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Nonlinear Optics.pdf
Nonlinear Opt
4. Nonlinear Optics
157
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This chapter provides a brief introduction into the
basic nonlinear-optical phenomena and discusses
some of the most significant recent advances and
breakthroughs in nonlinear optics, as well as novel
applications of nonlinear-optical processes and
devices.
Nonlinear optics is the area of optics that
studies the interaction of light with matter in
the regime where the response of the material
system to the applied electromagnetic field is
nonlinear in the amplitude of this field. At low
light intensities, typical of non-laser sources, the
properties of materials remain independent of
the intensity of illumination. The superposition
principle holds true in this regime, and light waves
can pass through materials or be reflected from
boundaries and interfaces without interacting with
each other. Laser sources, on the other hand, can
provide sufficiently high light intensities to modify
the optical properties of materials. Light waves
can then interact with each other, exchanging
momentum and energy, and the superposition
principle is no longer valid. This interaction of light
waves can result in the generation of optical fields
at new frequencies, including optical harmonics of
incident radiation or sum- or difference-frequency
signals.
4.1 Nonlinear Polarization
and Nonlinear Susceptibilities ............... 159
4.2 Wave Aspects of Nonlinear Optics ........... 160
4.3
Second-Order Nonlinear Processes ......... 161
4.3.1 Second-Harmonic Generation........ 161
4.3.2 Sum- and Difference-Frequency
Generation and Parametric
Amplification............................... 163
4.4 Third-Order Nonlinear Processes ............ 164
4.4.1 Self-Phase Modulation ................. 165
4.4.2 Temporal Solitons......................... 166
4.4.3 Cross-Phase Modulation ............... 167
4.4.4 Self-Focusing............................... 167
4.4.5 Four-Wave Mixing........................ 169
4.4.6 Optical Phase Conjugation ............. 169
4.4.7 Optical Bistability and Switching .... 170
4.4.8 Stimulated Raman Scattering......... 172
4.4.9 Third-Harmonic Generation
by Ultrashort Laser Pulses.............. 173
4.5 Ultrashort Light Pulses
in a Resonant Two-Level Medium:
Self-Induced Transparency
and the Pulse Area Theorem .................. 178
4.5.1
Interaction of Light
with Two-Level Media .................. 178
4.5.2 The Maxwell and Schrödinger
Equations for a Two-Level Medium 178
4.5.3 Pulse Area Theorem ...................... 180
4.5.4 Amplification
of Ultrashort Light Pulses
in a Two-Level Medium ................ 181
4.5.5 Few-Cycle Light Pulses
in a Two-Level Medium ................ 183
4.6 Let There be White Light:
Supercontinuum Generation.................. 185
4.6.1 Self-Phase Modulation,
Four-Wave Mixing,
and Modulation Instabilities
in Supercontinuum-Generating
Photonic-Crystal Fibers ................. 185
4.6.2 Cross-Phase-Modulation-Induced
Instabilities ................................. 187
4.6.3 Solitonic Phenomena in Media
with Retarded Optical Nonlinearity. 189
4.7 Nonlinear Raman Spectroscopy .............. 193
The Basic Principles ...................... 194
4.7.1
4.7.2 Methods of Nonlinear Raman
Spectroscopy ............................... 196
4.7.3 Polarization Nonlinear Raman
Techniques .................................. 199
4.7.4 Time-Resolved Coherent
Anti-Stokes Raman Scattering........ 201
4.8 Waveguide Coherent Anti-Stokes
Raman Scattering ................................. 202
4.8.1 Enhancement of Waveguide CARS
in Hollow Photonic-Crystal Fibers... 202
158 Part A Basic Principles and Materials
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4.8.2 Four-Wave Mixing and CARS
in Hollow-Core Photonic-Crystal
Fibers ......................................... 205
4.9 Nonlinear Spectroscopy
with Photonic-Crystal-Fiber Sources....... 209
4.9.1 Wavelength-Tunable Sources and
4.11 High-Order Harmonic Generation .......... 219
4.11.1 Historical Background ................... 219
4.11.2 High-Order-Harmonic Generation
in Gases ...................................... 220
4.11.3 Microscopic Physics ...................... 222
