I. INTRODUCTION

Section:

ChooseTop of pageABSTRACTI. INTRODUCTION <<II. EXPERIMENTAL RESULTSIII. NUMERICAL SIMULATION...IV. DISCUSSIONV. CONCLUSIONS AND OUTLOO...SUPPLEMENTARY MATERIALREFERENCESIn the past decade, there is an ever-growing effort to utilize multimode optical fibers (MMFs) in a wide range of photonic applications such as optical fiber communication,1–31. D. J. Richardson, J. M. Fini, and L. E. Nelson, “Space-division multiplexing in optical fibres,” Nat. Photonics 7, 354–362 (2013). https://doi.org/10.1038/nphoton.2013.942. M. Plöschner, T. Tyc, and T. Čižmár, “Seeing through chaos in multimode fibres,” Nat. Photonics 9, 529–535 (2015). https://doi.org/10.1038/nphoton.2015.1123. B. J. Puttnam, G. Rademacher, and R. S. Luís, “Space-division multiplexing for optical fiber communications,” Optica 8, 1186–1203 (2021). https://doi.org/10.1364/optica.427631 high energy fiber lasers,4–64. L. G. Wright, D. N. Christodoulides, and F. W. Wise, “Spatiotemporal mode-locking in multimode fiber lasers,” Science 358, 94–97 (2017). https://doi.org/10.1126/science.aao08315. U. Teğin, E. Kakkava, B. Rahmani, D. Psaltis, and C. Moser, “Spatiotemporal self-similar fiber laser,” Optica 6, 1412–1415 (2019). https://doi.org/10.1364/OPTICA.6.0014126. H. Haig, P. Sidorenko, A. Dhar, N. Choudhury, R. Sen, D. Christodoulides, and F. Wise, “Multimode Mamyshev oscillator,” Opt. Lett. 47, 46–49 (2022). https://doi.org/10.1364/ol.447208 quantum optics,7–117. H. Defienne, M. Barbieri, I. A. Walmsley, B. J. Smith, and S. Gigan, “Two-photon quantum walk in a multimode fiber,” Sci. Adv. 2, e1501054 (2016). https://doi.org/10.1126/sciadv.15010548. N. H. Valencia, S. Goel, W. McCutcheon, H. Defienne, and M. Malik, “Unscrambling entanglement through a complex medium,” Nat. Phys. 16, 1112–1116 (2020). https://doi.org/10.1038/s41567-020-0970-19. S. Leedumrongwatthanakun, L. Innocenti, H. Defienne, T. Juffmann, A. Ferraro, M. Paternostro, and S. Gigan, “Programmable linear quantum networks with a multimode fibre,” Nat. Photonics 14, 139–142 (2020). https://doi.org/10.1038/s41566-019-0553-910. H. Cao, S.-C. Gao, C. Zhang, J. Wang, D.-Y. He, B.-H. Liu, Z.-W. Zhou, Y.-J. Chen, Z.-H. Li, S.-Y. Yu et al., “Distribution of high-dimensional orbital angular momentum entanglement over a 1 km few-mode fiber,” Optica 7, 232–237 (2020). https://doi.org/10.1364/optica.38140311. K. Sulimany and Y. Bromberg, “All-fiber source and sorter for multimode correlated photons,” Npj Quantum Inf. 8, 4 (2022). https://doi.org/10.1038/s41534-021-00515-x nonlinear optics,12–1812. A. Mafi, “Pulse propagation in a short nonlinear graded-index multimode optical fiber,” J. Lightwave Technol. 30, 2803–2811 (2012). https://doi.org/10.1109/jlt.2012.220821513. L. G. Wright, D. N. Christodoulides, and F. W. Wise, “Controllable spatiotemporal nonlinear effects in multimode fibres,” Nat. Photonics 9, 306–310 (2015). https://doi.org/10.1038/nphoton.2015.6114. K. Krupa, A. Tonello, B. M. Shalaby, M. Fabert, A. Barthélémy, G. Millot, S. Wabnitz, and V. Couderc, “Spatial beam self-cleaning in multimode fibres,” Nat. Photonics 11, 237–241 (2017). https://doi.org/10.1038/nphoton.2017.3215. O. Tzang, A. M. Caravaca-Aguirre, K. Wagner, and R. Piestun, “Adaptive wavefront shaping for controlling nonlinear multimode interactions in optical fibres,” Nat. Photonics 12, 368–374 (2018). https://doi.org/10.1038/s41566-018-0167-716. K. Krupa, A. Tonello, A. Barthélémy, T. Mansuryan, V. Couderc, G. Millot, P. Grelu, D. Modotto, S. A. Babin, and S. Wabnitz, “Multimode nonlinear fiber optics, a spatiotemporal avenue,” APL Photonics 4, 110901 (2019). https://doi.org/10.1063/1.511943417. U. Teğin, B. Rahmani, E. Kakkava, N. Borhani, C. Moser, and D. Psaltis, “Controlling spatiotemporal nonlinearities in multimode fibers with deep neural networks,” APL Photonics 5, 030804 (2020). https://doi.org/10.1063/1.513813118. H. Pourbeyram, P. Sidorenko, F. O. Wu, N. Bender, L. Wright, D. N. Christodoulides, and F. Wise, “Direct observations of thermalization to a Rayleigh–Jeans distribution in multimode optical fibres,” Nat. Phys. 18, 685 (2022). https://doi.org/10.1038/s41567-022-01579-y and imaging.19–2619. T. Čižmár and K. Dholakia, “Exploiting multimode waveguides for pure fibre-based imaging,” Nat. Commun. 