Theory of optical propagation and diffraction

We start with Maxwell's equations which describe the interaction between an electric field vector E (E_x , E_y , E_z ) and a magnetic field vector H (H_x , H_y , H_z ).

$$\begin{eqnarray}\begin{array}{rcl} \nabla \times {\boldsymbol{E}} & = & -\mu \frac{\partial {\boldsymbol{H}} }{\partial t} \\ \nabla \times {\boldsymbol{H}} & = & \varepsilon \frac{\partial {\boldsymbol{E}} }{\partial t} \\ \nabla \bullet \varepsilon {\boldsymbol{E}} & = & 0 \\ \nabla \bullet \mu {\boldsymbol{H}} & = & 0 \\\end{array}\end{eqnarray} \tag{ 1.1 }$$

This assumes that there is no free charge; μ is the permeability and ε the permittivity of the propagation medium. Both electric and magnetic field vectors are functions of both position and time, represented by their Cartesian components and the time derivative.

The equations can be combined by taking the cross-product of the first two time-derivative equations, which results in the wave equations.

$$\begin{eqnarray}\begin{array}{ccccccccccccccc} {{\nabla }^{2}}{\boldsymbol{E}} -\frac{{{n}^{2}}}{{{c}^{2}}}\frac{{{\partial }^{2}}{\boldsymbol{E}} }{\partial {{t}^{2}}} & = & 0 \\ {{\nabla }^{2}}{\boldsymbol{H}} -\frac{{{n}^{2}}}{{{c}^{2}}}H & = & 0 \\\end{array}\end{eqnarray} \tag{ 1.2 }$$

We assume that the medium is linear, isotropic, non-dispersive and homogeneous. That is, ε and μ are constant. We define ε₀ and μ₀ as the constants in a vacuum. The refractive index is $n={{(\frac{\varepsilon }{{{\varepsilon }_{0}}})}^{\frac{1}{2}}}$ and the velocity of propagation in a vacuum is $c=\frac{1}{\sqrt{{{\varepsilon }_{0}}{{\mu }_{0}}}}$. The vector equations are obeyed by both the electric and magnetic fields and there is a scalar equation that is obeyed by each component of the fields. The expression for the x-component of the electric field equation becomes

$$\begin{eqnarray}&&{{\nabla }^{2}}{{E}_{x}}-\frac{{{n}^{2}}}{{{c}^{2}}}\frac{{{\partial }^{2}}{{E}_{x}}}{\partial {{t}^{2}}}=0.\end{eqnarray} \tag{ 1.3 }$$

Now we can generalize to any scalar field component u(x, y, x, t)

$$\begin{eqnarray}&&{{\nabla }^{2}}u(x,y,x,t)-\frac{{{n}^{2}}}{{{c}^{2}}}\frac{{{\partial }^{2}}u(x,y,x,t)\;~~}{\partial {{t}^{2}}}=0.\end{eqnarray} \tag{ 1.4 }$$

Reducing the vector equations to scalar equations has consequences. If a medium is inhomogeneous where ε depends on position, a coupling occurs between the components of the fields and all components will not necessarily satisfy the same wave equation. Furthermore, at an interface between two media, the boundary conditions must be satisfied over a small region where an error occurs between the vector results and the scalar approximation. This is the crux of the study of Fourier optics. By making more simplifying assumptions about the propagation conditions, which result in simpler and more solvable equations, errors in the details appear. By limiting the errors, we can use simpler expressions and many useful mathematical tools to provide practical results.

Instead of using the Cartesian coordinates x, y, z, we can combine the spatial coordinates into one three-dimensional coordinate, s. Let us assume that we have a monochromatic wave that must satisfy the wave equation. The wave number k is given by $k=\frac{2\pi }{\lambda }$ where λ is the wavelength. And, where the velocity in the homogeneous dielectric medium is ν, the wavelength is defined as $\lambda =\frac{c}{n\nu }$. One solution that represents a traveling wave is

$$\begin{eqnarray}&&\;u(s,t)=\operatorname{Re}\left[ U(s){\rm exp} \left( -{\rm j}2\pi vt \right) \right]=A(s){\rm cos} \left[ 2\pi vt-\phi (s) \right].\end{eqnarray} \tag{ 1.5 }$$

A(s) is the wave amplitude and ϕ(s) is the wavefront phase. Knowing that the Laplacian is

$$\begin{eqnarray}&&{{\nabla }^{2}}u=\sum \limits_{\,i=1}^{3} \frac{{{\partial }^{2}}u}{\partial s_{i}^{2}}.\end{eqnarray} \tag{ 1.6 }$$

