EE485 Introduction to Photonics Laser Operation 1. Rate equations 2. Gain media 3. Steady-state laser operation 4. Laser line broadening 5. Pulsed operation

Reading: Pedrotti3, Chapter 26 Ref: Verdeyen, “Laser Electronics,” 3rd ed. Sec. 9.4-9.5

Lih Y. Lin

Transition Cross Section

2

Effective absorption and emission cross sections of ytterbium-doped germanosilicate glass, as used in the cores of ytterbium-doped fibers.

21 ( ') '/B g h cσ = ν νStimulated emission cross section

Absorption cross section

2 1( / )abs g gσ = σ

Exercise: The cross section σ, for a transition from the ground state to an excited state that is resonant with an EM field of λ = 808 nm, for a neodymium (Nd) atom doped into a YAG (yttrium aluminum garnet) crystal is ~3 × 10−20 cm2. Assume that the dopant density of Nd in the YAG crystal is 1020 atoms/cm3 and that the YAG crystal itself is transparent to 808-nm light. A diode laser with λ = 808 nm is used to pump an Nd:YAG laser rod. (a) Estimate the small-signal absorption coefficient for the Nd:YAG crystal. (b) Estimate the depth of penetration into to the Nd:YAG crystal at which the intensity of the diode laser beam attenuates to 1/e of its initial value. (c) One can define saturation intensity 𝐼𝑆,𝑎𝑎𝑎 for an absorptive medium as the intensity for which the loss coefficient α is reduced by a factor of 2 from its small-signal value. Find the expression for 𝐼𝑆,𝑎𝑎𝑎.

Absorption coefficient

2 1( )N Nα = −σ −dI Idz

= −α

Lih Y. Lin

Four-Level System 3

3 3 3 0( )P P

P

dN IN N Ndt h

σ= −κ − −

ν2

32 3 2 2 2 1( )'

dN IN N N Ndt h

σ= κ − κ − −

ν

131 3 21 2 10 1 2 1( )

'dN IN N N N Ndt h

σ= κ + κ − κ + −

ν

030 3 20 2 10 1 3 0( )P P

P

dN IN N N N Ndt h

σ= κ + κ + κ + −

ν

Rate equations:

( )0 1 2 3 0d N N N Ndt

+ + + =

3 32 31 30κ = κ + κ + κ 2 21 20κ = κ + κ

Gain coefficient 2 1( ) invN N Nγ = σ − = σ

dI Idz

= γ The gain coefficient depends on intensity I since the population inversion is, in general, dependent on the intensity.

Lih Y. Lin

Exercise: 4-level amplifying system

4 4

κ32 = 108/s, κ21 = 1000/s, κ10 = 108/s, κ30 = κ31 = κ20 ~ 0 σP = 3x10-19 cm2, σ = 10-18 cm2 λ30 = 400 nm, λ21 = 600 nm (in free space) NT = 1.5x1026/m3

(a) Rate equations for the population densities of the levels. (b) Steady-state small-signal population inversion as a function of the pump intensity. (c) Pump intensity IP required to sustain a steady-state population inversion. (d) Pump intensity IP required to sustain a small-signal gain coefficient of 0.01/cm. (e) Pump intensity IP required to sustain a small-signal gain coefficient of 1/cm. (f) Compare N0 to N1, N2, and N3 for the pump intensities in (d) and (e). (g) Use the ideal four-level gain medium relation together with the definition of the effective density under undepleted pump approximation, estimate the pump intensity required to sustain a small-signal gain coefficient of 0.01/cm and 1/cm. Compare these results to those obtained in (d) and (e).

Lih Y. Lin

Steady-State Ring Laser Output

5

Consider a ring laser cavity. Mirrors M1 and M2 have reflectances R1 = R2. Let the gain medium be homogeneously broadened and have length L = 10 cm and a saturation intensity (at the lasing frequency) of Is = 2000 W/cm2. (a) Find the threshold gain coefficient if R1 = R2 = 1, R3 = 0.95, and T3=0.05. (b) If the small-signal gain coefficient is twice the threshold value, find the intensity

of the output. (c) Assuming that the laser beam has cross sectional area A = 0.1 cm2, find the

output power of the laser. (d) Assuming that the overall efficiency of this laser system is 5%, find the pump

power required to operate this laser system.

Lih Y. Lin

Exercise: Four-Mirror Ring Laser Cavity

6

Analyze the intensity change during a round trip in a four-mirror ring laser cavity.

