Abstract

A novel dual-view polarization-resolved pulsed holographic system for particle measurements is presented. Both dual-view configuration and polarization-resolved registration are well suited for particle holography. Dual-view registration improves the accuracy in the detection of 3D position and velocities, and polarization-resolved registration provides polarization information about individual particles. The necessary calibrations are presented, and aberrations are compensated for by mapping the positions in the two views to positions in a global coordinate system. The system is demonstrated on a sample consisting of 7 μm spherical polystyrene particles dissolved in water in a cuvette. The system is tested with different polarizations of the illumination. It is found that the dual view improves the accuracy significantly in particle tracking. It is also found that by having polarization-resolved holograms, it is possible to separate naturally occurring sub-micrometer particles from the larger, 7 μm seeding particles.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2018 (3)

2017 (5)

X. Liu, Y. Yang, L. Han, and C.-S. Guo, “Fiber-based lensless polarization holography for measuring Jones matrix parameters of polarization-sensitive materials,” Opt. Express 25, 7288–7299 (2017).
[Crossref]

Q.-Y. Yue, Z.-J. Cheng, L. Han, Y. Yang, and C.-S. Guo, “One-shot time-resolved holographic polarization microscopy for imaging laser-induced ultrafast phenomena,” Opt. Express 25, 14182–14191 (2017).
[Crossref]

L. Han, Z.-J. Cheng, Y. Yang, B.-Y. Wang, Q.-Y. Yue, and C.-S. Guo, “Double-channel angular-multiplexing polarization holography with common-path and off-axis configuration,” Opt. Express 25, 21877–21886 (2017).
[Crossref]

F. Merola, P. Memmolo, L. Miccio, R. Savoia, M. Mugnano, A. Fontana, G. D’Ippolito, A. Sardo, A. Iolascon, A. Gambale, and P. Ferraro, “Tomographic flow cytometry by digital holography,” Light Sci. Appl. 6, e16241 (2017).
[Crossref]

H. Zhang, M. Zhai, J. Sun, Y. Zhou, D. Jia, T. Liu, and Y. Zhang, “Discrimination between spheres and spheroids in a detection system for single particles based on polarization characteristics,” J. Quant. Spectrosc. Radiat. Transfer 187, 62–75 (2017).
[Crossref]

2016 (1)

2015 (5)

2014 (1)

A. Wang, T. G. Dimiduk, J. Fung, S. Razavi, I. Kretzschmar, K. Chaudhary, and V. N. Manoharan, “Using the discrete dipole approximation and holographic microscopy to measure rotational dynamics of non-spherical colloidal particles,” J. Quant. Spectrosc. Radiat. Transfer 146, 499–509 (2014).
[Crossref]

2013 (2)

N. A. Buchmann, C. Atkinson, and J. Soria, “Ultra-high-speed tomographic digital holographic velocimetry in supersonic particle-laden jet flows,” Meas. Sci. Technol. 24, 024005 (2013).
[Crossref]

Y. S. Bae, J. I. Song, and D. Y. Kim, “Volumetric reconstruction of Brownian motion of a micrometer-size bead in water,” Opt. Commun. 309, 291–297 (2013).
[Crossref]

2012 (1)

2011 (1)

2010 (1)

2008 (1)

J. Soria and C. Atkinson, “Towards 3C-3D digital holographic fluid velocity vector field measurement—tomographic digital holographic PIV (Tomo-HPIV),” Meas. Sci. Technol. 19, 074002 (2008).
[Crossref]

2007 (1)

2006 (1)

2004 (1)

T. Colomb, E. Cuche, F. Montfort, P. Marquet, and C. Depeursinge, “Jones vector imaging by use of digital holography: simulation and experimentation,” Opt. Commun. 231, 137–147 (2004).
[Crossref]

2003 (1)

2002 (1)

1998 (1)

K. T. Chan and Y. J. Li, “Pipe flow measurement by using a side-scattering holographic particle imaging technique,” Opt. Laser Technol. 30, 7–14 (1998).
[Crossref]

Amer, E.

D. Khodadad, E. Amer, P. Gren, E. Melander, E. Hällstig, and M. Sjödahl, “Single-shot dual-polarization holography: measurement of the polarization state of a magnetic sample,” Proc. SPIE 9660, 96601E (2015).
[Crossref]

Andersson, M.

Atkinson, C.

