## A concomitant and complete set of nonvolatile all-optical logic gates based on hybrid spatial solitons |

Optics Express, Vol. 22, Issue 6, pp. 6934-6947 (2014)

http://dx.doi.org/10.1364/OE.22.006934

Acrobat PDF (2256 KB)

### Abstract

We theoretically demonstrate the realization of a complete canonical set of all-optical logic gates (AND, OR, NOT), with a persistent (stored) output, by combining propagative spatial solitons in a photorefractive crystal and dissipative cavity solitons in a downstream broad-area vertical cavity surface emitting laser (VCSEL). The system uses same-color, optical-axis aligned input and output channels with fixed readout locations, while switching from one gate to another is achieved by simply varying the potential applied to the photorefractive crystal. The inputs are Gaussian beams launched in the photorefractive crystal and the output is a bistable, persistent soliton in the VCSEL with a ’robust’ eye diagram and large signal-to-noise ratio (SNR). Fast switching and intrinsic parallelism suggest that high bit flow rates can be obtained.

© 2014 Optical Society of America

## 1. Introduction

1. O. Firstenberg, T. Peyronel, Q. Liang, A. V. Gorshkov, M. D. Lukin, and V. Vuletic, “Attractive photons in a quantum nonlinear medium,” Nature **502**, 71–76 (2013). [CrossRef] [PubMed]

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4. T. Kanna and M. Lakshmanan, “Exact soliton solutions of coupled nonlinear Schrödinger equations: Shape-changing collisions, logic gates, and partially coherent solitons,” Phys. Rev. E **67**, 046617(2003). [CrossRef]

5. S. Barland, J. R. Tredicce, M. Brambilla, L. A. Lugiato, S. Balle, M. Giudici, T. Maggipinto, L. Spinelli, G. Tissoni, T. Knödl, M. Miller, and R. Jäger, “Cavity solitons as pixels in semiconductor microcavities,” Nature **419**, 699–702 (2002). [CrossRef] [PubMed]

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10. T. Maggipinto, M. Brambilla, G. K. Harkness, and W. J. Firth, “Cavity solitons in semiconductor microresonators: Existence, stability, and dynamical properties,” Phys. Rev. E **62**, 8726–8739 (2000). [CrossRef]

11. R. Landauer, “Irreversibility and heat generation in the computing process,” IBM J. Res. Dev. **5**, 183–191 (1961). [CrossRef]

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13. L. Columbo, C. Rizza, M. Brambilla, F. Prati, and G. Tissoni, “Controlling cavity solitons by means of photorefractive soliton electro-activation,” Opt. Lett. **37**, 4696–4698 (2012). [CrossRef] [PubMed]

14. R. McLeod, K. Wagner, and S. Blair, “(3+1)-dimensional optical soliton dragging logic,” Phys. Rev. A **52**, 3254–3278 (1995). [CrossRef] [PubMed]

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16. A. Piccardi, A. Alberucci, U. Bertolozzo, S. Residori, and G. Assanto, “Soliton gating and switching in liquid crystal light valve,” Appl. Phys. Lett. **96**, 071104 (2010). [CrossRef]

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13. L. Columbo, C. Rizza, M. Brambilla, F. Prati, and G. Tissoni, “Controlling cavity solitons by means of photorefractive soliton electro-activation,” Opt. Lett. **37**, 4696–4698 (2012). [CrossRef] [PubMed]

## 2. Model for CS formation in a coupled PRC-VCSEL system

13. L. Columbo, C. Rizza, M. Brambilla, F. Prati, and G. Tissoni, “Controlling cavity solitons by means of photorefractive soliton electro-activation,” Opt. Lett. **37**, 4696–4698 (2012). [CrossRef] [PubMed]

*holding beam*(HB).

*λ*= 0.5

*μ*m, (ii) a readout at the near-infrared wavelength

*λ*= 0.85

*μ*m. In the writing phase, when the PRC is not coupled with the VCSEL, a refractive index modulation is encoded in form of two solitonic waveguides exploiting the photorefractive effect in the PRC. In the readout phase two Gaussian beams are launched through the PRC and linearly propagate through the previously imprinted refractive index landscape. The latter can be modified on the scale of tens of nanoseconds [20

20. N. Sapiens, A. Weissbrod, and A. J. Agranat, “Fast electroholographic switching,” Opt. Lett. **34**, 353–355 (2009). [CrossRef] [PubMed]

*V*. In particular the guiding or (anti–guiding) character of the solitonic channels is determined by the sign of

_{e}*V*. Depending on the resulting phase and intensity profile at the exit of the PRC a CS can be switched on in the VCSEL. We observe that, contrary to the case discussed in [13

_{e}**37**, 4696–4698 (2012). [CrossRef] [PubMed]

*λ*= 0.85

*μ*m) is injected into the VCSEL to sustain stable CSs emission after the extinction of the Gaussian beams.

