Single Slit Diffraction and Huygen's Principle

[bluevires]

Hey guys, We are going through single-slit diffraction recently in Physics,
I understand that using Huygen's Principle, we can treat the slit as a series of segmented wave sources, and they will interfere with each other based on their phase difference because of the path difference they traveled, this phase difference will be 0 at the central maximum position, at which point the waves interact constructively,
however, here's where the confusion starts for me,
If they are all wave sources on the same wavefront, why aren't they coherent in terms of phase difference? websites and textbook always says this

the amplitudes of the segments will have a constant phase displacement from each other, and will form segments of a circular arc when added as vectors. In this way, the single slit intensity can be constructed.

What is this vector they are adding exactly?
what does the direction represent? what does the magnitude represent? why is it circular? what does this have to do with the phase difference of the waves after they interact?

They formulated the formula for intensity of the waves based on the angle away from the central axis and the slit width as well as the wavelength of the wave being diffracted, (see the picture)

if so, how do we find the central maximum intensity? since the angle is 0.

Sorry guys I'm just really confused right now, I would appreciate it very much if you can help me out.

 

[jtbell]

What is this vector they are adding exactly?

It's a phasor ("phase vector"). Its magnitude is the amplitude of a wave, and its direction represents the phase of the wave. If you have a wave or oscillation given by

$$y = A \cos (\omega t + \phi)$$

the phasor has magnitude A and direction $\phi$. You can add waves by adding their corresponding phasors. The resultant phasor has the amplitude and phase of the resultant wave.

To add together a lot of waves from small sections of the slit, you add a lot of tiny phasors together. Graphically, you put the phasors head to tail, just like adding regular vectors. Each phasor's direction differs only slightly from its "neighbors", so the resulting chain of phasors forms a pretty smooth curve, in this case part of a circle.

 

[astrokreng]

Hi there,

I can understand your confusion about single-slit diffraction and Huygen's Principle. Let me try to clarify it for you.

Firstly, Huygen's Principle states that every point on a wavefront can be considered as a source of secondary waves, which then interfere to form the new wavefront. In the case of single-slit diffraction, we can think of the slit as a series of mini wave sources that are all in phase with each other at the central maximum. This means that the waves coming from each mini source will interfere constructively at the central maximum, resulting in a bright spot.

Now, to address your question about coherence, it is true that the waves coming from the slit are not completely coherent in terms of their phase difference. This is because the waves coming from different mini sources will have different path lengths, resulting in a slight phase difference. However, this phase difference is small enough that it does not affect the overall interference pattern.

Next, let's talk about the vector addition. The vector represents the amplitude and phase of each mini wave source. When we add them together, we are essentially adding their amplitudes and phases to get the overall amplitude and phase of the wave at that point. The direction of the vector represents the direction of the wave, and the magnitude represents the intensity of the wave.

As for why the vector addition results in a circular arc, it is because the waves coming from the mini sources are all in phase at the central maximum, but as we move away from the central maximum, the phase difference between the waves increases. This results in the interference pattern forming a circular arc.

To find the intensity at the central maximum, you can use the formula you mentioned, which takes into account the angle, slit width, and wavelength. At the central maximum, the angle is 0, so the intensity will be at its maximum value.

I hope this helps to clarify things for you. Let me know if you have any further questions. Keep up the good work in your physics studies!