Transcript CHAPTER 3---- Laser Amplifiers

CHAPTER 3---- Laser Amplifiers
Chapter 3
Laser Amplifiers
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CHAPTER 3---- Laser Amplifiers
Concept of the laser amplifier
Pump
atoms
Ouput
photons
Input
photons
Laser amplifier
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CHAPTER 3---- Laser Amplifiers
Optical Regeneration
Ideal analog amplification
• Faithfully reproduces input signal with minimal
distortion
• Can be used as a linear repeater by periodically
boosting
• optical power
• Can be used in nonlinear region as a level clamping
amplifier
• Single amplifier can be used as a multichannel
amplifier
• ideally with minimal crosstalk and distortion
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CHAPTER 3---- Laser Amplifiers
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CHAPTER 3---- Laser Amplifiers
Real Amplifier
•
•
•
•
•
•
Gain
Bandwidth
Phase shift
Power source
Nonlinearity and gain saturation
Noise
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CHAPTER 3---- Laser Amplifiers
Optical Amplifier
Pump
atoms
Ouput
photons
Input
photons
Laser amplifier
Figure 5.1-1 The laser amplifier. An external power source (called a pump) excites
the active medium (represented by a collection of atoms), producing a population
inversion. Photons interact with the atoms; when stimulated emission is more
prevalent than absorption, the medium acts as a coherent amplifier.
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CHAPTER 3---- Laser Amplifiers
Optical Amplifier Physics
An atomic system with two energy levels can
◆ absorb light
◆ amplify light
◆ spontaneously emit light
Stimulated Spontaneous
Absorption emission
emission
Stimulated and spontaneous emission are achieved
by pumping the amplifier electrically or optically.
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CHAPTER 3---- Laser Amplifiers
Amplifier
+d

Input
light
Output
light
z
0
z+dz
d
Figure 13.1-1 The photon-flux density  (photons/cm2-s) entering an
incremental cylinder containing excited atoms grows to +d after length dz.
Wi   ( ) d  NWi dz
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……(13.1-3)
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d ( z )
  ( ) ( z )
dz
2
 ( )  N ( )  N
g ( )
8 tsp
 ( z )   (0) exp[ ( ) z ]
G ( )  exp[ ( )d ]
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CHAPTER 3---- Laser Amplifiers
Homogeneous and inhomogeneous
broadening
•
To describe the distribution of the emitted intensity versus the frequency
v, we define a lineshape function g(v):

 g ( )d  1

•
•
•
→g(v)dv can be considered as a priori probability that a given
spontaneous emission 2→1 will result in a photon whose frequency is
between v and v+dv
→Both the emission and the absorption are described by the same
lineshape function g(v)
→g(v) can be measured by measuring the profile of the absorption
spectrum for the transition 1→2
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CHAPTER 3---- Laser Amplifiers
Lorentzian lineshape
 / 2
g ( ) 
(  0 ) 2  ( / 2) 2
The gain coefficient is then also Lorentzain with the same width, i.e.,
( / 2) 2
 ( )   ( 0 ) 
(  0 ) 2  ( / 2) 2
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CHAPTER 3---- Laser Amplifiers
Homogeneous Broadening
Radiated field e(t )  E0 e
 t /
 / 2   1
E0 i (0 i / 2)t
cos(0t )  [e
 e i (0 i / 2)t ]
2
Field decay rate

Fourier Transform E ( )   e(t )e  it dt 
0
E0
i
i
[

]
2 0    i / 2 0    i / 2
At the vicinity of the resonant frequency w0
E ( ) 2 
1
(0   ) 2  ( / 2) 2
Coresponded curves are called Lorentzian
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CHAPTER 3---- Laser Amplifiers

1
Linewidth  

2 
Lineshape function g ( ) 
 
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1


2 [(  0 ) 2  ( / 2) 2 ]
( u1   l1   cu1   cl1 )
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CHAPTER 3---- Laser Amplifiers
 ( )
 ( 0 )
1

0
0

Figure 13.1-2 Gain coefficient ( ) of
a Loretzian-lineshape laser amplifier.
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CHAPTER 3---- Laser Amplifiers
Amplifier phase shift
 ( ) 
  0
 ( )

