Abstract
We demonstrate a new guiding regime termed endlessly monoradial, in the proposed annular core photonic crystal fiber (ACPCF), whereby only modes of the fundamental radial order are supported by the fiber at all input wavelengths. This attribute is of high interest for applications that require the stable and broadband guiding of monoradial (i.e. doughnut shaped) cylindrical vector beams and vortex beams carrying orbital angular momentum. We further show that one can significantly tailor the chromatic dispersion and optical nonlinearities of the waveguide through proper optimization of the photonic crystal microstructured cladding. The analytical investigation of the remarkable modal properties of the ACPCF is validated by fullvector simulations. As an example, we performed simulations of the nonlinear fiber propagation of short femtosecond pulses at 835 nm center wavelength and kilowattlevel peak power, which indicate that the ACPCF represents a promising avenue to investigate the supercontinuum generation of optical vortex light. The proposed fiber design has potential applications in spacedivision multiplexing, optical sensing and superresolution microscopy.
Introduction
Orbital angular momentum (OAM) beams, aka optical vortices, and cylindrical vector beams (CVB) exhibit an annular intensity profile that possesses a zero onaxis intensity due to a phase singularity and a polarization singularity, respectively^{1,2}. Optical vortex beams may carry both spin and orbital angular momenta owing to circular polarization and a helically varying phase front (described by a phase term exp(ilθ) in the transverse plane, where θ is the azimuthal coordinate and l refers to the topological charge), respectively. The peculiar doughnut shaped intensity distribution as well as the theoretically infinite topological states and the inhomogeneous polarization have allowed researchers to push the frontiers of optical physics and explore new modalities in superresolution imaging^{3,4,5}, laser material processing^{6,7,8}, optical trapping^{9,10}, sensing^{11,12} and spacedivision multiplexing (SDM)^{13,14}, to name a few.
One wellestablished application of vortex beams pertains to stimulated emission depletion (STED) superresolution microscopy where an annular beam is used together with a coaligned Gaussian beam in order to breach the diffraction limit^{15}. Another intriguing area of research relates to probing of chiral lightmatter interactions with potential applications in molecular spectroscopy^{16}. These applications of vortex beams would benefit from the versatility of a broadband coherent source of OAM light^{17}. In this regard, researchers have explored freespace methods for the generation of broadband vortex light, including: nonlinear crystals^{18,19}, nonlinear gases^{20}, nanostructured metamaterials^{21} and wideband qplates^{22,23}. However these freespace methods still harbours some technical limitations in the continuum’s wavelength coverage or in spatial dispersion. An alternate approach pertains to supercontinuum generation in nonlinear optical fibers which could extend the spectral coverage of the ensuing optical vortex beams.
To this end, photonic crystal fibers (PCF) have demonstrated key optical properties such as endlessly singlemode guiding, high nonlinearities and chromatic dispersion engineering, making it an ideal medium for supercontinuum generation^{24,25}. Prior research was conducted on PCF and photonic bandgap fiber designs towards SDM transmission applications^{26,27}. A recent study proposed a ring core chalcogenide glass photonic crystal fiber design with promising numerical results of supercontinuum generation in the infrared^{28}. A fiber based supercontinnum generation of the highorder (l = 8) OAM beam was also experimentally demonstrated in the nearinfrared spectrum using a 3.5 m long aircore fiber^{29}. Both previous studies however did not specifically address the issue of the broadband stability of vortex beams guided in the fundamental radial order, which is critical to ensure high purity of the monoannular CVB and OAM beams at the fiber output across the widest spectral range possible.
In this work, we uncover a novel waveguiding regime in the annular core photonic crystal fiber (ACPCF) thereby supporting the endlessly monoradial (EMR) guiding regime, where the fiber enforces wavelength independent “doughnutshaped” monoannular guided modes. The latter property is vital as it opens the possibility to achieve the broadest and purest fiber supercontinuum vortex light supported by fiber eigenmodes of the fundamental radial order. The parameterization of the ACPCF is presented below along with the theory that allows the optimization of its modal properties, including the special EMR guiding regime. These results are validated through fullvector finite element method (FEM) calculations (using COMSOL Multiphysics package) with perfectly matched layer boundary conditions. Further, we theoretically study optimized designs of nonlinear ACPCF towards the supercontinuum generation in the visiblenearinfrared spectrum by the numerical solution of the generalized nonlinear Schrödinger equation via the split step Fourier method.
