Abstract

The performance of wavelength-based photonic analog-to-digital converters (ADCs) is theoretically analyzed in terms of resolution and bandwidth as well as of noise tolerance. The analysis applies to any photonic ADC in which the analog input signal is converted into the wavelength of an optical carrier, but special emphasis is put on the spectrometerlike setup in which the wavelength is mapped to a spatial spot position. The binary output signals are then retrieved by an array of fan-out diffractive optical elements that redirect the beam onto the correct detectors. In particular, the case when the input signal controls the wavelength directly such that it will chirp in frequency during each sampling pulse or interval is studied. This chirping obviously broadens the spot on the diffractive optical element array; the effect of this broadening on noise tolerance and comparator accuracy is analytically analyzed, and accurate numerical calculations of the probability of error are presented.

© 2006 Optical Society of America

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References

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  1. R. H. Walden, "Analog-to-digital converter survey and analysis," IEEE J. Sel. Areas Commun. 17, 539-550 (1999).
    [CrossRef]
  2. S.-I. Oda and A. Maruta, "A novel quantization scheme by slicing a supercontinuum spectrum for all-optical analog-to-digital conversion," IEEE Photon. Technol. Lett. 17, 465-467 (2005).
    [CrossRef]
  3. H. Sakata, "Photonic analog-to-digital conversion by use of nonlinear Fabry-Perot resonators," Appl. Opt. 40, 240-248 (2001).
    [CrossRef]
  4. M. J. Hayduk, R. J. Bussjager, and M. A. Getbehead, "Photonic analog-to-digital conversion techniques using semiconductor saturable absorbers," in Enabling Photonic Technologies for Aerospace Applications II, E. W. Taylor and A. R. Pirich, eds., Proc. SPIE 4042, 54-60 (2000).
  5. H. Taylor, "An optical analog-to-digital converter--design and analysis," IEEE J. Quantum Electron. 15, 210-216 (1979).
    [CrossRef]
  6. J. Stigwall and S. Galt, "Interferometric analog-to-digital conversion scheme," IEEE Photon. Technol. Lett. 17, 468-470 (2005).
    [CrossRef]
  7. P. E. Pace and D. D. Styer, "High-resolution encoding process for an integrated optical analog-to-digital converter," Opt. Eng. 33, 2638-2645 (1994).
    [CrossRef]
  8. B. Jalali and Y. M. Xie, "Optical folding-flash analog-to-digital converter with analog encoding," Opt. Lett. 20, 1901-1903 (1995).
    [CrossRef] [PubMed]
  9. M. Currie, "Optical quantization of microwave signals via distributed phase modulation," IEEE J. Lightwave Technol. 23, 827-833 (2005).
    [CrossRef]
  10. Y. Tsunoda and J. W. Goodman, "Combined optical AD conversion and page composition for holographic memory applications," Appl. Opt. 16, 2607-2609 (1977).
    [CrossRef] [PubMed]
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  12. M. Johansson, B. Lofving, S. Hard, L. Thylen, M. Mokhtari, U. Westergren, and C. Pala, "Study of an ultrafast analog-to-digital conversion scheme based on diffractive optics," Appl. Opt. 39, 2881-2887 (2000).
    [CrossRef]
  13. C. Pala, L. Thylen, M. Mokhtari, and U. Westergren, "A high-speed electro-optical analog-to-digital converter principle," in Proceedings of the IEEE International Symposium on Circuits and Systems (Institute of Electrical and Electronics Engineers, 2001), Vol. 1, pp. 432-435.
  14. H. Zmuda, M. J. Hayduk, R. Bussjager, and E. N. Toughlian, "Wavelength-based analog-to-digital conversion," in Photonics for Space and Radiation Environments II, F. Berghmans and E. W. Taylor, eds., Proc. SPIE 4547, 134-145 (2002).
  15. S. Galt, A. Magnusson, and S. Hard, "Dynamic demonstration of diffractive optic analog-to-digital converter scheme," Appl. Opt. 42, 264-270 (2003).
    [CrossRef] [PubMed]
  16. S. Hisatake, K. Shibuya, and T. Kobayashi, "Ultrafast traveling-wave electro-optic deflector using domain-engineered LiTaO3 crystal," Appl. Phys. Lett. 87, 1-3 (2005).
    [CrossRef]
  17. P. W. Juodawlkis, J. C. Twichell, G. E. Betts, J. J. Hargreaves, R. D. Younger, J. L. Wasserman, F. J. O'Donnell, K. G. Ray, and R. C. Williamson, "Optically sampled analog-to-digital converters," IEEE Trans. Microwave Theory Tech. 49, 1840-1853 (2001).
    [CrossRef]
  18. J. Stigwall and J. Bengtsson, "Design of array of diffractive optical elements with inter-element coherent fan-outs," Opt. Express 12, 5675-5683 (2004).
    [CrossRef] [PubMed]
  19. O. A. Lavrova, L. Rau, and D. J. Blumenthal, "10-Gb/s agile wavelength conversion with nanosecond tuning times using a multisection widely tunable laser," IEEE J. Lightwave Technol. 20, 712-717 (2002).
    [CrossRef]
  20. G. P. Agrawal, Fiber-Optic Communication Systems (Wiley, 2002).
    [CrossRef]

