Abstract

We consider the basic properties and features of beams with a defined type of polarization symmetry, in particular beams formed by Laguerre–Gauss modes of the second-order. On the whole the polarization structure of such beams does not have a certain polarization state. We have shown the approach to such beams’ formation with the use of cube-corner reflectors, with the faces having a special dielectric interference coating.

© 2013 Optical Society of America

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  1. P. I. Lamekin and K. G. Predko, “Change of the polarization structure of axial polarized light beams by lens systems,” Opt. Spectrosc. 60, 137–142 (1986).
  2. J. P. McGuire and R. A. Chipman, “Diffraction image formation in optical systems with polarization. I: formulation and example,” J. Opt. Soc. Am. A. 7, 1614–1626 (1990).
    [CrossRef]
  3. R. A. Chipman, “Mechanics of polarization ray tracing,” Opt. Eng. 34, 1636–1645 (1995).
    [CrossRef]
  4. E. F. Ishchenko and A. L. Sokolov, “Spatially depolarized laser radiation,” Quantum Electron. 34, 91–94 (2004).
    [CrossRef]
  5. A. L. Sokolov, “Theory of polarization-non-uniform laser radiation,” Proc. SPIE 4316, 112–120 (2000).
    [CrossRef]
  6. A. L. Sokolov, “Method of polarization-ray matrix,” Laser Technol. Opt. Electron. 3–4, 98–105 (1993).
  7. E. F. Ishchenko and A. L. Sokolov, Polarization Optics (Physmathlit, 2012), pp. 324–421.
  8. A. L. Sokolov, “A technique for the calculation of the natural waves of a cavity with polarization nonuniform elements,” Opt. Spectrosc. 83, 930–936 (1997).
  9. A. L. Sokolov and V. V. Murashkin, “Diffraction polarization optical elements with radial symmetry,” Opt. Spectrosc. 111, 859–865 (2011).
    [CrossRef]
  10. V. V. Korotaev and E. D. Pankov, “Polarization properties of cube-corner reflectors,” Opt. Mech. Ind. 1, 9–12 (1981).
  11. J. J. Degnan, “Millimeter accuracy satellite laser ranging: a review” in Contributions of Space Geodesy to Geodynamics: Technology, Vol. 25, AGU Geodynamics Series (AGU, 1993), pp. 133–162.
  12. D. Arnold, “Cross section of ILRS satellites,” ILRS Technical Workshop, Koetzting, Germany, October2003.
  13. M. A. Sadovnikov and A. L. Sokolov, “Spatial polarization structure of radiation formed by a retroreflector with nonmetalized faces,” Opt. Spectrosc. 107, 201–206 (2009).
    [CrossRef]
  14. K. Crabtree and R. Chipman, “Polarization conversion cube-corner retroreflector,” Appl. Opt. 49, 5882–5890 (2010).
    [CrossRef]
  15. A. L. Sokolov, “Laser beams with periodic polarization properties,” Opt. Spectrosc. 104, 124–125 (2008).
    [CrossRef]
  16. J. P. McGuire and R. A. Chipman, “Polarization aberrations,” Appl. Opt. 33, 5080–5100 (1994).
    [CrossRef]
  17. A. L. Sokolov, “Polarization aberration of laser radiation,” Opt. Spectrosc. 89, 469–475 (2000).
    [CrossRef]
  18. M. Shribak, S. Inoue, and R. Oldenbourg, “Polarization aberration caused by differential transmission and phase shift in high-numerical-aperture lenses: theory, measurement and rectification,” Opt. Eng. 41, 943–954 (2002).
    [CrossRef]
  19. A. L. Sokolov, “Polarization aberration of radiation at the lens focus,” Tech. Phys. Lett. 31, 756–758 (2005).
    [CrossRef]
  20. V. N. Kuryatov and A. L. Sokolov, “Polarization inhomogeneities of a ring resonator and nonreciprocity of counter propagating waves,” Quantum Electron. 32, 324–328 (2002).
    [CrossRef]
  21. D. L. Golovashkin, L. L. Doskolovich, N. L. Kazanskiy, and A. V. Volkov, Methods of Computer Optics (Physmathlit., 2007).
  22. A. V. Nesterov, V. G. Niziev, and A. L. Sokolov, “Laser radiation with axial-symmetry polarization,” Vestnik MEI 2, 76–79 (1999).
  23. A. V. Nesterov and V. G. Niziev, “Laser beams with axially symmetric polarization,” J. Phys. D 33, 1817–1822 (2000).
    [CrossRef]
  24. R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901 (2003).
    [CrossRef]
  25. D. Rozas, C. Law, and G. Swartzlander, “Propogation dynamics of optical vortices,” J. Opt. Soc. Am. B 14, 3054–3065 (1997).
    [CrossRef]
  26. V. V. Kotlyar, A. A. Almazov, S. N. Khonina, V. A. Soifer, K. Jefimovs, and J. Turunen, “Generation of phase singularity through diffracting a plane or Gaussian beam by a spiral phase plate,” J. Opt. Soc. Am. A 22, 849–861 (2005).
    [CrossRef]
  27. J. P. McGuire and R. A. Chipman, “Analysis of spatial pseudodepolarizers in imaging systems,” Opt. Eng. 29, 1478–1484 (1990).