4.11.4 Macroscopic Physics...................... 225
Progress in Nonlinear Spectroscopy 209
4.12 Attosecond Pulses:
4.9.2 Photonic-Crystal Fiber Frequency
Shifters ....................................... 210
4.9.3 Coherent Anti-Stokes Raman
Scattering Spectroscopy
with PCF Sources .......................... 211
4.9.4 Pump-Probe Nonlinear
Absorption Spectroscopy
using Chirped Frequency-Shifted
Light Pulses
from a Photonic-Crystal Fiber ........ 213
4.10 Surface Nonlinear Optics, Spectroscopy,
Measurement and Application ............... 227
4.12.1 Attosecond Pulse Trains
and Single Attosecond Pulses......... 227
4.12.2 Basic Concepts
for XUV Pulse Measurement ........... 227
4.12.3 The Optical-Field-Driven XUV Streak
Camera Technique ........................ 230
4.12.4 Applications of Sub-femtosecond
XUV Pulses: Time-Resolved
Spectroscopy of Atomic Processes ... 234
4.12.5 Some Recent Developments........... 236
and Imaging ........................................ 216
References .................................................. 236
Although the observation of most nonlinear-optical
phenomena requires laser radiation, some classes of
nonlinear-optical effects were known long before the
invention of the laser. The most prominent examples of
such phenomena include Pockels and Kerr electrooptic
effects [4.1], as well as light-induced resonant absorp-
tion saturation, described by Vavilov [4.2, 3]. It was,
however, only with the advent of lasers that systematic
studies of optical nonlinearities and the observation of
a vast catalog of spectacular nonlinear-optical phenom-
ena became possible.
In the first nonlinear-optical experiment of the laser
era, performed by Franken et al. in 1961 [4.4], a ruby-
laser radiation with a wavelength of 694.2 nm was
used to generate the second harmonic in a quartz crys-
tal at the wavelength of 347.1 nm. This seminal work
was followed by the discovery of a rich diversity
of nonlinear-optical effects, including sum-frequency
generation, stimulated Raman scattering, self-focusing,
optical rectification, four-wave mixing, and many others.
While in the pioneering work by Franken the efficiency
of second-harmonic generation (SHG) was on the or-
−8, optical frequency doublers created by early
der of 10
1963 provided 20%–30% efficiency of frequency con-
version [4.5, 6]. The early phases of the development
and the basic principles of nonlinear optics have been
reviewed in the most illuminating way in the classi-
cal books by Bloembergen [4.7] and Akhmanov and
Khokhlov [4.8], published in the mid 1960s.
Over the following four decades, the field of nonlin-
ear optics has witnessed an enormous growth, leading
to the observation of new physical phenomena and giv-
ing rise to novel concepts and applications. A systematic
introduction into these effects along with a comprehen-
sive overview of nonlinear-optical concepts and devices
can be found in excellent textbooks by Shen [4.9],
Boyd [4.1], Butcher and Cotter [4.10], Reintjes [4.11]
and others. One of the most recent up-to-date reviews of
the field of nonlinear optics with an in-depth discussion
of the fundamental physics underlying nonlinear-optical
interactions was provided by Flytzanis [4.12]. This
chapter provides a brief introduction into the main
nonlinear-optical phenomena and discusses some of the
most significant recent advances in nonlinear optics, as
well as novel applications of nonlinear-optical processes
and devices.
Nonlinear Optics
4.1 Nonlinear Polarization and Nonlinear Susceptibilities
159
4.1 Nonlinear Polarization and Nonlinear Susceptibilities
Nonlinear-optical effects belong to a broader class of
electromagnetic phenomena described within the gen-
eral framework of macroscopic Maxwell equations. The
Maxwell equations not only serve to identify and classify
nonlinear phenomena in terms of the relevant nonlinear-
optical susceptibilities or, more generally, nonlinear
terms in the induced polarization, but also govern the
nonlinear-optical propagation effects. We assume the
absence of extraneous charges and currents and write
the set of Maxwell equations for the electric, E(r, t),
and magnetic, H(r, t), fields in the form
,
(4.1)
(4.2)
(4.3)
∇ × E = − 1
∂ B
∂t
c
∇ × B = 1
∂ D
∂t
c
∇ · D = 0 ,
∇ · B = 0 .
