Phase-Only Hologram (POH) Concepts

• Phase-Only Holograms (POHs) are optical encodings that use spatially varying phase with fixed amplitude, making them pivotal in computer-generated holography.
• Optimization methods like conjugate-gradient and L-BFGS tackle the nonlinear, nonconvex inverse problem inherent in reconstructing desired intensity patterns.
• Recent advances integrate error diffusion, physics-informed neural networks, and compression techniques to enhance applications in beam shaping, displays, and spatiotemporal multiplexing.

A phase-only hologram (POH), also called a phase-only computer-generated hologram or kinoform in some contexts, is a hologram in which the optical field is encoded only through spatially varying phase while the amplitude is fixed, typically to unity. In a standard formulation for a phase-only spatial light modulator (SLM), the field immediately after modulation is $U_0(x,y)=e^{j\phi(x,y)}$, and in the far-field/Fraunhofer regime the reconstructed field is $U_f(u,v)=\mathcal{F}\{U_0(x,y)\}$ with observed intensity $I_f(u,v)=|U_f(u,v)|^2$ (Singh et al., 19 Feb 2026). POHs are central in computer-generated holography because practical LCoS and related SLMs are naturally phase-modulating devices, but the phase-only constraint makes the inverse mapping from a desired reconstruction to a realizable hologram highly nonlinear, nonconvex, and sensitive to propagation model, quantization, and hardware artifacts (Soner et al., 2020).

1. Definition and optical formulation

A POH differs from a complex hologram by retaining only phase and discarding direct amplitude control. Several papers express the complex field generically as $A\exp(j\phi)$, while the phase-only case is written as $\exp(j\phi)$ with unit amplitude (Jiao et al., 2018, Bowman et al., 2017). In digital storage and coding pipelines, the phase values may be represented as a grayscale image; one study states that each pixel in a phase-only hologram has an intensity value ranging from $0$ to $2\pi$ and that the hologram can be regarded as a gray-scale intensity image (Jiao et al., 2018).

Two propagation regimes dominate the literature summarized here. In the Fraunhofer regime, the POH forward model is a Fourier transform, so the basic inverse problem is: given a target intensity $I^T(u,v)$, find $\phi(x,y)$ such that

$$I_f(u,v)=\left|\mathcal{F}\{e^{j\phi(x,y)}\}\right|^2 \approx I^T(u,v).$$

This formulation is explicit in recent far-field work on exact FFT-based training and in optimization studies under Fraunhofer diffraction (Singh et al., 19 Feb 2026, Sha et al., 2022). In Fresnel settings, the hologram is generated and reconstructed through convolution with the Fresnel impulse response

$U_f(u,v)=\mathcal{F}\{U_0(x,y)\}$0

after which a complex hologram may be converted to a phase-only hologram, for example by error diffusion (Jiao et al., 2018).

The phase-only restriction is physically simple but optically severe. A conventional phase-only hologram can be written as

$U_f(u,v)=\mathcal{F}\{U_0(x,y)\}$1

which removes the true hologram-plane amplitude $U_f(u,v)=\mathcal{F}\{U_0(x,y)\}$2 from $U_f(u,v)=\mathcal{F}\{U_0(x,y)\}$3 (Shimobaba et al., 2019). This is why reconstructed amplitude is difficult to control in phase-only CGH, why many tasks are formulated as inverse problems, and why POHs remain a distinct research object rather than a trivial special case of complex holography (Ishii et al., 2021, Mukherjee et al., 2022).

2. Inverse problem and optimization methods

The core POH inverse problem is globally coupled. In far-field diffraction every SLM pixel influences the entire reconstructed intensity distribution, and under a phase-only constraint the target intensity must be formed indirectly through global phase interference. One recent formulation writes the map explicitly as

$U_f(u,v)=\mathcal{F}\{U_0(x,y)\}$4

and emphasizes that this makes the optimization highly nonlinear and nonconvex; classical approaches such as Gerchberg–Saxton can stagnate in local minima or depend strongly on initialization (Singh et al., 19 Feb 2026).

A large part of POH research therefore concerns optimization strategies. Conjugate-gradient minimization has been used to design phase-only SLM patterns that reproduce complex target fields in a chosen measure region of the Fourier plane. In that formulation the cost is based on the overlap between normalized target and output fields, and the headline numerical result is a calculated fidelity $U_f(u,v)=\mathcal{F}\{U_0(x,y)\}$5 for an $U_f(u,v)=\mathcal{F}\{U_0(x,y)\}$6-like pattern with $U_f(u,v)=\mathcal{F}\{U_0(x,y)\}$7 light-usage efficiency; uncorrected experimental realizations reached $U_f(u,v)=\mathcal{F}\{U_0(x,y)\}$8 and $U_f(u,v)=\mathcal{F}\{U_0(x,y)\}$9 light efficiency (Bowman et al., 2017). This is an important correction to the common misconception that POHs can only control intensity: in a restricted output region, they can be optimized for simultaneous amplitude-and-phase control.

