Abstract

By using a photonic bandgap (PBG) fiber bundle and a monochrome CCD camera, we experimentally demonstrate an all-fiber spectrometer. A total of 100 Bragg fibers that have complementary and overlapping bandgaps are chosen to compose the fiber bundle. A monochrome CCD is then used to capture the binned image. To reconstruct the test spectrum from a single CCD image, we develop an algorithm based on pseudoinversion of the spectrometer transmission matrix. We demonstrate that the peak center wave length can always be reconstructed within several percent of its true value regardless of the peak width or position, and that, although the widths of the individual Bragg fiber bandgaps are quite large (60180nm), the spectroscopic system has a resolution limit of 30nm. Moreover, we conclude that, by minimizing system errors, the resolution can be further improved down to several nanometers in width. Finally, we report fabrication of PBG fiber bundles containing hundreds of fibers using a two-stage drawing technique. This method constitutes a very promising approach toward industrial-strength fabrication of all-fiber spectrometers.

© 2010 Optical Society of America

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References

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  1. S. M. Ramasamy, V. Venkatasubrmanian, and S. Anbazhagan, “Reflectance spectra of minerals and their discrimination using Thematic Mapper, IRS and SPOT multispectral data,” Int. J. Remote Sens. 14, 2935–2970 (1993).
    [CrossRef]
  2. G. Vane and A. F. H. Goetz, “Terrestrial imaging spectroscopy,” Remote Sens. Environ. 24, 1–29 (1988).
    [CrossRef]
  3. A. Rosselet, W. Graff, U. P. Wild, and R. Gshwind, “Persistent spectral hole burning used for spectrally high-resolved imaging of the sun,” Proc. SPIE 2480, 205–212 (1995).
    [CrossRef]
  4. D. L. Farkas, B. T. Ballow, G. W. Fisher, W. Niu, and E. S. Wachman, “Microscopic and mesoscopic spectral bio-imaging,” Proc. SPIE 2678, 200–206 (1996).
    [CrossRef]
  5. M. Vilaseca, J. Pujol, and M. Arjona, “Multispectral system for reflectance reconstruction in the near-infrared region,” Appl. Opt. 45, 4241–4253 (2006).
    [CrossRef] [PubMed]
  6. M. Vilaseca, J. Pujol, and M. Arjona, “Spectral-reflectance reconstruction in the near-infrared region by use of conventional charge-coupled-device camera measurements,” Appl. Opt. 42, 1788–1798 (2003).
    [CrossRef] [PubMed]
  7. H. Suto, “Chalcogenide fiber bundle for 3D spectroscopy,” Infrared Phys. Technol. 38, 93–99 (1997).
    [CrossRef]
  8. B. Lienert, J. Porter, and S. K. Sharma, “Simultaneous measurement of spectra at multiple ranges using a single spectrometer,” Appl. Opt. 48, 4762–4766 (2009).
    [CrossRef] [PubMed]
  9. J. Y. Hardeberg, F. Schmitt, and H. Brettel, “Multispectral image capture using a liquid crystal tunable filter,” Opt. Eng. 41, 2532–2548 (2002).
    [CrossRef]
  10. J. Y. Hardegerg, F. Schmitt, H. Brettel, J. Crettez, and H. Maitre, “Multispectral image acquisition and simulation of illuminant changes,” in Colour Imaging: Vision and Technology, L.W.MacDonald and M.R.Luo, eds (Wiley, 1999), pp. 145–164.
  11. M. H. Kasari, “Spectral vision system for measuring color images,” J. Opt. Soc. Am. A 16, 2352–2362 (1999).
    [CrossRef]
  12. R. J. Bell, Introductory Fourier Transform Spectroscopy(Academic, 1972), pp 56–60.
  13. F. Hase, T. Blumenstock, and C. Paton-Walsh, “Analysis of the instrumental line shape of high-resolution Fourier transform IR spectrometers with gas cell measurements and new retrieval software,” Appl. Opt. 38, 3417–3422 (1999).
    [CrossRef]
  14. C. Bernardo and D. W. T. Griffith, “Fourier transform spectrometer instrumental lineshape (ILS) retrieval by Fourier deconvolution,” J. Quant. Spectrosc. Radiat. Transfer 95, 141–150 (2005).
    [CrossRef]
  15. M. Skorobogatiy and N. Guo, “Bandwidth enhancement by differential mode attenuation in multimode photonic crystal Bragg fibers,” Opt. Lett. 32, 900–902 (2007).
    [CrossRef] [PubMed]
  16. A. Dupuis, N. Guo, B. Gauvreau, A. Hassani, E. Pone, F. Boismenu, and M. Skorobogatiy, “Guiding in the visible with “colorful” solid-core Bragg fibers,” Opt. Lett. 32, 2882–2884 (2007).
    [CrossRef] [PubMed]
  17. B. Gauvreau, N. Guo, K. Schicker, K. Stoeffler, F. Boismenu, A. Ajji, R. Wingfield, C. Dubois, and M. Skorobogatiy, “Color-changing and color-tunable photonic bandgap fiber textiles,” Opt. Express 16, 15677–15693 (2008).
    [CrossRef] [PubMed]
  18. S. G. Johnson, M. Ibanescu, M. Skorobogatiy, O. Weisberg, T. D. Engeness, M. Solljacic, S. A. Jacobs, J. D. Joannopoulos, and Y. Fink, “Low-loss asymptotically single-mode propagation in large-core OmniGuide fiber,” Opt. Express 9, 748–779(2001).
    [CrossRef] [PubMed]
  19. M. Skorobogatiy and A. Dupuis, “Ferroelectric all-polymer hollow Bragg fibers for terahertz guidance,” Appl. Phys. Lett. 90, 113514 (2007).
    [CrossRef]
  20. Y. Gao, N. Guo, B. Gauvreau, M. Rajabian, O. Skorobogata, E. Pone, O. Zabeida, L. Martinu, C. Dubois, and M. Skorobogatiy, “Consecutive solvent evaporation and co-rolling technique for polymer multilayer hollow fiber preform fabrication,” J. Mater. Res. 21, 2246–2254(2006).
    [CrossRef]

