China
Africa
Explore chapters and articles related to this topic
Nonlinear Optics and Supercontinuum Generation in Photonic Crystal Fibers
Published in Narendra Kumar, Bhuvneshwer Suthar, Advances in Photonic Crystals and Devices, 2019
where ΩS=ωP−ωS=ωAS−ωP. The parameter β2n is the 2nth derivative of β(ω). This special case (i.e., ω1=ω2=ωP) is known as degenerate FWM. The significant FWM occurs only if the phase mismatch nearly vanishes. The FWM is the underlying mechanism of many nonlinear effects including SPM, cross-phase modulation, and MI.
Quadratic, Bilinear, and Sesquilinear Forms
Published in Leslie Hogben, Richard Brualdi, Anne Greenbaum, Roy Mathias, Handbook of Linear Algebra, 2006
where a,b,c,d,e,f∈ℝ. The solution set is a conic section, namely an ellipse, hyperbola, parabola, or a degenerate form of those. The analysis of this equation depends heavily on the quadratic form ax12+bx1x2+cx22, which is represented in the standard basis of A=[ab/2b/2c]. If the solution of the quadratic equation above represents a nondegenerate conic section, then its type is determined by the sign of 4ac – b2. More precisely, the conic is an ellipse, hyperbola, or parabola if 4ac – b2 is positive, negative, or zero, respectively.
Birefringent Ray Trace
Published in Russell A. Chipman, Wai-Sze Tiffany Lam, Garam Young, Polarized Light and Optical Systems, 2018
Russell A. Chipman, Wai-Sze Tiffany Lam, Garam Young
Depending on the type of anisotropic material, different symbols and subscripts label the types eigenmodes as tabulated in Tables 19.1 and 19.2. Isotropic materials are a special case with degenerate modes. When refracting into an isotropic material, both refracted modes, s and p, share the same Poynting vector direction S^, the same propagation direction k^, and the same refractive index; hence, these modes are degenerate. Thus, for isotropic refraction, the s- and p-modes can be combined and treated as a single mode labeled i, denoting an isotropic mode. In uniaxial materials, the two modes are labeled o for the ordinary and e for the extraordinary modes. In biaxial materials, the two modes are distinguished by the associated refractive index of the ray, the mode with the higher index being the slow-mode and the other mode being the fast-mode. In isotropic optically active materials, the two modes are the right and left circularly polarized modes. Note the symbols for modes are in lowercase. A list of parameters needed by the polarization ray trace for each ray segment is presented in Table 19.2. Many of these parameters were first introduced in Chapters 9 and 10, but birefringent interfaces need additional parameters.
Vibronic mean-field and perturbation theory for Jahn-Teller and pseudo-Jahn-Teller molecules
Published in Molecular Physics, 2021
Consider for a moment the special case of an accidental degeneracy between diabatic normal coordinate frequencies , so that all four vibrational coordinates are degenerate. In this case, an orthogonal coordinate transformation can be applied that eliminates one linear coupling term without introducing any new diabatic cross-terms, where . With this coordinate system, the Hamiltonian is separable into an Jahn-Teller problem involving and an uncoupled spectator e mode . The exact solutions are factorable into the form a significant simplication from a fully correlated wavefunction.
Explicit multi-material topology optimization embedded with variable-size movable holes using moving morphable bars
Published in Engineering Optimization, 2021
Xuan Wang, Kai Long, Zeng Meng, Bo Yu, Changzheng Cheng
As a special case, when the connecting structure is composed of a single material (namely m = 1), the interpolation scheme will degenerate into a simple one: Then the element stiffness matrix of the eth element for two-dimensional linear elasticity can be expressed as with . and are the strain–displacement matrix and constitutive stress–strain matrix, respectively. The interpolated elasticity modulus (constant) in is taken out of the integration for linear elasticity.
Distributionally robust optimization for sequential decision-making
Published in Optimization, 2019
Zhi Chen, Pengqian Yu, William B. Haskell
Let be the set of histories at time t, given by and for all . A history dependent randomized decision rule is a mapping , where is the probability simplex on the set of actions . A Markovian randomized decision rule is a mapping . A deterministic decision rule can be regarded as a special case of a randomized decision rule in which the probability distribution on the set of actions is degenerate. The sets of history dependent randomized decision rules, Markovian randomized decision rules and Markovian deterministic decision rules are denoted by , and , respectively. A strategy is a sequence of decision rules for the entire time horizon, i.e. where and K designates a class of decision rules (K=HR, MR, MD). We denote the set of all policies of class K by .