N. A. Buchmann, C. Atkinson, and J. Soria, “Ultra-high-speed tomographic digital holographic velocimetry in supersonic particle-laden jet flows,” Meas. Sci. Technol. 24, 024005 (2013).
[Crossref]

J. Soria and C. Atkinson, “Towards 3C-3D digital holographic fluid velocity vector field measurement—tomographic digital holographic PIV (Tomo-HPIV),” Meas. Sci. Technol. 19, 074002 (2008).
[Crossref]

Bae, Y. S.

Y. S. Bae, J. I. Song, and D. Y. Kim, “Volumetric reconstruction of Brownian motion of a micrometer-size bead in water,” Opt. Commun. 309, 291–297 (2013).
[Crossref]

Beghuin, D.

Bohren, C. F.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 2008).

Buchmann, N. A.

N. A. Buchmann, C. Atkinson, and J. Soria, “Ultra-high-speed tomographic digital holographic velocimetry in supersonic particle-laden jet flows,” Meas. Sci. Technol. 24, 024005 (2013).
[Crossref]

Byeon, H.

H. Byeon, T. Go, and S. J. Lee, “Digital stereo-holographic microscopy for studying three-dimensional particle dynamics,” Opt. Lasers Eng. 105, 6–13 (2018).
[Crossref]

Byeon, H. J.

Caprio, G. D.

Chan, K. T.

K. T. Chan and Y. J. Li, “Pipe flow measurement by using a side-scattering holographic particle imaging technique,” Opt. Laser Technol. 30, 7–14 (1998).
[Crossref]

Chaudhary, K.

A. Wang, T. G. Dimiduk, J. Fung, S. Razavi, I. Kretzschmar, K. Chaudhary, and V. N. Manoharan, “Using the discrete dipole approximation and holographic microscopy to measure rotational dynamics of non-spherical colloidal particles,” J. Quant. Spectrosc. Radiat. Transfer 146, 499–509 (2014).
[Crossref]

Cheng, Z.-J.

Cheong, F. C.

Colomb, T.

T. Colomb, E. Cuche, F. Montfort, P. Marquet, and C. Depeursinge, “Jones vector imaging by use of digital holography: simulation and experimentation,” Opt. Commun. 231, 137–147 (2004).
[Crossref]

T. Colomb, P. Dahlgren, D. Beghuin, E. Cuche, P. Marquet, and C. Depeursinge, “Polarization imaging by use of digital holography,” Appl. Opt. 41, 27–37 (2002).
[Crossref]

Coppola, G.

Cuche, E.

T. Colomb, E. Cuche, F. Montfort, P. Marquet, and C. Depeursinge, “Jones vector imaging by use of digital holography: simulation and experimentation,” Opt. Commun. 231, 137–147 (2004).
[Crossref]

T. Colomb, P. Dahlgren, D. Beghuin, E. Cuche, P. Marquet, and C. Depeursinge, “Polarization imaging by use of digital holography,” Appl. Opt. 41, 27–37 (2002).
[Crossref]

D’Ippolito, G.

F. Merola, P. Memmolo, L. Miccio, R. Savoia, M. Mugnano, A. Fontana, G. D’Ippolito, A. Sardo, A. Iolascon, A. Gambale, and P. Ferraro, “Tomographic flow cytometry by digital holography,” Light Sci. Appl. 6, e16241 (2017).
[Crossref]

Dahlgren, P.

Depeursinge, C.

T. Colomb, E. Cuche, F. Montfort, P. Marquet, and C. Depeursinge, “Jones vector imaging by use of digital holography: simulation and experimentation,” Opt. Commun. 231, 137–147 (2004).
[Crossref]

T. Colomb, P. Dahlgren, D. Beghuin, E. Cuche, P. Marquet, and C. Depeursinge, “Polarization imaging by use of digital holography,” Appl. Opt. 41, 27–37 (2002).
[Crossref]

Dimiduk, T. G.

A. Wang, T. G. Dimiduk, J. Fung, S. Razavi, I. Kretzschmar, K. Chaudhary, and V. N. Manoharan, “Using the discrete dipole approximation and holographic microscopy to measure rotational dynamics of non-spherical colloidal particles,” J. Quant. Spectrosc. Radiat. Transfer 146, 499–509 (2014).
[Crossref]

Ferraro, P.

F. Merola, P. Memmolo, L. Miccio, R. Savoia, M. Mugnano, A. Fontana, G. D’Ippolito, A. Sardo, A. Iolascon, A. Gambale, and P. Ferraro, “Tomographic flow cytometry by digital holography,” Light Sci. Appl. 6, e16241 (2017).
[Crossref]

P. Memmolo, L. Miccio, M. Paturzo, G. D. Caprio, G. Coppola, P. A. Netti, and P. Ferraro, “Recent advances in holographic 3D particle tracking,” Adv. Opt. Photon. 7, 713–755 (2015).
[Crossref]

Fontana, A.

