pump transition probability,

Steady-State Laser Pumping and Population Inversion

Four-level pumping analysis

E1

E2

E3

E4 43 (fast)

32

(slow)

21 (fast)

Wp

pumping

21

2

32

3

42

4221332442

2

NNNNNN

dt

dN

33

342

2143

32

212 NNN

342

2143

32

21

where

If 1, 32 NN : population inversion on the 3 2 transition

In a good laser system, 042 so that the level 4 will relax primarily into the level 3.

32

21

condition for population inversion 1

32

21

3

2

N

N

In a good laser system, 43 NN 433 so that

Rate equations for level 2 and 3

3

3

43

433132443

3 )(

NN

NNdt

dN

at steady state

4

43

33 NN

E1

E2

E3

E4 43 (fast)

32

(slow)

21 (fast)

Wp

pumping

Fluorescent quantum efficiency

rad

rad

3

43

4

34

43

The number of fluorescent photons spontaneously emitted on the laser transition

divided by the number of pump photons absorbed on the pump transitions when

the laser material is below threshold

Fraction of the total atoms excited to

level 4 relax directly into the level 3

Fraction of the total decay out of level

3 is purely radiative decay to level2

Four level population inversion

radprad

radp

W

W

N

NN

/211

)1(

43

23

4321 NNNNN

In a good laser system, 43 rad

radp

radp

radp

radp

W

W

W

W

N

NN

1)1(1

)1(23

, 0. 3-level system

0 1 2 3

-N

0

N

N

Wprad

4-level system

Nth

Laser gain saturation analysis

E0

E1

E2

E3 32

21

1

Wp

pumping

N0N

N1

N2

N3

Wsig

stimulated

Pumping transition

NWNWNNWdt

dNppp

pump

0303 )(

0NWR ppp Effective pumping rate:

quantum efficiency E3 E2

22122 )( NNNWR

dt

dNsigp

Rate equations for laser levels 1 and 2:

11221121 )( NNNNW

dt

dNsig

20212

p

sig

sigR

W

WN

21201

21

1)(

p

sig

sigR

W

WN

21201

1

2)(

Gain saturation behavior

effsig

sig

p

WN

WRNNN

1

1

/)(1

1

0

2120121

2111221

Small signal or unsaturated

population inversion 2

21

1

21

2110 1

pp RRN

Effective recovery time

201

211

eff

20

12 1

eff

or

If 20 0, 2 21 2

12211

1)(

sig

pW

RN

• Population inversion requires 21 1.

•

• If 1 0, eff 2.

• The saturation intensity of the inverted population is independent of Rp.

)/1( 2120 pRN

Wave Propagation in an Atomic Medium

Wave equation in a laser medium

0),,(/122 zyxEiat m

Ohmic loss atomic

susceptibility

0)(/12

2

2

zEi

dz

dat

Plane wave approximation

“Free-space” propagation constant:

m

nn

c

2

Propagation factor:

/122 iat

zeEzE 0)(

/)()(1/1 iiiii at

1/, iatUsually,

0)()(2

1

2)(

2

1)(

2

1

2)(

2

1)(

2

11

mmii

vii

iii

Propagation of a +z traveling wave

zzitiEtzE mm 00 )()(expRe),(

The effects of ohmic losses and atomic transition are included.

Phase shift

by atomic transition

Gain by atomic transiton

+ ohmic loss

22222

22222

22

22222

22

22)(

a

a

a

a

a

a

at

ai

a

ia

i

a

)(

)(

atomic phase-shift contribution zm )(

a

Propagation factors

zz mtot )(),(

2

total phase shift

“free-space”

Contribution

z

2

0

22222

22

/21

/)(2

)(

a

a

a

am

a

a

atomic gain zm )(

2

0

/21)(

a

m

= 𝑛 𝜔𝜔

𝑐𝑧

Single-Pass Laser Amplification

Gain medium 0z Lz

Laser gain formulas

Complex amplitude gain: LLiE

LEg mm 0)()(exp

)0(

)()(

Total phase shift amplitude

gain or loss

Lorenzian transition line shape: 2

0

/)(21)(

aa

Midband value

Power or intensity gain: LgI

LIG m 0

2)(2exp

)0(

)()(

)(2

1

Power gain:

2

0

/)(21

1exp)(

aav

LG

Power gain in decibels (dB):

)(34.4

)(ln34.4)(log10)( 10

v

LGGG a

dB

Amplification bandwidth and gain narrowing

3-dB amplifier bandwidth: 3)(

33

adB

adBG

Amplifier phase shift

)(2

)(),(

L

v

LLz mtotTotal phase shift:

2

10 /)(21

/)(2

log20

)()(2)(

aa

aaadBm

a

am

e

GLL

Atomic transition phase shift:

-2 0 2

)(G

aa /)(2

dB3

Saturation Intensities in Laser Materials

Saturation of the population difference

sateff IIN

WNN

/1

1

1

100

INIdz

dIm 2Traveling wave:

Stimulated transition cross-section

Population difference:

sat

mm

IzIz

dz

zdI

zI /)(1

2)(2

)(

)(

1 0

002 Nm where

L

m

II

IIsat

dzdIII

out

in 002

11

00 ln2ln GLI

II

I

Im

sat

inout

in

out

where )2exp( 00 LG m

unsaturated

power gain

sat

out

sat

in

sat

inout

in

out

GI

IGG

I

IGG

I

IIG

I

IG

)1(exp

)1(exp

exp

00

0

Overall power gain:

G

G

GI

I

sat

in 0ln1

1

G

G

G

G

I

I

sat

out 0ln1

and

Power extraction and available power

G

GIIII satinoutextr

0ln

satmsatsatG

avail ILGIG

GII

00

0

12lnlnlim