Abstract

We consider how vectorial aspects (polarization) of light propagation can be implemented and their origin within a Feynman path integral approach. A key part of this scheme is in generalizing the standard optical path length integral from a scalar to a matrix quantity. Reparametrization invariance along the rays allows a covariant formulation where propagation can take place along a general curve. A general gradient index background is used to demonstrate the scheme. This affords a description of classical imaging optics when the polarization aspects may be varying rapidly and cannot be neglected.

© 2021 Optical Society of America

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References

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  1. C. Misner, K. Thorne, and J. Wheeler, Gravitation (W. H. Freeman, 1973).
  2. D. Gloge and D. Marcuse, “Formal quantum theory of light rays,” J. Opt. Soc. Am. 59, 1629–1631 (1969).
    [Crossref]
  3. G. W. Forbes and M. A. Alonso, “What on earth is a ray and how can we use them best?” Proc. SPIE 3482, 22–31 (1998).
    [Crossref]
  4. A. V. Semichaevsky and M. E. Testorf, “Anything optical rays cannot do?” Proc. SPIE 4436, 56–67 (2001).
    [Crossref]
  5. A. V. Gitin, “Huygens-Feynman-Fresnel principle as the basis of applied optics,” Appl. Opt. 52, 7419–7434 (2013).
    [Crossref]
  6. M. Mout, M. Wick, F. Bociort, J. Petschulat, and P. Urbach, “Simulating multiple diffraction in imaging systems using a path integration method,” Appl. Opt. 55, 3847–3853 (2016).
    [Crossref]
  7. J. Babington, “Ray-wave duality in classical optics: crossing the Feynman bridge,” Opt. Lett. 43, 5591–5594 (2018).
    [Crossref]
  8. Z. Wan, Z. Wang, X. Yang, Y. Shen, and X. Fu, “Digitally tailoring arbitrary structured light of generalized ray-wave duality,” Opt. Express 28, 31043–31056 (2020).
    [Crossref]
  9. Z. Wang, Y. Shen, D. Naidoo, X. Fu, and A. Forbes, “Astigmatic hybrid Su(2) vector vortex beams: towards versatile structures in longitudinally variant polarized optics,” Opt. Express 29, 315–329 (2021).
    [Crossref]
  10. Y. Shen, X. Yang, D. Naidoo, X. Fu, and A. Forbes, “Structured ray-wave vector vortex beams in multiple degrees of freedom from a laser,” Optica 7, 820–831 (2020).
    [Crossref]
  11. Y. Shen, Z. Wang, X. Fu, D. Naidoo, and A. Forbes, “Su(2) Poincaré sphere: a generalized representation for multidimensional structured light,” Phys. Rev. A 102, 031501 (2020).
    [Crossref]
  12. M. Chaichian, Path Integrals in Physics, Series in mathematical and computational physics (IOP Publishing, 2001), Vol. I and II.
  13. Y. Dimant and S. Levit, “Path integrals for light propagation in dielectric media,” J. Opt. Soc. Am. B 27, 899–903 (2010).
    [Crossref]
  14. J. I. Gersten and A. Nitzan, “Path-integral approach to electromagnetic phenomena in inhomogeneous systems,” J. Opt. Soc. Am. B 4, 293–298 (1987).
    [Crossref]
  15. C. R. Ocier, C. A. Richards, D. A. Bacon-Brown, Q. Ding, R. Kumar, T. J. Garcia, J. van de Groep, J.-H. Song, A. J. Cyphersmith, A. Rhode, A. N. Perry, A. J. Littlefield, J. Zhu, D. Xie, H. Gao, J. F. Messinger, M. L. Brongersma, K. C. Toussaint, L. L. Goddard, and P. V. Braun, “Direct laser writing of volumetric gradient index lenses and waveguides,” Light Sci. Appl. 9, 196 (2020).
    [Crossref]
  16. E. Mottola, “Functional integration over geometries,” J. Math. Phys. 36, 2470–2511 (1995).
    [Crossref]
  17. H. Kleinert, Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets (World Scientific, 2009).
  18. L. Schulman, Techniques and Applications of Path Integration, Dover Books on Physics (Dover Publications, 2005).
  19. M. Born and E. Wolf, Principles of Optics (Cambridge University, 2020).
  20. S. M. Carroll, “Lecture notes on general relativity,” arXiv:gr-qc/9712019 (1997).
  21. M. V. Berry and K. E. Mount, “Semiclassical approximations in wave mechanics,” Rep. Prog. Phys. 35, 315–397 (1972).
    [Crossref]
  22. E. W. Marchand, “Ray tracing in cylindrical gradient-index media,” Appl. Opt. 11, 1104–1106 (1972).
    [Crossref]
  23. T. Alieva and M. J. Bastiaans, “Dynamic and geometric phase accumulation by Gaussian-type modes in first-order optical systems,” Opt. Lett. 33, 1659–1661 (2008).
    [Crossref]
  24. J.-M. Souriau, “Construction explicite de l’indice de maslov. applications,” in Group Theoretical Methods in Physics, A. Janner, T. Janssen, and M. Boon, eds. (Springer, 1976), pp. 117–148.
  25. C. He, J. Chang, Q. Hu, J. Wang, J. Antonello, H. He, S. Liu, J. Lin, B. Dai, D. S. Elson, P. Xi, H. Ma, and M. J. Booth, “Complex vectorial optics through gradient index lens cascades,” Nat. Commun. 10, 4264 (2019).
    [Crossref]
  26. X.-F. Qian and J. H. Eberly, “Entanglement and classical polarization states,” Opt. Lett. 36, 4110–4112 (2011).
    [Crossref]
  27. X.-F. Qian, A. N. Vamivakas, and J. H. Eberly, “Entanglement limits duality and vice versa,” Optica 5, 942–947 (2018).
    [Crossref]
  28. A. Holleczek, A. Aiello, C. Gabriel, C. Marquardt, and G. Leuchs, “Classical and quantum properties of cylindrically polarized states of light,” Opt. Express 19, 9714–9736 (2011).
    [Crossref]
  29. Y. Shen, I. Nape, X. Yang, X. Fu, M. Gong, D. Naidoo, and A. Forbes, “Creation and control of high-dimensional multi-partite classically entangled light,” Light Sci. Appl. 10, 50 (2021).
    [Crossref]
  30. D. Sugic and M. R. Dennis, “Singular knot bundle in light,” J. Opt. Soc. Am. A 35, 1987–1999 (2018).
    [Crossref]
  31. J. W. Dalhuisen and D. Bouwmeester, “Twistors and electromagnetic knots,” J. Phys. A 45, 135201 (2012).
    [Crossref]
  32. M. R. Dennis and M. A. Alonso, “Gaussian mode families from systems of rays,” J. Phys. Photon. 1, 025003 (2019).
    [Crossref]

