Abstract

We analyze the performance of a dispersion instrument in which light is multiplexed both in the entrance and exit slit positions. This double multiplexing scheme allows one to recover both Fellgett’s advantage and the high throughput advantage normally attributed only to interferometric spectrometers. The spectrometer’s performance is evaluated for a number of binary cyclic coding schemes. Optical limitations on doubly multiplexed instruments are discussed, and we show that such spectrometers compare favorably with Michelson interferometric spectrometers. Some first results obtained with a laboratory pilot model are presented.

© 1970 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. R. N. Ibbett, D. Aspinall, J. F. Grainger, Appl. Opt. 7, 1089 (1968).
    [CrossRef] [PubMed]
  3. J. A. Decker, M. O. Harwit, Appl. Opt. 7, 2205 (1968).
    [CrossRef] [PubMed]
  4. N. J. A. Sloane, T. Fine, P. G. Phillips, M. O. Harwit, Appl. Opt. 8, 2103 (1969).
    [CrossRef] [PubMed]
  5. P. Fellgett, J. Phys. 19, 187 (1958).
  6. G. A. Vanasse, H. Sakai, in Progress in Optics, E. Wolf, Ed. (Wiley-Interscience, New York, 1967), Vol. 7, p. 298.
  7. P. Jacquinot, J. Opt. Soc. Amer. 44, 761 (1954).
    [CrossRef]
  8. A. Girard, Appl. Opt. 2, 79 (1963), as quoted by J. T. Houghton, S. D. Smith in Infra-red Physics (Clarendon Press, Oxford, 1966), p. 225.
    [CrossRef]
  9. L. Mertz, Transformations in Optics (John Wiley & Sons, Inc., New York, 1965), p. 64.
  10. D. A. Laws, Opt. Acta 9, 69 (1962).
    [CrossRef]

1969 (1)

1968 (2)

1963 (1)

1962 (1)

D. A. Laws, Opt. Acta 9, 69 (1962).
[CrossRef]

1958 (1)

P. Fellgett, J. Phys. 19, 187 (1958).

1954 (1)

P. Jacquinot, J. Opt. Soc. Amer. 44, 761 (1954).
[CrossRef]

1949 (1)

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

Aspinall, D.

Decker, J. A.

Fellgett, P.

P. Fellgett, J. Phys. 19, 187 (1958).

Fine, T.

Girard, A.

Golay, M. J. E.

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

Grainger, J. F.

Harwit, M. O.

Ibbett, R. N.

Jacquinot, P.

P. Jacquinot, J. Opt. Soc. Amer. 44, 761 (1954).
[CrossRef]

Laws, D. A.

D. A. Laws, Opt. Acta 9, 69 (1962).
[CrossRef]

Mertz, L.

L. Mertz, Transformations in Optics (John Wiley & Sons, Inc., New York, 1965), p. 64.

Phillips, P. G.

Sakai, H.

G. A. Vanasse, H. Sakai, in Progress in Optics, E. Wolf, Ed. (Wiley-Interscience, New York, 1967), Vol. 7, p. 298.

Sloane, N. J. A.

Vanasse, G. A.

G. A. Vanasse, H. Sakai, in Progress in Optics, E. Wolf, Ed. (Wiley-Interscience, New York, 1967), Vol. 7, p. 298.

Appl. Opt. (4)

J. Opt. Soc. Amer. (2)

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

P. Jacquinot, J. Opt. Soc. Amer. 44, 761 (1954).
[CrossRef]

J. Phys. (1)

P. Fellgett, J. Phys. 19, 187 (1958).

Opt. Acta (1)

D. A. Laws, Opt. Acta 9, 69 (1962).
[CrossRef]

Other (2)

G. A. Vanasse, H. Sakai, in Progress in Optics, E. Wolf, Ed. (Wiley-Interscience, New York, 1967), Vol. 7, p. 298.

L. Mertz, Transformations in Optics (John Wiley & Sons, Inc., New York, 1965), p. 64.

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Figures (2)

Fig. 1
Fig. 1

Schematic diagram of doubly multiplexed spectrometer.

