Appl. Phys. B 81, 225–229 (2005)
Applied Physics B
DOI: 10.1007/s00340-005-1815-4
Lasers and Optics
L. TKESHELASHVILI1,2
K. BUSCH1,2, ✉
Nonlinear three-wave interaction
in photonic crystals
1 Institut
für Theorie der Kondensierten Materie, Universität Karlsruhe, 76128 Karlsruhe, Germany
of Physics and College of Optics & Photonics: CREOL & FPCE, University of Central Florida,
Orlando, FL 32816, USA
2 Department
Received: 31 January 2005
Published online: 15 July 2005 • © Springer-Verlag 2005
We present a multi-scale analysis of nonlinear
three-wave-interaction processes in photonic crystals. Based
on photonic Bloch functions as carrier waves, we derive the
effective nonlinear coupled wave equations that govern pulse
propagation in these systems and obtain the corresponding
effective photonic crystal parameters directly from photonic
band-structure computations. As an illustration, we show how
hitherto inaccessible radiation-conversion processes such as
wave-front reversal of optical pulses can be realized. Furthermore, we describe a novel regime of nonlinear three-wave interaction in photonic crystals associated with the nearly degenerate case and show how these results may be utilized to study
experimentally certain problems from plasma physics and hydrodynamics in the context of nonlinear photonic crystals.
ABSTRACT
PACS
1
42.70.Qs; 42.65.Ky; 42.65.−k
Introduction
Over the past two decades, photonic crystals (PCs)
[1–3] have developed into a novel class of optical materials largely because the flexibility in design of these
two-dimensional (2D) or three-dimensional (3D) periodic
dielectric arrays allows one to tailor the photonic dispersion
relation and associated mode structure to almost any need.
For instance, recent experiments have verified earlier theoretical predictions that 3D PCs such as the inverse opals [4, 5]
exhibit photonic band gaps (PBGs), frequency ranges over
which ordinary linear propagation is forbidden irrespective of
direction. The existence of these complete PBGs leads to inhibition of spontaneous emission for atomic transition frequencies deep in the PBG [1] and leads to strongly non-Markovian
effects such as fractional localization of the atomic population for atomic transition frequencies in close proximity to a
complete PBG [6, 7].
For a number of applications such as guided light in planar waveguide structures and most nonlinear wave mixing
experiments, it is sufficient to obtain control over the propagation of light within the plane of propagation. Corresponding
✉ Fax: +1-(407)-823-5112, E-mail: kbusch@physics.ucf.edu
high-quality 2D PCs can be manufactured through planar microstructuring techniques [8–11]. In the linear regime, PBGs
in 2D PCs offer novel passive optical guiding characteristics
through the engineering of defects such as microcavities and
waveguides and their combination into functional elements
such as wavelength add–drop filters [12, 13] as well as broadband bends, beam splitters, and waveguide crossings [14] .
Similarly, the incorporation of nonlinear materials into PBG
structures creates the possibility for novel solitary-wave propagation for frequencies inside the PBG [15–18].
As compared to the numerous applications of PBGs, wave
propagation in linear and nonlinear PCs for frequencies inside
photonic bands has received far less attention. However, the
discovery of super-refractive phenomena [19] such as the superprism effect in higher-dimensional PCs [20] greatly adds
to the rich physics of wave propagation in PCs. Similarly, in
the context of nonlinear optical phenomena it is the tailoring
of photonic dispersion relations and mode structures through
judiciously designed PCs that allows us to explore regimes for
parameters such as group velocity, group-velocity dispersion,
and effective nonlinearities that, hitherto, have been virtually
inaccessible. Any successful experimental exploration of this
huge parameter space has to be accompanied by a quantitative analysis in order to identify the most interesting cases
and help to interpret the data. To date, only a few works have
been carried out for either Kerr nonlinearities [15, 16] or χ (2)
nonlinearities [21] in PCs. Moreover, the approximations involved often seriously limit the applicability of these theories
to real PCs. For instance, the recent investigation of secondharmonic generation in 2D PCs [21, 22] failed to reproduce
the well-known results for the limiting case of homogeneous
materials.
In the present paper, we study the problem of sumfrequency generation in 2D PCs. Based on a multi-scale analysis, we derive the effective coupled nonlinear equations of
motion that govern the propagation of pulses in these systems.
