Octave-wide supercontinuum generation of light-carrying orbital angular momentum

Nonlinear frequency generation of light-carrying orbital angular momentum (OAM), which facilitates realization of on-demand, frequency-diverse optical vortices, would have utility in fields such as super-resolution microscopy, space-division multiplexing and quantum hyper-entanglement. In bulk media, OAM beams primarily differ in spatial phase, so the nonlinear overlap integral for self-phase matched χ processes remains the same across the 4-fold degenerate subspace of beams (formed by different combinations of spin and orbital angular momentum) carrying the same OAM magnitude. This indistinguishable nature of nonlinear coupling implies that supercontinuum generation, which substantially relies on self/cross-phase modulation, and Raman soliton shifting of ultrashort pulses typically results in multimode outputs that do not conserve OAM. Here, using specially designed optical fibers that support OAM modes whose group velocity can be tailored, we demonstrate Raman solitons in OAM modes as well as the first supercontinuum spanning more than an octave (630 nm to 1430 nm), with the entire spectrum in the same polarization as well as OAM state. This is fundamentally possible because spin-orbit interactions in suitably designed fibers lead to large effective index and group velocity splitting of modes, and this helps tailoring nonlinear mode selectivity such that all nonlinearly generated frequencies reside in modes with high spatial mode purity. © 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

This special case notwithstanding, a generalized methodology to nonlinearly frequencyconvert, or obtain spectrally diverse, OAM beams with high spatial coherence and mode purity has not been possible, though doing so would be of great utility to a variety of applications that require on-demand OAM beams of different colors, for which the only broadband solutions require independent generation of white light that is then transformed with a wideband mode converter [18,19].
The electric field, E(r, φ, z) of an OAM mode in optical fibers is given by: where F(r) represents the radial distribution of the electric field (which is substantially similar for modes of the same L ), L is the topological charge associated with an OAM of  L per photon, φ is the azimuthal angle, σ ± denotes the (circular) polarization states associated with a spin angular momentum (SAM) of ± per photon, and β is the propagation constant, related to the effective index, n eff , of the mode by 2 / eff n β π λ = (λ is the free-space wavelength). Note the subscripts of β, representing two degenerate spin-orbit aligned states (SOA) where the sign of L and σ are the same, and two degenerate spin-orbit anti-aligned states (SOAA) where these two quantities are opposite in sign, respectively. In bulk media, for a given magnitude L of OAM, β SOA = β SOAA , and modes with the four combinations of SAM and OAM are four-fold degenerate (in β or n eff , and hence also in group velocity, group velocity dispersion (GVD) and higher-order dispersion terms). Third-order nonlinear coupling between different modes is governed by the field overlap integral where the subscript j E and k E denote the normalized fields associated with two different modes. From Eq. (2), it is immediately apparent that nonlinear interactions would have similar strengths when coupling modes of the same or opposite sign of L since their radial field profiles are nearly identical. This implies that nonlinear scattering (due to Raman or self/cross-phase modulation) from a "pump" mode in L would occur, with equal probabilities, to a mode with the same L as well as a mode with the opposite topological charge of −L . In addition, as the pulses propagate, their envelopes continue to temporally overlap due to the aforementioned four-fold degeneracy, enabling the nonlinear interaction to build up. This explains why nonlinearly generated supercontinua of OAM beams in bulk media have, thus far, resulted in multimode outputs.

Nonlinear evolution of OAM beams in fibers
The situation is dramatically different in an optical fiber. Due to the confinement potential of the waveguide, SOA and SOAA modes have different propagation constants -a result of spinorbit interactions in the presence of dielectric anisotropies [20]. We have previously shown that this spin-orbit effect can be exacerbated by fiber design, and lifting this degeneracy (i.e. making |β SOA −β SOAA |) large) avoids linear mode mixing during fiber propagation through lengths as long as 13 km [21]. In addition, angular momentum conservation rules dictate that even the degenerate orthogonally polarized modes do not mix in the linear regime [22]. Thus, in a suitably designed optical fiber, the four-fold degeneracy of OAM modes depicted in Eq.
(1) reduces to two 2-fold-degenerate subspaces, and coupling even within the doubly degenerate modes is inhibited. The nonlinear overlap integral (Eq. (2), however, remains the same, and linear mode stability does not guarantee nonlinear mode selectivity. Here, we show that the aforementioned degeneracy-lifting criteria that enabled stable linear behavior of OAM modes in fibers also facilitates controllable nonlinear interactions for ultrafast pulses of OAM beams in fibers. Figure 1(a) illustrates this effect: for a spin-orbit aligned pump ( , ) σ + + L th temporally ov nonlinear con with the pump (arising from known result beams [23]. T light launche generated freq Fig

