1. Introduction

In the path towards micro-scale photonic integration, photonic crystals (PhCs) arise as a very promising technology as they allow control of lightwave propagation on a very small scale. All-optical switches using nonlinear 2×2 directional couplers fabricated by means of many different kinds of technologies have already been demonstrated [1–3]. However, large coupling regions and high optical powers are needed to achieve switching when using these technologies. A drastic reduction of both the size and the optical power consumption of conventional photonic devices based on nonlinear effects may be feasible due to the strong confinement of the electromagnetic field and the low group velocity of the guided modes in PhC structures, as outlined in [4,5].

In this paper, a novel all-optical switching structure based on a PhC directional coupler created by placing in proximity two parallel PhC waveguides is proposed and discussed. In a directional coupler, light confined into one of the waveguides jumps to the other waveguide after propagating a distance known as coupling length L_c owing to the different propagation constants of the even and odd modes of the coupler. The coupling length is directly related to the coupling coefficient (κ) that characterizes the coupling strength of the device, so a larger coupling coefficient translates to a shorter coupling length. The value of κ can be tuned properly by changing the radius of the central row of rods between the waveguides, or more generally, by changing the effective index of refraction of the region between waveguides, as it was previously stated in [6]. This property can be used to switch an optical signal between the two outputs of the coupler by modifying optically the region between waveguides. The simulation results show that the all-optical switch may be implemented by means of a PhC coupler with a length of approximately 1.1 mm and it can be controlled by an optical signal with an optical power of 1.56 W.

2. Principle of operation

The structure shown in Fig. 1 is considered to analyze the performance of the proposed switch. It consists of a two-dimensional (2D) hexagonal pattern of high refractive-index rods (n_H=3.46) with radius r=0.2a, a being the lattice constant, embedded in a low refractive-index material (n_L=1.45) [6]. It is assumed that the high-index dielectric rods have a not negligible nonlinear Kerr coefficient. For this structure a bandgap opens from 0.26387 to 0.35987 in [a/λ] units for TM modes. The directional coupler is created by removing two parallel rows of rods in the ΓK direction that act as waveguides and are separated by a single row of defects with a reduced radius r_c=0.7r, as depicted in Fig. 1. This row of reduced radius r_c rods is another waveguide that can be used for the control signal of the device as will be shown below. It may be easy to find many alternative structures with different configurations of waveguides and materials that may work as well as a switch with the same mechanism that is explained below so this is just an instance of a possible configuration.

 

Fig. 1. Schematic of the PhC directional coupler used as switching structure.

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The band diagram of the PhC coupler obtained using a plane wave expansion (PWE) method is shown in Fig. 2. It can be observed that three different guided modes appear in the bandgap. The modes in the higher frequency range have even and odd symmetries with respect to the plane equidistant from the axis of the waveguides and correspond to the supermodes of the coupler. When an input port of the coupler is excited, both the even and the odd modes propagate and there is a periodic transfer of power between waveguides. Figure 3(a) shows the electric field distribution inside the PhC coupler, at the normalized frequency of 0.3281, obtained by means of a finite-difference time-domain (FDTD) method when input I ₁ is excited in the linear regime. It can be seen that, for this length (L) of the directional coupler, the coupler is in the cross state, although it could also be designed to be in the bar state.

When a nonlinear regime is imposed by a specifically designed power level in the control signal (that excites the control mode corresponding to the one in the low frequency region of the bandgap), the state of the directional coupler may be inverted from the cross to the bar state (as depicted in Fig. 3(b)); or vice versa, if the coupler were in the bar state in the linear regime it may switch to the cross state in the nonlinear regime. Thus, this mode is used to control the performance of the switch and operates in a different range of the frequencies of the coupled modes. This control signal has even symmetry and it is strongly confined in the central row of rods, as it can be deduced from its electric field distribution obtained using FDTD and shown in Fig. 3(c) for the normalized frequency of 0.28. Therefore, the central row of rods that separates the waveguides in the coupling region behaves as another waveguide and it is used to steer the control signal of the switch.

 

Fig. 2. Band diagram of the PhC coupler: control signal (solid curve), coupler even mode (dashed curve) and odd coupler mode (dotted curve). Insets: transverse pattern of the modulus of the electric field amplitudes.

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Fig. 3. FDTD electric field distribution inside the coupler f=0.3281 [a/λ], (a) in the linear regime (1.3 MB movie) and (b) in the non-linear regime (1.6 MB movie); (c) control signal used to tune κ, f=0.285 [a/λ].

