Photonic crystal wires for optical parametric oscillators in isotropic media

Abstract

We investigate four wave mixing in photonic crystal wire microresonators realized in an isotropic medium. One-dimensional optical parametric oscillators are numerically analyzed by solving Maxwells equations in all dimensions and including material dispersion as well as nonlinear polarization.

Full text

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DOI:10.1007/s00340-004-1504-8
Appl. Phys.B (2004)
Lasers and Optics
AppliedPhysicsB
a.difalco
c.conti
g.assantou
Photonic crystal wires for optical parametric
oscillators in isotropic media
NOOEL-Nonlinear Optics and OptoElectronics Laboratory, National Institute for the Physics of the Matter
(INFM), University “Roma Tre”, Via della Vasca Navale 84, 00186 Rome, Italy
Received: 12 February 2004/
Revised version: 18 February 2004
Published online: 7 April 2004• © Springer-Verlag 2004
ABSTRACTWeinvestigatefourwavemixinginphotoniccrystal
wire microresonators realized in an isotropic medium. One-
dimensional optical parametric oscillators are numerically ana-
lyzed by solving Maxwell’s equations in all dimensions and
including material dispersion as well as nonlinear polarization.
PACS 42.65.Yj; 42.70.Qs; 42.82.Gw
1Introduction
Theinterestofphotoniccrystals (PC)relies mostly
on the design versatility afforded by such structures in vari-
ousmaterials foruseinmicro-andnano-opticsandintegrated
applications. The engineering of the dispersion relation in
PC-based devices can be achieved by periodic alternation of
high and low refractive index materials in one, two or three
dimensions, and has been extensively studied and demon-
strated [1–4]. Several linear and nonlinear phenomena have
been proposed for exploitation in PC structures, spanning
fromBragg reflectors and bistable gates to filters, delay lines,
waveguides,switches, lasers[5–11]. In thepastfewyears, by
introducing suitably located defects in the periodic pattern,
researchers have been able to obtain high quality (Q-) fac-
tors for the modes resonating in PC structures, up to values
Q > 104[12]. More recently, the features of optical modes
near the band-edges have been examined in two dimensional
(2D-)PCexhibiting comparably high Q-factors [13].
One of the simplest PC geometries is the PC-wire,
i.e., a strip of material with a limited number of holes
able to open a band-gap in the transmission characteristic.
Among them, air-bridge PC-wires are particularly interest-
ing and potentially amenable to the highest degree of circuit
integration [14].
In this paper we show for the first time how the micro-
resonant structure of an air-bridge PC-wire can be exploited
to efficiently confine light and obtain gain and optical para-
metric oscillation of the cavity modes via four wave mixing
(FWM), exploiting the cubic response of isotropic media.
u Fax: +39-065579078, E-mail: assanto@ele.uniroma3.it
In the next section we describe the model and its numeri-
cal implementation, while in Sect. 3 we perform the modal
analysis of the structure. Finally, in Sect. 4 we explore the
oscillatory behaviour of the PC-wire subject to high power
pumping. We will show how the oscillation dynamics relates
to the pumped mode, in terms of both symmetry constraints
and quality factors.
2 Numerical Model
Optical oscillations throughout energy exchange
between PC modes via nonlinear four wave mixing can be
studied by means of a finite-difference time-domain (FDTD)
code. Maxwell’s equations were solved with no approxima-
tions, except for discretization, in both the three spatial coor-
dinates and time. An isotropic nonlinear medium was mod-
eled by introducing a Lorentz oscillator yielding the cubic
polarization P:
∇ ×E = −µ0∂H
∇ ×H = ε0∂E
∂2P
∂t2+2γ0∂P
∂t
∂t+∂P
∂t
∂t+ω2
0f(P)P = ε0(εs−1) ˆ ω2
0E
(1)
where f(P) = 1 (P2= P·P) represents a linear single-pole
dispersive response. To describe an isotropic Kerr-like mate-
rial we chose f(P) = [1+(P/P0)2]−3/2as in [15]. As long as
the ratio P/P0is small, a standard Kerr response is retrieved;
as its size becomes appreciable, however, the model is able to
account for higher order terms (χ(5)etc.). For the integration
we employed a nonlinear generalization of the L-DIM1 ap-
proach for dispersive media [16]. To enforce electromagnetic
fieldcontinuityattheboundariesbetweendifferentmediaand
prevent spurious reflections, we adopted the Yee’s grid and
uniaxialperfectly matched layers(UPML)[17].Thesourcing
wasarrangedthroughoutatotal-field/scattered-field (TF/SF)
approach, allowing both single-cycle (SC) and continuous
wave (cw) excitations, the latter mimicked by an “mnm”
pulse. The “mnm” pulse allows for the avoiding of undesired
spectraldistortions bytrailing andtailing withm cycles while
holdingthepeakvalueforn cycles,respectively[18].Alinear
waveguide provided the input to the nonlinear portion of the

