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
Taken from Lasers and Electro-Optics: Fundamentals and Engineering by Christopher Davis, Cambridge University Press, 1996
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
Taken from Lasers and Electro-Optics: Fundamentals and Engineering by Christopher Davis, Cambridge University Press, 1996
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
Taken from Lasers and Electro-Optics: Fundamentals and Engineering by Christopher Davis, Cambridge University Press, 1996
5
Helium-Neon Laser: Pumping by Collision
Taken from Lasers and Electro-Optics: Fundamentals and Engineering by Christopher Davis, Cambridge University Press, 1996
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
Taken from Lasers and Electro-Optics: Fundamentals and Engineering by Christopher Davis, Cambridge University Press, 1996
7
Helium-Neon Lasers
Taken from Lasers and Electro-Optics: Fundamentals and Engineering by Christopher Davis, Cambridge University Press, 1996
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
Laser Amplification Medium
Laser Amplification Medium
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
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.