2011 (1)

A. L. Sokolov and V. V. Murashkin, “Diffraction polarization optical elements with radial symmetry,” Opt. Spectrosc. 111, 859–865 (2011).
[CrossRef]

2010 (1)

2009 (1)

M. A. Sadovnikov and A. L. Sokolov, “Spatial polarization structure of radiation formed by a retroreflector with nonmetalized faces,” Opt. Spectrosc. 107, 201–206 (2009).
[CrossRef]

2008 (1)

A. L. Sokolov, “Laser beams with periodic polarization properties,” Opt. Spectrosc. 104, 124–125 (2008).
[CrossRef]

2005 (2)

2004 (1)

E. F. Ishchenko and A. L. Sokolov, “Spatially depolarized laser radiation,” Quantum Electron. 34, 91–94 (2004).
[CrossRef]

2003 (1)

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901 (2003).
[CrossRef]

2002 (2)

V. N. Kuryatov and A. L. Sokolov, “Polarization inhomogeneities of a ring resonator and nonreciprocity of counter propagating waves,” Quantum Electron. 32, 324–328 (2002).
[CrossRef]

M. Shribak, S. Inoue, and R. Oldenbourg, “Polarization aberration caused by differential transmission and phase shift in high-numerical-aperture lenses: theory, measurement and rectification,” Opt. Eng. 41, 943–954 (2002).
[CrossRef]

2000 (3)

A. V. Nesterov and V. G. Niziev, “Laser beams with axially symmetric polarization,” J. Phys. D 33, 1817–1822 (2000).
[CrossRef]

A. L. Sokolov, “Polarization aberration of laser radiation,” Opt. Spectrosc. 89, 469–475 (2000).
[CrossRef]

A. L. Sokolov, “Theory of polarization-non-uniform laser radiation,” Proc. SPIE 4316, 112–120 (2000).
[CrossRef]

1999 (1)

A. V. Nesterov, V. G. Niziev, and A. L. Sokolov, “Laser radiation with axial-symmetry polarization,” Vestnik MEI 2, 76–79 (1999).

1997 (2)

A. L. Sokolov, “A technique for the calculation of the natural waves of a cavity with polarization nonuniform elements,” Opt. Spectrosc. 83, 930–936 (1997).

D. Rozas, C. Law, and G. Swartzlander, “Propogation dynamics of optical vortices,” J. Opt. Soc. Am. B 14, 3054–3065 (1997).
[CrossRef]

1995 (1)

R. A. Chipman, “Mechanics of polarization ray tracing,” Opt. Eng. 34, 1636–1645 (1995).
[CrossRef]

1994 (1)

1993 (1)

A. L. Sokolov, “Method of polarization-ray matrix,” Laser Technol. Opt. Electron. 3–4, 98–105 (1993).

1990 (2)

J. P. McGuire and R. A. Chipman, “Diffraction image formation in optical systems with polarization. I: formulation and example,” J. Opt. Soc. Am. A. 7, 1614–1626 (1990).
[CrossRef]

J. P. McGuire and R. A. Chipman, “Analysis of spatial pseudodepolarizers in imaging systems,” Opt. Eng. 29, 1478–1484 (1990).