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Applied Physics B – Lasers and Optics
FIGURE 1
sourcing interface, D1 and D2 the photodetectors, a the PC lattice constant
Sketch of the structure. In the top panel, TF/SF indicates the
structure.Suchanin-couplingarrangementnotonlypreserves
thevalidity oftheTF/SFimplementation, butalso effectively
holds the adopted UPML boundaries even at high fluencies,
whenan index changecould alter theinvariance of the refrac-
tivedistribution [17].
Figure 1 is a sketch of the geometry: a one-dimensional
photonic crystal contains holes of radius 0.35a, with a =
450nm the PC lattice constant (i.e., the distance between
neighboring holes). The 4 µm-long strip, 450nm wide and
270nm thick, ensures single-mode propagation in the in-
put waveguide and throughout the frequency range of rel-
evance. D1 and D2 indicate the placement of virtual pho-
todetectors to monitor electric fields and powers. The en-
tire simulation window was sized 4×1.2×1.2µm3, and
the spatial discretization provided ∆x = ∆y = 20nm and
∆z ? 16nm. The time-step was set to ∆t = 0.008fs. For the
parameters of the Lorentz oscillator, we took εs= 11.7045,
ω0= 1.1406×1016rad/s,
3.802×108s−1. Such values mimic the dispersion relation
of Al0.1Ga0.9As, yielding an effective index n = 3.3258 at
λ = 1.55µm [19]. Finally, we chose P0= 1 (C/m2) to get
aKerrcoefficientn2? 10−18(m2/W),asdetermined byeval-
uating self phase modulation by numerical integration of (1).
The latter is a conservative value if compared to n2in poly-
mers,liquid crystals,semiconductors [20–22].
ˆ ω0= 1.0995×1015rad/s, γ0=
3Linear response
In order to infer the band structure of the PC-
wire, we launched an SC pulse in the transverse-electric (y)
linear polarization. Such pulsed input encompasses a broad
spectrum; hence, it can excite all the modes with compati-
ble symmetry features. After excitation, the y-component of
the electric field (Ey) was acquired in both D1 and D2 and
Fourier transformed to yield the spectra of Fig. 2. In either
positions within the wire a sharp band-gap is visible and cen-
tered around λ = 1500nm, owing to the chosen ratio r/a.
The stars above the top panel in Fig. 2 mark three wave-
FIGURE 2
(lower panel) by taking the square moduli of the Fourier transforms of the
electric field
Normalized spectra obtained in D1 (upper panel) and D2
lengths λ1= 1µm, λ2= 1.118µm, λ3= 1.695µm at which
we performed numerical experiments to characterize the de-
vice transmission. To this extent we carried out simulations
with cw excitations tovisualize the field(Ey) distribution and
symmetry, as shown in Figs. 3 throughout 5 corresponding
to cross-sections yz in x = Lx/2 (a) and xz in y = Ly/2 (b),
respectively. The limited number of holes does not allow for
FIGURE 3
plane; b Eyin the xz plane
y-component of the electric field for λ = 1µm. a Eyin the yz

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DI FALCO et al. Photonic crystal wires for optical parametric oscillators in isotropic media
FIGURE 4
yz plane; b Eyin the xz plane
y-component of the electric field for λ = 1.186 µm. a Eyin the
FIGURE 5
yz plane; b Eyin the xz plane
y-component of the electric field for λ = 1.695 µm. a Eyin the
theapplicationofastandardfrequencyanalysis;however,this
field mapping enables us to characterize the excited modes
at the wavelengths of interest and where the electric field is
mainly localized. The differing spatial distributions (at λ1,
λ2and λ3) correspond to unequal excitability and transmis-
sion of the wire. Feeding the PC at λ1, the transverse electric
field shows maxima and minima well distributed among air
(holes) and the medium, in both the horizontal (Fig. 3a) and
the vertical (Fig. 3b) sections. λ1, in fact, lies well outside
the band-gap displayed in Fig. 2. At λ2, conversely, the pump
is located at the high PC band-edge (in frequency): Ey is
more intense in the empty portions of the structure (Fig. 4).
The opposite case is displayed in Fig. 5 with reference to
a pump wavelength (λ3) at the lower band-edge: the field is
mostly localized within the medium. To calculate the low-
power (linear) transmission, we launched Pin= 1W in the
feeding waveguide for the three cases above, and obtained
T(λ1) = 21.3%. T(λ2) = 51%, T(λ3) = 97%, respectively, for
the ratios between output (e.g. in D2) and input powers. Such
limited values originate from the impedance mismatch be-
tween guided-wave and PC- modes, inevitable when a uni-
formwaveguideisattached to aperiodicstructure[23–25].
The Q-factors of each involved mode need also be taken
into careful consideration. However, it is hardly useful to cal-
culate the unloaded Q’s, as often evaluated in literature by
exciting the resonant mode with a proper intra-cavity source.
This approach, in fact, does not provide a sound estimate of
the actual device response when supplying energy from out-
side, as it would be in most real applications. Therefore, we
evaluated the loaded Q-factors, inclusive of coupling and ra-
diation losses.
By fitting the excitation time with the function of 1−
exp[−(ω0/2Q)t], we computed Q1= 20, Q2= 28, Q3= 15
atthethreewavelengths,respectively,ingoodagreementwith
the existing literature [14,26,27]. As anticipated above, such
differing Q’s (and T’s) relate to the unequal spectra gathered
in D1and D2forthesameexcitation (seeFig. 2).
4 Nonlinear dynamics
The main idea behind a PC optical parametric os-
cillator (OPO) stems from the large pump fluencies available
in the micro-cavity. The energy can then be re-distributed via
FWM between all the Bloch modes of thestructure, with effi-
ciencies dependingonsymmetry constraintsand Q-factors of
both pump and modes. The coupled mode theory in the time
domain, adopted in [28] to treat quadratic parametric oscilla-
tors,canbeutilizedforthestudyofparametricoscillationsvia
degenerate FWM. The oscillation process is characterized by
athresholdpower,whichdependsonthe Q’sofboththepump
and themodes as:
Pth=
ω
2|g|Q√Q+Q−
where the two generated frequencies ω+ and ω− are such
that 2ω = ω++ω− and Q± are the corresponding quality
factors, respectively. g is the relevant three-dimensional over-
lap integral between the modal field profiles and the spa-
tial distribution of the nonlinearity. If enough power is made
available, the newly generated frequencies play the role of
,
(2)

