All-optical switches, unidirectional flow, and logic gates with discrete solitons in waveguide arrays

U. Al Khawaja; S. M. Al-Marzoug; H. Bahlouli

doi:10.1364/OE.24.011062

1. Introduction

The use of optical pulses to transfer data in optical fibers is dominating the electronic means of data transfer due to many appealing advantages [1–3]. Most importantly, orders of magnitude higher band widths have become available. These advantages are, however, downgraded by the relatively slow electronic-based data processing. This has triggered extensive research to perform the so-called all-optical data processing [4,5]. Many proposals have been suggested and realised all-optical switching, routing, logic gates, and even fundamental mathematical operations [6–15]. We also mention the all-optical soliton switch that is based on a soliton interaction in a multilayered kerr medium [16]. The key element in this optical device is the inhomogeneity of the nonlinear medium, that is the existence of interfaces in the multilayered kerr medium. Among the different systems proposed to realise all-optical processing, the simplest and most realistic system is waveguide arrays [17]. The latest technological advances in femto-second lasers allow to fabricate hundreds of these arrays in a one-centimeter-width substrate of nonlinear material with flexibility in modulating their separation along the length of the sample [18,19]. This has opened the door wide to investigate a host of previously-predicted nonlinear phenomena [20–22]. Of particular importance, the solitonic solutions of this nonlinear discrete system have been investigated with an interest in finding fundamental differences with the continuum solitons [23,24]. Indeed, in some cases a sharp contrast between the behavior of discrete and continuum solitons was found such as in Refs. [25–27]. While continuum solitons collide elastically, discrete solitons may go through completely inelastic collisions that are, even more drastically, accompanied by symmetry breaking in which the total momentum is not conserved [25]. The discreteness in the spacial localization causes such effects by producing an effective so-called Peierls-Nabarro potential, which in turn leads to kinetic energy losses [28].

Scattering continuum solitons off reflectionless potentials has been shown to exhibit sharp transitions in the transport coefficients from full (quantum) reflection to full transmission [29, 30]. It has been argued that this may find application as an optical switch [30, 31] and later shown that with two such potential wells unidirectional flow may be obtained [32]. Motivated by the above-mentioned technological advances in nonlinear waveguide arrays and the unique properties of discrete solitons, the main objective of the present work is to show that all-optical operations such as switching, filtering, unidirectional flow, and most importantly logical operations are all possible in waveguide arrays. The proposal put forward here is to scatter a signal soliton off single and double reflectionless effective potential wells realised by modulating the coupling coefficients between the wave guides through modulating their separations. The unique feature of this work is the injection of a stationary soliton, which we denote as the control soliton at the minimum of the potential well. It turns out that the power of the control soliton will be the only parameter that transforms this simple device into a multi-purpose device operating as a switch, filter, diode, or a logic gate depending on the selected parameters. The trapping of the control soliton by the potential well prevents it from contributing to the reflection or transmission signals from which the output is taken. This results in clean and robust scattering process with comfortable parameter ranges over which the same behaviour is maintained ensuring the experimental feasibility of our proposal.

We start in section II by introducing the discrete nonlinear Schrödinger equation (DNLSE) that governs the propagation of discrete solitons in the waveguide array. We show how the effective reflectionless potentials will be introduced by suitably modulating the separations between the waveguides, then we describe our numerical approach. In Section 3, we present the possible all-optical operations based on the scattering outcomes. We conclude in Section 4 by a summary of the main results and discussion of the potential realisation of the applications.

2. Theoretical model and numerical procedure

2.1. Theoretical model

Propagation of solitons in a one-dimensional array of N waveguides with focusing nonlinearity can be described, in the tight-binding approximation, by the following DNLSE for the normalised mode amplitude ψ_n,

i \frac{\partial ψ_{n}}{\partial z} + C_{n, n - 1} ψ_{n - 1} + C_{n, n + 1} ψ_{n + 1} + γ {| ψ_{n} |}^{2} ψ_{n} = 0,

where n is an integer number associated with the waveguide channel, z is the propagation distance, C_n,m are the coupling coefficients between different waveguide channels n and m, and γ is the strength of the focusing nonlinearity.

