1. Introduction

Recently, there has been a resurgence of interest in multi-mode optical fibers (MMFs) due to the unique properties of higher-order states of light, both inside and outside the waveguide. Pure modes, and even coherent superpositions of modes, can provide the structured light fields necessary for applications such as atomic acceleration and trapping [1,2], as well as biological imaging [3]. Inside the fiber, large effective areas make higher order modes (HOMs) ideal for building fiber lasers [4], intermodal interactions facilitate otherwise impossible nonlinear optical phenomena [5,6], and the sheer abundance of states allows for increase in data capacity for telecommunications [7,8].

For all of these applications, there is an underlying need to begin with a Gaussian beam (the typical profile of a given laser source), excite the desired mode or modes within the fiber, and depending on the application, convert back to a Gaussian – all with high conversion efficiency and low loss. Such conversions can be achieved by converting the fundamental mode to a target HOM and back within the fiber via devices such as long period gratings (LPGs) [9], fused couplers [10], and multi-mode interference devices [11]. In-fiber conversion methods offer high conversion efficiency (>99.9% in the case of LPGs), can be low-loss, and extremely broadband [12], but typically require the fiber to have a single-moded core which renders these devices unsuitable for high power operation due to self-phase modulation, dielectric breakdown, and other detrimental effects from high intensity in the fundamental mode. Alternatively, one can manipulate the amplitude and phase of the light in free-space using elements such as axicons [13,14], phase- [15] or q- [16] plates, and computer-generated holograms (CGHs) [17,18], and then couple directly into the target HOM. Such free-space conversion systems typically have higher loss and lower conversion efficiency, but may be better suited for high power applications. Moreover, free-space methods using CGHs allow for rapid testing of multiple modes and wavelengths without the need for physically fabricating new devices.

In this work, we consider two free-space methods for input and output coupling: axicons, and binary phase plates (BPPs). We demonstrate pure excitation of the LP_0,9 mode in commercial coreless step-index multi-mode fiber with > 15 dB parasitic mode suppression using both a free-space BPP and an axicon coupling system, each implemented by a spatial light modulator (SLM). Reciprocal systems are used to convert the LP_0,9 mode back into a Gaussian-like beam (M² < 1.25) with low loss (1.30 ± 0.1 dB for the BPP, 0.85 ± 0.1 dB for the axicon). The relative merits of each coupling technique are discussed along with the extension of these methods to other modes and other fiber designs. All the experiments were conducted using stock commercial components and fibers, demonstrating that the unique properties of HOMs can be exploited without costly infrastructure or complicated fiber design.

2. Fiber modes and their free-space analogues

For the purposes of this work, we consider the set of stable, azimuthally-invariant LP_0,m fiber modes. In the case of a step-index, single-clad multi-mode fiber, the solution for each mode is of the form$J_0(k_{t}r)$within the guiding region, where the transverse wave vector is defined by$k_t^2=k_0^2{n}_1^2-{\beta }^2$, where $k_0=2\pi /\lambda$, n₁ is the refractive index of the guiding region, and β is the propagation constant for the mode [19]. Outside the guiding region, the field evanescently decays, thus each mode can be accurately described as a truncated version of a Bessel beam [20]. We have previously shown that despite this truncation, LP_0,m modes have many of the same properties as Bessel beams in free space, including diffraction-resistance, and self-healing [21]. It therefore follows that many of the same free space techniques used to create Bessel beams could apply to excitation of LP_0,m modes in fiber, or conversion of fiber modes into Gaussian beams.

The Bessel beam is an oscillatory function with a few key properties relevant to this work. The simplest model of a Bessel beam is plane waves interfering on a cone, as discussed in [21]. We can define the cone angle as$\theta \approx k_t/\beta$, which increases with increasing mode order (m, in LP_0,m). Since an axicon is a conical lens which creates this condition of interfering plane waves on a cone [22], we can see there is a direct proportionality between base angle of the axicon and the mode order one aims to excite.

Another key property to consider is that$J_0(k_{t}r)$ is a real, oscillatory function and accordingly, the phase can only have binary values of 0 or π. This is ideal for free-space conversion because this phase pattern can be imparted with a simple structure – such as a binary phase plate. Lastly, the envelope of the Bessel-beam’s amplitude decays as$\sqrt{2/(\pi k_{t}r)}$ [23], considerably different than a Gaussian envelope. We therefore expect that the lack of overlap between these two envelope functions will pose the fundamental limit for conversion efficiency and loss for free-space excitation of these modes.

Moving away from simple phase structures, it is well-known that arbitrary amplitude and phase sculpting of any optical beam is possible with two complex phase structures separated by a propagation distance, and this method has been used for fiber mode conversion [18]. However, such structures are highly complex and possible to create only using CGHs such as spatial light modulators (SLM). In this work, we utilize SLMs for mode conversion, but we intentionally keep the phase structures as simple as possible thus restricting our study to devices that can be physically manufactured in silica or other dielectrics. This retains the utility of these schemes for high power applications where low loss and high power tolerance are of paramount importance.

