Transcript CHAPTER 4---

CHAPTER 4----LASER
Chapter 4
Laser
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LASERS
In 1958 Arthur Schawlow, together with Charles Townes, showed how
to extend the principle of the maser to the optical region. He shared
the 1981 Nobel Prize with Nicolaas Bloembergen. Maiman
demonstrated the first successful operation of the ruby laser in 1960.
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LASERS
an oscillator is an amplifier with positive feedback
Two conditions for an oscillation:
1. Gain greater than loss: net gain
2. Phase shift in a round trip is a multiple of 2π
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Stable condition: gain = loss
Gain
Loss
0
Power
Steady-state
power
If the initical amplifier gain is greater than the loss, oscillation may initiate.
The amplifier then satuates whereupon its gain decreases. A steady-state
condition is reached when the gain just equals the loss.
An oscillator comprises:
◆ An amplifier with a gain-saturation mechanism
◆ A feedback system
◆ A frequency-selection mechanism
◆ An output coupling scheme
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Light amplifier with positive feedback
Pout  gPin
Pin
Gain medium (e.g. 3level system w
population inversion)
When the gain exceeds the roundtrip
losses, the system goes into oscillation
+

g
+
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LASERS
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Amplified once
Initial photon
Gain medium
(e.g. 3-level
system w
population
inversion)
Reflected
Output
Amplified
twice
Reflected
Amplified
Again
Light
Amplification through
Stimualted
Emission
Radiation
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Partially
reflecting
Mirror
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Mirror
Active medium
Partially
transmitting
mirror
d
Laser
output
A laser consists of an optical amplifier (employing
an active medium) placed within an optical
resonator. The output is extracted through a
partially transmitting mirror.
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Optical amplification and feedback
★ Gain medium
The laser amplifier is a distributed-gain device
characterized by its gain coefficient
2
 0 ( )  N 0 ( )  N 0
g ( )
8 tsp
5.1-43 Small signal
Gain Coefficient
 0 ( )
 ( ) 
1   / s ( )
5.1-42 Saturated
Gain Coefficient
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  0
 ( ) 
 ( )

Phase-shift Coefficient
(Lorentzian Lineshape)
Figure 5.1-5 Spectral dependence of the gain and phase-shift
coefficients for an optical amplifier with Lorentzian lineshape function
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Optical Feedback-Optical Resonator
Feedback and Loss: The optical resonator
A Fabry-Perot resonator, comprising two mirrors separated by a distance d, contains
the medium (refractive index n). Travel through the medium introduces a phase shift
per unit length equal to the wavenumber
k
2
c
In traveling a round trip through a resonator of length d, the photon-flux density is
reduced by the factor R1R2exp(-2asd). The overall loss in one round trip can
therefore be described by a total effective distributed loss coefficient ar, where
exp(2a r d )  R1R2 exp(2a s d )
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Loss coefficient
a r  a s  a m1  a m 2
a m1 
1
1
ln
2d R1
a m2 
1
1
ln
2d R2
a m  a m1  a m 2 
Photon lifetime
1
1
ln
2d R1 R2
1
p 
arc
ar represents the total loss of energy (or number of photons)
per unit length, arc represents the loss of photons per second
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 q  q F , q  1, 2,...,
 
F
F
, F  c / 2d

F
 2 p F
ard
F 
Resonator
response
c
2d

 q 1
q
 q 1

Resonator modes are separated by the frequency
 F  c / 2d and have linewidths    F / F  1/ 2 p.
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Conditions for laser oscillation
Gain condition: Laser threshold
 0 ( )  a r
Threshold Gain
Condition
N 0  Nt
where
ar
Nt 
 ( )
or
Nt 
1
c p ( )
8 tsp 1
Nt  2
 c  p g ( )
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Threshold Population
Difference
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For a Lorentzian lineshape function, as
g ( 0 )  2 / 
2 2 tsp
Nt  2
 c p
If the transition is limited by lifetime broadening with a decay time tsp
2a r
2
Nt  2
 2
 c p

As a numerical example, if 01 mm, p=1 ns, and the refractive index n=1,
we obtain Nt=2.1×107 cm-3
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Conditions for laser oscillation(2)
Phase condition: Laser Frequencies
2kd  2 ( )d  2 q, q  1, 2……
Frequency Pulling
c   0

