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Digital holography

Digital holography is an advanced optical imaging technique that records the interference pattern between an object wave and a reference wave using digital sensors, such as CCD or CMOS cameras, and reconstructs the full complex amplitude (intensity and phase) of the light field through numerical computation, enabling label-free, three-dimensional (3D) visualization and quantitative measurements of phase shifts without physical lenses.^[1] This method surpasses traditional photography by capturing the entire wavefront of a scene, allowing post-processing refocusing, parallax, and depth perception in reconstructed images.^[2] The foundations of digital holography trace back to Dennis Gabor's 1948 invention of holography for electron microscopy, which laid the groundwork for wavefront reconstruction, though initial implementations relied on analog photographic plates. Significant advancements occurred in the 1960s with the advent of lasers, enabling off-axis holography by Emmett Leith and Juris Upatnieks, which separated object and reference beams to improve image quality. The digital era began in the late 1960s with computer-generated holograms (CGH) proposed by Adolf W. Lohmann and colleagues, and further evolved in the 1990s with the first experimental demonstration of digital recording using a CCD sensor by U. Schnars and W. Jüptner in 1994, alongside high-resolution digital sensors and computational power allowing real-time numerical reconstruction via algorithms like the Fresnel transform or angular spectrum propagation.^[3]^[1] Key principles involve coherent light illumination, interferometric recording of the hologram, and phase retrieval techniques—such as phase-shifting holography (PSH) for precise phase extraction or off-axis configurations to avoid twin-image artifacts—yielding quantitative data on optical path length, refractive index, and surface topography.^[3] Digital holography finds diverse applications across scientific and industrial domains, including digital holographic microscopy (DHM) for non-invasive 3D imaging of biological cells and tissues, revealing morphological changes with sub-micrometer resolution.^[1] In manufacturing, it supports non-destructive testing (NDT) for defect detection in aerospace components and surface metrology in automotive parts, achieving nanometer-scale precision.^[2] Environmental monitoring benefits from its use in analyzing plankton dynamics or cloud microstructures in oceanic and atmospheric studies.^[4] Recent developments integrate deep learning for accelerated reconstruction, super-resolution enhancement beyond diffraction limits, and multimodal imaging (e.g., combining phase with fluorescence), with emerging trends in lensless configurations and optical diffraction tomography (ODT) for thick-sample 3D refractive index mapping, as demonstrated in clinical diagnostics.^[5]

Fundamentals

Principles of digital holography

Digital holography records the interference pattern formed by the superposition of an object wave, scattered from an illuminated object, and a reference wave on a digital sensor, such as a charge-coupled device (CCD) or complementary metal-oxide-semiconductor (CMOS) array, supplanting the photographic emulsion of conventional holography.^[6] This digital capture preserves the intensity distribution of the interfering waves, facilitating numerical reconstruction of the object's amplitude and phase information to yield a three-dimensional wavefront representation.^[7] Coherent light sources, particularly lasers, are indispensable, providing the necessary spatial and temporal coherence to generate stable, high-contrast interference fringes essential for accurate recording.^[6] In a basic setup, a coherent beam from the laser is divided by a beam splitter into an object beam that illuminates the target and a reference beam that propagates directly to the sensor plane, where the two waves interfere to form the hologram.^[7] The recorded intensity at sensor coordinates (x, y) is mathematically expressed as

I(x,y) = \left| O(x,y) + R(x,y) \right|^2,

where O(x,y) denotes the complex amplitude of the object wave and R(x,y) the reference wave.^[7] Expanding this yields

I(x,y) = \left| O(x,y) \right|^2 + \left| R(x,y) \right|^2 + O(x,y) R^*(x,y) + O^*(x,y) R(x,y),

with the first two terms forming the zero-frequency (DC) component, the third term the virtual (conjugate) image, and the fourth the real image; these can be separated in the spatial frequency domain during reconstruction.^[7] Distinct from analog holography, which employs continuous recording media for optical reconstruction, digital holography introduces pixelation due to the discrete sampling of the sensor array, constraining spatial resolution to the Nyquist frequency—half the inverse of the pixel pitch—to prevent aliasing of high-frequency fringes.^[8] This discretization necessitates computational methods for hologram reconstruction, enabling digital manipulation but imposing limits on the fidelity of fine details compared to analog's potentially higher resolution.^[7] Common approaches, like the off-axis configuration, aid in isolating the image terms by shifting them away from the DC region in the Fourier domain.^[7]

