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Chapter 4: Problem 6

Fresnel Diffraction from Two Pinholes. Show that the Fresnel diffraction pattern from two pinholes separated by a distance \(2 a\), i.e., \(p(x, y)=[\delta(x-a)+\delta(x+a)] \delta(y)\), at an observation distance \(d\) is the periodic pattern, \(I(x, y)=(2 / \lambda d)^{2} \cos ^{2}(2 \pi a x / \lambda d)\).

Short Answer

Expert verified
The diffraction pattern is a cosine square function: \(I(x, y) = (2/\lambda d)^{2} \cos^{2}(2 \pi a x/\lambda d)\).

Step by step solution

01

Understanding the Problem Statement

We are given two pinholes separated by a distance of \(2a\) in the \(x\)-direction, expressed by the function \(p(x,y) = [\delta(x-a) + \delta(x+a)] \delta(y)\). Our goal is to derive the Fresnel diffraction pattern at a distance \(d\), which is given as \(I(x, y) = \left(\frac{2}{\lambda d}\right)^{2} \cos^{2}\left(\frac{2 \pi a x}{\lambda d}\right)\).
02

Writing the Fresnel Diffraction Integral

Using the Fresnel diffraction formula, the intensity at a point \((x, y)\) at distance \(d\) from the apertures is determined by the wave contribution from each pinhole, formulated as:\[E(x, y) = \frac{e^{ikd}}{i \lambda d} \iint A(x', y') e^{\frac{i \pi}{\lambda d}((x-x')^2 + (y-y')^2)} \, dx' \, dy'\]where \(A(x', y')\) is the aperture function. Here, \(x' = a\) and \(x' = -a\) are the locations of the pinholes.
03

Calculating Contribution from Each Pinhole

The contribution of the pinholes is calculated using:For \(x' = a\), \(p(x',y') = \delta(x'-a)\delta(y')\):\[E_a(x, y) = \frac{e^{ikd}}{i \lambda d} \cdot e^{\frac{i \pi}{\lambda d}(x-a)^2}\]For \(x' = -a\), \(p(x',y') = \delta(x'+a)\delta(y')\):\[E_{-a}(x, y) = \frac{e^{ikd}}{i \lambda d} \cdot e^{\frac{i \pi}{\lambda d}(x+a)^2}\]
04

Adding the Contributions and Simplifying

The total electric field, \(E(x, y)\), at the observation plane is the sum of contributions:\[E(x, y) = E_a(x, y) + E_{-a}(x, y) = \frac{e^{ikd}}{i \lambda d} \left( e^{\frac{i \pi}{\lambda d}(x-a)^2} + e^{\frac{i \pi}{\lambda d}(x+a)^2}\right)\]Using the identity for cosine, we rewrite this sum as:\[E(x, y) = \frac{2 e^{ikd}}{i \lambda d} e^{\frac{i \pi x^2}{\lambda d}} \cos\left(\frac{2 \pi ax}{\lambda d}\right)\]
05

Calculating the Intensity Pattern

The intensity \(I(x, y)\) is the square of the magnitude of \(E(x, y)\):\[I(x, y) = \left| E(x, y) \right|^2 = \left| \frac{2 e^{ikd}}{i \lambda d} e^{\frac{i \pi x^2}{\lambda d}} \cos\left(\frac{2 \pi ax}{\lambda d}\right) \right|^2\]This simplifies to:\[I(x, y) = \left(\frac{2}{\lambda d}\right)^2 \cos^2\left(\frac{2 \pi ax}{\lambda d}\right)\]which matches the given pattern.

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Key Concepts

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

Diffraction pattern
In the realm of optical physics, the diffraction pattern reveals how light bends around obstacles or openings. When light encounters an obstacle or a small opening, such as a slit or pinhole, it doesn't travel in just a straight line. Instead, it spreads out, creating a ripple effect on the way light waves travel. This rippling effect is what we call diffraction.

The Fresnel diffraction pattern, a specific type of diffraction, forms when light waves pass close to an object, and produces an intricate pattern of bright and dark bands or fringes. This pattern arises from the interference of light waves that have been bent around the obstacle. Essentially, the pattern is a visual representation of wave interference, showing regions of constructive and destructive interference.

Understanding these patterns is crucial for interpreting results in experiments like the pinhole experiment, where light's behavior offers insights into the nature of wave physics.
Pinhole experiment
The pinhole experiment is an excellent demonstration of wave interference and diffraction. By passing light through a small circular opening, we can observe how light behaves as a wave. In the particular case of two pinholes separated by a distance, an intriguing diffraction pattern emerges at the observation plane.

In this setup, the light waves travelling through each pinhole interfere with one another. Given a separation of distance in the x-direction, this causes the waves to generate a periodic pattern that is analyzed to study the principles of light-wave interference. This experiment provides a clear visual representation of Fresnel diffraction.

