Abstract

A general family of scalar structured Gaussian beams naturally emerges from a consideration of families of rays. These ray families, with the property that their transverse profile is invariant upon propagation (except for a global rescaling), have two parameters, the first giving a position on an ellipse naturally represented by a point on a ray-family analog of the Poincaré sphere (familiar from polarization optics), and the other determining the position of a curve traced out on this Poincaré sphere. This construction naturally accounts for the well-known families of Gaussian beams, including Hermite–Gaussian, Laguerre–Gaussian, and generalized Hermite–Laguerre–Gaussian beams, but is far more general, opening the door for the design of a large variety of propagation-invariant beams. This ray-based description also provides a simple explanation for many aspects of these beams, such as “self-healing” and the Gouy and Pancharatnam–Berry phases. Further, through a conformal mapping between a projection of the Poincaré sphere and the physical space of the transverse plane of a Gaussian beam, the otherwise hidden geometric rules behind the beam’s intensity distribution are revealed. While the treatment is based on rays, a simple prescription is given for recovering exact solutions to the paraxial wave equation corresponding to these rays.

© 2017 Optical Society of America

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References

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2017 (1)

M. R. Dennis and M. A. Alonso, “Swings and roundabouts: optical Poincaré spheres for polarization and Gaussian beams,” Philos. Trans. R. Soc. Lond. A 375, 20150441 (2017).
[Crossref]

2016 (2)

N. D. Bareza and N. Hermosa, “Subluminal group velocity and dispersion of Laguerre Gauss beams in free space,” Sci. Rep. 6, 26842 (2016).
[Crossref]

F. Bouchard, J. Harris, H. Mand, R. W. Boyd, and E. Karimi, “Observation of subluminal twisted light in vacuum,” Optica 3, 351–354 (2016).
[Crossref]

2015 (1)

V. Potoček and S. M. Barnett, “Generalized ray optics and orbital angular momentum carrying beams,” New J. Phys. 17, 103034 (2015).
[Crossref]

2014 (2)

L. Gao, L. Shao, B.-C. Chen, and E. Betzig, “3D live fluorescence imaging of cellular dynamics using Bessel beam plane illumination microscopy,” Nat. Protocols 9, 1083–1101 (2014).
[Crossref]

Z. Yang, M. Prokopas, J. Nylk, C. Coll-Lladó, F. J. Gunn-Moore, D. E. K. Ferrier, T. Vettenburg, and K. Dholakia, “A compact Airy beam light sheet microscope with a tilted cylindrical lens,” Biomed. Opt. Express 5, 3434–3442 (2014).
[Crossref]

2013 (1)

2012 (1)

G. Milione, S. Evans, D. A. Nolan, and R. R. Alfano, “Higher order Pancharatnam–Berry phase and the angular momentum of light,” Phys. Rev. Lett. 108, 190401 (2012).
[Crossref]

2011 (2)

M. Woerdemann, C. Alpmann, and C. Denz, “Optical assembly of microparticles into highly ordered structures using Ince–Gaussian beams,” Appl. Phys. Lett. 98, 111101 (2011).
[Crossref]

G. Milione, H. I. Sztul, D. A. Nolan, and R. R. Alfano, “Higher-order Poincaré sphere, stokes parameters, and the angular momentum of light,” Phys. Rev. Lett. 107, 053601 (2011).
[Crossref]

2010 (4)

S. J. M. Habraken and G. Nienhuis, “Geometric phases in astigmatic optical modes of arbitrary order,” J. Math. Phys. 51, 082702 (2010).
[Crossref]

F. O. Fahrbach, P. Simon, and A. Rohrbach, “Microscopy with self-reconstructing beams,” Nat. Photonics 4, 780–785 (2010).
[Crossref]

S. J. M. Habraken and G. Nienhuis, “Geometric phases in higher-order transverse optical modes,” Proc. SPIE 7613, 76130F (2010).
[Crossref]

S. J. M. Habraken and G. Nienhuis, “Universal description of geometric phases in higher-order optical modes bearing orbital angular momentum,” Opt. Lett. 35, 3535–3537 (2010).
[Crossref]

2008 (4)

M. A. Bandres, “Accelerating parabolic beams,” Opt. Lett. 33, 1678–1680 (2008).
[Crossref]

K.-S. Lee and J. P. Rolland, “Bessel beam spectral-domain high-resolution optical coherence tomography with micro-optic axicon providing extended focusing range,” Opt. Lett. 33, 1696–1699 (2008).
[Crossref]

M. V. Berry and K. T. McDonald, “Exact and geometrical optics energy trajectories in twisted beams,” J. Opt. A 10, 035005 (2008).
[Crossref]

J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics 2, 675–678 (2008).
[Crossref]

2007 (2)

G. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99, 213901 (2007).
[Crossref]

M. Anguiano-Morales, A. Martínez, M. D. Iturbe-Castillo, S. Chávez-Cerda, and N. Alcalá-Ochoa, “Self-healing property of a caustic optical beam,” Appl. Opt. 46, 8284–8290 (2007).
[Crossref]

2006 (1)

E. J. Botcherby, R. Juškaitis, and T. Wilson, “Scanning two photon fluorescence microscopy with extended depth of field,” Opt. Commun. 268, 253–260 (2006).
[Crossref]

2005 (2)

2004 (3)

2003 (1)

