Abstract

We carry out the recursion-relation analysis for the optical pumping scheme to polarize nuclei by a sequence of short laser pulses and show that such analysis agrees very well with the numerical solution of density matrix equations. A small difference between the results obtained by the numerical solution of density matrix equations and recursion-relation analysis arises from the contribution of the very small populations remaining in excited states when the next pulse arrives, and the difference becomes smaller if the pulse interval is chosen to be much longer than the spontaneous lifetime of the excited states.

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  1. K. S. E. Eikema, J. Walz, and T. W. Hänsch, “Continuous wave coherent Lyman-α radiation,” Phys. Rev. Lett. 83, 3828–3831(1999).
    [CrossRef]
  2. M. Scheid, D. Kolbe, F. Markert, T. W. Hänsch, and J. Walz, “Continuous-wave Lyman-α generation with solid-state lasers,” Opt. Express 17, 11274–11280 (2009).
    [CrossRef] [PubMed]
  3. R. Mahon, T. J. McIlrath, and D. Koopman, “Nonlinear generation of Lyman-alpha radiation,” Appl. Phys. Lett. 33, 305–307 (1978).
    [CrossRef]
  4. D. Cotter, “Tunable narrow-band coherent VUV source for the Lyman-alpha region,” Opt. Commun. 31, 397–400 (1979).
    [CrossRef]
  5. R. Wallenstein, “Generation of narrowband tunable VUV radiation at the Lyman-α wavelength,” Opt. Commun. 33, 119–122(1980).
    [CrossRef]
  6. G. Hilber, A. Lago, and R. Wallenstein, “Broadly tunable vacuum-ultraviolet/extreme-ultraviolet radiation generated by resonant third-order frequency conversion in krypton,” J. Opt. Soc. Am. B 4, 1753–1764 (1987).
    [CrossRef]
  7. J. P. Marangos, N. Shen, H. Ma, M. H. R. Hutchinson, and J. P. Connerade, “Broadly tunable vacuum-ultraviolet radiation source employing resonant enhanced sum-difference frequency mixing in krypton,” J. Opt. Soc. Am. B 7, 1254–1259(1990).
    [CrossRef]
  8. I. D. Setija, H. G. C. Werij, O. J. Luiten, M. W. Reynolds, T. W. Hijmans, and J. T. Walraven, “Optical cooling of atomic hydrogen in a magnetic trap,” Phys. Rev. Lett. 70, 2257–2260 (1993).
    [CrossRef] [PubMed]
  9. T. Nakajima, “A scheme to polarize the nuclear-spin of atoms by a sequence of short laser pulses: application to muonium,” Opt. Express 18, 27468–27480 (2010).
    [CrossRef]
  10. P. D. de Réotier and A. Yaouanc, “Muon spin rotation and relaxation in magnetic materials,” J. Phys. Condens. Matter 9, 9113–9166 (1997).
    [CrossRef]
  11. P. Strasser, K. Nagamine, T. Matsuzaki, K. Ishida, Y. Matsuda, K. Itahashi, and M. Iwasaki, “Muon spectroscopy with unstable nuclei,” J. Phys. G Nucl. Part. Phys. 29, 2047–2049 (2003).
    [CrossRef]
  12. J. P. Miller, E. de Rafael, and B. L. Roberts, “Muon (g-2) experiment and theory,” Rep. Prog. Phys. 70, 795–881 (2007).
    [CrossRef]
  13. S. Nagamiya, K. Nagamine, O. Hashimoto, and T. Yamazaki, “Negative-muon spin rotation at the oxygen site in paramagnetic MnO+,” Phys. Rev. Lett. 35, 308–311 (1975).
    [CrossRef]
  14. P. Bakule, Y. Matsuda, Y. Miyake, P. Straser, K. Shimomura, S. Makimura, and K. Nagamine, “Slow muon experiment by laser resonant ionization method at RIKEN-RAL muon facility,” Spectrochim. Acta Part B 58, 1019–1030 (2003).
    [CrossRef]
  15. P. Bakule and E. Morenzoni, “Generation and application of slow polarized muons,” Contemp. Phys. 45, 203–225(2004).
    [CrossRef]
  16. D. Fick, G. Grawert, and I. M. Turkiewicz, “Nuclear physics with polarized heavy ions,” Phys. Rep. 214, 1–111 (1992).
    [CrossRef]

