# Linearity and Optimum-Sampling in Photon-Counting Digital Holographic Microscopy

^{1}

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## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Optimum Sampling Conditions

_{N}= 2). Changing f in simulations means that the pixel size of the detector is changing. From Figure 5, it can be seen that the highest beating amplitude occurs when the input signal and the detector’s pixel are in phase. Also, note that some of the patterns depicted in Figure 5 resemble patterns in the recorded hologram, shown in Figure 1. Of course, with the increase of the sampling rate, the beatings fade out. Figure 6 shows a decrease in the beating effect for higher frequencies ($f\ge 2.5{f}_{N}$).

#### 2.2. Experimental Setup

^{2}, pixel size of 7.4 μm and full well capacity is 40,000 electrons. The PCD is a homemade one, based on a single-photon avalanche photodiode (SPAD) (model SAP500, LC) selected for low noise, cooled to −23 °C, and operated in Geiger mode. The detector features a maximum count rate of 10 Mcps and a dark count rate of 21 cps; for more details, see [27]. A small aperture was placed before the SPAD to increase the resolution. We used apertures of 1 μm and 2 μm which determined the resolution. Hologram recordings are made with the PCD during motorized movement of the translation stage. Thus, the pixel size is defined by the product of the speed at which the PCD moves and the integration time convoluted with the aperture size. Each line of a hologram was taken separately, always in the same direction, and during the transition to the next line, the recording was stopped. The maximum size of holograms taken with moving PCD was 3000 × 3000 pixels, and the recording time was 6 h. For smaller sizes, the recording time was shorter. The CCD camera recordings are used for comparison.

## 3. Results

#### 3.1. Nonlinear Recording

#### 3.2. Thin Metallic Filmbuckling

_{x}, assumed to be the same for every line, is deduced from simple Fourier analysis. Then each line j is fitted separately one after another with fitting parameters: A

_{j}the offset, B

_{j}the amplitude, and δ

_{j}the fundamental fringe shift of the line. For the start value regarding fitting the specific δ

_{j}, the previous value, δ

_{j}

_{−1}is taken. In this way, we avoid the phase jumps and obtain δ

_{j}value for each line. These values are smooth along the lines, except maybe at the region where the object is located. The overall smooth function of the shift, ∆(j), is then retrieved from the fitting of the δ

_{j}for j = 1, 2, …, N. In this fitting procedure, more weight is given to the lines on the edges, i.e., the regions of the hologram outside the object. An example of finding ∆(j) as polynomial $\Delta \left(j\right)={\displaystyle \sum}_{k=0}^{4}{c}_{k}{j}^{k}$ is shown in Figure 11.

## 4. Discussion

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Image plane hologram of the 1951 USAF resolution target recorded using the experimental setup described in Section 3. The inset shows the signal along one line of the hologram.

**Figure 3.**Numerical model of the signal digitization at Nyquist frequency. The lowest curve represents the original signal, the other curves from the bottom to the top are recordings for the phase shifts 0, π/8, π/4, 3π/8, and π/2.

**Figure 5.**The beating effect shown for the sampling frequencies around Nyquist frequency. The graphs show signals for f = 2.00, 2.05, 2.10, 2.20, 2.30, and 2.50 (from the bottom to the top).

**Figure 6.**The signals for the sampling frequencies above Nyquist show a decrease in beatings. From the bottom to the top: f = 2.05, 3.05, 4.05, 5.05, and 6.05.

**Figure 8.**Twyman-Green experimental setup. He-Ne, Helium-neon; M1and M2, mirrors; OBJ1-3, microscope objectives; PIN, pinhole; L1 and L2, lenses; BS1-2, beam splitters; AT1-2, neutral density filters; O, object; POL, polarizer; AP, aperture; PCD, single-photon counting detector; XY1, manual translation stage; and XY2, motorized translation stage.

**Figure 9.**(

**a**–

**d**): FT spectra recorded with CCD using ND filters on both beams with OD = 0, 1.0, 2.0 and 3.0, respectively. (

**e**–

**h**): FT spectra recorded with PCD using ND filters on both beams with OD = 0, 1.0, 2.0 and 3.0, respectively.

**Figure 10.**(

**a**–

**d**): FT spectra recorded with CCD using ND filters to attenuate the object beam with OD = 0, 0.6, 1.3, and 2.0, respectively. (

**e**–

**h**): FT spectra recorded with PCD using ND filters to attenuate the object beam with OD = 0, 0.6, 1.3, and 2.0, respectively.

**Figure 13.**Object reconstruction from the hologram (

**a**,

**c**), and object reconstruction from the interferogram (

**b**,

**d**). Pictures (

**a**,

**b**) depict the end of the telephone cord buckle, and (

**c**,

**d**) depict a poke in the thin film that caused inflation of the structure.

**Figure 14.**(

**a**): Phase picture of the object shown in Figure 13b, and (

**b**): phase values along the red line.

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**MDPI and ACS Style**

Demoli, N.; Abramović, D.; Milat, O.; Stipčević, M.; Skenderović, H.
Linearity and Optimum-Sampling in Photon-Counting Digital Holographic Microscopy. *Photonics* **2022**, *9*, 68.
https://doi.org/10.3390/photonics9020068

**AMA Style**

Demoli N, Abramović D, Milat O, Stipčević M, Skenderović H.
Linearity and Optimum-Sampling in Photon-Counting Digital Holographic Microscopy. *Photonics*. 2022; 9(2):68.
https://doi.org/10.3390/photonics9020068

**Chicago/Turabian Style**

Demoli, Nazif, Denis Abramović, Ognjen Milat, Mario Stipčević, and Hrvoje Skenderović.
2022. "Linearity and Optimum-Sampling in Photon-Counting Digital Holographic Microscopy" *Photonics* 9, no. 2: 68.
https://doi.org/10.3390/photonics9020068