# Multi-Incidence Holographic Profilometry for Large Gradient Surfaces with Sub-Micron Focusing Accuracy

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## Abstract

**:**

## 1. Introduction

## 2. Measurement System

_{1}= 125 mm). This beam, after reflection from mirror M, passes through a second afocal system including an imaging lens IL (f

_{IL}= 200 mm) and a microscope objective MO (Mitutoyo Infinity-Corrected Long Working Distance Objective, 50×, NA 0.75). Distance from L1 to IL is equal to f

_{1}+ f

_{IL}. The beam exiting MO illuminates the measured surface. The reflected object beam goes back through the afocal imaging system and is directed by optical wedge W2 towards the CCD camera (Basler Pilot, resolution 2456 × 2058, pixel size 3.45 μm). The CCD sensor is set at the distance f

_{IL}, which realizes optical conjugation between object and image plane (CCD). The path of the second beam, consisting of a reference of an interferometer, is shorter and much less complicated. The flat wavefront is reflected by a mirror MR directly to the CCD camera. The azimuth of the linear polarization of the reference beam is adjusted by the HP while the polarizer P limits its power. Object and reference beams are combined by the optical wedge W2. The polarization elements allow matching polarization of beams and obtain optimal contrast of the resulting interference pattern. The tilt of the MR is used to introduce a high spatial frequency of interference fringes required to separate the object beam from the zero-order and its conjugate.

## 3. Improved Algorithm for Fast LSF Evaluation

**k**and consequently, ${\mathit{f}}_{p}=[{f}_{px},{f}_{py}]$ is the spatial frequency of illumination plane wave, f

^{p}_{zp}is the corresponding longitudinal frequency, and p is the number of illumination waves. By using shifted frequency coordinates $\mathit{f}+{\mathit{f}}_{p}$ and after analytical manipulations in the algorithm, Equation (1) can be rewritten in the form:

_{x}N

_{y}/4 discrete positions). After summation and before inverse FFT, zero padding is realized to have the full frequency range of the MIDHP system. In this way, there is no reduction in the sampling rate. This part of the algorithm is referred to as the padding part.

## 4. Illumination Strategies for Large Gradient Objects

## 5. High Accuracy Autofocusing Method

_{d}= −1 μm. Two micro focusing objects of spherical shape were simulated, first with low (maximum slope 2.8°, maximum height 1.3 μm) and second with high gradient (maximum slope 17.5°, maximum height 8 μm). Reconstruction errors are illustrated in Figure 6a,b for low and high NA, respectively. These error maps are obtained by subtraction of the defocused object reconstruction from the reconstruction for best focus. Illumination components of employed geometrically frequency illumination scanning are along the y-axis with angles shown in Figure 3b. Thus, for both cases, stronger errors are along this axis. Figure 6c presents the standard deviation (STD) of shape reconstruction error objects as a function of defocus error. It can be noticed that the reconstruction error is considerably larger for objects with higher NA.

**x**: =

_{Ω}**x**∈Ω. The proposed approach compares shape calculated using LSF and the TTEA [21]. The TTEA enables shape reconstruction for objects of continuous shape with low NA and large object illumination angle. It was shown that for objects having NA smaller than 0.05, the approximation provides accurate shape results [21]. For the parabolic focusing object of refractive index 1.5, the NA of 0.05 corresponds to the maximum object slope of 5.7° [25]. Thus, within the focusing algorithm using the TTEA, the shapes are calculated for all P illumination wave components and for the selected area having a continuous surface of low gradient.

_{d}is given by:

_{rec}is the shape reconstructed using the LSF, STD is standard deviation, while

_{p}is the angle of wave vector of illumination wave, and Ψ

_{p}is unwrapped phase of p object wave. In this work, illumination angles are contained in y-z plane, thus the components of transverse vector

**x**are

_{S}**x**

_{S}= 0 and ${y}_{S}=\mathrm{tan}({\theta}_{p}){z}_{TTEA(p)}(y)$, respectively.

_{d}, (iii) setting phase to zero at focus point

**x**for all wave fields, (iv) calculation of focus measure F(z

_{f}_{p}), (v) finding minimum of focus measure within search range Z

_{s}, (vi) focus correction of object waves. In the initialization step focus point

**x**, focus area

_{f}**x**, and focus depth search range Z

_{Ω}_{s}are set. Steps (ii)–(iv) are executed for each considered depth z

_{d}∈ Z

_{s}. In step (ii) all object wavefields are refocused, then the phase of all waves is adjusted to have zero phase at the focus point, next the Equations (4) and (5) are evaluated to find the focus measure. In step (v) minimum of focus measure is found and finally in (vi) the focus of all object waves is corrected for the evaluated focus depth.

