New PO Theory - Revision history

Kazem Sabet: /* Conventional Physical Optics (GO-PO) */

2018-07-23T05:43:45Z

‎Conventional Physical Optics (GO-PO)

Kazem Sabet: /* Conventional Physical Optics (GO-PO) */

2018-07-23T05:43:03Z

‎Conventional Physical Optics (GO-PO)

Kazem Sabet: /* Conventional Physical Optics (GO-PO) */

2018-07-23T05:31:20Z

‎Conventional Physical Optics (GO-PO)

Kazem Sabet: /* Conventional Physical Optics (GO-PO) */

2018-07-23T05:30:19Z

‎Conventional Physical Optics (GO-PO)

Kazem Sabet at 05:29, 23 July 2018

2018-07-23T05:29:02Z

Kazem Sabet: /* Conventional Physical Optics (GO-PO) */

2018-07-23T05:27:03Z

‎Conventional Physical Optics (GO-PO)

Kazem Sabet: /* Conventional Physical Optics (GO-PO) */

2018-07-23T05:22:35Z

‎Conventional Physical Optics (GO-PO)

Kazem Sabet: /* Iterative Physical Optics (IPO) */

2018-07-23T04:53:00Z

‎Iterative Physical Optics (IPO)

Kazem Sabet: /* Calculating Near & Far Fields In PO */

2018-07-23T04:51:40Z

‎Calculating Near & Far Fields In PO

Kazem Sabet: /* Calculating Near & Far Fields In PO */

2018-07-23T04:50:47Z

‎Calculating Near & Far Fields In PO

@@ Line 32: / Line 32: @@
 where &theta; is the incident angle between the propagation vector of the incident field and the normal to the surface and <math>\eta_0 = 120\pi \; \Omega</math> is the intrinsic impedance of the free space.
-It can easily be shown that
+From the surface impedance boundary condition, it can easily be shown that
 :<math> \mathbf{M(r)} = -Z_s \mathbf{\hat{n}\times} \mathbf{J(r)}  </math>
 In the case of an impedance-matched surface (Z<sub>s</sub> =  &eta;<sub>0</sub>), one can write:

← Older revision		Revision as of 05:43, 23 July 2018
Line 31:		Line 31:

	where θ is the incident angle between the propagation vector of the incident field and the normal to the surface and <math>\eta_0 = 120\pi \; \Omega</math> is the intrinsic impedance of the free space.		where θ is the incident angle between the propagation vector of the incident field and the normal to the surface and <math>\eta_0 = 120\pi \; \Omega</math> is the intrinsic impedance of the free space.
		+
		+	It can easily be shown that
		+
		+	:<math> \mathbf{M(r)} = -Z_s \mathbf{\hat{n}\times} \mathbf{J(r)} </math>
		+

	In the case of an impedance-matched surface (Z<sub>s</sub> = η<sub>0</sub>), one can write:		In the case of an impedance-matched surface (Z<sub>s</sub> = η<sub>0</sub>), one can write:

@@ Line 42: / Line 42: @@
 </table>
-Two special limiting cases of an impedance surface are perfect electric conductor (PEC) and perfect magnetic conductor (PMC) surface. For a PEC surface, Z<sub>s</sub> = 0, R|| = R⊥ = -1, and one can write:
+Two special limiting cases of an impedance surface are perfect electric conductor (PEC) and perfect magnetic conductor (PMC) surface. For a PEC surface, Z<sub>s</sub> = 0, R<sub>||</sub> = R<sub>&perp;</sub> = -1, and one can write:
 :<math> \mathbf{J(r)} = 2 \mathbf{\hat{n} \times H(r)} </math>
@@ Line 48: / Line 48: @@
 :<math> \mathbf{M(r)} = 0 </math>
-while for a PMC surface, Z<sub>s</sub> = &infin;, R|| = R⊥ = 1, and one can write:
+while for a PMC surface, Z<sub>s</sub> = &infin;, R<sub>||</sub> = R<sub>&perp;</sub> = 1, and one can write:
 :<math> \mathbf{J(r)} = 0 </math>

@@ Line 42: / Line 42: @@
 </table>
-Two special limiting cases of an impedance surface are perfect electric conductor (PEC) and perfect magnetic conductor (PMC) surface. For a PEC surface, Z<sub>s</sub> = 0, and one can write:
+Two special limiting cases of an impedance surface are perfect electric conductor (PEC) and perfect magnetic conductor (PMC) surface. For a PEC surface, Z<sub>s</sub> = 0, R|| = R⊥ = -1, and one can write:
 :<math> \mathbf{J(r)} = 2 \mathbf{\hat{n} \times H(r)} </math>
@@ Line 48: / Line 48: @@
 :<math> \mathbf{M(r)} = 0 </math>
-while for a PMC surface, Z<sub>s</sub> = &infin;, and one can write:
+while for a PMC surface, Z<sub>s</sub> = &infin;, R|| = R⊥ = 1, and one can write:
 :<math> \mathbf{J(r)} = 0 </math>

