## Quantum properties of optical field in photonic band gap structures

Optics Express, Vol. 9, Issue 9, pp. 454-460 (2001)

http://dx.doi.org/10.1364/OE.9.000454

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### Abstract

A theoretical analysis of the quantum behaviour of radiation field’s propagation in photonic band gaps structures is performed. In these initial calculations we consider linear inhomogeneous and nondispersive media.

© Optical Society of America

## 1. Introduction

1. E. Yablonovitch and T.J. Gmitter, “Photonic band structure: the face-centered-cubic case” Phys. Rev. Lett. **63**, 1950–1953 (1989). [CrossRef] [PubMed]

^{2}+k

^{2}

*ε*(

*z*,

*ω*)]f(z,ω)=0, it does not have closed solution in the general form and for the general case [1

1. E. Yablonovitch and T.J. Gmitter, “Photonic band structure: the face-centered-cubic case” Phys. Rev. Lett. **63**, 1950–1953 (1989). [CrossRef] [PubMed]

2. C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, *Photons and Atoms* (JohnWiley & Sons, 1997). [CrossRef]

*V*is normalization constant and it is linked to the quantization volumes.

## 2. PBG structure.

*ε*=Cost., the

*Â*-potential operator is of the form :

*C*

_{ω}is a normalization constant and

*â*is independent of z. We observe that

*â*≠

*â*

^{+}instead

*â*=

*â*

^{+}. Operators

*â*and

*â*

^{+}satisfy the well know boson commutation relations:

*â*

_{j}and

*a*

2. C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, *Photons and Atoms* (JohnWiley & Sons, 1997). [CrossRef]

*U*

_{11}=

*F*;

*U*

_{12}=

*G*and det

*U͇*=1. Matrix

*U͇*could be easily linked to the Transmission matrix

*T͇*:

*F*and

*G*will depend on the particular shape of region between 1 and 2 half spaces. Using definition (5), for the permittivity coefficient, the

*F*and

*G*coefficients are

*B*

_{2}=|

*z*

_{3}-

*z*

_{2}|,

*ε*

_{2}constituted of N periodic regions (a real PBG structure), as reported in fig. 2.

4. Jon M. Bendickson, J. P. Dowling, and M. Scalora, “Analytic expression for the electromagnetic mode density in finite, one-dimensional, photonic band-gap structures”, Phys. Rev. E **53**, 4107–4121 (1996). [CrossRef]

_{N}(β)=Sin(Nβ)/Sin(β) is the modified Chebyshev function, β is the Bloch phase and in this particular case it’s defined as

*ω̃*is normalized to midgap frequency (≡ω/ω

_{0}).

## 3. Correlation Functions.

*Â*, in the following form [5

5. T. Gruner and D.G. Welsch, “Quantum-optical input-output relations for dispersive and lossy multilayer dielectric plates” Phys. Rev. A **54**, 1661–1677 (1996). [CrossRef] [PubMed]

*Â*is defined as

*Â*

^{(-)}(

*z*,

*ω*)=

*Â*

^{(+)}(

*z*,-

*ω*)

^{†}and obviously

*Â*

_{j}(

*z*,

*ω*)=

*Â*

_{j}(

*z*,

*ω*)

^{†}. If we use identical assumptions, used for the potential operator, for the electric field operator, we find the output photon-number density, i.e. the correlation function 〈

^{st}and the 3

^{rd}one) [6]:

*N*

_{1}(

*ω*)=

*N*

_{1out}(ω)/

*N*

_{1in}(

*ω*), and

*N*

_{3}(

*ω*)=

*N*

_{3out}(

*ω*)/

*N*

_{1in}(

*ω*), in the further hypothesis of irradiating the dielectrics from one side (the input field

*3*- is in the vacuum state):

*t*

_{11}|

^{2}=|

*G*/

*F*|

^{2}. Similar calculations, performed for to the

*N*

_{3out}(

*ω*) field, give the following results:

*t*

_{21}|

^{2}=|1/

*F*|

^{2}.

*N*

_{1}(

*ω*) as a function of the frequency and as a function of the thickness B

_{2}of the layer. Figure 4 represents

*N*

_{3}(

*ω*)as a function of the frequency and as a function of the thickness B

_{2}of the layer.

