Vortex topographic microscopy for full-field reference-free imaging and testing

Petr Bouchal; Lenka Štrbková; Zbyněk Dostál; Zdeněk Bouchal

doi:10.1364/OE.25.021428

1. Introduction

Under the ongoing technological progress, the characterization of surface topography has become increasingly important for a variety of applications in both science and industry. The particular attention has been focused on optical methods because they provide noncontact operation, full-field measurement capability, and fast data acquisition. The optical interferometric methods have been established as well-proven and widely developed measurement techniques. In these methods, the surface topography is reconstructed from the interference fringes created by waves reflected from a testing object and a high-quality reference surface. Analysis of the interference fringes differs depending on whether the quasi-monochromatic or broadband illumination is used. Methods utilizing quasi-monochromatic light are typically based on the phase shifting interferometry [1–6], which requires acquisition of four or more frames. These techniques provide excellent measurements with high precision and low noise yet they are not able to map depths greater than a quarter of the wavelength for the sake of ambiguity in determining the fringe order. When larger measurement ranges are required, methods of white light vertical scanning interferometry [7,8] are commonly deployed. The local surface height is determined from the best focus position obtained by identifying maximum position of the white light interference fringes during vertical scanning through the entire measured range. Besides localization of the fringe envelope, the phase of the fringes can also be processed by phase-shifting methods to improve resolution [9]. In focus variation methods, the vertical scanning is combined with the focus localization enabled by a small depth of focus of microscope objective [10]. In recent years, experiments utilizing unique properties of special optical beams have also emerged in surface topography. A binary diffractive element with vortex axicon and vortex lens functions was used as a simple topography analyzer and its operation tested in the phase edge measurement [11]. An extensive effort was devoted to the development of optical vortex interferometer implemented in modified Michelson and Mach-Zehnder configurations. In this interferometer, regular vortex lattices were created by interference of three or more plane waves. In the measurements, the positions and phase singularities of individual vortices were examined and used in the phase reconstruction procedures [12–15]. Optical vortices emerging in a speckle pattern were successfully used as sensitive indicators of local speckle displacements [16]. The self-imaging effect caused by the interference of two non-diffractive beams was used to measure the liquid mirror curvature [17]. The specific properties of Bessel beams were utilized in self-referenced systems for inspection of cylindrical and conical surfaces [18].

In this paper, imaging and testing of reflective samples is demonstrated by original measurement strategy that has not yet been used in surface topography, to our knowledge. Information on the local depth of the sample is encoded into a set of defocused waves that are reflected at precisely defined lateral positions and then transformed to pairs of optical vortices with different topological charges. By the self-interference of optical vortices coming from the same wave, a lattice of image spots is created. The individual image spots take the form of a double-helix point spread function (PSF) whose angular rotation is used to determine the local depth. In this way, the benefits of interference techniques are delivered to a specific focus variation method, providing the possibility of a precise measurement in the extremely large range of depths. In recent years, the depth estimation based on the rotation of defocused image has been widely developed [19–23] and successfully deployed in the photo-activation microscopy. The potential of this method has been demonstrated in experiments of super-resolution fluorescence microscopy enabling a three-dimensional single molecule localization with nanoscale precision [24,25]. The double-helix PSF was also successfully deployed in the measurement of flow fields [26]. Here, the vortex localization is applied in a new way and its optical performance is further enhanced. By optimizing parameters of the spiral phase mask (SPM) used in experiments, a pair of non-diffractive optical vortices is generated for each light spot created on the measured surface by the structured illumination system. The interference of the vortices forms an array of spatially invariant double-helix PSFs rotating differently due to local changes of defocusing. In this technique, the created PSFs are laterally isoplanatic and their shape and size remain unchanged over the entire range of measured depths.

The developed method was implemented in a standard microscope Nikon Eclipse E600 whose illumination and imaging parts were modified. The structured illumination was accomplished using a micro-hole plate (MHP) forming an array of mutually uncorrelated light spots on the measured surface. The splitting of each defocused wave into two optical vortices with precisely defined topological charges was carried out in the additional imaging module using the SPM fabricated by electron beam lithography. The proposed vortex topography technique was tested in the calibration measurements, where both accuracy and precision of the method were determined. In the range of 11 µm exceeding the depth of focus of microscope objective up to 19 times, the surface depth was measured with the precision and accuracy better than 30 nm and 100 nm, respectively. The lateral measurement positions were determined with the precision better than 10 nm. Applicability of the proposed vortex topography technique was tested in the measurement of smooth and periodic reflective surfaces. Profiling of the top surface of a bearing ball was performed in full-field imaging mode. Local depths were determined from a single CCD record mapping the surface in approximately 550 measuring positions appearing in the field of view of the microscope objective. The relief of a reflection phase grating was reconstructed using a stack of CCD frames obtained by sequentially rotating the MHP. Because the individual holes were arranged in the arms of Archimedean spiral, a quasi-continuous imaging of the phase grating was possible. To compare the full-field imaging mode with scanning operation of the system, the top surface of the bearing ball was reconstructed also from stack of images captured while rotating the MHP.

