Abstract

Starshade external occulters are a leading technology that provide the starlight suppression needed to directly image and spectroscopically characterize Earth-sized exoplanets in the habitable zone of nearby stars. A high-priority technology area identified in need of development for a future starshade mission is the development and validation of high-fidelity optical models to predict the performance of a full-scale starshade. We present the generalization of an algorithm to formulate the Fresnel diffraction equation as a one-dimensional integral around the edge of an arbitrary binary diffraction screen. Our edge integral provides an efficient method for capturing diffraction over a large range of size scales and is computationally superior to standard two-dimensional codes. We also present a novel method to implement wavefront errors with the edge integral. This paper provides the derivation of the algorithms and their validation with comparisons to known solutions and results from standard Fresnel propagation codes.

© 2018 Optical Society of America

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References

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  1. W. Cash, “Detection of Earth-like planets around nearby stars using a petal-shaped occulter,” Nature 442, 51–53 (2006).
    [Crossref]
  2. N. Siegler, “Exoplanet exploration program technology plan appendix: 2016,” (Jet Propulsion Laboratory Publications, 2016).
  3. S. R. W. Group, “Starshade readiness working group recommendation to astrophysics division director,” (Jet Propulsion Laboratory Publications, 2016).
  4. M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999).
  5. W. Cash, “Analytic modeling of starshades,” Astrophys. J. 738, 76–88 (2011).
    [Crossref]
  6. G. A. Maggi, “Sulla propagazione libera e perturbata delle onde luminose in un mezzo isotropo,” Ann. Mat. IIa 16, 21–48 (1888).
  7. A. Rubinowicz, “Die Beugungswelle in der Kirchhoffschen Theorie der Beugungserscheinungen,” Ann. Phys. Lpz. 358, 257–278 (1917).
    [Crossref]
  8. A. Dubra and J. A. Ferrari, “Diffracted field by an arbitrary aperture,” Am. J. Phys. 67, 87–92 (1999).
    [Crossref]
  9. A. Schoch, “Betrachtungen über das Schallfeld einer Kolbenmembran,” Akust. Z. 6, 318–326 (1941).
  10. W. Cash, “External occulters for direct exoplanet studies,” in The Direct Detection of Planets and Circumstellar Disks in the 21st Century (Spirit of Bernard Lyot, 2007).
  11. T. Glassman, A. Johnson, A. Lo, D. Dailey, H. Shelton, and J. Vogrin, “Error analysis on the NWO starshade,” Proc. SPIE 7731, 773150 (2010).
    [Crossref]
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  13. K. Miyamoto and E. Wolf, “Generalization of the Maggi-Rubinowicz theory of the boundary diffraction wave—Part I,” J. Opt. Soc. Am. 52, 615–625 (1962).
    [Crossref]
  14. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C. The Art of Scientific Computing (Cambridge University, 1992).
  15. G. E. Sommargren and H. J. Weaver, “Diffraction of light by an opaque sphere. 1: description and properties of the diffraction pattern,” Appl. Opt. 29, 4646–4657 (1990).
    [Crossref]
  16. G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge University, 1922).
  17. R. Soummer, L. Pueyo, A. Sivaramakrishnan, and R. J. Vanderbei, “Fast computation of Lyot-style coronagraph propagation,” Opt. Express 15, 15935–15951 (2007).
    [Crossref]
  18. Y. Kim, D. Sirbu, M. Galvin, N. J. Kasdin, and R. J. Vanderbei, “Experimental study of starshade at flight Fresnel numbers in the laboratory,” Proc. SPIE 9904, 99043G (2016).
    [Crossref]
  19. D. Sirbu, N. J. Kasdin, and R. J. Vanderbei, “Diffractive analysis of limits of an occulter experiment,” Proc. SPIE 9143, 91432P (2014).
    [Crossref]
  20. J. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).
  21. S. S. Chesnokov, V. P. Kandidov, V. I. Shmalhausen, and V. V. Shuvalov, “Numerical/optical simulation of laser beam propagation through atmospheric turbulence,” , DTIC Document (1995).