2021 (2)

Z. Wang, Y. Shen, D. Naidoo, X. Fu, and A. Forbes, “Astigmatic hybrid Su(2) vector vortex beams: towards versatile structures in longitudinally variant polarized optics,” Opt. Express 29, 315–329 (2021).
[Crossref]

Y. Shen, I. Nape, X. Yang, X. Fu, M. Gong, D. Naidoo, and A. Forbes, “Creation and control of high-dimensional multi-partite classically entangled light,” Light Sci. Appl. 10, 50 (2021).
[Crossref]

2020 (4)

Y. Shen, X. Yang, D. Naidoo, X. Fu, and A. Forbes, “Structured ray-wave vector vortex beams in multiple degrees of freedom from a laser,” Optica 7, 820–831 (2020).
[Crossref]

Y. Shen, Z. Wang, X. Fu, D. Naidoo, and A. Forbes, “Su(2) Poincaré sphere: a generalized representation for multidimensional structured light,” Phys. Rev. A 102, 031501 (2020).
[Crossref]

C. R. Ocier, C. A. Richards, D. A. Bacon-Brown, Q. Ding, R. Kumar, T. J. Garcia, J. van de Groep, J.-H. Song, A. J. Cyphersmith, A. Rhode, A. N. Perry, A. J. Littlefield, J. Zhu, D. Xie, H. Gao, J. F. Messinger, M. L. Brongersma, K. C. Toussaint, L. L. Goddard, and P. V. Braun, “Direct laser writing of volumetric gradient index lenses and waveguides,” Light Sci. Appl. 9, 196 (2020).
[Crossref]

Z. Wan, Z. Wang, X. Yang, Y. Shen, and X. Fu, “Digitally tailoring arbitrary structured light of generalized ray-wave duality,” Opt. Express 28, 31043–31056 (2020).
[Crossref]

2019 (2)

M. R. Dennis and M. A. Alonso, “Gaussian mode families from systems of rays,” J. Phys. Photon. 1, 025003 (2019).
[Crossref]

C. He, J. Chang, Q. Hu, J. Wang, J. Antonello, H. He, S. Liu, J. Lin, B. Dai, D. S. Elson, P. Xi, H. Ma, and M. J. Booth, “Complex vectorial optics through gradient index lens cascades,” Nat. Commun. 10, 4264 (2019).
[Crossref]

2018 (3)

2016 (1)

2013 (1)

2012 (1)

J. W. Dalhuisen and D. Bouwmeester, “Twistors and electromagnetic knots,” J. Phys. A 45, 135201 (2012).
[Crossref]

2011 (2)

2010 (1)

2008 (1)

2001 (1)

A. V. Semichaevsky and M. E. Testorf, “Anything optical rays cannot do?” Proc. SPIE 4436, 56–67 (2001).
[Crossref]

1998 (1)

G. W. Forbes and M. A. Alonso, “What on earth is a ray and how can we use them best?” Proc. SPIE 3482, 22–31 (1998).
[Crossref]

1995 (1)

E. Mottola, “Functional integration over geometries,” J. Math. Phys. 36, 2470–2511 (1995).
[Crossref]

1987 (1)

1972 (2)

M. V. Berry and K. E. Mount, “Semiclassical approximations in wave mechanics,” Rep. Prog. Phys. 35, 315–397 (1972).
[Crossref]

E. W. Marchand, “Ray tracing in cylindrical gradient-index media,” Appl. Opt. 11, 1104–1106 (1972).
[Crossref]

1969 (1)

Aiello, A.

Alieva, T.

Alonso, M. A.

M. R. Dennis and M. A. Alonso, “Gaussian mode families from systems of rays,” J. Phys. Photon. 1, 025003 (2019).
[Crossref]

G. W. Forbes and M. A. Alonso, “What on earth is a ray and how can we use them best?” Proc. SPIE 3482, 22–31 (1998).
[Crossref]

Antonello, J.

C. He, J. Chang, Q. Hu, J. Wang, J. Antonello, H. He, S. Liu, J. Lin, B. Dai, D. S. Elson, P. Xi, H. Ma, and M. J. Booth, “Complex vectorial optics through gradient index lens cascades,” Nat. Commun. 10, 4264 (2019).
[Crossref]

Babington, J.

Bacon-Brown, D. A.

C. R. Ocier, C. A. Richards, D. A. Bacon-Brown, Q. Ding, R. Kumar, T. J. Garcia, J. van de Groep, J.-H. Song, A. J. Cyphersmith, A. Rhode, A. N. Perry, A. J. Littlefield, J. Zhu, D. Xie, H. Gao, J. F. Messinger, M. L. Brongersma, K. C. Toussaint, L. L. Goddard, and P. V. Braun, “Direct laser writing of volumetric gradient index lenses and waveguides,” Light Sci. Appl. 9, 196 (2020).
[Crossref]

Bastiaans, M. J.

Berry, M. V.

M. V. Berry and K. E. Mount, “Semiclassical approximations in wave mechanics,” Rep. Prog. Phys. 35, 315–397 (1972).
[Crossref]

Bociort, F.

Booth, M. J.

C. He, J. Chang, Q. Hu, J. Wang, J. Antonello, H. He, S. Liu, J. Lin, B. Dai, D. S. Elson, P. Xi, H. Ma, and M. J. Booth, “Complex vectorial optics through gradient index lens cascades,” Nat. Commun. 10, 4264 (2019).
[Crossref]

Born, M.

M. Born and E. Wolf, Principles of Optics (Cambridge University, 2020).

Bouwmeester, D.

J. W. Dalhuisen and D. Bouwmeester, “Twistors and electromagnetic knots,” J. Phys. A 45, 135201 (2012).
[Crossref]

Braun, P. V.

