Full Poincaré beams II: partial polarization

doi:10.1364/OE.20.009357

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Full Poincaré beams II: partial polarization

Amber M. Beckley, Thomas G. Brown, and Miguel A. Alonso »View Author Affiliations

Optics Express, Vol. 20, Issue 9, pp. 9357-9362 (2012)
http://dx.doi.org/10.1364/OE.20.009357

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Abstract

Optical fields whose coherence and/or polarization properties appear to change under propagation have intrigued researchers for many years. We describe and experimentally demonstrate a class of optical fields whose polarization content at any transverse plane spans a disk-like region within the Poincaré sphere. When examined through a paraxial focal region, the disk rotates under propagation, spanning all possible states of polarization. We map the change in Stokes parameters through focus for each case, comparing experiment with the theoretical predictions.

1. Introduction

In a recent publication [1

A. M. Beckley, T. G. Brown, and M. A. Alonso, “Full Poincaré beams,” Opt. Express 18, 10777–10785 (2010). [CrossRef] [PubMed]

], we introduced a family of beams constructed from a coaxial superposition of a fundamental Gaussian mode and a spiral-phase Laguerre-Gauss mode having orthogonal polarizations. The polarization on the beam maps onto the complete surface of the Poincaré sphere according to a stereographic mapping, and propagation corresponds to a rigid rotation of the sphere. For this reason, we referred to these beams as full Poincaré (FP) beams. Similar beams have also recently been reported, with emphasis on tailoring the Poincaré sphere coverage [2

G. M. Lerman, L. Stern, and U. Levy, “Generation and tight focusing of hybridly polarized vector beams,” Opt. Express 18, 27650–27657 (2010). [CrossRef]

, 3

H. Chen, J. Hao, B-F Zhang, J. Xu, J. Ding, and H-T Wang, “Generation of vector beam with space-variant distribution of both polarization and phase,” Opt. Lett. 36, 3179–3181 (2011) [CrossRef]

], using them to modify a focal region [4

W. Han, W. Cheng, and Q. Zhan, “Flattop focusing with full Poincaré beams under low numerical aperture illumination,” Opt. Lett. 36, 1605–1607 (2011). [CrossRef] [PubMed]

], or analyzing the geometric phase intrinsic to their phase and polarization structure [5

J. C. Gutiérrez-Vega, “Pancharatnam-Berry phase of optical systems,” Opt. Lett. 36, 1143–1145 (2011). [CrossRef] [PubMed]

]. The angular momentum of this type of beam has also been studied, with an emphasis on particle trapping [6

I. Mokhun, R. Brandel, and Ju. Viktorovskaya, “Angular momentum of electromagnetic field in areas of polarization singularities,” Ukr. J. Phys. Opt. 7, 63–73 (2006). [CrossRef]

, 7

I. Mokhun and R. Khrobatin, “Shift of application point of angular momentum in the area of elementary polarization singularity,” J. Opt. A: Pure and Appl. Opt. 10, 1–10 (2008). [CrossRef]

In this paper, we present beams that explore the interior of the Poincaré sphere by examining the superposition of uncorrelated full Poincaré beams having different angular momentum. One such superposition spans all possible polarization states during propagation by sweeping out the full volume of the sphere. It has an axisymmetric irradiance profile that is preserved under propagation and is built on the analytical formulation of the original full Poincaré beam. The beams can be created by a space-variant birefringence present in a stress engineered optical element.

2. Fully correlated full Poincaré beams

As was the case in Ref. [1

A. M. Beckley, T. G. Brown, and M. A. Alonso, “Full Poincaré beams,” Opt. Express 18, 10777–10785 (2010). [CrossRef] [PubMed]

], we employ Laguerre-Gauss (LG) beams, particularly the two lowest order members of this basis. However, other types of beams can also be used. 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:

U_{00} (r) = \frac{u_{0}}{ξ (z)} exp [i k z - \frac{ρ^{2}}{w_{0}^{2} ξ (z)}],

(1)

where

ρ = \sqrt{x^{2} + y^{2}}

, u₀ is the beam’s amplitude at the origin, w₀ is the beam waist, and

ξ (z) = 1 + i \frac{2 z}{k w_{0}^{2}} = 1 + i \frac{z}{z_{R}},

(2)

with

z_{R} = k w_{0}^{2} / 2

being the Rayleigh range. Similarly, the lowest order LG beams with unit azimuthal angular momentum can be written as

U_{0 \pm 1} (r) = \sqrt{2} \frac{x \pm i y}{w_{0} ξ (z)} U_{00} (r) = 2 u_{0} \frac{ρ}{w (z)} exp {i [\pm ϕ - ϕ_{ξ} (z)]} exp [i k z - \frac{ρ^{2}}{w_{0}^{2} ξ (z)}],

(3)

where ϕ = arctan(x,y) is the azimuthal coordinate, w(z) = w₀|ξ(z)| is the z-dependent beam’s width, and ϕξ(z) = arg[ξ(z)] is the Gouy phase.

