Nonlinear Mixing in Optical Signal Processing

Nonlinear Mixing in Optical Signal Processing Nourin Ahmed

Refat Kibria

Department of Computer Science & Engineering Shahjalal University of Science & Technology Sylhet, Bangladesh [email protected]

Department of Computer Science & Engineering Shahjalal University of Science & Technology Sylhet, Bangladesh [email protected]

Abstract—Kerr-effect based Nonlinear Optical Mixing is reviewed. The superiority of Four Wave Mixing over SPM and XPM in high speed data transmission is discussed. The applications of nonlinear mixing and use of Four Wave Mixing in optical signal processing in recent past is reviewed in the paper. Keywords—Kerr-effect, SPM, XPM, FWM, Optical Signal Processing

1.

INTRODUCTION

Signal processing in electronic domain provides much less processing speed compared to signal processing in optical domain [1]. To enable real time signal processing the photonic processing of an input RF signal has been investigated for past three decades [2]. Generally the input RF signal is modulated on optical carrier signal to enable the processing in optical domain [3]. The nonlinear properties of optical fiber are being used for high speed processing of optical signal [4-8]. Many recent observations have promoted the field of nonlinear effect as an interesting area of academic research. The aim of this paper is to give a review of fiber nonlinearities and discuss the characteristics and applications of Kerr-Nonlinear effect. More emphasis will be put on Four Wave Mixing (FWM) to point out why the FWM effect is preferable for signal processing in ultrafast communication systems [9]. 2.

NONLINEAR MIXING

In optics intensity-dependent phenomena are addressed as nonlinear properties of optics. These phenomena appear under two possible sceneries, either for the change in refractive index of the medium with optical intensity or for inelasticscattering phenomenon [8]. The power dependence of the refractive index is responsible for Kerr-effect [10]. Kerrnonlinearity manifests itself in three different effects known as Self-Phase Modulation(SPM), Cross-Phase Modulation(XPM) and Four Wave Mixing(FWM) [6, 8]. In this paper we are going to discuss about Kerr Nonlinearities and particular emphasis will be put on FWM to promote its advantage in optical signal processing over SPM and XPM. 2.1 Self Phase Modulation Light beam with high optical intensity changes its own phase by changing the effective refractive index of the medium; this effect is referred to as self-phase modulation [6, 9, 11]. Phase change of an optical field is governed by the following equation,

Φ = nk0L = (n+n2|E|2)k0L

(1)

Here n is the refractive index, k0 = 2π/λ and L is the fiber length. The second term of the right side of the equation represents the intensity-dependent nonlinear phase shift ΦNL = n2k0L|E|2 which is due to self-phase modulation [6, 10, 12]. The main effect of SPM is to broaden the spectrum of optical pulses propagating through the fiber [13]. The pulse shape is also affected if SPM and Group Velocity Dispersion(GVD) act together to influence the optical pulse [8, 11]. SPM can modulate amplitude of the pulse[8]. Other application of SPM includes all optical data regeneration technique utilizing spectral broadening characteristic [13, 14], wavelength conversion [15]. SPM based all-optical regenerators are simple because the regenerators don’t need either of pump or probe optical source [5]. 2.2 Cross Phase Modulation When multiple light beams with different frequencies propagate together through optical fiber, changes in refractive index made by one light beam affects other light beams in the fiber. This nonlinear effect is known as cross-phase modulation [6, 8, 10, 16]. As described in [8], in order to understand XPM effect we can start by considering a WDM system with two channels. For such a system the electric field is given by, E(r,t) = ½{E1exp(-iω1t) + E2exp(-iω2t) + c.c}

(2)

The nonlinear polarization is given by: PNL(r,t) = ε0 εNL E(r,t)

(3)

εNL = ¾ χ(3) |E(r,t)|2 = χeff |E(r,t)|2

(4)

