Resonant-state-expansion Born approximation for waveguides with dispersion

The resonant-state expansion (RSE) Born approximation, a rigorous perturbative method developed for electrodynamic and quantum mechanical open systems, is further developed to treat waveguides with a Sellmeier dispersion. For media that can be described by these types of dispersion over the relevant frequency range, such as optical glass, I show that the perturbed RSE problem can be solved by diagonalizing a second-order eigenvalue problem. In the case of a single resonance at zero frequency, this is simplified to a generalized eigenvalue problem. Results are presented using analytically solvable planar waveguides and parameters of borosilicate BK7 glass, for a perturbation in the waveguide width. The efficiency of using either an exact dispersion over all frequencies or an approximate dispersion over a narrow frequency range is compared. I included a derivation of the RSE Born approximation for waveguides to make use of the resonances calculated by the RSE, an RSE extension of the well-known Born approximation.


I. INTRODUCTION
Fundamental to scattering theory, the Born approximation consists of taking the incident field in place of the total field as the driving field at each point inside the scattering potential, it was first discovered by Born and presented in Ref. [1]. The Born approximation gave an expression for the differential scattering cross section in terms of the Fourier transform of the scattering potential. An important feature of this appearance of the Fourier transform is the availability of the inverse Fourier transform operation for the inverse scattering problem. The Born approximation is only valid for weak scatterers. In this paper I apply the RSE Born approximation Ref. [2,3] to optical fibers or general open waveguide systems, which allows an arbitrary number of resonantstates (RSs) to be taken into account for scattering and transmission perturbative calculations.
Optical fibers provide a well-controlled optical path which can carry light over potentially very long distances of several hundred to thousands of kilometers since the light is contained within the fiber by total internal reflection. Fibers are clearly important for telecommunication since the light may be modulated to carry information. It has recently been reported that a hollow-core photonic-band gap fiber yielded a record combination of low loss and wide bandwidth [4]. Beyond telecommunications waveguides are used in integrated optical circuits [5], and terabit chip-to-chip interconnects [6].
Critical to understanding the response of a waveguide to being optically driven are its resonant states (RSs). The concept of RSs was first conceived and used by Gamow in 1928 in order to describe mathematically the process of radioactive decay, specifically the escape from the nuclear potential of an α particle by tunneling. Mathematically this corresponded to solving Schrödinger's equation for outgoing boundary conditions (BCs). These states have complex frequency ω with negative imaginary part meaning their time dependence exp(−iωt) decays exponentially, thus giving an explanation for the exponential decay law of nuclear physics. The consequence of this exponential decay with time is that the further from the decaying system at a given instant of time, the greater the wave amplitude. An intuitive way of understanding this divergence of wave amplitude with distance is to notice that waves that are further away have left the system at an earlier time when less of the particle probability density had leaked out. There already exists numerical techniques for finding eigenmodes, such as the finite element method (FEM) and finite difference in time domain (FDTD) method to calculate resonances in open cavities. However, determining the effect of perturbations, which break the symmetry, presents a significant challenge as these popular computational techniques need large computational resources to model high-quality modes. Also these methods generate spurious solutions, which would damage the accuracy of the RSE Born approximation if included in the perturbation expansion basis.
In order to calculate the resonances of open systems we have recently developed the resonant-state expansion. Such an approach was not previously available due to the lack of a normalization for resonant states. I derived the normalization of resonant states as a contribution to Ref. [9], and the generalization of the normalisation to dispersive media I derived straight forwardly at the time of my rigorous derivation. The problem of normalising resonant states stems from the fact that RSs with complex frequencies have wave functions which are exponentially growing in space away from system, making a volume integral over all space as would be done for a hermitian system a meaningless exercise (see Appendix F). However the S -wave normalization was previously available and analytically correct but numerically unstable as I showed analytically in Ref [3].
