January 1, 2014 / Vol. 39, No. 1 / OPTICS LETTERS
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Surface profilometry with binary axicon-vortex and lens-vortex optical elements D. Wojnowski,1 E. Jankowska,2 J. Masajada,2,* J. Suszek,1 I. Augustyniak,2 A. Popiolek-Masajada,2 I. Ducin,1 K. Kakarenko,1 and M. Sypek1 1 2
Faculty of Physics, Warsaw University of Technology, Koszykowa 75, 00-662 Warsaw, Poland
Institute of Physics, Wroclaw University of Technology, Wyspianskiego 27, 50-370 Wroclaw, Poland *Corresponding author: [email protected] Received September 17, 2013; revised November 22, 2013; accepted November 22, 2013; posted November 25, 2013 (Doc. ID 197565); published December 23, 2013
We report on the interesting effect observed with the diffractive binary element, which matches the property of an axicon and vortex lens. Binary phase coding simplifies the manufacturing process and gives additional advantages for metrology purposes. Under laser beam illumination, our element produces two waves: converging and diverging. Both waves carry a single optical vortex. We show that this special diffractive element can be used to set up a simple surface profilometer. © 2013 Optical Society of America OCIS codes: (050.1380) Binary optics; (050.1970) Diffractive optics; (050.4865) Optical vortices; (180.3170) Interference microscopy. http://dx.doi.org/10.1364/OL.39.000119
Diffractive optical elements are thin phase elements that operate by means of interference and diffraction to produce an arbitrary distribution of light. To produce the high-quality diffractive optical elements for visible light, the tools for drawing details with resolution better than 100 nm are necessary. The main reason for this is diffraction efficiency. To achieve 100% diffraction efficiency, the profile of each detail must be drawn continuously. In practice, the continuous profile is divided into steps (levels). The more levels, the better efficiency of the diffractive element. Manufacturers usually offer from two (binary elements) up to 16 level elements with theoretical diffraction efficiency from 41% to 99%. For some applications, the typical drawback of the binary elements (e.g., diffraction efficiency of 41% for −1 and 1 diffractive orders) becomes useful. In this Letter, we describe the diffraction element, which combines an axicon and a vortex lens (AV element). We planned the element for enhancing the measurement methodology reported in [1,2]. For the test of principles, it was manufactured in the simplest way (i.e., as a binary element). But working with it changed our approach. This binary element revealed some possibilities that might be useful for metrology. According to the classical definition by McLeod [3], an axicon is an optical element that images a point into a line segment along the optical axis. The length of the line can vary from a couple of millimeters up to several meters, depending on the design of the axicon. The simplest axicon is manufactured as refractive cones. To get better imaging, various modifications have been proposed [4,5]. The spiral phase plate is an optical element whose optical thickness increases in the azimuthal direction, around the plate center; thus having a transmission function Tθ expfimθg; m integer; θ azimuthal coordinate defined in the plate plane. It is a well-known element that transforms the Gaussian beam into an output vortex beam, i.e., beam carrying an optical vortex [6,7]. The spiral phase plate is also known as a vortex lens [8,9]. 0146-9592/14/010119-04$15.00/0
In [2] the four-level diffractive element combining the lens and the vortex lens of topological charge 2 was used to find the subwavelength phase step position with an accuracy of 60 nm. The focal length of the element was 38 mm, and the system was working with 633 nm He–Ne laser light. Obviously, the step position was found with resolution beyond the diffraction limit. In [1] the optical vortex beam was used for the quality inspection of the high-aspect-ratio micro-samples (HARMS) elements. HARMS elements consist of a matrix of wells or pillars whose lateral size is 30–70 μm and depth is 300–1000 μm. A simple optical system enabled fast and ambiguity-free quality inspection. The new diffractive element combining the axicon and vortex lens was designed for enhancing the potential of HARMS inspection method. Simple experiments show that it also can be used in the other way. When the binary AV element is illuminated, two waves emerge from the system (Fig. 1). Both beams carry an optical vortex. For a high-quality binary element, no zero-order beam should be detected. A series of binary AV elements was fabricated in order to verify the far-field intensity distributions. The manufactured elements had topological charges m 1 and m 2 and were designed for a wavelength λ 632.8 nm. The numerical design consisted of two steps: calculation of the annular linear axicon [4] forming a focal section on the distance from f 1 47 mm to f 2 51 mm [Eq. (1)]
Fig. 1. When illuminated by Gaussian beam, the AV element produces a diverging and a converging beam. Both beams carry the optical vortex. © 2014 Optical Society of America
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and multiplication by spiral phase distribution [Eq. (2)] with topological charge of m 1 or m 2. The height of the phase step was optimized for the required wavelength, knowing the type of photo-resist. The plates were exposed based on numerical design on matrices of 8192 × 8192 pixels, with a sampling of 1 μm. The design parameters of the modeled and/or manufactured structures were as follows: the radius of the aperture was R 4 mm. The equation of the phase distribution of the axicon was q r 2 · f 22 − f 21 ∕R2 f 21 −2π · : φr λ f 22 − f 21 ∕R2
second starts approximately at twice a focal length from the AV element and extends to infinity. At this area, the interference between two vortex beams is clearly visible as it is shown at the first (z 20 mm) and last (z 80 mm) figures in Figs. 4 and 5. No such effects are present in the image of kinoform AV element (Fig. 3).
