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2 Spatial Transformation: Propagation of a Multimode Laser Beam

12.3

507

Amplitude Transformation: Laser Amplification

procedure involving steps (a), (b), (c),

and (d): (a) Starting with the multimode

laser beam characterized by given values of

W0x , Mx2 , and z0x , one defines the embedded Gaussian beam with w0x D W0x =Mx and

beam-waist at the location of the multimode

beam-waist. (b) One then calculates the propagation of the embedded Gaussian beam

through the optical system by, e.g., using

the ABCD law of Gaussian-beam propagation. (c) At any location within the optical

system, the wavefront radius of curvature of

the multimode beam will then coincide with

that of the embedded Gaussian beam. This

means, in particular, that any waist will have

the same location for the two beams. (d) The

spot-size parameter, Wx , of the multimode

beam, at any location, will then be given by

Wx .z/ D Mx wx .z/.

12.3. AMPLITUDE

TRANSFORMATION:

LASER AMPLIFICATION.6

8/

Example 12.1. Focusing of a multimode Nd:YAG beam

by a thin lens Consider a multimode beam from a repetitively pulsed Nd:YAG laser, at Š 1.06 m wavelength,

such as used for welding or cutting metallic materials.

The near-field transverse-intensity profile may be taken

to be approximately Gaussian with a diameter (FWHM)

of D D 4 mm, while the M 2 factor may be taken to be

40. We want now to see what happens when the beam

is focused by a spherical lens of f D 10 cm focal length.

We assume that the waist of this multimode beam coincides with the output mirror, this being a plane mirror. We

will also assume that the lens is located very near to this

mirror so that the waist of the multimode beam and hence

of the embedded Gaussian beam can be taken to coincide

with the lens location. For a Gaussian intensity profile,

the spot size parameter of the input beam, W D W0 , is

then related to the beam diameter, D, by the condition

We get W0 D D=Œ2 ln 21=2 Š

exp 2.D=2W0 /2 D .1=2/. p

3.4 mm, so that w0 D W0 = M 2 Š 0.54 mm. According

to Eq. (4.7.26), since the Rayleigh range corresponding to this spot size, zR D w20 = Š 85 cm, is much

larger than the focal length of the lens, the waist formed

beyond the lens will approximately be located at the lens

focus. From Eq. (4.7.28), the spot-size of the embedded Gaussian beam at this focus, w0f , is then given by

w0f Š f = w0 Š 63 m and the

p spot-size parameter of

the multimode beam by W0f D M 2 w0f Š 400 m.

In this section we consider the rateequation treatment of a laser amplifier. We

assume that a plane wave of uniform intensity I enters (at z D 0) a laser amplifier extending

for a length l along the z direction. We limit our considerations to the case where the incoming

laser beam is in the form of a pulse (pulse amplification) while we refer elsewhere.8/ for the

amplification of a c.w. beam (steady-state amplification).

We consider first the case of an amplifier medium working on a four-level scheme and

, where 1 and are the

further assume that pulse duration, p , is such that 1

p

lifetime of the lower and upper levels of the amplifier medium, respectively. In this case the

population of the lower level of the amplifier can be set equal to zero. This is perhaps the

most relevant case to consider as it would apply, for instance, to the case of a Q-switched

laser pulse from a Nd: YAG laser being amplified. We will also assume that pumping to the

amplifier upper-level and subsequent spontaneous decay can be neglected during the passage

of the pulse and that the transition is homogeneously-broadened. Under these conditions and

with the help of Eq. (2.4.17) [in which we set F D I=hv], the rate of change of population

inversion N.t, z/ at a point z within the amplifier can be written as

@N

D

@t

WN D

NI

s

(12.3.1)

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12

Propagation, Amplification, Frequency Conversion, Pulse Compression

where

s D .hv=

/

(12.3.2)

is the saturation energy fluence of the amplifier [see Eq. (2.8.29)]. Note that a partial derivative

is required in Eq. (12.3.1) since we expect N to be a function of both z and t, i.e., N D N.t, z/,

on account of the fact that I D I.t, z/. Note also that Eq. (12.3.1) can be solved for N.t/ to

yield

N.1/ D N0 exp . = s/

(12.3.3)

where N0 D N. 1/ is the amplifier’s upper-level population before the arrival of the pulse,

as established by the combination of pumping and spontaneous decay, and where

Z

.z/ D

C1

I.z, t/ dt

(12.3.4)

1

is the total fluence of the laser pulse.