_{d}= −1, 0 [μm]). Figure 7a,b shows the difference between shapes reconstructed using LSF and TTEA for two defocus values. For TTEA the most off axis angle is selected for p = 1 (θ

_{p}= −29°) is selected to visualize the focus errors obtained for one of the beams. It is showed that for proper focus (z

_{d}= 0 μm) the phase of all illuminations agrees and there is no difference between approaches taken for shape reconstruction. However, for the defocus error there is a significant difference between the results provided with different approaches. It is worth noting that for different illumination angles the obtained differences have similar distributions. For investigated defocus the difference is no larger than 100 nm. Final plots are presented in Figure 7c. It shows focus measures as a function of defocus error, which were calculated for two different objects (maximum slope 2.8° and 17.5°). The sizes of the area considered are 33 × 33 μm

^{2}for the high NA object, and 69 × 69 μm

^{2}for the low NA. Figure 7c shows normalized focus measures to the smallest obtained value. The simulations show that the proposed focus measure has a distinct minimum. The yellow rectangle is added to indicate a region where the normalized focus measure F grows to 2. It shows that this value corresponds to a focus error smaller than 1 μm. The usefulness of the minimum is investigated in the experimental Section 6.

## 6. Experimental Results

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**(

**a**) Scheme of the measurement system, (

**b**) details of the aperture A (view from illumination direction), (

**c**) visualization of the test object.

**Figure 3.**Frequency distribution of illumination beams in (

**a**) equally angle spaced strategy; (

**b**) geometrically frequency spaced strategy. The circle shows the NA limit of the imaging system.

**Figure 4.**Numerical LSFs for a 3D micro focusing object when using different illumination scanning strategies: (

**a**) horizontal and (

**b**) vertical cross-section for equally angle spaced strategy, respectively; (

**c**) horizontal and (

**d**) vertical cross-section for geometrically frequency spaced strategy.

**Figure 5.**Comparison of the shapes of the simulated and reconstructed object: (

**a**) shape difference map and (

**b**) corresponding horizontal cross-section for equally angle spaced strategy, (

**c**) shape difference map and (

**d**) horizontal cross-section for geometrically frequency spaced strategy.

**Figure 6.**The difference of shapes obtained using LSF for defocus z

_{d}= −1 μm for object of maximum slope 2.8° (NA = 0.025) (

**a**) and 17.5° (NA = 0.15) (

**b**). The standard deviation of shape reconstruction error as a function of defocus error for two different objects (

**c**). The calculations are performed for the geometrically frequency illumination scanning (Figure 3b).

**Figure 7.**The difference of shapes obtained using LSF and TTEA for θ

_{p}= −29° for defocus z

_{d}= −1 μm (

**a**) and 0 μm (

**b**). The focus measurement for two different objects (

**c**); the calculations performed for the geometrical frequency illumination scanning (Figure 3b).

**Figure 8.**Example of experimental data: (

**a**) recorded hologram, (

**b**) phase, and (

**c**) Fourier spectrum of the object in linear scale.

**Figure 9.**The focus measurement for: (

**a**) equally angle spaced strategy, and (

**b**) geometrically frequency spaced strategy.

**Figure 10.**Results of the topography reconstruction of the micro mold with the equally angle spaced strategy: (

**a**) 2D map, (

**b**) horizontal cross-section (A-A), (

**c**) vertical cross-section (B-B), and (

**d**) diagonal cross-section (C-C).

**Figure 11.**Influence of focusing distance on reconstructed shape: (

**a**) topography reconstruction at z = −1.96 µm, (

**b**) topography reconstruction at z = −3.96 µm, (

**c**) difference in vertical cross-sections of the reconstructed object at z = −0.96 µm and −1.96 µm, and (

**d**) difference in vertical cross-sections of the reconstructed object at −0.96 and −3.96 µm.

**Figure 12.**Results of the topography reconstruction of the mold with geometrically frequency spaced strategy: (

**a**) 2D map, (

**b**) horizontal cross-section (A-A), (

**c**) vertical cross-section (B-B), and (

**d**) diagonal cross-section (C-C).

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**MDPI and ACS Style**

Idicula, M.S.; Kozacki, T.; Józwik, M.; Mitura, P.; Martinez-Carranza, J.; Choo, H.-G.
Multi-Incidence Holographic Profilometry for Large Gradient Surfaces with Sub-Micron Focusing Accuracy. *Sensors* **2022**, *22*, 214.
https://doi.org/10.3390/s22010214

**AMA Style**

Idicula MS, Kozacki T, Józwik M, Mitura P, Martinez-Carranza J, Choo H-G.
Multi-Incidence Holographic Profilometry for Large Gradient Surfaces with Sub-Micron Focusing Accuracy. *Sensors*. 2022; 22(1):214.
https://doi.org/10.3390/s22010214

**Chicago/Turabian Style**

Idicula, Moncy Sajeev, Tomasz Kozacki, Michal Józwik, Patryk Mitura, Juan Martinez-Carranza, and Hyon-Gon Choo.
2022. "Multi-Incidence Holographic Profilometry for Large Gradient Surfaces with Sub-Micron Focusing Accuracy" *Sensors* 22, no. 1: 214.
https://doi.org/10.3390/s22010214