We present two sets of equations to describe nonlinear pulse propagation in a birefringent fiber Bragg grating. The first set uses a coupled-mode formalism to describe light in or near the photonic band gap of the grating. The second set is a pair of coupled nonlinear Schroedinger equations. We use these equations to examine viable switching experiments in the presence of birefringence. We show how the birefringence can both aid and hinder device applications.
© Optical Society of America
It is well known that in a transparent medium with a periodic variation in its optical properties there are ranges of frequencies (stop-gaps, photonic band gaps) at which light cannot propagate . However, in the presence of a Kerr nonlinearity, the intensity of the wave can sufficiently modify the nonlinear index of refraction to shift the wave out of the gap and permit pulse formation and propagation . Solitary wave solutions have been called ‘gap solitons’ if their center frequency lies in the stop gap, and ‘Bragg grating solitons’ if their center frequency lies outside the stop gap, but near enough that the dispersion due to the periodicity is dominant in stabilizing the solitary wave.
Theoretical studies in the field typically employ the nonlinear coupled mode equations (NCME). Much work has been devoted to discovering and exploring solutions to these equations, and to limiting cases which can lead to simpler equations. Notably, if the frequency content of an incident pulse is outside the photonic bandgap, then it has been shown that solutions to the NCME reduce to those described by the nonlinear Schroedinger equation (NLSE) . The key difference between the Bragg grating soliton NLSE and the usual optical fiber NLSE is that in the former the dispersion is caused by the periodic medium. This induced dispersion can be several orders of magnitude larger than the material dispersion of a fiber.
Because of the manner in which the NCME scales into the NLSE, a great deal of the qualitative behavior of a gap soliton can be understood in the context of the NLSE. The NLSE supports soliton solutions, and those solitons exhibit a robust balance between the dispersion and the nonlinearity of the medium. Since the dispersion of a periodic medium can be very high, a Bragg grating soliton excited in an optical fiber requires large intensities. But interaction lengths are correspondingly reduced. For instance, a Bragg grating soliton length in a representative experiment is only about 2cm . A second effect of the grating’s dispersion relation is that a pulse’s group velocity can range between zero and the speed of light in the uniform medium, leading to a host of interesting effects .
To date, however, the effect of optical waveguide birefringence on gap and Bragg grating soliton properties has been largely overlooked. Existing work includes a 1993 paper by Samir et al. that investigated the continuous wave NCME in the presence of birefringence (NBCME) , and the proposal and observation of the operation of an all-optical AND gate by other workers [6,7].
In this paper we investigate some effects of birefringence for gap solitons and Bragg grating solitons. We do this by numerically integrating the relevant coupled-mode and Schroedinger equations in two viable experiment regimes. The first is the all-optical AND gate mentioned above; although observed in the laboratory, no NBCME simulations of this device have yet been reported. The second regime is a generalization to grating geometries of the switching device, proposed by Barad and Silberberg , based on an elliptically polarized soliton. Here we find that the intrinsic birefringence of the medium upsets the operation of the device.
We organize this paper as follows. In section 2 we sketch the theory of birefringence in a periodic, nonlinear medium, and present both a NBCME and a nonlinear, birefringent Schroedinger equation (NBSE). In section 3 we describe the all-optical AND gate, and present numerical simulations of the device. In section 4 we describe the elliptically polarized soliton and present numerical simulations.
In this section we present the NBCME and NBSE relevant to a nonlinear, birefringent medium. We assume that an effective 1D analysis will suffice, as is usually the case in single mode fibers, and we begin with Maxwell’s equations in a presence of a dielectric tensor that is a function of only one Cartesian component ε=ε(z); we choose the (x,y) coordinates such that the tensor is diagonal,
Neglecting magnetic effects by setting the permeability μ equal to that of free space, μ=μ0, we can then define an index of refraction associated with polarization along the x and y axes,
where ε0 is the permittivity of free space. We seek transverse fields that depend only on the coordinate z,
for which Maxwell’s equations reduce to
To include nonlinearity, we consider a Kerr nonlinear polarization,
with i, j, k, l = x, y. In the absence of birefringence a number of methods have been used to derive the relevant coupled mode and Schroedinger equations from (4,5). The method we use to generalize to the birefringent case closely follows the multiple scales analysis of de Sterke et al. . In this paper we quote our results and leave the details of the derivation to be presented in a forthcoming publication.
