Abstract

We present the analytical design of an imaging spectrometer based on the three-concentric-mirror (Offner) configuration. The approach presented allows for the rapid design of this class of system. Likewise, high-optical-quality spectrometers are obtained without the use of aberration-corrected gratings, even for high speeds. Our approach is based on the calculation of both the meridional and the sagittal images of an off-axis object point. Thus, the meridional and sagittal curves are obtained in the whole spectral range. Making these curves tangent to each other for a given wavelength results in a significant decrease in astigmatism, which is the dominant residual aberration. RMS spot radii less than 5 µm are obtained for speeds as high as f/2.5 and a wavelength range of 0.41.0 µm. A design example is presented using a free interactive optical design tool.

© 2006 Optical Society of America

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References

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  1. N. Gat, "Imaging Spectroscopy using tunable filters: a review," Proc. SPIE 4056, 50-64 (2000). G. A. Shaw, H. K. Burke, "Spectral imaging for remote sensing", Lincoln Laboratory Journal 14,3-28, (2003).
    [CrossRef]
  2. P. Mouroulis and M. McKerns, "Pushbroom imaging spectrometer with high spectroscopic data fidelity: experimental demonstration," Opt. Eng. 39,808-816 (2000).
    [CrossRef]
  3. D. Kwo, G. Lawrence, and M. Chrisp, "Design of a grating spectrometer from a 1:1 Offner mirror system," Proc. SPIE 818,275-279 (1987).
  4. M.P. Chrisp, Convex diffraction grating imaging spectrometer, U.S. Patent 5,880,834.
  5. W.J. Smith, Modern Optical Engineering (McGraw-Hill, Inc., New York, 1990).
  6. H. Beutler, "The theory of the concave grating," J. Opt. Soc. Am. 35, 311-350 (1945).W. T.Welford, "Aberration theory of gratings and grating mountings" in Progress in Optics, E. Wolf, ed., Vol. IV, 241-280, North-Holland, Amsterdam (1965).
    [CrossRef]
  7. M.C. Hutley, Diffraction Gratings (Academic Press, London, 1982).
  8. G.A. Korn and T.M. Korn, Mathematical Handbook for Scientists and Engineers, second edition (Dover Publications, Inc., Mineola, New York, 2000), Chap. 17.1.
  9. A copy of the macro used in this paper can be obtained by contacting the authors.
  10. J.M. Howard, "Optical Design using computer graphics," Appl. Opt. 40,3225-3231 (2001).
    [CrossRef]
  11. OSLO is a registered trademark of Lambda Research Corporation, 80 Taylor Street, P.O. Box 1400, Littleton, Mass. 01460.
  12. C. Davis, J. Bowles, R. Leathers, D. Korwan, TV Downes,W. Snyder,W. Rhea,W. Chen, J. Fisher, P. Bissett, and R.A. Reisse, "Ocean PHILLS Hyperspectral Imager: Design, Characterization, and Calibration," Opt. Express 10,210-221 (2002).
    [PubMed]

2003 (1)

N. Gat, "Imaging Spectroscopy using tunable filters: a review," Proc. SPIE 4056, 50-64 (2000). G. A. Shaw, H. K. Burke, "Spectral imaging for remote sensing", Lincoln Laboratory Journal 14,3-28, (2003).
[CrossRef]

2002 (1)

2001 (1)

2000 (1)

P. Mouroulis and M. McKerns, "Pushbroom imaging spectrometer with high spectroscopic data fidelity: experimental demonstration," Opt. Eng. 39,808-816 (2000).
[CrossRef]

1987 (1)

D. Kwo, G. Lawrence, and M. Chrisp, "Design of a grating spectrometer from a 1:1 Offner mirror system," Proc. SPIE 818,275-279 (1987).

Bissett, P.

Bowles, J.

Chen, W.

Chrisp, M.

D. Kwo, G. Lawrence, and M. Chrisp, "Design of a grating spectrometer from a 1:1 Offner mirror system," Proc. SPIE 818,275-279 (1987).