,
t
D = E+ 4π
Here, B = H+ 4π M, where M is the magnetic dipole
polarization, c is the speed of light, and
(4.4)
J(ζ)dζ ,
−∞
(4.5)
where J is the induced current density. Generally, the
equation of motion for charges driven by the electromag-
netic field has to be solved to define the relation between
the induced current J and the electric and magnetic
fields. For quantum systems, this task can be fulfilled
by solving the Schrödinger equation. In Sect. 4.5 of this
chapter, we provide an example of such a self-consistent
analysis of nonlinear-optical phenomena in a model
two-level system. Very often a phenomenological ap-
proach based on the introduction of field-independent
or local-field-corrected nonlinear-optical susceptibilities
can provide an adequate description of nonlinear-optical
processes.
Formally, the current density J can be represented
as a series expansion in multipoles:
(P −∇ · Q)+ c (∇ × M) ,
J = ∂
∂t
(4.6)
where P and Q are the electric dipole and electric
quadrupole polarizations, respectively. In the electric
dipole approximation, we keep only the first term on
the right-hand side of (4.6). In view of (4.5), this gives
the following relation between the D, E, and P vectors:
D = E+ 4π P.
We now represent the polarization P as a sum
P = PL + Pnl ,
(4.8)
where PL is the part of the electric dipole polarization
linear in the field amplitude and Pnl is the nonlinear part
of this polarization.
The linear polarization governs linear-optical phe-
nomena, i. e., it corresponds to the regime where the
optical properties of a medium are independent of the
field intensity. The relation between PL and the electric
field E is given by the standard formula of linear optics:
PL =
χ(1)(t − t
)dt
,
)E(t
(4.9)
where χ(1)(t) is the time-domain linear susceptibility
tensor. Representing the field E and polarization PL in
the form of elementary monochromatic plane waves,
E = E (ω) exp (ikr − ωt)+ c.c.
and
PL = PL(ω) exp
ikr − ωt
+ c.c. ,
we take the Fourier transform of (4.9) to find
PL(ω) = χ(1)(ω)E(ω) ,
where
χ(1)(ω) =
χ(1)(t) exp(iωt)dt .
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1
(4.10)
(4.11)
(4.12)
(4.13)
In the regime of weak fields, the nonlinear part of the
polarization Pnl can be represented as a power-series
expansion in the field E:
χ(2)(t − t1, t − t2) : E(t1)E(t2)dt1 dt2
Pnl =
+
χ(3)(t − t1, t − t2, t − t3)
...E(t1)E(t2)E(t3)dt1 dt2 dt3 + . . . ,
(4.14)
where χ(2) and χ(3) are the second- and third-order
nonlinear susceptibilities.
of plane monochromatic waves,
Representing the electric field in the form of a sum
E =
Ei(ωi) exp(ikir − ωit)+ c.c. ,
(4.15)
i
we take the Fourier transform of (4.14) to arrive at
(4.7)
Pnl(ω) = P(2)(ω)+ P(3)(ω)+ . . . ,
(4.16)
160 Part A Basic Principles and Materials
where
P(2)(ω) = χ(2)(ω; ωi , ω j) : E(ωi)E(ω j) ,
P(3)(ω) = χ(3)(ω; ωi , ω j , ωk)
χ(2)(ω; ωi , ω j) = χ(2)(ω = ωi + ω j)
=
χ(2)(t1, t2) exp[i(ωit1 + ω jt2)]dt1 dt2
...E(ωi)E(ω j)E(ωk) ,
(4.17)
ω 1, k 1
(4.18)
(4.19)
ω 2, k 2
(2)
ω 3, k 3
is the second-order nonlinear-optical susceptibility and
χ(3)(ω; ωi , ω j , ωk) = χ(3)(ω = ωi + ω j + ωk)
=
χ(3)(t1, t2, t3)
exp[i(ωit1 + ω jt2 + ωkt3)]dt1 dt2 dt3
(4.20)
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2
is the third-order nonlinear-optical susceptibility.
