Abstract

In a recent paper, Mansuripur et al. indicated and numerically verified the generation of the helical wavefront of optical beams using a conical-shape reflector. Because the optical reflection is largely free from chromatic aberrations, the conical reflector has an advantage of being able to manipulate the helical wavefront with broadband light such as white light or short light pulses. In this study, we introduce geometrical understanding of the function of the conical reflector using the spatially-dependent geometric phase, or more specifically, the spin redirection phase. We also present a theoretical analysis based on three-dimensional matrix calculus and elucidate relationships of the spin, orbital, and total angular momenta between input and output beams. These analyses are very useful when designing other optical devices that utilize spatially-dependent spin redirection phases. Moreover, we experimentally demonstrate the generation of helical beams from an ordinary Gaussian beam using a metallic conical-shape reflector.

© 2012 OSA

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References

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  1. M. V. Berry, “Quantal phase factors accompanying adiabatic changes,” Proc. R. Soc. London A 392, 45–57 (1984).
    [CrossRef]
  2. A. Shapere and F. Wilczek, eds., Geometric Phases in Physics (World Scientific, Singapore, 1989).
  3. R. Y. Chiao and Y.-S. Wu, “Manifestations of Berry’s topological phase for the photon,” Phys. Rev. Lett. 57, 933–936 (1986).
    [CrossRef] [PubMed]
  4. M. V. Berry, “Interpreting the anholonomy of coiled light,” Nature 326, 277–278 (1987).
    [CrossRef]
  5. A. Tomita and R. Y. Chiao, “Observation of Berry’s topological phase by use of an optical fiber,” Phys. Rev. Lett. 57, 937–940 (1986).
    [CrossRef] [PubMed]
  6. M. Kitano, T. Yabuzaki, and T. Ogawa, “Comment on Observation of Berry’s topological phase by use of an optical fiber,” Phys. Rev. Lett. 58, 523–523 (1987).
    [CrossRef] [PubMed]
  7. T. F. Jordan, “Quantum phases from reflections,” Phys. Rev. Lett. 60, 1584–1584 (1988).
    [CrossRef] [PubMed]
  8. R. Y. Chiao, A. Antaramian, K. M. Ganga, H. Jiao, S. R. Wilkinson, and H. Nathel, “Observation of a topological phase by means of a nonplanar Mach-Zehnder interferometer,” Phys. Rev. Lett. 60, 1214–1217 (1988).
    [CrossRef] [PubMed]
  9. M. Segev, R. Solomon, and A. Yariv, “Manifestation of Berry’s phase in image-bearing optical beams,” Phys. Rev. Lett. 69, 590–592 (1992).
    [CrossRef] [PubMed]
  10. E. J. Galvez and C. D. Holmes, “Geometric phase of optical rotators,” J. Opt. Soc. Am. A 16, 1981–1985 (1999).
    [CrossRef]
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  12. M. V. Berry, “The adiabatic phase and Pancharatnam’s phase for polarized light,” J. Mod. Opt. 34, 1401–1407 (1987).
    [CrossRef]
  13. R. Bhandari, “Polarization of light and topological phases,” Phys. Rep. 281, 1–64 (1997).
    [CrossRef]
  14. Z. Bomzon, G. Biener, V. Kleiner, and E. Hasman, “Space-variant pancharatnam-berry phase optical elements with computer-generated subwavelengths graings,” Opt. Lett. 27, 1141–1143 (2002).
    [CrossRef]
  15. 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] [PubMed]
  16. H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75, 826–829 (1995).
    [CrossRef] [PubMed]
  17. A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412, 313–316 (2001).
    [CrossRef] [PubMed]
  18. L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96, 163905 (2006).
    [CrossRef] [PubMed]
  19. M. Mansuripur, A. R. Zakharian, and E. M. Wright, “Spin and orbital angular momenta of light reflected from a cone,” Phys. Rev. A 84, 033813 (2011).
    [CrossRef]
  20. M. Mansuripur, A. R. Zakharian, and E. M. Wright, “Spin-to-orbital angular momentum exchange via reflection from a cone,” Proc. SPIE 8097, 809716 (2011).
    [CrossRef]
  21. C. Konz and G. Benford, “Geometric absorption of electromagnetic angular momentum,” Opt. Comm. 226, 249–254 (2003).
    [CrossRef]
  22. T. A. Nieminen, “Comment on Geometric absorption of electromagnetic angular momentum,” Opt. Comm. 235, 227–229 (2004).
    [CrossRef]

