Abstract

Theoretical and experimental studies of photon-echo modulation for the four σ transitions of the R1 line in ruby with an on-axis magnetic field are described. Echoes originating from the Cr3+ ground-state 4A2 (−3/2) spin level (−3/2 echo) are studied for the first time. Unlike ±½ echoes, the −3/2-echo modulation is found to be dominated by Cr–Al interactions occurring outside the 13 inner-shell ions. The +½-echo modulation is observed to be in major disagreement with the theory using previously determined superhyperfine parameters. Regression-analysis studies indicate that this cannot be due solely to parameter errors. Finite nuclear linewidths and/or optical pumping effects are shown to be a plausible source of the disagreement.

© 1986 Optical Society of America

Full Article  |  PDF Article

Corrections

A. Szabo, "On-axis photon-echo modulation in ruby: erratum," J. Opt. Soc. Am. B 3, 1323-1323 (1986)
https://www.osapublishing.org/josab/abstract.cfm?uri=josab-3-10-1323

References

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  1. E. L. Hahn, “Spin echoes,” Phys. Rev. 80, 580–594 (1950).
    [CrossRef]
  2. W. B. Mims, in Electron Paramagnetic Resonance, S. Geschwind, ed. (Plenum, New York, 1972), p. 263.
  3. N. A. Kurnitt, I. D. Abella, S. R. Hartmann, “Observation of a photon echo,” Phys. Rev. Lett. 15, 567–568 (1964).
    [CrossRef]
  4. C. K. N. Patel, R. E. Slusher, “Photon echoes in gases,” Phys. Rev. Lett. 20, 1087–1089 (1968); B. Bolger, J. C. Diels, “Photon echoes in Cs vapor,” Phys. Lett. 28A, 401–402 (1968).
    [CrossRef]
  5. L. G. Rowan, E. L. Hahn, W. B. Mims, “Electron-spin-echo envelope modulation,” Phys. Rev. A 137, 61–71 (1965). For a recent review of electron-paramagnetic-resonance echo modulation, see L. Kevan, in Time Domain Electron Spin Resonance, L. Kevan, R. N. Schwartz, eds. (Wiley, New York, 1979), pp. 279–341.
  6. L. Q. Lambert, A. Compaan, I. D. Abella, “Modulation and fast decay of photon echoes in ruby,” Phys. Lett. 30A, 153–154 (1969); L. Q. Lambert, “Effects of superhyperfine interactions on photon-echo behavior in dilute ruby,” Phys. Rev. B 7, 1834–1846 (1973).
    [CrossRef]
  7. D. Grischkowsky, S. R. Hartmann, “Behavior of electron-spin-echoes and photon echoes in high fields,” Phys. Rev. B 2, 60–74 (1970).
    [CrossRef]
  8. S. Meth, S. R. Hartmann, “Photon echo modulation in ruby,” Opt. Commun. 24, 100–104 (1978).
    [CrossRef]
  9. Y. C. Chen, K. Chiang, S. R. Hartman, “Spectroscopic and relaxation character of the 3P0–3H4transition in LaF3:Pr3+ measured by photon echoes,” Phys. Rev. B 21, 40–47 (1980), and references therein.
    [CrossRef]
  10. J. B. W. Morsink, D. A. Wiersma, “Photon echoes in the 3P0–3H4transition in Pr3+:LaF3,” Chem. Phys. Lett. 65, 105–108 (1967).
    [CrossRef]
  11. P. F. Liao, S. R. Hartmann, “Determination of Cr–Al hyper-fine and electric quadrupole interaction parameters in ruby using spin-echo electron-nuclear double resonance,” Phys. Rev. B 8, 69–80 (1973).
    [CrossRef]
  12. N. Laurance, E. C. McIrvine, J. Lambe, “Aluminum hyperfine interactions in ruby,” J. Phys. Chem. Solids 23, 515–531 (1962).
    [CrossRef]
  13. E. A. Whittaker, S. R. Hartmann, “Hyperfine structure of the 1D2–3H4levels of Pr3+:LaF3with the use of photon echo modulation spectroscopy,” Phys. Rev. B 26, 3617–3621 (1982).
    [CrossRef]
  14. A. Compaan, “Concentration-dependent photon-echo decay in ruby,” Phys. Rev. B 5, 4450–4465 (1972).
    [CrossRef]
  15. P. F. Liao, P. Hu, R. Leigh, S. R. Hartmann, “Photon-echo nuclear double resonance and its application in ruby,” Phys. Rev. A 9, 332–340 (1974). The frequency scales in Fig. 6 of this paper are incorrectly drawn. 1 MHz should be subtracted from the values shown.
    [CrossRef]
  16. The ground-state point-dipole value for B and the derived value A, for the G A1 set, in Ref. 8 are incorrect. The correct values (Ref. 8) are B= 20.411 (10.411) kHz/nm3 and A= 0.693 (0.988) MHz. Also, the coordinates of one of the I atoms should be −0.2753, 0, −0.0564 (0.139, 0, −0.057) nm. These corrections have a negligible effect on the calculations of Ref. 8.
  17. P. E. Jessop, A. Szabo, “Single frequency cw dye laser operation in the 690–700 nm gap,” IEEE J. Quantum Electron. QE-16, 812–813 (1980).
    [CrossRef]
  18. A. Szabo, M. Kroll, “Electric field induced shifting of optical holes, fluorescence line narrowing and free induction decay in ruby,” Opt. Commun. 18, 224–225 (1976); “Stark-induced optical transients in ruby,” Opt. Lett. 2, 10–12 (1978); S. Nakanishi, O. Tamura, T. Muramoto, T. Hashi, “Observation of various photon echoes and FID in ruby by Stark switching technique,” J. Phys. Soc. Jpn. 45, 1437–1438 (1978).
    [CrossRef] [PubMed]
  19. D. W. Marquardt, “An algorithm for least-squares estimation of nonlinear parameters,” J. Soc. Indust. Appl. Math. 11, 431–441 (1963).
    [CrossRef]
  20. In this paper, we use the term “dephasing” in the context of a stochastic process. In some earlier work [e.g., D. Grischkowsky, S. R. Hartmann, “Echo behavior in ruby,” Phys. Rev. Lett. 20, 41–43 (1968)] this term was also used in connection with modulation effects. Current usage implies stochastic processes.
    [CrossRef]
  21. S. Meth, “Photon echo modulation and photon echo relaxation in ruby,” Ph.D. dissertation (Columbia University, New York, 1977) (unpublished).
  22. S. Lee, C. M. Brodbeck, “Dynamic nuclear polarization method of investigating the correlated ESR and NMR c-axis variation effect in single crystals,” Phys. Rev. B 17, 3484–3491 (1978).
    [CrossRef]
  23. We have experimentally confirmed the results of Ref. 8. Also, our computer code gave identical echo-modulation curves.
  24. R. M. Shelby, R. M. Macfarlane, “Frequency-dependent optical dephasing in the stoichiometric material EuP5O14,” Phys. Rev. Lett. 45, 1098–1101 (1980).
    [CrossRef]
  25. R. G. DeVoe, A. Szabo, S. C. Rand, R. G. Brewer, “Ultraslow optical dephasing of LaF3:Pr3+,” Phys. Rev. Lett. 42, 1560–1563 (1979).
    [CrossRef]
  26. A. Szabo, “Spin dependence of optical dephasing in ruby: the frozen core,” Opt. Lett. 8, 486–487 (1983).
    [CrossRef] [PubMed]
  27. A. Szabo, J. Chrostowski, “Stimulated on-axis photon echo modulation in ruby,” in Coherence and Quantum Optics V, L. Mandel, E. Wolf, eds. (Plenum, New York, 1984), pp. 301–308.
  28. A. H. Silver, T. Kushida, J. Lambe, “Nuclear magnetic dipole coupling in Al2O3,” Phys. Rev. 125, 1147–1149 (1962).
    [CrossRef]
  29. W. B. Mims, “Amplitudes of superhyperfine frequencies displayed in the electron-spin-echo envelope,” Phys. Rev. B 6, 3543–3545 (1972).
    [CrossRef]

