1. Introduction

Since their discovery more than a decade ago [1, 2], photonic crystals (PCs) have attracted a rapidly growing research interest. These optical structures have a spatially periodic dielectric constant (index of refraction), which induces striking changes of the photon density of states. In particular, the ability of PCs to induce a band gap in the frequencies of the electromagnetic modes that can propagate in such structures has leaded to advances in the basic understanding of light-matter interactions, as well as to optoelectronic devices with new and improved functionality. Furthermore, inserting structural defects in PCs not only fundamentally changes their physical properties but also leads to an unprecedented flexibility in the ability to control and manipulate light. For instance, by introducing line (1D) or point (0D) defects in the periodic structure of a PC, one can guide light around sharp corners with negligible losses [3, 4] and create ultra-small laser nanocavities [5] or efficient optical isolators [6].

To employ PCs in advanced technologies, e.g. for ultra-fast optical functions and all-optical switching, it is crucially important to have dynamic tunability of their properties. In recent years, several promising schemes for achieving this feature have been proposed. For example, tunability of the photonic band-gap (PBG) has been obtained by modulating the PC’s refractive index through the electro-optic effect [7] or through temperature-induced changes in the PC’s refractive index [8]. One promising approach, which is suitable for ultra-fast devices, is to employ intrinsic optical nonlinearities in the PC material. Thus, experimental studies have recently demonstrated both the ultrafast response of semiconductor-based PC waveguides as well as enhanced nonlinearity effects [9, 10], which are specific to PC structures. Furthermore, it recently has been predicted that such nonlinear optical PC devices can be used to achieve unique functionalities, namely switching [11–13], all-optical control of their enhanced dispersive properties [14, 15], or optical limiting and higher harmonics generation [13].

In this paper, we introduce a numerical method that allows us to calculate the field profile and dispersion curves of waveguide modes, as well as the modes and resonant frequencies corresponding to resonant cavities; in both cases these structural defects are embedded in nonlinear PCs. Furthermore, as an example, we apply this numerical method to investigate the optical response of a channel drop filter consisting of a resonant cavity side-coupled to a waveguide PC whose optical properties can be tuned by means of a pump beam.

2. Numerical method

In this section, we describe the numerical method we used to find the photonic band structure (PBS) of nonlinear PCs as well as the frequencies and the corresponding modes associated to structural defects in nonlinear PCs, namely a plane wave expansion (PWE) method [16], modified to self-consistently include the changes in the refractive index caused by the nonlinear effects [17]. We first present the method that is applied to a PC without defects and then discuss the modifications that are required when describing structural defects, namely the supercell approach. A generic geometry of a PC with 1D and 0D defects is shown in Fig. 1.

 

Fig. 1. Schematics of a 1D and 0D defects in a 2D PC with square symmetry (a is the lattice pitch). The two rectangles represent the supercells used to find the defect modes and the corresponding mode frequencies. In the lower panels, the electric field (TM polarization) of the 0D cavity mode (right) and 1D waveguide mode (left); the mode frequency is ω_r .

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To begin with, we consider the propagation of a monochromatic electromagnetic field in a medium with a periodic distribution of the dielectric constant, ε(r)=ε(r+R), with R a lattice vector. Then, the electromagnetic field distribution is governed by the following wave equation,

where H(r) and ω are the magnetic field and wave frequency, respectively. Using Bloch’s theorem, we express the field as H(r)=H_k (r)e ^ik·r, where k is the Bloch wavevector and H_k (r) is a periodic field. By Fourier expanding H_k (r) and ε ^-1(r), H_K (r)=∑G∑λ=1,2 h ^λ Ge ^iG ·^rê ^λ G and ε ^-1(r)=∑G${\epsilon }_{\mathbf{G}}^{-1}$ e ^iG·r, and inserting them in Eq. (1), one yields the eigenvalue matrix equation:

Here, the matrix elements M _Gλ,G′λ′=${\epsilon }_{\mathbf{G}-\mathbf{G}\prime }^{-1}$ [(k+G)×ê ^λ ′_G′, with G a reciprocal lattice vector, $h_{\mathbf{G}}^{\lambda }$ and ${\epsilon }_{\mathbf{G}}^{-1}$ are Fourier coefficients, and ${ê}_{\mathbf{G}}^{\lambda }$ are unit vectors orthogonal on k+G. In a 2D geometry, in the particular cases that correspond to TE or TM polarizations, in Eq. (2) there is only one summation, as for the TE (TM) polarization there is only one unit vector ê_G, oriented perpendicular (parallel) to the plane of the PC.

The eigenvalue matrix equation (2) is solved for k in the first Brillouin zone; the eigenvalues {ω_n } give the PBS whereas the eigenvectors {$h_{\mathbf{G}}^{\lambda }$ }, expressed in the k-space, determine the propagating eigenmodes. The field distribution H_k (r) in the position space is then obtained by Fourier transforming these coefficients. Finally, the electric field is calculated by using the relation iωε0ε (r)E(r)=-∇×H(r).

