## Local field effects in multicomponent media

Optics Express, Vol. 1, Issue 6, pp. 152-159 (1997)

http://dx.doi.org/10.1364/OE.1.000152

Acrobat PDF (354 KB)

### Abstract

We investigate local-field effects in nonlinear optical materials composed of two species of atoms. One species of atom is assumed to be near resonance with an applied field and is modeled as a two-level system while the other species of atom is assumed to be in the linear regime. If the near dipole-dipole interaction between two-level atoms is negligible, the usual local- field enhancement of the field is obtained. For the case in which near-dipole-dipole interactions are significant due to a high density of two-level atoms, local-field effects associated with the presence of a optically linear material component lead to local-field enhancement of the near dipole-dipole interaction, intrinsic cooperative decays, and coherence exchange processes.

© Optical Society of America

## 1. Introduction

1. 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]

2. F. A. Hopf, C. M. Bowden, and W. Louisell, “Mirrorless optical bistability with the use of the local-field correction,”Phys. Rev. A **29**, 2591 (1984). [CrossRef]

3. M. E. Crenshaw and C. M. Bowden, “Quasiadiabatic Following Approximation for a Dense Medium of Two-Level Atoms,” Phys. Rev. Lett. **69**, 3475 (1992). [CrossRef] [PubMed]

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

7. M. P. Hehlen, H. U. Güdel, Q. Shu, J. Rai, S. Rai, and S. C. Rand, “Cooperative Bistability in Dense, Excited Atomic Systems” Phys. Rev. Lett. **73**, 1103 (1994). [CrossRef] [PubMed]

^{3}) demonstrated intrinsic optical bistability (IOB) due to near dipole- dipole (NDD) interactions between Yb

^{3+}ions in a Cs

_{3}Y

_{2}Br

_{9}crystal. In light of this example in which more than one species of particles is sufficiently dense to require the application on the LLFC, we investigate the local-field effect for optically nonlinear materials which have more than one dense polarizable component. We find that local field effects in dense multicomponent nonlinear media lead to local-field enhancement effects, local cooperative decays, and coherence exchange processes.

## 2. The Lorentz local-field condition

1. 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**

_{L}=

**E**+

**E**

_{i}is written as the sum of the average macroscopic field

**E**and

**E**

_{i}, the internal field due to the polarization of the near dipoles within a volume of the order of a cubic wavelength. The internal field within this volume is calculated in two pieces. At the smallest length scale

*d*, of the order of the typical intermolecular spacing, the effects of retardation are negligible and the near field component

**E**

_{near}of the internal field is calculated by taking the actual contribution of individual dipoles in the static limit, Fig. 2. It is well known that

**E**

_{near}is identically zero for a cubic lattice as a consequence of symmetry. However, it can be seen in Fig. 2 that cubic symmetry is broken if more than one species of polarizable particle is present. To account for the non-cubic symmetry, we write

**E**

_{near}= ∑

_{i}

*s*

_{Li}**P**

_{i}, where

*s*is the structure factor and

_{Li}**P**

_{i}is the partial polarization for species

*i*.

**E**

_{P}to the internal field arises from the transition from microscopic variables to macroscopic variables. This transition occurs on length scales of order

*l*, where

*d*≪

*l*≪ λ. The procedure is detailed in Ref. [1

1. 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**

_{P}= - (4

*π*/3)∑

_{i}

**P**

_{i}upon performing the integration.

**P**

_{i}is the partial polarization of species

*i*,

*s*are structure factors for the different species, and

_{Li}*η*= 1 + 3

_{Li}*s*/4

_{Li}*π*. We see that the polarization that appears in the Lorentz local field condition is the sum of the partial polarizations of the constituents of the material. This simple result has profound implications for the nonlinear dynamics of dense multicomponent media because the macroscopic polarization couples the nonlinear interaction of one component with the electromagnetic field to the dynamics of other components.

