Question

A \(5.000-\mathrm{cm}\) -wide diffraction grating with 200 grooves is used to resolve two closely spaced lines (a doublet) in a spectrum. The doublet consists of two wavelengths, \(\lambda_{\mathrm{a}}=\) \(629.8 \mathrm{nm}\) and \(\lambda_{\mathrm{b}}=630.2 \mathrm{nm} .\) The light illuminates the entire grating at normal incidence. Calculate to four significant digits the angles \(\theta_{1 \mathrm{a}}\) and \(\theta_{1 \mathrm{~b}}\) with respect to the normal at which the first-order diffracted beams for the two wavelengths, \(\lambda_{\mathrm{a}}\) and \(\lambda_{\mathrm{b}}\), respectively, will be reflected from the grating. Note that this is not \(0^{\circ}\) What order of diffraction is required to resolve these two lines using this grating?

Short answer

The angles for the first-order diffracted beams at the given wavelengths of \(629.8\,\text{nm}\) and \(630.2\,\text{nm}\) are approximately \(\theta_{1a} = 8.372^\circ\) and \(\theta_{1b} = 8.380^\circ\). The minimum order of diffraction required to resolve these two lines for the given grating is \(m = 1\).

Step-by-step solution

Understanding the grating equation

The grating equation is given by: \[d(\sin{\theta_m} + \sin{\theta_i}) = m\lambda\] In our case, the light is illuminating the grating at normal incidence, so \(\theta_i = 0\). Therefore, the grating equation simplifies to: \[d\sin{\theta_m} = m\lambda\]

Finding the grating spacing

We are given that there are 200 grooves in a \(5.000\,\text{cm}\)-wide grating. To find the grating spacing \(d\), we will divide the width of the grating by the number of grooves. \[d = \frac{5.000\,\text{cm}}{200} = \frac{5.000 \times 10^{-2}\,\text{m}}{200} = 2.500 \times 10^{-4} \,\text{m}\]

Finding the angles for the first-order diffracted beams

Now we will use the grating equation to find the angles \(\theta_{1a}\) and \(\theta_{1b}\) for the first-order diffraction (\(m = 1\)) and the given wavelengths \(\lambda_a = 629.8\,\text{nm}\) and \(\lambda_b = 630.2\,\text{nm}\). For \(\theta_{1a}\): \[2.500 \times 10^{-4} \,\text{m} \sin{\theta_{1a}} = (1) (629.8 \times 10^{-9}\,\text{m})\] \[\sin{\theta_{1a}} = \frac{(1) (629.8 \times 10^{-9}\,\text{m})}{2.500 \times 10^{-4} \,\text{m}}\] \[\theta_{1a} = \arcsin{(\frac{(1) (629.8 \times 10^{-9}\,\text{m})}{2.500 \times 10^{-4} \,\text{m}})} = 0.14601 \,\text{rad}\] Now converting radians to degrees, we get: \[\theta_{1a} = 0.14601 \,\text{rad} \times \frac{180^\circ}{\pi} = 8.3715 ^\circ\] For \(\theta_{1b}\): \[2.500 \times 10^{-4} \,\text{m} \sin{\theta_{1b}} = (1) (630.2 \times 10^{-9}\,\text{m})\] \[\sin{\theta_{1b}} = \frac{(1) (630.2 \times 10^{-9}\,\text{m})}{2.500 \times 10^{-4} \,\text{m}}\] \[\theta_{1b} = \arcsin{(\frac{(1) (630.2 \times 10^{-9}\,\text{m})}{2.500 \times 10^{-4} \,\text{m}})} = 0.14624 \,\text{rad}\] Now converting radians to degrees, we get: \[\theta_{1b} = 0.14624 \,\text{rad} \times \frac{180^\circ}{\pi} = 8.3797 ^\circ\] So, to four significant digits, the angles are \(\theta_{1a} = 8.372^\circ\) and \(\theta_{1b} = 8.380^\circ\).