So far the RSE has been applied to non dispersive systems of different dimensionality [7][8][9][10][11]. However, almost all realistic systems have a relevant frequency dispersion of the refractive index. I have recently found [3,12] that an Ohm's law dispersion, i.e. a term in the susceptibility scaling at the inverse frequency, can be introduced to the RSE while keeping its linearity. We found in Ref. [12] this dispersion can be a reasonable approximation for some materials over a limited range of wavelengths such as SHOTT BK7 glass over the optical range. In this work I generalize the RSE approach for waveguides detailed in Ref. [7] to systems constructed from dispersive media. Specifically I treat dispersion fully described by the Sellmeier equation, using a method similar to Muljarov et al in Ref. [13] for nano particles obeying Drude-Lorentz dispersion, however in this case generalised to inclined geometry. I also treat dispersion linear in wavelength squared, a generalisation of my work in Ref. [3,12] to inclined geometry. I find the generalization of my Ohm's law approach to inclined geometry to be greatly superior to the generalization of the full dispersion treatment. This is most likely due to the unphysicality of the full dispersion at high and low frequencies, which are beyond the fitting range of the dispersion models. In order to make use of the RSs, I derive the RSE Born approximation for waveguides [2,3], a theory for finding the field outside of a waveguide being internally or externally driven (see Appendix G).
The paper is organized as follows, Sec. II outlines the general recursive solution of Maxwell's equation using Green's function. Sec. III develops the solution in Sec. II into a perturbation theory for waveguides with linear dispersion in wavelength squared. Sec. IV develops the solution in Sec. II into a perturbation theory for waveguides with Sellmeier dispersion. Sec. VI demonstrates and compares the two approaches for including dispersion into the RSE. The perturbation considered corresponds to a narrowing of the waveguide and is discussed in Sec. VI C. In Sec. VI D I show the results using the simple dispersive approximation BK7. In Sec. VI E I use the full Sellmeier dispersion of BK7. Finally, we discuss the comparison and performance of the two methods in Sec. VI F, in Sec. VII, I derive the equations for the RSE Born approximation for planar waveguides. In Appendix C, D, E, and F I generalize my part of the proof of the normalization of RSs to open waveguide systems.

II. RESONANT STATE EXPANSION FOR WAVEGUIDES AND NON-NORMAL INCIDENCE
In this section I develop the general recursive solution to the problem of calculating the resonant states of a perturbed waveguide. The recursive solution requires the Green's function (GF) of the unperturbed waveguide, and a perturbation which is retaining the translational invariance along the waveguide.
I first consider a waveguide of thickness 2a in vacuum with translational invariance in one direction, having the dielectric constant where ǫ ω is the frequency dependent relative permittivity (RP) of the wave guide with the angular frequency ω. r are the coordinates normal to the waveguide, which for a cylindrical waveguide is given in polar coordinateṡ r = (ρ, θ) and for a planar waveguideṙ is Cartesian x. I assume a relative permeability of µ = 1 throughout this work. The electric fieldÊ satisfies Maxwell's equation, For cylindrical and planar waveguides, due to translational invariance in the z direction I first assume then prove thatÊ can be factorised as followŝ in which p is the wave vector along the translationally invariant direction. For the component E(ṙ) of the electric field, Eq. (2) transforms to a one-dimensional (1D) wave equation in the planar case or a 2D wave-equation for the cylindrical waveguide to Here L is a linear operator not dependent on z orε ω (ṙ). The form of L is derived in Appendix C, the derivation shows that the trial solution Eq. (3) is correct since it demonstrates thatÊ can be factorised by separation of variables. In principle L could be any linear operator independent of both z andε ω (ṙ), and hence the treatment of waveguides in this paper is readily applicable to quantum mechanics or acoustics. The non-normal incidence, characterized by p = 0, is treated here. The previously used spectral representation of the GF in the frequency domain contains a cut for p = 0, which can be removed by mapping the problem onto the complex normal wave-vector space k, as demonstrated in Ref. [7]. The relation between k, p, and ω is defined by us to be ω 2 /c 2 = k 2 + p 2 , k and p are orthogonal by Pythagorean identity. I follow also here the approach used in Ref. [7] and formulate the RSE in the complex k-plane, for which the spectral representation of the GF of an infinite planar system with an in-plane momentum p = 0 written in the spectral form where E n (ṙ) is the electric field of a RS, defined as an eigensolution of Eq. (4) with an arbitrary profile ofε ω (ṙ) within a region |ṙ| < a, satisfying the outgoing wave boundary conditions at infinity. The dyadic product of the E n (ṙ) fields is denoted by ⊗. The in-plane eigenfrequency k n are the poles inĜ k (ṙ,ṙ ′ ). By definition ω 2 n /c 2 = k 2 n + p 2 where ω 2 n is the eigenfrequency corresponding to the eigenstate E n (ṙ). For a study of the derivation of such spectral representations see Appendix D.