Although our full analysis is more general, in the particular results we quote below we restrict ourselves to a weak grating as well as a weak nonlinearity. This is certainly sufficient for most Bragg gratings in optical fibers. We assume the grating to be cosinusoidally varying, with a grating spacing d, and an index contrast δn so that
d The uniform background indices of refraction are n0x,y. The structure of the nonlinear susceptibility χijkl is assumed to be that of an isotropic medium, which limits the validity of our equations to a weakly birefringent medium, but has the advantage of casting our equations in a form familiar from the optical fiber literature. This restriction is not essential, and can be removed by using a nonlinear susceptibility derived for a birefringent medium.
As is usual in calculations of this sort, the equations derived represent the spatio-temporal evloution of slowly-varying envelope functions. Although in heuristic analyses these envelope functions are often taken to modulate plane waves at the carrier frequency, the spirit of our approach is to use the Bloch functions of the grating as a more accurate spatial carrier basis. However, in the weak-grating limit, near the Bragg frequency, our approach reduces to the heuristic approach.
We first present the NBCME. We remind the reader that the coupled mode equations are designed to describe situations where the frequency of the light is at or near the Bragg frequency of the grating, so reflected waves are expected. Hence we find there are four appropriate fields: X+, X-, Y+, Y-. They represent the forward and backward going amplitudes of the x and y polarizations of the pulse respectively. In the following equations, the fields have been normalized such that their square-modulus represents power.
Here the field variables modulate complex combinations of the Bloch functions at the edges of the x and y bandgaps such that they represent travelling waves . The nonlinear αx,y coefficient is
where i=x,y; n2 is the nonlinear index of refraction; k0 is the Bragg wavevector; Aeff is an effective cross-sectional area in the (x,y) plane associated with the problem. The other two coefficients can be determined from αI because for a weakly birefringent optical fiber they are in the ratio
The birefringence coefficient is defined as
The grating coefficient is defined as
and, since we neglect any underlying material dispersion, the group velocities in the limit of no grating are
The inclusion of birefringence in (7) results in a dispersion relation that is different for the x and y polarizations. Figure 1 shows a typical such dispersion relation. The x and y stop bands are centered about their respective Bragg frequencies,
which causes them to be separated. This separation is important for the dynamics of the optical AND gate discussed in section three.
We now turn our attention to the nonlinear Schroedinger equations for a birefringent medium. Without birefringence, it has been shown that, for pulse widths not too narrow and center frequencies not too deep in the gap, the nonlinear coupled mode equations scale into a nonlinear Schroedinger equation ; the NLSE is known to describe the dynamics of fields at carrier frequencies far from the gap . The same arguments hold in the presence of birefringence except that the resulting NBSE involves two coupled equations
The field variables X and Y describe the amplitude of envelope functions that modulate carrier Bloch functions at a specified k, belonging to either the upper or lower band, and are normalized such that their square modulus represents power; the birefringence parameter,
measures the distance between the x and y bands at the k of interest in figure 1. The nonlinear coefficients close to the bandgap, in the presence of a weak birefringence are given by
where the velocity coefficients ωi’=dωi/dk; the velocity fraction ρi=ωi’/vi; and the other quantities are defined following (8). The remaining nonlinear coefficients can be determined for a weakly birefringent optical fiber using the ratio
The velocity coefficients and the dispersion coefficients ωi”=d2ωi/dk2 characterize the dispersion relation at the carrier Bloch functions of interest, and result from the grating. From figure 1 it is clear that ωi’ can take on any value between ±c/n0i because the dispersion relation approaches that of a uniform medium far from the Bragg frequencies. However, (14) is valid only when the grating dispersion is significantly larger than the underlying, neglected, material dispersion.
