Lecture 9: Laser oscillators



Theory of laser oscillation
Laser output characteristics
Pulsed lasers
References: This lecture follows the materials from Fundamentals of Photonics, 2nd ed.,
Saleh & Teich, Chapter 15. Also from Photonic Devices, Jia-Ming Liu,
Chapter 11.
1
Intro




There are a wide variety of lasers, covering a spectral range
from the soft X-ray (few nm) to the far infrared (hundreds of
m), delivering output powers from microwatts (or lower) to
terawatts, operating from continuous wave (CW) to
femtosecond (even attosecond) pulses, and having spectral
linewidths from just a few hertz to many terahertz.
The gain media utilized include plasma, free electrons, ions,
atoms, molecules, gases, liquids, solids, etc.
The sizes range from microscopic, of the order of 10 m3
(recently down to the order of sub m3 for so-called
nanolasers), to gigantic, of an entire building, to stellar, of
astronomical dimensions.
An optical gain medium can amplify an optical field through
stimulated emission.
2
Intro






The laser is an optical oscillator.
It comprises a resonant optical amplifier whose output is fed
back to the input with matching phase.
The oscillation process can be initiated by the presence at the
amplifier input of even a small amount of noise that contains
frequency components lying within the bandwidth of the
amplifier.
This input is amplified and the output is fed back to the input,
where it undergoes further amplification.
The process continues until a large output is produced.
The increase of the signal is ultimately limited by saturation
of the amplifier gain, and the system reaches a steady state in
which an output signal is created at the frequency of the
resonant amplifier.
3
Laser oscillators



In a practical laser device, it is generally necessary to have
certain positive optical feedback in addition to optical
amplification provided by a gain medium.
This requirement can be met by placing the gain medium in
an optical resonator. The optical resonator provides selective
feedback to the amplified optical field.
In many lasers the optical feedback is provided by placing the
gain medium inside a “Fabry-Perot” cavity, formed by using
two mirrors or highly reflecting surfaces
Gain medium
reflectivity (R1 ~ 100 %)
Light output (laser)
R2 < 100 %
4
4
Intro

Two conditions must be satisfied for oscillation to occur:
 The amplifier gain must be greater than the loss in the
feedback system s.t. net gain is incurred in a round trip
through the feedback loop.
 The total phase shift in a single round trip must be a
multiple of 2 s.t. the feedback input phase matches the
phase of the original input.
If these conditions are satisfied, the system becomes unstable and
oscillation begins.
5
Intro



As the power in the oscillator grows, the amplifier gain
saturates and decreases below its initial value.
A stable condition is reached when the reduced gain is equal
to the loss.
The gain then just compensates the loss s.t. the cycle of
amplification and feedback is repeated without change and
steady-state oscillation prevails.
gain
loss
Steady-state
power
Oscillator
power
6
Intro



Because the gain and phase shift are functions of
frequency, the two oscillation conditions are satisfied
only at one or several frequencies, which are the
resonance frequencies of the oscillator.
The useful output is extracted by coupling a portion of
the power out of the oscillator.
An oscillator comprises:




An amplifier with a gain-saturation mechanism
A feedback system
A frequency-selection mechanism
An output coupling scheme
7
Intro





The laser is an oscillator in which the amplifier is the
pumped active medium.
Gain saturation is a basic property of laser amplifiers.
Feedback is enabled by placing the active medium in
an optical resonator, which in its simplest form
reflects the light back and forth between its mirrors.
Frequency selection is jointly attained by the resonant
amplifier and the resonator, which admits only certain
modes.
Output coupling is attained by making one of the
resonator mirrors partially transmitting.
8
Theory of laser oscillation
9
Laser amplification




The laser amplifier is a narrowband coherent amplifier of
light.
Amplification is attained by stimulated emission from an
atomic or molecular system with a transition whose
population is inverted (i.e. the upper energy level is more
populated than the lower).
The amplifier bandwidth is determined by the linewidth of
the atomic transition, or by an inhomogeneous broadening
mechanism (e.g. defects and strains and impurities in host
solids)
The laser amplifier is a distributed-gain device characterized
by its gain coefficient (gain per unit length) (), which
governs the rate at which the photon-flux density  (or the
optical intensity I = h) increases.
10
Small-signal gain coefficient

When the photon-flux density  is small, the gain coefficient
is
2
c
 0 ( )  N 0 ( )  N 0
ĝ( )
2 2
8 n   sp
where N0 = equilibrium population density difference (density of
atoms in the upper energy state minus that in the lower state).
Assumes degeneracy of the upper laser level equals that of
the lower laser level (i.e. g1=g2). N0 increases with
increasing pumping rate.
 () = transition cross section (e() = a() = ())
 sp = spontaneous lifetime
 g() = transition lineshape
11
Saturation photon-flux density


As the photon-flux density increases, the amplifier enters a
region of nonlinear operation. It saturates and its gain
decreases.
The amplification process then depletes the initial population
difference N0, reducing it to
N
N0
1  /  s ( )
for a homogeneously broadened medium, where
s ( ) 
1
 s ( )
Saturation photon-flux density
s saturation time constant, which depends on the decay times
of the energy levels involved. s ≈ sp for four-level pumping,
s = 2sp for three-level pumping
12
Saturated gain coefficient

The gain coefficient of the saturated amplifier is therefore
reduced to (for homogeneous broadening)
 ( )  N ( ) 

