1. Introduction

Intermodally phase-matched processes and harmonic generation in optical fibers have a long history. It was recognized long ago that because of the long interaction length allowed in fibers nonlinear processes can be very efficient, even though the absolute values of the nonlinear constants are extremely low for silica glass.

Observation of high-order guided modes in a standard multimode fiber through a nonlinear process of four-wave mixing was first performed by Stolen and coworkers [1], where Stokes and Antistokes components were generated in different guided modes. Third, and more surprisingly, second harmonic generation in telecom fibers was observed in the 80’s with efficiencies up to several percent [2, 3, 4]. No explanation for these observations was offered [5] until it was realized that it was the photosensitivity of the germanium-doped glass which was responsible for both the effective second-order nonlinearity and the high efficiency of the process [6, 7]. Currently, intermodally phase-matched second harmonic generation in planar waveguide structures is being explored as a potential source of new frequency components [8].

 

Fig. 1. Phase-matching for the THG process in a weakly guiding fiber with core-cladding index difference of 0.01. Guided modes are located between core and cladding index curves and radiation modes comprise a continuum below the cladding line. Cherenkov-type phase matching is schematically shown at the top.

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We do not expect the second-order nonlinear processes to be efficient in a centrosymmetric media, such as silica glass comprising the core and the cladding of a typical fiber. Moreover, phase-matched generation of third harmonic (TH) is also impossible in a weakly guiding fiber due to the small index difference between the core and the cladding, as illustrated in Fig. 1. Phase matching condition for third harmonic generation (THG) process requires the equality of effective indices in the fundamental and TH wavelengths (rectangles in Fig. 1). However, because of the dispersion of the glass any guided mode in the fiber, located between the core and cladding index curves will phase match to a radiation mode which is not strictly guided. Indeed such a process of Cherenkov-type phase-matched harmonic generation was observed and described previously [9, 10], and practical applications for wavelength conversion based on this process are currently being explored [11].

To allow for intermodally phase-matched harmonic generation in silica waveguides, the corecladding index difference must be increased. In this regard, the recently developed microstructured fibers with high air-fill factor (cobweb fibers) provide the appropriate structure [12]. Harmonic generation in these fibers have been observed with both 1550 nm [13] and 800 nm [14] femtosecond pumping. In the latter case wavelengths as short as 260 nm were obtained. In both cases it was found that the visible or UV light is generated in one or several higher-order guided modes of the fiber, the output is strongly dependent on the input pump polarization and the harmonic spectral peak positions are independent on the pump wavelength. The latter fact strongly indicates that phase-matching wavelengths are determined by the fiber structure and the harmonics originate from the different spectral regions of the fundamental which itself undergoes complicated nonlinear dynamics and solitonic behaviour since the dispersion at the fundamental wavelength is typically anomalous.

In what follows we investigate in detail the process of THG in cobweb fibers. The modeling details and the results of numerical calculations are discussed in Section 2, and the experimental data is shown in Section 3. Conclusions are drawn in Section 4.

2. Guided mode modeling details

Exact determination of phase-matching wavelengths and far-field intensity distributions at the fiber output require precise modeling of the transverse structure of the fiber. Scanning electron microscope (SEM) images of the fibers modeled are shown in Fig. 2. Fibers were cleaved and gold coated to obtain high-contrast SEM images. High-resolution scans, Figs. 2(b) and 2(e) were used to determine the glass-air boundaries of the central guiding core and to build a model structure corresponding to each fiber, Figs. 2(c) and 2(f). The model, thus, consisted of the central glass guiding region incorporating the full dispersion of silica surrounded by air with the refractive index of 1. Results indicated that electric field of the guided modes does not penetrate too far into the six submicron support pellicles and therefore we truncated those at some length to minimize the computational window.

We performed fully vectorial computation of guided modes for these structures for a range of the visible and fundamental wavelengths. A map of modal effective indices for 2.5µm core fiber in the visible region of the spectrum is shown in Fig. 3. The large index difference between the silica core and the air cladding allows for an unprecedented richness of the modal structure of these fibers. All guided modes are situated below the dispersion curve of silica. Fundamental (lowest order) modes of two orthogonal eigen-polarizations have the highest effective index and higher-order modes possess progressively lower effective indices. In Fig. 3 also shown are the lowest-order quasi-linearly polarized modes at the fundamental wavelength shifted into the visible by three photons, e.g. 1500 nm⇒500 nm, to show phase-matching for the THG process. Phase-matching occurs where the curves for the wavelength-shifted fundamental modes intersect the higher-order modal curves of the visible modes. It can be seen that the curves intersect at a large angle which will lead to the high spectral selectivity of the THG process. The phase-matching bandwidth associated with each intersection point will be much narrower than the bandwidth of the fundamental pump pulse.