2. Principle of reference-free vortex array topography

The proposed vortex topography is based on the self-interference of optical vortices originating from the defocused light waves carrying information on the local depths of a reflecting sample being tested. As a result of interference, the double-helix PSFs are created, and their angular rotations indicate the sample depths at precisely determined lateral positions. The principle of the method can be discussed in more detail using the scheme in Fig. 1. The measured sample is illuminated by an array of mutually uncorrelated light spots that are focused at the focal plane of the microscope objective (MO) (implementation of structured illumination is shown in Fig. 2). The light reflected from the sample is captured and transformed by the MO. The SPM located in the back focal plane of the MO is then illuminated by a set of plane or spherical waves, depending on whether they originate from light reflected from in focus or out of focus sample areas. The wavefront of each wave impinging on the SPM is divided into two annular zones in which a spiral phase modulation with different topological charges is applied. After performing optical Fourier transform of the light transmitted through the SPM, an image lattice composed of two-lobe PSFs is created in the back focal plane of the tube lens (TL). The optical performance of the method is enhanced by telecentricity and spatial invariance of the generated PSFs obtained by an optimized design of the SPM. These properties of the PSFs are crucial for achieving high accuracy and extended measurement range and can be described in a simplified computational model including generation and self-interference of optical vortices.

Fig. 1 Simplified scheme of the vortex topography demonstrating transformation of defocused waves to optical vortices, whose interference forms a lattice of double-helix PSFs. The angular rotation of the individual PSFs indicates the local depth of the tested sample (MO-microscope objective, SPM-spiral phase mask, TL-tube lens).

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Fig. 2 Scheme of experimental setup for reference-free vortex array topographic microscopy. Illumination path: LED–light emitting diode, MHP–micro-hole plate (rotated by stepper motor in quasi-continuous surface reconstruction), CL–collector lens, M–mirror, BS–non-polarizing beam splitter. Imaging path: PT–piezoelectric transducer (used in calibration measurement), MO–microscope objective, L1, L2–Fourier lenses, SPM–spiral phase mask, TL–tube lens, CCD–charge coupled device with detected array of double-helix PSFs.

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The shaping of the PSF is explained using one of the beams of the illuminating array which is focused to the point $(x_{0}, y_{0})$ of the front focal plane of the MO. When the light is regularly reflected from the surface located at the out of focus position $Δ z_{0}$ , a virtual point source with the coordinates $(x_{0}, y_{0}, 2 Δ z_{0})$ is created. The wave emanating from this point is transformed by the MO, hence spherical or plane wave impinges on the SPM when $Δ z_{0} \neq 0$ or $Δ z_{0} = 0$ , respectively. Using a paraxial approximation, the complex amplitude of this wave evaluated at the back focal plane of the MO can be written as

ψ_{0} = a_{0} \exp (- i \frac{π Δ z_{0} {| {\vec{r}}_{0 ⊥} |}^{2}}{λ f_{0}^{2}}) \exp (i \frac{2 π {\vec{r}}_{0 ⊥} \cdot {\vec{r}}_{⊥}}{λ f_{0}}),

where

{\vec{r}}_{0 ⊥} = (x_{0}, y_{0})

and

{\vec{r}}_{⊥} = (x, y)

are the position vectors at the sample and SPM planes, and

λ

and

f_{0}

denote the wavelength and the focal length of the MO, respectively. After passing through the SPM with the transparency

t

, the light wave is captured by the TL performing the optical Fourier transform. The complex amplitude of light in the back focal plane of the TL then can be written as

ψ \propto a_{0} \int \int_{- \infty}^{\infty} t (x, y) \exp (- i \frac{π Δ z_{0} {| {\vec{r}}_{⊥} |}^{2}}{λ f_{0}}) \exp (\frac{i 2 π {\vec{r}}_{⊥} \cdot {\vec{R}}_{⊥}^{}}{λ f_{T}}) d {\vec{r}}_{⊥},

where

{\vec{R}}_{⊥}^{} = (X, Y)

,

X = x_{i} - M x_{0}

,

Y = y_{i} - M y_{0}

, and

x_{i}, y_{i}

and

f_{T}

denote the image coordinates at the detector plane and the focal length of the TL, respectively, and

M = - f_{T} / f_{0}

is the magnification of the 4f optical system composed of the MO and the TL. Using the polar coordinates,

x = r \cos φ,

y = r \sin φ

and

X = R \cos Φ,

Y = R \sin Φ,

the complex amplitude (2) can be rewritten as

ψ \propto a_{0} \int_{0}^{2 π} \int_{0}^{\infty} t (r, φ) \exp (- i \frac{π Δ z_{0} r^{2}}{λ f_{0}^{2}}) \exp [\frac{i 2 π r R \cos (φ - Φ)}{λ f_{T}}] r d r d φ .

