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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 22, Iss. 7 — Apr. 7, 2014
  • pp: 8349–8366
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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, and Shuwen Yang  »View Author Affiliations


Optics Express, Vol. 22, Issue 7, pp. 8349-8366 (2014)
http://dx.doi.org/10.1364/OE.22.008349


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Abstract

A type of tunable femtosecond soliton logic gate based on fiber Raman Self-Frequency Shift (SFS) is studied in this paper. The Raman SFSs of femtosecond solitons governed by the Newton’s cradle mechanism in logic gate are analyzed with an Improved Split-Step Fast Fourier Transform (ISSFFT) algorithm. The impact factors of the solitonic pulse frequency shift and temporal time shift, which are included the Third-Order Dispersion (TOD) effect, are investigated. The existing theoretical equation of SFS is modified into a new expression for this type of soliton logic gate. A lower switching power and the small size of the soliton logic gate device is designed to realize the logic functions of AND, NOT, and XOR. The results demonstrate that the logic gate based on SFS is belonged to the asynchronous system and can be achieved with Milli-Watt switching power and good extinction ratio. ISSFFT is effective and accurately to analyze higher-order dispersive and nonlinear effects in the logic gates.

© 2014 Optical Society of America

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

1. J. H. Kim, Y. M. Jhon, Y. T. Byun, S. Lee, D. H. Woo, and S. H. Kim, “All-optical xor gate using semiconductor optical amplifiers without additional input beam,” IEEE Photon. Technol. Lett. 14, 1436–1438 (2002). [CrossRef]

,2

2. K. Chan, C.-K. Chan, L. K. Chen, and F. Tong, “Demonstration of 20-gb/s all-optical xor gate by four-wave mixing in semiconductor optical amplifier with rz-dpsk modulated inputs,” IEEE Photon. Technol. Lett. 16, 897–899 (2004). [CrossRef]

] are highly switching efficiency, lowly power consumption, easily integration [3

3. J. M. Dailey, S. K. Ibrahim, R. J. Manning, R. P. Webb, S. Lardenois, G. D. Maxwell, and A. J. Poustie, “42.6 gbit/s fully integrated all-optical xor gate,” Electron. Lett. 45, 1047–1049 (2009). [CrossRef]

], etc. Assisted with an interferometer, the transmission rates of a SOA-MZI all-optical gates had been reported to 160Gb/s [4

4. R. Webb, R. Manning, G. Maxwell, and A. Poustie, “40 gbit/s all-optical xor gate based on hybrid-integrated mach-zehnder interferometer,” Electron. Lett. 39, 79–81 (2003). [CrossRef]

, 5

5. S. Randel, A. M. de Melo, K. Petermann, V. Marembert, and C. Schubert, “Novel scheme for ultrafast all-optical xor operation,” J. Lightwave Technol. 22, 2808–2815 (2004). [CrossRef]

]. Even functional operation of logical OR (XOR), NOR (NXOR) and the other multi-functional operations have realized in some studies [6

6. I. Kang, M. Rasras, L. Buhl, M. Dinu, S. Cabot, M. Cappuzzo, L. Gomez, Y. Chen, S. Patel, and N. Dutta, “All-optical xor and xnor operations at 86.4 gb/s using a pair of semiconductor optical amplifier mach-zehnder interferometers,” Opt. Express 17, 19062–19066 (2009). [CrossRef]

, 7

7. Y. Feng, X. Zhao, L. Wang, and C. Lou, “High-performance all-optical or/nor logic gate in a single semiconductor optical amplifier with delay interference filtering,” Appl. Opt. 48, 2638–2641 (2009). [CrossRef] [PubMed]

]. 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

8. C. Yu, L. Christen, T. Luo, Y. Wang, Z. Pan, L.-S. Yan, and A. E. Willner, “All-optical xor gate using polarization rotation in single highly nonlinear fiber,” IEEE Photon. Technol. Lett. 17, 1232–1234 (2005). [CrossRef]

,9

9. J. Wang, Q. Sun, and J. Sun, “All-optical 40 gbit/s csrz-dpsk logic xor gate and format conversion using four-wave mixing,” Opt. Express 17, 12555–12563 (2009). [CrossRef] [PubMed]

], and controlled polarization dependent [10

10. B.-E. Olsson and P. A. Andrekson, “Polarization-independent all-optical and-gate using randomly birefringent fiber in a nonlinear optical loop mirror,” in “Optical Fiber Communication Conference and Exhibit, 1998. OFC’98., Technical Digest,” (IEEE), pp. 375–376.