2005 (4)

J. Stigwall and S. Galt, "Interferometric analog-to-digital conversion scheme," IEEE Photon. Technol. Lett. 17, 468-470 (2005).
[CrossRef]

S.-I. Oda and A. Maruta, "A novel quantization scheme by slicing a supercontinuum spectrum for all-optical analog-to-digital conversion," IEEE Photon. Technol. Lett. 17, 465-467 (2005).
[CrossRef]

M. Currie, "Optical quantization of microwave signals via distributed phase modulation," IEEE J. Lightwave Technol. 23, 827-833 (2005).
[CrossRef]

S. Hisatake, K. Shibuya, and T. Kobayashi, "Ultrafast traveling-wave electro-optic deflector using domain-engineered LiTaO3 crystal," Appl. Phys. Lett. 87, 1-3 (2005).
[CrossRef]

2004 (1)

2003 (1)

2002 (1)

O. A. Lavrova, L. Rau, and D. J. Blumenthal, "10-Gb/s agile wavelength conversion with nanosecond tuning times using a multisection widely tunable laser," IEEE J. Lightwave Technol. 20, 712-717 (2002).
[CrossRef]

2001 (2)

H. Sakata, "Photonic analog-to-digital conversion by use of nonlinear Fabry-Perot resonators," Appl. Opt. 40, 240-248 (2001).
[CrossRef]

P. W. Juodawlkis, J. C. Twichell, G. E. Betts, J. J. Hargreaves, R. D. Younger, J. L. Wasserman, F. J. O'Donnell, K. G. Ray, and R. C. Williamson, "Optically sampled analog-to-digital converters," IEEE Trans. Microwave Theory Tech. 49, 1840-1853 (2001).
[CrossRef]

2000 (1)

1999 (1)

R. H. Walden, "Analog-to-digital converter survey and analysis," IEEE J. Sel. Areas Commun. 17, 539-550 (1999).
[CrossRef]

1995 (1)

1994 (1)

P. E. Pace and D. D. Styer, "High-resolution encoding process for an integrated optical analog-to-digital converter," Opt. Eng. 33, 2638-2645 (1994).
[CrossRef]

1979 (1)

H. Taylor, "An optical analog-to-digital converter--design and analysis," IEEE J. Quantum Electron. 15, 210-216 (1979).
[CrossRef]

1977 (1)

Agrawal, G. P.

G. P. Agrawal, Fiber-Optic Communication Systems (Wiley, 2002).
[CrossRef]

Bengtsson, J.

Betts, G. E.