1983 (1)

1982 (1)

E. A. Whittaker, S. R. Hartmann, “Hyperfine structure of the 1D2–3H4levels of Pr3+:LaF3with the use of photon echo modulation spectroscopy,” Phys. Rev. B 26, 3617–3621 (1982).
[CrossRef]

1980 (3)

P. E. Jessop, A. Szabo, “Single frequency cw dye laser operation in the 690–700 nm gap,” IEEE J. Quantum Electron. QE-16, 812–813 (1980).
[CrossRef]

Y. C. Chen, K. Chiang, S. R. Hartman, “Spectroscopic and relaxation character of the 3P0–3H4transition in LaF3:Pr3+ measured by photon echoes,” Phys. Rev. B 21, 40–47 (1980), and references therein.
[CrossRef]

R. M. Shelby, R. M. Macfarlane, “Frequency-dependent optical dephasing in the stoichiometric material EuP5O14,” Phys. Rev. Lett. 45, 1098–1101 (1980).
[CrossRef]

1979 (1)

R. G. DeVoe, A. Szabo, S. C. Rand, R. G. Brewer, “Ultraslow optical dephasing of LaF3:Pr3+,” Phys. Rev. Lett. 42, 1560–1563 (1979).
[CrossRef]

1978 (2)

S. Lee, C. M. Brodbeck, “Dynamic nuclear polarization method of investigating the correlated ESR and NMR c-axis variation effect in single crystals,” Phys. Rev. B 17, 3484–3491 (1978).
[CrossRef]

S. Meth, S. R. Hartmann, “Photon echo modulation in ruby,” Opt. Commun. 24, 100–104 (1978).
[CrossRef]

1976 (1)

A. Szabo, M. Kroll, “Electric field induced shifting of optical holes, fluorescence line narrowing and free induction decay in ruby,” Opt. Commun. 18, 224–225 (1976); “Stark-induced optical transients in ruby,” Opt. Lett. 2, 10–12 (1978); S. Nakanishi, O. Tamura, T. Muramoto, T. Hashi, “Observation of various photon echoes and FID in ruby by Stark switching technique,” J. Phys. Soc. Jpn. 45, 1437–1438 (1978).
[CrossRef] [PubMed]

1974 (1)

P. F. Liao, P. Hu, R. Leigh, S. R. Hartmann, “Photon-echo nuclear double resonance and its application in ruby,” Phys. Rev. A 9, 332–340 (1974). The frequency scales in Fig. 6 of this paper are incorrectly drawn. 1 MHz should be subtracted from the values shown.
[CrossRef]