C. R. Ocier, C. A. Richards, D. A. Bacon-Brown, Q. Ding, R. Kumar, T. J. Garcia, J. van de Groep, J.-H. Song, A. J. Cyphersmith, A. Rhode, A. N. Perry, A. J. Littlefield, J. Zhu, D. Xie, H. Gao, J. F. Messinger, M. L. Brongersma, K. C. Toussaint, L. L. Goddard, and P. V. Braun, “Direct laser writing of volumetric gradient index lenses and waveguides,” Light Sci. Appl. 9, 196 (2020).
[Crossref]

Brongersma, M. L.

C. R. Ocier, C. A. Richards, D. A. Bacon-Brown, Q. Ding, R. Kumar, T. J. Garcia, J. van de Groep, J.-H. Song, A. J. Cyphersmith, A. Rhode, A. N. Perry, A. J. Littlefield, J. Zhu, D. Xie, H. Gao, J. F. Messinger, M. L. Brongersma, K. C. Toussaint, L. L. Goddard, and P. V. Braun, “Direct laser writing of volumetric gradient index lenses and waveguides,” Light Sci. Appl. 9, 196 (2020).
[Crossref]

Carroll, S. M.

S. M. Carroll, “Lecture notes on general relativity,” arXiv:gr-qc/9712019 (1997).

Chaichian, M.

M. Chaichian, Path Integrals in Physics, Series in mathematical and computational physics (IOP Publishing, 2001), Vol. I and II.

Chang, J.

C. He, J. Chang, Q. Hu, J. Wang, J. Antonello, H. He, S. Liu, J. Lin, B. Dai, D. S. Elson, P. Xi, H. Ma, and M. J. Booth, “Complex vectorial optics through gradient index lens cascades,” Nat. Commun. 10, 4264 (2019).
[Crossref]

Cyphersmith, A. J.

C. R. Ocier, C. A. Richards, D. A. Bacon-Brown, Q. Ding, R. Kumar, T. J. Garcia, J. van de Groep, J.-H. Song, A. J. Cyphersmith, A. Rhode, A. N. Perry, A. J. Littlefield, J. Zhu, D. Xie, H. Gao, J. F. Messinger, M. L. Brongersma, K. C. Toussaint, L. L. Goddard, and P. V. Braun, “Direct laser writing of volumetric gradient index lenses and waveguides,” Light Sci. Appl. 9, 196 (2020).
[Crossref]

Dai, B.

C. He, J. Chang, Q. Hu, J. Wang, J. Antonello, H. He, S. Liu, J. Lin, B. Dai, D. S. Elson, P. Xi, H. Ma, and M. J. Booth, “Complex vectorial optics through gradient index lens cascades,” Nat. Commun. 10, 4264 (2019).
[Crossref]

Dalhuisen, J. W.

J. W. Dalhuisen and D. Bouwmeester, “Twistors and electromagnetic knots,” J. Phys. A 45, 135201 (2012).
[Crossref]

Dennis, M. R.

M. R. Dennis and M. A. Alonso, “Gaussian mode families from systems of rays,” J. Phys. Photon. 1, 025003 (2019).
[Crossref]

D. Sugic and M. R. Dennis, “Singular knot bundle in light,” J. Opt. Soc. Am. A 35, 1987–1999 (2018).
[Crossref]

Dimant, Y.

Ding, Q.

C. R. Ocier, C. A. Richards, D. A. Bacon-Brown, Q. Ding, R. Kumar, T. J. Garcia, J. van de Groep, J.-H. Song, A. J. Cyphersmith, A. Rhode, A. N. Perry, A. J. Littlefield, J. Zhu, D. Xie, H. Gao, J. F. Messinger, M. L. Brongersma, K. C. Toussaint, L. L. Goddard, and P. V. Braun, “Direct laser writing of volumetric gradient index lenses and waveguides,” Light Sci. Appl. 9, 196 (2020).
[Crossref]

Eberly, J. H.

Elson, D. S.

C. He, J. Chang, Q. Hu, J. Wang, J. Antonello, H. He, S. Liu, J. Lin, B. Dai, D. S. Elson, P. Xi, H. Ma, and M. J. Booth, “Complex vectorial optics through gradient index lens cascades,” Nat. Commun. 10, 4264 (2019).
[Crossref]

Forbes, A.

Y. Shen, I. Nape, X. Yang, X. Fu, M. Gong, D. Naidoo, and A. Forbes, “Creation and control of high-dimensional multi-partite classically entangled light,” Light Sci. Appl. 10, 50 (2021).
[Crossref]

Z. Wang, Y. Shen, D. Naidoo, X. Fu, and A. Forbes, “Astigmatic hybrid Su(2) vector vortex beams: towards versatile structures in longitudinally variant polarized optics,” Opt. Express 29, 315–329 (2021).
[Crossref]

Y. Shen, Z. Wang, X. Fu, D. Naidoo, and A. Forbes, “Su(2) Poincaré sphere: a generalized representation for multidimensional structured light,” Phys. Rev. A 102, 031501 (2020).
[Crossref]

Y. Shen, X. Yang, D. Naidoo, X. Fu, and A. Forbes, “Structured ray-wave vector vortex beams in multiple degrees of freedom from a laser,” Optica 7, 820–831 (2020).
[Crossref]

Forbes, G. W.

G. W. Forbes and M. A. Alonso, “What on earth is a ray and how can we use them best?” Proc. SPIE 3482, 22–31 (1998).
[Crossref]

Fu, X.

Gabriel, C.

Gao, H.

C. R. Ocier, C. A. Richards, D. A. Bacon-Brown, Q. Ding, R. Kumar, T. J. Garcia, J. van de Groep, J.-H. Song, A. J. Cyphersmith, A. Rhode, A. N. Perry, A. J. Littlefield, J. Zhu, D. Xie, H. Gao, J. F. Messinger, M. L. Brongersma, K. C. Toussaint, L. L. Goddard, and P. V. Braun, “Direct laser writing of volumetric gradient index lenses and waveguides,” Light Sci. Appl. 9, 196 (2020).
[Crossref]

Garcia, T. J.

