Abstract

A cost-effective yet robust and versatile dual-channel spectral comparator is presented. The silica-on-silicon planar-waveguide integrated device includes two holographic Bragg-grating reflectors (HBRs) with complementary spectral transfer functions. Output comprises projections of input signal spectra onto the complementary spectral channels. Spectral comparators may be useful in optical code-division multiplexing, optical packet decoding, spectral target recognition, and the identification of molecular spectra. HBRs may be considered to be mode-specific photonic crystals.

© 2004 Optical Society of America

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References

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  1. T. W. Mossberg, “Planar holographic optical processing devices,” Opt. Lett. 26, 414–416 (2001).
    [CrossRef]
  2. T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. 15, 1277–1294 (1997).
    [CrossRef]
  3. A. Othonos, “Fiber Bragg gratings,” Rev. Sci. Instrum. 68, 4309–4341 (1997).
    [CrossRef]
  4. R. Adar, C. H. Henry, R. C. Kistler, R. F. Kazarinov, “Polarization independent narrowband Bragg reflection gratings made with silica-on-silicon waveguides,” Appl. Phys. Lett. 60, 1779–1781 (1992).
    [CrossRef]
  5. D. Wiesmann, R. Germann, G. L. Bona, C. David, D. Erni, H. Jackel, “Add–drop filter based on apodized surface-corrugated gratings,” J. Opt. Soc. Am. B 20, 417–223 (2003).
    [CrossRef]
  6. A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984).
  7. D. Labilloy, H. Benisty, C. Weisbuch, T. F. Krauss, D. Cassagne, C. Jouanin, R. Houdre, U. Oesterle, V. Bardinal , “ Diffraction efficiency and guided light control by two-dimensional photonic-bandgap lattices,” IEEE J. Quantum Electron. 35, 1045–1052 (1999).
    [CrossRef]
  8. A. Giorgio, A. G. Perri, M. N. Armenise, “Modeling of fully etched waveguiding photonic bandgap structures,” IEEE J. Quantum Electron. 38, 630–639 (2002).
    [CrossRef]
  9. D. M. Pustai, A. Sharkawy, S. Shi, G. Jin, J. Murakowski, D. W. Prather, “Characterization and analysis of photonic crystal coupled waveguides,” J. Microlith. Microfab. Microsyst. 2, 292–298 (2003).
  10. S. Rowson, A. Chelnokov, J. M. Lourtioz, “Two-dimensional photonic crystals in macroporous silicon: from midinfrared (10 μm) to telecommunication wavelengths (1.3–1.5 μm),” J. Lightwave Technol. 17, 1989–1995 (1999).
    [CrossRef]
  11. D. Iazikov, C. Greiner, T. W. Mossberg, “Effective gray-scale in lithographically-scribed planar holographic Bragg reflectors,” Appl. Opt. 43, 1149–1155 (2004).
    [CrossRef]
  12. C. Greiner, D. Iazikov, T. W. Mossberg, “Lithographically-fabricated planar holographic Bragg reflectors,” J. Lightwave Technol. 22, 136–145 (2004).
    [CrossRef]
  13. D. Iazikov, C. Greiner, T. W. Mossberg, “Apodizable integrated filters for coarse WDM and FTTH-type applications,” J. Lightwave Technol. (to be published).
  14. T. W. Mossberg, M. G. Raymer, “Optical code division multiplexing,” Opt. Photon. News 12, 50–55 (2001).
    [CrossRef]
  15. K. Takiguchi, T. Shibata, M. Itoh, “Encoder/decoder on planar lightwave circuit for time-spreading/wavelength-hopping optical CDMA,” Electron. Lett. 38, 469–470 (2002).
    [CrossRef]
  16. C. H. Lee, S. Zhong, X. Lin, J. F. Young, Y. J. Chen, “Planar lightwave circuit design for programmable complementary spectral keying encoder and decoder,” Electron. Lett. 35, 1813–1815 (1999).
    [CrossRef]
  17. H. Tsuda, H. Takenouchi, T. Ishii, K. Okamoto, T. Goh, K. Sato, A. Hirano, T. Kurokawa, C. Amano, “Spectral encoding and decoding of 10 Gbit/s femtosecond pulses using high resolution arrayed-waveguide grating,” Electron. Lett. 35, 1186–1188 (1999).
    [CrossRef]

