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PRL 96, 163904 (2006)
PHYSICAL REVIEW LETTERS
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Localized Polaritons and Second-Harmonic Generation
in a Resonant Medium with Quadratic Nonlinearity
D. V. Skryabin,1 A. V. Yulin,1 and A. I. Maimistov2
1
Centre for Photonics and Photonic Materials, Department of Physics, University of Bath, Bath BA2 7AY, United Kingdom
2
Moscow Engineering Physics Institute, Kashirskoe shosse 31, Moscow 115409, Russia
(Received 3 February 2006; published 27 April 2006)
We derive model equations for the propagation of ultrashort pulses in materials with resonant linear and
quadratic nonlinear responses and find approximate soliton solutions describing all-bright and dark-bright
polaritons. We report the specific phase matching condition for efficient 2nd harmonic generation, which
involves detuning from the resonance. We also demonstrate that the 2nd harmonic emission by the
polaritonic pulses can lead to reduction of their group velocity, having zero as a theoretical limit. Our
analytical results are supported by numerical simulations.
DOI: 10.1103/PhysRevLett.96.163904
PACS numbers: 42.65.Tg, 42.65.Ky
Studies of materials engineered by inclusion of metal
nanostructures into dielectric hosts is one of the most
active areas of the current fundamental research in optics
underpinning ongoing technological revolution in optoelectronics and all-optical processing [1]. Applications of
such materials include subwavelength photonic circuits
[2], negative refractive index optics [3], nonlinear optics
[4,5], and more. The linear and nonlinear responses of
these materials are dominated by localized surface plasmons, exhibiting pronounced resonances in the extinction
spectra. Basic nonlinear processes such as secondharmonic generation and the Raman effect have all been
observed in plasmonic materials and continue to attract a
significant research effort [1,4,5].
In this Letter we present a theoretical study of short
pulse propagation in resonant materials with quadratic
nonlinearity. The conceptual novelty of our approach,
which goes beyond applications in plasmonics, is that we
are considering the cases when either the fundamental
wave or the 2nd harmonic is in resonance with the intrinsic
material oscillations. This enhances the second order susceptibility beyond the validity of the standard expansion of
the material polarization into the power series of the interacting harmonics [6] and requires the development of a
new approach. Methods of slowing the pulses of light
and related fundamental problems have been another area
of sustained recent interest spanning various approaches
ranging from classical to quantum optics [7]. Analytical
and numerical results presented below demonstrate intrinsically the nonlinear method of slowing light down
and bringing it to a stop in plasmonic materials, where
the soliton effects and 2nd harmonic emission play paramount roles.
The essence of the phenomenological model used below
is that the nanoinclusions are considered as the artificial
classically behaving ‘‘atoms.’’ Providing that the size of
the atoms is much less then the wavelength, the response of
the material to the electromagnetic waves can be described
using the average resonant polarization, Pr , which obeys
0031-9007=06=96(16)=163904(4)$23.00
the Lorentz equation, see, e.g., [6],
2
@2t Pr 20 @t Pr !20 fPr 0 1
1 !0 E:
(1)
Here !0 is the resonant frequency of the surface plasmon
and 0 is the decay rate. fPr Pr 2 P2r , where the
2 term characterizes quadratic nonlinearity of the oscillators. Linear susceptibility at the frequency ! created by
2
2
the resonant polarization is given by 1 1
1 !0 !0 1
2i0 !0 !2 1 , where 1
1 lim!0 !1 . The electric
field E obeys the wave equation
@2z E 1 2
@t E 0 @2t Pr Ph :
c2
(2)
Here Ph is the nonresonant linear polarization of the host
material and 0 0 1=c2 . The idealization tacitly used
here is that all nanoinclusions are identical, so that their
resonant frequencies are the same. Below we will assume
that the spectral width of the pulse is greater than the width
of plasmonic resonance and that the propagation distances
are such that the effects of losses can be disregarded, i.e.,
0 0. Inclusion of losses into our model under these
conditions does not qualitatively alter the soliton propagation regimes and 2nd harmonic emission discussed below.
