Abstract

We developed a method for making quantitative characterizations of bi-grid rotating modulation collimators (RMC’s) that are used in a Fourier transform x-ray imager. With appropriate choices of the collimator spacings, this technique can be implemented with a beam-expanded He–Ne laser to simulate the plane wave produced by a point source at infinity even though the RMC’s are diffraction limited at the He–Ne wavelength of 632.8 nm. The expanded beam passes through the grid pairs at a small angle with respect to their axis of rotation, and the modulated transmission through the grids as the RMC’s rotate is detected with a photomultiplier tube. In addition to providing a quantitative characterization of the RMC’s, the method also produces a measured point response function and provides an end-to-end check of the imaging system. We applied our method to the RMC’s on the high-energy imaging device (HEIDI) balloon payload in its preflight configuration. We computed the harmonic ratios of the modulation time profile from the laser measurements and compared them with theoretical calculations, including the diffraction effects on irregular grids. Our results indicate the 25-in. (64-cm) x-ray imaging optics on HEIDI are capable of achieving images near the theoretical limit and are not seriously compromised by imperfections in the grids.

© 1996 Optical Society of America

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  1. H. S. Hudson, K. T. Strong, B. R. Dennis, D. Zous, M. Inda, T. Kosugi, T. Sakao, “Impulsive behavior in solar soft x-radiation,” Astrophys. J. 422, L25–L27 (1994).
    [Crossref]
  2. S. Masuda, T. Kosugi, H. Hara, S. Tsuneta, Y. Ogawara, “Loop-top impulsive hard x-ray source of a solar flare as evidence of magnetic reconnection,” Nature (London) 371, 495–497 (1994).
    [Crossref]
  3. T. Sakao, “Characteristics of solar flare hard x-ray sources as revealed with the hard x-ray telescope aboard the Yohkoh satellite,” Ph.D. dissertation (University of Tokyo, Tokyo, Japan1994).
  4. B. R. Dennis, R. P. Lin, R. C. Canfield, C. J. Crannell, A. G. Emslie, G. D. Holman, H. S. Hudson, G. J. Hurford, J. C. Ling, N. W. Madden, R. Ramaty, “The High Energy Solar Spectroscopic Imager—HESSI,” in Missions to the Sun, D. M. Rust, ed., Proc. SPIE2804 (1996).
  5. C. J. Crannell, B. R. Dennis, L. E. Orwig, E. J. Schmahl, F. L. Lang, R. Starr, J. P. Norris, M. E. Greene, G. J. Hurford, W. N. Johnson, K. S. Wood, “A balloon-borne payload for imaging hard x-rays and gamma rays from solar flares,” in Proceedings of the AIAA International Balloon Technology Conference (American Institute of Aeronautics and Astronautics, New York, 1991).
  6. M. Pelling, P. T. Feffer, K. Hurley, S. R. Kane, R. P. Lin, S. McBride, J. H. Primbsch, D. M. Smith, K. Youseffi, G. Zimmer, F. Cotin, J. M. Lavigne, G. Rouaix, S. Slassi, G. Vedrenne, R. Pehl, C. Cork, P. Luke, N. Madden, D. Malone, “A high resolution gamma-ray and hard X-ray spectrometer (HIREGS) for long duration balloon flights,” in EUV, X-Ray, and Gamma-Ray Instrumentation for Astronomy III, O. H. Siegmund, ed. Proc. SPIE1743, 408–421 (1992).
  7. K. Makishima, “Instrumental performance of the SXT collimator,” in Hinotori Symposium on Solar Flares (Institute of Space and Astronautical Science, Tokyo, 1982), pp. 120–129.
  8. K. Makishima, Y. Ogawara, S. Miyamoto, N. Muranaka, M. Oda, “Alignment of modulation collimator grids using optical diffraction patterns,” ISAS Research Note 29 (Institute of Space and Astronautical Science, Megura, Japan, 1977).
  9. J. M. Cowley, A. F. Moodie, “Fourier images. I. The point source,” Proc. Phys. Soc. London Sect. B 70, 486–496 (1957).
    [Crossref]
  10. J. M. Cowley, Diffraction Physics (Elsevier, New York, 1981).
  11. H. W. Schnopper, R. I. Thompson, S. Watt, “Predicted performance of a rotating modulation collimator for locating celestial hard x-ray sources,” Space Sci. Rev. 8, 534–542 (1968).
    [Crossref]
  12. M. J. Murphy, “The virtues of positive-definite reconstruction of x-ray and gamma-ray images,” Nucl. Instrum. Methods A 290, 551–558 (1990).
    [Crossref]
  13. S. Tsuneta, “Image processing software system of the x-ray telescope aboard the Hinotori spacecraft,” Ann. Tokyo Astron. Observatory 20, 1–50 (1984).
  14. L. N. Mertz, G. H. Nakano, J. R. Kilner, “Rotational aperture synthesis for x-rays,” J. Opt. Soc. Am. A 3, 2167–2170 (1986).
    [Crossref]
  15. J. R. Kilner, G. H. Nakano, “Design studies for x-ray and gamma ray rotation modulation collimators,” in EUV, X-ray and Gamma-Ray Instrumentation for Astronomy and Atomic Physics, C. J. Hailey, ed., Proc. SPIE1159, 27–33 (1989).
  16. J. A. Högbom, “Aperture synthesis with a non-regular distribution of interferometer baselines,” Astron. Astrophys. Suppl. Ser. 15, 417–426 (1974).
  17. S. J. Wernecke, L. R. D’Addario, “Maximum entropy image reconstruction,” IEEE Trans. Comput. C 26, 351–364 (1977).
    [Crossref]
  18. B. Frieden, “Statistical models for the image restoration problem,” Comput. Graphics Image Process. 12, 40–59 (1980).
    [Crossref]
  19. C. A. Lindsey, “Effects of diffraction in multiple-grid telescopes for x-ray astronomy,” J. Opt. Soc. Am. 68, 1708–1715 (1978).
    [Crossref]
  20. E. Bar-Ziv, “Effect of diffraction on the moiré image. I. Theory,” J. Opt. Soc. Am. A 2, 371–379 (1985).
    [Crossref]

1994 (2)

H. S. Hudson, K. T. Strong, B. R. Dennis, D. Zous, M. Inda, T. Kosugi, T. Sakao, “Impulsive behavior in solar soft x-radiation,” Astrophys. J. 422, L25–L27 (1994).
[Crossref]

S. Masuda, T. Kosugi, H. Hara, S. Tsuneta, Y. Ogawara, “Loop-top impulsive hard x-ray source of a solar flare as evidence of magnetic reconnection,” Nature (London) 371, 495–497 (1994).
[Crossref]

1990 (1)

M. J. Murphy, “The virtues of positive-definite reconstruction of x-ray and gamma-ray images,” Nucl. Instrum. Methods A 290, 551–558 (1990).
[Crossref]

1986 (1)

1985 (1)

1984 (1)

S. Tsuneta, “Image processing software system of the x-ray telescope aboard the Hinotori spacecraft,” Ann. Tokyo Astron. Observatory 20, 1–50 (1984).