(1)
This was combined with the spiral phase distribution of the expression φθ imθ:
(2)
Finally, the resulting phase patterns were binarized. The elements were fabricated using the electron-beam lithography (EBL) method on a JENOPTIK ZBA20 machine. A rectangular electron beam with a variable stamp size was used with a voltage of 20 kV. The nominal resolution of the writing machine was 0.2 μm for a single line and 0.4 μm for periodical structures. The size of the rectangular e-beam stamp was variable between 0.2 and 6.3 μm independently in the x and y directions. The available positioning of the beam was 0.01 μm. Figure 2 shows the profile of one of the fabricated structures (topological charge m 1). Figures 3 and 4 show the phase and intensity distributions calculated for hypothetical structures with a continuous phase profile (Fig. 3) and for a manufactured binary phase profile (Fig. 4). Figure 5 shows the phase and intensity profile for the optical vortex having topological charge m 2, which is calculated for binary AV. At the focal plane, the converging beam creates a small spot, and the diverging beam is widely spread (Fig. 1). There are two areas, at some distance from AV element, where both beams possess a similar wavefront curvature. The first area is closed behind the AV element. The
Fig. 2. Measurements of the AV element of m 1. (a) A photograph of the central part of the plate seen with an optical microscope. (b) An AFM scan of a fragment of the structure (the center of the scanned fragment is located approximately 3.14 mm from the center of the element). (c) A profile of the scanned fragment.
Fig. 3. AV element manufactured as kinoform will generate only the converging beam. Figure shows the numerical simulations; z denotes the distance between the element and the observation plane.
Fig. 4. Binary AV element for m 1 will produce converging and diverging beams. This effect is easily visible in the area where both beams have a comparable radius of curvature, i.e., close (z 20 mm) and far from the AV element (z 80 mm). At z 20 mm and z 80 mm, the amplitude and phase distribution reveals characteristic features for the overlapping vortex beams. Figure shows the numerical simulations.
Fig. 5. Binary AV element for m 2 works in a similar way as the element shown in Fig. 4. However, now we can see characteristic features for the overlapping vortex beams having double topological charge (for z 20 and z 80 mm).
January 1, 2014 / Vol. 39, No. 1 / OPTICS LETTERS
Fig. 6. Binary AV element of charge m 1. The experimental results reveal the same features as the numerical simulations (Fig. 4).
