Abstract

We present a study of the properties of the transversal “spin angular momentum” and “orbital angular momentum” operators. We show that the “spin angular momentum” operators are generators of spatial translations that depend on helicity and frequency and that the “orbital angular momentum” operators generate transformations that are a sequence of this kind of translation and rotation. We give some examples of the use of these operators in light–matter interaction problems. Their relationship with the helicity operator allows us to involve electromagnetic duality symmetry in the analysis. We also find that simultaneous eigenstates of the three “spin” operators and parity define a type of standing mode that has recently been singled out for the interaction of light with chiral molecules. With respect to the relationship between “spin angular momentum,” polarization, and total angular momentum, we show that, except for the case of a single plane wave, the total angular momentum of the field is decoupled from its vectorial degrees of freedom even in the regime in which the paraxial approximation holds. Finally, we point out a relationship between the three “spin” operators and the spatial part of the Pauli–Lubanski four vector.

© 2014 Optical Society of America

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  1. A. Messiah, Quantum Mechanics (Dover, 1999).
  2. J. D. Jackson, Classical Electrodynamics (Wiley, 1998).
  3. S. J. Van Enk and G. Nienhuis, “Commutation rules and eigenvalues of spin and orbital angular momentum of radiation fields,” J. Mod. Opt. 41, 963–977 (1994).
    [CrossRef]
  4. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).
  5. R. Jáuregui and S. Hacyan, “Quantum-mechanical properties of Bessel beams,” Phys. Rev. A 71, 033411 (2005).
    [CrossRef]
  6. S. M. Barnett, “Rotation of electromagnetic fields and the nature of optical angular momentum,” J. Mod. Opt. 57, 1339–1343 (2010).
    [CrossRef]
  7. K. Y. Bliokh, M. Alonso, E. Ostrovskaya, and A. Aiello, “Angular momenta and spin-orbit interaction of nonparaxial light in free space,” Phys. Rev. A 82, 063825 (2010).
    [CrossRef]
  8. K. Y. Bliokh, A. Y. Bekshaev, and F. Nori, “Dual electromagnetism: helicity, spin, momentum and angular momentum,” New J. Phys. 15, 033026 (2013).
    [CrossRef]
  9. C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Photons and Atoms—Introduction to Quantum Electrodynamics (Wiley-Interscience, 1997).
  10. M. G. Calkin, “An invariance property of the free electromagnetic field,” Am. J. Phys. 33, 958–960 (1965).
    [CrossRef]
  11. D. Zwanziger, “Quantum field theory of particles with both electric and magnetic charges,” Phys. Rev. 176, 1489–1495 (1968).
    [CrossRef]
  12. W.-K. Tung, Group Theory in Physics (World Scientific, 1985).
  13. I. Fernandez-Corbaton, X. Zambrana-Puyalto, and G. Molina-Terriza, “Helicity and angular momentum: a symmetry-based framework for the study of light-matter interactions,” Phys. Rev. A 86, 042103 (2012).
    [CrossRef]
  14. I. Fernandez-Corbaton, X. Zambrana-Puyalto, N. Tischler, X. Vidal, M. L. Juan, and G. Molina-Terriza, “Electromagnetic duality symmetry and helicity conservation for the macroscopic Maxwell’s equations,” Phys. Rev. Lett. 111, 060401 (2013).
    [CrossRef]
  15. X. Zambrana-Puyalto, X. Vidal, M. L. Juan, and G. Molina-Terriza, “Dual and anti-dual modes in dielectric spheres,” Opt. Express 21, 17520–17530 (2013).
    [CrossRef]
  16. I. Fernandez-Corbaton and G. Molina-Terriza, “Role of duality symmetry in transformation optics,” Phys. Rev. B 88, 085111 (2013).
    [CrossRef]
  17. S. J. v. Enk and G. Nienhuis, “Spin and orbital angular momentum of photons,” Europhys. Lett. 25, 497–501 (1994).
    [CrossRef]
  18. C. Cohen-Tannoudji, B. Diu, and F. Laloe, Quantum Mechanics, 1st ed. (Wiley, 1991), Vol. 1.
  19. J. J. Sakurai, Modern Quantum Mechanics (Revised Edition), 1st ed. (Addison Wesley, 1993).
  20. R. P. Cameron, S. M. Barnett, and A. M. Yao, “Optical helicity, optical spin and related quantities in electromagnetic theory,” New J. Phys. 14, 053050 (2012).
    [CrossRef]
  21. Y. Tang and A. E. Cohen, “Optical chirality and its interaction with matter,” Phys. Rev. Lett. 104, 163901 (2010).
    [CrossRef]
  22. A. T. O’Neil, I. MacVicar, L. Allen, and M. J. Padgett, “Intrinsic and extrinsic nature of the orbital angular momentum of a light beam,” Phys. Rev. Lett. 88, 053601 (2002).
    [CrossRef]
  23. M. Padgett and R. Bowman, “Tweezers with a twist,” Nat. Photonics 5, 343–348 (2011).
    [CrossRef]
  24. P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill Kogakusha, 1953).
  25. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, 1941).
  26. The shift in the integration interval due to this change is irrelevant because the argument inside the integral is 2π-periodic in the integration variable.
  27. R. Penrose and W. Rindler, Spinors and Space-Time: Spinor and Twistor Methods in Space-Time Geometry (Cambridge University, 1986), Vol. 2.
  28. K. Y. Bliokh and F. Nori, “Characterizing optical chirality,” Phys. Rev. A 83, 021803 (2011).
    [CrossRef]