Next we derive a differential equation describing the temporal and spatial variation of

intensity I. To do this we first write an expression for the rate of change of e.m. energy within

unit volume of the amplifier. For this we refer to Fig. 12.1 where an elemental volume of the

amplifier medium of length dz and cross-section S is indicated by the shaded area. We can

then write

Â Ã

Â Ã

Â Ã

@

@

@

@

D

C

C

(12.3.5)

@t

@t 1

@t 2

@t 3

where .@ =@t/1 accounts for stimulated emission and absorption in the amplifier, .@ =@t/2 for

the amplifier loss (e.g., scattering losses), and .@ =@t/3 for the net photon flux which flows

into the volume. With the help again of Eq. (2.4.17) ŒF D I=hv we obtain

Â

@

@t

Ã

D WNhv D NI

(12.3.6)

1

FIG. 12.1. Rate of change of the photon energy contained in an elemental volume of length dz and cross sectional

area S of a laser amplifier.

12.3

509

Amplitude Transformation: Laser Amplification

and from Eqs. (2.4.17) and (2.4.32) we obtain

Â Ã

@

D Wa Na hv D

@t 2

˛I

(12.3.7)

where Na is the density of the loss centers, while Wa is the absorption rate, and ˛ the absorption coefficient associated with the loss centers. To calculate .@ =@t/3 , we refer again to

Fig. 12.1, and note that .@ =@t/3 Sdz is the rate of change of e.m. energy in this volume

due to the difference between the incoming and outgoing laser power. We can then write

.@ =@t/3 Sdz D SŒI.t, z/ I.t, z C dz/, which readily gives

Â Ã

@I

@

dz

(12.3.8)

dz D

@t 3

@z

Equation (12.3.5), with the help of Eqs. (12.3.6)–(12.3.8) and with the observation that

.@ =@t/ D .@I=c@t/, gives

1 @I

@I

C

D NI

c @t

@z

˛I

(12.3.9)

This equation, together with Eq. (12.3.1), completely describes the amplification process.

Note that Eq. (12.3.9) has the usual form of a time-dependent transport equation.

Equations (12.3.1) and (12.3.9) must be solved with the appropriate boundary and initial

conditions. As the initial condition we take N.0, z/ D N0 , where N0 is the amplifier upper-level

population before the arrival of the laser pulse. The boundary condition is obviously established by the intensity I0 .t/ of the light pulse injected into the amplifier, i.e., I.t, 0/ D I0 .t/.

For negligible amplifier losses (i.e., neglecting the term ˛l), the solution to Eqs. (12.3.1)

and (12.3.9) can be written as

8

2

39 1

Z

<

=

I0 . 0 / d 0 = s 5

(12.3.10)

I.z, / D I0 . / 1 Œ1 exp. gz/ exp 4

:

;

1

where D t .z=c/ and where g D N0 is the unsaturated gain coefficient of the amplifier.

From Eqs. (12.3.1) and (12.3.9), we can also obtain a differential equation for the total

fluence of the pulse, .z/, given by Eq. (12.3.4). Thus, we first integrate bothÁsides of

R C1

Eq. (12.3.1) with respect to time, from t D 1 to t D C1, to obtain

1 NIdt= s D N0

N.C1/ D N0 Œ1 exp.

= s/, where Eq. (12.3.3) has been used. We then integrate both

sides of Eq. (12.3.9) with respect

to time, on the same time interval, and use the above

Á

R C1

expression for

and

the fact that I.C1, z/ D I. 1, z/ D 0. We obtain

NIdt=

s

1

d

D g s Œ1

dz

exp.

= s/

Again neglecting amplifier losses, Eq. (12.3.11) gives

Ä

Â Ã

in

.l/ D s ln 1 C exp

s

˛

1 G0

(12.3.11)

(12.3.12)

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Propagation, Amplification, Frequency Conversion, Pulse Compression

where G0 D exp.gl/ is the unsaturated gain of the amplifier and in is the energy fluence of the

input beam. As a representative example the ratio = s is plotted in Fig. 12.2 versus in = s

for G0 D 3. Note that, for in

s , Eq. (12.3.12) can be approximated as

.l/ D G0

(12.3.13)

in

and the output fluence increase linearly with the input fluence (linear amplification regime).

Equation (12.3.13) is also plotted in Fig. 12.2 as a dashed straight line starting from the origin.

