Abstract

We investigate the linear propagation of Gaussian-apodized solutions to the paraxial wave equation in free-space and first-order optical systems. In particular, we present complex coordinate transformations that yield a very general and efficient method to apply a Gaussian apodization (possibly with initial phase curvature) to a solution of the paraxial wave equation. Moreover, we show how this method can be extended from free space to describe propagation behavior through nonimaging first-order optical systems by combining our coordinate transform approach with ray transfer matrix methods. Our framework includes several classes of interesting beams that are important in applications as special cases. Among these are, for example, the Bessel–Gauss and the Airy–Gauss beams, which are of strong interest to researchers and practitioners in various fields.

© 2012 Optical Society of America

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References

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    [CrossRef]
  2. T. M. Pritchett and A. D. Trubatch, “A differential formulation of diffraction theory for the undergraduate optics course,” Am. J. Phys. 72, 1026–1034 (2004).
    [CrossRef]
  3. M. V. Berry and N. L. Balazs, “Non-spreading wave packets,” Am. J. Phys. 47, 264–267 (1979).
    [CrossRef]
  4. M. Levy, Parabolic Equation Methods for Electromagnetic Wave Propagation, IEE Electromagnetic Waves Series (Institution of Electrical Engineers, 2000).
  5. W. L. Siegmann and D. Lee, “Aspects of three-dimensional parabolic equation computations,” Comput. Math. Appl. 11, 853–862 (1985), Special Issue on Computational Ocean Acoustics.
    [CrossRef]
  6. P. W. Milonni and J. H. Eberly, Laser Physics (Wiley, 2010).
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    [CrossRef]
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    [CrossRef]
  11. J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
    [CrossRef]
  12. D. McGloin and K. Dholakia, “Bessel beams: diffraction in a new light,” Contemp. Phys. 46, 15–28 (2005).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  20. Z. Mei, D. Zhao, X. Wei, F. Jing, and Q. Zhu, “Propagation of Bessel-modulated Gaussian beams through a paraxial ABCD optical system with an annular aperture,” Optik 116, 521–526 (2005).
    [CrossRef]
  21. N. Zhou and G. Zeng, “Propagation properties of Hermite–cosine–Gaussian beams through a paraxial optical ABCD system with hard-edge aperture,” Opt. Commun. 232, 49–59 (2004).
    [CrossRef]
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    [CrossRef]
  25. X. Liu and K.-H. Brenner, “Minimal optical decomposition of ray transfer matrices,” Appl. Opt. 47E88–E98 (2008).
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  29. L. W. Casperson, “Synthesis of Gaussian beam optical systems,” Appl. Opt. 20, 2243–2249 (1981).
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  30. G. R. Fowles, Introduction to Modern Optics, 2nd ed. (Dover, 1989).
  31. J. W. Goodman, Introduction to Fourier Optics, McGraw-Hill Physical and Quantum Electronics Series (Roberts, 2005).
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  34. National Institute of Standards and Technology, Digital Library of Mathematical Functions (29August2011), http://dlmf.nist.gov/ .
  35. E. Jones, T. Oliphant, and P. Peterson, “SciPy: open source scientific tools for Python” (2001), http://www.scipy.org .
  36. J. D. Hunter, “Matplotlib: a 2D graphics environment,” Comput. Sci. Eng. 9, 90–95 (2007).
    [CrossRef]

2009

2008

2007

2005

D. McGloin and K. Dholakia, “Bessel beams: diffraction in a new light,” Contemp. Phys. 46, 15–28 (2005).
[CrossRef]

J. C. Gutiérrez-Vega and M. A. Bandres, “Helmholtz–Gauss waves,” J. Opt. Soc. Am. A 22, 289–298 (2005).
[CrossRef]

Z. Mei, D. Zhao, X. Wei, F. Jing, and Q. Zhu, “Propagation of Bessel-modulated Gaussian beams through a paraxial ABCD optical system with an annular aperture,” Optik 116, 521–526 (2005).
[CrossRef]

