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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 23 — Nov. 5, 2012
  • pp: 25718–25743
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Nonlinear distortion of optical pulses by self-produced free carriers in short or highly lossy silicon-based waveguides

Hagen Renner  »View Author Affiliations


Optics Express, Vol. 20, Issue 23, pp. 25718-25743 (2012)
http://dx.doi.org/10.1364/OE.20.025718


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Abstract

An explicit analytical solution for the asymmetric attenuation of optical pulses by self-produced free carriers in silicon waveguides is derived. It allows us to quantify the pulse distortion and to calculate explicitly the free-carrier density and the nonlinear phase shifts caused by the Kerr effect and by free-carrier refraction. We show that omitting two-photon absorption (TPA) as a cause of attenuation and accounting only for free-carrier absorption (FCA) as done in the derivation appropriately models the pulse propagation in short or highly lossy silicon-based waveguides such as plasmonic waveguides with particular use for high-energy input pulses. Moreover, this formulation is also aimed at serving as a tool in discussing the role of FCA in its competition with TPA when used for continuum generation or pulse compression in low-loss silicon waveguides. We show that sech-shaped intensity pulses maintain their shape independently of the intensity or pulse width and self-induced FCA may act as an ideal limiter on them. Pulse propagation under self-induced free-carrier absorption exhibits some features of superluminal propagation such as fast or even backward travelling. We find that input pulses need to have a sufficiently steep front slope to be compressible at all and illustrate this with the FCA-induced pulse broadening for Lorentzian-shaped input pulses.

© 2012 OSA

1. Introduction

Silicon technology provides a promising platform for Integrated Optics since the high index contrast can be used to realize a great variety of useful components and the maturity achieved for the CMOS technology can be drawn on [1

1. S. Fathpour and B. Jalali (Eds.), Silicon Photonics for Telecommunications and Biomedicine (CRC Press, Taylor & Francis, Boca Raton, 2011). [CrossRef]

3

3. G. T. Reed and A. P. Knights, Silicon Photonics: An Introduction (John Wiley, West Sussex, 2004). [CrossRef]

]. Compact and stable lowloss wavelength-selective linear components acting as filters and demultiplexers have been fabricated, amongst them ring resonators, interferometers, arrayed waveguide gratings and diffraction gratings [4

4. W. Bogaerts, S. K. Selvaraja, P. Dumon, J. Brouckaert, K. De Vos, D. Van Thourhout, and R. Baets, “Silicon-on-insulator spectral filters fabricated with CMOS technology,” IEEE J. Sel. Top. Quantum Electron. 16, 33–44 (2010). [CrossRef]

]. However, if the interesting non-linear properties of silicon waveguides [5

5. H. K. Tsang and Y. Liu, “Nonlinear optical properties of silicon waveguides,” Semicond. Sci. Technol. 23, 064007 (2008). [CrossRef]

8

8. M. A. Foster, A. C. Turner, M. Lipson, and A. L. Gaeta, “Nonlinear optics in photonic nanowires,” Opt. Express 16, 1300–1320 (2008). [CrossRef] [PubMed]

] shall be taken advantage of, for which high intensity levels are required, free carriers generated by two-photon absorption (TPA) often limit the range of useful operation. For instance, free-carrier absorption (FCA) limits the achievable spectral broadening when using the Kerr effect for continuum generation [9

9. P. Koonath, D. R. Solli, and B. Jalali, “Limiting nature of continuum generation in silicon,” Appl. Phys. Lett. 93, 091114 (2008). [CrossRef]

,10

10. P. T. S. DeVore, D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, “Stimulated supercontinuum generation extends broadening limits in silicon,” Appl. Phys. Lett. 100, 101111 (2012). [CrossRef]

]. In silicon Raman amplifiers, FCA puts an upper limit on the absolute Stokes power which can be amplified at most [11

11. H. Renner, “Upper limit for the amplifiable Stokes power in saturated silicon waveguide Raman amplifiers,” in 7th International Conference on Group IV Photonics (GFP), Beijing, China, 1–3 Sept. 2010, paper P1.15.

], and even for low Stokes-power levels FCA restricts the maximum possible gain [12

12. R. Claps, V. Raghunathan, D. Dimitropoulos, and B. Jalali, “Influence of nonlinear absorption on Raman amplification in silicon waveguides,” Opt. Express 12, 2774–2780 (2004). [CrossRef] [PubMed]

17

17. H. Renner and M. Krause, “Maximal total gain of non-tapered silicon-on-insulator Raman amplifiers,” in Optical Amplifiers and Their Applications/Coherent Optical Technologies and Applications, OSA Technical Digest Series(CD) (Optical Society of America, 2006), paper OMD2.

]. On the other hand, the attenuation by free carriers has successfully been exploited for compressing and shaping optical pulses in [18

18. E. K. Tien, F. Qian, N. S. Yuksek, and O. Boyraz, “Influence of nonlinear loss competition on pulse compression and nonlinear optics in silicon,” Appl. Phys. Lett. 91, 201115 (2007). [CrossRef]

, 19

19. E. K. Tien, N. S. Yuksek, F. Qian, and O. Boyraz, “Effect of TPA and FCA interplay on pulse compression in silicon,” in 20th Annual Meeting of the IEEE Lasers and Electro-Optics Society 2007, Lake Buena Vista, Fl, paper ThY2, 2007. [CrossRef]

].

In a number of papers dealing with non-linear optical pulse propagation in silicon waveguides [9

9. P. Koonath, D. R. Solli, and B. Jalali, “Limiting nature of continuum generation in silicon,” Appl. Phys. Lett. 93, 091114 (2008). [CrossRef]

, 10

10. P. T. S. DeVore, D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, “Stimulated supercontinuum generation extends broadening limits in silicon,” Appl. Phys. Lett. 100, 101111 (2012). [CrossRef]

, 18

18. E. K. Tien, F. Qian, N. S. Yuksek, and O. Boyraz, “Influence of nonlinear loss competition on pulse compression and nonlinear optics in silicon,” Appl. Phys. Lett. 91, 201115 (2007). [CrossRef]

21

21. L. Yin and G. P. Agrawal, “Impact of two-photon absorption on self-phase modulation in silicon waveguides,” Opt. Lett. 32, 2031–2033 (2007). [CrossRef] [PubMed]

] the discussion of the physics is based on the helpful and enlightening concept of an interplay or competition of FCA and TPA, and the individual contributions from TPA, Kerr effect, FCA and free-carrier refraction (FCR) are partially studied in isolation from the others. While the well-known analytical solution [21

21. L. Yin and G. P. Agrawal, “Impact of two-photon absorption on self-phase modulation in silicon waveguides,” Opt. Lett. 32, 2031–2033 (2007). [CrossRef] [PubMed]

, 22

22. Q. Lin, O. J. Painter, and G. P. Agrawal, “Nonlinear optical phenomena in silicon waveguides: modeling and applications,” Opt. Express 15, 16604–16644 (2007). [CrossRef] [PubMed]

] for pulse propagation in the presence of TPA and Kerr effect eases this discussion considerably, no such tool for describing explicitly the effects of self-produced free-carriers on the non-linear distortion of short optical pulses in silicon waveguides is available up to now [22

22. Q. Lin, O. J. Painter, and G. P. Agrawal, “Nonlinear optical phenomena in silicon waveguides: modeling and applications,” Opt. Express 15, 16604–16644 (2007). [CrossRef] [PubMed]

32

32. I. D. Rukhlenko, M. Premaratne, and G. P. Agrawal, “Nonlinear silicon photonics: analytical tools,” IEEE J. Sel. Top. Quantum Electron. 16, 200–215 (2010). [CrossRef]

]. This lack of an analytical tool as the corresponding counterpart in this discussion of FCA versus TPA is addressed in this paper.

In order to derive analytical results for the action of FCA including FCR and the Kerr effect, we intentionally omit the effect of TPA as an attenuation while keeping its function of generating free carriers. We will prove in Section 4 that this is an appropriate approximation for describing the propagation of optical pulses in short or highly lossy silicon-based waveguides such as silicon-metal plasmonic waveguides. In numerical simulations for the latter type of waveguides [30

30. I. D. Rukhlenko, M. Premaratne, and G. P. Agrawal “Nonlinear propagation in silicon-based plasmonic waveguides from the standpoint of applications,” Opt. Express 19, 206–217 (2011). [CrossRef] [PubMed]

] it was observed that self-induced FCA may have a significantly stronger effect on the pulse propagation than TPA has. However, we also propose this model to serve as a tool in discussing the competition of FCA and TPA in low-loss silicon waveguides such as those based on silicon-on-insulator technology. As a result we will arrive at explicit closed-form formulations for the evolution of the pulse intensity, free-carrier concentration, Kerr-induced phase shift and the phase shift caused by FCR, and we will draw a number of general conclusions on their principal behavior in dependence of the input pulse properties.

We show in Section 5.5 that sech-shaped intensity pulses maintain their original shape and width during propagation all along the waveguide independently of the peak intensity and pulse duration. We find that self-induced FCA is an ideal limiter for sech intensity pulses, as they remain undistorted but cannot exceed a certain peak-intensity level at the waveguide output. The behavior of pulses travelling under self-induced FCA turns out to have some similarities with superluminal pulse propagation.

Further, regarding FCA as a means for pulse compression [18

18. E. K. Tien, F. Qian, N. S. Yuksek, and O. Boyraz, “Influence of nonlinear loss competition on pulse compression and nonlinear optics in silicon,” Appl. Phys. Lett. 91, 201115 (2007). [CrossRef]

, 19

19. E. K. Tien, N. S. Yuksek, F. Qian, and O. Boyraz, “Effect of TPA and FCA interplay on pulse compression in silicon,” in 20th Annual Meeting of the IEEE Lasers and Electro-Optics Society 2007, Lake Buena Vista, Fl, paper ThY2, 2007. [CrossRef]

], we discuss in Section 6 the requirements on a pulse’s shape to be compressible at all, show the sech pulse to indicate a compressibility limit and illustrate by means of the Lorentzian pulse that a widespread intensity profile causes the pulse even to be broadened by FCA instead of being compressed.

Finally, we calculate in Section 7 the attenuation a weak probe pulse experiences from the free carriers left by a preceding strong pump pulse and compare it with the CW operation. Interestingly, at high intensities the pulsed pump may exert a higher aggregate FCA attenuation on the probe pulse if the latter follows closely enough.

Since the full problem of optical pulse propagation in silicon waveguides taking both TPA and FCA into account can not be solved analytically, exact results have only been obtained purely numerically [9

9. P. Koonath, D. R. Solli, and B. Jalali, “Limiting nature of continuum generation in silicon,” Appl. Phys. Lett. 93, 091114 (2008). [CrossRef]

, 10

10. P. T. S. DeVore, D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, “Stimulated supercontinuum generation extends broadening limits in silicon,” Appl. Phys. Lett. 100, 101111 (2012). [CrossRef]

, 18

18. E. K. Tien, F. Qian, N. S. Yuksek, and O. Boyraz, “Influence of nonlinear loss competition on pulse compression and nonlinear optics in silicon,” Appl. Phys. Lett. 91, 201115 (2007). [CrossRef]

32

32. I. D. Rukhlenko, M. Premaratne, and G. P. Agrawal, “Nonlinear silicon photonics: analytical tools,” IEEE J. Sel. Top. Quantum Electron. 16, 200–215 (2010). [CrossRef]

]. Some methods provide a simplification of the mathematical treatment with the benefit of deeper physical insight. For instance, the variational method in [33

33. S. Roy, S. K. Bhadra, and G. P. Agrawal, “Femtosecond pulse propagation in silicon waveguides: variational approach and its advantages,” Opt. Commun. 281, 5889–5893 (2008). [CrossRef]

] gets rid of the time integration, but a set of differential equations with respect to the propagation distance remains to be solved numerically without accounting for the pulse distortion caused by the self-produced free carriers. The explicit analytical results given in [34

34. I. D. Rukhlenko, M. Premaratne, C. Dissanayake, and G. P. Agrawal, “Nonlinear pulse evolution in silicon waveguides: an approximate analytic approach,” J. Lightwave Technol. 27, 3241–3248 (2009). [CrossRef]

] take the FCA-induced self-distortion partially into account in the sense of a successive approximation method, however, restricting itself to weak pulse distortions only. In both methods the introduced error has to be interpreted mathematically. In contrast to that, the present model explicitly omits the physical effect of TPA-induced pulse attenuation but fully accounts for the attenuation (FCA) and phase shift (FCR) induced by the TPA-generated free carriers as well as for the Kerr effect, while from that point on all the mathematical formulations carried out are exact. Hence, the error introduced here is unambiguously interpretable physically in terms of the omission of TPA what, as already mentioned, even allows us to identify the propagation of high-energy optical pulses in short or highly lossy silicon-based waveguides to be the physically realistic corresponding situation of useful application.

