Abstract
In this work, multi-incident digital holographic profilometry for microscale measurements is presented. This technique assembles the set of object fields from captured holograms for generation of the longitudinal scanning function (LSF). Numerical propagation is used for refocusing, and thus, the LSF can be determined at any given plane along the optical axis. The LSF takes maximum value for in focus object points, which are used to obtain full-field height distribution of the sample. This principle is the base of proposed measurement technique. Three capturing holograms strategies, which give control over the shape of the LSF, unambiguous measurement range, axial resolution, and noise immunity, are discussed. The conclusions of this work are supported by numerical and experimental results.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Photonics microstructures have become critical components in many devices that are employed in research, industry and everyday life. Since those elements are gaining more and more demands, there is a necessity of accurate shape characterization. Currently such inspection can be carried out by techniques such as digital holographic microscopy (DHM) [1].
Conventional DHM allows retrieving amplitude and phase information of an object under study from a single hologram. Retrieved phase, as obtained from the numerical process, is a discontinuous distribution within the interval (-π, π], which is known as the wrapped phase [2]. In order to characterize the surface profile, the extracted phase must be unwrapped and then converted to the corresponding topography distribution, which is typically carried out by the Thin Element Approximation [3]. Thus, DHM enables high resolution and contactless full-field surface measurements in short time. Unfortunately, conventional DHM is not suitable for characterizing objects with high numerical aperture (NA) or discontinuities [3–5]. In order to overcome this limitation, numerical techniques can be applied to conventional DHM for improving the shape characterization of objects with high NA [3–6]. However, shape discontinuities larger than λ/4 (reflection mode) or λ/2 (transmission mode), where λ is the wavelength, cannot be recovered when employing one-single illumination direction or one wavelength because of the 2π ambiguity problem [7].
Multi-wavelength techniques can be employed for measuring the topography of discontinuous objects. These techniques use a set of holograms that have been acquired by using different wavelengths. Subtraction of two reconstructed phase maps of selected wavelengths allows generating a phase map with artificial wavelength. The generated phase map has a larger unambiguous measurement range (UMR) than using a single wavelength [8,9]. In this way, the employment of more wavelengths opens the possibility of enlarging the UMR to millimeters with high axial resolution [9–13]. However, the multi-wavelength approach might suffer from some drawbacks as having low wavelength tuning range and source instabilities. Moreover, the optical system is subjected to chromatic aberrations due to the employment of different wavelengths [13,14] and phase noise can be amplified due to the phase map subtraction [15].
The 2π ambiguity problem can be solved by using spatial frequency diversity as well. Optical contouring offers a contactless full field measurement of a topography distribution by using two holograms captured at different illumination angles [16–18]. The extracted phases from the holograms are subtracted from each other generating the corresponding contour lines, which encode the height distribution. This procedure delivers UMR of up to millimeters with a simpler experimental set-up than multi-wavelength approaches [18]. However, the axial resolution of this technique is limited due to the employment of two illumination angles that must be close to each other (δθ < 0.1°) [16]. In order to overcome the limitations of optical contouring, Ref. [14] proposed the multi-angle interferometry (MAI) method. By employing several holograms captured with different illumination angles and the optical contouring principle, it was shown that the MAI can enhance the axial resolution. Nevertheless, the allowed angular span to be employed is very small (∼0.5° -1°), which still limits the axial resolution of the MAI method.
Surface topography can be also retrieved when the measured object is illuminated from multiple directions at the same time, as shown in coherence holography (CH) [19–21]. CH uses a monochromatic, fully incoherent extended source for generating large spatial frequency diversity. The multiple uncorrelated beams generate the longitudinal complex spatial coherence function (LCSCF), which modulates the visibility of fringes along the optical axis as a function of the optical path difference between the interferometer arms [20–23]. CH allows scanning the topography of the object by displacing the object arm [21,24] or modulating the amplitude of the extended source by using an amplitude spatial light modulator (aSLM) [22,23,25]. Amplitude modulation of the source is employed for scanning the object over the optical axis without mechanical movement [22,25], but comes at the cost of reducing the UMR. CH uses incoherent sources with a larger angular span than MAI, and thus, the axial resolution is significantly larger. However, the angular size of the employed source is restricted to the paraxial approximation [24], i.e. small NA of the source.
