Quantum Electrodynamical Basis for Wave. Propagation through Photonic Crystal

Transcription

1 Adv. Studies Theor. Phys., Vo. 6, 01, no. 3, Quantum Eectrodynamica Basis for Wave Propagation through Photonic Crysta 1 N. Chandrasekar and Har Narayan Upadhyay Schoo of Eectrica and Eectronics Engineering SASTRA University Thanjavur , India 1 nchandra@ece.sastra.edu hnu@ece.sastra.edu Abstract Quantum eectrodynamica mode of eectromagnetic fied interaction with inear dieectric is considered to find the Hamitonian for eectromagnetic fieds in photonic crysta. Equations of motion for fied operators in one-dimensiona photonic crysta are determined. Eectric fied distribution is obtained from the expectation vaue of eectric fied operator using coherent states of fied. Emergence of photonic bandgap due to periodic structure of photonic crysta is discussed. Keywords: Hamitonian, Quantum eectrodynamics, Photonic crysta, Photonic bandgap, coherent states 1 Introduction The subject of photonic crystas is deveoping rapidy in recent years because of nove devices that are expected from these artificia materias [5,6]. They offer contros over the fow of ight which are not possibe with other types of photonic devices. Photonic crystas are constructed by introducing a periodic variation in refractive index in dieectric materia in one, two or three dimensions. The periodic variation in refractive index is simiar to periodic potentia experienced by eectrons in semiconductor crysta and consequenty creates the photonic

2 130 N. Chandrasekar and H. N. Upadhyay bandgaps for eectromagnetic radiation, which means radiation with frequencies faing in the bandgap cannot exist inside the photonic crysta. This property of photonic bandgap can be used for bandgap guidance aong defect ines in photonic crysta, which gives greater contro over ight fow and aows reaization of nove photonic devices. The propagation of eectromagnetic waves through photonic crystas is governed by the master equation [6], which is the eigenvaue equation for eectric fied in photonic crysta. In this etter we derive the cassica eectric fied configuration in one-dimensiona photonic crysta starting from microscopic approach. Foowing the references [3,4] we construct the Hamitonian for eectromagnetic fied in periodic medium from quantum eectrodynamica principes. Both fied and medium are modeed as quantized osciators. To move from microscopic to macroscopic scae, we eiminate medium degree of freedom and use adiabatic-foowing and continuum approximations which resuts in a Hamitonian expressed in terms of medium susceptibiity and fied operators. Expectation vaue of eectric fied inside the photonic crysta is found using eectric fied operator and taking the fied states as coherent states. Coherent states are the states with minimum uncertainty condition and it is coser in its properties to cassica eectromagnetic fieds. The bandgap can be observed in photonic crysta when eectric fied is found to be zero for certain range of frequencies for a distances and time. Our approach gives a different perspective for the wave propagation through photonic crysta and it can be easiy generaized to two and three dimensiona photonic crystas. For photonic crystas with sma refractive index contrast, this approach gives the quantum properties of fieds in the photonic crystas.. Hamitonian for Eectromagnetic Fied in Medium In quantum eectrodynamica approach for eectromagnetic fieds interacting with dieectric medium, both eectromagnetic fied and medium are modeed as quantized harmonic osciators and the interaction between them given by the dipoe interaction [1,,7]. The microscopic tota Hamitonian is given as H = hω a a + hω b b ih ( hab e h a b e ) (1) λ ik rn * * ik rn b n n n n n * where a and a denote creation and destruction operators respectivey for fied mode, b n and b denote creation and destruction operators respectivey for the osciators of the medium, and ω and ω b are the frequencies of fied mode and the resonance frequency of the medium osciator respectivey. The λ term in summation sign indicate summation over poarizations of fied modes. The couping between fied mode and medium osciator is given as interaction between fied mode and medium dipoe and determined by the term h ( / ) 1/ ˆ ˆ = πω h V μbp ekλ where ˆp and e ˆk λ are unit vectors in the directions of dipoe moment and k ˆ respectivey. The evoution of fied modes is given by