Substitution of the two frequency field in the nonlinear polarization yields, PNL(r,t) = ½ [ PNL(ω1) exp(-i ω1t) + PNL(ω2) exp(-i ω2t) + PNL(2ω1 - ω2 ) exp{-i(2ω1 - ω2)t} + PNL(2ω2 - ω1 ) exp{-i(2ω2 - ω1)t}] +c.c

(5)

Here nonlinear contribution to the refractive index given by the terms, PNL(ω1) = χeff (|E1|2 + 2|E2|2) E1

(6)

PNL(ω2) = χeff (|E2|2 + 2|E1|2) E2

(7)

Substituting Equation (6) and Equation (7) into Equation (8) we get, PNL(r,t) = ½ χeff [ (|E1|2 + 2|E2|2) E1 exp(-i ω1t)

(8)

+ (|E2|2 + 2|E1|2) E2 exp(-i ω2t) ] Now the nonlinear contribution to the refractive index can be formulated as: PNL(ωj) = ε0 εjNL Ej

(9)

P(ωj) = PL(ωj) + PNL(ωj) = ε0 εjNL Ej

(10)

εj = εjL + εjNL = (nj + Δnj)2

(11)

Here nj = refractive index; Δnj = induced index change by 3rd order nonlinear effect;

2.3 Four Wave Mixing(FWM) Refractive index dependency on the intensity of the optical beam produces another type of nonlinear effect which gives rise to signals at new frequencies such as 2ωi−ωj and ωi+ωj −ωk. This is known as Four Wave Mixing [6, 8, 10, 12, 2426]. In order to understand the origin of FWM we can start by considering the process described in [26], so we can consider the total electric field of a signal as a summation of ‘n’ plane waves as: E = ∑p=1n Ep cos(ωpt - kpz )

(17)

Here the propagation parameters are defined as, Ep = amplitude; ωp = frequency; kp = optical field propagation constant; The nonlinear polarization is given by, PNL = ε0 χ(3) E3

(18)

Putting the total electric field in nonlinear polarization equation,

Since Δnj « nj, the nonlinear part of the refractive index is: Δnj ≈ εjNL ⁄ 2 njL = n2 (|Ej|2 + 2|E3-j|2)

Applications of Cross phase modulation effect have been illustrated for ultra-fast optical switching[17], pulse compression [19], pulse retiming[20, 21] and wavelength conversion [22, 23].

(12)

Here n2 is the nonlinear refractive index coefficient which value is given by, n2 = 3⁄8n χ(3) (13) Nonlinear phase acquired by the signal due to refractive index change of the medium is given by:

PNL = ε0 χ(3) ∑p=1n ∑q=1n ∑r=1n Ep cos(ωpt - kpz ) Eq cos(ωqt - kqz ) Er cos(ωrt - krz )

(19)

By expansion we get, PNL = ¾ ε0 χ(3) ∑np=1 (Ep2 + 2 ∑q≠p Ep Eq) Ep cos(ωpt - kpz )

(14)

+ ¼ ε0 χ(3) ∑np=1 Ep3 cos(3ωpt - 3kpz )

(15)

+ ¾ ε0 χ(3) ∑np=1 ∑q≠p Ep2 Eq cos{(2ωpt - ωq )t - (2 kp - kq )z}

Putting j = 2 we get the nonlinear phase shift due to XPM effect,

+ ¾ ε0 χ(3) ∑np=1 ∑q≠p Ep2 Eq cos{(2ωpt + ωq )t + (2 kp+ kq )z}

ΦjNL = βjNL z = ωjzn2 / c (|Ej|2 + 2|E3-j|2)

NL

Φ2

2

2

= ω2zn2 / c (|E2| + 2|E1| )

(16)

The factor 2 indicates that XPM effect is twice as effective as SPM for the same intensity [9, 12, 17]. In presence of fiber chromatic dispersion, cross phase modulation results in amplitude modulation [18]. XPM effect depends the wavelength spacing between different channels and increasing the spacing can significantly reduce the effect in WDM [8]. Theoretically, XPM can impose a power limit of 0.1 mW per channel for a 100-channels system[12].

