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 P. Piot, PHYS430-530, NIU FA2018

Michelson Interferometer Case of a monochromatic plane wave Total intensity at the detector is Case of two wave packets? P. Piot, PHYS430-530, NIU FA2018

Michelson interferometer II Consider two arbitrary waveforms the total field is The total intensity is then P. Piot, PHYS430-530, NIU FA2018

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 P. Piot, PHYS430-530, NIU FA2018

Example of a Gaussian wavepacket Consider a Gaussian wavepacket Its power spectrum is The degree of coherence is given by So that P. Piot, PHYS430-530, NIU FA2018

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 P. Piot, PHYS430-530, NIU FA2018

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 P. Piot, PHYS430-530, NIU FA2018

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 2 2 + 𝐷 2 1/2 ≃𝐷(1+ 𝑦± ℎ 2 2 2 +…) So the latter equation hold assuming 𝐷≫𝑦 and 𝐷≫ℎ. P. Piot, PHYS430-530, NIU FA2018

Case of an extended source The fields are And the total field is found via Total intensity is P. Piot, PHYS430-530, NIU FA2018

Case of an extended source II Assume the phases of emis- sion are random: The we get the time-averaged intensity Of the form P. Piot, PHYS430-530, NIU FA2018