## Birefringent Fourier-transform imaging spectrometer

Optics Express, Vol. 12, Issue 22, pp. 5368-5374 (2004)

http://dx.doi.org/10.1364/OPEX.12.005368

Acrobat PDF (225 KB)

### Abstract

Fourier-transform imaging spectrometers offer important advantages over other spectral imaging modalities, such as, a wider free spectral range, higher spectral resolutions and, in low-photon-flux conditions, higher signal-to-noise ratios can be achieved. Unfortunately, for application in harsh environments, deployment of Fourier-transform instruments based on traditional moving-mirror interferometers is problematic due to their inherent sensitivity to vibration. We describe a new Fourier-transform imaging spectrometer, based on a scanning birefringent interferometer. This system retains the advantages of traditional Fourier transform instruments, but is inherently compact and insensitive to vibration. Furthermore, the precision requirements of the movement can be relaxed by typically two orders of magnitude in comparison to a traditional two-beam interferometer. The instrument promises to enable application of Fourier-transform imaging spectrometry to applications, such as airborne reconnaissance and industrial inspection, for the first time. Example spectral images are presented.

© 2004 Optical Society of America

## 1. Introduction

4. M.J. Persky, “A review of space infrared Fourier transform spectrometers for remote sensing,” Rev. Sci. Instrum. **66**, 4763–4797 (1995). [CrossRef]

4. M.J. Persky, “A review of space infrared Fourier transform spectrometers for remote sensing,” Rev. Sci. Instrum. **66**, 4763–4797 (1995). [CrossRef]

6. J W Brault, “New approach to high-precision Fourier-transform spectrometer design,” Appl. Opt. **35**, pp2891–2896 (1996) [CrossRef] [PubMed]

7. G. Zahn, K. Oka, T. Ishigaki, and N. Baba, “Birefringent imaging spectrometer,” Appl. Opt. **41**, 734–738 (2002). [CrossRef]

10. J. Genest, P. Tremblay, and A. Villemaire, “Throughput of tilted interferometers,” App. Opt. **37**,21, pp. 4819–4822. 1998 [CrossRef]

11. M. Hashimoto and S. Kawata, “Multichannel Fourier-transform infrared spectrometer,” Appl. Opt. **31**, 6096–6101 (1992). [CrossRef] [PubMed]

## 2. Fourier-transform imaging spectrometer technique

### 2.1. Principle of operation

12. M.J. Padgett and A.R. Harvey, “A static Fourier-transform spectrometer based on Wollaston prisms,” Rev. Sci. Instrum. **66**, 2807–2811 (1995) [CrossRef]

14. S. Prunet, B. Journet, and G. Fortunato, “Exact calculation of the optical path difference and description of a new birefringent interferometer,” Opt. Eng. **38**, 983–990 (1999). [CrossRef]

*d*is the displacement of the ray from the centerline of the prism,

*b*=(

*n*) is the birefringence of the Wollaston prism material where

_{o}-n_{e}*n*and

_{o}*n*are the ordinary and extraordinary refractive indices and

_{e}*θ*is the prism wedge angle. The path difference for consecutive transmission through two Wollaston prisms with equal splitting angles, with the optic axes aligned as shown in Fig. 1, can be easily shown to be

*h*is the offset of the centers of the two Wollaston prisms.

_{OPL}. Translation of the second Wollaston prism so as to vary

*h*introduces a time varying path difference between the two components, enabling an interferogram to be recorded as a function of prism displacement. Whereas the mirror in a Michelson interferometer is scanned along the optical axis, here the Wollaston prism is scanned transversely to produce the temporal interferogram.

### 2.2. The Fourier-transform relationship

_{1}and τ

_{2}are amplitude transmittances of the two polarisers, then the amplitude of the optical field at a particular pixel at an optical angular frequency

*ω*, in a small interval of optical frequencies d

*ω*, and at a prism displacement,

*h*, is given by

*ε*

_{o}(

*ω*) is the amplitude of the light incident upon the first polarizer,

*t*is time and

*c*is the speed of light. The sign of the second term is positive if the polarisers are co-aligned and negative if they are crossed. The intensity of the recombined beams is then

*h*for all frequencies yields:

*I*is a bias term and

_{o}*I*(

*ω*)=

*e*

_{o}(ω)

^{2}is the spectral intensity at frequency ω. The second term in equation 5 is the interferogram which is the Fourier cosine transform of the spectrum with 2

*b*tan

*θ*acting as the constant of proportionality with respect to the length variable

*h*. Inverse Fourier transformation of the interferogram yields

*I*(

*ω*).

*Δh*of the Wollaston prism such that the Nyquist criterion for the short-wavelength cutoff of the system spectral response is obeyed. Discrete inverse Fourier transformation of the sampled interferogram yields the spectral distribution, which is sampled at the values:

*σ*=

*ω*/2

*π*=1/

*λ*,

*N*is the number of samples in the interferogram and

*i*ranges from 0 to

*N*/2. Given appropriate functional forms of

*b*(

*σi*) equation 6 can be solved for

*σi*.