1973 (1)

P. F. Liao, S. R. Hartmann, “Determination of Cr–Al hyper-fine and electric quadrupole interaction parameters in ruby using spin-echo electron-nuclear double resonance,” Phys. Rev. B 8, 69–80 (1973).
[CrossRef]

1972 (2)

A. Compaan, “Concentration-dependent photon-echo decay in ruby,” Phys. Rev. B 5, 4450–4465 (1972).
[CrossRef]

W. B. Mims, “Amplitudes of superhyperfine frequencies displayed in the electron-spin-echo envelope,” Phys. Rev. B 6, 3543–3545 (1972).
[CrossRef]

1970 (1)

D. Grischkowsky, S. R. Hartmann, “Behavior of electron-spin-echoes and photon echoes in high fields,” Phys. Rev. B 2, 60–74 (1970).
[CrossRef]

1969 (1)

L. Q. Lambert, A. Compaan, I. D. Abella, “Modulation and fast decay of photon echoes in ruby,” Phys. Lett. 30A, 153–154 (1969); L. Q. Lambert, “Effects of superhyperfine interactions on photon-echo behavior in dilute ruby,” Phys. Rev. B 7, 1834–1846 (1973).
[CrossRef]

1968 (2)

C. K. N. Patel, R. E. Slusher, “Photon echoes in gases,” Phys. Rev. Lett. 20, 1087–1089 (1968); B. Bolger, J. C. Diels, “Photon echoes in Cs vapor,” Phys. Lett. 28A, 401–402 (1968).
[CrossRef]

In this paper, we use the term “dephasing” in the context of a stochastic process. In some earlier work [e.g., D. Grischkowsky, S. R. Hartmann, “Echo behavior in ruby,” Phys. Rev. Lett. 20, 41–43 (1968)] this term was also used in connection with modulation effects. Current usage implies stochastic processes.
[CrossRef]

1967 (1)

J. B. W. Morsink, D. A. Wiersma, “Photon echoes in the 3P0–3H4transition in Pr3+:LaF3,” Chem. Phys. Lett. 65, 105–108 (1967).
[CrossRef]

1965 (1)

L. G. Rowan, E. L. Hahn, W. B. Mims, “Electron-spin-echo envelope modulation,” Phys. Rev. A 137, 61–71 (1965). For a recent review of electron-paramagnetic-resonance echo modulation, see L. Kevan, in Time Domain Electron Spin Resonance, L. Kevan, R. N. Schwartz, eds. (Wiley, New York, 1979), pp. 279–341.

1964 (1)

N. A. Kurnitt, I. D. Abella, S. R. Hartmann, “Observation of a photon echo,” Phys. Rev. Lett. 15, 567–568 (1964).
[CrossRef]

1963 (1)

D. W. Marquardt, “An algorithm for least-squares estimation of nonlinear parameters,” J. Soc. Indust. Appl. Math. 11, 431–441 (1963).
[CrossRef]

1962 (2)

N. Laurance, E. C. McIrvine, J. Lambe, “Aluminum hyperfine interactions in ruby,” J. Phys. Chem. Solids 23, 515–531 (1962).
[CrossRef]

A. H. Silver, T. Kushida, J. Lambe, “Nuclear magnetic dipole coupling in Al2O3,” Phys. Rev. 125, 1147–1149 (1962).
[CrossRef]

1950 (1)

E. L. Hahn, “Spin echoes,” Phys. Rev. 80, 580–594 (1950).
[CrossRef]

Abella, I. D.

L. Q. Lambert, A. Compaan, I. D. Abella, “Modulation and fast decay of photon echoes in ruby,” Phys. Lett. 30A, 153–154 (1969); L. Q. Lambert, “Effects of superhyperfine interactions on photon-echo behavior in dilute ruby,” Phys. Rev. B 7, 1834–1846 (1973).
[CrossRef]

N. A. Kurnitt, I. D. Abella, S. R. Hartmann, “Observation of a photon echo,” Phys. Rev. Lett. 15, 567–568 (1964).
[CrossRef]

Brewer, R. G.

R. G. DeVoe, A. Szabo, S. C. Rand, R. G. Brewer, “Ultraslow optical dephasing of LaF3:Pr3+,” Phys. Rev. Lett. 42, 1560–1563 (1979).
[CrossRef]

Brodbeck, C. M.

S. Lee, C. M. Brodbeck, “Dynamic nuclear polarization method of investigating the correlated ESR and NMR c-axis variation effect in single crystals,” Phys. Rev. B 17, 3484–3491 (1978).
[CrossRef]

Chen, Y. C.

Y. C. Chen, K. Chiang, S. R. Hartman, “Spectroscopic and relaxation character of the 3P0–3H4transition in LaF3:Pr3+ measured by photon echoes,” Phys. Rev. B 21, 40–47 (1980), and references therein.
[CrossRef]

Chiang, K.

Y. C. Chen, K. Chiang, S. R. Hartman, “Spectroscopic and relaxation character of the 3P0–3H4transition in LaF3:Pr3+ measured by photon echoes,” Phys. Rev. B 21, 40–47 (1980), and references therein.
[CrossRef]

Chrostowski, J.

