Abstract

We study theoretically and experimentally the synthesis of arbitrary time-domain pulse shapes, using short laser pulses scattered by holograms stored in a spectrally selective hole-burning material. In general, writing holograms in spectrally selective materials results in cross talk and interference between different frequencies because of Kramers–Kronig dispersion relations. We discuss different ways to exclude the cross talk that disturbs faithful reproduction of the desired time-domain pulse shapes. In particular, we show that one can exclude the cross talk by writing holograms in a way that simulates a time-domain offset of the object pulse. To confirm our theoretical considerations we carry out experiments by writing persistent spectral hole-burning holograms with a tunable dye laser in an organic dye–polymer system at low temperature. By reading out the time-domain response with subpicosecond white-light pulses we demonstrate the feasibility of spectral synthesis of light pulses with complicated amplitude and phase properties on the time scale of hundreds of picoseconds with a subpicosecond time resolution.

© 1995 Optical Society of America

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  1. A. A. Gorokhovskii, R. K. Kaarli, and L. A. Rebane, "Hole burning in the contour of a pure electronic line in a Shpolskii system," JETP Lett. 20, 216–219 (1974); B. M. Kharlamov, R. I. Personov, and L. A. Bykovskaya, "Stable gap in absorption spectra of solid solutions of organic molecules by laser irradiation," Opt. Commun. 12, 191–194 (1974).
    [CrossRef]
  2. L. A. Rebane, A. A. Gorokhovskii, and J. V. Kikas, "Lowtemperature spectroscopy of organic molecules in solids by photochemical hole burning," Appl. Phys. B 29, 235–250 (1982); J. Friedrich and D. Haarer, "Photochemical hole burning: spectroscopic study of relaxation processes in polymers and glasses," Angew. Chem. Int. Ed. Engl. 23, 113–140 (1984).
    [CrossRef]
  3. W. E. Moerner, ed., Persistent Spectral Hole Burning: Science and Applications, Vol. 44 of Topics in Current Physics (Springer-Verlag, Berlin, 1988), and references therein.
    [CrossRef]
  4. K. Holliday and U. P. Wild, "Spectral hole burning," in Molecular Luminescence Spectroscopy. Part 3, S. G. Schulman, ed., Vol. 77 of Chemical Analysis Series (Wiley, New York, 1993), p. 149.
  5. K. K. Rebane, Impurity Spectra of Solids (Plenum, New York, 1970).
  6. A. Renn, A. J. Meixner, U. P. Wild, and F. A. Burkhalter, "Holographic detection of photochemical holes," Chem. Phys. 93, 157–162 (1985).
    [CrossRef]
  7. A. Renn and U. P. Wild, "Spectral hole burning and hologram storage," Appl. Opt. 26, 4040–4042 (1987); A. J. Meixner, A. Renn, and U. P. Wild, "Spectral hole burning and holography. I. Transmission and holographic detection of spectral holes," J. Chem. Phys. 91, 6728–6736 (1989); U. P. Wild, A. Renn. C. De Caro, and S. Bernet, "Spectral hole burning and molecular computing," Appl. Opt. 29, 4329–4331 (1990).
    [CrossRef] [PubMed]
  8. A. Renn, A. J. Meixner, and U. P. Wild, "Spectral hole burning and holography. II. Diffraction properties of two spectrally adjacent holgorams," J. Chem. Phys. 92, 2748–2755 (1990).
    [CrossRef]
  9. T. W. Mossberg, "Time-domain frequency-selective optical data storage," Opt. Lett. 7, 77–79 (1982).
    [CrossRef] [PubMed]
  10. N. W. Carlson, L. J. Rothberg, A. G. Yodh, W. R. Babbitt, and T. W. Mossberg, "Storage and time reversal of light pulses using photon echoes," Opt. Lett. 8, 483–485 (1983).
    [CrossRef] [PubMed]
  11. A. K. Rebane, R. K. Kaarli, and P. M. Saari, "Hole burning by coherent sequences of picosecond pulses," JETP Lett. 38, 383–386 (1983); A. Rebane, R. Kaarli, P. Saari, A. Anijalg, and K. Timpmann, "Photochemical time-domain holography of weak picosecond pulses," Opt. Commun. 47, 173–176 (1983); A. Rebane and R. Kaarli, "Picosecond pulse shaping by photochemical time domain holography," Chem. Phys. Lett. 101, 317–319 (1983).
    [CrossRef]
  12. P. Saari and A. Rebane, "Time- and space-domain holography of pulsed light fields in a spectrally photo-active medium," Proc. Estonian Acad. Sci. Phys. Math. 33, 322–332 (1984); P. Saari, R. Kaarli, and A. Rebane, "Picosecond timeand space-domain holography by photochemical hole burning," J. Opt. Soc. Am. B 3, 527–533 (1986); A. Rebane, "Coherent recall and time–space holography in impurity systems with photochemical hole-burning," Ph.D. dissertation (Institute of Physics, Estonian Academy of Sciences, Tartu, Estonia, 1985).
    [CrossRef]
  13. A. Rebane, J. Aaviksoo, and J. Kuhl, "Storage and time reversal of femtosecond light signals via persistent spectral hole burning holography," Appl. Phys. Lett. 54, 93–95 (1989); A. Rebane and J. Feinberg, "Time-resolved holography," Nature 351, 378–380 (1991); H. Gygax, A. Rebane, and U. P. Wild, "Stark effect in dye-doped polymers studied by photochemically accumulated photon echo," J. Opt. Soc. Am. B 10, 1149–1158 (1993).
    [CrossRef]
  14. Yu. T. Mazurenko, "Holography of wave packets," Appl. Phys. B 50, 101–114 (1990).
    [CrossRef]
  15. A. M. Weiner, D. E. Leaird, D. H. Reitze, and E. G. Paek, "Femtosecond spectral holography," IEEE J. Quantum Electron. 28, 2251–2261 (1992).
    [CrossRef]
  16. M. Mitsunaga, R. Yano, and N. Uesugi, "Spectrally programmed stimulated photon echo," Opt. Lett. 16, 264–266 (1991).
    [CrossRef] [PubMed]
  17. H. S˜najalg, A. Gorokhovskii, R. Kaarli, V. Palm, M. Rätsep, and P. Saari, "Optical pulse shaping by filters based on spectral hole burning," Opt. Commun. 71, 377–380 (1989).
    [CrossRef]
  18. H. Schwoerer, D. Erni, A. Rebane, and U. P. Wild, "Subpicosecond pulse shaping via spectral hole burning," Opt. Commun. 107, 123–128 (1994).
    [CrossRef]
  19. A. Rebane, S. Bernet, A. Renn, and U. P. Wild, "Holography in frequency selective media: hologram phase and causality," Opt. Commun. 86, 7–13 (1991).
    [CrossRef]
  20. S. Bernet, B. Kohler, A. Rebane, A. Renn, and U. P. Wild, "Holography in frequency selective media. II. Controlling the diffraction efficiency," J. Lumin. 53, 215–218 (1992).
    [CrossRef]
  21. S. Bernet, B. Kohler, A. Rebane, A. Renn, and U. P. Wild, "Spectral hole burning and holography. V. Asymmetric diffraction from thin holograms," J. Opt. Soc. Am. B 9, 987–991 (1992).
    [CrossRef]
  22. S. Bernet, "Phasenkontrollierte Holographie in frequenzselektiven Materialien," Ph.D. dissertation Diss ETH Nr. 10292 (Swiss Federal Institute of Technology, Zurich, 1993).
  23. L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms (Wiley, New York, 1975).
  24. J. H. Eberly, S. R. Hartmann, and A. Szabo, "Propagation narrowing in the transmission of a light pulse through a spectral hole," Phys. Rev. A 23, 2502–2506 (1981); M. D. Crisp, "Propagation of small-area pulses of coherent light through a resonant medium," Phys. Rev. A 1, 1604–1611 (1970).
    [CrossRef]
  25. A. Rebane, "Associative space-and-time domain recall of picosecond light signals via photochemical hole burning holography," Opt. Commun. 65, 175–178 (1988); "Associative recall of time- and space-domain holograms in spectrally selective photo-active medium," Proc. Estonian Acad. Sci. Phys. Math. 37, 89–92 (1988).
    [CrossRef]
  26. D. E. Vakman and L. A. Vainstein, "Amplitude, phase, frequency—fundamental concepts of oscillation theory," Sov. Phys. Usp. 20, 1002–1016 (1977).
    [CrossRef]
  27. T. W. Mossberg, R. Kachru, and S. R. Hartmann, "Echoes in gaseous media: a generalized theory of rephasing phenomena," Phys. Rev. A 20, 1976–1996 (1979).
    [CrossRef]
  28. A. J. Meixner, A. Renn, S. E. Bucher, and U. P. Wild, "Spectral hole burning in glasses and polymer films: the Stark effect," J. Phys. Chem. 90, 6777–6785 (1986).
    [CrossRef]
  29. T. H. Chyba, L. J. Wang, L. Mandel, and R. Simon, "Measurement of the Pancharatnam phase for a light beam," Opt. Lett. 13, 562–564 (1988).
    [CrossRef] [PubMed]
  30. S. O. Elyutin, S. M. Zakharov, and E. M. Manykin, "Theory of formation of photon echo pulses," Sov. Phys. JETP 49, 421–431 (1979).
  31. V. A. Zuikov, V. V. Samartsev, and R. G. Usmanov, "Correlation of the shape of photon echo signals with the shape of excitation pulses," JETP Lett. 32, 270–274 (1980).

