## Full Poincaré beams

Optics Express, Vol. 18, Issue 10, pp. 10777-10785 (2010)

http://dx.doi.org/10.1364/OE.18.010777

Acrobat PDF (988 KB)

### Abstract

We explore the behavior of a class of fully correlated optical beams that span the entire surface of the Poincaré sphere. The beams can be constructed from a coaxial superposition of a fundamental Gaussian mode and a spiral-phase Laguerre-Gauss mode having orthogonal polarizations. When the orthogonal polarizations are right and left circular, the coverage extends from one pole of the sphere to the other in such a way that concentric circles on the beam map onto parallels on the Poincaré sphere and radial lines map onto meridians. If the beam waist parameters match, the map is stereographic and the beam propagation corresponds to a rigid rotation about the pole. We present an experimental example of how a symmetrically stressed window can produce these beams and show that the predicted rotation indeed occurs when moving through the beams’ focus.

© 2010 Optical Society of America

## 1. Introduction

4. D. G. Hall, “Vector-beam solutions of Maxwells wave equation,” Opt. Lett. **21**, 9–11 (1996), http://www.opticsinfobase.org/abstract.cfm?URI=ol-21-1-9. [CrossRef] [PubMed]

5. R. H. Jordan and D. G. Hall, “Free-space azimuthal paraxial wave equation: the azimuthal Bessel-Gauss beam solution,” Opt. Lett. **19**, 427 (1994), http://www.opticsinfobase.org/abstract.cfm?URI=ol-19-7-427. [CrossRef] [PubMed]

6. P. L. Greene and D. G. Hall, “Focal shift in vector beams,” Opt. Express **4**, 411–419 (1999), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-4-10-411. [CrossRef] [PubMed]

1. K. S. Youngworth and T. G. Brown, “Focusing of high numerical aperture cylindrical vector beams,” Opt. Express **7**, 77–87 (2000), http://wwww.opticsinfobase.org/oe/abstract.cfm?URI=oe-7-2-77. [CrossRef] [PubMed]

14. W. Chen and Q. Zhan, “Numerical study of an apertureless near field scanning optical microscope probe under radial polarization illumination,” Opt. Express **15**, 4106–4111 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-7-4106. [CrossRef] [PubMed]

17. K. Venkatakrishnan and B. Tan, “Interconnect microvia drilling with a radially polarized laser beam,” J. Micromech. Microeng. **16**, 2603 (2006). [CrossRef]

18. N. Moore and M. A. Alonso, “Closed-form formula for Mie scattering of nonparaxial analogues of Gaussian beams,” Opt. Express **16**, 5926–5933 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-8-5926. [CrossRef] [PubMed]

*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

*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:

*ρ*= √

*x*

^{2}+

*y*

^{2},

*u*

_{0}is the beam’s amplitude at the origin,

*w*

_{0}is the beam waist, and

*z*

_{R}=

*k*

*w*

_{0}

^{2}/2 is the Rayleigh range. Similarly, the lowest order LG beam with unit azimuthal angular momentum can be written as

**ê**

_{1}and

**ê**

_{2}are two arbitrary orthogonal unit polarization vectors with no

*z*components. It is straightforward to see that these beams have axially-symmetric intensity profiles:

*w*(

*z*) =

*w*

_{0}|

*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*= (2

*m*+ 1)

*π*/4, the beam profile is flat (i.e. quartic) on axis.

*ϕ*(

_{ξ}*z*) = arg[

*ξ*(

*z*)]. The prefactor in parentheses determines the polarization at a given point. For any

*z*, the dimensionless parameter

*= √2 tan*ρ ^

*γρ*/

*w*regulates the relative amounts of each of the two polarizations, so that near the axis, the polarization tends towards

**ê**

_{1}, while away from it, it tends towards

**ê**

_{2}. 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.

