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REPORTS Soliton Self-Frequency Shift Cancellation in Photonic Crystal Fibers D. V. Skryabin,* F. Luan, J. C. Knight, P. St. J. Russell We report the cancellation of the soliton self-frequency shift in a silica-core photonic crystal fiber with a negative dispersion slope. Numerical and experimental results show that stabilization of the soliton wavelength is accompanied by exponential amplification of the red-shifted Cherenkov radiation emitted by the soliton. The spectral recoil from the radiation acts on the soliton to compensate for the Raman frequency shift. This phenomenon may find applications in the development of a family of optical parametric amplifiers. Solitary waves, or solitons, are self-localized regions of energy or matter that occur in a large variety of fundamental processes in many natural and artificially created nonlinear systems, such as the sea surface, nerve fibers, superconducting transmission lines, optical fibers, elementary particles, and outer space (1). Optical solitons are self-localized pulses or beams of light with temporal or spatial dispersion suppressed by the action of a nonlinear medium in which they propagate (2, 3). Soon after the experimental observations of temporal solitons in dispersive optical fibers (4 ), it was found that fiber solitons can exhibit a strong self-frequency shift (5, 6 ). That is, a short (⬍1 ps) soliton propagating in a Raman-active medium like silica is continuously red-shifted because the lowfrequency end of the soliton spectrum experiences Raman gain at the expense of the high-frequency end. Although this has been considered an immutable feature of subpicosecond soliton propagation in silica optical fibers, we show theoretically and experimentally that the soliton self-frequency shift can be canceled in a silica-core fiber with a negative dispersion slope. Cancellation of the frequency shift arises because of the exponential amplification and subsequent saturation of the new radiation band red-shifted with respect to the soliton and emitted by the soliton itself through the Cherenkov mechanism. Cherenkov radiation (7–9) by a charged particle propagating in a dispersive medium occurs when the particle travels at speeds faster than the phase velocity of light (10). Equivalently, one can say that the Cherenkov effect takes place if the wavenumber of the wave created by the particle becomes smaller than the wavenumber of the dispersive wave Department of Physics, University of Bath, Bath BA2 7AY, UK. *To whom correspondence should be addressed. Email: [email protected] for the same frequency. Then, the wavenumber matching condition is satisfied and radiation is emitted under some angle to the direction of the particle motion (9). Because propagation in fibers is restricted to one spatial direction, time is the only other relevant coordinate. The angle in the spacetime plane, which any dispersive wave makes with the soliton trajectory, characterizes the frequency of the wave relative to the soliton frequency. One of the most important properties of idealized optical solitons is their robustness against perturbations (11). The soliton locally modifies the dispersion characteristics so that there are no dispersive waves with real frequencies having wavenumbers close to the soliton ones. This prohibits energy leakage from the soliton to the waves and back. However, when the group velocity dispersion (GVD) of the fiber changes significantly over the spectral bandwidth of the pulse, the soliton loses its immunity to dispersive waves, because a critical frequency or set of frequencies can be found, such that the wavenumbers of the waves are matched with the solitonic wavenumbers (12). This leads to Cherenkov radiation, and the soliton is no longer “ideal,” but becomes a quasi-soliton coupled to dispersive waves (13). This effect gets more pronounced for shorter pulses and steeper GVD slopes. In what follows GVD will be denoted as 2(), defined as the second derivative of wavenumber k with respect to angular frequency . Recently developed photonic-crystal fibers (PCFs) (14–16) give rise to previously unattainable optical characteristics. One of the most prominent features of