A. Szabo, J. Chrostowski, “Stimulated on-axis photon echo modulation in ruby,” in Coherence and Quantum Optics V, L. Mandel, E. Wolf, eds. (Plenum, New York, 1984), pp. 301–308.

Compaan, A.

A. Compaan, “Concentration-dependent photon-echo decay in ruby,” Phys. Rev. B 5, 4450–4465 (1972).
[CrossRef]

L. Q. Lambert, A. Compaan, I. D. Abella, “Modulation and fast decay of photon echoes in ruby,” Phys. Lett. 30A, 153–154 (1969); L. Q. Lambert, “Effects of superhyperfine interactions on photon-echo behavior in dilute ruby,” Phys. Rev. B 7, 1834–1846 (1973).
[CrossRef]

DeVoe, R. G.

R. G. DeVoe, A. Szabo, S. C. Rand, R. G. Brewer, “Ultraslow optical dephasing of LaF3:Pr3+,” Phys. Rev. Lett. 42, 1560–1563 (1979).
[CrossRef]

Grischkowsky, D.

D. Grischkowsky, S. R. Hartmann, “Behavior of electron-spin-echoes and photon echoes in high fields,” Phys. Rev. B 2, 60–74 (1970).
[CrossRef]

In this paper, we use the term “dephasing” in the context of a stochastic process. In some earlier work [e.g., D. Grischkowsky, S. R. Hartmann, “Echo behavior in ruby,” Phys. Rev. Lett. 20, 41–43 (1968)] this term was also used in connection with modulation effects. Current usage implies stochastic processes.
[CrossRef]

Hahn, E. L.

L. G. Rowan, E. L. Hahn, W. B. Mims, “Electron-spin-echo envelope modulation,” Phys. Rev. A 137, 61–71 (1965). For a recent review of electron-paramagnetic-resonance echo modulation, see L. Kevan, in Time Domain Electron Spin Resonance, L. Kevan, R. N. Schwartz, eds. (Wiley, New York, 1979), pp. 279–341.

E. L. Hahn, “Spin echoes,” Phys. Rev. 80, 580–594 (1950).
[CrossRef]

Hartman, S. R.

Y. C. Chen, K. Chiang, S. R. Hartman, “Spectroscopic and relaxation character of the 3P0–3H4transition in LaF3:Pr3+ measured by photon echoes,” Phys. Rev. B 21, 40–47 (1980), and references therein.
[CrossRef]

Hartmann, S. R.

E. A. Whittaker, S. R. Hartmann, “Hyperfine structure of the 1D2–3H4levels of Pr3+:LaF3with the use of photon echo modulation spectroscopy,” Phys. Rev. B 26, 3617–3621 (1982).
[CrossRef]

S. Meth, S. R. Hartmann, “Photon echo modulation in ruby,” Opt. Commun. 24, 100–104 (1978).
[CrossRef]

P. F. Liao, P. Hu, R. Leigh, S. R. Hartmann, “Photon-echo nuclear double resonance and its application in ruby,” Phys. Rev. A 9, 332–340 (1974). The frequency scales in Fig. 6 of this paper are incorrectly drawn. 1 MHz should be subtracted from the values shown.
[CrossRef]

P. F. Liao, S. R. Hartmann, “Determination of Cr–Al hyper-fine and electric quadrupole interaction parameters in ruby using spin-echo electron-nuclear double resonance,” Phys. Rev. B 8, 69–80 (1973).
[CrossRef]

D. Grischkowsky, S. R. Hartmann, “Behavior of electron-spin-echoes and photon echoes in high fields,” Phys. Rev. B 2, 60–74 (1970).
[CrossRef]

In this paper, we use the term “dephasing” in the context of a stochastic process. In some earlier work [e.g., D. Grischkowsky, S. R. Hartmann, “Echo behavior in ruby,” Phys. Rev. Lett. 20, 41–43 (1968)] this term was also used in connection with modulation effects. Current usage implies stochastic processes.
[CrossRef]

N. A. Kurnitt, I. D. Abella, S. R. Hartmann, “Observation of a photon echo,” Phys. Rev. Lett. 15, 567–568 (1964).
[CrossRef]

Hu, P.

P. F. Liao, P. Hu, R. Leigh, S. R. Hartmann, “Photon-echo nuclear double resonance and its application in ruby,” Phys. Rev. A 9, 332–340 (1974). The frequency scales in Fig. 6 of this paper are incorrectly drawn. 1 MHz should be subtracted from the values shown.
[CrossRef]

Jessop, P. E.

P. E. Jessop, A. Szabo, “Single frequency cw dye laser operation in the 690–700 nm gap,” IEEE J. Quantum Electron. QE-16, 812–813 (1980).
[CrossRef]

Kroll, M.

A. Szabo, M. Kroll, “Electric field induced shifting of optical holes, fluorescence line narrowing and free induction decay in ruby,” Opt. Commun. 18, 224–225 (1976); “Stark-induced optical transients in ruby,” Opt. Lett. 2, 10–12 (1978); S. Nakanishi, O. Tamura, T. Muramoto, T. Hashi, “Observation of various photon echoes and FID in ruby by Stark switching technique,” J. Phys. Soc. Jpn. 45, 1437–1438 (1978).
[CrossRef] [PubMed]

Kurnitt, N. A.