21. E. DelRe, A. Ciattoni, and E. Palange, “Role of charge saturation in photorefractive dynamics of micron-sized beams and departure from soliton behavior,” Phys. Rev. E **73**, 017601 (2006). [CrossRef]

22. A. Ciattoni, E. DelRe, A. Marini, and C. Rizza, “Wiggling and bending-free micron-sized solitons in periodically biased photorefractives,” Opt. Express **16**, 10867–10872 (2008). [CrossRef] [PubMed]

23. M. Brambilla, L. A. Lugiato, F. Prati, L. Spinelli, and W. J. Firth, “Spatial soliton pixels in semiconductor devices,” Phys. Rev. Lett. **79**, 2042–2045 (1997). [CrossRef]

24. L. Spinelli, G. Tissoni, M. Brambilla, F. Prati, and L. A. Lugiato, “Spatial solitons in semiconductor microcavities,” Phys. Rev. A **58**, 2542–2559 (1998). [CrossRef]

### Writing phase

*λ*=0.5

*μ*m) is launched into an externally biased paraelectric crystal where charges are photo-generated. The charges spatio-temporal dynamics due to diffusion and the applied potential modify the quasi static electric field inside the crystal thus influencing in turn light propagation through the dependence of the refractive index by its intensity. This photorefractive effect is characterized by typical time scale of few seconds.

*ρ*(

*x*,

*z*,

*t*) in the PRC due to optical photo-excitation and spatial redistribution is given by the charge continuity equation [21

21. E. DelRe, A. Ciattoni, and E. Palange, “Role of charge saturation in photorefractive dynamics of micron-sized beams and departure from soliton behavior,” Phys. Rev. E **73**, 017601 (2006). [CrossRef]

22. A. Ciattoni, E. DelRe, A. Marini, and C. Rizza, “Wiggling and bending-free micron-sized solitons in periodically biased photorefractives,” Opt. Express **16**, 10867–10872 (2008). [CrossRef] [PubMed]

*∂*,

_{x}*∂*),

_{z}*ϕ*satisfies the Poisson equation ∇

^{2}

*ϕ*= −

*ρ*/(

*ε*

_{0}

*ε*) where

_{r}*ε*

_{0}is the vacuum permittivity and

*ε*is the relative static permittivity,

_{r}*μ*and

*q*are the electron mobility and charge respectively,

*K*is the Boltzmann constant,

_{B}*T*is the crystal temperature and is the electron density, where

_{PR}*N*and

_{a}*N*are the acceptor and donor impurity density, respectively. Note that

_{d}*β*is the rate of thermal excitation of electrons,

*γ*is the electron-ionized trap recombination rate,

*S*= 1 +

*ρ*/(

*qN*),

_{a}*χ*=

*γN*,

_{a}/β*Q*= 1 + |

*E*|

_{PR}/E_{b}^{2},

*E*(

_{PR}*x*,

*z*,

*t*) is the slowly-varying amplitude of the optical electric field polarized along

*x*(at wavelength

*λ*=0.5

_{PR}*μ*m) and

*E*is the amplitude of the uniform background illumination. Furthermore, the optical field dynamics is described by [21

_{b}21. E. DelRe, A. Ciattoni, and E. Palange, “Role of charge saturation in photorefractive dynamics of micron-sized beams and departure from soliton behavior,” Phys. Rev. E **73**, 017601 (2006). [CrossRef]

22. A. Ciattoni, E. DelRe, A. Marini, and C. Rizza, “Wiggling and bending-free micron-sized solitons in periodically biased photorefractives,” Opt. Express **16**, 10867–10872 (2008). [CrossRef] [PubMed]

*n*is the uniform background refractive index,

_{PR}*k*= 2

_{PR}*πn*,

_{PR}/λ_{PR}*g*is the effective electro-optic coefficient and

*δn*is the refractive index change associated with the standard quadratic electro-optic response of paraelectric crystals.

*δn*has the form of two solitonic waveguides, encoded in this phase. This can be achieved by applying a potential

*V*=10

_{s}*V*and injecting into the PRC two Gaussians of shape given by

*E*(

_{PR,s}*x*,

*z*= 0) =

*E*× 1.06 (

_{b}*e*

^{−(x−X0)2}/2

*σ*

^{2}+

*e*

^{−(x+X0)2}/2

*σ*

^{2}), where

*X*

_{0}=20

*μ*m,

*σ*≃10

*μ*m [25

25. M. Segev and A. J. Agranat, “Spatial solitons in centrosymmetric photorefractive media,” Opt. Lett. **22**, 1299–1301 (1997). [CrossRef]

### Readout phase

*λ*=0.85

*μ*m) are launched and propagate in the PRC. During propagation they acquire an intensity and phase modulation that depends on the refractive index landscape imprinted in the writing phase and on the values of the electroactivation potential. The resulting optical field at the exit of the PRC, because of the coupled PRC/VCSEL configuration, is injected into the driven VCSEL where it represents the CS addressing field.

