Abstract

In this paper, the imaging performance of an Offner concentric imaging spectrometer is analyzed when the spectrometer entrance slit is disposed arbitrarily on the plane that is parallel to the grating grooves and contains the common center of curvature. Astigmatism-corrected designs are obtained for off-plane incidence on the grating if one point on the slit is located on the Rowland circle of the primary mirror. In this case, the combined system of primary mirror plus diffraction grating provides two astigmatic line images oriented parallel and orthogonal to the plane of diffraction, with the former located on the same plane as the slit. Consequently, these images can be brought to a single focus on this plane by the tertiary mirror if its radius of curvature is chosen properly. In addition, coma aberration is simultaneously removed. These results can be applied to the design of two-mirror or three-mirror spectrometers, generalizing the concept of the best imaging circle and providing solutions to get anastigmatic imaging for two object points and two wavelengths.

© 2011 Optical Society of America

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References

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  1. M. P. Chrisp, “Convex diffraction grating imaging spectrometer,” U.S. patent 5,880,834 (9 March 1999).
  2. X. Prieto-Blanco, C. Montero-Orille, B. Couce, and R. de la Fuente, “Analytical design of an Offner imaging spectrometer,” Opt. Express 14, 9156–9168 (2006).
    [CrossRef] [PubMed]
  3. R. L. Lucke, “Out-of-plane dispersion in an Offner spectrometer,” Opt. Eng. 46, 073004 (2007).
    [CrossRef]
  4. X. Prieto-Blanco, C. Montero-Orille, H. González-Nuñez, M. D. Mouriz, E. L. Lago, and R. de la Fuente, “The Offner imaging spectrometer in quadrature,” Opt. Express 18, 12756–12769(2010).
    [CrossRef] [PubMed]
  5. H. G. Beutler, “The theory of the concave grating,” J. Opt. Soc. Am. 35, 311–350 (1945).
    [CrossRef]
  6. T. Namioka, “Theory of the concave grating. I,” J. Opt. Soc. Am. 49, 446–460 (1959).
    [CrossRef]
  7. W. Werner, “The geometric optical aberration theory of diffraction gratings,” Appl. Opt. 6, 1691–1699 (1967).
    [CrossRef] [PubMed]
  8. C. H. F. Velzel, “A general theory of the aberrations of diffraction gratings and gratinglike optical instruments,” J. Opt. Soc. Am. 66, 346–353 (1976).
    [CrossRef]
  9. S. Morozumi, “Aberration theory of diffraction gratings,” Optik 53, 75–88 (1979).
  10. S. D. Brorson and H. A. Haus, “Diffraction gratings and geometrical optics,” J. Opt. Soc. Am. B 5, 247–248 (1988).
    [CrossRef]
  11. X. Prieto-Blanco, C. Montero-Orille, H. González-Núñez, M. D. Mouriz, E. López Lago, and R. de la Fuente, “Imaging with classical spherical diffraction gratings: the quadrature configuration,” J. Opt. Soc. Am. A 26, 2400–2409 (2009).
    [CrossRef]
  12. G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers, 2nd ed. (Dover, 2000), Chap. 17.

2010 (1)

2009 (1)

2007 (1)

R. L. Lucke, “Out-of-plane dispersion in an Offner spectrometer,” Opt. Eng. 46, 073004 (2007).
[CrossRef]

2006 (1)

2000 (1)

G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers, 2nd ed. (Dover, 2000), Chap. 17.

1988 (1)

1979 (1)

S. Morozumi, “Aberration theory of diffraction gratings,” Optik 53, 75–88 (1979).

1976 (1)

1967 (1)

1959 (1)

1945 (1)

Beutler, H. G.

Brorson, S. D.

Chrisp, M. P.

M. P. Chrisp, “Convex diffraction grating imaging spectrometer,” U.S. patent 5,880,834 (9 March 1999).

Couce, B.

de la Fuente, R.

González-Nuñez, H.

González-Núñez, H.

Haus, H. A.

Korn, G. A.

G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers, 2nd ed. (Dover, 2000), Chap. 17.

Korn, T. M.

G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers, 2nd ed. (Dover, 2000), Chap. 17.

Lago, E. L.

Lago, E. López

Lucke, R. L.

R. L. Lucke, “Out-of-plane dispersion in an Offner spectrometer,” Opt. Eng. 46, 073004 (2007).
[CrossRef]

Montero-Orille, C.

Morozumi, S.

S. Morozumi, “Aberration theory of diffraction gratings,” Optik 53, 75–88 (1979).

Mouriz, M. D.

Namioka, T.

Prieto-Blanco, X.

Velzel, C. H. F.

Werner, W.

Appl. Opt. (1)

J. Opt. Soc. Am. (3)

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

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

Opt. Eng. (1)

R. L. Lucke, “Out-of-plane dispersion in an Offner spectrometer,” Opt. Eng. 46, 073004 (2007).
[CrossRef]

Opt. Express (2)

Optik (1)

S. Morozumi, “Aberration theory of diffraction gratings,” Optik 53, 75–88 (1979).

Other (2)

G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers, 2nd ed. (Dover, 2000), Chap. 17.

M. P. Chrisp, “Convex diffraction grating imaging spectrometer,” U.S. patent 5,880,834 (9 March 1999).

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Figures (9)

Fig. 1
Fig. 1

Offner spectrometer. S, slit; M 1 , primary concave mirror; G, convex diffraction grating; M 3 , tertiary concave mirror.