P. W. Juodawlkis, J. C. Twichell, G. E. Betts, J. J. Hargreaves, R. D. Younger, J. L. Wasserman, F. J. O'Donnell, K. G. Ray, and R. C. Williamson, "Optically sampled analog-to-digital converters," IEEE Trans. Microwave Theory Tech. 49, 1840-1853 (2001).
[CrossRef]

Blumenthal, D. J.

O. A. Lavrova, L. Rau, and D. J. Blumenthal, "10-Gb/s agile wavelength conversion with nanosecond tuning times using a multisection widely tunable laser," IEEE J. Lightwave Technol. 20, 712-717 (2002).
[CrossRef]

Bussjager, R.

H. Zmuda, M. J. Hayduk, R. Bussjager, and E. N. Toughlian, "Wavelength-based analog-to-digital conversion," in Photonics for Space and Radiation Environments II, F. Berghmans and E. W. Taylor, eds., Proc. SPIE 4547, 134-145 (2002).

Bussjager, R. J.

M. J. Hayduk, R. J. Bussjager, and M. A. Getbehead, "Photonic analog-to-digital conversion techniques using semiconductor saturable absorbers," in Enabling Photonic Technologies for Aerospace Applications II, E. W. Taylor and A. R. Pirich, eds., Proc. SPIE 4042, 54-60 (2000).

Currie, M.

M. Currie, "Optical quantization of microwave signals via distributed phase modulation," IEEE J. Lightwave Technol. 23, 827-833 (2005).
[CrossRef]

Galt, S.

J. Stigwall and S. Galt, "Interferometric analog-to-digital conversion scheme," IEEE Photon. Technol. Lett. 17, 468-470 (2005).
[CrossRef]

S. Galt, A. Magnusson, and S. Hard, "Dynamic demonstration of diffractive optic analog-to-digital converter scheme," Appl. Opt. 42, 264-270 (2003).
[CrossRef] [PubMed]

J. Stigwall, S. Galt, and S. Hard, "Experimental evaluation of an ultra-fast free space optical analog-to-digital conversion scheme using a tunable semiconductor laser," in Photonics Europe 2004: Metrology, Sensing, and Biophotonics, Proc. SPIE 5466, 123-130 (2004).

Getbehead, M. A.

M. J. Hayduk, R. J. Bussjager, and M. A. Getbehead, "Photonic analog-to-digital conversion techniques using semiconductor saturable absorbers," in Enabling Photonic Technologies for Aerospace Applications II, E. W. Taylor and A. R. Pirich, eds., Proc. SPIE 4042, 54-60 (2000).

Goodman, J. W.

Hard, S.

S. Galt, A. Magnusson, and S. Hard, "Dynamic demonstration of diffractive optic analog-to-digital converter scheme," Appl. Opt. 42, 264-270 (2003).
[CrossRef] [PubMed]

M. Johansson, B. Lofving, S. Hard, L. Thylen, M. Mokhtari, U. Westergren, and C. Pala, "Study of an ultrafast analog-to-digital conversion scheme based on diffractive optics," Appl. Opt. 39, 2881-2887 (2000).
[CrossRef]

J. Stigwall, S. Galt, and S. Hard, "Experimental evaluation of an ultra-fast free space optical analog-to-digital conversion scheme using a tunable semiconductor laser," in Photonics Europe 2004: Metrology, Sensing, and Biophotonics, Proc. SPIE 5466, 123-130 (2004).

Hargreaves, J. J.

P. W. Juodawlkis, J. C. Twichell, G. E. Betts, J. J. Hargreaves, R. D. Younger, J. L. Wasserman, F. J. O'Donnell, K. G. Ray, and R. C. Williamson, "Optically sampled analog-to-digital converters," IEEE Trans. Microwave Theory Tech. 49, 1840-1853 (2001).
[CrossRef]

Hayduk, M. J.