@@ Line 32: / Line 32: @@
 where &theta; is the incident angle between the propagation vector of the incident field and the normal to the surface and <math>\eta_0 = 120\pi \; \Omega</math> is the intrinsic impedance of the free space.
-In the case of an impedance-matched surface, one can write:
+In the case of an impedance-matched surface (Z<sub>s</sub> =  &eta;<sub>0</sub>), one can write:
 :<math> R_{\|} = -R_{\perp} = \frac{1-\cos\theta} {1+\cos\theta}  </math>
@@ Line 42: / Line 42: @@
 </table>
-Two special limiting cases of an impedance surface are perfect electric conductor (PEC) and perfect magnetic conductor (PMC) surface. For a PEC surface, Z = 0,  &alpha; = 1, and one can write:
+Two special limiting cases of an impedance surface are perfect electric conductor (PEC) and perfect magnetic conductor (PMC) surface. For a PEC surface, Z<sub>s</sub> = 0, and one can write:
 :<math> \mathbf{J(r)} = 2 \mathbf{\hat{n} \times H(r)} </math>
 :<math> \mathbf{M(r)} = 0 </math>
-while for a PMC surface, Z = &infin;,  &alpha; = -1, and one can write:
+while for a PMC surface, Z<sub>s</sub> = &infin;, and one can write:
 :<math> \mathbf{J(r)} = 0 </math>
 :<math> \mathbf{M(r)} = -2 \mathbf{\hat{n} \times E(r)} </math>
 A major difficulty encountered in determining the PO currents of the scatterer is identification of lit and shadowed facets. Determination of lit and shadowed regions for simple, stand-alone, convex objects is rather simple. Denoting the incidence direction from a source to a point on the scatterer by the unit vector '''k''', the point is considered lit if '''n.k'''&lt; 0, and shadowed if '''n.k'''&gt; 0. These conditions, however, are only valid if there is a direct line of sight (LOS) between the source and the centroid of the cell under consideration. They cannot predict if there are any obstructing objects in the path of the incident beam or ray. For simple convex objects, a Geometrical Optics (GO) approach can be used to finds the optical LOS lines and determine the lit and shadowed areas on the object. The conventional PO can then be used to find the electric and magnetic surface currents.

@@ Line 25: / Line 25: @@
 where it is assumed that both the incident electric field and incident magnetic field have been decomposed into two parallel and perpendicular polarizations and R<sub>||</sub> and R<sub>&perp;</sub> denote the reflection coefficients at the interface between air and the impedance surface for the cases of parallel and perpendicular polarizations, respectively. These reflection coefficients are given by:
 :<math> R_{\perp} = \frac{Z_s\cos\theta  - \eta_0} {Z_s\cos\theta  + \eta_0} </math>
 where &theta; is the incident angle between the propagation vector of the incident field and the normal to the surface and <math>\eta_0 = 120\pi \; \Omega</math> is the intrinsic impedance of the free space.

