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4 Frequency Conversion: Second-Harmonic Generation and Parametric Oscillation

4 Frequency Conversion: Second-Harmonic Generation and Parametric Oscillation

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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. 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


In the above expression c.c. means the complex conjugate of the other term appearing in the
brackets and
k! D

n! !


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
2! D ."0 d=2/ E .z, !/ expŒj.2! t

2k! z/ C c.c.


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


The quantity PNL also contains a term at frequency ! D 0 which leads to development of a dc voltage across the
crystal (optical rectification).



Propagation, Amplification, Frequency Conversion, Pulse Compression

k2! D

2n2! !


is the propagation constant of a wave at frequency 2!. The physical origin of SHG can thus be
traced back to the fact that, as a result of the nonlinear relation Eq. (12.4.2), the e.m. wave at
the fundamental frequency ! will beat with itself to produce a polarization at 2!. A comparison of Eq. (12.4.5) with Eq. (12.4.6) reveals a very important condition that must be satisfied
if this process is to occur efficiently, viz., that the phase velocity of the polarization wave
. P D 2!=2k! / be equal to that of the generated e.m. wave . E D 2!=k2! /. This condition
can thus be written as
k2! D 2k!


In fact, if this condition is not satisfied, the phase of the polarization wave at coordinate
z D l into the crystal, 2k! l, will be different from that, k2! l, of the wave generated at z D 0
which has subsequently propagated to z D l. The difference in phase, .2k! k2! /l, would then
increase with distance l and the generated wave, being driven by a polarization which does not
have the appropriate phase, will then not grow cumulatively with distance l. Equation (12.4.8)
is therefore referred to as the phase-matching condition. Note that, according to Eqs. (12.4.4)
and (12.4.7), equation (12.4.8) implies that
n2! D n!


Now, if the polarization directions of E! and PNL (and hence of E2! / were indeed the
same [as implied by Eq. (12.4.2)] condition Eq. (12.4.9) could not be satisfied owing to the
dispersion . n D n2! n! / of the crystal. This would then set a severe limit to the crystal
length lc over which PNL can give contributions which keep adding cumulatively to form the
second harmonic wave. This length lc (the coherence length) must in fact correspond to the
distance over which the polarization wave and the SH wave get out of phase with each other
by an amount . This means that k2! lc 2k! lc D , from which, with the help of Eqs. (12.4.4)
and (12.4.7), one gets
lc D

4 n


where D 2 c=! is the wavelength in vacuum of the fundamental wave. Taking, as an example, Š 1 m and n D 10 2 , we get lc D 25 m. Note that, at this distance into the crystal,
the polarization wave becomes 180ı out of phase compared to the SH wave and the latter
begins to decrease with increased distance rather than continuing to grow. Since, as seen in
the previous example, lc is usually very small, only a very small fraction of the incident power
can then be transformed into the second harmonic wave.
At this point it is worth pointing out another useful way of visualizing the SHG process,
in terms of photons rather than fields. First we write the relation between the frequency of the
fundamental (!) and second-harmonic .!SH / wave, viz.,
!SH D 2!




Frequency Conversion: Second-Harmonic Generation

If we now multiply both sides of Eqs. (12.4.11) and (12.4.8) by „, we get
„!SH D 2„!


„k2! D 2„k!


