This paper has two purposes: 1) to clarify the relationship between the quantum theory of radiation, where the electromagnetic field-expansion coefficients satisfy commutation relations, and the semiclassical theory, where the electromagnetic field is considered as a definite function of time rather than as an operator; and 2) to apply some of the results in a study of amplitude and frequency stability in a molecular beam maser. In 1), it is shown that the semiclassical theory, when extended te take into account both the effect of the field on the molecules and the effect of the molecules on the field, reproduces almost quantitatively the same laws of energy exchange and coherence properties as the quantized field theory, even in the limit of one or a few quanta in the field mode. In particular, the semiclassical theory is shown to lead to a prediction of spontaneous emission, with the same decay rate as given by quantum electrodynamics, described by the Einstein A coefficients. In 2), the semiclassical theory is applied to the molecular beam maser. Equilibrium amplitude and frequency of oscillation are obtained for an arbitrary velocity distribution of focused molecules, generalizing the results obtained previously by Gordon, Zeiger, and Townes for a singel-velocity beam, and by Lamb and Helmer for a Maxwellian beam. A somewhat surprising result is obtained; which is that the measurable properties of the maser, such as starting current, effective molecular Q, etc., depend mostly on the slowest 5 to 10 per cent of the molecules. Next we calculate the effect of amplitude and frequency of oscillation, of small systematic perturbations. We obtain a prediction that stability can be improved by adjusting the system so that the molecules emit all their energy h Ω to the field, then reabsorb part of it, before leaving the cavity. In general, the most stable operation is obtained when the molecules are in the process of absorbing energy from the radiation as they leave the cavity, most unstable when they are still emitting energy at that time. Finally, we consider the response of an oscillating maser to randomly time-varying perturbations. Graphs are given showing predicted response to a small superimposed signal of a frequency near the oscillation frequency. The existence of "noise enhancing" and "noise quieting" modes of operation found here is a general property of any oscillating system in which amplitude is limited by nonlinearity.

Comparison of quantum and semiclassical radiation theories with application to the beam maser

Recent advances in the field of nonlinear optical phenomena are reviewed with particular empphasis placed on such topics as parametric oscillation self-focusing and trapping of laser beams, and stimulated Raman, Rayleigh, and Brillouin scattering. The optical frequency radiation is treated classically in terms of the amplitudes and phases of the electromagnetic fields. The interactions of light waves in a mterial are then formulated in terms of Maxwell's equations and the electric dipole approximation. In this method, non-linear susceptibility tensors are introdueed which relate the induced dipole moment to a power series expansion in field strengths. The tensor nature and the frequency dependence of the nonlinearity coefficients are considered. The various experimental, observations are described and interpreted in terms of this formalism.

https://ieeexplore.ieee.org/iel5/5/31089/01447053.pdf

Nonlinear optics

Structural polymers are susceptible to damage in the form of cracks, which form deep within the structure where detection is difficult and repair is almost impossible. Cracking leads to mechanical degradation of fibre-reinforced polymer composites; in microelectronic polymeric components it can also lead to electrical failure. Microcracking induced by thermal and mechanical fatigue is also a long-standing problem in polymer adhesives. Regardless of the application, once cracks have formed within polymeric materials, the integrity of the structure is significantly compromised. Experiments exploring the concept of self-repair have been previously reported, but the only successful crack-healing methods that have been reported so far require some form of manual intervention. Here we report a structural polymeric material with the ability to autonomically heal cracks. The material incorporates a microencapsulated healing agent that is released upon crack intrusion. Polymerization of the healing agent is then triggered by contact with an embedded catalyst, bonding the crack faces. Our fracture experiments yield as much as 75% recovery in toughness, and we expect that our approach will be applicable to other brittle materials systems (including ceramics and glasses).

Autonomic healing of polymer composites

We describe a method to estimate the capacity limit of fiber-optic communication systems (or ?fiber channels?) based on information theory. This paper is divided into two parts. Part 1 reviews fundamental concepts of digital communications and information theory. We treat digitization and modulation followed by information theory for channels both without and with memory. We provide explicit relationships between the commonly used signal-to-noise ratio and the optical signal-to-noise ratio. We further evaluate the performance of modulation constellations such as quadrature-amplitude modulation, combinations of amplitude-shift keying and phase-shift keying, exotic constellations, and concentric rings for an additive white Gaussian noise channel using coherent detection. Part 2 is devoted specifically to the "fiber channel.'' We review the physical phenomena present in transmission over optical fiber networks, including sources of noise, the need for optical filtering in optically-routed networks, and, most critically, the presence of fiber Kerr nonlinearity. We describe various transmission scenarios and impairment mitigation techniques, and define a fiber channel deemed to be the most relevant for communication over optically-routed networks. We proceed to evaluate a capacity limit estimate for this fiber channel using ring constellations. Several scenarios are considered, including uniform and optimized ring constellations, different fiber dispersion maps, and varying transmission distances. We further present evidences that point to the physical origin of the fiber capacity limitations and provide a comparison of recent record experiments with our capacity limit estimation.

