Abstract

We propose a measurement protocol and parameter estimation algorithm to recover the powers and relative phases of each of the vector modes present at the output of an optical fiber that supports the HE11, TE01, HE21, and TM01 modes. The measurements consist of polarization filtered near-field intensity images that are easily implemented with standard off-shelf components. We demonstrate the accuracy of the method on both simulated and measured data from a recently demonstrated fiber that supports stable orbital angular momentum states.

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  2. C. N. Alexeyev, A. V. Volyar, and M. A. Yavorsky, “Vortex-preserving weakly guiding anisotropic twisted fibres,” J. Opt. A-Pure Appl. Op.6(5), S162–S165 (2004).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  5. “Launched Power Distribution Measurement Procedure for Graded-Index Multimode Fiber Transmitters (FOTP-203),” ANSI/TIA/EIA-455-203-2001, (2001).
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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  10. M. Skorobogatiy, C. Anastassiou, S. G. Johnson, O. Weisberg, T. D. Engeness, S. A. Jacobs, R. U. Ahmad, and Y. Fink, “Quantitative characterization of higher-order mode converters in weakly multimoded fibers,” Opt. Express11(22), 2838–2847 (2003).
    [CrossRef] [PubMed]
  11. B. A. Alvi, M. Asif, and A. Israr, “Measurement of complex mode amplitudes in multimode fiber,” in SPIE OPTO, 82840K–82840K (2012).
    [CrossRef]
  12. T. Kaiser, D. Flamm, S. Schrter, and M. Duparr, “Complete modal decomposition for optical fibers using CGH-based correlation filters,” Opt. Express17(11), 9347–9356 (2009).
    [CrossRef] [PubMed]
  13. J. W. Nicholson, A. D. Yablon, S. Ramachandran, and S. Ghalmi, “Spatially and spectrally resolved imaging of modal content in large-mode-area fibers,” Opt. Express16(10), 7233–7243 (2008).
    [CrossRef] [PubMed]
  14. J. W. Nicholson, A. D. Yablon, J. M. Fini, and M. D. Mermelstein, “Measuring the modal content of large-mode-area fibers,” IEEE J. Sel. Top. Quantum Electron.15(1), 61–70 (2009).
    [CrossRef]
  15. D. N. Schimpf, R. A. Barankov, and S. Ramachandran, “Cross-correlated (C2) imaging of fiber and waveguide modes,” Opt. Express19(14), 13,008–13,019 (2011).
    [CrossRef]
  16. V. Inavalli and N. Viswanathan, “Switchable vector vortex beam generation using an optical fiber,” Opt. Commun.283(6), 861–864 (2010).
    [CrossRef]
  17. A. Snyder and J. Love, Optical waveguide theory, vol. 190 (Springer, 1983).
  18. J. J. Kapany and N. S. Burke, Optical Waveguides (Academic Press, 1972).
  19. A. V. Volyar and T. A. Fadeeva, “Vortex nature of optical fiber modes: I. structure of the natural modes,” Tech. Phys. Lett.22, 330–332 (1996).
  20. R. J. Black and L. Gagnon, Optical Waveguide Modes: Polarization, Coupling and Symmetry, 1st ed. (McGraw-Hill Professional, 2010).

2012 (3)

2011 (1)

D. N. Schimpf, R. A. Barankov, and S. Ramachandran, “Cross-correlated (C2) imaging of fiber and waveguide modes,” Opt. Express19(14), 13,008–13,019 (2011).
[CrossRef]

2010 (1)

V. Inavalli and N. Viswanathan, “Switchable vector vortex beam generation using an optical fiber,” Opt. Commun.283(6), 861–864 (2010).
[CrossRef]

2009 (4)

2008 (1)

2005 (2)

2004 (1)

C. N. Alexeyev, A. V. Volyar, and M. A. Yavorsky, “Vortex-preserving weakly guiding anisotropic twisted fibres,” J. Opt. A-Pure Appl. Op.6(5), S162–S165 (2004).
[CrossRef]

2003 (2)

A. Volyar, “Algebra and geometry of optical vortices: measuring of a mode spectrum in optical fibres,” in First International Conference on Advanced Optoelectronics and Lasers, 2003. Proceedings of CAOL 2003, 1, 30–35 (2003).

