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We demonstrate an experimental technique that allows a mapping of vectorial nonlinear-optical processes in multimode photonic-crystal fibers (PCFs). Spatial and polarization modes of PCFs are selectively excited in this technique by varying the tilt angle of the input beam and rotating the polarization of the input field. Intensity spectra of the PCF output plotted as a function of the input field power and polarization then yield mode-resolved maps of nonlinear-optical interactions in multimode PCFs, facilitating the analysis and control of nonlinear-optical transformations of ultrashort laser pulses in such fibers.
©2006 Optical Society of America
1. Introduction
Photonic-crystal fibers (PCFs) [1, 2] have proved to be a powerful tool for ultrafast nonlinear optics and femtosecond laser technologies [3]. Dispersion [4] and spatial field [5] profiles in this type of waveguides can be optimized for efficient nonlinear-optical spectral transformations of ultrashort pulses by modifying the core--cladding geometry of the fiber. Enhanced nonlinear-optical interactions of ultrashort pulses in PCFs are at the heart of frequency-comb expanders for optical metrology [6 – 8], systems for carrier--envelope phase stabilization in ultrafast optics [9, 10], sources of correlated photon pairs in quantum optics [11], fiber-optic frequency shifters for optical parametric chirped-pulse amplification [12], and novel compact and efficient broadband light sources [13, 14] and frequency converters [15] for biomedical optics [16], coherent spectroscopy [17] and nonlinear microscopy [18].
In multimode PCFs, nonlinear-optical interactions can involve guided modes with different dispersion properties, spatial field profiles, and polarization patterns [19]. The vectorial nature of nonlinear-optical processes becomes especially important for PCFs with a form anisotropy, leading to new interesting and often complicated scenarios of spectral and temporal transformations of ultrashort pulses [20, 21]. The vector sensitive wave-number matching conditions have been discussed in the latter work. These multimode interactions in PCFs give rise to additional frequency components and bands in output spectra, often contributing very significantly to the emission of radiation with a very broad and smooth spectrum at the output of the fiber [22]. Such high-quality spectra are, however, often difficult to understand because of the nontrivial scenarios of mode interaction and multimode wave mixing. As a result, designing PCFs for well-controlled multimode spectral transformation of ultrashort pulses is a challenging problem.
In this work, we demonstrate an experimental approach that facilitates the analysis and allows mode-selective mapping and control of ultrafast nonlinear-optical processes in multimode PCFs. We show that vectorial nonlinear-optical interactions in PCFs can be resolved and controlled by varying the polarization of the input field and changing the tilt angle of the fiber with respect to the axis of the input beam. The mode structure and the spectral properties of the PCF output are controlled through polarization-sensitive phase matching between dispersive waves and solitons. The tilt angle of the input beam is shown to control the mode structure of the frequency-shifted PCF output, allowing the generation of well-resolved fundamental and higher order guided modes in the visible. Rotation of the polarization vector of the linearly polarized input field with a fixed tilt angle of the input beam is shown to switch the central wavelength of Cherenkov emission in PCF output. The maps of PCF output spectral intensity plotted as a function of the input field power and polarization are modeled by using the generalized nonlinear Schrödinger equation (GNSE) and are explained in terms of polarization-sensitive phase matching between dispersive waves and solitons.
2. Photonic-crystal fibers
Photonic-crystal fibers were fabricated of fused silica with the use of the standard technology, described in detail elsewhere [1, 2]. An SEM image of the central part of the PCF is shown in the inset to Fig. 1. Propagation constants β of PCF modes, their field intensity profiles, and wavelength dependences of effective mode areas were calculated by numerically solving the Maxwell equations for transverse field components in the cross section of the PCF using a modification of the technique based on polynomial expansions of the fields and the two-dimensional refractive index profile in the cross section of the fiber [23]. The properties of PCF modes calculated with the use of this approach were then plugged into the generalized nonlinear Schrödinger equation [24], which served to model the spectral and temporal evolution of laser pulses in the fiber.
Fig. 1. Group-velocity dispersion for the fundamental (1) and second-order (2) modes of the photonic-crystal fiber shown in the inset.
