The analysis of all-optical logic gates based with tunable femtosecond soliton self-frequency shift

Ming Xu; Yan Li; Tiansheng Zhang; Jun Luo; Jianhua Ji; Shuwen Yang

doi:10.1364/OE.22.008349

1. Introduction

According to material composition and working principles, all-optical logic gates can be categorized into three types: Semiconductor Optical Amplifier (SOA)-based gates, fiber nonlinear effects (mainly cross-phase modulation and four-wave mixing effects)-based gates and optical solitons-based gates. The merits of SOA-based all-optical logic gates [1,2] are highly switching efficiency, lowly power consumption, easily integration [3], etc. Assisted with an interferometer, the transmission rates of a SOA-MZI all-optical gates had been reported to 160Gb/s [4, 5]. Even functional operation of logical OR (XOR), NOR (NXOR) and the other multi-functional operations have realized in some studies [6, 7]. But dynamic instabilities of SOA carriers in logic, which introduce additional noises, have so long saturated relaxation time (≈ 100ps) that its response speed is limited. Therefore, a integrated component, based on cascaded multi-SOA-based optical logic gate, may have unbearable response time and can not be applied to the ultra-fast, ultra-short pulse transmission system. The large volume, complex structure and serious polarization dependent also limit its application. The logic gates based on fiber nonlinear effects can overcome above drawbacks because of simple structure, easily integration, fast response (about fs) [8,9], and controlled polarization dependent [10]. But this kind of logic gates have very serious dispersion and complex nonlinearly chirp noise for too long length of fiber.

Optical logic gates based on solitons are taken advantage of its inherent particle properties and associated logical value with changes of solitonic location, time and frequency through the interaction of solitons [11–13]. There are many kinds of optical soliton logic gates, including soliton-dragging logical gates based on self-trapping mechanism, which pulses are maintained with Manakov solitons for polarization effect [14–16], band gap-soliton logic gates in fiber bragg grating [17, 18] or in the special saturated medium [13, 19, 20]. For the stably output values of the soliton logic gates, soliton parameters can be controlled precisely in soliton logic gates. So they are very suitable for cascaded multi-functional logic operations in high speed systems. It is disappointing that the optical soliton logic gates have high switching power (at least an order of w to kw) to real system and seldom run on power level of the milliwatt or microwatt [20].

Soliton self-frequency shift is a phenomenon of frequency red shifting arisen from intra-soliton Raman Scattering Effects (RSE) [21, 22]. Since the soliton SFS in quartz optical fibers was observed in 1986 [23], there are many applications, such as supercontinuum spectrum generation [24], optical comb-like device [25], all-optical tunable delay line [26] and so on [27]. Though soliton SFS can realize optical- intensity dependent switching [28], the optical logic gates based on SFS are rarely involved in studying. The reason may be that it requires in-phase superposition of the fields at the input coupler and essentially kills real prospect of using in a real system. Recently researchers show that soliton SFS can be controlled and tunable in time and frequency domains in Photonic Crystal Fibers (PCF) [29–31]. Inspired by these works in this paper, a tunable unit is designed to check and mark the working frequency to complete the multi-logic function after continue frequency shifting of solitons in PCF.

The soliton logic gates based on fiber Raman SFS are studied in the remaining parts of the paper. The frequency shift and time shift of pulse in soliton logic gates are calculated theoretically, and the existing theoretical equation of SFS is modified into the Eq. (4) with Improved Split-Step Fast Fourier Transform (ISSFFT) method. Then a assembly device (shown in Fig. 7) is designed to achieve logic functions of AND, NOT, and XOR (shown in Table 1) in this soliton logic gate of SFS.

Table 1:. Truth table of SFS-based soliton logic gate of AND, XOR, and NOT

View Table

2. Analysis of the solitonic pulse frequency shift and time shift with ISSFFT and modification of the existing theoretical equation in SFS soliton logic gates

2.1. Analysis of the pulse frequency shift and time shift in SFS soliton logic gates

When the pulse width is narrower than 5ps or the optical pulse is femtosecond scale, the frequency shift induced by Raman scattering can not be ignored. The propagation field of pulse is described by the higher-order nonlinear Schrödinger equation [21]:

i \frac{\partial u}{\partial ξ} - sgn (β_{2}) \frac{1}{2} \frac{\partial^{2} u}{\partial τ^{2}} + {| u |}^{2} u = i δ_{3} \frac{\partial^{3} u}{\partial τ^{3}} - i s \frac{\partial}{\partial τ} ({| u |}^{2} u) + τ_{R} u \frac{\partial {| u |}^{2}}{\partial τ}

where u, ξ and τ are denoted that normalized amplitude envelope of field, the normalized transmission distance and the normalized delay time, respectively. Parameters β₃, s and τ_R are described as the third-order dispersion, the self-steeping and Raman scattering effect, respectively. δ₃ = β₃/(6 |β₂| T₀), s = 1/(ω₀ T₀) and τ_R = T_R/T₀. ω₀ is the center wavelength of the pulse, T₀ is the pulse width, β₂ is the second-order dispersion parameter which describes the group velocity dispersion (GVD), β₃ is TOD parameter and T_R is relates to the Raman response. In the soliton transmission systems, the constrained relationship between pulse width and peak power of Nth-order solitons is

