1 Taken from Lasers and Electro-Optics: Fundamentals and Engineering by Christopher Davis, Cambridge...

1

Taken from Lasers and Electro-Optics: Fundamentals and Engineering by Christopher Davis, Cambridge University Press, 1996

Ruby Laser

Crystal structure of sapphire: -Al2O3 (aluminum oxide). The shaded atoms make up a unit cell of the structure. The aluminum atom inside the dashed hexagonal prism experiences an almost cubic field symmetry from the oxygen atoms on the prism.

Schematic energy level diagram for ruby – Cr3+ ions in sapphire.

2

Ruby Laser: Absorption Spectra


Absorption coefficient and absorption cross-section as a function of wavelength for pink ruby. These absorption spectra are slightly different depending on whether the incident polarized light being absorbed is linearly polarized with its electric vector parallel, or perpendicular, to the c symmetry axis of the crystal.

Detailed absorption spectrum of pink ruby in the 686 – 702 nm region showing the absorption peaks corresponding to the R1 and R2 components of the ruby laser transition.

3

Ruby Laser

Simple electrical circuit for driving a flashlamp

Schematic energy level diagram of three- and four-level lasers


4

Ruby Laser: Pumping

Schematic arrangement of Maiman’s original ruby laser

Elliptical reflector arrangement for optical pumping a laser crystal by a linear flashlamp


5

Helium-Neon Laser: Pumping by Collision


Calculated variation of energy transfer cross-section for a collision between two atomic species as a function of the energy discrepancy E∞. The probability of excitation transfer is linearly dependent on the cross-section

6

Helium-Neon Laser: Energy Level Diagram


7

Helium-Neon Lasers


Schematic arrangement of the first gas laser.

Typical schematic design of a modern laser.

8

Introduction to Optical Electronics

Quantum (Photon) Optics (Ch 12)

Resonators (Ch 10)

Electromagnetic Optics (Ch 5)

Wave Optics (Ch 2 & 3)

Ray Optics (Ch 1)

Photons & Atoms (Ch 13)

Laser Amplifiers (Ch 14)

Lasers (Ch 15) Photons in Semiconductors (Ch 16)

Semiconductor Photon Detectors (Ch 18)

Semiconductor Photon Sources (Ch 17)

Optics Physics Optoelectronics

9

Putting it all togetherTheory of Laser Oscillation

Laser Amplification Medium

Optical Resonator

Laser

+

=

10

Population DifferenceDepletion of the steady-state population difference

0.1s

1s

10s

Wi

N0

N02

Po

pu

latio

n D

iffe

ren

ce

10 2 2

21

22 1

21

0

1

( )

1

1

( )

s

s i

i

N

N

N

R

W

W

201

21 2

1

2

1R

2R 1iW

nrspt

11

Population Inversion2 1N N N

Population Difference

Steady-State Difference

Saturation Time Constant*

Four-Level Laser

Three-Level Laser

0

1 s i

NN

W

0 1sp a

sp

t N WN

t W

1sp

ssp

t

t W

0

1 s i

NN

W

0

1

1a sp

sp

N t WN

t W

2

1sp

ssp

t

t W

*What is the small-signal approximation?

12

Amplifier Nonlinearity Gain Coefficient

0

2

0 0 0

( )( )

1 / ( )

where ( ) ( ) ( )8

s

sp

N N gt

Note: 0() is called the small-signal gain coefficient. Why?