Nevertheless, the lack of high axial resolution in MAI and CH, unlike the multi-wavelength approach, prevents the employment of these techniques in DHM. Hence, the goal of this paper is to introduce a new DHM technique that combines the capturing approach from MAI and the imaging principle of multi-beam illumination of CH for implementing it as a numerical algorithm. This allows obtaining the following challenging features: (i) high axial resolution measurement; (ii) large UMR from a few captured holograms; and (iii) flexibility for controlling axial resolution and UMR. The technique is based on sequential capture of holograms with varying illumination direction. Then the corresponding set of complex fields is recovered and is propagated along the z-axis. The propagated fields are normalized, and corresponding spatial carriers are removed. The resulting set of complex fields is then linearly superimposed to generate a function similar to the LCSCF. The absolute value of this new LSF has its maximum at in focus object points. Sweeping the propagation distance, the described procedure enables to carry out a point-wise reconstruction of the topography of a surface with increased UMR and high axial resolution at the same time. Moreover, in this work three different strategies for capturing the corresponding holograms are discussed. It will be shown that the capturing strategies give control over different shapes of the LSF, UMR, axial resolution, and noise suppression.
2. Operation principle
This work employs a digital holographic system using a modified Twyman-Green interferometer (TGI) presented in Fig. 1. The first modification consists of a beam splitter (BS) that is rotated 90°. The BS divides the collimated beam that comes from lens L1. Unlike the common configuration, one of the divided beams goes directly to the camera (blue arrows). This beam is the reference plane and its complex amplitude can be simply described as ur = Arexp[iφr], where φr is a constant phase. The second beam illuminates a phase only SLM, which displays a blaze grating. This is done with the aim to modulate the direction of the reflected first diffraction order. In this work, we assume modulation of the beam in x direction. The modulated beam is defined as uill(x) = exp(ikixx+φo), where kix = ksinαi, αi is the illumination angle, k = 2π/λ, and φo is constant. The wavefront uill travels back to the BS and is reflected from the object to the camera. The second important modification into the TGI is that an imaging system (IS) is inserted between the object and the BS. The IS is placed in such a way that: (1) the top surface of the object and the camera are at the conjugates planes π and π’’ of IS, and (2) the SLM and the top surface of the object are placed at the conjugates planes π and π’ of IS, which is presented in Fig. 1. Since the top surface of the measured object is placed in the plane of focus π(x,y), only this region of the object appears sharp at the camera plane. Moreover, the image of the SLM is projected onto the in focus surface of the object, which is shown in the inset of Fig. 1. Distance between the object and IS is S1, while the separation between IS and camera and IS and the SLM is S2. Selected configuration allows projecting the wavefront uill over the surface of the sample. The wavefront is then modulated by the shape of the sample and reflected back to the focus plane π. Modulation of the uill by the depth object is shown in the inset of Fig. 1.
The optical field that arrives to the plane π can be expressed as
where Φ is the phase that encrypts information about the height distribution Δz(x,y) of the object in respect to the plane π. The optical field u is imaged at the camera plane by IS. However, it should be noted that the captured image of defocused regions is going to be subjected to transversal shift and diffraction. The exact amount of shifting and diffraction depend on the illumination angle and local height between the surface and the plane of focus. On the other hand, regions of the image that are in focus do not suffer this transversal shift and the phase value over those regions is determined by the illumination wave, which is known.Hologram of the recombined object and reference beams can be expressed as
The plane of focus of the extracted field Uo can be changed by means of numerical propagation techniques to a new focus plane (x’, y’). In this way, the information about object height contained in ΦB can be separated from generally unknown component Φ(x’, y’) for all object points that are now in focus. Numerical field propagation is carried out by means of the Angular Spectrum (AS) method [26] as follows
3. Shape reconstruction algorithm using the LSF
The described methodology gives the possibility of rendering the shape of an object by calculating the 3D distribution of the LSF. Figure 2 depicts the algorithm that is employed for obtaining the reconstruction topography of the object and it can be described as steps:
- 1) complex fields are extracted from the captured hologram using the FT method [29].
