1. Introduction

Due to promising new devices and applications, there has recently been a growing number of theoretical and experimental activities related to photonic crystals. Photonic crystals are periodic arrangements of dielectric scatterers. A substantial property of these structures is the appearance of band-gaps, i.e. a frequency range exists within which wave propagation is exponentially attenuated and waves directed onto the crystal are reflected. For an overview see e.g. [1,2].

In general, the transmission of light through a photonic crystal depends on the geometry and the index of refraction of the dielectric materials. If one seeks to dynamically control the optical signal, photonic crystal structures then offer two possibilities: The first one is to change the geometry of the band-gap structure e.g. by means of piezo-active materials. A second possibility is the variation of the index of refraction by means of a nonlinear photonic material and a strong pump pulse [4,3] (optical switching). Focusing on the latter, we will in the following analyze the dynamics of the optical switching process in a photonic band-gap switching device. It consists of a sequence of dielectric layers with alternating indices of refraction which - via a Kerr nonlinearity -nonlinearly depend on the light intensity. The transmission of light through the structure is illustrated in Fig. 1. A probe beam whose frequency is chosen to be outside the band-gap close to the band edge will then pass the multilayer structure. However, when we change the dielectric constant ϵ_h , the transmission curve is shifted and the probe beam no longer passes the structure but instead is nearly completely reflected.

 

Figure 1. Transmission versus frequency for a one-dimensional multilayer structure. The multilayer structure consists of 29 layers with a low dielectric constant ϵ_l of 1.0 and a high dielectric constant ϵ_h of 2.0 (green curve) and 2.1 (blue curve), respectively. The width of one layer is d = λ ₀/(4 ∙ n̄), where n̄ = 1.21 is the averaged index of refraction of two consecutive layers. The arrow indicates the frequency f_probe = 0.8745 ∙ c/λ ₀ of the probe beam, where c is the speed of light.

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In our set-up, the change of ϵ_h is realized by means of a second sufficiently strong pump pulse. The principle of this optical switching has been demonstrated by Scalora and Tran on the basis of a one-dimensional (1D) model. There, both the probe and the pump pulse are propagating in the same direction such that the strong pump pulse overlays the probe beam. Here, we present an alternative set-up with a cross-geometry and take into account the propagation and dynamic two-dimensional (2D) light field variations. The underlying configuration is shown in Fig. 2. The photonic switch consists of two crossed waveguides with a nonlinear photonic band-gap structure in the center. This cross geometry represents in comparison to the one-dimensional arrangement a more realistic configuration for an optical switch. In particular, the pump beam propagates perpendicular to the direction of the probe beam. In our cross-waveguide optical switch the overlay of the probe beam by the pump beam is significantly reduced - a fact which may become important in potential technical applications of all-optical switching devices. Note that the multilayer is structured periodicly only in the direction of the probe beam and there is no further restriction with respect to the frequency of the pump pulse.

 

Figure 2. Schematic representation of the geometry of the nonlinear photonic band-gap switch. On the left, there is the inlet of the probe beam, on the bottom the inlet of the pump pulse. The boundaries are perfectly conducting. Besides the finite extension in y-direction, the multilayer structure in the center corresponds to the multilayer structure described in Fig. 1. In addition, we have a Kerr nonlinearity of χ ₃ = 0.001 in the layers with the high dielectric constant (blue stripes).

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Presently, most of the numerical methods for the simulation of photonic chrystals are based either on a treatment of the field equations in the (ω, $\overrightarrow{k}$ ) domain or in the (ω,$\overrightarrow{r}$ ) domain [2,5]. Another possibility is the use of the slowly varying envelope approximation [4]. All these methods are based on a fixed frequency. When nonlinearity and multiple frequencies are incorporated, these methods quickly become rather complex. To overcome these difficulties, we directly solve Maxwell’s Equations by numerical integration in the (t, $\overrightarrow{r}$ ) domain. The direct integration strategy has previously been used with nonlinear photonic crystals by Tran in the case of one spatial dimension and by Reineix in a linear 2D study [6]. To model the photonic switching the method has been extended to incorporate the 2D nonlinear spatio-temporal variations of the transmission properties of the photonic chrystal structure. All simulations in this paper are based on this method. To obtain results in the Fourier domain such as Fig. 1, a Fourier decomposition is performed on the simulated data in the time domain.