]. 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

11. R. Radhakrishnan, M. Lakshmanan, and J. Hietarinta, “Inelastic collision and switching of coupled bright solitons in optical fibers,” Phys. Rev. E 56, 2213–2216 (1997). [CrossRef]

13

13. 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. 81, 3335–3337 (2002). [CrossRef]

]. 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

14. M. N. Islam, C. E. Soccolich, and D. A. Miller, “Low-energy ultrafast fiber soliton logic gates,” Opt. Lett. 15, 909–911 (1990). [CrossRef] [PubMed]

16

16. K. Steiglitz, “Time-gated manakov spatial solitons are computationally universal,” Phys. Rev. E 63, 016608(2001). [CrossRef]

], band gap-soliton logic gates in fiber bragg grating [17

17. D. Taverner, N. Broderick, D. Richardson, M. Ibsen, and R. Laming, “All-optical and gate based on coupled gap-soliton formation in a fiber bragg grating,” Opt. Lett. 23, 259–261 (1998). [CrossRef]

, 18

18. Y. P. Shapira and M. Horowitz, “Optical and gate based on soliton interaction in a fiber bragg grating,” Opt. Lett. 32, 1211–1213 (2007). [CrossRef] [PubMed]

] or in the special saturated medium [13

13. 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. 81, 3335–3337 (2002). [CrossRef]

, 19

19. J. Scheuer and M. Orenstein, “All-optical gates facilitated by soliton interactions in a multilayered kerr medium,” JOSA B 22, 1260–1267 (2005). [CrossRef]

, 20

20. S. V. Serak, N. V. Tabiryan, M. Peccianti, and G. Assanto, “Spatial soliton all-optical logic gates,” IEEE Photon. Technol. Lett. 18, 1287–1289 (2006). [CrossRef]

]. 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

20. S. V. Serak, N. V. Tabiryan, M. Peccianti, and G. Assanto, “Spatial soliton all-optical logic gates,” IEEE Photon. Technol. Lett. 18, 1287–1289 (2006). [CrossRef]

].

Soliton self-frequency shift is a phenomenon of frequency red shifting arisen from intra-soliton Raman Scattering Effects (RSE) [21

21. G. Agrawal, Nonlinear Fiber Optics Principles and Applications (Electronic Industry Press, 2002).

, 22

22. J. P. Gordon, “Theory of the soliton self-frequency shift,” Opt. Lett. 11, 662–664 (1986). [CrossRef] [PubMed]

]. Since the soliton SFS in quartz optical fibers was observed in 1986 [23

23. F. M. Mitschke and L. F. Mollenauer, “Discovery of the soliton self-frequency shift,” Opt. Lett. 11, 659–661 (1986). [CrossRef] [PubMed]

], there are many applications, such as supercontinuum spectrum generation [24

24. C. L. Hagen, J. W. Walewski, and S. T. Sanders, “Generation of a continuum extending to the midinfrared by pumping zblan fiber with an ultrafast 1550-nm source,” IEEE Photon. Technol. Lett. 18, 91–93 (2006). [CrossRef]

], optical comb-like device [25

25. C. Xu and X. Liu, “Photonic analog-to-digital converter using soliton self-frequency shift and interleaving spectral filters,” Opt. Lett. 28, 986–988 (2003). [CrossRef] [PubMed]

], all-optical tunable delay line [26

26. S. Oda and A. Maruta, “All-optical tunable delay line based on soliton self-frequency shift and filtering broadened spectrum due to self-phase modulation,” Opt. Express 14, 7895–7902 (2006). [CrossRef] [PubMed]

] and so on [27

27. J. H. Lee, J. van Howe, C. Xu, and X. Liu, “Soliton self-frequency shift: experimental demonstrations and applications,” IEEE J. Select. Topics in Quantum Electron. 14, 713–723 (2008). [CrossRef]

]. Though soliton SFS can realize optical- intensity dependent switching [28

28. J. Lucek and K. Blow, “Optical-intensity dependent switching using soliton self-frequency shift,” Electron. Lett. 27, 882–884 (1991). [CrossRef]

