GB2544601A - A method and apparatus for determining the electromagnetic resonances of a system - Google Patents

Abstract

A method of obtaining resonance information characteristic of an optical system including at least one dispersive material having a dispersion value is disclosed. The method determines the resonant information of a system and outputs this information via an output module when a threshold range is reached on comparison of sensing optical parameters with at least one sensed optical parameter. The resonance information is determined by sensing and storing an optical parameter of the system so as to determine a provisional resonant state. A spectral representation is provided based on the provisional resonant states. A matrix of a perturbation is provided that includes the dispersion value based upon the spectral representation and the provisional resonant states and/or the sensed optical parameter. A linearization method is used to solve a matrix eigenvalue problem based on the matrix elements and the resonance information of the system is generated.

Description

A method and apparatus for determining the electromagnetic resonances of a system

This invention relates to a method and apparatus for determining the electromagnetic resonances of a system, in particular for a system that comprises dispersive materials.

It is known that optical systems can be characterised by their resonance. Resonances are a cornerstone of physics and the concept of resonant states (RSs) is a mathematically rigorous way of treating them. Formally, RSs are the optical eigenmodes of the system i.e. the eigen-solutions of Maxwell’s wave equation, which satisfy the outgoing wave boundary conditions. In optical systems the RS eigenfrequencies ωη are generally complex, which physically reflects the fact that the energy leaks out of the system. The real part Re(oon) gives the position of the resonance, while the imaginary part ΙπΊ(ωη) gives its half width at half maximum, also determining the quality factor of the resonance as Qn = \Re(o)n)/2Ιτη(ωη)\,

It is known to use computational methods for calculating optical systems in electrodynamics. For example, the finite difference in time domain (FDTD) and the finite element method (FEM) are widely implemented.

In FDTD, RSs can be found by fitting the calculated time evolution by a sum of RSs. Only RSs which have been excited in the simulation are visible, and the fitting procedure does not uniquely determine the number of RSs.

In contrast, the FEM instead determines RSs one by one, iteratively solving a non-linear equation with unknown analytics. Therefore, using the FEM makes it impractical, if not impossible to verify that all RSs within a complex frequency range have been found.

When using the FEM and FDTD, spurious solutions come from the spatial discretisation and the non-ideality of the PMLs which can be perfect only in 1 dimensional problems. The other known methods including FEM and FDTD use an artificial discretisation in space or in time/frequency domain which limit their accuracy.

The problems associated with calculating optical systems can be avoided by using a known rigorous perturbative method. This resonant state expansion (RSE) method treats perturbations of open optical systems of arbitrary strength and shape. This method has been shown to be advantageous over other established computational methods in electrodynamics, such as FDTD and FEM in terms of accuracy and efficiency. This is because the RSE is numerically exact, i.e. its accuracy is limited by the basis truncation only. This is in contrast to FDTD, FEM and other popular techniques that use spatial grids and perfectly matched layers. Furthermore the RSE produces, in a single calculation, all RSs originating from the basis states. Therefore all RSs, in a frequency range of interest may be determined. A further benefit of the RSE is that it reduces the solution of Maxwell’s wave equation to a discrete linear eigenvalue problem, which guarantees that no spurious solutions occur within a frequency range of interest. The RSE uses a natural discretisation in the frequency domain provided by the RSs of the original system.

The RSE has been formulated for different dimensionality and geometries. However, the RSE known to date requires non-dispersive materials.

At the same time, all realistic systems, even dielectrics such as glass, have a significant frequency dispersion of their electromagnetic properties.

Therefore, embodiments of the present invention are intended to address at least some of the above described problems and desires. In particular there is provided a method and apparatus for determining the resonance of a dispersive system that is accurate and efficient whilst retaining the desirable attributes of the known RSE.