Historical development

Holography originated with Dennis Gabor's invention in 1948, developed as a technique to enhance the resolution of electron microscopes through in-line wavefront reconstruction using coherent illumination from a mercury arc lamp filtered for coherence. Although groundbreaking, Gabor's method produced noisy images due to the twin-image problem and limited light sources, restricting practical applications until the laser's emergence. The field was revitalized in 1962 by Emmett Leith and Juris Upatnieks at the University of Michigan, who introduced off-axis holography using a helium-neon laser to separate the reference and object beams, enabling clear three-dimensional reconstructions of complex objects like a toy train. This laser-based approach transformed holography from a theoretical concept into a viable optical imaging tool, laying the foundation for subsequent digital advancements.^[9] The shift to digital holography began in the early 1990s, driven by electronic detectors and personal computing capabilities that allowed replacement of wet photographic processing with numerical reconstruction. In 1994, Ulf Schnars and Werner Jüptner achieved the first practical digital holograms by recording off-axis interferograms directly onto a charge-coupled device (CCD) sensor with 1024×1024 pixel resolution and reconstructing the complex wavefront via Fourier transform on a standard computer, demonstrating deformation measurements in hologram interferometry. This innovation eliminated chemical development, enabling faster iteration and quantitative phase retrieval, though early CCDs suffered from limited dynamic range and quantum efficiency for holographic fringes. A pivotal milestone came in 1997 with Ichiro Yamaguchi and Tong Zhang's introduction of phase-shifting digital holography, which recorded four interferograms at quarter-wave phase shifts to suppress noise and improve phase accuracy, facilitating single-color phase maps with subwavelength precision.^[6]^[10] Post-2000, digital holography gained widespread adoption in microscopy for label-free quantitative phase imaging of biological samples, exemplified by Tong Zhang and Ichiro Yamaguchi's 1998 work on three-dimensional microscopy using phase-shifting techniques to visualize microscopic specimens without axial scanning.^[11] By the 2010s, advances in parallel computing, such as GPU-accelerated algorithms, transformed reconstruction speeds; for instance, CUDA-based implementations reduced processing time for 1-megapixel holograms from seconds to milliseconds, enabling real-time volumetric imaging in dynamic systems.^[12] Sensor technology evolved concurrently, progressing from low-resolution, slow-readout CCDs (e.g., 512×512 pixels at 10 frames per second in the 1990s) to high-speed CMOS arrays in the 2020s, offering over 100 megapixels, global shutter operation, and color sensitivity for multispectral holography, thus supporting applications in high-throughput metrology and biomedical imaging.^[13] This progression was profoundly influenced by exponential growth in computing power, as described by Moore's Law, which doubled transistor density roughly every two years, slashing the computational burden of Fresnel or angular spectrum propagation from hours on early workstations to near-instantaneous on modern hardware and enabling sophisticated iterative phase retrieval methods previously infeasible.^[13]

Recording Techniques

Off-axis configuration

The off-axis configuration in digital holography employs a reference beam tilted at an angle relative to the object beam, enabling the interference pattern to be captured on a digital sensor such as a CCD or CMOS array in a single exposure. This setup directs the object beam, scattered from the illuminated specimen, onto the sensor plane, where it overlaps with the off-axis reference beam, forming a hologram that encodes both amplitude and phase information. Originating from the analog holography work of Leith and Upatnieks, this geometry was adapted for digital recording to facilitate numerical reconstruction without the need for multiple exposures.^[6] The mathematical foundation relies on the interference of the object wave O(x, y) and the reference wave R(x, y), typically a plane wave given by R(x, y) = \exp[i 2\pi (u_0 x + v_0 y)], where u_0 and v_0 represent the carrier frequencies induced by the tilt angle. The recorded hologram intensity is

I(x, y) = |O(x, y) + R(x, y)|^2 = |O|^2 + |R|^2 + O R^* + O^* R,

which expands into the zero-order term (|O|^2 + |R|^2), the virtual image term (O R^*), and the real (twin) image term (O^* R). The Fourier transform H(u, v) = \mathcal{F}\{I(x, y)\} spatially separates these components: the zero-order term centers at (0, 0), the virtual image at (u_0, v_0), and the twin image at (-u_0, -v_0), preventing overlap provided the carrier frequency exceeds the bandwidth of the object spectrum. To reconstruct the object wave, the desired term is isolated via frequency-domain filtering, such as

\tilde{O}(x, y) \approx \mathcal{F}^{-1} \left\{ H(u, v) \cdot \rect\left(\frac{u - u_0}{\Delta u}\right) \right\},

where \rect is a rectangular window function with width \Delta u encompassing the virtual image spectrum, followed by a phase adjustment to recover the complex field.^[14] This configuration offers key advantages, including single-shot recording suitable for dynamic or moving objects, and compatibility with standard coherent illumination sources without requiring phase modulation hardware. However, it incurs a resolution penalty, as the carrier frequency occupies sensor bandwidth, typically limiting the effective spatial resolution to about one-third of the sensor's Nyquist frequency due to the need to allocate space for the separated terms.^[6] In practice, the off-axis angle is optimized to position the carrier frequency at approximately 1/4 to 1/8 of the sensor's sampling rate, ensuring separation without excessive overlap or aliasing while maximizing usable bandwidth; for example, angles of 1° to 5° are common for visible wavelengths and typical sensor pixel sizes around 5–10 μm. Precise alignment is critical, as misalignment can cause spectrum overlap, degrading reconstruction quality.^[15]

