Abstract

We propose multiwavelength in-line digital holography with wavelength-multiplexed phase-shifted holograms and arbitrary symmetric phase shifts. We use phase-shifting interferometry selectively extracting wavelength information to reconstruct multiwavelength object waves separately from wavelength-multiplexed monochromatic images. The proposed technique obtains systems of equations for real and imaginary parts of multiwavelength object waves from the holograms by introducing arbitrary symmetric phase shifts. Then, the technique derives each complex amplitude distribution of each object wave selectively and analytically by solving the two systems of equations. We formulate the algorithm in the case of an arbitrary number of wavelengths and confirm its validity numerically and experimentally in the cases where the number of wavelengths is two and three.

© 2017 Optical Society of America

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References

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2016 (1)

2015 (3)

X. Quan, K. Nitta, O. Matoba, P. Xia, and Y. Awatsuji, “Phase and fluorescence imaging by combination of digital holographic microscopy and fluorescence microscopy,” Opt. Rev. 22(2), 349–353 (2015).
[Crossref]

T. Tahara, R. Mori, S. Kikunaga, Y. Arai, and Y. Takaki, “Dual-wavelength phase-shifting digital holography selectively extracting wavelength information from wavelength-multiplexed holograms,” Opt. Lett. 40(12), 2810–2813 (2015).
[Crossref] [PubMed]

T. Tahara, R. Mori, Y. Arai, and Y. Takaki, “Four-step phase-shifting digital holography simultaneously sensing dual-wavelength information using a monochromatic image sensor,” J. Opt. 17, 125707 (2015).

2013 (2)

2012 (1)

2011 (2)

2009 (1)

A. Alfalou and C. Brosseau, “Optical image compression and encryption methods,” Adv. Opt. Photonics 1(3), 589–636 (2009).
[Crossref]

2008 (1)

P. Ferraro, S. Grilli, L. Miccio, D. Alfieri, S. De Nicola, A. Finizio, and B. Javidi, “Full color 3-D imaging by digital holography and removal of chromatic aberrations,” J. Disp. Technol. 4(1), 97–100 (2008).
[Crossref]

2006 (1)

2002 (2)

2000 (1)

S. Murata and N. Yasuda, “Potential of digital holography in particle measurement,” Opt. Laser Technol. 32(7-8), 567–574 (2000).
[Crossref]

1999 (1)

1997 (1)

1984 (1)

1974 (1)

1967 (1)

J. W. Goodman and R. W. Lawrence, “Digital image formation from electronically detected holograms,” Appl. Phys. Lett. 11(3), 77–79 (1967).
[Crossref]

1948 (1)

D. Gabor, “A new microscopic principle,” Nature 161(4098), 777–778 (1948).
[Crossref] [PubMed]

Alfalou, A.

A. Alfalou and C. Brosseau, “Optical image compression and encryption methods,” Adv. Opt. Photonics 1(3), 589–636 (2009).
[Crossref]

Alfieri, D.

P. Ferraro, S. Grilli, L. Miccio, D. Alfieri, S. De Nicola, A. Finizio, and B. Javidi, “Full color 3-D imaging by digital holography and removal of chromatic aberrations,” J. Disp. Technol. 4(1), 97–100 (2008).
[Crossref]

Andrés, P.

Arai, Y.

T. Tahara, R. Mori, S. Kikunaga, Y. Arai, and Y. Takaki, “Dual-wavelength phase-shifting digital holography selectively extracting wavelength information from wavelength-multiplexed holograms,” Opt. Lett. 40(12), 2810–2813 (2015).
[Crossref] [PubMed]

T. Tahara, R. Mori, Y. Arai, and Y. Takaki, “Four-step phase-shifting digital holography simultaneously sensing dual-wavelength information using a monochromatic image sensor,” J. Opt. 17, 125707 (2015).

Araiza-Esquivel, M. A.

Awatsuji, Y.

X. Quan, K. Nitta, O. Matoba, P. Xia, and Y. Awatsuji, “Phase and fluorescence imaging by combination of digital holographic microscopy and fluorescence microscopy,” Opt. Rev. 22(2), 349–353 (2015).
[Crossref]

Barada, D.

Brangaccio, D. J.

Brosseau, C.

A. Alfalou and C. Brosseau, “Optical image compression and encryption methods,” Adv. Opt. Photonics 1(3), 589–636 (2009).
[Crossref]

Bruning, J. H.

