Drake G.W.F. (editor) Handbook of Atomic, Molecular, and Optical Physics

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1026 Part F Quantum Optics

where t

, t

are the mirror transmissivities at the right

and left mirrors, respectively. The uniform intensity

approximation implies that I

= I

−

= I/2and

out

sat

[

(ν)/g

−1

]

. (70.18)

Suppose one of the mirrors is perfectly transmitting,

so that t

= 0andt

= t > 0. Furthermore, if the re-

ﬂectivity r of the transmitting mirror is close to unity,

then g

∼

(1/2)(1−r) = (1/2)(t +s),wheres is the

fraction of the incident beam power that is scattered or

absorbed at the output mirror. Then

out

∼

sat



(ν)

t +s

−1



(70.19)

and it follows that the optimal output coupling, i. e., the

transmissivity that maximizes the output intensity, is

opt



(ν)s−s . (70.20)

This output coupling gives the output intensity

max

out

= I

sat





(ν) −



s/2



. (70.21)

(ν)I

sat

is the largest possible power per unit volume

extractable as output laser radiation at the frequency ν.

More generally, when mirror reﬂectivities are not

necessarily close to unity, I

= I

−

and both I

and I

−

vary with the axial coordinate z. In this more general

case, (70.14)and(70.15) are replaced by

g(ν, z) =

(ν)

1+[I

(z) + I

−

(z)]/I

sat

(70.22)

and

= g(ν, z)I

−

=−g(ν, z)I

−

, (70.23)

where it is assumed that g

(ν) is independent of z,and

that all cavity loss processes occur at the mirrors. The

solution of these equations with the boundary condi-

tions I

−

(L) =r

(L), I

(0) =r

−

(0) for mirrors at

z = 0andz = L gives, for I

out

= t

−

(0) +t

(L),the

formula [70.3,6]

out

= I

sat







√



√



1−

√





(ν) +ln

√



(70.24)

Analysis of this result gives an optimal output coup-

ling that reduces to (70.19) in the limit t+s 1. Curves

for optimal output coupling and I

out

as a function of

and s are given by Rigrod [70.6]. These results are

based on several assumptions and approximations: the

gain medium is assumed to be homogeneously broad-

ened and to saturate according to the formula (70.14);

and I

sat

are taken to be constant throughout the

medium; the ﬁeld is approximated as a plane wave;

ﬁeld loss processes occur only at the mirrors; and in-

terference between the left- and right-going waves is

ignored.

Interference of the counterpropagating waves in

a standing wave, single-mode laser modiﬁes the gain

saturation formula (70.14) as follows:

g(ν, z) =

(ν)

1+(2I

sat

) sin

(70.25)

in the case of small output coupling, where I

∼

−

assumed in (70.18). (The general case of arbitrary output

coupling with spatial interference of counterpropagating

waves is somewhat complicated and is not considered

here.) The sin

kz term is responsible for spatial hole

burning: “holes” are “burned” in the curve of g(ν, z)

versus z at points where sin

kz is largest. This spatially

dependent saturation acts to reduce the output intensity,

typically by as much as about 30%, compared with the

case where interference of counterpropagating waves

is absent or ignored. Spatial hole burning tends to be

washed out by atomic motion in gas lasers, and is absent

entirely in pure traveling wave ring lasers. If complete

spatial hole burning based on (70.25) is assumed, the

output intensity is

out

sat



(ν)

−



(ν)



(70.26)

in the case where one mirror is perfectly reﬂecting.

In inhomogeneously broadened media, the gain co-

efﬁcientis obtained by integrating the contributions from

all possible values of ν

. The different contributions sat-

urate differently, depending on the detuning of ν

from

the cavity mode frequency ν. If spatial hole burning and

power broadening (see Sect. 69.3) are ignored, then, to

a good approximation, the gain saturates as

g(ν) =

(ν)

√

1+I/I

sat

(70.27)

in typical inhomogeneously broadened media.

Oscillation on a single longitudinal mode (see

Sect. 70.3) may be realized simply by making the cav-

ity length L small enough that the mode spacing c/2nL

(70.28) exceeds the spectral width of the gain curve.

This is possible in many gas lasers where the spectral

Part F 70.2

Laser Principles 70.2 Continuous Wave, Single-Mode Operation 1027

width is small, and in semiconductor lasers, where L

is very small. More generally the gain clamping condi-

tion g(ν) = g

implies that the cavity mode frequency

having the largest small-signal gain g

(ν) saturates the

gain g(ν) down to the threshold value g

, while the

gain at all other mode frequencies then lies below g

In other words, the gain clamping condition implies

single-mode oscillation. However, this conclusion as-

sumes homogeneous broadening and also that spatial

hole burning is unimportant, so that the gain satu-

rates uniformly throughout the cavity. High pressure

gas lasers, where the line broadening is due primar-

ily to collisions and therefore is homogeneous, tend

to oscillate on a single mode because spatial hole

burning is largely washed out by atomic motion. On

the other hand, homogeneously broadened solid state

lasers can be multi-mode as a consequence of spatial

hole burning.