**ê**

_{1,2}= (

**x̂**± i

**ŷ**)/√2. It is easy to show that the normalized Stokes parameters are then given by

*S*

_{3}is independent of

*ϕ*. Further, we know that

*S*

_{3}/

*S*

_{0}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

*S*

_{3}axis. It is then easy to show from Eq. (7) that

*= tan(*ρ ¯

*χ*′/2). Further, the fact that

*S*

_{1}/

*S*

_{0}and

*S*

_{2}/

*S*

_{0}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

**ê**

_{2}.

**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)

**ê**

_{1}is right-hand circular (RHC) and

**ê**

_{2}is left-hand circular (LHC), b)

**ê**

_{1}is LHC and

**ê**

_{2}is RHC, and c)

**ê**

_{1}is vertical and

**ê**

_{2}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

19. A. K. Spilman and T. G. Brown, “Stress birefringent, space-variant wave plates for vortex illumination,” Appl. Opt. **26**, 61–66 (2007), http://www.opticsinfobase.org/ao/abstract.cfm?URI=ao-46-1-61. [CrossRef]

20. A. K. Spilman and T. G. Brown, “Stress-induced Focal Splitting,” Opt. Express **15**, 8411–8421 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-13-8411. [CrossRef] [PubMed]

*μ*m 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:

**= (**

*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

19. A. K. Spilman and T. G. Brown, “Stress birefringent, space-variant wave plates for vortex illumination,” Appl. Opt. **26**, 61–66 (2007), http://www.opticsinfobase.org/ao/abstract.cfm?URI=ao-46-1-61. [CrossRef]

*U*

_{01}of the theoretical construct, as is shown in the experimental results that follow.

*U*

_{01}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

*U*

_{00}and

*U*

_{01}portions. For the example shown, the

*U*

_{00}waist was found to be 0.181±0.002 mm, and the

*U*

_{01}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

*U*

_{01}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.

*μ*m) 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).

*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.

*s*

_{1}| < 0.05, we were able to carry out a linear regression and compute the slope of the contour of zero

*s*

_{1}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*. 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

_{R}*z*= 165 mm, close to the Rayleigh range predicted by the Gaussian beam fit.

_{R}## 4. Discussion

23. E. G. Sauter, “Gaussian beams and the Poincare sphere,” Microwave Opt. Technol. Lett. **4**, 485–486 (1991). [CrossRef]

**ê**

_{1,2}is a linear basis.

*U*

_{00}and

*U*

_{01}with the same width parameter

*w*

_{0}. However, there are many other possible choices of fields that would lead to FP beams. For example, we could have chosen

*U*

_{00}and

*U*

_{01}to have different

*w*

_{0}, 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

*U*

_{00}becomes wider than

*U*

_{01}.

*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*for

_{nm}*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.

22. N. Moore and M. A. Alonso, “Closed-form bases for the description of monochromatic, strongly focused, electromagnetic fields,” J. Opt. Soc. Am. A **29**, 2211–2218 (2009), http://www.opticsinfobase.org/abstract.cfm?uri=josaa-26-7-1754. [CrossRef]

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

*q*of the dipoles’ origin causes them to be directional. For

*kq*≫ 1, this expression reduces to Eq. (4) with

*w*

_{0}= √2

*q*/

*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

18. N. Moore and M. A. Alonso, “Closed-form formula for Mie scattering of nonparaxial analogues of Gaussian beams,” Opt. Express **16**, 5926–5933 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-8-5926. [CrossRef] [PubMed]

*ϕ*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.

_{ξ}## Acknowledgements

## References and links

1. | K. S. Youngworth and T. G. Brown, “Focusing of high numerical aperture cylindrical vector beams,” Opt. Express |

2. | Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photon. |

3. | D. P. Biss and T. G. Brown, “Cylindrical vector beam focusing through a dielectric surface,” Opt. Express |

4. | D. G. Hall, “Vector-beam solutions of Maxwells wave equation,” Opt. Lett. |

5. | R. H. Jordan and D. G. Hall, “Free-space azimuthal paraxial wave equation: the azimuthal Bessel-Gauss beam solution,” Opt. Lett. |

6. | P. L. Greene and D. G. Hall, “Focal shift in vector beams,” Opt. Express |

7. | C. J. R. Sheppard and P. Török, “Electromagnetic field in the focal region of an electric dipole wave,” Optik |