Description of the Annularcore Photonic Crystal Fiber Design
The proposed ACPCF structure shown in Fig. 1(a) resembles that of a standard hexagonal lattice PCF described via air hole diameter (d), pitch (Λ) and number of rings of air holes (N). The key difference though, is that in the case of the ACPCF waveguiding occurs within a “ring” of six missing holes; while in the standard PCF the optical mode is guided around a missing center hole.
Assuming that the holey cladding can be modeled as a homogeneous material of tunable refractive index via the fill ratio d/Λ, one realizes that the modal properties of the ACPCF can be investigated by means of the functionally comparable allsolid ringcore fiber^{30}. Therefore the ACPCF exhibits analogous waveguiding features such as a fundamental HE_{11} mode showing an annular intensity profile (Fig. 1(b)) along with other cylindrical vector modes TE_{0m}, HE_{2m}, TM_{0m} and other higher order mode EH_{1m} and HE_{3m} with radial order m = 1. In Fig. 1(c) we present the intensity and phase distributions of OAM_{±11} modes supported in a ACPCF through the coherent superposition of hybrid modes: OAM_{±11} = HE_{21(even)} \(\mp \) i HE_{21(odd)}. We note that other higherorder OAM modes can be supported by the fiber such as: OAM_{±21} = HE_{31(even)} \(\mp \) i HE_{31(odd)} and OAM_{±21} = EH_{11(even)} ± i EH_{11(odd)}.
The light guiding properties of optical fibers are commonly modelled via the dimensionless Vnumber. In the case of the ACPCF we define this parameter as \(V=(2\pi b/\lambda )(\sqrt{{n}_{core}^{2}{n}_{clad}^{2}})\) where the effective core radius is b = 2Λ/\(\sqrt{3}\), the core refractive index corresponds to that of the solid glass (n_{core} = n_{glass}), and the refractive index of the holey cladding is defined by the effective index of the fundamental spacefilling mode (n_{clad} = n_{FSM}) in the triangular lattice of airholes. Based on the latter analytical formalism, the ACPCF enables single mode operation for V < 2.405, as displayed by standard optical fibers and allsolid ringcore fibers^{30}. For applications previously outlined, it is crucial to prevent the onset of the first biannular eigenmode with m = 2, namely the HE_{12} mode, which occurs for V ≥ V_{cut}; where V_{cut} = 3.832 is the HE_{12} mode cutoff. In that regard, the most prominent feature of the ACPCF is its ability for endlessly monoradialorder (EMR) guiding, in which case modes with fundamental radial order m = 1 are strictly supported by the fiber at all input wavelengths (Fig. 2(a)). The threshold of this special waveguiding regime was identified through FEM calculations for relative hole diameters less than 0.35 (namely d/Λ < 0.35). The ability to enforce modes with m = 1 inside an EMRguiding ACPCF helps to mitigate issues related to mode coupling with undesired higherradialorder modes (m ≥ 2). The latter feature is also desirable, among others, in SDM applications using CVB and OAM beams where mux/demux operations generally assume the coaxial alignment of modes in the fundamental radial order^{13}.
The full list of possible monoradial (m = 1) modes supported in the EMR regime are: HE_{11}, TE_{01}, HE_{21}, TM_{01}, EH_{11} and HE_{31}. Figure 2(b) demonstrates that the cutoff conditions for these modes of interest along with that of the HE_{12} mode, as predicted by the analytical model^{30}, are in good quantitative agreement with the fullvector FEM numerical calculations. We recall that the fundamental HE_{11} mode has no cutoff. Small deviations between the analytical description and FEM simulations of the ACPCF can be attributed to the fact that ringcore fibers and ACPCF fibers are not identical in shape due to the inhomogeneous photonic crystal cladding present in the latter type of fiber, and also because the ringcore fiber model assumes the same refractive index in the inner and outer claddings (while this condition is not exactly met in the case of the ACPCF). The procedure used for the determination of the modal cutoffs is detailed in the Methods section.
