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Optics formulas

Remember that: $ \vec{a} = \mathbf{a} $, $\dot{a}= \frac{da}{dt}$ and $\Delta a = a_{final} - a_{initial} = a_{f} - a_{i} = a_{2} - a_{1} $. The $\propto$ symbol is read as "is proportional to". While $\hat{a} = \frac{\vec{a}}{\Vert \vec{a} \Vert}$ is the unit vector.

Symbols

name	symbol
amplitude	$$A$$
slit width	$$a$$
light velocity	$$c = 299\ 792\ 458\ \mathrm{m/s}$$
distance	$$d$$
energy	$$E$$
focal length, frequency	$$f$$
Planck constant	$$\begin{split} h &= 6.626 \times 10^{-34}\ \mathrm{J \cdot s} \\ &= 4.136 \times 10^{-15}\ \mathrm{eV \cdot s} \end{split}$$
intensity	$$I$$
distance from diffraction slits to screen	$$L$$
angular magnification (or amplification)	$$M$$
lateral magnification, diffraction number of order	$$m$$
near point approximation	$$NP$$
refractive index	$$n$$
optical power	$$P$$
curvature radius (spherical mirror)	$$r$$
object distance	$$s$$
image distance	$$s^{\prime}$$
speed of light in a medium	$$v$$
object height	$$y$$
image height	$$y^{\prime}$$
phase difference	$$\delta$$
wavelength	$$\lambda$$
subtended angle	$$\theta$$
phase difference of several waves	$$\phi$$

Geometrical optics

name	equation
refractive index	$$n = \frac{c}{v}$$
reflection law	$$\theta_{i} = \theta_{r}$$
Snell's law of refraction	$$n_{1} \sin \theta_{1} = n_{2} \sin \theta_{2}$$ $$\frac{\sin \theta_{1}}{v_{1}} = \frac{\sin \theta_{2}}{v_{2}}$$
reflected intensity	$$I = \left( \frac{n_{1} - n_{2}}{n_{1} + n_{2}} \right)^{2} I_{0}$$
total internal reflection critical angle	$$\sin \theta_{c} = \frac{n_{2}}{n_{1}}$$
Malus law	$$I = I_{0} \cos^{2} \theta$$
Brewster angle, polarization angle	$$\tg \theta_{p} = \frac{n_{2}}{n_{1}}$$

Spherical mirror

name	equation
mirror equation	$$\frac{1}{f} = \frac{1}{s} + \frac{1}{s^{\prime}}$$
focal length of a mirror	$$f = \frac{1}{2}r$$
lateral magnification	$$m = \frac{y^{\prime}}{y} = - \frac{s^{\prime}}{s}$$
lateral magnification using curvature radius	$$\frac{y^{\prime}}{y} = - \frac{r/2}{s - \left( r/2 \right)}$$

Lenses

name	equation
refraction on a single surface	$$\frac{n_{1}}{s} + \frac{n_{2}}{s^{\prime}} = \frac{n_{2} - n_{1}}{r}$$
magnification due to a refracting surface	$$m = \frac{y^{\prime}}{y} = - \frac{n_{1} s^{\prime}}{n_{2}s}$$
lens maker equation	$$\frac{1}{f} = \left( n - 1 \right) \left( \frac{1}{r_{1}} - \frac{1}{r_{2}} \right)$$
thin lens equation	$$\frac{1}{f} = \frac{1}{s} + \frac{1}{s^{\prime}}$$
optical power	$$P = \frac{1}{f}$$
effective focal length of two lenses	$$\frac{1}{f_{ef}} = \frac{1}{f_{1}} + \frac{1}{f_{2}}$$
optical power of two lenses	$$P_{ef} = P_{1} + P_{2}$$
subtended angle for the eye	$$\theta = \frac{y}{s}$$
subtended angle for near point	$$\theta = \frac{y}{NP}$$
subtended angle with lens	$$\theta = \frac{y}{f}$$
lens angular magnification (or amplification)	$$M = \frac{\theta}{\theta_{o}} = \frac{NP}{f}$$
microscope objective lens lateral magnification	$$m_{o} = \frac{y^{\prime}}{y} = - \frac{L}{f_{o}}$$
ocular lens angular magnification	$$M_{e} = \frac{x_{pp}}{f_{e}}$$
microscope angular magnification	$$M = m_{o}M_{e} = - \frac{L}{f_{o}}\frac{x_{pp}}{f_{e}}$$
telescope angular magnification	$$M = \frac{\theta_{e}}{\theta_{o}} = - \frac{f_{o}}{f_{e}}$$

Physical optics

Two slit interference

name	equation
phase difference due to the difference in the optical path traveled	$$\delta = \frac{\Delta r}{\lambda} 2\pi = \frac{\Delta r}{\lambda} 360\ ^{\circ}\mathrm{C}$$
two-slit interference maxima	$$d \sin \theta_{\max} = m\lambda,\quad m = 0,1,2, \ldots$$
two-slit interference minima	$$d \sin \theta_{\min} = \left( m - \frac{1}{2} \right)\lambda,\quad m = 1,2,3, \ldots$$
two-slit phase difference	$$\frac{\delta}{2 \pi}= \frac{d \sin \theta}{\lambda}$$
distance on the screen to the m-th bright fringe	$$y_{m} = m \frac{\lambda L}{d}$$
intensity as a function of phase difference	$$I = 4 I_{0} \cos^{2} \frac{1}{2} \delta$$

One slit diffraction

name	equation
intensity null values	$$a\sin \theta = m \lambda, \qquad m = 1,2,3, \ldots$$
zero intensity minima	$$\tg \theta_{1} = \frac{y_{1}}{L}$$
diffraction intensity of a slit	$$I = I_{0} \left( \frac{\sin \frac{1}{2} \phi}{\frac{1}{2} \phi} \right)^{2}$$

Quantum optics

name	equation
equation for photon energy	$$E = hf = \frac{hc}{\lambda}$$
hc product	$$hc = 1240\ \mathrm{eV \cdot nm}$$
photon energy	$$\vert \Delta E \vert = hf$$
emitted wavelength	$$\lambda = \frac{c}{f} = \frac{hc}{\vert \Delta E \vert}$$

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