## Abstract

We derive a perturbative solution to the nonlinear Schrödinger equation to include the effect of a fiber Bragg grating whose bandgap is much smaller than the pulse bandwidth. The grating generates a slow dispersive wave which may be computed from an integral over the unperturbed solution if nonlinear interaction between the grating and unperturbed waves is negligible. Our approach allows rapid estimation of large grating continuum enhancement peaks from a single nonlinear simulation of the waveguide without grating. We apply our method to uniform and sampled gratings, finding good agreement with full nonlinear simulations, and qualitatively reproducing experimental results.

© 2006 Optical Society of America

## 1. Introduction

Fiber Bragg gratings (FBGs) are periodic modulations of the fiber core refractive index that form a one dimensional photonic band gap over which guided light is strongly reflected. FBGs are also known to have substantial dispersion for transmitted light [1]. It was suggested as early as 1970 that such photonic bandgap dispersion could affect and enhance nonlinear optical processes [2]. Over the years, numerous nonlinear effects exploiting this dispersion have been studied [3]. Examples include Bragg solitons [4], optical self-switching and gap solitons [5], and enhanced harmonic generation [6, 7, 8]. In all of these processes, the bandwidth of the transmitted light is similar to or less than the grating bandgap. However, grating dispersion is also expected to influence more extreme nonlinear effects such as continuum generation, [9] that can generate optical spectrum far in excess of a Bragg grating photonic bandgap. In particular, recent experiments have shown that when a continuum is generated using femtosecond pulses in the presence of a fiber Bragg grating, spectral peaks, some greater than an order of magnitude above the surrounding spectrum, appear in the transmitted continuum near the grating [10, 11]. These grating enhancements may have significant impact in frequency metrology, where they can coherently amplify an arbitrary portion of a frequency comb, greatly increasing the signal to noise of beat notes between the comb and other frequency-stabilized light sources [12]. We have simulated the qualitative features of these grating-induced enhancements with a nonlinear Schrödinger equation (NLSE) that included the grating transmission response [10]. The enhancement was predominantly on one side of the grating bandgap (the long wavelength side) and could be as large as 20dB above the surrounding continuum. While NLSE simulations provide a direct tool to compute the enhancement from a given grating, they provide little physical insight and can be very time consuming due to the necessity for a very fine time/frequency grid to accommodate the disparate grating (~0.1THz) and continuum (>10THz) bandwidths.

In this work we show that highly nonlinear propagation in the presence of a fiber grating, or other narrowband spectral feature, may be estimated by treating the grating as a perturbation which affects only a small fraction of the pulse bandwidth. Therefore, unlike previous treatments of nonlinear effects in gratings [4–7], our perturbative approach requires that the pulse bandwidth be much greater than the grating bandwidth. Since the continuum bandwidth is very large, even strong FBGs can satisfy this assumption, and we will show that our approach gives accurate results for such gratings. In our treatment, the continuum acts as a source term for the grating induced E-field. The large dispersion near the bandgap generates a slow grating-induced dispersive wave, which can grow significantly as it is fed by the expanding continuum spectrum. When nonlinear interaction between the grating-induced and continuum waves is neglected, the growth of the grating-induced field can be computed from a z integral (along the fiber) over the nonlinear propagation in the absence of the grating. Our approach is valid in a band of frequencies centered at the Bragg wavelength (typically a few times the photonic bandgap), where the dispersive effect of the grating dominates the nonlinear interaction between the grating-induced and continuum waves. Since this is where the spectral enhancements occur, our method is expected to be a good estimator of the maximum size of the enhancement peaks. Away from the bandgap, we also require that the grating dispersion be weak enough relative to the fiber dispersion that continuum generation is minimally altered, since these changes can make the unperturbed solution inaccurate. Both of these assumptions follow from a sufficiently large ratio of continuum to grating bandwidths (>100 in our case). We apply our method to both strong and weak gratings satisfying this condition, and find agreement with full NLSE simulations as well as experiment. In particular, our approach shows that the enhancement is related to the change in propagation constant due to the grating, giving enhancement predominantly on the long wavelength side of the grating.

