Abstract

Based on the plane-wave angular spectrum representation, we derive a formal expression for any light fields propagating in biaxial crystals, and particularly, present an effective numerical method to investigate the propagation behavior for a Gaussian light beam. Unlike uniaxial crystals, we observe the intriguing formation, repulsion and disappearance of vortex pairs, as the refractive indices deviate slightly and gradually from the uniaxial limit. In the Minkowski angular momentum picture, we also investigate the orbital angular momentum dynamics for both left- and right-handed circularly polarized components. Of further interest is the revelation of nonconservation of the angular momentum within the light field during the spin-orbit interactions, and the optical torque per photon that the light exerts on the biaxial crystal is quantified. We interpret these interesting phenomena by the weakly broken rotational invariance of biaxial crystals. The self-consistency of our theory is confirmed by the balance equation describing the conservation law of total angular momentum of filed and crystal in the Minkowski picture.

© 2012 OSA

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2011

A. M. Yao and M. J. Padgett, “Orbital angular momentum: origins, behavior and applications,” Adv. Opt. Photon. 3(2), 161–204 (2011).
[CrossRef]

L. Marrucci, E. Karimi, S. Slussarenko, B. Piccirillo, E. Santamato, E. Nagali, and F. Sciarrino, “Spin-to-orbital conversion of the angular momentum of light and its classical and quantum applications,” J. Opt. 13(6), 064001 (2011).
[CrossRef]

A. Picón, A. Benseny, J. Mompart, and G. F. Calvo, “Spin and orbital angular momentum propagation in anisotropic media: theory,” J. Opt. 13(6), 064019 (2011).
[CrossRef]

I. Skab, Y. Vasylkiv, I. Smaga, and R. Vlokh, “Spin-to-orbital momentum conversion via electro-optic Pockels effect in crystals,” Phys. Rev. A 84(4), 043815 (2011).
[CrossRef]

2010

2009

2008

2007

2006

U. Leonhardt, “Optics: momentum in an uncertain light,” Nature 444(7121), 823–824 (2006).
[CrossRef] [PubMed]

A. Voylar, V. Shvedov, T. A. Fadeyeva, A. S. Desyatnikov, D. N. Neshev, W. Krolikowski, and Y. S. Kivshar, “Generation of single-charge optical vortices with an uniaxial crystal,” Opt. Express 14(9), 3724–3729 (2006).
[CrossRef] [PubMed]

L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96(16), 163905 (2006).
[CrossRef] [PubMed]

2005

M. V. Berry, M. R. Jeffrey, and M. Mansuripur, “Orbital and spin angular momentum in conical diffraction,” J. Opt. A, Pure Appl. Opt. 7(11), 685–690 (2005).
[CrossRef]

2004

A. Ciattoni and C. Palma, “Anisotropic beam spreading in uniaxial crystals,” Opt. Commun. 231(1-6), 79–92 (2004).
[CrossRef]

2003

2002

2001

V. N. Belyi, T. A. King, N. S. Kazak, N. A. Khjlo, E. G. Katranji, and A. A. Ryzhevich, “Methods of formation and nonlinear conversion ofBessel optical vortices,” Proc. SPIE 4403, 229–240 (2001).
[CrossRef]

W. She and W. Lee, “Wave coupling theory of linear electrooptic effect,” Opt. Commun. 195(1-4), 303–311 (2001).
[CrossRef]

A. Ciattoni, B. Crosignani, and P. Di Porto, “Vectorial theory of propagation in uniaxially anisotropic media,” J. Opt. Soc. Am. A 18(7), 1656–1661 (2001).
[CrossRef] [PubMed]

1999

N. S. Kazak, N. A. Khilo, and A. A. Ryzhevich, “Generation of Bessel light beams under the conditions of internal conical refraction,” Quantum Electron. 29(11), 1020–1024 (1999).
[CrossRef]

M. A. Dreger, “Optical beam propagation in biaxial crystals,” J. Opt. A, Pure Appl. Opt. 1(5), 601–616 (1999).
[CrossRef]

1992

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(11), 8185–8189 (1992).
[CrossRef] [PubMed]

1986

W. Gough, “The angular momentum of radiatlon,” Eur. J. Phys. 7(2), 81–87 (1986).
[CrossRef]

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(11), 8185–8189 (1992).
[CrossRef] [PubMed]

Barnett, S. M.