2004 (2)

2003 (2)

D. Wiesmann, R. Germann, G. L. Bona, C. David, D. Erni, H. Jackel, “Add–drop filter based on apodized surface-corrugated gratings,” J. Opt. Soc. Am. B 20, 417–223 (2003).
[CrossRef]

D. M. Pustai, A. Sharkawy, S. Shi, G. Jin, J. Murakowski, D. W. Prather, “Characterization and analysis of photonic crystal coupled waveguides,” J. Microlith. Microfab. Microsyst. 2, 292–298 (2003).

2002 (2)

K. Takiguchi, T. Shibata, M. Itoh, “Encoder/decoder on planar lightwave circuit for time-spreading/wavelength-hopping optical CDMA,” Electron. Lett. 38, 469–470 (2002).
[CrossRef]

A. Giorgio, A. G. Perri, M. N. Armenise, “Modeling of fully etched waveguiding photonic bandgap structures,” IEEE J. Quantum Electron. 38, 630–639 (2002).
[CrossRef]

2001 (2)

T. W. Mossberg, M. G. Raymer, “Optical code division multiplexing,” Opt. Photon. News 12, 50–55 (2001).
[CrossRef]

T. W. Mossberg, “Planar holographic optical processing devices,” Opt. Lett. 26, 414–416 (2001).
[CrossRef]

1999 (4)

S. Rowson, A. Chelnokov, J. M. Lourtioz, “Two-dimensional photonic crystals in macroporous silicon: from midinfrared (10 μm) to telecommunication wavelengths (1.3–1.5 μm),” J. Lightwave Technol. 17, 1989–1995 (1999).
[CrossRef]

C. H. Lee, S. Zhong, X. Lin, J. F. Young, Y. J. Chen, “Planar lightwave circuit design for programmable complementary spectral keying encoder and decoder,” Electron. Lett. 35, 1813–1815 (1999).
[CrossRef]

H. Tsuda, H. Takenouchi, T. Ishii, K. Okamoto, T. Goh, K. Sato, A. Hirano, T. Kurokawa, C. Amano, “Spectral encoding and decoding of 10 Gbit/s femtosecond pulses using high resolution arrayed-waveguide grating,” Electron. Lett. 35, 1186–1188 (1999).
[CrossRef]

D. Labilloy, H. Benisty, C. Weisbuch, T. F. Krauss, D. Cassagne, C. Jouanin, R. Houdre, U. Oesterle, V. Bardinal , “ Diffraction efficiency and guided light control by two-dimensional photonic-bandgap lattices,” IEEE J. Quantum Electron. 35, 1045–1052 (1999).
[CrossRef]

1997 (2)

T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. 15, 1277–1294 (1997).
[CrossRef]

A. Othonos, “Fiber Bragg gratings,” Rev. Sci. Instrum. 68, 4309–4341 (1997).
[CrossRef]

1992 (1)

R. Adar, C. H. Henry, R. C. Kistler, R. F. Kazarinov, “Polarization independent narrowband Bragg reflection gratings made with silica-on-silicon waveguides,” Appl. Phys. Lett. 60, 1779–1781 (1992).
[CrossRef]

Adar, R.

R. Adar, C. H. Henry, R. C. Kistler, R. F. Kazarinov, “Polarization independent narrowband Bragg reflection gratings made with silica-on-silicon waveguides,” Appl. Phys. Lett. 60, 1779–1781 (1992).
[CrossRef]

Amano, C.

H. Tsuda, H. Takenouchi, T. Ishii, K. Okamoto, T. Goh, K. Sato, A. Hirano, T. Kurokawa, C. Amano, “Spectral encoding and decoding of 10 Gbit/s femtosecond pulses using high resolution arrayed-waveguide grating,” Electron. Lett. 35, 1186–1188 (1999).
[CrossRef]

Armenise, M. N.

A. Giorgio, A. G. Perri, M. N. Armenise, “Modeling of fully etched waveguiding photonic bandgap structures,” IEEE J. Quantum Electron. 38, 630–639 (2002).
[CrossRef]

Bardinal, V.