1980 (1)

B. Frieden, “Statistical models for the image restoration problem,” Comput. Graphics Image Process. 12, 40–59 (1980).
[Crossref]

1978 (1)

1977 (1)

S. J. Wernecke, L. R. D’Addario, “Maximum entropy image reconstruction,” IEEE Trans. Comput. C 26, 351–364 (1977).
[Crossref]

1974 (1)

J. A. Högbom, “Aperture synthesis with a non-regular distribution of interferometer baselines,” Astron. Astrophys. Suppl. Ser. 15, 417–426 (1974).

1968 (1)

H. W. Schnopper, R. I. Thompson, S. Watt, “Predicted performance of a rotating modulation collimator for locating celestial hard x-ray sources,” Space Sci. Rev. 8, 534–542 (1968).
[Crossref]

1957 (1)

J. M. Cowley, A. F. Moodie, “Fourier images. I. The point source,” Proc. Phys. Soc. London Sect. B 70, 486–496 (1957).
[Crossref]

Bar-Ziv, E.

Canfield, R. C.

B. R. Dennis, R. P. Lin, R. C. Canfield, C. J. Crannell, A. G. Emslie, G. D. Holman, H. S. Hudson, G. J. Hurford, J. C. Ling, N. W. Madden, R. Ramaty, “The High Energy Solar Spectroscopic Imager—HESSI,” in Missions to the Sun, D. M. Rust, ed., Proc. SPIE2804 (1996).

Cork, C.

M. Pelling, P. T. Feffer, K. Hurley, S. R. Kane, R. P. Lin, S. McBride, J. H. Primbsch, D. M. Smith, K. Youseffi, G. Zimmer, F. Cotin, J. M. Lavigne, G. Rouaix, S. Slassi, G. Vedrenne, R. Pehl, C. Cork, P. Luke, N. Madden, D. Malone, “A high resolution gamma-ray and hard X-ray spectrometer (HIREGS) for long duration balloon flights,” in EUV, X-Ray, and Gamma-Ray Instrumentation for Astronomy III, O. H. Siegmund, ed. Proc. SPIE1743, 408–421 (1992).

Cotin, F.

M. Pelling, P. T. Feffer, K. Hurley, S. R. Kane, R. P. Lin, S. McBride, J. H. Primbsch, D. M. Smith, K. Youseffi, G. Zimmer, F. Cotin, J. M. Lavigne, G. Rouaix, S. Slassi, G. Vedrenne, R. Pehl, C. Cork, P. Luke, N. Madden, D. Malone, “A high resolution gamma-ray and hard X-ray spectrometer (HIREGS) for long duration balloon flights,” in EUV, X-Ray, and Gamma-Ray Instrumentation for Astronomy III, O. H. Siegmund, ed. Proc. SPIE1743, 408–421 (1992).

Cowley, J. M.

J. M. Cowley, A. F. Moodie, “Fourier images. I. The point source,” Proc. Phys. Soc. London Sect. B 70, 486–496 (1957).
[Crossref]

J. M. Cowley, Diffraction Physics (Elsevier, New York, 1981).

Crannell, C. J.

B. R. Dennis, R. P. Lin, R. C. Canfield, C. J. Crannell, A. G. Emslie, G. D. Holman, H. S. Hudson, G. J. Hurford, J. C. Ling, N. W. Madden, R. Ramaty, “The High Energy Solar Spectroscopic Imager—HESSI,” in Missions to the Sun, D. M. Rust, ed., Proc. SPIE2804 (1996).

C. J. Crannell, B. R. Dennis, L. E. Orwig, E. J. Schmahl, F. L. Lang, R. Starr, J. P. Norris, M. E. Greene, G. J. Hurford, W. N. Johnson, K. S. Wood, “A balloon-borne payload for imaging hard x-rays and gamma rays from solar flares,” in Proceedings of the AIAA International Balloon Technology Conference (American Institute of Aeronautics and Astronautics, New York, 1991).

D’Addario, L. R.

S. J. Wernecke, L. R. D’Addario, “Maximum entropy image reconstruction,” IEEE Trans. Comput. C 26, 351–364 (1977).
[Crossref]

Dennis, B. R.

H. S. Hudson, K. T. Strong, B. R. Dennis, D. Zous, M. Inda, T. Kosugi, T. Sakao, “Impulsive behavior in solar soft x-radiation,” Astrophys. J. 422, L25–L27 (1994).
[Crossref]

C. J. Crannell, B. R. Dennis, L. E. Orwig, E. J. Schmahl, F. L. Lang, R. Starr, J. P. Norris, M. E. Greene, G. J. Hurford, W. N. Johnson, K. S. Wood, “A balloon-borne payload for imaging hard x-rays and gamma rays from solar flares,” in Proceedings of the AIAA International Balloon Technology Conference (American Institute of Aeronautics and Astronautics, New York, 1991).

B. R. Dennis, R. P. Lin, R. C. Canfield, C. J. Crannell, A. G. Emslie, G. D. Holman, H. S. Hudson, G. J. Hurford, J. C. Ling, N. W. Madden, R. Ramaty, “The High Energy Solar Spectroscopic Imager—HESSI,” in Missions to the Sun, D. M. Rust, ed., Proc. SPIE2804 (1996).

Emslie, A. G.

B. R. Dennis, R. P. Lin, R. C. Canfield, C. J. Crannell, A. G. Emslie, G. D. Holman, H. S. Hudson, G. J. Hurford, J. C. Ling, N. W. Madden, R. Ramaty, “The High Energy Solar Spectroscopic Imager—HESSI,” in Missions to the Sun, D. M. Rust, ed., Proc. SPIE2804 (1996).

Feffer, P. T.