M. Padgett, S. M. Barnett, and R. Loudon, “The angular momentum of light inside a dielectric,” J. Mod. Opt. 10, 1555–1562 (2003).

S. M. Barnett, “Optical angular-momentum flux,” J. Opt. B Quantum Semiclassical Opt. 4(2), S7–S16 (2002).
[CrossRef]

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(11), 8185–8189 (1992).
[CrossRef] [PubMed]

Belyi, V. N.

V. N. Belyi, T. A. King, N. S. Kazak, N. A. Khjlo, E. G. Katranji, and A. A. Ryzhevich, “Methods of formation and nonlinear conversion ofBessel optical vortices,” Proc. SPIE 4403, 229–240 (2001).
[CrossRef]

Benseny, A.

A. Picón, A. Benseny, J. Mompart, and G. F. Calvo, “Spin and orbital angular momentum propagation in anisotropic media: theory,” J. Opt. 13(6), 064019 (2011).
[CrossRef]

Berry, M. V.

M. V. Berry, M. R. Jeffrey, and M. Mansuripur, “Orbital and spin angular momentum in conical diffraction,” J. Opt. A, Pure Appl. Opt. 7(11), 685–690 (2005).
[CrossRef]

Brasselet, E.

Calvo, G. F.

A. Picón, A. Benseny, J. Mompart, and G. F. Calvo, “Spin and orbital angular momentum propagation in anisotropic media: theory,” J. Opt. 13(6), 064019 (2011).
[CrossRef]

G. F. Calvo and A. Picón, “Spin-induced angular momentum switching,” Opt. Lett. 32(7), 838–840 (2007).
[CrossRef] [PubMed]

Chen, L.

Ciattoni, A.

Cincotti, G.

Crosignani, B.

Desyatnikov, A.

Desyatnikov, A. S.

Di Porto, P.

Dreger, M. A.

M. A. Dreger, “Optical beam propagation in biaxial crystals,” J. Opt. A, Pure Appl. Opt. 1(5), 601–616 (1999).
[CrossRef]

Dreischuh, A.

Fadeyeva, T. A.

Gough, W.

W. Gough, “The angular momentum of radiatlon,” Eur. J. Phys. 7(2), 81–87 (1986).
[CrossRef]

Izdebskaya, Y.

Izdebskaya, Y. V.

Jeffrey, M. R.

M. V. Berry, M. R. Jeffrey, and M. Mansuripur, “Orbital and spin angular momentum in conical diffraction,” J. Opt. A, Pure Appl. Opt. 7(11), 685–690 (2005).
[CrossRef]

Karimi, E.

L. Marrucci, E. Karimi, S. Slussarenko, B. Piccirillo, E. Santamato, E. Nagali, and F. Sciarrino, “Spin-to-orbital conversion of the angular momentum of light and its classical and quantum applications,” J. Opt. 13(6), 064001 (2011).
[CrossRef]

Katranji, E. G.

V. N. Belyi, T. A. King, N. S. Kazak, N. A. Khjlo, E. G. Katranji, and A. A. Ryzhevich, “Methods of formation and nonlinear conversion ofBessel optical vortices,” Proc. SPIE 4403, 229–240 (2001).
[CrossRef]

Kazak, N. S.

