Theory of Wood’s Anomalies

Abstract

Discovered by Wood in 1902, grating anomalies have fascinated specialists of optics for more than one century. Long after the first interpretation given by Rayleigh, Fano has suggested that the origin of anomalies could be found in the excitation of surface waves. This chapter describes the quantitative phenomenological theory of Wood’s anomalies developed in the 1970s, based on the interpretation given by Fano and on the macroscopic laws of electromagnetics. This theory leads to a formula giving the efficiency of gratings in the region of anomaly and predicts the phenomenon of total absorption of light by a grating.

Appendix 1: Electromagnetic modelling in Two Dimensions

We consider an interface $\Sigma$ separating two homogeneous regions R$_{1}$ and R$_{2}$. The Maxwell equations write:

$$\begin{aligned} \nabla \times \mathbf{ E}&= -\partial \mathbf{ B} /\partial t,\end{aligned}$$

(2.56)

$$\begin{aligned} \nabla \times \mathbf{ H}&= \mathbf{ j} _c +\partial \mathbf{ D} /\partial t, \end{aligned}$$

(2.57)

where E and H are the electric and magnetic fields, D and B are the electric and magnetic inductions, and $\mathbf{ j} _{c}$ is the conduction current density. In the harmonic regime, the fields and currents have a sinusoidal behaviour in time. A function of space and time $f\left({\mathbf{ r} ,t} \right)$ with sinusoidal behaviour in time can be written as:

$$\begin{aligned} f\left({\mathbf{ r} ,t} \right)=a\left(\mathbf{ r} \right)\cos \left({\omega t-\varphi \left(\mathbf{ r} \right)} \right), \end{aligned}$$

(2.58)

with a and $\varphi$ being real functions of space called amplitude and phase, $\omega$ being the frequency. Such a function is classically represented by its complex amplitude $\tilde{f}\left(\mathbf{ r} \right)$ independent of time, and Eq. (2.58) is re-written in the form:

$$\begin{aligned} f({\mathbf{ r} },t)={\text{ Re}}\left\{ a({\mathbf{ r} })\exp \left({-i\omega t+i\varphi ({\mathbf{ r} })} \right)\right\} ={\text{ Re}}\left\{ a({\mathbf{ r} })\exp \left(i\varphi ({\mathbf{ r} })\right)\exp ({-i\omega t} )\right\} . \end{aligned}$$

(2.59)

The expression $a\left(\mathbf{ r} \right)\exp \left({i\varphi \left(\mathbf{ r} \right)} \right)$ being called complex amplitude of f and denoted by $\tilde{f}$, Eq. (2.59) can be written:

$$\begin{aligned} f({\mathbf{ r} },t)={\text{ Re}}\left\{ {\tilde{f}}({\mathbf{ r} })\exp (-i\omega t)\right\} . \end{aligned}$$

(2.60)

Thus the function f deduces from its complex amplitude $\tilde{f}$ by multiplying by $\exp \left({-i\omega t} \right)$ then by taking the real part of the product. It is straightforward to show that the multiplication of $f\left({\mathbf{ r} ,t} \right)$ by a real function $u\left(\mathbf{ r} \right)$ results in a multiplication of the complex amplitude by the same function and conversely, and that the complex amplitude of $\partial f/\partial t$ is equal to $-i\omega \tilde{f}$. Thus, Maxwell’s equations can be written using the complex amplitudes of the field and, for simplicity, the complex amplitudes of the fields and current are denoted using the same names and symbols as the fields and current themselves, in such a way that harmonic Maxwell equations can be written:

$$\begin{aligned} \nabla \times \mathbf{ E}&= i\omega \mathbf{ B} ,\end{aligned}$$

(2.61)

$$\begin{aligned} \nabla \times \mathbf{ H}&= \mathbf{ j} _c -i\omega \mathbf{ D} . \end{aligned}$$

(2.62)

In addition to Maxwell equations, constitutive relations allow one to express the electromagnetic properties of the materials. In contrast with Maxwell equations, they are not rigorous (except in vacuum). Assuming that a material is non-magnetic, homogeneous, isotropic and linear, these relations can be written:

$$\begin{aligned} \mathbf{ B} =\mu _0 \mathbf{ H} ,\quad \mathbf{ D} =\varepsilon _0 \varepsilon ^{\prime }\mathbf{ E} ,\quad \mathbf{ j} _\mathbf{ c} =\sigma \mathbf{ E} , \end{aligned}$$

(2.63)

with $\mu _0 =4 \cdot \pi \cdot 10^{-7}$ being the permeability of vacuum and $\varepsilon _0 =1/\left({36\pi 10^{9}} \right)$ being the permittivity of vacuum. The parameters $\varepsilon ^{\prime }$ and $\sigma$ denote the relative dielectric permittivity and the conductivity of the material respectively. Using the relations $\mathbf{ D} =\varepsilon _0 \mathbf{ E} +\mathbf{ P} ,\quad \mathbf{ P} =\varepsilon _0 \chi \mathbf{ E}$, with $\mathbf{ P}$ electric polarization density and $\chi$ electric susceptibility, it turns out that $\varepsilon ^{\prime }=1+\chi$.

