Lasers | Physics - Wyatt's Notes

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.

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: $\theta = \lambda/(\pi w_0)$

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).

flowchart TD
    A[21_Lasers 17] --> B[Key Concepts]
    A --> C[Core Principles]
    A --> D[Practical Applications]
    B --> E[Fundamental definitions]
    C --> F[Design patterns]
    D --> G[Real-world usage]

Lasers produce coherent light through stimulated emission, where an incoming photon triggers an excited atom to emit an identical photon. Population inversion, where more atoms are in excited than ground states, is essential for amplification. The optical cavity provides feedback, selecting specific frequencies and directions. The result is light that is monochromatic, coherent, and highly directional. Different gain media produce different wavelengths: helium-neon for red, argon-ion for blue-green, semiconductor diodes for infrared. Mode locking produces ultrashort pulses, while Q-switching produces high-energy pulses. Lasers enable precision measurements, surgery, and telecommunications.

• Coherence Theory — The coherence properties of laser light (temporal and spatial) are central to interferometry and holography applications.
• Fourier Optics — Gaussian beam optics and the Abbe limit relate laser beam quality to diffraction-limited resolution.
• Electromagnetic Waves — The wave equation and Poynting vector provide the electromagnetic foundation for laser cavity mode analysis.
• Calculus
• Linear Algebra
• Vector Calculus