Application One: Diffraction

Problem Introduction

  • what is diffraction: inteference patterns that light makes passing through apertures

  • how to make:

    • light from a distant source, which means the aperture plain is a wavefront i.e. wave have the same phase at all points of the aperture

    • plain with apertures

    • image plain at some distance

  • assumptions:

    • light is an oscollating E/M field

    • light is monochromatic (only one frequency)

    • light of the aperture plain as:

    E0e2πiνt

where E0 is strength of the field on the aperture plain and ν is frequency (monochromatic)

Near-Field vs. Far-Field

Distance between the aperture plain and the image plain (measured relative to the wavelength) determines diffraction

  • near-field Fresnel diffraction

  • far-field Fraunhofer diffraction

Huygens Principle

Each point on a wavefront can be regarded as a source i.e. all the sources on the wavefront will be integrated

Question: what is the strength of the wave on point P?

../../_images/app_diffraction_01.png

Solutioin

After introducing coordinates on the aperture plain, it is clear that the main effect in light going from X to P over a certain distance is: change in phase. The distance r can be represented as rλ cycles, which means a phase change of 2πrλ. Therefore the light magnitude of P resulted from a tiny source of point x of the aperture plain is:

dE=E0e2πiνte2πirλdx

therefore the total field is the integral over the aperture:

EP=E0e2πiνte2πirλdx=E0e2πiνte2πirλdx=E0e2πiνt+e2πirλA(x)dx

where r (not t) depends on x and A(x) is the aperture function:

A(x)={1xaperture0otherwise

with Fraunhofer approximation:

../../_images/app_diffraction_02.png
r=r0xsinθ

The integral can be re-written as:

+e2πirλA(x)dx=e2πixsinθr0λA(x)dx=e2πir0λ+e2πixpA(x)dx

where axillary variable p:

p=sinθλ
EP=C+e2πixpA(x)dx=CF1A(p)

Conclusion: for far-field diffraction, the intensity of the light is the magnitude of the inverse Fourier transform of the aperture function.

Back to Fourier Transform.