Experime
We probe the ( Fig. 2 Fig. 4  wave-plate es in the launched polarization state as well as the orthogonal polarization state, respectively. Spatial integration of the camera intensity pattern across the two images reveals that the power ratio in the launched, versus the orthogonal polarization bins remains 6 dB (i.e. 75% power remain in the launched polarization state) across the spectrum [ Fig. 5(c)]. This confirms the polarization (hence SAM) preserving behavior of the nonlinear process. Within the launched polarization bin, the relative modal content in SOA and SOAA is found by utilizing the fact that the L = 14 OAM state (converted from L = 8 SOA) diffracts more than the L = 2 state (converted from L = 8 SOAA) in free-space, as shown in Fig. 5(b); hence, the relative powers in the desired mode with respect to other modes is found by spatial integration of the four spatially separated regions (Region 1 corresponding to L = 14 and Region 3 corresponding to L = 2, and Region 2 and 4 composed of power in all other parasitic modes). Using this method, we find that the mode purity is better than 13 dB (>95%) across the spectral bandwidth [ Fig. 5(d)]. This confirms that the dominant OAM content at all the nonlinearly generated frequencies is same as the launched pump state. We additionally confirm that both OAM and polarization are preserved across the generated supercontinuum when L = + 7; σ + state is used as the pump, indicating that the phenomenon is not dependent on OAM charge of the pump. The measurement technique to discern mode purity above rests on two assumptions, both of which we show to be valid: (a) the intensity profiles of modes of the same, first, radial order (the primary mode orders used for the pump and probed in this experiment) need to be similar across different order of L . This is substantially true for the air core fibers used in our experiments. In contrast to free-space or bulk media, the intensity profiles of modes with different L depends weakly on L , because the high index contrast of the waveguide offering confinement plays the dominant role here. (b) power in other radial mode orders is assumed to be negligible. This validity of this assumption arises from the fact that intensity line cuts of the near field profiles all OAM modes exiting from our fiber [see Fig. 5(e)] match very well with simulated intensity profiles for our fiber -we find that an intensity overlap integral between the two yield 98% coincidence. Moreover, theoretically constructed intensity profiles assuming incoherent addition with other radial orders shows that obtaining such a high overlap would have meant that the other radial mode orders would have a maximum of 0.5% power, which is indeed negligible.

Discussion, summary and conclusions
These experiments reveal an interesting and highly useful attribute of nonlinear optics with OAM fiber modes -all nonlinear products arising from ultrafast pulse nonlinear effects conserve both polarization and OAM. Recall that this is fundamentally due to the effect of spin-orbit interactions in the air-core fiber, which results in group-velocity walk-off between spin-orbit aligned and anti-aligned states. Hence, self-phase matched nonlinear effects, such as Raman scattering or supercontinuum generation, with OAM beams can achieve high spatial coherence only in media that offer optical confinement with cylindrical symmetry, such as optical fibers or whispering gallery modes in ring resonators, but not in bulk media, where non-exclusive nonlinear coupling occurs.  in an optical f nonlinear op de size, as is t r optics (eithe avelength of c optics in any s l is independen , is an order of mmonly used f nua produced ble in conventio le spatially sel shifting and su abler is a suita actions and disp greater than an OAM state. rs is not only u ch as spectrally also provides d-wave nonline output using q-pla d camera at diffe a). (b) Representa r mode-conversion to the orthogonal p length, in launche ted = L 8 mode ity profile of the s fiber enables c ptics. This has typically done er on-chip or choice -hence spectral range nt of the effec f magnitude lar for supercontin by OAM beam onal fibers. lective and co upercontinuum ably designed persive behavi n octave) freq Hence, thirduseful for the g y-diverse supe an attractive w ear optics. ate, bandpass filter erent wavelengths ative output mode n using the q-plate polarization across ed polarization bin at 650 nm [mode ame mode.
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