1994 (1)

H. Schwoerer, D. Erni, A. Rebane, and U. P. Wild, "Subpicosecond pulse shaping via spectral hole burning," Opt. Commun. 107, 123–128 (1994).
[CrossRef]

1992 (3)

S. Bernet, B. Kohler, A. Rebane, A. Renn, and U. P. Wild, "Holography in frequency selective media. II. Controlling the diffraction efficiency," J. Lumin. 53, 215–218 (1992).
[CrossRef]

A. M. Weiner, D. E. Leaird, D. H. Reitze, and E. G. Paek, "Femtosecond spectral holography," IEEE J. Quantum Electron. 28, 2251–2261 (1992).
[CrossRef]

S. Bernet, B. Kohler, A. Rebane, A. Renn, and U. P. Wild, "Spectral hole burning and holography. V. Asymmetric diffraction from thin holograms," J. Opt. Soc. Am. B 9, 987–991 (1992).
[CrossRef]

1991 (2)

M. Mitsunaga, R. Yano, and N. Uesugi, "Spectrally programmed stimulated photon echo," Opt. Lett. 16, 264–266 (1991).
[CrossRef] [PubMed]

A. Rebane, S. Bernet, A. Renn, and U. P. Wild, "Holography in frequency selective media: hologram phase and causality," Opt. Commun. 86, 7–13 (1991).
[CrossRef]

1990 (2)

Yu. T. Mazurenko, "Holography of wave packets," Appl. Phys. B 50, 101–114 (1990).
[CrossRef]

A. Renn, A. J. Meixner, and U. P. Wild, "Spectral hole burning and holography. II. Diffraction properties of two spectrally adjacent holgorams," J. Chem. Phys. 92, 2748–2755 (1990).
[CrossRef]

1989 (2)

H. S˜najalg, A. Gorokhovskii, R. Kaarli, V. Palm, M. Rätsep, and P. Saari, "Optical pulse shaping by filters based on spectral hole burning," Opt. Commun. 71, 377–380 (1989).
[CrossRef]

A. Rebane, J. Aaviksoo, and J. Kuhl, "Storage and time reversal of femtosecond light signals via persistent spectral hole burning holography," Appl. Phys. Lett. 54, 93–95 (1989); A. Rebane and J. Feinberg, "Time-resolved holography," Nature 351, 378–380 (1991); H. Gygax, A. Rebane, and U. P. Wild, "Stark effect in dye-doped polymers studied by photochemically accumulated photon echo," J. Opt. Soc. Am. B 10, 1149–1158 (1993).
[CrossRef]

1988 (2)

A. Rebane, "Associative space-and-time domain recall of picosecond light signals via photochemical hole burning holography," Opt. Commun. 65, 175–178 (1988); "Associative recall of time- and space-domain holograms in spectrally selective photo-active medium," Proc. Estonian Acad. Sci. Phys. Math. 37, 89–92 (1988).
[CrossRef]

T. H. Chyba, L. J. Wang, L. Mandel, and R. Simon, "Measurement of the Pancharatnam phase for a light beam," Opt. Lett. 13, 562–564 (1988).
[CrossRef] [PubMed]

1987 (1)

1986 (1)

A. J. Meixner, A. Renn, S. E. Bucher, and U. P. Wild, "Spectral hole burning in glasses and polymer films: the Stark effect," J. Phys. Chem. 90, 6777–6785 (1986).
[CrossRef]

1985 (1)

A. Renn, A. J. Meixner, U. P. Wild, and F. A. Burkhalter, "Holographic detection of photochemical holes," Chem. Phys. 93, 157–162 (1985).
[CrossRef]

1984 (1)

P. Saari and A. Rebane, "Time- and space-domain holography of pulsed light fields in a spectrally photo-active medium," Proc. Estonian Acad. Sci. Phys. Math. 33, 322–332 (1984); P. Saari, R. Kaarli, and A. Rebane, "Picosecond timeand space-domain holography by photochemical hole burning," J. Opt. Soc. Am. B 3, 527–533 (1986); A. Rebane, "Coherent recall and time–space holography in impurity systems with photochemical hole-burning," Ph.D. dissertation (Institute of Physics, Estonian Academy of Sciences, Tartu, Estonia, 1985).
[CrossRef]

1983 (2)