V. N. Belyi, T. A. King, N. S. Kazak, N. A. Khjlo, E. G. Katranji, and A. A. Ryzhevich, “Methods of formation and nonlinear conversion ofBessel optical vortices,” Proc. SPIE 4403, 229–240 (2001).
[CrossRef]

N. S. Kazak, N. A. Khilo, and A. A. Ryzhevich, “Generation of Bessel light beams under the conditions of internal conical refraction,” Quantum Electron. 29(11), 1020–1024 (1999).
[CrossRef]

Khilo, N. A.

N. S. Kazak, N. A. Khilo, and A. A. Ryzhevich, “Generation of Bessel light beams under the conditions of internal conical refraction,” Quantum Electron. 29(11), 1020–1024 (1999).
[CrossRef]

Khjlo, N. A.

V. N. Belyi, T. A. King, N. S. Kazak, N. A. Khjlo, E. G. Katranji, and A. A. Ryzhevich, “Methods of formation and nonlinear conversion ofBessel optical vortices,” Proc. SPIE 4403, 229–240 (2001).
[CrossRef]

King, T. A.

V. N. Belyi, T. A. King, N. S. Kazak, N. A. Khjlo, E. G. Katranji, and A. A. Ryzhevich, “Methods of formation and nonlinear conversion ofBessel optical vortices,” Proc. SPIE 4403, 229–240 (2001).
[CrossRef]

Kivshar, Y.

Kivshar, Y. S.

Krolikowski, W.

Lee, W.

W. She and W. Lee, “Wave coupling theory of linear electrooptic effect,” Opt. Commun. 195(1-4), 303–311 (2001).
[CrossRef]

Leonhardt, U.

U. Leonhardt, “Optics: momentum in an uncertain light,” Nature 444(7121), 823–824 (2006).
[CrossRef] [PubMed]

Loudon, R.

M. Padgett, S. M. Barnett, and R. Loudon, “The angular momentum of light inside a dielectric,” J. Mod. Opt. 10, 1555–1562 (2003).

Loussert, C.

Mansuripur, M.

M. V. Berry, M. R. Jeffrey, and M. Mansuripur, “Orbital and spin angular momentum in conical diffraction,” J. Opt. A, Pure Appl. Opt. 7(11), 685–690 (2005).
[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(16), 163905 (2006).
[CrossRef] [PubMed]

Marrucci, L.

L. Marrucci, E. Karimi, S. Slussarenko, B. Piccirillo, E. Santamato, E. Nagali, and F. Sciarrino, “Spin-to-orbital conversion of the angular momentum of light and its classical and quantum applications,” J. Opt. 13(6), 064001 (2011).
[CrossRef]

L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96(16), 163905 (2006).
[CrossRef] [PubMed]

Melamed, T.

Mompart, J.

A. Picón, A. Benseny, J. Mompart, and G. F. Calvo, “Spin and orbital angular momentum propagation in anisotropic media: theory,” J. Opt. 13(6), 064019 (2011).
[CrossRef]

Nagali, E.

L. Marrucci, E. Karimi, S. Slussarenko, B. Piccirillo, E. Santamato, E. Nagali, and F. Sciarrino, “Spin-to-orbital conversion of the angular momentum of light and its classical and quantum applications,” J. Opt. 13(6), 064001 (2011).
[CrossRef]

Neshev, D. N.

Padgett, M.

M. Padgett, S. M. Barnett, and R. Loudon, “The angular momentum of light inside a dielectric,” J. Mod. Opt. 10, 1555–1562 (2003).

Padgett, M. J.

Palma, C.

Paparo, D.

L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96(16), 163905 (2006).
[CrossRef] [PubMed]

Piccirillo, B.

L. Marrucci, E. Karimi, S. Slussarenko, B. Piccirillo, E. Santamato, E. Nagali, and F. Sciarrino, “Spin-to-orbital conversion of the angular momentum of light and its classical and quantum applications,” J. Opt. 13(6), 064001 (2011).
[CrossRef]

Picón, A.

