Abstract

A general treatment of the transfer function (TF) of spectroscopic systems is presented. The TF is expressed as a function of the delay time but not of the spatial frequency. Second, a new method for measuring the TF of spectroscopic systems with a sinusoidally modulated spectrum (SMS) is proposed, and the TF of a prism spectroscope is measured to verify the effectiveness of the method. As this new method is similar to methods of measuring the TF of ordinary imaging systems by using a spatial sine grating, the TF of spectroscopic systems is obtained from measurements of the contrast in the spectrogram of the SMS. This SMS is produced by means of a Michelson interferometer with white light. The period of the SMS can be varied by changing the path difference between two reflecting mirrors of the interferometer.

© 1981 Optical Society of America

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References

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  1. A. Lohmann, “Contrast transfer in the grating spectrograph,” Opt. Acta 6, 175–185 (1959).
    [Crossref]
  2. K. Goto and S. Morozumi, “Fourier transform formulation of optical imaging applied to the grating mounting,” Appl. Opt. 10, 764–768 (1971).
    [Crossref] [PubMed]
  3. H. Kanamori and K. Kozima, “Correction of a spectral image formed by a plane-grating monochromator by means of optical transfer functions—a partially coherent case,” Jpn. J. Appl. Phys. 14-1, 199–200 (1975).
  4. T. Katayama and A. Takahashi, “Optical transfer function for concave grating spectrometer,” Mem. Def. Acad. Math. Phys. Chem. Eng. Yokosuka, Jpn. 7, 1055–1069 (1967).
  5. T. Katayama and A. Takahashi, “Optical transfer function of concave grating spectrometer based on wave optical method,” Jpn. J. Appl. Phys. 9, 1509–1516 (1970).
    [Crossref]
  6. H. Kanamori and K. Kozima, “Measurement of optical transfer functions and correction of images in spectroscopic systems,” in Applications of Holography and Optical Data Processing, E. Marom, A. A. Friesem, and E. Wiener-Avnear, eds. (Pergamon, Oxford, 1977), pp. 635–640.
  7. K. Kozima and H. Kanamori, “Measurement of optical transfer functions of spectroscopic systems (II),” Radio Image Inform. 6, 133–138 (1976) (in Japanese).
  8. K. Murata, “Instruments for the measuring of optical transfer functions,” in Progress in Optics V, E. Wolf, ed. (North-Holland, Amsterdam, 1965), pp. 199–245.
  9. P. A. Jansson, “Method for determining the response function of a high-resolution infrared spectrometer,” J. Opt. Soc. Am. 60, 184–191 (1970).
    [Crossref]
  10. W. H. Steel, “The defocused image of sinusoidal gratings,” Opt. Acta 3, 65–74, (1956).
    [Crossref]
  11. K. D. Mielenz, “Spectroscope slit images in partially coherent light,” J. Opt. Soc. Am. 57, 66–74 (1967).
    [Crossref]
  12. C. Delisle, M. Brochu, and J. M. St-Arnaud, “Etude de l’effet desparamètres d’appareil sur la décroissance de la visibilité du spectre cannelé,” Can. J. Phys. 49, 2237–2249 (1971).
    [Crossref]

1976 (1)

K. Kozima and H. Kanamori, “Measurement of optical transfer functions of spectroscopic systems (II),” Radio Image Inform. 6, 133–138 (1976) (in Japanese).

1975 (1)

H. Kanamori and K. Kozima, “Correction of a spectral image formed by a plane-grating monochromator by means of optical transfer functions—a partially coherent case,” Jpn. J. Appl. Phys. 14-1, 199–200 (1975).

1971 (2)

K. Goto and S. Morozumi, “Fourier transform formulation of optical imaging applied to the grating mounting,” Appl. Opt. 10, 764–768 (1971).
[Crossref] [PubMed]

C. Delisle, M. Brochu, and J. M. St-Arnaud, “Etude de l’effet desparamètres d’appareil sur la décroissance de la visibilité du spectre cannelé,” Can. J. Phys. 49, 2237–2249 (1971).
[Crossref]

1970 (2)

P. A. Jansson, “Method for determining the response function of a high-resolution infrared spectrometer,” J. Opt. Soc. Am. 60, 184–191 (1970).
[Crossref]

T. Katayama and A. Takahashi, “Optical transfer function of concave grating spectrometer based on wave optical method,” Jpn. J. Appl. Phys. 9, 1509–1516 (1970).
[Crossref]

1967 (2)

K. D. Mielenz, “Spectroscope slit images in partially coherent light,” J. Opt. Soc. Am. 57, 66–74 (1967).
[Crossref]

T. Katayama and A. Takahashi, “Optical transfer function for concave grating spectrometer,” Mem. Def. Acad. Math. Phys. Chem. Eng. Yokosuka, Jpn. 7, 1055–1069 (1967).

1959 (1)

A. Lohmann, “Contrast transfer in the grating spectrograph,” Opt. Acta 6, 175–185 (1959).
[Crossref]

1956 (1)

W. H. Steel, “The defocused image of sinusoidal gratings,” Opt. Acta 3, 65–74, (1956).
[Crossref]

Brochu, M.

