Theory of Pulsed Four-Wave-Mixing in One-dimensional Silicon Photonic Crystal Slab Waveguides
We present a comprehensive theoretical analysis and computational study of four-wave mixing (FWM) of optical pulses co-propagating in one-dimensional silicon photonic crystal waveguides (Si-PhCWGs). Our theoretical analysis describes a very general set-up of the interacting optical pulses, namely we consider nondegenerate FWM in a configuration in which at each frequency there exists a superposition of guiding modes. We incorporate in our theoretical model all relevant linear optical effects, including waveguide loss, free-carrier (FC) dispersion and FC absorption, nonlinear optical effects such as self- and cross-phase modulation (SPM, XPM), two-photon absorption (TPA), and cross-absorption modulation (XAM), as well as the coupled dynamics of FCs and optical field. In particular, our theoretical analysis based on the coupled-mode theory provides rigorously derived formulae for linear dispersion coefficients of the guiding modes, linear coupling coefficients between these modes, as well as the nonlinear waveguide coefficients describing SPM, XPM, TPA, XAM, and FWM. In addition, our theoretical analysis and numerical simulations reveal key differences between the characteristics of FWM in the slow- and fast-light regimes, which could potentially have important implications to the design of ultra-compact active photonic devices.
pacs:78.67.Pt, 78.20.Bh, 42.65.Wi, 42.70.Qs, 42.65.Ky
One of the most promising applications of photonics is the development of ultra-compact optical interconnects for chip-to-chip and even intra-chip communications. The driving forces behind research in this area are the perceived limitations at high frequency of currently used copper interconnects hmh01pieee (), combined with a rapidly increasing demand to move huge amounts of data within increasingly more confined yet increasingly intricate communication architectures. An approach showing great potential towards developing optical interconnects at chip scale is based on high-index contrast optical waveguides, such as silicon photonic waveguides (Si-PhWGs) implemented on the silicon-on-insulator material platform lll00apl (); apc02ptl (). Among key advantages provided by this platform are the increased potential for device integration facilitated by the enhanced confinement of the optical field achievable in high-index contrast photonic structures, as well as the particularly large optical nonlinearity of silicon, which makes it an ideal material for active photonic devices. Many of the basic device functionalities required in networks-on-chip have in fact already been demonstrated using Si-PhWGs, including parametric amplification cdr03oe (); edo04oe (); rjl05n (); fts06n (); lov10np (), optical modulation cir95ptl (); ljl04n (); xsp05n (), pulse compression cph06ptl (); m08ptl (), supercontinuum generation bkr04oe (); hcl07oe (), pulse self-steepening plo09ol (), modulational instability pco06ol (), and four-wave mixing (FWM) fys05oe (); edo05oe (); fts07oe (); zpm10np (); dog12oe (); for a review of optical properties of Si-PhWGs see opd09aop (). However, since the parameter space of Si-PhWGs is rather limited, there is little room to engineer their optical properties.
A promising solution to this problem has its roots in the advent of photonic crystals (PhCs) in the late 80’s y87 (); j87 (). Thus, by patterning an optical medium in a periodic manner, with the spatial periods of the pattern being comparable to the operating optical wavelength, the optical properties of the resulting medium can be modified and engineered to a remarkable extent. Following this approach, a series of photonic devices have been demonstrated using PhCs, including optical waveguides and bends mck96prl (); lch98s (); bfy99el (); mck99prb (); cn00prb (), optical micro-cavities fvf97nat (); pls99s (); aas03nat (); rsl04nat (); ysh04nat (), and optical filters fvj99prb (); cmi01apl (). One of the most effective approaches to affect the optical properties of PhCs is to modify the group-velocity (GV), , of the propagating modes. Unlike the case of waves propagating in regular optical media, whose GV can hardly be altered, by varying the geometrical parameters of PhCs one can tune the corresponding GV over many orders of magnitude. Perhaps the most noteworthy implication of the existence of optical modes with significantly reduced GV, the so-called slow-light sj04nm (); k08nphot (); b08nphot (), is that both linear and nonlinear optical effects can be dramatically enhanced in the slow-light regime nys01prl (); sjf02josab (); pbo03ol (); pbo04oe (); vbh05n (); myp06ol (); cmg09nphot (); rls11prb ().
