Introduction

The Physical Optics method is a high frequency technique for calculating the $\mathbf{E}$ and $\mathbf{H}$ fields scattered by objects.

The general scattering problem consists of a known incident field incident on a scatterer of some known geometry and material properties. The output is the total radiated fields, which can be divided into the incident fields $\mathbf{E}_{inc}$ and $\mathbf{H}_{inc}$ and the scattered fields $\mathbf{E}_{scat}$ and $\mathbf{H}_{scat}$.

$$ \mathbf{E} = \mathbf{E}_{inc} + \mathbf{E}_{scat} $$ $$ \mathbf{H} = \mathbf{H}_{inc} + \mathbf{H}_{scat} $$

For a thin scattering surface, the scattered fields can be further divided into reflected and transmitted fields:

$$ \mathbf{E}_{scat} = \mathbf{E}_{ref} + \mathbf{E}_{trans} $$ $$ \mathbf{H}_{scat} = \mathbf{H}_{ref} + \mathbf{H}_{trans} $$

The scattered fields can be taken as entirely due to electric and magnetic currents, $\mathbf{J}$ and $\mathbf{M}$, induced on the surface(s) of the scatterer by the incident fields.

The Physical Optics Approximation

The Physical Optics approximation consists of assuming that the surface electric and magnetic currents at a specific point on a curved scatterer are the same as those on an infinite plane tangent to scattering surface at that point. Each of the incident, reflected and transmitted fields contribute to the surface currents.

On a perfectly electrially conducting (PEC) plane surface, the transmitted fields vanish. The reflected $\mathbf{E}_{ref}$ field at the scattering surface is equal to the incident field, but with opposite phase, while the reflected $\mathbf{H}_{ref}$ field has the same phase as the incident field.

Thus for a perfectly conducting scatterering plane, these currents are given by:

$$ \mathbf{J} = \mathbf{\hat{n}} \times (\mathbf{H}_{inc} + \mathbf{H}_{ref}) = 2 \mathbf{\hat{n}} \times \mathbf{H}_{inc} $$ $$ \mathbf{M} = - \mathbf{\hat{n}} \times (\mathbf{E}_{inc} + \mathbf{E}_{ref}) = 0 $$

The currents across the surface of a curved scatterer are thus calculated by dividing the scatterer into planar facets that are tangent to the surface at their center, and calculating the currents at the center of each facet as if it were an infinite plane.

These currents are only present in the region of the scatterer directly illuminated by the source of the incident radiation field. An additional approximation within the Physical Optics method for PEC scatterers is that the currents on portions of the scatterer that are shadowed from the source (under geometric ray optics) are exactly zero.

Calculation of fields at a point

Once the electric and magnetic currents induced at each point across the scatterer have been calculated, it is then necessary to calculated the field due to those currents at some distant point $\mathbf{r}$. This is carried out using source representation of the field that is a solution to the Helmholtz equation for monochromatic fields in isotropic media due to a source Hertzian dipole, and integrating over the current distribution induced on the scatterer surface.

For the electric component of the field, this is given by:

$$ \mathbf{E}(\mathbf{r}) = - k Z_m \int_{S} \Big( 1 + \frac{1}{k^2}\mathbf{\nabla}\mathbf{\nabla} \Big) G(\mathbf{r}, \mathbf{r_s}) \cdot \mathbf{J}(\mathbf{r_s}) d s_q + \int_{S} \mathbf{M}(\mathbf{r_s}) \times \mathbf{\nabla} G(\mathbf{r}, \mathbf{r_q}) d s $$

And for the magnetic component

$$ \mathbf{H}(\mathbf{r}) = - k Z_m \int_{S} \Big( 1 + \frac{1}{k^2}\mathbf{\nabla}\mathbf{\nabla} \Big) G(\mathbf{r}, \mathbf{r_s}) \cdot \mathbf{M}(\mathbf{r_s}) d s_q + \int_{S} \mathbf{J}(\mathbf{r_s}) \times \mathbf{\nabla} G(\mathbf{r}, \mathbf{r_q}) d s $$

where:

$\mathbf{r}_s$ is a point on the source scatterer.

$S$ is the source scatterer to be integrated over.

$d s_q$ is an area element of the source scatterer surface centered at $\mathbf{r}_s$

$k = \frac{2 \pi c}{n \lambda}$ is the wave number.

$Z_m = \sqrt{\frac{\mu_r \mu_0}{\epsilon_r \epsilon_0}}$ is the impedance of the background media

$G = \frac{e^{-i k R}}{4 \pi R}$ is the freespace Green's function

$\mathbf{R} = \mathbf{r_s} - \mathbf{r}$ is the vector from the target point to the source point

$R = |\mathbf{R}| = |\mathbf{r_s} - \mathbf{r}|$ is the distance between the observation point and the point on the source scatterer.

$\mathbf{\hat{R}} $ is the unit vector in the direction of

Which yields:

$$ \mathbf{E} = \frac{k^2}{4\pi} \int_S Z \Bigg[J \Bigg( \frac{-i}{kR} - \frac{1}{(kR)^2} + \frac{i}{(kR)^3}\Bigg) + (\mathbf{J}\cdot\mathbf{\hat{R}})\mathbf{\hat{R}}\Bigg( \frac{i}{kR} + \frac{3}{(kR)^2} - \frac{3i}{(kR)^3} \Bigg)\Bigg] - \mathrm{M}\times\mathrm{\hat{R}}\Bigg(\frac{i}{kR}+\frac{1}{(kR)^2}\Bigg) e^{-ikR} ds $$

$$ \mathbf{H} = \frac{k^2}{4\pi} \int_S \frac{1}{Z} \Bigg[M \Bigg( \frac{-i}{kR} - \frac{1}{(kR)^2} + \frac{j}{(kR)^3}\Bigg) + (\mathbf{M}\cdot\mathbf{\hat{R}})\mathbf{\hat{R}}\Bigg( \frac{i}{kR} + \frac{3}{(kR)^2} - \frac{3i}{(kR)^3} \Bigg)\Bigg] + \mathrm{J}\times\mathrm{\hat{R}}\Bigg(\frac{i}{kR}+\frac{1}{(kR)^2}\Bigg) e^{-ikR} ds $$

