Associative memories based on message-bearing optical modes in phase-conjugate resonators

We describe in generic terms a class of associative memory that is based on (1) an oscillating message-bearing optical mode in a resonator containing a multimessage hologram, (2) image-selective (discriminatory) amplification of the desired picture in a photorefractive crystal, and (3) phase-conjugate reflection

Theory of laser oscillation in resonators with photorefractive gain

A theory for oscillation in an optical resonator with photorefractive gain was formulated. The threshold conditions for the oscillation were also obtained. The result, applicable to a whole class of new devices, is a prediction for an oscillation frequency different from that of the pump beam

Determinantal and eigenvalue inequalities for matrices with numerical ranges in a sector

Let A = \pmatrix A_{11} & A_{12} \cr A_{21} & A_{22}\cr\pmatrix \in M_n, where $A_{11} \in M_m$ with $m \le n/2$, be such that the numerical range of $A$ lies in the set \{e^{i\varphi} z \in \IC: |\Im z| \le (\Re z) \tan \alpha\}, for some $\varphi \in [0, 2\pi)$ and $\alpha \in [0, \pi/2)$. We obtain the optimal containment region for the generalized eigenvalue $\lambda$ satisfying \lambda \pmatrix A_{11} & 0 \cr 0 & A_{22}\cr\pmatrix x = \pmatrix 0 & A_{12} \cr A_{21} & 0\cr\pmatrix x \quad \hbox{for some nonzero} x \in \IC^n, and the optimal eigenvalue containment region of the matrix $I_m - A_{11}^{-1}A_{12} A_{22}^{-1}A_{21}$ in case $A_{11}$ and $A_{22}$ are invertible. From this result, one can show $|\det(A)| \le \sec^{2m}(\alpha) |\det(A_{11})\det(A_{22})|$. In particular, if $A$ is a accretive-dissipative matrix, then $|\det(A)| \le 2^m |\det(A_{11})\det(A_{22})|$. These affirm some conjectures of Drury and Lin.Comment: 6 pages, to appear in Journal of Mathematical Analysi

Canonical forms, higher rank numerical range, convexity, totally isotropic subspace, matrix equations

Results on matrix canonical forms are used to give a complete description of the higher rank numerical range of matrices arising from the study of quantum error correction. It is shown that the set can be obtained as the intersection of closed half planes (of complex numbers). As a result, it is always a convex set in $\mathcal C$. Moreover, the higher rank numerical range of a normal matrix is a convex polygon determined by the eigenvalues. These two consequences confirm the conjectures of Choi et al. on the subject. In addition, the results are used to derive a formula for the optimal upper bound for the dimension of a totally isotropic subspace of a square matrix, and verify the solvability of certain matrix equations.Comment: 10 pages. To appear in Proceedings of the American Mathematical Societ

Optical tracking filter using transient energy coupling

Transient energy coupling between two coherent beams occurs in dynamic holographic media with local and noninstantaneous responses. The physical origin of the effect is described, and an optical tracking filter based on the effect is demonstrated in a photorefractive Bi12SiO20 crystal

Real time image subtraction and "exclusive or" operation using a self-pumped phase conjugate mirror

Real time "exclusive or" operation with an interferometer using a self-pumped phase conjugate mirror is reported. Also, results of image subtraction and intensity inversion are shown

Conversion of optical path length to frequency by an interferometer using photorefractive oscillation

Frequency detuning effects in photorefractive oscillators are used in a new type of (passive) interferometry which converts optical path length changes to frequency shifts. Such an interferometer is potentially more accurate than conventional interferometers which convert optical path length changes to phase or intensity changes