光技術基礎-波動光学(49)

～途中計算～

途中計算は、こちら。 \[ \begin{align*} {r_0}^{\prime} + r^{\thinspace \prime} &= z_0 + \frac{\left( x_0 - \xi \right)^2}{2 z_0} + \frac{\left( y_0 - \eta \right)^2}{2 z_0} + z + \frac{\left( x - \xi \right)^2}{2 z} + \frac{\left( y - \eta \right)^2 }{2 z} \\ &= z_0 + \frac{{x_0}^2}{2 z_0} - \frac{x_0 \xi}{z_0}+ \frac{{\xi}^2}{2 z_0} + \frac{{y_0}^2}{2 z_0} - \frac{y_0 \eta}{z_0} + \frac{{\eta}^2}{2 z_0} + z + \frac{x^2}{2 z} - \frac{x \xi}{z} + \frac{{\xi}^2}{2 z} + \frac{y^2 }{2 z} - \frac{y \eta}{z} + \frac{{\eta}^2 }{2 z} \\ &= z_0 + z + \frac{{x_0}^2}{2 z_0} + \frac{x^2}{2 z} + \frac{{y_0}^2}{2 z_0} + \frac{y^2}{2 z} - \xi \left( \frac{x_0}{z_0} + \frac{x}{z} \right) - \eta \left( \frac{y_0}{z_0} + \frac{y}{z} \right) + \frac{{\xi}^2}{2} \left( \frac{1}{z_0} + \frac{1}{z} \right) + \frac{{\eta}^2}{2} \left( \frac{1}{z_0} + \frac{1}{z} \right) \\ &= z_0 + z + \frac{{x_0}^2}{2 z_0} + \frac{x^2}{2 z} + \frac{{y_0}^2}{2 z_0} + \frac{y^2}{2 z} - \xi \frac{x_0 z + x z_0}{z_0 z} - \eta \frac{y_0 z + y z_0 }{z_0 z} + \frac{{\xi}^2}{2} \left( \frac{1}{z_0} + \frac{1}{z} \right) + \frac{{\eta}^2}{2} \left( \frac{1}{z_0} + \frac{1}{z} \right) \\ &= z_0 + z + \frac{{x_0}^2}{2 z_0} + \frac{x^2}{2 z} + \frac{{y_0}^2}{2 z_0} + \frac{y^2}{2 z} - \xi \frac{x_0 z + x z_0}{z_0 + z} \frac{z_0 + z}{z_0 z} - \eta \frac{y_0 z + y z_0 }{z_0 + z} \frac{z_0 + z}{z_0 z} + \frac{{\xi}^2}{2} \left( \frac{1}{z_0} + \frac{1}{z} \right) + \frac{{\eta}^2}{2} \left( \frac{1}{z_0} + \frac{1}{z} \right) \\ &= z_0 + z + \frac{{x_0}^2}{2 z_0} + \frac{x^2}{2 z} + \frac{{y_0}^2}{2 z_0} + \frac{y^2}{2 z} - \xi \xi_m \left( \frac{1}{z_0} + \frac{1}{z} \right) - \eta \eta_m \left( \frac{1}{z_0} + \frac{1}{z} \right) + \frac{{\xi}^2}{2} \left( \frac{1}{z_0} + \frac{1}{z} \right) + \frac{{\eta}^2}{2} \left( \frac{1}{z_0} + \frac{1}{z} \right) \\ &= z_0 + z + \frac{{x_0}^2}{2 z_0} + \frac{x^2}{2 z} + \frac{{y_0}^2}{2 z_0} + \frac{y^2}{2 z} - \frac{{\xi_m}^2}{2} \left( \frac{1}{z_0} + \frac{1}{z} \right) - \frac{{\eta_m}^2}{2} \left( \frac{1}{z_0} + \frac{1}{z} \right) \\ &\quad\quad\quad + \frac{{\xi_m}^2}{2} \left( \frac{1}{z_0} + \frac{1}{z} \right) + \frac{{\eta_m}^2}{2} \left( \frac{1}{z_0} + \frac{1}{z} \right) - \xi \xi_m \left( \frac{1}{z_0} + \frac{1}{z} \right) - \eta \eta_m \left( \frac{1}{z_0} + \frac{1}{z} \right) + \frac{{\xi}^2}{2} \left( \frac{1}{z_0} + \frac{1}{z} \right) + \frac{{\eta}^2}{2} \left( \frac{1}{z_0} + \frac{1}{z} \right) \\ &= z_0 + z + \frac{{x_0}^2}{2 z_0} + \frac{x^2}{2 z} + \frac{{y_0}^2}{2 z_0} + \frac{y^2}{2 z} - \frac{1}{2} \left( \frac{x_0 z + x z_0}{z_0 + z} \right)^2 \frac{z_0 + z}{z_0 z} - \frac{1}{2} \left( \frac{y_0 z + y z_0 }{z_0 + z} \right)^2 \frac{z_0 + z}{z_0 z} \\ &\quad\quad\quad + \frac{1}{2} \left( {\xi}^2 - 2 \xi \xi_m + {\xi_m}^2 \right) \left( \frac{1}{z_0} + \frac{1}{z} \right) + \frac{1}{2} \left( {\eta}^2 - 2 \eta \eta_m + {\eta_m}^2 \right) \left( \frac{1}{z_0} + \frac{1}{z} \right) \\ &= z_0 + z + \frac{{x_0}^2}{2 z_0} + \frac{x^2}{2 z} + \frac{{y_0}^2}{2 z_0} + \frac{y^2}{2 z} - \frac{1}{2 \left( z_0 + z \right)} \frac{{x_0}^2 z^2 +2 x_0 x z_0 z + x^2 {z_0}^2}{z_0 z} - \frac{1}{2 \left( z_0 + z \right)} \frac{{y_0}^2 z^2 + 2 y_0 y z_0 z + y^2 {z_0}^2 }{z_0 z} \\ &\quad\quad\quad + \frac{1}{2} \left( \xi - \xi_m \right)^2 \left( \frac{1}{z_0} + \frac{1}{z} \right) + \frac{1}{2} \left( \eta - \eta_m \right)^2 \left( \frac{1}{z_0} + \frac{1}{z} \right) \\ &= z_0 + z + \frac{1}{2 \left( z_0 + z \right)} \left( \frac{{x_0}^2 \left( z_0 + z \right)}{z_0} - \frac{{x_0}^2 z^2 + 2 x_0 x z_0 z + x^2 {z_0}^2}{z_0 z} + \frac{x^2 \left( z_0 + z \right)}{z} \right) + \frac{1}{2 \left( z_0 + z \right)} \left( \frac{{y_0}^2 \left( z_0 + z \right)}{z_0} - \frac{{y_0}^2 z^2 + 2 y_0 y z_0 z + y^2 {z_0}^2}{z_0 z} + \frac{y^2 \left( z_0 + z \right)}{z} \right)\\ &\quad\quad\quad + \frac{\left( \xi - \xi_m \right)^2 + \left( \eta - \eta_m \right)^2}{2} \left( \frac{1}{z_0} + \frac{1}{z} \right) \\ &= z_0 + z + \frac{1}{2 \left( z_0 + z \right)} \frac{{x_0}^2 z_0 z + {x_0}^2 z^2 - {x_0}^2 z^2 - 2 x_0 x z_0 z - x^2 {z_0}^2 + x^2 {z_0}^2 + x^2 z_0 z}{z_0 z} + \frac{1}{2 \left( z_0 + z \right)} \frac{{y_0}^2 z_0 z + {y_0}^2 z^2 - {y_0}^2 z^2 - 2 y_0 y z_0 z - y^2 {z_0}^2 + y^2 {z_0}^2 + y^2 z_0 z}{z_0 z} \\ &\quad\quad\quad + \frac{\left( \xi - \xi_m \right)^2 + \left( \eta - \eta_m \right)^2}{2} \left( \frac{1}{z_0} + \frac{1}{z} \right) \\ &= z_0 + z + \frac{1}{2 \left( z_0 + z \right)} \frac{{x_0}^2 z_0 z - 2 x_0 x z_0 z + x^2 z_0 z}{z_0 z} + \frac{1}{2 \left( z_0 + z \right)} \frac{{y_0}^2 z_0 z - 2 y_0 y z_0 z + y^2 z_0 z}{z_0 z} + \frac{\left( \xi - \xi_m \right)^2 + \left( \eta - \eta_m \right)^2}{2} \left( \frac{1}{z_0} + \frac{1}{z} \right) \\ &= z_0 + z + \frac{{x_0}^2 - 2 x_0 x + x^2}{2 \left( z_0 + z \right)} + \frac{{y_0}^2 - 2 y_0 y + y^2}{2 \left( z_0 + z \right)} + \frac{\left( \xi - \xi_m \right)^2 + \left( \eta - \eta_m \right)^2}{2} \left( \frac{1}{z_0} + \frac{1}{z} \right) \\ &= z_0 + z + \frac{\left( x_0 - x \right)^2}{2 \left( z_0 + z \right)} + \frac{\left( y_0 - y \right)^2}{2 \left( z_0 + z \right)} + \frac{\left( \xi - \xi_m \right)^2 + \left( \eta - \eta_m \right)^2}{2} \left( \frac{1}{z_0} + \frac{1}{z} \right) \\ &= z_0 + z + \frac{\left( x_0 - x \right)^2 + \left( y_0 - y \right)^2}{2 \left( z_0 + z \right)} + \frac{\left( \xi - \xi_m \right)^2 + \left( \eta - \eta_m \right)^2}{2} \left( \frac{1}{z_0} + \frac{1}{z} \right) \end{align*} \]

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