One of the most important nonlinear optical process, as far as nonlinear optics applications are concerned, is FWM. In the generic case, it consists of the combination of two photons with frequencies, and , belonging to two pump continuous-waves (CWs) or pulses, followed by the generation of a pair of photons with frequencies and . The energy conservation requires that . In practice, however, an easier to implement FWM configuration is usually employed, namely degenerate FWM. In this case one uses just one pump with frequency, , the generated photons belonging to a signal () and an idler () beam; in this case the conservation of the optical energy is expressed as: . Among the most important applications of degenerate FWM it is noteworthy to mention optical amplification, wavelength generation and conversion, phase conjugation, generation of squeezed states, and supercontinuum generation. While FWM has been investigated theoretically and experimentally in PhC waveguides myk10oe (); ssv10oe (); meg10oe (); jli11oe (); csl12oe () and long-period Bragg waveguides dog12oe (); lzd14ol (), a comprehensive theory of FWM in silicon PhC waveguides (Si-PhCWGs), which rigorously incorporates in a unitary way all relevant linear and nonlinear optical effects as well as the influence of photogenerated free-carriers (FCs) on the pulse dynamics is not available yet.
In this article we introduce a rigorous theoretical model that describes FWM in Si-PhCWGs. Our model captures the influence on the FWM process of linear optical effects, including waveguide loss, FC dispersion (FCD) and FC absorption (FCA), nonlinear optical effects such as self- and cross-phase modulation (SPM, XPM), two-photon absorption (TPA), and cross-absorption modulation (XAM), as well as the mutual interaction between FCs and optical field. We also illustrate how our model can be applied to investigate the characteristics of FWM in the slow- and fast-light regimes, showing among other things that by incorporating the effects of FCs on the optical pulse dynamics new physics emerge. One noteworthy example in this context is that the well-known linear dependence of FCA on is replaced in the slow-light regime by a power-law dependence.
The remaining of the paper is organized as follows. In the next section we present the optical properties of the PhC waveguide considered in this work. Then, in Sec. III, we develop the theory of pulsed FWM in Si-PhCWGs whereas the particular case of degenerate FWM is analyzed in Sec. IV. Then, in Sec. V, we apply these theoretical tools to explore the physical conditions in which efficient FWM can be achieved. The results are subsequently used, in Sec. VI, to study via numerical simulations the main properties of pulsed FWM in Si-PhCWGs. We conclude our paper by summarizing in the last section the main findings of our article and discussing some of their implications to future developments in this research area. Finally, an averaged model that can be used in the case of broad optical pulses is presented in an Appendix.
Ii Description of the photonic crystal waveguide
In this section we present the geometrical and material properties of the PhC waveguide considered in this work, as well as the physical properties of its optical modes. Thus, our Si-PhCWG consists of a one-dimensional (1D) waveguide formed by introducing a line defect in a two-dimensional (2D) honeycomb-type periodic lattice of air holes in a homogeneous slab made of silicon (a so-called W1 PhC waveguide). The line defect is oriented along the -axis, which is chosen to coincide with one of the symmetry axes of the crystal, and is created by filling in a row of holes [see Fig. 1(a)]. The slab height is and the radius of the holes is , where is the lattice constant, whereas the index of refraction of silicon is .
