Abstract

We present high efficiency, low noise electrically-controlled Fresnel phase zone plates that were made by creating ring-shaped 180° ferroelectric domains in a lithium niobate wafer. The primary focal lengths of these lenses ranged from 5 to 43 cm, and the light-gathering efficiency was over 37%, very close to the maximum theoretical value of 40.5%.

© 2004 Optical Society of America

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References

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  1. M. DiDomenico and S. H. Wemple, “Oxygen-Octahedra Ferroelectrics. I. Theory of Electro-optical and Nonlinear optical Effects,” J. Appl. Phys. 40, 720–734 (1969).
    [Crossref]
  2. Q. Chen, Y. Chiu, D. N Lambeth, T. E. Schlesinger, and D. D. Stancil, “Guided-wave electro-optic beam deflector using domain reversal in LiTaO3,” J. Lightwave Technol. 12, 1401–1404 (1994).
    [Crossref]
  3. D. A. Scrymgeour, A. Sharan, V. Gopalan, K. T. Gahagan, J. L. Casson, R. Sander, J. M. Robinson, F. Muhammad, P. Chandramani, and F. Kiamilev, “Cascaded electro-optic scanning of laser light over large angles using domain microengineered ferroelectrics,” Appl. Phys. Lett. 81, 3140–3142 (2002).
    [Crossref]
  4. J. L. Casson, L. Wang, N. J. C. Libatique, R. K. Jain, D. A. Scrymgeour, V. Gopalan, K. T. Gahagan, R. K. Sander, and J. M Robinson, “Near-IR tunable laser with an integrated LiTaO3 electro-optic deflector,” Appl. Opt. 41, 6416–6419 (2002).
    [Crossref] [PubMed]
  5. Y. Chiu, V. Gopalan, M. J. Kawas, T. E. Schlesinger, D. D. Stancil, and W. P. Risk, “Integrated Optical Device with Second-Harmonic Generator, Electro-optic Lens and Electrooptic Scanner in LiTaO3,” J. Lightwave Technol. 17, 462–465 (1999).
    [Crossref]
  6. R. W. Eason, A. J. Boyland, S. Mailis, and P. G. R. Smith, “Electro-optically controlled beam deflection for grazing incidence geometry on a domain-engineered interface in LiNbO3,” Opt. Commun. 197, 201–207 (2001).
    [Crossref]
  7. R. S. Cudney, H. M. Escamilla, and L. A. Ríos, “Electrically controllable diffuser made from randomly-poled lithium niobate,” J. Opt. Soc. Am. B 21, 1797–1803 (2004).
    [Crossref]
  8. J. S. Patel and K. Rastani, “Electrically controlled polarization-independent liquid-crystal Fresnel lens arrays,” Opt. Lett. 16, 532–534 (1991).
    [Crossref] [PubMed]
  9. Y. H. Fan, H. Ren, and S. T. Wu, “Switchable Fresnel lens using polymer stabilized liquid crystals,” Opt. Express 11, 3080–3086 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-23-3080.
    [Crossref] [PubMed]
  10. H. Ren, Y. H. Fan, and S. T. Wu, “Tunable Fresnel lens using nanoscale polymer-dispersed liquid crystals,” Appl. Phys. Lett. 83, 1515–1517 (2003).
    [Crossref]
  11. T. Tatebayashi, T. Yamamoto, and H. Sato, “Dual focal point electro-optic lens with a Fresnel-zone plate on a PLZT ceramic,” Appl. Opt. 31, 2770–2775 (1992).
    [Crossref] [PubMed]
  12. V. Ya. Shur, E. L. Rumyantsev, R. G. Batchko, G. D. Miller, M. M. Fejer, and R. L. Byer, “Domain kinetics in the formation of a periodic domain structure in lithium niobate,” Phys. Solid State 41, 1681–1687 (1999).
    [Crossref]
  13. J. W. Goodman, Introduction to Fourier Optics, 2nd edition (McGraw-Hill, New York, 1996).
  14. G. J. Edwards and M. Lawrence, “A temperature dependent dispersion for congruently grown lithium niobate,” Opt. Quantum Electron. 16, 373–374 (1984).
    [Crossref]
  15. M. Jazbinsek and M. Zgonik, “Material tensor parameters of LiNbO3 relevant for electro-and elasto-optics,” Appl. Phys. B 74, 407–414 (2002).
    [Crossref]
  16. M. Müller, E. Soergel, M.C. Wengler, and K. Buse, “Light deflection from ferroelectric domain boundaries,” Appl. Phys. B 78, 367–370 (2004).
    [Crossref]

