Inter-modal phase-matched third harmonic generation has been demonstrated in an exposed-core microstructured optical fiber. Our fiber, with a partially open core having a diameter of just 1.85 µm, shows efficient multi-peak third-harmonic generation between 500 nm and 530 nm, with a maximum visible-wavelength output of 0.96 μW. Mode images and simulations show strong agreement, confirming the phase-matching process and polarization dependence. We anticipate this work will lead to tailorable and tunable visible light sources by exploiting the open access to the optical fiber core, such as depositing thin-film coatings in order to shift the phase matching conditions.
© 2016 Optical Society of America
Microstructured optical fibers (MOFs) are a class of optical fiber that is constructed of micro- or even nanometer-size longitudinal air holes that run along the entire fiber length [1–3]. Perhaps the simplest MOF design is the suspended-core fiber where a central glass core is surrounded by a small number of air holes, typically three, which is a close approximation to a glass rod suspended in air [4–6] and allows guidance in very small cores due to the high index contrast between glass and air. Core diameters as small as 800 nm have been achieved by using fused silica glass . Even smaller diameters have been achieved using soft glasses with higher refractive indices, such as 420 nm in lead-silicate glass , 480 nm in tellurite glass , and 450 nm in bismuth glass . These extreme dimensions provide new opportunities for nonlinear interactions, such as supercontinuum generation [11–13], given that the nonlinear parameter is inversely proportional to the effective area of the propagating mode .
Third harmonic generation (THG) is a phase matched nonlinear process that is typically achieved in crystalline structures by controlling polarization within a birefringent material in order to compensate the material dispersion . In optical fibers this is not possible and thus inter-modal phase matching, typically of the fundamental mode to modes of higher order, is usually required in order to counter the material and waveguide dispersion. THG was first demonstrated in step-index fibers by Gabriagues in 1983, though the phase matching conditions were not specifically engineered or analyzed . Inter-modal phase matched THG has since been demonstrated in configurations such as highly germanium doped fibers [17–19], tapered silica microfibers [20,21], silica suspended-core fiber [22–24], and argon-filled hollow-core photonic crystal fiber . Furthermore, broadband THG has been demonstrated in tapered silica fibers . Alternatively, weak third harmonics can be generated outside of phase matching conditions directly into the fundamental mode by using highly nonlinear material such as tellurite glass, which has been demonstrated for both solid step-index fibers  and suspended-core fibers .
In this paper we demonstrate inter-modal phase-matched THG in a new class of suspended core fiber referred to as an exposed-core fiber (ECF), where part of the cladding has been removed in order to provide direct access to the evanescent field of the core [28,29]. Such fibers have recently been demonstrated in a host of sensing related experiments such as fluorescence sensing , fiber Bragg grating refractometry , rubidium spectroscopy , and multimode  and Fabry-Perot  interferometry. Nonlinear light generation in these fibers, in particular THG, will open up new opportunities for tuning and tailoring modal and pulse dispersion and thus the nonlinear interaction, as the open core can easily be modified via post-processing (e.g. coating) with a wide range of materials.
2. Third harmonic generation experiment
Fused silica glass exposed-core fibers, with a continuous length of 80 meters, was fabricated using a ultrasonic drilled preform as detailed in a previous publication (cross-sectional scanning electron microscope (SEM) image shown in Fig. 1) . The fiber has an outer diameter of 145 µm [Fig. 1(a)] and an effective core diameter of 1.85 µm [Fig. 1(b)], using the definition of core diameter in .
To demonstrate THG in the ECF we used a 1560 nm femtosecond laser (Toptica, FemtoFiber pro IRS, 80 MHz, <40 fs pulses, average power 220 mW) as the pump source. We controlled both the input pump power and polarization using a combination of a rotating half wave plate and a polarizer. The pump beam was then focused into the exposed-core fiber using an aspheric lens (Thorlabs, C230TM-C). A short length of fiber was used (3.3 mm) in order to reduce the probability of contamination of the exposed-core (e.g. dust) leading to increased fiber loss. However, no increase in fiber loss was measured during the course of the experiment. The output was imaged using a 40X microscope objective and a camera in order to confirm coupling into the fiber core. Coupling efficiencies into the ECF core were in the range of 15-20%. The variation was primarily due to adjustment of the half-wave plate during the experiment, which slightly alters the pump beam path, and the consequent impact on coupling into the small 1.85 µm core. Once optimal coupling was achieved the output was coupled into a graded-index multi-mode fiber using a 4X microscope objective and connected to an optical spectrum analyzer (ANDO 6315A). The output spectra of the third harmonic (TH) signals for vertical input polarization are shown in Fig. 2, while the inset shows the corresponding transmitted input infra-red (IR) spectra.