A. K. Rebane, R. K. Kaarli, and P. M. Saari, "Hole burning by coherent sequences of picosecond pulses," JETP Lett. 38, 383–386 (1983); A. Rebane, R. Kaarli, P. Saari, A. Anijalg, and K. Timpmann, "Photochemical time-domain holography of weak picosecond pulses," Opt. Commun. 47, 173–176 (1983); A. Rebane and R. Kaarli, "Picosecond pulse shaping by photochemical time domain holography," Chem. Phys. Lett. 101, 317–319 (1983).
[CrossRef]

N. W. Carlson, L. J. Rothberg, A. G. Yodh, W. R. Babbitt, and T. W. Mossberg, "Storage and time reversal of light pulses using photon echoes," Opt. Lett. 8, 483–485 (1983).
[CrossRef] [PubMed]

1982 (2)

T. W. Mossberg, "Time-domain frequency-selective optical data storage," Opt. Lett. 7, 77–79 (1982).
[CrossRef] [PubMed]

L. A. Rebane, A. A. Gorokhovskii, and J. V. Kikas, "Lowtemperature spectroscopy of organic molecules in solids by photochemical hole burning," Appl. Phys. B 29, 235–250 (1982); J. Friedrich and D. Haarer, "Photochemical hole burning: spectroscopic study of relaxation processes in polymers and glasses," Angew. Chem. Int. Ed. Engl. 23, 113–140 (1984).
[CrossRef]

1981 (1)

J. H. Eberly, S. R. Hartmann, and A. Szabo, "Propagation narrowing in the transmission of a light pulse through a spectral hole," Phys. Rev. A 23, 2502–2506 (1981); M. D. Crisp, "Propagation of small-area pulses of coherent light through a resonant medium," Phys. Rev. A 1, 1604–1611 (1970).
[CrossRef]

1980 (1)

V. A. Zuikov, V. V. Samartsev, and R. G. Usmanov, "Correlation of the shape of photon echo signals with the shape of excitation pulses," JETP Lett. 32, 270–274 (1980).

1979 (2)

S. O. Elyutin, S. M. Zakharov, and E. M. Manykin, "Theory of formation of photon echo pulses," Sov. Phys. JETP 49, 421–431 (1979).

T. W. Mossberg, R. Kachru, and S. R. Hartmann, "Echoes in gaseous media: a generalized theory of rephasing phenomena," Phys. Rev. A 20, 1976–1996 (1979).
[CrossRef]

1977 (1)

D. E. Vakman and L. A. Vainstein, "Amplitude, phase, frequency—fundamental concepts of oscillation theory," Sov. Phys. Usp. 20, 1002–1016 (1977).
[CrossRef]

1974 (1)

A. A. Gorokhovskii, R. K. Kaarli, and L. A. Rebane, "Hole burning in the contour of a pure electronic line in a Shpolskii system," JETP Lett. 20, 216–219 (1974); B. M. Kharlamov, R. I. Personov, and L. A. Bykovskaya, "Stable gap in absorption spectra of solid solutions of organic molecules by laser irradiation," Opt. Commun. 12, 191–194 (1974).
[CrossRef]

Aaviksoo, J.

A. Rebane, J. Aaviksoo, and J. Kuhl, "Storage and time reversal of femtosecond light signals via persistent spectral hole burning holography," Appl. Phys. Lett. 54, 93–95 (1989); A. Rebane and J. Feinberg, "Time-resolved holography," Nature 351, 378–380 (1991); H. Gygax, A. Rebane, and U. P. Wild, "Stark effect in dye-doped polymers studied by photochemically accumulated photon echo," J. Opt. Soc. Am. B 10, 1149–1158 (1993).
[CrossRef]

Allen, L.

L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms (Wiley, New York, 1975).

Babbitt, W. R.

Bernet, S.

S. Bernet, B. Kohler, A. Rebane, A. Renn, and U. P. Wild, "Holography in frequency selective media. II. Controlling the diffraction efficiency," J. Lumin. 53, 215–218 (1992).
[CrossRef]

S. Bernet, B. Kohler, A. Rebane, A. Renn, and U. P. Wild, "Spectral hole burning and holography. V. Asymmetric diffraction from thin holograms," J. Opt. Soc. Am. B 9, 987–991 (1992).
[CrossRef]

A. Rebane, S. Bernet, A. Renn, and U. P. Wild, "Holography in frequency selective media: hologram phase and causality," Opt. Commun. 86, 7–13 (1991).
[CrossRef]

S. Bernet, "Phasenkontrollierte Holographie in frequenzselektiven Materialien," Ph.D. dissertation Diss ETH Nr. 10292 (Swiss Federal Institute of Technology, Zurich, 1993).

Bucher, S. E.

A. J. Meixner, A. Renn, S. E. Bucher, and U. P. Wild, "Spectral hole burning in glasses and polymer films: the Stark effect," J. Phys. Chem. 90, 6777–6785 (1986).
[CrossRef]

Burkhalter, F. A.

A. Renn, A. J. Meixner, U. P. Wild, and F. A. Burkhalter, "Holographic detection of photochemical holes," Chem. Phys. 93, 157–162 (1985).
[CrossRef]

Carlson, N. W.

Chyba, T. H.

Eberly, J. H.

J. H. Eberly, S. R. Hartmann, and A. Szabo, "Propagation narrowing in the transmission of a light pulse through a spectral hole," Phys. Rev. A 23, 2502–2506 (1981); M. D. Crisp, "Propagation of small-area pulses of coherent light through a resonant medium," Phys. Rev. A 1, 1604–1611 (1970).
[CrossRef]

L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms (Wiley, New York, 1975).

Elyutin, S. O.

S. O. Elyutin, S. M. Zakharov, and E. M. Manykin, "Theory of formation of photon echo pulses," Sov. Phys. JETP 49, 421–431 (1979).

Erni, D.

H. Schwoerer, D. Erni, A. Rebane, and U. P. Wild, "Subpicosecond pulse shaping via spectral hole burning," Opt. Commun. 107, 123–128 (1994).
[CrossRef]

Gorokhovskii, A.

H. S˜najalg, A. Gorokhovskii, R. Kaarli, V. Palm, M. Rätsep, and P. Saari, "Optical pulse shaping by filters based on spectral hole burning," Opt. Commun. 71, 377–380 (1989).
[CrossRef]

Gorokhovskii, A. A.

L. A. Rebane, A. A. Gorokhovskii, and J. V. Kikas, "Lowtemperature spectroscopy of organic molecules in solids by photochemical hole burning," Appl. Phys. B 29, 235–250 (1982); J. Friedrich and D. Haarer, "Photochemical hole burning: spectroscopic study of relaxation processes in polymers and glasses," Angew. Chem. Int. Ed. Engl. 23, 113–140 (1984).
[CrossRef]

A. A. Gorokhovskii, R. K. Kaarli, and L. A. Rebane, "Hole burning in the contour of a pure electronic line in a Shpolskii system," JETP Lett. 20, 216–219 (1974); B. M. Kharlamov, R. I. Personov, and L. A. Bykovskaya, "Stable gap in absorption spectra of solid solutions of organic molecules by laser irradiation," Opt. Commun. 12, 191–194 (1974).
[CrossRef]

Hartmann, S. R.

