## Abstract

We demonstrate frequency redshifting and blueshifting of dispersive waves at group velocity horizons of solitons in fibers. The tunnelling probability of waves that cannot propagate through the fiber-optical solitons (horizons) is measured and described analytically. For shifts up to two times the soliton spectral width, the waves frequency shift with probability exceeding 90% rather than tunnelling through the soliton in our experiment. We also discuss key features of fiber optical Cherenkov radiation such as high efficiency and large bandwidth within this framework.

© 2012 Optical Society of America

## 1. Introduction

Since the prediction and demonstration of fiber-optical solitons three decades ago [1, 2], the manipulation of light by optical pulses in fibers is an active field of research with applications e.g. in optical communication and switching [3–9]. The interaction of dispersive waves with solitons via the optical Kerr effect is called cross phase modulation (XPM) [10, 11], which has extensively been studied in the context of soliton communication systems [12]. This interaction is normally limited because of the relatively large difference in group velocity. In fact, solitons are known to penetrate each other, regaining their shape, energy, and velocity after a transient spectral change during collision [13,14]. The interaction imprints a phase and position shift onto the solitons. A dispersive wave normally interacts with the soliton in a similar way, acquiring a transient spectral shift and a permanent phase and position shift [12].

This situation changes, however, if the group velocities become comparable. The dispersive wave may be slowed or accelerated by the soliton such that it cannot pass over the soliton anymore. Amongst other effects, this leads to a permanent frequency shift of the dispersive wave. The invention of photonic crystal fibers [15, 16] and development of laser technology now allows engineering of the fiber dispersion such that this regime is easily accessible. Novel effects such as fiber optic Cherenkov radiation [17–19], wave trapping [20–22], and frequency shifts at artificial event horizons [23] were discovered. In the latter two effects, the dispersive ‘probe‘ wave, copropagating with the pulse, but at a distinct spectral location, is frequency shifted. The critical dependence on group velocity characterizes these effects compared to other nonlinear conversion effects such as four wave mixing, Brillouin-, or Raman scattering.

The frequency shift can be induced by a non-stationary evolution of the soliton. In the case of light trapping, the soliton evolves non-stationary, because it is continuously slowed under the Raman-induced soliton-self frequency shift. Dispersive waves are trapped behind the soliton, resulting in a quasi-continuous process of frequency shifting, a cascade of frequency shifts much smaller than the pulse spectral width. In this way the total frequency shift is due to the soliton frequency shift and more than 50nm spectral shift have been obtained experimentally [24].

In the case of optical event horizons, however, frequency shifts of dispersive waves develop at stationary solitons. Waves slightly faster than the soliton are prevented from passing over it due to the XPM-induced refractive index increase. Hence they are shifted within a single collision with the soliton during which the soliton velocity is constant, i.e. the Raman effect is negligible [25]. The position where the index change has slowed the wave to the speed of the soliton defines a turning point for the light, a ‘group velocity horizon’, as the propagation is changing from subluminal outside the soliton to superluminal under the soliton. The soliton induces an effective space-time geometry for light, which is an exact analogue to the astronomical event horizon associated with black and white holes [23, 26]. As a result, this leads to a permanent frequency shift of the dispersive wave [23]. Effectively, the soliton creates an analogue gravity in the lab. If the dispersive wave is in its vacuum state, it is expected that vacuum modes convert to photon pairs, a manifestation of the Hawking effect in this system [23,26–28]. In the context of nonlinear optics the group velocity horizons are interesting, because they allow up as well as downconversion. As we will discuss, they can be used for optical switching, optical delays, and dispersion management [8, 22]. In the first demonstration of fiber optical event horizons, frequency shifts were small and only blue shifting was demonstrated [23].

In this paper we demonstrate red and blue shifting of dispersive waves at (stationary) optical event horizons for the first time. Hence we investigate tunnelling of waves through the soliton, which crucially determines the efficiency of the interaction. Finally we reveal that this interaction is in one class with other nonlinear fiber optics effects such as ‘Cherenkov‘ radiation. As a result we show that frequency shifts are not limited by the pulse spectral width, as might be expected from a naive Fourier-bandwidth argument or a photon picture.

First we introduce an analytical theory of scattering of light at solitons in fibers, including frequency shifts and wave tunnelling. The model gives a characteristic probability for the wave to tunnel through the soliton. Secondly, we present a set of measurements of observed frequency shifts and compare the measured efficiency to the analytical model. Thirdly, we discuss the results and give an interpretation to fiber optic Cherenkov radiation based on these findings, before we conclude.