Phase-shifting holography

Phase-shifting holography is a recording technique in digital holography that recovers the full complex amplitude of the object wave by sequentially modulating the phase of the reference beam and capturing multiple interferograms on a detector array.^[10] This method, introduced in the late 1990s, enables in-line configurations that avoid the spatial separation required in off-axis setups, thereby maximizing the use of the sensor's field of view.^[10] The phase modulation is typically achieved using devices such as piezoelectric transducers (PZT) for mechanical shifting or liquid crystal elements for electro-optic control, allowing precise control over the reference wave's phase.^[10] In the standard four-step implementation, four intensity patterns I_k(x,y) are recorded corresponding to phase shifts \delta_k = k \pi / 2 for k = 0, 1, 2, 3.^[10] The complex amplitude \Gamma(x,y) of the object wave is then reconstructed at the recording plane via the formula:

\Gamma(x,y) = \frac{1}{4} \left[ I_0(x,y) - I_2(x,y) + i \left( I_3(x,y) - I_1(x,y) \right) \right]

where the imaginary unit is i.^[10] This direct computation yields both the amplitude and phase without the zero-order and twin-image artifacts inherent in single-exposure methods, provided the phase steps are accurately calibrated.^[10] Variants of the phase-shifting approach address specific challenges, such as recording dynamic scenes with reduced exposures. Two-step methods, which use phase shifts of 0 and \pi, enable complex amplitude recovery with only two interferograms, making them suitable for objects with moderate motion where full stability across four steps is impractical.^[16] For environments with noisy data or imprecise phase calibration, least-squares optimization algorithms minimize errors by iteratively fitting the recorded intensities to the expected phase-shifting model, improving phase accuracy under low signal-to-noise conditions. Compared to off-axis holography, which supports single-shot recording as an alternative, phase-shifting techniques offer superior light efficiency by utilizing the entire detector bandwidth without dedicating portions to carrier frequencies, resulting in a higher signal-to-noise ratio—typically 2 to 4 times better in low-photon regimes. This efficiency stems from the absence of spatial multiplexing, allowing denser packing of holographic information. Despite these benefits, phase-shifting holography demands a vibration-isolated setup to maintain phase stability across multiple sequential exposures, limiting its use in turbulent or highly dynamic scenarios.^[17] Real-time adaptations overcome this by incorporating spatial phase modulation, such as through pixelated phase masks or spatial light modulators that impose the required shifts across sub-regions of the detector in a single exposure, enabling parallel recording for video-rate holography.^[18]

Frequency-shifting holography

Frequency-shifting holography, also known as heterodyne holography, involves introducing a frequency shift between the object and reference beams during recording to enable temporal demodulation of the interference pattern. This is typically achieved by shifting the frequency of the reference beam using an acousto-optic modulator (AOM), which generates a beat frequency that modulates the recorded intensity over time.^[19]^[20] The mathematical foundation relies on the time-varying intensity captured at the detector, expressed as

I(x,y,t) = |O|^2 + |R|^2 + 2 \operatorname{Re} \left\{ O R^* \exp(i 2\pi f t) \right\},

where O and R are the complex object and reference fields, respectively, and f is the frequency shift. Demodulation of this signal to extract the complex amplitude O can be performed using techniques such as the Hilbert transform or lock-in amplification, which isolate the phase and amplitude components from the temporal modulation.^[21]^[20] This approach offers high sensitivity to dynamic phase changes, making it suitable for real-time measurements, and suppresses the unwanted twin-image artifact without requiring spatial carrier frequencies or off-axis geometries. It is particularly advantageous for vibration analysis, where the temporal beat allows precise tracking of minute displacements in oscillating objects.^[22]^[23] In practice, implementations often use single-pixel detectors combined with raster scanning for high-resolution mapping or full-field detection with high-speed cameras to capture the temporal evolution at rates up to 10 kHz, enabling applications in dynamic imaging. Unlike phase-shifting holography, which relies on discrete phase steps, frequency-shifting enables continuous, real-time demodulation through the frequency domain. Typical shift frequencies range from kHz to hundreds of MHz, depending on the modulator and detector capabilities.^[24]^[25]

