Abstract

The flux flow equation of Burkhard and Shealy is a simplified equation which can be used to evaluate the energy flux density at the image plane for a general optical system. Since the flux flow equation is based on the differential geometry of the wave front passing through the system, the energy flux density at the image plane can be computed by tracing a single ray through the system and using the flux flow equation. This technique has been used to calculate the meridional section of the point spread function of Wolter I x-ray telescopes and thin-film multilayered optics. Results, which have been obtained by the flux flow ray tracing method for the point spread function of several Wolter I x-ray telescopes and hybrid x-ray telescopes using convexed thin-film multilayered optics located near the primary focus, are compared with the rms blur circle results and the point spread function results obtained by conventional ray tracing techniques.

© 1986 Optical Society of America

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References

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  1. Military Standardization Handbook-Optical Design, MIL-HDBK-141 (Defense Supply Agency, Washington, DC, 1962), Chap. 5.
  2. W. T. Welford, Aberrations of the Symmetrical Optical System (Academic, New York, 1974), p. 93.
  3. R. J. Gagnon, “Spot Diagram of Maximum Sharpness,” J. Opt. Soc. Am. 58, 1160 (1968).
  4. J. D. Mangus, J. H. Underwood, “Optical Design of a Glancing Incidence X-Ray Telescope,” Appl. Opt. 8, 95 (1969).
  5. L. P. VanSpeybroeck, R. C. Chase, “Design Parameters of Paraboloid–Hyperboloid Telescopes for X-Ray Astronomy,” Appl. Opt. 11, 440 (1972).
  6. R. C. Chase, L. P. VanSpeybroeck, “Wolter-Schwarzchild Telescopes for X-Ray Astronomy,” Appl. Opt. 12, 1042 (1973).
  7. R. C. Chase, J. K. Silk, “Ellipsoid–Hyperboloid X-Ray Imaging Instrument for Laser-Pellet Diagnostics,” Appl. Opt. 14, 2096 (1975).
  8. J. K. Silk, “A Grazing Incidence Microscope for X-Ray Imaging Applications,” Ann. N.Y. Acad. Sci. 342, 116 (1980).
  9. J. W. Foreman, “Computation of rms Spot Radii by Ray Tracing,” Appl. Opt. 13, 2585 (1974).
  10. T. B. Andersen, “Evaluating rms Spot Radii by Ray Tracing,” Appl. Opt. 21, 1241 (1982).
  11. J. W. Foreman, G. W. Hunt, E. K. Lawson, “Analytical Study of the Imaging Characteristics of the Goddard ATM X-Ray Telescope,” Space Support Division, Sperry Rand Corp., Huntsville, AL (Sept.1969).
  12. D. L. Shealy, A. Kassim, S. Chao, “Extended Range X-Ray Telescope: X-Ray Microscope Design,” Final Report Submitted to Marshall Space Flight Center, Contract NAS8-34728 (July1982).
  13. D. L. Shealy, “Analysis of NOAA-MSFC GOES X-Ray Telescope,” Final Report Submitted to Marshall Space Flight Center, Contract H-34373B (Aug.1979).
  14. S. Ramage, M. V. Zombeck, “Off-Axis Behavior of the High Resolution Mirror Assembly,” AXAF Interim Report SAO-AXAF-83-014, Smithsonian Astrophysical Observatory (May1983).
  15. W. Werner, “Imaging Properties of Wolter I Type X-Ray Telescopes,” Appl. Opt. 16, 764 (1977).
  16. D. G. Burkhard, D. L. Shealy, “Simplified Formula for the Illuminance in an Optical System,” Appl. Opt. 20, 897 (1981).
  17. D. L. Shealy, D. G. Burkhard, “Heat Flux Contours on a Plane for Parallel Radiation Specularly Reflected from a Cone, a Hemisphere and a Paraboloid,” Int. J. Heat Mass Transfer 16, 281 (1973).
  18. D. G. Burkhard, D. L. Shealy, “Equation for the Intensity of Acoustic Rays Deflected by an Object in a Variable Velocity Medium,” J. Acoust. Soc. Am. 56, 1327 (1974).
  19. D. L. Shealy, “Analytical Illuminance and Caustic Surface Calculations in Geometrical Optics,” Appl. Opt. 15, 2588 (1976).
  20. D. G. Burkhard, D. L. Shealy, “Specular Aspheric Surface to Obtain a Specified Irradiance from Discrete or Continuous Line Source Radiation: Design,” Appl. Opt. 14, 1279 (1975).
  21. P. W. Rhodes, D. L. Shealy, “Refractive Optical Systems for Irradiance Redistribution of Collimated Radiation: Their Design and Analysis,” Appl. Opt. 19, 3545 (1980).
  22. R. B. Hoover, S. Chao, D. L. Shealy, “Design and Analysis of Spectral Slicing X-Ray Telescope Systems,” Proc. Soc. Photo-Opt. Instrum. Eng. 563, 280 (1985).