- 2) The constant term φr - φo is removed from the phase of the retrieved fields for generating the complex fields Uo (Eq. (3)). This can be done by adding a constant value to the retrieved phase such that the region in focus has an average value equal to zero, i.e. Φ(xf,yf) = 0.
- 3) Complex fields Uo are propagated with the AS method using the distance z (Eq. (5)).
- 4) Spatial carriers are removed from propagated fields (Eq. (6)).
- 5) Amplitudes ALF(x,y,z) are set to one.
- 6) Resulting complex fields ULF(x,y,z) are summed as shown in Eq. (8) to obtain μ(x,y,z).
- 7) Propagation distance is increased z = z+δz and steps 3 to 6 are repeated.
- 8) The values of z having maximum value of μ are found and stored as a distribution of a render shape.
4. Optimization of the LSF
4.1 Solutions of the LSF
The depth scanning property of the LSF can be employed to retrieve the surface of an object. However, the way of capturing those holograms for carrying out an efficient rendering of the object topography was not discussed yet. The proper capturing strategy of the holograms has impact on UMR, number of measurements, measurement of axial resolution or the shape of the LSF, which can introduce measurement errors due to amplitude of the side lobes of μ. Therefore, in this section we describe the solutions of Eq. (8) for selected capturing strategies, which provide control over the mentioned properties of the LSF.
For studying these capturing strategies, it is assumed that the object is an infinite mirror that runs parallel to the plane π, and is placed to a constant distance Δz. Selected object allows defining the corresponding parameters as follows: kiz = kcosαi, Δz(x,y) = Δz, and Φ(x,y) = 2kizΔz. Hence, the propagated field after carrier removal is given solely by ULF(Δz - z) = exp(i2kiz(Δz - z)). Introducing this field into Eq. (8) yields
4.1.1 Equally frequency spaced strategy
As shown in Eq. (9), the LSF is a result of superimposing a set of periodic functions with longitudinal spatial frequencies kiz, which depends on the illumination angles. Thus, the properties of the LSF can be derived when considering specific selection of the values of kiz. The first spacing strategy is based on the selection of illumination angles when the components kiz are equally spaced, i.e.
where kizn is within the frequency range [kzmin, kzmax], Δkz = kzmax - kzmin, and kzmin = kcos(αmax) and kmax = kcos(αmin). Introducing Eq. (10) into Eq. (9) yieldsFor the case of a small span of longitudinal frequencies such that Δkz<<1, the following approximation can be made: Δkz≈Δαsinαmin, where Δα = αmax-αmin, and thus, zres = π/2Δαsinαmin, and UMR = 2(N-1)zres, which is similar to the solution proposed by Ref. [14].
