1. Introduction

Highly nonlinear photonic-crystal fibers (PCFs) [1, 2] suggest an attractive platform for the creation of compact and efficient frequency shifters and wavelength converters for spectroscopic, microscopic, and bioimaging applications [3–10], optical communication [11, 12], as well as ultrafast science and technology [13, 14]. The most widely used strategies for frequency shifting in PCFs employ supercontinuum radiation [15–17], sideband generation through four-wave mixing (FWM) [18, 19], and soliton transformations of ultrashort pulses, including soliton self-frequency shift (SSFS) [20, 21] and dispersive-wave emission by solitons [22]. A unique flexibility of dispersion and nonlinearity management provided by PCFs [23] is the key advantage for the development of practical fiber sources capable of generating light pulses within a broad range of wavelengths and pulse widths for a wide variety of applications. Advanced PCF technologies enable fabrication of a remarkable variety of fiber cross-section geometries (see Refs. 1, 2, 24 for a review). For solid-core PCFs, the desired dispersion profile is typically engineered [23, 25, 26] by choosing an appropriate cladding geometry, as well as by varying the sizes and the shape of the fiber core. As shown by recent work [27–36], PCF dispersion engineering strategies can be enhanced through a careful design of air-hole defects in the fiber core. This approach has been recently employed for the design of ultraflattened group-velocity dispersion (GVD) profiles in PCFs [28], including large-mode-area microstructure fibers [29, 32], as well as for the control of the fiber nonlinearity and gain [30]. PCF nanomanagement strategies for the optimization of supercontinuum generation have been outlined by Frosz et al. [33]. PCFs with a solid core modified with a ring-shaped array of nanosize air-hole defects have been shown [34] to allow a precise management of dispersion profiles of guided modes for an efficient nonlinear-optical frequency conversion of femtosecond laser pulses, resulting in the generation of signals at the desired central wavelengths at the fiber output.

Here, we show that fiber dispersion and nonlinearity management based on a modification of the PCF core with an air hole helps optimize PCF components for a stable SSFS and subpetahertz FWM sideband generation. A high sensitivity of the SSFS to the power of the input pulse often causes serious difficulties in SSFS-based optical schemes, as input power fluctuations are transformed in this regime into unwanted variations in the central wavelength and the timing jitter of the frequency-shifted pulse at the output of the fiber [36]. Several physical factors, such as the high-order dispersion, diffraction-induced wavelength dependence of the effective mode area, and waveguide loss, have been demonstrated to limit the SSFS after a certain propagation length, helping to reduce wavelength uncertainties and timing jitter of frequency-shifted solitons at the fiber output [36–40]. However, even with all these mechanisms put to work, variations in the central wavelength of frequency shifted solitons and their timing jitter cannot be completely suppressed, which limits application of soloton PCF frequency shifters in practical systems for microspectroscopy and bioimaging. In a highly nonlinear PCF with two zero-GVD points, spectral recoil of an optical soliton by a red-shifted dispersive-wave (Cherenkov) radiation, as shown by Skryabin et al. [41] (see also Refs. 42, 43), can stabilize the soliton frequency shift. In this work, we implement this strategy of SSFS stabilization relative to pump power variations using a highly nonlinear PCF with an air-hole-modified core. Moreover, an air-hole-modified-core PCF with appropriate dispersion and nonlinearity parameters will be shown to provide efficient generation of Stokes and anti-Stokes FWM sidebands shifted from the pump wavelength by more than 3000 cm^-1 (corresponding to ~0.1 PHz), thus offering an attractive source for nonlinear Raman microspectroscopy.

