1. Introduction

Solitons are, amongst all nonlinear phenomena, perhaps the most intriguing. They have been observed in many different physical systems, including fluid dynamics, plasma physics, and nonlinear optics [1,2]. Solitons are localized structures which do not broaden because the nonlinearity balances the dispersion of the medium. Of particular interest is the interaction of solitons with dispersion which is induced by complex photonic geometries, such as photonic crystals [3], waveguide arrays [4,5], nanowires [6], or microresonators [7].

In optics solitons have been studied in many geometries, the most common being uniform optical fibers, in which the Kerr nonlinearity of the glass balances the dispersion of the fiber [8–10]. Another optical geometry is that of a fiber Bragg grating (FBG), a one-dimensional periodic structure (1D photonic crystal). While this geometry still relies on the Kerr nonlinearity of the glass, the grating now provides the dispersion. This dispersion is orders of magnitude larger than that of the fiber, so that the latter is essentially irrelevant. Using the grating dispersion adds three novel elements: first, the dispersion is easily tailored by adjusting the properties of the grating, such as strength, length, and resonant wavelength. Second, the soliton velocity can vary in principle between 0 and c/n, where n is the effective mode index. Third, because the dispersion is so large the relevant length scales shrinks proportionally. Therefore, solitons, referred to in this context as gap solitons [11,12], form over centimeter length scales, which makes them suited for integrated optical circuits and devices. Another consequence of the stronger dispersion is that gap solitons require high optical intensities. The first experimental study of gap solitons in 1996 was subsequently followed up by more careful studies, and by the demonstration that gap solitons are well-suited for slow light experiments since the broadening effect of dispersion, which is unavoidable in linear experiments, is eliminated in the nonlinear regime.

Here we report the observation, both experimentally and in simulations, of the quantization of the high-intensity transmission of a 1D photonic crystal, implemented in our case by an FBG. We show the quantization to be a novel manifestation of the inherent discreteness of solitons (which in our experiments are gap solitons). Wave propagation in 1D photonic crystals, including FBGs, can often be described, at least approximately, by the nonlinear Schrödinger equation (NLSE), which is well understood [8–10]. It has soliton solutions which are bell shaped, and which, at a fixed dispersion, have one degree of freedom: they can be narrow with high peak intensity, or broad with low peak intensity, but all have the same area, the product of the field amplitude and the width. This fixed area introduces an intrinsic element of quantization. Here we report a novel manifestation of this quantization in the transmission of high-intensity light pulses through a Fiber Bragg Grating (FBG): the short wavelength edge of the bandgap acquires a characteristic staircase shape at high optical powers, which we link to the excitation and propagation of gap-solitons.

The paper is structured as follows. Section 2 discusses optics in FBGs and puts forth a simple, cw-based model on the effect of nonlinearity on light propagation inside such a grating. Section 3 describes our experimental setup and the numerical model. Section 4 contains experimental and numerical results of our experiments, which invalidate the earlier, simple model. Instead an explanation in terms of gap-solitons is included, which is simplified further and then linked to the NLSE model. Section 5 uses these results to show that the quantization of the NLSE-soliton area is the driving force behind our results, which is demonstrated by experimental and numerical data. An upper limit for the quantum of energy is derived from those findings and shown to be in good agreement with simulations. Section 6 underlines that, although self-induced quantization is not a new phenomenon, our findings differ considerably from pervious scenarios. Section 7 concludes the paper.

2. Optics in Fiber Bragg Gratings

An FBG is a single mode optical fiber with a weak, periodic modulation of the core’s refractive index along its propagation direction. The periodicity resonantly couples the forward and backward propagating modes around the Bragg wavelength λ_B = 2nΛ with n the effective mode index, and Λ the grating period. The coupling leads to a (one dimensional) bandgap centered at λ_B with a half-width of$\Delta \lambda =(\Delta n/n){\lambda }_B$, where $\Delta n$ is the amplitude of the refractive index modulation. Light with a wavelength λ inside this bandgap cannot propagate in the grating and is reflected. The wavelength λ is usually expressed as the detuning $\delta =2\pi n({\lambda }^{-1}-{\lambda }_B{}^{-1})$, the deviation from the Bragg resonance. In terms of this parameter the bandgap extends from –κ < δ < κ, where the coupling strength $\kappa =\pi \Delta n/{\lambda }_B$ is proportional to the modulation contrast. Fields with large detuning ($\left|\delta \right|\gg \kappa$) are essentially unaffected by the grating. For wavelengths between these extremes, the coupling leads to a reduction in the group velocity, vanishing at the gap edges. As the group velocity thus depends on wavelength, FBGs lead to dispersion, which is so strong that any intrinsic dispersion of the uniform fiber can be neglected. On the short wavelength side of the bandgap, the upper band edge $(\delta >\kappa )$, the group velocity increases with decreasing wavelength, thus the dispersion is anomalous. In contrast, on the long-wavelength side, the lower band edge $(\delta <-\kappa )$, the dispersion is normal.

