Optical vortex (OV) beams have null-intensity singular points, and the intensities in the region surrounding the singular point are quite low. This low intensity region influences the position detection accuracy of phase singular point, especially for high-order OV beam. In this paper, we propose a new method for solving this problem, called the phase-slope-combining correlation matching method. A Shack-Hartmann wavefront sensor (SH-WFS) is used to measure phase slope vectors at lenslet positions of the SH-WFS. Several phase slope vectors are combined into one to reduce the influence of low-intensity regions around the singular point, and the combined phase slope vectors are used to determine the OV position with the aid of correlation matching with a pre-calculated database. Experimental results showed that the proposed method works with high accuracy, even when detecting an OV beam with a topological charge larger than six. The estimated precision was about 0.15 in units of lenslet size when detecting an OV beam with a topological charge of up to 20.
© 2015 Optical Society of America
Light beams having optical vortices (OVs) [1–4] have been attracting much interest because of their non-zero topological charge, or orbital angular momentum. The phase structure of an OV has a helical waveform that varies from of 0 to 2nπ (where n is the topological charge), and the center is a singular point with zero amplitude and undefined phase. OV beams have found many applications, such as information encoding , manipulation of small particles , stimulated emission depletion microscopy , and optical measurement . We expect that high-order OV beams will be even more useful because they produce more torque force, which is advantageous for optical manipulation applications such as rotating small particles [9, 10], and have more degrees-of-freedom for encoding information [11, 12]. Some applications, such as adaptive control of trapped particles, require extremely accurate confirmation of the position of the singular point in an OV beam, and therefore, it is important to develop a fast, real-time measurement technique that can satisfy this requirement.
Several methods have been proposed for determining the position of a singular point in an OV beam. The most widely used method is interferometry , which has high measurement precision and spatial resolution. However, this measurement technique is very sensitive to the environmental conditions, such as vibrations and turbulence. Relatively complicated optics are also required. Recently, a Shack-Hartmann wavefront sensor (SH-WFS) has been used to overcome these problems [14, 15]. The SH-WFS simply consists of a lens array and an image sensor and can easily obtain the phase slope vectors of an incoming wavefront. Several SH-WFS-based methods have been proposed to detect singular point positions [2, 16–24]. One typical method, called the contour sum method, was proposed by Fried . The presence and absence of phase singularities in a light beam is determined according to an evaluation value calculated from of the phase slope vectors. Some improvements have been made to reduce noise [16–19]. Another method, which is called the branch point potential method, was reported by Murphy [20, 21]. In this method, the phase slope vectors are first rotated by 90 degrees. However, the spatial resolution of measurements is limited by the lenslet size of the lenslet array in the SH-WFS. To realize sub-lenslet size spatial resolution, we have proposed the correlation matching method (CMM) [22, 23]. This method involves calculating evaluation values of phase slope vectors along a closed-path connecting the centers of 2 × 2 neighboring lenslets and comparing the evaluation values with a set of pre-calculated reference values. By using this method, a detection accuracy better than 0.1 in units of the lenslet size can be achieved. We have also proposed a hybrid centroid method to reduce the distribution error owing to the positional relationship , and an extended contour method to simplify the implementation . These methods work well when detecting a low-order OV (small topological charge); however, when detecting a high-order OV (large topological charge), the detection accuracy becomes worse due to the intrinsic low-intensity dark regions existing around the singular point.
Figure 1 shows how this dark region affects the Hartmanngrams, revealing the spiral phase patterns created by a phase modulation device like a phase plate or spatial light modulator (SLM) used to generate the OV beam (top row), the Hartmanngrams captured by a SH-WFS (middle row), and the intensity distributions captured by an image sensor (bottom row). The phase values for generating the OV beam were wrapped in the interval from 0 to 2π radians, and were represented by gray-scale brightnesses. Both the Hartmanngrams and the intensity distributions were recorded at the conjugate plane of the phase modulation device while a collimated uniform intensity beam was made incident on the device. The dark regions of low intensity existing around the singular points are clearly seen. Within these regions, the spots of the Hartmanngrams are deformed, and in the worst case, one or more spots disappear. Therefore, the phase slope vectors around the singular point involve large error or may even be impossible to calculate, strongly affecting the precision of the OV detection.
