lect9 laser oscillators(3)

7/27/2019 Lect9 Laser Oscillators(3)

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Lecture 9: Laser oscillators

Theory of laser oscillation

Laser output characteristics

Pulsed lasers

References: This lecture follows the materials from Fundamentals of Photonics, 2nded.,Saleh & Teich, Chapter 15. Also from Photonic Devices, Jia-Ming Liu,

Chapter 11.


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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) toterawatts, operating from continuous wave (CW) tofemtosecond (even attosecond) pulses, and having spectrallinewidths 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 m3for so-callednanolasers), to gigantic, of an entire building, to stellar, ofastronomical dimensions.

An optical gain medium can amplify an optical field throughstimulated emission. 2


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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 theamplifier.

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 saturationof the amplifier gain, and the system reaches a steady state in

which an output signal is created at the frequency of theresonant amplifier. 3


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Laser oscillators

In a practical laser device, it is generally necessary to havecertain positive opticalfeedbackin 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 selectivefeedback to the amplified optical field.

In many lasers the optical feedback is provided by placing the

gain medium inside aFabry-Perot

cavity, formed by usingtwo mirrors or highly reflecting surfaces

reflectivity (R1

~ 100 %) R 2

< 100 %

Gain mediumLight output (laser)


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Intro

Two conditions must be satisfied for oscillation to occur:

The amplifier gain must be greater than the loss in thefeedback system s.t. net gain is incurred in a round tripthrough the feedback loop.

The total phase shift in a single round trip must be amultiple of 2 s.t. the feedback input phase matches thephase of the original input.

If these conditions are satisfied, the system becomes unstable and

oscillation begins.

5


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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 equalto the loss.

The gain then just compensates the loss s.t. the cycle of

amplification and feedback is repeated without change andsteady-state oscillation prevails.

6

gain

loss

Oscillatorpower

Steady-state

power


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Intro

Because the gain andphase shift are functions offrequency, the two oscillation conditions are satisfiedonly at one or several frequencies, which are theresonance frequencies of the oscillator.

The useful output is extracted by coupling a portion ofthe 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


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Intro

The laser is anoscillator in which theamplifier is thepumped active medium.

Gain saturation is a basic property of laser amplifiers. Feedback is enabled by placing the active medium in

anoptical resonator, which in its simplest formreflects 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


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Theory of laser oscillation

9


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Laser amplification The laser amplifier is a narrowband coherent amplifier of

light.

Amplification is attained by stimulated emission from anatomic or molecular system with a transition whosepopulation 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 broadeningmechanism (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) (), whichgoverns the rate at which the photon-flux density (or theoptical intensity I = h) increases. 10


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Small-signal gain coefficient When the photon-flux density is small, thegain 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 ofthe lower laser level (i.e. g1=g2). N0 increases withincreasing pumping rate.

() = transition cross section (e() = a() = ())

sp = spontaneous lifetime

g() = transition lineshape11

0 () N0() N0 c2

8n22spg()


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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 homogeneouslybroadened medium, where

12

N N0

1/s ()

s () 1s()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


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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

()


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Gain coefficient and phase-shift coefficient for a laser amplifierwith a Lorentzian lineshape function

Phase-shift

coefficient

()

gaincoefficient

()


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Optical resonators

Optical feedback is attained by placing the active medium inan optical resonator.

A Fabry-Perot resonator, comprising two mirrors separatedby a distance d, contains the active medium (refractive indexn).

Travel through the medium introduces a phase shift per unitlength equal to the wavenumber k = 2n/c

The resonator sustains only frequencies that correspond to around-trip phase shift that is a multiple of 2.

15

k 2d2n

c 2d q2 q = 1, 2,


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Fabry-Perot resonators

Onlystanding 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

d

refractive index n

or 2kd = 2q


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R1 R2

Pout

R1

R2

Pout

Pout

Pout Pout Pout

Pout Pout

Resonant optical cavities

Bragggrating

Fiber/waveguideringresonator


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Resonant optical cavities

A linearcavity with two end mirrors is known as a Fabry-Perotcavity because it takes the form of a Fabry-Perot

interferometer. In the case of semiconductor diodes, thediode end facets form the two end mirrors.

