Abstract

The Rayleigh–Sommerfeld theory is applied to diffraction of a spherical wave by a grating. The grating equation is obtained from the aberration-free diffraction pattern, and its aberrations are shown to be the same as the conventional aberrations obtained by using Fermat’s principle. These aberrations are shown to be not associated with the diffraction process. Moreover, it is shown that the irradiance distribution of a certain diffraction order is the Fraunhofer diffraction pattern of the grating aperture as a whole aberrated by the aberration of that order.

© 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);also, J. E. Harvey, “Fourier treatment of near-field scalar theory,” Am. J. Phys. 47, 974–980 (1979).
    [CrossRef] [PubMed]
  2. V. N. Mahajan, “Aberrations of diffracted wave fields. I. Optical imaging,” J. Opt. Soc. Am. A 17, 2216–2222 (2000).
    [CrossRef]
  3. D. J. Schroeder, Astronomical Optics (Academic, New York, 1987), Chap. 14.A comprehensive treatment of gratings is given in this book. However, the aberration function given by Eq. (14.2.1) is for a reflection grating, although Figure 14.1 is for a transmission grating. The correct expression for a transmission grating is obtained if the sign of the first term in each square bracket is made minus [D. J. Schroeder, Professor Emeritus, Department of Physics and Astronomy, Beloit College, Beloit, Wisconsin 53511 (personal communication, 2000)].
  4. W. T. Welford, “Aberration theory of gratings and grating mountings,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1965), Vol. IV, Chap. 6.
  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.
  6. 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.
  7. H. Osterberg, L. W. Smith, “Closed solutions of Rayleigh’s diffraction integrals for axial points,” J. Opt. Soc. Am. A 51, 1050–1054 (1961).
    [CrossRef]
  8. V. N. Mahajan, “Axial irradiance and optimum focusing of laser beams,” Appl. Opt. 22, 3042–3053 (1983).
    [CrossRef] [PubMed]
  9. V. N. Mahajan, Optical Imaging and Aberrations. Part I: Ray Geometrical Optics (SPIE, Bellingham, Wash., 1998), Sec. 5.2.
  10. V. N. Mahajan, Optical Imaging and Aberrations. Part I: Ray Geometrical Optics (SPIE, Bellingham, Wash., 1998), Sec. 3.2.
  11. V. N. Mahajan, “Comparison of geometrical and diffraction point-spread functions,” in International Conference on Optics and Optoelectronics ’98, K. Singh, O. P. Nijhawan, A. K. Gupta, A. K. Musla, eds., Proc. SPIE3729, 434–445 (1998).
    [CrossRef]
  12. M. V. R. K. Murty, “Use of convergent and divergent illumination with plane gratings,” J. Opt. Soc. Am. 52, 768–773 (1962).
    [CrossRef]

2000 (1)

1983 (1)

1978 (1)

1962 (1)

1961 (1)

H. Osterberg, L. W. Smith, “Closed solutions of Rayleigh’s diffraction integrals for axial points,” J. Opt. Soc. Am. A 51, 1050–1054 (1961).
[CrossRef]

Goodman, J. W.

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.

Mahajan, V. N.

V. N. Mahajan, “Aberrations of diffracted wave fields. I. Optical imaging,” J. Opt. Soc. Am. A 17, 2216–2222 (2000).
[CrossRef]

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 Geometrical Optics (SPIE, Bellingham, Wash., 1998), Sec. 5.2.

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

V. N. Mahajan, “Comparison of geometrical and diffraction point-spread functions,” in International Conference on Optics and Optoelectronics ’98, K. Singh, O. P. Nijhawan, A. K. Gupta, A. K. Musla, eds., Proc. SPIE3729, 434–445 (1998).
[CrossRef]

Murty, M. V. R. K.

Osterberg, H.

H. Osterberg, L. W. Smith, “Closed solutions of Rayleigh’s diffraction integrals for axial points,” J. Opt. Soc. Am. A 51, 1050–1054 (1961).
[CrossRef]

Schroeder, D. J.

D. J. Schroeder, Astronomical Optics (Academic, New York, 1987), Chap. 14.A comprehensive treatment of gratings is given in this book. However, the aberration function given by Eq. (14.2.1) is for a reflection grating, although Figure 14.1 is for a transmission grating. The correct expression for a transmission grating is obtained if the sign of the first term in each square bracket is made minus [D. J. Schroeder, Professor Emeritus, Department of Physics and Astronomy, Beloit College, Beloit, Wisconsin 53511 (personal communication, 2000)].

Shack, R. V.

Smith, L. W.

H. Osterberg, L. W. Smith, “Closed solutions of Rayleigh’s diffraction integrals for axial points,” J. Opt. Soc. Am. A 51, 1050–1054 (1961).
[CrossRef]

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.

Welford, W. T.