Figure 1 displays the wavelength dependences of the group-velocity dispersion (GVD) D = -2πcλ -2 β (2), where λ is the radiation wavelength, c is the speed of light in vacuum, and β (2) = ∂2 β/∂ω 2 for the fundamental (curve 1) and one of the higher order (curve 2) PCF modes. Both modes provide anomalous dispersion for 800-nm Ti: sapphire laser pulses employed in our experiments, allowing formation of solitons inside the PCF. Scenarios of pulse evolution in these two modes are, however, completely different, as demonstrated by the results of GNSE-based simulations in Figs. 2(a) and 2(b). For the fundamental mode, the nonlinear-optical transformation of a femtosecond pulse in the PCF yields a broad and continuous spectrum stretching from approximately 510 to 1050 nm (Fig. 2(a)). For the higher order mode, on the other hand, the output spectrum features a gap (Fig. 2(b)) in the range of wavelengths from 620 to 670 nm, with an intense spectral line emitted near 470 nm.
Fig. 2. The spectral intensity of the field in the fundamental (a) and second-order (b) modes at the output of a PCF with a length of 7 cm and the cross-section structure shown in the inset to Fig. 1. Results of simulations performed by using the generalized nonlinear Schrödinger equation for an input pulse with a peak power of 7 kW.
These differences in pulse evolution in the fundamental and a higher order PCF spatial mode, visualized by GNSE-based numerical simulations, can be understood through a more detailed analysis of dispersive-wave emission by solitons and four-wave mixing (FWM) processes in the considered fiber modes. In both modes, the input laser field, falling within the region of anomalous dispersion, is coupled into solitons. These solitons experience a continuous red shifting as they propagate along the fiber. This effect is induced by the retarded part of the nonlinear response of the fiber material, resulting in the Raman-type amplification of the long-wavelength part of the pulse spectrum at the expense of the depletion of its high-frequency wing – phenomenon known as soliton self-frequency shift (SSFS) [24]. High-order dispersion, on the other hand, induces wave-matching resonances between solitons and dispersive waves, giving rise to intense blue-shifted emission in PCF output spectra [25 – 27]. The central wavelength of dispersive-wave emission, which takes place in the regime of Cherenkov radiation, is controlled by phase matching between the parent soliton and the emitted dispersive wave. The phase-matching condition providing a resonant energy exchange between a soliton with a propagation constant λs and a central wavelength λ 0 and a dispersive wave with a propagation constant λ and a central wavelength λ d is written as [26] δs = βs(λ 0) - β(λd) = 0. In Figs. 3a and 3b, we plot the propagation-constant mismatch δs calculated as a function of radiation wavelength for the fundamental and the higher order spatial modes of the PCF, respectively. It can be seen from the results of these calculations presented in Fig. 3(a) that, for the fundamental PCF mode, multiple solitons, generated by the 800-nm input laser pulse and shifted toward longer wavelengths due to the Raman effect (Fig. 2(a)), can emit phase-matched dispersive waves in the 550 -- 600 nm wavelength range. Along with the spectral broadening of the solitonic and nonsolitonic parts of the field, these soliton-dynamics phenomena give rise to a broad continuous radiation spectrum at the output of the fiber (Fig. 2(a)).
Fig. 3. (a, b) Propagation-constant mismatch δs = βs (λ 0) - β(λd) between a soliton with a central wavelength λ 0 and a dispersive wave with a central wavelength λ d as a function of the wavelength λ d for the fundamental (a) and second-order (b) spatial modes of the PCF with the cross-section structure shown in the inset to Fig. 1. The central wavelength of the soliton is λs = 820 nm (1), 850 nm (2), and 880 nm (3). (c) Propagation-constant mismatch δβFWM = β(ωa) + β(ωs) - 2 β (ωp) for the FWM process 2ω p = ωs + ω a in the second-order PCF mode. The pump wavelength is (1) 650 nm and (2) 630 nm.
For the higher order mode, red-shifted solitons produced by the 800-nm pump field are phase-matched with dispersive waves in the 450 -- 500 nm wavelength region (Fig. 3(b)). Cherenkov emission in this mode is thus noticeably blue-shifted relative to dispersive waves emitted in the fundamental mode (cf. Figs. 2(a) and 2(b)). In Fig. 3(c), we demonstrate that some of the spectral components of the field in the second-order PCF spatial mode can serve as a pump for phase-matched FWM 2ω p = ωs + ω a, where ω p, ωs, and ω a are the frequencies of the pump, Stokes, and anti-Stokes fields, respectively. The phase-matching condition for this process is written as δβFWM = β(ωa)+β(ωs)-2β(ωp)=0, where β(ω) is the propagation constant of the field spectral component with frequency ω in a given fiber mode. For the second-order spatial mode of the considered type of PCF with GVD shown by curve 2 in Fig. 1, the phase-matching condition δβFWM = 0 is satisfied for pump wavelength falling within the 630--650-nm wavelength range (Fig. 3(c)), giving rise to a 1200--1250-nm Stokes field and a 430--450-nm anti-Stokes signal. This parametric process, as shown by the results of pulse-dynamics simulations in Fig. 2(b), gives rise to an intense emission band in the infrared part of the spectrum and facilitates the generation of bright blue-shifted spectral components in the visible. All the above outlined features of spectral transformation dynamics in the fundamental and higher order spatial modes are clearly observed in the experiments presented in Section 4 below in this paper, allowing PCF modes to be distinguished not only by their spatial intensity profiles, but also by typical spectra of PCF output.
