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Lecture : 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.
2
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
3
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
44
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
reflectivity (R1 ~ 100 %) R2 < 100 %
Gain mediumLight output (laser)
5
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
6
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.
6
gain
loss
Oscillatorpower
Steady-state power
7
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
8
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
9
Theory of laser oscillation
9
10
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
11
Small-signal gain coefficient When the photon-flux density is small, the gain coefficient
is
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
0 ( ) N0 ( ) N0c2
8n2 2 sp
g( )
12
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
for a homogeneously broadened medium, where
12
N N0
1 /s ( )
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 = 2sp for three-level pumping
13
Saturated gain coefficient The gain coefficient of the saturated amplifier is therefore
reduced to (for homogeneous broadening)
The laser amplification process also introduces a phase shift. When the lineshape is Lorentzian with linewidth ,
The amplifier phase shift per unit length is
13
( ) N ( ) 0 ( )1 /s ( )
g( ) / 2( 0 )2 ( / 2)2
( ) 0
( )
1414
Gain coefficient and phase-shift coefficient for a laser amplifier with a Lorentzian lineshape function
Phase-shift coefficient()
gain coefficient()
15
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 = 2n/c
The resonator sustains only frequencies that correspond to a round-trip phase shift that is a multiple of 2.
15
k 2d 2nc
2d q2 q = 1, 2, …
1616
Fabry-Perot resonators
• Only standing waves at discrete wavelengths exist in the cavity.=> the laser wavelengths must match the cavity resonance wavelengths.
The resonance condition: 2nd = q
where q is an integer (=1, 2, …), known as the longitudinal mode order, k = 2n/2n/c
drefractive index n
or 2kd = 2q
1717
R1 R2
Pout
R1
R2Pout
Pout
Pout Pout Pout
Pout Pout
Resonant optical cavities
Bragg grating
Fiber/waveguidering resonator
1818
Resonant optical cavities
A linear cavity with two end mirrors is known as a Fabry-Perot 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 cavitywith a standing oscillating field.
A folded cavity can also be a non-Fabry-Perot ring cavitythat 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.
19
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:
where lRT is the round-trip optical path length (=2nd for Fabry-Perot cavities). 19
TF lRT
c
2020
• The modes along the cavity axis is referred to as longitudinal modes.• Many ’s may satisfy the resonance condition => multimode cavity
The longitudinal mode spacing (free-spectral range):
= 2 / 2nd
intensity
q q-2q+2
q+1 q-1
…..…..
Longitudinal mode spacing
2121
e.g. A semiconductor laser diode has a cavity length 400 mwith a refractive index of 3.5. The peak emission wavelength fromthe device is 0.8 m. Determine the longitudinal mode orderand the frequency spacing of the neighboring modes.
• The longitudinal mode order q = 2nd/ ~ 3500• The frequency spacing q = c/2nd ~ 100 GHz
• The longitudinal mode frequencies:
= q = qc/2nd
• The mode spacing (free-spectral range) in frequency unit:
q = c/2nd
Longitudinal mode spacing
2222
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(-2sd)
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(-2rd) = R1R2exp(-2sd)
2323
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))
2424
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
= 1/2p
The cavity quality factor Q at resonance frequency q is Q = q × (energy stored in the resonator/average power dissipation)
= q p = q/
2525
The finesse of the resonator
F ≈ q/where q = c/2nd
When the resonator losses are small and the finesse is large
F ≈ /(rd)
q q q
q c/2nd
Photon lifetime and resonator linewidth
2626
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
27
Gain Condition: Laser threshold The initiation of laser oscillation requires that the small-
signal gain coefficient be greater than the loss coefficient
Or, the gain be greater than the loss. Translates this to the population difference
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
0 ( ) r
N0 0 ( ) ( )
r
( ) Nt
28
Gain condition: Laser threshold
r may be written in terms of the photon lifetime,
Thus, Nt is given as
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
r n
c p
Nt n
c p ( )
29
Threshold population difference By using the transition cross section
we find another expression for the threshold population difference,
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
( ) c2
8n2 2 sp
g( )
Nt 8n3 2
c3
sp
p
1g( )
30
Threshold population difference The minimum population difference for oscillation at the central
frequency 0 turns out to be
Nt is directly proportional to the linewidth . If the transition is limited by lifetime broadening with a decay time sp,
and = 1/2sp
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
Nt 2n3 2
c3
2 sp
p
Nt 2n3 2
c3 p
2n2 2r
c2
3131
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 = 2q, 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
3232
Frequency pulling
The laser oscillation frequencies fall near the cold-resonator modes – they are pulled slightly toward the atomic resonance central frequency 0.
