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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 5 — Feb. 27, 2012
  • pp: 5538–5546
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Efficient frequency shifting of dispersive waves at solitons

Amol Choudhary and Friedrich König  »View Author Affiliations


Optics Express, Vol. 20, Issue 5, pp. 5538-5546 (2012)
http://dx.doi.org/10.1364/OE.20.005538


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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 OSA

1. Introduction

Since the prediction and demonstration of fiber-optical solitons three decades ago [1

1. A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion,” Appl. Phys. Lett. 23, 142–144 (1973). [CrossRef]

, 2

2. L. F. Mollenauer, R. H. Stolen, and J. G. Gordon, “Experimental observation of picosecond pulse narrowing and solitons in optical fibers,” Phys. Rev. Lett. 45, 1095–1098 (1980). [CrossRef]

], 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

3. K. Kitayama, Y. Kimura, and S. Seikai, “Fiber-optic logic gate,” Appl. Phys. Lett. 46, 317–319 (1985). [CrossRef]

9

9. M. Nazarathy, Z. Zalevsky, A. Rudnitsky, B. Larom, A. Nevet, M. Orenstein, and B. Fischer, “All-optical linear reconfigurable logic with nonlinear phase erasure,” J. Opt. Soc. Am. A 26, A21–A39 (2009). [CrossRef]

]. The interaction of dispersive waves with solitons via the optical Kerr effect is called cross phase modulation (XPM) [10

10. S. Akhmanov, A. Sukhorukov, and A. Chirkin, “Nonstationary phenomena and spacetime analogy in nonlinear optics,” Sov. Phys. JETP 28, 748–757 (1969).

, 11

11. M. N. Islam, L. F. Mollenauer, R. H. Stolen, J. R. Simpson, and H. T. Shang, “Cross-phase modulation in optical fibers,” Opt. Lett. 12, 625–627 (1987). [CrossRef] [PubMed]

], which has extensively been studied in the context of soliton communication systems [12

12. J. P. Gordon, “Dispersive perturbations of solitons of the nonlinear Schrodinger-equation,” J. Opt. Soc. Am. B 9, 91–97 (1992). [CrossRef]

]. 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

13. V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Sov. Phys. JETP 34, 62–69 (1972).

,14

14. J. R. Taylor, Optical Solitons Theory and Experiment (Cambridge Press, 2005).

]. 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

12. J. P. Gordon, “Dispersive perturbations of solitons of the nonlinear Schrodinger-equation,” J. Opt. Soc. Am. B 9, 91–97 (1992). [CrossRef]

].

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

15. J. C. Knight, T. A. Birks, P. S. Russell, and D. M. Atkin, “All-silica single-mode optical fiber with photonic crystal cladding,” Opt. Lett. 21, 1547–1549 (1996). [CrossRef] [PubMed]

, 16

16. P. Russell, “Photonic crystal fibers,” Science 299, 358–362 (2003). [CrossRef] [PubMed]

] 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

17. P. K. A. Wai, C. R. Menyuk, Y. C. Lee, and H. H. Chen, “Nonlinear pulse propagation in the neighborhood of the zero-dispersion wavelength of monomode optical fibers,” Opt. Lett. 11, 464–466 (1986). [CrossRef] [PubMed]

19

19. L. Tartara, I. Cristiani, and V. Degiorgio, “Blue light and infrared continuum generation by soliton fission in a microstructured fiber,” Appl. Phys. B 77, 307–311 (2003). [CrossRef]

], wave trapping [20

20. N. Nishizawa and T. Goto, “Characteristics of pulse trapping by ultrashort soliton pulse in optical fibers across zerodispersion wavelength,” Opt. Express 10, 1151–1159 (2002). [PubMed]

22

22. A. Efimov, A. Yulin, D. Skryabin, J. C. Knight, N. Joly, F. Omenetto, A. J. Taylor, and P. Russell, “Interaction of an optical soliton with a dispersive wave,” Phys. Rev. Lett. 95, 213902 (2005). [CrossRef] [PubMed]

], and frequency shifts at artificial event horizons [23

23. T. G. Philbin, C. Kuklewicz, S. Robertson, S. Hill, F. König, and U. Leonhardt, “Fiber-optical analog of the event horizon,” Science 319, 1367–1370 (2008). [CrossRef] [PubMed]

] 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

24. S. Hill, C. E. Kuklewicz, U. Leonhardt, and F. König, “Evolution of light trapped by a soliton in a microstructured fiber,” Opt. Express 1713588–13600 (2009). [CrossRef] [PubMed]

].