At higher input fluences, however, increases with in at a lower rate than that predicted by

Eq. (12.3.13) i.e., amplifier saturation begins to occur. For in

s (deep saturation regime)

Eq. (12.3.12) can be approximated to

.l/ D

in

C gl

s

(12.3.14)

Equation (12.3.14) has also been plotted in Fig. 12.2 as a dashed straight line. Note that

Eq. (12.3.14) shows that, for high input fluences, the output fluence is linearly dependent on

the length l of the amplifier. Since s gl D N0 lhv, one then realizes that every excited atom

undergoes stimulated emission and thus contributes its energy to the beam. Such a condition

obviously represents the most efficient conversion of stored energy to beam energy, and for

this reason amplifier designs operating in the saturation regime are used wherever practical.

It should be pointed out again that the previous equations have been derived for an amplifier having an ideal four-level scheme. For a quasi-three-level scheme, on the other hand, one

can see from the considerations developed in Sect. 7.2.2 that Eq. (12.3.1) still applies provided

that s is now given by

s D hv=. e

C

a/

(12.3.15)

where e and a are the effective cross-sections for stimulated emission and absorption,

respectively. One can also show that Eq. (12.3.9) still applies provided that is replaced by e .

FIG. 12.2. Output laser energy fluence

versus input fluence in for a laser amplifier with a small signal gain

G0 D 3. The energy fluence is normalized to the laser saturation fluence s D h = .

12.3

511

Amplitude Transformation: Laser Amplification

It then follows that Eq. (12.3.12) still remain valid provided that s is given by Eq. (12.3.15)

and G0 given by G0 D exp e N0 l. Similar considerations can be made for an amplifier operating on a four-level scheme when the pulse duration becomes much shorter than the lifetime

of the lower level of the transition. In this case the population driven to the lower level by

stimulated emission remains in this level during the pulse and one can show that Eq. (12.3.12)

still remains valid provided that is replaced by e , and s is given by Eq. (12.3.15), where

a is the effective absorption cross-section of the lower level of the transition.

If amplifier losses cannot be neglected, the above picture has to be modified somewhat.

In particular the output fluence .l/ does not continue increasing with input fluence, as in

Fig. 12.2, but reaches a maximum and then decreases. This can be understood by noting

that, in this case, the output as a function of amplifier length tends to grow linearly due to

amplification [at least for high input fluences, see Eq. (12.3.14)] and to decrease exponentially

due to loss [on account of the term ˛ in Eq. (12.3.11)]. The competition of these two terms

then gives a maximum for the output fluence . For ˛

g this maximum value of the output

fluence, m , turns out to be

m

Š g s =˛

(12.3.16)

It should be noted, however, that, since amplifier losses are typically quite small, other phenomena usually limit the maximum energy fluence that can be extracted from an amplifier. In

fact, the limit is usually set by the amplifier damage fluence d (10 J=cm2 is a typical value

for a number of solid-state media). From Eq. (12.3.14) we then get the condition

Š gl s <

d

(12.3.17)

Another limitation to amplifier performance arises from the fact that the unsaturated

gain G0 D exp.gl/ must not be made too high, otherwise two undesirable effects can occur in

the amplifier: (1) parasitic oscillations, (2) amplified spontaneous emission (ASE). Parasitic

oscillation occurs when the amplifier starts lasing by virtue of some internal feedback which

will always be present to some degree, (e.g., due to the amplifier end faces). The phenomenon

of ASE has already been discussed in Sect. 2.9.2. Both these phenomena tend to depopulate

the available inversion and hence decrease the laser gain. To minimize parasitic oscillations

one should avoid elongated amplifiers and in fact ideally use amplifiers with roughly equal

dimensions in all directions. Even in this case, however, parasitic oscillations set an upper

limit .gl/max to the product of gain coefficient, g, with amplifier length, l, i.e.,

gl < .gl/max

(12.3.18)

where, for typical cases, .gl/max may range between three and five. The threshold for ASE

has already been given in Sect. 2.9.2 [see Eq. (2.9.4a), for a Lorentzian line]. For an amplifier

in the form of a cube (i.e., for ˝ Š 1) and for a unitary fluorescence quantum yield we get

G Š 8 [i.e., gl Š 2.1] which is comparable to that established by parasitic oscillations. For

smaller values of solid angle ˝, which are more typical, the value of G for the onset of ASE

is expected to increase [Eq. (2.9.4a)]. Hence parasitic oscillations, rather than ASE, usually

determine the maximum gain that can be achieved.