This model is not integrable [23] but it admits numerical stable solitonic solutions such as the on-site (OS) and inter-site (IS) solitons [17]. Both are localised sech-like modes but differ in being either localised on one waveguide (OS) or being localised between two successive waveguides (IS). Due to the presence of the Peierls-Nabarro effective potential, there are no movable exact solutions to this model [23]. However, moving stationary solitons may preserve their integrity to a large extent apart from some kinetic energy loss in terms of background radiation. For the dynamical processes we aim at studying here such an approximate solitonic behaviour will be adequate; the processing time will be considerably smaller than that for the soliton to cause changes in its shape or speed considerably.

The main reason for choosing waveguide arrays to perform all-optical processes is that effective potential wells can be realised simply by modulating the separation between the waveguides. It was found experimentally, that the coupling strength between waveguides decreases exponentially with increase in their separation [33,34]. This fact has indeed been used to study the scattering of linear pulses off reflectionless potentials [35,36]. Considering a modulation of the coupling constants, through their separation, in the following form

C_{n, n + 1} = C + V (n - 1), C_{n, n + 1} = C + V (n + 1),

and substituting in Eq. (1), we obtain a DNLSE with an effective potential

\begin{array}{l} i \frac{\partial ψ_{n}}{\partial z} & + & C ψ_{n - 1} + C ψ_{n + 1} \\ + & V (n - 1) ψ_{n - 1} + V (n + 1) ψ_{n + 1} \\ + & γ {| ψ_{n} |}^{2} ψ_{n} = 0 . \end{array}

It is essential for the potential to be of the reflectionless type. This will guarantee the required sharp transitions of the transport behaviour from full reflection to full transmission and the absence of background radiation. Exact solitonic solutions of integrable models provide such a reflectionless potential. In the present case, we use the integrable Ablowitz-Ladik model [23]

i \frac{\partial ψ_{n}}{\partial z} + (ψ_{n - 1} + ψ_{n + 1}) (C + {| ψ_{n} |}^{2}) = 0,

to construct the reflectionless potential from its exact soliton solution

ψ_{n}^{AL} = \sqrt{C} sinh (μ) sech [μ (n - n_{0})] exp (i β z),

with β = 2C cosh(μ), μ is the inverse width of the soliton, and n₀ corresponds to the location of the soliton peak. Following Eq. (2), the coupling is modulated as follows

C_{n, n \pm 1} = C + {| ψ_{n \pm 1}^{AL} |}^{2},

and Eq. (3) becomes

\begin{array}{l} i \frac{\partial ψ_{n}}{\partial z} & = & - C (ψ_{n - 1} + ψ_{n + 1}) \\ - & {| ψ_{n - 1}^{AL} |}^{2} ψ_{n - 1} - {| ψ_{n + 1}^{AL} |}^{2} ψ_{n + 1} - γ {| ψ_{n} |}^{2} ψ_{n} . \end{array}

Clearly, the effective potential is a sech²-like modulation in an otherwise constant coupling. This can be achieved by a corresponding reduction in the separation of the wave guides according to the exponential law found in the experiment of [33].

Manipulating the dispersion coefficient, C, breaks the hermiticity of the hamiltonian corresponding to Eq. (7) while energy conservation requires that the coupling coefficients between adjacent waveguides be symmetric. The usual remedy to this problem is the usage of the following symmetrised coupling coefficients [37]

C_{n, n \pm 1}^{S} = \sqrt{(C + {| ψ_{n} |}^{2}) (C + {| ψ_{n \pm 1} |}^{2})} .

We found an alternative procedure by introducing an n-dependent strength of the nonlinearity that transforms Eq. (7) to an integrable form. The modulated nonlinearity strength has to be introduced in accordance with the integrability conditions found in [38] for the continuum case, namely

γ_{n, n \pm 1} = γ_{0} / C_{n, n \pm}^{2}

, where γ₀ is an arbitrary constant. We have verified numerically that the two procedures lead to a hermitian hamiltonian by checking the conservation of the soliton norm and energy. Here, we adopted the first procedure and hence Eq. (7) becomes

i \frac{\partial ψ_{n}}{\partial z} = - C_{n - 1}^{S} ψ_{n - 1} - C_{n + 1}^{S} ψ_{n + 1} - γ {| ψ_{n} |}^{2} ψ_{n},

where

C_{n \pm 1}^{S} = \sqrt{(C + {| ψ_{n}^{AL} |}^{2}) (C + {| ψ_{n \pm 1}^{AL} |}^{2})} .

