Abstract

A new spectrometer design is presented in which the angular dispersion with respect to wave number is nearly constant. The spectrometer is a type of grism, a series combination of grating and prism, in which the constant parts of the dispersion terms add to one another but the slopes of the dispersions tend to cancel one another. A systematic method is presented for optimizing the grating and prism parameters. Expressions are given for the blaze efficiency and effective groove width of a transmission grating with generalized triangular grooves. A cross-dispersion technique is presented, eliminating overlapping grating orders. A design example is given for the visible region from 0.45 to 0.80 μm with essentially constant wave-number dispersion and a peak transmission of approximately 95%. This grism is well suited for measuring channeled spectra as generated by an optical stellar interferometer.

© 1990 Optical Society of America

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References

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  1. W. H. Steel, Interferometry, 2nd ed. (Cambridge U. Press, Cambridge, 1987).
  2. M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1971).
  3. D. J. Schroeder, Astronomical Optics (Academic, San Diego, 1987).
  4. Z. Knittl, Optics of Thin Films (Wiley, New York, 1976).
  5. I. H. Malitson, “Interspecimen comparison of the refractive index of fused silica,” J. Opt. Soc. Am. 55, 1205–1209 (1965).
    [CrossRef]
  6. W. A. Traub, N. P. Carleton, M. G. Lacasse, “Oblique diffraction grating equations,” J. Opt. Soc. Am. A (to be published).

1965 (1)

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1971).

Carleton, N. P.

W. A. Traub, N. P. Carleton, M. G. Lacasse, “Oblique diffraction grating equations,” J. Opt. Soc. Am. A (to be published).

Knittl, Z.

Z. Knittl, Optics of Thin Films (Wiley, New York, 1976).

Lacasse, M. G.

W. A. Traub, N. P. Carleton, M. G. Lacasse, “Oblique diffraction grating equations,” J. Opt. Soc. Am. A (to be published).

Malitson, I. H.

Schroeder, D. J.

D. J. Schroeder, Astronomical Optics (Academic, San Diego, 1987).

Steel, W. H.

W. H. Steel, Interferometry, 2nd ed. (Cambridge U. Press, Cambridge, 1987).

Traub, W. A.

W. A. Traub, N. P. Carleton, M. G. Lacasse, “Oblique diffraction grating equations,” J. Opt. Soc. Am. A (to be published).

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1971).

J. Opt. Soc. Am. (1)

Other (5)

W. A. Traub, N. P. Carleton, M. G. Lacasse, “Oblique diffraction grating equations,” J. Opt. Soc. Am. A (to be published).

W. H. Steel, Interferometry, 2nd ed. (Cambridge U. Press, Cambridge, 1987).

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1971).

D. J. Schroeder, Astronomical Optics (Academic, San Diego, 1987).

Z. Knittl, Optics of Thin Films (Wiley, New York, 1976).

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Figures (14)

Fig. 1
Fig. 1

Schematic diagram showing the appearance of spatially dispersed channeled spectrum from a prism, grating, and grism. On the left, a channeled spectrum is shown being produced by a Michelson spectral interferometer, although a Michelson stellar interferometer or even a Fabry–Perot interferometer would perform as well. On the right-hand side, the channeled spectrum intensities I(x) are shown as a function of focal-plane spatial coordinate x. The labels R, G, and B refer to equispaced wave numbers identifying intensity maxima, respectively, in the red, green, and blue regions of the channeled spectrum. (a) A prism disperses the channeled spectrum, more in the blue than in the red, so that the red fringes are crowded together. (b) A grating produces crowding of the blue fringes. (c) A grism disperses the spectrum so that the channeled spectrum fringes are equally spaced across the detector.

Fig. 2
Fig. 2

(a) Thin prism schematic showing beam deflection βp by a prism of index np and vertex angle E. Small deflection angles correspond to long wavelengths, as indicated. (b) Transmission grating schematic, for a grating with index ng, groove spacing σ, input angle α, and output angle βg. The refracted beam is of the order of m = 0; the diffracted beams form positive and negative orders lying below and above this direction, respectively. Small deflection angles, with respect to m = 0, correspond to short wavelengths, for all orders.

Fig. 3
Fig. 3

Grism schematic, showing ray paths from air (n = 1), to prism (np), to grating (ng), and back to air. The prism vertex angle is E. At the surface of the grating, the major and minor facet angles are δ and , respectively. The diffracted ray exits the grating at angle β. The sign convention is that counterclockwise angles are positive; thus at the input face of the prism, angles A and B are negative as drawn. Dashed lines indicate the directions of the prism normal (PN), grating normal (GN), and facet normal (FN).

