Abstract

An analytic theory describing the effects of diffraction and aberrations on single-pixel imaging performed by temporally modulating illumination light is presented. This method encodes spatial information using sinusoidal temporal modulations that are chirped in frequency across the extent of an illumination line focus. With some approximations, a point spread function relationship as a function of defocus or other aberrations is found for both spatially coherent and incoherent cases. The theory is validated through experiments and simulations, including measurement of the transverse and longitudinal optical transfer function, and confirmation of insensitivity to aberrations and significant optical scattering after encoding of spatial information through temporal modulation.

© 2012 Optical Society of America

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References

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  1. J. J. Art and M. B. Goodman, “Rapid-scanning confocal microscopy,” Methods Cell Biol. 38, 47–77 (1993).
    [CrossRef]
  2. R. Carriles, D. N. Schafer, K. E. Sheetz, J. J. Field, R. Cisek, V. Barzda, A. W. Sylvester, and J. A. Squier, “Invited review article: imaging techniques for harmonic and multiphoton absorption fluorescence microscopy,” Rev. Sci. Instrum. 80, 081101 (2009).
    [CrossRef]
  3. S. H. Yue, M. N. Slipchenko, and J. X. Cheng, “Multimodal nonlinear optical microscopy,” Laser Photon. Rev. 5, 496–512 (2011).
    [CrossRef]
  4. J. Bewersdorf, R. Pick, and S. W. Hell, “Multifocal multiphoton microscopy,” Opt. Lett. 23, 655–657 (1998).
    [CrossRef]
  5. G. J. Brakenhoff, J. Squier, T. Norris, A. C. Bliton, M. H. Wade, and B. Athey, “Real-time two-photon confocal microscopy using a femtosecond, amplified Ti:sapphire system,” J. Microsc. 181, 253–259 (1996).
    [CrossRef]
  6. A. H. Buist, M. Muller, J. Squier, and G. J. Brakenhoff, “Real time two-photon absorption microscopy using multi point excitation,” J. Microsc. 192, 217–226 (1998).
    [CrossRef]
  7. V. Andresen, A. Egner, and S. W. Hell, “Time-multiplexed multifocal multiphoton microscope,” Opt. Lett. 26, 75–77 (2001).
    [CrossRef]
  8. E. Chandler, E. Hoover, J. Field, K. Sheetz, W. Amir, R. Carriles, S. Y. Ding, and J. Squier, “High-resolution mosaic imaging with multifocal, multiphoton photon-counting microscopy,” Appl. Opt. 48, 2067–2077 (2009).
    [CrossRef]
  9. E. E. Hoover, M. D. Young, E. V. Chandler, A. Luo, J. J. Field, K. E. Sheetz, A. W. Sylvester, and J. A. Squier, “Remote focusing for programmable multi-layer differential multiphoton microscopy,” Biomed. Opt. Express 2, 113–122 (2011).
    [CrossRef]
  10. K. E. Sheetz, E. E. Hoover, R. Carriles, D. Kleinfeld, and J. A. Squier, “Advancing multifocal nonlinear microscopy: development and application of a novel multibeam Yb:KGd(WO4)2oscillator,” Opt. Express 16, 17574–17584 (2008).
    [CrossRef]
  11. G. Futia, P. Schlup, D. G. Winters, and R. A. Bartels, “Spatially chirped modulation imaging of absorption and fluorescent objects on single-element optical detector,” Opt. Express 19, 1626–1640 (2011).
    [CrossRef]
  12. P. Schlup, G. Futia, and R. A. Bartels, “Lateral tomographic spatial frequency modulated imaging,” Appl. Phys. Lett. 98, 211115 (2011).
    [CrossRef]
  13. E. E. Hoover, J. Field, D. Winters, M. Young, E. Chandler, J. Speirs, J. Lapenna, S. Kim, S.-Y. Ding, R. A. Bartels, J. Wang, and J. Squier, “Eliminating the scattering ambiguity in multifocal, multimodal, multiphoton imaging systems,” J. Biophotonics 12, 1–12 (2012).
    [CrossRef]
  14. M. Gu, Advanced Optical Imaging Theory (Springer-Verlag, 2000).
  15. J. W. Goodman, Introduction to Fourier Optics (Roberts, 2004).
  16. C.-L. Hsieh, Y. Pu, R. Grange, and D. Psaltis, “Digital phase conjugation of second harmonic radiation emitted by nanoparticles in turbid media,” Opt. Express 18, 12283–12290 (2010).
    [CrossRef]