In this paper, we propose a new method, which is called the phase-slope-combining correlation matching method (PSC-CMM), to solve the dark-region problem and thus obtain the precise position of a singular point in an OV beam. This paper is organized as follows. In Section 2, we describe our proposed method. In Section 3, we describe the optical setup used for performing proof-of-principle experiments. Section 4 presents the results and discussion. Section 5 summarizes the paper.
2. Phase-slope-combining correlation matching method
In this section, we describe our phase-slope-combining correlation matching method (PSC-CMM), which provides an effective solution to the dark-region problem mentioned above. The proposed PSC-CMM is a modified version of our conventional correlation matching method (CMM) . Figure 2 shows a flow chart of the PSC-CMM, which mainly consists of six steps: dark-region analysis, slope calculation, slope combination, circulation calculation, correlation matching, and coordinate conversion.
2.1 Dark-region analysis
For simplicity, we consider the case of a SH-WFS consisting of a square-grid lenslet array and an image sensor. From the Hartmanngram I(ξ, η) obtained by the SH-WFS, the location and size of the dark region are confirmed with the help of an intensity sum map.
The intensity sum map can be simply obtained by summing the intensities of the pixels within each lenslet area. For the (i, j)-th lenslet area, the intensity sum Isum can be calculated by
The maximum numbers of lenslet areas within the dark region in the x- and y-directions, Nx and Ny, are given by
Figure 3 shows an example of a Hartmanngram (a) and its intensity sum map (b). The position and the size of the dark-region are easily confirmed from the intensity sum map.
2.2 Slope calculation
In this step, phase slope vectors, which correspond to the local tilting angles of the wavefront in each lenslet -area, are calculated according to the principle of the SH-WFS  by
2.3 Slope combination
The phase slope vectors within the dark region include large error. Our purpose is to reduce the influence of the dark region. To realize this, we combine several phase slope vectors and form a new combined phase slope vector so that each combined phase slope vector is a reasonable representation for the combined lenslet area.
In this step, we calculate the geometric center of the dark-region and set a 2 × 2 array of identical square combined lenslet areas around the geometric center. Then the initial points of the combined lenslet areas and the combination size are determined for the phase slope combination calculation.
The geometric center (xdc, ydc) of the dark-region, in units of lenslet size, can be calculated by
The combination size No is obtained by,Eq. (4).
Figure 4 schematically shows the coordinate system and symbols. In Fig. 4, the areas enclosed by thin-line squares are lenslet areas, and the open circles denote the crosspoint of each 2 × 2 lenslet area. The lenslet areas within the dark region are indicated by shading, and the geometric center of the dark region is denoted by the circle with a plus inside. The nearest crosspoint to the geometric center is indicated by the solid-square with a white circle inside. The area enclosed by the thick-line square is the combined lenslet area.
To make it more clear, in Fig. 5, we give some examples of assumed dark region shapes. In Fig. 5(a), the dark region has only one lenslet area. Taking the upper-left crosspoint of the original (non-combined) lenslet areas as (0, 0), we get, xdc = 2.5 and ydc = 2.5, and thus, the nearest crosspoint is io = 3 and jo = 3. The number of lenslet areas in the dark region is Nx = 1 in the x-direction and Ny = 1 in the y-direction, and hence the size of the combined lenslet-area is No = 2. Similarly, we can obtain the combination size and the nearest crosspoint as shown in Figs. 5(b) and 5(c).
As shown in Fig. 5, it is obvious that the dark region’s lenslet areas may not be equally allocated onto the nearest four combined lenslet areas, but the allocation according to the above process is optimized for the case where the SH-WFS device is used. Also it must be mentioned that although the initial crosspoint for forming the combined lenslet area is not unique, the final result is not affected much.
After obtaining the center (io, jo) of the dark region and the combination size, the combined phase slope vector for the (I, J)-th combined lenslet-area can be calculated byFigure 6 shows a schematic diagram of the slope combination process for each combined lenslet area.
Once the combined-phase slope vectors are obtained, the CMM processes can be applied . The only difference is that the contour for the circulation calculation is a closed path connecting the centers of the 2 × 2 combined lenslet areas. Therefore, the final OV position results need a coordinate conversion from the combined lenslet domain to the original lenslet domain.
