Transcript Ruby Laser Crystal structure of sapphire: -Al2O3 (aluminum oxide). The shaded atoms make up a unit cell of the structure.

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.
Taken from Lasers and Electro-Optics: Fundamentals and Engineering by Christopher Davis, Cambridge University Press, 1996
1
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.
Taken from Lasers and Electro-Optics: Fundamentals and Engineering by Christopher Davis, Cambridge University Press, 1996
2
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
3
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
4
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
Taken from Lasers and Electro-Optics: Fundamentals and Engineering by Christopher Davis, Cambridge University Press, 1996
5
Helium-Neon Laser: Energy Level Diagram
Taken from Lasers and Electro-Optics: Fundamentals and Engineering by Christopher Davis, Cambridge University Press, 1996
6
Helium-Neon Lasers
Schematic arrangement of the
first gas laser.
Typical schematic design
of a modern laser.
Taken from Lasers and Electro-Optics: Fundamentals and Engineering by Christopher Davis, Cambridge University Press, 1996
7
Introduction to Optical Electronics
Semiconductor Photon Detectors (Ch 18)
Semiconductor Photon Sources (Ch 17)
Lasers (Ch 15)
Laser Amplifiers (Ch 14)
Photons in Semiconductors
(Ch 16)
Photons & Atoms (Ch 13)
Quantum (Photon) Optics (Ch 12)
Resonators (Ch 10)
Electromagnetic Optics (Ch 5)
Wave Optics (Ch 2 & 3)
Ray Optics (Ch 1)
Optics
Physics
Optoelectronics
8
Putting it all together
Theory of Laser Oscillation
Laser Amplification Medium
+
Optical Resonator
=
Laser
9
Population Difference
Depletion of the steady-state population difference
Population Difference
N0
N0
2
0.1
s
1
10
s
s
Wi
2
R2
Wi 1
1
R1
 21  2
tsp  nr
1
 20
N
N0
1   sWi
  
N 0  R2 2 1  1 
  21 
 2 
 s   2   1 1  
  21 
Wi   ( )  ( )
10
Population Inversion
N  N2  N1
Population
Difference
Four-Level Laser
N0
N
1   sWi
Three-Level Laser
N0
N
1   sWi
Steady-State
Difference
N0 
N0 
tsp NaW
1  tspW
N a  tspW  1
1  tspW
Saturation Time
Constant*
s 
s 
tsp
1  tspW
2tsp
1  tspW
*What is the small-signal approximation?
11
Amplifier Nonlinearity
Gain Coefficient
 ( ) 
 0 ( )
1   / s ( )
2
where  0 ( )  N 0 ( )  N 0
g ( )
8 tsp
Note: 0() is called the smallsignal gain coefficient. Why?
0
1
0.5
0.01
0.1
1
10
100
s
12
Amplifier Nonlinearity
Gain
s
   z    z     (0)  (0) 


ln
  ln
  0z
s
s   s
s 

10
d
12
8
Y  X e 0d
 0d
 ln Y  Y   ln X  X    0 d
Output Y
6
4
2
  0
 d 
where X 
and Y 
s
s
YX
0
1
2
3
Input X
4
0
5
e 0 d
6
d
5
4
Gain
Gain
0
7
 (d ) Y

 (0) X
6
s
3
2
0.001
0.01
Input
0.1
0
1
s
10
13
Saturable Absorbers
Output Y  (d )
Transmittance =
 
Input
X  (0)
where  ( )  0 (i.e., attenuation)
0
0.8
d
0.6
Transmittance Y X
0.7
0.5
0.4
0.3
e 0 d
0.2
0.001
0.01
0.1
Input X
1
0
10
s
14
Saturated Gain Coefficient
Saturated Gain Coefficient
0
0
2
s
10
s
10
s
small-signal
region
 ( ) 
 0 ( )
1   / s ( )
 0 ( )  N 0  ( )
2
 N0
g ( )
8 tsp
1
s
  s  ( )
large-signal
region
ln
 ( z)  ( z)
 (0)  (0)

 ln

 0z
s
s
s
s
 ( ) z
small-signal:  ( z)   (0) e 0
large-signal:  ( z)   (0)   0s z
15
Gain Coefficient
Gain Coefficient
Inhomogeneously Broadened Medium
  
 s

0

0
1
16
Laser Amplification Medium
Laser Amplification Medium
 0 ( )
2
 ( ) 
where  0 ( )  N0 ( )  N0
g ( )
1   / s ( )
8 tsp
  / 2 
 ( )   ( 0 )
2
2
  0     / 2 
2
Lorentzian:
0
 2
where  ( 0 )  N 
 4 t 
sp





 ( ) 
  0
 ( )