2012 (1)

E. E. Hoover, J. Field, D. Winters, M. Young, E. Chandler, J. Speirs, J. Lapenna, S. Kim, S.-Y. Ding, R. A. Bartels, J. Wang, and J. Squier, “Eliminating the scattering ambiguity in multifocal, multimodal, multiphoton imaging systems,” J. Biophotonics 12, 1–12 (2012).
[CrossRef]

2011 (4)

2010 (1)

2009 (2)

E. Chandler, E. Hoover, J. Field, K. Sheetz, W. Amir, R. Carriles, S. Y. Ding, and J. Squier, “High-resolution mosaic imaging with multifocal, multiphoton photon-counting microscopy,” Appl. Opt. 48, 2067–2077 (2009).
[CrossRef]

R. Carriles, D. N. Schafer, K. E. Sheetz, J. J. Field, R. Cisek, V. Barzda, A. W. Sylvester, and J. A. Squier, “Invited review article: imaging techniques for harmonic and multiphoton absorption fluorescence microscopy,” Rev. Sci. Instrum. 80, 081101 (2009).
[CrossRef]

2008 (1)

2001 (1)

1998 (2)

A. H. Buist, M. Muller, J. Squier, and G. J. Brakenhoff, “Real time two-photon absorption microscopy using multi point excitation,” J. Microsc. 192, 217–226 (1998).
[CrossRef]

J. Bewersdorf, R. Pick, and S. W. Hell, “Multifocal multiphoton microscopy,” Opt. Lett. 23, 655–657 (1998).
[CrossRef]

1996 (1)

G. J. Brakenhoff, J. Squier, T. Norris, A. C. Bliton, M. H. Wade, and B. Athey, “Real-time two-photon confocal microscopy using a femtosecond, amplified Ti:sapphire system,” J. Microsc. 181, 253–259 (1996).
[CrossRef]

1993 (1)

J. J. Art and M. B. Goodman, “Rapid-scanning confocal microscopy,” Methods Cell Biol. 38, 47–77 (1993).
[CrossRef]

Amir, W.

Andresen, V.

Art, J. J.

J. J. Art and M. B. Goodman, “Rapid-scanning confocal microscopy,” Methods Cell Biol. 38, 47–77 (1993).
[CrossRef]

Athey, B.

G. J. Brakenhoff, J. Squier, T. Norris, A. C. Bliton, M. H. Wade, and B. Athey, “Real-time two-photon confocal microscopy using a femtosecond, amplified Ti:sapphire system,” J. Microsc. 181, 253–259 (1996).
[CrossRef]

Bartels, R. A.

E. E. Hoover, J. Field, D. Winters, M. Young, E. Chandler, J. Speirs, J. Lapenna, S. Kim, S.-Y. Ding, R. A. Bartels, J. Wang, and J. Squier, “Eliminating the scattering ambiguity in multifocal, multimodal, multiphoton imaging systems,” J. Biophotonics 12, 1–12 (2012).
[CrossRef]

P. Schlup, G. Futia, and R. A. Bartels, “Lateral tomographic spatial frequency modulated imaging,” Appl. Phys. Lett. 98, 211115 (2011).
[CrossRef]

G. Futia, P. Schlup, D. G. Winters, and R. A. Bartels, “Spatially chirped modulation imaging of absorption and fluorescent objects on single-element optical detector,” Opt. Express 19, 1626–1640 (2011).
[CrossRef]

Barzda, V.

R. Carriles, D. N. Schafer, K. E. Sheetz, J. J. Field, R. Cisek, V. Barzda, A. W. Sylvester, and J. A. Squier, “Invited review article: imaging techniques for harmonic and multiphoton absorption fluorescence microscopy,” Rev. Sci. Instrum. 80, 081101 (2009).
[CrossRef]

Bewersdorf, J.

Bliton, A. C.

