Lecture 11 – Fourier Transforms III The return of the Fourier Transform

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Lecture 11 – Fourier Transforms III The return of the Fourier Transform

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Lecture 11 – Fourier Transforms III The return of the Fourier Transform

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Lecture 11 – Fourier Transforms III

The return of the Fourier Transform

- Cosine form of Fourier Transforms
- Applications of Fourier Transforms
- The Dirac delta function

where

where

Definition of Fourier Transforms

Complexity, Symmetry and the Cosine Transform

So we can say:-

If f(x) is real and even then

We know that the Fourier transform from x space into k space can be written as:-

We also know that we can write

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

If f(x) is even then only the first integral above is non-zero and F(k) will be real

If f(x) is odd then only the second integral above is non-zero and F(k) will be complex

Example 5: Repeat Example 1 using Fourier cosine transform formula above.

Find F(k) for this function

Diffraction through an aperture, now ky0

Application – Diffraction in Optics

Plane wave, only kx0

It can be shown that the vertical components of the wavevector (ksin) are produced by the Fourier transform of the function that defines the aperture f(y).

The diffraction pattern is proportional to the square of the Fourier transform of f(y):

Dirac Delta Function

Introduction

I’m thinking of a single square wave pulse of area 1 unit (‘unity’).

What will happen if I slowly decrease the width for a constant area ?

What will happen as the width tends to zero?

We’re left with a spike of

zero width and infinite height. This is called a delta function.

Dirac Delta Function

The delta function d(x) has the value zero everywhere except at x = 0 where its value is infinitely large in such a way that its total integral is 1.

The Dirac delta function d(x) is very useful in many areas of physics. It is not an ordinary function, in fact properly speaking it can only live inside an integral.

d(x) is a spike centred at x = 0

d(x – x0) is a spike centred at x = x0

Dirac Delta Function

The product of the delta function d(x – x0) with any function f(x) is zero except where x ~ x0.

Formally, for any function f(x)

Example: What is ?

Dirac Delta Function

The product of the delta function d(x – x0) with any function f(x) is zero except where x ~ x0.

Formally, for any function f(x)

Examples

(a) find

(b) find

(c) find the FT of

Example

What is the diffraction pattern produced by two very narrow slits separated by a distance d?

Describe system by two delta functions at y=d/2

f(y)=(y-d/2)+(y+d/2)

y

d

D

Conclusions

- FT of even and odd functions – cosine form of FT
- Applications of FTs
- The Dirac delta function