Modified theory of physical optics

Yusuf Z. Umul

doi:10.1364/OPEX.12.004959

Optics Express
Vol. 12,
Issue 20,
pp. 4959-4972
(2004)
•https://doi.org/10.1364/OPEX.12.004959

Modified theory of physical optics

Yusuf Z. Umul

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Yusuf Z. Umul, "Modified theory of physical optics," Opt. Express 12, 4959-4972 (2004)

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Abstract

A new procedure for calculating the scattered fields from a perfectly conducting body is introduced. The method is defined by considering three assumptions. The reflection angle is taken as a function of integral variables, a new unit vector, dividing the angle between incident and reflected rays into two equal parts is evaluated and the perfectly conducting (PEC) surface is considered with the aperture part, together. This integral is named as Modified Theory of Physical Optics (MTPO) integral. The method is applied to the reflection and edge diffraction from a perfectly conducting half plane problem. The reflected, reflected diffracted, incident and incident diffracted fields are evaluated by stationary phase method and edge point technique, asymptotically. MTPO integral is compared with the exact solution and PO integral for the problem of scattering from a perfectly conducting half plane, numerically. It is observed that MTPO integral gives the total field that agrees with the exact solution and the result is more reliable than that of classical PO integral.

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Fig. 1. Scattered fields from a perfectly conducting surface and an aperture continuation

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Fig. 2 Reflection geometry from a perfectly conducting half plane

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Fig. 3. Transmission geometry for the modified theory of physical optics

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Fig. 4. Regions for scattered fields in a perfectly conducting half plane

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Fig. 5. Reflected and diffracted fields from perfectly conducting half plane (PO and exact solution)

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Fig. 6. Reflected and diffracted fields from perfectly conducting half plane [MTPO (β = ϕ ₀) and exact solution]

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Fig. 7. Reflected and diffracted fields from perfectly conducting half plane (MTPO and exact solution)

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Equations (66)

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{\vec{J}}_{es} = {\vec{n}}_{1} \times {\vec{H}}_{t} |_{S_{1}}

{\vec{J}}_{es} = {\vec{n}}_{2} \times {\vec{H}}_{i} |_{S_{2}}, {\vec{J}}_{ms} = - {\vec{n}}_{2} \times {\vec{E}}_{i} |_{S_{2}}

{\vec{n}}_{1} = \cos (u + α) \vec{t} + \sin (u + α) \vec{n}

{\vec{n}}_{2} = \cos (v + α) \vec{t} - \sin (v + α) \vec{n}

{\vec{E}}_{t} = {\vec{E}}_{is} + {\vec{E}}_{rs}

{\vec{E}}_{is} = - \frac{jω μ_{0}}{4 π} \iint_{S_{2}} {\vec{n}}_{2} \times {\vec{H}}_{i} |_{S_{2}} \frac{e^{- jk R_{2}}}{R_{2}} dS' + \frac{1}{4 π} \iint_{S_{2}} \nabla \times ({\vec{n}}_{2} \times {\vec{E}}_{i} |_{S_{2}} \frac{e^{- jk R_{2}}}{R_{2}}) dS'

{\vec{E}}_{rs} = - \frac{jω μ_{0}}{4 π} \iint_{S_{1}} {\vec{n}}_{1} \times {\vec{H}}_{t} |_{S_{1}} \frac{e^{- jk R_{1}}}{R_{1}} dS'

{\vec{H}}_{t} = {\vec{H}}_{is} + {\vec{H}}_{rs}

{\vec{H}}_{is} = \frac{1}{4 π} \iint_{S_{2}} \nabla \times ({\vec{n}}_{2} \times {\vec{H}}_{i} |_{S_{2}} \frac{e^{- jk R_{2}}}{R_{2}}) dS' + \frac{jω ε}{4 π} \iint_{S_{2}} {\vec{n}}_{2} \times {\vec{E}}_{i} |_{S_{2}} \frac{e^{- jk R_{2}}}{R_{2}} dS'