Other work frames POH synthesis as direct loss minimization under Fraunhofer propagation. A limited-memory BFGS formulation with cross-entropy loss reports that L-BFGS + CE converged fastest and reached MSE of about $I_f(u,v)=|U_f(u,v)|^2$0, whereas the other successful methods converged to about $I_f(u,v)=|U_f(u,v)|^2$1; in the same study, Adam + CE did not converge (Sha et al., 2022). A different gradient-based line replaces direct phase parameterization by an auxiliary complex variable $I_f(u,v)=|U_f(u,v)|^2$2 and defines the phase-only hologram as

$I_f(u,v)=|U_f(u,v)|^2$3

This retains a unit-modulus hologram while allowing optimization to proceed in Cartesian coordinates with Wirtinger calculus. The method reports faster convergence and better PSNR than previous phase-only gradient methods, with one timing example giving $I_f(u,v)=|U_f(u,v)|^2$4 s to reach PSNR $I_f(u,v)=|U_f(u,v)|^2$5 dB versus $I_f(u,v)=|U_f(u,v)|^2$6 s for WFPF + Adam and $I_f(u,v)=|U_f(u,v)|^2$7 s for WFPF (Uchiyama et al., 2024).

These methods share a common structure: a propagation operator, a phase-only constraint, and an intensity- or field-domain loss. What changes across methods is the parameterization of the feasible set, the optimization geometry, and whether the controlled object is a single intensity plane, a complex field in a measure region, or a multiplane target.

3. Phase-only encoding, complex-field approximation, and quantization

Because a POH cannot assign amplitude directly, many methods aim to approximate complex modulation while preserving phase-only hardware compatibility. A classical route is error diffusion. Starting from a complex Fresnel hologram $I_f(u,v)=|U_f(u,v)|^2$8, the method forces each processed pixel to unit amplitude, computes the complex error, and diffuses it to neighboring pixels with weights

$I_f(u,v)=|U_f(u,v)|^2$9

yielding a non-iterative conversion from complex hologram to POH (Jiao et al., 2018).

A simpler complex-amplitude encoding method binarizes the hologram-plane amplitude $A\exp(j\phi)$0 into $A\exp(j\phi)$1, assigns the original phase on “on” pixels, and assigns a canceling phase $A\exp(j\phi)$2 on “off” pixels: $A\exp(j\phi)$3 In the reported tests, the alternate $A\exp(j\phi)$4 pattern outperformed checkerboard and random off-pixel assignments, and the method achieved reconstruction quality close to double-phase holography while giving light-efficiency ratios $A\exp(j\phi)$5 and $A\exp(j\phi)$6 in two image examples; for TEM$A\exp(j\phi)$7 beam generation it reached amplitude PSNR $A\exp(j\phi)$8 dB versus $A\exp(j\phi)$9 dB for DPH, with $\exp(j\phi)$0 (Shimobaba et al., 2019).

A more formal complex-field reconstruction on phase-only hardware is double-phase holography. In visible dielectric meta-optics, the target field

$\exp(j\phi)$1

is decomposed using

$\exp(j\phi)$2

and the two phase terms are assigned to complementary checkerboard masks (Mukherjee et al., 2022). In the reported experiments, DPH reconstruction SSIM was $\exp(j\phi)$3 versus $\exp(j\phi)$4 for a simple phase-only hologram on one target portion, and $\exp(j\phi)$5 versus $\exp(j\phi)$6 on another, with the paper describing the improvement as about $\exp(j\phi)$7 over phase-only holograms (Mukherjee et al., 2022).

Quantization itself is also an active POH topic. Sympathetic quantisation argues that nearest-neighbour phase quantization is physically mismatched because a perturbation at one hologram pixel affects every replay pixel. Its soft sympathetic quantisation variant modifies rotationally symmetric phase-only pixel pairs together so that replay perturbations are biased toward phase-like rather than intensity-like distortion. For the considered single-transform algorithms such as OSPR and STTM, the paper reports that MSE was reduced to under $\exp(j\phi)$8 of traditional approaches and SSIM improved by more than $\exp(j\phi)$9 (Christopher et al., 2020). This directly challenges the assumption that POH quantization is a purely local post-processing step.