2011 (2)

M. Mansuripur, A. R. Zakharian, and E. M. Wright, “Spin and orbital angular momenta of light reflected from a cone,” Phys. Rev. A 84, 033813 (2011).
[CrossRef]

M. Mansuripur, A. R. Zakharian, and E. M. Wright, “Spin-to-orbital angular momentum exchange via reflection from a cone,” Proc. SPIE 8097, 809716 (2011).
[CrossRef]

2006 (1)

L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96, 163905 (2006).
[CrossRef] [PubMed]

2004 (1)

T. A. Nieminen, “Comment on Geometric absorption of electromagnetic angular momentum,” Opt. Comm. 235, 227–229 (2004).
[CrossRef]

2003 (1)

C. Konz and G. Benford, “Geometric absorption of electromagnetic angular momentum,” Opt. Comm. 226, 249–254 (2003).
[CrossRef]

2002 (1)

2001 (1)

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412, 313–316 (2001).
[CrossRef] [PubMed]

1999 (1)

1997 (1)

R. Bhandari, “Polarization of light and topological phases,” Phys. Rep. 281, 1–64 (1997).
[CrossRef]

1995 (1)

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75, 826–829 (1995).
[CrossRef] [PubMed]

1992 (2)

M. Segev, R. Solomon, and A. Yariv, “Manifestation of Berry’s phase in image-bearing optical beams,” Phys. Rev. Lett. 69, 590–592 (1992).
[CrossRef] [PubMed]

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] [PubMed]

1988 (2)

T. F. Jordan, “Quantum phases from reflections,” Phys. Rev. Lett. 60, 1584–1584 (1988).
[CrossRef] [PubMed]

R. Y. Chiao, A. Antaramian, K. M. Ganga, H. Jiao, S. R. Wilkinson, and H. Nathel, “Observation of a topological phase by means of a nonplanar Mach-Zehnder interferometer,” Phys. Rev. Lett. 60, 1214–1217 (1988).
[CrossRef] [PubMed]

1987 (3)

M. V. Berry, “Interpreting the anholonomy of coiled light,” Nature 326, 277–278 (1987).
[CrossRef]

M. Kitano, T. Yabuzaki, and T. Ogawa, “Comment on Observation of Berry’s topological phase by use of an optical fiber,” Phys. Rev. Lett. 58, 523–523 (1987).
[CrossRef] [PubMed]

M. V. Berry, “The adiabatic phase and Pancharatnam’s phase for polarized light,” J. Mod. Opt. 34, 1401–1407 (1987).
[CrossRef]

1986 (2)

A. Tomita and R. Y. Chiao, “Observation of Berry’s topological phase by use of an optical fiber,” Phys. Rev. Lett. 57, 937–940 (1986).
[CrossRef] [PubMed]

R. Y. Chiao and Y.-S. Wu, “Manifestations of Berry’s topological phase for the photon,” Phys. Rev. Lett. 57, 933–936 (1986).
[CrossRef] [PubMed]

1984 (1)

M. V. Berry, “Quantal phase factors accompanying adiabatic changes,” Proc. R. Soc. London A 392, 45–57 (1984).
[CrossRef]

1956 (1)

S. Pancharatnam, “Generalized theory of interference, and its applications,” Proc. Ind. Acad. Sci. A 44, 247–262 (1956).

Allen, L.

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] [PubMed]

Antaramian, A.

R. Y. Chiao, A. Antaramian, K. M. Ganga, H. Jiao, S. R. Wilkinson, and H. Nathel, “Observation of a topological phase by means of a nonplanar Mach-Zehnder interferometer,” Phys. Rev. Lett. 60, 1214–1217 (1988).
[CrossRef] [PubMed]

Beijersbergen, M. W.