Solutions of the system (1) and (2) represent an intrinsically coupled material and field excitations, which are
often referred to as polaritons. We focus our study on
finding the localized solitary wave solutions of the system,
which are likely to serve as natural attractors for pump
pulses and play a profound role in 2nd harmonic generation. Interest in solitary solutions to models with quadratic
nonlinearities has been sustained during the last decade,
see [8] for a review. In particular, the experimental observations of two-frequency temporal solitons were reported
in [9]. Reference [10] reported single-frequency bright
solitons in the systems similar to Eqs. (1) and (2), but
with purely cubic nonlinearity in the polarization.
Reference [11] studied the case when the equation for the
polarization is linear, while the nonresonant cubic nonlinearity is included into the wave equation. Soliton solutions
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© 2006 The American Physical Society
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PHYSICAL REVIEW LETTERS
PRL 96, 163904 (2006)
of Eqs. (1) and (2) having zero carrier frequency have also
been studied [12]. Because of the zero frequency assumption these solutions have no or little physical meaning in
the optical spectral range, where the experimental efforts
are directed.
In order to develop an analytical approach relevant in the
optical range of frequencies, it is natural to use the slowly
varying approximation. We use substitutions Pr P1 eik1 zi!t P2 eik2 zi2!t c:c: and E A1 eik1 zi!t A2 eik2 zi2!t c:c:, where kj kj!, j 1; 2 and retain
only the first derivatives of Pj , Aj and slowly oscillating
nonlinear terms. An effect of Ph 0 is that the dispersion
k! of the host material is different from the free space
one, k !=c. The resulting set of equations is
2 2
2ikj @z v1
j @t Aj 0 j ! Pj ;
(3)
2i!@t P1 !20 !2 P1 2!20 2 P2 P1 eiz
0 1 !20 A1 ;
(4)
4i!@t P2 !20 4!2 P2 !20 2 P21 eiz 0 1 !20 A2 ;
(5)
where j 1; 2, 2k1 k2 is the wave number mismatch due to the host dispersion, vj vj! and v! @! k1 is the frequency dependence of the group velocity.
The simplest and most studied situation is when !0 is
detuned far from the both frequencies ! and 2!. In this
case, the time derivatives in Eqs. (4) and (5) can be
neglected and material equations solved perturbatively,
which leads to the second order susceptibility defined as
2
2 2
2
2 1
2 ! 2
where
1 1 ! =!0 1 4! =!0 ,
2
1 2
2
1 lim!0 !1 22 0 0 1 c . If either ! or
2! are close to !0 the nonlinearity gets resonantly enhanced and the above expression for 2 losses its validity.
We switched to a dimensionless formulation of the
problem through the substitutions A1 z; t A0 A~1 ~
z; ~t,
z; ~teiz , P1 z; t P0 P~1 ~
z; ~t, P2 z; t A2 z; t A0 A~2 ~
P0 P~2 ~
z; ~teiz , where all the tilde variables are dimension1
2
less. Here z~ z1
t t 1 =20 v1 n1 , 0 1 A0 P0 , ~
2
z=v1 =0 , and 0 !=!0 , so that the time is scaled to an
optical cycle. Lastly, n1;2 are the host refractive indices at
! and 2!, respectively.
First we consider a resonance with the fundamental
wave, i.e., ! is close to !0 . This situation closely corresponds to the recent experiments on the second-harmonic
generation from arrays of gold nanoparticles embedded
inside glass and irradiated with femtosecond pulses [5].
In this case, the time derivative in Eq. (5) can be neglected,
which gives P~2 16 P~21 13 A~2 , where 2
1 A0 =
1
21 characterizes the strength of the nonlinearity. Detuning from the second resonance !20 4!2 was approximated with 3!20 . The resulting dimensionless system is
i@z A1 P1 ;
(6)
i@z w1 @t A2 n1 2
P 11 A2 ;
3n2 1
1
1
i@t P1 1 P1 2 jP1 j2 P1 A2 P1 A1 :
3
3
2
(7)
(8)
In Eqs. (6)–(8) we dropped the tildes for brevity. 1 1 !=!0 is the dimensionless detuning from the first
resonance and 1=w 2v1 v2 n21 =v2 1
1 characterizes the group velocity mismatch of the two fields. 11 1
1 2n
3n2 is the wave number mismatch which includes
contributions from the dispersion of the host material (1st
term), 1 2n21 0 v1 =1
1 , and from the response of the
resonant inclusions at 2! only (2nd term).