N. A. Kurnitt, I. D. Abella, S. R. Hartmann, “Observation of a photon echo,” Phys. Rev. Lett. 15, 567–568 (1964).
[CrossRef]

Kushida, T.

A. H. Silver, T. Kushida, J. Lambe, “Nuclear magnetic dipole coupling in Al2O3,” Phys. Rev. 125, 1147–1149 (1962).
[CrossRef]

Lambe, J.

A. H. Silver, T. Kushida, J. Lambe, “Nuclear magnetic dipole coupling in Al2O3,” Phys. Rev. 125, 1147–1149 (1962).
[CrossRef]

N. Laurance, E. C. McIrvine, J. Lambe, “Aluminum hyperfine interactions in ruby,” J. Phys. Chem. Solids 23, 515–531 (1962).
[CrossRef]

Lambert, L. Q.

L. Q. Lambert, A. Compaan, I. D. Abella, “Modulation and fast decay of photon echoes in ruby,” Phys. Lett. 30A, 153–154 (1969); L. Q. Lambert, “Effects of superhyperfine interactions on photon-echo behavior in dilute ruby,” Phys. Rev. B 7, 1834–1846 (1973).
[CrossRef]

Laurance, N.

N. Laurance, E. C. McIrvine, J. Lambe, “Aluminum hyperfine interactions in ruby,” J. Phys. Chem. Solids 23, 515–531 (1962).
[CrossRef]

Lee, S.

S. Lee, C. M. Brodbeck, “Dynamic nuclear polarization method of investigating the correlated ESR and NMR c-axis variation effect in single crystals,” Phys. Rev. B 17, 3484–3491 (1978).
[CrossRef]

Leigh, R.

P. F. Liao, P. Hu, R. Leigh, S. R. Hartmann, “Photon-echo nuclear double resonance and its application in ruby,” Phys. Rev. A 9, 332–340 (1974). The frequency scales in Fig. 6 of this paper are incorrectly drawn. 1 MHz should be subtracted from the values shown.
[CrossRef]

Liao, P. F.

P. F. Liao, P. Hu, R. Leigh, S. R. Hartmann, “Photon-echo nuclear double resonance and its application in ruby,” Phys. Rev. A 9, 332–340 (1974). The frequency scales in Fig. 6 of this paper are incorrectly drawn. 1 MHz should be subtracted from the values shown.
[CrossRef]

P. F. Liao, S. R. Hartmann, “Determination of Cr–Al hyper-fine and electric quadrupole interaction parameters in ruby using spin-echo electron-nuclear double resonance,” Phys. Rev. B 8, 69–80 (1973).
[CrossRef]

Macfarlane, R. M.

R. M. Shelby, R. M. Macfarlane, “Frequency-dependent optical dephasing in the stoichiometric material EuP5O14,” Phys. Rev. Lett. 45, 1098–1101 (1980).
[CrossRef]

Marquardt, D. W.

D. W. Marquardt, “An algorithm for least-squares estimation of nonlinear parameters,” J. Soc. Indust. Appl. Math. 11, 431–441 (1963).
[CrossRef]

McIrvine, E. C.

N. Laurance, E. C. McIrvine, J. Lambe, “Aluminum hyperfine interactions in ruby,” J. Phys. Chem. Solids 23, 515–531 (1962).
[CrossRef]

Meth, S.

S. Meth, S. R. Hartmann, “Photon echo modulation in ruby,” Opt. Commun. 24, 100–104 (1978).
[CrossRef]

S. Meth, “Photon echo modulation and photon echo relaxation in ruby,” Ph.D. dissertation (Columbia University, New York, 1977) (unpublished).

Mims, W. B.

W. B. Mims, “Amplitudes of superhyperfine frequencies displayed in the electron-spin-echo envelope,” Phys. Rev. B 6, 3543–3545 (1972).
[CrossRef]

L. G. Rowan, E. L. Hahn, W. B. Mims, “Electron-spin-echo envelope modulation,” Phys. Rev. A 137, 61–71 (1965). For a recent review of electron-paramagnetic-resonance echo modulation, see L. Kevan, in Time Domain Electron Spin Resonance, L. Kevan, R. N. Schwartz, eds. (Wiley, New York, 1979), pp. 279–341.

W. B. Mims, in Electron Paramagnetic Resonance, S. Geschwind, ed. (Plenum, New York, 1972), p. 263.

Morsink, J. B. W.

J. B. W. Morsink, D. A. Wiersma, “Photon echoes in the 3P0–3H4transition in Pr3+:LaF3,” Chem. Phys. Lett. 65, 105–108 (1967).
[CrossRef]

Patel, C. K. N.

C. K. N. Patel, R. E. Slusher, “Photon echoes in gases,” Phys. Rev. Lett. 20, 1087–1089 (1968); B. Bolger, J. C. Diels, “Photon echoes in Cs vapor,” Phys. Lett. 28A, 401–402 (1968).
[CrossRef]

Rand, S. C.

R. G. DeVoe, A. Szabo, S. C. Rand, R. G. Brewer, “Ultraslow optical dephasing of LaF3:Pr3+,” Phys. Rev. Lett. 42, 1560–1563 (1979).
[CrossRef]

Rowan, L. G.

