## Analytical design of an Offner imaging spectrometer

Optics Express, Vol. 14, Issue 20, pp. 9156-9168 (2006)

http://dx.doi.org/10.1364/OE.14.009156

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### 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 *µ*m are obtained for speeds as high as *f/2.5* and a wavelength range of 0.4*–*1.0 µm. A design example is presented using a free interactive optical design tool.

© 2006 Optical Society of America

## 1. Introduction

1. N. Gat, “Imaging Spectroscopy using tunable filters: a review,” Proc. SPIE4056, 50–64 (2000). G. A. Shaw and H. K. Burke, “Spectral imaging for remote sensing”, Lincoln Laboratory Journal14, 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]

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

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]

## 2. Theory of the design

*θ*is the incidence angle,

*θ*

^{′}the diffraction angle for the maximum of order

*m*, λ the wavelength and

*p*the number of grooves by unit of length. The usual sign convention has been used (see for example [5]). The meridional image of an off-axis object point produced by a reflective grating satisfies[7]

*R*is the curvature radius of the grating;

*r*and

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]

*m*=0). It is advisable that Rowland’s condition is satisfied because it ensures a meridional image free of coma[6

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]

*m*≠0) if object point is on the x-axis.

*I*

_{M}) meridional images laying on such circles. The object lies on the first Rowland circle if the angle between CO and the reference ray is

*π*/2. That follows from the geometric property which holds that a triangle is rectangle if its hypotenuse is the diameter of its circumscribed circle. Taking the quadrangle AOCG we get

_{1}<0. Furthermore the distance CO verifies

*φ̄*=

*φ̄*

^{′}=

*θ*

_{2}+2

*θ*

_{1}=

*θ̄*

_{3}=0, that is, the reference ray in both the object and image space and the z-axis are parallel. Such a particular solution leads to the design proposed by Chrisp et al. [4].

*φ̄*

^{′}. After some algebra we obtain a cubic equation in tan

*φ̄*

^{′}:

*φ̄*

^{′}. Moreover for typical values of

*θ̄*

_{3}(|

*θ̄*

_{3}|<25°)

*CI*. However we also have to determine the tilt angle of the image plane. We choose the image plane to be tangent to the meridional curve at

*α*in Fig. 5. To obtain

*α*we first find

*β*, the angle between the tangent to the image curves and the position vector

*C⃗I*

_{M}at

*CI*

_{M}and

*φ*

^{′}are the polar coordinates of the meridional curve we have (see for example [8])

*α*=

*β*+

*φ*

^{′}. Thus combining Eq. (14), (19) and (22) we obtain

^{-}, λ

^{+}) and subtract them:

*h*

_{spec}is the size of the spectral image and the last approximation can be made because of the small values which

*φ*

^{′}takes. The combination of these last Eqs. provides

*R*

_{2}can be regarded as a scaling factor).

_{3}and after choosing the desired specifications of the spectrometer (spectral range, image size, f-number and size of the device) the above Eqs. allow to obtain its geometric parameters. Namely, both curvature and aperture radius of elements, grating density, both slit and image locations, and image plane tilting. On the other hand the value of

_{3}is fixed through the vignetting analysis. This is considered in the following section.

## 3. Vignetting analysis

*x*

_{i}and

*z*

_{i}as a function of

*θ̄*

_{3}is long and tedious. Moreover the Eqs. derived do not allow to calculate

10. J.M. Howard, “Optical Design using computer graphics,” Appl. Opt. **40**, 3225–3231 (2001). [CrossRef]

*θ̄*

_{3}which leads to no vignetting. This also allows for rapid design. Some results using such a software are shown in the following Section.

## 4. Design procedure

*θ̄*

_{3}. Then the Eqs. derived above are used in the indicated order. As soon as all the parameters are determined calculation of vignetting is carried out. If vignetting exists then |

*θ̄*

_{3}| must be increased, the procedure starts again, and so on. The procedure ends when |

*θ̄*

_{3}| is high enough to avoid vignetting. As we have mentioned in the previous section calculation of vignetting is not direct. So we have implemented the design procedure in the macro language of OSLO-EDU, a free version of OSLO lens design software[11]. OSLO provides a macro language (CCL) for building interactive interfaces as well as the conventional tools for lens design.