G. J. Brakenhoff, J. Squier, T. Norris, A. C. Bliton, M. H. Wade, and B. Athey, “Real-time two-photon confocal microscopy using a femtosecond, amplified Ti:sapphire system,” J. Microsc. 181, 253–259 (1996).
[CrossRef]

Brakenhoff, G. J.

A. H. Buist, M. Muller, J. Squier, and G. J. Brakenhoff, “Real time two-photon absorption microscopy using multi point excitation,” J. Microsc. 192, 217–226 (1998).
[CrossRef]

G. J. Brakenhoff, J. Squier, T. Norris, A. C. Bliton, M. H. Wade, and B. Athey, “Real-time two-photon confocal microscopy using a femtosecond, amplified Ti:sapphire system,” J. Microsc. 181, 253–259 (1996).
[CrossRef]

Buist, A. H.

A. H. Buist, M. Muller, J. Squier, and G. J. Brakenhoff, “Real time two-photon absorption microscopy using multi point excitation,” J. Microsc. 192, 217–226 (1998).
[CrossRef]

Carriles, R.

Chandler, E.

E. E. Hoover, J. Field, D. Winters, M. Young, E. Chandler, J. Speirs, J. Lapenna, S. Kim, S.-Y. Ding, R. A. Bartels, J. Wang, and J. Squier, “Eliminating the scattering ambiguity in multifocal, multimodal, multiphoton imaging systems,” J. Biophotonics 12, 1–12 (2012).
[CrossRef]

E. Chandler, E. Hoover, J. Field, K. Sheetz, W. Amir, R. Carriles, S. Y. Ding, and J. Squier, “High-resolution mosaic imaging with multifocal, multiphoton photon-counting microscopy,” Appl. Opt. 48, 2067–2077 (2009).
[CrossRef]

Chandler, E. V.

Cheng, J. X.

S. H. Yue, M. N. Slipchenko, and J. X. Cheng, “Multimodal nonlinear optical microscopy,” Laser Photon. Rev. 5, 496–512 (2011).
[CrossRef]

Cisek, R.

R. Carriles, D. N. Schafer, K. E. Sheetz, J. J. Field, R. Cisek, V. Barzda, A. W. Sylvester, and J. A. Squier, “Invited review article: imaging techniques for harmonic and multiphoton absorption fluorescence microscopy,” Rev. Sci. Instrum. 80, 081101 (2009).
[CrossRef]

Ding, S. Y.

Ding, S.-Y.

E. E. Hoover, J. Field, D. Winters, M. Young, E. Chandler, J. Speirs, J. Lapenna, S. Kim, S.-Y. Ding, R. A. Bartels, J. Wang, and J. Squier, “Eliminating the scattering ambiguity in multifocal, multimodal, multiphoton imaging systems,” J. Biophotonics 12, 1–12 (2012).
[CrossRef]

Egner, A.

Field, J.

E. E. Hoover, J. Field, D. Winters, M. Young, E. Chandler, J. Speirs, J. Lapenna, S. Kim, S.-Y. Ding, R. A. Bartels, J. Wang, and J. Squier, “Eliminating the scattering ambiguity in multifocal, multimodal, multiphoton imaging systems,” J. Biophotonics 12, 1–12 (2012).
[CrossRef]

E. Chandler, E. Hoover, J. Field, K. Sheetz, W. Amir, R. Carriles, S. Y. Ding, and J. Squier, “High-resolution mosaic imaging with multifocal, multiphoton photon-counting microscopy,” Appl. Opt. 48, 2067–2077 (2009).
[CrossRef]

Field, J. J.

E. E. Hoover, M. D. Young, E. V. Chandler, A. Luo, J. J. Field, K. E. Sheetz, A. W. Sylvester, and J. A. Squier, “Remote focusing for programmable multi-layer differential multiphoton microscopy,” Biomed. Opt. Express 2, 113–122 (2011).
[CrossRef]

R. Carriles, D. N. Schafer, K. E. Sheetz, J. J. Field, R. Cisek, V. Barzda, A. W. Sylvester, and J. A. Squier, “Invited review article: imaging techniques for harmonic and multiphoton absorption fluorescence microscopy,” Rev. Sci. Instrum. 80, 081101 (2009).
[CrossRef]

Futia, G.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (Roberts, 2004).

