Multi-incidence digital holographic profilometry with high axial resolution and enlarged measurement range

Juan Martinez-Carranza; Juan Martinez-Carranza; Marta Mikuła-Zdańkowska; Marta Mikuła-Zdańkowska; Michał Ziemczonok; Tomasz Kozacki

doi:10.1364/OE.385743

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 u_r= A_rexp[iφ_r], where φ_r 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 u_ill(x) = exp(ik_ixx+φ_o), where k_ix= ksinα_i, α_i is the illumination angle, k = 2π/λ, and φ_o is constant. The wavefront u_ill 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 u_ill 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 u_ill by the depth object is shown in the inset of Fig. 1.

Fig. 1. Schematic representation of the modified Twyman-Green interferometer used in this work. The arrows in the inset show the generation of the object beam. BS – beam splitter, L -lens, IS – imaging system.

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The optical field that arrives to the plane π can be expressed as

(1)$$u(x,y) = {A_0}(x,y)\exp [{i({{k_{ix}}x + \Phi (\Delta z(x,y)) + {\varphi_o}} )} ],$$

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

(2)$$I(x,y) = {I_o} + {I_r} + 2\sqrt {{I_o}{I_r}} \cos ({{k_{ix}}x + \Phi (\Delta z(x,y)) + {\varphi_r} - {\varphi_o}} ).$$

The proposed method is based on capturing several holograms each with different fringe density as done in MAI [14] but using the SLM for angular modulation instead. Reconstruction of the object topography requires extracting the complex object field from Eq. (2). After neglecting constant terms, the object field is given by

(3)$${U_o}(x,y) = {A_o}(x,y)\exp [{i({{k_{ix}}x + \Phi (\Delta z(x,y))} )} ].$$

At this point, it is convenient to express the measured phase Φ as a sum of two components

(4)$$\Phi ({\Delta z(x,y)} )= {k_{iz}}\Delta z(x,y) + {\Phi _B}(\Delta z(x,y)),$$

where k_iz=kcosα_x is the longitudinal component of the illuminating wavevector k_i. The first term in Eq. (4) corresponds to the change of phase due to the longitudinal component of the illuminating plane wave that travels from the plane π to the objects’ surface. The second term corresponds to the change of phase due to the longitudinal component of the reflected wavevector k_r. Note that for the coordinates (x_f, y_f), which are coordinates at the region in focus, the phase is zero, i.e. Δz(x_f,y_f) = Φ(x_f, y_f) = 0, and thus Φ_B(x_f, y_f) = 0.

The plane of focus of the extracted field U_o 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 Φ_B 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

(5)$${ {AS[{{U_o}(x,y)} ]} |_z} = IFT\left[ {FT[{{U_o}({\Delta z(x,y)} )} ]{e^{iz\sqrt {{k^2} - k_x^2 - k_y^2} }}} \right],$$

where FT and IFT are the Fourier transform and its inverse operation, respectively, k_x and k_y are the angular spatial frequencies, and z is the propagation distance. Introducing Eq. (4) into Eq. (5), the propagated field is calculated as follows

(6)$$U^{\prime}(x^{\prime},y^{\prime},z) = {e^{ - i{k_{ix}}x^{\prime}}}\left\{ {IFT\left[ {FT[{{A_o}(x,y){e^{i\Phi ({\Delta z(x,y)} )}}} ]{e^{iz\sqrt {{k^2} - {{({k_x^{} - {k_{ix}}} )}^2} - k_y^2} }}} \right]} \right\}.$$

Note that this operation would be equivalent to moving the object towards the focus plane. When multiplying the right-hand side of Eq. (6) by the term exp[i(k_ixx’-k_izz)], the propagated field can be rewritten as

(7)$${U_{LF}}(x^{\prime},y^{\prime},\Delta z(x^{\prime},y^{\prime}) - z) = {A_{LF}}(x^{\prime},y^{\prime},\Delta z(x^{\prime},y^{\prime}) - z){e^{i({{\Phi _{LF}}(x^{\prime},y^{\prime},\Delta z(x^{\prime},y^{\prime}) - z) - {k_{iz}}z} )}},$$

where U_LF is the propagated field with removed spatial carrier and A_LF, Φ_LF are its corresponding amplitude and phase. The amplitudes A_LF are set to one (A_LF(x,y,Δz-z) = 1) and Eq. (5) – (7) are applied to each retrieved field U_o for the same propagation distance z. The obtained set of fields U_LF are averaged. This emulates the imaging principle of CH [20]. Hence, the LSF is given as

