CRE Lambda-parameter

also: CRE $ \Lambda$ -parameter

When a scalar beam passes through a CRE, its gravity center shifts. The magnitude of this shift is defined as CRE Lambda-parameter. Thus defined, the $ \Lambda$ -parameter depends on the scalar beam wavelength $ \lambda$ and not on other beam characteristics as diameter, divergence or intensity profile; it does not depend on the position of measurement too.

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The center of gravity of the incident beam is along the axis $ \{0, 0, Z\}$ and the center of gravity of the passed through CRE beam is along the axis $ \lbrace\Lambda, 0, Z\rbrace$ ; $ \{X, Y, Z\}$ is the CRE coordinate system.

The CRE $ \Lambda$ -parameter can be measured also by the radius of the Poggendorff dark ring. The later varies with the position of measurement and equals the $ \Lambda$ -parameter only in the isomomentum plane. This alternative method allows the scalar beam to be replaced by a linearly polarized beam.

The isomomentum plane for the beam after CRE can be real or virtual. It is real when it is positioned after the CRE output base, and virtual in the opposite case. The $ \Lambda$ -parameter can be measured directly by detector only in real isomomentum plane. The Poggendorff dark ring is resolved when the waist radius $ \omega_{0}$ of the beam before CRE is much smaller than the $ \Lambda$ -parameter. The resolution is defined as $ R = \Lambda/\omega_{0}$ . The maximum resolution of this method is

$\displaystyle R_{max}=\frac{\pi}{2\lambda}n_{2}\kappa C_{A} ; $

here $ \kappa$ is the crystal conicity, n2 is the intermediate refraction index and CA is the clear aperture of the CRE input base. In the usual experimental setup the incident beam is laser beam with isomomentum plane in the resonator cavity. To increase the resolution of the alternative method, the incident beam is transformed by lens to a beam before in such manner, that the isomomentum plane of the beam before is imaged in the vicinity of the output base of CRE. After the “shift of the focus” $ \Delta$ caused by CRE, the isomomentum plane of the beam after is real. This situation is presented below:

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If the isomomentum plane of the beam after is virtual, it can be imaged by a lens on a screen or detector. The resolution of the method in this case (the figure below) is always lower than $ R_{max}$ . (The lens optical magnification preserves the ratio $ \Lambda/\omega_{0}$ .)

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May 24, 2006
Author: T Kirilov