LASER Physics and Optical Amplification

The physical foundation of LASER (Light Amplification by Stimulated Emission of Radiation) was established in 1917 by Albert Einstein through thermodynamic analysis of atoms in equilibrium with blackbody radiation.

1. The Three Fundamental Radiative Transitions

Consider an atomic system with two discrete energy levels $E_1$ (lower state) and $E_2$ (upper state), separated by energy $h\nu = E_2 - E_1$, immersed in an electromagnetic radiation field of spectral energy density $\rho(\nu)$:

1. Induced (Stimulated) Absorption: An atom in level 1 absorbs a photon of energy $h\nu$ and transitions to level 2. The rate of absorption is: $$R_{\text{abs}} = B_{12} N_1 \rho(\nu)$$ where $B_{12}$ is Einstein's coefficient of stimulated absorption.
2. Spontaneous Emission: An atom in level 2 spontaneously drops to level 1 without any external trigger, emitting a photon of energy $h\nu$ in a random direction and phase. The transition rate is: $$R_{\text{spont}} = A_{21} N_2$$ where $A_{21}$ is Einstein's coefficient of spontaneous emission (equal to the inverse spontaneous lifetime $1/\tau_{\text{sp}}$).
3. Stimulated (Induced) Emission: An incident photon of energy $h\nu$ perturbs an atom in excited level 2, inducing it to drop to level 1 and emit a second photon that is an **identical clone** of the incident photon (identical frequency, wavevector $\mathbf{k}$, phase, and polarization state). The rate is: $$R_{\text{stim}} = B_{21} N_2 \rho(\nu)$$

2. Thermodynamic Derivation of Einstein Relations

In thermodynamic equilibrium at absolute temperature $T$, the rate of upward transitions must equal the rate of downward transitions (principle of detailed balance): $$R_{\text{abs}} = R_{\text{spont}} + R_{\text{stim}} \implies B_{12} N_1 \rho(\nu) = A_{21} N_2 + B_{21} N_2 \rho(\nu)$$ Solving for the spectral radiation density: $$\rho(\nu) = \frac{A_{21} N_2}{B_{12} N_1 - B_{21} N_2} = \frac{A_{21} / B_{21}}{\left(\frac{B_{12}}{B_{21}}\right) \frac{N_1}{N_2} - 1}$$ From Maxwell-Boltzmann statistics, the equilibrium population ratio of non-degenerate states is: $$\frac{N_1}{N_2} = \frac{g_1 e^{-E_1/k_B T}}{g_2 e^{-E_2/k_B T}} = \frac{g_1}{g_2} e^{(E_2 - E_1)/k_B T} = \frac{g_1}{g_2} e^{h\nu / k_B T}$$ Substituting into $\rho(\nu)$: $$\rho(\nu) = \frac{A_{21} / B_{21}}{\left(\frac{B_{12} g_1}{B_{21} g_2}\right) e^{h\nu / k_B T} - 1}$$ Comparing this directly with Planck’s Blackbody Radiation Formula: $$\rho(\nu) = \frac{8\pi h \nu^3}{c^3} \frac{1}{e^{h\nu / k_B T} - 1}$$ Equating corresponding terms yields the fundamental **Einstein Relations**: $$g_1 B_{12} = g_2 B_{21} \quad \implies \quad B_{12} = B_{21} \quad (\text{for } g_1 = g_2)$$ $$\frac{A_{21}}{B_{21}} = \frac{8\pi h \nu^3}{c^3}$$

3. Physical Consequence

The probability of stimulated emission is strictly identical to the probability of stimulated absorption ($B_{21} = B_{12}$). The ratio of spontaneous to stimulated emission scales as $\nu^3$, explaining why optical and UV lasers are vastly harder to construct than microwave masers.

Under thermal equilibrium at any temperature $T > 0$, the Boltzmann distribution mandates: $$\frac{N_2}{N_1} = e^{-h\nu / k_B T} < 1 \implies N_1 > N_2$$ Because $N_1 > N_2$, stimulated absorption always exceeds stimulated emission ($B_{12} N_1 \rho > B_{21} N_2 \rho$). Any optical beam propagating through such a medium is attenuated exponentially: $I(z) = I_0 e^{-\alpha z}$. To achieve optical amplification (gain), we must create a non-equilibrium condition where the excited state population exceeds the lower state: $$N_2 > \left(\frac{g_2}{g_1}\right) N_1 \quad \implies \quad \textbf{Population Inversion}$$