J. H. Eberly, S. R. Hartmann, and A. Szabo, "Propagation narrowing in the transmission of a light pulse through a spectral hole," Phys. Rev. A 23, 2502–2506 (1981); M. D. Crisp, "Propagation of small-area pulses of coherent light through a resonant medium," Phys. Rev. A 1, 1604–1611 (1970).
[CrossRef]

T. W. Mossberg, R. Kachru, and S. R. Hartmann, "Echoes in gaseous media: a generalized theory of rephasing phenomena," Phys. Rev. A 20, 1976–1996 (1979).
[CrossRef]

Holliday, K.

K. Holliday and U. P. Wild, "Spectral hole burning," in Molecular Luminescence Spectroscopy. Part 3, S. G. Schulman, ed., Vol. 77 of Chemical Analysis Series (Wiley, New York, 1993), p. 149.

Kaarli, R.

H. S˜najalg, A. Gorokhovskii, R. Kaarli, V. Palm, M. Rätsep, and P. Saari, "Optical pulse shaping by filters based on spectral hole burning," Opt. Commun. 71, 377–380 (1989).
[CrossRef]

Kaarli, R. K.

A. K. Rebane, R. K. Kaarli, and P. M. Saari, "Hole burning by coherent sequences of picosecond pulses," JETP Lett. 38, 383–386 (1983); A. Rebane, R. Kaarli, P. Saari, A. Anijalg, and K. Timpmann, "Photochemical time-domain holography of weak picosecond pulses," Opt. Commun. 47, 173–176 (1983); A. Rebane and R. Kaarli, "Picosecond pulse shaping by photochemical time domain holography," Chem. Phys. Lett. 101, 317–319 (1983).
[CrossRef]

A. A. Gorokhovskii, R. K. Kaarli, and L. A. Rebane, "Hole burning in the contour of a pure electronic line in a Shpolskii system," JETP Lett. 20, 216–219 (1974); B. M. Kharlamov, R. I. Personov, and L. A. Bykovskaya, "Stable gap in absorption spectra of solid solutions of organic molecules by laser irradiation," Opt. Commun. 12, 191–194 (1974).
[CrossRef]

Kachru, R.

T. W. Mossberg, R. Kachru, and S. R. Hartmann, "Echoes in gaseous media: a generalized theory of rephasing phenomena," Phys. Rev. A 20, 1976–1996 (1979).
[CrossRef]

Kikas, J. V.

L. A. Rebane, A. A. Gorokhovskii, and J. V. Kikas, "Lowtemperature spectroscopy of organic molecules in solids by photochemical hole burning," Appl. Phys. B 29, 235–250 (1982); J. Friedrich and D. Haarer, "Photochemical hole burning: spectroscopic study of relaxation processes in polymers and glasses," Angew. Chem. Int. Ed. Engl. 23, 113–140 (1984).
[CrossRef]

Kohler, B.

S. Bernet, B. Kohler, A. Rebane, A. Renn, and U. P. Wild, "Spectral hole burning and holography. V. Asymmetric diffraction from thin holograms," J. Opt. Soc. Am. B 9, 987–991 (1992).
[CrossRef]

S. Bernet, B. Kohler, A. Rebane, A. Renn, and U. P. Wild, "Holography in frequency selective media. II. Controlling the diffraction efficiency," J. Lumin. 53, 215–218 (1992).
[CrossRef]

Kuhl, J.

A. Rebane, J. Aaviksoo, and J. Kuhl, "Storage and time reversal of femtosecond light signals via persistent spectral hole burning holography," Appl. Phys. Lett. 54, 93–95 (1989); A. Rebane and J. Feinberg, "Time-resolved holography," Nature 351, 378–380 (1991); H. Gygax, A. Rebane, and U. P. Wild, "Stark effect in dye-doped polymers studied by photochemically accumulated photon echo," J. Opt. Soc. Am. B 10, 1149–1158 (1993).
[CrossRef]

Leaird, D. E.

A. M. Weiner, D. E. Leaird, D. H. Reitze, and E. G. Paek, "Femtosecond spectral holography," IEEE J. Quantum Electron. 28, 2251–2261 (1992).
[CrossRef]

Mandel, L.

Manykin, E. M.

S. O. Elyutin, S. M. Zakharov, and E. M. Manykin, "Theory of formation of photon echo pulses," Sov. Phys. JETP 49, 421–431 (1979).

Mazurenko, Yu. T.

Yu. T. Mazurenko, "Holography of wave packets," Appl. Phys. B 50, 101–114 (1990).
[CrossRef]

Meixner, A. J.

A. Renn, A. J. Meixner, and U. P. Wild, "Spectral hole burning and holography. II. Diffraction properties of two spectrally adjacent holgorams," J. Chem. Phys. 92, 2748–2755 (1990).
[CrossRef]

A. J. Meixner, A. Renn, S. E. Bucher, and U. P. Wild, "Spectral hole burning in glasses and polymer films: the Stark effect," J. Phys. Chem. 90, 6777–6785 (1986).
[CrossRef]

A. Renn, A. J. Meixner, U. P. Wild, and F. A. Burkhalter, "Holographic detection of photochemical holes," Chem. Phys. 93, 157–162 (1985).
[CrossRef]

Mitsunaga, M.

Mossberg, T. W.

Paek, E. G.

A. M. Weiner, D. E. Leaird, D. H. Reitze, and E. G. Paek, "Femtosecond spectral holography," IEEE J. Quantum Electron. 28, 2251–2261 (1992).
[CrossRef]

Palm, V.

H. S˜najalg, A. Gorokhovskii, R. Kaarli, V. Palm, M. Rätsep, and P. Saari, "Optical pulse shaping by filters based on spectral hole burning," Opt. Commun. 71, 377–380 (1989).
[CrossRef]

Rätsep, M.

H. S˜najalg, A. Gorokhovskii, R. Kaarli, V. Palm, M. Rätsep, and P. Saari, "Optical pulse shaping by filters based on spectral hole burning," Opt. Commun. 71, 377–380 (1989).
[CrossRef]

Rebane, A.