We observe a series of well-separated peaks in the spectral distribution of the output power between 500 nm and 530 nm with a power maximum power at 509 nm. The spectral shape is preserved for different input powers for the THG wavelengths, indicating the absence of other nonlinear effects influencing the phase matching process. Even though ultra-short optical pulses are used here, no nonlinear effects are observed in the domain of the IR pump wavelength (i.e. shape of the spectral power distribution remains constant), indicating separation between the zero-dispersion wavelength of the group velocity dispersion, and the pump and TH wavelengths. In particular, no other generated wavelengths were measured above the detection noise limit (−80 dBm) for the range 350-1750 nm.
The integration (total power) of the spectra in Fig. 2 is shown in Fig. 3 against the femtosecond laser power that was coupled into the fiber core. In particular, the experimentally measured power displays a cubic relationship with input power, adding evidence that the visible spectra in Fig. 2 results from third harmonic generation. From Fig. 3, the maximum efficiency of converting coupled power into THG was 3.1x10−5 (0.96 μW generated from 30.5 mW coupled power) while the maximum overall system efficiency was 4.5x10−6 (210 mW total fs laser power). This compares well with previously reported values of overall system THG conversion efficiency, such as in silica microfibers (e.g. 2.5x10−7  and 2x10−6 ) and highly germanium-doped step-index fibers (3x10−7 ).
The exposed-core fiber structure is highly asymmetric (Fig. 1) and thus it is expected that polarization of the input beam should have a significant impact on the phase-matching conditions and thus the resulting TH-signal. Figure 4 shows the TH-spectra for two orthogonal input polarizations, corresponding to the orientations shown in Fig. 1(b). It can be seen that the TH-signal for vertically orientated polarization is over one order of magnitude stronger than for the horizontal polarization (coupled power v- and h-pol.: 29.0mW and 30.8mW). As will be explained in the following section, this can be a result of both phase matching conditions and modal overlap. Note also that for the horizontal polarization there are two distinct peaks rather than a series of peaks seen in the vertical case, which correspond to two phase matched visible modes.
3. Numerical modeling
We simulated the different propagating modes using finite element modeling (COMSOL v5) on the structure indicated by the white dashed box of Fig. 1(b) in order to understand the phase matching conditions responsible for the measured TH-signals. A rectangular perfectly matched layer (PML) boundary condition was applied to prevent spurious modes from forming at the termination of the ECF struts. The dispersion values for the relevant eigenmodes were calculated in the range of the THG [450-550 nm, Fig. 5(a)] and the IR pumping range [1500-1600 nm, Fig. 5(b)]. As a result of the PML boundary conditions the calculated effective indices include an imaginary component, indicating how well confined the mode is to the fiber core. In the visible (THG) wavelength region only modes for which the imaginary part was less than 10−3 (approx. 100 dB/mm) are shown in Fig. 5(a). Also note that four other modes exist for the IR wavelengths, below neff = 1.25.
Figure 5(a) shows that there is a high density of modes that exist at the TH-wavelengths, providing high probability for inter-modal phase matching. This is expected given the high numerical aperture of the air-clad fiber geometry. In addition to phase matching, THG also requires polarization and modal overlap between the modes at the pump and generated wavelengths. To a first approximation, the THG modes should exhibit peak intensities at the center of the fiber core with a polarization matching that of the mode at the fundamental wavelength. All such modes that are close in effective index (< 1.0%) to the fundamental mode are highlighted in Fig. 5(a) (purple: horizontal modes, green: vertical modes). There are two such TH-modes that have predominantly vertically orientated polarization at the center [Figs. 6(b) and 6(c)] and two that are predominantly horizontally polarized at the center [Figs. 6(f) and 6(g)], which matches the polarization of the corresponding mode at the fundamental wavelength [v: Fig. 6(a), h: Fig. 6(e)].