The family of FP beams defined in Ref. [1

A. M. Beckley, T. G. Brown, and M. A. Alonso, “Full Poincaré beams,” Opt. Express 18, 10777–10785 (2010). [CrossRef] [PubMed]

] is a linear combination of a Gaussian beam and a LG beam. Here we will give a slightly more general definition:

\begin{array}{l} E_{P}^{\pm} (r; γ, ϕ_{0}, {\hat{e}}_{1}, {\hat{e}}_{2}) & = & cos γ {\hat{e}}_{1} U_{00} (r) + exp (i ϕ_{0}) sin γ {\hat{e}}_{2} U_{0 \pm 1} (r) \\ = & ({\hat{e}}_{1} + exp {i [\pm ϕ + ϕ_{0} - ϕ_{ξ} (z)]} \bar{ρ} {\hat{e}}_{2}) cos γ U_{00} (r), \end{array}

(4)

where ê₁ and ê₂ are two arbitrary orthogonal unit polarization vectors (

{\hat{e}}_{1}^{*} \cdot {\hat{e}}_{2} = 0

) with no z components, γ is a parameter that regulates the relative amounts of each mode, ϕ₀ is a constant phase, and

\bar{ρ} = \sqrt{2} tan γ ρ / w (z)

. The intensity profile of these beams is axially symmetric and its shape is invariant under propagation, except for a global scaling:

\begin{array}{l} {| E_{P}^{\pm} (r, γ, ϕ_{0}, {\hat{e}}_{1}, {\hat{e}}_{2}) |}^{2} & = & {cos}^{2} γ {| U_{00} (r) |}^{2} + {sin}^{2} γ {| U_{0 \pm 1} (r) |}^{2} \\ = & \frac{u_{0}^{2} {cos}^{2} γ}{{| ξ (z) |}^{2}} (1 + {\bar{ρ}}^{2}) exp (- \frac{{\bar{ρ}}^{2}}{{tan}^{2} γ}) . \end{array}

(5)

Note that the parameter γ regulates the shape of intensity profile of the beam. As discussed in Ref. [1

A. M. Beckley, T. G. Brown, and M. A. Alonso, “Full Poincaré beams,” Opt. Express 18, 10777–10785 (2010). [CrossRef] [PubMed]

], at each plane of constant z, the polarization distribution corresponds to a stereographic projection of the Poincaré sphere, and the only effects of propagation in z on this mapping are a scaling (equal to that of the intensity profile) and a rigid rotation. Let us, from now on, consider the case when the two polarizations are given by

{\hat{e}}_{1, 2} = (\hat{x} \mp i \hat{y}) / \sqrt{2}

(right- and left-circular, respectively). It is easy to show that the normalized Stokes parameters are then given by

\frac{S_{1}}{S_{0}} = \frac{2 \bar{ρ} cos [ϕ \pm (ϕ_{0} - ϕ_{ξ})]}{1 + {\bar{ρ}}^{2}}, \frac{S_{2}}{S_{0}} = \mp \frac{2 \bar{ρ} sin [ϕ \pm (ϕ_{0} - ϕ_{ξ})]}{1 + {\bar{ρ}}^{2}}, \frac{S_{3}}{S_{0}} = \frac{1 - {\bar{ρ}}^{2}}{1 + {\bar{ρ}}^{2}},

(6)

The rigid rotation of the polarization distribution results from the fact that the Gouy phase ϕξ(z) = arctan(z/z_R) varies from −π/2 to π/2 as z varies from −∞ (i.e. many Rayleigh ranges before the focal plane) to ∞ (i.e. many Rayleigh ranges after the focal plane).

