Higher-order modes (HOMs) are known to be scalably stable in step-index multimode fibers [1,2], and have been used for rare-earth-doped high-power fiber amplifiers [3]. The high confinement of these modes, in spite of being large in mode area, is also useful for tailoring dispersion [4,5] for high-power nonlinear applications [6]. The guidance region for HOMs in fibers has typically been defined by either a fluorine-doped outer cladding or an air-clad microstructured cladding [2]. For fluorine-doped fibers, the dispersion tunabilty is limited by the number of guided modes, which depends on manufacturing constraints in achievable refractive index step contrasts. On the other hand, air-clad microstructured fibers may not have the azimuthal symmetry required for stably guiding very-high-order modes. Given recent advances in efficient free-space excitation of vortices and HOMs [7,8], it is worth considering whether a very simple fiber structure for HOM guidance could be realized, using low-index polymer jackets, conventionally used for multimode pump propagation in high-power fiber lasers.

Here, we demonstrate, for the first time to the best of our knowledge, stable HOM (${\mathrm{HE}}_{1,m}$ modes) propagation in a simple fiber structure. The guiding region is defined by a silica core surrounded by a low-index polymer cladding (Efiron PC373). Modes ranging from ${\mathrm{HE}}_{\mathrm{1,12}}$ to ${\mathrm{HE}}_{\mathrm{1,22}}$ (effective areas up to $2000{\mathrm{\mu m}}^2$) were found to be stable ($>10.9\mathrm{dB}$ mode purity compared to other mode groups) over 15.6 m of this fiber.

The fabricated fiber has a pure-silica core with a diameter of 100 μm, surrounded by a low-index polymer jacket of 62 μm thickness. The simulated dispersion for a selection of ${\mathrm{HE}}_{1,m}$ modes in this fiber are shown in Fig. 1(a). The refractive index profile of the fiber is shown (blue trace) in Fig. 1(b). For reference, a fictitious fiber of identical dimensions, but with a down doped fluorine region instead of low-index polymer is also shown (red dashed trace). The simulated dispersion at 1064 nm for different modes of the two fibers is shown in Fig. 1(c). The index step for the polymer fiber is approximately 4 times larger than that possible with a fluorine down doped region, thus the modal cutoff occurs at a much higher mode order (${\mathrm{HE}}_{\mathrm{1,41}}$ instead of ${\mathrm{HE}}_{\mathrm{1,20}}$). This leads to anomalous dispersion at 1064 nm as large as $7\times$ that possible in the all-glass HOM fiber. The effective area of the modes between ${\mathrm{HE}}_{\mathrm{1,10}}$ and ${\mathrm{HE}}_{\mathrm{1,40}}$ ranges from 1600 to $2000{\mathrm{\mu m}}^2$. Both these properties are of great interest in nonlinear applications.

 

Fig. 1. (a) Simulated dispersion for a selection of ${\mathrm{HE}}_{1,m}$ modes in the fabricated fiber. (b) Refractive index profile of fabricated polymer coated fiber and fictious fluorine fiber. (c) Simulated dispersion at 1064 nm as a function of mode order for the two fibers.

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The experimental setup is shown in Fig. 2(a). The source used for mode imaging is a fiber Bragg grating stabilized laser diode at 1048 nm ($\mathrm{FWHM}=0.8\mathrm{nm}$). Quantitative mode purity measurements are performed by frequency-domain cross-correlation imaging (${\mathrm{fC}}^2$) using a 10xx nm tunable external cavity diode laser (ECL) [9], or time-domain cross-correlation imaging ($C^2$) using a 1064 nm LED with 2.8 nm FWHM [10,11]. Modes are excited using a spatial light modulator (SLM) that encodes the desired spatial phase on the linearly polarized incident Gaussian beam before the light is coupled into the fiber under test (FUT) [8].

 

Fig. 2. (a) Experimental setup for higher-order mode excitation. The source is at 1048 nm ($\mathrm{FWHM}=0.08\mathrm{nm}$) for images, a 10xx nm tunable ECL for ${\mathrm{fC}}^2$, or a $\mathrm{FWHM}=2.8\mathrm{nm}$ filtered LED centered at 1064 nm for $C^2$, single-mode fiber (SMF), spatial light modulator (SLM), fiber under test (FUT), polarizing beam displacing prism (PBDP). (b)–(g) Imaged modes after 15.6 m of propagation.

