In this paper, the implementation of an all-optical logic gate based on a Mach-Zehnder interferometer (MZI) configuration is addressed with underlying nonlinear slot-waveguides. In order to reduce power consumption requirements, different ring-resonator structures are introduced in the arms of the MZI. A nonlinear Transfer Matrix Method is developed and used to analyze the response of the nonlinear MZI in order to optimize power requirements with maximum bit rates. The numerical analysis shows that a reduction in the switching power from 2.5 W to less than 5 mW can be achieved by a proper design of the ring-resonator structures introduced in the MZI arms. In addition, it is shown that the logic gate can handle bit rates higher than 60 Gbit/s.
© 2007 Optical Society of America
The implementation of active optical devices whose transfer function can be dynamically controlled by external inputs is a key issue within the research of photonic integrated circuits. One of the main challenges is the implementation of optical logic gates as basic building blocks of more complex devices, since advanced optical functionalities (optical switching, routing, wavelength conversion, optical regeneration) may be easily developed by properly cascading these building blocks. In addition, all-optical technology (optical external inputs) has the great advantage over other techniques of providing an extremely high bandwidth, which would allow for the building of all-optical devices to be used in optical networks with bit rates of more than 40 Gbit/s per wavelength. The control of an optical response by an external optical signal requires the presence of a material with an adequate nonlinear response. In this work, the instantaneous Kerr effect  (nonlinear index n2), which occurs in widely used semiconductors such as GaAs, is considered. This effect is also present in a material formed by a distribution of Silicon nanocrystals (Si-nc) embedded in a silica host . This nonlinear material, which hereinafter will be referred to as Si-nc/SiO2, will be considered to be responsible for the nonlinear response that makes possible the switching of the all-optical logic gate addressed in this work.
At high optical intensity levels the nonlinear Kerr effect leads to an increase in the effective refractive index of the underlying materials. This implies an additional phase shift in the traveling signal that can be exploited to implement an optical logic gate. By employing a Mach-Zehnder interferometer (MZI) structure, the phase modulation induced by the nonlinearity can be transformed into amplitude variations. One of the constraints that may arise is the high power level required due to the small Kerr coefficients of conventional materials. Therefore, different mechanisms for confining light tightly in the nonlinear medium should be implemented. Owing to this reason, a slot waveguide is employed in order to achieve a strong confinement of light in the central nonlinear SiO2/Si-nc layer as it can be observed from the transverse field profile for TE-like polarization shown in Fig. 1 .
Figure 2 shows the two different configurations of MZI all-optical XOR logic gates that are considered in this work. Resonant structures are included in the MZI arms to enhance the nonlinear effects, as will be shown below. Light launched into the Control port is split into two halves that propagate through each of the arms of the MZI. If no pump signal is launched in the Data A/B ports, the nonlinear effects can be neglected and the phase shift undertaken by the idle signal is the same in each of the resonating structures of the MZI arms, which should be exactly identical. By making one of the MZI arms longer than the other (asymmetric MZI) the Control signal at the MZI output may be inhibited (logic “0” output) if a π phase shift between the signals is achieved at a designed wavelength. On the other hand, if a high-intensity signal is launched in one of the Data ports, the nonlinear effects cannot be further neglected, and an additional nonlinear phase shift will be induced in the Control signal in one of the MZI arms, leading to a power increase of the Control signal at the output of the device (logic “1” output). Thus, the nonlinear Kerr effect is responsible for switching the logic gate from a “0” output (in linear regime) to a “1” output. Moreover, if high-intensity signals are launched simultaneously in ports Data A and Data B, the nonlinear phase shift induced in each MZI arm will be the same so the Control signal at the output will be inhibited (logic “0” output”) as in the linear case, thus achieving the required XOR logic performance. In order to distinguish between the Control and the Data signals at the output of the device, different wavelengths are employed for each of them.