2016 (1)

Y. Kim, D. Sirbu, M. Galvin, N. J. Kasdin, and R. J. Vanderbei, “Experimental study of starshade at flight Fresnel numbers in the laboratory,” Proc. SPIE 9904, 99043G (2016).
[Crossref]

2014 (1)

D. Sirbu, N. J. Kasdin, and R. J. Vanderbei, “Diffractive analysis of limits of an occulter experiment,” Proc. SPIE 9143, 91432P (2014).
[Crossref]

2012 (1)

2011 (1)

W. Cash, “Analytic modeling of starshades,” Astrophys. J. 738, 76–88 (2011).
[Crossref]

2010 (1)

T. Glassman, A. Johnson, A. Lo, D. Dailey, H. Shelton, and J. Vogrin, “Error analysis on the NWO starshade,” Proc. SPIE 7731, 773150 (2010).
[Crossref]

2007 (1)

2006 (1)

W. Cash, “Detection of Earth-like planets around nearby stars using a petal-shaped occulter,” Nature 442, 51–53 (2006).
[Crossref]

1999 (1)

A. Dubra and J. A. Ferrari, “Diffracted field by an arbitrary aperture,” Am. J. Phys. 67, 87–92 (1999).
[Crossref]

1990 (1)

1962 (1)

1941 (1)

A. Schoch, “Betrachtungen über das Schallfeld einer Kolbenmembran,” Akust. Z. 6, 318–326 (1941).

1917 (1)

A. Rubinowicz, “Die Beugungswelle in der Kirchhoffschen Theorie der Beugungserscheinungen,” Ann. Phys. Lpz. 358, 257–278 (1917).
[Crossref]

1888 (1)

G. A. Maggi, “Sulla propagazione libera e perturbata delle onde luminose in un mezzo isotropo,” Ann. Mat. IIa 16, 21–48 (1888).

Born, M.

M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999).

Cady, E.

Cash, W.

W. Cash, “Analytic modeling of starshades,” Astrophys. J. 738, 76–88 (2011).
[Crossref]

W. Cash, “Detection of Earth-like planets around nearby stars using a petal-shaped occulter,” Nature 442, 51–53 (2006).
[Crossref]

W. Cash, “External occulters for direct exoplanet studies,” in The Direct Detection of Planets and Circumstellar Disks in the 21st Century (Spirit of Bernard Lyot, 2007).

Chesnokov, S. S.

S. S. Chesnokov, V. P. Kandidov, V. I. Shmalhausen, and V. V. Shuvalov, “Numerical/optical simulation of laser beam propagation through atmospheric turbulence,” , DTIC Document (1995).

Dailey, D.

T. Glassman, A. Johnson, A. Lo, D. Dailey, H. Shelton, and J. Vogrin, “Error analysis on the NWO starshade,” Proc. SPIE 7731, 773150 (2010).
[Crossref]

Dubra, A.

A. Dubra and J. A. Ferrari, “Diffracted field by an arbitrary aperture,” Am. J. Phys. 67, 87–92 (1999).
[Crossref]

Ferrari, J. A.

A. Dubra and J. A. Ferrari, “Diffracted field by an arbitrary aperture,” Am. J. Phys. 67, 87–92 (1999).
[Crossref]

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C. The Art of Scientific Computing (Cambridge University, 1992).

Galvin, M.

Y. Kim, D. Sirbu, M. Galvin, N. J. Kasdin, and R. J. Vanderbei, “Experimental study of starshade at flight Fresnel numbers in the laboratory,” Proc. SPIE 9904, 99043G (2016).
[Crossref]

Glassman, T.

T. Glassman, A. Johnson, A. Lo, D. Dailey, H. Shelton, and J. Vogrin, “Error analysis on the NWO starshade,” Proc. SPIE 7731, 773150 (2010).
[Crossref]

Goodman, J.

J. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).

Group, S. R. W.

S. R. W. Group, “Starshade readiness working group recommendation to astrophysics division director,” (Jet Propulsion Laboratory Publications, 2016).

Johnson, A.

T. Glassman, A. Johnson, A. Lo, D. Dailey, H. Shelton, and J. Vogrin, “Error analysis on the NWO starshade,” Proc. SPIE 7731, 773150 (2010).
[Crossref]

Kandidov, V. P.