3, 1027 (2012). https://doi.org/10.1038/ncomms202420. Y. Choi, C. Yoon, M. Kim, T. D. Yang, C. Fang-Yen, R. R. Dasari, K. J. Lee, and W. Choi, “Scanner-free and wide-field endoscopic imaging by using a single multimode optical fiber,” Phys. Rev. Lett. 109, 203901 (2012). https://doi.org/10.1103/physrevlett.109.20390121. R. Y. Gu, R. N. Mahalati, and J. M. Kahn, “Design of flexible multi-mode fiber endoscope,” Opt. Express 23, 26905–26918 (2015). https://doi.org/10.1364/oe.23.02690522. A. M. Caravaca-Aguirre and R. Piestun, “Single multimode fiber endoscope,” Opt. Express 25, 1656–1665 (2017). https://doi.org/10.1364/oe.25.00165623. N. Borhani, E. Kakkava, C. Moser, and D. Psaltis, “Learning to see through multimode fibers,” Optica 5, 960–966 (2018). https://doi.org/10.1364/optica.5.00096024. I. T. Leite, S. Turtaev, D. E. Boonzajer Flaes, and T. Čižmár, “Observing distant objects with a multimode fiber-based holographic endoscope,” APL Photonics 6, 036112 (2021). https://doi.org/10.1063/5.003836725. S. Resisi, S. M. Popoff, and Y. Bromberg, “Image transmission through a dynamically perturbed multimode fiber by deep learning,” Laser Photonics Rev. 15, 2000553 (2021). https://doi.org/10.1002/lpor.20200055326. S.-Y. Lee, V. J. Parot, B. E. Bouma, and M. Villiger, “Confocal 3D reflectance imaging through multimode fiber without wavefront shaping,” Optica 9, 112–120 (2022). https://doi.org/10.1364/optica.446178 The main challenge of MMF-based technologies is that the transverse modes of the fiber are distorted by inter-modal interference, mode coupling, and modal dispersion. The information delivered by an MMF is, therefore, often scrambled, yielding at its output complex spatiotemporal speckle patterns. Nevertheless, the complexity associated with propagation in MMFs can, in fact, be harnessed for a wide range of applications.2727. H. Cao and Y. Eliezer, “Harnessing disorder for photonic device applications,” Appl. Phys. Rev. 9, 011309 (2022). https://doi.org/10.1063/5.0076318 For example, MMFs were utilized for realizing all-fiber spectrometers,28,2928. B. Redding and H. Cao, “Using a multimode fiber as a high-resolution, low-loss spectrometer,” Opt. Lett. 37, 3384–3386 (2012). https://doi.org/10.1364/ol.37.00338429. B. Redding, M. Alam, M. Seifert, and H. Cao, “High-resolution and broadband all-fiber spectrometers,” Optica 1, 175–180 (2014). https://doi.org/10.1364/optica.1.000175 for advanced optical cryptography protocols,30,3130. Y. Bromberg, B. Redding, S. M. Popoff, N. Zhao, G. Li, and H. Cao, “Remote key establishment by random mode mixing in multimode fibers and optical reciprocity,” Opt. Eng. 58, 016105 (2019). https://doi.org/10.1117/1.oe.58.1.01610531. L. V. Amitonova, T. B. H. Tentrup, I. M. Vellekoop, and P. W. H. Pinkse, “Quantum key establishment via a multimode fiber,” Opt. Express 28, 5965–5981 (2020). https://doi.org/10.1364/oe.380791 and for optical implementations of neural networks.3232. U. Teğin, M. Yıldırım, İ. Oğuz, C. Moser, and D. Psaltis, “Scalable optical learning operator,” Nat. Comput. Sci. 1, 542–549 (2021). https://doi.org/10.1038/s43588-021-00112-0Recently, MMFs were used for implementing reconfigurable linear operators by carefully shaping the wavefront coupled to the fiber using a spatial light modulator.9,339. S. Leedumrongwatthanakun, L. Innocenti, H. Defienne, T. Juffmann, A. Ferraro, M. Paternostro, and S. Gigan, “Programmable linear quantum networks with a multimode fibre,” Nat. Photonics 14, 139–142 (2020). https://doi.org/10.1038/s41566-019-0553-933. M. W. Matthès, P. Del Hougne, J. De Rosny, G. Lerosey, and S. M. Popoff, “Optical complex media as universal reconfigurable linear operators,” Optica 6, 465–472 (2019). https://doi.org/10.1364/optica.6.000465 As MMFs have a lot in common with complex media,3434. H. Cao, A. P. Mosk, and S. Rotter, “Shaping the propagation of light in complex media,” Nat. Phys. 18, 994–1007 (2022). https://doi.org/10.1038/s41567-022-01677-x these demonstrations were inspired by pioneering works in the microwave domain, in which tunable metasurface reflect-arrays were placed inside chaotic cavities to tailor microwave fields3535. M. Dupré, P. Del Hougne, M. Fink, F. Lemoult, and