Optimization of Fiber Parameters for Supercontinuum Generation in the VisibleNearInfrared
In this section, a fused silica glass ACPCF is optimized for parameters relevant to supercontinuum generation (SCG) in fiber by means of guided CVB and OAM modes: high optical nonlinearity, high modal effective index separation and nearzero chromatic dispersion at the input pump wavelength of 835 nm. Figure 3 shows the value of the effective mode area (A_{eff}) and nonlinear parameter (γ) of a silica ACPCF operating inside the EMR waveguiding regime (d/Λ < 0.35) at λ = 835 nm for the HE_{21} mode. We note that in this section we have limited our discussion to the HE_{21} mode for concision since its modal properties very closely mirror that of the TE_{01} and TM_{01} modes, and also because the even and odd HE_{21} modes serve as the basis set to create the OAM_{±11} modes of interest. In our numerical analysis we have defined the threshold for modal cutoff as the level where less than 40% of the optical power is guided within the core region of the fiber (a detailed description of this criterion for modal cutoff is presented in the Methods section). Inexistent data corresponding to regions of modal cutoffs have duly been identified in Figs 3 to 5.
One can observe in Fig. 3(a) that the mode confinement, i.e. value of A_{eff}, decreases with increase in the d/Λ ratio, and increases with the hole pitch (Λ). Since the nonlinear parameter is defined as γ = 2πn_{2}/(λA_{eff}), where n_{2} is the nonlinear refractive index of the glass, we observe the reverse relationship for γ as a function of d/Λ and Λ. Hence the optimization of optical nonlinearities (i.e. magnitude of γ) can be achieved with fiber structures presenting aspect ratios near the edge of the EMR regime (d/Λ = 0.35) and for smaller hole periods. However, lowering the value of Λ eventually creates a very small core region that forces the guided mode to have a significant fraction of its power leaking into the cladding region and to eventually cutoff. Figure 3(b) displays the nonlinear parameter (γ) values for silica ACPCF in the EMR at λ = 835 nm for the HE_{21} mode. We note that the value of γ increases with the d/Λ ratio, while decreases with Λ as expected from the inverse relationship with the effective mode area. In particular, high nonlinearities can be achieved for normalized hole diameters in the 0.2 < d/Λ < 0.35 range and for submicron periods (Λ < 1 μm).
Moreover, the fiber is designed for the stable propagation of monoradial OAM beams, which depends on lifting the modal degeneracy of the constituent hybrid HE/EH cylindrical vector modes^{28,31,32}. A common rule of thumb is to maintain a minimum intermodal separation of the effective indices of Δn_{eff} ≥ 10^{−4} between adjacent vector modes so as to promote their fiber transmission stability^{14,32}. Figure 4(a,b) shows the value of the HE_{21} effective mode index (n_{eff}) and its minimum intermodal separation (Δn_{eff}) with respect to the adjacent vector modes TE_{01} and TM_{01}. Figure 4(a) indicates that the value of n_{eff} increases almost linearly with the pitch Λ, ostensibly because a larger core radius (b = 2Λ/\(\sqrt{3}\)) promotes a better Efield overlap within the highrefractiveindex annular ring.
Figure 4(b) indicates that a parametric space with sufficient intermodal separation is found in the region roughly bounded by (0.3 ≤ d/Λ ≤ 0.35) and (0.8 ≤ Λ ≤ 1.6 μm). Similarly, another interesting region for the stable propagation of the HE_{21} mode can be found for small fiber geometries with (0.18 ≤ d/Λ ≤ 0.25) and (0.3 ≤ Λ ≤ 0.7 μm), although this second region is located near the modal cutoff.
Chromatic Dispersion Engineering
The chromatic dispersion plays an important role in supercontinuum generation as it determines the extent to which spectral components of a short pulse will travel with different velocities. The undesirable effect of group velocity chromatic dispersion (GVD) is the temporal broadening of the optical pulse that ultimately affects the phasematching and excitation of optical nonlinearities in the fiber. The GVD in the fiber is commonly described by the dispersion parameter (D) in units of ps/(kmnm):
where n_{eff} is the wavelengthdependent effective refractive index of the propagating mode of interest in the singlematerial fiber (silica in this case). We note that the material contribution to the total chromatic dispersion was here taken into account by implementing the Sellmeier equation of pure fused silica in our simulations. Similar to what was achieved with regular PCFs^{33,34}, we demonstrate below the ability to engineer the chromatic dispersion of CVBs and OAM modes in an ACPCF through the precise tuning of its microstructure.
