## Abstract

We investigate optical parametric oscillations via four-wave mixing in a dielectric photonic crystal. Using a fully vectorial 3D time-domain approach, including both dispersion and Kerr nonlinear polarization, we analyze the response of an inverted opal. The results demonstrate the feasibility of parametric sources in isotropic media arranged in photonic band-gap geometries.

©2004 Optical Society of America

Micro-cavities are emerging as excellent candidates for exploring both fundamental effects and applications of nonlinear optics [1]. Frequency generation and Raman processes have been observed at very low threshold powers in isotropic materials [2]. Large Q-factors and high index contrast, in fact, enable concentration of electromagnetic energy, thus supporting oscillations driven by parametric interactions and allowing for energy exchange and gain.

In a cavity filled with an isotropic material, coupled mode theory (CMT) [3, 4, 5] allows to show that a pump beam of angular frequency *ω* can generate two frequencies *ω*
_{±}, with *ω*
_{+}+*ω*
_{-}=2*ω*, via parametric oscillation at powers above the threshold:

being *Q,Q*
_{+},*Q*
_{-}, the cavity Q-factors at *ω,ω*
_{+} and *ω*
_{-} respectively, and g the pertinent three-dimensional overlap integral between the involved mode profiles and the spatial distribution of the nonlinearity. High Qs obviously favor parametric effects.

As underlined in Ref. [2], however, harmonic generation in simply-shaped cavities -such as spheres or rings- is dictated by both the regular distribution of the modes and the stringent phase matching constraints. Basic nonlinear mechanisms such as four-wave mixing (FWM) in elementary geometries, therefore, can be penalized.

Photonic crystals (PC) [6, 7, 8, 9, 10, 11, 12, 13] appear to be a very promising solution to overcome such limitations. In fact, by realizing cavities with PC, i.e., with a three-dimensional (3D) periodic and high-contrast arrangement of the refractive index, it is possible to tailor the distribution of states and obtain large Q-factors (see e.g., Refs. [14, 15]). Moreover, owing to the 3D periodicity, the phase-matching constraints are significantly alleviated by the presence of a large number of wave-vectors in the reciprocal lattice. At variance with simply-shaped cavities, design and engineering of such systems can only be performed numerically, particularly if material dispersion and nonlinearity have to be accounted for. Furthermore, the involved nonlinear interactions are strongly dependent on the spatial symmetries of the modes, which, in turn, are significantly affected by the truncated periodicity of finite systems [9].

In this Paper, using a suitably parallelized algorithm, we investigate the use of a 3D PC in exploiting optical parametric oscillations in isotropic media with a third-order nonlinearity.

Following the CMT analysis, denoting with *a*
_{±} the amplitudes of the generated frequencies (normalized to unit energy), above threshold:

Δ*p* measures the excess pumping, which is transferred to other generated frequencies, as in standard optical parametric oscillators (OPO) (see Refs. [3, 5, 16]) in cubic materials. If *Pin* is the power excitation at the pump frequency *ω*, $\Delta p=\sqrt{{P}_{\mathit{in}}\u2044{P}_{\mathit{th}}}-1$. Equation (2) shows that, if the cavity has a complex mode distribution as in a finite PC, the oscillation spectrum will not exhibit symmetric sidebands, but will strongly depend of the Q-factors (and hence the coupling losses) of the excited modes. Moreover, additional frequencies will be pumped by FWM processes (e.g., *ω*+*ω*
_{+}-*ω*
_{-}). With reference to the generation of a specific ω2, the energy stored in the cavity is:

where we introduced the density of states (DOS)*ρ*(*ω*) of the cavity [9]. In deriving Eq. (3) we considered that, if *Q*
_{2}≡*Q*(*ω*
_{2}) is sufficiently high, the relevant modes are those in proximity of *ω*
_{2}.

If the DOS varies smoothly (see Ref. [19]), Eq. (3) states that the energy is roughly proportional to *ρ*(*ω*
_{2}). Hence, owing to FWM, we expect the output spectrum at high fluences to resemble the DOS, thus enabling the *nonlinear spectroscopy* of PC-microresonators. This holds valid also for those modes excited in the non-stationary regime, i.e., not pumped by a specific frequency-mixing process and relying on transient parametric fluorescence. Since their dynamics is strongly linked to the time-scale of the nonlinearity, they need be modeled in conjunction with a non-instantaneous nonlinear response. This picture of parametric wave mixing in a PC microcavity gets even more involved when considering not only the degenerate interaction (2*ω=ω*
_{+}+*ω*
_{-}), but also processes of the type*ω=ω*_{a}
+*ω*_{b}
+*ω*_{c}
. Hence a comprehensive numerical approach is required.

