## Pulse compression in adiabatically tapered silicon photonic wires |

Optics Express, Vol. 22, Issue 6, pp. 6296-6312 (2014)

http://dx.doi.org/10.1364/OE.22.006296

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### Abstract

We present a comprehensive analysis of pulse compression in adiabatically tapered silicon photonic wire waveguides (Si-PhWWGs), both at telecom (*λ* ∼ 1.55 μm) and mid-IR (*λ* ≳ 2.1 μm) wavelengths. Our theoretical and computational study is based on a rigorous model that describes the coupled dynamics of the optical field and photogenerated free carriers, as well as the influence of the physical and geometrical parameters of the Si-PhWWGs on these dynamics. We consider both the soliton and non-soliton pulse propagation regimes, rendering the conclusions of this study relevant to a broad range of experimental settings and practical applications. In particular, we show that by engineering the linear and nonlinear optical properties of Si-PhWWGs through adiabatically varying their width, one can achieve more than 10× pulse compression in millimeter-long waveguides. The inter-dependence between the pulse characteristics and compression efficiency is also discussed.

© 2014 Optical Society of America

## 1. Introduction

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*n*

_{Si}≈ 3.4) and that of the cladding (usually air,

*n*

_{air}= 1) and the subwavelength nature of the cross-sectional area of the waveguide, variations in the transverse size of the waveguide as small as only a few percent of the operating wavelength can induce changes of the mode propagation constant,

*β*, which are large enough to significantly affect the temporal and spectral properties of pulses that propagate in such photonic wires. In particular, by simply varying the waveguide width, one can readily design Si-PhWWGs whose frequency dispersion changes between normal dispersion, where the second-order dispersion coefficient,

*β*

_{2}(

*ω*) =

*β″*(

*ω*) > 0, and anomalous dispersion (

*β*

_{2}< 0), or waveguides that possess multiple zero-dispersion points, defined by

*β*

_{2}(

*ω*) = 0. Equally important, the large intrinsic third-order nonlinearity of Si, in conjunction with the strong field confinement achievable in high-index contrast waveguides, make it possible to attain strong nonlinear pulse reshaping at low optical power. More specifically, the dispersion length,

*L*, and the nonlinear length,

_{D}*L*

_{nl}, of Si-PhWWGs can be more than four orders of magnitude smaller than in silica fibers [27

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

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43. J. B. Driscoll, N. Ophir, R. R. Grote, J. I. Dadap, N. C. Panoiu, K. Bergman, and R. M. Osgood, “Width-modulation of Si photonic wires for quasi-phase-matching of four-wave-mixing: experimental and theoretical demonstration,” Opt. Express **20**, 9227–9242 (2012). [CrossRef] [PubMed]

44. S. Lavdas, J. B. Driscoll, H. Jiang, R. R. Grote, R. M. Osgood, and N. C. Panoiu, “Generation of parabolic similaritons in tapered silicon photonic wires: comparison of pulse dynamics at telecom and mid-infrared wavelengths,” Opt. Lett. **38**, 3953–3956 (2013). [CrossRef] [PubMed]

*β*

_{2}changes sign during pulse propagation. While the first approach can be employed only at relatively large peak pulse power, namely in the soliton propagation regime, the latter one can be used to compress pulses whose power is below the soliton formation threshold as well. Our theoretical model rigorously describes the effects of the adiabatic variation of the cross-section of Si-PhWWGs on the pulse dynamics by fully accounting for the influence of this variation on the linear and nonlinear optical coefficients of the waveguide. For the sake of completeness, we consider the pulse dynamics at the optical communications wavelength,

*λ*∼ 1.55 μm, and at mid-infrared wavelengths,

*λ*≳ 2.1 μm.