## 2. Perturbative Solution of NLSE

In our previous work [10], continuum generation within a fiber grating was modeled using a NLSE in which the dispersion operator was modified to include the grating dispersion:

$$\phantom{\rule{3.2em}{0ex}}+{\int}_{-\infty}^{\infty}{\int}_{-\infty}^{\infty}K\left(\omega -{\omega}_{1}\right)A({\omega}_{1},z)A({\omega}_{2},z){A}^{*}\left({\omega}_{1}+{\omega}_{2}-\omega ,z\right)d{\omega}_{1}d{\omega}_{2}$$

Here A is the amplitude of the E-field, *D*(*ω*) = *β*(*ω*)-*β*(*ω*
_{0})-*β*
_{1}(*ω*
_{0})(*ω*-*ω*
_{0}) is the fiber dispersion operator, β and β_{1} are the fiber propagation constant and its first frequency derivative at ω_{0}, the center frequency of the pulse, and *K(ω)* is the Fourier transform of the time dependent nonlinear response function of the third-order susceptibility [13]. The grating is included by adding a term to the fiber dispersion operator, δβ_{fbg}, which is the difference between the fiber and FBG dispersion operators: *δβ*_{fbg}
=*D*_{fbg}
-*D*, where *D*_{fbg}
(*ω*) = *β*_{fbg}
(*ω*)-*β*_{fbg}
(*ω*
_{0})-*β*
_{1,fbg}(*ω*
_{0})(*ω*-*ω*
_{0}). The grating response is treated like an atomic resonance, localized to every point within the grating, and characterized by the propagation constant β_{fbg}. In [10] β_{fbg} was approximated by an ideal, uniform 1D photonic bandgap. Here, we take β_{fbg} from the FBG transmission coefficient *t*_{fpg}
computed using coupled mode theory: *t*_{fbg}
(*ω*)=exp(*iβ*_{fbg}
(*ω*
*L*), where L is the length of the grating. This choice of β_{fbg} gives the exact, causal result for linear propagation through the grating, and allows us to estimate the nonlinear response arising in non-uniform gratings. The real part of δβ_{fbg} describes the large dispersion (and low group velocity) of transmitted light. For strong gratings, δβ_{fbg} has a maximum value of ~κ=πδnη/λ_{Bragg}, where δn is the grating index modulation, η is the core-mode overlap, and λ_{Bragg} is the Bragg wavelength [14]. Strong gratings are defined by κL>>1. The imaginary part of δβ_{fbg} represents the reduced transmission due to reflection for inband frequencies and is roughly zero outside the bandgap for the apodized gratings that we consider here. Figure 1(a) shows a typical δβ_{fbg}.

In our perturbative approach we first compute the unperturbed nonlinear solution A_{0} when δβ_{fbg}=0. The effect of the grating enters as a perturbation: *A* ≈ *A*
_{0}+*A*
_{1}. When this is substituted into Eq. (1), and the purely A_{0} terms are cancelled out, A_{1} satisfies:

$$+{\int}_{-\infty}^{\infty}{\int}_{-\infty}^{\infty}K\left(\omega -{\omega}_{1}\right){A}_{0}({\omega}_{1},z){A}_{0}({\omega}_{2},z){A}_{1}*\left({\omega}_{1}+{\omega}_{2}-\omega ,z\right)d{\omega}_{1}d{\omega}_{2}+\dots $$

