## Abstract

In this paper we present an all-optical approach allowing the realization of logic gates and other building blocks of a processing unit. The modules have dimensions of only few microns, operation rate of tens of Tera Hertz, low power consumption and high energetic efficiency. The operation principle is based upon construction of unconventional wave guiding nano-photonic structures which do not include non-linear materials or interactions. The devices developed and presented in this paper include logic diffractive phase detector, generalized diffractive phase detector, logic gates as AND, OR and NOT, amplitude modulator and analog adder/ subtractor.

© 2005 Optical Society of America

## 1. Introduction

There is a recognized need for all-optical data processing done by devices with very small response times, consumption of relatively low light intensity and small dimensions. The existing approaches for realizing all-optical interactions are mainly based upon non-linear optics. The main disadvantage of these approaches is that strong optical power is required in order to obtain a non-linear interaction. In addition, since the non-linear coefficient is usually rather small, long interaction lengths are used. Another disadvantage is that non-linear materials are rather expensive and do not fit easily into integrated optical circuits and chips [1–5]. On the other hand the advantage of non-linear optics approach is that it allows using the amplitude modulation format. Thus it is possible to perform logic functions between two signals of different wavelength and due to the non-linear interaction the relative phase difference is not important, only the synchronization between the pulses trains is relevant (which is far more tolerant to deviations).

Another recently introduced interesting approach, which demonstrates the operation of basic all-optical devices, is based on multi mode interference (MMI) [6]. Although this approach is modular and simpler for optical integration, it is still not complete enough to cope with the large variety of building blocks required in order to realize an all-optical processor. Its additional disadvantage is that the demonstrated devices are still large and thus cannot be used for the realization of VLSI (Very Large Scale of Integration) all-optical circuits [7–10]. Note that having a larger device means slower operation speed since the light requires a longer time to travel through the device. Thus the MMI approach leads to slower modules as well.

A new approach suggested by the present authors [11] solves the above-mentioned problems by providing a full set of all-optical data processing modules that may be joined to build an all-optical processor. The suggested approach utilizes light propagation through a linear-medium. The operation principle involves MMI and conversion done in specially constructed waveguides and metallic coating inserted inside the interference regions. The modules have very small dimensions (only a few cubic micrometers for a logic AND/OR gate). Consequently, an all-optical processor may be similar in dimensions to an electronic VLSI circuit. Furthermore, the operation rates may approach a few tens of THz, being more than four orders of magnitude faster than electronic VLSI circuits. The technology used to make these modules is simple and inexpensive, using existing E-beam lithography and electro plating instrumentation. The devices presented in this paper facilitate the realization of various logic functions that may be used in a fast RAM module, a femto-second pulse generator, a light amplitude/phase modulator, triggers, encoder/decoder, an optical switch, an analog/digital or digital/analog converter, and in other data processing components.

In section 2, we present the theory and the numerical model of these new devices and their principle of operation. In section 3, we describe numerical simulations of various all-optical elements. In section 4, we present initial experimental results and in section 5, we conclude the paper.

## 2. Theory and modeling

#### 2.1 General description

The idea behind the present approach is associated with the following [11]: When two or more light beams of the same wavelength propagate in free space (linear medium propagation), an interaction between the beams takes place at a point (region) of intersection between the light fields of the beams. This results in interference, namely the summation of the light fields (which is a linear function of the input field) at the point of intersection. The beams then continue their propagation along respective axes with unaffected input field properties as if there was no interaction. According to the conventional approaches, in order to achieve an interaction between the beams that will affect the beams or their propagation properties, the beam interaction must occur in a non-linear medium.

The technique described in this paper provides appropriate light coupling into and out of a waveguide (made from materials that do not exhibit optical non-linear effects), to thereby obtain at the output a desired phase or phase modulation, and/or to obtain at the output a desired amplitude modulation of the input field using reference light beams. This operational principle is based on the provision that a non-uniform spatial energy distribution of a light field is resulting from interaction between the light components propagating in a waveguide, although the waveguide medium itself is linear and the diffraction effect is a linear effect. Such non-uniformity of the light field is created due to the total internal reflection of light at the edges (walls) of the waveguide. This desired reflection is obtained by proper metallic coating of the walls of the interaction regions using the electro plating procedure. In materials having high refraction index (as Silicon) the required internal reflection can be obtained even without the need for special metallic coating.