2010

2009

2007

J. P. Miller, E. de Rafael, and B. L. Roberts, “Muon (g-2) experiment and theory,” Rep. Prog. Phys. 70, 795–881 (2007).
[CrossRef]

2004

P. Bakule and E. Morenzoni, “Generation and application of slow polarized muons,” Contemp. Phys. 45, 203–225(2004).
[CrossRef]

2003

P. Strasser, K. Nagamine, T. Matsuzaki, K. Ishida, Y. Matsuda, K. Itahashi, and M. Iwasaki, “Muon spectroscopy with unstable nuclei,” J. Phys. G Nucl. Part. Phys. 29, 2047–2049 (2003).
[CrossRef]

P. Bakule, Y. Matsuda, Y. Miyake, P. Straser, K. Shimomura, S. Makimura, and K. Nagamine, “Slow muon experiment by laser resonant ionization method at RIKEN-RAL muon facility,” Spectrochim. Acta Part B 58, 1019–1030 (2003).
[CrossRef]

1999

K. S. E. Eikema, J. Walz, and T. W. Hänsch, “Continuous wave coherent Lyman-α radiation,” Phys. Rev. Lett. 83, 3828–3831(1999).
[CrossRef]

1997

P. D. de Réotier and A. Yaouanc, “Muon spin rotation and relaxation in magnetic materials,” J. Phys. Condens. Matter 9, 9113–9166 (1997).
[CrossRef]

1993

I. D. Setija, H. G. C. Werij, O. J. Luiten, M. W. Reynolds, T. W. Hijmans, and J. T. Walraven, “Optical cooling of atomic hydrogen in a magnetic trap,” Phys. Rev. Lett. 70, 2257–2260 (1993).
[CrossRef] [PubMed]

1992

D. Fick, G. Grawert, and I. M. Turkiewicz, “Nuclear physics with polarized heavy ions,” Phys. Rep. 214, 1–111 (1992).
[CrossRef]

1990

1987

1980

R. Wallenstein, “Generation of narrowband tunable VUV radiation at the Lyman-α wavelength,” Opt. Commun. 33, 119–122(1980).
[CrossRef]

1979

D. Cotter, “Tunable narrow-band coherent VUV source for the Lyman-alpha region,” Opt. Commun. 31, 397–400 (1979).
[CrossRef]

1978

R. Mahon, T. J. McIlrath, and D. Koopman, “Nonlinear generation of Lyman-alpha radiation,” Appl. Phys. Lett. 33, 305–307 (1978).
[CrossRef]

1975

S. Nagamiya, K. Nagamine, O. Hashimoto, and T. Yamazaki, “Negative-muon spin rotation at the oxygen site in paramagnetic MnO+,” Phys. Rev. Lett. 35, 308–311 (1975).
[CrossRef]

Bakule, P.

P. Bakule and E. Morenzoni, “Generation and application of slow polarized muons,” Contemp. Phys. 45, 203–225(2004).
[CrossRef]

P. Bakule, Y. Matsuda, Y. Miyake, P. Straser, K. Shimomura, S. Makimura, and K. Nagamine, “Slow muon experiment by laser resonant ionization method at RIKEN-RAL muon facility,” Spectrochim. Acta Part B 58, 1019–1030 (2003).
[CrossRef]

Connerade, J. P.

Cotter, D.

D. Cotter, “Tunable narrow-band coherent VUV source for the Lyman-alpha region,” Opt. Commun. 31, 397–400 (1979).
[CrossRef]

de Rafael, E.