2004

N. Zhou and G. Zeng, “Propagation properties of Hermite–cosine–Gaussian beams through a paraxial optical ABCD system with hard-edge aperture,” Opt. Commun. 232, 49–59 (2004).
[CrossRef]

T. M. Pritchett and A. D. Trubatch, “A differential formulation of diffraction theory for the undergraduate optics course,” Am. J. Phys. 72, 1026–1034 (2004).
[CrossRef]

2000

A. Belafhal and L. Dalil-Essakali, “Collins formula and propagation of Bessel-modulated Gaussian light beams through an ABCD optical system,” Opt. Commun. 177, 181–188 (2000).
[CrossRef]

1996

M. Santarsiero, “Propagation of generalized Bessel–Gauss beams through ABCD optical systems,” Opt. Commun. 132, 1–7 (1996).
[CrossRef]

1989

1987

J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987).
[CrossRef]

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef]

F. Gori, G. Guattari, and C. Padovani, “Bessel–Gauss beams,” Opt. Commun. 64, 491–495 (1987).
[CrossRef]

1986

1985

W. L. Siegmann and D. Lee, “Aspects of three-dimensional parabolic equation computations,” Comput. Math. Appl. 11, 853–862 (1985), Special Issue on Computational Ocean Acoustics.
[CrossRef]

T. Takenaka, M. Yokota, and O. Fukumitsu, “Propagation of light beams beyond the paraxial approximation,” J. Opt. Soc. Am. A 2, 826–829 (1985).
[CrossRef]

1982

R. Grella, “Fresnel propagation and diffraction and paraxial wave equation,” J. Opt. 13, 367–374 (1982).
[CrossRef]

1981

1979

1977

1973

1970

1954

Agrawal, G. P.

Arnaud, J. A.

J. A. Arnaud, Beam and Fiber Optics (Academic, 1976), Chap. 2.

Balazs, N. L.

M. V. Berry and N. L. Balazs, “Non-spreading wave packets,” Am. J. Phys. 47, 264–267 (1979).
[CrossRef]

Bandres, M. A.

Belafhal, A.

A. Belafhal and L. Dalil-Essakali, “Collins formula and propagation of Bessel-modulated Gaussian light beams through an ABCD optical system,” Opt. Commun. 177, 181–188 (2000).
[CrossRef]

Berry, M. V.

M. V. Berry and N. L. Balazs, “Non-spreading wave packets,” Am. J. Phys. 47, 264–267 (1979).
[CrossRef]

Brenner, K.-H.

Casperson, L. W.

Christodoulides, D. N.

M. S. Mills, G. A. Siviloglou, N. Efremidis, T. Graf, E. M. Wright, J. V. Moloney, and D. N. Christodoulides are preparing a manuscript to be called “Optical bullets with hydrogen-like symmetries.”

Collins, S. A.

Dalil-Essakali, L.

A. Belafhal and L. Dalil-Essakali, “Collins formula and propagation of Bessel-modulated Gaussian light beams through an ABCD optical system,” Opt. Commun. 177, 181–188 (2000).
[CrossRef]

Dholakia, K.

D. McGloin and K. Dholakia, “Bessel beams: diffraction in a new light,” Contemp. Phys. 46, 15–28 (2005).
[CrossRef]

Durnin, J.

J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987).
[CrossRef]

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef]

Eberly, J. H.

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef]

P. W. Milonni and J. H. Eberly, Laser Physics (Wiley, 2010).

Efremidis, N.

M. S. Mills, G. A. Siviloglou, N. Efremidis, T. Graf, E. M. Wright, J. V. Moloney, and D. N. Christodoulides are preparing a manuscript to be called “Optical bullets with hydrogen-like symmetries.”

Felsen, L. B.

Fowles, G. R.