The laser amplification process also introduces a phase shift.
When the lineshape is Lorentzian with linewidth ,
ĝ( ) 

 0 ( )
1  / s ( )
 / 2 
(   0 )2  ( / 2)2
The amplifier phase shift per unit length is
 0
 ( ) 
 ( )

13
Gain coefficient and phase-shift coefficient for a laser amplifier
with a Lorentzian lineshape function
gain
coefficient
()



Phase-shift
coefficient
()


14
Optical resonators




Optical feedback is attained by placing the active medium in
an optical resonator.
A Fabry-Perot resonator, comprising two mirrors separated
by a distance d, contains the active medium (refractive index
n).
Travel through the medium introduces a phase shift per unit
length equal to the wavenumber k = 2n/c
The resonator sustains only frequencies that correspond to a
round-trip phase shift that is a multiple of 2.
2 n
k  2d 
2d  q2
c
q = 1, 2, …
15
Fabry-Perot resonators
refractive index n
d
• Only standing waves at discrete wavelengths exist in the cavity.
=> the laser wavelengths must match the cavity resonance wavelengths.
The resonance condition: 2nd = q 
or
2kd = 2q
where q is an integer (=1, 2, …), known as the longitudinal
mode order, k = 2n/2n/c
16
Resonant optical cavities
Pout
R1
R1
R2
Pout
Pout
R2
Pout
Pout
Pout
Fiber/waveguide
ring resonator
Pout
Pout
Bragg grating
17
17
Resonant optical cavities

A linear cavity with two end mirrors is known as a FabryPerot cavity because it takes the form of a Fabry-Perot
interferometer. In the case of semiconductor diodes, the
diode end facets form the two end mirrors.

A folded cavity can simply be a folded Fabry-Perot cavity
with a standing oscillating field.

A folded cavity can also be a non-Fabry-Perot ring cavity
that supports two independent oscillating fields traveling in
opposite directions (clockwise, counterclockwise). Ring
cavity can be made of multiple mirrors in free space, or in the
form of fiber/waveguide-based devices.

The optical cavity can also comprise a distributed Bragg
grating with distributed feedback. Distributed Feedback
(DFB) diode lasers are the most common single-mode laser
diodes for optical communications.
18
18
Resonant optical cavities



In a ring cavity, an intracavity field completes one round trip
by circulating inside the cavity in only one direction. The
two contrapropagating fields that circulate in opposite
directions in a ring cavity are independent of each other even
when they have the same frequency.
In a Fabry-Perot cavity, an intracavity field has to travel the
length of the cavity twice in opposite directions to complete a
round trip.
The time it takes for an intracavity field to complete one
round trip in the cavity is called the round-trip time, TF:
lRT
TF 
c
where lRT is the round-trip optical path length (=2nd for
Fabry-Perot cavities).
19
19
Longitudinal mode spacing
• The modes along the cavity axis is referred to as longitudinal modes.
• Many ’s may satisfy the resonance condition => multimode cavity
intensity
q+1
q+2
…..
q-1
q
q-2

…..

The longitudinal mode spacing (free-spectral range):
 = 2 / 2nd
20
20
Longitudinal mode spacing
• The longitudinal mode frequencies:
 = q = qc/2nd
• The mode spacing (free-spectral range) in frequency unit:
q = c/2nd
e.g. A semiconductor laser diode has a cavity length 400 m
with a refractive index of 3.5. The peak emission wavelength from
the device is 0.8 m. Determine the longitudinal mode order
and the frequency spacing of the neighboring modes.
• The longitudinal mode order q = 2nd/ ~ 3500
• The frequency spacing q = c/2nd ~ 100 GHz
21
21
Resonator losses


The resonator also contributes to losses. Absorption and
scattering of light in the medium introduces a power loss per
unit length (attenuation coefficient s)
In traveling a round trip through a resonator of length d, the
photon-flux density is reduced by the factor
R1R2 exp(-2sd)
where R1 and R2 are the reflectances of the two mirrors

The overall power loss in one round trip can be described by a
total effective distributed loss coefficient r
exp(-2rd) = R1R2exp(-2sd)
22
Loss coefficients
r = s + m1 + m2
m1 = (1/2d) ln(1/R1)
m2 = (1/2d) ln(1/R2)
where m1 and m2 represent the contributions of mirrors 1
and 2.

The contribution from both mirrors
m = m1 + m2 = (1/2d) ln(1/(R1R2))
23
Photon lifetime and resonator linewidth

Define photon lifetime (cavity lifetime) p as the 1/e-power
lifetime for photons inside the cavity of refractive index n:
exp(-r pc/n) = exp(-1)


p = n/rc
The resonator linewidth (FWHM)  is inversely
proportional to the cavity lifetime
 = 1/2p

The cavity quality factor Q at resonance frequency q is
Q = q × (energy stored in the resonator/average power dissipation)
24
= q p = q/
Photon lifetime and resonator linewidth
q c/2nd

q
q

q

The finesse of the resonator
F ≈ q/

where q = c/2nd
When the resonator losses are small and the finesse is large
F ≈ /(rd)
25
Conditions for laser oscillation

Two conditions must be satisfied for the laser to oscillate
(lase):
 The gain condition determines the minimum population
difference, and thus the pumping threshold required for
lasing
 The phase condition determines the frequency (or
frequencies) at which oscillation takes place
26
Gain Condition: Laser threshold