J. P. Miller, E. de Rafael, and B. L. Roberts, “Muon (g-2) experiment and theory,” Rep. Prog. Phys. 70, 795–881 (2007).
[CrossRef]

de Réotier, P. D.

P. D. de Réotier and A. Yaouanc, “Muon spin rotation and relaxation in magnetic materials,” J. Phys. Condens. Matter 9, 9113–9166 (1997).
[CrossRef]

Eikema, K. S. E.

K. S. E. Eikema, J. Walz, and T. W. Hänsch, “Continuous wave coherent Lyman-α radiation,” Phys. Rev. Lett. 83, 3828–3831(1999).
[CrossRef]

Fick, D.

D. Fick, G. Grawert, and I. M. Turkiewicz, “Nuclear physics with polarized heavy ions,” Phys. Rep. 214, 1–111 (1992).
[CrossRef]

Grawert, G.

D. Fick, G. Grawert, and I. M. Turkiewicz, “Nuclear physics with polarized heavy ions,” Phys. Rep. 214, 1–111 (1992).
[CrossRef]

Hänsch, T. W.

M. Scheid, D. Kolbe, F. Markert, T. W. Hänsch, and J. Walz, “Continuous-wave Lyman-α generation with solid-state lasers,” Opt. Express 17, 11274–11280 (2009).
[CrossRef] [PubMed]

K. S. E. Eikema, J. Walz, and T. W. Hänsch, “Continuous wave coherent Lyman-α radiation,” Phys. Rev. Lett. 83, 3828–3831(1999).
[CrossRef]

Hashimoto, O.

S. Nagamiya, K. Nagamine, O. Hashimoto, and T. Yamazaki, “Negative-muon spin rotation at the oxygen site in paramagnetic MnO+,” Phys. Rev. Lett. 35, 308–311 (1975).
[CrossRef]

Hijmans, T. W.

I. D. Setija, H. G. C. Werij, O. J. Luiten, M. W. Reynolds, T. W. Hijmans, and J. T. Walraven, “Optical cooling of atomic hydrogen in a magnetic trap,” Phys. Rev. Lett. 70, 2257–2260 (1993).
[CrossRef] [PubMed]

Hilber, G.

Hutchinson, M. H. R.

Ishida, K.

P. Strasser, K. Nagamine, T. Matsuzaki, K. Ishida, Y. Matsuda, K. Itahashi, and M. Iwasaki, “Muon spectroscopy with unstable nuclei,” J. Phys. G Nucl. Part. Phys. 29, 2047–2049 (2003).
[CrossRef]

Itahashi, K.

P. Strasser, K. Nagamine, T. Matsuzaki, K. Ishida, Y. Matsuda, K. Itahashi, and M. Iwasaki, “Muon spectroscopy with unstable nuclei,” J. Phys. G Nucl. Part. Phys. 29, 2047–2049 (2003).
[CrossRef]

Iwasaki, M.

P. Strasser, K. Nagamine, T. Matsuzaki, K. Ishida, Y. Matsuda, K. Itahashi, and M. Iwasaki, “Muon spectroscopy with unstable nuclei,” J. Phys. G Nucl. Part. Phys. 29, 2047–2049 (2003).
[CrossRef]

Kolbe, D.

Koopman, D.

R. Mahon, T. J. McIlrath, and D. Koopman, “Nonlinear generation of Lyman-alpha radiation,” Appl. Phys. Lett. 33, 305–307 (1978).
[CrossRef]

Lago, A.

Luiten, O. J.

I. D. Setija, H. G. C. Werij, O. J. Luiten, M. W. Reynolds, T. W. Hijmans, and J. T. Walraven, “Optical cooling of atomic hydrogen in a magnetic trap,” Phys. Rev. Lett. 70, 2257–2260 (1993).
[CrossRef] [PubMed]

Ma, H.

Mahon, R.