(8)$$\mu ({x^{\prime},y^{\prime},\Delta z(x^{\prime},y^{\prime}) - z} )= \frac{1}{N}\left|{\sum\limits_{{\alpha_i}} {{U_{LF}}(x^{\prime},y^{\prime},\Delta z(x^{\prime},y^{\prime}) - z)} } \right|,$$

where N is the number of captured holograms. In order to describe the mechanism behind our technique, let’s assume a surface that is not at the plane of focus. Thus, the captured images of this defocused surface are subjected to transversal shift and diffraction that depend on the illumination angle and the distance between the surface and the plane of focus. When obtaining the fields U_LF, the corresponding phases are given as Φ_LF(x’, y’, Δz-z) - k_izz ≠ 0. Note that the phase values of U_LF depend on the illumination angle, and therefore they are different from each other. Large differences among the phases of the fields U_LF decrease the value of μ. When applying numerical propagation in the proper direction, propagated waves converge towards the focus plane and thus, the phase of the fields U_LF is reduced progressively, which increases the value of μ. Focusing condition will be reached when Φ_LF(x’, y’, Δz-z) - k_izz = 0, increasing the value of μ to its maximum value, which is one. This means that the corresponding height has been found. Similar principle for digital focusing has been presented in Refs. [27,28]. It has to be noted that our method is point wise, and therefore, there is no mechanism to modify the lateral resolution of the system. Notably, the employment of the AS method allows changing the position of the plane (x’, y’) along the optical axis enabling axial scanning of the object without changing the axial position of the object during the reconstruction process.

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 φ_r - φ_o is removed from the phase of the retrieved fields for generating the complex fields U_o (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(x_f,y_f) = 0.
3) Complex fields U_o are propagated with the AS method using the distance z (Eq. (5)).
4) Spatial carriers are removed from propagated fields (Eq. (6)).
5) Amplitudes A_LF(x,y,z) are set to one.
6) Resulting complex fields U_LF(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.

Fig. 2. Shape reconstruction algorithm with the employment of the LSF.

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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: k_iz= kcosα_i, Δz(x,y) = Δz, and Φ(x,y) = 2k_izΔz. Hence, the propagated field after carrier removal is given solely by U_LF(Δz - z) = exp(i2k_iz(Δz - z)). Introducing this field into Eq. (8) yields

(9)$$\mu (\Delta z - z) = \frac{1}{N}\left|{\sum\limits_{i = 1}^N {\exp (2i{k_{izn}}(\Delta z - z))\; } } \right|.$$

The obtained LSF is the base for developing different capturing strategies of holograms. In this work three strategies are considered: (1) equal frequency spacing, (2) geometrical frequency spacing, and (3) equal angle spacing.

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 k_iz, which depends on the illumination angles. Thus, the properties of the LSF can be derived when considering specific selection of the values of k_iz. The first spacing strategy is based on the selection of illumination angles when the components k_iz are equally spaced, i.e.

(10)$${k_{izn}} = {k_{z\max }} - \frac{{n\Delta {k_z}}}{{N - 1}},$$

where k_izn is within the frequency range [k_zmin, k_zmax], Δk_z = k_zmax - k_zmin, and k_zmin= kcos(α_max) and k_max= kcos(α_min). Introducing Eq. (10) into Eq. (9) yields

(11)$${\mu _{eq}}({\Delta z - z,\Delta {k_z},N} )= \left|{\frac{{{e^{2i{k_{z\max }}(\Delta z - z)}}}}{N}\sum\limits_{n = 0}^{N - 1} {\exp \left( { - 2i\frac{{n\Delta {k_z}}}{{N - 1}}(\Delta z - z)} \right)} } \right|.$$

Equation (11) can be regarded as the discrete FT of a rectangular gate function, which means that the shape of μ_eq is a sinc function [23,25]. The axial resolution of the synthesized LSF is z_res= π/2Δk_z [21,30]. Thus, high axial resolution can be obtained when the span of longitudinal frequencies [k_zmin,k_zmax] is large [21,23,30]. However, this sampling strategy generates periodicity over the z-axis. The period of μ_eq is determined by UMR = (N-1)π/Δk_z. Hence, the equal frequency spacing can expand the UMR by increasing the number of captured holograms or reducing Δk_z; however, the latter decreases the axial resolution.

For the case of a small span of longitudinal frequencies such that Δk_z<<1, the following approximation can be made: Δk_z≈Δαsinα_min, where Δα = α_max-α_min, and thus, z_res= π/2Δαsinα_min, and UMR = 2(N-1)z_res, 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 [k_zmin,k_zmax] 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 k_iz based on the geometrical series. Under this approach, the sampling of k_izn is defined as

(12)$${k_{izn}} = \left[ {{k_{\max }} - \frac{{\Delta {k_z}}}{{{R^0}}},{k_{\max }} - \frac{{\Delta {k_z}}}{{{R^1}}},\ldots ,{k_{\max }} - \frac{{\Delta {k_z}}}{{{R^{N - 2}}}},{k_{\max }}} \right]\; ,$$

where R is an integer geometric progression factor. Introducing Eq. (12) into Eq. (9) yields