According to a first aspect of the invention there is provided a method of obtaining resonance information characteristic of an optical system including at least one dispersive material having a dispersion value, the method comprising the steps of: a. sensing at least one optical parameter of the system; b. storing the parameter; c. determining provisional resonant states of the system; d. determining a spectral representation of the optical system using the provisional resonant states; e. generating matrix elements of a perturbation including the dispersion value based upon the determined provisional resonant states and the determined spectral representation and/or the at least one sensed optical parameter; f. using a linearization method to solve a matrix eigenvalue problem based on the matrix elements; g. generating resonance spectral information of the system and sensing further optical parameters based upon the resonance spectral information; h. comparing the further sensing optical parameters with the at least one sensed optical parameters ; and in the case that the sensing optical parameters equate to the sensed parameters to within a predefined threshold range i. determining the resonance information of the system; and j. outputting the resonance information via and output module.

In the case that the sensing optical parameters do not equate to the at least one sensed parameters to within the predefined threshold range, matrix elements of a further perturbation may be generated and may include the dispersion value based upon the determined provisional resonant states and the determined spectral representation and/or the at least one sensed optical parameter and repeating steps f to h.

The method may further comprise choosing an appropriate perturbation based upon the at least one sensed optical parameter.

The perturbation may be a predefined approximate perturbation value.

The resonance information may be determined when the modelled data coincides within a predefined threshold based upon the sensed data.

The linearization method may comprise a linear resonant state expansion which reduces Maxwell’s wave equation to a linear matrix eigenvalue problem.

The system may have at least one dimension of space that is unbounded.

The system may be homogenous.

The resonant states may be determined analytically using a predetermined algorithm.

The at least one sensed optical parameter may comprise one or more of temperature, pressure, stiffness, refractive index or electric field.

The method may use a dispersive system to determine basis states.

All of the resonances of the system in a desired frequency range may be generated.

The method may further comprise solving a Maxwell wave equation and reducing a solution to a discrete linear matrix eigenvalue problem.

The physical dispersion contribution of the matrix formed by the matrix elements of the perturbation may be provided by a dispersive relative permittivity tensor in the complex frequency plane.

The spectral representation may comprise a contribution from additional sum rules determined dependent upon the dispersive relative permittivity tensor, the sum being taken over all the resonant states required.

The additional sum rules may relate to the poles of the dispersive relative permittivity.

Using the sum rules may obtain additional representations of a Green’s Function for each pole in the dispersive relative permittivity. A linear matrix eigenvalue problem may be provided dependent upon the additional sum rules and spectral representation.

The spectral representation may use normalised resonant states.

The method may, further comprise solving a perturbed Maxwell’s wave equation for the given optical system wherein an unperturbed dispersive relative permittivity may be replaced by a perturbed dispersive relative permittivity.

The method may further comprise converting the perturbed wave equation into a matrix equation by expanding the perturbed resonant states into the basis of the unperturbed ones.

The method may use new spectral representations of the Green’s function, following from the sum rules.

The method may further comprise subsequently providing a linear matrix eigenvalue problem where the perturbation matrix represents the change of the dispersive relative permittivity for a physical dispersion.

The method may further comprise solving the linear matrix eigenvalue problem to generate information on the resonant states of the optical system.

The same pole frequencies of the dispersive relative permittivity may be required in the perturbed and unperturbed state.

The method may further comprise taking a limit to zero weight of the pole in the dispersion in the unperturbed system having no such pole present in its dispersion of permittivity such that the pole-related resonant states may have frequencies equal to the pole frequency and such that the refractive indices of the resonant states may have separate discrete values.

The method may use a non-dispersive system as basis.

Additional resonant states may be created due to the poles of the dispersion by the non-linearity of the resulting generalised eigenvalue problem. A finite number of poles may be determined.

The solution of Maxwell’s equation with the dispersive material described by poles of the relative permittivity may be expressed as a polynomial matrix equation.

The order of the polynomial may be given by 2J+1 with J pairs of Lorentz poles in the permittivity tensor of the perturbed system.

The method may further comprise using a linearization method to solve the polynomial eigenvalue problem and extending the basis of unperturbed resonant states by a factor of 2J+1.

In a further embodiment of the invention there is provided a method of generating the optimised values of an optical system using the before-mentioned method in a double step perturbation scheme, wherein a first step perturbation and a second step perturbation may be implemented.