23. M. Brambilla, L. A. Lugiato, F. Prati, L. Spinelli, and W. J. Firth, “Spatial soliton pixels in semiconductor devices,” Phys. Rev. Lett. **79**, 2042–2045 (1997). [CrossRef]

24. L. Spinelli, G. Tissoni, M. Brambilla, F. Prati, and L. A. Lugiato, “Spatial solitons in semiconductor microcavities,” Phys. Rev. A **58**, 2542–2559 (1998). [CrossRef]

*E*(

_{V}*x*,

*t*) is the slowly-varying amplitude of the optical electric field polarized along

*x*(at wavelength

*λ*=0.85

_{V}*μ*m) and scaled to the characteristic amplitude

*E*

_{0},

*τ*is the photon decay time,

_{p}*θ*is the cavity detuning between

*ω*= 2

*πc/λ*and the closest cavity resonance,

_{V}*C*is the gain-to-loss ratio,

*α*is the linewidth enhancement factor,

*N*is the carriers density scaled to the transparency value

*N*

_{0},

*k*= 2

_{V}*πn*and

_{V}/λ_{V}*n*is the background refractive index. In the readout phase, where the PRC and VCSEL are coupled, the injected field

_{V}*E*is the the sum of the field at the exit of the PRC and the

_{I}*holding beam E*∈

_{HB}*ℛ*suitably normalized:

*E*

_{0}is associated with the saturation intensity

*L*is the length of the region filled by the active medium (generally

_{A}*I*

_{0}∼ 10KW/cm

^{2}), and

*T*is the mirror transmissivity. In the carrier density equation,

*I*is the pump current,

_{p}*τ*is the carriers density decay time and we neglect radiative decay and carriers diffusion.

_{e}**16**, 10867–10872 (2008). [CrossRef] [PubMed]

24. L. Spinelli, G. Tissoni, M. Brambilla, F. Prati, and L. A. Lugiato, “Spatial solitons in semiconductor microcavities,” Phys. Rev. A **58**, 2542–2559 (1998). [CrossRef]

26. In all simulations, we used the following parameters: L_{x}= 200μ m, L_{PR}= 1mm, L_{V}= 2μ m, χ= 10^{4}, N_{a}= 3.04 · 10^{22}m^{−3}, N_{d}= 101 · N_{a}, n_{PR}= 2.4, g= 0.13m^{4}C^{−2}, ε_{r}= 3 · 10^{4}, α= 5, θ= −2, C= 0.45, I_{p}= 2, τ_{p}= 11.7ps, n_{V}= 3.5, τ_{e}= 1ns, R= 1 − T= 0.996 and L_{A}= 50nm,

**58**, 2542–2559 (1998). [CrossRef]

*D*device such as a VCSEL, we reduced it to a single transverse dimension to deal with the more simple case of 1

*D*solitonic waveguides in the PRC. This is also equivalent of assuming a rectangular VCSEL [27

27. S. Gronenborn, J. Pollmann-Retsch, P. Pekarski, M. Miller, M. Strösser, J. Kolb, H. Mönch, and P. Loosen, “High-power VCSELs with a rectangular aperture,” Appl. Phys B **105**, 783–792 (2011). [CrossRef]

*V*.

_{e}## 3. INPUT and OUTPUT channels definition. Logic operations

*A*and

*B*are represented by two Gaussian beams launched through the imprinted waveguides in the PC at the positions

*X*=−10

_{A}*μ*m and

*X*=10

_{B}*μ*m with

*σ*=10

*μ*m,

*t*

_{1}−

*t*

_{0}≃10 ns,

*ϕ*=

*π*/2 being the phase

*ϕ*referred to the phase of the HB. We assume that

*u*

_{0}≠ 0 in

*X*(

_{A}*X*) corresponds to bit ’1’ in

_{B}*A*(

*B*), while

*u*

_{0}=0 in

*X*(

_{A}*X*) corresponds to bit ’0’ in

_{B}*A*(

*B*). The input beams propagate through the previously imprinted solitonic pattern electroactivated by

*V*. At the exit of the PRC the resulting refractive index profile is approximated by [28

_{e}28. E. DelRe, M. Tamburrini, and A. J. Agranat, “Soliton electro-optic effects in paraelectrics,” Opt. Lett., **25**, 963–965 (2000). [CrossRef]

*V*. In particular for negative values of

_{e}*V*, the radiation is confined around the position

_{e}*X*=0

*μ*m. As an example in Fig. 2 we plot the numerically calculated refractive index

*δn*(

*x*,

*z*) for

*V*= ±37.5 V. Depending of the choice of

_{e}*V*the induced modulation of the phase and intensity profile at the PRC exit may or may not switch on a CS in the VCSEL [23

_{e}23. M. Brambilla, L. A. Lugiato, F. Prati, L. Spinelli, and W. J. Firth, “Spatial soliton pixels in semiconductor devices,” Phys. Rev. Lett. **79**, 2042–2045 (1997). [CrossRef]

*X*=0

_{Out}*μ*m, so that the occurrence of a CS in that position is taken as a ’1’ and ’0’ otherwise.

**37**, 4696–4698 (2012). [CrossRef] [PubMed]

*X*up to few microns distant from the device center

_{Out}*X*= 0

*μ*m. This would not cause any readout error in the realistic hypothesis that the pick-up area of the readout system (e.g. a photodetector) centered at

*X*=0

*μ*m is typically ∼ 100

*μ*m

^{2}.

*V*in the scheme just described, and in this sense we say that they are concomitant.