The defect line breaks the discrete translational symmetry of the photonic system along the -axis, so that the optical modes of the waveguide are invariant only to discrete translation along the -axis j08book (). Moreover, based on experimental considerations, we restrict our analysis to in-plane wave propagation, namely the wave vector, , lies in the plane. The component, on the other hand, can be restricted to the first Brillouin zone, , which is an immediate consequence of the Bloch theorem. Under these circumstances, we determined numerically the photonic band structure of the system and the guiding optical modes of the waveguide using MPB, a freely available code based on the plane-wave expansion (PWE) method jj01oe (). To be more specific, we used a supercell with size of along the -, -, and -axis, respectively, the corresponding step size of the computational grid being , , and , respectively. Figure 1(b) summarizes the results of these calculations. Thus, the waveguide has two fundamental TE-like optical guiding modes located in the band-gap of the unperturbed PhC, one -even and the other one -odd.
In order to better understand the physical properties of the optical guiding modes, we plot in Fig. 2 the profile of the magnetic field , which is its only nonzero component in the symmetry plane. These field profiles, calculated for several values of , show that although the optical field is primarily confined at the location of the defect (waveguide), for some values of it is rather delocalized in the transverse direction. This field delocalization effect is particularly strong in the spectral domains where the modal dispersion curves are relatively flat, namely in the so-called slow-light regime, and increases when the group index of the mode, defined as , increases.
The dispersion effects upon pulse propagation in the waveguide are characterized by the waveguide dispersion coefficients, defined as . In particular, the first-order dispersion coefficient is related to the pulse GV via , whereas the second-order dispersion coefficient, , quantifies the GV dispersion (GVD) as well as pulse broadening effects. The wavelength dependence of the first four dispersion coefficients, determined for both guided modes, is presented in Fig. 3, the shaded areas indicating the spectral regions of slow-light. For the sake of clarity, we set the corresponding threshold to , that is the slow-light regime is defined by . As it can be seen in Fig. 3, the even mode possesses two slow-light regions, one located at the band-edge () and the other one at , i.e. , whereas the odd mode contains only one such spectral domain located at the band-edge (). Moreover, the even mode can have both positive and negative GVD, the zero-GVD point being at , whereas the odd mode has normal GVD () throughout. Since usually efficient FWM can only be achieved in the anomalous GVD regime (), we will assume that the interacting pulses propagate in the even mode unless otherwise is specified.
Iii Derivation of the Mathematical Model
This section is devoted to the derivation of a system of coupled-mode equations describing the co-propagation of a set of mutually interacting optical pulses in a Si-PhCWG, as well the influence of photogenerated FCs on the pulse evolution. We will derive these coupled-mode equations in the most general setting, namely the nondegenerate FWM, then show how they can be applied to a particular case most used in practice, the so-called degenerate FWM configuration. Our derivation follows the general approach used to develop a theoretical model for pulse propagation in silicon waveguides with uniform cross-section cpo06jqe () and Si-PhCWGs pmw10jstqe ().
iii.1 Optical modes of photonic crystal waveguides
In the presence of an external perturbation described by the polarization, , the electromagnetic field of guiding modes with frequency, , is described by the Maxwell equations, which in the frequency domain can be written in the following form:
where is the magnetic permeability, which in the case of silicon and other nonmagnetic materials can be set to , is the dielectric constant of the PhC, and and are the electric and magnetic fields, respectively. In our case, is the sum of polarizations describing the refraction index change induced by photogenerated FCs and nonlinear (Kerr) effects.
In order to understand how the modes of the PhC waveguide are affected by external perturbations, let us consider first the unperturbed system, that is . Thus, let us assume that, at the frequency , the unperturbed PhC waveguide has guiding modes. It follows then from the Bloch theorem that the fields of these modes can be written as:
where is the th mode propagation constant and () denotes forward (backward) propagating modes. Here, we consider that the harmonic time dependence of the fields was chosen as . The mode amplitudes and are periodic along the -axis, with period . Moreover, the forward and backward propagating modes obey the following symmetry relations:
where the symbol “” denotes complex conjugation. As such, one only has to determine either the forward or the backward propagating modes.