## 2. Theoretical models for pulse propagation in subwavelength tapered Si waveguides

### 2.1. Theoretical model for propagation of ultrashort optical pulses

38. N. C. Panoiu, X. Liu, and R. M. Osgood, “Self-steepening of ultrashort pulses in silicon photonic nanowires,” Opt. Lett. **34**, 947–949 (2009). [CrossRef] [PubMed]

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46. N. C. Panoiu, J. F. McMillan, and C. W. Wong, “Theoretical analysis of pulse dynamics in silicon photonic crystal wire waveguides,” IEEE J. Sel. Top. Quantum Electron. **16**, 257–266 (2010). [CrossRef]

*u*(

*z*,

*t*) is the pulse envelope, measured in

*z*and

*t*are the distance along the Si-PhWWGs and time, respectively,

*β*(

_{n}*z*) =

*d*is the

^{n}β/dω^{n}*n*th order dispersion coefficient,

*κ*(

*z*) measures the overlap between the optical mode and the (Si) active area of the waveguide,

*v*(

_{g}*z*) is the group-velocity,

*δn*(

*z*) and

*α*(

*z*) are the FC-induced refractive index change and losses, respectively, and are given by

*N*is the FC density,

*m*

_{0}the electron mass, and

*μ*= 850 cm

_{e}^{2}/(V s) [

*μ*= 210 cm

_{h}^{2}/(V s)] the electron (hole) mobility. The nonlinear properties of the waveguide are described by the nonlinear coefficient,

*i.e.*the characteristic response time of the nonlinearity,

*τ*(

_{s}*z*) =

*∂*ln

*γ*(

*z*)/

*∂ω*, where

*A*(

*z*) and Γ(

*z*) are the cross-sectional area and the effective third-order susceptibility of the waveguide, respectively. Nonlinear optical effects higher than the third-order are not considered here; however, at high peak power they might become important. Note that in Eq. (1) linear losses are neglected because the loss length is significantly larger than the waveguide lengths considered in our study,

*L*< 10 cm. Specifically, the loss coefficient of Si waveguides is

*α*< 0.4 dB/cm [47

_{i}47. M. P. Nezhad, O. Bondarenko, M. Khajavikhan, A. Simic, and Y. Fainman, “Etch-free low loss silicon waveguides using hydrogen silsesquioxane oxidation masks,” Opt. Express **19**, 18827–18832 (2011). [CrossRef] [PubMed]

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*t*is the FC relaxation time. In our analysis we considered

_{c}*t*= 1 ns [29

_{c}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. Vlassov, “Engineering nonlinearities in nanoscale optical systems: physics and applications in dispersion-engineered silicon nanophotonic wires,” Adv. Opt. Photon. **1**, 162–235 (2009). [CrossRef]

### 2.2. Semi-analytical model based on the method of moments

## 3. Dispersion maps of linear and nonlinear parameters of tapered silicon waveguides

*h*= 250 nm, whereas its width,

*w*, varies adiabatically along the waveguide. More exactly, this means that the condition

*λ*and

*n*

_{eff}are the wavelength and effective refractive index of the mode, respectively. The Si core is assumed to be buried in silica.

*w*, the higher-order dispersion coefficients,

*β*, being determined by simply taking the corresponding derivatives wrt

_{n}*λ*. A similar procedure was used to determine the frequency dispersion of the other optical coefficients of the tapered Si-PhWWG, namely

*γ*(

*λ*),

*κ*(

*λ*), and

*τ*(

_{s}*λ*). Polynomial interpolation was used to calculate the values of these coefficients when the waveguide width was different from our 51 chosen values (see also [44

44. S. Lavdas, J. B. Driscoll, H. Jiang, R. R. Grote, R. M. Osgood, and N. C. Panoiu, “Generation of parabolic similaritons in tapered silicon photonic wires: comparison of pulse dynamics at telecom and mid-infrared wavelengths,” Opt. Lett. **38**, 3953–3956 (2013). [CrossRef] [PubMed]

*β*

_{2}(

*λ*) = 0, depicted in Fig. 1(b) as a black contour. Note that in the soliton regime the waveguide nonlinearity is relatively large, as per Fig. 2(a), because this regime is achieved for small

*w*, when the optical field is strongly confined in the Si core. This means that these Si photonic wires readily provide the main ingredients needed for soliton pulse compression in dispersion-varying optical guiding devices. Moreover, since we investigate the compression of pulses down to just a few hundreds of femtoseconds, the influence of third-order dispersion (TOD) and frequency dispersion of the nonlinearity must be accounted for. The dispersion map of the TOD coefficient,

*β*

_{3}(

*λ*), is shown in Fig. 1(c), whereas the dispersive properties of the nonlinearity dispersion are illustrated in Figs. 2(c) and 2(d).