Capacity Limits of Optical Fiber Networks

Contrary to the well known diffraction limit, the fluorescence microscope is in principle capable of unlimited resolution. The necessary elements are spatially structured illumination light and a nonlinear dependence of the fluorescence emission rate on the illumination intensity. As an example of this concept, this article experimentally demonstrates saturated structured-illumination microscopy, a recently proposed method in which the nonlinearity arisesfromsaturationoftheexcitedstate.Thismethodcanbeused in a simple, wide-field (nonscanning) microscope, uses only a single, inexpensive laser, and requires no unusual photophysical properties of the fluorophore. The practical resolving power is determined by the signal-to-noise ratio, which in turn is limited by photobleaching. Experimental results show that a 2D point resolution of <50 nm is possible on sufficiently bright and photostable samples.

/pdf/nonlinear-structured-illumination-microscopy-wide-field-8qgfmfd6e0.pdf

Nonlinear structured-illumination microscopy: Wide-field fluorescence imaging with theoretically unlimited resolution

A simple, rigorous geometrical representation for the Schrodinger equation is developed to describe the behavior of an ensemble of two quantum‐level, noninteracting systems which are under the influence of a perturbation. In this case the Schrodinger equation may be written, after a suitable transformation, in the form of the real three‐dimensional vector equation dr/dt=ω×r, where the components of the vector r uniquely determine ψ of a given system and the components of ω represent the perturbation. When magnetic interaction with a spin ½ system is under consideration, ``r'' space reduces to physical space. By analogy the techniques developed for analyzing the magnetic resonance precession model can be adapted for use in any two‐level problems. The quantum‐mechanical behavior of the state of a system under various different conditions is easily visualized by simply observing how r varies under the action of different types of ω. Such a picture can be used to advantage in analyzing various MASER‐type devi...

Geometrical Representation of the Schrödinger Equation for Solving Maser Problems

We propose a new theoretical model for the light‐induced migration of charges which mediates the ’’photorefractive effect’’ (light‐induced refractive index change) in barium titanate and other crystals. We also present experimental results of various effects of this light‐induced charge migration in a single‐domain crystal of barium titanate, specifically, (1) energy transfer between two intersecting optical beams, (2) optical four‐wave mixing and optical‐beam phase conjugation, (3) erasure of spatial patterns of photorefractive index variations, and (4) photoconductivity. The theoretical model predicts the observed dependences of these effects on (1) beam intensities, directions, and polarizations, (2) crystal orientation, and (3) on an externally applied dc electric field. Time dependences of transients as well as steady‐state magnitudes are predicted. In this model, identical charges migrate by hopping between adjacent sites, with a hopping rate proportional to the total light intensity at the starting site. The net hopping rate varies with the local electric potential that is calculated self‐consistently from the charge migration pattern. In barium titanate the charges are positive with a density of (1.90.2) ×1016 cm−3 at 514 nm. The origin of the charges and sites is at present unknown. The hopping rate constant determined from optical beam interactions is used to predict the observed photoconductivity of 1.3×10−10 cm Ω−1 W−1 at 514 nm.

Photorefractive effects and light‐induced charge migration in barium titanate

We describe a nonlinear method for generating, nearly instantaneously, a time-reversed replica of any monochromatic-beam wave pattern The method employs the interaction of the incident beam, of arbitrary wave front, with counter-propagating plane “pump” waves in a homogeneous, transparent, nonlinear medium Media are shown to exist in which time-reversed waves can be generated with high efficiency using available laser pump sources

Generation of time-reversed wave fronts by nonlinear refraction*

We demonstrate a phase-conjugating mirror that has a continuous-wave power reflectivity much greater than unity (gain ~100). This mirror uses nonresonant degenerate four-wave mixing in a single crystal of barium titanate (BaTiO3). With our mirror we have (1) observed cw self-oscillation in an optical resonator formed by this mirror and a normal mirror, (2) demonstrated a cw oscillator that, in spite of phase-distorting material placed inside the resonator, will always emit a TEM00 mode, and (3) demonstrated an optical image amplifier. This mirror will work at any visible wavelength and with weak (milliwatt or weaker) pump beams.

Robert W. Hellwarth

Papers

Geometrical Representation of the Schrödinger Equation for Solving Maser Problems

Photorefractive effects and light‐induced charge migration in barium titanate

Generation of time-reversed wave fronts by nonlinear refraction*

Phase-conjugating mirror with continuous-wave gain

Stimulated Raman Scattering From Organic Liquids