The GF satisfies the equation In the present work, we consider the refractive profile (RP) having the property whereε(ṙ) if the frequency independent part ofε ω (ṙ). If we assumeε ω (ṙ) to be discontinuous through the boundary of the system then by substituting Eq. (5) into Eq. (6), convoluting with a finite function, and letting k → ∞, we obtain the sum rule (see Appendix E) which allows us to re-write the Green's function a second way aŝ I now consider an arbitrary perturbation ∆ε ω (ṙ) of the dielectric constant inside the layer |ṙ| < a. I use the Eq. (6) to solve the perturbed problem, which is the solution to the equation Note that the perturbed modes E ν (ṙ) satisfy Eq. (4) withε ω (ṙ) replaced byε ω (ṙ) + ∆ε ω (ṙ) and the outgoing boundary conditions with ω n replaced by ω. I show the solution of this equation perturbatively for two different types of dispersion in the following two sections.

III. SINGLE SELLMEIER RESONANCE AT ZERO FREQUENCY
For a Sellmeier dispersion with a single resonance at zero frequency (SRZ), I write the perturbation in Eq. (10) as again present only inside the layer |ṙ| < a. It is then possible to linearise the RSE by using the different forms of the Green's function, similar to Ref. [13], Eq. (5) and Eq. (9) for the different components of the perturbation in Eq. (10), which results in the following relationship between unperturbed and perturbed modes The perturbed mode E(ṙ) satisfies Eq. (4) withε ω (ṙ) replaced byε ω (ṙ) + ∆ε ω (ṙ) and outgoing boundary conditions with the wave vector k. In the interior region |ṙ| < a which contains the perturbation, the perturbed RSs E ν of wavenumbers κ ν can be expanded into the unperturbed ones, exploiting the completeness of the latter which follows from Eq. (A6) (see also Appendix E), Substituting this expansion into Eq. (13) and equating coefficients at the same basis functions E n results in the matrix equation where and With the substitution c nν = b nν √ k n , Eq. (16) can be rewritten as care should be taken to take the sign of √ k a consistently between matrix elements. Eq. (19) is linear in κ ν and can be solved by libraries for generalized linear matrix eigenvalue problems. In the absence of dispersion, S nm = 0, and Eq. (19) reverts back to the expression for nondispersive waveguides [7]. In the absence of p, pa = 0, we see that Eq. (19) reverts to an expression for dispersive perturbation to nano-particles [12].

IV. SELLMEIER DISPERSION
In this section I develop the existing RSE for wave guides into a perturbation theory for waveguides with Sellmeier dispersion, following a similar approach to Ref. [13]. The Sellmeier dispersion is the sum of a set of Lorentzians in frequency with poles q j on the real axis. The starting point for this derivation is Eq. (10), the recursive solution for the perturbed problem derived in Sec. II.
In this section, we consider the RP and perturbation, with j numbering the resonances at frequencies Ω j having oscillator strengthsσ j . We introduce an effective resonance wave vectorq j asq 2 j = Ω 2 j /c 2 − p 2 and re-write Eq. (21) as, In the Appendix A I use my method of [3], which was first adapted for full dispersion of 3D nano-particles by E. A. Muljarov in Ref. [13], to derive for non-vanishinĝ σ j the sum rule which allows us to write the Green's function aŝ which has the useful (k ±q j ) in the numerator for reducing the order of the perturbation matrix problem through cancellation with perturbation denominators which would otherwise have lead to high order polynomial eigen problems.