H. Schwoerer, D. Erni, A. Rebane, and U. P. Wild, "Subpicosecond pulse shaping via spectral hole burning," Opt. Commun. 107, 123–128 (1994).
[CrossRef]

S. Bernet, B. Kohler, A. Rebane, A. Renn, and U. P. Wild, "Holography in frequency selective media. II. Controlling the diffraction efficiency," J. Lumin. 53, 215–218 (1992).
[CrossRef]

S. Bernet, B. Kohler, A. Rebane, A. Renn, and U. P. Wild, "Spectral hole burning and holography. V. Asymmetric diffraction from thin holograms," J. Opt. Soc. Am. B 9, 987–991 (1992).
[CrossRef]

A. Rebane, S. Bernet, A. Renn, and U. P. Wild, "Holography in frequency selective media: hologram phase and causality," Opt. Commun. 86, 7–13 (1991).
[CrossRef]

A. Rebane, J. Aaviksoo, and J. Kuhl, "Storage and time reversal of femtosecond light signals via persistent spectral hole burning holography," Appl. Phys. Lett. 54, 93–95 (1989); A. Rebane and J. Feinberg, "Time-resolved holography," Nature 351, 378–380 (1991); H. Gygax, A. Rebane, and U. P. Wild, "Stark effect in dye-doped polymers studied by photochemically accumulated photon echo," J. Opt. Soc. Am. B 10, 1149–1158 (1993).
[CrossRef]

A. Rebane, "Associative space-and-time domain recall of picosecond light signals via photochemical hole burning holography," Opt. Commun. 65, 175–178 (1988); "Associative recall of time- and space-domain holograms in spectrally selective photo-active medium," Proc. Estonian Acad. Sci. Phys. Math. 37, 89–92 (1988).
[CrossRef]

P. Saari and A. Rebane, "Time- and space-domain holography of pulsed light fields in a spectrally photo-active medium," Proc. Estonian Acad. Sci. Phys. Math. 33, 322–332 (1984); P. Saari, R. Kaarli, and A. Rebane, "Picosecond timeand space-domain holography by photochemical hole burning," J. Opt. Soc. Am. B 3, 527–533 (1986); A. Rebane, "Coherent recall and time–space holography in impurity systems with photochemical hole-burning," Ph.D. dissertation (Institute of Physics, Estonian Academy of Sciences, Tartu, Estonia, 1985).
[CrossRef]

Rebane, A. K.

A. K. Rebane, R. K. Kaarli, and P. M. Saari, "Hole burning by coherent sequences of picosecond pulses," JETP Lett. 38, 383–386 (1983); A. Rebane, R. Kaarli, P. Saari, A. Anijalg, and K. Timpmann, "Photochemical time-domain holography of weak picosecond pulses," Opt. Commun. 47, 173–176 (1983); A. Rebane and R. Kaarli, "Picosecond pulse shaping by photochemical time domain holography," Chem. Phys. Lett. 101, 317–319 (1983).
[CrossRef]

Rebane, K. K.

K. K. Rebane, Impurity Spectra of Solids (Plenum, New York, 1970).

Rebane, L. A.

L. A. Rebane, A. A. Gorokhovskii, and J. V. Kikas, "Lowtemperature spectroscopy of organic molecules in solids by photochemical hole burning," Appl. Phys. B 29, 235–250 (1982); J. Friedrich and D. Haarer, "Photochemical hole burning: spectroscopic study of relaxation processes in polymers and glasses," Angew. Chem. Int. Ed. Engl. 23, 113–140 (1984).
[CrossRef]

A. A. Gorokhovskii, R. K. Kaarli, and L. A. Rebane, "Hole burning in the contour of a pure electronic line in a Shpolskii system," JETP Lett. 20, 216–219 (1974); B. M. Kharlamov, R. I. Personov, and L. A. Bykovskaya, "Stable gap in absorption spectra of solid solutions of organic molecules by laser irradiation," Opt. Commun. 12, 191–194 (1974).
[CrossRef]

Reitze, D. H.

A. M. Weiner, D. E. Leaird, D. H. Reitze, and E. G. Paek, "Femtosecond spectral holography," IEEE J. Quantum Electron. 28, 2251–2261 (1992).
[CrossRef]

Renn, A.

S. Bernet, B. Kohler, A. Rebane, A. Renn, and U. P. Wild, "Holography in frequency selective media. II. Controlling the diffraction efficiency," J. Lumin. 53, 215–218 (1992).
[CrossRef]

S. Bernet, B. Kohler, A. Rebane, A. Renn, and U. P. Wild, "Spectral hole burning and holography. V. Asymmetric diffraction from thin holograms," J. Opt. Soc. Am. B 9, 987–991 (1992).
[CrossRef]

A. Rebane, S. Bernet, A. Renn, and U. P. Wild, "Holography in frequency selective media: hologram phase and causality," Opt. Commun. 86, 7–13 (1991).
[CrossRef]

A. Renn, A. J. Meixner, and U. P. Wild, "Spectral hole burning and holography. II. Diffraction properties of two spectrally adjacent holgorams," J. Chem. Phys. 92, 2748–2755 (1990).
[CrossRef]

A. Renn and U. P. Wild, "Spectral hole burning and hologram storage," Appl. Opt. 26, 4040–4042 (1987); A. J. Meixner, A. Renn, and U. P. Wild, "Spectral hole burning and holography. I. Transmission and holographic detection of spectral holes," J. Chem. Phys. 91, 6728–6736 (1989); U. P. Wild, A. Renn. C. De Caro, and S. Bernet, "Spectral hole burning and molecular computing," Appl. Opt. 29, 4329–4331 (1990).
[CrossRef] [PubMed]

A. J. Meixner, A. Renn, S. E. Bucher, and U. P. Wild, "Spectral hole burning in glasses and polymer films: the Stark effect," J. Phys. Chem. 90, 6777–6785 (1986).
[CrossRef]

A. Renn, A. J. Meixner, U. P. Wild, and F. A. Burkhalter, "Holographic detection of photochemical holes," Chem. Phys. 93, 157–162 (1985).
[CrossRef]

Rothberg, L. J.

S˜najalg, H.

H. S˜najalg, A. Gorokhovskii, R. Kaarli, V. Palm, M. Rätsep, and P. Saari, "Optical pulse shaping by filters based on spectral hole burning," Opt. Commun. 71, 377–380 (1989).
[CrossRef]

Saari, P.

H. S˜najalg, A. Gorokhovskii, R. Kaarli, V. Palm, M. Rätsep, and P. Saari, "Optical pulse shaping by filters based on spectral hole burning," Opt. Commun. 71, 377–380 (1989).
[CrossRef]

P. Saari and A. Rebane, "Time- and space-domain holography of pulsed light fields in a spectrally photo-active medium," Proc. Estonian Acad. Sci. Phys. Math. 33, 322–332 (1984); P. Saari, R. Kaarli, and A. Rebane, "Picosecond timeand space-domain holography by photochemical hole burning," J. Opt. Soc. Am. B 3, 527–533 (1986); A. Rebane, "Coherent recall and time–space holography in impurity systems with photochemical hole-burning," Ph.D. dissertation (Institute of Physics, Estonian Academy of Sciences, Tartu, Estonia, 1985).
[CrossRef]

Saari, P. M.

A. K. Rebane, R. K. Kaarli, and P. M. Saari, "Hole burning by coherent sequences of picosecond pulses," JETP Lett. 38, 383–386 (1983); A. Rebane, R. Kaarli, P. Saari, A. Anijalg, and K. Timpmann, "Photochemical time-domain holography of weak picosecond pulses," Opt. Commun. 47, 173–176 (1983); A. Rebane and R. Kaarli, "Picosecond pulse shaping by photochemical time domain holography," Chem. Phys. Lett. 101, 317–319 (1983).
[CrossRef]

Samartsev, V. V.

V. A. Zuikov, V. V. Samartsev, and R. G. Usmanov, "Correlation of the shape of photon echo signals with the shape of excitation pulses," JETP Lett. 32, 270–274 (1980).

Schwoerer, H.