respectively. For energy to be conserved in the SHG process, we must have dI 2! =dz D
dI ! =dz, where I2! and I! are the intensities of the waves at the two frequencies. With
the help of Eq. (12.4.12a) we get dF2! =dz D .1=2/dF! =dz, where F2! and F! are the photon fluxes of the two waves. From this equation we can then say that, whenever, in the SHG
process, one photon at frequency 2! is produced, correspondingly two photons at frequency
! disappear. Thus the relation Eq. (12.4.12a) can be regarded as a statement of conservation
of photon energy. Remembering that „k is the photon momentum, Eq. (12.4.12b) is then seen
to correspond to the requirement that photon momentum is also conserved in the process.
We now reconsider the phase-matching condition Eq. (12.4.9) to see how it can be satisfied in a suitable, optically anisotropic, crystal..18, 19/ To understand this we will first need
to make a small digression to explain the propagation behavior of waves in an anisotropic
crystal, and also to show how the simple nonlinear relation Eq. (12.4.2) should be generalized
for anisotropic media.
In an anisotropic crystal it can be shown that, for a given direction of propagation, there
are two linearly polarized plane waves that can propagate with different phase velocities. Corresponding to these two different polarizations one can then associate two different refractive
indices, the difference of refractive index being referred to as the crystal’s birefringence. This
behavior is usually described in terms of the so-called index ellipsoid which, for a uniaxial
crystal, is an ellipsoid of revolution around the optic axis (the z axis of Fig. 12.5). Given this
ellipsoid, the two allowed directions of linear polarization and their corresponding refractive
indices are found as follows: Through the center of the ellipsoid one draws a line in the direction of beam propagation (line OP of Fig. 12.5) and a plane perpendicular to this line. The
intersection of this plane with the ellipsoid is an ellipse. The direction of the two axes of the
ellipse then give the two polarization directions and the length of each semiaxis gives the
refractive index corresponding to that polarization. One of these directions is necessarily perpendicular to the optic axis and the wave having this polarization is referred to as the ordinary
wave. Its refractive index, no , can be seen from the figure to be independent of the direction
of propagation. The wave with the other direction of polarization is referred to as the extraordinary wave and the corresponding index, ne .Â/, depends of the angle  and ranges in value
from that of the ordinary wave n0 (when OP is parallel to z) to the value ne , referred to as the
extraordinary index, which occurs when OP is perpendicular to z. Note now that one defines a
positive uniaxial crystal as corresponding to the case ne > no while a negative uniaxial crystal
corresponds to the case ne < no . An equivalent way to describe wave propagation is through
the so-called normal (index) surfaces for the ordinary and extraordinary waves (Fig. 12.6). In
this case, for a given direction of propagation OP and, for either ordinary or extraordinary
waves, the length of the segment between the origin O and the point of interception of the ray
OP with the surface gives the refractive index of that wave. The normal surface for the ordinary wave is thus a sphere, while the normal surface for the extraordinary wave is an ellipsoid
of revolution around the z axis. In Fig. 12.6 the intersections of these two normal surfaces
with the y-z plane are indicated for the case of a positive uniaxial crystal.



Propagation, Amplification, Frequency Conversion, Pulse Compression

FIG. 12.5. Index ellipsoid for a positive uniaxial crystal.

FIG. 12.6. Normal (index) surface for both the ordinary and extraordinary waves (for a positive uniaxial crystal).

After this brief discussion of wave propagation in anisotropic crystals, we now return to
the problem of the induced nonlinear polarization. In general, in an anisotropic medium, the
scalar relation Eq. (12.4.2) does not hold and a tensor relation needs to be introduced. First,
we write the electric field E! .r, t/ of the e.m. wave at frequency ! and at a given point r and
the nonlinear polarization vector at frequency 2!, P2!
NL .r, t/ in the form
E! .r, t/ D .1=2/ŒE! .r, !/ exp.j!t/ C c.c.


.r, 2!/ exp.2j!t/ C c.c.
P 2!
NL .r, t/ D .1=2/ŒP


A tensor relation can then be established between P2! .r, 2!/ and E! .r, !/. In fact, the
second harmonic polarization component, along, e.g., the i-direction of the crystal, can be
written as
i D

j,k D 1,2,3

2! ! !
"0 dijk
Ej Ek




Frequency Conversion: Second-Harmonic Generation

Note that Eq. (12.4.14) is often written in condensed notation as
i D


m "0 dim




where m runs from one to six. The abbreviated field notation is that .EE/1 Á E12 Á Ex2 , Á
.EE/2 Á E22 Á Ey2 , .EE/3 Á E32 Á Ez2 , .EE/4 Á 2E2 E3 Á 2Ey Ez , .EE/5 Á 2E1 E3 Á
2Ex Ez , and .EE/6 Á 2E1 E2 Á 2Ex Ey , where both the 1, 2, 3 and the x, y, z notation for
axes have been indicated. Note that, expressed in matrix form, dim is a 3 6 matrix that
operates on the column vector .EE/m . Depending on the crystal symmetry, some of the values
of the dim matrix may be equal and some may be zero. For the 42m
point group symmetry,
which includes the important nonlinear crystals of the KDP type and the chalcopyrite semiconductors, only d14 , d25 , and d36 are non-zero and these three d coefficients, are themselves
equal. Therefore only one coefficient, e.g., d36 , needs to be specified, and one can write
Px D 2"0 d36 Ey Ez