G. Lerosey, “Wave-field shaping in cavities: Waves trapped in a box with controllable boundaries,” Phys. Rev. Lett. 115, 017701 (2015). https://doi.org/10.1103/PhysRevLett.115.017701 and realize linear microwave transformations.3636. P. del Hougne and G. Lerosey, “Leveraging chaos for wave-based analog computation: Demonstration with indoor wireless communication signals,” Phys. Rev. X 8, 041037 (2018). https://doi.org/10.1103/physrevx.8.041037 These works highlight new opportunities for controlling electromagnetic fields by manipulating the propagation medium itself rather than the incident wavefronts, such as realizing coherent perfect absorption,3737. M. F. Imani, D. R. Smith, and P. del Hougne, “Perfect absorption in a disordered medium with programmable meta-atom inclusions,” Adv. Funct. Mater. 30, 2005310 (2020). https://doi.org/10.1002/adfm.202005310 achieving optimal channel diversity,3838. P. Del Hougne, M. Fink, and G. Lerosey, “Optimally diverse communication channels in disordered environments with tuned randomness,” Nat. Electron. 2, 36–41 (2019). https://doi.org/10.1038/s41928-018-0190-1 and developing programmable analog differentiators.3939. J. Sol, D. R. Smith, and P. Del Hougne, “Meta-programmable analog differentiator,” Nat. Commun. 13, 1713 (2022). https://doi.org/10.1038/s41467-022-29354-w We have recently adopted this concept to the optical domain and demonstrated all-fiber spatial light modulation by manipulating the conformation of an MMF to control the wavefront at its output.4040. S. Resisi, Y. Viernik, S. M. Popoff, and Y. Bromberg, “Wavefront shaping in multimode fibers by transmission matrix engineering,” APL Photonics 5, 036103 (2020). https://doi.org/10.1063/1.5136334 Since propagation in complex media is typically wavelength-sensitive, manipulation of the propagation medium also tailors the spectrum of the propagating fields, as demonstrated in the microwave37,39,4137. M. F. Imani, D. R. Smith, and P. del Hougne, “Perfect absorption in a disordered medium with programmable meta-atom inclusions,” Adv. Funct. Mater. 30, 2005310 (2020). https://doi.org/10.1002/adfm.20200531039. J. Sol, D. R. Smith, and P. Del Hougne, “Meta-programmable analog differentiator,” Nat. Commun. 13, 1713 (2022). https://doi.org/10.1038/s41467-022-29354-w41. J. Sol, A. Alhulaymi, A. D. Stone, and P. Del Hougne, “Reflectionless programmable signal routers,” Sci. Adv. 9, eadf0323 (2023). https://doi.org/10.1126/sciadv.adf0323 and optical domains.4242. E. Small, O. Katz, Y. Guan, and Y. Silberberg, “Spectral control of broadband light through random media by wavefront shaping,” Opt. Lett. 37, 3429–3431 (2012). https://doi.org/10.1364/ol.37.003429 Inspired by these works, in this Letter, we utilize the complex wavelength-dependent interference in MMFs to realize all-fiber reconfigurable spectral filters by manipulating the fiber’s conformation.In this work, we utilize the complex wavelength-dependent interference in MMFs to realize all-fiber reconfigurable spectral filters, by manipulating the fiber’s conformation. To this end, we apply computer-controlled local bends at multiple positions along the fiber in a configuration we call fiber piano.4040. S. Resisi, Y. Viernik, S. M. Popoff, and Y. Bromberg, “Wavefront shaping in multimode fibers by transmission matrix engineering,” APL Photonics 5, 036103 (2020). https://doi.org/10.1063/1.5136334 The bends change the local propagation constants of the fiber’s guided modes and induce mode coupling, thus changing the fiber’s wavelength-dependent transmission matrix. The curvatures of the bends are determined by an optimization algorithm that finds a configuration of bends that maximizes the overlap between the output spectrum and a target spectrum. The main advantage of using the fiber piano for spectral filtering is that its resolution is inversely proportional to the fiber length, allowing us to demonstrate filters with a 5 pm resolution using a 150 m long fiber. Moreover, in contrast to grating-based reconfigurable filters, the resolution of the fiber piano is not limited by the operating wavelength range, opening the door for realizing high-resolution reconfigurable spectral filters operating over broad bandwidths.