M. J. Hayduk, R. J. Bussjager, and M. A. Getbehead, "Photonic analog-to-digital conversion techniques using semiconductor saturable absorbers," in Enabling Photonic Technologies for Aerospace Applications II, E. W. Taylor and A. R. Pirich, eds., Proc. SPIE 4042, 54-60 (2000).

H. Zmuda, M. J. Hayduk, R. Bussjager, and E. N. Toughlian, "Wavelength-based analog-to-digital conversion," in Photonics for Space and Radiation Environments II, F. Berghmans and E. W. Taylor, eds., Proc. SPIE 4547, 134-145 (2002).

Hisatake, S.

S. Hisatake, K. Shibuya, and T. Kobayashi, "Ultrafast traveling-wave electro-optic deflector using domain-engineered LiTaO3 crystal," Appl. Phys. Lett. 87, 1-3 (2005).
[CrossRef]

Jalali, B.

Johansson, M.

Juodawlkis, P. W.

P. W. Juodawlkis, J. C. Twichell, G. E. Betts, J. J. Hargreaves, R. D. Younger, J. L. Wasserman, F. J. O'Donnell, K. G. Ray, and R. C. Williamson, "Optically sampled analog-to-digital converters," IEEE Trans. Microwave Theory Tech. 49, 1840-1853 (2001).
[CrossRef]

Kobayashi, T.

S. Hisatake, K. Shibuya, and T. Kobayashi, "Ultrafast traveling-wave electro-optic deflector using domain-engineered LiTaO3 crystal," Appl. Phys. Lett. 87, 1-3 (2005).
[CrossRef]

Lavrova, O. A.

O. A. Lavrova, L. Rau, and D. J. Blumenthal, "10-Gb/s agile wavelength conversion with nanosecond tuning times using a multisection widely tunable laser," IEEE J. Lightwave Technol. 20, 712-717 (2002).
[CrossRef]

Lofving, B.

Magnusson, A.

Maruta, A.

S.-I. Oda and A. Maruta, "A novel quantization scheme by slicing a supercontinuum spectrum for all-optical analog-to-digital conversion," IEEE Photon. Technol. Lett. 17, 465-467 (2005).
[CrossRef]

Mokhtari, M.

M. Johansson, B. Lofving, S. Hard, L. Thylen, M. Mokhtari, U. Westergren, and C. Pala, "Study of an ultrafast analog-to-digital conversion scheme based on diffractive optics," Appl. Opt. 39, 2881-2887 (2000).
[CrossRef]

C. Pala, L. Thylen, M. Mokhtari, and U. Westergren, "A high-speed electro-optical analog-to-digital converter principle," in Proceedings of the IEEE International Symposium on Circuits and Systems (Institute of Electrical and Electronics Engineers, 2001), Vol. 1, pp. 432-435.

Oda, S.-I.

S.-I. Oda and A. Maruta, "A novel quantization scheme by slicing a supercontinuum spectrum for all-optical analog-to-digital conversion," IEEE Photon. Technol. Lett. 17, 465-467 (2005).
[CrossRef]

O'Donnell, F. J.

P. W. Juodawlkis, J. C. Twichell, G. E. Betts, J. J. Hargreaves, R. D. Younger, J. L. Wasserman, F. J. O'Donnell, K. G. Ray, and R. C. Williamson, "Optically sampled analog-to-digital converters," IEEE Trans. Microwave Theory Tech. 49, 1840-1853 (2001).
[CrossRef]

Pace, P. E.

P. E. Pace and D. D. Styer, "High-resolution encoding process for an integrated optical analog-to-digital converter," Opt. Eng. 33, 2638-2645 (1994).
[CrossRef]

Pala, C.

