Wave Optics and Diffraction Part 3 Flashcards
What is the fraunhofer limit?
That the distance to the detector screen D is much greater than the aperture size w.
For a circular aperture, what is the y vector equal to on the aperture screen in polar coordinates?
(scosα, ssinα, 0), where s is the length and theta the angle.
For a circular aperture, what is the x vector equal to on the detector screen in polar coordinates?
(ρcosФ, ρsinФ, D)
What do we need to do to find the diffraction pattern for the circular aperture?
Take the fourier transform of the aperture function, as u(x,t) is proportional to this.
What is the change in area dA equal to in polar coordinates for the circular aperture?
dA = s dsdα
What is a(s, α) equal to?
a(s, α) = 1 if s
What is k equal to for the circular aperture and how is it calculated?
k = kx/|x| ~= k/D(ρcosФ, ρsinФ, D)
What is cosФcosα+sinФsinα equal to?
cos(α-Ф)
How do we compute the fourier transform of a(s, α)?
Put it in the fourier transform function and compute k*y and sub this in.
What is a Bessel Function?
J0(z) = 1/2π * integral from 0 to 2π of exp(-izcos(α’)) dα’
What is the bessel function defined in terms of?
xJ0(x) = d/dx[xJ1(x)], so can integrate first one to get rid of differential on right sided
How can we use the Bessel Function to solve the fourier transform of a(s, α)?
- Find that it is equal to 2π*the bessel function
- Change the variables in the integral version of bessel function and do the integral
What can we assume for the angle in the solution?
θ ~ ρ/D
What is the angular separation of diffraction patterns which is resolvable?
2θ
What is the equation for the angular separation resolvable in the circular aperture problem?
2θ = 122*λ/2ω
What are diffraction grating used for?
To separate light of different wavelengths/frequencies.
What is the function a(y) equal to for a diffraction grating?
integral of a(y-y’)*g(y’) dy’
What is g(y’) equal to for the diffraction grating?
sum from -N to N of 𝛿(y’-mle1(hat))
How can we use the convolution theorem for the diffraction grating transform?
a(k) = a(k)(slit)*g(k)
What is the intensity proportional to?
|u|^2 which is proportional to |a(k)|^2
How do we compute the transform of g(k)?
- 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
What do we find after working out the transform of g(y)?
Is a geometric progression: g(k) = (sin(klx1/D)*(N+1/2))/sin(klx1/2D)
How do we find dx1, the separation of the fringes? What do we find?
dx1*kl/2D = π, rearrange this. Find that since k depends on the wavelength, the spacing of the fringes does too.
What is the equation for the criterion for resolving two wavelengths?
(λ1-λ2)/λ1 > 1/(2N+1)
