1. Introduction

The Stokes parameters have played a central role in the description and measurement of optical polarization over the last century. The projection of the Stokes parameters on the Poincaré sphere is a useful way to visualize polarization, particularly for those beams that may be described as unconventional polarization states. For example, so-called cylindrical vector beams (e.g. radial or azimuthal polarized beams) [1–3] are fully correlated solutions to the vector paraxial wave equation that maintain a spatially inhomogeneous polarization distribution under propagation and which, when projected on to the Poincaré sphere, span the equator.

Hall [4], Jordan and Hall [5], and Greene and Hall [6] described the paraxial behavior of azimuthally polarized beams. Their work implicitly described the paraxial behavior of radially polarized beams due to the homology in the free space solutions of magnetic and electric polarizations. These beams have also been applied to problems in high numerical aperture focusing and microscopy [1, 7–12], interactions with nanostructures (e.g. plasmonics) [14–16], laser machining [17] to name just a few studies. They have also opened opportunities for new approaches to analytical models of propagation and scattering [18].

Given the interesting and potentially useful properties of azimuthal and radial polarizations, it is interesting to ask the following question: Can we create an optical field that covers the entire Poincaré sphere? Is such coverage conserved under propagation? To answer this question, we present an analytic model based on a superposition of Laguerre-Gauss modes and show that, for certain beam conditions, the coverage is indeed complete and conserved. These beams are referred to here as full Poincaré (FP) beams. We then show how an approximation to these beams can be created experimentally by exploiting the stress birefringence distribution present in a symmetrically stressed optical element. Throughout this work, we will assume monochromatic, fully polarized beams. In a follow-up article, we will consider the case of partially coherent, partially polarized FP beams, which cover not only the complete surface but also the interior of the Poincaré sphere.

2. A simple family of full Poincaré beams

FP beams can be generated through combinations of the elements of several different bases. In this work, we employ Laguerre-Gauss (LG) beams, particularly the two lowest order members of this basis. For simplicity, we choose the origin to be at the focus of these beams, and the z axis to coincide with their main direction of propagation. Since we are working within the paraxial regime, each electromagnetic LG beam can be written as the product of the corresponding scalar LG beam and a constant vector perpendicular to z. The lowest order scalar LG beam is the Gaussian beam:

where ρ = √x² + y², u₀ is the beam’s amplitude at the origin, w₀ is the beam waist, and

where z_R = kw₀²/2 is the Rayleigh range. Similarly, the lowest order LG beam with unit azimuthal angular momentum can be written as

A family of FP beams is defined as

where ê₁ and ê₂ are two arbitrary orthogonal unit polarization vectors with no z components. It is straightforward to see that these beams have axially-symmetric intensity profiles:

where w(z) = w₀|ξ(z)| is the z-dependent beam’s width. Therefore, amongst other things, the parameter γ regulates the intensity profile of the beam, which is invariant under propagation up to a global scaling. Note that, for γ = (2m + 1)π/4, the beam profile is flat (i.e. quartic) on axis.

The fact that the polarizations at each point of any transverse plane cover the surface of the Poincaré sphere can be seen from writing Eq. (4) as

where ϕ_ξ(z) = arg[ξ(z)]. The prefactor in parentheses determines the polarization at a given point. For any z, the dimensionless parameter $\widehat{\rho }$ = √2 tan γρ/w regulates the relative amounts of each of the two polarizations, so that near the axis, the polarization tends towards ê₁, while away from it, it tends towards ê₂. Both polarizations are added in equal amounts at rings of radius w/(√2 tan γ). Since the width of the beam is w, it is desirable to choose γ such that tan γ is of the order of unity, so that most polarizations are represented within the region where the intensity of the beam is significant. The azimuthal angle determines the relative phase of the two polarizations. It is easy to see from Eq. (6) that the two effects of the propagation distance z on the polarization distribution are a global scaling according to w(z) and a rotation according to ϕ_ξ(z), so that from the waist plane to the far zone, the distribution of polarizations experiences a rotation of π/2.

