Diffraction — Question 10

Question 10

A diffraction measurement separates wavelength and aperture uncertainties

A far-field measurement uses $W\simeq2L\lambda/a$. The screen distance is $L=2.00\,\mathrm m$, the full central width is $W=12.0\,\mathrm{mm}$, and the calibrated slit width is $a=0.200\,\mathrm{mm}$. Independent standard relative uncertainties in W, a, and L are 1.0 percent, 0.5 percent, and 0.2 percent.

1. Infer the wavelength and derive its first-order fractional differential.

2. Calculate the standard uncertainty in wavelength using independent uncertainty propagation.

Question 10 - Solution

A diffraction measurement separates wavelength and aperture uncertainties

(a) Method and calculation

$$\lambda=\frac{Wa}{2L}\Longrightarrow\boxed{\lambda=600\,\mathrm{nm}}.$$$$\boxed{\frac{d\lambda}{\lambda}=\frac{dW}{W}+\frac{da}{a}-\frac{dL}{L}}.$$

(b) Interpretation and further calculation

$$\left(\frac{\sigma_\lambda}{\lambda}\right)^2=0.010^2+0.005^2+0.002^2=0.000129.$$$$\boxed{\sigma_\lambda=6.81\,\mathrm{nm},\quad\lambda=(600\pm7)\,\mathrm{nm}}.$$

Physical check

The minus sign in the differential does not subtract a variance; independent uncertainty contributions add in quadrature.