Optical Modeling of Nano-Structured Materials and Devices

mathematics or on quantum mechanics. List of All Articles 16 4.3 General Solution to Wave Equation Mukul Agrawal Noticing that f j t (t) forms an orthonormal set of functions, we get ( ∂ 2 ∂x2 + ∂ 2 ∂y2 + ∂ 2 ∂ z2 ) f j xyz(x,y,z) =−εμ0ωj f j xyz(x,y,z) (15) Hence the space dependent separable part of the eigen function of the operator i ∂ ∂ t , i.e. the f j xyz(x,y,z), should be an eigen function of the operator ( ∂ 2 ∂x2 + ∂ 2 ∂y2 + ∂ 2 ∂ z2 )with an eigen value of−εμ0ωj . The eigen value problem 15 is also known as the time independent wave equation. This also goes by the name of Helmholtz equation. The ortho-”normal” functions f j xyz(x,y,z) are commonly known as “normal modes” or the eigen modes of the system being considered. In unbound homogeneous system that we are considering here, finding normal mores of the system is a actually a trivial problem. In inhomogeneous system where we have many material boundaries, problem of solving an equivalent time-dependent wave equation (this we would study latter) reduces to finding the normal modes of the equivalent time independent wave equation (which is basically an eigen value problem). Further we would mostly be interested in systems that are discreetly homogeneous. So system is made up many locally homogeneous materials put-together. In such systems time-independent wave equation in inhomogeneous systems can often be reduced to combination of boundary conditions and bunch of time-independent wave equation for homogeneous systems. Since normal modes of homogeneous wave equation can be written trivially, problem reduces to find a linear combination of these so that the linear combination satisfies all boundary conditions. This entire linear combination would then form the normal mode of the inhomogeneous system. Such problems would be discussed latter in this article. I just outlined the details here to give a birds-eye-view of what we actually want to do with these equations. General Solution to Time Dynamic Problem Problem of solving the source free wave equation 13 can be worded as follows. Given a space dependent field profile at time t = 0 we want to calculate the field profiles at any other instance of time. Since,−i ∂ ∂x is an Hermitian operator so is the operator (−i ∂ ∂x)(−i ∂ ∂x). Hence the Laplacian L ≡ ( ∂ 2 ∂x2 + ∂ 2 ∂y2 + ∂ 2 ∂ z2 ) is also an Hermitian operator. So its eigen functions forms a complete set of orthonormal basis for expanding any space dependent function (no time dependence). First part of solving the problem is to find the eigen functions of the Laplacian operator. One can easily see that operator L commutes with all of the three operators −i ∂ ∂x , −i ∂ ∂y and −i ∂ ∂ z . Hence one should be able to find at least one complete set of common eigen vectors for all four operators. Following the same approach as indicated above one can easily see that f j xyz(x,y,z) = exp(ikxx)exp(ikyy)exp(ikzz) forms such a set. These are eigen functions of Laplacian with eigen values of −(k2 x + k2 y + k2 z ). Now we have seen that if f j xyz(x,y,z) are the eigen functions of Laplacian with an eigen value of −(k2 x + k2 y + k2 z ) then any field profile of the form ∑ j A j f j xyz(x,y,z) f j t (t)ê would satisfy the wave equation 13. Here A j are arbitrary space-time independent complex valued constants and f j t (t) = exp(−iω jt) are the orthonormal eigen functions of the operator List of All Articles 17 4.4 Plane Waves and Relation to Fourier Transforms Mukul Agrawal i ∂ ∂ t . Here ω 2 j = (k 2 x + k 2 y + k 2 z )/(εμ0). Hence, at t = 0, field profiles that satisfy the timedependent wave equation 13, should be given by ∑ j A j f j xyz(x,y,z)ê. In the problem statement, these fields are given as a boundary condition. Hence, the second part of the problem is to expand the space dependent field profiles given at t = 0 in terms of f j xyz(x,y,z). This would give us values of the constants A j. Once this is done, the field profiles at any other instance of time is given by ∑ j A j f j xyz(x,y,z)exp(−iω jt)ê with ωj = (k2 x + k2 y + k2 z )/(εμ0). This completes solution to the problem. Because of non-dispersion we have assumed, even the most general pulse form would travel at phase velocity without distortion (but diffraction can not be avoided). Dispersion Relation Assuming harmonic time oscillations ~E ∼ ~E exp(−iωt), ρext ∼ ρext exp(−iωt) and ~jext ∼ ~jext exp(−iωt), we get ~∇( ρext ε )−~∇2~E− εμ0ω~E = iμ0ω~jext And for source free case ~∇2~E =−εμ0ω~E Further assuming harmonic space oscillations ~E ∼ ~E exp(i~k.~r) we get ~∇( ρext ε )+~k.~k~E− εμ0ω~E = iμ0ω~jext And for source free case we get the dispersion relation