Lecture 4

More Convolution, Diffraction, and

Reciprocal Space

Recap….

• Magnitude and Phase• The Uncertainty Principle• Convolution

• Convolution• Diffraction• Reciprocal space and spatial frequency• 2D Fourier transforms

Outline of Lecture 4

Problems classes this week are in:

B11 (Monday and Friday)B15 (Thursday)

What do I need to know for the class test

next week?• Class Test I will comprise four “Section A”-type questions (see www.nottingham.ac.uk/~ppzpjm/F32SMS)

• Topics: Fourier series, Fourier transforms, delta functions, convolution, Parseval’s theorem, conjugate variables (time-frequency, position-momentum etc…), response of filters (low pass, band pass, high pass);

• Open book.

• If there are formulae given on the front page of the test paper, consider that they might be there for a reason…

So how do we convolve two functions?

Is there not an easier way of convolving two functions? The integral seems tricky to calculate and the graphical method is laborious.

Convolution theorem:The Fourier transform of the convolution of two functions is 2 times the product of the Fourier transforms of the individual functions:

FT (f g) = 2 F(k)G(k)

Extremely powerful theorem

Impulse response and convolution.

The response of a system (optical, audio, electrical, mechanical, etc..) to an arbitrary signal f(t) is the convolution of f(t) with the impulse response of the system.

f(t)f(t)

tt

f(t) may be represented as a series of impulses of varying height. System responds to each of these in a characteristic fashion (impulse response).To get response to ‘stream’ of impulses (i.e. f(t)) convolve f(t) with impulse response function.

Impulse response and convolution.

Can also deconvolve if we know the impulse response (or point spread) function. (HST before corrective optics).

?? How do you think it was possible to evaluate the point spread function for the Hubble telescope?

Impulse response and convolution: Audio signals

Remember that convolution holds for a vast range of systems.Another example – audio signals.

Record impulse response of each environment. Then convolve with given signal to recreate charateristic acoustics of concert hall, cavern, or recording studio…

Large concert hall Ice cavern Recording studio

Impulse response and convolution: Audio signals

Now, take a recording….

and convolve this with the impulse response functions on the previous slide…

Single slit diffraction

?? Sketch the (Fraunhofer) diffraction pattern you’d expect for a single slit whose transmission function is as shown below.

f(x)

x

1

Fraunhofer diffraction - limiting case where:- light appoaching the diffracting object is parallel and monochromatic;- compared to the size of the diffracting object, the image plane is located at a large distance from the object.

?? If the slit is widened, the central spot in the diffraction pattern will: (a) narrow, (b) widen, (c) stay the same ?

“The nature of light is a subject of no material importance to the concerns of life or to the practice of the arts, but it is in many other respects extremely interesting.”

Thomas Young ( (1773 – 1829))

“The most beautiful experiment”

“We choose to examine a phenomenon which is impossible, absolutely impossible, to explain in any classical way, and which has in it the heart of quantum mechanics. In reality, it contains the only mystery."

RP Feynmann (1961)

Double slit diffraction

One electron at a time…..

Convolution and the double slit diffraction

patternh(x)

x

b

-a +a

??The function h(x) is a convolution of two functions – sketch them.

f(x)

xb

g(x)

+a-a

Convolution and the double slit diffraction pattern

f(x)

xb

?? Sketch F(k)

g(x)

+a-a

?? Sketch G(k)

Convolution and the and the double slit diffraction patterndouble slit diffraction pattern

Don’t get confused between the modulus of the Fourier transform (|F(w)|)

and the Fourier transform itself.

G(k)

+a-a

g(x) = cos (ax)

g(x) = sin (ax)

G(k)

+a-a

PURELY REAL

PURELY IMAGINARY

Convolution and the double slit diffraction pattern

Fourier transform of (f g)= ( 2) F(k)G(k)

g(x)f(x)

xb

= ??

Convolution and the double slit diffraction pattern

?? What is the effect on the slit pattern of (a) narrowing the slits? (b) changing the separation of the slits ?

http://www.physics.nwu.edu/ugrad/vpl/optics/diffraction.html

Reciprocal space and

spatial frequencies

Just as we can build up a complex waveform from a variety of sinusoids of different amplitudes and phases, so too can we generate an image from a Fourier integral.