C. R. Ocier, C. A. Richards, D. A. Bacon-Brown, Q. Ding, R. Kumar, T. J. Garcia, J. van de Groep, J.-H. Song, A. J. Cyphersmith, A. Rhode, A. N. Perry, A. J. Littlefield, J. Zhu, D. Xie, H. Gao, J. F. Messinger, M. L. Brongersma, K. C. Toussaint, L. L. Goddard, and P. V. Braun, “Direct laser writing of volumetric gradient index lenses and waveguides,” Light Sci. Appl. 9, 196 (2020).
[Crossref]

Gersten, J. I.

Gitin, A. V.

Gloge, D.

Goddard, L. L.

C. R. Ocier, C. A. Richards, D. A. Bacon-Brown, Q. Ding, R. Kumar, T. J. Garcia, J. van de Groep, J.-H. Song, A. J. Cyphersmith, A. Rhode, A. N. Perry, A. J. Littlefield, J. Zhu, D. Xie, H. Gao, J. F. Messinger, M. L. Brongersma, K. C. Toussaint, L. L. Goddard, and P. V. Braun, “Direct laser writing of volumetric gradient index lenses and waveguides,” Light Sci. Appl. 9, 196 (2020).
[Crossref]

Gong, M.

Y. Shen, I. Nape, X. Yang, X. Fu, M. Gong, D. Naidoo, and A. Forbes, “Creation and control of high-dimensional multi-partite classically entangled light,” Light Sci. Appl. 10, 50 (2021).
[Crossref]

He, C.

C. He, J. Chang, Q. Hu, J. Wang, J. Antonello, H. He, S. Liu, J. Lin, B. Dai, D. S. Elson, P. Xi, H. Ma, and M. J. Booth, “Complex vectorial optics through gradient index lens cascades,” Nat. Commun. 10, 4264 (2019).
[Crossref]

He, H.

C. He, J. Chang, Q. Hu, J. Wang, J. Antonello, H. He, S. Liu, J. Lin, B. Dai, D. S. Elson, P. Xi, H. Ma, and M. J. Booth, “Complex vectorial optics through gradient index lens cascades,” Nat. Commun. 10, 4264 (2019).
[Crossref]

Holleczek, A.

Hu, Q.

C. He, J. Chang, Q. Hu, J. Wang, J. Antonello, H. He, S. Liu, J. Lin, B. Dai, D. S. Elson, P. Xi, H. Ma, and M. J. Booth, “Complex vectorial optics through gradient index lens cascades,” Nat. Commun. 10, 4264 (2019).
[Crossref]

Kleinert, H.

H. Kleinert, Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets (World Scientific, 2009).

Kumar, R.

C. R. Ocier, C. A. Richards, D. A. Bacon-Brown, Q. Ding, R. Kumar, T. J. Garcia, J. van de Groep, J.-H. Song, A. J. Cyphersmith, A. Rhode, A. N. Perry, A. J. Littlefield, J. Zhu, D. Xie, H. Gao, J. F. Messinger, M. L. Brongersma, K. C. Toussaint, L. L. Goddard, and P. V. Braun, “Direct laser writing of volumetric gradient index lenses and waveguides,” Light Sci. Appl. 9, 196 (2020).
[Crossref]

Leuchs, G.

Levit, S.

Lin, J.

C. He, J. Chang, Q. Hu, J. Wang, J. Antonello, H. He, S. Liu, J. Lin, B. Dai, D. S. Elson, P. Xi, H. Ma, and M. J. Booth, “Complex vectorial optics through gradient index lens cascades,” Nat. Commun. 10, 4264 (2019).
[Crossref]

Littlefield, A. J.

C. R. Ocier, C. A. Richards, D. A. Bacon-Brown, Q. Ding, R. Kumar, T. J. Garcia, J. van de Groep, J.-H. Song, A. J. Cyphersmith, A. Rhode, A. N. Perry, A. J. Littlefield, J. Zhu, D. Xie, H. Gao, J. F. Messinger, M. L. Brongersma, K. C. Toussaint, L. L. Goddard, and P. V. Braun, “Direct laser writing of volumetric gradient index lenses and waveguides,” Light Sci. Appl. 9, 196 (2020).
[Crossref]

Liu, S.

C. He, J. Chang, Q. Hu, J. Wang, J. Antonello, H. He, S. Liu, J. Lin, B. Dai, D. S. Elson, P. Xi, H. Ma, and M. J. Booth, “Complex vectorial optics through gradient index lens cascades,” Nat. Commun. 10, 4264 (2019).
[Crossref]

Ma, H.

C. He, J. Chang, Q. Hu, J. Wang, J. Antonello, H. He, S. Liu, J. Lin, B. Dai, D. S. Elson, P. Xi, H. Ma, and M. J. Booth, “Complex vectorial optics through gradient index lens cascades,” Nat. Commun. 10, 4264 (2019).
[Crossref]

Marchand, E. W.

Marcuse, D.

Marquardt, C.

Messinger, J. F.

C. R. Ocier, C. A. Richards, D. A. Bacon-Brown, Q. Ding, R. Kumar, T. J. Garcia, J. van de Groep, J.-H. Song, A. J. Cyphersmith, A. Rhode, A. N. Perry, A. J. Littlefield, J. Zhu, D. Xie, H. Gao, J. F. Messinger, M. L. Brongersma, K. C. Toussaint, L. L. Goddard, and P. V. Braun, “Direct laser writing of volumetric gradient index lenses and waveguides,” Light Sci. Appl. 9, 196 (2020).
[Crossref]

Misner, C.

C. Misner, K. Thorne, and J. Wheeler, Gravitation (W. H. Freeman, 1973).

Mottola, E.

E. Mottola, “Functional integration over geometries,” J. Math. Phys. 36, 2470–2511 (1995).
[Crossref]

Mount, K. E.

M. V. Berry and K. E. Mount, “Semiclassical approximations in wave mechanics,” Rep. Prog. Phys. 35, 315–397 (1972).
[Crossref]

Mout, M.

Naidoo, D.