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

25. S. Robertson and U. Leonhardt, “Frequency shifting at fiber-optical event horizons: the effect of Raman deceleration,” Phys. Rev. A 81, 063835 (2010). [CrossRef]

]. 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

23. T. G. Philbin, C. Kuklewicz, S. Robertson, S. Hill, F. König, and U. Leonhardt, “Fiber-optical analog of the event horizon,” Science 319, 1367–1370 (2008). [CrossRef] [PubMed]

, 26

26. W. G. Unruh, “Experimental black-hole evaporation,” Phys. Rev. Lett. 46, 1351–1353 (1981). [CrossRef]

]. As a result, this leads to a permanent frequency shift of the dispersive wave [23

23. T. G. Philbin, C. Kuklewicz, S. Robertson, S. Hill, F. König, and U. Leonhardt, “Fiber-optical analog of the event horizon,” Science 319, 1367–1370 (2008). [CrossRef] [PubMed]

]. 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

23. T. G. Philbin, C. Kuklewicz, S. Robertson, S. Hill, F. König, and U. Leonhardt, “Fiber-optical analog of the event horizon,” Science 319, 1367–1370 (2008). [CrossRef] [PubMed]

,26

26. W. G. Unruh, “Experimental black-hole evaporation,” Phys. Rev. Lett. 46, 1351–1353 (1981). [CrossRef]

28

28. S. M. Hawking, “Particle creation by black-holes,” Commun. Math. Phys. 43, 199–220 (1975). [CrossRef]

]. 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

8. A. Demircan, Sh. Amiranashvili, and G. Steinmeyer, “Controlling light by light with an optical event horizon,” Phys. Rev. Lett. 106, 163901 (2011). [CrossRef] [PubMed]

, 22

22. A. Efimov, A. Yulin, D. Skryabin, J. C. Knight, N. Joly, F. Omenetto, A. J. Taylor, and P. Russell, “Interaction of an optical soliton with a dispersive wave,” Phys. Rev. Lett. 95, 213902 (2005). [CrossRef] [PubMed]

]. In the first demonstration of fiber optical event horizons, frequency shifts were small and only blue shifting was demonstrated [23

23. T. G. Philbin, C. Kuklewicz, S. Robertson, S. Hill, F. König, and U. Leonhardt, “Fiber-optical analog of the event horizon,” Science 319, 1367–1370 (2008). [CrossRef] [PubMed]

].

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

29. G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, 2006).

]. Here we consider copropagation of two optical fields: the soliton and the dispersive ‘probe’ wave. Their slowly varying amplitudes are given as A1(z,t) for the probe wave and A2(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

30. D. V. Skryabin and A. V. Yulin, “Theory of generation of new frequencies by mixing of solitons and dispersive waves in optical fibers,” Phys. Rev. E 72, 016619 (2005). [CrossRef]

]. With these approximations the NLSE becomes (see e.g. [24

24. S. Hill, C. E. Kuklewicz, U. Leonhardt, and F. König, “Evolution of light trapped by a soliton in a microstructured fiber,” Opt. Express 1713588–13600 (2009). [CrossRef] [PubMed]

, 29

29. G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, 2006).

, 31

31. V. E. Lobanov and A. P. Sukhorukov, “Total reflection, frequency, and velocity tuning in optical pulse collision in nonlinear dispersive media,” Phys. Rev. A , 82, 033809 (2010). [CrossRef]

, 32

32. N. N. Rosanov, N. V. Vysotina, and A. N. Shatsev, “Forward light reflection from a moving inhomogeneity,” JETP Lett. 93, 308–312 (2011). [CrossRef]