1986 (1)

P. I. Lamekin and K. G. Predko, “Change of the polarization structure of axial polarized light beams by lens systems,” Opt. Spectrosc. 60, 137–142 (1986).

1981 (1)

V. V. Korotaev and E. D. Pankov, “Polarization properties of cube-corner reflectors,” Opt. Mech. Ind. 1, 9–12 (1981).

Almazov, A. A.

Arnold, D.

D. Arnold, “Cross section of ILRS satellites,” ILRS Technical Workshop, Koetzting, Germany, October2003.

Chipman, R.

Chipman, R. A.

R. A. Chipman, “Mechanics of polarization ray tracing,” Opt. Eng. 34, 1636–1645 (1995).
[CrossRef]

J. P. McGuire and R. A. Chipman, “Polarization aberrations,” Appl. Opt. 33, 5080–5100 (1994).
[CrossRef]

J. P. McGuire and R. A. Chipman, “Diffraction image formation in optical systems with polarization. I: formulation and example,” J. Opt. Soc. Am. A. 7, 1614–1626 (1990).
[CrossRef]

J. P. McGuire and R. A. Chipman, “Analysis of spatial pseudodepolarizers in imaging systems,” Opt. Eng. 29, 1478–1484 (1990).

Crabtree, K.

Degnan, J. J.

J. J. Degnan, “Millimeter accuracy satellite laser ranging: a review” in Contributions of Space Geodesy to Geodynamics: Technology, Vol. 25, AGU Geodynamics Series (AGU, 1993), pp. 133–162.

Dorn, R.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901 (2003).
[CrossRef]

Doskolovich, L. L.

D. L. Golovashkin, L. L. Doskolovich, N. L. Kazanskiy, and A. V. Volkov, Methods of Computer Optics (Physmathlit., 2007).

Golovashkin, D. L.

D. L. Golovashkin, L. L. Doskolovich, N. L. Kazanskiy, and A. V. Volkov, Methods of Computer Optics (Physmathlit., 2007).

Inoue, S.

M. Shribak, S. Inoue, and R. Oldenbourg, “Polarization aberration caused by differential transmission and phase shift in high-numerical-aperture lenses: theory, measurement and rectification,” Opt. Eng. 41, 943–954 (2002).
[CrossRef]

Ishchenko, E. F.

E. F. Ishchenko and A. L. Sokolov, “Spatially depolarized laser radiation,” Quantum Electron. 34, 91–94 (2004).
[CrossRef]

E. F. Ishchenko and A. L. Sokolov, Polarization Optics (Physmathlit, 2012), pp. 324–421.

Jefimovs, K.

Kazanskiy, N. L.

D. L. Golovashkin, L. L. Doskolovich, N. L. Kazanskiy, and A. V. Volkov, Methods of Computer Optics (Physmathlit., 2007).

Khonina, S. N.

Korotaev, V. V.

V. V. Korotaev and E. D. Pankov, “Polarization properties of cube-corner reflectors,” Opt. Mech. Ind. 1, 9–12 (1981).

Kotlyar, V. V.

Kuryatov, V. N.

V. N. Kuryatov and A. L. Sokolov, “Polarization inhomogeneities of a ring resonator and nonreciprocity of counter propagating waves,” Quantum Electron. 32, 324–328 (2002).
[CrossRef]

Lamekin, P. I.

P. I. Lamekin and K. G. Predko, “Change of the polarization structure of axial polarized light beams by lens systems,” Opt. Spectrosc. 60, 137–142 (1986).

Law, C.

Leuchs, G.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901 (2003).
[CrossRef]

McGuire, J. P.

J. P. McGuire and R. A. Chipman, “Polarization aberrations,” Appl. Opt. 33, 5080–5100 (1994).
[CrossRef]

J. P. McGuire and R. A. Chipman, “Diffraction image formation in optical systems with polarization. I: formulation and example,” J. Opt. Soc. Am. A. 7, 1614–1626 (1990).
[CrossRef]

J. P. McGuire and R. A. Chipman, “Analysis of spatial pseudodepolarizers in imaging systems,” Opt. Eng. 29, 1478–1484 (1990).