G. R. Fowles, Introduction to Modern Optics, 2nd ed. (Dover, 1989).

Fukumitsu, O.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, McGraw-Hill Physical and Quantum Electronics Series (Roberts, 2005).

Gori, F.

F. Gori, G. Guattari, and C. Padovani, “Bessel–Gauss beams,” Opt. Commun. 64, 491–495 (1987).
[CrossRef]

Graf, T.

M. S. Mills, G. A. Siviloglou, N. Efremidis, T. Graf, E. M. Wright, J. V. Moloney, and D. N. Christodoulides are preparing a manuscript to be called “Optical bullets with hydrogen-like symmetries.”

Grella, R.

R. Grella, “Fresnel propagation and diffraction and paraxial wave equation,” J. Opt. 13, 367–374 (1982).
[CrossRef]

Guattari, G.

F. Gori, G. Guattari, and C. Padovani, “Bessel–Gauss beams,” Opt. Commun. 64, 491–495 (1987).
[CrossRef]

Gutiérrez-Vega, J. C.

Hunter, J. D.

J. D. Hunter, “Matplotlib: a 2D graphics environment,” Comput. Sci. Eng. 9, 90–95 (2007).
[CrossRef]

Jing, F.

Z. Mei, D. Zhao, X. Wei, F. Jing, and Q. Zhu, “Propagation of Bessel-modulated Gaussian beams through a paraxial ABCD optical system with an annular aperture,” Optik 116, 521–526 (2005).
[CrossRef]

Lee, D.

W. L. Siegmann and D. Lee, “Aspects of three-dimensional parabolic equation computations,” Comput. Math. Appl. 11, 853–862 (1985), Special Issue on Computational Ocean Acoustics.
[CrossRef]

Levy, M.

M. Levy, Parabolic Equation Methods for Electromagnetic Wave Propagation, IEE Electromagnetic Waves Series (Institution of Electrical Engineers, 2000).

Liu, X.

McGloin, D.

D. McGloin and K. Dholakia, “Bessel beams: diffraction in a new light,” Contemp. Phys. 46, 15–28 (2005).
[CrossRef]

McLeod, J.

Mei, Z.

Z. Mei, D. Zhao, X. Wei, F. Jing, and Q. Zhu, “Propagation of Bessel-modulated Gaussian beams through a paraxial ABCD optical system with an annular aperture,” Optik 116, 521–526 (2005).
[CrossRef]

Miceli, J. J.

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef]

Mills, M. S.

M. S. Mills, G. A. Siviloglou, N. Efremidis, T. Graf, E. M. Wright, J. V. Moloney, and D. N. Christodoulides are preparing a manuscript to be called “Optical bullets with hydrogen-like symmetries.”

Milonni, P. W.

P. W. Milonni and J. H. Eberly, Laser Physics (Wiley, 2010).

Moloney, J. V.

M. S. Mills, G. A. Siviloglou, N. Efremidis, T. Graf, E. M. Wright, J. V. Moloney, and D. N. Christodoulides are preparing a manuscript to be called “Optical bullets with hydrogen-like symmetries.”

Padovani, C.

F. Gori, G. Guattari, and C. Padovani, “Bessel–Gauss beams,” Opt. Commun. 64, 491–495 (1987).
[CrossRef]

Pattanayak, D. N.

Pritchett, T. M.

T. M. Pritchett and A. D. Trubatch, “A differential formulation of diffraction theory for the undergraduate optics course,” Am. J. Phys. 72, 1026–1034 (2004).
[CrossRef]

Santarsiero, M.

M. Santarsiero, “Propagation of generalized Bessel–Gauss beams through ABCD optical systems,” Opt. Commun. 132, 1–7 (1996).
[CrossRef]

Shin, S. Y.

Siegman, A. E.

Siegmann, W. L.

