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intensity distribution at aperture stop - POP versus analytical solution


Dear Zemaxers, 

I have a question regarding the intensity distribution of a Gaussian at the stop aperture calculated by POP, versus the analytical solution of I(rho) = e^(-2*G*rho^2), where rho is the normalized radius coordinate. 
I have a stop aperture diameter of 6.0 mm, thus a radius of 3.0 mm.

The Gaussian beam has a waist size (i.e. radius) of 1.65822 mm. This gives me rho = 0.55274.

So, at rho = 0.55274 the intensity has dropped to 1/e^2. This leads me to calculate the apodization factor G:

G = 1/rho^2 = 3.27309

With this calculated apodization factor, I can calculate the intensity at the edge of the aperture (at rho = 1). 
I(rho = 1) = e^-(2*3.27309) = 1,43 *10^-3.

However, when I look at the intensity profile (cross-X display) calculated by POP I see that at r = 3.0 mm an intensity of  3.0*10^-4

Where does this deviation come from?

See also the image attached:

Note that beam size of the pilot beam at surface 2 (that is the stop aperture of diameter 6.0 mm) is 1,6582 mm. 
I sampled with 1024x1024. See the POP settings:
 

Finally, I present a screen shot of the lens data editor

 
Could someone explain where the difference come from of the intensity value at the edge calculated via POP, and the analytical solution of I(rho) = e^(-2*G*rho^2)?

Thank you very much! Any help is much appreciated.

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Best answer by Jeff.Wilde 3 May 2022, 01:30

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There are a couple of points to consider here.  First, POP ignores the system aperture, otherwise the beam irradiance would fall to zero in the stop plane for any radius greater than the stop value (although a hard aperture can be added to any surface to truncate the beam, but that’s separate from the system aperture).  So, when using POP the apodization factor doesn’t have much meaning.  However, to compare theory to the POP result, let’s use the correct math.  For a peak irradiance of one (I0 = 1) the following equation governs the irradiance profile as the beam propagates away from the waist (see https://en.wikipedia.org/wiki/Gaussian_beam):

 

In your example, w0 = 0.00505 mm and w = 1.6582 mm (beam radius at the designated stop surface).  So, at this surface the irradiance evaluated at a radius of r = 3.0 mm is given by  (w0/w)^2 * exp(-2*r^2/w^2) = (0.00505/1.6582)^2 * exp[-2*3.0^2/1.6582^2] = 1.332e-08.  Now, if we go to POP, and select the Peak Irradiance option with a value of one, then we see that at r=3.0 mm the irradiance is precisely this same value.

 

 

Dear Jeff Wilde,

 

Thank you for this clear answer. 
I noticed that I did not have set the intensity irradiance correctly as you already pointed out. Changing it indeed to ‘Peak Irradiance’ and setting it to 1, gave me the same results. 

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