Abstract

A family of plane curves is developed which can diffract incident parallel rays to a point focus. These curves, termed diffractoidal curves, are rotated around an axis to produce surfaces of revolution correspondingly termed diffractoids, whose imaging properties for sources at infinity are studied by ray tracing in a few examples. The paraboloid emerges as a limiting case of the diffractoid. A comparison is made between the stigmatic focusing properties of the diffractoid and the toroidal grating.

© 1978 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. J. H. Underwood, W. M. Neupert, Solar Phys. 35, 241 (1974).
    [CrossRef]
  2. M. Stedman, B. Gale, in Proceedings of X-Ray Optics Symposium, Milliard Space Science Laboratory of University College London, St. Mary (1973), pp. 122–136.
  3. R. J. Speer, Space Sci. Instrum. 2, 463 (1976).
  4. M. V. R. K. Murty, J. Opt. Soc. Am. 50, 923 (1960).
    [CrossRef]
  5. C. W. Gear, Numerical Initial Value Problems in Ordinary Differential Equations (Prentice-Hall, Englewood Cliffs, N.J., 1971), p. 158.
  6. H. T. Davis, Introduction to Nonlinear Differential and Integral Equations (Dover, New York, 1962), pp. 51–53.
  7. S. O. Kastner, W. M. Neupert, J. Opt. Soc. Am. 53, 1180 (1963).
    [CrossRef]
  8. S. O. Kastner, W. M. Neupert, M. Swartz, Astrophys. J. 191, 261 (1974).
    [CrossRef]
  9. M. P. Nakada, R. D. Chapman, W. M. Neupert, R. J. Thomas, Solar Phys. 43, 337 (1975).
    [CrossRef]
  10. J. A. Samson, Techniques of Vacuum Ultraviolet Spectroscopy (Wiley-Interscience, New York, 1967), p. 25.

1976 (1)

R. J. Speer, Space Sci. Instrum. 2, 463 (1976).

1975 (1)

M. P. Nakada, R. D. Chapman, W. M. Neupert, R. J. Thomas, Solar Phys. 43, 337 (1975).
[CrossRef]

1974 (2)

S. O. Kastner, W. M. Neupert, M. Swartz, Astrophys. J. 191, 261 (1974).
[CrossRef]

J. H. Underwood, W. M. Neupert, Solar Phys. 35, 241 (1974).
[CrossRef]

1963 (1)

1960 (1)

Chapman, R. D.

M. P. Nakada, R. D. Chapman, W. M. Neupert, R. J. Thomas, Solar Phys. 43, 337 (1975).
[CrossRef]

Davis, H. T.

H. T. Davis, Introduction to Nonlinear Differential and Integral Equations (Dover, New York, 1962), pp. 51–53.

Gale, B.

M. Stedman, B. Gale, in Proceedings of X-Ray Optics Symposium, Milliard Space Science Laboratory of University College London, St. Mary (1973), pp. 122–136.

Gear, C. W.

C. W. Gear, Numerical Initial Value Problems in Ordinary Differential Equations (Prentice-Hall, Englewood Cliffs, N.J., 1971), p. 158.

Kastner, S. O.

S. O. Kastner, W. M. Neupert, M. Swartz, Astrophys. J. 191, 261 (1974).
[CrossRef]

S. O. Kastner, W. M. Neupert, J. Opt. Soc. Am. 53, 1180 (1963).
[CrossRef]

Murty, M. V. R. K.

Nakada, M. P.

M. P. Nakada, R. D. Chapman, W. M. Neupert, R. J. Thomas, Solar Phys. 43, 337 (1975).
[CrossRef]

Neupert, W. M.

M. P. Nakada, R. D. Chapman, W. M. Neupert, R. J. Thomas, Solar Phys. 43, 337 (1975).
[CrossRef]

S. O. Kastner, W. M. Neupert, M. Swartz, Astrophys. J. 191, 261 (1974).
[CrossRef]

J. H. Underwood, W. M. Neupert, Solar Phys. 35, 241 (1974).
[CrossRef]

S. O. Kastner, W. M. Neupert, J. Opt. Soc. Am. 53, 1180 (1963).
[CrossRef]

Samson, J. A.

J. A. Samson, Techniques of Vacuum Ultraviolet Spectroscopy (Wiley-Interscience, New York, 1967), p. 25.

Speer, R. J.

R. J. Speer, Space Sci. Instrum. 2, 463 (1976).

Stedman, M.

M. Stedman, B. Gale, in Proceedings of X-Ray Optics Symposium, Milliard Space Science Laboratory of University College London, St. Mary (1973), pp. 122–136.

Swartz, M.

S. O. Kastner, W. M. Neupert, M. Swartz, Astrophys. J. 191, 261 (1974).
[CrossRef]

Thomas, R. J.

M. P. Nakada, R. D. Chapman, W. M. Neupert, R. J. Thomas, Solar Phys. 43, 337 (1975).
[CrossRef]

Underwood, J. H.