W. L. Siegmann and D. Lee, “Aspects of three-dimensional parabolic equation computations,” Comput. Math. Appl. 11, 853–862 (1985), Special Issue on Computational Ocean Acoustics.
[CrossRef]

Siviloglou, G. A.

M. S. Mills, G. A. Siviloglou, N. Efremidis, T. Graf, E. M. Wright, J. V. Moloney, and D. N. Christodoulides are preparing a manuscript to be called “Optical bullets with hydrogen-like symmetries.”

Takenaka, T.

Trubatch, A. D.

T. M. Pritchett and A. D. Trubatch, “A differential formulation of diffraction theory for the undergraduate optics course,” Am. J. Phys. 72, 1026–1034 (2004).
[CrossRef]

Wei, X.

Z. Mei, D. Zhao, X. Wei, F. Jing, and Q. Zhu, “Propagation of Bessel-modulated Gaussian beams through a paraxial ABCD optical system with an annular aperture,” Optik 116, 521–526 (2005).
[CrossRef]

Wright, E. M.

M. S. Mills, G. A. Siviloglou, N. Efremidis, T. Graf, E. M. Wright, J. V. Moloney, and D. N. Christodoulides are preparing a manuscript to be called “Optical bullets with hydrogen-like symmetries.”

Wünsche, A.

Yariv, A.

A. Yariv, Optical Electronics, The Holt, Rinehart, and Winston Series in Electrical Engineering (Saunders College, 1991).

Yokota, M.

Zauderer, E.

Zeng, G.

N. Zhou and G. Zeng, “Propagation properties of Hermite–cosine–Gaussian beams through a paraxial optical ABCD system with hard-edge aperture,” Opt. Commun. 232, 49–59 (2004).
[CrossRef]

Zhao, D.

Z. Mei, D. Zhao, X. Wei, F. Jing, and Q. Zhu, “Propagation of Bessel-modulated Gaussian beams through a paraxial ABCD optical system with an annular aperture,” Optik 116, 521–526 (2005).
[CrossRef]

Zhou, N.

N. Zhou and G. Zeng, “Propagation properties of Hermite–cosine–Gaussian beams through a paraxial optical ABCD system with hard-edge aperture,” Opt. Commun. 232, 49–59 (2004).
[CrossRef]

Zhu, Q.

Z. Mei, D. Zhao, X. Wei, F. Jing, and Q. Zhu, “Propagation of Bessel-modulated Gaussian beams through a paraxial ABCD optical system with an annular aperture,” Optik 116, 521–526 (2005).
[CrossRef]

Am. J. Phys.

T. M. Pritchett and A. D. Trubatch, “A differential formulation of diffraction theory for the undergraduate optics course,” Am. J. Phys. 72, 1026–1034 (2004).
[CrossRef]

M. V. Berry and N. L. Balazs, “Non-spreading wave packets,” Am. J. Phys. 47, 264–267 (1979).
[CrossRef]

Appl. Opt.

Comput. Math. Appl.

W. L. Siegmann and D. Lee, “Aspects of three-dimensional parabolic equation computations,” Comput. Math. Appl. 11, 853–862 (1985), Special Issue on Computational Ocean Acoustics.
[CrossRef]

Comput. Sci. Eng.

J. D. Hunter, “Matplotlib: a 2D graphics environment,” Comput. Sci. Eng. 9, 90–95 (2007).
[CrossRef]

Contemp. Phys.

D. McGloin and K. Dholakia, “Bessel beams: diffraction in a new light,” Contemp. Phys. 46, 15–28 (2005).
[CrossRef]

J. Opt.

R. Grella, “Fresnel propagation and diffraction and paraxial wave equation,” J. Opt. 13, 367–374 (1982).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Commun.