]):
A1z+β1A1t+i2β22A1t2=iγ(|A1|2+r|A2|2)A1
(1)
Here β1 and β2 are the first and second derivative of the propagation constant β(ω) at a frequency ωm of the same group velocity u=β11 as the soliton and close to that of the field A1 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 = tz/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 |A2(τ)|2 = P0 sech2(τ/T0), where P0 and T0 are the soliton amplitude and length. We arrive at:
2A1τ22iβ1β2A1ζ2rγP0β2sech2(τ/T0)A1=0.
(2)
We introduce the frequency of a wave in the reference frame moving with the group velocity of the soliton, ω′ = ωβu = ω{1 – n(ω,I(τ))/(β1c)}. The refractive index n has a nonlinear contribution from the soliton intensity I(τ) due to the Kerr effect. Finally, we use the ansatz A1 = 𝒜 exp((ζ,τ)+iωmζ), where ϕ is the optical phase and ωm is the moving frame frequency at the group velocity matched lab frequency ωm. The nonlinear Schrödinger equation becomes:
2A1τ22β2[β1Ω+rγP0sech2(τ/T0)]A1=0,
(3)
where we define Ω′ = ω′ – ωm. 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 β2 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

23. T. G. Philbin, C. Kuklewicz, S. Robertson, S. Hill, F. König, and U. Leonhardt, “Fiber-optical analog of the event horizon,” Science 319, 1367–1370 (2008). [CrossRef] [PubMed]

]. 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

33. L. D. Landau and E. M. Lifshitz, Quantum Mechanics3, (Butterworth-Heinemann, 1981).

]. In the variables of fiber optics these coefficients are:
T=11+ξR=ξ1+ξ,ξ={cos2(π/21B)sinh2(πΩT0):B<1cosh2(π/2B1)sinh2(πΩT0):B1
(4)
where Ω = ωωm 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 ΩT0, the ratio of detuning to soliton bandwidth, and B=8rγP0T02/β2, the normalized barrier height, only. B > 0 (B < 0) implies a potential barrier (pit). For B = 1 the reflectivity has the sech2-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. rγP0) or the dispersion length ( T02/|β2|) 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,ΩT0 > 1):
B=1+[2πln3+2ΩT0]2.
(5)
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 β2 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.

Fig. 1 Probe waves reflecting off a soliton: Contours of constant reflectivity (10% increments) as a function of the Barrier height B and the detuning ΩT0. High reflectivity is not limited to small detunings (ΩT0 ≪ 1).

3. Tunnelling experiment

To test this model we set up an experiment. We use fundamental solitons ( N2=1=γP0T02/|β2|) such that B=16|β2s|β2=16Dsλs2|D|λ2, where D(Ds) and λ(λs) 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 λm. As is indicated in the inset in Fig. 4, the integral over the dispersion curve, which is the group delay, vanishes between λm and λs. Hence, for the photonic crystal fiber (pcf) we used (NL-1.5–670, NKT Photonics, Inc.), we obtain B = 22.4 with dispersions of −256ps/(nm km) and 144ps/(nm km) for probe (λp = 532nm) and soliton (λs ∼ 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

34. Details of this technique will be published elsewhere.

].

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 ηint 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 ηint to the total efficiency η:
η=Rηint,ηint=LcνδngLνβ2Ω,
(6)
where ν is the pulse repetition rate and δng is the difference in group indices of pulse and probe wave.

In Fig. 2 the contributions R (blue) and ηint (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, ΩT0 ≈ 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.

Fig. 2 Contributions to the total mode conversion efficiency as a function of the frequency shift of the probe wave (Eq. (4) and (6)). Dashed: Reflectivity R of the soliton (Eq. (4)). Green: Fraction ηint of the probe light colliding with the soliton. Red: total efficiency η = Rηint.

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

25. S. Robertson and U. Leonhardt, “Frequency shifting at fiber-optical event horizons: the effect of Raman deceleration,” Phys. Rev. A 81, 063835 (2010). [CrossRef]

].

Fig. 3 Two spectra of the blue and the red shifted probe light, initially at λ = 532nm (notch-filtered in output, grey area). The soliton was tuned to 846 and 825nm, respectively. Spectral shifts of −12nm (blue) and +13nm (red) are observed. The expected reflectivities are 98% and 55%, respectively, according to Eq. (4).
Fig. 4 Location of shifted probe spectra for different soliton wavelengths. The solid line is the prediction from the dispersion curve as shown in the inset. Wavelengths between which the integral (shaded area) of the dispersion parameter D vanishes are group velocity matched.
Fig. 5 Measurement of mode conversion efficiency as a function of total frequency shift of probe wave. Red: tunnelling model η (Eq. (6)). Efficient conversion (R > 90%) occurs over more than ±12 THz and is described by the tunnelling model. The grey curve, for comparison, shows a typical input pulse spectrum.

