Abstract

We analyze the bulk elastic transformation of volume holograms as a general approach for three-dimensional pupil engineering. The physical relationship between transformation and the resulting point spread function is discussed by deriving the corresponding analytical expressions. For affine transformations, an analytical solution is directly possible. However, for nonaffine transformations, the analytical solution is not straightforward and we must turn to quasi-analytical solutions using the approximation of the stationary phase. Transformational volume holography offers richer design flexibility and real-time adjustment capabilities for imaging systems.

© 2014 Optical Society of America

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  1. H. Owen, D. E. Battey, M. J. Pelletier, and J. B. Slater, “New spectroscopic instrument based on volume holographic optical elements,” Proc. SPIE 2406, 260–267 (1995).
  2. G. Barbastathis and D. J. Brady, “Multidimensional tomographic imaging using volume holography,” Proc. IEEE 87, 2098–2120 (1999).
    [CrossRef]
  3. A. Sinha, W. Sun, T. Shih, and G. Barbastathis, “Volume holographic imaging in transmission geometry,” Appl. Opt. 43, 1533–1551 (2004).
    [CrossRef]
  4. Y. Luo, I. K. Zervantonakis, S. B. Oh, R. D. Kamm, and G. Barbastathis, “Spectrally resolved multidepth fluorescence imaging,” J. Biomed. Opt. 16, 096015 (2011).
    [CrossRef]
  5. H. Gao, J. M. Watson, J. S. Stuart, and G. Barbastathis, “Design of volume hologram filters for suppression of daytime sky brightness in artificial satellite detection,” Opt. Express 21, 6448–6458 (2013).
    [CrossRef]
  6. K. Tian, T. Cuingnet, Z. Li, W. Liu, D. Psaltis, and G. Barbastathis, “Diffraction from deformed volume holograms: perturbation theory approach,” J. Opt. Soc. Am. A 22, 2880–2889 (2005).
    [CrossRef]
  7. H. Gao and G. Barbastathis, “Design of volume holograpic imaging point spread functions using multiple point deformations,” in Digital Holography and Three-Dimensional Imaging (Optical Society of America, 2012), paper DTu3C.6.
  8. H. Gao and G. Barbastathis, “Design and optimization of point spread functions in volume holographic imaging systems,” in Digital Holography and Three-Dimensional Imaging (Optical Society of America, 2013), paper DTh3A.6.
  9. H. Gao, “Design and transformation of three-dimensional pupils: diffractive and subwavelength,” Ph.D. dissertation (Massachusetts Institute of Technology, 2014).
  10. U. Leonhardt, “Optical conformal mapping,” Science 312, 1777–1780 (2006).
    [CrossRef]
  11. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
    [CrossRef]
  12. G. Barbastathis, F. Mok, and D. Psaltis, “Non-volatile readout of shift multiplexed holograms,” U.S. patent5,978,112 (November2, 1999).
  13. J. T. Gallo and C. M. Verber, “Model for the effects of material shrinkage on volume holograms,” Appl. Opt. 33, 6797–6804 (1994).
    [CrossRef]
  14. P. Hariharan, Optical Holography: Principles, Techniques, and Applications, 2nd ed. (Cambridge University, 1996).
  15. D. H. R. Vilkomerson and D. Bostwick, “Some effects of emulsion shrinkage on a holograms image space,” Appl. Opt. 6, 1270–1272 (1967).
    [CrossRef]
  16. R. C. Hibbeler, Mechanics of Materials (Prentice-Hall, 2010).
  17. V. A. Borovikov, Uniform Stationary Phase Method (Institution of Electrical Engineers, 1994).
  18. H. Cheng, Advanced Analytic Methods in Applied Mathematics, Science, and Engineering (LuBan, 2007).
  19. J. C. Cooke, “Stationary phase in two dimensions,” IMA J. Appl. Math. 29, 25–37 (1982).
    [CrossRef]
  20. V. A. Borovikov and B. Y. Kinber, Geometrical Theory of Diffraction (Institution of Electrical Engineers, 1994).
  21. O. M. Conde, J. Pérez, and M. F. Cátedra, “Stationary phase method application for the analysis of radiation of complex 3-D conducting structures,” IEEE Trans. Antennas Propag. 49, 724–731 (2001).
    [CrossRef]
  22. G. L. James, Geometrical Theory of Diffraction for Electromagnetic Waves (The Institution of Engineering and Technology, 1979).
  23. J. W. Goodman, Introduction to Fourier Optics (Roberts, 2005).
  24. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables (Dover, 1965).