## 3. Dense two-level atoms embedded in a linearly polarizable host

2. F. A. Hopf, C. M. Bowden, and W. Louisell, “Mirrorless optical bistability with the use of the local-field correction,”Phys. Rev. A **29**, 2591 (1984). [CrossRef]

9. R. Friedberg, S. R. Hartmann, and J. T. Manassah, “Effect of local-field correction on a strongly pumped resonance,” Phys. Rev. A **40**, 2446 (1989). [CrossRef] [PubMed]

10. M. E. Crenshaw, M. Scalora, and C. M. Bowden, “Ultrafast Intrinsic Optical Swithcin in a Dense Medium of Two-Level Atoms,” Phys. Rev. Lett. **68**, 911 (1992). [CrossRef] [PubMed]

3. M. E. Crenshaw and C. M. Bowden, “Quasiadiabatic Following Approximation for a Dense Medium of Two-Level Atoms,” Phys. Rev. Lett. **69**, 3475 (1992). [CrossRef] [PubMed]

*in vacuo*. However, citing the example of the (HGSR

^{3}) experiment, we note that materials for which the resonant atoms are sufficiently dense for NDD effects to be significant often contain more than one polarizable component, even though only the one near resonance with the applied field is of primary interest. Such a material can be described macroscopically as a dense collection of two-level atoms embedded in a linear dielectric. But in order to self-consistently apply the LLFC, one adopts the atomistic description of two species of atoms embedded

*in vacuo*; one species has a dipole-allowed transition that is nearly resonant with the incident laser field and is modeled as a two-level quantum system while the other species is far from resonance and is modeled as a linearly polarizable particle. Then the total polarization is the sum of the partial polarizations of the two-level atoms and the linearly polarizable particles and Eq. 1 becomes [6]

*ε*

_{L},

*ε*, 𝑝, 𝑝

^{lin}, and 𝑝

^{res}where

^{lin}and 𝑝

^{res}are the envelope functions of the partial polarization due to the linear particles and the two-level atoms, respectively. Also,

*α*is the linear polarizability and

*N*

_{α}is the number density of linear systems. The Clausius-Mossotti-Lorentz-Lorenz (CMLL) relation

*ε*=

*n*

^{2}of the background. Employing the CMLL relation, the local field can be written as

*r*

_{21}=

*ρ*

_{21}

^{iωpt},

*ρ*

_{21}is the off-diagonal density matrix element,

*ω*=

*ρ*

_{22}-

*ρ*

_{11}is the inversion,

*μ*is the transition dipole moment, and Δ =

*ω*

_{p}-

*ω*

_{0}is the detuning from resonance.

^{res}= 2

*NμR*

_{21}is the nonlinear polarization and ∊ = 4

*πηNμ*

^{2}/(3

*ħ*) is the NDD interaction parameter, a measure of the interaction between near dipoles. We have adopted the convention of using italicized upper-case Roman letters for the macroscopic, spatially averaged, atomic variables in the rotating frame of reference:

*R*

_{21}= 〈

*r*

_{21}〉

_{sp},

*R*

_{12}= 〈

*r*

_{12}〉

_{sp}, and

*W*=

*R*

_{22}-

*R*

_{11}= 〈

*r*

_{22}〉

_{sp}- 〈

*r*

_{11}〉

_{sp}. Here, 〈⋯〉

_{sp}corresponds to a spatial average over a volume of the order of a resonance wavelength cubed. Damping has been introduced phenomenologically, where

*γ*

_{∥}= 1/

*T*

_{1}is the population relaxation rate,

*γ*

_{⊥}= 1/

*T*

_{2}is the dipole dephasing rate, and

*W*

_{eq}is the population difference at thermal equilibrium.

*T*

_{1}and

*T*

_{2}are the familiar longitudinal and transverse relaxation times, respectively.