The guiding modes can be orthogonalized, the most commonly used normalization convention being
where is the power carried by the th mode. This mode power is related to the mode energy contained in one unit cell of the PhC waveguide, , via the relation:
are the electric and magnetic energy of the mode, respectively, and is the volume of the unit cell. Note that in Eq. (5) we used the fact that the mode contains equal amounts of electric and magnetic energy.
It should be stressed that the waveguide modes defined by Eqs. (2) are exact solutions of the Maxwell equations (1) with , and thus they should not be confused with the so-called local modes of the waveguide. The latter modes correspond to waveguides whose optical properties vary adiabatically with , on a scale comparable to the wavelength and have been used to describe, e.g., wave propagation in tapered waveguides s70tmtt () or pulse propagation in 1D long-period Bragg gratings sse02jmo ().
iii.2 Perturbations of the photonic crystal waveguide
Due to the photogeneration of FCs and nonlinear optical effects, the dielectric constant of Si-PhCWGs undergoes a certain local variation, , upon the propagation of optical pulses in the waveguide. The corresponding perturbation polarization, in Eq. (1b), can be divided in two components according to the physical effects they describe: the linear change of the dielectric constant via generation of FCs and the nonlinearly induced variation of the index of refraction.
Assuming an instantaneous response of the medium, the linear contribution to , , is written as:
where cpo06jqe ():
Here, is the intrinsic loss coefficient of the waveguide and is the characteristic function of the domain where FCs can be generated, namely in the domain occupied by Si and otherwise. Based on the Drude model, the FC-induced change of the index of refraction, , and FC losses, , are given by sb87jqe ():
Here, is the charge of the electron, () is the electron (hole) mobility, () is the conductivity effective mass of the electrons (holes), with the mass of the electron, and () is the induced variation of the electrons (holes) density (in what follows, we assume that ).
The nonlinear contribution to , , is described by a third-order nonlinear susceptibility, , and can be written as:
The real part of the susceptibility describes parametric optical processes such as SPM, XPM, and FWM, while the imaginary part of corresponds to TPA and XAM. Note that in this study we neglect the stimulated Raman scattering effect as it is assumed that the frequencies of the interacting pulses do not satisfy the condition required for an efficient, resonant Raman interaction.
Since silicon belongs to the crystallographic point group the susceptibility tensor has 21 nonzero elements, of which only 4 are independent, namely, , , , and b08book (). In addition, the frequency dispersion of the nonlinear susceptibility can be neglected as we consider optical pulses with duration of just a few picoseconds or larger. As a consequence, the Kleinman symmetry relations imply that . Moreover, experimental studies have shown that zlp07apl () within a broad frequency range. Therefore, the nonlinear optical effects considered here can be described by only one element of the tensor .
Because of fabrication considerations, in many instances the waveguide is not aligned with any of the crystal principal axes and as such these axes are different from the coordinate axes in which the optical modes are calculated. Therefore, one has to transform the tensor from the crystal principal axes into the coordinate system in which the optical modes are calculated cpo06jqe (),
where is the nonlinear susceptibility in the crystal principal axes and is the rotation matrix that transforms one coordinate system into the other. In our case, is the matrix describing a rotation with around the -axis (see Fig. 1).
iii.3 Coupled-mode equations for the optical field
In order to derive the system of coupled-mode equations describing pulsed FWM in Si-PhCWGs we employ the conjugated form of the Lorentz reciprocity theorem cpo06jqe (); mpw03pre (); ks07jqe (); s83book (). To this end, let us consider two solutions of the Maxwell equations (1), and , which correspond to two different spatial distribution of the dielectric constant, and , respectively. If we insert the vector , defined as , in the integral identity:
where is the transverse section at position, , and is the boundary of , and use the Maxwell equations, we arrive at the following relation:
Let us consider now a nondegenerate FWM process in which two pulses at carrier frequencies and interact and generate two optical pulses at carrier frequencies and , with the energy conservation expressed as . Then, in the Lorentz reciprocity theorem given by Eq. (III.3) we choose as the first set of fields a mode of the unperturbed waveguide (), which corresponds to the frequency , where is one of the carrier frequencies , , , or :
where and is an integer, , , with being the number of guiding modes at the frequency . In Eqs. (14), and in what follows, a bar over a symbol means that the corresponding quantity is evaluated at one of the carrier frequencies.