M. Pelling, P. T. Feffer, K. Hurley, S. R. Kane, R. P. Lin, S. McBride, J. H. Primbsch, D. M. Smith, K. Youseffi, G. Zimmer, F. Cotin, J. M. Lavigne, G. Rouaix, S. Slassi, G. Vedrenne, R. Pehl, C. Cork, P. Luke, N. Madden, D. Malone, “A high resolution gamma-ray and hard X-ray spectrometer (HIREGS) for long duration balloon flights,” in EUV, X-Ray, and Gamma-Ray Instrumentation for Astronomy III, O. H. Siegmund, ed. Proc. SPIE1743, 408–421 (1992).

Frieden, B.

B. Frieden, “Statistical models for the image restoration problem,” Comput. Graphics Image Process. 12, 40–59 (1980).
[Crossref]

Greene, M. E.

C. J. Crannell, B. R. Dennis, L. E. Orwig, E. J. Schmahl, F. L. Lang, R. Starr, J. P. Norris, M. E. Greene, G. J. Hurford, W. N. Johnson, K. S. Wood, “A balloon-borne payload for imaging hard x-rays and gamma rays from solar flares,” in Proceedings of the AIAA International Balloon Technology Conference (American Institute of Aeronautics and Astronautics, New York, 1991).

Hara, H.

S. Masuda, T. Kosugi, H. Hara, S. Tsuneta, Y. Ogawara, “Loop-top impulsive hard x-ray source of a solar flare as evidence of magnetic reconnection,” Nature (London) 371, 495–497 (1994).
[Crossref]

Högbom, J. A.

J. A. Högbom, “Aperture synthesis with a non-regular distribution of interferometer baselines,” Astron. Astrophys. Suppl. Ser. 15, 417–426 (1974).

Holman, G. D.

B. R. Dennis, R. P. Lin, R. C. Canfield, C. J. Crannell, A. G. Emslie, G. D. Holman, H. S. Hudson, G. J. Hurford, J. C. Ling, N. W. Madden, R. Ramaty, “The High Energy Solar Spectroscopic Imager—HESSI,” in Missions to the Sun, D. M. Rust, ed., Proc. SPIE2804 (1996).

Hudson, H. S.

H. S. Hudson, K. T. Strong, B. R. Dennis, D. Zous, M. Inda, T. Kosugi, T. Sakao, “Impulsive behavior in solar soft x-radiation,” Astrophys. J. 422, L25–L27 (1994).
[Crossref]

B. R. Dennis, R. P. Lin, R. C. Canfield, C. J. Crannell, A. G. Emslie, G. D. Holman, H. S. Hudson, G. J. Hurford, J. C. Ling, N. W. Madden, R. Ramaty, “The High Energy Solar Spectroscopic Imager—HESSI,” in Missions to the Sun, D. M. Rust, ed., Proc. SPIE2804 (1996).

Hurford, G. J.

B. R. Dennis, R. P. Lin, R. C. Canfield, C. J. Crannell, A. G. Emslie, G. D. Holman, H. S. Hudson, G. J. Hurford, J. C. Ling, N. W. Madden, R. Ramaty, “The High Energy Solar Spectroscopic Imager—HESSI,” in Missions to the Sun, D. M. Rust, ed., Proc. SPIE2804 (1996).

C. J. Crannell, B. R. Dennis, L. E. Orwig, E. J. Schmahl, F. L. Lang, R. Starr, J. P. Norris, M. E. Greene, G. J. Hurford, W. N. Johnson, K. S. Wood, “A balloon-borne payload for imaging hard x-rays and gamma rays from solar flares,” in Proceedings of the AIAA International Balloon Technology Conference (American Institute of Aeronautics and Astronautics, New York, 1991).

Hurley, K.

M. Pelling, P. T. Feffer, K. Hurley, S. R. Kane, R. P. Lin, S. McBride, J. H. Primbsch, D. M. Smith, K. Youseffi, G. Zimmer, F. Cotin, J. M. Lavigne, G. Rouaix, S. Slassi, G. Vedrenne, R. Pehl, C. Cork, P. Luke, N. Madden, D. Malone, “A high resolution gamma-ray and hard X-ray spectrometer (HIREGS) for long duration balloon flights,” in EUV, X-Ray, and Gamma-Ray Instrumentation for Astronomy III, O. H. Siegmund, ed. Proc. SPIE1743, 408–421 (1992).

Inda, M.

H. S. Hudson, K. T. Strong, B. R. Dennis, D. Zous, M. Inda, T. Kosugi, T. Sakao, “Impulsive behavior in solar soft x-radiation,” Astrophys. J. 422, L25–L27 (1994).
[Crossref]

Johnson, W. N.

C. J. Crannell, B. R. Dennis, L. E. Orwig, E. J. Schmahl, F. L. Lang, R. Starr, J. P. Norris, M. E. Greene, G. J. Hurford, W. N. Johnson, K. S. Wood, “A balloon-borne payload for imaging hard x-rays and gamma rays from solar flares,” in Proceedings of the AIAA International Balloon Technology Conference (American Institute of Aeronautics and Astronautics, New York, 1991).

Kane, S. R.

M. Pelling, P. T. Feffer, K. Hurley, S. R. Kane, R. P. Lin, S. McBride, J. H. Primbsch, D. M. Smith, K. Youseffi, G. Zimmer, F. Cotin, J. M. Lavigne, G. Rouaix, S. Slassi, G. Vedrenne, R. Pehl, C. Cork, P. Luke, N. Madden, D. Malone, “A high resolution gamma-ray and hard X-ray spectrometer (HIREGS) for long duration balloon flights,” in EUV, X-Ray, and Gamma-Ray Instrumentation for Astronomy III, O. H. Siegmund, ed. Proc. SPIE1743, 408–421 (1992).

Kilner, J. R.

L. N. Mertz, G. H. Nakano, J. R. Kilner, “Rotational aperture synthesis for x-rays,” J. Opt. Soc. Am. A 3, 2167–2170 (1986).
[Crossref]

J. R. Kilner, G. H. Nakano, “Design studies for x-ray and gamma ray rotation modulation collimators,” in EUV, X-ray and Gamma-Ray Instrumentation for Astronomy and Atomic Physics, C. J. Hailey, ed., Proc. SPIE1159, 27–33 (1989).

Kosugi, T.

H. S. Hudson, K. T. Strong, B. R. Dennis, D. Zous, M. Inda, T. Kosugi, T. Sakao, “Impulsive behavior in solar soft x-radiation,” Astrophys. J. 422, L25–L27 (1994).
[Crossref]

S. Masuda, T. Kosugi, H. Hara, S. Tsuneta, Y. Ogawara, “Loop-top impulsive hard x-ray source of a solar flare as evidence of magnetic reconnection,” Nature (London) 371, 495–497 (1994).
[Crossref]

Lang, F. L.

