## Nonlinear distortion of optical pulses by self-produced free carriers in short or highly lossy silicon-based waveguides |

Optics Express, Vol. 20, Issue 23, pp. 25718-25743 (2012)

http://dx.doi.org/10.1364/OE.20.025718

Acrobat PDF (1335 KB)

### 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

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

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]

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]

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]

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]

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]

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]

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. 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]

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]

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. 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]

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]

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]

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]

**91**, 201115 (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]

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]

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]

## 2. Pulse propagation in silicon waveguides

*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. 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. 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. 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]

*E*(

*z*,

*t*)exp[

*i*(

*β*

_{0}

*z*−

*ωt*)] through the silicon waveguide, where

*ω*is the optical angular frequency and

*β*

_{0}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,

*β*

_{r}the two-photon absorption (TPA) coefficient,

*β*

_{i}= −2

*k*

_{0}

*n*

_{2}, where

*k*

_{0}= 2

*π*/

*λ*and

*n*

_{2}is the Kerr coefficient,

*σ*

_{r}the coefficient of the free-carrier absorption (FCA), and

*σ*

_{i}is the efficiency of the free-carrier refraction (FCR) or free-carrier plasma dispersion effect [21

**32**, 2031–2033 (2007). [CrossRef] [PubMed]

*λ*= 1.55

*μ*m, where [21

**32**, 2031–2033 (2007). [CrossRef] [PubMed]

*n*

_{2}≃ 6 × 10

^{−18}m

^{2}/W and thus

*β*

_{i}= −5 × 10

^{−12}m/W,

*β*

_{r}≃ 5 × 10

^{−12}m/W, and further

*σ*

_{r}= 1.45 × 10

^{−21}m

^{2}and

*σ*

_{i}= 1.1 × 10

^{−20}m

^{2}. 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*= 10

^{8}m/s.

*β*

_{2}. For a pulse duration

*t*

_{0}the latter determines the dispersion length

*μ*m, modern silicon nanowires exhibit a GVD parameter |

*β*

_{2}| well below 10 ps

^{2}/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. 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]

*β*

_{2}| is below 5 ps

^{2}/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]

*L*

_{D}as those considered here GVD may be neglected as we do so in the present formulation by setting

*β*

_{2}= 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.

*μ*m

^{2}the parameters may be closer to bulk silicon, in nanowire waveguides the proper effective areas have to be incorporated to modify the nonlinear parameters [22

**15**, 16604–16644 (2007). [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]

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]

**15**, 16604–16644 (2007). [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]

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]

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]

**19**, 206–217 (2011). [CrossRef] [PubMed]

**15**, 16604–16644 (2007). [CrossRef] [PubMed]

**17**, 2298–2318 (2009). [CrossRef]

**32**, 2031–2033 (2007). [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]

**19**, 206–217 (2011). [CrossRef] [PubMed]

*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

*β*

_{r}and the GVD term by setting

*β*

_{2}= 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, where we assume that an optical pulse with intensity

*I*(

*z*= 0,

*t*) =

*I*

_{0}(

*t*) and phase

*ϕ*(

*z*= 0,

*t*) =

*ϕ*

_{0}(

*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

*β*

_{r}term, we keep the

*β*

_{i}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

*σ*

_{r}and

*σ*

_{r}terms.

*t*

_{P}(

*z*) − Δ

*t*,

*t*

_{P}(

*z*) + Δ

*t*]. For a pulse launched at the waveguide input

*z*= 0 at time

*t*=

*t*

_{P}(0) and travelling at the group velocity

*v*along the waveguide,

*t*

_{P}(

*z*) =

*t*

_{P}(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 2Δ

*t*is much smaller than the characteristic decay time

*τ*

_{eff}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 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 Since in the moment we are interested only in the evolution of an optical pulse with a duration much shorter than the time scale

*τ*

_{eff}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).