Afoldedcavity 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 inopposite directions (clockwise, counterclockwise). Ringcavity can be made of multiple mirrors infree space, or in theform offiber/waveguide-based devices.

The optical cavity can also comprise a distributed Bragggrating with distributed feedback. Distributed Feedback(DFB) diode lasers are the most common single-mode laserdiodes for optical communications.


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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 oppositedirections in a ring cavity are independent of each other evenwhen 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 around 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 forFabry-Perot cavities). 19

TFlRT

c


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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


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e.g. A semiconductor laser diode has a cavity length 400 mwith a refractive index of 3.5. The peak emission wavelength from

the device is 0.8 m. Determine the longitudinal mode orderand thefrequency 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


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Resonator losses

The resonator also contributes to losses. Absorption andscattering of light in the medium introduces apower 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 atotal effective distributed loss coefficientr

exp(-2r

d) = R1

R2

exp(-2s

d)


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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 1and 2.

The contribution from both mirrors

m = m1 + m2 = (1/2d) ln(1/(R1R2))


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Photon lifetime and resonator linewidth Definephoton lifetime (cavity lifetime) p as the 1/e-power

lifetime for photons inside the cavity of refractive index n:

exp(-rpc/n) = exp(-1)

p = n/rc

The resonator linewidth (FWHM) is inverselyproportional to the cavity lifetime

= 1/2p

The cavity quality factorQ at resonance frequency qis

Q = q (energy stored in the resonator/average power dissipation)= qp = q/


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Thefinesse of the resonator

F q

/where q= c/2nd

When the resonator losses are small and thefinesse is large

F /(rd)

q q q

q c/2ndPhoton lifetime and resonator linewidth


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Conditions for laser oscillation

Two conditions must be satisfied for the laser to oscillate(lase):

The gain condition determines the minimumpopulationdifference, and thus thepumping thresholdrequired for

lasing Thephase condition determines the frequency (or

frequencies) at which oscillation takes place


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Gain Condition: Laser threshold The initiation of laser oscillation requires that the small-

signal gain coefficientbe greater than the loss coefficient

Or, the gain be greater than the loss.

Translates this to the population difference

where Nt is thethreshold population difference. Nt, which

is proportional to r, determines the minimum pumping rate

Rt for the initiation of laser oscillation. 27

0 () r

N00 ()

() r

()Nt


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Gain condition: Laser threshold rmay be written in terms of the photon lifetime,

Thus, Nt is given as

The threshold population density difference is thereforedirectly proportional to rand inversely proportional to p.Higher loss (shorter photon lifetime) requires more vigorouspumping to attain lasing.

28

r n

cp

Nt n

cp()


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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 readilyattained, at the frequency where the lineshape function islargest, i.e., at its central frequency = 0.

For a Lorentzian lineshape function, g(0) = 2/29

() c2

8n22spg()

Nt8n32

c3sp

p

1

g()


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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 toattain oscillation is a simple function of the frequency and the photonlifetime p. Laser oscillation becomes more difficult to attain as the

frequency increases.

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Nt2n3

2

c32sp

p

Nt2n32

c3p

2n22rc2


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Phase condition: laser frequencies

Thephase condition of oscillation requires that the phaseshift 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()dis small, then the laser modes are given by the cold (orpassive) cavity modes.

In general, 2()dgives rise to a set of oscillation frequencies

q that are slightly displaced from the cold-resonatorfrequencies q. The cold-resonator modal frequencies are allpulledslightly

toward the central frequency of the atomic transitionfrequency pulling or mode pulling


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Frequency pulling

The laser oscillation frequencies fall near the cold-resonatormodes they are pulled slightly toward the atomic resonance

central frequency 0.