2009 (1)

2008 (1)

2007 (3)

2006 (2)

Y. Gao, N. Guo, B. Gauvreau, M. Rajabian, O. Skorobogata, E. Pone, O. Zabeida, L. Martinu, C. Dubois, and M. Skorobogatiy, “Consecutive solvent evaporation and co-rolling technique for polymer multilayer hollow fiber preform fabrication,” J. Mater. Res. 21, 2246–2254(2006).
[CrossRef]

M. Vilaseca, J. Pujol, and M. Arjona, “Multispectral system for reflectance reconstruction in the near-infrared region,” Appl. Opt. 45, 4241–4253 (2006).
[CrossRef] [PubMed]

2005 (1)

C. Bernardo and D. W. T. Griffith, “Fourier transform spectrometer instrumental lineshape (ILS) retrieval by Fourier deconvolution,” J. Quant. Spectrosc. Radiat. Transfer 95, 141–150 (2005).
[CrossRef]

2003 (1)

2002 (1)

J. Y. Hardeberg, F. Schmitt, and H. Brettel, “Multispectral image capture using a liquid crystal tunable filter,” Opt. Eng. 41, 2532–2548 (2002).
[CrossRef]

2001 (1)

1999 (3)

F. Hase, T. Blumenstock, and C. Paton-Walsh, “Analysis of the instrumental line shape of high-resolution Fourier transform IR spectrometers with gas cell measurements and new retrieval software,” Appl. Opt. 38, 3417–3422 (1999).
[CrossRef]

J. Y. Hardegerg, F. Schmitt, H. Brettel, J. Crettez, and H. Maitre, “Multispectral image acquisition and simulation of illuminant changes,” in Colour Imaging: Vision and Technology, L.W.MacDonald and M.R.Luo, eds (Wiley, 1999), pp. 145–164.