Usually, the result is an alternating series of light and dark bands, forming the classic pattern known as the interference pattern. It's a practical way to illustrate the wave nature of light, emphasizing how complex phenomena can be broken down by understanding basic physical principles like wave interference and diffraction.
Wave interference
Wave interference is at the heart of understanding optical phenomena like diffraction patterns. When waves meet, they interact in ways that can reinforce or weaken each other. This interaction is known as interference.

There are two main types of interference: constructive and destructive. Constructive interference occurs when two waves align perfectly in phase (their crests and troughs match), amplifying the resulting wave. Destructive interference happens when waves are out of phase (one wave's crest meets another's trough), cancelling each other out to create a minimal or null effect.

In the pinhole experiment, the light from the two pinholes interferes, meaning that depending on their phase alignment, certain points will see bright (constructive interference) or dark (destructive interference) spots. This periodic fluctuation of intensity manifests as a diffraction pattern, vividly illustrating the principles of wave physics.
Optical physics
Optical physics utilizes principles from classical and quantum physics to explain how light behaves. It covers topics like the wave-particle duality of light, refraction, reflection, and diffraction.

In optical physics, understanding why light behaves like it does in phenomena like the diffraction pattern from a pinhole experiment is essential. The experiment captures the wave aspect of light, showcasing its ability to bend around obstacles and create interference patterns 鈥� a concept that underlies many other optical devices and applications.

Optical physics not only helps in designing lenses and enhancing visual technologies but also aids in precision measurements in scientific research and industry. By understanding the basic principles of diffraction and wave interference, students and professionals can deepen their knowledge of how light interacts with different materials and technologies, shaping modern advancements.

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Most popular questions from this chapter

Image of a Phase Object. An imaging system has an impulse response function \(h(x, y)=\) \(\operatorname{rect}(x) \delta(y)\). If the input wave is $$ f(x, y)= \begin{cases}\exp \left(j \frac{\pi}{2}\right) & \text { for } x>0 \\\ \exp \left(-j \frac{\pi}{2}\right) & \text { for } x \leq 0\end{cases} $$ determine and sketch the intensity \(|g(x, y)|^{2}\) of the output wave \(g(x, y)\). Verify that even though the intensity of the input wave \(|f(x, y)|^{2}=1\), the intensity of the output wave is not uniform.

Ring Aperture. (a) A focused single-lens imaging system, with magnification \(M=1\) and focal length \(f=\) \(100 \mathrm{~cm}\) has an aperture in the form of a ring $$ p(x, y)= \begin{cases}1, & a \leq \sqrt{x^{2}+y^{2}} \leq b, \\ 0, & \text { otherwise, }\end{cases} $$ where \(a=5 \mathrm{~mm}\) and \(b=6 \mathrm{~mm}\). Determine the transfer function \(H\left(\nu_{x}, \nu_{y}\right)\) of the system and sketch its cross section \(H\left(v_{x}, 0\right)\). The wavelength \(\lambda=1 \mu \mathrm{m}\). (b) If the image plane is now moved closer to the lens so that its distance from the lens becomes \(d_{2}=25 \mathrm{~cm}\), with the distance between the object plane and the lens \(d_{1}\) as in \((a)\), use the ray-optics approximation to determine the impulse response function of the imaging system \(h(x, y)\) and sketch \(h(x, 0)\).

Optical Correlation. A transparency with an amplitude transmittance given by \(f(x, y)=\) \(f_{1}(x-a, y)+f_{2}(x+a, y)\) is Fourier transformed by a lens and the intensity is recorded on a transparency (hologram). The hologram is subsequently illuminated with a reference wave and the reconstructed wave is Fourier transformed with a lens to generate the function \(g(x, y)\). Derive an expression relating \(g(x, y)\) to \(f_{1}(x, y)\) and \(f_{2}(x, y)\). Show how the correlation of the two functions \(f_{1}(x, y)\) and \(f_{2}(x, y)\) may be determined with this system.

Blurring a Sinusoidal Grating. An object \(f(x, y)=\cos ^{2}(2 \pi x / a)\) is imaged by a defocused single-lens imaging system whose impulse response function \(h(x, y)=1\) within a square of width \(D\), and \(=0\) elsewhere. Derive an expression for the distribution of the image \(g(x, 0)\) in the \(x\) direction. Derive an expression for the contrast of the image in terms of the ratio \(D / a\). The contrast \(=(\max -\min ) /(\max +\min )\), where max and min are the maximum and minimum values of \(g(x, 0)\).

In Prob, 4.1-3, if \(f(x, y)\) is a circularly symmetric function with a maximum spatial frequency of 200 lines/mm, determine the angle of the cone within which the wave directions are confined. Assume that \(\lambda=633 \mathrm{~nm}\).
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