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] [PubMed]

Benford, G.

C. Konz and G. Benford, “Geometric absorption of electromagnetic angular momentum,” Opt. Comm. 226, 249–254 (2003).
[CrossRef]

Berry, M. V.

M. V. Berry, “Interpreting the anholonomy of coiled light,” Nature 326, 277–278 (1987).
[CrossRef]

M. V. Berry, “The adiabatic phase and Pancharatnam’s phase for polarized light,” J. Mod. Opt. 34, 1401–1407 (1987).
[CrossRef]

M. V. Berry, “Quantal phase factors accompanying adiabatic changes,” Proc. R. Soc. London A 392, 45–57 (1984).
[CrossRef]

Bhandari, R.

R. Bhandari, “Polarization of light and topological phases,” Phys. Rep. 281, 1–64 (1997).
[CrossRef]

Biener, G.

Bomzon, Z.

Chiao, R. Y.

R. Y. Chiao, A. Antaramian, K. M. Ganga, H. Jiao, S. R. Wilkinson, and H. Nathel, “Observation of a topological phase by means of a nonplanar Mach-Zehnder interferometer,” Phys. Rev. Lett. 60, 1214–1217 (1988).
[CrossRef] [PubMed]

A. Tomita and R. Y. Chiao, “Observation of Berry’s topological phase by use of an optical fiber,” Phys. Rev. Lett. 57, 937–940 (1986).
[CrossRef] [PubMed]

R. Y. Chiao and Y.-S. Wu, “Manifestations of Berry’s topological phase for the photon,” Phys. Rev. Lett. 57, 933–936 (1986).
[CrossRef] [PubMed]

Friese, M. E. J.

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75, 826–829 (1995).
[CrossRef] [PubMed]

Galvez, E. J.

Ganga, K. M.

R. Y. Chiao, A. Antaramian, K. M. Ganga, H. Jiao, S. R. Wilkinson, and H. Nathel, “Observation of a topological phase by means of a nonplanar Mach-Zehnder interferometer,” Phys. Rev. Lett. 60, 1214–1217 (1988).
[CrossRef] [PubMed]

Hasman, E.

He, H.

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75, 826–829 (1995).
[CrossRef] [PubMed]

Heckenberg, N. R.

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75, 826–829 (1995).
[CrossRef] [PubMed]

Holmes, C. D.

Jiao, H.

R. Y. Chiao, A. Antaramian, K. M. Ganga, H. Jiao, S. R. Wilkinson, and H. Nathel, “Observation of a topological phase by means of a nonplanar Mach-Zehnder interferometer,” Phys. Rev. Lett. 60, 1214–1217 (1988).
[CrossRef] [PubMed]

Jordan, T. F.

T. F. Jordan, “Quantum phases from reflections,” Phys. Rev. Lett. 60, 1584–1584 (1988).
[CrossRef] [PubMed]

Kitano, M.

M. Kitano, T. Yabuzaki, and T. Ogawa, “Comment on Observation of Berry’s topological phase by use of an optical fiber,” Phys. Rev. Lett. 58, 523–523 (1987).
[CrossRef] [PubMed]

Kleiner, V.

Konz, C.

C. Konz and G. Benford, “Geometric absorption of electromagnetic angular momentum,” Opt. Comm. 226, 249–254 (2003).
[CrossRef]

Mair, A.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412, 313–316 (2001).
[CrossRef] [PubMed]

Mansuripur, M.

M. Mansuripur, A. R. Zakharian, and E. M. Wright, “Spin-to-orbital angular momentum exchange via reflection from a cone,” Proc. SPIE 8097, 809716 (2011).
[CrossRef]

M. Mansuripur, A. R. Zakharian, and E. M. Wright, “Spin and orbital angular momenta of light reflected from a cone,” Phys. Rev. A 84, 033813 (2011).
[CrossRef]

Manzo, C.

L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96, 163905 (2006).
[CrossRef] [PubMed]

Marrucci, L.