The first step perturbation may involve determining the first resonant states generated from the first step perturbation of the system with a predetermined analytically solvable simple system as an unperturbed system providing the first resonant states.

The method may further comprise determining the second resonant states of a modified system using the first resonant states as basis.

The method may further comprise determining a merit function from the second resonant states.

The method may further comprise subsequently varying the geometry and the material of the modified system and repeating the calculation of the second resonant states generated using the second step perturbation so as to maximise the merit function using a predetermined iterative maximisation algorithm.

The method may further comprise comparing the value of second resonant state with a target value and on reaching the target value, stopping the iterative maximisation algorithm.

Subsequently the optical quantities from the second resonant states may be calculated.

The optical quantities may be at least one or more of the scattering, absorption or extinction cross-section and may further comprise determining a scattering matrix from input to output states.

Drude or Drude-Lorentz approximations may be applied.

The generalised Drude-Lorentz model may be used for the permittivity tensor, including arbitrary finite number of Lorentz poles.

The basis selection of resonant states may be compared to a predetermined criteria value.

Only states with basis satisfying the condition |πΓ(ωη)ωη| < o)max may be utilised.

In a further embodiment of the invention there is provided an apparatus configured to obtain resonance information characteristic of an optical system including at least one dispersive material, the apparatus including: a sensor for sensing information relating to at least one optical parameter of the system; a device configured to receive the at least one optical parameter; a storing means located within the device for storing the information relating to the optical parameter; a processor configured to generate resonance information in accordance with any preceding claim; and an output module for generating a signal indicative of the resonance information determined by the processor.

The output module may be one or more of a printer, monitor or speakers.

The sensor may be in communication with the device to enable the transfer of sensed information to the memory of the device.

The apparatus may further comprise the sensor being one or more of a thermometer, optical power meter or field sensor.

Whilst the invention has been described above it extends to any inventive combination of the features set out above, or in the following description, drawings or claims. For example, any features described in relation to any one aspect of the invention is understood to be disclosed also in relation to any other aspect of the invention.

The invention will now be described, by way of example only, with reference to the accompanying drawings, in which:-

Figure 1a is a real part of the refractive index ηΓ(ω) of gold;

Figure 1 b is an imaginary part of the refractive index ηΓ(ω) of gold;

Figure 2a shows the RS energies Άωη of the unperturbed system and perturbed system (gold to sand);

Figure 2b shows the relative difference between the RSE and exact eigenenergies, for different values of ioma3e as given (gold to sand);

Figure 3a shows the RS energies ho>nof the unperturbed system and perturbed system (sand to gold);

Figure 3b shows the relative error of the RS energies for linear and nonlinear RSE and for different values of Mmax (sand to gold);

Figure 4 shows a double step calculation and optimisation method;

Figure 5 shows a flow diagram the method according to the invention; and

Figure 6 shows a schematic of the apparatus according to the invention.

The invention is a generalization of the resonant state expansion to arbitrary dispersive materials. The RSs of the system of frequency ωη are solutions of Maxwell’s wave equation

(1)

To describe a physical dispersion of the materials, the dispersive relative permittivity (RP) tensor ε (r, ω) in the complex frequency plane ω is expressed as

(2) where ^(r) is the high-frequency value of the RP and Ω, are the resonance frequencies and poles of the RP determining the dispersion, with the weight tensors 0“;(r) corresponding to generalized conductivities of the medium at these resonances. The Lorentz reciprocity theorem requires that all tensors in Eq. (2) are symmetric, and the causality principle requires that ε*(τ,ω) = ε(τ,ω*) .

Therefore, for a physically relevant dispersion, each pole of the RP with a positive real part of Ωj has a partner at Ω_;· = -Ω;· with a*_j = aj, while poles with zero real part of Qj have real σ;· . As an example, the Ohm’s law conductivity corresponds to the sum Eq. (2) replaced by a single term with Ω0 = 0 and <r0(r) being the dc conductivity tensor. The Drude model of metals consists of two poles with Ω0 = 0, Ω-ι = -ίγ, and σ0(r) = The Drude-

Lorentz model introduces additional poles at ω = Ω] with j = ±2, ±3, . . . and complex conductivities σ;·.