Cheng, Y.-Y.

De Nicola, S.

P. Ferraro, S. Grilli, L. Miccio, D. Alfieri, S. De Nicola, A. Finizio, and B. Javidi, “Full color 3-D imaging by digital holography and removal of chromatic aberrations,” J. Disp. Technol. 4(1), 97–100 (2008).
[Crossref]

Desse, J. M.

J. M. Desse, P. Picart, and P. Tankam, “Sensor influence in digital 3λ holographic interferometry,” Meas. Sci. Technol. 22(6), 064005 (2011).
[Crossref]

Ferraro, P.

P. Ferraro, S. Grilli, L. Miccio, D. Alfieri, S. De Nicola, A. Finizio, and B. Javidi, “Full color 3-D imaging by digital holography and removal of chromatic aberrations,” J. Disp. Technol. 4(1), 97–100 (2008).
[Crossref]

Finizio, A.

P. Ferraro, S. Grilli, L. Miccio, D. Alfieri, S. De Nicola, A. Finizio, and B. Javidi, “Full color 3-D imaging by digital holography and removal of chromatic aberrations,” J. Disp. Technol. 4(1), 97–100 (2008).
[Crossref]

Gabor, D.

D. Gabor, “A new microscopic principle,” Nature 161(4098), 777–778 (1948).
[Crossref] [PubMed]

Gallagher, J. E.

Goodman, J. W.

J. W. Goodman and R. W. Lawrence, “Digital image formation from electronically detected holograms,” Appl. Phys. Lett. 11(3), 77–79 (1967).
[Crossref]

Grilli, S.

P. Ferraro, S. Grilli, L. Miccio, D. Alfieri, S. De Nicola, A. Finizio, and B. Javidi, “Full color 3-D imaging by digital holography and removal of chromatic aberrations,” J. Disp. Technol. 4(1), 97–100 (2008).
[Crossref]

Hayasaki, Y.

Herriott, D. R.

Hoshiba, T.

Javidi, B.

Kato, J.

Kawai, H.

Kiire, T.

Kikunaga, S.

Kim, N.

Lancis, J.

Lawrence, R. W.

J. W. Goodman and R. W. Lawrence, “Digital image formation from electronically detected holograms,” Appl. Phys. Lett. 11(3), 77–79 (1967).
[Crossref]

Leclercq, M.

Li, W. N.

Martínez-León, L.

Matoba, O.

X. Quan, K. Nitta, O. Matoba, P. Xia, and Y. Awatsuji, “Phase and fluorescence imaging by combination of digital holographic microscopy and fluorescence microscopy,” Opt. Rev. 22(2), 349–353 (2015).
[Crossref]

Matsumura, T.

Miccio, L.

P. Ferraro, S. Grilli, L. Miccio, D. Alfieri, S. De Nicola, A. Finizio, and B. Javidi, “Full color 3-D imaging by digital holography and removal of chromatic aberrations,” J. Disp. Technol. 4(1), 97–100 (2008).
[Crossref]

Mori, R.

T. Tahara, R. Mori, Y. Arai, and Y. Takaki, “Four-step phase-shifting digital holography simultaneously sensing dual-wavelength information using a monochromatic image sensor,” J. Opt. 17, 125707 (2015).

T. Tahara, R. Mori, S. Kikunaga, Y. Arai, and Y. Takaki, “Dual-wavelength phase-shifting digital holography selectively extracting wavelength information from wavelength-multiplexed holograms,” Opt. Lett. 40(12), 2810–2813 (2015).
[Crossref] [PubMed]

Murata, S.

S. Murata and N. Yasuda, “Potential of digital holography in particle measurement,” Opt. Laser Technol. 32(7-8), 567–574 (2000).
[Crossref]

Nitta, K.

X. Quan, K. Nitta, O. Matoba, P. Xia, and Y. Awatsuji, “Phase and fluorescence imaging by combination of digital holographic microscopy and fluorescence microscopy,” Opt. Rev. 22(2), 349–353 (2015).
[Crossref]

Ohzu, H.

Piao, M.-L.

Picart, P.

Quan, X.

X. Quan, K. Nitta, O. Matoba, P. Xia, and Y. Awatsuji, “Phase and fluorescence imaging by combination of digital holographic microscopy and fluorescence microscopy,” Opt. Rev. 22(2), 349–353 (2015).
[Crossref]

Rosenfeld, D. P.

Shi, C.-X.