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 ηint 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 ΩT0 = 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

23. T. G. Philbin, C. Kuklewicz, S. Robertson, S. Hill, F. König, and U. Leonhardt, “Fiber-optical analog of the event horizon,” Science 319, 1367–1370 (2008). [CrossRef] [PubMed]

] 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.

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.

References and links

1.

A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion,” Appl. Phys. Lett. 23, 142–144 (1973). [CrossRef]

2.

L. F. Mollenauer, R. H. Stolen, and J. G. Gordon, “Experimental observation of picosecond pulse narrowing and solitons in optical fibers,” Phys. Rev. Lett. 45, 1095–1098 (1980). [CrossRef]

3.

K. Kitayama, Y. Kimura, and S. Seikai, “Fiber-optic logic gate,” Appl. Phys. Lett. 46, 317–319 (1985). [CrossRef]

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A. Demircan, Sh. Amiranashvili, and G. Steinmeyer, “Controlling light by light with an optical event horizon,” Phys. Rev. Lett. 106, 163901 (2011). [CrossRef] [PubMed]

9.

M. Nazarathy, Z. Zalevsky, A. Rudnitsky, B. Larom, A. Nevet, M. Orenstein, and B. Fischer, “All-optical linear reconfigurable logic with nonlinear phase erasure,” J. Opt. Soc. Am. A 26, A21–A39 (2009). [CrossRef]

10.

S. Akhmanov, A. Sukhorukov, and A. Chirkin, “Nonstationary phenomena and spacetime analogy in nonlinear optics,” Sov. Phys. JETP 28, 748–757 (1969).

11.

M. N. Islam, L. F. Mollenauer, R. H. Stolen, J. R. Simpson, and H. T. Shang, “Cross-phase modulation in optical fibers,” Opt. Lett. 12, 625–627 (1987). [CrossRef] [PubMed]

12.

J. P. Gordon, “Dispersive perturbations of solitons of the nonlinear Schrodinger-equation,” J. Opt. Soc. Am. B 9, 91–97 (1992). [CrossRef]

13.

V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Sov. Phys. JETP 34, 62–69 (1972).

14.

J. R. Taylor, Optical Solitons Theory and Experiment (Cambridge Press, 2005).

15.

J. C. Knight, T. A. Birks, P. S. Russell, and D. M. Atkin, “All-silica single-mode optical fiber with photonic crystal cladding,” Opt. Lett. 21, 1547–1549 (1996). [CrossRef] [PubMed]

16.

P. Russell, “Photonic crystal fibers,” Science 299, 358–362 (2003). [CrossRef] [PubMed]

17.

P. K. A. Wai, C. R. Menyuk, Y. C. Lee, and H. H. Chen, “Nonlinear pulse propagation in the neighborhood of the zero-dispersion wavelength of monomode optical fibers,” Opt. Lett. 11, 464–466 (1986). [CrossRef] [PubMed]

18.

N. Akhmediev and M. Karlsson, “Cherenkov radiation emitted by solitons in optical fibers,” Phys. Rev. A 51, 2602–2607 (1995). [CrossRef] [PubMed]

19.

L. Tartara, I. Cristiani, and V. Degiorgio, “Blue light and infrared continuum generation by soliton fission in a microstructured fiber,” Appl. Phys. B 77, 307–311 (2003). [CrossRef]

20.

N. Nishizawa and T. Goto, “Characteristics of pulse trapping by ultrashort soliton pulse in optical fibers across zerodispersion wavelength,” Opt. Express 10, 1151–1159 (2002). [PubMed]

21.

A. V. Gorbach and D. V. Skryabin, “Light trapping in gravity-like potentials and expansion of supercontinuum spectra in photonic-crystal fibres,” Nat. Photonics 1, 653–656 (2007). [CrossRef]

22.

A. Efimov, A. Yulin, D. Skryabin, J. C. Knight, N. Joly, F. Omenetto, A. J. Taylor, and P. Russell, “Interaction of an optical soliton with a dispersive wave,” Phys. Rev. Lett. 95, 213902 (2005). [CrossRef] [PubMed]

23.