P_{N} = \frac{N^{2} | β_{2} |}{γ T_{0}^{2}}

. Therefore the highly nonlinear effects in PCF were utilized to reduce the power of solitons in [32, 33].

The frequency shift in per unit length arisen from RSE of pulse is roughly about τ⁻⁴, which is found by G. P. Gordon from (1) through averaging method [22]. The Gordon’s formula is valid for soliton durations larger than the Raman response time scale (typically 50 fs in silica). There is a more accurate expression in Ref. [21] through perturbation methods without considering the TOD effects and the self-steepening effects. SFS induced by Raman scattering can be expressed by the next formula:

Δ ω_{R} (z) = - 8 | β_{2} | T_{R} z / (15 T_{0}^{4})

The negative sign shows that the angular’s frequency of carrier is reduced and the frequency spectrum of soliton is shifted toward longer wavelengths (red shifting). The expression (2), which is included in GVD’s factors, is consistent with the findings of Gordon obtained under the condition of normalization in ordinary single mode optical fiber. The differential equation of the frequency shift arisen from Raman scattering in the dispersion-managed soliton system has been gotten in Ref. [34]:

\frac{d Δ f}{d z} = - \frac{τ_{R}}{{T_{0}}^{2}} (\frac{\sqrt{2} ln 2}{π}) \frac{γ P_{0}}{{(1 + S^{2})}^{1 / 2}}

where P₀ and S are represented as the peak power of the input pulse and the dispersion management strength respectively. Δf is frequency shifting in Hz. It is obvious that Eq. (3) is also consistent with Eq. (2) as the dispersion management strength S = 0 and N = 1 (the fundamental soliton or first order soliton) in the dispersion-managed system. The other similarly analytical results have been deduced by Zheltikov [35] in about 20cm length of PCF, in which soliton width is about 30 fs and is short to Raman oscillation period of 78.5 fs.

Gordon’s formula and the other analytical treatments are only considered the dynamic characteristics of solitons in the adiabatic region, where this self-localized pulse of energy is not be dissipated. It is shown that many dispersive waves, such as Cherenkov radiation in dispersive media escaped from main soliton wave’s field [36], would interact and dissipate the main body of soliton, which can acquire an acceleration or deceleration. In addition to fundamental solitons, the higher-order N-solitons are oscillating periodically to restore their shape at distances that are multiples of periods of the fundamental soliton. Higher-order linear and nonlinear effects, such as the self-steepening and TOD, cause fission of N-solitons. There exists a Satsuma-Yajima regime [30], when the initial N-soliton is split into N distinct fundamental solitons. The fission is governed by the Newton’s cradle mechanism [37] in the case of TOD, which explains that fission of N-solitons would emit strong dispersive radiation. The self-steepening effects would decelerate slightly the fundamental soliton SFS if pulse width is greater than 30fs soliton width [36, 38]. The SFS relationship remains τ⁻⁴. A bad situation, of which the self-steepening effects lead to soliton remarkably decayed, would be happened in the high-order solitons system [21].

Thus a three-step scheme is designed to easily control and mark over the trajectory of soliton. Firstly, the fundamental soliton which is lower power than higher order of solitons, is considered as logic working pulse to avoid the Satsuma-Yajima effects. Secondly, the Gordon’s formula is modified into a relatively precise expression included TOD effect through simulating of ISSFFT method, which effectively solves the Eq. (1) when the soliton width is larger than the 30 fs [21]. Thirdly, a tunable Fabry-perot filter named the Sliding-Frequency Guiding Filter (SFGF) [39], which is narrow band frequency filter, is applied to handle the outputs of logic gate. The central frequency of SFGF is gradually shifted monotonically from one position to the next along the transmission line in a nearly complete quenching of both Gordon-Haus and amplitude jitter of solitons. The difference of sliding frequency between the soliton and filter frequency is designed exactly matching quantity of SFS which is gotten from preceding expression of simulation.