We now consider a perturbation ∆ε ω (ṙ) of the RP inside the layer |ṙ| < a. Similar to the previous section we use the Eq. (6) to solve the perturbed problem, where we take ∆ε ω (ṙ) of the same form as Eq. (22). Using both forms of the Green's function [13], Eq. (9) and Eq. (24) in Eq. (25), yields which results in the following relationship between unperturbed and perturbed modes The perturbed mode E(ṙ) satisfies Eq. (4) withε ω (ṙ) replaced byε ω (ṙ) + ∆ε ω (ṙ) and outgoing boundary conditions with the wave vector k. In the interior region |ṙ| < a which contains the perturbation, the perturbed RSs E ν of wavenumbers κ ν can be expanded into the unperturbed ones, exploiting the completeness of the latter which follows from Eq. (A6) (see also Appendix E), Substituting this expansion into Eq. (27) and equating coefficients at the same basis functions E n results in the matrix equation where and is the matrix of the perturbation in the basis of unperturbed RSs. Introducing the abbreviation which is of second order in κ ν . We can write this matrix problem compactly as To solve this matrix problem for a basis of size N , I follow Ref. [14] and use the first companion linearization, defining where I is the N -by-N identity matrix. with the corresponding vector We solve R(κ ν )z = 0 using the generalized eigenvalue solver from the numerical algorithms group (NAG) C++ library. We then take the first N components of z as the eigenvector b ν .

V. POSSIBLE EXPLANATION FOR THE DIFFERENCE IN ORDER BETWEEN THE TWO METHODS
Here I go beyond the algebraic mathematics presented so far and discuss the fundamental topological differences between the two RSE approaches that I have developed.
The RSE perturbation theory presented here is an injective continuous mapping function except in the neighbourhood of the finite number of poles such that, is mapped tô in the following way where [ε j ω ω 2 ] (poles) and [∆ε j ω ω 2 ] (poles) have poles in the same position in frequency space while [ε ω ω 2 ] (reg) and [ε ω ω 2 + ∆ε ω ω 2 ] (reg) are regular functions.
The mapping in Eq. (38) is between two spaces which are topologically equivalent to non-dispersive spaces. Therefore, the mapping problem is mathematically equivalent to non-dispersive RSE perturbation theory and so the order of the eigenvalue problem is not increased upon that of the non-dispersive problem.
However, in the case of mapping by Eq. (39) [ε j ω ω 2 ] (poles) contains poles in frequency space and so cannot be continuously deformed into [ε ω ω 2 ] (reg) hence the map given by Eq. (39) is not equivalent to a non-dispersive mapping and so the order of the RSE eigenvalue problem is increased upon that of the nondispersive problem.
Formally A and B are topologically equivalent if there is a homeomorphism mapping orbits of A to orbits of B homeomorphically, and preserving orientation of the orbits.

VI. APPLICATION TO A PLANAR WAVEGUIDE
In this section I discuss the application of the Sellmeier waveguide RSE to an effectively 1D planar waveguide system, translationally invariant in the Cartesian z and y directions, described by a scalar RP, i.e.,ε ω (x) =1ε ω (x), ∆ε ω (x) =1∆ε ω (x). As unperturbed system I use a homogeneous planar wave guide of half width a, so that A. Unperturbed resonant states The solutions of Eq. (4), which satisfy the outgoingwave boundary conditions in TE polarization take the form [7] where the eigenvalues k n satisfy the secular equation with I use here an integer index n which takes even (odd) values for symmetric (anti-symmetric) RSs, respectively. The normalization constants A n and B n are found from the continuity of E n across the boundaries and the normalization condition found in Appendix F. In order to arrive at the normalization condition I consider an effectively 2D system and I takeĒ(ṙ, k) as being the analytic continuation into k space of E n (ṙ) and A being an arbitrary cross-section of the translational invariant waveguide to arrive at (see Appendix F) n dṙ outside the system in free space Maxwell's equation simplifies (see Appendixes F and G), therefore by extending A outside the system and denoting its circumference L A I can write, I made the necessary assumption thatε ω is a (real) symmetric matrix or a scalar so that (defined to be in this case) E ·ε ωn E n = E n ·ε ωn E and non dispersive at high frequencies (see Appendix E and F).
