FIGURE CP33.73 shows two nearly overlapped intensity peaks of the sort you might produce with a diffraction grating (see Figure 33.9b). As a practical matter, two peaks can just barely be resolved if their spacing Δy equals the width w of each peak, where w is measured at half of the peak’s height. Two peaks closer together than w will merge into a single peak. We can use this idea to understand the resolution of a diffraction grating. a. In the small-angle approximation, the position of the m=1 peak of a diffraction grating falls at the same location as the m=1 fringe of a double slit: y1=λL/d. Suppose two wavelengths differing by Δλ pass through a grating at the same time. Find an expression for Δy, the separation of their first-order peaks.

Verified step by step guidance

1. Identify the given variables and constants: λ (wavelength), L (distance from the grating to the screen), d (spacing between slits in the grating), and Δλ (difference in wavelengths).
2. Use the formula for the position of the first-order peak (m=1) for a diffraction grating, which is given by y1 = λL/d. This formula is derived from the small-angle approximation and the interference condition for a diffraction grating.
3. Calculate the position of the first-order peak for each wavelength, λ and λ + Δλ. For the first wavelength, the position is y1 = λL/d. For the second wavelength, the position is y1' = (λ + Δλ)L/d.
4. Find the separation between the two first-order peaks, Δy, by subtracting the position of the first peak from the position of the second peak: Δy = y1' - y1 = ((λ + Δλ)L/d) - (λL/d).
5. Simplify the expression for Δy to find the final formula: Δy = ΔλL/d. This expression represents the separation between the first-order peaks of two wavelengths differing by Δλ when passed through a diffraction grating.

Key Concepts

Diffraction Grating

A diffraction grating is an optical component with a periodic structure that splits and diffracts light into several beams. The angles at which these beams are observed depend on the wavelength of the light and the spacing of the grating lines. The grating equation, d sin(θ) = mλ, relates the grating spacing (d), the angle of diffraction (θ), the order of the peak (m), and the wavelength (λ). Understanding this concept is crucial for analyzing how different wavelengths produce distinct peaks.

Resolution in Optics

Resolution in optics refers to the ability to distinguish between two closely spaced objects or peaks. In the context of diffraction gratings, two peaks can be resolved if their separation (Δy) is at least equal to the width (w) of each peak at half its maximum height. This principle is essential for determining the minimum spacing required for clear observation of multiple wavelengths, which is particularly relevant when analyzing overlapping diffraction patterns.

Small-Angle Approximation

The small-angle approximation is a simplification used in physics when angles are small enough that sin(θ) can be approximated by θ (in radians). This approximation is particularly useful in diffraction problems where the angles involved are small, allowing for easier calculations of peak positions. In the context of the question, applying this approximation helps derive the relationship between the wavelengths and their corresponding peak separations in a diffraction grating setup.