Stern, A.

Sugisaka, J.

Tahara, T.

T. Tahara, R. Mori, S. Kikunaga, Y. Arai, and Y. Takaki, “Dual-wavelength phase-shifting digital holography selectively extracting wavelength information from wavelength-multiplexed holograms,” Opt. Lett. 40(12), 2810–2813 (2015).
[Crossref] [PubMed]

T. Tahara, R. Mori, Y. Arai, and Y. Takaki, “Four-step phase-shifting digital holography simultaneously sensing dual-wavelength information using a monochromatic image sensor,” J. Opt. 17, 125707 (2015).

Tajahuerce, E.

Takaki, Y.

Tankam, P.

J. M. Desse, P. Picart, and P. Tankam, “Sensor influence in digital 3λ holographic interferometry,” Meas. Sci. Technol. 22(6), 064005 (2011).
[Crossref]

Watanabe, E.

White, A. D.

Wyant, J. C.

Xia, P.

X. Quan, K. Nitta, O. Matoba, P. Xia, and Y. Awatsuji, “Phase and fluorescence imaging by combination of digital holographic microscopy and fluorescence microscopy,” Opt. Rev. 22(2), 349–353 (2015).
[Crossref]

Yamaguchi, I.

Yasuda, N.

S. Murata and N. Yasuda, “Potential of digital holography in particle measurement,” Opt. Laser Technol. 32(7-8), 567–574 (2000).
[Crossref]

Yatagai, T.

Zhang, T.

Adv. Opt. Photonics (1)

A. Alfalou and C. Brosseau, “Optical image compression and encryption methods,” Adv. Opt. Photonics 1(3), 589–636 (2009).
[Crossref]

Appl. Opt. (5)

Appl. Phys. Lett. (1)

J. W. Goodman and R. W. Lawrence, “Digital image formation from electronically detected holograms,” Appl. Phys. Lett. 11(3), 77–79 (1967).
[Crossref]

J. Disp. Technol. (1)

P. Ferraro, S. Grilli, L. Miccio, D. Alfieri, S. De Nicola, A. Finizio, and B. Javidi, “Full color 3-D imaging by digital holography and removal of chromatic aberrations,” J. Disp. Technol. 4(1), 97–100 (2008).
[Crossref]

J. Opt. (1)

T. Tahara, R. Mori, Y. Arai, and Y. Takaki, “Four-step phase-shifting digital holography simultaneously sensing dual-wavelength information using a monochromatic image sensor,” J. Opt. 17, 125707 (2015).

J. Opt. Soc. Am. A (1)

Meas. Sci. Technol. (1)

J. M. Desse, P. Picart, and P. Tankam, “Sensor influence in digital 3λ holographic interferometry,” Meas. Sci. Technol. 22(6), 064005 (2011).
[Crossref]

Nature (1)

D. Gabor, “A new microscopic principle,” Nature 161(4098), 777–778 (1948).
[Crossref] [PubMed]

Opt. Express (1)

Opt. Laser Technol. (1)

S. Murata and N. Yasuda, “Potential of digital holography in particle measurement,” Opt. Laser Technol. 32(7-8), 567–574 (2000).
[Crossref]

Opt. Lett. (6)

Opt. Rev. (1)

X. Quan, K. Nitta, O. Matoba, P. Xia, and Y. Awatsuji, “Phase and fluorescence imaging by combination of digital holographic microscopy and fluorescence microscopy,” Opt. Rev. 22(2), 349–353 (2015).
[Crossref]

Other (4)

T.-C. Poon and J.-P. Liu, eds., Introduction to Modern Digital Holography with MATLAB (Cambridge University, 2014).

T. Tahara, S. Kikunaga, Y. Arai, and Y. Takaki, “Phase-shifting interferometry capable of selectively extracting multiple wavelength information and color three-dimensional imaging using a monochromatic image sensor,” in Proceedings of Optics and Photonics Japan (OPJ, 2013), paper 13aE9.

T. Tahara, S. Kikunaga, Y. Arai, and Y. Takaki, “Phase-shifting interferometry capable of selectively extracting multiple wavelength information and its applications to sequential and parallel phase-shifting digital holography,” in Digital Holography and Three-Dimensional Imaging 2014 (DH), OSA Technical Digest (online) (Optical Society of America, 2014), paper DM3B.4.

