1. Introduction

Hollow light beams are broadly defined as beams with an intensity minimum (zero in the ideal case) along the beam axis. Such beams are used to guide and trap cold atoms [1, 2], confine Bose--Einstein condensate [3], and create optical traps [4, 5] and tweezers [6, 7] for the manipulation of micro-objects, including biological species. Hollow beams with appropriate polarization patterns can also generate optical vortices [8]. Light beams of this type can be generated either directly, by using specifically designed laser cavities [9], phase-front-modifying phase masks [10], or capillary waveguides [11], or indirectly, through the application of computer-generated holograms [12, 13]. A high efficiency of doughnut-beam generation can be achieved by using a liquid-crystal cell [14].

In this work, we demonstrate that high-quality frequency-tunable hollow beams can be efficiently generated through a nonlinear-optical transformation of ultrashort laser pulses in a photonic-crystal fiber (PCF). Fibers of this type [15, 16] have recently gained a broad application in optical technologies for the creation of highly efficient sources of supercontinuum radiation [17, 18], frequency-comb expanders [19, 20], and wavelength converters [21]. Here we show that, simultaneously with efficient frequency conversion, PCFs can transform a spatial profile of an input laser beam, permitting, in particular, the generation of high-quality hollow beams in the frequency-shifted output, thus suggesting attractive fiberoptic solutions for atom optics, biophotonics, and microparticle manipulation. In experiments presented below in this paper, femtosecond pulses of unamplified Ti: sapphire laser radiation are coupled into solitons in a high-order PCF mode, producing a hollow beam at the output of the PCF through the resonant emission of phase-matched dispersive waves in the wavelength range around 600 nm.

2. A few-mode photonic-crystal fiber

The cross-section view of the PCF used in our experiments is shown in Fig. 1(a). The fiber was made of fused silica using the standard PCF technology [15, 16]. The fiber core diameter is about 3 μm. The ratio of the air-hole diameter d to the period of the structure in the cladding Λ for this PCF is d/Λ ≈ 0.33, allowing the existence of a few guided modes in the fiber core within the wavelength range studied in this work. The number of modes supported by this PCF is, however, much less than that typical of PCFs with large diameters of air holes in the cladding (corresponding to large d/Λ ratios), which are often referred to as highly nonlinear PCFs. Such high-d/Λ PCFs have been shown earlier to allow the generation of frequency-shifted hollow-beam higher order modes as a part of their supercontinuum output [22]. However, a careful spectral filtering is necessary to separate those hollow-beam modes from the rest of the supercontinuum. Even with spectral filtering, the contrast of hollow beams generated by high-d/Λ PCFs is limited because of radiation in the fundamental and other guided modes emitted in the same wavelength range. By contrast, the PCF used in this work supports only a few guided modes. This prevents distortions of the hollow-beam PCF output due to the generation of other high order modes in the same spectral range, giving rise to a clean hollow beam at the output of the fiber spectrally isolated from the long-wavelength part of the field propagating in the PCF.

To analyze dispersion properties of PCF modes [Fig. 1(b)], their field intensity profiles [Figs. 2(a)–2(c)], and wavelength dependences of effective mode areas [Fig. 2(d)], we numerically solved 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]. In Figs. 2(a)–2(c), we present the spatial field intensity profiles for the fundamental and a high-order mode of the PCF. While the fundamental mode [Fig. 2(a)] displays a typical bell-shaped beam profile with the maximum achieved along the fiber axis, for some of the higher order modes, the field intensity vanishes along the beam axis, giving rise to a hollow-beam structure of the intensity profile [Figs. 2(b), and 2(c)] with a diameter of the central dark area of less than 1 μm. With the symmetry of guided modes controlled by the symmetry of the waveguide structure, PCFs offer a unique flexibility in tailoring intensity profiles and polarization patterns of hollow-beam modes. In particular, for the considered PCF design, hollow-beam modes arising as a superposition of doublets of higher-order guided modes [24] can display either radial [(a superposition of HE₂₁- and TM₀₁-type modes, shown in Fig. 2(b)] or spiral [a superposition of TE₀₁- and HE₂₁-type modes, illustrated in Fig. 2(c)] polarization patterns, suggesting a variety of strategies for atom guiding, optical trapping and laser tweezing.