Z. Wang, Y. Shen, D. Naidoo, X. Fu, and A. Forbes, “Astigmatic hybrid Su(2) vector vortex beams: towards versatile structures in longitudinally variant polarized optics,” Opt. Express 29, 315–329 (2021).
[Crossref]

Y. Shen, I. Nape, X. Yang, X. Fu, M. Gong, D. Naidoo, and A. Forbes, “Creation and control of high-dimensional multi-partite classically entangled light,” Light Sci. Appl. 10, 50 (2021).
[Crossref]

Y. Shen, X. Yang, D. Naidoo, X. Fu, and A. Forbes, “Structured ray-wave vector vortex beams in multiple degrees of freedom from a laser,” Optica 7, 820–831 (2020).
[Crossref]

Y. Shen, Z. Wang, X. Fu, D. Naidoo, and A. Forbes, “Su(2) Poincaré sphere: a generalized representation for multidimensional structured light,” Phys. Rev. A 102, 031501 (2020).
[Crossref]

Nape, I.

Y. Shen, I. Nape, X. Yang, X. Fu, M. Gong, D. Naidoo, and A. Forbes, “Creation and control of high-dimensional multi-partite classically entangled light,” Light Sci. Appl. 10, 50 (2021).
[Crossref]

Nitzan, A.

Ocier, C. R.

C. R. Ocier, C. A. Richards, D. A. Bacon-Brown, Q. Ding, R. Kumar, T. J. Garcia, J. van de Groep, J.-H. Song, A. J. Cyphersmith, A. Rhode, A. N. Perry, A. J. Littlefield, J. Zhu, D. Xie, H. Gao, J. F. Messinger, M. L. Brongersma, K. C. Toussaint, L. L. Goddard, and P. V. Braun, “Direct laser writing of volumetric gradient index lenses and waveguides,” Light Sci. Appl. 9, 196 (2020).
[Crossref]

Perry, A. N.

C. R. Ocier, C. A. Richards, D. A. Bacon-Brown, Q. Ding, R. Kumar, T. J. Garcia, J. van de Groep, J.-H. Song, A. J. Cyphersmith, A. Rhode, A. N. Perry, A. J. Littlefield, J. Zhu, D. Xie, H. Gao, J. F. Messinger, M. L. Brongersma, K. C. Toussaint, L. L. Goddard, and P. V. Braun, “Direct laser writing of volumetric gradient index lenses and waveguides,” Light Sci. Appl. 9, 196 (2020).
[Crossref]

Petschulat, J.

Qian, X.-F.

Rhode, A.

C. R. Ocier, C. A. Richards, D. A. Bacon-Brown, Q. Ding, R. Kumar, T. J. Garcia, J. van de Groep, J.-H. Song, A. J. Cyphersmith, A. Rhode, A. N. Perry, A. J. Littlefield, J. Zhu, D. Xie, H. Gao, J. F. Messinger, M. L. Brongersma, K. C. Toussaint, L. L. Goddard, and P. V. Braun, “Direct laser writing of volumetric gradient index lenses and waveguides,” Light Sci. Appl. 9, 196 (2020).
[Crossref]

Richards, C. A.

C. R. Ocier, C. A. Richards, D. A. Bacon-Brown, Q. Ding, R. Kumar, T. J. Garcia, J. van de Groep, J.-H. Song, A. J. Cyphersmith, A. Rhode, A. N. Perry, A. J. Littlefield, J. Zhu, D. Xie, H. Gao, J. F. Messinger, M. L. Brongersma, K. C. Toussaint, L. L. Goddard, and P. V. Braun, “Direct laser writing of volumetric gradient index lenses and waveguides,” Light Sci. Appl. 9, 196 (2020).
[Crossref]

Schulman, L.

L. Schulman, Techniques and Applications of Path Integration, Dover Books on Physics (Dover Publications, 2005).

Semichaevsky, A. V.

A. V. Semichaevsky and M. E. Testorf, “Anything optical rays cannot do?” Proc. SPIE 4436, 56–67 (2001).
[Crossref]

Shen, Y.

Song, J.-H.

C. R. Ocier, C. A. Richards, D. A. Bacon-Brown, Q. Ding, R. Kumar, T. J. Garcia, J. van de Groep, J.-H. Song, A. J. Cyphersmith, A. Rhode, A. N. Perry, A. J. Littlefield, J. Zhu, D. Xie, H. Gao, J. F. Messinger, M. L. Brongersma, K. C. Toussaint, L. L. Goddard, and P. V. Braun, “Direct laser writing of volumetric gradient index lenses and waveguides,” Light Sci. Appl. 9, 196 (2020).
[Crossref]

Souriau, J.-M.

J.-M. Souriau, “Construction explicite de l’indice de maslov. applications,” in Group Theoretical Methods in Physics, A. Janner, T. Janssen, and M. Boon, eds. (Springer, 1976), pp. 117–148.

Sugic, D.

Testorf, M. E.

A. V. Semichaevsky and M. E. Testorf, “Anything optical rays cannot do?” Proc. SPIE 4436, 56–67 (2001).
[Crossref]

Thorne, K.

C. Misner, K. Thorne, and J. Wheeler, Gravitation (W. H. Freeman, 1973).

Toussaint, K. C.

C. R. Ocier, C. A. Richards, D. A. Bacon-Brown, Q. Ding, R. Kumar, T. J. Garcia, J. van de Groep, J.-H. Song, A. J. Cyphersmith, A. Rhode, A. N. Perry, A. J. Littlefield, J. Zhu, D. Xie, H. Gao, J. F. Messinger, M. L. Brongersma, K. C. Toussaint, L. L. Goddard, and P. V. Braun, “Direct laser writing of volumetric gradient index lenses and waveguides,” Light Sci. Appl. 9, 196 (2020).
[Crossref]

Urbach, P.

Vamivakas, A. N.

van de Groep, J.

C. R. Ocier, C. A. Richards, D. A. Bacon-Brown, Q. Ding, R. Kumar, T. J. Garcia, J. van de Groep, J.-H. Song, A. J. Cyphersmith, A. Rhode, A. N. Perry, A. J. Littlefield, J. Zhu, D. Xie, H. Gao, J. F. Messinger, M. L. Brongersma, K. C. Toussaint, L. L. Goddard, and P. V. Braun, “Direct laser writing of volumetric gradient index lenses and waveguides,” Light Sci. Appl. 9, 196 (2020).
[Crossref]

Wan, Z.

Wang, J.

C. He, J. Chang, Q. Hu, J. Wang, J. Antonello, H. He, S. Liu, J. Lin, B. Dai, D. S. Elson, P. Xi, H. Ma, and M. J. Booth, “Complex vectorial optics through gradient index lens cascades,” Nat. Commun. 10, 4264 (2019).
[Crossref]

Wang, Z.

