The initial theoretical finding that eventually led to laser development was Einstein’s prediction, based upon statistical considerations, that the energy of quanta of light be given by Planck’s constant times the frequency of the light. A new theoretical development based upon Weyl’s gauge field theory predicts that photon energies are quantized with the energy given by N2hν. Such quantization of photon energy changes the character of the photon from the Einstein photon that does not have a quantum number. Photon energy that includes a quantum number means that for a given energy the frequency may have more than one value. Conversely, photons of a given frequency may be found that have more energy than the Einstein photon. Further, the phat photons, all at a given frequency will have energy proportional to the number of phat photons and N2. For these phat photons the electric field strength, which causes breakdown in optical fibers or air, depends linearly on N. Thus, more energy may be transmitted using phat photons of higher quantum numbers than increasing the number of photons of lesser quantum numbers while still keeping the electric field below the breakdown level. Further, while the stimulated and spontaneous emission probabilities are proportional to 1/N2 the Rayleigh scattering cross section diminishes by 1/N8. This reduction in the scattering cross section means that a laser emitting phat photons with N<1 will lose less energy traveling through the Earth’s atmosphere than lasers using N=1. This reduction in energy losses through the atmosphere means increased efficiency for Earth based beamed applications. This presentation discusses the fundamental theory, emission probabilities, and cross section calculations.
The initial theoretical finding that eventually led to laser development was Einstein’s prediction, based upon statistical considerations, that the energy of quanta of light be given by Planck’s constant times the frequency of the light. A new theoretical development based upon Weyl’s gauge field theory predicts that photon energies are quantized with the energy given by N2hν. Such quantization of photon energy changes the character of the photon from the Einstein photon that does not have a quantum number. Photon energy that includes a quantum number means that for a given energy the frequency may have more than one value. Conversely, photons of a given frequency may be found that have more energy than the Einstein photon. Further, the phat photons, all at a given frequency will have energy proportional to the number of phat photons and N2. For these phat photons the electric field strength, which causes breakdown in optical fibers or air, depends linearly on N. Thus, more energy may be transmitted using phat photons of higher quantum numbers than increasing the number of photons of lesser quantum numbers while still keeping the electric field below the breakdown level. Further, while the stimulated and spontaneous emission probabilities are proportional to 1/N2 the Rayleigh scattering cross section diminishes by 1/N8. This reduction in the scattering cross section means that a laser emitting phat photons with N<1 will lose less energy traveling through the Earth’s atmosphere than lasers using N=1. This reduction in energy losses through the atmosphere means increased efficiency for Earth based beamed applications. This presentation discusses the fundamental theory, emission probabilities, and cross section calculations.
The idea of the photon has stirred the imaginations of physicists ever since 1905 when Einstein originally proposed the use of light quanta to explain the photoelectric effect. This concept is formalized in the quantum theory of radiation, which has had unfailing success in explaining the interaction of electromagnetic radiation with matter, seemingly limited only by the ability of physicists to perform the indicated calculations. Nevertheless, it has its conceptual problems—various infinities and frequent misinterpretations. Consequently an increasing number of workers are asking, “to what extent is the quantized field really necessary and useful?” In fact the experimental results of the photoelectric effect were explained by G. Wentzel in 1927 without the quantum theory of radiation. Similarly most electro-optic phenomena such as stimulated emission, reaction of the emitted field on the emitting atom, resonance fluorescence, and so on, do not require the quantization of the field for their explanation. As we will see, these processes can all be quantitatively explained and physically understood in terms of the semiclassical theory of the matter–field interaction in which the electric field is treated classically while the atoms obey the laws of quantum mechanics. The quantized field is fundamentally required for accurate descriptions of certain processes involving fluctuations in the electromagnetic field: for example, spontaneous emission, the Lamb shift, the anomalous magnetic moment of the electron, and certain aspects of blackbody radiation. (The Compton effect also fits here, but see later under references 8b and c.) Here we will outline how the photon concept originated and developed, where it is not required and is often misused, and finally where it plays an essential role in the understanding of physical phenomena. In our discussion we will attempt to give a logically consistent definition of the word “photon”—a statement far more necessary than one might think, for so many contradictory uses exist of this elusive beast. In particular consider the original coining of the word by G. N. Lewis: “[because it appears to spend] only a minute fraction of its existence as a carrier of radiant energy, while the rest of the time it remains an important structural element within the atom…, I therefore take the liberty of proposing for this hypothetical new atom which is not light but plays an essential part in every process of radiation, the name photon!” (our exclamation point). Clearly the present usage of the word is very different.
The theory of stimulated emission was put forwarded by A. Einstein (1917), which predicted that as light passed through a substance it could stimulate the emission of more light. In 1954, this phenomenonwas used by Charles Townes andArthur Schawlow in the construction of a microwave amplification by stimulated emission of radiation (maser) using ammonia gas and microwave radiation. Later, this operating principle of maser was put forth to the optical frequencies by them from the Bell Telephone laboratories in 1958, which led to the realization of the device, known as light amplification by stimulated emission of radiation (laser; Townes and Schawlow, 1958). A laser is a quantum optical device that produces a nearly monochromatic (single wavelength), and coherent beam of light by exciting atoms to a higher energy level and causing them to radiate their energy in phase.