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This PhC coupler is very different from couplers implemented by using conventional index-guided optical waveguides. The mechanism that explains the switching behavior of this directional coupler is the following: the refractive index of the row of rods between both waveguides changes owing to the Kerr nonlinearity when a high power level signal is launched as a control signal, thus it becomes possible both to tune L and vary the power ratio between the PhC coupler outputs. If the length of the coupler L and the peak power of the control signal P are designed properly, the response of the directional coupler can be reversed. That is, if the coupler is in the cross state in the linear regime as in Fig. 3(a) (low level of the control signal) at a given frequency, it will switch to the bar state (Fig. 3(b)) because of the effect of the high optical power of the control signal in the index of refraction between the arms of the coupler. Thus a PhC switch controlled by the optical signal that propagates through the central row of rods in a different range of frequencies is achieved. For the sake of simplicity in terms of computational cost, it should be pointed out that the results shown in Fig. 3(b) have been obtained using an overestimated nonlinear refractive index change (δn=0.05n_H) in the central row of rods. It can be seen in Fig. 3(b) that an exchange of light power between the arms of the coupler still appears instead of being inhibited as in conventional nonlinear directional couplers. The number of jumps per unit of length decreases due to the reduction in the coupling strength.

Furthermore, it should also be noticed that the lowest frequency mode of the structure used as control signal is strongly confined in the rods between waveguides (see Fig. 3(c)) and it has a flat dispersion relation, and therefore, a low group velocity near the band edge (see Fig. 2). Owing to these two reasons, nonlinear effects are expected to be very strong in the high-index region between waveguides, even if a relatively low optical power is injected [4].

3. Performance analysis

Apart from the PWE method mentioned in the preceding section, a nonlinear computational scheme based on the finite element method (FEM) [7] developed by the authors has been used to analyze the performance of the proposed PhC switch. The FEM is a well known numerical technique that allows obtaining approximate solutions to boundary-value problems of mathematical physics. The region under study is divided into small elements, triangles of three nodes, in which the Maxwell equations and periodic boundary conditions are satisfied. The use of a mesh of triangles results in a high flexibility to model the contours of the geometry, in this case the high-index cylinders of the PhC pattern. Moreover, each of the elements has a dielectric constant independent of the others so material discontinuities can be easily addressed. Another of the advantages of the FEM is that it uses sparse matrices during the calculations, so its computational cost in memory and processor scales linearly with the number of nodes. Finally, the main reason to use FEM is that the dielectric constant in each element does not need to be constant and it may have a linear change in the refractive index. This feature has been used to implement a nonlinear iterative method (reported in [8]) in order to simulate the nonlinear behavior of the structure and the nonlinear dispersion diagram has been calculated [9]. The refractive index of the rods between waveguides is modulated with the field pattern of the control signal owing the Kerr nonlinearity.

When the control signal is applied, the refractive index of the rods between waveguides increases and the even mode shifts slightly to lower frequencies, whereas the odd mode remains almost unchanged, so κ diminishes a quantity Δκ. If the condition Δκ·L=π/2 is satisfied, light switches from one output to the other.

In order to illustrate the performance of the directional coupler as a switch some nonlinear calculations have been carried. Figure 4(a) shows the coupler length L required to switch the outputs as a function of the operational frequencies of a data signal that propagates through the device. An optical power of 1.56 W has been assumed for the control signal, as discussed in the next paragraph. For instance, if a normalized frequency of 0.3281 is considered for the data signal, the length required to switch the device is L≈1.1 mm for a period a=511.5 nm. The bandwidth of the switched data signal for the calculated L is approximately 0.5 nm FWHM, as it can be seen in Fig. 4(b) where the bandwidth as a function of the central frequency of the data signal in normalized units is presented. The chosen normalized frequency gives a wide enough working bandwidth for the data signal. Smaller lengths for the device, keeping constant the optical power of the control signal, may be achieved at the expense of decreasing the available bandwidth of the data signal that can be switched. Studies of the insertion losses or the extinction ratio of this structure have not been performed exhaustively but the scenario does not change considerably from that presented in a previous publication [6].