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

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]

^{+}and λ

^{-}is very different from each other another design wavelength should be selected.

*Slider Window*is the interactive window to input the specifications. Particularly angle

*θ̄*

_{3}can be modified until no vignetting occurs. In this case it does for |

*θ̄*

_{3}|>13.437° but to allow for building tolerances it is safer to let |

*θ̄*

_{3}|=15°. Whether or not vignetting occurs is checked in the

*Lens Drawing*window. The rays corresponding to the light which is diffracted more towards the object are plotted in this window (λ

^{+}=1

*µm*in our example). Finally the

*Spot Diagram Analysis*window shows the image spots for this wavelength and for three field points (0 mm, 3.08 mm and 4.4 mm). The Airy’s disk boundary is also plotted as a black circle. The optical quality of the design is obvious regarding this figure. A completer analysis reveals that the worst RMS spot radius for both wavelengths considered (0.4 µm, 0.7 µm, and 1.0 µm) and for the three field points mentioned is 4.74 µm which is close to the diffraction limit (3.18 µm for λ

^{+}=1

*µm*). A spectral resolution less than 0.9 nm can be achieved with such a spot size and for the given spectral range. Moreover if we decrease |

*θ*

_{3}| to 13.5° (which is in the range to avoid vignetting) the worst RMS spot radius is reduced to 3.68

*µ*m and the spectral resolution to less than 0.7 nm. We stress that in both cases we have slightly z-shifted the image plane to get more rounded spots. This shift changes the spot radii very little, actually it increases slightly the worst RMS value. The design prescription for |

*θ̄*

_{3}|=15° is given in Table 2. We

**39**, 808–816 (2000). [CrossRef]

*µ*m.

*θ̄*

_{3}| is the smaller angle that leads to no vignetting. It can be seen from this figure that reasonable results are obtained even for very fast speeds. For example a worst RMS spot radius of ~12

*µm*is obtained for

*f*/2.0 and

*m*=1. This spot radius is similar to the RMS value of the commercially available VS-15 spectrometer which uses an aberration corrected grating[12

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]

## 5. Conclusions

## Acknowledgments

## References and links

1. | N. Gat, “Imaging Spectroscopy using tunable filters: a review,” Proc. SPIE4056, 50–64 (2000). G. A. Shaw and H. K. Burke, “Spectral imaging for remote sensing”, Lincoln Laboratory Journal14, 3–28, (2003). [CrossRef] |

2. | P. Mouroulis and M. McKerns, “Pushbroom imaging spectrometer with high spectroscopic data fidelity: experimental demonstration,” Opt. Eng. |

3. | D. Kwo, G. Lawrence, and M. Chrisp, “Design of a grating spectrometer from a 1:1 Offner mirror system,” Proc. SPIE |

4. | M.P. Chrisp, |

5. | W.J. Smith, |

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 |

7. | M.C. Hutley, |

8. | G.A. Korn and T.M. Korn, |

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. |

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 |

**OCIS Codes**

(050.1950) Diffraction and gratings : Diffraction gratings

(120.6200) Instrumentation, measurement, and metrology : Spectrometers and spectroscopic instrumentation

(220.1000) Optical design and fabrication : Aberration compensation

(220.4830) Optical design and fabrication : Systems design

**ToC Category:**

Instrumentation, Measurement, and Metrology

**History**

Original Manuscript: May 15, 2006

Revised Manuscript: August 18, 2006

Manuscript Accepted: August 21, 2006

Published: October 2, 2006

**Citation**

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)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-20-9156

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### References

- 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]
- P. Mouroulis and M. McKerns, "Pushbroom imaging spectrometer with high spectroscopic data fidelity: experimental demonstration," Opt. Eng. 39,808-816 (2000). [CrossRef]
- 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).
- 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.
- J.M. Howard, "Optical Design using computer graphics," Appl. Opt. 40,3225-3231 (2001). [CrossRef]
- OSLO is a registered trademark of Lambda Research Corporation, 80 Taylor Street, P.O. Box 1400, Littleton, Mass. 01460.
- 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]

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