# In diffraction, what exactly is causing the obliquity factor?

As I understand it, this is a factor we introduce (sometimes as $$\cos\theta$$, sometimes as $$\frac{1}{2}(1+\cos\theta)$$) to correct behaviour at an aperture when Huygens principle is applied (e.g. prevent backwards propagation). This matches observation.

From Goodman's Introduction to Fourier Optics:

The last property, namely the obliquity factor, has no simple "quasi-physical" ex- planation, but arises in slightly different forms in all the theories of diffraction. It is perhaps expecting too much to find such an explanation. After all, there are no material sources within the aperture; rather, they all lie on the rim of the aperture. Therefore the Huygens-Fresnel principle should be regarded as a relatively simple mathematical construct that allows us to solve diffraction problems without paying attention to the physical details of exactly what is happening at the edges of the aperture.

It is the part in bold that I am trying to understand / pay attention to.

Blindly applying "every point of a wavefront acts as a spherical emitter", it is not clear to me what is happening at the edges of the aperture that is causing this fall-off in amplitude. I suppose the Goodman quote suggests there isn't an 'easy' reason / intuition, but could someone point me towards some reason?

Would appreciate this a lot!

• "can be derived mathemtically" may not be true?? Jun 3 at 16:53
• @user45664 ah right, my bad! Even better question then! Jun 3 at 17:26
• K. 622 Where did the image come from? I was wondering if there is an actual photo that had those blurry areas that spread out at different angles. Or is this just a created image. Jun 3 at 18:00
• it is not trivial to derive the obliquity factor but it is done here in some detail for paraxial waves, it is more complicated for the general case. Jun 3 at 18:24
• @BillAlsept just Wikipedia (article on diffraction) Jun 4 at 11:30

## 1 Answer

The obliquity factor is not needed to eliminate the backward wave. See my

Huygens' Principle geometric derivation and elimination of the wake and backward wave

https://www.nature.com/articles/s41598-021-99049-7

Especially see the Notes in the Supplementary Information (SI). The backward wave is eliminated by an additional backward wave of opposite polarity (or sign) under the proper conditions. Eg. 'As for the backward wave, a source propagates both a forward wave and a backward wave when it is stationary, but it propagates only the forward wave front when it is advancing with a speed equal to the propagation speed of the wave fronts'. Think of an impulsively excited planar source.