 ( )   q
2 
or
c   0
  q 
 ( )
2 
   q'   q
c  q  0
  q 
 ( q )
2 
'
q
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
   q  ( q  0 )

'
q
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Laser Frequencies
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The laser oscillation frequencies fall near the coldresonator modes; they are pulled slightly toward the
atomic resonance central frequency 0.
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Characteristics of the laser output
Internal Photon-Flux Density
Gain Clamping
 0 ( ) /[1   / s ( )]  a r
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Laser
turn-on
Time
 0 ( )
Steady
state
ar Loss coefficient
(u) Gain coefficient
0
s ( )
10
s ( )
10s ( )
Photon-flux density
Determination of the steady-state laser photon-flux density .
The smaller the loss, the greater the value of .
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Steady Photon Density
 0 ( )
 1],  0 ( )  a r
ar
  0,  0 ( )  a r
  s ( )[
Since
 0 ( )  N0 ( )
  s ( )(
a r  Nt ( )
N0
 1), N 0  N t
Nt
  0, N 0  N t
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Steady-State Laser
Internal Photon-Flux
Density
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Steady-state values of the population difference N, and the laser internal
photon-flux density , as functions of N0 (the population difference in the
absence of radiation; N0, increases with the pumping rate R).
Output photon-flux density
0 
Optical Intensity of Laser Output
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T
2
h T 
I0 
2
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Optimization of the output photon-flux density
1
1
1
ln  
ln(1  T )
2d R1
2d
From
a m1 
We obtain
1
a r  a s  a m2 
ln(1  T )
2d
g0
1
0  sT [
 1], g0  2 0 ( )d , L  2(a s  a m 2 )d
2
L  ln(1  T )
When, T
1
use the approximation
Then
ln(1  T )  T
Top  ( g0 L)1/ 2  L
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Spectral Distribution
Determined both by the atomic lineshape and by the
resonant modes
M
B
F
Number of Possible
Laser Modes
Linewidth   ?
Schawlow-Townes limit
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 0 ( )
Gain

ar Loss
B


0
F
Resonator modes
  ……M
1
2

Allowed modes
Figure 5.3-3 (a) Laser oscillation can occur only at frequencies for which
the gain coefficient is greater than the loss coefficient (stippled region). (b)
Oscillation can occur only within  of the resonator modal frequencies
(which are represented as lines for simplicity of illustration ).
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Homogeneously Broadened Medium
 ( ) 
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 0 ( )
1  i 1 j / s ( j )
M
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Inhomogeneously Broadened Medium
 0 ( )
 ( )
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 s
c
2d
ar

Frequency

 q 1  q  q 1 
(a)
(b)
Figure 5.3-6 (a) Laser oscillation occurs in an inhomogeneously broadened medium by each
mode independently burning a hole in the overrall spectral gain profile. (b) Spectrum of a
typical inhomogeneously broadened multimode gas laser.
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Hole burning in a Doppler-broadened medium
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Hole burning in a Doppler-broadened medium
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Spatial distribution and polarization
Spatial distribution
x,y
Spherical
mirror
Spherical
mirror
Laser
intensity
The laser output for the (0,0) tansverse mode of a sphericalmirror resonator takes the form of a Gaussian beam.
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 0,0
 1,1
TEM0,0
a11
a00
B11
B00