8. | S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing Light to a Tighter Spot,” Opt. Commun. |

9. | R. Dorn, S. Quabis, and G. Leuchs “Sharper Focus for a Radially Polarized Light Beam,” Phys. Rev. Lett. |

10. | C. J. R. Sheppard and A. Choudhury, “Annular pupils, radial polarization, and superresolution,” Appl. Opt. |

11. | R. Borghi, M. Santarsiero, and M. A. Alonso, “Highly focused spirally polarized beams,” J. Opt. Soc. Am. A |

12. | G. Lerman and U. Levy, “Effect of radial polarization and apodization on spot size under tight focusing conditions,” Opt. Express |

13. | Q. Zhan, “Evanescent Bessel beam generation via surface plasmon resonance excitation by a radially polarized beam,” Opt. Lett. |

14. | W. Chen and Q. Zhan, “Numerical study of an apertureless near field scanning optical microscope probe under radial polarization illumination,” Opt. Express |

15. | K. J. Moh, X.-C. Yuan, J. Bu, S. W. Zhu, and Bruce Z. Gao, “Radial polarization induced surface plasmon virtual probe for two-photon fluorescence microscopy,” Opt. Lett. |

16. | N. Hayazawa, “Focused Excitation of Surface Plasmon Polaritons Based on Gap-Mode in Tip-Enhanced Spectroscopy,” Jpn. J. Appl. Phys. |

17. | K. Venkatakrishnan and B. Tan, “Interconnect microvia drilling with a radially polarized laser beam,” J. Micromech. Microeng. |

18. | N. Moore and M. A. Alonso, “Closed-form formula for Mie scattering of nonparaxial analogues of Gaussian beams,” Opt. Express |

19. | A. K. Spilman and T. G. Brown, “Stress birefringent, space-variant wave plates for vortex illumination,” Appl. Opt. |

20. | A. K. Spilman and T. G. Brown, “Stress-induced Focal Splitting,” Opt. Express |

21. | A. K. Spilman, A. M. Beckley, and T. G. Brown, “Focal splitting and optical vortex structure induced by stress birefringence,” Proc. SPIE, |

22. | N. Moore and M. A. Alonso, “Closed-form bases for the description of monochromatic, strongly focused, electromagnetic fields,” J. Opt. Soc. Am. A |

23. | E. G. Sauter, “Gaussian beams and the Poincare sphere,” Microwave Opt. Technol. Lett. |

**OCIS Codes**

(260.1440) Physical optics : Birefringence

(260.5430) Physical optics : Polarization

**History**

Original Manuscript: March 17, 2010

Revised Manuscript: April 14, 2010

Manuscript Accepted: April 14, 2010

Published: May 10, 2010

**Virtual Issues**

Unconventional Polarization States of Light (2010) *Optics Express*

**Citation**

Amber M. Beckley, Thomas G. Brown, and Miguel A. Alonso, "Full Poincare beams," Opt. Express **18**, 10777-10785 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-10-10777