Single-mode oscillation in inhomogeneously broad-

ened media is generally more difﬁcult to achievebecause

of spectral hole burning (see Sect. 69.3), which makes

the simple gain clamping argument inapplicable. How-

ever, single-mode oscillation can be enforced in any case

by introducing, in effect, an additional loss mechanism

for all mode frequencies except one. This is commonly

done with a Fabry–Perot etalon having a free spectral

range that is large compared with the spectral width

of the gain curve. By choosing the tilt angle appro-

priately, a particular resonant frequency of the etalon

can be brought close to the center of the gain curve

while all other resonance frequencies lie outside the gain

bandwidth.

Laser oscillation at a ﬁxed polarization can likewise

be achieved by discriminating against the orthogonal

polarization, as is done when Brewster windows are

employed.

Laser oscillation, in general, does not occur precisely

at one of the allowed cavity mode frequencies. Associ-

ated with the sin kz dependence of the intracavity ﬁeld

is the condition kL = Nπ,or

ν = N

2nL

(N an integer),

(70.28)

for the cavity mode frequencies in the plane wave ap-

proximation. If the gain medium of refractive index n

does not ﬁll the entire length L between the mirrors,

then (70.28) must be replaced by

ν =

Nc/2

n +(L −)

(70.29)



[n(ν) −1]ν = ν

−ν, (70.30)

where  (≤ L) is the length of the gain medium and

= Nc/2L is an empty cavity mode frequency. The

laser oscillation frequency will therefore be different,

in general, from any of the allowed empty-cavity mode

frequencies. If the refractive index n(ν) is attributable

primarily to the lasing transition, as opposed to the host

material, or other nonlasing transitions, then the follow-

ing relation between n(ν) and g(ν) maybeusedinthe

case of homogeneous broadening [70.3]:

n(ν) −1 =−

4π

−ν

δν

g(ν) , (70.31)

where λ

, ν

,andδν

are the wavelength, frequency, and

homogeneous linewidth (HWHM), respectively, of the

lasing transition. This implies that

ν =

δν

+ν

δν

+δν

(70.32)

for the laser oscillation frequency ν,where

δν

≡

cg(ν)

4πL

(70.33)

is the cavity bandwidth. Thus, the actual lasing fre-

quency is not simply one of the allowed empty-cavity

frequencies ν

, but rather is “pulled” away from ν

toward the center of the gain proﬁle. This is called

frequency pulling.

If spatial hole burning is absent or ignored, then

g(ν) = g

in steady state oscillation, and the cavity

bandwidth δν

= cg

/4πL is largest for lossy cavities.

Most lasers fall into the “good cavity” category, that is

δν

 δν

,sothat(70.32) may be approximated by

∼

+(ν

−ν

)δν

/δν

. (70.34)

Similar results apply to inhomogeneously broadened

lasers. For a Doppler broadened medium, for instance,

the frequency pulling formula is

∼

+(ν

−ν

)(δν

/δν

)



4ln2/π (70.35)

for good cavities. These results show that frequency

pulling is most pronounced in lasers with large peak

gain coefﬁcients and narrow gain proﬁles, as observed

experimentally.

Spectral hole burning leads to especially interesting

consequences in Doppler broadened gas lasers. Since

two traveling waves propagating in opposite (±z)di-

rections will strongly saturate spectral packets of atoms

Part F 70.2

1028 Part F Quantum Optics

with oppositely Doppler shifted frequencies, a stand-

ing wave ﬁeld will burn two holes on opposite sides

of the peak of the Doppler proﬁle. When the mode

frequency is exactly at the center of the Doppler pro-

ﬁle, however, the two holes merge, the ﬁeld now being

able to saturate strongly only those atoms with zero vel-

ocity along the z-direction. In this case, since the ﬁeld

“feeds” off a single spectral packet of more strongly

saturated atoms, there is a dip in the output power

compared with the case when the mode frequency is

detuned from line center. This dip in the output power

at line center is called the Lamb dip. It can be used

to determine whether a gas laser is Doppler broadened,

and more importantly, to stabilize the laser frequency

to the center of the dip. Lamb dip frequency sta-

bilization employs a feedback circuit to control the

bias voltage across a piezoelectric element used to

vary the cavity length, and thereby sweep the laser

frequency.