Amplifier gain coefficient
Laser oscillation modes
Cold-resonator modes
q
33
Laser output characteristics
33
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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 = 2q, 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
3535
Laser oscillation: the unsaturated gain must exceed the loss
• sub-threshold (incoherent emission)
• Threshold (oscillation begins,start to emit coherentlight)
• above-threshold(increase incoherent output power)
loss (assumeconstant)gain
gain < loss gain = loss
loss loss
gain > loss
36
Laser oscillation
36
Loss r
Allowed modes
Resonator modes
q
• Laser oscillation can occur only at frequencies for which thesmall-signal gain coefficient exceeds the loss coefficient. Only a finite number of oscillation frequencies (1, 2, …, m) are possible.
3737
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 .
s
Laser turn-on
r loss coefficient
steady-state
Gain saturation
3838
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() > r
= 0, 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 photon-flux density for laser photons traveling in a single direction is thus /2. Spontaneous emission noise is neglected.
Gain clamping
39
Steady-state internal photon-flux density As 0() = N0() and r = Nt(), the steady-state internal
photon-flux density can be written as
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
s ( ) N0
Nt
1
0
N0 > Nt
N0 ≤ Nt
40
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 steady-state value of N then saturates, clamping at the value Nt [just as 0() is clamped at r]. Above threshold, is proportion to N0 – Nt.
40
N
N0Nt
Nt
Pumping rate
Population difference
N0Nt
s
Pumping rate
Photon-fluxdensity
2Nt
41
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
The corresponding optical intensity of the laser output I0 is
The laser output power is
where A is the cross-sectional area of the laser beam41
o T 2
Io hT 2
Po I0A
42
Internal photon-number density The steady-state number of photons per unit volume inside
the resonator Np is related to the steady-state internal photon-flux 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
where Nps = s()n/c is the photon-number density saturation value.
42
10
tspp N
NNN N0 > Nt
43
Internal photon-number density Using the relations
s() = 1/s(), r = (), r = n/cp and () = N() = Nt(), we can write steady-state photon number density as
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) .
s
ptp NNN
0 N0 > Nt
44
Internal photon-number density Upon ideal pumping conditions in a four-level laser system, s ≈ sp and N0 ≈ Rsp, where R is the rate (s-1 cm-3) at which
atoms are pumped. We can rewrite the steady-state photon-number density as
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.
tp
p RRN
R > Rt
45
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
If there are loss mechanisms other than through the output lasermirror, the output photon flux can be written as
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
o (R Rt )V
o e(R Rt )V
R > Rt
46
Output photon flux and efficiency If the useful light exits only through mirror 1,
If, T = 1 – R1 << 1, the extraction efficiency
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
46
e m1
r
c2nd
p ln 1R1
e p
TF
T
P0 h0 eh (R Rt )V
47
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
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
47
c Po
Pp
s dPo
dPp
4848
Laser optical output vs. pumpingLi
ght o
utpu
t (po
wer
)
Pumping
Incoherent emission
Coherentemission(Lasing)
Threshold pumping
s
4949
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
50
Spectral distribution
50
Loss r
Allowed modes
Resonator modes
q
• Laser oscillation can occur only at frequencies for which thesmall-signal gain coefficient exceeds the loss coefficient. Only a finite number of oscillation frequencies (1, 2, …, m) are possible.
