1. Introduction

In stimulated Raman scattering (SRS), pump photons of frequency ${\omega }_{\text{P}}$ are scattered into Stokes photons of frequency ${\omega }_{\text{S}}={\omega }_{\text{P}}-{\Omega }_{\text{R}}$ where ${\Omega }_{\text{R}}$ is the Raman frequency shift. This process is accompanied by the excitation of molecules—the scatterers—into a higher rotational or vibrational state. The synchronously oscillating scatterers form a coherence wave with frequency ${\Omega }_{\text{R}}$ and phase velocity ${\Omega }_{\text{R}}/{\beta }_{\text{C}}$, where ${\beta }_{\text{C}}={\beta }_{\text{P}}-{\beta }_{\text{S}}$ and ${\beta }_{\text{P}}$ and ${\beta }_{\text{S}}$ are the propagation constants of the pump and Stokes waves.

If an excited molecule scatters a pump photon while decaying to the ground state, a blue-shifted anti-Stokes (AS) photon of frequency ${\omega }_{\text{AS}}={\omega }_{\text{P}}+{\Omega }_{\text{R}}$ is generated. This phenomenon can for example be used to generate narrowband radiation in the deep and vacuum ultraviolet spectral region [1]. Furthermore, CARS spectroscopy [2] and Raman frequency combs—which can span several octaves [3]—also rely on the generation of an AS signal. Phase-coherent frequency combs even provide a method for generating trains of ultra-short pulses [4, 5], which can be especially interesting when pumping with a continuous wave laser [6, 7].

Efficient AS generation—which is crucial for all these applications—in general requires phase-matching, i.e., the wavevector mismatch $\Delta \beta =({\beta }_{\text{P}}-{\beta }_{\text{S}})-({\beta }_{\text{AS}}-{\beta }_{\text{P}})={\beta }_{\text{C}}-({\beta }_{\text{AS}}-{\beta }_{\text{P}})$ must be zero. This condition is not automatically satisfied since the group velocity dispersion is typically non-zero. In free space, this mismatch is commonly eliminated using oblique beams [2]. Other techniques include using dispersion-compensating mirrors in a cavity [7], or birefringent waveguides [8].

Gas-filled photonic crystal fibers (PCF) provide huge advantages for SRS applications compared to free-space systems, including exceptionally low thresholds [9], and pressure-controllable dispersion [10]. If one wishes, however, to establish a system based on single-pass propagation through a fiber, neither oblique beams nor dispersive mirrors can be used to achieve phase-matched AS generation.

Here we present a fiber-based method for obtaining phase-matching for efficient AS generation, making use of higher-order modes (HOMs) in a hollow-core photonic crystal fiber (HC-PCF) [11, 12]. This method works because the dispersion profiles of HOMs differ considerably from that of the fundamental LP₀₁ mode. The technique involves arranging a combination of modes (or modal mixtures) in the pump, Stokes and AS signals that approximately satisfies phase-matching, and then fine-tuning for efficient AS generation by adjusting the gas pressure.

2. Experimental set-up and techniques

2.1 Basic set-up

A schematic of the experimental set-up is shown in Fig. 1. The pump source is a 1064 nm microchip laser delivering pulses of duration 1.8 ns (FWHM) at a repetition rate of 1 kHz. Its Gaussian beam is projected onto a spatial light modulator (SLM) [13], and then coupled into a hydrogen-filled kagomé-PCF. By adjusting the SLM, a controllable mixture of fundamental and HOMs is excited in the fiber. Depending on the experimental conditions (gas pressure, pump energy and pump mode) SRS-induced first and second vibrational AS signals are observed in different HOMs at the fiber output. After passing through appropriate bandpass filters for spectral selection, the signal power is measured and its near-field pattern imaged by a CCD camera.

 

Fig. 1 Schematic of the experimental set-up. λ/2: half-wave plate; PBS: polarizing beam splitter; BP: bandpass filter.

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2.2 Properties of gas-filled kagomé-PCF

The fiber used in the experiment is a 1.2 m long, broadband-guiding kagomé HC-PCF filled with hydrogen gas. A scanning electron micrograph of its structure is shown in Fig. 2. The effective refractive index of the LP_pm mode guided in a kagomé-PCF can be written to good accuracy as [10, 14]:

 

Fig. 2 Scanning electron micrograph of the kagomé-PCF used in the experiments.