2013 (1)

2011 (1)

Y. Luo, I. K. Zervantonakis, S. B. Oh, R. D. Kamm, and G. Barbastathis, “Spectrally resolved multidepth fluorescence imaging,” J. Biomed. Opt. 16, 096015 (2011).
[CrossRef]

2006 (2)

U. Leonhardt, “Optical conformal mapping,” Science 312, 1777–1780 (2006).
[CrossRef]

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
[CrossRef]

2005 (1)

2004 (1)

2001 (1)

O. M. Conde, J. Pérez, and M. F. Cátedra, “Stationary phase method application for the analysis of radiation of complex 3-D conducting structures,” IEEE Trans. Antennas Propag. 49, 724–731 (2001).
[CrossRef]

1999 (1)

G. Barbastathis and D. J. Brady, “Multidimensional tomographic imaging using volume holography,” Proc. IEEE 87, 2098–2120 (1999).
[CrossRef]

1995 (1)

H. Owen, D. E. Battey, M. J. Pelletier, and J. B. Slater, “New spectroscopic instrument based on volume holographic optical elements,” Proc. SPIE 2406, 260–267 (1995).

1994 (1)

1982 (1)

J. C. Cooke, “Stationary phase in two dimensions,” IMA J. Appl. Math. 29, 25–37 (1982).
[CrossRef]

1967 (1)

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables (Dover, 1965).

Barbastathis, G.

H. Gao, J. M. Watson, J. S. Stuart, and G. Barbastathis, “Design of volume hologram filters for suppression of daytime sky brightness in artificial satellite detection,” Opt. Express 21, 6448–6458 (2013).
[CrossRef]

Y. Luo, I. K. Zervantonakis, S. B. Oh, R. D. Kamm, and G. Barbastathis, “Spectrally resolved multidepth fluorescence imaging,” J. Biomed. Opt. 16, 096015 (2011).
[CrossRef]

K. Tian, T. Cuingnet, Z. Li, W. Liu, D. Psaltis, and G. Barbastathis, “Diffraction from deformed volume holograms: perturbation theory approach,” J. Opt. Soc. Am. A 22, 2880–2889 (2005).
[CrossRef]

A. Sinha, W. Sun, T. Shih, and G. Barbastathis, “Volume holographic imaging in transmission geometry,” Appl. Opt. 43, 1533–1551 (2004).
[CrossRef]

G. Barbastathis and D. J. Brady, “Multidimensional tomographic imaging using volume holography,” Proc. IEEE 87, 2098–2120 (1999).
[CrossRef]

H. Gao and G. Barbastathis, “Design and optimization of point spread functions in volume holographic imaging systems,” in Digital Holography and Three-Dimensional Imaging (Optical Society of America, 2013), paper DTh3A.6.

G. Barbastathis, F. Mok, and D. Psaltis, “Non-volatile readout of shift multiplexed holograms,” U.S. patent5,978,112 (November2, 1999).

H. Gao and G. Barbastathis, “Design of volume holograpic imaging point spread functions using multiple point deformations,” in Digital Holography and Three-Dimensional Imaging (Optical Society of America, 2012), paper DTu3C.6.

Battey, D. E.

H. Owen, D. E. Battey, M. J. Pelletier, and J. B. Slater, “New spectroscopic instrument based on volume holographic optical elements,” Proc. SPIE 2406, 260–267 (1995).

Borovikov, V. A.

V. A. Borovikov and B. Y. Kinber, Geometrical Theory of Diffraction (Institution of Electrical Engineers, 1994).

V. A. Borovikov, Uniform Stationary Phase Method (Institution of Electrical Engineers, 1994).

Bostwick, D.

Brady, D. J.

G. Barbastathis and D. J. Brady, “Multidimensional tomographic imaging using volume holography,” Proc. IEEE 87, 2098–2120 (1999).
[CrossRef]

Cátedra, M. F.