For sufficiently high optical intensities I, nonlinear effects become important. For silica-based glass we need to include the Kerr effect which increases the refractive index linearly with the intensity according to $n(I)=n+n_{2}I$, where the Kerr constant $n_2=2.7\cdot {10}^{-20}{m}^2/W$ in glass [4]. The Kerr effect is transient on a femtosecond timescale, much shorter than the shortest, picosecond timescale in our experiment. It is therefore considered as instantaneous. Since the refractive index increases with intensity and λ_B = 2nΛ, the Bragg wavelength, and thus the bandgap, shift to longer wavelengths with increasing intensity. As illustrated in Fig. 1(a) , light tuned just inside the upper edge of the bandgap, is reflected at low intensities, but should enter the grating at high intensities. Similarly, light tuned just outside the lower edge is transmitted at low intensities but reflected at high intensities [13].

 

Fig. 1 (a) Effect of a positive Kerr nonlinearity in a simple cw-model on the measurement of the transmission versus detuning (lower scale) and wavelength (upper scale). A fixed laser wavelength, tuned just inside the grating’s upper band edge, is reflected al low intensities and might be expected to be transmitted at high intensities. (b) Schematic of the experimental setup. Laser pulses coupled into the FBG and the transmission is measured with a power meter. Time-resolved transmission is measured with a fast sampling oscilloscope. Longitudinal strain applied to the FBG can be used to tune the grating with respect to the laser.

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We report here that this simple continuous wave description, although correct at the lower band edge, is incomplete at the upper band edge [14–16]. The reason is that at high intensities, solitons form inside the grating, which changes the transmission profoundly. When the nonlinearity is positive, as it is here, solitons only exist when the dispersion is anomalous. Indeed, though gap solitons exist for all frequencies inside the photonic bandgap [14], those at the upper edge of the bandgap have the lowest intensity and are therefore experimentally most easily accessible. These gap solitons, travel through the grating and thus mediate its transmission for frequencies inside the bandgap . Gap solitons have been studied both theoretically [17,11,12], and experimentally [18,19]. While experimental work to date has concentrated on the characteristics of single solitons, here we observe and discuss the emission of multiple solitons.

3. Experimental setup and numerical model

Our samples were fabricated by inscribing the periodic modulation into the photosensitive core of a Nufern PS1060 fiber by exposure to a sinusoidal 244 nm UV diffraction pattern [20,21]. As the fiber consists of silica-based glass it has low loss, a suitable Kerr-type nonlinearity, and withstands high optical powers [19]. Our FBG is 70 mm long, with 10 mm raised cosine apodization regions on side, and 10 mm grating free ends on either side of the grating, which serve to glue to grating to coupling stages. The apodization lowers the threshold for soliton formation but does otherwise not affect the observed dynamics as verified by [22,19]. The grating strength κ = 450 m⁻¹ was estimated from the width of the FBG’s transmission spectrum using a spectrometer. Measuring the grating extinction is difficult because of instrument limitations but it is far over −40 dB in the centre of the gap.

A sketch of our experimental setup is shown in Fig. 1(b): a Q-switched Nd:YAG microchip laser emits pulses at a wavelength of λ = 1064.3 nm, a repetition rate of 3-4 kHz, with a full width at half maximum of 700 ps, an energy of 8 µJ, about 4 µJ of which is coupled into the FBG, leading to a maximum input peak power of approximately 5 kW. The spectral width is approximately 2 pm, close to transform limited, with the trailing edge being slightly longer than the leading edge [19].

The detuning is controlled either by applying strain to the FBG, changing λ_B, or by varying the laser’s repetition rate, slightly changing the laser’s wavelength, but leaving other properties unchanged. Strain tuning is achieved by mounting the output facet of the grating onto a motorized translation stage, and has a range of approximately 1 nm with a resolution of 0.5 pm. The known width of the grating spectrum is used to calibrate the relationship between strain and detuning. Repetition rate tuning has a smaller range of 50 pm, with a better resolution of 0.15 pm.

Transmission and reflection from the grating were measured using Si-based power meters and a fast sampling oscilloscope with 22 ps response time. A series of measurements with varying value for the detunings thus allows the measurement of transmission spectra $T(\delta )$ and of time-resolved transmission spectra$P(\delta ,t)$.

