Q12Engineering Physics

The fundamental requirement for any laser to function is the achievement of a population inversion—a non-equilibrium state where more atoms reside in an excited, high-energy state ($N_2$) than in a lower energy state ($N_1$). Once this is established, an incident photon can trigger stimulated emission, amplifying the light wave. However, nature fiercely resists this unnatural state, primarily through the process of spontaneous emission, where atoms randomly decay and lose their energy. When attempting to build lasers that operate at very high frequencies (such as deep ultraviolet, X-rays, or Gamma rays), this natural resistance becomes an almost insurmountable physical barrier.

Einstein's Coefficients and the ${\nu }^3$ Dependency

The root of the problem lies in the fundamental quantum mechanical relationship derived by Albert Einstein in 1916 between the probability of spontaneous emission and stimulated emission. The ratio of the Einstein A coefficient ($A_{21}$, representing spontaneous emission) to the Einstein B coefficient ($B_{21}$, representing stimulated emission) is given by the equation:

Where $h$ is Planck's constant, $c$ is the speed of light, and crucially, $\nu$ is the frequency of the transition. This equation reveals a devastating dependency: the rate of spontaneous emission is directly proportional to the cube of the frequency (${\nu }^3$).

The Power Requirement Problem

Because X-rays have frequencies thousands to millions of times higher than visible light, the $A_{21}$ coefficient becomes astronomically large. Consequently, the lifetime of an atom in an X-ray excited state is extraordinarily short—often on the scale of femtoseconds ($10^{-15}$ seconds) or less. As soon as an atom is pumped to the upper state, it instantly decays spontaneously, firing off a random, incoherent X-ray photon before an incident photon can arrive to stimulate it.

To maintain a population inversion against this massive avalanche of spontaneous decay, the pumping power must exceed the decay rate. The pumping power required scales roughly as ${\nu }^4$. While a visible light laser can be pumped with a small electrical discharge or a flashlamp, an X-ray laser requires utterly colossal bursts of energy—such as detonating a nuclear weapon or using highly massive, kilometer-long Free Electron Lasers (FELs) at national accelerator facilities. Therefore, compact, conventional X-ray lasers remain physically impractical.