R. Mahon, T. J. McIlrath, and D. Koopman, “Nonlinear generation of Lyman-alpha radiation,” Appl. Phys. Lett. 33, 305–307 (1978).
[CrossRef]

Makimura, S.

P. Bakule, Y. Matsuda, Y. Miyake, P. Straser, K. Shimomura, S. Makimura, and K. Nagamine, “Slow muon experiment by laser resonant ionization method at RIKEN-RAL muon facility,” Spectrochim. Acta Part B 58, 1019–1030 (2003).
[CrossRef]

Marangos, J. P.

Markert, F.

Matsuda, Y.

P. Strasser, K. Nagamine, T. Matsuzaki, K. Ishida, Y. Matsuda, K. Itahashi, and M. Iwasaki, “Muon spectroscopy with unstable nuclei,” J. Phys. G Nucl. Part. Phys. 29, 2047–2049 (2003).
[CrossRef]

P. Bakule, Y. Matsuda, Y. Miyake, P. Straser, K. Shimomura, S. Makimura, and K. Nagamine, “Slow muon experiment by laser resonant ionization method at RIKEN-RAL muon facility,” Spectrochim. Acta Part B 58, 1019–1030 (2003).
[CrossRef]

Matsuzaki, T.

P. Strasser, K. Nagamine, T. Matsuzaki, K. Ishida, Y. Matsuda, K. Itahashi, and M. Iwasaki, “Muon spectroscopy with unstable nuclei,” J. Phys. G Nucl. Part. Phys. 29, 2047–2049 (2003).
[CrossRef]

McIlrath, T. J.

R. Mahon, T. J. McIlrath, and D. Koopman, “Nonlinear generation of Lyman-alpha radiation,” Appl. Phys. Lett. 33, 305–307 (1978).
[CrossRef]

Miller, J. P.

J. P. Miller, E. de Rafael, and B. L. Roberts, “Muon (g-2) experiment and theory,” Rep. Prog. Phys. 70, 795–881 (2007).
[CrossRef]

Miyake, Y.

P. Bakule, Y. Matsuda, Y. Miyake, P. Straser, K. Shimomura, S. Makimura, and K. Nagamine, “Slow muon experiment by laser resonant ionization method at RIKEN-RAL muon facility,” Spectrochim. Acta Part B 58, 1019–1030 (2003).
[CrossRef]

Morenzoni, E.

P. Bakule and E. Morenzoni, “Generation and application of slow polarized muons,” Contemp. Phys. 45, 203–225(2004).
[CrossRef]

Nagamine, K.

P. Bakule, Y. Matsuda, Y. Miyake, P. Straser, K. Shimomura, S. Makimura, and K. Nagamine, “Slow muon experiment by laser resonant ionization method at RIKEN-RAL muon facility,” Spectrochim. Acta Part B 58, 1019–1030 (2003).
[CrossRef]

P. Strasser, K. Nagamine, T. Matsuzaki, K. Ishida, Y. Matsuda, K. Itahashi, and M. Iwasaki, “Muon spectroscopy with unstable nuclei,” J. Phys. G Nucl. Part. Phys. 29, 2047–2049 (2003).
[CrossRef]

S. Nagamiya, K. Nagamine, O. Hashimoto, and T. Yamazaki, “Negative-muon spin rotation at the oxygen site in paramagnetic MnO+,” Phys. Rev. Lett. 35, 308–311 (1975).
[CrossRef]

Nagamiya, S.

S. Nagamiya, K. Nagamine, O. Hashimoto, and T. Yamazaki, “Negative-muon spin rotation at the oxygen site in paramagnetic MnO+,” Phys. Rev. Lett. 35, 308–311 (1975).
[CrossRef]

Nakajima, T.

Reynolds, M. W.

I. D. Setija, H. G. C. Werij, O. J. Luiten, M. W. Reynolds, T. W. Hijmans, and J. T. Walraven, “Optical cooling of atomic hydrogen in a magnetic trap,” Phys. Rev. Lett. 70, 2257–2260 (1993).
[CrossRef] [PubMed]

Roberts, B. L.