@@ Line 104: / Line 104: @@
 The induced electric and magnetic surface currents on each point of the scatterer object can be calculated from the Magnetic and Electric Field Integral Equations (MFIE &amp; EFIE):
 :<math> \mathbf{J(r)} = \mathbf{\hat{n}\times} \begin{bmatrix} 1-R_{||} & 0 \\ 0 & 1-R_{\perp} \end{bmatrix} \cdot \begin{bmatrix} \mathbf{H_{||}^{tot}(r)} \\ \mathbf{H_{\perp}^{tot}(r)} \end{bmatrix}  </math>
@@ Line 118: / Line 110: @@
 :<math> \mathbf{M(r)} = -\mathbf{\hat{n}\times} \begin{bmatrix} 1+R_{||} & 0 \\ 0 & 1+R_{\perp} \end{bmatrix} \cdot \begin{bmatrix} \mathbf{E_{||}^{tot}(r)} \\ \mathbf{E_{\perp}^{tot}(r)} \end{bmatrix}  </math>
 The shadowing phenomenon can be attributed to near-field interaction of surface currents. The current on the lit region produces a scattered field in the forward direction that is almost equal and out of phase with the incident wave. Hence, the sum of the scattered field and incident field over the shadowed region almost cancel each other, giving rise to a very small field there. This suggests that keeping track of multiple scattering can take care of shadowing problems automatically. In addition, the effects of multiple scattering can be readily accounted for by an iterative PO approach to be formulated next.
-−
-−
-<!--[[File:PO9(1).png]]-->
-−
-where '''R''' ='''r''' - '''r'''', R = |'''R'''|, and
-−
-:<math>\mathbf{ \hat{R} = \frac{R}{|R|} = \frac{r-r'}{|r-r'|} }</math>
-<!--[[File:PO11.png]]-->
-−
-The shadowing phenomenon can indeed be attributed to near-field interaction of surface currents. The current on the lit region produces a scattered field in the forward direction that is almost equal and out of phase with the incident wave. Hence, the sum of the scattered field and incident field over the shadowed region almost cancel each other, giving rise to a very small field there. This suggests that keeping track of multiple scattering can take care of shadowing problems automatically. In addition, the effects of multiple scattering can be readily accounted for by an iterative PO approach to be formulated next.
 The starting point for the iterative PO solution is the above MFIE and EFIE integral equations. To the first (zero-order) approximation, we can write
 :<math> \mathbf{J^{(0)}(r)} = \mathbf{\hat{n}\times} \begin{bmatrix} 1-R_{||} & 0 \\ 0 & 1-R_{\perp} \end{bmatrix} \cdot \begin{bmatrix} \mathbf{H_{||}^{inc}(r)} \\ \mathbf{H_{\perp}^{inc}(r)} \end{bmatrix}  </math>
@@ Line 140: / Line 118: @@
 :<math> \mathbf{M^{(0)}(r)} = -\mathbf{\hat{n}\times} \begin{bmatrix} 1+R_{||} & 0 \\ 0 & 1+R_{\perp} \end{bmatrix} \cdot \begin{bmatrix} \mathbf{E_{||}^{inc}(r)} \\ \mathbf{E_{\perp}^{inc}(r)} \end{bmatrix}  </math>
 which are the conventional PO currents. However, this approximation does not formally recognize the lit and shadowed areas. Instead of identifying the exact boundaries of the lit and shadowed areas over a complex target, a simple condition is used first to find the primary shadowed areas. Then, through PO iterations all shadowed areas are determined automatically. When calculating the field on the scatterer for every source point, a primary shadowing condition given by '''n.k'''&lt; 0 is examined. In complex scatterer geometries, there are shadowed points in concave regions where '''n.k'''&gt; 0, but the correct shadowing is eventually achieved through the iteration of the PO currents. Therefore, in computation of the above equations, only the contribution of the points that satisfy the following condition are considered:
@@ Line 150: / Line 125: @@
 At the subsequent iterations, the higher order PO currents are given by:
 :<math> \mathbf{J^{(n)}(r)} = \mathbf{\hat{n}\times} \begin{bmatrix} 1-R_{||} & 0 \\ 0 & 1-R_{\perp} \end{bmatrix} \cdot \begin{bmatrix} \mathbf{H_{||}^{(n-1)}(r)} \\ \mathbf{H_{\perp}^{(n-1)}(r)} \end{bmatrix}  </math>
@@ Line 164: / Line 131: @@
 :<math> \mathbf{M^{(n)}(r)} = -\mathbf{\hat{n}\times} \begin{bmatrix} 1+R_{||} & 0 \\ 0 & 1+R_{\perp} \end{bmatrix} \cdot \begin{bmatrix} \mathbf{E_{||}^{(n-1)}(r)} \\ \mathbf{E_{\perp}^{(n-1)}(r)} \end{bmatrix}  </math>
 For most practical applications, iterations up to the second order is sufficient. The iterative solution will not only account for double-bounce scattering over the lit regions but it also removes the lower order currents erroneously placed over concave shadowed areas.

← Older revision		Revision as of 04:51, 23 July 2018
Line 88:		Line 88:


−	where k_0 = \frac{2\pi}{\lambda_0} \text{ and } Z_0 = 1/Y_0 = \eta_0 </math>, '''R''' ='''r''' - '''r'''', R = \|'''R'''\|, and	+	where <math>k_0 = \frac{2\pi}{\lambda_0} \text{ and } Z_0 = 1/Y_0 = \eta_0 </math>, '''R''' ='''r''' - '''r'''', R = \|'''R'''\|, and

	:<math>\mathbf{ \hat{R} = \frac{R}{\|R\|} = \frac{r-r'}{\|r-r'\|} }</math>		:<math>\mathbf{ \hat{R} = \frac{R}{\|R\|} = \frac{r-r'}{\|r-r'\|} }</math>

← Older revision		Revision as of 04:50, 23 July 2018
Line 88:		Line 88:


−	where ~~<math> R=\|r-r'\| \text{, }~~ k_0 = \frac{2\pi}{\lambda_0} \text{ and } Z_0 = 1/Y_0 = \eta_0 </math>.	+	where k_0 = \frac{2\pi}{\lambda_0} \text{ and } Z_0 = 1/Y_0 = \eta_0 </math>, '''R''' ='''r''' - '''r'''', R = \|'''R'''\|, and
		+
		+	:<math>\mathbf{ \hat{R} = \frac{R}{\|R\|} = \frac{r-r'}{\|r-r'\|} }</math>