{\vec{H}}_{rs} = \frac{1}{4 π} \iint_{S_{1}} \nabla \times ({\vec{n}}_{1} \times {\vec{H}}_{t} |_{S_{1}} \frac{e^{- jk R_{1}}}{R_{1}}) dS'

{\vec{H}}_{i} = - \frac{E_{i}}{Z_{0}} ({\vec{e}}_{x} \sin ϕ_{0} - {\vec{e}}_{y} \cos ϕ_{0}) e^{jk (x \cos ϕ_{0} + y \sin ϕ_{0})}

{\vec{E}}_{r} = {\vec{e}}_{z} E_{r} e^{jk (x \cos β - y \sin β)}

E_{r} = - E_{i} e^{jkx' (\cos ϕ_{0} - \cos β)}

{\vec{J}}_{MTPO} = \frac{E_{i}}{Z_{0}} [\cos u - \cos (u + β + ϕ_{0})] e^{jkx' \cos ϕ_{0}} {\vec{e}}_{z}

{\vec{J}}_{PO} = {\vec{e}}_{z} 2 \frac{E_{i}}{Z_{0}} \sin ϕ_{0} e^{jkx' \cos ϕ_{0}}

{\vec{J}}_{MTPO} = \frac{2 E_{i}}{Z_{0}} \sin (\frac{β + ϕ_{0}}{2}) e^{jkx' \cos ϕ_{0}} {\vec{e}}_{z}

{\vec{E}}_{rs} = - {\vec{e}}_{z} \frac{jk E_{i}}{2 π} \int_{x' = 0}^{\infty} \int_{z' = - \infty}^{\infty} e^{jkx' \cos ϕ_{0}} \frac{e^{- jk R_{1}}}{R_{1}} \sin (\frac{β + ϕ_{0}}{2}) dx' dz'

R_{1} = \sqrt{{(x - x')}^{2} + y^{2} + {(z - z')}^{2}} .

\int_{c} e^{- jkchα} dα = \frac{π}{j} H_{0}^{(2)} (kR)

{\vec{E}}_{rs} = - {\vec{e}}_{z} \frac{k E_{i}}{2} \int_{x' = 0}^{\infty} e^{jkx' \cos ϕ_{0}} H_{0}^{(2)} (kR) \sin (\frac{β + ϕ_{0}}{2}) dx'

{\vec{E}}_{ieq} = - {\vec{e}}_{z} E_{i} e^{jk (x \cos ϕ_{0} - y \sin ϕ_{0})}

{\vec{H}}_{ieq} = - \frac{E_{i}}{Z_{0}} ({\vec{e}}_{x} \sin ϕ_{0} + {\vec{e}}_{y} \cos ϕ_{0}) e^{jk (x \cos ϕ_{0} - y \sin ϕ_{0})}

{\vec{J}}_{MTPO} = - \frac{2 E_{i}}{Z_{0}} \sin (\frac{β + ϕ_{0}}{2}) e^{jkx' \cos ϕ_{0}} {\vec{e}}_{z}

{\vec{E}}_{is} \approx {\vec{e}}_{z} \frac{k E_{i}}{\sqrt{2 π}} e^{j \frac{π}{4}} \int_{x' = - \infty}^{0} e^{jkx' \cos ϕ_{0}} \frac{e^{- jk R_{1}}}{\sqrt{k R_{1}}} \sin \frac{ϕ_{0} + β}{2} dx'

{\vec{E}}_{s} = {\vec{E}}_{rs} + {\vec{E}}_{is}

{\vec{E}}_{rs} \approx - {\vec{e}}_{z} \frac{k E_{i}}{\sqrt{2 π}} e^{j \frac{π}{4}} \int_{x' = 0}^{\infty} e^{jkx' \cos ϕ_{0}} \frac{e^{- jk R_{1}}}{\sqrt{k R_{1}}} \sin (\frac{β + ϕ_{0}}{2}) dx'

