Question 24.7: A Diffraction Grating Goal Calculate different-order princip...
A Diffraction Grating
Goal Calculate different-order principal maxima for a diffraction grating.
Problem Monochromatic light from a helium-neon laser (\lambda=632.8 \mathrm{~nm}) is incident normally on a diffraction grating containing 6.00 \times 10^{3} lines / \mathrm{cm}. Find the angles at which one would observe the first-order maximum, the second-order maximum, and so forth.
Strategy Find the slit separation by inverting the number of lines per centimeter, then substitute values into Equation 24.12.
d\,\sin\theta_{\mathrm{bright}}=\,m\lambda\;\;\;\;\;\;\;\;m=0,\,1,2,\,.\,.\,\,. (24.12)
Invert the number of lines per centimeter to obtain the slit separation:
d=\frac{1}{6.00 \times 10^{3} \mathrm{~cm}^{-1}}=1.67 \times 10^{-4} \mathrm{~cm}=1.67 \times 10^{3} \mathrm{~nm}
Substitute m=1 into Equation 24.12 to find the sine of the angle corresponding to the first-order maximum:
\sin \theta_{1}=\frac{\lambda}{d}=\frac{632.8 \mathrm{~nm}}{1.67 \times 10^{3} \mathrm{~nm}}=0.379
Take the inverse sine of the preceding result to find \theta_{1} :
\theta_{1}=\sin ^{-1} 0.379=22.3^{\circ}
Repeat the calculation for m=2 :
\begin{aligned} \sin \theta_{2} & =\frac{2 \lambda}{d}=\frac{2(632.8 \mathrm{~nm})}{1.67 \times 10^{3} \mathrm{~nm}}=0.758 \\ \theta_{2} & =49.3^{\circ} \end{aligned}
Repeat the calculation for m=3 :
\sin \theta_{3}=\frac{3 \lambda}{d}=\frac{3(632.8 \mathrm{~nm})}{1.67 \times 10^{3} \mathrm{~nm}}=1.14
Because \sin \theta can’t exceed one, there is no solution for \theta_{3}.
Remarks The foregoing calculation shows that there can only be a finite number of principal maxima. In this case, only zeroth-, first-, and second-order maxima would be observed.