4.1.2 Geometrically frequency spaced strategy
Optimal hologram capturing enables large UMR and high axial resolution. Moreover, it is required that the longitudinal frequency range [kzmin,kzmax] will be scanned with a small number of measurements without modifying the UMR or axial resolution. These requirements can be fulfilled using a sampling strategy of kiz based on the geometrical series. Under this approach, the sampling of kizn is defined as
4.1.3 Equally angle spaced strategy
This section analyzes an alternative solution for the hologram capturing when the employed angles are equally spaced. Under this consideration, Eq. (9) becomes
Numerical evaluation of Eq. (15) and (18) are carried out assuming that λ = 532 nm, Δz = 0, which means that the object is placed at the focus plane of the optical system. The angular scanning ranges are given by [0°,15°], [0°, 30°], [0°, 45°] and [0°, 90°] and angular scanning step of δα=1°. The obtained results are presented in Fig. 3. The blue solid line in Fig. 3 represent the solution provided by Eq. (15) while the red solid line is the solution provided by Eq. (18). Figures 3(a-c) show that the larger is the angular range, the larger the similarity between the analytical and numerical solutions of the LSF. Notably, Fig. 3(d) demonstrates that numerical and analytical solutions of the LSFs match exactly for the angular range [0°,90°]. However, this range is difficult to obtain experimentally due to the limited NA of the IS. The results shown in Fig. 3 highlight the fact that the equally spaced angle strategy allows obtaining a solution that is totally unknown in classical CH [15,16,26].
The solutions of Eq. (15) and (18) in Fig. 3 are also compared with the scanning function μeq (Eq. (11)), which are shown by the dash dotted green line. The functions μeq were obtained using N = 15, 30, 45, and 90, and a frequency spacing of δkz = 0.0265 μm−1, 0.0527 μm−1, 0.0769 μm−1, and 0.1314 μm−1 in Fig. 3(a-d), respectively. When comparing the axial resolution of the computed LSFs in Fig. 3, it can be observed that zres are approximately the same for each range, which is pointed out by gray field. Calculation of the axial resolution gives zres≈3.9 μm, 1 μm, 0.5 μm, and 0.26 μm for the scanning ranges [0°,15°], [0°, 30°], [0°, 45°] and [0°,90°], respectively. Similarly to the equal frequency spacing, a special case of μM is presented when considering that nδα<<1. With this condition, the cosine in Eq. (15) can be written as: cos(αmin+nδα)≈cosαmin-nδαsinαmin. When introducing this approximation into Eq. (15), the solution proposed by Ref. [14] is obtained.
4.2 UMR and noise immunity
In this section, we analyze UMR and noise immunity of the LSF. Numerical simulations of the three spacing strategies that utilize the same wavelength and Δz, as in previous simulation, are carried out. Moreover, the angular scanning range is given by [0°, 45°] and thus, Δkz = 3.3761 μm−1. The results depicted in Fig. 4 present the different shapes that the LSF can take depending of the capturing strategy. Two LSFs were generated with equal frequency spacing strategy by using N = 9 and 28, with a frequency spacing of δkz = 0.4324 μm−1 and 0.1235 μm−1, and thus UMR = 7.44 μm and 25.13 μm, respectively. The corresponding functions μeq are plotted with the blue line in Fig. 4(a) and (b). Three LSFs for geometric frequency spacing strategy were simulated using R = 2, 3, and 4, and N = 5. The obtained functions μG are depicted with the red line in Fig. 4(a-c), respectively. The calculated UMR are: 7.44 μm; 25.13 μm; and 59.55 μm. Finally, the LSF that corresponds to the equally angle spaced strategy was generated with δθ = 5° and N = 10, which is shown by the black line in Fig. 4(c). Since the same angular range is used for generating all LSFs, the axial resolution is zres ≈ 0.5 μm. The vertical black dash-dotted lines in Fig. 4 indicate the periodicity of the peaks for the generated LSFs. As can be seen in Fig. 4(a), and b), the UMRs of the scanning functions μeq and μG are the same for each case. Figure 4(b) shows that the equal frequency spacing is able to obtain a large UMR but at the cost of large number of holograms. On the contrary, the geometric frequency spacing is capable of enlarging the UMR by changing the geometric factor R while keeping constant the number of holograms, as presented in Fig. 4(a)-(c). However, increasing the factor R leads to larger side lobes of μG, as seen in Fig. 4(a) and (b). The amplitude of those side lobes could make the detection of local maxima difficult, and thus, induce errors in the reconstruction of object topography. In order to avoid noise in the shape of the LSF it is established that maximum amplitude of the side lobes cannot be larger than the amplitude of the scanning function evaluated at μ(z = UMR + zres/2). Or, in other words, the maximum amplitude of the side lobes cannot be larger than 75% of the LSF maximum. The horizontal dash-black line in the plots shows the maximum allowed level of amplitude of the side lobes. For the case of μM presented in Fig. 4(c), it can be seen that this LSF has side lobes with small amplitude over whole propagation range. Moreover, the presence of a second maximum is not observed for this strategy. This is possible by solely employing N larger than the geometric frequency spacing.