Equations (6)–(8) are well suited for studies of secondharmonic generation by the ultrashort pulses with a spectrum in the proximity of the resonance. There are two
distinct nonlinear mechanisms in Eqs. (6)–(8). One is the
effective cubic nonlinearity in the P1 equation, which is
induced by 2
1 0 and is primarily responsible for the
nonlinear phase rotation of P1 and A1 . The second nonlinear mechanism in Eqs. (6)–(8) is related to the parametric terms A2 P1 and P21 . The parametric effect couples
the second-harmonic photons with the material excitations
at the fundamental frequency. The jP1 j2 P1 term is formally
analogous to, but physically different from, the effective
Kerr nonlinearity of the fundamental field in the offresonant and phase mismatched 2nd harmonic generation
[8]. In our case, the cubic nonlinearity appears in the
material excitations instead of the optical field. Therefore
the 2nd harmonic of the optical field A2 can be assumed to
be exactly zero and the system remains nonlinear. On the
other hand, in the off-resonant case, the effective Kerr
nonlinearity of A1 is provided by a small (1=jj) but
nonzero value of A2 [8].
The 2nd harmonic is small if the wave number mismatch
j11 j is large. Putting A2 0, we find that in the leading
order, A1 and P1 obey
i@z A1 ’ P1 ;
1
1
i@t P1 1 P1 ’ 2 jP1 j2 P1 A1 : (9)
3
2
Equation (9) is exactly the system considered in [10] to
describe optical resonance in nanostructured materials
with intrinsic Kerr nonlinearity. Adopting the approach
of [10] we find the approximate localized polariton (LP)
solutions:
A1 rei
i1 t O2
11 ;
t z=v;
(10)
r
P1 p e3i
i1 t O2
11 :
2v
p
p
Here
r2 12v 2v2 sech 2v,
p
arctanth v=2 . The parameter v > 0 characterizes
the field amplitude and, simultaneously, the velocity shift
of the nonlinear polariton relative to the linear group
velocity v1 . Larger values of v result in larger soliton
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PRL 96, 163904 (2006)
amplitudes. By analyzing dispersive properties of the linearized version of Eq. (9) one can show existence of the
expected polaritonic band gap [6,11]. Therefore the above
solutions can be understood as gap solitons.
The simple expression for the second harmonic is found
by neglecting the derivatives in the left-hand side of Eq. (7)
(valid for j11 j 1) giving
A2 n1 2
P O3
11 :
3n2 11 1
(11)
Direct numerical modeling of Eqs. (6)–(8) initialized with
the approximate soliton solutions (10) and (11), confirms
the validity of the latter to a very good accuracy. The same
solution is also excited with the pump only at the fundamental frequency, see Figs. 1(a)–1(c). Note that the width
of the pump pulses in our modeling is taken to be of the
order of 10 dimensionless units. This means that the pulse
covers around ten optical periods, which puts us into a
regime where the slowly varying approximation is
applicable.
In the off-resonant system the temporal solitary waves
found for the large wave number mismatch can be varied to
that of the perfectly matched situation [8]. Simulation of
Eqs. (6)–(8) reveals a different scenario. Figures 1(d)–1(f)
show the evolution of the fundamental frequency pump
pulse for 11 10, 1 0. One can see that the localized
component of the second harmonic is still strong, but it
acquires a noticeable tail of radiation emerging in the
direction of the nonsolitonic remnant of the 2nd harmonic
signal. As we will demonstrate below, the amplitude of this
tail is exponentially small in 11 and therefore cannot be
described using a perturbation expansion in the inverse
powers of the latter. Our method of finding the tail relies
on the fact that Eq. (7) for the 2nd harmonic can be
integrated exactly.