], 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

29. A. Bendahmane, O. Vanvincq, A. Mussot, and A. Kudlinski, “Control of the soliton self-frequency shift dynamics using topographic optical fibers,” Opt. Lett. 38, 3390–3393 (2013). [CrossRef] [PubMed]

31

31. K. Zhe, Y. Jin-Hui, L. Sha, X. Song-Lin, Y. Bin-Bin, S. Xin-Zhu, and Y. Chong-Xiu, “Six-bit all-optical quantization using photonic crystal fiber with soliton self-frequency shift and pre-chirp spectral compression techniques,” Chin. Phy. B. 22, 114211 (2013). [CrossRef]

]. 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

table-icon
View This 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

21. G. Agrawal, Nonlinear Fiber Optics Principles and Applications (Electronic Industry Press, 2002).

]:
iuξsgn(β2)122uτ2+|u|2u=iδ33uτ3isτ(|u|2u)+τRu|u|2τ
(1)
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 τR 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 τR = TR/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 TR 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 PN=N2|β2|γT02. Therefore the highly nonlinear effects in PCF were utilized to reduce the power of solitons in [32

32. B. Memarzadeh Isfahani, T. Ahamdi Tameh, N. Granpayeh, and A. R. Maleki Javan, “All-optical nor gate based on nonlinear photonic crystal microring resonators,” JOSA B 26, 1097–1102 (2009). [CrossRef]

, 33

33. P. Andalib and N. Granpayeh, “All-optical ultracompact photonic crystal and gate based on nonlinear ring resonators,” JOSA B 26, 10–16 (2009). [CrossRef]

].

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

34. T. Lakoba and D. Kaup, “Influence of the raman effect on dispersion-managed solitons and their interchannel collisions,” Opt. Lett. 24, 808–810 (1999). [CrossRef]

]:
dΔfdz=τRT02(2ln2π)γP0(1+S2)1/2
(3)
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

35. A. Zheltikov, “Perturbative analytical treatment of adiabatically moderated soliton self-frequency shift,” Phys. Rev. E 75, 037603 (2007). [CrossRef]

] 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

36. D. Skryabin, F. Luan, J. Knight, and P. S. J. Russell, “Soliton self-frequency shift cancellation in photonic crystal fibers,” Science 301, 1705–1708 (2003). [CrossRef] [PubMed]

], 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

30. R. Driben and B. A. Malomed, “Generation of tightly compressed solitons with a tunable frequency shift in raman-free fibers,” Opt. Lett. 38, 3623–3626 (2013). [CrossRef] [PubMed]

], when the initial N-soliton is split into N distinct fundamental solitons. The fission is governed by the Newton’s cradle mechanism [37

37. R. Driben, B. Malomed, A. Yulin, and D. Skryabin, “Newton’s cradles in optics: From n-soliton fission to soliton chains,” Phys. Rev. A 87, 063808 (2013). [CrossRef]

] 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

36. D. Skryabin, F. Luan, J. Knight, and P. S. J. Russell, “Soliton self-frequency shift cancellation in photonic crystal fibers,” Science 301, 1705–1708 (2003). [CrossRef] [PubMed]

, 38

38. A. A. Voronin and A. M. Zheltikov, “Soliton self-frequency shift decelerated by self-steepening,” Opt. Lett. 33, 1723–1725 (2008). [CrossRef] [PubMed]

]. 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

21. G. Agrawal, Nonlinear Fiber Optics Principles and Applications (Electronic Industry Press, 2002).

].

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

21. G. Agrawal, Nonlinear Fiber Optics Principles and Applications (Electronic Industry Press, 2002).

], 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

40. O. V. Sinkin, R. Holzlhner, J. Zweck, and C. R. Menyuk, “Optimization of the split-step fourier method in modeling optical-fiber communications systems,” J. Lightwave Technol. 21, 61 (2003). [CrossRef]

]. The symmetrized split-step Fourier method [21

21. G. Agrawal, Nonlinear Fiber Optics Principles and Applications (Electronic Industry Press, 2002).

], 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 κz)3. This algorithm is able to amplify the nonlinear effects such as four-wave mixing (FWM) [41

41. M. Ablowitz and H. Segur, “Solitons, nonlinear evolution equations and inverse scattering. by m. j,” J. Fluid Mech 244, 721–725 (1992).