The initiation of laser oscillation requires that the smallsignal gain coefficient be greater than the loss coefficient
 0 ( )   r

Or, the gain be greater than the loss.
Translates this to the population difference
 0 ( )  r
N0 

 Nt
 ( )  ( )
where Nt is the threshold population difference. Nt, which
is proportional to r, determines the minimum pumping rate
Rt for the initiation of laser oscillation.
27
Gain condition: Laser threshold

r may be written in terms of the photon lifetime,
n
r 
c p

Thus, Nt is given as
Nt 

n
c p ( )
The threshold population density difference is therefore
directly proportional to r and inversely proportional to p.
Higher loss (shorter photon lifetime) requires more vigorous
pumping to attain lasing.
28
Threshold population difference

By using the transition cross section
c2
 ( ) 
ĝ( )
2 2
8 n   sp
we find another expression for the threshold population
difference,
8 n 3 2  sp 1
Nt 
c 3  p ĝ( )


The threshold is lowest, and thus lasing is most readily
attained, at the frequency where the lineshape function is
largest, i.e., at its central frequency  = 0.
For a Lorentzian lineshape function, g(0) = 2/
29
Threshold population difference

The minimum population difference for oscillation at the central
frequency 0 turns out to be
2 n3 2 2 sp
Nt 
c3
p


Nt is directly proportional to the linewidth .
If the transition is limited by lifetime broadening with a decay time sp,
and  = 1/2sp
3 2
2 2
2 n 
2 n   r
Nt  3

c p
c2

This shows that the minimum threshold population difference required to
attain oscillation is a simple function of the frequency  and the photon
lifetime p. Laser oscillation becomes more difficult to attain as the
frequency increases.
30
Phase condition: laser frequencies

The phase condition of oscillation requires that the phase
shift of the laser light completing a cavity round-trip must be
a multiple of 2
2kd + 2()d = 2q, q = 1,2,…



If the contribution arising from the active laser atoms 2()d
is small, then the laser modes are given by the “cold” (or
passive) cavity modes.
In general, 2()d gives rise to a set of oscillation frequencies
q’ that are slightly displaced from the cold-resonator
frequencies q.
The cold-resonator modal frequencies are all pulled slightly
toward the central frequency of the atomic transition –
frequency pulling or mode pulling
31
Frequency pulling





q




Amplifier
gain coefficient
Cold-resonator
modes
Laser oscillation
modes
The laser oscillation frequencies fall near the cold-resonator
modes – they are pulled slightly toward the atomic resonance
central frequency 0.
32
Laser output
characteristics
33
Laser power

A laser pumped above the threshold exhibits a small-signal
gain coefficient 0() that is greater than the loss coefficient r.
0() > r

Laser oscillation may then begin, provided that the phase
condition is satisfied.
2kd + 2()d = 2q, q = 1,2,…


As the photon-flux density  inside the resonator increases, the
gain coefficient () begins to uniformly drop (for
homogeneously broadened media)
() = 0() / (1 + /s())
As long as the gain coefficient remains larger than the loss
coefficient, the photon flux continues to grow.
34
Laser oscillation: the unsaturated gain must exceed the loss
gain
loss (assume
constant)
loss
• sub-threshold
(incoherent emission)



gain < loss
loss
gain = loss
• Threshold
(oscillation begins,
start to emit coherent
light)
gain > loss
• above-threshold
(increase in
coherent
output
power)
35
Laser oscillation

Loss r

q

Resonator
 modes

 
Allowed
 modes
• Laser oscillation can occur only at frequencies for which the
small-signal gain coefficient exceeds the loss coefficient. Only a finite
36
number of oscillation frequencies (1, 2, …, m) are possible.
Gain saturation
Laser
 turn-on







steady-state
r loss coefficient



 s
At the moment the laser lases,  = 0 so that () = 0().
As the oscillation builds up in time, the increase in  causes ()
to drop through gain saturation.
When  reaches r, the photon-flux density ceases its growth
and steady-state conditions are attained.
The smaller the loss, the greater the values of .
37
Gain clamping


Gain clamping at the value of the loss.
The steady-state laser internal photon-flux density is
therefore determined by equating the saturated gain
coefficient to the loss coefficient
() = 0() / (1 + /s()) = r

 = s() (0()/r – 1),
= 0,

0() > r
0() ≤ r
This is the mean number of laser photons per second crossing
a unit area in both directions – laser photons traveling in both
directions contribute to the saturation process. The photonflux density for laser photons traveling in a single direction is
thus /2. Spontaneous emission noise is neglected.
38
Steady-state internal photon-flux density



As 0() = N0() and r = Nt(), the steady-state internal
photon-flux density  can be written as
 N0 
  s ( ) 1
 Nt 
N0 > Nt
 0
N0 ≤ Nt
Below threshold, the laser photon-flux density is zero. Any
increase in the pumping rate is manifested as an increase in
the spontaneous-emission photon flux, but there is no
sustained oscillation.
Above threshold, the steady-state internal laser photon-flux
density is directly proportional to the initial population
difference N0, and therefore increases with the pumping rate
R.
39
Steady-state internal photon-flux density


