Abstract

We demonstrate, both analytically and experimentally, that a broad class of vector bottle beams can be synthesized by passing a Gaussian beam through a uniaxial c-cut crystal. The polarization state and shape of the optical bottle can be easily reconfigured by tuning the input beam polarization and the crystal parameters.

© 2012 Optical Society of America

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References

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  1. J. Arlt and M. J. Padgett, “Generation of a beam with a dark focus surrounded by regions of higher intensity: the optical bottle beam,” Opt. Lett. 25, 191–193 (2000).
    [CrossRef]
  2. T. Kuga, Y. Torii, N. Shiokawa, and T. Hirano, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78, 4713–4716 (1997).
    [CrossRef]
  3. R. Ozeri, L. Khaykovich, and N. Davidson, “Long spin relaxation times in a single-beam blue-detuned optical trap,” Phys. Rev. A 59, R1750–R1753 (1999).
    [CrossRef]
  4. D. McGloin, G. C. Spalding, H. Melville, W. Sibbett, and K. Dholakia, “Applications of spatial light modulators in atom optics,” Opt. Express 11, 158–166 (2003).
    [CrossRef]
  5. L. Isenhower, W. Williams, A. Dally, and M. Saffman, “Atom trapping in an interferometrically generated bottle beam trap,” Opt. Lett. 34, 1159–1161 (2009).
    [CrossRef]
  6. V. G. Shvedov, A. V. Rode, Y. Izdebskaya, A. Desyatnikov, W. Krolikowski, and Yu. S. Kivshar, “Selective trapping of multiple particles by volume speckle field,” Opt. Express 18, 3137–3142 (2010).
    [CrossRef]
  7. V. G. Shvedov, C. Hnatovsky, A. V. Rode, and W. Krolikowski, “Robust trapping and manipulation of airborne particles with a bottle beam,” Opt. Express 19, 17350–17356 (2011).
    [CrossRef]
  8. P. Zhang, Z. Zhang, J. Prakash, S. Huang, D. Hernandez, M. Salazar, D. N. Christodoulides, and Z. Chen, “Trapping and transporting aerosols with a single optical bottle beam generated by moiré techniques,” Opt. Lett. 36, 1491–1493(2011).
    [CrossRef]
  9. V. G. Shvedov, C. Hnatovsky, N. Shostka, A. V. Rode, and W. Krolikowski, “Optical manipulation of particle ensembles in air,” Opt. Lett. 37, 1934–1936 (2012).
    [CrossRef]
  10. C. Alpmann, M. Esseling, P. Rose, and C. Denz, “Holographic optical bottle beams,” Appl. Phys. Lett. 100, 111101(2012).
    [CrossRef]
  11. K. T. Gahagan and G. A. Swartzlander, “Trapping of low-index microparticles in an optical vortex,” J. Opt. Soc. Am. B. 15, 524–534 (1998).
    [CrossRef]
  12. V. R. Daria, P. J. Rodrigo, and J. Glückstad, “Dynamic array of dark optical traps,” Appl. Phys. Lett. 84, 323–325(2004).
    [CrossRef]
  13. B. P. S. Ahluwalia, W. C. Cheong, X.-C. Yuan, L.-S. Zhang, S.-H. Tao, J. Bu, and H. Wang, “Design and fabrication of a double-axicon for generation of tailorable self-imaged three-dimensional intensity voids,” Opt. Lett. 31, 987–989 (2006).
    [CrossRef]
  14. Y. Kozawa and S. Sato, “Optical trapping of micrometer-sized dielectric particles by cylindrical vector beams,” Opt. Express 18, 10828–10833 (2010).
    [CrossRef]
  15. V. Shvedov, C. Hnatovsky, N. Eckerskorn, A. Rode, and W. Krolikowski, “Polarization-sensitive photophoresis,” Appl. Phys. Lett. 101, 051106 (2012).
    [CrossRef]
  16. L. Huang, H. Guo, J. Li, L. Ling, B. Feng, and Z. Y. Li, “Optical trapping of gold nanoparticles by cylindrical vector beam,” Opt. Lett. 37, 1694–1696 (2012).
    [CrossRef]
  17. A. Ciattoni, B. Crosingani, and P. Di Porto, “Vectorial theory of propagation in uniaxially anisotropic media,” J. Opt. Soc. Am. A 18, 1656–1661 (2001).
    [CrossRef]
  18. A. V. Volyar and T. A. Fadeeva, “Generation of singular beams in uniaxial crystals,” Opt. Spectrosc. 94, 235–244 (2003).
    [CrossRef]
  19. E. Brasselet, Ya. Izdebskaya, V. Shvedov, A. S. Desyatnikov, W. Krolikowski, and Yu. Kivshar, “Dynamics of optical spin-orbit coupling in uniaxial crystals,” Opt. Lett. 34, 1021–1023 (2009).
    [CrossRef]
  20. T. A. Fadeyeva, V. G. Shvedov, Ya. V. Izdebskaya, A. Volyar, E. Brasselet, D. N. Neshev, A. S. Desyatnikov, W. Krolikowski, and Y. S. Kivshar, “Spatially engineered polarization states and optical vortices in uniaxial crystals,” Opt. Express 18, 10848–10863 (2010).
    [CrossRef]
  21. C. Hnatovsky, V. Shvedov, W. Krolikowski, and A. Rode, “Revealing local field structure of focused ultrashort pulses,” Phys. Rev. Lett. 106, 123901 (2011).
    [CrossRef]
  22. C. Hnatovsky, V. G. Shvedov, W. Krolikowski, and A. Rode, “Materials processing with a tightly focused femtosecond laser vortex pulse,” Opt. Lett. 35, 3417–3419 (2010).
    [CrossRef]

