3.4 フレネル領域
フレネル領域（距離$r$），あるいは円すいホーン（開口径$D$，軸長$L$）に対しては，
\begin{gather}
t=\frac{D^2}{8\lambda} \left( \frac{1}{r} + \frac{1}{L}\right)
\end{gather}
を導入して，積分項は，
\begin{gather}
I_{m \pm 1,n} \equiv
\int_0^1 J_{m \pm 1} \left( \bar{\chi}_{mn} \bar{\rho} \right)
J_{m \pm 1} (u \bar{\rho}) e^{-j 2\pi t \bar{\rho}^2} \bar{\rho} d\bar{\rho}
\end{gather}
このとき，
\begin{gather}
\bar{\VEC{F}}_{mn} = \frac{1+\cos \theta}{2}
\Big( \bar{N}_x \VEC{a}_\xi + \bar{N}_y \VEC{a}_\eta \Big)
\end{gather}
ここで，
\begin{eqnarray}
\bar{N}_x
&=& j^{m-1} A_{mn} \pi a\bar{\chi}_{mn}
\left\{ I_{m-1,n} \ \begin{matrix} \cos \\ \sin \end{matrix} (m-1) \phi
- \ell I_{m+1,n} \ \begin{matrix} \cos \\ \sin \end{matrix} (m+1) \phi \right\}
\\
\bar{N}_y
&=& -j^{m-1} A_{mn} \pi a\bar{\chi}_{mn}
\nonumber \\
&&\cdot \left\{ I_{m-1,n} \ \begin{matrix} \sin \\ -\cos \end{matrix} (m-1) \phi
+ \ell I_{m+1,n} \ \begin{matrix} \sin \\ -\cos \end{matrix} (m+1) \phi \right\}
\end{eqnarray}
これより，
\begin{eqnarray}
&&\bar{N}_x \cos \phi + \bar{N}_y \sin \phi
\nonumber \\
&=& j^{m-1} A_{mn} \pi a\bar{\chi}_{mn}
\nonumber \\
&&\cdot \left[
\left\{ I_{m-1,n} \ \begin{matrix} \cos \\ \sin \end{matrix} (m-1) \phi
- \ell I_{m+1,n} \ \begin{matrix} \cos \\ \sin \end{matrix} (m+1) \phi \right\} \cos \phi \right.
\nonumber \\
&&\left.
- \left\{ I_{m-1,n} \ \begin{matrix} \sin \\ -\cos \end{matrix} (m-1) \phi
+ \ell I_{m+1,n} \ \begin{matrix} \sin \\ -\cos \end{matrix} (m+1) \phi \right\} \sin \phi \right]
\end{eqnarray}
上式の$[ \ \ ]$の中は，
\begin{eqnarray}
&&I_{m-1,n} \left\{ \begin{matrix} \cos \\ \sin \end{matrix} (m-1) \phi \cos \phi
- \begin{matrix} \sin \\ -\cos \end{matrix} (m-1) \phi \sin \phi \right\}
\nonumber \\
&&- \ell I_{m+1,n} \left\{ \begin{matrix} \cos \\ \sin \end{matrix} (m+1) \phi \cos \phi
+ \begin{matrix} \sin \\ -\cos \end{matrix} (m+1) \phi \sin \phi \right\}
\nonumber \\
&=& I_{m-1,n} \ \begin{matrix} \cos \\ \sin \end{matrix} m \phi
- \ell I_{m+1,n} \ \begin{matrix} \cos \\ \sin \end{matrix} m \phi
\nonumber \\
&=& \Big( I_{m-1,n} - \ell I_{m+1,n} \Big) \begin{matrix} \cos \\ \sin \end{matrix} m \phi
\end{eqnarray}
また，
\begin{eqnarray}
&&-\bar{N}_x \sin \phi + \bar{N}_y \cos \phi
\nonumber \\
&=& j^{m-1} A_{mn} \pi a\bar{\chi}_{mn}
\nonumber \\
&&\cdot \left[
-\left\{ I_{m-1,n} \ \begin{matrix} \cos \\ \sin \end{matrix} (m-1) \phi
- \ell I_{m+1,n} \ \begin{matrix} \cos \\ \sin \end{matrix} (m+1) \phi \right\} \sin \phi \right.
\nonumber \\
&&\left.
