CRE-filtered beam


Two identically oriented conerefringent elements CRE′ and CRE″ are arranged in series. CRE′ transforms the incident beam to an intermediate one, and CRE″ transforms the intermediate beam to a final one. The incident beam is scalar beam.

Different parts of the intermediate beam can be selected by filter and allowed to pass through CRE″. We are interested how the final beam changes. Both beams are compared in their isomomentum planes.

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In the experimental setup above the scalar beam is randomly polarized He-Ne laser beam. Lens1 increases the experimental resolution and images the isomomentum plane of the incident beam. The filter is placed in the position of the real isomomentum plane of the intermediate beam, which is between CRE′ and CRE″. The isomomentum plane of the final beam is virtual. Lens2 images the virtual isomomentum plane of the final beam on the screen; if CRE″ is removed from the setup then Lens2 can image also the real isomomentum plane of the intermediate beam. Taking into account the magnification and inversion of Lens2, we obtain the originals from the imaged patterns.

Without any filter, the originals (the ring patterns in the isomomentum planes of the intermediate and final beams) are presented below. The plane $\{X', Y'\}$ belongs to the CRE′ coordinate system.

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The intermediate and the final ring patterns are related by homothety transformation with origin $\{0, 0\}$ and scale coefficient $s = (\Lambda ' + \Lambda '' ) / \Lambda '$. The radius vectors $\mbox{\boldmath$\mathit{P_{I}}$}$ and $\mbox{\boldmath$\mathit{P_{F}}$}$ of two corresponding points $P_{I}$ and $P_{F}$ are related by $\mbox{\boldmath$\mathit{P_{F}}$} = s \mbox{\boldmath$\mathit{P_{I}}$}$. Two corresponding points have the same polarization axis.

The final beam can be considered as passed through one CRE with Lambda-parameter $\Lambda ' + \Lambda ''$.

When the Poggendorff pattern is resolved, the point $P_{I}$ from the intermediate beam must be more precisely determined. Two cases are experimentally realizable: with and without accounting for the fine splitting. When the fine splitting is accounted the point $P_{I}$ can be filtered from the inner or outer ring of the intermediate Poggendorff pattern. When the fine splitting is neglected the filtered point $P_{I}$ is doublet — it has parts from the inner ring and from the outer ring. If the experiment gives the Lloyd pattern, the filtered point $P_{I}$ in itself is an unresolved doublet.

CRE-filtered beam (fine splitting neglected)

The case of fine splitting is neglected and the effect of the filter will be interpreted with Lloyd patterns.

The experiment shows that if an arc $A_{I}$ from the intermediate ring is filtered, the final pattern is also arc, $A_{F}$ on the figure, and both arcs are homothety corresponding. In general, an arbitrary point from the intermediate ring is transformed to the homothety corresponding point from the final ring.

The point filter is a pinhole with radius comparable or a bit bigger than the calculated waist radius of the beam focused by Lens1. Different points can be selected by moving transversally the pinhole along the ring. The beam passed through the pinhole is named CRE-filtered beam and it is represented by the point $P_{I}$ in the figure below. Its homothety corresponding point is $P_{F}$. The blue arrows indicate the linear polarization axes.

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In quantum mechanical terms the beams represented by the points $P_{I}$ and $P_{F}$ are “pure” or polarized beams. The photons of these beams are in basis state of CRE″, considered as one Stern-Gerlach apparatus.

Consider the filter effect in the plane $\{X'', Y'', 0\}$ of CRE″ coordinate system when only point $P_{I}$ is filtered. Note that the Z″-axis is shifted to the gravity center of the CRE-filtered beam, that is the point $P_{I}$. The coordinate axes X″ and Y″ are marked with dashed lines in the above figure. This is equivalent of the passage of CRE-filtered beam through CRE″ and is presented in the left figure below.

Remove now CRE′ and Lens1 from the experimental setup and illuminate the pinhole by the same He-Ne laser, but now linearly polarized. The polarization axis of the laser beam is identically oriented to the polarization axis of the CRE-filtered beam represented by point $P_{I}$. This laser beam is identical to the CRE-filtered beam in regard to all variables characterizing one photon as fundamental particle: spin, wavelength and 3-momentum. Denote it as the “same” beam.

Compare the patterns generated by the CRE-filtered beam and the "same" one. As it is known from the internal conical refraction, the “same” beam generates Lloyd pattern with $sin^{2}(\phi - \pi/4)$ intensity distribution (presented for comparison in the right figure).

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The experiment demonstrates that in spite of having identical spin, wavelength and 3-momentum, the photons from one CRE-filtered beam and the photons from the “same” beam are not identical.


May 24, 2006
Author: T Kirilov