As the second set of fields we take those that propagate in the perturbed waveguide, at the frequency . These fields are written as a series expansion of the guiding modes at frequencies , , thus neglecting the frequency dispersion of the guiding modes and the radiative modes that might exist at the frequency . This approximation is valid as long as all interacting optical pulses have narrow spectra centered at the corresponding carrier frequencies, that is the physical situation considered in this work. In particular, this modal expansion becomes less accurate when any of the pulses propagates in the slow-light regime, as generally the smaller the GV of a mode is the larger its frequency dispersion is. Thus, the second set of fields are expanded as:
With the fields normalization used in Eqs. (15), the mode amplitudes , , are measured in units of . Note that since the optical pulses are assumed to be spectrally narrow, the mode amplitudes have negligible values except when the frequency lies in a narrow spectral domain centered at the carrier frequency, .
The dielectric constant in the two cases is and , where is the dielectric constant of the unperturbed PhC. If the material dispersion is neglected, . Inserting the fields given by Eqs. (14) and Eqs. (15) in Eq. (III.3), and neglecting the line integral in Eq. (III.3), which cancels for exponentially decaying guiding modes, one obtains the following set of coupled equations:
In Eq. (III.3) and what follows a prime symbol to a sum means that the summation is taken over all modes, except that with , , and . Moreover, in deriving the l.h.s. of Eq. (III.3) we used the orthogonality relation given by Eq. (4).
The time-dependent fields are obtained by integrating over all frequency components contained in the spectra of the system of interacting optical pulses:
where , and , are the positive and negative frequency parts of the spectrum, respectively.
Let us now introduce the envelopes of the interacting pulses in the time domain, , defined as the integral of the mode amplitudes taken over the part of the spectrum that contains only positive frequencies,
With this definition, the time-dependent fields given in Eqs. (18) become:
Following the same approach, the time-dependent polarization, too, can be decomposed in two components, which contain positive and negative frequencies, that is, it can be written as:
The next step of our derivation is to Fourier transform Eq. (III.3) in the time domain. To this end, we first expand the coefficients and in Taylor series, around the carrier frequency [note that according to Eq. (17), is frequency independent]:
It can be easily seen from this equation that the average of over one lattice cell of the PhC waveguide is equal to 1, i.e.,
Here and in what follows the tilde symbol indicates that the corresponding physical quantity has been averaged over a lattice cell of the waveguide. With this notation, Eqs. (23) become:
These relations show that is the th order dispersion coefficient of the waveguide mode characterized by the parameters , evaluated at .
We now multiply Eq. (III.3) by and integrate over the positive-frequency domain. These simple calculations lead to the time-domain coupled-mode equations for the field envelopes, :
The temporal width of the pulses considered in this analysis is much smaller as compared to the nonlinear electronic response time of silicon and therefore the latter can be approximated to be instantaneous. In addition, we assume that the spectra of the interacting pulses are narrow and do not overlap. Under these circumstances, the optical pulses can be viewed as quasi-monochromatic waves and their nonlinear interactions can be treated in the adiabatic limit. Separating the nonlinear optical effects contributing to the nonlinear polarization, one can express in the time domain this polarization as bc91book ():
This expression for the nonlinear polarization accounts for the fact that the nonlinear susceptibility is invariant to frequency permutations. The first term in Eq. (III.3) represents SPM effects of the pulse envelopes, the second and third terms describe the XPM between modes with the same frequency and XPM between pulses propagating at different frequencies, respectively, whereas the last term describes FWM processes.