Using photonic Bloch functions as carrier waves, we obtain
the corresponding effective PC parameters such as group velocities and effective nonlinearities directly from photonic
band-structure computations. We show that these 2D PCs
are well suited to the experimental investigation of radiationconversion processes studied theoretically by Zakharov [23]
and Kaup [24, 25], which until now have been realized in a
226
Applied Physics B – Lasers and Optics
very limited parameter range only. As an illustration, we discuss the realization of wave-front reversal of optical pulses
in a realistic 2D PC based on microstructured LiNbO3 . Furthermore, for the case of three-wave mixing with two nearly
degenerate frequencies, we obtain a class of solitary waves
which represent the optical analogue of certain solitary-wave
problems in nonlinear chains such as the Davydov α-helix
model, plasma physics, acoustics, and hydrodynamics [28–
30].
2
Multi-scale analysis of resonant interaction
We start our analysis by writing down the corresponding wave equation for TM-polarized radiation in a 2D
PC:
p (r )
4π
E(r , t) − 2 ∂t2 E(r , t) = 2 ∂t2 χ (2) (r )E 2 (r , t),
(1)
c
c
where c is the vacuum speed of light, = ∂x2 + ∂y2 denotes the 2D Laplace operator, and the nonlinear polarization
PNL (r , t) = χ (2) (r )E 2 (r , t) represents the nonlinear response
of the materials that constitute the PC. Since optical nonlinearities are generally quite weak, Eq. (1) should be solved
perturbatively taking into account that the effect of the nonlinearity accumulates only on time and spatial scales that are
much slower than the natural scales of the underlying linear
problem. Therefore, we formally replace the space and time
variables, r and t, with sets of independent spatial and temporal variables {rn ≡ µn r} and {tn ≡ µn t} (n = 0, 1, 2, . . .). For
instance, in this multi-scale approach [31] the time derivative
is replaced according to ∂t = ∂t0 + µ∂t1 + µ2 ∂t2 + . . ., and
analogous replacements of spatial and higher-order derivatives should be made. A corresponding expansion of the electromagnetic field
E(r , t) = µe1 ({ri }; {ti }) + µ2 e2 ({ri }; {ti }) + · · ·
(2)
leads to a hierarchy of equations which effectively separate
the different scales of the problem [31]. In the present case,
the fastest spatial scale, r0 , corresponds to the wavelength
of the radiation propagating in the linear PC, and the fastest
temporal scale, t0 , is associated with the optical period. As a
consequence, we find from the wave equation, Eq. (1), that,
on the fastest scale, the field e1 satisfies the linear equation of
motion within the PC:
p (r0 ) 2
0 e1 ({ri }; {ti }) −
∂ e1 ({ri }; {ti }) = 0,
(3)
c2 t0
where 0 = ∂x20 + ∂y20 . In view of the quadratic field dependence of the nonlinearity, Eq. (3) suggests a multi-frequency
ansatz of the form
e1 ({ri }; {ti }) =
3
Ai (r1 , . . . ; t1 , . . .) φi (r0 ) eiωαi t0
i=1
+c.c.,
(4)
which identifies the carrier waves φi (r0 ) = E αi (r0 ), i =
1, 2, 3, with the eigenmodes of the linear PC, of the Bloch
functions E αi (r0 ). Here, we have introduced the composite
index αi ≡ (n i , ki ) representing band index n i and wave vector ki of the Bloch functions. The equations of motion for the
slowly varying envelope functions Ai are obtained by considering the wave equation on the next higher scale:
(2) ∗
∂t + vα1 ∂r A1 = 4π iωα1 χeff
A2 A3 ,
(5)
(2) ∗
∂t + vα2 ∂r A2 = 4π iωα2 χeff
A1 A3 ,
(6)
(2)
∂t + vα3 ∂r A3 = 4π iωα3 χeff
A1 A2 ,
(7)
where the group velocities vαi as well as the effective nonlin(2)
are determined through the carrier waves:
earity χeff
c2
vαi = −i
(8)
d2r φi∗ (r )∂r φi (r ),
ωαi wsc
(2)
d2r χ (2) (r ) φ1∗ (r ) φ2∗ (r ) φ3 (r ).