O. M. Conde, J. Pérez, and M. F. Cátedra, “Stationary phase method application for the analysis of radiation of complex 3-D conducting structures,” IEEE Trans. Antennas Propag. 49, 724–731 (2001).
[CrossRef]

Cheng, H.

H. Cheng, Advanced Analytic Methods in Applied Mathematics, Science, and Engineering (LuBan, 2007).

Conde, O. M.

O. M. Conde, J. Pérez, and M. F. Cátedra, “Stationary phase method application for the analysis of radiation of complex 3-D conducting structures,” IEEE Trans. Antennas Propag. 49, 724–731 (2001).
[CrossRef]

Cooke, J. C.

J. C. Cooke, “Stationary phase in two dimensions,” IMA J. Appl. Math. 29, 25–37 (1982).
[CrossRef]

Cuingnet, T.

Gallo, J. T.

Gao, H.

H. Gao, J. M. Watson, J. S. Stuart, and G. Barbastathis, “Design of volume hologram filters for suppression of daytime sky brightness in artificial satellite detection,” Opt. Express 21, 6448–6458 (2013).
[CrossRef]

H. Gao, “Design and transformation of three-dimensional pupils: diffractive and subwavelength,” Ph.D. dissertation (Massachusetts Institute of Technology, 2014).

H. Gao and G. Barbastathis, “Design and optimization of point spread functions in volume holographic imaging systems,” in Digital Holography and Three-Dimensional Imaging (Optical Society of America, 2013), paper DTh3A.6.

H. Gao and G. Barbastathis, “Design of volume holograpic imaging point spread functions using multiple point deformations,” in Digital Holography and Three-Dimensional Imaging (Optical Society of America, 2012), paper DTu3C.6.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (Roberts, 2005).

Hariharan, P.

P. Hariharan, Optical Holography: Principles, Techniques, and Applications, 2nd ed. (Cambridge University, 1996).

Hibbeler, R. C.

R. C. Hibbeler, Mechanics of Materials (Prentice-Hall, 2010).

James, G. L.

G. L. James, Geometrical Theory of Diffraction for Electromagnetic Waves (The Institution of Engineering and Technology, 1979).

Kamm, R. D.

Y. Luo, I. K. Zervantonakis, S. B. Oh, R. D. Kamm, and G. Barbastathis, “Spectrally resolved multidepth fluorescence imaging,” J. Biomed. Opt. 16, 096015 (2011).
[CrossRef]

Kinber, B. Y.

V. A. Borovikov and B. Y. Kinber, Geometrical Theory of Diffraction (Institution of Electrical Engineers, 1994).

Leonhardt, U.

U. Leonhardt, “Optical conformal mapping,” Science 312, 1777–1780 (2006).
[CrossRef]

Li, Z.

Liu, W.

Luo, Y.

Y. Luo, I. K. Zervantonakis, S. B. Oh, R. D. Kamm, and G. Barbastathis, “Spectrally resolved multidepth fluorescence imaging,” J. Biomed. Opt. 16, 096015 (2011).
[CrossRef]

Mok, F.

G. Barbastathis, F. Mok, and D. Psaltis, “Non-volatile readout of shift multiplexed holograms,” U.S. patent5,978,112 (November2, 1999).

Oh, S. B.

Y. Luo, I. K. Zervantonakis, S. B. Oh, R. D. Kamm, and G. Barbastathis, “Spectrally resolved multidepth fluorescence imaging,” J. Biomed. Opt. 16, 096015 (2011).
[CrossRef]

Owen, H.

H. Owen, D. E. Battey, M. J. Pelletier, and J. B. Slater, “New spectroscopic instrument based on volume holographic optical elements,” Proc. SPIE 2406, 260–267 (1995).

Pelletier, M. J.

H. Owen, D. E. Battey, M. J. Pelletier, and J. B. Slater, “New spectroscopic instrument based on volume holographic optical elements,” Proc. SPIE 2406, 260–267 (1995).

Pendry, J. B.

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
[CrossRef]

Pérez, J.

O. M. Conde, J. Pérez, and M. F. Cátedra, “Stationary phase method application for the analysis of radiation of complex 3-D conducting structures,” IEEE Trans. Antennas Propag. 49, 724–731 (2001).
[CrossRef]

Psaltis, D.