Fig. 2
Fig. 2

Diagram of a convex grating, illustrating anastigmatic illumination by light reflected in a concentric concave mirror (not shown). O m and O s are the centers of the meridional and sagittal images provided by the mirror, respectively. Regarding the grating, these images can be considered as virtual objects.

Fig. 3
Fig. 3

Diagram of a convex grating showing image formation in the disposition discussed in Subsection 2A. Light reflected by the concave mirror (not shown) comes to the grating. In the drawing, some virtual rays are shown: meridional rays, which converge to the virtual image O m located at a distance R cos θ from the grating vertex, and sagittal rays converging to the virtual image O s , located at a distance R / cos θ from the vertex. These rays correspond to the virtual astigmatic source. The light diffracted from the grating appears to diverge from an astigmatic virtual image with focus at I m and I s . These points are located at distances R cos θ and R / cos θ from the grating vertex, respectively. Virtual rays diverging from these points are also shown.

Fig. 4
Fig. 4

Drawing of an Offner spectrometer showing the path of the chief ray for an arbitrary wavelength. The object point O is on the Rowland circle of the primary mirror, M 1 .

Fig. 5
Fig. 5

View of the (a) plane of incidence and (b) diffraction in the Offner spectrometer. O is the object point. I 1 m , I 1 s are the centers of its virtual meridional and sagittal images, respectively, provided by the primary mirror M 1 . O 3 m , O 3 s are the centers of the virtual meridional and sagittal images, respectively, provided by the convex grating G. They are the centers of virtual objects regarding the tertiary mirror M 3 . I is the final anastigmatic image.

Fig. 6
Fig. 6

View along the optical axis of an Offner spectrometer in an antisymmetric configuration. O is the design object point, O is its specular image, and I is its spectral image at the design wavelength.

Fig. 7
Fig. 7

Spot diagrams of the anastigmatic image in an antisymmetric Offner spectrometer arrangement. Units of the square boxes are in millimeters. (a)  γ = 0 ; (b)  γ = 15 ; (c)  γ = 30 ; (d)  γ = 45 ; (e)  γ = 60 ; (f)  γ = 75 ; (g)  γ = 85 . The spectrometer parameters are R 1 = R 3 = 100 mm , R 2 = 52 mm , g = 300 lines / mm , and f / 2.4 . Note that the design wavelength is different in each diagram.

Fig. 8
Fig. 8

Illustration of a configuration anastigmatic at two wavelengths. The view is along the optical axis.

Fig. 9
Fig. 9

Illustration of the off-center optimization process in an in-plane configuration.

Equations (31)

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U m ( x , y , z ) = C m s exp [ i 2 π λ F o + F i + m g λ S ( x , y ) ] d x d y ,
F o ( x , 0 , z ) = ( x x m ) 2 + ( z z m ) 2 .
F o ( x , y , z ) F o ( x , ρ ) = ( x x m ) 2 + ( ρ R + z m ) 2 ,
F ( x , y , z ; x , y , z ) = F o ( x , ρ ) + F i ( x , y , z ; x , y , z ) + m g λ S ( x , y ) = ( x x m ) 2 + ( ρ R + z m ) 2 + ( x x ) 2 + ( y y ) 2 + ( z z ) 2 + m g λ ( x cos ϕ + y sin ϕ ) .
F o y = 0 , F o x | V = x m r ,
F i x | V = x r , F i y | V = y r ,
x m r + x r = m g λ cos ϕ ,
y r = m g λ sin ϕ .
sin θ + sin θ cos α = m g λ cos ϕ ,
sin θ sin α = m g λ sin ϕ ,
F ( x , y , z ; x , y , z ) = F ( 0 , 0 , 0 ; x , y , z ) + 1 2 ( 2 F x 2 x 2 + 2 2 F x y x y + 2 F y 2 y 2 ) + = r + r + 1 2 ( 2 F x 2 x 2 + 2 2 F x y x y + 2 F y 2 y 2 ) + .
[ 2 F x 2 2 F y 2 ( 2 F x y ) 2 ] V = 0.
2 F o y 2 | V = 2 F o x y | V = 0 ,
2 F o x 2 | V = 1 r [ 1 z m R ( x m r ) 2 ] = cos θ ( cos θ r 1 R ) ,
2 F i x 2 | V = 1 r [ 1 z R ( x r ) 2 ] = 1 r cos θ R cos 2 α sin 2 θ r ,
2 F i y 2 | V = 1 r [ 1 z R ( y r ) 2 ] = 1 r cos θ R sin 2 α sin 2 θ r ,
2 F i x y | V = x y r 3 = sin α cos α sin 2 θ r .
cos θ ( cos θ r 1 R ) ( 1 sin 2 α sin 2 θ r cos θ R ) + cos θ ( 1 r cos θ R ) ( cos θ r 1 R ) = 0.
r = R / cos θ ,
cos 2 θ r + cos 2 θ r = cos θ + cos θ R .
r = R cos θ ,
cos 2 θ r + 1 r = cos θ + cos θ R .
r = R cos θ ,
r = R / cos θ .
R 1 sin θ 1 = R 2 sin θ 2 ,
R 2 sin θ 2 = R 3 sin θ 3 ,
sin θ 2 cos γ + sin θ 2 cos γ = m g λ ,
sin θ 2 sin γ + sin θ 2 sin γ = 0 ,
( H O R 1 ) 2 + ( R 1 2 R 2 ) 2 = 1
( H I R 3 ) 2 + ( R 3 2 R 2 ) 2 = 1 ,
sin θ 2 = m g λ d 2 cos γ .

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