2. Basic equations and numerical method

The propagation of electromagnetic waves in periodically structured dielectric media may be generally described by Maxwell’s Equations. Restricting ourselves to the TM-polarization¹, Maxwell’s equations in the two dimensional (2D) x - y plane reduce to²

and the divergence relation

where c is the speed of light and x, y, z label the Cartesian components of the fields. The interaction between the optical field and the nonlinear dielectric medium is modeled by a Kerr-type nonlinearity

The spatially varying dielectric constant ϵ = ϵ(x, y), the Kerr nonlinearity χ ₃ = χ ₃(x, y), and the nonlinear dielectric function ϵ_nl = ϵ_nl (x, y, E_z ) = ϵ(x, y) + χ ₃(x, y) ∙ $E_z^2$(x, y, t) represent the nonlinear photonic band-gap structure depicted in Fig. 2. Note that we have assumed possible dispersive effects to have no critical influence on the switching effects studied here. Further, we neglect any magnetic response in the material, i.e. B = H. Assuming the boundaries of the waveguides to be perfectly conducting leads to $\overrightarrow{n}$ × $\overrightarrow{E}$ = 0 and $\overrightarrow{n}$ ∙ $\overrightarrow{B}$ = 0, where $\overrightarrow{n}$ is a vector normal to the boundaries. Then the energy density of the fields reads

Equations (1)–(3) are solved by means of a finite volume solver [7]. The procedure is based on casting the equations in conservative form

which after integration in space and time results in

with an arbitrary domain G and $\overrightarrow{V}$ = 1/c ∙ (D_z , B_x , B_y ), $\overrightarrow{F}$ _x = (-H_y ,0,-E_z ) and $\overrightarrow{F}$ _y = (H_x , E_z ,0).

A discretization of the equations is done by decomposing the domain of interest into a sufficiently large number of subdomains. The field is approximated in any of the subdomains by means of suitable functions, e.g. polynoms. Using Eq. (7), one can calculate the field values $\overrightarrow{V}$ _n+1 at time t = t _n+1 = t_n + Δt from the existing values $\overrightarrow{V}$ _n at time t = t_n . We note that our numerical procedure is similar to the frequently used method of time domain finite differences [8]. It is, however, superior in handling distorted meshs.

3. Dynamic optical switching

In the waveguides of the photonic switch depicted in Fig. 2, light can generally propagate in various modes, see e.g. [9]. In our simulations we take the dominant TM ₁ modes (or superpositions of these modes) into account. The explicit functional form of these modes is described in the appendix. Besides the finite extension in y-direction, the multilayer sequence is periodically structured only along the x-direction. Thus, a photonic band-gap does not exist for all directions of the wave vector $\overrightarrow{k}$ = (k_x ,k_y ) of the incident probe beam. A well pronounced band gap is easily obtained, however, by choosing k_y sufficiently small with respect to k_x ³.

Before considering the more complicated situation of the dynamic nonlinear switching process, we first simulate the propagation of the cw probe beam in the waveguides and through the photonic band-gap structure (without application of the switching pulse). The frequency of the probe beam is adjusted to f = f_probe , marked by the arrow in Fig. 1, and the amplitude A_probe is set to 1.0. The resulting energy density distributions are shown in Fig. 3 and Fig. 4 for eh = 2.0 and ϵ_h = 2.1, respectively. As expected, the beam passes the multilayer structure in the case of ϵ_h = 2.0 and is exponentially attenuated in the case of ϵ_h = 2.1. In addition, the multilayer structure also serves as an effective waveguide in the center of the photonic switch where the waveguides are absent: In Fig. 3 and Fig. 4 the portion of the intensity which is being scattered towards the top and the bottom only amounts to ≈ 0.1%.

 

Figure 3. Snapshot of the probe beam with ϵ_h = 2.0. The frequency is adjusted to f = f_probe .