F. Merola, P. Memmolo, L. Miccio, R. Savoia, M. Mugnano, A. Fontana, G. D’Ippolito, A. Sardo, A. Iolascon, A. Gambale, and P. Ferraro, “Tomographic flow cytometry by digital holography,” Light Sci. Appl. 6, e16241 (2017).
[Crossref]

Fournel, T.

Fournier, C.

Fung, J.

A. Wang, T. G. Dimiduk, J. Fung, S. Razavi, I. Kretzschmar, K. Chaudhary, and V. N. Manoharan, “Using the discrete dipole approximation and holographic microscopy to measure rotational dynamics of non-spherical colloidal particles,” J. Quant. Spectrosc. Radiat. Transfer 146, 499–509 (2014).
[Crossref]

J. Fung, K. E. Martin, R. W. Perry, D. M. Kaz, R. McGorty, and V. N. Manoharan, “Measuring translational, rotational, and vibrational dynamics in colloids with digital holographic microscopy,” Opt. Express 19, 8051–8065 (2011).
[Crossref]

Gambale, A.

F. Merola, P. Memmolo, L. Miccio, R. Savoia, M. Mugnano, A. Fontana, G. D’Ippolito, A. Sardo, A. Iolascon, A. Gambale, and P. Ferraro, “Tomographic flow cytometry by digital holography,” Light Sci. Appl. 6, e16241 (2017).
[Crossref]

Gao, J.

Go, T.

H. Byeon, T. Go, and S. J. Lee, “Digital stereo-holographic microscopy for studying three-dimensional particle dynamics,” Opt. Lasers Eng. 105, 6–13 (2018).
[Crossref]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).

Gren, P.

D. Khodadad, E. Amer, P. Gren, E. Melander, E. Hällstig, and M. Sjödahl, “Single-shot dual-polarization holography: measurement of the polarization state of a magnetic sample,” Proc. SPIE 9660, 96601E (2015).
[Crossref]

Grier, D. G.

Guo, C.-S.

Hällstig, E.

D. Khodadad, E. Amer, P. Gren, E. Melander, E. Hällstig, and M. Sjödahl, “Single-shot dual-polarization holography: measurement of the polarization state of a magnetic sample,” Proc. SPIE 9660, 96601E (2015).
[Crossref]

Han, L.

Huffman, D. R.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 2008).

Iolascon, A.

F. Merola, P. Memmolo, L. Miccio, R. Savoia, M. Mugnano, A. Fontana, G. D’Ippolito, A. Sardo, A. Iolascon, A. Gambale, and P. Ferraro, “Tomographic flow cytometry by digital holography,” Light Sci. Appl. 6, e16241 (2017).
[Crossref]

Jeong, J.

Jia, D.

H. Zhang, M. Zhai, J. Sun, Y. Zhou, D. Jia, T. Liu, and Y. Zhang, “Discrimination between spheres and spheroids in a detection system for single particles based on polarization characteristics,” J. Quant. Spectrosc. Radiat. Transfer 187, 62–75 (2017).
[Crossref]

Katz, J.

Kaz, D. M.

Khodadad, D.

D. Khodadad, E. Amer, P. Gren, E. Melander, E. Hällstig, and M. Sjödahl, “Single-shot dual-polarization holography: measurement of the polarization state of a magnetic sample,” Proc. SPIE 9660, 96601E (2015).
[Crossref]

Kim, D. Y.

Y. S. Bae, J. I. Song, and D. Y. Kim, “Volumetric reconstruction of Brownian motion of a micrometer-size bead in water,” Opt. Commun. 309, 291–297 (2013).
[Crossref]

Kim, M. W.

Kim, S.-H.

Kim, Y.

Kostinski, A. B.

Kretzschmar, I.

A. Wang, T. G. Dimiduk, J. Fung, S. Razavi, I. Kretzschmar, K. Chaudhary, and V. N. Manoharan, “Using the discrete dipole approximation and holographic microscopy to measure rotational dynamics of non-spherical colloidal particles,” J. Quant. Spectrosc. Radiat. Transfer 146, 499–509 (2014).
[Crossref]

Krishnatreya, B. J.

Lee, S. J.