K. Tian, T. Cuingnet, Z. Li, W. Liu, D. Psaltis, and G. Barbastathis, “Diffraction from deformed volume holograms: perturbation theory approach,” J. Opt. Soc. Am. A 22, 2880–2889 (2005).
[CrossRef]

G. Barbastathis, F. Mok, and D. Psaltis, “Non-volatile readout of shift multiplexed holograms,” U.S. patent5,978,112 (November2, 1999).

Schurig, D.

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
[CrossRef]

Shih, T.

Sinha, A.

Slater, J. B.

H. Owen, D. E. Battey, M. J. Pelletier, and J. B. Slater, “New spectroscopic instrument based on volume holographic optical elements,” Proc. SPIE 2406, 260–267 (1995).

Smith, D. R.

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
[CrossRef]

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables (Dover, 1965).

Stuart, J. S.

Sun, W.

Tian, K.

Verber, C. M.

Vilkomerson, D. H. R.

Watson, J. M.

Zervantonakis, I. K.

Y. Luo, I. K. Zervantonakis, S. B. Oh, R. D. Kamm, and G. Barbastathis, “Spectrally resolved multidepth fluorescence imaging,” J. Biomed. Opt. 16, 096015 (2011).
[CrossRef]

Appl. Opt. (3)

IEEE Trans. Antennas Propag. (1)

O. M. Conde, J. Pérez, and M. F. Cátedra, “Stationary phase method application for the analysis of radiation of complex 3-D conducting structures,” IEEE Trans. Antennas Propag. 49, 724–731 (2001).
[CrossRef]

IMA J. Appl. Math. (1)

J. C. Cooke, “Stationary phase in two dimensions,” IMA J. Appl. Math. 29, 25–37 (1982).
[CrossRef]

J. Biomed. Opt. (1)

Y. Luo, I. K. Zervantonakis, S. B. Oh, R. D. Kamm, and G. Barbastathis, “Spectrally resolved multidepth fluorescence imaging,” J. Biomed. Opt. 16, 096015 (2011).
[CrossRef]

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

Opt. Express (1)

Proc. IEEE (1)

G. Barbastathis and D. J. Brady, “Multidimensional tomographic imaging using volume holography,” Proc. IEEE 87, 2098–2120 (1999).
[CrossRef]

Proc. SPIE (1)

H. Owen, D. E. Battey, M. J. Pelletier, and J. B. Slater, “New spectroscopic instrument based on volume holographic optical elements,” Proc. SPIE 2406, 260–267 (1995).

Science (2)

U. Leonhardt, “Optical conformal mapping,” Science 312, 1777–1780 (2006).
[CrossRef]

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
[CrossRef]

Other (12)

G. Barbastathis, F. Mok, and D. Psaltis, “Non-volatile readout of shift multiplexed holograms,” U.S. patent5,978,112 (November2, 1999).

R. C. Hibbeler, Mechanics of Materials (Prentice-Hall, 2010).

V. A. Borovikov, Uniform Stationary Phase Method (Institution of Electrical Engineers, 1994).

H. Cheng, Advanced Analytic Methods in Applied Mathematics, Science, and Engineering (LuBan, 2007).

H. Gao and G. Barbastathis, “Design of volume holograpic imaging point spread functions using multiple point deformations,” in Digital Holography and Three-Dimensional Imaging (Optical Society of America, 2012), paper DTu3C.6.

H. Gao and G. Barbastathis, “Design and optimization of point spread functions in volume holographic imaging systems,” in Digital Holography and Three-Dimensional Imaging (Optical Society of America, 2013), paper DTh3A.6.

H. Gao, “Design and transformation of three-dimensional pupils: diffractive and subwavelength,” Ph.D. dissertation (Massachusetts Institute of Technology, 2014).

V. A. Borovikov and B. Y. Kinber, Geometrical Theory of Diffraction (Institution of Electrical Engineers, 1994).

G. L. James, Geometrical Theory of Diffraction for Electromagnetic Waves (The Institution of Engineering and Technology, 1979).

J. W. Goodman, Introduction to Fourier Optics (Roberts, 2005).

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables (Dover, 1965).