3 Quantum eectrodynamica basis for wave propagation 131 Heisenberg s equations of motion for the operators a, and bn with eimination of medium degrees of freedom from these equations. In order to find the Hamitonian for the fied in terms of macroscopic medium parameter ike dieectric constant or dieectric susceptibiity adiabatic-foowing approximation and continuum approximation are made. Adiabatic-foowing approximation assumes that a quantum mechanica system remain in eigenstate that evoves sowy in time and in continuum approximation we consider infinitesima voume consisting of arge number of osciators so that the fied operator is spatiay averaged. The equation of motion for spatiay averaged fied operator a is obtained as da = i(1 πχ ) ω a () dt and the corresponding Hamitonian is 1 (1 πχ)( h ωa a ) (3) H = + The approximations made in arriving at this Hamitonian imit its appicabiity in finding quantum properties to medium which is rarer and is weaky couped to the fied. However it can used to obtain cassica fied distribution in medium if we use coherent states of fieds. For structured media ike photonic crysta susceptibiity χ becomes a periodic function of position. Eectric Fied Distribution in Photonic Crysta Coherent states are the quantum mechanica equivaent of cassica monochromatic eectromagnetic waves [,7]. The cassica eectric fied distribution in a photonic crysta can be evauated as the expectation vaue of quantum mechanica eectric fied operator using coherent states. Considering one dimensiona photonic crysta with periodicity p the susceptibiity becomes a periodic function of z, i.e., χ = χ( z+ p) and coherent state of fied is given as n 1 α α = exp( α ) n (4) n= 0 n! i with a α = αα and α = α e θ. The expectation vaue of the eectric fied operator 1 hω ( ) ( ) ˆ(, ) [ i k z ω t i k z ω Ezt = i ae t ae ] (5) ε with coherent states is obtained as

4 13 N. Chandrasekar and H. N. Upadhyay ˆ ˆ hω Ezt (, ) = α Eα = α sin[(1 πχ) ωt k( zz ) θ] (6) ε Here k is a periodic function of same period as that of χ and equation (6) gives the eectric fied distribution in the photonic crysta. The procedure can be easiy extended to two and three dimensiona cases and dispersion can be incuded by taking χ as a function of ω. The terms χ and k inside the sine function cause the fuctuations in the eectric fied at the interface of two media and it is possibe that with these two terms of same periodicity, the argument of the sine function can be made equa to nπ where n = 0, ± 1, ±,... for same frequencies. This resuts in zero eectric fied for these frequencies which means a bandgap is formed for a periodic structure. With the expectation vaue of square of eectric fied given as hω E ( z, t) = {1+ 4α sin[(1 πχ) ωt k z θ ]} ε the fuctuations in eectric fied can be determined as (7) 1/ 1/ hω Δ E = ( Δ E) = ε This is same as for vacuum state which means it contains ony the noise of the vacuum. (8) 3 Concusions The eectric fied distribution inside a photonic crysta or any structured media can be determined the Hamitonian constructed from microscopic approach with adiabatic-foowing and continuum approximations. The form of Hamitonian is simpe in that it consists of medium susceptibiity and fied operators. The expectation vaue of eectric fied operators with coherent states of fied taken gives the fied distribution inside the media. When the media is rarer or the refractive index contrast in photonic crysta is smaer the evoution equations of fied operators describe the quantum properties of the fied. References [1] M. Born, E.Wof, Principes of Optics, sixth ed., Pergamon, Oxford, [] Christopher C. Gerry and Peter L. Knight, Introductory Quantum Optics, Cambridge University Press, Cambridge, 005. [3] M.E. Crenshaw, Microscopic foundations of macroscopic quantum optics, Phys. Rev. A 67 (003)

5 Quantum eectrodynamica basis for wave propagation 133 [4] M.E. Crenshaw, Quantum eectrodynamic foundations of continuum eectrodynamics, Phys. Lett. A 336 (005) [5] J.D. Joannopouos, Pierre R. Vieneuve and Shanhui Fan, Nature 386 (1997) [6] J.D. Joannopouos, S.G. Johnson, J.N. Winn, and R.D. Meade, Photonic Crystas: Moding the Fow of Light, Second ed., Princeton University Press, (008). [7] Rodney Loudon,The Quantum Theory of Light, Third ed., Oxford University Press, 000.