*I*and express the squared intensity as

*I*

^{2}(

*z*,

*t*) =

*∂R*(

*z*,

*t*)/

*∂t*=:

*Ṙ*(

*z*,

*t*), We integrate this equation over time

*t*, 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 implying

*z*=

*v*(

*x̃*−

*ỹ*)/2 and

*t*= (

*x̃*+

*ỹ*)/2, and obtain This ordinary differential equation is easily solved by separation of the variables

*x̃*and

*R*(or using [37]), where, after re-inserting the original space and time coordinates, the integration constant

*C*(

*ỹ*) =

*C*(

*t*−

*z/v*) is used to fit

*R*(

*z*,

*t*) to its initial value

*R*

_{0}(

*t*) =

*R*(

*z*= 0,

*t*) at the waveguide input

*z*= 0, where and

*I*

_{0}(

*t*) =

*I*(

*z*= 0,

*t*) is the input pulse intensity at

*z*= 0. We obtain where defines two different effective interaction lengths for

*m*= 1 and

*m*= 2. Further, we have where

*Q*

_{0}=

*Q*(

*z*= 0) =

*R*

_{0}(

*t*→ ∞), and the rightmost term is the limit for

*Q*

_{0}→ ∞ 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

*Q*

_{0}at the input, no matter what shape they have and how much they are distorted along the waveguide by the self-produced free carriers.

*z*. The pulse intensity

*I*(

*z*,

*t*) follows directly from Eqs. (8) and (15) as

*R*

_{0}(

*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

**93**, 091114 (2008). [CrossRef]

**100**, 101111 (2012). [CrossRef]

**91**, 201115 (2007). [CrossRef]

**32**, 2031–2033 (2007). [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]

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.

## 3. Nonlinear phase shifts

*N*(

*z*,

*t*) =

*γR*(

*z*,

*t*) from (7a), After inserting the analytical solutions Eqs. (18) and (15) for

*I*(

*z*,

*t*) and

*R*(

*z*,

*t*) using the coordinate transformation

*z*=

*v*(

*x̃*−

*ỹ*)/2 and

*t*−

*z/v*=

*ỹ*, the integration is easily done by means of the integral relations 2.313.1 and 2.314 given in [39]. Fitting the

*ỹ*-dependent integration constant properly for a given phase function

*ϕ*

_{0}(

*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*

_{Kerr}(

*z*,

*t*) and

*ϕ*

_{FCR}(

*z*,

*t*) will be given in Sections 3.1 and 3.2, respectively. We will also discuss the non-linear frequency shift Δ

*ω*(

*z*,

*t*) = −

*∂ϕ/∂t*which represents a spectral red shift for

*∂ϕ/∂t*> 0 and a blue shift for

*∂ϕ/∂t*< 0 [9

**93**, 091114 (2008). [CrossRef]

**100**, 101111 (2012). [CrossRef]

**91**, 201115 (2007). [CrossRef]

**15**, 16604–16644 (2007). [CrossRef] [PubMed]

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

*η*=

*γσ*

_{r}/2

*α*. Like the intensity does, the Kerr phase vanishes for

*t*= ±∞. For negligibly small free-carrier absorption

*ηR*

_{0}→ 0, it reduces to the well-known result for nonlinear pulse propagation in optical fibers [41], 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.

*t*→ −∞), where

*ηR*

_{0}is still small, the Kerr phase shift is approximately described by Eq. (22), which after sufficiently large propagation distances

*z*≫ 1/

*α*becomes

*ϕ*

_{Kerr}(

*z*,

*t*) = −

*β*

_{i}

*I*

_{0}(

*t*−

*z/v*)/2

*α*. In the trailing edge, however,

*ηR*

_{0}approaches its constant asymptotic value

*ηQ*

_{0}and the Kerr phase becomes proportional to Eq. (22) times a factor different from unity. For example,

