Diffraction by a black half plane: Modified theory of physical optics approach

Yusuf Z. Umul

doi:10.1364/OPEX.13.007276

Optics Express
Vol. 13,
Issue 19,
pp. 7276-7287
(2005)
•https://doi.org/10.1364/OPEX.13.007276

Diffraction by a black half plane: Modified theory of physical optics approach

Yusuf Z. Umul

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Yusuf Z. Umul, "Diffraction by a black half plane: Modified theory of physical optics approach," Opt. Express 13, 7276-7287 (2005)

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Abstract

The scattered fields from a black half plane which absorbs all the incoming electromagnetic energy are evaluated by defining a new modified theory of physical optics surface current. This current eliminates the reflected fields, coming from the first stationary point of the reflection integral and only creates a reflected diffracted field. The incident scattered fields are found from the same integral, written for the perfectly conducting half plane. The scattered fields are evaluated by using the stationary phase method and edge point technique. The evaluated fields are plotted numerically.

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Fig. 1. Reflection geometry from a PEC half plane

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Fig. 2. Comparison of the PO and MTPO integrals for the places of the stationary points

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Fig. 3. Comparison of MTPO and incident diffracted fields

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Fig. 4. Comparison of MTPO and incident total fields

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Fig. 5. MTPO and incident total diffracted fields

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Fig. 6. MTPO and incident total fields

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Fig. 7 MTPO total diffracted fields

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Equations (56)

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E_{rz} = - e^{j \frac{π}{4}} \frac{{kE}_{i}}{\sqrt{2 π}} \int_{0}^{\infty} e^{jkx' \cos ϕ_{0}} \frac{e^{- jkR}}{\sqrt{kR}} \sin (\frac{β + ϕ_{0}}{2}) dx'

I (k, \vec{r}) = \int_{0}^{\infty} f (\vec{r}, x') e^{- jkg (\vec{r}, x')} dx'

f (\vec{r}, x') = - e^{j \frac{π}{4}} \frac{{kE}_{i}}{\sqrt{2 π}} \frac{\sin (\frac{β + ϕ_{0}}{2})}{\sqrt{kR}}

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

\cos ϕ_{0} - \cos β = 0

β_{s 1} = ϕ_{0}, β_{s 2} = 2 π - ϕ_{0}

I \approx f (x_{s}) e^{- j [kg (x_{s}) + \frac{π}{4}]} \sqrt{\frac{2 π}{kg'' (x_{s})}}

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

E_{rz} = e^{j \frac{π}{4}} \frac{{kE}_{i}}{\sqrt{2 π}} \int_{0}^{\infty} e^{jkx' \cos ϕ_{0}} \frac{e^{- jkR}}{\sqrt{kR}} \sin (\frac{β - ϕ_{0}}{2}) dx'

I \approx - \frac{f (x_{e})}{jkg' (x_{e})} e^{- jkg (x_{e})}

E_{rds} = \frac{E_{i}}{\sqrt{2 π}} \frac{\cos \frac{ϕ - ϕ_{0}}{2}}{\cos ϕ + \cos ϕ_{0}} \frac{e^{- jkρ}}{\sqrt{kρ}}

{\vec{J}}_{MTPO} = {\vec{e}}_{z} \frac{2 E_{i}}{Z_{0}} \sin \frac{β + ϕ_{0}}{2} 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}}

u_{A} = u_{i} + u_{di}

u_{S} = u_{r} + u_{dr} .