M. H. Kasari, “Spectral vision system for measuring color images,” J. Opt. Soc. Am. A 16, 2352–2362 (1999).
[CrossRef]

1997 (1)

H. Suto, “Chalcogenide fiber bundle for 3D spectroscopy,” Infrared Phys. Technol. 38, 93–99 (1997).
[CrossRef]

1996 (1)

D. L. Farkas, B. T. Ballow, G. W. Fisher, W. Niu, and E. S. Wachman, “Microscopic and mesoscopic spectral bio-imaging,” Proc. SPIE 2678, 200–206 (1996).
[CrossRef]

1995 (1)

A. Rosselet, W. Graff, U. P. Wild, and R. Gshwind, “Persistent spectral hole burning used for spectrally high-resolved imaging of the sun,” Proc. SPIE 2480, 205–212 (1995).
[CrossRef]

1993 (1)

S. M. Ramasamy, V. Venkatasubrmanian, and S. Anbazhagan, “Reflectance spectra of minerals and their discrimination using Thematic Mapper, IRS and SPOT multispectral data,” Int. J. Remote Sens. 14, 2935–2970 (1993).
[CrossRef]

1988 (1)

G. Vane and A. F. H. Goetz, “Terrestrial imaging spectroscopy,” Remote Sens. Environ. 24, 1–29 (1988).
[CrossRef]

1972 (1)

R. J. Bell, Introductory Fourier Transform Spectroscopy(Academic, 1972), pp 56–60.

Ajji, A.

Anbazhagan, S.

S. M. Ramasamy, V. Venkatasubrmanian, and S. Anbazhagan, “Reflectance spectra of minerals and their discrimination using Thematic Mapper, IRS and SPOT multispectral data,” Int. J. Remote Sens. 14, 2935–2970 (1993).
[CrossRef]

Arjona, M.

Ballow, B. T.

D. L. Farkas, B. T. Ballow, G. W. Fisher, W. Niu, and E. S. Wachman, “Microscopic and mesoscopic spectral bio-imaging,” Proc. SPIE 2678, 200–206 (1996).
[CrossRef]

Bell, R. J.

R. J. Bell, Introductory Fourier Transform Spectroscopy(Academic, 1972), pp 56–60.

Bernardo, C.

C. Bernardo and D. W. T. Griffith, “Fourier transform spectrometer instrumental lineshape (ILS) retrieval by Fourier deconvolution,” J. Quant. Spectrosc. Radiat. Transfer 95, 141–150 (2005).
[CrossRef]

Blumenstock, T.

Boismenu, F.

Brettel, H.

J. Y. Hardeberg, F. Schmitt, and H. Brettel, “Multispectral image capture using a liquid crystal tunable filter,” Opt. Eng. 41, 2532–2548 (2002).
[CrossRef]

J. Y. Hardegerg, F. Schmitt, H. Brettel, J. Crettez, and H. Maitre, “Multispectral image acquisition and simulation of illuminant changes,” in Colour Imaging: Vision and Technology, L.W.MacDonald and M.R.Luo, eds (Wiley, 1999), pp. 145–164.

Crettez, J.

J. Y. Hardegerg, F. Schmitt, H. Brettel, J. Crettez, and H. Maitre, “Multispectral image acquisition and simulation of illuminant changes,” in Colour Imaging: Vision and Technology, L.W.MacDonald and M.R.Luo, eds (Wiley, 1999), pp. 145–164.

Dubois, C.

B. Gauvreau, N. Guo, K. Schicker, K. Stoeffler, F. Boismenu, A. Ajji, R. Wingfield, C. Dubois, and M. Skorobogatiy, “Color-changing and color-tunable photonic bandgap fiber textiles,” Opt. Express 16, 15677–15693 (2008).
[CrossRef] [PubMed]

Y. Gao, N. Guo, B. Gauvreau, M. Rajabian, O. Skorobogata, E. Pone, O. Zabeida, L. Martinu, C. Dubois, and M. Skorobogatiy, “Consecutive solvent evaporation and co-rolling technique for polymer multilayer hollow fiber preform fabrication,” J. Mater. Res. 21, 2246–2254(2006).
[CrossRef]

Dupuis, A.