 

Fig. 1. (a) An SEM image of the photonic-crystal fiber. (b) Group-velocity dispersion (GVD) as a function of radiation wavelength for the fundamental (1) and hollow-beam (2) PCF modes.

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3. The laser system

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 (Millennia VS, Spectra-Physics). A Brewster-cut Ti: sapphire crystal with a length of 2.3 mm was 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 were 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 – 60 fs, an energy up to 5 nJ at a pulse repetition rate of 100 MHz and a central wavelength of 800 nm. A 40× 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 PCF output was collimated with an identical lens and was studied with an Ando spectrum analyzer.

 

Fig. 2. (a)–(c) Transverse field intensity profiles for (a) the fundamental and (b, c) hollow-beam PCF modes corresponding to a superposition of (b) HE₂₁- and TM₀₁-type modes and (c)TE₀₁- and HE₂₁-type modes. (d) The effective mode area for (1) the fundamental (inset 1) and (2) hollow-beam (inset 2) modes of the PCF calculated as a function of the radiation wavelength.

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4. Results and discussion

The fundamental wavelength of Ti: sapphire laser radiation falls within the range of normal dispersion for the fundamental mode of our PCF [curve 1 in Fig. 1(b)], as the group-velocity dispersion (GVD) vanishes at the wavelength of 830 nm for this mode. For the hollow-beam mode, shown in Figs. 2(b), 2(c), the zero-GVD wavelength is 760 nm, providing anomalous dispersion for 800-nm Ti: sapphire laser pulses [curve 2 in Fig. 1(b)]. Accordingly, the fundamental and hollow-beam modes of the PCF provide radically different regimes of nonlinear-optical spectral transformation for Ti: sapphire laser pulses. With an input beam aligned exactly along the axis of the PCF, the laser field was coupled into the fundamental mode of the fiber. In this experimental geometry, the beam at the output of the fiber had a typical bell-shaped profile [Fig. 3(a)]. The spectral transformation of Ti: sapphire laser pulses in this regime was dominated by self-phase-modulation-induced spectral broadening [Fig. 3(b)], resulting in a compact output spectrum, which stretches from 560 to 820 nm with an input laser power of 200 mW.

Similar to standard fibers, the efficiency of radiation coupling into higher order PCF modes is increased by tilting the input beam with respect to the fiber axis. Using this geometry, we were able to produce the hollow-beam PCF mode shown in Figs. 2(b), 2(c). In this mode, Ti: sapphire laser pulses sense anomalous dispersion and tend to generate optical solitons. These solitons undergo a continuous frequency down-shift due to the Raman effect as they propagate through the fiber╍phenomenon known as soliton self-frequency shift (SSFS) [25]. The red shifted peaks resulting from this process are readily observed in PCF output spectra presented in Fig. 4(b).

High-order dispersion tends to perturb solitons, making them unstable with respect to emission of nonsolitonic, dispersive waves. As shown by Wai et al. [26], as well as by Akhmediev and Karlsson [27], instabilities of solitons induced by third- and fourth-order dispersion are manifested as radiation at specific frequencies where the soliton wave number is in resonance with the wave number of a dispersive wave. In PCFs, such dispersive-wave emission has been shown [28] to be one of the key physical mechanisms behind supercontinuum generation and frequency up-conversion of ultrashort laser pulses. In the experiments presented here, dispersive-wave emission is observed as a prominent spectral feature centered at 600 nm in Fig. 4(b). The far-field pattern of this dispersive-wave emission [Fig. 4(a)] displays a characteristic structure of the hollow-beam PCF mode [Figs. 2(b), 2(c)] with the field intensity vanishing along the beam axis and the beam profile featuring a spatial symmetry with the sixth-order rotation axis, inherited from the fiber structure.