Aiming at demonstrating the feasibility of OPO with PC in isotropic media, we performed a fully vectorial computation of nonlinear Maxwell equations with an FDTD approach for dispersive materials [20]. A nonlinear Lorentz oscillator yielded the polarization **P** in the regions where material is present:

$$\nabla \times \mathbf{H}={\epsilon}_{0}{\partial}_{t}\mathbf{E}+{\partial}_{t}\mathbf{P}$$

$${\partial}_{t}^{2}\mathbf{P}+2{\gamma}_{0}{\partial}_{t}\mathbf{P}+{\omega}_{0}^{2}f\left(P\right)\mathbf{P}={\epsilon}_{0}\left({\epsilon}_{s}-1\right){\omega}_{0}^{2}\mathbf{E}.$$

The Yee’s grid [21] was employed to enforce continuity at the boundaries between different media, and uniaxial phase-matched layers (UPML) were adopted. [22] The algorithm is a nonlinear generalization of the L-DIM1 scheme used in Ref. [20] (details will be given elsewhere).

We carefully specified the form of the Lorentz oscillator *f*(*P*), with *f*(*P*)=1 describing a linear single-pole dispersive medium (*P*
^{2}=**P**·**P**). For an isotropic material, we used *f*(*P*)=[1+(*P/P*
_{0})^{2}]^{-3/2} as in Ref. [23], with *P*
_{0}(${P}_{0}^{2}$=(9/2)(*ε*_{s}
-1)^{3}
${\epsilon}_{0}^{3/2}$
${\mathit{\mu}}_{0}^{1/2}$/*ε*_{s}*n*
_{2}) a measure of the nonlinearity linked to the Kerr coefficient *n*
_{2}. This provides a simple way to describe non-instantaneous FWM and higher order nonlinearities, [24] distinguishing our approach from previous ones (see e.g., Ref. [25]). For small ratios *P/P*
_{0}, *f* produces a Kerr response, but compared to the standard Kerr *f* (*P*)=1+*χ*
^{P2}, the resulting algorithm is stable even near the Courant limit [22]. This approach accounts for all the FWM interactions of the type *ω=ω*_{a}
+*ω*_{b}
+*ω*_{c}
, but also higher order processes (e.g., *χ*
^{(5)}), in principle up to an infinite number of frequencies, the leading effect being cubic.

Because of the significant resources needed for the numerics, the code was parallelized. ^{1} We computed the response of an inverted opal PC: a Face-Centered-Cubic (FCC) lattice (of period *a*) of air-spheres (radius *r*=0.3535*a*) embedded in a dielectric (see inset of Fig. 1). The FCC is among the simplest structures admitting a complete photonic-band-gap (PBG) [26, 27]. The PC, of size 8×8×8*µm*
^{3}, was placed in air and excited by a 2*µm*-waist linearly y-polarized Gaussian beam, numerically implemented through a total field/scattered field layer [22]. The FCC lattice, for an index 3.5 as in Si or GaAs, has a complete band-gap around the normalized frequency *a/λ*=0.8, with λ the wavelength. To obtain a gap near λ=1500*nm* we chose *a*=1200*nm*, and the parameters of the single pole dispersion were taken as *ε*_{s}
≅11.971, *ω*
_{0}=1.1×10^{16} and *γ*
_{0}=2×10^{5} (MKS units), yielding an index≅3.5. An *n*
_{2}=1.5×10^{-17}
*m*
^{2}
*W*
^{-1} [28] gives *P*
_{0}≅1*Cm*
^{-2}. The integration domain was discretized with *dx*≅*dy*≅*dz*≅30*nm* and temporal steps *dt*=0.02 *fs*, allowing more than 40 points at each wavelength and runs with 30000 steps in time (the spectral resolution is of the order of 10*nm* at λ=1500*nm*).