Figures 6 and 7 show experimental results for AV element having topological charge 1 or 2, respectively. Due to calculation time, Figs. 4 and 5 cover central areas of Figs. 6 and 7. The binary AV element can be used as a surface topography analyzer, which has a simple design. The sample is located at the focal plane of the converging beam. Since the focused spot is small, the beam is sensitive for the small sample details. The diverging beam illuminates the large sample area, and the small details do not have measurable influence on it. At the observation plane, both beams interfere. In the experiment, the system shown in Fig. 8 was used. To analyze the interferograms taken in our system, the carrier frequency method proposed by Takeda et al. [10] was applied. This method worked well for the vortex microscopy [11]. We scanned the sample with the phase step, which introduces the 3π-phase jump. In the vicinity of the edge, the vortices are cancelled (Fig. 9). When the phase jump is different than π (calculated modulo 2π), the vortices behave in a different way, which is illustrated in Fig. 10, showing the results of numerical modeling. Please note that higher sensitivities in the z direction should be possible for a Fresnel lens combined with an OV (FV element). Thus we calculated edge detection with a binary FV plate (focal length f 50 mm; topological charge m 1). Figure 10 shows phase distributions at the plane positioned 50 mm after the focal plane of the
Fig. 7. Binary AV element of charge m 2. The experimental results reveal the same features as the numerical simulations (Fig. 5).
121
Fig. 8. Measurement system is based on the Mach–Zehnder interferometer scheme.
plate. Edge detection with a binary FV element is certainly possible down to the 0.23π high phase step. It may be possible even for lower steps; however, it shall be difficult. We also confirmed that application of the AV element into profilometry is possible. At first we measured a highquality optical wedge with the wedge angle 90 s of arc. Shifting the wedge by 3 mm introduced no change in the vortex position, but the phase image around vortex points was changing (the total phase shift introduced by this movement exceeds the wavelength of the laser beam). In our system, we could observe this phase change with sensitivity of λ∕40. This was done by subtracting the subsequent phase images. Next, the optical wedge with damaged area was measured. In the damaged area, the wedge was extra-polished to make a valley. The wedge geometry was measured before, and the results were published in [12]. Figure 11 shows the exemplary experimental data. The focused spot diameter was about 15 μm. The scanned distance shown in Fig. 11 was 10 μm wide. At this diameter, the slope angle difference at the center of the beam at first position and at the last position was smaller than 2°. Nevertheless, its influence on the vortex points positions was measurable. Numerical simulations are comparable to this result (see Fig. 12).
Fig. 9. Phase distribution and interferograms measured while scanning through the phase step, which introduces a phase shift of ∼3π. (a) The beam center is positioned −5 μm from the step. (b) The beam center is positioned at the step. (c) The beam center is positioned 5 μm from the step. There are characteristic fork-like fringes indicating the presence of optical vortices in the interferograms (a) and (c). These fork-like fringes disappear in the interferogram (b).
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Fig. 10. Phase distributions at the plane positioned 50 mm behind the focal plane of the FV element, where the phase step was introduced. The Δ parameter denotes the distance between the phase-step and z axis. The h parameter denotes the phase difference introduced by the step.
Fig. 11. Phase maps measured while scanning the damaged optical wedge. The wedge was shifted by 2 μm for each next measurement. The scanning beam center lies at the area of the valley. It is well seen that vortices come apart while scanning the wedge.
Fig. 12. Phase distributions at the plane positioned 50 mm after the focal plane of the FV element, where the phase valley was introduced. The Δ parameter denotes the distance between the center of the valley and z axis.