Py D 2"0 d36 Ez Ex


Pz D 2"0 d36 Ex Ey


where the z-axis is again taken along the optic axis of the uniaxial crystal. The nonlinear
optical coefficients, the symmetry class, the transparency range, and the damage threshold of
some selected nonlinear materials are indicated in Table 12.1. Except for cadmium germanium
arsenate and AgGaSe2 , which are commonly used around the 10μm range, all the other crystals listed are used in the near UV to near IR range. The table includes the more recent crystals
of KTP (potassium titanyl phosphate), and BBO (beta-barium borate), the former being commonly used for second harmonic generation at, e.g., the Nd:YAG wavelength. The nonlinear
d-coefficients are normalized to that of KDP, whose actual value is d36 Š 0.5 10 12 m=V.
Following this digression on the properties of anisotropic media we can now go on to
show how phase matching can be achieved for the particular case of a crystal of 42m
group symmetry. From Eq. (12.4.16) we note that, if Ez D 0, only Pz will be non-vanishing
and will thus tend to generate a second-harmonic wave with a non-zero z-component. We
recall (see Fig. 12.5) that a wave with Ez D 0 is an ordinary wave while a wave with Ez ¤ 0 is
an extraordinary wave. Thus an ordinary wave at the fundamental frequency ! tends, in this
case, to generate an extraordinary wave at 2!. To satisfy the phase-matching condition one
can then propagate the fundamental wave at an angle Âm to the optic axis, in such a way that
ne .2!, Âm / D no .!/


This can be better understood with the help of Fig. 12.7 which shows the intercepts of the
normal surfaces no .!/ and ne .2!, Â/ with the plane containing the z axis and the propagation
direction. Note that, since crystals usually show a normal dispersion, one has no .!/ < no .2!/,
while for a negative uniaxial crystal one has ne .2!/ < no .2!/, where, as a short-hand notation (see Fig. 12.7), we have set ne .2!/ Á ne .2!, 90ı / and no .2!/ Á ne .2!, 0/. Thus the
ordinary circle, corresponding to the wave at frequency, intersects the extraordinary ellipse,



Propagation, Amplification, Frequency Conversion, Pulse Compression
TABLE 12.1. Nonlinear optical coefficients for selected materials



Lithium iodate

NH4 H2 PO4
CsH2 AsO4

Lithium niobate






BaB2 O4

d coefficient
(relative to KDP)
d36 D d14 D 1
d36 D d14 D 0.92
d36 D d14 D 1.2
d36 D d14 D 0.92
d31 D d32 D d24
d15 D 12.7
d31 D 12.5
d22 D 6.35
d31 D 13
d32 D 10
d33 D 27.4
d24 D 15.2
d15 D 12.2
d22 D 4.1


. m/

.GW=cm2 /

4N 2m
4N 2m
4N 2m
4N 2m













d36 D d14 D 538

4N 2m




d36 D d14 D 66

4N 2m



FIG. 12.7. Phase-matching angle Âm for type I second-harmonic generation in a negative uniaxial crystal.

corresponding to the wave at frequency 2!, at some angle Âm . For light propagating at
this angle Âm to the optic axis (i.e., for all ray directions lying in a cone around the z axis,
with cone angle Âm /, Eq. (12.4.17) is satisfied and hence the phase-matching condition is
It should be noted that for this intersection to occur at all it is necessary for ne .2!, 90ı / to be less than
no .!/, otherwise the ellipse for ne .2!/ (see Fig 12.7) will lie wholly outside the circle for no .!/. Thus
ne .2!, 90ı / D ne .2!/ < no .!/ < no .2!/, which shows that crystal birefringence no .2!/ ne .2!/ must be larger
than crystal dispersion no .2!/ no .!/.