M. Johansson, B. Lofving, S. Hard, L. Thylen, M. Mokhtari, U. Westergren, and C. Pala, "Study of an ultrafast analog-to-digital conversion scheme based on diffractive optics," Appl. Opt. 39, 2881-2887 (2000).
[CrossRef]

C. Pala, L. Thylen, M. Mokhtari, and U. Westergren, "A high-speed electro-optical analog-to-digital converter principle," in Proceedings of the IEEE International Symposium on Circuits and Systems (Institute of Electrical and Electronics Engineers, 2001), Vol. 1, pp. 432-435.

Rau, L.

O. A. Lavrova, L. Rau, and D. J. Blumenthal, "10-Gb/s agile wavelength conversion with nanosecond tuning times using a multisection widely tunable laser," IEEE J. Lightwave Technol. 20, 712-717 (2002).
[CrossRef]

Ray, K. G.

P. W. Juodawlkis, J. C. Twichell, G. E. Betts, J. J. Hargreaves, R. D. Younger, J. L. Wasserman, F. J. O'Donnell, K. G. Ray, and R. C. Williamson, "Optically sampled analog-to-digital converters," IEEE Trans. Microwave Theory Tech. 49, 1840-1853 (2001).
[CrossRef]

Sakata, H.

Shibuya, K.

S. Hisatake, K. Shibuya, and T. Kobayashi, "Ultrafast traveling-wave electro-optic deflector using domain-engineered LiTaO3 crystal," Appl. Phys. Lett. 87, 1-3 (2005).
[CrossRef]

Stigwall, J.

J. Stigwall and S. Galt, "Interferometric analog-to-digital conversion scheme," IEEE Photon. Technol. Lett. 17, 468-470 (2005).
[CrossRef]

J. Stigwall and J. Bengtsson, "Design of array of diffractive optical elements with inter-element coherent fan-outs," Opt. Express 12, 5675-5683 (2004).
[CrossRef] [PubMed]

J. Stigwall, S. Galt, and S. Hard, "Experimental evaluation of an ultra-fast free space optical analog-to-digital conversion scheme using a tunable semiconductor laser," in Photonics Europe 2004: Metrology, Sensing, and Biophotonics, Proc. SPIE 5466, 123-130 (2004).

Styer, D. D.

P. E. Pace and D. D. Styer, "High-resolution encoding process for an integrated optical analog-to-digital converter," Opt. Eng. 33, 2638-2645 (1994).
[CrossRef]

Taylor, H.

H. Taylor, "An optical analog-to-digital converter--design and analysis," IEEE J. Quantum Electron. 15, 210-216 (1979).
[CrossRef]

Thylen, L.

M. Johansson, B. Lofving, S. Hard, L. Thylen, M. Mokhtari, U. Westergren, and C. Pala, "Study of an ultrafast analog-to-digital conversion scheme based on diffractive optics," Appl. Opt. 39, 2881-2887 (2000).
[CrossRef]

C. Pala, L. Thylen, M. Mokhtari, and U. Westergren, "A high-speed electro-optical analog-to-digital converter principle," in Proceedings of the IEEE International Symposium on Circuits and Systems (Institute of Electrical and Electronics Engineers, 2001), Vol. 1, pp. 432-435.

Toughlian, E. N.

H. Zmuda, M. J. Hayduk, R. Bussjager, and E. N. Toughlian, "Wavelength-based analog-to-digital conversion," in Photonics for Space and Radiation Environments II, F. Berghmans and E. W. Taylor, eds., Proc. SPIE 4547, 134-145 (2002).

Tsunoda, Y.

Twichell, J. C.

P. W. Juodawlkis, J. C. Twichell, G. E. Betts, J. J. Hargreaves, R. D. Younger, J. L. Wasserman, F. J. O'Donnell, K. G. Ray, and R. C. Williamson, "Optically sampled analog-to-digital converters," IEEE Trans. Microwave Theory Tech. 49, 1840-1853 (2001).
[CrossRef]

Walden, R. H.

R. H. Walden, "Analog-to-digital converter survey and analysis," IEEE J. Sel. Areas Commun. 17, 539-550 (1999).
[CrossRef]

Wasserman, J. L.