To design all-optical logic gates, first it is important to engineer the transverse cross-section of the nonlinear waveguides (in our case, the slot waveguide filled with Si-nc/SiO2) as it was done in Ref  in order to obtain high-power densities in the nonlinear material, which will be allocated inside the slot (see Fig. 1). The inverse of the field confinement can be estimated by obtaining the effective area (A eff) of the mode. Once A eff is calculated for a given configuration, the power-dependent phase shift induced by a given section (of length L) of a straight slot waveguide can be estimated as φ = 2π/λ∙(2n 2 P/A eff)/q∙L, where P is the power of the input signal and λ is the free space wavelength. The bracket in the previous expression can be understood as the nonlinear increment in the refractive index (Δn) of the material, which induces a q times smaller shift in the effective index of the propagating mode. The factor of 2 included in the expression of Δn is because we are considering the cross phase modulation (XPM) induced by a pump signal into a second signal, called idle signal. These signals are launched respectively into the Data A/B and Control ports of the logic gates shown in Fig. 2. If propagation losses are neglected, the required power to induce a nonlinear phase shift of π (switching from “0” to “1”) in a straight lossless waveguide section (MZI arm) is Pπ = q∙λA eff/4Ln2. This amount of power is still very high (2.5 Watts for a length L=500 μm) despite of the strong field confinement in the slot waveguide. Thus, the use of slow-wave or resonant structures needs to be considered as a very promising way to reduce power requirements [5–6]. In this work, the use of ring-resonator (RR) structures (see Fig. 2) is addressed in order to enhance the efficiency of nonlinear effects and reduce power requirements. Specifically, we consider two types of RR structures, which are named as Type I and Type II in Fig. 2. It should be mentioned that a similar study could be performed by cascading Fabry-Perot cavities made by partial reflectors between straight waveguides .
2. Analysis of nonlinear ring resonator structures
A ring-resonator (RR) structure, as those used in the MZI arms of the logic gates shown in Fig. 2, consists basically of a given number of RRs coupled to one or more straight waveguides used as input-output ports. Light coupled to a RR recirculates in it, leading to an increase in the effective group delay at the resonant frequencies . This is due to the increase of round trips performed by the signal in the RR before reaching the output port. Additionally, this recirculation leads to a coherent overlap of the signal launched at different instants of time, which gives rise to a big increase of the intensity of the light confined in the resonator and thus the nonlinear effects can be enhanced.
A photonic structure incorporating coupled RRs can be easily described, as shown in Fig. 3, as a set of ideal couplers and waveguides whose behavior at each frequency can be described by means of scattering matrices. A simple modeling of linear RR structures is that based on the transfer matrix method (TMM), in which the scattering matrices are transformed into transfer matrices. Therefore, it is straightforward to relate the input signals to the output signals for a given frequency . For that purpose, the effective index in the waveguide and the coupling parameters should be known. The effect of propagation losses due to the absorption of the material, the waveguide sidewall roughness or the curvature of the waveguides can also be considered in the model. To include all these effects in the modeling they should be previously estimated by using more complex modeling but in more simple structures as it is done in .
The TMM works in the frequency domain so results are obtained in the stationary regime. When pulse propagation is considered, the spectrum of the pulse is simply multiplied by the frequency response of the RR structure and, by the inverse Fourier transform, the time response can be easily obtained. This model fails if the underlying material is considered to be nonlinear because the effective index is a function of the intensity. In that case the following self-consistent iterative method, widely used in a number of numerical approaches [11–13], can be employed to obtain the transfer function denoted by H in Eq. (1). In each iteration an approximation of the real H is obtained. It can be observed that H is a function dependent on the structural parameters, the wavelength and the intensity of the signal. First, a linear simulation is performed for each wavelength with the linear effective index of the underlying waveguides (I 0=0) in order to obtain the transfer function and the optical intensity in each of the RRs. The result of the intensity at the frequency of the pump signal is employed to modify the refractive index in each of the rings adding the corresponding Δn. This change in the effective refractive index of the RR waveguides leads to a shift of resonances so the spectral behaviour and the intensity distribution must be recalculated for the whole structure. Thus, another linear simulation is performed in order to obtain the new distribution of intensity in each of the rings. This process is iteratively repeated and finishes when the calculated refractive index of two successive iterations hardly changes.
This iterative method converges to the solution if the nonlinearity is sufficiently weak. After each iteration and in order to improve the convergence, the new refractive index is calculated as a weighted sum of the old value and the new theoretical in order to slow down the refractive index change between iterations. Although this mechanism increases the convergence time, it permits the algorithm to converge for higher nonlinear variations.