+ 6/4 ε0 χ(3) ∑np=1 ∑q > p Er>q Ep Eq Er { cos(( ωp + ωq + ωr)t - (kp + kq + kr)z + cos(( ωp + ωq - ωr)t - (kp + kq - kr)z + cos(( ωp - ωq + ωr)t - (kp - kq + kr)z + cos(( ωp - ωq - ωr)t - (kp - kq - kr)z } (20) The first term includes the intensity dependent refractive index terms representing the effect of SPM and XPM. And the last term represents the Four Wave Mixing phenomenon [12]. An illustration of FWM process in HNLF is shown in the figure 1 below. The term Four Wave Mixing comes from the fact that three waves with the frequencies ωi , ωj, and ωk are combined to generate a fourth wave at a frequency ωi ± ωj ± ωk [8, 25, 26].

transmission rate. To meet this requirement only XPM and FWM effects are considered. But XPM is bit rate dependent and causes serious distortion in the signal. In case of FWM, it is bit rate independent thus suitable for photonic processing in ultrafast transmission [9]. A theoretical comparison between SPM, XPM and FWM is shown on Table 1 below. Nonline ar Phenom enon’s

Bit rate Origin

Fig. 1. Illustration of FWM process in Highly Nonlinear Fiber(HNLF)[25]

Effects

Study of Four Wave Mixing effect has resulted in many application of optical signal processing, such as reducing quantum noise through squeezing [27], highly efficient and broadband wavelength conversion [28],wavelength conversion in W-type soft glass fiber [29], highly nonlinear crystal fiber[30] and HNLF [31]. Recent researches on FWM process present us with time spectral convolution based photonic pattern recognition [32], remote transmitter correlation using FWM [33]and optical subtraction with phase management [34].

Shape of broade ning Channel Spacin g

Kerr Nonlinearities SPM

XPM

FWM

Dependent Nonlinear susceptibility

Dependent

Independent

Nonlinear susceptibility

Nonlinear susceptibility

Phase shift due to pulse itself only

Phase shift is due to copropagating pulses

New waves are generated

Symmetric

May be symmetric or asymmentric

----------------

No effect

Increases on decreasing the spacing

Increases on decreasing the spacing

TABLE 1 : PERFORMANCE COMPARISON OF SPM,XPM AND FWM [9]

4. 3.

DISCUSSION

Four wave mixing in an instantaneous which is independent to bit rate and data format of the signal hence preserves both phase and amplitude information [35].For high bit rate systems the effect of SPM and XPM are significant but FWM is critically dependant on the channel spacing[36] and fiber dispersion [12]. This characteristic of FWM makes its use in optical signal processing much desirable compared to SPM or XPM [9]. In SPM, as we have reviewed already, the intensity of the optical signal itself modulates its own refractive index which results in its own phase modulation. The resulting timedependent change, or modulation of the phase, leads to spectral broadening or frequency chirping. This effect is important to consider for systems operating at 10 Gb/s or above, or for systems with high transmitted power and lowerbit-rate. Because for these systems SPM can significantly increase pulse broadening effects or chromatic dispersion [8]. XPM effect is similar but it involves two signal beams and the spectral broadening is double of SPM effect [10]. And FWM transfers energy between waves thus it doesn’t modulates refractive index of the medium [35]. For single WDM channel SPM induced nonlinear dispersion can provide some advantage. But modern communication applications face challenges for maintaining multiple channels with ultrafast

CONCLUSION

The aim of this paper was to give a review of Kerr-Effect based optical nonlinearities to the readers. SPM, XPM and FWM effects were discussed. Recent applications of the effects in optical signal processing have been discussed. The advantage of Four Wave Mixing over SPM and XPM for ultrafast communication systems has been pointed out.