u ∣_{S +} - u ∣_{S -} = u_{i}

{(u_{S} + u_{i} + u_{di}) ∣}_{S +} - (u_{S} + u_{di}) ∣_{S -} = u_{i}

{(u_{S} + u_{di}) ∣}_{S +} = (u_{S} + u_{di}) ∣_{S -}

u_{S} ∣_{S \mp} = - u_{di} ∣_{S \mp}

{\vec{J}}_{es} = {\vec{n} \times {\vec{H}}_{T} ∣}_{S} = 0

{\vec{J}}_{ms} = - {\vec{n} \times {\vec{E}}_{T} ∣}_{S} = 0

{\vec{J}}_{es} = {\vec{n}}_{1} \times {\vec{H}}_{T} ∣_{S} \neq 0

{\vec{J}}_{ms} = - {\vec{n}}_{1} \times {\vec{E}}_{T} ∣_{S} \neq 0

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

{\vec{J}}_{ms} = - E_{i} ({\vec{e}}_{x} \cos \frac{β - ϕ_{0}}{2} - {\vec{e}}_{y} \sin \frac{β - ϕ_{0}}{2}) \sin \frac{β - ϕ_{0}}{2} e^{jkx' \cos ϕ_{0}}

{\vec{J}}_{es} = \vec{n} \times {\vec{H}}_{T} ∣_{S} = 0

\vec{n} \times {\vec{E}}_{T} ∣_{S} = 0

{\vec{J}}_{es} = {{\vec{n}}_{1} \times {\vec{H}}_{T} ∣}_{S} \neq 0

{\vec{n}}_{1} \times {\vec{E}}_{T} ∣_{S} = 0

{\vec{J}}_{es} = {\vec{e}}_{z} \frac{{2 E}_{i}}{Z_{0}} \sin \frac{β + ϕ_{0}}{2} \sin \frac{β - ϕ_{0}}{2} e^{jkx' \cos ϕ_{0}}

E_{sz} = - \frac{{kE}_{i}}{\sqrt{2 π}} e^{j \frac{π}{4}} \int_{0}^{\infty} e^{jkx' \cos ϕ_{0}} \sin \frac{β - ϕ_{0}}{2} \sin \frac{β + ϕ_{0}}{2} \frac{e^{- jkR}}{\sqrt{kR}} dx'

\vec{n} \times {\vec{H}}_{T} ∣_{S} = 0

{\vec{J}}_{ms} = - \vec{n} \times {\vec{E}}_{T} ∣_{S} = 0

\vec{n} \times {\vec{H}}_{T} ∣_{S} = 0

{\vec{J}}_{ms} = - \vec{n} \times {\vec{E}}_{T} ∣_{S} \neq 0

{\vec{J}}_{ms} = - 2 E_{i} ({\vec{e}}_{x} \cos \frac{β - ϕ_{0}}{2} - {\vec{e}}_{y} \sin \frac{β - ϕ_{0}}{2}) \sin \frac{β - ϕ_{0}}{2} e^{jkx' \cos ϕ_{0}}

\vec{F} = \frac{ε_{0}}{4 π} \underset{s}{\int\int} {\vec{J}}_{ms} \frac{e^{- jkR}}{R} dS' .

{\vec{E}}_{s} = \frac{e^{- j \frac{π}{4}}}{2 \sqrt{2 π}} \int_{0}^{\infty} \nabla \times ({\vec{J}}_{ms} \frac{e^{- jkR}}{\sqrt{kR}}) dx'

jk \frac{e^{- jkR}}{\sqrt{kR}} (\frac{\partial R}{\partial x} \sin \frac{β - ϕ_{0}}{2} + \frac{\partial R}{\partial y} \cos \frac{β - ϕ_{0}}{2}) = jk \frac{e^{- jkR}}{\sqrt{kR}} \sin \frac{β + ϕ_{0}}{2}

E_{sz} = - \frac{{kE}_{i}}{\sqrt{2 π}} e^{j \frac{π}{4}} \int_{0}^{\infty} e^{jkx' \cos ϕ_{0}} \sin \frac{β - ϕ_{0}}{2} \sin \frac{β + ϕ_{0}}{2} \frac{e^{- jkR}}{\sqrt{kR}} dx'