For an exponential decay of coupling in terms of separation such as

C_{n, n \pm 1}^{S} = C exp (1 - \frac{D_{n, n \pm 1}}{D_{0}}),

where D_n,n±1 is the separation between waveguides n and n ± 1, and D₀ and C are positive constants, the separation between waveguides that gives rise to an effective potential is obtained by inverting the last equation, namely

D_{n, n \pm 1} = D_{0} [1 - log (\frac{C_{n, n \pm 1}^{S}}{C})] .

This practical relation can be used to design specific effective potentials. For an effective potential well, Eq. (10) shows that

C_{n \pm 1}^{S} > C

for all n, i.e., an upward profile above the constant background C. Therefore, the equivalent profile of waveguide separations should be a downward profile below the constant separations of D₀. Explicitly, an effective single potential well can be obtained, from Eq. (10), with the modulated coupling constants

\begin{array}{l} C_{n \pm 1}^{S} & = & C {[1 + {sinh}^{2} (μ) {sech}^{2} μ (n - n_{0})] \\ \times & {[1 + {sinh}^{2} (μ) {sech}^{2} μ (n \pm 1 - n_{0})]}}^{1 / 2} . \end{array}

This is achieved by the separations’ profile given by Eq. (12), namely

\begin{array}{l} D_{n, n \pm 1} & = & D_{0} [1 - \frac{1}{2} log [1 + {sinh}^{2} (μ) {sech}^{2} (μ (n - n_{0}))] \\ - & \frac{1}{2} log [1 + {sinh}^{2} (μ) {sech}^{2} (μ (n \pm 1 - n_{0}))]] . \end{array}

Similarly, a double potential well is obtained by generalizing Eq. (10) as follows

\begin{array}{l} C_{n, n \pm 1}^{S} & = & [(C + {| ψ_{1, n}^{AL} |}^{2} + {| ψ_{2, n}^{AL} |}^{2}) \\ \times & {(C + {| ψ_{1, n \pm 1}^{AL} |}^{2} + {| ψ_{2, n \pm 1}^{AL} |}^{2})]}^{1 / 2}, \end{array}

where

ψ_{1, 2}^{AL}

are two exact solitonic solutions centered at different waveguides which take the form

ψ_{i, n}^{AL} = \sqrt{C} sinh (μ_{i}) sech [μ_{i} (n - n_{i})] exp (i β_{i} z), i = 1, 2,

with β_i = 2C cosh(μ_i), μ_i is the inverse width of the i-th soliton, and n_i corresponds to the location of the i-th soliton’s peak.

2.2. Numerical procedure

Typically, stationary solutions of Eq. (1) are first obtained using the Newton-Raphson or iterative methods. For the real-time evolution the fourth-order Runge-Kutta method is used with a stationary soliton as the initial profile. Once a stationary soliton is given an initial speed, it starts to move generally with deceleration, as mentioned above. The time-dependent soliton speed is shown in Fig. 1 for different initial soliton speeds and coupling constants. Clearly for coupling strengths C = 0.35 and C = 0.4 the motion of the soliton is far from being similar to that of a free soliton moving with a constant speed. For C = 0.45 and C = 0.5 the motion is considerably smoother especially for the large values of the initial speed where we also noticed that there is a constant velocity reduction. Therefore, to exploit the solitonic feature efficiently we conduct the numerical investigations in the rest of this paper using C = 0.45, with initial speed range v ∼ 0.2 – 0.3, and time less than about 300. Furthermore, one can verify that with larger values of coupling the difference in the free energy of the OS and IS solitons will be small. Therefore, the results we obtain here will not change considerably when the OS or IS solitons are used.

Fig. 1 Soliton speed versus time for different coupling strength and initial speeds.