II. EXPERIMENTAL RESULTS

Section:

ChooseTop of pageABSTRACTI. INTRODUCTIONII. EXPERIMENTAL RESULTS <<III. NUMERICAL SIMULATION...IV. DISCUSSIONV. CONCLUSIONS AND OUTLOO...SUPPLEMENTARY MATERIALREFERENCESThe experimental setup is depicted in Fig. 1(a). A broadband fiber-coupled LED source is coupled to a single mode fiber (SMF) to obtain spatially coherent broadband light. The light is then coupled to a 1 m-long step-index MMF with a numerical aperture (NA) of 0.22 and a core diameter of 50 μm. The output facet of the MMF is spliced to another SMF that is coupled to a spectrometer. The projection of the mode of the single-mode fibers on the modes of the multimode fiber sets the modal decomposition of the input and output fields. The measured spectrum is determined by the complex wavelength-dependent interference of the fiber modes at the splicing point. We apply computer-controlled deformations to the fiber by placing 28 piezoelectric actuators along the fiber, with a 1.5 cm separation between adjacent actuators [Figs. 1(b) and 1(c)]. The response time of the actuators is 45 ms (see Seq. B in the supplementary material for more details). We limit the minimal radius of curvature the actuators apply to 1 cm to avoid significant bending loss. As different deformations of the fiber yield different spectra at the output of the SMF, we search for the bend configuration that yields the required spectrum using a simulated annealing (SA) optimization algorithm4343. R. A. Rutenbar, “Simulated annealing algorithms: An overview,” IEEE Circuits Devices Mag. 5, 19–26 (1989). https://doi.org/10.1109/101.17235 (see Seq. A in the supplementary material for additional information).To demonstrate spectral shaping, we realize a tunable bandpass filter by maximizing the output intensity at different spectral bands [Fig. 2(a)]. The full-width-half maximum (FWHM) bandwidth of each band is 245 pm, determined by the spectral bandwidth of the MMF.2828. B. Redding and H. Cao, “Using a multimode fiber as a high-resolution, low-loss spectrometer,” Opt. Lett. 37, 3384–3386 (2012). https://doi.org/10.1364/ol.37.003384 Defining the enhancement factor as the peak-to-background ratio in the obtained spectra, we achieve enhancements in the range of 13–30. The efficiency of the system, defined by the ratio of output and input intensities at the chosen spectral band, is 0.06 and is mostly limited by the small number of actuators we use. Roughly 50% of the loss is due to the macro bends induced by the actuators, which couple light to the clad of the MMF. Out of the remaining power, roughly 15% is coupled to the output SMF. The coupling efficiency in the SMF could be significantly improved by using more actuators, as discussed in Sec. . We measured a return loss of −47 dB with an optical vector analyzer (Luna OVA 5000). The optimized pattern remains stable for 10 h (see Seq. C the supplementary material for more details).To demonstrate the versatility of the spectral shaping, we also implement a dual bandpass filter, with two peaks separated by 1.5 nm and a FWHM of 241 pm for each peak [see Fig. 2(b)]. Here we observe a factor of two lower enhancements, as the optimized wavelengths are separated by more than the spectral correlation width and are, therefore, uncorrelated. Bandpass filters with a bandwidth wider than the spectral correlation width will suffer from a similar reduction in the observed enhancements and will, therefore, require more degrees of control, unless long-range correlations are present in the MMF.4444. C. W. Hsu, A. Goetschy, Y. Bromberg, A. D. Stone, and H. Cao, “Broadband coherent enhancement of transmission and absorption in disordered media,” Phys. Rev. Lett. 115, 223901 (2015). https://doi.org/10.1103/physrevlett.115.223901Since the spectral bandwidth of the fiber is inversely proportional to the fiber’s length,28,2928. B. Redding and H. Cao, “Using a multimode fiber as a high-resolution, low-loss spectrometer,” Opt. Lett. 37, 3384–3386 (2012). https://doi.org/10.1364/ol.37.00338429. B. Redding, M. Alam, M. Seifert, and H. Cao, “High-resolution and broadband all-fiber spectrometers,” Optica 1, 175–180 (2014). https://doi.org/10.1364/optica.1.000175 it is, in principle, possible to realize bandpass filters with extremely narrow linewidths by using long multimode fibers. However, in practice, the linewidths we could achieve were limited by the 60 pm resolution of the spectrometer used in the optimization process. We, therefore, replaced the spectrometer and the LED source with a tunable laser at telecom wavelengths that can tune the wavelength in the C-band with a spectral resolution of 1 pm and an InGaAs camera that images the output facet of the MMF [Fig. 1(d)]. We fix the laser at a specific wavelength and apply the same optimization process described earlier to enhance the intensity at a chosen square region of interest (ROI) on the camera. We set the width of the ROI to be equal to the FWHM of the intensity autocorrelation of the speckle pattern recorded by the camera. The ROI sets the modal decomposition of the measured output field. After the optimization, we scan the wavelength of the laser and observe the transmitted spectrum. The FWHM of the measured transmission peaks is inversely proportional to the fiber’s length. Using a 150 m long fiber, we realize bandpass filters with an FWHM of 5 pm and enhancement factors in the range of 5–7 (inset of Fig. 3).