As an example, consider the case when the two polarizations are circular, e.g. ê_1,2 = (x̂ ± iŷ)/√2. It is easy to show that the normalized Stokes parameters are then given by

That is, in this case, S₃ is independent of ϕ. Further, we know that S₃/S₀ equals the sine of χ, the angle from the equator in the Poincaré sphere. Let us define χ′ = π/2 − χ as the angle from the S₃ axis. It is then easy to show from Eq. (7) that $\overline{\rho }$ = tan(χ′/2). Further, the fact that S₁/S₀ and S₂/S₀ at z = 0 are proportional to cos ϕ and −sin ϕ, respectively, implies that ψ, the azimuthal angle in the Poincaré sphere, equals −ϕ. Therefore, in this case, the polarization distribution over the transverse plane at z = 0 is just a stereographic projection of the Poincaré sphere from the south (left-hand circular polarization) pole. At any other z, this projection just rotates and expands. Even when ê_1,2 do not correspond to circular polarizations but to some other pair of orthonormal polarizations, the beam’s polarization distribution at any transverse plane covers the whole surface of the Poincaré sphere according to a stereographic projection, this time from the point in the sphere corresponding to ê₂.

Note that this rigid (up to a radial scaling) rotation of the Stokes parameter distribution upon propagation does not imply (necessarily) a rigid rotation of the beam’s electric field distribution, as the polarization vectors ê_1,2 are themselves not rotating. There are, however, special cases for which the electric field pattern itself does rotate rigidly. These cases happen precisely when ê_1,2 correspond to circular polarizations. The animation in Fig. 1 shows the evolution of the polarization patterns when a) ê₁ is right-hand circular (RHC) and ê₂ is left-hand circular (LHC), b) ê₁ is LHC and ê₂ is RHC, and c) ê₁ is vertical and ê₂ is horizontal. In these movies, a green (red) ellipsoid indicates right-(left-)handedness, while a blue line indicates linear polarization. Notice that in a), the scaled pattern does rotate rigidly by π/2 between the waist plane and far field, but in the opposite sense as the Stokes parameter distribution. Similarly, in b) the pattern rotates rigidly and in the same sense as the Stokes parameter distribution, but only by π/6. The evolution in c), on the other hand, does not correspond to a rigid rotation of the electric field pattern.

3. Experimental Beam Generation

A symmetrically stressed optical window–hereafter referred to as a stress–engineered optical (SEO) element-illuminated by a circularly polarized beam forms a good approximation to the LG beams described in the previous section. The use of these elements in creating both scalar phase vortices and cylindrical vector beams have been described in detail elsewhere [19, 20]. An optical window (diameter 12.5 mm, thickness 8 mm, BK7 glass) is stressed using a thermal compression procedure in which an outer metal ring is fashioned with a hole about 25 μm smaller than the outer diameter of the window. Material is removed at 120° positions to create three contact regions. The high thermal expansion coefficient of the metal ring allows the insertion of the glass window at about 300°C; after cooling, the SEO shows a stress distribution of trigonal symmetry that, near the center of the window, follows a power law model in which the birefringence increases as a linear function of radius and the direction of the fast axis precesses with the symmetry of the stress. The space-variant Jones matrix then has the following form:

in which ρ = (ρ,ϕ) is the window coordinate, m denotes the order of the stress (in our case, m = 3) and ℙ is the pseudorotation matrix. Also, c is a constant proportional to the external applied force; for m = 3, it is simply the rate of change of the phase retardance at the center of the SEO. When illuminated with circularly polarized light, the first term describes an apodization of the incident polarization while the second gives a complementary apodization of the orthogonal polarization. It follows that, if g(ρ) represents an input apodization, the transmitted field can be represented as

 

Fig. 1. (Media 1) Evolution of the distribution of polarization ellipses for FP beams where ê₁ and ê₂ are, respectively, (a) RHC and LHC, (b) LHC and RHC, and (c) vertical and horizontal. In all movies, the radial distance is scaled by w(z) and the angle between the x axis and the purple line represents ϕ_ξ(z).

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When illuminated with a Gaussian beam, the transmitted polarization distribution then consists of the superposition of a central, circularly polarized lobe and a annular beam of the opposite circular polarization. [19–21]. The annular beam has a phase vortex and therefore approximates U₀₁ of the theoretical construct, as is shown in the experimental results that follow.

 

Fig. 2. Experimental arrangement, in which the SEO element is placed in front of a 400 mm focal length lens (L). The illumination source polarization is controlled through a combination linear polarizer/quarter wave plate (P), and passed through an analyzer (A) placed in front of the camera.

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The experimental arrangement was as shown in Fig. 2. A diode pumped, frequency doubled Nd:YAG laser (wavelength 532 nm) was spatially filtered, collimated, and aligned with the center of an SEO element. The diameter of the entering beam was adjusted so as to partially illuminate the first half-wave ring. The window was placed in the Fourier transform plane (front focal plane) of a 400 mm focal length lens, and the focal region was characterized directly with a CCD camera.