2013

K. Y. Bliokh, A. Y. Bekshaev, and F. Nori, “Dual electromagnetism: helicity, spin, momentum and angular momentum,” New J. Phys. 15, 033026 (2013).
[CrossRef]

I. Fernandez-Corbaton, X. Zambrana-Puyalto, N. Tischler, X. Vidal, M. L. Juan, and G. Molina-Terriza, “Electromagnetic duality symmetry and helicity conservation for the macroscopic Maxwell’s equations,” Phys. Rev. Lett. 111, 060401 (2013).
[CrossRef]

X. Zambrana-Puyalto, X. Vidal, M. L. Juan, and G. Molina-Terriza, “Dual and anti-dual modes in dielectric spheres,” Opt. Express 21, 17520–17530 (2013).
[CrossRef]

I. Fernandez-Corbaton and G. Molina-Terriza, “Role of duality symmetry in transformation optics,” Phys. Rev. B 88, 085111 (2013).
[CrossRef]

2012

R. P. Cameron, S. M. Barnett, and A. M. Yao, “Optical helicity, optical spin and related quantities in electromagnetic theory,” New J. Phys. 14, 053050 (2012).
[CrossRef]

I. Fernandez-Corbaton, X. Zambrana-Puyalto, and G. Molina-Terriza, “Helicity and angular momentum: a symmetry-based framework for the study of light-matter interactions,” Phys. Rev. A 86, 042103 (2012).
[CrossRef]

2011

M. Padgett and R. Bowman, “Tweezers with a twist,” Nat. Photonics 5, 343–348 (2011).
[CrossRef]

K. Y. Bliokh and F. Nori, “Characterizing optical chirality,” Phys. Rev. A 83, 021803 (2011).
[CrossRef]

2010

Y. Tang and A. E. Cohen, “Optical chirality and its interaction with matter,” Phys. Rev. Lett. 104, 163901 (2010).
[CrossRef]

S. M. Barnett, “Rotation of electromagnetic fields and the nature of optical angular momentum,” J. Mod. Opt. 57, 1339–1343 (2010).
[CrossRef]

K. Y. Bliokh, M. Alonso, E. Ostrovskaya, and A. Aiello, “Angular momenta and spin-orbit interaction of nonparaxial light in free space,” Phys. Rev. A 82, 063825 (2010).
[CrossRef]

2005

R. Jáuregui and S. Hacyan, “Quantum-mechanical properties of Bessel beams,” Phys. Rev. A 71, 033411 (2005).
[CrossRef]

2002

A. T. O’Neil, I. MacVicar, L. Allen, and M. J. Padgett, “Intrinsic and extrinsic nature of the orbital angular momentum of a light beam,” Phys. Rev. Lett. 88, 053601 (2002).
[CrossRef]