2. Pulse propagation in silicon waveguides

Pulse propagation along waveguide direction z and time t is assumed to be governed by the system of the nonlinear partial integro-differential equations [21

21. L. Yin and G. P. Agrawal, “Impact of two-photon absorption on self-phase modulation in silicon waveguides,” Opt. Lett. 32, 2031–2033 (2007). [CrossRef] [PubMed]

, 22

22. Q. Lin, O. J. Painter, and G. P. Agrawal, “Nonlinear optical phenomena in silicon waveguides: modeling and applications,” Opt. Express 15, 16604–16644 (2007). [CrossRef] [PubMed]

, 24

24. E. Dulkeith, Y. A. Vlasov, X. Chen, N. C. Panoiu, and R. M. Osgood Jr., “Self-phase-modulation in submicron silicon-on-insulator photonic wires,” Opt. Express 14, 5524–5534 (2006). [CrossRef] [PubMed]

27

27. S. V. Afshar and T. M. Monro, “A full vectorial model for pulse propagation in emerging waveguides with subwavelength structures part I: Kerr nonlinearity,” Opt. Express 17, 2298–2318 (2009). [CrossRef]

]
Ez+1vEt+iβ222Et2=α2Eβr+iβi2|E|2Eσr+iσi2NE,
(1)
N(z,t)=γtI2(z,t)G(tt)dt,
(2)
where the electrical field propagates according to E(z, t)exp[i(β0zωt)] through the silicon waveguide, where ω is the optical angular frequency and β0 the real propagation constant of the waveguide mode, N = N(z,t) is the free-carrier concentration, v the group velocity of the waveguide mode, α the linear power attenuation coefficient, βr the two-photon absorption (TPA) coefficient, βi = −2k0n2, where k0 = 2π/λ and n2 is the Kerr coefficient, σr the coefficient of the free-carrier absorption (FCA), and σi is the efficiency of the free-carrier refraction (FCR) or free-carrier plasma dispersion effect [21

21. L. Yin and G. P. Agrawal, “Impact of two-photon absorption on self-phase modulation in silicon waveguides,” Opt. Lett. 32, 2031–2033 (2007). [CrossRef] [PubMed]

]. Throughout the paper we consider pulse propagation at the telecom wavelength λ = 1.55μm, where [21

21. L. Yin and G. P. Agrawal, “Impact of two-photon absorption on self-phase modulation in silicon waveguides,” Opt. Lett. 32, 2031–2033 (2007). [CrossRef] [PubMed]

] the Kerr coefficient is n2 ≃ 6 × 10−18m2/W and thus βi = −5 × 10−12m/W, βr ≃ 5 × 10−12m/W, and further σr = 1.45 × 10−21m2 and σi = 1.1 × 10−20m2. For the linear losses α we choose a conservative estimate of 1 dB/cm if not stated otherwise. The group velocity v of the fundamental mode propagating along the silicon waveguide is assumed to be v = 108m/s.

Group-velocity dispersion (GVD) is represented by the third term on the left-hand side of Eq. (1) containing the GVD parameter β2. For a pulse duration t0 the latter determines the dispersion length LD=t02/|β2| after which GVD may have caused a significant distortion of the pulse. In the wavelength range around 1.55 μm, modern silicon nanowires exhibit a GVD parameter |β2| well below 10 ps2/m while waveguides with a larger cross section come along with even smaller values [26

26. J. I. Dadap, N. C. Panoiu, X. G. Chen, I. W. Hsieh, X. P. Liu, C. Y. Chou, E. Dulkeith, S. J. McNab, F. N. Xia, W. M. J. Green, L. Sekaric, Y. A. Vlasov, and R. M. Osgood, “Nonlinear-optical phase modification in dispersion-engineered Si photonic wires,” Opt. Express 16, 1280–1299 (2008). [CrossRef] [PubMed]

, 29

29. R. M. Osgood, N. C. Panoiu, J. I. Dadap, X. Liu, X. Chen, I. W. Hsieh, E. Dulkeith, W. M. J. Green, and Y. A. Vlasov, “Engineering nonlinearities in nanoscale optical systems: physics and applications in dispersion-engineered silicon nanophotonic wires,” Adv. Opt. Photon. 1, 162–235 (2009). [CrossRef]

]. For the silicon-based plasmonic waveguides [30

30. I. D. Rukhlenko, M. Premaratne, and G. P. Agrawal “Nonlinear propagation in silicon-based plasmonic waveguides from the standpoint of applications,” Opt. Express 19, 206–217 (2011). [CrossRef] [PubMed]

] considered here |β2| is below 5 ps2/m for thicknesses above 50 nm. Thus, for both waveguide types and picosecond pulses the dispersion length is well above 10 cm. On the other hand, silicon waveguide structures have been designed which exhibit chromatic dispersion as low as that in silica optical fibers over a wide wavelength range with a dispersion length for picosecond pulses of tens of meters [35

35. L. Zhang, Q. Lin, Y. Yue, Y. Yan, R. G. Beausoleil, and A. E. Willner, “Silicon waveguide with four zero-dispersion wavelengths and its application in on-chip octave-spanning supercontinuum generation,” Opt. Express 20, 1685–1690 (2012). [CrossRef] [PubMed]

]. For waveguides significantly shorter than LD as those considered here GVD may be neglected as we do so in the present formulation by setting β2 = 0. This step is justified even more as the analytical results derived here by neglecting the effect of TPA as a source of pulse attenuation will be shown to apply to short waveguide lengths or to highly lossy waveguides which, in turn, will usually be kept short as well to avoid unnecessarily high linear losses. However, it should be taken into account that pulses significantly shortened by self-induced FCA and distorted in phase by FCR and the Kerr effect during propagation may have a wider spectrum than the input pulse and will possibly be prone to experience a larger GVD.

Stimulated Raman scattering may be neglected for the piscosecond pulses considered here which may have a narrow spectral bandwidth on the order of 1 THz or less, i.e., scarcely spanning the Raman frequency shift of 15.6 THz in silicon [22

22. Q. Lin, O. J. Painter, and G. P. Agrawal, “Nonlinear optical phenomena in silicon waveguides: modeling and applications,” Opt. Express 15, 16604–16644 (2007). [CrossRef] [PubMed]

]. Thus, for all the situations considered here the principal form of the pulse-evolution equation is similar [27

27. S. V. Afshar and T. M. Monro, “A full vectorial model for pulse propagation in emerging waveguides with subwavelength structures part I: Kerr nonlinearity,” Opt. Express 17, 2298–2318 (2009). [CrossRef]

] and only the linear and nonlinear parameters have to be properly adjusted [21

21. L. Yin and G. P. Agrawal, “Impact of two-photon absorption on self-phase modulation in silicon waveguides,” Opt. Lett. 32, 2031–2033 (2007). [CrossRef] [PubMed]

, 25

25. I.-W. Hsieh, X. Chen, J. I. Dadap, N. C. Panoiu, R. M. Osgood, S. J. McNab, and Y. A. Vlasov, “ Ultrafast-pulse self-phase modulation and third-order dispersion in Si photonic wire-waveguides,” Opt. Express 14, 12380–12387 (2006). [CrossRef] [PubMed]

, 26

26. J. I. Dadap, N. C. Panoiu, X. G. Chen, I. W. Hsieh, X. P. Liu, C. Y. Chou, E. Dulkeith, S. J. McNab, F. N. Xia, W. M. J. Green, L. Sekaric, Y. A. Vlasov, and R. M. Osgood, “Nonlinear-optical phase modification in dispersion-engineered Si photonic wires,” Opt. Express 16, 1280–1299 (2008). [CrossRef] [PubMed]

, 30

30. I. D. Rukhlenko, M. Premaratne, and G. P. Agrawal “Nonlinear propagation in silicon-based plasmonic waveguides from the standpoint of applications,” Opt. Express 19, 206–217 (2011). [CrossRef] [PubMed]

].

The complex field amplitude in Eq. (1) can be written E(z,t)=I(z,t)×exp[iϕ(z,t)], where I(z,t) is the intensity and ϕ(z,t) the phase of the electrical field. After neglecting the attenuation caused by TPA by omitting the term containing βr and the GVD term by setting β2 = 0 as discussed in the Introduction and below Eqs. (1) and (2), respectively, we may separate Eq. (1) into the following differential equations for the intensity I(z,t) and the phase ϕ(z,t) of the propagating field,
Iz+1vItαIσrNI,
(5)
ϕz+1vϕt=βi2Iσi2N,
(6)
where we assume that an optical pulse with intensity I(z = 0, t) = I0(t) and phase ϕ(z = 0, t) = ϕ0(t) has been launched at the waveguide input at z = 0. Although we have neglected the attenuation of the wave by the TPA process itself by dropping the βr term, we keep the βi term as it describes the strong influence of the Kerr effect on the phase chirp of the pulse which may be highly sensitive to interference experiments and which may cause a non-linear frequency shift in the pulse. The action of the TPA-generated free carriers on the intensity attenuation and on the phase chirp is kept fully included by the σr and σr terms.

We consider a short propagating pulse essentially confined to a time interval [tP(z) − Δt, tP(z) + Δt]. For a pulse launched at the waveguide input z = 0 at time t = tP(0) and travelling at the group velocity v along the waveguide, tP(z) = tP(0) + z/v designates the time at which the center of this time slot passes the waveguide position z. If now this maximal pulse width 2Δt is much smaller than the characteristic decay time τeff of the free carriers, the pulse itself only experiences the accumulation of free carriers, and will have travelled on already when the free-carrier concentration starts to perceptibly decrease again. Thus the free-carrier density can be approximated by
N(z,t){γR(z,t),ttP(z)+Δt,γQ(z)G[ttP(z)],t>tP(z)+Δt.
(7a) (7b)
where the upper line applies for times before and during the travelling pulse, while the lower line stands for times after the pulse has already passed the position z, and
R(z,t)=tI2(z,t)dt,Q(z)=R(z,t)==+I2(z,t)dt.
(8)
Since in the moment we are interested only in the evolution of an optical pulse with a duration much shorter than the time scale τeff of the free-carrier decay function G(t), we focus on a time interval around the pulse only and approximate N(z,t) ≃ γR(z,t) as in Eq. (7a).

In order to arrive at explicit analytical results, we multiply the whole equation (5) by I and express the squared intensity as I2(z,t) = ∂R(z,t)/∂t =: (z,t),
12R˙z+12vR˙t=αR˙γσrRR˙
(9)
We integrate this equation over time t,
12Rz+12vRt=αRγσr2R2+K(z)
(10)
where the integration constant K(z) is independent of t. Since Eq. (10) must vanish all along the waveguide before any pulse has been launched we have K(z) = 0 for all z. To solve this partial differential equation (10), we introduce two new independent time coordinates
x˜=t+z/v,y˜=tz/v
(11)
implying z = v()/2 and t = ( + )/2, and obtain
1vRx˜=αRγσr2R2.
(12)
This ordinary differential equation is easily solved by separation of the variables and R (or using [37

37. E. Kamke, Differentialgleichungen. Lösungsmethoden und Lösungen: Gewöhnliche Differentialgleichungen I. (B. G. Teubner, Stuttgart, 1983), p. 298, Eq. (1.34).

]),
R=2αC(y˜)exp(2αvx˜)γσr
(13)
where, after re-inserting the original space and time coordinates, the integration constant C() = C(tz/v) is used to fit R(z,t) to its initial value R0(t) = R(z = 0, t) at the waveguide input z = 0, where
R0(t)=R(z=0,t)=tI02(t)dt
(14)
and I0(t) = I(z = 0, t) is the input pulse intensity at z = 0. We obtain
R(z,t)=R0(tz/v)exp(2αz)1+γσrLeff(2)(z)R0(tz/v)
(15)
where
Leff(m)(z)=[1exp(mαz)]/(mα)
(16)
defines two different effective interaction lengths for m = 1 and m = 2. Further, we have
Q(z)=Q0exp(2αz)1+γσrLeff(2)(z)Q0exp(2αz)γσrLeff(2)(z),
(17)
where Q0 = Q(z = 0) = R0(t → ∞), and the rightmost term is the limit for Q0 → ∞ for z > 0. This quantity is of interest since γQ(z) gives us the peak value of the free-carrier density right behind the pulse when it has just passed the position z, and which afterwards decays as in Eq. (7b). Interestingly, Q(z) has the same dependence on z for all pulses with the same Q0 at the input, no matter what shape they have and how much they are distorted along the waveguide by the self-produced free carriers.

Thus, Eqs. (15) and (17) with Eq. (7) already give us the free-carrier concentration before, during and after the optical pulse at position z. The pulse intensity I(z,t) follows directly from Eqs. (8) and (15) as I(z,t)=R(z,t)/t,
I(z,t)=I0(tz/v)exp(αz)1+γσrLeff(2)(z)R0(tz/v).
(18)
The fact that R0(t) in Eq. (18) is monotonously increasing with time causes an attenuation increasing towards the rear part of the pulse during propagation as already observed in numerical studies [9

9. P. Koonath, D. R. Solli, and B. Jalali, “Limiting nature of continuum generation in silicon,” Appl. Phys. Lett. 93, 091114 (2008). [CrossRef]

, 10

10. P. T. S. DeVore, D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, “Stimulated supercontinuum generation extends broadening limits in silicon,” Appl. Phys. Lett. 100, 101111 (2012). [CrossRef]

, 18

18. E. K. Tien, F. Qian, N. S. Yuksek, and O. Boyraz, “Influence of nonlinear loss competition on pulse compression and nonlinear optics in silicon,” Appl. Phys. Lett. 91, 201115 (2007). [CrossRef]

21

21. L. Yin and G. P. Agrawal, “Impact of two-photon absorption on self-phase modulation in silicon waveguides,” Opt. Lett. 32, 2031–2033 (2007). [CrossRef] [PubMed]

, 31

31. N. Suzuki, “FDTD analysis of two-photon absorption and free-carrier absorption in Si high-index-contrast waveguides,” J. Lightwave Technol. 25, 2495–2501 (2007). [CrossRef]

] and seen experimentally [38

38. T. K. Liang, L. R. Nunes, M. Tsuchiya, K. S. Abedin, T. Miyazaki, D. Van Thourhout, W. Bogaerts, P. Dumon, R. Baets, and H. K. Tsang, “Nonlinear self-distortion of picosecond optical pulses in silicon wire waveguides,” in Conference on Lasers and Electro-Optics (CLEO) 2006, 21–26 May 2006, Long Beach, paper JThC44.

].

To summarize, in this Section we have derived explicit analytical expressions for the intensity of short optical pulses propagating in silicon waveguides and for the concentration of TPA-generated free-carriers under omission of TPA as an attenuation effect.