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Our numerical procedure is summarised as follows. First, we find the stationary solution, ψ^ST, of the homogeneous DNLSE, i.e. Eq. (1) with constant coupling. Then we evolve in real time a soliton that has the aforementioned stationary soliton profile and moves with an initial speed, v, namely ψ^ST e^ivz. A reflectionless single or double potential well is constructed using n-dependent coupling coefficients according to Eq. (10) or Eq. (15). The control soliton is a stationary soliton that is injected at the minimum of the potential well and has the same profile as that of the signal soliton, rψ^ST, apart from the power control parameter r. For a soliton initially moving to the right and located at n₀ and generally two potentials located at n₁ and n₂, such that n₀ < n₁ < n₂, the transport coefficients are defined as follows: reflection $R = \sum_{1}^{n_{1} - δ n} {| ψ_{n} |}^{2} / \sum_{1}^{N} {| ψ_{n} |}^{2}$ , transmission $T = \sum_{n_{2} + δ n}^{N} {| ψ_{n} |}^{2} / \sum_{1}^{N} {| ψ_{n} |}^{2}$ , and trapping $L = \sum_{n_{1} - δ n}^{n_{2} + δ n} {| ψ_{n} |}^{2} / \sum_{1}^{N} {| ψ_{n} |}^{2}$ , where N is the number of waveguides and δn is roughly equal to the width of the soliton in order to avoid the inclusion of the tails of the trapped soliton with the reflected or transmitted ones. For the soliton moving from the right to the left, the expressions for T and R should be interchanged. A preliminary investigation of the scattering outcomes in terms of the potential and soliton parameters including potential depth, width, location, soliton initial speed, phase, and type, gives an idea of the ranges of parameters for which a useful application could be obtained as shown in the next section.

3. All-optical applications

3.1. Switches and filters

Sharp transitions from full reflection to full transmission characterise the scattering of continuum solitons off reflectionless potentials [29, 30]. We verify in the present work that discrete solitons show a similar behaviour. The transport coefficients of a soliton scattered off a reflectionless potential well are shown in Fig. 2. The sharp transitions at a critical soliton speed or amplitude can be exploited to design a filter or a switch device. The critical speed or amplitude depend on the depth and width of the potential well. Requesting such a device to function as a filter or a switch will thus be restricted to these critical values. We found that a much more flexible setup can be obtained by injecting a stationary soliton through the potential well which we denote as the control soliton, as sketched in Fig. 3. Depending on the power of the control soliton the critical speed for full transmission transition can be decreased or increased. This creates, for instance, the possibility of transforming a reflected signal to transmitted one just by injecting the control soliton in the potential well. In this manner, switching or filtering is done with an independent signal and without any modulation on the input signal. Figure 4 shows that by increasing the power of the control soliton, the critical speed decreases. The presence of the control soliton effectively decreases the depth of the potential well working as a bridge over which the input soliton crosses the potential well.

Fig. 2 Transmission coefficients versus soliton initial velocity and amplitude. Reflection (R) is shown with the solid (red) curve, transmission (T) with dashed (green) curve, and trapping (L) with dotted (black) curve. Coupling used: C_n = C + sinh(1.5) sech(1.5(n – 32)) with C = 0.45 in the v-curves and C ranging between 0.45 and 0.39 in the A-curve. The parameter A is the amplitude of the numerically-found stationary soliton. Signal and control solitons are found numerically by solving the time-independent version of Eq. (1) with C = 0.45 and γ = 1 and initial location at 22 and 32, respectively. The total number of wave guides is N = 63. The amplitude of the control soliton is multiplied by the power control parameter r = 0.16.

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Fig. 3 Schematic figure showing the proposal of injecting a control soliton into a potential well in order to control the scattering of the signal soliton.

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Fig. 4 Soliton scattering off a single potential well for different control soliton powers. Left: Reflectance in terms of soliton initial speed for different values of r. Right: critical speed, v_c, for quantum reflection versus power control parameter r. All other parameters are the same as in Fig. 2.

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3.2. Unidirectional flow

Unidirectional flow of continuum solitons has been shown to occur when a soliton is scattered by two reflectionless potential wells with slightly different depths [32]. Since, as we found above, the presence of a control soliton effectively amounts to reducing the depth of the potential well, we expect the unidirectional flow to be realised and triggered solely by the control solitons. Indeed, this is what we find by scattering a soliton off two similar reflectionless potential wells of equal depth and width and separated by a finite distance. The symmetry is broken by injecting a control soliton only in one of the potential wells. Clearly, Fig. 5 shows that unidirectional flow is obtained within a velocity range of v ∈ [0.1375, 0.1550]. The polarity of this diode can be reversed by injecting the control soliton in the other potential well.