Here we have shown only one of the many nonlinear terms involving A_{1} and A_{0}. The key to our perturbation approximation is that we neglect nonlinear interaction between the grating-induced wave A_{1} and the unperturbed continuum wave A_{0}, keeping only the first two terms on the right hand side. The A_{1}-A_{0} interaction is small for two reasons: Firstly, because A_{1} depends on δβ_{fbg}, it is non-negligible only near the grating photonic bandgap. Since the grating bandwidth is much smaller than the continuum bandwidth (~1/100), one of the integrations in the A_{1}-A_{0} Kerr terms is smaller by roughly the ratio of these bandwidths. Secondly, the grating field amplitude A_{1} is small because the grating dispersion spreads the grating induced light A_{1} in the time domain. As shown below (see Fig. 1(d)), these assumptions imply that A_{1} has a small temporal overlap with A_{0}. Note that we retain the term δβ_{fbg}A_{1}. As previously stated, our approximation requires a narrow band grating resonance. However, this still allows for strong Bragg gratings with δβ_{fbg}L>>1 as long as their bandwidth is much less than the continuum bandwidth, making the A_{1}-A_{0} interaction negligible. In strong gratings, δβ_{fbg}A_{1} may be the dominant term in Eq. (2) for frequencies near the bandgap. As a result, A_{1} can become much larger than A_{0}, and this requires that the dispersive effect of the grating on A_{1} be included through δβ_{fbg}A_{1}. Note also that even though A_{1} may become larger than A_{0} near the bandgap, we may still neglect the higher order nonlinear terms in Eq. (2), since these depend on an integral over A_{1}, which will still be small because of the small spectral extent of A_{1}.

With the nonlinear interaction eliminated, the solution to Eq. (2) may be written as an integral over A_{0} (using the substitution A_{1} = *ae*
^{i(D+δβfbg
)z}). Substituting this solution into *A* ≈ *A*
_{0}+*A*
_{1}, the approximate solution to Eq. 1 is then:

Once A_{0} (ω,z) is computed from the NLSE without a grating present, Eq. (3) can then be used to estimate A(ω,L), the full continuum at L generated in the presence of the grating. Equation (3) is exact if there is no Kerr nonlinearity, i.e., A_{0} propagates linearly. The grating-induced peaks arise only when new frequencies are generated by the Kerr nonlinearity, making A_{0} increase along z near the Bragg wavelength. In the time domain, these nonlinearly generated frequencies are slowed by the grating dispersion and build up in a dispersive wave that has little interaction with the continuum pulse.

## 3. Comparison with full NLSE simulations and experiment

We now apply Eq. (3) to continuum generation in specific FBGs. In the case of a very weak grating, δβ_{fbg}L<<1, and we expect a weak enhancement: A_{1}<<A_{0}. Equation (3) gives a continuum intensity of:

Since the A_{0} terms are slowly varying in frequency, the frequency dependence of the enhancement should follow Re{δβ_{fbg}(ω)} in the region just outside of the bandgap where Im{δβ_{fbg}(ω)}=0. The sign and shape of the enhancement will depend on the phase of ${\mathit{\text{iA}}}_{0}^{*}$${\int}_{0}^{L}$
*A*
_{0}
*dz*. We did not examine very weak gratings, however weak enhancements (A_{1}<A_{0}, or equivalently a peak less than 3dB) roughly proportional to Re{δβ_{fbg}(ω)} were observed in ref [11]. These gratings had κL>1, but δβ_{fbg}(ω)L<1 for wavelengths a few tenths of a nm from the minimum and maximum values of the grating features.