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All guided ${\mathrm{HE}}_{1,m}$ modes were selectively excited and imaged in a 15.6 m long FUT with a coiling radius of 13 cm [output images in Figs. 2(b)–2(g)]. Modes below ${\mathrm{HE}}_{\mathrm{1,11}}$ [Figs. 2(b) and 2(c)] exhibit distributed mode coupling, as apparent from the mode images. Cut-back experiments confirmed that mode images more closely resembled simulated modes at shorter lengths decreased. In this class of large-mode-area (LMA), step-index multimode fibers, mode stability is largely determined by the effective index ($n_{\text{eff}}$) separation between a given ${\mathrm{HE}}_{1,m}$ mode and its nearest neighbor antisymmetric mode—as these modes are preferentially coupled by bends in the fiber [1]. Separation in $n_{\text{eff}}$ increases with radial mode order (m); thus we expect that higher modes orders propagate more stably—in keeping with the “pure” mode images measured for ${\mathrm{HE}}_{\mathrm{1,14}}$ and ${\mathrm{HE}}_{\mathrm{1,17}}$ [Figs. 2(d) and 2(e)]. However, above ${\mathrm{HE}}_{\mathrm{1,24}}$ [Figs. 2(f) and 2(g)] the mode images start to degrade. In order to better understand the origin of this modal degradation for very high mode orders, and explain why even the “pure” modes have a noncircular first ring, cross-correlation measurements were performed [9–11]. Before discussing the cross-correlation measurements, we will first consider modes in high-index waveguides.

Mode classification within the scalar approximation leads to azimuthally symmetric ${\mathrm{LP}}_{0,m}$ modes, also called ${\mathrm{HE}}_{1,m}$ modes in the full vectorial description. This uniformly (e.g., linearly) polarized approximation gradually breaks down as the field strength at a large refractive index step increases, in any waveguide [12–14], because the radial and azimuthal components of the electric field must satisfy different boundary conditions. Consequently, the electric fields of the ${\mathrm{HE}}_{1,m}$ modes become quasi-radially polarized, while another set of vector modes, the ${\mathrm{EH}}_{1,m}$ modes, become quasi-azimuthally polarized. Since the polymer-clad fibers discussed herein have especially high-index contrast, we expect the modes of these fibers to show significant polarization nonuniformities. Figures 3(a)–3(d) show the simulated intensity distribution overlaid with polarization projections, for a lower- and higher-order mode pairs comprising the almost degenerate ${\mathrm{HE}}_{1,m}$ and ${\mathrm{EH}}_{1,m-1}$ modes for our fiber, simulated using a full-vectorial mode solver described in [15]. For lower mode orders, the LP approximation holds well, but for higher mode orders, the polarization, and even the intensity distribution along the azimuth, becomes nonuniform (these high-order modes have been named “bow-tie” modes [13,14]). Stated differently, as mode order increases (and the mode field increasingly encounters the high-index step boundary), the ${\mathrm{HE}}_{1,m}$ modes’ polarization extinction ratio (PER), the power ratio between the maximum and minimum linear polarization projection of the mode, decreases dramatically [Fig. 3(e)]. Likewise, even assuming the most perfect phase structured free-space coupling setup as much as 3 dB of power would be coupled into the undesired ${\mathrm{EH}}_{\mathrm{1,16}}$ mode with respect to the intended ${\mathrm{HE}}_{\mathrm{1,17}}$ mode, for example. Figure 3(f) summarizes this calculation using overlap integrals, and shows that, even with perfect scalar mode excitation, other parasitic modes are increasingly excited as mode order increases (note that our simulations in Fig. 3 assume an air-silica guide because mode excitation and imaging is performed on a cleaved fiber input/output, where the guidance structure is effectively silica-air). These calculations qualitatively match the observations illustrated in Figs. 2(d) and 2(e), which show progressively more distorted output mode images as mode order is increased, in spite of the fact that $n_{\text{eff}}$ separations increase with mode order, which would have predicted more stable guidance for higher mode orders. Thus, while high-index contrasts easily enabled by glass-polymer-clad structures provide expansive dispersion tailoring opportunities, the number of modes actually available for such tailoring may depend on the aforementioned vector effects, and quantitative mode purity analysis is needed to understand the subset of modes available.

 

Fig. 3. Simulated intensity distribution of four modes ($\mathrm{\Gamma }=0.6$), orange arrows indicate local polarization: (a) ${\mathrm{HE}}_{1,3}$, (b) ${\mathrm{HE}}_{\mathrm{1,17}}$, (c) ${\mathrm{EH}}_{1,2}$, (d) ${\mathrm{EH}}_{\mathrm{1,16}}$. (e) Simulated PER. (f) Simulation of most parasitic modal content in the fiber relative to targeted ${\mathrm{HE}}_{1,m}$ mode.