III. NUMERICAL SIMULATIONS

Section:

ChooseTop of pageABSTRACTI. INTRODUCTIONII. EXPERIMENTAL RESULTSIII. NUMERICAL SIMULATION... <<IV. DISCUSSIONV. CONCLUSIONS AND OUTLOO...SUPPLEMENTARY MATERIALREFERENCES

To compare the performance of our system with its optimal limit, we numerically simulate the operation of the fiber piano by computing the transmission spectra through a MMF with controlled local macro bends. We model the fiber piano by numerically propagating scalar waves through curved segments of a MMF, separated by straight segments. Each curved segment simulates a macro bend induced by an actuator, where the travel of the actuator corresponds to a change in the radius of curvature of the segment. We use 44 discrete values for the radii of curvatures in the range of 1–8 cm, which roughly corresponds to the range of curvatures induced by the actuators in the experiment.

To find the field at the output of the fiber, we calculate the transmission matrix (TM) of each segment and propagate the input field by multiplying it by the TMs of all the segments (see Seq. D in the supplementary material). To calculate the TM of the curved segments, we use “BPM-Matlab,”4545. M. Veettikazhy, A. Kragh Hansen, D. Marti, S. Mark Jensen, A. Lykke Borre, E. Ravn Andresen, K. Dholakia, and P. Eskil Andersen, “BPM-Matlab: An open-source optical propagation simulation tool in MATLAB,” Opt. Express 29, 11819–11832 (2021). https://doi.org/10.1364/oe.420493 an open source optical propagation simulation in MATLAB that implements a finite difference beam propagation method for monochromatic light. We first compute the TM of the 44 curved segments to be used in the simulation by propagating each guided mode of the fiber and projecting the output field on the guided mode basis. Propagation in the straight fiber segment is modeled by a diagonal TM, whose elements, which represent the phase accumulated by each guided mode, are computed by solving the Helmholtz equation for the fiber’s index profile. The goal of the simulation is to find the radius of curvature of each bend, out of the 44 possible radii, that focuses the light at the distal end of the MMF at a chosen wavelength. To this end, we use the same SA optimization algorithm used in the experiment. Once the optimized radii of curvature are obtained, we compute the output spectrum by simulating propagation through the same configuration of bends and the same input field at different wavelengths.We start by simulating propagation through a fiber piano with the same number of actuators as in our experiments, 28 curved segments separated by straight segments. The fiber we simulate is a 2 m-long step-index MMF with a numerical aperture of 0.22 and a core diameter of 50 μm. We excite the 50 first modes of the fiber, which correspond to the roughly 50 modes we excite in the experiment (see Fig. S4 in the supplementary material), with equal amplitudes and random relative phases. We achieve an enhancement of 15 and a FWHM of 350 pm [the blue curve in Fig. 4(a)]. Since we simulate the propagation over a bandwidth that is only slightly wider than the bandwidth of the fiber, here we define the enhancement as the spatial peak-to-background instead of the spectral peak-to-background measured in the experiments. We attribute the difference between the experimental values obtained for the same fiber and the same wavelength range (enhancement of 7 and FWHM of 650 pm) to mode coupling that may increase the fiber’s spectral bandwidth4646. K.