(13)$${\mu _G}({\Delta z - z,N,R} )= \left|{\frac{{\exp ({2i{k_{z\max }}(\Delta z - z)} )}}{N}\left( {1 + \sum\limits_{n = 0}^{N - 2} {\exp \left( {2i(\Delta z - z)\frac{{\Delta {k_z}}}{{{R^n}}}} \right)} } \right)} \right|.$$

Equation (13) cannot be associated to a close form expression, like Eq. (11), but the period of μ_G can be known by solely taking the exponential function with the lowest frequency. Hence, the UMR of μ_G is given by

(14)$$UMR = \frac{{\pi R{}^{N - 2}}}{{\Delta {k_z}}}.$$

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

(15)$${\mu _M}({\Delta z - z} )= \left|{\sum\limits_{n = 0}^{N - 1} {\exp ({2ik\cos ({{\alpha_{\min }} + n\delta \alpha } )(\Delta z - z)} )} } \right|,$$

where δα is the angle spacing. In order to derive a solution for Eq. (15), we are going to substitute the sum by an integral as follows

(16)$${\mu _M}({\Delta z - z} )= \left|{\int_{{\alpha_{\min }}}^{{\alpha_{\max }}} {\exp ({2i(\Delta z - z)k\cos \alpha } )} d\alpha } \right|.$$

Since we are interested in representing Eq. (16) in terms of k_iz instead of the scanning angle α, the change of variable k_z=kcosα is introduced into Eq. (16), which yields

(17)$${\mu _M}(\Delta z - {z_0}) = \left|{\int_{ - \infty }^\infty {\frac{{rect[{({k_z} - {a_o})/\Delta {k_z}} ]}}{{\sqrt {{k^2} - k_z^2} }}\exp [{i2(\Delta z - z){k_z}} ]} d{k_z}} \right|.$$

where a_o=Δk_z/2. Note that Eq. (17) is similar to the Fourier transform for obtaining a Bessel function of first kind and zero order J₀. An approximated solution of Eq. (17) can be given as

(18)$${\mu _M}({\Delta z - z} )= |{{J_0}({2\Delta {k_z}(\Delta z - z)} )+ i{H_0}({2\Delta {k_z}(\Delta z - z)} )} |,$$

where H₀ is the Struvel function of zero order. Notably, as the value of Δk_z approaches to k, Eq. (17) converges to its exact solution.

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].

Fig. 3. Evaluation of the obtained solutions for μ_M and μ_eq when employing the angular ranges: a) [0°,15°]; b) [0°,30°]; c) [0°,45°]; and d) [0°,90°].

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The solutions of Eq. (15) and (18) in Fig. 3 are also compared with the scanning function μ_eq (Eq. (11)), which are shown by the dash dotted green line. The functions μ_eq were obtained using N = 15, 30, 45, and 90, and a frequency spacing of δk_z= 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 z_res are approximately the same for each range, which is pointed out by gray field. Calculation of the axial resolution gives z_res≈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 μ_M is presented when considering that nδα<<1. With this condition, the cosine in Eq. (15) can be written as: cos(α_min+nδα)≈cosα_min-nδαsinα_min. 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, Δk_z= 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 δk_z= 0.4324 μm⁻¹ and 0.1235 μm⁻¹, and thus UMR = 7.44 μm and 25.13 μm, respectively. The corresponding functions μ_eq 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 μ_G 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 z_res ≈ 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 μ_eq and μ_G 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 μ_G, 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 + z_res/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 μ_M 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.

Fig. 4. Comparison of the shapes of the LSFs μ_eq and μ_M using different parameter N with μ_G when employing constant N and: a) R = 2, b) R = 3, and c) R = 5.

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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 L_F1 and L_F2 (f_L1 = f_L2 = 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 L_AS (WD = 50 mm, f_AS = 200 mm) and a microscope objective MO (Mitutoyo plan apo, infinity corrected, WD = 5.2 mm, f_MO = 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 (f_MO), 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 (M_REF) 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 L_AS. 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.

Fig. 5. Experimental MI-DHM system based on the Twyman-Green configuration for surface measurement. The inset to the right shows the geometry of the test object. HP – half wave plate, BS – beam splitter, SF – spatial filter, P – polarizer, ND - neutral-density filter, C – collimator, SLM – spatial light modulator, L – lens, F – filter, M – mirror, W – optical wedge, MO – microscope objective, OB – object.

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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 δk_z= 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 μ_eq, μ_G and μ_M, 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 μ_eq and μ_G 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 μ_G increases. When analyzing μ_M, 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 μ_M are smaller than the side lobes of μ_G and slightly larger than the side lobes of μ_eq. 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.