1994

S. J. v. Enk and G. Nienhuis, “Spin and orbital angular momentum of photons,” Europhys. Lett. 25, 497–501 (1994).
[CrossRef]

S. J. Van Enk and G. Nienhuis, “Commutation rules and eigenvalues of spin and orbital angular momentum of radiation fields,” J. Mod. Opt. 41, 963–977 (1994).
[CrossRef]

1968

D. Zwanziger, “Quantum field theory of particles with both electric and magnetic charges,” Phys. Rev. 176, 1489–1495 (1968).
[CrossRef]

1965

M. G. Calkin, “An invariance property of the free electromagnetic field,” Am. J. Phys. 33, 958–960 (1965).
[CrossRef]

Aiello, A.

K. Y. Bliokh, M. Alonso, E. Ostrovskaya, and A. Aiello, “Angular momenta and spin-orbit interaction of nonparaxial light in free space,” Phys. Rev. A 82, 063825 (2010).
[CrossRef]

Allen, L.

A. T. O’Neil, I. MacVicar, L. Allen, and M. J. Padgett, “Intrinsic and extrinsic nature of the orbital angular momentum of a light beam,” Phys. Rev. Lett. 88, 053601 (2002).
[CrossRef]

Alonso, M.

K. Y. Bliokh, M. Alonso, E. Ostrovskaya, and A. Aiello, “Angular momenta and spin-orbit interaction of nonparaxial light in free space,” Phys. Rev. A 82, 063825 (2010).
[CrossRef]

Barnett, S. M.

R. P. Cameron, S. M. Barnett, and A. M. Yao, “Optical helicity, optical spin and related quantities in electromagnetic theory,” New J. Phys. 14, 053050 (2012).
[CrossRef]

S. M. Barnett, “Rotation of electromagnetic fields and the nature of optical angular momentum,” J. Mod. Opt. 57, 1339–1343 (2010).
[CrossRef]

Bekshaev, A. Y.

K. Y. Bliokh, A. Y. Bekshaev, and F. Nori, “Dual electromagnetism: helicity, spin, momentum and angular momentum,” New J. Phys. 15, 033026 (2013).
[CrossRef]

Bliokh, K. Y.

K. Y. Bliokh, A. Y. Bekshaev, and F. Nori, “Dual electromagnetism: helicity, spin, momentum and angular momentum,” New J. Phys. 15, 033026 (2013).
[CrossRef]

K. Y. Bliokh and F. Nori, “Characterizing optical chirality,” Phys. Rev. A 83, 021803 (2011).
[CrossRef]

K. Y. Bliokh, M. Alonso, E. Ostrovskaya, and A. Aiello, “Angular momenta and spin-orbit interaction of nonparaxial light in free space,” Phys. Rev. A 82, 063825 (2010).
[CrossRef]

Bowman, R.

M. Padgett and R. Bowman, “Tweezers with a twist,” Nat. Photonics 5, 343–348 (2011).
[CrossRef]

Calkin, M. G.

M. G. Calkin, “An invariance property of the free electromagnetic field,” Am. J. Phys. 33, 958–960 (1965).
[CrossRef]

Cameron, R. P.

R. P. Cameron, S. M. Barnett, and A. M. Yao, “Optical helicity, optical spin and related quantities in electromagnetic theory,” New J. Phys. 14, 053050 (2012).
[CrossRef]

Cohen, A. E.

Y. Tang and A. E. Cohen, “Optical chirality and its interaction with matter,” Phys. Rev. Lett. 104, 163901 (2010).
[CrossRef]

Cohen-Tannoudji, C.

C. Cohen-Tannoudji, B. Diu, and F. Laloe, Quantum Mechanics, 1st ed. (Wiley, 1991), Vol. 1.

C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Photons and Atoms—Introduction to Quantum Electrodynamics (Wiley-Interscience, 1997).

Diu, B.

C. Cohen-Tannoudji, B. Diu, and F. Laloe, Quantum Mechanics, 1st ed. (Wiley, 1991), Vol. 1.

Dupont-Roc, J.