L. G. Rowan, E. L. Hahn, W. B. Mims, “Electron-spin-echo envelope modulation,” Phys. Rev. A 137, 61–71 (1965). For a recent review of electron-paramagnetic-resonance echo modulation, see L. Kevan, in Time Domain Electron Spin Resonance, L. Kevan, R. N. Schwartz, eds. (Wiley, New York, 1979), pp. 279–341.

Shelby, R. M.

R. M. Shelby, R. M. Macfarlane, “Frequency-dependent optical dephasing in the stoichiometric material EuP5O14,” Phys. Rev. Lett. 45, 1098–1101 (1980).
[CrossRef]

Silver, A. H.

A. H. Silver, T. Kushida, J. Lambe, “Nuclear magnetic dipole coupling in Al2O3,” Phys. Rev. 125, 1147–1149 (1962).
[CrossRef]

Slusher, R. E.

C. K. N. Patel, R. E. Slusher, “Photon echoes in gases,” Phys. Rev. Lett. 20, 1087–1089 (1968); B. Bolger, J. C. Diels, “Photon echoes in Cs vapor,” Phys. Lett. 28A, 401–402 (1968).
[CrossRef]

Szabo, A.

A. Szabo, “Spin dependence of optical dephasing in ruby: the frozen core,” Opt. Lett. 8, 486–487 (1983).
[CrossRef] [PubMed]

P. E. Jessop, A. Szabo, “Single frequency cw dye laser operation in the 690–700 nm gap,” IEEE J. Quantum Electron. QE-16, 812–813 (1980).
[CrossRef]

R. G. DeVoe, A. Szabo, S. C. Rand, R. G. Brewer, “Ultraslow optical dephasing of LaF3:Pr3+,” Phys. Rev. Lett. 42, 1560–1563 (1979).
[CrossRef]

A. Szabo, M. Kroll, “Electric field induced shifting of optical holes, fluorescence line narrowing and free induction decay in ruby,” Opt. Commun. 18, 224–225 (1976); “Stark-induced optical transients in ruby,” Opt. Lett. 2, 10–12 (1978); S. Nakanishi, O. Tamura, T. Muramoto, T. Hashi, “Observation of various photon echoes and FID in ruby by Stark switching technique,” J. Phys. Soc. Jpn. 45, 1437–1438 (1978).
[CrossRef] [PubMed]

A. Szabo, J. Chrostowski, “Stimulated on-axis photon echo modulation in ruby,” in Coherence and Quantum Optics V, L. Mandel, E. Wolf, eds. (Plenum, New York, 1984), pp. 301–308.

Whittaker, E. A.

E. A. Whittaker, S. R. Hartmann, “Hyperfine structure of the 1D2–3H4levels of Pr3+:LaF3with the use of photon echo modulation spectroscopy,” Phys. Rev. B 26, 3617–3621 (1982).
[CrossRef]

Wiersma, D. A.

J. B. W. Morsink, D. A. Wiersma, “Photon echoes in the 3P0–3H4transition in Pr3+:LaF3,” Chem. Phys. Lett. 65, 105–108 (1967).
[CrossRef]

Chem. Phys. Lett. (1)

J. B. W. Morsink, D. A. Wiersma, “Photon echoes in the 3P0–3H4transition in Pr3+:LaF3,” Chem. Phys. Lett. 65, 105–108 (1967).
[CrossRef]

IEEE J. Quantum Electron. (1)

P. E. Jessop, A. Szabo, “Single frequency cw dye laser operation in the 690–700 nm gap,” IEEE J. Quantum Electron. QE-16, 812–813 (1980).
[CrossRef]

J. Phys. Chem. Solids (1)

N. Laurance, E. C. McIrvine, J. Lambe, “Aluminum hyperfine interactions in ruby,” J. Phys. Chem. Solids 23, 515–531 (1962).
[CrossRef]

J. Soc. Indust. Appl. Math. (1)

D. W. Marquardt, “An algorithm for least-squares estimation of nonlinear parameters,” J. Soc. Indust. Appl. Math. 11, 431–441 (1963).
[CrossRef]

Opt. Commun. (2)

A. Szabo, M. Kroll, “Electric field induced shifting of optical holes, fluorescence line narrowing and free induction decay in ruby,” Opt. Commun. 18, 224–225 (1976); “Stark-induced optical transients in ruby,” Opt. Lett. 2, 10–12 (1978); S. Nakanishi, O. Tamura, T. Muramoto, T. Hashi, “Observation of various photon echoes and FID in ruby by Stark switching technique,” J. Phys. Soc. Jpn. 45, 1437–1438 (1978).
[CrossRef] [PubMed]

S. Meth, S. R. Hartmann, “Photon echo modulation in ruby,” Opt. Commun. 24, 100–104 (1978).
[CrossRef]

Opt. Lett. (1)

Phys. Lett. (1)

L. Q. Lambert, A. Compaan, I. D. Abella, “Modulation and fast decay of photon echoes in ruby,” Phys. Lett. 30A, 153–154 (1969); L. Q. Lambert, “Effects of superhyperfine interactions on photon-echo behavior in dilute ruby,” Phys. Rev. B 7, 1834–1846 (1973).
[CrossRef]

Phys. Rev. (2)

E. L. Hahn, “Spin echoes,” Phys. Rev. 80, 580–594 (1950).
[CrossRef]

A. H. Silver, T. Kushida, J. Lambe, “Nuclear magnetic dipole coupling in Al2O3,” Phys. Rev. 125, 1147–1149 (1962).
[CrossRef]

Phys. Rev. A (2)

L. G. Rowan, E. L. Hahn, W. B. Mims, “Electron-spin-echo envelope modulation,” Phys. Rev. A 137, 61–71 (1965). For a recent review of electron-paramagnetic-resonance echo modulation, see L. Kevan, in Time Domain Electron Spin Resonance, L. Kevan, R. N. Schwartz, eds. (Wiley, New York, 1979), pp. 279–341.

