Optics 430/530, week IX Superposition of plane waves

Optics 430/530, week IX Superposition of plane waves coherence This class notes freely use material from http://optics.byu.edu/BYUOpticsBook_2015.pdf P. Piot, PHYS430-530, NIU FA2018 1 Sum of two waves We now specialize to the case of two wave with equal amplitudes:

The phase velocities is given by The superimposed field is So that the optical intensity is P. Piot, PHYS430-530, NIU FA2018 2 Group velocity Consider the previous equation From the argument of the cosine we can define a velocity as

this is the group velocity which describes the velocity of the wave envelope Note that the phase velocity of the superimposed wave is P. Piot, PHYS430-530, NIU FA2018 3 Frequency spectrum of light In Physics it is common to decompose a temporal signal over the Frequency domain. Such a decomposition of the E field writes

The function is referred to as the Fourier transform of . The previous operation is actually called inverse Fourier transform. The Fourier transform is defined as P. Piot, PHYS430-530, NIU FA2018 4 Power spectrum We saw that the optical intensity

We can also write this intensity in term of Fourier transform is such a case it is called power spectrum Note that is not the Fourier transform of . P. Piot, PHYS430-530, NIU FA2018 5 Fourier transforms P. Piot, PHYS430-530, NIU FA2018

6 Parsevals theorem The Parseval theorem is a general theorem that states Consider the example of a modulated Gaussian pulse We have for the Fourier transform So that both the time integral and frequency integral give P. Piot, PHYS430-530, NIU FA2018 7

Wave propagation in a medium To understand the propagation of a wave packet in a given medium we follow the prescription: Decompose the wave packet in its Fourier components Propagate each of the Fourier component including the frequency dependent properties of the medium Do an inverse Fourier transform to find the wave packet temporal distribution Consider propagation of in a medium length . The Fourier component is transform accordingly to (phase added)

P. Piot, PHYS430-530, NIU FA2018 8 Wave propagation in a medium (II) In the previous equation describes the frequency-dependent properties of the medium Take an inverse Fourier transform to yield delay P. Piot, PHYS430-530, NIU FA2018

9 Dispersion Taylor Expand as Group Velocity Dispersion (GVD) Group Delay function

Specializing to the z axis: P. Piot, PHYS430-530, NIU FA2018 10 Effect of dispersion From previous equation so field is temporally shifted with an added delay related to the phase velocity P. Piot, PHYS430-530, NIU FA2018

11 Quadratic dispersion Consider a Gaussian wave packet in a medium with dispersion expanded to 2nd order. Its Fourier transform is Plugging the expansion gives Which reduces to P. Piot, PHYS430-530, NIU FA2018

12 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 13 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

14 Michelson interferometer II Consider two arbitrary waveforms the total field is The total intensity is then P. Piot, PHYS430-530, NIU FA2018 15 Degree of coherence function

Integrate previous equation over time And introduce the fluence The 3rd term is So that Degree of coherence function P. Piot, PHYS430-530, NIU FA2018 16