Amplifiergain coefficient

Laser oscillationmodes

Cold-resonatormodes

q


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Laser outputcharacteristics

33


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Laser power A laser pumped above the thresholdexhibits a small-signal

gain coefficient0() that is greaterthan the loss coefficient r.

0() > r

Laser oscillation may then begin, provided that thephasecondition is satisfied.

2kd + 2()d = 2q, q = 1,2,

As the photon-flux density inside the resonator increases, thegain coefficient() begins to uniformly drop (for

homogeneouslybroadened media)() = 0() / (1 + /s())

As long as the gain coefficient remains larger than the loss

coefficient, the photon flux continues to grow. 34


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Laser oscillation: the unsaturated gain must exceed the loss

sub-threshold(incoherent emission)

Threshold(oscillation begins,

start to emit coherent

light)

above-threshold(increase in

coherent

output

power)

loss (assumeconstant)gain

gain < loss gain = loss

loss loss

gain > loss


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Laser oscillation

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Loss r

Allowedmodes

Resonatormodes

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.


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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

Laserturn-on

rloss coefficient

steady-state

Gain saturation


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Gain clamping at the value of the loss. The steady-state laser internalphoton-flux density is

therefore determined by equating the saturated gaincoefficient 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 crossinga unit area in both directions laserphotons traveling in bothdirections contribute to the saturation process. The photon-flux density for laser photons traveling in a single direction isthus/2. Spontaneous emission noise is neglected.

Gain clamping


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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. Anyincrease in the pumping rate is manifested as an increase inthe spontaneous-emissionphoton 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 rateR.39

s () N0

Nt1

0

N0 > Nt

N0Nt


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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 tohalf 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 toN0 Nt.

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N

N0Nt

Nt

Pumping rate

Populationdifference

N0Nt

s

Pumping rate

Photon-fluxdensity

2Nt


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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-fluxdensity 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

IohT

2

PoI0A


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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) bythe simple relation

Np = n/c

The photon-number density corresponding to the steady-stateinternal photon-flux density in

where Nps = s()n/c is the photon-number density saturationvalue.

42

10

tspp N

N

NN N0 > Nt


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Internal photon-number density Using the relations

s() = 1/s(), r= (), r= n/cp and () = N() = Nt(),

we can writesteady-state photon number density as

Interpretation: (N0 Nt) is the population difference (per unitvolume) in excess of threshold, and (N0 Nt)/s represents therate at which photons are generated which, upon steady-stateoperation, 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 areemitted (1/s) to the rate at which they are lost (1/p) .

s

p

tp NNN

0 N0 > Nt


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Internal photon-number density Upon ideal pumping conditions in a four-level laser system,

ssp 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 lossrate Np/p is equal to the excess pumping rate R Rt.

t

p

pRR

N

R > Rt


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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 arisingfrom the extracted useful light to all of the total losses in the

resonator r. 45

o (R Rt)V

oe (R

Rt)V

R > Rt


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Output photon flux and efficiency If the useful light exits only through mirror 1,

If, T= 1 R1


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Output photon flux and efficiency Losses result from other sources as well, such as inefficiency

in the pumping process.

Thepower conversion efficiency c (also called the overallefficiency or wall-plug efficiency) is defined at the ratio of theoutput optical power Po to the supplied pump power Pp

Because the laser output power increases linearly with pumppower above threshold, the differential power-conversion

efficiency (also called the slope efficiency) is another measureof performance

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c

Po

Pp

sdPo

dPp


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Laser optical output vs. pumping

Lightoutput(power)

Pumping

Incoherent

emission

Coherentemission

(Lasing)

Threshold

pumping

s


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Spectral distribution The spectral distribution of the generated laser light is

determined both by the spectral lineshape of the activemedium (homogeneous or inhomogeneous broadened) and bythe resonator modes.

The gain condition0() > ris satisfied for alloscillation frequencies lying within a continuous spectralband of width B centered about the resonance frequency0. The bandwidth B increases with the spectral linewidth and the ratio 0(0)/r. The precise relation dependson the shape of the function 0().