There are two types of numerical methods to calculate the high-order nonlinear Schrödinger equation, which are the finite-difference method and the pseudo spectral method. Generally speaking, pseudo spectral methods are faster than the finite-difference method in condition of the same accuracy. The split-step Fourier algorithm is a pseudo-spectral method [21], whose the efficiency are depended on choosing suitable step sizes and time windows. Various criteria selection of step-sizes have been proposed for optimizing the split-step method in Ref. [40]. The symmetrized split-step Fourier method [21], whose step sizes are kept as a constant, can be accurate to the second order of the step size Δz. The third order or higher of the step size Δz are error terms, but the coefficient of the third order error term is a constant κ(Δz)³. This algorithm is able to amplify the nonlinear effects such as four-wave mixing (FWM) [41]. Logarithmic step size method can be used to analyze the nonlinear effects, which is based on the length of transmission fiber and the loss of optical pulse along the propagation distance to select the step size [42], but it has larger error term of the third order. The ISSFFT that presented here can choose step size adaptively, and is thus suitable for investigating Raman and other higher order nonlinear effects. Similar to the symmetrized split-step Fourier method, ISSFFT sets an error of the variant field values between two steps of size Δz and one step of size 2Δz under a specified standard error. When the error is calculated in the range of standard error, step sizes are kept constant, or step sizes are shortened to reduce the error. Therefore, the accuracy of the nonlinear effects (represented phases in the algorithm) is controlled by eliminated the third order error term κ(Δz)³, and calculation accuracy is improved by an order of magnitude.

Taking the symmetrized split-step Fourier method whose step size is set 0.001m as a reference, the single step size errors that are calculated by ISSFFT and the constant step size or fixed step size of FFT method are plotted in Fig. 1(a). It is clear that phase errors of both methods are increased along with the step size, but the single-step phase error of the fixed step size method is increased more rapidly than that of the ISSFFT. The higher-order errors of two methods along different transmission distances are also drawn in Fig. 1(b) while the step sizes of the two methods are same. It is shown that the higher-order error of the fixed step size method is about 0.0012 compared to 0.00036 of ISSFFT method in Fig. 1(b), though the amount of calculation (mainly considering the number of FFT) of the ISSFFT method is about 3/2 times of that the fixed step size method.

Fig. 1: The analysis on phase errors of ISSFFT and the fixed step size FFT method with the single step size errors along the different step sizes is shown in Fig. 1(a) and the higher order errors along the different transmission distances is shown in Fig. 1(b).

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The first- and second-order solitons which propagate 0.5m length of fiber (about 4 times periods of soliton) are calculated by ISSFFT method shown in Figs. 2 (a, b, c, and d) in the time and frequency domains. The pulse width is 50 fs, its wavelength is 1.55μm, and β₂ is −10.016ps²/km. The red dash line of the Fig. 2(a) is ignored the Raman scattering and the TOD effects, the black solid line is considered the Raman scattering only, and the blue dot line is considered both the Raman scattering and TOD effects. The TOD effect modulates the tail of soliton pulse into oscillation structure deeply in the time-domain shown in Fig. 2(a) and the soliton spectrum of SFS and TOD in blue color line are divided into two well-resolved frequency peaks in the frequency-domain shown in Fig. 2 (b), which will impact the soliton XOR-gate in later simulation. These results are very consistent with Ref. [21]. The suppression of TOD effect on soliton is also obvious. When the first-order soliton (the soliton power is 232.443w, the pulse width is 256 fs) propagates through a 5km standard fiber (β₂ is −20ps²/km and τ_R is 3 fs), the frequency shift of the soliton is 4.88THz (not considered the TOD effect). If considering the TOD effect, the frequency shift of soliton is 4.19THz (β₃ = 0.08ps³/km). The difference of frequency shift between two cases mentioned above is 0.69THz (which is about 14% of the total amount). Therefore, SFS can not be accurately calculated with the Eq. (2) and Eq. (2) must be modified in order to consider TOD effect.

Fig. 2: The evolutions of time and frequency domains about the first- and second-order of soliton simulated with ISSFFT method. The time-domain (a) and frequency-domain (b) of a first-order soliton pulse are calculated to show the effect of SFS and TOD. The fission of the second-order soliton in time-domain (c) and frequency domain (d) are caused by SFS.

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The fission and frequency red shifting of soliton caused by RSE are shown in Figs. 2(c and d) when a second-order soliton propagates in a fiber. The numerical parameters for the peak power and the Raman response of the first- and second- order solitons are 2.0703kW, T_R = 3 fs(τ_R = 0.106) and 8.2811kW, T_R = 0.28 fs(τ_R = 0.01), respectively. If the fiber loss (α = 0) is neglected, the full width of pulse at half maximum T_FWHM is 50 fs (T₀ ≈ 28.36 fs). The wavelength of the carrier is 1550nm. The TOD and the self-steepening effects are ignored (δ = 0 and s = 0). The transmission distance is 1m (about 8 times period of soliton). From Figs. 2 (c and d), the spectral width of the pulse is large enough that the Raman gain amplifies the low-frequency (red) spectral components of the pulse with high-frequency (blue) components of the same pulse when pulse width is in order of fs or shorter. It is obvious that the energy from blue components is continuously transferred to red components. The numerical results in this paper are consistent with Ref. [21], which further prove that our proposed method is reliable. We will show later that the peak power of the second-order soliton can be reduced to order of W by introducing the PCF. If more longer fiber link can be tolerated, the peak power can be reduced to order of mW.