Fig. 4
Fig. 4

Transmission grating geometry for case I with α and β. The facet spacing is σ, and the width of the unshadowed part of the major facet is b. In terms of the unshadowed length s, measured in the plane of the grating, the geometrical transmission factor is Tg = s/σ.

Fig. 5
Fig. 5

Case II with α and β. See Fig. 4.

Fig. 6
Fig. 6

Case III with α and β. See Fig. 4.

Fig. 7
Fig. 7

Case IV with α and β. See Fig. 4.

Fig. 8
Fig. 8

Index of refraction of 12 prism glasses and 2 grating resins. The glasses and resins are selected as described in the text.

Fig. 9
Fig. 9

Relationship between the face height of the grism (h) and the ratio of blue to red fringe spacing (r) for a range of glasses and grating groove frequencies. The purpose of this diagram is to determine which combinations of glass and groove frequency most closely approach the ideal grism condition, which is a value of r = 1 and a minimum value of h; this condition gives nearly constant dispersion and minimum physical size, respectively, for r and h. The prism vertex angle is 60 deg, and the angle of incidence is set for minimum deviation at the wavelength 0.58 μm. The best combination that is shown here is SF52, 100 grooves/mm; the next best is SF2, 75 grooves/mm.

Fig. 10
Fig. 10

Grism transmission efficiency in the blazed (m = 1) order, as a function of wavelength, for values of the major facet angle δ from 1 to 5 deg (dashed curves) and for right-angled grooves, i.e., = δ. The prism is SF2, with a 60-deg vertex, operated at minimum deviation for λ = 0.58 μm. The uppermost solid curve shows the results for an optimum groove shape, with δ = 3.3 deg and = 40 deg. Ignoring Fresnel reflection losses, the peak transmission here is 96.5%.

Fig. 11
Fig. 11

Normalized fringe spacing is shown as a function of wavelength. Fringe spacing is proportional to dispersion, as discussed in the text. Four curves are shown. (a) In the ideal case, fringe spacing is constant and independent of wavelength. (b) For the optimized grism, fringe spacing varies by only 16% across the spectrum. (c) A prism of SF2, with approximately twice the height of the grism, will have fringe spacings that vary by a factor of greater than 2.17. (d) A grating of approximately the same size as the grism will have fringe spacings varying by approximately a factor of 3.16.

Fig. 12
Fig. 12

The transmission efficiency BF of all grism orders from m = +5 to m = −5 is shown as a function of grism output angle β. Within each order, the plotted points are equally spaced in wave number, from λ = 0.45 to λ = 0.80 μm. Longer wavelengths are labeled with filled squares and shorter wavelengths with open squares. The blaze order is m = 1. Within this wavelength range, the only orders that overlap the blaze order are m = 0 and m = 2. Note the peaking of efficiency in the center of order m = 1, and the extremely low efficiencies at the same wavelengths in many other orders (m = −1, 0, 2, 3, 4). Also note the reversal of wavelengths in orders m = −4 to m = −1 and the folding over of the order m = −2 and m = −1.

Fig. 13
Fig. 13

Perspective sketch of a diffraction grating, showing an obliquely incident ray in the x < 0 half-space, diffracted by grating grooves z that are parallel to the z axis. The zero-order transmitted ray (x > 0) is shown; the corresponding reflected ray (x < 0) is not shown. For a given wavelength and incident direction (α, ϕ), all diffracted rays fall on the surface of a half-cone, as is shown. The cone axis is parallel to the grooves. In a grism, the incident rays will have a range of angles α = α(λ), generated by the dispersion of a preceding prism. The diffracted rays will have a cone angle η = η(λ) that varies slowly with wavelength within each order; see Fig. 14 for an example.

Fig. 14
Fig. 14

Focal-plane schematic, showing the effect of cross-dispersion. Order separation in the η direction is achieved by introducing a small twist ϕ of the grating with respect to the prism. This causes each spectral order m to tilt by angle Ψm in the detector plane. (Without cross-dispersion, the tilt Ψm is zero, and all orders overlap along the dispersion axis ρ.)

