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 (λ = 632.8 nm) is incident normally on a diffraction grating containing 6.00 × 10³ lines/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\qquad m=0,\pm,1,\pm,2,\ldots [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\times10^{-4}\,{\mathrm{cm}}\,=\,1.67\times10^{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:
\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}
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 θ can’t exceed 1, 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.