In order to check the validity of the frequency-domain nonlinear TMM described above, a rigorous time-domain algorithm has been developed . This method is based on Eq. (2), which stands for the impulse response of a nonlinear RR coupled to a waveguide. The phase shift θ that undertakes the wave in one round trip is dependent on the power of the input signal in nonlinear simulations. The parameters t and k correspond to the transmission and coupling coefficients of the coupler and TR is the transit time across the RR.
The implementation of this time-domain method is quite useful to study pulse propagation through RR structures but it may become unfeasible when the complexity of the considered structure increases, especially when nonlinear RRs of different sizes or optical properties are considered. This is due to the fact that variation in the transit time TR in each of the RR should be taken into account for a better accuracy in the results. For the sake of comparison, Fig. 4 shows the nonlinear phase shift as a function of the input optical power due to self-phase modulation obtained by applying both numerical methods (TMM and time-domain) to a simple structure consisting of a single nonlinear RR side-coupled to a straight waveguide. It can be clearly appreciated that both algorithms lead to the same results. It can also be observed that the introduction of a simple RR reduces the power requirements estimated in the preceding section as it was already studied in Ref . In the following sections the nonlinear TMM will be employed for more complex structures and the effect considered will be XPM. Additionally, the nonlinear TMM will be used to study the dispersive effects that determine the speed of the structure. Thus this nonlinear TMM method can be employed for the optimization of both, the power consumption and the maximum bit rates of the MZI logic gate. As it was mentioned above, previous estimations of the propagation parameters and the effective area should be performed. This is done at the beginning of the following section.
3. Design of the nonlinear RR structures to optimize the logic gate performance
3.1 Description of the structure.
This work focuses on the design of the RR-based structures responsible for the enhancement of the nonlinear effects taking place in the MZI-based logic gate depicted in Fig. 2. In the Type I structure, the arms of the MZI behave as an all-pass filter  that induces an extra phase delay in the transmitted signal when it is tuned at the resonance frequencies of the RRs whereas in Type II structure only the resonant frequencies can be transmitted. In the following, it is considered that the employed RRs are formed by a slot-waveguide filled with Si-nc in a silica matrix (see Fig. 1) with a radius of 20 μm. The free spectrum range, FSR, from one resonant wavelength to another is estimated to be 10.8 nm. This will be the spacing between the Data and Control signal used in the structure. Smaller radius, down to 2.5 μm, could be chosen using a special bend design  and would lead to an increase of the FSR between resonances and also the effective area might need to be recalculated. The chosen value is considered large enough to neglect the bend effects on the propagation mode. Anyway, if a bigger RR radius is desired the calculation procedure described below is still valid but results may differ from the ones shown due to the increase of the losses induced in the signal in each round trip.
It is assumed that the nonlinear Kerr coefficient of the low-index Si-nc/SiO2 material inside the slot is n2 = 1×10-12 cm2/W . The change in the effective refractive index of the propagating mode is calculated numerically to be q = 3.2 times smaller than the Δn. The slot waveguide can be properly engineered to have an effective area as small as 0.1 μm2  for TE-like polarization, owing to the strong confinement of the field inside the slot region. Hence, a power Pπ ≅ 9.87 W would be required to achieve a nonlinear π phase shift at λ = 1.55 μm in a lossless straight slot-waveguide section whose length is equal to one round trip in the RR. If the slot waveguide design chosen is not optimum and the real effective area differs (A′efff), all the power estimations shown in this paper should be multiplied by A′eff /0.1.
3.2 Results with one RR.
Figures 5, 6 and 7 show the results obtained with the nonlinear TMM for the simplest case in which only one RR is used for both Type I and Type II structures. The coupling of the RRs to the output ports has been designed in both structures in order to have a resonant intensity inside the RR that is 20 times higher than that launched in the input waveguides (k = 0.425 for Type I and k 1 = k 2 = 0.218 for Type II structure).