H. Schwoerer, D. Erni, A. Rebane, and U. P. Wild, "Subpicosecond pulse shaping via spectral hole burning," Opt. Commun. 107, 123–128 (1994).
[CrossRef]

Simon, R.

Szabo, A.

J. H. Eberly, S. R. Hartmann, and A. Szabo, "Propagation narrowing in the transmission of a light pulse through a spectral hole," Phys. Rev. A 23, 2502–2506 (1981); M. D. Crisp, "Propagation of small-area pulses of coherent light through a resonant medium," Phys. Rev. A 1, 1604–1611 (1970).
[CrossRef]

Uesugi, N.

Usmanov, R. G.

V. A. Zuikov, V. V. Samartsev, and R. G. Usmanov, "Correlation of the shape of photon echo signals with the shape of excitation pulses," JETP Lett. 32, 270–274 (1980).

Vainstein, L. A.

D. E. Vakman and L. A. Vainstein, "Amplitude, phase, frequency—fundamental concepts of oscillation theory," Sov. Phys. Usp. 20, 1002–1016 (1977).
[CrossRef]

Vakman, D. E.

D. E. Vakman and L. A. Vainstein, "Amplitude, phase, frequency—fundamental concepts of oscillation theory," Sov. Phys. Usp. 20, 1002–1016 (1977).
[CrossRef]

Wang, L. J.

Weiner, A. M.

A. M. Weiner, D. E. Leaird, D. H. Reitze, and E. G. Paek, "Femtosecond spectral holography," IEEE J. Quantum Electron. 28, 2251–2261 (1992).
[CrossRef]

Wild, U. P.

H. Schwoerer, D. Erni, A. Rebane, and U. P. Wild, "Subpicosecond pulse shaping via spectral hole burning," Opt. Commun. 107, 123–128 (1994).
[CrossRef]

S. Bernet, B. Kohler, A. Rebane, A. Renn, and U. P. Wild, "Holography in frequency selective media. II. Controlling the diffraction efficiency," J. Lumin. 53, 215–218 (1992).
[CrossRef]

S. Bernet, B. Kohler, A. Rebane, A. Renn, and U. P. Wild, "Spectral hole burning and holography. V. Asymmetric diffraction from thin holograms," J. Opt. Soc. Am. B 9, 987–991 (1992).
[CrossRef]

A. Rebane, S. Bernet, A. Renn, and U. P. Wild, "Holography in frequency selective media: hologram phase and causality," Opt. Commun. 86, 7–13 (1991).
[CrossRef]

A. Renn, A. J. Meixner, and U. P. Wild, "Spectral hole burning and holography. II. Diffraction properties of two spectrally adjacent holgorams," J. Chem. Phys. 92, 2748–2755 (1990).
[CrossRef]

A. Renn and U. P. Wild, "Spectral hole burning and hologram storage," Appl. Opt. 26, 4040–4042 (1987); A. J. Meixner, A. Renn, and U. P. Wild, "Spectral hole burning and holography. I. Transmission and holographic detection of spectral holes," J. Chem. Phys. 91, 6728–6736 (1989); U. P. Wild, A. Renn. C. De Caro, and S. Bernet, "Spectral hole burning and molecular computing," Appl. Opt. 29, 4329–4331 (1990).
[CrossRef] [PubMed]

A. J. Meixner, A. Renn, S. E. Bucher, and U. P. Wild, "Spectral hole burning in glasses and polymer films: the Stark effect," J. Phys. Chem. 90, 6777–6785 (1986).
[CrossRef]

A. Renn, A. J. Meixner, U. P. Wild, and F. A. Burkhalter, "Holographic detection of photochemical holes," Chem. Phys. 93, 157–162 (1985).
[CrossRef]

K. Holliday and U. P. Wild, "Spectral hole burning," in Molecular Luminescence Spectroscopy. Part 3, S. G. Schulman, ed., Vol. 77 of Chemical Analysis Series (Wiley, New York, 1993), p. 149.

Yano, R.

Yodh, A. G.

Zakharov, S. M.

S. O. Elyutin, S. M. Zakharov, and E. M. Manykin, "Theory of formation of photon echo pulses," Sov. Phys. JETP 49, 421–431 (1979).

Zuikov, V. A.

V. A. Zuikov, V. V. Samartsev, and R. G. Usmanov, "Correlation of the shape of photon echo signals with the shape of excitation pulses," JETP Lett. 32, 270–274 (1980).

Appl. Opt. (1)

Appl. Phys. B (2)

L. A. Rebane, A. A. Gorokhovskii, and J. V. Kikas, "Lowtemperature spectroscopy of organic molecules in solids by photochemical hole burning," Appl. Phys. B 29, 235–250 (1982); J. Friedrich and D. Haarer, "Photochemical hole burning: spectroscopic study of relaxation processes in polymers and glasses," Angew. Chem. Int. Ed. Engl. 23, 113–140 (1984).
[CrossRef]

Yu. T. Mazurenko, "Holography of wave packets," Appl. Phys. B 50, 101–114 (1990).
[CrossRef]

Appl. Phys. Lett. (1)

A. Rebane, J. Aaviksoo, and J. Kuhl, "Storage and time reversal of femtosecond light signals via persistent spectral hole burning holography," Appl. Phys. Lett. 54, 93–95 (1989); A. Rebane and J. Feinberg, "Time-resolved holography," Nature 351, 378–380 (1991); H. Gygax, A. Rebane, and U. P. Wild, "Stark effect in dye-doped polymers studied by photochemically accumulated photon echo," J. Opt. Soc. Am. B 10, 1149–1158 (1993).
[CrossRef]

Chem. Phys. (1)

A. Renn, A. J. Meixner, U. P. Wild, and F. A. Burkhalter, "Holographic detection of photochemical holes," Chem. Phys. 93, 157–162 (1985).
[CrossRef]

IEEE J. Quantum Electron. (1)

A. M. Weiner, D. E. Leaird, D. H. Reitze, and E. G. Paek, "Femtosecond spectral holography," IEEE J. Quantum Electron. 28, 2251–2261 (1992).
[CrossRef]

J. Chem. Phys. (1)

A. Renn, A. J. Meixner, and U. P. Wild, "Spectral hole burning and holography. II. Diffraction properties of two spectrally adjacent holgorams," J. Chem. Phys. 92, 2748–2755 (1990).
[CrossRef]

J. Lumin. (1)

S. Bernet, B. Kohler, A. Rebane, A. Renn, and U. P. Wild, "Holography in frequency selective media. II. Controlling the diffraction efficiency," J. Lumin. 53, 215–218 (1992).
[CrossRef]

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

J. Phys. Chem. (1)

A. J. Meixner, A. Renn, S. E. Bucher, and U. P. Wild, "Spectral hole burning in glasses and polymer films: the Stark effect," J. Phys. Chem. 90, 6777–6785 (1986).
[CrossRef]

JETP Lett. (3)

V. A. Zuikov, V. V. Samartsev, and R. G. Usmanov, "Correlation of the shape of photon echo signals with the shape of excitation pulses," JETP Lett. 32, 270–274 (1980).