3. The laser system
In experiments, we used a Ti: sapphire oscillator with an X-folded cavity, pumped with a 4-W second-harmonic output of a diode-pumped Nd: YVO4 laser (Millennia VS, Spectra-Physics). A Brewster-cut Ti: sapphire crystal with a length of 2.3 mm is placed at the center of the laser cavity between two focusing mirrors (Newport) with a focal length of 50 mm. Chirped mirrors and a prism pair are used for dispersion compensation. The separation of the prisms in the pair is 240 mm. Each of the chirped mirrors (Layertec, Germany) provides an average group-dispersion delay (GDD) of about 60 fs2 per bounce at 800 nm. Chirped mirrors in our laser cavity provide a flat GDD profile over a broad spectral band. The level of GDD is controlled by the prism separation and can be tuned from negative to positive values. Such a laser oscillator can deliver pulses with a typical temporal width of about 30 fs, an energy up to 5 nJ at a pulse repetition rate of 100 MHz and a central wavelength of 800 nm.
A 40x lens was used to couple laser radiation into a PCF with a length of 7 to 20 cm, placed on a three-dimensional translation stage. The coupling efficiency achieved with this lens was estimated as 30%. The PCF output was collimated with an identical lens and was studied with an Ando spectrum analyzer.
4. Experimental results and discussion
In the experimental part of this work, the spectral properties of radiation transmitted through the fused silica PCF with the cross-section structure shown in the inset to Fig. 1 were studied as a function of the pulse energy and polarization of input radiation and the tilt angle θ of the incident beam with respect to the fiber axis. Radiation spectra measured at the output of the PCF were plotted in the form of two-dimensional maps, where the spectral intensity is encoded in color (or can be represented by levels of gray scale), one of the axes corresponds to the radiation wavelength, while the second axis represents either the average input radiation power or the angle ϕ of the polarization vector of the input field with respect to the slow axis of the elliptically deformed PCF core.
With an input beam aligned exactly along the axis of the PCF (θ= 0), the laser field was coupled into the fundamental mode of the fiber. In this geometry, frequency-shifted components originating from nonlinear-optical processes in the fiber are also emitted in the fundamental mode, as indicated by characteristic output beam profiles shown in Figs. 4(a) and 4(b). For low input energies, blue-shifted frequency components in the PCF output are observed as isolated spectral features (Fig. 5). As the input energy is increased, both the infrared and the visible parts of radiation spectrum experience broadening due to self- and cross-phase modulation, as well as FWM, leading to the emission of supercontinuum at the level of input power on the order of 300 mW (Fig. 5).
The form anisotropy of the fiber removes the degeneracy from the doublet of fundamental modes with orthogonal polarizations. The modes of this doublet in a birefringent PCF display slightly different dispersion profiles [28], providing wave-matching resonances between solitons and dispersive waves at slightly different frequencies. This effect is observed in the two-dimensional map of Fig. 6, showing PCF output spectra measured for different ϕ, as switching of the central wavelength of dispersive-wave emission from 570 nm at ϕ = 0 to 530 nm at ϕ= 90°. This wavelength shift is easily noticeable visually in Figs. 4(a) and 4(b) as the change in the color of the fundamental-mode beam pattern at the output of the PCF.
Fig. 4. The beam profile of the frequency-shifted PCF output. The input beam is aligned with the PCF axis (θ= 0). The angle between the polarization vector of the input field and the slow axis of the fiber core is (a) ϕ= 0 and (b) ϕ= 90°. The input average power of laser pulses is 300 mW.
Fig. 5. The spectral intensity of the PCF output as a function of the radiation wavelength and the average power of the input field for θ= 0 and ϕ= 90°. The spectrum of the PCF output corresponding to an input average power of 320 mW is shown on the right.