3. Nonlinear phase shifts

In order to calculate how the self-produced free carriers affect the pulse phase, we similarly transform the differential equation (6) for the phase, with N(z,t) = γR(z,t) from (7a),
1vϕx˜=βi4Iσiγ4R.
(19)
After inserting the analytical solutions Eqs. (18) and (15) for I(z,t) and R(z,t) using the coordinate transformation z = v()/2 and tz/v = , the integration is easily done by means of the integral relations 2.313.1 and 2.314 given in [39

39. I. S. Gradstein and I. M. Ryshik, Table of Integrals, Series and Products (Academic Press, Boston, 1994) p. 104.

]. Fitting the -dependent integration constant properly for a given phase function ϕ0(t) = ϕ(z = 0, t) at the waveguide input z = 0 we can write the total pulse phase in original time-space coordinates in terms of the individual contributions from the Kerr and FCR effects as
ϕ(z,t)=ϕ0(tz/v)+ϕKerr(z,t)+ϕFCR(z,t).
(20)

Explicit expressions for ϕKerr(z,t) and ϕFCR(z,t) will be given in Sections 3.1 and 3.2, respectively. We will also discuss the non-linear frequency shift Δω(z,t) = −∂ϕ/∂t which represents a spectral red shift for ∂ϕ/∂t > 0 and a blue shift for ∂ϕ/∂t < 0 [9

9. P. Koonath, D. R. Solli, and B. Jalali, “Limiting nature of continuum generation in silicon,” Appl. Phys. Lett. 93, 091114 (2008). [CrossRef]

, 10

10. P. T. S. DeVore, D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, “Stimulated supercontinuum generation extends broadening limits in silicon,” Appl. Phys. Lett. 100, 101111 (2012). [CrossRef]

, 18

18. E. K. Tien, F. Qian, N. S. Yuksek, and O. Boyraz, “Influence of nonlinear loss competition on pulse compression and nonlinear optics in silicon,” Appl. Phys. Lett. 91, 201115 (2007). [CrossRef]

22

22. Q. Lin, O. J. Painter, and G. P. Agrawal, “Nonlinear optical phenomena in silicon waveguides: modeling and applications,” Opt. Express 15, 16604–16644 (2007). [CrossRef] [PubMed]

, 40

40. R. Dekker, N. Usechak, M. Först, and A. Driessen, “Ultrafast nonlinear all-optical processes in silicon-on-insulator waveguides,” J. Phys. D: Appl. Phys. 40, R249–R271 (2007). [CrossRef]

].

3.1. Kerr-induced phase shift

The Kerr-induced phase shift in Eq. (20) is obtained as
ϕKerr(z,t)=βiI0(tz/v)2αηR0(tz/v)[1+ηR0(tz/v)]×arctanh{αLeff(1)(z)ηR0(tz/v)[1+ηR0(tz/v)]1+αLeff(1)(z)ηR0(tz/v)}.
(21)
where η = γσr/2α. Like the intensity does, the Kerr phase vanishes for t = ±∞. For negligibly small free-carrier absorption ηR0 → 0, it reduces to the well-known result for nonlinear pulse propagation in optical fibers [41

41. G. P. Agrawal, Nonlinear Fiber Optics (Elsevier, Amsterdam, 2007).

],
ϕKerr(z,t)=βiI0(tz/v)Leff(1)(z)/2,
(22)
which is temporally symmetric for symmetric input pulses. Interestingly, this stringent conservation of symmetry between pulse intensity and Kerr phase is destroyed in the presence of free-carriers. This can be shown as follows.

At the leading edge of the propagating pulse (t → −∞), where ηR0 is still small, the Kerr phase shift is approximately described by Eq. (22), which after sufficiently large propagation distances z ≫ 1/α becomes ϕKerr(z,t) = −βi I0(tz/v)/2α. In the trailing edge, however, ηR0 approaches its constant asymptotic value ηQ0 and the Kerr phase becomes proportional to Eq. (22) times a factor different from unity. For example, ϕKerr(z,t → +∞) = −βi I0(tz/v)ln(4ηQ0)/(4αηQ0) if ηQ0 ≫ 1 in addition to z ≫ 1/α. Thus, for symmetric input pulses I0(t) the Kerr phase will evolve asymmetrically. In particular, even if the asymmetrically attenuated pulse I(z,t) maintains its temporal symmetry during propagation, as it will be shown to be possible for the form-stable sech pulse in Section 5.5, the Kerr phase shift may not keep this symmetry as we discuss now. Using Eq. (18) the Kerr phase at a given position z can also be written
ϕKerr(z,t)=βi2exp(αz)Leff(1)(z)I(z,t)h{u[R0(tz/v)]}g[R0(tz/v)].
(23)
where
h(u)=arctanh(u)u,u(R0)=αLeff(1)(z)ηR0(1+ηR0)1+αLeff(1)(z)ηR0,g(R0)=1+2αLeff(2)(z)ηR01+αLeff(1)(z)ηR0.
(24)
The way in which the Kerr phase ϕKerr(z,t) in Eq. (23) is distorted in comparison to the pulse intensity I(z,t) can be seen by investigating the monotonicity of the function g · h with respect to time t. Since arctanh(u) ≃ u for u ≃ 0 and d2arctanh(u)/du2 ≥ 0 the function h(u) is monotonously increasing with u starting from h(0) = 1. Obviously, u(R0) is monotonously increasing with R0 from u(0) = 0 since αLeff(1)(z)1, while g(R0) is monotonously increasing with R0 from g(0) = 1 because 2Leff(2)(z)Leff(1)(z). Finally, R0(tz/v) is monotonously increasing in t because of its definition (14). Therefore we know that the function g · h is monotonously increasing with respect to t. This property of g · h causes ϕKerr(z,t) to be elevated towards later times as compared to the propagating pulse intensity I(z,t). This behavior is caused by the fact that the Kerr phase already started to accumulate at the waveguide input where the intensity in the rear part of the pulse was even stronger than the current peak of the propagated pulse. Consequently the Kerr phase can only peak after or at the occurrence of an intensity peak. Thus, the latter will always be given a Kerr-induced red-shift ΔωKerr(z,t)=(βi/2)Leff(1)(z)exp(αz)I(z,t)(gh)/t0 which, however, is too lengthy to display here explicitly. The trailing pulse edge will get a blue shift after the Kerr phase has peaked for the last time.

3.2. Free-carrier induced phase shift

The additional phase shift induced by FCR in Eq. (20), i.e. by the refractive-index change in the presence of self-produced free carriers, is obtained as
ϕFCR(z,t)=σi2σr×ln{1+γσrLeff(2)(z)R0(tz/v)}
(25)
Since γσr ≥ 0, the logarithmic function is always non-negative and ϕFCR(z,t) ≤ 0 as σi/σr > 0. The FCR-induced phase shift is zero for t → −∞, decreases monotonously with time and reaches its maximum (negative) value at t → +∞, which is ϕFCR = −σi ln(1 + ηQ0)/2σr for z ≫ 1/2α. Thus, FCR always causes a blue shift, i.e. its corresponding frequency shift ΔωFCR(z,t) = −∂ϕFCR(z,t)/∂t is non-negative. The latter can be written
ΔωFCR(z,t)=σi2×γLeff(2)(z)I02(tz/v)1+γσrLeff(2)(z)R0(tz/v)
(26)
=σiγ2×exp(+2αz)Leff(2)(z)I2(z,t)[1+γσrLeff(2)(z)R0(tz/v)],
(27)
and its time dependence at a position z is proportional to the squared intensity I2(z,t) of the travelling pulse times a monotonously increasing function of t. This means that ΔωFCR(z,t) can only peak after or at the occurrence of a peak in I(z,t). For smoothly single-peaked pulses this lag may make the net blue shift less efficient in the spectrum of the complete pulse. The logarithmic function in Eq. (25) results from the fact that the accumulation of FCR phase at the end of the pulse is mitigated by the simultaneous FCA-induced attenuation of the rear part of the pulse as already observed in numerical simulations [9

9. P. Koonath, D. R. Solli, and B. Jalali, “Limiting nature of continuum generation in silicon,” Appl. Phys. Lett. 93, 091114 (2008). [CrossRef]

]. Consequently, the FCR phase shift is smaller than its upper limit ϕFCR=σiγLeff(2)(z)R0(tz/v)/2 obtained for σr → 0 in Eq. (25), i.e., when free carriers would fictitiously cause FCR only and no FCA. The corresponding upper limit for the frequency blue shift would the be ΔωFCR=σiγLeff(2)(z)I02(tz/v)/2.

Summarizing, the Kerr- and FCR-induced phase shifts can be expressed explicitly as well. For symmetric input pulses any asymmetry in the total phase arises from the asymmetric history of FCA attenuation across the pulse when contributed by the Kerr effect, and from the accumulation of free carriers towards the end of the pulse when contributed by FCR.

4. Interpretation of omitting TPA as a cause of attenuation

We now illustrate how omitting TPA as an attenuation as done in this paper may be interpreted to model FCA distortion of short optical pulses in short or highly lossy silicon-based waveguides with particular use for high input pulse energies. This is in agreement with the experimental observation of a predominance of FCA over TPA in pulses propagating large energies [18

18. E. K. Tien, F. Qian, N. S. Yuksek, and O. Boyraz, “Influence of nonlinear loss competition on pulse compression and nonlinear optics in silicon,” Appl. Phys. Lett. 91, 201115 (2007). [CrossRef]

]. A direct comparison of these two attenuation mechanisms is hampered by their different temporal behavior. While TPA is an instantaneous effect, FCA only accumulates during the passage of the pulse. Therefore we define a shape function
f0(ξ)=f0(t/t0)=I0(t)/I^0
(28)
of the input pulse with a peak value of unity, where ξ = t/t0, and t0 determines the temporal width of the pulse. Thus, any input pulse I0(t) = Î0f0(t/t0) is solely described by its peak intensity Î0, its duration t0, and a dimensionless shape function f0(ξ). Accordingly, we define
F0(ξ)=ξf02(ξ)dξ,W0=t0I^0J,J=+f0(ξ)dξ,
(29)
such that R0(t)=t0I^02F0(t/t0), Q0=t0I^02F0(), and W0 is the input pulse energy per unit cross-sectional area. To compare first the local relevance of TPA versus FCA we consider the ratio αTPA/αFCA of the non-linear attenuation due to TPA and FCA, respectively, for general input pulses,
αTPAαFCA=βrI^0f0(t/t0)σrγt0I^02F0(t/t0)=1W0×βrJf0(t/t0)σrγF0(t/t0).
(30)
Since by definition f0(ξ), F0(ξ) and J depend on neither the peak intensity Î0 nor the pulse width t0, Eq. (30) shows that the ratio of TPA to FCA attenuation at any time t of the pulse duration is inversely proportional to the pulse energy W0, and the relevance of TPA against FCA locally reduces more and more for increasing W0.

However, in order to see the more important aggregate effect of TPA and FCA after propagation through the waveguide, we re-write the pulse intensity evolution Eq. (18) in a normalized way,
f(z,t)=I(z,t)I^0exp(αz)=f0(ξz/vt0)1+ΘFCAF0(ξz/vt0),
(31)
where
ΘFCA=σrγt0I^02Leff(2)(z)
(32)
is a dimensionless distortion parameter due to FCA. On the other hand, in a model accounting for TPA only [21

21. L. Yin and G. P. Agrawal, “Impact of two-photon absorption on self-phase modulation in silicon waveguides,” Opt. Lett. 32, 2031–2033 (2007). [CrossRef] [PubMed]

, 22

22. Q. Lin, O. J. Painter, and G. P. Agrawal, “Nonlinear optical phenomena in silicon waveguides: modeling and applications,” Opt. Express 15, 16604–16644 (2007). [CrossRef] [PubMed]

], i.e., when σr = 0 or γ = 0 in Eq. (67) in Appendix A, pulse propagation can be described by
f(z,t)=I(z,t)I^0exp(αz)=f0(ξz/vt0)1+ΘTPAf0(ξz/vt0),
(33)
where the dimensionless distortion parameter due to TPA is
ΘTPA=βrI^0Leff(1)(z).
(34)

An important result of the present paper is derived in Appendix A: If TPA as a source of attenuation is found to be too small to affect pulse propagation perceptibly in a TPA-only model according to Eq. (33) then it can be neglected all the more when solving the full problem actually admitting of both TPA and FCA, and the FCA-only formulation presented in this paper applies. Therefore, such a consideration of FCA and TPA in separate models allows us to formulate two criteria for identifying physically reasonable situations in which the omission of TPA is justified while nevertheless the pulses are significantly distorted by self-produced free carriers, i.e., when ΘTPA be small while ΘFCA remain finite.

The first condition requires
ΘTPA=βrI^0Leff(1)1,
(35)
which ensures that the denominator in Eq. (33) is close to constant and unity since f0(ξ) ≤ 1, and the pulse remains unaffected by TPA. Trivially, Eq. (35) is fulfilled for small input intensities representing the quasi-linear case in which TPA can be neglected and our formulation applies, but no FCA-induced distortion arises at the same time either. In the more interesting case of non-vanishing input peak intensities Î0, Eq. (35) is equivalent with requiring a small TPA-effective propagation length Leff(1), i.e., short or highly lossy waveguides for which
Leff(1)=1exp(αz)α=ΘTPAβrI^01βrI^0.
(36)
In accordance with Appendix A the conditions displayed in Eq. (35) and hence in Eq. (36) which have been derived from a simple TPA-only model are already sufficient to justify omitting TPA and applying our FCA-only model, no matter whether FCA has a significant effect or not.

