1. Coherence (chapter 8)
Coherence theory is the study of correlation that exist between different parts of a light field
Two type of coherences:
Temporal coherence: correlation between
Spatial coherence: correlation between
Temporal coherence can be measured with a Michelson interferometer
Spatial coherence can be measured with a two-slit interferometer
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2. Michelson Interferometer
Case of a monochromatic plane wave
Total intensity at the detector is
Case of two wave packets?
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3. Michelson interferometer II
Consider two arbitrary waveforms the total field is
The total intensity is then
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4. Degree of coherence function
Integrate previous equation over time
And introduce the fluence
The 3rd term is
So that
Use of Parseval’s theorem
Degree of coherence
function
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5. Example of a Gaussian wavepacket
Consider a Gaussian wavepacket
Its power spectrum is
The degree of coherence is given by
So that
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6. Coherence time and fringe visibility
𝜏 𝑐
Cohrence time is defined as the amount of delay necessary to cause 𝛾(𝜏) to approach. A somewhat arbitrary definition is
To the coherence time one can associate a coherence length
The fringe visibility is defined as
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7. Fourier Spectroscopy
The output signal of a Michelson interferometer for a pulsed signal
If we take the Fourier transform we get
So a measurement of the autocorrelation provides the spectrum of the signal. This is the essence of Fourier spectroscopy a widely used technique
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8. Spatial coherence
Consider the double slits “spatial” interferometer and a point source
This is similar to the Michelson interferometer except. Note theTaylor expansion 𝑑 1,2 𝑦 = 𝑦± ℎ 𝐷 2 1/2 ≃𝐷(1+ 𝑦± ℎ …)
So the latter equation hold assuming 𝐷≫𝑦 and 𝐷≫ℎ.
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9. Case of an extended source
The fields are
And the total field is found via
Total intensity is
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10. Case of an extended source II
Assume the phases of emis- sion are random:
The we get the time-averaged intensity
Of the form
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