M. Skorobogatiy, C. Anastassiou, S. G. Johnson, O. Weisberg, T. D. Engeness, S. A. Jacobs, R. U. Ahmad, and Y. Fink, “Quantitative characterization of higher-order mode converters in weakly multimoded fibers,” Opt. Express11(22), 2838–2847 (2003).
[CrossRef] [PubMed]

1996 (1)

A. V. Volyar and T. A. Fadeeva, “Vortex nature of optical fiber modes: I. structure of the natural modes,” Tech. Phys. Lett.22, 330–332 (1996).

Ahmad, R. U.

Alexeyev, C. N.

C. N. Alexeyev, A. V. Volyar, and M. A. Yavorsky, “Vortex-preserving weakly guiding anisotropic twisted fibres,” J. Opt. A-Pure Appl. Op.6(5), S162–S165 (2004).
[CrossRef]

Alvi, B. A.

B. A. Alvi, M. Asif, and A. Israr, “Measurement of complex mode amplitudes in multimode fiber,” in SPIE OPTO, 82840K–82840K (2012).
[CrossRef]

Anastassiou, C.

Asif, M.

B. A. Alvi, M. Asif, and A. Israr, “Measurement of complex mode amplitudes in multimode fiber,” in SPIE OPTO, 82840K–82840K (2012).
[CrossRef]

Barankov, R. A.

D. N. Schimpf, R. A. Barankov, and S. Ramachandran, “Cross-correlated (C2) imaging of fiber and waveguide modes,” Opt. Express19(14), 13,008–13,019 (2011).
[CrossRef]

Black, R. J.

R. J. Black and L. Gagnon, Optical Waveguide Modes: Polarization, Coupling and Symmetry, 1st ed. (McGraw-Hill Professional, 2010).

Bolle, C.

Bozinovic, N.

Burke, N. S.

J. J. Kapany and N. S. Burke, Optical Waveguides (Academic Press, 1972).

Burrows, E. C.

Dimarcello, F. V.

Duparr, M.

Engeness, T. D.

Esmaeelpour, M.

Essiambre, R.-J.

Fadeeva, T. A.

A. V. Volyar and T. A. Fadeeva, “Vortex nature of optical fiber modes: I. structure of the natural modes,” Tech. Phys. Lett.22, 330–332 (1996).

Fini, J. M.

J. W. Nicholson, A. D. Yablon, J. M. Fini, and M. D. Mermelstein, “Measuring the modal content of large-mode-area fibers,” IEEE J. Sel. Top. Quantum Electron.15(1), 61–70 (2009).
[CrossRef]

Fink, Y.

Flamm, D.

Fleming, J.

Gagnon, L.

R. J. Black and L. Gagnon, Optical Waveguide Modes: Polarization, Coupling and Symmetry, 1st ed. (McGraw-Hill Professional, 2010).

Ghalmi, S.

Gnauck, A. H.

Golowich, S.

Inavalli, V.

V. Inavalli and N. Viswanathan, “Switchable vector vortex beam generation using an optical fiber,” Opt. Commun.283(6), 861–864 (2010).
[CrossRef]

Inavalli, V. V. G.

Israr, A.

B. A. Alvi, M. Asif, and A. Israr, “Measurement of complex mode amplitudes in multimode fiber,” in SPIE OPTO, 82840K–82840K (2012).
[CrossRef]

Jacobs, S. A.

Johnson, S. G.

Kaiser, T.

Kapany, J. J.

J. J. Kapany and N. S. Burke, Optical Waveguides (Academic Press, 1972).

Kristensen, P.

Lingle, R.

Love, J.