PCFs is the range of unusual dispersion curves that have been demonstrated (17, 18), which have a profound effect on the nonlinear optics within the fiber (18–23). To reach the balance between dispersion and nonlinearity required for the solitonic regime, one needs to operate at a frequency where the GVD is anomalous, i.e., 2() ⬍ 0. We used a PCF with anom- alous GVD over a finite band of frequencies, rather than a semi-infinite band as in conventional fibers (Fig. 1A). The dispersion in the standard telecom fiber (SMF28) is anomalous at wavelengths longer than 1.3 m, whereas the PCF dispersion is anomalous between 0.6 and 1.3 m. We have calculated the dependence of the Cherenkov resonances on the soliton central frequency from the wavenumber matching conditions (12, 21) (Fig. 1B). There are two branches of Cherenkov radiation. One branch is blue-shifted and the other one is red-shifted with respect to the soliton frequency, whereas for the conventional fiber there is only a blue-shifted branch. The blueshifted radiation arises because there is a region of where the slope of 2() is positive and 2() changes its sign from negative to positive. Similarly, the existence of the frequency range where 2() changes from positive to negative, i.e., its slope is negative, is a crucial condition for observation of the red-shifted radiation. Such a frequency range is absent in conventional fibers and in PCFs with larger cores, and its presence in our PCF is a direct result of the very small core diameter of 1.2 m. It has been shown theoretically (12) that solitons emitting Cherenkov radiation lose energy slowly (i.e., nonexponentially) with increasing propagation distance z, transferring it to the resonant dispersive wave. To conserve the overall energy of the photons, the carrier soliton frequency gets shifted in the spectral direction opposite to that of the radiation. This is the so-called spectral recoil effect (2, 12). It is important to note for the following that the analysis of (12) disregards the soliton self-frequency shift due to the Raman effect, which is a process in which photons cascade their energy to photons at a larger wavelength through scattering from optical phonons. However, the Raman response of the silica accounts for about 20% of the overall nonlinear response (2) and is a strong effect in the solitonic propagation regime, so it cannot be neglected. The frequency of the Cherenkov radiation approaches the central soliton frequency close to the zero GVD wavelengths (Fig. 1B). Energy exchange between the soliton and the resonant dispersive wave is expected to reach its maximum in these regions, where energy is fed into the wave from the most intense central part of the soliton spectrum. This is because the amplitude of the emitted radiation is primarily determined by the spectral amplitude of the soliton at the radiation frequency r. The spectral amplitude at the frequency r for an ideal soliton with the central frequency s is proportional to 1/{exp[⫺(r ⫺ s)/ 2)] ⫹ exp[(r ⫺ s)/2]}, where is the soliton duration (24). Thus, for values of r approaching s, the intensity of the emitted wave increases exponentially. www.sciencemag.org SCIENCE VOL 301 19 SEPTEMBER 2003 1705 REPORTS The Raman scattering generates a red shift in the soliton carrier frequency, which is directly proportional to the propagation distance z (2). Therefore, as the soliton approaches the redshifted zero GVD point with a negative GVD slope, the intensity of the red-shifted branch of the Cherenkov radiation is expected to increase exponentially in z. The exponential growth of the radiation should saturate, however, as the soliton recoils against the radiation toward the blue side of the spectrum. Thus, there exists the possibility of a balance between the red Raman self-frequency shift and the blue recoil on the soliton from the red-shifted Cherenkov radiation. This balance will then produce frequencylocked solitary pulses with a growing tail of Cherenkov radiation. To verify the above considerations we carried out a series of numerical experiments modeling propagation of optical solitons in the PCF. We used the generalized nonlinear Schrödinger (NLS) equation with the linear dispersion fitted to the