As the input beam is tilted with respect to the fiber axis, the efficiency of radiation coupling to higher order spatial modes increases. This makes it possible to generate a blue-shifted dispersive-wave PCF output in a higher order guided mode of the fiber with the central wavelength of this signal controlled by the dispersion of the fiber mode. With θ = 10°, this type of dispersive-wave emission is observed for one of the polarizations of the input field (Fig. 7(a)). While the input field polarized at ϕ = 90° to the slow axis of the fiber is still mainly coupled into solitons emitting blue-shifted dispersive waves in the fundamental mode, observed as an elliptical green-color elliptical spot (Fig. 7(b)), the input field with the orthogonal polarization produces blue-color output emission with a beam shape corresponding to the second-order PCF spatial mode (Fig. 7(a)). The two-dimensional map of the spectral intensity of this signal measured as a function of radiation wavelength and the average power of the input field is shown in Fig. 8. The spectrum of the high-order-mode PCF output, as can be seen from Fig. 8, features a central band, corresponding to a spectrally broadened pump, as well as spectrally isolated bands in the visible and in the infrared. These experimental findings agree well with our theoretical analysis for the second-order PCF spatial mode (Figs. 2(b), 3(b), 3(c)), which predicts wave-matching resonances between solitons and dispersive waves (Fig. 3(b)) and phase-matched parametric FWM decay for spectral components within the 630--650-nm wavelength range (Fig. 3(c)).
Fig. 6. The spectral intensity of the PCF output as a function of the radiation wavelength and polarization angle ϕ for θ= 0 and an input average power of 180 mW. The spectrum of the PCF output corresponding to the polarization angle ϕ = 90° is shown on the right.
Fig. 7. The beam profile of the frequency-shifted PCF output. The angle between the input beam and the PCF axis is θ= 10°. The angle between the polarization vector of the input field and the slow axis of the fiber core is (a) ϕ= 0 and (b) ϕ = 90°. The input average power of laser pulses is 300 mW.
Fig. 8. The spectral intensity of the PCF output as a function of the radiation wavelength and the average power of the input field for θ = 10° and ϕ = 0. The spectrum of the PCF output corresponding to an input average power of 360 mW is shown on the right.
Fig. 9. The spectral intensity of the PCF output as a function of the radiation wavelength and polarization angle ϕ for θ= 10° and an input average power of 350 mW. The spectrum of the PCF output corresponding to the polarization angle ϕ = 90° is shown on the right.
The two-dimensional map presented in Fig. 9 shows how the central wavelength of dispersive-wave emission can be switched by rotation polarization of the input field. While for small ϕ, PCF output is predominantly produced in a higher order guided mode with the blue-shifted spectral component centered around 450 nm, for the polarization angle ϕ approaching 90°, all-fundamental-mode nonlinear-optical processes start to play a dominant role. In the latter case, the central wavelength of dispersive-wave emission is shifted to 530--540 nm – the wavelength region typical of the wave-matching soliton--dispersive-wave resonances in the fundamental mode of the PCF (Figs. 2(a), 3(a)). The PCF output spectrum in Fig. 9 recovered each time the polarization of the input field was rotated by π. This π periodicity of spectral variations in ϕ observed in our experiments confirms that these variations, as well as the change in the spatial mode seen in Fig. 7, are truly polarization effects. In agreement with theoretical predictions, nonlinear-optical transformation of ultrashort pulses in the higher order PCF mode is accompanied by the generation of an isolated frequency component in the infrared part of the spectrum. This component is clearly seen as a well-resolved spectral feature in 2D maps shown in Figs. 8 and 9.
5. Conclusion
We have demonstrated in this work that ultrafast vectorial nonlinear-optical processes in multimode PCFs can be conveniently mapped by plotting the intensity spectra of the PCF output as a function of the input field power and polarization. Nonlinear-optical interactions occurring in various polarization and spatial modes of PCFs are resolved by varying the polarization of the input field and changing the tilt angle of the fiber with respect to the axis of the input beam. The main tendencies observed in our experiments in the maps of PCF output spectral intensity plotted as a function of the input field power and polarization have been adequately explained in terms of the model based on the generalized nonlinear Schrödinger equation. The presented mapping technique thus facilitates the analysis and control of nonlinear-optical transformations of ultrashort laser pulses in multimode PCFs, allowing the generation of frequency-tunable radiation in the visible and infrared at the output of the fiber and permitting a mode-controlled transformation of ultrashort laser pulses into supercontinuum radiation.
Acknowledgments
This study was supported in part by the Russian Foundation for Basic Research, the Russian Federal Research and Technology Program (contract no. 02.434.11.2010), INTAS (projects nos. 03-51-5037 and 03-51-5288), the US Civilian Research and Development Foundation for the Independent States of the Former Soviet Union (award no. RP2-2558), National Key Basic Research Special Foundation (project no. 2003CB314904), National Nature Science Foundation of China (project no. 60278003), and National High-Technology Program of China (project no. 2003AA311010).
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