The second-order nonlinear polarization defined
by (4.17) gives rise to three-wave mixing processes,
optical
rectification and linear electrooptic effect.
In particular, setting ωi = ω j = ω0
in (4.17) and
(4.19), we arrive at ω = 2ω0, which corresponds to
second-harmonic generation, controlled by the nonlin-
= χ(2)(2ω0; ω0, ω0). In a more
ear susceptibility χ(2)
SHG
general case of
three-wave mixing process with
ωi = ω1 = ω j = ω2, the second-order polarization de-
fined by (4.17) can describe sum-frequency generation
(SFG) ωSF = ω1 + ω2 Fig. 4.1 or difference-frequency
generation (DFG) ωDF = ω1 − ω2, governed by the
4.2 Wave Aspects of Nonlinear Optics
In the electric dipole approximation, the Maxwell equa-
tions (4.1–4.4) yield the following equation governing
the propagation of light waves in a weakly nonlinear
medium:
∇ × (∇ × E)− 1
c2
∂2 E
∂t2
− 4π
c2
∂2 PL
∂t2
= 4π
c2
∂2 Pnl
∂t2
.
(4.21)
Fig. 4.1 Sum-frequency generation ω1 + ω2 = ω3 in a me-
dium with a quadratic nonlinearity. The case of ω1 = ω2
corresponds to second-harmonic generation
= χ(2)(ωSF; ω1, ω2) and
nonlinear susceptibilities χ(2)
SFG
χ(2)
DFG
= χ(2)(ωDF; ω1,−ω2), respectively.
The third-order nonlinear polarization defined by
(4.18) is responsible for four-wave mixing (FWM),
stimulated Raman scattering, two-photon absorption,
and Kerr-effect-related phenomena,
including self-
phase modulation (SPM) and self-focusing. For the
particular case of third-harmonic generation, we set
ωi = ω j = ωk = ω0 in (4.18) and (4.20) to obtain
ω = 3ω0. This type of nonlinear-optical interaction, in
accordance with (4.18) and (4.20), is controlled by
= χ(3)(3ω0; ω0, ω0, ω0).
the cubic susceptibility χ(3)
THG
A more general, frequency-nondegenerate case can cor-
respond to a general type of an FWM process. These
and other basic nonlinear-optical processes will be con-
sidered in greater details in the following sections.
E (r, t) = Re
axis, we represent the field E in (4.21) by
eA (z, t) exp (ikz− ωt)
and write the nonlinear polarization as
ikpz− ωt
Pnl (r, t) = Re
ep Pnl (z, t) exp
,
(4.22)
(4.23)
The nonlinear polarization, appearing on the right-hand
side of (4.21), plays the role of a driving source, inducing
an electromagnetic wave with the same frequency ω as
the nonlinear polarization wave Pnl(r, t). Dynamics of
a nonlinear wave process can be then thought as a result
of the interference of induced and driving (pump) waves,
controlled by the dispersion of the medium.
Assuming that the fields have the form of quasi-
monochromatic plane waves propagating along the z-
where k and A(z, t) are the wave vector and the envelope
of the electric field, k p and Pnl(z, t) are the wave vector
and the envelope of the polarization wave.
If the envelope A(z, t) is a slowly varying func-
tion over the wavelength, |∂2 A/∂z2| |k∂A/∂z|, and
∂2 Pnl/∂t2 ≈ −ω2 Pnl, (4.21) is reduced to [4.9]
∂A
∂z
+ 1
u
∂A
∂t
= 2πiω2
kc2 Pnl exp (i∆kz) ,
(4.24)
where u = (∂k/∂ω)−1
∆k = kp − k is the wave-vector mismatch.
is
the group velocity and
In the following sections,
this generic equation
of slowly varying envelope approximation (SVEA)
Nonlinear Optics
4.3 Second-Order Nonlinear Processes
161
will be employed to analyze the wave aspects of
the basic second- and third-order nonlinear-optical
phenomena.