2. Laser setup and photonic-crystal fiber

Two types of fused silica PCFs with an air-hole-modified core (Figs. 1(a), 1(b), 2(a), 2(b)) were fabricated for the purposes of our experiments. The frequency profile of fiber dispersion was tailored by varying the diameters of the air holes in the cladding and the pitch of the cladding structure, as well as by changing the size of the air hole at the central part of the fiber. PCF of the first type had a core diameter of 2.3 µm with a diameter of a central air hole of 0.9 µm (Figs. 1(a), 1(b)), providing a dispersion profile with both positive and negative slopes with two zero-GVD points around 710 and 1020 nm (Fig. 1(c)), suitable for SSFS cancellation due to the spectral recoil of the soliton by its red-shifted dispersive-wave radiation. PCF of the second type (Figs. 2(a), 2(b)) had a core with a diameter of 2.9 µm modified with a central air hole with a diameter of 0.8 µm. Dispersion profile for this type of PCF (Fig. 2(c)) displays only one zero-GVD point at 740 nm, allowing transformation of 800- nm Ti: sapphire laser pulses into frequency-shifted solitons emitting dispersive waves only in the visible part of the spectrum.

Nonlinear experiments were performed with the use of a Ti: sapphire oscillator with an X-folded cavity, pumped with a 4-W second-harmonic output of a diode-pumped Nd: YVO₄ laser. This laser oscillator can deliver pulses with a typical temporal width of about 30 – 50 fs, an energy up to 5 nJ at a pulse repetition rate of 100 MHz, and a central wavelength close to 800 nm. These laser pulses were transmitted through an optical isolator and were coupled by an aspheric lens into the PCF, which is placed on a high-precision three-dimensional translation stage. Radiation coming out of the fiber was collimated with an identical aspheric lens and was studied with an Ando spectrum analyzer.

 

Fig. 1. Scanning electron-microscope images (a, b) and the dispersion parameter β ₂=∂ ² β/∂ ω ² and group-velocity dispersion D=-2πcλ ^-2 β ⁽²⁾ as a function of the wavelength (c) of the photonic-crystal fiber of the first type.

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Fig. 2. Scanning electron-microscope images (a, b) and the group-velocity dispersion as a function of the wavelength (c) of the photonic-crystal fiber of the second type.

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3. Results and discussion

For the PCF of the first type, the central wavelength of Ti: sapphire laser pulses, as can be seen from Fig. 1(c), lies within the range of anomalous dispersion (β ₂=∂ ² β/∂ ω ²<0, with β being the relevant mode propagation constant) on the positive slope of fiber dispersion profile (β ₃=∂ ³ β/∂ ω ³ > 0). Laser pulses with a sufficient peak power therefore tend to evolve toward solitons as they propagate through the PCF. Figure 3(a) presents a typical map of PCF output spectra measured for different values of the average laser power. The abscissa axis in this figure shows the nominal average laser power measured right after the laser oscillator. Only a few percent of this power was launched into the PCF core, yielding light pulses with an energy of up to 100 – 150 pJ inside the fiber. The initial pulse width was estimated as 50 fs. Formation of solitonic features is clearly seen in the experimental spectra of PCF output (Fig. 3(a)) and is verified (Figs. 3(b), 4(a), 4(b)) by the numerical solution of the generalized nonlinear Schrödinger equation (GNSE) [44].

 

Fig. 3. The spectrally resolved output signal from a 50-cm PCF measured (a) and calculated with the use of the GNSE (b) as a function of the average laser power (a) and the pump energy (b).

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Fig. 4. PCF output signal calculated with the use of the GNSE (a) as a function of radiation wavelength and the pulse propagation distance z along the fiber and (b) as a function of retarded time and the distance z for a pump energy of 95 pJ.