S. S. Chesnokov, V. P. Kandidov, V. I. Shmalhausen, and V. V. Shuvalov, “Numerical/optical simulation of laser beam propagation through atmospheric turbulence,” , DTIC Document (1995).

Kasdin, N. J.

Y. Kim, D. Sirbu, M. Galvin, N. J. Kasdin, and R. J. Vanderbei, “Experimental study of starshade at flight Fresnel numbers in the laboratory,” Proc. SPIE 9904, 99043G (2016).
[Crossref]

D. Sirbu, N. J. Kasdin, and R. J. Vanderbei, “Diffractive analysis of limits of an occulter experiment,” Proc. SPIE 9143, 91432P (2014).
[Crossref]

Kim, Y.

Y. Kim, D. Sirbu, M. Galvin, N. J. Kasdin, and R. J. Vanderbei, “Experimental study of starshade at flight Fresnel numbers in the laboratory,” Proc. SPIE 9904, 99043G (2016).
[Crossref]

Lo, A.

T. Glassman, A. Johnson, A. Lo, D. Dailey, H. Shelton, and J. Vogrin, “Error analysis on the NWO starshade,” Proc. SPIE 7731, 773150 (2010).
[Crossref]

Maggi, G. A.

G. A. Maggi, “Sulla propagazione libera e perturbata delle onde luminose in un mezzo isotropo,” Ann. Mat. IIa 16, 21–48 (1888).

Miyamoto, K.

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C. The Art of Scientific Computing (Cambridge University, 1992).

Pueyo, L.

Rubinowicz, A.

A. Rubinowicz, “Die Beugungswelle in der Kirchhoffschen Theorie der Beugungserscheinungen,” Ann. Phys. Lpz. 358, 257–278 (1917).
[Crossref]

Schoch, A.

A. Schoch, “Betrachtungen über das Schallfeld einer Kolbenmembran,” Akust. Z. 6, 318–326 (1941).

Shelton, H.

T. Glassman, A. Johnson, A. Lo, D. Dailey, H. Shelton, and J. Vogrin, “Error analysis on the NWO starshade,” Proc. SPIE 7731, 773150 (2010).
[Crossref]

Shmalhausen, V. I.

S. S. Chesnokov, V. P. Kandidov, V. I. Shmalhausen, and V. V. Shuvalov, “Numerical/optical simulation of laser beam propagation through atmospheric turbulence,” , DTIC Document (1995).

Shuvalov, V. V.

S. S. Chesnokov, V. P. Kandidov, V. I. Shmalhausen, and V. V. Shuvalov, “Numerical/optical simulation of laser beam propagation through atmospheric turbulence,” , DTIC Document (1995).

Siegler, N.

N. Siegler, “Exoplanet exploration program technology plan appendix: 2016,” (Jet Propulsion Laboratory Publications, 2016).

Sirbu, D.

Y. Kim, D. Sirbu, M. Galvin, N. J. Kasdin, and R. J. Vanderbei, “Experimental study of starshade at flight Fresnel numbers in the laboratory,” Proc. SPIE 9904, 99043G (2016).
[Crossref]

D. Sirbu, N. J. Kasdin, and R. J. Vanderbei, “Diffractive analysis of limits of an occulter experiment,” Proc. SPIE 9143, 91432P (2014).
[Crossref]

Sivaramakrishnan, A.

Sommargren, G. E.

Soummer, R.

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C. The Art of Scientific Computing (Cambridge University, 1992).

Vanderbei, R. J.

Y. Kim, D. Sirbu, M. Galvin, N. J. Kasdin, and R. J. Vanderbei, “Experimental study of starshade at flight Fresnel numbers in the laboratory,” Proc. SPIE 9904, 99043G (2016).
[Crossref]

D. Sirbu, N. J. Kasdin, and R. J. Vanderbei, “Diffractive analysis of limits of an occulter experiment,” Proc. SPIE 9143, 91432P (2014).
[Crossref]

R. Soummer, L. Pueyo, A. Sivaramakrishnan, and R. J. Vanderbei, “Fast computation of Lyot-style coronagraph propagation,” Opt. Express 15, 15935–15951 (2007).
[Crossref]

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C. The Art of Scientific Computing (Cambridge University, 1992).