_{e}### 3.1. Logic operation: AND

*V*=−37.5 V. For this value of the applied potential the field amplitude and phase at the PRC exit is such that a CS is created and persists (

_{e}*t*≫

*t*

_{1}) only if

*u*

_{0}≠ 0 for both the two beams, thus reproducing the truth table of the AND operator. Only the constructive interference of the two input beams that are deviated towards the PRC center by the electroactivated waveguides is able to generate an injected field amplitude |

*E*| that locally brings the system beyond the CSs separatrix. In Fig. 3 we plot the field amplitude |

_{I}*E*| in the PRC (bottom panel), the field amplitude |

_{PR}*E*| injected into the VCSEL (black line in the upper panel) for

_{I}*t*

_{0}≤

*t*≤

*t*

_{1}, and the field amplitude in the VCSEL at steady state |

*E*| (red line in the upper panel). The two different initial conditions correspond to a (1, 1) input (left) and to a (0, 1) input (right). The (1, 0) case is spatially symmetrical with respect to the (0, 1) case.

_{V}### 3.2. Logic operation: OR

*V*=−42 V as shown in Fig. 4. For this value of the applied potential, the phase and amplitude profile of the field at the PRC exit associated with a single input Gaussian beam is sufficient to switch on a stable CS, thus reproducing the truth table of the OR operator. As in the previous case, the two input beams sum up constructively during propagation, but even a single input beam injected into the VCSEL becomes a local perturbation capable to switch a CS on.

_{e}### 3.3. Logic operation: NOT (and XOR)

*V*= −40 V as shown in Fig. 5. The amplitude and phase landscape at the exit of the PRC is such that in presence of an input beam, a stable CS forms in the VCSEL only in absence of the other input beam, thus reproducing the truth table of the NOT operator. In this case the electroactivated waveguides are such that the two input beams interfere destructively during propagation, while a single beam is sufficient to switch a CS on as in the OR gate. To realize a NOT A (NOT B), the value of the input B (A) is always set to ’1’. We observe that if we use this scheme as a binary operator, thus letting both inputs A and B to assume values ’0’ or ’1’, the gate obtained works as a XOR. The latter combined with the AND and the OR logic gates makes a FULL ADDER, i.e. the digital circuit used to add two binary numbers in any computer [29].

_{e}## 4. Discussion. Logic gates performance

*V*. The AND gate appears to be most sensitive to the

_{e}*V*values and can be achieved within an interval of amplitude 0.3 V (for the input beams parameters used in Figs. 3 and 5), which is anyway not critical for a properly stabilized electronic control.

_{e}*E*|

^{SC}^{2}that affects the refractive index modulations

*δn*and then the solitonic propagation (self-bending) [22

**16**, 10867–10872 (2008). [CrossRef] [PubMed]

30. E. DelRe, B. Crosignani, and P. Di Porto, “Chapter 3 Photorefractive solitons and their underlying nonlocal physics,” Prog. Opt. **53**, 153–200 (2009). [CrossRef]

**16**, 10867–10872 (2008). [CrossRef] [PubMed]

*μ*m) [30

30. E. DelRe, B. Crosignani, and P. Di Porto, “Chapter 3 Photorefractive solitons and their underlying nonlocal physics,” Prog. Opt. **53**, 153–200 (2009). [CrossRef]

*X*= 0

*μ*m we chose as pick-up location [13

**37**, 4696–4698 (2012). [CrossRef] [PubMed]

10. T. Maggipinto, M. Brambilla, G. K. Harkness, and W. J. Firth, “Cavity solitons in semiconductor microresonators: Existence, stability, and dynamical properties,” Phys. Rev. E **62**, 8726–8739 (2000). [CrossRef]

31. X. Hachair, L. Furfaro, J. Javaloyes, M. Giudici, S. Balle, J. Tredicce, G. Tissoni, L. A. Lugiato, M. Brambilla, and T. Maggipinto, “Cavity-solitons switching in semiconductor microcavities,” Phys. Rev. A **72**, 013815 (2005). [CrossRef]

20. N. Sapiens, A. Weissbrod, and A. J. Agranat, “Fast electroholographic switching,” Opt. Lett. **34**, 353–355 (2009). [CrossRef] [PubMed]

*μ*m

^{2}that corresponds to a potential flux ∼100

*GigaOps/*(s× mm

^{2}).

*A*(we remind that the input

*B*is always equal to ’1’ in the NOT A operator). A similar analysis for a solitary semiconductor laser has been recently reported in a case where the direct digital modulation was included in the pump current [32

32. M. Ahmed and M. Yamada, “Effect of intensity noise of semiconductor lasers on the digital modulation characteristics and the bit error rate of optical communication systems,” J. Appl. Phys. **104**, 013104 (2008). [CrossRef]

*X*were superimposed. We note that in order to account for CS drifting towards its steady state location, we plotted the field intensity |

_{Out}*E*|

_{V,m}^{2}in the VCSEL mediated over an interval of 20

*μ*m centered at

*X*= 0

*μ*m. Each operation cycle consists in: 1) assign a value ’1’ or ’0’ to the input A by injecting a pulse or not into the PRC, 2) let the VCSEL evolve towards the steady state exhibiting a CS or not and then let the output OUT acquire a logical ’1’ or ’0’ value respectively, 3) reset of the output OUT to the ’0’ level. The reset is obtained by reducing to zero the HB for less than a nanosecond.