N. Zhou and G. Zeng, “Propagation properties of Hermite–cosine–Gaussian beams through a paraxial optical ABCD system with hard-edge aperture,” Opt. Commun. 232, 49–59 (2004).
[CrossRef]

M. Santarsiero, “Propagation of generalized Bessel–Gauss beams through ABCD optical systems,” Opt. Commun. 132, 1–7 (1996).
[CrossRef]

F. Gori, G. Guattari, and C. Padovani, “Bessel–Gauss beams,” Opt. Commun. 64, 491–495 (1987).
[CrossRef]

A. Belafhal and L. Dalil-Essakali, “Collins formula and propagation of Bessel-modulated Gaussian light beams through an ABCD optical system,” Opt. Commun. 177, 181–188 (2000).
[CrossRef]

Opt. Express

Opt. Lett.

Optik

Z. Mei, D. Zhao, X. Wei, F. Jing, and Q. Zhu, “Propagation of Bessel-modulated Gaussian beams through a paraxial ABCD optical system with an annular aperture,” Optik 116, 521–526 (2005).
[CrossRef]

Phys. Rev. Lett.

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef]

Other

P. W. Milonni and J. H. Eberly, Laser Physics (Wiley, 2010).

M. Levy, Parabolic Equation Methods for Electromagnetic Wave Propagation, IEE Electromagnetic Waves Series (Institution of Electrical Engineers, 2000).

M. S. Mills, G. A. Siviloglou, N. Efremidis, T. Graf, E. M. Wright, J. V. Moloney, and D. N. Christodoulides are preparing a manuscript to be called “Optical bullets with hydrogen-like symmetries.”

A. E. Siegman. Lasers (University Science, 1986).

National Institute of Standards and Technology, Digital Library of Mathematical Functions (29August2011), http://dlmf.nist.gov/ .

E. Jones, T. Oliphant, and P. Peterson, “SciPy: open source scientific tools for Python” (2001), http://www.scipy.org .

G. R. Fowles, Introduction to Modern Optics, 2nd ed. (Dover, 1989).

J. W. Goodman, Introduction to Fourier Optics, McGraw-Hill Physical and Quantum Electronics Series (Roberts, 2005).

A. Yariv, Optical Electronics, The Holt, Rinehart, and Winston Series in Electrical Engineering (Saunders College, 1991).

J. A. Arnaud, Beam and Fiber Optics (Academic, 1976), Chap. 2.

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Figures (6)

Fig. 1.
Fig. 1.

Creating a conical wave with an axicon lens. The incoming beam on the left is refracted, and the wave vectors after refraction all lie on the surface of a cone. In the diamond-shaped Bessel region behind the axicon, the refracted waves form a conical superposition.

Fig. 2.
Fig. 2.

Evolution of the absolute value of the complex amplitude of a Gaussian profile during propagation in free space. Here we denote r=x2+y2. The numerical values used for this simulation are k=0.5, w0=10.0, and R(0)=0.0.

Fig. 3.
Fig. 3.

Illustration of the evolution of spot width and radius of curvature of a Gaussian beam during propagation. Here w0=w(0) is called the beam waist. The CBP q(z) is defined in terms of the real spot width and radius of curvature by 1q(z)=1R(z)+i2kw2(z). The numerical values used for this illustration are k=0.5, w0=10.0, and R(0)=0.0.

Fig. 4.
Fig. 4.

A first-order optical system can be described by a single ABCD matrix by multiplying the ABCD matrices of the individual components.

Fig. 5.
Fig. 5.

Propagation through a canonical first-order optical system (ABCD system) consisting of a thin lens L1 with focal length f1, free-space propagation over a distance B0, and a second thin lens L2 with focal length f2. The B component in the ray transfer matrix corresponds to an effective propagation distance in free space, while the focal distances of the thin lenses are given by 1f1=1AB and 1f2=1DB, respectively.

Fig. 6.
Fig. 6.