]. 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

42. A. Nakamura, “A direct method of calculating periodic wave solutions to nonlinear evolution equations. ii. exact one-and two-periodic wave solution of the coupled bilinear equations,” Journal of the Physical Society of Japan 48, 1365–1370 (1980). [CrossRef]

], 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 κz)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).

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

21. G. Agrawal, Nonlinear Fiber Optics Principles and Applications (Electronic Industry Press, 2002).

]. 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 τR 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.

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 ΔωTOD 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, τR 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 ΔωTOD 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. ΔωTODT07 is described by the fitting function y = ax−7 in Fig. 3(a). It can be similarly found that ΔωTODZ5/2 when z is changed from 1.065km to 6.065km in Fig. 3(b). ΔωTODβ34/5 in Fig. 3(c) is fitted with the same conditions (the power of the first-order soliton is 232.443W, β2 is −20ps2/km, τR 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 ΔωTOD with T0, z and β3, which are shown in (a), (b) and (c), respectively.

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).

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

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, TR 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 (Δωz), 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

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

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
Eout=10α202(Ein1+Ein2)
(5)
where parameter α is power attenuation coefficient of coupler, Ein1 and Ein2 are pulse energy of the signal 1 and 2, and Eout 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

21. G. Agrawal, Nonlinear Fiber Optics Principles and Applications (Electronic Industry Press, 2002).

]:
Pout=Eout2=(12P0)2=12P0
(6)
Pout=Eout2=(12(P0+P0))2=2P0
(7)
where the first order of soliton power is P0 and Ein1=Ein2=P0. 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.
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.

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.

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 Ps (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:
Pout0=Eout2=(122P0)2=P0
(8)
Pout1=Eout2=(12(2P0+Ps))2=P0+2P0Ps+12Ps
(9)

For example, the power of the first-order soliton P0 is about 6.74W and Ps is 1W, we obtain that Pout0 and Pout1 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 Pout0 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 Pout1 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:
(υυ0)T0=(193.1THz192.8THz)0.2556ps=0.07668.
(10)
According to Eq. (2), the theoretical red-shifting value of the first-order soliton in the normalized form as
(υυ0)T0=8|β2|TRz15T03=8×20×0.002×0.0615×0.25563=0.07665.
(11)
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.

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 Pout1 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

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)

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 ΔτTOD = 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.

References and links

1.

J. H. Kim, Y. M. Jhon, Y. T. Byun, S. Lee, D. H. Woo, and S. H. Kim, “All-optical xor gate using semiconductor optical amplifiers without additional input beam,” IEEE Photon. Technol. Lett. 14, 1436–1438 (2002). [CrossRef]

2.

K. Chan, C.-K. Chan, L. K. Chen, and F. Tong, “Demonstration of 20-gb/s all-optical xor gate by four-wave mixing in semiconductor optical amplifier with rz-dpsk modulated inputs,” IEEE Photon. Technol. Lett. 16, 897–899 (2004). [CrossRef]

3.

J. M. Dailey, S. K. Ibrahim, R. J. Manning, R. P. Webb, S. Lardenois, G. D. Maxwell, and A. J. Poustie, “42.6 gbit/s fully integrated all-optical xor gate,” Electron. Lett. 45, 1047–1049 (2009). [CrossRef]

4.

R. Webb, R. Manning, G. Maxwell, and A. Poustie, “40 gbit/s all-optical xor gate based on hybrid-integrated mach-zehnder interferometer,” Electron. Lett. 39, 79–81 (2003). [CrossRef]

5.

S. Randel, A. M. de Melo, K. Petermann, V. Marembert, and C. Schubert, “Novel scheme for ultrafast all-optical xor operation,” J. Lightwave Technol. 22, 2808–2815 (2004). [CrossRef]

6.

I. Kang, M. Rasras, L. Buhl, M. Dinu, S. Cabot, M. Cappuzzo, L. Gomez, Y. Chen, S. Patel, and N. Dutta, “All-optical xor and xnor operations at 86.4 gb/s using a pair of semiconductor optical amplifier mach-zehnder interferometers,” Opt. Express 17, 19062–19066 (2009). [CrossRef]

7.

Y. Feng, X. Zhao, L. Wang, and C. Lou, “High-performance all-optical or/nor logic gate in a single semiconductor optical amplifier with delay interference filtering,” Appl. Opt. 48, 2638–2641 (2009). [CrossRef] [PubMed]

8.