C. Delisle, M. Brochu, and J. M. St-Arnaud, “Etude de l’effet desparamètres d’appareil sur la décroissance de la visibilité du spectre cannelé,” Can. J. Phys. 49, 2237–2249 (1971).
[Crossref]

Delisle, C.

C. Delisle, M. Brochu, and J. M. St-Arnaud, “Etude de l’effet desparamètres d’appareil sur la décroissance de la visibilité du spectre cannelé,” Can. J. Phys. 49, 2237–2249 (1971).
[Crossref]

Goto, K.

Jansson, P. A.

Kanamori, H.

K. Kozima and H. Kanamori, “Measurement of optical transfer functions of spectroscopic systems (II),” Radio Image Inform. 6, 133–138 (1976) (in Japanese).

H. Kanamori and K. Kozima, “Correction of a spectral image formed by a plane-grating monochromator by means of optical transfer functions—a partially coherent case,” Jpn. J. Appl. Phys. 14-1, 199–200 (1975).

H. Kanamori and K. Kozima, “Measurement of optical transfer functions and correction of images in spectroscopic systems,” in Applications of Holography and Optical Data Processing, E. Marom, A. A. Friesem, and E. Wiener-Avnear, eds. (Pergamon, Oxford, 1977), pp. 635–640.

Katayama, T.

T. Katayama and A. Takahashi, “Optical transfer function of concave grating spectrometer based on wave optical method,” Jpn. J. Appl. Phys. 9, 1509–1516 (1970).
[Crossref]

T. Katayama and A. Takahashi, “Optical transfer function for concave grating spectrometer,” Mem. Def. Acad. Math. Phys. Chem. Eng. Yokosuka, Jpn. 7, 1055–1069 (1967).

Kozima, K.

K. Kozima and H. Kanamori, “Measurement of optical transfer functions of spectroscopic systems (II),” Radio Image Inform. 6, 133–138 (1976) (in Japanese).

H. Kanamori and K. Kozima, “Correction of a spectral image formed by a plane-grating monochromator by means of optical transfer functions—a partially coherent case,” Jpn. J. Appl. Phys. 14-1, 199–200 (1975).

H. Kanamori and K. Kozima, “Measurement of optical transfer functions and correction of images in spectroscopic systems,” in Applications of Holography and Optical Data Processing, E. Marom, A. A. Friesem, and E. Wiener-Avnear, eds. (Pergamon, Oxford, 1977), pp. 635–640.

Lohmann, A.

A. Lohmann, “Contrast transfer in the grating spectrograph,” Opt. Acta 6, 175–185 (1959).
[Crossref]

Mielenz, K. D.

Morozumi, S.

Murata, K.

K. Murata, “Instruments for the measuring of optical transfer functions,” in Progress in Optics V, E. Wolf, ed. (North-Holland, Amsterdam, 1965), pp. 199–245.

St-Arnaud, J. M.

C. Delisle, M. Brochu, and J. M. St-Arnaud, “Etude de l’effet desparamètres d’appareil sur la décroissance de la visibilité du spectre cannelé,” Can. J. Phys. 49, 2237–2249 (1971).
[Crossref]

Steel, W. H.

W. H. Steel, “The defocused image of sinusoidal gratings,” Opt. Acta 3, 65–74, (1956).
[Crossref]

Takahashi, A.

T. Katayama and A. Takahashi, “Optical transfer function of concave grating spectrometer based on wave optical method,” Jpn. J. Appl. Phys. 9, 1509–1516 (1970).
[Crossref]

T. Katayama and A. Takahashi, “Optical transfer function for concave grating spectrometer,” Mem. Def. Acad. Math. Phys. Chem. Eng. Yokosuka, Jpn. 7, 1055–1069 (1967).

Appl. Opt. (1)

Can. J. Phys. (1)

C. Delisle, M. Brochu, and J. M. St-Arnaud, “Etude de l’effet desparamètres d’appareil sur la décroissance de la visibilité du spectre cannelé,” Can. J. Phys. 49, 2237–2249 (1971).
[Crossref]

J. Opt. Soc. Am. (2)

Jpn. J. Appl. Phys. (2)

H. Kanamori and K. Kozima, “Correction of a spectral image formed by a plane-grating monochromator by means of optical transfer functions—a partially coherent case,” Jpn. J. Appl. Phys. 14-1, 199–200 (1975).

T. Katayama and A. Takahashi, “Optical transfer function of concave grating spectrometer based on wave optical method,” Jpn. J. Appl. Phys. 9, 1509–1516 (1970).
[Crossref]

Mem. Def. Acad. Math. Phys. Chem. Eng. Yokosuka, Jpn. (1)

T. Katayama and A. Takahashi, “Optical transfer function for concave grating spectrometer,” Mem. Def. Acad. Math. Phys. Chem. Eng. Yokosuka, Jpn. 7, 1055–1069 (1967).

