Abstract

A new method for optically encoding the spectral output of multislit spectrometers is proposed. It is based on sequential measurements of the total intensity of light in combinations of selected spectral bands. The resulting encoded optical information is obtained as a set of simultaneous linear algebraic equations, and spectrum reconstruction is accomplished through the use of matrix inversion techniques. This encoding method avoids the problems associated with frequency transform encoding, because it is not based on the usual Fourier or Fresnel transforms. Rather, it makes use of the theory of simultaneous linear algebraic equations.

© 1968 Optical Society of America

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References

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  1. M. J. E. Golay, J. Opt. Soc. Amer. 39, 437 (1949).
    [CrossRef]
  2. P. Fellgett, Cambridge Univ. Ph.D. thesis (1951); quoted by L. Mertz, Transformations in Optics (John Wiley & Sons, Inc., New York, 1965), p. 1; J. Phys. Radium 19, 187 (1958).
  3. P. Jacquinot, J. Opt. Soc. Amer. 44, 761 (1954); Rep. Progr. Phys. 23, 267 (1960).
    [CrossRef]
  4. J. F. Grainger, J. Ring, J. H. Stell, J. Phys. 28, suppl. 3–4, p. C2–44 (1967).
  5. J. F. Grainger, Univ. of Hull (England) Reports, Contract AF61 (052)–751 (unpublished): Annual Summary Report (1965), and Final Scientific Report (1966).
  6. L. Mertz, see Ref. 2, Chaps. I and II.
  7. M. J. E. Golay, J. Opt. Soc. Amer. 41, 468 (1951).
    [CrossRef]
  8. A. Girard, Appl. Opt. 2, 79 (1963); quoted by L. Mertz, Ref. 2, p. 84.
    [CrossRef]
  9. A. Labeyrie, Comp. Rend. B265, 119 (1967).
  10. S. Prydz, Appl. Opt. 7, 211 (1968).
    [CrossRef] [PubMed]
  11. R. N. Ibbett, D. Aspinall, J. F. Grainger, Appl. Opt. 7, 1089 (1968).
    [CrossRef] [PubMed]
  12. S. Kaczmarz, H. Steinhaus, Theorie der Orthogonalreihen (Chelsea Publishing Company, New York, 1951), p. 132.
  13. J. L. Walsh, Amer. J. Math. 45, 5 (1923).
    [CrossRef]
  14. See, e.g., F. B. Hildebrand, Methods of Applied Mathematics (Prentice-Hall, Inc., New York, 1952), p. 12.
  15. T. Muir, W. H. Metzler, A Treatise on the Theory of Determinants (Longmans, Green and Company, Inc., New York, 1933), p. 487.
  16. L. B. W. Jolley, Summation of Series (Dover Publications, Inc., New York, 1961).

1968 (2)

1967 (2)

A. Labeyrie, Comp. Rend. B265, 119 (1967).

J. F. Grainger, J. Ring, J. H. Stell, J. Phys. 28, suppl. 3–4, p. C2–44 (1967).

1963 (1)

1954 (1)

P. Jacquinot, J. Opt. Soc. Amer. 44, 761 (1954); Rep. Progr. Phys. 23, 267 (1960).
[CrossRef]

1951 (1)

M. J. E. Golay, J. Opt. Soc. Amer. 41, 468 (1951).
[CrossRef]

1949 (1)

M. J. E. Golay, J. Opt. Soc. Amer. 39, 437 (1949).
[CrossRef]

1923 (1)

J. L. Walsh, Amer. J. Math. 45, 5 (1923).
[CrossRef]

Aspinall, D.

Fellgett, P.

P. Fellgett, Cambridge Univ. Ph.D. thesis (1951); quoted by L. Mertz, Transformations in Optics (John Wiley & Sons, Inc., New York, 1965), p. 1; J. Phys. Radium 19, 187 (1958).

Girard, A.

Golay, M. J. E.

M. J. E. Golay, J. Opt. Soc. Amer. 41, 468 (1951).
[CrossRef]

M. J. E. Golay, J. Opt. Soc. Amer. 39, 437 (1949).
[CrossRef]

Grainger, J. F.

R. N. Ibbett, D. Aspinall, J. F. Grainger, Appl. Opt. 7, 1089 (1968).
[CrossRef] [PubMed]

J. F. Grainger, J. Ring, J. H. Stell, J. Phys. 28, suppl. 3–4, p. C2–44 (1967).

J. F. Grainger, Univ. of Hull (England) Reports, Contract AF61 (052)–751 (unpublished): Annual Summary Report (1965), and Final Scientific Report (1966).

Hildebrand, F. B.

See, e.g., F. B. Hildebrand, Methods of Applied Mathematics (Prentice-Hall, Inc., New York, 1952), p. 12.

Ibbett, R. N.

Jacquinot, P.

P. Jacquinot, J. Opt. Soc. Amer. 44, 761 (1954); Rep. Progr. Phys. 23, 267 (1960).
[CrossRef]

Jolley, L. B. W.

L. B. W. Jolley, Summation of Series (Dover Publications, Inc., New York, 1961).

Kaczmarz, S.

S. Kaczmarz, H. Steinhaus, Theorie der Orthogonalreihen (Chelsea Publishing Company, New York, 1951), p. 132.

Labeyrie, A.