Wheeler, J.

C. Misner, K. Thorne, and J. Wheeler, Gravitation (W. H. Freeman, 1973).

Wick, M.

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Cambridge University, 2020).

Xi, P.

C. He, J. Chang, Q. Hu, J. Wang, J. Antonello, H. He, S. Liu, J. Lin, B. Dai, D. S. Elson, P. Xi, H. Ma, and M. J. Booth, “Complex vectorial optics through gradient index lens cascades,” Nat. Commun. 10, 4264 (2019).
[Crossref]

Xie, D.

C. R. Ocier, C. A. Richards, D. A. Bacon-Brown, Q. Ding, R. Kumar, T. J. Garcia, J. van de Groep, J.-H. Song, A. J. Cyphersmith, A. Rhode, A. N. Perry, A. J. Littlefield, J. Zhu, D. Xie, H. Gao, J. F. Messinger, M. L. Brongersma, K. C. Toussaint, L. L. Goddard, and P. V. Braun, “Direct laser writing of volumetric gradient index lenses and waveguides,” Light Sci. Appl. 9, 196 (2020).
[Crossref]

Yang, X.

Zhu, J.

C. R. Ocier, C. A. Richards, D. A. Bacon-Brown, Q. Ding, R. Kumar, T. J. Garcia, J. van de Groep, J.-H. Song, A. J. Cyphersmith, A. Rhode, A. N. Perry, A. J. Littlefield, J. Zhu, D. Xie, H. Gao, J. F. Messinger, M. L. Brongersma, K. C. Toussaint, L. L. Goddard, and P. V. Braun, “Direct laser writing of volumetric gradient index lenses and waveguides,” Light Sci. Appl. 9, 196 (2020).
[Crossref]

Appl. Opt. (3)

J. Math. Phys. (1)

E. Mottola, “Functional integration over geometries,” J. Math. Phys. 36, 2470–2511 (1995).
[Crossref]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

J. Opt. Soc. Am. B (2)

J. Phys. A (1)

J. W. Dalhuisen and D. Bouwmeester, “Twistors and electromagnetic knots,” J. Phys. A 45, 135201 (2012).
[Crossref]

J. Phys. Photon. (1)

M. R. Dennis and M. A. Alonso, “Gaussian mode families from systems of rays,” J. Phys. Photon. 1, 025003 (2019).
[Crossref]

Light Sci. Appl. (2)

Y. Shen, I. Nape, X. Yang, X. Fu, M. Gong, D. Naidoo, and A. Forbes, “Creation and control of high-dimensional multi-partite classically entangled light,” Light Sci. Appl. 10, 50 (2021).
[Crossref]

C. R. Ocier, C. A. Richards, D. A. Bacon-Brown, Q. Ding, R. Kumar, T. J. Garcia, J. van de Groep, J.-H. Song, A. J. Cyphersmith, A. Rhode, A. N. Perry, A. J. Littlefield, J. Zhu, D. Xie, H. Gao, J. F. Messinger, M. L. Brongersma, K. C. Toussaint, L. L. Goddard, and P. V. Braun, “Direct laser writing of volumetric gradient index lenses and waveguides,” Light Sci. Appl. 9, 196 (2020).
[Crossref]

Nat. Commun. (1)

C. He, J. Chang, Q. Hu, J. Wang, J. Antonello, H. He, S. Liu, J. Lin, B. Dai, D. S. Elson, P. Xi, H. Ma, and M. J. Booth, “Complex vectorial optics through gradient index lens cascades,” Nat. Commun. 10, 4264 (2019).
[Crossref]

Opt. Express (3)

Opt. Lett. (3)

Optica (2)

Phys. Rev. A (1)

Y. Shen, Z. Wang, X. Fu, D. Naidoo, and A. Forbes, “Su(2) Poincaré sphere: a generalized representation for multidimensional structured light,” Phys. Rev. A 102, 031501 (2020).
[Crossref]

Proc. SPIE (2)

G. W. Forbes and M. A. Alonso, “What on earth is a ray and how can we use them best?” Proc. SPIE 3482, 22–31 (1998).
[Crossref]

A. V. Semichaevsky and M. E. Testorf, “Anything optical rays cannot do?” Proc. SPIE 4436, 56–67 (2001).
[Crossref]

Rep. Prog. Phys. (1)

M. V. Berry and K. E. Mount, “Semiclassical approximations in wave mechanics,” Rep. Prog. Phys. 35, 315–397 (1972).
[Crossref]

Other (7)

C. Misner, K. Thorne, and J. Wheeler, Gravitation (W. H. Freeman, 1973).

H. Kleinert, Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets (World Scientific, 2009).

L. Schulman, Techniques and Applications of Path Integration, Dover Books on Physics (Dover Publications, 2005).

M. Born and E. Wolf, Principles of Optics (Cambridge University, 2020).

S. M. Carroll, “Lecture notes on general relativity,” arXiv:gr-qc/9712019 (1997).

M. Chaichian, Path Integrals in Physics, Series in mathematical and computational physics (IOP Publishing, 2001), Vol. I and II.

J.-M. Souriau, “Construction explicite de l’indice de maslov. applications,” in Group Theoretical Methods in Physics, A. Janner, T. Janssen, and M. Boon, eds. (Springer, 1976), pp. 117–148.

Data Availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Figures (4)

Fig. 1.
Fig. 1. Basic construction and notation used for the path integral description. ${\dot X^a}(t)$ is the tangent vector to the curve (red line) followed by an optical ray in a background refractive index $n({X^a})$ . ${V^a}(t) $ and ${U^a}(t) $ are orthogonal vectors that represent directions of transverse polarization.
Fig. 2.
Fig. 2. Quadratic GRIN fiber rod and the discrete multiple lens relay.
Fig. 3.
Fig. 3. Gouy phase for an initial Gaussian beam wavepacket that is a ground-state eigen-function of the reduced form path integral.
Fig. 4.
Fig. 4. Longitudinal component of field propagation as described by parallel propagation along sinusoidal geodesics.