2012 (4)

V. G. Shvedov, C. Hnatovsky, N. Shostka, A. V. Rode, and W. Krolikowski, “Optical manipulation of particle ensembles in air,” Opt. Lett. 37, 1934–1936 (2012).
[CrossRef]

C. Alpmann, M. Esseling, P. Rose, and C. Denz, “Holographic optical bottle beams,” Appl. Phys. Lett. 100, 111101(2012).
[CrossRef]

V. Shvedov, C. Hnatovsky, N. Eckerskorn, A. Rode, and W. Krolikowski, “Polarization-sensitive photophoresis,” Appl. Phys. Lett. 101, 051106 (2012).
[CrossRef]

L. Huang, H. Guo, J. Li, L. Ling, B. Feng, and Z. Y. Li, “Optical trapping of gold nanoparticles by cylindrical vector beam,” Opt. Lett. 37, 1694–1696 (2012).
[CrossRef]

2011 (3)

2010 (4)

2009 (2)

2006 (1)

2004 (1)

V. R. Daria, P. J. Rodrigo, and J. Glückstad, “Dynamic array of dark optical traps,” Appl. Phys. Lett. 84, 323–325(2004).
[CrossRef]

2003 (2)

A. V. Volyar and T. A. Fadeeva, “Generation of singular beams in uniaxial crystals,” Opt. Spectrosc. 94, 235–244 (2003).
[CrossRef]

D. McGloin, G. C. Spalding, H. Melville, W. Sibbett, and K. Dholakia, “Applications of spatial light modulators in atom optics,” Opt. Express 11, 158–166 (2003).
[CrossRef]

2001 (1)

2000 (1)

1999 (1)

R. Ozeri, L. Khaykovich, and N. Davidson, “Long spin relaxation times in a single-beam blue-detuned optical trap,” Phys. Rev. A 59, R1750–R1753 (1999).
[CrossRef]

1998 (1)

K. T. Gahagan and G. A. Swartzlander, “Trapping of low-index microparticles in an optical vortex,” J. Opt. Soc. Am. B. 15, 524–534 (1998).
[CrossRef]

1997 (1)

T. Kuga, Y. Torii, N. Shiokawa, and T. Hirano, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78, 4713–4716 (1997).
[CrossRef]

Ahluwalia, B. P. S.

Alpmann, C.

C. Alpmann, M. Esseling, P. Rose, and C. Denz, “Holographic optical bottle beams,” Appl. Phys. Lett. 100, 111101(2012).
[CrossRef]

Arlt, J.

Brasselet, E.

Bu, J.

Chen, Z.

Cheong, W. C.

Christodoulides, D. N.

Ciattoni, A.

Crosingani, B.

Dally, A.

Daria, V. R.

V. R. Daria, P. J. Rodrigo, and J. Glückstad, “Dynamic array of dark optical traps,” Appl. Phys. Lett. 84, 323–325(2004).
[CrossRef]

Davidson, N.

R. Ozeri, L. Khaykovich, and N. Davidson, “Long spin relaxation times in a single-beam blue-detuned optical trap,” Phys. Rev. A 59, R1750–R1753 (1999).
[CrossRef]

Denz, C.