## 4. Soliton pulse compression in dispersion-varying tapered silicon waveguides

*λ*= 1.55 μm in a tapered Si-PhWWG. The longitudinal

*z*-variation of the width profile of the waveguide is assumed to be

*w*(

*z*) =

*w*

_{in}+ (

*w*

_{out}−

*w*

_{in}) tanh(

*az*)/tanh(

*aL*), where

*w*

_{in}= 700 nm and

*w*

_{out}= 660 nm, are the initial and final width of the waveguide, respectively,

*L*= 9 cm is the waveguide length, and

*a*= 80 m

^{−1}. This choice of the taper parameters ensures that the adiabaticity criterion,

*λa*≪ 1, is clearly satisfied and that the pulse propagates throughout into the soliton regime,

*β*

_{2}< 0. The width of the input pulse is

*τ*= 180 fs and the input peak power,

*P*

_{0}= 1.4 W, which means that the initial value of the soliton number,

*N*, defined as

_{s}*N*= 10.

_{s}*z*< 20 mm) a significant pulse compression is observed beyond

*z*≃ 20 mm. It can be seen, however, that the pulse shape does not change periodically with

*z*, which is due to the influence of effects other than the GVD and SPM. Specifically, TPA induces strong pulse decay whereas the TOD leads to pulse breakup. The effect of TPA on the pulse dynamics is particularly strong in the initial propagation stage during which, as Fig. 3(c) suggests, most of the pulse energy is absorbed by photogenerated FCs. Nevertheless, a maximum of more than fivefold pulse compression is achieved at

*z*≃ 60 mm, which corresponds to a pulse duration of ∼ 36 fs. A common feature of pulses compressed via this method, which can also be seen in our simulations for

*z*≳ 60 mm, is the generation of pedestals at the edges of the pulse.

*λ*= 2.1 μm. The tapered wire considered in this case is defined by

*w*

_{in}= 850 nm,

*w*

_{out}= 750 nm,

*L*= 3 cm, and

*a*= 80 m

^{−1}, the pulse parameters being

*τ*= 180 fs and

*P*

_{0}= 2.07 W (

*N*= 9). Unlike the telecom case, the pulse decay is negligible at mid-IR wavelengths, its peak amplitude in fact increasing considerably. This substantial increase of the peak power, illustrated in Fig. 3(d), is explained by the fact that although part of the pulse energy is absorbed via TPA, the pulse undergoes significant compression as well, i.e., by more than 10× at

_{s}*z*= 25 mm, which should obviously result in increased peak power.

*λ*= 1.55 μm the FC density decreases monotonously with

*z*, at

*λ*= 2.1 μm there is a certain propagation distance,

*z*≃ 20 mm, at which a maximum amount of FCs is generated. This distance is roughly equal to the distance at which maximum pulse compression is observed. To understand this difference in pulse dynamics, note that at a given

*z*the peak FC density can be estimated from Eq. (2) to be

*A*, and

*v*vary only slightly with

_{g}*z*, the main contribution to Δ

*N*(

*z*) comes from the

*z*-variation of the factor

*P*

^{2}(

*z*)

*τ*(

*z*) ∼

*P*(

*z*)

*E*(

*z*). At mid-IR the pulse energy loss is rather small and therefore the maximum amount of FCs is generated when the peak power reaches its maximum, that is, when the pulse compression is maximum as well.

*z*≃ 30 mm at

*λ*= 1.55 μm and

*z*≃ 10 mm at

*λ*= 2.1 μm) as beyond this distance the optical pulse begins to split up. Our simulations (not shown here) suggest that, as expected, the distance over which the predictions of the full and semi-analytical models agree decreases with increasing

*P*

_{0}, since by increasing the nonlinearity one induces stronger pulse distortion. At both wavelengths, the pulse profile follows a similar evolution: initially the pulse broadens and then its width gradually decreases. To quantify the pulse compression after it breaks up, we determined its width by calculating the FWHM of the main pulse, as per the green curves in Fig. 4. It can be seen that, contrary to the predictions of the semi-analytical model, the main pulse is compressed significantly, which proves the effectiveness of tapered Si-PhWWGs in optical pulse reshaping applications.