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High-order fiber dispersion induces instabilities of these solitons with respect to the emission of dispersive waves – phenomenon that is often referred to as Cherenkov radiation by optical solitons [45, 46]. The central wavelength of radiation emitted as a result of this process is controlled [41, 42, 45, 46] by phase matching between the soliton and a dispersive wave $\delta \beta =\frac{\beta \left(\omega \right)-{\beta }_s\left({\omega }_s\right)\approx \sum _{m=2}^M{\beta }_m{\left(\omega -{\omega }_s\right)}^m}{\frac{m!-\gamma P}{2}},\mathrm{where} {\beta }_m=\frac{{\partial }^m\beta }{\partial {\omega }^m|{\omega }_s},\gamma$, γ is the nonlinear coefficient, and P is the soliton peak power. In the upper panel of Fig. 5, we plot the parameter δβ calculated as a function of radiation wavelength for the fiber dispersion profile shown in Fig. 1(c) with M = 9. At the initial stage of pulse evolution in the PCF (propagation distance z ranging from 2 to 20 cm in Fig. 4), the central frequency of a soliton lies on the positive slope of the fiber dispersion profile (β ₃ > 0). By virtue of phase-matching condition δβ = 0, dispersive waves emitted at this stage of pulse evolution, as explained in earlier work [41, 42], are predominantly blue-shifted with respect to the parent soliton (see Fig. 5). The red-shifted branch of soliton–dispersive-wave phase matching in this regime does not exist or lies far in the infrared and is very weak. Blue-shifted dispersive-wave radiation is observed in GNSE simulations (Figs. 3(b), 4(a)) as an intense spectral component centered at the wavelength λ _b ≈ 550 nm, dictated by phase matching δβ(λ _b) = 0 between a soliton centered at an early stage of pulse propagation at approximately 810 nm (seen in the maps in Figs. 3(a), 3(b), 4(a)) and the blue-shifted dispersive wave. In the time domain, this blue-shifted dispersive-wave radiation is seen as a clearly resolved feature that branches off the soliton part of the field around z ≈ 2 cm and gets dispersed by normal fiber dispersion in the process of field evolution. The central wavelength of blue-shifted radiation detected in experiments (550 nm, see Fig. 2(a)) agrees very well with the predictions of GNSE simulations, as well as with the wavelength determined from the phase-matching condition δβ(λ _b) = 0 (see Fig. 5).

 

Fig. 5. The lower panel shows experimental (1) and theoretical (2) output spectra of a 50-cm PCF for an input pulse energy of 95 pJ. The upper panel presents the phase mismatch δβ (3) between a soliton centered at 810 nm and the blue-shifted dispersive wave and (4) between a soliton with the central wavelength of 960 nm and the red-shifted dispersive wave. The dashed vertical lines show the phase-matching wavelengths λ _b and λ _r for the blue- and red-shifted dispersive waves.

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The Raman effect induces a continuous red shift of a soliton [44, 47, 48]. A continuously red-shifting solitonic part of the field is clearly seen in both experimental and theoretical spectra of the PCF output (Figs. 3(a), 3(b), 4(a)) as a well-resolved feature that sweeps over the range of wavelengths from 800 to 960 nm as the pump pulse energy changes from 10 to 90 pJ. In the time-domain, fiber dispersion translates the wavelength shift into a time delay. As a result, the red-shifted soliton accumulates a time delay of about 2 ps at the output of a 50-cm piece of PCF (Fig. 4(b)). For a fixed initial soliton pulse width and with given fiber dispersion and nonlinearity, the soliton frequency shift is controlled by the input peak power and the fiber length. As the central wavelength of a soliton is shifted to the region of a negative slope of the fiber dispersion profile, where β ₃<0(λ > 890 nm in Fig. 1(c)), phase matching between a soliton and a dispersive wave, as first pointed out by Skryabin et al. [41], enables efficient generation of dispersive waves with wavelengths that are longer than the central wavelength of the soliton (see the upper panel in Fig. 5). At this stage of field evolution (z > 20 cm in Fig. 4), the soliton starts to radiate red-shifted dispersive waves, observed as a spectral component centered at λ _r ≈ 1180 nm in both experimental and theoretical maps in Figs. 3(a), 3(b), and 4(a). The central wavelength of the red-shifted radiation observed in experimental spectra also agrees well with the wavelength dictated by phase matching δβ(λ _r) = 0 between the soliton with the central wavelength of 960 nm and the red-shifted dispersive wave (Fig. 5). In the time domain, the red-shifted dispersive-wave radiation becomes clearly visible (Fig. 4(b)) as a dispersed part of the field adjacent to the red-shifted soliton branch. In this regime of soliton frequency shifting, intense high- and low-frequency branches of dispersive-wave radiation, as can be seen from Figs. 3(a), 3(b), 4(a), are simultaneously observed in the spectrum of PCF output (see also Ref. 49). Discrepancy between the experimental PCF output spectrum (curve 1 in Fig. 5) and theoretical predictions (curve 2 in Fig. 5) is mainly attributed under conditions of our experiments to the loss of the pump field due to the coupling between the guided and leaky PCF modes [50], which is not included in our GNSE-based model. The contribution of optical nonlinearity of atmospheric air in the central hole in the core of the fiber [51–53] was negligible for the considered fiber design. Inclusion of this part of nonlinearity into our model changed the results of simulations for the output spectra by less than 1%.