The relevant nonlinear effects in the equations can be classified into phase modulation terms and phase conjugation or energy exchange terms. Phase modulation terms involve one component of the electric field seeing an enhanced index of refraction due to the intensity of the other components of the field. The term in the NBSE describes an enhanced index for the X field due to its own intensity (self phase modulation). The term enhances the index for the X field due to the intensity of the Y field (cross phase modulation). However, there are two types of cross phase modulation in the NBCME (7). The first is cross phase modulation between different directions of the same polarization (i.e. X+ being affected by the intensity of X-). The second is cross phase modulation between different polarizations (i.e. X+ being affected by the intensity of Y±). The energy exchange terms are those that couple to the field conjugates, including the αpc terms in the NBSE and the corresponding terms in the NBCME. If the birefringence is high, then the quickly varying exp(±iδt) term will cause the effect of these energy exchange terms to vanish.
3. Optical AND Gate
In 1993 Lee and Ho  proposed an all-optical AND gate based on the NBSE for a birefringent nonlinear grating structure. In 1998 this effect was observed by Taverner et al. . In this section we present numerical simulations of the Taverner experiment with the NBCME (7), using similar experimental parameters. To achieve a clearly recognizable result for our simulations we use unchirped pulses and slightly higher powers than those employed in the experiment.
The concept behind the AND gate can be understood with reference to figure 1. The particular dispersion relation has ny<nx so the y Bragg frequency is slightly higher than the x Bragg frequency. We tune our input pulse near the upper edge of the x gap so that both the x and y polarizations are within their photonic band gap, but the y polarization is more deeply within its gap. If either the x or the y polarizations are sent into the system alone, they are sufficiently deep within their band-gaps that they experience almost 100% reflection. However, when both pulses are present, the cross phase modulation between the polarizations can shift one or both of them out of the gap. Clearly this behavior is that of an AND gate.
The parameters used in the numerical simulation are as follows. The grating is 8cm long and is apodized by one half-period of a sinusoid. The x polarization index is n0x=1.5; the birefringence is n0x-n0y=4×10-6; the grating index contrast is δn=2.75×10-5; the nonlinear index is n2=0.23×10-15cm2/W; the pulse width is 2ns; the pulse peak intensity is 10GW/cm2 for each polarization. The long pulse width in this experiment gives a narrow frequency spectrum. A smaller pulse width would broaden the spectrum so that, even if the central pulse frequency was shifted out of the gap, a significant portion of the wings might be left inside.
Figure 2 shows the AND gate nature of the device. When the x polarization or y polarization is injected alone, the output is zero. However, when both pulses are present, they help each other to pass through the grating. Both polarizations experience significant pulse compression due to their high intensity.
A convenient way of visualizing the operation of this device is by charting the field distribution within the grating as a function of time. This task is performed by two quicktime movies. The first movie (figure 3) plots the situation where the x or y pulse is injected into the system alone. Both polarizations are plotted within the same graph for the sake of comparison. The x pulse penetrates deeper into the structure because its frequency content is farther from its Bragg frequency, but neither pulse transmits through the grating.
By contrast, the movie in figure 4 plots the situation where the x and y pulses are injected together into the grating. The dynamics in this case are rather rich. At first the x polarization penetrates slightly more deeply into the grating and, via cross phase modulation, drags the y pulse along with it. But once the y pulse has been drawn into the grating it cannot leave the neighborhood of the x pulse because, without the cross phase modulation, it would experience strong Bragg reflection. One then observes an oscillation in the intensities of the x and y pulses due to two competing effects: the x pulse experiences less Bragg reflection, so it can penetrate more easily into the grating; but the y pulse finds itself trapped by the x pulse, and can thus more effectively build up its energy. In the end both pulses transmit through the structure, but the transmitted y pulse is much more intense (figure 2).
This device is also feasible in the absence of birefringence. In this case we still inject an ‘x’ and ‘y’ polarization each with a given power P0. Again the pulses are tuned near the edge of the stop gap so that individually they experience nearly 100% reflection. However, when the two polarizations are injected simultaneously, the isotropic system sees this as being one pulse of power 2P0. This allows us to cast the familiar nonlinear switching scenarios for gap solitons into a different geometry. We present such a simulation in the following movie (figure 5). In this case the AND gate operates, but the dynamics are different. The initial energy builds into a gap soliton which flushes out of the system, but then the remaining energy is insufficient to form a soliton, so it is reflected.