5. Experimental system
In this section, experimental system and sample are described. The experimental system, which is presented in Fig. 5, is a multi-incidence DHM (MI-DHM) based on the TGI presented in Fig. 1. The beam generated by a laser light source (Nd-YAG, λ = 532 nm) is split into reference (blue arrow) and object (red arrow) beam by the cube beam splitter (BS). The object beam is filtered by a spatial filter (SF) and transformed into a plane wave by the collimating lens (C). Then, it travels along the optical axis and through the second BS and is reflected from the surface of the phase only spatial light modulator (SLM). The polarizing optics (half wave plate HP, polarizer P) ensure precise orientation of the polarization, which allows phase modulation of the illuminating beam. As shown in Fig. 5, the SLM is slightly tilted with the aim to align the first diffraction order onto the optical axis. The SLM is responsible for tilting the illumination beam in x direction up to 40° (in the object plane) with the angular resolution of 0.1°. This is made by introducing specialized generated phase mask into the SLM. The modulated beam by the SLM goes through a telecentric system built from lenses LF1 and LF2 (fL1 = fL2 = 122 mm). At the Fourier plane of this system, an amplitude filter (F) is placed, which assures that only the first diffraction order passes through the telecentric system. The filtered beam is directed to an afocal imaging system by the mirror M and the optical wedge (W). The afocal imaging system consists of a lens LAS (WD = 50 mm, fAS = 200 mm) and a microscope objective MO (Mitutoyo plan apo, infinity corrected, WD = 5.2 mm, fMO = 4 mm, NA = 0.75, 50×) and its transverse magnification is ∼49.9×. The illumination object beam is reflected from the measured object surface (OB). A surface from the object will be placed in the focal length of MO (fMO), which will be our reference surface. The positioning of a such surface is assured by a motorized linear stage. The reflected wave comes back through the afocal imaging system and is reflected from W directly towards CMOS camera (JAI GO-5101M PGE, 2464 × 2056px, 3.45 × 3.45µm, sensor Sony IMX264). The reference wave goes through the neutral filter (ND) and is directed to the spatial filter (SF) and collimating lens (C) by the mirror (M). The collimated reference wave is reflected by a mirror (MREF) and goes through the polarizer and W to the CMOS camera. Selected configuration of the measurement system allows introducing arbitrary spatial carrier to the fringe pattern by tilting the reference mirror. The reference and object beams interfere producing a fringe pattern, which is registered by CMOS camera placed directly in the back focal length of the LAS. Frame rate of the camera is given by 10 fps (which was limited by the hologram intensity at the camera), resulting in single hologram acquisition time of 0.1 s.
The test objects used in experimental verification of our technique are two microelements of 5 steps each, as shown in the right inset of Fig. 5. These objects have been fabricated with 0.5 µm and 4 µm height difference between subsequent steps, respectively. They have been fabricated using the three dimensional two-photon photolithography (also called direct laser writing), in which two-photon absorption leads to solidification of a liquid polymer near the focus of a femtosecond-pulsed laser beam [31]. The test objects are made out of IP-L 780 polymer on top of a regular glass coverslip using Photonic Professional GT workstation (Nanoscribe GmbH) equipped with a piezo scanning stage and 100× microscope objective with NA = 1.4. Once the writing process is complete, the unexposed photoresist is chemically removed. Moreover, the samples have been sputter coated with ∼50 nm layer of gold (Q150R, Quorum Technologies) in order to increase reflectivity.