C. Yu, L. Christen, T. Luo, Y. Wang, Z. Pan, L.-S. Yan, and A. E. Willner, “All-optical xor gate using polarization rotation in single highly nonlinear fiber,” IEEE Photon. Technol. Lett. 17, 1232–1234 (2005). [CrossRef]

9.

J. Wang, Q. Sun, and J. Sun, “All-optical 40 gbit/s csrz-dpsk logic xor gate and format conversion using four-wave mixing,” Opt. Express 17, 12555–12563 (2009). [CrossRef] [PubMed]

10.

B.-E. Olsson and P. A. Andrekson, “Polarization-independent all-optical and-gate using randomly birefringent fiber in a nonlinear optical loop mirror,” in “Optical Fiber Communication Conference and Exhibit, 1998. OFC’98., Technical Digest,” (IEEE), pp. 375–376.

11.

R. Radhakrishnan, M. Lakshmanan, and J. Hietarinta, “Inelastic collision and switching of coupled bright solitons in optical fibers,” Phys. Rev. E 56, 2213–2216 (1997). [CrossRef]

12.

O. V. Kolokoltsev, R. Salas, and V. Vountesmeri, “All-optical phase-independent logic elements based on phase shift induced by coherent soliton collisions,” J. Lightwave Technol. 20, 1048 (2002). [CrossRef]

13.

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. 81, 3335–3337 (2002). [CrossRef]

14.

M. N. Islam, C. E. Soccolich, and D. A. Miller, “Low-energy ultrafast fiber soliton logic gates,” Opt. Lett. 15, 909–911 (1990). [CrossRef] [PubMed]

15.

M. W. Chbat, B. Hong, M. N. Islam, C. E. Soccolich, and P. R. Prucnal, “Ultrafast soliton-trapping and gate,” J. Lightwave Technol. 10, 2011–2016 (1992). [CrossRef]

16.

K. Steiglitz, “Time-gated manakov spatial solitons are computationally universal,” Phys. Rev. E 63, 016608(2001). [CrossRef]

17.

D. Taverner, N. Broderick, D. Richardson, M. Ibsen, and R. Laming, “All-optical and gate based on coupled gap-soliton formation in a fiber bragg grating,” Opt. Lett. 23, 259–261 (1998). [CrossRef]

18.

Y. P. Shapira and M. Horowitz, “Optical and gate based on soliton interaction in a fiber bragg grating,” Opt. Lett. 32, 1211–1213 (2007). [CrossRef] [PubMed]

19.

J. Scheuer and M. Orenstein, “All-optical gates facilitated by soliton interactions in a multilayered kerr medium,” JOSA B 22, 1260–1267 (2005). [CrossRef]

20.

S. V. Serak, N. V. Tabiryan, M. Peccianti, and G. Assanto, “Spatial soliton all-optical logic gates,” IEEE Photon. Technol. Lett. 18, 1287–1289 (2006). [CrossRef]

21.

G. Agrawal, Nonlinear Fiber Optics Principles and Applications (Electronic Industry Press, 2002).

22.

J. P. Gordon, “Theory of the soliton self-frequency shift,” Opt. Lett. 11, 662–664 (1986). [CrossRef] [PubMed]

23.

F. M. Mitschke and L. F. Mollenauer, “Discovery of the soliton self-frequency shift,” Opt. Lett. 11, 659–661 (1986). [CrossRef] [PubMed]

24.

C. L. Hagen, J. W. Walewski, and S. T. Sanders, “Generation of a continuum extending to the midinfrared by pumping zblan fiber with an ultrafast 1550-nm source,” IEEE Photon. Technol. Lett. 18, 91–93 (2006). [CrossRef]

25.

C. Xu and X. Liu, “Photonic analog-to-digital converter using soliton self-frequency shift and interleaving spectral filters,” Opt. Lett. 28, 986–988 (2003). [CrossRef] [PubMed]

26.

S. Oda and A. Maruta, “All-optical tunable delay line based on soliton self-frequency shift and filtering broadened spectrum due to self-phase modulation,” Opt. Express 14, 7895–7902 (2006). [CrossRef] [PubMed]

27.

J. H. Lee, J. van Howe, C. Xu, and X. Liu, “Soliton self-frequency shift: experimental demonstrations and applications,” IEEE J. Select. Topics in Quantum Electron. 14, 713–723 (2008). [CrossRef]

28.