For backwards propagation, we change the Green's functon to $G_{r} = \frac{e^{i k R}}{4 \pi R}$, yielding:

$$ \mathbf{E} = \frac{k^2}{4\pi} \int_S Z \Bigg[J \Bigg( \frac{-i}{kR} + \frac{1}{(kR)^2} + \frac{j}{(kR)^3}\Bigg) + (\mathbf{J}\cdot\mathbf{\hat{R}})\mathbf{\hat{R}}\Bigg( \frac{i}{kR} - \frac{3}{(kR)^2} - \frac{3i}{(kR)^3} \Bigg)\Bigg] + \mathrm{M}\times\mathrm{\hat{R}}\Bigg(\frac{i}{kR}+\frac{1}{(kR)^2}\Bigg) e^{ikR} ds$$

$$ \mathbf{H} = \frac{k^2}{4\pi} \int_S \frac{1}{Z} \Bigg[M \Bigg( \frac{-i}{kR} + \frac{1}{(kR)^2} + \frac{j}{(kR)^3}\Bigg) + (\mathbf{M}\cdot\mathbf{\hat{R}})\mathbf{\hat{R}}\Bigg( \frac{i}{kR} - \frac{3}{(kR)^2} - \frac{3i}{(kR)^3} \Bigg)\Bigg] - \mathrm{J}\times\mathrm{\hat{R}}\Bigg(\frac{i}{kR}+\frac{1}{(kR)^2}\Bigg) e^{ikR} ds $$

Far fields

The far field is defined by the limit $$\mathbf{E}_{far} = \lim_{r\rightarrow\infty} \mathbf{E}(\mathbf{r}) k r e^{ikr}$$

$$\mathbf{H}_{far} = \lim_{r\rightarrow\infty} \mathbf{H}(\mathbf{r}) k r e^{ikr}$$

where $r$ = $|\mathbf{r}|$.

Inserting the above equations for forward propagation gives:

$$ \mathbf{E}_{far} = -\frac{ik^2}{4\pi} \int_S \Bigg(Z \Big(\mathbf{J} - (\mathbf{J}\cdot\mathbf{\hat{r}})\mathbf{\hat{r}}\Big) - \mathbf{\hat{r}} \times \mathbf{M} \Bigg) e^{ik\mathbf{r}_s\cdot\mathbf{\hat{r}}} ds $$

$$ \mathbf{H}_{far} = -\frac{ik^2}{4\pi} \int_S \Bigg(\frac{1}{Z} \Big(\mathbf{M} - (\mathbf{M}\cdot\mathbf{\hat{r}})\mathbf{\hat{r}}\Big) + \mathbf{\hat{r}} \times \mathbf{J} \Bigg) e^{ik\mathbf{r}_s\cdot\mathbf{\hat{r}}} ds $$

For backward propagation, the farfield is given by:

$$ \mathbf{E}_{far} = -\frac{ik^2}{4\pi} \int_S \Bigg(Z \Big(\mathbf{J} - (\mathbf{J}\cdot\mathbf{\hat{r}})\mathbf{\hat{r}}\Big) + \mathbf{\hat{r}} \times \mathbf{M} \Bigg) e^{-ik\mathbf{r}_s\cdot\mathbf{\hat{r}}} ds $$

$$ \mathbf{H}_{far} = -\frac{ik^2}{4\pi} \int_S \Bigg(\frac{1}{Z} \Big(\mathbf{M} - (\mathbf{M}\cdot\mathbf{\hat{r}})\mathbf{\hat{r}}\Big) - \mathbf{\hat{r}} \times \mathbf{J} \Bigg) e^{-ik\mathbf{r}_s\cdot\mathbf{\hat{r}}} ds $$

The PO Simulation Process

Physical optics simulations of a system of scatterings proceeds by:

A source of electrical and magnetic currents is defined¸ either directly, or from a specified $\mathbf{E}$ and $\mathbf{H}$ field using a specified boundary condition (PEC is the default choice).
The fields and currents on the first scatterer due to the source are calculated. First $\mathbf{E}_{inc}$ and $\mathbf{H}_{inc}$ fields due to the source are calculated at each grid point on the target scatterer. From these fields, the electric and magnetic currents, $\mathbf{J}$ and $\mathbf{M}$, induced on the scatterer are determined.
The fields and currents on the second scatterer due to the currents on the first scatterer are calculated. This process is repeated for each scatterer in the system.
The output fields are calculated from the currents induced on the final scatterer.

In general, the fields on each scattering and output plane can be calculated due to multiple sources by simply adding the fields from each individual source.

Material Properties

For a scatterer other than a perfect electrical conductor, the reflected $\mathbf{H}_{ref}$ field at the surface is no longer equal to the incident $\mathbf{H}_{inc}$ field, and the reflected $\mathbf{E}_{ref}$ field no longer exactly cancels the incident $\mathbf{E}_{inc}$ field. The transmitted fields are also no longer generally zero.

The reflection and transmission coefficients in general vary with angle, and also differ for incident plane waves that are polarized perpendicular and parallel to the plane defined by the incident and reflected rays.

Limitations of the PO approximation