Since the double-helix PSFs are formed by the interference of two optical vortices, the SPM splits each defocused wave into two radial zones, in which spiral phase modulation with different topological charges

l_{1}

and

l_{2}

is applied,

t (r, φ) = \sum_{m = 1}^{2} t_{m} (r) \exp (i l_{m} φ)

. Using the Jacobi-Anger expansion and the orthogonality of trigonometric functions, the integration over

φ

can be performed, resulting in

ψ \propto 2 π a_{0} \sum_{m = 1}^{2} i^{l_{m}} \exp (i l_{m} Φ) \int_{0}^{\infty} t_{m} (r) J_{l_{m}} (\frac{2 π r R}{λ f_{T}}) \exp (- i \frac{π Δ z_{0} r^{2}}{λ f_{0}^{2}}) r d r,

where

J_{l}

denotes the Bessel function of the first kind and

l

-th order. To achieve a spatial invariance of the PSFs, the optical vortices are generated from thin annular zones with the radii

r_{1}

and

r_{2}

(r_{2} > r_{1})

. In the simplified calculation model, the radial zones are represented by the Dirac delta function,

t_{m} (r) = δ (r - r_{m})

. In experiments, the double-helix PSFs composed of two lobes are conveniently used to estimate the local depths. The required double-helix PSFs result from the interference of optical vortices with the difference of the topological charges given as

Δ l = | l_{1} - l_{2} | = 2

. As the size of the PSFs scales with the order of the Bessel function, the topological charges

l_{1} = 1

and

l_{2} = - 1

are preferably used to obtain the smallest image spots. With this choice, the PSF defined as

I = {| ψ |}^{2}

is obtained in the form

I = {(2 π a_{0})}^{2} [\sum_{j = 1}^{2} J_{1}^{2} (\frac{2 π R r_{j}}{λ f_{T}}) + 2 J_{1} (\frac{2 π R r_{1}}{λ f_{T}}) J_{1} (\frac{2 π R r_{2}}{λ f_{T}}) \cos (2 Φ + κ Δ z_{0})],

where

R = {[{(x_{i} - M x_{0})}^{2} + {(y_{i} - M y_{0})}^{2}]}^{1 / 2},

Φ = arc \tan (\frac{y_{i} - M y_{0}}{x_{i} - M x_{0}}),

κ = \frac{π (1 - Q^{2}) N A_{e f f}^{2}}{λ}, Q = \frac{r_{1}}{r_{2}},

and

N A_{e f f} = r_{2} / f_{0}

is an effective numerical aperture of the MO. The PSF represents the image of a virtual source with the coordinates

(x_{0}, y_{0}, 2 Δ z_{0})

that provides information on the local depth of the tested surface. The light spot is formed by two overlapping rings given by the sum terms in Eq. (5), which are cosine modulated by the interference term. The PSF is suitably described by the polar coordinates

R

and

Φ

defined in the coordinate system with the origin

(M x_{0}, M y_{0})

determined by the lateral position of the virtual source and the magnification of the 4f system. As a result of interference, two bright lobes of the PSF are created whose maxima are directed to the angles

Φ = 0

and

π

, when

Δ z_{0} = 0

. If the light is reflected from the surface in the out of focus position

Δ z_{0}

, the PSF is rotated and maxima of its lobes lie in the directions

Φ = - κ Δ z_{0} / 2

and

π - κ Δ z_{0} / 2

. The sensitivity of the PSF rotation to the measured depth can be assessed by the depth period

Λ = 2 π / κ

related to the rotation angle

π

. This parameter can be controlled by the effective numerical aperture and the ratio of the radii of the SPM slits,

Λ = 2 λ / [(1 - Q^{2}) N A_{e f f}^{2}]

. By determining the angular rotation and the position of the PSF center, the local depth

Δ z_{0}

of the tested surface is known at the measuring position

(x_{0}, y_{0})

. When the surface is mapped at different lateral positions, the PSFs are shifted while maintaining their shape unchanged (property known as isoplanacy). In addition, the shape and size of the PSF do not change even when the out of focus measurement is performed because only the angular rotation of the PSF depends on

Δ z_{0} .

Since the position of the PSF center given by the origin of the shifted coordinate system is independent of

Δ z_{0},

the telecentricity of the measurement is assured. The spatial invariance and the telecentricity of the PSF are clearly demonstrated by Eq. (5) obtained with the slits of the SPM given by the Dirac delta function. These properties are maintained even when the annular zones with a finite width are used and phase is sampled into discrete levels. To investigate the method with the parameters used in real experiments, the advanced computational model was developed. The complex amplitude

ψ

providing the PSF was calculated by the transmission function

t

that was modified to simulate properties of the SPM implemented by electron-beam lithography. The annular zones with a finite width and eight discrete levels of the spiral phase were used. Instead of the Fresnel transform, the exact calculation of the Kirchhoff diffraction integral was performed. In order to assess the precision limits given by the Fisher information theory, the Cramér-Rao lower bound (CRLB) was calculated by processing simulated or measured double-helix PSFs.

To obtain the simple relationship between the width of the annular zones of the SPM and the range of measured depths, the approximate calculations adopting the concept of non-diffractive vortex beams were performed [23]. In this approach, the applicable range of the measured depths was determined as $L ≅ λ f_{0}^{2} / (r_{2} Δ r)$ , where $r_{2}$ and $Δ r$ are parameters of the SPM denoting the middle radius of the larger annular zone and the width of the zones, respectively. If the measurement is performed within this range of depths, the shape and size of the PSFs remain almost unchanged approaching well the theoretical predictions carried out with the narrow rings given by the Dirac delta function. When the measurements are made beyond the specified range, the PSFs are blurred and their lobes became deformed. These effects complicate the numerical processing of measured data and deteriorate both the accuracy and the precision of the depth reconstruction.