T. G. Philbin, C. Kuklewicz, S. Robertson, S. Hill, F. König, and U. Leonhardt, “Fiber-optical analog of the event horizon,” Science 319, 1367–1370 (2008). [CrossRef] [PubMed]

24.

S. Hill, C. E. Kuklewicz, U. Leonhardt, and F. König, “Evolution of light trapped by a soliton in a microstructured fiber,” Opt. Express 1713588–13600 (2009). [CrossRef] [PubMed]

25.

S. Robertson and U. Leonhardt, “Frequency shifting at fiber-optical event horizons: the effect of Raman deceleration,” Phys. Rev. A 81, 063835 (2010). [CrossRef]

26.

W. G. Unruh, “Experimental black-hole evaporation,” Phys. Rev. Lett. 46, 1351–1353 (1981). [CrossRef]

27.

S. M. Hawking, “Black-hole explosions,” Nature 248, 30–31 (1974). [CrossRef]

28.

S. M. Hawking, “Particle creation by black-holes,” Commun. Math. Phys. 43, 199–220 (1975). [CrossRef]

29.

G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, 2006).

30.

D. V. Skryabin and A. V. Yulin, “Theory of generation of new frequencies by mixing of solitons and dispersive waves in optical fibers,” Phys. Rev. E 72, 016619 (2005). [CrossRef]

31.

V. E. Lobanov and A. P. Sukhorukov, “Total reflection, frequency, and velocity tuning in optical pulse collision in nonlinear dispersive media,” Phys. Rev. A , 82, 033809 (2010). [CrossRef]

32.

N. N. Rosanov, N. V. Vysotina, and A. N. Shatsev, “Forward light reflection from a moving inhomogeneity,” JETP Lett. 93, 308–312 (2011). [CrossRef]

33.

L. D. Landau and E. M. Lifshitz, Quantum Mechanics3, (Butterworth-Heinemann, 1981).

34.

Details of this technique will be published elsewhere.

35.

H. Tu and S. A. Boppart, “Optical frequency up-conversion by supercontinuum-free widely-tunable fiber-optic Cherenkov radiation,” Opt. Express 179858–9872 (2009). [CrossRef] [PubMed]

36.

H. Tu and S. A. Boppart, “Ultraviolet-visible non-supercontinuum ultrafast source enabled by switching single silicon strand-like photonic crystal fibers,” Opt. Express 1717983–17988 (2009). [CrossRef] [PubMed]

37.

G. Q. Chang, L. J. Chen, and F. X. Kärtner, “Highly efficient Cherenkov radiation in photonic crystal fibers for broadband visible wavelength generation,” Opt.Lett. 35, 2361–2363, (2010). [CrossRef] [PubMed]

38.

G. Q. Chang, L. J. Chen, and F. X. Kärtner, “Fiber-optic Cherenkov radiation in the few-cycle regime,” Opt. Express 19, 6635–6647 (2011). [CrossRef] [PubMed]

OCIS Codes
(060.7140) Fiber optics and optical communications : Ultrafast processes in fibers
(190.4370) Nonlinear optics : Nonlinear optics, fibers
(190.5530) Nonlinear optics : Pulse propagation and temporal solitons

ToC Category:
Nonlinear Optics

History
Original Manuscript: September 26, 2011
Revised Manuscript: December 30, 2011
Manuscript Accepted: February 5, 2012
Published: February 22, 2012

Citation
Amol Choudhary and Friedrich König, "Efficient frequency shifting of dispersive waves at solitons," Opt. Express 20, 5538-5546 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-5-5538