3. Input coupling

In order to efficiently excite a given HOM, we structure the phase of an incoming Gaussian beam using an SLM (Hamamatsu X10468-08), and then image the beam into the fiber under test (FUT). The setups used for exciting modes with a binary phase plate (BPP) and axicon are shown in Figs. 1(a) and 1(b), respectively. In each case, the light source is an external cavity diode laser (ECL, ~100 KHz linewidth, λ = 1050 nm), which is coupled to single-mode fiber (SMF). The beam is horizontally polarized (the preferred orientation of the liquid crystals on the SLM), and imaged onto the surface of the SLM (1/e² width is 3.3 mm).

 

Fig. 1 (a) Binary phase plate (BPP) input coupling setup; SLM phase profile includes a BPP designed for LP_0,9 (diameter = 3.8 mm) and a lens (f = 465 mm) as shown on the left-hand side of the figure; the mode in the far field of the SLM is shown inset (contrast and brightness enhanced for visibility); (b) Axicon input coupling setup; the setup upstream of the SLM is identical to (a); SLM phase profile is an axicon with angle α = 0.787° as shown on the left-hand side of the figure; Bessel-Gauss beam in the focal plane of the axicon and the beam in the focal plane of lens f₂ are shown inset (contrast and brightness enhanced for visibility); (c) Facet image of the fiber under test (FUT); (d) Refractive index profile of the FUT; Modal purity is characterized via frequency-domain C² imaging (fC²).

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A BPP imparts concentric rings of alternating 0 and π phase onto the incident Gaussian beam. The radial positions of the phase reversals match the phase profile of the target HOM, such that the phase transfer function is given by $\phi (r)={\tan }^{-1}(\mathrm{Im}[f(r)]/\mathrm{Re}[f(r)])$, where f(r) is the electric field of the target fiber mode. The structured beam is imaged into the FUT where it has high overlap with the target mode. The hologram on the SLM combines a BPP matching the LP_0,9 mode in the FUT (diameter = 3.8 mm), and a parabolic phase profile corresponding to a lens of focal length f = 465 mm. The lens on the SLM works with lens f₁ (f = 8 mm) to image the Gaussian with the imparted binary phase into the FUT (magnification M = 75.2). In the far field of the lens on the SLM [inset Fig. 1(a)], the beam has a ring reminiscent of the far field of a Bessel beam.

An axicon is essentially a conical lens [22], with base angle α, and accordingly has an effective focal length, f_axicon. The phase transfer function of the device is given by$\phi (r)=-k_{0}rtan[si{n}^{-1}(nsin[\alpha ])-\alpha ]$, where n is the refractive index of the material. In the focal plane, the Gaussian becomes a Bessel-Gaussian beam with of form $\psi (r)=J_0(\kappa r)exp[-{(r/W_z)}^2]$, where κ is the spatial frequency of the Bessel beam given by $\kappa =k_0\alpha (n-1)$, n is the refractive index of the axicon, r is the radial coordinate, and W_z is the 1/e beam width of the field at the focal plane. In order to match the spatial frequency of the Bessel-Gauss beam in free-space to the spatial frequency of the target mode we require$\kappa M=k_t$ where M is the magnification of an imaging system used to relay the beam in the FUT, and the fiber is assumed to be step index. However, this relationship requires one to know the propagation constant of the mode, β, in order to evaluate the transverse wave vector, k_t. Alternatively, one can make the assumption that for proper coupling, the m^th zero of the truncated Bessel function J₀^(m), corresponding to the LP_0,m mode, must coincide with the cladding radius, R_c, when imaged onto the fiber facet. This condition leads to the relation

The axicon input coupling setup is shown in Fig. 1(b). The hologram is a conical phase profile corresponding to a physical axicon with angle α = 0.787°. In the focal plane of the axicon, the beam is Bessel-Gaussian as expected [left inset Fig. 1(b)]. Lenses f₂ (f = 200 mm) and f₁ (f = 6 mm) are used as a telescope to image the Bessel-Gauss beam into the FUT (measured magnification M = 33.9). In the far field of f₂, the beam once again resembles the far field of a Bessel beam.

The key difference between the BPP and axicon setups is the plane being imaged into the fiber. In the binary case, the beam has highest overlap with the target mode in the plane where the binary phase is applied, therefore the lenses image the surface of the SLM. In the axicon case, the beam must propagate a certain focal length (determined by the beam size and the axicon angle) before it becomes a Bessel-Gauss beam and correspondingly has high overlap with the target mode; therefore, the telescope must image the axicon’s focal plane. For the binary case, the image plane is static even as the phase plate is changed, therefore arbitrary modes can be excited by changing the phase plate, with no need for realignment of the system. In the axicon case, the angle must be adjusted to match the spatial frequency of the target mode [Eq. (1)], but changing this angle also changes the focal length of the axicon and thus the system must be realigned. Alternatively, a single axicon with fixed α can be used to excite arbitrary modes with high purity, provided M and W₀ are changed accordingly.