The above semiclassical laser theory suggests that

cw laser radiation should be perfectly monochromatic,

since the amplitude and phase of the ﬁeld given by (70.7)

are time independent. However, when quantum electro-

dynamical considerations are built into laser theory, it

is found that spontaneous emission, which adds to the

number of photons put in the lasing mode by stimu-

lated emission, causes a phase diffusion that results in

a Lorentzian linewidth (HWHM)

∆ν =

−N

8πhν

out

(δν

)

, (70.36)

where P

out

is the output laser power. This is the

Schawlow–Townes linewidth, which implies a fun-

damentally quantum mechanical, ﬁnite linewidth that

persists no matter how small various sources of “tech-

nical noise”, such as mirror jitter, are made. Although

the Schawlow–Townes linewidth has been observed in

highly stabilized gas lasers, it is negligible compared

with technical noise in conventional lasers. But in semi-

conductor lasers, L is very small and consequently δν

large, and quantum noise associated with the Schawlow–

Townes formula can be the dominant contribution to the

laser linewidth.

However, the 10–100 MHz linewidths typically ob-

served in semiconductor lasers are too large to be

explained by the Schawlow–Townes formula (70.36),

and two modiﬁcations to this formula are necessary, each

of them involving a multiplication of the Schawlow–

Townes linewidth ∆ν by a certain factor:

∆ν → ∆ν





1+α



K∆ν,

(70.37)

where α is called the “Henry α parameter” and is as-

sociated with a coupling between phase and intensity

ﬂuctuations above the laser threshold [70.7]. Values

of α between about 4 and 6 are typical in semicon-

ductor injection lasers, and consequently, the correction

to the Schawlow-Townes linewidth due to the Henry

α parameter is substantial. The K factor [70.8–10]

arises as a consequence of the deviation from the

spatially uniform intracavity intensity assumed in the

derivation of the Schawlow–Townes formula [70.11,

12]. (Intracavity intensities along the optical axis are

approximately uniform only in the case of low out-

put couplings.) The fundamental quantum mechanical

linewidth under consideration can be associated with

vacuum ﬁeld ﬂuctuations, which, according to gen-

eral ﬂuctuation–dissipation ideas, will increase as the

cavity loss increases. This explains why the “Peter-

mann K factor” deviates increasingly from unity as

the output coupling (cavity loss) increases. Values of

K between 1 and 2 appear to be typical for lossy,

stable resonators [70.13, 14], but much larger values

are possible for unstable resonators [70.15](seealso

Siegman [70.2]).

70.3 Laser Resonators

The assumption that the complex ﬁeld amplitude E (r, t)

is slowly varying in z compared with exp(ikz) leads to

the paraxial wave equation (70.10)

∇

E +2ik

∂E

∂z

= 0

(70.38)

for a monochromatic ﬁeld in vacuum. If E (x, y, z) sat-

isﬁes the paraxial wave equation and is speciﬁed in the

plane (x, y, z = 0), it follows that

E (x, y, z) =−

λz





E (x



, y



, 0)

×e

ik[(x−x



)

+(y−y



)

]/2z

. (70.39)

Thus, in the case of a laser resonator, the ﬁeld E (x, y, L)

at the mirror at z = L is related to the ﬁeld E (x, y, 0) at

Part F 70.3

Laser Principles 70.3 Laser Resonators 1029

the mirror at z = 0by

E (x, y, L) =−

λL





E (x



, y



, 0)

×e

ik[(x−x



)

+(y−y



)

]/2L

≡





K(x, y; x



, y



)E (x



, y



, 0).

(70.40)

Similarly, the ﬁeld at z = 0 after one round trip pass

through the resonator is

E (x, y, 0)





K(x, y; x



, y



)E (x



, y



, L)





K(x, y; x



, y



)





× K(x



, y





, y



)E (x



, y



, 0)

≡





K(x, y;x



, y



)E (x



, y



, 0).

(70.41)

By deﬁnition, a mode of the resonator is a ﬁeld dis-

tribution that does not change on successive round-trip

passes through the resonator. More precisely, since an

empty cavity is assumed, a mode will be such that the

ﬁeld spatial pattern remains the same except for a con-

stant decrease in amplitude per pass. A longitudinal

mode is deﬁned by the value of k in exp(ikz), whereas

a transverse mode is deﬁned by the corresponding (x, y)

dependence and satisﬁes the integral equation

γ E (x, y, z) =





K(x, y;x



, y



)E (x



, y



, z),

(70.42)

where z deﬁnes any plane between the mirrors and

|γ | < 1. Iterative numerical solutions of this equation

for laser resonator modes were ﬁrst discussed by Fo x

and Li [70.16].

Laser resonators may be classiﬁed as stable or un-

stable according to whether a paraxial ray traced back

and forth through the resonator remains conﬁned in the

resonator or escapes. This leads to the condition

0 ≤ g

≤ 1 (70.43)

for stability, where the g parameters are deﬁned in terms

of the (spherical) mirror curvatures R

and the mirror

separation L by g

≡1−L/R

. R

is deﬁned as positive

or negative depending on whether the mirror i is concave

or convex, respectively, with respect to the interior of

the resonator. Plane–parallel mirrors (R

→∞) can be

used, but they are difﬁcult to keep aligned and have much

larger diffractive losses.

