## Configurational disorder and the local field effects in nonlinear optical systems

Optics Express, Vol. 1, Issue 6, pp. 169-174 (1997)

http://dx.doi.org/10.1364/OE.1.000169

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### Abstract

We argue that in nonlinear optical systems with atoms randomly distributed in crystals or amorphous hosts one should go beyond the Clausius-Mossoti limit in order to take into account the effect of local field fluctuations induced by *configurational* disorder in atom position. This effect is analyzed by means of a random local mean field approach with neglect of correlations between dipole moments of different atoms. The formalism is applied to 3-level Λ type systems with quantum coherence possessing an absorptionless index of refraction and lasing without inversion. We show that the effect of configurational fluctuations results in the suppression of the atom susceptibility compared with the predictions based on the Clausius-Mossoti equation.

© Optical Society of America

## 1. Introduction

*E*(

*t*) being the time dependent macroscopic electric field and

*P*the macroscopic polarization of the system which is determined self consistently from the equation

*n*is the atom concentration and α is a single atom nonlinear polarizability. Eqs.(1) and (2) result in the Clausius-Mossoti (CM) equation for the macroscopic susceptibility χ

_{DD}.

_{e}≡ 4

*πnα*is the susceptibility of non-interacting atoms. It has been shown [1, 2, 4

4. R. Friedberg, S.R. Hartmann, and J.T. Manassah, “Frequency Shift in Emission and Absorption by Resonant systems of Two-Level Atoms”, Phys. Rep. **7**, 101 (1973). [CrossRef]

5. C.M. Bowden and J.P. Dowling, “Near Dipole-Dipole Effects in Dense Media: Generalized Maxwell-Bloch Equations”, Phys. Rev. A **47**, 1247 (1993). [CrossRef] [PubMed]

*E*(

*t*). In particular, in gases Eq.(1) corresponds to the account of hard core interactions [4

4. R. Friedberg, S.R. Hartmann, and J.T. Manassah, “Frequency Shift in Emission and Absorption by Resonant systems of Two-Level Atoms”, Phys. Rep. **7**, 101 (1973). [CrossRef]

6. J.T. Manassah, “Statistical Quantum Electrodynamics of Resonant Atoms”, Phys. Rep. **101**, 359 (1983). [CrossRef]

5. C.M. Bowden and J.P. Dowling, “Near Dipole-Dipole Effects in Dense Media: Generalized Maxwell-Bloch Equations”, Phys. Rev. A **47**, 1247 (1993). [CrossRef] [PubMed]

*P*(

*ω*),

*E*(

*ω*), and

*E*(

_{L}*ω*) in Eq. (2) are the Fourier components of their time dependent analogs.

7. Y. Ben-Aryeh, C.M. Bowden, and J.C. Englund, “Intrinsic Optical Bistability in Collection of Spatially Distributed Two-Level Atoms”, Phys.Rev. A **34**, 3917 (1986). [CrossRef] [PubMed]

8. R. Friedberg, S. R. Hartman, and J.T. Manassah, “Effect of Local Field Correction on a Strongly Pumped Resonance”, Phys. Rev. A **40**, 2446 (1989). [CrossRef] [PubMed]

9. J.J. Maki, M.S. Malcuit, J.E. Sipe, and R.W. Boyd, “Linear and Nonlinear Optical Measurements of the Lorentz Local Field”, Phys. Rev. Lett. **67**, 972 (1991). [CrossRef] [PubMed]

10. J.P. Dowling and C.M. Bowden, “Near Dipole-Dipole Effects in Lasing without Inversion: An Enhancement of Gain and Absorptionless Index of Refraction”, Phys. Rev. Lett. **70**, 1421 (1993). [CrossRef] [PubMed]

11. A. Manka, J. P. Dowling, and C.M. Bowden, “Piezophotonic Switching Due to Local Field Effects in a Coherently Prepared Medium of Three-Level Atoms”, Phys. Rev. Lett. **73** , 1789 (1994). [CrossRef] [PubMed]

12. R.R. Mosley, B.D. Sinclair, and M.H. Dunn, “Local Field Effect in the Three-level Atom”, Opt. Commun. **108**, 247 (1994). [CrossRef]

13. C.M. Bowden, S. Singh, and G. Agrawal, “Laser instabilities and chaos in inhomogeneously broadened dense media”, J. Mod. Opt. **42**, 101 (1995). [CrossRef]

11. A. Manka, J. P. Dowling, and C.M. Bowden, “Piezophotonic Switching Due to Local Field Effects in a Coherently Prepared Medium of Three-Level Atoms”, Phys. Rev. Lett. **73** , 1789 (1994). [CrossRef] [PubMed]