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Applied Physics B – Lasers and Optics
(secondary) pumps, and the PC density of states ([4]) will
be reproduced in the output spectrum. To verify and explore
such features, we performed simulations in cw increasing
the power at the pump wavelengths already investigated in
the linear regime. We achieved a spectral resolution ∆ f ?
1.11×1012s−1by recording the time-varying signals at the
output (D2) for intervals as long as 900fs. Figures 6 through-
out 8 show the normalized square modulus of the Fourier
transforms of the electric field |F (Ey)|2, at the output (log
scale on the vertical axes) for the cases above, comparing
linear (Pin= 1W) and nonlinear (Pin= 2kW) responses. It
should be noted in passing that the power needed to ob-
serve oscillations not only depends on the Q-factors of the
excited modes, but it also scales with the material n2. As
the pump wavelength is tuned, the OPO spectra markedly
change, as apparent in the figures. Their widths are linked to
the pertinent Q’s and the distribution of neighboring states,
with higher Q’s corresponding to better light confinement in
the resonator and, hence, more efficient nonlinear response
and energy exchange via FWM. While the wavelength λ1
FIGURE 6
line: Pin= 1 W; dashed line: Pin= 2kW
|F (Ey)|2at the output of the wire (D2) for λ1(Q ? 20). Solid
FIGURE 7
line: Pin= 1 W; dashed line: Pin= 2kW
|F (Ey)|2at the output of the wire (D2) for λ2(Q ? 28). Solid
FIGURE 8
line: Pin= 1 W; dashed line: Pin= 2kW
|F (Ey)|2at the output of the wire (D2) for λ3(Q ? 15). Solid
(with Q = 20 and T = 0.21) represents an intermediate be-
havior of the wire-OPO, λ2 (with Q = 28 and T = 0.51)
yields a broader spectrum (Fig. 7) and λ3(Q = 15 and T =
0.97) a narrow output which does not extend beyond the
band-gap region (Fig. 8). In the latter case, in fact, the low
Q penalizes the nonlinear response while allowing a better
transmission.
Notice that the secondary peaks associated to the pump
(solid line) in Figs. 7 and 8 at lower and higher frequencies,
respectively, are due to the non-monochromatic “mnm” exci-
tation,thespectrumofwhichoverlapswithothermodesinthe
initial stageoftheinteraction.
5 Conclusions
We investigated optical oscillators based on four
wavemixing in one-dimensionalPCwires.To suchanextent,
we resorted to a fully parallelized three-dimensional FDTD
code including material dispersion and a cubic nonlinearity.
In this novel geometry we demonstrate that optical paramet-
ric oscillations can be efficiently obtained even in isotropic
media encompassing a Kerr nonlinear response. The inter-
play between resonant, dispersive and nonlinear properties
in such PC-wires links the mode excitation to the pertinent
quality factors and the overall transmission. The ease of tai-
loring both these aspects in PC-wires is a key factor in ob-
taining a good tunability of the OPO with pump power, and
underlines the relevance of proper trade-offs in the device
design.
These results stress the richness afforded by photonic
crystals in enhancing resonant nonlinear interactions, and
pave the way to an entire new generation of highly integrated
optical parametric sources where a careful design could com-
pensatealimited nonlinearity.
ACKNOWLEDGEMENTS
We acknowledge partial support
from INFM-Initiative Parallel Computing, the Tronchetti-Provera Foun-
dation and the Italian Electronic and Electrical Association (AEI). We
thank Prof. S. Riva Sanseverino (University of Palermo) for enlightening
discussions.

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DI FALCO et al. Photonic crystal wires for optical parametric oscillators in isotropic media
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