C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Photons and Atoms—Introduction to Quantum Electrodynamics (Wiley-Interscience, 1997).

Enk, S. J. v.

S. J. v. Enk and G. Nienhuis, “Spin and orbital angular momentum of photons,” Europhys. Lett. 25, 497–501 (1994).
[CrossRef]

Fernandez-Corbaton, I.

I. Fernandez-Corbaton and G. Molina-Terriza, “Role of duality symmetry in transformation optics,” Phys. Rev. B 88, 085111 (2013).
[CrossRef]

I. Fernandez-Corbaton, X. Zambrana-Puyalto, N. Tischler, X. Vidal, M. L. Juan, and G. Molina-Terriza, “Electromagnetic duality symmetry and helicity conservation for the macroscopic Maxwell’s equations,” Phys. Rev. Lett. 111, 060401 (2013).
[CrossRef]

I. Fernandez-Corbaton, X. Zambrana-Puyalto, and G. Molina-Terriza, “Helicity and angular momentum: a symmetry-based framework for the study of light-matter interactions,” Phys. Rev. A 86, 042103 (2012).
[CrossRef]

Feshbach, H.

P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill Kogakusha, 1953).

Grynberg, G.

C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Photons and Atoms—Introduction to Quantum Electrodynamics (Wiley-Interscience, 1997).

Hacyan, S.

R. Jáuregui and S. Hacyan, “Quantum-mechanical properties of Bessel beams,” Phys. Rev. A 71, 033411 (2005).
[CrossRef]

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (Wiley, 1998).

Jáuregui, R.

R. Jáuregui and S. Hacyan, “Quantum-mechanical properties of Bessel beams,” Phys. Rev. A 71, 033411 (2005).
[CrossRef]

Juan, M. L.

I. Fernandez-Corbaton, X. Zambrana-Puyalto, N. Tischler, X. Vidal, M. L. Juan, and G. Molina-Terriza, “Electromagnetic duality symmetry and helicity conservation for the macroscopic Maxwell’s equations,” Phys. Rev. Lett. 111, 060401 (2013).
[CrossRef]

X. Zambrana-Puyalto, X. Vidal, M. L. Juan, and G. Molina-Terriza, “Dual and anti-dual modes in dielectric spheres,” Opt. Express 21, 17520–17530 (2013).
[CrossRef]

Laloe, F.

C. Cohen-Tannoudji, B. Diu, and F. Laloe, Quantum Mechanics, 1st ed. (Wiley, 1991), Vol. 1.

MacVicar, I.

A. T. O’Neil, I. MacVicar, L. Allen, and M. J. Padgett, “Intrinsic and extrinsic nature of the orbital angular momentum of a light beam,” Phys. Rev. Lett. 88, 053601 (2002).
[CrossRef]

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

Messiah, A.

A. Messiah, Quantum Mechanics (Dover, 1999).

Molina-Terriza, G.

X. Zambrana-Puyalto, X. Vidal, M. L. Juan, and G. Molina-Terriza, “Dual and anti-dual modes in dielectric spheres,” Opt. Express 21, 17520–17530 (2013).
[CrossRef]

I. Fernandez-Corbaton, X. Zambrana-Puyalto, N. Tischler, X. Vidal, M. L. Juan, and G. Molina-Terriza, “Electromagnetic duality symmetry and helicity conservation for the macroscopic Maxwell’s equations,” Phys. Rev. Lett. 111, 060401 (2013).
[CrossRef]

I. Fernandez-Corbaton and G. Molina-Terriza, “Role of duality symmetry in transformation optics,” Phys. Rev. B 88, 085111 (2013).
[CrossRef]

I. Fernandez-Corbaton, X. Zambrana-Puyalto, and G. Molina-Terriza, “Helicity and angular momentum: a symmetry-based framework for the study of light-matter interactions,” Phys. Rev. A 86, 042103 (2012).
[CrossRef]

Morse, P. M.

P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill Kogakusha, 1953).

Nienhuis, G.

S. J. v. Enk and G. Nienhuis, “Spin and orbital angular momentum of photons,” Europhys. Lett. 25, 497–501 (1994).
[CrossRef]

S. J. Van Enk and G. Nienhuis, “Commutation rules and eigenvalues of spin and orbital angular momentum of radiation fields,” J. Mod. Opt. 41, 963–977 (1994).
[CrossRef]

Nori, F.