To develop the dispersive RSE, we assume that the Green Function (GF) of Maxwell’s wave equation has an infinite countable number of simple poles in the complex frequency plane and therefore has the following spectral representation

(3) in which the sum is taken over all RSs, and ® denotes the dyadic product of vectors. The spectral representation Eq. (3) requires that the RSs are normalized according to

(4) where Fn = (r · V)En , V is an arbitrary simply connected volume with a boundary surface Sv enclosing the inhomogeneity of the system, and the derivative dids is taken along the outer surface normal. Note that for static modes (ωη = 0), the volume of integration can be extended to the full space and the surface term vanishes, since the electric field in such modes decay sufficiently fast outside the system. All other modes instead have the electric field exponentially growing outside the system, so that the normalization has to be evaluated for finite V.

Using this expression, together with the form of the RP in Eq. (2), provides the closure relation

(5) and new sum rules

(6) which are related to the poles of the RP. Using these sum rules we found additional spectral representations Gjuof the GF, one for each pole in the RP.

(7) where

(8)

The sum rule in Eq. (6) for Ω0 = 0 holds also without dispersion, and is due to the constant term, such as (r) in Eq. (2), which is always present in the RP.

These new additional sum rules and spectral representations enable the dispersive RSE to be written as a linear matrix eigenvalue problem as follows.

Firstly a perturbed Maxwell’s wave equation equivalent to Eq. (1) is solved, with the unperturbed RP έ(τ,ω) replaced by a perturbed one, έ(τ,ω) + Δε(Γ,ω). The electric field E(r)and the eigenfrequency ω of a perturbed RS can be determined by using the unperturbed GF in the different representations Eq. (7) for the corresponding terms of the RP, yielding

(9) (10)

This integral equation is then converted to a matrix equation by expanding the perturbed RS into the basis of unperturbed ones,

(11) then using the expansions Eq. (7) of the GF, and equating the coefficients at different basis functions En(r). The result is a linear eigenvalue problem

(12) with the perturbation matrix

(13)

This is the linear dispersive RSE. The perturbation matrix Vnm(w) represents the change Δε(Γ, ω) of the RP for any physical dispersion described by Eq. (2).

The above linear dispersive RSE requires the same pole frequencies of the RP in the perturbed and unperturbed system. In a general situation, the perturbation can introduce new pole frequencies, which increases the size of the basis of RSs owing to the additional countable infinite number of RSs approaching each pole of the RP. Such a situation, where poles of a given frequency have a finite weight in the perturbed system but zero weight in the unperturbed system, can be treated by considering the limit to zero weight in the unperturbed system. In this limit, the pole-related RSs have frequencies equal to the pole frequency, but refractive indices having separate discrete values.

An alternative approach which uses a non-dispersive system as basis, and creates the additional RSs due to the poles of the dispersion by the nonlinearity of the resulting generalized eigenvalue problem has also been developed. This approach is detailed here. Assuming that the unperturbed i(r) has no dispersion, the only valid sum rule in Eq. (6) is the one with Qj = 0 which provides only one alternative representation of the GF: (r,r') with Ι/Κη°(ω) = 1 in Eq. (7). Replacing &amp;ω by in Eq. (10) results in the non-linear eigenvalue problem

(14)

For a finite number of poles in the RP, Eq. (14) can be written as a polynomial matrix equation. The order of the polynomial is given by the number of poles at non- zero frequency in Eq. (2), which is 2J + 1 in the case of a Drude-Lorentz model with J pairs of Lorentz poles. Such a polynomial eigenvalue problem can be solved by linearization, extending the basis of unperturbed RSs by a factor of 2J + 1.

Figure 1a and 1b provides an example of the Drude (lower thicker line) and Drude- Lorentz with two pairs of CPs (thin upper line) models for the fit of the measured dispersion of gold using known fit parameters and used below for illustration of the dispersive RSE. The real part is shown in figure 1a and the imaginary part is shown in Figure 1 b. The error bars are the measured values. These models are used below for illustration of the dispersive RSE.

