Diffraction by a Double Slit — Question 4

Question 4

The diffraction envelope shifts true maxima inward from nominal orders

Let $I(\beta)=4I_s(\sin\beta/\beta)^2\cos^2(r\beta)$ with $r=d/a\gg1$. Consider a nominal positive order $\beta_m=m\pi/r$ strictly within the central envelope. Write the nearby true maximum as $\beta_m+\delta$, with $|r\delta|\ll1$.

1. Differentiate the logarithm of intensity to obtain the nonzero-intensity stationarity condition.

2. Keeping the leading large-r term, derive the shift and establish its sign for $0<\beta_m<\pi$.

Question 4 - Solution

The diffraction envelope shifts true maxima inward from nominal orders

(a) Method and calculation

$$\frac12\frac{d\ln I}{d\beta}=\cot\beta-\frac1\beta-r\tan(r\beta).$$$$\boxed{\cot\beta-1/\beta-r\tan(r\beta)=0}.$$

(b) Interpretation and further calculation

$$\tan(r\beta_m+r\delta)\simeq r\delta\Longrightarrow\boxed{\delta\simeq\frac{\cot\beta_m-1/\beta_m}{r^2}}.$$$$\cot\beta_m<1/\beta_m\quad(0<\beta_m<\pi)\Longrightarrow\boxed{\delta<0}.$$

Physical check

The approximation requires the nominal order not to be too close to an envelope zero, where the envelope derivative becomes large.