12. R.R. Mosley, B.D. Sinclair, and M.H. Dunn, “Local Field Effect in the Three-level Atom”, Opt. Commun. **108**, 247 (1994). [CrossRef]

14. O. Kocharovskaya, “Amplification and Lasing without Inversion”, Phys. Rep. **219**, 175 (1992). [CrossRef]

15. M.O. Scully, “From Lasers and Masers to Phaseonium and Phasers”, Phys. Rep. **219**, 191 (1992). [CrossRef]

16. M. Fleischhauer, C.H. Keitel, M.O. Scully, C. Su, and S.-Y. Zhu, “Resonantly Enhanced Refractive Index without Absorption via Atomic Coherence”, Phys. Rev. A **46**, 1468 (1992). [CrossRef] [PubMed]

10. J.P. Dowling and C.M. Bowden, “Near Dipole-Dipole Effects in Lasing without Inversion: An Enhancement of Gain and Absorptionless Index of Refraction”, Phys. Rev. Lett. **70**, 1421 (1993). [CrossRef] [PubMed]

11. A. Manka, J. P. Dowling, and C.M. Bowden, “Piezophotonic Switching Due to Local Field Effects in a Coherently Prepared Medium of Three-Level Atoms”, Phys. Rev. Lett. **73** , 1789 (1994). [CrossRef] [PubMed]

*optically active atoms are frozen in random positions in a crystal or amorphous host*. We will show that in such a situation the effect of configurational fluctuations associated with near dipole-dipole interactions leads to the suppression of the refractive index compared to the predictions based on CM equation.

## 2. Local mean field equations

*P*in Eq.(2) can be written as

*n*=

*N*/

*V*

_{0}

*is a single atom dipole moment operator; the angular brackets denote the quantum statistical average at given random atom positions and the overbar denotes the configurational average over the atom positions. In fact,*μ ^

_{i}*m*is the classical dipole moment associated with the

_{i}*i*-th atom located at point

*r*. It is induced by the interaction of the

_{i}*i*-th atom with the other atoms and with the applied electric field.

*α*. However, in disordered systems there are limitations on the applicability of mean field theory due to configurational fluctuations in atom positions resulting in fluctuations of the local field

*E*and the local polarization

_{Li}*m*. This is especially important in systems with the sign-changeable dipole-dipole interaction. In order to estimate qualitatively the effect of configurational fluctuations we will apply the local mean field formalism developed in the theory of spin glasses[17

_{i}17. K. Binder and A.P. Young, “Spin Glasses”, Rev. Mod. Phys. **58**,801 (1986). [CrossRef]

18. M.W. Klein, C. Held, and E. Zuroff, “Dipole Interactions Among Polar Defects: a Self-Consistent Theory with Application to OH- Impurities in KCL”, Phys. Rev. B **13**, 3576 (1976). [CrossRef]

20. B.E. Vugmeister and M.D. Glinchuk, “Dipole Glass and Ferroelectricity in Random Site electric Dipole Systems” Rev. Mod. Phys. **62**, 993 (1990). [CrossRef]

21. B.E. Vugmeister and H. Rabitz, “Effect of Local Field Fluctuations on Orientational in Random Site Dipole Systems”, J. Stat. Phys. **88**, 477 (1997). [CrossRef]

*E*is the local mean field experienced by the i-th atom due to dipole-dipole interactions with its neighbors in the presence of the external electric field. In the retar-dationless approximation (i.e., the distance between the atoms is less than the resonance wavelength)

_{iL}*E*can be written as

_{iL}4. R. Friedberg, S.R. Hartmann, and J.T. Manassah, “Frequency Shift in Emission and Absorption by Resonant systems of Two-Level Atoms”, Phys. Rep. **7**, 101 (1973). [CrossRef]

6. J.T. Manassah, “Statistical Quantum Electrodynamics of Resonant Atoms”, Phys. Rep. **101**, 359 (1983). [CrossRef]

*J*, Eqs.(5) and (6) can be solved with the use of computer simulation techniques. The analytical solution below invokes an additional approximation of the distribution function of the local fields.