H. Byeon, T. Go, and S. J. Lee, “Digital stereo-holographic microscopy for studying three-dimensional particle dynamics,” Opt. Lasers Eng. 105, 6–13 (2018).
[Crossref]

H. J. Byeon, K. W. Seo, and S. J. Lee, “Precise measurement of three-dimensional positions of transparent ellipsoidal particles using digital holographic microscopy,” Appl. Opt. 54, 2106–2112 (2015).
[Crossref]

Lee, S.-H.

Li, Y. J.

K. T. Chan and Y. J. Li, “Pipe flow measurement by using a side-scattering holographic particle imaging technique,” Opt. Laser Technol. 30, 7–14 (1998).
[Crossref]

Liu, T.

H. Zhang, M. Zhai, J. Sun, Y. Zhou, D. Jia, T. Liu, and Y. Zhang, “Discrimination between spheres and spheroids in a detection system for single particles based on polarization characteristics,” J. Quant. Spectrosc. Radiat. Transfer 187, 62–75 (2017).
[Crossref]

Liu, X.

Manoharan, V. N.

A. Wang, T. G. Dimiduk, J. Fung, S. Razavi, I. Kretzschmar, K. Chaudhary, and V. N. Manoharan, “Using the discrete dipole approximation and holographic microscopy to measure rotational dynamics of non-spherical colloidal particles,” J. Quant. Spectrosc. Radiat. Transfer 146, 499–509 (2014).
[Crossref]

J. Fung, K. E. Martin, R. W. Perry, D. M. Kaz, R. McGorty, and V. N. Manoharan, “Measuring translational, rotational, and vibrational dynamics in colloids with digital holographic microscopy,” Opt. Express 19, 8051–8065 (2011).
[Crossref]

Marquet, P.

T. Colomb, E. Cuche, F. Montfort, P. Marquet, and C. Depeursinge, “Jones vector imaging by use of digital holography: simulation and experimentation,” Opt. Commun. 231, 137–147 (2004).
[Crossref]

T. Colomb, P. Dahlgren, D. Beghuin, E. Cuche, P. Marquet, and C. Depeursinge, “Polarization imaging by use of digital holography,” Appl. Opt. 41, 27–37 (2002).
[Crossref]

Martin, K. E.

McGorty, R.

Melander, E.

D. Khodadad, E. Amer, P. Gren, E. Melander, E. Hällstig, and M. Sjödahl, “Single-shot dual-polarization holography: measurement of the polarization state of a magnetic sample,” Proc. SPIE 9660, 96601E (2015).
[Crossref]

Memmolo, P.

F. Merola, P. Memmolo, L. Miccio, R. Savoia, M. Mugnano, A. Fontana, G. D’Ippolito, A. Sardo, A. Iolascon, A. Gambale, and P. Ferraro, “Tomographic flow cytometry by digital holography,” Light Sci. Appl. 6, e16241 (2017).
[Crossref]

P. Memmolo, L. Miccio, M. Paturzo, G. D. Caprio, G. Coppola, P. A. Netti, and P. Ferraro, “Recent advances in holographic 3D particle tracking,” Adv. Opt. Photon. 7, 713–755 (2015).
[Crossref]

Meng, H.

Merola, F.

F. Merola, P. Memmolo, L. Miccio, R. Savoia, M. Mugnano, A. Fontana, G. D’Ippolito, A. Sardo, A. Iolascon, A. Gambale, and P. Ferraro, “Tomographic flow cytometry by digital holography,” Light Sci. Appl. 6, e16241 (2017).
[Crossref]

Miccio, L.

F. Merola, P. Memmolo, L. Miccio, R. Savoia, M. Mugnano, A. Fontana, G. D’Ippolito, A. Sardo, A. Iolascon, A. Gambale, and P. Ferraro, “Tomographic flow cytometry by digital holography,” Light Sci. Appl. 6, e16241 (2017).
[Crossref]

P. Memmolo, L. Miccio, M. Paturzo, G. D. Caprio, G. Coppola, P. A. Netti, and P. Ferraro, “Recent advances in holographic 3D particle tracking,” Adv. Opt. Photon. 7, 713–755 (2015).
[Crossref]

Montfort, F.

T. Colomb, E. Cuche, F. Montfort, P. Marquet, and C. Depeursinge, “Jones vector imaging by use of digital holography: simulation and experimentation,” Opt. Commun. 231, 137–147 (2004).
[Crossref]

Mugnano, M.

