Maxwell equations Fourier transform

By deriving or computing the Maxwell equation in the frame of a cylindrical geometry, it is possible to determine the modal structure for any refractive index shape. In this paragraph we are going to give a more intuitive model to determine the number of modes to be propagated. The refractive index profile allows to determine w and the numerical aperture NA = sin (3), as dehned in equation 2. The near held (hber output) and far field (diffracted beam) are related by a Fourier transform relationship Far field = TF(Near field). 

The following development is devoted to obtaining an expression for B in terms of more familiar quantities. To this end, consider the Fourier transform of Maxwell s equations (in the time independent case)... 

You will notice that this is the expression for a Maxwell model (see Equation 4.25). From Equations (4.121) to (4.125) we have applied a Fourier transform and confirmed that a Maxwell model fits at least this portion of the theory of linear viscoelasticity. The simple expression for the relationship between J (co) and G (co) allows an interesting comparison to be performed. Suppose we take our equations for a Maxwell model and apply Equation (4.108) to transform the response to an oscillating strain into the response for an oscillating stress. This requires careful use of simple algebra to give... 

The Fourier transform S (w, b) is found by solving the Maxwell equations. In the case of a fast particle with charge ze moving with constant speed, the components ( , b) and < ft(w, b) of its electric field ( Jv and <9fc b) are given by the formulas148... 

The next most familiar part of the picture is the upper right-hand corner. This i s the domain of classical applied mathematics and mathematical physics where the linear partial differential equations live. Here we find Maxwell s equations of electricity and magnetism, the heat equation, Schrodinger s wave equation in quantum mechanics, and so on. These partial differential equations involve an infinite continuum of variables because each point in space contributes additional degrees of freedom. Even though these systems are large, they are tractable, thanks to such linear techniques as Fourier analysis and transform methods. 

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