Past exam of the physics A course of the University of Cambridge / 2022 / Paper 2 / Section B / B7 / c

Take the quoted separation to mean adjacent slit centre spacing $d=50\mu \mathrm{m}$. For three equal very narrow slits at $-d,0,d$, the Fraunhofer diffraction complex amplitude in the transverse direction isWith $I_1$ denoting one narrow slit's contribution, the light intensity isThe quotient is interpreted by its limit where both numerator and denominator vanish. At small angles on a screen at $R=1\mathrm{m}$, $\sin \theta \simeq X/R$ and $\beta =\pi X/(10\mathrm{mm})$. Principal interference fringes occur atBetween successive principal maxima there are two zeros, at $X=(m+1/3)(10\mathrm{mm})$ and $X=(m+2/3)(10\mathrm{mm})$. Halfway between them is a smaller maximum $I=I_1$, one ninth of a principal peak. The pattern is symmetric about $X=0$. The positions quoted in millimetres use the paraxial approximation; exactly $d\sin {\theta }_m=m\lambda$ and $X_m=R\tan {\theta }_m$.

A slit of width $w$ has the single-slit diffraction complex amplitude factor $\sin \alpha /\alpha$, where $\alpha =\pi w\sin \theta /\lambda$. Thus for $w=25\mu \mathrm{m}=d/2$,The interference factor keeps the same zeros and the same nominal principal-order positions as in part (i), while the single-slit diffraction wave envelope suppresses off-axis wave intensity. Its zeros areAt nominal principal diffraction orders $\sin {\theta }_m=m\lambda /d$, the wave envelope factor is $[\sin (m\pi /2)/(m\pi /2){]}^2$. ConsequentlyAll nonzero even principal orders are missing. The central single-slit diffraction maximum extends between the envelope zeros near $X=\pm 20\mathrm{mm}$ and includes the central and the $m=\pm 1$ principal orders. Because there are only three slits, the maxima of the product can shift slightly from the nominal nonzero-order positions; the quoted values are the exact product at those positions. The lower panel of the figure shows the full product.

Both panels normalize to their own central wave intensity. An absolute comparison would require the unspecified widths and transmitted power of the initially “narrow” slits; the normalized pattern change is fully determined.