The second condition requires that FCA shall still perceptibly affect the pulse and thus ΘFCA shall remain finite even when ΘTPA becomes very small. This latter condition requires
ΘFCAΘTPA=σrγt0I^02Leff(2)βrI^0Leff(1)1
(37)
In order to translate these two conditions Eqs. (35) and (37) into requirements on the waveguide and on the launching conditions, we combine ΘTPA and ΘFCA of Eqs. (32) and (34) to eliminate I^02 and use Eq. (29) for W0 to write
Leff(1)(z)=σrγt0βr2Ξ(αz)ΘTPA2ΘFCA,W0=JβrΞ(αz)σrγΘFCAΘTPA,
(38)
where Ξ(αz)=Leff(1)(z)/Leff(2)(z)=2/[1+exp(αz)]=2/[2αLeff(1)(z)] can range only between unity and two, 1 = Ξ(αz ≪ 1) ≤ Ξ(αz) ≤ Ξ(αz ≫ 1) = 2. Thus, a vanishing small TPA-distortion parameter ΘTPA and a finite and hence significantly larger FCA-distortion parameter ΘFCA require short TPA-effective propagation lengths Leff(1) and large input pulse energies W0, and the two conditions in Eqs. (35) and (37) read
Leff(1)(z)σrγt0βr2Ξ(αz)ΘFCA,W0JβrΞ(αz)ΘFCAσrγ.
(39)
Each of these two conditions is equivalent with requiring ΘTPA ≪ 1 according to Eq. (35) as long as the desired finite ΘFCA is realized via Eq. (32). Now aiming at decreasing ΘTPA while keeping ΘFCA fixed and finite, we have to take into account that the non-linear parameters of silicon are not available to being varied significantly. Although the pulse duration t0 may vary in a wide range it can not be arbitrarily increased as we assume t0τeff. Finally, the pulse shape determining J is reserved to the specific application. Therefore we re-arrange Eqs. (38) as a function of the linear attenuation α,
z=2αarctanh(ασrγt02βr2ΘTPA2ΘFCA),W0=J2βr2ΘFCA+ασrγt0ΘTPA22βrσrγΘTPA
(40)
where, depending on the choice of z, the ratio ΘTPA2/ΘFCA=2βr2tanh(αz/2)/(ασrγt0) can range only within 0ΘTPA2/ΘFCA2βr2/(ασrγt0) independently of the input peak intensity Î0. Hence, two extreme situations to realize the conditions in Eq. (39) can be distinguished.

First, if the attenuation α cannot be changed (e.g. when using the same waveguide technology), then the waveguide length would have to be small and the input pulse energy to be large according to
zσrγt0ΘTPA2βr2ΘFCAσrγt0βr2ΘFCA,W0JβrΘFCAσrγΘTPAJβrΘFCAσrγ,forαfixed,z1/α.
(41)
The parameter values given in Section 2 yield σrγ/βr2=1.13 m/ns and βr/σrγ = 0.177 Wns/μm2. For ΘFCA = 1, ΘTPA = 0.01, t0 = 0.1ns and J=π for a Gaussian input pulse (see Section 5.1) we obtain z ≃ 11.3μm and W0 = 31.4 Wns/μm2 corresponding to a peak intensity of Î0 = 177W/μ2=17.7GW/cm2. This means, for Gaussian 0.1ns input pulses, ΘTPA never exceeds 0.01 up to an input intensity of Î0 = 177W/μ2, at which ΘFCA already reaches unity. For a more relaxed TPA suppression of ΘTPA = 0.05 the results are z ≃ 283μm and W0 = 6.27 Wns/μm2 corresponding to Î0 = 35.4W/μ2=3.54GW/cm2.

Second, for a fixed geometric length z, on the other hand, the conditions in Eq. (39) require an increased attenuation α ≫ 2/z compelling the arctanh function in Eq. (40) to approach unity [as tanh(αz/2) → 1], and both the linear losses and the input pulse energy are required to be large enough according to
α2βr2ΘFCAσrγt0ΘTPA22βr2ΘFCAσrγt0,W02JβrΘFCAσrγΘTPA2JβrΘFCAσrγ,forzfixed,α2/z,
(42)
ensuring that even the maximum possible effective propagation lengths Leff(1)(z1/α)=2Leff(2)(z1/α)=1/α are small. With the parameters of Section 2 and ΘFCA = 1, ΘTPA = 0.01, t0 = 0.1ns and J=π for the Gaussian pulse again, we find α ≃ 0.177/μm according to 768dB/mm and W0 = 62.7 Wns/μm2 corresponding to Î0 = 354W/μ2=35.4GW/cm2. For ΘTPA = 0.05 the results are α ≃ 7.08/mm according to 30.7dB/mm and W0 = 12.5 Wns/μm2 corresponding to Î0 = 70.8W/μ2=7.08GW/cm2.

The requirement of large linear attenuation is frequently met in silicon-based plasmonic waveguides with their typically high linear propagation losses as the mode field is partially guided in metal layers [1

1. S. Fathpour and B. Jalali (Eds.), Silicon Photonics for Telecommunications and Biomedicine (CRC Press, Taylor & Francis, Boca Raton, 2011). [CrossRef]

]. Pulse propagation in metal-silicon-metal waveguides has been proposed in [30

30. I. D. Rukhlenko, M. Premaratne, and G. P. Agrawal “Nonlinear propagation in silicon-based plasmonic waveguides from the standpoint of applications,” Opt. Express 19, 206–217 (2011). [CrossRef] [PubMed]

] to be modelled by a set of integro-differential equations analogous to Eqs. (1) and (2) with the non-linear parameters modified by a plasmonic attenuation factor. This factor becomes negligibly small in the silver-silicon-silver waveguide analyzed there for silicon layers thicker than 150nm, and the non-linear parameters approach the values valid for silicon waveguides. The propagation losses, however, are still high and only start to decrease below 1dB/μm, restricting the effective propagation lengths to a few micrometers. For example, Leff(1)(z)2Leff(2)(z)1/α4μm at 150nm silicon thickness. In this latter case ΘTPA ≤ 0.01 for input peak intensities up to 500W/μm2 at which intensity level a significant FCA-distortion with ΘFCA = 2.82 is reached when t0 = 0.1ns. Thus, a practical approach could be as follows. If the error caused by a TPA-distortion parameter of ΘTPA ≤ 0.01 has been found to be tolerable for the desired application and the silicon-based waveguide has such a high attenuation of α = 0.25/μm= 1.086dB/μm, then all calculations using input peak intensities up to 500W/μm2 can be carried out by the present FCA-only formulation without checking the accuracy again. It should be noted that a smaller FCA-distortion parameter ΘFCA then merely relaxes the requirements on W0 and z or α in Eqs. (41) or (42) while the validity of the FCA-only formulation is maintained. We will illustrate the range of applicability of our FCA-only formulation in Section 5.1.

Summarizing we have shown that our FCA-only model for pulse propagation in silicon-based waveguides applies as soon as the condition in Eq. (35) or (36) is met, i.e. for short or highly lossy waveguides. A significant distortion by FCA is then possible for high-energy input pulses while TPA remains negligible.

5. Propagation of pulses with various input shapes

We now consider input pulses I0(t) = Î0f(t/t0) of different shapes f(t/t0), which may experience different distortions of intensity and phase.

5.1. Gaussian input pulse

Assuming a Gaussian input pulse of duration 2t0, we write
I0(t)=I^0exp(t2/t02),R0(t)=π/8t0I^02[1+erf(2t/t0)]
(43)
and obtain J=π and Q0=π/2t0I^02. The evolution for Gaussian input pulses along the silicon waveguide according to Eqs. (18) is shown in Fig. 1. It can clearly be seen that the rear part of the pulse is attenuated more and more causing a temporally asymmetric pulse shape as already observed experimentally in Ref. [38

38. T. K. Liang, L. R. Nunes, M. Tsuchiya, K. S. Abedin, T. Miyazaki, D. Van Thourhout, W. Bogaerts, P. Dumon, R. Baets, and H. K. Tsang, “Nonlinear self-distortion of picosecond optical pulses in silicon wire waveguides,” in Conference on Lasers and Electro-Optics (CLEO) 2006, 21–26 May 2006, Long Beach, paper JThC44.

] and calculated numerically by a full model including both FCA and TPA, e.g., in [14

14. A. Liu, H. Rong, M. Paniccia, O. Cohen, and D. Hak, “Net optical gain in a low loss silicon-oninsulator waveguide by stimulated Raman scattering” Opt. Express 12, 4261–4267 (2004). [CrossRef] [PubMed]

, 20

20. R. Dekker, A. Driessen, T. Wahlbrink, C. Moormann, J. Niehusmann, and M. Först, “Ultrafast Kerr-induced all-optical wavelength conversion in silicon waveguides using 1.55 μ femtosecond pulses,” Opt. Express 14, 8336–8346 (2006). [CrossRef] [PubMed]

]. The error function from Eq. (43) in the pulse evolution Eq. (18) makes the trailing edge of the Gaussian input pulse flatter and flatter while the remaining front edge becomes steeper and steeper. The plots in Figs. 1(b), 1(d), 1(f) and 1(h) are versus the reference time tz/v for a better visualization of the distortion with respect to the input pulse.

Fig. 1 (a) Pulse intensity I(z,t) at waveguide input z = 0 (dotted red curve) and after z = 1, 10, 20 and 50mm (black solid curves) propagation distance in a silicon waveguide with α = 1dB/cm for a Gaussian input pulse [2t0 = 0.2ns in Eq. (43)] with input peak intensity Î0 = 3W/μm2. The green dash-dotted curve indicates the theoretical temporal peak intensity of an undistorted pulse at z = vt in the presence of linear attenuation exp(−αz) only. The center times of the pulses at positions z = 1, 10, 20 and 50 mm are 0.01, 0.1, 0.2 and 0.5 ns, respectively. (b) Same as (a) in a reference time frame tz/v for better visualization of asymmetric pulse attenuation. (c) Non-linear phase shifts ϕKerr (dashed red curve), ϕFCR (dotted green curve), ϕ = ϕKerr + ϕFCR (solid black curve), and normalized free-carrier density N′ = 10−15cm3 × N with N = γR (blue dash-dotted curve) for the pulse after z = 50mm propagation distance. (d) Total non-linear phase shifts ϕ = ϕKerr + ϕFCR for the pulses shown in (b). (e)–(h) Same as (a)–(d), respectively, for Î0 = 30W/μm2.

Interestingly, in Fig. 1(e), the FCA-induced attenuation of the rear part of the pulse is so fast that the attenuated pulse seems to move backwards in time and to peak earlier at the waveguide end than at the input. This seeming superluminal propagation behavior [42

42. R. W. Boyd and D. J. Gauthier, “Controlling the velocity of light pulses,” Science 326, 1074–1077 (2009). [CrossRef] [PubMed]

] will be discussed in more detail using the analytical expressions which can be derived for the sech pulse in Section 5.5.

The Kerr phase ϕKerr plotted by red dashed lines in Figs. 1(c) and 1(g) has a temporal asymmetry with the slope steeper in the leading edge than in the trailing edge. Thus the Kerr effect results in a spectral extension ΔωKerr deeper into the ”red” than into the ”blue”. The FCR phase shift ϕFCR (green dotted lines) decreases monotonously with time. It can be seen comparing Figs. 1(c) and 1(g) for the propagation distance z =50mm, that the FCR-induced phase shift ϕFCR grows much stronger than N does when increasing Î0 from 3 to 30W/μm2, because ϕFCR is accumulating all along the waveguide [see Figs. 1(d) and 1(h)] and has already seen the larger carrier densities N near the input of the waveguide. The sum of ϕKerr and ϕFCR shown by black solid lines in Figs. 1(c–d) and 1(g–f) first forms a positive peak, decreases afterwards and becomes negative, thus causing a redshift in the front part of the pulse and a blueshift in the rear part.

5.2. Effect of omitting TPA illustrated with the Gaussian input pulse

We use the Gaussian input pulse to discuss the meaning and the range of applicability of our FCA-only formulation of pulse propagation in silicon-based waveguides according to Section 4. In Fig. 2, red dashed lines show the normalized intensity of the Gaussian input pulse with t0 = 0.1ns which is launched with different peak intensities Î0 at the waveguide input z = 0. Green dotted lines represent the pulse intensities after a propagation distance z calculated by solving numerically the full problem Eq. (67) taking both FCA and TPA into account. The black solid lines stand for the pulse intensities obtained from our analytical FCA-only formulation Eq. (18) omitting TPA. Note that all intensities are normalized by dividing them by the product Î0 exp(−αz) to better visualize the distortion and the introduced error, i.e., in the absence of any non-linear effects all curves would coincide with that for the input intensity.

Fig. 2 Interpretation of omitting TPA. See explanation in the main text of Section 5.2.

Figure 2(a) shows the results for a typical silicon-on-insulator waveguide with a conservative loss estimate of α = 1dB/cm and a length of 10cm, i.e. for a long low-loss waveguide. Obviously omitting TPA causes an error of more than 80 percent for 30W/μm2 for which case a TPA-distortion parameter ΘTPA as large as 5.87 obtained from Eq. (34) prohibits the omission of TPA from the outset. Even for an input peak intensity as small as 0.3W/μm2 TPA still depresses the pulse intensity by about 6 percent while FCA has almost no effect. In this situation the FCA-only formulation presented in this paper as well as the TPA-only model of Refs. [21

21. L. Yin and G. P. Agrawal, “Impact of two-photon absorption on self-phase modulation in silicon waveguides,” Opt. Lett. 32, 2031–2033 (2007). [CrossRef] [PubMed]

, 22

22. Q. Lin, O. J. Painter, and G. P. Agrawal, “Nonlinear optical phenomena in silicon waveguides: modeling and applications,” Opt. Express 15, 16604–16644 (2007). [CrossRef] [PubMed]

], are actually restricted to serve as a mere tool in estimating the individual contributions of FCA and TPA, e.g., in studying their competition in pulse compression [18

18. E. K. Tien, F. Qian, N. S. Yuksek, and O. Boyraz, “Influence of nonlinear loss competition on pulse compression and nonlinear optics in silicon,” Appl. Phys. Lett. 91, 201115 (2007). [CrossRef]

, 19

19. E. K. Tien, N. S. Yuksek, F. Qian, and O. Boyraz, “Effect of TPA and FCA interplay on pulse compression in silicon,” in 20th Annual Meeting of the IEEE Lasers and Electro-Optics Society 2007, Lake Buena Vista, Fl, paper ThY2, 2007. [CrossRef]

] or continuum generation [9

9. P. Koonath, D. R. Solli, and B. Jalali, “Limiting nature of continuum generation in silicon,” Appl. Phys. Lett. 93, 091114 (2008). [CrossRef]

, 10

10. P. T. S. DeVore, D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, “Stimulated supercontinuum generation extends broadening limits in silicon,” Appl. Phys. Lett. 100, 101111 (2012). [CrossRef]

]. Thus, the results shown in Fig. 1 illustrate the contribution from FCA only while TPA actually would add another concurrent non-negligible contribution to the pulse distortion which would have to be calculated numerically by the full model of Eq. (67) as done in Fig. 2(a).