*ϕ*

_{Kerr}(

*z*,

*t*→ +∞) = −

*β*

_{i}

*I*

_{0}(

*t*−

*z/v*)ln(4

*ηQ*

_{0})/(4

*αηQ*

_{0}) if

*ηQ*

_{0}≫ 1 in addition to

*z*≫ 1/

*α*. Thus, for symmetric input pulses

*I*

_{0}(

*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 where

*ϕ*

_{Kerr}(

*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 d

^{2}arctanh(

*u*)/d

*u*

^{2}≥ 0 the function

*h*(

*u*) is monotonously increasing with

*u*starting from

*h*(0) = 1. Obviously,

*u*(

*R*

_{0}) is monotonously increasing with

*R*

_{0}from

*u*(0) = 0 since

*g*(

*R*

_{0}) is monotonously increasing with

*R*

_{0}from

*g*(0) = 1 because

*R*

_{0}(

*t*−

*z/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

*ϕ*

_{Kerr}(

*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

### 3.2. Free-carrier induced phase shift

*γσ*

_{r}≥ 0, the logarithmic function is always non-negative and

*ϕ*

_{FCR}(

*z*,

*t*) ≤ 0 as

*σ*

_{i}/

*σ*

_{r}> 0. The FCR-induced phase shift is zero for

*t*→ −∞, decreases monotonously with time and reaches its maximum (negative) value at

*t*→ +∞, which is

*ϕ*

_{FCR}= −

*σ*

_{i}ln(1 +

*ηQ*

_{0})/2

*σ*

_{r}for

*z*≫ 1/2

*α*. Thus, FCR always causes a blue shift, i.e. its corresponding frequency shift Δ

*ω*

_{FCR}(

*z*,

*t*) = −

*∂ϕ*

_{FCR}(

*z*,

*t*)/

*∂t*is non-negative. The latter can be written and its time dependence at a position

*z*is proportional to the squared intensity

*I*

^{2}(

*z*,

*t*) of the travelling pulse times a monotonously increasing function of

*t*. This means that Δ

*ω*

_{FCR}(

*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

**93**, 091114 (2008). [CrossRef]

*σ*

_{r}→ 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

## 4. Interpretation of omitting TPA as a cause of attenuation

**91**, 201115 (2007). [CrossRef]

*ξ*=

*t*/

*t*

_{0}, and

*t*

_{0}determines the temporal width of the pulse. Thus, any input pulse

*I*

_{0}(

*t*) =

*Î*

_{0}

*f*

_{0}(

*t/t*

_{0}) is solely described by its peak intensity

*Î*

_{0}, its duration

*t*

_{0}, and a dimensionless shape function

*f*

_{0}(

*ξ*). Accordingly, we define such that

*W*

_{0}is the input pulse energy per unit cross-sectional area. To compare first the

*local*relevance of TPA versus FCA we consider the ratio

*α*

_{TPA}/

*α*

_{FCA}of the non-linear attenuation due to TPA and FCA, respectively, for general input pulses, Since by definition

*f*

_{0}(

*ξ*),

*F*

_{0}(

*ξ*) and

*J*depend on neither the peak intensity

*Î*

_{0}nor the pulse width

*t*

_{0}, 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

*W*

_{0}, and the relevance of TPA against FCA locally reduces more and more for increasing

*W*

_{0}.

*aggregate*effect of TPA and FCA after propagation through the waveguide, we re-write the pulse intensity evolution Eq. (18) in a normalized way, where is a dimensionless distortion parameter due to FCA. On the other hand, in a model accounting for TPA only [21

**32**, 2031–2033 (2007). [CrossRef] [PubMed]

**15**, 16604–16644 (2007). [CrossRef] [PubMed]

*σ*

_{r}= 0 or

*γ*= 0 in Eq. (67) in Appendix A, pulse propagation can be described by where the dimensionless distortion parameter due to TPA is

_{TPA}be small while Θ

_{FCA}remain finite.