χeff =
(9)
wsc
In deriving Eqs. (5)–(9), we have assumed that energy
and crystal momentum conservation, ωα1 + ωα2 = ωα3 and
are satisfied in order to obtain optimal
k1 + k2 = k3 + G,
phase matching between the three interacting waves. The expression for the group velocity, Eq. (8), is exact and follows
from first-order k · p perturbation theory [32–34]. The multibranch nature of photonic band structures in 2D or 3D PCs
together with the possibility of Umklapp processes through
allows us to satisfy these
nonzero reciprocal lattice vectors G
conservation laws rather easily.
Equations (2) and (4)–(9) describe the evolution of pulses
in a PC with χ (2) nonlinearities, where envelope functions Ai
propagate in an effective homogeneous medium whose characteristics such as group velocity, Eq. (8), and effective nonlinearity, Eq. (9), are determined through the Bloch functions that
act as carrier waves. In particular, it is worth noting that these
equations of motion are independent with respect to a phase
transformation E αi → exp ψαi E αi of the carrier waves. Furthermore, in the limiting case of a homogeneous medium, the
carrier waves become plane waves and the equations of motion
reduce to well-known results [35]. Most importantly, the effective PC parameters may be calculated from photonic bandstructure theory and, in contrast to homogeneous media, may
take on values that are virtually inaccessible otherwise. For
instance, the existence of flat bands in 2D or 3D photonic band
structures allows us to engineer the group velocities entering
Eqs. (5)–(7) to be substantially different from each other [33].
This flexibility of designing structures with low group velocities may lead to strongly enhanced second-harmonic generation, as has been demonstrated recently [36]. Furthermore, the
nonuniform field distribution of the Bloch functions within a
unit cell [37] may be utilized to tune the values of the effec(2)
tive nonlinearity χeff
over a wide range around the bulk values
of the materials that constitute the PC. In addition, the symmetry properties of the Bloch functions [22] leads to certain
(2)
(see Eq. (9)).
selection rules for the effective nonlinearity χeff
The above considerations are of paramount importance
when attempting to utilize nonlinear three-wave interaction
in PCs. For instance, Zakharov [23] has shown that even if
the three waves are not collinear, Eqs. (5)–(7) exhibit a decay
instability for certain ranges of group velocities and effective
nonlinearities. Unfortunately, these parameter ranges are not
easily accessible using homogeneous or quasi-phase-matched
TKESHELASHVILI
et al. Nonlinear three-wave interaction in photonic crystals
nonlinear materials. As a consequence, only within 2D or
3D PCs may efficient wavelength-conversion applications between the various telecommunication windows be realized.
3
Wave-front reversal in photonic crystals
To illustrate the rich physics associated with nonlinear three-wave interaction in PCs, we explicitly show how
the multi-branch dispersion relation of a PC allows us to
realize wave-front reversal in optical systems. In fact, due
to difficulties in fulfilling energy and momentum conservation laws in the dispersion relation of light waves in optical fibers, wave-front reversal caused by a three-wave interaction has never been observed in optics. Instead, the first
realization of this effect has recently been achieved using
dipolar spin wave pulses in ferromagnetic films [38]. We consider a square lattice (lattice constant a) of air pores (radius
r/a = 0.3) in LiNbO3 ( ≈ 4.99, χ (2) = 1.94 × 10−7 esu). In
Fig. 1, we show the corresponding photonic band structure of
the linear system. In order to realize wave-front reversal in
this system, the carrier-wave frequencies and wave vectors of
the incident signal pulse (ωs , ks ), reversed pulse (ωr , kr ), and
pumping pulse (ωp , kp ) have to satisfy energy and momentum
conservation and the pumping frequency ωp must be close
to double the signal and reversed pulse frequencies, respectively, ωp ≈ 2ωs ≈ 2ωr . Furthermore, since the reversed pulse
should propagate in the backward direction, we must require
vr ≈ −vs and kr ≈ −ks or, equivalently, |kp | |ks |, |kr | [38].