6. Experimental results
In this section, the experimental verification of our technique for topography reconstruction is presented. For this, three experiments are performed: (i) generation of the shapes of the LSFs for the different spacing strategies; (ii) reconstruction of microelements with small and large height difference by employing the same geometric spacing strategy; and (iii) high resolution topography reconstruction with the equal angle spacing strategy.
In our first experiment, the sample with 4 µm height difference between the steps is examined. The surface 5 of the sample, which is the tallest part of the structure, is placed at the focus plane of the imaging system. The topography of this surface will be reconstructed. Three sets of holograms were captured within the angular range [0°,25°]. The first set of holograms is captured by employing the equally frequency spaced strategy with N = 17, and a frequency spacing δkz = 0.0692 μm−1. The second set of holograms is captured with the geometrically frequency spaced strategy by using the parameters N = 6, and R = 2. The values of the employed angles are: 0°, 6.2°, 8.8°, 12.4°, 17.6°, and 25°. The third set of holograms corresponds to the equally angle spaced strategy when using N = 10 and δθ ≈ 2.8°. The complex object fields are extracted and processed as described in section 3. Figure 6 presents the experimental calculation of the corresponding LSFs, which were obtained by taking values of μ at the central pixel point of surface 5. The blue, red, and yellow plots in Fig. 6 depict μeq, μG and μM, respectively. The obtained LSFs have similar axial resolution since the angular range is the same for all the spacing strategies, as shown in the inset of Fig. 6. The plots in this figure show that the UMR of μeq and μG will be defined by the distance between local peaks with amplitude > 75%. Those peaks can be observed at z = 0, and 45.9 μm in Fig. 6, which are marked by the vertical dash-line. The experimental value is in agreement with the theoretical UMRs of both strategies. However, the peaks at z = 45.9 μm are smaller than the peaks at z = 0 μm. The reduction of the second maximum is due to propagation of the optical fields. These two spacing strategies enable large UMR but the geometric spacing achieves this with a reduced number of holograms. However, the amplitude of the side lobes of μG increases. When analyzing μM, it can be noticed that this LSF does not have a second maximum as the other LSFs within the propagation range. Moreover, amplitudes of the side lobes of μM are smaller than the side lobes of μG and slightly larger than the side lobes of μeq. The trade off this strategy relies in capturing more holograms than the geometric frequency spaced strategy but less than the equally frequency spaced strategy. Hence, Fig. 6 shows that the theoretical and experimental shapes of the LSFs are in agreement, which validates the utility of our algorithm.
The second experiment can be divided in two parts: first, the topography of the tall multi-step object is retrieved using the angular range [0°, 25°] and the same parameters for geometric spacing approach as in the first experiment. The surface five is placed at the plane of focus of the optical system (z = 0 μm). The propagation step is δz = 0.25 μm for scanning the z-axis range [1 μm, -27 μm]. The results of the surface reconstruction are shown in Fig. 7(a-c). Figure 7(a) shows the full field reconstruction of the sample. Figure 7(b) depicts the horizontal cross sections of the object. Figure 7(c) presents the vertical cross sections of the object. The depicted cross sections show that the shape of the object has been recovered in great detail with steps being clearly separated from each other. Artifacts are mostly visible at the edges of each surface, which originate from the numerical propagation errors. The horizontal lines in Fig. 7(b-c) present the heights of the multi-step object. Finally, the heights of the object from the substrate to the highest surface are approximately 5.3 μm, 9.7 μm, 13.6 μm, 18.5 μm, 22.4 μm and, 26.6 μm. The application of our reconstruction algorithm shows that the total height of the object is around 26.6 μm and the height different between each step is approximately 4 μm. The obtained measurements are in agreement with the fabrication parameters.
