Abstract

We propose a lower-cost and practical active scanning optical scanning Fourier ptychographic microscopy (OSFPM). Featured is a simple setup of Galvo mirrors capable of scanning large-sized objects. The active scanning laser beam is projected onto the sample in a circular pattern to form multiple lower-resolution images. With multiple lower-resolution images, a higher-resolution image is subsequently reconstructed. The OSFPM is able to more precisely control the overlap of the incident light illumination as compared to that in conventional LED-based or other laser-based scanning FPM systems. The proposed microscope is also suitable for applications where a larger size of the object needs to be imaged with efficient illumination.

© 2020 Optical Society of America

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    [Crossref]

2019 (4)

A. Pan, K. Wen, and B. Yao, “Linear space-variant optical cryptosystem via Fourier ptychography,” Opt. Lett. 44, 2032–2035 (2019).
[Crossref]

A. Pan, C. Zuo, Y. Xie, M. Lei, and B. Yao, “Vignetting effect in Fourier ptychographic microscopy,” Opt. Lasers Eng. 120, 40–48 (2019).
[Crossref]

A. C. Chan, J. Kim, A. Pan, H. Xu, D. Nojima, C. Hale, S. Wang, and C. Yang, “Parallel fourier ptychographic microscopy for high-throughput screening with 96 cameras (96 eyes),” Sci. Rep. 9, 11114 (2019).
[Crossref]

H. Zhang, L. Wang, W. Zhou, Z. Hu, P. Tsang, and T. Poon, “Fourier ptychography: effectiveness of image classification,” Proc. SPIE 11205, 112050G (2019).
[Crossref]

2018 (3)

2017 (1)

A. Pan, Y. Zhang, T. Zhao, Z. Wang, D. Dan, M. Lei, and B. Yao, “System calibration method for Fourier ptychographic microscopy,” J. Biomed. Opt. 22, 096005 (2017).
[Crossref]

2016 (1)

2015 (2)

2014 (4)

2013 (2)

G. Zheng, R. Horstmeyer, and C. Yang, “Wide-field, high-resolution Fourier ptychographic microscopy,” Nat. Photonics 7, 739–745 (2013).
[Crossref]

X. Ou, R. Horstmeyer, C. Yang, and G. Zheng, “Quantitative phase imaging via Fourier ptychographic microscopy,” Opt. Lett. 38, 4845–4848 (2013).
[Crossref]

2010 (1)

2009 (1)

A. M. Maiden and J. M. Rodenburg, “An improved ptychographical phase retrieval algorithm for diffractive imaging,” Ultramicroscopy 109, 1256–1262 (2009).
[Crossref]

2008 (2)

J. M. Rodenburg, “Ptychography and related diffractive imaging methods,” Adv. Imaging Electron. Phys. 150, 87–184 (2008). ___
[Crossref]

T.-C. Poon, “On the fundamentals of optical scanning holography,” Am. J. Phys. 76, 738–745 (2008).
[Crossref]

2007 (1)

B. Ma, T. Zimmermann, M. Rohde, S. Winkelbach, F. He, W. Lindenmaier, and K. E. Dittmar, “Use of autostitch for automatic stitching of microscope images,” Micron 38, 492–499 (2007).
[Crossref]

2000 (1)

B. Hecht, B. Sick, U. P. Wild, V. Deckert, R. Zenobi, O. J. Martin, and D. W. Pohl, “Scanning near-field optical microscopy with aperture probes: fundamentals and applications,” J. Chem. Phys. 112, 7761–7774 (2000).
[Crossref]

1995 (1)

T.-C. Poon, K. B. Doh, B. W. Schilling, M. H. Wu, K. K. Shinoda, and Y. Suzuki, “Three-dimensional microscopy by optical scanning holography,” Opt. Eng. 34, 1338–1345 (1995).
[Crossref]

1982 (1)

Barbastathis, G.

Bian, L.

Bian, Z.

Bianco, V.

V. Bianco, P. Memmolo, M. Leo, S. Montresor, C. Distante, M. Paturzo, P. Picart, B. Javidi, and P. Ferraro, “Strategies for reducing speckle noise in digital holography,” Light Sci. Appl. 7, 1–16 (2018).
[Crossref]

Candes, E. J.

E. J. Candes, Y. C. Eldar, T. Strohmer, and V. Voroninski, “Phase retrieval via matrix completion,” SIAM Rev. 57, 225–251 (2015).
[Crossref]

Chan, A. C.