Various schemes exist to evaluate the line integral limit in Eq. (47) such as analytic methods in Ref. [9,11] or numerically extending the surface into a non-reflecting, absorbing, perfectly matched layer where it vanishes. Hence, I derive from Eq. (47) the relevant normalization condition for the planar waveguide systems, which I will use for the numerical demonstration, Here in Eq. (48) the first integral is taken over an arbitrary simply connected line enclosing the inhomogeneity of the system and the center of the coordinates used, and the second term is evaluated at its end points. The coefficients in Eq. (43) take the form and M n = ǫ ω (−1) n 2 sin(2q n a) q n + 4a , where ω 2 n /c 2 = k 2 n + p 2 . Around each pole of ε ω the secular equation Eq. (44) has a countable infinite number of solutions, each one creating a RS. An analytic approximation to these solutions is given in Appendix B, which are used as starting values for the numerical solution of Eq. (44) to determine these RSs closely spaced in the complex frequency plane.

B. BK7 Sellmeier dispersion
In our numerical examples I use dispersion parameters describing the common borosilicate glass SCHOTT BK7. Its refractive index n r in the optical frequency range is well described by the Sellmeier expression The resulting RP is real and is shown in Fig. 1 The deviations of this fit over the fitted range are around 10 −4 , as shown in Fig. 1.

C. Unperturbed and perturbed system
The unperturbed system I consider in this work as example is a planar waveguide of width 2a = 2 µm. I consider a size perturbation, narrowing the waveguide by 10 percent, as shown in Fig. 2 i.e., with a wave vector in the medium below a suitably chosen maximum k max (N ), this follows the approach of Ref. [13]. The motivation for the choice of basis selection criteria is the analogy between the RSE and a Fourier expansion. As I increase k n √ ǫ ωn , I increase the number of oscillations in the field as can be seen from Eq. (43). These increasingly oscillating fields go into the basis and similar to Fourier series, increase the resolution of the composite field generated by the expansion.

D. Single Sellmeier resonance at zero frequency
Here I show results of the SRZ yielding the linear eigenvalue problem Eq. (19). For the unperturbed and perturbed system I use a RP given by the SRZ Eq. (55). The refractive index of the unperturbed and perturbed system is compared in Fig. 3 with the Sellmeier expression Eq. (53). I can see that for the chosen waveguide width, the fitted wavelength range corresponds to 3.6 < aω/c < 5. The RS frequencies are given in Fig. 3(b) for the unperturbed system ω n = c k 2 n + p 2 and for the perturbed system ̟ ν = c κ 2 ν + p 2 . To compare ̟ ν   Fig. 3(c). I find that as I increase N , ∆ decreases proportional to N −3 , similar to the findings for the nondispersive RSE [8][9][10], and values in the 10 −6 range are reached for N = 200. This is actually smaller than the relative error due to the SRZ approximation of the Sellmeier dispersion (see Fig. 1).
The evolution of the perturbed RSs wavenumbers ̟ ν with the in-plane wave vector p is shown in Fig. 4. I can distinguish [7] the waveguide (WG) and anti-waveguide (AWG) modes, which have real ̟ ν , and the Fabry-Pérot (FP) modes which have a finite imaginary part representing their losses. Increasing p, the FP modes split into WG and AWG mode at the bifurcation point at which pc = ̟ ν , at which k = 0 i.e. at grazing incidence of the external field. The relative error ∆ of ̟ ν is given in Fig. 5 as function of p. I can see that ∆ has generally a weak dependence on p, except close to the bifurcation points, where the error of the FP and AWG mode is significantly increased. Under closer examination I see further smaller discontinuities as function of p, which correspond to a change of the basis states included according to Eq. (56) for fixed N = 50. These results indicates that the RSE is able to reproduce all relevant RSs with a good accuracy.

E. Sellmeier Dispersion
Here I show results of the Sellmeier RSE yielding the quadratic eigenvalue problem Eq. (33).