E_{tz} = \frac{{kE}_{i}}{\sqrt{2 π}} e^{j \frac{π}{4}} (\int_{- \infty}^{\infty} e^{jkx' \cos ϕ_{0}} \sin \frac{β + ϕ_{0}}{2} \frac{e^{- jkR}}{\sqrt{kR}} dx' - \int_{0}^{\infty} e^{jkx' \cos ϕ_{0}} \sin \frac{β - ϕ_{0}}{2} \sin \frac{β + ϕ_{0}}{2} \frac{e^{- jkR}}{\sqrt{kR}} dx')

E_{is} {= e}^{jkρ \cos (ϕ - ϕ_{0})} u (- ξ_{i}) - \frac{e^{- j \frac{π}{4}}}{2 \sqrt{2 π} \cos \frac{ϕ - ϕ_{0}}{2}} \frac{e^{- jkρ}}{\sqrt{kρ}}

E_{dr} = \frac{e^{- j \frac{π}{4}}}{2 \sqrt{2 π}} \frac{e^{- jkρ}}{\sqrt{kρ}}

E_{dr} = \frac{e^{- j \frac{π}{4}}}{2 \sqrt{2 π}} (1 - \frac{1}{\cos \frac{ϕ - ϕ_{0}}{2}}) \frac{e^{- jkρ}}{\sqrt{kρ}}

E_{dtc} = \frac{e^{- j \frac{π}{4}}}{2 \sqrt{2 π}} (\frac{sgn (- ξ_{i})}{\cos \frac{ϕ_{0}}{2}} - \frac{1}{\cos \frac{ϕ - ϕ_{0}}{2}}) \frac{e^{- jkρ}}{\sqrt{kρ}}

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

E_{di} = - \hat{F} (- \sqrt{2 kρ} \cos \frac{ϕ - ϕ_{0}}{2}) e^{jkρ \cos (ϕ - ϕ_{0})}

E_{diu} = - F (∣ ξ_{i} ∣) sgn (ξ_{i}) e^{jkρ \cos (ϕ - ϕ_{0})}

F (x) = \frac{e^{j \frac{π}{4}}}{\sqrt{π}} \int_{x}^{\infty} e^{{- jt}^{2}} dt .

E_{Su} = E_{i} e^{jkρ \cos (ϕ - ϕ_{0})} F (ξ_{i}) + \frac{E_{i} e^{- j \frac{π}{4}}}{2 \sqrt{2 π} \cos (\frac{ϕ_{0}}{2})} \frac{e^{- jkρ}}{\sqrt{kρ}} sgn (- ξ_{i})

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

E_{sz} = e^{jkρ \cos (ϕ - ϕ_{0})} F (- \sqrt{2 kρ} \cos \frac{ϕ - ϕ_{0}}{2})

E_{Sz} = e^{jkρ \cos (ϕ - ϕ_{0})} F (∣ \sqrt{2 kρ} \cos \frac{ϕ - ϕ_{0}}{2} ∣) sgn (- \sqrt{2 kρ} \cos \frac{ϕ - ϕ_{0}}{2}) .

E_{dr} = \frac{e^{- j \frac{π}{4}}}{2 \sqrt{2 π}} \frac{e^{- j 2 kρ \cos^{2} \frac{ϕ_{0}}{2}}}{\sqrt{2 kρ} \cos \frac{ϕ_{0}}{2}} e^{jkρ \cos ϕ_{0}} sgn (- ξ_{i})

E_{dr} = - \hat{F} (- \sqrt{2 kρ} \cos \frac{ϕ_{0}}{2}) e^{jkρ \cos ϕ_{0}} sgn (- ξ_{i})

E_{dr} = - F (∣ \sqrt{2 kρ} \cos \frac{ϕ_{0}}{2} ∣) sgn (- \sqrt{2 kρ} \cos \frac{ϕ_{0}}{2}) sgn (- ξ_{i}) e^{jkρ \cos ϕ_{0}}

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Equations (56)

Optics Express