A. Snyder and J. Love, Optical waveguide theory, vol. 190 (Springer, 1983).

McCurdy, A. H.

Mermelstein, M. D.

J. W. Nicholson, A. D. Yablon, J. M. Fini, and M. D. Mermelstein, “Measuring the modal content of large-mode-area fibers,” IEEE J. Sel. Top. Quantum Electron.15(1), 61–70 (2009).
[CrossRef]

Monberg, E.

Mumtaz, S.

Nicholson, J. W.

J. W. Nicholson, A. D. Yablon, J. M. Fini, and M. D. Mermelstein, “Measuring the modal content of large-mode-area fibers,” IEEE J. Sel. Top. Quantum Electron.15(1), 61–70 (2009).
[CrossRef]

J. W. Nicholson, A. D. Yablon, S. Ramachandran, and S. Ghalmi, “Spatially and spectrally resolved imaging of modal content in large-mode-area fibers,” Opt. Express16(10), 7233–7243 (2008).
[CrossRef] [PubMed]

Peckham, D. W.

Ramachandran, S.

Randel, S.

Ryf, R.

Schimpf, D. N.

D. N. Schimpf, R. A. Barankov, and S. Ramachandran, “Cross-correlated (C2) imaging of fiber and waveguide modes,” Opt. Express19(14), 13,008–13,019 (2011).
[CrossRef]

Schrter, S.

Sierra, A.

Skorobogatiy, M.

Snyder, A.

A. Snyder and J. Love, Optical waveguide theory, vol. 190 (Springer, 1983).

Viswanathan, N.

V. Inavalli and N. Viswanathan, “Switchable vector vortex beam generation using an optical fiber,” Opt. Commun.283(6), 861–864 (2010).
[CrossRef]

Viswanathan, N. K.

Volyar, A.

A. Volyar, “Algebra and geometry of optical vortices: measuring of a mode spectrum in optical fibres,” in First International Conference on Advanced Optoelectronics and Lasers, 2003. Proceedings of CAOL 2003, 1, 30–35 (2003).

Volyar, A. V.

C. N. Alexeyev, A. V. Volyar, and M. A. Yavorsky, “Vortex-preserving weakly guiding anisotropic twisted fibres,” J. Opt. A-Pure Appl. Op.6(5), S162–S165 (2004).
[CrossRef]

A. V. Volyar and T. A. Fadeeva, “Vortex nature of optical fiber modes: I. structure of the natural modes,” Tech. Phys. Lett.22, 330–332 (1996).

Weisberg, O.

Winzer, P. J.

Wisk, P.

Yablon, A. D.

J. W. Nicholson, A. D. Yablon, J. M. Fini, and M. D. Mermelstein, “Measuring the modal content of large-mode-area fibers,” IEEE J. Sel. Top. Quantum Electron.15(1), 61–70 (2009).
[CrossRef]

J. W. Nicholson, A. D. Yablon, S. Ramachandran, and S. Ghalmi, “Spatially and spectrally resolved imaging of modal content in large-mode-area fibers,” Opt. Express16(10), 7233–7243 (2008).
[CrossRef] [PubMed]

Yan, M. F.

Yavorsky, M. A.

C. N. Alexeyev, A. V. Volyar, and M. A. Yavorsky, “Vortex-preserving weakly guiding anisotropic twisted fibres,” J. Opt. A-Pure Appl. Op.6(5), S162–S165 (2004).
[CrossRef]

First International Conference on Advanced Optoelectronics and Lasers, 2003. Proceedings of CAOL 2003 (1)

A. Volyar, “Algebra and geometry of optical vortices: measuring of a mode spectrum in optical fibres,” in First International Conference on Advanced Optoelectronics and Lasers, 2003. Proceedings of CAOL 2003, 1, 30–35 (2003).