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For quantitative measurements, we employ a polarization-resolved variant of the ${\mathrm{fC}}^2$ technique presented in [9] [see setup in Fig. 2(a)]. An ${\mathrm{fC}}^2$ trace shows the power of excitation at different temporal delays, and since temporal delays map directly to mode order, mode purity is simply deduced by comparing the highest peak with the parasitic peaks at other temporal delays. Moreover, digital mode reconstructions help identify the mode order causing the parasitic peaks in the trace. Three example traces are shown in Figs. 4(a)–4(c). In Fig. 4(a), we attempt to excite the ${\mathrm{HE}}_{1,5}$ mode, peaks after 200 ps are spurious and are caused by reflections in the system. The desired mode appears at a delay of about 95 ps, as expected; however, the temporal response around this peak is significantly wider than the expected delay to the neighboring ${\mathrm{HE}}_{1,m}$ modes, which is about 16 ps. This indicates that the mode is not pure and that distributed coupling has occurred to other modes, as suggested by the mode images in Figs. 2(b) and 2(c). In Fig. 4(b), the peak at a delay of 92 ps is the mode group containing the targeted ${\mathrm{HE}}_{\mathrm{1,17}}$ mode, and the most parasitic mode is suppressed by 17 dB. The peak is narrow compared to the resolution of the ${\mathrm{fC}}^2$ system, and also significantly narrower than the peak for the ${\mathrm{HE}}_{1,5}$ mode, which is evidence of negligible distributed mode coupling between mode groups. Figure 4(c) shows the ${\mathrm{fC}}^2$ measurement when exciting the ${\mathrm{HE}}_{\mathrm{1,30}}$ mode (appears at delay of 214 ps), and in this case, the nearest parasitic mode group has 8.2 dB lower power. This lower purity [compared to that for the ${\mathrm{HE}}_{\mathrm{1,17}}$ mode of Fig. 4(b)] is consistent with the observation that the images of modes of very high order were degraded [Figs. 2(f) and 2(g)]. Since their ${\mathrm{fC}}^2$ peaks are narrow, mode propagation appears to be stable, and hence we speculate that the cause for the lower measured purity is impure excitation with our SLMs (due to vector effects discussed in context of Fig. 3). Similar measurements were performed thrice for each of the modes from ${\mathrm{HE}}_{1,5}$ to ${\mathrm{HE}}_{\mathrm{1,33}}$, and are summarized in Fig. 4(d). The red crosses are individual measurements and the blue line denotes the average result. The modes are divided into three groups. The green region represents “pure” modes, the blue region illustrates modes suffering from distributed mode coupling, and the red region are the modes that we suspect to be free from distributed coupling but not purely excited.

 

Fig. 4. (a) Three examples of ${\mathrm{fC}}^2$ traces: (a) ${\mathrm{HE}}_{1,5}$, (b) ${\mathrm{HE}}_{\mathrm{1,17}}$, (c) ${\mathrm{HE}}_{\mathrm{1,30}}$. (d) Modal purity as a function of targeted mode order. Each red cross represents mode purity to most parasitic mode found from a ${\mathrm{fC}}^2$ trace. The blue line is the average for each mode. Representative mode images in the three regimes are shown in Fig. 2.

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The ${\mathrm{fC}}^2$ measurements, however, do not reveal some subtleties related to mode purity in these fibers. For instance, when the ${\mathrm{HE}}_{\mathrm{1,17}}$ mode is excited, ${\mathrm{fC}}^2$ measurements reveal parasitic modes to be below $-17\mathrm{dB}$. This is normally a very high-purity mode that would not have a distorted first high-intensity ring, as the mode intensity image in Fig. 2(e) shows. This apparent discrepancy is addressed by considering that HOMs in a step-index fiber are divided into mode groups that are (almost) degenerate in group index. That is, the ${\mathrm{HE}}_{1,m}$ and ${\mathrm{EH}}_{1,m-1}$ modes of given radial order m, form a mode group. The delay between modes within a group is typically less than the temporal resolution of the ${\mathrm{fC}}^2$ system, meaning that ${\mathrm{fC}}^2$ measurements probe the purity between mode groups rather than individual modes. To analyze the modal content within a mode group higher resolution (time-domain) $C^2$ measurements are needed.

