Abstract

In a conventional grating spectrograph consisting of a single entrance slit, a grating, and a multichannel (imaging) detector, considerable light throughput advantage can be realized by replacement of the single entrance slit with a mask. This replacement can yield a signal-to-noise ratio increase because of increased light collection over an extended area of the mask when compared with a single slit. The mask produces a spectrum on the detector, which is the convolution of the mask pattern and the spectral distribution of the light source. To retrieve the spectrum, the spectrum has to be inverted. In special cases in which emission spectra are superimposed on weak backgrounds, the signal-to-noise advantage is preserved through the inversion process. Thus this technique is valuable in the observation of light sources that are produced by atomic or molecular emissions such as aurora, airglow, some interstellar emission, or laboratory spectra. Considerable signal-to-noise advantages can also be realized when the background noise of the imaging detector is not negligible. The spectral mixing of the light from the mask on the detector causes high photon fluxes on the detector, which tend to swamp the detector noise. This is a particularly important advantage in the application of CCD’s as detectors because they can have significant background noise. The technique was demonstrated by computer simulations and laboratory tests.

© 1993 Optical Society of America

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References

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    [CrossRef] [PubMed]
  2. M. J. E. Golay, “Static multislit spectrometry and its application to the panoramic display of infrared spectra,” J. Opt. Soc. Am. 41, 468–472 (1951).
    [CrossRef] [PubMed]
  3. A. Girard, “Nouveaux Dispositifs de Spectroscopie à Grande Luminosité,” Opt. Acta 7, 81–97 (1960).
    [CrossRef]
  4. A. Girard, “Spectromètre à grilles,” Appl. Opt. 2, 79–87 (1963).
    [CrossRef]
  5. A. Girard, “Spectromètre à grilles,” J. Phys. (Paris) 24, 139–141 (1963).
  6. L. Mertz, N. O. Young, J. Armitage, “Mock interferometry,” in Proceedings of the Conference on Optical Instruments and Techniques, K. J. Habell, ed. (Wiley, New York, 1963).
  7. L. Mertz, Transformations in Optics (Wiley, New York, 1965).
  8. M. Harwit, N. J. A. Sloane, Hadamard Transform Optics (Academic, New York, 1979).
  9. G. K. Skinner, “X-ray imaging with coded masks,” Sci. Am. 295(2), 84–89 (1988).
    [CrossRef]
  10. S. R. Gottesman, E. E. Fenimore, “New family of binary arrays for coded aperture imaging,” Appl. Opt. 28, 4344–4352 (1989).
    [CrossRef] [PubMed]
  11. J. A. Decker, “Experimental realization of the multiplex advantage with a Hadamard-transform spectrometer,” Appl. Opt. 10, 510–514 (1971).
    [CrossRef] [PubMed]
  12. S. Miyamoto, “Hadamard transform x-ray telescope,” Space Sci. Instrum. 3, 473–481 (1977).
  13. C. L. Lawson, R. J. Hanson, Solving Least Squares Problems (Prentice-Hall, Englewood Cliffs, N.J., 1974).
  14. G. Strang, Linear Algebra and Its Applications (Academic, Orlando, Fla., 1980).
  15. M. Zukic, D. G. Torr, “Multiple reflectors as narrow-band and broadband vacuum ultraviolet filters,” Appl. Opt. 31, 1588–1596 (1992).
    [CrossRef] [PubMed]
  16. S. C. Chakrabarti, “Extreme and far ultraviolet emissions from the Polar Cap,” J. Geophys. Res. 91, 8065–8072 (1986).
    [CrossRef]
  17. R. L. Gattinger, National Research Council of Canada, Ottawa, Ontario K1A 0R6, Canada (personal communication, 1989).

1992 (1)

1989 (1)

1988 (1)

G. K. Skinner, “X-ray imaging with coded masks,” Sci. Am. 295(2), 84–89 (1988).
[CrossRef]

1986 (1)

S. C. Chakrabarti, “Extreme and far ultraviolet emissions from the Polar Cap,” J. Geophys. Res. 91, 8065–8072 (1986).
[CrossRef]

1977 (1)

S. Miyamoto, “Hadamard transform x-ray telescope,” Space Sci. Instrum. 3, 473–481 (1977).

1971 (1)

1963 (2)

A. Girard, “Spectromètre à grilles,” Appl. Opt. 2, 79–87 (1963).
[CrossRef]

A. Girard, “Spectromètre à grilles,” J. Phys. (Paris) 24, 139–141 (1963).

1960 (1)

A. Girard, “Nouveaux Dispositifs de Spectroscopie à Grande Luminosité,” Opt. Acta 7, 81–97 (1960).
[CrossRef]

1951 (1)

1949 (1)

Armitage, J.