The spatial non-uniformity is translated into temporal modulation of the input light field, for example following the phase relations between the input beam and the reference beam. Thus, the basic device is actually an optical waveguide structure configured to define one or more optical waveguide units, each waveguide unit having a linear-medium interaction zone for light components created by multiple reflections of input light in the waveguide, and input and output aperture arrangements. Note that since the suggested modules operate due to interference they must be operated using the same light source.

#### 2.2 Numerical modeling

The wave equation for a two dimensional waveguide in the y-z plane may be written as:

where E is the electrical field, μ. is the relative permeability, μ_{0} is the permeability constant, ε is the relative dielectric constant and ε_{0} is the dielectric constant. ω is the radial frequency of light.

This differential equation can be solved by finding eigen values and functions, which represent the basic possible multi modes:

where ${{\mathrm{\beta}}_{\mathrm{m}}}^{2}$ are the eigen values, A_{m} are the coefficients of the different modes and E_{m}(y) is the field profile dependence of the m^{th} mode along the y direction. The numerical solution for Eq.1 can be obtained by the following approximations:

$$\frac{{\partial}^{2}E}{\partial {z}^{2}}=\frac{{E}_{32}-2{E}_{22}+{E}_{12}}{\Delta {z}^{2}}$$

$${E}_{32}=\left(-\mu \bullet {\mu}_{0}\bullet \epsilon \bullet {\epsilon}_{0}\bullet {\omega}^{2}{E}_{22}-\frac{{E}_{23}-2{E}_{22}+{E}_{21}}{\Delta {y}^{2}}\right)\bullet \Delta {z}^{2}+2{E}_{22}-{E}_{12}$$

where E_{ij}, ∆y and ∆z can be seen in Fig. 1. In order to solve numerically these equations we need to assume the following boundary conditions:

$${E}_{21}\cdots {E}_{2M}=\mathrm{exp}\left(-\mathrm{ik}2\mathrm{\Delta z}\right)\bullet \mathrm{cos}(\pi \bullet \frac{\mid M/2-m\mid}{D})$$

where k=2π/λ and λ is the optical wavelength and D is a constant related to the dimensions of the waveguide. Note that the smaller the designed device is, the faster it will operate since the time that will take light to travel from its input channel to the output channel (where the desired result is obtained) will be shorter. Assuming that L is the length of the device and c is the speed of light, the operation rate equals to:

where N is an integer number, *ν*
_{opt} the optical frequency and λ the optical wavelength.

We use this numerical model in order to solve the interference patterns obtained at the output of the designed structures.

## 3. Simulations and operation principle

In order to explain the operation principle of the all-optical processing unit we first describe several constituent building blocks. Note that each one of the devices introduced below has an area of just few square microns, and the simulations were performed at an optical wavelength of λ=0.85μm. Reducing the wavelength reduces the dimensions of the various devices. In addition, the wave guiding channels were assumed to have a diameter of one optical wavelength. Further reducing this diameter to half a wavelength scales down the length of the devices by a factor of 4 and the width by a factor of 2. The length units designated in the simulations are microns. Also note that the initial simulations were done following the numerical model described in the previous section. However, the results were verified using industrial numerical softwares such as FEMLAB and R-Soft. Note also that the devices described below operate while the information is modulated in a binary phase format.