Reconstruction and Processing

Numerical reconstruction algorithms

Numerical reconstruction algorithms in digital holography simulate the diffraction of light from the recorded hologram to retrieve the complex wavefront of the object, enabling visualization of amplitude and phase without physical optics. These algorithms rely on scalar diffraction theory under the paraxial approximation and are implemented computationally using fast Fourier transforms (FFTs) for efficiency. The two primary approaches are the Fresnel diffraction method, suitable for short propagation distances, and the angular spectrum method, which offers greater flexibility for varying distances.^[26] The Fresnel approximation models near-field propagation and forms the basis for many early digital holography reconstructions. The complex field at a distance z from the hologram plane is given by

U(x, y, z) = \frac{e^{ikz}}{i \lambda z} \iint H(\xi, \eta) \exp\left[i \frac{\pi}{\lambda z} \left( (x - \xi)^2 + (y - \eta)^2 \right) \right] d\xi \, d\eta,

where H(\xi, \eta) is the recorded hologram intensity, k = 2\pi / \lambda is the wave number, and \lambda is the wavelength. In discrete form, for a sampled hologram on a pixel grid, this integral is evaluated using a single 2D FFT preceded and followed by multiplication with quadratic phase factors, allowing efficient computation on digital hardware. This method was pioneered for digital holograms by Schnars and Jüptner, who demonstrated its use for reconstructing CCD-recorded holograms.^[6]^[26] For reconstructions over larger distances or to minimize approximation errors in the Fresnel kernel, the angular spectrum method decomposes the hologram into its plane-wave spectrum and propagates each component independently. The field is propagated as

U(x, y, z) = \mathcal{F}^{-1} \left\{ \mathcal{F}\{H(x, y)\} \cdot \exp(i k_z z) \right\},

where \mathcal{F} denotes the 2D Fourier transform, and k_z = \sqrt{k^2 - k_x^2 - k_y^2} is the longitudinal wave vector component, with k_x and k_y as transverse spatial frequencies. This approach requires two 2D FFTs and is valid within the paraxial regime but avoids the $1/z singularity of the Fresnel method, making it preferable for focal plane analysis or tilted geometries. It has been widely adopted in digital holography for its numerical stability and exactness in handling evanescent waves when k_z becomes imaginary.^[26]^[27] In off-axis recording configurations, the hologram contains distinct spectral components for the virtual (desired) image, real (twin) image, and zero-order term, enabling separation through spatial filtering in the Fourier domain or selective phase manipulation. Twin-image suppression is achieved by isolating the virtual image term via bandpass filtering around the carrier frequency, preventing overlap with the conjugate term. Advanced techniques for in-line holograms, such as iterative phase retrieval, further mitigate twin-image issues but are computationally intensive.^[6]^[26] Key error sources in these algorithms include aliasing from limited pixel resolution in the sensor, which undersamples high spatial frequencies and distorts the reconstructed field; this is addressed by zero-padding the hologram array to at least twice its original size before FFT to satisfy the Nyquist criterion and prevent circular convolution artifacts. Pixel size also limits the maximum reconstruction distance via the relation z_{\max} \approx N \Delta \xi^2 / \lambda, where N is the number of pixels and \Delta \xi the pixel pitch. The overall computational complexity scales as O(N^2 \log N) for an N \times N hologram due to the FFT operations, though GPU acceleration can reduce practical times to milliseconds for typical resolutions.^[28]^[29]

Digital processing methods

Digital processing methods in holography build upon numerical reconstruction to refine the recovered wavefront, addressing imperfections introduced during recording and propagation. These techniques enhance image quality by mitigating artifacts and improving quantitative accuracy, enabling applications in high-resolution imaging.^[30] Noise mitigation is essential in digital holography due to the granular speckle pattern arising from coherent light interference, which degrades contrast and detail in reconstructed images. A widely adopted approach is Wiener filtering, an adaptive method that suppresses multiplicative speckle noise while preserving edges by estimating the signal-to-noise ratio locally. The filtered amplitude is given by