1985 (1)

R. B. Hoover, S. Chao, D. L. Shealy, “Design and Analysis of Spectral Slicing X-Ray Telescope Systems,” Proc. Soc. Photo-Opt. Instrum. Eng. 563, 280 (1985).

1982 (1)

1981 (1)

1980 (2)

P. W. Rhodes, D. L. Shealy, “Refractive Optical Systems for Irradiance Redistribution of Collimated Radiation: Their Design and Analysis,” Appl. Opt. 19, 3545 (1980).

J. K. Silk, “A Grazing Incidence Microscope for X-Ray Imaging Applications,” Ann. N.Y. Acad. Sci. 342, 116 (1980).

1977 (1)

1976 (1)

1975 (2)

1974 (2)

D. G. Burkhard, D. L. Shealy, “Equation for the Intensity of Acoustic Rays Deflected by an Object in a Variable Velocity Medium,” J. Acoust. Soc. Am. 56, 1327 (1974).

J. W. Foreman, “Computation of rms Spot Radii by Ray Tracing,” Appl. Opt. 13, 2585 (1974).

1973 (2)

R. C. Chase, L. P. VanSpeybroeck, “Wolter-Schwarzchild Telescopes for X-Ray Astronomy,” Appl. Opt. 12, 1042 (1973).

D. L. Shealy, D. G. Burkhard, “Heat Flux Contours on a Plane for Parallel Radiation Specularly Reflected from a Cone, a Hemisphere and a Paraboloid,” Int. J. Heat Mass Transfer 16, 281 (1973).

1972 (1)

1969 (1)

1968 (1)

Andersen, T. B.

Burkhard, D. G.

D. G. Burkhard, D. L. Shealy, “Simplified Formula for the Illuminance in an Optical System,” Appl. Opt. 20, 897 (1981).

D. G. Burkhard, D. L. Shealy, “Specular Aspheric Surface to Obtain a Specified Irradiance from Discrete or Continuous Line Source Radiation: Design,” Appl. Opt. 14, 1279 (1975).

D. G. Burkhard, D. L. Shealy, “Equation for the Intensity of Acoustic Rays Deflected by an Object in a Variable Velocity Medium,” J. Acoust. Soc. Am. 56, 1327 (1974).

D. L. Shealy, D. G. Burkhard, “Heat Flux Contours on a Plane for Parallel Radiation Specularly Reflected from a Cone, a Hemisphere and a Paraboloid,” Int. J. Heat Mass Transfer 16, 281 (1973).

Chao, S.

R. B. Hoover, S. Chao, D. L. Shealy, “Design and Analysis of Spectral Slicing X-Ray Telescope Systems,” Proc. Soc. Photo-Opt. Instrum. Eng. 563, 280 (1985).

D. L. Shealy, A. Kassim, S. Chao, “Extended Range X-Ray Telescope: X-Ray Microscope Design,” Final Report Submitted to Marshall Space Flight Center, Contract NAS8-34728 (July1982).

Chase, R. C.

Foreman, J. W.

