Quantum properties of optical field in photonic band gap structures

S. Severini; C. Sibilia; M. Bertolotti; M. Scalora; C. Bowden

doi:10.1364/OE.9.000454

1. Introduction

Photonic band gap (PBG) structures have been extensively studied during these last years [1], due to the possibility of handling light. The propagation can occur in 1D,2D, or 3D periodical structures and gives rise to gaps in the transmission as for electron energy in crystals. In what follows we consider a 1D PBG.

The propagation in inhomogeneous materials is described by the following equation: [∇²+k² ε(z,ω)]f(z,ω)=0, it does not have closed solution in the general form and for the general case [1]. Of course this problem remains even when we work in a quantum domain. In the absence of absorption and dispersion, in the quantum domain, we have the operator equation

[\frac{\partial^{2}}{\partial z^{2}} + \frac{ω^{2}}{c^{2}} ε (z)] \hat{A} (z, ω) = 0

In the previous equation we refer to an angular frequency of the vector potential operator; the time-independent potential operator given by

\hat{A} (z) = \int_{0}^{\infty} \hat{A} (z, ω) e^{- i ω t} d ω

and the electric field operator is:

\hat{E} (z) = \int_{0}^{\infty} i ω \hat{A} (z, ω) e^{- i ω t} d ω

These operators satisfy the well-known canonical commutation relation [2]:

[\hat{A} (z), \hat{E} (z^{'})] = \frac{i ℏ}{V} δ (z - z^{'})

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

2. PBG structure.

We are interested to the study of Eq.(1) when e(z) is function of the propagation variable z, for example in the particular case of 1 D photonic band gap (PBG) structures. Consider the dielectric permittivity function given by:

ε (z) = \sum_{j = 1}^{M - 1} r e c t_{B j} (z - \frac{z_{i + 1} - z_{i}}{2}) ε_{j}

where B_j =|z _i+1-z_i | and z₁ =-∞z_N =+∞. j is an index related to the number of interfaces, and M M is an index which defines the space: i.e. M=1 is the semi-space on the left side of the structure. No absorption and dispersion are considered.

Let us consider the simplest case in which M=4. In this situation we have:

Fig. 1. In this figure the three homogeneous regions are represented, in which the permittivity ε(z) is subdivided.

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In homogeneous regions, i.e. where ε=Cost., the Â-potential operator is of the form :

\hat{A} (z, ω) = C_{ω} \hat{a} e^{i k z} + H . c .

where 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:

[{\hat{a}}_{j +}, {\hat{a}}_{j -}] = 0; [{\hat{a}}_{j +} (x, ω), {\hat{a}}_{i +}^{+} (x^{'}, ω^{'})] = δ_{j, i} δ (x - x^{'}) δ (ω - ω^{'})

where we have considered two distinct fields, in region 1:

for z < z_{2} \Rightarrow \hat{A} (z, ω) = C_{ω} [{\hat{a}}_{1 +} e^{i k z} + {\hat{a}}_{1 -} e^{- i k z}] + H . c .

forward (1+) and backwards (1-) propagating fields, respectively. Subscripts j and I are related to modes of the radiation field. The same considerations apply to the other external region 3:

for z > z_{3} \Rightarrow \hat{A} (z, ω) = C_{ω} [{\hat{a}}_{3 +} e^{i k z} + {\hat{a}}_{3 -} e^{- i k z}] + H . c .