512

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Propagation, Amplification, Frequency Conversion, Pulse Compression

Example 12.2. Maximum energy which can be extracted

from an amplifier. It is assumed that the maximum

value of gl is limited by parasitic oscillations such that

.gl/2max Š 10 and the rather low gain coefficient of

g D 10 2 cm 1 is also assumed. For a damage energyfluence of the amplifier medium of d D 10 J=cm2 , we

get from Eq. (12.3.19) Em Š l MJ. It is however worth

noting that this represents an upper limit to the energy

since it would require a somewhat impractical amplifier

dimension of the order of lm Š .gl/m =g Š 3 m.

When both limits, due to damage,

Eq. (12.3.17), and parasitic oscillations,

Eq. (12.3.18), are taken into account, one

can readily obtain an expression for the maximum energy Em , that can be extracted from

an amplifier, as

Em D

2

d lm

D

2

2

d .gl/m =g

(12.3.19)

where lm is the maximum amplifier dimension (for a cubic amplifier) implied by

Eq. (12.3.18). Equation (12.3.19) shows that

Em is increased by decreasing the amplifier gain coefficient g. Ultimately, a limit to this

reduction in gain coefficient would be established by the amplifier losses ˛.

So far we have concerned ourselves mostly with the change of laser pulse energy as the

pulse passes through an amplifier. In the saturation regime, however, important changes in

both the temporal and spatial shape of the input beam also occur. The spatial distortions can

be readily understood with the help of Fig. 12.2. For an input beam with a bell-shaped transverse intensity profile (e.g., a Gaussian beam), the beam center, as a result of saturation, will

experience less gain than the periphery of the beam. Thus, the width of the beam’s spatial profile is enlarged as the beam passes through the amplifier. The reason for temporal distortions

can also be seen quite readily. Stimulated emission caused by the leading edge of the pulse

implies that some of the stored energy has already been extracted from the amplifier by the

time the trailing edge of the pulse arrives. This edge will therefore see a smaller population

inversion and thus experience a reduced gain. As a result, less energy is added to the trailing

edge than to the leading edge of the pulse, and this leads to considerable pulse reshaping.

The output pulse shape can be calculated from Eq. (12.3.10), and it is found that the amplified pulse may either broaden or narrow (or even remain unchanged), the outcome depending

upon the shape of the input pulse..7/

12.3.1. Examples of Laser Amplifiers: Chirped-Pulse-Amplification

One of the most important and certainly the most spectacular example of laser pulse

amplification is that of Nd:glass amplifiers used to produce pulses of high energy (10–100 kJ)

for laser fusion research..8/ Very large Nd:glass laser systems have, in fact, been built and operated at a number of laboratories throughout the world, the one having the largest output energy

being operated at the Lawrence Livermore National Laboratory in the USA (the NOVA laser).

Most of these Nd:glass laser systems exploit the master-oscillator power-amplifier (MOPA)

scheme. This scheme consists of a master oscillator, which generates a well controlled pulse

of low energy, followed by a series of power amplifiers, which amplify the pulse to high

energy. The clear aperture of the power amplifiers is increased along the chain to avoid optical damage as the beam energy increases. A schematic diagram of one of the ten arms of the

NOVA system is shown in Fig. 12.3. The initial amplifiers in the chain consist of phosphateglass rods (of 380 mm length and with a diameter of 25 mm for the first amplifiers, 50 mm

for the last). The final stage of amplification is achieved via face-pumped disk amplifiers (see

12.3

Amplitude Transformation: Laser Amplification

FIG. 12.3. Schematic layout of the amplification system, utilizing Nd:glass amplifiers, for one arm of the Nova

system [after ref.,.8/ by permission].

Fig. 6.3b) with large clear-aperture diameter (10 cm for the first amplifiers, 20 cm for the last).

Note the presence in Fig. 12.3 of Faraday isolators (see Fig. 7.23) whose purpose is to avoid

reflected light counter-propagating through the amplifier chain and thus damaging the initial

stages of the system. Note also the presence of spatial filters consisting of two lenses, in a

confocal arrangement (Fig. 11.10), with a pinhole at the common focus. These filters serve

the double purpose of removing the small-scale spatial irregularities of the beam, as well as

matching the beam profile between two consecutive amplifiers of different aperture. The laser

system of Fig. 12.3 delivers an output energy of 10 kJ in a pulse of duration down to 1 ns,

which gives a total energy of the 10-arms NOVA system of 100 kJ. Laser systems based on

this layout concept and delivering an overall output energy of 1 MJ are now being built in

USA (National Ignition Facility, NIF, Livermore) and in France (Megajoule project, Limeil)

[see also Sect. 9.2.2.2].