2.4 Circulation calculation
In this step, similarly to the CMM, we use the contour sum method to find the rough position of the singular point with a precision equal to the combined lenslet size.
The circulation value at the combined-cross-point (p’, q’) is calculated by Figure 7 shows the concept of circulation calculation, where the dashed line indicates a contour connecting the centers of the 2 × 2 combined lenslet areas.
By calculating the circulation at every crosspoint of the combined lenslet area, we can obtain a circulation map. Previous studies  showed that C(p’,q’) is a local maximum if there is a singular point within the area surrounded by the contour. Thus, the nearest crosspoint (p’m, q’m) to the singular point can be found by:
2.5 Correlation matching
To detect the position of the OV more precisely, in this step, the matching operation is performed using zero-mean normalized cross correlation (NCC) between the circulation values around the local peak of the circulation map and a set of pre-calculated reference values :
Therefore, the position of the singular point in the combined-lenslet coordinate system is obtained by
2.6 Coordinate conversion
The position of the OV obtained by Eq. (16) is expressed in the combined-lenslet coordinate system. The final position of the singular point in the OV beam in the ordinal-lenslet coordinate system is obtained by
3. Experimental setup
We built the experimental setup schematically shown in Fig. 8 to test our method. A laser beam with a wavelength of 633 nm was collimated after passing through a polarizer, a reflecting mirror, an objective lens, a pinhole and collimation lens. The collimated beam passed through an aperture (6 mm in diameter) and illuminated a liquid-crystal-on-silicon spatial light modulator (LCOS-SLM), on which various spiral phase patterns were displayed for generating high-quality OV beams . The intensity distribution of the beam on the LCOS-SLM was almost uniform. The incident collimated beam was transformed into an OV beam after being reflected back from the LCOS-SLM, and was then split into two by a beamsplitter (BS 2). One beam passed through a 4-f system composed of two lenses (lens 1 with f = 250 mm and lens 2 with f = 400 mm), was then split into two beams by another beamsplitter (BS 3), and these beams arrived at the SH-WFS and CCD 1, respectively. Both the SH-WFS and CCD 1 were set at the conjugate plane of the LCOS-SLM. The other beam split by BS 2 went through a third lens (lens 3, f = 250 mm) and was focused onto CCD 2, to check the quality of the OV beam in the far field region. We also added a blazed-grating-type phase onto the spiral phase pattern and used a spatial filter (3 mm in diameter) at the focal plane of lens 1 to separate the OV beam from the zero-order and other diffraction orders.
The LCOS-SLM used in our experiment was a phase-only modulation type SLM (Hamamatsu Photonics, X10468-01) with 792 × 600 pixels, and each pixel had dimensions 20 μm × 20 μm . The SH-WFS consisted of two elements: a square-grid lenslet array and a high-speed intelligent vision sensor (Hamamatsu Photonics, C8201) with 512 × 512 pixels and 20 μm × 20 μm pixel size . We designed the lenslet array to have 51 × 51 active lenslets, whose pitch and focal length are 200 μm and 11 mm, respectively. The CCD 1 and CCD 2 were image sensor (Lumenera, Lu135M) with 1392 × 1040 pixels and 4.65 μm × 4.65 μm pixel size.
4. Results and discussions
To test the performance of the experimental system, the intensity images of the OV beams generated by the LCOS-SLM were observed. Figure 9 shows an example of the spiral phase pattern displayed on the LCOS-SLM (a), the intensity distribution captured by CCD 1 at the conjugate plane of the LCOS-SLM (b), and the far-field pattern of the OV beam captured by CCD 2 (c). The spiral phase pattern in Fig. 9(a) was larger than 6 mm in diameter. In Fig. 9(b), we can see a clear dark region in the conjugate image, and in Fig. 9(c), an OV beam with a typical donut-shaped distribution having a constant-intensity circular contour is clearly visible.
We generated various OV beams of different topological charges and shifted them to different locations by displacing the spiral phase pattern on the LCOS-SLM. The Hartmanngrams were acquired by the SH-WFS, and then the positions of the singular points in the OV beams were calculated.