References [1] [2] [3] [4] [5] [6]

Y. Jianping, "Microwave Photonics," Lightwave Technology, Journal of, vol. 27, pp. 314-335, 2009. K. Wilner and A. P. van den Heuvel, "Fiber-optic delay lines for microwave signal processing," Proceedings of the IEEE, vol. 64, pp. 805-807, 1976. J. Capmany, B. Ortega, and D. Pastor, "A tutorial on microwave photonic filters," Lightwave Technology, Journal of, vol. 24, pp. 201-229, 2006. G. P. Agrawal, "Chapter 1 - Introduction," in Nonlinear Fiber Optics (Fourth Edition), G. P. Agrawal, Ed., ed San Diego: Academic Press, 2006, pp. 1-24. M. Matsumoto, "Fiber-Based All-Optical Signal Regeneration," Selected Topics in Quantum Electronics, IEEE Journal of, vol. 18, pp. 738-752, 2012. K. Thyagarajan and A. Ghatak, "Nonlinear Effects in Optical Fibers," in Fiber Optic Essentials, ed: Wiley-IEEE Press, 2007, pp. 205-220.

[7] [8]

[9]

[10] [11] [12] [13]

[14] [15]

[16] [17] [18] [19] [20] [21] [22]

G. P. Agrawal, "Chapter 8 - Optical Signal Processing," in Applications of Nonlinear Fiber Optics (Second Edition), Ed., ed Burlington: Academic Press, 2008, pp. 349-396. R. Ramaswami and K. N. Sivarajan, "Chapter 2- Propagation of Signals in Optical Fiber," in Optical Networks (Second Edition), R. Ramaswami and K. N. Sivarajan, Eds., ed San Francisco: Morgan Kaufmann, 2002, pp. 49-106. L. J. B.K.Mishra, Karishma Mhatre, "Performance analysis of third order nonlinearities in optical communication system," International Journal of Scientific & Engineering Research IJSER, vol. 4, Mar 2013 2013. G. P. Agrawal, "Chapter 2 - Pulse Propagation in Fibers," in Nonlinear Fiber Optics (Fourth Edition), G. P. Agrawal, Ed., ed San Diego: Academic Press, 2006, pp. 25-50. G. P. Agrawal, "Chapter 4 - Self-Phase Modulation," in Nonlinear Fiber Optics (Fourth Edition), G. P. Agrawal, Ed., ed San Diego: Academic Press, 2006, pp. 79-119. N. S. S. P. Singh, "Nonlinear effects in optical fibers: origin, management and applications," Progress In Electromagnetics Research, vol. 23, pp. 249-275, 2007. G. P. Agrawal and N. A. Olsson, "Self-phase modulation and spectral broadening of optical pulses in semiconductor laser amplifiers," Quantum Electronics, IEEE Journal of, vol. 25, pp. 2297-2306, 1989. M. Matsumoto, "Analysis of optical regeneration utilizing selfphase modulation in a highly nonlinear fiber," Photonics Technology Letters, IEEE, vol. 14, pp. 319-321, 2002. M. Chen, Tan Yan-lan, and Li Tian-song, "Wavelength conversion technique based on self phase modulation in highly nonlinear microstructure fiber," Optoelectronics Letters, vol. 6, pp. 278-280, 2010/07/01 2010. N. Kikuchi, K. Sekine, and S. Sasaki, "Analysis of cross-phase modulation (XPM) effect on WDM transmission performance," Electronics Letters, vol. 33, pp. 653-654, 1997. G. P. Agrawal, "Chapter 7 - Cross-Phase Modulation," in Nonlinear Fiber Optics (Fourth Edition), G. P. Agrawal, Ed., ed San Diego: Academic Press, 2006, pp. 226-273. S. Betti and M. Giaconi, "Analysis of the cross-phase modulation effect in WDM optical systems," Photonics Technology Letters, IEEE, vol. 13, pp. 43-45, 2001. K.-t. Chan and W.-h. Cao, "Enhanced soliton-effect pulse compression by cross-phase modulation in optical fibers," Optics Communications, vol. 178, pp. 79-88, 5/1/ 2000. K. S. Abedin, "Ultrafast pulse retiming by cross-phase modulation in an anomalous-dispersion polarization-maintaining fiber," Optics Letters, vol. 30, pp. 2979-2981, 2005/11/15 2005. R. R. Alfano, The Supercontinuum Laser Source, 2nd edition ed.: Springer, New York, 2006. B. C. Sarker, T. Yoshino, and S. P. Majumder, "All-optical wavelength conversion based on cross-phase modulation (XPM) in

