Compensation for phase distortions in nonlinear media by phase conjugation

We demonstrate theoretically that the distortion-correction property of phase-conjugate beams propagating in reverse through aberrating media is also operative when the indices of refraction of the media depend on the intensity. A necessary condition is that the phase-conjugate mirror that generates the reflected beam possess a unity (magnitude) "reflection" coefficient

Amplified reflection, phase conjugation, and oscillation in degenerate four-wave mixing

A number of new optical effects that result from degenerate four-wave mixing in transparent optical media are proposed and analyzed. The applications are relevant to time-reversed (phase-conjugated) propagation as well as to a new mode of parametric oscillation

Compensation for channel dispersion by nonlinear optical phase conjugation

It is proposed that the process of nonlinear optical phase conjugation can be utilized to compensate for channel dispersion and hence to correct for temporal pulse broadening. Specifically, a four-wave nonlinear interaction is shown to achieve pulse renarrowing. Spectral bandwidth constraints of the input pulse are presented for typical phase-conjugate interaction parameters

Observation of amplified phase-conjugate reflection and optical parametric oscillation by degenerate four-wave mixing in a transparent medium

We report on the observation of amplified reflection and optical parametric oscillation via degenerate four-wave mixing in a nonresonant medium. The process is mediated through the third-order nonlinear susceptibility in a transparent liquid medium, CS2. A collinear mixing geometry is utilized to obtain long interaction lengths and polarization discrimination is used to separate the pump and signal fields

Upper bounds on the k-forcing number of a graph

Given a simple undirected graph $G$ and a positive integer $k$, the $k$-forcing number of $G$, denoted $F_k(G)$, is the minimum number of vertices that need to be initially colored so that all vertices eventually become colored during the discrete dynamical process described by the following rule. Starting from an initial set of colored vertices and stopping when all vertices are colored: if a colored vertex has at most $k$ non-colored neighbors, then each of its non-colored neighbors becomes colored. When $k=1$, this is equivalent to the zero forcing number, usually denoted with $Z(G)$, a recently introduced invariant that gives an upper bound on the maximum nullity of a graph. In this paper, we give several upper bounds on the $k$-forcing number. Notable among these, we show that if $G$ is a graph with order $n \ge 2$ and maximum degree $\Delta \ge k$, then $F_k(G) \le \frac{(\Delta-k+1)n}{\Delta - k + 1 +\min{\{\delta,k\}}}$. This simplifies to, for the zero forcing number case of $k=1$, $Z(G)=F_1(G) \le \frac{\Delta n}{\Delta+1}$. Moreover, when $\Delta \ge 2$ and the graph is $k$-connected, we prove that $F_k(G) \leq \frac{(\Delta-2)n+2}{\Delta+k-2}$, which is an improvement when $k\leq 2$, and specializes to, for the zero forcing number case, $Z(G)= F_1(G) \le \frac{(\Delta -2)n+2}{\Delta -1}$. These results resolve a problem posed by Meyer about regular bipartite circulant graphs. Finally, we present a relationship between the $k$-forcing number and the connected $k$-domination number. As a corollary, we find that the sum of the zero forcing number and connected domination number is at most the order for connected graphs.Comment: 15 pages, 0 figure

Spatial convolution and correlation of optical fields via degenerate four-wave mixing

A nonlinear optical technique is described that performs, essentially instantaneously, the functions of spatial correlation and convolution of spatially encoded waves. These real-time operations are accomplished by mixing spatially dependent optical fields in the Fourier-transform plane of a lens system. The use of a degenerate four-wave mixing scheme eliminates (in the Fresnel approximation) phase-matching restrictions and (optical) frequency-scaling factors. Spatial bandwidth-gain considerations and numerical examples, as well as applications to nonlinear microscopy, are presented

Image phase compensation and real-time holography by four-wave mixing in optical fibers

It is proposed that real-time holography can be performed inside multimode fibers (or optical waveguides) using four-wave optical mixing. Of particular interest is the generation of complex-conjugate replicas of input fields for image transmission and compensation of propagation distortion. A theoretical analysis and a numerical estimate are presented

A theoretical and experimental investigation of the modes of optical resonators with phase-conjugate mirrors

We present an analysis of resonator properties for a cavity bounded by a phase conjugate mirror, which is generated by a degenerate four-wave nonlinear optical interaction. Using a ray matrix formalism to describe the conjugate mirror, resonator stability conditions are derived. Longitudinal and transverse mode characteristics are discussed. Results are compared with an experiment where laser oscillation was observed at 6943 Å using carbon disulfide as the nonlinear interacting medium comprising the phase conjugate mirror