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We derive the analytical expression of the Stokes parameters corresponding to a Gaussian beam propagating along the optical axis of a uniaxial crystal, pointing the simultaneous effects of anisotropy and diffraction out. The theoretical results are compared with experimental measurements at the output of a calcite crystal, showing a good agreement.
©2002 Optical Society of America
1 Introduction
Optical technologies using liquid crystals, electro-optic devices, polarimetric sensors, anisotropic fibers and fiber amplifiers focus the research interest on the evolution of the state of polarization of light through anisotropic media [1]–[6]. Generally, the change of the polarization state is discussed for plane-waves, as within the Jones formalism or the Mueller calculus. The evolution of the Stokes parameters through an anisotropic medium is a well-understood phenomenon and a general treatment is given in literature for both monochromatic [7]–[14] and quasi-monochromatic (partially polarized) plane waves [15],[16]. But the plane-wave analysis does not take the diffraction effects into account and it could be a serious shortcoming when the beam width is comparable with the wavelength. Many numerical and analytical methods have been proposed to evaluate the propagation of an optical beam through an unbounded anisotropic media [17],[21], but a comparison between computational and experimental results is lacking in literature.
In the present paper, we theoretically and experimentally investigate the evolution of the Stokes parameters associated with a laser beam propagating along the optical axis of a calcite crystal; due to the simultaneous effects of anisotropy and diffraction, the propagating field is not uniformly polarized, and the Stokes parameters are functions of the position vector at any transverse plane. The Stokes parameters are computed in Sec. 2, starting from the expression of the fundamental Gaussian beam inside a uniaxial medium given in Ref. [22], whereas the experimental setup and results are described in Sec. 3. The agreement between theory and experiment is such to confirm the validity of the theoretical approach.
The case of light propagation at any angle to the oprical axis can be treated in a similar way, but the algebra is somewhat more tedius.
2 Stokes parameters
The use of the Stokes parameters is a standard method to characterize the state of polarization of an optical field, by means of simple measurements of the intensity distribution. Referring to the electric field E (x, y, z) = Ex (x, y, z) êx + Ey (x, y, z) êy, the four Stokes parameters are defined as
where asterisk denotes the complex conjugate. The parameters s 0 and s 1 are straightforwardly evaluated by summing or subtracting the intensities of light transmitted by two polarizers which accept linear polarization along the x- or y-axis, respectively; the parameter s 2 is analogously obtained rotating the polarizer of 45o and 135o with respect to the x-axis, whereas s 3 is measured at the output of the same polarizer, when a further phase difference among Ex and Ey has been introduced with a π/2 compensator, i.e. a λ/4 plate with the slow axis lying along the y-axis [9].
We consider the propagation of a Gaussian beam along the optical axis z of a uniaxial crystal with ordinary no and extraordinary ne refractive indices; if the input beam is linearly polarized along the x-axis, the Cartesian components of the field inside the crystal have the following expressions [22]
where Qo (z) = + i(2z)/(k 0 no ) and Qe (z) = + i(2z no )/(k 0 ) are the complex propagation parameters associated to the ordinary and extraordinary components, respectively. Substituting Eqs. (2) into (1), we obtain a compact expression of the four Stokes parameters
with
and
The movie sequences of Fig. 1 show the evolution of the Stokes parameters for propagation distances up to z = 4zRo , being zRo = no /λ the Rayleigh distance in a homogeneous medium, with refraction index no . From an inspection of s 0 and s 1, the combined effects of anisotropy and diffraction is evident, as the beam loses its boundary cylindrical symmetry due to different diffraction lengths of the ordinary and extraordinary components [23]. On the other hand, from the animation of s 2 and s 3 one can notice that for z = zRe , being zRe = ne /λ the Rayleigh distance in a homogeneous medium with refraction index ne , the phase difference between Ex and Ey practically vanishes. In fact, both the moduli of s 2 and s 3 (that are zero at the input plane z = 0) at first grow with z and then decrease: in the plane z = zRe , s 3 vanishes almost everywhere, whereas | s 2 | is maximum, so that the beam is almost linearly polarized. Of course, due to the anisotropy, the polarization direction is different at any position.
Fig. 1. Animations of the evolution of the Stokes parameters associated with a Gaussian beam propagating along the optical axis of a calcite crystal, from the z = 0 to the z = 4zRo planes; the transverse coordinates are normalized with respect to the input spot size w 0. [Media 1] [Media 2] [Media 3] [Media 4]
3 Experiment
The laboratory setup is illustrated in Fig. 2. On the right side there is an Argon-Kripton laser tuned at the λ = 0.514μm line on the fundamental Gaussian mode, with 1mm spot size, 1mrad divergence and linear polarization. The beam is attenuated passing through the neutral density filter F, and, as the linear polarization is strongly requested, it is passed through the polarizer P1 in order to avoid possible spurious components. Afterwards, the beam is focused on the entrance facet of the calcite crystal by the lens L1, whose focal length is varied to change the beam input spot sizes w 0. The calcite crystal has refractive indices no = 1.658 and ne = 1.486, dimensions 1 × 1 × 2cm, and the optical axis coincident with the longest axis, along which the beam is made to propagate. The optical system constituted by the lens L2 and the objective O forms a magnified image of the exit facet of the crystal on the CCD detector (3.3 × 4.4mm, 576 × 768 pixels) of a Sony TV camera, that is connected to a PC to record and process the images. The rotating polarizer P2, can be oriented to transmit the components along the x-axis, or y-axis, or in the azimuth 45o or 135o, so that it allows us to measure the intensities necessary to compute the Stokes parameters s 0, s 1 and s 2. An additional λ/4 plate R, positioned before the polarizer P2, allows us to measure the intensities related to s 3; finally the polarizer P3 is used to attenuate the intensity on the CCD detector.
We report in Fig. 3 the experimental and the numerical results for the case of a Gaussian beam with input spot-size w 0 = 10μm, propagated for a distance z = 20zRo ; analogously, Fig. 4 shows the Stokes parameter for the case w 0 = 15μm and z = 8.5zRo . It is evident that the agreement between theory and measurements is pretty good. To have a deeper insight of the results achieved, in Fig. 5 we contrast the numerical and experimental intensity profiles of the Stokes parameters evaluated at the x = y plane (for s 0, s 1 and s 3) or at the x = 6w 0 plane (for s 2 that vanishes along the x, y bisector).
4 Summary
The combined effects of anisotropy and diffraction on a Gaussian beam propagating along the optical axis of a calcite crystal have been investigated, by means of numerical and experimental studies of the evolution of the Stokes parameters. The agreement between theory and experiment is good. We also show that the propagating beam is almost linearly polarized at the plane z = ne /λ, even though the polarization direction varies with the position vector.
Fig. 3. Numerical (left hand) and experimental (right hand) results for Stokes parameters of a Gaussian beam with input spot size w 0 = 10μm propagated for a distance z = 20zRo along the optical axis of a calcite crystal.
Fig. 5. Comparison between numerical and experimental results for the Stokes parameters of Fig. 3 evaluated for x = y or x = 6w 0. Solid line refers to experimental values and dashed line to numerical values.
Fig. 6. Comparison between numerical and experimental results for the Stokes parameters of Fig. 4 evaluated for x = y or x = 6w 0. Solid line refers to experimental values and dashed line to numerical values.
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