To determine the altered PBS of the PC, in the presence of an illumination pump beam with the frequency ω_p and power per unit length in the transverse direction, per unit cell, P, one proceeds as follows. First, one determines the PBS using the dielectric constant ε ⁰(r)≡ε (r), with P=0, i.e., the PBS of the linear PC. Then, by using an interpolation procedure, one numerically computes the wavevector ${\mathbf{k}}_{\mathbf{p}}^{\mathbf{0}}$ that corresponds to the frequency ω_p , the field distribution of the corresponding eigenmode, and its group velocity, ${\mathbf{v}}_{\mathbf{g}}^{\mathbf{0}}$ =∇_k ω(k=${\mathbf{k}}_{\mathbf{p}}^{\mathbf{0}}$ ). The group velocity can be calculated by using the Hellmann-Feynman theorem [18], h ^†·∇_k ℳ·h=[2ω(k)/c ²]∇_k ω(k), or by direct numerical differentiation of the frequency manifold ω(k).

Once the group velocity is calculated, the electric field in the pumping mode ${\mathbf{E}}^{\mathbf{0}}\left(\mathbf{r}\right)\equiv {\mathbf{E}}_{{\mathbf{k}}_{\mathbf{p}}^{\mathbf{0}}}(\mathbf{r})$ is rescaled such that the equation

is satisfied. Here, 〈·〉 means spatial averaging over the unit cell. Equation 3 can be readily derived from the relationship between the average of the Poynting vector and the average of the mode energy density, 〈S_k (r)〉=v_g (k)〈𝓤 _k(r)〉. After E ⁰(r) is calculated, one determines the change in the refractive index n of the Kerr material, δn ⁰(r)=ε0cnn2|E ⁰(r)|²/2, and then the new value of the dielectric constant, ${\epsilon }^1\left(\mathbf{r}\right)={\left[\sqrt{\epsilon \left(\mathbf{r}\right)}\delta n^0\left(\mathbf{r}\right)\right]}^2$. These steps are repeated until the variation of ε between two iterations is below some threshold, δε≡∑i|ε ^j+1(r_i )-ε ^j (ri)|/N<10^-8. Here, the sum is taken over the grid points of the mesh that covers the unit cell. Then, the final value of ε is used to calculate the PBS of the pumped nonlinear crystal.

When structural defects are introduced into the PC, this method is modified as follows. First, consider that a 1D defect is introduced into the PC, as shown in Fig. 1, with the optical power propagating in the waveguide mode being P. To compute the waveguide mode, one considers that the unit cell of the PC is a supercell whose transverse dimension is large enough to ensure that the whole waveguide mode is contained within. Then, the iterative procedure described before is used, with the only modification that in this case the Eq. (3) becomes

where N _l is the number of unit cells in the transverse direction and the average is taken over the supercell. A 0D defect is treated similarly. Generally, both for a pure periodic PC and for a PC with structural defects the algorithm converged in less than ten iterations; however, if the pump frequency was close to k=0, which corresponds to a region of low group velocity, the number of iterations required to reach convergence increased by about a factor of two.

3. Tunable channel drop filter

In what follows, we illustrate how the numerical method introduced in Section 2 can be used to analyze a tunable channel drop filter consisting of a resonant cavity side-coupled to a waveguide in a nonlinear PC. The linear version of this device has been recently demonstrated, both as a single- [19] as well as multi-frequency [20] operating device. We consider a 2D PC composed of a square lattice of circular rods made from nonlinear instantaneous Kerr dielectric with refractive index n=3.4, e.g., corresponding to Si or GaAs semiconductors, and Kerr coefficient n ₂=3·10^-16m²/W. For simplicity, the refractive index of the background medium is set to n=1. The structural parameter of the PC is r/a=0.18, where r and a are the rod radius and the lattice pitch, respectively. Note that the results presented here can be extended to PC slab waveguides, but the numerical analysis becomes much more computationally demanding. The waveguide PC is obtained by removing one row of rods, whereas the resonant cavity by removing one rod (see Fig. 1). The waveguide mode dispersion, calculated for the linear PC, as well as the corresponding resonant frequency of the cavity are shown in Fig. 2. Also shown in Fig. 2 is the result of the finite-difference time-domain (FDTD) simulation of the resonant CW excitation of the cavity side-coupled to the waveguide.