P. Hariharan, Optical Holography: Principles, Techniques, and Applications, 2nd ed. (Cambridge University, 1996).

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Figures (10)

Fig. 1.
Fig. 1.

System architecture for VH (top) recording and (bottom) probing processes.

Fig. 2.
Fig. 2.

Analysis approach for including bulk transformations in volume holographic imaging systems. Left and right figures show coordinates used for calculating the integrals without and with the transformation, respectively.

Fig. 3.
Fig. 3.

Shrinkage of the VH along (a) z and (e) x directions and the resulting PSFs when the compression ratio is (b) δ=0.1 and (f) δ=0.02, respectively. (c), (d) and (g), (h) Separation of two sinc functions in Eq. (9) and Eq. (11), which result in (b) and (f), respectively.

Fig. 4.
Fig. 4.

(a) Bending of the VH. (b) Resulting PSFs at different bending ratios.

Fig. 5.
Fig. 5.

Expressions of (left) f(x) and (right) real(exp[iΛϕ(x))]) for the one-dimensional integral of Eq. (20) at detector positions of (a), (b) x=0, (c), (d) x=3.2×104m, and (e), (f) x=3.1×104m. Note that only part of the integral range is shown. The bending ratio is γ=0.030 in these calculations.

Fig. 6.
Fig. 6.

PSF at the detector for a bent hologram calculated using (a) Eq. (27) (including the contributions only from poles) and (b) the stationary phase approximation combining four critical points, which is compared with the full numerical solution.

Fig. 7.
Fig. 7.

PSFs calculated from Parts 1–3 and combined values.

Fig. 8.
Fig. 8.

PSFs for twisting at different maximum twisting angles: (a) θm=0° (without deformation), (b) θm=0.5°, and (c) θm=1°. Note that in order to show sidelobes clearly, q(x,y) is plotted instead of the actual intensity, which is proportional to |q(x,y)|2.

Fig. 9.
Fig. 9.

Intuitive explanation of the PSF shape after twisting.

Fig. 10.
Fig. 10.

(a) PSF calculated using stationary phase approximation, and (b) its difference to the result calculated using full numerical approach (Fig. 8(b)). Maximum twisting angle is θm=0.5°.

Equations (42)