Figure 10 shows the measured results for OV beams of topological charges (a) n = 10 and (b) n = 20. The spiral phase pattern for generating OV beam was displaced along the x-direction (horizontal displacement) on the LCOS-SLM. In the graphs, the horizontal axes are the horizontal displacement of the spiral phase pattern in units of the LCOS-SLM’s pixel size, and the vertical axes are the measured positions in units of lenslet size. The top and bottom graphs are the x- and y-components of the measured positions, respectively. For comparison, we plot the measured positions calculated by PSC-CMM (blue diamond), CMM (red cross), and theoretical prediction (solid line) in the same graph.
As shown in Fig. 10, using the PSC-CMM, we can obtain good linearity of the measured positions with respect to the displacement applied. The measured x-position increases linearly with the amount of horizontal displacement, meanwhile the measured y-position maintains almost unchanged. However, we cannot find such a linear relationship using the CMM. In fact, the calculation using the CMM showed complete failure in some cases as indicated by small arrows in Fig. 10, owing to the information loss caused by the dark region. The position errors at such failure points indicated by the arrows are larger than 0.5 [lenslet size]. The position error is defined by the absolute difference between the measured position and theoretical predication. The ratios of the number of the failure points to the total number of the measurement points are approximately 35% and 70% for topological charges 10 and 20, respectively. In the top graph of Fig. 10(a), the slopes of the linear fitting plot are 0.161 for the PSC-CMM, and 0.260 for the CMM. Similarly in the top graph of Fig. 10(b), the slopes of the linear fitting plot are 0.159 for the PSC-CMM, and 0.198 for the CMM, respectively. Before the experiments, the positional relationship between the LCOS-SLM and the SH-WFS was confirmed. According to our experimental setup, one pixel displacement ( = 20 μm) on the LCOS-SLM plane corresponds to a 32 μm displacement on the SH-WFS plane, or 0.16 in units of lenslet size. Therefore, the slope obtained with the PSC-CMM agreed well with the theoretical prediction.
The relationship between the measurement error of the singular-point position and its topological charge was also investigated. For topological charges from 1 to 20, the phase pattern for generating a phase singularity was displaced two-dimensionally in steps of one pixel (20 μm) on the LCOS-SLM, and a total of 11 × 11 measurements were performed for each topological charge. For each measurement, we calculated the absolute difference between the measured value and the theoretical prediction. Figure 11 shows the measured position errors, which are the average of the absolute differences over all 11 × 11 positional measurements, versus the topological charges. An enlarged view of the portion denoted by the dashed square in Fig. 11(a) is given as Fig. 11(b). The blue solid bars are the errors for PSC-CMM, and red hashed bars are the errors for CMM. As can be seen from Fig. 11, compared with the conventional CMM, the PSC-CMM can greatly reduce the measurement error, especially when measuring phase singularity with large topological charge. The errors for PSC-CMM were small and close to those of CMM when the topological charge was one to five. The amount of error remained small even when the topological charge was varied up to 20. However, the CMM’s performance was worse when the charge was equal to or larger than six, and the CMM was unable to find the correct position of the singular point when the charge was larger than 10. Statistical analysis showed that the root-mean-square (RMS) errors over the measurements of the 11 × 11 positions and 20 charges were 0.76 [lenslet size] for CMM and 0.15 [lenslet size] for PSC-CMM.
We also checked the relationships between topological charge, dark-region size and combination size. Figure 12 show the dark-region sizes measured by CCD 1 (blue solid squares) and the combination sizes employed in the circulation calculation (red open circles) versus the topological charges of the OV beams. The dark-region sizes were obtained from analyzing the intensity distributions captured by CCD 1. It shows that the dark-region size of an OV beam is directly proportional to its topological charge. The dark-region size increased at a factor of 0.286 [lenslet size] per charge in the case of our experimental system. As shown in Fig. 12, two kinds of combination sizes could be applied for detecting the OV beams of topological charges 3, 4, 9, 10, 11, 17 and 18. This means that, for those topological charges, one combination size was applied for some OV locations, but for other OV locations another combination size was applied. In Fig. 12, the numerical values near the red open circles indicate the percentages of the combination sizes applied for detecting the OV beams of these topological charges.