Engeness, T. D.

Farkas, D. L.

D. L. Farkas, B. T. Ballow, G. W. Fisher, W. Niu, and E. S. Wachman, “Microscopic and mesoscopic spectral bio-imaging,” Proc. SPIE 2678, 200–206 (1996).
[CrossRef]

Fink, Y.

Fisher, G. W.

D. L. Farkas, B. T. Ballow, G. W. Fisher, W. Niu, and E. S. Wachman, “Microscopic and mesoscopic spectral bio-imaging,” Proc. SPIE 2678, 200–206 (1996).
[CrossRef]

Gao, Y.

Y. Gao, N. Guo, B. Gauvreau, M. Rajabian, O. Skorobogata, E. Pone, O. Zabeida, L. Martinu, C. Dubois, and M. Skorobogatiy, “Consecutive solvent evaporation and co-rolling technique for polymer multilayer hollow fiber preform fabrication,” J. Mater. Res. 21, 2246–2254(2006).
[CrossRef]

Gauvreau, B.

Goetz, A. F. H.

G. Vane and A. F. H. Goetz, “Terrestrial imaging spectroscopy,” Remote Sens. Environ. 24, 1–29 (1988).
[CrossRef]

Graff, W.

A. Rosselet, W. Graff, U. P. Wild, and R. Gshwind, “Persistent spectral hole burning used for spectrally high-resolved imaging of the sun,” Proc. SPIE 2480, 205–212 (1995).
[CrossRef]

Griffith, D. W. T.

C. Bernardo and D. W. T. Griffith, “Fourier transform spectrometer instrumental lineshape (ILS) retrieval by Fourier deconvolution,” J. Quant. Spectrosc. Radiat. Transfer 95, 141–150 (2005).
[CrossRef]

Gshwind, R.

A. Rosselet, W. Graff, U. P. Wild, and R. Gshwind, “Persistent spectral hole burning used for spectrally high-resolved imaging of the sun,” Proc. SPIE 2480, 205–212 (1995).
[CrossRef]

Guo, N.

Hardeberg, J. Y.

J. Y. Hardeberg, F. Schmitt, and H. Brettel, “Multispectral image capture using a liquid crystal tunable filter,” Opt. Eng. 41, 2532–2548 (2002).
[CrossRef]

Hardegerg, J. Y.

J. Y. Hardegerg, F. Schmitt, H. Brettel, J. Crettez, and H. Maitre, “Multispectral image acquisition and simulation of illuminant changes,” in Colour Imaging: Vision and Technology, L.W.MacDonald and M.R.Luo, eds (Wiley, 1999), pp. 145–164.

Hase, F.

Hassani, A.

Ibanescu, M.

Jacobs, S. A.

Joannopoulos, J. D.

Johnson, S. G.

Kasari, M. H.

Lienert, B.

Maitre, H.

J. Y. Hardegerg, F. Schmitt, H. Brettel, J. Crettez, and H. Maitre, “Multispectral image acquisition and simulation of illuminant changes,” in Colour Imaging: Vision and Technology, L.W.MacDonald and M.R.Luo, eds (Wiley, 1999), pp. 145–164.

Martinu, L.

Y. Gao, N. Guo, B. Gauvreau, M. Rajabian, O. Skorobogata, E. Pone, O. Zabeida, L. Martinu, C. Dubois, and M. Skorobogatiy, “Consecutive solvent evaporation and co-rolling technique for polymer multilayer hollow fiber preform fabrication,” J. Mater. Res. 21, 2246–2254(2006).
[CrossRef]

Niu, W.

D. L. Farkas, B. T. Ballow, G. W. Fisher, W. Niu, and E. S. Wachman, “Microscopic and mesoscopic spectral bio-imaging,” Proc. SPIE 2678, 200–206 (1996).
[CrossRef]

Paton-Walsh, C.

Pone, E.

