2 Dec 2008 22:51

## Re: efield vs. dpwr, field definition --> follow up on old thread

```On Dec 1, 2008, at 11:25 PM, Chad Husko wrote:
> real
> h5topng -Zc dkbluered -C epsilon.h5:data-new -d x.r-new   ex.band14.h5
> imaginary
> h5topng -Zc dkbluered -C epsilon.h5:data-new -d x.i-new   ex.band14.h5
>
> give basically the same results --> modes that do not look like they
> propagate.

I don't know what you mean by "modes that do not look like they
propagate".   Maybe the field is not oriented in the x direction for
the mode you are interested in?  It is usually easier to see what is
going on by looking at a scalar field like epsilon*|E|^2 first.

I have calculated modes of photonic crystal fibers many times with MPB
without a problem.

> Aeff = int ( int (    |Ex|^2 dy dz ))^2  /     int ( int (    |Ex|^4
> dy
> dz )) in octave/matlab?

Note that this definition of "effective area" was derived as a figure
of merit for Kerr nonlinearities in the scalar approximation for low
index contrasts.  If you are interested in Kerr nonlinearities in high-
contrast media like photonic-crystal fibers, this formula is incorrect
(except in certain limits).  If you are interested in other things
like bending losses, etcetera, be aware that formulas derived based on
this effective area are generally from the scalar low-contrast limit
and are inapplicable to photonic-crystal fibers. (In general, a lot of
formulas from the fiber literature, e.g. in the Snyder and Love book,
were derived in the scalar approximation, and it is dangerous to use
them blindly.)

The correct formula for effective area in high-contrast fibers, as a
figure of merit for Kerr nonlinearities (i.e. the parameter that goes
into the nonlinear Schrodinger equation), was derived by Tzolov in
1995; the formula and reference are in chapter 9 of our book (ab-
initio.mit.edu/book)

Note also that you can compute the effective area directly in MPB with
the field integration routines.  The following computes the effective
area for a band b for a photonic crystal fiber consisting of an index
n material with a constant Kerr coefficient, and air (with Kerr
coefficient zero), using Tzolov's formula:

(define (Aeff b)
(define (Aeff-integrand r eps E)
(* (/ eps 3) (/ (- eps 1) (- (sqr n) 1))
(+ (sqr (magnitude (vector3-dot E E)))
(* 2 (sqr (real-part (vector3-cdot E E)))))))
(define (zflux-integrand r E H)
(real-part (vector3-z (vector3-cross (vector3-conj E) H))))
(get-efield b)
(let ((E (field-copy cur-field)))
(get-epsilon)
(let ((denom (integrate-fields Aeff-integrand cur-field E)))
(get-hfield b)
(/ (sqr (integrate-fields zflux-integrand E cur-field))
denom))))

Note that Tzolov's formula reduces to the one you quoted in the scalar
limit (either low-contrast materials or propagation constant beta ->
infinity).

Steven
```

Gmane