Goodman, M. B.

J. J. Art and M. B. Goodman, “Rapid-scanning confocal microscopy,” Methods Cell Biol. 38, 47–77 (1993).
[CrossRef]

Grange, R.

Gu, M.

M. Gu, Advanced Optical Imaging Theory (Springer-Verlag, 2000).

Hell, S. W.

Hoover, E.

Hoover, E. E.

Hsieh, C.-L.

Kim, S.

E. E. Hoover, J. Field, D. Winters, M. Young, E. Chandler, J. Speirs, J. Lapenna, S. Kim, S.-Y. Ding, R. A. Bartels, J. Wang, and J. Squier, “Eliminating the scattering ambiguity in multifocal, multimodal, multiphoton imaging systems,” J. Biophotonics 12, 1–12 (2012).
[CrossRef]

Kleinfeld, D.

Lapenna, J.

E. E. Hoover, J. Field, D. Winters, M. Young, E. Chandler, J. Speirs, J. Lapenna, S. Kim, S.-Y. Ding, R. A. Bartels, J. Wang, and J. Squier, “Eliminating the scattering ambiguity in multifocal, multimodal, multiphoton imaging systems,” J. Biophotonics 12, 1–12 (2012).
[CrossRef]

Luo, A.

Muller, M.

A. H. Buist, M. Muller, J. Squier, and G. J. Brakenhoff, “Real time two-photon absorption microscopy using multi point excitation,” J. Microsc. 192, 217–226 (1998).
[CrossRef]

Norris, T.

G. J. Brakenhoff, J. Squier, T. Norris, A. C. Bliton, M. H. Wade, and B. Athey, “Real-time two-photon confocal microscopy using a femtosecond, amplified Ti:sapphire system,” J. Microsc. 181, 253–259 (1996).
[CrossRef]

Pick, R.

Psaltis, D.

Pu, Y.

Schafer, D. N.

R. Carriles, D. N. Schafer, K. E. Sheetz, J. J. Field, R. Cisek, V. Barzda, A. W. Sylvester, and J. A. Squier, “Invited review article: imaging techniques for harmonic and multiphoton absorption fluorescence microscopy,” Rev. Sci. Instrum. 80, 081101 (2009).
[CrossRef]

Schlup, P.

Sheetz, K.

Sheetz, K. E.

Slipchenko, M. N.

S. H. Yue, M. N. Slipchenko, and J. X. Cheng, “Multimodal nonlinear optical microscopy,” Laser Photon. Rev. 5, 496–512 (2011).
[CrossRef]

Speirs, J.

E. E. Hoover, J. Field, D. Winters, M. Young, E. Chandler, J. Speirs, J. Lapenna, S. Kim, S.-Y. Ding, R. A. Bartels, J. Wang, and J. Squier, “Eliminating the scattering ambiguity in multifocal, multimodal, multiphoton imaging systems,” J. Biophotonics 12, 1–12 (2012).
[CrossRef]

Squier, J.

E. E. Hoover, J. Field, D. Winters, M. Young, E. Chandler, J. Speirs, J. Lapenna, S. Kim, S.-Y. Ding, R. A. Bartels, J. Wang, and J. Squier, “Eliminating the scattering ambiguity in multifocal, multimodal, multiphoton imaging systems,” J. Biophotonics 12, 1–12 (2012).
[CrossRef]

E. Chandler, E. Hoover, J. Field, K. Sheetz, W. Amir, R. Carriles, S. Y. Ding, and J. Squier, “High-resolution mosaic imaging with multifocal, multiphoton photon-counting microscopy,” Appl. Opt. 48, 2067–2077 (2009).
[CrossRef]

A. H. Buist, M. Muller, J. Squier, and G. J. Brakenhoff, “Real time two-photon absorption microscopy using multi point excitation,” J. Microsc. 192, 217–226 (1998).
[CrossRef]

G. J. Brakenhoff, J. Squier, T. Norris, A. C. Bliton, M. H. Wade, and B. Athey, “Real-time two-photon confocal microscopy using a femtosecond, amplified Ti:sapphire system,” J. Microsc. 181, 253–259 (1996).
[CrossRef]

Squier, J. A.