J. W. Foreman, “Computation of rms Spot Radii by Ray Tracing,” Appl. Opt. 13, 2585 (1974).

J. W. Foreman, G. W. Hunt, E. K. Lawson, “Analytical Study of the Imaging Characteristics of the Goddard ATM X-Ray Telescope,” Space Support Division, Sperry Rand Corp., Huntsville, AL (Sept.1969).

Gagnon, R. J.

Hoover, R. B.

R. B. Hoover, S. Chao, D. L. Shealy, “Design and Analysis of Spectral Slicing X-Ray Telescope Systems,” Proc. Soc. Photo-Opt. Instrum. Eng. 563, 280 (1985).

Hunt, G. W.

J. W. Foreman, G. W. Hunt, E. K. Lawson, “Analytical Study of the Imaging Characteristics of the Goddard ATM X-Ray Telescope,” Space Support Division, Sperry Rand Corp., Huntsville, AL (Sept.1969).

Kassim, A.

D. L. Shealy, A. Kassim, S. Chao, “Extended Range X-Ray Telescope: X-Ray Microscope Design,” Final Report Submitted to Marshall Space Flight Center, Contract NAS8-34728 (July1982).

Lawson, E. K.

J. W. Foreman, G. W. Hunt, E. K. Lawson, “Analytical Study of the Imaging Characteristics of the Goddard ATM X-Ray Telescope,” Space Support Division, Sperry Rand Corp., Huntsville, AL (Sept.1969).

Mangus, J. D.

Ramage, S.

S. Ramage, M. V. Zombeck, “Off-Axis Behavior of the High Resolution Mirror Assembly,” AXAF Interim Report SAO-AXAF-83-014, Smithsonian Astrophysical Observatory (May1983).

Rhodes, P. W.

Shealy, D. L.

R. B. Hoover, S. Chao, D. L. Shealy, “Design and Analysis of Spectral Slicing X-Ray Telescope Systems,” Proc. Soc. Photo-Opt. Instrum. Eng. 563, 280 (1985).

D. G. Burkhard, D. L. Shealy, “Simplified Formula for the Illuminance in an Optical System,” Appl. Opt. 20, 897 (1981).

P. W. Rhodes, D. L. Shealy, “Refractive Optical Systems for Irradiance Redistribution of Collimated Radiation: Their Design and Analysis,” Appl. Opt. 19, 3545 (1980).

D. L. Shealy, “Analytical Illuminance and Caustic Surface Calculations in Geometrical Optics,” Appl. Opt. 15, 2588 (1976).

D. G. Burkhard, D. L. Shealy, “Specular Aspheric Surface to Obtain a Specified Irradiance from Discrete or Continuous Line Source Radiation: Design,” Appl. Opt. 14, 1279 (1975).

D. G. Burkhard, D. L. Shealy, “Equation for the Intensity of Acoustic Rays Deflected by an Object in a Variable Velocity Medium,” J. Acoust. Soc. Am. 56, 1327 (1974).

D. L. Shealy, D. G. Burkhard, “Heat Flux Contours on a Plane for Parallel Radiation Specularly Reflected from a Cone, a Hemisphere and a Paraboloid,” Int. J. Heat Mass Transfer 16, 281 (1973).

D. L. Shealy, “Analysis of NOAA-MSFC GOES X-Ray Telescope,” Final Report Submitted to Marshall Space Flight Center, Contract H-34373B (Aug.1979).

D. L. Shealy, A. Kassim, S. Chao, “Extended Range X-Ray Telescope: X-Ray Microscope Design,” Final Report Submitted to Marshall Space Flight Center, Contract NAS8-34728 (July1982).

Silk, J. K.

J. K. Silk, “A Grazing Incidence Microscope for X-Ray Imaging Applications,” Ann. N.Y. Acad. Sci. 342, 116 (1980).

R. C. Chase, J. K. Silk, “Ellipsoid–Hyperboloid X-Ray Imaging Instrument for Laser-Pellet Diagnostics,” Appl. Opt. 14, 2096 (1975).

Underwood, J. H.