A. Dupuis, N. Guo, B. Gauvreau, A. Hassani, E. Pone, F. Boismenu, and M. Skorobogatiy, “Guiding in the visible with “colorful” solid-core Bragg fibers,” Opt. Lett. 32, 2882–2884 (2007).
[CrossRef] [PubMed]

Y. Gao, N. Guo, B. Gauvreau, M. Rajabian, O. Skorobogata, E. Pone, O. Zabeida, L. Martinu, C. Dubois, and M. Skorobogatiy, “Consecutive solvent evaporation and co-rolling technique for polymer multilayer hollow fiber preform fabrication,” J. Mater. Res. 21, 2246–2254(2006).
[CrossRef]

Porter, J.

Pujol, J.

Rajabian, M.

Y. Gao, N. Guo, B. Gauvreau, M. Rajabian, O. Skorobogata, E. Pone, O. Zabeida, L. Martinu, C. Dubois, and M. Skorobogatiy, “Consecutive solvent evaporation and co-rolling technique for polymer multilayer hollow fiber preform fabrication,” J. Mater. Res. 21, 2246–2254(2006).
[CrossRef]

Ramasamy, S. M.

S. M. Ramasamy, V. Venkatasubrmanian, and S. Anbazhagan, “Reflectance spectra of minerals and their discrimination using Thematic Mapper, IRS and SPOT multispectral data,” Int. J. Remote Sens. 14, 2935–2970 (1993).
[CrossRef]

Rosselet, A.

A. Rosselet, W. Graff, U. P. Wild, and R. Gshwind, “Persistent spectral hole burning used for spectrally high-resolved imaging of the sun,” Proc. SPIE 2480, 205–212 (1995).
[CrossRef]

Schicker, K.

Schmitt, F.

J. Y. Hardeberg, F. Schmitt, and H. Brettel, “Multispectral image capture using a liquid crystal tunable filter,” Opt. Eng. 41, 2532–2548 (2002).
[CrossRef]

J. Y. Hardegerg, F. Schmitt, H. Brettel, J. Crettez, and H. Maitre, “Multispectral image acquisition and simulation of illuminant changes,” in Colour Imaging: Vision and Technology, L.W.MacDonald and M.R.Luo, eds (Wiley, 1999), pp. 145–164.

Sharma, S. K.

Skorobogata, O.

Y. Gao, N. Guo, B. Gauvreau, M. Rajabian, O. Skorobogata, E. Pone, O. Zabeida, L. Martinu, C. Dubois, and M. Skorobogatiy, “Consecutive solvent evaporation and co-rolling technique for polymer multilayer hollow fiber preform fabrication,” J. Mater. Res. 21, 2246–2254(2006).
[CrossRef]

Skorobogatiy, M.

Solljacic, M.

Stoeffler, K.

Suto, H.

H. Suto, “Chalcogenide fiber bundle for 3D spectroscopy,” Infrared Phys. Technol. 38, 93–99 (1997).
[CrossRef]

Vane, G.

G. Vane and A. F. H. Goetz, “Terrestrial imaging spectroscopy,” Remote Sens. Environ. 24, 1–29 (1988).
[CrossRef]

Venkatasubrmanian, V.

S. M. Ramasamy, V. Venkatasubrmanian, and S. Anbazhagan, “Reflectance spectra of minerals and their discrimination using Thematic Mapper, IRS and SPOT multispectral data,” Int. J. Remote Sens. 14, 2935–2970 (1993).
[CrossRef]

Vilaseca, M.

Wachman, E. S.

D. L. Farkas, B. T. Ballow, G. W. Fisher, W. Niu, and E. S. Wachman, “Microscopic and mesoscopic spectral bio-imaging,” Proc. SPIE 2678, 200–206 (1996).
[CrossRef]

Weisberg, O.

Wild, U. P.

A. Rosselet, W. Graff, U. P. Wild, and R. Gshwind, “Persistent spectral hole burning used for spectrally high-resolved imaging of the sun,” Proc. SPIE 2480, 205–212 (1995).
[CrossRef]

Wingfield, R.

Zabeida, O.