Introducing the constitutive relations in Maxwell equations yields:

$$\begin{aligned} \nabla \times \mathbf{ E}&= i\omega \mu _0 \mathbf{ H} ,\end{aligned}$$

(2.64)

$$\begin{aligned} \nabla \times \mathbf{ H}&= \left({\sigma -i\omega \varepsilon _0 \varepsilon ^{\prime }} \right)\mathbf{ E} . \end{aligned}$$

(2.65)

Defining the complex permittivity:

$$\begin{aligned} \varepsilon =\varepsilon ^{\prime }+i\sigma /\left({\omega \varepsilon _0 } \right)=1+\chi +i\sigma /\left({\omega \varepsilon _0 } \right), \end{aligned}$$

(2.66)

Equation (2.65) takes a form symmetrical to Eq. (2.64):

$$\begin{aligned} \nabla \times \mathbf{ H} =-i\omega \varepsilon _0 \varepsilon \mathbf{ E} . \end{aligned}$$

(2.67)

This equation can be expressed in the form:

$$\begin{aligned} \nabla \times \mathbf{ H} =-i\omega \varepsilon _0 \mathbf{ E} +\mathbf{ j} _\mathbf{ t} ,\text{ with}\, \mathbf{ j} _\mathbf{ t} =\sigma \mathbf{ E} -i\omega \varepsilon _0 \chi \mathbf{ E} =i\omega \varepsilon _0 \left({1-\varepsilon } \right)\mathbf{ E} =\mathbf{ j} _\mathbf{ c} +\mathbf{ j} _\mathbf{ b} , \end{aligned}$$

(2.68)

with $\mathbf{ j} _\mathbf{ t}$ total current density, including both the conduction current density $\mathbf{ j} _\mathbf{ c}$ and the bound current density $\mathbf{ j} _\mathbf{ b} =-i\omega \mathbf{ P} =-i\omega \varepsilon _0 \chi \mathbf{ E}$ resulting from the electric polarization. The optical index of a material is given by $\nu =\sqrt{\varepsilon }$.

Let us notice that by taking the divergence of Eqs. (2.64) and (2.67) and using $\nabla \cdot \left({\nabla \times \mathbf{ V} } \right)=0$, one can get the complementary couple of Maxwell equations in harmonic regime:

$$\begin{aligned} \nabla \cdot \mathbf{ H} =0,\quad \nabla \cdot \left({\varepsilon \mathbf{ E} } \right)=\nabla \cdot \left({\varepsilon ^{\prime }\mathbf{ E} } \right)=0. \end{aligned}$$

(2.69)

By combining Eqs. (2.64) and (2.67), one can obtain partial derivative equations for each field inside a homogeneous region. Introducing the value of H given by Eq. (2.64) in Eq. (2.67), we obtain:

$$\begin{aligned} \nabla \times \nabla \times \mathbf{ E} -k^{2}\mathbf{ E} =0,\quad \text{ with}\;k=\omega \sqrt{\varepsilon \varepsilon _0 \mu _0 }. \end{aligned}$$

(2.70)

In a homogeneous region, $\nabla \cdot \left({\varepsilon \mathbf{ E} } \right)=0$ entails that $\nabla \cdot \mathbf{ E} =0$ then using Eq. (2.69) and the vector relationship $\nabla \times \nabla \times \mathbf{ E} =\nabla \left({\nabla \cdot \mathbf{ E} } \right)-\nabla ^{2}\mathbf{ E}$ we get:

$$\begin{aligned} \nabla \mathbf{ E} +k^{2}\mathbf{ E} =0, \end{aligned}$$

(2.71)

and, following the same lines for the magnetic field:

$$\begin{aligned} \nabla \mathbf{ H} +k^{2}\mathbf{ H} =0. \end{aligned}$$

(2.72)

It is worth noting that Maxwell equations (Eqs. (2.64) and (2.67)) are valid in the sense of distributions. In other words, they include the boundary conditions at the limit between two homogeneous materials. In order to express them in an explicit form, one can recall that the surface density of V included in $\Delta \times \mathbf{ V}$ is equal to $\mathbf{ n} \times \left({\mathbf{ V} _+ -\mathbf{ V} _- } \right)$, with $\mathbf{ V} _+ -\mathbf{ V} _-$ being the jump of V across the interface in the direction of n [29]. It must be recalled that this surface term is the coefficient of a Delta distribution located on the surface. In Eqs. (2.64) and (2.67), the right-hand member contains the fields and thus they should not include distributive surface parts, thus the left-hand member satisfies the same property. We deduce that the tangential components of the electric and magnetic fields are continuous across an interface.