15
5
0
-1000 -500
0 500 1000
time, t
10
5
0
-1000 -500
15
5
0
-1000 -500
0 500 1000
time, t
(c) Localized
polariton
=100
z=10
5
0 500 1000
time, t
15
distance, z
10
10
0
-1000 -500
0 500 1000
time, t
15
distance, z
distance, z
15
distance, z
distance, z
distance, z
15
10
10
5
0
-1000 -500
=10
z=10
5
0
-1000 -500
0 500 1000
time, t
(f) Localized
polariton
10
0 500 1000
time, t
(i)
(j)
=3
z=10
=3
z=5
FIG. 1 (color online). Results of the numerical simulation of
Eqs. (6)–(8). 1st and 2nd rows show evolution of jA1 j and jA2 j,
respectively, in the t; z plane. The scale used is logarithmic. 3rd
row shows jA1 tj (full red line) and jA2 tj (dashed blue line) for
the propagation distances indicated by the white horizontal lines
in the
t; z plots. Initial conditions used in modeling are A1 p
0:13 secht=7:1 and A2 0. Parameters are (a) –(c) 11 100, (d)–(f) 11 10, (g)–(j) 11 3, and w 0:1, 1 0,
1, n1 =n2 1.
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Imposing condition A2 0 at z 0 we find A2 in terms
of variables z and :
n1 u i11 u Z 0
A2 e
P21 0 ; zei11 u d0 : (12)
i3n2
z=u
Here 1=u 1=w 1=v characterizes the mismatch of the
two velocity shifts. Before proceeding with further analysis, let us note, that whatever the properties of the function
under the integral are, the energy in the second-harmonic
field has little opportunity to grow if v is close to w. Simply
because the integration interval in this case is small. If
this interval is much greater than that in which the
sech function is substantially different from zero, then
the properties of the phase factor in the integrated function
are crucial. Using P1 from Eq. (10) we find that the overall
phase factor is expi6
21 11 =u0 ei . The
p
straight line approximation ’ v=2 for the soliton phase is satisfactory over the interval making the
dominant contribution to the integral, which gives ’
0 , where
p
(13)
3 2v 21 11 u:
The simplest limiting case allowing explicit calculation
of A2 , is obtained by assuming that all of the contributions into the phase cancel out, i.e., 0. Assuming,
for
0 and z ! 1, we are left
pwith
u >
R example,0 pthat
0
sech
f
2
arctanexp
2vg=
2v
d
p1
2v. This function is a kink which represents the tail of the
second harmonic left behind by the passing pump pulse. If
is made large, then the amplitude of the tail drops down
and it becomes oscillatory. Figs. 1(f) and 1(j) show the tail
structure of the second harmonic calculated numerically.
The amplitude of the tail, A1
2 , far from the center of the
pump pulse can be estimated by assuming that ! 1,
z= ! 1 and u > 0. Then the integration limits in Eq. (12)
can be replaced by 1, giving for an arbitrary ,
2n1 juj
p
sech
j
jA1
:
(14)
2
n2 2 2v
Equation (14) explicitly demonstrates that jA1
2 j decays
exponentially for increasing j1 j and j11 j. An important
result of this calculation is that it shows that even if j11 j is
large, one still can ensure exponential increase in the
efficiency of second-harmonic generation by tuning 1 to
minimize . The inverse is also true, i.e., if 1 is fixed then
by tuning 11 , e.g., with quasiphase matching, we can
achieve 0. In fact, the 0 condition is the phase
matching condition, which should be used in the resonant
material to achieve efficient energy transfer into the 2nd
harmonic from the short pump pulses. This condition also
explicitly includes nonlinear phase shifts, because v is
directly related to the pulse amplitude. It can be straightforwardly demonstrated, either form Eq. (13) or directly
from Eqs. (6)–(8) that without the nonlinear corrections
the phase matching condition is 11 21 =w 0. If
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PHYSICAL REVIEW LETTERS
1=w 0, i.e., the group velocities of the 1st and 2nd
harmonics are matched (which is hard, but possible to
achieve [9]), then the phase matching condition is 11 0. The latter is essentially the same condition as the one
derived in the off-resonant limit [6].