Vogrin, J.

T. Glassman, A. Johnson, A. Lo, D. Dailey, H. Shelton, and J. Vogrin, “Error analysis on the NWO starshade,” Proc. SPIE 7731, 773150 (2010).
[Crossref]

Watson, G. N.

G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge University, 1922).

Weaver, H. J.

Wolf, E.

Akust. Z. (1)

A. Schoch, “Betrachtungen über das Schallfeld einer Kolbenmembran,” Akust. Z. 6, 318–326 (1941).

Am. J. Phys. (1)

A. Dubra and J. A. Ferrari, “Diffracted field by an arbitrary aperture,” Am. J. Phys. 67, 87–92 (1999).
[Crossref]

Ann. Mat. IIa (1)

G. A. Maggi, “Sulla propagazione libera e perturbata delle onde luminose in un mezzo isotropo,” Ann. Mat. IIa 16, 21–48 (1888).

Ann. Phys. Lpz. (1)

A. Rubinowicz, “Die Beugungswelle in der Kirchhoffschen Theorie der Beugungserscheinungen,” Ann. Phys. Lpz. 358, 257–278 (1917).
[Crossref]

Appl. Opt. (1)

Astrophys. J. (1)

W. Cash, “Analytic modeling of starshades,” Astrophys. J. 738, 76–88 (2011).
[Crossref]

J. Opt. Soc. Am. (1)

Nature (1)

W. Cash, “Detection of Earth-like planets around nearby stars using a petal-shaped occulter,” Nature 442, 51–53 (2006).
[Crossref]

Opt. Express (2)

Proc. SPIE (3)

T. Glassman, A. Johnson, A. Lo, D. Dailey, H. Shelton, and J. Vogrin, “Error analysis on the NWO starshade,” Proc. SPIE 7731, 773150 (2010).
[Crossref]

Y. Kim, D. Sirbu, M. Galvin, N. J. Kasdin, and R. J. Vanderbei, “Experimental study of starshade at flight Fresnel numbers in the laboratory,” Proc. SPIE 9904, 99043G (2016).
[Crossref]

D. Sirbu, N. J. Kasdin, and R. J. Vanderbei, “Diffractive analysis of limits of an occulter experiment,” Proc. SPIE 9143, 91432P (2014).
[Crossref]

Other (8)

J. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).

S. S. Chesnokov, V. P. Kandidov, V. I. Shmalhausen, and V. V. Shuvalov, “Numerical/optical simulation of laser beam propagation through atmospheric turbulence,” , DTIC Document (1995).

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C. The Art of Scientific Computing (Cambridge University, 1992).

G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge University, 1922).

W. Cash, “External occulters for direct exoplanet studies,” in The Direct Detection of Planets and Circumstellar Disks in the 21st Century (Spirit of Bernard Lyot, 2007).

N. Siegler, “Exoplanet exploration program technology plan appendix: 2016,” (Jet Propulsion Laboratory Publications, 2016).

S. R. W. Group, “Starshade readiness working group recommendation to astrophysics division director,” (Jet Propulsion Laboratory Publications, 2016).

M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999).

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Figures (15)

Fig. 1.
Fig. 1.

Spherical wave originating from P0 passes through a transparent aperture in an opaque screen and is observed at P. The closed surface of integration is broken into three components, the transparent opening A, the non-illuminated screen B, and a large circle C centered on P.

Fig. 2.
Fig. 2.

Coordinate system used to set up the derivation of diffraction problems. The source is located at P0, the starshade is denoted as surface S and is located at the origin, and the observation point is at P.

Fig. 3.
Fig. 3.

On the left is the observation point outside the aperture, where the winding number is 0. In this case, the integral must be evaluated at each crossing of the aperture using Eq. (26). On the right is the observation point inside the aperture, where the winding number is 1. In this case, the inner radius is 0 and the integration is performed using Eq. (27).

Fig. 4.
Fig. 4.

Design of 5 cm diameter starshade mask to be used in the Princeton testbed. The hatched regions denote transparency. The starshade is supported by struts that extend radially from each petal tip and join an outer mask, which is also apodized.