T. Tahara, R. Otani, Y. Arai, and Y. Takaki, “Multiwavelength digital holography and phase-shifting interferometry selectively extracting wavelength information: phase-division multiplexing (PDM) of wavelengths,” in Holographic Materials and Optical Systems, I. Naydenova Ed. (InTech, 2017).

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Figures (8)

Fig. 1
Fig. 1 Basic concept of phase-shifting interferometry selectively extracting wavelength information. (a) Flow from recording to reconstruction. (b) Wavelength separation in the polar coordinate plane when applying arbitrary symmetric phase shifts.
Fig. 2
Fig. 2 An optical implementation and obtained holograms in the proposed digital holography.
Fig. 3
Fig. 3 The optical system and object wave set for numerical simulations, and numerically obtained hologram. (a) Optical setup with a piezo-driven mirror. (b) Amplitude and (c) phase distributions of the object wave. (d) Red-, (e) green-, and (f) blue-color components of (b). (g) One of the numerically obtained wavelength-multiplexed phase-shifted holograms.
Fig. 4
Fig. 4 Numerical results. Reconstructed amplitude images at the wavelengths of (a) 633 nm, (b) 532 nm, and (c) 473 nm and (d) the color-synthesized image. Phase images at (e) 633 nm, (f) 532 nm, and (g) 473 nm obtained from the holograms with a 50 nm shift of the mirror for each phase shift. Phase images at (h) 633 nm, (i) 532 nm, and (j) 473 nm obtained from the holograms with a 190 nm shift.
Fig. 5
Fig. 5 (a) RMSEs and (b) CCs of the reconstructed amplitude images, which are calculated for quantitative evaluations of the image quality.
Fig. 6
Fig. 6 Experimental results. (a) A photograph of the object that is illuminated by natural light. (b) One of wavelength-multiplexed phase-shifted holograms and its magnified image. Images reconstructed with Fig. 6(b) at the wavelengths of (c) 640 mm and (d) 532 mm. Images reconstructed by the proposed technique at the wavelengths of (e) 640 mm and (f) 532 mm, which are focused on the depth of 115 mm from the image sensor plane. Those at the wavelengths of (g) 640 mm and (h) 532 mm, and 145 mm distance from the image sensor plane. Color-synthesized images obtained from (i) Figs. 6(e) and 6(f), and (j) Figs. 6(g) and 6(h).
Fig. 7
Fig. 7 Schematic of the constructed three-wavelength digital holographic microscopy system.
Fig. 8
Fig. 8 Experimental results. Photographs of a specimen at (a) λ1, (b) λ2, and (c) λ3, and (d) its color-synthesized image. (e) – (g) Intensity images reconstructed from a wavelength-multiplexed in-line hologram and (h) its color-synthesized image. (i) – (k) Images obtained by the proposed microscopy and (l) its color-synthesized image. Magnified color-synthesized images in which focused planes are (m) 0 and (n) 10 mm from the image sensor one, respectively. Arrows shown in (n) indicate the focused images of the stained cell nuclei on a preparation of a mouse kidney cells. Rectangles shown in (i) – (l) correspond with the areas of (m) and (n).

Tables (1)

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Table 1 RMSEs of the reconstructed images.

Equations (35)