J. H. Underwood, W. M. Neupert, Solar Phys. 35, 241 (1974).
[CrossRef]

Astrophys. J. (1)

S. O. Kastner, W. M. Neupert, M. Swartz, Astrophys. J. 191, 261 (1974).
[CrossRef]

J. Opt. Soc. Am. (2)

Solar Phys. (2)

J. H. Underwood, W. M. Neupert, Solar Phys. 35, 241 (1974).
[CrossRef]

M. P. Nakada, R. D. Chapman, W. M. Neupert, R. J. Thomas, Solar Phys. 43, 337 (1975).
[CrossRef]

Space Sci. Instrum. (1)

R. J. Speer, Space Sci. Instrum. 2, 463 (1976).

Other (4)

J. A. Samson, Techniques of Vacuum Ultraviolet Spectroscopy (Wiley-Interscience, New York, 1967), p. 25.

M. Stedman, B. Gale, in Proceedings of X-Ray Optics Symposium, Milliard Space Science Laboratory of University College London, St. Mary (1973), pp. 122–136.

C. W. Gear, Numerical Initial Value Problems in Ordinary Differential Equations (Prentice-Hall, Englewood Cliffs, N.J., 1971), p. 158.

H. T. Davis, Introduction to Nonlinear Differential and Integral Equations (Dover, New York, 1962), pp. 51–53.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (10)

Fig. 1
Fig. 1

Geometry used to derive the differential equation of a curve which will diffract rays ī, parallel to the z axis, through the origin.

Fig. 2
Fig. 2

Comparison of the slopes, at the point (0,1), of diffractoidal curves (a) of type I and (b) of type II, as a function of the parameter A; the slope of the corresponding parabola is unity.

Fig. 3
Fig. 3

Comparison of diffractoidal curves [solution of Eq. (3)] for parameter values A = 0.010, 0.100, with the parabola (A = 0).

Fig. 4
Fig. 4

Images, in the plane z = 0 normal to its z axis, produced by a diffractoid of parameter A = 1 × 10−4 with ruling interval σo = 1 × 10−4 cm, corresponding to point focusing of 100-Å radiation.

Fig. 5
Fig. 5

Images, in the plane z = 0 normal to its z axis, produced by a diffractoid of parameter A = 9 × 10−4 with ruling interval σo = 1 × 10−4 cm, corresponding to point focusing of 300-Å radiation.

Fig. 6
Fig. 6

Images produced in the plane z = 0.10 by the same diffractoid used for Fig. 5.

Fig. 7
Fig. 7

Comparison of the factors cos2β and Rs/Rm for a diffractoid of parameter A = 9 × 10−4 as a function of axial position z; these factors would be equal in the case of a toroidal grating used for stigmatic focusing of a distant source. Their approximate equality shows that locally the diffractoid may be approximated by a toroidal grating of similar curvatures Rs, Rm.

Fig. 8
Fig. 8

A plot of the factor mλ/σ as a function of curvature ratio Rs/Rm and incident angle αst for the toroidal stigmatic focusing condition.

Fig. 9
Fig. 9

Variation of the diffraction angle βst with curvature ratio Rs/Rm for the toroidal stigmatic focusing condition.

Fig. 10
Fig. 10

Normalized image distance r′/Rs as a function of incident angle αst and diffracted angle βst for the toroidal stigmatic focusing condition.

Equations (14)

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

m λ σ o = sin ( 2 π θ n ) + sin ( θ r + π θ n ) = sin θ n ( cos θ r sin θ r tan θ n 1 ) .
sin θ n = [ 1 + ( d x d z ) 2 ] 1 / 2
m λ σ o = [ 1 + ( d x d z ) 2 ] 1 / 2 ( cos θ r + sin θ r d x d z 1 ) .
( x 2 r 2 A ) ( d x d z ) 2 2 x r ( 1 z r ) d x d z + z 2 r 2 2 z r A + 1 = 0 .
d x d z = [ x 2 ( 1 A ) A z 2 ] 1 { x [ ( x 2 + z 2 ) 1 / 2 z ] A 1 / 2 [ ( 2 A ) ( x 2 + z 2 ) 2 2 z ( x 2 + z 2 ) 3 / 2 ] 1 / 2 } .
d x d z = x 1 [ ( 1 A 1 / 2 ) ( x 2 + z 2 ) 1 / 2 z ] .
type I : d x d z = 1 A 1 / 2 ( 2 A ) 1 / 2 1 A , type II : d x d z = 1 A 1 / 2 .
sin α = [ 1 ( n 1 p o + n 3 r o ) 2 1 q o 2 ] 1 / 2 ,
sin β = ( m λ σ o cos δ ) sin α
H = n 1 cos δ cos β + n 3 cos δ sin β , J = sin δ , K = n 3 cos δ cos β n 1 cos δ sin β . {
x i = x o + H K ( z i z o ) y i = J K ( z i z o ) .
r m = R m cos 2 β cos α + cos β R m cos 2 α / r r s = R cos α + cos β R s / r } ,
ρ = [ 1 + x ( z 0 , x o ) ] 3 / 2 [ x ( z 0 , x o ) ] 1 / 2 ,
R m = x o 3 ( 1 + x o 2 ) 3 / 2 = ( 2 z 0 + 2 ) 3 / 2 .

Metrics