3. Experimental setup

The basic principle of reference-free vortex topographic microscopy was verified in the setup shown in Fig. 2. The developed system is based on a commercial microscope Nikon Eclipse E600 utilizing add-on illumination and imaging modules. These modules provide structured illumination of the sample and the transformation of reflected light waves into pairs of interfering optical vortices, respectively. The illumination module is composed of a LED (Thorlabs M625L3, peak wavelength 630 nm), a custom-made MHP attached to stepper motor (Thorlabs K10CR1/M) and a collector lens CL (Nikon second objective lens, $f = 200$ mm). The MHP was prepared by a laser micromachining system produced by Oxford lasers. Beryllium-bronze sheet was chosen as an optimal material for the fabrication of the MHP. The thickness of the material was 180 µm and the diameter of the individual holes was 22.3 µm. In the MHP testing, 30 holes located in different parts of the field of view were measured and the standard deviation of the diameter $\pm$ 2.3 µm was determined. The spacing of the holes was approximately 200 µm. The MHP contains 3800 holes arranged in the arms of Archimedean spirals with a constant pitch. An active LED area is projected onto the MHP and light transmitted through the individual holes is transferred to the sample using collector lens CL, mirror M, non-polarizing beam splitter BS and microscope objective MO (Nikon CFI Plan APO 60x/1.2 WI). With this critical illumination path, an array of mutually uncorrelated light spots is created at the sample. The images of the holes are reduced 60 times so that the in focus light spots are in size comparable to the Airy diffraction pattern of the MO. The light specularly reflected from the sample is captured by the MO and directed through the beam splitter BS and 4f system composed of lenses L1 and L2 (Nikon second objective lenses, $f_{1} = f_{2} = 200$ mm) toward the SPM placed at the front focal plane of the tube lens TL (Thorlabs AC-508-400-A). In the optical configuration used, the back focal plane of the MO is projected onto the SPM that modulates the spatial spectrum of light reflected from the sample. Since the back focal plane is projected with unitary magnification, conditions considered in theoretical modeling are preserved. The SPM was designed using Fisher information theory and custom made by electron beam lithography to perform both amplitude and phase modulation. The amplitude modulation is introduced by the opaque areas of the SPM, where the light is blocked by thin metal layer. In the transparent zones, the optical path increases with the azimuthal angle achieving the stroke equal to the wavelength $λ$ for the angle $2 π .$ Light waves impinging on the SPM are transmitted through two annular zones with the middle radii $r_{1} = 1.34$ mm and $r_{2} = 1.9$ mm and the width $Δ r = 0.25$ mm. In the zones of the SPM, a spiral phase modulation characterized by the topological charges $l_{1} = 1$ and $l_{2} = - 1$ is performed. The spiral phase does not change continuously and takes 8 discrete levels in segments with the angle $π / 4$ . By optical Fourier transform implemented by the TL, each wave impinging on the SPM is split to two non-diffractive vortex beams whose interference creates the double-helix PSF. In this way, an array of angularly rotated PSFs is created at a CCD (Ximea MR4021MC-BH) and used in the three-dimensional reconstruction of the tested surface. In calibration measurement, the sample was loaded on a piezoelectric transducer (Physik Instrumente P-621.ZCD) enabling the change of the measured depth in a controlled and high-precision manner.

4. Automated experiment execution and three-dimensional image reconstruction

All activities running during the measurement and system calibration were controlled by software for automated experiment execution created in LabView development environment. The developed software enables control of the piezoelectric transducer with loaded sample, the stepper motor with the MHP and CCD. To control the hardware distributed by different manufacturers, their libraries produced for LabView environment were applied. The control software can be divided into three main parts. In the first part, hardware is initialized and communication between computer and hardware is established. The second part performs manual or automated hardware control. Manual control enables independent adjustment of CCD, stepper motor and piezoelectric transducer. Through the manual control also the initial hardware parameters can be set for the automated measurement procedures. In automated regimes, CCD frame per step of either stepper motor or piezoelectric transducer can be captured. The last part of the software terminates the communication between the computer and hardware.