-P. Ho and J. M. Kahn, “Statistics of group delays in multimode fiber with strong mode coupling,” J. Lightwave Technol. 29, 3119–3128 (2011). https://doi.org/10.1109/jlt.2011.2165316 and system instabilities that may decrease the obtained peak-to-background ratios.4747. I. M. Vellekoop and A. P. Mosk, “Phase control algorithms for focusing light through turbid media,” Opt. Commun. 281, 3071–3080 (2008). https://doi.org/10.1016/j.optcom.2008.02.022It is instructive to compare the enhancement obtained by the fiber piano to the optimal enhancement one expects to achieve, for example, using a perfect spatial light modulator at the proximal end of the fiber. We, therefore, look for the input field that maximizes the peak-to-background ratio using the same bend configuration that was found by the optimization algorithm for some arbitrary input. To this end, we first calculate the TM of the bent fiber as described earlier. Next, by multiplying the Hermitian conjugate of the TM by the target pattern (a focused spot in the desired ROI), we get the complex input field that maximizes the intensity in the ROI.4747. I. M. Vellekoop and A. P. Mosk, “Phase control algorithms for focusing light through turbid media,” Opt. Commun. 281, 3071–3080 (2008). https://doi.org/10.1016/j.optcom.2008.02.022 Then, to find the output spectrum in the ROI, we numerically propagate this input field at different wavelengths. The optimal input field yields an enhancement of 45 and a FWHM of 320 pm [red curve in Fig. 4(a)]. We, therefore, conclude that with 28 actuators we could achieve ≈30% of the optimal enhancement.

To further explore the performance of the fiber piano under optimal conditions, we simulate the achievable enhancement in a loss-free system for an increasing number of fiber modes and actuators. Intuitively, the enhancement should rise as these two parameters are increased since they provide more degrees of freedom for the optimization process. The number of curved segments in the simulation is set by the number of actuators we simulate. The number of guided modes in the fiber is set by changing the core diameter while keeping the numerical aperture of the fiber fixed. We fix the total fiber length to 20 m and excite all the guided modes of the fiber with equal amplitudes and random relative phases. To neglect loss in the simulation, we force the transmission matrix of each segment to be unitary.

In Fig. 4(b), we depict the enhancement vs the number of actuators for fibers with different numbers of modes. As expected, the enhancement increases with the number of actuators [Fig. 4(b)]. The saturated enhanced values obtained by the fiber piano optimization [blue symbols in Fig. 4(c)] are higher than the enhancements obtained for phase-only modulation at the input of the fiber (yellow symbols) and close to the enhancements obtained by optimal amplitude and phase wavefront control at the input of the fiber. However, saturation is expected when the number of degrees of control, the actuators, approaches the number of guided modes of the fiber. Here we see the saturation is obtained at a much higher number of actuators, which indicates that the degrees of control in the simulation are not independent. Noticeably, as the number of modes grows, the simulation performance deteriorates compared with the theoretical limit. This difference is most probably due to the increased probability that the SA algorithm finds a local minimum when the dimension of the optimization problem increases.

