Source-linked AI summary
Practical algorithms for simulation and reconstruction of digital in-line holograms
Tatiana Latychevskaia, Hans-Werner Fink
TL;DR
The paper addresses practical simulation and reconstruction of in-line digital holograms for plane and spherical waves, including absorption and phase objects. It develops Fourier-based procedures and resolution criteria, showing that plane- and spherical-wave reconstruction procedures coincide under a stated relation, while approximation validity depends on propagation conditions.
Problem
Practical simulation and reconstruction of in-line holograms require suitable propagation, sampling, distance, and resolution criteria across plane- and spherical-wave recordings.
Method
The paper presents numerical reconstruction recipes based on Fourier propagation and evaluates Fresnel, angular-spectrum, and non-paraxial conditions for plane and spherical waves.
Results
The methods reconstruct holograms of absorbing and phase-shifting objects, and plane- and spherical-wave reconstruction procedures are identical when the stated relation is fulfilled.
Takeaways & Limitations
A digital hologram can be assigned a unique parameter defining its reconstruction, while wavelength-independent procedures apply to radiation exhibiting wave nature.
Takeaways & Limitations
The non-paraxial spherical-wave treatment is required when the incident wave spans larger angles because the paraxial approximation no longer applies.
Abstract
from arXiv · showhide
Here we present practical methods for simulation and reconstruction of in-line digital holograms recorded with plane and spherical waves. The algorithms described here are applicable to holographic imaging of an object exhibiting absorption as well as phase shifting properties. Optimal parameters, related to distances, sampling rate, and other factors for successful simulation and reconstruction of holograms are evaluated and criteria for the achievable resolution are worked out. Moreover, we show that the numerical procedures for the reconstruction of holograms recorded with plane and spherical waves are identical under certain conditions. Experimental examples of holograms and their reconstructions are also discussed.
1 Introduction
In-line holography is a simple, widely used arrangement now paired with digital detectors and numerical reconstruction. Digital holography routines commonly employ fast Fourier transforms.
- In-line holography uses a conceptually simple design without optical elements between the sample and detector.
- The approach has been used with light, electrons, X-rays, and other wave types.
- Digital holography records holograms with digital detectors and reconstructs them numerically.
- Digital holography simulation and reconstruction routines employ fast Fourier transforms.
2 Hologram Formation and Reconstruction
In-line holograms arise from interference between reference and object waves sharing an optical axis. Reconstruction normalizes the recorded intensity, propagates the hologram numerically, and recovers absorption and phase information.
- In-line holography records interference between an unscattered reference wave and an object wave scattered beyond the object.
- The recorded hologram contains reference intensity, object-wave intensity, and two cross terms that produce the observed interference pattern.
- Before reconstruction, the hologram is normalized using a background image recorded under identical conditions without the object.
- Subtracting one after normalization removes the constant background and reduces folding-fringe effects near hologram edges.
- Reconstruction multiplies the normalized hologram by the reference wave and back-propagates it using the Fresnel-Kirchhoff diffraction integral.
- The reconstructed perturbation is combined with one to obtain the transmission function, from which absorption and phase distributions are extracted.
3 In-line Holography with Plane Waves
Plane-wave hologram simulation and reconstruction use Fourier-based propagation, with Fresnel, convolution, and angular-spectrum formulations selected according to distance and approximation conditions. Resolution is governed by detected spatial frequencies and numerical aperture.
- Plane-wave propagation: Plane-wave holograms are simulated by propagating an object transmission function from the object plane to the detector using Fresnel-Kirchhoff diffraction.
- Large z-distance, Fresnel Approximation: For sufficiently large z-distance, the Fresnel approximation supports Fourier-transform-based simulation and reconstruction procedures.
- Large z-distance, Fresnel Approximation: The Fresnel propagation can be represented as convolution with a Fresnel function whose Fourier transform is calculated directly for correct sampling.
- Large z-distance, Fresnel Approximation: At very large distances, the Fraunhofer condition makes the scattered wave a Fourier transform of the object transmission function.
- Angular Spectrum Method: The angular spectrum method avoids propagation approximations and is valid within the classical resolution limit.
- Angular Spectrum Method: The angular-spectrum simulation and reconstruction use Fourier transforms, multiplication by a propagation factor, and inverse transforms to obtain the object perturbation.
- Resolution in In-line Holography with Plane Waves: Lateral resolution can be evaluated from the highest visible spatial frequency in a hologram’s Fourier spectrum.
- Resolution in In-line Holography with Plane Waves: Axial resolution is estimated as a depth of focus determined by the system’s numerical aperture.
4 In-line Holography with Spherical Waves
The spherical-wave section develops two-step Fourier-transform procedures for simulating and reconstructing in-line holograms, including non-paraxial propagation through the source plane. It specifies sampling, magnification, and resolution considerations for practical digital implementation.
- Wave propagation: Spherical-wave holograms are modeled by propagating the field from the source to the object and then from the object toward the detector using Fresnel-Kirchhoff diffraction.The incident and exit waves are defined relative to source, object, and detector coordinates.
- Numerical simulation: A single-Fourier-transform formulation is replaced by a two-step routine through the source plane to improve digital wave-propagation sampling.The two stages use an inverse Fourier transform and a Fourier transform, with spherical phase factors and coordinate transformations.