#### 3.1 Diffractive phase detector

The diffractive phase detector (DPD) is depicted in Fig. 2 This device provides a comparison between two input beams. The schematic sketch of the device is seen in Fig. 2(a). Numerical simulations are presented in Figs. 2(b) and 2(c). An input beam A should have phases of either 0 or π and an input beam B should have phases of +π/2 or -π/2. An input having binary phase modulation of 0 and π can produce this input phase distribution if a phase shift element of -π/2 is connected in front of input B. Thus in our treatment we may assume that both inputs have binary phase modulation of 0 and π. Having the equal phase distribution in both inputs produces an energy maximum at the left output facet [Fig. 2(b)]. When the phase relation is opposite, the maximal energy is obtained at the right output facet [Fig. 2(c)]. We will denote the left output as Outleft and the right one as Outright. Table 1 describes the four possibilities of the input phase combinations and their corresponding output. As one may see at each one of the outputs there are three possibilities. The amplitude can be zero or if it is not zero then the phases of the output corresponds to the phases of one of the inputs.

In our case φ_{0}=0 and φ_{1}=π. Since there are three output states instead of two, this is not yet a logic function but this building block is essential for the construction of such a function as shown below.

Note that the DPD can be used as an analog adder/ subtractor as well. For this module the two inputs should not have different phases but rather different amplitudes I_{A} and I_{B} and the obtained output will be (I_{A}+I_{B})/2 and (I_{A}-I_{B})/2 on the left and the right output facets respectively.

Also note that when we talk about phases we are referring to the relative and not the absolute phase. Since all devices are operated by the same light source, indeed, only the relative phase is becoming relevant and interference can take place.

The DPD has some similarity to the Hadamard transform known from the quantum communication [6]. The definition of this transform is:

where the matrix above is the Hadamard matrix, and Σ=A+B, ∆=A-B.

#### 3.2 Generalized diffractive phase detector

This device is schematically described in Fig. 3(a). The left and the right inputs are fixed reference beams while the central input is the information channel (time modulated optical flow of the input). The device operates for the three input possibilities of the central input channel: (i) its phase equals to the phase of the two reference beams, (ii) its phase is opposed to the phase of the reference beams and (iii) the input (information channel) is absent (its amplitude is zero). For case (i) the output energy is 1.5 times larger than for cases (ii) and (iii). This device can be connected to the output of the DPD, thereby converting its three output possibilities into only two. The device structure is seen in Fig. 3(b) and the numerical simulations of the obtained light distribution for the three possibilities of the input channel are seen in Figs. 3(c)–3(e).

A summary of the various possibilities of the Generalized DPD module can be seen in Table 2.

Note that the reference beam added to this module is coming from the same light source as the information input channels and thus interference can take place. The relative phases′ adjustment is done by proper numerical and experimental design of the lengths and shape of the wave guiding channels.

#### 3.3 Logic and/or gate

There are several possible schemes that may be used to realize a logic AND/ OR gate, and one is described here. The module includes three stages: The first block is the DPD, it's, output inserted into the Generalized DPD. The output of the generalized DPD is cascaded into another DPD block [see Fig. 4(a)]. First the phases of the two inputs (A and B) of Fig. 4(a) are compared using the DPD. If the phases are equal, the energy is output at the right output channel. Otherwise the output is obtained at the left channel. The right output is connected to the Generalized DPD module. For the three phase states combinations of the inputs A and B, the beam at the output of this module is obtained with equal energy and identical phase. In the fourth phase combination state of the inputs A and B, the output has 1.5 more the energy but identical phase state. This was seen in section 3.2. In order to realize logic gate that can be integrated with the next processing units the output of the gate must be in the same modulation format as its inputs, i.e. binary phase modulation. Thus, the purpose of the third module is to convert the difference of energies of the output into phase differences.

Applying subtraction operation to this output may convert the output to zero and one in amplitude (zero for three phase combinations of the input channels and 1 for the fourth one) or the same subtractor can be used for conversion of the amplitude modulated beam into a phase modulation with amplitude of 0.25 and phase of zero or π. Figure 4(b) shows the complete module. The results of the numerical simulations can be seen in Figs. 4(c)–4(f) where the output distribution is presented for the four possibilities of the input phase modulation. In this simulation the last stage subtractor was chosen such that logic one is obtained only in the left output facet of Fig. 4(d). For the other three combinations the left output facet has zero energy [Figs. 4(c), 4(e) and 4(f)].

