This problem follows one chain of reasoning from Green's theorem to a spectrometer's…
PHYSICS · Subjective · Fermi University Physics · Optics & Photonics · Interference and diffraction · Interference and diffraction
What an Instrument Can See, Ch Scalar diffraction theory, derived
Problem
This problem follows one chain of reasoning from Green's theorem to a spectrometer's resolving power. (a) Starting from Green's second identity applied to the field U(r) and the free-space Green function G=eikr/r centred on an observation point P inside a closed surface S, derive the Helmholtz-Kirchhoff integral theorem U(P)=4π1∮S[U∂n∂G−G∂n∂U]dS, with ∂/∂n the inward normal derivative into the volume containing P. State explicitly what has been assumed about U in reaching this result, and what has not yet been assumed about any screen or aperture. (b) Starting from the Fresnel diffraction integral obtained from the Rayleigh-Sommerfeld propagator by the paraxial expansion of r, show that when the Fresnel number of the aperture satisfies NF≪1 the residual quadratic phase can be dropped and the field a distance z away becomes the Fraunhofer transform of the aperture transmittance tA(x′,y′), U(x,y;z)∝∫tA(x′,y′)e−i2π(xx′+yy′)/(λz)dx′dy′. Apply this to a transmission grating of N identical slits of spacing Λ, illuminated at normal incidence, and write down the form of the diffracted amplitude as a function of spatial frequency νx=x/(λz) (a single-slit envelope multiplied by a Dirichlet-type interference factor). (c) From the width in νx of the mth-order peak of this interference factor, derive the grating's resolving power R=λ/Δλ=mN. (d) A spectrometer grating has an illuminated width W=60mm ruled at 900 lines per millimetre, used in second order (m=2) at normal incidence near λ=500nm to resolve two lines separated by Δλ=0.006nm. Find N, the resolving power R, and state by what factor R exceeds the resolving power λ/Δλ that the doublet requires.