The RS frequencies are given in Fig. 6(b) for the unperturbed system ω n = c k 2 n + p 2 and for the perturbed system ̟ ν = c κ 2 ν + p 2 . The relative error ∆ is given in Fig. 6(c). Also here I find that as we increase N , ∆ decreases proportional to N −3 , and values in the 10 −5 range are reached for N = 800.
The evolution of the perturbed RSs wavenumbers ̟ ν with the in-plane wave vector p is shown in Fig. 7 including the relative error ∆, and in Fig. 4 including the imaginary part ℑ(̟ ν ).
I can see that ∆ has generally a weak dependence on p, except for the bifurcation points, at which a Fabry-Perot mode splits into a waveguide and anti-waveguide mode.
These results indicate that with full Sellmeier dispersion the RSE is able to reproduce all relevant RSs with a good accuracy.

F. Performance Comparison
Here I compare the computational complexity of the SRZ RSE and the Sellmeier RSE. The corresponding eigenvalue problems Eq. (19) and Eq. (33) show that for the same number of basis states N , the SRZ RSE is a generalized eigenvalue problem of size N × N , while the Sellmeier RSE has a size 2N ×2N due to the quadratic nature of Eq. (33). Also I see from Fig. 8 that for pa = 5 the resonances closely associated to the Sellmeierq j poles, those eigen-frequencies found using the solutions to Eq. (B4) as a starting point for the Newton-Raphson search, contribute approximately 10% to the basis size thereby increasing the numerical complexity and reducing the efficiency.
In Fig. 9 I show a comparison between the average relative error in the perturbed resonances for pa = 5 found in the range 3 < aω < 5 versus the number of seconds required to diagonalise the perturbation eigenvalue problems using an Intel Core 2 Duo, 6M Cache, 3.16GHz, 1333MHz FSB processor connected to 4GB of RAM and using the NAG generalized eigenvalue problem solver software.
For the case of SRZ dispersion the relative error reaches the limit of about 5 × 10 −5 due to the RP approximation at N ≈ 70. Once the perturbation method using the full Sellmeier dispersion reaches this relative error I see that upon adding further basis states more noise in the relative error plot is generated and that the average relative error changes little. Fig. 9 suggests that the SRZ RSE is several orders of magnitude more efficient than the full Sellmeier RSE when considering the resonances over a small range of frequencies such as the small range used for optical communications.

VII. RSE BORN APPROXIMATION FOR PLANAR WAVEGUIDES
In this section I derive the RSE Born approximation for planar waveguide, the equivalent theory for effectively 2D systems is given in Appendix G.
In effectively one dimension Eq. (2) becomes in free space (taking c = 1), However, ∇ 2 θ p = −p 2 θ p , therefore  9: Comparison of the average relative error of the wave guide modes occurring in the range 3 < aω < 5 calculated using the two different perturbation schemes in this paper versus time required to diagonalise the perturbation matrix using an Intel Core 2 Duo, 6M Cache, 3.16GHz, 1333MHz FSB processor connected to 4GB of RAM and using the NAG generalised eigenvalue problem solver software Hence the free space GF equation is, which has the solution, The systems associated with G f s k and G k of Eq. (6) are related by the Dyson Equations perturbing back and forth with ∆ε ω (x) = ε ω (x) − 1 similar to Ref. [2], Combining Eq. (61) and Eq. (62) it is obtained similar to Ref. [2] Hence, in one dimension the RSE Born approximation can be greatly simplified by using Eq. (60) and the spectral GF Eq. (E7) in Eq. (63) to arrive at, where A n is defined as the Fourier transform, Interestingly in one dimension I do not require the far field approximation to make the simplification of the Green's function required to bring the RSE Born approximation to the form of Eq. (G9). Hence, in one dimension the RSE Born Approximation is valid everywhere outside of the slab and not just in the far field. I note that fast Fourier transforms are available for use upon Eq. (65).