4.3 Second-Order Nonlinear Processes
4.3.1 Second-Harmonic Generation
reference with z
In second-harmonic generation, a pump wave with a fre-
quency of ω generates a signal at the frequency 2ω
as it propagates through a medium with a quadratic
nonlinearity (Fig. 4.1). Since all even-order nonlinear
susceptibilities χ(n) vanish in centrosymmetric me-
dia, SHG can occur only in media with no inversion
symmetry.
Assuming that diffraction and second-order dis-
persion effects are negligible, we use (4.24)
for
a quadratically nonlinear medium with a nonlinear SHG
= χ(2) (2ω; ω, ω) to write a pair of
susceptibility χ(2)
SHG
coupled equations for the slowly varying envelopes of
the pump and second-harmonic fields A1 = A1(z, t) and
A2 = A2(z, t):
+ 1
u1
+ 1
u2
∗
1 A2 exp (i∆kz) ,
1 exp (−i∆kz) ,
= iγ1 A
= iγ2 A2
∂A1
∂z
∂A2
∂z
∂A1
∂t
∂A2
∂t
(4.25)
(4.26)
where
γ1 = 2πω2
1
k1c2
χ(2) (ω; 2ω,−ω)
and
γ2 = 4πω2
1
k2c2
χ(2)
SHG
(4.28)
are the nonlinear coefficients, u1 and u2 are the group
velocities of the pump and second-harmonic pulses,
respectively, and ∆k = 2k1− k2 is the wave-vector mis-
match for the SHG process.
If the difference between the group velocities of the
pump and second-harmonic pulses can be neglected for
a nonlinear medium with a given length and if the in-
tensity of the pump field in the process of SHG remains
much higher than the intensity of the second-harmonic
field, we set u1 = u2 = u and |A1|2 = |A10|2 = const.
in (4.25) and (4.26) to derive in the retarded frame of
= z and η = t − z/u
A2 (L) = iγ2 A2
10
sin
i∆kL
2
,
(4.29)
∆kL
2
∆kL
2
L exp
where L is the length of the nonlinear medium.
The intensity of the second-harmonic field is then
given by
I2 (L) ∝ γ 2
2 I 2
10
sin
∆kL
2
∆kL
2
2
L2,
(4.30)
where I10 is the intensity of the pump field.
Second-harmonic intensity I2, as can be seen from
(4.30) oscillates as a function of L Fig. 4.2 with a period
Lc = π/|∆k| = λ1(4|n1− n2|)
−1, where λ1 is the pump
wavelength and n1 and n2 are the values of the refractive
index at the frequencies of the pump field and its second
harmonic, respectively. The parameter Lc, defining the
length of the nonlinear medium providing the maximum
SHG efficiency, is referred to as the coherence length.
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Second-harmonic intensity (arb. units)
(4.27)
0.4
Lc2 = 2Lc1
0.3
0.2
0.1
0.0
0
Lc1
1
2
3
4
5
6
L/Lc
Fig. 4.2 Second-harmonic intensity as a function of the
length L of the nonlinear medium normalized to the coher-
ence length Lc for two values of Lc: (dashed line) Lc1 and
(solid line) Lc2 = 2Lc1
162 Part A Basic Principles and Materials
Although the solution (4.29) describes the simplest
regime of SHG, it is very instructive as it visualizes
the significance of reducing the wave-vector mismatch
∆k for efficient SHG. Since the wave vectors k1 and
k2 are associated with the momenta of the pump and
second-harmonic fields, p1 = k1 and p2 = k2, with
being the Planck constant, the condition ∆k = 0,
known as the phase-matching condition in nonlinear
optics, in fact, represents momentum conservation for
the SHG process, where two photons of the pump field
are requited to generate a single photon of the second
harmonic.
