Wave Optics and Diffraction Part 3

Question 1

What is the fraunhofer limit?

Answer 1

That the distance to the detector screen D is much greater than the aperture size w.

Question 2

For a circular aperture, what is the y vector equal to on the aperture screen in polar coordinates?

Answer 2

(scosα, ssinα, 0), where s is the length and theta the angle.

Question 3

For a circular aperture, what is the x vector equal to on the detector screen in polar coordinates?

Answer 3

(ρcosФ, ρsinФ, D)

Question 4

What do we need to do to find the diffraction pattern for the circular aperture?

Answer 4

Take the fourier transform of the aperture function, as u(x,t) is proportional to this.

Question 5

What is the change in area dA equal to in polar coordinates for the circular aperture?

Answer 5

dA = s dsdα

Question 6

What is a(s, α) equal to?

Answer 6

a(s, α) = 1 if s

Question 7

What is k equal to for the circular aperture and how is it calculated?

Answer 7

k = kx/|x| ~= k/D(ρcosФ, ρsinФ, D)

Question 8

What is cosФcosα+sinФsinα equal to?

Answer 8

cos(α-Ф)

Question 9

How do we compute the fourier transform of a(s, α)?

Answer 9

Put it in the fourier transform function and compute k*y and sub this in.

Question 10

What is a Bessel Function?

Answer 10

J0(z) = 1/2π * integral from 0 to 2π of exp(-izcos(α’)) dα’

Question 11

What is the bessel function defined in terms of?

Answer 11

xJ0(x) = d/dx[xJ1(x)], so can integrate first one to get rid of differential on right sided

Question 12

How can we use the Bessel Function to solve the fourier transform of a(s, α)?

Answer 12

• Find that it is equal to 2π*the bessel function

- Change the variables in the integral version of bessel function and do the integral

Question 13

What can we assume for the angle in the solution?

Answer 13

θ ~ ρ/D

Question 14

What is the angular separation of diffraction patterns which is resolvable?

Answer 14

2θ

Question 15

What is the equation for the angular separation resolvable in the circular aperture problem?

Answer 15

2θ = 122*λ/2ω

Question 16

What are diffraction grating used for?

Answer 16

To separate light of different wavelengths/frequencies.

Question 17

What is the function a(y) equal to for a diffraction grating?

Answer 17

integral of a(y-y’)*g(y’) dy’

Question 18

What is g(y’) equal to for the diffraction grating?

Answer 18

sum from -N to N of 𝛿(y’-mle1(hat))

Question 19

How can we use the convolution theorem for the diffraction grating transform?

Answer 19

a(k) = a(k)(slit)*g(k)

Question 20

What is the intensity proportional to?

Answer 20

|u|^2 which is proportional to |a(k)|^2

Question 21

How do we compute the transform of g(k)?

Answer 21

• Sub it into the equation and set y = mle1(hat) to get rid of kronecker delta
• Compute what k is equal to (k1 = x1k/D)
• Sub this into the equation

Question 22

What do we find after working out the transform of g(y)?

Answer 22

Is a geometric progression: g(k) = (sin(klx1/D)*(N+1/2))/sin(klx1/2D)

Question 23

How do we find dx1, the separation of the fringes? What do we find?

Answer 23

dx1*kl/2D = π, rearrange this. Find that since k depends on the wavelength, the spacing of the fringes does too.

Question 24

What is the equation for the criterion for resolving two wavelengths?

Answer 24

(λ1-λ2)/λ1 > 1/(2N+1)