Henceforth, we investigated the dependence of the GVD on the geometrical parameters (d and Λ) of a silica ACPCF optimized for supercontinuum generation at 835 nm input wavelength in the few femtoseconds regime. Our calculations in Fig. 5 indicate that low dispersion can be achieved inside a relatively large parameter space: (0.3 ≤ d/Λ ≤ 0.35) and (1 ≤ Λ ≤ 2 μm). Similarly, a smaller region of low GVD occurs near modal cutoff for (0.18 ≤ d/Λ ≤ 0.25) and (0.3 ≤ Λ ≤ 0.7 μm). Within these interesting regions of low absolute dispersion, inspection of Fig. 5(b) allows to further mitigate the effects of thirdorder dispersion by plotting the dispersion slope, which should also be minimized in order to enhance phasematching with optical nonlinearities. In that regard we observe low dispersion slope values for fiber geometries with (0.3 ≤ d/Λ ≤ 0.35) and (0.8 ≤ Λ ≤ 1.2 μm) that is colocated with a desirable region of low absolute GVD.
Figure 6 plots the chromatic dispersion as a function of wavelength for a number of potentially interesting ACPCF configurations. In particular, for a given fixed pitch value of Λ = 0.5 μm, Fig. 6(a) indicates that it is possible to go from a nearzero dispersion at λ = 835 nm to high negative dispersion (in the normal dispersion regime), by increasing the normalized hole diameter (d/Λ) from 0.20 to 0.32. On the other hand if one keeps the normalized hole diameter constant while increasing the pitch, Fig. 6(b) indicates that the GVD progressively shifts toward higher positive values in the direction of anomalous dispersion.
Based on the analysis of Fig. 6, we selected two promising designs of ACPCF (also keeping in consideration practical fiber fabrication limitations) named “Fiber 1” and “Fiber 2” for the supercontinuum generation with short optical pulses centered at λ = 835 nm. The structural parameters of the two optimized fiber designs and their principal modal properties are presented in Table 1. The β values in Table 1 denote the coefficients of the Taylor series expansion of the wavenumber β(ω) about the pulse’s center frequency (ω_{0})^{33}. Here we note that β_{1} (i.e. group velocity) was omitted since the calculations of the nonlinear pulse propagation in the next Section were performed in a moving time reference frame. In addition, we confirmed that the mode purity of the generated OAM_{±11} beams using both fiber designs remained very high (>96%) across the whole simulated range of wavelengths from 600 to 1200 nm. Details of the OAM mode purity calculations based on a modal field decomposition into spiral harmonics are provided in the Methods section.
Numerical Simulation of Supercontinuum Generation in ACPCF
The nonlinear pulse propagation in the fiber can be simulated by solving the generalized nonlinear Schrödinger equation (GNLSE)^{33,34}:
where, A = A(z, t) is the electric field envelop, α is the attenuation constant, β_{n} is the n^{th} order dispersion coefficient [see Table 1] about the center frequency ω_{0}, and γ is the nonlinear parameter given by:
In Eq. (3), n_{2} represents the nonlinear refractive index of fused silica, λ denotes the wavelength and A_{eff} is the effective mode area as defined by^{35}:
In Eq. (4), \({\overrightarrow{e}}_{v}(x,y,\omega )\) and \({\overrightarrow{h}}_{v}(x,y,\omega )\) respectively denote the transverse electric and magnetic vector field distributions which were here calculated via FEM simulations. The time derivative term on the right hand side of Eq. (2) includes the effect of dispersion owing to nonlinearity, associated with the phenomena of selfsteepening and optical shock. The first term in the definition of τ_{shock} below is the dominant contribution while the second term includes the effect of the frequencydependent effective mode area and nonlinear refractive index, as^{36}:
This equation can be rewritten as:
In this study we have utilized Eq. (6) to determine the shock time scale that takes into account the dependence of the optical shock time on the effective mode area, while we neglected the contribution from the frequency dependence of the nonlinear index n_{2}, since this contribution is negligibly small. The obtained values for τ_{shock} at the pump’s central wavelength (835 nm) are thus found to be 0.62 fs and 0.59 fs for Fiber 1 and Fiber 2, respectively. The total nonlinear response R(t) of the material, which includes both the instantaneous electronic and delayed ionic Raman contributions, is described as:
where in the case of silica glass we have f_{R} = 0.18 for the fractional contribution to the delayed Raman response, δ(t) is the Dirac function and h_{R} is the Raman response function which can be analytically modeled as:
where for silica^{37} we have: τ_{1} = 12.2 fs, τ_{2} = 32 fs, f_{b} = 0.21, τ_{b} = 96 fs.