If strain tuning is used, as for the spectra in Fig. 2(a) , the strain motion leads to a lateral misalignment of the power meter, which is adjusted at the lower edge, underestimating the transmitted power by ~10% at the upper edge. Simultaneously a 10% baseline transmission is observed, which is attributed to a secondary spectral laser peak at 1064.7 nm, which is unaffected by the grating. Because of the limited range of the repetition rate detuning, the strain stage is used to find the approximate measurement position, which involves moving the stage back and forth a few times. This introduces mechanical backlash and thus some uncertainty in the absolute position of the stage. This explains the different detunings for which gap solitons are first observed in Fig. 4(a) and 4(b). Each pulse inherently has a fixed detuning and does not interact with other pulses, hysteretic effects are of no importance in our experiment, as they would have in cw experiments. To avoid thermal effects [23], the grating is covered in thermally conductive paste and mounted on a metal platform, kept at constant temperature.

 

Fig. 2 (a) Measured and (b) calculated low- and high-intensity transmission spectra near the upper band edge for different incident pulse energies.

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Fig. 4 (a) Measured, and (b) calculated time resolved transmission spectra near the upper the band edge. Each vertical slice represents the optical power P(t) at the end of the grating for a given detuning. In (b), the calculated transmission spectrum has been overlaid. Input energy E = 2.5 µJ. Different positions on the detuning axes are caused by backlash of the strain stage.

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The propagation of high-intensity light pulses through a grating is described by the coupled mode equations [24,25]:

Coupled mode Eqs. (1) have been widely used to model nonlinear wave propagation in one-dimensional photonic crystals, and have discussed thoroughly before [13,11,12,18]. The boundary conditions are $A_{+}(t,z=0)=A_0(t)$,$A_{-}(t,z=L)=0$, $A_{-}(t=0,z=0)=0$, and $A_{+}(t=0,z=L)=0$, with the input pulse characterized by $A_0(t)$. The Transmission and reflection are measured at the ages as well: $T(t)=A_{+}(z=L,t)$, $R(t)=A_{-}(z=0,t)$, where $\kappa =0$, corresponding to the grating free ends of the fibre.

While Eqs. (1) are a very general description, close to the band edge $\left(\delta \approx \kappa \right)$ they can be approximated by the simpler NLSE [26,27]:

4. Results

The blue curve in Fig. 2(a) shows the measured transmission spectrum close to the upper band edge for low powers, where the nonlinear Kerr effect is negligible. The red curve is the corresponding result for high input powers, where the nonlinear effects are strong. At low intensities the transmission is low inside the bandgap and high outside the gap, with the regions separated by a steep band edge. As the power is increased the transmission spectrum changes qualitatively, taking a characteristic staircase shape. Numerical solutions of Eq. (1), shown in Fig. 2(b), exhibit behavior very similar to the experiments: the transmission spectrum has a pronounced band edge at low pulse energies, and is quantized at high energies. A simulation with an input energy of E = 6 µJ, which is higher than can be achieved in the experiment, is included, as well, showing that the number of steps increases and the detuning at which the first step appears decreases as a function of the input energy. This quantization, which sharply contrasts the expectation that the nonlinearity merely shifts the bandgap [see Fig. 1 (a)], is shown below to be intimately related to the properties of gap solitons.

At the lower band edge the dispersion is normal, and no new steps appear in the experiments or in the simulations. Light just outside the lower edge is transmitted at low intensities but reflects at high intensities, consistent with the absence of solitons at these frequencies.

The time-resolved transmission through the grating for different detunings is shown in Fig. 3 . It shows that the energy is transferred in well-defined pulses, gap solitons, and that the increase in transmission with detuning is associated with the transmission of additional gap solitons [28,29]. A better way to display such data is given in Fig. 4(a), with associated numerical results in Fig. 4(b): these contour plots show the instantaneous optical power transmitted by the grating versus time (vertical) and detuning (horizontal). The simulations, which have the transmission spectrum overlaid to ease interpretation, are consistent with the experiments and confirm that with increasing detuning a series of solitons appears. Each step of the transmission spectrum is located at the critical detuning where one additional soliton emerges. Just above a critical detuning, the corresponding soliton propagates slowly, but its speed approaches c/n as the detuning increases. For very large detunings, the initial pulse shape is regained, because the light and the grating no longer interact resonantly. We find that this behavior is generic to this system for sufficiently high input powers.

 

Fig. 3 Measured time-resolved transmission for different detunings near the upper band edge where gap solitons can form. Input energy E = 2.5 µJ.