J. Lucek and K. Blow, “Optical-intensity dependent switching using soliton self-frequency shift,” Electron. Lett. 27, 882–884 (1991). [CrossRef]

29.

A. Bendahmane, O. Vanvincq, A. Mussot, and A. Kudlinski, “Control of the soliton self-frequency shift dynamics using topographic optical fibers,” Opt. Lett. 38, 3390–3393 (2013). [CrossRef] [PubMed]

30.

R. Driben and B. A. Malomed, “Generation of tightly compressed solitons with a tunable frequency shift in raman-free fibers,” Opt. Lett. 38, 3623–3626 (2013). [CrossRef] [PubMed]

31.

K. Zhe, Y. Jin-Hui, L. Sha, X. Song-Lin, Y. Bin-Bin, S. Xin-Zhu, and Y. Chong-Xiu, “Six-bit all-optical quantization using photonic crystal fiber with soliton self-frequency shift and pre-chirp spectral compression techniques,” Chin. Phy. B. 22, 114211 (2013). [CrossRef]

32.

B. Memarzadeh Isfahani, T. Ahamdi Tameh, N. Granpayeh, and A. R. Maleki Javan, “All-optical nor gate based on nonlinear photonic crystal microring resonators,” JOSA B 26, 1097–1102 (2009). [CrossRef]

33.

P. Andalib and N. Granpayeh, “All-optical ultracompact photonic crystal and gate based on nonlinear ring resonators,” JOSA B 26, 10–16 (2009). [CrossRef]

34.

T. Lakoba and D. Kaup, “Influence of the raman effect on dispersion-managed solitons and their interchannel collisions,” Opt. Lett. 24, 808–810 (1999). [CrossRef]

35.

A. Zheltikov, “Perturbative analytical treatment of adiabatically moderated soliton self-frequency shift,” Phys. Rev. E 75, 037603 (2007). [CrossRef]

36.

D. Skryabin, F. Luan, J. Knight, and P. S. J. Russell, “Soliton self-frequency shift cancellation in photonic crystal fibers,” Science 301, 1705–1708 (2003). [CrossRef] [PubMed]

37.

R. Driben, B. Malomed, A. Yulin, and D. Skryabin, “Newton’s cradles in optics: From n-soliton fission to soliton chains,” Phys. Rev. A 87, 063808 (2013). [CrossRef]

38.

A. A. Voronin and A. M. Zheltikov, “Soliton self-frequency shift decelerated by self-steepening,” Opt. Lett. 33, 1723–1725 (2008). [CrossRef] [PubMed]

39.

P. Mamyshev and L. Mollenauer, “Stability of soliton propagation with sliding-frequency guiding filters,” Opt. Lett. 19, 2083–2085 (1994). [CrossRef] [PubMed]

40.

O. V. Sinkin, R. Holzlhner, J. Zweck, and C. R. Menyuk, “Optimization of the split-step fourier method in modeling optical-fiber communications systems,” J. Lightwave Technol. 21, 61 (2003). [CrossRef]

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M. Ablowitz and H. Segur, “Solitons, nonlinear evolution equations and inverse scattering. by m. j,” J. Fluid Mech 244, 721–725 (1992).

42.

A. Nakamura, “A direct method of calculating periodic wave solutions to nonlinear evolution equations. ii. exact one-and two-periodic wave solution of the coupled bilinear equations,” Journal of the Physical Society of Japan 48, 1365–1370 (1980). [CrossRef]

43.

C. Luo, M. Ibanescu, S. G. Johnson, and J. D. Joannopoulos, “Cerenkov radiation in photonic crystals,” Science 299, 368–371 (2003). [CrossRef] [PubMed]

OCIS Codes
(190.5530) Nonlinear optics : Pulse propagation and temporal solitons
(190.5650) Nonlinear optics : Raman effect
(200.4660) Optics in computing : Optical logic
(320.7140) Ultrafast optics : Ultrafast processes in fibers

ToC Category:
Optics in Computing

History
Original Manuscript: February 4, 2014
Revised Manuscript: March 4, 2014
Manuscript Accepted: March 5, 2014
Published: April 1, 2014

Citation
Ming Xu, Yan Li, Tiansheng Zhang, Jun Luo, Jianhua Ji, and Shuwen Yang, "The analysis of all-optical logic gates based with tunable femtosecond soliton self-frequency shift," Opt. Express 22, 8349-8366 (2014)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-7-8349