(1,1) modes
(0,0) modes
Laser
output
TEM1,1

Figure 5.3-8 The gains and losses for two transverse modes, say (0,0) and (1,1),
usually differ because of their different spatial distributions.
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Two Issues: Polarization, Unstable Resonators
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Mode Selection
Selection of
1. Laser Line
2. Transverse Mode
3. Polarization
4. Longitudinal Mode
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High
reflectance
mirror
Output mirror
Laser
output
Active medium
Prism
Aperture
Unwanted
line
Figure 5.3-9 A paticular atomic line may be selected by
the use of a prism placed inside the resonator. A
transverse mode may be selected by means of a
spatial aperture of carefully chosen shaped and size.
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Brewster
window
Active
midum
Brewster
window
Polarized
laser
output
qB
High
reflectance
mirror
Output
mirror
Figure 5.3-10 The use of Brewster windows in a gas laser provides a
linearly polarized laser beam. Light polarized in the plane of incidence
(the TM wave) is transmitted without reflection loss through a window
placed at the Brewster angle. The orthogonally polarized (TE) mode
suffers reflection loss and therefore does not oscillate.
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Selection of Longitudinal Mode
CHAPTER 4----LASER
Etalon
High reflectance
mirror
Active midum
d1
d
Output
mirror
Resonator loss
c/2d
Resonator mdoes
Etalon mdoes
c/2d1
Laser output
Figure 5.3-11 Longitudianl mode selection by the use of an intracavity etalon. Oscillation occurs
at frequencies where a mode of the resonator coincides with an etalon mode; both must, of
course, lie within the spectral window where the gain of the medium exceeds the loss.
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Multiple Mirror Resonators
(a)
(b)
(c)
Figure 5.3-12 Longitudinal mode selection by use of (a) two
coupled resonators (one passive and one active); (b) two
coupled active resonators; (c) a coupled resonator-interferometer.
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Characteristics of Common Lasers
Solid State Lasers: Ruby, Nd3+:YAG, Nd3+:Silica, Er3+:Fiber, Yb3+:Fiber
Gas Lasers: He-Ne, Ar+; CO2, CO, KF;
Liquid Lasers: Dye
Plasma X-Ray Lasers
Free Electron Lasers
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Pulsed Lasers
Method of pulsing lasers
External Modulator or Internal Modulator?
Modulator
Modulator
Average
power
CW power
t
t
(a)
(b)
Figure 5.4-1 Comparison of pulsed laser outputs achievable
with (a) an external modulator, and (b) an internal modulator
1. Gain switching
2. Q-Switching
3. Cavity Dumping 4.Mode Locking
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Gain Switching
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Q- Switching
Modulator
Loss
Gain
t
Laser
output
t
t
Figure 5.4-3 Q-switching.
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Cavity Dumping
Figure 5.4-4 Cavity dumping.
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Mode locking
• Laser modes coupling together
• Lock their phases to each other
CHAPTER 4----LASER
Rate equation for the photon-number density
dn
n
   NWi
dt
p
p
photon lifetime
Wi   ( )  cn ( )
From
 ( )  1/ c p N t
We have
Probability density for
induced absorption/emission
dn
n N n
 
dt
 p Nt  p
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Photon-Number
Rate Equation
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Rate equation for the Population Difference
For a three level system
dN2
N
 R  2  Wi ( N2  N1 )
dt
tsp
Note
N1  ( Na  N ) / 2, N2  ( N a  N ) / 2, N  N 2  N1
dN N0 N

  2Wi N
Then
dt
tsp tsp
Where the small signal population difference
Substituting
Wi  n / N t p
We have
dN N0 N
N n

 2
dt
tsp tsp
Nt  p
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N 0  2 Rtsp  N a
Population-difference rate
equation (Three-level system)
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Situation for the gain switching
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Situation for the Q-switching
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Dynamics of a Q-Switching process
Dynamics of the Q-switching process
Note the time relationship between the photon density and the
population inversion variations!
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Determination of the peak power, energy, width
and shape of the optical pulse
dn
N
n
 (  1)
dt
Nt
p
dN
N n
 2
dt
Nt  p
Dividing
dn 1 N t
 (  1)
dN 2 N
1
1
n  N t ln( N )  N  cons tan t
2
2
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Power
1
N 1
n  N t ln
 ( N  Ni )
2
Ni 2
1
c
P0  h A0  h cTAn  h T
Vn
2
2d
Peak pulse power
np 
N
N N
1
N i (1  t ln t  t )
2
Ni Ni Ni
Pp  h T
c
Vn p
2d
When Ni>>Nt
It is clear
np 
1
Ni
2
So
1
c
Pp  h T
VN i
2
2d
The larger the initial population inversion,
the higher the Q-switched pulse peak power.
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c. Pulse energy:
tf
Nf
c
c
dt
E  h T
V  n(t )dt h T
V  n(t )
dN
t
N
i
i
2d
2d
dN
Ni dN
1
c
E  h T
VN t p 
Nf N
2
2d
N
1
c
E  h T
VNt p ln i
2
2d
Nf
The final population difference Nf
Ni Ni  N f
ln