C. J. Crannell, B. R. Dennis, L. E. Orwig, E. J. Schmahl, F. L. Lang, R. Starr, J. P. Norris, M. E. Greene, G. J. Hurford, W. N. Johnson, K. S. Wood, “A balloon-borne payload for imaging hard x-rays and gamma rays from solar flares,” in Proceedings of the AIAA International Balloon Technology Conference (American Institute of Aeronautics and Astronautics, New York, 1991).

Lavigne, J. M.

M. Pelling, P. T. Feffer, K. Hurley, S. R. Kane, R. P. Lin, S. McBride, J. H. Primbsch, D. M. Smith, K. Youseffi, G. Zimmer, F. Cotin, J. M. Lavigne, G. Rouaix, S. Slassi, G. Vedrenne, R. Pehl, C. Cork, P. Luke, N. Madden, D. Malone, “A high resolution gamma-ray and hard X-ray spectrometer (HIREGS) for long duration balloon flights,” in EUV, X-Ray, and Gamma-Ray Instrumentation for Astronomy III, O. H. Siegmund, ed. Proc. SPIE1743, 408–421 (1992).

Lin, R. P.

M. Pelling, P. T. Feffer, K. Hurley, S. R. Kane, R. P. Lin, S. McBride, J. H. Primbsch, D. M. Smith, K. Youseffi, G. Zimmer, F. Cotin, J. M. Lavigne, G. Rouaix, S. Slassi, G. Vedrenne, R. Pehl, C. Cork, P. Luke, N. Madden, D. Malone, “A high resolution gamma-ray and hard X-ray spectrometer (HIREGS) for long duration balloon flights,” in EUV, X-Ray, and Gamma-Ray Instrumentation for Astronomy III, O. H. Siegmund, ed. Proc. SPIE1743, 408–421 (1992).

B. R. Dennis, R. P. Lin, R. C. Canfield, C. J. Crannell, A. G. Emslie, G. D. Holman, H. S. Hudson, G. J. Hurford, J. C. Ling, N. W. Madden, R. Ramaty, “The High Energy Solar Spectroscopic Imager—HESSI,” in Missions to the Sun, D. M. Rust, ed., Proc. SPIE2804 (1996).

Lindsey, C. A.

Ling, J. C.

B. R. Dennis, R. P. Lin, R. C. Canfield, C. J. Crannell, A. G. Emslie, G. D. Holman, H. S. Hudson, G. J. Hurford, J. C. Ling, N. W. Madden, R. Ramaty, “The High Energy Solar Spectroscopic Imager—HESSI,” in Missions to the Sun, D. M. Rust, ed., Proc. SPIE2804 (1996).

Luke, P.

M. Pelling, P. T. Feffer, K. Hurley, S. R. Kane, R. P. Lin, S. McBride, J. H. Primbsch, D. M. Smith, K. Youseffi, G. Zimmer, F. Cotin, J. M. Lavigne, G. Rouaix, S. Slassi, G. Vedrenne, R. Pehl, C. Cork, P. Luke, N. Madden, D. Malone, “A high resolution gamma-ray and hard X-ray spectrometer (HIREGS) for long duration balloon flights,” in EUV, X-Ray, and Gamma-Ray Instrumentation for Astronomy III, O. H. Siegmund, ed. Proc. SPIE1743, 408–421 (1992).

Madden, N.

M. Pelling, P. T. Feffer, K. Hurley, S. R. Kane, R. P. Lin, S. McBride, J. H. Primbsch, D. M. Smith, K. Youseffi, G. Zimmer, F. Cotin, J. M. Lavigne, G. Rouaix, S. Slassi, G. Vedrenne, R. Pehl, C. Cork, P. Luke, N. Madden, D. Malone, “A high resolution gamma-ray and hard X-ray spectrometer (HIREGS) for long duration balloon flights,” in EUV, X-Ray, and Gamma-Ray Instrumentation for Astronomy III, O. H. Siegmund, ed. Proc. SPIE1743, 408–421 (1992).

Madden, N. W.

B. R. Dennis, R. P. Lin, R. C. Canfield, C. J. Crannell, A. G. Emslie, G. D. Holman, H. S. Hudson, G. J. Hurford, J. C. Ling, N. W. Madden, R. Ramaty, “The High Energy Solar Spectroscopic Imager—HESSI,” in Missions to the Sun, D. M. Rust, ed., Proc. SPIE2804 (1996).

Makishima, K.

K. Makishima, “Instrumental performance of the SXT collimator,” in Hinotori Symposium on Solar Flares (Institute of Space and Astronautical Science, Tokyo, 1982), pp. 120–129.

K. Makishima, Y. Ogawara, S. Miyamoto, N. Muranaka, M. Oda, “Alignment of modulation collimator grids using optical diffraction patterns,” ISAS Research Note 29 (Institute of Space and Astronautical Science, Megura, Japan, 1977).

Malone, D.

M. Pelling, P. T. Feffer, K. Hurley, S. R. Kane, R. P. Lin, S. McBride, J. H. Primbsch, D. M. Smith, K. Youseffi, G. Zimmer, F. Cotin, J. M. Lavigne, G. Rouaix, S. Slassi, G. Vedrenne, R. Pehl, C. Cork, P. Luke, N. Madden, D. Malone, “A high resolution gamma-ray and hard X-ray spectrometer (HIREGS) for long duration balloon flights,” in EUV, X-Ray, and Gamma-Ray Instrumentation for Astronomy III, O. H. Siegmund, ed. Proc. SPIE1743, 408–421 (1992).

Masuda, S.

S. Masuda, T. Kosugi, H. Hara, S. Tsuneta, Y. Ogawara, “Loop-top impulsive hard x-ray source of a solar flare as evidence of magnetic reconnection,” Nature (London) 371, 495–497 (1994).
[Crossref]

McBride, S.

M. Pelling, P. T. Feffer, K. Hurley, S. R. Kane, R. P. Lin, S. McBride, J. H. Primbsch, D. M. Smith, K. Youseffi, G. Zimmer, F. Cotin, J. M. Lavigne, G. Rouaix, S. Slassi, G. Vedrenne, R. Pehl, C. Cork, P. Luke, N. Madden, D. Malone, “A high resolution gamma-ray and hard X-ray spectrometer (HIREGS) for long duration balloon flights,” in EUV, X-Ray, and Gamma-Ray Instrumentation for Astronomy III, O. H. Siegmund, ed. Proc. SPIE1743, 408–421 (1992).