Davis, C.

Downes, TV

Fisher, J.

Gat, N.

N. Gat, "Imaging Spectroscopy using tunable filters: a review," Proc. SPIE 4056, 50-64 (2000). G. A. Shaw, H. K. Burke, "Spectral imaging for remote sensing", Lincoln Laboratory Journal 14,3-28, (2003).
[CrossRef]

Howard, J.M.

Korwan, D.

Kwo, D.

D. Kwo, G. Lawrence, and M. Chrisp, "Design of a grating spectrometer from a 1:1 Offner mirror system," Proc. SPIE 818,275-279 (1987).

Lawrence, G.

D. Kwo, G. Lawrence, and M. Chrisp, "Design of a grating spectrometer from a 1:1 Offner mirror system," Proc. SPIE 818,275-279 (1987).

Leathers, R.

McKerns, M.

P. Mouroulis and M. McKerns, "Pushbroom imaging spectrometer with high spectroscopic data fidelity: experimental demonstration," Opt. Eng. 39,808-816 (2000).
[CrossRef]

Mouroulis, P.

P. Mouroulis and M. McKerns, "Pushbroom imaging spectrometer with high spectroscopic data fidelity: experimental demonstration," Opt. Eng. 39,808-816 (2000).
[CrossRef]

Reisse, R.A.

Rhea, W.

Snyder, W.

Appl. Opt. (1)

Lincoln Laboratory Journal (1)

N. Gat, "Imaging Spectroscopy using tunable filters: a review," Proc. SPIE 4056, 50-64 (2000). G. A. Shaw, H. K. Burke, "Spectral imaging for remote sensing", Lincoln Laboratory Journal 14,3-28, (2003).
[CrossRef]

Opt. Eng. (1)

P. Mouroulis and M. McKerns, "Pushbroom imaging spectrometer with high spectroscopic data fidelity: experimental demonstration," Opt. Eng. 39,808-816 (2000).
[CrossRef]

Opt. Express (1)

Proc. SPIE (1)

D. Kwo, G. Lawrence, and M. Chrisp, "Design of a grating spectrometer from a 1:1 Offner mirror system," Proc. SPIE 818,275-279 (1987).

Other (7)

M.P. Chrisp, Convex diffraction grating imaging spectrometer, U.S. Patent 5,880,834.

W.J. Smith, Modern Optical Engineering (McGraw-Hill, Inc., New York, 1990).

H. Beutler, "The theory of the concave grating," J. Opt. Soc. Am. 35, 311-350 (1945).W. T.Welford, "Aberration theory of gratings and grating mountings" in Progress in Optics, E. Wolf, ed., Vol. IV, 241-280, North-Holland, Amsterdam (1965).
[CrossRef]

M.C. Hutley, Diffraction Gratings (Academic Press, London, 1982).

G.A. Korn and T.M. Korn, Mathematical Handbook for Scientists and Engineers, second edition (Dover Publications, Inc., Mineola, New York, 2000), Chap. 17.1.

A copy of the macro used in this paper can be obtained by contacting the authors.

OSLO is a registered trademark of Lambda Research Corporation, 80 Taylor Street, P.O. Box 1400, Littleton, Mass. 01460.

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Figures (9)

Fig. 1.
Fig. 1.

Offner configuration for an imaging spectrometer. The object is at the top left. It is a slit parallel to y-axis. The grating lies upon the convex mirror and its grooves are also parallel to y-axis. The reference ray is plotted in dot-dashed line.

Fig. 2.
Fig. 2.

a) Virtual object and meridional image location on the Rowland circle. b) Schematic to calculate sagittal image location through the grating for a virtual object point.

Fig. 3.
Fig. 3.

Rowland circles for the concentric Offner spectrometer showing meridional image locations. Center of curvature of the three elements (C) and both object (O) and image (I M) locations are indicated.

Fig. 4.
Fig. 4.

Location of the sagittal images throughout the concentric Offner spectrometer. For comparison the location of the final meridional image is also shown.