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References

  1. J. H. Kim, Y. M. Jhon, Y. T. Byun, S. Lee, D. H. Woo, S. H. Kim, “All-optical xor gate using semiconductor optical amplifiers without additional input beam,” IEEE Photon. Technol. Lett. 14, 1436–1438 (2002). [CrossRef]
  2. K. Chan, C.-K. Chan, L. K. Chen, F. Tong, “Demonstration of 20-gb/s all-optical xor gate by four-wave mixing in semiconductor optical amplifier with rz-dpsk modulated inputs,” IEEE Photon. Technol. Lett. 16, 897–899 (2004). [CrossRef]
  3. J. M. Dailey, S. K. Ibrahim, R. J. Manning, R. P. Webb, S. Lardenois, G. D. Maxwell, A. J. Poustie, “42.6 gbit/s fully integrated all-optical xor gate,” Electron. Lett. 45, 1047–1049 (2009). [CrossRef]
  4. R. Webb, R. Manning, G. Maxwell, A. Poustie, “40 gbit/s all-optical xor gate based on hybrid-integrated mach-zehnder interferometer,” Electron. Lett. 39, 79–81 (2003). [CrossRef]
  5. S. Randel, A. M. de Melo, K. Petermann, V. Marembert, C. Schubert, “Novel scheme for ultrafast all-optical xor operation,” J. Lightwave Technol. 22, 2808–2815 (2004). [CrossRef]
  6. I. Kang, M. Rasras, L. Buhl, M. Dinu, S. Cabot, M. Cappuzzo, L. Gomez, Y. Chen, S. Patel, N. Dutta, “All-optical xor and xnor operations at 86.4 gb/s using a pair of semiconductor optical amplifier mach-zehnder interferometers,” Opt. Express 17, 19062–19066 (2009). [CrossRef]
  7. Y. Feng, X. Zhao, L. Wang, C. Lou, “High-performance all-optical or/nor logic gate in a single semiconductor optical amplifier with delay interference filtering,” Appl. Opt. 48, 2638–2641 (2009). [CrossRef] [PubMed]
  8. C. Yu, L. Christen, T. Luo, Y. Wang, Z. Pan, L.-S. Yan, A. E. Willner, “All-optical xor gate using polarization rotation in single highly nonlinear fiber,” IEEE Photon. Technol. Lett. 17, 1232–1234 (2005). [CrossRef]
  9. J. Wang, Q. Sun, J. Sun, “All-optical 40 gbit/s csrz-dpsk logic xor gate and format conversion using four-wave mixing,” Opt. Express 17, 12555–12563 (2009). [CrossRef] [PubMed]
  10. B.-E. Olsson, P. A. Andrekson, “Polarization-independent all-optical and-gate using randomly birefringent fiber in a nonlinear optical loop mirror,” in “Optical Fiber Communication Conference and Exhibit, 1998. OFC’98., Technical Digest,” (IEEE), pp. 375–376.
  11. R. Radhakrishnan, M. Lakshmanan, J. Hietarinta, “Inelastic collision and switching of coupled bright solitons in optical fibers,” Phys. Rev. E 56, 2213–2216 (1997). [CrossRef]
  12. O. V. Kolokoltsev, R. Salas, V. Vountesmeri, “All-optical phase-independent logic elements based on phase shift induced by coherent soliton collisions,” J. Lightwave Technol. 20, 1048 (2002). [CrossRef]
  13. 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]
  14. M. N. Islam, C. E. Soccolich, D. A. Miller, “Low-energy ultrafast fiber soliton logic gates,” Opt. Lett. 15, 909–911 (1990). [CrossRef] [PubMed]
  15. M. W. Chbat, B. Hong, M. N. Islam, C. E. Soccolich, P. R. Prucnal, “Ultrafast soliton-trapping and gate,” J. Lightwave Technol. 10, 2011–2016 (1992). [CrossRef]
  16. K. Steiglitz, “Time-gated manakov spatial solitons are computationally universal,” Phys. Rev. E 63, 016608(2001). [CrossRef]
  17. D. Taverner, N. Broderick, D. Richardson, M. Ibsen, R. Laming, “All-optical and gate based on coupled gap-soliton formation in a fiber bragg grating,” Opt. Lett. 23, 259–261 (1998). [CrossRef]
  18. Y. P. Shapira, M. Horowitz, “Optical and gate based on soliton interaction in a fiber bragg grating,” Opt. Lett. 32, 1211–1213 (2007). [CrossRef] [PubMed]