F. Merola, P. Memmolo, L. Miccio, R. Savoia, M. Mugnano, A. Fontana, G. D’Ippolito, A. Sardo, A. Iolascon, A. Gambale, and P. Ferraro, “Tomographic flow cytometry by digital holography,” Light Sci. Appl. 6, e16241 (2017).
[Crossref]

Netti, P. A.

Öhman, J.

Park, Y.

Paturzo, M.

Perry, R. W.

Pu, Y.

Razavi, S.

A. Wang, T. G. Dimiduk, J. Fung, S. Razavi, I. Kretzschmar, K. Chaudhary, and V. N. Manoharan, “Using the discrete dipole approximation and holographic microscopy to measure rotational dynamics of non-spherical colloidal particles,” J. Quant. Spectrosc. Radiat. Transfer 146, 499–509 (2014).
[Crossref]

Roichman, Y.

Sardo, A.

F. Merola, P. Memmolo, L. Miccio, R. Savoia, M. Mugnano, A. Fontana, G. D’Ippolito, A. Sardo, A. Iolascon, A. Gambale, and P. Ferraro, “Tomographic flow cytometry by digital holography,” Light Sci. Appl. 6, e16241 (2017).
[Crossref]

Savoia, R.

F. Merola, P. Memmolo, L. Miccio, R. Savoia, M. Mugnano, A. Fontana, G. D’Ippolito, A. Sardo, A. Iolascon, A. Gambale, and P. Ferraro, “Tomographic flow cytometry by digital holography,” Light Sci. Appl. 6, e16241 (2017).
[Crossref]

Schedin, S.

Seo, K. W.

Shaw, R. A.

Sjödahl, M.

Song, J. I.

Y. S. Bae, J. I. Song, and D. Y. Kim, “Volumetric reconstruction of Brownian motion of a micrometer-size bead in water,” Opt. Commun. 309, 291–297 (2013).
[Crossref]

Soria, J.

N. A. Buchmann, C. Atkinson, and J. Soria, “Ultra-high-speed tomographic digital holographic velocimetry in supersonic particle-laden jet flows,” Meas. Sci. Technol. 24, 024005 (2013).
[Crossref]

J. Soria and C. Atkinson, “Towards 3C-3D digital holographic fluid velocity vector field measurement—tomographic digital holographic PIV (Tomo-HPIV),” Meas. Sci. Technol. 19, 074002 (2008).
[Crossref]

Sun, J.

H. Zhang, M. Zhai, J. Sun, Y. Zhou, D. Jia, T. Liu, and Y. Zhang, “Discrimination between spheres and spheroids in a detection system for single particles based on polarization characteristics,” J. Quant. Spectrosc. Radiat. Transfer 187, 62–75 (2017).
[Crossref]

van Blaaderen, A.

van Oostrum, P.

Verrier, N.

Wang, A.

A. Wang, T. G. Dimiduk, J. Fung, S. Razavi, I. Kretzschmar, K. Chaudhary, and V. N. Manoharan, “Using the discrete dipole approximation and holographic microscopy to measure rotational dynamics of non-spherical colloidal particles,” J. Quant. Spectrosc. Radiat. Transfer 146, 499–509 (2014).
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H. Zhang, M. Zhai, J. Sun, Y. Zhou, D. Jia, T. Liu, and Y. Zhang, “Discrimination between spheres and spheroids in a detection system for single particles based on polarization characteristics,” J. Quant. Spectrosc. Radiat. Transfer 187, 62–75 (2017).
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H. Zhang, M. Zhai, J. Sun, Y. Zhou, D. Jia, T. Liu, and Y. Zhang, “Discrimination between spheres and spheroids in a detection system for single particles based on polarization characteristics,” J. Quant. Spectrosc. Radiat. Transfer 187, 62–75 (2017).
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H. Zhang, M. Zhai, J. Sun, Y. Zhou, D. Jia, T. Liu, and Y. Zhang, “Discrimination between spheres and spheroids in a detection system for single particles based on polarization characteristics,” J. Quant. Spectrosc. Radiat. Transfer 187, 62–75 (2017).
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Figures (9)