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q(x)=dxdzEp(x,z)ϵ(x,z)s(x,z)·exp(i2πxxλf2)exp[i2π(1x22f22)zλ],
q(x)=Lx·L·sinc[Lxλ(xsf1+xf2)]sinc[L2λ(xs2f12x2f22)].
q(x,y)=dxEp(x)ϵ(x)s(x)·exp[i2π(xf2,yf2,1x2+y22f22)·x],
x(2)=fx(2)(x,y,z),y(2)=fy(2)(x,y,z),z(2)=fz(2)(x,y,z).
T=(fx(2)xfx(2)yfx(2)zfy(2)xfy(2)yfy(2)zfz(2)xfz(2)yfz(2)z).
q(x,y)=dx|T|Ep(T·x)ϵ(x)s(x)·exp[i2π(xf2,yf2,1x2+y22f22)·(T·x)].
x(2)=x,z(2)=(1δ)z,
T=(1001δ)
q(x)=Lx·sinc[Lxλ(xsf1+xf2)]·(1δ)L·sinc[L2λ(xs2f12x2f22)].
T=(1δ001)
q(x)=(1δ)Lx·sinc[Lxλ((1+δ)xsf1+(1δ)xf2)]·L·sinc[L2λ(xs2f12x2f22)].
T=(1+νδ001δ),
q(x)=(1+νδ)Lx·(1δ)L·sinc[Lxλ((1νδ)xsf1+(1+νδ)xf2)]·sinc[L2λ(xs2f12x2f22)].
T=(cosθsinθsinθcosθ);
q(x)=Lx·L·sinc[Lxλ(2xsf1+cosθ(xf2xsf1)sinθ(xs22f12x22f22))]·sinc[Lλ(sinθ(xf2xsf1)+cosθ(xs22f12x22f22))].
T=(1α01),
q(x)=Lx·L·sinc[Lxλ(xf2+xsf1)]·sinc[Lλ((xs22f12x22f22)α(xsf1xf2))].
|T|=(RzRcosx|R||R|Rsinx|R|Rz|R|sinx|R|cosx|R|),
q(x)=dxdzrect(xLx)rect(zL)(zRR)·exp(i2πλ2xsxf1)·exp[i2πλ(xf2xsf1)(|R|RzRsinx|R|)]·exp[i2πλ(xs22f12x22f22)·(R+(zR)cosxR)].
q(x)=dxrect(xLx)[sin(πLζ)πζ+Lcos(πLζ)iπRζ+isin(πLζ)2π2Rζ2]exp(i2πλ2xsxf1)·exp[i2πλ(xf2xsf1)|R|sinx|R|]·exp[i2πλ(xs22f12x22f22)R(1cosxR)],
ζ=1λ(xf2xsf1)|R|Rsinx|R|+1λ(xs22f12x22f22)cosxR.
I=Cf(x)exp[iΛϕ(x)]dx,
I=exp[iΛϕ(x0)]1Λ2π|ϕ(x0)|·exp[isign(ϕ(x0))·π4]f(x0).
sin(πLζ)πζ+Lcos(πLζ)iπRζ+isin(πLζ)2π2Rζ2.
f(x)=12iπζ,
exp[iΛϕ(x)]=[exp(iπLζ)exp(iπLζ)]·exp(i2πλ2xsxf1)·exp[i2πλ(xf2xsf1)|R|sinx|R|]·exp[i2πλ(xs22f12x22f22)R(1cosxR)],
I=ejΛϕ(a)ϕ(a)jΛf(a)ejΛϕ(b)ϕ(b)jΛf(b){+icπexp[iΛϕ(xp)]ifϕ(xp)>0icπexp[iΛϕ(xp)]ifϕ(xp)<0,
abf(x)exp[iΛϕ(x)]dx=af(x)exp[iΛϕ(x)]dxbf(x)exp[iΛϕ(x)]dx,
I=exp(iΛϕ0){2πif(xp)Afin(Λ23ξ,Λ13β,Λ13q)+πA0Λ13Φ(Λ23ξ,Λ13q)πB0iΛ23Φ(Λ23ξ,Λ13q)},
A0=ξ1/4[f(x2)x2xp2|ϕ(x2)|+f(x1)x1xp2|ϕ(x1)|]2βf(xp)ξβ2,
B0=ξ1/4[f(x2)x2xp2|ϕ(x2)|f(x1)x1xp2|ϕ(x1)|]2f(xp)ξβ2.
I=exp(iΛϕ0)·{2πif(x0)Afin[Λ23(2ϕ)13α,Λ13β,Λ13q]+2π(2ϕ)13f0Λ13Φ[Λ23(2ϕ)13α,Λ13q]2πi(2ϕ)23f1Λ23Φ[Λ23(2ϕ)13α,Λ13q]},
θ(z)=zL/2θm,
T=(cos(2zLθm)sin(2zLθm)0sin(2zLθm)cos(2zLθm)0001),
q(x,y)=dxdydzcirc(x2+y2Rc)rect(zL)·exp[j2πxs(cos(2zLθm)xsin(2zLθm)y)λf1]·exp[jπxs2zλf12]exp[j2π2xsxλf1]·exp[j2πx(cos(2zLθm)xsin(2zLθm)y)λf1]·exp[j2πy(sin(2zLθm)x+cos(2zLθm)y)λf1]·exp[j2π(x2+y2)z2f22λ],
q(x,y)=dzrect(zL)πRc2jinc(Rcu2+v2)·exp[iπxs2zλf12]exp[i2πx2+y22f22zλ],
u=1λ[xscos(2zLθm)f1+2xsf1+xcos(2zLθm)f2+ysin(2zLθm)f2],
v=1λ[+xssin(2zLθm)f1xsin(2zLθm)f2+ycos(2zLθm)f2],
jinc(x)=J1(2πx)πx,
q(x,y)=πRc2·jinc[Rcλ(xsf1+xf2)2+(yf2)2]·L·sinc[Lλ(xs22f12x2+y22f22)].
u2+v2=1λ(xsf1+xf2)2+(yf2)2+1λ2xsyf1f2(xsf1+xf2)2+(yf2)22zLθm,
jinc(x){cos(πx)if|x|<0.32681π|πx|3cos(|2πx|3π4)if|x|0.3268.

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