In the following calculations, we neglect the H.c. specifications. The operators â _j and a ${â_{j}}^{+}$ inside the two external regions (j=1,3) are linked through a linear transformation:

{\hat{a}}_{3 +} = U_{11} {\hat{a}}_{1 +} + U_{12} {\hat{a}}_{1 -}

Coefficients of this transformation are determined starting from boundary conditions [3]. We omit tedious calculations for the specific case of interest [2]; but it’s interesting to point out the symmetries of this transformation: ${U_{22}}^{*}$ =U ₁₁=F; ${U_{21}}^{*}$ =U ₁₂=G and det U͇=1. Matrix U͇ could be easily linked to the Transmission matrix T͇ :

[\begin{matrix} a_{1 -} \\ a_{3 +} \end{matrix}] = \underline{\underline{T}} [\begin{matrix} a_{1 +} \\ a_{3 -} \end{matrix}] \Rightarrow

The coefficients 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

F = [\cos (k^{'} B_{2}) + i \frac{k^{2} - k^{' 2}}{2 k k^{'}} \sin (k^{'} B_{2})] \cdot e^{- i k B_{2}}

where B₂ =|z₃ -z₂ |, $k = \frac{ω}{c}$ , $k^{'} = \frac{ω}{c} \sqrt{ε_{2}} = \frac{ω}{c} n_{2} = k \cdot n_{2}$ .

We can also consider a more complicated structure, in which we have ε ₂ constituted of N periodic regions (a real PBG structure), as reported in 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.

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In this case, in the further hypothesis of considering quarter-wave stacks (optical path in every region it’s equal to each other and it’s a quarter of wave length), we have for the F and G coefficients [4]

F = 1 / {(\frac{4 n_{a} n_{b} {(n_{a} + n_{b})}^{2}}{(e^{- i π \tilde{ω}} - \frac{{(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 = {(\frac{(n_{a} - n_{b})}{4 n_{a} n_{b} {(n_{a} + n_{b})}^{3}} \cdot (e^{i π \tilde{ω}} - 1) Ξ_{N} (β))}^{*}

where Ξ_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

Cos (β) = \frac{Cos (π \tilde{ω}) - \frac{{(n_{a} - n_{b})}^{2}}{{(n_{a} + n_{b})}^{2}}}{4 n_{a} n_{b} {(n_{a} + n_{b})}^{2}}

and ω̃ is normalized to midgap frequency (≡ω/ω₀).

3. Correlation Functions.

We can write the potential operator Â, in the following form [5]:

{\hat{A}}_{j} (z, ω) = {\hat{A}}_{j}^{(+)} (z, ω) + {\hat{A}}_{j}^{(-)} (z, ω)

where the positive part of Â is defined as

{\hat{A}}_{j}^{(+)} (z, ω) = C_{ω} a_{j +} e^{i k z}

It is immediately evident that 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 〈 $Ê_{3}^{(+)}$ $Ê_{3}^{(-)}$ 〉, is directly linked to the following quantity

〈 {\hat{a}}_{3 +}^{+} {\hat{a}}_{3 +} 〉 = 〈 (F^{*} {\hat{a}}_{1 +}^{+} + G^{*} {\hat{a}}_{1 -}^{+}) (F {\hat{a}}_{1 +} + G {\hat{a}}_{1 -}) 〉 =

Using the following photon-number densities (number of photons per unit frequency) functions for the fields in the two regions (the 1^st and the 3^rd one) [6]:

N_{1 out} (ω) = 〈 {\hat{a}}_{1 -}^{+} {\hat{a}}_{1 -} 〉; N_{1 in} (ω) = 〈 {\hat{a}}_{1 +}^{+} {\hat{a}}_{1 +} 〉

if we consider 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):

N_{1 in} (ω) \geq 0, N_{3 in} (ω) = 0

and using the transmission matrix formalism (see Eq.(16)), we have:

N_{1 out} (ω) = 〈 {\hat{a}}_{1 -}^{+} {\hat{a}}_{1 -} 〉 = 〈 (t_{11}^{*} {\hat{a}}_{1 +}^{+} + t_{12}^{*} {\hat{a}}_{3 -}^{+}) (t_{11} {\hat{a}}_{1 +} + t_{12} {\hat{a}}_{3 -}) 〉 =

where |t ₁₁|²=|G/F|². Similar calculations, performed for to the N _3out(ω) field, give the following results:

N_{3 out} (ω) = ∣ t_{21} ∣^{2} N_{1 in} (ω)

where |t ₂₁|²=|1/F|².