Among stable conﬁgurations, the symmetric confo-

cal resonator with R

= R

= L has the smallest mode

spot sizes at the mirrors, while the concentric resonator

with R

= R

slightly greater than L/2, has the smallest

beam waist (Fig. 70.1). The widely used hemispherical

resonator (R

=∞, R

slightly greater than L) is rela-

tively easy to keep aligned and allows the spot size at

mirror 2 to be adjusted by slight changes in L.

The fundamental Gaussian beam modes of stable

resonators may be constructed from the free space solu-

tions of the paraxial wave equation. The most important

(lowest-order) solution for this purpose is

E (x, y, z)

−iφ(z)

ik(x

)/2R(z)

−(x

)/w

(z)



1+z

(70.44)

where A is a constant, φ(z) = tan

−1

(z/z

),andR(z),

w(z),andz

are the radius of curvature of surfaces of

constant phase, the spot size, and the Rayleigh range,

respectively. The confocal parameter, 2z

,isalsousedto

characterize Gaussian beams. Here z

≡ πw

/λ,where

is the spot size at the beam waist at z =0(Fig.70.1),

and R(z) and w(z) vary with the distance z as follows:

R(z) = z +z

/z ,w(z) =w



1+z

(70.45)

The divergence angle of a Gaussian beam (Fig. 70.1)is

given by θ =λ/πw

The ABCD law for Gaussian beams allows the ef-

fects of various optical elements on Gaussian beam

propagation to be calculated in a relatively simple fash-

ion [70.1–3]. For instance, a Gaussian beam incident on

a lens of focal length f at its waist is focused to a new

waistatadistanced = f/



1+ f



behind the lens,

(x, y)(x, y)

Intensity Intensity

Beam waist

z = 0

w(z)

√2

Fig. 70.1 Variation of the spot size w(z) of a Gaussian beam

with the propagation distance z from the beam waist

Part F 70.3

1030 Part F Quantum Optics

and the spot size at the new waist is



fλ

πw



1+ f

, (70.46)

which is approximately fλ/πw

= fθ for tight focusing.

This shows that a Gaussian beam can be focused to a very

small spot. On the other hand, beam expanders consist-

ing of two appropriately spaced lenses may be used to

increase the spot size by the ratio of the focal lengths.

Gaussian beam modes of laser resonators have radii

of curvature that match in magnitude those of the mir-

rors. The spot sizes at the mirrors and the location of

the beam waist with respect to the mirrors may be ex-

pressedintermsofλ, L,andg

. The empty-cavity

mode frequencies are given by



N +

cos

−1

√



(70.47)

where N is an integer. For a host medium of refrac-

tive index n, c/2L is replaced by c/2nL,and(70.47)

then generalizes the plane wave result (70.28) to ac-

count for both longitudinal and transverse effects in the

determination of the cavity mode frequencies.

The assumption of Gaussian modes presupposes that

the resonator mirrors are large enough to intercept the en-

tire beam without any “spillover”; i. e., that a  w

where w

are the spot sizes at the mirrors and a is an

effective mirror cross sectional radius. This implies that

the Fresnel number N

≡ a

/λL  1. Diffraction losses

generally increase with decreasing Fresnel numbers.

Higher-order Gaussian modes, where the Gaussian

functions of x and y in (70.44) are replaced by higher-

order Hermite polynomials, are often more difﬁcult to

realize than the fundamental lowest-order mode because

their larger spot sizes imply higher diffractive losses, and

beyond a certain mode order the spot sizes are too large

to satisfy the no-spillover condition for a pure Gaussian

mode.

It is not possible in general to write closed form

expressions for laser modes. There are at least two rea-

sons for this, the ﬁrst being that the gain medium cannot

in general be regarded as a simple amplifying element

that preserves the basic empty-cavity mode structure. In

low power gas lasers, the spatial variations of the gain

and refractive index are sufﬁciently mild that the las-

ing modes can be accurately described as Gaussians,

but more generally, there can be strong gain and index

variations that themselves play an important role in de-

termining the modes of the laser, as in “index-guided”

and “gain-guided” semiconductor lasers.

Secondly, the resonator structure itself may intro-

duce complications that preclude closed form solutions

even for the empty cavity. This is generally true, for in-

stance, of unstable resonators, where iterative numerical

solutions of the integral equation (70.42) are necessary

for accurate predictions of modes. In such computer

simulations, the mode structure must be determined self

consistently with the numerical solutions of density ma-

trix or, more commonly, rate equations for the gain

medium.

Unstable resonators, though inherently lossy, offer

some important advantages for high-gain lasers. Thus,

whereas the Gaussian modes of stable resonators typic-

ally involve very small beam spot sizes, the distinctly

non-Gaussian modes of unstable resonators can have

large mode volumes and make more efﬁcient use of

the available gain volume. Unstable resonators also tend

to yield higher output powers when they oscillate on

the lowest loss transverse mode, whereas in stable res-

onators higher output powers are generally associated

with multitransverse mode oscillation. For very high

power lasers, the fact that unstable resonators involve

all-reﬂective optics, as opposed to transmissive output

coupling, can be important in avoiding optical damage.