The FUT employed for these experiments is a 1.38-m segment of commercially available step-index single-clad multi-mode fiber (Thorlabs FG050LGA) with an index step Δn = 17.6 × 10⁻³, and a 50 μm guidance region diameter, which guides the first ten LP_0,m modes at λ = 1050 nm. An image of the fiber facet and the refractive index profile are shown in Figs. 1(c) and 1(d), respectively. As discussed in [24], effective index splitting in step-index multi-mode fibers increases with radial mode order, therefore we expect the excited modes to be stably guided with respect to their nearest neighbor LP_1,m modes for tens of meters.

Figure 2 shows the simulated performance of each coupling system. The mode profiles for the FUT were calculated using a scalar eigenmode solver, and the beam in the focal plane of the axicon was simulated with a Fresnel-regime propagator [25]. The two key metrics for input coupling are loss, and coupling purity. To examine loss, we calculate the overlap integral, η_0,m, between the incident field, ψ_i(r,ϕ), and the target mode, ψ_0,m(r,ϕ).

 

Fig. 2 Input coupling simulations; (a) Comparison of the simulated electric fields for a Gaussian beam converted by a BPP and the target LP_0,9 mode; (b) MPI as a function of mode order for the mode converted by BPP; (c) Comparison of the simulated electric fields for a Gaussian beam converted by an axicon and the target LP_0,9 mode; (d) Simulated MPI as a function of mode order for the mode converted by axicon; (e) Simulated overlap between the converted and target modes as a function of normalized Gaussian width for binary (red) and axicon (blue) conversion; (f) Incoherent sum of simulated MPI to all parasitic modes as a function of normalized Gaussian width for binary (red) and axicon (blue) conversion; Dashed red and blue lines in (e) and (f) are optimal widths which correspond to those used for the simulations in (a), (b), (c) and (d).

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Figure 2(a) shows the electric field profile of a Gaussian beam with imparted binary phase (red) for a BPP-based coupling system. The phase reversals of the converted mode match those of the target LP_0,9 mode (green), resulting in a 74.0% overlap between the modes. Figure 2(b) shows the MPI between the target mode and all other LP_0,m modes in the fiber. The most parasitic mode is LP_0,7 which is suppressed by 16 dB. It should be noted that in absence of an assumed tilt on the incoming beam, the simulated MPI suppression for LP_l≠0,n modes → ∞ because the overlap integral [Eq. (2)] is exactly zero. Therefore, the observation of higher order angular states in experiments corresponds to imperfect alignment of the system.

Figure 2(c) shows the electric field profile of the Bessel-Gauss beam at the focal plane of an axicon (blue), as compared to the target LP_0,9 mode for an axicon-based coupling system. The phase reversals are in sync, though we can see that the envelope of the Bessel-Gauss beam extends beyond the target mode. Despite this fact, the overlap between the two is 83.5%. MPI as a function of mode order is shown in Fig. 2(d). The most parasitic mode is LP_0,8, which is suppressed by 16.1 dB.

The effect of the input Gaussian size, W₀, on overlap is shown in Fig. 2(e). In the BPP case (red lines), there is a clear local maximum in overlap (η_0,9 = 74%) when W₀ = 0.83·R_c·M which is marked with a dashed red line in Fig. 2(e). The simulations in Figs. 2(a) and 2(b) correspond to this optimal point. Figure 2(f) shows an incoherent sum of the total parasitic mode content in the fiber, the “sum MPI,” for a given input size. Coherent crosstalk between the modes, which is ignored in this case, can have significant effects on the noise properties of the system, and the intensity distribution inside and outside the fiber [26]. Ignoring coherent effects does not give a full picture of the behavior of the system when in operation, but it does allow us to quantify the parasitic mode content which is ultimately the parameter we are trying to optimize.

In the binary case (red line), the sum MPI has a local minimum (−14.5 dB) which is nearly coincident with the maximum overlap in Fig. 2(e). The maximum overlap point was used for the simulations present in Figs. 2(a) and 2(b). For the experimental setup, W₀ = 1.65 mm, and R_c·M = 1.9 mm, therefore the corresponding normalized width is 0.87.

In the axicon case (blue), the overlap has a local maximum, but there is no minimum in the sum MPI. Instead, it monotonically decreases with increasing Gaussian width. This can be understood by considering the envelope of the Bessel-Gauss beam. If the width of the Bessel-Gauss beam, W_z, is large, then the Gaussian envelope is effectively a plane wave and the beam approaches a true Bessel-beam, leading to low MPI. In this limit, however, a large portion of the power exists outside the cladding of the FUT, and the overlap integral decreases accordingly. This poses an interesting tradeoff in loss and MPI, which is not present in the BPP case where sum MPI cannot fall below 15.65 dB. We choose W₀ = 2R_c·M as the operation point (dashed blue lines in both figures) as it is near the maximum overlap and the parasitic mode suppression at this point [Fig. 2(d)] is comparable to the BPP case. In the axicon experiments, W₀ = 1.65 mm, and R_c·M = 0.82 mm, therefore the normalized width is 2.01.