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We make use of the vibrational Raman transition Q(1) of ortho-H₂, which provides a Raman frequency shift of ${\Omega }_{\text{R}}/2\pi =125\text{THz}$ (4155 cm⁻¹) [16]. When pumped at 1064 nm this results in a Stokes signal at 1912 nm, a first AS signal at 737 nm and a second AS signal at 564 nm. The process begins once the first Stokes signal has reached threshold. In order to optimize generation of the AS signal, the hydrogen pressure, the pump energy and the pump mode mixture are varied. The highest pressure available in the gas-filling system was 25 bar, and the maximum pump pulse energy coupled into the fiber was 50 µJ.

2.3 Generating modal mixtures with SLM

The electronically-driven liquid crystal array in an SLM changes the local phase of the beam pixel-by-pixel. In order to excite a certain mode mixture in the fiber, its phase pattern is imprinted on the incoming Gaussian beam by means of the SLM [Fig. 3]. However, as the amplitude of the beam cannot be changed with the phase-only SLM used in the experiment, the desired mode mixture is not exactly reproduced in the fiber. Even so, since the field distribution reflected off the SLM is known, the mode mixture excited at the fiber input face can be calculated. This is done by calculating the complex amplitude distribution of the focused beam at the fiber input face using the Fraunhofer (far-field) approximation. Then the overlap between this field and the LP_pm fiber mode is determined numerically [10, 14], and the fraction f_pm of energy launched into the LP_pm mode determined. Note that, when calculating the SLM phase pattern of the mixture of LP₀₁ and LP₁₁ modes, we ignore the radial dependence, retaining only the correct azimuthal phase function. This has the advantage of making the SLM phase pattern independent of the beam radius of the incoming Gaussian beam [Fig. 3(a)].

 

Fig. 3 Phase patterns created on the SLM for exciting (a) a mixture of LP₀₁ and LP₁₁ modes, (b) a mixture of LP₀₁ and LP₀₂ modes within the kagomé fiber.

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3. Experimental results

3.1 Basic observations

When the pump light was launched purely in the fundamental mode, no significant AS signal was seen. The situation changed dramatically, however, when the pump beam contained a mixture of fundamental and LP₁₁ modes [Fig. 4(a)]. In this case, a first AS signal was generated in the LP₁₁ mode [Fig. 4(b)], appearing either as a double-lobed mode or a doughnut mode. A second AS signal was observed in a clean four-lobed LP₂₁ mode at pressures below ~20 bar [Fig. 4(c)], gradually switching to the LP₁₁ mode at higher pressures. At 25 bar (the highest pressure available in the gas system) the second AS signal was in a distorted LP₁₁ mode [Fig. 4(d)], which we attribute to the presence of some LP₂₁ mode. As explained later (Section 3.3), at even higher pressures we expect to see a clean LP₁₁ mode. When the pump was changed to a mixture of LP₀₁ and LP₀₂ modes, the AS signal switched to the LP₀₃ mode, as shown in Fig. 4(e).

 

Fig. 4 Intensity distributions of the modes at the fiber end-face: (a) pump in a 1:2 mixture of LP₀₁ and LP₁₁ modes; (b) first AS in the LP₁₁ mode; (c) second AS in the LP₂₁ mode (H₂-pressure 6 bar); (d) second AS in the LP₁₁ mode distorted by some LP₂₁ content (H₂-pressure 25 bar); (c) first AS in the LP₀₃ mode the pump being a mixture of LP₀₁ and LP₀₂ modes.

