The Double Slit with Delta function slits - A rigorous approach

The truest statement about Fourier Transforms is that they are complex. - A.G.S.

Two-slit interference

The interference of light arises from something called Huygen's Principle. It basically states the every point a beam of light reaches becomes a source of a spherical wave. This is seen in the image below (source Wikipedia)

Plane


The cylindrical waves will interfere with each other do to the difference in the path length to a point on the screen, \(\Delta\). We see immediately that if \(\alpha = 0\), the waves interfere constructively creating a bright spot. The condition for constructive interference is
\[ \Delta = n \lambda, \ n=0, 1, 2 ...\]
and for destructive interference it is,
\[\Delta = \left(n + \frac{1}{2}\right)\lambda, \ n=0, 1, 2 ... ,\]
where \(\lambda\) is the wavelength of the incident plane wave.

Schematic

Finding \(\Delta\)

The key value here is \(\Delta\). \(\alpha\) is the most obvious angle to work with, as it natrually defines the position of interest on the screen. However, \(\Delta\) is more naturally related to \(\theta\). To find \(\theta\), we note that
\[\frac{\pi}{2} = \theta + \gamma - \theta_2 \]
\[ \theta = \frac{\pi}{2} - \gamma + \theta_2 .\]
with a pretty good number of lines of geometry we find,
\[\gamma = \frac{\pi}{2} - \frac{\theta_1 - \theta_2}{2}\]
\[\Longrightarrow \theta = \frac{\theta_1 + \theta_2}{2}\]

\)\Delta\) may now be determined using the Law of Sines,

\[ \frac{\sin{(\theta)}}{\Delta} = \frac{\sin{(\phi)}}{d},\]

where \(\phi = \frac{\pi}{2} + \frac{\theta_1 - \theta_2}{2}\). Solving for \(\Delta\),

\[\Delta = \frac{d \sin{(\theta)}}{\sin{(\phi)}}\]
\[\Delta = \frac{d \sin{\left(\frac{\theta_1 + \theta_2}{2}\right)}}{\sin{\left(\frac{\pi}{2}+\frac{\theta_1 - \theta_2}{2}\right)}}\]
\[\Delta = \frac{d \sin{\left(\frac{\theta_1 + \theta_2}{2}\right)}}{\cos{\left(\frac{\theta_1 - \theta_2}{2}\right)}}.\]

This is a wonderfully familiar equation, and it is 100% correct. However, we want this in terms of \(\alpha\) or \(\theta\) as the variable. At this point, the approximation is often made that \(\alpha=\theta\) and the proverbial "they" blast through the problem. However, an interesting blog post that does not make. Instead, lets go through the math manually, and show why and when this approximation is valid. Using the equations,

\[\tan{\alpha} = \frac{y}{L}\]
\[\tan{\theta_{1}} = \tan{\alpha} + \frac{d}{2 L}\]
\[\tan{\theta_{2}} = \tan{\alpha} - \frac{d}{2 L},\]

we derive a form depending only on \(\alpha\),

\[\Delta = \frac{d \sin{\left(\frac{\arctan{\left(\tan{\alpha} + \frac{d}{2 L}\right)} + \arctan{\left(\tan{\alpha} - \frac{d}{2 L}\right)}}{2}\right)}}{\cos{\left(\frac{\arctan{\left(\tan{\alpha} + \frac{d}{2 L}\right)} - \arctan{\left(\tan{\alpha} - \frac{d}{2 L}\right)}}{2}\right)}}\]
Pleasant right! It's immediately obvious what it's behavior is (the author says facetiously). To simplify this, it is easy to show that \(\theta \approx \alpha\) for \(L>>d\) and \(\alpha<<1\) with a Taylor expansion. Simplifying we find

\[\Delta = d\sin{(\theta)} \]

Nulls in the brightness distribution on the screen are located at \(\theta_{null}\) s.t.
\[ \left(n+\frac{1}{2}\right)\frac{\lambda}{d} = \sin{(\theta)} \]

Relating \(\Delta\) to the phase of the waves

The two waves here have a phase difference that oscillates from \(0\) to \(2\pi\) as a function of \(\Delta\). This phase difference of waves is
\[ \varphi = \frac{2\pi \Delta}{\lambda}.\]
The phase difference determines how bright a spot on the screen will be. It will peak where \(\varrho=2\pi n\) and will at a null where \(\varrho=2\pi (n + 1/2)\). This function the sum of two cosines with a phase offset which results in the intensity distribution being,
\[I = \cos^2{\!\left( \frac{2\pi \Delta}{\lambda}\right)}\]

Comments

Post a Comment

Popular posts from this blog

A Quick Look at Spherical Coordinates

Image
Because astronomy works on the Celestial Sphere, spherical coordinates play a very important role in astronomy. Most of the time we can ignore the intricacies of physics in 3-dimensions and work in a 2D far-field regime. (The far-field regime is just fancy speak for "small-angles"). However, it can be useful to understand why spherical coordinates are they way they are. Here, we will quickly derive the spherical differential volume element. Differential, just means vanishingly small. To let the cat out of the bag, the spherical differential volume element is, \[ dV = r^2 \sin{\theta}\, dr\, d\theta \, d\phi .\] The figure above shows the coordinate system we will be working with. This form is commonly used in astronomy, and also in some physics. However, it is different from the way mathematicians define it. In mathematics the \(\theta\) and the \(\phi\) are often switched. The astronomy convention is the more commonly used one in physics and astronomy (sorry mathematica...
Keep reading

Coding and the Interwebs

Image
If there is one thing you should know about learning to code, it is that the internet is your friend . Learning how to find what you need on the internet is an important skill to learn. A lot of the skill comes with time, but there are a few things that can get you going. Someone has had the same problem you are having. There is an answer. Stack Overflow and Stack Exchange are miracles of the internet. Don't worry which is which unless you want to Occam's Razor - Try the simplest solution first, but avoid hackey solutions. How to ask a good question* A good format to follow for asking Google a question is language what you want to do Say I am working on some C++ code, and I want to read in columns from a file. Suppose these have multiple data types, some are strings while others are integers. Example: Alice 1 10 1e3 Bob 2 12 2e3 I would ask C++ how do I read in columns with multiple data types from a file . While this is doable, a Google search sh...
Keep reading

Astrobites

Image
Astrobites is an astro-ph digest developed by Harvard graduates students which now has an international author roll. What is astrobites ? In their own words, Astrobites is a daily astrophysical literature journal written by graduate students in astronomy. Our goal is to present one interesting paper per day in a brief format that is accessible to undergraduate students in the physical sciences who are interested in active research. Essentially, Astrobites takes new research in astronomy and astrophysics and digests it into something approachable. With posts written by graduate students from around the globe, it provides an excellent introduction to technical writing that is understandable for the lay person, yet still scientifically interesting to the expert. Check out some of the posts by the following authors: John Johnson Ben Cook Ben Montet Elisabeth Newton (co-founder) Chris Feasi (co-founder)
Keep reading