Figure 12 shows numerical effects of scanning a phasevalley described with equation: φr −34861
1 1 · r − 4; 485 · r 2 794 2 mm mm
(3)
for radius r ≤ 10 μm. This corresponds to average inclination of c.a. 5 degrees in a material with a refractive index of n 2.2. The slope angle difference of 2° seems to be large for optical measurements. However, this slope was detected at the scanning range of 10 μm, so the local angle difference between subsequent scanning steps was smaller than 30 min of arc. This corresponds to the average phase difference smaller than 2π∕50 for each scanning step of the 2 μm size. We presented successful application of a binary AV element for metrology purposes. Binary coding is not only easier to manufacture (compared to multilevel or continuous profile), but it has an advantage of giving interference between 1 and −1 diffractive orders, which is useful for metrology. With the simple element, we could sense the phase difference as small as π∕25. These results can be improved by careful AV or FV elements optimization for surface topography application. The system also can be used with no reference beam. Thus the system becomes much simpler, but its accuracy is lower mainly because interferometric vortex localization methods are more accurate than methods based on the intensity pattern inspection. The questions of fast and effective algorithms for surface geometry reconstructions from the results presented in Fig. 11 and measurements of soft phase objects (like biological objects) are the subject of intense research. This work was supported by the Polish Ministry of Scientific Research and Information Technology (grant N505463438), by the European Social Fund implemented under the Human Capital Operational Programme (POKL), project: “Preparation and Realization of Medical Physics Specialty” and by the National Laboratory of Quantum Technologies (project POIG. 02.02.00-00-003/ 08-00). We also thank Michal Makowski from the Warsaw University of Technology. References 1. J. Masajada, M. Leniec, E. Jankowska, H. Thienpont, H. Ottevaere, and V. Gomez, Opt. Express 16, 19179 (2008). 2. J. Masajada, M. Leniec, S. Drobczynski, H. Thienpont, and B. Kress, Opt. Express 17, 16144 (2009). 3. J. H. McLeod, J. Opt. Soc. Am. 44, 592 (1954). 4. J. Sochacki, A. Kolodziejczyk, Z. Jaroszewicz, and S. Bara, Appl. Opt. 31, 5326 (1992). 5. Z. Jaroszewicz, J. Sochacki, A. Kolodziejczyk, and L. R. Staronski, Opt. Lett. 18, 1893 (1993). 6. S. N. Khonina, V. V. Kotlyar, M. V. Shinkaryev, V. A. Soifer, and G. V. Uspleniev, J. Mod. Opt. 39, 1147 (1992). 7. L. Allen, M. J. Padgett, and M. Babiker, Progress in Optics, E. Wolf, ed. (Elsevier, 1999), Vol. 39, Chap. IV, pp. 291–372. 8. G. A. Swartzlander, Jr., Opt. Lett. 26, 497 (2001). 9. G. A. Swartzlander, Jr., Opt. Lett. 31, 2042 (2006). 10. M. Takeda, H. Ina, and S. Kobayashi, J. Opt. Soc. Am. 72, 156 (1982). 11. J. Masajada, A. Popiolek-Masajada, and I. Augustyniak, Proc. SPIE 8550, 85503B (2013). 12. A. Popiolek-Masajada, M. Borwinska, T. Przerwa-Tetmajer, and P. Kurzynowski, Opt. Laser Technol. 48, 503 (2013).
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Surface profilometry with binary axicon-vortex and lens-vortex optical elements D. Wojnowski,1 E. Jankowska,2 J. Masajada,2,* J. Suszek,1 I. Augustyniak,2 A. Popiolek-Masajada,2 I. Ducin,1 K. Kakarenko,1 and M. Sypek1 1 2
Faculty of Physics, Warsaw University of Technology, Koszykowa 75, 00-662 Warsaw, Poland
Institute of Physics, Wroclaw University of Technology, Wyspianskiego 27, 50-370 Wroclaw, Poland *Corresponding author: [email protected] Received September 17, 2013; revised November 22, 2013; accepted November 22, 2013; posted November 25, 2013 (Doc. ID 197565); published December 23, 2013
We report on the interesting effect observed with the diffractive binary element, which matches the property of an axicon and vortex lens. Binary phase coding simplifies the manufacturing process and gives additional advantages for metrology purposes. Under laser beam illumination, our element produces two waves: converging and diverging. Both waves carry a single optical vortex. We show that this special diffractive element can be used to set up a simple surface profilometer. © 2013 Optical Society of America OCIS codes: (050.1380) Binary optics; (050.1970) Diffractive optics; (050.4865) Optical vortices; (180.3170) Interference microscopy. http://dx.doi.org/10.1364/OL.39.000119
Diffractive optical elements are thin phase elements that operate by means of interference and diffraction to produce an arbitrary distribution of light. To produce the high-quality diffractive optical elements for visible light, the tools for drawing details with resolution better than 100 nm are necessary. The main reason for this is diffraction efficiency. To achieve 100% diffraction efficiency, the profile of each detail must be drawn continuously. In practice, the continuous profile is divided into steps (levels). The more levels, the better efficiency of the diffractive element. Manufacturers usually offer from two (binary elements) up to 16 level elements with theoretical diffraction efficiency from 41% to 99%. For some applications, the typical drawback of the binary elements (e.g., diffraction efficiency of 41% for −1 and 1 diffractive orders) becomes useful. In this Letter, we describe the diffraction element, which combines an