Wave Optics and Diffraction Part 2 Flashcards
What is the phase difference ΔФ equal to for two point sources at y and 0?
ΔФ = k(|x-y| - |x|)
How do you calculate the magnitude of a vector?
Square root of the sum of its squared components
What can we do with the magnitude of x-y to simplify it?
- Once expanded out, we get (|y|/|x|)^2, which can be neglected as it is negligible
- set the second part equal to t
What is |x-y| approximately equal to?
|x|(1 - (xy/|x|^2))
What is the phase difference ΔФ approximately equal to after expanding?
-k * (x*y)/|x|
What can we approximate |x-y(i)| to?
To D as the fringe spacings are negligible.
How can we separate the exponential in u(x,t) to make it into a useful form?
take the y out of the x-y part and make two exponentials with x in one and y in the other - x terms is the same for all sources but the y term is different in phase due to pinhole positions.
What is the general results for u(x,t)?
1/D * exp(ik(|x|-ct)) * sum from i=1 to N of A(i)exp(-iky(i))
What can we do for discrete pinholes?
Make the sum part an integral: integral of d^2y*exp(-iky) * a(y), where a(y) is the sum of A(i)𝛿(y-y(i))
What is u(x,t) proportional to?
u(x,t) ∝ a(k), the transform, where a(y) is a general aperture function which can define apertures finitely, and is equal to 1 inside the aperture and 0 outside.
Considering the example where the aperture is a pair of pinholes, what is a(y) equal to if the pinholes are separated by 2l?
a(y) = 𝛿(y-ly(hat)) + 𝛿(y+ly(hat))
What is the transform of a(y) equal to?
a(k) = exp(-ik(ly(hat)))+exp(ik(ly(hat))) = 2cos(k/|x| * l*x1))
How can we find the interference pattern using the transform?
- Know that I = |u|^2, and u ∝ a(k), so I ∝ cos^2(klx1/D)
- Can use this cosine function to sketch out the interference patterns on the screen by changing x values
In u(x,t), what does the A represent?
Amplitude of light emerging from each point source.
When considering a square aperture of side 2w, what do we first do to calculate the intensity on the detector screen?
- Compute a(k) by separating into y1 and y2
- y = (y1, y2, 0), so k.y = k1y1 + k2y2
- Sub this in, separate the integral and then do the integrals
- Find the trig solution
What is the definition of sinc(z)?
sinc(z) = sin(z)/z
What is the transform of a(y) for the square aperture proportional to?
4ω^2sinc(k1ω)sinc(k2ω)
How do we get an equation for x1?
k1 = k*x1/D k1ω = nπ
Use these two and rearrange for x1.
What does the interference pattern look like for a square aperture? How is this found?
Squares in x and y direction getting dimmer the further you go out - can be found using equation for x1 and equation for u
How do we examine the diffraction pattern from a single, slender slit?
Same as square but have height and width (rectangle).
How do we compute the transform for a single slit?
- Do the integral for y1 and y2 over entire aperture screen using -inf and inf, but then change to w and h when using cartesian coordinates.
- Do same as before: separate exponential terms and do the integrals, then change to trig form
What does a narrow slit in 1 direction and broad slit in the other direction mean for the fringes?
Broad fringes in 1 direction and narrow in the other.
What is the exponential form of sinθ and cosθ?
sinθ = exp(iθ)-exp(-iθ)/2i cosθ = exp(iθ)+exp(-iθ)/2
For 2 slits a distance 2l apart, what is the function a(y) equal to?
a(y+l e1(hat)) + a(y-l e1(hat)) = a(slit) * g(y)
What is the function g(y) equal to in the a(y) function for 2 slits?
g(y) = 𝛿(y+l e1(hat)) + 𝛿(y-l e1(hat))
What is the transform of g(y) equal to?
exp(-ikl e1(hat))+exp(-ik-l e1(hat)) = 2cos(k1*l)
What do we use to compute the transform of a(y)?
The convolution theorem.