[23]

[24] [25] [26] [27] [28]

[29]

[30]

[31]

[32] [33] [34]

[35]

[36]

a single-mode fiber and a Mach-Zehnder interferometer," Photonics Technology Letters, IEEE, vol. 14, pp. 340-342, 2002. M. Galili, L. K. Oxenlowe, H. C. Hansen Mulvad, A. T. Clausen, and P. Jeppesen, "Optical Wavelength Conversion by Cross-Phase Modulation of Data Signals up to 640 Gb/s," Selected Topics in Quantum Electronics, IEEE Journal of, vol. 14, pp. 573-579, 2008. K. T. Ajoy Ghatak, Introduction to fiber optics: Cambridge University PRess, 1997. R. Kibria and M. Austin, "All Optical Signal-Processing Techniques Utilizing Four Wave Mixing," Photonics, vol. 2, p. 200, 2015. C. W. Thiel, "Four-Wave Mixing and its Applications." G. P. Agrawal, "Chapter 10 - Four-Wave Mixing," in Nonlinear Fiber Optics (Fourth Edition), G. P. Agrawal, Ed., ed San Diego: Academic Press, 2006, pp. 368-423. S. Asimakis, P. Petropoulos, F. Poletti, J. Y. Leong, R. C. Moore, K. E. Frampton, et al., "Towards efficient and broadband fourwave-mixing using short-length dispersion tailored lead silicate holey fibers," Opt Express, vol. 15, pp. 596-601, Jan 22 2007. A. Camerlingo, F. Parmigiani, F. Xian, F. Poletti, P. Horak, W. H. Loh, et al., "Wavelength Conversion in a Short Length of a Solid Lead&Silicate Fiber," Photonics Technology Letters, IEEE, vol. 22, pp. 628-630, 2010. P. A. Andersen, T. Tokle, G. Yan, C. Peucheret, and P. Jeppesen, "Wavelength conversion of a 40-Gb/s RZ-DPSK signal using fourwave mixing in a dispersion-flattened highly nonlinear photonic crystal fiber," Photonics Technology Letters, IEEE, vol. 17, pp. 1908-1910, 2005. L. Ju Han, W. Belardi, K. Furusawa, P. Petropoulos, Z. Yusoff, T. M. Monro, et al., "Four-wave mixing based 10-Gb/s tunable wavelength conversion using a holey fiber with a high SBS threshold," Photonics Technology Letters, IEEE, vol. 15, pp. 440442, 2003. R. Kibria, B. Lam Anh, and M. W. Austin, "Nonlinear mixing based photonic correlator," in Microwave Photonics (MWP), 2012 International Topical Meeting on, 2012, pp. 30-33. R. Kibria, B. Lam, A. Mitchell, and M. W. Austin, "HNLF-Based Photonic Pattern Recognition Using Remote Transmitter," Photonics Technology Letters, IEEE, vol. 26, pp. 457-460, 2014. R. Kibria, L. A. Bui, A. Mitchell, and M. W. Austin, "(IPC) A Photonic Correlation Scheme Using FWM With Phase Management to Achieve Optical Subtraction," Photonics Journal, IEEE, vol. 5, pp. 5502209-5502209, 2013. K. S. Anupjeet Kaur, Bhawna Utreja, "Performance analysis of semiconductor optical amplifier using four wave mixing based wavelength Converter for all Optical networks," International Journal of Engineering, vol. 3, pp. 108-113, 2013. S. P. Singh, S. Kar, and V. K. Jain, "Novel Strategies for Reducing FWM Using Modified Repeated Unequally Spaced Channel Allocation," Fiber and Integrated Optics, vol. 23, pp. 415-437, 2004/01/01 2004.