IEEE J. Sel. Top. Quantum Electron. (1)

J. W. Nicholson, A. D. Yablon, J. M. Fini, and M. D. Mermelstein, “Measuring the modal content of large-mode-area fibers,” IEEE J. Sel. Top. Quantum Electron.15(1), 61–70 (2009).
[CrossRef]

J. Lightwave Technol. (1)

J. Opt. A-Pure Appl. Op. (1)

C. N. Alexeyev, A. V. Volyar, and M. A. Yavorsky, “Vortex-preserving weakly guiding anisotropic twisted fibres,” J. Opt. A-Pure Appl. Op.6(5), S162–S165 (2004).
[CrossRef]

Opt. Commun. (1)

V. Inavalli and N. Viswanathan, “Switchable vector vortex beam generation using an optical fiber,” Opt. Commun.283(6), 861–864 (2010).
[CrossRef]

Opt. Express (5)

Opt. Lett. (4)

SPIE OPTO (1)

B. A. Alvi, M. Asif, and A. Israr, “Measurement of complex mode amplitudes in multimode fiber,” in SPIE OPTO, 82840K–82840K (2012).
[CrossRef]

Tech. Phys. Lett. (1)

A. V. Volyar and T. A. Fadeeva, “Vortex nature of optical fiber modes: I. structure of the natural modes,” Tech. Phys. Lett.22, 330–332 (1996).

Other (4)

R. J. Black and L. Gagnon, Optical Waveguide Modes: Polarization, Coupling and Symmetry, 1st ed. (McGraw-Hill Professional, 2010).

“Launched Power Distribution Measurement Procedure for Graded-Index Multimode Fiber Transmitters (FOTP-203),” ANSI/TIA/EIA-455-203-2001, (2001).

A. Snyder and J. Love, Optical waveguide theory, vol. 190 (Springer, 1983).

J. J. Kapany and N. S. Burke, Optical Waveguides (Academic Press, 1972).

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Figures (10)

Fig. 1
Fig. 1

(a) Measured index profile (red) of the six-mode vortex fiber studied in this paper, along with the calculated radial wave functions F01 (black) and F11 (blue); (b) effective index spectrum of the LP11 modes, with calculated (solid) and measured (dashed) values shown.

Fig. 2
Fig. 2

Spatial polarization patterns of the Vector basis set.

Fig. 3
Fig. 3

Experimental setups. (a) A vortex state is excited by a polarization controller (Pol-Con) and microbend grating, and NFI images are taken in series by an InGaAs short-wave infrared (SWIR) camera after passing through a quarter-wave plate (QWP) and/or linear polarizer. (b) Setup for simultaneous imaging of all four polarization filtered images. Components are non-polarizing beam splitters (NPBS), polarizing beam displacing prisms (PBDP), and mirrors (M). (c) Sample image from setup (b).

Fig. 4
Fig. 4

Simulated polarization filtered NFI images for the example with dominant V 21 ( H E + ) state.

Fig. 5
Fig. 5

Simulated polarization filtered NFI images for the example with dominant TM01 state.

Fig. 6
Fig. 6

Parameter estimates for simulated example with dominant V 21 ( H E + ) state. (a) Mode power estimates; (b) phase estimates. In both, the green dots represent the estimates obtained for each of 100 noise realizations.

Fig. 7
Fig. 7

Parameter estimates for simulated example with dominant TM01 state. (a) Mode power estimates; (b) phase estimates. In both, the green dots represent the estimates obtained for each of 100 noise realizations.

Fig. 8
Fig. 8

Measured and fitted values of polarization filtered near-field intensity images.

Fig. 9
Fig. 9

(a) Power fraction in the fundamental, transverse, and HE21 pairs; (b) Power estimated in the HE1,1 modes by regression analysis (red crosses) and by coupling into a SMF (blue curve).

Fig. 10
Fig. 10

(a) Estimated power fraction of each of the two OAM states; (b) Estimated relative phase of the two OAM states.