L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96, 163905 (2006).
[CrossRef] [PubMed]

Nathel, H.

R. Y. Chiao, A. Antaramian, K. M. Ganga, H. Jiao, S. R. Wilkinson, and H. Nathel, “Observation of a topological phase by means of a nonplanar Mach-Zehnder interferometer,” Phys. Rev. Lett. 60, 1214–1217 (1988).
[CrossRef] [PubMed]

Nieminen, T. A.

T. A. Nieminen, “Comment on Geometric absorption of electromagnetic angular momentum,” Opt. Comm. 235, 227–229 (2004).
[CrossRef]

Ogawa, T.

M. Kitano, T. Yabuzaki, and T. Ogawa, “Comment on Observation of Berry’s topological phase by use of an optical fiber,” Phys. Rev. Lett. 58, 523–523 (1987).
[CrossRef] [PubMed]

Pancharatnam, S.

S. Pancharatnam, “Generalized theory of interference, and its applications,” Proc. Ind. Acad. Sci. A 44, 247–262 (1956).

Paparo, D.

L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96, 163905 (2006).
[CrossRef] [PubMed]

Rubinsztein-Dunlop, H.

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75, 826–829 (1995).
[CrossRef] [PubMed]

Segev, M.

M. Segev, R. Solomon, and A. Yariv, “Manifestation of Berry’s phase in image-bearing optical beams,” Phys. Rev. Lett. 69, 590–592 (1992).
[CrossRef] [PubMed]

Shapere, A.

A. Shapere and F. Wilczek, eds., Geometric Phases in Physics (World Scientific, Singapore, 1989).

Solomon, R.

M. Segev, R. Solomon, and A. Yariv, “Manifestation of Berry’s phase in image-bearing optical beams,” Phys. Rev. Lett. 69, 590–592 (1992).
[CrossRef] [PubMed]

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] [PubMed]

Tomita, A.

A. Tomita and R. Y. Chiao, “Observation of Berry’s topological phase by use of an optical fiber,” Phys. Rev. Lett. 57, 937–940 (1986).
[CrossRef] [PubMed]

Vaziri, A.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412, 313–316 (2001).
[CrossRef] [PubMed]

Weihs, G.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412, 313–316 (2001).
[CrossRef] [PubMed]

Wilczek, F.

A. Shapere and F. Wilczek, eds., Geometric Phases in Physics (World Scientific, Singapore, 1989).

Wilkinson, S. R.

R. Y. Chiao, A. Antaramian, K. M. Ganga, H. Jiao, S. R. Wilkinson, and H. Nathel, “Observation of a topological phase by means of a nonplanar Mach-Zehnder interferometer,” Phys. Rev. Lett. 60, 1214–1217 (1988).
[CrossRef] [PubMed]

Woerdman, J. P.

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] [PubMed]

Wright, E. M.

M. Mansuripur, A. R. Zakharian, and E. M. Wright, “Spin and orbital angular momenta of light reflected from a cone,” Phys. Rev. A 84, 033813 (2011).
[CrossRef]

M. Mansuripur, A. R. Zakharian, and E. M. Wright, “Spin-to-orbital angular momentum exchange via reflection from a cone,” Proc. SPIE 8097, 809716 (2011).
[CrossRef]

Wu, Y.-S.

R. Y. Chiao and Y.-S. Wu, “Manifestations of Berry’s topological phase for the photon,” Phys. Rev. Lett. 57, 933–936 (1986).
[CrossRef] [PubMed]

Yabuzaki, T.

M. Kitano, T. Yabuzaki, and T. Ogawa, “Comment on Observation of Berry’s topological phase by use of an optical fiber,” Phys. Rev. Lett. 58, 523–523 (1987).
[CrossRef] [PubMed]

Yariv, A.

M. Segev, R. Solomon, and A. Yariv, “Manifestation of Berry’s phase in image-bearing optical beams,” Phys. Rev. Lett. 69, 590–592 (1992).
[CrossRef] [PubMed]

Zakharian, A. R.