Equation (1.5) satisfies the time independent wave equation

$$\begin{eqnarray}&&\left( {{\nabla }^{2}}+{{k}^{2}} \right)=0.\end{eqnarray} \tag{ 1.7 }$$

There are many solutions to the wave equation. Waves can be traveling spherically from a source point. They can be plane waves where the plane is normal to the direction of propagation. Depending upon the boundary conditions and source of the radiation, they can take on many different forms. In 1678 Christian Huygens stated that any wavefront of a field that satisfies the wave equation can be considered to be made up of an infinite number of spherical source points that form a new wavefront as it propagates. See figure 1.1.

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Let us now assume that we have a propagating electromagnetic field impinging on an opaque screen where diffraction occurs due to some opening in the screen. We want to calculate the field at points behind the screen. To use the formulation so far described, we have to assert two conditions called the Kirchhoff boundary conditions. The first is that the surface of integration after the screen is entirely arbitrary and the field U and its derivative $\frac{\partial U}{\partial n}$ are exactly the same as they would be in the absence of the screen. The second condition is that the screen is really opaque, that is, in the geometrical shadow of the screen, the field U and its derivative $\frac{\partial U}{\partial n}$ are identically zero. These conditions are generally met if the opening in the screen (the aperture) is large compared to the wavelength.

Another simplification can be made, if the distance r₀₁ is large compared to the wavelength, the $\;({\rm j}k-\frac{1}{{{r}_{01}}})\;$ term in equation (jk). If the aperture is illuminated by a single spherical wave at point P₂,

$$\begin{eqnarray}&&U({{{\rm P}}_{1}})=\frac{A{\rm exp} ({\rm j}k{{r}_{21}})}{{{r}_{21}}}.\end{eqnarray} \tag{ 1.13 }$$

the field at P₀ reduces to the Fresnel–Kirchhoff diffraction formula:

$$\begin{eqnarray}&&U({{{\rm P}}_{0}})=\frac{A}{{\rm j}\lambda }\int \mathop \int \nolimits_{{\rm Aper}} \left\{ \frac{{\rm exp} \left[ {\rm j}k\left( {{r}_{21}}+{{r}_{01}} \right) \right]}{{{r}_{21}}{{r}_{01}}}\;\left[ \frac{{\rm cos} \left( {\boldsymbol{n}} ,{{{\boldsymbol{r}} }_{01}} \right)-{\rm cos} \left( {\boldsymbol{n}} ,{{{\boldsymbol{r}} }_{21}} \right)}{2}\; \right] \right\}{\rm d}S.\end{eqnarray} \tag{ 1.14 }$$

The integration is now over the aperture in the region around P₁. This equation is symmetrical between P₀ and P₂. The field at P₀ from the source at P₂ through the aperture would be the same as the field at P₂ from a point source at P₀. The fact that light and diffraction do not seem to care whether the beam is going left-to-right or right-to-left is known as the reciprocity theorem of Helmholtz:

$$\begin{eqnarray}&&U\left( {{{\rm P}}_{0}} \right)=\frac{A}{{\rm j}\lambda }\int \mathop \int \nolimits_{{\rm Aper}} U^{\prime} \left( {{{\rm P}}_{1}} \right)\frac{{\rm exp} \left[ {\rm j}k\left( {{r}_{01}} \right) \right]}{{{r}_{01}}}{\rm d}S.\end{eqnarray} \tag{ 1.15 }$$

where

$$\begin{eqnarray}&&U\prime\left( {{{\rm P}}_{1}} \right)=\frac{A}{{\rm j}\lambda }\frac{{\rm exp} \left[ {\rm j}k\left( {{r}_{21}} \right) \right]}{{{r}_{21}}}\;\left[ \frac{{\rm cos} \left( {\boldsymbol{n}} ,{{{\boldsymbol{r}} }_{01}} \right)-{\rm cos} \left( {\boldsymbol{n}} ,{{{\boldsymbol{r}} }_{21}} \right)}{2}\; \right].\end{eqnarray} \tag{ 1.16 }$$

The r vectors end in the aperture at P₁. As one moves across the aperture, the directions of the r vectors change and it appears that the field in the aperture is an infinite number of secondary spherical waves that match the Huygens principle. Fresnel made the Huygens assumption work along with Young's principle of interference and Kirchhoff showed that this property was a fundamental part of the nature of light itself.