E. J. Galvez, P. R. Crawford, H. I. Sztul, M. J. Pysher, P. J. Haglin, and R. E. Williams, “Geometric phase associated with mode transformations of optical beams bearing orbital angular momentum,” Phys. Rev. Lett. 90, 203901 (2003).
[Crossref]

2002 (3)

V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419, 145–147 (2002).
[Crossref]

M. A. Alonso and G. W. Forbes, “Stable aggregates of flexible elements give a stronger link between rays and waves,” Opt. Express 10, 728–739 (2002).
[Crossref]

W. T. Cathey and E. R. Dowski, “New paradigm for imaging system,” Appl. Opt. 41, 6080–6092 (2002).
[Crossref]

2001 (5)

2000 (1)

1999 (3)

1995 (2)

1993 (2)

R. Simon and N. Mukunda, “Bargmann invariant and the geometry of the Güoy effect,” Phys. Rev. Lett. 70, 880–883 (1993).
[Crossref]

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
[Crossref]

1992 (1)

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre–Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[Crossref]

1991 (1)

E. G. Abramochkin and V. G. Volostnikov, “Beam transformations and nontransformed beams,” Opt. Commun. 83, 123–135 (1991).
[Crossref]

1990 (2)

A. E. Siegman, “New developments in laser resonators,” Proc. SPIE 1224, 2–14 (1990).

T. F. Johnston, “M2 concept characterises beam quality,” Laser Focus World 26, 173–183 (1990).

1987 (2)

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[Crossref]

J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. 4, 651–654 (1987).
[Crossref]

1980 (1)

M. V. Berry and C. Upstill, “Catastrophe optics: morphologies of caustics and their diffraction patterns,” Prog. Opt. XVIII, 257–346 (1980).

1979 (1)

M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. 47, 264–267 (1979).
[Crossref]

1975 (1)

C. P. Boyer, E. G. Kalnins, and W. Miller, “Lie theory and separation of variables. 6. The equation iUt+Δ2U=0,” J. Math. Phys. 16, 499–511 (1975).
[Crossref]

1971 (2)

J. B. Keller and W. Streifer, “Complex rays with an application to Gaussian beams,” J. Opt. Soc. Am. 61, 40–43 (1971).
[Crossref]

G. A. Deschamps, “Gaussian beams as a bundle of complex rays,” Electron. Lett. 7, 684–685 (1971).
[Crossref]

1969 (1)

M. V. Berry, “Uniform approximation: a new concept in wave theory,” Sci. Prog. 57, 43–64 (1969).

1967 (1)

Yu. A. Kravtsov, “Complex ray and complex caustics,” Radiophys. Quantum Electron. 10, 719–730 (1967).
[Crossref]

1966 (1)

1890 (1)

L. R. Gouy, “Sur une propriètè nouvelle des ondes lumineuses,” C. R. Acad. Sci. Paris 110, 1251–1253 (1890).

Abramochkin, E. G.

E. G. Abramochkin and V. G. Volostnikov, “Generalized Gaussian beams,” J. Opt. A 6, S157–S161 (2004).
[Crossref]

E. G. Abramochkin and V. G. Volostnikov, “Beam transformations and nontransformed beams,” Opt. Commun. 83, 123–135 (1991).
[Crossref]

Agarwal, G. S.

G. S. Agarwal, “SU(2) structure of the Poincaré sphere for light beams with orbital angular momentum,” J. Opt. Soc. Am. B 16, 2914–2916 (1999).
[Crossref]

Alcalá-Ochoa, N.

Alfano, R. R.

G. Milione, S. Evans, D. A. Nolan, and R. R. Alfano, “Higher order Pancharatnam–Berry phase and the angular momentum of light,” Phys. Rev. Lett. 108, 190401 (2012).
[Crossref]

G. Milione, H. I. Sztul, D. A. Nolan, and R. R. Alfano, “Higher-order Poincaré sphere, stokes parameters, and the angular momentum of light,” Phys. Rev. Lett. 107, 053601 (2011).
[Crossref]

Allen, L.

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
[Crossref]

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre–Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[Crossref]

Alonso, M. A.

Alpmann, C.

M. Woerdemann, C. Alpmann, and C. Denz, “Optical assembly of microparticles into highly ordered structures using Ince–Gaussian beams,” Appl. Phys. Lett. 98, 111101 (2011).
[Crossref]

Anguiano-Morales, M.

Balazs, N. L.

M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. 47, 264–267 (1979).
[Crossref]

Bandres, M. A.

Bareza, N. D.

N. D. Bareza and N. Hermosa, “Subluminal group velocity and dispersion of Laguerre Gauss beams in free space,” Sci. Rep. 6, 26842 (2016).
[Crossref]

Barnett, S. M.

V. Potoček and S. M. Barnett, “Generalized ray optics and orbital angular momentum carrying beams,” New J. Phys. 17, 103034 (2015).
[Crossref]

Baumgartl, J.

J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics 2, 675–678 (2008).
[Crossref]

Beijersbergen, M. W.

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
[Crossref]

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre–Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[Crossref]

Berry, M. V.

M. V. Berry and K. T. McDonald, “Exact and geometrical optics energy trajectories in twisted beams,” J. Opt. A 10, 035005 (2008).
[Crossref]

M. V. Berry and C. Upstill, “Catastrophe optics: morphologies of caustics and their diffraction patterns,” Prog. Opt. XVIII, 257–346 (1980).

