Question

A diffraction grating with exactly 1000 lines per centimeter is illuminated by a He-Ne laser of wavelength \(633 \mathrm{nm}\). a) What is the highest order of diffraction that could be observed with this grating? b) What would be the highest order if there were exactly 10,000 lines per centimeter?

Short answer

Short answer: The highest orders of diffraction are (a) 15 and (b) 1.

Step-by-step solution

Write the formula for the diffraction grating

The formula for a diffraction grating is given by: \(d \sin{\theta} = m \lambda\) where: \(d\) is the distance between each grating line, \(\theta\) is the angle of diffraction, \(m\) is the order of diffraction, and \(\lambda\) is the wavelength of the light.

Find the distance between each grating line

We can calculate the distance between each grating line (\(d\)) using the number of lines per centimeter given in the problem. For part a), the grating has 1000 lines per centimeter. To find the distance, we can divide 1 centimeter by the number of lines: \(d_a = \frac{1}{1000} \ \mathrm{cm}\) For part b), the grating has 10,000 lines per centimeter. So, we can find the distance as: \(d_b = \frac{1}{10000} \ \mathrm{cm}\)

Convert wavelength and distance to the same units

We need to have the same units for both \(\lambda\) and \(d\). Since \(d\) is given in centimeters, let's convert the wavelength from nanometers to centimeters: \(\lambda = 633 \ \mathrm{nm} \times \frac{1}{10^7} \ \mathrm{cm}\) Now we have: \(\lambda = 6.33 \times 10^{-5} \ \mathrm{cm}\)

Calculate the highest order of diffraction for each grating

To find the highest order of diffraction, we need to look at the maximum value for \(\sin{\theta}\), which is 1. So, the maximum order of diffraction can be found by re-arranging the diffraction grating equation as follows: \(m_{max} = \frac{d \sin{\theta}}{\lambda}\) For part a): \(m_{max_a} = \frac{d_a}{\lambda} = \frac{\frac{1}{1000}\ \mathrm{cm}}{6.33 \times 10^{-5}\ \mathrm{cm}} = 15.8\) The maximum order of diffraction has to be an integer, so we round down to get: \(m_{max_a} = 15\) For part b): \(m_{max_b} = \frac{d_b}{\lambda} = \frac{\frac{1}{10000}\ \mathrm{cm}}{6.33 \times 10^{-5}\ \mathrm{cm}} = 1.58\) Again, we round down to get: \(m_{max_b} = 1\) So the highest orders of diffraction for the gratings are 15 in part a) and 1 in part b).

Key concepts

Order of Diffraction

Understanding the order of diffraction is essential when discussing the behavior of light as it encounters a diffraction grating.

Put simply, the order of diffraction, denoted by the integer value 'm', represents the series of bright lines observed as light bends (diffracts) around the edges of an obstacle or opening. These bright lines, or fringes, are a direct result of constructive interference between the waves of light. In the case of a diffraction grating, which consists of many closely spaced slits, the mth order corresponds to the mth set of bright fringes seen on either side of the central, zero-order fringe.

When light of a particular wavelength hits the grating, different orders represent the different angles at which light of that specific wavelength constructively interferes. The higher the order, the greater the angle from the original direction of the light. The zero order, or m=0, corresponds to light that does not change direction and the first-order, m=1, represents the first set of fringes adjacent to the central maximum.

It's important to note that not all orders may be observed. The maximum order of diffraction depends on the spacing of the grating and the wavelength of light, with narrower spacing and shorter wavelengths allowing for higher orders to be seen.

Wavelength Conversion

In various physics and engineering problems, such as those involving diffraction gratings, wavelengths must be converted to appropriate units to correctly apply the formulas and obtain accurate results.

Typically, light's wavelengths are measured in nanometers (nm) as they fall in the range of the electromagnetic spectrum that's perceivable by the human eye (approximately 400-700 nm). However, when using the diffraction grating formula, you often need to align the wavelength units with those used for the grating distance, usually meters (m) or centimeters (cm).

For wavelength conversion, remember key conversion factors, such as 1 meter equals 10^9 nanometers. When converting the given wavelength in nanometers to centimeters, as seen in the example problem, multiplying by \(10^{-7}\) effectively shifts the wavelength from the nanometer scale to the centimeter scale, which is required by the grating equation to correctly compute the order of diffraction. This step is a crucial part of problem-solving in optics to ensure the units used in calculations are consistent.

Grating Equation

The heart of understanding how a diffraction grating manipulates light lies in the grating equation:
\[d \sin{\theta} = m \lambda\]
Here, \(d\) represents the distance between adjacent slits (grating lines), \(\theta\) is the angle at which light is diffracted with respect to the original path, \(m\) indicates the order of diffraction, and \(\lambda\) is the wavelength of light. This simple yet foundational equation encapsulates the physics behind the grating's ability to separate light into its component wavelengths.

A grating characteristically produces several diffracted beams, each corresponding to a different order of diffraction. The angles at which these beams emerge are not arbitrary; they are dictated by the interplay of the grating's line spacing and the wavelength of the incoming light as per the grating equation. It is through this formula that one can calculate the maximum order predictable for a given setup, as demonstrated in the textbook exercise provided.

An important reminder when using the grating equation is that the value of \(\sin{\theta}\) cannot exceed one, as it's based on the sine function. This principle is fundamental in establishing the theoretical limit for the maximum order of diffraction that can be observed, forming a bridge between theoretical physics and empirical observation.