L. Mertz, N. O. Young, J. Armitage, “Mock interferometry,” in Proceedings of the Conference on Optical Instruments and Techniques, K. J. Habell, ed. (Wiley, New York, 1963).

Chakrabarti, S. C.

S. C. Chakrabarti, “Extreme and far ultraviolet emissions from the Polar Cap,” J. Geophys. Res. 91, 8065–8072 (1986).
[CrossRef]

Decker, J. A.

Fenimore, E. E.

Gattinger, R. L.

R. L. Gattinger, National Research Council of Canada, Ottawa, Ontario K1A 0R6, Canada (personal communication, 1989).

Girard, A.

A. Girard, “Spectromètre à grilles,” Appl. Opt. 2, 79–87 (1963).
[CrossRef]

A. Girard, “Spectromètre à grilles,” J. Phys. (Paris) 24, 139–141 (1963).

A. Girard, “Nouveaux Dispositifs de Spectroscopie à Grande Luminosité,” Opt. Acta 7, 81–97 (1960).
[CrossRef]

Golay, M. J. E.

Gottesman, S. R.

Hanson, R. J.

C. L. Lawson, R. J. Hanson, Solving Least Squares Problems (Prentice-Hall, Englewood Cliffs, N.J., 1974).

Harwit, M.

M. Harwit, N. J. A. Sloane, Hadamard Transform Optics (Academic, New York, 1979).

Lawson, C. L.

C. L. Lawson, R. J. Hanson, Solving Least Squares Problems (Prentice-Hall, Englewood Cliffs, N.J., 1974).

Mertz, L.

L. Mertz, Transformations in Optics (Wiley, New York, 1965).

L. Mertz, N. O. Young, J. Armitage, “Mock interferometry,” in Proceedings of the Conference on Optical Instruments and Techniques, K. J. Habell, ed. (Wiley, New York, 1963).

Miyamoto, S.

S. Miyamoto, “Hadamard transform x-ray telescope,” Space Sci. Instrum. 3, 473–481 (1977).

Skinner, G. K.

G. K. Skinner, “X-ray imaging with coded masks,” Sci. Am. 295(2), 84–89 (1988).
[CrossRef]

Sloane, N. J. A.

M. Harwit, N. J. A. Sloane, Hadamard Transform Optics (Academic, New York, 1979).

Strang, G.

G. Strang, Linear Algebra and Its Applications (Academic, Orlando, Fla., 1980).

Torr, D. G.

Young, N. O.

L. Mertz, N. O. Young, J. Armitage, “Mock interferometry,” in Proceedings of the Conference on Optical Instruments and Techniques, K. J. Habell, ed. (Wiley, New York, 1963).

Zukic, M.

Appl. Opt. (4)

J. Geophys. Res. (1)

S. C. Chakrabarti, “Extreme and far ultraviolet emissions from the Polar Cap,” J. Geophys. Res. 91, 8065–8072 (1986).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Phys. (Paris) (1)

A. Girard, “Spectromètre à grilles,” J. Phys. (Paris) 24, 139–141 (1963).

Opt. Acta (1)

A. Girard, “Nouveaux Dispositifs de Spectroscopie à Grande Luminosité,” Opt. Acta 7, 81–97 (1960).
[CrossRef]

Sci. Am. (1)

G. K. Skinner, “X-ray imaging with coded masks,” Sci. Am. 295(2), 84–89 (1988).
[CrossRef]

Space Sci. Instrum. (1)

S. Miyamoto, “Hadamard transform x-ray telescope,” Space Sci. Instrum. 3, 473–481 (1977).

Other (6)

C. L. Lawson, R. J. Hanson, Solving Least Squares Problems (Prentice-Hall, Englewood Cliffs, N.J., 1974).

G. Strang, Linear Algebra and Its Applications (Academic, Orlando, Fla., 1980).

L. Mertz, N. O. Young, J. Armitage, “Mock interferometry,” in Proceedings of the Conference on Optical Instruments and Techniques, K. J. Habell, ed. (Wiley, New York, 1963).

