The purpose of this paper is to describe a method for observing stimulated emission of gamma radiation. Stimulated emission of gamma radiation has never been observed. Stimulated emission of gamma radiation will be essential for developing gamma-ray lasers. The method uses the ‘π phase-shift induced transparency’ of gamma radiation, previously called the ‘gamma echo’. The gamma-echo technique is based on a modification of the time-differential Mössbauer spectroscopic method, i.e. the source is moved ‘instantaneously’ during the lifetime of the excited state. Then a resonant absorber becomes transparent to the resonant gamma radiation. There is a type of ‘self-stimulated’ emission in which the radiation ‘absorbed’, before the phase shift, is stimulated to emit after the phase shift. To observe stimulated emission of the resonant gamma radiation, one performs the same gamma-echo experiment except the resonant absorber is replaced by another source. This secondary source is chosen to be in resonance with the original source. Under these conditions there will be no stimulated emission, in the secondary source, before the π phase shift because the incident radiation is absorbed. However, after the phase shift, the secondary source becomes transparent to the incident radiation. If one uses a matched pair of Co57 sources and the secondary source is 2mCi, there will be about 3 nuclei in the secondary source, in the first excited state of Fe57, that are available to be stimulated during the 97.8ns lifetime of the primary source radiation.
A classical optical theory is given for the phenomenon of $\ensuremath{\gamma}$-ray quantum beats observed by Perlow in a study of the M\"ossbauer effect of a frequency-modulated source. The intensity $I({\ensuremath{\omega}}_{0}\ensuremath{-}{\ensuremath{\omega}}_{0}^{\ensuremath{'}},t)$ of the radiation from such a source transmitted through a resonant absorber is obtained as a function of ${\ensuremath{\omega}}_{0}\ensuremath{-}{\ensuremath{\omega}}_{0}^{\ensuremath{'}}$ and
t, where ${\ensuremath{\omega}}_{0}^{\ensuremath{'}}$ is the frequency of an absorber resonance, ${\ensuremath{\omega}}_{0}$ is the central frequency of the frequency-modulated source, and
t is the laboratory time. An average is taken over the unobserved initial formation time of the excited nuclear state in the source. When viewed at fixed ${\ensuremath{\omega}}_{0}\ensuremath{-}{\ensuremath{\omega}}_{0}^{\ensuremath{'}}$, the calculated intensity displays beats in the time spectrum, and when viewed at fixed
t, the intensity shows dispersion at appropriate values of ${\ensuremath{\omega}}_{0}\ensuremath{-}{\ensuremath{\omega}}_{0}^{\ensuremath{'}}$, responsible for the observed enhancement of intensity above background. The harmonic content of the quantum beats is calculated explicitly in the thin-absorber limit, and the observed linear variation about ${\ensuremath{\omega}}_{0}\ensuremath{-}{\ensuremath{\omega}}_{0}^{\ensuremath{'}}=0$ of the ratio of Fourier components
D2D1 is explained. The use of
D2D1 to measure small frequency shifts is analyzed by a statistical comparison with the method of switching between steepest points of the line, as used in gravitational-red-shift measurements. The variances are comparable. The effect on $I({\ensuremath{\omega}}_{0}\ensuremath{-}{\ensuremath{\omega}}_{0}^{\ensuremath{'}},t)$ of line broadening due to sample inhomogeneities is calculated for a Lorentz distribution of center frequencies in source and absorber, and a prescription is given for modifying the various terms in $I({\ensuremath{\omega}}_{0}\ensuremath{-}{\ensuremath{\omega}}_{0}^{\ensuremath{'}},t)$ accordingly. Finally, the effect of a distribution of phase and amplitude of the motion of the vibrating source is discussed.
Classical phase-modulation theory is applied to describe the interference of the recoil-free gamma-radiaton fields emitted by a M\"ossbauer source and a resonant absorber in forward-scattering geometry. During the exponential decay of the excited state in the source the resonant absorber develops and emits a coherent field, which interferes destructively with the source field. A stepwise phase change of the source field relative to the absorber field is introduced by a fast mechanical displacement of the source. As a result, an intense pulse of gamma rays is observed in the time dependence of the resonantly filtered radiation. This coherent transient phenomenon is called a gamma echo. The time reference of the gamma-echo experiment is the moment of formation of the excited state in the source. This is in contrast to earlier transient M\"ossbauer experiments in which the measurements were synchronized to the phase of the modulation and averaging occurred over all possible times of formation of the excited state. By choosing a suitable shape for the phase-modulation function in the echo experiment, short pulses of gamma radiation with desired duration and amplitude can be generated within a time frame defined by the lifetime of the excited nuclear state. Here, a detailed description of the experimental and theoretical aspects of stepwise phase modulation in a coincidence scheme is given. Theoretical results are presented in a general form, taking into account the experimental nonidealities. The properties of the mechanical drive system are discussed in detail. Gamma-echo data are presented. Good agreement between the theoretical calculations and the experimental data has been established.