Fig. 6. Experimental LSF for a single point object when using different spacing strategies and the angular range [0°, 25°].

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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.

Fig. 7. Results of the topography reconstruction of a) tall object (4 µm per step) and d) low object (0.5 µm per step); b) and e) show horizontal cross-sections A-A and B-B of the tall and low object, respectively; c) and f) show the vertical cross-sections C-C, D-D, E-E, F-F of tall and low object, respectively. The dotted lines indicate step heights.

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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.

Fig. 8. Topography reconstruction of the multi-step object with height difference of 0.5 µm per step using the equally angle spaced strategy: a) 2D reconstruction of the object, b) horizontal cross-section, and c) vertical cross-section. The dotted lines indicate step heights.

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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, z_res, 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 U_LF. 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 U_o (Eq. (3)) captured with the illumination angle α_i are removed, which generates the new field

(19)$${U_{lf}}({x,y} )= \exp [{i\Phi (\Delta z(x,y))} ].$$

The following on-axis scanning plane wave is generated

(20)$${U_s}(z )= \exp [{2i{k_{iz}}z} ].$$

Note that fields U_lf and U_s must have the same illumination angle. The fields U_lf and U_s are linearly superimposed for generating the numerical hologram

(21)$$I({x,y,\Delta z - z} )= 2[{1 + \cos ({\Phi ({\Delta z(x,y)} )- 2{k_{iz}}z} )} ].$$

Assuming that Φ(x,y) = 2k_izΔz(x,y) [23,32] and adding all the resulting intensities, the following hologram is obtained

(22)$${I_S}(x,y,\Delta z - z) = \sum\limits_{n = 1}^N {({1 + \cos ({2{k_{izn}}(\Delta z(x,y) - z)} )} )} .$$

Note that the different frequencies k_izn into Eq. (22) cause axial modulation of the intensity I_S, where the first maximum intensity I_S takes place at Δz(x,y)- z = 0. Thus, Eq. (22) can be rewritten as

(23)$${I_S}(x,y,\Delta z - {z_0}) = 1 + \Gamma (\Delta z(x,y) - z)\cos \left( {\arg \left\{ {\frac{1}{N}\sum\limits_{n = 1}^N {{e^{2i{k_{izn}}(\Delta z(x,y) - z)}}} } \right\}} \right),$$

Where

(24)$$\Gamma ({\Delta z - z} )= \frac{1}{N}\left|{\sum\limits_{n = 1}^N {\exp [{2i{k_{izn}}(\Delta z(x,y) - z)} ]} } \right|.$$

Notably, Eq. (9) and (24) are the same, and therefore, the LSF is obtained. In this case, the LSF modulates the fringe contrast along the optical axis. When extracting the LSF from Eq. (23), topography reconstruction can be carried out.

At this point, let’s assume that the SLM is able to modulate the illumination angles with continuous values then Eq. (24) becomes

(25)$$\Gamma ({\Delta z - z} )= \frac{1}{{{C_0}}}\left|{\int {I({k_{iz}})\exp [{2i{k_{iz}}(\Delta z(x,y) - z)} ]d{k_{iz}}} } \right|,$$

where C₀ is a normalization constant and I(k_zi) is the intensity of the wavefronts modulated by the SLM. The last expression is the modulus of the LCSCF from CH [20,23,32]. When introducing k_zi ≈ k(1-r²/2f²) into Eq. (A7), where r is the radial coordinate in the plane source and f is the focal length of the collimating lens, it can be shown that

(26)$$\Gamma (\Delta z - z) \approx \left|{\int {I(r)\exp \left[ { - ik(\Delta z - z)\frac{{{r^2}}}{{{f^2}}}} \right]rdr} } \right|.$$

This last expression is the LCSCF defined in Refs. [19,20,22,24,32].

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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Multi-incidence digital holographic profilometry with high axial resolution and enlarged measurement range

Abstract

1. Introduction

2. Operation principle

3. Shape reconstruction algorithm using the LSF

4. Optimization of the LSF

4.1 Solutions of the LSF

4.1.1 Equally frequency spaced strategy

4.1.2 Geometrically frequency spaced strategy

4.1.3 Equally angle spaced strategy

4.2 UMR and noise immunity

5. Experimental system

6. Experimental results

7. Summary

Appendix A. Relation of the LSF to the longitudinal coherence function of CH

Funding

Disclosures

References

Cited By

Figures (8)

Equations (26)

Optics Express

Juan Martinez-Carranza	https://orcid.org/0000-0001-8123-0008
Marta Mikuła-Zdańkowska	https://orcid.org/0000-0002-4451-2492
Michał Ziemczonok	https://orcid.org/0000-0002-4802-1105
Tomasz Kozacki	https://orcid.org/0000-0003-0804-2857