Entries by Charles Link

Fundamentals of the Diffraction Grating Spectrometer

Introduction In this article we will discuss the fundamentals of the diffraction grating spectrometer.  The operation of the instrument is based upon the textbook equations for the far-field interference (Fraunhofer case) that results from a plane wave incident on a diffraction grating.   It is rather remarkable how the standard textbook equations can be used to…

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…

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 paraboidal mirror:  When a…

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 for 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…