Abstract

We propose a new class of aperture-coded spectrometer that is optimized for the spectral characterization of diffuse sources. The instrument achieves high throughput and high spectral resolution by replacing the slit of conventional dispersive spectrometers with a more complicated spatial filter. We develop a general mathematical framework for deriving the required aperture codes and discuss several appealing code families. Experimental results validate the performance of the instrument.

© 2006 Optical Society of America

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References

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  1. D. J. Brady, "Multiplex sensors and the constant radiance theorem," Opt. Lett. 27, 16-18 (2002).
    [CrossRef]
  2. P. Jacquinot, "New developments in interference spectroscopy," Rep. Prog. Phys. 23, 267-312 (1960).
    [CrossRef]
  3. P. B. Fellgett, "The multiplex advantage," Ph.D. dissertation (University of Cambridge, Cambridge, UK, 1951).
  4. M. J. E. Golay, "Multislit spectrometry," J. Opt. Soc. Am. 39, 437-444 (1949).
    [CrossRef] [PubMed]
  5. 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]
  6. A. Girard, "Spectrometre a Grilles," Appl. Opt. 2, 79-87 (1963).
    [CrossRef]
  7. A. S. Hedayat, N. J. A. Sloane, and J. Stufken, Orthogonal Arrays: Theory and Applications (Springer-Verlag, 1999).
  8. R. N. Ibbett, D. Aspinall, and J. F. Grainger, "Real-time multiplexing of dispersed spectra in any wavelength region," Appl. Opt. 7, 1089-1093 (1968).
    [CrossRef] [PubMed]
  9. J. A. Decker and M. O. Harwitt, "Sequential encoding with multislit spectrometers," Appl. Opt. 7, 2205-2209 (1968).
    [CrossRef] [PubMed]
  10. J. A. Decker, "Experimental realization of the multiplex advantage with a Hadamard-transform spectrometer," Appl. Opt. 10, 510-514 (1971).
    [CrossRef] [PubMed]
  11. P. Hansen and J. Strong, "High resolution Hadamard transform spectrometer," Appl. Opt. 11, 502-506 (1972).
    [CrossRef] [PubMed]
  12. P. G. Phillips and D. A. Briotta, "Hadamard-transform spectrometry of the atmopheres of Earth and Jupiter," Appl. Opt. 13, 2233-2235 (1974).
    [PubMed]
  13. R. D. Swift, R. B. Wattson, J. A. Decker, R. Paganetti, and M. O. Harwitt, "Hadamard transform imager and imaging spectrometer," Appl. Opt. 15, 1595-1609 (1976).
    [CrossRef] [PubMed]
  14. M. O. Harwitt and N. J. A. Sloane, Hadamard Transform Optics (Academic, 1979).
  15. S. B. Mende, E. S. Claflin, R. L. Rairden, and G. R. Swenson, "Hadamard spectroscopy with a two-dimensional detecting array," Appl. Opt. 32, 7095-7105 (1993).
    [CrossRef] [PubMed]
  16. R. Riesenberg and U. Dillner, "HADAMARD imaging spectrometer with micro slit matrix," in Imaging Spectrometry V, M.R. Descour and S.S. Shen, eds., Proc. SPIE 3753, 203-213 (1999).
    [CrossRef]
  17. R. A. de Verse, R. M. Hammaker, and W. G. Fateley, "Realization of the Hadamard multiplex advantage using a programmable optical mask in a dispersive flat-field near-infrared spectrometer," Appl. Spectrosc. 54, 1751-1758 (2000).
    [CrossRef]
  18. R. Riesenberg, G. Nitzsche, and W. Voigt, "HADAMARD encoding and other optical multiplexing," VDI Ber. 1694, 345-350 (2002).
  19. P. Jacquinot, "How the search for a throughput advantage led to Fourier transform spectroscopy," Infrared Phys. 2-3, 99-101 (1984).
    [CrossRef]
  20. G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists, 6th ed. (Academic, 2005).
  21. D. J. Schroeder, Astronomical Optics (Academic, 1987).

2002

R. Riesenberg, G. Nitzsche, and W. Voigt, "HADAMARD encoding and other optical multiplexing," VDI Ber. 1694, 345-350 (2002).