K. Y. Bliokh, A. Y. Bekshaev, and F. Nori, “Dual electromagnetism: helicity, spin, momentum and angular momentum,” New J. Phys. 15, 033026 (2013).
[CrossRef]

K. Y. Bliokh and F. Nori, “Characterizing optical chirality,” Phys. Rev. A 83, 021803 (2011).
[CrossRef]

O’Neil, A. T.

A. T. O’Neil, I. MacVicar, L. Allen, and M. J. Padgett, “Intrinsic and extrinsic nature of the orbital angular momentum of a light beam,” Phys. Rev. Lett. 88, 053601 (2002).
[CrossRef]

Ostrovskaya, E.

K. Y. Bliokh, M. Alonso, E. Ostrovskaya, and A. Aiello, “Angular momenta and spin-orbit interaction of nonparaxial light in free space,” Phys. Rev. A 82, 063825 (2010).
[CrossRef]

Padgett, M.

M. Padgett and R. Bowman, “Tweezers with a twist,” Nat. Photonics 5, 343–348 (2011).
[CrossRef]

Padgett, M. J.

A. T. O’Neil, I. MacVicar, L. Allen, and M. J. Padgett, “Intrinsic and extrinsic nature of the orbital angular momentum of a light beam,” Phys. Rev. Lett. 88, 053601 (2002).
[CrossRef]

Penrose, R.

R. Penrose and W. Rindler, Spinors and Space-Time: Spinor and Twistor Methods in Space-Time Geometry (Cambridge University, 1986), Vol. 2.

Rindler, W.

R. Penrose and W. Rindler, Spinors and Space-Time: Spinor and Twistor Methods in Space-Time Geometry (Cambridge University, 1986), Vol. 2.

Sakurai, J. J.

J. J. Sakurai, Modern Quantum Mechanics (Revised Edition), 1st ed. (Addison Wesley, 1993).

Stratton, J. A.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, 1941).

Tang, Y.

Y. Tang and A. E. Cohen, “Optical chirality and its interaction with matter,” Phys. Rev. Lett. 104, 163901 (2010).
[CrossRef]

Tischler, N.

I. Fernandez-Corbaton, X. Zambrana-Puyalto, N. Tischler, X. Vidal, M. L. Juan, and G. Molina-Terriza, “Electromagnetic duality symmetry and helicity conservation for the macroscopic Maxwell’s equations,” Phys. Rev. Lett. 111, 060401 (2013).
[CrossRef]

Tung, W.-K.

W.-K. Tung, Group Theory in Physics (World Scientific, 1985).

Van Enk, S. J.

S. J. Van Enk and G. Nienhuis, “Commutation rules and eigenvalues of spin and orbital angular momentum of radiation fields,” J. Mod. Opt. 41, 963–977 (1994).
[CrossRef]

Vidal, X.

I. Fernandez-Corbaton, X. Zambrana-Puyalto, N. Tischler, X. Vidal, M. L. Juan, and G. Molina-Terriza, “Electromagnetic duality symmetry and helicity conservation for the macroscopic Maxwell’s equations,” Phys. Rev. Lett. 111, 060401 (2013).
[CrossRef]

X. Zambrana-Puyalto, X. Vidal, M. L. Juan, and G. Molina-Terriza, “Dual and anti-dual modes in dielectric spheres,” Opt. Express 21, 17520–17530 (2013).
[CrossRef]

Wolf, E.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

Yao, A. M.

R. P. Cameron, S. M. Barnett, and A. M. Yao, “Optical helicity, optical spin and related quantities in electromagnetic theory,” New J. Phys. 14, 053050 (2012).
[CrossRef]

Zambrana-Puyalto, X.