W. T. Welford, “Aberration theory of gratings and grating mountings,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1965), Vol. IV, Chap. 6.

Appl. Opt. (2)

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

V. N. Mahajan, “Aberrations of diffracted wave fields. I. Optical imaging,” J. Opt. Soc. Am. A 17, 2216–2222 (2000).
[CrossRef]

H. Osterberg, L. W. Smith, “Closed solutions of Rayleigh’s diffraction integrals for axial points,” J. Opt. Soc. Am. A 51, 1050–1054 (1961).
[CrossRef]

Other (7)

V. N. Mahajan, Optical Imaging and Aberrations. Part I: Ray Geometrical Optics (SPIE, Bellingham, Wash., 1998), Sec. 5.2.

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

V. N. Mahajan, “Comparison of geometrical and diffraction point-spread functions,” in International Conference on Optics and Optoelectronics ’98, K. Singh, O. P. Nijhawan, A. K. Gupta, A. K. Musla, eds., Proc. SPIE3729, 434–445 (1998).
[CrossRef]

D. J. Schroeder, Astronomical Optics (Academic, New York, 1987), Chap. 14.A comprehensive treatment of gratings is given in this book. However, the aberration function given by Eq. (14.2.1) is for a reflection grating, although Figure 14.1 is for a transmission grating. The correct expression for a transmission grating is obtained if the sign of the first term in each square bracket is made minus [D. J. Schroeder, Professor Emeritus, Department of Physics and Astronomy, Beloit College, Beloit, Wisconsin 53511 (personal communication, 2000)].

W. T. Welford, “Aberration theory of gratings and grating mountings,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1965), Vol. IV, Chap. 6.

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.

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Figures (1)

Fig. 1
Fig. 1

Geometry of grating diffraction. A beam focused at a point Po is incident from the left on a transmission grating whose lines (grooves) are parallel to the y axis. A diffracted beam is shown focused at a point Pi. The points Po and Pi lie at distances do and di, respectively, from the center O of the grating. The incident and the diffracted chief rays OPo and OPi make angles γo and γi with the plane of symmetry zx of the grating, and their projections OA and OB on this plane make angles α and β with the z axis.

Equations (26)

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U(r; z)=U(r; 0)h(r-r; z)dr,
h(r-r; z)=1λ1kl-izlexp(ikl)l
U(xp, yp)=G(xp, yp)[A(xp, yp)/s]exp(-iks),
G(xp, yp)=n=0N-1rect(xp/a) * δ(xp-nd)rect(yp/b),
rect(xp/a)=1,|xp|a/2=0,otherwise.
s=[do2+rp2-2rp·ro]1/2.
U(xi, yi)=-(i/λ)G(xp, yp)[A(xp, yp)/ls]×exp[ik(l-s)]dxpdyp,
l=[di2+rp2-2rp·ri]1/2.
ro=(xo, yo)=do(cos γo sin α, sin γo),
ri=(xi, yi)=di(cos γi sin β, sin γi).
l-s=(di-do)-(gxp+hyp)+W(xp, yp),
g=cos γi sin β-cos γo sin α,
h=sin γi-sin γo,
I(β; α)=(Io/λ2di2)G(xp, yp)×exp[-ik(gxp+hyp)]dxpdyp2=(Io/λ2di2)n=0N-1 exp(-ikngd)×-a/2a/2 exp(-ikgxp)dxp×-b/2b/2 exp(-ikhyp)dyp2=Ioabλdi2sin(Nkgd/2)sin kgd/22×sin(kga/2)kga/22sin(khb/2)khb/22.
d cos γo(sin β-sin α)=mλ,
Im(β; α)=Imsin[Nk(gd-mλ)/2]Nk(gd-mλ)/22sin(khb/2)khb/22,
Im=I0Nabλdi2sin(πma/d)πma/d2
W(xp, yp)=121di-1dorp2+Wa+Wc-181di3-1do3rp4+.
Wa(xp, yp)=-12cos2 γi sin2 βdi-cos2 γo sin2 αdoxp2+sin2 γidi-sin2 γodoyp2-12sin 2γi sin βdi-sin 2γo sin αdoxpyp,
Wc(xp, yp)=12cos γi sin βdi2-cos γo sin αdo2xp+sin γidi2-sin γodo2yprp2.
l-s=(di-do)-(sin β-sin α)xp+W(xp, yp),
d(sin β-sin α)=mλ,
W(xp, yp)=121di-1dorp2-12sin2 βdi-sin2 αdoxp2+12sin βdi2-sin αdo2xprp2-181di3-1do3rp4+.
Wc(xp, yp)=-(xprp2/di2)sin α.
Wc(xp, yp)=12sin β cos2 βdi2-sin α cos2 αdo2xp3+12sin βdi2-sin αdo2xpyp2.
Wc(xp, yp)=-(xprp2/di2)sin α+(xp3/di2)sin3 α.

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