Fresnel Incoherent Correlation Holography (FINCH)

• FINCH is a self-interference-based digital holography technique that splits each object's light into two modulated beams to encode phase and amplitude information for 3D imaging.
• It achieves up to 1.5 times higher lateral resolution than conventional direct imaging while extending depth of focus, crucial for biomedical optics and fluorescence microscopy.
• Recent implementations, including single-shot FINCH and PEAR-FINCH, employ advanced computational methods to enhance imaging speed and improve axial resolution.

Fresnel Incoherent Correlation Holography (FINCH) is an advanced self-interference-based digital holography method for three-dimensional imaging of objects illuminated by spatially incoherent light. Utilizing optical modulation and computational propagation, FINCH creates a hologram encoding phase and amplitude information by splitting the light from each object point into two differently modulated beams, which subsequently interfere on a detector. FINCH achieves notably higher lateral resolution and extended depth of focus compared to traditional direct imaging, making it suitable for applications requiring precise, volumetric imaging, especially in fields such as biomedical optics and fluorescence microscopy.

1. Core Principles of FINCH

FINCH operates by introducing a controlled phase modulation to the light emitted by an incoherent object such that self-interference is created without a true reference beam. Typically, a diffractive lens (either on an SLM or as a fixed optical element) splits each object point’s wavefront into two beams with different curvatures. These two beams traverse slightly different optical paths determined by lens focal properties and interfere at the detector array, producing a single intensity hologram that encodes complex field information.

Mathematically, for each object point, the detected intensity at a given transverse coordinate $\mathbf{u}$ can be modeled as: $$I(\mathbf{u}) = |\mathcal{F} \{ T(\mathbf{r}) e^{i\phi(\mathbf{r})}\}|^2,$$ where $T(\mathbf{r})$ is the object transmittance and $\phi(\mathbf{r})$ is the imposed random or quadratic phase (Singh, 2017).

To eliminate twin-image and bias terms and reconstruct the complex hologram, at least three phase-shifted intensity measurements are customarily taken using different modulations. Reconstruction is performed by Fresnel back-propagation computed numerically, yielding the object’s three-dimensional image.

2. Resolution Criteria and Imaging Advantages

In FINCH, lateral and axial resolutions are set by both the optical system’s numerical aperture (NA) and the digital reconstruction algorithms. Standard formulas, valid in both far-field (Fraunhofer) and near-field (Fresnel) regimes, are:

• Lateral resolution (Rayleigh criterion):

$$R_\text{lateral} = 0.61 \frac{\lambda}{\mathrm{NA}}$$

or by spectrum-based analysis,

$$R_\text{lateral}^{(\text{DP})} = \frac{\lambda}{\mathrm{NA}}$$

• Axial resolution:

$R_\text{axial} = 2\frac{\lambda}{\mathrm{NA}^2}$

where $\lambda$ is the wavelength.

FINCH can achieve a lateral resolution up to 1.5 times higher than that of direct imaging systems with the same NA, while the axial resolution is typically poorer but improvable via computational techniques (Latychevskaia, 2019, Arockiaraj et al., 2023). Numerical reconstruction bandwidth and pixel size also affect final resolution.

3. Implementations and Recent Developments

Single-Shot FINCH

Traditional FINCH requires multi-shot phase-shifting for complete hologram synthesis. Recent approaches, such as the use of randomly multiplexed bifocal binary Fresnel zone lenses (RMBDL), realize FINCH in a single camera shot, significantly increasing temporal resolution and enabling imaging of dynamic phenomena (Anand et al., 2019). The RMBDL element splits the object beam into collimated and focused components which self-interfere, with the resulting intensity hologram containing full reconstruction information.

Post-Processing Axial Resolution Engineering (“PEAR-FINCH”)

PEAR-FINCH is an advanced method for post-recording engineering of axial resolution and depth of focus. By recording a library of FINCH holograms with varied diffractive lens focal lengths, each encoding different axial responses and DOFs, one can synthesize a “Synthetic Object Hologram” (SOH) by combining selected holograms. The SOH enables customizable DOF, achieved in post-processing rather than being limited by a single optical configuration. Reconstruction proceeds in two steps:

1. Back-propagation of SOH using the Fresnel kernel:

$$IR_1 = \left| H_{SOH} \otimes Q\left(\frac{1}{z_r}\right) \right|$$

1. Deconvolution with a “Synthetic Point Spread Hologram” (SPSH), further cross-correlated using algorithms such as the Lucy-Richardson-Rosen algorithm (LRRA):

$$IR_2 = \left| \left( H_{SOH} \otimes Q\left(\frac{1}{z_r}\right) \right) \otimes \left( H_{SPSH} \otimes Q\left(\frac{1}{z_r}\right) \right) \right|$$

This produces superior axial response for multiplane objects (Gopinath et al., 30 Sep 2025).