A. K. Rebane, R. K. Kaarli, and P. M. Saari, "Hole burning by coherent sequences of picosecond pulses," JETP Lett. 38, 383–386 (1983); A. Rebane, R. Kaarli, P. Saari, A. Anijalg, and K. Timpmann, "Photochemical time-domain holography of weak picosecond pulses," Opt. Commun. 47, 173–176 (1983); A. Rebane and R. Kaarli, "Picosecond pulse shaping by photochemical time domain holography," Chem. Phys. Lett. 101, 317–319 (1983).
[CrossRef]

A. A. Gorokhovskii, R. K. Kaarli, and L. A. Rebane, "Hole burning in the contour of a pure electronic line in a Shpolskii system," JETP Lett. 20, 216–219 (1974); B. M. Kharlamov, R. I. Personov, and L. A. Bykovskaya, "Stable gap in absorption spectra of solid solutions of organic molecules by laser irradiation," Opt. Commun. 12, 191–194 (1974).
[CrossRef]

Opt. Commun. (4)

A. Rebane, "Associative space-and-time domain recall of picosecond light signals via photochemical hole burning holography," Opt. Commun. 65, 175–178 (1988); "Associative recall of time- and space-domain holograms in spectrally selective photo-active medium," Proc. Estonian Acad. Sci. Phys. Math. 37, 89–92 (1988).
[CrossRef]

H. S˜najalg, A. Gorokhovskii, R. Kaarli, V. Palm, M. Rätsep, and P. Saari, "Optical pulse shaping by filters based on spectral hole burning," Opt. Commun. 71, 377–380 (1989).
[CrossRef]

H. Schwoerer, D. Erni, A. Rebane, and U. P. Wild, "Subpicosecond pulse shaping via spectral hole burning," Opt. Commun. 107, 123–128 (1994).
[CrossRef]

A. Rebane, S. Bernet, A. Renn, and U. P. Wild, "Holography in frequency selective media: hologram phase and causality," Opt. Commun. 86, 7–13 (1991).
[CrossRef]

Opt. Lett. (4)

Phys. Rev. A (2)

J. H. Eberly, S. R. Hartmann, and A. Szabo, "Propagation narrowing in the transmission of a light pulse through a spectral hole," Phys. Rev. A 23, 2502–2506 (1981); M. D. Crisp, "Propagation of small-area pulses of coherent light through a resonant medium," Phys. Rev. A 1, 1604–1611 (1970).
[CrossRef]

T. W. Mossberg, R. Kachru, and S. R. Hartmann, "Echoes in gaseous media: a generalized theory of rephasing phenomena," Phys. Rev. A 20, 1976–1996 (1979).
[CrossRef]

Proc. Estonian Acad. Sci. Phys. Math. (1)

P. Saari and A. Rebane, "Time- and space-domain holography of pulsed light fields in a spectrally photo-active medium," Proc. Estonian Acad. Sci. Phys. Math. 33, 322–332 (1984); P. Saari, R. Kaarli, and A. Rebane, "Picosecond timeand space-domain holography by photochemical hole burning," J. Opt. Soc. Am. B 3, 527–533 (1986); A. Rebane, "Coherent recall and time–space holography in impurity systems with photochemical hole-burning," Ph.D. dissertation (Institute of Physics, Estonian Academy of Sciences, Tartu, Estonia, 1985).
[CrossRef]

Sov. Phys. JETP (1)

S. O. Elyutin, S. M. Zakharov, and E. M. Manykin, "Theory of formation of photon echo pulses," Sov. Phys. JETP 49, 421–431 (1979).

Sov. Phys. Usp. (1)

D. E. Vakman and L. A. Vainstein, "Amplitude, phase, frequency—fundamental concepts of oscillation theory," Sov. Phys. Usp. 20, 1002–1016 (1977).
[CrossRef]

Other (5)

S. Bernet, "Phasenkontrollierte Holographie in frequenzselektiven Materialien," Ph.D. dissertation Diss ETH Nr. 10292 (Swiss Federal Institute of Technology, Zurich, 1993).

L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms (Wiley, New York, 1975).

W. E. Moerner, ed., Persistent Spectral Hole Burning: Science and Applications, Vol. 44 of Topics in Current Physics (Springer-Verlag, Berlin, 1988), and references therein.
[CrossRef]

K. Holliday and U. P. Wild, "Spectral hole burning," in Molecular Luminescence Spectroscopy. Part 3, S. G. Schulman, ed., Vol. 77 of Chemical Analysis Series (Wiley, New York, 1993), p. 149.

K. K. Rebane, Impurity Spectra of Solids (Plenum, New York, 1970).

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Figures (8)

Fig. 1
Fig. 1

Geometry of the readout of a spectrally selective hologram. The SHB material is a parallel plate of thickness l positioned in the xy plane. The illuminating wave propagates in the positive direction of the z axis. The amplitude behind the hologram plate is characterized by two diffraction angles, ξx and ξy.

Fig. 2
Fig. 2

(a) Scheme for writing SHB holograms. The dye laser beam is split into reference and the object beams and is recombined at the sample positioned in an optical 4He bath cryostat. The delay of the object pulse with respect to the reference pulse is variable from 0 to 100 ps. One controls the phase between the object and the reference beams by rotating the second quarter-wave plate (see text). (b) Experimental setup to measure the time response of the holograms. A small fraction of the Ti:sapphire fundamental pulse is split off by a glass plate and is used as a reference in the cross correlator. The rest is focused into a block of quartz glass to generate a white-light continuum. The interference filter has a spectral window of 250 cm−1 centered at 635 nm. PVB, poly(vinyl butyral); PSHB, persistent SHB.

Fig. 3
Fig. 3

Dependence on the delay of the cross-correlation intensity of the signal diffracted in the +1 order. The SHB plate contains two adjacent holograms stored with a frequency interval of 7.3 cm−1. (a) – (e) Different cross-correlation traces corresponding to writing holograms with different relative phases (ϕ = 0°, 45°, 90°, 135°, 180°) set by rotation of the second quarter-wave plate. The peak at the zero delay in the experimental traces is an artifact that occurs because of residual scattering of the readout pulse on the sample surfaces and in the bulk of our sample.

Fig. 4
Fig. 4

Simulation of the frequency- and the spatial-domain structure of the hologram corresponding to the time-domain signal shown in Fig. 3. (The angle between the object and the reference beams is chosen to be smaller than in the experiment only for better visualization.) The theoretical (dashed) curve shown in Fig. 3 is obtained when the transmission function shown here is substituted into formula (10). OD, optical density.

Fig. 5
Fig. 5

Cross correlation of the fundamental pulse with the hologram signal with fixed time offset (15 ps) and constant phase but different intensity profiles in the frequency dimension. (a) The hole profile is constant in the 15 700–15 850-cm−1 interval. The diffracted signal appears only in the +1 order. (b), (c) The profile consists of 16 narrow holes burned at a constant interval of 7.3 cm−1. The signal appears in both (b) the positive and (c) the negative diffraction directions. The dashed curve is the simulated time-domain intensity obtained by Fourier transform of the frequency-domain amplitude profile.