*A*. We observe that, apart from the intrinsic time scale associated with the waveguides electro-activation and the formation of a CS in a VCSEL, the duration of each cycle is affected by the response time of the readout system and the frequency of the shutter or of any other field amplitude modulator used to reset the output. We arbitrarily consider here a total duration of a single cycle of ∼15 ns.

*E*and

_{V}*N*on the system performances by adding Langevin noise sources

*σf*(

*x*,

*t*) and

*σg*(

*x*,

*t*) into Eqs.(5) and (6) respectively (refer for example to [32

32. M. Ahmed and M. Yamada, “Effect of intensity noise of semiconductor lasers on the digital modulation characteristics and the bit error rate of optical communication systems,” J. Appl. Phys. **104**, 013104 (2008). [CrossRef]

*f*(

*x*,

*t*) and

*g*(

*x*,

*t*), with zero mean and

*δ*correlation in space and time describe stochastic processes such as as spontaneous emission or pump noise, and field (e.g. HB) fluctuations. To integrate the resulting stochastic differential equations, we used a standard numerical scheme based on modified version of the Runge-Kutta algorithm as described in details for instance in [33

33. M. San Miguel and R. Toral, “Stochastic effects in physical systems,” in *Nonlinear Phenomena and Complex Systems*, Vol. 5 of Instabilities and Nonequilibrium Structures VI (Kluwer Academic, 2000), pp. 35–127. [CrossRef]

*σ*(that are almost equally spaced on a log

_{10}scale).

*σ*= 0.001 and

*σ*= 0.03.

- The horizontal opening of the eye diagram, i.e. the interval where the output signal can be sampled with ’fidelity’, i.e. where the the ’1’ or ’0’ levels can be correctly identified, has a fixed duration starting after the extinction of the input beam at
*t*=*t*_{1}and lasting up to the start of the reset phase. Above the CS onset transient (less than ∼ 1 ns) it is essentially determined by the response time of the readout device (that we suppose of the order of few nanoseconds) and, instead, not limited by the stochastic dynamics. The widest part of the eye diagram, also called the ’sampling’ or ’decision’ time, occurs very soon after the extinction of the input signals and the eye width remains about the same for a time that can be in principle much longer than typical readout detector times. This almost fixed amplitude eye diagram is different from those associated with digital systems based on continuos signals, e.g. raised cosine pulses, where the eye opening strongly depend on the shape of the waveforms used to send multiple bits, or equivalently by the transmission bandwidth (see for e.g. Fig. 9 in [34]). A further estimation of the eye diagram quality, and in particular of the eye aperture level, is represented by the signal Q-factor defined as [3234. C. H. Wu, F. Tan, M. K. Wu, M. Feng, and N. Holonyak, “The effect of microcavity laser recombination lifetime on microwave bandwidth and eye-diagram signal integrity,” J. Appl. Phys.

**109**, 053112 (2011). [CrossRef]] where**104**, 013104 (2008). [CrossRef]*I*and_{j}*σ*,_{j}*j*= 0, 1 represent the average intensity and standard deviation of the ’0’ and ’1’ levels respectively. The calculated*Q*is high, having a maximum value of ≃ 200 for_{signal}*σ*= 0.001 and a minimum value of ≃ 5 for*σ*= 0.03, and this implies a low probability of incorrect identification of either ’1’ or ’0’ (bit error rate). - The amount of distortion that affects the ’1’ and the ’0’ output levels, which is directly linked to the SNR, increases with noise up to the ’closed’ eye condition reached for
*σ*= 0.1. The SNR is defined here as:*SNR*_{1}and*SNR*_{0}have their maximum values of ≃ 25 db for*σ*= 0.001 and their minimum values of ≃ 10 db for*σ*= 0.03. - As reported in the inset of Fig. 6(b), by increasing the noise level we also observe a decrease down to the intrinsic limit (i.e. the slow carriers dynamics) of ≃ 2 ns in the turn-on delay (TON) time, i.e. the time needed by the VCSEL to reach half of the intensity difference
*I*_{1}−*I*_{0}. This is due to an increase of the average field modulus |*E*| under addition of a noise process. Moreover even the maximum value of |_{I}*E*| in each point increases with_{I}*σ*. These two combined effects lead the system closer to the CS separatrix in a shorter time.An increased turn-on delay jitter (TOJ), i.e. the standard deviation of the turn-on delay, can be seen in the eye diagram by increasing the noise up to*σ*= 0.03. It increases up to a maximum value of ≃4 ns for*σ*= 0.01 and it decreases down to ≃ 1 ns for*σ*= 0.03 (see the inset of Fig. 6(b)). While the TOJ increment is associated with the increase in the amplitude of the noise induced fluctuations in the injected beam modulus and phase, a clear interpretation of the TOJ non monotonic dependence from*σ*is still object of our study. It might be ascribed to the fact that in absence of the addressing beams a noise amplitude of e.g.*σ*= 0.03 is sufficient to bring the system closer to the CS branch. In presence of the Gaussian beams this would change the system trajectories towards the CS branch respect to the cases with smaller*σ*and then strongly modify the turn-on delay characteristics and the associated jitter.