D. Labilloy, H. Benisty, C. Weisbuch, T. F. Krauss, D. Cassagne, C. Jouanin, R. Houdre, U. Oesterle, V. Bardinal , “ Diffraction efficiency and guided light control by two-dimensional photonic-bandgap lattices,” IEEE J. Quantum Electron. 35, 1045–1052 (1999).
[CrossRef]

Benisty, H.

D. Labilloy, H. Benisty, C. Weisbuch, T. F. Krauss, D. Cassagne, C. Jouanin, R. Houdre, U. Oesterle, V. Bardinal , “ Diffraction efficiency and guided light control by two-dimensional photonic-bandgap lattices,” IEEE J. Quantum Electron. 35, 1045–1052 (1999).
[CrossRef]

Bona, G. L.

Cassagne, D.

D. Labilloy, H. Benisty, C. Weisbuch, T. F. Krauss, D. Cassagne, C. Jouanin, R. Houdre, U. Oesterle, V. Bardinal , “ Diffraction efficiency and guided light control by two-dimensional photonic-bandgap lattices,” IEEE J. Quantum Electron. 35, 1045–1052 (1999).
[CrossRef]

Chelnokov, A.

Chen, Y. J.

C. H. Lee, S. Zhong, X. Lin, J. F. Young, Y. J. Chen, “Planar lightwave circuit design for programmable complementary spectral keying encoder and decoder,” Electron. Lett. 35, 1813–1815 (1999).
[CrossRef]

David, C.

Erdogan, T.

T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. 15, 1277–1294 (1997).
[CrossRef]

Erni, D.

Germann, R.

Giorgio, A.

A. Giorgio, A. G. Perri, M. N. Armenise, “Modeling of fully etched waveguiding photonic bandgap structures,” IEEE J. Quantum Electron. 38, 630–639 (2002).
[CrossRef]

Goh, T.

H. Tsuda, H. Takenouchi, T. Ishii, K. Okamoto, T. Goh, K. Sato, A. Hirano, T. Kurokawa, C. Amano, “Spectral encoding and decoding of 10 Gbit/s femtosecond pulses using high resolution arrayed-waveguide grating,” Electron. Lett. 35, 1186–1188 (1999).
[CrossRef]

Greiner, C.

Henry, C. H.

R. Adar, C. H. Henry, R. C. Kistler, R. F. Kazarinov, “Polarization independent narrowband Bragg reflection gratings made with silica-on-silicon waveguides,” Appl. Phys. Lett. 60, 1779–1781 (1992).
[CrossRef]

Hirano, A.

H. Tsuda, H. Takenouchi, T. Ishii, K. Okamoto, T. Goh, K. Sato, A. Hirano, T. Kurokawa, C. Amano, “Spectral encoding and decoding of 10 Gbit/s femtosecond pulses using high resolution arrayed-waveguide grating,” Electron. Lett. 35, 1186–1188 (1999).
[CrossRef]

Houdre, R.

D. Labilloy, H. Benisty, C. Weisbuch, T. F. Krauss, D. Cassagne, C. Jouanin, R. Houdre, U. Oesterle, V. Bardinal , “ Diffraction efficiency and guided light control by two-dimensional photonic-bandgap lattices,” IEEE J. Quantum Electron. 35, 1045–1052 (1999).
[CrossRef]

Iazikov, D.

Ishii, T.

H. Tsuda, H. Takenouchi, T. Ishii, K. Okamoto, T. Goh, K. Sato, A. Hirano, T. Kurokawa, C. Amano, “Spectral encoding and decoding of 10 Gbit/s femtosecond pulses using high resolution arrayed-waveguide grating,” Electron. Lett. 35, 1186–1188 (1999).
[CrossRef]

Itoh, M.

K. Takiguchi, T. Shibata, M. Itoh, “Encoder/decoder on planar lightwave circuit for time-spreading/wavelength-hopping optical CDMA,” Electron. Lett. 38, 469–470 (2002).
[CrossRef]

Jackel, H.

Jin, G.

D. M. Pustai, A. Sharkawy, S. Shi, G. Jin, J. Murakowski, D. W. Prather, “Characterization and analysis of photonic crystal coupled waveguides,” J. Microlith. Microfab. Microsyst. 2, 292–298 (2003).