3. Beams that span all states of polarization

It is possible to generate beams that span all the interior of the Poincaré sphere (i.e. all possible states of polarization) by simply superimposing incoherently (in equal amounts) two fully polarized Poincaré beams with different vorticities:

E_{FP} (r; γ, ϕ_{1}, ϕ_{2}, {\hat{e}}_{1}, {\hat{e}}_{2}) = a_{1} E_{P}^{+} (r; γ, ϕ_{1}, {\hat{e}}_{1}, {\hat{e}}_{2}) + a_{2} E_{P}^{-} (r; γ, ϕ_{2}, {\hat{e}}_{1}, {\hat{e}}_{2}),

(7)

where a₁ and a₂ are two uncorrelated stochastic variables satisfying 〈|a₁|²〉 = 〈|a₂|²〉, and ϕ₁ and ϕ₂ are two constant phases. This type of beam preserves the axially symmetric intensity profile of each of its components, given in Eq. (5). For the case

{\hat{e}}_{1, 2} = (\hat{x} \mp i \hat{y}) / \sqrt{2}

it is easy to show that its normalized Stokes parameters are given by the average of the two options for each parameter in Eqs. (6):

\frac{S_{1}}{S_{0}} = \frac{2 \bar{ρ} cos (ϕ - ϕ^{'}) cos (ϕ_{ξ} - \bar{ϕ})}{1 + {\bar{ρ}}^{2}}, \frac{S_{2}}{S_{0}} = \frac{2 \bar{ρ} cos (ϕ - ϕ^{'}) sin (ϕ_{ξ} - \bar{ϕ})}{1 + {\bar{ρ}}^{2}}, \frac{S_{3}}{S_{0}} = \frac{1 - {\bar{ρ}}^{2}}{1 + {\bar{ρ}}^{2}},

(8)

where ϕ̄ = (ϕ₁ +ϕ₂)/2 and ϕ′ = (ϕ₂ −ϕ₁)/2. That is, at any plane of constant z, the polarization distribution of the beam spans a unit disk containing the S₃ axis. The angle of this disk with respect to the S₁ axis is ϕξ − ϕ̄, which varies from −π/2 − ϕ̄ to π/2 − ϕ̄ as the beam propagates from z = −∞ to z = ∞. This is illustrated in the movie in Fig. 1 where ϕ̄ = −π/4 and z varies from −z_R to z_R; within this range, the disk sweeps an angle of π/2. (we chose this range for ease of matching to the experiments—extending the range from z = −∞ to z = ∞ sweeps out the entire sphere.) Notice, however, that each state of polarization corresponding to a point inside the Poincaré sphere actually corresponds to two physical spatial locations, at points that are mirror images about a plane containing the beam’s axis and at an angle ϕ′ with respect to the x axis. Therefore, this beam provides a complete mapping of the interior of the Poincaré sphere to each of two regions of the physical 3D space separated by this plane.

Fig. 1 The theoretical (left ( Media 1)) and experimental (right ( Media 2)) Poincaré sphere coverage of beams that span the sphere. Clicking on each figure will play a movie that shows the through-focus rotation of each disk about the S3 axis. In the example shown, the Rayleigh range is about 90 mm

In the experiment, we create the beam using a stress-engineered optical element (SEOE) illuminated with circularly polarized light to create the necessary LG states. These elements introduce a space-variant birefringence that can produce regular full Poincaré beams, as well as cylindrical vector beams and scalar phase vortices [8

A. K. Spilman and T. G. Brown, “Stress birefringent, space-variant wave plates for vortex illumination,” Appl. Opt. 26, 61–66 (2007). [CrossRef]

, 9

A. K. Spilman, A. M. Beckley, and T. G. Brown, “Focal splitting and optical vortex structure induced by stress birefringence” Proc. SPIE 6667, 666701 (2007).

]. For a description of how these elements are produced, see Ref.[1

A. M. Beckley, T. G. Brown, and M. A. Alonso, “Full Poincaré beams,” Opt. Express 18, 10777–10785 (2010). [CrossRef] [PubMed]

The space-dependent Jones matrix of a stress-engineered optical element under m-point stress is presented in [8

A. K. Spilman and T. G. Brown, “Stress birefringent, space-variant wave plates for vortex illumination,” Appl. Opt. 26, 61–66 (2007). [CrossRef]

, 10

A. K. Spilman and T. G. Brown, “Stress-induced focal splitting,” Opt. Express 15, 8411–8421 (2007). [CrossRef] [PubMed]

]. We use an SEOE under three-point stress which, if we calculate for m = 3, exhibits a Jones matrix approximately of the form

\underset{=}{J} (ρ) = cos (\frac{c ρ}{2}) \underset{=}{I} + i sin (\frac{c ρ}{2}) [\begin{matrix} cos (ϕ) & - sin (ϕ) \\ - sin (ϕ) & - cos (ϕ) \end{matrix}]

in which ρ = (ρ,ϕ) is the window polar coordinate, I̳ is the identity matrix and c is the linear rate of change of the phase retardance at the center of the three-fold symmetric SEOE.