A. C. Chan, J. Kim, A. Pan, H. Xu, D. Nojima, C. Hale, S. Wang, and C. Yang, “Parallel fourier ptychographic microscopy for high-throughput screening with 96 cameras (96 eyes),” Sci. Rep. 9, 11114 (2019).
[Crossref]

Chen, F.

Chen, Q.

Chung, J.

Dai, Q.

Dan, D.

A. Pan, Y. Zhang, T. Zhao, Z. Wang, D. Dan, M. Lei, and B. Yao, “System calibration method for Fourier ptychographic microscopy,” J. Biomed. Opt. 22, 096005 (2017).
[Crossref]

Dasari, R. R.

Deckert, V.

B. Hecht, B. Sick, U. P. Wild, V. Deckert, R. Zenobi, O. J. Martin, and D. W. Pohl, “Scanning near-field optical microscopy with aperture probes: fundamentals and applications,” J. Chem. Phys. 112, 7761–7774 (2000).
[Crossref]

Distante, C.

V. Bianco, P. Memmolo, M. Leo, S. Montresor, C. Distante, M. Paturzo, P. Picart, B. Javidi, and P. Ferraro, “Strategies for reducing speckle noise in digital holography,” Light Sci. Appl. 7, 1–16 (2018).
[Crossref]

Dittmar, K. E.

B. Ma, T. Zimmermann, M. Rohde, S. Winkelbach, F. He, W. Lindenmaier, and K. E. Dittmar, “Use of autostitch for automatic stitching of microscope images,” Micron 38, 492–499 (2007).
[Crossref]

Doh, K. B.

T.-C. Poon, K. B. Doh, B. W. Schilling, M. H. Wu, K. K. Shinoda, and Y. Suzuki, “Three-dimensional microscopy by optical scanning holography,” Opt. Eng. 34, 1338–1345 (1995).
[Crossref]

Dong, S.

Eldar, Y. C.

E. J. Candes, Y. C. Eldar, T. Strohmer, and V. Voroninski, “Phase retrieval via matrix completion,” SIAM Rev. 57, 225–251 (2015).
[Crossref]

Fan, Y.

Ferraro, P.

V. Bianco, P. Memmolo, M. Leo, S. Montresor, C. Distante, M. Paturzo, P. Picart, B. Javidi, and P. Ferraro, “Strategies for reducing speckle noise in digital holography,” Light Sci. Appl. 7, 1–16 (2018).
[Crossref]

Fienup, J. R.

García, J.

Granero, L.

Guo, K.

Hale, C.

A. C. Chan, J. Kim, A. Pan, H. Xu, D. Nojima, C. Hale, S. Wang, and C. Yang, “Parallel fourier ptychographic microscopy for high-throughput screening with 96 cameras (96 eyes),” Sci. Rep. 9, 11114 (2019).
[Crossref]

He, F.

B. Ma, T. Zimmermann, M. Rohde, S. Winkelbach, F. He, W. Lindenmaier, and K. E. Dittmar, “Use of autostitch for automatic stitching of microscope images,” Micron 38, 492–499 (2007).
[Crossref]

Hecht, B.

B. Hecht, B. Sick, U. P. Wild, V. Deckert, R. Zenobi, O. J. Martin, and D. W. Pohl, “Scanning near-field optical microscopy with aperture probes: fundamentals and applications,” J. Chem. Phys. 112, 7761–7774 (2000).
[Crossref]

Horstmeyer, R.

Hu, Z.

H. Zhang, L. Wang, W. Zhou, Z. Hu, P. Tsang, and T. Poon, “Fourier ptychography: effectiveness of image classification,” Proc. SPIE 11205, 112050G (2019).
[Crossref]

Javidi, B.

V. Bianco, P. Memmolo, M. Leo, S. Montresor, C. Distante, M. Paturzo, P. Picart, B. Javidi, and P. Ferraro, “Strategies for reducing speckle noise in digital holography,” Light Sci. Appl. 7, 1–16 (2018).
[Crossref]

Kim, J.

A. C. Chan, J. Kim, A. Pan, H. Xu, D. Nojima, C. Hale, S. Wang, and C. Yang, “Parallel fourier ptychographic microscopy for high-throughput screening with 96 cameras (96 eyes),” Sci. Rep. 9, 11114 (2019).
[Crossref]

Kuang, C.