If N0 is twice the threshold value Nt, the photon-flux density
is precisely equal to the saturation value s(), which is the
photon-flux density at which the gain coefficient decreases to
half its maximum value.
Laser oscillation occurs when N0 exceeds Nt. The steadystate value of N then saturates, clamping at the value Nt [just
as 0() is clamped at r]. Above threshold,  is proportion to
N0 – Nt.
Population
difference N
Photon-flux

density
s
Nt
Nt
Pumping rate
N0
2Nt
Nt
Pumping rate
N0
40
Output photon-flux density



Only a portion of the steady-state internal photon-flux density leaves the
resonator in the form of useful light.
The output photon-flux density 0 is that part of the internal photon-flux
density that propagates toward mirror 1 (/2) and is transmitted by it.
If the transmittance of mirror 1 is T, the output photon-flux density is
o  T


2
The corresponding optical intensity of the laser output I0 is
I o  h T

The laser output power is

2
Po  I 0 A
where A is the cross-sectional area of the laser beam
41
Internal photon-number density

The steady-state number of photons per unit volume inside
the resonator Np is related to the steady-state internal photonflux density  (for photons traveling in both directions) by
the simple relation
Np = n/c

The photon-number density corresponding to the steady-state
internal photon-flux density in
 N0 
N p  N p s 
 1
N0 > Nt
 Nt

where Nps = s()n/c is the photon-number density saturation
value.
42
Internal photon-number density

Using the relations
s() = 1/s(), r = (), r = n/cp and () = N() = Nt(),
we can write steady-state photon number density as
p
N p  N 0  N t 
s


N0 > Nt
Interpretation: (N0 – Nt) is the population difference (per unit
volume) in excess of threshold, and (N0 – Nt)/s represents the
rate at which photons are generated which, upon steady-state
operation, is equal to the rate at which photons are lost, Np/p.
The fraction p/s is the ratio of the rate at which photons are
emitted (1/s) to the rate at which they are lost (1/p) .
43
Internal photon-number density



Upon ideal pumping conditions in a four-level laser system,
s ≈ sp and N0 ≈ Rsp, where R is the rate (s-1 cm-3) at which
atoms are pumped.
We can rewrite the steady-state photon-number density as
Np
 R  Rt
R > Rt
p
where Rt = Nt/sp is the threshold value of the pumping rate.
=> Upon steady-state conditions, the overall photon-density loss
rate Np/p is equal to the excess pumping rate R – Rt.
44
Output photon flux and efficiency

If transmission through the laser output mirror is the only source
of resonator loss (which is accounted for in p), and V is the
volume of the active medium, the total output photon flux
(photons per second) is
o  (R  Rt )V

R > Rt
If there are loss mechanisms other than through the output laser
mirror, the output photon flux can be written as
o  e (R  Rt )V
where the extraction efficiency e is the ratio of the loss arising
from the extracted useful light to all of the total losses in the
resonator r.
45
Output photon flux and efficiency

If the useful light exits only through mirror 1,

If, T = 1 – R1 << 1, the extraction efficiency
c
1
 m1
e 

 p ln
R1
 r 2nd
e 
p
TF
T
where we have defined 1/TF = c/2nd, indicating that the
extraction efficiency e can be understood in terms of the
photon lifetime to its round-trip travel time, multiplied by the
mirror transmittance.
 The output laser power is
P0  h 0  e h (R  Rt )V
46
Output photon flux and efficiency


Losses result from other sources as well, such as inefficiency
in the pumping process.
The power conversion efficiency c (also called the overall
efficiency or wall-plug efficiency) is defined at the ratio of the
output optical power Po to the supplied pump power Pp
Po
c 
Pp

Because the laser output power increases linearly with pump
power above threshold, the differential power-conversion
efficiency (also called the slope efficiency) is another measure
of performance
dPo
s 
dPp
47
Light output (power)
Laser optical output vs. pumping
s
Coherent
emission
(Lasing)
Incoherent
emission
Threshold
pumping
Pumping
48
Spectral distribution

The spectral distribution of the generated laser light is
determined both by the spectral lineshape of the active
medium (homogeneous or inhomogeneous broadened) and by
the resonator modes.

The gain condition – 0() > r is satisfied for all
oscillation frequencies lying within a continuous spectral
band of width B centered about the resonance frequency
0. The bandwidth B increases with the spectral linewidth
 and the ratio 0(0)/r. The precise relation depends
on the shape of the function 0().

The phase condition – the oscillation frequency be one of
the resonator modal frequencies q (assuming mode
pulling is negligible). The FWHM linewidth of each
mode is  ≈ q/F
49
Spectral distribution

Loss r

q

Resonator
 modes

 
Allowed
 modes
• Laser oscillation can occur only at frequencies for which the
small-signal gain coefficient exceeds the loss coefficient. Only a finite
50
number of oscillation frequencies (1, 2, …, m) are possible.
Spectral distribution

The number of possible laser modes
M ≈ B/q

However, of these M possible modes, the number of modes
that actually carry optical power depends on the nature of the
lineshape broadening mechanism.

For an inhomogeneously broadened medium (e.g. HeNe,
Nd:glass) all M modes oscillate (albeit at different powers).