The birefringent case discussed above transmits 10% more energy, but these simulations do not represent an exhaustive engineering study of the AND gate device. It may be that for some parameters the isotropic device is more efficient. What is clear from these simulations is that birefringence can significantly modify the dynamics of the device.
4. Elliptically Polarized Soliton
In 1995 Barad and Silberberg proposed a nonlinear switch based on a perturbative solution to the NLSE in an isotropic medium . We begin by re-writing (14) in a frame moving with the pulse, which allows us to remove either the ∂/∂z or ∂/∂t term. It is convenient to discuss our equations in terms of a z evolution, so we convert the (isotropic) ω” term to β2 (=d2k/dω2) by using the appropriate Galilean transformation. We then re-write the equations in a circular basis by letting U = (X+iY)/√2 and V = (X-iY)√2 and find
where U and V refer to right and left circularly polarized fields respectively. The nonlinear coefficient α is related to αspm in (14) via α=αspm/ω’ (note that since we are using an isotropic medium we do not need to index αspm or ω' by polarization). The energy exchange terms have disappeared, so the only nonlinear effects are self and cross phase modulation. These terms can be considered an induced birefringence because, if |U|≠|V|, then the nonlinear terms in the two equations are unequal, so one of the polarizations will experience a different nonlinear refractive index. In the case of linearly polarized light this induced birefringence is trivial because one can always rotate to an appropriate basis where the two equations (18) reduce to a single NLSE. However, for general elliptical polarization the situation is more complicated.
To solve (18), Barad and Silberberg assumed that the incident pulse was mostly U polarized, in which case one can identify a circularly polarized soliton solution,
where the parameter η is related to the material parameters ω″ and α (see Agrawal  for details), and the desired pulse width. If to this is added a weak V polarized component, an approximate solution is found by taking (19) for U and
with s=1.562, and «6η (i.e. the V polarization is much weak than the U polarization). The phase accumulation in (20) is represented by ζ which is given by
The solution (19) and (20) represents an elliptically polarized soliton for which the azimuthal angle rotates at a fixed rate equal to (η–ζ (rad/unit length) . Each point in the pulse has its own polarization state which can be specified using the Stokes parameters or Jones calculus. The U and V components have different intensity profiles so, so the polarization state varies across the pulse. However, the azimuthal angle of the polarization is constant across the pulse. The important quality about this solution is that upon propagation the azimuthal angle rotates equally for each point on the soliton. We note that in all our simulations we inject strong right circularly polarized light with a perturbative left circularly polarized component. This means that the azimuthal angle of our elliptical polarization must range between ±π/4 but, in order to superimpose our graphs more clearly, we plot our azimuthal angle between (0, π/2).
In their experiments, Barad and Silberberg observed an azimuthal rotation of 0.5rad degrees using 2m of nearly isotropic fiber. They were limited by the fact that fiber dispersion is a material parameter so their soliton length scales were fixed once the pulse width was chosen. The same effect should be observable in a grating structure, but the interaction lengths should be much smaller, on the order of a few centimeters.
Our numerical experiments proceed by integrating equation (7) using Gaussian input pulses with similar parameters to the suggested solution (19) and (20) for varying values of intrinsic birefringence. We used Gaussian pulses because the proposed solutions would be difficult to inject directly into the system. The relevant material parameters are: n0=1.5; δn=1.67×10-4; n2=0.23×10-15W/cm2; pulse width = 85ps; right circularly polarized peak intensity = 9GW/cm2; left circularly polarized peak intensity = 0.5GW/cm2. For our system parameters, a right circularly polarized soliton would require a sech profile with a peak intensity of 9GW/cm2.
In figure 6 we plot azimuthal angle vs grating length for zero birefringence. This demonstrates that the polarization ellipse does, in fact, rotate upon propagation. The angle is determined by simulating the placement of a linear polarizer at the output and identifying the angle of maximum transmission.
