Problem 30 - University Physics with Modern Physics

Question

A laser beam of wavelength \(\lambda\) = 632.8 nm shines at normal incidence on the reflective side of a compact disc. (a) The tracks of tiny pits in which information is coded onto the CD are 1.60 \(\mu\)m apart. For what angles of reflection (measured from the normal) will the intensity of light be maximum? (b) On a DVD, the tracks are only 0.740 \(\mu\)m apart. Repeat the calculation of part (a) for the DVD.

Step-by-Step Solution

Answer

(a) CD: θ ≈ 23.3° for m=1, θ ≈ 52.0° for m=2; (b) DVD: θ ≈ 58.7° for m=1. Higher orders not possible for DVD.

Step 1: Understanding the Condition for Intensity Maximum

The condition for maximum intensity in reflection, i.e., constructive interference, is given by the diffraction grating formula:\[m \lambda = d \sin \theta\]where \(m\) is the order of the maximum, \( \lambda \) is the wavelength of the incident light (632.8 nm), \(d\) is the distance between tracks (1.60 \mu m for the CD and 0.740 \mu m for the DVD), and \(\theta\) is the angle of reflection from the normal.

Step 2: Calculating the Angle for Maximum Intensity for CD

For the CD, use the distance \(d = 1.60 \mu m = 1600 \ nm\). Now rearrange the equation for \(\sin \theta\):\[sin \theta = \frac{m \lambda}{d}\]For the first order maximum (\(m = 1\)), calculate \(\sin \theta\):\[\sin \theta = \frac{1 \times 632.8 \ nm}{1600 \ nm} = 0.3955\]Calculate \(\theta\) using \(\theta = \sin^{-1}(0.3955)\).

Step 3: Calculating Possible Orders for CD

Check possible orders by considering \(m = 2, 3, ...\) as long as \(\sin \theta \leq 1\). For example, for \(m = 2\):\[\sin \theta = \frac{2 \times 632.8}{1600} = 0.7910\]Calculate \(\theta = \sin^{-1}(0.7910)\) and continue finding \(\theta\) for higher \(m\) values.

Step 4: Repeat the Calculation for the DVD

Repeat the process for the DVD with track distance \(d = 0.740 \mu m = 740 \ nm\) and the same wavelength (632.8 nm). Start with \(m = 1\):\[\sin \theta = \frac{632.8}{740} = 0.8549\]Then, calculate \(\theta = \sin^{-1}(0.8549)\).

Step 5: Calculating Possible Orders for DVD

Similar to the CD, calculate possible orders for the DVD by checking higher\(m\) values until \(\sin \theta > 1\). For example, for \(m = 2\):\[\sin \theta = \frac{2 \times 632.8}{740} = 1.7098\]Since \(\sin \theta > 1\), only \(m = 1\) is valid for the DVD.

Key Concepts

Constructive Interference

When light waves meet at a point, they can combine to form a new wave pattern. This phenomenon is called interference. Constructive interference happens when the crests of two waves align together and amplify the wave. This results in a brighter light spot.
In the case of a diffraction grating, like the surface of a CD or a DVD, constructive interference occurs at specific angles. These angles are determined by the equation:

• \[ m \lambda = d \sin \theta \]

Here, \( m \) refers to the order of maximum, \( \lambda \) is the wavelength of the laser light, \( d \) is the distance between the grating lines, and \( \theta \) is the angle of reflection producing maximum brightness.

Laser Beam

A laser beam is a focused source of light characterized by its coherent and monochromatic properties. This means the laser light waves have a single wavelength and are aligned in phase, making them highly predictable and stable.
This uniformity is crucial in experiments where precise measurements are needed, like diffraction. When a laser beam hits a CD or DVD, its reflection patterns depend heavily on these properties. The beam interacts with the nanostructures on the disc's surface, gaining insights into how the light waves interfere with each other.
This interaction is key in tasks involving wavelength calculation and determining angles where the light intensity becomes maximal.

Wavelength Calculation

Calculating the wavelength in a diffraction grating setup involves using the observed angles of reflection where maximum brightness is noted. The formula is used:

• \[ m \lambda = d \sin \theta \]

Where \( \lambda \) is the unknown wavelength if the other values are known. The distance \( d \) is the spacing between the reflecting lines on the disc, and \( \theta \) is the angle observed where the light is brightest.
If you know the wavelength, you can rearrange the equation to find the different angles \( \theta \), which tell us where constructive interference occurs. This calculation helps understand how the disc's grating interacts with the particular laser light.

Angle of Reflection

Understanding the angle of reflection in the context of a diffraction grating is important for determining how light behaves when it encounters a surface with closely spaced lines, such as a CD or DVD. The angle of reflection is measured from an imaginary line perpendicular to the disc's surface, known as the normal.
Using the diffraction formula \( m \lambda = d \sin \theta \), we calculate this angle to find points of constructive interference, where the light's intensity is maximized. It's important to experiment with different values of \( m \) to find all possible angles \( \theta \) where brightness peaks.
Understanding these angles enables the design and production of optical storage media and informs the principles of optics technology in devices.