J. P. Miller, E. de Rafael, and B. L. Roberts, “Muon (g-2) experiment and theory,” Rep. Prog. Phys. 70, 795–881 (2007).
[CrossRef]

Scheid, M.

Setija, I. D.

I. D. Setija, H. G. C. Werij, O. J. Luiten, M. W. Reynolds, T. W. Hijmans, and J. T. Walraven, “Optical cooling of atomic hydrogen in a magnetic trap,” Phys. Rev. Lett. 70, 2257–2260 (1993).
[CrossRef] [PubMed]

Shen, N.

Shimomura, K.

P. Bakule, Y. Matsuda, Y. Miyake, P. Straser, K. Shimomura, S. Makimura, and K. Nagamine, “Slow muon experiment by laser resonant ionization method at RIKEN-RAL muon facility,” Spectrochim. Acta Part B 58, 1019–1030 (2003).
[CrossRef]

Straser, P.

P. Bakule, Y. Matsuda, Y. Miyake, P. Straser, K. Shimomura, S. Makimura, and K. Nagamine, “Slow muon experiment by laser resonant ionization method at RIKEN-RAL muon facility,” Spectrochim. Acta Part B 58, 1019–1030 (2003).
[CrossRef]

Strasser, P.

P. Strasser, K. Nagamine, T. Matsuzaki, K. Ishida, Y. Matsuda, K. Itahashi, and M. Iwasaki, “Muon spectroscopy with unstable nuclei,” J. Phys. G Nucl. Part. Phys. 29, 2047–2049 (2003).
[CrossRef]

Turkiewicz, I. M.

D. Fick, G. Grawert, and I. M. Turkiewicz, “Nuclear physics with polarized heavy ions,” Phys. Rep. 214, 1–111 (1992).
[CrossRef]

Wallenstein, R.

Walraven, J. T.

I. D. Setija, H. G. C. Werij, O. J. Luiten, M. W. Reynolds, T. W. Hijmans, and J. T. Walraven, “Optical cooling of atomic hydrogen in a magnetic trap,” Phys. Rev. Lett. 70, 2257–2260 (1993).
[CrossRef] [PubMed]

Walz, J.

M. Scheid, D. Kolbe, F. Markert, T. W. Hänsch, and J. Walz, “Continuous-wave Lyman-α generation with solid-state lasers,” Opt. Express 17, 11274–11280 (2009).
[CrossRef] [PubMed]

K. S. E. Eikema, J. Walz, and T. W. Hänsch, “Continuous wave coherent Lyman-α radiation,” Phys. Rev. Lett. 83, 3828–3831(1999).
[CrossRef]

Werij, H. G. C.

I. D. Setija, H. G. C. Werij, O. J. Luiten, M. W. Reynolds, T. W. Hijmans, and J. T. Walraven, “Optical cooling of atomic hydrogen in a magnetic trap,” Phys. Rev. Lett. 70, 2257–2260 (1993).
[CrossRef] [PubMed]

Yamazaki, T.

S. Nagamiya, K. Nagamine, O. Hashimoto, and T. Yamazaki, “Negative-muon spin rotation at the oxygen site in paramagnetic MnO+,” Phys. Rev. Lett. 35, 308–311 (1975).
[CrossRef]

Yaouanc, A.

P. D. de Réotier and A. Yaouanc, “Muon spin rotation and relaxation in magnetic materials,” J. Phys. Condens. Matter 9, 9113–9166 (1997).
[CrossRef]

Appl. Phys. Lett.

R. Mahon, T. J. McIlrath, and D. Koopman, “Nonlinear generation of Lyman-alpha radiation,” Appl. Phys. Lett. 33, 305–307 (1978).
[CrossRef]

Contemp. Phys.