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I( x,y: α 11 , α 21 ,, α N1 )=  I 0th ( x,y )+2 i=1 N A oi ( x,y ) A ri ( x,y )cos[ ϕ oi ( x,y ) α i1 ].
( I( x,y:0,,0 ) I( x,y: α 11 ,, α N1 ) I( x,y: α 12 ,, α N2 ) I( x,y: α 12N ,, α N2N ) )=( 1 1 0 1 cos α 11 sin α 11 1 cos α 12 sin α 12 ... 1 0 ... cos α N1 sin α N1 ... cos α N2 sin α N2 1 cos α 12N sin α 12N ... ... cos α N2N sin α N2N ) ×( I 0th ( x,y ) 2 A o1 ( x,y ) A r1 ( x,y )cos ϕ o1 ( x,y ) 2 A o1 ( x,y ) A r1 ( x,y )sin ϕ o1 ( x,y ) 2 A oN ( x,y ) A rN ( x,y )sin ϕ oN ( x,y ) ).
( I 1 ( x,y ) I 2 ( x,y ) I 3 ( x,y ) I 2N ( x,y ) I 2N+1 ( x,y ) )=( I( x,y:0,,0 ) I( x,y: α 11 ,, α N1 ) I( x,y: α 11 ,, α N1 ) I( x,y: α 1N ,, α NN ) I( x,y: α 1N ,, α NN ) ) =( 1 1 0 1 cos α 11 sin α 11 1 cos α 11 sin α 11 ... 1 0 ... cos α N1 sin α N1 ... cos α N1 sin α N1 1 cos α 1N sin α 1N ... ... cos α NN sin α NN 1 cos α 1N sin α 1N ... cos α NN sin α NN ) ×( I 0th ( x,y ) 2 A o1 ( x,y ) A r1 ( x,y )cos ϕ o1 ( x,y ) 2 A o1 ( x,y ) A r1 ( x,y )sin ϕ o1 ( x,y ) 2 A oN ( x,y ) A rN ( x,y )cos ϕ oN ( x,y ) 2 A oN ( x,y ) A rN ( x,y )sin ϕ oN ( x,y ) ).
R e k (x,y)= i=1 N A oi (x,y) A ri (x,y)cos ϕ oi (x,y)(1cos α ik ) (k=1 , ... , N) = 2 I 1 (x,y)[ I 2k (x,y)+ I 2k+1 (x,y)] 4 ,
I m k (x,y)= i=1 N A oi (x,y) A ri (x,y)sin ϕ oi (x,y)sin α ik = I 2k (x,y) I 2k+1 (x,y) 4 .
R e 1 ( x,y )= A o1 ( x,y ) A r1 ( x,y )cos ϕ o1 ( x,y )( 1cos α 11 ) + A o2 ( x,y ) A r2 ( x,y )cos ϕ o2 ( x,y )( 1cos α 21 ),
R e 2 ( x,y )= A o1 ( x,y ) A r1 ( x,y )cos ϕ o1 ( x,y )( 1cos α 12 ) + A o2 ( x,y ) A r2 ( x,y )cos ϕ o2 ( x,y )( 1cos α 22 ),
I m 1 ( x,y )= A o1 ( x,y ) A r1 ( x,y )sin ϕ o1 ( x,y )sin α 11 + A o2 ( x,y ) A r2 ( x,y )sin ϕ o2 ( x,y )sin α 21 ,
I m 2 ( x,y )= A o1 ( x,y ) A r1 ( x,y )sin ϕ o1 ( x,y )sin α 12 + A o2 ( x,y ) A r2 ( x,y )sin ϕ o2 ( x,y )sin α 22 .
U 1 ( x,y )= R e 1 ( x,y )( 1cos α 22 )R e 2 ( x,y )( 1cos α 21 ) A r1 ( x,y )[ ( 1cos α 11 )( 1cos α 22 )( 1cos α 12 )( 1cos α 21 ) ] +j I m 1 ( x,y )sin α 22 I m 2 ( x,y )sin α 21 A r1 ( x,y )[ sin α 11 sin α 22 sin α 12 sin α 21 ] ,
U 2 ( x,y )= R e 1 ( x,y )( 1cos α 12 )R e 2 ( x,y )( 1cos α 11 ) A r2 ( x,y )[ ( 1cos α 12 )( 1cos α 21 )( 1cos α 11 )( 1cos α 22 ) ] +j I m 1 ( x,y )sin α 12 I m 2 ( x,y )sin α 11 A r2 ( x,y )[ sin α 12 sin α 21 sin α 11 sin α 22 ] .
R e 1 ( x,y )= A o1 ( x,y ) A r1 ( x,y )cos ϕ o1 ( x,y )( 1cos α 11 ) + A o2 ( x,y ) A r2 ( x,y )cos ϕ o2 ( x,y )( 1cos α 21 ) + A o3 ( x,y ) A r3 ( x,y )cos ϕ o3 ( x,y )( 1cos α 31 ),