experimentally measured one (Fig. 1A), and a nonlinear response function that included both the instantaneous electronic and noninstantaneous Raman responses (18, 22, 23). We initialized this model with a single-soliton solution (2) of the ideal NLS equation having peak power ⬵ 215 W, duration 53 fs, and carrier frequency 2 ⫻ 250 THz. For this frequency, 2 ⬵ ⫺70 ps2/km and the derivative of 2 is ⫺0.25 ps3/km, and the soliton is expected to emit red-shifted radiation with an intensity much stronger than the intensity of the blue- shifted radiation, which has a larger detuning from the soliton frequency. Thus, the overall recoil effect is expected to push the soliton toward the blue side of the spectrum. The results of our modeling (Fig. 2, A and B) show the evolution of the absolute value of the electric field in (t ⫺ z/v, z) plane and the logarithm of the absolute value of the Fourier transform of the same field in the (, z) plane. Here t, z, and v represent time, coordinate along the fiber, and soliton group velocity, respectively. Up to ⬵2.5 m the pulse evolves as expected of Raman-shifted solitons (2). However, for z ⬎ 2.5 m, the pulse dynamics in the spectral and temporal domains undergo a dramatic change. The self-frequency shift of the soliton is suddenly Fig. 1. (A) GVD plots for the telecommunication fiber (SMF 28) and PCF used in our experiments. (Inset) Scanning electron micrograph of the PCF transverse section. The core diameter of the PCF is 1.2 m. (B) Dependence of the frequency of the Cherenkov resonances from the soliton frequencies for the telecom fiber and PCF. Two vertical dashed lines mark the zero GVD points in the PCF. The diagonal line marks the boundary where the radiation and soliton frequencies coincide. The radiation branches above/below this line are, respectively, blue-/red-shifted relative to the soliton carrier frequency. The top axes in (A) and (B) are marked in wavelength units. Fig. 2. Results of the numerical modeling showing spatiotemporal and spectral dynamics of the femtosecond soliton propagating down the PCF. (A) Spatiotemporal plot of the absolute value of the field. The gray shaded region appearing for z ⬎ 2.5 m is the Cherenkov radiation. The gray-scale mapping chosen for this plot exaggerates the strength of the radiation tail in order to show it clearly. For quantitative comparison of the soliton and radiation intensities, see fig. S1. (B) Plot of the logarithm of the absolute value of the Fourier transform of the field in the (, z) plane. White dashed line marks the zero GVD frequency with the negative slope. The radiation and solitonic parts of the field are located, respectively, to the left and the right of the white line. The top axis is marked in wavelength units. 1706 19 SEPTEMBER 2003 VOL 301 SCIENCE www.sciencemag.org REPORTS canceled almost completely. This happens at the expense of a bright new spectral line (Fig. 2B), which appears on the red side of the spectrum in the normal GVD (2 ⬎ 0) region. Simultaneously, the soliton acquires a pronounced tail of Cherenkov radiation (Fig. 2A). The radiation field needs less time than the soliton to reach a given distance z in the fiber, so the radiation propagates ahead of the soliton in the laboratory reference frame. It is clear that we can directly associate the radiation tail with the new red-shifted spectral line. From Fig. 2B it is seen that the radiation exists also for z ⬍ 2.5 m, but its amplitude is relatively small compared to the strong radiation line rapidly emerging and continuing to exist for z ⬎ 2.5 m. Taking the soliton frequency measured from our modeling, the radiation frequency can be computed directly from the wavenumber matching conditions; the corresponding points are marked by white diamonds (Fig. 2B). The excellent agreement between results obtained with these two methods strongly suggests that the new spectral band is indeed Cherenkov radiation emitted by the soliton. Thus, we have established that, after some propagation distance, the rate of energy transfer from the soliton to the red-shifted Cherenkov radiation builds up to a point when it balances, through the spectral recoil mechanism, the Fig. 3. (A) Experimentally measured spectral evolution of the