Re: Problem with integration of fields

On Thu, 20 Sep 2007, John Travers wrote:
> I'm having a problem with the integration of the electric field. I'm
> trying to calculate the effective area of an optical fiber. This is
> defined as:
>
> ( integral ( abs(E)^2 ) )^2
> ----------------------------------
>  integral ( abs(E)^4 )

I'm glad you figured out your problem.  Sorry I don't have time to be as 
active on this mailing list as I would like.

By the way, I should just make one general comment about this formula: it 
is only strictly valid in the limit of low index contrast, although many 
people blindly plug it in even for high-contrast fibers.

A typical use for the effective area is as a figure of merit for the 
strength of nonlinearities in the fiber.  The above formula is derived in 
the low-contrast scalar limit, but the correct generalization to 
high-contrast fibers is different and was derived in:
 	Tzolov et al, Optics Letters, vol. 20 (no. 5), p. 456 (1995).

In general, you have to be careful about using formulas from classic 
textbooks on optical fibers when studying high-contrast systems, because 
many of the simple formulas for bending loss, coupling, nonlinearity, 
etcetera were derived only for low-contrast fibers in the scalar 
approximation.

> When I do this I get values which are wrong by about a factor of
> 2-2.5. The following script shows this for a very simple example of a
> silica strand (1.25 micrometers in radius) surrounded by air. For this
> case the analytical solution can be found. For a wavelength of 1.06

You most definitely should be wary of these scalar-based formulas for 
micron-scale silica strands surrounded by air!  In such a high-contrast 
case, the scalar-based formulas give at best only qualitative information.

Regards,
Steven G. Johnson