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Alexander Prokhorov and Nikolay Basov developed a theory of the molecular maser. Their analysis combines a prepared molecular beam, stimulated emission and feedback from a resonant cavity. Oscillation begins only when the molecular radiation supplies more power than the cavity and load lose. Including saturation determines a finite steady amplitude. The calculation connects molecular energy levels with practical requirements for a stable microwave oscillator.^[ref]

A physicist.^[ref]

The 1955 paper by James Power Gordon, Herbert Jack Zeiger and Charles Hard Townes acknowledged Nikolay Basov and Aleksandr Prokhorov's independent proposal.^[ref] The latter authors' theory paper states that they presented the molecular-generator possibility in May 1952, cites their 1954 publication, and separately cites the Gordon–Zeiger–Townes experimental realization. These are contributions from independent groups.^[ref]

The following original model isolates amplification by inverted electric dipoles. Let $N=N_u-N_l$ be the excess number in the upper member of one equally degenerate resonant pair. Let $a$ be a slowly varying cavity amplitude with photon number $n=|a{|}^2$, and let $p$ describe the collective molecular dipole component in the amplifying phase convention. On exact resonance, stipulate the linear equationsThe dipole relaxation rate is $\Gamma$, and $g_0=|d_{ul}{\mathcal{E}}_{\mathrm{vac}}|/\hslash$ is a single-molecule coupling in the chosen mode normalization. Here $d_{ul}$ is an electric dipole moment matrix element, ${\mathcal{E}}_{\mathrm{vac}}$ is the field's zero-point amplitude, and $\hslash =h/(2\pi )$. Field overlap and dipole orientation enter this coupling. The characteristic polynomial for growth rate $\lambda$ isFor $N>0$, one root becomes positive precisely when $g_0^{2}N>\kappa \Gamma /2$. If dipole relaxation is fast enough to eliminate $p$, then $p\approx g_{0}Na/\Gamma$ andThus positive population inversion alone is insufficient: the gain rate $GN$ must exceed the passive decay rate. This reduction describes the onset near resonance; it does not predict output phase noise, actual transit dynamics or an exact historical threshold. Nikolay Basov and Aleksandr Prokhorov's molecular-generator theory likewise treats self-excitation as competition between molecular gain and resonator-plus-load losses.^[ref]

To include replenishment, suppose upper-state molecules enter at rate $F$ and both levels leave with residence rate $\gamma$. A deliberately simple ordinary differential equation model isOne downward transition removes two units of the difference $N_u-N_l$, so $\dot{N}=F-\gamma N-2GNn$. Below onset the unsaturated inversion is $N_0=F/\gamma$, givingFor fixed mode volume $V$, the equivalent number density condition is $N_0/V>\kappa /(GV)$. Raising selected beam intensity raises gain; increasing quality factor lowers the loss rate at fixed frequency. These statements hold with $G$, residence time and mode overlap fixed.

Above onset, the nonzero steady state hasThe inversion is clamped while excess supply supports output. A finite initial seed is understood: the noiseless equations leave $n=0$ exactly unchanged even above onset. Actual startup has spontaneous emission and field fluctuations.

Nikolay Basov analyzed coherent molecular amplification with Alexander Prokhorov. Their 1956 maser theory relates the generated frequency to both the molecular transition and the cavity response, showing why a molecular oscillator can serve as a precise frequency reference. Below the condition for self-sustained oscillation, the same system can amplify an external microwave input. Distinguishing an amplifier from an oscillator therefore depends on feedback and loss as well as molecular excitation.^[ref]

A physicist.^[ref]