In the two-dimensional case, the interface is invariant by translation along the z-axis. One can distinguish the two fundamental cases of polarization: s-polarization with the electric field $\mathbf{ E} = E\hat{\mathbf{ z} }$ and p-polarization with the magnetic field $\mathbf{ H} =H\hat{\mathbf{ z} }$. In both cases, the boundary-value problem becomes scalar. Projecting Eqs. (2.71) and (2.72) on the z-axis, we obtain:

$$\begin{aligned} \nabla ^{2}E+k^{2}E=0\;\text{ for} \text{ s-polarized} \text{ light} \text{ and}\;\nabla ^{2}H+k^{2}H=0\;\text{ for} \text{ p-polarized} \text{ light}. \end{aligned}$$

(2.73)

Thus the electric and magnetic fields satisfy a scalar Helmholtz equation.

As regards the boundary conditions on the interface, the continuity of the tangential component of the fields entails the continuity of E for s-polarization and the continuity of H for p-polarization. Since the partial derivative equation is of the second order, a second boundary condition is needed. For s-polarization, one can express the continuity of the tangential component of the magnetic field using Eq. (2.64). Bearing in mind the vector relation $\nabla \times \left({E\hat{\mathbf{ z} }} \right)=\nabla E\times \hat{\mathbf{ z} }$, we obtain:

$$\begin{aligned} \mathbf{ H} =\frac{1}{i\omega \mu _0 }\nabla E\times \hat{\mathbf{ z} }. \end{aligned}$$

(2.74)

Furthermore, on each side of the interface with normal vector $\mathbf{ n}$, the continuity of the tangential component of the magnetic field entails the continuity of $\mathbf{ n} \times \mathbf{ H}$, and from Eq. (2.74), it turns out that

$$\begin{aligned} \mathbf{ n} \times \mathbf{ H} =\frac{1}{i\omega \mu _0 }\mathbf{ n} \times \left({\nabla E\times \hat{\mathbf{ z} }} \right)=\frac{1}{i\omega \mu _0 }\nabla E\left({\mathbf{ n} \cdot \hat{\mathbf{ z} }} \right)-\hat{\mathbf{ z} }\left({\mathbf{ n} \cdot \nabla E} \right)=\frac{-1}{i\omega \mu _0 }\hat{\mathbf{ z} }\frac{\partial E}{\partial n}. \end{aligned}$$

(2.75)

It follows that the normal derivative of the electric field $\frac{\partial E}{\partial n}$ is continuous across the interface.

For p-polarization, the electric field can be expressed from Eq. (2.67):

$$\begin{aligned} \mathbf{ E} =\frac{1}{-i\omega \varepsilon \varepsilon _0 }\nabla \times \left({H\hat{\mathbf{ z} }} \right)=\frac{1}{-i\omega \varepsilon \varepsilon _0 }\nabla H\times \hat{\mathbf{ z} }, \end{aligned}$$

(2.76)

and thus the continuity of $\mathbf{ n} \times \mathbf{ E}$ leads to the continuity of $\frac{1}{\varepsilon }\frac{\partial H}{\partial n}$. This second boundary condition is slightly more complicated than that obtained for s-polarization, due to the fact that, in contrast with the permeability, the permittivity is not the same on the two sides of the interface. It can be noticed that this dissymmetry disappears for magnetic materials, for which $\frac{1}{\mu }\frac{\partial E}{\partial n}$ is continuous across the interface.

In conclusion, the boundary conditions across the interface can be written in 2D problems:

$$\begin{aligned}&E\;\text{ and}\;\partial E/\partial n\;\text{ are} \text{ continuous} \text{ for} \text{ s-polarized} \text{ light},\end{aligned}$$

(2.77)

$$\begin{aligned}&H\;\text{ and}\;\frac{1}{\varepsilon }\frac{\partial H}{\partial n}\;\text{ are} \text{ continuous} \text{ for} \text{ p-polarized} \text{ light}. \end{aligned}$$

(2.78)

The surface charge densities can be derived by taking the divergence of Eqs. (2.62) and (2.68). Bearing in mind that the surface density included in $\Delta \cdot \mathbf{ V}$ is equal to $\mathbf{ n} \cdot \left({\mathbf{ V} _+ -\mathbf{ V} _- } \right)$, with $\mathbf{ V} _+ -\mathbf{ V} _-$ being the jump of V in the direction of n across the interface [29], it comes out that:

$$\begin{aligned} \rho _c&= \mathbf{ n} \cdot \left({\mathbf{ D} }_{+} -{\mathbf{ D} }_{-} \right)\end{aligned}$$

(2.79)

$$\begin{aligned} \rho _t&= \varepsilon _0 \mathbf{ n} \cdot \left({\mathbf{ E} _{+} -\mathbf{ E} _{-}} \right), \end{aligned}$$

(2.80)

with $\rho _c$ and $\rho _t$ being the surface densities of free and total charges.