Fraunhofer Diffraction Flashcards
Why does Fraunhofer diffraction use lenses
so that the source and fringe pattern can both be at convinient distances from the aperture
What are the requirements for fraunhofer diffraction to be observed
the incoming and outgoing waves approach being planar over the diffracting apertures or obstacles
What is crucial for determining the resultant field for Fraunhofer diffraction
the phase of each contribution at the screen due to differences in path traversed
how can path difference be described for planar wavefronts passing through an aperture
linear function of the 2 aperture variables
what is the definitive mathematical criterion for fraunhofer diffraction
linearity in the aperture variables
what is the formula for the aperture size
D^2 = x0^2+y0^2
what is the formula for fraunhofer diffraction if we neglect the phase factors and explicitly write the aperture function
E (π₯1, π¦1) β double integral ββ to β exp {β ππ/π§(π₯0π₯1 + π¦0π¦1)} π΄ (π₯0, π¦0) πΈ(π₯0, π¦0)ππ₯0ππ¦0
What is notable about the fraunhofer diffraction formula
it is a fourier transform of the aperture field from one position x0 to another x1
How do you write the Fraunhofer diffraction in terms of the off-axis k vectors which are kx=kx1/z and ky=ky1/z
πΈ (ππ₯, ππ¦) β double integral from ββ to β exp{βπ ππ₯π₯ + ππ¦π¦} π΄(π₯, π¦) πΈ(π₯, π¦)ππ₯ππ¦
How can we treat an aperture (single slit) of width b
divide it into N coherent point sources each of extent Ξ΄y=b/N
What can we say about point sources along an aperture which is uniformly illuminated by a plane wave
the amplitude of each point source is proportional to Ξ΄y, they are a source of spherical waves which would all be in phase
How do you evaluate the intensity of received wave at a point on a screen
the sum of all the waves arriving from all point sources in the aperture
In a direction ΞΈ, the wave emitted from a point source at y is out of phase with the wave emitted from y=0 by how much
kysinΞΈ
What is the field of a wave emitted from a point source at y on an aperture
πΏπΈ = πΏπ¦ exp(βπππ¦π πππ)
What is the intensity I(π) on the screen for a single slit aperture of width b
πΌ (π) = πΌ0 [sin π½/π½]^2 = πΌ0 sinc^2π½
what is the formula for sincπ½
sinπ½/π½
What is the formula for π½ where the wave approaches the slit at an angle i
π½=1/2 kb sinπ = π/π π (sin i + sin π)
(in words) what is π½
half the phase shift at angle theta across the aperture b
In single slit diffraction what is happening at the first zero?
the ray from the centre of the slit exactly cancels the edge ray due to the Ξ»/2 path difference and so on across the half slit
What are the conditions for minima from single slit diffraction
I(π) = 0 when sinπ½=0
π½ = +-mpi
sinπ = mΞ»/b
what are the conditions for maxima from single slit diffraction
π½=0 or when we minimise sinc^2π½
sinπ½=0 or tanπ½=π½
Describe the fourier approach to fraunhofer diffraction from a slit
it is the fourier transform of a rect function which is the sinc function, hence the irradiance is sinc^2
what is the fourier transform pair of a delta function
1
what is the fourier transform pair of exp(iw0t)
2pi delta(w-w0)
what is the fourier transform pair of cos(w0t)
pi [delta(w-w0)+delta(w-w0)]
what is the fourier transform pair of sin(w0t)
pi/i[-delta(w-w0)+delta(w-w0)]
what is the fourier transform pair of rect(t/T) top hat
Tsinc(Tw/2)
what is the fourier transform pair of exp(-t^2/2T^2) gaussian
Tsqrt(2pi) exp(-T^2w^2/2)
What happens for Fraunhofer diffraction from 2 slits
the rapidly varying double-slit interference pattern is modulated by the single slit diffraction pattern which then functions as an envelope to the whole pattern
How can we think of two slit diffraction using the fourier method
the two slits are two rect functions
what is π½ for multi slit diffraction
π½ = ππ sin π/π
When the slit separation is a, what is πΌ for multislit diffraction
πΌ = ππ sin π/π
For 2 slit diffraction, with slit width b and separation a, what happens when a=b
the 2 slits become a single slit of width 2b
In diffraction patterns when does a missing order appear?
When the minimum of the single slit diffraction pattern overlaps a maximum of the multi slit interference
in terms of a and πΌ, what are the conditions for missing orders
π = (π/π) π or πΌ = (π/π) π½
Considering N long narrow parallel slits of width d and centre to centre separation a, how do we find the diffraction pattern
integrate over one slit N times
What is the N slit diffraction pattern
I(π) = πΌ0(sin^2 π½/π½^2)^2 (sin ππΌ/sin πΌ)^2
where I0 is the flux density in the normal direction from any slit
What is I(0) for multislit diffraction
N^2I(0)
For multislit diffraction, when do principle maxima occur
alpha = 0, Β±pi, Β±pi, β¦ or, equivalently, when π sin ππ = ππ, π = 0, Β±1, Β±2
What is a diffraction grating
an N slit device where N is very large and a/d is small
How many subsidary maxima are seen in the N slit diffraction pattern
N-2
How can we think of an N slit as a convolution
a convolution of a single slit with N delta functions
Describe diffraction from a rectangular aperture
The diffracted field is a sinc in x and y because the fourier transform of a rect function is a sinc
Describe diffraction from a circular aperture
it yields a diffracted airy pattern involving the bessel function Jn(u)
What is the radius of the airy disc
1.22Rπ/2a
R is dist to screen
a is aperture radius
