Review the elementary arguments for discussing single-slit Fraunhofer diffraction by pairing up portions of the wavefront in the aperture that are \(180^{\circ}\) out of phase to show that (a) a null is observed when the slit may be divided into two equal portions such that $$ \frac{\delta}{2}=\frac{a}{2} \sin \theta=\frac{\lambda}{2} $$ Prob. 10.4.1 (b) Similarly, nulls result when the aperture is divided up into \(2 n\) portions such that $$ \frac{a}{2 n} \sin \theta=\frac{\lambda}{2} $$ where \(n\) is an integer; \((c)\) the secondary maxima occur approximately when the aperture is divided up into \(2 n+1\) portions such that $$ \frac{a}{2 n+1} \sin \theta=\frac{\lambda}{2} $$ and that the intensity of these maxima, relative to the principal maximum, is approximately $$ \frac{1}{2}\left(\frac{1}{2 n+1}\right)^{2} . $$ How do you justify the factor of \(\frac{1}{2}\) ?

Step by step solution

01

Path difference and superposition

To determine the position of nulls and maxima in the diffraction pattern, we will use the superposition principle, which states that the resultant wave at a point is the vector sum of the waves from all the points of the aperture. We will also use the concept of path difference between the waves arriving at a point in the far-field from two different sources within the aperture. For constructive interference, the path difference between these two sources should be an integral multiple of the wavelength, and for destructive interference, it should be an odd multiple of half of the wavelength.

02

Determine the position of the first null

We will determine the position of the first null in the diffraction pattern, which occurs when the slit is divided into two equal portions: $$ \frac{\delta}{2}=\frac{a}{2} \sin \theta=\frac{\lambda}{2} $$ In this case, the path difference between the waves from the top and bottom halves of the slit is equal to half of the wavelength. The waves from these two halves interfere destructively, resulting in a null in the diffraction pattern.

03

Determine the position of other nulls

Next, we determine the position of nulls for the case when the aperture is divided up into \(2n\) portions: $$ \frac{a}{2n} \sin \theta=\frac{\lambda}{2} $$ For nulls to occur in these cases, the same condition of half-wavelength path difference must hold. Therefore, for each integer value of \(n\), a null appears in the diffraction pattern.

04

Determine the position of secondary maxima

We now determine the position of secondary maxima, which occur approximately when the aperture is divided up into \(2n+1\) portions: $$ \frac{a}{2n+1} \sin \theta=\frac{\lambda}{2} $$ In this case, the path differences between the waves from individual portions result in an almost perfect destructive interference, but not enough to create a null. This results in secondary maxima in the diffraction pattern.

05

Intensity of secondary maxima and the factor of 1/2

The intensity of the secondary maxima, relative to the principal maximum, can be derived from the interference of the two waves and is approximately given by: $$ \frac{1}{2}\left(\frac{1}{2n+1}\right)^{2} $$ The factor of \(\frac{1}{2}\) appears in this expression because the intensity of a wave is proportional to the square of its amplitude. Since each interfering wave contributes a half-wavelength path difference, their amplitudes add up to \(\frac{1}{2n+1}\) of the principal maximum, and the intensity is proportional to the square of this amplitude, thus leading to the factor of \(\frac{1}{2}\).

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Key Concepts

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

Path Difference

The concept of path difference is crucial when analyzing patterns produced by the Fraunhofer diffraction. It denotes the difference in distance that two coherent light rays travel from an aperture to a point of observation. When a wavefront passes through a single slit, different parts of the wavefront will travel different distances to reach the same point on a screen. If this path difference is a multiple of the wavelength \( \lambda \), we get constructive interference, where light waves amplify each other.

Conversely, when the path difference is an odd multiple of half the wavelength \( \lambda/2 \), destructive interference occurs, where waves cancel each other out, resulting in a dark fringe or a null. This is what happens when the single slit is divided into two equal portions, and each portion's path difference leads to destructive interference.

Superposition Principle

The superposition principle is a fundamental concept in physics that states the resultant wave at any point is the sum of the individual waves from all points in the aperture. In the context of Fraunhofer diffraction, each part of the slit acts as a separate source of waves. When light waves overlap, they superimpose on each other to create a pattern of varying brightness on a distant screen through constructive or destructive interference.

It is this principle that allows us to predict where the bright and dark fringes will appear in a diffraction pattern, depending on how the individual wave amplitudes align with each other. The principle applies not only to light but to all kinds of wave phenomena.

Destructive Interference

We encounter destructive interference in the context of Fraunhofer diffraction when two waves arrive at a point with a path difference of an odd number of half wavelengths \( (2n+1)\lambda/2 \), causing the waves to cancel each other out. This leads to a decrease in light intensity, creating dark bands or 'nulls' on the screen. These nulls become apparent when the slit is divided into an even number of equal portions, and the collective path differences cause a complete cancellation of light in those specific directions.

In the Fraunhofer diffraction pattern, the positions of these dark fringes can be calculated, and they indicate places where the light from different parts of the slit destructively interferes.

Constructive Interference

On the flip side, constructive interference takes place when the path differences between waves result in them arriving in phase, or separated by whole multiples of the wavelength \( n\lambda \). This synchronicity amplifies the light intensity, culminating in bright bands or 'maxima' on the observation screen.

In a single-slit Fraunhofer diffraction setup, bright fringes occur at angles \( \theta \) where the difference in path length corresponds to a whole number of wavelengths. The central maximum is the brightest because it represents the point where waves from all parts of the slit arrive in phase.

Diffraction Intensity

The term diffraction intensity refers to how bright or dark a point in the diffraction pattern is and is a direct outcome of the interference of the waves. According to the superposition principle, the total intensity at any given point is the square of the sum of the amplitudes of all waves arriving there.

Secondary maxima's relative intensity in a single-slit diffraction pattern, as compared to the primary maximum, is reduced due to the division of the aperture into an odd number \( (2n+1) \) of portions. As shown in the exercise, the intensity falls off with increasing \( n \) as \( (1/(2n+1))^2 \) and is halved due to the amplitude factor of the wavefront, thus justifying the \( 1/2 \) factor in the intensity equation.