## 5. Conclusions

*GigaOps/*(s× mm

^{2}. Robustness against fluctuations in the input parameters is assured by the fixed intensity and the dynamical stability of the output result (formation of a CS). This is also confirmed by the simulated eye diagram for the logical NOT that shows integrity against noise in terms of low signal distortion, almost constant eye aperture and limited jitter in the CS turn-on delay time.

## Acknowledgments

## References and links

1. | O. Firstenberg, T. Peyronel, Q. Liang, A. V. Gorshkov, M. D. Lukin, and V. Vuletic, “Attractive photons in a quantum nonlinear medium,” Nature |

2. | R. Keil, M. Heinrich, F. Dreisow, T. Pertsch, A. Tünnermann, S. Nolte, D. N. Christodulides, and A. Szameit, “All-optical routing and switching for three-dimensional photonic circuitry,” Sci. Rep. |

3. | H. Wei, Z. Wang, X. Tian, M. Käll, and H. Xu, “Cascaded logic gates in nanophotonic plasmon networks,” Nat. Commun. |

4. | T. Kanna and M. Lakshmanan, “Exact soliton solutions of coupled nonlinear Schrödinger equations: Shape-changing collisions, logic gates, and partially coherent solitons,” Phys. Rev. E |

5. | S. Barland, J. R. Tredicce, M. Brambilla, L. A. Lugiato, S. Balle, M. Giudici, T. Maggipinto, L. Spinelli, G. Tissoni, T. Knödl, M. Miller, and R. Jäger, “Cavity solitons as pixels in semiconductor microcavities,” Nature |

6. | P. Genevet, S. Barland, M. Giudici, and J. R. Tredicce, “Cavity soliton laser based on mutually coupled semiconductor microresonators,” Phys. Rev. Lett. |

7. | F. Pedaci, P. Genevet, S. Barland, M. Giudici, and J. R. Tredicce, “Positioning cavity solitons with a phase mask,” Appl. Phys. Lett. |

8. | F. Pedaci, G. Tissoni, S. Barland, M. Giudici, and J. Tredicce, “Mapping local defects of extended media using localized structures,” Appl. Phys. Lett. |

9. | M. Eslami and R. Kheradmand, “All optical logic gates based on cavity solitons with nonlinear gain,” Opt. Rev. |

10. | T. Maggipinto, M. Brambilla, G. K. Harkness, and W. J. Firth, “Cavity solitons in semiconductor microresonators: Existence, stability, and dynamical properties,” Phys. Rev. E |

11. | R. Landauer, “Irreversibility and heat generation in the computing process,” IBM J. Res. Dev. |

12. | A. Bérut, A. Arakelyan, A. Petrosyan, S. Ciliberto, R. Dillenschneider, and E. Lutz, “Experimental verification of Landauer’s principle linking information and thermodynamics,” Nature |

13. | L. Columbo, C. Rizza, M. Brambilla, F. Prati, and G. Tissoni, “Controlling cavity solitons by means of photorefractive soliton electro-activation,” Opt. Lett. |

14. | R. McLeod, K. Wagner, and S. Blair, “(3+1)-dimensional optical soliton dragging logic,” Phys. Rev. A |

15. | S. V. Serak, N. V. Tabiryan, M. Peccianti, and G. Assanto, “Spatial soliton all-optical logic gates,” IEEE Photonics Technol. Lett. |

16. | A. Piccardi, A. Alberucci, U. Bertolozzo, S. Residori, and G. Assanto, “Soliton gating and switching in liquid crystal light valve,” Appl. Phys. Lett. |

17. | A. Jacobo, D. Gomila, M. A. Matias, and P. Colet, “Logical operations with localized structures,” New J. Phys. |

18. | M. Peccianti, C. Conti, G. Assanto, A. De Luca, and C. Umeton, “All-optical switching and logic gating with spatial solitons in liquid crystals,” Appl. Phys. Lett. |

19. | H. Enderton, |

20. | N. Sapiens, A. Weissbrod, and A. J. Agranat, “Fast electroholographic switching,” Opt. Lett. |

21. | E. DelRe, A. Ciattoni, and E. Palange, “Role of charge saturation in photorefractive dynamics of micron-sized beams and departure from soliton behavior,” Phys. Rev. E |

22. | A. Ciattoni, E. DelRe, A. Marini, and C. Rizza, “Wiggling and bending-free micron-sized solitons in periodically biased photorefractives,” Opt. Express |

23. | M. Brambilla, L. A. Lugiato, F. Prati, L. Spinelli, and W. J. Firth, “Spatial soliton pixels in semiconductor devices,” Phys. Rev. Lett. |

24. | L. Spinelli, G. Tissoni, M. Brambilla, F. Prati, and L. A. Lugiato, “Spatial solitons in semiconductor microcavities,” Phys. Rev. A |

25. | M. Segev and A. J. Agranat, “Spatial solitons in centrosymmetric photorefractive media,” Opt. Lett. |