In general the GNLSE must be solved through numerical methods. To that end, the splitstep Fourier method is based on parsing the linear dispersive and nonlinear contributions of the GNLSE and solving them independently over small half discretization steps^{38,39} Based on the latter approach to solving Eq. (2), we performed simulations of supercontinuum generation in a 20 cm long silica ACPCF using the optimized parameters of Fiber 1 and Fiber 2 presented in Table 1.
Simulations were performed using an unchirped Gaussian pulse of τ_{FWHM} = 60 fs duration and 10 kW peak power (P_{p}). At λ = 835 nm center wavelength, Fiber 1 and Fiber 2 are both pumped in the normal dispersion regime (D = −30.1 and −12.56 ps/(km · nm), respectively) with nonlinear coefficient γ = 10.6 and 14.7 W^{−1} km^{−1}. We note that since the designed fiber has no form birefringence, the effect of polarization coupling were not considered in the solution to GNLSE. Figure 7(a) shows the spectral and temporal evolution of the input Gaussian pulse in Fiber 1. Initial evolution of the spectrum is characterized by selfphase modulation and normal dispersion as these phenomena together leads to substantial spectral and temporal broadening as well as a rapid decrease in peak power over a short distance as expected^{33}. The significant amount of normal dispersion in Fiber 1 limits the nonlinear spectral broadening of the input pulse, the full extent of which occurs within 3 centimeters propagation length, and is followed by the formation of additional redshifted components due to Raman scattering. Figure 7(b) also illustrates temporal and spectral broadening of the input pulse for Fiber 2 since it is similarly pumped in the normal dispersion regime. However, the pumping occurs closer to the zero dispersion wavelength and with a larger nonlinear coefficient. The latter conditions allow for a more rapid spectral broadening and significant power transfer towards the anomalous dispersion region. In this case, we thus observe complex soliton fission dynamics coupled with Raman scattering that results in dispersivewave generation and a fine structuring of the output pulse spectrum.
Another set of calculations was performed to simulate the output spectrum after 20 cm propagation in either Fiber 1 or Fiber 2, for various levels of peak power (1, 10, 100 kW) of a 60 fs input pulse. We note that experimental demonstrations of SCG in smallcore nonlinear fibers using comparable kWlevel femtosecond pulses have been demonstrated in the past^{25}. Figure 8(a) shows the corresponding spectral profile simulated for Fiber 1 where we observe that nonlinear spectral broadening dominates over the effects of chromatic dispersion such that an output spectrum spanning from 696 to 1058 nm at −20 dB from the top is obtained for 100 kW peak power. In the case of Fiber 2 in Fig. 8(b), the highest peak power is shown to translate into more power towards the anomalous dispersion regime such that the mechanisms of soliton fission and dispersive wave generation are more prominently displayed and lead to a spectral broadening spanning from 644 to 1219 nm at P_{p} = 100 kW.
Hence, Fiber 2 with structural parameters Λ = 0.5 μm and d/Λ = 0.2 represents the most promising fiber design for supercontinuum generation due to its higher optical nonlinearities and lower group velocity dispersion. However, fabrication of this design with very small holes of diameter d = 0.1 μm could pose practical problems, in which case the design of Fiber 1 will represent a viable alternate solution. The proposed fiber design is amenable to fabrication via the established stackanddraw technique. For applications in the neartofar infrared spectrum (including the Cband) for which hole diameters are relatively large (d ≥ 1 μm) we expect no particular fabrication issues. However for applications that include the visible range (as in the present study) the holey cladding can become deeply submicron in size, such that particular care must be taken throughout the fiber fabrication process starting from the preform creation to the fiber drawing under precisely controlled gas pressure. Tolerance of the fiber design to imperfections in the fabrication process can be inferred from the parametric study of the modal properties shown in Figs 3–5. We remark that because the point of operation in Fiber 2 is close to cutoff, this specific design has low tolerance to perturbations in the nominal values of the structural parameters Λ and d/Λ. In the case of Fiber 1, Fig. 4(b) indicates that the minimum modal separation (Δn_{eff}) remains above the 10^{−4} threshold even for changes in the structural parameters as large as 10%. Figures 3 and 5 similarly indicate that Fiber 1 retains a fair degree of tolerance to structural perturbations (as large as 5%) with respect to optimal optical nonlinearities and GVD.