On the other hand, Fig. 2(b) illustrates the situation for the highly lossy silicon-based plasmonic waveguide from [30

30. I. D. Rukhlenko, M. Premaratne, and G. P. Agrawal “Nonlinear propagation in silicon-based plasmonic waveguides from the standpoint of applications,” Opt. Express 19, 206–217 (2011). [CrossRef] [PubMed]

] already considered at the end of Section 4 with a 150nm thick silicon layer sandwiched between two silver layers. The high losses of α = 0.25/μm=1.086dB/μm limit the TPA-effective and the FCA-effective propagation length to no more than 4μm and 2μm, respectively. In this regard, the geometrical waveguide length of 20μm may be considered to be relatively large and not to terminate any non-linear propagation effects, and the linear loss accumulates up to almost 22 dB at the waveguide end. Thus, the results in Fig. 2(b) illustrate the situation required in Eqs. (42). Up to 500W/μm2 the TPA-distortion parameter stays below one percent, and the intensity curves obtained by the exact numerical method and by the present FCA-only model almost coincide even up to Î0 = 3kW/μm2 or 300GW/cm2 while FCA significantly distorts the pulse.

Figure 2(c) shows the results for the low-loss silicon-on-insulator waveguide with α = 1dB/cm again [as already used in Figs. 1 and 2(a)] when, however, its length of only 50μm is relatively small as compared to the maximum possible TPA- and FCA-effective propagation length of 43.5mm and 21.7mm, respectively, corresponding to the situation required by Eqs. (41). For instance, at Î0 = 100W/μm2 the pulse is significantly distorted by FCA, while ΘTPA is no larger than 2.5 percent. Again, the results obtained by the present FCA-only formulation are very close to those from the numerical integration of the exact equations (67) plotted by green dotted lines. Even up to Î0 = 1kW/μm2 or 100GW/cm2 the curves almost coincide although the TPA-distortion parameter ΘTPA already reaches 0.25 at this input intensity. This proves even more that requiring ΘTPA to be small is a sufficient criterion for the applicability of the FCA-only model developed in this paper.

5.3. Rectangular input pulse

In case of a rectangular input pulse of duration 2t0, we have
t<t0:I0(t)=0,R0(t)=0,|t|<t0:I0(t)=I^0,R0(t)=(t+t0)I^02,t>t0:I0(t)=0,R0(t)=2t0I^02,
(44)
while J = 2 and Q0=2t0I^02. The evolution of this pulse is shown in Fig. 3 where again a clear attenuation of the rear part of the pulse can be seen. Due to the absence of free carriers in front of the leading edge of the rectangular pulse all along the propagation, the intensity step there is only attenuated by the linear losses α and can be arbitrarily large by properly choosing the input intensity. This is in contrast to smoothly rising pulses such as the Gaussian or the shape-maintaining sech input pulse which we will discuss in more detail in Section 5.5. Since R0(tz/v) in the denominator of Eq. (18) is a continuous function of time tz/v, any intensity step in general pulses will maintain its relative height during propagation as compared to the intensity at neighboring times and will propagate at the group velocity v. We will return to this observation when discussing the resemblance to superluminal pulse propagation in Section 5.5.

Fig. 3 Same as Fig. 1 but for a rectangular input pulse [2t0 = 0.2ns in Eq. (44)].

The Kerr phase has a maximum just coinciding with the intensity maximum because the latter is infinitely peaked and Eq. (23) cannot act as a delay, while it decreases towards the end of the pulse. Thus, besides forming a frequency spike at either end of the pulse due to the phase steps there, the Kerr phase causes a pure blue shift all along the pulse even enhancing the blue shift already induced by FCR.

5.4. Lorentzian input pulse

For a Lorentzian input pulse of duration 2t0, we write
I0(t)=I^01+(t/t0)2,R0(t)=t0I^022[t0tt02+t2+arctan(tt0)+π2]
(45)
and obtain J = π and Q0=πt0I^02/2. The pulse evolution during propagation is illustrated in Fig. 4. For the propagating pulse intensity we observe a flipped temporal asymmetry as compared to the Gaussian pulse, see Figs. 4(f) and 1(f), although both pulses are exactly symmetric at the waveguide input. The trailing edge is now steeper than the leading edge. The asymmetry of the Kerr-induced phase shift is also inverted as compared to the Gaussian pulse as it is shown by the red dashed lines in Figs. 4(g) and 1(g). Here, in contrast to the Gaussian pulse, the Kerr-induced blue shift in the rear part is larger than the red shift of the leading edge including the peak.

Fig. 4 Same as Fig. 1 but for a Lorentzian input pulse [2t0 = 0.2ns in Eq. (45)].

5.5. Sech input pulse

A particularly interesting input shape is the sech pulse. For an assumed input pulse of duration 2t0, we have
I0(t)=I^0cosh(t/t0),R0(t)=t0I^02[1+tanh(t/t0)]
(46)
and find J = π and Q0=2t0I^02. The corresponding results for the propagating pulse are shown in Fig. 5, while the free-carrier density after the pulse can be calculated from Q(z) in Fig. 6 according to Eq. (7b). For this sech pulse, Eqs. (18) and (15) can be re-written
I(z,t)=I^0exp[M(z)αz]cosh[M(z)+(tz/v)/t0]=exp[M(z)αz]I0[t0M(z)+tz/v]
(47)
and
R(z,t)=t0I^02exp[2M(z)2αz]{1+tanh[M(z)+(tz/v)/t0]}=exp[2M(z)2αz]R0[t0M(z)+tz/v]
(48)
where
M(z)=12ln[1+2γσrt0I^02Leff(2)(z)].
(49)
Obviously, the shape and the duration of the sech pulse is maintained all along the propagation down the waveguide, while it is attenuated by a modified damping function
I[z,tpeak(z)]I^0=exp[M(z)αz]=exp(αz)1+2γσrt0I^02Leff(2)(z).
(50)
The attenuation profile Eq. (50) for the sech pulse of Eq. (47) along the propagation distance z is the same as that for the time-invariant intensity of the CW case given by Eq. (71) in Appendix B, when τeffI¯02 is replaced by t0I^02. This means, that the pulse peak intensity may be by a factor Î0/Ī0 = (τeff/t0)1/2 larger than the CW intensity to induce the same FCA self-attenuation at any position z. For an effective free-carrier lifetime of 10ns and a pulse duration of 100ps this factor amounts to 10.

Fig. 5 Same as Fig. 1 but for a sech-shaped input pulse [2t0 = 0.2ns in Eq. (46)].
Fig. 6 (a) Peaking time tpeak(z) of sech pulses along the propagation distance z for input peak intensities 0, 1, and 2 times the threshold intensity Iinv of Eq. (55) and a pulse duration of 2t0 = 0.2ns. (b) Q(z)/Q0 for sech pulses as a function of z for different input peak intensities and 2t0 = 0.2ns. (c) Upper limiting peak intensity Ilim(z) (black solid curves) for sech pulses of different durations 2t0, and z-dependent peak intensity Î0 exp [−M(z) −αz] (dashed red curves) of the travelling sech pulses for an input peak intensity Î0 = 10 W/μm2 and the given pulse lengths.

We may conclude from Eq. (47), that attenuation by self-induced FCA is an ideal limiter for sech pulses since for large intensities or large effective interaction lengths, i.e. for I^02Leff(2)(z)1/(2γσrt0), the output pulse will be purely sech but simultaneously be limited in strength to the asymptotic form
I(z,t)Ilim(z)cosh[M(z)+(tz/v)/t0],Ilim(z)=exp(αz)2γσrt0Leff(2)(z),
(51)
where Ilim(z) is the maximum possible peak intensity of a sech pulse of width 2t0 at a position z in the waveguide. Except of the waveguide input z = 0 where I(0, t) = I0(t) may assume any launched initial condition and thus Ilim(z = 0) → ∞, the sech-pulse intensity inside the waveguide is restricted by the upper limit Ilim(z) as illustrated in Fig. 6(c). This is in clear contrast to the rectangular input pulse of Section 5.3, where the front peak of the propagating pulse does not see any self-induced FCA and can thus be arbitrarily increased in proportion to the input intensity.

Note that unlike solitons [41

41. G. P. Agrawal, Nonlinear Fiber Optics (Elsevier, Amsterdam, 2007).

, 43

43. N. Akhmediev and A. Ankiewicz, Dissipative Solitons (Springer, Heidelberg, 2005). [CrossRef]

], the sech input pulse here does not need to fulfil any fixed relation between pulse duration and peak intensity to propagate without changing its shape. Any sech pulse propagating according to the present model described by Eqs. (5), (7a) and (8) will keep its original shape. In addition, any input pulse having an exponentially rising leading edge will at least asymptotically turn into a sech pulse after enough self-induced FCA has shaped it.

In Figs. 1, 4 and 5, i.e. for smooth pulses, it can be observed that the pulses despite having already propagated into the waveguide may peak earlier than the input pulse at z = 0 itself. For the sech pulse of Eq. (47) we can quantify this peaking time at an arbitrary position z explicitly as
tpeak(z)=zvt0M(z).
(52)
Its dependence on the propagation distance z is illustrated in Fig. 6(a). Its slope along z,
dtpeak(z)dz=1vpeak(z)=1vγσrt02I^02exp(2αz)1+2γσrt0I^02Leff(2)(z),
(53)
becomes zero at the position z = zinv where
zinv=12αln[γσrt0I^02(1+αvt0)α+γσrt0I^02].
(54)
Thus, if zinv > 0, then the apparent group velocity vpeak(z) = [dtpeak(z)/dz]−1 of the FCA-attenuated sech pulse seems to be ±∞ at z = zinv ± 0, negative for z < zinv and positive only for z > zinv (while, nevertheless, all energy or photons coming through unabsorbed up to this point have quite regularly been travelling at the group velocity v). Thus, zinv is the position in the waveguide where the pulse peaks first and from which the latter seems to travel into both directions afterwards, seemingly amplified towards the input and obviously attenuated towards the end of the waveguide. This point zinv shifts deeper into the waveguide with increasing peak intensity Î0 and pulse width t0, but no further than zinv ≤ ln(1 + αvt0)/2α. The seeming behavior of a group-velocity inversion requires a large enough input energy W0 (or peak intensity Î0 for a given t0) for which zinv ≥ 0,
W0=πt0I^0Winv=πt0Iinv=π/γσrv.
(55)
For the waveguide parameters given in Section 2, these conditions correspond to W0 ≥ 1.87 × 10−9Ws/μm2 or, e.g., to Î0 ≥ 5.95W/μm2 for t0 = 0.1ns. The peaking time of the propagating sech pulse as a function of the position z is plotted in Fig. 4(a) for different input intensities Î0 = 0 × Iinv, 1 × Iinv, and 2 × Iinv. It is easy to see that the pulse may peak inside the waveguide (z > 0) before the input pulse does when Eq. (55) is fulfilled.

At the waveguide input the velocity of the peak is vpeak(z=0)=v/(1vγσrt02I^02), where it is larger than v for input energies W0 up to Winv, while it becomes infinite and changes sign when W0 = Winv, and finally reduces to zero for further increasing input energies. Together with the observation made in Section 5.3 that any intensity step always propagates at the group velocity v, the occurrence of the peak of smooth pulses inside the waveguide before the peaking at the input and the seeming inversion of the peak velocity vpeak have some similarity with the superluminal pulse propagation in media with extreme group-velocity modifications [42

42. R. W. Boyd and D. J. Gauthier, “Controlling the velocity of light pulses,” Science 326, 1074–1077 (2009). [CrossRef] [PubMed]

]. However, it should be noted that in the present paper no special requirements have been put on the group velocity v and the seeming superluminal propagation behavior only results from the attenuation of the rear part of the pulse by self-induced free carriers.

An explicit expression for Kerr phase shift of the sech pulse can be given by inserting Eqs. (46) into (21) but it is too lengthy to display here. A temporal delay of the Kerr phase against the propagating pulse intensity profile I(z,t) was predicted from Eq. (23). Indeed, the peak of the intensity of the propagating sech pulse in Fig. 4(e) after 50mm propagation distance occurs 47ps earlier than the peak of the corresponding Kerr phase shift in Fig. 4(g).

Recalling the observation that a temporally asymmetric Kerr phase should originate from the FCA-induced asymmetry of the propagating pulse intensity [21

21. L. Yin and G. P. Agrawal, “Impact of two-photon absorption on self-phase modulation in silicon waveguides,” Opt. Lett. 32, 2031–2033 (2007). [CrossRef] [PubMed]

] it may seem paradoxical that the Kerr phase of the form-stable and hence always symmetric sech pulse also happens to become asymmetric. However, applying the general discussion of the destroyed symmetry relation between the propagating intensity profile and the Kerr phase from Section 4.1, it is clear that despite its maintained shape even the always symmetric sech pulse may develop an asymmetric Kerr phase as it is the history of the propagating pulse which determines the accumulated Kerr phase. While the trailing edge of the pulse contains the remainder of the peak of the input pulse, the Kerr phase could accumulate a significant amount there and thus the Kerr phase lags the intensity profile during propagation.