Thephase condition the oscillation frequency be one ofthe resonator modal frequenciesq(assuming modepulling is negligible). The FWHM linewidth of eachmode is

q/F


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Spectral distribution

50

Loss r

Allowedmodes

Resonatormodes

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.


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Spectral distribution The number of possible laser modes

M B/q

However, of these M possible modes, the number of modesthat actually carry optical power depends on the nature of thelineshape broadening mechanism.

For an inhomogeneouslybroadened medium (e.g. HeNe,

Nd:glass) all M modes oscillate (albeit at different powers).

For a homogeneouslybroadened medium (e.g.semiconductor) these modes compete, rendering fewer modes

(ideally single mode) to oscillate. 51


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Laser linewidth The approximate FWHM linewidth of each laser mode might be

expected to be the cavity resonance linewidth , but it turns out tobefar smallerthan this.

The oscillating mode width can be orders of magnitude narrowerthan 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 afundamental feature of lasers.

Almost all lasers have linewidths far wider than the Schawlow-

Townes limit as a result of extraneous effects such as acoustic andthermal fluctuations of the resonator mirrors, but the limit can beapproached in carefully controlled experiments.

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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 beingconsidered and Nsp ( 1) is the spontaneous emission factor.

The effect of spontaneous emission on the linewidth of an oscillating lasermode enters the above relation through the population densities of the

upper and the lower laser levels in the form of the spontaneous emissionfactor.

Because Nsp 1, the ultimate lower limit of the laser linewidth, which isknown as the Schawlow-Townes limit, is that given above with Nsp = 1.

53

ST2h()2

PoutNsp

h

2p2Pout

Nsp


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Homogeneously broadened medium Immediately after being turned on, all laser modes for

which the initial gain is greater than the loss begin togrow.

Photon-flux densities 1, 2, , M are created in theM modes.

Modes whose frequencies lie closest to the transition

central frequency 0 grow most quickly and acquirethe 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


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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 gainsaturates so that the central modes continue to grow while the peripheralmodes, for which the loss has become greater than the gain, areattenuated and eventually vanish. Ideally,only a single mode survives.


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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 powerwhile the more central modes continue to grow, albeit at aslower rate.

Ultimately, only a single surviving mode maintains a gain

equal to the loss, with the loss exceeding the gain for allother 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


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Spatial hole burning In practice, however, homogeneously broadened

lasers do indeed oscillate on multiple modesbecausethe different modes occupy different spatial portions ofthe active medium.

When oscillation on the most central mode is

established, the gain coefficient can still exceed theloss coefficient at those locations where the standing-wave electric field of the most central mode vanishes.

This phenomenon is calledspatial hole burning.

It allows another mode, whose peak fields are locatednear the energy nulls of the central mode, theopportunity to lase.

57


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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 thehomogeneously 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 interactwith different atoms.

Atoms whose lineshapes fail to coincide with any of the modes areignorant of the presence of photons in the resonator.

Their population difference is therefore not affected and the gainthey provide remains the small-signal (unsaturated) gain.

58


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Spectral hole burning Atoms whose frequencies coincide with modes deplete their

inverted population and their gain saturates, creating holes in thegain spectral profile.

This process is known asspectral 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 generallysustains fewer modes than spectral hole burning.

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Spatial distribution The spatial distribution of the emitted laser depends on the

geometry of the resonator and on the shape of the activemedium.

So far we have ignored transverse spatial effects by assumingthat the resonator is constructed of two parallel planar mirrorsof infinite extent and that the space between them is filledwith 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


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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 modesbelonging to two different transverse modes are displaced with

respect to each other by some fraction of

q.


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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 cancontribute to the output if it lies in the spectral band within which thesmall-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


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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 centralobstruction, 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 theircompetitive advantage in contributing to the laser oscillation.


