1. The Fourier Transform Some Fourier Transform theorems Scale: f(at )
Sum: f(t) + g(t)
Shift: f(t–a)
Complex conjugate: f*(t)
Derivative: f'(t)
Modulation: f(t)cos(w0 t)
The intensity and phase and the spectral intensity and phase
The 2D Fourier Transform
The relative importance of intensity and phase t w
Prof. Rick Trebino, Georgia Tech

2. The Fourier Transform and its Inverse

3. Scale Theorem The Fourier transform of a scaled function, f(at):
Proof:
Assuming a > 0, change variables: u = at, so dt = du / a
If a < 0, the limits flip when we change variables, introducing a minus sign, hence the absolute value.

4. The Scale Theorem in action
f(t)
F(w)
The Scale Theorem in action t w
Short pulse t w
The shorter the pulse, the broader the spectrum!
Medium- length pulse t w
This is the essence of the Uncertainty Principle!
Long pulse

5. The Fourier Transform of a sum of two functions
F(w) t w
g(t)
G(w) t w
F(w) + G(w)
Also, constants factor out.
The Fourier transform is a linear function of functions.
f(t)+g(t) t w

6. Shift Theorem

7. Application of the Shift Theorem
Suppose we’re measuring the spectrum of E(t), but a small fraction of its irradiance, say e, takes a longer path to the spectrometer.
The extra light has the field, √e E(t–a), where a is the extra path.
The measured spectrum is:
E(t)
Spectro-meter
e E(t-a)
Performing the Fourier Transform and using the Shift Theorem:

8. Application of the Shift Theorem (cont’d)
Neglecting e compared to √e and 1:
The contaminated spectrum will have ripples with a period of 2p/a.
And these ripples will have a surprisingly large amplitude.
If e = 1% (a seemingly small amount), these ripples will have an amazing-ly large amplitude of 2√e = 20%! And peak-to-peak, they’re 40%!

9. The Fourier Transform of the complex conjugate of a function
Proof:

10. Negative frequencies contain no additional information for real functions.
If a function is real (e.g., a light wave!), then f*(t) = f(t). So:
Using the result we just proved:
So, at –w, the real part of the Fourier transform of f is the same as at +w. And the imaginary part is just the negative of it.

11. Derivative Theorem
The Fourier transform of a derivative of a function, f’(t):
Proof:
Integrate by parts:
Remember that the function must be zero at ±∞, so the other term, [f(t) exp(-iwt)] vanishes. +∞ -∞

12. The Modulation Theorem: The Fourier Transform of E(t) cos(w0 t)
Example:
E(t) = exp(-t2) t
-w0 w0 w

13. The Fourier transform and its inverse are symmetrical: f(t) F(w) and F(t) f(w) (almost).
If f(t) Fourier transforms to F(w), then F(t) Fourier transforms to:
Rearranging:
Relabeling the integration variable from t to w’, we can see that we have an inverse Fourier transform:
This is why it is often said that f and F are a Fourier Transform Pair.

14. Fourier Transform Magnitude and Phase
As with any complex quantity, we can decompose f(t) and F(w) into their magnitude and phase.
f(t) can be written: f(t) = Mag{f(t)} exp[ -i Phase{f(t)}]
where Mag{f(t)}2 is often called the intensity, I(t), and Phase{f(t)} is called the phase, f(t). They’re the same quantities we’re used to for light waves (the minus signs are arbitrary).
Analogously, F(w) = Mag{F(w)} exp[ -i Phase{F(w)}]
The Mag{F(w)}2 is called the spectrum, S(w), and the Phase{F(w)} is called the spectral phase, j(w).
Just as both the intensity and phase are required to specify f(t), both the spectrum and spectral phase are required to specify F(w), and hence f(t).

15. Light has intensity and phase also.
We usually extract out the carrier frequency.
A light wave has the time-domain electric field:
Intensity
Phase
The minus signs are just conventions.
(neglecting the
negative-frequency
component)
Equivalently, vs. frequency:
Spectrum
Spectral
Phase
Knowledge of the intensity and phase or the spectrum and spectral phase is sufficient to determine the light wave.

16. Calculating the Intensity and the Phase
It’s easy to go back and forth between the electric field and the intensity and phase.
The intensity:
I(t) = |E(t)|2
The phase:
E(ti) Im
√I(ti)
-f(ti) Re
Equivalently,
f(t) = -Im{ ln[E(t)] }
We can do the same for the field in the frequency domain.

17. Complex Lorentzian and its Intensity and Phase
Real
component
Imaginary
component w
Real part
Imag part

18. Intensity and phase of a decaying exponential and its Fourier transform
exp(-2at)
Time domain:
Frequency domain:
(solid)

19. Intensity and Phase of a Gaussian
The Gaussian is real, so its phase is zero.
Time domain:
Frequency domain:
A Gaussian transforms
to a Gaussian
So the spectral phase is zero, too.

20. The spectral phase of a time-shifted pulse
Recall the Shift Theorem:
Time-shifted Gaussian pulse (with a flat phase):
So a time-shift simply adds some linear spectral phase to the pulse!

21. What is the spectral phase anyway?
It’s the absolute phase of each frequency in the wave-form. t
All of these frequencies have zero phase. So this pulse has:
j(w) = 0
Note that this wave-form sees constructive interference, and hence peaks, at t = 0.
And it has cancellation everywhere else. w1 w2 w3 w4 w5 w6

22. Now try a linear spectral phase: j(w) = aw.
By the Shift Theorem, a linear spectral phase is just a delay in time.
And this is what occurs! w1 w2 w3 w4 w5 w6
j(w1) = 0
j(w2) = 0.2 p
j(w3) = 0.4 p
j(w4) = 0.6 p
j(w5) = 0.8 p
j(w6) = p t

23. The spectral phase distinguishes a light bulb from a short pulse.
Intensity vs. time
Random phases
Light bulb
Time
Intensity vs. time
Ultrashort pulse!
Locked phases
Time

24. Fourier Transform with respect to space
If f(x) is a function of position, x
F {f(x)} = F(k) k
We refer to k as the spatial frequency.
Everything we’ve said about Fourier transforms between the t and w domains also applies to the x and k domains.

25. The 2D Fourier Transformx y
f(x,y)
F (2){f(x,y)} = F(kx,ky)
= f(x,y) exp[-i(kxx+kyy)] dx dy
If f(x,y) = fx(x) fy(y),
then the 2D FT splits into two 1D FT's.
But this doesn’t always happen.
F (2){f(x,y)}

26. A 2D Fourier Transform: a square functionx y
f(x,y)
Consider a square function in the xy plane:
f(x,y) = rect(x) rect(y)
The 2D Fourier Transform splits into the product of two 1D Fourier Transforms:
F {f(x,y)} = sinc(kx/2) sinc(ky/2)
This picture is an optical determination
of the Fourier Transform of the
square function!
F (2){f(x,y)}

27. Fourier Transform Magnitude and Phase
Rick
Linda
Pictures reconstructed
using the spectral phase
of the other picture
Mag{F [Linda]}
Phase{F [Rick]}
Mag{F [Rick]}
Phase{F {Linda]}
The phase of the Fourier transform (spectral phase) is much more important than the magnitude in reconstructing an image.