|U_f| = \frac{|U|}{\sqrt{1 + \frac{\sigma^2}{|U|^2}}},

where |U| is the original amplitude, |U_f| is the filtered amplitude, and \sigma^2 is the noise variance. This formulation minimizes mean square error under the assumption of stationary signal and additive white Gaussian noise post-logarithmic transformation of speckle. Evaluations show Wiener filtering reduces speckle contrast by up to 50% in holographic reconstructions compared to spatial averaging, though it may slightly blur fine features.^[31]^[32] Autofocus algorithms automate the determination of the optimal reconstruction distance, crucial for dynamic or unknown-depth samples in digital holography. These methods evaluate focus metrics across a range of propagation distances, selecting the plane that maximizes sharpness. A common phase metric optimization involves maximizing the variance of the intensity gradient in the reconstructed image, which peaks at focus due to enhanced edge definition and reduced blur. This criterion is robust to phase-only objects and computes efficiently via Sobel operators, achieving sub-micron accuracy in holographic microscopy setups with computation times under 1 second for 1 megapixel holograms.^[33] Aberration correction compensates for optical distortions, such as those from imperfect lenses or media, which introduce wavefront errors in the reconstructed hologram. Zernike polynomial fitting models these aberrations as a series expansion over the pupil, with coefficients estimated via least-squares optimization on a background region of the phase map. Low-order terms (e.g., defocus, astigmatism) dominate typical setups, and fitting converges in 5-10 iterations to reduce root-mean-square phase error below \lambda/20. For defocus adjustment, iterative angular spectrum propagation refines the correction by back-propagating the aberrated field and subtracting the estimated phase mask, enabling real-time compensation in microscopy without hardware changes.^[34]^[35] Multi-wavelength processing extends the unambiguity range of phase measurements beyond the single-wavelength limit of \lambda/2, vital for large-scale or discontinuous objects. By recording holograms at two wavelengths \lambda_1 and \lambda_2, a synthetic wavelength is generated as \lambda_\text{synth} = \frac{\lambda_1 \lambda_2}{|\lambda_1 - \lambda_2|}, which can reach millimeters for closely spaced lasers (e.g., 1-2 nm separation). The phase difference yields a coarse map unwrapped hierarchically with the finer single-wavelength data, improving depth resolution to 1-10 \mum over fields of view up to 1 cm. Although this approach extends the unambiguity range of phase measurements, it increases noise sensitivity compared to single-wavelength methods due to amplified phase errors from the larger synthetic wavelength, while maintaining computational simplicity.^[36]

Advanced Techniques

Hologram multiplexing

Hologram multiplexing in digital holography refers to methods that enable the recording of multiple holograms within a single physical medium or exposure, either simultaneously or sequentially, to capture multi-perspective views, multi-parameter data, or enhanced information density. This approach leverages the interference properties of light to overlay holograms without significant crosstalk, facilitating applications in multidimensional imaging. By separating reference beams or recording parameters, multiplexing expands the utility of digital holography beyond single-hologram limitations, often building on off-axis configurations for angular separation of beams.^[37] Spatial multiplexing achieves this by dividing the recording space into distinct regions or using angular or polarization separation of multiple reference beams, allowing simultaneous capture of diverse object fields. In angular multiplexing, reference beams at different angles create unique carrier frequencies in the hologram, enabling demultiplexing through selective filtering in the Fourier domain during numerical reconstruction. For instance, combining on-axis and off-axis reference beams with polarization control prevents unwanted interference, improving resolution by isolating low- and high-frequency components. Polarization multiplexing further enhances separation by encoding orthogonal states, as demonstrated in setups where dual illuminations yield synthesized images with extended field of view. Recent advancements include angular multiplexed holographic optical elements (MHOE) for wide-field digital holography, as demonstrated in 2024 setups.^[38]^[38]^[39]^[40] Temporal multiplexing involves sequential recording of holograms by varying parameters such as optical path length or phase shifts over time, suitable for dynamic or multi-wavelength scenarios. This technique records holograms at different instances, often using temporal frequency-division multiplexing (TFDM) to encode variations like wavelength changes, which are later separated via Fourier transforms along the temporal axis. In color holography, sequential exposures with red, green, and blue sources build full-color reconstructions, with high light efficiency achieved without spatial filters. TFDM has enabled rapid multiwavelength imaging, such as 3D shape measurements of moving objects at frame rates up to 20,000 fps.^[37]^[37]^[41] Wavelength multiplexing superimposes holograms recorded at distinct wavelengths, typically using RGB lasers (e.g., 633 nm red, 532 nm green, 473 nm blue), to capture spectrally diverse information in a single hologram. Reconstruction occurs channel-by-channel via phase-shifting interferometry, where multiple exposures with symmetric phase shifts allow analytical separation of object waves using matrix inversion techniques like LU decomposition. Challenges include chromatic dispersion, which causes wavelength-dependent focusing errors and crosstalk; these are mitigated by removing zero-order terms and optimizing phase shifts to avoid multiples of π, ensuring artifact-free 3D color images. This method supports full space-bandwidth product utilization for high-fidelity multiwavelength holography.^[42]^[42]^[42] In data storage applications, hologram multiplexing enables high-density holographic memory by overlaying numerous data pages in a volume medium, with spatial light modulators (SLMs) modulating input data onto the object beam for efficient recording. Techniques like angular, shift, or wavelength multiplexing store terabyte-scale capacities in crystals such as LiNbO₃, where SLMs generate phase- or amplitude-encoded pages, and demultiplexing retrieves them with low crosstalk. Eigenmode multiplexing with SLMs supports energy-efficient storage with theoretical densities exceeding 1 Tb/cm³ in volume holographic systems, enabling archival applications with long-term stability.^[43]^[44]

