![]() ![]() The calculation is straightforward and can be done easily by hand!Ī graph of the aperture function (showing the transmissivity): To combine more steps we simply add more sections of the integral. We found the intensity pattern for a single slit (cf. The aperture has a width of d /2 and is chosen to be symmetric around y =0. We choose d as the spacing of a line and a gap (the aperture).įirst the single slit. Now we set up the calculation for the Ronchi ruling (a set of equidistant black lines). The special role played by | x |= 1/ d is evident from the analytical result: Observe that the intensity ratio between the central and side peaks is close to the value of 4: We are ready to calculate the result for an aperture with n cycles ('slits'): Observe how a strong pair of peaks appears at | x |=1/ d. Now we add more slits gradually by extending the range of integration. Warning, the name changecoords has been redefined > plot(subs(d=10,log10(Af(x)^2)),x=-Pi/2.Pi/2,numpoints=500) Ī log-plot can also be produced directly using the plots-package: It is useful to look at a logarithmic representation of the amplitude: Note the behaviour of the result at | x |=1/d: a 0/0 expression results as for x =0, however the 1/ x factor yields some suppression of the amplitude compared to x =0. We set up the cos-squared profile such that the distance d contains the full width of a slit that includes the perfectly transmitting part up to the perfectly blocking parts on both sides (the zeroes of the cosine) at |y|=d/2 ![]() (9.10-9.11) from the reference we calculate the amplitude first for a 'single' slit. For a diffraction grating the aperture function is smooth: a cos-squared behaviour of the transmissivity as a function of separation y across the aperture is produced (using holography in modern times) and the periodicity is controlled by d. The distance parameter d plays the role of the spacing between the slits. To observe diffraction patterns for gratings and Ronchi rulings we calculate the Fourier transform of the respective aperture functions. The nonlinearity of the expression in theta becomes more apparent at even larger theta: We can look at the intensity pattern at larger angles: Vary the length parameters (wavelength and slit separation d ) and observe the change in the intesity pattern as a function of which is measured in radians. > Am:=theta->cos(Pi*d/lambda*sin(theta)) The amplitude as a function of the diffraction angle in radians: The aperture width (or slit separation for Young's experiment) should be at least several microns. We can think of the length unit as being microns, in which case a typical (yellow) wavelength equals 1/2. We use dimensionless quantities, and should use the displacements d which are larger than the wavelength. (where is the diffraction angle and the wavelength).įor illustration purposes we begin with the Fraunhofer diffraction pattern for a pair of narrow slits displaced by d (Young's expt.). This worksheet deals with the generation of diffraction patterns produced by various apertures illuminated by monochromatic light.įollowing Smith-Thomson: Optics (2nd ed.), chapter 9, we use as a convenient variable Fourier Optics: a study of diffraction patterns in the focal plane of a lens ![]()
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