 

Fig. 2. Modeling details of cobweb fibers: 1.5µm core fiber SEM image at low resolution, a); at high resolution, b); and the structure used for modeling, c). Same for 2.5µm core fiber, d), e), f).

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A close-up view of these phase-matching points is shown in Fig. 4. We observe, first, that because the real structure of the fiber modeled is not exactly symmetric, none of the modes are truly degenerate. Fundamental lowest-order modes split in two orthogonal polarizations and this accounts for the birefringence of the fiber. From Fig. 4 we also expect that for each polarization of the pump, the THG spectrum will consist of two doublets around 510 nm and 550 nm corresponding to modes 21, 22, 23, and 24 if counted sequentially from the lowest-order mode (single-lobe, quasi-linearly polarized). These doublets at two different polarizations are shifted with respect to each other by ~5 nm. For each mode in the doublet, both the effective index and the far-field intensity distribution are slightly different. The latter can, in fact, be observed experimentally, as shown later in Section 3.

Similar computations were performed for 1.5µm core fiber and the results are shown in Fig. 5. Here, a larger index range is displayed to demonstrate that even at this small core diameter this fiber is, in fact, multimode at the fundamental wavelength. Coupling pump light into the higher-order modes will result in phase-matched THG generation of very-high order modes in the blue-visible part of the spectrum. This process was also observed in our experiments.

3. Experimental results

Visible harmonics generated in cobweb fibers of various core sizes were characterized spectrally and spatially in the far field. For spectral measurements a wide-wavelength range optical spectrum analyzer (OSA) was used. The output tip of the fiber under study was imaged in free space to the input aperture of the OSA. Far-field intensity profiles of the harmonic modes were obtained with a high-resolution digital camera. In comparing the theoretical and experimental intensity distributions, one needs to keep in mind that the camera responsivity is nonlinear with light intensity. An optical parametric oscillator (OPO) was used to generate a train of 100 fs pump pulses at a 82 MHz repetition rate with a carrier wavelength tunable in the 1450–1650 nm range. 250 mW of average power was typically available for the experiments. Coupling into the fiber was achieved with a single aspheric lens with experimentally optimized focal length. Both Geltech and New Focus lenses with f~3 mm were found adequate for the experiments. Coupling efficiency was dependent on the fiber core size and was typically in the 30%–40% range.

 

Fig. 3. Modal effective indices in the visible (squares) for 2.5 µm core fiber. All the modes lie under the silica dispersion line (white curve). Fundamental quasi-linearly polarized modes have the highest indices which differ slightly for two orthogonal eigen-polarizations (red and black curves). Also shown are the fundamental modes at fundamental wavelength shifted by three photons into the visible (open circles, red and black curves).

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Fig. 4. Close-up view of the phase-matching points between the polarization-split fundamental modes at the fundamental wavelength and higher-order modes in the visible. Farfield profiles for four visible modes involved are also shown.

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Fig. 5. Modal effective indices in the visible for 1.5µm core fiber. Three-photon shifted modes at the fundamental wavelength are shown in black curves with open circles.

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Fibers with different central core diameters (1.5, 2.5, 3 and 4µm) were all observed to produce visible harmonics when pumped at the telecommunication wavelengths, however only fibers with 1.5µm and 2.5µm core diameters are discussed here. Because of slight unavoidable asymmetry in the core structure, our fibers always displayed strong birefringence with orthogonal polarization extinction exceeding 20 dB. This birefringence was reproduced well in our numerical modeling of fiber structures, as discussed in Section 2.

Power-dependent visible and infrared spectra at the output of a 2.5µm core fiber, 65 mm in length, are shown in Fig. 6, for two orthogonal eigen-polarizations. In the IR region, Fig. 6(b) and 6(d), a typical solitonic behaviour is observed with power-dependent Raman shifts in wavelength extending to beyond 1750 nm, the far edge of the instrument’s wavelength range. In both eigen-axes the IR spectra are nearly identical to each other. In the visible, on the other hand, Fig. 6(a) and 6(c), a pair of doublets in the vicinity of 510 nm and 550 nm is observed for each polarization of the pump. The positions of the peaks are shifted by ~5 nm between eigen-polarizations, as expected from theory. We must note that the harmonics around 510 nm are generated directly by the fundamental pulse, whereas the ones around 550 nm originate from the Raman-shifted soliton further down along the fiber length. The widths of the visible peaks are limited by the OSA resolution. More detailed spectral measurements reveal complicated power-dependent structure of the visible harmonics (this will be the subject of a separate publication.)

 

Fig. 6. Visible, a), c), and infrared, b), d) spectra at the output of 65 mm-long piece of 2.5µm core fiber for different input powers. Top and bottom figures correspond to two orthogonal eigen-axes of the fiber. Eigen-axes are marked in degrees of waveplate angular position in front of the fiber.