In the particular case of a pump signal (Data A/B) centered at the resonance wavelength of the RR (blue line in Figs. 5–7) the nonlinear effects increase very quickly because of the large build-up of intensity due to the recirculation of light. The nonlinear effects lead to a shift of the resonance frequency of the RR towards longer wavelengths due to the nonlinear increase in the dielectric constant of the underlying material. Then the recirculation of the pumping light decreases and the enhancement of the nonlinearity diminishes giving rise to the saturation of the nonlinear effect induced in the idle (Control) signal as it is shown in Fig. 5. It should be remarked that the saturation shown in Fig. 5 is due to effect of the wavelength shift of the RR response on the Data and Control signal. If the input light of the Data A/B signal is red detuned (red line in Figs. 5–7), it has initially a smaller enhancement of the nonlinear effects at low power levels but when the frequency response shifts the enhancement increases considerably. This fact gives rise to larger changes in the nonlinear phase shift induced in the Control signal before the saturation. In these examples, the Control signal wavelength is separated from the Data signal wavelength by a FSR.
The variation of the transmission intensity of the Control signal as a function of the input Data power, shown in Fig. 6, must also be considered although previous studies usually neglect it. Type I structure is ideally an all-pass filter but when a lossy RR is considered it induces larger losses at the resonant frequency, when light recirculates in the RR, than out of resonance. This is a very interesting behavior, since at the nonlinear regime, where maximum transmission is desired, the RR losses can be avoided due to the nonlinear detuning of the wavelength. On the other hand, Type II structure has no transmission out of the resonant frequency and induces a similar amount of losses in it. Thus, as it is shown in the overall transmission results (ratio between the power of the Control power at the output port and at the input port, P c-out/P c-in) depicted in Fig. 7, the performance of the Type I structure is clearly advantageous over the Type II structure and a transmission of 100π for only 0.2 W power in the Data signal could be achieved in the absence of losses. It must be observed that the output of the MZI structure has been designed to cancel at low power levels (linear regime) by making one MZI arm slightly longer than the other. Thus, the extinction relation between the linear and nonlinear regimes is clearly maximized and it would be theoretically infinite in the absence of noise.
As shown in Fig. 7, the transmission as a function of the Data power is also dependent on the input Data signal wavelength. Therefore, Fig. 8 shows a map of the transmission in the Control signal as a function of the power and the wavelength of the Data signal. The wavelength of the Control signal considered for each value of Data power and wavelength is the one with maximum transmission from all the wavelengths around the central resonance of the RR. This kind of maps will be employed in the following section in order to optimize the RR resonator structures for minimum input power requirements. It can be observed that the wavelengths longer than the resonant wavelength induce the maximum transmission with smaller power requirements and it is in this region where bistabilities may arise.
To evaluate properly the performance of the logic gates it is also important to estimate their bandwidth and maximum allowable speed. Previous studies conclude that the thinner is the resonance the higher is the nonlinear enhancement . Hence, a trade-off arises between the bandwidth of the device and the nonlinear enhancement. Additionally, it must be remarked that, when the nonlinear TMM is employed in this work, it is assumed that the Data and Control signals have a sufficiently thin spectrum. Therefore the bandwidth of the input light must be smaller than the transmission bandwidth of the RR structure . For short pulses and high bit rates it may not be true so estimations of the minimum pulse duration and the maximum bit rate must be considered.
In the particular case of Type I structure, which behaves as an all-pass filter, the bandwidth should be defined in a different way. For that purpose it is important to study the time delay induced by the RR structure in the input signal as a function of wavelength. Thus, the maximum spectral full-width at half-maximum (FWHM) of the input pulse should be smaller than the FWHM of the time-delay plot as a function of wavelength to avoid distortion. This estimation of bandwidth is completely equivalent to the estimation of the FWHM of the transmission response in structures of Type II. The bandwidths of the structures considered above are 0.3 nm for the Type I structure and 0.17 nm for Type II. It should be noted that the smaller bandwidth of the second structure is due to a smaller coupling coefficient. Comparing both types of structures, the output waveguide in Type II structure is an additional loss path for the signal recirculating inside the RR and therefore smaller coupling coefficients are needed to obtain the same amount of intensity build-up. This leads to thinner resonances and the bandwidth of the structure decreases reducing at the same time the maximum speed of the device.