Fig. 5.
Fig. 5.

Greypixel approximation in action on the boundary of a clear aperture. The value of each pixel is the area [0,1] of that pixel interior to the boundary. The black x’s define the edge of the occulter. The cyan line is a polynomial fit to the edge points passing through the pixel of interest. The vertical blue lines are where that polynomial enters and leaves the pixel. The area inside the pixel is calculated by integrating the polynomial between the vertical lines.

Fig. 6.
Fig. 6.

Simulation of a 50 mm diameter circular clear aperture as calculated by the edge integral, the 2D Fresnel code, and the analytical solution.

Fig. 7.
Fig. 7.

Absolute value of the difference in intensities computed by the edge algorithm and 2D Fresnel propagation code compared to the analytical solution for the simulation of a 50 mm diameter circular clear aperture.

Fig. 8.
Fig. 8.

Simulation of a 50 mm diameter circular opaque occulter as calculated by the edge integral, the 2D Fresnel code, and the analytical solution. The peak in the center of the pupil is the Spot of Arago.

Fig. 9.
Fig. 9.

Absolute value of the difference in intensities computed by the edge algorithm and 2D Fresnel propagation code compared to the analytical solution for the simulation of a 50 mm diameter circular opaque occulter.

Fig. 10.
Fig. 10.

Simulation of the normalized intensity (i.e., suppression) for the Princeton testbed using the starshade mask in Fig. 4, comparing the edge algorithm and 2D Fresnel propagation code.

Fig. 11.
Fig. 11.

Absolute value of the difference in intensities computed by the edge algorithm and 2D Fresnel propagation code for the simulation of Fig. 10.

Fig. 12.
Fig. 12.

Normalized intensity (top) and phase (bottom) for a 9 mm diameter circular opaque occulter at a distance of 50 m with Zernike modes (100 nm amplitude) of tip (left) and astigmatism (right). There is excellent agreement between the 1D Edge integral code and the 2D Fresnel code.

Fig. 13.
Fig. 13.

Transparent region of Princeton mask with top-hat phase perturbation of π/2 phase in the hatched region.

Fig. 14.
Fig. 14.

Simulation of the top-hat phase perturbation of Fig. 13 in one transparent region of the Princeton mask. The edge algorithm accurately captures phase perturbations even when the perturbations do not overlap the edge of the occulter.

Fig. 15.
Fig. 15.

Simulation of the Princeton testbed with atmospheric turbulence (Kolmogorov power spectrum with Fried parameter=1  cm) incident on the mask, as calculated by the edge integral and the 2D Fresnel code.

Tables (1)

Tables Icon

Table 1. Sample Starshade Architecture Parameters

Equations (56)