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3.2 Analysis of results

Figures 5(a)–5(c) explain the generation of the AS signals when the pump light is in a mixture of fundamental and LP₁₁ modes. The frequency conversion takes place in three stages. First, a stimulated Stokes signal is generated in the LP₀₁ mode, mediated by the coherence wave marked with a green arrow in Fig. 5(a). This signal then seeds the generation of a second coherence wave [purple arrow in Fig. 5(a)] that converts LP₁₁ pump photons to Stokes photons in the LP₀₁ mode. In the second stage, the first AS is generated [Fig. 5(b)]. No coherence wave is available to drive the red dashed transition towards the first AS in LP₀₁; neither the green nor the purple coherence wave fits. An AS signal can however be generated in the LP₁₁ mode using either the purple coherence wave in conjunction with the pump in the LP₀₁ mode or the green coherence wave in conjunction with the pump in the LP₁₁ mode. Note that this is only possible if the pump signal is simultaneously available in the two modes. If not, none of the two transitions will work because of the lack of either a suitable coherence wave or pump light in an appropriate mode. In the third stage a second AS signal can be generated in the LP₂₁ mode by re-using the purple coherence wave, or in the LP₁₁ by exploiting the green coherence wave [Fig. 5(c)]. Depending on the pressure, conversion to either the LP₂₁ or the LP₁₁ mode is preferred, as explained in Section 3.3.

 

Fig. 5 Diagrams showing the pump, Stokes, first and second AS frequencies together with the dispersion curves for the LP₀₁, LP₁₁, LP₂₁, LP₀₂ and LP₀₃ modes, plotted versus the wavevector difference between k₀ (the vacuum wavevector) and β (the propagation constant in the kagomé-PCF at zero pressure). Note that k₀ changes linearly with the optical frequency. (a) The pump energy is shared between the LP₀₁ and the LP₁₁ modes. A Stokes signal is generated in the LP₀₁ mode, accompanied by excitation of the green and the purple coherence waves. (b) The first AS signal in LP₀₁ mode is not phase-matched since neither the green nor the purple coherence wave can drive the red dashed transition. However, a first AS signal can be generated in the LP₁₁ mode using the purple coherence wave along with an LP₀₁ pump, or the green coherence wave along with an LP₁₁ pump. (c) Second AS signal is generated in the LP₂₁ mode using the purple coherence wave, or in the LP₁₁ mode using the green coherence wave. (d) With a pump shared between the LP₀₁ and the LP₀₂ modes, Stokes generation in the LP₀₁ mode is accompanied by the creation of the yellow and the green coherence waves. The yellow coherence wave is re-used for the conversion of the LP₀₂ pump to an LP₀₃ AS signal.

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The appearance of the first AS signal in the LP₀₃ mode can be explained in a similar manner by means of the dispersion curves for modes of higher radial order [Fig. 5(d)]. The pump is a mixture of LP₀₁ and LP₀₂ modes. The Stokes is again generated in the LP₀₁ mode, but this time a coherence wave is generated by scattering from the LP₀₂ pump mode to the LP₀₁ Stokes mode (yellow arrow). This coherence wave is phase-matched for conversion of the LP₀₂ pump to an AS signal in the LP₀₃ mode.

3.3 Pressure-dependence

Generation of the AS signals is affected by the core refractive index, which changes with pressure [Eq. (1)]. This results, for example, in switching of the second AS signal from the LP₂₁ mode at low pressure to the LP₁₁ mode at high pressure. In Fig. 6 the phase-mismatch calculated using the dispersion curves is plotted versus the pressure. Efficient AS generation is not only possible when the phase-mismatch is zero, but can also occur for small amounts of phase-mismatch by nonlinear phase-locking [17–19]. This allows the phase-mismatch to be compensated by SRS-induced nonlinear phase-shifts of pump, Stokes and anti-Stokes signals. It is even possible that the AS signal will be stronger in the presence of a slight phase-mismatch because parametric gain suppression can occur at the exact phase-matching point [20] (gain suppression occurs when the “pump+Stokes” and “anti-Stokes+pump” coherence waves destructively interfere, resulting in loss of Stokes and anti-Stokes signals). As the phase-mismatch increases, however, AS generation via phase-locking becomes more and more inefficient, eventually fading away. Consequently, efficient AS generation is only possible close to the phase-matching pressure.

 

Fig. 6 Phase-mismatch versus pressure, (a) for conversion to a first AS signal in the LP₀₁ (blue), LP₀₃ (red) or LP₁₁ mode (green), (b) for conversion to a second AS signal in LP₁₁ (purple) or LP₂₁ mode (cyan). The experimental pressure was limited to 25 bar.

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Note, however, that pressure changes influence not only the dispersion but also the gain of the Raman medium and the dephasing time T₂ of the coherence waves [21]. This makes the pressure dependence of the AS signal highly complicated. We now present a qualitative explanation of the observations.