axicon and a vortex lens (AV element). We planned the element for enhancing the measurement methodology reported in [1,2]. For the test of principles, it was manufactured in the simplest way (i.e., as a binary element). But working with it changed our approach. This binary element revealed some possibilities that might be useful for metrology. According to the classical definition by McLeod [3], an axicon is an optical element that images a point into a line segment along the optical axis. The length of the line can vary from a couple of millimeters up to several meters, depending on the design of the axicon. The simplest axicon is manufactured as refractive cones. To get better imaging, various modifications have been proposed [4,5]. The spiral phase plate is an optical element whose optical thickness increases in the azimuthal direction, around the plate center; thus having a transmission function Tθ expfimθg; m integer; θ azimuthal coordinate defined in the plate plane. It is a well-known element that transforms the Gaussian beam into an output vortex beam, i.e., beam carrying an optical vortex [6,7]. The spiral phase plate is also known as a vortex lens [8,9]. 0146-9592/14/010119-04$15.00/0
In [2] the four-level diffractive element combining the lens and the vortex lens of topological charge 2 was used to find the subwavelength phase step position with an accuracy of 60 nm. The focal length of the element was 38 mm, and the system was working with 633 nm He–Ne laser light. Obviously, the step position was found with resolution beyond the diffraction limit. In [1] the optical vortex beam was used for the quality inspection of the high-aspect-ratio micro-samples (HARMS) elements. HARMS elements consist of a matrix of wells or pillars whose lateral size is 30–70 μm and depth is 300–1000 μm. A simple optical system enabled fast and ambiguity-free quality inspection. The new diffractive element combining the axicon and vortex lens was designed for enhancing the potential of HARMS inspection method. Simple experiments show that it also can be used in the other way. When the binary AV element is illuminated, two waves emerge from the system (Fig. 1). Both beams carry an optical vortex. For a high-quality binary element, no zero-order beam should be detected. A series of binary AV elements was fabricated in order to verify the far-field intensity distributions. The manufactured elements had topological charges m 1 and m 2 and were designed for a wavelength λ 632.8 nm. The numerical design consisted of two steps: calculation of the annular linear axicon [4] forming a focal section on the distance from f 1 47 mm to f 2 51 mm [Eq. (1)]
Fig. 1. When illuminated by Gaussian beam, the AV element produces a diverging and a converging beam. Both beams carry the optical vortex. © 2014 Optical Society of America
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and multiplication by spiral phase distribution [Eq. (2)] with topological charge of m 1 or m 2. The height of the phase step was optimized for the required wavelength, knowing the type of photo-resist. The plates were exposed based on numerical design on matrices of 8192 × 8192 pixels, with a sampling of 1 μm. The design parameters of the modeled and/or manufactured structures were as follows: the radius of the aperture was R 4 mm. The equation of the phase distribution of the axicon was q r 2 · f 22 − f 21 ∕R2 f 21 −2π · : φr λ f 22 − f 21 ∕R2
second starts approximately at twice a focal length from the AV element and extends to infinity. At this area, the interference between two vortex beams is clearly visible as it is shown at the first (z 20 mm) and last (z 80 mm) figures in Figs. 4 and 5. No such effects are present in the image of kinoform AV element (Fig. 3).
(1)
This was combined with the spiral phase distribution of the expression φθ imθ:
(2)
Finally, the resulting phase patterns were binarized. The elements were fabricated using the electron-beam lithography (EBL) method on a JENOPTIK ZBA20 machine. A rectangular electron beam with a variable stamp size was used with a voltage of 20 kV. The nominal resolution of the writing machine was 0.2 μm for a single line and 0.4 μm for periodical structures. The size of the rectangular e-beam stamp was variable between 0.2 and 6.3 μm independently in the x and y directions. The available positioning of the beam was 0.01 μm. Figure 2 shows the profile of one of the fabricated structures (topological charge m 1). Figures 3 and 4 show the phase and intensity distributions calculated for hypothetical structures with a continuous phase profile (Fig. 3) and for a manufactured binary phase profile (Fig. 4). Figure 5 shows the phase and intensity profile for the optical vortex having topological charge m 2, which is calculated for binary AV. At the focal plane, the converging beam creates a small spot, and the diverging beam is widely spread (Fig. 1). There are two areas, at some distance from AV element, where both beams possess a similar wavefront curvature. The first area is closed behind the AV element. The
Fig. 2. Measurements of the AV element of m 1. (a) A photograph of the central part of the plate seen with an optical microscope. (b) An AFM scan of a fragment of the structure (the center of the scanned fragment is located approximately 3.14 mm from the center of the element). (c) A profile of the scanned fragment.