_{ij}22. P.W. Miloni and P.L. Knight, ”Retardation in the Resonant Interaction of Two Identical Atoms”, Phys. Rev. A **10**, 1076 (1974). [CrossRef]

6. J.T. Manassah, “Statistical Quantum Electrodynamics of Resonant Atoms”, Phys. Rep. **101**, 359 (1983). [CrossRef]

23. Y. Ben-Aryeh and S. Ruschin, “Cooperative Decay of a Linear Chain of Molecules Including Explicit Spatial Dependence”, Physica A **88**, 362 (1977). [CrossRef]

7. Y. Ben-Aryeh, C.M. Bowden, and J.C. Englund, “Intrinsic Optical Bistability in Collection of Spatially Distributed Two-Level Atoms”, Phys.Rev. A **34**, 3917 (1986). [CrossRef] [PubMed]

*m*are oriented along the axis

_{i}*x*and

**n**

_{ij}=

**r**

_{ij}/

*r*where

_{ij}**r**

_{ij}=

**r**

_{i}-

**r**

_{j}is the radius-vector separating atoms i and j;

*E*in Eq.(6) is the x-component of the local electric field.

_{iL}*E*in Eq.(6) by

_{iL}*E*, and the inner spherical Lorentz surface giving rise to the Lorentz local field

_{dep}*E*=

*E*+

_{ex}*E*.

_{dep}_{i}

*m*/

_{i}*V*

_{0}is a self-averaging variable equal to macroscopic polarization

*P*. The polarization

*P*defined by Eq.(2),(5) can be written in terms of the distribution function

*f*(

*e*) = ̅

*δ*(

*e*-

*e*) of the local field e;

_{i}*δ*(

*z*) is the Dirac delta-function. For small values of E i.e., for linear response with respect to the macroscopic field (but not the local field) one can neglect the dependence of polarizability

*α*on

*E*and

*P*and we obtain

*f*(

*e*) →

*δ*(

*e*). Also

*κ*=

_{e}*χ*in linear systems where the susceptibility does not depend on the applied field. However, as we will show below, taking account of the finite width of

_{e}*f*(

*e*) caused by the configurational fluctuations in atom positions is important for properly estimating the effect of the dipole-dipole interaction on the macroscopic susceptibility

*χ*in nonlinear systems.

_{DD}## 3. Macroscopic susceptibility in systems with coherence

**73** , 1789 (1994). [CrossRef] [PubMed]

_{e}of non-interacting atoms has been shown[16

16. M. Fleischhauer, C.H. Keitel, M.O. Scully, C. Su, and S.-Y. Zhu, “Resonantly Enhanced Refractive Index without Absorption via Atomic Coherence”, Phys. Rev. A **46**, 1468 (1992). [CrossRef] [PubMed]

*C*= 2

*πμ*

^{2}

*n*/

*ħγ*which characterizes the strength of the dipole-dipole interaction.

*γ*is the characteristic rate of dissipative processes. The frequency dependence of

*χ′*and

_{e}*χ′′*is presented in Fig. 1 for C=4.6 and the values of other parameter used in Ref. [11

_{e}**73** , 1789 (1994). [CrossRef] [PubMed]

*ω*-

*ω*

_{0})/

*γ*≈ -0.59, where

*ω*is the frequency the incident light and

*ω*

_{0}is the resonance frequency, the susceptibility

*χ′*reaches its maximum value ≈ 3 and

_{e}*χ′′*≈ 0. In this situation the CM equation predicts a significant enhancement[11

_{e}**73** , 1789 (1994). [CrossRef] [PubMed]

_{DD}compared with

*χ*.

_{e}*f*(

*e*) in the assumption that

*m′′*≈ 0. This assumption is true in the vicinity of Δ ≈ -0.59 where χ′

_{i}_{e}peaks and χ′′

_{e}≈ 0. The calculation of

*f*(

*e*) has been performed in the self-consistent manner[18

18. M.W. Klein, C. Held, and E. Zuroff, “Dipole Interactions Among Polar Defects: a Self-Consistent Theory with Application to OH- Impurities in KCL”, Phys. Rev. B **13**, 3576 (1976). [CrossRef]

19. H. Margenau, “Pressure Broadenning of Spectral Lines”, Phys. Rev. **43**, 129 (1933). [CrossRef]

*M*satisfies the equation

*M*which is the order parameter of the nonequilibrium spin glass state characterizing the noncoherent oscillations of the atom dipole moments [24

24. B.E. Vugmeister and H. Rabitz, “Nonequilibrium Spin Glass State in Nonlinear Optical Systems with Coherence”, Phys. Lett. **232**,129 (1997). [CrossRef]

*χ*calculated with the use of Eqs.(11),(13),(14) and the explicit form of single atom polarizability

_{DD}*α*[16

16. M. Fleischhauer, C.H. Keitel, M.O. Scully, C. Su, and S.-Y. Zhu, “Resonantly Enhanced Refractive Index without Absorption via Atomic Coherence”, Phys. Rev. A **46**, 1468 (1992). [CrossRef] [PubMed]

*χ*given by the CM equation. One can see that effect of configurational disorder results in significant suppression of the susceptibility compared with the predictions based on the CM equation. Note, however, that for the more exact estimation of

_{DD}*χ*at frequencies of external field

_{DD}*ω*≠

*ω*

_{0}one need to calculate the single atom polarizability

*α*(

*ω*) being a subject of strong oscillating local field at frequency

*ω*

_{0}. The results of such calculations will be reported elsewhere.