The waveguide mode dispersion curve in Fig. 2 was determined by using the supercell approach discussed in Section 2, the size of the supercell being N_l ×1, with N_l =9. In the PWE, a number of 1024 plane waves was used. In the case of the cavity mode, an N_p ×N_p , with N_p =5, supercell was used, the number of plane waves in the PWE being 4096. We analyzed three cases, in which the resonant cavity was placed one, two, and three rows apart from the waveguide PC. The values of the quality factors Q that correspond to these cases are illustrated in Fig. 3, and were computed by using FDTD calculations. This figure shows that as the cavity is placed closer to the waveguide PC, Q decreases, due to increased coupling losses. Also, the resonant frequency ω_r increases, which is explained by a slight decrease in the effective refractive index of the cavity mode. Finally, as the distance between the cavity and the waveguide PC changes from two to three rows there is hardly any change in Q, a behavior that is explained by the fact that the overlap between the cavity and waveguide modes remains almost unchanged.

 

Fig. 2. Left, mode dispersion of the waveguide PC (red line), at P=0; the blue line corresponds to the resonant frequency of the cavity, also at P=0. Right, FDTD simulation of propagation of a low power CW, at ${\omega }_r^0$ , in the waveguide PC side-coupled to the cavity.

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Fig. 3. Spectral densities corresponding to resonant cavities placed one row (Q ₁), two rows (Q ₂), and three rows (Q ₃) from the waveguide PC.

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Next, we consider addition of a pump beam, and, in particular, that a mode with optical power P and frequency ω_p propagates in the waveguide PC. As a result, the refractive index in the region covered by the waveguide mode changes due to the induced Kerr effect. This change leads to a variation of the coupling between the cavity and the waveguide, as well as the resonant frequency ω_r . To describe quantitatively these effects, we used the numerical method previously introduced and proceeded as follows. We first numerically determined the waveguide mode corresponding to a given power P and then, from the field distribution, we computed the change in the refractive index in the region of the cavity. Then, by using this distribution of the refractive index in the spatial domain occupied by the cavity, we computed the corresponding resonant frequency ω_r . Note that in these calculations we considered that the pump frequency ω_p is far from the resonant frequency ω_r , and therefore the waveguide mode is not affected by the presence of the cavity (very close to the resonance this assumption no longer holds). Moreover, as the grids of the supercells used to compute the waveguide and cavity modes do not coincide, we used a bilinear interpolation procedure to determine the latter from the former one. These calculations were repeated for different pump powers P, for several pump frequencies ω_p . The results we obtained are summarized in Fig. 4. The waveguide mode dispersion, calculated at pump powers P=400W/µm and P=900W/µmand pump frequencies (expressed in normalized units of c/2π a) ω_p =0.33 and ω_p =0.35, respectively, are shown in Fig. 4(a). Note that, although in the latter case the power P is more than twice as much as the power in the former case, the mode frequency shift is almost the same. This behavior is explained by the reduced group velocity at ω_p =0.33 (0.295c at ω_p =0.33 as compared to 0.426c at ω_p =0.35). Thus, as Eq. (4) suggests, for a certain power P the electric field in the waveguide mode is inverse proportional to the group velocity, so that waveguide modes that correspond to lower group velocities induce stronger nonlinear effects. The dependence of the resonant frequency ω_r on the power P, calculated for the case in which the cavity is placed one and two rows apart from the waveguide, is illustrated in Fig. 4(b). Regarding this figure, we mention that at ω_p =0.33 we investigated a smaller range of powers P, because in this case at large powers the numerical algorithm converges at a slower rate; again, this is due to the reduced group velocity. Notice that at low powers, if the cavity is placed closer to the waveguide PC, ω_r shifts to higher values, which is in agreement with the behavior seen in Fig. 3. In addition, if the cavity is closer to the waveguide PC, its resonant frequency ω_r depends stronger on the power P. Furthermore, Fig. 4(b) shows that at a given power the shift in ω_r is much larger if the cavity is closer to the waveguide, a consequence of an increased variation of the refractive index in the region covered by the cavity mode.

 

Fig. 4. Mode dispersion curves of the waveguide PC, determined at different pump power P and pump frequency ω_p (a); in (b), resonant frequency ω_r vs. power P, determined for ω_p =0.33 (—) and ω_p =0.35 (-·-·-). The cavity is placed one row (red curves) and two rows (blue curves) apart from the waveguide PC.

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4. Discussion and conclusions

In conclusion, we have introduced a numerical method that can be used to compute the frequencies and the corresponding modes associated with 1D and 0D structural defects in nonlinear PCs. The method was applied to characterize the optical response of a device consisting of a resonant cavity side-coupled to a waveguide, both embedded in a nonlinear PC. Furthermore, note that the optical power required to induce a specific variation in the operating frequency of the device can be reduced if materials with large Kerr nonlinearity, e.g. InSb or ZnSe-based semiconductors or polymers are used. Finally, instead of the waveguide PC discussed here, one can use waveguide PCs whose mode dispersion is specially engineered, so as to have an extremely low group velocity. For instance, using coupled-resonator optical waveguides [21], one can achieve group velocities less than 0.01c, and thus a large decrease in the operating power.