*N*

_{1}(

*ω*/

*ω*

_{0}), is plotted as a function of the normalized frequency

*ω*/

*ω*

_{0}=

*ωB*

_{2}

*n*

_{2}/

*c*. In green the plot of the photon number densities of the transmitted outgoing field over the incoming field,

*N*

_{3}(

*ω*/

*ω*

_{0}), as a function of the normalized frequency

*ω*/

*ω*

_{0}=

*ωB*

_{2}

*n*

_{2}/c is presented,

*B*

_{2}is the dielectric thickness and

*n*

_{2}is the refractive index (≈3 in our example). In this simulation N=1 (single layer).

*N*

_{1}(

*ω*/

*ω*

_{0}), is plotted as a function of the normalized frequency

*ω*/

*ω*

_{0}. In green the plot of the photon number densities of the transmitted outgoing field over the incoming field,

*N*

_{3}(

*ω*/

*ω*

_{0}), as a function of the normalized frequency

*ω*/

*ω*

_{0}is presented,

*B*

_{2}is the dielectric thickness and

*n*

_{2}is the refractive index (≈3 in our example). In this simulation a quarter-wave stack has been considered, and

*ω*

_{0}=2πc/λ

_{0}. The refractive index of each layer is n

_{a}=1, n

_{b}=2, and the number of cells (see fig. 2) is N=3 (multi-layer material: PBG structure). We can observe as the photon number density follows the classical transmission spectrum of the layered structure [4

4. Jon M. Bendickson, J. P. Dowling, and M. Scalora, “Analytic expression for the electromagnetic mode density in finite, one-dimensional, photonic band-gap structures”, Phys. Rev. E **53**, 4107–4121 (1996). [CrossRef]

## 4. States symmetries.

*φ*〉} is an ortho-normal set. If we try to calculate the average on this (general) state, of equation 23, we obtain:

*n*,

*m*,

*s*,

*p*are positive integer numbers. If we are in a state in which we have

*N*photons (in total), the general expression (24) gives the following state:

*n*+

*m*+

*s*≤

*N*. The number of such ortho-normal states is (6+11

*N*+6

*N*

^{2}+

*N*

^{3})/6. If we consider the further condition (25), the number of ortho-normal states will became (2+N)

^{2}/4 and the general

*N*photon state is:

*n*,

*m*and

*N*are positive integers. Replacing

*N*with 2

*N*’, and renaming

*N*’ by

*N*, equation (27) becomes:

*N*photon field in the system.

## 6. Conclusion.

## References and links

1. | E. Yablonovitch and T.J. Gmitter, “Photonic band structure: the face-centered-cubic case” Phys. Rev. Lett. |

2. | C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, |

3. | C. Cohen-Tannoudji, B. Diu, and F. Laloe, |

4. | Jon M. Bendickson, J. P. Dowling, and M. Scalora, “Analytic expression for the electromagnetic mode density in finite, one-dimensional, photonic band-gap structures”, Phys. Rev. E |

5. | T. Gruner and D.G. Welsch, “Quantum-optical input-output relations for dispersive and lossy multilayer dielectric plates” Phys. Rev. A |

6. | L. Mandel and E. Wolf, |

**OCIS Codes**

(160.1190) Materials : Anisotropic optical materials

(270.0270) Quantum optics : Quantum optics

**ToC Category:**

Research Papers

**History**

Original Manuscript: September 6, 2001

Published: October 22, 2001

**Citation**

Sergio Severini, Concita Sibilia, Mario Bertolotti, Michael Scalora, and Charles Bowden, "Quantum properties of optical field in photonic band gap structures," Opt. Express **9**, 454-460 (2001)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-9-9-454

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### References

- E.Yablonovitch , T.J. Gmitter, "Photonic band structure: the face-centered-cubic case" Phys. Rev. Lett. 63, 1950-1953 (1989). [CrossRef] [PubMed]
- C. Cohen-Tannoudji, J. Dupont-Roc, G. Grynberg, Photons and Atoms (JohnWiley & Sons, 1997). [CrossRef]
- C. Cohen-Tannoudji, B. Diu, F. Laloe, Quantum Mechanics (JohnWiley & Sons, 1977)
- Jon M. Bendickson, J. P. Dowling, M. Scalora, "Analytic expression for the electromagnetic mode density in finite, one-dimensional, photonic band-gap structures," Phys. Rev. E 53, 4107-4121 (1996). [CrossRef]
- T. Gruner and D.G. Welsch, "Quantum-optical input-output relations for dispersive and lossy multiplayer dielectric plates" Phys. Rev. A 54, 1661-1677 (1996). [CrossRef] [PubMed]
- L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, 1995).

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