Entries by Charles Link

Maxwell’s Equations in Magnetostatics and Solving with the Curl Operator

Introduction: Maxwell’s equation in differential form ## \nabla \times \vec{B}=\mu_o \vec{J}_{total}+\mu_o \epsilon_o \dot{\vec{E}}  ##  with ## \dot{\vec{E}}=0 ## comes up quite frequently in magnetostatic problems.  In addition, the equation from the magnetic pole model ## \vec{B}=\mu_o \vec{H}+ \mu_o \vec{M}  ## comes up quite often as well.  It seems the textbooks are somewhat lacking in a…

Learn an Integral Result from Parseval’s Theorem

Introduction: In this Insight article,  Parseval’s theorem will be applied to a sinusoidal signal that lasts a finite period of time.  It will be shown that it necessarily follows that ## (\frac{\sin(2 x_o)}{x_o})( \frac{\pi}{2})=\int\limits_{-\infty}^{+\infty} \frac{\sin(x-x_o) \sin(x+x_o)}{(x-x_o)(x+x_o)} \, dx   ##. Note:  This integral result was computed by this author in 2009.  I have not done…

The Diffraction Limited Spot Size with Perfect Focusing

Purpose The purpose of this Insights article is to give the reader a brief introduction to the principles behind diffraction-limited focusing. The reader is assumed to be somewhat familiar with the concepts of diffraction theory and the Huygens sources that are used to compute a diffraction pattern. Perfect Focusing by paraboloidal mirror When a collimated…

Learn Frenet Equations in 2-D Which Result in the Cornu Spiral

This Insights article is intended as an introduction to the Frenet-Serret equations by showing an interesting application that results from a case in two-dimensional motion. The two-dimensional result for this case=the Cornu Spiral has some interesting associated mathematics that also arises in solutions of certain diffraction equations in optics. Frenet-Serret equations: The Frenet-Serret equations are…

Fabry-Perot and Michelson Interferometry: A Fundamental Approach

Fabry-Perot Effect: The Fabry-Perot effect is usually treated in most optics textbooks as the interference that results from multiple reflections of the two interfaces of a dielectric slab using a single incident beam. Michelson Interferometer: Meanwhile, the interference in the Michelson interferometer is usually described as the interference of a partially transmitted beam through the…