2004 (2)

R. S. Cudney, H. M. Escamilla, and L. A. Ríos, “Electrically controllable diffuser made from randomly-poled lithium niobate,” J. Opt. Soc. Am. B 21, 1797–1803 (2004).
[Crossref]

M. Müller, E. Soergel, M.C. Wengler, and K. Buse, “Light deflection from ferroelectric domain boundaries,” Appl. Phys. B 78, 367–370 (2004).
[Crossref]

2003 (2)

2002 (3)

D. A. Scrymgeour, A. Sharan, V. Gopalan, K. T. Gahagan, J. L. Casson, R. Sander, J. M. Robinson, F. Muhammad, P. Chandramani, and F. Kiamilev, “Cascaded electro-optic scanning of laser light over large angles using domain microengineered ferroelectrics,” Appl. Phys. Lett. 81, 3140–3142 (2002).
[Crossref]

J. L. Casson, L. Wang, N. J. C. Libatique, R. K. Jain, D. A. Scrymgeour, V. Gopalan, K. T. Gahagan, R. K. Sander, and J. M Robinson, “Near-IR tunable laser with an integrated LiTaO3 electro-optic deflector,” Appl. Opt. 41, 6416–6419 (2002).
[Crossref] [PubMed]

M. Jazbinsek and M. Zgonik, “Material tensor parameters of LiNbO3 relevant for electro-and elasto-optics,” Appl. Phys. B 74, 407–414 (2002).
[Crossref]

2001 (1)

R. W. Eason, A. J. Boyland, S. Mailis, and P. G. R. Smith, “Electro-optically controlled beam deflection for grazing incidence geometry on a domain-engineered interface in LiNbO3,” Opt. Commun. 197, 201–207 (2001).
[Crossref]

1999 (2)

V. Ya. Shur, E. L. Rumyantsev, R. G. Batchko, G. D. Miller, M. M. Fejer, and R. L. Byer, “Domain kinetics in the formation of a periodic domain structure in lithium niobate,” Phys. Solid State 41, 1681–1687 (1999).
[Crossref]

Y. Chiu, V. Gopalan, M. J. Kawas, T. E. Schlesinger, D. D. Stancil, and W. P. Risk, “Integrated Optical Device with Second-Harmonic Generator, Electro-optic Lens and Electrooptic Scanner in LiTaO3,” J. Lightwave Technol. 17, 462–465 (1999).
[Crossref]

1994 (1)

Q. Chen, Y. Chiu, D. N Lambeth, T. E. Schlesinger, and D. D. Stancil, “Guided-wave electro-optic beam deflector using domain reversal in LiTaO3,” J. Lightwave Technol. 12, 1401–1404 (1994).
[Crossref]

1992 (1)

1991 (1)

1984 (1)

G. J. Edwards and M. Lawrence, “A temperature dependent dispersion for congruently grown lithium niobate,” Opt. Quantum Electron. 16, 373–374 (1984).
[Crossref]

1969 (1)

M. DiDomenico and S. H. Wemple, “Oxygen-Octahedra Ferroelectrics. I. Theory of Electro-optical and Nonlinear optical Effects,” J. Appl. Phys. 40, 720–734 (1969).
[Crossref]

Batchko, R. G.

V. Ya. Shur, E. L. Rumyantsev, R. G. Batchko, G. D. Miller, M. M. Fejer, and R. L. Byer, “Domain kinetics in the formation of a periodic domain structure in lithium niobate,” Phys. Solid State 41, 1681–1687 (1999).
[Crossref]

Boyland, A. J.

R. W. Eason, A. J. Boyland, S. Mailis, and P. G. R. Smith, “Electro-optically controlled beam deflection for grazing incidence geometry on a domain-engineered interface in LiNbO3,” Opt. Commun. 197, 201–207 (2001).
[Crossref]

Buse, K.

M. Müller, E. Soergel, M.C. Wengler, and K. Buse, “Light deflection from ferroelectric domain boundaries,” Appl. Phys. B 78, 367–370 (2004).
[Crossref]

Byer, R. L.

