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Optics Express

  • Editor: J. H. Eberly
  • Vol. 9, Iss. 9 — Oct. 22, 2001
  • pp: 454–460
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Quantum properties of optical field in photonic band gap structures

S. Severini, C. Sibilia, M. Bertolotti, M. Scalora, and C. Bowden  »View Author Affiliations


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

Photonic band gap (PBG) structures have been extensively studied during these last years [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]

], 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

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

]. 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

[2z2+ω2c2ε(z)]Â(z,ω)=0
(1)

Â(z)=0Â(z,ω)eiωtdω
(2)

and the electric field operator is:

Ê(z)=0iωÂ(z,ω)eiωtdω
(3)

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

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

]:

[Â(z),Ê(z)]=iVδ(zz)
(4)

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)=j=1M1rectBj(zzi+1zi2)εj
(5)

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.

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

Â(z,ω)=Cωâeikz+H.c.
(6)

where C ω is a normalization constant and â is independent of z. We observe that ââ+ instead â=â+. Operators â and â+ satisfy the well know boson commutation relations:

[âj+,âj]=0;[âj+(x,ω),âi++(x,ω)]=δj,iδ(xx)δ(ωω)
(7)

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

forz<z2Â(z,ω)=Cω[â1+eikz+â1eikz]+H.c.
(8)

forz>z3Â(z,ω)=Cω[â3+eikz+â3eikz]+H.c.
(9)

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

â3+=U11â1++U12â1
â3=U21â1++U22â1
(10)

Coefficients of this transformation are determined starting from boundary conditions [3

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

]. We omit tedious calculations for the specific case of interest [2

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

]; but it’s interesting to point out the symmetries of this transformation: U22*=U 11=F; U21*=U 12=G and det =1. Matrix could be easily linked to the Transmission matrix :

[a1a3+]=T¯¯[a1+a3]
T¯¯=1F*(G*11G)
(11)

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(kB2)+ik2k22kksin (kB2)]·eikB2
G=ik2k22kksin (kB2)
(12)

where B2 =|z3 -z2 |, k=ωc, k=ωcε2=ωcn2=k·n2.

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.

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

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]

]

F=1/(4nanb(na+nb)2(eiπω˜(nanb)2(na+nb)2)ΞN(β)4nanb(na+nb)2ΞN1(β))*
(13)
G=((nanb)4nanb(na+nb)3·(eiπω˜1)ΞN(β))*
(14)

where ΞN(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)=Cos(πω˜)(nanb)2(na+nb)24nanb(na+nb)2
(15)

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

3. Correlation Functions.

We can write the potential operator 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]

]:

Âj(z,ω)=Âj(+)(z,ω)+Âj()(z,ω)
(16)

where the positive part of  is defined as

Âj(+)(z,ω)=Cωaj+eikz
(17)

It is immediately evident that Â(-)(z,ω)=Â(+)(z,-ω) and obviously  j (z,ω)= 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 〈 Ê3(+) Ê3() 〉, is directly linked to the following quantity

â3++â3+=(F*â1+++G*â1+)(Fâ1++Gâ1)=
=F2â1++â1++(F21)â1+â1
(18)

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

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

]:

N1out(ω)=â1+â1;N1in(ω)=â1++â1+
N3out(ω)=â3++â3+N3in(ω)=â3+â3
(19)

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):

N1in(ω)0,N3in(ω)=0
(20)

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

N1out(ω)=â1+â1=(t11*â1+++t12*â3+)(t11â1++t12â3)=
=t112N1in(ω)
(21)

where |t 11|2=|G/F|2. Similar calculations, performed for to the N 3out(ω) field, give the following results:

N3out (ω)=t212N1in(ω)
(22)

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).
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).

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).

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 na=1, nb=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]

].

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).

4. States symmetries.

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

n̂3++n̂1=n̂1++n̂3
(23)

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
(24)

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
(25)

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:

φN=n,m,s,N(n+m+s)
(26)

where n+m+sN. 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:

φN=n,m,N2m,N2n,n,m[0,N2]
(27)

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

φ2N=n,m,Nm,Nn,n,m[0,N]
(28)

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).

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

  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).

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