The peculiar property of this process is that – under certain
conditions – the leading front in the incident pulse becomes
the trailing front of the reversed pulse, i.e. the time profile
of the reversed pulse is inverted with respect to the incident
pulse. In particular, the inversion of the time profile happens
only if the three-wave interaction is nonstationary. This is the
case when the pumping pulse duration tp is small compared
to all other characteristic time scales of the problem:
Here, ts is the duration of the incident pulse, and T =
L p /|vs | is the time necessary for the incident pulse to traverse
the pumping region which extends over a length L p in its propagation direction. In the limiting case when the pumping region has a (Dirac) δ-function-like shape and vp ≈ 0, i.e. when
Ap = Cδ(r ) with a suitable constant C, the problem can be
solved rather straightforwardly. In fact, when the signal pulse
As = As (r − vs t) interacts with such a pump pulse, Eq. (6) in
the moving reference frame with group velocity vr reads
∂
Ar = 4π iωr C χeff(2) As (r − 2vs t) δ(r − vs t),
∂t
(11)
where vr ≈ −vs has been used. The solution of Eq. (11) in
the original reference frame is
Ar = 4π iωr C χeff(2) As (−r + vs t).
(12)
0.8
From Eq. (12) one can see that the reversed pulse Ar has
the inverted time profile of the signal pulse As . This result
remains valid if the pumping region is finite but sufficiently
small [38–40].
In our 2D PC, a careful group-theoretical analysis [22]
of the mode symmetries reveals that all these requirements
can, for instance, be fulfilled when pumping at frequencies in
ωp a/2π c =
the seventh band near the -point (kp = 0.002 X,
0.7327) and by using signal and reversed pulses at frequen ωs a/2π c = 0.3659
cies in the second band (ks = 0.535 X,
and kr = 0.533 X, ωr a/2π c = 0.3664), respectively (indicated through arrows in Fig. 1). The corresponding effective
(2)
= 0.63 × 10−7 esu, which is somewhat
nonlinearity is χeff
lower than the bulk value of LiNbO3 . However, the very low
group velocity vp ≈ 0.01c of the pumping pulse along with the
low group velocities vs ≈ vr ≈ 0.43c for signal and reversed
pulses, respectively, will guarantee an enhanced efficiency
of the nonlinear process. Besides the fundamental interest in
phase conjugation of optical pulses via nonlinear three-wave
processes, the wave-front reversal discussed above may have
applications in information processing. Further applications
may include the all-optical realization of storing, retrieving,
and processing of information that has recently been suggested in the context of plasma physics [41].
0.7
4
0.6
Equations (5)–(7) and the associated results of
Zakharov [23] and Kaup [24, 25] do not apply to the case
of frequency conversion between different channels in a
given telecommunication window, because the narrow channel spacing requires us to consider dispersion effects for the
two pulses that are close in frequency and wave vector (nearly
degenerate regime). This failure of the dispersion-less case
suggests that we have to carry the multi-scale analysis presented above one order further. To facilitate this, we consider
the frequencies ωα2 and ωα3 as side bands of a central fre and rewrite the wave vectors as k3 = k + k1 /2,
quency ω(k)
k2 = k − k1 /2, where |k1 |a ∼ µ and a is the lattice constant
of the 2D PC. We then have from frequency conservation,
tp ts T.
ω a ⁄ 2π c
227
(10)
0.5
0.4
0.3
0.2
0.1
0
X
Γ
X
M
Γ
FIGURE 1 Photonic band structure for a square lattice of air pores (r/a =
0.3) in LiNbO3 ( ≈ 4.99). The arrows indicate the positions of the carrierwave frequencies for signal, reversed, and pumping pulses for realizing wavefront reversal in this photonic crystal. See text for details
Long-wave/short-wave interaction
ωα1 = ωα3 − ωα2 =
∂ω(k)
· k1 + O(µ2 ),
∂ k
(13)
228
Applied Physics B – Lasers and Optics
that the velocities of all three interacting waves coincide
v = v1 = v2 = v3 and that the long wave represented by ωα1
appears on another scale than the short wave represented by
Here, we should note that these conditions which are
ω(k).
necessary for obtaining true χ (2) effects can indeed be realized
in a suitably engineered PC and constitute a novel regime of
nonlinear three-wave interaction in optics, as will be shown
below.
In order to highlight the features of this novel regime, we
employ a modified multi-frequency ansatz which reflects the
fact that we wish to consider the interaction between two highfrequency waves that is mediated through a low-frequency
wave that we assume to be initially on the noise level:
e1 ({rn }; {tn }) = µA1 (r1 , . . . ; t1 , . . .) φ1 (r0 ) eiωα1 t0
+
3
5
A j (r1 , . . . ; t1 , . . .) φ j (r0 ) eiωα j t0
j=2
+ c.c.
