Moire pattern and interference of plane EM waves

[Nikitin]

Homework Statement

Look at the Moire pattern in "Oppgave 5" (assignment 5) in the attached pdf file. The Moire pattern you see is supposed to represent the interference between two plane EM waves whose directional $\vec{k}$ vectors form an angle $\theta$ with each other.

Anyway, the main question is: "What is the angle $\theta$ between the patterns if the distance between the interference-strips is $\Lambda = 2mm$ and the wavelength of the light is $\lambda = 550nm$.

Homework Equations

The Attempt at a Solution

According to the solutions manual, the distance between the interference-strips is given by $\Delta \vec{k} = \vec{k_2} - \vec{k_1}$

I do not understand how this is possible. Can somebody explain? This is the only thing I struggle with understanding.

 

[haruspex]

According to the solutions manual, the distance between the interference-strips is given by $\Delta \vec{k} = \vec{k_2} - \vec{k_1}$

That doesn't seem right for two reasons. First that gives a vector, not a scalar. Did you mean d = ||$\Delta \vec{k}$||? Secondly, it surely depends on λ.
Draw a diagram with two sets of parallel lines representing the wave peaks at some instant. Identify the regions of constructive and destructive interference. With a little geometry, find the relationship between d, θ and λ.

 

[Nikitin]

I'm sorry, I was so tired yesterday I didn't explain it. What I meant was:

$\frac{2 \pi}{\Lambda} = |\Delta k|$

thanks for the reply, I'll try that :)

 

[Nikitin]

OK I got it now, thanks. Turns out I was just set astray by the solutions manual as this was really easy...

 

[paranormal]

The Moire pattern is created when two similar patterns are overlaid on top of each other, resulting in an interference pattern. In this case, the two patterns are created by two plane EM waves with different directional vectors, forming an angle θ with each other. The distance between the interference-strips is a result of the interference between these two waves.

The equation Δk = k2 - k1 represents the difference between the two directional vectors of the waves. This difference is also known as the wave vector difference and is directly related to the angle θ between the two waves. The larger the angle θ, the larger the difference in the wave vectors and therefore, the larger the distance between the interference-strips.

To calculate the angle θ, we can use the equation Δk = 2π/λ, where λ is the wavelength of the light. By rearranging this equation, we get θ = arcsin(Δkλ/2π).

Substituting the given values of Δk = 2π/Λ and λ = 550nm, we get θ = arcsin((2π/Λ)λ/2π) = arcsin(λ/Λ) = arcsin(550nm/2mm) = 0.125°

Therefore, the angle θ between the patterns is approximately 0.125°. This means that the two waves are almost parallel to each other, resulting in a small difference in their wave vectors and a small distance between the interference-strips.