L. Mertz, Transformations in Optics (Wiley, New York, 1965).

M. Harwit, N. J. A. Sloane, Hadamard Transform Optics (Academic, New York, 1979).

R. L. Gattinger, National Research Council of Canada, Ottawa, Ontario K1A 0R6, Canada (personal communication, 1989).

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Figures (12)

Fig. 1
Fig. 1

Schematic of a single-slit spectrometer and a Hadamard spectrometer. In the single-slit spectrometer the slit’s image is shown at two different wavelengths to illustrate dispersion. The Hadamard spectrometer is distinguished by a widened and coded entrance aperture; each transparent slit produces the same spectrum as a single slit but offset in position.

Fig. 2
Fig. 2

Seven-element (1 1 1 0 1 0 0) coded aperture spectrograph. Contributions to detector element 5, for example, are indicated by arrows representing spectral intensities X1, X3, X4, and X5, each diffracted by a differing amount. (X1 is diffracted the most leftward.) The array shows how each spectral intensity contributes to the detector elements according to the mask pattern.

Fig. 3
Fig. 3

Simulation of spectrometer performance in the presence of Poisson photon-counting noise. The error bars are the average and standard deviation of 50 statistical trials: Results are shown for a, a single slit; b, a 15-element coded aperture; c, a 63-element coded aperture.

Fig. 4
Fig. 4

Hadamard mask. Transmissive elements are shown in black and opaque elements in white.

Fig. 5
Fig. 5

Hadamard mask image in zero order (without grating) displayed on the CCD.

Fig. 6
Fig. 6

Deconvolution of laboratory Hg spectrum: a, the single-slit spectrum with background subtracted; b, the same graph but for the 63-element coded aperture; c, the deconvolution of b; d, the single-slit Hg spectrum with the background subtracted as measured at high intensity in a previous experiment. The three lines in d are, left to right, 5770, 5461, and 4358 Å. different position and have different spacing than in c because of differences in experimental setup.

Fig. 7
Fig. 7

Spectral image of the Hadamard mask through the spectrometer with the Ne discharge lamp as the source.

Fig. 8
Fig. 8

Same as Fig. 6 but for a laboratory Ne spectrum that spans the spectral range, left to right, at 7439–5853 Å. (Wavelength bin numbers are 50–90 in d.)

Fig. 9
Fig. 9

Chakrabarti16 nightside auroral UV spectrum at 830–1260 Å. The Lyman-α line at 1216 Å is very strong and went off scale on the original plot. For simulation purposes the line has been reduced in strength to a convenient value.

Fig. 10
Fig. 10

Simulation of spectrometer performance in the presence of 10 photons/pixel/slit of detector noise and Poisson photon-counting noise. The solid lines are the same spectrum as in Fig. 9 but as a function of the wavelength bin number. The error bars are the average and standard deviation of 25 statistical trials: a, the input spectrum alone; b, results for a single slit; c, a 15-element coded aperture; d, a 63-element coded aperture.

Fig. 11
Fig. 11

Gattinger17 computer-modeled daytime auroral UV spectrum, 1190–1450 Å.

Fig. 12
Fig. 12

Same as Fig. 10 but with the Gattinger UV spectrum as input.

Equations (17)

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X 1 1 1 1 0 1 0 0 X 2 0 1 1 1 0 1 0 X 3 0 0 1 1 1 0 1 X 4 1 0 0 1 1 1 0 X 5 0 1 0 0 1 1 1 X 6 1 0 1 0 0 1 1 X 7 1 1 0 1 0 0 1 S 1 S 2 S 3 S 4 S 5 S 6 S 7
S j = i = 1 n a i j I i .
X i = 2 n + 1 j = 1 n 2 ( a i j - 0.5 ) S j .
S j = i = 1 n a i j I i + e j .
X i = I i + 2 n + 1 j = 1 n 2 ( a i j - 0.5 ) e j
X i - I i = 2 n + 1 j = 1 n 2 ( a i j - 0.5 ) e j .
E = [ ( X i - I i ) 2 ] 1 / 2 = 2 n + 1 ( j = 1 n e j 2 ¯ ) 1 / 2 ,
E q = 2 n + 1 ( n + 1 2 i = 1 n I i ) 1 / 2 .
E q = 2 n + 1 ( n + 1 2 j = 1 n I c ) 1 / 2 = ( 2 n n + 1 I c ) 1 / 2 .
E q = 2 n + 1 ( n + 1 2 I s ) 1 / 2 = ( 2 n + 1 I s ) 1 / 2 .
E d c = 2 n + 1 ( j = 1 n d c 2 ) 1 / 2 = 2 n n + 1 d c .
η = ψ ,
G ( η - ψ ) ,
g i i = 1 σ i .
ψ = ( ˜ t ˜ ) - 1 ˜ t η ˜ ,
˜ = G ,
η ˜ = G η .

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