2.2. Numerical analysis of SFS and modification of the existing theoretical equation in soliton logic gate

The relationships between SFS with the transmission distance z and the pulse width T₀ are analyzed in Eq. (1) by ISSFFT. The Δω_TOD is the difference between the total amount of frequency shift Δω (considering the TOD effect) and Δω₁ (not considering the TOD effect). The first-order solitons (the power is 232.443W) propagate through the standard optical fiber (here the parameter β₂ is −20ps²/km, τ_R is 3 fs, β₃ is 0.08ps³/km and the length of fiber is 5.13km), the width of the pulse T₀ (the half of width at 1/e amplitude) is changed from 256 fs to 380 fs. The relationship between Δω_TOD and T₀ is shown in Fig. 3(a). The solid line is the simulating datum of ISSFFT, and the dashed line is the math fitting curve. The datum line and the fitting line are well consistent each other. $Δ ω_{TOD} \propto T_{0}^{- 7}$ is described by the fitting function y = ax⁻⁷ in Fig. 3(a). It can be similarly found that Δω_TOD ∝ Z^5/2 when z is changed from 1.065km to 6.065km in Fig. 3(b). $Δ ω_{TOD} \propto β_{3}^{4 / 5}$ in Fig. 3(c) is fitted with the same conditions (the power of the first-order soliton is 232.443W, β₂ is −20ps²/km, τ_R is 3 fs, T₀ is 256 fs, and z is 5.13km) except changing the value of parameters β₃ from 0.01ps³/km to 0.13ps³/km.

Fig. 3: The math fitting curves of the relationships between Δω_TOD with T₀, z and β₃, which are shown in (a), (b) and (c), respectively.

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To sum up, we can get $Δ ω_{TOD} = k T_{0}^{- 7} Z^{5 / 2} β_{3}^{4 / 5}$ , where k is related with β₂, τ_R, the pulse power, the optical wavelength and so on. We get the value of k is 7.9 in this paper parameters. Therefore, the equation of calculating the total frequency shift included the TOD effect is:

Δ ω = - 8 | β_{2} | T_{R} z / (15 T_{0}^{4}) + 7.9 T_{0}^{- 7} Z^{5 / 2} β_{3}^{4 / 5}

Eq. (4) shows that the TOD effect restrains the soliton SFS. In order to get larger frequency shift, TOD must be compensated in logic gates. This viewpoint is very different from the results of previous researches.

2.3. Investigations of pulse time’s delay arisen from SFS

The pulse delay of soliton in the time-domain is corresponded to the SFS of the soliton in the frequency-domain shown in Fig. 4 with ISSFFT. Fig. 4(a) is the time-domain of the first-order soliton without RSE and TOD effects. The solitons are delayed in the time-domain considered the Raman scatting and the TOD effects in Fig. 4(b). As soliton evolutes in the non-adiabatic zone, the energy of soliton is dissipated in form of dispersive waves such as Cherenkov radiation. Taking the first-order soliton (T₀ is 256 fs) through the fiber (β₂ is −20 ps²/km and β₃ is 0.08ps³/km) into account, we observe the soliton delay in the time-domain with changing the distance z from 106.5m to 606.5m.

Fig. 4: The pulse delay arisen from SFS in the time-domain. The first-order solitons without SFS and TOD effects have stably characteristic of transmission in (a), but there are time’s delay caused by the SFS and the TOD effects in (b).

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The relationship between Δτ and the distance z is shown in Fig. 5(a) described with Δτ = k₁Z², which is fitted by the function y = ax². k₁ is related with the fiber nonlinear parameters, the pulse power, the optical wavelength and so on. When these factors are determined, k₁ can be determined and kept unchanged. Under the conditions of the numerical calculation in this paper, the value of k₁ is 55.6, and the equation of the soliton time delay is: Δτ = 55.6Z². When β₃ is varied from 0.01ps³/km to 0.13ps³/km and the first-order soliton have the same parameters of T₀ = 256 fs at 5.13km, the relationship between Δτ_TOD and β₃ is shown in Fig. 5(b). The function of math fitting y = ax in Fig. 5(b) can be described into Δτ_TOD = k₂β₃. Under the conditions of the numerical calculation in this paper, the value of k₂ is 17.3. Then the pulse time’s delay caused by TOD is Δτ_TOD = 17.3β₃. The TOD effect is shown a sort of restraining on soliton SFS and to slow down the soliton pulse delay, which are consisted with Ref. [36].

