Abstract

We present our results on measurement of the modulation transfer function (MTF) of a given spectrometer by using the sinusoidally modulated spectrum from a Michelson interferometer with white light. We studied the MTF by varying the periodicity of the spectral fringes produced by the interferometer. Experimental data are fitted to a theoretical model to derive the spectral slit width from the measured MTF of the spectrometer.

© 1999 Optical Society of America

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References

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  1. K. Iizuka, Engineering Optics, Vol. 35 in Springer Series in Optical Sciences (Springer-Verlag, Berlin, 1985), Chap. 10.
  2. K. Gotto, S. Morozumi, “Fourier transform formulation of optical imaging applied to the grating mounting,” Appl. Opt. 10, 764–768 (1971).
    [CrossRef]
  3. H. Kanamori, 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, A. Takashi, “Optical transfer function of concave grating spectrometer based on wave optical method,” Jpn. J. Appl. Phys. 9, 1509–1516 (1970).
    [CrossRef]
  5. H. Kanamori, K. Kozima, “Measurement of optical transfer functions and correction of images in spectroscopic systems,” in Application of Holography and Optical Data Processing, E. Marom, A. A. Friessen, E. Wiener-Avnear, eds. (Pergamon, New York, 1977), p. 635.
    [CrossRef]
  6. K. Kozima, H. Kanamori, O. Matsuda, “Direct measurement of optical transfer function of spectroscopic systems,” Jpn. J. Appl. Phys. 17, 1271–1274 (1978).
    [CrossRef]
  7. V. J. Coates, H. Hausdorff, “Interferometric method of measuring the spectral slit width of spectrometers,” J. Opt. Soc. Am. 45, 425–430 (1955).
    [CrossRef]
  8. S. Brodersen, “Interferometric frequency calibration of infrared spectrometers,” J. Opt. Soc. Am. 46, 255–258 (1956).
    [CrossRef]
  9. V. N. Kumar, D. N. Rao, “Determination of the instrument function of a grating spectrometer using white light interferometry,” Appl. Opt. 36, 4535–4539 (1997).
    [CrossRef] [PubMed]
  10. H. Fujiwara, “Transfer function of spectroscopic systems using a sinusoidally modulated spectrum,” J. Opt. Soc. Am. 71, 238–242 (1981).
    [CrossRef]
  11. V. N. Kumar, D. N. Rao, “Using interference in the frequency domain for precise determination of thickness and refractive indices of normal dispersive materials,” J. Opt. Soc. Am. B 12, 1559–1563 (1995); V. N. Kumar, D. N. Rao, “A compact interferometric system for simultaneous measurement of refractive index and thickness,” Commun. Instrum. 5, 209–216 (1997).
    [CrossRef]
  12. L. Mandel, E. Wolf, “Spectral coherence and the concept of cross-spectral purity,” J. Opt. Soc. Am. 66, 529–535 (1976).
    [CrossRef]
  13. D. S. Goodman, “Basic optical instruments,” in Methods of Experimental Physics, Geometrical and Instrumental Optics, D. Malacara, ed. (Academic, New York, 1988), Vol. 25, p. 204.
  14. D. J. Schroeder, “Diffraction grating instruments,” in Methods of Experimental Physics, N. Carleton, ed. (Academic, New York, 1974).
    [CrossRef]
  15. F. Kneubuhl, “Diffraction grating spectroscopy,” Appl. Opt. 8, 505–519 (1969).
    [CrossRef] [PubMed]
  16. V. N. Kumar, “Spectral interferometry: a study of the degree of coherence in the space-frequency domain and the applications,” Ph.D. dissertation (University of Hyderabad, Hyderabad, India, 1997).
  17. J. E. Stewart, “Polarization interferometer for the determination of spectral modulation transfer functions of monochromators,” Appl. Opt. 6, 1523–1525 (1967).
    [CrossRef] [PubMed]
  18. J. E. Stewart, “Spurious spectral resolution and the spectral transfer function of monochromators,” Appl. Opt. 4, 609–612 (1965).
    [CrossRef]
  19. K. D. Mielenz, “Spectroscope slit images in partially coherent light,” J. Opt. Soc. Am. 57, 66–74 (1967).
    [CrossRef]

1997 (1)

1995 (1)

1981 (1)

1978 (1)

K. Kozima, H. Kanamori, O. Matsuda, “Direct measurement of optical transfer function of spectroscopic systems,” Jpn. J. Appl. Phys. 17, 1271–1274 (1978).
[CrossRef]

1976 (1)

1975 (1)

H. Kanamori, 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 (1)

1970 (1)

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

1969 (1)

1967 (2)

1965 (1)

1956 (1)

1955 (1)

Brodersen, S.

Coates, V. J.

Fujiwara, H.

Goodman, D. S.

D. S. Goodman, “Basic optical instruments,” in Methods of Experimental Physics, Geometrical and Instrumental Optics, D. Malacara, ed. (Academic, New York, 1988), Vol. 25, p. 204.

Gotto, K.

Hausdorff, H.

Iizuka, K.

K. Iizuka, Engineering Optics, Vol. 35 in Springer Series in Optical Sciences (Springer-Verlag, Berlin, 1985), Chap. 10.

Kanamori, H.

