Geometrical Optics Summary Notes

• Model the behaviour of light when hitting various surfaces and combinations of surfaces with the laws of reflection and refraction. Find all angles and draw a verbose diagram showing them.

  • Apply the law of reflection (angle of incidence equals angle of reflection) and Snell’s law for refraction ($n_1\sin ({\theta }_1)=n_2\sin ({\theta }_2)$), where n is the refractive index). Draw a detailed diagram indicating incident, reflected, and refracted angles, normal line, and relevant surface interfaces.

• Find a critical angle of total internal reflection.

  • Use Snell’s law. Set the angle of refraction to 90 degrees (since total internal reflection occurs when the refracted ray is along the boundary). Rearrange Snell’s law to find the critical angle: ${\theta }_c={\sin }^{-1}(\frac{n_2}{n_1})$, where $n_1$ is the refractive index of the denser medium, and $n_2$ is that of the less dense medium.

• Find the image of an object of a given size and given distance away from a concave or convex mirror with a given radius of curvature.

  • Use the mirror equation ($\frac{1}{f}=\frac{1}{v}+\frac{1}{u}$) and the magnification equation ($m=\frac{-v}{u}$), where $f$ is the focal length (half the radius of curvature for a mirror), $v$ is the image distance, and $u$ is the object distance. The signs of $v$ and $u$ depend on the convention being used.

• Find the focal length of a given convex-concave lens, or biconvex, where both radii are known.

  • Use the lens maker’s equation: $\frac{1}{f}=(n-1)\times (\frac{1}{R_1}-\frac{1}{R_2})$, where $n$ is the refractive index of the lens material, $R_1$ and $R_2$ are the radii of curvature of the two surfaces (sign conventions apply).

• Calculate interference patterns (maxima count, minima count, etc.) based on wavelength, distance of the screen away from the slits, as well as the first interference maximum’s distance from the central maximum.

  • Apply the formula for double-slit interference: $d\times \sin (\theta )=m\times \lambda$, where $d$ is the separation of the slits, $\theta$ is the angle to the maxima/minima, $\lambda$ is the wavelength, and $m$ is the order of the maxima/minima. Use geometric relations to find $\theta$ based on the given distances.

• Calculate the spacing between successive maxima/minima near the central fringe on a screen for a given wavelength, distance from screen, and slit separation distance.

  • Use the formula $\Delta y=\frac{\lambda \ast L}{d}$, where $\Delta y$ is the spacing between fringes, $\lambda$ is the wavelength, $L$ is the distance to the screen, and $d$ is the slit separation. This formula applies near the central maximum where the path difference is small.

• Calculate the intensities of light transmitted between various interfaces.

  • Use the Fresnel equations, which give the reflection and transmission coefficients. These coefficients depend on the angle of incidence, the polarisation of light, and the refractive indices of the two media at the interface. The intensity of transmitted light is then calculated using the transmission coefficient.