Lasers

17.1 Conditions for Lasing

A laser requires three conditions:

1. Population inversion: $N_2 > N_1$ for the lasing transition (between levels 2 and 1), achieved by pumping.
2. Stimulated emission dominance: The stimulated emission rate must exceed the absorption rate: $N_2 > N_1$.
3. Optical feedback: A resonant cavity ( two mirrors) provides positive feedback.

Threshold condition: The gain per round trip must exceed the losses:

$R_1 R_2\,e^{2gL} \geq 1$

Where $R_1, R_2$ are mirror reflectivities, $g$ is the gain coefficient, and $L$ is the cavity length.

The threshold gain:

$g_{\text{th} = -\frac{1}{2L}\ln(R_1 R_2) = \alpha_i + \alpha_m}$

Where $\alpha_i$ is the internal loss and $\alpha_m = -\ln(R_1 R_2)/(2L)$ is the mirror loss.

17.2 Laser Types

17.3 Gaussian Beam Optics

The fundamental mode (\text{TEM_}{00}) of a laser cavity is a Gaussian beam:

$E(r, z) = E_0\frac{w_0}{w(z)}\exp\!\left(-\frac{r^2}{w(z)^2}\right)\exp\!\left(-i\left[kz + \frac{kr^2}{2R(z)} - \zeta(z)\right]\right)$

Beam parameters:

• Beam waist: $w_0$ (minimum spot size)
• Rayleigh range: $z_R = \pi w_0^2/\lambda$
• Beam radius: $w(z) = w_0\sqrt{1 + (z/z_R)^2}$
• Radius of curvature: $R(z) = z[1 + (z_R/z)^2]$
• Divergence angle: $\theta = \lambda/(\pi w_0)$

Common Pitfalls (Additional)

1. Coherence length limits interferometer arm difference: In a Michelson interferometer, the path difference must not exceed the coherence length $l_c = \lambda^2/\Delta\lambda$ for fringes to be visible. White light fringes are visible only for near-zero path difference ($l_c \sim 1.5\,\mu$M), while laser fringes remain visible for path differences of many metres.

2. The Abbe limit is not a fundamental limit: Techniques such as STED (stimulated emission depletion), PALM (photoactivated localisation microscopy), and SIM (structured illumination microscopy) can achieve resolutions well below the Abbe limit of $\lambda/(2\text{NA})$. The 2014 Nobel Prize in Chemistry was awarded for super-resolution microscopy.

3. Gaussian beams do not have sharp edges: Unlike geometrical optics rays, Gaussian beams have no well-defined edge. The beam radius $w$ is defined as the $1/e^2$ intensity radius ($\sim 86.5\%$ of the peak). The power contained within $w$ is $1 - e^{-2} \approx 86.5\%$ of the total, not 100%.

4. Spatial filtering with a pinhole: A pinhole of diameter $d$ in the focal plane of a lens acts as a low-pass spatial filter with cutoff frequency $f_c = d/(\lambda f)$. The transmitted beam approaches a Gaussian profile (Airy pattern central maximum), which is why spatial filtering is used to “clean up” laser beams.

5. Polarisation and Brewster”s angle: At Brewster’s angle, the reflected beam is purely $s$-polarised, not the transmitted beam. The transmitted beam has reduced $s$-component and becomes partially $p$-polarised. Complete polarisation of the transmitted beam requires many interfaces (pile-of-plates polariser).

Problems (Additional)