Figure 3 shows a representative cross section of the irradiance profiles of the RHC and LHC components at the beam waist. To provide good quantitative comparison with the theory, the beam waist of the Gaussian portion was first obtained using a single-parameter least-squared fit to the data. The relative amplitude of the U₀₁ beam was then deduced using a similar optimization. Finally, a separate least-squares fit was carried out in order to estimate the difference in beam waist parameters between the U₀₀ and U₀₁ portions. For the example shown, the U₀₀ waist was found to be 0.181±0.002 mm, and the U₀₁ waist 0.165±0.002 mm. The two are therefore sufficiently close to be able to compare with the salient features of the theory; indeed, the two main experimental difficulties were: (1) some rotational asymmetry in the U₀₁ beam, something that is seen in the asymmetry of the plot and in the experimental profiles, and; (2) asymmetry in the irradiance about the beam waist, indicating perhaps some residual spherical aberration introduced by the window.

 

Fig. 3. Cross section of the irradiance profile near the beam waist. The total irradiance is shown by the (black) solid line. Dots (red) denote the left hand circular component; Crosses (blue) denote the right hand circular component. The dashed lines represent the best fit Gaussian and Laguerre-Gauss functions, as well as their sum. The beam size of the Gaussian was measured to 0.181±0.002 mm.

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The beam coverage over the Poincare sphere may be verified by the usual measurement of the Stokes parameters. In our case, irradiance images were taken using an Imaging Source 480 × 640 format CCD sensor (pixel spacing of 5.6 μm) controlled by IC Capture™, in which we could control for the gamma of the camera and assure that no pixels were saturated. Images were taken for the usual linear states: vertical, horizontal, + 45°, − 45°; circular measurements were taken with the usual sequence of a quarter wave plate followed by an analyzer. The quality of components was verified using a commercial polarimeter (ThorLabs).

Figure 4 provides a side by side comparison of the normalized Stokes paramaters predicted by the theory and those measured in our experiment. Despite the fact that our stressed window produces only an approximation of the idealized LG superposition, two key properties in the ideal beam can be seen in the experiment. The first is the rotation of S1 and S2 with defocus, in which the rotation is π/4 through the Rayleigh range. The second is that the beam maintains full coverage of the Poincaré sphere from the beam waist to the edge of the Rayleigh range. Our translation equipment and detector size made quantitative observation outside the Rayleigh range difficult; qualitatively, we were able to observe the rigid rotation predicted by the theory. There is some degradation of the beam approaching the edge of the Rayleigh range; we suspect that this is due to residual asymmetries in the illumination, evident both in the Stokes maps and in the irradiance profile shown in Fig. 3.

To quantitatively test the rate of rotation against that predicted by the theory, Stokes maps were acquired over a range of 44 beam positions on either side of the Gaussian beam waist. By selecting a region within 20 pixels of the beam center of the image, and searching out points satisfying |s₁| < 0.05, we were able to carry out a linear regression and compute the slope of the contour of zero s₁ near the beam center. If the phase reference (controlled by the window orientation) is set so that the slope is zero at z = 0, then the measured slope will simply be tan(ϕ_ξ) = z/z_R. The measured data is shown in Fig. 5; the dashed line is a linear least-squared fit to the center region whose inverse slope is z_R = 165 mm, close to the Rayleigh range predicted by the Gaussian beam fit.

 

Fig. 4. Normalized Stokes maps of the beam sampled at four longitudinal positions within the Rayleigh range. Left: Simulation, based on an ideal superposition with the beam sampled at (a) z = 0; (b) z = 0.3z_R; (c) z = 0.67z_R; (d) z = z_R; Right: Experimental Stokes maps for the beam of Fig. 3 (a) z = 0; (b) z = 60 mm (0.36z_R); (c) z = 110 mm (0.67z_R); (d) z = 170mm (1.03z_R).

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Fig. 5. The measured value of tan(ϕ_ξ) deduced from the zeros of s₁ plotted as a function of propagation distance z. The linear range is the region over which the experimental beam matches the evolution predicted by LG beam propagation. The dashed line is a linear fit to the central region.