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

(2+k2)U=0.
U(P)=14πS(UUnUUn)dS,
on  A:U=U(i),Un=U(i)n,on  B:U=0,Un=0,
U(i)=Aeikrr,U(i)n=Aeikrr(1rik)cos(n,r),
(1rik)ik,(1sik)ik.
cos(n,r)1,cos(n,s)1.
U(P)=ikA2πS1rseikreiksdS,
r2=(ξx0)2+(ηy0)2+z02,s2=(ξx)2+(ηy)2+z2.
rz0+(ξx0)2+(ηy0)22z0,sz+(ξx)2+(ηy)22z.
z03π4λ[(ξx0)2+(ηy0)2]2,z3π4λ[(ξx)2+(ηy)2]2.
U(x,y,z)=ikA2πzz0eik(z+z0)eik2z0[(ξx0)2+(ηy0)2]×eik2z[(ξx)2+(ηy)2]dξdη.
U(x,y,z)=C(z0,z)eik2f(ϕ)dξdη,
f(ϕ)=1z0[(ξx0)2+(ηy0)2]+1z[(ξx)2+(ηy)2].
a(z0z0+z),
b(zz0+z).
(x,y)=a(x,y),
(x0,y0)=b(x0,y0).
xcξ(x+x0)=ξ(ax+bx0),
ycη(y+y0)=η(ay+by0).
f(ϕ)x=1z0[xc+x0(b1)+ax]2+1z[xc+bx0+x(a1)]2=1z0[xcax0+ax]2+1z[xc+bx0bx]2.
f(ϕ)x=xc2(1z0+1z)+x02z0+z+x2z0+z2xx0z0+z=xc2(z0+zzz0)+(xx0)2z0+z=xc2az+(xx0)2z0+z.
f(ϕ)=xc2+yc2az+(xx0)2+(yy0)2z0+z.
U(x,y,z)=ikA2πzz0eik(z+z0)eik2(z+z0)[(xx0)2+(yy0)2]×eik2az(xc2+yc2)dxcdyc.
U(r,ϕ,z)=ikA2πzz0eik(z+z0)eikr22(z+z0)eikρ22azρdρdθ.
U(r,ϕ,z)=12πz0z0+zeik(z+z0)eikr22(z+z0)02πeikρ(θ)22azdθ|ρiρo,
U12π02πeik2azρi2dθ12π02πeik2azρo2dθ.
U112π02πeik2azρo2dθ.
f(x)dxif(xi+1+xi2)(xi+1xi),
f(θ)=exp{ik2az[xc(θ)2+yc(θ)2]}
U(x,y)U0(ξ,η)exp{2πixξ+yηλz}dξdη,
F{E}=Ejk=m=0N1n=0N1emnexp{2πijm+knN}.
NΔxΔξ=λz.
npad=λzDtelfθ,
I=inλ|Ei|2wiinλwi.
U(s,ϕ,z)=eikziλzeiks22zA(ρ)U0(ρ,θ)eikρ22zeikρszcos(θϕ)ρdρdθ.
U(s,z)=U0eikzeik2z(s2+R2)[V0(u,v)iV1(u,v)],
Vn(u,v)=m=0(1)m(vu)n+2mJn+2m(v),
u=kR2z;v=kRsz.
U(s,z)=U0eikzeik2z(s2+R2)[W2(u,v)+iW1(u,v)],
Wn(u,v)=m=0(1)m(vu)(n+2m)Jn+2m(v).
U(x,y,z)F[U0(ξ,η)M(ξ,η)eik2z(ξ2+η2)],
A(α)=f(x)ei2παxdx.
U(p)U0(x)eik2z(xp)2eik2z0x2dxeik2zp2U0eik2azx2eikzpxdx,
eik2zp2F[U0eik2azx2](pλz),
U(p)eik2zp2U0eik2azx2eikzpx(eiϕ(x))dxeik2zp2F[U0eik2azx2eiϕ(x)](pλz).
U(p)eik2zp2U0(x)eik2azx2eikzpx[A(α)ei2παxdα]dxeik2zp2A(α)U0(x)eik2azx2ei2πx(pλzα)dxdαeik2zp2A(α)F[U0eik2azx2](pλzα)dαeik2zp2[A*F(U0eik2azx2)](pλz),
Uedge(p)eik2(z0+z)p2eik2azx2dx,
eik2(z0+z)p2G(p),
U0eik2azx2=Uedge(p)eik2zp2eikzpxdx=eik2(z0+z)p2G(p)eik2zp2ei2πλzpxdx=F1[eik2(z0+z)p2G(p)eik2zp2]=F1[eik2zap2G(p)].
U(p)eik2zp2F{eiϕ(x)F1[eik2zap2G(p)]}.
F1[A*B]=F1[A]·F1[B],
U(P)ikzU0eikρ22zeikρ48z3ρdρdθ.
U(P)ikzU0eikρ22z(1ikρ48z3)ρdρdθ=ikzU0eikρ22zρdρdθ+k28z4U0eikρ22zρ5dρdθ=U0eikρ22zdθ+k28z4U0eikρ22zρ5dρdθ.
ΔUikρ48z3U0eikρ22zdθ.
U(P)ikU0eikρ22z(1z+ρ22z)ρdρdθ=ikU0eikρ22z(1zρ22z3(1+ρ22z2))ρdρdθ=ikzU0eikρ22zρdρdθik2z3U0eikρ22zρ31+ρ22z2dρdθ=U0eikρ22zdθik2z3U0eikρ22zρ31+ρ22z2dρdθ.
ΔUρ22z2U0eikρ22zdθ.