Mertz, L. N.

Miyamoto, S.

K. Makishima, Y. Ogawara, S. Miyamoto, N. Muranaka, M. Oda, “Alignment of modulation collimator grids using optical diffraction patterns,” ISAS Research Note 29 (Institute of Space and Astronautical Science, Megura, Japan, 1977).

Moodie, A. F.

J. M. Cowley, A. F. Moodie, “Fourier images. I. The point source,” Proc. Phys. Soc. London Sect. B 70, 486–496 (1957).
[Crossref]

Muranaka, N.

K. Makishima, Y. Ogawara, S. Miyamoto, N. Muranaka, M. Oda, “Alignment of modulation collimator grids using optical diffraction patterns,” ISAS Research Note 29 (Institute of Space and Astronautical Science, Megura, Japan, 1977).

Murphy, M. J.

M. J. Murphy, “The virtues of positive-definite reconstruction of x-ray and gamma-ray images,” Nucl. Instrum. Methods A 290, 551–558 (1990).
[Crossref]

Nakano, G. H.

L. N. Mertz, G. H. Nakano, J. R. Kilner, “Rotational aperture synthesis for x-rays,” J. Opt. Soc. Am. A 3, 2167–2170 (1986).
[Crossref]

J. R. Kilner, G. H. Nakano, “Design studies for x-ray and gamma ray rotation modulation collimators,” in EUV, X-ray and Gamma-Ray Instrumentation for Astronomy and Atomic Physics, C. J. Hailey, ed., Proc. SPIE1159, 27–33 (1989).

Norris, J. P.

C. J. Crannell, B. R. Dennis, L. E. Orwig, E. J. Schmahl, F. L. Lang, R. Starr, J. P. Norris, M. E. Greene, G. J. Hurford, W. N. Johnson, K. S. Wood, “A balloon-borne payload for imaging hard x-rays and gamma rays from solar flares,” in Proceedings of the AIAA International Balloon Technology Conference (American Institute of Aeronautics and Astronautics, New York, 1991).

Oda, M.

K. Makishima, Y. Ogawara, S. Miyamoto, N. Muranaka, M. Oda, “Alignment of modulation collimator grids using optical diffraction patterns,” ISAS Research Note 29 (Institute of Space and Astronautical Science, Megura, Japan, 1977).

Ogawara, Y.

S. Masuda, T. Kosugi, H. Hara, S. Tsuneta, Y. Ogawara, “Loop-top impulsive hard x-ray source of a solar flare as evidence of magnetic reconnection,” Nature (London) 371, 495–497 (1994).
[Crossref]

K. Makishima, Y. Ogawara, S. Miyamoto, N. Muranaka, M. Oda, “Alignment of modulation collimator grids using optical diffraction patterns,” ISAS Research Note 29 (Institute of Space and Astronautical Science, Megura, Japan, 1977).

Orwig, L. E.

C. J. Crannell, B. R. Dennis, L. E. Orwig, E. J. Schmahl, F. L. Lang, R. Starr, J. P. Norris, M. E. Greene, G. J. Hurford, W. N. Johnson, K. S. Wood, “A balloon-borne payload for imaging hard x-rays and gamma rays from solar flares,” in Proceedings of the AIAA International Balloon Technology Conference (American Institute of Aeronautics and Astronautics, New York, 1991).

Pehl, R.

M. Pelling, P. T. Feffer, K. Hurley, S. R. Kane, R. P. Lin, S. McBride, J. H. Primbsch, D. M. Smith, K. Youseffi, G. Zimmer, F. Cotin, J. M. Lavigne, G. Rouaix, S. Slassi, G. Vedrenne, R. Pehl, C. Cork, P. Luke, N. Madden, D. Malone, “A high resolution gamma-ray and hard X-ray spectrometer (HIREGS) for long duration balloon flights,” in EUV, X-Ray, and Gamma-Ray Instrumentation for Astronomy III, O. H. Siegmund, ed. Proc. SPIE1743, 408–421 (1992).

Pelling, M.

M. Pelling, P. T. Feffer, K. Hurley, S. R. Kane, R. P. Lin, S. McBride, J. H. Primbsch, D. M. Smith, K. Youseffi, G. Zimmer, F. Cotin, J. M. Lavigne, G. Rouaix, S. Slassi, G. Vedrenne, R. Pehl, C. Cork, P. Luke, N. Madden, D. Malone, “A high resolution gamma-ray and hard X-ray spectrometer (HIREGS) for long duration balloon flights,” in EUV, X-Ray, and Gamma-Ray Instrumentation for Astronomy III, O. H. Siegmund, ed. Proc. SPIE1743, 408–421 (1992).

Primbsch, J. H.

M. Pelling, P. T. Feffer, K. Hurley, S. R. Kane, R. P. Lin, S. McBride, J. H. Primbsch, D. M. Smith, K. Youseffi, G. Zimmer, F. Cotin, J. M. Lavigne, G. Rouaix, S. Slassi, G. Vedrenne, R. Pehl, C. Cork, P. Luke, N. Madden, D. Malone, “A high resolution gamma-ray and hard X-ray spectrometer (HIREGS) for long duration balloon flights,” in EUV, X-Ray, and Gamma-Ray Instrumentation for Astronomy III, O. H. Siegmund, ed. Proc. SPIE1743, 408–421 (1992).

Ramaty, R.

B. R. Dennis, R. P. Lin, R. C. Canfield, C. J. Crannell, A. G. Emslie, G. D. Holman, H. S. Hudson, G. J. Hurford, J. C. Ling, N. W. Madden, R. Ramaty, “The High Energy Solar Spectroscopic Imager—HESSI,” in Missions to the Sun, D. M. Rust, ed., Proc. SPIE2804 (1996).

Rouaix, G.

M. Pelling, P. T. Feffer, K. Hurley, S. R. Kane, R. P. Lin, S. McBride, J. H. Primbsch, D. M. Smith, K. Youseffi, G. Zimmer, F. Cotin, J. M. Lavigne, G. Rouaix, S. Slassi, G. Vedrenne, R. Pehl, C. Cork, P. Luke, N. Madden, D. Malone, “A high resolution gamma-ray and hard X-ray spectrometer (HIREGS) for long duration balloon flights,” in EUV, X-Ray, and Gamma-Ray Instrumentation for Astronomy III, O. H. Siegmund, ed. Proc. SPIE1743, 408–421 (1992).

