A reflection grating is required which can resolve wavelengths as close as 0.002 nm in second order for the spectral region around 350 nm.The grating is to be installed in an instrument where light from the entrance slit is incident normal on the grating. If the manufacturer provides rulings over a 10 cm grating width, determine

Question

A reflection grating is required which can resolve wavelengths as close as 0.002 nm in second order for the spectral region around 350 nm.The grating is to be installed in an instrument where light from the entrance slit is incident normal on the grating. If the manufacturer provides rulings over a 10 cm grating width, determine

a) the minimum number of grooves/cm required
b) the optimum blaze angle
c) the angle of diffraction where irradiance is maximum

Answer 1

To determine the required specifications for the reflection grating, we can use the grating equation:

mλ = d(sin(θ_i) ± sin(θ_d))

where:
- m is the diffraction order
- λ is the wavelength of the light
- d is the spacing between the grooves of the grating
- θ_i is the angle of incidence
- θ_d is the angle of diffraction

Let's calculate the required specifications step-by-step:

a) Minimum number of grooves/cm required:

To resolve wavelengths as close as 0.002 nm in the second order, we need to find the spacing between the grooves (d). The second-order diffraction equation can be written as:

2λ = d(sin(θ_i) + sin(θ_d))

In this equation, we need to find d. We can rearrange the equation as follows:

d = 2λ / (sin(θ_i) + sin(θ_d))

Since the incident light is normal to the grating (θ_i = 0), the equation simplifies to:

d = 2λ / sin(θ_d)

Given λ = 0.002 nm and θ_d = 350 nm, we can substitute these values into the equation:

d = 2(0.002) / sin(350)

Note: I assume the values for λ and θ_d have the same units, either nm or cm. We'll use nm in this example.

Let's calculate d:

d = 2(0.002) / sin(350)

We can use a scientific calculator or online calculator to find the value of sin(350). After calculating sin(350), substitute it into the equation to get the value of d.

b) Optimum blaze angle:

The blaze angle is the angle at which the grooves of the grating are cut to optimize the diffraction efficiency for a specific wavelength. The optimum blaze angle can be calculated using the following formula:

tan(θ_blaze) = (m * λ) / (d * cos(θ_i))

Since we are interested in the second order, m = 2. Given the values of λ and d from the previous calculations, as well as the incident angle being normal (θ_i = 0), we can calculate the blaze angle (θ_blaze) using the formula:

tan(θ_blaze) = (2 * λ) / (d * cos(0))

Simplifying further:

tan(θ_blaze) = (2 * λ) / d

Substitute the values of λ and d to calculate the blaze angle.

c) Angle of diffraction where irradiance is maximum:

The angle of diffraction where irradiance is maximum is given by the equation:

sin(θ_max) = m * λ / d

For the second order (m = 2) and given the values of λ and d, we can calculate the angle of diffraction where irradiance is maximum (θ_max) using the equation:

sin(θ_max) = (2 * λ) / d

Substitute the values of λ and d to calculate θ_max.

Remember to convert the angles to degrees if necessary.

By calculating these values, you can determine the required specifications for the reflection grating: minimum number of grooves/cm, optimum blaze angle, and angle of diffraction where irradiance is maximum.