Sylvester, A. W.

E. E. Hoover, M. D. Young, E. V. Chandler, A. Luo, J. J. Field, K. E. Sheetz, A. W. Sylvester, and J. A. Squier, “Remote focusing for programmable multi-layer differential multiphoton microscopy,” Biomed. Opt. Express 2, 113–122 (2011).
[CrossRef]

R. Carriles, D. N. Schafer, K. E. Sheetz, J. J. Field, R. Cisek, V. Barzda, A. W. Sylvester, and J. A. Squier, “Invited review article: imaging techniques for harmonic and multiphoton absorption fluorescence microscopy,” Rev. Sci. Instrum. 80, 081101 (2009).
[CrossRef]

Wade, M. H.

G. J. Brakenhoff, J. Squier, T. Norris, A. C. Bliton, M. H. Wade, and B. Athey, “Real-time two-photon confocal microscopy using a femtosecond, amplified Ti:sapphire system,” J. Microsc. 181, 253–259 (1996).
[CrossRef]

Wang, J.

E. E. Hoover, J. Field, D. Winters, M. Young, E. Chandler, J. Speirs, J. Lapenna, S. Kim, S.-Y. Ding, R. A. Bartels, J. Wang, and J. Squier, “Eliminating the scattering ambiguity in multifocal, multimodal, multiphoton imaging systems,” J. Biophotonics 12, 1–12 (2012).
[CrossRef]

Winters, D.

E. E. Hoover, J. Field, D. Winters, M. Young, E. Chandler, J. Speirs, J. Lapenna, S. Kim, S.-Y. Ding, R. A. Bartels, J. Wang, and J. Squier, “Eliminating the scattering ambiguity in multifocal, multimodal, multiphoton imaging systems,” J. Biophotonics 12, 1–12 (2012).
[CrossRef]

Winters, D. G.

Young, M.

E. E. Hoover, J. Field, D. Winters, M. Young, E. Chandler, J. Speirs, J. Lapenna, S. Kim, S.-Y. Ding, R. A. Bartels, J. Wang, and J. Squier, “Eliminating the scattering ambiguity in multifocal, multimodal, multiphoton imaging systems,” J. Biophotonics 12, 1–12 (2012).
[CrossRef]

Young, M. D.

Yue, S. H.

S. H. Yue, M. N. Slipchenko, and J. X. Cheng, “Multimodal nonlinear optical microscopy,” Laser Photon. Rev. 5, 496–512 (2011).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

P. Schlup, G. Futia, and R. A. Bartels, “Lateral tomographic spatial frequency modulated imaging,” Appl. Phys. Lett. 98, 211115 (2011).
[CrossRef]

Biomed. Opt. Express (1)

J. Biophotonics (1)

E. E. Hoover, J. Field, D. Winters, M. Young, E. Chandler, J. Speirs, J. Lapenna, S. Kim, S.-Y. Ding, R. A. Bartels, J. Wang, and J. Squier, “Eliminating the scattering ambiguity in multifocal, multimodal, multiphoton imaging systems,” J. Biophotonics 12, 1–12 (2012).
[CrossRef]

J. Microsc. (2)

G. J. Brakenhoff, J. Squier, T. Norris, A. C. Bliton, M. H. Wade, and B. Athey, “Real-time two-photon confocal microscopy using a femtosecond, amplified Ti:sapphire system,” J. Microsc. 181, 253–259 (1996).
[CrossRef]

A. H. Buist, M. Muller, J. Squier, and G. J. Brakenhoff, “Real time two-photon absorption microscopy using multi point excitation,” J. Microsc. 192, 217–226 (1998).
[CrossRef]

Laser Photon. Rev. (1)

S. H. Yue, M. N. Slipchenko, and J. X. Cheng, “Multimodal nonlinear optical microscopy,” Laser Photon. Rev. 5, 496–512 (2011).
[CrossRef]

Methods Cell Biol. (1)

J. J. Art and M. B. Goodman, “Rapid-scanning confocal microscopy,” Methods Cell Biol. 38, 47–77 (1993).
[CrossRef]

Opt. Express (3)