To illustrate the invention and evaluate its convergence, Figs. 2a, 2b, 3a and 3b show the transverse magnetic (TM) eigenmodes with I = 1 (I is the orbital number) of spheres made of a dispersive material (gold) or a non-dispersive material (sand, nr = 1.5) in vacuum, and perturbations which transform gold to sand in Figure 2a and 2b and sand to gold in Figure 3a and 3b.

Figure 2a, shows the RS energies Λωηοί the unperturbed system and the perturbed system for 1-1 TM modes and a sphere radius of R=200nm. The perturbed energies are calculated exactly (squares) and using the linear RSE Eq.(12) (crosses) for h(omax = 200eV. Figure 2b shows the relative difference between the RSE and exact eigenenergies, for different values of o)max as given. The refractive index of gold was modelled with the Drude-Lorentz model with two pairs of CPs and known parameters, while nr-1.5 was used for sand.

Figure 3a shows the real part of the refractive index ηΓ(ω) of gold as measured (green error bars), as approximated by the Drude Lorentz model with two pairs of CPs (thin red lines) and as approximated by the Dude model (thick blue lines). Figure 3b shows the imaginary part of the refractive index η^ω) of gold as measured (green error bars), as approximated by the Drude Lorentz model with two pairs of CP (thin red lines) and as approximated by the Dude model (thick blue lines).

The eigenmodes of the sand and gold spheres in vacuum were taken in the analytic form and normalized according to Eq. (4). The radius of the sphere R = 200 nm is chosen such that both Drude and Drude- Lorentz approximations of the gold dispersion shown in Figure 1 are valid for the frequency of the fundamental surface plasmon (SP) mode shown in Figs. 2 and 3 by arrows. A finite number N of RSs are selected for the RSE basis, including only RSs satisfying the condition\ητ(ωη)ωη\ < oimax. This excludes RSs having a wavevector in the medium above ω^χ/(.ι which is the case of large ωη , and of large |ηΓ(ωη)| close to the poles. This basis selection can be optimized in the future. The RSE results for the perturbed eigenmodes are compared with the analytic solutions, and the relative errors are shown in Figs. 2(b) and 3(c) for different umax as given, demonstrating a high accuracy given the strong perturbation. For the present geometry N is approximately proportional to ojmax, with N = 456 for ft6)max= 200 eV. The observed 1/^3 convergence to the exact solution is comparable to the non- dispersive RSE.

Going from gold to sand (Figure 2(a)) the RSE reproduces the RSs of the non-dispersive sand sphere, and additionally produces a number of quasidegenerate RSs at the Drude and Lorentz poles. These RSs are present since in the linear RSE the same poles frequencies of the RP are present before and after perturbation. Poles which have zero weight in the perturbed system, as in the example here, still lead to a series of RSs, with frequencies at the pole position, but corresponding to different refractive indices, as exemplified in the inset of Figure 3(a). For the sphere geometry, they can be calculated analytically by taking the limit of the pole weight to zero in the secular equation. A perturbation which creates a finite weight of the pole then lifts the degeneracy of these RSs as exemplified in Figure 3.

To illustrate the non-linear dispersive RSE Eq. (14), we show results for the Drude dispersion of the perturbed system, for which Eq. (14) is a quadratic matrix problem. For the same basis cut-off uw as used for the linear dispersive RSE, the energies of the Fabry-Perot RSs are reproduced with a similar accuracy, see Figure 3(b). However, the SP mode has about 2 orders of magnitude larger error and modes around the Drude pole are also having an orders of magnitude larger error as shown in the inset of Figure 3(b). This can be understood as a result of the additional basis states in the non-linear RSE coming from the matrix nonlinearity, so they are generally less suited to describe the RSs close to the poles compared to the pole RSs used in the basis of the linear RSE.