D. J. Brady, "Multiplex sensors and the constant radiance theorem," Opt. Lett. 27, 16-18 (2002).
[CrossRef]

2000

1999

R. Riesenberg and U. Dillner, "HADAMARD imaging spectrometer with micro slit matrix," in Imaging Spectrometry V, M.R. Descour and S.S. Shen, eds., Proc. SPIE 3753, 203-213 (1999).
[CrossRef]

1993

1984

P. Jacquinot, "How the search for a throughput advantage led to Fourier transform spectroscopy," Infrared Phys. 2-3, 99-101 (1984).
[CrossRef]

1976

1974

P. G. Phillips and D. A. Briotta, "Hadamard-transform spectrometry of the atmopheres of Earth and Jupiter," Appl. Opt. 13, 2233-2235 (1974).
[PubMed]

1972

1971

1968

1963

1960

P. Jacquinot, "New developments in interference spectroscopy," Rep. Prog. Phys. 23, 267-312 (1960).
[CrossRef]

1951

1949

Arfken, G. B.

G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists, 6th ed. (Academic, 2005).

Aspinall, D.

Brady, D. J.

Briotta, D. A.

P. G. Phillips and D. A. Briotta, "Hadamard-transform spectrometry of the atmopheres of Earth and Jupiter," Appl. Opt. 13, 2233-2235 (1974).
[PubMed]

Claflin, E. S.

de Verse, R. A.

Decker, J. A.

Dillner, U.

R. Riesenberg and U. Dillner, "HADAMARD imaging spectrometer with micro slit matrix," in Imaging Spectrometry V, M.R. Descour and S.S. Shen, eds., Proc. SPIE 3753, 203-213 (1999).
[CrossRef]

Fateley, W. G.

Fellgett, P. B.

P. B. Fellgett, "The multiplex advantage," Ph.D. dissertation (University of Cambridge, Cambridge, UK, 1951).

Girard, A.

Golay, M. J. E.

Grainger, J. F.

Hammaker, R. M.

Hansen, P.

Harwitt, M. O.

Hedayat, A. S.

A. S. Hedayat, N. J. A. Sloane, and J. Stufken, Orthogonal Arrays: Theory and Applications (Springer-Verlag, 1999).

Ibbett, R. N.

Jacquinot, P.

P. Jacquinot, "How the search for a throughput advantage led to Fourier transform spectroscopy," Infrared Phys. 2-3, 99-101 (1984).
[CrossRef]

P. Jacquinot, "New developments in interference spectroscopy," Rep. Prog. Phys. 23, 267-312 (1960).
[CrossRef]

Mende, S. B.

Nitzsche, G.

R. Riesenberg, G. Nitzsche, and W. Voigt, "HADAMARD encoding and other optical multiplexing," VDI Ber. 1694, 345-350 (2002).

Paganetti, R.

Phillips, P. G.

P. G. Phillips and D. A. Briotta, "Hadamard-transform spectrometry of the atmopheres of Earth and Jupiter," Appl. Opt. 13, 2233-2235 (1974).
[PubMed]

Rairden, R. L.

Riesenberg, R.

R. Riesenberg, G. Nitzsche, and W. Voigt, "HADAMARD encoding and other optical multiplexing," VDI Ber. 1694, 345-350 (2002).

R. Riesenberg and U. Dillner, "HADAMARD imaging spectrometer with micro slit matrix," in Imaging Spectrometry V, M.R. Descour and S.S. Shen, eds., Proc. SPIE 3753, 203-213 (1999).
[CrossRef]

Schroeder, D. J.

D. J. Schroeder, Astronomical Optics (Academic, 1987).

Sloane, N. J. A.

A. S. Hedayat, N. J. A. Sloane, and J. Stufken, Orthogonal Arrays: Theory and Applications (Springer-Verlag, 1999).

M. O. Harwitt and N. J. A. Sloane, Hadamard Transform Optics (Academic, 1979).

Strong, J.

Stufken, J.

A. S. Hedayat, N. J. A. Sloane, and J. Stufken, Orthogonal Arrays: Theory and Applications (Springer-Verlag, 1999).

Swenson, G. R.

Swift, R. D.

Voigt, W.

R. Riesenberg, G. Nitzsche, and W. Voigt, "HADAMARD encoding and other optical multiplexing," VDI Ber. 1694, 345-350 (2002).

Wattson, R. B.

Weber, H. J.

G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists, 6th ed. (Academic, 2005).

Appl. Opt.