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Figure 7 shows a series of spectra for one of the eigen-axes of the fiber with varying the pump wavelength. We observe, most importantly, that the positions of the THG spectral peaks do not change with varying the pump wavelength. This fact suggests that indeed the phase-matching conditions are dictated by the fiber structure as discussed in Section 2. The relative intensities of the THG peaks are also easily explained by the dynamics of the fundamental spectra. For example, as the OPO is tuned toward longer wavelengths, Raman shifting of the fundamental soliton to a wavelength which will phase-match to the visible around 550 nm will occur at lower input powers. As a result, 550 nm doublet is stronger in 1550 nm pumping case. On the other hand, the 510 nm dublet is rather weak in this case since these lines are generated by the pump spectrum directly and the latter is tuned a little further to the red than would be optimal.

Theoretical and experimental far-field intensity distributions are compared in Fig. 8. As expected, experimental modes decrease in wavelength with mode order (right-to-left in the figure, bottom row). Modes 23 and 24 were spectrally separated using an angle-tuned narrow bandpass filter. Modes 21 and 22, however could not be reliably separated and therefore only a combined picture is shown in Fig. 8(g). In comparing theoretical and experimental intensity distributions one needs to keep in mind that (i) the camera responsivity is a nonlinear function of light intensity, and (ii) the numerical computations used planar projection in the far field, whereas the experimental profiles were measured through a lens. Arguably, the agreement between the experiment and the numerical analysis is very good.

Similar data were obtained for a 1.5µm fiber, 40 mm long, as shown in Fig. 9. Again, fundamental spectra, Fig. 9(b), (d) are very similar in two eigen-axes of the fiber and reveal even stronger soliton self-frequency shifting. Visible spectra, Fig 9(a), (c), consist of well separated peaks which correspond to the intermodal phase-matching wavelengths for the THG process. Previously reported polarization dependence of visible harmonics [15] can be understood from these spectra. Indeed, the dominant spectral component at 116° eigenaxis is the 615 nm line, whereas the 540 nm line is the strongest at the 71° axis. Different THG efficiencies in two eigen-axes of the fiber can be traced to the difference in modal overlap integral values.

 

Fig. 7. THG, a) and IR, b) spectra at the output of 2.5µm core fiber for eigen-axis 79° and different pump wavelengths: 1450 nm, top, 1500 nm, middle, and 1550 nm bottom.

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Fig. 8. Theoretical, top row, and experimental, bottom row, far-field intensity profiles of THG modes at the output of 2.5µm core fiber. Modes 21 and 22 could not be separated experimentally. Note that orientation of the experimental modes will match that of theoretical if all the experimental modes are rotated ~30° counterclockwise.

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Fig. 9. Visible, a), c), and infrared, b), d) spectra at the output of 40 mm long piece of 1.5µm core fiber for varying input powers. Similar to Fig. 6, top and bottom figures correspond to two orthogonal eigen-axes of the fiber.

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Fig. 10. Theoretical, top row, and experimental, bottom row, far-field intensity profiles of THG modes at the output of 1.5µm core fiber. Figures a), b), e), and f) correspond to THG generation from higher-order mode at the fundamental wavelength; profiles shown in c), d), g), and h) originate from the fundamental mode.

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Far-field profiles for 1.5µm fiber are shown in Fig. 10. Here, using offset pumping, we can excite either fundamental or higher-order modes at the pump wavelength of 1500 nm. In the former case, the strongest harmonic modes are shown in Figs. 10(g) and 10(h), for two different input polarizations and should be compared to the theoretical profiles, Figs. 10(c) and 10(d). If, however, higher-order modes at the pump wavelength are excited, the modes shown in Figs. 10(e) and 10(f) result. These compare well to the computed profiles for modes 39 and 48 in the visible. Again, the orientation of the experimental modes will match that of theoretical if a small counterclockwise rotation is applied to all experimental modes.

4. Conclusion

Observation of intermodally phase-matched harmonic generation in high-delta microstructured fibers is unique in that this process is not possible in typical weakly guiding fibers. In this work, numerical and experimental results were found to match very well in both spectral and far-field intensity distribution measurements. It is clear, that similar effects are expected to be observed in tapered fibers, however they have not been reported so far. Although typically obtained conversion efficiencies are well below a percent, methods to improve these are currently being developed. The complete description of the femtosecond THG in cobweb fibers shall include the full dynamics of the pulse at the fundamental wavelength and the influence of the cross-phase modulation on the phase-matching [16], as well as the dispersion and the group-velocity mismatch. This detailed analysis of the process will be the subject of a separate publication.