0.01 0.1 1 10 100

s

1

0.5

0

13

Amplifier Nonlinearity Gain

0

0

(0) (0)ln ln

ln ln

0where and

s s s s

s s

z zz

Y Y X X d

dX Y

( )Gain

(0)

d Y

X

0 1 2 3 4 5 6Input X 0s

2

4

6

8

10

12

tuptuOY

d

s

0.001 0.01 0.1 1 10Input 0s

2

3

4

5

6

7

niaGd

00de

0d

Y X

0dY X e

14

0.001 0.01 0.1 1 10Input X 0s

0.2

0.3

0.4

0.5

0.6

0.7

0.8

ecnattimsnarTYXd

0

Saturable Absorbers( )

Transmittance = (0)

where ( ) 0 (i.e., attenuation)

Output Y d

Input X

0de

15

s10

s10s

00

2

Saturated Gain Coefficient

Sa

tura

ted

Ga

in C

oe

ffic

ien

t

small-signalregion

0 0

2

0

( ) ( )

( )8

1( )

sp

ss

N

N gt

0 ( )( ) (0) zz e

0( ) (0) sz z

0

( ) ( ) (0) (0)ln ln

s s s s

z zz

large-signalregion

small-signal:

large-signal:

0 ( )( )

1 / ( )s

16

Gain CoefficientInhomogeneously Broadened Medium

0

0 1

Gai

n C

oeff

icie

nt

s

17

0

0

0

0



20

0 0 0

( )( ) where ( ) ( ) ( )

1 / ( ) 8s sp

N N gt

2

0 2 2

0

2

0

/ 2Lorentzian: ( ) ( )

/ 2

where ( )4 sp

Nt

0( ) ( )

0

0

18

Optical Resonator

Optical Resonator

2RTPS 2 q k

c

1 2

1 1Loss Coefficient ln

2r s d

RR

1Photon Lifetime p

rc

Finesse 2 p Frd

F

max

2 2

II =

1 2 / sin / F F

q1 q q1

I

F F

2F

c

d

q1q 1q

Resonatorresponse

I

19

Conditions for Laser Oscillations

• Gain Condition: Laser Threshold

• Phase Condition: Laser Frequencies

0Threshold Ga n: )i ( r

0 00

1Since where

( )

( )r

tp

tNN Nc

N

Round-Trip Phase: 2 2 ( ) 2 1,2,3...k d d q q

20

Exercise 15.1-1Threshold of a Ruby Laser

a) At the line center of the 0 = 694.3-nm transition, the absorption coefficient

of ruby in thermal equilibrium (i.e., without pumping) at

T = 300 K is (0) = - (0) ≈ 0.2 cm-1. If the concentration of Cr3+ ions

responsible for the transition is Na = 1.58 x 1019 cm-3, determine the

transition cross section 0 = (0).

b) A ruby laser makes use of a 10-cm-long ruby rod (refractive index

n = 1.76) of cross-sectional area 1 cm2 and operates on this transition at 0

= 694.3 nm. Both of its ends are polished and coated so that each has a

reflectance of 80%. Assuming that there are no scattering or other

extraneous losses, determine the resonator loss coefficient r and the

resonator lifetime p.

c) As the laser is pumped, (0) increases from its initial thermal equilibrium

value of -0.2 cm-1 and changes sign, thereby providing gain. Determine the

threshold population difference Nt for laser oscillation.

21

s10

s 10s

0Saturated Gain Coefficient

(Photon-Flux Density)

r Loss Coefficient

Steady State

Laser Turn-On

Time

() Gain Coefficient

00

0

( )( )( ) 1 ,

( )0,

rsr

r

00

0

( ) 1 ,

0,

tst

t

NN N

NN N

Steady-state: ( ) r

22

Steady-State Population Difference

Nt2Nt

s

Pho

ton

Flu

x D

ensi

ty

N0

N

N0Nt

Nt

Pop

ulat

ion

Dif

fere

nce

Pumping Rate Pumping Rate

00

0

( )( )( ) 1 ,

( )0,

rsr

r

23

Output Flux Density vs. Transmittance

/ 2 , output photon-flux density2o

T

Laser

0

0

2

( )1 11

2 2

2 ( )11

2 2 ln(1 )

o sr

ss m

d

d

T T

TT

0.1 0.2 0.3 0.4

0.1

0.2

Transmittance

Out

put

Pho

ton-

Flu

x D

ens

ity

24

Characteristics of Laser Output

00

00

1 ,

1 ,

s tt

s tt

c

NN N

N

NN N

N

n

n

n

Internal Photon-Number Density

Output Photon Flux & Efficiency 0 e tR R V

1

1

1

where

1ln

2

if 1- 1

me

r

p

p

F

c

d

T

R

T T R

25

Laser Oscillations

0

0

0 ( )

