Abstract

The concept of aberrations of diffracted waves is revisited by using the Rayleigh–Sommerfeld theory of diffraction, and it is pointed out that these aberrations are in a class by themselves; they are only deceptively similar to the aberrations of a rotationally symmetric imaging system. Although an exact Fourier-transform expression can be written for the diffracted wave field, its numerical calculation is cumbersome because of the dependence of the aberrations of the diffracted wave on the observation point. This is true regardless of whether the diffracted wave field is observed on a plane or a hemisphere. It is shown that the Fresnel and Fraunhofer approximations, which neglect these aberrations and thereby simplify the calculations, are valid in imaging applications.

© 2000 Optical Society of America

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References

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  1. J. E. Harvey, R. V. Shack, “Aberrations of diffracted wave fields,” Appl. Opt. 17, 3003–3009 (1978).
    [CrossRef] [PubMed]
  2. J. E. Harvey, “Fourier treatment of near-field scalar theory,” Am. J. Phys. 47, 974–980 (1979).
    [CrossRef]
  3. J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978), Sec. 10.5.
  4. J. E. Harvey, C. L. Vernold, A. Krywonos, P. L. Thompson, “Diffracted radiance: a fundamental quantity in non-paraxial scalar diffraction theory,” Appl. Opt. 38, 6469–6481 (1999). Only a short summary of the earlier work is given in this paper.
    [CrossRef]
  5. H. Osterberg, L. W. Smith, “Closed solutions of Rayleigh’s diffraction integrals for axial points,” J. Opt. Soc. Am. 51, 1050–1054 (1961).
    [CrossRef]
  6. V. N. Mahajan, “Axial irradiance and optimum focusing of laser beams,” Appl. Opt. 22, 3042–3053 (1983).
    [CrossRef] [PubMed]
  7. A. Sommerfeld, Optics (Academic, New York, 1972), Vol. 4, pp. 199–201; substitute Eq. (8) on p. 201 into Eq. (6) on p. 199.
  8. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), pp. 43–44; substitute Eq. (3-23) on p. 44 into Eq. (3-15) on p. 43.
  9. V. N. Mahajan, Optical Imaging and Aberrations. Part I: Ray Geometrical Optics (SPIE, Bellingham, Wash., 1998), Sec. 3.5.
  10. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1986, Sec. 5.1.
  11. V. N. Mahajan, Optical Imaging and Aberrations. Part I: Ray Geometrical Optics (SPIE, Bellingham, Wash., 1998), Chap. 7.
  12. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), p. 60.
  13. V. N. Mahajan, Optical Imaging and Aberrations. Part I: Ray Geoemtrical Optics (SPIE, Bellingham, Wash., 1998), Sec. 3.2.

1999 (1)

1983 (1)

1979 (1)

J. E. Harvey, “Fourier treatment of near-field scalar theory,” Am. J. Phys. 47, 974–980 (1979).
[CrossRef]

1978 (1)

1961 (1)

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1986, Sec. 5.1.

Gaskill, J. D.

J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978), Sec. 10.5.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), p. 60.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), pp. 43–44; substitute Eq. (3-23) on p. 44 into Eq. (3-15) on p. 43.

Harvey, J. E.

Krywonos, A.

Mahajan, V. N.

V. N. Mahajan, “Axial irradiance and optimum focusing of laser beams,” Appl. Opt. 22, 3042–3053 (1983).
[CrossRef] [PubMed]

V. N. Mahajan, Optical Imaging and Aberrations. Part I: Ray Geoemtrical Optics (SPIE, Bellingham, Wash., 1998), Sec. 3.2.

V. N. Mahajan, Optical Imaging and Aberrations. Part I: Ray Geometrical Optics (SPIE, Bellingham, Wash., 1998), Chap. 7.

V. N. Mahajan, Optical Imaging and Aberrations. Part I: Ray Geometrical Optics (SPIE, Bellingham, Wash., 1998), Sec. 3.5.

Osterberg, H.

Shack, R. V.

Smith, L. W.

Sommerfeld, A.

A. Sommerfeld, Optics (Academic, New York, 1972), Vol. 4, pp. 199–201; substitute Eq. (8) on p. 201 into Eq. (6) on p. 199.

Thompson, P. L.

Vernold, C. L.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1986, Sec. 5.1.

Am. J. Phys. (1)

J. E. Harvey, “Fourier treatment of near-field scalar theory,” Am. J. Phys. 47, 974–980 (1979).
[CrossRef]

Appl. Opt. (3)

J. Opt. Soc. Am. (1)

Other (8)

J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978), Sec. 10.5.

A. Sommerfeld, Optics (Academic, New York, 1972), Vol. 4, pp. 199–201; substitute Eq. (8) on p. 201 into Eq. (6) on p. 199.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), pp. 43–44; substitute Eq. (3-23) on p. 44 into Eq. (3-15) on p. 43.

V. N. Mahajan, Optical Imaging and Aberrations. Part I: Ray Geometrical Optics (SPIE, Bellingham, Wash., 1998), Sec. 3.5.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1986, Sec. 5.1.

V. N. Mahajan, Optical Imaging and Aberrations. Part I: Ray Geometrical Optics (SPIE, Bellingham, Wash., 1998), Chap. 7.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), p. 60.