Fig. 5.
Fig. 5.

Location of both meridional and sagittal final images for a wavelength λ another than the central one ( λ ¯ ). The meridional image lies on its own Rowland’s circle. Both meridional and sagittal curves are drawn.

Fig. 6.
Fig. 6.

Locations (1, 2, and 3) where vignetting may appear for small θ̄3 angles

Fig. 7.
Fig. 7.

Steps for carrying out the spectrometer design according to the theory presented in Section 2. The equation number used at each step is indicated.

Fig. 8.
Fig. 8.

Outputs of the implementation in OSLO-EDU of the design procedure. We show the results for the spectrometer whose specifications are in Table 1.

Fig. 9.
Fig. 9.

Worst RMS spot radius versus f-number for m=±1 diffraction orders (solid lines) and the specifications from Table 1. The diffraction limit (dashed line) is also represented for wavelength λ+=1000 nm. Note that diffraction limits the minimum spot size for f/#>2.7.

Tables (2)

Tables Icon

Table 1. Specifications of the spectrometer to be designed.

Tables Icon

Table 2. Prescription data

Equations (29)

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sin θ + sin θ = m λ p ,
cos 2 θ r + cos 2 θ r M = cos θ + cos θ R
1 r + 1 r S = cos θ + cos θ R .
r = R cos θ r M = R cos θ .
1 r = cos ( φ θ ) R cos φ
1 r S = cos ( r S θ ) R cos φ S .
K = sin θ tan φ = sin θ tan φ S .
π 2 2 θ 1 + ( π θ 2 ) + π 2 + φ = 2 π φ = θ 2 + 2 θ 1 .
CO = R 1 sin θ 1 = R 2 sin θ 2 .
π 2 2 θ 3 + ( π θ 2 ) + π 2 + φ M = 2 π φ M = θ 2 + 2 θ 3
CI M = R 2 sin θ 2 = R 3 sin θ 3 .
sin θ 2 tan φ = sin θ 2 tan φ S .
CI S = CI M cos ( φ M φ S )
Astig = CI S sin ( φ M φ S ) .
φ ̅ = φ ̅ S = φ ̅ M = θ ̅ 2 2 θ ̅ 3 .
CI ̅ = CI ̅ M = CI ̅ S .
d Astig d λ λ ̅ = 0 d ( φ M φ S ) d θ 2 θ ̅ 2 = 0 ,
d φ M d θ 2 = 1 2 tan θ 3 tan θ 2
d φ S d θ 2 = sin φ S cos φ S tan θ 2 .
tan ( φ ̅ + 2 θ ̅ 3 ) 2 tan θ ̅ 3 + sin 2 φ ̅ 2 = 0 .
sin 3 θ ̅ 3 cos θ ̅ 3 + tan φ ̅ tan θ ̅ 3 cos ( 2 θ ̅ 3 ) tan 2 φ ̅ + 1 + 2 sin 2 θ ̅ 3 2 tan 3 φ ̅ = 0 .
tan φ ̅ n = sin 3 θ ̅ 3 cos θ 3 + tan θ ̅ 3 cos ( 2 θ ̅ 3 ) tan 2 φ ̅ n 1 + 1 + 2 sin 2 θ ̅ 3 2 tan 3 φ ̅ n 1 ,
tan β = CI M d φ M d CI M CI ̅ = CI ̅ d φ M d θ 2 θ ̅ 2 d CI M d θ 2 θ ̅ 2 = tan θ ̅ 2 2 tan θ ̅ 3 ,
α = φ ̅ arctan ( sin 2 φ ̅ 2 ) .
( sin θ 2 + sin θ 2 ) = mp ( λ + λ ) = mp Δ λ .
R 2 ( sin θ 2 + sin θ 2 ) = CI M + CI M h spec ,
R 2 = h spec mp Δ λ
( x i 2 + z i 2 ) 1 2 > R 2 i = 1 , 2
( x 3 2 + z 3 2 ) 1 2 <min ( R 1 , R 3 ) .

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