Evolution of the absolute value of the complex amplitude (|A|) of a Bessel–Gauss beam profile during propagation in free space. The propagation characteristics due to dispersion are described by the change of the CBP of the Gaussian factor (broadening of the Gaussian beam width) as well as the complex coordinate transforms (modulation of the Bessel profile). Here we denote r=x2+y2. The numerical values used for this simulation are α=0.5, k=0.5, w0=10.0, and R(0)=0.0.

Equations (59)

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

iuz+12kx2u=0,
x2=j=1d2xj2
02πexp(iα(xcosθ+ysinθ))exp(inθ)dθexp(α2z2k)=2πinJn(αr)exp(inϕ)exp(α2z2k),
G(x;z)=1(1+z/q0)d/2exp(ikx22q(z)).
1q(z)=1R(z)+i2kw2(z),
v(x;z)=G(x;z)U(x˜;z˜),
iUz˜+12kx˜2U=0
x˜j(xj;z)=xj1+z/q0,j=1,2,,d,z˜(z)=z1+z/q0.
v(x,zout)=(ki2πB)d/2v(x,zin)exp(ik2B(Ax22x·x+Dx2))dx1dx2
v(x,0)=u(x,0)exp(ikx22qin),
v(x,zout)=1(A+Bqin)d/2exp(ikx22qout)u(xA+B/qin,BA+B/qin),
1qout=Cqin+DAqin+B.
iUz(x˜;z˜)+12kx2U(x˜;z˜)+iq(z)x,xU(x˜;z˜)=0.
0=iUz˜z˜z+k=122Ux˜k2(j=12(x˜kxj)2)+Uz˜(j=122z˜xj2+ixjq(z)z˜xj)+k=122Uz˜x˜kx˜kxjz˜xj+l=12l=12k12Ux˜kx˜l(j=12x˜kxjx˜lxj)+k=12Ux˜k(ix˜kz+j=12(2x˜kxj2+ixjq(z)x˜kxj)).
z˜z=j=12(x˜kxj)2,k=1,2.
z˜z=(x˜jxj)2=g2(z),j=1,2
x˜j=g(z)xj+cj(z),j=1,2
z˜(z)=g2(z)dz.
idg(z)dzxj+dcj(z)dz+ixjq(z)g(z)=0,j=1,2.
dg(z)dz=1q0+zg(z),
x˜j(xj;z)=xj1+z/q0+cj,j=1,2,z˜(z)=q02(q0+z)2+c3.
x˜j(xj;z)=xj1+z/q0,j=1,2,z˜(z)=z1+z/q0.
{iu(x;z)z=12kx2u(x;z),u(x;0)=u0(x).
x˜j=xj1+z/q0,z˜(z)=z1+z/q0,
G(x;z)=1(1+z/q0)d/2exp(ikx22q(z))
{iG(x;z)z=12kx2G(x;z),G(x;0)=exp(ikx22q0).
{iv(x;z)z=12kx2v(x;z),v(x;0)=exp(ikx22q0)u0(x)
v(x;z)=G(x;z)U(x˜(x;z);z˜(x;z)),
u(x,y,0)=2πinJn(αx2+y2)exp(inθ)=02πexp(inϕ)exp(iα(xcos(ϕ)+ysin(ϕ)))dϕ
u(x,y,z)=2πinJn(αx2+y2)exp(inθ)exp(iα2z/(2k))=02πexp(inϕ)exp(iα(xcos(ϕ)+ysin(ϕ)))dϕ×exp(iα2z/(2k)),
u(x,y,0)=2πinexp(ik(x2+y2)/(2q0))Jn(αx2+y2)exp(inθ)
u(x,y,z)=2πin11+z/q0exp(ik(x2+y2)2q(z))Jn(αx2+y21+z/q0)exp(inθ)exp(iα2z2k(1+z/q0)),