M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. 47, 264–267 (1979).
[Crossref]

M. V. Berry, “Uniform approximation: a new concept in wave theory,” Sci. Prog. 57, 43–64 (1969).

Betzig, E.

L. Gao, L. Shao, B.-C. Chen, and E. Betzig, “3D live fluorescence imaging of cellular dynamics using Bessel beam plane illumination microscopy,” Nat. Protocols 9, 1083–1101 (2014).
[Crossref]

Blum, H.

H. Blum, “A transformation for extracting new descriptors of shape,” in Models for the Perception of Speech and Visual Form, W. Wathen-Dunn, ed. (MIT, 1967), pp. 362–380.

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999), pp. 116–141.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999), pp. 25–37.

Botcherby, E. J.

E. J. Botcherby, R. Juškaitis, and T. Wilson, “Scanning two photon fluorescence microscopy with extended depth of field,” Opt. Commun. 268, 253–260 (2006).
[Crossref]

Bouchard, F.

Boyd, R. W.

Boyer, C. P.

C. P. Boyer, E. G. Kalnins, and W. Miller, “Lie theory and separation of variables. 6. The equation iUt+Δ2U=0,” J. Math. Phys. 16, 499–511 (1975).
[Crossref]

Broky, J.

G. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99, 213901 (2007).
[Crossref]

Calvo, G. F.

Cathey, W. T.

Chávez-Cerda, S.

Chen, B.-C.

L. Gao, L. Shao, B.-C. Chen, and E. Betzig, “3D live fluorescence imaging of cellular dynamics using Bessel beam plane illumination microscopy,” Nat. Protocols 9, 1083–1101 (2014).
[Crossref]

Christodoulides, D. N.

G. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99, 213901 (2007).
[Crossref]

Chuu, C.-S.

Coll-Lladó, C.

Courtial, J.

Crawford, P. R.

E. J. Galvez, P. R. Crawford, H. I. Sztul, M. J. Pysher, P. J. Haglin, and R. E. Williams, “Geometric phase associated with mode transformations of optical beams bearing orbital angular momentum,” Phys. Rev. Lett. 90, 203901 (2003).
[Crossref]

Dennis, M. R.

M. R. Dennis and M. A. Alonso, “Swings and roundabouts: optical Poincaré spheres for polarization and Gaussian beams,” Philos. Trans. R. Soc. Lond. A 375, 20150441 (2017).
[Crossref]

J. D. Ring, C. J. Howls, and M. R. Dennis, “Incomplete Airy beams: finite energy from a sharp spectral cutoff,” Opt. Lett. 38, 1639–1641 (2013).
[Crossref]

Denz, C.

M. Woerdemann, C. Alpmann, and C. Denz, “Optical assembly of microparticles into highly ordered structures using Ince–Gaussian beams,” Appl. Phys. Lett. 98, 111101 (2011).
[Crossref]

Deschamps, G. A.

G. A. Deschamps, “Gaussian beams as a bundle of complex rays,” Electron. Lett. 7, 684–685 (1971).
[Crossref]

Dholakia, K.

Z. Yang, M. Prokopas, J. Nylk, C. Coll-Lladó, F. J. Gunn-Moore, D. E. K. Ferrier, T. Vettenburg, and K. Dholakia, “A compact Airy beam light sheet microscope with a tilted cylindrical lens,” Biomed. Opt. Express 5, 3434–3442 (2014).
[Crossref]

J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics 2, 675–678 (2008).
[Crossref]

V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419, 145–147 (2002).
[Crossref]

Dogariu, A.

G. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99, 213901 (2007).
[Crossref]

Dowski, E. R.

Durnin, J.

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[Crossref]

J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. 4, 651–654 (1987).
[Crossref]

Eberly, J. H.

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[Crossref]

Evans, S.

G. Milione, S. Evans, D. A. Nolan, and R. R. Alfano, “Higher order Pancharatnam–Berry phase and the angular momentum of light,” Phys. Rev. Lett. 108, 190401 (2012).
[Crossref]

Fahrbach, F. O.

F. O. Fahrbach, P. Simon, and A. Rohrbach, “Microscopy with self-reconstructing beams,” Nat. Photonics 4, 780–785 (2010).
[Crossref]

Feng, S.

Ferrier, D. E. K.

Forbes, G. W.

Galvez, E. J.

E. J. Galvez, P. R. Crawford, H. I. Sztul, M. J. Pysher, P. J. Haglin, and R. E. Williams, “Geometric phase associated with mode transformations of optical beams bearing orbital angular momentum,” Phys. Rev. Lett. 90, 203901 (2003).
[Crossref]

E. J. Galvez and C. D. Holmes, “Geometric phase of optical rotators,” J. Opt. Soc. Am. A 16, 1981–1985 (1999).
[Crossref]

Gao, L.

L. Gao, L. Shao, B.-C. Chen, and E. Betzig, “3D live fluorescence imaging of cellular dynamics using Bessel beam plane illumination microscopy,” Nat. Protocols 9, 1083–1101 (2014).
[Crossref]

Garcés-Chávez, V.

V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419, 145–147 (2002).
[Crossref]

Gouy, L. R.

L. R. Gouy, “Sur une propriètè nouvelle des ondes lumineuses,” C. R. Acad. Sci. Paris 110, 1251–1253 (1890).

Gunn-Moore, F. J.

Gutiérrez-Vega, J. C.

Habraken, S. J. M.