The dispersive RSE calculates the properties of the system of interest as perturbation of a known system. It is therefore particularly suited to be applied for optimization problems exploring the parameter space starting from a known configuration. For each dimensionality of the system, a simple system exists, such as the sphere for 3D systems, for which the RSs can be calculated analytically. To implement an optimization problem, a double step perturbation can be used. The first step determines the RSs of the system to be optimized (STO) using the RSE with a suited analytically solvable simple system (ASS) as unperturbed system, resulting in the RSs 1. During the optimization, the RSs 1 are used as basis for a RSE to calculate the RSs (RSs 2) of a modified system (MS). From the RSs 2, a merit function is calculated. For example the field strength for a specific frequency range over a specific spatial region, or the difference between a required and the actual temporal field response function. The geometry and material of the MS is then varied and the RSE calculation of RSs 2 is repeated in order to maximize the merit function using known iterative maximization algorithms. Once the target or maximum merit is reached, the iteration is stopped and any important derived quantities such as field distributions, scattering cross sections and others are calculated from the RSs 2. This concept is shown in a flowchart in Figure 4.

The dispersive RSE was implemented for a three-dimensional system, using a homogeneous sphere as the un-perturbed system. The materials considered were non-dispersive glass, and gold described either by a Drude-model or a Drude Lorenz model with 2 pairs of Lorenz oscillators. The method of generating the RS information is shown in Figure 5.

To find the analytical RSs, the secular equation was solved using a Newton-Raphson algorithm. The starting values for this algorithm close to the RP poles were determined using analytical approximations of the secular equation.

The matrix elements Vnm in Eq. (13) were calculated analytically, using the analytic form of the solutions of a sphere. The generalized matrix eigenvalue problem Eq.(12) was solved in a computer using algorithms implementing Gauss-Jordan elimination with pivoting and diagonalization of a non-symmetric complex matrix.

The nonlinear matrix eigenvalue problem Eq. (14) was solved by linearization, and the resulting linear matrix eigenvalue problem was solved as the generalized matrix eigenvalue problem stated above.

The calculations described above were implemented in the C++ programming language.

The increased speed and accuracy of the dispersive RSE can be implemented in an apparatus 1 as shown in Figure 6, which is a physical properties calculating apparatus comprising an input unit such as a sensor 2 for detecting the real world values, for example values of temperature, pressure, stiffness, refractive index, electric field etc. The data is then fed into a computational device 3 which comprises a memory 5 and a processor 4. The device also requires a mouse and a keyboard (not shown); and an output unit 6 such as a monitor and a printer, or speakers. The sensor values are used by the device 3 to determine the physical properties required as input parameters for the RSE. The output of the RSE provides important information for determining the optical properties of the system. This output may be provided as a signal.

The method is extremely versatile and can be applied in varying optical systems, for example nanoparticles can be characterised and optimised using their optical properties. The design and optimisation of waveguides and resonators can be achieved using this method. Also the design and optimisation of optical biosensors, including plasmonic and dielectric resonators can be realised using this method. There are many further applications of the method where it is required to provide parameters for optical systems containing dispersive materials.

The sensed data is used to compare with the model simulated data using the RSE. Then the model geometry is changed in order to produce a simulated data equal to the measured sensed data. The sensed data is therefore used as a target value. In this way, the geometry of the system can be determined from the sensed data. This is achieved by generating the resonant states of the modelled system through the eigenvalues and eigenvectors of the matrix problem which is dependent on the sensed values through the fitted and optimised parameters of the modelled system, since the simulated data is compared to the sensed data.

Alternatively, the sensed data can be used to inform the optimization algorithm on the sensitivity and noise in the data. The optimization loop can then optimize the sensitivity of the simulated system given the noise and selection of sensed data.

For avoidance of doubt, resonance information may refer to the resonant states of the optical system and any observable optical quantities of interest following from it. It is noted that converting Maxwell’s wave equation for a given optical system into a linear matrix eigenvalue problem on the basis of unperturbed resonant states of a simpler system, enables the system to determine all relevant eigenmodes. These eigenmodes enable the efficient calculation of the optical properties of the system, such as absorption, scattering or extinction. Therefore, other parameters may also be determined using the method of the invention.

