# Fourier Transform Holography: A Lensless Imaging Technique, Its Principles and Applications

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## Abstract

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## 1. Introduction

#### 1.1. Phase Problem

#### 1.2. Fourier Transform Holography (FTH)

## 2. Principles of FTH

#### 2.1. Hologram Formation

#### 2.2. Reconstruction of the Sample Distribution from the Hologram

#### 2.3. Optimal Parameters

#### 2.3.1. Distance between the Object and the Reference

#### 2.3.2. The Reconstructed Field of View

#### 2.3.3. The Detector Parameters

#### 2.3.4. Reference Size and Resolution

## 3. Types of References

#### 3.1. Point-like References

#### 3.1.1. Single Point-like Reference

**Figure 3.**The first experimental realisation of Fourier transform holography (FTH) by Stroke in 1965. (

**a**) Schematics of the light-optical experiment and (

**b**) reconstruction of the sample. The recording and reconstruction were performed using a 632.8 Å laser light. P

_{n}are the points of the object, θ is the incident angle, Σ

_{Pn}and Σ

_{R}are the two spherical wavefronts, z is the optical axis, and ξ and x are the axes in the object and hologram planes, respectively. Reprinted from G. W. Stroke, “Lensless Fourier-transform method for optical holography”, Applied Physics Letters 6 (10), 201–203 (1965) [11], with the permission of AIP Publishing.

#### 3.1.2. Multiple Point-like References

#### 3.2. Extended References

#### 3.2.1. References in the Form of Geometrical Shapes

**Figure 6.**Holography with extended reference by autocorrelation linear differential operation (HERALDO) light-optical experimental results; photograph of the “cameraman” was printed on a slide. (

**a**) Amplitude distribution of the sample: object with a thin-slit reference. (

**b**) Measured intensity pattern (Fourier transform hologram). (

**c**) Reconstruction obtained by calculating the inverse Fourier transform of the product $i2\pi {K}_{y}{I}_{\mathrm{H}}({K}_{x},{K}_{y})$ and (

**d**) the magnified region of (

**c**) showing the upper reconstruction. Adapted with permission from M. Guizar-Sicairos and J. R. Fienup, “Direct image reconstruction from a Fourier intensity pattern using HERALDO”, Optics Letters 33 (22), 2668–2670 (2008) [77], © The Optical Society.

#### 3.2.2. Uniformly Redundant Array (URA) Reference

#### 3.2.3. Arbitrary Extended Reference

#### 3.2.3.1. Reconstruction by Deconvolution

#### 3.2.3.2. Reconstruction via System of Linear Equations

## 4. FTH with Different Types of Waves

#### 4.1. Light

#### 4.2. Electrons

#### 4.3. X-rays and Extreme Ultraviolet

## 5. X-ray FTH Applications

#### 5.1. Imaging of Magnetic Domains—Spectro-Holography

#### 5.2. Time-Resolved Imaging

#### 5.3. Biological Imaging

#### 5.4. Three-Dimensional Imaging

## 6. FTH and Other Coherent Imaging Techniques

**Figure 18.**Comparison between reconstructions obtained from experimental light optical Fourier transform (FT) and Gabor-type holograms. (

**a**) Direct optical image of the object (an Australian gnat placed on a rectangular based pyramid) obtained using a He–Ne laser source. (

**b**) Arrangement for Fourier transform holography (FTH). (

**c**) Magnitude of the reconstructed distribution obtained from the FT hologram. (

**d**) Arrangement for Gabor in-line holography. (

**e**) Image of the object reconstructed from the Gabor in-line hologram. Scale bar in (

**a**,

**e**) is 200 μm. Reprinted from S. G. Podorov, A. I. Bishop, D. M. Paganin, K. M. Pavlov, “Mask-assisted deterministic phase-amplitude retrieval from a single far-field intensity diffraction pattern: Two experimental proofs of principle using visible light”, Ultramicroscopy 111 (7), 782–787 (2011) [75], with permission from Elsevier.

**Figure 19.**Schematic diagram of the X-ray holographic microscopy setup, consisting of a holography mask support, a movable object (sample) support, and a charge coupled device (CCD) detector. The membrane with the optical elements (mask), i.e., the object and reference holes, is fixed in the centre of the X-ray beam. A second membrane, which provides support for the object, can be moved freely in a plane perpendicular to the beam. Reprinted from D. Stickler, R. Fromter, H. Stillrich, C. Menk, C. Tieg, S. Streit-Nierobisch, M. Sprung, C. Gutt, L. M. Stadler, O. Leupold, G. Grubel, and H. P. Oepen, Applied Physics Letters 96 (4), 042501 (2010) [51], with the permission of AIP Publishing.