Super-resolution techniques

Super-resolution techniques in digital holography aim to surpass the diffraction limit imposed by the numerical aperture of the imaging system and the pixel size of the detector, typically achieving 2-4× improvement beyond the Nyquist limit, though gains are constrained by noise levels and object complexity.^[45] These methods leverage either optical modifications to extend the captured spatial frequency spectrum or computational algorithms to synthesize higher-resolution reconstructions from multiple measurements.^[45] Optical super-resolution approaches, such as structured illumination microscopy (SIM) integrated with digital holography, use spatial light modulators (SLMs) or digital micromirror devices (DMDs) to generate periodic illumination patterns that shift higher spatial frequencies of the object into the detectable passband of the optical system.^[46] By recording multiple holograms under different illumination phases and orientations—typically three to nine frames—the spectra are combined to form a synthetic aperture with an effective numerical aperture up to twice that of conventional setups, enabling resolution enhancements of approximately 2×.^[45] For instance, fringe patterns projected via an SLM allow distinction of features separated by 1.4 μm in biological samples, as demonstrated in early implementations.^[46] Iterative reconstruction algorithms incorporating sparsity constraints further refine these optically extended spectra, promoting solutions where the object is represented in a sparse basis (e.g., wavelets) to suppress artifacts and noise.^[45] Computational super-resolution methods process sequences of holograms captured with sub-pixel shifts, either by mechanical translation of the sample, source, or sensor, to interpolate finer sampling grids. Pixel super-resolution (PSR) techniques, such as the shift-and-add algorithm, align and average multiple low-resolution holograms to yield a high-resolution composite, effectively achieving a numerical aperture of ~0.5 and spatial resolution of ~0.6 μm using 36 (6×6) sub-pixel shifted holograms.^[47] These approaches draw from digital processing tools like registration and optimization to minimize misalignment errors. Deconvolution-based methods adapt the Lucy-Richardson algorithm to holographic reconstructions by iteratively estimating the object from the recorded intensity, using the hologram's point spread function (PSF) as the kernel to reverse blurring effects. The update rule for the estimated object f^{(k+1)} at iteration k is given by:

f^{(k+1)}(x) = f^{(k)}(x) \cdot \left( \frac{g(x)}{h \ast f^{(k)}(x)} \ast h(-x) \right),

where g(x) is the observed hologram intensity, h(x) is the PSF, and \ast denotes convolution; regularization terms, such as entropy, are often added to stabilize convergence in noisy holographic data.^[48] This yields super-resolved phase and amplitude images with up to 2× lateral resolution improvement in single-shot configurations.^[48] Recent developments as of 2024 include deep learning-based super-resolution models for digital holography, providing benchmarked improvements in quantitative phase and intensity reconstruction.^[49] Sparsity-based computational methods employ compressive sensing principles, solving an \ell_1-minimization problem to reconstruct sparse objects from under-sampled holographic measurements, which is particularly effective for lensless or off-axis setups. The optimization typically minimizes \|f\|_1 subject to \|Af - g\|_2 \leq \epsilon, where A is the forward sensing model incorporating holographic propagation, f is the sparse object representation, g is the measurement, and \epsilon accounts for noise; iterative solvers like gradient descent or alternating direction methods yield phase retrieval beyond the detector's pixel limit.^[50] Seminal applications in digital holographic microscopy have extrapolated low-resolution holograms (e.g., 300×300 pixels) to higher ones (1000×1000 pixels), improving resolution from 5.9 μm to 1.8 μm for sparse specimens like cells.^[45] These techniques excel for objects with sparse frequency content but require prior knowledge of sparsity bases and can be computationally intensive.^[50]