Further, for the avoidance of doubt the spectral representation is a mathematical spectral representation.

Various modifications to the principles described above would suggest themselves to the skilled person. For example, for more general shapes (i.e. not a sphere), the matrix elements are instead calculated by numerical integration over the domain of perturbation, using well-known integration algorithms. Symmetries of the geometry are used to reduce the numerical complexity.

The dispersive RSE can be applied to 1, 2 or 3 dimensions, as required.

Claims (50)

1. A method of obtaining resonance information characteristic of an optical system including at least one dispersive material having a dispersion value, the method comprising the steps of: a. sensing at least one optical parameter of the system; b. storing the parameter; c. determining provisional resonant states of the system; d. determining a spectral representation of the optical system using the provisional resonant states; e. generating matrix elements of a perturbation including the dispersion value based upon the determined provisional resonant states and the determined spectral representation and/or the at least one sensed optical parameter; f. using a linearization method to solve a matrix eigenvalue problem based on the matrix elements; g. generating resonance spectral information of the system and sensing further optical parameters based upon the resonance spectral information; h. comparing the further sensing optical parameters with the at least one sensed optical parameters ; and in the case that the sensing optical parameters equate to the sensed parameters to within a predefined threshold range i. determining the resonance information of the system; and j. outputting the resonance information via an output module.

2. A method according to claim 1, wherein in the case that the sensing optical parameters do not equate to the at least one sensed parameters to within the predefined threshold range, generating matrix elements of a further perturbation including the dispersion value based upon the determined provisional resonant states and the determined spectral representation and/or the at least one sensed optical parameter and repeating steps f to h.

3. A method according to claim 1 or claim 2, further comprising choosing an appropriate perturbation based upon the at least one sensed optical parameter.

4. A method according to any preceding claim, wherein the perturbation is a predefined approximate perturbation value.

5. A method according to any preceding claim, wherein the resonance information is determined when the modelled data coincides within a predefined threshold based upon the sensed data.

6. A method according to any preceding claim, wherein the linearization method comprises a linear resonant state expansion which reduces Maxwell’s wave equation to a linear matrix eigenvalue problem.

7. A method according to any preceding claim, wherein the system has at least one dimension of space that is unbounded.

8. A method according to any preceding claim, wherein the system is homogenous.

9. A method according to any preceding claim, wherein the resonant states are determined analytically using a predetermined algorithm.

10. A method according to any preceding claim, wherein the at least one optical parameter comprises one or more of temperature, pressure, stiffness, refractive index or electric field.

11. A method according to any preceding claim, wherein the method uses a dispersive system to determine basis states.

12. A method according to any preceding claim, wherein all of the resonances of the system in a desired frequency range are generated.

13. A method according to any preceding claim, wherein the physical dispersion contribution of the matrix formed by the matrix elements of the perturbation is provided by a dispersive relative permittivity tensor in the complex frequency plane.

14. A method according to any preceding claim, wherein the spectral representation comprises a contribution from additional sum rules determined dependent upon the dispersive relative permittivity tensor, the sum being taken over all the resonant states required.

15. A method according to claim 14, wherein the additional sum rules relate to the poles of the dispersive relative permittivity.

16. A method according to claim 14 or 15, wherein using the sum rules obtains additional representations of a Green’s Function for each pole in the dispersive relative permittivity.

17. A method according to any of claims 14 to 16, wherein a linear matrix eigenvalue problem is provided dependent upon the additional sum rules and spectral representation.

18. A method according to any preceding claim, wherein the spectral representation uses normalised resonant states.

19. A method according to any preceding claim, further comprising solving a perturbed Maxwell’s wave equation for the given optical system wherein an unperturbed dispersive relative permittivity is replaced by a perturbed dispersive relative permittivity.

20. A method according to claim 19, further comprising converting the perturbed wave equation into a matrix equation by expanding the perturbed resonant states into the basis of the unperturbed ones.

21. A method according to claim 20 when dependent on claim 14 further comprising using new spectral representations of the Green’s function, based upon the sum rules.