*z*-dependence of the chirp,

*C*(

*z*), center frequency shift, Ω(

*z*), and temporal shift,

*T*(

*z*), calculated as function of the input pulse peak power

*P*

_{0}and width

*τ*. The corresponding results are presented in Figs. 5 and 6, respectively. Note that these numerical simulations were performed for a range of pulse parameters and propagation distance where the semi-analytical model agrees well with the rigorous one, so that these parameters have a meaningful physical interpretation. Figures 5 and 6 show that, except for relatively small peak power, the chirp is positive and increases monotonously with

*z*, that is the positive chirp generated by SPM is larger than the negative one generated by the anomalous GVD. As a result, the initial propagation stage in which the pulse broadens is followed by a monotonous pulse compression. This is the expected behavior of a pulse that propagates in the anomalous GVD regime, the

*z*-dependence of the pulse chirp explaining why the pulse compression is preceded by an initial pulse broadening. Thus, initially the pulse broadens under the influence of the GVD, but as the positive chirp induced by the SPM increases, the GVD begins to have an opposite effect, namely it compresses the pulse. As the pulse is compressed, the GVD and the peak power increase. Therefore, the SPM and, consequently, the rate at which the chirp increases become larger, too, which further increases the efficiency of the pulse compression process. A similar monotonous increase with the propagation distance, albeit extremely small (

*T*≃ 1 fs), is shown by the pulse temporal shift. This effect, whose magnitude is comparable to

*τ*, is due to the frequency dispersion of the nonlinearity and the TOD. Finally, the center frequency of the pulse is redshifted, the magnitude of this redshift increasing with

_{s}*z*. As expected, the magnitude of the frequency shift increases with the peak power,

*P*

_{0}(the strength of the nonlinearity dispersion increases with

*P*

_{0}) and decreases with the pulse width,

*τ*(TOD increases as

*τ*decreases).

*τ*and

*P*

_{0}. The results of this investigation are summarized in Fig. 7. Thus, comparing the results presented in the top and bottom panels in Fig. 7, one can see that the pulse compression is much more efficient at mid-IR, namely a shorter propagation distance is needed to achieve a certain compression factor. This can be easily explained using the characteristics of the pulse evolution we just discussed: the TPA is much weaker at mid-IR, which means that the positive SPM-induced chirp and hence the pulse compression is much larger in this frequency domain. Moreover, the maps plotted in Fig. 7 show that the compression factor increases with

*P*

_{0}but decreases with

*τ*. This finding has a simple explanation, namely whereas the SPM is proportional to

*P*

_{0}, the strength of GVD effects is inverse proportional to

*τ*

^{2}. Note, however, that if only waveguide tapering effects are considered then both SPM and GVD increase with the propagation distance, as for the two tapers considered in this section

*γ*and |

*β*

_{2}| increase with

*z*.

*β*

_{2}| decreases. Indeed, under these circumstances, for the soliton number

*τ*, decreases, the GVD coefficient |

*β*

_{2}| must decrease as well. In tapered Si-PhWWGs, however, the increase of the GVD coefficient can be offset by the increase of

*γ*(

*γ*increases as the waveguide width decreases, as per Fig. 2), so that soliton pulse compression can be achieved even in dispersion-increasing tapered waveguides.

*P*

_{0}= 1.4 W (

*P*

_{0}= 2.07 W) in a tapered Si-PhWWGs with

*w*

_{in}= 700 nm (

*w*

_{in}= 850 nm),

*w*

_{out}= 660 nm (

*w*

_{out}= 750 nm), and

*L*= 90 mm (

*L*= 30 mm), the wavelength being

*λ*= 1.55 μm (

*λ*= 2.1 μm). In both cases

*τ*= 180 fs and

*a*= 80 m

^{−1}. The

*z*-dependence of the pulse width, as well as that of

*L*and

_{D}*L*

_{nl}, are depicted in Fig. 8 with dotted lines. The solid lines in this figure correspond to the pulse propagation in the inverted taper, namely to the case in which the roles of

*w*

_{in}and

*w*

_{out}are interchanged while the pulse parameters are kept unchanged. As a result, in the first case the GVD dispersion increases (in absolute value), whereas the latter one corresponds to decreasing dispersion. Nevertheless, in both cases, the soliton is significantly compressed, as per Figs. 8(a) and 8(c). A closer look at the interplay between the two main characteristic lengths defining the soliton dynamics leads us to the same conclusion. More specifically, Figs. 8(b) and 8(d) show that for both tapers

*L*>

_{D}*L*

_{nl}, which means that the nonlinear effects are stronger. A consequence of this fact is that, as discussed above, the total chirp is positive. As the pulse propagates in the anomalous GVD regime, this results in pulse compression.

