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2 Spatial Transformation: Propagation of a Multimode Laser Beam

# 2 Spatial Transformation: Propagation of a Multimode Laser Beam

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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 21=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.

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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)
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

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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).