Opt. Acta (2)

W. H. Steel, “The defocused image of sinusoidal gratings,” Opt. Acta 3, 65–74, (1956).
[Crossref]

A. Lohmann, “Contrast transfer in the grating spectrograph,” Opt. Acta 6, 175–185 (1959).
[Crossref]

Radio Image Inform. (1)

K. Kozima and H. Kanamori, “Measurement of optical transfer functions of spectroscopic systems (II),” Radio Image Inform. 6, 133–138 (1976) (in Japanese).

Other (2)

K. Murata, “Instruments for the measuring of optical transfer functions,” in Progress in Optics V, E. Wolf, ed. (North-Holland, Amsterdam, 1965), pp. 199–245.

H. Kanamori and K. Kozima, “Measurement of optical transfer functions and correction of images in spectroscopic systems,” in Applications of Holography and Optical Data Processing, E. Marom, A. A. Friesem, and E. Wiener-Avnear, eds. (Pergamon, Oxford, 1977), pp. 635–640.

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Figures (5)

Fig. 1
Fig. 1

Experimental setup for measuring the TF of a prism spectroscope. A halogen lamp is used as a light source of broad spectral width. A Michelson interferometer serves to produce the SMS on the entrance slit of the spectroscope. The period of the SMS can be varied by displacing the movable mirror M2. The tilt of the mirror M2 can be corrected with reference spectral lines introduced into the spectroscope. At the output plane of the spectroscope, the spectrogram of the SMS is constructed and recorded with the scanning photodetector combined with a narrow slit. L1, L2, L3, L4, and L5 are lenses; BS1 and BS2 are beam splitters; C is a compensator; SP is a photodetector; and A is a dc amplifier.

Fig. 2
Fig. 2

Illustration for evaluating the phase TF by means of the iteration method. The spectral density at ω0, with a maximum at tn (n = integers), oscillates with gradually diminishing amplitude as the delay time increases. The envelope (dotted curve) indicates the decrease in values of the MTF with increasing td.

Fig. 3
Fig. 3

Records of spectrograms of the SMS for path differences between two mirrors of the Michelson interferometer given by (a) ld = 20 μm, (b) 40 μm, and (c) 140 μm. The slit width of the entrance slit of the spectroscope was chosen to be 50 μm.

Fig. 4
Fig. 4

Dependence of the modulation TF on the wavelength. The cases (a) and (b) correspond to widths of the entrance slit w = 50 μm and w = 100 μm, respectively. The parameter indicates the path difference ld between two mirrors of the Michelson interferometer: crosses, ld = 20 μm; open squares, 40 μm; solid circles, 60 μm; triangles, 100 μm; and open circles, 140 μm.

Fig. 5
Fig. 5

Dependence of the modulation TF on the path difference ld between two mirrors of the Michelson interferometer. The slit width is the same as in Fig. 4. Three kinds of marks, solid circles, crosses, and solid triangles, correspond to the measured modulation TF at wavelengths λ = 450 nm, λ = 520 nm, and λ = 600 nm, respectively.

Equations (22)

Equations on this page are rendered with MathJax. Learn more.

g ( ω ) = f ( ω - ω ) h ( ω ) d ω .
H ( t ) = h ( ω ) e i ω t d ω , F ( t ) = f ( ω ) e i ω t d ω , G ( t ) = g ( ω ) e i ω t d ω ,
G ( t ) = F ( t ) H ( t ) .
H ( 0 ) = h ( ω ) d ω = 1.
F ( 0 ) = f ( ω ) d ω = f s ( ω ) ( 1 + cos ω t d ) d ω .
f ( ω ) = f s ( ω ) ( 1 + cos ω t d )
Δ ω = 2 π t d
g ( ω ) = h ( ω ) f s ( ω - ω ) d ω + cos ω t d h ( ω ) f s ( ω - ω ) cos ω t d d ω + sin ω t d h ( ω ) f s ( ω - ω ) sin ω t d d ω .
g ( ω ) = f s ( ω ) [ 1 + H c ( t d ) cos ω t d + H s ( t d ) sin ω t d ] ,
H c ( t d ) = Re H ( t d ) ,             H s ( t d ) = Im H ( t d ) ,
H ( t d ) = [ H c 2 ( t d ) + H s 2 ( t d ) ] 1 / 2
ϕ ( t d ) = tan - 1 H s ( t d ) H c ( t d ) ,
g ( ω ) = f s ( ω ) { 1 + H ( t d ) cos [ ω t d - ϕ ( t d ) ] } ,
C = g max - g min g max + g min ,
g max = f s ( ω ) [ 1 + H ( t d ) ] for ω t d - ϕ = 2 n π , g min = f s ( ω ) [ 1 - H ( t d ) ] for ω t d - ϕ = ( 2 n + 1 ) π ,
C = H ( t d ) .
ϕ ( t n ) = ω 0 t n - 2 n π .
ϕ ( t n + Δ t n ) - ϕ ( t n ) Δ t n d ϕ ( t n ) d t n
ω 0 Δ t n - 2 π Δ t n d ϕ ( t n ) d t n .
ϕ ( t n ) = 0 t n d ϕ ( t ) d t d t .
ϕ ( t n ) = ω 0 n = 1 n Δ t n - 1 - 2 n π ,
V = w D 2 λ f > 1 ,