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Spatial distribution In a homogeneous broadened laser, the strongest mode tendsto suppress the gain for the other modes, but spatial holeburning can permit a few longitudinal modes to oscillate.

Transverse modes can have substantially different spatialdistributions 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 cancoexist without competition because they draw their energyfrom different atoms. Partial spatial overlap betweendifferent 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 modebecause it hasthe smallest beam diameter and can be focused to the smallest

spot size. Oscillation on higher-order modes can be desirablefor ur oses such as eneratin lar e o tical ower.


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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-mirrorresonator, the two polarization modes of the same l and mhave the same spatial distributions.

If the resonator and the active medium provide equal gains

and losses for both polarizations, the laser will oscillate onthe two modes simultaneously, independently, and with thesame intensity. The laser output is then unpolarized.

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Pulsed lasers

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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 asfemtoseconds; the durations can be furthercompressed to the attosecondregime by making use ofnonlinear-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 intensitiesreach 10 TW/cm2.

Some lasers can only be operated in a pulsed mode as CWoperation cannot be sustained.


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Methods of pulsing lasers The most direct method of obtaining pulsed

light from a laser is to use a CW laser inconjunction with an external modulator thattransmits 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.


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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 andreleased during the on-time.

Energy may be stored either in the resonator, in the form oflight that is periodically permitted to escape, or in the atomic

system, in the form of a population inversion that is releasedperiodically 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 byCW lasers.

Four common methods used for the internal modulation oflaser light are: gain switching, Q-switching, cavity dumpingand mode locking.

Here, we only focus onmode 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 therange 700 1050 nm, and with individual pulses asshort 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 arepetition rate of 80 MHz. Mode-locked lasers find applications including time-

resolved measurements, imaging, metrology,

communications, materials processing, and clinicalmedicine.

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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 notbased on the transientdynamics of a laser. Instead, a mode-locked laser operates in adynamic steady state.

A laser can oscillate on many longitudinal modes, with frequenciesthat are equally separated by the Fabry-Perot intermodal spacingq= c/2nd.

Although these modes normally oscillate independently (they arethen called free-running modes), external means can be used tocouple 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 frequencieswith locked phase constructively interfere.


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Mode locking The mode-locking operation is accomplished by a nonlinearoptical elementknown as themode locker that is placedinside the laser cavity, typically near one end of the cavity if

the laser has the configuration of a linear cavity.

In thefrequency domain, mode locking is a process thatgenerates 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 togetherthrough nonlinear interactions among the mode fields.

72

Light output (laser)

Modelocker


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Mode locking In the time domain, the mode-locking process can beunderstood as aregenerative pulse-generating processbywhich 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 resemblesthat of apulse-shaping optical shutter that opens periodicallyin synchronism with the arrival at the mode locker of the laserpulse circulating in the cavity.

Consequently, the output of a mode-locked laser is a train ofregularly spaced pulses of identical pulse envelope.

73

d

Mode locker 2d


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Mode locking: two modes The simplest case of multimode oscillation is when there areonly two oscillating longitudinal modes of frequencies 1 and2.

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

74

E(t) E1ei1 (t)e

i1t E2ei2 (t)e

i2t

I(t) E(t)2

E1

2 E222 2E1E2cos (12 )t1(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 characteristictime scale that is shorter than 2/(1-2), the beat note of thetwo frequencies cannot be observed. In this case, the output

of the laser has a constant intensity that is the incoherentsumof the intensities of the individual modes.

This situation represents the ordinary multimode oscillation

of 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) definedby the beat frequency.

The modulation depth of this intensity profile depends on theratio between E1 and E2. When E1 = E2, the modulation depthis 100% with Imin = 0.

In this case, I(t) resembles a train of periodic pulses thathave a duty cycle of 50% and a peak intensity of twice theincoherent sum of the intensities.

This is coherent mode beatingbetween two oscillatingmodes.

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Coherent mode beating between two modes

77

time

Intensity

Incoherent

sum oftheintensities

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, wemay 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 ofmode q, and Aqis its complex envelope.