B

2F

c

d

1 ... M

allowed modes

Number of possible modes: F

BM

Each mode's FWHM F

F

- Lossr

Resonator modes

- Gain

26

Exercise 15.2-1Number of Modes in a Gas Laser

A Doppler-broadened gas laser has a gain coefficient with a Gaussian spectral profile given by

where is the FWHM linewidth.

• Derive an expression for the allowed oscillation band B as a function of D and the ration 0(0)/r where r is the loss coefficient.

• A He-Ne laser has a Doppler linewidth D = 1.5 GHz and a midband

gain coefficient 0(0) = 2 x 10-3 cm-1. The length of the laser

resonator is d = 100 cm, and the reflectances of the mirrors are 100% and 97% (all other resonator losses are negligible). Assuming that the refractive index n = 1, determine the number of laser modes M.

202

( )

20 0 0( ) ( ) De

8ln 2D D

27

Homogeneously Broadened Medium

0

1

( )( )

1 / ( )M

j s jj

0

0

0

0

0

0

0 ( )

r

1 ... M

0 ( ) 0 ( )

( ) ( )

28

Inhomogeneously Broadened Medium

q1 q q1 q

Typical Doppler

0 ( )

r

s

( ) 0 ( )

r

s

( )

29

Doppler Broadening

Laser Line (atomic)

Transverse Mode

Polarization

BrewsterWindow

30

0

0

Longitudinal Mode Selection

Etalon

d1

d

12

c

d

2

c

d

Resonator Modes

Etalon Modes

Laser Output

Gain

31

How to Pulse Lasers

Modulator

Peak Power

AveragePower

t

Modulator

32

Pulsed Lasers

Gain Switching tt

Gain

Loss

t

LaserOutput

Pump

Q-SwitchingModulatedabsorber

t

LaserOutput

t

Loss

Gain

33

Gain Switched Laser

34

Q-Switching

35

Pulsed Lasers

Cavity Dumping

t

LaserOutput

t

Gain

Loss

MirrorTransmittance

Mode LockingOptical

Modulator

36

Mode-Locked Laser

M = 5 M = 15 M = 25

TF

FT

M

FT

M

MI

22

2

sin /( , ) | |

sin /F

F

M t TI t z A

t T

0

2 ( / )

2 ( / ) 2 /

( , ) where 0, 1, 2...

( / ) where ( / )

q

F

j t z c

qq

j t z c j t Tq

q

U z t A e q

t z c e t z c A e

A A

37

Exercise 15.4-3Demonstration of Pulsing by Mode Locking

Write a computer program to plot the intensity I(t)=|A(t)|2 of a wave whose envelope A(t) is given by the sum

Assume that the number of modes M = 11 and use the following choices for the complex coefficients Aq.

a) Equal magnitudes and equal phases.

b) Magnitudes that obey the Gaussian spectral profile|Aq| = exp[-1/2 (q/5)2] and equal phases.

c) Equal magnitudes and random phases (obtain the phases by using a random number generator to produce a random variable uniformly distributed between 0 and 2.

2( ) exp( )q

q F

jq tA t A

T

38

1 2 3 4

2

4

6

8

10

12

1 2 3 4

20

40

60

80

1 2 3 4

20

40

60

80

100

120

(a) Equal magnitudes and equal phases.

(b) Magnitudes that obey the Gaussian spectral profile and equal phases.

(c) Equal magnitudes and random phases (obtain the phases by using a random number generator to produce a random variable uniformly distributed between 0 and 2.

1 Taken from Lasers and Electro-Optics: Fundamentals and Engineering by Christopher Davis, Cambridge...

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Transcript of 1 Taken from Lasers and Electro-Optics: Fundamentals and Engineering by Christopher Davis, Cambridge...