A. Picón, A. Benseny, J. Mompart, and G. F. Calvo, “Spin and orbital angular momentum propagation in anisotropic media: theory,” J. Opt. 13(6), 064019 (2011).
[CrossRef]

G. F. Calvo and A. Picón, “Spin-induced angular momentum switching,” Opt. Lett. 32(7), 838–840 (2007).
[CrossRef] [PubMed]

Ryzhevich, A. A.

V. N. Belyi, T. A. King, N. S. Kazak, N. A. Khjlo, E. G. Katranji, and A. A. Ryzhevich, “Methods of formation and nonlinear conversion ofBessel optical vortices,” Proc. SPIE 4403, 229–240 (2001).
[CrossRef]

N. S. Kazak, N. A. Khilo, and A. A. Ryzhevich, “Generation of Bessel light beams under the conditions of internal conical refraction,” Quantum Electron. 29(11), 1020–1024 (1999).
[CrossRef]

Santamato, E.

L. Marrucci, E. Karimi, S. Slussarenko, B. Piccirillo, E. Santamato, E. Nagali, and F. Sciarrino, “Spin-to-orbital conversion of the angular momentum of light and its classical and quantum applications,” J. Opt. 13(6), 064001 (2011).
[CrossRef]

Sciarrino, F.

L. Marrucci, E. Karimi, S. Slussarenko, B. Piccirillo, E. Santamato, E. Nagali, and F. Sciarrino, “Spin-to-orbital conversion of the angular momentum of light and its classical and quantum applications,” J. Opt. 13(6), 064001 (2011).
[CrossRef]

She, W.

Shvedov, V.

Shvedov, V. G.

Skab, I.

I. Skab, Y. Vasylkiv, I. Smaga, and R. Vlokh, “Spin-to-orbital momentum conversion via electro-optic Pockels effect in crystals,” Phys. Rev. A 84(4), 043815 (2011).
[CrossRef]

Slussarenko, S.

L. Marrucci, E. Karimi, S. Slussarenko, B. Piccirillo, E. Santamato, E. Nagali, and F. Sciarrino, “Spin-to-orbital conversion of the angular momentum of light and its classical and quantum applications,” J. Opt. 13(6), 064001 (2011).
[CrossRef]

Smaga, I.

I. Skab, Y. Vasylkiv, I. Smaga, and R. Vlokh, “Spin-to-orbital momentum conversion via electro-optic Pockels effect in crystals,” Phys. Rev. A 84(4), 043815 (2011).
[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(11), 8185–8189 (1992).
[CrossRef] [PubMed]

Tinkelman, I.

Vasylkiv, Y.

I. Skab, Y. Vasylkiv, I. Smaga, and R. Vlokh, “Spin-to-orbital momentum conversion via electro-optic Pockels effect in crystals,” Phys. Rev. A 84(4), 043815 (2011).
[CrossRef]

Vlokh, R.

I. Skab, Y. Vasylkiv, I. Smaga, and R. Vlokh, “Spin-to-orbital momentum conversion via electro-optic Pockels effect in crystals,” Phys. Rev. A 84(4), 043815 (2011).
[CrossRef]

Volyar, A. V.

Voylar, A.

Weber, H.

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(11), 8185–8189 (1992).
[CrossRef] [PubMed]

Yao, A. M.

Adv. Opt. Photon.

Eur. J. Phys.

W. Gough, “The angular momentum of radiatlon,” Eur. J. Phys. 7(2), 81–87 (1986).
[CrossRef]

J. Mod. Opt.

M. Padgett, S. M. Barnett, and R. Loudon, “The angular momentum of light inside a dielectric,” J. Mod. Opt. 10, 1555–1562 (2003).

