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.

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References

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  1. E.Yablonovitch , 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, G. Grynberg, Photons and Atoms (JohnWiley & Sons, 1997).
    [CrossRef]
  3. C. Cohen-Tannoudji, B. Diu, F. Laloe, Quantum Mechanics (JohnWiley & Sons, 1977)
  4. 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]
  5. 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]
  6. L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, 1995).

Other (6)

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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Figures (6)

Fig. 1.
Fig. 1.

In this figure the three homogeneous regions are represented, in which the permittivity ε(z) is subdivided.

Fig. 2.
Fig. 2.

Extension of the previous calculations to a real PBG structure. Now ε2 is z-dependent, and it consists of N regions in which we have n=constant.

Fig. 3.
Fig. 3.

Photon number densities ratio of the reflected outgoing field over the incoming field, N 1(ω)=N 1out (ω)/N 1in (ω), as a function of frequency and dielectric thickness. ω is of the order of 1014 s-1 and B 2 is of the order of 10-6 m. In this simulation N=1 (one layer).

Fig. 4.
Fig. 4.

Photon number densities ratio of the transmitted outgoing field over the incoming field, N 3(ω)=N 3out (ω)/N 1in (ω), as a function of frequency and dielectric thickness. ω is of order 1014 s-1 and B 2 is of order 10-6 m. In this simulation N=1 (one layer).

Fig. 5.
Fig. 5.

In red: Photon number densities of the reflected outgoing field over the incoming field, N 1(ω/ω0 ), as a function of the normalized frequency ω/ω0 =ωB2n2 /c. In green: Photon number densities of the transmitted outgoing field over the incoming field, N 3(ω/ω0 ), as a function of the normalized frequency ω/ω0 =ωB2n2 /c. B2 is the dielectric thickness and n2 is the refractive index (≈3 in our example). In this simulation N=1 (single layer).

Fig. 6.
Fig. 6.

In red: Photon number densities of the reflected outgoing field and incoming field, N 1(ω/ωn ), as a function of the normalized frequency ω̃=ω/ω0. In green: Photon number densities of the transmitted outgoing field and incoming field, N 3(ω/ωn ), as a function of the normalized frequency ω̃. In this quarter-wave stack, ω0 =2πc/λ0. In this simulation na=1, nb=2 and the number of cells (see fig. 2) is N=3 (multi-layer material: PBG structure).

Equations (34)

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[ 2 z 2 + ω 2 c 2 ε ( z ) ] A ̂ ( z , ω ) = 0
A ̂ ( z ) = 0 A ̂ ( z , ω ) e i ω t d ω
E ̂ ( z ) = 0 i ω A ̂ ( z , ω ) e i ω t d ω
[ A ̂ ( z ) , E ̂ ( z ) ] = i V δ ( z z )
ε ( z ) = j = 1 M 1 r e c t B j ( z z i + 1 z i 2 ) ε j
A ̂ ( z , ω ) = C ω a ̂ e i k z + H . c .
[ a ̂ j + , a ̂ j ] = 0 ; [ a ̂ j + ( x , ω ) , a ̂ i + + ( x , ω ) ] = δ j , i δ ( x x ) δ ( ω ω )
for z < z 2 A ̂ ( z , ω ) = C ω [ a ̂ 1 + e i k z + a ̂ 1 e i k z ] + H . c .
for z > z 3 A ̂ ( z , ω ) = C ω [ a ̂ 3 + e i k z + a ̂ 3 e i k z ] + H . c .
a ̂ 3 + = U 11 a ̂ 1 + + U 12 a ̂ 1
a ̂ 3 = U 21 a ̂ 1 + + U 22 a ̂ 1
[ a 1 a 3 + ] = T ¯ ¯ [ a 1 + a 3 ]
T ¯ ¯ = 1 F * ( G * 1 1 G )
F = [ cos ( k B 2 ) + i k 2 k 2 2 k k sin ( k B 2 ) ] · e i k B 2
G = i k 2 k 2 2 k k sin ( k B 2 )
F = 1 / ( 4 n a n b ( n a + n b ) 2 ( e i π ω ˜ ( n a n b ) 2 ( n a + n b ) 2 ) Ξ N ( β ) 4 n a n b ( n a + n b ) 2 Ξ N 1 ( β ) ) *
G = ( ( n a n b ) 4 n a n b ( n a + n b ) 3 · ( e i π ω ˜ 1 ) Ξ N ( β ) ) *
Cos ( β ) = Cos ( π ω ˜ ) ( n a n b ) 2 ( n a + n b ) 2 4 n a n b ( n a + n b ) 2
A ̂ j ( z , ω ) = A ̂ j ( + ) ( z , ω ) + A ̂ j ( ) ( z , ω )
A ̂ j ( + ) ( z , ω ) = C ω a j + e i k z
a ̂ 3 + + a ̂ 3 + = ( F * a ̂ 1 + + + G * a ̂ 1 + ) ( F a ̂ 1 + + G a ̂ 1 ) =
= F 2 a ̂ 1 + + a ̂ 1 + + ( F 2 1 ) a ̂ 1 + a ̂ 1
N 1 out ( ω ) = a ̂ 1 + a ̂ 1 ; N 1 in ( ω ) = a ̂ 1 + + a ̂ 1 +
N 3 out ( ω ) = a ̂ 3 + + a ̂ 3 + N 3 in ( ω ) = a ̂ 3 + a ̂ 3
N 1 in ( ω ) 0 , N 3 in ( ω ) = 0
N 1 out ( ω ) = a ̂ 1 + a ̂ 1 = ( t 11 * a ̂ 1 + + + t 12 * a ̂ 3 + ) ( t 11 a ̂ 1 + + t 12 a ̂ 3 ) =
= t 11 2 N 1 in ( ω )
N 3 out ( ω ) = t 21 2 N 1 in ( ω )
n ̂ 3 + + n ̂ 1 = n ̂ 1 + + n ̂ 3
φ = n , m , s , p
p + n = m + s
φ N = n , m , s , N ( n + m + s )
φ N = n , m , N 2 m , N 2 n , n , m [ 0 , N 2 ]
φ 2 N = n , m , N m , N n , n , m [ 0 , N ]

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