- Numerical simulation: Simulation applies a Fourier transform to the object transmission function, multiplies by a simulated propagation factor, inverse-transforms the product, and takes its squared magnitude.The simulated hologram size is the object-area size multiplied by the magnification factor.
- Numerical reconstruction: Reconstruction reverses the simulation sequence: the normalized hologram is inverse-transformed, multiplied by a conjugate spherical phase factor, and Fourier-transformed to obtain the object transmission perturbation.The reconstructed object area is the hologram size divided by the magnification factor.
- Sampling and resolution: For spherical-wave holography, hologram-plane pixels equal object-plane pixels multiplied by magnification, while practical resolution can be estimated from the highest observed Fourier frequency.The resolution expression uses the detected highest-frequency pixel, hologram size, and magnification factor.
5 Relationship between Holograms Recorded with Plane Respectively Spherical Waves
The paper identifies conditions under which holograms recorded with spherical and plane waves can use equivalent numerical reconstruction procedures. A dimensionless parameter α can characterize holograms across changes in wavelength, screen size, or source-detector distance.
- Algorithmic relationship: Plane- and spherical-wave reconstruction algorithms share inverse Fourier transformation, spherical-phase multiplication, and Fourier transformation.The paper states that the reconstruction steps are similar regardless of wavefront shape.
- Algorithmic relationship: A spherical-wave hologram can be reconstructed as a plane-wave hologram, or vice versa, when the specified sampling relation between the two representations is fulfilled.The equivalence follows from equality of the relevant phase terms under the stated condition.
- Parameterization: Holograms recorded with different wavelengths, screen sizes, or source-detector distances can be uniquely reconstructed when α remains constant.This statement applies to a thin object in one plane or reconstruction at a selected plane within an object distribution.
- Experimental example: The parameter α = 4.046 · 10^-3 is assigned to the tungsten-tip hologram illustrated in Figure 2.The figure records the hologram with λ = 532 nm, Z = 1060 mm, and z = 1.4 mm.
6 Optimal Parameters
The optimal-parameter section derives a sampling condition for the spherical phase term. This condition is used to select experimental parameters for correctly sampled numerical reconstruction.
- Sampling condition: Correct sampling of the spherical phase term requires satisfying the derived condition at Nyquist or higher frequency.The sampling variables are related to pixel numbers and the digital Fourier-domain sampling interval.
- Parameter selection: The derived sampling condition allows selection of optimal experimental parameters.The paper presents this as the practical consequence of the sampling analysis.
7 Conclusions
The paper presents wavelength-independent numerical recipes for reconstructing in-line holograms recorded with plane or spherical waves. It also assigns each hologram a unique reconstruction parameter and supports objects with absorption and phase-shifting properties.
- Conclusions: The proposed reconstruction methods apply to in-line holograms recorded with plane and spherical waves and can handle absorbing or phase-shifting objects.The methods are described as simple numerical recipes and wavelength-independent.
- Conclusions: Each digital hologram can be assigned a uniquely defined parameter that determines its digital reconstruction.The conclusion summarizes the paper's parameterization result.
A Apodization Cosine Filter
The hologram is apodized with a cosine filter that reduces edge intensity to zero, minimizing edge effects introduced by digital Fourier transformation.
- A Apodization Cosine Filter: The normalized hologram is multiplied by an apodization cosine filter before reconstruction.The filter smooths intensity at the hologram edges down to zero.
- A Apodization Cosine Filter: Apodization minimizes edge effects and folding fringes caused by digital Fourier transformation.Subtracting the background contribution also leaves the interference term approaching zero near hologram edges.
- A Apodization Cosine Filter: The cosine filter is defined piecewise by radial distance from the image center, with a transition region between inner and outer thresholds.Its intensity and central amplitude profile are illustrated in Figure 3.
B From Analytical to Fast Fourier Transform (FFT)
The analytical Fourier transform is converted into an FFT procedure by specifying centered sampling in both domains and imposing a phase-consistency condition.
- B From Analytical to Fast Fourier Transform (FFT): The analytical Fourier transform connects the spatial and Fourier domains and provides the basis for the centered transform formula.The inverse transform is derived analogously.
- B From Analytical to Fast Fourier Transform (FFT): The fast Fourier transform is expressed using pixel indices m, n, p, and q.These indices identify sampled pixels in the two domains.
- B From Analytical to Fast Fourier Transform (FFT): The input distribution is sampled at centered points in (x, y) coordinates, with corresponding samples defined in the Fourier domain.The sampling intervals Δ1 and Δ2 depend on the sizes of the related domains.
- B From Analytical to Fast Fourier Transform (FFT): The discrete phase term matches the analytical phase term when the stated sampling condition is fulfilled.This condition links the discrete FFT representation to the centered analytical transform.
C κ-coordinates
The κ-coordinate transformation remaps hologram data from detector coordinates using the system’s numerical aperture and can be reversed similarly.
- C κ-coordinates: The maximum κx value is determined from the emission-vector coordinates.The detector size S × S determines the pixel size in κ-coordinates.
- C κ-coordinates: The transformation constructs κ-coordinate arrays and assigns each κ pixel a value from a corresponding detector-coordinate pixel.This provides the discrete remapping from H(X, Y) to κ-space.
- C κ-coordinates: The transformation depends only on numerical aperture, determined by source-to-detector distance and detector size.The resulting appearance resembles a fish-eye effect, and the reverse transformation is performed similarly.