Geometric Optics

6.1 Reflection and Refraction

Law of Reflection: The angle of incidence equals the angle of reflection: $\theta_i = \theta_r$ (both measured from the normal).

Snell”s Law: $n_1 \sin\theta_1 = n_2 \sin\theta_2$.

Derivation of Snell’s law from Fermat’s principle. The optical path length from point $A$ in Medium 1 to point $B$ in medium 2 via a point on the interface at $x$ is:

$\Lambda(x) = n_1\sqrt{a^2 + x^2} + n_2\sqrt{b^2 + (d - x)^2}$

Setting $d\Lambda/dx = 0$:

$n_1 \frac{x}{\sqrt{a^2 + x^2}} = n_2 \frac{d - x}{\sqrt{b^2 + (d-x)^2}}$

Which gives $n_1 \sin\theta_1 = n_2 \sin\theta_2$. $\blacksquare$

6.2 Total Internal Reflection

When light travels from a denser to a rarer medium ($n_1 \gt n_2$), total internal reflection occurs When $\theta_1 \geq \theta_c$ where:

$\sin\theta_c = \frac{n_2}{n_1}$

Evanescent wave. Beyond the critical angle, the transmitted field decays exponentially:

$E_t \propto e^{-\kappa x}$

Where $\kappa = \frac{2\pi}{\lambda}\sqrt{n_1^2 \sin^2\theta_1 - n_2^2}$.

6.3 The Thin Lens Equation

$\frac{1}{s} + \frac{1}{s'} = \frac{1}{f}$

Where $s$ is the object distance, $s'$ is the image distance, and $f$ is the focal length.

Sign convention (Cartesian): Distances are positive in the direction of light propagation. $f \gt 0$ For converging lenses, $f \lt 0$ for diverging.

Magnification:

$M = -\frac{s'}{s}$

Negative $M$ indicates an inverted image.

6.4 Lensmaker’s Equation

For a thin lens with radii of curvature $R_1$ and $R_2$:

$\frac{1}{f} = (n - 1)\left(\frac{1}{R_1} - \frac{1}{R_2}\right)$

6.5 Matrix Optics (Ray Transfer Matrix)

A paraxial ray is described by a vector $\begin{pmatrix} y \\ \theta \end{pmatrix}$ where $y$ is the Height and $\theta$ is the angle with the optical axis.

Free space propagation by distance $d$:

$M_{\mathrm{prop} = \begin{pmatrix} 1 & d \\ 0 & 1 \end{pmatrix}}$

Thin lens of focal length $f$:

$M_{\mathrm{lens} = \begin{pmatrix} 1 & 0 \\ -1/f & 1 \end{pmatrix}}$

System matrix: The overall transformation is the product of individual matrices (applied in Reverse order): $M_{\mathrm{sys} = M_n \cdots M_2 M_1}$.

6.6 Mirror Equation

For a spherical mirror of radius $R$ (with $R \gt 0$ for concave, $R \lt 0$ for convex):

$\frac{1}{s} + \frac{1}{s'} = \frac{2}{R}$

The focal length is $f = R/2$. The magnification is $M = -s'/s$ (negative for inverted images).

Derivation from the law of reflection. A ray from the object at height $h$ striking the mirror At height $y$ reflects such that $\theta_i = \theta_r$. In the paraxial approximation ($y \ll R$), Applying the law of reflection and the small-angle approximation $\sin\theta \approx \theta$:

$\frac{h}{s} + \frac{h'}{s'} = \frac{2y}{R}$

Dividing through by $y$ and using $h/s = y/s$, $h'/s' = y/s'$ (paraxial rays) yields the mirror Equation. $\blacksquare$

6.7 Optical Instruments

Magnifying glass. Angular magnification when the image is at the near point $D$:

$M = 1 + \frac{D}{f}$

Compound microscope. Total magnification:

$M_{\mathrm{total} = -\frac{L}{f_o} \cdot \frac{D}{f_e}}$

Where $L$ is the tube length, $f_o$ is the objective focal length, and $f_e$ is the eyepiece focal Length.

Refracting telescope. Angular magnification:

$M = -\frac{f_o}{f_e}$

For large magnification, the objective should have a long focal length and the eyepiece a short one. The length of the telescope tube is approximately $f_o + f_e$.

Reflecting telescope. A concave primary mirror replaces the objective lens. The Cassegrain design Uses a secondary convex mirror to redirect the focus behind the primary. Advantages: no chromatic Aberration, lighter and cheaper for large apertures, and easier support structures.