Lee, J.

Lei, M.

A. Pan, C. Zuo, Y. Xie, M. Lei, and B. Yao, “Vignetting effect in Fourier ptychographic microscopy,” Opt. Lasers Eng. 120, 40–48 (2019).
[Crossref]

A. Pan, Y. Zhang, K. Wen, M. Zhou, J. Min, M. Lei, and B. Yao, “Subwavelength resolution Fourier ptychography with hemispherical digital condensers,” Opt. Express 26, 23119–23131 (2018).
[Crossref]

A. Pan, Y. Zhang, T. Zhao, Z. Wang, D. Dan, M. Lei, and B. Yao, “System calibration method for Fourier ptychographic microscopy,” J. Biomed. Opt. 22, 096005 (2017).
[Crossref]

Leo, M.

V. Bianco, P. Memmolo, M. Leo, S. Montresor, C. Distante, M. Paturzo, P. Picart, B. Javidi, and P. Ferraro, “Strategies for reducing speckle noise in digital holography,” Light Sci. Appl. 7, 1–16 (2018).
[Crossref]

Li, X.

Lindenmaier, W.

B. Ma, T. Zimmermann, M. Rohde, S. Winkelbach, F. He, W. Lindenmaier, and K. E. Dittmar, “Use of autostitch for automatic stitching of microscope images,” Micron 38, 492–499 (2007).
[Crossref]

Lu, H.

Ma, B.

B. Ma, T. Zimmermann, M. Rohde, S. Winkelbach, F. He, W. Lindenmaier, and K. E. Dittmar, “Use of autostitch for automatic stitching of microscope images,” Micron 38, 492–499 (2007).
[Crossref]

Ma, Y.

Maiden, A. M.

A. M. Maiden and J. M. Rodenburg, “An improved ptychographical phase retrieval algorithm for diffractive imaging,” Ultramicroscopy 109, 1256–1262 (2009).
[Crossref]

Martin, O. J.

B. Hecht, B. Sick, U. P. Wild, V. Deckert, R. Zenobi, O. J. Martin, and D. W. Pohl, “Scanning near-field optical microscopy with aperture probes: fundamentals and applications,” J. Chem. Phys. 112, 7761–7774 (2000).
[Crossref]

Memmolo, P.

V. Bianco, P. Memmolo, M. Leo, S. Montresor, C. Distante, M. Paturzo, P. Picart, B. Javidi, and P. Ferraro, “Strategies for reducing speckle noise in digital holography,” Light Sci. Appl. 7, 1–16 (2018).
[Crossref]

Micó, V.

Min, J.

Montresor, S.

V. Bianco, P. Memmolo, M. Leo, S. Montresor, C. Distante, M. Paturzo, P. Picart, B. Javidi, and P. Ferraro, “Strategies for reducing speckle noise in digital holography,” Light Sci. Appl. 7, 1–16 (2018).
[Crossref]

Nojima, D.

A. C. Chan, J. Kim, A. Pan, H. Xu, D. Nojima, C. Hale, S. Wang, and C. Yang, “Parallel fourier ptychographic microscopy for high-throughput screening with 96 cameras (96 eyes),” Sci. Rep. 9, 11114 (2019).
[Crossref]

Ou, X.

Pan, A.

A. C. Chan, J. Kim, A. Pan, H. Xu, D. Nojima, C. Hale, S. Wang, and C. Yang, “Parallel fourier ptychographic microscopy for high-throughput screening with 96 cameras (96 eyes),” Sci. Rep. 9, 11114 (2019).
[Crossref]

A. Pan, K. Wen, and B. Yao, “Linear space-variant optical cryptosystem via Fourier ptychography,” Opt. Lett. 44, 2032–2035 (2019).
[Crossref]

A. Pan, C. Zuo, Y. Xie, M. Lei, and B. Yao, “Vignetting effect in Fourier ptychographic microscopy,” Opt. Lasers Eng. 120, 40–48 (2019).
[Crossref]

A. Pan, Y. Zhang, K. Wen, M. Zhou, J. Min, M. Lei, and B. Yao, “Subwavelength resolution Fourier ptychography with hemispherical digital condensers,” Opt. Express 26, 23119–23131 (2018).
[Crossref]

A. Pan, Y. Zhang, T. Zhao, Z. Wang, D. Dan, M. Lei, and B. Yao, “System calibration method for Fourier ptychographic microscopy,” J. Biomed. Opt. 22, 096005 (2017).
[Crossref]

Paturzo, M.