femtosecond optical pulse propagating down the PCF. (B) Corresponding numerical modeling. The top axes are marked in wavelength units. Cherenkov radiation band appears to the left of the vertical dashed line marking the zero GVD frequency. Raman-induced soliton self-frequency shift. After this point, no further spectral shift of the soliton occurs. After the balance has been reached, the radiation contains ⬃ 50% of the initial soliton energy. For z ⬎ 2.5 m, the length of the radiation tail increases (Fig. 2A), leading to a nonexponential decay of the solitonic part of the field. The latter, however, remains strongly localized during this process (fig. S1). To verify the above numerical results, we carried out a series of experiments in which we launched optical pulses into the PCF. The laser source used in the experiments was a mode-locked Ti:sapphire system emitting 200-fs pulses at a wavelength of 0.86 m. The fiber length was progressively reduced from 4 to 0.5 m, and the output spectrum was recorded for each length with an optical spectrum analyzer. Figure 3A shows measurement results for a peak pump power of 230 W. These initial conditions correspond to the fourth-order solitonic solution of the ideal NLS equation (2). For the chosen pump wavelength, the dispersion slope is still positive, but both red and blue radiation fields are very weak, because of the large detuning from the soliton frequency. Therefore, the initial soliton dynamics is dominated by the Raman effect. Initially, the pulse splits into two Raman-shifting solitons, and some residual radiation, which retains the pump fre- quency. The more intense soliton acquires the stronger Raman shift. It passes the minimum point of 2() and enters the spectral region with negative dispersion slope, where the red-shifted Cherenkov frequency quickly approaches the central part of the soliton spectrum. This leads to the exponential amplification of the red-shifted Cherenkov radiation, which stabilizes the soliton frequency at ⬵2 ⫻ 236 THz, corresponding to the wavelength of ⬵1.27 m, through the recoil mechanism. Figure 3B shows the corresponding modeling results: They are in excellent agreement with experiment. Note that the blobs in the low middle part of the experimental plot (Fig. 3A) are an artifact of the numerical interpolation of the data. Thus, our numerical and experimental results show that we have identified and observed an efficient mechanism to amplify the Cherenkov radiation emitted by optical solitons in PCFs with negative dispersion slopes. Saturation of this amplification, in its turn, leads to the cancellation of the soliton selffrequency shift. The detuning between the solitonic pump and the Cherenkov radiation in the frequency-locking regime is sensitive to the value of the GVD slope, which is determined by the PCF geometry. This suggests the possibility of tailoring the latter to achieve a desired output frequency, and thus, of developing a family of optical parametric amplifiers based on the solitonic Cherenkov radiation. Methods to control the frequency of the Cherenkov radiation emitted by a charged particle inside a photonic crystal by varying the crystal geometry have also been recently suggested (25). The effect of the soliton self-frequency shift cancellation described above is very robust and has been observed for a broad range of pump frequencies and powers. The most likely reason why it has not been previously identified is that in telecommunication fibers, for commonly used frequencies, the radiation is always blue-shifted, so that the recoil and Raman effects act in the same spectral direction. The same is true for the recent experiments with PCFs with larger core diameters (18, 21). In these experiments, the blueshifted radiation was a dominant feature of generated spectra, when pulses were launched in the proximity of the zero GVD point with positive slope. However, Raman and recoil effects quickly pull the emergent solitons away from the zero GVD point and into the region of the less steep positive or practically flat GVD slopes, which leads to the decay rather than amplification of the radiation intensity with propagation distance. Results of this work show how novel and unexpected effects can be discovered in photonic crystal fibers due to the unique combination of their dispersive and nonlinear properties. www.sciencemag.org SCIENCE VOL 301 19 SEPTEMBER 2003 1707 REPORTS References and Notes 1. A. Scott, Nonlinear Science: Emergence and Dynamics of Coherent Structures (Oxford Univ. Press, Oxford, 1999). 