J. Opt.

L. Marrucci, E. Karimi, S. Slussarenko, B. Piccirillo, E. Santamato, E. Nagali, and F. Sciarrino, “Spin-to-orbital conversion of the angular momentum of light and its classical and quantum applications,” J. Opt. 13(6), 064001 (2011).
[CrossRef]

A. Picón, A. Benseny, J. Mompart, and G. F. Calvo, “Spin and orbital angular momentum propagation in anisotropic media: theory,” J. Opt. 13(6), 064019 (2011).
[CrossRef]

J. Opt. A, Pure Appl. Opt.

M. V. Berry, M. R. Jeffrey, and M. Mansuripur, “Orbital and spin angular momentum in conical diffraction,” J. Opt. A, Pure Appl. Opt. 7(11), 685–690 (2005).
[CrossRef]

M. A. Dreger, “Optical beam propagation in biaxial crystals,” J. Opt. A, Pure Appl. Opt. 1(5), 601–616 (1999).
[CrossRef]

J. Opt. B Quantum Semiclassical Opt.

S. M. Barnett, “Optical angular-momentum flux,” J. Opt. B Quantum Semiclassical Opt. 4(2), S7–S16 (2002).
[CrossRef]

J. Opt. Soc. Am. A

Nature

U. Leonhardt, “Optics: momentum in an uncertain light,” Nature 444(7121), 823–824 (2006).
[CrossRef] [PubMed]

Opt. Commun.

W. She and W. Lee, “Wave coupling theory of linear electrooptic effect,” Opt. Commun. 195(1-4), 303–311 (2001).
[CrossRef]

A. Ciattoni and C. Palma, “Anisotropic beam spreading in uniaxial crystals,” Opt. Commun. 231(1-6), 79–92 (2004).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev. A

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(11), 8185–8189 (1992).
[CrossRef] [PubMed]

I. Skab, Y. Vasylkiv, I. Smaga, and R. Vlokh, “Spin-to-orbital momentum conversion via electro-optic Pockels effect in crystals,” Phys. Rev. A 84(4), 043815 (2011).
[CrossRef]

Phys. Rev. E Stat. Nonlin. Soft Matter Phys.

A. Ciattoni, G. Cincotti, and C. Palma, “Angular momentum dynamics of a paraxial beam in a uniaxial crystal,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 67(3), 036618 (2003).
[CrossRef] [PubMed]

Phys. Rev. Lett.

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Supplementary Material (1)

» Media 1: AVI (124 KB)     

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Figures (5)

Fig. 1
Fig. 1

(a) The schematic diagram of our demonstration. (b) For biaxial crystals, the rotational invariance is weakly broken, while (c) for uniaxial crystals, it holds.

Fig. 2
Fig. 2

A comparison of numerical and analytical results. (a) and (b) show the moduli of E+ and E, the circularly polarized components of Gaussian beam propagating in uniaxial crystal, as a function of transverse radius r. (c) and (d) show the absolute errors of our numerical method for |E+| and |E|, respectively.

Fig. 3
Fig. 3

The moduli of E+ (upper row) and E (lower row) at a fixed propagating distance z = 2000μm for various deviations of Δn. (a) and (f): Δn = 0, (b) and (g): Δn = 0.00001, (c) and (h): Δn = 0.00005, (d) and (i): Δn = 0.0001, (e) and (j): Δn = 0.01. The x and y coordinates are in unit of μm. We observe the intriguing formation, repulsion and disappearance of vortex pairs, as shown in the lower row. (Media 1).

Fig. 4
Fig. 4

Spiral spectra of the emergent light field carried by (a) left-handed component, W l + , and (b) right-handed one, W l , as a function of the crystal length Z, where l denotes the OAM number. The deviation is fixed at Δn = 0.0001 and the other parameters used for calculation are the same as those in Fig. 3. The dashed lines in the uniaxial case are also plotted for comparison.

Fig. 5
Fig. 5

(a) Spin-orbit angular momentum dynamics within the light field emerging from the biaxial crystal; (b) Conservation law of total angular momentum of the light field and biaxial crystal. All parameters used in calculation are the same as those in Fig. 4.