Jouanin, C.

D. Labilloy, H. Benisty, C. Weisbuch, T. F. Krauss, D. Cassagne, C. Jouanin, R. Houdre, U. Oesterle, V. Bardinal , “ Diffraction efficiency and guided light control by two-dimensional photonic-bandgap lattices,” IEEE J. Quantum Electron. 35, 1045–1052 (1999).
[CrossRef]

Kazarinov, R. F.

R. Adar, C. H. Henry, R. C. Kistler, R. F. Kazarinov, “Polarization independent narrowband Bragg reflection gratings made with silica-on-silicon waveguides,” Appl. Phys. Lett. 60, 1779–1781 (1992).
[CrossRef]

Kistler, R. C.

R. Adar, C. H. Henry, R. C. Kistler, R. F. Kazarinov, “Polarization independent narrowband Bragg reflection gratings made with silica-on-silicon waveguides,” Appl. Phys. Lett. 60, 1779–1781 (1992).
[CrossRef]

Krauss, T. F.

D. Labilloy, H. Benisty, C. Weisbuch, T. F. Krauss, D. Cassagne, C. Jouanin, R. Houdre, U. Oesterle, V. Bardinal , “ Diffraction efficiency and guided light control by two-dimensional photonic-bandgap lattices,” IEEE J. Quantum Electron. 35, 1045–1052 (1999).
[CrossRef]

Kurokawa, T.

H. Tsuda, H. Takenouchi, T. Ishii, K. Okamoto, T. Goh, K. Sato, A. Hirano, T. Kurokawa, C. Amano, “Spectral encoding and decoding of 10 Gbit/s femtosecond pulses using high resolution arrayed-waveguide grating,” Electron. Lett. 35, 1186–1188 (1999).
[CrossRef]

Labilloy, D.

D. Labilloy, H. Benisty, C. Weisbuch, T. F. Krauss, D. Cassagne, C. Jouanin, R. Houdre, U. Oesterle, V. Bardinal , “ Diffraction efficiency and guided light control by two-dimensional photonic-bandgap lattices,” IEEE J. Quantum Electron. 35, 1045–1052 (1999).
[CrossRef]

Lee, C. H.

C. H. Lee, S. Zhong, X. Lin, J. F. Young, Y. J. Chen, “Planar lightwave circuit design for programmable complementary spectral keying encoder and decoder,” Electron. Lett. 35, 1813–1815 (1999).
[CrossRef]

Lin, X.

C. H. Lee, S. Zhong, X. Lin, J. F. Young, Y. J. Chen, “Planar lightwave circuit design for programmable complementary spectral keying encoder and decoder,” Electron. Lett. 35, 1813–1815 (1999).
[CrossRef]

Lourtioz, J. M.

Mossberg, T. W.

Murakowski, J.

D. M. Pustai, A. Sharkawy, S. Shi, G. Jin, J. Murakowski, D. W. Prather, “Characterization and analysis of photonic crystal coupled waveguides,” J. Microlith. Microfab. Microsyst. 2, 292–298 (2003).

Oesterle, U.

D. Labilloy, H. Benisty, C. Weisbuch, T. F. Krauss, D. Cassagne, C. Jouanin, R. Houdre, U. Oesterle, V. Bardinal , “ Diffraction efficiency and guided light control by two-dimensional photonic-bandgap lattices,” IEEE J. Quantum Electron. 35, 1045–1052 (1999).
[CrossRef]

Okamoto, K.

H. Tsuda, H. Takenouchi, T. Ishii, K. Okamoto, T. Goh, K. Sato, A. Hirano, T. Kurokawa, C. Amano, “Spectral encoding and decoding of 10 Gbit/s femtosecond pulses using high resolution arrayed-waveguide grating,” Electron. Lett. 35, 1186–1188 (1999).
[CrossRef]

Othonos, A.

A. Othonos, “Fiber Bragg gratings,” Rev. Sci. Instrum. 68, 4309–4341 (1997).
[CrossRef]

Perri, A. G.

A. Giorgio, A. G. Perri, M. N. Armenise, “Modeling of fully etched waveguiding photonic bandgap structures,” IEEE J. Quantum Electron. 38, 630–639 (2002).
[CrossRef]

Prather, D. W.