A. Labeyrie, Comp. Rend. B265, 119 (1967).

Mertz, L.

L. Mertz, see Ref. 2, Chaps. I and II.

Metzler, W. H.

T. Muir, W. H. Metzler, A Treatise on the Theory of Determinants (Longmans, Green and Company, Inc., New York, 1933), p. 487.

Muir, T.

T. Muir, W. H. Metzler, A Treatise on the Theory of Determinants (Longmans, Green and Company, Inc., New York, 1933), p. 487.

Prydz, S.

Ring, J.

J. F. Grainger, J. Ring, J. H. Stell, J. Phys. 28, suppl. 3–4, p. C2–44 (1967).

Steinhaus, H.

S. Kaczmarz, H. Steinhaus, Theorie der Orthogonalreihen (Chelsea Publishing Company, New York, 1951), p. 132.

Stell, J. H.

J. F. Grainger, J. Ring, J. H. Stell, J. Phys. 28, suppl. 3–4, p. C2–44 (1967).

Walsh, J. L.

J. L. Walsh, Amer. J. Math. 45, 5 (1923).
[CrossRef]

Amer. J. Math. (1)

J. L. Walsh, Amer. J. Math. 45, 5 (1923).
[CrossRef]

Appl. Opt. (3)

Comp. Rend. (1)

A. Labeyrie, Comp. Rend. B265, 119 (1967).

J. Opt. Soc. Amer. (3)

P. Jacquinot, J. Opt. Soc. Amer. 44, 761 (1954); Rep. Progr. Phys. 23, 267 (1960).
[CrossRef]

M. J. E. Golay, J. Opt. Soc. Amer. 39, 437 (1949).
[CrossRef]

M. J. E. Golay, J. Opt. Soc. Amer. 41, 468 (1951).
[CrossRef]

J. Phys. (1)

J. F. Grainger, J. Ring, J. H. Stell, J. Phys. 28, suppl. 3–4, p. C2–44 (1967).

Other (7)

J. F. Grainger, Univ. of Hull (England) Reports, Contract AF61 (052)–751 (unpublished): Annual Summary Report (1965), and Final Scientific Report (1966).

L. Mertz, see Ref. 2, Chaps. I and II.

S. Kaczmarz, H. Steinhaus, Theorie der Orthogonalreihen (Chelsea Publishing Company, New York, 1951), p. 132.

P. Fellgett, Cambridge Univ. Ph.D. thesis (1951); quoted by L. Mertz, Transformations in Optics (John Wiley & Sons, Inc., New York, 1965), p. 1; J. Phys. Radium 19, 187 (1958).

See, e.g., F. B. Hildebrand, Methods of Applied Mathematics (Prentice-Hall, Inc., New York, 1952), p. 12.

T. Muir, W. H. Metzler, A Treatise on the Theory of Determinants (Longmans, Green and Company, Inc., New York, 1933), p. 487.

L. B. W. Jolley, Summation of Series (Dover Publications, Inc., New York, 1961).

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Figures (1)

Fig. 1
Fig. 1

Portion of encoder disk at spectometer exit focal plane position.

Equations (15)

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f = 2 V c ν ,
a 1 , 1 x 1 + a 1 , 2 x 2 + + a 1 , j x j + + a 1 , m x m = I 1 a 2 , 1 x 1 + a 2 , 2 x 2 + + a 2 , j x j + + a 2 , m x m = I 2 . . . . . . . . . . . . . . . a i , 1 x 1 + a i , 2 x 2 + + a i , j x j + + a i , m x m = I i . . . . . . . . . . . . . . . a m , 1 x 1 + a m , 2 x 2 + + a m , j x j + + a m , m x m = I m ;
[ a i , j ] { x j } = { I i } ,
{ x j } = [ a i , j ] - 1 { I i } ,
a i , j = 0 , i j = 1 , i = j ,
a i , j = 1 , i j = 0 , i = j .
10 11 00 11 00 11 00 01 11 00 11 00 11 00 ,
[ A B C D E E A B C D D E A B C C D E A B B C D E A ] ,
A = [ 10 01 ] , B = [ 11 11 ] , C = [ 00 00 ] , D = [ 11 11 ] , E = [ 00 00 ] .
= k = 1 m / 2 - 1 | A + B α k + C α k + C α 2 k + D α 2 k + | ,
= k = 0 m / 2 - 1 | 1 + 2 α k + 2 α 2 k + 2 α 3 k + 2 α [ ( m - 4 ) / 2 ] k | .
= m / 2 k = 1 m / 2 - 1 { 1 + 2 cos ( θ k ) + 2 i sin ( θ k ) + 2 cos 3 ( θ k ) + 2 i sin 3 ( 0 k ) + 2 i sin [ ( m - 4 ) / 2 ] ( θ k ) } ,
= m / 2 k = 1 m / 2 - 1 { 1 - 2 [ cos ( θ k / 2 ) - i sin ( θ k / 2 ) ] · sin ( θ k / 2 ) × [ sin θ k ] - 1 } ,
= m / 2 k = 1 m / 2 - 1 i tan ( θ k / 2 ) .
= ( m / 2 ) 2 .

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