R e 2 ( x,y )= A o1 ( x,y ) A r1 ( x,y )cos ϕ o1 ( x,y )( 1cos α 12 ) + A o2 ( x,y ) A r2 ( x,y )cos ϕ o2 ( x,y )( 1cos α 22 ) + A o3 ( x,y ) A r3 ( x,y )cos ϕ o3 ( x,y )( 1cos α 32 ),
R e 3 ( x,y )= A o1 ( x,y ) A r1 ( x,y )cos ϕ o1 ( x,y )( 1cos α 13 ) + A o2 ( x,y ) A r2 ( x,y )cos ϕ o2 ( x,y )( 1cos α 23 ) + A o3 ( x,y ) A r3 ( x,y )cos ϕ o3 ( x,y )( 1cos α 33 ),
I m 1 ( x,y )= A o1 ( x,y ) A r1 ( x,y )sin ϕ o1 ( x,y )sin α 11 + A o2 ( x,y ) A r2 ( x,y )sin ϕ o2 ( x,y )sin α 21 + A o3 ( x,y ) A r3 ( x,y )sin ϕ o3 ( x,y )sin α 31 ,
I m 2 ( x,y )= A o1 ( x,y ) A r1 ( x,y )sin ϕ o1 ( x,y )sin α 12 + A o2 ( x,y ) A r2 ( x,y )sin ϕ o2 ( x,y )sin α 22 + A o3 ( x,y ) A r3 ( x,y )sin ϕ o3 ( x,y )sin α 32 ,
I m 3 ( x,y )= A o1 ( x,y ) A r1 ( x,y )sin ϕ o1 ( x,y )sin α 13 + A o2 ( x,y ) A r2 ( x,y )sin ϕ o2 ( x,y )sin α 23 + A o3 ( x,y ) A r3 ( x,y )sin ϕ o3 ( x,y )sin α 33 .
A o1 ( x,y )cos ϕ o1 ( x,y )= C 4 Re'( x,y ) C 3 Re''( x,y ) A r1 ( x,y )( C 1 C 4 C 2 C 3 ) ,
A o2 ( x,y )cos ϕ o2 ( x,y )= ( 1cos α 32 )R e 1 ( x,y )( 1cos α 31 )R e 2 ( x,y ) A r2 ( x,y ) C 3 C 1 A o1 ( x,y ) A r1 ( x,y )cos ϕ o1 ( x,y ) A r2 ( x,y ) C 3 ,
A o3 ( x,y )cos ϕ o3 ( x,y )= R e 3 ( x,y )( 1cos α 13 ) A o1 ( x,y ) A r1 ( x,y )cos ϕ o1 ( x,y ) A r3 ( x,y )( 1cos α 33 ) ( 1cos α 23 ) A o2 ( x,y ) A r2 ( x,y )cos ϕ o2 ( x,y ) A r3 ( x,y )( 1cos α 33 ) ,
A o1 ( x,y )sin ϕ o1 ( x,y )= S 4 Im'( x,y ) S 3 Im''( x,y ) A r1 ( x,y )( S 1 S 4 S 2 S 3 ) ,
A o2 ( x,y )sin ϕ o2 ( x,y )= sin α 32 I m 1 ( x,y )sin α 31 I m 2 ( x,y ) A r2 ( x,y ) S 3 S 1 A o1 ( x,y ) A r1 ( x,y )sin ϕ o1 ( x,y ) A r2 ( x,y ) S 3 ,
A o3 ( x,y )sin ϕ o3 ( x,y )= I m 3 ( x,y )sin α 13 A o1 ( x,y ) A r1 ( x,y )sin ϕ o1 ( x,y ) A r3 ( x,y )sin α 33 sin α 23 A o2 ( x,y ) A r2 ( x,y )sin ϕ o2 ( x,y ) A r3 ( x,y )sin α 33 ,
Re'( x,y )=( 1cos α 32 )R e 1 ( x,y )( 1cos α 31 )R e 2 ( x,y ) ,
Re''( x,y )=( 1cos α 33 )R e 1 ( x,y )( 1cos α 31 )R e 3 ( x,y ),
C 1 =( 1cos α 11 )( 1cos α 32 )( 1cos α 12 )( 1cos α 31 ),
C 2 =( 1cos α 11 )( 1cos α 33 )( 1cos α 13 )( 1cos α 31 ),
C 3 =( 1cos α 21 )( 1cos α 32 )( 1cos α 22 )( 1cos α 31 ),
C 4 =( 1cos α 21 )( 1cos α 33 )( 1cos α 23 )( 1cos α 31 ),
Im'( x,y )=sin α 32 I m 1 ( x,y )sin α 31 I m 2 ( x,y ),
Im''( x,y )=sin α 33 I m 1 ( x,y )sin α 31 I m 3 ( x,y ),
S 1 =sin α 11 sin α 32 sin α 12 sin α 31 ,
S 2 =sin α 11 sin α 33 sin α 13 sin α 31 ,
S 3 =sin α 21 sin α 32 sin α 22 sin α 31 ,
S 4 =sin α 21 sin α 33 sin α 23 sin α 31 .

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