Fourier and Optics

What does a Fourier series allow you to do?

Find the frequencies present in a function of time, f(t)

How do you find the frequencies present in a function f(t) when f(t) is periodic?

Use a Fourier series

How do you find the frequencies present in a function f(t) when f(t) is non-periodic?

Use a Fourier transform

What does a Fourier series represent?

Any periodic function as an infinite sum of sine and cosine waves

What is the definition of a Fourier series?

If f(x) is an even function, which, if any, out of a_0, a_n, b_n are zero?

b_n = 0 for even functions

If f(x) is an odd function, which, if any, out of a_0, a_n, b_n are zero?

a₀ = 0, a_n = 0 for odd functions

If the function f(x) is even, then is this integral odd or even?

Overall even because even function*even function = even function

If the function f(x) is odd, then is this integral odd or even?

Overall even because odd function*odd function=even function

What can this be simplified to?

What does this graph plot?

This graph plots the harmonics on the x axis (first, second, third harmonics etc) against the amplitude of each harmonic on the y axis

What is the definition of a Fourier transform for a function of time?

What is the definition of a Fourier transform for a function of distance?

What is the equation for wavenumber, k?

k = 2π/λ

How do you measure the width of the sinc function?

Measure the the distance between the two points where the function crosses 0, width = Δ(ωτ)

What is the physical significance of Fourier transforms?

Tells us how much of each frequency component we would need to use to construct the original voltage pulse

What do Fourier transforms tell us?

The frequency components present in any signal and their amplitudes

How are the width of the original function f(x) and its Fourier transform F(k) related (i.e. what’s their proportionality like)?

Width of the original function f(x) and its Fourier transform F(k) are inversely proportional

Describe the relationship shown here

The broader f(t), the narrower F(w) and vice versa

For a pulse of light, what are the main frequencies present in the range of?

What is the condition for symmetry A for a Fourier transform?

f(x) is generally a real function but may have odd or even but possibly neither symmetry

What are the three conditions in symmetry B for a Fourier transform?

If f(x) is even

If f(x) is odd

If f(x) is real and even

What is the equation for the Fourier transform for f(x) when f(x) is real and even?

What is the equation involving cos and sin and i for the Fourier transform?

What are 4 applications of Fourier transforms?

Diffraction of light

Nuclear physics - scattering of electrons

Optical fibre data transmission

MP3 encoding

How does MP3 encoding work with Fourier transforms?

Transforms time to frequency - allows components that are too quiet to be audible or masked by louder nearby frequencies to be discarded 

What is the Dirac delta function equation for a spike centred at x=0?

WHat is the Dirac delta function for a spike centred at x=x₀?

What is the Dirac delta function B?

What are the 4 manual steps for calculating convolution and what is the definition?

1. The signal function is given by g(x) and the system response function is given by f(x)

2. Introduce a dummy variable u which enters the two functions as: f(u) and g(x-u)

3. For a given value of x calculate the overlap between the two function f(u) and g(x-u)

   1. This is achieved by keeping x constant and integrating the product of the two functions between -∞ and + ∞ with respect to the variable u

   2. In the figure the overlap will depend on the form of the two functions in the shaded region where they are both non-zero

   3. As x varies the functions slide across each other 

4. This will give a new function c(x) which gives the detected signal as a function of x and is a convolution of the functions f(x) and g(x)

   1. A convolution function is therefore defined by the image

What is the definition of convolution?

A mathematical process which takes two input functions f(x) and g(x) and produces a third output function

What is the convolution theorem?

Tells us that if C(k) is the Fourier transform of the convoluted function then C(k)=√(2π)F(k)G(k)

How do you find f(x) using c(x), g(x), and the result for C(k)?

What is C(k) given by for 2 different Gaussian functions?

What is the equation for 1/Δ?

What is the condition for small aperture diffraction?

When light passes through a small aperture it is diffracted IF the wavelength is comparable or larger than the aperture size

What is Fraunhofer diffraction?

Where the waves incident on and leaving the aperture can be modelled as a plane

What is Fresnel diffraction?

Where the waves incident on and leaving the aperture CANNOT be modelled as a plane

What is the Fraunhofer diffraction pattern described by? (without? lenses - i think)

What is the Fraunhofer diffraction pattern described by? (with lenses)

What are the 5 steps for finding the Fraunhofer diffraction pattern produced by an aperture?

Write down the function f(x) that describes the transmission of light through the screen - the aperture function. For a 2D aperture f(x,y)

Calculate the Fourier transform f(x) to give F(k(x))

Calculate  |F(k_x)|²

Substitute in for k_x = (2πsinθ)/λ

Replace sinθ with x/f or s/f

Compare a square slit to a triangular slit for diffraction

Triangular slit produces a wider central maxima but weak subsidiary maxima

What is the equation for intensity for a square slit?

What is the equation for intensity for a triangular slit?

Under what condition can you consider an aperture as effectively one dimensional?

if the size in the other direction is much larger than the wavelength of light

What does this equation mean?

It is the equation for 2D diffraction

What is the equation for 2D diffraction?

What is the equation for the width of an aperture in the x direction?

What is the equation for the diffraction by a rectangular aperture?

f(x,y)=g(x)h(y)

where g(x) and h(y) are both top hat functions of the widths 2a and 2b

What does diffraction by a circular aperture depend on?

The first order Bessel function J1()

Where do the first zeros of the Bessel function occur?

For arguments ~1.22π

Where do the first zeros of the sinc function occur?

At +-π

What is the effect of diffraction on spatial resolution of a telescope?

Diffraction as light passes through a telescope may prevent closely spaced objects from being separately resolved 

For two very narrow slits, what is the equation for the intensity of fringes?

What is the equation for fringe spacing for two very narrow slits?

What is the equation for the intensity for two very narrow slits? and what does each component represent?

What is the equation for the intensity for diffractoin by an array of N identical slits and what do the components mean?

What is the diffraction grating equation and what do the letters mean?

d is

What is the equation for the diffraction grating width of main maxima?