C. Alpmann, M. Esseling, P. Rose, and C. Denz, “Holographic optical bottle beams,” Appl. Phys. Lett. 100, 111101(2012).
[CrossRef]

Desyatnikov, A.

Desyatnikov, A. S.

Dholakia, K.

Di Porto, P.

Eckerskorn, N.

V. Shvedov, C. Hnatovsky, N. Eckerskorn, A. Rode, and W. Krolikowski, “Polarization-sensitive photophoresis,” Appl. Phys. Lett. 101, 051106 (2012).
[CrossRef]

Esseling, M.

C. Alpmann, M. Esseling, P. Rose, and C. Denz, “Holographic optical bottle beams,” Appl. Phys. Lett. 100, 111101(2012).
[CrossRef]

Fadeeva, T. A.

A. V. Volyar and T. A. Fadeeva, “Generation of singular beams in uniaxial crystals,” Opt. Spectrosc. 94, 235–244 (2003).
[CrossRef]

Fadeyeva, T. A.

Feng, B.

Gahagan, K. T.

K. T. Gahagan and G. A. Swartzlander, “Trapping of low-index microparticles in an optical vortex,” J. Opt. Soc. Am. B. 15, 524–534 (1998).
[CrossRef]

Glückstad, J.

V. R. Daria, P. J. Rodrigo, and J. Glückstad, “Dynamic array of dark optical traps,” Appl. Phys. Lett. 84, 323–325(2004).
[CrossRef]

Guo, H.

Hernandez, D.

Hirano, T.

T. Kuga, Y. Torii, N. Shiokawa, and T. Hirano, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78, 4713–4716 (1997).
[CrossRef]

Hnatovsky, C.

Huang, L.

Huang, S.

Isenhower, L.

Izdebskaya, Y.

Izdebskaya, Ya.

Izdebskaya, Ya. V.

Khaykovich, L.

R. Ozeri, L. Khaykovich, and N. Davidson, “Long spin relaxation times in a single-beam blue-detuned optical trap,” Phys. Rev. A 59, R1750–R1753 (1999).
[CrossRef]

Kivshar, Y. S.

Kivshar, Yu.

Kivshar, Yu. S.

Kozawa, Y.

Krolikowski, W.

V. Shvedov, C. Hnatovsky, N. Eckerskorn, A. Rode, and W. Krolikowski, “Polarization-sensitive photophoresis,” Appl. Phys. Lett. 101, 051106 (2012).
[CrossRef]

V. G. Shvedov, C. Hnatovsky, N. Shostka, A. V. Rode, and W. Krolikowski, “Optical manipulation of particle ensembles in air,” Opt. Lett. 37, 1934–1936 (2012).
[CrossRef]

V. G. Shvedov, C. Hnatovsky, A. V. Rode, and W. Krolikowski, “Robust trapping and manipulation of airborne particles with a bottle beam,” Opt. Express 19, 17350–17356 (2011).
[CrossRef]

C. Hnatovsky, V. Shvedov, W. Krolikowski, and A. Rode, “Revealing local field structure of focused ultrashort pulses,” Phys. Rev. Lett. 106, 123901 (2011).
[CrossRef]

V. G. Shvedov, A. V. Rode, Y. Izdebskaya, A. Desyatnikov, W. Krolikowski, and Yu. S. Kivshar, “Selective trapping of multiple particles by volume speckle field,” Opt. Express 18, 3137–3142 (2010).
[CrossRef]

T. A. Fadeyeva, V. G. Shvedov, Ya. V. Izdebskaya, A. Volyar, E. Brasselet, D. N. Neshev, A. S. Desyatnikov, W. Krolikowski, and Y. S. Kivshar, “Spatially engineered polarization states and optical vortices in uniaxial crystals,” Opt. Express 18, 10848–10863 (2010).
[CrossRef]

C. Hnatovsky, V. G. Shvedov, W. Krolikowski, and A. Rode, “Materials processing with a tightly focused femtosecond laser vortex pulse,” Opt. Lett. 35, 3417–3419 (2010).
[CrossRef]

E. Brasselet, Ya. Izdebskaya, V. Shvedov, A. S. Desyatnikov, W. Krolikowski, and Yu. Kivshar, “Dynamics of optical spin-orbit coupling in uniaxial crystals,” Opt. Lett. 34, 1021–1023 (2009).
[CrossRef]

Kuga, T.