26. | In all simulations, we used the following parameters: L |

27. | S. Gronenborn, J. Pollmann-Retsch, P. Pekarski, M. Miller, M. Strösser, J. Kolb, H. Mönch, and P. Loosen, “High-power VCSELs with a rectangular aperture,” Appl. Phys B |

28. | E. DelRe, M. Tamburrini, and A. J. Agranat, “Soliton electro-optic effects in paraelectrics,” Opt. Lett., |

29. | L. M. Surhone, M. T. Timpledon, and S. F. Marseken, |

30. | E. DelRe, B. Crosignani, and P. Di Porto, “Chapter 3 Photorefractive solitons and their underlying nonlocal physics,” Prog. Opt. |

31. | X. Hachair, L. Furfaro, J. Javaloyes, M. Giudici, S. Balle, J. Tredicce, G. Tissoni, L. A. Lugiato, M. Brambilla, and T. Maggipinto, “Cavity-solitons switching in semiconductor microcavities,” Phys. Rev. A |

32. | M. Ahmed and M. Yamada, “Effect of intensity noise of semiconductor lasers on the digital modulation characteristics and the bit error rate of optical communication systems,” J. Appl. Phys. |

33. | M. San Miguel and R. Toral, “Stochastic effects in physical systems,” in |

34. | C. H. Wu, F. Tan, M. K. Wu, M. Feng, and N. Holonyak, “The effect of microcavity laser recombination lifetime on microwave bandwidth and eye-diagram signal integrity,” J. Appl. Phys. |

**OCIS Codes**

(190.5330) Nonlinear optics : Photorefractive optics

(190.5970) Nonlinear optics : Semiconductor nonlinear optics including MQW

(190.6135) Nonlinear optics : Spatial solitons

**ToC Category:**

Optics in Computing

**History**

Original Manuscript: November 22, 2013

Revised Manuscript: January 13, 2014

Manuscript Accepted: February 3, 2014

Published: March 18, 2014

**Citation**

L. L. Columbo, C. Rizza, M. Brambilla, F. Prati, and G. Tissoni, "A concomitant and complete set of nonvolatile all-optical logic gates based on hybrid spatial solitons," Opt. Express **22**, 6934-6947 (2014)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-6-6934

