Someone argues: "A slit of full width is illuminated by light of wavelength

PHYSICS · Subjective · Fermi University Physics · Optics & Photonics · Interference and diffraction · Interference and diffraction

What an Instrument Can See, Ch Scalar diffraction theory, derived

Problem

Someone argues: "A slit of full width w=2 mmw = 2\,\mathrm{mm} is illuminated by light of wavelength λ=500 nm\lambda = 500\,\mathrm{nm}, and a diffraction pattern is observed on a screen at z=400 mmz = 400\,\mathrm{mm}. Since a diffraction pattern is seen, the Fraunhofer formula applies directly: I(θ)=I0 sinc2 ⁣(πwsin⁡θλ),I(\theta) = I_0\,\mathrm{sinc}^2\!\left(\frac{\pi w \sin\theta}{\lambda}\right), so the first minima occur where sin⁡θ=λ/w=2.5×10−4\sin\theta = \lambda/w = 2.5\times10^{-4}, giving bright and dark fringes spaced by about λz/w=0.1 mm\lambda z/w = 0.1\,\mathrm{mm} on the screen." Find the error in this reasoning, using the Fresnel number of the slit to justify the conclusion, and state (without carrying out the full Fresnel integral) what kind of pattern is actually present on the screen instead.

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