Equations (75)

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C ( ν ) = S ( ν ) [ 1 + V cos ( 2 π ν / Q + ϕ ) ] .
Q = l 1 .
β p ( n p 1 ) E ,
d β p / d ν E d n p / d ν .
d n / d λ D p / λ 3
d n / d ν D p ν .
d β p / d ν E D p ν .
n g sin α sin β g = m λ / σ ,
β g n g α m λ / σ = n g α m / ( σ ν ) ,
d β g / d ν m / ( σ ν 2 ) .
β ( ν ) β p + β g ,
β ( ν ) d β / d ν E D p ν + m σ 1 ν 2 .
ν 0 = ( 2 m E D p σ ) 1 / 3 .
β 0 3 m σ ν 0 2 .
β ( ν ) β 0 [ 2 3 ( ν ν 0 ) + 1 3 ( ν 0 ν ) 2 ] .
ν + = ν 0 2 , ν = ν 0 / 2 .
β ( ν + ) 1.11 β 0 , β ( ν ) 1.14 β 0 .
n σ δ σ δ + m λ 0 ,
δ m / [ σ ν 0 ( n g 1 ) ] ,
D p 0.02 μ m 2 ,
λ 0 0.60 μ m , ν 0 = 1.67 μ m 1 ,
E 1 rad ,
m = 1 ,
n g 1.50.
σ 22 μ m groove spacing ,
β 0 2.9 deg / μ m 1 dispersion ,
δ 3.2 deg facet tilt .
( ν + ν ) β 0 = ν 0 β 0 / 2 1 / 2 3.4 deg .
α = δ + ϑ ,
β = δ + ϑ .
n g sin α sin β = m λ / σ ,
BF = T g ( sin γ γ ) 2 .
γ = π b λ ( n g sin ϑ sin ϑ ) ,
, δ 0 , α , β ,
, δ 0 , α , β ,
, δ 0 , α , β ,
, δ 0 , α , β .
b σ = cos cos ( δ ) ,
b σ = cos β cos ( β δ ) ,
b σ = cos α cos ( α δ ) ,
b σ = cos α cos ( α δ ) + cos β cos ( β δ ) cos cos ( δ ) .
T g = s / σ .
T g = b σ cos ( α δ ) cos α .
T s ( A , B ) = 1 [ sin ( A B ) sin ( A + B ) ] 2 ,
T p ( A , B ) = 1 [ tan ( A B ) tan ( A + B ) ] 2 .
T s = T s ( A , B ) T s ( C , D ) T s ( ϑ , ϑ ) ,
T p = T p ( A , B ) T p ( C , D ) T p ( ϑ , ϑ ) .
T F = ( T s + T p ) / 2 .
δ F = 2 π n 1 t ν cos F .
U s = n 1 cos F ,
X s = n 0 cos A cos δ F ,
W s = n 2 cos B cos δ F ,
Y s = n 0 cos A n 2 cos B sin δ F / U s ,
V s = U s sin δ F .
T s = 1 ( X s W s ) 2 + ( Y s V s ) 2 ( X s + W s ) 2 + ( Y s + V s ) 2 .
U p = n 1 / cos F ,
X p = ( n 0 / cos A ) cos δ F ,
W p = ( n 2 / cos B ) cos δ F ,
Y p = ( n 0 / cos A ) ( n 2 / cos B ) sin δ F / U p ,
V p = U p sin δ F ,
T p = 1 ( X p W p ) 2 + ( Y p V p ) 2 ( X p + W p ) 2 + ( Y p + V p ) 2 .
h = w det ( f / n o ) ( β + β ) cos β 0 .
r = β + β ,
sin A = n p ( ν 0 ) sin ( E / 2 ) ,
n g sin α = ( n p 2 sin 2 A ) 1 / 2 sin E sin A cos E .
sin η = n g sin α sin ϕ ,
n g sin α cos ϕ cos η sin ρ = m λ / σ .
B = C = E / 2 ,
sin A = n p ( λ 0 ) sin ( E / 2 ) .
x = f [ sin E cos ( E / 2 ) Δ n p m σ Δ λ λ σ Δ m ] / cos ρ 0 ,
y = f sin E cos ( E / 2 ) ϕ Δ n p .
ϕ = cos ( E / 2 ) sin E Δ y ( 1,2 ) / f Δ n p ( 1,2 ) ,
tan Ψ m = Δ y m ( 1,2 ) Δ x m ( 1,2 ) .
ϕ = Ψ m [ 1 m cos ( E / 2 ) Δ λ σ sin E Δ n p ] .
ϕ 2.81 Ψ 1 .

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