0
0
17
Optical Resonator
Optical Resonator
I=
Imax
1   2F /   sin 2  /  F 
2
2

RTPS  2 q k 
c

1
1
Loss Coefficient  r   s 
ln
2d RR
1 2
Photon Lifetime  p 
Finesse F 
I
F 
I
Resonator
response
q
1
 q 1
c
2d

 2  p  F
r d
 
q
q
q
1
 q 1
1
r c
F
F

18
Conditions for Laser Oscillations
• Gain Condition: Laser Threshold
Threshold Gain:  0 ( )  r
Since  0    N0  
 N 0  Nt
r
1
where Nt 

 ( ) c p  ( )
• Phase Condition: Laser Frequencies
Round-Trip Phase: 2 k d  2 ( ) d  2 q q  1, 2,3...
19
Exercise 15.1-1
Threshold 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.
20
Saturated Gain Coefficient
Laser Turn-On
0
Time
Steady State
r Loss Coefficient
() Gain Coefficient
s
10
s
10
s
 (Photon-Flux Density)
Steady-state:  ( )  r 

  0 ( ) 

(

)
 1 ,  0 ( )   r


   s  r
  ( )  
0
r

0,

 N0 
 1 , N 0  N t
s ( ) 
 
N
 t
 N N
0
t

0,
21
Steady-State Population Difference

Photon
Flux Density
Population
Difference
N
Nt
s
N0
Nt
Nt
Pumping Rate
2Nt
N0
Pumping Rate

  0 ( ) 
 1 ,  0 ( )   r
s ( ) 
 

 r
  ( )  
0
r

0,
22
Output Flux Density vs. Transmittance

1
2
1
2
  0 ( )
o  T   T s 
 r
/2

 1



2  0 ( ) d
1
 T s 
 1
2
 2  s   m 2  d  ln(1  T ) 
o 
Output Photon-Flux Density
Laser
T 
2
, output photon-flux density
0.2
0.1
0.1
0.2
0.3
0.4
Transmittance
23
Characteristics of Laser Output
Internal Photon-Number Density
n  c
N

 ns  0  1 , N 0  N t
 Nt

 N0 
 ns 
 1 , N 0  N t
 Nt

Output Photon Flux & Efficiency
0  e  R  Rt V
where e 


 m1
r
c
1
 p ln
2d
R1
p
TF
T
if
T
 1-R 1  1
24
Laser Oscillations
 0 ( )
- Gain

r - Loss
B
0
F 
Resonator modes
1 ...
c
2d
M
allowed modes
Number of possible modes: M 
B
F
Each mode's FWHM   
F
F
25
Exercise 15.2-1
Number of Modes in a Gas Laser
A Doppler-broadened gas laser has a gain coefficient with a Gaussian
spectral profile given by
 0 ( )   0 ( 0 ) e
where

(  0 )2
2 D2
 D  8ln 2  D
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.
26
Homogeneously Broadened Medium
 0 ( )
 0 ( )
 0 ( )
r
 ( )
 ( )
0
0
0
1 ...  M
 ( ) 
 0 ( )
1   j 1 j / s ( j )
M
27
Inhomogeneously Broadened Medium
Typical
Doppler
 0 ( )
 0 ( )
 ( )
 ( )
 s
 s
r
r
q 1
q
q 1
q
28
Doppler Broadening
Laser Line (atomic)
Transverse Mode
Brewster
Window
Polarization
29
Longitudinal Mode Selection
Gain
Etalon
d
0
c
2d
Resonator Modes
c
2d1
Etalon Modes
Laser Output
30
How to Pulse Lasers
Modulator
Modulator
Peak Power
Average
Power
t
31
Pulsed Lasers
Gain
Gain Switching
t
Loss
Pump
t
Laser
Output
t
Q-Switching
Modulated
absorber
Loss
Gain
t
Laser
Output
t
32
Gain Switched Laser
33
Q-Switching
34
Pulsed Lasers
Gain
Loss
Cavity Dumping
t
Mirror
Transmittance
Laser
Output
t
Mode Locking
Optical
Modulator
35
Mode-Locked Laser
U ( z, t )   Aq e
j 2 q ( t  z / c )
q
 A (t  z / c)e j 2 0 (t  z / c )
where q  0, 1, 2...
where A (t  z / c)   Aq e j 2 t /TF
q
2
sin
M  t / TF 

2
I (t , z ) | A |
sin 2  t / TF 
TF
MI
TF
M
TF
M
M=5
M = 15
M = 25
36
Exercise 15.4-3
Demonstration 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
A(t )   Aq exp(
q
jq 2 t
)
TF
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.
37
120
(a) Equal magnitudes and equal phases.
100
80
60
40
20
1
2
3
4
1
2
3
4
1
2
3
4
80
(b) Magnitudes that obey the Gaussian
spectral profile and equal phases.
60
40
20
(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.
12
10
8
6
4
2
38