P. F. Liao, P. Hu, R. Leigh, S. R. Hartmann, “Photon-echo nuclear double resonance and its application in ruby,” Phys. Rev. A 9, 332–340 (1974). The frequency scales in Fig. 6 of this paper are incorrectly drawn. 1 MHz should be subtracted from the values shown.
[CrossRef]

Phys. Rev. B (7)

E. A. Whittaker, S. R. Hartmann, “Hyperfine structure of the 1D2–3H4levels of Pr3+:LaF3with the use of photon echo modulation spectroscopy,” Phys. Rev. B 26, 3617–3621 (1982).
[CrossRef]

A. Compaan, “Concentration-dependent photon-echo decay in ruby,” Phys. Rev. B 5, 4450–4465 (1972).
[CrossRef]

P. F. Liao, S. R. Hartmann, “Determination of Cr–Al hyper-fine and electric quadrupole interaction parameters in ruby using spin-echo electron-nuclear double resonance,” Phys. Rev. B 8, 69–80 (1973).
[CrossRef]

D. Grischkowsky, S. R. Hartmann, “Behavior of electron-spin-echoes and photon echoes in high fields,” Phys. Rev. B 2, 60–74 (1970).
[CrossRef]

Y. C. Chen, K. Chiang, S. R. Hartman, “Spectroscopic and relaxation character of the 3P0–3H4transition in LaF3:Pr3+ measured by photon echoes,” Phys. Rev. B 21, 40–47 (1980), and references therein.
[CrossRef]

W. B. Mims, “Amplitudes of superhyperfine frequencies displayed in the electron-spin-echo envelope,” Phys. Rev. B 6, 3543–3545 (1972).
[CrossRef]

S. Lee, C. M. Brodbeck, “Dynamic nuclear polarization method of investigating the correlated ESR and NMR c-axis variation effect in single crystals,” Phys. Rev. B 17, 3484–3491 (1978).
[CrossRef]

Phys. Rev. Lett. (5)

In this paper, we use the term “dephasing” in the context of a stochastic process. In some earlier work [e.g., D. Grischkowsky, S. R. Hartmann, “Echo behavior in ruby,” Phys. Rev. Lett. 20, 41–43 (1968)] this term was also used in connection with modulation effects. Current usage implies stochastic processes.
[CrossRef]

R. M. Shelby, R. M. Macfarlane, “Frequency-dependent optical dephasing in the stoichiometric material EuP5O14,” Phys. Rev. Lett. 45, 1098–1101 (1980).
[CrossRef]

R. G. DeVoe, A. Szabo, S. C. Rand, R. G. Brewer, “Ultraslow optical dephasing of LaF3:Pr3+,” Phys. Rev. Lett. 42, 1560–1563 (1979).
[CrossRef]

N. A. Kurnitt, I. D. Abella, S. R. Hartmann, “Observation of a photon echo,” Phys. Rev. Lett. 15, 567–568 (1964).
[CrossRef]

C. K. N. Patel, R. E. Slusher, “Photon echoes in gases,” Phys. Rev. Lett. 20, 1087–1089 (1968); B. Bolger, J. C. Diels, “Photon echoes in Cs vapor,” Phys. Lett. 28A, 401–402 (1968).
[CrossRef]

Other (5)

W. B. Mims, in Electron Paramagnetic Resonance, S. Geschwind, ed. (Plenum, New York, 1972), p. 263.

The ground-state point-dipole value for B and the derived value A, for the G A1 set, in Ref. 8 are incorrect. The correct values (Ref. 8) are B= 20.411 (10.411) kHz/nm3 and A= 0.693 (0.988) MHz. Also, the coordinates of one of the I atoms should be −0.2753, 0, −0.0564 (0.139, 0, −0.057) nm. These corrections have a negligible effect on the calculations of Ref. 8.

A. Szabo, J. Chrostowski, “Stimulated on-axis photon echo modulation in ruby,” in Coherence and Quantum Optics V, L. Mandel, E. Wolf, eds. (Plenum, New York, 1984), pp. 301–308.

S. Meth, “Photon echo modulation and photon echo relaxation in ruby,” Ph.D. dissertation (Columbia University, New York, 1977) (unpublished).

We have experimentally confirmed the results of Ref. 8. Also, our computer code gave identical echo-modulation curves.

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Figures (11)

Fig. 1
Fig. 1

Energy levels of the optical R1 line (693.4 nm) of Cr3+ in ruby in a magnetic field (|| C3 axis) showing four circularly polarized transitions studied. The ground-state level crossing at 2.069 kG was used for absolute field calibration.

Fig. 2
Fig. 2

Experimental setup for photon-echo studies.