Measured data were processed and reconstructed in Matlab. Algorithm in Matlab was developed for the automated evaluation of the angular rotation and the lateral position of the double-helix PSFs obtained from the experimental data. The algorithm includes several steps, which are schematically shown in Fig. 3. The input for the algorithm was formed by TIFF images gained in the experiments, containing the lattice of double-helix PSFs. The double-helix PSFs encode the local depths of the sample and the lateral measurement positions. The images were firstly pre-processed in order to obtain reliable data for the reconstruction. Each image was normalized by scaling the pixel values to a fixed range from 0 to 1. Subsequently, thresholding was performed for each image, while producing a binary mask. To cope with the non-uniform intensity over the image, the local adaptive thresholding was used. The double-helix PSFs were segmented from the background by multiplying the binary mask with the original image. A set of morphological operations were performed for the purpose of filtering out PSFs containing artifacts. PSFs located on the border of the image were excluded from the image as well. Remaining PSFs were labeled as objects and their centers of the mass were determined. To retrieve the axial and lateral positions of the sample, the image subregion (20 x 20 pixels) was cropped from the original image around each detected center of the mass. Each subregion contained a single PSF and was thresholded while creating a binary mask. By multiplying the binary mask with the image subregion, the images containing two segmented PSF lobes were obtained. Segmented PSF lobes were labeled and their individual centers of the mass were detected allowing for the calculation of the angular rotation of the double-helix PSF. The angular rotation was subsequently converted to axial position (local depth of the tested sample). The reconstruction was performed on PC with processor Intel^® Core^TM i5–2320, 8 GB RAM, 6 MB cache and processor frequency 3.00 GHz. In the procedure, the CCD images having the size of 2048x2048 pixels and containing approximately 550 double-helix PSFs were used. The execution time of one image was 16.4 seconds.

Fig. 3 Workflow of the evaluation of angular rotation and lateral position of the double-helix PSF: (a) input TIFF image (b) zoomed part of the input image with the array of double-helix PSFs, (c) binary mask obtained by local adaptive thresholding of the input image, (d) labeled double-helix PSFs, (e) segmented double-helix PSFs with detected centroids, (f) cropped image subregion containing a single PSF, (g) binary mask obtained by thresholding, (h) labeled PSF lobes, (i) segmented PSF lobes with centroids determining the angular rotation of the double-helix PSF.

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5. Results and discussion

In realized experiments, the basic principle of self-interference vortex topography was successfully verified. Although the effort was not primarily focused on exploring the optical limits of the method, the accuracy and precision of the used configuration were assessed with promising results. To demonstrate the application potential of the developed technique, full-field image reconstruction of the top surface of a bearing ball with a cap height exceeding the depth of focus of the MO more than 10 times was successfully realized from a single CCD record. Using a stack of CCD frames obtained by gradually rotating the MHP, three-dimensional imaging of a reflection phase grating was performed. For the comparison of full-field and scanning operation of the system additional experiment with bearing ball was performed, providing a high resolution surface reconstruction.

Calibration and accuracy of measurement

The surface depth is determined by the angular rotation of the double-helix PSF generated from the light reflected at the measuring position. The theoretical dependence of the rotation angle $Φ$ on the depth $Δ z_{0}$ is linear and follows from Eq. (5). In the experiments, the linearity is maintained in a good approximation, but the slope and intercept of a regression line must be specified by the calibration measurement.

In the system calibration, plane mirror loaded on a piezoelectric transducer was used as a sample. Before measuring, the piezoelectric transducer was calibrated interferometrically to provide ground truth axial positions. In the measurement, the plane mirror was illuminated by light spots created by the MHP and a lattice of double-helix PSFs was generated. Using the piezoelectric transducer, the mirror scanned axially in the range of 11 µm with 300 nm step, hence the surface was mapped at 37 different depth positions. This measurement procedure was repeated 16 times. The calibration data were obtained by evaluating the rotation of nearly 600 PSFs collected from all parts of the entire field of view of 130x130 µm². By processing the individual double-helix PSFs, the dependence of the PSF rotation on the depth set by the piezoelectric transducer was obtained by linear fitting the measured data. The calibration curve is shown in Fig. 4(a), where its slope and intercept are also specified. The angular rotations of the recorded PSFs were converted to the depths by means of the calibration curve and the accuracy of measurement was evaluated using the ground truth depths provided by the piezoelectric transducer. Results obtained are shown in Fig. 4(b), where the bars and the error bars represent the mean error (ME) and the root mean square error (RMSE) of the recovered depth, respectively. The depth measurements in the range of 11 µm were carried out with ME and RMSE better than 50 nm and 100 nm, respectively.

Fig. 4 Calibration of the system and evaluation of measuring accuracy. (a) Measured angular rotation of the double-helix PSF plotted against the ground truth depth provided by the interferometrically calibrated piezoelectric transducer and the regression line specified by the slope and intercept. (b) Accuracy of the measurement assessed in the depth range of 11 µm and represented by the mean error ME (bars) and the root mean square error RMSE (error bars).

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When testing reflective surfaces in following experiments, the angular rotations of the PSFs recorded in different lateral positions were converted to the local depths using the calibration curve.

Residual flatness and precision of measurement

The proposed vortex topography benefits from the spatially invariant double-helix PSF generated using a specially designed SPM. The spatial invariance ensures that the PSF is shift-invariant (isoplanatic) over the entire field of view of the MO and rotates when measuring depths, while maintaining the shape and size unchanged. To demonstrate the unique properties of the PSF, the residual flatness (RF) was evaluated using a plane mirror as a sample. The flat surface of the mirror was repeatedly measured at 9 positions equidistantly set in the range of 9.6 µm by the piezoelectric transducer. At adjusted mirror positions, image array composed of 550 double-helix PSFs was captured and the angular rotation of each PSF was automatically evaluated in the developed software. Using the calibration curve, the depths at the individual lateral positions were determined and the fitting plane was created, enabling tilt elimination of the measured surface.