iu(x,y,z)z+(2x2+2y2)u(x,y,z)=0,
U(x,y,z)=w1w2(w12+4iz)1/2(w22+4iz)1/2×exp(x2w12+4izy2w22+4iz)×im(α14izw12+4iz)m/2Hm(x(α1+w12(α14iz)(w12+4iz))1/2)×in(α24izw22+4iz)n/2Hn(y(α2+w22(α24iz)(w22+4iz))1/2).
v(x,y,z)=w02w02+4izexp(x2+y2w02+4iz)w1w2(w12+4iz)1/2(w22+4iz)1/2exp(x2w12+4izy2w22+4iz)×im(w12w02+4iz(w12w02)w12w02+4iz(w12+w02))m/2Hm(w02xw02+4iz((α1+w12)(w02+4iz)2(α12w02+4iz(α12w02))(w12w02+4iz(w12+w02)))1/2)×in(w22w02+4iz(w22w02)w22w02+4iz(w22+w02))n/2Hn(w02yw02+4iz((α1+w22)(w02+4iz)2(α22w02+4iz(α22w02))(w22w02+4iz(w22+w02)))1/2).
w04(w02+4iz)2exp(2(x2+y2)w02+4iz).
Aout(x)=Ain(x)exp(ikx22f).
u(x,zout)=(ki2πB)d/2u(x,zin)exp(ik2B(Ax22x·x+Dx2))dx1dx2.
u(x,zout)=(ki2πB)d/2exp(ik2B(D1)x2)u(x,zin)exp(ik2B(A1)x2)×exp(ik2B(x22x·x+x2))dx1dx2.
(ABCD)=(101f11)second lens(1B01)free space(101f11)first lens.
1f1=1AB,1f2=1DB.
1qout=C+D/qinA+B/qin.
Aout=Ain(A+B/qin)d/2.
v(x,z0)=1(1+z0q0)d/2exp(ikx22q0)u(x,z0)
v(x,z1)=1(1+z0q0)d/2exp(ikx22q1)u(x,z0),
1q1=1q01f1=1q0+A1B.
v(x,z2)=(1+z0q1)d/2(1+z0q0)d/21(1+z0+Bq1)d/2exp(ikx22(q1+B))×u(x1+B/q1,z0+B1+B/q1).
v(x,z3)=(1+z0q1)d/2(1+z0q0)d/21(1+z0+Bq1)d/2exp(ikx22q3)×u(x1+B/q1,z0+B1+B/q1),
1q3=1q1+B+D1B=(D1)B+(Bq0+Aq0)(B2+Aq0B)=Cq0+DAq0+B.
iuz=12kx2u
v(x,0)=u(x,0)exp(ikx22qin).
v(x,zout)=1(A+Bqin)d/2exp(ikx22qout)×u(xA+B/qin,BA+B/qin),
1qout=Cqin+DAqin+B,
v(x,zout)=(ki2πB)d/2Rdu(x,0)exp(ik2B(Ax22x·x+Dx2))dx.
u(x,0)=exp(S3i3+S(δ+x)iκ)Ai(δ+xκ),
u(x,z)=exp((S+z2κ2k)3i3+(S+z2κ2k)(δ+xz(2S+z2κ2k)2κk)iκ)×Ai(δ+xκz(2S+z2κ2k)2κ2k).
u(x,zout)=exp((S+B2κ2k(A+B/q0))3i3)×exp((S+B2κ2k)(δ(A+B/q0)+xB(2S+B2κ2k(A+B/q0))2κk)iκ(A+B/q0))×Ai(δ(A+B/q0)+xκ(A+B/q0)B(2S+B2κ2k(A+B/q0))2κ2k(A+B/q0))×11+B/q1exp(ikx22Cq0+DAq0+B),
v(x,0)=11+z0q0exp(ik(x2+y2)2q0)×Jn(αx2+y2)exp(inθ),
v(x,zout)=1A+Bq0exp(ik(x2+y2)2Cq0+DAq0+B)×Jn(αx2+y2A+Bq0)exp(inθ)exp(iα2B2k(A+Bq0)).

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