Fourier theory We know a lot about waves at a single  : n( , v p ( , R(  absorption(  … Analyze arbitrary in terms of these, because Fourier told us that…. How does a complicated optical pulse reflect off a surface?

Fourier Series Example f(t) c n or c(  )

Fourier Theory Summary Fourier Series: (f periodic or defined over (0,T) Fourier Transforms: (f nonperiodic, all time) with c n  c(  ) and m  0   Optic’s choice of sign for f(t):

Discrete vs continuous f(  )

Fourier Transform Example f(t) c n or c(  )

FTs you should get to know!

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

Widths in t, 

“Uncertainty principle” in QM: related to time-frequency widths in waves 101 waves

Uncertainty principle E(t) N functions added, equally spaced in frequency 11 waves101 waves:  is much wider

Power spectrum 11 waves101 waves Inverse FT: Does the same power spectrum give the same f(t)? Power spectrum of cos, sin

Uncertainty principle 101 waves Why is it an inequality?

Importance of phase in f(  ) 11 waves101 waves Try: Linear phase function:  t  is the same. Pulse is shifted

Importance of phase 11 waves101 waves Try: Quadratic dependence:  t  is much bigger for the same 

Importance of phase 11 waves101 waves Random dependence:  t  is infinite (noise)  t  is much bigger for the same 

Summary: Importance of phase 11 waves101 waves Why is it an inequality? f(t) changes greatly with phase  The shortest is had only for  constant or linear. All others will make  comes entirely from |f(  )|, which has no phase information.

Carrier frequency-envelope principle 11 waves101 waves The FT f(  ) is the FT of ___ centered at ____. The width  is the width of ____ Optical pulses are often a steady (“carrier”) wave at multiplied by an envelope function

Which pulse f(t) will have f(  ) centered around the highest frequencies? a) b) c) Which f(t) will have the greatest width  in f(  ) 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

Fourier theory and delta functions