The RF was obtained by comparing the measured data with the values given by the best fit plane. The results obtained are presented in Fig. 5. In Fig. 5(a), the RMSE for the RF values obtained by evaluating all 550 double-helix PSFs of the image array is demonstrated in dependence on the axial position $Δ z_{0}$ of the plane mirror. Throughout the whole range of measured depths, the RMSE better than 70 nm was obtained. The spatial changes of the RF are shown in Figs. 5(b)–5(e) for the mirror position $Δ z_{0} = 1.2$ µm. In Fig. 5(b), the three-dimensional distribution of the RF in the field of view of the MO is shown, together with the color-coded RF values projected to the planes $x_{0} - y_{0}$ , $x_{0} - Δ z_{0}$ and $y_{0} - Δ z_{0}$ [Figs. 5(c)–5(e)].

Fig. 5 Evaluation of the residual flatness (RF) at 9 different axial positions of a plane mirror within the range of 9.6 µm. (a) RMSE obtained by processing 550 double-helix PSFs of the image array captured in each axial position of the plane mirror. (b) Three-dimensional distribution of the RF in the field of view of the MO for axial position of the plane mirror $Δ z_{0} = 1.2$ µm. (c)-(e) Projections of the color-coded RF to $x_{0} - y_{0},$ $x_{0} - Δ z_{0}$ and $y_{0} - Δ z_{0}$ planes.

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In the realized experiments, the precision of surface reconstruction was tested with a plane mirror scanning axially in the range of 11 µm. The mirror loaded on the piezoelectric transducer was sequentially placed in 10 positions and a stack of 100 images was recorded in each of them. In the individual image arrays, representative double-helix PSFs located in central part and all corners of the field of view were selected. Using the developed software, the lateral positions of the PSFs were determined and their angular rotations converted to the depths by means of the calibration curve. By processing a stack of 100 records, the standard deviations $σ_{x}, σ_{y}$ and $σ_{z}$ were determined for the lateral and axial coordinates in each position of the mirror. The same procedure was repeated 5 times in order to get the standard deviations of the independent measurements. The results obtained are shown in Figs. 6(a)–6(c). The standard deviation of the depth better than 30 nm was achieved in the whole axial range of 11 µm. The lateral coordinates of the depth measuring positions were determined with the standard deviations better than 11 nm.

Fig. 6 Demonstration of the measurement precision evaluated in 10 axial positions equidistantly spaced in the axial range of 11 µm. (a), (b) Standard deviations of the measured lateral positions. (c) Standard deviation of the measured depth. The standard deviations $σ_{x},$ $σ_{y}$ and $σ_{z}$ were obtained by processing 100 images recorded at each axial position. Error bars correspond to 5 independent measurements.

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6. Demonstration of practical applicability

Application potential of the vortex topographic microscopy was tested in different types of experiments involving full-field reconstruction of a deep smooth surface and scanning imaging of a periodic surface.

Full-field three-dimensional image reconstruction

When measuring a surface whose depth varies slowly, light spots created by the MHP cover the surface with sufficient density so that the full-field reconstruction can be made from a single CCD record. More importantly, the generated double-helix PSFs are spatially invariant allowing for successful reconstruction even when the depth of the surface significantly exceeds the depth of field of the MO. These advantages of the method were fully utilized in the experiment demonstrating reconstruction of the top surface of a bearing ball. The cap height of the surface was 6 µm, hence the axial range of the measurement exceeded the depth of field of the MO more than 10 times. The experimental results are presented in Fig. 7. The conventional image in Fig. 7(a) was obtained by focusing on the peripheral part of the surface marked by the arrows. When recording the surface image in Fig. 7(b), the structured illumination was used and conventional image spots were created without using the SPM. Image spots in the central part of the field of view are significantly blurred, which clearly shows that the cap height is out of the depth of field of the MO. Inserting the MHP and the SPM into the illumination and imaging path, the array of double-helix PSFs was generated and captured by the CCD. The array of image spots obtained for the dashed line area (c) in Fig. 7(a) is illustrated in Fig. 7(c). Since the PSFs are spatially invariant, their shape and size remain unchanged regardless of whether they originate from in focus or out of focus areas of the surface. The height profile of the surface was reconstructed from angular rotations of the PSFs included in a single CCD record. Surface reconstruction with color-coded height in both $x_{0} - y_{0}$ projection and three-dimensional visualization are shown in Fig. 7(d) and Fig. 7(e). The same procedure was repeated for 5 different angular positions of the MHP to obtain independent data used to determine the standard deviation of the measurement. Reconstructed local depths of the surface were fitted by a sphere enabling estimation of the ball diameter. Although the spherical cap was mapped only in the area of 130x130 µm² given by the field of view of the MO, the diameter of the fitting sphere 1.962 $\pm$ 0.015 mm corresponds well to the value 1.99 $\pm$ 0.01 mm provided by the manufacturer.