Nf
Nt
1
c
E  h T
V p ( Ni  N f )
2
2d
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d. Pulse width:
A rough estimation of the pulse width is the ratio
of the pulse energy to the peak pulse power.
 pulse   p
Ni / Nt  N f / Nt
Ni / Nt  ln( Ni / Nt )  1
When Ni>>Nth and Ni>>Nf
 pulse   p
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The shorter the photon life time,
the shorter the Q-switched pulses.
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Techniques for Q-switching
1. Mechanical rotating mirror method:
Q-switching principle: rotating the cavity mirror results in
the cavity losses high and low, so the Q-switching is obtained.
Advantages: simple, inexpensive.
Disadvantages: very slow, mechanical vibrations.
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2. Electro-optic Q-switching
Disadvantages:
complicate and expensive
Advantages:
very fast and stable.
Pockels effect: applying electrical field in a uniaxial crystal results in
additional birefringence, which changes the polarization of light when
passing through it.
Q-switching principle: placing an electro-optic crystal between crossed
polarizers comprises a Pockels switch. Turning on and off the electrical
field results in high and low cavity losses.
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Electro-optic Q-switch operated at (a) quarter-wave and (b) half-wave retardation voltage
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3. Acousto-optic Q-switching
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Diffracted light
Incident light
q
q
L
Sound
Piezoelectric transducer
Transmitted light
RF
Bragg scattering: due to existence of the acoustic wave, light
changes its propagation direction.
Q-switching principle: through switching on and off of the acoustic
wave the cavity losses is modulated.
Advantages: works even for long wavelength lasers.
Disadvantages: low modulation depth and slow.
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4. Saturable absorber Q-switching
What’s a saturable absorber?
a 
a0
1
I
Is
Absorption coefficient of the material is
reversely proportional to the light intensity.
Is: saturation intensity.
Saturable absorber Q-switching:
Insertion a saturable absorber in the
laser cavity, the Q-switching will be
automatically obtained.
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General characteristics of laser Q-switching
• Pulsed laser output:
– Pulse duration – related to the photon lifetime.
– Pulse energy - related to the upper level lifetime.
• Laser operation mode:
– Single or multi-longitudinal modes.
• Active verses passive Q-switching methods:
– Passive: simple, economic, pulse jitter and intensity fluctuations.
– Active: stable pulse energy and repetition, expensive.
• Comparison with chopped laser beams:
– Energy concentration in time axis.
• Function of gain medium
– Energy storage
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Laser mode-locking
Aims:
1. Familiarize with the principle of laser mode-locking.
2. Familiarize with different techniques of achieving laser
Mode-locking.
Outlines:
1.
2.
3.
4.
5.
Principle of laser mode-locking.
Methods of laser mode-locking.
Active mode-locking.
Passive mode-locking.
Transform-limited pulses.
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Principle of laser mode-locking
1. Lasing in inhomogeneously broadened lasers:
i) Laser gain and spectral hole-burning.
ii) Cavity longitudinal mode frequencies.
iii) Multi-longitudinal mode operation.
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2. Laser multimode operation:
Single mode lasers:
E (t )  E0 cos 0t   (t )
N
Multimode lasers:
E (t )   Ei cos i t   i (t )
i 1
Mode-frequency separations:
~
c
nL
Phase relation between modes: Random and independent!
Total laser intensity fluctuates with time !
The mean intensity of a
multimode laser remains
constant, however, its
instant intensity varies with
time.
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3. Effect of mode-locking:
(i) Supposing that the phases of all modes are locked together:
 i (t )   0  0
(ii) Supposing that all modes have the same amplitude:
Ei  E0
purely for the convenience of the
mathematical analysis
(iii) Under the above two conditions, the total electric field
of the multimode laser is:
N
i i t 
E (t )  Re  Ei e 
where
 i 1

c
 N  1
 i   0  i 
 c  c 

2 
L

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0 is the frequency of the central mode, N is the number
of modes in the laser, c is the mode frequency separation.
i is the frequency of the i-th mode.
Calculating the summation yields:
 c t 
sin N

2 

E(t )  E 0
cos 0 t


t

c 
sin

2


Note this is the optical
field of the total laser
Emission !
The optical filed can be thought to consist of a carrier wave of
frequency 0 that amplitude modulated by the function
sin (Nx )
AN (x ) 
sin (x )
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4. Characteristics of the mode-locked lasers:
The intensity of the laser field is:
  c t 
sin 2  N