Examples of photon number densities ratio are presented in figures 3 and 4 for one single layer. Figure 3 represents N ₁(ω) as a function of the frequency and as a function of the thickness B₂ of the layer. Figure 4 represents N ₃(ω)as a function of the frequency and as a function of the thickness B₂ of the layer.

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).

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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).

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In Figure 5 the photon number densities of the reflected outgoing field over the incoming field, N ₁(ω/ω₀ ), is plotted as a function of the normalized frequency ω/ω₀ =ωB₂n₂ /c. In green the plot of the 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 is presented, B₂ is the dielectric thickness and n₂ is the refractive index (≈3 in our example). In this simulation N=1 (single layer).

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).

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In Figure 6 the photon number densities of the reflected outgoing field over the incoming field, N ₁(ω/ω₀ ), is plotted as a function of the normalized frequency ω/ω₀ . In green the plot of the 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 is presented, B₂ is the dielectric thickness and n₂ is the refractive index (≈3 in our example). In this simulation a quarter-wave stack has been considered, and ω₀ =2πc/λ₀. 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].

Fig. 6. In red: Photon number densities of the reflected outgoing field and incoming field, N ₁(ω/ω_n ), 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/ω_n ), as a function of the normalized frequency ω̃. In this quarter-wave stack, ω₀ =2πc/λ₀. In this simulation n_a=1, n_b=2 and the number of cells (see fig. 2) is N=3 (multi-layer material: PBG structure).

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4. States symmetries.

Starting from U (or T) matrix, we obtain in the general case, the following property:

{\hat{n}}_{3 +} + {\hat{n}}_{1 -} = {\hat{n}}_{1 +} + {\hat{n}}_{3 -}

that is the conservation energy relation for the total optical system, involving the 4 fields. The Hilbert Space describing the system, has the following base vector (Fock state):

∣ φ 〉 = ∣ n, m, s, p 〉

where labels in the ket are the photon numbers of modes 1-,1+,3-,3+, respectively. The set {|φ〉} is an ortho-normal set. If we try to calculate the average on this (general) state, of equation 23, we obtain:

p + n = m + s

where 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:

∣ φ_{N} 〉 = ∣ n, m, s, N - (n + m + s) 〉

where n+m+s≤N. The number of such ortho-normal states is (6+11N+6N 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:

∣ φ_{N} 〉 = ∣ n, m, \frac{N}{2} - m, \frac{N}{2} - n 〉, n, m \in [0, \frac{N}{2}] \cap ℵ

where the numbers n,m and N are positive integers. Replacing N with 2N’, and renaming N’ by N, equation (27) becomes:

∣ φ_{2 N} 〉 = ∣ n, m, N - m, N - n 〉, n, m \in [0, N] \cap ℵ

This is the structure of the 2N photon field in the system.

6. Conclusion.

These calculations show the study of the quantum correlation behaviour of the optical field in the space regions external to the PBG structures. The transfer matrix formalism has been applied to follow the output behaviour of photon number density of a quantum field exiting from the PBG structure. The knowledge of these properties will be of interest for implementation of quantum computing and quantum gates. These calculations are only the starting analysis of the interesting quantum properties of the inhomogeneous media. We’ll extend the study of the state’s symmetries and the correlation functions from linear to nonlinear cases.

The authors thank professor R. Horak for his useful advice and clarifications.

References and links

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]

3. C. Cohen-Tannoudji, B. Diu, and F. Laloe, Quantum Mechanics (JohnWiley & Sons, 1977)

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]

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]

6. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, 1995).

Quantum properties of optical field in photonic band gap structures

Abstract

1. Introduction

2. PBG structure.

3. Correlation Functions.

4. States symmetries.

6. Conclusion.

References and links

Cited By

Figures (6)

Equations (34)

Optics Express