## 4. Conclusion

## 5. Acknowledgements

## References

1. | J.D. Jackson, |

2. | C. Kittel, |

3. | H.A. Lorentz, |

4. | R. Friedberg, S.R. Hartmann, and J.T. Manassah, “Frequency Shift in Emission and Absorption by Resonant systems of Two-Level Atoms”, Phys. Rep. |

5. | C.M. Bowden and J.P. Dowling, “Near Dipole-Dipole Effects in Dense Media: Generalized Maxwell-Bloch Equations”, Phys. Rev. A |

6. | J.T. Manassah, “Statistical Quantum Electrodynamics of Resonant Atoms”, Phys. Rep. |

7. | Y. Ben-Aryeh, C.M. Bowden, and J.C. Englund, “Intrinsic Optical Bistability in Collection of Spatially Distributed Two-Level Atoms”, Phys.Rev. A |

8. | R. Friedberg, S. R. Hartman, and J.T. Manassah, “Effect of Local Field Correction on a Strongly Pumped Resonance”, Phys. Rev. A |

9. | J.J. Maki, M.S. Malcuit, J.E. Sipe, and R.W. Boyd, “Linear and Nonlinear Optical Measurements of the Lorentz Local Field”, Phys. Rev. Lett. |

10. | J.P. Dowling and C.M. Bowden, “Near Dipole-Dipole Effects in Lasing without Inversion: An Enhancement of Gain and Absorptionless Index of Refraction”, Phys. Rev. Lett. |

11. | A. Manka, J. P. Dowling, and C.M. Bowden, “Piezophotonic Switching Due to Local Field Effects in a Coherently Prepared Medium of Three-Level Atoms”, Phys. Rev. Lett. |

12. | R.R. Mosley, B.D. Sinclair, and M.H. Dunn, “Local Field Effect in the Three-level Atom”, Opt. Commun. |

13. | C.M. Bowden, S. Singh, and G. Agrawal, “Laser instabilities and chaos in inhomogeneously broadened dense media”, J. Mod. Opt. |

14. | O. Kocharovskaya, “Amplification and Lasing without Inversion”, Phys. Rep. |

15. | M.O. Scully, “From Lasers and Masers to Phaseonium and Phasers”, Phys. Rep. |

16. | M. Fleischhauer, C.H. Keitel, M.O. Scully, C. Su, and S.-Y. Zhu, “Resonantly Enhanced Refractive Index without Absorption via Atomic Coherence”, Phys. Rev. A |

17. | K. Binder and A.P. Young, “Spin Glasses”, Rev. Mod. Phys. |

18. | M.W. Klein, C. Held, and E. Zuroff, “Dipole Interactions Among Polar Defects: a Self-Consistent Theory with Application to OH- Impurities in KCL”, Phys. Rev. B |

19. | H. Margenau, “Pressure Broadenning of Spectral Lines”, Phys. Rev. |

20. | B.E. Vugmeister and M.D. Glinchuk, “Dipole Glass and Ferroelectricity in Random Site electric Dipole Systems” Rev. Mod. Phys. |

21. | B.E. Vugmeister and H. Rabitz, “Effect of Local Field Fluctuations on Orientational in Random Site Dipole Systems”, J. Stat. Phys. |

22. | P.W. Miloni and P.L. Knight, ”Retardation in the Resonant Interaction of Two Identical Atoms”, Phys. Rev. A |

23. | Y. Ben-Aryeh and S. Ruschin, “Cooperative Decay of a Linear Chain of Molecules Including Explicit Spatial Dependence”, Physica A |

24. | B.E. Vugmeister and H. Rabitz, “Nonequilibrium Spin Glass State in Nonlinear Optical Systems with Coherence”, Phys. Lett. |

**OCIS Codes**

(190.0190) Nonlinear optics : Nonlinear optics

(270.2500) Quantum optics : Fluctuations, relaxations, and noise

**ToC Category:**

Focus Issue: Local field effects

**History**

Original Manuscript: August 21, 1997

Published: September 15, 1997

**Citation**

B. Vugmeister, Alexei Bulatov, and Herschel Rabitz, "Configurational disorder and the local field effects
in nonlinear optical systems," Opt. Express **1**, 169-174 (1997)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-1-6-169

