A tilted plane reference wave, polarized by the same polarizer, interfered with the speckle pattern to record the complex image of the speckle pattern field through off-axis holography in the V polarization state. To measure T fw, the forward TM, we coupled the focal spot into the MMF through input spatial channels on the proximal side and imaged the speckle pattern exiting from output spatial channels on the distal side with another identical objective and a tube lens (f = 30 cm) onto an InGaAs camera (OW1.7-VS-CL-LP-640, Raptor Photonics) with a vertically oriented, linear polarizer (LP) placed in front of it. 22 The angular spectrum of the spot exceeded the NA of the MMF to ensure efficient population of high-order modes. The MMF theoretically supports ∼550 guided modes per linear polarization. An offset phase ramp was applied to the SLM to block unmodulated light from entering the fiber. A laser beam (λ = 1550 nm and linewidth < 100 kHz) was linearly polarized in a vertical (V) polarization state, reflected on a phase-only spatial light modulator (SLM, Model P192-HDMI, Meadowlark Optics) in the same polarization state, and then focused by using an objective lens (Mitutoyo Plan Apo NIR Infinity Corrected) with a numerical aperture (NA) of 0.4 into a 2.5 μm full-width at half maximum (FWHM) spot on the facet of the step-index MMF with 105 µm core diameter and a NA of 0.22 (FG105LCA, Thorlabs). (4) and (5), we measured the monochromatic TMs, T fw and T 2X, of a 1-m-long MMF randomly coiled with a minimum radius of curvature of 23 mm, using the setup shown in Fig. 8–11 However, the more general underlying transmission symmetry of bi-directional light transmission through complex systems and its implications have not been explicitly demonstrated and discussed. Digital optical phase conjugation (DOPC) has been well established for focusing and imaging through complex or disordered media, including multimode fibers (MMFs). 6,7 Optical phase conjugation is a well-known consequence of this symmetry in loss-free systems, whereby an original light distribution is replicated by reversing the propagation direction of the detected field while conjugating its wave-front. This symmetry not only underlies the behavior of common optical components, such as polarizers, beam splitters, and wave-plates, but also engenders surprising physical phenomena in complex systems such as coherent backscattering (or weak localization) and Anderson localization. 3–5 In the linear regime, this suggests a definite relation, or symmetry, between the forward and the backward transmission when interchanging the source and detector. The bi-directional transmission through photonic systems is governed by the universal Lorentz reciprocity (or the Helmholtz reciprocity), which states that light propagating along a reversed path experiences the exact same transmission coefficient as in the forward direction, independent of the path complexity 1,2 or the presence of loss. These insights may inform the development of new imaging techniques through complex media and coherent control of waves in photonic systems. Furthermore, we show how focusing through the MMF with digital optical phase conjugation is compromised by system loss since time reversibility relies on power conservation. This symmetry impedes straightforward MMF calibration from proximal measurements of the round-trip TM. We measured phase-corrected TMs of forward and round-trip propagation in a single polarization state through a looped 1 m-long step-index optical multimode fiber (MMF) to experimentally verify a transpose relationship between the forward and backward transmission. ![]() Here, we demonstrate the optical reciprocal nature of complex media by exploring their TM properties. The coherent transmission matrix (TM) is a convenient method to characterize wave transmission through general media. Reciprocity is a fundamental principle of wave physics and directly relates to the symmetry in the transmission through a system when interchanging the input and output.
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