*l*=

*l*

_{r}+

*il*

_{i}into real and imaginary components, the generalized Bloch equations (8) and (9) can be written as

*i*) The term involving the product

*l*

_{r}

*WR*

_{21}is bilinear in the atomic variables and can be interpreted as an intrinsic frequency modulation or inversion-dependent detuning from resonance. The presence of the host medium enhances the inversion-dependent detuning (nonlinear Lorentz frequency shift) that is due to the NDD interaction. This is especially significant because NDD effects are only important at sufficiently high densities and large oscillator strengths. For example, Friedberg, Hartmann, and Manassah [9

9. R. Friedberg, S. R. Hartmann, and J. T. Manassah, “Effect of local-field correction on a strongly pumped resonance,” Phys. Rev. A **40**, 2446 (1989). [CrossRef] [PubMed]

*γ*

_{⊥}. This density threshold condition cannot be controlled in a vapor of two-level systems because the dephasing rate increases in direct proportion to the density due to collisional broadening. For two-level systems in a dispersionless dielectric, the threshold condition

*l*∊ > 4

*γ*

_{⊥}can be satisfied by a smaller value of ∊, because the inversion- dependent detuning is enhanced in condensed matter and the homogeneous linewidth is no longer restricted to the formula for a collisionally broadened vapor. Significantly, it is in a similar case that intrinsic optical bistability, analyzable by a two-level model, was observed experimentally [7

7. M. P. Hehlen, H. U. Güdel, Q. Shu, J. Rai, S. Rai, and S. C. Rand, “Cooperative Bistability in Dense, Excited Atomic Systems” Phys. Rev. Lett. **73**, 1103 (1994). [CrossRef] [PubMed]

*ii*) There is an enhancement of the magnitude of the field that drives the atoms. This effect is present for a dilute embedding of atoms in a dielectric [5, 6].

*iii*) The imaginary part of the enhancement factor appears as a coefficient, along with the NDD parameter, of bilinear products of the macroscopic atomic variables in two terms of Eqs. (10) and (11). These terms correspond to local cooperative decay effects representing the interaction of near dipoles mediated by the imaginary component of the dielectric function of the host medium. Because the effects of NDD interactions can be manifested in films that are significantly thinner than a vacuum wavelength, the detrimental effects, absorption and heating, associated with the imaginary part of the index of refraction can be mitigated for dense media. Then, for a thin film of a strongly dispersive dielectric containing a dense collection of two- level atoms, one can expect local cooperative decay effects to play a significant role in the dynamics.

## 4. The generalized Bloch-Drude model

*ω*

_{p}is defined by the relation

*πN*

_{b}

*e*

^{2}/

*m*,

*N*

_{b}, is the number density of oscillators,

*ω*

_{b}is the resonance frequency of the oscillators, and

*X̄*=

*mμX*/

*eħ*is a normalized coordinate. The coupling between the equations of motion for the two species of particles represents a coherence exchange that occurs because the dynamics of the two species of particles are coupled through the electromagnetic field as described by the LLFC.

*l*= 1 + (

*ω*

^{2}-

*iωγ*). Now we have the local field enhancement factor, and also the linear dielectric function, in terms of the physical parameters of the harmonic oscillator model. In this case, the imaginary part of the local field enhancement factor, and thus the intrinsic cooperative decay of the two-level atoms, can be seen to be a consequence of the normal decay processes of the linear oscillators.

## 5. Summary

## References

1. | C. M. Bowden and J. P. Dowling, “Near-dipole-dipole effects in dense media: Generalized Maxwell-Bloch equations,” Phys. Rev. A |

2. | F. A. Hopf, C. M. Bowden, and W. Louisell, “Mirrorless optical bistability with the use of the local-field correction,”Phys. Rev. A |

3. | M. E. Crenshaw and C. M. Bowden, “Quasiadiabatic Following Approximation for a Dense Medium of Two-Level Atoms,” Phys. Rev. Lett. |