When illuminated with a beam with right-hand circular polarization (ê₁) and appropriate input beam apodization g(ρ) (e.g. a Gaussian beam of width ≈ 1.5/c or less), the transmitted field is a reasonable approximation to

E_{P}^{+}

defined in Eq. (4) and included in the first term of Eq. (7),

E_{t} (ρ) = g (ρ) [cos (\frac{c ρ}{2}) {\hat{e}}_{1} + i e^{i ϕ} sin (\frac{c ρ}{2}) {\hat{e}}_{2}] \approx E_{P}^{+} (r; γ, π / 2, {\hat{e}}_{1}, {\hat{e}}_{2}) .

(9)

To create the second term in Eq. (7),

E_{P}^{-}

, a half-wave of phase retardance is introduced before and after the SEOE. The first half-wave of retardance changes the handedness of the input polarization, creating a phase vortex of opposite sign and exchanges ê₁ and ê₂. The second half-wave returns the polarizations, but leaves the phase vortex sign unchanged, giving a transmitted field:

E_{t} (ρ) = g (ρ) [cos (\frac{c ρ}{2}) {\hat{e}}_{1} + i e^{- i(ϕ + 4 ϕ_{H})} sin (\frac{c ρ}{2}) {\hat{e}}_{2}] \approx E_{P}^{-} (r; γ, π / 2 - 4 ϕ_{H}, {\hat{e}}_{1}, {\hat{e}}_{2}),

(10)

where ϕ_H is the angle between the x axis (chosen to coincide with the direction of a stress point or the SEOE) and the fast axis of the two half-wave plates.

In an experimental procedure similar to that presented in Ref. [1

A. M. Beckley, T. G. Brown, and M. A. Alonso, “Full Poincaré beams,” Opt. Express 18, 10777–10785 (2010). [CrossRef] [PubMed]

], images were taken of the horizontal, vertical, 45°, 135° and right/left circular components of the resulting beams using an analyzer (quarter-wave plate and linear polarizer) and CCD. Difference images normalized to the total irradiance then yield the Stokes parameters. An Imaging Source 480 x 640 format CCD sensor with 5.6μm pixel spacing was used, which was operated using the IC Capture^TM software to control for gamma and pixel saturation. A schematic of the experimental setup is shown in Fig. 2. A diode pumped, frequency doubled Nd:YAG laser (λ = 532nm) is spatially-filtered, collimated and right-hand circularly polarized using a linear polarizer and quarter-wave plate. The beam is magnified to size match the SEOE, so that part of the first half wave ring of retardance is illuminated. The SEOE is located in the front focal plane (the Fourier transform plane) of a long focal length lens (f = 300mm). This creates the standard full Poincaré beam with a beam waist at the focal plane. The Fourier transformation changes ϕ₀ from π/2 to 0 at the focal plane. The incoherent superposition of the two beams can be achieved in a number of ways. Here we chose time multiplexing realized by placing liquid crystal retarders (VAN type Boulder Vision Optik) before and after the stressed element and summing sequential exposures of the on and off states of the liquid crystals. This half-wave of retardance switches the handedness of the polarization, changing the state from

E_{P}^{+}

E_{P}^{-}

. This is equivalent to the incoherent sum of these two beams.

Fig. 2 The experimental setup for partially polarized FP beam combinations. Illumination laser light is filtered and collimated, polarized (P), phase apodized by the stress-engineered optical element (SEOE), focused by a slow lens (f = 300mm), and analyzed (A) then measured on a CCD. For beams that span the sphere, two half-wave liquid crystal retarders (LC-1,2) convert the beam and vortex states. For beams of a changing degree of polarization, LC-2 is removed.