### 2.3. Advantages of this technique

*b*tan

*θ*, in equations (4–5) can be considered as an optical gearing ratio,

*G*, of the system; that is to say, a displacement

*h*of the Wollaston prism introduces an optical path difference

*hG*. The value of

*G*is a design parameter, but is typically less than 10

^{-2}, whereas for a Michelson interferometer,

*G*=2. This small value of

*G*for a birefringent interferometer greatly reduces the refinement required of the moving parts. Firstly, the required accuracy of the scanning mechanism is reduced from, λ/20 for a Michelson interferometer, to λ/(10G) (typically about 10λ) for a birefringent interferometer; a greater than 200-fold reduction. The sensitivity to vibration in the direction of movement of the Wollaston prism is also reduced by the same factor in comparison to the sensitivity to vibration of a Michelson interferometer. For vibrations in other directions, and for air currents, the near-collinear propagation of the interfering beams means that these disturbances are common mode and have negligible effect. This contrasts strongly with Michelson interferometers that require enclosed and isolated instruments to ensure adequate performance.

13. A.R. Harvey, “Determination of the optical constants of thin films in the visible by dispersive Fourier transform spectroscopy,” Rev. of Sci. Instr. **69**, pp3649–3658 (1998) [CrossRef]

## 3. Experiment and results

*n*≈0.175) Wollaston prisms with a wedge angle of 1.5°. The prism dimensions were such that

_{o}-n_{e}*NΔ*was limited to a maximum value of 10 mm giving a limiting spectral resolution of 110 cm

_{h}^{-1}. The second Wollaston prism was mounted on a motorized translation stage; computer control of the stage motion, image acquisition and all data processing was accomplished with

*Labview*software [17

17. Labview virtual instrumentation software for personal computers, (National Instruments, 2003), http://www.ni.com/labview/.

^{-1}(1000-400 nm). An example image, taken from the included movie file, of five spectral calibration tiles [18] imaged through the interferometer is shown in Fig. 2(a). The hyperbolic fringes arise from the variation of path difference with field angle as was described in Section 2.1. As can be seen from the movie, temporal scanning of the mirror produces a time-sequential interferogram at each pixel, as a function of Wollaston prism displacement; an example interferogram is shown in Fig. 2(b). Note that the horizontal scale, in mm, is the displacement of the Wollaston prism; the equivalent displacement of the mirror in a Michelson interferometer would be some 200-fold smaller. Inverse Fourier transformation of the complete data cube of interferograms produces a spectrum for each pixel; that is, the spectral data cube. An example narrow-band image of the calibration tiles calculated from the spectral data cube and normalized to give albedo is shown in Fig. 2(c). The associated movie file shows the complete spectral data cube as an animation. The spectral albedo data cube of two plants appears as a movie file in Fig. 2(d). Several features are readily apparent; in particular, the distressed leaf of the plant on the right exhibits high contrast with the healthy leaves from the same plant for optical frequencies close to 16,000 cm

^{-1}and the so-called chlorophyll edge is very noticeable as a rapid increase in albedo close to 14,000 cm

^{-1}.

## 5. Conclusions

## Acknowledgments

## References and Links

1. | A.R. Harvey, J. Beale, A.H. Greenaway, T.J. Hanlon, and J. Williams, “Technology options for imaging spectrometry” in |

2. | P.J. Miller and A.R. Harvey, “Signal to noise analysis of various imaging systems” in |

3. | S.P. Davis, M.C. Abrams, and J.W. Brault, |

4. | M.J. Persky, “A review of space infrared Fourier transform spectrometers for remote sensing,” Rev. Sci. Instrum. |

5. | A.R. Harvey and D.W. Fletcher-Holmes, “Imaging Apparatus,” Patent WO2004 005870 A1, (2004). |

6. | J W Brault, “New approach to high-precision Fourier-transform spectrometer design,” Appl. Opt. |

7. | G. Zahn, K. Oka, T. Ishigaki, and N. Baba, “Birefringent imaging spectrometer,” Appl. Opt. |

8. | R. Heintzmann, K.A. Lidke, and T.M. Jovin, “Double-pass Fourier transform imaging spectroscopy,” Optics Express , |

9. | L. J. Otten, A. D. Meigs, B. A. Jones, P. Prinzing, and D. S. Fronterhouse, “Payload Qualification and Optical Performance Test Results for the MightySat II.1 Hyperspectral Imager,” Proc. SPIE. |

10. | J. Genest, P. Tremblay, and A. Villemaire, “Throughput of tilted interferometers,” App. Opt. |

11. | M. Hashimoto and S. Kawata, “Multichannel Fourier-transform infrared spectrometer,” Appl. Opt. |

12. | M.J. Padgett and A.R. Harvey, “A static Fourier-transform spectrometer based on Wollaston prisms,” Rev. Sci. Instrum. |