Fig. 1.
Fig. 1. General scattering problem. A particle is illuminated from below by the field E ¯ i and generates a scattered field E ¯ s . E ¯ i is linearly polarized at an angle α . The scattering planes for each camera are also shown, orange for camera 1 and blue for camera 2.
Fig. 2.
Fig. 2. Polarization-resolved reference waves. (a) Configuration of the reference waves in the aperture plane; the arrows indicate the polarization orientation of each reference wave. (b) Corresponding spatial frequency content of a polarization-resolved hologram. In this example, a square aperture is used, thus the square lobes in the spatial frequency domain.
Fig. 3.
Fig. 3. Orientation of the local coordinate systems and a visualization of how the particles are elongated along the optical axis of each camera.
Fig. 4.
Fig. 4. Flowcharts for (a) calibration process and (b) evaluation process. The results E 1 and E 2 are used in the evaluation of measurements as shown.
Fig. 5.
Fig. 5. Sketch of the experimental setup. Sample cuvette is imaged by the dual-view polarization-resolved holographic system. M, mirrors; L, lenses; T, 5 × telescope lens; BS, beam splitters; FC, fiber couplers; A, apertures; and BD, beam dump. Inset (a) shows the side view of the sample cell volume and (b) shows the configuration of the fiber ends in the aperture plate.
Fig. 6.
Fig. 6. Polarization calibration. Resulting polarization angle β for the two cavities as a function of the illumination polarization α . (a), (b) Result for camera 1; (c), (d) result for camera 2. The standard deviation is 0.6°.
Fig. 7.
Fig. 7. Estimated field of curvature on camera 1 and camera 2, respectively, from the correction functions E 1 and E 2 , respectively. The lateral field of view is 5.4 mm for each camera.
Fig. 8.
Fig. 8. Particle tracking using both global and single camera coordinates for different illumination polarizations. Each row is for a certain α , while each column is for detection with either camera 1 or camera 2 or global coordinates. All tracks for each row are estimated from the same set of holograms. It is observed that far more particles are detected in (b) and (d) than in (a) and (e), respectively.
Fig. 9.
Fig. 9. Polarization ratio angle β for the particles in Fig. 8. The first column is all the particles recorded by camera 1, the second column is all particles recorded by camera 2, and the third and fourth columns are particles recorded by camera 1 and camera 2, respectively, that are successfully mapped to the global coordinate system. Each row corresponds to a specific illumination polarization; (a)–(d)  α = 0 ° , (e)–(h)  α = 90 ° , and (i)–(l)  α = 45 ° .

Equations (16)

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E ¯ s = ( | E s | exp ( i φ s ) | E s | exp ( i φ s ) )
R ¯ = ( | R | exp ( i φ R ) 0 ) ,
R ¯ = ( 0 | R | exp ( i φ R ) ) ,
I = | E ¯ s + R ¯ | 2 = | E s | 2 + | E s | 2 + | R | 2 + | R | 2 + E s R * + E s * R + E s R * + E s * R ,
U = E s R * = | E s | | R | exp [ i ( φ s φ R ) ] ,
U = E s R * = | E o | | R | exp [ i ( φ s φ R ) ] .
E ^ s = U R ^ | R ^ | 2 = | E ^ s | exp [ i ( φ s α ) ] ,
E ^ s = U R ^ | R ^ | 2 = | E ^ s | exp [ i ( φ s α ) ] ,
E ^ ( x , y , z D + Δ z ) = F 2 D 1 { F 2 D { E ^ ( x , y , z D ) } exp [ i 2 π f z Δ z ] } ,
β = arctan ( | E ^ ( x , y , z ) | | E ^ ( x , y , z ) | ) ,
g : ( x 2 ( 2 ) , y 2 ( 2 ) , z 2 ( 2 ) ) ( x 2 ( 1 ) , y 2 ( 1 ) , z 2 ( 1 ) ) ,
f 1 : ( x ˜ 1 ( 1 ) , y ˜ 1 ( 1 ) , z ˜ 1 ( 1 ) ) ( x 1 ( 1 ) , y 1 ( 1 ) , z 1 ( 1 ) ) ,
f 2 : ( x ˜ 2 ( 2 ) , y ˜ 2 ( 2 ) , z ˜ 2 ( 2 ) ) ( x 2 ( 2 ) , y 2 ( 2 ) , z 2 ( 2 ) ) ,
PC = exp ( [ ( x 1 ( 1 ) x 2 ( 1 ) σ x ) 2 + ( y 1 ( 1 ) y 2 ( 1 ) σ y ) 2 + ( z 1 ( 1 ) z 2 ( 1 ) σ z ) 2 ] ) ,
z 1 ( 1 ) ( x , y ) = z ˜ 1 ( 1 ) E 1 ( x , y , z ) ,
z 2 ( 2 ) ( x , y ) = z ˜ 2 ( 2 ) E 2 ( x , y , z ) .