Conclusion
In summary, we have studied the design and simulation of the triangularlattice annular core photonic crystal fiber (ACPCF) enabling both the stable broadband guidedtransmission and supercontinuum generation of optical vortex beams in fiber. The analytical investigation, supported by numerical simulations, shows that a novel waveguiding regime is possible in the ACPCF in which the fiber strictly supports modes of the fundamental radial order at all wavelengths. This special regime, here called endlessly monoradial, occurs when the photonic crystal structure obeys the condition d/Λ < 0.35. Similarly to standard PCFs, we also show that the ACPCF can be tailored to enhance the optical nonlinerarities as well as engineer the chromatic dispersion of the fiber. The unique waveguiding features of the proposed design thus make it an ideal medium to study monoannular beams, namely cylindrical vector beams and orbital angular momentum beams, within a linear or nonlinear broadband regime solely limited by the transparency window of the host material. In particular, we demonstrate through numerical solutions of the generalized nonlinear Schrödinger equation that properly optimized designs of the endlessly monoradial ACPCF can support the supercontinuum generation of stable optical vortex beams. The work is relevant to research in the topical areas of spacedivision multiplexing, superresolution microscopy and optical sensing via structured light.
Methods
We here describe the method of validation via FEMbased calculations of the ACPCF’s modal cutoffs with the reported fullvector analytical theory detailed in^{30} for the functionally equivalent ringcore fiber (in an effectiveindex model). Here we note that in the design of ACPCF in Fig. 1(a) the first ring of air holes are missing, whilst in a standard PCF it is the central hole that is missing. A proposed definition for the effective core radius (b) of an ACPCF is b = 2Λ/\(\sqrt{3}\), thus exactly twice the value proposed for the PCF in^{40}. The ACPCF is further defined by its core refractive index n_{core}, (i.e. refractive index of the host material) and the effective refractive index of the cladding as defined by the fundamental spacefilling mode (n_{FSM}) in the triangular lattice of air holes of diameter d and period Λ.
In this work, we obtained the effective index of the fundamental spacefilling mode (Fig. 9(b)) by means of the fullvector FEM eigenmode solution of the infinite triangular lattice of air holes (where periodic boundary conditions were implemented on all sides of the unit cell in Fig. 9(a)). For validation purposes, the numerical results were compared with a reported empirical model in^{40} as depicted in Fig. 9(b).
The condition of cutoff frequency for different eigenmodes (TE_{0m}, TM_{0m}, HE_{v,m} and EH_{v,m}) of the ring core fiber are analytically defined in Eqs 9–12, respectively^{30}:
here, J_{v} and N_{v} are Bessel functions of the first and second kind, v is the azimuthal order, m is the radial order, V_{0} is the cutoff normalized frequency with
where \(\rho =b/a\mathrm{=(2}{\rm{\Lambda }}/\sqrt{3})/(d\mathrm{/2)}\), n_{0} = n_{core}/n_{FSM} and the inner cladding radius a = d/2. Of particular interest is the numerical solution of Eq. (11) for the first higherradial order HE _{12} mode which provides the cutoff condition for monoradial order guiding.
The numerical solution of Eqs (9–12) using the above analytical formalism are compared with the cutoff frequencies obtained from the exact FEM simulation in Fig. 2(b). In the FEM simulations, the cutoff condition of all the modes is quantitatively defined based on the optical power fraction (f_{p}) guided within the transverse region of interest (i.e. fiber core) as given by.
where S_{z} denotes the Poynting vector and the core area is bounded by (0 ≤ r ≤ b). Through an iterative study, a power fraction threshold of f_{p} = 40% within the core area was identified as the criterion for modal cutoff.
Therefore all physical modes numerically found with f_{p} > 40% are considered as coreguided in our analysis; while those modes whose power fraction is below threshold are systematically rejected (i.e. cutoff). Figure 10 shows exemplar profiles of eigenmodes supported by an ACPCF that are exactly at cutoff (f_{p} = 40%).