22. A method according to claim 21, further comprising subsequently providing a linear matrix eigenvalue problem where the perturbation matrix represents the change of the dispersive relative permittivity for a physical dispersion.

23. A method according to claim 22, further comprising solving the linear matrix eigenvalue problem to generate information on the resonant states of the optical system.

24. A method according to claim 23, wherein the same pole frequencies of the dispersive relative permittivity are required in the perturbed and unperturbed state.

25. A method according to claim 24, wherein the method further comprises taking a limit to zero weight of the pole in the dispersion in the unperturbed system having no such pole present in its dispersion of permittivity such that the pole-related resonant states have frequencies equal to the pole frequency and such that the refractive indices of the resonant states have separate discrete values.

26. A method according to claim 1, wherein the method uses a non-dispersive system as basis.

27. A method according to claim 26 wherein additional resonant states are created due to the poles of the dispersion by the non-linearity of the resulting generalised eigenvalue problem.

28. A method according to claim 27, wherein a finite number of poles are determined.

29. A method according to claim 26 or 28, wherein the solution of Maxwell’s equation with the dispersive material described by poles of the relative permittivity is expressed as a polynomial matrix equation.

30. A method according to claim 29, wherein the order of the polynomial is given by 2J+1 with J pairs of Lorentz poles in the permittivity tensor of the perturbed system.

31. A method according to claim 30, further comprising using a linearization method to solve the polynomial eigenvalue problem and extending the basis of unperturbed resonant states by a factor of 2J+1.

32. A method of generating the optimised values of an optical system using the method of any preceding claim in a double step perturbation scheme, wherein a first step perturbation and a second step perturbation is implemented.

33. A method according to claim 32, wherein the first step perturbation involves determining the first resonant states generated from the first step perturbation of the system with a predetermined analytically solvable simple system as an unperturbed system providing the first resonant states.

34. A method according to claim 33, further comprising determining the second resonant states of a modified system using the first resonant states as basis.

35. A method according to claim 34, further comprising determining a merit function from the second resonant states.

36. A method according to claim 35, further comprising subsequently varying the geometry and the material of the modified system and repeating the calculation of the second resonant states generated using the second step perturbation so as to maximise the merit function using a predetermined iterative maximisation algorithm.

37. A method according to claim 36, further comprising comparing the value of second resonant state with a target value and on reaching the target value, stopping the iterative maximisation algorithm.

38. A method according to claim 37, further comprising subsequently calculating the optical quantities from the second resonant states.

39. A method according to claim 38, wherein the optical quantities are at least one or more of the scattering, absorption or extinction cross-section.

40. A method according to claim 39, comprising determining a scattering matrix from input to output states.

41. A method according to any preceding claim, wherein Drude or Drude-Lorentz approximations are applied.

42. A method according to any preceding claim, wherein the generalised Drude-Lorentz model is used for the permittivity tensor, including arbitrary finite number of Lorentz poles.

43. A method according to any preceding claim, wherein the basis selection of resonant states are compared to a predetermined criteria value.

44. A method according to any preceding claim, wherein only states with basis satisfying the condition |πΓ(ωη)ωη| < 6)ma3e are utilised.

45. An apparatus configured to obtain resonance information characteristic of an optical system including at least one dispersive material, the apparatus including: a sensor for sensing information relating to at least one optical parameter of the system; a device configured to receive the at least one optical parameter; a storing means located within the device for storing the information relating to the optical parameter; a processor configured to generate resonance information in accordance with any preceding claim; and an output module for generating a signal indicative of the resonance information determined by the processor.

46. An apparatus according to claim 45, wherein the output module being one or more of a printer, monitor or speakers.

47. An apparatus according to claim 45 or 46, wherein the sensor is in communication with the device to enable the transfer of sensed information to the memory of the device.

48. An apparatus according to any of claims 45 to 47 further comprising the sensor being one or more of a thermometer, optical power meter or field sensor.

49. An assembly as hereinbefore described in reference to the accompanying drawings.

50. A method as hereinbefore described in reference to the accompanying drawings.