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### References

- J.D. Jackson, Classical Electrodynamics (Wiley, New York, 1975).
- C. Kittel, Solid State Theory ( Dover Publ., New York, 1986).
- H.A. Lorentz, The Theory of Electrons (Teubner, Leipzig, 1909)
- R. Friedberg, S.R. Hartmann and J.T. Manassah, "Frequency Shift in Emission and Absorption by Resonant systems of Two-Level Atoms", Phys. Rep. 7, 101 (1973). [CrossRef]
- C.M. Bowden and J.P. Dowling, "Near Dipole-Dipole Effects in Dense Media: Generalized Maxwell-Bloch Equations", Phys. Rev. A 47, 1247 (1993). [CrossRef] [PubMed]
- J.T. Manassah, "Statistical Quantum Electrodynamics of Resonant Atoms", Phys. Rep. 101, 359 (1983). [CrossRef]
- Y. Ben-Aryeh, C.M. Bowden and J.C. Englund, "Intrinsic Optical Bistability in Collection of Spatially Distributed Two-Level Atoms", Phys.Rev. A 34, 3917 (1986). [CrossRef] [PubMed]
- R. Friedberg and S. R. Hartman, and J.T. Manassah, "Eect of Local Field Correction on a Strongly Pumped Resonance", Phys. Rev. A 40, 2446 (1989). [CrossRef] [PubMed]
- J.J. Maki, M.S. Malcuit, J.E. Sipe, and R.W. Boyd, "Linear and Nonlinear Optical Measurements of the Lorentz Local Field", Phys. Rev. Lett. 67, 972 (1991). [CrossRef] [PubMed]
- J.P.Dowling and C.M. Bowden, "Near Dipole-Dipole Eects in Lasing without Inversion: An Enhancement of Gain and Absorptionless Index of Refraction", Phys. Rev. Lett. 70, 1421 (1993). [CrossRef] [PubMed]
- A. Manka, J. P. Dowling,and C.M. Bowden, "Piezophotonic Switching Due to Local Field Eects in a Coherently Prepared Medium of Three-Level Atoms", Phys. Rev. Lett. 73 , 1789 (1994). [CrossRef] [PubMed]
- [CrossRef]
- C.M. Bowden, S. Sinch, and G. Agraval, "Laser instabilities and chaos in inhomogeneously broadened dense media", J. Mod. Opt. 42, 101 (1995). [CrossRef]
- O. Kocharovskaya, "Amplication and Lasing without Inversion", Phys. Rep. 219, 175 (1992). [CrossRef]
- M.O. Scully, "From Lasers and Masers to Phaseonium and Phasers", Phys. Rep. 219, 191 (1992). [CrossRef]
- M. Fleischhauer, C.H. Keitel, M.O. Scully, C. Su, and S.-Y. Zhu, "Resonantly Enhanced Refractive Index without Absorption via Atomic Coherence", Phys. Rev. A 46, 1468 (1992). [CrossRef] [PubMed]
- K. Binder and A.P. Young, "Spin Glasses", Rev. Mod. Phys. 58,801 (1986). [CrossRef]
- M.W. Klein, C. Held, and E. Zuro, "Dipole Interactions Among Polar Defects: a Self-Consistent Theory with Application to OH Impurities in KCL", Phys. Rev. B 13, 3576 (1976). [CrossRef]
- H. Margenau, "Pressure Broadenning of Spectral Lines", Phys. Rev. 43, 129 (1933). [CrossRef]
- B.E. Vugmeister and M.D. Glinchuk, "Dipole Glass and Ferroelectricity in Random Site electric Dipole Systems" Rev. Mod. Phys. 62, 993 (1990). [CrossRef]
- B.E.Vugmeister and H. Rabitz, "Eect of Local Field Fluctuations on Orientational in Random Site Dipole Systems", J. Stat. Phys. 88, 477 (1997). [CrossRef]
- P.W. Miloni and P.L. Knight, "Retardation in the Resonant Interaction of Two Identical Atoms", Phys. Rev. A 10, 1076 (1974). [CrossRef]
- Y. Ben-Aryeh and S. Ruschin, "Cooperative Decay of a Linear Chain of Molecules Including Explicit Spatial Dependence", Physica A 88, 362 (1977). [CrossRef]
- B.E. Vugmeister and H. Rabitz, "NonequilibriumSpin Glass State in Nonlinear Optical Systems with Coherence", Phys. Lett. 232,129 (1997). [CrossRef]
- R.R. Mosley, B.D. Sinclair, and M.H. Dunn, "Local Field Eect in the Three-level Atom", Opt. Commun. 108, 247 (1994).

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