V. Bianco, P. Memmolo, M. Leo, S. Montresor, C. Distante, M. Paturzo, P. Picart, B. Javidi, and P. Ferraro, “Strategies for reducing speckle noise in digital holography,” Light Sci. Appl. 7, 1–16 (2018).
[Crossref]

Picart, P.

V. Bianco, P. Memmolo, M. Leo, S. Montresor, C. Distante, M. Paturzo, P. Picart, B. Javidi, and P. Ferraro, “Strategies for reducing speckle noise in digital holography,” Light Sci. Appl. 7, 1–16 (2018).
[Crossref]

Pohl, D. W.

B. Hecht, B. Sick, U. P. Wild, V. Deckert, R. Zenobi, O. J. Martin, and D. W. Pohl, “Scanning near-field optical microscopy with aperture probes: fundamentals and applications,” J. Chem. Phys. 112, 7761–7774 (2000).
[Crossref]

Poon, T.

H. Zhang, L. Wang, W. Zhou, Z. Hu, P. Tsang, and T. Poon, “Fourier ptychography: effectiveness of image classification,” Proc. SPIE 11205, 112050G (2019).
[Crossref]

Poon, T.-C.

T.-C. Poon, “On the fundamentals of optical scanning holography,” Am. J. Phys. 76, 738–745 (2008).
[Crossref]

T.-C. Poon, K. B. Doh, B. W. Schilling, M. H. Wu, K. K. Shinoda, and Y. Suzuki, “Three-dimensional microscopy by optical scanning holography,” Opt. Eng. 34, 1338–1345 (1995).
[Crossref]

L. Wang, Q. Song, H. Zhang, Y. Xin, and T.-C. Poon, “Optical scanning Fourier ptychographic microscopy,” in Digital Holography and Three-Dimensional Imaging (Optical Society of America, 2019), paper W3A-10.

Ramchandran, K.

Rodenburg, J. M.

A. M. Maiden and J. M. Rodenburg, “An improved ptychographical phase retrieval algorithm for diffractive imaging,” Ultramicroscopy 109, 1256–1262 (2009).
[Crossref]

J. M. Rodenburg, “Ptychography and related diffractive imaging methods,” Adv. Imaging Electron. Phys. 150, 87–184 (2008). ___
[Crossref]

Rohde, M.

B. Ma, T. Zimmermann, M. Rohde, S. Winkelbach, F. He, W. Lindenmaier, and K. E. Dittmar, “Use of autostitch for automatic stitching of microscope images,” Micron 38, 492–499 (2007).
[Crossref]

Schilling, B. W.

T.-C. Poon, K. B. Doh, B. W. Schilling, M. H. Wu, K. K. Shinoda, and Y. Suzuki, “Three-dimensional microscopy by optical scanning holography,” Opt. Eng. 34, 1338–1345 (1995).
[Crossref]

Shinoda, K. K.

T.-C. Poon, K. B. Doh, B. W. Schilling, M. H. Wu, K. K. Shinoda, and Y. Suzuki, “Three-dimensional microscopy by optical scanning holography,” Opt. Eng. 34, 1338–1345 (1995).
[Crossref]

Shiradkar, R.

Sick, B.

B. Hecht, B. Sick, U. P. Wild, V. Deckert, R. Zenobi, O. J. Martin, and D. W. Pohl, “Scanning near-field optical microscopy with aperture probes: fundamentals and applications,” J. Chem. Phys. 112, 7761–7774 (2000).
[Crossref]

Situ, G.

So, P. T.

Song, Q.

L. Wang, Q. Song, H. Zhang, Y. Xin, and T.-C. Poon, “Optical scanning Fourier ptychographic microscopy,” in Digital Holography and Three-Dimensional Imaging (Optical Society of America, 2019), paper W3A-10.

Strohmer, T.

E. J. Candes, Y. C. Eldar, T. Strohmer, and V. Voroninski, “Phase retrieval via matrix completion,” SIAM Rev. 57, 225–251 (2015).
[Crossref]

Sun, J.

Suo, J.

Suzuki, Y.