These nonlinear calculations have been performed assuming that the control signal at the normalized frequency of a/λ≈0.285 induces a refraction index increment of δn=0.0012n_H that causes a nonlinear cross-phase modulation of the data signal. The change in the refraction index is obtained by the expression |δn=2·1.5·n ₂·P/A·(v_g|^u/v _g|^c), where (v _g|^u/v _g|^c) is the relation of the group velocity in a conventional axially uniform waveguide (v _g|^u) and that of the center waveguide of the switch (v _g|^c). The factor of 2 is introduced because cross-phase modulation induces an index change twice as strong as self-phase modulation [10] and the factor 1.5 because the longitudinal confinement of the mode is not uniform (as may be seen in Fig. 3(c)). If the nonlinear Kerr coefficient is n ₂=1.5×10^-13 cm²/W and assuming an effective area of the mode of A _ef=0.5 µm², the peak power of the control signal must be P=1.56 W. The group velocity of the control signal is estimated to be 0.03c from Fig. 5, which depicts the group velocity of the control signal as a function of frequency. It can be seen that a group velocity of the same order of magnitude can be obtained in a wide range of frequencies (3 nm) if the lattice constant of the structure is designed to center the data frequency signal at λ=1.55 µm.

 

Fig. 4. (a) Coupler length L and (b) maximum switched channel bandwidth as function of the operational frequency of the switch for a peak power of 1.56 W in the control signal. The bandwidth is calculated as the FWHM.

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Fig. 5. Group velocity of the control signal as a function of frequency. It can be seen that the control signal may have a large spectral bandwidth calculated in this example for a lattice period a=511.5 µm.

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The reduced dimensions of photonic crystal structures make them very sensitive to tolerances in the fabrication processes. The tolerances with respect to the periods of the lattice are small and usually do not affect considerably. On the other hand, tolerances in the radius of the rods are more detrimental and may induce losses as well as a shift of the frequency in the modes. Another detrimental effect characteristic of the directional couplers is that a slight error in the rods dimensions can easily change the relation of power between the outputs of the coupler. The longer the device is, the more sensitive it will become. Thus, in the linear regime light will not exit completely from one of the outputs and it will be split with different percentages between the two outputs (the power extinction ratio between output ports will not be zero or infinite). This problem can be easily solved by working with two power levels in the control signal. Hence, there is not a linear regime state but two different power states inducing two different nonlinear refractive changes. With a “low” power level the coupling strength of the device can be adjusted such that light of a given frequency is directed to one of the output ports. On the other hand, by means of a higher power level (“high” level) the coupling strength can be adjusted to reverse the behavior of the device and light of the considered frequency will exit from the other output port. Further implications of the fabrication tolerances will be taken into account in future work.

In order to check the results obtained with the nonlinear FEM method, the same calculations were performed in the same way as in [4] using a PWE method. It should be mentioned that the PWE method is linear so nonlinear effects can only be estimated by changing uniformly the refractive index of the central rods of the coupler by δn. Hence, the results obtained from this approach are more inaccurate and a bit more optimistic, as it can be seen in Fig. 6 that shows the switch length L as a function of the peak optical power of the control signal for the normalized operational frequency of 0.3281 [a/λ]. It can be observed that the necessary L is directly proportional to the inverse of the power of the control signal (L×P=constant=C). On the other hand, the bandwidth of a directional coupler is inversely proportional to its length, so the available bandwidth of the data signal switched will grow proportionally to the optical power of the control signal.

 

Fig. 6. Dependence of the switch length L as a function of the peak power of the control signal at the normalized operational frequency of 0.3281 [a/λ].

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4. Conclusion

The performance of a novel architecture for an all-optical switch based on a PhC directional coupler has been proposed and demonstrated. This structure may become a key building-block of a larger and more complex switching device. The mechanism explained in this paper may be easily extrapolated to other configurations differing in the permitivities of the materials employed or the configurations of the guiding defects. The use of PhC technology makes it possible to modify the refractive index of the region between PhC waveguides employing an strongly-confined optical signal, which is an added advantage over switching approaches based on conventional dielectric waveguide couplers. Moreover, the extreme confinement of light in small areas and the small group velocities of guided modes in PhCs enhances their sensitivity to nonlinear effects resulting in lower optical power consumption levels. The small size of the PhC switch may permit its integration in microscale photonic circuits. Results show that the switch may be implemented by means of a coupler with a length of approximately 1.1 mm and can be controlled by an optical signal with 1.56 W of peak optical power. Future work will deal with the practical implementation of the proposed switch.