P. W. Juodawlkis, J. C. Twichell, G. E. Betts, J. J. Hargreaves, R. D. Younger, J. L. Wasserman, F. J. O'Donnell, K. G. Ray, and R. C. Williamson, "Optically sampled analog-to-digital converters," IEEE Trans. Microwave Theory Tech. 49, 1840-1853 (2001).
[CrossRef]

Westergren, U.

M. Johansson, B. Lofving, S. Hard, L. Thylen, M. Mokhtari, U. Westergren, and C. Pala, "Study of an ultrafast analog-to-digital conversion scheme based on diffractive optics," Appl. Opt. 39, 2881-2887 (2000).
[CrossRef]

C. Pala, L. Thylen, M. Mokhtari, and U. Westergren, "A high-speed electro-optical analog-to-digital converter principle," in Proceedings of the IEEE International Symposium on Circuits and Systems (Institute of Electrical and Electronics Engineers, 2001), Vol. 1, pp. 432-435.

Williamson, R. C.

P. W. Juodawlkis, J. C. Twichell, G. E. Betts, J. J. Hargreaves, R. D. Younger, J. L. Wasserman, F. J. O'Donnell, K. G. Ray, and R. C. Williamson, "Optically sampled analog-to-digital converters," IEEE Trans. Microwave Theory Tech. 49, 1840-1853 (2001).
[CrossRef]

Xie, Y. M.

Younger, R. D.

P. W. Juodawlkis, J. C. Twichell, G. E. Betts, J. J. Hargreaves, R. D. Younger, J. L. Wasserman, F. J. O'Donnell, K. G. Ray, and R. C. Williamson, "Optically sampled analog-to-digital converters," IEEE Trans. Microwave Theory Tech. 49, 1840-1853 (2001).
[CrossRef]

Zmuda, H.

H. Zmuda, M. J. Hayduk, R. Bussjager, and E. N. Toughlian, "Wavelength-based analog-to-digital conversion," in Photonics for Space and Radiation Environments II, F. Berghmans and E. W. Taylor, eds., Proc. SPIE 4547, 134-145 (2002).

Appl. Opt. (4)

Appl. Phys. Lett. (1)

S. Hisatake, K. Shibuya, and T. Kobayashi, "Ultrafast traveling-wave electro-optic deflector using domain-engineered LiTaO3 crystal," Appl. Phys. Lett. 87, 1-3 (2005).
[CrossRef]

IEEE J. Lightwave Technol. (2)

O. A. Lavrova, L. Rau, and D. J. Blumenthal, "10-Gb/s agile wavelength conversion with nanosecond tuning times using a multisection widely tunable laser," IEEE J. Lightwave Technol. 20, 712-717 (2002).
[CrossRef]

M. Currie, "Optical quantization of microwave signals via distributed phase modulation," IEEE J. Lightwave Technol. 23, 827-833 (2005).
[CrossRef]

IEEE J. Quantum Electron. (1)

H. Taylor, "An optical analog-to-digital converter--design and analysis," IEEE J. Quantum Electron. 15, 210-216 (1979).
[CrossRef]

IEEE J. Sel. Areas Commun. (1)

R. H. Walden, "Analog-to-digital converter survey and analysis," IEEE J. Sel. Areas Commun. 17, 539-550 (1999).
[CrossRef]

IEEE Photon. Technol. Lett. (2)

S.-I. Oda and A. Maruta, "A novel quantization scheme by slicing a supercontinuum spectrum for all-optical analog-to-digital conversion," IEEE Photon. Technol. Lett. 17, 465-467 (2005).
[CrossRef]

J. Stigwall and S. Galt, "Interferometric analog-to-digital conversion scheme," IEEE Photon. Technol. Lett. 17, 468-470 (2005).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