Several strategies have been developed to solve the
phase-matching problem for SHG. The most practi-
cally significant solutions include the use of birefringent
nonlinear crystals [4.13, 14], quasi-phase-matching in
periodically poled nonlinear materials [4.15, 16] and
waveguide regimes of nonlinear interactions with the
phase mismatch related to the material dispersion com-
pensated for by waveguide dispersion [4.7]. Harmonic
generation in the gas phase, as demonstrated by Miles
and Harris [4.17], can be phase-matched through an
optimization of the content of the gas mixture. Fig-
ure 4.3 illustrates phase matching in a birefringent
crystal. The circle represents the cross section of the
refractive-index sphere n0(ω) for an ordinary wave at
the pump frequency ω. The ellipse is the cross sec-
tion of the refractive-index ellipsoid ne(2ω) for an
extraordinary wave at the frequency of the second har-
monic 2ω. Phase matching is achieved in the direction
where n0ω = ne(2ω), corresponding to an angle θpm
with respect to the optical axis c of the crystal in
Fig. 4.3.
When the phase-matching condition ∆k = 0 is satis-
fied, (4.29) and (4.30) predict a quadratic growth of the
P
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3
c
θ
ηo(ω)
ηe(2ω)
Fig. 4.3 Phase-matching second-harmonic generation in
a birefringent crystal
Pump amplitude
(arb. units)
Second-harmonic amplitude
(arb. units)
1.0
0.8
0.6
0.4
0.2
0.0
0
1
2
3
1.0
0.8
0.6
0.4
0.2
0.0
4
z/zsh
Fig. 4.4 The amplitudes of the pump and second-harmonic
fields as functions of the normalized propagation distance
z/zsh with zsh = [γρ10(0)]−1
second-harmonic intensity as a function of the length
L of the nonlinear medium. This scaling law holds
true, however, only as long as the second-harmonic
intensity remains much less than the pump intensity.
As |A2| becomes comparable with |A1|, depletion of
the pump field has to be taken into consideration.
To this end, we introduce the real amplitudes ρ j and
phases ϕ j of the pump and second-harmonic fields,
, with j = 1, 2. Then, assuming that
A j = ρ j exp
u1 = u2 = u and γ1 = γ2 = γ , we derive from (4.25)
and (4.26)
iϕ j
ρ1 (η, z) = ρ10 (η) sech [γρ10 (η) z] ,
ρ2 (η, z) = ρ10 (η) tanh [γρ10 (η) z] .
(4.31)
(4.32)
show that
(4.31) and (4.32)
the
The solutions
entire energy of
the pump field in the phase-
matching regime can be transferred to the second
the pump field becomes depleted
harmonic. As
Fig. 4.4,
the second-harmonic field
saturates.
the growth of
Effects related to the group-velocity mismatch be-
come significant when the length of the nonlinear
medium L exceeds the length Lg = τ1/|u
|,
−1
1
where τ1 is the pulse width of the pump field. The length
Lg characterizes the walk-off between the pump and
second-harmonic pulses caused by the group-velocity
mismatch. In this nonstationary regime of SHG, the
amplitude of the second harmonic in the constant-pump-
− u
−1
2
field approximation is given by
A2 (z, t) = iγ2
z
t − z/u2 + ξ
A2
10
0
× exp (−i∆kξ) dξ .
u
−1
2
− u
−1
1
(4.33)
Group-velocity mismatch may lead to a considerable
increase in the pulse width of the second harmonic
τ2. For L Lg, the second harmonic pulse width,
τ2 ≈ |u
|z, scales linearly with the length of the
nonlinear medium and is independent of the pump pulse
width.