## 2. Tunnelling model

Waves propagating in a single-mode optical fiber are described by the nonlinear Schrödinger equation (NLSE) [29]. Here we consider copropagation of two optical fields: the soliton and the dispersive ‘probe’ wave. Their slowly varying amplitudes are given as *A*_{1}(*z*,*t*) for the probe wave and *A*_{2}(*z*,*t*) for the soliton. We focus on the principal effects of group velocity dispersion and nonlinearity. We assume a soliton unaffected by higher order effects as well as a weak probe wave with negligible backaction on the soliton. Numerical treatment including these effects can be found in [30]. With these approximations the NLSE becomes (see e.g. [24, 29, 31, 32]):

*β*

_{1}and

*β*

_{2}are the first and second derivative of the propagation constant

*β*(

*ω*) at a frequency

*ω*

*of the same group velocity $u={\beta}_{1}^{-1}$ as the soliton and close to that of the field*

_{m}*A*

_{1}representing the probe.

*γ*is the fiber nonlinearity and

*r*is a factor accounting for the reduction in cross-phase modulation due to conditions such as the relative polarization orientation or mode size mismatch. We change to the coordinates

*τ*=

*t*–

*z*/

*u*and

*ζ*=

*z*/

*u*of the moving frame of the soliton with velocity

*u*. Hence, we neglect self-phase modulation of the weak probe and assume a soliton, unaffected by the probe, of the form |

*A*

_{2}(

*τ*)|

^{2}=

*P*

_{0}sech

^{2}(

*τ*/

*T*

_{0}), where

*P*

_{0}and

*T*

_{0}are the soliton amplitude and length. We arrive at:

*ω*′ =

*ω*–

*β*

*u*=

*ω*{1 –

*n*(

*ω*,

*I*(

*τ*))/(

*β*

_{1}

*c*)}. The refractive index

*n*has a nonlinear contribution from the soliton intensity

*I*(

*τ*) due to the Kerr effect. Finally, we use the ansatz

*A*

_{1}= 𝒜

*exp*(

*iϕ*(

*ζ*,

*τ*)+

*i*

*ω*′

_{m}*ζ*), where

*ϕ*is the optical phase and

*ω*′

*is the moving frame frequency at the group velocity matched lab frequency*

_{m}*ω*

*. The nonlinear Schrödinger equation becomes:*

_{m}*ω*′ –

*ω*′

*. Equation (3) is formally identical to the Schrödinger equation in quantum mechanics. This analogy allows us to investigate quantum mechanical potential problems with classical nonlinear fiber optics. As*

_{m}*β*

_{2}can either be positive or negative, we can realize attractive or repulsive potentials. Input and converted modes are connected: the waves follow contours of constant

*ω*′ (moving frame frequency) as this is a conserved quantity in Eq. (3) [23]. Note that

*ω*′ is conserved even in the presence of higher order dispersion acting on the probe. Therefore, the waves are shifted in frequency in the laboratory frame by an amount that depends on the soliton velocity and the refractive index profile of the fiber mode only. In addition to the modes let us consider the efficiency of the process, i.e. the probability of conversion from the input mode to the output mode. The scattering at solitons represents a spatially constant one-dimensional potential, for which the transmission and reflection coefficients,

*T*and

*R*, are known [33]. In the variables of fiber optics these coefficients are:

*ω*–

*ω*

*is the detuning of the probe wave from the group velocity matched frequency. The transmission through the barrier therefore is determined by the two parameters Ω*

_{m}*T*

_{0}, the ratio of detuning to soliton bandwidth, and $B=8r\gamma {P}_{0}{T}_{0}^{2}/{\beta}_{2}$, the normalized barrier height, only.

*B*> 0 (

*B*< 0) implies a potential barrier (pit). For

*B*= 1 the reflectivity has the sech

^{2}-form of the pulse and the interaction decreases exponentially with detuning. At large

*B*, the effective wave number becomes complex (cf. Eq. (3)) and the dispersive wave can only pass the soliton by tunnelling. Accordingly, tunneling takes place if the effective nonlinearity (i.e.

*r*

*γ*

*P*

_{0}) or the dispersion length ( ${T}_{0}^{2}/\left|{\beta}_{2}\right|$) is large. Figure 1 is a contour plot of the reflectivity of a soliton (

*r*= 2, copolarized). For

*B*< 0, the probe wave experiences anomalous dispersion and reflection is limited to detunings of much less than the spectral width of the soliton. In the case of

*B*> 0, however, effective reflection can be achieved for large detunings. The contour for 90% reflection is given by (