Equations (32)

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2 E(E)+ k 0 2 εE=0,
ε=[ n x 2 0 0 0 n y 2 0 0 0 n z 2 ],
E( r ,z)= d 2 k exp(i k r ) E ˜ ( k ,z),
E ˜ ( k ,z)=( c 1 c 3 c 5 )exp( λ 1 z)+( c 2 c 4 c 5 )exp( λ 2 z),
λ 1 , λ 2 = 1 2 ( K± L ),
c 1 = 1 L [ ( 1 β γ ) k x k y E ˜ y (0)+( M L ) E ˜ x (0) 2 ], c 2 = 1 L [ ( 1 β γ ) k x k y E ˜ y (0)+( M+ L ) E ˜ x (0) 2 ],
c 3 = 1 L [ ( 1 α γ ) k x k y E ˜ x (0)( M+ L ) E ˜ y (0) 2 ], c 4 = 1 L [ ( 1 α γ ) k x k y E ˜ x (0)( M L ) E ˜ y (0) 2 ],
c 5 = λ 1 p ( k x c 1 + k y c 3 ), c 6 = λ 2 p ( k x c 2 + k y c 4 ),
α= k 0 2 n x 2 , β= k 0 2 n y 2 , γ= k 0 2 n z 2 , p=γ( k x 2 + k y 2 ),
K=(α+β)( 1+ α γ ) k x 2 ( 1+ β γ ) k y 2 ,
L= (αβ) 2 + ( 1 α γ ) 2 k x 4 + ( 1 β γ ) 2 k y 4 +2(αβ)( 1 α γ ) k x 2 +2(βα)( 1 β γ ) k y 2 +2( 1 α γ )( 1 β γ ) k x 2 k y 2 ,
M=(αβ)+( 1 α γ ) k x 2 ( 1 β γ ) k y 2 .
E( r ,0)=exp( r 2 2 s 2 ) e ^ + ,
( E + E )= 1 2 [ 1 i 1 i ]( E x E y ).
E ˜ (0)= s 2 2π exp( k 2 s 2 2 ) e ^ + .
W ± ( Z + )= 1 W z0 0 rdr 0 2π dϕ | E ± (r,ϕ, Z + ) | 2 ,
E ± (r,ϕ, Z + )= l= + E l ± (r, Z + ) e ilϕ ,
E l ± (r, Z + )= 1 2π 0 2π E ± (r,ϕ, Z + ) e ilϕ dϕ.
W l ± ( Z + )= 1 W z0 0 2πr | E l ± (r, Z + ) | 2 dr .
d L f dt = S F n ^ dS + V gdr,
F ij = ε imn x m T nj , g i = ε imn T mn ,
T ij = ε 0 E i (εE) j + 1 μ 0 B i B j 1 2 ( ε 0 EεE+ 1 μ 0 BB ) δ ij .
g=[ T yz T zy T zx T xz T xy T yx ]= ε 0 [ E y E z ( n z 2 n y 2 ) E z E x ( n x 2 n z 2 ) E x E y ( n y 2 n x 2 ) ],
d L fz dt =Φ( 0 + )Φ( Z )+ V g z dr ,
E (r,ϕ,z)= l= + ( E x,l (r,z) e ^ x + E y,l (r,z) e ^ y )exp(ilϕ) ,
L x = θ x × n x k 0 ×r=l, L y = θ y × n y k 0 ×r=l ,
L + (z)= l l W l + l W l + ,
L (z)= l l W l l W l .
J f (z)= l (l+1) W l + (z) + l (l1) W l (z) .
g z = E x E y ( n y 2 n x 2 )0.
J f (z)+ J c (z)= J f (0),
J c (z)= 1 η V g z dr = 1 η ε 0 ( n x 2 n y 2 ) 0 0 2π 0 z 1 2 Re( E x E y * )drdϕdz ,

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