The setup for performing higher-resolution time-domain $C^2$ measurements is shown in Fig. 2(a). Due to the computationally intensive nature of $C^2$ data acquisition, only the center portion of the ${\mathrm{HE}}_{1,m}$ mode images (center spot and first ring) were included in the analysis. In postprocessing, the envelope of each pixel and polarization was found separately, and the envelopes for all pixels for a given polarization were then added together and normalized by the peak. Figure 5(a) illustrates $C^2$ traces for the ${\mathrm{HE}}_{\mathrm{1,12}}$ mode, which, based on excitation purity predictions [Fig. 3(e)], is substantially uniformly polarized like an ${\mathrm{LP}}_{\mathrm{0,12}}$ mode, and hence is expected to be excited with high purity. In either polarization bins there is only a single Gaussian envelope at the same delay, confirming the aforementioned theoretically expected behavior. The ${\mathrm{HE}}_{\mathrm{1,17}}$ mode [Fig. 5(b)], on the other hand, becomes harder to excite with a uniform polarization setup, and the excitation purity is expected to be only $\sim 3\mathrm{dB}$ [see Fig. 3(f)]. In this case each mode within the mode group (${\mathrm{HE}}_{\mathrm{1,17}}$ and ${\mathrm{EH}}_{\mathrm{1,16}}$) projects into orthogonal polarization bins, and the measured (1.45 ps) and simulated (1.58 ps) delays between the individual modes match well. The excitation of these two modes is also predicted in Fig. 3(f). The two modes being distinct and narrow also confirms that these modes remain pure and stable while propagating. For even higher mode orders the envelopes always have several peaks in each polarization projection [Fig. 5(c); target mode: ${\mathrm{HE}}_{\mathrm{1,26}}$ mode]. The envelopes are broader than the theoretically deduced 2.7 ps group-delay difference between the target mode and the closest mode within the group (${\mathrm{EH}}_{\mathrm{1,25}}$). This indicates that higher azimuthal order modes, e.g., ${\mathrm{HE}}_{3,m-1}$, ${\mathrm{HE}}_{5,m-2}$, ${\mathrm{EH}}_{3,m-2}$, etc. (each of these modes have a simulated delay of a few picoseconds relative to ${\mathrm{EH}}_{\mathrm{1,25}}$) may have been additionally excited since the need for encoding higher spatial frequencies on the SLM may cause undersampling. The dispersion of the modes can be calculated from the width of the envelopes and the dispersion in the reference arm. This was performed for each pixel, and the results are shown for different mode orders in Fig. 5(d). For the lower-order modes, where a single mode was excited, the agreement is good. This validates the assertion that distributed mode coupling did not take place along the fiber. However, the error bars increase with increasing mode order, and for modes larger than ${\mathrm{HE}}_{\mathrm{1,24}}$ the dispersion is overestimated. This likely is due to the difficulty in accurately fitting the $C^2$ envelope in the presence of numerous peaks, as seen in Fig. 5(c).

 

Fig. 5. (a) Three examples of $C^2$ traces target mode are: (a) ${\mathrm{HE}}_{\mathrm{1,12}}$, (b) ${\mathrm{HE}}_{\mathrm{1,17}}$, (c) ${\mathrm{HE}}_{\mathrm{1,26}}$. (d) Dispersion at 1064 nm for different mode orders. Black circles are average over all pixels, error bars denote a single standard deviation, and blue crosses are simulated results.

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Thus, we believe that, given our current excitation method, there is an amount of ${\mathrm{EH}}_{1,m-1}$ content being excited when targeting a ${\mathrm{HE}}_{1,m}$ mode in the bowtie regime. This is in agreement with the $C^2$ measurements shown in Fig. 5(b), which showed the presence of two modes that both propagated stably in the fiber. Furthermore, this is also evident from the noncircular first ring observed in the mode images of Figs. 2(b)–2(g), caused by interference between these two modes. Therefore, excitation of pure modes in high-index contrast fibers would require an excitation scheme that offers both spatial polarization and phase tailoring. Given the existence of devices such as q-plates [16] and continual developments in the field of micro-optics that can achieve many SLM functionalities in a compact, manufacturable, high-power tolerant platform, we believe that such field-profile tailoring can be practically realized. Another concern with such fibers is that the guiding region and the physical edge of the glass fiber are the same—hence care is needed when cleaving or polishing the fiber. Nevertheless, quantitative interferometric measurement techniques allow us to confirm, independent of our ability to excite the modes cleanly, that HOM propagation is stable in these fibers, and dispersion design is enhanced by the presence of many more modes than are available in all-glass waveguiding structures.

In summary, we demonstrate stable LMA propagation in a simple fiber comprising of only silica drawn with a low-index polymer jacket. Modes ranging from ${\mathrm{HE}}_{\mathrm{1,12}}$ to ${\mathrm{HE}}_{\mathrm{1,22}}$, with mode areas ranging $1700–2000{\mathrm{\mu m}}^2$, and dispersion-zeroes ranging from 1037 to 835 nm, were found to be stable ($\gtrsim 11\mathrm{dB}$ pure compared to other mode groups) over 15.6 m of propagation in this fiber. The simplicity of this fiber design, which could potentially be scaled to even larger $A_{\text{eff}}$, may be beneficial for building doped fiber lasers or achieving enhanced dispersion tunability for high-energy fiber nonlinear phenomena. Although such structures are the only way, to the best of our knowledge, to achieve large dispersion tunability in large-mode-area fibers, future work would clarify if the inherent simplicity offered by this design is, however, offset by the need for a more complicated excitation.