T. Kuga, Y. Torii, N. Shiokawa, and T. Hirano, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78, 4713–4716 (1997).
[CrossRef]

Li, J.

Li, Z. Y.

Ling, L.

McGloin, D.

Melville, H.

Neshev, D. N.

Ozeri, R.

R. Ozeri, L. Khaykovich, and N. Davidson, “Long spin relaxation times in a single-beam blue-detuned optical trap,” Phys. Rev. A 59, R1750–R1753 (1999).
[CrossRef]

Padgett, M. J.

Prakash, J.

Rode, A.

V. Shvedov, C. Hnatovsky, N. Eckerskorn, A. Rode, and W. Krolikowski, “Polarization-sensitive photophoresis,” Appl. Phys. Lett. 101, 051106 (2012).
[CrossRef]

C. Hnatovsky, V. Shvedov, W. Krolikowski, and A. Rode, “Revealing local field structure of focused ultrashort pulses,” Phys. Rev. Lett. 106, 123901 (2011).
[CrossRef]

C. Hnatovsky, V. G. Shvedov, W. Krolikowski, and A. Rode, “Materials processing with a tightly focused femtosecond laser vortex pulse,” Opt. Lett. 35, 3417–3419 (2010).
[CrossRef]

Rode, A. V.

Rodrigo, P. J.

V. R. Daria, P. J. Rodrigo, and J. Glückstad, “Dynamic array of dark optical traps,” Appl. Phys. Lett. 84, 323–325(2004).
[CrossRef]

Rose, P.

C. Alpmann, M. Esseling, P. Rose, and C. Denz, “Holographic optical bottle beams,” Appl. Phys. Lett. 100, 111101(2012).
[CrossRef]

Saffman, M.

Salazar, M.

Sato, S.

Shiokawa, N.

T. Kuga, Y. Torii, N. Shiokawa, and T. Hirano, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78, 4713–4716 (1997).
[CrossRef]

Shostka, N.

Shvedov, V.

V. Shvedov, C. Hnatovsky, N. Eckerskorn, A. Rode, and W. Krolikowski, “Polarization-sensitive photophoresis,” Appl. Phys. Lett. 101, 051106 (2012).
[CrossRef]

C. Hnatovsky, V. Shvedov, W. Krolikowski, and A. Rode, “Revealing local field structure of focused ultrashort pulses,” Phys. Rev. Lett. 106, 123901 (2011).
[CrossRef]

E. Brasselet, Ya. Izdebskaya, V. Shvedov, A. S. Desyatnikov, W. Krolikowski, and Yu. Kivshar, “Dynamics of optical spin-orbit coupling in uniaxial crystals,” Opt. Lett. 34, 1021–1023 (2009).
[CrossRef]

Shvedov, V. G.

Sibbett, W.

Spalding, G. C.

Swartzlander, G. A.

K. T. Gahagan and G. A. Swartzlander, “Trapping of low-index microparticles in an optical vortex,” J. Opt. Soc. Am. B. 15, 524–534 (1998).
[CrossRef]

Tao, S.-H.

Torii, Y.

T. Kuga, Y. Torii, N. Shiokawa, and T. Hirano, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78, 4713–4716 (1997).
[CrossRef]

Volyar, A.

Volyar, A. V.

A. V. Volyar and T. A. Fadeeva, “Generation of singular beams in uniaxial crystals,” Opt. Spectrosc. 94, 235–244 (2003).
[CrossRef]

Wang, H.

Williams, W.

Yuan, X.-C.

Zhang, L.-S.

Zhang, P.

Zhang, Z.

Appl. Phys. Lett. (3)

C. Alpmann, M. Esseling, P. Rose, and C. Denz, “Holographic optical bottle beams,” Appl. Phys. Lett. 100, 111101(2012).
[CrossRef]

V. R. Daria, P. J. Rodrigo, and J. Glückstad, “Dynamic array of dark optical traps,” Appl. Phys. Lett. 84, 323–325(2004).
[CrossRef]

V. Shvedov, C. Hnatovsky, N. Eckerskorn, A. Rode, and W. Krolikowski, “Polarization-sensitive photophoresis,” Appl. Phys. Lett. 101, 051106 (2012).
[CrossRef]

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

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

K. T. Gahagan and G. A. Swartzlander, “Trapping of low-index microparticles in an optical vortex,” J. Opt. Soc. Am. B. 15, 524–534 (1998).
[CrossRef]

Opt. Express (5)