S. J. M. Habraken and G. Nienhuis, “Geometric phases in higher-order transverse optical modes,” Proc. SPIE 7613, 76130F (2010).
[Crossref]

S. J. M. Habraken and G. Nienhuis, “Geometric phases in astigmatic optical modes of arbitrary order,” J. Math. Phys. 51, 082702 (2010).
[Crossref]

S. J. M. Habraken and G. Nienhuis, “Universal description of geometric phases in higher-order optical modes bearing orbital angular momentum,” Opt. Lett. 35, 3535–3537 (2010).
[Crossref]

Haglin, P. J.

E. J. Galvez, P. R. Crawford, H. I. Sztul, M. J. Pysher, P. J. Haglin, and R. E. Williams, “Geometric phase associated with mode transformations of optical beams bearing orbital angular momentum,” Phys. Rev. Lett. 90, 203901 (2003).
[Crossref]

Hanssen, J. L.

Harris, J.

Hermosa, N.

N. D. Bareza and N. Hermosa, “Subluminal group velocity and dispersion of Laguerre Gauss beams in free space,” Sci. Rep. 6, 26842 (2016).
[Crossref]

Holmes, C. D.

Howls, C. J.

Iturbe-Castillo, M. D.

Johnston, T. F.

T. F. Johnston, “M2 concept characterises beam quality,” Laser Focus World 26, 173–183 (1990).

Juškaitis, R.

E. J. Botcherby, R. Juškaitis, and T. Wilson, “Scanning two photon fluorescence microscopy with extended depth of field,” Opt. Commun. 268, 253–260 (2006).
[Crossref]

Kalnins, E. G.

C. P. Boyer, E. G. Kalnins, and W. Miller, “Lie theory and separation of variables. 6. The equation iUt+Δ2U=0,” J. Math. Phys. 16, 499–511 (1975).
[Crossref]

Karimi, E.

Keller, J. B.

Kogelnik, H.

Kravtsov, Y. A.

Y. A. Kravtsov and Y. A. Orlov, Caustics, Catastrophes and Wave Fields, 2nd ed. (Springer, 1999).

Kravtsov, Yu. A.

Yu. A. Kravtsov, “Complex ray and complex caustics,” Radiophys. Quantum Electron. 10, 719–730 (1967).
[Crossref]

Lee, K.-S.

Li, T.

Luneburg, R. K.

R. K. Luneburg, Mathematical Theory of Optics (University of California, 1966), pp. 246–257.

Mand, H.

Martínez, A.

Mazilu, M.

J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics 2, 675–678 (2008).
[Crossref]

McDonald, K. T.

M. V. Berry and K. T. McDonald, “Exact and geometrical optics energy trajectories in twisted beams,” J. Opt. A 10, 035005 (2008).
[Crossref]

McGloin, D.

V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419, 145–147 (2002).
[Crossref]

Melville, H.

V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419, 145–147 (2002).
[Crossref]

Meyrath, T. P.

Miceli, J. J.

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[Crossref]

Milione, G.

G. Milione, S. Evans, D. A. Nolan, and R. R. Alfano, “Higher order Pancharatnam–Berry phase and the angular momentum of light,” Phys. Rev. Lett. 108, 190401 (2012).
[Crossref]

G. Milione, H. I. Sztul, D. A. Nolan, and R. R. Alfano, “Higher-order Poincaré sphere, stokes parameters, and the angular momentum of light,” Phys. Rev. Lett. 107, 053601 (2011).
[Crossref]

Miller, W.

C. P. Boyer, E. G. Kalnins, and W. Miller, “Lie theory and separation of variables. 6. The equation iUt+Δ2U=0,” J. Math. Phys. 16, 499–511 (1975).
[Crossref]

Mukunda, N.

R. Simon and N. Mukunda, “Bargmann invariant and the geometry of the Güoy effect,” Phys. Rev. Lett. 70, 880–883 (1993).
[Crossref]

Nienhuis, G.

S. J. M. Habraken and G. Nienhuis, “Geometric phases in astigmatic optical modes of arbitrary order,” J. Math. Phys. 51, 082702 (2010).
[Crossref]

S. J. M. Habraken and G. Nienhuis, “Geometric phases in higher-order transverse optical modes,” Proc. SPIE 7613, 76130F (2010).
[Crossref]

S. J. M. Habraken and G. Nienhuis, “Universal description of geometric phases in higher-order optical modes bearing orbital angular momentum,” Opt. Lett. 35, 3535–3537 (2010).
[Crossref]

Nolan, D. A.

G. Milione, S. Evans, D. A. Nolan, and R. R. Alfano, “Higher order Pancharatnam–Berry phase and the angular momentum of light,” Phys. Rev. Lett. 108, 190401 (2012).
[Crossref]

G. Milione, H. I. Sztul, D. A. Nolan, and R. R. Alfano, “Higher-order Poincaré sphere, stokes parameters, and the angular momentum of light,” Phys. Rev. Lett. 107, 053601 (2011).
[Crossref]

Nye, J. F.

J. F. Nye, Natural Focusing and Fine Structure of Light (IoP, 1999).

Nylk, J.

O’Neil, A. T.

A. T. O’Neil and M. J. Padgett, “Axial and lateral trapping efficiency of Laguerre–Gaussian modes in inverted optical tweezers,” Opt. Commun. 193, 45–50 (2001).
[Crossref]

Orlov, Y. A.

Y. A. Kravtsov and Y. A. Orlov, Caustics, Catastrophes and Wave Fields, 2nd ed. (Springer, 1999).

Padgett, M. J.