In Fig. 11(b), it can be observed that the average absolute difference for PSC-CMM exhibits an S-shaped distribution, achieving local minima at topological charges of 4, 11, and 17. We consider that this result can be explained by the fact that two combination sizes were applied for detecting the OV beams of these topological charges.
The intrinsic dark region in an OV beam prevents accurate measurement of the singular point position. In this paper, we have proposed a new SH-WFS-based phase-slope-combining correlation matching method (PSC-CMM) to solve the dark-region problem and to realize precise measurement. The main steps in the method, which can be considered as an extension of the conventional CMM, are: analyzing the dark-region’s size and location from a Hartmanngram obtained by the SH-WFS; calculating the phase slope vectors; grouping several phase slope vectors into one according to the dark-region’s size and location; calculating circulations from the combined phase slope vectors; and determining the singular point position from the circulation distribution. The experimental results showed that the proposed method exhibits good performance especially for detecting high-order OV beams. The performance was almost the same as the conventional CMM when testing OV beams of topological charge less than six. When the topological charge was equal to or larger than six, the PSC-CMM maintained its performance, whereas the performance of the CMM gradually declined. Statistical analysis showed that the precision of the PSC-CMM was about 0.15, in units of lenslet size, when measuring OV beams having topological charges up to 20.
It is important for optical manipulation and information encoding to realize accurate determination of topological charge of optical vortex. Additionally, as shown in reference 12, an OV beam having multiple singular points could be used for deformation sensing, which must accurately measure the position and the topological charge. Therefore, the further research along this direction is that extends the SH-WFS to be able to determine accurate topological charges and positions of a multiple OV beam.
We gratefully thank A. Hiruma, and T. Hara for their support and encouragement throughout this study, and Y. Takiguchi and N. Matsumoto for their useful discussions.
References and links
1. J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. Lond. A Math. Phys. Sci. 336(1605), 165–190 (1974). [CrossRef]
3. E. O. Le Bigot, W. J. Wild, and E. J. Kibblewhite, “Branch point reconstructors for discontinuous light phase functions,” Proc. SPIE 3381, 76–87 (1998). [CrossRef]
4. D. L. Fried, “Branch point problem in adaptive optics,” J. Opt. Soc. Am. A 15(10), 2759–2768 (1998). [CrossRef]
5. J. Wu, H. Li, and Y. Li, “Encoding information as orbital angular momentum states of light for wireless optical communications,” Opt. Eng. 46(1), 019701 (2007). [CrossRef]
6. J. E. Curtis, B. A. Koss, and D. G. Grier, “Dynamic holographic optical tweezers,” Opt. Commun. 207(1–6), 169–175 (2002). [CrossRef]
7. E. Auksorius, B. R. Boruah, C. Dunsby, P. M. P. Lanigan, G. Kennedy, M. A. A. Neil, and P. M. W. French, “Stimulated emission depletion microscopy with a supercontinuum source and fluorescence lifetime imaging,” Opt. Lett. 33(2), 113–115 (2008). [CrossRef] [PubMed]
8. W. Wang, Y. Qiao, R. Ishijima, T. Yokozeki, D. Honda, A. Matsuda, S. G. Hanson, and M. Takeda, “Constellation of phase singularities in a speckle-like pattern for optical vortex metrology applied to biological kinematic analysis,” Opt. Express 16(18), 13908–13917 (2008). [CrossRef] [PubMed]
9. M. Padgett and R. Bowman, “Tweezers with a twist,” Nat. Photonics 5(6), 343–348 (2011). [CrossRef]