P. Bakule and E. Morenzoni, “Generation and application of slow polarized muons,” Contemp. Phys. 45, 203–225(2004).
[CrossRef]

J. Opt. Soc. Am. B

J. Phys. Condens. Matter

P. D. de Réotier and A. Yaouanc, “Muon spin rotation and relaxation in magnetic materials,” J. Phys. Condens. Matter 9, 9113–9166 (1997).
[CrossRef]

J. Phys. G Nucl. Part. Phys.

P. Strasser, K. Nagamine, T. Matsuzaki, K. Ishida, Y. Matsuda, K. Itahashi, and M. Iwasaki, “Muon spectroscopy with unstable nuclei,” J. Phys. G Nucl. Part. Phys. 29, 2047–2049 (2003).
[CrossRef]

Opt. Commun.

D. Cotter, “Tunable narrow-band coherent VUV source for the Lyman-alpha region,” Opt. Commun. 31, 397–400 (1979).
[CrossRef]

R. Wallenstein, “Generation of narrowband tunable VUV radiation at the Lyman-α wavelength,” Opt. Commun. 33, 119–122(1980).
[CrossRef]

Opt. Express

Phys. Rep.

D. Fick, G. Grawert, and I. M. Turkiewicz, “Nuclear physics with polarized heavy ions,” Phys. Rep. 214, 1–111 (1992).
[CrossRef]

Phys. Rev. Lett.

S. Nagamiya, K. Nagamine, O. Hashimoto, and T. Yamazaki, “Negative-muon spin rotation at the oxygen site in paramagnetic MnO+,” Phys. Rev. Lett. 35, 308–311 (1975).
[CrossRef]

K. S. E. Eikema, J. Walz, and T. W. Hänsch, “Continuous wave coherent Lyman-α radiation,” Phys. Rev. Lett. 83, 3828–3831(1999).
[CrossRef]

I. D. Setija, H. G. C. Werij, O. J. Luiten, M. W. Reynolds, T. W. Hijmans, and J. T. Walraven, “Optical cooling of atomic hydrogen in a magnetic trap,” Phys. Rev. Lett. 70, 2257–2260 (1993).
[CrossRef] [PubMed]

Rep. Prog. Phys.

J. P. Miller, E. de Rafael, and B. L. Roberts, “Muon (g-2) experiment and theory,” Rep. Prog. Phys. 70, 795–881 (2007).
[CrossRef]

Spectrochim. Acta Part B

P. Bakule, Y. Matsuda, Y. Miyake, P. Straser, K. Shimomura, S. Makimura, and K. Nagamine, “Slow muon experiment by laser resonant ionization method at RIKEN-RAL muon facility,” Spectrochim. Acta Part B 58, 1019–1030 (2003).
[CrossRef]

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Figures (6)

Fig. 1
Fig. 1

(a) Level scheme. A sequence of right-circularly polarized short pulses pump the muonium ( μ + e ) atoms in the ground 1 s state to the 2 p state. Energy values in the parentheses are those for the hydrogen atoms. (b) Level scheme using the coupled basis for the excited states with a clear distinction of magnetic sublevels. (c) Level scheme using the uncoupled basis for the excited states. Dipole interactions and hyperfine interactions in the excited states are depicted by the long upward arrows and double-ended arrows, respectively. The short up and down arrows by the ket vectors represent the nuclear-spin orientations.

Fig. 2
Fig. 2

Schematic description of the branching ratios from the (a) spin-up and (b) spin-down excited states to the lower states using the uncoupled basis for the excited states. Note that the branching ratios are normalized to unity for each excited state.

Fig. 3
Fig. 3

(a) Comparison of the change of nuclear-spin polarization by a sequence of short laser pulses obtained by solving the recursion relation (horizontal thin solid lines) and the density matrix equations (taken from Ref. [9]). The five 10.2 eV pulses are turned on at 0, 5, 10, 15, and 20 ns with a time interval of 5 ns . The initial population distribution is N 0 ( 0 ) = N 1 ( 0 ) = N 6 ( 0 ) = N 10 ( 0 ) = 1 / 4 . (b) Similar to graph (a) but with a time interval of 10 ns , and accordingly the five 10.2 eV pulses are turned on at 0, 10, 20, 30, and 40 ns .