*f*

_{0}(

*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

*Î*

_{0}, Eq. (35) is equivalent with requiring a small TPA-effective propagation length

_{FCA}shall remain finite even when Θ

_{TPA}becomes very small. This latter condition requires In order to translate these two conditions Eqs. (35) and (37) into requirements on the waveguide and on the launching conditions, we combine Θ

_{TPA}and Θ

_{FCA}of Eqs. (32) and (34) to eliminate

*W*

_{0}to write where

*αz*≪ 1) ≤ Ξ(

*αz*) ≤ Ξ(

*αz*≫ 1) = 2. Thus, a vanishing small TPA-distortion parameter Θ

_{TPA}and a finite and hence significantly larger FCA-distortion parameter Θ

_{FCA}require short TPA-effective propagation lengths

*W*

_{0}, and the two conditions in Eqs. (35) and (37) read Each of these two conditions is equivalent with requiring Θ

_{TPA}≪ 1 according to Eq. (35) as long as the desired finite Θ

_{FCA}is realized via Eq. (32). Now aiming at decreasing Θ

_{TPA}while keeping Θ

_{FCA}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

*t*

_{0}may vary in a wide range it can not be arbitrarily increased as we assume

*t*

_{0}≪

*τ*

_{eff}. 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*, where, depending on the choice of

*z*, the ratio

*Î*

_{0}. Hence, two extreme situations to realize the conditions in Eq. (39) can be distinguished.

*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 The parameter values given in Section 2 yield

*β*

_{r}/

*σ*

_{r}

*γ*= 0.177 Wns/

*μ*m

^{2}. For Θ

_{FCA}= 1, Θ

_{TPA}= 0.01,

*t*

_{0}= 0.1ns and

*z*≃ 11.3

*μ*m and

*W*

_{0}= 31.4 Wns/

*μ*m

^{2}corresponding to a peak intensity of

*Î*

_{0}= 177W/

*μ*

^{2}=17.7GW/cm

^{2}. This means, for Gaussian 0.1ns input pulses, Θ

_{TPA}never exceeds 0.01 up to an input intensity of

*Î*

_{0}= 177W/

*μ*

^{2}, at which Θ

_{FCA}already reaches unity. For a more relaxed TPA suppression of Θ

_{TPA}= 0.05 the results are

*z*≃ 283

*μ*m and

*W*

_{0}= 6.27 Wns/

*μ*m

^{2}corresponding to

*Î*

_{0}= 35.4W/

*μ*

^{2}=3.54GW/cm

^{2}.

*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 ensuring that even the maximum possible effective propagation lengths

_{FCA}= 1, Θ

_{TPA}= 0.01,

*t*

_{0}= 0.1ns and

*α*≃ 0.177/

*μ*m according to 768dB/mm and

*W*

_{0}= 62.7 Wns/

*μ*m

^{2}corresponding to

*Î*

_{0}= 354W/

*μ*

^{2}=35.4GW/cm

^{2}. For Θ

_{TPA}= 0.05 the results are

*α*≃ 7.08/mm according to 30.7dB/mm and

*W*

_{0}= 12.5 Wns/

*μ*m

^{2}corresponding to

*Î*

_{0}= 70.8W/

*μ*

^{2}=7.08GW/cm

^{2}.

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

**19**, 206–217 (2011). [CrossRef] [PubMed]

*μ*m, restricting the effective propagation lengths to a few micrometers. For example,

_{TPA}≤ 0.01 for input peak intensities up to 500W/

*μ*m

^{2}at which intensity level a significant FCA-distortion with Θ

_{FCA}= 2.82 is reached when

*t*

_{0}= 0.1ns. Thus, a practical approach could be as follows. If the error caused by a TPA-distortion parameter of Θ

_{TPA}≤ 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/

*μ*m= 1.086dB/

*μ*m, then all calculations using input peak intensities up to 500W/

*μ*m

^{2}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 Θ

_{FCA}then merely relaxes the requirements on

*W*

_{0}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.

## 5. Propagation of pulses with various input shapes

*I*

_{0}(

*t*) =

*Î*

_{0}

*f*(

*t/t*

_{0}) of different shapes

*f*(

*t/t*

_{0}), which may experience different distortions of intensity and phase.

### 5.1. Gaussian input pulse

*t*

_{0}, we write and obtain

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.

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. 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]

*t*−

*z/v*for a better visualization of the distortion with respect to the input pulse.