Fig. 3. AV element manufactured as kinoform will generate only the converging beam. Figure shows the numerical simulations; z denotes the distance between the element and the observation plane.
Fig. 4. Binary AV element for m 1 will produce converging and diverging beams. This effect is easily visible in the area where both beams have a comparable radius of curvature, i.e., close (z 20 mm) and far from the AV element (z 80 mm). At z 20 mm and z 80 mm, the amplitude and phase distribution reveals characteristic features for the overlapping vortex beams. Figure shows the numerical simulations.
Fig. 5. Binary AV element for m 2 works in a similar way as the element shown in Fig. 4. However, now we can see characteristic features for the overlapping vortex beams having double topological charge (for z 20 and z 80 mm).
January 1, 2014 / Vol. 39, No. 1 / OPTICS LETTERS
Fig. 6. Binary AV element of charge m 1. The experimental results reveal the same features as the numerical simulations (Fig. 4).
Figures 6 and 7 show experimental results for AV element having topological charge 1 or 2, respectively. Due to calculation time, Figs. 4 and 5 cover central areas of Figs. 6 and 7. The binary AV element can be used as a surface topography analyzer, which has a simple design. The sample is located at the focal plane of the converging beam. Since the focused spot is small, the beam is sensitive for the small sample details. The diverging beam illuminates the large sample area, and the small details do not have measurable influence on it. At the observation plane, both beams interfere. In the experiment, the system shown in Fig. 8 was used. To analyze the interferograms taken in our system, the carrier frequency method proposed by Takeda et al. [10] was applied. This method worked well for the vortex microscopy [11]. We scanned the sample with the phase step, which introduces the 3π-phase jump. In the vicinity of the edge, the vortices are cancelled (Fig. 9). When the phase jump is different than π (calculated modulo 2π), the vortices behave in a different way, which is illustrated in Fig. 10, showing the results of numerical modeling. Please note that higher sensitivities in the z direction should be possible for a Fresnel lens combined with an OV (FV element). Thus we calculated edge detection with a binary FV plate (focal length f 50 mm; topological charge m 1). Figure 10 shows phase distributions at the plane positioned 50 mm after the focal plane of the
Fig. 7. Binary AV element of charge m 2. The experimental results reveal the same features as the numerical simulations (Fig. 5).
121
Fig. 8. Measurement system is based on the Mach–Zehnder interferometer scheme.
plate. Edge detection with a binary FV element is certainly possible down to the 0.23π high phase step. It may be possible even for lower steps; however, it shall be difficult. We also confirmed that application of the AV element into profilometry is possible. At first we measured a highquality optical wedge with the wedge angle 90 s of arc. Shifting the wedge by 3 mm introduced no change in the vortex position, but the phase image around vortex points was changing (the total phase shift introduced by this movement exceeds the wavelength of the laser beam). In our system, we could observe this phase change with sensitivity of λ∕40. This was done by subtracting the subsequent phase images. Next, the optical wedge with damaged area was measured. In the damaged area, the wedge was extra-polished to make a valley. The wedge geometry was measured before, and the results were published in [12]. Figure 11 shows the exemplary experimental data. The focused spot diameter was about 15 μm. The scanned distance shown in Fig. 11 was 10 μm wide. At this diameter, the slope angle difference at the center of the beam at first position and at the last position was smaller than 2°. Nevertheless, its influence on the vortex points positions was measurable. Numerical simulations are comparable to this result (see Fig. 12).
Fig. 9. Phase distribution and interferograms measured while scanning through the phase step, which introduces a phase shift of ∼3π. (a) The beam center is positioned −5 μm from the step. (b) The beam center is positioned at the step. (c) The beam center is positioned 5 μm from the step. There are characteristic fork-like fringes indicating the presence of optical vortices in the interferograms (a) and (c). These fork-like fringes disappear in the interferogram (b).