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Principle of Fourier transform holography (FTH) with point-like reference. (

**a**) Scheme for Fourier hologram acquisition. (

**b**) Sample distribution: object and reference aperture. D is the extent (diameter) of the object, and L is the distance between the object and the reference. (

**c**) Diffraction pattern of the reference alone. (

**d**) Diffraction pattern of the object and the reference together—Fourier transform hologram. (

**e**) Sample distribution (amplitude) reconstructed by calculating the Fourier transform of the Fourier transform hologram, exhibiting two centro-symmetric reconstructions at the reference position.

**Figure 2.**Overview of the different types of references for Fourier transform holography: (

**a**) single point-like reference, (

**b**) multiple point-like references, (

**c**) object inside a squared-aperture support, (

**d**) extended reference, (

**e**) uniformly redundant array (URA) reference, and (

**f**) arbitrary reference. The sizes and positions of the references are not to scale for purposes of presentation.

**Figure 4.**Fourier transform holography with multiple references. (

**a**) Scanning electron micrograph (SEM) of the sample. (

**b**) Diffraction pattern acquired with coherent soft X-rays (λ = 1.58 nm, E = 780 eV). Enlargement shows details of the interference pattern characteristics for the five reference sources. (

**c**) The reconstruction obtained by calculating the Fourier transform of the recorded hologram shown in (

**b**). Reprinted from W. F. Schlotter, R. Rick, K. Chen, A. Scherz, J. Stohr, J. Luning, S. Eisebitt, C. Gunther, W. Eberhardt, O. Hellwig, I. McNulty, “Multiple reference Fourier transform holography with soft x rays”, Applied Physics Letters 89 (16), 163112 (2006) [40], with the permission of AIP Publishing.

**Figure 5.**Fourier transform holography with an object placed inside a squared-aperture support. (

**a**) Object distribution. (

**b**) Distribution of the sample, where the object is placed inside a squared-aperture support that is at least twice as large as the object. (

**c**) Diffraction pattern (Fourier transform hologram) of the sample shown in (

**b**). (

**d**) Reconstructed distribution.

**Figure 7.**Fourier transform holography with uniformly redundant array (URA). (

**a**) Experimental arrangement and the steps of the reconstruction procedure. (

**b**) Scanning electron micrograph (SEM) of the sample, exhibiting the structure of the URA; scalebar is 2 µm. Reproduced from S. Marchesini, S. Boutet, A. E. Sakdinawat, M. J. Bogan, S. Bajt, A. Barty, H. N. Chapman, M. Frank, S. P. Hau-Riege, A. Szoke, C. W. Cui, D. A. Shapiro, M. R. Howells, J. C. H. Spence, J. W. Shaevitz, J. Y. Lee, J. Hajdu, M. M. Seibert, “Massively parallel X-ray holography”, Nature Photonics 2 (9), 560–563 (2008) [91], with permission from Springer Nature.

**Figure 8.**Fourier transform holography with arbitrary references. (

**a**) Reconstruction of a cluster of gold balls from soft X-ray transmission diffraction patterns using deconvolution and a Wiener filter; one fast Fourier transform (FFT) was used. Constant $\mathrm{\Phi}=100$ in Equation (23) produced the best results. (

**b**) Scanning electron microscopy image of the same cluster (upper one), the reference (lower one), and their relative positions. (

**c**) Reconstruction obtained by using an iterative algorithm to reduce the effect of missing data due to a beam stop, 10 times FFT used. The diameter of each gold ball is about 50 nm. Reprinted from H. He, U. Weierstall, J. C. H. Spence, M. Howells, H. A. Padmore, S. Marchesini, H. N.Chapman, “Use of extended and prepared reference objects in experimental Fourier transform X-ray holography”, Applied Physics Letters 85 (13), 2454–2456 (2004) [98], with the permission of AIP Publishing.

**Figure 9.**Holography with arbitrary references. (

**a**) The scanning electron microscopy (SEM) image of the sample. (

**b**) The corresponding diffraction pattern with central shaded areas indicating regions that have been set to zero for the reconstruction. (

**c**) The reference function estimated from the SEM image along with the assumed object area indicated by the white rectangle. (

**d**) The object intensity reconstructed with the iterative linear retrieval using Fourier transforms (ILRUFT) method. The scale bars in (

**a**), (

**c**), and (

**d**) are 500 nm and the scale bar in (

**b**) is 0.1 µm

^{−1}. Reproduced from A. V. Martin, A. J. D’Alfonso, F. Wang, R. Bean, F. Capotondi, R. A. Kirian, E. Pedersoli, L. Raimondi, F. Stellato, C. H. Yoon, H. N. Chapman, “X-ray holography with a customizable reference”, Nature Communications 5, 4661 (2014) [92], with permission from Springer Nature.