Tables (1)

Tables Icon

Table 1. Derived Optical Parameters for GRIN Material

Equations (108)

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S [ X , X ˙ ] = t I t F d t n 2 δ a b X ˙ a X ˙ b .
S [ X , X ˙ , Λ ] = 1 2 λ t I t F d t 1 Λ ( t ) δ a b X ˙ a X ˙ b + Λ ( t ) λ 2 n 2 ,
S [ X , P , Λ ] = t I t F d t P a X ˙ a H ( X , P , Λ ) ,
H ( X , P , Λ ) λ Λ ( t ) ( δ a b P a P b n 2 ) / 2.
δ t = ϵ ,
δ X a = ϵ X ˙ a ,
δ P a = ϵ P ˙ a ,
δ Λ = d ( ϵ Λ ) d t .
δ S [ X , P , Λ ] δ Λ ( t ) = λ ( δ a b P a P b n 2 ) / 2 = 0 ,
H ( X , P , Λ ) = 0.
( λ 2 δ a b a b + n 2 ) ϕ ( X ) = 0.
Λ ( t ) ( λ 2 δ a b a b + n 2 ) Φ ( X , t ) = i Φ ( X , t ) t ,
H ^ ( X , i , Λ ) Φ ( X , t ) = i λ Φ ( X , t ) t .
X F , t F | X I , t I Λ = [ d X ( t ) d P ( t ) ] I F e i λ S [ X , P , Λ ] ,
= [ d X ( t ) ] I F e i λ S [ X , X ˙ , Λ ] .
X F , t F | X I , t I = [ d X ( t ) d P ( t ) Λ ( t ) ] I F e i λ S [ X , P , Λ ] ,
= [ d X ( t ) d P ( t ) ] I F e i λ t I t F d t P a X ˙ a × δ ( δ a b P a P b n 2 ) .
𝟙 = Δ FP [ d ϵ ] δ [ F ( Λ ( ϵ ) ] ,
G = [ d ϵ ] .
Δ FP = det [ δ F δ ϵ ] F = 0 ,
= det [ d ( Λ ) d t ] Λ = 1 ,
= det [ d d t ] .
X F , t F | X I , t I = I F [ d X ( t ) d P ( t ) d Λ ( t ) ] I F 𝟙 e i λ S [ X , P , Λ ] ,
= I F [ d X ( t ) d P ( t ) d Λ ( t ) ] I F Δ FP [ d ϵ ] δ [ F ( Λ ( ϵ ) ] e i λ S [ X , P , Λ ] ,
= [ d ϵ ] I F [ d X ( t ) ] I F e i λ S [ X , X ˙ , Λ = 1 ] ( det [ d d t ] ) .
X F , t F | X I , t I = [ d X d P d Λ d ϵ ] I F G Δ FP δ [ F ( Λ ( ϵ ) ] e i λ S [ X , P , Λ ] ,
= [ d X d Λ d ϵ ] I F G Δ FP δ [ F ( Λ ( ϵ ) ] e i λ S [ X , X ˙ , Λ ] .
( λ 2 δ a b a b + n 2 ) K ( X , X ) = δ 3 ( X X ) .
K ( X , X ) = d 3 P ( 2 π ) 3 e i P a ( X X ) a P 2 n 2 .
K ( X , X ) = X | 1 ( λ 2 δ a b a b + n 2 ) | X .
K ( X , X ) = 0 d T X | e T ( λ 2 δ a b a b + n 2 ) | X ,
= 0 d T X ( 0 ) X ( T ) [ d X ( t ) ] e i ( 2 λ ) 0 T d t ( δ a b X ˙ a X ˙ b + n 2 ) .
T := t F t I ,
0 d T X F , t F | X I , t I = X F | X I
= 0 d T [ d X ] I F Δ FP e i λ S [ X , X ˙ , Λ = 1 ] .
Φ ( X F , t F ) | Φ ( X I , t F ) X F , t F | X I , t I .
V a ( X F , t F ) | V b ( X I , t I ) X F , t F | X I , t I a b .
( λ 2 ) a b + n 2 δ a b ) E b = 0 ,
( λ 2 n 2 ) a b + δ a b ) B b = 0.
{ ( λ 2 2 + n 2 ) δ a b + ( λ b ln n 2 ) ( λ a ) + ( λ 2 a b ln n 2 ) } E b = 0 ,
Δ a b E E b 0 ,
{ ( λ 2 2 + n 2 ) δ a b + ( λ b ln n 2 ) ( λ a ) δ a b ( λ c ln n 2 ) ( λ c ) } B b = 0 ,
Δ a b M B b 0.
H ^ a b E λ Λ ( t ) Δ a b E ,
H ^ a b M λ Λ ( t ) Δ a b M .
H ^ a b E ( X , , Λ ) E b ( X , t ) = i λ E a ( X , t ) t ,
H ^ a b M ( X , , Λ ) B b ( X , t ) = i λ B a ( X , t ) t .
S [ X , X ˙ , Λ ] S a b [ X , X ˙ , Λ ] .
S a b E [ X , P , Λ ] = t I t F d t P a X ˙ b H a b E ,
S a b M [ X , P , Λ ] = t I t F d t P a X ˙ b H a b M ,
H a b E = λ Λ ( t ) 2 [ ( P 2 n 2 ) δ a b + 2 i λ Γ b P a + λ 2 a Γ b ] ,
H a b M = λ Λ ( t ) 2 [ ( P 2 n 2 ) δ a b + 2 i λ Γ b P a 2 i λ Γ c P c δ a b ] ,
Γ a := 1 2 a ln n 2 .
X F , t F | X I , t I a b E = [ d X d P d Λ d ϵ ] I F G Δ FP δ [ F ( X a ( ϵ ) ] × P { e i λ S a b E [ X , P , Λ ] } ,
X F , t F | X I , t I a b M = [ d X d P d Λ d ϵ ] I F G Δ FP δ [ F ( X a ( ϵ ) ] × P { e i λ S a b M [ X , P , Λ ] } .
X F , t F | X I , t I a b E = [ d X d Λ d ϵ ] I F G Δ FP δ [ F ( X a ( ϵ ) ] × P { e i λ S a b E [ X , X ˙ , Λ ] } ,
X F , t F | X I , t I a b M = [ d X d Λ d ϵ ] I F G Δ FP δ [ F ( X a ( ϵ ) ] × P { e i λ S a b M [ X , X ˙ , Λ ] } ,