For a homogeneously broadened medium (e.g.
semiconductor) these modes compete, rendering fewer modes
(ideally single mode) to oscillate.
51
Laser linewidth





The approximate FWHM linewidth of each laser mode might be
expected to be the cavity resonance linewidth , but it turns out to
be far smaller than this.
The oscillating mode width can be orders of magnitude narrower
than the cavity mode linewidth.
It is limited by the so-called Schawlow-Townes linewidth, which
decreases inversely as the optical power.
This linewidth-narrowing effect is caused by the coherent nature
of the stimulated emission and is a fundamental feature of lasers.
Almost all lasers have linewidths far wider than the SchawlowTownes limit as a result of extraneous effects such as acoustic and
thermal fluctuations of the resonator mirrors, but the limit can be
approached in carefully controlled experiments.
52
Schawlow-Townes relation

A detailed analysis taking into account spontaneous emission yields the
Schawlow-Townes relation for the linewidth of a laser mode in terms of
the laser parameters:
 ST


2 h ( )2
h

N sp 
N sp
2
Pout
2 p Pout
where Pout is the output power of the laser mode being
considered and Nsp (≥ 1) is the spontaneous emission factor.
The effect of spontaneous emission on the linewidth of an oscillating laser
mode enters the above relation through the population densities of the
upper and the lower laser levels in the form of the spontaneous emission
factor.
Because Nsp ≥ 1, the ultimate lower limit of the laser linewidth, which is
known as the Schawlow-Townes limit, is that given above with Nsp = 1.
53
Homogeneously broadened medium




Immediately after being turned on, all laser modes for
which the initial gain is greater than the loss begin to
grow.
Photon-flux densities 1, 2, …, M are created in the
M modes.
Modes whose frequencies lie closest to the transition
central frequency 0 grow most quickly and acquire
the highest photon-flux densities.
These photons interact with the medium and reduce
the gain by depleting the population difference. The
saturated gain is
 0 ( )
 ( ) 
M
j
1 
j1  s ( j )
54
Growth of oscillation in an ideal homogeneously broadened medium


r






• Immediately following laser turn-on, all modal frequencies for which the smallsignal gain coefficient exceeds the loss coefficient begin to grow, with the
central modes growing at the highest rate. After a transient the gain
saturates so that the central modes continue to grow while the peripheral
modes, for which the loss has become greater than the gain, are
attenuated and eventually vanish. Ideally, only a single mode survives.
55
Homogeneously broadened medium




Because the gain coefficient is reduced uniformly, for
modes sufficiently distant from the line center the loss
becomes greater than the gain. These modes lose power
while the more central modes continue to grow, albeit at a
slower rate.
Ultimately, only a single surviving mode maintains a gain
equal to the loss, with the loss exceeding the gain for all
other modes.
Under ideal steady-state conditions, the power in this
preferred mode remains stable, while laser oscillation at all
other modes vanishes.
The surviving mode has the frequency lying closest to 0
(but not necessarily equal to 0).
56
Spatial hole burning




In practice, however, homogeneously broadened
lasers do indeed oscillate on multiple modes because
the different modes occupy different spatial portions of
the active medium.
When oscillation on the most central mode is
established, the gain coefficient can still exceed the
loss coefficient at those locations where the standingwave electric field of the most central mode vanishes.
This phenomenon is called spatial hole burning.
It allows another mode, whose peak fields are located
near the energy nulls of the central mode, the
opportunity to lase.
57
Inhomogeneously broadened medium






In an inhomogeneously broadened medium, the gain represents the
composite envelope of gains of different species of atoms.
The situation immediately after laser turn-on is the same as in the
homogeneously broadened medium.
Modes for which the gain is larger than the loss begin to grow and
the gain decreases.
If the spacing between the modes is larger than the width  of the
constituent atomic lineshape functions, different modes interact
with different atoms.
Atoms whose lineshapes fail to coincide with any of the modes are
ignorant of the presence of photons in the resonator.
Their population difference is therefore not affected and the gain
they provide remains the small-signal (unsaturated) gain.
58
Spectral hole burning






Atoms whose frequencies coincide with modes deplete their
inverted population and their gain saturates, creating “holes” in the
gain spectral profile.
This process is known as spectral hole burning.
This process of saturation by hole burning progresses
independently for the different modes until the gain is equal to the
loss for each mode in steady state.
Modes do not compete because they draw power from different,
rather than shared, atoms.
Many modes oscillate independently, with the central modes
burning deeper holes and growing larger.
The number of modes is typically larger than that in
homogeneously broadened media as spatial hole burning generally
sustains fewer modes than spectral hole burning.
59
Spatial distribution



The spatial distribution of the emitted laser depends on the
geometry of the resonator and on the shape of the active
medium.
So far we have ignored transverse spatial effects by assuming
that the resonator is constructed of two parallel planar mirrors
of infinite extent and that the space between them is filled
with the active medium.
In this idealized geometry the laser output is a plane wave
propagating along the axis of the resonator. But this planarmirror resonator is highly sensitive to misalignment.
60
Spatial distribution









Laser resonators usually have spherical mirrors.
The spherical-mirror resonator supports a Gaussian beam.
A laser using a spherical-mirror resonator may therefore give rise
to an output that takes the form of a Gaussian beam.
The spherical-mirror resonator supports a set of transverse electric
and magnetic modes denoted TEMl,m,q.
Each pair of indexes (l, m) defines a transverse mode with an
associated spatial distribution.
The (0, 0) transverse mode is the Gaussian beam.
Modes of a higher l and m form Hermite-Gaussian beams.
For a given (l, m), the index q defines a number of longitudinal
modes of the same spatial distribution but of different frequencies
q, which are separated by the longitudinal-mode spacing q =
c/2nd, regardless of l and m.
The resonance frequencies of two sets of longitudinal modes
belonging to two different transverse modes are displaced with
respect to each other by some fraction of q.
61
Spatial distribution


