Stigmatic Optics

General diffraction integral for converging freeform vectorial fields

“…In figure 1 we observe that equation (2) considers only plane waves, then [k x , k y , k z ] are given just by real values. Therefore, acknowledging the Malus-Dupin theorem [29], we can get the propagation vector of a wavefront w(x, y) as the normal vector towards w(x, y), where ∂ x w ≡ ∂w/∂x is the partial derivative of w respect to x and ∂ y w ≡ ∂w/∂y is the partial derivative of w respect to y. k x , k y and k z are the cosine directors in x, y and z directions, respectively, of the wave vector k. In order to express the integration in terms of x and y instead of k x , and k y . The Jacobian can be computed using equation (3), and changing the integrands [30],…”

Wavelength sensitivity of analytical stigmatic surfaces

“…The approach is to replace the last surface by an aspherical surface calculated by means of the general equation of stigmatic sequential optical systems. [1][2][3][4] Once the method is applied, a set of discrete values both for the radial and axial coordinates are obtained. These points define where all rays should refract one last time to achieve stigmatism.…”

“…These points define where all rays should refract one last time to achieve stigmatism. As mentioned, the procedure in [1][2][3][4] is formulated for a single wavelength, hence a different set of points is obtained for each considered wavelength. The number of obtained points corresponds to the the aperture mapping.…”

“…Recently, R. G. González-Acuña, Chaparro-Romo and J. C. Gutiérrez-Vega presented a general formula to design stigmatic rotationally symmetric sequential optical systems. [1][2][3][4] Given any rotationally symmetric set of surfaces comprising a lens-based optical system, the method is able to analytically find the points in space where rays should last refract in order to achieve stigmatism, by applying Snell's Law and Fermat Principle, without paraxial or numerical approximations. The only condition imposed is that all rays must have the same optical path and the authors use the vector form of the refraction law to derive the equations.…”

Twist families of $L$-space knots, their genera, and Seifert surgeries

“…Hence K n (r n ) must be a torus bundle for all integers n. Therefore for each integer n, the knot K n is a genus one fibered knot [15,Corollary 8.23], and so it is either a trefoil or the figure eight knot [6,19]. Since c is not a meridian of K and does not bound a disk disjoint from K, this contradicts that the same knot can only appear finitely many times in the twist family {K n } [37, Theorem 3.2].…”