The second part of this experiment consists in retrieving the topography of the multi-step object with small height differences by using the geometric frequency spacing of the previous experiment. For the second reconstruction, a smaller propagation step δz = 0.1 μm was used for scanning the z-axis range [-1 μm, 9 μm]. The results are presented in Fig. 7(d)-(f). Notably, the employment of the captured holograms and our algorithm are sufficient to reconstruct the topography of this microstructure, as shown in Fig. 7(d). The horizontal and vertical cross sections depicted in Fig. 7(e) and (f) allow observing the height of each surface of the object more evidently. However, it can be observed that the edges are not well retrieved due to the low resolution of the selected angular range. In this case, the heights of the object are approximately 5 μm, 5.6 μm, 6 μm, 7.19 μm, 7.5 μm, and 8.1μm for the multi-steps.
Finally, with the aim to test the resolution capabilities of our technique, we measured the low multi-step object again but using the equally angle spaced strategy with a wider-angle range [0°, 40°]. The angle separation is set to δθ = 8°, and thus, 11 holograms were captured. Propagation step is set to δz = 0.1 μm for scanning the range [-1 μm, 9 μm]. The results of this reconstruction are presented in Fig. 8. Figure 8(a) shows the full field reconstruction of the object. Comparison between Fig. 7(d) and Fig. 8(a) shows some differences between the topography reconstructions. The cross sections in Fig. 8(b) and c) give the opportunity to visualize in detail those differences. In those figures, it can be noticed that the roughness of the recovered surfaces has diminished as a result of averaging more fields. As well, the sharpness of the object at the edges has been increased due to the employment of a larger angular scanning range. Moreover, the horizontal lines in Fig. 8(b and c) show the heights of this multi-step object. The heights of the structures are the same as in the previous reconstruction.
7. Summary
In this work, we have developed a new technique that employs MAI and CH principles for topography reconstruction in DHM. The technique uses complex fields retrieved from the holograms for generating the LSF. The use of numerical propagation allows refocusing the reconstructed complex fields, and thus, the shape of the LSF can be generated over the z-axis without the necessity of mechanical movement of the object. The generated LSF is then analyzed in order to retrieve the topography of an object. It is shown that the proposed technique is able to process holograms with large illumination angles, and therefore it is capable of obtaining high resolution, overcoming the inherent limitations of MAI and CH. The design of three hologram capturing strategies gives the possibility of obtaining a large UMR from a few measurements without affecting the axial resolution. The experimental reconstruction of two multi-step objects with different step heights supports the theoretical analysis and reconstruction algorithm of our approach. Hence, it is proven that our method is suitable to apply in DHM, something that it has not been done for techniques of this type until now. It is important to note that the calculation of the LSF depends on the model of a flat surface that is parallel to the plane of focus. When considering a surface with a significant tilt, the shape of the LSF will be modified. In consequence, parameters such as UMR, zres, and amplitude of the side lobes will be modified, and thus, the solution presented here for the LSF might not be optimal for retrieving the profile of a tilted surface. Disadvantage of our technique is the need to capture several frames, which increases the measurement time.
Appendix A. Relation of the LSF to the longitudinal coherence function of CH
This work has proven that numerical generation of the LSF is possible by employing the linear superposition of the propagated optical fields ULF. However, this is not the only way to obtain the LSF. The LSF can be generated by incoherent superposition of processed digital holograms. For this, amplitude and spatial carrier of the object complex field Uo (Eq. (3)) captured with the illumination angle αi are removed, which generates the new field
The following on-axis scanning plane wave is generated Note that fields Ulf and Us must have the same illumination angle. The fields Ulf and Us are linearly superimposed for generating the numerical hologram Assuming that Φ(x,y) = 2kizΔz(x,y) [23,32] and adding all the resulting intensities, the following hologram is obtainedAt this point, let’s assume that the SLM is able to modulate the illumination angles with continuous values then Eq. (24) becomes
Funding
National Science Centre, Poland (2015/17/B/ST8/02220); Statutory funds of Warsaw University of Technology.
Disclosures
The authors declare no conflicts of interest.
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