Optical formulas

The emitted wavelength of an LED is set by the band gap, so $E\approx E_g$. SI: $h=6.63\times 10^{-34}$ J s, $c=3\times 10^8$ m/s, $1\text{eV}=1.6\times 10^{-19}$ J.

$\Delta \lambda$ is the spectral width of the source (order-of-magnitude relation). Clear interference fringes need the path difference to be smaller than $L_c$.

Light of wavelength longer than ${\lambda }_c$ is not absorbed and gives no photocurrent. Silicon ($E_g\approx 1.12$ eV): ${\lambda }_c\approx 1.1\mu$m.

For a detector without internal gain, $0\le \eta \le 1$. Avalanche gain is counted separately.

$I_p$ is the photocurrent and $P$ the incident optical power. Applies for $\lambda <{\lambda }_c$; responsivity falls to zero beyond the cut-off. $M$ is the multiplication factor of an avalanche photodiode ($M=1$ for an ordinary photodiode).

The photocurrent follows optical power, not field amplitude, so the electrical power in the load $R_L$ varies as the square of the optical power. The bar denotes an average over many optical cycles.

$N$ fringes counted; $\lambda$ is the wavelength in the medium of the path; the mirror moves along the beam (Michelson, light passes twice).

A transparent plate of index $n$ and thickness $t$ (or a gas cell of length $t$ whose index changes from 1 to $n$) placed normal to the beam in one arm shifts $N$ fringes. Michelson double pass, surrounding medium of index 1, $\lambda$ the vacuum wavelength.

Step-index fibre. $n_1$ core index, $n_2$ cladding index, $n_0$ index of the outside medium (1 for air); ${\theta }_{\max }$ is the acceptance half-angle measured from the fibre axis; ${\theta }_c$ is the critical angle at the core-cladding boundary.

$a$ is the core radius. Step-index fibre.

$n_{\text{eff}}$ is the effective index of the guided mode and $\Lambda$ the grating period. Strain or temperature changes $\Lambda$ and $n_{\text{eff}}$, which shifts the reflected wavelength ${\lambda }_B$.

$\epsilon$ is the molar absorptivity at the chosen wavelength (usually L mol${}^{-1}$ cm${}^{-1}$), $c$ the concentration (mol/L) and $l$ the path length (cm). Holds for a dilute, non-scattering solution, monochromatic light, and no association or reaction of the absorbing species.

$T$ as a fraction; $T\%$ is the percentage transmittance.

The ion starts from rest, is accelerated through potential $V$, then moves on a circle of radius $r$ in a uniform magnetic field $B$ perpendicular to its velocity. SI units.

The ion starts from rest, is accelerated through $V$ and drifts a field-free length $L$. Flight time grows as $\sqrt{m/q}$.