Fig. 4
Fig. 4

Same as Fig. 3a, but with an initial population distribution of N 0 ( 0 ) = 1 / 4 , N 1 ( 0 ) = 0 , N 6 ( 0 ) = 1 / 4 , N 10 ( 0 ) = 1 / 2 .

Fig. 5
Fig. 5

Evolution of nuclear-spin polarization as a function of pulse number. Results at the intensities of 10 7 , 5 × 10 7 , 10 8 , and 2 × 10 8 W / cm 2 are shown by the open (filled) circles, squares, up triangles, and down triangles, respectively, for the initial popu lation distribution of N 0 ( 0 ) = N 1 ( 0 ) = N 6 ( 0 ) = N 10 ( 0 ) = 1 / 4 ( N 0 ( 0 ) = 1 / 4 , N 1 ( 0 ) = 0 , N 6 ( 0 ) = 1 / 4 , N 10 ( 0 ) = 1 / 2 ). Data points are connected as a guide to the eye.

Fig. 6
Fig. 6

Same as Fig. 3a, but the hyperfine coupling has been fictitiously removed from the excited states. The initial population distribution is N 0 ( 0 ) = N 1 ( 0 ) = N 6 ( 0 ) = N 10 ( 0 ) = 1 / 4 .

Equations (16)

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N g ( m 1 ) = ( N 0 ( m 1 ) N 1 ( m 1 ) N 6 ( m 1 ) N 10 ( m 1 ) ) ,
N e ( m ) = ( N 2 ( m ) N 3 ( m ) N 4 ( m ) N 5 ( m ) N 7 ( m ) N 8 ( m ) N 9 ( m ) N 11 ( m ) ) ,
N 2 ( m ) = 0 ,
N 3 ( m ) = 2 3 sin 2 ( θ 2 ) N 1 ( m 1 ) ,
N 4 ( m ) = 0 ,
N 5 ( m ) = 1 3 sin 2 ( θ 2 ) N 1 ( m 1 ) ,
N 7 ( m ) = 1 3 sin 2 ( θ 2 ) N 0 ( m 1 ) + 1 3 sin 2 ( θ 2 ) N 6 ( m 1 ) ,
N 8 ( m ) = 1 6 sin 2 ( θ 2 ) N 0 ( m 1 ) + 1 6 sin 2 ( θ 2 ) N 6 ( m 1 ) ,
N 9 ( m ) = 1 2 sin 2 ( θ 2 ) N 0 ( m 1 ) + 1 2 sin 2 ( θ 2 ) N 6 ( m 1 ) ,
N 11 ( m ) = sin 2 ( θ 2 ) N 10 ( m 1 ) ,
N e ( m ) = ( N 3 ( m ) sin 2 α N 3 ( m ) cos 2 α N 5 ( m ) sin 2 α N 5 ( m ) cos 2 α N 7 ( m ) N 8 ( m ) cos 2 β + N 9 ( m ) sin 2 β N 8 ( m ) sin 2 β + N 9 ( m ) cos 2 β N 11 ( m ) ) .
N g ( m ) = [ 1 sin 2 ( θ 2 ) ] N g ( m 1 ) + R N e ( m ) ,
R = ( 1 6 1 6 1 3 1 3 1 3 1 6 1 2 0 0 2 3 0 1 3 0 0 0 0 1 6 1 6 1 3 1 3 1 3 1 6 1 2 0 2 3 0 1 3 0 1 3 2 3 0 1 ) .
P = P up P down P up + P down ,
P up = 1 2 N 0 ( n max ) + 1 2 N 6 ( n max ) + N 10 ( n max )
P down = 1 2 N 0 ( n max ) + N 1 ( n max ) + 1 2 N 6 ( n max ) .

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