Interference with polarized light

Transcription

1 Interference with polarized light Summary of the previous lecture (see lecture 3 - slides 12 to 25) With polarized light E 1 et E 2 are complex amplitudes: E 1 + E 2 e iϕ 2 = E E Re(E 1 * E 2 e iϕ ) No interference for orthogonal polarizations For linear polarizations, interference term = 2 E 1.E 2 cosϕ To obtain interference between ordinary and extraordinary waves, the birefringent medium must placed between polarizer and analyzer

2 Summary of the previous lecture BEST POLARIZATION CONDITION FOR INTERFERENCE USING BIREFRINGENT MEDIA To maximize both contrast and intensity, the birefringent medium should be placed between a polarizer P and an analyzer A such that : 1. P and A parallel and at 45 of the neutral axes 2. P and A orthogonal and at 45 of the neutral axes P and A orthogonal is usually a better choice because the contrast is always maximum even if the neutral axes are not at 45

3 Summary of the previous lecture Example 1: Observation of fringes with a small angle Wollaston prism x y θ Small angle θ we neglect the ray deviation δ=2(ne-no)xθ Straight fringes // y axis Dark fringe (if P A) for x=0 P A

4 Summary of the previous lecture Application to the measurement of the phase shift of a wave plate: Babinet compensator We add the unknown phase plate with its axes parallel to the Wollaston s axes The plate induces a translation of the fringes by δ plate /λ*period of the fringes Unknown plate P A

5 Summary of the previous lecture Example 2: a parallel birefringent plate illuminated with white light Each wavelength is transmitted with an intensity depending on the corresponding phase shift (ratio of OPD δ to λ) If the OPD is small enough (less than 2,5µm), there will be a color that is characteristic of each value of OPD (given by Newton s color scale) Easy to observe in polarized light because δ=(n e -n o )e is small observation on the slide projector with layers of scotch tape Sensitive color (purple = equal amount of blue and red) useful to detect OPD variation by less than 100nm

6 Summary of the previous lecture

7 Summary of the previous lecture Interference with white light : When the OPD is large, no interference color is visible and the transmitted light must be observed with a spectrometer. We then observe a Channeled spectrum Observation with spectrometer dispersion of the different λ P//A: dark fringes if δ=λ/2+kλ P A: dark fringes si δ=kλ

8 From this slide on: New material about interference Conditions on coherence Reminder of the standard coherence requirements (for unpolarized waves) Temporal coherence: condition on the spectral width of the source (standard criterion: finesse λ/δλ>4δ max /λ) Spatial coherence : condition on the size of the source - only for amplitude splitting interference - surface of localization (intersection of the two splitted rays): best location to observe interference, spatial coherence easiest to fulfil - Standard criterion: variation of OPD over the size of the source smaller than λ/4

9 Calculating the Optical Path Difference Start with the following observation look at the interference color of a parallel birefringent plate between polarizer and analyzer: From above (normal incidence) From the side (oblique incidence) Move around the sample to look from the same incidence but from different sides Do you see the same interference color?

10 How is this done in non polarized interferometers? some examples: Young s interferometer (non localized fringes) The parallel glass plate (fringes localized at infinity) Michelson s interferometer with an angle between mirrors Calculating the Optical Path Difference General method in the case of amplitude splitting interference (localized fringes): determine the surface of localization of the fringes calculate the optical path difference from an initial wavefront before splitting up to the surface of localization

11 How is it different if the paths are inside an anisotropic medium? Two possible approaches: calculate the optical path length along the ordinary and extraordinary light rays and take into account the radial velocities calculate the optical path length along the wave normals (i.e. the k vectors) and take into account the phase velocities, i.e. the indices EASIER, especially in the case of a parallel plate

12 Calculation of the Optical Path Difference (OPD) for a parallel plate as a function of the angle of incidence Fringes localized at infinity calculation of the path lengths up to a plane wavefront common to both paths (careful not to forget the part of the path in air!) Calculation along the k vectors: we construct the refracted k vectors using the index surfaces

13 Constructing the refracted k vectors using the index surfaces Review of the isotropic case Uniaxial case D e

14 Application to the calculation of the optical path difference between ordinary and extraordinary i 1 1 n o O H n e >n o N o n e N e

15 Construction of the wavefronts i Σin O H Σextraord N o J e δ i J o Σord I N e K δ= J e K= J e J o sini δ = e N e N o Valid for any orientation of the optical axis, as long as the plate is parallel!