2. A. Hasegawa, M. Matsumoto, Optical Solitons in Fibers (Springer, Berlin, 2003). 3. G. I. Stegeman, M. Segev, Science 286, 1518 (1999). 4. L. F. Mollenauer, R. H. Stolen, J. P. Gordon, Phys. Rev. Lett. 45, 1095 (1980). 5. E. M. Dianov et al., JETP Lett. 41, 294 (1985). 6. F. M. Mitschke, L. F. Mollenauer, Opt. Lett. 11, 659 (1986). 7. P. A. Cherenkov, Dokl. Akad. Nauk SSSR 2, 451 (1934). 8. S. Vavilov, Dokl. Akad. Nauk SSSR 2, 457 (1934). 9. I. Frank, I. Tamm, Dokl. Akad. Nauk SSSR 14, 109 (1937). 10. Cherenkov radiation at speeds below the light threshold has also been recently reported for a spatially extended system of electric dipoles created by a femtosecond optical pulse (26). 11. V. E. Zakharov, A.B. Shabat, Sov. Phys. JETP 34, 62 (1971). 12. N. Akhmediev, M. Karlsson, Phys. Rev. A 51, 2602 (1995). 13. For numerous reasons such as, e.g., Cherenkov radiation or dissipation, most if not all solitons observed in nature are not the “ideal” ones. To stress the importance of the nonideal features of the solitons, the term “quasi-solitons” has been introduced and widely used over the last decade [see, e.g., (27)]. 14. J. C. Knight, T. A. Birks, P. St. J. Russell, D. M. Atkin, Opt. Lett. 21, 1547 (1996). 15. J. C. Knight, J. Broeng, T. A. Birks, P. St. J. Russell, Science 282, 1476 (1998). 16. R. F. Cregan et al., Science 285, 1537 (1999). 17. J. C. Knight et al., IEEE Photon. Technol. Lett. 12, 807 (2000). 18. W. H. Reeves et al., Nature 424, 511 (2003). 19. J. K. Ranka, R. S. Windeler, A. J. Stenz, Opt. Lett. 25, 25 (2000). 20. W. J. Wadsworth et al., Electron. Lett. 36, 53 (2000). 21. J. Herrmann et al., Phys. Rev. Lett. 88, 173901 (2002). 22. A. L. Gaeta, Opt. Lett. 11, 924 (2002). 23. J. M. Dudley et al., J. Opt. Soc. Am. B 19, 765 (2002). The Anatomy of the World’s Largest Extinct Rodent Marcelo R. Sánchez-Villagra,1* Orangel Aguilera,2 Inés Horovitz3 Phoberomys is reported to be the largest rodent that ever existed, although it has been known only from isolated teeth and fragmentary postcranial bones. An exceptionally complete skeleton of Phoberomys pattersoni was discovered in a rich locality of fossil vertebrates in the Upper Miocene of Venezuela. Reliable body mass estimates yield ⬃700 kilograms, more than 10 times the mass of the largest living rodent, the capybara. With Phoberomys, Rodentia becomes one of the mammalian orders with the largest size range, second only to diprotodontian marsupials. Several postcranial features support an evolutionary relationship of Phoberomys with pakaranas from the South American rodent radiation. The associated fossil fauna is diverse and suggests that Phoberomys lived in marginal lagoons and wetlands. Phoberomys belongs to the Caviomorpha, a diverse and endemic group of South American rodents that includes arboreal, cursorial, and fossorial forms and that ranges today in size between ⬃200 g and ⬃50 kg (1). The evolution of caviomorphs is recorded in a rich but geographically biased fossil record. The southern portion of South America contains most of the record (2); hence, discoveries in the northern tropics are of special significance. The Urumaco Formation in northwestern Venezuela contains one of the few examples of a diverse fauna of Upper Miocene vertebrates in the continent. Recent explorations resulted in the discovery of additional vertebrates in the upper and middle members of this formation, including the rodent reported here (table S1). Old and new discoveries make Urumaco one of the best-documented 1 Universität Tübingen, Spezielle Zoologie, Auf der Morgenstelle 28, D-72076 Tübingen, Germany. 2Universidad Nacional Experimental Francisco de Miranda, CICBA, Complejo Docente Los Perozos, Carretera Variante Sur, Coro, 4101, Estado Falcón, Venezuela. 