P. W. Juodawlkis, J. C. Twichell, G. E. Betts, J. J. Hargreaves, R. D. Younger, J. L. Wasserman, F. J. O'Donnell, K. G. Ray, and R. C. Williamson, "Optically sampled analog-to-digital converters," IEEE Trans. Microwave Theory Tech. 49, 1840-1853 (2001).
[CrossRef]

Opt. Eng. (1)

P. E. Pace and D. D. Styer, "High-resolution encoding process for an integrated optical analog-to-digital converter," Opt. Eng. 33, 2638-2645 (1994).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Other (5)

M. J. Hayduk, R. J. Bussjager, and M. A. Getbehead, "Photonic analog-to-digital conversion techniques using semiconductor saturable absorbers," in Enabling Photonic Technologies for Aerospace Applications II, E. W. Taylor and A. R. Pirich, eds., Proc. SPIE 4042, 54-60 (2000).

G. P. Agrawal, Fiber-Optic Communication Systems (Wiley, 2002).
[CrossRef]

J. Stigwall, S. Galt, and S. Hard, "Experimental evaluation of an ultra-fast free space optical analog-to-digital conversion scheme using a tunable semiconductor laser," in Photonics Europe 2004: Metrology, Sensing, and Biophotonics, Proc. SPIE 5466, 123-130 (2004).

C. Pala, L. Thylen, M. Mokhtari, and U. Westergren, "A high-speed electro-optical analog-to-digital converter principle," in Proceedings of the IEEE International Symposium on Circuits and Systems (Institute of Electrical and Electronics Engineers, 2001), Vol. 1, pp. 432-435.

H. Zmuda, M. J. Hayduk, R. Bussjager, and E. N. Toughlian, "Wavelength-based analog-to-digital conversion," in Photonics for Space and Radiation Environments II, F. Berghmans and E. W. Taylor, eds., Proc. SPIE 4547, 134-145 (2002).

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Figures (6)

Fig. 1
Fig. 1

Schematic setup of a spectrometerlike quantizer when the input signal is controlling the wavelength of a tunable laser. The wavelength is in turn converted into a spatial position in the spectrum plane by the grating and the focusing lens and then encoded into binary form by an array of beam-splitting diffractive optical elements.

Fig. 2
Fig. 2

(a), (b) Spot intensity profiles Is (x). (c) LSB bit function q LSB(x) (dashed line) together with the resultant LSB output, g LSB(x) = qLSB (x) ⊗ Is (x) (solid curves with shaded noise regions) for wx = 2Δx. Here the sinusoidal approximation is practically identical to the exact solution. (d) g LSB(x) with acceptable noise regions for wx = 0.5Δx, together with the Gaussian approximation (dashed–dotted curve).

Fig. 3
Fig. 3

Error of the two approximations shown, at x = Δx∕2, as functions of wx.

Fig. 4
Fig. 4

Required detector output SNR versus beam radius. Two-bit error probability P for N = 6 bits (solid curves). The SNR calculated from J noise is also plotted (dotted curve).

Fig. 5
Fig. 5

Maximum spectral broadening as a function of temporal apodization window wt. Transform-limited (dashed curve), chirp-limited (dotted curve), and total (solid curve) broadenings are plotted for f = 1 GHz and Δν = 20 GHz.

Fig. 6
Fig. 6

(a) Maximum frequency f versus tuning range Δν (or Δλ) for a range of resolutions N (number of bits), directly driven without S&H. (b) Sample rate M versus tuning range for a range of resolutions with η = 0.5, with S&H. In both cases, wx ∕Δx = 2 and λ0 = 1550 nm.

