## 5 Calculated Relationships for Particular Cases of Optical Circuits  
If a circular diaphragm with diameter \(2a\) is the source of radiation, whose center is located at the point O\({}^{\prime}\) (\(x_{0}\), \(y_{0}\), 0) (see Figure 2), and the normal to the diaphragm aperture makes an angle \(\alpha_{0}\) with the \(y\)-axis, then in Relationships (17) and (36), the quantities \(E^{\prime}_{\theta^{\prime}}\) and \(E^{\prime}_{\psi^{\prime}}\) for the case of polarization in question will be determined by the expressions:  
\[ \begin{split} E^{\prime}_{\theta^{\prime}\parallel}&=-i \frac{\exp\left(ikr^{\prime}\right)}{r^{\prime}}\cdot\frac{a\sin\psi^{\prime} }{\sin\theta^{\prime}}J_{1}(ka\sin\theta^{\prime});E^{\prime}_{\psi^{\prime} \parallel}=-i\frac{\exp\left(ikr^{\prime}\right)}{r^{\prime}}\cdot\frac{a\cos \theta^{\prime}\cos\psi^{\prime}}{\sin\theta^{\prime}}J_{1}(ka\sin\theta^{ \prime});\\ E^{\prime}_{\theta^{\prime}\perp}&=-i\frac{\exp\left( ikr^{\prime}\right)}{r^{\prime}}\cdot\frac{a\cos\psi^{\prime}}{\sin\theta^{ \prime}}J_{1}(ka\sin\theta^{\prime});\;E^{\prime}_{\psi^{\prime}\perp}=-i\frac{ \exp\left(ikr^{\prime}\right)}{r^{\prime}}\cdot\frac{a\cos\theta^{\prime} \sin\psi^{\prime}}{\sin\theta^{\prime}}J_{1}(ka\sin\theta^{\prime}),\end{split}  \tag{47}\]where \(a\) is the diaphragm radius, \(J_{1}\) is the Bessel function of the first order. Similar relationships can be obtained for a mirror diaphragm, where instead of the coordinates of the system denoted by a single stroke, it is necessary to write coordinates with two strokes. By substituting Relationship (47) in Expressions (17) and (36), and carrying out transformations similar to those given above, one can obtain expressions for calculating the field on the beam splitter surface. In the case when the beam splitter is not completely illuminated (represents itself a round \"spot\", described on the surface of the beam splitter by cylindrical coordinates \(\rho\) and \(\varphi\)), it is convenient to represent the source field by the cylindrical components of the vector \(E_{\rho}\) and \(E_{\varphi}\). On the surface \(y=0\), the expressions for these components have the form:  
\[ \begin{split} E_{\rho}&=\frac{\overline{r}^{\prime }}{\overline{r}^{\prime\prime}}\left[\sin\varphi\sin\alpha_{0}+\frac{ \overline{y}^{\prime}(\rho+x_{0}\sin\varphi)}{\left(\overline{r}^{\prime} \right)^{2}}\right]E_{\theta^{\prime}}^{\prime}+\frac{\overline{r}^{\prime \prime}(-x_{0}\cos\alpha_{0}+y_{0}\sin\alpha_{0})\cos\varphi}{\rho^{2}\cos^{2 }\varphi+\left(\overline{x}\right)^{2}}E_{\psi^{\prime}}^{\prime};\\ E_{\varphi}&=\frac{\overline{r}^{\prime}\cos\varphi }{\overline{r}^{\prime\prime}}\left[\sin\alpha_{0}+\frac{\overline{y}^{ \prime}\,x_{0}}{\left(\overline{r}^{\prime}\right)^{2}}\right]E_{\theta^{ \prime}}^{\prime}+\frac{\overline{r}^{\prime\prime}(\rho\cos\alpha_{0}+x_{0} \sin\varphi\cos\alpha_{0}-y_{0}\sin\varphi\sin\alpha_{0})}{\rho^{2}\cos^{2} \varphi+\left(\overline{x}\right)^{2}}E_{\psi^{\prime}}^{\prime},\end{split}  \tag{48}\]  
where \(\overline{r}^{\prime}=\sqrt{\rho^{2}+2\rho x_{0}\sin\varphi+x_{0}^{2}+y_{0}^{ 2}}\); \(\overline{x}^{\prime}=(\rho\sin\varphi+x_{0})\cos\alpha_{0}-y_{0}\sin\alpha_{0}\);  
\[\overline{y}^{\prime}=-(\rho\sin\varphi+x_{0})\sin\alpha_{0}-y_{0}\cos\alpha_ {0};\ \overline{r}^{\prime}=\sqrt{\left(\overline{r}^{\prime}\right)^{2}-\left( \overline{y}^{\prime}\right)^{2}}. \tag{49}\]  
The same relationships can be obtained by expanding the vector \(E\) written in a spherical coordinate system associated with a mirror diaphragm. To do this, we need to replace the notations with one stroke by two strokes in Expressions (48) and (49), and replace \(x_{0}\), \(y_{0}\), \(\alpha_{0}\) by values of \(x_{0}^{\prime\prime},y_{0}^{\prime\prime},-(\alpha_{0}+2\alpha)\) from Expression (3), respectively. Figure 2: Special case of optical circuits (the source of radiation is the diaphragm).