Question

For the grating in Problem $7.13$, calculate the wavelengths of first- and second-order diffraction at reflective angles of (a) $25°$and (b) $0°$. Assume the incident angle is $45°$.

Short answer

$$\begin{gathered} \mathrm{a}){𝝀}_1=12.5\mathrm{\mu m} \\ {𝝀}_2=6.27\mathrm{\mu m} \\ \mathrm{b}){𝝀}_1=7.85\mathrm{\mu m} \\ {𝝀}_2=3.92\mathrm{\mu m} \end{gathered}$$

Step-by-step solution

Given information

The given values are

$$\begin{gathered} lines/mm=90 \\ i=45° \\ a)r=25° \\ b)r=0° \end{gathered}$$

To obtain d

$$\begin{gathered} d=\frac{1\mathrm{mm}}{90\mathrm{lines}} \\ d=\frac{1\times {10}^{-3}\mathrm{m}}{90\mathrm{lines}} \\ d=1.11\times {10}^{-5}\mathrm{m} \\ d=11.1\times {10}^{-6}\mathrm{m} \\ d=11.1\mathrm{\mu m} \end{gathered}$$

Part a) Step 1: Calculations for first order diffraction.

Formula

$n𝝀=d(\sin i+\sin r)$

By putting values in the formula

$$\begin{gathered} n=1 \\ 1\times {𝝀}_1=11.1\mathrm{\mu m}(\sin 45°+\sin 25°) \\ {𝝀}_1=11.1\mathrm{\mu m}(\sin 45°+\sin 25°) \\ {𝝀}_1=1.25\times {10}^{-5} \\ {𝝀}_1=12.5\mathrm{\mu m} \end{gathered}$$

Part a) Step 2: Calculations for second order diffraction.

Formula

$n𝝀=d(\sin i+\sin r)$

By putting values

localid="1646387695674" $$\begin{gathered} n=2 \\ 2\times {𝝀}_2=11.1\mathrm{\mu m}(\sin 45°+\sin 25°) \\ {𝝀}_2=\frac{11.1\mathrm{\mu m}(\sin 45°+\sin 25°)}{2} \\ {𝝀}_2=6.27\times {10}^{-6} \\ {𝝀}_2=6.27\mathrm{\mu m} \end{gathered}$$

Part b) Step 1: Calculations for first order diffraction.

Formula

$n𝝀=d(\sin i+\sin r)$

By putting values

localid="1646387727980" $$\begin{gathered} n=1 \\ 1\times {𝝀}_1=11.1\mathrm{\mu m}(\sin 45°+\sin 0°) \\ {𝝀}_1=11.1\mathrm{\mu m}(\sin 45°+\sin 0°) \\ {𝝀}_1=7.85\times {10}^{-6} \\ {𝝀}_1=7.85\mathrm{\mu m} \end{gathered}$$

Part b) Step 2: Calculations for second order diffraction.

Formula

$n𝝀=d(\sin i+\sin r)$

By putting values

localid="1646387762274" $$\begin{gathered} n=2 \\ 2\times {𝝀}_2=11.1\mathrm{\mu m}(\sin 45°+\sin 0°) \\ {𝝀}_2=\frac{11.1\mathrm{\mu m}(\sin 45°+\sin 0°)}{2} \\ {𝝀}_2=3.92\times {10}^{-6} \\ {𝝀}_2=3.92\mathrm{\mu m} \end{gathered}$$