Equations (48)

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

q LSB ( x ) = mod { ceil [ ( x / Δ x ) / 2 ] , 2 } ,
I s ( x ) = 2 w x π exp ( 2 x 2 w x     2 ) ,
g LSB ( x ) = q LSB ( x ) I s ( x ) .
q LSB ( x ) = 1 2 + 2 π n = 1 , 3 , 5 , 1 n sin ( n π x 2 Δ x ) .
I s ( k x ) = [ I s ( x ) ] = exp ( k x     2 w x     2 8 ) = exp ( π 2 n 2 w x     2 32 Δ x 2 ) .
g LSB ( x ) = 1 2 + 2 π n = 1 , 3 , 5 , 1 n sin ( n π x 2 Δ x ) ×  exp ( - π 2 n 2 w x     2 32 Δ x 2 ) .
g LSB ( x ) 1 2 + 2 π sin ( π x 2 Δ x ) exp ( - π 2 w x     2 32 Δ x 2 ) = 1 2 + A 1 sin ( π x 2 Δ x ) g LSB sine ( x ) .
J noise sine < g LSB sine ( x = Δ x / 2 ) 1 / 2
A 1 sin ( π 4 ) = 2 2 A 1 = 2 π exp ( - π 2 w x     2 32 Δ x 2 ) .
w x     sine < 4 Δ x π [ 2 ln ( 2 π J noise ) ] 1 / 2 .
g LSB ( x ) 1 1 2 exp [ 1 2 ( 2 x w x + 1 2 ) 2 ] g LSB Gauss ( x )
J noise Gauss < g LSB Gauss ( x = Δ x / 2 ) 1 2
1 2 1 2 exp [ 1 2 ( Δ x w x + 1 2 ) 2 ]
w x Gauss < Δ x - 2 ln ( 1 2 J noise ) - 1 .
g k ( x ) = g k i ( x ) + X = q k ( x ) I s ( x ) + X ,
f x ( y ) = 1 σ 2 π exp ( y 2 2 σ 2 ) ,
SNR = 0.25 / σ 2 .
P k 0 ( x ) = p [ g k ( x ) < 0.5 ] = Φ [ 0.5 g k i ( x ) σ ] ,
P k err ( x ) = min [ P k 0 , P k 1 ] = 1 2 | 1 2 P k 0 ( x ) | .
P ( x ) = 1 k s [ 1 P k err ( x ) ]
P = x min x max P ( x ) d x x max - x min .
ν ( t ) = ν 0 + Δ ν 2 sin ( 2 π f t ) ,
a ( t ) = exp ( t 2 w t 2 )
w ν   trans = 1 π w t .
ν t = π Δ ν f ,
w ν   chirp = ν t w t = π Δ ν f w t .
w ν   tot = [ w v trans 2 + w v chirp 2 ] 1 / 2 = [ ( 1 / π w t ) 2 + ( πΔν f w t ) 2 ] 1 / 2 .
w t ( w v trans 2 + w v chirp 2 ) = 0 ,
2 π 2 Δ ν 2 f 2 w t 2 π 2 w t 3 = 0
w t opt = 1 π 1 / Δ ν f .
w v trans opt = 1 π w t = Δ ν f = w v chirp opt .
w v tot min = 2 Δ ν f .
ν = Δ ν m Δ x x .
w ν = w x Δ ν m Δ x > 2 Δ ν f ,
m < Δ ν 2 f w x Δ x ,
f < Δ ν 2 m 2 ( w x Δ x ) 2 ,
Δ ν > 2 m 2 f ( Δ x w x ) .
J noise sine < 2 π exp ( π 2 m 2 f 16 Δ ν ) ;
J noise Gauss < 1 2 1 2 exp [ 1 2 ( 1 m Δ ν 2 f + 1 2 ) 2 ] .
w t Δ t 2 = η 2 M ,
w ν = 1 π w t 2 M π η .
w ν = w x Δ ν m Δ x > 2 M π η ,
m < π η Δ ν 2 M w x Δ x .
M < π η Δ ν 2 m w x Δ x ,
Δ ν > 2 m M π η Δ x w x .
SNR   = P sig P shot + P therm ,
P in = 2 R [ SNR ( k B T / R L ) F n Δ f ] 1 / 2 .
f LSB = m f max = 2 N f max .

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