It is clear from the numerical modeling that for small
values of , the tail of the second harmonic starts to be a
dominant feature, see Figs. 1(g)–1(i). In this regime important observations can be made about the dynamics of
the optical and material excitations at the fundamental
frequency. In the modeling we observed that despite the
relatively small ’s used, i.e., when the 2nd harmonic is
fully delocalized, the sufficiently strong initial pulse still
evolves into the quasisolitonic structure in A1 and P1 fields.
The structure in Fig. 1(g) is clearly solitonic because the
linear nonsolitonic pulse would have a propagation velocity equal to v1 and propagate along a vertical trajectory in
our (z; t) plots, see Fig. 1, while the excited pulse moves
under the angle, see Fig. 1(g), as prescribed by Eq. (10) for
finite values of 1=v. This quasisoliton losses its energy and
momentum into the tail of the second harmonic until it
completely disperses by emitting vertically propagating
linear waves, see Figs. 1(g)–1(j). Note, that the drop in
the pulse amplitude is accompanied by a change in its
velocity [v ! 0 leads to the almost horizontal trajectory
in Fig. 1(g)], as predicted by Eq. (10). The physical group
2 1
velocity of the LP is given by v1 1 1
1 =v2n1 .
Therefore, for v ! 0, the true group velocity also tends
to zero. Therefore, emission of the 2nd harmonic by the
LPs can be considered as a method for the efficient reduction of light’s group velocity. P21 decays slower with v ! 0
than A1 , see Eq. (10). Therefore, the relative energy balance within the quasisolitonic part of the field is changing
in favor of the 2nd harmonic with z increasing.
Finally, we briefly describe the case, when the resonance
happens at the second harmonic, i.e., 2! is close to !0 .
This case closely matches the conditions of the experiment
[4], where the 2nd harmonic generation in the femtosecond
regime has been observed from a quasiphase matched glass
sample with embedded ellipsoidal silver nanoparticles.
Analytical advancement here is possible in the weak excitation limit, 1. Neglecting the time derivative in the
equation for P1 and using the perturbation series in we
find P~1 13 A~1 29 P~2 A~1 O2 . Here the detuning
from the first resonance !20 !2 was approximated by
3!20 =4. The resulting system of dimensionless equations
with dropped tildes is
1
A 1 P2 A1 ;
12 1
n
i@z w@t A2 1 A2 1 P2 ;
2n2
1
i@t P2 2 P2 1 A21 A2 :
4
i@z A1 (15)
Here 2 12 !!0 and 1 =18. The only nonlinearity
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in Eq. (15) is the parametric one, corresponding to the
conversion of the ! photons into the material excitations
at 2!, which are subsequently converted to 2! photons. To
find soliton solutions we again consider the limit j1 j 1.
Then, A2 can be disregarded in the leading order and the
equations for A1 and P2 can be integrated analytically,
which gives
p
A1 C v sechv1 Ceiz=12i2 t O2
1 ;
P2 iC tanhv1 Ceiz=6i22 t O2
1 ;
A2 n1 P2 =2n2 1 O3
1 ;
(16)
t z=v:
Here C and v > 0 are the soliton parameters. Thus in the
soliton regime, the optical field and the material response
at 2! are the dark pulses, while the pump is the bright
pulse. Because of the sign difference in the definitions of and , the physical group velocity of the solutions (16) is
greater than the velocity v1 . Thus, in this case, the nonlinearity serves as the light accelerator.
In summary, we derived model equations for the 2nd
harmonic generation in the ultrashort pulse regime for the
quadratically nonlinear materials exhibiting resonance response close to either the pump or second-harmonic frequencies. We found approximate quasisolitonic solutions
and described their role in generation of the second harmonic. We also derived conditions for efficient 2nd harmonic generation and demonstrated that the energy
emission from the quasisolitons into the 2nd harmonic is
able to significantly slow down their group velocity.
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