Fig. 5: The math fitting curves of the relationships among Δτ_TOD and z, β₃, in which (a) is the relationship between Δτ_TOD and z and (b) is the relationship between Δτ_TOD and β₃.

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The research on the soliton pulse delay has great significance to the femtosecond high speed optical logic gate. For example, the logic states of NOT-gate based on capture of soliton are achieved depending on whether the controlled pulse can arrive within a specified time slot.

2.4. The relationship between power and frequency shift of femtosecond solitonic pulses

The frequency shifts resulted from RSE are different with different power, and the logic functions can be implemented by using received signals from different frequency channels. It needs to investigate the relationship between different power of pulses with the frequency shift for realization of the logic gate functions in SFS soliton logic gate. The simulation is carried out by ISSFFT in PCF transmission system in condition of 160Gb/s bit rates. The parameters of PCF are as follows: β₂ is −20ps²/km, β₃ is 0.08ps³/km, T_R is 2 fs (3 fs), the nonlinear parameter γ is 45.38W⁻¹km⁻¹, the length is 60 m, the pulse width of soliton is 255.6 fs (the duty cycle is 0.0720992), and the power of the first-order soliton in PCF is 6.7W which is very less than the power of the soliton in the G.652 fiber.

It is shown from Fig. 6 that the frequency shift of femtosecond soliton is increased linearly with z (Δω ∝ z), and the amounts of SFS of 2P₀ soliton is much larger than that of P₀ (almost an order of magnitude). Therefore, you can increase the value of soliton power to obtain the different widely of frequency shift and to reduce the length of the transmission fiber.

Fig. 6: The relationships between SFS with different powers and parameters of Raman response along with distance of PCF in SFS-based soliton logic gate

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3. Operation of SFS-based all-optical soliton logic AND, NOT, and XOR gates

3.1. SFS-based all-optical logic AND gate

Fig. 7: SFS-based soilton logic AND gate

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Using results of above analysis in given parameters, the SFS of solitons can realize the logic AND, NOT and XOR operation. The scheme of logic AND gate is shown in Fig. 7. The signal 1 and signal 2 are “00 11”, “01 01” respectively while “1” means the incident pulse of the first-order soliton, whose power is P₀. Field energy of soliton after the coupler can be given by

E_{out} = \frac{10^{\frac{- α}{20}}}{\sqrt{2}} (E_{in 1} + E_{in 2})

where parameter α is power attenuation coefficient of coupler, E_in₁ and E_in₂ are pulse energy of the signal 1 and 2, and E_out is the energy of output optical pulse after the coupler. Assuming the power attenuation coefficient is 0, the Eq. (5) can be simplified. The two series of pulses are coupled together and the output powers of coupler would be 0, 0.5P₀, 0.5P₀ and 2P₀ considered combinations of 0 and 1 from signals. For example, when the inputs of two signal are combined into group of “01”, “10” and “11” respectively, the Eq. (5) can be derived as follows [21]:

P_{out} = E_{out}^{2} = {(\frac{1}{\sqrt{2}} \sqrt{P_{0}})}^{2} = \frac{1}{2} P_{0}

P_{out} = E_{out}^{2} = {(\frac{1}{\sqrt{2}} (\sqrt{P_{0}} + \sqrt{P_{0}}))}^{2} = 2 P_{0}

where the first order of soliton power is P₀ and

E_{in 1} = E_{in 2} = \sqrt{P_{0}}

. As the optical pulses of different power have different frequency shifts in Fig. 7, the truth table of AND gate can be obtained in Table 1. When power of pulse is 0.5P₀, the frequency of pulse through PCF moves to υ₁ (but pulse may not be maintained a soliton, the shape of pulses during transmission would change). When power of pulse is 2P₀, the frequency of pulse moves to υ₂. If detecting optical power with frequency υ₂ in detection port, the signal whose frequency is υ₂ with only two input ports can be detected with a designed filter. Here, We design the difference of sliding frequency between the input soliton and filter’s frequency exactly matching self-frequency shift which comes from Eq. (4). If the tunable Fabry-perot filter named as SFGF are exactly tuned to frequency of υ₂ at export, the pulse in υ₂ are gotten to implement the logic AND operation in Fig. 7.

In order to optimize the parameters of logic gates more effectively and avoid cross talk, we can first determine the type of logic gate. Then v₁ and v₂ frequencies and the PCF parameters are designed according to Table 1 and Eq. (4).