A second class of laser amplifiers which has revolutionized the laser field in terms of

focusable beam intensity, relies on the Chirped Pulse Amplification (CPA) concept.9/ and is

used to amplify picosecond or femtosecond laser pulses. At such short pulse durations, in

fact, the maximum energy which can be obtained from an amplifier depends on the onset,

either of self-focusing, which is related to the beam peak power, or multi-photon-ionization,

which is related to the beam peak intensity. To overcome these limitations, one can adopt a

technique, already used in radar technology, of pulse expansion (or pulse-stretching), before

amplification, followed by pulse compression, to its original shape, after the amplification

process. In this way the peak power and hence the peak intensity of the pulse, in the amplifier

chain, may be reduced by a few orders of magnitude .103 –104 /. This allows a corresponding

increase in the maximum energy which can be safely extracted from a given amplifier. Pulse

expansion is achieved via an optical system which provides, e.g., a positive group-delay dispersion (GDD). In this way the pulse may be considerably expanded in time while acquiring

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Propagation, Amplification, Frequency Conversion, Pulse Compression

a positive frequency sweep (see Sect. 12.5.2 and Appendix G). The amplified pulse is then

passed through an optical system having negative GDD (see Sect. 12.5.1). The effect of this

second dispersive element is then to compensate the frequency sweep introduced by the first

and so restore the initial shape of the pulse entering the amplifier chain.

Figure 12.4 illustrates a commonly used lay-out for a Ti:sapphire CPA. In the figure,

P1 , P2 , and P3 are three polarizers which transmit light whose field is polarized in the plane of

the figure (horizontally-polarized light) while reflecting light with field polarized orthogonal

to the figure (vertically-polarized light). The combination of the =2-plate and Faraday-rotator

(F.R.) is such as to transmit, without rotation, light traveling from right to left and to rotate, by

90ı , the polarization of light traveling from left to right (see Fig. 7.24). Low-energy . 1 nJ/,

high-repetition rate .f Š 80 MHz/, horizontally-polarized, femtosecond pulses, emitted by a

Ti:sapphire mode-locked oscillator, are sent to the CPA. They are thus transmitted by polarizer

P2 , do not suffer polarization-rotation on passing through the =2-plate-F.R. combination, are

then transmitted by polarizer P1 , and thus sent to the pulse stretcher (whose lay-out will be

discussed in Sect. 12.5.2). Typical expansion of the retroreflected pulse from the stretcher may

be by a factor of 5, 000, e.g. from 100 fs to 500 ps. The expanded pulses are then transmitted

by polarizer P1 , undergo a 90ı polarization rotation in the F.R.- =2-plate combination and

are reflected by polarizer P2 . With the help of polarizer P3 , the expanded pulses are then

injected into a so-called regenerative amplifier which consists of a Ti:sapphire amplifier and a

Pockels cell (P.C.) located in a three-mirror (M1 , M2 , and M3 ) folded resonator. The Pockels

cell is oriented so as to produce a static =4 retardation. The cavity Q is thus low before the

pulse arrival and the regenerative amplifier is below the oscillation threshold. In this situation,

any injected pulse become horizontally polarized after a double passage through the P.C.,

and is thus transmitted by polarizer P3 toward the Ti:sapphire amplifier. After returning from

FIG. 12.4. Amplification of femtosecond laser pulses via a Ti:sapphire regenerative amplifier and the chirped-pulseamplification technique.