K. Kozima, H. Kanamori, O. Matsuda, “Direct measurement of optical transfer function of spectroscopic systems,” Jpn. J. Appl. Phys. 17, 1271–1274 (1978).
[CrossRef]

H. Kanamori, 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, K. Kozima, “Measurement of optical transfer functions and correction of images in spectroscopic systems,” in Application of Holography and Optical Data Processing, E. Marom, A. A. Friessen, E. Wiener-Avnear, eds. (Pergamon, New York, 1977), p. 635.
[CrossRef]

Katayama, T.

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

Kneubuhl, F.

Kozima, K.

K. Kozima, H. Kanamori, O. Matsuda, “Direct measurement of optical transfer function of spectroscopic systems,” Jpn. J. Appl. Phys. 17, 1271–1274 (1978).
[CrossRef]

H. Kanamori, 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, K. Kozima, “Measurement of optical transfer functions and correction of images in spectroscopic systems,” in Application of Holography and Optical Data Processing, E. Marom, A. A. Friessen, E. Wiener-Avnear, eds. (Pergamon, New York, 1977), p. 635.
[CrossRef]

Kumar, V. N.

Mandel, L.

Matsuda, O.

K. Kozima, H. Kanamori, O. Matsuda, “Direct measurement of optical transfer function of spectroscopic systems,” Jpn. J. Appl. Phys. 17, 1271–1274 (1978).
[CrossRef]

Mielenz, K. D.

Morozumi, S.

Rao, D. N.

Schroeder, D. J.

D. J. Schroeder, “Diffraction grating instruments,” in Methods of Experimental Physics, N. Carleton, ed. (Academic, New York, 1974).
[CrossRef]

Stewart, J. E.

Takashi, A.

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

Wolf, E.

Appl. Opt. (5)

J. Opt. Soc. Am. (5)

J. Opt. Soc. Am. B (1)

Jpn. J. Appl. Phys. (3)

K. Kozima, H. Kanamori, O. Matsuda, “Direct measurement of optical transfer function of spectroscopic systems,” Jpn. J. Appl. Phys. 17, 1271–1274 (1978).
[CrossRef]

H. Kanamori, 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, A. Takashi, “Optical transfer function of concave grating spectrometer based on wave optical method,” Jpn. J. Appl. Phys. 9, 1509–1516 (1970).
[CrossRef]

Other (5)

H. Kanamori, K. Kozima, “Measurement of optical transfer functions and correction of images in spectroscopic systems,” in Application of Holography and Optical Data Processing, E. Marom, A. A. Friessen, E. Wiener-Avnear, eds. (Pergamon, New York, 1977), p. 635.
[CrossRef]

K. Iizuka, Engineering Optics, Vol. 35 in Springer Series in Optical Sciences (Springer-Verlag, Berlin, 1985), Chap. 10.

D. S. Goodman, “Basic optical instruments,” in Methods of Experimental Physics, Geometrical and Instrumental Optics, D. Malacara, ed. (Academic, New York, 1988), Vol. 25, p. 204.

D. J. Schroeder, “Diffraction grating instruments,” in Methods of Experimental Physics, N. Carleton, ed. (Academic, New York, 1974).
[CrossRef]

V. N. Kumar, “Spectral interferometry: a study of the degree of coherence in the space-frequency domain and the applications,” Ph.D. dissertation (University of Hyderabad, Hyderabad, India, 1997).

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Figures (2)

Fig. 1
Fig. 1

Schematic of the experimental setup for measuring the MTF and the spectral slit width of a grating spectrometer: L, lamp; P, pinhole; CL, condenser lens; A, aperture; BS, beam splitter; M1, M2, mirrors; L1, lens; CM, collimating mirror; G, grating; FM, focusing mirror; PMT, photomultiplier tube.

Fig. 2
Fig. 2

Variation of the MTF as a function of the path delay between the interfering beams for different equal values of the entrance- and exit-slit widths: (a) δx 1 = δx 2 = 50 µm, (b) 100 µm, (c) 200 µm, (d) 400 µm. Dots, experimental data; continuous lines, theoretical fit where Eq. (8) is used for (a) and (b) and Eq. (6) is used for (c) and (d). The dotted curve in (c) is for the theoretical fit where Eq. (8) is used.

Tables (2)

Tables Icon

Table 1 Fitted Values of the MTF (Fig. 1) when Eq. (8) is Used

Tables Icon

Table 2 Values of the Path Delay Δmin where MTF goes to Zero, the Corresponding Number of Fringes (N0) over the Entire Spectral Region and Calculated Spectral Slit Width δν sp in cm-1 for Different Slit Widths δxa

Equations (11)

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Sν=1/2S0ν1+Reμ12νcos2πντ+θ,
sν=1/2  S0ν1+cos2πντσν-νdν,
στ= σνexp2πiντdν.
sν=1/2S0ν1+|σ0τ|cos2πντ-ϕτ,
V=smaxν-sminνsmaxν+sminν=|σ0τ|,
sν, δνsp, τ=1/2S0ν×1+|σ0τ|sinπδνspτπδνspτcos2πντ.
σ0τ=|σ0τ|sin cπδνspτ.
δxδλsp=md cos θ,
σ0τ=|σ0τ|sin c2πδνspτ.
δλsp=δxP,
σ0τ=|σ0τ|sin cπδνspτsin cπδνspτ-y,

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