Wave Optics and Diffraction Part 2

Question 1

What is the phase difference ΔФ equal to for two point sources at y and 0?

Answer 1

ΔФ = k(|x-y| - |x|)

Question 2

How do you calculate the magnitude of a vector?

Answer 2

Square root of the sum of its squared components

Question 3

What can we do with the magnitude of x-y to simplify it?

Answer 3

• Once expanded out, we get (|y|/|x|)^2, which can be neglected as it is negligible
• set the second part equal to t

Question 4

What is |x-y| approximately equal to?

Answer 4

|x|(1 - (xy/|x|^2))

Question 5

What is the phase difference ΔФ approximately equal to after expanding?

Answer 5

-k * (x*y)/|x|

Question 6

What can we approximate |x-y(i)| to?

Answer 6

To D as the fringe spacings are negligible.

Question 7

How can we separate the exponential in u(x,t) to make it into a useful form?

Answer 7

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.

Question 8

What is the general results for u(x,t)?

Answer 8

1/D * exp(ik(|x|-ct)) * sum from i=1 to N of A(i)exp(-iky(i))

Question 9

What can we do for discrete pinholes?

Answer 9

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))

Question 10

What is u(x,t) proportional to?

Answer 10

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.

Question 11

Considering the example where the aperture is a pair of pinholes, what is a(y) equal to if the pinholes are separated by 2l?

Answer 11

a(y) = 𝛿(y-ly(hat)) + 𝛿(y+ly(hat))

Question 12

What is the transform of a(y) equal to?

Answer 12

a(k) = exp(-ik(ly(hat)))+exp(ik(ly(hat))) = 2cos(k/|x| * l*x1))

Question 13

How can we find the interference pattern using the transform?

Answer 13

• 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

Question 14

In u(x,t), what does the A represent?

Answer 14

Amplitude of light emerging from each point source.

Question 15

When considering a square aperture of side 2w, what do we first do to calculate the intensity on the detector screen?

Answer 15

• 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

Question 16

What is the definition of sinc(z)?

Answer 16

sinc(z) = sin(z)/z

Question 17

What is the transform of a(y) for the square aperture proportional to?

Answer 17

4ω^2sinc(k1ω)sinc(k2ω)

Question 18

How do we get an equation for x1?

Answer 18

k1 = k*x1/D
k1ω = nπ

Use these two and rearrange for x1.

Question 19

What does the interference pattern look like for a square aperture? How is this found?

Answer 19

Squares in x and y direction getting dimmer the further you go out - can be found using equation for x1 and equation for u

Question 20

How do we examine the diffraction pattern from a single, slender slit?

Answer 20

Same as square but have height and width (rectangle).

Question 21

How do we compute the transform for a single slit?

Answer 21

• 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

Question 22

What does a narrow slit in 1 direction and broad slit in the other direction mean for the fringes?

Answer 22

Broad fringes in 1 direction and narrow in the other.

Question 23

What is the exponential form of sinθ and cosθ?

Answer 23

sinθ = exp(iθ)-exp(-iθ)/2i
cosθ = exp(iθ)+exp(-iθ)/2

Question 24

For 2 slits a distance 2l apart, what is the function a(y) equal to?

Answer 24

a(y+l e1(hat)) + a(y-l e1(hat)) = a(slit) * g(y)

Question 25

What is the function g(y) equal to in the a(y) function for 2 slits?

Answer 25

g(y) = 𝛿(y+l e1(hat)) + 𝛿(y-l e1(hat))

Question 26

What is the transform of g(y) equal to?

Answer 26

exp(-ikl e1(hat))+exp(-ik-l e1(hat)) = 2cos(k1*l)

Question 27

What do we use to compute the transform of a(y)?

Answer 27

The convolution theorem.