10. T. Otsu, T. Ando, Y. Takiguchi, Y. Ohtake, H. Toyoda, and H. Itoh, “Direct evidence for three-dimensional off-axis trapping with single Laguerre-Gaussian beam,” Sci. Rep. 4, 4579 (2014). [CrossRef] [PubMed]
11. J. Wang, J. Yang, I. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012). [CrossRef]
12. S. Sato, I. Fujimoto, T. Kurihara, and S. Ando, “Remote Six-axis Deformation Sensing with Optical Vortex Beams,” Proc. SPIE 6877, 68770I (2008). [CrossRef]
13. C. Rockstuhl, A. A. Ivanovskyy, M. S. Soskin, M. G. Salt, H. P. Herzig, and R. Dandliker, “High-resolution measurement of phase singularities produced by computer-generated holograms,” Opt. Commun. 242(1–3), 163–169 (2004). [CrossRef]
14. D. R. Neal, J. Copland, and D. Neal, “Shack-Hartmann sensor precision and accuracy,” Proc. SPIE 4779, 148–160 (2002). [CrossRef]
15. A. Chernyshov, U. Sterr, F. Riehle, J. Helmcke, and J. Pfund, “Calibration of a Shack-Hartmann sensor for absolute measurements of wavefronts,” Appl. Opt. 44(30), 6419–6425 (2005). [CrossRef] [PubMed]
16. F. A. Starikov, G. G. Kochemasov, S. M. Kulikov, A. N. Manachinsky, N. V. Maslov, A. V. Ogorodnikov, S. A. Sukharev, V. P. Aksenov, I. V. Izmailov, F. Y. Kanev, V. V. Atuchin, and I. S. Soldatenkov, “Wavefront reconstruction of an optical vortex by a Hartmann-Shack sensor,” Opt. Lett. 32(16), 2291–2293 (2007). [CrossRef] [PubMed]
17. F. A. Starikov, V. P. Aksenov, V. V. Atuchin, I. V. Izmailov, F. Y. Kanev, G. G. Kochemasov, A. V. Kudryashov, S. M. Kulikov, Y. I. Malakhov, A. N. Manachinsky, N. V. Maslov, A. V. Ogorodnikov, I. S. Soldatenkov, and S. A. Sukharev, “Wave front sensing of an optical vortex and its correction in the close-loop adaptive system with bimorph mirror,” Proc. SPIE 6747, 67470P (2007). [CrossRef]
18. F. A. Starikov, V. P. Aksenov, V. V. Atuchin, I. V. Izmailov, F. Y. Kanev, G. G. Kochemasov, S. M. Kulikov, A. N. Manachinsky, N. V. Maslov, A. V. Ogorodnikov, I. S. Soldatenkov, and S. A. Sukharev, “Correction of vortex laser beams in a closed-loop adaptive system with bimorph mirror,” Proc. SPIE 7131, 71311G (2008). [CrossRef]
22. C. Huang, H. Huang, H. Toyoda, T. Inoue, and H. Liu, “Correlation matching method for high-precision position detection of optical vortex using Shack-Hartmann wavefront sensor,” Opt. Express 20(24), 26099–26109 (2012). [CrossRef] [PubMed]
23. H. Huang, C. Huang, H. Toyoda, and T. Inoue, “Correlation matching method for optical vortex detection using Shack–Hartmann wavefront sensor,” in 2013 Conference on Lasers and Electro-Optics Pacific Rim, (Optical Society of America 2013), paper WO4–2 (June 2013).
24. C. Huang, H. Zhang, H. Huang, H. Toyoda, T. Inoue, and H. Liu, “Error reduction method for singularity point detection using Shack–Hartmann wavefront sensor,” Opt. Commun. 311, 163–171 (2013). [CrossRef]
25. H. Huang, J. Luo, Y. Matsui, H. Toyota, and T. Inoue, “Optical vortex position detection with Shack-Hartmann wavefront sensor using extended closed contour method,” Proc. SPIE 9379, 9379(2015).
26. N. Matsumoto, T. Ando, T. Inoue, Y. Ohtake, N. Fukuchi, and T. Hara, “Generation of high-quality higher-order Laguerre-Gaussian beams using liquid-crystal-on-silicon spatial light modulators,” J. Opt. Soc. Am. A 25(7), 1642–1651 (2008). [CrossRef] [PubMed]
27. T. Inoue, H. Tanaka, N. Fukuchi, M. Takumi, N. Matsumoto, T. Hara, N. Yoshida, Y. Igasaki, and Y. Kobayashi, “LCOS spatial light modulator controlled by 12-bit signals for optical phase-only modulator,” Proc. SPIE 6487(64870Y), 64870Y (2007). [CrossRef]
28. H. Huang, T. Inoue, and T. Hara, “Adaptive aberration compensation system using a high-resolution liquid crystal on silicon spatial light modulator,” Proc. SPIE 7156(71560F), 71560F (2008). [CrossRef]