V. Ya. Shur, E. L. Rumyantsev, R. G. Batchko, G. D. Miller, M. M. Fejer, and R. L. Byer, “Domain kinetics in the formation of a periodic domain structure in lithium niobate,” Phys. Solid State 41, 1681–1687 (1999).
[Crossref]

Casson, J. L.

J. L. Casson, L. Wang, N. J. C. Libatique, R. K. Jain, D. A. Scrymgeour, V. Gopalan, K. T. Gahagan, R. K. Sander, and J. M Robinson, “Near-IR tunable laser with an integrated LiTaO3 electro-optic deflector,” Appl. Opt. 41, 6416–6419 (2002).
[Crossref] [PubMed]

D. A. Scrymgeour, A. Sharan, V. Gopalan, K. T. Gahagan, J. L. Casson, R. Sander, J. M. Robinson, F. Muhammad, P. Chandramani, and F. Kiamilev, “Cascaded electro-optic scanning of laser light over large angles using domain microengineered ferroelectrics,” Appl. Phys. Lett. 81, 3140–3142 (2002).
[Crossref]

Chandramani, P.

D. A. Scrymgeour, A. Sharan, V. Gopalan, K. T. Gahagan, J. L. Casson, R. Sander, J. M. Robinson, F. Muhammad, P. Chandramani, and F. Kiamilev, “Cascaded electro-optic scanning of laser light over large angles using domain microengineered ferroelectrics,” Appl. Phys. Lett. 81, 3140–3142 (2002).
[Crossref]

Chen, Q.

Q. Chen, Y. Chiu, D. N Lambeth, T. E. Schlesinger, and D. D. Stancil, “Guided-wave electro-optic beam deflector using domain reversal in LiTaO3,” J. Lightwave Technol. 12, 1401–1404 (1994).
[Crossref]

Chiu, Y.

Y. Chiu, V. Gopalan, M. J. Kawas, T. E. Schlesinger, D. D. Stancil, and W. P. Risk, “Integrated Optical Device with Second-Harmonic Generator, Electro-optic Lens and Electrooptic Scanner in LiTaO3,” J. Lightwave Technol. 17, 462–465 (1999).
[Crossref]

Q. Chen, Y. Chiu, D. N Lambeth, T. E. Schlesinger, and D. D. Stancil, “Guided-wave electro-optic beam deflector using domain reversal in LiTaO3,” J. Lightwave Technol. 12, 1401–1404 (1994).
[Crossref]

Cudney, R. S.

DiDomenico, M.

M. DiDomenico and S. H. Wemple, “Oxygen-Octahedra Ferroelectrics. I. Theory of Electro-optical and Nonlinear optical Effects,” J. Appl. Phys. 40, 720–734 (1969).
[Crossref]

Eason, R. W.

R. W. Eason, A. J. Boyland, S. Mailis, and P. G. R. Smith, “Electro-optically controlled beam deflection for grazing incidence geometry on a domain-engineered interface in LiNbO3,” Opt. Commun. 197, 201–207 (2001).
[Crossref]

Edwards, G. J.

G. J. Edwards and M. Lawrence, “A temperature dependent dispersion for congruently grown lithium niobate,” Opt. Quantum Electron. 16, 373–374 (1984).
[Crossref]

Escamilla, H. M.

Fan, Y. H.

Fejer, M. M.

V. Ya. Shur, E. L. Rumyantsev, R. G. Batchko, G. D. Miller, M. M. Fejer, and R. L. Byer, “Domain kinetics in the formation of a periodic domain structure in lithium niobate,” Phys. Solid State 41, 1681–1687 (1999).
[Crossref]

Gahagan, K. T.

D. A. Scrymgeour, A. Sharan, V. Gopalan, K. T. Gahagan, J. L. Casson, R. Sander, J. M. Robinson, F. Muhammad, P. Chandramani, and F. Kiamilev, “Cascaded electro-optic scanning of laser light over large angles using domain microengineered ferroelectrics,” Appl. Phys. Lett. 81, 3140–3142 (2002).
[Crossref]

J. L. Casson, L. Wang, N. J. C. Libatique, R. K. Jain, D. A. Scrymgeour, V. Gopalan, K. T. Gahagan, R. K. Sander, and J. M Robinson, “Near-IR tunable laser with an integrated LiTaO3 electro-optic deflector,” Appl. Opt. 41, 6416–6419 (2002).
[Crossref] [PubMed]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd edition (McGraw-Hill, New York, 1996).