Experimentally, MPI is measured using a polarization-resolved variant of the frequency domain imaging (fC²) technique presented in [27]. The laser was swept 10 nm for each measurement around a center wavelength of 1050 nm.

For each fC² measurement, interference between the modes in the FUT and an external reference is captured by collecting camera images as a function of wavelength. A pixel-by-pixel Fourier transform is then applied to convert from frequency to the time domain. Each mode manifests as a separate peak in the measurement so that the relative power in each mode can be determined. The spatial information is still preserved, so each peak can be reconstructed to determine the mode identity. More details about this technique can be found in [27] and [28].

Figure 3(a) shows a fC² trace for BPP excitation of LP_0,9, and the corresponding mode image is shown (contrast and brightness enhanced for visibility) in Fig. 3(b). The large peak in each fC² trace at 15.8 ps corresponds to the target LP_0,9 mode, and surrounding peaks correspond to parasitic modes. There are some spurious peaks in each data set which we believe correspond to a Fabry-Perot reflection from the camera cover glass (peaks at 2.2 and 2.9 ps), a back reflection (22.4 ps), and a mode-hopping frequency from the ECL (26.0 ps). The parasitic suppression for this measurement is 16.5 dB, and the most parasitic mode is LP_0,7, as predicted by simulations [Fig. 2(b)].

 

Fig. 3 (a) Sample fC² trace for BPP excitation of LP_0,9; (b) LP_0,9 mode image for BPP excitation, contrast and brightness enhanced for visibility. (c) Sample fC² trace for axicon excitation of LP_0,9; (d) LP_0,9 mode image for axicon excitation, contrast and brightness enhanced for visibility.

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An fC² measurement and corresponding mode image for axicon input coupling are shown in Figs. 3(c) and 3(d) respectively. Here the parasitic suppression is 17.1 dB and the most parasitic mode is LP_0,8 in keeping with the simulations [Fig. 2(d)].

The component losses for the system were measured to form a power budget. We anticipate some loss due to the SLM, partially from imperfect reflection from the broadband coating, and partially due to imperfect diffraction efficiency. The measured losses due to the SLM were 1.1 dB for the BPP case, and 1.3 dB for the axicon case. To avoid this loss, one could fabricate a phase plate or use a physical axicon with an anti-reflection coating.

The coupling loss, as measured from after the SLM to after the FUT, is 1.8 dB for the BPP case and 2.9 dB for the axicon case. Each input coupling setup was optimized to increase mode purity rather than coupling efficiency, we therefore expect that the loss could be reduced with further optimization, particularly for the axicon case. There is a small loss due to the polarizer at the end of the FUT. Some of this loss is due to the polarization extinction ratio (PER) of the beam exiting the fiber. PER, defined as the maximum extinction ratio between orthogonal polarizations of the beam, is typically > 25 dB in a single-mode system and limited only by the polarizer. In the multimode case, however, the polarization state of the ensemble of modes in the test fiber evolve differently based on the length and coiling of the fiber, leading to degradation in the PER. For the excitation schemes considered here, the PER is > 10 dB, thus the loss measured before and after the polarizer (1.2 dB for the BPP case, 1.0 dB for the axicon) is mostly due to the insertion loss of the polarizer itself.

Figure 4 shows the alignment tolerances for the BPP and axicon input coupling setups. In Fig. 4(a), the size of the BPP is adjusted and fC² measurements are performed to measure the corresponding parasitic mode suppression for each case. Changing the size of the phase plate changes the spatial frequency of the phase imparted on the beam, and as a result, power coupled to the parasitic LP_0,10 and LP_0,8 modes increases. Mode images for different BPP sizes are inset in the figure (contrast and brightness enhanced for visibility). Even in the extreme cases where the parasitic modes are not suppressed at all, the mode profile still contains a center spot and 8 rings, highlighting the importance of mode purity measurement techniques such as fC² for accurately determining parasitic mode content. From the data, the tolerance for phase plate size is roughly ± 1% without significant (< 3 dB) degradation of the parasitic mode suppression.

 

Fig. 4 Parasitic mode suppression as measured by fC² imaging for various changes to the system alignment; Mode images for different conditions inset, contrast and brightness enhanced for visibility; (a) Suppression as a function of BPP size; (b) Suppression as a function of FUT offset with respect to the coupling lens for BPP excitation; (c) Suppression as a function of axicon angle; (d) Suppression as a function of FUT offset for axicon excitation.

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In Fig. 4(b), the FUT is intentionally offset with respect to the coupling lens and fC² measurements are made for the BPP input coupling case. The most parasitic modes as the system is misaligned are LP_1,8 and LP_1,9 due to the effective tilt imparted by offsetting the FUT. The tolerance to alignment for an MPI degradation of ~3 dB, is ± 0.3 μm.