12.3

Amplitude Transformation: Laser Amplification

the amplifier, the pulse is transmitted by polarizer P3 , and, after again double-passing the

P.C., becomes vertically polarized and thus reflected out of the cavity by polarizer P3 . So,

in this double transit through the regenerative amplifier, very little amplification is obtained

for the output pulse. If however, while the pulse is between the polarizer and mirror M1 , a

=4-voltage is applied to the P.C., the cell becomes equivalent to a =2-plate, and the pulse

does not change its polarization state after each double passage through the cell. Therefore,

the pulse gets trapped in the regenerative amplifier and, on each pass through the amplifying

medium, it is amplified. After a suitable number of round-trips in the cavity (typically 15–20),

the pulse energy reaches its maximum value and is then extracted from the cavity by applying

an additional =4 voltage to the P.C.. In this case, in fact, after a double pass through the cell,

the pulse becomes vertically polarized and is reflected by polarizer P3 back in the direction

of the incoming pulses. This, high-energy, vertically-polarized pulse is reflected by polarizer

P2 , does not suffer polarization-rotation on passing through the /2-plate-F.R. combination,

and is reflected by polarizer P1 toward the pulse compressor (whose lay-out will be discussed

in Sect. 12.5.1). The retroreflected beam from the compressor then consists of a train of high

energy pulses, each with a duration approximately equal to that of the original pulses emitted

by the oscillator, and with a repetition rate equal to that at which the Ti:sapphire amplifier is

pumped (1–10 kHz, usually by the second-harmonic green-beam of a repetitively Q-switched

Nd:YLF laser).

Systems of this type, exploiting the CPA technique, have allowed the development of

lasers with ultra-high peak-power..10/ For instance, using Ti:sapphire active media, table-top

CPA systems with peak power of 20 TW have already been demonstrated while systems

with peak powers approaching 100 TW (e.g., 2 J in a 20 fs pulse) are under construction. The largest peak power, so far achieved by exploiting the CPA technique, is actually

1.25 PW .1 PW D 1015 W/,.11/ obtained using a chain of amplifiers taken from one arm of

the NOVA laser (so as to obtain an amplified pulse with 580 J energy and 460 fs duration). The peak intensity obtained by focusing these ultra-high-power pulses is extremely high

.1019 –1020 W=cm2 /, representing an increase of four to five orders of magnitude compared to

that available before introducing the CPA technique. When these ultra-high intensity beams

interact with a solid target or with a gas, a highly ionized plasma is obtained and a completely

new class of nonlinear optical phenomena is produced. Applications of these high intensities cover a broad area of science and technology including ultrafast x-ray and high-energy

electron sources, as well as novel fusion concepts and plasma astrophysics..12/

A third class of amplifier, widely used in optical fiber communications, is represented by

the Er-doped optical-fiber amplifier (EDFA)..13/ This amplifier is diode-pumped either in the

980 nm or 1480 nm pump bands of the ErC ion [see Fig. 9.4] and is used to amplify pulses at

wavelengths corresponding to the so-called third transmission window of silica optical fibers

. Š 1550 nm/. Since, usually, the pulse repetition rate of a communication system is very

high . GHz/ and the upper-state lifetime of ErC is very long ( 10 ms, see Table 9.4) the

saturation behavior of the ErC population is a cumulative result of many laser pulses, i.e.,

determined by the average beam intensity. The rate-equation treatment of this type of amplification can then be made in terms of average beam intensity and, in principle, is very simple.

Complications however arise from several factors, namely: (1) The ErC system works on an

almost pure three-level scheme (see Sect. 9.2.4) and therefore, the effective cross-sections of

stimulated-emission and absorption, both covering a large spectral bandwidth, must be taken

into account. (2) Transverse variation, within the fiber, of both the ErC population profile and

515

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Propagation, Amplification, Frequency Conversion, Pulse Compression

the intensity profile of the propagating mode, must be taken into account. (3) Account must

also be taken of the simultaneous presence of bidirectional noise arising from amplified spontaneous emission (ASE). We therefore make no attempt to cover this subject in any detail,

limiting ourselves to pointing out that a vast literature exists,.14/ that very high small-signal

gain (up to 50 dB), relatively large saturated average output powers . 100 mW/, and low

noise are achieved via these amplifiers. Thus, Erbium-doped fiber amplifiers must be considered a major break-through in the field of optical fiber communications, with applications

regarding both long-haul systems as well as distribution networks.

12.4. FREQUENCY CONVERSION: SECOND-HARMONIC

GENERATION AND PARAMETRIC OSCILLATION.1, 15/

In classical linear optics one assumes that the induced dielectric polarization of a medium

is linearly related to the applied electric field, i.e.,

P D "0 E

(12.4.1)

where is the dielectric susceptibility. With the high electric fields involved in laser beams

the above linear relation is no longer a good approximation, and further terms in which P is

related to higher-order powers of E must also be considered. This nonlinear response can lead

to an exchange of energy between e.m. waves at different frequencies.