Gopalan, V.

Jain, R. K.

Jazbinsek, M.

M. Jazbinsek and M. Zgonik, “Material tensor parameters of LiNbO3 relevant for electro-and elasto-optics,” Appl. Phys. B 74, 407–414 (2002).
[Crossref]

Kawas, M. J.

Kiamilev, F.

D. A. Scrymgeour, A. Sharan, V. Gopalan, K. T. Gahagan, J. L. Casson, R. Sander, J. M. Robinson, F. Muhammad, P. Chandramani, and F. Kiamilev, “Cascaded electro-optic scanning of laser light over large angles using domain microengineered ferroelectrics,” Appl. Phys. Lett. 81, 3140–3142 (2002).
[Crossref]

Lambeth, D. N

Q. Chen, Y. Chiu, D. N Lambeth, T. E. Schlesinger, and D. D. Stancil, “Guided-wave electro-optic beam deflector using domain reversal in LiTaO3,” J. Lightwave Technol. 12, 1401–1404 (1994).
[Crossref]

Lawrence, M.

G. J. Edwards and M. Lawrence, “A temperature dependent dispersion for congruently grown lithium niobate,” Opt. Quantum Electron. 16, 373–374 (1984).
[Crossref]

Libatique, N. J. C.

Mailis, S.

R. W. Eason, A. J. Boyland, S. Mailis, and P. G. R. Smith, “Electro-optically controlled beam deflection for grazing incidence geometry on a domain-engineered interface in LiNbO3,” Opt. Commun. 197, 201–207 (2001).
[Crossref]

Miller, G. D.

V. Ya. Shur, E. L. Rumyantsev, R. G. Batchko, G. D. Miller, M. M. Fejer, and R. L. Byer, “Domain kinetics in the formation of a periodic domain structure in lithium niobate,” Phys. Solid State 41, 1681–1687 (1999).
[Crossref]

Muhammad, F.

D. A. Scrymgeour, A. Sharan, V. Gopalan, K. T. Gahagan, J. L. Casson, R. Sander, J. M. Robinson, F. Muhammad, P. Chandramani, and F. Kiamilev, “Cascaded electro-optic scanning of laser light over large angles using domain microengineered ferroelectrics,” Appl. Phys. Lett. 81, 3140–3142 (2002).
[Crossref]

Müller, M.

M. Müller, E. Soergel, M.C. Wengler, and K. Buse, “Light deflection from ferroelectric domain boundaries,” Appl. Phys. B 78, 367–370 (2004).
[Crossref]

Patel, J. S.

Rastani, K.

Ren, H.

Ríos, L. A.

Risk, W. P.

Robinson, J. M

Robinson, J. M.

D. A. Scrymgeour, A. Sharan, V. Gopalan, K. T. Gahagan, J. L. Casson, R. Sander, J. M. Robinson, F. Muhammad, P. Chandramani, and F. Kiamilev, “Cascaded electro-optic scanning of laser light over large angles using domain microengineered ferroelectrics,” Appl. Phys. Lett. 81, 3140–3142 (2002).
[Crossref]

Rumyantsev, E. L.

V. Ya. Shur, E. L. Rumyantsev, R. G. Batchko, G. D. Miller, M. M. Fejer, and R. L. Byer, “Domain kinetics in the formation of a periodic domain structure in lithium niobate,” Phys. Solid State 41, 1681–1687 (1999).
[Crossref]

Sander, R.

D. A. Scrymgeour, A. Sharan, V. Gopalan, K. T. Gahagan, J. L. Casson, R. Sander, J. M. Robinson, F. Muhammad, P. Chandramani, and F. Kiamilev, “Cascaded electro-optic scanning of laser light over large angles using domain microengineered ferroelectrics,” Appl. Phys. Lett. 81, 3140–3142 (2002).
[Crossref]

Sander, R. K.

Sato, H.

Schlesinger, T. E.

Y. Chiu, V. Gopalan, M. J. Kawas, T. E. Schlesinger, D. D. Stancil, and W. P. Risk, “Integrated Optical Device with Second-Harmonic Generator, Electro-optic Lens and Electrooptic Scanner in LiTaO3,” J. Lightwave Technol. 17, 462–465 (1999).
[Crossref]

Q. Chen, Y. Chiu, D. N Lambeth, T. E. Schlesinger, and D. D. Stancil, “Guided-wave electro-optic beam deflector using domain reversal in LiTaO3,” J. Lightwave Technol. 12, 1401–1404 (1994).
[Crossref]

Scrymgeour, D. A.