IV. DISCUSSION

Section:

ChooseTop of pageABSTRACTI. INTRODUCTIONII. EXPERIMENTAL RESULTSIII. NUMERICAL SIMULATION...IV. DISCUSSION <<V. CONCLUSIONS AND OUTLOO...SUPPLEMENTARY MATERIALREFERENCESThe spectral resolution of any wavelength-selective technology is fundamentally limited by the maximal path length difference that the light can experience in the device. Specifically, in reconfigurable arbitrary-shaped spectral filters, which are based on diffracting the light with a grating and applying a spectral modulation using a programmable array of pixels, the path length difference scales linearly with the distance between the grating and the programmable pixel array. This poses a resolution-footprint trade-off. For spectrometers, it has been shown that this trade-off can be mitigated by dispersing the light with long multimode fibers instead of gratings. Using a 100 m long fiber coiled to a 3 in. spool, a 1 pm resolution at 1500 nm was demonstrated, beyond the performance of even 1-m long bench-top grating spectrometers.2929. B. Redding, M. Alam, M. Seifert, and H. Cao, “High-resolution and broadband all-fiber spectrometers,” Optica 1, 175–180 (2014). https://doi.org/10.1364/optica.1.000175 Our fiber piano technology adopts this approach for reconfigurable spectral filters and opens new opportunities for realizing filters with a spectral resolution on a picometer scale.A unique advantage of the fiber piano technology is that it disentangles the number of required degrees of control from the ratio of the operational wavelength range and the spectral resolution. In grating-based filters, this ratio must be smaller than the number of resolvable diffraction-limited spots on the programmable array and smaller than the number of pixels in the array. The ratio of the operational bandwidth and spectral resolution is, therefore, bounded to a few thousand. In contrast, for the fiber piano, this ratio is limited by the wavelength range over which the device can be calibrated and the fiber length, but not by the number of actuators. Typical programmable spectral filters such as waveshapers and wavelength selective switches support bandwidths of about 5 THz from C-band to L-band and a resolution of about 10 GHz. In comparison, we demonstrated a resolution of 0.6 GHz (5 pm at 1550 nm) and proved feasibility in operating in the visible and telecom spectral ranges using the same MMF. A table comparing the fiber piano to wavelength selective switches and to five other tunable optical filter technologies is presented in the supplementary material Seq. E.

The number of actuators does set a limit on the observed peak-to-background ratios of the filter rather than on its bandwidth. We experimentally observed peak-to-background ratios of 13 dB and estimated an optimal ratio of 21 dB for 1500 actuators. State-of-the-art programmable filters can exhibit much higher ratios, approaching 40 dB. The limited extinction ratio is a typical disadvantage of disorder-based technologies. Nevertheless, certain applications can benefit from the high resolution-bandwidth product of the fiber piano and tolerate the relatively low extinction ratios. For example, a nonlinear interaction after the fiber piano can dramatically increase the extinction ratio of the filter, while optical gain can compensate for low transmissions.

V. CONCLUSIONS AND OUTLOOK

Section:

ChooseTop of pageABSTRACTI. INTRODUCTIONII. EXPERIMENTAL RESULTSIII. NUMERICAL SIMULATION...IV. DISCUSSIONV. CONCLUSIONS AND OUTLOO... <<SUPPLEMENTARY MATERIALREFERENCES

We experimentally realized all-fiber spectral shaping by applying computer-controlled deformations to a multimode fiber. As a proof of concept of this approach, we demonstrated a tunable bandpass filter with an FWHM of 245 pm and a dual-wavelength filter with the same spectral width. Next, we showed the possibility of achieving narrower bandpass filters using longer MMF and achieved FWHM bandwidths as low as 5 pm. Finally, we ran numerical simulations to test the performance of the fiber under optimal conditions. We show that with the fiber piano, it is possible to achieve peak-to-background ratios that are above 70% of the ratios obtained for optimal wavefront control. The efficiency of the experimental system, which is 0.06, is currently a drawback, yet numerical simulation suggests that it is possible to achieve efficiencies approaching 0.5 using a larger number of actuators.

In the high resolution configuration, where a camera that images the distal end of the fiber is used to select one spatial channel, one can simplify the detection method by replacing the camera with an SMF that is spliced at the distal end of the MMF along with a telecom photodiode. Our method demonstrates a first step toward realizing an all-fiber modulator for tailoring the spatial-spectral waveform at the output of multimode fibers, a key ingredient in a wide range of applications that are based on multimode fibers.