In the simulation of Fig. 4 we have presented a logic gate with amplitude modulation in its output. Indeed as we mentioned before in order to cascade several modules it is important to keep the same modulation format which is a phase format in our case. However, we presented an amplitude modulation since it is much better visualized. In any case the purpose of the third module that was to shift the output of the second module from being 1 and 1.5 to be -0.25 and 0.25 (phase modulation format), was demonstrated also for a different shift to 0 and 0.5 (for the output channel).

The logic gate described in this subsection operates using linear effects such as interference and diffraction which fulfill the law of superposition of fields. However, the logic AND operation is not a linear operation. Thus, how can we realize this operation the way we do? The preliminary answer is described in Fig. 4(g). The Generalized DPD module has two constant inputs, i.e. reference beams (let us denote their effect to the output port as C) and an input channel (the central one) which we will refer to as the information input. This channel has three states: (i) amplitude of zero, (ii) amplitude of one with phase of zero and (iii) amplitude of one and a phase of π. Those three states are described in the vector diagram of Fig. 4(g). In case that only the vector C exists in the output port (the input in the information channel is zero) its length equals to one and its phase is -12 degrees in comparison to the dashed black line. In case that the information beam has amplitude of one and a phase of zero (its effect to the output port is denoted by the dashed green line) then the overall vector is the one described by the green solid line. This vector has also amplitude of one and a phase of about +12 degrees in comparison to the black dashed line. When the information beam has amplitude of one but phase of π, the resulted overall vector is the solid blue line which has the length of 1.5. Thus, the diagram of Fig. 4(g) exactly describes Table 2 where the three input states of the information channel are converted into two output states but this time with relative error of 12 degrees. Thus, by allowing small relative error in the phase (12 degrees) one may fulfill Table 2 and therefore the logic operation. Although this error is small it will interfere with cascading large number of such devices. In order to solve this, additional element having a threshold function must be introduced. Such an element and additional relevant modules are to be discussed and explored in the next paper, which is currently in preparation.

#### 3.4 Logic NOT gate for amplitude modulated beams

Two variations for the logic NOT gate are demonstrated and simulated in Fig. 5. Figure 5(a) presents the schematic sketch of the device. The middle waveguide unit is a typical optical waveguide used as the reference according to which the NOT operation is performed. The phase of light propagating through such a waveguide unit is not affected. The left and right waveguide units are configured according to two examples of the phase inversion used to provide a change of phase of the input light at the output. This phase inversion can be achieved by passing the input light through a region of the waveguide that has a different refraction index, width or length. As a result, the phase inversion occurs at the output (as seen in the right waveguide where the phase inversion occurs due to width variation). The left waveguide unit is configured generally similar to the middle reference unit but it has a core region of refraction index that is different than what is used in the reference central wave guiding channel. Figure 5(b) presents the numerical simulations of the device of Fig. 5(a). Its central channel is the reference beam in comparison to which we wish to invert the phase. In the left output channel we inverted the phase by changing the refraction index of that channel and in the right channel the phase inversion was obtained due to diffraction (via width variations).

#### 3.5 Amplitude modulator

An amplitude modulator may be realized using the DPD. If a reference beam is connected to one of the inputs of the DPD [Fig. 2(a)], the amplitude of the left output is maximal for opposite phases of reference and information beams and the amplitude of the right output is maximal when the reference and the information beams are in phase. Thus, the right output channel is an amplitude modulated beam and the left output is an inverted amplitude modulated beam.

#### 3.6 Analog adder/subtractor

This module is constructed out of three waveguides that are combined into a wider interaction region, in which the output channel has the width equal to that of the input waveguides [see Fig. 6(a)]. The central input is denoted by A and the other two outer inputs by B. If the phase of A plus π/2 equals to the phase of B then the obtained output C equals to A+B. If the phase of A plus π/2 is opposed to the phase of B, then the output C equals to A-B. To realize this operation one also needs |A|>|B|. Figures 6(b) and 6(c) present simulation results of this device. In Fig. 6(b) one may see the case where the output equals to A+B while in Fig. 6(c) one obtains A-B (for the case of A=B).