S a b E [ X , X ˙ , Λ ] = 1 2 t I t F d t ( 1 λ Λ ( t ) X ˙ c X ˙ c + λ Λ ( t ) n 2 ) δ a b + i λ Γ b X ˙ a + λ 2 a Γ b ,
= S [ X , X ˙ , Λ ] δ a b + i λ t I t F d t Γ b X ˙ a + O ( λ 2 ) ,
S a b M [ X , X ˙ , Λ ] = 1 2 t I t F d t ( 1 λ Λ ( t ) X ˙ c X ˙ c + λ Λ ( t ) n 2 ) δ a b + i λ Γ b X ˙ a i λ Γ c X ˙ c δ a b ,
= S [ X , X ˙ , Λ ] δ a b + i λ t I t F d t ( Γ b X ˙ a Γ c X ˙ c δ a b ) + O ( λ 2 ) .
E a ( X ) = e a ( X ) e 2 π i λ S ( X ) ,
B a ( X ) = b a ( X ) e 2 π i λ S ( X ) ,
u a e a / ( | e | ) ,
v a b a / ( | b | ) .
g a b = n 2 δ a b ,
Γ b c a = 1 2 g a d ( c g d b + b g d c d g b c ) ,
= 1 2 ( δ b a c ln n 2 + δ c a b ln n 2 δ b c a ln n 2 ) .
a V b a V b + Γ a c b V c .
D X ˙ V a X ˙ c c V a = t V a + Γ b c a V b X ˙ c = 0.
V a ( t F ) = P { exp ( t I t F d t Γ b c a X ˙ c ) } V b ( t I ) .
V a ( t F ) = P { exp ( t I t F d t M b a ( t ) ) } V b ( t I )
= m = 0 ( 1 ) m m ! t I t F d t m t I t F d t 1 P { M c a ( t m ) M b d ( t 1 ) } V b ( t I )
:= { δ b a + ( t I t F d t 1 M b a ( t 1 ) ) + ( t I t F d t 2 t I t 2 d t 1 M c a ( t 2 ) M b c ( t 1 ) ) + ( t I t F d t 1 M b a ( t 1 ) ) } V b ( t I ) .
d ( n V a ) d t = X ˙ a ( n V b ) b ln n ,
n ( t F ) V a ( t F ) = P { exp ( t I t F d t X ˙ a b ln n ) } n ( t I ) V b ( t I ) = P { exp ( t I t F d t X ˙ a Γ b ) } n ( t I ) V b ( t I ) .
V a ( t F ) = β P { exp ( t I t F d t α ( X ) ( X ˙ a X b X ˙ b X a ) ) } V b ( t I ) ,
= β P { exp ( t I t F d t α ( X ) J a b ) } V b ( t I ) ,
E a ( X ) = P { e i λ S a b ( X ) u b ( X ) } ,
B a ( X ) = P { e i λ S a b ( X ) v b ( X ) } .
X F , t F | X I , t I a b = P { e t I t F d t X ˙ a Γ b + i λ S a b [ X , X ˙ ] det ( D ) } ,
n 2 ( x , z ) = n 0 2 ( 1 x 2 / R 2 ) ,
X F , t F | X I , t I = I F [ d X ( t ) ] I F e i λ S [ X , X ˙ , Λ = 1 ] ( det [ d d t ] ) ,
S [ X , X ˙ , Λ = 1 ] = 1 2 λ t I t F d t ( x ˙ 2 + z ˙ 2 + λ 2 n 0 2 ( 1 x 2 / R 2 ) ) .
z ( t ) = z ( t ) class . + Z ( t ) .
z ( t ) class . = ( t t I ) λ n 0 + z I .
[ d Z ( t ) ] exp ( i 2 λ 2 d t Z ˙ 2 ) = ( det [ d d t 2 ] ) 1 / 2 .
x F , z F | x I , z I = [ d x ( t ) ] I F e i 2 λ S [ x , x ˙ ] ,
S [ x , x ˙ ] = n 0 2 z I z F d z ( x ˙ 2 x 2 / R 2 ) .
x F , z F | x I , z I = 1 ( 2 π i ) ( n 0 λ R sin ( z F z I ) / R ) × exp [ i n 0 4 λ R sin ( z F z I ) / R × [ ( x F 2 + x I 2 ) cos ( z F z I ) / R 2 x F x I ] ] .
( z F z I ) / R = m π ,
x F , z F | x I , z I = D e t ( S ) exp [ i Δ f + i Δ n ] ,
Det ( S ) = ( n 0 2 π λ R | sin ( z F z I ) / R | )
Δ n = ( n 0 4 λ R sin ( z F z I ) / R ) × [ ( x F 2 + x I 2 ) cos ( z F z I ) / R 2 x F x I ] ,
Δ f = π 2 ( 2 + 2 m ) .
ψ ( x I , z I ) = 1 ( 2 π A ) 1 / 2 exp ( x I 2 / 4 A ) .
ψ ( x F , z F ) = 1 ( 2 π F ) 1 / 2 exp ( x F 2 / 4 F ) e i θ ,
F := A [ cos 2 ( z F z I R ) + 2 A n 0 λ R sin 2 ( z F z I R ) ] ,
θ = tan 1 ( 2 A n 0 λ R cot ( z F z I ) / R ) ) n 0 x F 2 2 λ R cot ( z F z I ) / R ) ( A F 1 ) π 2 ( 2 + 2 m ) ,
θ = z R π ( 1 + m ) .
S r r E [ X , X ˙ ] = S [ X , X ˙ , Λ = 1 ] + i λ t I t F d t Γ r r ˙ ,
S z r E [ X , X ˙ ] = S [ X , X ˙ , Λ = 1 ] + i λ t I t F d t Γ r z ˙ ,
t I t F d t Γ r r ˙ = t I t F d t r ˙ r ln n = ln n ( t F ) n ( t I ) ,
t I t F d t Γ r z ˙ = t I t F d t z ˙ r ln n = z I z F d z 2 r / R 2 1 r 2 / R 2 .
r ( z ) = r I cos z / R ,
z I = 0 ,
κ := r I / R .
t I t F d t Γ r z ˙ = 2 ( 1 κ 2 ) 1 / 2 tan 1 ( sin z / R ( 1 κ 2 ) 1 / 2 ) := Θ .

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