TEM0,0
1,1
0,0
B1,1
B0,0
(1,1) modes


(0,0) modes
TEM1,1
The gains and losses for two transverse modes, e.g., (0,0) and (1,1),
usually differ because of their different spatial distributions. A mode can
contribute to the output if it lies in the spectral band within which the
small-signal gain coefficient exceeds the loss coefficient. There can be
multiple longitudinal modes for each transverse mode.
62
Spatial distribution





Because of their different spatial distributions, different
transverse modes undergo different gains and losses.
The (0, 0) Gaussian mode is the most confined about the
optical axis and therefore suffers the least diffraction loss at
the boundaries of the mirrors.
The (1, 1) mode vanishes at points on the optical axis. Thus,
if the laser mirror were blocked by a small central
obstruction, the (1,1) mode would be completely unaffected,
whereas the (0,0) mode would suffer significant loss.
Higher-order modes occupy a larger volume and therefore
can have larger gain.
This difference between the losses and/or gains of different
transverse modes in different geometries determine their
competitive advantage in contributing to the laser oscillation.
63
Spatial distribution





In a homogeneous broadened laser, the strongest mode tends
to suppress the gain for the other modes, but spatial hole
burning can permit a few longitudinal modes to oscillate.
Transverse modes can have substantially different spatial
distributions so that they can readily oscillate simultaneously.
A mode whose energy is concentrated in a given transverse
spatial region saturates the atomic gain in that region, thereby
burning a spatial hole there.
Two transverse modes that do not spatially overlap can
coexist without competition because they draw their energy
from different atoms. Partial spatial overlap between
different transverse modes and atomic migrations (as in
gases) allow for mode competition.
Lasers are often designed to operate on a single transverse
mode. This is usually the (0, 0) Gaussian mode because it has
the smallest beam diameter and can be focused to the smallest
spot size. Oscillation on higher-order modes can be desirable
for purposes such as generating large optical power.
64
Polarization




Each (l, m, q) mode has two degrees of freedom,
corresponding to two independent orthogonal polarizations.
These two polarizations are regarded as two independent
modes.
Because of the circular symmetry of the spherical-mirror
resonator, the two polarization modes of the same l and m
have the same spatial distributions.
If the resonator and the active medium provide equal gains
and losses for both polarizations, the laser will oscillate on
the two modes simultaneously, independently, and with the
same intensity. The laser output is then unpolarized.
65
Pulsed lasers
66
Pulsed lasers





It is sometimes desirable to operate lasers in a pulsed mode as
the optical power can be greatly increased when the output
pulse has a limited duration.
Lasers can be made to emit optical pulses with durations as
short as femtoseconds; the durations can be further
compressed to the attosecond regime by making use of
nonlinear-optical techniques.
Maximum pulse-repetition rates reach more than 100 GHz.
Maximum pulse energies reach from fJ to MJ, while peak
powers extend to more than 10 MW and peak intensities
reach 10 TW/cm2.
Some lasers can only be operated in a pulsed mode as CW
operation cannot be sustained.
67

Methods of pulsing lasers


The most direct method of obtaining pulsed
light from a laser is to use a CW laser in
conjunction with an external modulator that
transmits the light only during selected short
time intervals.
This method has two drawbacks:


the scheme is inefficient as it blocks energy
during the off-time of the pulse train.
the peak power of the pulse cannot exceed the
steady power of the CW source.
68

Methods of pulsing lasers





More efficient pulsing schemes are based on turning the laser
itself on and off by means of an internal modulation process,
designed so that energy is stored during the off-time and
released during the on-time.
Energy may be stored either in the resonator, in the form of
light that is periodically permitted to escape, or in the atomic
system, in the form of a population inversion that is released
periodically by allowing the system to oscillate.
These schemes permit short laser pulses to be generated with
peak powers far in excess of the constant power delivered by
CW lasers.
Four common methods used for the internal modulation of
laser light are: gain switching, Q-switching, cavity dumping
and mode locking.
Here, we only focus on mode locking.
69

Example of mode-locked lasers




Ti:sapphire is a popular mode-locked laser.
With the ability to tune the center wavelength over the
range 700 – 1050 nm, and with individual pulses as
short as 10-fs duration
A commercial version of this laser readily delivers 50nJ pulses of duration 10 fs and peak power 1 MW, at a
repetition rate of 80 MHz.
Mode-locked lasers find applications including timeresolved measurements, imaging, metrology,
communications, materials processing, and clinical
medicine.
70
Mode locking






Mode locking is the most important technique for the generation of
repetitive, ultrashort laser pulses.
The principle of mode locking is not based on the transient
dynamics of a laser. Instead, a mode-locked laser operates in a
dynamic steady state.
A laser can oscillate on many longitudinal modes, with frequencies
that are equally separated by the Fabry-Perot intermodal spacing
q = c/2nd.
Although these modes normally oscillate independently (they are
then called free-running modes), external means can be used to
couple them and lock their phases together.
The modes can then be regarded as the components of a Fourierseries expansion of a periodic function of time of period TF = 1/q
= 2nd/c, which constitute a periodic pulse train.
The multiple monochromatic waves of equally spaced frequencies
with locked phase constructively interfere.
71

