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I have seen that it is often stated that Babinet's principle is only valid in the far field limit/Fraunhofer condition as it makes use of the linearity of the fourier transform.

However couldn't you use the linearity of any integral (even the Kirchoff-Fresnel integral) to still use Babinet's principle? I don't see how it relies on the linearity of the phase as in the Fraunhofer condition.

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I think it depends how you define Babinet's principle. The fact that the electric field due to diffraction through an apeture summed with the electric field due to diffraction through the compliment of that aperture (around an equivalent obstacle) is equal to the electric field as if no obstructions were present is always true (even in the Fresnel regime).

However only when applied to Fraunhofer diffraction (not Fresnel etc) does Babinet's principle imply that the aperture and its compliment will result in the exact same intensity distribution (except at 0 angle). I think this is because this result relies on positions/angles where the electric field of the unobstructed ray is exactly 0, so this is only possible in the far-field limit or with aid of focussing lenses (hence Fraunhofer limit).

More information is available p531 of Hecht: Optics.

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