Coded Aperture Imaging (CAI) Framework

FINCH optimized within CAI incorporates computational phase mask synthesis (TAP-GSA) and advanced deconvolution (LR2A) to improve light throughput, SNR, and axial resolution, approaching those of direct imaging while retaining super-resolution capabilities (Arockiaraj et al., 2023). PSF engineering avoids resolution loss due to finite pinhole size in training, allowing the PSF to reflect only the NA limit rather than the physical aperture.

4. Computational and Mathematical Framework

The FINCH reconstruction uses ensemble or space averaging to extract spatial correlations in the recorded hologram. Output intensity fluctuations are calculated, and the cross-covariance provides access to the object’s spatial structure: $$\langle \Delta I_o(0)\Delta I_c(u) \rangle = \left| \int T(r) \exp[i k_z(r)] \exp[-i2\pi u\cdot r] dr \right|$$ (Singh, 2017).

Angular spectrum or Fresnel propagation kernels govern numerical back-propagation. The phase-space treatment (Wigner distribution formalism) predicts relationships and stationarity properties essential for reconstruction efficiency (Bhalavi et al., 2021).

5. Practical Applications and System Configurations

FINCH is widely implemented for three-dimensional imaging, non-destructive testing, phase object visualization, fluorescence microscopy, and tissue morphology studies. Its high lateral resolution and extended DOF enable imaging of fine structure and multiplane features not simultaneously accessible in conventional systems.

Compact single-shot configurations utilizing geometric phase lenses (GP lens) and space-division multiplexed polarization cameras provide fast, robust, and cost-effective FINCH systems (Liang et al., 2018). Lensless, interferenceless variants using coded phase masks further reduce setup complexity, eliminate mechanical calibration, and facilitate multiplane imaging (Kumar et al., 2017).

Hybrid correlation holography with single-pixel detectors enables background-free imaging through intensity cross-covariance, suitable for low-light or scattering environments. The ability to reconstruct vectorial/polarized objects extends FINCH’s capabilities to polarization-sensitive modalities (Singh, 2017).

6. Comparative Evaluation and Limitations

FINCH’s inherent super-resolution and extended DOF come with trade-offs primarily in axial resolution, light throughput, and sensitivity to twin image/bias artifacts. Standard FINCH’s need for multiple phase-shifted exposures increases data acquisition time and sensitivity to environmental disturbances, mitigated by recent single-shot and computational innovations (Anand et al., 2019, Arockiaraj et al., 2023).

Lensless interferenceless approaches (e.g., LI-COACH) forgo two-wave interference, offering resilience to vibration and alignment errors, with comparable resolution to lens-based imaging when a coded phase mask and PSH library are engineered (Kumar et al., 2017).

Limitations remain: calibration of PSH libraries is needed; SLM characteristics (pixel pitch, active area) constrain field-of-view; and multi-step reconstructions may reduce SNR if not properly deconvolved.

7. Future Directions and Research Trends

Research is focused on improving axial resolution, light throughput, and SNR, with computational post-processing (PEAR-FINCH) and sophisticated phase mask engineering (TAP-GSA, LR2A) making real-time, super-resolved, multiplane imaging feasible. Further extensions include combining FINCH with coded aperture schemes for quantitative phase imaging and large field-of-view acquisition, benefitting biological tissue examination and dynamic sample imaging.

Innovations in structured illumination, single-pixel detection, and machine learning-based reconstructions are being explored to further enhance robustness, simplicity, and imaging quality (Mandal et al., 2021). Continued progress in phase-space modeling and digital propagation algorithms is expected to yield even more precise, customizable, and computationally efficient FINCH implementations suited for demanding scientific and industrial imaging applications.