#### 3.7 Logic NOT gate for amplitude modulated beams

The NOT element is capable of producing output beam when the input beam is not input into the device. Figure 6 presents the element which includes three input ports and one output port. The information is input in the middle input port while in the two external input ports reference beam is inserted. In this configuration, the simulation results shown in Fig. 6 represent exactly a logic NOT gate (previously presented in Fig. 5).

#### 3.8 Tolerances and system configuration

The devices described in this paper are based upon interference. Thus, there should be sensitivity to the wavelength. In this subsection we wish to show that such sensitivity does not interfere with the fact that the operation rate is tens of THz which means a wideband signal. In Fig. 7 we explore the sensitivity of the DPD module. Figure 7(a) presents the field distribution obtained when the operation wavelength was equal to the wavelength for which the device was designed. In Fig. 7(b) the same device was tested with wavelength which is 0.67 of the design wavelength (deviation of approximately 30%). For both cases identical field distribution was obtained. Since the preferred operation wavelength is λ=0.85μm (frequency of 3.5E14) deviation of 30% leads to possible bandwidth of more than several tens of THz which is compatible with our assumption for the required operation rates.

Another important topic that should be addressed is related to the fact that in order to perform interference, the various beams should originate from the same light source. Our intension is to use a VCSEL or a laser diode connected to the optical chip and operating all the devices on the chip. From energetic consumption considerations it appears that a source of few mW (optical and not electronic power) can operate more than 2500 logic gates. Thus, the proposed operational scheme is as described by Fig. 8 where the same single laser source is used in order to provide the reference beams as well as the modulated information signals to all the logic devices on the chip.

## 4. Experimental investigation

Figure 9 depicts some initial experimental results performed in a water bath in which the optical waves were replaced by water waves having similar propagation equations. The reason for this type of experiment is that water fulfills the same wave equation as light waves, but the wavelength is much larger and thus the construction of the device is significantly simpler. Note that if the water is shallow enough the non-linear effects are negligible and the obtained results may be considered to reliably represent the results anticipated in an optical waveguide. For the experiment, we constructed a module which is similar to the DPD but with a single central output. Special eccentric ellipses were designed and fabricated. The ellipses were connected to pedals such that when rotated by a motor the pedals were pressed and perfect sinusoidal water waves were generated in the bath. Placing the ellipses in different relative angles allowed control of the relative phases of the generated water waves.

The module was constructed by building the proper water interaction region inside the bath. In-phase or out of phase input waves were generated by aligning the relative angle of the ellipses. The wavelength of the water waves was 4cm. In the upper part of Fig. 9, one may see the picture of the experimental setup. In the lower part one may see the comparison between the numerically anticipated results and the obtained distribution. Matching can be seen not only in the wave distribution but also in the number of wave cycles until the proper interaction is obtained. The results of the water experiment matched nicely the optical simulation, with the optical wavelength replaced (and thus scaled up) with the 4cm water waves wavelength.

The matching between the experiment and the numerical simulation presented at the bottom of Fig. 9 is for two inputs having equal phases. The lower right part of Fig. 9 depicts the experimentally obtained water wave interference. The lower left part is a zoom view on the interference region of the lower right part, where the waves are focused into a spot. In the lower left part of the figure, one may see the converging spherical wave front before the focus spot and the diverging wave front just after it.

## 5. Conclusions

In this paper, we present a novel approach that allows realization of an all-optical processing unit using several all-optical building blocks. The suggested approach has several significant advantages: an operation principle involving interaction within linear media, thus inexpensive for construction and modular for integration; small and compact modules leading to dimensions not much larger than those obtained in electronic VLSI circuitry, fast operation rates of few tens of THz and low power consumption.

The presented building blocks were numerically investigated and tested in preliminary experimental set.

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