Side-by-side maps of the measured and theoretical Stokes parameters are displayed in Fig. 3. The z locations sampled were chosen for ease of measurement in the experimental setup, and the simulation images were created to match experimental conditions in the value of γ, Rayleigh range, beam size and phase variables. We also display the Poincaré sphere coverage of the experimental Stokes parameters in Fig. 1–each figure is linked to a movie showing the through focus rotation of the disk. Points of low irradiance on the image produce nonphysical (DoP > 1) results—we have color-coded those points in yellow in order to permit easy viewing of the disk inside the sphere. For the orientation of the wave retarders used in this experimental realization, ϕ̄ = −π/4.

Fig. 3 The experimental (left) and simulated (right) realizations of beams that span the sphere at (a) −1.1z_R, (b) the focal plane and (c) 1.1z_R. The simulation is matched to the beam size, value of γ, Rayleigh range and ϕ̄ = ϕ′ = −π/4.

4. Other constructions

This type of incoherent superposition of beams can be used to achieve other interesting polarization distributions. For example, consider a similar construction, but where the basic polarizations of the second FP beam are exchanged:

E_{var} (r; γ, ϕ_{1}, ϕ_{2}, {\hat{e}}_{1}, {\hat{e}}_{2}) = a_{1} E_{P}^{+} (r; γ, ϕ_{1}, {\hat{e}}_{1}, {\hat{e}}_{2}) + a_{2} E_{P}^{-} (r; γ, ϕ_{2}, {\hat{e}}_{2}, {\hat{e}}_{1}),

(11)

where, again, a₁ and a₂ are uncorrelated stochastic variables with equal second moments. For the same choice of circular polarizations as before, the normalized Stokes parameters become

\frac{S_{1}}{S_{0}} = \frac{2 \bar{ρ} cos (ϕ + ϕ^{'}) cos (ϕ_{ξ} - \bar{ϕ})}{1 + {\bar{ρ}}^{2}}, \frac{S_{2}}{S_{0}} = - \frac{2 \bar{ρ} sin (ϕ + ϕ^{'}) cos (ϕ_{ξ} - ϕ^{'})}{1 + {\bar{ρ}}^{2}}, \frac{S_{3}}{S_{0}} = 0,

(12)

That is, at any plane of constant z, the polarizations in this beam span a disk over the S₁–S₂ plane, of radius |cos(ϕξ − ϕ̄)| and centered at the origin. Therefore, for ϕ̄ = π/2, the beam at its waist is fully unpolarized but it gains polarization off-axis upon propagation until it becomes fully polarized in the far field at the ρ̄ = 1 cone. On the other hand, for ϕ̄ = 0, the beam achieves maximum degrees of polarization at the waist plane and it tends toward a fully unpolarized beam upon propagation.

5. Concluding remarks

Optical fields whose coherence and/or polarization properties appear to change under propagation have intrigued researchers for many years. As sources of unconventionally polarized illumination, we believe they are likely to produce a variety of exotic effects in scattering, trapping, and quantum optics [11

T. G. Brown and Q. Zhan, “Introduction: Unconventional Polarization States of Light Focus Issue,” Opt. Express 18, 10775–10776 (2010). [CrossRef] [PubMed]

]. For example, If such a beam were to illuminate small scatterers, the scattered light from particles of various locations within the beam could possibly give information about their location within the beam.

The cases discussed here are intended to be instructive examples of what could represent a larger category of partially coherent, partially polarized vector fields whose statistical behavior appears to change under propagation and whose description differs from the way polarization is generally used in optical systems. Fields such as an FP beam are especially interesting because they show the irradiance profile of an ordinary beam while exhibiting an evolution in the polarization that maps onto a rotation of the Poincar’e sphere relative to the beam coordinates. For the beams described in this paper, the sphere collapses to a disk that either rotates or expands/contracts with propagation through focus.

Acknowledgments

The work was supported in part by the National Science Foundation ( PHY-1068325) and Rochester Precision Optics. Opinions set forth in this paper are those of the authors and do not necessarily reflect the views of the National Science Foundation.