Optical sectioning and depth extension

Optical sectioning in digital holography leverages the numerical propagation of the recorded complex field to computationally refocus on specific axial planes, enabling the isolation of in-focus features from out-of-focus blur without mechanical adjustments. This technique exploits the phase information encoded in the hologram to reconstruct the wavefront at various depths, providing a non-invasive method for volumetric imaging. The axial resolution achievable through such propagation is approximated by δz ≈ λ / (n (1 - cos θ)), where λ is the wavelength, n is the refractive index of the medium, and θ represents the illumination angle, which determines the effective numerical aperture for depth discrimination. This resolution is particularly enhanced in off-axis configurations where angular diversity improves the separation of planes. As of 2025, trillion-frame-rate all-optical sectioning enables ultrafast 3D holographic imaging for applications in dynamic processes.^[51] To extend the depth-of-focus beyond the inherent limitations of single-plane reconstruction, multi-plane imaging algorithms simultaneously compute focused images across multiple axial positions by applying shifted diffraction kernels during propagation. Wavefront coding further augments this by introducing a phase mask in the optical path, which encodes depth information into a single hologram, allowing post-processing to yield an extended depth-of-field image with reduced sensitivity to defocus. Synthetic aperture approaches, involving sample or illumination scanning to acquire multiple angular holograms, can also synthesize a larger effective aperture, thereby broadening the axial range while maintaining resolution. These methods collectively enable imaging of extended volumes, such as in biological tissues, by compensating for the shallow depth-of-field typical in high-magnification setups. Holographic optical sectioning advances volumetric reconstruction through tomographic techniques that compile angularly diverse holograms into a 3D model, employing the inverse Radon transform or filtered back-projection to recover the object's refractive index distribution from projections. This process treats each hologram as a 2D projection, integrating them to resolve fine axial structures with resolutions down to sub-micrometer scales in transparent samples. However, challenges arise in thick specimens due to multiple scattering, which distorts the phase information and degrades reconstruction fidelity, often limiting penetration depth to tens of micrometers in turbid media. Recent integrations of light-sheet illumination with digital holography mitigate these issues by confining excitation to thin planes, reducing scattering contributions and extending effective imaging depth to hundreds of micrometers while preserving high axial resolution.^[52]

Applications

In microscopy and interferometry

Digital holographic microscopy (DHM) integrates digital holography with optical microscopy to enable quantitative phase imaging of microscopic samples, providing both amplitude and phase information from a single hologram recording. In DHM, the interference pattern between the object wave scattered from the sample and a reference wave is captured by a digital sensor, allowing numerical reconstruction of the complex wavefront. This approach facilitates non-invasive, label-free visualization of transparent specimens, such as biological cells, by retrieving the optical path length differences that manifest as phase shifts. Common-path setups in DHM enhance system stability by propagating the object and reference beams along the same optical path, minimizing phase disturbances from environmental vibrations and thermal fluctuations. These configurations, such as off-axis interferometers using lateral shearing or point diffraction, achieve phase stability below 1 nm over extended periods, making them ideal for high-precision microscopy. Phase retrieval in DHM typically employs numerical propagation algorithms, often enabled by phase-shifting holography for accurate phase demodulation, to map cell thickness or refractive index variations via the relation \Delta n = \frac{\phi \lambda}{2\pi t}, where \phi is the retrieved phase, \lambda is the wavelength, and t is the cell thickness. For instance, this has been applied to quantify thickness fluctuations in erythrocytes, revealing dynamic changes on the order of 37 nm during live-cell imaging.^[53] In interferometry, digital holography combines with white-light interferometry to extend the measurement range beyond the limitations of monochromatic setups, fusing high-resolution phase data from DH with broadband coherence for absolute thickness profiling. This hybrid approach enables quantitative mapping of thin films with sub-wavelength accuracy over larger depths.^[54] Dynamic phase measurements in DHM further support real-time tracking of live cells, capturing morphological changes without markers. Transmission-mode DHM excels for transparent biological samples due to its simplicity and low aberration, while reflection modes offer higher resolution for opaque materials but introduce trade-offs in signal strength and complexity.^[54] Key advantages of DHM in these domains include its non-destructive nature and ability to perform quantitative analysis without staining, enabling studies of cell dynamics and material integrity. However, challenges persist, particularly speckle noise in biological samples, which arises from coherent illumination and degrades phase accuracy, often requiring optical or numerical suppression techniques like multi-look processing to achieve significant contrast improvements. Resolution in transmission setups is typically limited to the sensor pixel size (several microns laterally), while reflection modes can enhance axial sensitivity but amplify noise from scattering.