VanSpeybroeck, L. P.

Welford, W. T.

W. T. Welford, Aberrations of the Symmetrical Optical System (Academic, New York, 1974), p. 93.

Werner, W.

Zombeck, M. V.

S. Ramage, M. V. Zombeck, “Off-Axis Behavior of the High Resolution Mirror Assembly,” AXAF Interim Report SAO-AXAF-83-014, Smithsonian Astrophysical Observatory (May1983).

Ann. N.Y. Acad. Sci. (1)

J. K. Silk, “A Grazing Incidence Microscope for X-Ray Imaging Applications,” Ann. N.Y. Acad. Sci. 342, 116 (1980).

Appl. Opt. (11)

Int. J. Heat Mass Transfer (1)

D. L. Shealy, D. G. Burkhard, “Heat Flux Contours on a Plane for Parallel Radiation Specularly Reflected from a Cone, a Hemisphere and a Paraboloid,” Int. J. Heat Mass Transfer 16, 281 (1973).

J. Acoust. Soc. Am. (1)

D. G. Burkhard, D. L. Shealy, “Equation for the Intensity of Acoustic Rays Deflected by an Object in a Variable Velocity Medium,” J. Acoust. Soc. Am. 56, 1327 (1974).

J. Opt. Soc. Am. (1)

Proc. Soc. Photo-Opt. Instrum. Eng. (1)

R. B. Hoover, S. Chao, D. L. Shealy, “Design and Analysis of Spectral Slicing X-Ray Telescope Systems,” Proc. Soc. Photo-Opt. Instrum. Eng. 563, 280 (1985).

Other (6)

J. W. Foreman, G. W. Hunt, E. K. Lawson, “Analytical Study of the Imaging Characteristics of the Goddard ATM X-Ray Telescope,” Space Support Division, Sperry Rand Corp., Huntsville, AL (Sept.1969).

D. L. Shealy, A. Kassim, S. Chao, “Extended Range X-Ray Telescope: X-Ray Microscope Design,” Final Report Submitted to Marshall Space Flight Center, Contract NAS8-34728 (July1982).

D. L. Shealy, “Analysis of NOAA-MSFC GOES X-Ray Telescope,” Final Report Submitted to Marshall Space Flight Center, Contract H-34373B (Aug.1979).

S. Ramage, M. V. Zombeck, “Off-Axis Behavior of the High Resolution Mirror Assembly,” AXAF Interim Report SAO-AXAF-83-014, Smithsonian Astrophysical Observatory (May1983).

Military Standardization Handbook-Optical Design, MIL-HDBK-141 (Defense Supply Agency, Washington, DC, 1962), Chap. 5.

W. T. Welford, Aberrations of the Symmetrical Optical System (Academic, New York, 1974), p. 93.

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Figures (6)

Fig. 1
Fig. 1

Wolter I x-ray telescope configuration.

Fig. 2
Fig. 2

Schematic of a hybrid x-ray telescope with a convex thin-film multilayer optic.

Fig. 3
Fig. 3

Meridional section of the PSF of the S056 Wotler I telescope for field angles of (a) 5 and (b) 10 min of arc. A conventional ray tracing technique was used.

Fig. 4
Fig. 4

Meridional section of the PSF of the AXAF # 1 Wolter I telescope for field angles of (a) 16 and (b) 20 min of arc. A conventional ray tracing techique was used.

Fig. 5
Fig. 5

Meridional section of the PSF for the S056 Wolter I telescope for field angles of (a) 5 and (b) 10 min of arc. The flux flow ray tracing technique was used.

Fig. 6
Fig. 6

Meridional section of the PSF of the AXAF # 1 Wolter I telescope for field angles of (a) 16 and (b) 20 min of arc. The flux flow ray tracing technique was used.

