# Complex Wavefront Shaping through a Multi-Core Fiber
## 2 Phase Distortion Compensation in a Multi-Core Fiber  
A multi-core fiber (MCF) consists of about 10,000 single-mode cores with a diameter of 350 \(\upmu\)m. Modulating the input phase of each core by the spatial light modulator (SLM) at the proximal side, the MCF is thus functioned as a phased-array waveguide for tailored light field delivery. However, due to the optical path difference between individual cores, a plane-wave illumination at the proximal facet results in a random speckle pattern at the distal output. DOPC [23; 24] is implemented to compensate the phase distortion [16; 17; 18]. The phase delay of each core is measured at the distal side using off-axis holography, and the conjugated phase is then displayed on the SLM to precompensate for the phase distortion. The experimental setup is shown in Figure 1. The laser beam from a 532-nm diode-pumped solid-state laser (Verdi, Coherent Inc., Santa Clara, CA, USA) is split by the polarizing beamsplitter, the intensity ratio of both beams can be adjusted by rotating the half-wave plate. One of the beams is coupled into the single-mode fiber (SMF) as the reference beam, and the quarter-wave plate is used to isolate the back-reflected light. The other beam is expanded and illuminates the liquid crystal panel of the SLM (PLUTO, Holoeye Photonics AG, Munich, Germany). For higher diffraction efficiency, a blazed grating is displayed on the SLM. The phase modulated light passes through a 4-f telescope system (L2-L3) and the first diffraction order is filtered out by the spatial filter (iris diaphragm, Thorlabs, Germany) [25; 26]. The diameter of the beam is further magnified by a microscope system (L4, MO1) and the phase modulation \(\phi_{m}(x,y)\) is thus projected on the proximal MCF (FIGH-10-350S, Fujikura, Japan) facet. Assuming the intensity distribution of the in-coupling light is homogeneous, the phase delay in the MCF is \(\Delta\phi(x,y)\), the light field at the distal facet can be expressed as  
\[E_{d}=e^{j(\phi_{m}+\Delta\phi)}. \tag{1}\]  
The distal facet is imaged on the CMOS camera (CAM2) by a microscope system (MO2, L5) and interfered with the reference beam which is slightly tilted for off-axis holography [27; 28]. To measure the phase delay \(\Delta\phi(x,y)\), a Gaussian beam is illuminated on the proximal facet. The phase at the distal facet is thus reconstructed from the off-axis hologram by the angular spectrum method [29]. The measured phase delay is conjugated and displayed on the SLM to precompensate for the phase distortion. \[\phi_{m}^{\prime}(x,y)=-[\Delta\phi(x,y)+c], \tag{2}\]  
where \(c\) is a constant. The phase delays of cores are thus compensated at the distal facet, and the compensated light field is  
\[E_{d}^{\prime}=e^{j(\phi_{m}^{\prime}+\Delta\phi)}=e^{-jc}. \tag{3}\]As the phase delays can be compensated by the DOPC, it is thus possible for simple wavefront shaping through the MCF. For instance, a focus can be generated in the distal far-field with a tailored Fresnel lens. For a core centered at \((x,y)\) in the coordinate system of the SLM, the phase modulation value for the Fresnel lens \(\phi_{f}(x,y)\) is  
\[\phi_{f}(x,y)=mod\left(\frac{\pi aM}{2f\lambda}[(x-x_{0})^{2}+(y-y_{0})^{2}],2 \pi\right), \tag{4}\]  
where \(a\) is the SLM pixel pitch, \(M\) is the microscopic magnification rate from the SLM and to the proximal fiber facet, \(f\) is the focal length, and \((x_{0},y_{0})\) is the center position of the SLM. Figure 2a demonstrates the light field at 600 \(\upmu\)m away from the distal fiber facet with and without DOPC when a tailored Fresnel lens \(\phi_{f}(x,y)\) is displayed on the SLM. A high-quality focus with a 29 dB peak-to-background ratio (PBR) and close to diffraction limit full-width half maximum (FWHM) is achieved. The peak-to-background ratio (PBR) for an amplitude image \(A(x,y)\) is determined as  
\[PBR=\frac{mean(A(x,y)>P/2)}{mean(A(x,y)\leq P/2)}, \tag{5}\]  
where the \(P\) is the peak amplitude of the image.