51
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
52
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 Schawlow-Townes 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
53
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:
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
ST 2h ( )2
Pout
Nsp h
2 p2Pout
Nsp
54
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
54
( ) 0 ( )
1 j
s ( j )j1
M
5555
Growth of oscillation in an ideal homogeneously broadened medium
r
• Immediately following laser turn-on, all modal frequencies for which the small-signal gain coefficient exceeds the loss coefficient begin to grow, with thecentral modes growing at the highest rate. After a transient the gain saturates so that the central modes continue to grow while the peripheralmodes, for which the loss has become greater than the gain, are attenuated and eventually vanish. Ideally, only a single mode survives.
56
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
57
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 standing-wave 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
58
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
59
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
60
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 mirrorsof 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 planar-mirror resonator is highly sensitive to misalignment.
60
61
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.
62
Spatial distribution
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
0,0
1,1
(0,0) modes(1,1) modes
TEM0,0
TEM1,1
B1,1
B0,0
63
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 atthe 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.
64
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, therebyburning 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.
65
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
66
Pulsed lasers
66
67
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.
68
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.
69
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 atomicsystem, 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.
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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 50-nJ pulses of duration 10 fs and peak power 1 MW, at a repetition rate of 80 MHz.
Mode-locked lasers find applications including time-resolved measurements, imaging, metrology, communications, materials processing, and clinical medicine.
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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 Fourier-series 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.
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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 ifthe 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.
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Light output (laser)M
ode
lock
er
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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.
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dMode locker 2d
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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
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
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E(t) E1ei1(t )ei1t E2e
i2 (t )ei2t
I(t) E(t) 2 E1
2 E222 2E1E2 cos (1 2 )t 1(t)2 (t)
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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 oscillationof a CW laser.
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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.
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Coherent mode beating between two modes
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time
Intensity
Incoherent sum of the intensities
Max.=(1+1)2
(Assume E1 = E2 for 100% modulation depth)
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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:
where q = 0 + qq, 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.
U(z, t) Aq expq i2q (t nz
c)
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Properties of a mode-locked pulse train Substituting the q = 0 + qq into U(z, t), we obtain
where the complex envelope A(t)
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.
U(z, t) A(t nzc
)exp i20 (t nzc
)
A(t) Aqq exp iq2 t
TF
TF 1
q
2ndc
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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.
The optical intensity I(t, z) = |A(t-nz/c)|2
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A(t) A exp iq2 tTF
qS
S
A xq
qS
S
A xS1 xS
x 1 A x
S12 x
S12
x12 x
12
A(t) A sin(M t / TF )sin( t / TF )
I(t, z) A 2 sin2[M (t nz / c) / TF ]sin2[ (t nz / c) / TF ]
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Intensity of periodic pulse train
81M = 20, TF = 300, I = 1
time
intensity
Max. = 202
TF
TF/M
Incoherent sum
M
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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 mode-locked 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
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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.
The transmitted pulses are separated by the distance 2d83
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Properties of a mode-locked pulse train
Characteristic properties of a mode-locked pulse train
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Temporal period
Spatial period
Mean intensity
Pulse duration
Pulse length
Peak intensity
2nd/c
2d
I
pulse=TF/M=1/
dpulse= 2d/M
Ip = MI
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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.
85Mode locker
d2d
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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. ultrashortpulses cannot be obtained by mode locking.
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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.
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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.
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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.
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Diode-Pumped Solid-State Ultrafast laser-Coherent Vitesse 800
Specification
Ref: http://www.coherent.com/Products/index.cfm?1439/Vitesse-Family
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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
Pump: green laser-532 nm
Cavity schematic: Passive mode-locking