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The flatness of the steps in Fig. 2 indicates that each soliton carries a similar energy, even though the width, peak power, and speed of each soliton vary with detuning. Below we show that this observation reflects the fixed area of NLSE solitons. A related, but conceptually easier, argument applies if the detuning is kept constant but the input energy is varied. Figure 5 shows the transmitted versus the incident energy for different grating strengths and (fixed) detunings as found by solving Eqs. (1). It demonstrates that the transmission increases in regular steps as the input energy is increased. The height of the steps depends only weakly on the detuning of the incident light or on the grating strength, which is varied by a factor of 4, whereas the step height only varies by ~20%. In Section 5 this is shown to be consistent with analytic approximations. Figure 5 shows only numerical data since the experiment is difficult because of the lack of fine tuneability of the incident power. These results are reminiscent of soliton fission or modulational instability in the NLSE [30,31]. In these processes a pulse of peak power P and a width T₀ decays into an integer number N(P;T₀) of solitons, while the remaining energy disperses. Inside the bandgap, only solitons can propagate, whereas dispersive waves cannot. Thus the inherent discreteness of the fission manifests in the transmission staircase. When the detuning is varied at fixed power, as in our experiments, steps occur because of variations of the soliton excitation efficiency [14–16].

 

Fig. 5 Calculated transmission curves (output energy vs. input energy) close to the upper band edge for different detunings and grating strengths.

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Below we estimate the energy of each soliton, to find the approximate magnitudes of the steps in Fig. 2 and Fig. 3(b). The essence of the calculation is that we obtain the area of each pulse from the NLSE and the grating properties. Combining this with limitations of the minimum soliton width, we find an upper limit for the pulse energy, which is in fair agreement with experimentally and numerically measured step heights.

5. Derivation of transmission quanta

From our simulations we conclude that the solitons emerge close to the upper band edge (δ_out ≈κ) even though the incident field is deep inside the band gap (δ_in < κ) by measuring the centre frequency of the spectrum of the transmitted pulses, which is shifted with respect to the incident pulse. Our analysis suggests that this is caused by spectral broadening of the pulse in front part of the grating, of which the frequency components closest to the band edge can penetrate further into the grating and are thus preferably transmitted. This unexpected behavior allows us to use the NLSE as an approximation, whose solitary solutions are considerably easier to handle than those of Eqs. (1) and still give surprisingly precise insight into the underlying physics. The pulse area is defined through $F^{\text{NLSE}}={\displaystyle {\int }_{-\infty }^{\infty }\left|u\left(t\right)\right|dt}={\displaystyle {\int }_{-\infty }^{\infty }\sqrt{P\left(t\right)}dt}$. For solitons it takes the value of $F_0^{\text{NLSE}}=\pi \sqrt{\left|{\beta }_2\right|/\Gamma }$, where β₂ is the (quadratic) dispersion and Γ is a nonlinear parameter. In terms of the grating parameters in Eq. (1) and Eq. (2), these expression for the area read as

In our simulations of Eq. (1), where we calculate the fields everywhere inside the fiber, we confirm to great precision that each soliton has an area F₀. This is illustrated in Fig. 6(a) , which shows the evolution of the pulse area in units of F₀, for a detuning where one soliton is excited (blue curve), a value of F/F₀ ≈0.98. For a detuning where three solitons are excited (red curve), one of which is very slow, we find 2.8<F/F₀ <3.1. Figure 6(b) and 6(c) contain corresponding snapshots of the power distribution in the grating and show the grating profile $\kappa (z)$. Animations of the temporal pulse evolution are available with Fig. 6(b) (Media 1) and Fig. 6(c) (Media 2). The integration is carried out for 15 mm < z < 65 mm, for which κ(z) is constant and the solitons are thus unaffected by the apodization. The residual pulse area at t > 2 ns is attributed to the field which is back-reflected by the end of the grating. Its low power levels cannot sustain a soliton.

 

Fig. 6 (a) Evolution of the normalized soliton area [see Eq. (3)] as a function of the time, for a single transmitted soliton for E_in = 6 µJ (blue, δ = 0.67) and a triplet of transmitted solitons, at the edge of the transition from two to three solitons (red, δ = 0.81). (b) Power distribution inside the FBG for δ = 0.67, t ≈1.2 ns (Media 1). (c) Power distribution inside the FBG for δ = 0.81, t ≈1.0 ns (Media 2).