{\vec{E}}_{t} \approx {\vec{e}}_{z} \frac{k E_{i}}{\sqrt{2 π}} e^{j \frac{π}{4}} (\int_{x' = - \infty}^{0} e^{jkx' \cos ϕ_{0}} \frac{e^{- jk R_{1}}}{\sqrt{k R_{1}}} \sin \frac{ϕ_{0} + β}{2} dx' - \int_{x' = 0}^{\infty} e^{jkx' \cos ϕ_{0}} \frac{e^{- jk R_{1}}}{\sqrt{k R_{1}}} \sin (\frac{β + ϕ_{0}}{2}) dx')

g (x') = ρ \cos γ + x' (\cos β - \cos ϕ_{0})

\frac{\partial g}{\partial x'} = - ρ \sin γ \frac{dγ}{dx'} - x' \sin β \frac{dβ}{dx'} + \cos β - \cos ϕ_{0}

γ = π - β \mp ϕ

ρ \sin γ \frac{dγ}{dx'} = - x' \sin β \frac{dβ}{dx'}

β_{s} = ϕ_{0}

\frac{\partial g}{\partial x'} ∣_{s} = \cos β_{s} - \cos ϕ_{0} = 0

\frac{\partial^{2} g}{\partial {x'}^{2}} |_{s} = \frac{\sin^{2} ϕ_{0}}{l}

l = R_{1} |_{s} = ρ \cos γ_{s}

g (x') \approx l + \frac{1}{2} \frac{\sin^{2} ϕ_{0}}{l} {(x' - {x'}_{s})}^{2}

f ({x'}_{s}) \approx \pm \frac{k E_{i}}{\sqrt{2 π}} e^{j \frac{π}{4}} \frac{\sin ϕ_{0}}{\sqrt{kl}}

E_{r, i} \approx \pm \frac{k E_{i} \sin ϕ_{0}}{\sqrt{2 π}} e^{j \frac{π}{4}} \frac{e^{- jkl}}{\sqrt{kl}} \int_{- \infty}^{\infty} e^{- jk \frac{\sin^{2} ϕ_{0}}{l} (x' - {x'}^{2})} dx'

\int_{- \infty}^{\infty} e^{- \frac{y^{2}}{2}} dy = \sqrt{2 π} .

E_{r} \approx - E_{i} e^{jkρ \cos (ϕ + ϕ_{0})}

E_{i} \approx E_{i} e^{jkρ \cos (ϕ - ϕ_{0})}

{\vec{E}}_{d} \approx \mp {\vec{e}}_{z} \frac{1}{jk} \frac{f (0)}{g' (0)} e^{- jkg (0)}

g (0) = ρ,

g' (0) = - (\cos ϕ + \cos ϕ_{0})

f (0) = - \frac{k E_{i}}{2 \sqrt{2 π}} e^{j \frac{π}{4}} \frac{\cos (\frac{ϕ - ϕ_{0}}{2})}{\sqrt{kρ}}

{\vec{E}}_{rd} = {\vec{e}}_{z} \frac{E_{i}}{2 \sqrt{2 π}} \frac{e^{- jkρ}}{\sqrt{kρ}} \frac{\cos (\frac{ϕ - ϕ_{0}}{2})}{\cos ϕ + \cos ϕ_{0}} e^{- j \frac{π}{4}}

D_{rd} = \frac{e^{- j \frac{π}{4}}}{2 \sqrt{2 π}} \frac{\cos (\frac{ϕ - ϕ_{0}}{2})}{\cos ϕ + \cos ϕ_{0}}

g (0) = ρ,

g' (0) = - (\cos ϕ + \cos ϕ_{0})

f (0) = - \frac{k E_{i}}{2 \sqrt{2 π}} e^{j \frac{π}{4}} \frac{\cos (\frac{ϕ + ϕ_{0}}{2})}{\sqrt{kρ}}

{\vec{E}}_{id} = - {\vec{e}}_{z} \frac{E_{i}}{2 \sqrt{2 π}} \frac{e^{- jkρ}}{\sqrt{kρ}} \frac{\cos (\frac{ϕ + ϕ_{0}}{2})}{\cos ϕ + \cos ϕ_{0}} e^{- j \frac{π}{4}}