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

5. | D. Marcuse, |

6. | N. Bloembergen, |

7. | M. P. Hehlen, H. U. Güdel, Q. Shu, J. Rai, S. Rai, and S. C. Rand, “Cooperative Bistability in Dense, Excited Atomic Systems” Phys. Rev. Lett. |

8. | M. Born and E. Wolf, |

9. | R. Friedberg, S. R. Hartmann, and J. T. Manassah, “Effect of local-field correction on a strongly pumped resonance,” Phys. Rev. A |

10. | M. E. Crenshaw, M. Scalora, and C. M. Bowden, “Ultrafast Intrinsic Optical Swithcin in a Dense Medium of Two-Level Atoms,” Phys. Rev. Lett. |

11. | L. Allen and J. H. Eberly, |

12. | M. Sargent III, M. O. Scully, and W. E. Lamb Jr., |

**OCIS Codes**

(030.1670) Coherence and statistical optics : Coherent optical effects

(270.0270) Quantum optics : Quantum optics

(270.1670) Quantum optics : Coherent optical effects

**ToC Category:**

Focus Issue: Local field effects

**History**

Original Manuscript: August 21, 1997

Published: September 15, 1997

**Citation**

Michael Crenshaw, Kay Sullivan, and Charles Bowden, "Local Field Effects in Multicomponent Media," Opt. Express **1**, 152-159 (1997)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-1-6-152

Sort: Journal | Reset

### References

- C. M. Bowden and J. P. Dowling, Phys. Rev. A 47, 1247 (1993); 49, 1514 (1994). [CrossRef] [PubMed]
- F. A. Hopf, C. M. Bowden, W. Louisell, "Mirrorless optical bistability with the use of the local-field correction,"Phys. Rev. A 29, 2591 (1984). [CrossRef]
- M. E. Crenshaw and C. M. Bowden, "Quasiadiabatic Following Approximation for a Dense Medium of Two-Level Atoms," Phys. Rev. Lett. 69, 3475 (1992). [CrossRef] [PubMed]
- A. S. Manka, J. P. Dowling, C. M. Bowden, and M. Fleishhauer, "Piezophotonic Switching Due to Local Field Eects in a Coherently Prepared Medium of Three-Level Atoms," Phys. Rev. Lett. 73, 1789 (1994). [CrossRef] [PubMed]
- D. Marcuse, Principles of Quantum Electronics, (Academic Press, Orlando, FL, 1980), pg. 307.
- N. Bloembergen, Nonlinear Optics (W. A. Benjamin, New York, 1964).
- M.P. Hehlen, H. U. Gudel, Q. Shu, J. Rai, S. Rai, and S. C. Rand, "Cooperative Bistability in Dense, Excited Atomic Systems" Phys. Rev. Lett. 73, 1103 (1994). [CrossRef] [PubMed]
- M. Born and E. Wolf, Principles of Optics, Sixth Ed., (Pergamon Press, Oxford, 1991).
- R. Friedberg, S. R. Hartmann, J. T. Manassah, "Effect of local-field correction on a strongly pumped resonance," Phys. Rev. A 40, 2446 (1989). [CrossRef] [PubMed]
- M. E. Crenshaw, M. Scalora, and C. M. Bowden, "Ultrafast Intrinsic Optical Swithcin in a Dense Medium of Two-Level Atoms," Phys. Rev. Lett. 68, 911 (1992). [CrossRef] [PubMed]
- L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms, (Wiley, New York, 1975), (republished by Dover, NY, 1987).
- M.Sargent III, M. O. Scully, and W.E. Lamb, Jr., Laser Physics, (Addison-Wesley, NY, 1987).

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

### Supplementary Material

» Media 1: GIF (8 KB)

» Media 2: GIF (6 KB)

» Media 3: GIF (11 KB)

» Media 4: GIF (12 KB)

» Media 5: GIF (20 KB)

» Media 6: GIF (19 KB)

« Previous Article | Next Article »

OSA is a member of CrossRef.