Sakao, T.

H. S. Hudson, K. T. Strong, B. R. Dennis, D. Zous, M. Inda, T. Kosugi, T. Sakao, “Impulsive behavior in solar soft x-radiation,” Astrophys. J. 422, L25–L27 (1994).
[Crossref]

T. Sakao, “Characteristics of solar flare hard x-ray sources as revealed with the hard x-ray telescope aboard the Yohkoh satellite,” Ph.D. dissertation (University of Tokyo, Tokyo, Japan1994).

Schmahl, E. J.

C. J. Crannell, B. R. Dennis, L. E. Orwig, E. J. Schmahl, F. L. Lang, R. Starr, J. P. Norris, M. E. Greene, G. J. Hurford, W. N. Johnson, K. S. Wood, “A balloon-borne payload for imaging hard x-rays and gamma rays from solar flares,” in Proceedings of the AIAA International Balloon Technology Conference (American Institute of Aeronautics and Astronautics, New York, 1991).

Schnopper, H. W.

H. W. Schnopper, R. I. Thompson, S. Watt, “Predicted performance of a rotating modulation collimator for locating celestial hard x-ray sources,” Space Sci. Rev. 8, 534–542 (1968).
[Crossref]

Slassi, S.

M. Pelling, P. T. Feffer, K. Hurley, S. R. Kane, R. P. Lin, S. McBride, J. H. Primbsch, D. M. Smith, K. Youseffi, G. Zimmer, F. Cotin, J. M. Lavigne, G. Rouaix, S. Slassi, G. Vedrenne, R. Pehl, C. Cork, P. Luke, N. Madden, D. Malone, “A high resolution gamma-ray and hard X-ray spectrometer (HIREGS) for long duration balloon flights,” in EUV, X-Ray, and Gamma-Ray Instrumentation for Astronomy III, O. H. Siegmund, ed. Proc. SPIE1743, 408–421 (1992).

Smith, D. M.

M. Pelling, P. T. Feffer, K. Hurley, S. R. Kane, R. P. Lin, S. McBride, J. H. Primbsch, D. M. Smith, K. Youseffi, G. Zimmer, F. Cotin, J. M. Lavigne, G. Rouaix, S. Slassi, G. Vedrenne, R. Pehl, C. Cork, P. Luke, N. Madden, D. Malone, “A high resolution gamma-ray and hard X-ray spectrometer (HIREGS) for long duration balloon flights,” in EUV, X-Ray, and Gamma-Ray Instrumentation for Astronomy III, O. H. Siegmund, ed. Proc. SPIE1743, 408–421 (1992).

Starr, R.

C. J. Crannell, B. R. Dennis, L. E. Orwig, E. J. Schmahl, F. L. Lang, R. Starr, J. P. Norris, M. E. Greene, G. J. Hurford, W. N. Johnson, K. S. Wood, “A balloon-borne payload for imaging hard x-rays and gamma rays from solar flares,” in Proceedings of the AIAA International Balloon Technology Conference (American Institute of Aeronautics and Astronautics, New York, 1991).

Strong, K. T.

H. S. Hudson, K. T. Strong, B. R. Dennis, D. Zous, M. Inda, T. Kosugi, T. Sakao, “Impulsive behavior in solar soft x-radiation,” Astrophys. J. 422, L25–L27 (1994).
[Crossref]

Thompson, R. I.

H. W. Schnopper, R. I. Thompson, S. Watt, “Predicted performance of a rotating modulation collimator for locating celestial hard x-ray sources,” Space Sci. Rev. 8, 534–542 (1968).
[Crossref]

Tsuneta, S.

S. Masuda, T. Kosugi, H. Hara, S. Tsuneta, Y. Ogawara, “Loop-top impulsive hard x-ray source of a solar flare as evidence of magnetic reconnection,” Nature (London) 371, 495–497 (1994).
[Crossref]

S. Tsuneta, “Image processing software system of the x-ray telescope aboard the Hinotori spacecraft,” Ann. Tokyo Astron. Observatory 20, 1–50 (1984).

Vedrenne, G.

M. Pelling, P. T. Feffer, K. Hurley, S. R. Kane, R. P. Lin, S. McBride, J. H. Primbsch, D. M. Smith, K. Youseffi, G. Zimmer, F. Cotin, J. M. Lavigne, G. Rouaix, S. Slassi, G. Vedrenne, R. Pehl, C. Cork, P. Luke, N. Madden, D. Malone, “A high resolution gamma-ray and hard X-ray spectrometer (HIREGS) for long duration balloon flights,” in EUV, X-Ray, and Gamma-Ray Instrumentation for Astronomy III, O. H. Siegmund, ed. Proc. SPIE1743, 408–421 (1992).

Watt, S.

H. W. Schnopper, R. I. Thompson, S. Watt, “Predicted performance of a rotating modulation collimator for locating celestial hard x-ray sources,” Space Sci. Rev. 8, 534–542 (1968).
[Crossref]

Wernecke, S. J.

S. J. Wernecke, L. R. D’Addario, “Maximum entropy image reconstruction,” IEEE Trans. Comput. C 26, 351–364 (1977).
[Crossref]

Wood, K. S.

C. J. Crannell, B. R. Dennis, L. E. Orwig, E. J. Schmahl, F. L. Lang, R. Starr, J. P. Norris, M. E. Greene, G. J. Hurford, W. N. Johnson, K. S. Wood, “A balloon-borne payload for imaging hard x-rays and gamma rays from solar flares,” in Proceedings of the AIAA International Balloon Technology Conference (American Institute of Aeronautics and Astronautics, New York, 1991).

Youseffi, K.

M. Pelling, P. T. Feffer, K. Hurley, S. R. Kane, R. P. Lin, S. McBride, J. H. Primbsch, D. M. Smith, K. Youseffi, G. Zimmer, F. Cotin, J. M. Lavigne, G. Rouaix, S. Slassi, G. Vedrenne, R. Pehl, C. Cork, P. Luke, N. Madden, D. Malone, “A high resolution gamma-ray and hard X-ray spectrometer (HIREGS) for long duration balloon flights,” in EUV, X-Ray, and Gamma-Ray Instrumentation for Astronomy III, O. H. Siegmund, ed. Proc. SPIE1743, 408–421 (1992).

Zimmer, G.