Figure 4(c) shows parasitic mode suppression as a function of axicon angle. The most parasitic modes are LP_0,8 and LP_0,10 as the angle deviates from the optimum value. The angle tolerance is roughly ± 1%, as with the binary case, without significant (~3 dB) MPI degradation. Offset alignment tolerance is measured for the axicon case in the same fashion as for the BPP case [Fig. 4(d)], and the alignment tolerance for ~3 dB of MPI degradation is ± 0.15 μm. Comparing the two, the axicon is more sensitive to offset misalignment than the phase plate. The increased sensitivity for the axicon case may be a result of the physical lens f₂ incorporated into the system rather than an intrinsic tradeoff between the two coupling methods.

4. Output coupling

If we design a system to shape a Gaussian beam and couple into a given HOM, we expect that, reciprocally, the same HOM sent backwards through the system will be converted to a near-Gaussian beam. We therefore continue to use the BPP and axicon setups shown in Figs. 1(a) and 1(b) to excite the LP_0,9 mode in the fiber, and then build the respective reciprocal system after the FUT to convert the mode back into a Gaussian. The output coupling setups are nearly identical to those shown in Fig. 1, with the difference being that the physical lens f₂ in the axicon coupling setup [Fig. 1(b)] is replaced by a lens on the SLM itself. The system is reciprocal, therefore we expect a SLM-based lens could also be used for the input coupling setup, however, optimization through minimization of MPI proved more difficult than optimization by maximizing power in the central portion of the output-coupled beam.

For the BPP output coupling setup, the LP_0,9 mode exiting the FUT is imaged onto a second SLM (Hamamatsu X10468-07). A quarter-wave plate, half-wave plate and a polarizer are used to optimize the polarization for the SLM. The hologram on the SLM combines a LP_0,9 BPP (diameter = 4.12 mm), and a lens (f = 540 mm). The axicon output coupling setup uses the same polarization optics, and images the mode to a plane prior to the SLM. The distance between the imaging plane and the SLM corresponds to the focal length of the axicon. The hologram combines an axicon with angle α = 0.6° and a lens (f = 310 mm).

For input coupling, the performance metrics are loss and MPI. In output coupling, the beam is in free-space so neither of these metrics are relevant. Instead, we can use the M² beam quality parameter as a metric for how Gaussian-like the beam is [29].

Figure 5 shows simulations for the performance of the output coupling system. In Fig. 5(a) the intensity profile of LP_0,9 as converted by a BPP (red) is shown in comparison to a fitted Gaussian (green). On a linear scale, the two look similar, but on a log scale [Fig. 5(b)] it becomes apparent that some of the intensity is in rings surrounding the center portion of the beam. Figures 5(c) and 5(d) show the corresponding simulated performance of the axicon coupling system with linear and log scales, respectively.

 

Fig. 5 Simulated Gaussian-like intensity profiles of a converted LP_0,9 mode; (a) BPP conversion on a linear scale; (b) BPP conversion on a log scale; (c) Axicon conversion on a linear scale; (d) Axicon conversion on a log scale. The “qualitatively Gaussian point” (QGP), as defined in the text, is marked in each figure, and a zoomed-in plot of the QGP region is inset in each top right corner.

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It is well known that a phase plate alone cannot improve the M² of a beam [30], so despite the seemingly Gaussian appearance of the output-coupled beams, the M² for each remains large. If we combine a spatial filter with the phase plate, however, the M² can be reduced to near unity, albeit with some added loss. In our case, this is achieved by imposing a circular aperture around the center spot of the simulated output coupled beam, and then propagating into the far field to determine the M² value. The aperture must be appropriately sized to truncate the rings without impinging on the center spot, else knife-edge diffraction causes the M² to increase. When the aperture is appropriately sized, the beam is qualitatively a Gaussian, hence we refer to this condition as the “qualitatively Gaussian point” (QGP). The M² of the beam at the QGP and the loss due to the aperture are the metrics we use to assess the performance of the output coupling systems. The QGP is marked in each of the simulations in Fig. 5. For BPP output coupling, the simulated M² of the filtered beam at the QGP is 1.05, and the simulated loss due to the aperture is 1.25 dB (75% transmitted). For axicon output coupling, the simulated M² of the filtered beam at the QGP is 1.04, and the simulated loss due to the aperture is 0.66 dB (85.9% transmitted).

To measure the M² of the converted beam, we use the setup shown in Fig. 6(a). The full beam in the focal plane of the lens on the SLM [shown for the BPP case with linear and gamma-altered (Γ = 1/5) scales in Figs. 6(b) and 6(c), respectively] has a strong, Gaussian-like center spot, with surrounding rings, in keeping with the simulations [Figs. 5(a) and 5(b)]. The beam is incident on an adjustable iris which acts as a spatial filter. The transmitted portion of the beam is focused down and we translate a camera (Thorlabs DCC1645) through the focus, recording images of the beam at different points. The second-moment width in the x and y directions are calculated for each frame, and the beam width as a function of z is fit to parabola [Eq. (3)] in order to determine M² in each direction (M_x² and M_y²).