Sort: Year | Journal | Reset

### References

- K. S. Youngworth, and T. G. Brown, “Focusing of high numerical aperture cylindrical vector beams,” Opt. Express 7, 77–87 (2000), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-7-2-77. [CrossRef] [PubMed]
- Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photon. 1, 1–57 (2009), http://www.opticsinfobase.org/abstract.cfm?URI=aop-1-1-1 and references therein. [CrossRef]
- D. P. Biss, and T. G. Brown, “Cylindrical vector beam focusing through a dielectric surface,” Opt. Express 9, 490–497 (2001), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-9-10-490. [CrossRef] [PubMed]
- D. G. Hall, “Vector-beam solutions of Maxwell’s wave equation,” Opt. Lett. 21, 9–11 (1996), http://www.opticsinfobase.org/abstract.cfm?URI=ol-21-1-9. [CrossRef] [PubMed]
- R. H. Jordan, and D. G. Hall, “Free-space azimuthal paraxial wave equation: the azimuthal Bessel-Gauss beam solution,” Opt. Lett. 19, 427 (1994), http://www.opticsinfobase.org/abstract.cfm?URI=ol-19-7-427. [CrossRef] [PubMed]
- P. L. Greene, and D. G. Hall, “Focal shift in vector beams,” Opt. Express 4, 411–419 (1999), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-4-10-411. [CrossRef] [PubMed]
- C. J. R. Sheppard, and P. Török, “Electromagnetic field in the focal region of an electric dipole wave,” Optik (Stuttg.) 104, 175–177 (1997).
- S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing Light to a Tighter Spot,” Opt. Commun. 179, 1 (2000). [CrossRef]
- R. Dorn, S. Quabis, and G. Leuchs, “Sharper Focus for a Radially Polarized Light Beam,” Phys. Rev. Lett. 91, 233901 (2003). [CrossRef] [PubMed]
- C. J. R. Sheppard, and A. Choudhury, “Annular pupils, radial polarization, and superresolution,” Appl. Opt. 43, 4322–4327 (2004), http://www.opticsinfobase.org/abstract.cfm?URI=ao-43-22-4322. [CrossRef] [PubMed]
- R. Borghi, M. Santarsiero, and M. A. Alonso, “Highly focused spirally polarized beams,” J. Opt. Soc. Am. A 22, 1420–1431 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=josaa-22-7-1420. [CrossRef]
- G. Lerman, and U. Levy, “Effect of radial polarization and apodization on spot size under tight focusing conditions,” Opt. Express 16, 4567–4581 (2008), http://www.opticsinfobase.org/abstract.cfm?URI=oe-16-7-4567. [CrossRef] [PubMed]
- Q. Zhan, “Evanescent Bessel beam generation via surface plasmon resonance excitation by a radially polarized beam,” Opt. Lett. 31, 1726–1728 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=ol-31-11-1726. [CrossRef] [PubMed]
- W. Chen, and Q. Zhan, “Numerical study of an apertureless near field scanning optical microscope probe under radial polarization illumination,” Opt. Express 15, 4106–4111 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-7-4106. [CrossRef] [PubMed]
- K. J. Moh, X.-C. Yuan, J. Bu, S. W. Zhu, and Z. Bruce, “Gao, “Radial polarization induced surface plasmon virtual probe for two-photon fluorescence microscopy,” Opt. Lett. 34, 971–973 (2009), http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-34-7-971. [CrossRef] [PubMed]
- N. Hayazawa, “Focused Excitation of Surface Plasmon Polaritons Based on Gap-Mode in Tip-Enhanced Spectroscopy,” Jpn. J. Appl. Phys. 46, 7995 (2007). [CrossRef]
- K. Venkatakrishnan, and B. Tan, “Interconnect microvia drilling with a radially polarized laser beam,” J. Micromech. Microeng. 16, 2603 (2006). [CrossRef]
- N. Moore, and M. A. Alonso, “Closed-form formula for Mie scattering of nonparaxial analogues of Gaussian beams,” Opt. Express 16, 5926–5933 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-8-5926. [CrossRef] [PubMed]
- A. K. Spilman, and T. G. Brown, “Stress birefringent, space-variant wave plates for vortex illumination,” Appl. Opt. 26, 61–66 (2007), http://www.opticsinfobase.org/ao/abstract.cfm?URI=ao-46-1-61. [CrossRef]
- A. K. Spilman, and T. G. Brown, “Stress-induced Focal Splitting,” Opt. Express 15, 8411–8421 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-13-8411. [CrossRef] [PubMed]
- 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).
- N. Moore, and M. A. Alonso, “Closed-form bases for the description of monochromatic, strongly focused, electromagnetic fields,” J. Opt. Soc. Am. A 29, 2211–2218 (2009), http://www.opticsinfobase.org/abstract.cfm?uri=josaa-26-7-1754. [CrossRef]
- E. G. Sauter, “Gaussian beams and the Poincare sphere,” Microw. Opt. Technol. Lett. 4, 485–486 (1991) . [CrossRef]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.