70.4 Photon Statistics

Optical ﬁelds may be characterized and distinguished

by their photon statistical properties (Chapt. 78 and

[70.17–21]). In a photon counting experiment, the num-

ber of photons registered at a photodetector during a time

interval T is measured and used to infer the probability

(T ) that n photons are counted in a time interval T.If

the probability of counting a photon at the time t in the

interval dt is denoted αI(t) dt,whereI(t) is the intensity

of the ﬁeld and α is a factor depending on the micro-

scopic details of the photoelectric process and on the

phototube geometry, then it may be shown from largely

classical considerations that [70.3,17–21]

(T ) =











I(t



)





−α





I(t



)



(70.48)

Part F 70.4

Laser Principles 70.5 Multi-Mode and Pulsed Operation 1031

where the average ···is over the intensity variations

during the counting interval. This is Mandel’s formula.

In the simplest case of constant intensity, P

follows the

Poisson distribution:

= n

−n

/n! , (70.49)

where n = αIT. Since a single-mode laser ﬁeld can

be thought of approximately as a “classical stable

wave” [70.20,21], it is not surprising that its photocount

distribution is found both theoretically and experimen-

tally to satisfy (70.49), which is characteristic of a coher-

ent state of the ﬁeld (Chapt. 78). A thermal light source,

by contrast, follows the Bose–Einstein distribution,

= n

/(1 +n)

n+1

, (70.50)

if the time interval T is short compared with the coher-

ence time of the light; i.e., if T∆ν 1, where ∆ν is the

bandwidth. If T∆ν  1, P

is again Poissonian.

Thus, if a quasi-monochromatic beam of light is

made from a natural source by spatial and spectral

ﬁltering, it has measurably different photon counting

statistics from a single-mode laser beam of exactly the

same bandwidth and average intensity. Laser radiation

approaches the ideal coherent state that, of all possible

quantum mechanical states of the ﬁeld, most closely re-

sembles the “classical stable wave”. These differences

are exhibited in other experiments, such as the meas-

urement of intensity correlations of the Brown–Twiss

type [70.3, 17–21]. In such experiments, thermal pho-

tons have a statistical tendency to arrive in pairs (photon

bunching), whereas the photons from a laser arrive

independently.

70.5 Multi-Mode and Pulsed Operation

Multi-mode laser theory is generally much more compli-

cated than single-mode theory, particularly in the case

of inhomogeneous broadening with both spectral and

spatial hole burning. In certain situations, however, con-

siderable simpliﬁcation is possible. For instance, when

the cavity-mode frequency spacing is small compared

with the homogeneous linewidth δν

, the gain tends

to saturate homogeneously, and the total output power

on all modes is well described by the Rigrod analysis

outlined in Sect. 70.2.

Pulsed laser operation adds the further complication

of temporal variations to the cw theory outlined in the

preceding sections. It is possible, nevertheless, to under-

stand some of the most important types of multi-mode

and pulsed operation using relatively simple models.

One method of obtaining short, high power laser

pulses is Q-switching. In a very lossy cavity, the gain

can be pumped to large values before the threshold con-

dition is met and gain saturation occurs. If the cavity

loss is suddenly decreased, there will be a rapid build-

up of intensity because the small-signal gain is far above

the (now reduced) threshold value. The switching of the

cavity loss is called Q-switching, the “quality factor” Q

being deﬁned as ν/2δν

. This type of Q-switching pro-

duces intense pulses of duration typically in the range

10–100 ns, as dictated by the fact that a light pulse must

make several passes through the gain cell for ampli-

ﬁcation. Q-switching requires that the gain medium

be capable of retaining a population inversion over

a time much larger than the Q-switched pulse dura-

tion, and in particular that the spontaneous emission

lifetime be relatively long. The pulse duration can be re-

duced to approximately a single cavity round trip time

by Q-switching from low reﬂectivity mirrors to 100%

reﬂectivities, and then switching the reﬂectivity of the

outcoupling mirror from 100% to 0% at the peak of

the ampliﬁed pulse. In addition to this pulsed transmis-

sion mode is cavity dumping, where both mirrors have

nominally 100% reﬂectivity and the intracavity power

is “dumped” by an acousto-optic or electro-optic intra-

cavity element that deﬂects the light out of the cavity.

The pulse duration achieved in this way is again roughly

a round trip time. Cavity dumping can be employed with

cw lasers and does not require the long energy storage

times necessary for ordinary Q-switching.