V. N. Mahajan, Optical Imaging and Aberrations. Part I: Ray Geoemtrical Optics (SPIE, Bellingham, Wash., 1998), Sec. 3.2.

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Figures (3)

Fig. 1
Fig. 1

Geometry of wave propagation for determining the complex amplitude in a z plane from its knowledge in the z=0 plane. PP=l, OO=z, and d=OP.

Fig. 2
Fig. 2

Geometry of wave propagation from exit pupil of an imaging system to an image plane. The Gaussian image plane lies at a distance zg from the plane of the exit pupil, and observation is made in a defocused image plane at a distance zi. θpi=θp-θi, and θpg=θp-θg.

Fig. 3
Fig. 3

Observation of a diffracted wave on a hemisphere passing through a Gaussian image point Pg with its center of curvature at the center Op of the exit pupil. Note that ri is parallel to rg only when the point of observation Pi lies in the tangential plane, i.e., when it lies on the arc intersected by the plane containing the z axis and passing through Pg.

Tables (1)

Tables Icon

Table 1 Aberrations of a Diffracted Wave for an On-Axis and an Off-Axis Point Object

Equations (36)

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U(r; z)=U(r; 0)h(r-r; z)dr,
h(r-r; z)=1λ1kl-izlexp(ikl)l
l=(z2+|r-r|2)1/2
=z+12z(r2+r2-2r·r)-18z3(r-r)4+.
U(r; z)=-iλzexpikz+r22zexp(ikr2/2z)×U(r; 0)exp-2πiλzr·rdrforz3k|r-r|max4/8(Fresnel),
U(r; z)=-iλzexpikz+r22zU(r; 0)×exp-2πiλzr·rdrforzkrmax2/2(Fraunhofer).
W(rp; rg)=j=0n=1m=0 2j+manmrg2j+mrpn cosm θpg,
W(rp, θpg; rg)= 0a40rp4+ 1a31rgrp3 cos θpg+ 2a22rg2rp2 cos2 θpg+ 2a20rg2rp2+ 3a11rg3rp cos θpg,
Uex(rp; rg)=P(rp; rg)exp(-iks)s,
P(rp; rg)=A(rp; rg)exp[ikW(rp; rg)]
s=(zg2+|rg-rp|2)1/2
Ui(ri; zi; rg)=1λP(rp;rg)1kl-i×zi exp[ik(l-s)]l2sdrp,
l=(zi2+|ri-rp|2)1/2
l-s=(zi-zg)+12ri2zi-rg2zg-18ri4zi3-rg4zg3-1zirp·ri-zizgrg+Wd(rp; ri; rg),
Wd(rp; ri; rg)=121zi-1zgrp2-181zi3-1zg3rp4+12rp3rizi3cos θpi-rgzg3cos θpg-12rp2ri2zi3cos2 θpi-rg2zg3cos2 θpg-14ri2zi3-rg2zg3rp2+12rpri3zi3cos θpi-rg3zg3cos θpg.
Ui(ri; zi; rg)=f(rp; rg; ri)×exp-2πiλzirp·ri-zizgrgdrp,
f(rp; rg; ri)=A(rp; rg)exp{ik[W(rp; rg)+Wd(rp; ri)]}×1kl-iziλl2s
Ui(ri; zi; rg)=-iλzi×expik(zi-zg)+12ri2zi-rg2zg×P(rp; rg)expik21zi-1zgrp2×exp-2πiλzirp·ri-zizgrgdrp
for1zi31zg3+λa4(1-4)(Fresnel),
Ui(ri; zi; rg)=-iλziexpik(zi-zg)+12ri2zi-rg2zgP(rp; rg)×exp-2πiλzirp·ri-zizgrgdrp
for1zi1zg+λ4a2(1-2)(Fraunhofer),
l-s=(di-dg)-rp·ridi-rgdg+Wd(rp; ri; rg),
di=(zi2+ri2)1/2,
dg=(zg2+rg2)1/2
Wd(rp; ri; rg)=121di-1dgrp2-181di3-1dg3rp4+12rp3ridi3cos θpi-rgdg3cos θpg-12rp2ri2di3cos θpi2-rg2dg3cos θpg2+
Ui(ri; zi; rg)=f(rp; rg; ri)×exp-2πiλrp·ridi-rgdgdrp,
l=(l02+rp2-2rp·ri)1/2,
s=(l02+rp2-2rp·rg)1/2,
l0=(zi2+ri2)1/2=(zg2+rg2)1/2
l-s=-1l0rp·(ri-rg)+Wd(rp; ri; rg),
Us(ri; zi; rg)=f(rp; ri; rg)×exp-2πiλl0rp·(ri-rg)drp,
Wd(rp; ri; rg)=12l03[rp3(ri cos θpi-rg cos θpg)-rp2(ri2 cos2 θpi-rg2 cos2 θpg)]+.
U(rp; 0)=exp(2πiρ0·rp),
ρ0=1λ(α0, β0).
U(r; z; ρ0)=zλexpikz+r22z× exp[ikWd(rp)]1l21kl-i×exp-2πiλzrp·(r-λzρ0)drp,
Wd(rp; r)=l-z+r22z-2r·rp=rp22z+18z3(r-rp)4+.

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