Fig. 7 Reconstruction of the top surface of bearing ball in vortex topographic microscopy. (a) Conventional bright-field image focused on a peripheral part of the field of view. (b) Conventional image using structured illumination whose blurred spots in the central part are out of focus of the MO. (c) Array of double-helix PSFs generated with the MHP and the SPM in illumination and imaging path, respectively. The PSFs are spatially invariant and their shape and size are preserved in the entire field of view (here demonstrated for dotted square area in (a)). (d) Color-coded height map of the surface. (e) Three-dimensional visualization of the reconstructed surface.

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Three-dimensional imaging with high spatial resolution

When measuring specimens requiring high spatial resolution, angular scanning of a specially designed MHP can be fully utilized. Since the holes are arranged in the arms of Archimedean spiral, a nearly continuous mapping of the measured surface can be achieved when sufficiently fine angular step is used during rotation of the MHP. In the realized experiment, the rotation of the MHP is carried out by means of a stepper motor which allows changing the number of frames recorded during one full turn of the MHP. In this way, the lateral sampling can be adapted to the measured surface profile. Imaging with a different number of frames taken during one turn of the MHP is demonstrated in Fig. 8. In Fig. 8(a), a reference bright-field image of the USAF resolution target is shown, recorded under conditions of the conventional microscopy. When the MHP was inserted into the illumination path, a full-field record with lateral sampling insufficient to image the USAF resolution target was obtained [Fig. 8(b)]. The image of the USAF target in which low-frequency groups are resolved was obtained from a stack of 30 frames taken during one full turn of the MHP carried out with the step of 12° [Fig. 8(c)]. With the angular step 1° providing a stack of 360 frames, the resolution of the USAF target comparable to the reference bright-field image was obtained [Fig. 8(d)]. The images demonstrated in Figs. 8(b)–8(d) were obtained by conventional imaging of the amplitude object. If the experiment is performed with the SPM allowing depth estimation, the demonstrated mechanism of variable lateral sampling can also be utilized in surface topography measurements.

Fig. 8 Images of the 1951 USAF resolution target reconstructed using different conditions of the structured illumination. (a) Reference bright-field image obtained by conventional microscopy. (b) Insufficiently sampled full-field image taken with the MHP in a fixed position. (c) Low-resolution image reconstructed from 30 frames taken during one full turn of the MHP performed with the step of 12°. (d) High-resolution image reconstructed from 360 frames taken during one full turn of the MHP performed with the step of 1°.

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Reconstruction using the angularly scanning MHP was successfully demonstrated in the experiment with a reflection phase grating (Fig. 9). In Fig. 9(a), a conventional image of the grating is shown together with a cropped area illustrating the reconstruction of the grating profile [Fig. 9(b)]. In this picture, several PSFs were superimposed numerically with the bright-field grating image to demonstrate how the angular orientation of the individual PSFs varies across the grating grooves. To increase the density of the measuring points covering the grating surface, the MHP was rotated with the step of 1°. With one full turn of the MHP a stack of 360 frames was obtained, providing approximately $1.6 \times 10^{5}$ double-helix PSFs. Determining the angular rotation of the PSFs, the grating surface was quantified in the corresponding lateral positions. Principle of the MHP scanning and grating reconstruction are demonstrated in Visualization 1. The reconstructed grating is shown in Fig. 9(c) by a color-coded height map and in Fig. 9(d) by a three-dimensional visualization. Evaluating the grating profile at five different parts of the field of view, the stroke and the period of the grating were determined as 711 $\pm$ 63 nm and 6.46 $\pm$ 0.38 µm, respectively. For better documentation of three-dimensional grating reconstruction, orthogonal slice planes through the volumetric data are presented in Visualization 2.

Fig. 9 Reconstruction of the phase diffraction grating in vortex topographic microscopy. (a) Conventional bright-field image. (b) The double-helix PSFs superimposed numerically with bright-field image of two grating grooves. Rotation of the PSFs indicates height changes across the grating profile. (c) Color-coded height map of the phase grating. (d) Three-dimensional visualization of the phase grating. In demonstrations (c) and (d) the phase grating was reconstructed at $1.6 \times 10^{5}$ lateral positions.

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To demonstrate the high-resolution reconstruction also in extended axial range, the top surface of the bearing ball was recorded in scanning mode. When recording the data, one whole revolution of the MHP was carried out with the step of 1°. In this way, a stack of 360 image arrays with 550 double-helix PSFs in each was obtained. By processing the data in the developed software, the depth of the surface was successfully mapped approximately in $1.8 \times 10^{5}$ lateral positions. Comparison of full-field reconstruction obtained by interpolation of 500 points successfully measured in single CCD frame [marked by circles in Fig. 10(a)] and reconstruction of all measured points is shown by color-coded height maps in Fig. 10(a) and Fig. 10(b). Subtracting the bearing ball image obtained by interpolation of quasi-continuous reconstruction [Fig. 10(b)] from the single-frame reconstruction [Fig. 10(a)], the RMSE of residual distances better than 100 nm was obtained in the entire field of view. Figure 10(c) demonstrates three-dimensional visualization of the height map from Fig. 10(b). The three-dimensional image reconstruction is further supported by Visualization 3 presenting orthogonal slice planes through the volumetric data.