Finally, the FCR-induced phase and frequency shifts of the sech pulse follow directly from Eqs. (25) to (27) by inserting the corresponding expression for I0(tz/v), I(z,t) and R0(tz/v) from Eqs. (46) to (47). The FCR frequency shift may be written
ΔωFCR(z,t)=σiγLeff(2)(z)I^02exp[M(z)]cosh[M(z)]+cosh[M(z)+2(tz/v)/t0].
(56)
Thus, the FCR-induced phase ϕFCR of the sech pulse, which is the dominant non-linear phase shift in Figs. 4(g) and 4(h), causes a maximum spectral blueshift
ΔωFCR(max)(z)=σiσrt0×1+2γσrLeff(2)(z)t0I^0211+2γσrLeff(2)(z)t0I^02+1,
(57)
which increases with input peak intensity and effective propagation length but never exceeds ΔωFCRσi/σrt0. This maximum blueshift occurs at time
tΔωFCR(max)(z)=zvt04ln[1+2γσrLeff(2)(z)t0I^02]
(58)
=zvt0M(z)2=tpeak(z)+t0M(z)2,
(59)
i.e., it lags the intensity peak at time tpeak(z) given by Eq. (52) by a delay t0M(z)/2 steadily growing up to t0ln(1+2ηt0I^02)/4 in agreement with the discussion below Eq. (27). Thus, the maximum FCR-induced blue shift occurs later and later with respect to the pulse peak. Interestingly, even the relative spread of the peaking times of the intensity and of the frequency shift grows with intensity and pulse width itself as [tΔωFCR(max)(z)tpeak(z)]/t0=ln[1+2γσrLeff(2)(z)t0I^02]/4. The FCR blueshift at the intensity peak t = tpeak(z) becomes
ΔωFCR[z,tpeak(z)]=σiγLeff(2)(z)I^022[1+σrγt0I^02Leff(2)(z)],
(60)
and stabilizes at its maximum value σi/(2σrt0) for large effective propagation distances and input intensities, i.e. at half of the limit of Eq. (57).

6. Asymptotic compressibility of pulses by self-induced free-carrier absorption

The self-attenuation of the pulse’s tail by the accumulating free carriers has been proposed as a means for compressing optical pulses in silicon waveguides, i.e., for reducing the pulse duration [18

18. E. K. Tien, F. Qian, N. S. Yuksek, and O. Boyraz, “Influence of nonlinear loss competition on pulse compression and nonlinear optics in silicon,” Appl. Phys. Lett. 91, 201115 (2007). [CrossRef]

, 19

19. E. K. Tien, N. S. Yuksek, F. Qian, and O. Boyraz, “Effect of TPA and FCA interplay on pulse compression in silicon,” in 20th Annual Meeting of the IEEE Lasers and Electro-Optics Society 2007, Lake Buena Vista, Fl, paper ThY2, 2007. [CrossRef]

]. Naturally, the discussion of pulse compressibility depends on the detailed definition of the pulse width, especially when the compression goes along with a change of the pulse shape. Here we compare pulses by normalizing and shifting them to make their peaks coincide. The spreading of the edges will then be seen to be a sufficiently unique criterion for the simple pulses considered here.

Some estimates about the necessary predominance of FCA over TPA for the compression of pulses have already been made in [19

19. E. K. Tien, N. S. Yuksek, F. Qian, and O. Boyraz, “Effect of TPA and FCA interplay on pulse compression in silicon,” in 20th Annual Meeting of the IEEE Lasers and Electro-Optics Society 2007, Lake Buena Vista, Fl, paper ThY2, 2007. [CrossRef]

]. Here, however, we show that even in the absence of the attenuation by TPA only pulses with certain features of the initial pulse shape can be compressed.

The compression of optical pulses by self-induced FCA [18

18. E. K. Tien, F. Qian, N. S. Yuksek, and O. Boyraz, “Influence of nonlinear loss competition on pulse compression and nonlinear optics in silicon,” Appl. Phys. Lett. 91, 201115 (2007). [CrossRef]

, 19

19. E. K. Tien, N. S. Yuksek, F. Qian, and O. Boyraz, “Effect of TPA and FCA interplay on pulse compression in silicon,” in 20th Annual Meeting of the IEEE Lasers and Electro-Optics Society 2007, Lake Buena Vista, Fl, paper ThY2, 2007. [CrossRef]

] relies on the attenuation of the rear part of the pulse by the accumulated free carriers. However, the form stability of the sech pulses discussed in Section 5.5 raises the question what shape features are required to make a pulse compressible.

Obviously, after the original trailing edge and the pulse peak have been attenuated by FCA, the remainder of the propagating pulse will be comprised of the former leading edge and the shape of the latter becomes most important for the evolving pulse shape. To discuss this in more detail we refer to the normalized presentation of the pulse evolution in Eq. (31) where f0(ξ) and F0(ξ) depend only on the shape but neither on the peak intensity Î0 nor the duration t0 of the pulse. If the relative slope (∂f0/∂t)/f0 of the leading edge increases and hence the latter becomes steeper and steeper towards negative times t → −∞, or the intensity is even zero before a certain time t < 0, the pulse may be compressible at least asymptotically for large FCA-distortion parameters ΘFCA. This may be visualized as follows. After the rear part of a pulse has been taken away by self-induced FCA the pulse width may be judged by re-normalizing the attenuated pulse, e.g., to its now-reduced power or peak intensity. If then the remaining leading edge is more flat because it has a smaller relative slope it will be found to stretch further towards negative times and the re-normalized pulse appears to be broadened. Vice versa, if the relative slope of the remaining leading edge is found to be larger and the latter has become steeper then the pulse may have been compressed.

This is obvious for the Gaussian pulse in Eq. (43) which is compressible since (f0/t)/f0=2t/t02 and the leading edge becomes steeper and steeper for t → −∞, where the pulse energy is not widely enough distributed to prevent compression [see Fig. 7(a)]. In other words, the mechanism which cuts away the rear part of pulse faces a steeper and steeper relative slope of the remaining front part of the pulse as it progresses towards the leading edge. The rectangular pulse of Eq. (44) is even easier to compress since it has an infinitely steep front edge and is zero for t < −t0, compare Fig. 7(b). Thus, for pulses having an intensity step in the leading edge down to zero such as the rectangular pulse in Section 5.3, pulse compression by self-produced free carriers is most efficient.

Fig. 7 Pulse compressibility: Normalized pulse intensity after 10 mm propagation distance (solid black curves), where arrows indicate the increase of the input peak intensities Î0 from 5 to 50 to 500 W/μm2. Dashed red curves indicate the input intensity for (a) Gaussian, (b) rectangular, (c) sech, and (d) Lorentzian input pulse.

On the other hand, the relative front slope of the Lorentzian pulse (∂f0)/∂t/f0 ≃ −2/t decreases monotonously towards t → −∞. The more energy is cut away from the rear end the more important becomes the remaining energy spreading far into the front edge, and self-induced FCA even broadens the Lorentzian pulse as shown in Fig. 7(d).

The limiting shape between pulse compression and broadening is the exponentially rising front edge since its relative slope is the same everywhere and cutting away some rear part of the pulse by self-produced free carriers asymptotically shapes the pulse into the sech form. Thus any pulse having an exponentially rising leading edge can at least asymptotically be compressed or broadened until it has assumed the sech shape. The resulting incompressible sech pulse, then, remains stable and does not change its shape as shown in Section 5.5, see Fig. 7(c). Moreover, the sech pulse can be compressed even less if TPA is present. TPA tends to broaden the pulse [18

18. E. K. Tien, F. Qian, N. S. Yuksek, and O. Boyraz, “Influence of nonlinear loss competition on pulse compression and nonlinear optics in silicon,” Appl. Phys. Lett. 91, 201115 (2007). [CrossRef]

,21

21. L. Yin and G. P. Agrawal, “Impact of two-photon absorption on self-phase modulation in silicon waveguides,” Opt. Lett. 32, 2031–2033 (2007). [CrossRef] [PubMed]

] while FCA would tend to attenuate its rear part and shape it back into sech as the leading edge remains exponential.

The FCA-distortion parameter ΘFCA(z)=γσrt0I^02Leff(2)(z) in Eq. (31) approaches ηt0I^02 for large propagation distances z ≫ 1/(2α), after which the pulse distortion comes to an end and the pulse shape stabilizes. Thus, pulse distortion is eased when η is large, i.e., when α is small or t0I^02 is large. For zero attenuation α → 0, the pulse distortion parameter ΘFCA(z)=γσrt0I^02z may increase arbitrarily large along z. Thus if a pulse has a compressible shape such as the rectangular or the Gaussian one, small linear losses α help achieving highly compressed pulses. Equation (31) also implies that a pulse with a given input shape at z = 0 will go through the same distortion no matter what pulse duration t0 or peak intensity Î0 it has, as long as ΘFCA(z) reaches the same final value at the waveguide output. For ΘFCA(z) = 0 the pulse shape remains unchanged.

Thus, one of the particular advantages of the present FCA-only formulation is that it allows us to check easily whether a pulse is compressible by self-induced FCA at all, even when TPA actually has such a significant effect on the pulse distortion that it may not be neglected. In contrast to this, with the alternative and more accurate full model the lack of analytical solutions would frustrate the attempt for such an explicit judgement, and the numerical simulations as the only left alternative would end up with less clear insight.

The principal tendency of TPA to broaden a pulse even in the presence of FCA can clearly be seen in Fig. 2(a). As compared to the FCA-only results from Eq. (18) shown by solid black lines, the full inclusion of TPA according to Eq. (67) flattens the propagating pulse down to the green dotted curvesd which essentially have lost the pronounced peak and are therefore significantly broader. Thus, if a pulse is found to be not compressible in a FCA-only model then it is even less so after inclusion of TPA.

7. Attenuation and phase shift of a weak probe pulse following an intense pump pulse

We now consider a weak second pulse, which we call the ”probe pulse”, following an intense first pulse, which we call the ”pump pulse”, in a temporal distance T at nearly the same wavelength and group velocity. Its pulse duration shall again be much shorter than the free-carrier lifetime τeff. The knowledge of this attenuation may be useful in experiments where pulsed instead of CW operation is chosen to mitigate FCA such as pulsed Raman experiments [13

13. T. K. Liang and H. K. Tsang, “Pulsed-pumped silicon-on-insulator waveguide Raman amplifier,” in Proceedings of International Conference on Group IV Photonics, 29 Sept.–1 Oct. 2004, paper WA4.

, 14

14. A. Liu, H. Rong, M. Paniccia, O. Cohen, and D. Hak, “Net optical gain in a low loss silicon-oninsulator waveguide by stimulated Raman scattering” Opt. Express 12, 4261–4267 (2004). [CrossRef] [PubMed]

, 44

44. O. Boyraz and B. Jalali, “Demonstration of a silicon Raman laser,” Opt. Express 12, 5269 – 5273 (2004). [CrossRef] [PubMed]

, 45

45. B. Jalali, V. Raghunathan, D. Dimitropoulos, and Boyraz, “Raman-based silicon photonics,” IEEE J. Sel. Top. Quantum Electron. 12, 412–421 (2006).

], pulse compression [18

18. E. K. Tien, F. Qian, N. S. Yuksek, and O. Boyraz, “Influence of nonlinear loss competition on pulse compression and nonlinear optics in silicon,” Appl. Phys. Lett. 91, 201115 (2007). [CrossRef]

, 19

19. E. K. Tien, N. S. Yuksek, F. Qian, and O. Boyraz, “Effect of TPA and FCA interplay on pulse compression in silicon,” in 20th Annual Meeting of the IEEE Lasers and Electro-Optics Society 2007, Lake Buena Vista, Fl, paper ThY2, 2007. [CrossRef]

] or continuum generation [9

9. P. Koonath, D. R. Solli, and B. Jalali, “Limiting nature of continuum generation in silicon,” Appl. Phys. Lett. 93, 091114 (2008). [CrossRef]

, 10

10. P. T. S. DeVore, D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, “Stimulated supercontinuum generation extends broadening limits in silicon,” Appl. Phys. Lett. 100, 101111 (2012). [CrossRef]

]. Because of its small intensity assumed here the probe pulse does not suffer any non-linear self-distortion. However, it experiences an excess attenuation and phase shift caused by the free carriers generated by the strong pump pulse ahead, which has been launched a time T ≫ Δt earlier. Both the free-carrier-induced attenuation and the phase shift vary only slightly across the short probe pulse as its duration is assumed much shorter than the free-carrier lifetime. We may write the field amplitude of the probe pulse e(z,t) = e0(tz/v)exp{FCR(z) − [αz + aFCA(z)]/2}, where e(z = 0, t) = e0(t) is the input field amplitude of the probe pulse at z = 0 and
aFCA(z)=0zαFCA(z)dz,ψFCR(z)=σi2σraFCA(z)
(61)
are the accumulated attenuation and phase, respectively, induced via FCA and FCR by the free-carriers generated by the pump pulse ahead. With Eqs. (7b) and (17) the local FCA-induced attenuation coefficient is
αFCA(z)=σrN[z,tP(z)+T]=σrγQ(z)G(T)=G(T)σrγQ0exp(2αz)1+γσrQ0Leff(2)(z).
(62)
Comparison with the analogous attenuation for the probe pulse caused by the free carriers generated by a CW pump as derived in Eq. (71) in Appendix B reveals that pulsed and CW operation result in the same probe-pulse attenuation αFCA(z) = FCA(z) at all positions z only if G(T) = 1/2 and Q0=t0I^02F0()=2τeffI¯02, where Ī0 is the input intensity in case of CW operation at z = 0 as defined in Appendix B. For a Gaussian input pump pulse and an exponential decay according to Eq. (4), this happens for Î0 = (2/π)1/4(2τexp/t0)1/2Ī0 ≃ 1.26 (τexp/t0)1/2Ī0 and T = ln(2) × τexp = 0.69 τexp. For a discussion of sech-shaped pump pulses see Section 5.5.

The accumulated attenuation the probe pulse will have experienced after following the pump pulse all along the waveguide length L follows from Eqs. (61) and (62),
aFCA(L)=G(T)ln[1+σrγQ0Leff(2)(L)].
(63)
For a sufficiently long waveguide L ≫ 1/2α, it approaches the constant asymptotic value aFCAG(T)ln(1 + ηQ0). In even longer waveguides the probe pulse finally travels in a region where the preceding pump pulse had no longer been strong enough to leave a perceptible amount of free carriers. The attenuation can be arbitrarily reduced by increasing the time T between the two pulses according to the detailed shape of the decay function G(t).