It is straightforward to see from Figs. 3(a) and 3(b) that, for input pulse energies exceeding 60 pJ, the central wavelength of the red-shifted soliton becomes insensitive to variations in the input peak energy and stays constant within a broad range of input pulse energies at least up to 100 pJ. Within this range of pump energies, the branch representing the red-shifted soliton in the maps of Figs. 3(a) and 3(b) forms a nearly ideally horizontal plateau. This stabilization of the SSFS relative to variations in the pump power is due to the spectral recoil of a soliton by the red-shifted dispersive wave, which exactly compensates, as first shown by Skryabin et al. [41], for the SSFS induced by the Raman effect, thus stabilizing the SSFS in a highly nonlinear fiber as a function of the pump peak power. This effect makes PCF-based soliton frequency shifters attractive sources for nonlinear microspectroscopy and bioimaging and enables creation of efficient PCF wavelength converters for optical communication technologies. In particular, the soliton frequency shift in a PCF demonstrated in this work can be used as a source of a frequency-stabilized Stokes field in coherent Raman microspectroscopy [3–10], providing an access to the fingerprint region of Raman transitions in molecules of biological significance.

 

Fig. 6. The spectrally resolved output signal of a 50-cm PCF of the second (a) and first (b) type measured as a function of the pump energy. Higher order guided mode is studied for the first type PCF. The spatial intensity profile for this mode is presented in the inset.

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PCF of the second type features only one zero-GVD point at 740 nm (Fig. 2(c)). Similar to the first-type PCF, this fiber also provides anomalous dispersion for 800-nm radiation, allowing Ti: sapphire laser pulses to be coupled into frequency-shifting solitons. However, in contrast to the first-type PCF, dispersion profile of the second-type PCF permits dispersive-wave emission by solitons only in the visible part of the spectrum. The map of the spectrally resolved fiber output measured for the second-type PCF as a function of the input average laser power, shown in Fig. 6(a), confirms this expectation. This map visualizes red-shifting solitons generated in the fiber by Ti: sapphire laser pulses. The number of solitons increases with the growth in the input laser power. Each soliton excited in the second-type PCF, as can be also seen from Fig. 6(a), emits excessive energy only in the visible part of the spectrum, where the fiber dispersion is normal. At high input laser powers, these branches of blue-shifted emission merge together into a supercontinuum-like broadband spectrum. The fundamental soliton carries the highest peak power, undergoing the largest frequency shift (down to 1090 nm in Fig. 6(a)). This frequency shift is decelerated [36–40, 54] by the growing GVD (Fig. 2(c)), increasing mode area, pulse self-steepening, and fiber loss. All these effects partially stabilize the SSFS against fluctuations in the input laser power. However, as can be seen from the comparison of Figs. 3(a) and 6(a), the spectral recoil of an optical soliton by a red-shifted dispersive-wave in the first-type PCF enables SSFS stabilization within a much broader range of input laser powers. Quantitatively, in the regime of a minimum sensitivity of the SSFS to fluctuations in the input laser power, a 5% variance of the input laser power translated into a 0.5% timing jitter of the soliton fiber output for the first-type PCF and a 2% timing jitter for the second-type PCF. It should be noted, however, that a frequency shifter based on the first-type PCF in the regime of SSFS stabilization, rigorously speaking, is not frequency-tunable, as the stabilized SSFS depends only on fiber parameters and cannot be controlled by varying parameters of the input laser field unless the second, controlling laser pulse is applied.