− u
−1
2
−1
1
4.3.2 Sum- and Difference-Frequency
Generation and Parametric
Amplification
In sum-frequency generation Fig. 4.1, two laser fields
with frequencies ω1 and ω2 generate a nonlinear
signal at the frequency ω3 = ω1 + ω2 in a quadrati-
cally nonlinear medium with a nonlinear susceptibility
= χ(2) (ω3; ω1, ω2). In the first order of dis-
χ(2)
SFG
the coupled equations for slowly
persion theory,
varying envelopes of the laser fields A1 = A1(z, t) and
A2 = A2(z, t) and the nonlinear signal A3 = A3(z, t) are
written as
+ 1
u1
+ 1
u2
+ 1
u3
∂A1
∂t
∂A2
∂t
∂A3
∂t
∂A1
∂z
∂A2
∂z
∂A3
∂z
where
= iγ1 A3 A
∗
2 exp (i∆kz) ,
= iγ2 A3 A
∗
1 exp (i∆kz) ,
= iγ3 A1 A2 exp (−i∆kz) ,
γ1 = 2πω2
1
k1c2
γ2 = 2πω2
2
k2c2
γ3 = 2πω2
3
k3c2
χ(2) (ω1; ω3,−ω2) ,
χ(2) (ω2; ω3,−ω1) ,
χ(2)
SFG
(4.34)
(4.35)
(4.36)
(4.37)
(4.38)
(4.39)
are the nonlinear coefficients, u1, u2, and u3 and k1, k2,
and k3 are the group velocities and the wave vectors of
the fields with frequencies ω1, ω2, and ω3, respectively,
and ∆k = k1 + k2 − k3 is the wave-vector mismatch for
the SFG process.
As long as the intensity of the sum-frequency field
remains much less than the intensities of the laser fields,
the amplitudes of the laser fields can be can be as-
sumed to be given functions of t, A1(z, t) = A10(t) and
Nonlinear Optics
4.3 Second-Order Nonlinear Processes
163
z
A2(z, t) = A20(t), and the solution of (4.36) yields
− u
−1
1
A3 (z, t) = iγ3
t − z/u3 + ξ
−1
3
A10
0
t − z/u3 + ξ
× A20
× exp (−i∆kξ) dξ.
− u
−1
3
−1
2
u
u
(4.40)
−1
1
|L, ∆31 ≈ |u
The efficiency of frequency conversion, as can be seen
from (4.40) is controlled by the group delays ∆21 ≈
|u
−
− u
−1
−1
|L between the pulses involved in the SFG process.
−1
2
3
u
2
In particular, the laser fields cease to interact with each
other when the group delay ∆21 starts to exceed the
pulse width of the faster laser pulse.
|L, and ∆32 ≈ |u
− u
−1
3
−1
1
In difference-frequency generation (DFG), two input
fields with frequencies ω1 and ω2 generate a nonlinear
signal at the frequency ω3 = ω1− ω2. This process is of
considerable practical significance as it can give rise to
intense coherent radiation in the infrared range. In the
limiting case of ω1 ≈ ω2, this type of nonlinear-optical
interaction corresponds to optical rectification, which
has been intensely used over the past two decades for
the generation of terahertz radiation.
If the field at the frequency ω1 is strong and re-
mains undepleted in the process of nonlinear-optical
interaction, A1(z, t) = A10(t), the set of coupled equa-
tions governing the amplitudes of the remaining two
fields in the stationary regime is written as
P
a
r
t
A
4
.
3
+ 1
u2
+ 1
u3
∂A2
∂t
∂A3
∂t
= iγ2 A1 A
= iγ3 A1 A
∗
3 exp (i∆kz) ,
2 exp (−i∆kz) ,
∗
∂A2
∂z
∂A3
∂z
where,
γ2 = 2πω2
2
k2c2
γ3 = 2πω2
3
k3c2
χ(2) (ω2; ω1,−ω3) ,
χ(2) (ω3; ω1,−ω2)
are the nonlinear coefficientsm and ∆k = k1− k2− k3 is
the wave-vector mismatch for the DFG process.
With no signal at ω3 applied at the input of the non-
linear medium, A3(0, t) = 0, the solution to (4.41) and
(4.42) in the stationary regime is given by [4.12]
sinh (κz)
cosh (κz)+ i
,
A2 (z) = A2 (0)
A3 (z) = iA2 (0) sinh (κz) ,
∆k
2κ
(4.41)
(4.42)
(4.43)
(4.44)
(4.45)
(4.46)
164 Part A Basic Principles and Materials
where
κ2 = 4γ2γ∗
3
|A1|2 − (∆k)2 .