Usually, the DOS is calculated in infinitely extended structures by a plane-wave expansion and neglecting material dispersion (see, e.g., Refs. [26, 29]). Hence, such standard approach cannot be applied to our dispersive and finite-extent device. In particular while the finiteness could be dealt with adopting the supercell method [8], the inclusion of arbitrary dispersion would require a very demanding frequency-domain analysis. Therefore, to determine the states of the FCC-PC-cavity we resorted to a time-domain approach. ^{2} A low-power (1*n*^{W}
) single-cycle pulse [30] excited the PC along the Γ*X* direction and the *E*^{y}
component of the transmitted signal was analyzed just after it. The resulting spectrum is shown in Fig. 1, where peaks correspond to concentrations of states (taken aside the low-frequency oscillations, due to the finiteness of the structure [9]) compatibly with symmetry constraints. The band structure of this medium encompasses a PBG around 1500*nm* and a pseudo-gap around 2400*nm*. Using these results, in order to cw pump the nonlinear parametric processes we picked λ≅1336*nm* corresponding to *a/λ*=0.898, i.e. close to a state by the upper edge of the PBG (in frequency), as marked by the star in Fig. 1. The 600 *f s* quasi-cw excitation was realized by an *mnm pulse* with spectrum well peaked at the carrier frequency [30]. The *mnm pulse* is a sinusoidal signal at the carrier (*ω*), with a constant amplitude for n periods, and smooth trailing and tailing edges of *m* periods. It allows the implementation of a quasi-CW (narrow-band) signal without a spurious frequency content. However, since simulations run for hundred fs, the results could be observed with pulses of similar duration.

Figure 2 displays the *E*^{y}
spectral density obtained in a low-symmetry point at the center of the PC, for a *y*-polarized pump propagating along the Γ*X* direction. The insets show the generated frequencies as the input power is increased. Large output bandwidths are attained, with no oscillations at frequencies within the PBG, a smooth profile in the large wavelength region and several peaks above the PBG upper-edge. Each peak corresponds to a region with a large DOS [26]. Resonances slightly shift at high fluences owing to self- and cross-phase modulation [10, 11].

It is noteworthy that the spectrum is non-symmetric because of the non-trivial (and engineer-able) distribution of states and Q-factors (see Eq. (2)). While the oscillation spectrum resembles the DOS (see Eq. (3) and Fig. 3 in Ref. [26]), small wavelengths are favoured by higher Q’s. Even for the simplest interaction 2*ω=ω*
_{+}+*ω*
_{-}, *ω*
_{+} and *ω*
_{-} need not be symmetrically located with respect to the pump. Such asymmetries, as well as the spectrum dependence on power, were previously observed through cubic parametric amplification in fibers,[16] and are clearly accentuated in dispersive PC microcavities.

Even the spectrum observed outside the nano-structured cavity will be strongly affected by the frequency and polarization dependence of the Q-factors, being in general different than inside the PC and markedly sensitive to pump polarization and direction of propagation [5].

While a comprehensive study of such a complex phenomenon beyond our scope here, it is enlightening to examine the output spectrum for the case *P*=1.2*MW*, as graphed in Fig. 3. Large wavelengths are substantially emitted due to lower Q-factors, while lower λ’s, although more efficiently generated (see Fig. 2), are not well out-coupled. Hence, the generation of a specific frequency would require an appropriate out-coupling.

Our analysis, stemming from a complete solution of the model represented by dispersive and nonlinear Maxwell equations, shows that PC micro-cavities hold great promises for the realization of compact frequency generators.

It is worth emphasizing that the existence of a complete PBG cannot be stated by linear scattering experiments, e.g., by varying the incidence angle for a given input frequency. Due to symmetry constraints, in fact, [9] not all the modes can be coupled-in. Potential approaches include the implantation of active atoms and the analysis of the spontaneous emission as in Ref. [31]; the frequency mixing we investigated numerically enlightens the fundamental role of the nonlinear response in revealing the basic properties of a PC. Furthermore, even though in a classical frame, our study points to the tailoring of the FWM fluorescence in nano-structured systems, which might pave the way to fundamental developments such as the production of new quantum states [32].

In conclusions, we carried out complete simulations of 3D Maxwell equations for OPO’s based on FWM in FCC photonic crystals. Nonlinear spectroscopy of such devices effectively candidates for investigating the fundamental properties of PC, and fully vectorial 3D simulations demonstrate that regions of significant mode-concentration and high Q-factors are available and aid the onset of parametric oscillations. Novel generations of tailored parametric sources employing isotropic materials in highly integrated PBG geometries can be envisaged.

## Acknowledgements

We acknowledge support from INFM-“Initiative Parallel Computing”, the Tronchetti-Provera Foundation and the Italian Electronic and Electrical Association (AEI).

## Footnotes

^{1} | The program runs on the IBM-SP4 system at the Italian Interuniversity Consortium for Advanced Calculus (CINECA), as well as on the NOMAD-BEOWULF cluster at NOOEL |

^{2} | It must also be noticed that -at best- the frequency domain analysis is of order N log(N), while the time-domain is of order N, with N the dimension of the problem. |

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