Y. Gao, N. Guo, B. Gauvreau, M. Rajabian, O. Skorobogata, E. Pone, O. Zabeida, L. Martinu, C. Dubois, and M. Skorobogatiy, “Consecutive solvent evaporation and co-rolling technique for polymer multilayer hollow fiber preform fabrication,” J. Mater. Res. 21, 2246–2254(2006).
[CrossRef]

Appl. Opt. (4)

Appl. Phys. Lett. (1)

M. Skorobogatiy and A. Dupuis, “Ferroelectric all-polymer hollow Bragg fibers for terahertz guidance,” Appl. Phys. Lett. 90, 113514 (2007).
[CrossRef]

Infrared Phys. Technol. (1)

H. Suto, “Chalcogenide fiber bundle for 3D spectroscopy,” Infrared Phys. Technol. 38, 93–99 (1997).
[CrossRef]

Int. J. Remote Sens. (1)

S. M. Ramasamy, V. Venkatasubrmanian, and S. Anbazhagan, “Reflectance spectra of minerals and their discrimination using Thematic Mapper, IRS and SPOT multispectral data,” Int. J. Remote Sens. 14, 2935–2970 (1993).
[CrossRef]

J. Mater. Res. (1)

Y. Gao, N. Guo, B. Gauvreau, M. Rajabian, O. Skorobogata, E. Pone, O. Zabeida, L. Martinu, C. Dubois, and M. Skorobogatiy, “Consecutive solvent evaporation and co-rolling technique for polymer multilayer hollow fiber preform fabrication,” J. Mater. Res. 21, 2246–2254(2006).
[CrossRef]

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

J. Quant. Spectrosc. Radiat. Transfer (1)

C. Bernardo and D. W. T. Griffith, “Fourier transform spectrometer instrumental lineshape (ILS) retrieval by Fourier deconvolution,” J. Quant. Spectrosc. Radiat. Transfer 95, 141–150 (2005).
[CrossRef]

Opt. Eng. (1)

J. Y. Hardeberg, F. Schmitt, and H. Brettel, “Multispectral image capture using a liquid crystal tunable filter,” Opt. Eng. 41, 2532–2548 (2002).
[CrossRef]

Opt. Express (2)

Opt. Lett. (2)

Proc. SPIE (2)

A. Rosselet, W. Graff, U. P. Wild, and R. Gshwind, “Persistent spectral hole burning used for spectrally high-resolved imaging of the sun,” Proc. SPIE 2480, 205–212 (1995).
[CrossRef]

D. L. Farkas, B. T. Ballow, G. W. Fisher, W. Niu, and E. S. Wachman, “Microscopic and mesoscopic spectral bio-imaging,” Proc. SPIE 2678, 200–206 (1996).
[CrossRef]

Remote Sens. Environ. (1)

G. Vane and A. F. H. Goetz, “Terrestrial imaging spectroscopy,” Remote Sens. Environ. 24, 1–29 (1988).
[CrossRef]

Other (2)

J. Y. Hardegerg, F. Schmitt, H. Brettel, J. Crettez, and H. Maitre, “Multispectral image acquisition and simulation of illuminant changes,” in Colour Imaging: Vision and Technology, L.W.MacDonald and M.R.Luo, eds (Wiley, 1999), pp. 145–164.

R. J. Bell, Introductory Fourier Transform Spectroscopy(Academic, 1972), pp 56–60.

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Figures (9)

Fig. 1
Fig. 1

(a) Transmission spectra of five typical Bragg fibers used in the fiber bundle. (b) Distribution of the fiber transmission bandwidth as a function of the fiber bandgap center wavelength for all the Bragg fibers in a bundle. Inset, photo of a Bragg fiber cross section showing a solid core surrounded by a periodic multilayer reflector.

Fig. 2
Fig. 2

Fiber bundle spectrometer. Top, schematic of the spectrometer. Light from the illuminant is launched into the fiber bundle; the image is taken by the monochrome CCD. Bottom, when the broadband light is launched into the fiber bundle, the output is a mosaic of colors selected by the individual Bragg fibers.