13. | A.R. Harvey, “Determination of the optical constants of thin films in the visible by dispersive Fourier transform spectroscopy,” Rev. of Sci. Instr. |

14. | S. Prunet, B. Journet, and G. Fortunato, “Exact calculation of the optical path difference and description of a new birefringent interferometer,” Opt. Eng. |

15. | R.F. Horton, “Optical design for a High Etendue Imaging Fourier Transform Spectrometer” in |

16. | M. Françon and S. Mallick, |

17. | Labview virtual instrumentation software for personal computers, (National Instruments, 2003), http://www.ni.com/labview/. |

18. | Labsphere Inc, 231 Shaker Street, POB 70, North Sutton, NH 03260, USA. |

**OCIS Codes**

(120.0280) Instrumentation, measurement, and metrology : Remote sensing and sensors

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

(260.1440) Physical optics : Birefringence

(300.6190) Spectroscopy : Spectrometers

(300.6300) Spectroscopy : Spectroscopy, Fourier transforms

**ToC Category:**

Research Papers

**History**

Original Manuscript: August 19, 2004

Revised Manuscript: October 19, 2004

Published: November 1, 2004

**Citation**

Andrew Harvey and David Fletcher-Holmes, "Birefringent Fourier-transform imaging spectrometer," Opt. Express **12**, 5368-5374 (2004)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-22-5368

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### References

- A.R. Harvey, J. Beale, A.H. Greenaway, T.J. Hanlon and J. Williams, �??Technology options for imaging spectrometry�?? in Imaging Spectrometry VI, Descour & Shen, Proc. SPIE 4132, 13-24 (2000).
- P.J. Miller and A.R. Harvey, �??Signal to noise analysis of various imaging systems�?? in Biomarkers and Biological Spectral Imaging, Bearman, Bornhop & Levenson, Proc. SPIE 4259, 16-21 (2001).
- S.P.Davis, M.C.Abrams and J.W.Brault, Fourier Transform Spectrometry (Academic Press, 2001).
- M.J.Persky, �??A review of space infrared Fourier transform spectrometers for remote sensing,�?? Rev. Sci. Instrum. 66, 4763-4797 (1995). [CrossRef]
- A.R. Harvey and D.W. Fletcher-Holmes, �??Imaging Apparatus,�?? Patent WO2004 005870 A1, (2004).
- J W Brault, �??New approach to high-precision Fourier-transform spectrometer design,�?? Appl. Opt. 35, pp. 2891-2896 (1996) [CrossRef] [PubMed]
- G. Zahn, K. Oka, T. Ishigaki and N. Baba, �??Birefringent imaging spectrometer,�?? Appl. Opt. 41, 734-738 (2002). [CrossRef]
- R. Heintzmann, K.A. Lidke and T.M. Jovin, �??Double-pass Fourier transform imaging spectroscopy,�?? Optics Express, 12, pp 753-763 (2004) [CrossRef] [PubMed]
- L. J. Otten, A. D. Meigs, B. A. Jones, P. Prinzing, and D. S. Fronterhouse, �??Payload Qualification and Optical Performance Test Results for the MightySat II.1 Hyperspectral Imager,�?? Proc. SPIE. 3498, pp. 231-238 (1998) [CrossRef]
- J. Genest, P. Tremblay, and A. Villemaire, �??Throughput of tilted interferometers,�?? App. Opt. 37,21, pp. 4819-4822. 1998 [CrossRef]
- M. Hashimoto and S. Kawata, �??Multichannel Fourier-transform infrared spectrometer,�?? Appl. Opt. 31, 6096-6101 (1992). [CrossRef] [PubMed]
- M.J. Padgett and A.R. Harvey, �??A static Fourier-transform spectrometer based on Wollaston prisms,�?? Rev. Sci. Instrum. 66, 2807-2811 (1995) [CrossRef]
- A.R.Harvey, �??Determination of the optical constants of thin films in the visible by dispersive Fourier transform spectroscopy,�?? Rev. of Sci. Instr. 69, pp. 3649-3658 (1998) [CrossRef]
- S. Prunet, B. Journet and G. Fortunato, �??Exact calculation of the optical path difference and description of a new birefringent interferometer,�?? Opt. Eng. 38, 983-990 (1999). [CrossRef]
- R.F. Horton, �??Optical design for a High Etendue Imaging Fourier Transform Spectrometer�?? in Imaging Spectrometry II, Descour & Mooney, Proc. SPIE 2819, 300-315 (1996).
- M. Françon and S. Mallick, Polarization Interferometers Applications in Microscopy and Macroscopy (Wiley-Interscience, 1972).
- Labview virtual instrumentation software for personal computers, (National Instruments, 2003), <a href= "http://www.ni.com/labview/.">http://www.ni.com/labview/.</a>
- Labsphere Inc, 231 Shaker Street, POB 70, North Sutton, NH 03260, USA.

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