Fourier theory

We know a lot about waves at a single w:n( )w , vp( )w , R( ), w absorption( )w …

Analyze arbitrary in terms of these, because Fourier told us that….

,E r t

How does a complicated optical pulse reflect off a surface?

,E r t

Fourier Series Example

4 4 4 4( ) sin 2 sin 6 sin 10 sin 14 ...

3 5 7f t t t t t

2 2 6 6 10 102 2 2 2 2 2( ) ...

3 3 5 5i t i t i t i t i t i ti i i i i i

f t e e e e e e

f(t)

cn or c(w)

Fourier Theory Summary

1( ) ( )

2i tf t f e d

1( ) ( )

2i tf f t e dt

0( ) im tm

m

f t c e

0

0

1( )

Tm ti

mc f t e dtT

Fourier Series: (f periodic or defined over (0,T)

Fourier Transforms: (f nonperiodic, all time)

( ) ( ) i t

m

f t c e

0

1( ) ( )

Ti tc f t e dt

T with cn c(w)

and mw0 w

Optic’s choice of sign for f(t): ( ), to matchi t i kx te e

http://phet.colorado.edu/en/simulation/fourier

Discrete vs continuous f(w)

Fourier Transform Example

f(t)

cn or c(w)

2( ) 1 cos 2

ic

FTs you should get to know!

FTs you should get to know!

Gaussian

Gaussian doesn’t have “ringing” in the FT!

Widths in t, w

“Uncertainty principle” in QM:related to time-frequency widths in

waves

101 waves

11 or approx: t O t

Uncertainty principle

E(t)

N functions added, equally spaced in frequency

11 waves 101 waves: Dw is much wider

Power spectrum

11 waves 101 waves

2f

Inverse FT: Does the same power spectrum give the same f(t)?

1( ) ( )

2i tf t f e d

Power spectrum of cos, sin

Uncertainty principle

101 waves

Why is it an inequality?

11 or approx: t O t

http://phet.colorado.edu/en/simulation/fourier

Importance of phase in f(w)

11 waves 101 waves

( ) a

1 Im[ ( )]( ) tan

Re[ ( )]

f

f

Try: Linear phase function:

( ) ( )iag e g Dt is the same.Pulse is shifted

( ) 0

( ) from adding closely spaced cos( ( ))i if t t

Importance of phase

11 waves 101 waves

2( ) a Try: Quadratic dependence:

2

( ) ( )iag e g

( ) 0

( ) from adding closely spaced cos( ( ))i if t t

Dt is much bigger for the same !Dw

Importance of phase

11 waves 101 waves

( ) rand

1 Im[ ( )]( ) tan

Re[ ( )]

f

f

Random dependence:

( ) 0

Dt is infinite (noise)

Dt is much bigger for the same !Dw

Summary: Importance of phase

11 waves 101 waves

Why is it an inequality?

11 or approx: t O t

f(t) changes greatly with phase ( ). f w The shortest is had only for ( ) = f w constant or linear.

All others will make

Dw comes entirely from |f(w)|, which has no phase information.

1 t

1 t

Carrier frequency-envelope principle

11 waves 101 waves

The FT f(w) is the FT of ___ centered at ____. The width Dw is the width of ____

Optical pulses are often a steady (“carrier”) wave at multiplied by an envelope function

( ) ( ) cos( )

( ) ( )sin( )

( ) ( ) i t

f t g t t

f t g t t

f t g t e

Which pulse f(t) will have f(w) centered around the highest frequencies?

a) b) c)

Which f(t) will have the greatest width Dw in f(w) around its central frequency?

a) b) c)

Compare the “ringing” in the FT of

rectangular pulse envelope

triangular pulse envelope

Gaussian pulse envelope

sinc pulse envelope

Fourier theory

1 1( ) [ ( )] ( )

2i tf t FT f f e d

1( ) [ ( )] ( )

2i tf FT f t f t e dt

Fourier theory and delta functions

( )ot t

( ) ( )of t t t du

( )ot t du

( )FT t

( )oFT t t

1( ) [ ( )] ( )

2i tf FT f t f t e dt

1 1( ) [ ( )] ( )

2i tf t FT f f e d

Fourier theory and delta functions

FT oi te

-1FT FTo ot t t t