The numerical algorithm of ISSFFT is applied to prove the mathematical model of Table 1 in Figs. 8 and 9. when the initial pulse of the signal in Figs. 8(a) and 8(b) is sent respectively, the power of pulse is reduced to 0.5P₀ (about 800mW) and the width is spread as drawn in Figs. 8(c) and 8(d), in which the combination signals are “01” or “10”, respectively. (Note : the time axis of Fig. 8(c) doubles that of Figs. 8(a) and 8(e)). But its center frequency υ₁, which is about 193.1THz in Table 1, is not changed. When the combination signals are “11” in Figs. 8(e) and 8(f), the power after coupler is 2P₀ (about 24W), the width through the fiber becomes narrow, and there is a certain time’s delay. The width of the spectrum is wider than the incident signal and SFS is increased significantly. The center frequency is moved from 193.1THz to 189.5THz, and the difference of υ₁ and υ₂ is up to 4.6THz as υ₂ is 189.5THz in Table 1.

Fig. 8: The numerical results of signals in SFS-based soliton logic AND gate. The initial input signal pulse (a) and its spectrum (b) is sent into the logic AND gate. The pulse of input “01”, “10” and its spectrum after PCF are simulated in (c) and (d), respectively. The pulse of input “11” and its spectrum after PCF are simulated in (e) and (f), respectively. The logic ouput of “0” and “1” after SFGF are simulated in (g) and (h), respectively.

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Fig. 9: The numerical results of data flow in SFS-based all-optical soliton logic AND gate. The output of coupler, PCF and SFGF are simulated in (a), (b) and (c), respectively.

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The optical pulse with center frequency υ₁ has a very weak channel crosstalk against υ₂. Considering a SFGF whose bandwidth is 1.2THz, its center frequency should tuned to υ₂, where the center frequency is 189.5THz. If the combination signals are “01” or “10” and logic AND output is “0” in Fig. 8(g), the results of simulation are shown that the optical power after the optical filter is about 56.4nW (−42.5dBm). But the optical power after the optical filter is about 6.9W (38.3dBm) when the logic AND gate output is “1” in Fig. 8(h). The pulse width in output of the logic “0” can be greatly expanded than the logic “1”. The all-optical logic AND gate has a nice performance of the extinction ratio (about 80.8dB).

Furthermore, we simulate the data flow running in the logic AND gate in Fig. 9. If the input signal 1 and signal 2 is “10010101” and “10101111” respectively, the results of logic AND operation of two signals should be “10000101.” The whole pulses has time’s delay of two bits in Fig. 9 for the red-shifting in the frequency domain and time’s delay in the time domain. The time’s delay about Δτ = k₁Z² makes the logic bits breaking away the original sequence. But each pulse has same delay, clock of signal can be extracted from the signal itself. Therefore, SFS-based logic gates have advantage of that it can applicable for asynchronization systems.

We also observe obvious Cherenkov radiation in Fig. 9(b). Because power pulse is less than the first-order soliton minimum requirements P₀, 0.5P₀ can not maintain the solitonic shape and it decayes into dispersive waves after PCF transmission. The pulse broadening caused by the dispersion waves makes the pulse disappear eventually and power spread into the surrounding to influence the neighbor bits. When the power of pulse after coupler is approximate 2P₀ in combination of “11”, there are exist obviously red shifting of the spectrum. The width of pulses become narrow and the peak power of pulses go higher, which are not equal each other due to the nonlinear effects of peripheral power residues. For only a part of frequency of pulse through the filter, the peak power will be weakened. The appropriate bandwidth of optical filter makes appropriate output power. In order to make logic gates can be cascaded to the next gate, we can choose a suitable bandwidth of optical filter which makes the output power of logical “1” is about P₀. The similar treatments can also be done in the other cascade logic gate. The eye diagram of the random sequence in length of 2⁷ −1 is shown in Fig. 9(c), and the SFS-based all-optical soliton logic AND gate has a good performance. The Q factor is approximately 19.0135, and the minimum bit error rate is about 3.9335e – 81 with further proved by a commercial software package of an Optiwaves System 7.0 in Fig. 9(c).

It is very important to control the Cherenkov radiation. Methods to control the Cherenkov radiation by varying the crystal geometry inside PCF have been suggested in Ref. [43]. We have designed a three-step scheme by adopting first-order soliton and SFGF in SFS-based soliton logic gate to restrain the soliton dissipation and dispersive waves led from the Cherenkov radiation. There have shown good extinction ratio in logic AND gate. More effective methods to optimize system or to accurately quantify in restraining Cherenkov radiation is our next key studying in practical systems, especially in cascaded logic gates systems.

