Abstract

The collimated optical beam in a grating spectrometer may be circular or elliptical in cross section, so that different parts of the beam illuminate different numbers of grooves on the grating. Here we estimate the consequent loss in spectral resolution relative to that obtained with a beam that illuminates a fixed number of grooves. The effect reduces the intrinsic resolving power of the spectrometer by ∼15%, exclusive of other contributions such as finite entrance-slit width.

© 2000 Optical Society of America

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References

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  1. M. C. Hutley, Diffraction Gratings (Academic, London, 1982), pp. 22, 31, 141.
  2. H. Nubbenmeyer, B. Wende, “Instrumental functions of a 5-m echelle spectrometer with diffraction-limited resolving power,” Appl. Opt. 16, 2708–2710 (1977).
  3. P. Jaquinot, B. Rozien-Dossier, “Apodisation,” in Progress in Optics, Vol. 3, E. Wolf, ed. (North-Holland, Amsterdam, 1964), pp. 35, 122.
  4. E. Hecht, 1998, Optics, (Addison Wesley Longman, Reading, Mass., 1998), pp. 443, 649.

1977 (1)

Hecht, E.

E. Hecht, 1998, Optics, (Addison Wesley Longman, Reading, Mass., 1998), pp. 443, 649.

Hutley, M. C.

M. C. Hutley, Diffraction Gratings (Academic, London, 1982), pp. 22, 31, 141.

Jaquinot, P.

P. Jaquinot, B. Rozien-Dossier, “Apodisation,” in Progress in Optics, Vol. 3, E. Wolf, ed. (North-Holland, Amsterdam, 1964), pp. 35, 122.

Nubbenmeyer, H.

Rozien-Dossier, B.

P. Jaquinot, B. Rozien-Dossier, “Apodisation,” in Progress in Optics, Vol. 3, E. Wolf, ed. (North-Holland, Amsterdam, 1964), pp. 35, 122.

Wende, B.

Appl. Opt. (1)

Other (3)

P. Jaquinot, B. Rozien-Dossier, “Apodisation,” in Progress in Optics, Vol. 3, E. Wolf, ed. (North-Holland, Amsterdam, 1964), pp. 35, 122.

E. Hecht, 1998, Optics, (Addison Wesley Longman, Reading, Mass., 1998), pp. 443, 649.

M. C. Hutley, Diffraction Gratings (Academic, London, 1982), pp. 22, 31, 141.

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Figures (2)

Fig. 1
Fig. 1

Geometry for the calculation of an area A N corresponding to a segment of the beam illuminating exactly N grooves. Only one quadrant of the grating face projected onto the incident beam is shown. The maximum beam height is D = 2Y, and the maximum beam width is 2X. The horizontal lines with vertical spacing d′ represent the edges of the grooves projected onto the incident beam. The narrow, hatched rectangle represents the area A N /4.

Fig. 2
Fig. 2

Comparison of the line-shape function f 1(γ, N) from Eq. (13) in the text with the corresponding degraded line-shape function g 1(γ, N) calculated from Eq. (12). Both curves are shown for N = 1000 but are nearly independent of N for N ≳ 100.

Tables (1)

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Table 1 Effects of Elliptical Beam Shape on Grating Spectrometer Resolution

Equations (23)

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N=D/d cos α=D/d,
ϕ=2π/λdsin α+sin β,
fγ, N.
 dγfγ, N=1,
A=0Xdx4yx.
A=N=2,4,N AN,
AN=4yNxN-xN+2,
yN=Nd/2,
xN=X1-yN/Y21/2, 0<N<N,
xN+2=0,
N+2d/2>Y>Nd/2.
gγ, N=N=2,4,NAN/Afγ, N.
f1γ, N=sinNγ/sinγ2/πN.
δγ=0.886π/N.
f2γ, N=a+bγ-a/b<γ<0+a-bγ0<γ<a/b, =0|γ|>a/b
f3γ, N=exp-γ2/2σ2/2π σ,
f4γ, N=1/δγ|γ|<δγ/2, =0|γ|>δγ/2.
Iβ/I0=sinc2πDβ/λ,
δβ=0.886λ/D.
mλ=dsin α+sin β,
mδλ=d cos βδβ.
R=2 tan β/δβ.
δβ=1.02λ/D.

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