This work examines the physical and conceptual evolution of the light quantum from Planck's blackbody theory to the theoretical and experimental developments that led to the quantization of the electromagnetic field. The present study shows that the decisive transition occurred in Einstein's quantum theory of radiation (1916-1917). Absorption, stimulated emission, and spontaneous emission were formulated as elementary probabilistic mechanisms whose statistical balance alone reproduces the blackbody spectrum. In particular, spontaneous emission requires the emission of a light quantum, thereby implicitly proving its physical necessity before its theoretical status was clarified. At the same time, the already existing term photon began to acquire a stable usage following Lewis's 1926 proposal and became increasingly associated with Einstein's light quantum. By the mid-1920s, the central problem had shifted from whether light quanta were physically required to how radiation could be incorporated into the emerging quantum-mechanical formalism. This transition marks a key stage, illustrating how initial debates about the existence of light quanta gave way to their integration into a comprehensive theoretical structure. The resulting asymmetry between the novel quantum description of matter and the still-classical description of radiation, called into question by the phenomenon of spontaneous emission, identifies the physical problem that led to the quantization of the electromagnetic field.
Radiation can change the energy state of matter via three optical processes: spontaneous emission, absorption and stimulated emission. In spontaneous emission, an excited atomic or molecular state decays to a lower energy state with the emission of a photon of energy equal to the difference in the quantum state energies. Photon absorption is the reverse process, with a photon causing excitation from a lower to an excited quantum state. Stimulated emission was first proposed by Einstein in a paper published in 1917. He attempted to balance spontaneous emission and photon absorption, but could not get the correct balance without the postulated process of stimulated emission (see Section 4.2). With stimulated emission, an excited quantum state is stimulated to decay to a lower quantum state by an impinging photon. The existing photon survives with another identical photon being created (so that energy is conserved). The rates at which radiative transitions occur can be calculated using quantum mechanics. The time-dependent Schrodinger equation is used with the system wavefunction comprising time-varying linear combinations of the two wavefunctions for the lower and upper quantum states. In semi-classical treatments, an atom is treated quantum mechanically with the radiation field treated classically. There is a perturbation in energy associated with, say, the electric field of the radiation interacting with the electric dipole of the atom, though other interactions can be significant if the electric dipole interaction energy is small. In absorption, for example, a differential equation representing the change in ‘weighting’ of the wavefunction from the lower to upper state is obtained. A ‘time constant’ for the speed of change represents the rate of the radiative process. We examine the quantum mechanical nature of radiation absorption in Section 10.1 before discussing (Section 10.3) another important parameter arising with radiative transitions between discrete bound quantum states: the lineshape function. Quantum Theory of the Atom–Radiation Interaction An expression for the Einstein B-coefficient for photo-absorption is derived using quantum mechanics in this section. Einstein determined the detailed balance relationship between the rates of spontaneous emission, photo-absorption and stimulated emission when light interacts with an atom by considering an atomic system of two energy levels which are in equilibrium with a black-body radiation field. We use a semi-classical treatment where the atom is treated quantum mechanically and the radiation field is treated as a classical oscillating electric field.
Classical electromagnetics in free space leaves no room for accommodation of a light particle. To underline this, it is sufficient to remember that Einstein in order to reach his lightquantum (energy-quantum) hypothesis had to mix classical theoretical physics with a piece of experimental information that defies description in classical terms. In his 1905-paper Einstein concluded [60]:Monochromatic radiation of low density (within the range of validity of Wien’s radiation formula) behaves thermodynamically as though it consisted of a number of mutually independent energy quanta of magnitude hν. The introduction of light quanta in the Wien regime (the high-frequency part of the Planck blackbody radiation law) is the first step toward the concept of electromagnetic radiation as a Bose gas of photons [180, 39, 68, 69]. Although a photon gas obeys Bose-Einstein statistics for all frequencies the statistical independence assumed by Einstein in 1905 is not true in general. Einstein used Boltzmann statistics when he conjectured the existence of light quanta. In the Wien regime Boltzmann and Bose-Einstein statistics lead to the same conclusion since the mean number of particles (photons) with a given energy is much less than one. The circumstance that the light quanta are not statistically independent in Bose-Einstein counting finds its explanation in quantum mechanics. Hence, the statistical correlation is induced by the requirement of totally symmetric wave functions.
Introduction The preceding chapters have focused on atomic transitions induced by applied radiation. That is, there is an optical field illuminating the atoms, and their response is calculated. It is implicitly assumed that an excited atom left alone will spontaneously return to the ground state, and conserve energy by emitting light that satisfies E 2 − E 1 = ħ ω 21 in accordance with the Planck and Bohr pictures. But this spontaneous emission process is not addressed quantitatively in these earlier chapters. Spontaneous emission is one of the most pervasive phenomena of atomic physics, and yet its origins remain among the least well-understood. The number of misconceptions is enormous, and they are to be found in textbooks, journal articles, and lecture notes. Because the natural decay of an excited atom adds energy to what is usually an empty mode of the electromagnetic field, the process is fundamentally quantum mechanical. Classical or even semi-classical descriptions cannot be assured to give correct answers. This is not to say that such discussion is to be avoided, but only that caution is needed in interpreting the results. Toward the end of the nineteenth century, many experimental results made it clear that classical mechanics needed to be supplemented by new theories. The first venture into the new physics was Planck's hypothesis that the energy of classical radiating oscillators like those discussed in Chap. 1 would have to be quantized into integer multiples of ħ ω. This notion led to his famous formula for the spectrum of thermal radiation from a non-reflective object (black body), and this formula describes a radiation field in thermodynamic equilibrium with its environment. The result of Planck's hypothesis in the context of thermal equilibrium plays a vital role in the discussion of spontaneous emission. Einstein A - and B -coefficients Soon after Bohr proposed a model of atomic structure consisting of discrete energy levels, connected by radiation whose energy satisfied Δ E = ħω when atoms made transition between them, Einstein published the first serious attempt to connect the absorption and emission processes. Absorption, he reasoned, could occur only in the presence of applied radiation, whereas spontaneous decay occurred without the application of any field.