I. Fernandez-Corbaton, X. Zambrana-Puyalto, N. Tischler, X. Vidal, M. L. Juan, and G. Molina-Terriza, “Electromagnetic duality symmetry and helicity conservation for the macroscopic Maxwell’s equations,” Phys. Rev. Lett. 111, 060401 (2013).
[CrossRef]

X. Zambrana-Puyalto, X. Vidal, M. L. Juan, and G. Molina-Terriza, “Dual and anti-dual modes in dielectric spheres,” Opt. Express 21, 17520–17530 (2013).
[CrossRef]

I. Fernandez-Corbaton, X. Zambrana-Puyalto, and G. Molina-Terriza, “Helicity and angular momentum: a symmetry-based framework for the study of light-matter interactions,” Phys. Rev. A 86, 042103 (2012).
[CrossRef]

Zwanziger, D.

D. Zwanziger, “Quantum field theory of particles with both electric and magnetic charges,” Phys. Rev. 176, 1489–1495 (1968).
[CrossRef]

Am. J. Phys.

M. G. Calkin, “An invariance property of the free electromagnetic field,” Am. J. Phys. 33, 958–960 (1965).
[CrossRef]

Europhys. Lett.

S. J. v. Enk and G. Nienhuis, “Spin and orbital angular momentum of photons,” Europhys. Lett. 25, 497–501 (1994).
[CrossRef]

J. Mod. Opt.

S. J. Van Enk and G. Nienhuis, “Commutation rules and eigenvalues of spin and orbital angular momentum of radiation fields,” J. Mod. Opt. 41, 963–977 (1994).
[CrossRef]

S. M. Barnett, “Rotation of electromagnetic fields and the nature of optical angular momentum,” J. Mod. Opt. 57, 1339–1343 (2010).
[CrossRef]

Nat. Photonics

M. Padgett and R. Bowman, “Tweezers with a twist,” Nat. Photonics 5, 343–348 (2011).
[CrossRef]

New J. Phys.

R. P. Cameron, S. M. Barnett, and A. M. Yao, “Optical helicity, optical spin and related quantities in electromagnetic theory,” New J. Phys. 14, 053050 (2012).
[CrossRef]

K. Y. Bliokh, A. Y. Bekshaev, and F. Nori, “Dual electromagnetism: helicity, spin, momentum and angular momentum,” New J. Phys. 15, 033026 (2013).
[CrossRef]

Opt. Express

Phys. Rev.

D. Zwanziger, “Quantum field theory of particles with both electric and magnetic charges,” Phys. Rev. 176, 1489–1495 (1968).
[CrossRef]

Phys. Rev. A

I. Fernandez-Corbaton, X. Zambrana-Puyalto, and G. Molina-Terriza, “Helicity and angular momentum: a symmetry-based framework for the study of light-matter interactions,” Phys. Rev. A 86, 042103 (2012).
[CrossRef]

K. Y. Bliokh, M. Alonso, E. Ostrovskaya, and A. Aiello, “Angular momenta and spin-orbit interaction of nonparaxial light in free space,” Phys. Rev. A 82, 063825 (2010).
[CrossRef]

R. Jáuregui and S. Hacyan, “Quantum-mechanical properties of Bessel beams,” Phys. Rev. A 71, 033411 (2005).
[CrossRef]

K. Y. Bliokh and F. Nori, “Characterizing optical chirality,” Phys. Rev. A 83, 021803 (2011).
[CrossRef]

Phys. Rev. B

I. Fernandez-Corbaton and G. Molina-Terriza, “Role of duality symmetry in transformation optics,” Phys. Rev. B 88, 085111 (2013).
[CrossRef]

Phys. Rev. Lett.

I. Fernandez-Corbaton, X. Zambrana-Puyalto, N. Tischler, X. Vidal, M. L. Juan, and G. Molina-Terriza, “Electromagnetic duality symmetry and helicity conservation for the macroscopic Maxwell’s equations,” Phys. Rev. Lett. 111, 060401 (2013).
[CrossRef]

Y. Tang and A. E. Cohen, “Optical chirality and its interaction with matter,” Phys. Rev. Lett. 104, 163901 (2010).
[CrossRef]

A. T. O’Neil, I. MacVicar, L. Allen, and M. J. Padgett, “Intrinsic and extrinsic nature of the orbital angular momentum of a light beam,” Phys. Rev. Lett. 88, 053601 (2002).
[CrossRef]

Other

P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill Kogakusha, 1953).