 

Fig. 6 (a) Schematic of the M² measurement setup; (b) Experimentally stitched mode image of the converted mode from the BPP output coupling setup; (c) Image of the converted mode with increased gamma contrast (Γ = 1/5); (d) Experimental intensity line cut of in the x direction as a function of z; (e) W_x²(z) measurement with parabolic fit corresponding to M_x² = 8.05; (f) Experimental intensity line cut of in the y direction as a function of z; (g) W_y²(z) measurement with parabolic fit corresponding to M_y² = 9.65.

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For these measurements, it is key that the lens or lenses used focus the beam tightly enough that the size changes significantly, but not so tightly that the camera cannot resolve the structure in the beam. Accordingly, a 300 mm focal length, 2-inch clear aperture lens and a 40 mm focal length, 1-inch clear aperture lens are used for focusing when the beam is large. A single 25 mm focal length 1-inch clear aperture lens is sufficient for smaller beams.

The bit depth of the camera is only 8 bits, which is not necessarily sufficient to resolve low intensity structure in the beam. To address this issue, multiple exposures (one un-saturated, two increasingly saturated) are taken for each image position and stitched together in post processing to effectively increase the camera’s bit depth.

Figures 6(d)-6(g) show a sample M² measurement for BPP conversion with a relatively wide iris diameter. The relative loss before and after the iris is measured with a power meter (HP 81521B); for this iris diameter, the loss is 0.8 ± 0.1 dB (83% transmission). The uncertainty in this and subsequent measurements arises from power fluctuations due to MPI in the FUT and back reflections due to the extremely long coherence length of the ECL. Figure 6(d) and 6(f) show line cuts in the x and y directions respectively through the centroid of the beam as it propagates through focus. Figures 6(e) and 6(g) show the calculated beam widths W_x²(z) and W_y²(z) for each z position. The data points (red dots) are fit to a parabola (black line), and the curvature coefficient of the fit [Eq. (3)] is used to calculate M_x² and M_y².

Figures 7(a) and 7(b) show simulations (red line, blue line) and data (green crosses, black circles) for the M² of the beam as a function of the power transmitted through a circular aperture for BPP and axicon output coupling. Three general regions are apparent. When the aperture is wide, most of the higher spatial frequencies in the beam are allowed through, and thus the M² is large – comparable to the M² of the LP_0,9 beam alone. The rightmost inset in each figure shows a gamma-altered image of the beam (Γ = 1/5) in this regime.

 

Fig. 7 M² as a function of power transmitted through the iris for (a) BPP output coupling; (b) Axicon output coupling; Red and blue lines correspond to simulations, and markers indicate experimental measurements; Mode images with increased gamma-contrast (Γ = 1/5) for various iris diameters are insets.

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When the aperture is sized so that only the center portion of the beam is transmitted (the QGP), M² is approximately unity. The center inset in each figure corresponds to the measured QGP, and accordingly the image shows a Gaussian spot with no surrounding rings. The measured loss due to the aperture at the QGP for BPP conversion is 1.7 ± 0.1 dB (68% transmission), and the measured M² was 1.17 in the x direction and 1.24 in the y direction. For axicon output coupling, the loss at the QGP was 0.85 ± 0.1 dB (74% transmission), and the measured M² was 1.1 in the x direction and 1.15 in the y direction. Note that the rectangular-symmetric pattern of small spots surrounding the Gaussian center at the QGP [center inset Fig. 7(b)] are artifacts from overexposure during the stitching process (as described above).

There is a power fraction offset between the simulated and measured QGP of approximately 7% for the BPP case and 4% for the axicon case. However, we note that, if the simulations are shifted by this constant offset, such that the simulated QGP matches the measurement, then in each case, the rest of the data points fit the simulated curves very well. One possible reason for this offset might be that parasitic mode content was not included in the simulations. In the experiments, the small amount of power in parasitic modes is not Gaussian-like in the far field of the output coupling system (since it is tailored to a different mode), which results in extra power outside the iris at the QGP. Confirming the reason for the observed offset is a subject for future research.

To measure the alignment tolerance for the system, we deliberately alter the phase structure on the SLM from the optimum condition by either translating the entire structure (lens and BPP or axicon) with respect to the incoming beam, or by changing the size of the BPP or the angle of the axicon. For small offsets, the center portion of the beam remains Gaussian, but the beam diffracts off the phase plate at a different angle than for the optimum alignment. Instead of using a physical iris we define a “virtual iris” in post-processing with a fixed diameter corresponding to the QGP for the aligned case. The virtual iris tracks the centroid of the beam as misalignment increases. This method accurately captures the change in the loss as a function of misalignment, but is not as ideal for measuring absolute loss numbers compared to a physical iris and power meter. This is because the absolute loss measurements with this method are subject to error from the finite sampling of the beam due to the camera’s pixels and the significantly lower dynamic range of the camera versus a power meter.