We next considered a strong apodized grating: index modulation, δn=0.002, L=3cm, λ_{Bragg}=980nm, and |δβ_{fbg}|L~50 near ω_{Bragg}. To simplify the computation, the grating response was computed from 967 to 993nm with a very fine wavelength resolution to capture the rapid phase variations near the bandgap. Outside this range, δβ_{fbg} was small enough that we simply set it to zero, thus avoiding the need to compute the grating response over the entire continuum bandwidth. This step was justified since we are only interested in the continuum within a few nms of the bandgap. This value of δβ_{fbg} was used for both the approximate computation and, when added to fiber dispersion operator D, for the full NLSE simulations as well. The NLSE simulation used a Gaussian pulse of width 50fs, pulse energy 4nJ, and a fiber with dispersion zero near 1375nm, and dispersion, effective area, and nonlinear coefficient of 7ps/nmkm, 13μm^{2}, and 10.6 W^{-1}km^{-1} respectively, near 1580nm. Figure 1(a) shows Re{δβ_{fbg}(ω)} (dashed) and Im{δβ_{fbg}(ω)} (solid). Figure 1(b) shows the continuum computed from the full NLSE simulation (Eq. (1), filled circles plus dotted line), and from the approximate solution (Eq. (3), solid). It also shows the unperturbed continuum without a grating present (dashed). The filled circles are the actual NLSE simulated points, showing that the grid spacing was barely sufficient to resolve the grating features. Substitution of A_{1} and A_{0} into Eq. (2) shows that the linear terms dominate the nonlinear terms over ~4nm around the Bragg wavelength, roughly consistent with the range over which there is agreement with the NLSE simulation. The inset of Fig. 1(b) shows the full NLSE simulations with and without grating on a larger wavelength scale. The grating changes the entire continuum to some extent, which can most easily be seen where the continuum is weak near 1100nm. The continuum is plotted on a log scale and little power is near these minima. Such changes are small enough for our approach to remain accurate. For comparison, Fig. 1(c) shows a measured response from a similar grating using the experimental setup of [10]. The enhancement is clearly on the long wavelength side in both cases. The peak near 987nm is associated with spectral structure in the experimentally observed linear grating response near 986nm [10] and is not included in the simulations. The oscillations in Fig. 1(b) are absent, most likely because of additional nonlinear fiber and pulse to pulse variations that can smooth a continuum spectrum. Oscillations were previously observed in other gratings [8]. Figure 1(d) shows the time dependence of the E-field intensity, computed from the full NLSE (dotted), the approximate solution (solid), and the full solution without a grating (dashed). The time scale is logarithmic to aid comparison. The continuum wave is roughly the same in each case, extending to 0.5ps. The grating-induced wave extends to beyond 30ps.

To test the applicability of our technique with a more complex grating, we simulated a sampled grating, whose index modulation is switched on and off with period Λ_{samp}, much larger than the grating period. Such gratings reproduce the unsampled grating transmission response at evenly spaced frequency intervals proportional to 1/Λ_{samp}. We therefore expect enhancements at multiple wavelengths from such a grating. The grating parameters were: L=3cm; δn=0.002; Λ_{samp}=375μm; duty cycle 25%; λ_{Bragg}=1250nm. Figure 2(b) shows that we again had good agreement with the full NLSE within the simulation resolution. Figure 2(c) shows a measured response for a similar grating resonance near 1120nm [15] and also shows multiple enhancement peaks. Full and approximate NLSE simulations at 1120nm were in agreement but gave little enhancement. Clearly, the enhancement peaks for strong gratings depend on the spectral phase of the unperturbed continuum envelope A_{0}, as was shown for weak gratings above. It is known that continuum generation is very sensitive to pulse and fiber parameters. With a sufficiently precise model of the unperturbed continuum generation we would expect better agreement on the exact Bragg wavelengths at which enhancement peaks are experimentally observed.

## 4. Conclusion

We have shown that the large spectral peaks observed near a Bragg resonance when continuum is generated in the Bragg grating can be described by treating the grating as a perturbation to the nonlinear propagation. The continuum generation in the fiber without grating is computed once as a function of z in the fiber, and then combined with the grating transmission response to arrive at the grating-modified continuum without further solution of the NLSE. Thus, the response near any grating may be computed from an integral over a single computation of the full nonlinear propagation, enabling rapid (~100x faster) simulation of the change in continuum generation near a Bragg resonance. Such grating enhancements have shown great potential to improve signal to noise ratio in frequency metrology applications using fiber continuum combs.

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