Shorter pulses can be realized by mode-locking,

where the phases of N longitudinal cavity modes are

locked together. In the simplest model, assuming equal

amplitudes and phases of the individual modes, the net

ﬁeld amplitude is proportional to

X(t) =

(N−1)/2



n=−(N−1)/2

sin[(ω

+n∆)t +φ

]

= X

sin(ω

t +φ

)

sin(N∆t/2)

sin(∆t/2)

(70.51)

The temporal variation described by this function for

large N is a train of “spikes” of amplitude NX

at times

= 2πm/∆, m = 0, ±1, ±2, ···, the width of each

spike being 2π/(N∆). In the case of a mode-locked laser,

Part F 70.5

1032 Part F Quantum Optics

∆ = 2π(c/2L), and the output ﬁeld consists of a train

of pulses separated in time by T = 2π/∆ =2L/c.The

peak amplitude of the spikes is proportional to N,and

the duration of each spike is approximately 2L/cN.

The maximum number N

max

of modes that can ac-

tually be phase-locked is limited by the spectral width

∆ν

of the gain curve:

max

∆ν

c/2L

∆ν

. (70.52)

Similarly, the shortest pulse duration is

min

max

∆ν

. (70.53)

Mode-locking thus requires a gain bandwidth large com-

pared with the cavity mode spacing, and the shortest and

most intense mode-locked pulse trains are obtained in

gain media having the largest gain bandwidths. Trains

of picosecond pulses are routinely obtained with li-

quid dye and solid gain media having gain bandwidths

∆ν

≈ 10

−1

or more.

Various techniques, employing acoustic or electro-

optic modulation or saturable absorbers, are used to

achieve mode-locking [70.1–3]. The different methods

all rely basically on the fact that a modulation of the gain

or loss at the mode separation frequency c/2L tends to

cause the different modes to oscillate in phase. Such

a modulation is achieved “passively” when a saturable

absorber is placed in the laser cavity: the multi-mode in-

tensity oscillates with a beat frequency that is impressed

on the saturated loss coefﬁcient. Dye lasers, with their

large gains across a broad range of optical frequencies,

are often employed to generate mode-locked picosecond

pulses.

Ultrashort pulse generation is possible with

additional nonlinear or frequency chirping tech-

niques [70.22]. The colliding-pulse laser [70.23]is

a three-mirror ring laser in which two mode-locked

pulse trains propagate in opposite senses and overlap

in a very thin (≈ 10 µm) saturable absorber placed in

the ring in addition to the gain cell. The cavity loss is

least when the two pulses synchronize to produce the

highest intensity, and therefore the lowest loss coefﬁ-

cient in the saturable absorber. The short length of the

absorbing cell forces the pulses to overlap within a very

small distance and therefore to produce very short pulses

(10 µm/c ≈30 fs pulse duration with c/L ≈ 100 MHz

repetition rate).

Another method of ultrashort pulse generation relies

on frequency chirping; i. e., a time dependent shift of the

frequency of an optical pulse [70.24–26]. In a medium

(e.g., a glass ﬁber) with linear and nonlinear refractive

index coefﬁcients n

and n

, the refractive index is

n = n

I , (70.54)

so that there is an instantaneous phase shift φ(t) that de-

pends on the instantaneous intensity. Therefore, as I(t)

increases toward the peak intensity of the pulse, φ(t)

increases (assuming n

> 0), whereas φ(t) decreases

as I(t) decreases from its peak value. The frequency

shift

φ(t) is such that the instantaneous frequency of the

pulse is smaller at the leading edge and larger at the trail-

ing edge of the pulse (Fig. 70.2), resulting in a stretching

of the pulse bandwidth.

Following this spectral broadening by the nonlin-

ear medium, the pulse can be compressed in time by

means of a frequency dependent delay line such that

the smaller frequencies, say, are delayed more than the

higher frequencies. The trailing edge of the pulse can

therefore “catch up” to the leading edge, resulting in

a shorter pulse whose duration is given by the inverse

of the chirp bandwidth. The delay line can be realized

with a pair of diffraction gratings (see Fig. 70.2). Using

this pulse compression technique, 40 fs ampliﬁed pulses

from a colliding pulse laser have been compressed to

8 fs, corresponding to about four optical cycles [70.27].

In chirped-pulsed-ampliﬁcation (CPA)lasers [70.28]

a laser pulse is chirped, temporally stretched, and then

passed through an ampliﬁer. The lengthening in time

of the pulse prior to ampliﬁcation allows greater en-

ergy extraction from the ampliﬁer. After ampliﬁcation,

Glass fiber

Chirped pulse

Chirped

input pulse

Output pulse

Grating

Blue

Grating

Red

Fig. 70.2a,b Nonlinear pulse compression by frequency

chirping. In

(a), the nonlinear refractive index of a glass

ﬁber results in a time dependent frequency of the trans-

mitted pulse, and in

(b) a pair of diffraction gratings is

used to produce frequency dependent path delays such as

to temporally compress the pulse

Part F 70.5

Laser Principles 70.7 Recent Developments 1033

pulse compression is performed with a grating pair.