3.2. SFS-based all-optical logic NOT gates

The structure of SFS-based logic NOT gates is similar with the logic AND gates, inputs of one port (such as signal 2) just are a sequence of all constant ‘1’ whose power are 2P₀. The power of signal 1 is P_s (centre of frequency of input pulses are 193.1THz). According to Eq. (5), when the input signal is “0” and “1” in Table 1, the pulse power after coupler is as follows:

P_{out 0} = E_{out}^{2} = {(\frac{1}{\sqrt{2}} \sqrt{2 P_{0}})}^{2} = P_{0}

P_{out 1} = E_{out}^{2} = {(\frac{1}{\sqrt{2}} (\sqrt{2 P_{0}} + \sqrt{P_{s}}))}^{2} = P_{0} + \sqrt{2 P_{0} P_{s}} + \frac{1}{2} \sqrt{P_{s}}

For example, the power of the first-order soliton P₀ is about 6.74W and P_s is 1W, we obtain that P_out₀ and P_out₁ is 6.74W and 10.91W (1.5P₀) after coupler, respectively. If the input signal 1 is injected “0”, the center frequency of pulse with shape of a soliton in power P_out₀ moves to υ₁ about 192.8THz in PCF shown by Fig. 10(a). When the input signal 1 is injected “1”, the center frequency of the power P_out₁ moves to υ₂ about 191.36THz (Fig. 10(d)) and has a relatively large delay through the PCF. Then the filter center frequency is tuned to υ₁ in the port of receiver, the logic NOT gate can be achieved. Here we can also prove that the results of numerical simulation is correct as follows:

(υ - υ_{0}) T_{0} = (193.1 THz - 192.8 THz) 0.2556 p s = 0.07668 .

According to Eq. (2), the theoretical red-shifting value of the first-order soliton in the normalized form as

(υ - υ_{0}) T_{0} = \frac{8 | β_{2} | T_{R} z}{15 T_{0}^{3}} = \frac{8 \times 20 \times 0.002 \times 0.06}{15 \times {0.2556}^{3}} = 0.07665 .

It is shown that our numerical results of the first-order soliton red-shifting in Eq. (10) agree well with the theoretical value in Eq. (11) as not considering TOD effect.

Fig. 10: The numerical results of all-optical logic NOT gate. The spectrum of input “0” and “1” after PCF are simulated in (a) and (b), respectively. The output of PCF and SFGF in logic NOT gate is “01101010” and “10010101” simulated in (c) and (d), respectively.

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If the input signal of data flow is “01101010”, the output signal of data flow should be “10010101” for the operation of logic NOT. The results of numerical simulation are shown in Figs. 10(c) and 10(d), respectively. The pulse with power P₀ generates little delay to work-off its original position, and the pulses with power P_out₁ have delay of a logical bit(Fig. 10(c)). The center frequency of two different power pulses differ in 1.44THz. The narrowband filter should be choosen smartly after transmission as we select the center frequency of the SFGF (192.8THz) and a bandwidth of 0.8THz with a lot of numerical calculations. Shown from the final results of output in Fig. 10(d), the device fully realizes logic NOT function. The pulse at the third logical bit in Fig. 10(d) appears lower power (about 364mW), it is due to the interference of pulses. The logic NOT gate has 9dB extinction ratio for the difference between 34.6dBm and 25.6dBm. Using a random sequence of length of (2⁷ – 1), the eye diagram of the logic NOT gate is given in Fig. 10(d). The value of Q factor is about 3.22652. The minimum bit error rate is about 6.01e – 4. Compared with AND gate, performance of NOT gate have deteriorated for the interference of frequencies (extinction ratio of AND gate is about 80.8 dB).

3.3. SFS-based all-optical soliton logic XOR gates

The structure of logic XOR gate is also similar with the logic AND gate, but the powers of two input pulses are 2P₀ shown in Table 1. When combinations of input signals are “01” or “10”, the power after coupler is $P_{out} = E_{out}^{2} = {(\frac{1}{\sqrt{2}} (\sqrt{2 P_{0}} + 0))}^{2} = P_{0}$ , which is formed as the first-order soliton with moving the center of the frequency to 192.8THz. When combination of input signal is “11”, the output of power is $P_{out} = E_{out}^{2} = {(\frac{1}{\sqrt{2}} (\sqrt{2 P_{0}} + \sqrt{2 P_{0}}))}^{2} = 4 P_{0}$ (about 26.96W) after coupler. The pulse is formed into the second-order soliton with changing the peak power to 77.1W and shifting the center frequency to 158.57THz shown in Fig. 11(b). The time’s delay is about five logical bits (31.25ps) after PCF. There are still power residues between the frequency (192 THz – 194 THz), which will interfere all-optical XOR gates decision. When the input signal 1 and 2 of data flow is “10010011” and “01100110” respectively, the output should be “11110101” for XOR gate. The results of numerical simulation are shown in Fig. 11(c). The central frequency and the bandwidth of SFGF is υ₁ (192.8 THz) and 1THz respectively. Output of the filter is just the result of two signals XOR operation. But the power of output pulse bits is different, especially in the last pulse bit which has lower power compared with other bits and has greatly broadened width. It is shown that the previous pulse of the lower power bit is exactly the second-order soliton at the output of coupler. The second-order soliton has a larger delay than the first-order soliton due to the effect of nonlinear modulation such as XPM. The logic XOR gate extinction ratio is about 6dB. Using a random sequence of length of (2⁷ – 1), the minimum bit error rate is about 4.01e – 3. Therefore, selecting the appropriate decision threshold will be a key factor for the all-optical XOR gates in the practical application.