Mode locking


The mode-locking operation is accomplished by a nonlinear
optical element known as the mode locker that is placed
inside the laser cavity, typically near one end of the cavity if
the laser has the configuration of a linear cavity.
In the frequency domain, mode locking is a process that
generates a train of short laser pulses by locking multiple
longitudinal laser modes in phase.
The function of the mode locker in the frequency domain is
thus to lock the phases of the oscillating modes together
through nonlinear interactions among the mode fields.
Mode locker

Light output (laser)
72
Mode locking


In the time domain, the mode-locking process can be
understood as a regenerative pulse-generating process by
which a short pulse circulating inside the laser cavity is
formed when the laser reaches steady state.
The action of the mode locker in the time domain resembles
that of a pulse-shaping optical shutter that opens periodically
in synchronism with the arrival at the mode locker of the laser
pulse circulating in the cavity.
Consequently, the output of a mode-locked laser is a train of
regularly spaced pulses of identical pulse envelope.
Mode locker
d
2d
73
Mode locking: two modes


The simplest case of multimode oscillation is when there are
only two oscillating longitudinal modes of frequencies 1 and
2.
The total laser field at a fixed location is
E(t)  E1ei1 (t )ei1t  E2 ei 2 (t )ei2t


where E1 and E2 are the amplitudes of the field amplitudes
and 1 and 2 are the phases.
With all the phase information included in 1 and 2, E1 and
E2 are positive real quantities.
The intensity of the laser is
I(t)  E(t)  E12  E222  2E1E2 cos  (1   2 )t  1 (t)   2 (t)
2
74
Mode locking: two modes



In general, the phases can vary with time.
If 1(t) and 2(t) vary randomly with time on a characteristic
time scale that is shorter than 2/(1-2), the beat note of the
two frequencies cannot be observed. In this case, the output
of the laser has a constant intensity that is the incoherent sum
of the intensities of the individual modes.
This situation represents the ordinary multimode oscillation
of a CW laser.
75
Mode locking: two modes

If 1 and 2 are time independent, the laser intensity becomes
periodically modulated with a period of 2/(1-2) defined
by the beat frequency.

The modulation depth of this intensity profile depends on the
ratio between E1 and E2. When E1 = E2, the modulation depth
is 100% with Imin = 0.

In this case, I(t) resembles a train of periodic “pulses” that
have a duty cycle of 50% and a peak intensity of twice the
incoherent sum of the intensities.

This is coherent mode beating between two oscillating
modes.
76
Coherent mode beating between two modes
Intensity
Max.
=(1+1)2
Incoherent
sum of
the
intensities
time
(Assume E1 = E2 for 100% modulation depth)
77
Properties of a mode-locked pulse train

If each of the laser modes is approximated by a uniform plane
wave propagating in the z direction with a velocity c/n, we
may write the total complex wavefunction of the field in the
form of a sum:

nz 
U(z, t)   Aq exp i2 q (t  )

c 
q



where q = 0 + qq, q = 0, ±1, ±2, … is the frequency of
mode q, and Aq is its complex envelope.
Here we assume that the q = 0 mode coincides with the
central frequency 0 of the atomic lineshape.
The magnitude |Aq| may be determined from knowledge of
the spectral profile of the gain and the resonator loss.
As the modes interact with different groups of atoms in an
inhomogeneously broadened medium, their phases arg{Aq}
are random and statistically independent.
78

Properties of a mode-locked pulse train

Substituting the q = 0 + qq into U(z, t), we obtain

nz
nz 
U(z, t)  A(t  )exp i2 0 (t  )

c
c 
where the complex envelope A(t)
 iq2 t 
A(t)   Aq exp 

 TF 
q
1
2nd
TF 

 q
c


The complex envelope A(t) is a periodic function of the period TF,
and A(t-nz/c) is a periodic function of z of period (c/n)TF = 2d.
If the magnitudes and phases of the complex coefficients Aq are
properly chosen, A(t) may be made to take the form of periodic
narrow pulses.
79

Properties of a mode-locked pulse train

Consider, for example, M modes (q = 0, ±1, … ±S, s.t. M =
2S+1), whose complex coefficients are all equal, Aq = A, q =
0, ±1, …, ±S.
S
S1
S
 iq2 t 
x
x

x
q
A(t)  A  exp 

A
x

A

A


x 1
 TF 
qS
qS
S
S
1
2
1
2
x
x x
S

1
2
1
2
sin(M  t / TF )
A(t)  A
sin( t / TF )

The optical intensity I(t, z) = |A(t-nz/c)|2
sin 2 [M  (t  nz / c) / TF ]
I(t, z)  A
sin 2 [ (t  nz / c) / TF ]
2
80
Intensity of periodic pulse train
intensity
TF
Max.
= 202
TF/M
M
Incoherent sum
time
M = 20, TF = 300, I = 1
81
Properties of a mode-locked pulse train

The shape of the mode-locked laser pulse train is
therefore dependent on the number of modes M, which
is proportional to the atomic linewidth or .

If M ≈ /q, then pulse = TF/M ≈ 1/.

The pulse duration pulse is therefore inversely
proportional to the atomic linewidth .


Because can be quite large, very narrow modelocked laser pulses can be generated.
The ratio between the peak and mean intensities is
equal to the number of modes M, which can also be
quite large.
82
Properties of a mode-locked pulse train


The period of the pulse train is TF = 2nd/c. This is just
the time for a single round trip of reflection within the
resonator.
The repetition rate of the pulses = 1/TF = c/2nd = q

The light in a mode-locked laser can be regarded as a
single narrow pulse of photons reflecting back and
forth between mirrors of the resonator.