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

*ϕ*

_{Kerr}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 Δ

*ω*

_{Kerr}deeper into the ”red” than into the ”blue”. The FCR phase shift

*ϕ*

_{FCR}(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

*ϕ*

_{FCR}grows much stronger than

*N*does when increasing

*Î*

_{0}from 3 to 30W/

*μ*m

^{2}, because

*ϕ*

_{FCR}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

*ϕ*

_{Kerr}and

*ϕ*

_{FCR}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

*t*

_{0}= 0.1ns which is launched with different peak intensities

*Î*

_{0}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

*Î*

_{0}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.

*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/

*μ*m

^{2}for which case a TPA-distortion parameter Θ

_{TPA}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/

*μ*m

^{2}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

**32**, 2031–2033 (2007). [CrossRef] [PubMed]

**15**, 16604–16644 (2007). [CrossRef] [PubMed]

**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]

**93**, 091114 (2008). [CrossRef]

**100**, 101111 (2012). [CrossRef]

**19**, 206–217 (2011). [CrossRef] [PubMed]

*α*= 0.25/

*μ*m=1.086dB/

*μ*m limit the TPA-effective and the FCA-effective propagation length to no more than 4

*μ*m and 2

*μ*m, respectively. In this regard, the geometrical waveguide length of 20

*μ*m 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/

*μ*m

^{2}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

*Î*

_{0}= 3kW/

*μ*m

^{2}or 300GW/cm

^{2}while FCA significantly distorts the pulse.

*α*= 1dB/cm again [as already used in Figs. 1 and 2(a)] when, however, its length of only 50

*μ*m 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

*Î*

_{0}= 100W/

*μ*m

^{2}the pulse is significantly distorted by FCA, while Θ

_{TPA}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

*Î*

_{0}= 1kW/

*μ*m

^{2}or 100GW/cm

^{2}the curves almost coincide although the TPA-distortion parameter Θ

_{TPA}already reaches 0.25 at this input intensity. This proves even more that requiring Θ

_{TPA}to be small is a sufficient criterion for the applicability of the FCA-only model developed in this paper.

### 5.3. Rectangular input pulse

*t*

_{0}, we have while

*J*= 2 and

*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

*R*

_{0}(

*t*−

*z/v*) in the denominator of Eq. (18) is a continuous function of time

*t*−

*z/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.

### 5.4. Lorentzian input pulse

*t*

_{0}, we write and obtain

*J*=

*π*and

### 5.5. Sech input pulse

*t*

_{0}, we have and find

*J*=

*π*and

*Q*(

*z*) in Fig. 6 according to Eq. (7b). For this sech pulse, Eqs. (18) and (15) can be re-written and where 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 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

*Î*

_{0}/

*Ī*

_{0}= (

*τ*

_{eff}/

*t*

_{0})

^{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.

*I*

_{lim}(

*z*) is the maximum possible peak intensity of a sech pulse of width 2

*t*

_{0}at a position

*z*in the waveguide. Except of the waveguide input

*z*= 0 where

*I*(0,

*t*) =

*I*

_{0}(

*t*) may assume any launched initial condition and thus

*I*

_{lim}(

*z*= 0) → ∞, the sech-pulse intensity inside the waveguide is restricted by the upper limit

*I*

_{lim}(

*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.

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

*z*= 0 itself. For the sech pulse of Eq. (47) we can quantify this peaking time at an arbitrary position

*z*explicitly as Its dependence on the propagation distance

*z*is illustrated in Fig. 6(a). Its slope along

*z*, becomes zero at the position

*z*=

*z*

_{inv}where Thus, if

*z*

_{inv}> 0, then the apparent group velocity

*v*

_{peak}(

*z*) = [d

*t*

_{peak}(

*z*)/d

*z*]

^{−1}of the FCA-attenuated sech pulse seems to be ±∞ at

*z*=

*z*

_{inv}± 0, negative for

*z*<

*z*

_{inv}and positive only for

*z*>

*z*

_{inv}(while, nevertheless, all energy or photons coming through unabsorbed up to this point have quite regularly been travelling at the group velocity