Opt. Lett. (8)

E. Brasselet, Ya. Izdebskaya, V. Shvedov, A. S. Desyatnikov, W. Krolikowski, and Yu. Kivshar, “Dynamics of optical spin-orbit coupling in uniaxial crystals,” Opt. Lett. 34, 1021–1023 (2009).
[CrossRef]

C. Hnatovsky, V. G. Shvedov, W. Krolikowski, and A. Rode, “Materials processing with a tightly focused femtosecond laser vortex pulse,” Opt. Lett. 35, 3417–3419 (2010).
[CrossRef]

P. Zhang, Z. Zhang, J. Prakash, S. Huang, D. Hernandez, M. Salazar, D. N. Christodoulides, and Z. Chen, “Trapping and transporting aerosols with a single optical bottle beam generated by moiré techniques,” Opt. Lett. 36, 1491–1493(2011).
[CrossRef]

V. G. Shvedov, C. Hnatovsky, N. Shostka, A. V. Rode, and W. Krolikowski, “Optical manipulation of particle ensembles in air,” Opt. Lett. 37, 1934–1936 (2012).
[CrossRef]

L. Isenhower, W. Williams, A. Dally, and M. Saffman, “Atom trapping in an interferometrically generated bottle beam trap,” Opt. Lett. 34, 1159–1161 (2009).
[CrossRef]

J. Arlt and M. J. Padgett, “Generation of a beam with a dark focus surrounded by regions of higher intensity: the optical bottle beam,” Opt. Lett. 25, 191–193 (2000).
[CrossRef]

L. Huang, H. Guo, J. Li, L. Ling, B. Feng, and Z. Y. Li, “Optical trapping of gold nanoparticles by cylindrical vector beam,” Opt. Lett. 37, 1694–1696 (2012).
[CrossRef]

B. P. S. Ahluwalia, W. C. Cheong, X.-C. Yuan, L.-S. Zhang, S.-H. Tao, J. Bu, and H. Wang, “Design and fabrication of a double-axicon for generation of tailorable self-imaged three-dimensional intensity voids,” Opt. Lett. 31, 987–989 (2006).
[CrossRef]

Opt. Spectrosc. (1)

A. V. Volyar and T. A. Fadeeva, “Generation of singular beams in uniaxial crystals,” Opt. Spectrosc. 94, 235–244 (2003).
[CrossRef]

Phys. Rev. A (1)

R. Ozeri, L. Khaykovich, and N. Davidson, “Long spin relaxation times in a single-beam blue-detuned optical trap,” Phys. Rev. A 59, R1750–R1753 (1999).
[CrossRef]

Phys. Rev. Lett. (2)

T. Kuga, Y. Torii, N. Shiokawa, and T. Hirano, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78, 4713–4716 (1997).
[CrossRef]

C. Hnatovsky, V. Shvedov, W. Krolikowski, and A. Rode, “Revealing local field structure of focused ultrashort pulses,” Phys. Rev. Lett. 106, 123901 (2011).
[CrossRef]

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Figures (7)

Fig. 1.
Fig. 1.

(a) Setup for the generation of a vector beam based on the propagation of a circularly polarized Gaussian beam through a uniaxial c -cut crystal; (b) simulated longitudinal intensity distribution of the vector beam in the focal region of L; (c) simulated and (d) experimental transverse intensity distribution of the vector beam in the focal region of L. Two spatially separated foci are clearly observed in (b)–(d). The distance in (b)–(d) is counted along z from the point located halfway between the two foci. The scale is the same for (b)–(d). (e) Polarization maps of the vector beam in the planes shown in (c) and (d).

Fig. 2.
Fig. 2.

(a) Setup for the separation of the x -polarization state of the vector beam shown in Fig. 1; (b) simulated longitudinal intensity distribution of the x -polarization state in the focal region of L; (c) simulated and (d) experimental transverse intensity distribution of the x -polarization state in the focal region of L.

Fig. 3.
Fig. 3.

(a) Setup for the separation of the x -polarization state when the vector beam is transmitted through a quarter-wave plate placed after the crystal; (b) simulated longitudinal intensity distribution of the x -polarization state in the focal region of L; (c) simulated and (d) experimental transverse intensity distribution of the x -polarization state in the focal region of L.

Fig. 4.
Fig. 4.

(a) Setup for the separation of the y -polarization state when the vector beam shown in Fig. 3; (b) simulated longitudinal intensity distribution of the y -polarization state in the focal region of L; (c) simulated and (d) experimental transverse intensity distribution of the y -polarization state in the focal region of L.

