Nonlinear frequency mixing as a means to coherently convert light to new frequencies is widely used in many branches of optics. This process requires momentum conservation through phase matching (PM). In free-space optics, PM is achieved through angle tuning the medium with respect to the incoming light—here we explore an in-fiber analogue: PM using spatial modes of the fiber. We demonstrate over two octaves (400–1700 nm) of coherent spectral translation generated by intermodal four-wave mixing between subsets of 11 different Bessel-like fiber modes. These interactions are facilitated by the unique mode-coupling resistance of this subset of azimuthally symmetric, zero orbital angular momentum fiber modes. Their stability allows overcoming previous limitations of multimode nonlinear-optical systems imposed by mode coupling, hence enabling long interaction lengths, large effective mode areas, and a highly multimode basis set with which a new degree of freedom for versatile PM can be obtained.
© 2015 Optical Society of America
Parametric nonlinear-optical interactions are inherent to many important scientific fields and applications [1–7], largely because the interacting photons preserve phase information. Consequently, this also means that they are constrained by phase matching (equivalently wavevector matching or momentum conservation). In bulk optics, angle tuning is employed to achieve phase matching (PM), but since waveguides transmit light primarily in one direction (wavevector), PM may be achieved via control of the spectral curvature of the guided mode’s wavevector. Indeed, the study of nonlinear optics in guided waves has long been the study of group velocity dispersion (GVD) engineering achieved by, for instance, high-confinement waveguides in fibers , chalcogenide [9,10] or silicon waveguides , photonic bandgaps , ring resonators , or gas-filled waveguides . Thus, guided-wave nonlinear-optical devices, in spite of enabling better beam quality, efficiency, compactness, and the potential for monolithic integration, have been constrained (in bandwidth, tunability, and/or power) compared to free-space nonlinear systems—bulk, solid state optical parametric oscillators and amplifiers (OPOs/OPAs) persist as the mainstay of high-power tunable sources today.
As illustrated in Fig. 1(a), PM is achieved in bulk nonlinear materials by tuning the angle (wavevector) of the incident beams with respect to the crystal’s optical axis. An analogous functionality can be realized in fiber with higher-order modes (HOMs)—in the ray optics picture [Fig. 1(b)], each mode travels with a unique bounce angle, which determines its effective index (, related to a wave’s longitudinal wavevector or propagation constant , given by ) and dispersion. Exciting different HOMs in a fiber is, in terms of PM, like tuning the angle of incidence in crystals (albeit with discrete angles, as modes are discrete states). Viewed in terms of a wavevector diagram [Fig. 1(c)], a multimode waveguide offers several more paths for momentum conservation in nonlinear optics than does a traditional singlemode waveguide or multimode waveguide operating in a single selected mode.
We specifically consider the azimuthally symmetric modes of a fiber. These resemble free-space diffraction-resistant Bessel beams , which can be understood as plane waves in cylindrical coordinates. Using a simple step index multimode fiber, modes can propagate stably for tens of meters without degradation (50 m demonstrated ) even in the presence of hundreds of other modes. Furthermore, their stability increases with radial mode order . This property overcomes the major limitations of previous studies of intermodal nonlinearities in few-moded fibers [18,19]: (a) unstable mode propagation leads to uncontrolled mixing [20–22], thus negating the long interaction length one aims to achieve in guided-wave geometries; (b) addressing this stability problem by limiting the number of modes available for nonlinear interaction [23–25] vastly constrains the PM possibilities (one would need a multitude of discrete modes—bounce angles—to approximate the continuous angle tuning capability of bulk optics). Our highly multimode fiber provides for a seemingly arbitrarily large set of discrete modes—we demonstrate coherent interactions between 11 modes—that are all stable, and hence retain the advantages of singlemode fiber waveguides (interaction length, no beam walk off, efficiency, integration, ultrawide transmission bandwidth, large-scale manufacturability) while also providing for the one degree of freedom (angle tuning) that has been successfully exploited primarily in bulk crystals thus far.
The PM condition in four-wave mixing (FWM) requires that the net phase mismatch between all four interacting photons be zero (neglecting a small nonlinear phase accumulation). It can be shown that this condition is upheld if the effective indices for the four energy-matched photons at their respective wavelengths fall along a straight line (see Supplement 1, S.4). In the monomode case, a straight line in effective index is equivalently the near-zero dispersion condition . The mode in our fiber has at the pump wavelength of 1064 nm . Figure 2(a) shows that between 700 and 1400 nm, versus for this mode is well approximated by a straight line. Outside this wavelength range, no monomode PM is therefore expected. However, this straight line, when extended to shorter wavelengths, crosses the effective index curves of other higher-order modes, implying that PM is possible between modes.