At each reflection from the output mirror, a fraction of
the photons is transmitted in the form of a pulse of
light.
83

The transmitted pulses are separated by the distance 2d
Properties of a mode-locked pulse train

Characteristic properties of a mode-locked
pulse train
Temporal period 2nd/c
Pulse duration
pulse=TF/M=1/
Spatial period
2d
Pulse length
dpulse= 2d/M
Mean intensity
I
Peak intensity
Ip = MI
84
Properties of a mode-locked pulse train

The mode-locked laser pulse reflects back and forth between
the mirrors of the resonator. Each time it reaches the output
mirror it transmits a short optical pulse. The transmitted
pulses are separated by the distance 2d and travel with
velocity c. The switch opens only when the pulse reaches it
and only for the duration of the pulse. The periodic pulse
train is therefore unaffected by the presence of the switch.
Other wave patterns suffer losses and are not permitted to
oscillate.
Mode locker
d
2d
85
Properties of a mode-locked pulse train

E.g. Consider a Nd3+:glass laser operating at 0 = 1.05 m. It has
a refractive index n = 1.5 and a linewidth  = 7 THz.

The pulse duration pulse = 1/ ≈ 140 fs and the pulse length dpulse
≈ 42 m.

If the resonator has a length d = 15 cm, the mode separation is 
= c/2nd = 1 GHz, which means that M = q = 7000 modes.

The peak intensity is therefore 7000 times greater than the average
intensity.

In media with broad linewidths, mode locking is generally more
advantageous than Q-switching for obtaining short pulses.
Gas lasers generally have narrow atomic linewidths, s.t. ultrashort
pulses cannot be obtained by mode locking.

86
Methods of mode locking







We consider active mode locking and passive mode locking.
Suppose that an optical switch controlled by an external applied signal is placed
inside the resonator, which blocks the light at all times, except when the pulse is
about to cross it, whereupon it opens for the duration of the pulse.
As the pulse itself is permitted to pass, it is not affected by the presence of the
switch and the pulse train continues uninterrupted.
In the absence of phase locking, the individual modes have different phases that
are determined by the random conditions at the onset of their oscillation.
If the phases happen, by accident, to take on equal values, the sum of the modes
will form a giant pulse that would not be affected by the presence of the switch.
Any other combination of phases would form a field distribution that is totally or
partially blocked by the switch, which adds to the losses of the system. Therefore,
in the presence of the switch, only when the modes have equal phases can lasing
occur.
The laser waits for the “lucky accident” of such phases, but once the
oscillations start, they continue to be locked.
87
Methods of mode locking






A passive switch such as saturable absorber may also be used to attain
mode locking.
A saturable absorber is a medium whose absorption coefficient decreases
as the intensity of the light passing through it increases.
It thus transmits intense pulses with relatively little absorption while
absorbing weak ones.
Oscillation can therefore occur only when the phases of the different
modes are related to each other in such a way that they form an intense
pulse that can then pass through the switch.
Semiconductor saturable-absorber mirrors, which are saturable absorbers
operating in reflection, are in widespread use. The more intense the light,
the greater the reflection. They work for 800 – 1600nm wavelengths, fs to
ns pulse durations, and power levels from mW to hundreds of W.
Saturable absorbers can also produce Q-switched modelocking, in which
the laser emits collections of modelocked pulses within a Q-switching
envelope.
88
Methods of mode locking






Passive mode locking can also be implemented by means of Kerr-lens
mode locking, which relies on a nonlinear-optical phenomenon in which
the refractive index of a material changes with optical intensity.
A Kerr medium, such as the gain medium itself, or a material placed
within the laser cavity, acts as a lens with a focal length inversely
proportional to the intensity. (refractive index change n  light intensity
I)
By placing an aperture at a proper position within the cavity, the Kerr lens
reduces the area of the laser mode for high intensities s.t. the light passes
through the aperture.
Alternatively, the reduced modal area in the gain medium can be used to
increase its overlap with the strongly focused pump beam, thereby
increasing the effective gain.
The Kerr-lens approach is inherently broadband because of the parametric
nature of the process.
The rapid recovery inherent in passive mode locking generally leads to
shorter optical pulses than can be attained with active mode locking.
89
Diode-Pumped Solid-State Ultrafast laser
-Coherent Vitesse 800
Specification
Ref: http://www.coherent.com/Products/index.cfm?1439/Vitesse-Family
90
Cavity schematic: Passive mode-locking
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Starter: initiate mode locking by perturbing the cavity. (changing the cavity length by
shaking a piece of glass to initiate lasing for a set of cavity longitudinal modes）
Self-mode-locking: Ti:Sapphire itself serves as both the laser medium and Kerr-lens
Long cavity and angle-cut crystal (usually brewster angle): prevent etalon effects;
preserve large M (number of longitudinal modes); increase peak power; shorten pulse
width
Slit: blocks the CW wide beam and forces energy into mode-locked lasing.- shorter
pulse(the CW components beam size is wider than the pulse (mode-locked) beam size. )
NDM: negative-dispersion mirror serves as additional dispersion compensation，to
prevent pulse broadening
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Pump: green laser-532 nm