Fig. 11: The numerical results of all-optical logic XOR gate. The second-order soliton and its spectrum after PCF is simulated in (a) and (b). The output of SFGF in XOR operation is “11110101” shown in (c)

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It is important to consider the relationships among the length of PCF, amount of frequency shift, power of soliton, and filter frequency of detection in designing a SFS-based soliton logic gate. Table 1 and Fig. 6 show that the power of pulse is a major issue that determines the frequency shift in approximate interval of PCF length 40 – 200m. When the logic gates operate on logic function of AND and NOR in this area of length, the frequency difference of two channels may have several THZ or even 10THZ. Then the narrow bandwidth of SFGF is able to distinguish two different channel frequencies easily. The frequency of crosstalk and Cherenkov radiation almost does not affect the performance of logic gates. Numerical studies also show that the two logic gates have a very good extinction ratio. Since there are fissions of second-order soliton in XOR, the XOR performance is worse than the others.

4. Conclusion

A femtosecond SFS-based soliton logic gate is demonstrated to achieve functions of logical operation through detection of different frequencies with a tunable SFGF in Table 1. A three-step scheme is designed to easily control and mark over the trajectory of soliton and to restrain the Cherenkov radiation. When the width of soliton is larger than 30 fs, the relatively precise expressions governed by the Newton’s cradle mechanism are simulated with ISSFFT, in which SFS of soliton is given by Eq. (4) and time’s delay of soliton is given by Δτ_TOD = k₂β₃. The tunable SFGF is applied to handle the output of logic gate in asynchronous way, which difference of frequency with soliton is designed exactly matching SFS from Eq. (4). This type logic gates is simple to structure and easy to implement for highly nonlinear effect of PCF. It reduces switch power of the soliton pulse to Milliwatt and makes logic gates more closer to practical application. The SFS based soliton logic gates have good performance in the extinction ratio that AND, NOT and XOR gates are about 80.8dB, 9dB and 6dB, respectively. The clock signal of this asynchronous logic gates can be extracted from the signal itself, which is very suitable for network transmission.

Acknowledgments

This work is supported by Shenzhen Key Lab of ACIP.

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The analysis of all-optical logic gates based with tunable femtosecond soliton self-frequency shift

Abstract

1. Introduction

2. Analysis of the solitonic pulse frequency shift and time shift with ISSFFT and modification of the existing theoretical equation in SFS soliton logic gates

2.1. Analysis of the pulse frequency shift and time shift in SFS soliton logic gates

2.2. Numerical analysis of SFS and modification of the existing theoretical equation in soliton logic gate

2.3. Investigations of pulse time’s delay arisen from SFS

2.4. The relationship between power and frequency shift of femtosecond solitonic pulses

3. Operation of SFS-based all-optical soliton logic AND, NOT, and XOR gates

3.1. SFS-based all-optical logic AND gate

3.2. SFS-based all-optical logic NOT gates

3.3. SFS-based all-optical soliton logic XOR gates

4. Conclusion

Acknowledgments

References and links

Cited By

Figures (11)

Tables (1)

Equations (11)

Optics Express

Logical Inputs		AND			XOR			NOT
port 1	port 2	Power after Couple	PCF Output Frequency	Probe Port υ₂	Power after Couple	PCF Output Frequency	Probe Port υ₁	Power after Couple		PCF Output Frequency	Probe Port υ₁
0	0	NO	NO	0	NO	NO	0	0	P₀	υ₁	1
0	1	0.5P₀	υ₁	0	P₀	υ₁	1	0	P₀	υ₁	1
1	0	0.5P₀	υ₁	0	P₀	υ₁	1	1	$\frac{3}{2} P_{0}$	υ₂	0
1	1	2P₀	υ₂	1	4P₀	υ₂	0	1	$\frac{3}{2} P_{0}$	υ₂	0