Fig. 3
Fig. 3

(a) Normalized spectral response of a CCD array. (b) Typical monochromatic near-linear response of a CCD array ( λ = 560 nm ).

Fig. 4
Fig. 4

Setup for the spectrometer calibration measurement.

Fig. 5
Fig. 5

Spectra reconstruction using PBG fiber-bundle-based spectrometer. (a) Reconstructed spectra of six 25 nm wide peaks. (b) Reconstructed spectra of four 40 nm wide bell-shaped spectra. The black dashed lines are the test spectra resolved by another monochromator; the red solid curves are the spectra reconstructed by the fiber bundle spectrometer. The gray area indicates the error level.

Fig. 6
Fig. 6

Properties of the reconstructed spectra as a function of the number of singular values used in the inversion algorithm. No noise is present in the system. (a) Dependence of the spectral shape of a reconstructed peak on N σ . (b) Width of a reconstructed peak as a function of N σ . (c) Reconstruction error as a function of N σ .

Fig. 7
Fig. 7

Effect of noise on the quality of reconstruction. Examples of the reconstructed spectra for several particular realizations of noise with amplitudes (i) δ = 0 , (ii) δ = 0.005 , (iii) δ = 0.05 , and (iv) δ = 0.1 .

Fig. 8
Fig. 8

Effect of noise on the reconstruction algorithm. (a) Average reconstruction error and its statistical deviation as a function of the noise amplitude. Inset, optimal number of singular values needed to minimize the reconstruction error. (b) Average width of a reconstructed peak and its statistical deviation as a function of the noise level.

Fig. 9
Fig. 9

Cross section of a fiber bundle fabricated using the two-stage drawing technique. The bundle is illuminated with a wideband halogen lamp source. At the fiber bundle output, a mosaic of colors is visible as the white light is split into the various spectral components.

Equations (10)

Equations on this page are rendered with MathJax. Learn more.

C n = λ min λ max I ( λ ) A n F n ( λ ) S ( λ ) O n ( λ ) d λ ,
C n = λ min λ max I ( λ ) T n ( λ ) d λ = i = 0 , λ i = λ min + i Δ λ N = ( λ max λ min ) / Δ λ T n ( λ i ) λ i λ i + Δ λ I ( λ ) d λ = i = 0 N I i · T n i .
[ C 1 C 100 ] = [ T 1 , 1 T 1 , N T 100 , 1 T 100 , N ] [ I 1 I N ] ,
C i λ ( 1 100 ) = T ( 1 100 , i λ ) · I i λ .
( T ) 100 × N = ( U ) 100 × 100 ( S ) 100 × N ( V T ) N × N , U T U = 1 ; V T V = 1 ; S = diag ( σ 1 , σ 2 , σ 3 , σ 100 ) , σ 1 > σ 2 > σ 3 > > σ 100 > 0 ,
( T ) 100 × N = ( U ) 100 × N σ ( S ) N σ × N σ ( V T ) N σ × N , σ 1 , σ 2 , , σ N σ 0 ,
( T ) N × 100 1 = ( V ) N × N σ ( S ) N σ × N σ 1 ( U T ) N σ × 100 .
Error ( N σ ) = min λ ( I N σ reconstr . ) / max λ ( I N σ reconstr . ) ,
( C ) 100 × 1 = ( T ) 100 × N ( I ) N × 1 , ( I N σ reconstr . ) N × 1 = ( V ) N × N σ ( S ) N σ × N σ 1 ( U T ) N σ × 100 ( C ) 100 × 1 .
( C ) 100 × 1 = ( T ) 100 × N ( I ) N × 1 , ( C noise ) 100 × 1 = ( C ) 100 × 1 · ( 1 + δ · ( η ) 100 × 1 ) ; η random [ 0.5 , 0.5 ] , ( I N σ reconstr . ) N × 1 = ( V ) N × N σ ( S ) N σ × N σ 1 ( U T ) N σ × 100 ( C noise ) 100 × 1 .

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