D. A. Scrymgeour, A. Sharan, V. Gopalan, K. T. Gahagan, J. L. Casson, R. Sander, J. M. Robinson, F. Muhammad, P. Chandramani, and F. Kiamilev, “Cascaded electro-optic scanning of laser light over large angles using domain microengineered ferroelectrics,” Appl. Phys. Lett. 81, 3140–3142 (2002).
[Crossref]

J. L. Casson, L. Wang, N. J. C. Libatique, R. K. Jain, D. A. Scrymgeour, V. Gopalan, K. T. Gahagan, R. K. Sander, and J. M Robinson, “Near-IR tunable laser with an integrated LiTaO3 electro-optic deflector,” Appl. Opt. 41, 6416–6419 (2002).
[Crossref] [PubMed]

Sharan, A.

D. A. Scrymgeour, A. Sharan, V. Gopalan, K. T. Gahagan, J. L. Casson, R. Sander, J. M. Robinson, F. Muhammad, P. Chandramani, and F. Kiamilev, “Cascaded electro-optic scanning of laser light over large angles using domain microengineered ferroelectrics,” Appl. Phys. Lett. 81, 3140–3142 (2002).
[Crossref]

Shur, V. Ya.

V. Ya. Shur, E. L. Rumyantsev, R. G. Batchko, G. D. Miller, M. M. Fejer, and R. L. Byer, “Domain kinetics in the formation of a periodic domain structure in lithium niobate,” Phys. Solid State 41, 1681–1687 (1999).
[Crossref]

Smith, P. G. R.

R. W. Eason, A. J. Boyland, S. Mailis, and P. G. R. Smith, “Electro-optically controlled beam deflection for grazing incidence geometry on a domain-engineered interface in LiNbO3,” Opt. Commun. 197, 201–207 (2001).
[Crossref]

Soergel, E.

M. Müller, E. Soergel, M.C. Wengler, and K. Buse, “Light deflection from ferroelectric domain boundaries,” Appl. Phys. B 78, 367–370 (2004).
[Crossref]

Stancil, D. D.

Y. Chiu, V. Gopalan, M. J. Kawas, T. E. Schlesinger, D. D. Stancil, and W. P. Risk, “Integrated Optical Device with Second-Harmonic Generator, Electro-optic Lens and Electrooptic Scanner in LiTaO3,” J. Lightwave Technol. 17, 462–465 (1999).
[Crossref]

Q. Chen, Y. Chiu, D. N Lambeth, T. E. Schlesinger, and D. D. Stancil, “Guided-wave electro-optic beam deflector using domain reversal in LiTaO3,” J. Lightwave Technol. 12, 1401–1404 (1994).
[Crossref]

Tatebayashi, T.

Wang, L.

Wemple, S. H.

M. DiDomenico and S. H. Wemple, “Oxygen-Octahedra Ferroelectrics. I. Theory of Electro-optical and Nonlinear optical Effects,” J. Appl. Phys. 40, 720–734 (1969).
[Crossref]

Wengler, M.C.

M. Müller, E. Soergel, M.C. Wengler, and K. Buse, “Light deflection from ferroelectric domain boundaries,” Appl. Phys. B 78, 367–370 (2004).
[Crossref]

Wu, S. T.

Yamamoto, T.

Zgonik, M.

M. Jazbinsek and M. Zgonik, “Material tensor parameters of LiNbO3 relevant for electro-and elasto-optics,” Appl. Phys. B 74, 407–414 (2002).
[Crossref]

Appl. Opt. (2)

Appl. Phys. B (2)

M. Jazbinsek and M. Zgonik, “Material tensor parameters of LiNbO3 relevant for electro-and elasto-optics,” Appl. Phys. B 74, 407–414 (2002).
[Crossref]

M. Müller, E. Soergel, M.C. Wengler, and K. Buse, “Light deflection from ferroelectric domain boundaries,” Appl. Phys. B 78, 367–370 (2004).
[Crossref]

Appl. Phys. Lett. (2)