**Figure 10.**Fourier transform holography using light in an optical laboratory. (

**a**) The detector part of the experimental setup. The collector lens C2 is placed downstream from the sample S. The charge-coupled device (CCD) detector D is placed at the back focal plane of the lens. The incident beam is focused on the CCD and the far-field diffraction pattern is formed in the CCD plane. (

**b**) The recorded Fourier transform hologram, sampled with 489 × 508 pixels at a pixel size of 9.6 × 7.5 µm

^{2}. (

**c**) The autocorrelation of the sample distribution—an insect wing and a small additional scatterer (the presence of the small scatterer was not intentional), obtained using the Fourier transform of the hologram. The scalebar is 1 mm. Reproduced from P. Thibault, I. C. Rankenburg, “Optical diffraction microscopy in a teaching laboratory”, American Journal of Physics 75 (9), 827–832 (2007) [16], with the permission of the American Association of Physics Teachers.

**Figure 11.**Three-dimensional coloured Fourier transform holography using light. Left: experimental scheme. M—mirror; BE—beam-expander; and BS—beam-splitter. Right: reconstructions of the hologram recorded with (

**a**) λ

_{1}= 632.8 nm, (

**b**) λ

_{2}= 532 nm, and (

**c**) λ

_{3}= 473 nm. (

**d**) Fused colour reconstructed object. (

**e**) Part magnification of (

**d**). (

**f**) Photograph of the coloured object used in the experiment. Adapted from J. L. Zhao, H. Z. Jiang, J. L. Di, “Recording and reconstruction of a color holographic image by using digital lensless Fourier transform holography”, Optics Express 16 (4), 2514–2519 (2008) [101], © 2008 Optica Publishing Group.

**Figure 12.**Imaging phase vortices in an electron beam using Fourier transform holography. (

**a**) Experimental arrangement. (

**b**) Fork-shaped pattern (grating) in a 150-nm-thick Si

_{3}N

_{4}membrane with an intensity transmittance of about 44%. (

**c**) Fork-shaped grating and the irradiation area of about 10 times the grating size for this electron holography. (

**d**) Defocused small-angle electron diffraction (SmAED) pattern as exhibiting electron holograms of the fork-shaped grating. Four holograms on both sides of the optical axis are recorded in a single interferogram. The upper-left inset is the FT of the defocused SmAED pattern and the lower-right inset represents the enlarged interference pattern inside the ring-shaped spot of the right-hand side. (

**e**,

**f**) are the amplitude and phase distributions, respectively, of the vortex beams reconstructed from the first-order SmAED ring-shaped spots. Adapted from K. Harada, K. Shimada, Y. A. Ono, “Electron holography for vortex beams”, Applied Physics Express 13 (3), 032003 (2020), doi: 10.35848/1882-0786/ab7059 [111].

**Figure 13.**Fourier transform holography with soft X-rays. A coherent X-ray beam illuminates the zone plate (shown in profile). The undeviated wave illuminates the specimen, and the focused wave serves as a reference source. A charge-coupled device (CCD) detector records the interference between the specimen and reference waves. The X-rays that are not scattered by the specimen are blocked by the beam stop to prevent the saturation of the CCD. From I. McNulty, J. Kirz, C. Jacobsen, E. H. Anderson, M. R. Howells, D. P. Kern, “High-resolution imaging by Fourier-transform X-ray holography”, Science 256 (5059), 1009–1012 (1992) [20]; Reprinted with permission from AAAS.

**Figure 14.**Fourier transform holography of a random magnetic domain structure in a Co–Pt multilayer film using soft X-rays (λ = 1.59 nm, E = 778 eV). (

**a**) Scheme of the experimental setup. Monochromatised and circular polarised X-rays are illuminating the mask–sample structure after spatial coherence filtering. The object and reference beam are defined by the mask, and the resulting hologram is recorded on a CCD detector. The lower inset shows the geometry and a scanning electron microscopy (SEM) image of the sample structure. The scale bar in the SEM image is 2.0 µm. The top inset shows a scanning transmission X-ray microscopy (STXM) image of the magnetic structure illuminated through the sample aperture. The field of view is 1.5 µm. (

**b**) Hologram recorded with right circular polarisation X-rays. Intensity is represented in a logarithmic grey scale. (

**c**) Two-dimensional fast Fourier transformation of the hologram in (

**b**). (

**d**) Zoomed-in image, obtained by subtracting the Fourier transformations of opposite-helicity holograms, showing the magnetisation map of the sample; the diameter of the shown area is 1.5 µm. Reproduced from S. Eisebitt, J. Lüning, W. F. Schlotter, M. Lorgen, O. Hellwig, W. Eberhardt, J. Stohr, “Lensless imaging of magnetic nanostructures by X-ray spectro-holography”, Nature 432 (7019), 885–888 (2004) [22], with permission from Springer Nature.