Opt. Lett. (2)

Rev. Sci. Instrum. (1)

R. Carriles, D. N. Schafer, K. E. Sheetz, J. J. Field, R. Cisek, V. Barzda, A. W. Sylvester, and J. A. Squier, “Invited review article: imaging techniques for harmonic and multiphoton absorption fluorescence microscopy,” Rev. Sci. Instrum. 80, 081101 (2009).
[CrossRef]

Other (2)

M. Gu, Advanced Optical Imaging Theory (Springer-Verlag, 2000).

J. W. Goodman, Introduction to Fourier Optics (Roberts, 2004).

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Figures (4)

Fig. 1.
Fig. 1.

SPIFI optical setup.

Fig. 2.
Fig. 2.

(a) Simulated and (b) experimentally measured spatially coherent illumination SPIFI image of 20 line pairs per mm ruling as a function of defocus. The in-focus plane is denoted with a solid white line. Planes that are integer multiples of the Talbot length away from the in-focus plane are denoted with dashed white lines. Planes that are odd integer multiples of half the Talbot length away from the in-focus plane are denoted with dashed black lines.

Fig. 3.
Fig. 3.

(a) Simulated and (b) experimentally measured spatially coherent illumination SPIFI optical transfer function. The dashed white lines are plotted at the inverse Talbot length and negative one times the inverse Talbot length for each transverse spatial frequency.

Fig. 4.
Fig. 4.

Fringe visibility of the fundamental spatial frequency of a 20 line pairs per mm Ronchi ruling as a function of the number of 127 μm thick layers of paraffin present in the collection system.

Equations (42)

Equations on this page are rendered with MathJax. Learn more.