Finding the order of diffraction required to resolve the lines

Resolving power of a grating is given by: \[R = mN\] where \(R\) is the resolving power, \(m\) is the order of diffraction, and \(N\) is the total number of grooves. We are given that there are 200 grooves in the grating. To find the minimum order of diffraction \(m\) that can resolve the two lines, we can use the shortest wavelength \(\lambda_a\) and the wavelength difference \(\Delta\lambda = \lambda_b - \lambda_a\) as follows: \[R = \frac{\lambda_a}{\Delta\lambda}\] \[mN = \frac{\lambda_a}{\Delta\lambda}\] \[m = \frac{\lambda_a}{\Delta\lambda \times N}\] \[m = \frac{629.8 \times 10^{-9}\,\text{m}}{(630.2 - 629.8) \times 10^{-9}\,\text{m} \times 200}\] \[m \approx 0.998\] Since the order of diffraction must be an integer, we will round \(m\) up to the next integer. Therefore, the minimum order of diffraction required to resolve the two lines is \(m = 1\).

Key concepts

Grating Equation

The grating equation forms the foundation for understanding how a diffraction grating manipulates light to produce patterns of dark and light, referred to as diffraction patterns. This relationship is given by the formula:
\[d(\sin{\theta_m} + \sin{\theta_i}) = m\lambda\]
Here, \(d\) represents the distance between adjacent grooves on the grating, \(\theta_i\) is the incident angle of light, \(\theta_m\) is the angle of the m-th order maximum, and \(m\) is the order of the diffraction pattern. For normal incidence, when the light strikes the grating perpendicularly, \(\theta_i\) is zero. This simplifies the grating equation to
\[d\sin{\theta_m} = m\lambda\]
By manipulating this equation, we can solve for the angles at which various orders of diffracted light will appear. This application is crucial when analyzing the spectral composition of light or resolving closely spaced spectral lines.

Spectral Lines Resolution

The ability of a diffraction grating to separate closely spaced wavelengths of light, known as spectral lines, is characterized by its resolution. This is quantified by the minimum distance between two wavelengths that can still be distinguished as separate. The precision of this resolution is illustrated when resolving a doublet, two very close spectral lines.
The resolving power, fundamental to the grating's resolution capability, depends on both the order of the diffraction pattern (\(m\)) and the total number of grooves illuminated on the grating (\(N\)). For a pair of wavelengths, if the angle of diffraction for each is distinct enough, they can be resolved. In practice, the higher the order of diffraction, the better the grating can separate the lines. Physically, this separation involves diffracting the light into different angles depending on its wavelength, a principle laid out by the grating equation.

Resolving Power

Resolving power is a measure of a diffraction grating's ability to distinguish two closely spaced wavelengths. It is defined as the inverse of the smallest wavelength difference that can be distinguished for a given wavelength, and is described by the formula:
\[R = mN\]
Here, \(R\) represents the resolving power, \(m\) is the order of diffraction, and \(N\) is the number of lines or grooves. It is evident from the equation that resolving power increases linearly with the order of diffraction and the number of grooves. This concept is critical because it dictates the grating's precision in resolving fine details within a spectrum. To achieve the resolution of a doublet, for example, the correct order of diffraction must be chosen based on the resolving power and the difference between the wavelengths.

Order of Diffraction

The order of diffraction refers to the series of maxima observed in a diffraction pattern. Each order corresponds to a multiple of wavelengths that constructively interfere to form a bright fringe. Mathematically, it is represented by the integer value \(m\) in the grating equation and the resolving power formula. The first order (\(m = 1\)) is often the most distinct and easiest to identify, but higher orders lead to greater separation between diffracted beams and thus better resolution. To ascertain the minimum order necessary to resolve a doublet, we must consider the resolving power equation and the fine difference between the two wavelengths. It's important to realize that the grating must have a sufficiently high number of lines and be illuminated by the proper order of light, in tandem with the grating equation, to achieve the desired resolving capability.