Once it has been checked that the pulse spectrum width, (independently if it is a RZ or NRZ signal) is smaller than the bandwidth of the structure, it should be taken into account the dispersive effects of the structure in order to evaluate its maximum bit rate. Once again, the time delay as a function of frequency is employed as it is shown in Eq. (3). The dispersion-induced time spreading of a Gaussian pulse traveling along the RR structure can be evaluated by use of the following expression where second and third order of dispersion is considered :
where σ and σ0 are the RMS time width of the input and output Gaussian pulses, respectively, τd is the time delay, ω is the radian frequency and finally B is the bit rate.
As a result the maximum bit rate that can be achieved with the previously studied structures is 22 Gbit/s for Type I structure and 16 Gbit/s for Type II. These results can be further improved as it is done in the following section for Type I structures. Type II structures are no further considered because are disadvantageous in the transmission and the speed as it has been shown above.
3.3 Optimized Design of Type I structures with a single RR.
The coupling coefficients of the RR to the side waveguide can be optimized in order to achieve high-speed devices with a small switching power. At low power levels the transmission is theoretically cancelled if the arms of the MZI are properly designed. Thus only noise would be measured in an experimental set-up. When a data signal is launched into the device the transmission T=P c-out/P c-in of the Control signal increases with the input Data power as it is shown in Fig. 7 until it reaches a saturation point where T increases no more. It can be observed that there are curves of T in which initially the slope is high but the final value of T is small and vice versa. Thus, we need to establish some criterion that allows us to obtain the structure that provides the best performance for the logic gate.
We will make use of some typical values that we have obtained when characterizing photonic circuits in an experimental set-up in our lab. For instance, we assume that insertion losses from input to output fibers are 19 dB. The optical noise measured at the output of the circuit is considered to be -40 dBm and the power of the Control signal launched into the circuit is 0 dBm. In order to differentiate between the “0” and “1” logic states at the gate output, an extinction ratio of 10 dB may be considered high enough. Thus, to achieve the switching from the “0” to the “1” state the power of the Control signal collected by the output fiber in the experimental set-up should be of -30 dBm. This means that the transmission T of the Control signal should reach a value of -11 dB to switch the device. The configuration that needs a smaller value of the Data signal power to reach T = -11 dB will be the optimum one from a power consumption point of view. The constraints imposed by the considered experimental set-up may be further relaxed with proper coupling structures that reduce the input-output coupling losses to the fiber but it is out of the scope of this study.
Table 1 shows the transmission T (due to the discretization of the numerical method, the values of T are a bit higher than the target -11 dB considered above) as well as the required power P of the Data signal for a certain wavelength at which P gets its minimum. These values have been obtained from the calculation of maps relating the wavelength, the Data signal power and the transmission T such as those shown in Fig. 8 for different values of the coupling parameter k for a type I structure made with only one RR. It is important to highlight the dependence of the optimum coupling coefficient with the propagation losses induced in the RR. Therefore, a different optimum design should be implemented depending on propagation losses.
For instance, let us assume that propagation losses in our RR are of 20 dB/cm. Then, a coupling constant k between 0.22 and 0.56 can be selected, since for this range of values the power required to switch the logic gate is below 10 mW, which is a quite low value. The higher the coupling constant is, the bigger the maximum bit rate of the device will be, but at the same time the switching power will increase. Thus, there is a trade off between the power consumption and maximum allowable bit rate, which depends strongly on the coupling constant k. For example, a logic gate including a Type I structure with a single RR, losses of 20 dB/cm and k = of 0.73 may easily reach a speed higher than 80 Gbit/s with a switching peak power of 24 mW, whereas the same device with a coupled RR with k = 0.22 requires a switching power of only 2 mW but the maximum bit rate is reduced below 1 Gbit/s.
The coupling coefficients shown in Table 1 can be related to the coupling of power coefficients by squaring them; kp = k 2. The coupling of power may be smaller in real round RR due to the high confinement of the signal in the center of the slot waveguide. Coupling coefficients up to 0.7 has been obtained from simulations with the Beam Propagation Method. Special coupling designs that may increase these coupling coefficients are out of the scope of this study but it can be achieved by distorting the RR with the racetrack configuration that enlarges the coupling of the waveguide to the resonant structure.
As mentioned before, smaller coupling coefficients k lead to longer time delays due to the increase of the recirculation time and, at the same time, to a higher intensity build-up and smaller switching powers. In the following section, where more than one single RR is considered, the coupling coefficients cannot be employed to compare the structures in power requirements and bandwidth. Therefore, the overall time delay will be considered as the critical parameter for the comparison of different structures.