I demonstrate the computational accuracy of the RSE Born approximation in Fig. 10 where I calculate the transmission defined as For comparison the analytic GF is found by solving Maxwell's wave equation in one dimension with a source of plane waves while making use of Maxwell's boundary conditions. The system treated is the unperturbed system in Sec. VI D with pa = 5. From Fig. 10 we can see that unlike the standard Born approximation the RSE Born approximation is valid over an arbitrarily wide range of k depending only on the basis size N used. Furthermore we see that as the basis size increases the RSE Born approximation converges to the exact solution. The absolute error in the RSE Born approximation is approximately reduced by an order of magnitude each time the basis size is doubled. Absolute errors of 10 −8 − 10 −5 are seen in the k range shown for basis size N = 401.

VIII. SUMMARY
In this work I have extended the RSE to media having a simple dispersion linear in wavelength squared. This dispersion has a single pole at zero frequency and does not introduce an additional dynamic degree of freedom as it would be the case for a more general Sellmeier model of the material response. This property allows to keep the simplicity of the RSE formulation, therefore retaining the advantage of the RSE in computational efficiency discussed in Ref. 9.
Furthermore, in this work I have also extended the RSE to waveguides obeying the Sellmeier equation for glasses. In order to do this I have reduced the order of the eigenvalue problem to second order by using sum rules as was done in [13].
To make use of the RSs generated by the RSE I derived the RSE Born approximation for effectively 1D and 2D waveguides.
My conclusion is that the simple dispersion treatment is more efficient than using the full Sellmeier dispersion. This is because the dispersion only has to be correctly reproduced over a narrow part of the optical frequency range and therefore it is inefficient to include the full and at some frequencies unphysical Sellmeier dispersion.
ǫ ω → ∞ I find from Eq. (45) that q n → √ ǫ ω ω n /c, so that Eq. (44) is given by −e iqna + (−1) n e −iqna = 0, and thus sin( √ ǫ ω ω n a/c) = 0 for even n cos( √ ǫ ω ω n a/c) = 0 for odd n (B1) In the limit k n → ±q j to the poles of the RP given in Eq. (22) I find and solutions of Eq. (B1) are given by The solutions of Eq. (B2) with k n close to ±q j are given by the analytic solution of the quadratic formula (σ j − 2γ n )k 2 n ± 2γ nqj k n + σ j p 2 = 0 (B4) I use the solutions of Eq. (B4) as starting points for the Newton-Raphson search for RSs. The RSs of the unperturbed system (see Sec. VI C) with Sellmeier dispersion Eq. (53) are given in Fig. 11. I note that close to the lowest resonance frequency of the dispersion at Ω j a/c ≈ 0.6174, there are a large number of RSs approaching the pole from smaller frequencies. These RSs arise because the refractive index is diverging to positive infinity on the low frequency side, allowing for a countable infinite number of WG and AWG modes to form. σ j is negative for all the resonances in Eq. (53). For planar systems L is given in Ref. 7 as For effectively two-dimensional systems with one direction of translational invariance L can be calculated as follows. Due to translational invariance in the direction z, the direction of propagation I can write the full field as,Ê whereṙ give the coordinates in the plane perpendicular to z. Therefore using standard vector identities I can write I make use of the following identities, where e z is the unit vector in the z direction so as to simplify Eq. (C3) to Hence I see that, So L is a linear operator independent of the RP and z as required. Therefore the eigenvalue problems derived in Sec. III and Sec. IV are valid for planar and cylindrical wave guides. Also the derivation of L demonstrates I can use separation of variables in Eq. (2) to arrive at Eq. (C2).
In the case of a GF made up of degenerate modes the proof of Eq. (D13) is modified by making use of orthogonality of the degenerate modes to choose D(ṙ) such that, for m = n and where state m is degenerate with n.
Hence I obtain Appendix E: Derivation of sum rule and completeness Here I exactly repeat my derivation of the sum rule and completeness of the GF which I made for Ref. [3] but in more detail.
In order to simplify the RSE Born approximation we require an appropriate spectral form of the GF which is different from the one already proven in Appendix D. To obtain this correct form I start with the GF valid inside the scatterer only, gives for k → ∞, since throughout the derivation in this appendix we are considering the limit where k → ∞ at whichε k (ṙ) = ε(ṙ), i.e. the system is non-dispersive at high frequencies.