We have evaluated the modal purity of the OAM_{±11} beams generated in the Fiber 1 and Fiber 2 designs optimized for supercontinuum generation. To do so we performed the projection of the transverse field distribution u(ρ, θ) onto the “spiral harmonics” \(exp(\,i\ell \theta )\) where the variable \(\ell \) denote any \(\ell \)th order OAM harmonic integer. We subsequently computed the energy \({C}_{\ell }\) transmitted within each \(\ell \)th OAM harmonic via^{26,41}:
Finally, the normalized power weight (\({P}_{\ell }\)) for each \(\ell \)th OAM harmonic (i.e. topological charge number) contained in the modal field under test is written as:
Figure 11(a) shows a typical “OAM spectrum” obtained using Eqs (14 and 15) and taking into account the harmonic range from \(\ell \) = −10 to \(\ell \) = +10 of the OAM_{−11} mode generated in Fiber 1 at 835 nm wavelength. The OAM spectrum indicates a very high (99%) \(\ell \) = −1 modal purity as expected, with residual peaks located at ±6 charges about the main harmonic ostensibly related to the sixfold symmetry of the photonic crystal structure. To verify that the high OAM mode purity is broadband, we computed in Fig. 11(b) the OAM_{±11} modes purities inside both optimized fibers within the range of simulated wavelengths from 600 to 1200 nm. One observes that the modal purity in Fiber 1 increases with the wavelength, while it decreases for Fiber 2. We attribute this behavior to the fact that Fiber 1 has a larger hole size (0.442 μm) compared to Fiber 2 (0.1 μm) such that modal confinement decreases with the wavelength in Fiber 2, whilst slightly improves in Fiber 1 for larger wavelengths. A similar relationship with the modal confinement was also observed for a different type of OAM fiber^{26}.
References
Yao, A. M. & Padgett, M. J. Orbital angular momentum: origins, behavior and applications. Advances in Optics and Photonics 3, 161–204 (2011).
Zhan, Q. Cylindrical vector beams: from mathematical concepts to applications. Advances in Optics and Photonics 1, 1–57 (2009).
Snoeyink, C. & Wereley, S. Singleimage farfield subdiffraction limit imaging with axicon. Optics letters 38, 625–627 (2013).
Li, L. & Li, F. Beating the rayleigh limit: Orbitalangularmomentumbased superresolution diffraction tomography. Physical Review E 88, 033205 (2013).
Liu, K. et al. Superresolution radar imaging based on experimental oam beams. Applied Physics Letters 110, 164102 (2017).
Duocastella, M. & Arnold, C. B. Bessel and annular beams for materials processing. Laser & Photonics Reviews 6, 607–621 (2012).
Weber, R. et al. Effects of radial and tangential polarization in laser material processing. Physics Procedia 12, 21–30 (2011).
Hamazaki, J. et al. Opticalvortex laser ablation. Optics express 18, 2144–2151 (2010).
Padgett, M. & Bowman, R. Tweezers with a twist. Nature Photonics 5, 343 (2011).
Gecevičius, M., Drevinskas, R., Beresna, M. & Kazansky, P. G. Single beam optical vortex tweezers with tunable orbital angular momentum. Applied Physics Letters 104, 231110 (2014).
Brullot, W., Vanbel, M. K., Swusten, T. & Verbiest, T. Resolving enantiomers using the optical angular momentum of twisted light. Science advances 2, e1501349 (2016).
Niederriter, R., Siemens, M. E. & Gopinath, J. Fiber optic sensors based on orbital angular momentum. In Lasers and ElectroOptics (CLEO), 2015 Conference on, 1–2 (IEEE, 2015).
Willner, A. E. et al. Optical communications using orbital angular momentum beams. Advances in Optics and Photonics 7, 66–106 (2015).
Rusch, L. A., Rad, M., Allahverdyan, K., Fazal, I. & Bernier, E. Carrying data on the orbital angular momentum of light. IEEE Communications Magazine 56, 219–224 (2018).
Yu, W. et al. Superresolution deep imaging with hollow bessel beam sted microscopy. Laser & Photonics Reviews 10, 147–152 (2016).
Cameron, R. P., Götte, J. B., Barnett, S. M. & Yao, A. M. Chirality and the angular momentum of light. Phil. Trans. R. Soc. A 375, 20150433 (2017).
Wildanger, D., Rittweger, E., Kastrup, L. & Hell, S. W. Sted microscopy with a supercontinuum laser source. Optics express 16, 9614–9621 (2008).
Neshev, D. N., Dreischuh, A., Maleshkov, G., Samoc, M. & Kivshar, Y. S. Supercontinuum generation with optical vortices. Optics Express 18, 18368–18373 (2010).
Tokizane, Y., Oka, K. & Morita, R. Supercontinuum optical vortex pulse generation without spatial or topologicalcharge dispersion. Optics express 17, 14517–14525 (2009).