4. Learned POH generation and physics-informed neural methods

Recent work increasingly treats POH synthesis as a learned inverse problem. A notable physics-in-the-loop formulation uses a transformer that maps target intensity directly to a phase-only SLM field and trains end-to-end through exact FFT-based Fraunhofer propagation: $0$0 with the intensity loss

$0$1

For $0$2 targets, the model used a coarse $0$3 patch tokenization, $0$4, and $0$5 trainable parameters. Reported final energy-scaled MSE was $0$6 on train, $0$7 on validation, and $0$8 on test; the same model showed out-of-distribution generalization to unseen MNIST digits and hand-drawn targets, and a separate $0$9 single-target demonstration improved from $2\pi$0 to $2\pi$1 to $2\pi$2 during coarse training and refinement (Singh et al., 19 Feb 2026). The paper’s main conceptual claim is that self-attention matches the global coupling of Fourier diffraction and that coarse patch tokenization acts as an implicit spectral regularizer.

Other neural approaches operate in hologram space rather than image space. A U-Net-based method refines a precomputed phase-only CGH generated by scaled diffraction and a random phase-free method. For standard phase-only CGH, the proposed network improved average test performance to PSNR $2\pi$3 dB and SSIM $2\pi$4, versus $2\pi$5 dB and $2\pi$6 for 10-iteration GS; for the paper’s bleached phase CGH representation, the gains were larger, reaching $2\pi$7 dB and $2\pi$8 versus $2\pi$9 dB and $I^T(u,v)$0 for GS, with runtime reduced from $I^T(u,v)$1 ms to $I^T(u,v)$2 ms per CGH (Ishii et al., 2021).

A different line uses conditional generative models but on a reduced proxy manifold. Instead of learning arbitrary full-resolution POHs, one study learns $I^T(u,v)$3 complex spatial-frequency coefficient matrices that are converted into physical phase-only holograms by a deterministic blazed-grating rule. On measured in-distribution test intensities, mean errors were $I^T(u,v)$4 for cVAE, $I^T(u,v)$5 for cGAN, $I^T(u,v)$6 for cVAE + forward loss, and $I^T(u,v)$7 for an expert system; on synthetic out-of-distribution targets, cVAE + forward loss was best at $I^T(u,v)$8 (Gladrow, 2019). This is a useful reminder that “learned POH generation” can mean either direct phase generation or inference inside a restricted hologram family.

End-to-end generative modeling of POHs has also appeared. A VQ-VAE framework named Holo-VQVAE decodes latent codes directly into phase maps $I^T(u,v)$9, propagates them with the angular spectrum method using $\phi(x,y)$0, and supervises training in the reconstruction domain. On MNIST, Fashion-MNIST, and CelebA-HQ, the reported Holo-VQVAE reconstruction scores were PSNR/SSIM $\phi(x,y)$1, $\phi(x,y)$2, and $\phi(x,y)$3, compared with $\phi(x,y)$4, $\phi(x,y)$5, and $\phi(x,y)$6 for a Gaussian-latent Holo-VAE baseline (Park et al., 2024). The paper positions this as direct hologram-domain generation rather than the older “image generator + hologram converter” pipeline.

5. Compression, calibration, and system integration

POHs are unusually difficult to compress because their information resides in high-frequency fringe structure. A practical demonstration uses standard JPEG at quality factor = 1 on $\phi(x,y)$7 phase-only holograms. Direct JPEG produces compressed files of about 142–144 KB from an original of approximately 1024 KB, with compression ratios near 7:1, but reconstructed images degrade severely. A four-layer CNN applied patch-by-patch to the decompressed hologram restores quality: for Cameraman at 0.5 m, reconstructed-image PSNR/SSIM improved from 19.10 dB / 0.1651 to 28.86 dB / 0.6036; for Pepper at 0.5 m, from 18.92 dB / 0.2007 to 29.88 dB / 0.6767 (Jiao et al., 2018).

For wearable displays, compression becomes a system requirement rather than a storage convenience. A head-mounted holographic display study estimates that a monochrome, 60 fps, Full HD, 2-eye POH HMHD system would require about 10 TFLOPS/W, while commercially available mobile processors are cited at < 200 GFLOPS/W; the uncompressed POH rate is 1.85 Gbit/s, whereas realistic wireless operation is in the 60–200 Mbit/s range (Soner et al., 2020). Under those constraints, the paper proposes a PCM-POH codec with progressive quantization and per-pixel rate-distortion control. It also reports that 2D phase unwrapping does not help net compression performance, because the dynamic-range increase dominates the gain in smoothness.

Learned POH compression has begun to appear as well. One benchmark compares vanilla MLP, SIREN, FilmSIREN, and the pretrained image VAE TAESD. The strongest result is that SIREN with 4,899 parameters on $\phi(x,y)$8 patches achieves 40% compression with 42.29 ± 2.45 dB hologram-space PSNR and 34.54 dB PSNR, 0.96 SSIM, and 0.10 LPIPS in reconstructed 3D images; the same study reports that TAESD with 2.2M parameters struggles on POH compression (Peng et al., 9 Jul 2025).

