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Homework Statement:

This is not so much a homework question but a conceptual one. I have written a numerical program for the Vectorial Kirchhoff Integral which relates incident fields to the diffracted fields (see relevant equations). I want to treat the case of a circular aperture of radius where the aperture radius ##(a\gg \lambda)## where ##lambda## is the incident wavelength. I have created flux patterns that corroborate known formulas in both the far and near field. I have then integrated to find total flux through cubes of varying sizes (all the way from the far field to the near field) surrounding the aperture. I was hoping for power conservation and that is what I got for most part EXCEPT for in the near field I'm getting what looks like an initial evanescent wave. Is this correct? I'm finding papers that talk about initial evanescent waves in the near field for subwavelength apertures but not for suprawavelength apertures (the case I am specifically treating).
Here is such a paper:
https://www.osapublishing.org/ao/abstract.cfm?uri=ao34173055
Could someone clarify if there are initial evanescent waves in the near field for suprawavelength diffraction? (and possibly post a source)
Relevant Equations:

For suprawavelength apertures I believe the following is accurate
##\vec{E}\left(\vec{r} , z > 0 \right) = \nabla \times 2\int_{aperture} \left[\hat{z} \times \vec{E}_0 \right] \times \nabla G \left(\vec{r} , \vec{r}' \right) \,da'##
##\vec{H}\left(\vec{r} , z > 0 \right) = \nabla \times 2\int_{aperture} \left[\hat{z} \times \vec{H}_0 \right] \times \nabla G \left(\vec{r} , \vec{r}' \right) \,da'##
There are different approximation schemes to the formulas above depending on if we're in the near field or far field.
According to the textbook (Zangwill) the accepted expression for electric field in the far field is
##\vec{E} \left(r, \theta, \phi \right) = \frac{i}{2} E_0 \left(ka \right)^2 \left( \frac{e^{ikr}}{kr}\right) \left[ \frac{2 J_1 \left(k a \sin \theta \right)}{ka \sin \theta}\right]\left(\sin \phi \hat{\theta} + \cos \phi \cos \theta \hat{\phi} \right)##
Accordingly I believe the formulas for ##\vec{H}##
##\vec{H} \left(r, \theta, \phi \right) = \frac{i}{2} H_0 \left(ka \right)^2 \left( \frac{e^{ikr}}{kr}\right) \left[ \frac{2 J_1 \left(k a \sin \theta \right)}{ka \sin \theta}\right]\left(\cos \phi \cos \theta \hat{\theta} + \sin \phi \hat{\phi} \right)##
After numerically evaluating the above integrals (in Cartesian because that is ultimately what a computer uses) I can calculate the time averaged Poynting vector numerically ##\vec{S} = \frac{1}{2} \text{Re} \left( \vec{E} \times \vec{H}^*\right)## with little difficulty.
In the near field we have (using scalar diffraction theory so this might look weird)
##E_z \left( \vec{r} \right) = \frac{E_0}{i \lambda} \int_{aperture} \frac{e^{ikr}}{r} \frac{z}{r} \, da'##
##S_z = \frac{1}{2 Z_0} \left E_z \right^2 ##
We have two different accepted formulas for the far field and near field respectively. I want a numerical program that works for both, furthermore I want to use it to calculate power through the aperture after confirming it in the far field vs near field.
I start off by treating the far field first ,where Fresnel Number ##F \ll 1##, because that is what most people think of when they think about diffraction. Because the Kirchhoff Integral produces cylindrically symmetric zflux patterns I will post cross sectional slices.
Far Field accepted formula
Far Field Numerical
Near Field Accepted Numerical
Near Field Numerical 2 (My numerical program that is supposed to be general)
As we can see there is a strong correspondence amongst the accepted formula and my numerical formulas.
Now when I graph power flux through various cubes surrounding the aperture
I get substantially up to 10% more power in the near field
My question is:
Should this be expected in the suprawavelength diffracting limit? I know that in the subwavelength diffracting limit there are initial evanescent waves
https://www.osapublishing.org/ao/abstract.cfm?uri=ao34173055
What about in the suprawavelength diffracting limit?
Edit: My units on the vertical axis of the last graph are wrong. They should have units of Poynting Vector times area not just Poynting vector. I have to think of a good way to normalize this.
I start off by treating the far field first ,where Fresnel Number ##F \ll 1##, because that is what most people think of when they think about diffraction. Because the Kirchhoff Integral produces cylindrically symmetric zflux patterns I will post cross sectional slices.
Far Field accepted formula
Far Field Numerical
Near Field Accepted Numerical
Near Field Numerical 2 (My numerical program that is supposed to be general)
As we can see there is a strong correspondence amongst the accepted formula and my numerical formulas.
Now when I graph power flux through various cubes surrounding the aperture
I get substantially up to 10% more power in the near field
My question is:
Should this be expected in the suprawavelength diffracting limit? I know that in the subwavelength diffracting limit there are initial evanescent waves
https://www.osapublishing.org/ao/abstract.cfm?uri=ao34173055
What about in the suprawavelength diffracting limit?
Edit: My units on the vertical axis of the last graph are wrong. They should have units of Poynting Vector times area not just Poynting vector. I have to think of a good way to normalize this.
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