m(x1,t)=w(t){1+cos[2π(κx1+fc)t]}/2,
|m(x1,t)|2=|w(t)|2{3+2cos[2π(κx1+fc)t]+cos[4π(κx1+fc)t]}/4.
hi(x,y;Δzi)=Midi2λ2Pi(xi,yi)exp[ik2(1di)2Mi2Δzi(xi2+yi2)]exp[ikdi(xix+yiy)]dxidyi,
Hi(fx,fy;Δzi)=Midi2λ2Pi(λdifx,λdify)exp[iπλMi2Δzi(fx2+fy2)].
I^1+(x1,y1,f)=I04κW(f)f|u(x1)|2δ(y1)δ[x1(ffc)/κ],
I^2+,exc(x2,y2,f)=I0Mi24κW(f)f|u(Mix2)|2δ(Miy2)δ[Mix2+(ffc)/κ].
S^1+(f)=γI0Mi4W(κxMi)x|u(Mix)|2|g(x,0)|2,
s1+(t)=γI0Mi4exp(i2πfcfxMiκ)|w(fxMiκ)|2G(fx),
S^(f)=γI^2(x2,y2,f)dx2dy2.
s(t)I2(x2,y2,t)|hc(x2+Mcx3,y2+Mcy3,dc;Δzc)|2|dx2dy2dx3dy3.
his(x,y;Δzi)=W(Miκx)δ(y)x,y|hi(Mix,Miy;Δzi)|2.
I^1+(x1,y1,f)=I04κW(f)f|u(ffcκ)|2δ(y1)δ[x1ffcκ].
I^2+,exc(x2,y2,f)=I04κW(f)f[|u[(ffc)/κ]|2|hi[Mix2+(ffc)/κ,Miy2;Δzi]|2].
S^1+(f)=γI^2+,exc(x2,y2,f)|g(x2,y2)|2dx2dy2.
S^1+(f)=γI0Mi4W(Miκx)x{|u(Mix)|2[|hi(Mix,Miy;Δzi)|2x,y|g(x,y)|2]y=0},
S^1+(f)γI0Mi4{his(x,y)x,y[|u(Mix)|2|g(x,y)|2]}y=0,
s1+(t)=γI04Miexp(i2πfcfxMiκ)|w(fxκMi)|2Hi(fxMi,fyMi;Δzi)G(fx,fy)dfy,
e1(x1,y1,t)=e0w(t)u(x1)δ(y1){2+exp[i2π(κx1+fc)t]+exp[i2π(κx1+fc)t]}/4.
vβ(x2,y2,t)=u(x1)δ(y1)exp[iβ2π(κx1+fc)t]hi(x1+Mix2,y1+Miy2;Δzi)dx1dy1,
I2,exc(x2,y2,t)=I0|w(t)|2|[2v0(x2,y2,t)+v1(x2,y2,t)+v1(x2,y2,t)]/4|2,
I2+,exc(x2,y2,t)=I08|w(t)|2[v0*(x2,y2,t)v1(x2,y2,t)+v0(x2,y2,t)v1*(x2,y2,t)],
Hi(fx,fy)=Midi2λ2Pi(0,λdify)exp[iπλMi2Δzi(fx2+fy2)].
I2+,exc(x2,y2,t)=I0Mi4di2λ2|w(t)|2cos(πλMi2Δzi(κt)2)×|hiy(Miy2)|2|u(Mix2)|2exp[i2π(κMix2+fc)t],
|w(t)|2=Midi2λ2|w(t)|2cos[πλMi2Δzi(κt)2],
I^2+,exc(x2,y2,f)=I0Mi24W(f)f|u(Mix2)|2hiy(Miy2)δ[κMix2+(ffc)/κ],
Hcsd(fx)=Midi2λ2|w(fxMiκ)|2cos(πλΔzifx2).
S^1+(f)=γI0Mi4hcsd(x)x|u(Mix)|2|g(x)|2,
Hcsd(fx)=Midi2λ2|w(fxMiκ)|2cos(2πΔzizt(fx)).
s(t)=γ|u(x,y)|2M(x,y,t)|g(x,y)|2dxdy.
s(t)=γ|hi(xx,yy)|2|g(x,y)|2dxdyM(x,y,t)|u(x,y)|2dxdy.
s(t)=γ|u(x,y)|2M(x,y,t)[|g(x,y)|2x,y|hi(x,y)|2]dxdy.
I2+,exc(x2,y2,t)=I08|w(t)|2[v0*(x2,y2,t)v1(x2,y2,t)+v0(x2,y2,t)v1*(x2,y2,t)],
vβ(x2,y2,t)=u(x1)δ(y1)exp[iβ2π(κx1+fc)t]hi(x1+Mix2,y1+Miy2;Δzi)dx1dy1.
V^β(x2,y2,f)=u(x1)δ(y1)δ[fβ(κx1+fc)]hi(x1+Mix2,y1+Miy2;Δzi)dx1dy1.
V^β(x2,y2,f)={κ1u[(ffc)/κ]hi[(ffc)/κ+Mix2,Miy2;Δzi]ifβ=1u(Mix2)hiy(Miy2;Δzi)δ(f)ifβ=0κ1u[(ffc)/κ)]hi[(ffc)/κ+Mix2,Miy2;Δzi]ifβ=1.
I^2+,exc(x2,y2,f)=I08W(f)f[V^0*(x2,y2,f)V^1(x2,y2,f)+V^0(x2,y2,f)V^1*(x2,y2,f)].
I^2+,exc(x2,y2,f)=I04κW(f)f{u(ffcκ)R[hiy*(Miy2;Δzi)hi(ffcκ+Mix2,Miy2;Δzi)]u(Mix2)},
S^1+(f)=γI04κW(f)f{u(ffcκ)×R[hiy*(Miy2;Δzi)hi(ffcκ+Mix2,Miy2;Δzi)]×u(Mix2)|g(x2,y2)|2dx2dy2}.
S^1+(f)=γI0Mi4W(Miκx)x(u(Mix){R[hiy*(Miy;Δzi)hi(Mix,Miy;Δzi)]x,y[u(Mix)|g(x,y)|2]})y=0.
S^1+(f)=γI0Mi4u(Mix){hcs(x,y;Δzi)x,y[u(Mix)|g(x,y)|2]}y=0,
hcs(x,y;Δzi)=W(Miκx)xR{hiy*(Miy;Δzi)hi(Mix,Miy;Δzi)}.
S^1+(f)=γI0Mi4{hcs(x,y;Δzi)x,y[u(Mix)|g(x,y)|2]}y=0,

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