A. T. O’Neil and M. J. Padgett, “Axial and lateral trapping efficiency of Laguerre–Gaussian modes in inverted optical tweezers,” Opt. Commun. 193, 45–50 (2001).
[Crossref]

M. J. Padgett and J. Courtial, “Poincaré-sphere equivalent for light beams containing orbital angular momentum,” Opt. Lett. 24, 430–432 (1999).
[Crossref]

Penrose, R.

R. Penrose, The Road to Reality: a Complete Guide to the Laws of the Universe (Jonathan Cape, 2004), p. 45.

Potocek, V.

V. Potoček and S. M. Barnett, “Generalized ray optics and orbital angular momentum carrying beams,” New J. Phys. 17, 103034 (2015).
[Crossref]

Prokopas, M.

Pysher, M. J.

E. J. Galvez, P. R. Crawford, H. I. Sztul, M. J. Pysher, P. J. Haglin, and R. E. Williams, “Geometric phase associated with mode transformations of optical beams bearing orbital angular momentum,” Phys. Rev. Lett. 90, 203901 (2003).
[Crossref]

Raizen, M. G.

Ring, J. D.

J. D. Ring, C. J. Howls, and M. R. Dennis, “Incomplete Airy beams: finite energy from a sharp spectral cutoff,” Opt. Lett. 38, 1639–1641 (2013).
[Crossref]

J. D. Ring, “Incomplete catastrophes and paraxial beams,” Ph.D. thesis (University of Bristol, 2013).

Rohrbach, A.

F. O. Fahrbach, P. Simon, and A. Rohrbach, “Microscopy with self-reconstructing beams,” Nat. Photonics 4, 780–785 (2010).
[Crossref]

Rolland, J. P.

Sasnett, M. W.

M. W. Sasnett, “Propagation of multimode laser beams—the M2 factor,” in The Physics and Technology of Laser Resonators, D. R. Hall and P. E. Jackson, eds. (Hilger, 1989), pp. 132–142.

Schreck, F.

Shao, L.

L. Gao, L. Shao, B.-C. Chen, and E. Betzig, “3D live fluorescence imaging of cellular dynamics using Bessel beam plane illumination microscopy,” Nat. Protocols 9, 1083–1101 (2014).
[Crossref]

Sibbett, W.

V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419, 145–147 (2002).
[Crossref]

Siegman, A. E.

A. E. Siegman, “New developments in laser resonators,” Proc. SPIE 1224, 2–14 (1990).

A. E. Siegman, Lasers (University Science, 1986), Chap. 16.

Simon, P.

F. O. Fahrbach, P. Simon, and A. Rohrbach, “Microscopy with self-reconstructing beams,” Nat. Photonics 4, 780–785 (2010).
[Crossref]

Simon, R.

R. Simon and N. Mukunda, “Bargmann invariant and the geometry of the Güoy effect,” Phys. Rev. Lett. 70, 880–883 (1993).
[Crossref]

Siviloglou, G.

G. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99, 213901 (2007).
[Crossref]

Spreeuw, R. J. C.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre–Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[Crossref]

Streifer, W.

Subbarao, D.

Sztul, H. I.

G. Milione, H. I. Sztul, D. A. Nolan, and R. R. Alfano, “Higher-order Poincaré sphere, stokes parameters, and the angular momentum of light,” Phys. Rev. Lett. 107, 053601 (2011).
[Crossref]

E. J. Galvez, P. R. Crawford, H. I. Sztul, M. J. Pysher, P. J. Haglin, and R. E. Williams, “Geometric phase associated with mode transformations of optical beams bearing orbital angular momentum,” Phys. Rev. Lett. 90, 203901 (2003).
[Crossref]

Upstill, C.

M. V. Berry and C. Upstill, “Catastrophe optics: morphologies of caustics and their diffraction patterns,” Prog. Opt. XVIII, 257–346 (1980).

van der Veen, H. E. L. O.

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
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Vettenburg, T.

Volostnikov, V. G.

E. G. Abramochkin and V. G. Volostnikov, “Generalized Gaussian beams,” J. Opt. A 6, S157–S161 (2004).
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E. G. Abramochkin and V. G. Volostnikov, “Beam transformations and nontransformed beams,” Opt. Commun. 83, 123–135 (1991).
[Crossref]

Williams, R. E.

E. J. Galvez, P. R. Crawford, H. I. Sztul, M. J. Pysher, P. J. Haglin, and R. E. Williams, “Geometric phase associated with mode transformations of optical beams bearing orbital angular momentum,” Phys. Rev. Lett. 90, 203901 (2003).
[Crossref]

Wilson, T.

E. J. Botcherby, R. Juškaitis, and T. Wilson, “Scanning two photon fluorescence microscopy with extended depth of field,” Opt. Commun. 268, 253–260 (2006).
[Crossref]

Winful, H. G.

Woerdemann, M.

M. Woerdemann, C. Alpmann, and C. Denz, “Optical assembly of microparticles into highly ordered structures using Ince–Gaussian beams,” Appl. Phys. Lett. 98, 111101 (2011).
[Crossref]

Woerdman, J. P.

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
[Crossref]

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre–Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[Crossref]

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999), pp. 25–37.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999), pp. 116–141.