**Figure 15.**Ultrafast high-harmonic nanoscopy. (

**a**) A magnetic sample (Co–Pd multilayers) is excited with a femtosecond laser pulse and probed with a circularly polarised high-harmonic pulse (with a wavelength of 21 nm) over a variable time delay. A quantitative real-space image is reconstructed from the diffraction pattern of the high-harmonic beam for each time delay between the pump and probe pulses. (

**b**) The ratio of images obtained from opposite high-harmonic generation helicities provides a map of the magnetic contrast isolated from nonmagnetic contributions, i.e., femtosecond snapshot of the spin structures at a given time delay between pump and probe pulses. (

**c**) Plot of the spatially averaged, normalised magnetisation within the field of view as a function of time delay. Adapted from S. Zayko, O. Kfir, M. Heigl, M. Lohmann, M. Sivis, M. Albrecht, M. C. Ropers, “Ultrafast high-harmonic nanoscopy of magnetization dynamics”, Nature Communications 12 (1), 6337 (2021) [125].

**Figure 16.**Femtosecond X-ray Fourier holography imaging of free-flying nanoparticles. The reconstruction of the mimivirus from the hologram is shown in the bottom right corner. Reproduced from T. Gorkhover, A. Ulmer, K. Ferguson, M. Bucher, F. Maia, J. Bielecki, T. Ekeberg, M. F. Hantke, B. J. Daurer, C. Nettelblad, J. Andreasson, A. Barty, P. Bruza, S. Carron, D. Hasse, J. Krzywinski, D. S. D. Larsson, A. Morgan, K. Muhlig, M. Muller, K. Okamoto, A. Pietrini, D. Rupp, M. Sauppe, G. van der Schot, M. Seibert, J. A. Sellberg, M. Svenda, M. Swiggers, N. Timneanu, D. Westphal, G. Williams, A. Zani, H. N. Chapman, G. Faigel, T. Moller, J. Hajdu, and C. Bostedt, “Femtosecond X-ray Fourier holography imaging of free-flying nanoparticles”, Nature Photonics 12 (3), 150–155 (2018) [68], with permission from Springer Nature.

**Figure 17.**Extracting 3D information from a single X-ray Fourier transform hologram. (

**a**,

**b**) Scanning electron microscopy image of the test sample made from 3D platinum structures. The side view of the sample (

**a**) shows the structured ramp which extends above the object hole. The ramp is deposited with a 45° inclination on a gold mask. The small pinhole seen on the upper left represents the reference source, whereas the large circular aperture seen at the bottom right constitutes the object’s aperture. The top view (

**b**) shows different platinum structures deposited on a Si

_{3}N

_{4}membrane at the bottom of the object aperture. (

**c**) Real part of the reconstructed object wave field corresponding to different longitudinal displacements from the substrate. The reconstruction on the left corresponds to the mask plane. The reconstructions shown in the middle and on the right are obtained by numerical propagation of the reconstructed wave field upstream by 6 μm and 9 μm, respectively. Focused features are indicated by the black arrows. Adapted from J. Geilhufe, C. Tieg, B. Pfau, C. M. Gunther, E. Guehrs, S. Schaffert, and S. Eisebitt, “Extracting depth information of 3-dimensional structures from a single-view X-ray Fourier-transform hologram”, Optics Express 22 (21), 24959 – 24969 (2014) [32], © 2014 Optica Publishing Group.

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**MDPI and ACS Style**

Mustafi, S.; Latychevskaia, T.
Fourier Transform Holography: A Lensless Imaging Technique, Its Principles and Applications. *Photonics* **2023**, *10*, 153.
https://doi.org/10.3390/photonics10020153

**AMA Style**

Mustafi S, Latychevskaia T.
Fourier Transform Holography: A Lensless Imaging Technique, Its Principles and Applications. *Photonics*. 2023; 10(2):153.
https://doi.org/10.3390/photonics10020153

**Chicago/Turabian Style**

Mustafi, Sara, and Tatiana Latychevskaia.
2023. "Fourier Transform Holography: A Lensless Imaging Technique, Its Principles and Applications" *Photonics* 10, no. 2: 153.
https://doi.org/10.3390/photonics10020153