We experimentally investigate this concept by pumping a continuum in our fiber sample [Fig. 2(b)]. This allows for spontaneous interrogation of thousands of PM conditions in a single spectrum. Our test fiber is a double-clad design with a photosensitive core, pure silica inner cladding (where the majority of the energy of the Bessel-like modes resides), and fluorine down-doped outer cladding (Supplement 1, S.2). A long period fiber grating (LPG) converts the amplified laser input into the mode (Supplement 1, S.1). In Fig. 3 we show the measured spectral evolution of the continuum as a function of fiber length via cutback. At the shortest fiber lengths, the output spectrum exhibits sidebands due to spontaneous modulation instability, and this evolves into a conventional, flat, octave-spanning continuum due to the interplay between Raman scattering and FWM [27,28]. More notably, the spectrum exhibits a series of discrete visible peaks beyond the continuum.
We characterized the modal content of the spectrum using a set of 10 nm bandpass filters and appropriate imaging cameras (Supplement 1, S.1). The continuum region from 700 to 1400 nm is in the mode, consistent with monomode continuum [Figs. 3(a) and 3(b)]. The discrete peaks in the visible spectral range monotonically increase in radial order from at 678 nm to at 453 nm [Fig. 3(c)–3(l)]. As shown in Fig. 4(a), these discrete peaks agree with the theoretical predictions in Fig. 2(a). The visible radiation results in a multicolored far-field output [inset Fig. 2(b)] comprising a superposition of many rings (consistent with the far-field patterns of Bessel beams ). For further analysis of modal content, see Supplement 1, S.4.
To understand the origin of the discrete peaks, we consider all valid processes for which the frequency of any participating photon occurs within of our measured peaks [to account for the slight discrepancy between experiment and simulation as shown in Fig. 4(a)]. We assume that one of the FWM pump photons is from our pump laser (, 1064 nm) as this is the most powerful spectral feature. We find multiple valid FWM transitions that could lead to an anti-Stokes photon at the observed wavelengths. To identify the dominant transitions amongst them, we calculate a modified effective area for FWM () for each valid process to determine their relative efficiencies (inversely proportional to , ):4(b)], the most efficient interactions follow a simple rule that holds for all the discrete peaks: 4(c)]: each discrete peak is generated, and then, along with the pump laser, drives a FWM transition to create a peak in the next higher mode order. The stability of the Bessel-like modes in the fiber is underscored by the existence of 11 resolvable mode orders over two octaves.
It is important to note that, given the dispersive characteristics of the pump () mode [Fig. 2(a)], this result would not be possible in a single mode of our fiber—only by extending the basis to multiple modes are we able to overcome the dispersion limitations and generate very high frequencies. The increase in generated bandwidth alone is not the novel aspect of this work, however, as experiments in supercontinuum generation have already shown that dispersion engineering and exploitation of ultrafast effects (such as soliton self-frequency shift and Cherenkov radiation) can result in wide, flat spectra. Instead, the availability of more modes only enhances the PM possibilities, by enabling an additional degree of freedom to obtain momentum conservation in addition to the conventional dispersion and ultrafast capabilities of nonlinear systems.
In summary, we demonstrate intermodal FWM, and associated coherent spectral translations over two octaves (400–1700 nm) and 11 interacting spatial modes (), using azimuthally symmetric fiber modes that behave like Bessel beams. We show that this uniquely stable set of modes offers a variety of momentum conservation paths, akin to angle control in bulk nonlinear crystals. The demonstration of spontaneous conversion to an ensemble of wavelengths and modes may have implications for building hyper-entangled sources for quantum communications. While the conversion bandwidth for these processes is narrow in the current implementation, the proper combination of pump modes can lead to roughly octave wide spectral translations with conversion bandwidths exceeding 100 nm . Another spin-off benefit is that these modes can be designed to be very large in mode area  (Supplement 1, S.3), thereby yielding a power-scalable  nonlinear-optical medium. Thus, intermodal nonlinearities with Bessel beams in fibers may enable truly monolithically integrated nonlinear-optical devices that combine the advantages of the bulk and guided-wave worlds.
Air Force Office of Scientific Research (AFOSR) (FA9550-14-1-0165); Defense Advanced Research Projects Agency (DARPA) (W911NF-12-1-0323, W911NF-13-1-0103); Office of Naval Research (ONR) (N00014-11-1-0098, N00014-11-1-0133).
The authors acknowledge L. Yan for several discussions and contributions to this work and R. Harrington for photographing the far-field output of our fiber. We also thank B. Samson and K. Wei (Nufern) for manufacturing the fiber, as well as B. Pati (Q-Peak) and S. Alam (Univ. Southampton) for assisting with the construction of our pump source.
See Supplement 1 for supporting content.
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