References and links

1.	A. M. Beckley, T. G. Brown, and M. A. Alonso, “Full Poincaré beams,” Opt. Express 18, 10777–10785 (2010). [CrossRef] [PubMed]
2.	G. M. Lerman, L. Stern, and U. Levy, “Generation and tight focusing of hybridly polarized vector beams,” Opt. Express 18, 27650–27657 (2010). [CrossRef]
3.	H. Chen, J. Hao, B-F Zhang, J. Xu, J. Ding, and H-T Wang, “Generation of vector beam with space-variant distribution of both polarization and phase,” Opt. Lett. 36, 3179–3181 (2011) [CrossRef]
4.	W. Han, W. Cheng, and Q. Zhan, “Flattop focusing with full Poincaré beams under low numerical aperture illumination,” Opt. Lett. 36, 1605–1607 (2011). [CrossRef] [PubMed]
5.	J. C. Gutiérrez-Vega, “Pancharatnam-Berry phase of optical systems,” Opt. Lett. 36, 1143–1145 (2011). [CrossRef] [PubMed]
6.	I. Mokhun, R. Brandel, and Ju. Viktorovskaya, “Angular momentum of electromagnetic field in areas of polarization singularities,” Ukr. J. Phys. Opt. 7, 63–73 (2006). [CrossRef]
7.	I. Mokhun and R. Khrobatin, “Shift of application point of angular momentum in the area of elementary polarization singularity,” J. Opt. A: Pure and Appl. Opt. 10, 1–10 (2008). [CrossRef]
8.	A. K. Spilman and T. G. Brown, “Stress birefringent, space-variant wave plates for vortex illumination,” Appl. Opt. 26, 61–66 (2007). [CrossRef]
9.	A. K. Spilman, A. M. Beckley, and T. G. Brown, “Focal splitting and optical vortex structure induced by stress birefringence” Proc. SPIE 6667, 666701 (2007).
10.	A. K. Spilman and T. G. Brown, “Stress-induced focal splitting,” Opt. Express 15, 8411–8421 (2007). [CrossRef] [PubMed]
11.	T. G. Brown and Q. Zhan, “Introduction: Unconventional Polarization States of Light Focus Issue,” Opt. Express 18, 10775–10776 (2010). [CrossRef] [PubMed]

OCIS Codes
(260.1440) Physical optics : Birefringence
(260.5430) Physical optics : Polarization

ToC Category:
Coherence and Statistical Optics

History
Original Manuscript: January 3, 2012
Revised Manuscript: February 6, 2012
Manuscript Accepted: February 8, 2012
Published: April 9, 2012

Citation
Amber M. Beckley, Thomas G. Brown, and Miguel A. Alonso, "Full Poincaré beams II: partial polarization," Opt. Express 20, 9357-9362 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-9-9357

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References

A. M. Beckley, T. G. Brown, and M. A. Alonso, “Full Poincaré beams,” Opt. Express18, 10777–10785 (2010). [CrossRef] [PubMed]
G. M. Lerman, L. Stern, and U. Levy, “Generation and tight focusing of hybridly polarized vector beams,” Opt. Express18, 27650–27657 (2010). [CrossRef]
H. Chen, J. Hao, B-F Zhang, J. Xu, J. Ding, and H-T Wang, “Generation of vector beam with space-variant distribution of both polarization and phase,” Opt. Lett.36, 3179–3181 (2011) [CrossRef]
W. Han, W. Cheng, and Q. Zhan, “Flattop focusing with full Poincaré beams under low numerical aperture illumination,” Opt. Lett.36, 1605–1607 (2011). [CrossRef] [PubMed]
J. C. Gutiérrez-Vega, “Pancharatnam-Berry phase of optical systems,” Opt. Lett.36, 1143–1145 (2011). [CrossRef] [PubMed]
I. Mokhun, R. Brandel, and Ju. Viktorovskaya, “Angular momentum of electromagnetic field in areas of polarization singularities,” Ukr. J. Phys. Opt.7, 63–73 (2006). [CrossRef]
I. Mokhun and R. Khrobatin, “Shift of application point of angular momentum in the area of elementary polarization singularity,” J. Opt. A: Pure and Appl. Opt.10, 1–10 (2008). [CrossRef]
A. K. Spilman and T. G. Brown, “Stress birefringent, space-variant wave plates for vortex illumination,” Appl. Opt.26, 61–66 (2007). [CrossRef]
A. K. Spilman, A. M. Beckley, and T. G. Brown, “Focal splitting and optical vortex structure induced by stress birefringence” Proc. SPIE6667, 666701 (2007).
A. K. Spilman and T. G. Brown, “Stress-induced focal splitting,” Opt. Express15, 8411–8421 (2007). [CrossRef] [PubMed]
T. G. Brown and Q. Zhan, “Introduction: Unconventional Polarization States of Light Focus Issue,” Opt. Express18, 10775–10776 (2010). [CrossRef] [PubMed]

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