M. Pelling, P. T. Feffer, K. Hurley, S. R. Kane, R. P. Lin, S. McBride, J. H. Primbsch, D. M. Smith, K. Youseffi, G. Zimmer, F. Cotin, J. M. Lavigne, G. Rouaix, S. Slassi, G. Vedrenne, R. Pehl, C. Cork, P. Luke, N. Madden, D. Malone, “A high resolution gamma-ray and hard X-ray spectrometer (HIREGS) for long duration balloon flights,” in EUV, X-Ray, and Gamma-Ray Instrumentation for Astronomy III, O. H. Siegmund, ed. Proc. SPIE1743, 408–421 (1992).

Zous, D.

H. S. Hudson, K. T. Strong, B. R. Dennis, D. Zous, M. Inda, T. Kosugi, T. Sakao, “Impulsive behavior in solar soft x-radiation,” Astrophys. J. 422, L25–L27 (1994).
[Crossref]

Ann. Tokyo Astron. Observatory (1)

S. Tsuneta, “Image processing software system of the x-ray telescope aboard the Hinotori spacecraft,” Ann. Tokyo Astron. Observatory 20, 1–50 (1984).

Astron. Astrophys. Suppl. Ser. (1)

J. A. Högbom, “Aperture synthesis with a non-regular distribution of interferometer baselines,” Astron. Astrophys. Suppl. Ser. 15, 417–426 (1974).

Astrophys. J. (1)

H. S. Hudson, K. T. Strong, B. R. Dennis, D. Zous, M. Inda, T. Kosugi, T. Sakao, “Impulsive behavior in solar soft x-radiation,” Astrophys. J. 422, L25–L27 (1994).
[Crossref]

Comput. Graphics Image Process. (1)

B. Frieden, “Statistical models for the image restoration problem,” Comput. Graphics Image Process. 12, 40–59 (1980).
[Crossref]

IEEE Trans. Comput. C (1)

S. J. Wernecke, L. R. D’Addario, “Maximum entropy image reconstruction,” IEEE Trans. Comput. C 26, 351–364 (1977).
[Crossref]

J. Opt. Soc. Am. (1)

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

Nature (London) (1)

S. Masuda, T. Kosugi, H. Hara, S. Tsuneta, Y. Ogawara, “Loop-top impulsive hard x-ray source of a solar flare as evidence of magnetic reconnection,” Nature (London) 371, 495–497 (1994).
[Crossref]

Nucl. Instrum. Methods A (1)

M. J. Murphy, “The virtues of positive-definite reconstruction of x-ray and gamma-ray images,” Nucl. Instrum. Methods A 290, 551–558 (1990).
[Crossref]

Proc. Phys. Soc. London Sect. B (1)

J. M. Cowley, A. F. Moodie, “Fourier images. I. The point source,” Proc. Phys. Soc. London Sect. B 70, 486–496 (1957).
[Crossref]

Space Sci. Rev. (1)

H. W. Schnopper, R. I. Thompson, S. Watt, “Predicted performance of a rotating modulation collimator for locating celestial hard x-ray sources,” Space Sci. Rev. 8, 534–542 (1968).
[Crossref]

Other (8)

J. M. Cowley, Diffraction Physics (Elsevier, New York, 1981).

J. R. Kilner, G. H. Nakano, “Design studies for x-ray and gamma ray rotation modulation collimators,” in EUV, X-ray and Gamma-Ray Instrumentation for Astronomy and Atomic Physics, C. J. Hailey, ed., Proc. SPIE1159, 27–33 (1989).

T. Sakao, “Characteristics of solar flare hard x-ray sources as revealed with the hard x-ray telescope aboard the Yohkoh satellite,” Ph.D. dissertation (University of Tokyo, Tokyo, Japan1994).

B. R. Dennis, R. P. Lin, R. C. Canfield, C. J. Crannell, A. G. Emslie, G. D. Holman, H. S. Hudson, G. J. Hurford, J. C. Ling, N. W. Madden, R. Ramaty, “The High Energy Solar Spectroscopic Imager—HESSI,” in Missions to the Sun, D. M. Rust, ed., Proc. SPIE2804 (1996).

C. J. Crannell, B. R. Dennis, L. E. Orwig, E. J. Schmahl, F. L. Lang, R. Starr, J. P. Norris, M. E. Greene, G. J. Hurford, W. N. Johnson, K. S. Wood, “A balloon-borne payload for imaging hard x-rays and gamma rays from solar flares,” in Proceedings of the AIAA International Balloon Technology Conference (American Institute of Aeronautics and Astronautics, New York, 1991).

M. Pelling, P. T. Feffer, K. Hurley, S. R. Kane, R. P. Lin, S. McBride, J. H. Primbsch, D. M. Smith, K. Youseffi, G. Zimmer, F. Cotin, J. M. Lavigne, G. Rouaix, S. Slassi, G. Vedrenne, R. Pehl, C. Cork, P. Luke, N. Madden, D. Malone, “A high resolution gamma-ray and hard X-ray spectrometer (HIREGS) for long duration balloon flights,” in EUV, X-Ray, and Gamma-Ray Instrumentation for Astronomy III, O. H. Siegmund, ed. Proc. SPIE1743, 408–421 (1992).

K. Makishima, “Instrumental performance of the SXT collimator,” in Hinotori Symposium on Solar Flares (Institute of Space and Astronautical Science, Tokyo, 1982), pp. 120–129.

K. Makishima, Y. Ogawara, S. Miyamoto, N. Muranaka, M. Oda, “Alignment of modulation collimator grids using optical diffraction patterns,” ISAS Research Note 29 (Institute of Space and Astronautical Science, Megura, Japan, 1977).

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Figures (13)

Fig. 1
Fig. 1

Schematic of the RMC’s in the HEIDI telescope. The grid pairs were coaligned, forming RMC’s within the telescope assembly, which was designed to point within a small fraction of a degree (labeled θ) of an x-ray source and rotated about the axis of symmetry (labeled Z). An x-ray detector was situated behind the rear grid of each RMC and measured the x-ray flux from the source as modulated by the rotating grid pair. The x-ray detector does not rotate with the grid pair. HV, high voltage.

Fig. 2
Fig. 2

Theoretical modulated signal that would be measured in one complete rotation about the RMC’s axis of symmetry, observing an off-axis point source with an RMC that has a grid pitch of 1.25 mm and a grid separation of 5.2 m. The maximum count rate in the plot was normalized to unity and offset to zero mean.