Fig. 10 (a) Color-coded height map representing full-field reconstruction of the bearing ball obtained by interpolation of 500 points (marked by circles) measured in single CCD frame. (b) Color-coded height map of the bearing ball reconstructed from 360 images recorded in scanning mode. (c) Three-dimensional visualization of the height map (b). In demonstrations (b) and (c) the bearing ball was reconstructed at $1.8 \times 10^{5}$ lateral positions.

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The demonstrated experiments are just selected application examples of the vortex topographic microscopy whose potential can be further developed in several directions. Although the optical surface topography has been widely deployed, the proposed technique can be advantageously incorporated due to its versatility and extremely high robustness. With the same microscope objective, the measurement accuracy and the range of measured depths can be operatively altered according to application requirements. In the experiments performed, the demonstrated axial resolution did not reach the standards common in the phase-shifting interference topography. However, since the optical performance of the interference topography is maintained in a very small depth range corresponding to a quarter of wavelength, the vortex topography can be considered a good way to increase the range of measured depths without scanning, provided a slight reduction in the axial resolution is accepted.

Although the vortex topography utilizes light interference, high implementation requirements that are common in interferometry are not applied. The robustness of the system is ensured by a common-path geometry in which the measured depths are obtained by self-interference of two vortex beams coming from the same sample point. Another advantage of the self-interference vortex technique is determination of the depth from the image rotation that eliminates problems with identifying interference fringes appearing in other interference methods. Vortex topographic microscopy can also be further developed as a good alternative to white light scanning interferometry and focal variation methods in which an increased range of measured depths is required while accepting a lower accuracy. Unlike the standard methods using lateral and axial scanning, the vortex topography provides an extremely large range of depths even in a single wide-field image. The depth range can be further enhanced by the controlled axial shifting of the sample and the reconstruction of the surface by merging several image recordings. Significant technical improvements of the system can be achieved by using a microlens array attached to the MHP according to the design of Yokogawa spinning disk. By this way the light efficiency of the system can be apparently increased. To accelerate the reconstruction of the surfaces requiring angular scanning of the MHP, a high-speed CCD can be deployed.

7. Conclusion

We present the original concept of vortex topographic microscopy that allows the full-field imaging and testing of reflective samples without using a reference wave. The method is based on the efficient connection of the structured illumination with appropriately adapted and upgraded vortex localization imaging that previously enabled super-resolution in photoactivation microscopy. The experimental demonstrations were carried out using a conventional microscope with modified illumination and imaging paths.

In the developed technique, the surface is mapped in the array of measuring points whose number and positions can be adjusted depending on the geometry of the sample being measured. The depth of the surface is reconstructed from the angular rotation of the double-helix PSFs created in individual measuring positions by the self-interference of optical vortices. The optical performance of the method is favorably enhanced by the use of a spiral phase mask generating non-diffractive optical vortices that provide spatially invariant PSFs. With this unique feature, the phase can be reconstructed in the extremely large range of depths, maintaining high precision and accuracy. In the calibration measurement, the depth of the surface was reconstructed in the range exceeding the depth of field of the microscope objective up to 19 times, while maintaining the precision better than 30 nm. Lateral positions, where the depth was measured, were determined with the precision better than 10 nm. Applicability of the developed technique was demonstrated by full-field reconstruction of a deep smooth surface and scanning imaging of a periodic surface.

A potential of the vortex topographic microscopy lies in delivering interferometric precision to a long-range focus variation measurement implemented without the need of a perfect reference surface or a vertical scanning. The vortex self-interference results in the shaping of fringe-free double-helix PSFs, hence the depth estimation avoids problems with unresolved interference fringes or ambiguity in their numbering. The demonstrated proof of principle experiments can be further developed to provide a compromise between the precision and the range of measured depths or to adapt to the specific requirements of imaging and testing applications. Quantitative phase imaging of specimens placed on reflective substrates or topography of diffusion surfaces are still remaining a challenge.

Funding

Grant Agency of the Czech Republic (No. 15-14612S); Central European Institute of Technology (CZ.1.05/1.1.00/02.0068).

Acknowledgments

The authors thank T. Axman from Central European Institute of Technology, Brno University of Technology for fabrication of micro-hole plate and V. Kolařík and members of his group from Institute of Scientific Instruments of the Czech Academy of Sciences in Brno for realization of the spiral masks.

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Name	Description
Visualization 1	Depth measurement using angular scanning of the micro-hole plate.
Visualization 2	Three-dimensional visualization of the grating reconstruction.
Visualization 3	Three-dimensional visualization of the bearing ball reconstruction.

Vortex topographic microscopy for full-field reference-free imaging and testing

Abstract

1. Introduction

2. Principle of reference-free vortex array topography

3. Experimental setup

4. Automated experiment execution and three-dimensional image reconstruction

5. Results and discussion

Calibration and accuracy of measurement

Residual flatness and precision of measurement

6. Demonstration of practical applicability

Full-field three-dimensional image reconstruction

Three-dimensional imaging with high spatial resolution

7. Conclusion

Funding

Acknowledgments

References and links

Supplementary Material (3)

Cited By

Figures (10)

Equations (8)

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