Lih Y. Lin 7

Two-mirror Linear Laser Cavity Threshold gain

1 2

1 1ln2th L R R

γ = α +

Output intensity

01 22

1 2 2 1

1(2 ) ln

2 (1 )(1 / )S

out

LR RT II

R R R R

γ

− =

− +

Exercise: Assume saturation intensity IS = 10 W/cm2 and α = 0. Design a laser cavity by picking a high reflectivity R1 for the back mirror and a cavity length L, then write a computer program to plot Iout versus R2 for R2 ranging from 0.1 to 1. Do this consider two situations: (1) The laser medium is pumped at a rate that leads to a small-signal gain γ0 coefficient 5 times of the threshold gain (therefore it’s a function of R2). (2) The laser medium is pumped at a rate that leads to a fixed small-signal gain coefficient γ0 5 times of the threshold gain value when R2= 0.2. Discuss the difference in the results.

Lih Y. Lin

Homogeneous and Inhomogeneous Broadening

8

2 20

( )2 ( ) ( / 2)

H

H

g ∆νν =

π ν − ν + ∆ν 2

04ln(2)[( )/ ]2

4ln(2)( ) D

D

g e− ν−ν ∆ν ν = π∆ν

1/2

028 ln(2)B

Dk T

Mc ∆ν = ν

Exercise: A typical Ar+ laser has a cavity length of 1 m and a center wavelength λ0 = 488 nm. The atomic mass of an Ar atom is M = 6.64x10-26 kg. Assume the temperature of the gas under the operating condition is 3000 K. How many longitudinal modes can exist within the FWHM of the inhomogeneous broadened lineshape function?

Lih Y. Lin

Relaxation Oscillation

9

Laser turn-on and approach to steady state.

22 2 2 2 2 2 2 2'p p P

dN IR N N R N cN Ndt h

σ= − κ − = − κ − σ

ν

' PI h cN= ν

2gP

P Pc

VdN N cN Ndt V

= −Γ + σ :Γ Cavity loss rate / :g cV V Ratio of the gain cell length to the perimeter of the laser cavity

18 210 cm−σ =8 110 s−Γ =

7 12 10 s−κ =

0 / 2thγ γ =

/ 0.3g cV V =

Lih Y. Lin

Q-Switching

10

Ref: (1) Verdeyen, “Laser Electronics,” Prentice Hall (2) Siegman , “Lasers,” University Science Books

Laser Q-switching techniques

Lih Y. Lin

Exercise: Q-Switched Laser

11

Assume a ruby laser (λ = 694.3 nm) to be pumped to four times threshold. For simplicity we assume equal degeneracy of the lasing states even though that is not true for ruby. We also assume a residual attenuation of 0.1 cm-1 by the switch medium even when it is in its high transmission state. The laser is operated at Q-switching mode. Calculate: (a) Threshold gain, inversion density and total # of

inverted atoms at threshold (b) Maximum photon # in the cavity (c) Photon lifetime in the cavity (d) Output power at the maximum of the pulse

1P P

P th

dN N ndt n

= − τ

2 P

th P

dn n Ndt n

= − ⋅τ

Lih Y. Lin

Mode-Locked Laser

12

0 0( ) (t)exp(i )E t E t= ω

[ ]( 1)/2

0( 1)/2

( ) ( )exp i( ) ( )N

n c nN

E t E t n t t+ −

− −= ω + ω + φ∑

00

1 exp( )( )1 exp( )

i t c

c

iN tE t E ei t

ω − ω=

− ω2 2 2

02

( ) sin ( / 2)( )2 2 sin ( / 2)

c

c

E t E N tI tt

ω= =

η η ω

2 /RTT d c= τ =

1Pt∆ ≈

∆ν

Lih Y. Lin

Mode-Locking

13

Out of phase

RANDOM phase for all the laser modes Irradiance vs. Time

Time Time

Out of phase Out of phase In phase

LOCKED phases for all the laser modes

Lih Y. Lin

Bandwidth vs. Pulsewidth

14

narrow spectrum

continuous wave (CW)

broader spectrum

pulses (mode-locked)

broadest spectrum

shortest pulses

bandwidth ∆ν

duration ∆τ

∆ν∆τ = const.

Lih Y. Lin

Various Types of Laser Output

15

Pow

er

Time

Pow

er

Time

cw cw ML

Pow

er

Time

Q-switch Pow

erTime

Q-sw.ML