Grating, second-order spectrum and interference

[erinec]

Homework Statement

What is the minimum number of lines per centimeter that a grating must have if there is to be no second-order spectrum for any visible wavelength? (Let visible region extend from 400 nm to 570 nm.) Hint: Draw a diagram showing the relative positions of the rays corresponding to the first-order blue spectrum, the first order red spectrum, and the edge of the second-order spectrum.

Homework Equations

dsin(theta) = m(lambda)

The Attempt at a Solution

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[togahockey15]

did you ever find a solution to this?

 

[nlightner]

The minimum number of lines per centimeter for a grating to eliminate the second-order spectrum can be calculated using the equation dsin(theta) = m(lambda), where d is the spacing between the lines, theta is the angle of incidence, m is the order of the spectrum, and lambda is the wavelength of light. In order to eliminate the second-order spectrum, we want to find the maximum value of d that satisfies this equation for all visible wavelengths from 400 nm to 570 nm.

To do this, we can use the shortest wavelength (400 nm) and the largest value of m (2) to find the minimum value of d. Plugging in these values, we get d = lambda/(2sin(theta)) = 400 nm / (2sin(90)) = 200 nm. This means that the spacing between lines must be at least 200 nm in order to eliminate the second-order spectrum for all visible wavelengths.

To visualize this, we can draw a diagram with the first-order blue spectrum, the first-order red spectrum, and the edge of the second-order spectrum. The angle of incidence for the first-order blue spectrum is smaller than the angle for the first-order red spectrum, and the angle for the second-order spectrum is even larger. Therefore, the lines on the grating must be spaced closely enough to only allow the first-order spectra to pass through, but not far enough apart to allow the second-order spectrum to form. This is why the minimum spacing is determined by the shortest wavelength and the largest value of m.

In conclusion, in order to eliminate the second-order spectrum for all visible wavelengths, a grating must have a minimum of 200 lines per centimeter.