Frequency Conversion: Second-Harmonic Generation


It should be noted that, if Âm ¤ 90ı , the
Example 12.3. Calculation of the phase-matching angle
phenomenon of double refraction will occur
for a negative uniaxial crystal. With reference to
in the crystal, i.e., the direction of the energy
Fig. 12.7, we label the horizontal-axis as the y-axis. If
flow for the extraordinary (SH) beam will be
we then let z and y represent the cartesian coordinates of
at an angle slightly different from Âm . Thus
general point of the ellipse describing the extraordinary
the fundamental and SH beams will travel
index, ne .2!, Â/, one can write
in slightly different directions (although satisfying the phase-matching condition). For a
fundamental beam of finite transverse dimeno 2
e 2
sions this will put an upper limit on the interaction length in the crystal. This limitation
where, as a short-hand notation, we have set no2 D no .2!/
can be overcome if it is possible to operate
and and ne2 D ne .2!/. If the coordinates z and y are now
with Âm D 90ı , i.e., ne .2!, 90ı / D no .!/.
expressed as a function of ne .2!, Â/ and of the angle Â,
This is referred to as the 90ı phase matching
the previous equation transforms to
condition. Since ne and no generally undergo
different changes with temperature, it turns
Œne .2!, Â/2 2
Œne .2!, Â/2
out that 90ı phase matching condition can,
sin  D 1
in some cases, be reached by changing the
crystal temperature. To summarize the above
For  D Âm , Eq. (12.4.17) must hold. Substitution of this
discussion, we can say that phase matchequation into the equation above then gives
ing is possible in a (sufficiently birefringent)
negative uniaxial crystal when an ordinary
 o Ã2
 o Ã2
ray at ! [Ex beam of Eq. (12.4.16c)] com2
sin2 Âm D 1
bines with an ordinary ray at ! [Ey beam of
Eq. (12.4.16c)] to give an extraordinary ray
at 2!, or, in symbols, o! C o! ! e2! . This
where, again as a short-hand notation we have set
is referred to as type I second-harmonic genno1 D no .!/. This last equation can be solved for sin2 Âm
eration. In a negative uniaxial crystal another
to obtain
scheme for phase-matched SHG, called type
no1 2
no2 2
II, is also possible. In this case an ordinary
wave at ! combines with an extraordinary
sin2 Âm D o Á2 2 o Á2 D 1o Á2
wave ! at to give an extraordinary wave at
2!, or, in symbols, o! C e! ! e2! .
Second-harmonic generation is currently used to provide coherent sources at new wavelengths. The nonlinear crystal may be
placed either outside or inside the cavity of the laser producing the fundamental beam. In
the latter case one takes advantage of the greater e.m. field strength inside the resonator to
increase the conversion efficiency. Very high conversion efficiencies (approaching 100%) have
been obtained with both arrangements. Among the most frequent applications of SHG are frequency doubling the output of a Nd:YAG laser (thus producing a green beam, D 532 nm,
from an infrared one, D 1.064 m) and generation of tunable UV radiation (down to
Š 205 nm) by frequency doubling a tunable dye laser. In both of these cases either cw
More generally, interactions in which the polarizations of the two fundamental waves are the same are termed type
I (e.g., also e! C e! ! o2! ), and interactions in which the polarization of the fundamental waves are orthogonal
are termed type II.



Propagation, Amplification, Frequency Conversion, Pulse Compression

or pulsed laser sources are used. The nonlinear crystals most commonly used as frequency
doublers for Nd:YAG lasers are KTP and ˇ BaB2 O4 (BBO), while BBO, due to its more
extended transparency toward the UV, is particularly used when a SH beam at UV wavelengths
down to 200 nm have to be generated. Efficient frequency conversion of infrared radiation
from CO2 or CO lasers is often produced in chalcopyrite semiconductors (e.g., CdGeAs2 ). Parametric Oscillation
We now go on to discuss the process of parametric oscillation. We begin by noticing that
the previous ideas introduced in the context of SHG can be readily extended to the case of two
incoming waves at frequencies !1 and !2 combining to give a wave at frequency !3 D !1 C!2
(sum-frequency generation). Harmonic generation can in fact be thought of as a limiting case
of sum-frequency generation with !1 D !2 and !3 D 2!. The physical picture is again very
similar to the SHG case: By virtue of the nonlinear relation Eq. (12.4.2) between PNL and the
total field EŒE D E!1 .z, t/ C E!2 .z, t/, the wave at !1 will beat with that at !2 , to give a
polarization component at !3 D !1 C !2 . This will then radiate an e.m. wave at !3 . Thus for
sum-frequency generation we can write
„!1 C „!2 D „!3


