When laser light of wavelength 632.8 nm passes through a diffraction grating, the first bright spots occur at ±17.8° from the central maximum. (a) What is the line density (in lines/cm) of this grating? (b) How many additional bright spots are there beyond the first bright spots, and at what angles do they occur?

Verified step by step guidance

1. Step 1: Start with the diffraction grating equation: $d\; \backslash ,\; \backslash (\backslash sin\backslash )\; \backslash (\backslash theta\backslash )\; =\; m\; \backslash (\backslash lambda\backslash )$, where $d$ is the grating spacing (distance between adjacent lines), $\backslash (\backslash theta\backslash )$ is the diffraction angle, $m$ is the order of the bright spot, and $\backslash (\backslash lambda\backslash )$ is the wavelength of the light. For the first bright spot ($m\; =\; 1$), substitute $\backslash (\backslash lambda\backslash )\; =\; 632.8\; \backslash ,\; \backslash (\backslash text\{nm\}\backslash )\; =\; 632.8\; \backslash (\backslash times\backslash )\; 10^\{-9\}\; \backslash ,\; \backslash (\backslash text\{m\}\backslash )$ and $\backslash (\backslash theta\backslash )\; =\; 17.8^\backslash (\backslash circ\backslash )$ into the equation.
2. Step 2: Rearrange the equation to solve for $d$: $d\; =\; \backslash (\backslash frac\{m\; \backslash lambda\}\{\backslash sin\; \backslash theta\}\backslash )$. Substitute the known values for $m$, $\backslash (\backslash lambda\backslash )$, and $\backslash (\backslash sin\backslash )\; \backslash (\backslash theta\backslash )$ to calculate $d$. Once $d$ is found, the line density of the grating is given by $\backslash (\backslash text\{line\; density\}\backslash )\; =\; \backslash (\backslash frac\{1\}\{d\}\backslash )$. Convert the result to lines/cm by multiplying by $10^2$.
3. Step 3: To determine how many additional bright spots occur, note that the maximum diffraction order $m$ is limited by the condition $\backslash (\backslash sin\backslash )\; \backslash (\backslash theta\backslash )\; \backslash (\backslash leq\backslash )\; 1$. Rearrange the diffraction equation to find the maximum order: $m\_\{\backslash (\backslash text\{max\}\backslash )\}\; =\; \backslash (\backslash lfloor\backslash )\; \backslash (\backslash frac\{d\}\{\backslash lambda\}\backslash )\; \backslash (\backslash rfloor\backslash )$, where $\backslash (\backslash lfloor\backslash )\; \backslash (\backslash cdot\backslash )\; \backslash (\backslash rfloor\backslash )$ represents the floor function. Substitute the value of $d$ and $\backslash (\backslash lambda\backslash )$ to calculate $m\_\{\backslash (\backslash text\{max\}\backslash )\}$.
4. Step 4: The total number of bright spots is given by $2\; \backslash (\backslash times\backslash )\; m\_\{\backslash (\backslash text\{max\}\backslash )\}\; +\; 1$ (including the central maximum). Subtract the first-order bright spots to find the number of additional bright spots beyond the first order.
5. Step 5: To find the angles of the additional bright spots, use the diffraction equation $\backslash (\backslash sin\backslash )\; \backslash (\backslash theta\backslash )\; =\; \backslash (\backslash frac\{m\; \backslash lambda\}\{d\}\backslash )$ for each order $m$ (from 2 to $m\_\{\backslash (\backslash text\{max\}\backslash )\}$). Solve for $\backslash (\backslash theta\backslash )$ for each order, ensuring that $\backslash (\backslash sin\backslash )\; \backslash (\backslash theta\backslash )\; \backslash (\backslash leq\backslash )\; 1$. Record the angles for both positive and negative diffraction orders.

Key Concepts

Diffraction Grating

A diffraction grating is an optical component with a periodic structure that disperses light into its constituent wavelengths. It works by exploiting the wave nature of light, causing interference patterns when light waves overlap. The angle at which bright spots (maxima) appear is determined by the grating equation, which relates the wavelength of light, the angle of diffraction, and the spacing between the grating lines.

Grating Equation

The grating equation, given by d sin(θ) = mλ, relates the wavelength (λ) of light to the angle of diffraction (θ) and the grating spacing (d). Here, m is the order of the maximum, which can be positive or negative. This equation is essential for calculating the line density of the grating and determining the angles of additional bright spots beyond the first order.

Line Density

Line density refers to the number of lines per unit length on a diffraction grating, typically expressed in lines/cm. It is the inverse of the grating spacing (d), which is the distance between adjacent lines. A higher line density results in more closely spaced diffraction maxima, affecting the angles at which bright spots appear and the overall resolution of the grating.