For both the fundamental and hollow-beam modes of the PCF [Figs. 3(a), 4(a)], the blue-shifted part of output emission [observed as yellowish part of the beam in Figs. 3(a), 4(a)] has a much more compact spatial profile than the longer wavelength part of the spectrum (the reddish part of the beam). This observation is fully consistent with the wavelength dependence of the effective mode area calculated for the fundamental [curve 1 in Fig. 2(d)] and hollow-beam [curve 2 in Fig. 2(d)] modes of our PCF using the polynomial expansion technique. For longer wavelengths, as can be seen from Fig. 2(d), diffraction tends to make the fiber modes less compact. As a result, the yellow spot, representing the blue-shifted part of the PCF output, is much stronger confined to the central part of the output beam pattern [Figs. 3(a), 4(a)] than the reddish part of the beam, which represents the frequency-nonshifted part of output emission with frequencies close to the central frequency of the input pulse.

Resonant energy exchange between a soliton with a central frequency ω _s and a dispersive wave with a frequency ω _s + Ω can occur [27] when the propagation constant of the soliton β _s is matched with the propagation constant β of the dispersive wave, δβ= β - β _s = 0. In the regimes where the third-order dispersion plays a much more significant role than the other high-order dispersion terms, the condition of wave matching between the soliton and the dispersive wave is written as δβ = β ₂Ω²/2 + β ₃/Ω³/6 - γP/2 = 0, where β_m =∂^m β/∂ω^m is the mth-order dispersion, γ is the nonlinear coefficient, and P is the peak power. The Kerr effect, as can be seen from the expression for δβ, allows the frequency Ω to be tuned by varying the peak power P. In Fig. 5, we demonstrate such a wavelength tunability of the hollow-beam PCF output by presenting the PCF output spectra measured for two different powers of 40-fs Ti: sapphire laser pulses. As the power of the input pulse increases, the central wavelength of the dispersive wave is noticeably blue-shifted. The inset to 5 illustrates the tunability of the hollow-beam output by showing the wavelength shift of the hollow-beam output relative to the central wavelength of the input pulse (800 nm) measured as a function of the input peak power. Careful dispersion profile engineering, on the other hand, makes it possible, as is also seen from the above expression δβ, to control the tunability range of hollow-beam dispersive-wave PCF output by varying the ratios of PCF dispersion parameters β_m to the nonlinear coefficient γ.

The spectral intensity of the 600-nm hollow-beam component in the PCF output spectrum, as shown by curve 2 in Fig. 5, can be a factor of 1.7 higher than the spectral intensity around the central wavelength of the input field, indicating a high efficiency of beam-profile transformation provided by the PCF. The efficiency of radiation power coupling into the hollow-beam PCF output, defined as the average power of the hollow-beam output to the input average power, is estimated as 8 and 15% for input powers of 200 and 360 mW, respectively.

 

Fig. 3. Spectral transformation of 60-fs Ti: sapphire laser pulses in the fundamental mode of the PCF: (a) the far-field beam profile of the PCF output and (b) the spectral intensity of the PCF output as a function of the radiation wavelength and the average power of the input field. The fiber length is 120 cm.

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Fig. 4. Spectral and beam-profile transformation of 60-fs Ti: sapphire laser pulses in the hollow-beam mode of the PCF: (a) the far-field beam profile of the PCF output and (b) the spectral intensity of the PCF output as a function of the radiation wavelength and the average power of the input field. The fiber length is 120 cm.

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Fig. 5. The spectra of the hollow-beam PCF output measured with two different input peak powers of 40-fs Ti: sapphire laser pulses: (1) 3 kW and (2) 30 kW. The inset shows the wavelength shift ∣Δλ∣ of the hollow-beam output relative to the central wavelength of the input pulse measured as a function of the input peak power.

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5. Conclusion

We have demonstrated that high-quality spectrally isolated hollow beams can be produced through a nonlinear-optical transformation of Ti: sapphire laser pulses in a higher order mode of a specifically designed few-mode PCF. Instead of a doughnut shape, typical of hollow beams produced by other methods, the far-field image of the hollow-beam PCF output features perfect sixth-order rotation symmetry, dictated by the symmetry of the PCF structure. Another important difference of hollow beams generated by PCFs is that, near the fiber end, such beams can have a central dark core less than a micron in diameter, which offers important advantages for optical tweezers and atom guides. Finally, the frequency of the PCF-generated hollow beam can be tuned by varying the input beam parameters, making PCFs a convenient and flexible tool for hollow-beam-based atom optics, microparticle manipulation and biophotonics.