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Figure 7 shows the experimentally inferred pulse area. In the experiments we can only measure the intensity of the light at the grating free end of the fiber as a function of time, which has to be converted into the pulse area defined in Eq. (3), using an approximation of the propagation speed of the pulses, which is c/n in the region of measurement, thus $F\approx \raisebox{1ex}{$c$}\!\left/ \!\raisebox{-1ex}{$n$}\right.{\displaystyle \int P{(t)}^{1/2}dt}$. Moreover, as discussed, the pulses are somewhat distorted as they leave the grating. Nevertheless we confirmed with good precision that each pulse has an area approximate area of F₀, as can be seen in Fig. 7.

 

Fig. 7 Normalized pulse area $F=\raisebox{1ex}{$c$}\!\left/ \!\raisebox{-1ex}{$n$}\right.{\displaystyle \int P{\left(t\right)}^{1/2}dt}$ [see Eq. (3)] as a function of the detuning. The pulse area is determined experimentally from the intensity of the light at the end of the FBG. Data is inferred from the measurement presented in Fig. 4 (a).

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Now that the cause for the observed quantization is understood, we have to relate it to the observed quantization of the transmission spectra, which is a measure of the pulse energy, not the pulse area. The relation of the two quantities depends on the soliton width. Unfortunately, we do not have an accurate estimate for this parameter. The soliton width is mainly determined by the decay rate of its wings, which essentially behave as in a linear medium as they have low intensity. The decay rate cannot exceed κ (the value it takes at the center of the band gap), but has no lower limit. Therefore, we take the decay rate to be κ/α, where α > 1, and the width of the soliton is thus approximately α/κ.. From measurements of pulse widths in the experiment and simulation, we find that α is in the order of unity. We note that we cannot use exact nonlinear solutions to Eqs. (1) or Eq. (2) since the precise frequency of the transmitted pulses is not sufficiently well known. Thus we can only make an approximate statement about an upper limit for the pulse energy quantum, not a precise prediction.

Using the estimated spatial width of the soliton $\alpha /\kappa$, and the known pulse area we find $P_{\text{max}}\approx {\alpha }^{-2}\left(2\kappa /3\gamma \right)$, using the expression for F₀ from Eq. (3), implying that the nonlinear shift of the grating spectrum is somewhat smaller than the width of the gap. With these quantities known, and the observed soliton speed of approximately c/n, the upper limit for the quantum of energy is

In our experiments the energy per soliton varies between 0.5 µJ and 0.75 µJ, whereas in the simulations it varies between 0.76 µJ and 1.05 µJ [see Fig. 3(b)]. Both findings are consistent Eq. (4). This equation also predicts that, at this level of the theory, the step size depends neither on the detuning δ, nor on the grating strength κ, consistent with Fig. 3(b).

6. Other systems with self-induced quantization

A stepwise transmission function is not unique to nonlinear FBGs. Bistable and multistable systems, such as nonlinear photonic crystals, also exhibit such transmission spectra [32,33], though their steps are associated with different metastable cw solutions and may, in principle, take any value. Wright et al numerically showed spatial soliton emission in nonlinear, planar waveguides with cw excitation [34]. Quantization of energy is also observed in soliton lasers, related to the excitation of dissipative solitons [35,36]. In both scenarios the solitons have no degree of freedom, and it is thus not surprising that they carry the same energy. In contrast, even though in our work the soliton energy is roughly constant as well, a range of different solitons is involved, as can for example be seen by their large variation of propagation speeds.

Quantized transmission is also characteristic of self-induced transparency, which occurs in the coherent, resonant interaction between light pulses and two-level atoms [37,38] or semiconductors [39]. Such systems only support pulses with integer multiples of a fundamental area. Here the quantization originates from Rabi oscillations of electrons, thus from the quantum mechanics of the atoms. Stepwise transmission also occurs in transport properties of mesoscopic systems [40,41]. There the quantization is related to the number of transmission channels, controlled by the channel’s geometry. Our work differs in two respects: it is classical and the quantization does not rely on existing transport channels—rather, the channels, here gap solitons, are generated nonlinearly at high intensities.

7. Conclusion

In conclusion, we have shown in experiment and simulation, that the high-intensity transmission of an FBG exhibits quantization, where the n^th step is associated with the transmission of n solitons. We believe this is the first demonstration of nonlinearly self-induced quantization based on the action of gap-solitons in a periodic lattice.

The quantization is governed by a constraint of the pulse area, which can only take integer multiples of a fundamental area quanta F₀, although other pulse parameters may vary. We further derived an approximation for the energy of those solitons, which is consequently quantized, as well.

We believe that this type of behavior is generic for systems in which solitons propagate through a medium with a forbidden frequency gap, opening perspectives for experiments on related systems, such as photonic crystals or waveguide arrays.