Appl. Opt. 13,

P. G. Phillips and D. A. Briotta, "Hadamard-transform spectrometry of the atmopheres of Earth and Jupiter," Appl. Opt. 13, 2233-2235 (1974).
[PubMed]

Appl. Spectrosc.

Infrared Phys.

P. Jacquinot, "How the search for a throughput advantage led to Fourier transform spectroscopy," Infrared Phys. 2-3, 99-101 (1984).
[CrossRef]

J. Opt. Soc. Am.

Opt. Lett.

Proc. SPIE

R. Riesenberg and U. Dillner, "HADAMARD imaging spectrometer with micro slit matrix," in Imaging Spectrometry V, M.R. Descour and S.S. Shen, eds., Proc. SPIE 3753, 203-213 (1999).
[CrossRef]

Rep. Prog. Phys.

P. Jacquinot, "New developments in interference spectroscopy," Rep. Prog. Phys. 23, 267-312 (1960).
[CrossRef]

VDI Ber. 1694,

R. Riesenberg, G. Nitzsche, and W. Voigt, "HADAMARD encoding and other optical multiplexing," VDI Ber. 1694, 345-350 (2002).

Other

G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists, 6th ed. (Academic, 2005).

D. J. Schroeder, Astronomical Optics (Academic, 1987).

P. B. Fellgett, "The multiplex advantage," Ph.D. dissertation (University of Cambridge, Cambridge, UK, 1951).

A. S. Hedayat, N. J. A. Sloane, and J. Stufken, Orthogonal Arrays: Theory and Applications (Springer-Verlag, 1999).

M. O. Harwitt and N. J. A. Sloane, Hadamard Transform Optics (Academic, 1979).

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Figures (12)

Fig. 1
Fig. 1

Schematic of a traditional slit-based spectrometer. Internal optical elements form an image of the slit at wavelength-dependent locations on the detector plane. Spectral resolution comes from the capability for spatially distinguishing these different locations. As such, the width of a spectral channel is proportional to the width of the input slit.

Fig. 2
Fig. 2

Definition of the coordinate systems used throughout the paper.

Fig. 3
Fig. 3

Aperture pattern for an independent column code based on harmonic functions. The codes were chosen such that the transmission has physically realizable values in the interval [0, 1]. Note that the pattern is continuous vertically but discrete horizontally.

Fig. 4
Fig. 4

Aperture pattern for an independent column code based on Legendre polynomials. The codes were chosen such that the transmission has physically realizable values in the interval [0, 1]. Note that the pattern is continuous vertically but discrete horizontally.

Fig. 5
Fig. 5

Aperture pattern for an independent column code based on a Hadamard S matrix. The codes were chosen such that the transmission has physically realizable values in the interval [0, 1]. Note that the pattern is discrete both horizontally and vertically.

Fig. 6
Fig. 6

Aperture pattern for an orthogonal column code (in conjunction with processing of the measured intensity) based on a row-doubled Hadamard matrix. The codes were chosen such that the transmission has physically realizable values in the interval [0, 1]. Note that the pattern is discrete both horizontally and vertically.

Fig. 7
Fig. 7

Raw intensity image captured at the focal plane. The smile distortion is clearly visible. The spectral source has only sharp spectral lines, so the image contains only a few, crisp images of the mask pattern.

Fig. 8
Fig. 8

Corrected intensity image after smile distortion was removed via software processing. The leftmost edge of the sharp mask image in Fig. 7 was fit to a parabola to determine the amount of shift to be applied to each row of the image.

Fig. 9
Fig. 9

Comparison between the reconstructed spectrum from a mask based on H ^ 40 and one based on a slit aperture. The mask aperture clearly captures significantly more light, while maintaining an equivalent spectral resolution.

Fig. 10
Fig. 10

Comparison of reconstructed spectra from row-doubled Hadamard masks of various orders (n = 40, 32, 24, 16, 12). The system throughput increases as the mask order increases, as expected. The codes increase the system throughput without affecting spectral resolution.

Fig. 11
Fig. 11

Throughput gain achieved by order-N masks compared to slits of equivalent height. Theoretically, the gain should scale as N∕2. We observe approximately N∕4. For reasons discussed in the text, we attribute this discrepancy to the MTF of the optical system. Note that even with this discrepancy, a mask of moderate complexity is capable of increasing throughput by an order of magnitude without sacrificing spectral resolution.

