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Suppose you cover one of two slits with a microscope slide and the other with nothing. If the slide has thickness \(1 \mathrm{~mm}\), show that monochromatic light of wavelength \(5000 \AA\) gets a retardation in one slit of about 1000 wavelengths relative to the other. If the double-slit pattern is not to wash out, the light must be fairly monochromatic. How narrow a band of wavelengths (in angstroms) is required so that the relative phase shift of the two slits varies by less than 180 deg from one edge of the wavelength band to the other? How could you use this fact to measure the bandwidth of a spectral line? (What would you measure and plot versus what, and how would you obtain the bandwidth from the plot?)

Step by step solution

01

Calculate Retardation

First, determine the retardation caused by the glass slide. The formula for retardation \( \Delta \) is \( \Delta = t(n-1) \), where \( t = 1 \text{ mm} = 10^{6} \text{ Ã…} \) and \( n \) is the refractive index of glass, approximately \( 1.5 \). Substituting the values: \( \Delta = 10^6 (1.5 - 1) = 5 \times 10^5 \text{ Ã…} \).

02

Convert Retardation to Wavelengths

To find the number of wavelengths by which one path is retarded relative to the other, divide the retardation by the wavelength: \( \text{Wavelength shift} = \frac{5 \times 10^5 \text{ Ã…}}{5000 \text{ Ã…}} = 100 \text{ wavelengths} \).

03

Calculate Wavelength Bandwidth

The phase shift \( \phi \) is given by \( \phi = \frac{2\pi \Delta}{\lambda} \). We want the variance in phase \( \Delta \phi = 2\pi \) over the band, corresponding to a range \( \Delta \lambda \) in wavelength, such that \( \frac{2\pi(t(n-1))}{\lambda^2} \Delta \lambda = \pi \). Simplifying gives: \( \Delta \lambda = \frac{\lambda^2}{2t(n-1)} = \frac{(5000)^2}{2 \times 10^6 \times 0.5} = 25 \text{ Ã…} \).

04

Use the Bandwidth to Measure Spectral Line

To measure the bandwidth of a spectral line, plot the intensity of the interference pattern against the wavelength. Measure the spectral line width by determining the base width of the peaks in the distribution, as the intensity pattern becomes less defined when the wavelength of light extends beyond the calculated bandwidth.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Double-Slit Experiment

The double-slit experiment is a fundamental demonstration in the study of wave interference. It shows how light behaves both as a wave and a particle. When light passes through two closely spaced slits, it creates an interference pattern on a screen behind the slits. This pattern consists of a series of bright and dark fringes. The bright fringes occur where the waves from the two slits reinforce each other (constructive interference), and dark fringes occur where they cancel each other out (destructive interference).

• The setup involves a barrier with two parallel slits and a screen to observe the interference pattern.
• It demonstrates the principle of superposition, where the amplitude of the resulting wave is the sum of the amplitudes of the individual waves.
• This experiment supports the wave theory of light, as the wave nature of light is necessary to explain the observed patterns.

To perform the double-slit experiment practically, one often uses lasers to provide coherent light, which simplifies the observation of interference patterns.

Wave Retardation

Wave retardation occurs when a wavefront slows down due to the passage through an optical medium. This is characterized by delay in the phase of the wave, leading to a shift in the interference pattern compared to waves that have not slowed down. Retardation occurs due to the differences in path length or refraction when light passes through materials with varying optical densities.

• Retardation can be calculated using the formula: \( \Delta = t(n-1) \), where \( t \) is the thickness of the medium and \( n \) is its refractive index.
• In the given problem, a glass slide with a thickness of \( 1 \text{ mm} \) causes a retardation by approximately 100 wavelengths.
• It is essential in controlling the phase relationship between waves, particularly in applications like anti-reflective coatings and optical instruments.

Wave retardation affects the resulting interference patterns, making it crucial to understand in experiments involving light interference.

Monochromatic Light

Monochromatic light consists of a single wavelength or color. It is ideal for experiments like the double-slit, as it ensures coherent wavefronts, leading to clear and stable interference patterns. Lasers are often used as sources of monochromatic light due to their ability to emit light at precise and narrow wavelengths.

• By using monochromatic light, you eliminate the spectral distribution that can blur interference patterns.
• It simplifies calculations in experiments, as the wavelength remains constant, allowing for accurate analysis of phase differences.
• Monochromatic light sources are crucial in spectroscopy, lasers, and other areas of optical studies.

To maintain a clear interference pattern, it is important that the light source does not have variations in intensity or wavelength over time.

Spectral Bandwidth Measurement

Spectral bandwidth refers to the range of wavelengths present in a light source. Measuring the bandwidth involves determining how broad this range is, which influences the resultant interference pattern. A narrow bandwidth is essential to ensure that interference effects are visible.

• The bandwidth can be determined by observing the intensity pattern and measuring where the fringes start to blur.
• In the given problem, we ensure a variation of less than 180 degrees in phase from one edge of the wavelength band to the other.
• Plotting the intensity versus wavelength allows us to visualize the bandwidth of a spectral line efficiently.

A narrow spectral bandwidth helps in precise measurements in experiments and applications like fiber-optic communications and precision spectroscopy.