*z*-profile of the linear taper is defined by

*w*(

*z*) =

*w*

_{in}− (

*w*

_{in}−

*w*

_{out})

*z/L*. The input and output widths as well as the waveguide length were the same for both types of tapers: at

*λ*= 1.55 μm,

*w*

_{in}= 700 nm,

*w*

_{out}= 660 nm, and

*L*= 90 mm, whereas at

*λ*= 2.1 μm,

*w*

_{in}= 850 nm,

*w*

_{out}= 750 nm, and

*L*= 30 mm. The pulse peak power was

*P*

_{0}= 1.4 W (

*P*

_{0}= 2.07 W) at

*λ*= 1.55 μm (

*λ*= 2.1 μm), whereas in both cases

*τ*= 180 fs and

*a*= 80 m

^{−1}. The results of our analysis, summarized in Fig. 9, suggest that whereas the pulse width follows a similar evolution as it propagates in the two tapers, specific differences lead us to conclude that the hyperbolic taper is more efficient for pulse compression. Interestingly enough, similar conclusions were reached when Gaussian pulses with the same power and width were used. This finding can be explained by the fact that in the case of the hyperbolic taper there is a more rapid transition to the waveguide region with large nonlinearity, as compared to the case of the linear taper, which means that the generated positive chirp that induces pulse compression is larger in the former case.

## 5. Pulse compression below the soliton power threshold

*β*

_{2}< 0 the chirp induced by the GVD is negative and hence reduces the positive chirp generated by the SPM. One possible solution to this problem, which can be easily implemented by using adiabatically tapered Si-PhWWGs, is to first propagate the pulse in a section of the waveguide with large normal GVD until the pulse accumulates a large positive chirp and then propagate this chirped pulse in a waveguide section with large anomalous GVD, where the pulse is compressed. In a practical setting, this scheme can be implemented by simply using a tapered Si-PhWWG whose width is varied in such a way that the waveguide has normal and anomalous GVD within its input and output sections, respectively [see also Fig. 1(b)]. It should be mentioned that this pulse compression method can be applied to pulses with low peak power as well, as it does not require that the pulse propagates in the soliton regime. The larger the power, however, the larger the compression factor will be because a larger positive chirp would be generated within the waveguide section with normal GVD.

*γ′*= 0,

*β*

_{3}= 0, and

*τ*= 0, Eqs. (8b) and (8c) become (see also [49]): where the pulse energy,

_{s}*E*, is conserved upon propagation. This system of equations shows that when the pulse propagates in the waveguide section with normal GVD,

*β*

_{2}> 0, both terms in the rhs of Eq. (6b) contribute to the increase of the chirp. When subsequently the pulse propagates in the waveguide section with anomalous GVD,

*β*

_{2}< 0, the rhs of Eq. (6a) is negative, and therefore the optical pulse is compressed. This system of equations also suggests that the larger is the chirp generated in the first section of the waveguide the larger will be the overall pulse compression factor.

*w*

_{in}= 850 nm and

*w*

_{out}= 610 nm, at

*λ*=1.55 μm, and

*w*

_{in}=980 nm and

*w*

_{out}= 735 nm, at

*λ*= 2.1 μm. In both cases

*a*= 80 m

^{−1},

*L*= 90 mm, and

*τ*= 180 fs, whereas the pulse power was

*P*

_{0}= 90 mW at

*λ*= 1.55 μm and

*P*

_{0}= 100 mW at

*λ*= 2.1 μm. This choice of waveguide and pulse parameters means that

*N*(

_{s}*z*= 0) = 0.895 < 1 at

*λ*= 1.55 μm and

*N*(

_{s}*z*= 0) = 0.908 < 1 at

*λ*= 2.1 μm. While solitons do not exist in the normal GVD regime, the soliton number allows us to compare the compression factor that can be achieved by the two methods, at specific peak powers.

*λ*= 1.55 μm, where the TPA is large, the pulse propagates in a lossy medium and therefore its width initially increases, before undergoing compression in the waveguide section with

*β*

_{2}< 0, namely beyond

*z*≃ 40 mm. By contrast, at

*λ*= 2.1 μm the TPA is weak, so that the width of the pulse remains approximately constant for

*z*≲ 30 mm and then begins to decrease. Since the peak power is relatively small in this case, the amount of generated FCs is small as well (compare the FC-densities in Figs. 3 and 10) and consequently the

*z*-dependence of FCs is similar to that of the pulse power.