In addition, we can take the portion of the beam that is transmitted through the virtual iris and use a two-dimensional Fourier transform to shift the beam into the far field. If we measure the width of the beam in the near field, W_nf, and the width of the beam in the far field, W_ff, we can calculate the M² of the beam

Figure 8 shows the behavior of the output coupling systems as the phase plate is altered from the optimal condition. The outer edges of the figure have mode images (log scale) as insets, showing the full beam for various misalignment conditions. Figure 8(a) shows the relative loss through the aperture (red), and the M² of the transmitted beam (purple) as a function of BPP size. Output coupling for the binary case seems less sensitive to phase plate size as compared to input coupling (Fig. 4) – a change in size of nearly 3% results in negligible additional loss (0.55 dB). Figure 8(b) shows the same measurements (loss and M²) as a function of lateral offset of the phase plate.

 

Fig. 8 Calculated loss at the QGP with respect to the optimal condition, and M² as a function of various changes in system alignment; Inset: stitched mode images for different conditions (log scale image for visibility) (a) Relative loss and M² as a function of BPP diameter; (b) Relative loss and M² as a function of BPP offset; (c) Relative loss and M² as a function of axicon angle; (d) Relative loss and M² as a function of axicon offset.

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Figure 8(c) shows the relative loss through the aperture (blue), and the M² of the transmitted beam (green) as a function of axicon angle. A deviation in angle of approximately 3% corresponds to an additional loss of 0.46 dB. Figure 8(d) shows the same measurements (loss and M²) as a function of lateral offset of the axicon and lens. Axicon output coupling is very sensitive to lateral misalignment, even more so than the BPP case.

We also remove the iris and couple the converted beams into a SMF. For high power applications where free-space mode conversion excels, SMF coupling is not possible due to dielectric breakdown and massive nonlinear distortion – however, for the purposes of demonstration, high SMF coupling efficiency underscores the quality of the beam. We measure 2.1 ± 0.1 dB of coupling loss between the full beam and the output of the SMF (Corning, HI980) for BPP output coupling, and 1.3 ± 0.1 dB loss for the axicon. The simulated overlaps between each output coupled beam and the LP_0,1 mode of the fiber give 1.4 dB and 0.99 dB loss, respectively. The small discrepancies between the simulations and measurements are typical of SMF coupling.

Apart from the loss due to an aperture, or SMF coupling, the major source of loss for output coupling is the SLM. The measured loss due to the SLM was 1.6 dB for both the BPP and the axicon. This loss could be mitigated by using physical optics rather than SLMs.

5. Extension to other modes and fibers

5.1 Extension to other modes

All of the simulations and data presented until this point have been for the LP_0,9 mode of the fiber under test (Thorlabs, FG050LGA). Near λ = 1 μm, this fiber guides the first 10 LP_0,m modes – though we expect that modes below LP_0,5 do not propagate stably due to nearest neighbor coupling which is typical for lower order modes in step index MMFs [25].

Figures 9(a) and 9(b) show the results of input coupling to other HOMs in the FUT. The red and blue curves in 9(a) and 9(b) represent the simulated MPI value for the most parasitic mode with BPP and axicon input coupling respectively. The crosses in each figure are measurements of parasitic mode suppression determined from fC² traces. The insets show example mode images from the excitation experiments (brightness and contrast are enhanced for visibility). Allowing for some variability (on the order of 1 dB) in suppression due to alignment drift and measurement noise, there is very good agreement between the simulation and measurement. Furthermore, the mode suppression is fairly constant as a function of mode order meaning that these coupling methods can be used to target any mode.

 

Fig. 9 Parasitic mode suppression as a function of mode order measured by fC² for (a) BPP input coupling, (b) axicon input coupling; simulations delineated with solid line, measurements are marked with crosses, example mode images (brightness and contrast enhanced for visibility) inset; (c) Simulated loss at the QGP as a function of mode order for axicon and BPP output coupling, measurements from section 4 shown with markers.

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Figure 9(c) shows simulations of the loss at the QGP as a function of mode order (the red line corresponds to BPP, blue to axicon). The markers correspond to the measurements described in section 4. Again, the loss is fairly flat with respect to mode order. For the axicon case, there is a slight roll off as the mode order increases – this is likely due to the fact that as a mode approaches cutoff, more of its power sits in the outer-cladding, and the solution less resembles a truncated Bessel-beam. The roll off effect is very small (0.17 dB difference between LP_0,5 and LP_0,10), therefore both methods are capable of converting all stably guided modes in the FUT to Gaussian-like beams with low loss. Furthermore, simulations show that the M² of the output beam at QGP is below 1.1 for all modes.

