6.
The Fourier Transform of ( t ) is 1. ( ) And the Fourier Transform of 1 is ( ): t ( t ) t
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The Fourier transform of exp( i 0 t ) The function exp( i 0 t ) is the essential component of Fourier analysis. It is a pure frequency. F {exp( i 0 t )} exp( i 0 t ) t t Re Im
8.
The Fourier transform of cos( t ) cos( 0 t ) t
The Fourier transform of a scaled function, f ( at ) :
If a < 0 , the limits flip when we change variables, introducing a minus sign, hence the absolute value. Assuming a > 0 , change variables: u = at Proof:
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The Scale Theorem in action f ( t ) F ( ) Short pulse Medium- length pulse Long pulse The shorter the pulse, the broader the spectrum! This is the essence of the Uncertainty Principle! t t t
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The Fourier Transform of a sum of two functions Also, constants factor out. f ( t ) g ( t ) t t t F ( ) G ( ) f ( t )+ g ( t ) F ( ) + G ( )
The Fourier transform of a derivative of a function, f’(t) :
Proof:
Integrate by parts:
17.
The Modulation Theorem: The Fourier Transform of E ( t ) cos( 0 t ) If E ( t ) = ( t ) , then: 0 - 0
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The Fourier transform and its inverse are symmetrical: f ( t ) F ( ) and F ( t ) f ( ) (almost). If f(t) Fourier transforms to F ( ), then F(t) Fourier transforms to: Relabeling the integration variable from t to ’, 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.” Rearranging:
As with any complex quantity, we can decompose f ( t ) and F ( ) 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, ( t ) . They’re the same quantities we’re used to for light waves.
Analogously, F ( ) = Mag { F ( ) } exp[ - i Phase { F ( ) } ]
The Mag { F ( ) } 2 is called the spectrum, S ( ) , and the Phase { F ( ) } is called the spectral phase, ( ) .
Just as both the intensity and phase are required to specify f ( t ) , both the spectrum and spectral phase are required to specify F ( ) , and hence f ( t ) .
20.
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: (t) = -Im{ ln[ E (t) ] } The phase: Equivalently, (t i ) Re Im E(t i ) √ I(t i ) I(t) = |E(t)| 2
Time domain: Frequency domain: So the spectral phase is zero, too. A Gaussian transforms to a Gaussian
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The spectral phase of a time-shifted pulse Recall the Shift Theorem: So a time-shift simply adds some linear spectral phase to the pulse! Time-shifted Gaussian pulse (with a flat phase):
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What is the spectral phase anyway? The spectral phase is the abs phase of each frequency in the wave-form. 0 All of these frequencies have zero phase. So this pulse has: ( ) = 0 Note that this wave-form sees constructive interference, and hence peaks, at t = 0 . And it has cancellation everywhere else. 1 2 3 4 5 6 t
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Now try a linear spectral phase: ( ) = a . t ( 1 ) = 0 ( 2 ) = 0.2 ( 3 ) = 0.4 ( 4 ) = 0.6 ( 5 ) = 0.8 ( 6 ) = 1 2 3 4 5 6 By the Shift Theorem, a linear spectral phase is just a delay in time. And this is what occurs!
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The spectral phase distinguishes a light bulb from an ultrashort pulse.
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Complex Lorentzian and its Intensity and Phase 0 Imaginary component Real component 0 a Real part Imag part
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Intensity and Phase of a decaying exponential and its Fourier transform Time domain: Frequency domain: (solid)
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Light has intensity and phase also. A light wave has the time-domain electric field: Intensity Phase Equivalently, vs. frequency: Spectral Phase (neglecting the negative-frequency component) Spectrum Knowledge of the intensity and phase or the spectrum and spectral phase is sufficient to determine the light wave. The minus signs are just conventions. We usually extract out the carrier frequency.
If f ( x ) is a function of position, We refer to k as the “spatial frequency.” Everything we’ve said about Fourier transforms between the t and domains also applies to the x and k domains. x k
The 2D Fourier Transform splits into the product of two 1D Fourier Transforms:
F { f ( x,y )} = sinc( k x /2 ) sinc( k y /2 )
This picture is an optical determination
of the Fourier Transform of the
square function!
F (2) { f ( x,y )} x y f ( x,y )
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Fourier Transform Magnitude and Phase Pictures reconstructed using the spectral phase of the other picture The phase of the Fourier transform (spectral phase) is much more important than the magnitude in reconstructing an image. Mag{ F [Linda]} Phase{ F [Rick]} Mag{ F [Rick]} Phase{ F {Linda]} Rick Linda
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