*vs*. the propagation distance. Thus, it can be seen that the pulse broadens while it propagates in the section of the waveguide with normal GVD (the zero-GVD point is located at

*z*= 30 mm) and subsequently undergoes significant compression. In particular, the pulse width decreases by 29% at

*λ*= 1.55 μm and by 27% at

*λ*= 2.1 μm, after a total propagation distance of 90 mm. The input pulse width is

*τ*= 180 fs and the input power is

*P*

_{0}= 90 mW at

*λ*= 1.55 μm and

*P*

_{0}= 100 mW at

*λ*= 2.1 μm, which ensures that

*N*(

_{s}*z*= 0) < 1, namely the peak power of the input pulse is smaller than the soliton formation threshold. However, as the pulse approaches the zero-GVD point,

*N*increases considerably since

_{s}*β*

_{2}becomes very small (

*N*→ ∞ as |

_{s}*β*

_{2}| → 0). After the pulse passes through the zero-GVD point

*N*begins to decrease and becomes again less than 1 in the final propagation stage.

_{s}*L*= 120 mm, is summarized in Fig. 12. Note that we chose the input power in these simulations such that the soliton number remains smaller than 1 for the most part of the propagation. The main conclusion illustrated by these plots is that pulse compression can be achieved for a broad spectrum of pulse widths and powers. For all values of the pulse parameters one requires a certain propagation distance (waveguide length) in order to achieve pulse compression, as initially the pulse broadens. It can also be seen that the compression factor depends rather weakly on the pulse parameters and the operation wavelength. In particular, for a given propagation distance, the compression factor increases with the pulse power and decreases with the pulse width. This conclusion can be derived from the dependence on the pulse parameters of the SPM and GVD chirps, namely, as suggested by Eq. (6b), the SPM chirp increases with the pulse power whereas the GVD chirp increases when the pulse width decreases.

## 6. Conclusion

## Appendix: Mathematical formulation of the semi-analytical model

*z*-derivatives of the pulse parameters defined in Eqs. (3a)–(3e) and inserting Eq. (1), in which the first two terms in the rhs are neglected, into the resulting integrals, simple but lengthy mathematical manipulations lead to the following system of nonlinear ordinary differential equations that describes the evolution of the pulse parameters:

*u*(

*z*,

*t*). For secant-hyperbolic pulses given by Eq. (4a)

*δ*= 12/

*π*

^{2}, so that this system reads,

*δ*= 2 in the case of Gaussian pulses given by Eq. (4b), so that (7) becomes,

## Acknowledgments

## References and links

1. | E. B. Treacy, “Optical pulse compression with diffraction gratings,” IEEE J. Quantum Electron. |

2. | O. E. Martinez, J. P. Gordon, and R. L. Fork, “Negative group-velocity dispersion using refraction,” J. Opt. Soc. Am. A |

3. | A. M. Weiner, “Femtosecond pulse shaping using spatial light modulators,” Rev. Sci. Instrum. |

4. | G. Imeshev, A. Galvanauskas, D. Harter, M. A. Arbore, M. Proctor, and M. M. Fejer, “Engineerable femtosecond pulse shaping by second-harmonic generation with Fourier synthetic quasi-phase-matching gratings,” Opt. Lett. |

5. | N. J. Doran and D. Wood, “Nonlinear-optical loop mirror,” Opt. Lett. |

6. | K. Smith, N. J. Doran, and P. G. J. Wigley, “Pulse shaping, compression, and pedestal suppression employing a nonlinear-optical loop mirror,” Opt. Lett. |

7. | M. D. Pelusi, Y. Matsui, and A. Suzuki, “Pedestal suppression from compressed femtosecond pulses using a nonlinear fiber loop mirror,” IEEE J. Quantum Electron. |

8. | W. J. Tomlinson, R. H. Stolen, and C. V. Shank, “Compression of optical pulses chirped by self-phase modulation in fibers,” J. Opt. Soc. Am. B |

9. | M. Nisoli, S. De Silvestri, O. Svelto, R. Szipocs, K. Ferencz, C. Spielmann, S. Sartania, and F. Krausz, “Compression of high-energy laser pulses below 5 fs,” Opt. Lett. |

10. | J. T. Manassah, “Pulse compression of an induced-phase-modulated weak signal,” Opt. Lett. |