- \left\{ I_{m-1,n} \ \begin{matrix} \sin \\ -\cos \end{matrix} (m-1) \phi
+ \ell I_{m+1,n} \ \begin{matrix} \sin \\ -\cos \end{matrix} (m+1) \phi \right\} \cos \phi \right]
\end{eqnarray}
上式の$[ \ \ ]$の中は，
\begin{eqnarray}
&&-I_{m-1,n} \left( \begin{matrix} \cos \\ \sin \end{matrix} (m-1) \phi \sin \phi
+ \begin{matrix} \sin \\ -\cos \end{matrix} (m-1) \phi \cos \phi \right)
\nonumber \\
&&- \ell I_{m+1,n} \left( - \begin{matrix} \cos \\ \sin \end{matrix} (m+1) \phi \sin \phi
+ \begin{matrix} \sin \\ -\cos \end{matrix} (m+1) \phi \cos \phi \right)
\nonumber \\
&=& -I_{m-1,n} \ \begin{matrix} \sin \\ -\cos \end{matrix} m \phi
- \ell I_{m+1,n} \ \begin{matrix} \sin \\ -\cos \end{matrix} m \phi
\nonumber \\
&=& -\Big( I_{m-1,n} + \ell I_{m+1,n} \Big) \begin{matrix} \sin \\ -\cos \end{matrix} m \phi
\end{eqnarray}
したがって，
\begin{align}
&\bar{N}_x \cos \phi + \bar{N}_y \sin \phi
= j^{m-1} A_{mn} \pi a\bar{\chi}_{mn}
\Big( I_{m-1,n} - \ell I_{m+1,n} \Big) \begin{matrix} \cos \\ \sin \end{matrix} m \phi
\\
&-\bar{N}_x \sin \phi + \bar{N}_y \cos \phi
= -j^{m-1} A_{mn} \pi a\bar{\chi}_{mn}
\Big( I_{m-1,n} + \ell I_{m+1,n} \Big) \begin{matrix} \sin \\ -\cos \end{matrix} m \phi
\end{align}
上式の積分項を計算するため，ベッセル関数の不定積分公式
\begin{align}
&\int z J_{\nu }(\alpha z) J_{\nu }(\beta z) dz
\nonumber \\
&= \frac{z}{\alpha ^2 - \beta ^2}
\left\{ \beta J_{\nu }(\alpha z) J_{\nu }'(\beta z)
- \alpha J_{\nu }'(\alpha z) J_{\nu }(\beta z) \right\} \ \ \ \ \
(\alpha \neq \beta )
\end{align}
を次のように変形する．
\begin{eqnarray}
&&\int \Big\{ J_{\nu -1}(\alpha z) J_{\nu -1}(\beta z)
\pm J_{\nu +1}(\alpha z) J_{\nu +1}(\beta z) \Big\} z \ dz
\nonumber \\
&=& \frac{z}{\alpha ^2 - \beta ^2}
\Big\{ \beta J_{\nu -1}(\alpha z) J_{\nu -1}'(\beta z)
- \alpha J_{\nu -1}'(\alpha z) J_{\nu -1}(\beta z)
\nonumber \\
&&\pm \beta J_{\nu +1}(\alpha z) J_{\nu +1}'(\beta z)
\mp \alpha J_{\nu +1}'(\alpha z) J_{\nu +1}(\beta z) \Big\}
\nonumber \\
&=& \frac{z}{\alpha ^2 - \beta ^2}
\left[ \beta J_{\nu -1}(\alpha z)
\left\{ \frac{\nu -1}{\beta z} J_{\nu -1}(\beta z) - J_{\nu }(\beta z) \right\} \right.
\nonumber \\
&&- \alpha \left\{ \frac{\nu -1}{\alpha z} J_{\nu -1}(\alpha z) - J_{\nu }(\alpha z) \right\}
J_{\nu -1}(\beta z)
\nonumber \\
&&\pm \beta J_{\nu +1}(\alpha z) \left\{ J_{\nu }(\beta z)
- \frac{\nu +1}{\beta z} J_{\nu +1}(\beta z) \right\}
\nonumber \\
&&\left.