Hansinger, P. et al. White light generated by femtosecond optical vortex beams. JOSA B 33, 681–690 (2016).
Zhao, Z., Wang, J., Li, S. & Willner, A. E. Metamaterialsbased broadband generation of orbital angular momentum carrying vector beams. Optics letters 38, 932–934 (2013).
Rumala, Y. S. et al. Tunable supercontinuum light vector vortex beam generator using a qplate. Optics letters 38, 5083–5086 (2013).
Gecevicius, M. et al. Toward the generation of broadband optical vortices: extending the spectral range of a qplate by polarizationselective filtering. JOSA B 35, 190–196 (2018).
Birks, T. A., Knight, J. C. & Russell, P. S. J. Endlessly singlemode photonic crystal fiber. Optics letters 22, 961–963 (1997).
Dudley, J. M., Genty, G. & Coen, S. Supercontinuum generation in photonic crystal fiber. Reviews of modern physics 78, 1135 (2006).
Li, H. et al. Broadband orbital angular momentum transmission using a hollowcore photonic bandgap fiber. Optics letters 41, 3591–3594 (2016).
Zhang, H. et al. The orbital angular momentum modes supporting fibers based on the photonic crystal fiber structure. Crystals 7, 286 (2017).
Yue, Y. et al. Octavespanning supercontinuum generation of vortices in an As 2S 3 ring photonic crystal fiber. Optics letters 37, 1889–1891 (2012).
Prabhakar, G., Gregg, P., Rishøj, L. & Ramachandran, S. Infiber monomode octavespanning oam supercontinuum. In Lasers and ElectroOptics (CLEO), 2016 Conference on, 1–2 (IEEE, 2016).
Brunet, C. et al. Vector mode analysis of ringcore fibers: Design tools for spatial division multiplexing. Journal of Lightwave Technology 32, 4046–4057 (2014).
Ramachandran, S., Gregg, P., Kristensen, P. & Golowich, S. On the scalability of ring fiber designs for oam multiplexing. Optics express 23, 3721–3730 (2015).
Ung, B. et al. Fewmode fiber with inverseparabolic gradedindex profile for transmission of oamcarrying modes. Optics Express 22, 18044–18055 (2014).
Agrawal, G. P. Nonlinear fiber optics. In Nonlinear Science at the Dawn of the 21st Century, 195–211 (Springer, 2000).
Dudley, J. M. & Taylor, J. R. Supercontinuum generation in optical fibers. (Cambridge University Press, 2010).
Afshar, S. & Monro, T. M. A full vectorial model for pulse propagation in emerging waveguides with subwavelength structures part i: Kerr nonlinearity. Optics express 17, 2298–2318 (2009).
Blow, K. J. & Wood, D. Theoretical description of transient stimulated raman scattering in optical fibers. IEEE Journal of Quantum Electronics 25, 2665–2673 (1989).
Lin, Q. & Agrawal, G. P. Raman response function for silica fibers. Optics letters 31, 3086–3088 (2006).
Deiterding, R., Glowinski, R., Oliver, H. & Poole, S. A reliable splitstep fourier method for the propagation equation of ultrafast pulses in singlemode optical fibers. Journal of Lightwave Technology 31, 2008–2017 (2013).
Muslu, G. & Erbay, H. A splitstep fourier method for the complex modified kortewegde vries equation. Computers & Mathematics with Applications 45, 503–514 (2003).
Saitoh, K. & Koshiba, M. Empirical relations for simple design of photonic crystal fibers. Optics express 13, 267–274 (2005).
Torner, L., Torres, J. P. & Carrasco, S. Digital spiral imaging. Optics express 13, 873–881 (2005).
Acknowledgements
This work was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) and the Fonds de recherche du QuebecNature et technologies (FRQNT).
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M.S. and B.U. contributed to the optical fiber design and cowrote the manuscript. All authors contributed to the analysis and discussion of the data and reviewed the manuscript. B.U. conceived the idea and provided primary supervision of the work.
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Sharma, M., Pradhan, P. & Ung, B. Endlessly monoradial annular core photonic crystal fiber for the broadband transmission and supercontinuum generation of vortex beams. Sci Rep 9, 2488 (2019). https://doi.org/10.1038/s41598019395271
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DOI: https://doi.org/10.1038/s41598019395271
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