11. | G. P. Agrawal, P. L. Baldeck, and R. R. Alfano, “Optical wave breaking and pulse compression due to cross-phase modulation in optical fibers,” Opt. Lett. |

12. | L. F. Mollenauer, R. H. Stolen, J. P. Gordon, and W. J. Tomlinson, “Extreme picosecond pulse narrowing by means of soliton effect in single-mode optical fibers,” Opt. Lett. |

13. | K. A. Ahmed, K. C. Chan, and H. F. Liu, “Femtosecond pulse generation from semiconductor laser using the soliton effect compression technique,” IEEE J. Sel. Top. Quantum Electron. |

14. | M. D. Pelusi and H. F. Liu, “Higher order soliton pulse compression in dispersion-decreasing optical fibers,” IEEE J. Quantum Electron. |

15. | A. A. Amorim, M. V. Tognetti, P. Oliveira, J. L. Silva, L. M. Bernardo, F. X. Kartner, and H. M. Crespo, “Sub-two-cycle pulses by soliton self-compression in highly nonlinear photonic crystal fibers,” Opt. Lett. |

16. | A. C. Peacock, “Mid-IR soliton compression in silicon optical fibers and fiber tapers,” Opt. Lett. |

17. | S. V. Chernikov, E. M. Dianov, D. J. Richardson, and D. N. Payne, “Soliton pulse compression in dispersion-decreasing fiber,” Opt. Lett. |

18. | M. Nakazawa, E. Yoshida, H. Kubota, and Y. Kimura, “Generation of a 170 fs, 10 GHz transform-limited pulse train at 1.55 μm using a dispersion decreasing, erbium-doped active soliton compressor,” Electron. Lett. |

19. | J. Hu, B. S. Marks, C. R. Menyuk, J. Kim, T. F. Carruthers, B. M. Wright, T. F. Taunay, and E. J. Friebele, “Pulse compression using a tapered microstructure optical fiber,” Opt. Express |

20. | M. L. V. Tse, P. Horak, J. H. V. Price, F. Poletti, F. He, and D. J. Richardson, “Pulse compression at 1.06 μm in dispersion-decreasing holey fibers,” Opt. Lett. |

21. | A. C. Peacock, “Soliton propagation in tapered silicon core fibers,” Opt. Lett. |

22. | L. Tong, R. Gattass, J. Ashcom, S. He, J. Lou, M. Shen, I. Maxwell, and E. Mazur, “Subwavelength-diameter silica wires for low-loss optical wave guiding,” Nature |

23. | M. Foster, A. Gaeta, Q. Cao, and R. Trebino, “Soliton-effect compression of supercontinuum to few-cycle durations in photonic nanowires,” Opt. Express |

24. | A. Blanco-Redondo, C. Husko, D. Eades, Y. Zhang, J. Li, T. F. Krauss, and B. J. Eggleton, “Observation of soliton compression in silicon photonic crystals,” Nat. Commun. |

25. | K. K. Lee, D. R. Lim, H. C. Luan, A. Agarwal, J. Foresi, and L. C. Kimerling, “Effect of size and roughness on light transmission in a Si/SiO |

26. | R. U. Ahmad, F. Pizzuto, G. S. Camarda, R. L. Espinola, H. Rao, and R. M. Osgood, “Ultracompact corner-mirrors and T-branches in silicon-on-insulator,” IEEE Photon. Technol. Lett. |

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

28. | 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 |

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. Vlassov, “Engineering nonlinearities in nanoscale optical systems: physics and applications in dispersion-engineered silicon nanophotonic wires,” Adv. Opt. Photon. |

30. | X. Chen, N. C. Panoiu, I. W. Hsieh, J. I. Dadap, and R. M. Osgood, “Third-order dispersion and ultrafast-pulse propagation in silicon wire waveguides,” IEEE Photon. Technol. Lett. |

31. | M. Mohebbi, “Silicon photonic nanowire soliton-effect compressor at 1.5 μm,” IEEE Photon. Technol. Lett. |

32. | N. C. Panoiu, X. Chen, and R. M. Osgood, “Modulation instability in silicon photonic nanowires,” Opt. Lett. |

33. | M. A. Foster, A. C. Turner, J. E. Sharping, B. S. Schmidt, M. Lipson, and A. L. Gaeta, “Broad-band optical parametric gain on a silicon photonic chip,” Nature |