2 

I (t )  E02
  c t 
sin 2 

2


The output of a mode-locked laser consists of a series of pulses.
The time separation between two pulses is determined by RT
and the pulse width of each pulse is tp.
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5. Properties of mode-locked pulses:
i) The pulse separation RT:
  c t 
Sin 2 
0
 2 
 RT
2
2L


 c
c
c t  2
The round-trip time of the cavity!
ii) The peak power: Considering sin a  a when a is small,
E 2 (0)  N 2 E02
N times of the average power. N: number of modes.
The more the modes the higher the peak power of the
Mode-locked pulses.
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iii)The individual pulse width:
  c t 
sin  N
0
2 

N
 a
 c
t p 
2
N c
2
1
t p 

 a  a
Narrower as N increases.
t p 
a: bandwidth of
the gain profile.
 RT
N
The mode locked pulse width is reversely proportional to
the gain band width, so the broader the gain profile, the shorter
are the mode locked pulses.
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Techniques of laser mode-locking
Active mode-locking:
Actively modulating the gain or loss of a laser cavity in a periodic way,
usually at the cavity repetition frequency c/2nL to achieve mode-locking.
Amplitude modulation:
A modulator with a transmission function of


 2t  
 
T  1   1  cos


  RT  

is inserted in the laser cavity to modulate the light. Where  is the
modulation strength and  < 0.5. Under the influence of the modulation
phases of the lasing modes become synchronized and as a consequence
become mode-locked.
Operation mechanism of the technique:
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Time domain analysis:
Consider the extreme case where a shutter is inside the cavity, and
the opens only for a short time every second. Is the cavity round
trip time. In this case only a pulse with pulse width narrower than
the opening time can survive in the cavity, all the CW type of
operation will be blocked by the shutter. To have a pulse moving
in the cavity the phase of all lasing modes must be synchronized.
The laser modes will arrange themselves to realize such a state.
Shutter losses
It is a natural competition and the fittest will survive.
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Frequency domain analysis:
Sidebands generation
Amplitude modulation
Em (t )   m sin (mt  m )
Electrical filed of each mode
 m   0 (1   cos t )
Amplitude of each mode is modulated




E m (t )   0 sin ( m t   m )  sin ( m   )t   m   sin ( m   )t   m 
2
2


Sidebands are generated by the modulation
m-
m
Without modulation
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m m+
After amplitude modulation
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In the case of a multimode laser
As all modes are modulated by the same frequency, the
sidebands of one mode will drive its adjacent modes, and
as a consequence, all modes will oscillate with locked phase.
From both the time domain and the frequency domain analysis it
is easy to understand why the modulation frequency must be
exactly the cavity longitudinal mode separation frequency.
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Passive mode-locking:
Inserting an appropriately selected saturable absorber inside the laser
cavity. Through the mutual interaction between light, saturable absorber
and gain medium to automatically achieve mode locking.
A typical passive mode locking laser configuration:
laser medium
saturable
absorber
Mechanism of the mode-locking:
i) Interaction between saturable absorber and laser gain:
Survival takes all!
ii)Balance between the pulse shortening and pulse broadening:
Final pulse width.
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Transform limited pulses
Gaussian pulses:
In our analysis we have assumed
that En=E0
The real gain line has a Gaussian
profile,which results in that the lasing
mode amplitudes have a Gaussian
distribution.
A Gaussian gain line shape function
a Gaussian
mode-locked pulse intensity variation, namely a Gaussian pulse.
1

 
2





t

 



(
)
E t  E0 
exp 
 
1



2
ln
2



 0 

  2(ln 2)2

Fundamentals of Photonics
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



2

  exp (i t )
0

 

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CHAPTER 4----LASER
Intensity of the pulse:
    
I (t )  

2
ln
2

  0 
2
 

exp  
1
 
  2(ln 2)2




2




Gaussian intensity profile!
Transform limited pulses:
If the product of pulse width and spectral bandwidth of a Gaussian
pulse equals 0.441, then the Gaussian pulse is called a transform
limited pulse as in this case the pulse width is purely determined
by the Fourier transformation of the pulse spectral distribution.
t p  L  0.441
For transform limited
Gaussian pulses!
tp: pulse width, L: spectral bandwidth
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激光振荡器
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均匀加宽激光器模式竞争
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非均匀加宽激光器模式竞争
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多普勒加宽增益饱和
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兰姆凹陷
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脉冲泵浦调Q
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