In 3D imaging and metrology

Digital holography facilitates 3D imaging by capturing and reconstructing scenes with full parallax, enabling viewers to experience natural depth perception through head motion. Multi-view holography synthesizes holograms from multiple angular perspectives, incorporating both horizontal and vertical parallax to generate immersive 3D representations without the need for special eyewear. This approach leverages computational rendering to combine elemental images into a single hologram, providing continuous viewpoint changes and enhanced visual realism in displays. Numerical reconstruction algorithms in digital holography allow for parallax correction during the display process, compensating for distortions in large-scale holograms to maintain accurate spatial relationships. Techniques such as angular spectrum propagation enable the efficient computation of wavefronts for extended parallax ranges, ensuring high-fidelity 3D visualization even for complex scenes. Hologram multiplexing can briefly support multi-view data integration by encoding multiple perspectives into a single recording medium, though primary reliance is on numerical methods for parallax adjustment.^[55] In metrology, digital holography excels in non-contact 3D surface profiling and deformation measurement, offering sub-wavelength precision for industrial quality assurance. Holographic interferometry maps surface deformations by comparing phase maps from sequential holograms, with a sensitivity of λ/2 per fringe—approximately 316 nm for a 633 nm laser—allowing detection of minute structural changes in materials under load. This full-field technique visualizes out-of-plane displacements across entire surfaces, as demonstrated in analyses of micro-deformations on engineered structures.^[56] Digital speckle pattern interferometry (DSPI), a variant of digital holography, addresses rough or diffuse surfaces by correlating speckle patterns to quantify in-plane and out-of-plane deformations with resolutions from nanometers to micrometers. DSPI enables real-time visualization of defects, vibrations, and mode shapes in non-destructive testing, outperforming traditional sensors for fragile components due to its contactless nature.^[57] Industrial applications of digital holography in 3D metrology span automotive and aerospace sectors, where non-contact profiling ensures precision in high-stakes components like turbine blades and engine parts. Systems achieve sub-micron accuracy over fields of view up to several square centimeters, enabling inline inspection during manufacturing to detect deviations as small as a few micrometers without halting production. For instance, multi-wavelength digital holography integrates into CNC machines for topography mapping of rough surfaces, optimizing processes like milling in automotive assembly.^[58] Recent advances feature portable holographic sensors for on-site metrology, facilitating inspections in field environments without laboratory setups. These compact devices, often based on off-axis digital holography, perform double-exposure interferometry to characterize defects, such as subsurface flaws in automotive tires under mechanical stress, with quantitative fringe analysis matching mechanical caliper measurements. Such portability extends metrology to remote aerospace maintenance, enhancing efficiency in non-destructive evaluations.

In biomedical and flow visualization

Digital holography enables label-free, quantitative phase imaging of dynamic biological processes, such as in vivo cell dynamics, by capturing phase shifts induced by cellular refractive index variations without invasive staining. In studies of sperm motility, partially spatially coherent digital holographic microscopy (PSC-DHM) has been employed to differentiate healthy sperm from those under stress conditions like oxidative damage or cryopreservation, achieving classification accuracies up to 85.6% via integration with deep neural networks. This approach quantifies motility parameters, revealing progressive motility rates of 73.9% in controls versus 2.4% under hydrogen peroxide stress, providing insights into fertility assessment through non-destructive 3D tracking of sperm trajectories.^[59] Similarly, for neural activity, digital holographic imaging detects nanometer-scale tissue deformations associated with cortical firing in vivo, offering sub-millisecond temporal resolution and ~30 μm spatial resolution through intact cranium up to 2 mm depth, as demonstrated in models of electrical stimulation and seizures.^[60] Integration with endoscopy enhances these capabilities, with flexible ultrathin fiber bundle probes enabling holographic 3D imaging of unstained tissues at 0.85 μm lateral resolution and 14 μm axial resolution over depths of 400–1200 μm, facilitating minimally invasive visualization of structures like intestinal villi in confined biomedical environments.^[61] In flow visualization, holographic particle image velocimetry (HPIV) leverages digital holography to reconstruct instantaneous 3D velocity fields in turbulent flows by recording in-line holograms of tracer particles across dual exposures, deriving velocities from particle displacements with spatial resolutions down to 1 mm over fields of view ~100 mm. This technique tracks Lagrangian particles in complex, time-resolved flows, such as free-surface jets, where it matches established velocity profiles while capturing three-component dynamics essential for understanding turbulence structures. Applications extend to multiphase systems, where digital holography quantifies droplet sizes, velocities, and interactions in reactive flows, such as combustion or chemical reactions, by processing holograms to resolve particle positions and morphologies in dense, scattering environments. Post-2020 advances have incorporated artificial intelligence to accelerate hologram analysis in biomedical contexts, with convolutional neural networks like U-Net and GAN-based models enabling super-resolution reconstruction and noise suppression, achieving up to 98.96% accuracy in cancer cell classification from phase images.^[62] In multiphase reactive flows, recent digital holography implementations have improved 3D particle tracking for non-reactive and reactive systems, enhancing velocimetry in sprays and bubbly flows through optimized reconstruction algorithms that handle high particle densities. Key challenges include high-speed recording for transient events, necessitating frame rates exceeding 1.5 kHz to counter phase noise from tissue decorrelation in milliseconds, and mitigating scattering in turbid media, where dynamic imaging requires coherence matching and spatial filtering to maintain contrast in biological or fluid samples obscured by multiple scattering.

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