On the hardware side, calibration and artifact suppression are equally central. A recent interferometric camera-in-the-loop method addresses zero-order diffraction in phase-only CGH by embedding a correction beam into the phase-only hologram: $\phi(x,y)$9 Using a pixel-wise phase map retrieved in a plane conjugate to the SLM, the method reports up to 99% suppression of zero-order diffraction intensity without loss of image quality or field of view; once calibrated, the correction can be applied to any hologram, with estimated overhead < 5 ms on CPU and down to $$I_f(u,v)=\left|\mathcal{F}\{e^{j\phi(x,y)}\}\right|^2 \approx I^T(u,v).$$0s-scale when integrated into GPU hologram generation (Cerioni et al., 28 Jan 2026). This illustrates a broader systems point: practical POHs are defined as much by calibration, compression, and display pipeline compatibility as by the inverse algorithm that first produced them.

6. Application domains and emerging extensions

POHs now support substantially more than single-plane scalar image reconstruction. In precision beam shaping for optical trapping and ultracold atoms, conjugate-gradient POH design has been used to generate Laguerre–Gaussian-like beams, ring and square lattices with phase winding, graphene-like patterns, flat-top beams with structured phase, and Gaussian lines with phase gradients (Bowman et al., 2017). In another Fourier-optics pipeline for 3D reconstruction from point clouds and MRI-derived volumes, the best reported POH result was AP SP-3D-AS-POH with 2D median filtering, reaching PSNR 30.35 on volumetric brain reconstruction (London, 8 Aug 2025).

The scalar character of conventional POHs has also been generalized. A polarization-insensitive phase hologram illuminated with an azimuthally polarized vector beam can synthesize momentum-space polarization states, including $$I_f(u,v)=\left|\mathcal{F}\{e^{j\phi(x,y)}\}\right|^2 \approx I^T(u,v).$$1-linear, $$I_f(u,v)=\left|\mathcal{F}\{e^{j\phi(x,y)}\}\right|^2 \approx I^T(u,v).$$2-linear, left-handed circular, right-handed circular, and radial polarization, using scalar phase maps such as a $$I_f(u,v)=\left|\mathcal{F}\{e^{j\phi(x,y)}\}\right|^2 \approx I^T(u,v).$$3-phase-step map, vortex phase map, and multi-zone $$I_f(u,v)=\left|\mathcal{F}\{e^{j\phi(x,y)}\}\right|^2 \approx I^T(u,v).$$4-phase map (Ren, 2022). The key result is that polarization control need not require a polarization-sensitive hologram if the illumination is itself vector-structured.

Spatiotemporal generalization is similarly explicit. A phase-only LC-SLM in a pulse-shaper geometry can encode a target spatial–spectral complex field

$$I_f(u,v)=\left|\mathcal{F}\{e^{j\phi(x,y)}\}\right|^2 \approx I^T(u,v).$$5

through a phase-only pattern

$$I_f(u,v)=\left|\mathcal{F}\{e^{j\phi(x,y)}\}\right|^2 \approx I^T(u,v).$$6

with amplitude encoded by diffraction-efficiency control and higher orders removed by an iris (Cao et al., 2024). The reported demonstrations include fundamental and higher-order spatiotemporal Bessel wavepackets, crystal-like and quasi-crystal-like structures, and flat-top spatiotemporal wavepackets.

POHs have also become a platform for multiplexed planar optics. In angle-multiplexed Holo-Printing, each diffraction order reconstructs a different printing image; the paper experimentally encoded 25 gray scale printings into 25 angles and 8 gray scale videos into 8 angles with a phase-only SLM, then extended the idea to a simulated 324 channel gradient metasurface by multiplexing angles, polarizations, and wavelengths simultaneously (Wei et al., 2022). A related but more volumetric development uses free-space communication modes derived from the singular value decomposition of the source-to-target coupling operator to synthesize complex fields across multiple axial planes. The desired source field is

$$I_f(u,v)=\left|\mathcal{F}\{e^{j\phi(x,y)}\}\right|^2 \approx I^T(u,v).$$7

after which an Arrizón Type 3 phase-only encoding and 4f filtering implement the field on a single phase-only SLM (Angelis et al., 16 Mar 2026). The reported experiments reconstruct both intensity and phase across multiple planes, including arbitrary structured phase singularity patterns.

Taken together, these results show that the phrase “phase-only hologram” no longer denotes only a wrapped phase map for a single far-field intensity pattern. It now encompasses exact-physics inverse design, complex-field approximation strategies, learned generators, compression and calibration pipelines for wearable or display hardware, and extensions to polarization, spacetime, angle multiplexing, and multiplane coherent field synthesis. This suggests that POH has become less a narrow encoding format than a general constrained wavefront-design paradigm grounded in unit-modulus optical hardware.