Over the pressure range used in the experiments (1 to 25 bar) the phase-mismatch for the first AS signal [Fig. 6(a)] is considerably larger when it is in the LP₀₁ (blue line) than when it is in the LP₀₃ (red line) or the LP₁₁ mode (green line). This explains the experimental observation that the AS signal appears in the two HOMs. Indeed, the phase-mismatch is small enough for both LP₁₁ and LP₀₃ modes to allow nonlinear phase-locking (as mentioned above), and thus efficient AS generation. Note that in the presence of phase-locking, pressure and phase-mismatch play only secondary roles. Which of the two HOMs is preferentially excited can be controlled by altering the modal mixture of the pump light. If the pump contains a mixture of LP₀₁ and LP₀₂ modes, the AS signal is generated in the LP₀₃ mode. If however it contains the LP₀₁ and the LP₁₁ modes, the AS signal appears in the LP₁₁ mode.

In contrast, the phase-matching conditions for second AS generation [Fig. 6(b)] change considerably with pressure, as does the observed mode. At lower pressures, the phase-mismatch is smaller for the LP₂₁ mode (cyan line) and a four-lobed mode is observed. The higher the pressure, the better is the phase-matching for the second AS in the LP₁₁ mode (purple line). As a result, competition between the LP₂₁ and the LP₁₁ modes is observed at pressures above ~20 bar in the experiments.

3.4 Optimizing the anti-Stokes signal

According to the explanation given in the paragraphs above, it is clear that efficient generation of the first AS signal in the LP₁₁ mode depends strongly on the presence of both LP₀₁ and the LP₁₁ modes at the pump frequency. In order to investigate this further, we used the SLM to vary the ratio of the two pump modes while keeping the total coupled pump energy constant. We define the conversion efficiency as the AS output energy divided by the pump output energy in the absence of SRS; this takes account of coupling and fiber losses at the pump wavelength. The conversion efficiency is plotted in Fig. 7 against the fraction f₁₁ of pump energy in the LP₁₁ mode.

 

Fig. 7 Conversion efficiency from pump to first AS in the LP₁₁ mode (output AS energy over output pump energy in the absence of SRS) plotted against the fraction f₁₁ of pump energy in the LP₁₁ mode. The fiber length was 1.2 m. The red curve (left-hand y-axis) is for a total coupled pump energy of 11 μJ at a pressure of 25 bar, and the blue curve (right-hand y-axis) for 15 μJ and 15 bar.

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It can be seen in Fig. 7 that conversion to the AS is efficient only for an appropriate pump mode mixture, approaching zero when the pump light is in a pure LP₀₁ or LP₁₁ mode (the non-zero experimental values for f₁₁ = 0 or f₁₁ = 1 can be explained by coupling of a small fraction of the pump energy into undesired modes). The AS conversion efficiency drops drastically at values of f₁₁ greater than 0.6. Since in this range the pump is mainly in the LP₁₁ mode, the gain for the LP₀₁ Stokes signal is considerably reduced [12]. As a consequence the threshold for the generation of Stokes in the LP₀₁ mode is no longer reached, resulting in suppression of the AS signal.

As the pressure is increased from 15 bar (blue curve) to 25 bar (red curve), the conversion efficiency increases considerably—this was explained in Section 3.3. The maximum AS conversion efficiency reached for a fiber length of 1.2 m is 5.3%, obtained for a pump energy of 11 µJ, a pressure of 25 bar and f₁₁ = 0.14. The trend indicates that even higher AS conversion efficiencies should be attainable at higher pressures.

4. Conclusions

Launching mixtures of fundamental and higher-order modes offers an elegant and efficient means of achieving phase-matched anti-Stokes generation by SRS in gas-filled HC-PCF. First and second AS signals, carried by HOMs, can be generated with conversion efficiencies as high as 5.3% for the first AS. By changing the pressure and the modal mixture of the pump, the first and second AS signals can be manipulated and optimized. The technique is highly versatile and can be used with almost any combination of pump wavelength, Raman-active gas and HC-PCF. It may prove useful for highly sensitive CARS spectroscopy of trace gases.