Here we assume that the q = 0 mode coincides with thecentral frequency 0 of the atomic lineshape.

The magnitude |Aq| may be determined from knowledge ofthe spectral profile of the gain and the resonator loss.

As the modes interact with different groups of atoms in aninhomogeneouslybroadened medium, their phases arg{Aq}

are random and statistically independent.

U(z, t) Aqexpq

i2q (tnz

c)


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Properties of a mode-locked pulse train Substituting the q= 0 + qqinto 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(tnz

c

)exp i20 (tnz

c

)

A(t) Aqq

exp iq2t

TF

TF 1

q

2nd

c


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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 iq2t

TF

qS

S

A xqqS

S

AxS1 xS

x 1A

xS1

2 xS1

2

x1

2 x

1

2

A(t) Asin(Mt/ TF)

sin(t/ TF)

I(t,z) A2sin2[M(t nz / c) / TF]

sin

2

[

(t nz

/c

) /T

F]


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Intensity of periodic pulse train

81

M = 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, whichis proportional to the atomic linewidth or .

If M /q, then pulse = TF/M 1/.

The pulse duration pulse is therefore inverselyproportional 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 isequal to the number of modes M, which can also be

quite large.82


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Properties of a mode-locked pulse train Theperiod of the pulse train is TF = 2nd/c. This is just

the time for a single round trip of reflection within theresonator.

The repetition rate of the pulses = 1/TF = c/2nd = q

The light in a mode-locked laser can be regarded as asingle narrow pulse of photons reflecting back and

forth between mirrors of the resonator.

At each reflection from the output mirror, a fraction ofthe photons is transmitted in the form of a pulse oflight.

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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 transmittedpulses are separated by the distance 2d and travel withvelocity c. The switch opens only when the pulse reaches itand 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

d

2d


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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 averageintensity.

In media with broad linewidths, mode locking is generally moreadvantageous 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 consideractive mode locking andpassive mode locking.

Suppose that an optical switch controlled by an external applied signal is placedinside 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 theswitch 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 orpartially 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 lasingoccur.

The laser waits for the lucky accident of such phases, but once the

oscillations start, they continue to be locked.87


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Methods of mode locking A passive switch such assaturable absorber may also be used to attain

mode locking.

A saturable absorber is a medium whose absorption coefficient decreasesas 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 intensepulse 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 whichthe laser emits collections of modelocked pulses within a Q-switchingenvelope.

88

h d f d l ki


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Methods of mode locking Passive mode locking can also be implemented by means ofKerr-lens

mode locking, which relies on a nonlinear-optical phenomenon in whichthe refractive index of a material changes with optical intensity.

A Kerr medium, such as the gain medium itself, or a material placedwithin the laser cavity, acts as a lens with a focal length inverselyproportional to the intensity. (refractive index change n light intensityI)

By placing an aperture at a proper position within the cavity, the Kerr lensreduces the area of the laser mode for high intensities s.t. the light passesthrough the aperture.

Alternatively, the reduced modal area in the gain medium can be used toincrease its overlap with the strongly focused pump beam, therebyincreasing the effective gain.

The Kerr-lens approach is inherently broadband because of the parametricnature of the process.

The rapid recovery inherent in passive mode locking generally leads toshorter optical pulses than can be attained with active mode locking.

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Diode-Pumped Solid-State Ultrafast laser

C h t Vit 800


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-Coherent Vitesse 800Specification

Ref: http://www.coherent.com/Products/index.cfm?1439/Vitesse-Family

C it h ti P i d l ki


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Starter: initiate mode locking by perturbing the cavity. (changing the cavity length byshaking 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.- shorterpulse(the CW components beam size is wider than the pulse (mode-locked) beam size. )

NDM: negative-dispersion mirror serves as additional dispersion compensationtoprevent pulse broadening

Pump: green laser-532 nm

Cavity schematic: Passive mode-locking

lect9 laser oscillators(3)

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