Fig. 3
Fig. 3

(a) Computed map for a steady flux of x rays from a hypothetical point source incident on an ideal 25-in. (64-cm) collimator for one full rotation of the collimator about its axis. The collimator was assumed to be ideal and the (steady) source was taken to have the position used to make the modulation pattern of Fig. 2. We made a vertical shift in the map amplitudes to force the map to have zero mean. Except for the vertical shift, this figure and Fig. 2 are mutually compatible, and given the assumption of a steady source, either one could be used to derive the other. (b) Image of (a) postprocessed with the standard clean algorithm used in radio astronomy to remove the sidelobes.

Fig. 4
Fig. 4

Schematic of the RMC and electronics showing the experimental setup used in our calibration measurements. The beam from the He–Ne laser was directed through the eyepiece of a small Cassegrain telescope that expanded the beam. The plane wave that was generated intersected the front grid of the RMC at an angle of 12 ft. (3.6 m) relative to its rotation axis, labeled Z. The light intensity that passes through both the front and the rear grids was measured with a PMT. The processing of the PMT signal is discussed in Section 3.

Fig. 5
Fig. 5

For a point source located off the rotation axis, labeled Z, of the RMC’s, the rear grid appears fixed and the front grid undergoes simple harmonic motion in both the x and the y directions centered around the Z axis. As discussed in the text, the motion in the x direction produces modulation that can be analyzed in one dimension.

Fig. 6
Fig. 6

Plot of the diffraction factor for the fundamental and the third harmonic versus wavelength for the RMC used in our calibration procedure (with a pitch of 1.25 mm and a grid separation of 5.2 m). FUND, fundamental.

Fig. 7
Fig. 7

Modulation profiles with derived empirical PSF and theoretical PSF: (a) raw data produced by the laser calibration system and (b) regularized data after correction for the variable rate of modulation caused by rotation of the collimator and after a best fit to the source position.

Fig. 8
Fig. 8

(a) Empirical PSF deduced from Fig. 7(b). Compare this with the ideal PSF shown in Fig. 3(a). The rounded shape of the central peak was produced by the reversed sign of the third harmonic, one of the effects of diffraction. (b) Theoretical PSF with the fundamental, second, and third harmonics determined from analytic fits to the curve in Fig. 7(a) and including the diffraction factors. Comparison of this PSF with the empirical PSF in (a) shows only slight differences, mainly attributable to grid irregularities and the neglect of higher harmonics.

Fig. 9
Fig. 9

Apparent positions of the point source for different locations of the mask apertures. The apparent relative position of the point source varies by approximately 2 in. (5 cm) or approximately 10% of the grid pitch of 25 in. (64 cm), depending on the location of the hole in the mask. The position shift was correlated angularly with the position of the mask hole.

Fig. 10
Fig. 10

Histogram of the measured variation in pitch across one of the HEIDI grids. The distribution in grid pitch is of the order of 10% of the mean grid pitch of 1.25 mm.

Fig. 11
Fig. 11

Diffraction factor for the fundamental and third harmonics for a range of pitches near 1.250 mm. This plot demonstrates the strong dependence of the diffraction factor on grid pitch.

Fig. 12
Fig. 12

Computed harmonics as a function of hypothetical reduction in slit width by shadowing, including the effects of diffraction and pitch irregularities. The amplitudes of the harmonics inferred from the regularization of the laser modulation signal are shown as horizontal bands bordered by dashed lines. The curves labeled 102 and 51 are for illumination of the full grid (102 slats) and for the inner half, respectively. The curves labeled 51N are for 51 illuminated slats with the addition of 0.020-mm noise to the aperture transmission function. The dark curves labeled 51N*S include the effect of bandwidth smearing, possibly caused by twists and grid curvature, totaling 25% of the pitch. The left vertical line shows the maximum reduction of slit width caused by the 12 ft (3.6 m) offset of the source, and the right vertical line shows the inferred effective reduction of slit width, where the computed harmonics (heavy curves) agree with the observations (dashed lines).

Fig. 13
Fig. 13

Schematic of proposed improvements to laser calibration setup. The collimator we propose to use consists of an objective lens, a pinhole for spatial filtering, and a large parabolic mirror instead of an inverted Cassegrain telescope. The parallel, large-area beam reflected from the parabolic mirror will then be reflected onto the front grid of the RMC with a large-area planar mirror that is computer controlled. CAL, calibration.

Tables (1)

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Table 1 Effect of Thickness on Harmonic Content

Equations (18)

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x = ( R / p ) * sin ( θ θ 0 ) ,
T ( ϕ ) = [ s t * tan ( ϕ ) ] / p ,
D = cos ( π L h 2 λ / p 2 ) ,
Φ = A * cos [ 2 π ( R / p ) sin ( θ θ 0 ) Δ ] ,
q ( x ) = h F h cos ( 2 π h x / p ) , h = 1 , 3 , 5 , ,
q ( x ) = n F n exp ( 2 i π n x / d ) n = M , M + 1 , , M ,
ψ n ( x ) = exp ( i k L ) F n exp ( i π L λ n 2 / d 2 ) exp ( 2 i π n x / d ) ,
I ( x ) = F 0 2 + n 0 cos ( π L λ n 2 / d 2 ) F n exp ( 2 i π n x / d ) + n 0 exp ( i π n x / d ) m 0 G m G m + n * ,
Φ ( δ ) = d / 2 d / 2 d x q ˆ ( x ) I ( x ) .
Φ ( δ ) = d m l F ˆ l m F m F l * exp ( i π L λ m 2 l 2 / d 2 ) × exp ( 2 i π m l δ / d ) .
Φ h ideal ( δ ) = 4 d | F ˆ h F h * | F 0 cos ( 2 π h δ / p ) cos ( π L λ h 2 / p 2 ) .
F n = ( 1 / 2 π i ) k = 1 N [ exp ( 2 i π n a k / d ) exp ( 2 i π n b k / d ) ] / n , n 0 ,
F 0 = ( 1 / d ) q ( x ) d x = effective transparency ,
Φ ( δ ) = n Φ n exp ( 2 i π n δ / d ) .
Φ n = d F ˆ n * m G m G m + n * ,
G m = F m exp ( i π L λ m 2 / d 2 ) .
Φ n = d F ˆ n * m F m F m + n * = d F ˆ n * F n , λ d 2 / L ,
Φ n = d F ˆ n * m = F m F m + n * cos ( n + 2 m n π L λ / d 2 ) .

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