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Appl. Phys. Lett. (1)

M. Woerdemann, C. Alpmann, and C. Denz, “Optical assembly of microparticles into highly ordered structures using Ince–Gaussian beams,” Appl. Phys. Lett. 98, 111101 (2011).
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Biomed. Opt. Express (1)

C. R. Acad. Sci. Paris (1)

L. R. Gouy, “Sur une propriètè nouvelle des ondes lumineuses,” C. R. Acad. Sci. Paris 110, 1251–1253 (1890).

Electron. Lett. (1)

G. A. Deschamps, “Gaussian beams as a bundle of complex rays,” Electron. Lett. 7, 684–685 (1971).
[Crossref]

J. Math. Phys. (2)

C. P. Boyer, E. G. Kalnins, and W. Miller, “Lie theory and separation of variables. 6. The equation iUt+Δ2U=0,” J. Math. Phys. 16, 499–511 (1975).
[Crossref]

S. J. M. Habraken and G. Nienhuis, “Geometric phases in astigmatic optical modes of arbitrary order,” J. Math. Phys. 51, 082702 (2010).
[Crossref]

J. Opt. A (2)

E. G. Abramochkin and V. G. Volostnikov, “Generalized Gaussian beams,” J. Opt. A 6, S157–S161 (2004).
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M. V. Berry and K. T. McDonald, “Exact and geometrical optics energy trajectories in twisted beams,” J. Opt. A 10, 035005 (2008).
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J. Opt. Soc. Am. (2)

J. B. Keller and W. Streifer, “Complex rays with an application to Gaussian beams,” J. Opt. Soc. Am. 61, 40–43 (1971).
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J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. 4, 651–654 (1987).
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J. Opt. Soc. Am. A (5)

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G. S. Agarwal, “SU(2) structure of the Poincaré sphere for light beams with orbital angular momentum,” J. Opt. Soc. Am. B 16, 2914–2916 (1999).
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Laser Focus World (1)

T. F. Johnston, “M2 concept characterises beam quality,” Laser Focus World 26, 173–183 (1990).

Nat. Photonics (2)

F. O. Fahrbach, P. Simon, and A. Rohrbach, “Microscopy with self-reconstructing beams,” Nat. Photonics 4, 780–785 (2010).
[Crossref]

J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics 2, 675–678 (2008).
[Crossref]

Nat. Protocols (1)

L. Gao, L. Shao, B.-C. Chen, and E. Betzig, “3D live fluorescence imaging of cellular dynamics using Bessel beam plane illumination microscopy,” Nat. Protocols 9, 1083–1101 (2014).
[Crossref]

Nature (1)

V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419, 145–147 (2002).
[Crossref]

New J. Phys. (1)

V. Potoček and S. M. Barnett, “Generalized ray optics and orbital angular momentum carrying beams,” New J. Phys. 17, 103034 (2015).
[Crossref]

Opt. Commun. (4)

E. G. Abramochkin and V. G. Volostnikov, “Beam transformations and nontransformed beams,” Opt. Commun. 83, 123–135 (1991).
[Crossref]

A. T. O’Neil and M. J. Padgett, “Axial and lateral trapping efficiency of Laguerre–Gaussian modes in inverted optical tweezers,” Opt. Commun. 193, 45–50 (2001).
[Crossref]

E. J. Botcherby, R. Juškaitis, and T. Wilson, “Scanning two photon fluorescence microscopy with extended depth of field,” Opt. Commun. 268, 253–260 (2006).
[Crossref]

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
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Opt. Express (2)

Opt. Lett. (10)

G. F. Calvo, “Wigner representation and geometric transformations of optical orbital angular momentum spatial modes,” Opt. Lett. 30, 1207–1209 (2005).
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M. A. Bandres and J. C. Gutiérrez-Vega, “Ince–Gaussian beams,” Opt. Lett. 29, 144–146 (2004).
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M. A. Bandres, “Accelerating parabolic beams,” Opt. Lett. 33, 1678–1680 (2008).
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K.-S. Lee and J. P. Rolland, “Bessel beam spectral-domain high-resolution optical coherence tomography with micro-optic axicon providing extended focusing range,” Opt. Lett. 33, 1696–1699 (2008).
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S. J. M. Habraken and G. Nienhuis, “Universal description of geometric phases in higher-order optical modes bearing orbital angular momentum,” Opt. Lett. 35, 3535–3537 (2010).
[Crossref]

J. D. Ring, C. J. Howls, and M. R. Dennis, “Incomplete Airy beams: finite energy from a sharp spectral cutoff,” Opt. Lett. 38, 1639–1641 (2013).
[Crossref]

D. Subbarao, “Topological phase in Gaussian beam optics,” Opt. Lett. 20, 2162–2164 (1995).
[Crossref]

M. J. Padgett and J. Courtial, “Poincaré-sphere equivalent for light beams containing orbital angular momentum,” Opt. Lett. 24, 430–432 (1999).
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J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, and S. Chávez-Cerda, “Alternative formulation for invariant optical fields: Mathieu beams,” Opt. Lett. 25, 1493–1495 (2000).
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S. Feng and H. G. Winful, “Physical origin of the Gouy phase shift,” Opt. Lett. 26, 485–487 (2001).
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Optica (1)

Philos. Trans. R. Soc. Lond. A (1)

M. R. Dennis and M. A. Alonso, “Swings and roundabouts: optical Poincaré spheres for polarization and Gaussian beams,” Philos. Trans. R. Soc. Lond. A 375, 20150441 (2017).
[Crossref]