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4. Discussion

It should be emphasized that the stereographic projection of the Poincaré sphere described here is onto a real, physical plane, and should therefore not be confused with that described by Sauter [23], which describes the mapping of a different Poincaré sphere onto a complex plane describing the focal point and width of a Gaussian beam.

The proof of principle experiments described here are intended to show how one may reasonably approximate an ideal Poincaré beam using a stress engineered optical element. With refinement, we believe it will be possible to explore a wider variety of beams−for example, the modification of the experiment described here by following the window with a quarter wave plate would allow the creation of beams for which ê_1,2 is a linear basis.

The FP beams discussed so far are a superposition of confocal and coaxial LG modes U₀₀ and U₀₁ with the same width parameter w₀. However, there are many other possible choices of fields that would lead to FP beams. For example, we could have chosen U₀₀ and U₀₁ to have different w₀, or to be axially displaced with respect to each other. These changes would not have altered the fact that all polarizations are represented in (at least) some transverse planes. However, the polarization distribution would not correspond to a stereographic projection of the Poincaré sphere, the intensity profile would no longer just scale upon propagation, and the rotation of the distribution of the Stokes parameters upon propagation might be at different angular rates depending on the radius. Further, the coverage of the Poincaré sphere would be incomplete at planes of constant z where U₀₀ becomes wider than U₀₁.

Similarly, different coaxial combinations of LG beams could be used, as long as they have different azimuthal angular momenta and different radial dependences. The stereographic projection property would not hold in general, however. Further, when the angular momenta of the two modes differ by an integer N greater than unity, each point in the Poincaré sphere would be represented N times, at equally spaced angles, at any transverse plane of the field. A similar repetition, although in the radial direction, would result if the fields have multiple radial zeroes, i.e. if we use modes U_nm for n ≠ 0. In general, the combination of any two paraxial beams which tend to zero at different places, whose relative phase varies by at least a full cycle, and whose relative amplitude and phase variations change in different directions, can be used to construct FP beams.

The FP beams presented here rely on the use of the paraxial approximation. It turns out that they can also be written as exact solutions of Maxwell’s equations through the use of recently proposed nonparaxial generalizations of the LG basis [22], defined in terms of multipolar fields whose origin is shifted to an imaginary position. In particular, the FP beam whose axial polarization is circular can be constructed in terms of electric and magnetic dipolar fields centered at imaginary positions. The electric field distributions for electric and magnetic dipolar fields can be written as:

where p is a vector indicating the direction of the dipoles. The FP beam can be written as

where the imaginary displacement of magnitude q of the dipoles’ origin causes them to be directional. For kq ≫ 1, this expression reduces to Eq. (4) with w₀ = √2q/k and circular polarization vectors. This construction is useful for studying the scattering of FP beams off spherical obstacles (either on-axis or off-axis), through the use of a recent closed-form generalization of the Lorenz-Mie scattering theory [18].

The experiments described here make use of a birefringence function in a stressed window that, while a good approximation to the LG superposition described here, contains residual modes that are higher order radial functions. The phase irregularity evident in the Stokes maps (S1 and S2) is likely due to interference with these weak higher order components. However, the rate of rotation of the Stokes distribution is insensitive to these details, as is evident from the comparison of the experiment with the idealized beam.

FP beams could be of great interest for investigations of small particle scattering. For particles that are smaller than the wavelength, scattering can be accurately described by Rayleigh’s theory, where the scattered field’s polarization is simply related to that of the incident field at the particle’s position. Therefore, if the incident field is a FP beam, the measurement of the state of polarization of the scattered field would give accurate information about the transverse location of the particle within the beam.

An intriguing variation of this concept is in polarization engineering of few-mode fibers, in which a lowest order Gaussian-like mode is superimposed with a low order doughnut mode. In this case, the rotation ϕ_ξ would be linear in z and would rotate according to the intermodal dispersion; a graded index fiber would rotate very slowly while a step index fiber would have a rapid rotation due to the higher modal dispersion. Still another interesting variation is the superposition of two such modes having a small frequency difference, resulting in a rotation about the poles with an angular velocity corresponding to the frequency difference of the modes.

As mentioned in the introduction, this work considers only fully polarized monochromatic beams. In a follow-up article, we will consider the theory and experimental implementation of partially polarized beams which cover surfaces such as spheroids or disks inside the Poincaré sphere, and which can even span the complete interior of this sphere upon propagation. If such a beam were to be used in the particle localization application mentioned earlier, the state of polarization of the field scattered by the particle would give information about both the particle’s transverse and longitudinal locations.