Fig. 6
Fig. 6

Cross correlation of the fundamental pulse with the hologram signal with fixed time offset 5 ps and constant intensity in the 15 700–15 850-cm−1 frequency interval but different phase function profiles in the frequency dimension. (a) The phase is constant. The diffracted signal appears only in the + 1 order. (b) The phase function is ε(ν) = 0.45 πcm−1. The signal appears only in the +1 order. (c) The phase function is ε(ν) = − 0.45π cm−1. The signal appears only in the − 1 order.

Fig. 7
Fig. 7

Synthesis of pulse trains in the +1 diffraction direction with different amplitude and phase functions in frequency dimension. (a) The intensity profile consists of 16 narrow holes burned at a constant interval of 7.3 cm−1. The phase function changes by π for each fourth hole. (b) The intensity profile is the same as in (a), but the phase function changes according to a more complicated function (see text).

Fig. 8
Fig. 8

Theoretical (a) amplitude and (b) phase functions used to generate the pulse sequence shown in Fig. 7(b). (c) Part of the experimental cross-correlation trace (solid curve) shown in comparison with the theoretically calculated time-domain intensity function (dashed curve).

Equations (27)

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| E in ( t ) | h Γ ZPL | d | ,
T ( ν , x , y ) = { 0 < T ( ν , x , y ) 1 if | x | a , | y | b and ν min ν ν max 1 if | x | a , | x | b and ν min > ν or ν > ν max 0 otherwise .
α ( ν , x , y ) = ln [ T ( ν , x , y ) ] l .
α ( ν ) = 0 g ( ν ) γ ( ν ν ) d ν .
E out ( x , y ) exp ( i 2 π ν 0 t ) = K ( ν 0 , x , y ) E in ( x , y ) [ exp { i 2 π ν 0 [ t ( l / c ) ] } ,
K ( ν , x , y ) = [ T ( ν , x , y ) ] 1 / 2 exp ( i Ĥ { ln [ T ( ν , x , y ) ] 1 / 2 } ) ,
Ĥ [ f ( ν ) ] = 1 π f ( ν ) ν ν d ν .
α ( ν , x , y ) = α 0 ( ν ) Δ α ( ν , x , y ) ,
K ( ν , x , y ) = K 0 + κ ( 1 + i Ĥ ) [ Δ α ( ν , x , y ) ] ,
E out ( ξ x , ξ y ) = ν 0 c L a a b b E in ( x , y ) K ( ν 0 , x , y ) × exp [ i 2 π ν 0 c ( x sin ξ x + y sin ξ y ) ] d x d y ,
I out ( ξ x ) | K 0 | 2 sin ( 2 π a ν 0 sin ξ x c ) 2 ( π ν 0 sin ξ x c ) 2 + κ 2 | a a ( 1 + i Ĥ ) [ Δ α ( ν , x ) ] × exp [ i 2 π ν 0 c x sin ξ x ] d x | 2 + 2 Re { κ K 0 sin ( 2 π a ν 0 sin ξ x c ) ( π ν 0 sin ξ x c ) × a a ( 1 + i Ĥ ) [ Δ α ( ν , x ) ] × exp ( i 2 π ν 0 c x sin ξ x ) d x } .
Δ α ( ν , x ) = s 1 ( ν , x ) { 1 + cos [ 2 π ( x / Λ 1 ) + ϕ 1 ( ν , x ) ] } + s 2 ( ν , x ) { 1 + cos [ 2 π ( x / Λ 2 ) + ϕ 2 ( ν , x ) ] } ,
s 1 ( ν , x ) s 2 ( ν , x ) = 0 .
E ( ± 1 ) ( ξ x ) ( 1 + i Ĥ ) { s 1 ( ν , x ) exp [ ± i ϕ 1 ( ν , x ) ] } × exp [ i 2 π ν 0 x c ( sin ξ c ν 0 Λ 1 ) ] d x + ( 1 + i Ĥ ) { s 2 ( ν , x ) exp [ ± i ϕ 2 ( ν , x ) ] } × exp [ i 2 π ν 0 x c ( sin ξ c ν 0 Λ 2 ) ] d x .
I ( interf ) ( ξ x ) ( { s 1 ( ν 0 , x ) exp [ ± i ϕ 1 ( ν 0 , x ) ] } × exp [ i 2 π ν 0 x c ( sin ξ c ν 0 Λ 1 ) ] d x ) * × Ĥ { s 2 ( ν , x ) exp [ ± i ϕ 2 ( ν , x ) ] } × exp [ i 2 π ν 0 x c ( sin ξ c ν 0 Λ 2 ) ] d x .
I ( interf ) ( ξ x ) ( { s 1 ( ν 0 , x + x ) exp [ ± i ϕ 2 ( ν 0 , x + x ) ] } * × Ĥ { s 2 ( ν , x ) exp [ ± i ϕ 2 ( ν , x ) ] } d x ) × exp [ i 2 π ν 0 x c ( sin ξ ± sin ξ 1 ) ] d x .
I ( interf ) ( ξ x ) ( { s 1 ( ν 0 ) exp [ ± i ϕ 1 ( ν 0 ) ] } ) * × Ĥ { s 2 ( ν ) exp [ ± i ϕ 2 ( ν ) ] } δ ( ξ + ξ ) .
Ĥ ( cos Ω t cos ω 0 t ) = Ĥ ( cos ω 0 t ) cos Ω t .
Ĥ { s ( ν ) exp [ i ϕ ( ν ) ] } = Ĥ { exp [ i ϕ ( ν ) ] } s ( ν ) .
I ( interf ) ( ξ x ) [ s 1 ( ν 0 ) s 2 ( ν 0 ) exp [ ± i ϕ 1 ( ν 0 ) ] × Ĥ { exp [ ± i ϕ 2 ( ν ) ] } δ ( ξ + ξ 1 ) = 0 .
Ĥ [ exp ( i ɛ i 2 π ν τ ) ] = { i exp ( i ɛ i 2 π ν τ ) if τ > 0 i exp ( i ɛ i 2 π ν τ ) if τ < 0 .
j = 1 N S j ( ν ) | S p ( ν ) | = | S p ( t ) exp [ i 2 π ( ν ν 0 ) t ] d t | .
E ( ± 1 ) ( t ) = ( j = 1 N s j ( ν ) ( 1 + i Ĥ ) { exp [ ± i ϕ j ( ν ) ] } ) × exp ( i ν t ) d ν δ ( ξ ξ 1 ) .
E ( 1 ) ( t ) = [ j = 1 N s j ( ν ) 2 exp ( i ɛ j ) ] × exp [ i ν ( t τ ) ] d ν S p ( t τ ) .
E ( 1 ) ( t ) = 0 .
E ( 1 ) ( t ) S p ( t ) Θ ( t ) E ( 1 ) ( t ) S p ( t ) Θ ( t ) ,
Θ ( t ) = { 1 if t 0 0 if t < 0 .

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