*B*≫ 1,Ω

*T*

_{0}> 1): This is a remarkable result. In the photon picture, the mode conversion may be thought of as an annihilation of a photon in the input mode and creation of a photon in the output mode, which has a different laboratory frequency. The energy difference has to come from the pulse. In the simplest way, a photon of one of the modes of the pulse is annihilated and another photon is generated with the corresponding energy difference. This is a simple four-wave mixing process well known in third order nonlinear materials. Hence, we would expect the spectral width of the pulse to severely limit the frequency shift as it is the maximum energy difference between the modes that can be compensated for. For most pulses, e.g. solitons, the spectrum exponentially decreases away from the carrier and an exponential decrease in frequency conversion is expected. Equation (5), however, states that efficient conversion is possible even for very large detunings, provided

*B*is sufficiently large. The required height of the barrier

*B*increases quadratically rather than exponentially. In principle, condition 5 can always be fulfilled in a medium with very small magnitude group velocity dispersion

*β*

_{2}at the probe frequency. Therefore, the mode conversion is a collective effect of the modes of the soliton and the probe rather than a phasematched mixing of only four modes.

## 3. Tunnelling experiment

To test this model we set up an experiment. We use fundamental solitons (
${N}^{2}=1=\gamma {P}_{0}{T}_{0}^{2}/\left|{\beta}_{2}\right|$) such that
$B=16\frac{\left|{\beta}_{2s}\right|}{{\beta}_{2}}=16\frac{{D}_{\text{s}}{\lambda}_{\text{s}}^{2}}{\left|D\right|{\lambda}^{2}}$, where *D*(*D*_{s}) and *λ*(*λ*_{s}) are the dispersion (in ps/nm/km) and wavelength of the probe (soliton). To achieve large *B* we use a highly nonlinear photonic crystal fiber that exhibits anomalous dispersion in the infrared for soliton creation. We choose the probe wavelength shorter than the soliton wavelength beyond the zero dispersion point of the fiber, where there is a group-velocity matched wavelength *λ** _{m}*. As is indicated in the inset in Fig. 4, the integral over the dispersion curve, which is the group delay, vanishes between

*λ*

*and*

_{m}*λ*

*. Hence, for the photonic crystal fiber (pcf) we used (NL-1.5–670, NKT Photonics, Inc.), we obtain*

_{s}*B*= 22.4 with dispersions of −256ps/(nm km) and 144ps/(nm km) for probe (

*λ*

*= 532nm) and soliton (*

_{p}*λ*

*∼ 840nm), respectively. The group index of the fiber was determined approximately using a simple silica strand model which reproduced the group velocity matching condition as observed [34].*

_{s}The solitons are generated using a 50-fs Titanium:Sapphire laser with a repetition rate of 81MHz (Trestles100, Del Mar Photonics, Inc.). Using a dichroic mirror, we also couple a continuous probe laser at 532nm wavelength (Verdi V6, Coherent Inc.) into the fiber. In order to change the probe detuning, we move the group velocity matched wavelength by tuning the pulsed laser in wavelength in small increments. At the end of the 1.35-m long fiber, the probe wave is analyzed using an optical spectrum analyzer (6315A, Yokogawa Ltd.). The pulse spectrum was taken with a compact CCD spectrometer.

The use of a continuous probe wave removes the technical difficulty of synchronizing two pulses. As a drawback, however, only the small fraction *η*_{int} of the probe light interacts with the pulse in the finite length of fiber *L*. Thus the probe is weak and the soliton recoil is negligible. Hence the conversion efficiency *R* (Eq. (4)) is reduced by *η*_{int} to the total efficiency *η*:

*ν*is the pulse repetition rate and

*δ*

*n*

*is the difference in group indices of pulse and probe wave.*

_{g}In Fig. 2 the contributions *R* (blue) and *η*_{int} (green) to the total efficiency *η* (red) are displayed according to the model, in which the experimental parameters were used (identical to Fig. 5). The larger the detuning of the probe from the group velocity of the soliton, the more light collides with the soliton (green). For small detunings there is negligible tunnelling and the probe is nearly perfectly reflected. At a detuning of approximately 12 THz, tunnelling sets in and rapidly increases until the probe light is no longer reflected at about 25 THz (blue). According to Fig. 5, Ω*T*_{0} ≈ 2, i.e. frequency shifting is expected up to a detuning of twice the soliton bandwidth. The resulting efficiency is displayed in red. The curves are slightly asymmetric because of the higher order dispersion in the fiber.

Figure 3 displays two example spectra of the frequency shifted probe light. The probe is (in the moving frame) reflected off the soliton, whose group velocity is set by its center wavelength in the dispersive fiber. By tuning the soliton wavelength we can realize situations where the probe wave is faster than the soliton and overtakes it and vice versa. These spectra correspond to a ∼ +13 nm /∼ −12 nm spectral shift of the probe. The spectral width and structure depends on the detailed pulse shape, which is affected by Raman interactions and higher order dispersion [25].