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Figure 4. Snapshot of the probe beam with ϵ_h = 2.1. The frequency is adjusted to f = f_probe .

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While in Fig. 4 we have artificially changed ϵ_h , the change of the dielectric properties is now provided by a sufficiently strong pump pulse. For specifity, we assume a Gaussian shaped superposition of the basic TM ₁ modes (see appendix) resulting in a pulse whose intensity smoothly rises and falls. The frequency of the pulse is chosen as f_pump = 0.483 and its width amounts σ = 5.0 ∙ T_pump , where T_pump = 1/f_pump . Its amplitude is set to A_pump = 14.0. In the absence of the probe beam one obtains the results illustrated as snapshots in Fig. 5 and Fig. 6 which show the spatial distribution of the pump pulse as it passes the multilayer structure and the distribution of the induced nonlinear refractive index ϵ_nl , respectively. Besides a small portion which is reflected, the main intensity of the pulse passes the photonic switch. In Fig. 5 one can clearly see that in the photonic band gap structure most of its energy is spatially confined to the center, resembling the initial shape of the incident pulse. As a consequence, the region with a strong induced change of the (nonlinear) dielectric constant ϵ_nl is thus also located at the center of the switch. In addition, the distribution of ϵ_nl is wave shaped, according to the wave length of the pump pulse. In passing we note that this wave length differs between the waveguide and the area of the photonic band-gap structure. Hence we have a dynamically induced 2D photonic band-gap structure. Note further that due to the transverse shape of the pump pulse the number of dielectric layers which will effectively contribute to a switching process is reduced in comparison to the static change of the dielectric constant assumed in Fig. 4. In Fig. 5 the loss due to scattering of the pump pulse in the waveguide amounts to ≈ 0.4% with respect to the incident pump pulse.

 

Figure 5. Snapshot of the pump pulse at the time when its maximum passes the nonlinear photonic band-gap structure.

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Figure 6. Snapshot of the induced nonlinear dielectric constant ϵ_nl .

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In order to study the dynamic optical switching process, we now apply both the cw probe beam and the pump pulse of Fig.3 and Fig. 5. The dynamics of the resulting optical switching process is shown in the animation below (Fig. 7). The upper left corner of Fig. 7 shows the spatio-temporal variation of the probe beam. Note that for an improved visualization of the switching effects a frequency filter is applied at the frequency of the pump pulse. The effects of the pump pulse are represented in the upper right corner, which shows the dynamics of the dielectric constant ϵ_nl . The time scale is given in units of [λ ₀/c]. For optical wave lengths, this corresponds to 10^-15 to 10^-14 seconds.

At time t = 0, the intensity of the pump pulse has not yet reached its maximum, and the change of the ϵ_nl is small. Hence, the probe beam passes the photonic switch. The animation also shows that the multilayer structure also causes a resonator effect reminiscent of a distributed feedback arrangement used e.g. in semiconductor lasers. With increasing time the intensity of the pump pulse rises and ϵ_nl increases. As a consequence, the band edge is dynamically shifted such that the propagation of the probe beam is significantly disturbed. At t = 10 the maximum of the pump pulse is reached an the transmitted intensity of the probe beam is continously reduced until t ≈ 20. In the follwing, with the pump pulse having passed the structure, the probe beam returns to his initial intensity.

 

Figure 7. Animation of the switching process. The animation at the left shows the spatio-temporal variation of the energy density of the probe beam. The sequence at the right visualizes the dynamically changing spatial distribution of the (nonlinear) dielectric constant. [Media 1]

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4. Conclusion

We have presented a numerical simulation of a two dimensional photonic band-gap switch. Our simulations are based on a full vector analysis of Maxwell’s Equations in the time domain using a finite volume time domain integration method which we found to reliably produce results for 2D photonic chrystals. Our nonlinear 2D photonic band-gap switch consists of two crossed waveguides with a nonlinear photonic band-gap structure in the overlap region. With this configuration an effective all optical switching of a cw beam by an optical pulse is demonstrated. Due to the perpendicular directions of propagation of the probe beam and the pump pulse the undesirable overlay of the two signals is strongly reduced in our nonlinear 2D photonic switch.