(14)
In the reference frame moving with the velocity v , we then
obtain the following equation of motion in the third order of
multi-scale analysis:
∂ Ah
1 ∂ 2 ω ∂ 2 Ah
(2)
+ 4π χeff
Al Ah = 0,
+
∂t
2 ∂ki ∂k j ∂ηi ∂η j
(15)
∂ Al
∂|Ah |2
(2)
v ·
− 4π χeff
= 0,
∂t
∂ η
(16)
i
novel regime allows us to study optically certain problems
regarding turbulence and wave collapses in plasma physics
and hydrodynamics and even astrophysics that are difficult
to investigate experimentally in their respective realms [26,
28–30, 42, 43]. The design of a suitable set of parameters
for a nonlinear 2D PC would be similar to and – in view of
the fewer constraints that have to be satisfied – even easier
than the design of the wave-front-reversal system discussed
above. Depending on the system under consideration, it may
then be necessary also to include the effects of Kerr nonlinearities in our analysis presented above, which can be done in
a straightforward manner.
where the envelope A1 (η, t) ≡ Al (η, t) of the low-frequency
wave (long wave) is real, while the envelopes A2 (η, t) =
A3 (η, t) ≡ Ah (η, t) of the high-frequency waves (short
waves) are complex and depend on the variable η = r − v t
of the moving frame. In addition, the dispersion tensor
p
∂ 2 ω(k)/∂k
i ∂k j may be obtained through second-order k · perturbation theory from photonic band-structure calculations
(2)
[33]. It should be noted that in the nearly degenerate case χeff
can be made real through an appropriate choice of the phase
transformation E α j → exp ψα j E α j of the carrier waves. Equations (15) and (16), which describe a long-wave/short-wave
interaction, have been studied in the context of plasma physics
[28] and hydrodynamics [27], where it has been emphasized
that their realization is a genuine result of a multi-branch dispersion relation. However, to our knowledge, Eqs. (15) and
(16) have not been derived in the context of optics before.
Equations (15) and (16) represent a generalized nonlinear
Schrödinger equation (NLSE) model. Since the long-wave
amplitude function Al is real, the solutions of Eq. (16) are very
similar to the solutions of the Korteveg–de Vries equation.
Similarly, the solutions for the complex short-wave amplitude
function Ah are very similar to the solutions of the NLSE.
In one dimension, Eqs. (15) and (16) exhibit novel soliton
solutions [28], while to our knowledge there does not exist an
analytical solution for 2D or 3D. For detailed discussions of
these equations, we refer the reader to Refs. [26, 28, 42].
We want to emphasize that this nearly degenerate case
of nonlinear three-wave interaction that is represented by
Eqs. (15) and (16) is very distinct from the degenerate case of
second-harmonic generation in PCs. As a consequence, this
Conclusions
In conclusion, we have presented a multi-scale
analysis of nonlinear three-wave interaction in 2D PCs. We
have shown that the multi-branch nature of the linear PC dispersion relation allows us to access parameter regimes for
group velocities and effective nonlinearities which have hitherto been out of reach. As a consequence, numerous radiationconversion effects discussed theoretically by Zakharov [23]
and Kaup [24, 25] can now be realized experimentally. In particular, we have discussed the realization of wave-front reversal of optical pulses in realistic LiNbO3 -based PCs with potential applications in PC-based telecommunication devices and
optical information processing. Furthermore, we have shown
that, in the nearly degenerate case, there exists a novel regime
of nonlinear interaction which we have illustrated through the
derivation of a long-wave/short-wave interaction that has not
been discussed in optics before. The resulting potential of
nonlinear three-wave interaction in PCs to analyze problems
of molecular chains, plasma physics, acoustics, and hydrodynamics in an optical context using sophisticated optical
measurement technology may prove highly advantageous to
these fields.
ACKNOWLEDGEMENTS We thank Peter Wölfle for illuminating discussions and acknowledge financial support from the Deutsche Forschungsgemeinschaft (DFG) under Bu 1107/2-2 and Bu 1107/2-3 (EmmyNoether program) and the DFG Forschungszentrum Center for Functional
Nanostructures (CFN) at the University of Karlsruhe within project A1.3.
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