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### References

- O. Firstenberg, T. Peyronel, Q. Liang, A. V. Gorshkov, M. D. Lukin, V. Vuletic, “Attractive photons in a quantum nonlinear medium,” Nature 502, 71–76 (2013). [CrossRef] [PubMed]
- R. Keil, M. Heinrich, F. Dreisow, T. Pertsch, A. Tünnermann, S. Nolte, D. N. Christodulides, A. Szameit, “All-optical routing and switching for three-dimensional photonic circuitry,” Sci. Rep. 1(94) 1–6 (2011). [CrossRef]
- H. Wei, Z. Wang, X. Tian, M. Käll, H. Xu, “Cascaded logic gates in nanophotonic plasmon networks,” Nat. Commun. 2(387), 1–5 (2011). [CrossRef]
- T. Kanna, M. Lakshmanan, “Exact soliton solutions of coupled nonlinear Schrödinger equations: Shape-changing collisions, logic gates, and partially coherent solitons,” Phys. Rev. E 67, 046617(2003). [CrossRef]
- S. Barland, J. R. Tredicce, M. Brambilla, L. A. Lugiato, S. Balle, M. Giudici, T. Maggipinto, L. Spinelli, G. Tissoni, T. Knödl, M. Miller, R. Jäger, “Cavity solitons as pixels in semiconductor microcavities,” Nature 419, 699–702 (2002). [CrossRef] [PubMed]
- P. Genevet, S. Barland, M. Giudici, J. R. Tredicce, “Cavity soliton laser based on mutually coupled semiconductor microresonators,” Phys. Rev. Lett. 101, 123905 (2008). [CrossRef] [PubMed]
- F. Pedaci, P. Genevet, S. Barland, M. Giudici, J. R. Tredicce, “Positioning cavity solitons with a phase mask,” Appl. Phys. Lett. 89, 221111 (2006). [CrossRef]
- F. Pedaci, G. Tissoni, S. Barland, M. Giudici, J. Tredicce, “Mapping local defects of extended media using localized structures,” Appl. Phys. Lett. 93, 111104 (2008). [CrossRef]
- M. Eslami, R. Kheradmand, “All optical logic gates based on cavity solitons with nonlinear gain,” Opt. Rev. 19, 242–246 (2012). [CrossRef]
- T. Maggipinto, M. Brambilla, G. K. Harkness, W. J. Firth, “Cavity solitons in semiconductor microresonators: Existence, stability, and dynamical properties,” Phys. Rev. E 62, 8726–8739 (2000). [CrossRef]
- R. Landauer, “Irreversibility and heat generation in the computing process,” IBM J. Res. Dev. 5, 183–191 (1961). [CrossRef]
- A. Bérut, A. Arakelyan, A. Petrosyan, S. Ciliberto, R. Dillenschneider, E. Lutz, “Experimental verification of Landauer’s principle linking information and thermodynamics,” Nature 483, 187–190 (2012). [CrossRef]
- L. Columbo, C. Rizza, M. Brambilla, F. Prati, G. Tissoni, “Controlling cavity solitons by means of photorefractive soliton electro-activation,” Opt. Lett. 37, 4696–4698 (2012). [CrossRef] [PubMed]
- R. McLeod, K. Wagner, S. Blair, “(3+1)-dimensional optical soliton dragging logic,” Phys. Rev. A 52, 3254–3278 (1995). [CrossRef] [PubMed]
- S. V. Serak, N. V. Tabiryan, M. Peccianti, G. Assanto, “Spatial soliton all-optical logic gates,” IEEE Photonics Technol. Lett. 18, 1287–1289 (2006). [CrossRef]
- A. Piccardi, A. Alberucci, U. Bertolozzo, S. Residori, G. Assanto, “Soliton gating and switching in liquid crystal light valve,” Appl. Phys. Lett. 96, 071104 (2010). [CrossRef]
- A. Jacobo, D. Gomila, M. A. Matias, P. Colet, “Logical operations with localized structures,” New J. Phys. 14, 013040 (2012). [CrossRef]
- M. Peccianti, C. Conti, G. Assanto, A. De Luca, C. Umeton, “All-optical switching and logic gating with spatial solitons in liquid crystals,” Appl. Phys. Lett. 81, 3335–3337 (2002). [CrossRef]
- H. Enderton, A Mathematical Introduction to Logic (Academic2001).
- N. Sapiens, A. Weissbrod, A. J. Agranat, “Fast electroholographic switching,” Opt. Lett. 34, 353–355 (2009). [CrossRef] [PubMed]
- E. DelRe, A. Ciattoni, E. Palange, “Role of charge saturation in photorefractive dynamics of micron-sized beams and departure from soliton behavior,” Phys. Rev. E 73, 017601 (2006). [CrossRef]
- A. Ciattoni, E. DelRe, A. Marini, C. Rizza, “Wiggling and bending-free micron-sized solitons in periodically biased photorefractives,” Opt. Express 16, 10867–10872 (2008). [CrossRef] [PubMed]
- M. Brambilla, L. A. Lugiato, F. Prati, L. Spinelli, W. J. Firth, “Spatial soliton pixels in semiconductor devices,” Phys. Rev. Lett. 79, 2042–2045 (1997). [CrossRef]
- L. Spinelli, G. Tissoni, M. Brambilla, F. Prati, L. A. Lugiato, “Spatial solitons in semiconductor microcavities,” Phys. Rev. A 58, 2542–2559 (1998). [CrossRef]
- M. Segev, A. J. Agranat, “Spatial solitons in centrosymmetric photorefractive media,” Opt. Lett. 22, 1299–1301 (1997). [CrossRef]
- In all simulations, we used the following parameters: Lx= 200μ m, LPR= 1mm, LV= 2μ m, χ= 104, Na= 3.04 · 1022m−3, Nd= 101 · Na, nPR= 2.4, g= 0.13m4C−2, εr= 3 · 104, α= 5, θ= −2, C= 0.45, Ip= 2, τp= 11.7ps, nV= 3.5, τe= 1ns, R= 1 − T= 0.996 and LA= 50nm, EHB/(E0T)=0.77
- S. Gronenborn, J. Pollmann-Retsch, P. Pekarski, M. Miller, M. Strösser, J. Kolb, H. Mönch, P. Loosen, “High-power VCSELs with a rectangular aperture,” Appl. Phys B 105, 783–792 (2011). [CrossRef]
- E. DelRe, M. Tamburrini, A. J. Agranat, “Soliton electro-optic effects in paraelectrics,” Opt. Lett., 25, 963–965 (2000). [CrossRef]
- L. M. Surhone, M. T. Timpledon, S. F. Marseken, XOR Gate (Betascript, 2010).
- E. DelRe, B. Crosignani, P. Di Porto, “Chapter 3 Photorefractive solitons and their underlying nonlocal physics,” Prog. Opt. 53, 153–200 (2009). [CrossRef]
- X. Hachair, L. Furfaro, J. Javaloyes, M. Giudici, S. Balle, J. Tredicce, G. Tissoni, L. A. Lugiato, M. Brambilla, T. Maggipinto, “Cavity-solitons switching in semiconductor microcavities,” Phys. Rev. A 72, 013815 (2005). [CrossRef]
- M. Ahmed, M. Yamada, “Effect of intensity noise of semiconductor lasers on the digital modulation characteristics and the bit error rate of optical communication systems,” J. Appl. Phys. 104, 013104 (2008). [CrossRef]
- M. San Miguel, R. Toral, “Stochastic effects in physical systems,” in Nonlinear Phenomena and Complex Systems, Vol. 5 of Instabilities and Nonequilibrium Structures VI (Kluwer Academic, 2000), pp. 35–127. [CrossRef]
- C. H. Wu, F. Tan, M. K. Wu, M. Feng, N. Holonyak, “The effect of microcavity laser recombination lifetime on microwave bandwidth and eye-diagram signal integrity,” J. Appl. Phys. 109, 053112 (2011). [CrossRef]

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