Fig. 2
Fig. 2

Results from laboratory pilot model showing mercury green line. The broken line is for N2 = 49 measurements and the solid line for 2N − 1 = 13 measurements. All measurements have the same integration time.

Tables (1)

Tables Icon

Table I Comparison of Total Mean Square Error for Three Grating Spectrometers in Estimating N Unknowns

Equations (34)

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η i , j = r = 1 N s = 1 N i , r ψ r , s χ j , s + ν i , j ,
ψ ( N 1 ) , , ψ 1 , ψ 0 , ψ 1 , , ψ N 1
η = ψ ¯ χ T + ν ,
ψ = ( ψ 0 ψ 1 ψ N + 1 ψ 1 ψ 0 ψ N ψ N 1 ψ N 2 ψ 0 ) .
E ( ν i , j ν k , l ) = σ 2 δ i , k δ j , l ,
σ 2 = K M / T ,
ψ ˆ t = i , j α t , i j η i , j ,
i = 1 N j = 1 N α t , i , j r = u + 1 N i , r χ j , r u = δ t , u for u 0 ,
i = 1 N j = 1 N α t , i , j r = 1 u + N i , r χ j , r u = δ t , u for u < 0.
t = N + 1 N 1 σ t 2 t = N + 1 N 1 E ( ψ ˆ t ψ t ) 2 = σ 2 t , i , j ( α t . i , j ) 2 .
Ω = ( ω 11 ω 12 ω 1 N ω 21 ω 22 ω 2 N ω N 1 ω N 2 ω N N ) 1 η ( χ T ) 1
ψ t = { 1 N t r = t + 1 N ω r , r t if t 0 1 N t r = 1 N + t ω r , r t if t < 0.
σ t 2 = 16 ( N + 1 ) x ( N 2 1 N | t | + 1 ) σ 2 16 σ 2 ( N | t | ) N 2 for N   large ,
ψ ( N 1 ) / 2 , , ψ 0 , , ψ ( N 1 ) / 2
σ total 2 = t = ( N 1 ) / 2 ( N 1 ) / 2 σ t 2 .
σ total 2 = σ 2 [ ( 22.18 / N ) ( 40.72 / N 2 ) + 0 ( 1 / N 3 ) ] .
σ total 2 t = ( N 1 ) / 2 ( N 1 ) / 2 16 σ 0 2 ( W | t | ) W 2 ,
L G = a ω = ( L / d ) ( G m λ / b R ) 2 ,
L I = A Ω = A ( 2 π / R ) .
L M S a ω a cos α ( 8 / R ) 1 2 1 / 12 ,
d / f = ( m λ / b ) ( 1 / R cos α ) 5 × 10 4 .
L M S ( 200 ) · 1 2 · 8 / ( 3 × 10 3 ) 1 2 1 12 0.4 cm 2 sr .
( b / m ) cos α = d λ / d α λ f / R d ,
G = f / cos α = b d R d / m λ .
L = L d 2 = ( L / d ) ( G m λ / b R ) 2 .
sin α sin β = m λ / b
sin ( α + δ ) sin ( β + δ ) = m λ / b ,
δ = ( 1 / cos β ) { δ cos α + ( δ 2 / 2 ) [ ( sin β / cos 2 β ) cos 2 α sin α ] } .
( sin α sin β ) ( cos δ 1 ) + ( cos α cos β ) sin δ = 0 ,
( cos α cos β ) = ( m λ / b ) [ ( sin δ / 2 ) + ( sin 3 δ / 8 ) + ] for δ 1.
sin ( α + δ ) sin ( β + δ ) = m ( λ + Δ λ ) / b .
Δ λ λ = ( sin 2 δ 2 + sin 4 δ 8 ) + + sin δ sin δ 2 + sin δ sin 3 δ 8 + .
Δ λ / λ = δ ( δ δ ) / 2 .
R = λ / Δ λ = 8 / δ 2 .

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