In order to see what is gained by using pulse operation instead of CW, we now seek to find out up to which input peak intensities Î0 of the pump pulse the aggregate attenuation for the probe pulse at the waveguide output z = L is smaller than in the CW case, i.e.,
aFCA(L)<a¯FCA(L).
(64)
The aggregate FCA attenuation āFCA(L) for the CW operation is given by Eq. (72) in Appendix B. Thus, the relation in Eq. (64) is fulfilled when
Q0=t0F0()I^02<[1+2σrγLeff(2)(L)τeffI¯02]1/[2G(T)]1σrγLeff(2)(L)
(65)
It is easy to see that for G(T) = 1/2 this relation is true for Q0<2τeffI¯02 independently of the waveguide length. Thus, the input peak intensity of the pump pulse can be up to a factor I^0/I¯02τeff/t0F0() larger than the CW pump intensity before the subsequent probe pulse suffers the same FCA-induced attenuation. For G(T) = 1/4, this relation becomes length dependent, Q0<4τeffI¯02[1+σrγτeffI¯02Leff(2)(L)]. In waveguides of short length L pulse-induced FCA for the probe pulse is weaker than CW-induced FCA as long as I^0/I¯0<2τeff/t0F0() while long waveguides leave even more room, namely I^0/I¯0<2[τeff(1+ητeffI¯02)/t0F0()]1/2, as the latter limit increases with I¯02. In case of an exponential decay function according to Eq. (4), G(T) = 1/2 and G(T) = 1/4 corresponds to T = ln(2) × τexp and T = ln(4) × τexp, respectively. In both cases the pump pulse can have a peak intensity larger than the CW pump intensity in proportion to τeff/t0 before causing more FCA for the probe pulse.

The situation is different if the probe pulse follows the intense pump pulse very closely and Eq. (65) has to be fulfilled for G(Tτeff) ≃ 1. For short enough waveguides or low intensities, this condition reqiures I^0/I¯0<τeff/t0F0(). However, for sufficiently long waveguides and large intensities, the admissible ratio I^0/I¯0<[2τeff/ηt0F0()I¯0]1/2 for guaranteeing the condition in Eq. (64) decreases for an increasing CW input intensity and asymptotically becomes zero for Ī0 → ∞. In particular, if
I¯0>1t0F0()×2[τefft0F0()]σrγLeff(2)(L),
(66)
the input peak intensity Î0 of the pump pulse must be smaller than the input CW intensity Ī0 to exert a lower accumulated FCA on the closely following probe pulse.

This inverted behavior can be understood as follows. For the same input intensity the CW pump is attenuated by self-induced FCA more rapidly than the pulsed pump, see for example the discussion below Eq. (50) for the sech pulse. When the CW pump is exhausted after a certain propagation distance it can no longer generate enough free carriers to attenuate the probe pulse. The pulsed pump with the same input peak intensity, however, has a much longer reach to produce free carriers and the probe pulse following closely experiences a larger aggregate FCA when propagating to the end of the waveguide.

The upper limit on G(T) above which pulse operation is no longer prior to CW operation for all input intensities is G(T) = 1/2. Since for large input intensities exp[aFCA(L)]I^0G(T) and exp[a¯FCA(L)]I¯01/2, aFCA(L) will always be able to exceed āFCA(L) for G(T) > 1/2 even for Î0 = Ī0, if only these two input intensities are large enough. Vice versa, for G(T) < 1/2 pulsed operation may reduce the aggregate FCA-induced attenuation on a following probe pulse. For an exponential free-carrier decay according to Eq. (4) this is true when the probe pulse follows in a temporal distance larger than T > 0.693τexp.

For the phase ψFCR(z) of the probe pulse analogous conclusions can be drawn using Eq. (61).

8. Conclusions

In conclusion we have derived explicit analytical solutions for the propagation of optical pulses and their attenuation by self-produced free carriers in silicon waveguides allowing us to quantify the pulse distortion and to calculate explicitly the free-carrier density and the nonlinear phase shifts caused by the Kerr effect and by free-carrier refraction. We have shown that the omission of two-photon absorption as a cause of attenuation as done in the derivation and accounting only for free-carrier absorption appropriately models the situation in short or highly lossy silicon-based waveguides especially for high-energy input pulses. On the other hand this formulation may also serve as a tool in understanding the role of TPA and FCA in the interplay during propagation when aiming at continuum generation or pulse compression in low-loss waveguides. Sech-shaped intensity pulses have been shown to maintain their shape independently of the intensity or pulse width with a resemblance to superluminal light propagation.

9. Appendix A

In this Appendix we show that TPA as a source of attenuation can be neglected in a full model actually admitting of both TPA and FCA, as soon as it has been found to be negligible already in a simpler TPA-only model. The well-known analytical solution of the latter allows us to make this decision of omitting TPA explicitly. The propagation equation analogous to Eqs. (5), (7a) and (8) resulting from the full problem described by Eq. (1) with TPA included but GVD neglected may be written
Iz+1vIt=2vIx˜=(α+σrN)IβrI2,N(z,t)=γtI2(z,t)dt,
(67)
where N = N(z,t) = N [v()/2, ( + )/2] is understood according to Eq. (11). Using [37

37. E. Kamke, Differentialgleichungen. Lösungsmethoden und Lösungen: Gewöhnliche Differentialgleichungen I. (B. G. Teubner, Stuttgart, 1983), p. 298, Eq. (1.34).

] and assuming I(0, t) = Î0f0(t/t0) at the waveguide input with f0(t/t0) ≤ 1 as in Eq. (28), we derive the solution accounting for the accumulated non-linear attenuation by TPA and FCA after propagation distance z,
I(z,t)=I^0f0[(tz/v)/t0]exp(αz)U(z,t)1+βrI^0L˜eff(1)(z,t)f0[(tz/v)/t0],
(68)
where
U(z,t)=exp{σr0zN[ζ,t+(ζz)/v]dζ},N(z,t)=γtI2(z,t)dt.
(69)
Importantly,
L˜eff(1)(z,t)=0zexp(αχ)U(χ,t)dχ
(70)
is a TPA-effective propagation length modified by the presence of U(χ, t) in the integral as compared to the original definition (16). The system of Eqs. (68)(70) is complete to describe pulse propagation in the presence of both TPA and FCA. Although N can not be given explicitly, we will later use its property of always being non-negative, N ≥ 0.

For the moment, we only focus on the denominator in Eq. (68). If FCA is mathematically switched off by setting γ = 0, then N ≡ 0, U(z,t) ≡ 1, and L˜eff(1)(z,t)Leff(1)(z)=[1exp(αz)]/α becomes the TPA-effective propagation length according to (16), and hence the TPA-only model of Eqs. (33) and (34) is obtained. We assume further that, within this TPA-only model, Leff(1)(z) has already been found to be small enough that the denominator can be approximated by unity because the condition in Eq. (35) be valid. If now FCA would be taken into account by setting γ > 0 again, then N > 0, and consequently U(z,t) < 1 would ensure that L˜eff(1)(z,t) becomes even smaller than Leff(1)(z), and approximating the denominator in Eq. (68) by unity would be justified even more. Finally, since this denominator is the only place where βr enters explicitly the complete system of the modelling equations (68)(70), approximating it by unity is equivalent with setting βr = 0 in the whole model and hence with fully ignoring TPA as a source of attenuation. This becomes obvious when checking that Eqs. (68) through (70) with the denominator set to unity exactly solve the FCA-only problem Eq. (5). The range of applicability of our FCA-only formulation is illustrated in Section 5.1.

Thus, we have shown that it is sufficient to check in a simple TPA-only model as described by Eqs. (33) and (34) whether the TPA-distortion parameter ΘTPA is small enough that TPA can be neglected in the full problem of Eq. (67) actually admitting of both TPA and FCA, and hence whether the FCA-only model presented here applies.

10. Appendix B

In this Appendix we derive the solution of the problem described by Eqs. (5) and (2) under CW operation. In this case we have ∂/∂t = 0, and I(z,t) = Ī(z), N(z,t) = (z) = γτeffĪ2(z) with the initial conditions Ī0 = Ī(z = 0) and N¯0=N¯(z=0)=γτeffI¯02, where τeff is defined by Eq. (3). For example, we have τeff = τexp for the exponential decay function in Eq. (4). We derive the CW intensity Ī(z) and the local FCA coefficient FCA(z) = σr(z) from Eq. (5) by separation of variables,
I¯(z)=I¯0exp(αz)1+2σrγτeffI¯02Leff(2)(z),α¯FCA(z)=σrγτeffI¯02exp(2αz)1+2σrγτeffI¯02Leff(2)(z).
(71)

The latter results in an accumulated attenuation for a weak probe pulse after propagation to the waveguide output z = L,
a¯FCA(L)=0Lα¯FCA(z)dz=12ln[1+2σrγτeffI¯02Leff(2)(L)].
(72)
These results allow us to compare the pulse propagation with the CW situation in Sections 5.5 and 7.

Acknowledgments

This work was funded by the Deutsche Forschungsgemeinschaft (DFG) under Grant FOR 653.

References and links

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W. Bogaerts, S. K. Selvaraja, P. Dumon, J. Brouckaert, K. De Vos, D. Van Thourhout, and R. Baets, “Silicon-on-insulator spectral filters fabricated with CMOS technology,” IEEE J. Sel. Top. Quantum Electron. 16, 33–44 (2010). [CrossRef]

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H. K. Tsang and Y. Liu, “Nonlinear optical properties of silicon waveguides,” Semicond. Sci. Technol. 23, 064007 (2008). [CrossRef]

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A. R. Motamedi, A. H. Nejadmalayeri, A. Khilo, F. X. Kärtner, and E. P. Ippen, “Ultrafast nonlinear optical studies of silicon nanowaveguides,” Opt. Express 20, 4085–4101 (2012). [CrossRef] [PubMed]

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P. Mehta, N. Healy, R. Slavik, R. Watts, J. Sparks, T. Day, P. Sazio, J. Badding, and A. Peacock, “Nonlinearities in silicon optical fibers,” in Optical Fiber Communication Conference, OSA Technical Digest (CD) OFC 2011, Los Angeles, (Optical Society of America, 2011), paper OThS3.

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P. Koonath, D. R. Solli, and B. Jalali, “Limiting nature of continuum generation in silicon,” Appl. Phys. Lett. 93, 091114 (2008). [CrossRef]

10.

P. T. S. DeVore, D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, “Stimulated supercontinuum generation extends broadening limits in silicon,” Appl. Phys. Lett. 100, 101111 (2012). [CrossRef]

11.

H. Renner, “Upper limit for the amplifiable Stokes power in saturated silicon waveguide Raman amplifiers,” in 7th International Conference on Group IV Photonics (GFP), Beijing, China, 1–3 Sept. 2010, paper P1.15.

12.

R. Claps, V. Raghunathan, D. Dimitropoulos, and B. Jalali, “Influence of nonlinear absorption on Raman amplification in silicon waveguides,” Opt. Express 12, 2774–2780 (2004). [CrossRef] [PubMed]

13.

T. K. Liang and H. K. Tsang, “Pulsed-pumped silicon-on-insulator waveguide Raman amplifier,” in Proceedings of International Conference on Group IV Photonics, 29 Sept.–1 Oct. 2004, paper WA4.

14.

A. Liu, H. Rong, M. Paniccia, O. Cohen, and D. Hak, “Net optical gain in a low loss silicon-oninsulator waveguide by stimulated Raman scattering” Opt. Express 12, 4261–4267 (2004). [CrossRef] [PubMed]

15.

T. K. Liang and H. K. Tsang; “Role of free carriers from two-photon absorption in Raman amplification in silicon-on-insulator waveguides,” Appl. Phys. Lett. 84, 2745–2747 (2004). [CrossRef]

16.

H. Renner, M. Krause, and E. Brinkmeyer, “Maximal gain and optimal taper design for Raman amplifiers in silicon-on-insulator Waveguides,” in Integrated Photonics Research and Applications Topical Meeting, San Diego, California, April 11–13. Joint IPRA/NPIS Oral Session: Frontiers in Nanophotonics (paper JWA3), (2005).

17.

H. Renner and M. Krause, “Maximal total gain of non-tapered silicon-on-insulator Raman amplifiers,” in Optical Amplifiers and Their Applications/Coherent Optical Technologies and Applications, OSA Technical Digest Series(CD) (Optical Society of America, 2006), paper OMD2.

18.

E. K. Tien, F. Qian, N. S. Yuksek, and O. Boyraz, “Influence of nonlinear loss competition on pulse compression and nonlinear optics in silicon,” Appl. Phys. Lett. 91, 201115 (2007). [CrossRef]

19.

E. K. Tien, N. S. Yuksek, F. Qian, and O. Boyraz, “Effect of TPA and FCA interplay on pulse compression in silicon,” in 20th Annual Meeting of the IEEE Lasers and Electro-Optics Society 2007, Lake Buena Vista, Fl, paper ThY2, 2007. [CrossRef]

20.

R. Dekker, A. Driessen, T. Wahlbrink, C. Moormann, J. Niehusmann, and M. Först, “Ultrafast Kerr-induced all-optical wavelength conversion in silicon waveguides using 1.55 μ femtosecond pulses,” Opt. Express 14, 8336–8346 (2006). [CrossRef] [PubMed]

21.

L. Yin and G. P. Agrawal, “Impact of two-photon absorption on self-phase modulation in silicon waveguides,” Opt. Lett. 32, 2031–2033 (2007). [CrossRef] [PubMed]

22.

Q. Lin, O. J. Painter, and G. P. Agrawal, “Nonlinear optical phenomena in silicon waveguides: modeling and applications,” Opt. Express 15, 16604–16644 (2007). [CrossRef] [PubMed]

23.

X. Chen, N. C. Panoiu, and R. M. Osgood, “Theory of Raman-mediated pulsed amplification in silicon-wire waveguides,” IEEE J. Quantum Electron. 42, 160–170 (2006). [CrossRef]

24.

E. Dulkeith, Y. A. Vlasov, X. Chen, N. C. Panoiu, and R. M. Osgood Jr., “Self-phase-modulation in submicron silicon-on-insulator photonic wires,” Opt. Express 14, 5524–5534 (2006). [CrossRef] [PubMed]

25.