Murashkin, V. V.

A. L. Sokolov and V. V. Murashkin, “Diffraction polarization optical elements with radial symmetry,” Opt. Spectrosc. 111, 859–865 (2011).
[CrossRef]

Nesterov, A. V.

A. V. Nesterov and V. G. Niziev, “Laser beams with axially symmetric polarization,” J. Phys. D 33, 1817–1822 (2000).
[CrossRef]

A. V. Nesterov, V. G. Niziev, and A. L. Sokolov, “Laser radiation with axial-symmetry polarization,” Vestnik MEI 2, 76–79 (1999).

Niziev, V. G.

A. V. Nesterov and V. G. Niziev, “Laser beams with axially symmetric polarization,” J. Phys. D 33, 1817–1822 (2000).
[CrossRef]

A. V. Nesterov, V. G. Niziev, and A. L. Sokolov, “Laser radiation with axial-symmetry polarization,” Vestnik MEI 2, 76–79 (1999).

Oldenbourg, R.

M. Shribak, S. Inoue, and R. Oldenbourg, “Polarization aberration caused by differential transmission and phase shift in high-numerical-aperture lenses: theory, measurement and rectification,” Opt. Eng. 41, 943–954 (2002).
[CrossRef]

Pankov, E. D.

V. V. Korotaev and E. D. Pankov, “Polarization properties of cube-corner reflectors,” Opt. Mech. Ind. 1, 9–12 (1981).

Predko, K. G.

P. I. Lamekin and K. G. Predko, “Change of the polarization structure of axial polarized light beams by lens systems,” Opt. Spectrosc. 60, 137–142 (1986).

Quabis, S.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901 (2003).
[CrossRef]

Rozas, D.

Sadovnikov, M. A.

M. A. Sadovnikov and A. L. Sokolov, “Spatial polarization structure of radiation formed by a retroreflector with nonmetalized faces,” Opt. Spectrosc. 107, 201–206 (2009).
[CrossRef]

Shribak, M.

M. Shribak, S. Inoue, and R. Oldenbourg, “Polarization aberration caused by differential transmission and phase shift in high-numerical-aperture lenses: theory, measurement and rectification,” Opt. Eng. 41, 943–954 (2002).
[CrossRef]

Soifer, V. A.

Sokolov, A. L.

A. L. Sokolov and V. V. Murashkin, “Diffraction polarization optical elements with radial symmetry,” Opt. Spectrosc. 111, 859–865 (2011).
[CrossRef]

M. A. Sadovnikov and A. L. Sokolov, “Spatial polarization structure of radiation formed by a retroreflector with nonmetalized faces,” Opt. Spectrosc. 107, 201–206 (2009).
[CrossRef]

A. L. Sokolov, “Laser beams with periodic polarization properties,” Opt. Spectrosc. 104, 124–125 (2008).
[CrossRef]

A. L. Sokolov, “Polarization aberration of radiation at the lens focus,” Tech. Phys. Lett. 31, 756–758 (2005).
[CrossRef]

E. F. Ishchenko and A. L. Sokolov, “Spatially depolarized laser radiation,” Quantum Electron. 34, 91–94 (2004).
[CrossRef]

V. N. Kuryatov and A. L. Sokolov, “Polarization inhomogeneities of a ring resonator and nonreciprocity of counter propagating waves,” Quantum Electron. 32, 324–328 (2002).
[CrossRef]

A. L. Sokolov, “Polarization aberration of laser radiation,” Opt. Spectrosc. 89, 469–475 (2000).
[CrossRef]

A. L. Sokolov, “Theory of polarization-non-uniform laser radiation,” Proc. SPIE 4316, 112–120 (2000).
[CrossRef]

A. V. Nesterov, V. G. Niziev, and A. L. Sokolov, “Laser radiation with axial-symmetry polarization,” Vestnik MEI 2, 76–79 (1999).

A. L. Sokolov, “A technique for the calculation of the natural waves of a cavity with polarization nonuniform elements,” Opt. Spectrosc. 83, 930–936 (1997).