Fig. 5.
Fig. 5.

(a) Setup for the generation of a vector beam based on the propagation of a linearly polarized Gaussian beam through a uniaxial c -cut crystal; (b) simulated longitudinal intensity distribution of the vector beam in the focal region of L; (c) simulated and (d) experimental transverse intensity distribution of the vector beam in the focal region of L.

Fig. 6.
Fig. 6.

(a) Setup for the separation of the x -polarization state of the vector beam shown in Fig. 5; (b) simulated longitudinal intensity distribution of the x -polarization state in the focal region of L; (c) simulated and (d) experimental transverse intensity distribution of the x -polarization state in the focal region of L.

Fig. 7.
Fig. 7.

(a) Setup for the separation of the y -polarization state of the vector beam shown in Fig. 5; (b) simulated and (c) experimental transverse intensity distribution of the state y -polarization in the focal region of L.

Equations (21)

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

( 2 4 i π n o λ z ) E = α ( E ) ,
E ( s ) = 1 2 ( c + u + ( 1 ) s c v ) Φ ( s ) ,
( 2 4 i π n ( s ) λ z ) Φ ( s ) = 0 .
G ( s ) = ( E / ξ ( s ) ) exp ( r 2 / ( w 0 2 ξ ( s ) ) )
E ( input ) ± = ( E / ξ ) exp ( u v / ( w 0 2 ξ ) ) c ± ,
Φ ( 1 ) ( + ) = G ( 1 ) d u ; Φ ( 2 ) ( + ) = G ( 2 ) d u
Φ ( 1 ) ( ) = G ( 1 ) d v ; Φ ( 2 ) ( ) = G ( 2 ) d v
E ( cryst ) ( + ) = E ( 1 ) ( + ) + E ( 2 ) ( + ) , E ( cryst ) ( ) = E ( 2 ) ( ) E ( 1 ) ( ) ,
E ( cryst ) ( + ) = 1 2 ( c + u ( G ( 1 ) + G ( 2 ) ) c v ( G ( 1 ) G ( 2 ) ) ) d u E ( cryst ) ( ) = 1 2 ( c v ( G ( 1 ) + G ( 2 ) ) c + u ( G ( 1 ) G ( 2 ) ) ) d v ,
E ( cryst ) ( + ) = 1 2 [ c + ( G ( 1 ) + G ( 2 ) ) c v 2 ( ( u v + w 0 2 ξ ( 1 ) ) G ( 1 ) ( u v + w 0 2 ξ ( 2 ) ) G ( 2 ) ) ] ,
E ( cryst ) ( ) = 1 2 [ c ( G ( 1 ) + G ( 2 ) ) c + u 2 ( ( u v + w 0 2 ξ ( 1 ) ) G ( 1 ) ( u v + w 0 2 ξ ( 2 ) ) G ( 2 ) ) ] .
E ( cryst ) ( ± ) = 1 2 [ c ± ( G ( 1 ) + G ( 2 ) ) c r 2 ( ( r 2 + w 0 2 ξ ( 1 ) ) G ( 1 ) ( r 2 + w 0 2 ξ ( 2 ) ) G ( 2 ) ) exp ( ± 2 i φ ) ] .
E ( output ) ( ± ) = 1 2 ( c ± ( G z + δ + G z δ ) c ( Ψ z + δ Ψ z δ ) exp ( ± 2 i φ ) ) ,
I ( G z + δ + G z δ ) ( G z + δ * + G z δ * ) + ( Ψ z + δ Ψ z δ ) ( Ψ z + δ * Ψ z δ * ) ,
E x = 1 2 ( ( G z + δ + G z δ ) ( Ψ z + δ Ψ z δ ) ) exp ( ± 2 i φ ) ,
E x = 1 2 ( G z + δ + G z δ ) .
E y = 1 2 ( Ψ z + δ Ψ z δ ) exp ( ± 2 i φ ) .
E ( input ) = 1 2 ( E ( input ) + + E ( input ) ) = ( E / ξ ) exp ( u v / ( w 0 2 ξ ) ) e x .
E x = 1 2 ( ( G z + δ + G z δ ) ( Ψ z + δ Ψ z δ ) cos 2 φ ) ,
E y = 1 2 ( Ψ z + δ Ψ z δ ) sin 2 φ .
I E x E x * + E y E y * .

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