A radically different regime of frequency conversion can be achieved in the regime where the nonlinear transformation of a laser field is dominated by four-wave mixing, resulting in the generation of intense Stokes and anti-Stokes sidebands ω ₀±Ω in the spectrum of the laser field with an input central frequency ω ₀. To demonstrate this regime of frequency conversion in a PCF with an air-hole-modified core, we have chosen to work with a higher order mode of the first-type fiber with a beam profile shown in the inset to Fig. 6(b). Such a two-lobe beam profile is typically needed for an improved spatial resolution in microscopic measurements using the stimulated emission depletion (STED) technique [55], as well as recently proposed promising modifications of coherent anti-Stokes Raman scattering (CARS) with specifically engineered beam profiles of the pump and Stokes fields [56]. The map of the spectrally resolved fiber output measured as a function of the input average laser power (Fig. 6(b)) display well-pronounced sidebands, which initially show up at f = (2π)^-1Ω ≈ 0.1 PHz, corresponding to ν = f/c ≈ 3000 cm^-1, and tend to broaden as the input laser power is increased. These tendencies in the spectral dynamics of the laser field can be understood in terms of an elementary model of FWM in a nonlinear fiber [44], which predicts parametric amplification of Stokes and anti-Stokes sidebands ω ₀±Ω in the spectrum of a light field with a maximum gain achieved at ${\Omega }_0={\left(\frac{2\gamma P}{\mid {\beta }_2\mid }\right)}^{\frac{1}{2}},$ where γ is the nonlinear coefficient and P ₀ is the peak power. With our experimental parameters γ ≈ 70 km^-1W^-1, β ₂ ≈ -0.002 ps²/m, and P ₀ ≈ 2.5 kW, we find Ω₀ ≈ 0.6 PHz, which agrees well with experimental observations. Subpetahertz sideband generation in PCFs, demonstrated by these experiments, suggests an attractive approach for the creation of fiber-format pulse shapers for nonlinear Raman microspectroscopy, where ν ≈ 3000-cm^-1 sidebands can give an access to sub-10-fs molecular vibrations and ultrafast transient processes. Such high-ν sidebands also offer much promise for fiber-based single-beam CARS microscopy in the fingerprint region of molecular vibrations, including CARS probing of ≈ 2850-cm^-1 symmetric CH₂ stretch, which is of key significance for CARS bioimaging [57, 58] and CARS endoscopy [59].

4. Conclusion

We have shown in this paper that fiber dispersion and nonlinearity management strategy based on a modification of a PCF core with an air hole facilitates optimization of PCF components for a stable soliton self-frequency shift and subpetahertz sideband generation through four-wave mixing. Spectral recoil of an optical soliton by a red-shifted dispersive wave, generated through a soliton instability induced by high-order fiber dispersion, has been shown to stabilize the SSFS in a highly nonlinear PCF with an air-hole-modified core relative to pump power variations. A fiber with a 2.3-µm-diameter core modified with a 0.9-µmdiameter air hole has been used to demonstrate a robust SSFS of unamplified 50-fs Ti: sapphire laser pulses to a central wavelength of about 960 nm, which remains insensitive to variations in the pump pulse energy within the range from 60 to at least 100 pJ. An air-hole-modified- core PCF with appropriate dispersion and nonlinearity parameters has been shown to provide efficient generation of Stokes and anti-Stokes FWM sidebands shifted from the pump wavelength by more than 3000 cm^-1 (~0.1 PHz), thus offering an attractive source for nonlinear Raman microspectroscopy and bioimaging.