T.-C. Poon, K. B. Doh, B. W. Schilling, M. H. Wu, K. K. Shinoda, and Y. Suzuki, “Three-dimensional microscopy by optical scanning holography,” Opt. Eng. 34, 1338–1345 (1995).
[Crossref]

Tian, L.

Tsang, P.

H. Zhang, L. Wang, W. Zhou, Z. Hu, P. Tsang, and T. Poon, “Fourier ptychography: effectiveness of image classification,” Proc. SPIE 11205, 112050G (2019).
[Crossref]

Voroninski, V.

E. J. Candes, Y. C. Eldar, T. Strohmer, and V. Voroninski, “Phase retrieval via matrix completion,” SIAM Rev. 57, 225–251 (2015).
[Crossref]

Waller, L.

Wang, L.

H. Zhang, L. Wang, W. Zhou, Z. Hu, P. Tsang, and T. Poon, “Fourier ptychography: effectiveness of image classification,” Proc. SPIE 11205, 112050G (2019).
[Crossref]

L. Wang, Q. Song, H. Zhang, Y. Xin, and T.-C. Poon, “Optical scanning Fourier ptychographic microscopy,” in Digital Holography and Three-Dimensional Imaging (Optical Society of America, 2019), paper W3A-10.

Wang, S.

A. C. Chan, J. Kim, A. Pan, H. Xu, D. Nojima, C. Hale, S. Wang, and C. Yang, “Parallel fourier ptychographic microscopy for high-throughput screening with 96 cameras (96 eyes),” Sci. Rep. 9, 11114 (2019).
[Crossref]

Wang, Z.

A. Pan, Y. Zhang, T. Zhao, Z. Wang, D. Dan, M. Lei, and B. Yao, “System calibration method for Fourier ptychographic microscopy,” J. Biomed. Opt. 22, 096005 (2017).
[Crossref]

Wen, K.

Wild, U. P.

B. Hecht, B. Sick, U. P. Wild, V. Deckert, R. Zenobi, O. J. Martin, and D. W. Pohl, “Scanning near-field optical microscopy with aperture probes: fundamentals and applications,” J. Chem. Phys. 112, 7761–7774 (2000).
[Crossref]

Winkelbach, S.

B. Ma, T. Zimmermann, M. Rohde, S. Winkelbach, F. He, W. Lindenmaier, and K. E. Dittmar, “Use of autostitch for automatic stitching of microscope images,” Micron 38, 492–499 (2007).
[Crossref]

Wu, M. H.

T.-C. Poon, K. B. Doh, B. W. Schilling, M. H. Wu, K. K. Shinoda, and Y. Suzuki, “Three-dimensional microscopy by optical scanning holography,” Opt. Eng. 34, 1338–1345 (1995).
[Crossref]

Xie, Y.

A. Pan, C. Zuo, Y. Xie, M. Lei, and B. Yao, “Vignetting effect in Fourier ptychographic microscopy,” Opt. Lasers Eng. 120, 40–48 (2019).
[Crossref]

Xin, H.

Xin, Y.

L. Wang, Q. Song, H. Zhang, Y. Xin, and T.-C. Poon, “Optical scanning Fourier ptychographic microscopy,” in Digital Holography and Three-Dimensional Imaging (Optical Society of America, 2019), paper W3A-10.

Xu, H.

A. C. Chan, J. Kim, A. Pan, H. Xu, D. Nojima, C. Hale, S. Wang, and C. Yang, “Parallel fourier ptychographic microscopy for high-throughput screening with 96 cameras (96 eyes),” Sci. Rep. 9, 11114 (2019).
[Crossref]

Yang, C.

A. C. Chan, J. Kim, A. Pan, H. Xu, D. Nojima, C. Hale, S. Wang, and C. Yang, “Parallel fourier ptychographic microscopy for high-throughput screening with 96 cameras (96 eyes),” Sci. Rep. 9, 11114 (2019).
[Crossref]

J. Chung, H. Lu, X. Ou, H. Zhou, and C. Yang, “Wide-field Fourier ptychographic microscopy using laser illumination source,” Biomed. Opt. Express 7, 4787–4802 (2016).
[Crossref]

R. Horstmeyer and C. Yang, “A phase space model of Fourier ptychographic microscopy,” Opt. Express 22, 338–358 (2014).
[Crossref]

X. Ou, R. Horstmeyer, C. Yang, and G. Zheng, “Quantitative phase imaging via Fourier ptychographic microscopy,” Opt. Lett. 38, 4845–4848 (2013).
[Crossref]

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Proc. SPIE (1)

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Figures (7)

Fig. 1.
Fig. 1. Iteration process of FPM. In the iteration, the sampling rate of the initial guess of high-resolution object is higher than the captured raw low-resolution images, resulting in increased spatial bandwidth. FT, Fourier transform; IFT, inverse Fourier transform.

