Laser Physics & Coherent Light
Laser Physics & Coherent Light
A laser (Light Amplification by Stimulated Emission of Radiation) produces coherent, directional, near-monochromatic light by coupling an optical gain medium to a resonant cavity. Stimulated emission — predicted by Einstein in 1917 — occurs when a photon of energy $h\nu$ induces an excited atom to emit an identical photon, building up a coherent field when population inversion ($N_2>N_1$) is maintained.
A Fabry-Perot cavity of length $L$ supports longitudinal modes at $\nu_q=qc/(2nL)$, spaced by $\Delta\nu_\text{FSR}=c/(2nL)$. Transverse Gaussian modes TEM$_{mn}$ govern beam profile. Beam quality $M^2\geq1$ measures closeness to the diffraction limit.
Laser Rate Equations
$$\dot{N}_2=R_p-N_2/\tau-\sigma c\phi N_2,\quad \dot{\phi}=\sigma c\phi N_2-\phi/\tau_c+\beta N_2/\tau.$$
Threshold Condition
Lasing begins when round-trip gain equals loss: $e^{2g_\text{th}L}=1/\sqrt{R_1R_2}\,e^{2\alpha L}$. Above threshold output power grows linearly with pump power.
Example 1
He-Ne cavity: $L=30\,\text{cm}$, $n=1$. Find longitudinal mode spacing.
Solution: $\Delta\nu_\text{FSR}=c/(2L)=3\times10^8/0.6\approx500\,\text{MHz}$.
Example 2
Why is population inversion necessary for lasing?
Solution: Without inversion, absorption dominates stimulated emission and the medium attenuates rather than amplifies the field.
Practice
- Distinguish spontaneous from stimulated emission.
- What is population inversion and how is it achieved in a 4-level laser?
- How does cavity length determine the longitudinal mode spectrum?
- What does $M^2=1$ signify about a laser beam?
Show Answer Key
1. Spontaneous emission: an excited atom decays randomly, emitting a photon in a random direction with random phase (incoherent). Stimulated emission: an incoming photon triggers decay, producing a clone photon with the same frequency, phase, direction, and polarization (coherent amplification). Stimulated emission is the basis of laser operation.
2. Population inversion: more atoms in the upper laser level than the lower ($N_2 > N_1$). In a 4-level laser: pump excites atoms from ground (0) to level 3, which rapidly decays (non-radiatively) to upper laser level 2. Lasing occurs on 2→1 transition. Level 1 rapidly decays to ground. Since level 1 is nearly empty, inversion ($N_2 > N_1$) is easily achieved.
3. Longitudinal modes: standing waves satisfying $L = m\lambda/(2n)$, $m$ integer. Mode spacing: $\Delta\nu = c/(2nL)$. Longer cavity → smaller mode spacing → more modes within the gain bandwidth. Single-mode operation requires an intracavity etalon or short cavity to select one mode.
4. $M^2 = 1$ means the beam is an ideal Gaussian (diffraction-limited). $M^2 > 1$ indicates a non-ideal beam that diverges $M^2$ times faster than the diffraction limit. It is a beam quality factor: the ratio of the beam's divergence-waist product to that of an ideal Gaussian. Lower $M^2$ → tighter focus, better beam quality.