D. M. Pustai, A. Sharkawy, S. Shi, G. Jin, J. Murakowski, D. W. Prather, “Characterization and analysis of photonic crystal coupled waveguides,” J. Microlith. Microfab. Microsyst. 2, 292–298 (2003).

Pustai, D. M.

D. M. Pustai, A. Sharkawy, S. Shi, G. Jin, J. Murakowski, D. W. Prather, “Characterization and analysis of photonic crystal coupled waveguides,” J. Microlith. Microfab. Microsyst. 2, 292–298 (2003).

Raymer, M. G.

T. W. Mossberg, M. G. Raymer, “Optical code division multiplexing,” Opt. Photon. News 12, 50–55 (2001).
[CrossRef]

Rowson, S.

Sato, K.

H. Tsuda, H. Takenouchi, T. Ishii, K. Okamoto, T. Goh, K. Sato, A. Hirano, T. Kurokawa, C. Amano, “Spectral encoding and decoding of 10 Gbit/s femtosecond pulses using high resolution arrayed-waveguide grating,” Electron. Lett. 35, 1186–1188 (1999).
[CrossRef]

Sharkawy, A.

D. M. Pustai, A. Sharkawy, S. Shi, G. Jin, J. Murakowski, D. W. Prather, “Characterization and analysis of photonic crystal coupled waveguides,” J. Microlith. Microfab. Microsyst. 2, 292–298 (2003).

Shi, S.

D. M. Pustai, A. Sharkawy, S. Shi, G. Jin, J. Murakowski, D. W. Prather, “Characterization and analysis of photonic crystal coupled waveguides,” J. Microlith. Microfab. Microsyst. 2, 292–298 (2003).

Shibata, T.

K. Takiguchi, T. Shibata, M. Itoh, “Encoder/decoder on planar lightwave circuit for time-spreading/wavelength-hopping optical CDMA,” Electron. Lett. 38, 469–470 (2002).
[CrossRef]

Takenouchi, H.

H. Tsuda, H. Takenouchi, T. Ishii, K. Okamoto, T. Goh, K. Sato, A. Hirano, T. Kurokawa, C. Amano, “Spectral encoding and decoding of 10 Gbit/s femtosecond pulses using high resolution arrayed-waveguide grating,” Electron. Lett. 35, 1186–1188 (1999).
[CrossRef]

Takiguchi, K.

K. Takiguchi, T. Shibata, M. Itoh, “Encoder/decoder on planar lightwave circuit for time-spreading/wavelength-hopping optical CDMA,” Electron. Lett. 38, 469–470 (2002).
[CrossRef]

Tsuda, H.

H. Tsuda, H. Takenouchi, T. Ishii, K. Okamoto, T. Goh, K. Sato, A. Hirano, T. Kurokawa, C. Amano, “Spectral encoding and decoding of 10 Gbit/s femtosecond pulses using high resolution arrayed-waveguide grating,” Electron. Lett. 35, 1186–1188 (1999).
[CrossRef]

Weisbuch, C.

D. Labilloy, H. Benisty, C. Weisbuch, T. F. Krauss, D. Cassagne, C. Jouanin, R. Houdre, U. Oesterle, V. Bardinal , “ Diffraction efficiency and guided light control by two-dimensional photonic-bandgap lattices,” IEEE J. Quantum Electron. 35, 1045–1052 (1999).
[CrossRef]

Wiesmann, D.

Yariv, A.

A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984).

Yeh, P.

A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984).

Young, J. F.

C. H. Lee, S. Zhong, X. Lin, J. F. Young, Y. J. Chen, “Planar lightwave circuit design for programmable complementary spectral keying encoder and decoder,” Electron. Lett. 35, 1813–1815 (1999).
[CrossRef]

Zhong, S.