*v*). Thus,

*z*

_{inv}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

*z*

_{inv}shifts deeper into the waveguide with increasing peak intensity

*Î*

_{0}and pulse width

*t*

_{0}, but no further than

*z*

_{inv}≤ ln(1 +

*αvt*

_{0})/2

*α*. The seeming behavior of a group-velocity inversion requires a large enough input energy

*W*

_{0}(or peak intensity

*Î*

_{0}for a given

*t*

_{0}) for which

*z*

_{inv}≥ 0, For the waveguide parameters given in Section 2, these conditions correspond to

*W*

_{0}≥ 1.87 × 10

^{−9}Ws/

*μ*m

^{2}or, e.g., to

*Î*

_{0}≥ 5.95W/

*μ*m

^{2}for

*t*

_{0}= 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

*Î*

_{0}= 0 ×

*I*

_{inv}, 1 ×

*I*

_{inv}, and 2 ×

*I*

_{inv}. 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.

*v*for input energies

*W*

_{0}up to

*W*

_{inv}, while it becomes infinite and changes sign when

*W*

_{0}=

*W*

_{inv}, 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

*v*

_{peak}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]

*v*and the seeming superluminal propagation behavior only results from the attenuation of the rear part of the pulse by self-induced free carriers.

*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).

**32**, 2031–2033 (2007). [CrossRef] [PubMed]

*I*

_{0}(

*t*−

*z/v*),

*I*(

*z*,

*t*) and

*R*

_{0}(

*t*−

*z/v*) from Eqs. (46) to (47). The FCR frequency shift may be written Thus, the FCR-induced phase

*ϕ*

_{FCR}of the sech pulse, which is the dominant non-linear phase shift in Figs. 4(g) and 4(h), causes a maximum spectral blueshift which increases with input peak intensity and effective propagation length but never exceeds Δ

*ω*

_{FCR}≤

*σ*

_{i}/

*σ*

_{r}

*t*

_{0}. This maximum blueshift occurs at time i.e., it lags the intensity peak at time

*t*

_{peak}(

*z*) given by Eq. (52) by a delay

*t*

_{0}

*M*(

*z*)/2 steadily growing up to

*t*=

*t*

_{peak}(

*z*) becomes and stabilizes at its maximum value

*σ*

_{i}/(2

*σ*

_{r}

*t*

_{0}) 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

**91**, 201115 (2007). [CrossRef]

*20th Annual Meeting of the IEEE Lasers and Electro-Optics Society 2007*, Lake Buena Vista, Fl, paper ThY2, 2007. [CrossRef]

*20th Annual Meeting of the IEEE Lasers and Electro-Optics Society 2007*, Lake Buena Vista, Fl, paper ThY2, 2007. [CrossRef]

**91**, 201115 (2007). [CrossRef]

*20th Annual Meeting of the IEEE Lasers and Electro-Optics Society 2007*, Lake Buena Vista, Fl, paper ThY2, 2007. [CrossRef]

*f*

_{0}(

*ξ*) and

*F*

_{0}(

*ξ*) depend only on the shape but neither on the peak intensity

*Î*

_{0}nor the duration

*t*

_{0}of the pulse. If the relative slope (

*∂f*

_{0}/

*∂t*)/

*f*

_{0}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 Θ

_{FCA}. 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.

*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*< −

*t*

_{0}, 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.

*∂f*

_{0})/

*∂t*/

*f*

_{0}≃ −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).

**91**, 201115 (2007). [CrossRef]

**32**, 2031–2033 (2007). [CrossRef] [PubMed]

*z*≫ 1/(2

*α*), 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

*α*→ 0, the pulse distortion parameter

*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

*t*

_{0}or peak intensity

*Î*

_{0}it has, as long as Θ

_{FCA}(

*z*) reaches the same final value at the waveguide output. For Θ

_{FCA}(

*z*) = 0 the pulse shape remains unchanged.