The Ti :sapphire (Ti :Al

) ampliﬁer is particularly

attractive for femtosecond CPA because of its very large

spectral width and high saturation ﬂuence.

The wavelength dependence of the linear refractive

index n

in (70.54) results in a group velocity

= c[n

−λdn

/dλ]

−1

(70.55)

that, if dn

/dλ<0, is such that higher frequencies prop-

agate more rapidly than lower frequencies. Assuming

> 0, on the other hand, the nonlinear part of the in-

dex causes a delay of higher frequencies with respect

to lower ones, as discussed above. This leads to soliton

solutions of the wave equation, such that the oppos-

ing effects of the linear and nonlinear dispersion are

balanced and the pulse propagates without distortion.

Soliton lasers, with pulse durations ranging from pico-

seconds down to ≈100 fs, depending on the ﬁber length,

have been made with solid state lasers and intracavity

optical ﬁbers [70.29].

70.6 Instabilities and Chaos

Mode-locked pulses and solitons exemplify an ordered

dynamics, as opposed to the erratic and seemingly ran-

dom intensity ﬂuctuations that are sometimes observed

in the output of a laser. In fact, it is possible, under certain

circumstances, for laser oscillation to exhibit determin-

istic chaos; i. e., an effectively random behavior that

can nevertheless be described by purely deterministic

equations of motion [70.30–34].

Lasers are nonlinear and dissipative systems, and as

such exhibit essentially all the modes of behavior charac-

teristic of such systems. It was shown by Haken [70.35]

that (70.6)and(70.11), with ∆ = 0 and with pumping

and ﬁeld loss terms included, can be put into the form

of the Lorenz model for chaos. For a single-mode, ho-

mogeneously broadened ring laser, the Lorenz model

instability requires a “bad cavity” in the sense that

the ﬁeld loss rate is larger than the sum of the ho-

mogeneous linewidth of the lasing transition and the

population decay rate. It also requires the gain medium

to be pumped at least nine times above the threshold gain

value, a condition sometimes referred to as a “second

laser threshold.”

When the single-mode laser equations correspond-

ing to the Lorenz model are generalized to the case

of inhomogeneous broadening, an instability occurs at

small–signal gain values much less than nine times

threshold. In particular, the cw output of the laser gives

way to an oscillatory intensity, even though the pump-

ing and loss terms in the equations are assumed to be

time independent. As the pumping and loss parameters

are varied, this self-pulsing instability can give way

to chaotic behavior, and numerical studies of the set

of coupled matter–ﬁeld equations reveal period doub-

ling, two-frequency, and intermittency routes to chaos

in different regimes [70.30]. This unstable behavior

of single-mode inhomogeneously broadened lasers was

ﬁrst discovered experimentally and analyzed by Casper-

son[70.36–38] for low pressure, 3.51 µmHe

−

Xe lasers.

Arecchi et al. [70.39] reported the ﬁrst observation and

characterization of chaotic behavior in a laser system; by

modulating the cavity loss of a CO

laser they observed

a period doubling route to chaos as the modulation

frequency was varied.

Extensive experimental and theoretical work on un-

stable and chaotic behavior in a wide variety of other

laser systems has been reported [70.30–34, 40], in-

cluding work aimed at the control of chaotic laser

oscillation by the so-called occasional proportional

feedback technique [70.41, 42]. Instabilities of single

transverse mode dynamics have also been studied, es-

pecially in connection with spontaneous spatial pattern

formation [70.40].

70.7 Recent Developments

Recent developments in the basic physics of lasers

include the application of cavity QED techniques

to produce a single-atom laser [70.43] that emits

< 10

photons/s, and a two-photon laser [70.44]that

operates on the basis of ampliﬁcation on a two-photon

transition. Recent progress in the development of ul-

trashort pulses includes the generation of attosecond

pulses by high-order harmonic generation [70.45, 46],

and the application of such pulses to a measurement of

the photoionization time of Auger electrons [70.47].

Part F 70.7

1034 Part F Quantum Optics

References

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70.2 A. E. Siegman: Lasers (University Science Books, Mill

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70.3 P. W. Milonni, J. H. Eberly: Lasers (Wiley, New York

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70.4 P. W. Milonni: Phys. Rep. 25,1(1976)

70.5 B. DiBartolo: Optical Interactions in Solids (Wiley,

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70.6 W. W. Rigrod: J. Appl. Phys. 36, 27 (1965)

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(1979)

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70.13 W. A. Hamel, J. P. Woerdman: Phys. Rev. Lett 64,

1506 (1990)

70.14 M. A. Eijkelenborg, A. M. Lindberg, M. S. Thijssen,

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70.15 K.-J. Cheng, P. Mussche, A. E. Siegman: IEEE J.

