Now for something less intuitive:

• Single-slit interference
• Equation: nλ = a sin θ

Let’s jump in!

What pattern does diffraction through a slit result in on a screen?
You might think that it causes an even spread of intensity (since no interference seems to occur), BUT an INTERFERENCE PATTERN actually shows up.

There are light bands called MAXIMA & dark bands called MINIMA.

The intensities at the screen can be shown by this graph:

Why does interference occur in single-slit diffraction?
This is to do with the WAVELETS.

The wavelets produced between edges of the slit can be said to interfere.

Let’s look at the maxima & minima:

There is a bright maxima at the centre of the screen, followed by a repeating pattern of maxima & minima going to the sides.

Since the middle is a bright maxima, we will consider the dark MINIMA to count the number of repetitions in the pattern.

The position of a minimum away from the middle is known as its ORDER.

For example,

• The first minimum has an order of 1, counted from the centre.
• The next minimum further away has an order of 2.

Generally, we say that a minimum has an order of n, where n = 1,2,3,…

The minima come from the DESTRUCTIVE interference between wavelets leaving the diffraction slit.

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There is a relationship between the:

• Order
• Wavelength
• Slit size
• Angle of the point on the screen from the centre

Once again, this is all due to PATH DIFFERENCE.

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What formula can we use to relate the angle & the order of the minima?

Here’s a video with the full proof:
https://www.khanacademy.org/science/physics/light-waves/interference-of-light-waves/v/single-slit-interference

Here’s a brief summary:
Consider this diagram:

In real life (for light diffraction), the difference in angle between the points to the wall is very small, & can be considered to be negligible.

Thus, we can approximate that the angles are both the same θ.

We can pair every point on the top of the edge with a point exactly a/2 below it:

Via trigonometry,

For destructive interference to occur at any point, the path difference of 2 waves to that point must be a whole ODD number of λ/2.

However, EVEN numbers of n are also accepted, if we pair up points a/4 apart:

This simplifies to our formula!

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How do we apply this?
With this formula, we can:

• Find the width of a slit, given the other information
• Find the angle between the centre line & the first minimum, given the other information
• Calculate the wavelength of light, given the other information
• Calculate the order of a certain minimum, given the other information
• Find the total number of minima possible for a given setup

The first 4 are simple enough – just a matter of substituting the correct values.

We’ll take a deeper look at the last one:

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How do you find the total number of minima possible for a given setup?

Given:

• a slit of width “a”
• light of wavelength λ

What is the total number of minima formed?

STEP 1: Identify the correct formula & list out the information	Since it’s a single slit,
STEP 2: Substitute the given information. Obtain an expression for n in terms of sin θ
STEP 3: Finding the range of n	Remember a few things: We consider the angle from the centre to the very edge: 0 < θ < 90 Thus, 0 < sin θ < 1 Thus, we can find the range of n: 0 < n < a/λ
STEP 4: Round down where appropriate	n needs to be a WHOLE number. So, given a maximum value larger than n, you can conclude that n is the largest whole number smaller than a/λ.
STEP 5: Consider the question carefully! What are they looking for?	If you need to find the TOTAL number of minima, this value of n must be multiplied by 2, since there are 2 minima for every order. If asked to find the MAXIMUM ORDER of minima, just leave the answer as n.

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