The experimentally recorded mode images of the TH-signal, after filtering of the IR pump wavelengths, are shown in Figs. 6(d) and 6(h). Comparison between the simulated vertically polarized mode images and the measured THG mode shows that the mode labeled by v2 in Fig. 5(a) is responsible for the inter-modal phase matching. The difference in effective index between the IR and THG wavelengths is Δneff = 0.0023 (0.17%), which is likely attributed to errors associated with recording the SEM image and importing it into COMSOL, rather than an actual significant phase difference. A potential contribution of a pump induced nonlinear phase shift has been calculated to be less than 0.7% and thus can be neglected here.
Understanding the THG-process in the horizontal polarization is more complex considering there are two peaks present in the TH-spectrum (Fig. 4). The measured mode image [Fig. 6(h)] exhibits features from both of the two modes shown in Figs. 6(f) and 6(g). In any case, the multi-lobed structure of the measured mode images in Figs. 6(d) and 6(h), and the strong agreement with simulations particularly for the vertical polarization, provides strong evidence that the visible emission observed in Fig. 2 is indeed inter-modal phase-matched third harmonic generation.
4. Discussion and conclusions
We have demonstrated inter-modal phase-matched third harmonic generation in an exposed-core microstructured optical fiber. Our fiber has a core diameter of only 1.85 µm leading to a strong field confinement for both the fundamental and third harmonic waves. We observe efficient multi-peak third-harmonic generation between 500 nm and 530 nm with a maximum generated average power at visible wavelengths of 0.96 μW. This corresponds to a total system efficiency of 4.5x10−6, comparable to previous reports of THG in microfibers. The experimental results agree well with modal simulations, including the mode patterns, effect of polarization, and phase difference between the IR and THG wavelengths.
The use of an exposed-core structure will open up opportunities in tuning and tailoring nonlinear light generation in fibers using post-processing steps that would otherwise be difficult to achieve in conventional enclosed microstructured optical fiber geometries. In future, thin-film coatings can be deposited along the entire exposed core of the fiber in order to tailor the modal and pulse dispersion and hence the phase-matching conditions. Alternatively, the open channel can be filled with liquids in order to enhance the nonlinearity, though care would need to be taken to ensure guidance is still achieved. Such techniques may be used for improving the THG efficiency or to allow the generation of light at specific wavelengths and within specific spatial modes that otherwise could not be achieved by controlling only the fiber fabrication (i.e. preform geometry and fiber drawing conditions), thus balancing out production inaccuracies via post-fabrication processing. The open channel also allows for accessing ultrafast excitations propagating in the core (e.g. solitons and dispersive waves) via the evanescent field, which can be useful for investigating the spatial dependence of nonlinear processes and might enable new schemes for pump-probe spectroscopy.
This work may provide a path for the reverse process of triple photon generation (TPG) for quantum optics applications in a fiber integrated device. The challenge of inter-modal phase-matching for TPG is to efficiently launch light into the higher order modes of an optical fiber. The exposed-core fiber offers a potential solution by advancing recently demonstrated surface relief fibre Bragg gratings (FBGs) . By adapting such FBGs as grating-couplers, side launching with an appropriate angle could be used to satisfy the phase-matching condition for launching directly into specific higher order modes. The ability to tailor the dispersion as described above may also be critical as the inter-modal phase matching condition may be adjusted for optimal operation at wavelengths available from commercial high peak power sources (e.g. 532 nm).
European Commission through the Seventh Framework Programme (PIIF-GA-2013-623248). ARC Georgina Sweet Laureate Fellowship; ARC Centre of Excellence for Nanoscale Biophotonics (CE14010003); The Australian Defence Science and Technology Group (under the Signatures, Materials and Energy Corporate Enabling Research Program); The OptoFab node of the Australian National Fabrication Facility utilizing Commonwealth and South Australian State Government funding; German Science Foundation (DFG, SCHM 2655/3-1); Free State of Thuringia (Fkz: 2015-0021).
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