We repeated this experiment with various pulse wavelengths to map out the frequency shifting as a function of detuning. Figure 4 shows the measured shifted wavelength of the probe wave as a function of soliton wavelength, i.e. different soliton velocities. We also show the wavelength that we expect the probe wave to shift to, determined by the fiber dispersion (inset) and the conservation of *ω*′, including higher order dispersion, as a red solid line. The figure shows that the center of the probe spectrum shifts as expected according to the fiber dispersion. Thus the soliton propagates approximately with constant velocity, unaffected by higher order effects. The efficiency of reflection and frequency shifting depends on the interaction efficiency *η*_{int} and the soliton reflectivity *R*. The former is a dispersive property, while the latter depends on the refractive index barrier. To find the total observed efficiency, we integrated the shifted spectra to find the shifted power and normalized to the power at the initial probe wavelength of 532nm, measured separately. For the spectra of Fig. 3, *η* = 1.210^{−4} (*η* = 1.110^{−4}), which results in an observed reflectivity *R* = 96.3% (*R* = 58.3%) for this particular blue (red) shifting.

Results of the efficiency measurements are presented in Fig. 5. Different frequency shifts of the probe wave were observed for different soliton wavelengths. The frequency shift was inferred from the pulse wavelength and the fiber dispersion (solid line in Fig. 4). Note that the total efficiency is generally low and vanishes for zero detuning, because only a small fraction of the probe wave interacts (relation 6). The inferred reflectivity *R*, however, is significant (>10%) for frequency shifts up to ±20 THz (±19 nm).

We also show the total efficiency *η*. The only adjustable parameter in these curves is *r* = 1.7, the effective cross-phase modulation strength. Note that *r* also includes effects such as coupling of light to higher order fiber modes and deviations from the amplitude required for a perfect *N* = 1 soliton. The agreement with the experimental data is very good and shows that the reflectivity of the soliton can be described by our tunnelling model. We also inserted a copy of a typical soliton input spectrum into the figure for comparison. According to Eq. (5), 90% reflectivity occurs at a detuning Ω*T*_{0} = 2.0. The frequency shift thus can exceed the spectral width of the soliton considerably before eventually decreasing. In our experiment, the reflectivity does not decrease for a frequency shift about two times the soliton spectral width, beyond which it rapidly decreases. For higher values of the barrier *B* an even wider range of frequency shifts is expected. Note that in [23] the barrier height was *B* = 0.85, a regime described by the other branch in Eq. (4). Our experiment is thus performed in a novel regime with a 25 times stronger barrier.

Let us consider another interaction of a pulse with a dispersive wave in this framework: fiber optical Cherenkov radiation [19,35–38]. Cherenkov radiation is described as resonant coupling of soliton and dispersive waves [17,18,30]. The condition for this coupling is that the dispersive wave shares the frequency in the frame moving at the velocity of the pulse (*ω*′_{disp}) with the pulse (*ω*′* _{s}*), i.e.

*ω*′

_{disp}=

*ω*′

*. Using the previous analysis this corresponds to mode conversion of a soliton photon at its own optical event horizon. This conversion is inherently efficient, because the photons of a soliton are perfectly group velocity matched to the soliton. Again, the frequency shift is not limited by the spectral width of the soliton and can easily exceed an octave [36].*

_{s}## 4. Conclusion

In conclusion, we observed frequency up and downshifting of dispersive waves at stationary solitons. This is an important case, because of possible applications in ultrashort pulse switching and manipulation. For example, the frequency shifts can be reversed by another collision with a second soliton at the same wavelength. In effect this would lead to an overall all-optical delay of the dispersive wave determined by the separation of the two solitons. In addition, the shifted spectra are conjugated, leading effectively to dispersion compensation between the two collisions. Hence, an incoming series of waves would be reversed in their order by a (single) collision.

In the experiment we investigated for the first time the tunnelling of waves through a group velocity horizon in fibers. The frequency shifting of dispersive waves interacting with pulses via cross-phase modulation can exceed the spectral width of the pulses without loss of efficiency. We demonstrated an experiment with > 90% efficiency over twice the pulse spectral width for frequency red and blueshifting.

The interaction can be described precisely and efficiently by a fully analytical model if higher order dispersion does not considerably affect the soliton propagation.

We also offer an explanation of fiber-optical Cherenkov radiation (FOCR) through the novel interaction with a group velocity horizon. Key features such as efficiency and bandwidth of the FOCR are recognized to resemble each other. Further theoretical work could further unify the two effects.

## Acknowledgments

The authors are thankful for discussions with U. Leonhardt and Th. Philbin. The optical spectrum analyzer was kindly provided by the Photonics innovation center at St. Andrews through C. Rae. Support from the European Commission is acknowledged through an Erasmus Mundus Fellowship.

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