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

*T*at nearly the same wavelength and group velocity. Its pulse duration shall again be much shorter than the free-carrier lifetime

*τ*

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

**91**, 201115 (2007). [CrossRef]

*20th Annual Meeting of the IEEE Lasers and Electro-Optics Society 2007*, Lake Buena Vista, Fl, paper ThY2, 2007. [CrossRef]

**93**, 091114 (2008). [CrossRef]

**100**, 101111 (2012). [CrossRef]

*T*≫ Δ

*t*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*) =

*e*

_{0}(

*t*−

*z/v*)exp{

*iψ*

_{FCR}(

*z*) − [

*αz*+

*a*

_{FCA}(

*z*)]/2}, where

*e*(

*z*= 0,

*t*) =

*e*

_{0}(

*t*) is the input field amplitude of the probe pulse at

*z*= 0 and 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 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

*α*

_{FCA}(

*z*) =

*ᾱ*

_{FCA}(

*z*) at all positions

*z*only if

*G*(

*T*) = 1/2 and

*Ī*

_{0}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

*Î*

_{0}= (2/

*π*)

^{1/4}(2

*τ*

_{exp}/

*t*

_{0})

^{1/2}

*Ī*

_{0}≃ 1.26 (

*τ*

_{exp}/

*t*

_{0})

^{1/2}

*Ī*

_{0}and

*T*= ln(2) ×

*τ*

_{exp}= 0.69

*τ*

_{exp}. For a discussion of sech-shaped pump pulses see Section 5.5.

*L*follows from Eqs. (61) and (62), For a sufficiently long waveguide

*L*≫ 1/2

*α*, it approaches the constant asymptotic value

*a*

_{FCA}≃

*G*(

*T*)ln(1 +

*ηQ*

_{0}). 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*).

*Î*

_{0}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., The aggregate FCA attenuation

*ā*

_{FCA}(

*L*) for the CW operation is given by Eq. (72) in Appendix B. Thus, the relation in Eq. (64) is fulfilled when It is easy to see that for

*G*(

*T*) = 1/2 this relation is true for

*G*(

*T*) = 1/4, this relation becomes length dependent,

*L*pulse-induced FCA for the probe pulse is weaker than CW-induced FCA as long as

*G*(

*T*) = 1/2 and

*G*(

*T*) = 1/4 corresponds to

*T*= ln(2) ×

*τ*

_{exp}and

*T*= ln(4) ×

*τ*

_{exp}, respectively. In both cases the pump pulse can have a peak intensity larger than the CW pump intensity in proportion to

*G*(

*T*≪

*τ*

_{eff}) ≃ 1. For short enough waveguides or low intensities, this condition reqiures

*Ī*

_{0}→ ∞. In particular, if the input peak intensity

*Î*

_{0}of the pump pulse must be smaller than the input CW intensity

*Ī*

_{0}to exert a lower accumulated FCA on the closely following probe pulse.

*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

*a*

_{FCA}(

*L*) will always be able to exceed

*ā*

_{FCA}(

*L*) for

*G*(

*T*) > 1/2 even for

*Î*

_{0}=

*Ī*

_{0}, 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*> 0.693

*τ*

_{exp}.

## 8. Conclusions

## 9. Appendix A

*N*=

*N*(

*z*,

*t*) =

*N*[

*v*(

*x̃*−

*ỹ*)/2, (

*x̃*+

*ỹ*)/2] is understood according to Eq. (11). Using [37] and assuming

*I*(0,

*t*) =

*Î*

_{0}

*f*

_{0}(

*t/t*

_{0}) at the waveguide input with

*f*

_{0}(

*t/t*

_{0}) ≤ 1 as in Eq. (28), we derive the solution accounting for the accumulated non-linear attenuation by TPA and FCA after propagation distance

*z*, where Importantly, 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.