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Fig. 10. Phase distributions at the plane positioned 50 mm behind the focal plane of the FV element, where the phase step was introduced. The Δ parameter denotes the distance between the phase-step and z axis. The h parameter denotes the phase difference introduced by the step.
Fig. 11. Phase maps measured while scanning the damaged optical wedge. The wedge was shifted by 2 μm for each next measurement. The scanning beam center lies at the area of the valley. It is well seen that vortices come apart while scanning the wedge.
Fig. 12. Phase distributions at the plane positioned 50 mm after the focal plane of the FV element, where the phase valley was introduced. The Δ parameter denotes the distance between the center of the valley and z axis.
Figure 12 shows numerical effects of scanning a phasevalley described with equation: φr −34861
1 1 · r − 4; 485 · r 2 794 2 mm mm
(3)
for radius r ≤ 10 μm. This corresponds to average inclination of c.a. 5 degrees in a material with a refractive index of n 2.2. The slope angle difference of 2° seems to be large for optical measurements. However, this slope was detected at the scanning range of 10 μm, so the local angle difference between subsequent scanning steps was smaller than 30 min of arc. This corresponds to the average phase difference smaller than 2π∕50 for each scanning step of the 2 μm size. We presented successful application of a binary AV element for metrology purposes. Binary coding is not only easier to manufacture (compared to multilevel or continuous profile), but it has an advantage of giving interference between 1 and −1 diffractive orders, which is useful for metrology. With the simple element, we could sense the phase difference as small as π∕25. These results can be improved by careful AV or FV elements optimization for surface topography application. The system also can be used with no reference beam. Thus the system becomes much simpler, but its accuracy is lower mainly because interferometric vortex localization methods are more accurate than methods based on the intensity pattern inspection. The questions of fast and effective algorithms for surface geometry reconstructions from the results presented in Fig. 11 and measurements of soft phase objects (like biological objects) are the subject of intense research. This work was supported by the Polish Ministry of Scientific Research and Information Technology (grant N505463438), by the European Social Fund implemented under the Human Capital Operational Programme (POKL), project: “Preparation and Realization of Medical Physics Specialty” and by the National Laboratory of Quantum Technologies (project POIG. 02.02.00-00-003/ 08-00). We also thank Michal Makowski from the Warsaw University of Technology. References 1. J. Masajada, M. Leniec, E. Jankowska, H. Thienpont, H. Ottevaere, and V. Gomez, Opt. Express 16, 19179 (2008). 2. J. Masajada, M. Leniec, S. Drobczynski, H. Thienpont, and B. Kress, Opt. Express 17, 16144 (2009). 3. J. H. McLeod, J. Opt. Soc. Am. 44, 592 (1954). 4. J. Sochacki, A. Kolodziejczyk, Z. Jaroszewicz, and S. Bara, Appl. Opt. 31, 5326 (1992). 5. Z. Jaroszewicz, J. Sochacki, A. Kolodziejczyk, and L. R. Staronski, Opt. Lett. 18, 1893 (1993). 6. S. N. Khonina, V. V. Kotlyar, M. V. Shinkaryev, V. A. Soifer, and G. V. Uspleniev, J. Mod. Opt. 39, 1147 (1992). 7. L. Allen, M. J. Padgett, and M. Babiker, Progress in Optics, E. Wolf, ed. (Elsevier, 1999), Vol. 39, Chap. IV, pp. 291–372. 8. G. A. Swartzlander, Jr., Opt. Lett. 26, 497 (2001). 9. G. A. Swartzlander, Jr., Opt. Lett. 31, 2042 (2006). 10. M. Takeda, H. Ina, and S. Kobayashi, J. Opt. Soc. Am. 72, 156 (1982). 11. J. Masajada, A. Popiolek-Masajada, and I. Augustyniak, Proc. SPIE 8550, 85503B (2013). 12. A. Popiolek-Masajada, M. Borwinska, T. Przerwa-Tetmajer, and P. Kurzynowski, Opt. Laser Technol. 48, 503 (2013).