(4.47)
Away from the phase-matching condition, the amplifica-
tion of a weak signal is achieved only when the intensity
of the pump field exceeds a threshold,
2
I1 > Ith = n1n2n3c3 (∆k)2
ω2ω3
χ(2)
32π3
DFG
,
(4.48)
where we took
χ(2)(ω2; ω1,−ω3) ≈ χ(2)(ω3; ω1,−ω2) = χ(2)
Above, this threshold, the growth in the intensity I2
DFG
.
of a weak input signal is governed by
I2 (z) = I2 (0)
γ
2
|A10|2
γ∗
κ2
3
sin2 (κz)+ 1
.
(4.49)
P
a
r
t
A
4
.
4
This type of three-wave mixing is often referred to as
optical parametric amplification. A weak input field, re-
ferred to as the signal field (the field with the amplitude
A2 in our case), becomes amplified in this type of pro-
cess through a nonlinear interaction with a powerful
pump field (the undepleted field with the amplitude A1
in the case considered here). In such a scheme of optical
parametric amplification, the third field (the field with
the amplitude A3) is called the idler field.
We now consider the regime of optical paramet-
ric amplification ω1 = ω2 + ω3 where the pump, signal
and idler pulses are matched in their wave vectors and
group velocities. Introducing the real amplitudes ρ j
and phases ϕ j of the pump, signal, and idler fields,
, where j = 1, 2, 3, assuming that
A j = ρ j exp
γ2 = γ3 = γ in (4.35) and (4.36), A1(z, t) = A10(t) and
A3(0, t) = 0, we write the solution for the amplitude of
the signal field as [4.18]
iϕ j
A2 (η, z) = A20 (η) cosh [γρ10 (η) z] .
(4.50)
The idler field then builds up in accordance with
A3 (η, z) = A
∗
20
(η) exp [iϕ10 (η)] sinh [γρ10 (η) z] .
(4.51)
As can be seen from (4.50), optical parametric ampli-
fication preserves the phase of the signal pulse. This
property of optical parametric amplification lies at the
heart of the principle of optical parametric chirped-pulse
amplification [4.19], allowing ultrashort laser pulses to
be amplified to relativistic intensities. It also suggests
a method of efficient frequency conversion of few-cycle
field waveforms without changing the phase offset be-
tween their carrier frequency and temporal envelope,
making few-cycle laser pulses a powerful tool for the
investigation of ultrafast electron dynamics in atomic
and molecular systems.
In the nonstationary regime of optical paramet-
ric amplification, when the pump, signal, and idler
fields propagate with different group velocities, useful
and important qualitative insights into the phase rela-
tions between the pump, signal, and idler pulses can
be gained from energy and momentum conservation,
ω1 = ω2+ ω3 and k1 = k2+ k3. These equalities dictate
the following relations between the frequency deviations
δω j in the pump, signal, and idler fields ( j = 1, 2, 3):
(4.52)
(4.53)
(4.54)
δω1 = δω2+ δω3
and
δω1/u1 = δω2/u2 + δω3/u3 .
In view of (4.52) and (4.53), we find
δω2 = q2δω1
and
(4.55)
In the
3 ), q3 = 1− q2.
−1
δω3 = q3δω1,
where q2 = (u
− u
− u
−1
−1
−1
3 )/(u
1
2
case of
a
linearly chirped pump,
ϕ1(t) = α1t2/2,
the phases of the signal and idler
pulses are given by ϕm(t) = αmt2/2, where αm = qm α1,
m = 2, 3. With qm 1, the chirp of the signal and idler
pulses can thus considerably exceed the chirp of the
pump field.
4.4 Third-Order Nonlinear Processes
Optical nonlinearity of the third order is a univer-
sal property, found in any material regardless of its
spatial symmetry. This nonlinearity is the lowest-
order nonvanishing nonlinearity for a broad class of
centrosymmetric materials, where all the even-order
nonlinear susceptibilities are identically equal to zero
for symmetry reasons. Third-order nonlinear processes
include a vast variety of four-wave mixing processes,
which are extensively used for frequency conversion of
laser radiation and as powerful methods of nonlinear
spectroscopy. Frequency-degenerate, Kerr-effect-type
phenomena constitute another important class of third-
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