A. L. Sokolov, “Method of polarization-ray matrix,” Laser Technol. Opt. Electron. 3–4, 98–105 (1993).

E. F. Ishchenko and A. L. Sokolov, Polarization Optics (Physmathlit, 2012), pp. 324–421.

Swartzlander, G.

Turunen, J.

Volkov, A. V.

D. L. Golovashkin, L. L. Doskolovich, N. L. Kazanskiy, and A. V. Volkov, Methods of Computer Optics (Physmathlit., 2007).

Appl. Opt. (2)

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

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

J. P. McGuire and R. A. Chipman, “Diffraction image formation in optical systems with polarization. I: formulation and example,” J. Opt. Soc. Am. A. 7, 1614–1626 (1990).
[CrossRef]

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

J. Phys. D (1)

A. V. Nesterov and V. G. Niziev, “Laser beams with axially symmetric polarization,” J. Phys. D 33, 1817–1822 (2000).
[CrossRef]

Laser Technol. Opt. Electron. (1)

A. L. Sokolov, “Method of polarization-ray matrix,” Laser Technol. Opt. Electron. 3–4, 98–105 (1993).

Opt. Eng. (3)

R. A. Chipman, “Mechanics of polarization ray tracing,” Opt. Eng. 34, 1636–1645 (1995).
[CrossRef]

M. Shribak, S. Inoue, and R. Oldenbourg, “Polarization aberration caused by differential transmission and phase shift in high-numerical-aperture lenses: theory, measurement and rectification,” Opt. Eng. 41, 943–954 (2002).
[CrossRef]

J. P. McGuire and R. A. Chipman, “Analysis of spatial pseudodepolarizers in imaging systems,” Opt. Eng. 29, 1478–1484 (1990).

Opt. Mech. Ind. (1)

V. V. Korotaev and E. D. Pankov, “Polarization properties of cube-corner reflectors,” Opt. Mech. Ind. 1, 9–12 (1981).

Opt. Spectrosc. (6)

M. A. Sadovnikov and A. L. Sokolov, “Spatial polarization structure of radiation formed by a retroreflector with nonmetalized faces,” Opt. Spectrosc. 107, 201–206 (2009).
[CrossRef]

A. L. Sokolov, “Polarization aberration of laser radiation,” Opt. Spectrosc. 89, 469–475 (2000).
[CrossRef]

P. I. Lamekin and K. G. Predko, “Change of the polarization structure of axial polarized light beams by lens systems,” Opt. Spectrosc. 60, 137–142 (1986).

A. L. Sokolov, “A technique for the calculation of the natural waves of a cavity with polarization nonuniform elements,” Opt. Spectrosc. 83, 930–936 (1997).

A. L. Sokolov and V. V. Murashkin, “Diffraction polarization optical elements with radial symmetry,” Opt. Spectrosc. 111, 859–865 (2011).
[CrossRef]

A. L. Sokolov, “Laser beams with periodic polarization properties,” Opt. Spectrosc. 104, 124–125 (2008).
[CrossRef]

Phys. Rev. Lett. (1)

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901 (2003).
[CrossRef]

Proc. SPIE (1)

A. L. Sokolov, “Theory of polarization-non-uniform laser radiation,” Proc. SPIE 4316, 112–120 (2000).
[CrossRef]

Quantum Electron. (2)

E. F. Ishchenko and A. L. Sokolov, “Spatially depolarized laser radiation,” Quantum Electron. 34, 91–94 (2004).
[CrossRef]

V. N. Kuryatov and A. L. Sokolov, “Polarization inhomogeneities of a ring resonator and nonreciprocity of counter propagating waves,” Quantum Electron. 32, 324–328 (2002).
[CrossRef]

Tech. Phys. Lett. (1)

A. L. Sokolov, “Polarization aberration of radiation at the lens focus,” Tech. Phys. Lett. 31, 756–758 (2005).
[CrossRef]

Vestnik MEI (1)

A. V. Nesterov, V. G. Niziev, and A. L. Sokolov, “Laser radiation with axial-symmetry polarization,” Vestnik MEI 2, 76–79 (1999).