Fig. 5 Diode behaviour shown with soliton scattering off a double potential well from both directions. Solid curves correspond to scattering from the left and dashed curves correspond to scattering from the right shown with reflection (R) in red, transmission (T) in green and trapping (T) in black. The two potential wells are centerd at 30 and 34 with total waveguides N = 63, the power control parameter used is r = 0.3 and all other parameters are the same as in Fig. 2.

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3.3. Logic gates

The same setup of the unidirectional flow can be used to obtain the logic operations AND, OR, NAND, NOR. The device functions in the desired manner just by reducing the power of the control solitons to almost half that of the unidirectional flow. The input signals of the logic gates are taken as 1 if there is a control soliton in the potential well and 0 otherwise, such that 11 corresponds to two equal control solitons in the two potential wells, 01(10) corresponds to a control soliton in the right (left) potential well, and 00 corresponds to no solitons in the potential wells. The output is taken from the scattered soliton which can be reflected, transmitted, or trapped. This setup is shown schematically in Fig. 3. Taking the output from a number of waveguides and not a single specific one makes the performance more robust against perturbations. The transport coefficients of the four possibilities are shown in Fig. 6. The AND and OR gates are accounted for by taking the output signal to be the transmitted soliton (green curves) while the NAND and NOR gates correspond to the reflected soliton (red curves). The AND gate is achieved within the velocity range v ∈ [0.175, 0.21] where transmission takes place only for the 11 input. Within the same range, the NAND gate functions where reflection takes place for 01, 10, and for 00, but not 11. The OR gate functions with the transmitted soliton within the velocity range v ∈ [0.21, 0.225] where transmittance takes place for 11, 01, 10, but not for 00. The NOR gate functions within the same velocity window where reflection takes place only for 00 state. Table 1 summarizes all these different situations. Sample spacio-temporal plots are shown in Fig. 7. The upper panel corresponds to initial soliton speed v = 0.19 which is within the the velocity window of the AND and NAND gates and the lower panel corresponds to v = 0.215 corresponding to the OR and NOR gates, as shown in Fig. 6. For the first case, the soliton transmits only when the two control solitons are present, i.e. the input is 11, this is the AND gate. Simultaneously, the soliton is reflected only for the 00, 10, 01 inputs which correspond to the NAND gate. For the lower panel, transmission takes place for 01,10,11, which corresponds to the OR gate. The reflection clearly corresponds to the NOR gate.

Table 1. Logic gates obtained by scattering a soliton off a double potential well with two control solitons. Input corresponds to the presence (1) or nonpresence (0) of a control soliton such that, for instance, 10 corresponds to a control soliton in the left well and no soliton in the right well. Output is taken from the reflected or transmitted signal soliton. The two potential wells are separated by 4 waveguides and the power control parameter used is r = 0.16.

View Table

Fig. 6 Logic gates performance shown with soliton scattering off a double potential wells behaving. Red curves correspond to reflection (R), green curves correspond to transmission (T), and black curves correspond to trapping (L). Filled circles correspond to the presence of control solitons in both wells (11). Up and lower triangles correspond to the presence of a control soliton in the left or right well (10 or 01), respectively. Empty circles correspond to the absence of control solitons from both wells (00). The two potentials are separated by 4 waveguides, the power control parameter used is r = 0.16, total number of waveguides is N = 124, and all other parameters are the same as in Fig. 2.

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Fig. 7 Density plots corresponding to two selected points from Fig. 6 showing the AND and NAND gates, with initial signal soliton speed v = 0.19, in the upper panel and the OR and NOR gates, with initial signal soliton speed v = 0.215, in the lower panel. In each subfigure waveguides range from 1 to 124 and time ranges from 0 to 375. All other parameters are the same as those of Fig. 6.

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4. Conclusions and discussion

We have shown that all-optical operations such as unidirectional flow, switiching, and logic gating are all possible with discrete optical solitons in waveguide arrays. The separations between the waveguides can be set in a pre-calculated manner such that an effective potential is formed. Designing waveguides with separations, such as those given by Fig. 8, effective reflections potential wells will be obtained. The flow of solitons across the potential wells is controlled by another high-intensity soliton trapped by the potential well. The presence, in presence, and power of the control soliton determine the outcome of the scattering process.

