Multi-Plane Spatially Resolved Phase Structuring Using Optical Communication Modes

Abstract

We present a deterministic framework for three-dimensional beam shaping that enables versatile control of intensity and phase, pixel-by-pixel, across multiple axial planes. Conventional multi-plane holographic techniques typically rely on iterative optimization and mitigate inter-plane crosstalk through phase randomization, introducing speckle noise and thereby limiting deterministic phase control. Here, target fields are synthesized as a linear superposition of free-space communication modes obtained from the singular value decomposition of a coupling operator connecting a source plane to multiple target planes. Because these modes form orthogonal and energy-efficient transmission channels between the source and receiving spaces, their superposition yields volumetric wavefields with enforced phase coherence and reduced inter-plane crosstalk, without iterative refinement. We experimentally demonstrate high-fidelity reconstruction of intensity and phase profiles across multiple planes using a single phase-only spatial light modulator, including arbitrary structured phase singularity patterns. The proposed approach establishes communication-mode optics as a practical and physically grounded framework for multi-plane beam shaping, particularly in applications where phase structure and coherence across depth are essential.

Figures (15)

• Figure 1: Communication modes and their coupling strengths associated with a transverse source plane and a set of three transverse receiving planes equally spaced from each other. (a) Dimensions of the source and receiving spaces, parametrized as listed in Table \ref{['tab_param']} for $n = 3$ receiving planes. (b) Coupling strengths in order of decreasing magnitude computed at $\lambda$ = 532 nm (black solid line). For comparison, the coupling strengths associated with each receiving plane separately are also shown. Each $n$-th receiving plane supports a total of 100 strongly coupled modes, which are classified into two sub-categories: intrinsic and extrinsic modes. After the strong modes, each plane also supports partially coupled modes. Normalized squared amplitude of some modes. For the first plane: (c)$j = 1$ (extrinsic mode); (d)$j = 4$, (e)$j = 100$ (intrinsic modes); (f)$j = 101$ (partially coupled mode). For the second plane: (g)$j = 121$ (extrinsic mode); (h)$j = 136$ (intrinsic mode), (i)$j = 221$ (partially coupled mode); For the third plane: (j)$j = 260$ (intrinsic mode). In each sub-figure the source eigenfunction is shown on the left and its resulting wave in the $xz$ plane on the right.
• Figure 2: Simulated structuring of arbitrary light wave profiles. (a) Coupling strengths for different separation distances $L$ between the receiving planes in the configuration of Fig. \ref{['Fig_modes_n_2']}(a). The inset depicts the coupling strengths on a logarithmic scale. (b) For $L = 0.5L_0$, projection of a target intensity profile consisting of the binary digits '1', '2' and '3' onto the receiving basis set (blue circles). (c) Calculated source function computed from the first 350 well-coupled modes (strong and partially coupled modes). (d) Calculated resulting wave from the source function in all three receiving planes. The dashed white squares highlight the size of the target planes. (e) Structuring phase profiles: the target consists of the same three binary digits with a phase of $\pi/2$ radians inside each digit’s contour and $3\pi/2$ radians outside. Amplitude is set to zero along each singularity contour (see Fig. S5). Projection onto the receiving basis set (blue circles) and calculated source function computed from the first 350 well-coupled modes (bottom). (f) Calculated phase and intensity (in dB) of the resulting wave in all three receiving planes.
• Figure 3: Optical reconstruction of the light waves using a phase-only reflective SLM. (a) Owing to its reactive near-field components and high amplitude source amplitudes originated from its diffraction orders, the source function cannot be directly encoded into a phase-only CGH. (b) The wave solution propagated from the source function to the plane $z = L_0$ was encoded instead as this Fresnel field is fully radiative and with no contributions from higher diffraction orders. The dashed white square highlights the aperture size. (c) The wave solution is interpolated to the utilized SLM display resolution ($801 \times 801$ pixels) and is converted into a phase mask using the deterministic hologram generation algorithm of Ref. Arrizon:07. The algorithm encodes the wave solution on the first diffraction order of the phase mask in the spatial frequency domain. (d) Optical setup: a green laser is expanded and collimated before impinging on the SLM. After the SLM, a 4$f$ system is employed to recover the wave solution at the front focal plane of the second lens. An iris is placed at the Fourier plane to filter the first diffraction order while blocking the SLM unmodulated wave (zeroth diffraction order). The intensity distributions of the resulting wave are recorded using a CCD camera mounted on a translational stage. The phase profiles are reconstructed using the method of single-beam multiple-intensity reconstruction.
• Figure 4: Optical reconstruction of the light waves using a phase-only reflective SLM. (a) Measured intensity and (b) retrieved phase distributions for the intensity-only target patterns of Fig. \ref{['Fig_svd_structuring_sim']}(d) at the three receiving planes. The reconstructed intensity profiles exhibit high fidelity, high contrast, and low speckle noise. (c) Measured retrieved phase and (d) intensity distributions for the target patterns of the example of Fig. \ref{['Fig_svd_structuring_sim']}(f). The digit's contours correspond to zero-amplitude lines where phase jumps of $\pi$ are imposed, forming two-dimensional phase-singularity sheets across all planes. Quantitative reconstruction metrics are reported in Tables 2 and 3. Scale bars (vertical white lines) represent 0.1 mm. The second and third planes are longitudinally spaced by 68.4 mm and 136.8 mm from the first plane. Inset figures show the simulated intensity and phase distributions.
• Figure 5: Structuring high-resolution and multi-plane complex amplitude wavefront profiles. (a) The number of supported strong modes in each receiving plane is increased by the array size. The same configuration of Fig. \ref{['Fig_modes_n_2']}(a) with $L = L_0$ and receiving planes composed of $41 \times 41$ receiving points, resulting in 400 strong modes in each plane. (b) Projection of the target patterns (checkerboard and sequence of digits '123') as intensity profiles and (c) as phase profiles. The inset figure shows the required source function computed from the first 912 well-coupled modes. Simulated resulting wave at the receiving planes and measured reconstructed profiles when the target patterns are (d) intensity-only profiles and (e) phase profiles (with the same target values as in Fig. \ref{['Fig_svd_structuring_sim']}). Scale bars (vertical white lines) represent 0.1 mm. The measured planes are longitudinally spaced by 136.8 mm.