Fig. 3
Fig. 3

Theory versus experiment for −½ echo at 1.405 kG. MD, 18%; dephasing time, 5.19 μsec. Vertical scale for all plots in arbitrary units. For clarity, not all experimental points are shown in this and later plots.

Fig. 4
Fig. 4

Theory versus experiment for −3/2 echo at 1.740 kG. MD, 23% (375 Al ions); dephasing time, 5.06 μsec.

Fig. 5
Fig. 5

Theory versus experiment for −3/2 echo at 2.272 kG. MD, 25% (375 Al ions); dephasing time, 6.51 μsec. For clarity, the data have been split into three time regions: 0–3, 3–6, and 6–9 μsec, as indicated on the plots. The curves are arbitrarily displaced vertically.

Fig. 6
Fig. 6

Theoretical plot for +3/2 echo at 4.0 kG. Dephasing time, infinite. Vertical scale in arbitrary units. The dc level indicates the time-independent part of the echo.

Fig. 7
Fig. 7

Theory versus experiment for +½ echo at 1.420 kG. MD, 265%; dephasing time, 2.38 μsec.

Fig. 8
Fig. 8

Theory versus experiment for +½ echo at 1.600 kG. MD, 81%; dephasing time, 4.16 μsec.

Fig. 9
Fig. 9

Regression-analysis fit of theory to experiment for +½ echo at 1.420 kG. Excited-state superhyperfine parameters are regressed for 12 nearest-neighbor Al (28 parameters) plus slope and height of theoretical (log) plot. MD, 5.6%; see Table 1.

Fig. 10
Fig. 10

Regression-analysis fit of theory to experiment for +½ echo at 1.420 kG. Ground-state populations are regressed for 12 nearest-neighbor Al (24 parameters) plus height and slope of theoretical (log) plot. MD, 41%; see Table 2.

Fig. 11
Fig. 11

Regression-analysis fit of theory to experiment for +½ echo at 1.420 kG. Ground and excited linewidths are regressed for 12 nearest-neighbor Al (8 parameters) plus height and slope of theoretical (log) plot. For a given Al ion set, all ground (excited) linewidths are assumed to be the same. MD, 17%; see Table 3.

Tables (3)

Tables Icon

Table 1 Excited-State Superhyperfine Parameters Obtained by Regression-Analysis Fit of +½-Echo Data at 1.42 kG

Tables Icon

Table 2 Ground-State Occupation Probabilities Obtained by Regression-Analysis Fit of +½-Echo Data at 1.42 kG

Tables Icon

Table 3 Nuclear Linewidths Obtained by Regression-Analysis Fit of +½-Echo Data at 1.42 kG

Equations (19)

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H n = - γ n H I z + S z [ ( A + B z ) I z + B t I α ] + Q exp [ - i θ 1 I β ) ] exp [ - ( i θ 2 I α ) ] × exp [ - ( i θ 3 I z ) ] [ I z 2 + ( η / 3 ) ( I α 2 - I β 2 ) ] × c . c .
B z = B ( 3 z 2 - r 2 ) / r 5
B t = 3 B z ( x 2 + y 2 ) 1 / 2 / r 5 ,
I t = A I 0 exp [ - ( 4 t / T 2 ) ] ,
MD = ( 1 / N ) N ( I exp - I t ) / I t × 100 % ,
H eff = ( 2 π / γ ) { [ ( γ / 2 π ) H - ( A + B z ) S z ] z + [ B t S z ] t } ,
Tr C = K , L , M , N = 1 6 N K P K L M N cos ( ω K L M N t ) ,
Tr C = [ 1 / ( 2 I + 1 ) ] K , L , M , N = 1 6 exp [ - ( Ω K M + Ω L N ) t ] × P K L M N cos ( ω K L M N t ) ,
E ( 2 t ) = E ( 0 ) Re j Tr C j ,
Tr C = [ 1 / ( 2 I + 2 ) ] K , L , M , N = 1 6 P K L M N cos ( ω K L M N t ) ,
P K L M N = W K L W M L * W M N W K N *
ω K L M N = - 1 [ ( E K - E M ) g + ( E L - E N ) e ] = ω K M g + ω L N e .
T 1 [ P K L M N cos ( ω K L M N t ) ] = P K L M N cos ( ω K L M N t )
T 2 P K L M N * = P K L M N = T 3 P K L M N * .
C 0 ( 1 / 6 ) K , L = 1 6 P K K L L .
C 1 ( t ) ( 1 / 3 ) L = 1 , N = L + 1 5 , 6 ( K = 1 6 P K L K N ) cos ( ω L N e t ) .
C 2 ( t ) ( 1 / 3 ) K = 1 , M = K + 1 5 , 6 ( L = 1 6 P K L M L ) cos ( ω K M g t ) .
C 3 r ( t ) ( 2 / 3 ) Re [ K = 1 , M = K + 1 , L = 1 , N = L + 1 5 , 6 , 5 , 6 P K L M N × cos ( ω K M g t ) cos ( ω L N e t ) ] ,
C 3 i ( t ) - ( 2 / 3 ) Im [ K = 1 , M = K + 1 , L = 1 , N = L + 1 5 , 6 , 5 , 6 P K L M N × sin ( ω K M g t ) sin ( ω L N e t ) ] .

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