\mp \alpha \left\{ J_{\nu }(\alpha z) - \frac{\nu +1}{\alpha z} J_{\nu +1}(\alpha z) \right\}
J_{\nu +1}(\beta z) \right]
\nonumber \\
&=& \frac{z}{\alpha ^2 - \beta ^2}
\Big\{ - \beta J_{\nu -1}(\alpha z) J_{\nu }(\beta z)
+ \alpha J_{\nu }(\alpha z) J_{\nu -1}(\beta z)
\nonumber \\
&&\pm \beta J_{\nu +1}(\alpha z) J_{\nu }(\beta z)
\mp \alpha J_{\nu }(\alpha z) J_{\nu +1}(\beta z) \Big\}
\nonumber \\
&=& \frac{z}{\alpha ^2 - \beta ^2}
\Big[ \alpha J_{\nu }(\alpha z)
\big\{ J_{\nu -1}(\beta z) \mp J_{\nu +1}(\beta z) \big\}
\nonumber \\
&&- \beta \big\{ J_{\nu -1}(\alpha z) \mp J_{\nu +1}(\alpha z) \big\}
J_{\nu }(\beta z) \Big]
\end{eqnarray}
上側符号については，
\begin{align}
&\int \Big\{ J_{\nu -1}(\alpha z) J_{\nu -1}(\beta z)
+ J_{\nu +1}(\alpha z) J_{\nu +1}(\beta z) \Big\} z \ dz
\nonumber \\
&= \frac{2z}{\alpha ^2 - \beta ^2}
\Big\{ \alpha J_{\nu }(\alpha z) J_{\nu }'(\beta z)
- \beta J_{\nu }'(\alpha z) J_{\nu }(\beta z) \Big\}
\end{align}
一方，下側符号については，
\begin{align}
&\int \Big\{ J_{\nu -1}(\alpha z) J_{\nu -1}(\beta z)
- J_{\nu +1}(\alpha z) J_{\nu +1}(\beta z) \Big\} z \ dz
\nonumber \\
&= \frac{z}{\alpha ^2 - \beta ^2}
\Big\{ \alpha J_{\nu }(\alpha z) \frac{2\nu}{\beta z} J_{\nu }(\beta z)
- \beta \frac{2\nu}{\alpha z} J_{\nu }(\alpha z) J_{\nu }(\beta z) \Big\}
\nonumber \\
&= \frac{2\nu}{\alpha \beta} J_{\nu }(\alpha z) J_{\nu }(\beta z)
\end{align}
したがって，定積分は，
\begin{eqnarray}
&&I_{m-1,n} + I_{m+1,n}
\nonumber \\
&=& \int_0^1 \Big( J_{m-1} \left( \bar{\chi}_{mn} \bar{\rho} \right) J_{m-1} (u \bar{\rho})
+ J_{m+1} \left( \bar{\chi}_{mn} \bar{\rho} \right) J_{m+1} (u \bar{\rho}) \Big)
\bar{\rho} d\bar{\rho}
\nonumber \\
&=& \frac{2}{\bar{\chi}_{mn}^2 - u^2}
\Big[ \bar{\rho} \big\{ \bar{\chi}_{mn} J_m (\bar{\chi}_{mn} \bar{\rho}) J_m'(u \bar{\rho})
- u J_m'(\bar{\chi}_{mn} \bar{\rho}) J_m (u \bar{\rho}) \big\} \Big]_0^1
\nonumber \\
&=& \frac{2}{\bar{\chi}_{mn}^2 - u^2}
\Big\{ \bar{\chi}_{mn} J_m (\bar{\chi}_{mn}) J_m'(u) - u J_m'(\bar{\chi}_{mn}) J_m (u) \Big\}
\end{eqnarray}
また，
\begin{eqnarray}
&&I_{m-1,n} - I_{m+1,n}
\nonumber \\
&=& \int_0^1 \Big( J_{m-1} \left( \bar{\chi}_{mn} \bar{\rho} \right) J_{m-1} (u \bar{\rho})
- J_{m+1} \left( \bar{\chi}_{mn} \bar{\rho} \right) J_{m+1} (u \bar{\rho}) \Big)
\bar{\rho} d\bar{\rho}
\nonumber \\
&=& \frac{2m}{\bar{\chi}_{mn} u} \Big[ J_m (\bar{\chi}_{mn} \bar{\rho}) J_m(u \bar{\rho}) \Big]_0^1
\nonumber \\
&=& \frac{2m}{\bar{\chi}_{mn} u} J_m (\bar{\chi}_{mn}) J_m(u)
\end{eqnarray}
TE$_{mn}$モードのとき，
$\bar{\chi}_{mn} = \chi_{mn}'$，$J_m'(\chi_{mn}') = 0$，$\ell=1$ より，
\begin{eqnarray}
I_{m-1,n} -\ell I_{m+1,n}
&=& I_{m-1,n} - I_{m+1,n}