which, according to a description in terms of photons rather than fields, implies that one
photon at !1 and one photon at !2 disappear while a photon at !3 is created. We therefore
expect the photon momentum to be also conserved in the process, i.e.,
„ k1 C „ k2 D „ k3


where the relationship is put in its general form, with the k denoted by vectors. Equation (12.4.18b), which expresses the phase-matching condition for sum-frequency generation, can be seen to be a straightforward generalization of that for SHG [compare with
Eq. (12.4.12b)].
Optical parametric generation is in fact just the reverse of sum-frequency generation.
Here, instead, a wave at frequency !3 (the pump frequency) generates two waves (called the
idler and signal waves) at frequencies !1 and !2 , in such a way that the total photon energy
and momentum is conserved, i.e.,
„!3 D „!1 C „!2


„ k3 D „ k1 C „ k2


The physical process occurring in this case can be visualized as follows. Imagine first that a
strong wave at !3 and a weak wave at !1 are both present in the nonlinear crystal. As a result
of the nonlinear relation Eq. (12.4.2), the wave at !3 will beat with the wave at !1 to give
a polarization component at !3 !1 D !2 . If the phase-matching condition Eq. (12.4.19b)
is satisfied, a wave at !2 will thus build up as it travels through the crystal. Then the total
E field will in fact be the sum of three fields ŒE D E!1 .z, t/ C E!2 .z, t/ C E!3 .z, t/ and
the wave at !2 will in turn beat with the wave at !3 to give a polarization component at



Frequency Conversion: Second-Harmonic Generation

FIG. 12.8. Schematic diagram of an optical parametric oscillator.

!3 !2 D !1 . This polarization will cause the !1 wave to grow also. Thus power will be
transferred from the beam at !3 to those at !1 and !2 , and the weak wave at !1 which was
assumed to be initially present will be amplified. From this picture one sees a fundamental
difference between parametric generation and SHG. In the latter case only a strong beam at the
fundamental frequency is needed for the SHG process to occur. In the former case, however, a
weak beam at !1 is also needed and the system behaves like an amplifier at frequency !1 (and
!2 ). In practice, however, the weak beam need not be supplied by an external source (such
as another laser) since it is generated, internally to the crystal, as a form of noise (so-called
parametric noise). One can then generate coherent beams from this noise in a way analogous to
that used in a laser oscillator. Thus, the nonlinear crystal, which is pumped by an appropriately
focused pump beam, is placed in an optical resonator (Fig. 12.8). The two mirrors (1 and 2)
of this parametric oscillator have high reflectivity (e.g., R1 D 1 and R2 Š 1) either at !1 only
(singly resonant oscillator, SRO) or at both !1 and !2 (doubly resonant oscillator, DRO). The
mirrors are ideally transparent to the pump beam. Oscillation will start when the gain arising
from the parametric effect just exceeds the losses of the optical resonator. Some threshold
power of the input pump beam is therefore required before oscillation will begin. When this
threshold is reached, oscillation occurs at both !1 and !2 , and the particular pair of values of
!1 and !2 is determined by the two Eq. (12.4.19). For instance, with type I phase matching
involving an extraordinary wave at !3 and ordinary waves at !1 and !2 (i.e., e!3 ! o!1 Co!2 /,
Eq. (12.4.19b) would give
!3 ne .!3 , Â/ D !1 no .!1 / C !2 no .!2 /


For a given Â, i.e., for a given inclination of the nonlinear crystal with respect to the cavity
axis, Eq. (12.4.20) provides a relation between !1 and !2 which, together with the relation
Eq. (12.4.19a), determines the values of both !1 and !2 . Phase-matching schemes of both type
I and type II (e.g., e!3 ! o!1 C e!2 for a negative uniaxial crystal) are possible and tuning
can be achieved by changing either the crystal inclination (angle tuning) or its temperature
(temperature tuning). As a final comment, we note that, if the gain from the parametric effect
is large enough, one can dispense with the mirrors altogether, and an intense emission at !1
and !2 , grows from parametric noise in a single pass through the crystal. This behavior is
often referred to as superfluorescent parametric emission and such a device is referred to as
an optical parametric generator (OPG).
Singly resonant and doubly resonant optical parametric oscillators have both been used.
Doubly resonant parametric oscillation has been achieved with both c.w. and pulsed pump