Phys. Rev. A (1)

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre–Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[Crossref]

Phys. Rev. Lett. (6)

G. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99, 213901 (2007).
[Crossref]

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[Crossref]

G. Milione, H. I. Sztul, D. A. Nolan, and R. R. Alfano, “Higher-order Poincaré sphere, stokes parameters, and the angular momentum of light,” Phys. Rev. Lett. 107, 053601 (2011).
[Crossref]

R. Simon and N. Mukunda, “Bargmann invariant and the geometry of the Güoy effect,” Phys. Rev. Lett. 70, 880–883 (1993).
[Crossref]

E. J. Galvez, P. R. Crawford, H. I. Sztul, M. J. Pysher, P. J. Haglin, and R. E. Williams, “Geometric phase associated with mode transformations of optical beams bearing orbital angular momentum,” Phys. Rev. Lett. 90, 203901 (2003).
[Crossref]

G. Milione, S. Evans, D. A. Nolan, and R. R. Alfano, “Higher order Pancharatnam–Berry phase and the angular momentum of light,” Phys. Rev. Lett. 108, 190401 (2012).
[Crossref]

Proc. SPIE (2)

A. E. Siegman, “New developments in laser resonators,” Proc. SPIE 1224, 2–14 (1990).

S. J. M. Habraken and G. Nienhuis, “Geometric phases in higher-order transverse optical modes,” Proc. SPIE 7613, 76130F (2010).
[Crossref]

Prog. Opt. (1)

M. V. Berry and C. Upstill, “Catastrophe optics: morphologies of caustics and their diffraction patterns,” Prog. Opt. XVIII, 257–346 (1980).

Radiophys. Quantum Electron. (1)

Yu. A. Kravtsov, “Complex ray and complex caustics,” Radiophys. Quantum Electron. 10, 719–730 (1967).
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Sci. Prog. (1)

M. V. Berry, “Uniform approximation: a new concept in wave theory,” Sci. Prog. 57, 43–64 (1969).

Sci. Rep. (1)

N. D. Bareza and N. Hermosa, “Subluminal group velocity and dispersion of Laguerre Gauss beams in free space,” Sci. Rep. 6, 26842 (2016).
[Crossref]

Other (10)

J. D. Ring, “Incomplete catastrophes and paraxial beams,” Ph.D. thesis (University of Bristol, 2013).

R. Penrose, The Road to Reality: a Complete Guide to the Laws of the Universe (Jonathan Cape, 2004), p. 45.

H. Blum, “A transformation for extracting new descriptors of shape,” in Models for the Perception of Speech and Visual Form, W. Wathen-Dunn, ed. (MIT, 1967), pp. 362–380.

R. K. Luneburg, Mathematical Theory of Optics (University of California, 1966), pp. 246–257.

M. W. Sasnett, “Propagation of multimode laser beams—the M2 factor,” in The Physics and Technology of Laser Resonators, D. R. Hall and P. E. Jackson, eds. (Hilger, 1989), pp. 132–142.

J. F. Nye, Natural Focusing and Fine Structure of Light (IoP, 1999).

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999), pp. 116–141.

Y. A. Kravtsov and Y. A. Orlov, Caustics, Catastrophes and Wave Fields, 2nd ed. (Springer, 1999).

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999), pp. 25–37.

A. E. Siegman, Lasers (University Science, 1986), Chap. 16.

Supplementary Material (7)

NameDescription
» Supplement 1: PDF (820 KB)      Supplementary material.
» Visualization 1: MOV (1926 KB)      Poincaré sphere describing the shape of the elliptic orbits of rays.
» Visualization 2: MOV (1747 KB)      Ray positions at a transverse plane for increasing z, where different orbits are indicated by different colors.
» Visualization 3: MOV (4410 KB)      Connection between a curve over the Poincaré sphere and elliptic orbits of rays. The bottom left figure shows the rays for the beam in question, while the one on the right shows the rays for the equivalent beam with opposite OAM.
» Visualization 4: MOV (5340 KB)      Construction of the medial axes of the Poincaré path and their mapping onto the caustics on the physical disk.
» Visualization 5: MOV (10063 KB)      Construction of a HG and a LG beam and the transition from one to the other through a rotation of the Poincaré sphere, yielding HLG beams.
» Visualization 6: MOV (1549 KB)      Ray-optical description of the Pancharatnam-Berry phase corresponding to a transformation tracing a closed path on the Poincaré sphere. Note that the final (LG) beam is identical in shape to the initial one, but the ray orbits are rotated.

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Figures (11)

Fig. 1.
Fig. 1.

Ray orbit in real space and on the Poincaré sphere. (a) The straight rays sweep out a hyperboloid whose cross sections at any constant z are ellipses with the same eccentricity and orientation. The green curve is a normal to the rays. The length of the orange ray segment must be an integer multiple of the wavelength. (b) The eccentricity and orientation of the ellipse correspond to a point on the Poincaré sphere that has Cartesian coordinates s 1 , s 2 , and s 3 , given by the analogs of the Stokes parameters.

Fig. 2.
Fig. 2.

Point s in the PED maps to an ellipse with foci f ± in the physical disk. The ellipse’s major and minor axes have lengths 2 cos 1 2 θ and 2 | sin 1 2 θ | , respectively, equal to the sides of the gray rectangle. Note that any rectangle in which the ellipse is inscribed is itself inscribed in the unit circle.

Fig. 3.
Fig. 3.