3 Department of Organismic Biology, Ecology, and Evolution, 621 Young Drive South, University of California, Los Angeles, CA 90095–1606, USA. *To whom correspondence should be addressed. Email: [email protected] 1708 tropical Miocene fossil fauna of vertebrates in the world after La Venta in Colombia (3). The Urumaco Formation is characterized by diverse faunal associations in continental (savannas), freshwater (swamps and rivers), estuarine (brackish), and marine (coastal lagoon, salt marsh, and sandy littoral) environments (table S1). Each assemblage can be correlated with a distinctive sedimentary environment. The following facies are apparent: shallow-water marine sediments rich in mollusks and fishes; brackish water rich in marine catfish; and swampy paleoenvironments rich in crocodilians and gavialids, in freshwater and marine turtles, and in freshwater catfish. These general sequences repeat several times in the outcrop (4). The skeleton reported here was found in brown shales interbedded with thin layers of coal. Two specimens of Phoberomys pattersoni Mones 1980 (5) provide the basis for this report. One consists of an almost complete associated skeleton (Fig. 1A). The skull is poorly preserved and consists of a deformed palate with the upper molariform series and most of the dentaries, with molariform teeth and fragments of the incisors. An additional shattered partial skull, preserving most of the occipital 24. Temporal profile of the amplitude of an ideal fiber soliton is given by secant hyperbolic: sech(t/) ⫽ 2/(et/ ⫹ e⫺t/). It is, however, not commonly known that the Fourier transform of the sechfunction is again a sech-function. This can be checked, e.g., using Mathematica 4.0 or in (28). 25. C. Luo, M. Ibanescu, S. G. Johnson, J. D. Joannopoulos, Science 299, 368 (2003). 26. T. E. Stevens, J. K. Wahlstrand, J. Kuhl, R. Merlin, Science 291, 627 (2001). 27. V. E. Zakharov, E. A. Kuznetsov, JETP 86, 1035 (1998). 28. W. Feller, An Introduction to Probability Theory and Its Applications (Wiley, New York, 1966), vol. 2, p. 503. Supporting Online Material www.sciencemag.org/cgi/content/full/301/5640/1705/ DC1 Fig. S1 27 June 2003; accepted 13 August 2003 and portions of the basicranial region, was also collected (Fig. 1B). Based on the degree of tooth wear and sutural fusion, we estimate that the specimens were adults at the time of death. The proximal epiphysis of the tibia and the distal epiphysis of the ulna are not fused to the diaphysis. However, it is possible that the animal was an adult, because no sutures can be recognized in the occipital region. In the pakarana Dinomys, probably the closest extant relative of Phoberomys, the epiphyses of long bones fuse late in ontogeny, some during adulthood (6). A description of the anatomy of the postcranial skeleton of P. pattersoni is presented in the supporting online material. Allocation of the specimens to P. pattersoni is secured based on two diagnostic features of the last upper molar (5): the narrowing of the posteriormost portion at the level of the last three prisms, and the size (mesiodistal length: 41 mm; width: 20.7 mm) and relative proportions of this tooth. Based solely on tooth dimensions, P. pattersoni is slightly smaller than P. insolita and P. lozanoi, which have a M3 with a mesiodistal length of 47 and 48 mm, respectively. Phoberomys, together with the genera Neoepiblema and Eusigmomys, belongs to the fossil Family Neoepiblemidae, distributed in Argentina, Chile, Brazil, and Venezuela (7). Of all other species of Neoepiblemidae, cranial remains of only Neoepiblema ambrosettianus have been reported to date (7). This animal had a prominent sagittal crest, absent in P. pattersoni. Based on fragmentary dental remains, Phoberomys (and therefore the neoepiblemids) has been classified either with the chinchillas and viscachas (8), with the pakarana (9), or as the sister group to both (10). We plotted a set of 13 postcranial characters on a preexisting phylogenetic tree based on molecular data and found that several postcranial features support the association of Phoberomys with the pakarana (Fig. 2). This position for Phoberomys was the one that required the least number of steps. P. pattersoni is reported to have been the size of a rhinoceros (1, 11, 12). This rough estimate, based on isolated teeth, can be 19 SEPTEMBER 2003 VOL 301 SCIENCE www.sciencemag.org