1. 3-Level vs 4-Level Pumping Schemes

• 3-Level Laser (e.g., Ruby Laser): Pumping excites ground state atoms ($E_1$) into a broad pump band ($E_3$). Fast non-radiative decay transfers atoms to a metastable laser level ($E_2$). Laser action occurs between $E_2 \to E_1$. Disadvantage: Because the lower laser level $E_1$ is the ground state containing nearly all atoms, more than $50\%$ of the total atomic population must be excited just to reach transparency ($N_2 = N_1$). This requires immense pump threshold power.
• 4-Level Laser (e.g., Nd:YAG, He-Ne): Laser action occurs between metastable state $E_3$ and an excited lower laser level $E_2$ situated well above the ground state ($E_2 - E_1 \gg k_B T$). Advantage: In thermal equilibrium, level $E_2$ is essentially empty ($N_2 \approx 0$). Thus, even a very weak pump creates population inversion ($N_3 > N_2$), resulting in orders-of-magnitude lower threshold pump power and enabling continuous-wave (CW) operation.

2. The Laser Cavity Threshold Gain Condition

An active gain medium of length $L$ with small-signal gain coefficient $\gamma_0$ is placed inside an optical resonator bounded by two mirrors of reflectances $R_1$ and $R_2$. For a beam of intensity $I$ completing one full round-trip of length $2L$:

1. Amplification through gain medium: $e^{2 \gamma_0 L}$
2. Internal scattering and absorption losses: $e^{-2 \alpha_s L}$
3. Mirror reflections: $R_1 R_2$

For steady-state sustained laser oscillation, the round-trip gain must equal the round-trip loss: $$R_1 R_2 e^{2(\gamma_{\text{th}} - \alpha_s)L} = 1$$ Taking the natural logarithm: $$2(\gamma_{\text{th}} - \alpha_s)L + \ln(R_1 R_2) = 0$$ $$\gamma_{\text{th}} = \alpha_s + \frac{1}{2L} \ln\left(\frac{1}{R_1 R_2}\right)$$ Laser oscillation initiates the instant the pumping rate drives the gain $\gamma_0$ above this critical threshold $\gamma_{\text{th}}$.

An optical resonator provides the positive optical feedback necessary to build coherent oscillations and determines the spectral and spatial coherence of the laser output.

1. Longitudinal (Axial) Cavity Modes

For constructive interference of waves reflecting back and forth between two flat parallel mirrors separated by cavity length $L$, the cavity must support an integral number $m$ of half-wavelengths: $$2 L = m \lambda_m = m \frac{c}{n \nu_m} \implies \nu_m = m \frac{c}{2 n L}, \quad m \in \mathbb{Z}$$ The frequency spacing between two adjacent longitudinal modes is constant: $$\Delta \nu = \nu_{m+1} - \nu_m = \frac{c}{2 n L}$$ For a He-Ne laser with $L = 30\text{ cm}$, $\Delta \nu = \frac{3 \times 10^8 \text{ m/s}}{2(0.30 \text{ m})} = 500\text{ MHz}$.

2. Transverse Electromagnetic Modes ($\text{TEM}_{mn}$)

Diffraction at mirror apertures and beam divergence create transverse intensity variations described by Hermite-Gaussian modes: $$I_{mn}(x, y) = I_0 \left[ H_m\left(\frac{\sqrt{2}x}{w}\right) \right]^2 \left[ H_n\left(\frac{\sqrt{2}y}{w}\right) \right]^2 e^{-2(x^2 + y^2)/w^2}$$ The fundamental mode $\text{TEM}_{00}$ has a purely Gaussian profile ($H_0(\xi) = 1$) with zero null lines, providing diffraction-limited beam divergence.

3. Survey of Practical Laser Systems

Laser Type	Active Medium	Wavelength	Pumping	Operation
Helium-Neon (He-Ne)	Neutral He + Ne gas mixture (10:1)	$632.8\text{ nm}$ (Red)	Electric DC discharge	Continuous Wave (CW)
Ruby Laser	$\text{Cr}^{3+}$ in $\text{Al}_2\text{O}_3$ crystal	$694.3\text{ nm}$ (Deep Red)	Xenon flashtube	Pulsed (Q-switched)
Nd:YAG	$\text{Nd}^{3+}$ in Yttrium Aluminum Garnet	$1064\text{ nm}$ (IR) / $532\text{ nm}$	Laser diode arrays / Flashlamp	CW / High-power pulsed
Carbon Dioxide ($\text{CO}_2$)	Molecular $\text{CO}_2 + \text{N}_2 + \text{He}$	$10.6 \ \mu\text{m}$ (Far IR)	RF / Electrical discharge	High power industrial (kW)
Semiconductor Diode	GaAs / InGaAsP p-n junction	$405 - 1550\text{ nm}$	Direct electrical current injection	High efficiency ($>50\%$) CW