Fig. 12
Fig. 12

Comparison of a small spectral peak as reconstructed by a mask based on H ^ 40 and a slit. The SNR gain with the mask is consistent with the measured throughput gain.

Equations (109)

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( Δ λ Δ x )
G A Ω .
I ( x , y ) = d λ d x d y H ( x , x , y , y ; λ ) T ( x , y ) × S ( x , y ; λ ) .
H ( x , x , y , y ; λ )
T ( x , y )
S ( x , y ; λ )
( x , y )
H ( x , y ; λ ) = δ ( y y ) δ { x [ x + α ( λ λ c ) ] }
λ c
x = 0
I ( x , y ) = d x T ( x , y ) S ( x , y ; x x α + λ c ) .
T ( x , y ) = δ ( x )
I ( x , y ) = S ( 0 , y ; λ c x α ) .
S mean ( λ ) d x d y S ( x , y ; λ ) .
T ˜ ( x , y )
y
E ( x , x ) = y min y max d y T ˜ ( x , y ) I ( x , y ) = y min y max d y d x T ˜ ( x , y ) T ( x , y ) S ( x , y ; x x α + λ c ) .
S ( x , y ; λ )
y
S ( x , y ; λ ) I ( y ) S ( x ; λ ) .
E ( x , x ) y min y max d y d x T ˜ ( x , y ) T ( x , y ) I ( y ) S ( x ; x x α + λ c ) .
T ( x , y )
T ˜ ( x , y )
y min y max d y T ˜ ( x , y ) T ( x , y ) I ( y ) = β δ ( x x ) ,
E ( x , x ) β d x δ ( x x ) S ( x ; x x α + λ c ) β S ( x ; x x α + λ c ) .
x
E ( x , x )
S mean ( λ )
x = λ α + x
S mean ( λ c λ ) d x d x δ [ x ( λ α + x ) ] E ( x , x )
d x S ( x ; λ c λ ) .
y min y max d y T ˜ x ( y ) T x ( y ) I ( y ) = β δ ( x x ) ,
I ( y )
T ˜
T ˜
x″
δ ( x x )
δ x , x
x
I ( x , y ) = d x T ( x , y ) S ( x x α ) .
x
| T x
{ | T x }
x
| I x
T x | I x
| I x
| T x ( T x | | T x )
λ x , x = ( x x ) / α + λ c
x
S ( x , λ x , x )
T x | I x
I ( y )
( y min = Y , y max = Y )
y
Y Y d y T ˜ x ( y ) T x ( y ) = β δ x , x ,
*
T x , T ˜ x { cos ( m y π Y ) } , m *
T x
T x
T ˜ x
T x { 1 2 [ 1 + cos ( m y π Y ) ] } , m * .
T ˜ x { 2 cos ( m y π Y ) } , m * .
m = 1 64
P n ( y ) = 1 2 n m = 0 n / 2 ( 1 ) m ( n m ) ( 2 n 2 m n ) y n 2 m ,
( a b ) = a ! ( a b ) ! b ! .
T x , T ˜ x { P m ( y Y ) } , m * .
T x { 1 2 [ 1 + P m ( y Y ) ] } , m * .
T ˜ x { 2 P m ( y Y ) } , m * .
m = 1 64
H n ( : , m )
H n ( m , : )
H n
T x , T ˜ x { H n ( : , m ) } , m * ; m n
T x { 1 2 [ 1 H n ( : , m ) ] } , m * ; m n .
T ˜ x { 2 H n ( : , m ) } , m * ; m n .
H n ( m , : )
H n ( m , : ) [ 1 / 2 [ 1 + H n ( m , : ) ] 1 / 2 [ 1 H n ( m , : ) ] ] .
H n
H ^ n
T x , T ˜ x { H ^ n ( : , m ) } , m * ; m n
( m * )
( m n )
I ( x , y )
y
H ( x , x , y , y ; λ ) = δ ( y y ) δ { x [ x + α ( λ λ c ) ] }
Δ x
Δ x
λ c
F / #
F / #
Δ λ 0.1 3   nm
160   ms
Δ Λ 775 900   nm
δ λ 0 .65   nm
36   μm
H ^ 40
( 36   μm )
H ^ n
( n = 40 , 32 , 24 , 16 , 12 )
N / 2
2
23.7
7.0
23.7 / 7.0 3.4
10.3
10.3 3.2
H ^ 40
H ^ 40

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