08-16-2017, 10:23 PM
Abstract
Since the recent demonstration of chip-scale, siliconbased,
photonic devices, silicon photonics provides a viable and
promising platform for modern nonlinear optics. The development
and improvement of such devices are helped considerably by theoretical
predictions based on the solution of the underlying nonlinear
propagation equations. In this paper, we review the approximate
analytical tools that have been developed for analyzing active and
passive silicon waveguides. These analytical tools provide the much
needed physical insight that is often lost during numerical simulations.
Our starting point is the coupled-amplitude equations that
govern the nonlinear dynamics of two optical waves interacting inside
a silicon-on-insulator waveguide. In their most general form,
these equations take into account not only linear losses, dispersion,
and the free-carrier and Raman effects, but also allow for the
tapering of the waveguide. Employing approximations based on
physical insights, we simplify the equations in a number of situations
of practical interest and outline techniques that can be used
to examine the influence of intricate nonlinear phenomena as light
propagates through a silicon waveguide. In particular, propagation
of single pulse through a waveguide of constant cross section
is described with a perturbation approach. The process of Raman
amplification is analyzed using both purely analytical and
semianalytical methods. The former avoids the undepleted-pump
approximation and provides approximate expressions that can be
used to discuss intensity noise transfer from the pump to the signal
in silicon Raman amplifiers. The latter utilizes a variational
formalism that leads to a system of nonlinear equations that governs
the evolution of signal parameters under the continuous-wave
pumping. It can also be used to find an optimum tapering profile
of a silicon Raman amplifier that provides the highest net gain for
a given pump power.
Index Terms Free-carrier absorption (FCA), integrated optics,
Kerr effect, nonlinear optics, optical pulse propagation, Raman
effect, silicon photonics, silicon Raman amplifiers, silicon waveguides,
two-photon absorption (TPA), waveguide tapering.
I. INTRODUCTION
THE problem of light propagation through silicon waveguides
with characteristic lateral dimensions of the order
of 1 m has been extensively studied in recent years, both experimentally
[1] [11] and theoretically [12] [22], because of
its immense practical applications [23] [27]. The main reason
why silicon is considered a promising photonics material stems
from its relatively strong nonlinear interaction with external
electromagnetic fields whose wavelengths lie in the transparent
infrared region beyond 1.1 m. Since this region includes the
telecommunication window near 1.55 m, a multitude of nonlinear
optical effects inside silicon waveguides can be used for
diverse beneficial applications. Moreover, these nonlinear interactions
can be further enhanced by employing silicon-oninsulator
(SOI) waveguides in which a tight-mode confinement
provides large optical intensities even at moderate input power
levels. Therefore, it is not surprising that, to date, almost all
physical properties of silicon have found applications in different
nonlinear SOI-based photonic devices [26], [28] [30]. For
example, stimulated Raman scattering (SRS), which is particularly
strong in silicon [31] [33], is employed to make optical amplifiers
[34] [42],modulators [43], and Raman lasers [44] [52].
The Kerr effect is successfully applied for optical phase modulation
[53], [54], soliton formation [6], and supercontinuum
generation [29], [55] [57]. The phenomenon of four-wave mixing
by itself, or in combination with SRS, has been used to
make broadband frequency converters [58] [66]. Although twophoton
absorption (TPA) by itself is undesirable, it has been
demonstrated that TPA-induced free-carrier generation and thermooptic
effects are suitable for all-optical switching [67] [71],
modulation [72], and pulse compression [73], [74]; they can
also be used for autocorrelation measurements [75], [76]. The
natural compatibility of SOI technology with the existing silicon
manufacturing process opens up wide possibilities for utilizing
these and other useful functionalities in fabricating photonic
integrated circuits.
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