5.2 Extension to other fibers

So far, we have only considered a simple step-index multimode fiber that is commercially available (Thorlabs, FG050LGA). In some instances, it is necessary to employ a fiber with a more complicated geometry. For example, HOM lasers [4,13] often use double-clad fibers with single-moded cores and LPGs for input coupling so that SMF pump combiners can still be used, conversion efficiency is maximized, and loss is negligible. This is only possible at the input because the power is still fairly low – the output must be converted using free-space means. For this reason, it is necessary to consider the effect of core geometry on coupling efficiency using free-space mode converters.

Figure 10 shows simulation results for axicon and BPP coupling for double-clad fibers with different core designs. Figure 10(a) is a schematic of a typical double-clad fiber with a silica inner-cladding (50-μm diameter), and a fluorine-down-doped outer-cladding with an index step of 17.6 × 10⁻³ (the same index step as Thorlabs FG050LGA). The index step between core and inner cladding, Δn, varies from zero (no core), to 18.8 × 10⁻³. The radius of the core also necessarily changes with Δn in order to keep the V number of the central core constant at 2.1 (to ensure SM operation near λ = 1 μm in the core) [19]. For each fiber, a scalar modesolver is used to calculate the mode profile for the LP_0,9 mode and Fresnel propagation is used to simulate input and output coupling.

 

Fig. 10 (a) Schematic for a double clad fiber, core V number is 2.1, Δn is variable; (b) Simulated parasitic mode suppression for input coupling as a function of Δn, measurements from section 3 shown with markers; (c) Simulated loss at the QGP as a function of Δn, measurements from section 4 shown with markers.

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Figure 10(b) shows parasitic mode suppression for input coupling as a function of Δn, and 10(c) shows output coupling loss at the QGP as a function Δn. In each figure, the red curve corresponds to BPPs, and the blue to axicons. The Δn = 0 case corresponds to the fiber used for the experiments described in sections 3 and 4.

The BPP performance for both input and output coupling is relatively invariant with change in fiber design. Parasitic mode suppression actually improves somewhat (~2 dB) as Δn increases, while loss changes negligibly. This lack of dependence on fiber design is to be expected because the BPP is designed to match the phase of the target HOM, regardless of the spatial field profile.

Axicon performance suffers significantly when a core is introduced. With a core, the mode solution is a superposition of different Bessel functions, rather than a simple truncated zero^th order Bessel beam. As a result, a Bessel-Gauss beam created by an axicon for input coupling will have significant overlap with multiple modes in the fiber, and the parasitic mode suppression will suffer. Reciprocally, the axicon will have poor performance for output coupling the HOM in a fiber with solutions that do not resemble truncated Bessel beams.

For fibers where the core to cladding index step is greater than 7 × 10⁻³, the axicon will have worse performance than the BPP. Therefore, if a core is necessary for the application, then it may be better to use BPPs for output or input coupling. It should be noted that the simulations here assume a constant V number for the core which does not necessary cover the entire parameter space. It may be possible to design the core in such a way that the mode solution for at least one mode has high overlap with a truncated Bessel beam, and thus axicon coupling would have better performance.

6. Conclusion

We have demonstrated two methods of shaping laser beams in free space using simple phase structures, in order to couple in and out of higher order modes in multi-mode optical fibers. Both binary phase plates (BPPs) and axicons can excite modes with high purity (>15 dB parasitic mode suppression) or convert modes to near-Gaussian beams (M² < 1.25). The relative advantages of the BPP coupling system are that (1) when BPPs are implemented by computer-controlled means (such as a spatial light modulator), mode order can be changed in real time (~60 Hz refresh rate) without realignment of the system, facilitating rapid testing in the laboratory environment; (2) BPP performance is independent of fiber design so they are compatible with double-clad fibers, or any other multi-mode fiber. The main advantages of axicon coupling are the lower loss with respect to BPPs (0.85 ± 0.1 dB vs. 1.75 ± 0.1 dB), and the commercial availability of axicons. By tuning the magnification and alignment of the imaging system at the output or input of the fiber, a single axicon can be used to couple into any mode of the fiber. Furthermore, axicons are essentially lenses, so the power-tolerance should be consistent with other bulk optics. Axicons have best performance with coreless fibers where the mode solutions are true truncated Bessel beams. Adding a core to the fiber will increase coupling loss and MPI, though it may be possible to carefully design the core such that this increase is negligible.

The free space techniques described in this work provide an attractive alternative to using LPGs, which require precise design and fabrication for each mode that must be accessed. That said, it is worth noting that none of these free-space techniques can approach the low loss and high purities offered by LPGs – instead, they serve to address two issues with the use of LPGs in multi-moded systems, at least for high-power applications: (a) the inability to use LPGs for output coupling, considering that conversion back to a small core mode would defeat the purpose of using large mode area HOMs, and (b) the need for specialty fiber designs. The methods shown greatly facilitate exploiting the beneficial properties of HOMs, since only commercially available components are needed.