To confirm that the azimuthal profile of the pulse is flat in the absence of birefringence we present a quicktime movie (figure 7) in which we plot the evolution of the Gaussian pulse with grating length increasing from 1cm to 7.8cm. The first movie frame is the 1cm case, from which it can be seen that the blue circles representing azimuthal angle are definitely not flat. However, as the length increases to about 4cm, the azimuthal profile does indeed become quite flat, and remains quite flat for the duration of the movie. At the same time, the Gaussian pulse evolves into a soliton of the sort described by (19) and (20).
Since we are injecting a Gaussian pulse we would expect some deviation in the wings. This is why the azimuthal profile is only ‘quite flat’ rather than completely flat. The azimuthal curve is only plotted in regions where the pulse intensity is greater than 10% of its initial peak intensity; at very low intensities numerical instability can cause fluctuations in the calculated azimuthal angle.
We now turn to the anisotropic situation by letting the intrinsic birefringence of the grating range between 10-7 and 3×10-5 which encompasses experimentally observed values for the intrinsic birefringence in a fiber Bragg grating. The major effect of the birefringence is to cause walk-off between the x and y polarizations, and hence between the constitutive quantities of the U and V fields (see definitions just before (18)). In some cases the nonlinear interactions can keep the pulses together , but our simulations indicate that significant walk-off occurs.
The following movie (figure 8) shows the evolution of a Gaussian input pulse with changing birefringence. The simulations have been performed with a grating length of 4cm. The film demonstrates that the azimuthal profile of the pulse remains flat until the birefringence reaches about 10-6. Past this point the pulse walk-off begins to dominate and the azimuthal profile quickly degenerates. The reader is reminded that the ±π/4 range of azimuthal angle has been converted to a range of (0,π/2) which is why the profile appears to sharply change direction.
The situation is slightly worse if we integrate the anisotropic coupled mode equations (7) with similar parameters. Figure 9 presents some results of numerical integration of the NBCME. The simulations are performed in the lab frame, so the time quoted is a lag time between the initial light injection and the output. For an intrinsic birefringence of 10-6 the azimuthal profile remains relatively flat, but at 1.6×10-6 the profile is destroyed. The effect of the birefringence is more marked with the NBCME because it takes careful account of the reflected waves. As the birefringence increases the two pulses begin to experience both group velocity mismatch, and different amounts of Bragg reflection. These differences combine to destroy the elliptically polarized soliton more quickly than in the NBSE case. Hence, care must be taken in using the NBSE near the gap in place of the more accurate NBCME.
We have not yet investigated the situation where smaller intrinsic birefringences are included in longer devices. It is possible that in such cases the nonlinear terms will be able to compensate for a small walk-off.
In this paper we have presented nonlinear coupled mode equations and nonlinear Schroedinger equations to describe light propagating in a nonlinear, birefringent, periodic medium, at or near the photonic band gap. These equations were used to simulate the operation of two proposed gap soliton devices in the presence of birefringence.
We first studied an all optical AND gate, which used x and y polarizations as input bits. The numerical simulations revealed rich dynamics in the interaction between the polarizations. It is as yet unclear how the device could be made most efficient, but it seems promising that the intrinsic birefringence can be effectively exploited. An AND gate geometry for an isotropic fiber was studied and shown to be feasible. The isotropic geometry would be easier to engineer if the intrinsic birefringences that accompany fiber Bragg grating growth can be controlled.
A nonlinear switch based on an elliptically polarized soliton in an isotropic medium was studied to determine whether the results would generalize to a birefringent system. It was found that, at intrinsic birefringences which induce significant pulse walk-off, the device begins to fail.
To conclude, we point out that fiber Bragg gratings do indeed have an intrinsic birefringence which can be relevant for gap soliton experiments. This birefringence need not be harmful, but it can greatly alter the dynamics of pulse propagation. The equations presented in section 2 of this paper allow one to describe birefringence, and hopefully reveal new effects in these structures.
This work was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) and Photonic Research Ontario. The authors thank Dr. Ben Eggleton of Lucent Technologies for useful discussions.
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