In this section we will consider some of the effects produced by a nonlinear polarization

term that is proportional to the square of the electric field. The two effects that we will consider

are: (1) Second-harmonic generation (SHG), in which a laser beam at frequency ! is partially

converted, in the nonlinear material, to a coherent beam at frequency 2! [as first shown by

Franken et al..16/ ]. (2) Optical parameter oscillation (OPO), in which a laser beam at frequency

!3 causes the simultaneous generation, in the nonlinear material, of two coherent beams at

frequency !1 and !2 such that !1 C !2 D !3 [as first shown by Giordmaine and Miller.17/ ].

With the high electric fields available in laser beams the conversion efficiency of both these

processes can be very high (approaching 100% in SHG). Therefore, these techniques are

increasingly used to generate new coherent waves at different frequencies from that of the

incoming wave.

12.4.1. Physical Picture

We will first introduce some physical concepts using the simplifying assumption that the

induced nonlinear polarization PNL is related to the electric field E of the e.m. wave by a scalar

equation, i.e.,

PNL D 2ε0 dE2

(12.4.2)

where d is a coefficient whose dimension is the inverse of an electric field. The physical origin of Eq. (12.4.2) resides in the nonlinear deformation of the outer, loosely bound, electrons

We use 2ε0 dE 2 rather than dE 2 (as often used in other textbooks) to make d conform to increasingly accepted

practice.

12.4

517

Frequency Conversion: Second-Harmonic Generation

of an atom or atomic system when subjected to high electric fields. This is analogous to a

breakdown of Hooke’s law for an extended spring, resulting in the restoring force no longer

being linearly dependent on the displacement from equilibrium. A comparison of Eqs. (12.4.2)

and (12.4.1) shows that the nonlinear polarization term becomes comparable to the linear one

for an electric field E Š =d. Since Š 1, we see that .1=d/ represents the field strength

for which the linear and nonlinear terms become comparable. At this field strength, a sizable

nonlinear deformation of the outer electrons must occur and .1=d/ is then expected to be of

the order of the electric field that an electronic charge produces at a distance corresponding

to a typical atomic dimension a, i.e., .1=d/ Š e=4 ε0 a2 [thus .1=d/ Š 1011 V=m for a Š

0.1 nm]. We note that d must be zero for a centrosymmetric material, such as a centrosymmetric crystal or the usual liquids and gases. For symmetry reasons, in fact, if we reverse the

sign of E, the sign of the total polarization Pt D P C PNL must also reverse. Since, however,

PNL / dE2 , this can only occur if d D 0. From now on we will therefore confine ourselves to

a consideration of non-centrosymmetric materials. We will see that the simple Eq. (12.4.2) is

then able to account for both SHG and OPO.

12.4.1.1. Second-Harmonic Generation

We consider a monochromatic plane wave of frequency ! propagating along some direction, denoted as the z-direction, within a nonlinear crystal, the origin of the z-axis being taken

at the entrance face of the crystal. For a plane wave of uniform intensity we can write the

following expression for the electric field E! .z, t/ of the wave

E! .z, t/ D .1=2/ fE.z, !/ expŒj.! t

k! z/ C c.c.g

(12.4.3)

In the above expression c.c. means the complex conjugate of the other term appearing in the

brackets and

k! D

!

n! !

D

c!

c

(12.4.4)

where c! is the phase velocity, in the crystal, of a wave of frequency !, n! is the refractive

index at this frequency, and c is the velocity of light in vacuum. Substitution of Eq. (12.4.3)

into Eq. (12.4.2) shows that PNL contains a term oscillating at frequency 2!, namely,

˚ 2

PNL

2! D ."0 d=2/ E .z, !/ expŒj.2! t

«

2k! z/ C c.c.

(12.4.5)

Equation (12.4.5) describes a polarization wave oscillating at frequency 2! and with

a propagation constant 2k! . This wave is then expected to radiate at frequency 2!, i.e., to

generate an e.m. wave at the second harmonic (SH) frequency 2!. The analytical treatment,

given later, involves in fact substitution of this polarization in the wave equation for the e.m.

field. The radiated SH field can be written in the form

E2! .z, t/ D .1=2/ fE.z, 2!/ expŒj.2! t

k2! z/ C c.c.g

(12.4.6)

The quantity PNL also contains a term at frequency ! D 0 which leads to development of a dc voltage across the

crystal (optical rectification).