\nonumber \\
&=& \frac{2m}{\chi_{mn}'u} J_m (\chi_{mn}') J_m(u)
\\
I_{m-1,n} +\ell I_{m+1,n}
&=& I_{m-1,n} + I_{m+1,n}
\nonumber \\
&=& \frac{2\chi_{mn}'}{\chi_{mn}^{\prime 2} - u^2} J_m (\chi_{mn}') J_m'(u)
\end{eqnarray}
これより，
\begin{align}
&\bar{N}_x^{\TE} \cos \phi + \bar{N}_y^{\TE} \sin \phi
= j^{m-1} A_{[mn]} \pi a \frac{2m}{u} J_m (\chi_{mn}') J_m(u)
\ \begin{matrix} \cos \\ \sin \end{matrix} m \phi
\\
&-\bar{N}_x^{\TE} \sin \phi + \bar{N}_y^{\TE} \cos \phi
\nonumber \\
&= -j^{m-1} A_{[mn]} \pi a
\frac{2}{1-\left( \frac{u}{\chi_{mn}'} \right)^2} J_m (\chi_{mn}') J_m'(u)
\ \begin{matrix} \sin \\ -\cos \end{matrix} m \phi
\end{align}
よって，
\begin{gather}
\VEC{E}_{p[mn]}
= \frac{j}{\lambda} \ \frac{e^{-jkr}}{r} \VEC{F}_{[mn]} (\theta ,\phi)
\end{gather}
ここで，
\begin{eqnarray}
\VEC{F}_{[mn]} (\theta ,\phi)
&=& j^{m-1} A_{[mn]} \pi a J_m(\chi _{mn}') \sqrt{Z_{[mn]}}\frac{1}{2}
\nonumber \\
&&\cdot \left[ \left\{ 1+\frac{\beta _{[mn]}}{k} \cos \theta
+\Gamma \left( 1 - \frac{\beta _{[mn]}}{k} \cos \theta \right) \right\} \right.
\nonumber \\
&& \cdot \frac{2m}{u} J_m(u) \ \begin{matrix} \cos \\ \sin \end{matrix} m \phi \VEC{a}_\theta
\nonumber \\
&&- \left\{ \cos \theta +\frac{\beta _{[mn]}}{k} + \Gamma \left( \cos \theta - \frac{\beta _{[mn]}}{k} \right) \right\}
\nonumber \\
&&\frac{2J_m'(u)}{1-\left( \frac{u}{\chi_{mn}'} \right)^2}
\left. \cdot \begin{matrix} \sin \\ -\cos \end{matrix} m \phi \ \VEC{a}_\phi \right]
\end{eqnarray}
TMモード
一方，TM$_{mn}$モードのとき，
$\bar{\chi}_{mn} = \chi_{mn}$，$J_m(\chi_{mn}) = 0$，$\ell =-1$ より，
\begin{align}
&I_{m-1,n} -\ell I_{m+1,n}
= I_{m-1,n} + I_{m+1,n}
= - \frac{2u}{\chi_{mn}^2 - u^2} J_m' (\chi_{mn}) J_m(u)
\\
&I_{m-1,n} +\ell I_{m+1,n}
= I_{m-1,n} - I_{m+1,n}
= 0
\end{align}
また，
\begin{align}
&\bar{N}_x^{\TM} \cos \phi + \bar{N}_y^{\TM} \sin \phi
\nonumber \\
&= -j^{m-1} A_{(mn)} \pi a \chi_{mn}
\frac{\frac{2u}{\chi_{mn}^2}}{1-\left( \frac{u}{\chi_{mn}} \right)^2}
J_m' (\chi_{mn}) J_m(u)
\ \begin{matrix} \cos \\ \sin \end{matrix} m \phi
\\
&-\bar{N}_x^{\TM} \sin \phi + \bar{N}_y^{\TM} \cos \phi = 0
\end{align}
よって，
\begin{gather}
\VEC{E}_{p(mn)} = \frac{j}{\lambda} \ \frac{e^{-jkr}}{r} \VEC{F}_{(mn)} (\theta ,\phi)
\end{gather}
ここで，
\begin{eqnarray}
\VEC{F}_{(mn)} (\theta ,\phi)
&=& -j^{m-1} A_{(mn)} \pi a J_m' (\chi_{mn}) \frac{\frac{2u}{\chi_{mn}}}{1-\left( \frac{u}{\chi_{mn}} \right)^2}
J_m(u) \ \begin{matrix} \cos \\ \sin \end{matrix} m \phi \sqrt{Z_{(mn)}}
\nonumber \\
&&\cdot \frac{1}{2} \left\{ 1+\frac{k}{\beta _{(mn)}} \cos \theta
+\Gamma \left( 1 - \frac{k}{\beta _{(mn)}} \cos \theta \right) \right\} \VEC{a}_\theta
\nonumber
\end{eqnarray}