C. H. Lee, S. Zhong, X. Lin, J. F. Young, Y. J. Chen, “Planar lightwave circuit design for programmable complementary spectral keying encoder and decoder,” Electron. Lett. 35, 1813–1815 (1999).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

R. Adar, C. H. Henry, R. C. Kistler, R. F. Kazarinov, “Polarization independent narrowband Bragg reflection gratings made with silica-on-silicon waveguides,” Appl. Phys. Lett. 60, 1779–1781 (1992).
[CrossRef]

Electron. Lett. (3)

K. Takiguchi, T. Shibata, M. Itoh, “Encoder/decoder on planar lightwave circuit for time-spreading/wavelength-hopping optical CDMA,” Electron. Lett. 38, 469–470 (2002).
[CrossRef]

C. H. Lee, S. Zhong, X. Lin, J. F. Young, Y. J. Chen, “Planar lightwave circuit design for programmable complementary spectral keying encoder and decoder,” Electron. Lett. 35, 1813–1815 (1999).
[CrossRef]

H. Tsuda, H. Takenouchi, T. Ishii, K. Okamoto, T. Goh, K. Sato, A. Hirano, T. Kurokawa, C. Amano, “Spectral encoding and decoding of 10 Gbit/s femtosecond pulses using high resolution arrayed-waveguide grating,” Electron. Lett. 35, 1186–1188 (1999).
[CrossRef]

IEEE J. Quantum Electron. (2)

D. Labilloy, H. Benisty, C. Weisbuch, T. F. Krauss, D. Cassagne, C. Jouanin, R. Houdre, U. Oesterle, V. Bardinal , “ Diffraction efficiency and guided light control by two-dimensional photonic-bandgap lattices,” IEEE J. Quantum Electron. 35, 1045–1052 (1999).
[CrossRef]

A. Giorgio, A. G. Perri, M. N. Armenise, “Modeling of fully etched waveguiding photonic bandgap structures,” IEEE J. Quantum Electron. 38, 630–639 (2002).
[CrossRef]

J. Lightwave Technol. (3)

J. Microlith. Microfab. Microsyst. (1)

D. M. Pustai, A. Sharkawy, S. Shi, G. Jin, J. Murakowski, D. W. Prather, “Characterization and analysis of photonic crystal coupled waveguides,” J. Microlith. Microfab. Microsyst. 2, 292–298 (2003).

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

Opt. Lett. (1)

Opt. Photon. News (1)

T. W. Mossberg, M. G. Raymer, “Optical code division multiplexing,” Opt. Photon. News 12, 50–55 (2001).
[CrossRef]

Rev. Sci. Instrum. (1)

A. Othonos, “Fiber Bragg gratings,” Rev. Sci. Instrum. 68, 4309–4341 (1997).
[CrossRef]

Other (2)

A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984).

D. Iazikov, C. Greiner, T. W. Mossberg, “Apodizable integrated filters for coarse WDM and FTTH-type applications,” J. Lightwave Technol. (to be published).

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Figures (5)

Fig. 1
Fig. 1

(a) Schematic top view of a two-channel holographic Bragg reflector with channel waveguide inputs and etched diffractive contours. (b) Cross section showing slab-waveguide region and etched diffractive contours. Some contours of the actual devices are written as dashed lines for diffractive amplitude control. For the devices considered here the grating period Λ0.5 μm. (c) Photograph of fabricated two-channel spectral comparator. For purposes of demonstration the device photographed has no upper cladding.

Fig. 2
Fig. 2

Phase and amplitude apodization profiles employed for one of the spectral comparator’s two channels. Amplitude (phase) apodization is implemented through partial contour writing (spatial displacement of contours).

Fig. 3
Fig. 3

(a), (b) Measured spectral transfer functions of the device’s two channels. (c) Summed spectra of both channels.

Fig. 4
Fig. 4

Comparison of measured and designed spectral transfer functions. (a) Channel spectrum as designed without accounting for amplitude-apodization-induced effective waveguide refractive-index variations. (b) Measured channel spectrum. (c) Channel spectrum as simulated from fabricated device design and accounting for amplitude-apodization-induced effective waveguide refractive-index variations.

Fig. 5
Fig. 5

Schematic depiction of a spectral comparator based on the device described here that provides analog comparison of input signal spectra to reference spectra encoded on device channels 1 and 2. The polarity of the output signal indicates which of the comparator spectral channels most closely matches the unknown input signal spectra.

Equations (3)

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

ΔTcLeff Δn2q24Λ2n3,
ΔR=c2nL.
SΔTΔRLLeff Δn2q22Λ2n2.

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