*γ*= 0, then

*N*≡ 0,

*U*(

*z*,

*t*) ≡ 1, and

*γ*> 0 again, then

*N*> 0, and consequently

*U*(

*z*,

*t*) < 1 would ensure that

*β*

_{r}enters explicitly the complete system of the modelling equations (68)–(70), approximating it by unity is equivalent with setting

*β*

_{r}= 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.

_{TPA}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

*∂/∂t*= 0, and

*I*(

*z*,

*t*) =

*Ī*(

*z*),

*N*(

*z*,

*t*) =

*N̄*(

*z*) =

*γτ*

_{eff}

*Ī*

^{2}(

*z*) with the initial conditions

*Ī*

_{0}=

*Ī*(

*z*= 0) and

*τ*

_{eff}is defined by Eq. (3). For example, we have

*τ*

_{eff}=

*τ*

_{exp}for the exponential decay function in Eq. (4). We derive the CW intensity

*Ī*(

*z*) and the local FCA coefficient

*ᾱ*

_{FCA}(

*z*) =

*σ*

_{r}

*N̄*(

*z*) from Eq. (5) by separation of variables,

## Acknowledgments

## References and links

1. | S. Fathpour and B. Jalali (Eds.), |

2. | L. Pavesi and D. J. Lockwood (Eds.), |

3. | G. T. Reed and A. P. Knights, |

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. |

5. | H. K. Tsang and Y. Liu, “Nonlinear optical properties of silicon waveguides,” Semicond. Sci. Technol. |

6. | A. R. Motamedi, A. H. Nejadmalayeri, A. Khilo, F. X. Kärtner, and E. P. Ippen, “Ultrafast nonlinear optical studies of silicon nanowaveguides,” Opt. Express |

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. Express |

9. | P. Koonath, D. R. Solli, and B. Jalali, “Limiting nature of continuum generation in silicon,” Appl. Phys. Lett. |

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. |

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 |

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 |

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. |

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 |

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. |

19. | E. K. Tien, N. S. Yuksek, F. Qian, and O. Boyraz, “Effect of TPA and FCA interplay on pulse compression in silicon,” in |

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 |

21. | L. Yin and G. P. Agrawal, “Impact of two-photon absorption on self-phase modulation in silicon waveguides,” Opt. Lett. |

22. | Q. Lin, O. J. Painter, and G. P. Agrawal, “Nonlinear optical phenomena in silicon waveguides: modeling and applications,” Opt. Express |

23. | X. Chen, N. C. Panoiu, and R. M. Osgood, “Theory of Raman-mediated pulsed amplification in silicon-wire waveguides,” IEEE J. Quantum Electron. |

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 |

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 |

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 |

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 |

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 |

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. |

30. | I. D. Rukhlenko, M. Premaratne, and G. P. Agrawal “Nonlinear propagation in silicon-based plasmonic waveguides from the standpoint of applications,” Opt. Express |

31. | N. Suzuki, “FDTD analysis of two-photon absorption and free-carrier absorption in Si high-index-contrast waveguides,” J. Lightwave Technol. |

32. | I. D. Rukhlenko, M. Premaratne, and G. P. Agrawal, “Nonlinear silicon photonics: analytical tools,” IEEE J. Sel. Top. Quantum Electron. |

33. | S. Roy, S. K. Bhadra, and G. P. Agrawal, “Femtosecond pulse propagation in silicon waveguides: variational approach and its advantages,” Opt. Commun. |

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. |

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 |

36. | Y. Liu and H. K. Tsang, “Time dependent density of free carriers generated by two photon absorption in silicon waveguides,” Appl. Phys. Lett. |

37. | E. Kamke, |

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, |

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. |

41. | G. P. Agrawal, |

42. | R. W. Boyd and D. J. Gauthier, “Controlling the velocity of light pulses,” Science |

43. | N. Akhmediev and A. Ankiewicz, |

44. | O. Boyraz and B. Jalali, “Demonstration of a silicon Raman laser,” Opt. Express |

45. | B. Jalali, V. Raghunathan, D. Dimitropoulos, and Boyraz, “Raman-based silicon photonics,” IEEE J. Sel. Top. Quantum Electron. |

**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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