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Fig. 2.
Fig. 2. Optical scanning FPM experimental setup. M1, M2, and M3 are Microscopes, L1 to L4 are Lenses, L5 is a tube lens, BS is a beam splitter, GSM is Galvo scanning mirrors.

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Fig. 3.
Fig. 3. Scanning patterns of LED array and optical scanning setup in the Fourier domain. (a) Fourier spectra of 25 low-resolution images using Square Scan. (b) Fourier spectra of 19 low-resolution images using Circular Scan. (c) USAF target (digital) used for simulation. (d) Element 2, group 6 of the USAF target. (e) The line traces (blue line across d) of element 2, group 6. The red-dashed line is from square scan, the blue-dashed line is from circular scan, and the black solid line is ground truth. Note: Each circle in (a) and (b) indicates one scanning and one low-resolution image, and the red circle in the center corresponds to the raw NA of the miscoscope objective.

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Fig. 4.
Fig. 4. OSFPM experimental setup. The optical scanning Galvo mirrors are located in the upper-left corner. The Galvo mirrors are driven by an STM32 micro-controller board, and are capable of reflecting the laser beam at any angle by tilting themselves along the $ x $ and $ y $ axis. The imaged sample is a USAF target.

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Fig. 5.
Fig. 5. Imaging and recovery result of OSFPM. (a) Raw low-resolution image of 0 degree incident angle illumination. (b) Reconstruction result using 19 raw low-resolution images. (c) Collected 19 raw low-resolution images with circular scan patterns. (d) Line traces of element 2, group 6, and the ground truth of the USAF target. The ground truth was captured by a high NA Olympus microscope objective ($40 \times /0.65$).

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Fig. 6.
Fig. 6. Synthetic NA of optical scanning-based FPM. (a) Composition of multiple Layers. (b) Default NA of the imaging system. (c) Synthetic NA of Layer 2. (d) Synthetic NA of Layer 3. ${k_{{\rm cf}}}$ indicates the radius of raw NA of the imaging system. Dotted lines represent the effective synthetic NA. The three green and red dots indicate other Layers (e.g., Layer 4, 5, etc.).

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Fig. 7.
Fig. 7. Overlap of two adjacent circles: Circle-Circle Intersection Problem. ${k_{{\rm offset}}}$, distance between centers of adjacent small circles; ${k_{{\rm cf}}}$, radius of red small circle; ${l_C}$, length of chord.

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Equations (11)

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g n 0 ( x , y ) = ℑ − 1 { G n 0 ( k x , k y ) } = ℑ − 1 { O 0 ( k x − k x n , k y − k y n ) C ( k x , k y ) } ,
g n 1 ( x , y ) = I n g n 0 | g n 0 | .
O 0 1 ( k x − k x n , k y − k x n ) = O 0 ( k x − k x n , k y − k y n ) [ 1 − C ( k x , k y ) ] + G n 1 ( k x , k y ) .
E k = ∑ n = 1 N ⁡ ∑ x , y ⁡ { | g k , n 1 ( x , y ) | − I n ( x , y ) } 2 .
I k = | o k ( x , y ) | 2 = | ℑ − 1 { O k ( k x , k y ) } | 2 .
{ N A s y n = ( k r + k c f ) × λ L = 2 N A s y n = ( k r + k r + k c f ) × λ L = 3 .
N A s y n = ( L − 1 ) k r + k c f k c f N A o b j .
k t = 2 [ ( L − 2 ) k r + k c f ] sin ⁡ ( π n L ) .
n L = π / arcsin ( 1 2 k t ( L − 2 ) k r + k c f ) | min ( | k t − k c f | ) .
P o v e r l a p = S / ( π k o f f s e t 2 ) = ( l a r c × k c f − 1 2 k o f f s e t × l C ) / ( π k o f f s e t 2 ) .
P o v e r l a p = [ 2 arccos ( k o f f s e t 2 k c f ) − k o f f s e t k c f 1 − ( k o f f s e t 2 k c f ) 2 ] / π .

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