Tables (4)

Tables Icon

Table I Wolter I Telescope Mirror Surface Parameters for Skylab ATM Experiment S056 and AXAF Element #1 Systems

Tables Icon

Table II Convexed Spherical, Thin-Film Multilayer Mirror Parameters for a Hybrid X-Ray Telescope System Based on the S056 Wolter I Telescope System

Tables Icon

Table III Measures of Resolution for Wolter I X-Ray Telescopes

Tables Icon

Table IV Performance of a Hybrid X-Ray Telescope Formed by the S056 Wolter I and a Convex Spherical Mirror

Equations (26)

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F = σ ( 0 ) ρ 1 ρ 2 cos φ ( 3 ) r ( 1 c ) r * ( 1 c ) r ( 2 c ) r * ( 2 c ) r ( 1 p ) r * ( 1 p ) r ( 2 p ) r * ( 2 p ) ,
1 r ( s ) = 1 r ( i ) + 2 cos φ r ,
1 r ( s ) = 1 r ( i ) + 2 r cos φ ,
1 τ ( s ) = 1 τ ( i ) + 2 τ ,
Z P ( R p ) = R P 2 2 p - p 2 ,
Z H ( R H ) = c + a [ 1 + ( R H b ) 2 ] 1 / 2 ,
p = R P min tan ϑ m , c = f / 2 , a = c ( 2 cos 2 ϑ m - 1 ) , b = ( c 2 - a 2 ) 1 / 2 .
1 r * ( 1 c ) = 2 cos φ ( 1 ) ( p 2 + R P 2 ) 1 / 2 + 1 ( - r 01 ) ,
1 r ( 1 c ) = - 2 p 2 cos φ ( 1 ) ( p 2 + R P 2 ) 3 / 2 + 1 ( - r 01 ) ,
r ( 1 p ) = r ( 1 , 2 ) - r ( 1 c ) ;             r * ( 1 p ) = r ( 1 , 2 ) - r * ( 1 c ) ,
1 r * ( 2 c ) = 1 [ - r * ( 1 p ) ] - 2 cos φ ( 2 ) Z H R H ( 1 + Z H 2 ) 1 / 2 ,
1 r ( 2 c ) = 1 [ - r ( 1 p ) ] - 2 Z H * cos φ ( 2 ) ( 1 + Z H 2 ) 3 / 2 ,
Z H = a R H b ( b 2 + R H 2 ) 1 / 2 ,
Z H = a b ( b 2 + R H 2 ) 1 / 2 .
r ( 2 p ) = 2 ( 2 , 3 ) - r ( 2 c ) ;             r * ( 2 p ) = r ( 2 , 3 ) - r * ( 2 c ) ,
F = σ ( 0 ) ρ 1 ρ 2 ρ 3 cos φ ( 4 ) [ r ( 1 c ) r * ( 1 c ) r ( 2 c ) r * ( 2 c ) r ( 3 c ) r * ( 3 c ) r ( 1 p ) r * ( 1 p ) r ( 2 p ) r * ( 2 p ) r ( 3 p ) r * ( 3 p ) ] ,
M = - V / U = ( Z I - Z 03 ) / ( Z 03 - Z ¯ ) ,
Z ¯ = z 2 - x 2 A 2 z / A 2 x ,
( Z 3 - Z 03 + R 3 ) 2 + X 3 2 + Y 3 2 = R 3 2 ,
Z 03 = ( Z I + Z ¯ M ) / ( 1 + M ) .
1 U + 1 V = - 2 R 3 .
R 3 = 2 M ( Z ¯ - Z I ) / ( M 2 - 1 ) ,
1 r ( 3 c ) = 1 - r ( 2 p ) + 2 R 3 cos φ ( 3 ) ;             1 r * ( 3 c ) = 1 - r * ( 2 p ) + 2 cos φ ( 3 ) R 3 .
r ( 3 p ) = r ( 3 , 4 ) - r ( 3 c ) ;             r * ( 3 p ) = r ( 3 , 4 ) - r * ( 3 c ) ,
E i = Δ A i F ( x , y ) d x d y .
E i [ ( x 1 + x 2 ) / 2 ] = 0.5 ( x 2 - x 1 ) [ F ( x 1 ) + F ( x 2 ) ] ,

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