Ray families for a Poincaré path. (a) Two Poincaré paths (red and blue) over the surface of the Poincaré sphere with the same projection (black) onto the equatorial plane (the PED path), (b) family of elliptical orbits for the Poincaré path in (a). The inner and outer envelopes of this family form caustics. (c), (d) Rays corresponding to the loops in the (c) upper and (d) lower hemispheres, where colors identify orbits.

Fig. 4.
Fig. 4.

Medial axes of PED paths map to caustics in the physical disk. Given a PED path (thick black curve), one can find two medial axes as the loci of the centers of circles that touch this curve and the unit circle. The mapping of Z ( q ) = ± Z ( t ) , where t represents points along the medial axes, corresponds to curves of points q that are the caustics of the resulting fields. Note that the caustics in (b) correspond to those in Fig. 3(b). (See also Visualization 4.)

Fig. 5.
Fig. 5.

Beam amplitude profiles reconstructed from ray families. (a)–(c) Intensities and (d)–(f) real parts of U N for N = 15 , ϕ = π / 2 , and (a), (d)  θ = 0 ; (b), (e)  θ = π / 4 ; and (c), (f)  θ = π / 2 . The yellow circle indicates the limit of the physical disk, and the ray orbits are shown in green.

Fig. 6.
Fig. 6.

Rays for LG beams with N = 30 and n = 4 (so = 22 ). (a) PED and (b) physical disk. In (a), the inner black circle is the PED path, and the orange and blue circles are the two medial axes, which map onto the two caustics of the same colors shown in (b) along with some of the elliptical orbits (green). (c) Propagation of the ray family, (d) wave field intensity with caustics overlaid.

Fig. 7.
Fig. 7.

Rays for HG beams with m = 23 and n = 7 (so N = 30 ). (a) PED and (b) physical disk. In (a), the vertical black line is the PED path, and the orange and blue parabolas are its medial axes. These medial axes map onto the straight caustics of the same colors shown in (b) along with some of the elliptical orbits (green). (c) Propagation of the ray family, (d) wave field intensity with caustics overlaid.

Fig. 8.
Fig. 8.

PED and physical disk for HLG beams with N = 30 , n = 4 , and three different angles of rotation β in the Poincaré sphere. Also shown are the resulting intensity profiles and the ray-optical caustics overlaid. The ray families clearly correspond to different projections of a torus, and the brightest parts of the intensities occur in close proximity to the caustics.

Fig. 9.
Fig. 9.

Illustration of geometric phase as a cycling of orbits. (a) The transformation of a LG beam with positive OAM into one with negative OAM by following a meridional path in the s 1 s 3 plane over the Poincaré sphere. Five stages of this path are shown explicitly, including a HG beam at the equator. (b) The transformation of the beam back into its initial configuration through a meridional path now in the s 2 s 3 plane. Note that the ray configurations are rotated by π / 4 with respect to those on the left. While the final beam has the same shape as the initial one, the orbits (identified by color) are rotated, resulting on a geometric phase.

Fig. 10.
Fig. 10.

Relevant segments of the PED path and medial axes over a peripheral segment of the PED and the corresponding caustics over a peripheral segment of the physical disk, for (a) a general asymmetric Airy beam and (b) a parabolic beam.

Fig. 11.
Fig. 11.

(a) Poincaré (red curve) and its projected PED (black square) paths, (b) intensity profile for a beam whose real-space caustics are an octagon and an eight-pointed star.

Equations (13)

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v ( θ , ϕ ) = cos θ 2 ( cos ϕ 2 , sin ϕ 2 ) + i sin θ 2 ( sin ϕ 2 , cos ϕ 2 ) ,
Q ( τ ; θ , ϕ ) = Q 0 R [ v ( θ , ϕ ) exp ( i τ ) ] ,
P ( τ ; θ , ϕ ) = P 0 I [ v ( θ , ϕ ) exp ( i τ ) ] ,
Q + z P = Q 0 2 + z 2 P 0 2 R { v ( θ , ϕ ) exp [ i ( τ + ζ ) ] } ,
L 1 ( τ 2 ) L 1 ( τ 1 ) = τ 1 τ 2 P · d Q d τ d τ = Q 0 P 0 2 [ τ 2 τ 1 sin ( 2 τ 2 ) sin ( 2 τ 1 ) 2 cos θ ] .
Q 0 P 0 = ( N + 1 ) λ / π ,
Z ( f ± ) = ± Z ( s ) .
Ω = ( 2 n + 1 ) 2 π N + 1 , n = 0 , 1 , , N / 2 ,
U ( x ) k P 0 2 exp ( i π 4 ) A ( η ) cos θ ϕ η + i θ η U N ( x Q 0 , v ) × exp ( i { k L 2 ( N + 1 ) [ T sin ( 2 T ) cos θ 2 ] } ) d η ,
U N ( x ¯ , v ) = 1 N ! exp ( N + 1 2 ) ( N + 1 2 v · v ) N 2 × exp [ ( N + 1 ) | x ¯ | 2 ] H N ( 2 ( N + 1 ) v · v x ¯ · v ) ,
r = 2 N + 1 + 4 n ( N n ) N + 1
x = ± Q 0 1 + 1 r 2 2 = ± Q 0 2 m + 1 2 ( N + 1 ) ,
y = ± Q 0 1 1 r 2 2 = ± Q 0 2 n + 1 2 ( N + 1 ) .

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