Propagation, Amplification, Frequency Conversion, Pulse Compression

lasers. For c.w. excitation, threshold powers as low as a few milliwatts have been demonstrated. It should be noted, however, that the requirement for the simultaneous resonance of
both parametric waves in the same cavity generally leads to poor amplitude and frequency
stability of the output beams. Singly resonant parametric oscillation had, until relatively
recently, only been achieved using pulsed pump lasers since the threshold pump power for
the singly resonant case is much higher (by as much as two orders of magnitude) than
that of the doubly resonant case. However, with improved nonlinear crystals, c.w. oscillation is now readily achieved. Singly resonant oscillators produce a much more stable
output and, for this reason, is the most frequently used. Optical parametric oscillators producing coherent radiation from the visible to the near infrared .0.5–5 m/ are now well
developed, with the most successful devices being based on BBO, LBO and lithium niobate .LiNbO3 /. Optical parametric oscillators can also generate coherent radiation at longer
infrared wavelengths (to 14 m) using crystals such as silver-gallium selenide .AgGaSe2 /
and cadmium selenide (CdSe). Synchronous pumping of OPOs, using a mode-locked pump,
is also proving very attractive as a means of generating short pulses with very wide tunability. A notable feature of these devices is that their gain is determined by the peak power of
the pump pulse, so that thresholds corresponding to very low average powers (a few milliwatts) can be achieved even for a singly resonant oscillator. It should last be observed
that the efficiency of an OPO can be very high, approaching the theoretical 100% photon

12.4.2. Analytical Treatment
To arrive at an analytical description of both SHG and parametric processes, we need to
see how the nonlinear polarization [e.g., Eq. (12.4.2)], which acts as the source term to drive
the generated waves, is introduced into the wave equation. The fields within the material obey
Maxwell’s equations







r DD


r BD0


is the free-charge density. For the media of interest here we can assume the
magnetization M to be zero; thus



0M D



Losses within the material (e.g., scattering losses) can be simulated by the introduction of a
fictitious conductivity s such that





Frequency Conversion: Second-Harmonic Generation

Finally we can write


where PL is the linear polarization of the medium and is taken account, in the usual way,
by introducing the dielectric constant ". Upon applying the r operator to both sides of
Eq. (12.4.21a), interchanging the order of r and @=@t operators on the right-hand side of the
resulting equation, and making use of Eqs. (12.4.22), (12.4.21b), (12.4.23), and (12.4.24), we
@2 E
@2 PNL
C" 2 C
r r ED
Using the identity r r
get from Eq. (12.4.25)

E D r.r E/

r 2E



r 2 E and under the assumption that r E Š 0, we
1 @2 E
1 @2 PNL
D 2
c @t
"c @t2


where c D ." 0 / 1=2 is the phase velocity in the material. Equation (12.4.26) is the wave
equation with the nonlinear polarization term included. Note that the linear part of the medium
polarization has been transferred to the left-hand side of Eq. (12.4.26) and its effect is included
in the dielectric constant. The nonlinear part PNL has been kept on the right-hand side since
it will be shown to act as a source term for the waves being generated at new frequencies as
well as a loss term for the incoming wave. Confining ourselves to the simple scalar case of
plane waves propagating along the z-direction, one sees that Eq. (12.4.26) reduces to
@2 E



1 @2 E
1 @2 PNL
D 2
c @t
"c @t2


We now write the field of the wave at frequency !i as
E!i .z, t/ D .1=2/ fEi .z/ expŒj.!i t

ki z/ C c.c.g


where Ei is taken to be complex in general. Likewise, the amplitude of the nonlinear
polarization at frequency !i will be written as
˚ NL
ki z/ C c.c.
!i D .1=2/ Pi .z/ expŒj.!i t
Since Eq. (12.4.26a) must hold separately for each frequency component of the waves present
in the medium, Eqs. (12.4.27a) and (12.4.27b) can be substituted into the left- and right-hand
sides of Eq. (12.4.26a), respectively. Within the slowly varying amplitude approximation,
we can neglect the second derivative of Ei .z/, i.e., assume that d 2 Ei =dz2
ki .dEi =dz/.
Equation (12.4.26a) then yields
Ei D j
ni "0 c
ni "0 c