3.4 Structures with more than one RR.
When more than one RR is considered the analysis becomes more complicated. First of all, the coupling parameters of each of the RR must be properly designed in order to achieve the desired linear response. If the coupling coefficients between RRs are high enough the structure will have a multiple peak response in the time delay response as a function of the wavelength. These peaks are usually very thin in the spectrum and lead to a high dispersion, which reduces the bandwidth of the device. On the other hand, if the coupling between the RRs decreases sufficiently as they are further from the input/output waveguide, a single peak resonance may be obtained with the same overall time delay (nonlinear sensitivity) as a single RR structure but with a larger bandwidth and better dispersion response. Thus, a logic gate supporting higher bit rates (bandwith) may be obtained with the same switching power. In Table II, the results for different configurations with two and three RRs of Type I structure are shown.
If propagation losses of 10 dB/cm are considered, the maximum time delays induced by the RR structures are 21.2 ps, 21.5 ps, 26.33 ps and 15.81 ps, respectively. These values are in agreement with the required switching power, which reaches a minimum for the first 3RR structure (2 mW) and a maximum for the latter one (4 mW). The advantage of these multiple RR structures is the increase that can be obtained in the bandwidth of the device. Therefore, shorter pulses can be employed reaching higher bit rates. On the other hand, the constraints that may arise are the higher coupling coefficients that are required in the first stage and a smaller tolerance to fabrication deviations. The power requirements of the multiple RR structures shown in Table 2 are, at lower losses, smaller than in the single RR structure with k=0.56. If higher losses are considered, the influence of the RR with a smaller coupling on the output signal decreases, and the behavior of the multiple RR structure becomes more similar to an equivalent structure with only the first RR.
For the sake of comparison, Fig. 9 shows the normalized time delay induced by the first and the last RR structures described in Table 2. It can be clearly observed that the structure with 3 RRs permits a larger bandwidth and, therefore, support higher bit rates as shown in Fig. 10. Bit rates over 60 Gbit/s are feasible with the 3 RR structure whereas the limit in the 2 RR structure is below 30 Gbit/s.
Finally, it should be taken into account that the resonant increase of the intensity is different in each of the RRs of a structure with multiple RRs and, therefore, the nonlinear effects are different in each of them. This leads to a distortion in the response of the structure in the nonlinear regime which can be observed in Figs. 9 and 10 that have been calculated for the linear (solid line) and the nonlinear case (dotted line). It can be observed that the induced leads distortion to a reduction in the allowable bandwidth of the logic gate.
In conclusion, a nonlinear TMM has been implemented for the study and optimization of an all-optical XOR logic gate based on a MZI structure with RRs in its arms to enhance nonlinear effects. The proposed method starts from the estimation of simple structure parameters and can give as a result information of the power requirements and bandwidth limitations of complex structures considering only the nonlinear Kerr effect. Other physical processes such as two photon absorption or free carrier dispersion are out of the scope of this work.
The waveguide forming the MZI device studied is a slot waveguide in which the slot is filled with a Si-nc/SiO2 material that displays Kerr nonlinearity. The side-coupled structure (Type I) has shown to provide a better performance than the Type II structure, mainly due to its increase in the transmission as a function of power and to the higher intensity increase inside the RR. A reduction of power requirements from 2.5W to less than 5 mW can be obtained by the use of RR structures in the implementation of all-optical devices but at the expense of reducing the bandwidth of the devices. In the design of the structures a trade-off arises between the reduction of power and the increase of bandwidth and allowable bit rate. By including more coupled RRs in the Type I structure this trade-off can be overcome, although the design becomes more complex. In this paper, we have shown that data bit rates over 60 Gbit/s can be reached theoretically by use of the Type I structure with 3 coupled RRs in the arms of the MZI. From these results, it might be envisaged the implementation of all-optical logic gates based on the Kerr effect with very small size and low power consumption.
Francisco Cuesta-Soto acknowledges the Ministry of Education and Science for funding his grant. This work was partially funded by the European commission by the framework of the PHOLOGIC FP6 Project IST-NMP-Z-2-017158 and the Spanish MEC under SILPHONICS project 2005-07830.
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