Other (4)

D. L. Golovashkin, L. L. Doskolovich, N. L. Kazanskiy, and A. V. Volkov, Methods of Computer Optics (Physmathlit., 2007).

J. J. Degnan, “Millimeter accuracy satellite laser ranging: a review” in Contributions of Space Geodesy to Geodynamics: Technology, Vol. 25, AGU Geodynamics Series (AGU, 1993), pp. 133–162.

D. Arnold, “Cross section of ILRS satellites,” ILRS Technical Workshop, Koetzting, Germany, October2003.

E. F. Ishchenko and A. L. Sokolov, Polarization Optics (Physmathlit, 2012), pp. 324–421.

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Figures (5)

Fig. 1.
Fig. 1.

Formation of N -beams with the use of first-order modes: (a)  N R -beam, (b)  N A -beam, (c)  N M -beam, and (d)  N K -beam.

Fig. 2.
Fig. 2.

Formation of polarization structures: (a)  S A 1 -beam, (b)  S A 2 -beam, (c)  S B 1 -beam, and (d)  S B 2 -beam by coherent summation of the orthogonally polarized Laguerre–Gauss modes.

Fig. 3.
Fig. 3.

(a) Ray trace in a CCR and (b) a photograph of CCRs produced by RPC “Precision Systems and Instruments.”

Fig. 4.
Fig. 4.

Polarization structure of a beam reflected by CCR (0, 0, 0) with a special interference coating in (a), (b) the near field and (c), (d) far-field zones for (a), (c) the linear horizontal and (b), (d) linear vertical polarization of the incident light.

Fig. 5.
Fig. 5.

(a) Two rotated CCRs (0, 0, 0) with interference coating, (b) far-field diffraction pattern in the case of light reflection with horizontal linear polarization, and (c) far-field intensity distribution after reflection from two CCRs (solid line) and intensity distribution in the transverse section of the S A 1 -beam (dashed line). Here D is the CCR aperture, λ is the wavelength, and θ is the diffraction angle.

Tables (2)

Tables Icon

Table 1. Polarization Structures of Polarization-Symmetrical Beams in Near- and Far-Field Zones

Tables Icon

Table 2. FFDP of CCRs for Various Phase Shifts

Equations (16)

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

0 0 E x d x d y = 0 , 0 0 E y d x d y = 0 ,
0 0 | E x | 2 d x d y = 0 0 | E y | 2 d x d y .
M A = ( cos φ sin φ sin φ cos φ ) = 1 r ( x y y x ) ,
ψ A = ψ A + γ .
( E x ( φ ) E y ( φ ) ) = ( cos γ sin γ sin γ cos γ ) ( E x ( φ γ ) E y ( φ γ ) ) .
D N R = 1 r ( x y ) = ( cos φ sin φ ) ,
D N A = 1 r ( y x ) = ( sin φ cos φ )
D N R a = ( 1 0 ) , D N A a = ( 0 1 ) .
Γ = sin φ + tg α cos φ cos φ tg α sin φ = tg ( α + φ ) .
D N K = 1 r ( x y ) = ( cos φ sin φ ) , D N M = 1 r ( y x ) = ( sin φ cos φ ) ,
D N K a = ( cos 2 φ sin 2 φ ) , D N M a = ( sin 2 φ cos 2 φ ) ,
D S A 1 = 1 r 2 ( ( x 2 y 2 ) 2 x y ) = ( cos 2 φ sin 2 φ ) , D S A 2 = 1 r 2 ( 2 x y ( x 2 y 2 ) ) = ( sin 2 φ cos 2 φ ) .
D S B 1 = 1 r 2 ( ( x 2 y 2 ) 2 x y ) = ( cos 2 φ sin 2 φ ) , D S B 2 = 1 r 2 ( 2 x y ( x 2 y 2 ) ) = ( sin 2 φ cos 2 φ ) .
D S A 1 a = ( cos φ sin φ ) , D S A 2 a = ( sin φ cos φ ) ,
D S B 1 a = ( cos 3 φ sin 3 φ ) , D S B 2 a = ( sin 3 φ cos 3 φ ) .
2 θ m = λ D ( s + 1 2 ) .

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