Fig. 8 Waveguides separation, D_n,n+1, and the corresponding coupling C_n,n+1. The calibration values D₀ = 24μm and C₀ = 0.45mm⁻¹ were taken from the experiment of [33] for the λ = 543 nm pulse.

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Specifically, a waveguide array with a single potential well and one control soliton may be used as an optical switch or filter device. The power of the control soliton will determine whether a signal soliton will reflect off or transmit through the potential well, as shown by Fig. 4. It can also be used to filter certain soliton speeds or powers as they are transmitted through the well, as shown by Fig. 2. A waveguide with a double potential well with two control solitons can function as a diode with adjustable polarity or as a logic gate. It functions as a diode if a control soliton is injected through only one of the wells. This diode functions over an appreciable window of soliton initial speeds, as shown by Fig. 5. The polarity of the diode is controlled by selecting through which potential well to inject the control soliton. The same device functions as a logic gate but at a considerably different value of the control soliton power. All gates including AND, OR, NAND and NOR are obtained using the same physical device. The AND and NAND gates are obtained within a velocity range different from that of the OR and NOR gates, see Fig. 6 and Table 1.

Few remarks about the experimental feasibility of this proposal are in order. In the experiment of [33] waveguides of 100 mm length were prepared and their coupling strength was calibrated with waveguides separation. Coupling strengths decreased exponentially from 0.24 mm⁻¹ to about 0.05 mm⁻¹ for waveguide separations ranging from 14 μm to 36 μm (See Fig. 3 in [33]). This was used in the experiment of [36] to realize reflectionless scattering of linear pulses with a sample of 31 waveguides. We have used in our numerical calculations C = 0.45. In the units of mm⁻¹, such a value is obtained, according to the experiment of [33], with waveguides separation D₀ ≈ 24μm for signals of wavelength λ = 543 nm (Fig. 3b in [33]). In the present proposal, the number of waveguides within which the relevant dynamics takes place is about 30. With typical speed of order 0.2, waveguides of length 150 mm will be required which is not far from that used in the experiments. In Fig. 8 we plot the waveguides separation in μm and the corresponding coupling strength in mm⁻¹ corresponding to the potential well used in Fig. 2. Similarly, the corresponding waveguide separations for the double potential well can be calculated using Eqs. (12) and (15).

While the number of waveguides, the evolution time, and soliton powers, are all within the feasibility range of recent experiments [33,34,36], our proposal did not take into account soliton damping due to propagation losses. However, experiments of Refs. [34, 36] were performed with linear waves in the presence of propagation losses of about 0.4 dB/cm. It is therefore, expected that propagation losses will not critically hinder the realization of the predicted effects provided that the length of the waveguides is not much longer than the currently-used length of about 100 mm. Thus, we anticipate that with an experimental setup similar to that of [36], the all-optical operations described here can be realised.

One may consider utilising the present results to achieve processes involving algebraic computations such as addition and subtraction of coded numbers which is left to future work.

Acknowledgments

This project was funded by the National Plan for Science, Technology and Innovation (MAAR-IFAH) - King Abdulaziz City for Science and Technology - through the Science & Technology Unit at King Fahd University of Petroleum & Minerals (KFUPM) - the Kingdom of Saudi Arabia, award number 14-NAN667-04. S. Al-Marzoug Acknowledges the support of King Fahd University of Petroleum and Minerals under the project numbers RG1333-1 and RG1333-2. U. Al Khawaja acknowledges the support of UAEU-UPAR 2013 and UAEU-UPAR 2015 grants. We also express our appreciation for the support of KFUPM and SCTP during the progress of this work.

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All-optical switches, unidirectional flow, and logic gates with discrete solitons in waveguide arrays

Abstract

1. Introduction

2. Theoretical model and numerical procedure

2.1. Theoretical model

2.2. Numerical procedure

3. All-optical applications

3.1. Switches and filters

3.2. Unidirectional flow

3.3. Logic gates

4. Conclusions and discussion

Acknowledgments

References and links

Cited By

Figures (8)

Tables (1)

Equations (16)

Optics Express

Input	Output
	Transmission		Reflection
	v ∈ [0.175, 0.21]	v ∈ [0.21, 0.225]	v ∈ [0.175, 0.21]	v ∈ [0.21, 0.225]
00	0	0	1	1
01	0	1	1	0
10	0	1	1	0
11	1	1	0	0
GATE	AND	OR	NAND	NOR