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70.17 R. J. Glauber: Phys. Rev. 130, 2529 (1963)

70.18 R. J. Glauber: Phys. Rev. 131, 2766 (1963)

70.19 L. Mandel, E. Wolf: Rev. Mod. Phys. 37,231

(1965)

70.20 R. Loudon: The Quantum Theory of Light, 3rd edn.

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70.21 P. Meystre, M. Sargent III: Elements of Quantum

Optics, 2nd edn. (Springer, Berlin, Heidelberg 1991)

p. 32

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38,671(1981)

70.24 D. Grischkowsky, A. C. Balant: Appl. Phys. Lett 41,1

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70.25 L. F. Mollenauer, R. H. Stolen, J. P. Gordon,

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70.26 J. G. Fujimoto, A. M. Weiner, E. P. Ippen: Appl. Phys.

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70.27 C. V. Shank: Science 233, 1276 (1986) and references

therein

70.28 P. Maine, D. Strickland, P. Bado, M. Pessot,

G. Mourou: IEEE J. Quantum Electron. 24, 398 (1988)

70.29 L. F. Mollenauer, R. H. Stolen: Opt. Lett. 9, 13 (1984)

70.30 P. W. Milonni, M.-L. Shih, J. R. Ackerhalt: Chaos in

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70.31 D. K. Bandy, A. N. Oraevsky, J. R. Treddice: Special

issues of J. Opt. Soc. Am. 5, 5 (1988)

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70.35 H. Haken: Phys. Lett. A 53, 77 (1975)

70.36 L. W. Casperson: IEEE J. Quantum Electron,Vol.14

(Springer, Berlin, Heidelberg 1978) p. 756

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(Springer, Berlin, Heidelberg 1983)

70.38 P. Chenkosol, L. W. Casperson: J. Opt. Soc. Am. B

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Part F 70

1035

Types of Laser

71. Types of Lasers

The availability of coherent light sources (i. e.,

lasers) has revolutionized atomic, molecular,

and optical science. Since its invention in

1960, the laser has become the basic tool for

atomic and molecular spectroscopy and for

elucidating fundamental properties of optics

and optical interactions with matter. The unique

properties of laser light have spawned new

types of spectroscopy, as discussed in Chapt. 72

to Chapt. 80.

There are now literally hundreds of dif-

ferent types of lasers. However, only a few

of these are commercially available, and

lasers tailor made with operational prop-

erties optimized for speciﬁc applications

are often needed. Chapter 70 describes

the principles of laser operation leading to

speciﬁc output characteristics. This chap-

ter summarizes the current status of the

development of different types of lasers,

emphasizing those that are commercially

available. There are several ways to cat-

egorize types of lasers; for example, in

terms of spectral range, temporal char-

acteristics, pumping mechanism, or lasing

media.

71.1 Gas Lasers ...........................................1036

71.1.1 Neutral Atom Lasers................... 1036

71.1.2 Ion Lasers................................. 1036

71.1.3 Metal Vapor Lasers .................... 1037

71.1.4 Molecular Lasers .......................1037

71.1.5 Excimer Lasers .......................... 1038

71.1.6 Nonlinear Mixing ......................1038

71.1.7 Chemical Lasers.........................1039

71.2 Solid State Lasers.................................1039

71.2.1 Transition Metal Ion Lasers ......... 1040

71.2.2 Rare Earth Ion Lasers ................. 1040

71.2.3 Color Center Lasers ....................1042

71.2.4 New Types of Solid State

Laser Systems ...........................1043

71.2.5 Frequency Shifters.....................1043

71.3 Semiconductor Lasers...........................1043

71.4 Liquid Lasers .......................................1044

71.4.1 Organic Dye Lasers..................... 1044

71.4.2 Rare Earth Chelate Lasers ...........1045

71.4.3 Inorganic Rare Earth Liquid Lasers1045

71.5 Other Types of Lasers ...........................1045

71.5.1 X-Ray and Extreme UV Lasers......1045

71.5.2 Nuclear Pumped Lasers ..............1046

71.5.3 Free Electron Lasers ...................1046

71.6 Recent Developments...........................1046

References .................................................. 1048

For this chapter, the types of lasers are categorized in

terms of the lasing media, as shown in Table 71.1.

The variety of different laser types offers a wide

range of beam parameters. The spectral output of lasers

covers the X-ray to far IR regions as shown in Fig. 71.1.

Spectral linewidths can be as narrow as 20 Hz and

Table 71.1 Categories of lasers

Gas lasers Special lasers Solid State lasers Liquid lasers Miscellaneous

Atomic Metal vapor Transition metal ion Organic dye X-ray

Ionic Chemical Rare earth ion Rare earth chelate Nuclear pumped

Molecular Color center Inorganic solvents with Free electron

rare earth ions

Excimer Semiconductor

diffraction limited beam quality with coherence lengths

up to 10 m can be obtained. Temporal pulse widths of

a few femtoseconds have been generated. Some lasers

can produce peak powers of over 10

Wandaver-

age powers of 10

W with pulse energies greater than

J. The important operational characteristics, such

Part F 71