H. Ren, Y. H. Fan, and S. T. Wu, “Tunable Fresnel lens using nanoscale polymer-dispersed liquid crystals,” Appl. Phys. Lett. 83, 1515–1517 (2003).
[Crossref]

D. A. Scrymgeour, A. Sharan, V. Gopalan, K. T. Gahagan, J. L. Casson, R. Sander, J. M. Robinson, F. Muhammad, P. Chandramani, and F. Kiamilev, “Cascaded electro-optic scanning of laser light over large angles using domain microengineered ferroelectrics,” Appl. Phys. Lett. 81, 3140–3142 (2002).
[Crossref]

J. Appl. Phys. (1)

M. DiDomenico and S. H. Wemple, “Oxygen-Octahedra Ferroelectrics. I. Theory of Electro-optical and Nonlinear optical Effects,” J. Appl. Phys. 40, 720–734 (1969).
[Crossref]

J. Lightwave Technol. (2)

Q. Chen, Y. Chiu, D. N Lambeth, T. E. Schlesinger, and D. D. Stancil, “Guided-wave electro-optic beam deflector using domain reversal in LiTaO3,” J. Lightwave Technol. 12, 1401–1404 (1994).
[Crossref]

Y. Chiu, V. Gopalan, M. J. Kawas, T. E. Schlesinger, D. D. Stancil, and W. P. Risk, “Integrated Optical Device with Second-Harmonic Generator, Electro-optic Lens and Electrooptic Scanner in LiTaO3,” J. Lightwave Technol. 17, 462–465 (1999).
[Crossref]

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

Opt. Commun. (1)

R. W. Eason, A. J. Boyland, S. Mailis, and P. G. R. Smith, “Electro-optically controlled beam deflection for grazing incidence geometry on a domain-engineered interface in LiNbO3,” Opt. Commun. 197, 201–207 (2001).
[Crossref]

Opt. Express (1)

Opt. Lett. (1)

Opt. Quantum Electron. (1)

G. J. Edwards and M. Lawrence, “A temperature dependent dispersion for congruently grown lithium niobate,” Opt. Quantum Electron. 16, 373–374 (1984).
[Crossref]

Phys. Solid State (1)

V. Ya. Shur, E. L. Rumyantsev, R. G. Batchko, G. D. Miller, M. M. Fejer, and R. L. Byer, “Domain kinetics in the formation of a periodic domain structure in lithium niobate,” Phys. Solid State 41, 1681–1687 (1999).
[Crossref]

Other (1)

J. W. Goodman, Introduction to Fourier Optics, 2nd edition (McGraw-Hill, New York, 1996).

Supplementary Material (1)

» Media 1: GIF (244 KB)     

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Figures (5)

Fig. 1.
Fig. 1.

Fresnel zone plate. (a) Photoresist pattern (dark yellow rings), (b) 180° domain structure observed through crossed polarizers (image obtained before the wafer was annealed).

Fig. 2.
Fig. 2.

Arrays of zone plates recorded on a single LiNbO3 wafer. (a) No field is applied; (b) High voltage is applied (~2000 V).

Fig. 3.
Fig. 3.

Efficiency of the primary lens vs. applied voltage. λ=632.8 nm

Fig. 4.
Fig. 4.

Point spread function of one of the Fresnel lenses. (a) 3-dimensional plot; (b) intensity distribution along one direction. λ=632.8 nm, f 0=30 cm, lens diameter=5 mm.

Fig. 5.
Fig. 5.

(0.2 MB) Movie of the performance of a 3×3 array of Fresnel zone plates.

Equations (7)

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

Δ ϕ ( V ) = π n o 3 r 13 V λ ,
r m = f 0 λ 2 ( 2 m 1 ) ,
t ( r , Δ ϕ ) = cos Δ ϕ + i j = 0 a j { exp [ i π r 2 λ f 0 ( 2 j + 1 ) ] + exp [ i π r 2 λ f 0 ( 2 j + 1 ) ] } ,
a j = 2 π [ 2 j + 1 ] ( 1 ) j sin Δ ϕ .
η j = a j | 2 = ( 2 [ 2 j + 1 ] π sin Δ ϕ ) 2 .
η 0 exp = p ap p inc = A cos 2 ( V / V π / 2 ) + B sin 2 ( V / V π / 2 )
V π / 2 = λ 2 n 3 o r 13 .

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