Geometrical Optics

$$c$$ - the speed of light in vacuum

$$c=3*10^8 \frac{m}{s}$$

$$v$$ - the speed of light in a specific medium

$$n$$ - refractive index of a specific medium

$$n=\frac{c}{v}$$

$$\epsilon_r$$ - relative electric permitivity of a medium

$$\mu_r$$ - relative magnetic permitivity of a medium

$$v=\frac{c}{\sqrt{\epsilon_r*\mu_r}}$$

The law of relfection $$i=r$$

$$i$$ - incidence angle

$$r$$ - relfection/refraction angle

The law of refraction $$\frac{\sin i}{\sin r}=\frac{n_2}{n_1}$$

$$n_{21}=\frac{n_2}{n_1}$$ - relative index of refraction of two mediums

Total reflexion $$\sin l=\frac{n_2}{n_1}$$

$$l$$ - total reflexion limit angle



$$A=r_1+r_2$$

$$\delta$$ - deviation angle

$$\delta=i_1+i_2-A$$

Emergence condition $$A\le2l$$

$$n_{21}*\sin (A-l) \le \sin i_1$$

Minimum deviation condition: $$i_1=i_2 r_1=r_2$$

$$\delta_{min}=2(i-r)$$

$$r=\frac{A}{2}$$

$$i=\frac{A+\delta_{min}}{2}$$

$$n=\frac{\sin \frac{\delta_{min}+A}{2}}{\sin \frac{A}{2}}$$

Gaussian aproximation: for $$\alpha < 5^o$$

$$\tan \alpha \simeq \sin \alpha \simeq \alpha$$

$$\cos \alpha \simeq 1$$

Conjugate points formulas

$$x_1$$ distance object - diopter

$$x_2$$ distance diopter - image

$$R$$ radius of curvature of the diopter

$$\frac{n_2}{x_2}-\frac{n_1}{x_1}=\frac{n_2-n_1}{R}$$

$$n_2 \left (\frac{1}{R}-\frac{1}{x_2} \right)=n_1 \left (\frac{1}{R}-\frac{1}{x_1} \right)$$

$$f$$ focal distance

$$f_1=-\frac{n_1R}{n_2-n_1}$$

$$f_2=\frac{n_2R}{n_2-n_1}$$

$$\frac{f_1}{x_1}+\frac{f_2}{x_2}=1$$

Magnification $$\beta=\frac{y_1}{y_2}$$

$$y_1$$ dimension of the objecy

$$y_2$$ dimension of the image

Liniar transvers magnification for diopters $$\beta=\frac{x_2}{x_1}*\frac{n_1}{n_2}$$

Liniar transvers magnification for plane diopters $$\beta=1$$

Parallel faces slide



$$\delta=\frac{d \sin (i-r)}{\cos r}$$

Mirrors and lens

Optical instruments

Common refractive indexes