M. Mansuripur, A. R. Zakharian, and E. M. Wright, “Spin-to-orbital angular momentum exchange via reflection from a cone,” Proc. SPIE 8097, 809716 (2011).
[CrossRef]

M. Mansuripur, A. R. Zakharian, and E. M. Wright, “Spin and orbital angular momenta of light reflected from a cone,” Phys. Rev. A 84, 033813 (2011).
[CrossRef]

Zeilinger, A.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412, 313–316 (2001).
[CrossRef] [PubMed]

J. Mod. Opt. (1)

M. V. Berry, “The adiabatic phase and Pancharatnam’s phase for polarized light,” J. Mod. Opt. 34, 1401–1407 (1987).
[CrossRef]

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

Nature (2)

M. V. Berry, “Interpreting the anholonomy of coiled light,” Nature 326, 277–278 (1987).
[CrossRef]

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412, 313–316 (2001).
[CrossRef] [PubMed]

Opt. Comm. (2)

C. Konz and G. Benford, “Geometric absorption of electromagnetic angular momentum,” Opt. Comm. 226, 249–254 (2003).
[CrossRef]

T. A. Nieminen, “Comment on Geometric absorption of electromagnetic angular momentum,” Opt. Comm. 235, 227–229 (2004).
[CrossRef]

Opt. Lett. (1)

Phys. Rep. (1)

R. Bhandari, “Polarization of light and topological phases,” Phys. Rep. 281, 1–64 (1997).
[CrossRef]

Phys. Rev. A (2)

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] [PubMed]

M. Mansuripur, A. R. Zakharian, and E. M. Wright, “Spin and orbital angular momenta of light reflected from a cone,” Phys. Rev. A 84, 033813 (2011).
[CrossRef]

Phys. Rev. Lett. (8)

L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96, 163905 (2006).
[CrossRef] [PubMed]

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75, 826–829 (1995).
[CrossRef] [PubMed]

A. Tomita and R. Y. Chiao, “Observation of Berry’s topological phase by use of an optical fiber,” Phys. Rev. Lett. 57, 937–940 (1986).
[CrossRef] [PubMed]

M. Kitano, T. Yabuzaki, and T. Ogawa, “Comment on Observation of Berry’s topological phase by use of an optical fiber,” Phys. Rev. Lett. 58, 523–523 (1987).
[CrossRef] [PubMed]

T. F. Jordan, “Quantum phases from reflections,” Phys. Rev. Lett. 60, 1584–1584 (1988).
[CrossRef] [PubMed]

R. Y. Chiao, A. Antaramian, K. M. Ganga, H. Jiao, S. R. Wilkinson, and H. Nathel, “Observation of a topological phase by means of a nonplanar Mach-Zehnder interferometer,” Phys. Rev. Lett. 60, 1214–1217 (1988).
[CrossRef] [PubMed]

M. Segev, R. Solomon, and A. Yariv, “Manifestation of Berry’s phase in image-bearing optical beams,” Phys. Rev. Lett. 69, 590–592 (1992).
[CrossRef] [PubMed]

R. Y. Chiao and Y.-S. Wu, “Manifestations of Berry’s topological phase for the photon,” Phys. Rev. Lett. 57, 933–936 (1986).
[CrossRef] [PubMed]

Proc. Ind. Acad. Sci. A (1)

S. Pancharatnam, “Generalized theory of interference, and its applications,” Proc. Ind. Acad. Sci. A 44, 247–262 (1956).

Proc. R. Soc. London A (1)

M. V. Berry, “Quantal phase factors accompanying adiabatic changes,” Proc. R. Soc. London A 392, 45–57 (1984).
[CrossRef]

Proc. SPIE (1)

M. Mansuripur, A. R. Zakharian, and E. M. Wright, “Spin-to-orbital angular momentum exchange via reflection from a cone,” Proc. SPIE 8097, 809716 (2011).
[CrossRef]

Other (1)

A. Shapere and F. Wilczek, eds., Geometric Phases in Physics (World Scientific, Singapore, 1989).

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Figures (7)

Fig. 1
Fig. 1

Equiphase surface of helical beam with l = +1 propagating along z-axis.