I.-W. Hsieh, X. Chen, J. I. Dadap, N. C. Panoiu, R. M. Osgood, S. J. McNab, and Y. A. Vlasov, “ Ultrafast-pulse self-phase modulation and third-order dispersion in Si photonic wire-waveguides,” Opt. Express 14, 12380–12387 (2006). [CrossRef] [PubMed]

26.

J. I. Dadap, N. C. Panoiu, X. G. Chen, I. W. Hsieh, X. P. Liu, C. Y. Chou, E. Dulkeith, S. J. McNab, F. N. Xia, W. M. J. Green, L. Sekaric, Y. A. Vlasov, and R. M. Osgood, “Nonlinear-optical phase modification in dispersion-engineered Si photonic wires,” Opt. Express 16, 1280–1299 (2008). [CrossRef] [PubMed]

27.

S. V. Afshar and T. M. Monro, “A full vectorial model for pulse propagation in emerging waveguides with subwavelength structures part I: Kerr nonlinearity,” Opt. Express 17, 2298–2318 (2009). [CrossRef]

28.

M. D. Turner, T. M. Monro, and S. V. Afshar, “A full vectorial model for pulse propagation in emerging waveguides with subwavelength structures part II: Stimulated Raman Scattering,” Opt. Express 17, 11565–11581 (2009). [CrossRef] [PubMed]

29.

R. M. Osgood, N. C. Panoiu, J. I. Dadap, X. Liu, X. Chen, I. W. Hsieh, E. Dulkeith, W. M. J. Green, and Y. A. Vlasov, “Engineering nonlinearities in nanoscale optical systems: physics and applications in dispersion-engineered silicon nanophotonic wires,” Adv. Opt. Photon. 1, 162–235 (2009). [CrossRef]

30.

I. D. Rukhlenko, M. Premaratne, and G. P. Agrawal “Nonlinear propagation in silicon-based plasmonic waveguides from the standpoint of applications,” Opt. Express 19, 206–217 (2011). [CrossRef] [PubMed]

31.

N. Suzuki, “FDTD analysis of two-photon absorption and free-carrier absorption in Si high-index-contrast waveguides,” J. Lightwave Technol. 25, 2495–2501 (2007). [CrossRef]

32.

I. D. Rukhlenko, M. Premaratne, and G. P. Agrawal, “Nonlinear silicon photonics: analytical tools,” IEEE J. Sel. Top. Quantum Electron. 16, 200–215 (2010). [CrossRef]

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S. Roy, S. K. Bhadra, and G. P. Agrawal, “Femtosecond pulse propagation in silicon waveguides: variational approach and its advantages,” Opt. Commun. 281, 5889–5893 (2008). [CrossRef]

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I. D. Rukhlenko, M. Premaratne, C. Dissanayake, and G. P. Agrawal, “Nonlinear pulse evolution in silicon waveguides: an approximate analytic approach,” J. Lightwave Technol. 27, 3241–3248 (2009). [CrossRef]

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L. Zhang, Q. Lin, Y. Yue, Y. Yan, R. G. Beausoleil, and A. E. Willner, “Silicon waveguide with four zero-dispersion wavelengths and its application in on-chip octave-spanning supercontinuum generation,” Opt. Express 20, 1685–1690 (2012). [CrossRef] [PubMed]

36.

Y. Liu and H. K. Tsang, “Time dependent density of free carriers generated by two photon absorption in silicon waveguides,” Appl. Phys. Lett. 90, 211105 (2007). [CrossRef]

37.

E. Kamke, Differentialgleichungen. Lösungsmethoden und Lösungen: Gewöhnliche Differentialgleichungen I. (B. G. Teubner, Stuttgart, 1983), p. 298, Eq. (1.34).

38.

T. K. Liang, L. R. Nunes, M. Tsuchiya, K. S. Abedin, T. Miyazaki, D. Van Thourhout, W. Bogaerts, P. Dumon, R. Baets, and H. K. Tsang, “Nonlinear self-distortion of picosecond optical pulses in silicon wire waveguides,” in Conference on Lasers and Electro-Optics (CLEO) 2006, 21–26 May 2006, Long Beach, paper JThC44.

39.

I. S. Gradstein and I. M. Ryshik, Table of Integrals, Series and Products (Academic Press, Boston, 1994) p. 104.

40.

R. Dekker, N. Usechak, M. Först, and A. Driessen, “Ultrafast nonlinear all-optical processes in silicon-on-insulator waveguides,” J. Phys. D: Appl. Phys. 40, R249–R271 (2007). [CrossRef]

41.

G. P. Agrawal, Nonlinear Fiber Optics (Elsevier, Amsterdam, 2007).

42.

R. W. Boyd and D. J. Gauthier, “Controlling the velocity of light pulses,” Science 326, 1074–1077 (2009). [CrossRef] [PubMed]

43.

N. Akhmediev and A. Ankiewicz, Dissipative Solitons (Springer, Heidelberg, 2005). [CrossRef]

44.

O. Boyraz and B. Jalali, “Demonstration of a silicon Raman laser,” Opt. Express 12, 5269 – 5273 (2004). [CrossRef] [PubMed]

45.

B. Jalali, V. Raghunathan, D. Dimitropoulos, and Boyraz, “Raman-based silicon photonics,” IEEE J. Sel. Top. Quantum Electron. 12, 412–421 (2006).

OCIS Codes
(040.6040) Detectors : Silicon
(190.4390) Nonlinear optics : Nonlinear optics, integrated optics
(190.5970) Nonlinear optics : Semiconductor nonlinear optics including MQW
(190.7110) Nonlinear optics : Ultrafast nonlinear optics
(320.5520) Ultrafast optics : Pulse compression
(320.7110) Ultrafast optics : Ultrafast nonlinear optics
(250.5403) Optoelectronics : Plasmonics

ToC Category:
Nonlinear Optics

History
Original Manuscript: August 17, 2012
Revised Manuscript: September 27, 2012
Manuscript Accepted: September 27, 2012
Published: October 30, 2012

Citation
Hagen Renner, "Nonlinear distortion of optical pulses by self-produced free carriers in short or highly lossy silicon-based waveguides," Opt. Express 20, 25718-25743 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-23-25718


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References

  1. S. Fathpour and B. Jalali (Eds.), Silicon Photonics for Telecommunications and Biomedicine (CRC Press, Taylor & Francis, Boca Raton, 2011). [CrossRef]
  2. L. Pavesi and D. J. Lockwood (Eds.), Silicon Photonics (Springer-Verlag, Berlin, 2004).
  3. G. T. Reed and A. P. Knights, Silicon Photonics: An Introduction (John Wiley, West Sussex, 2004). [CrossRef]
  4. W. Bogaerts, S. K. Selvaraja, P. Dumon, J. Brouckaert, K. De Vos, D. Van Thourhout, and R. Baets, “Silicon-on-insulator spectral filters fabricated with CMOS technology,” IEEE J. Sel. Top. Quantum Electron.16, 33–44 (2010). [CrossRef]
  5. H. K. Tsang and Y. Liu, “Nonlinear optical properties of silicon waveguides,” Semicond. Sci. Technol.23, 064007 (2008). [CrossRef]
  6. A. R. Motamedi, A. H. Nejadmalayeri, A. Khilo, F. X. Kärtner, and E. P. Ippen, “Ultrafast nonlinear optical studies of silicon nanowaveguides,” Opt. Express20, 4085–4101 (2012). [CrossRef] [PubMed]
  7. P. Mehta, N. Healy, R. Slavik, R. Watts, J. Sparks, T. Day, P. Sazio, J. Badding, and A. Peacock, “Nonlinearities in silicon optical fibers,” in Optical Fiber Communication Conference, OSA Technical Digest (CD) OFC 2011, Los Angeles, (Optical Society of America, 2011), paper OThS3.
  8. M. A. Foster, A. C. Turner, M. Lipson, and A. L. Gaeta, “Nonlinear optics in photonic nanowires,” Opt. Express16, 1300–1320 (2008). [CrossRef] [PubMed]
  9. P. Koonath, D. R. Solli, and B. Jalali, “Limiting nature of continuum generation in silicon,” Appl. Phys. Lett.93, 091114 (2008). [CrossRef]
  10. P. T. S. DeVore, D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, “Stimulated supercontinuum generation extends broadening limits in silicon,” Appl. Phys. Lett.100, 101111 (2012). [CrossRef]
  11. H. Renner, “Upper limit for the amplifiable Stokes power in saturated silicon waveguide Raman amplifiers,” in 7th International Conference on Group IV Photonics (GFP), Beijing, China, 1–3 Sept. 2010, paper P1.15.
  12. R. Claps, V. Raghunathan, D. Dimitropoulos, and B. Jalali, “Influence of nonlinear absorption on Raman amplification in silicon waveguides,” Opt. Express12, 2774–2780 (2004). [CrossRef] [PubMed]
  13. T. K. Liang and H. K. Tsang, “Pulsed-pumped silicon-on-insulator waveguide Raman amplifier,” in Proceedings of International Conference on Group IV Photonics, 29 Sept.–1 Oct. 2004, paper WA4.
  14. A. Liu, H. Rong, M. Paniccia, O. Cohen, and D. Hak, “Net optical gain in a low loss silicon-oninsulator waveguide by stimulated Raman scattering” Opt. Express12, 4261–4267 (2004). [CrossRef] [PubMed]
  15. T. K. Liang and H. K. Tsang; “Role of free carriers from two-photon absorption in Raman amplification in silicon-on-insulator waveguides,” Appl. Phys. Lett.84, 2745–2747 (2004). [CrossRef]
  16. H. Renner, M. Krause, and E. Brinkmeyer, “Maximal gain and optimal taper design for Raman amplifiers in silicon-on-insulator Waveguides,” in Integrated Photonics Research and Applications Topical Meeting, San Diego, California, April 11–13. Joint IPRA/NPIS Oral Session: Frontiers in Nanophotonics (paper JWA3), (2005).
  17. H. Renner and M. Krause, “Maximal total gain of non-tapered silicon-on-insulator Raman amplifiers,” in Optical Amplifiers and Their Applications/Coherent Optical Technologies and Applications, OSA Technical Digest Series(CD) (Optical Society of America, 2006), paper OMD2.
  18. E. K. Tien, F. Qian, N. S. Yuksek, and O. Boyraz, “Influence of nonlinear loss competition on pulse compression and nonlinear optics in silicon,” Appl. Phys. Lett.91, 201115 (2007). [CrossRef]
  19. E. K. Tien, N. S. Yuksek, F. Qian, and O. Boyraz, “Effect of TPA and FCA interplay on pulse compression in silicon,” in 20th Annual Meeting of the IEEE Lasers and Electro-Optics Society 2007, Lake Buena Vista, Fl, paper ThY2, 2007. [CrossRef]
  20. R. Dekker, A. Driessen, T. Wahlbrink, C. Moormann, J. Niehusmann, and M. Först, “Ultrafast Kerr-induced all-optical wavelength conversion in silicon waveguides using 1.55 μ femtosecond pulses,” Opt. Express14, 8336–8346 (2006). [CrossRef] [PubMed]
  21. L. Yin and G. P. Agrawal, “Impact of two-photon absorption on self-phase modulation in silicon waveguides,” Opt. Lett.32, 2031–2033 (2007). [CrossRef] [PubMed]
  22. Q. Lin, O. J. Painter, and G. P. Agrawal, “Nonlinear optical phenomena in silicon waveguides: modeling and applications,” Opt. Express15, 16604–16644 (2007). [CrossRef] [PubMed]
  23. X. Chen, N. C. Panoiu, and R. M. Osgood, “Theory of Raman-mediated pulsed amplification in silicon-wire waveguides,” IEEE J. Quantum Electron.42, 160–170 (2006). [CrossRef]
  24. E. Dulkeith, Y. A. Vlasov, X. Chen, N. C. Panoiu, and R. M. Osgood, “Self-phase-modulation in submicron silicon-on-insulator photonic wires,” Opt. Express14, 5524–5534 (2006). [CrossRef] [PubMed]
  25. I.-W. Hsieh, X. Chen, J. I. Dadap, N. C. Panoiu, R. M. Osgood, S. J. McNab, and Y. A. Vlasov, “ Ultrafast-pulse self-phase modulation and third-order dispersion in Si photonic wire-waveguides,” Opt. Express14, 12380–12387 (2006). [CrossRef] [PubMed]
  26. J. I. Dadap, N. C. Panoiu, X. G. Chen, I. W. Hsieh, X. P. Liu, C. Y. Chou, E. Dulkeith, S. J. McNab, F. N. Xia, W. M. J. Green, L. Sekaric, Y. A. Vlasov, and R. M. Osgood, “Nonlinear-optical phase modification in dispersion-engineered Si photonic wires,” Opt. Express16, 1280–1299 (2008). [CrossRef] [PubMed]
  27. S. V. Afshar and T. M. Monro, “A full vectorial model for pulse propagation in emerging waveguides with subwavelength structures part I: Kerr nonlinearity,” Opt. Express17, 2298–2318 (2009). [CrossRef]
  28. M. D. Turner, T. M. Monro, and S. V. Afshar, “A full vectorial model for pulse propagation in emerging waveguides with subwavelength structures part II: Stimulated Raman Scattering,” Opt. Express17, 11565–11581 (2009). [CrossRef] [PubMed]
  29. R. M. Osgood, N. C. Panoiu, J. I. Dadap, X. Liu, X. Chen, I. W. Hsieh, E. Dulkeith, W. M. J. Green, and Y. A. Vlasov, “Engineering nonlinearities in nanoscale optical systems: physics and applications in dispersion-engineered silicon nanophotonic wires,” Adv. Opt. Photon.1, 162–235 (2009). [CrossRef]
  30. I. D. Rukhlenko, M. Premaratne, and G. P. Agrawal “Nonlinear propagation in silicon-based plasmonic waveguides from the standpoint of applications,” Opt. Express19, 206–217 (2011). [CrossRef] [PubMed]
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