Figure Gain coefficient and phase-shift coefficient for a
laser amplifier with a Lorentzian line-shape function
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CHAPTER 3---- Laser Amplifiers
Features of homogeneous broadening:
1. Each atom in the system has a common emitting spectrum
widthΔv.g(v) describes the response of any of the atoms, which are
indistinguishable
2. Due most often to the finite interaction lifetime of the absorbing and
emitting atoms
Mechanisms of homogeneous broadening:
1. The spontaneous lifetime of the exited state
2. Collision of an atom embedded in a crystal with a phonon
3. Pressure broadening of atoms in a gas
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CHAPTER 3---- Laser Amplifiers
Features of Inhomogeneous Broadening
1. Individual atoms are distinguishable, each having a slightly different
frequency.
2. The observed spectrum of spontaneous emission reflects the spread
in the individual transition frequencies (not the broadening due to the
finite lifetime of the excited state).
Typical Examples:
• The energy levels of ions presents as impurities in a host crystal.
• Random strain
• Crystal imperfection
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CHAPTER 3---- Laser Amplifiers
Rate Equation
Gain constant
 ( )  ( N 2  N1 )
if :  ( )  0
if :  ( )  0
Fundamentals of Photonics
c2
8 n  tspont
2
2
g ( )
N 2  N1
Amplification
N 2  N1
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Attenuation
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CHAPTER 3---- Laser Amplifiers
Ampilifying medium (N2>N1)
Output wave
(a)
Atoms in upper state 2
Atoms in lower state 1
Absorbing medium (N2<N1)
Output wave
(b)
(a) Amplification of a traveling electromagnetic wave in an inverted
population (N2>N1), and (b) attenuation in a absorbing medium (N2<N1).
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CHAPTER 3---- Laser Amplifiers
Population Inversion
Negative temperature
At thermal equilibrium
N2
 exp(h / k BT )
N1
case1: N2  N1
As usual, T>0
case2 : N2  N1
Negative temperature
Wave intensity grows
Population
Inversion
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exponentially!!
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CHAPTER 3---- Laser Amplifiers
Atomic rate equations
•1. Radiation-atom interaction:
• Stimulated emission
• Absorption
2. Population inversion and laser pumping:
g2
N2 
N1  0
g1
3. Lifetime of atoms in upper energy level:
N (t )  N 0 e
 t
 : lifetime of atoms in the upper energy level.
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CHAPTER 3---- Laser Amplifiers
Either the radiation-atom interaction, laser pumping and
energy decay change the population density distribution.
To describe in details the rates of these changes
Atomic rate equations
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CHAPTER 3---- Laser Amplifiers
Two level system
2
2
21
tsp
nr
1
1
20
Figure 13.2-1 Energy levels 1 and 2 and their decay times.
 21   211   201
(13.2-1)
 211   sp1   nr1
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CHAPTER 3---- Laser Amplifiers
R2
2
1
R1
Figure 13.2-2 Energy levels I and 2, together with
surrounding higher and lower energy levels.
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CHAPTER 3---- Laser Amplifiers
Rate equations in the absence of
amplifier radiation
dN 2
N
 R2  2
dt
2
……(13.2-2)
dN1
N N
  R1  1  2 ……(13.2-3)
dt
 1  21
N 0  R2 2 (1 
1
)  R1 1
 21
Steady-state population
difference (in absence of
amplifier radiation)
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CHAPTER 3---- Laser Amplifiers
For large No
• Large R1and R2
• Long 2 (but tsp which contributes to 2 through
21 must be sufficiently long so as to make the
radiative transition large)
• Short 1 if R1(2/21)R2
N 0  R2t sp  R1 1
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……13.2-4(a)
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CHAPTER 3---- Laser Amplifiers
Rate equations in the presence of
amplifier radiation
dN 2
N
 R2  2  N 2Wi  N1Wi
dt
2
dN1
N N
  R1  1  2  N 2Wi  N1Wi
dt
 1  21
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……(13.2-5)
……(13.2-6)
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CHAPTER 3---- Laser Amplifiers
N0
N
1   sWi
2
 s   2   1 (1  )
 21
Fundamentals of Photonics
……(13.2-7)
Steady-state population difference
(in absence of amplifier radiation)
……(13.2-8)
Saturation time constant
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s
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Saturation time constant
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CHAPTER 3---- Laser Amplifiers
Derivation of atomic rate equations
1. Four-level pumping schemes
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Figure 5.1-11 Energy levels and decay rates for a four-level system.
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CHAPTER 3---- Laser Amplifiers
1
N 0  R 2 (1  )
 21
Typically as
tsp
……(5.1-23)
 nr and 20
N 0  Rtsp
tsp
……(5.1-24)
 s  t sp
……(5.1-25)
N  Rtsp /(1  t spWi )
……(5.1-26)
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1
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CHAPTER 3---- Laser Amplifiers
If considering
N g  N1  N 2  N 3  N a and
Then
N1  N3  0
N g  Na  N2  Na  N
At that time, the pump rate R is a linearly decreasing
function of population difference, not independent of it.
R  ( N a  N )W
(5.1-26) becomes
N
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tsp N aW
1  tspWi  tspW
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CHAPTER 3---- Laser Amplifiers
N0
N
1   sWi
N0 
s 
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tsp N aW
1  tspW
tsp
1  tspW
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……(5.1-30)
……(5.1-31)
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CHAPTER 3---- Laser Amplifiers
Derivation of atomic rate equations
Three-level pumping scheme
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3
Short-lived level
Rapid decay 32
R Pump
2
Long-lived level
Laser
Wi 1
21
1
Ground state
Figure 5.1-12 Energy levels and decay rates for a three-level system.
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CHAPTER 3---- Laser Amplifiers
In the steady state, from (5.1-19) and (5.1-20), we have
0 R
Note
N2
 21
 N 2Wi  N1Wi
……(5.1-32)
N1  N 2  N a
N 0  2 R 21  N a  2 Rtsp  N a
 s  2 21  2tsp
From
Then
R  ( N1  N3 )W , N3  0
1
( N a  N )W
2
1
N1  ( N a  N )
2
We have R 
N  (2 Rtsp  N a ) /(1  2tspWi )
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CHAPTER 3---- Laser Amplifiers
We have
but now
N
N0
1   sWi
N0 
s 
Fundamentals of Photonics
N a (tspW  1)
1  tspW
2tsp
1  tspW
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……(5.1-38)
……(5.1-39)
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CHAPTER 3---- Laser Amplifiers
Examples of Laser Amplifiers
S
b
a
Gas
Gas
Rod
Flashlamp
c
d
Laser
diode Lens Nd3+:YAG rod
Laser
diode
Lens
Er3+ : silica fiber
Figure 5.1-13 Examples of electrical and optical pumping.
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CHAPTER 3---- Laser Amplifiers
Ruby
ev
Ruby
4
Energy
4F1
3
4F2
3
32
R1
2
2
Pump
694.3nm laser
1
1
0
Figure 5.1-14 Energy levels pertinent to the 694.3nm red ruby
transition. The three interacting levels are indicated in circles.
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CHAPTER 3---- Laser Amplifiers
Ruby
rod
Flashlamp
Flashlamp
Input
photons
Ouput
photons
Capacitor
Ruby rod
Elliptical mirror
Power
supply
(a)
(b)
Figure 5.1-15 The ruby laser amplifier. (a) Geometry used in the first laser
oscillator built by Maiman in 1960. (b) Cross sction of a high-efficiency geometry
using a rod-shaped flashlamp and a reflecting elliptical cylinder.
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CHAPTER 3---- Laser Amplifiers
Nd3+:YAG and Nd3+:Glass
ev
Nd3+:YAG
3
2
Energy
32
4F3/2
2
1.064um laser
Pump
1
1
4I11/2
0
4I9/2
0
Figure 5.1-16 Energy levels pertinent to the 1.064um Nd3+:YAG laser transition.
The energy levels for Nd3+:glass are similar but the absorption bands are broader.
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CHAPTER 3---- Laser Amplifiers
Er3+:Silica Fiber
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CHAPTER 3---- Laser Amplifiers
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CHAPTER 3---- Laser Amplifiers
Amplifier nonlinearity and gain saturation
From : N 
N0
, and , Wi   ( )
1   sWi
Have : N 
N0
1   / s ( )
1
2 s
  s ( ) 
g ( )
s ( )
8 tsp
 0 ( )