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References

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  13. V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Sov. Phys. JETP34, 62–69 (1972).
  14. J. R. Taylor, Optical Solitons Theory and Experiment (Cambridge Press, 2005).
  15. J. C. Knight, T. A. Birks, P. S. Russell, and D. M. Atkin, “All-silica single-mode optical fiber with photonic crystal cladding,” Opt. Lett.21, 1547–1549 (1996). [CrossRef] [PubMed]
  16. P. Russell, “Photonic crystal fibers,” Science299, 358–362 (2003). [CrossRef] [PubMed]
  17. P. K. A. Wai, C. R. Menyuk, Y. C. Lee, and H. H. Chen, “Nonlinear pulse propagation in the neighborhood of the zero-dispersion wavelength of monomode optical fibers,” Opt. Lett.11, 464–466 (1986). [CrossRef] [PubMed]
  18. N. Akhmediev and M. Karlsson, “Cherenkov radiation emitted by solitons in optical fibers,” Phys. Rev. A51, 2602–2607 (1995). [CrossRef] [PubMed]
  19. L. Tartara, I. Cristiani, and V. Degiorgio, “Blue light and infrared continuum generation by soliton fission in a microstructured fiber,” Appl. Phys. B77, 307–311 (2003). [CrossRef]
  20. N. Nishizawa and T. Goto, “Characteristics of pulse trapping by ultrashort soliton pulse in optical fibers across zerodispersion wavelength,” Opt. Express10, 1151–1159 (2002). [PubMed]
  21. A. V. Gorbach and D. V. Skryabin, “Light trapping in gravity-like potentials and expansion of supercontinuum spectra in photonic-crystal fibres,” Nat. Photonics1, 653–656 (2007). [CrossRef]
  22. A. Efimov, A. Yulin, D. Skryabin, J. C. Knight, N. Joly, F. Omenetto, A. J. Taylor, and P. Russell, “Interaction of an optical soliton with a dispersive wave,” Phys. Rev. Lett.95, 213902 (2005). [CrossRef] [PubMed]
  23. T. G. Philbin, C. Kuklewicz, S. Robertson, S. Hill, F. König, and U. Leonhardt, “Fiber-optical analog of the event horizon,” Science319, 1367–1370 (2008). [CrossRef] [PubMed]
  24. S. Hill, C. E. Kuklewicz, U. Leonhardt, and F. König, “Evolution of light trapped by a soliton in a microstructured fiber,” Opt. Express1713588–13600 (2009). [CrossRef] [PubMed]
  25. S. Robertson and U. Leonhardt, “Frequency shifting at fiber-optical event horizons: the effect of Raman deceleration,” Phys. Rev. A81, 063835 (2010). [CrossRef]
  26. W. G. Unruh, “Experimental black-hole evaporation,” Phys. Rev. Lett.46, 1351–1353 (1981). [CrossRef]
  27. S. M. Hawking, “Black-hole explosions,” Nature248, 30–31 (1974). [CrossRef]
  28. S. M. Hawking, “Particle creation by black-holes,” Commun. Math. Phys.43, 199–220 (1975). [CrossRef]
  29. G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, 2006).
  30. D. V. Skryabin and A. V. Yulin, “Theory of generation of new frequencies by mixing of solitons and dispersive waves in optical fibers,” Phys. Rev. E72, 016619 (2005). [CrossRef]
  31. V. E. Lobanov and A. P. Sukhorukov, “Total reflection, frequency, and velocity tuning in optical pulse collision in nonlinear dispersive media,” Phys. Rev. A, 82, 033809 (2010). [CrossRef]
  32. N. N. Rosanov, N. V. Vysotina, and A. N. Shatsev, “Forward light reflection from a moving inhomogeneity,” JETP Lett.93, 308–312 (2011). [CrossRef]
  33. L. D. Landau and E. M. Lifshitz, Quantum Mechanics3, (Butterworth-Heinemann, 1981).
  34. Details of this technique will be published elsewhere.
  35. H. Tu and S. A. Boppart, “Optical frequency up-conversion by supercontinuum-free widely-tunable fiber-optic Cherenkov radiation,” Opt. Express179858–9872 (2009). [CrossRef] [PubMed]
  36. H. Tu and S. A. Boppart, “Ultraviolet-visible non-supercontinuum ultrafast source enabled by switching single silicon strand-like photonic crystal fibers,” Opt. Express1717983–17988 (2009). [CrossRef] [PubMed]
  37. G. Q. Chang, L. J. Chen, and F. X. Kärtner, “Highly efficient Cherenkov radiation in photonic crystal fibers for broadband visible wavelength generation,” Opt.Lett.35, 2361–2363, (2010). [CrossRef] [PubMed]
  38. G. Q. Chang, L. J. Chen, and F. X. Kärtner, “Fiber-optic Cherenkov radiation in the few-cycle regime,” Opt. Express19, 6635–6647 (2011). [CrossRef] [PubMed]

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