  19. J. Scheuer, M. Orenstein, “All-optical gates facilitated by soliton interactions in a multilayered kerr medium,” JOSA B 22, 1260–1267 (2005). [CrossRef]
  20. S. V. Serak, N. V. Tabiryan, M. Peccianti, G. Assanto, “Spatial soliton all-optical logic gates,” IEEE Photon. Technol. Lett. 18, 1287–1289 (2006). [CrossRef]
  21. G. Agrawal, Nonlinear Fiber Optics Principles and Applications (Electronic Industry Press, 2002).
  22. J. P. Gordon, “Theory of the soliton self-frequency shift,” Opt. Lett. 11, 662–664 (1986). [CrossRef] [PubMed]
  23. F. M. Mitschke, L. F. Mollenauer, “Discovery of the soliton self-frequency shift,” Opt. Lett. 11, 659–661 (1986). [CrossRef] [PubMed]
  24. C. L. Hagen, J. W. Walewski, S. T. Sanders, “Generation of a continuum extending to the midinfrared by pumping zblan fiber with an ultrafast 1550-nm source,” IEEE Photon. Technol. Lett. 18, 91–93 (2006). [CrossRef]
  25. C. Xu, X. Liu, “Photonic analog-to-digital converter using soliton self-frequency shift and interleaving spectral filters,” Opt. Lett. 28, 986–988 (2003). [CrossRef] [PubMed]
  26. S. Oda, A. Maruta, “All-optical tunable delay line based on soliton self-frequency shift and filtering broadened spectrum due to self-phase modulation,” Opt. Express 14, 7895–7902 (2006). [CrossRef] [PubMed]
  27. J. H. Lee, J. van Howe, C. Xu, X. Liu, “Soliton self-frequency shift: experimental demonstrations and applications,” IEEE J. Select. Topics in Quantum Electron. 14, 713–723 (2008). [CrossRef]
  28. J. Lucek, K. Blow, “Optical-intensity dependent switching using soliton self-frequency shift,” Electron. Lett. 27, 882–884 (1991). [CrossRef]
  29. A. Bendahmane, O. Vanvincq, A. Mussot, A. Kudlinski, “Control of the soliton self-frequency shift dynamics using topographic optical fibers,” Opt. Lett. 38, 3390–3393 (2013). [CrossRef] [PubMed]
  30. R. Driben, B. A. Malomed, “Generation of tightly compressed solitons with a tunable frequency shift in raman-free fibers,” Opt. Lett. 38, 3623–3626 (2013). [CrossRef] [PubMed]
  31. K. Zhe, Y. Jin-Hui, L. Sha, X. Song-Lin, Y. Bin-Bin, S. Xin-Zhu, Y. Chong-Xiu, “Six-bit all-optical quantization using photonic crystal fiber with soliton self-frequency shift and pre-chirp spectral compression techniques,” Chin. Phy. B. 22, 114211 (2013). [CrossRef]
  32. B. Memarzadeh Isfahani, T. Ahamdi Tameh, N. Granpayeh, A. R. Maleki Javan, “All-optical nor gate based on nonlinear photonic crystal microring resonators,” JOSA B 26, 1097–1102 (2009). [CrossRef]
  33. P. Andalib, N. Granpayeh, “All-optical ultracompact photonic crystal and gate based on nonlinear ring resonators,” JOSA B 26, 10–16 (2009). [CrossRef]
  34. T. Lakoba, D. Kaup, “Influence of the raman effect on dispersion-managed solitons and their interchannel collisions,” Opt. Lett. 24, 808–810 (1999). [CrossRef]
  35. A. Zheltikov, “Perturbative analytical treatment of adiabatically moderated soliton self-frequency shift,” Phys. Rev. E 75, 037603 (2007). [CrossRef]
  36. D. Skryabin, F. Luan, J. Knight, P. S. J. Russell, “Soliton self-frequency shift cancellation in photonic crystal fibers,” Science 301, 1705–1708 (2003). [CrossRef] [PubMed]
  37. R. Driben, B. Malomed, A. Yulin, D. Skryabin, “Newton’s cradles in optics: From n-soliton fission to soliton chains,” Phys. Rev. A 87, 063808 (2013). [CrossRef]
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