(

)

Then
1   / s ( )
(5.1-41)
(5.1-42) Saturated Gain Coefficient
2
 0 ( )  N 0 ( )  N 0
g ( ) (5.1-43) Small-signal Gain Coefficient
8 sp
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CHAPTER 3---- Laser Amplifiers
Figure 5.1-17 Dependence of the normalized saturated gain
coefficient on the normalized photon-flux densit.
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CHAPTER 3---- Laser Amplifiers
Gain
 0
d

dz 1   / s
ln
……(5.1-44)
 ( z )  ( z )   (0)

 0z
 (0)
s
……(5.1-45)
[ln(Y )  Y ]  [ln( X )  X ]   0 d
where
……(5.1-46)
X   (0) / s , and , Y   (d ) / s
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CHAPTER 3---- Laser Amplifiers
(a) A nonlinear (saturated) amplifier. (b) Relation between the normalized output
photon-flux density Y and the normalized input photon-flux density X. For X<<1,
the gain Y/X=exp(r0d). For X>>1, the gain Y=X+r0d. (c) Gain as a function of the
input normalized photon-flux density X in an amplifier of length d when r0d=2.
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CHAPTER 3---- Laser Amplifiers
Saturable Absorbers
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Difference of gain saturation between
inhomogeneous and homogeneous media
 ( 0 )
0
1
1
Inhomogeneous
Homogeneous
0.5
0
102
101
1
10
10 2

s
Fig.5.1-20 Comparison of gain saturation in homogeneous and inhomogeneous broadened media
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CHAPTER 3---- Laser Amplifiers
Hole burning
Gain coefficient
1
 ( )
 ( )
 s

0
1

Figure 5.1-21 The gain coefficient of an inhomogeneously broadened medium is
locally saturated by a large flux density of monochromatic photons at frequency 1
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CHAPTER 3---- Laser Amplifiers
Gain saturation in homogeneously and
inhomogeneously broadened systems:
Spectral hole-burning
(Homogeneous)
(Inhomogeneous)
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CHAPTER 3---- Laser Amplifiers
In the homogeneously broadened lasers, when gain
saturation occurs, the entire gain curve saturates
proportionally. The stronger the saturation effect, the lower
the gain curve (or the smaller the gain coefficients).
In the inhomogeneously broadened lasers,
saturation at one particular frequency causes a reduction
in the gain profile only near that frequency. Effectively, a
hole is burned in the gain profile at the frequency---phenomenally it is called spectral hole burning. No effect it
will have on the gain at other frequencies!
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CHAPTER 3---- Laser Amplifiers
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