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Differential Jones and Mueller Matrices
Published in José J. Gil, Razvigor Ossikovski, Polarized Light and the Mueller Matrix Approach, 2022
José J. Gil, Razvigor Ossikovski
Eq. (9.39) shows that, because of the matrix logarithm being a multivalued function, the matrix N is not unique, but rather depends on the choice of the integer p, called order. The multiplicity of solutions for N for a given T is physically due to the periodicity of the birefringent (B) properties of the medium. Possibly the simplest example is that of the waveplate or linear retarder (see Chapter 4): its Jones matrix defines the retardance value only within 2πp where p is the waveplate order. To determine the value of p, a measurement at a different wavelength has to be performed. In the general case where both birefringent (B) and dichroic (D) properties are present simultaneously in the medium, the values of all elementary properties depend on the order p in an intricate manner. In the special case where only D properties are present in the medium, see Eq. (9.33), all B properties are zero at zero order (p = 0) and change their values to 2π p at orders p different from zero.
Differential Jones and Mueller matrices
Published in José J. Gil Pérez, Razvigor Ossikovski, Polarized Light and the Mueller Matrix Approach, 2017
José J. Gil Pérez, Razvigor Ossikovski
Equation 9.39 shows that because of the matrix logarithm being a multivalued function, the matrix N is not unique, but rather depends on the choice of the integer p, called order. The multiplicity of solutions for N for a given T is physically due to the periodicity of the birefringent (B) properties of the medium. Possibly the simplest example is that of the waveplate or linear retarder (see Chapter 4): its Jones matrix defines the retardance value only within 2π p, where p is the waveplate order. To determine the value of p, a measurement at a different wavelength has to be performed. In the general case where both birefringent (B) and dichroic (D) properties are present simultaneously in the medium, the values of all elementary properties depend on the order p in an intricate manner. In the special case where only D properties are present in the medium (see Equation 9.33), all B properties are zero at zero order (p = 0) and change their values to 2π p at orders p different from zero.
Conformal Mappings
Published in Vladimir Eiderman, An Introduction to Complex Analysis and the Laplace Transform, 2021
A multivalued function F(z) is single-valued and analytic on its Riemann surface, with the exception of branch points. So if a regular branch exists on a domain D, it must also be possible to place this domain somewhere in the Riemann surface, without cutting it or touching any branch points. The domain D must either fit entirely within one sheet, or straddle several sheets around their glue binding (like a carpet over adjacent stair steps).
Impact response of a viscoelastic plate made of a material with negative Poisson’s ratio1
Published in Mechanics of Advanced Materials and Structures, 2023
Since the function is a multivalued function with the branch points and then the inversion from the Laplace domain to the time domain according to the Mellin–Fourier formula (52) is valid only within the first sheet of the Riemann surface and therefore, the closed contour of integration should be comprised from the vertical segment the segments along the cuts of the lengthwise negative real semi-axis and arcs of circumferences, one of which () closes the branches of cuts, while other two () connects the branches of cuts with the vertical segment (Figure 3). Due to the Jordan lemma, the integrals along the curves at and at tent to zero.