Convoluting Eq. (E3) with arbitrary finite functions D which are only non-zero inside the resonator, gives (E4) and assuming the series are convergent as k → ∞ we get, and since this is true for all D it follows that Combining Eq. (E1) and Eq. (E6) yieldŝ Combining Eq. (E3) and Eq. (E6) leads to the closure relationε which expresses the completeness of the RSs, so that any function can be written as a superposition of RSs. If in the perturbed system some of the series are not convergent or are instead conditionally convergent then we will not arrive at the sum rule and completeness, in which case I expect that the RSE Born approximation will still give convergence to the exact solution but only if a valid spectral Green's function is used, such as Eq. (E1).
Appendix F: Derivation of the correct normalisation for effectively 2D waveguides Here I almost exactly repeat my derivations which I contributed to Ref. [9] using exactly the same method as for Maxwell's equations in order to prove in this section that the spectral representation, only valid inside the waveguide (similar to Ref. [3] for nano-particles), rigorously derived in Appendix C, D, and Ê G k (ṙ,ṙ ′ ) = n E n (ṙ) ⊗ E n (ṙ ′ ) 2k(k − k n ) , leads to the RS normalization condition Eq. (F8) for general waveguide equations. To do so, I consider an analytic continuation E(ṙ, k) of the vector wave function E n (ṙ) around the point k = k n in the complex k-plane (k n is the wavenumber of the given RS). I select the analytic continuation such that it satisfies the outgoing wave boundary condition and my waveguide equation (taking c = 1) −LE(ṙ, k)+ε(ṙ, ω)(k 2 +p 2 )E(ṙ, k) = (k 2 −k 2 n )σ(ṙ) (F2) with an arbitrary source term. The source σ(ṙ) has to be zero outside the cross-section of the inhomogeneity ofε ω (ṙ) for the field E(ṙ, k) to satisfy the outgoing wave boundary condition. It also has to be non-zero somewhere inside that cross-section, as otherwise E(ṙ, k) would be identical to E n (ṙ). It is further require that σ(ṙ) is normalized according to A E n (ṙ) · σ(ṙ) dṙ = 1 + δ kn,0 , The integral in Eq. (F3) is taken over an arbitrary crosssection A which includes all system inhomogeneity of ε ω (ṙ). I do not derive Eq. (F3), it is simply a convenient condition to put on the otherwise arbitrary spatial dependence of the source which lies only inside the waveguide scatterer. If I had made the condition anything else (which in my earliest proof I did) then algebra and cancellation would lead us back to the same result, however with more mathematical complexity and operations. The condition make the mathematics easier because it causes E → E n exactly and without proportionality constant appearing, equation (F3) ensures that the analytic continuation reproduces E n (ṙ) in the limit k → k n . Solving Eq. (F2) with the help of the GF and using its spectral representation Eq. (F1), I find: same approach can be taken in three dimensions to aid the determination of resonator structure. The reason Eq. (G16) takes the form it does can be seen more clearly by substituting the GF of Eq. (G12) into the first line of Eq. (F4) and taking the limit k → k n , while following the arguments in that section that, lim k→kn E(ṙ, k) = E n (ṙ) .
The 3D equivalent [2,3] of Eq. (G16) one day might be valuable information for solving the inverse emission problem from such things as black hole gravitational wave emitters, i.e. calculations of the poten-tial from the emission (decay) via fast inverse Fourier transform methods upon the set of A n (rk n ), particularly if the potentials of interest are rotating about a fixed axis so we know their orientation to some extent such as occurs for decaying magnetic nuclei as part of a non-magnetic crystaline compound placed inside a NMR (nuclear-magnetic-resonance) machine. Because k n are discrete values A n (rk n ) only give angular information when inverse Fourier transformed and so the inverse Fourier methods might have to be used self-consistently in conjuncture with the RSE perturbation theory and the values of k n . This is a highly speculative aside and might be a possible topic for future research.