Conical refraction healing after partially blocking the input beam

In conical refraction, when a focused Gaussian beam passes along one of the optic axes of a biaxial crystal, it is transformed into a pair of concentric bright rings at the focal plane. We demonstrate both theoretically and experimentally that this transformation is hardly affected by partially blocking the Gaussian input beam with an obstacle. We analyze the influence of the size of the obstruction both on the transverse intensity pattern of the beam and on its state of polarization, which is shown to be very robust.

Figure 1

Transverse intensity patterns and state of polarization at the focal plane of CR beams numerically calculated from Eqs. (1, 2, 3, 4, 5, 6, 7, 8) for a left-handed circularly polarized Gaussian input beam for (a) ${\rho }_0\equiv R_0/w_0=10$, (b) ${\rho }_0=1.5$, and (c) ${\rho }_0=0.92$. $R_0$ is the geometrical CR ring radius and $w_0$ the waist radius of the focused Gaussian input beam.

Figure 2

Experimental setup. A Gaussian beam is obtained from a diode laser coupled to a monomode fiber. A collimator (C) is used to change the beam waist radius $w_0$ of the Gaussian beam. A linear polarizer (LP) and a quarter wave plate (QWP) are used to fix the state of polarization of the Gaussian beam to circular. Azimuthal angular amplitude masks (AAM) characterized by their closing angle $\varphi$ are used to block a sector of the Gaussian beam. The obtained beam passes through the biaxial crystal (BC) along one of its optic axes and a CCD camera and an imaging lens (IL) record the transverse intensity pattern at different planes.

Figure 3

Transverse intensity patterns at the focal plane $\left(z=0\right)$ obtained with an obstruction of closing angle [(a) and (f)] $\varphi =0^{\circ }$, [(b) and (g)] $\varphi ={45}^{\circ }$, [(c) and (h)] $\varphi ={90}^{\circ }$, [(d) and (i)] $\varphi ={135}^{\circ }$, and [(e) and (j)] $\varphi ={180}^{\circ }$ placed before the biaxial crystal. [(a) and (f)] Intensity pattern obtained in the absence of obstruction. First row: Experimental results $\left({\rho }_0^{\exp }=10.75\right)$. Second row: Numerical calculations obtained from Eqs. (1, 2, 3, 4, 5, 6) $\left({\rho }_0^{\mathrm{th}}=10\right)$.

Figure 4

Numerically calculated 2D density plots of the azimuth $\epsilon$ (first row) and the ellipticity $\beta$ (second row) of conically refracted beams at the focal plane for ${\rho }_0=10$ when the input Gaussian is blocked by azimuthal obstructions of [(b) and (g)] $\varphi ={45}^{\circ }$, [(c) and (h)] $\varphi ={90}^{\circ }$, [(d) and (i)] $\varphi ={135}^{\circ }$, and [(e) and (j)] $\varphi ={180}^{\circ }$ placed before the biaxial crystal. [(a) and (f)] Two-dimensional density plots of $\epsilon$ and $\beta$ in the absence of the obstruction.

Figure 5

Experimental (second row) and numerically calculated with Eqs. (1, 2, 3, 4, 5, 6) (third row) transverse intensity patterns along the axial direction $z$ for an obstruction with $\varphi =45$ for ${\rho }_0=10$. The first row shows the numerically calculated transverse intensity patterns obtained in the absence of the obstruction. $z$ is measured in units of the Rayleigh range, which for the Gaussian beam used in our experiments $\left(w_0=44\mu \mathrm{m}\right)$ is $z_R=9.5\mathrm{mm}$.

Figure 6

Transverse intensity patterns at the focal plane $\left(z=0\right)$ obtained when an obstruction of angle [(b) and (g)] $\varphi ={45}^{\circ }$, [(c) and (h)] $\varphi ={90}^{\circ }$, [(d) and (i)] $\varphi ={135}^{\circ }$, and [(e) and (j)] $\varphi ={180}^{\circ }$ is placed before the biaxial crystal. [(a) and (f)] Transverse intensity pattern in the absence of obstruction. First row: Experimental results $\left({\rho }_0^{\exp }=1.04\right)$. Second row: Numerical calculations obtained from Eqs. (1, 2, 3, 4, 5, 6) $\left({\rho }_0^{\mathrm{th}}=0.92\right)$.

Figure 7

Two-dimensional density plots of the azimuth $\epsilon$ (first row) and the ellipticity $\beta$ (second row) of conically refracted beams at the focal plane for ${\rho }_0=0.92$ when the input Gaussian is blocked by azimuthal obstructions of [(b) and (g)] $\varphi ={45}^{\circ }$, [(c) and (h)] $\varphi ={90}^{\circ }$, [(d) and (i)] $\varphi ={135}^{\circ }$, and [(e) and (j)] $\varphi ={180}^{\circ }$ is placed before the biaxial crystal. [(a) and (f)] Two-dimensional density plots of $\epsilon \beta$ in the absence of obstruction.

Figure 8

Experimental (second row) and numerically calculated with Eqs. (1, 2, 3, 4, 5, 6) (third row) transverse intensity patterns along the axial direction $z$ for an obstruction with $\varphi =45$ for ${\rho }_0=0.92$. The first row shows the numerically calculated transverse intensity patterns obtained in the absence of the obstruction. $z$ is measured in units of the Rayleigh range, which for the Gaussian beam used in our experiments $\left(w_0=44\mu \mathrm{m}\right)$ is $z_R=9.5\mathrm{mm}$.