D_{id} = - \frac{e^{- j \frac{π}{4}}}{2 \sqrt{2 π}} \frac{\cos (\frac{ϕ + ϕ_{0}}{2})}{\cos ϕ + \cos ϕ_{0}}

D_{t} = D_{id} + D_{rd} = \frac{e^{- j \frac{π}{4}}}{\sqrt{2 π}} \frac{\cos (\frac{ϕ - ϕ_{0}}{2}) - \cos (\frac{ϕ + ϕ_{0}}{2})}{\cos ϕ + \cos ϕ_{0}}

E_{TPO} \approx E_{i} e^{jk (x \cos ϕ_{0} + y \sin ϕ_{0})} - \frac{k E_{i}}{2} \sin ϕ_{0} \int_{0}^{\infty} e^{jkx' \cos ϕ_{0}} H_{0}^{(2)} (k R_{1}) dx'

E_{MTPO} |_{β = ϕ_{0}} \approx \frac{k E_{i}}{2} (\int_{x' = - \infty}^{0} e^{jkx' \cos ϕ_{0}} H_{0}^{(2)} (k R_{1}) \sin ϕ_{0} dx' - \int_{0}^{\infty} e^{jkx' \cos ϕ_{0}} H_{0}^{(2)} (k R_{1}) \sin ϕ_{0} dx')

E_{TMTPO} = \frac{1}{2} [E_{i} (e^{jkρ \cos (ϕ - ϕ_{0})} - e^{jkρ \cos (ϕ + ϕ_{0})}) u (π - ϕ) + E_{TPO} + E_{R}]

u (π - ϕ) = {\begin{matrix} 1, ϕ \leq π \\ 0, ϕ \geq π \end{matrix}

E_{t} = 2 E_{i} \sum_{m = 1}^{\infty} e^{jm \frac{π}{4}} J_{\frac{m}{2}} (kρ) \sin \frac{m}{2} ϕ \sin \frac{m}{2} ϕ_{0}

E_{MTPO} \approx \frac{k E_{i}}{\sqrt{2 π}} e^{j \frac{π}{4}} (\int_{x' = - \infty}^{0} e^{jkx' \cos ϕ_{0}} \frac{e^{- jk R_{1}}}{\sqrt{k R_{1}}} \sin \frac{ϕ_{0} + β}{2} dx' - \int_{x' = 0}^{\infty} e^{jkx' \cos ϕ_{0}} \frac{e^{- jk R_{1}}}{\sqrt{k R_{1}}} \sin (\frac{β + ϕ_{0}}{2}) dx')

{\vec{H}}_{i} = {\vec{e}}_{z} H_{i} e^{jk (x \cos ϕ_{0} + y \sin ϕ_{0})}

{\vec{E}}_{i} = Z_{0} H_{i} (\sin ϕ_{0} {\vec{e}}_{x} + \cos ϕ_{0} {\vec{e}}_{y}) e^{jk (x \cos ϕ_{0} + y \sin ϕ_{0})}

{\vec{H}}_{r} = {\vec{e}}_{z} H_{r} e^{jk (x \cos β - y \sin β)}

{\vec{E}}_{r} = - Z_{0} H_{r} (\sin β {\vec{e}}_{x} + \cos β {\vec{e}}_{y}) e^{jk (x \cos β - y \sin β)}

H_{r} e^{jkx' \cos β} = H_{i} e^{jkx' \cos ϕ_{0}}

{\vec{J}}_{MTPO} = 2 H_{i} [{\vec{e}}_{x} \cos \frac{β - ϕ_{0}}{2} - {\vec{e}}_{y} \sin \frac{β - ϕ_{0}}{2}] e^{jkx' \cos ϕ_{0}}

{\vec{J}}_{PO} = {\vec{e}}_{x} 2 H_{i} e^{jkx' \cos ϕ_{0}}

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Optics Express