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, 1941).

The shift in the integration interval due to this change is irrelevant because the argument inside the integral is 2π-periodic in the integration variable.

R. Penrose and W. Rindler, Spinors and Space-Time: Spinor and Twistor Methods in Space-Time Geometry (Cambridge University, 1986), Vol. 2.

W.-K. Tung, Group Theory in Physics (World Scientific, 1985).

C. Cohen-Tannoudji, B. Diu, and F. Laloe, Quantum Mechanics, 1st ed. (Wiley, 1991), Vol. 1.

J. J. Sakurai, Modern Quantum Mechanics (Revised Edition), 1st ed. (Addison Wesley, 1993).

C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Photons and Atoms—Introduction to Quantum Electrodynamics (Wiley-Interscience, 1997).

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

A. Messiah, Quantum Mechanics (Dover, 1999).

J. D. Jackson, Classical Electrodynamics (Wiley, 1998).

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Figures (1)

Fig. 1.
Fig. 1.

Diagrams on the left represent the (a), (c), (e) transverse and (g) longitudinal intensity patterns of Gaussian-like monochromatic fields with different helicity content. The diagrams on the right show the effect that the application of transformations generated by S^ has on these fields. (a)–(f) Effect of exp(iβxS^x) on the transverse intensity pattern for (a), (b) field of well-defined helicity equal to 1, (c), (d) field of well-defined helicity equal to 1, and (e), (f) field containing both helicity components. (g) and (f) show the effect of exp(iβzS^z) on the longitudinal intensity pattern of a field containing both helicity components.

Equations (20)

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

Λ=J·P|P|.
S^F=dp(n^p,+n^p,)p|p|,
S^mFτ(p)=p|p|(i=13piSi|p|)Fτ(p)=p|p|(p|p|·S)Fτ(p),
Λ=J·P|P|=S·P|P|,
S^mFν(p)=pν|p|Fν(p).
S^mνdppν|p||ΨpνΨpν|=dpp|p|(|Ψp+Ψp+||ΨpΨp|).
[S^j,S^k]=0,[L^j,L^k]=ilεjkl(L^lS^l),[S^j,L^k]=ilεjklS^l.
S^=ΛP|P|.
Ψp¯ν¯|ΛP|P||Ψpν=pν|p|Ψp¯ν¯|Ψpν=pν|p|δp¯pν¯ν,
exp(iβ·ΛP/|P|)|Ψp=exp(i(β·p/|p|)Λ)|Ψp=D(β·p/|p|)|Ψp=D(θp)|Ψp,
D(α)|Ψ±=exp(iα)|Ψ±.
D(θp)|Ψp,te/tm=12(exp(iθp)|Ψp,+±exp(iθp)|Ψp,)
exp(iβxΛPx/|P|)|Ψω,ν=k=0[(iβxΛPx/|P|)2k(2k)!+(iβxΛPx/|P|)2k+1(2k+1)!]|Ψω,ν=k=0[(iβxPx/|P|)2k(2k)!+(iβxPx/|P|)2k+1Λ(2k+1)!]Λ2k|Ψω,ν=k=0(iβxPx/|P|)2k(2k)!|Ψω,ν+νk=1(iβxPx/|P|)2k+1(2k+1)!|Ψω,ν=exp(i(νβx/ω)Px)|Ψω,ν.
exp(iβ·ΛP/|P|)|Ψω,ν=exp(i(ν/ω)β·P)|Ψω,ν.
exp(iβ·ΛP/|P|)F^±(x,y,z)exp(iωt)=F^±(xβx/|p|,yβy/|p|,zβz/|p|)exp(iωt),
L^=JS^=JΛP|P|.
12(|Ψp,+±|Ψp,).
(x^+iy^)[cos(pz)isin(pz)]exp(iωt),
|Ψm=0dω0πdθsinθππdϕexp(imϕ)Rz(ϕ)Ry(θ)(c+(ω,θ)|Ψ[0,0,ω],++c(ω,θ)|Ψ[0,0,ω],).
Wk=ΛPk=ΛPk|P||P|=S^k|P|=S^kP0,

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