34. | X. Liu, R. M. Osgood, Y. A. Vlasov, and W. M. J. Green, “Mid-infrared optical parametric amplifier using silicon nanophotonic waveguides,” Nat. Photonics |

35. | O. Boyraz, P. Koonath, V. Raghunathan, and B. Jalali, “All optical switching and continuum generation in silicon waveguides,” Opt. Express |

36. | I. W. Hsieh, X. Chen, X. P. Liu, J. I. Dadap, N. C. Panoiu, C. Y. Chou, F. Xia, W. M. Green, Y. A. Vlasov, and R. M. Osgood, “Supercontinuum generation in silicon photonic wires,” Opt. Express |

37. | L. Yin, Q. Lin, and G. P. Agrawal, “Soliton fission and supercontinuum generation in silicon waveguides,” Opt. Lett. |

38. | N. C. Panoiu, X. Liu, and R. M. Osgood, “Self-steepening of ultrashort pulses in silicon photonic nanowires,” Opt. Lett. |

39. | H. Fukuda, K. Yamada, T. Shoji, M. Takahashi, T. Tsuchizawa, T. Watanabe, J. Takahashi, and S. Itabashi, “Four-wave mixing in silicon wire waveguides,” Opt. Express |

40. | R. Espinola, J. Dadap, R. M. Osgood Jr., S. McNab, and Y. Vlasov, “C-band wavelength conversion in silicon photonic wire waveguides,” Opt. Express |

41. | M. A. Foster, A. C. Turner, R. Salem, M. Lipson, and A. L. Gaeta, “Broad-band continuous-wave parametric wavelength conversion in silicon nanowaveguides,” Opt. Express |

42. | S. Zlatanovic, J. S. Park, S. Moro, J. M. C. Boggio, I. B. Divliansky, N. Alic, S. Mookherjea, and S. Radic, “Mid-infrared wavelength conversion in silicon waveguides using ultracompact telecom-band-derived pump source,” Nat. Photonics |

43. | J. B. Driscoll, N. Ophir, R. R. Grote, J. I. Dadap, N. C. Panoiu, K. Bergman, and R. M. Osgood, “Width-modulation of Si photonic wires for quasi-phase-matching of four-wave-mixing: experimental and theoretical demonstration,” Opt. Express |

44. | S. Lavdas, J. B. Driscoll, H. Jiang, R. R. Grote, R. M. Osgood, and N. C. Panoiu, “Generation of parabolic similaritons in tapered silicon photonic wires: comparison of pulse dynamics at telecom and mid-infrared wavelengths,” Opt. Lett. |

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

46. | N. C. Panoiu, J. F. McMillan, and C. W. Wong, “Theoretical analysis of pulse dynamics in silicon photonic crystal wire waveguides,” IEEE J. Sel. Top. Quantum Electron. |

47. | M. P. Nezhad, O. Bondarenko, M. Khajavikhan, A. Simic, and Y. Fainman, “Etch-free low loss silicon waveguides using hydrogen silsesquioxane oxidation masks,” Opt. Express |

48. | G. Li, J. Yao, Y. Luo, H. Thacker, A. Mekis, X. Zheng, I. Shubin, J.-H. Lee, K. Raj, J. E. Cunningham, and A. V. Krishnamoorthy, “Ultralow-loss, high-density SOI optical waveguide routing for macrochip interconnects,” Opt. Express |

49. | G. P. Agrawal, |

50. | J. Santhanam and G. P. Agrawal, “Raman-induced spectral shifts in optical fibers: general theory based on the moment method,” Opt. Commun. |

**OCIS Codes**

(130.4310) Integrated optics : Nonlinear

(230.4320) Optical devices : Nonlinear optical devices

(230.7380) Optical devices : Waveguides, channeled

(320.5520) Ultrafast optics : Pulse compression

(320.5540) Ultrafast optics : Pulse shaping

**ToC Category:**

Integrated Optics

**History**

Original Manuscript: December 30, 2013

Revised Manuscript: February 22, 2014

Manuscript Accepted: February 24, 2014

Published: March 10, 2014

**Citation**

Spyros Lavdas, Jeffrey B. Driscoll, Richard R. Grote, Richard M. Osgood, and Nicolae C. Panoiu, "Pulse compression in adiabatically tapered silicon photonic wires," Opt. Express **22**, 6296-6312 (2014)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-6-6296

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### References

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