Fig. 2
Fig. 2

Transverse profile of fundamental Gaussian beam (l = 0) and Laguerre Gaussian beam with l = ±2. (a) Intensity profile. (b) Phase profile. (c) Interferogram with a spherical wave.

Fig. 3
Fig. 3

(a) Two-dimensional (dihedral) corner reflector. Two perfect mirrors, MA and MB, constituting the corner reflector are represented by the vectors normal to their surfaces, mA and mB, respectively. (b) Conical reflector. This reflector is made as the solid of revolution generated by the rotation of the dihedral corner reflector around z-axis.

Fig. 4
Fig. 4

The relationship between input and output angular momentum vectors. Sz = Szez, Lz = Lzez, and Jz = Jzez represent z components of the spin, orbital, and total angular momentum vectors, respectively.

Fig. 5
Fig. 5

Reflection from the surface of conical reflector. (a) The changing of the wave vector during the reflection on the conical reflector. (b) Change of the modified wave vector k̃n on k̃-sphere.

Fig. 6
Fig. 6

Fabricated conical reflector. For the grinding process of the fabrication, there remains a tiny recess at the center of the conical reflector.

Fig. 7
Fig. 7

Experimental setup and results. (a) Michelson interferometer to observe the spiral interferogram of the helical mode. (b) Intensity distribution of the reflected wave from the conical reflector. (c), (d) Interferograms generated by the reflected beam from the conical reflector and the quasi-spherical wave. The area encircled by the green dashed line corresponds to the beam diffracted from the recess at the bottom of the conical reflector. There are some deformations of wavefronts. We could remove this artifact by using a perfect conical reflector without a recess.

Equations (23)

Equations on this page are rendered with MathJax. Learn more.

E ( x , t ) = E ˜ l ( r ) e i ( k x ω t ) e ± + c . c . = E 0 ( r ) e i κ ( l ϕ + k z ) e i ω t e ± + c . c . ,
J z = L Z , S z = h ¯ κ ( l + σ ) ,
= ( π , m ) = 2 m m T ,
= i = 1 N i = ( 1 ) N ( θ T , a T ) ,
0 = B A = A B = ( 1 0 0 0 1 0 0 0 1 ) ,
CR ( ϕ ) = ( ϕ , e z ) 0 ( ϕ , e z ) = ( cos 2 ϕ sin 2 ϕ 0 sin 2 ϕ cos 2 ϕ 0 0 0 1 ) ,
E ˜ out ( r ) = CR ( ϕ ) E ˜ in ( r ) .
E in ( x , t ) = E ˜ in ( r ) e i ( k z + ω t ) + c . c . = E 0 ( r ) e i ( l ϕ + k z + ω t ) e + c . c . ,
E out ( x , t ) = e i 2 σ ϕ E ˜ in ( r ) e i ( k z ω t ) + c . c . = E 0 ( r ) e i [ ( l + 2 σ ) ϕ + k z ω t ] e ± + c . c . = E ˜ l 2 σ ( x ) e i ω t e ± + c . c .
S z out = S z in ,
L z out = L z in + 2 S z in ,
J z out = J z in ,
σ ( t ) = σ k ( t ) k .
σ n = σ n k n k = ( 1 ) n σ 0 k n k ,
k ˜ n = ( 1 ) n k n k .
γ = σ Ω .
E ( x , t ) = E ˜ ( x ) e i ω t + c . c . ,
σ i E ˜ ( x ) × E ˜ * ( x ) | E ˜ ( x ) | 2 .
E ˜ ( x ) = E 0 e i k x ( e i ϕ 1 cos θ e i ϕ 2 sin θ 0 ) ,
σ = sin ( ϕ 2 ϕ 1 ) sin ( 2 θ ) e z ,
σ = i E ˜ ( x ) × E ˜ * ( x ) | E ˜ ( x ) | 2 = i E ˜ ( x ) × E ˜ * ( x ) | E ˜ ( x ) | 2 = σ .
σ k σ k ,
σ = σ k k .

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