Equations (20)

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E t ( r , θ , t ) = e i ω t i = 1 6 γ i e i ( r , θ ) ,
H E 11 ( e ) ( r , θ ) = F 01 ( r ) x ^ H E 11 ( o ) ( r , θ ) = F 01 ( r ) y ^ H E 21 ( e ) ( r , θ ) = ( x ^ cos θ y ^ sin θ ) F 11 ( r ) H E 21 ( o ) ( r , θ ) = ( x ^ sin θ + y ^ cos θ ) F 11 ( r ) T M 01 ( r , θ ) = ( x ^ cos θ + y ^ sin θ ) F 11 ( r ) T E 01 ( r , θ ) = ( x ^ sin θ y ^ cos θ ) F 11 ( r ) .
V 11 ( H E + ) ( r , θ ) = ( H E 11 ( e ) + i H E 11 ( o ) ) / 2 = ( x ^ + i y ^ ) F 01 / 2 V 11 ( H E ) ( r , θ ) = ( H E 11 ( e ) i H E 11 ( o ) ) / 2 = ( x ^ i y ^ ) F 01 / 2 V 21 ( H E + ) ( r , θ ) = ( H E 21 ( e ) + i H E 21 ( o ) ) / 2 = e i θ ( x ^ + i y ^ ) F 11 / 2 V 21 ( H E ) ( r , θ ) = ( H E 21 ( e ) i H E 21 ( o ) ) / 2 = e i θ ( x ^ i y ^ ) F 11 / 2 V 01 ( T + ) ( r , θ ) = ( T M 01 i T E 01 ) / 2 = e i θ ( x ^ + i y ^ ) F 11 / 2 V 01 ( T ) ( r , θ ) = ( T M 01 + i T E 01 ) / 2 = e i θ ( x ^ i y ^ ) F 11 / 2 .
L P 01 ( x ) ( r , θ ) = x ^ F 01 ( r ) L P 01 ( y ) ( r , θ ) = y ^ F 01 ( r ) L P 11 ( x + ) ( r , θ ) = x ^ e i θ F 11 ( r ) L P 11 ( x ) ( r , θ ) = x ^ e i θ F 11 ( r ) L P 11 ( y + ) ( r , θ ) = y ^ e i θ F 11 ( r ) L P 11 ( y ) ( r , θ ) = y ^ e i θ F 11 ( r ) .
e 1 = V 11 ( H E + ) ; e 3 = V 21 ( H E + ) ; e 5 = V 01 ( T + ) ; e 2 = V 11 ( H E ) ; e 4 = V 21 ( H E ) ; e 6 = V 01 ( T ) .
γ i = | γ i | e i ϕ i .
| P + E t ( r , θ ) | 2 = | γ 1 F 01 ( r ) + γ 3 e i θ F 11 + γ 5 e i θ F 11 ( r ) | 2 = | γ 1 | 2 F 01 ( r ) 2 + ( | γ 3 | 2 + | γ 5 | 2 ) F 11 ( r ) 2 + 2 ( γ 3 γ 5 * ) cos ( 2 θ ) F 11 ( r ) 2 2 ( γ 3 γ 5 * ) sin ( 2 θ ) F 11 ( r ) 2 + 2 ( γ 1 ( γ 3 + γ 5 ) * ) ) cos ( θ ) F 01 ( r ) F 11 ( r ) + 2 ( γ 1 ( γ 3 γ 5 ) * ) ) sin ( θ ) F 01 ( r ) F 11 ( r ) .
| P E t ( r , θ ) | 2 = | γ 2 | 2 F 01 ( r ) 2 + ( | γ 4 | 2 + | γ 6 | 2 ) F 11 ( r ) 2 + 2 ( γ 4 γ 6 * ) cos ( 2 θ ) F 11 ( r ) 2 + 2 ( γ 4 γ 6 * ) sin ( 2 θ ) F 11 ( r ) 2 + 2 ( γ 2 ( γ 6 + γ 4 ) * ) ) cos ( θ ) F 01 ( r ) F 11 ( r ) + 2 ( γ 2 ( γ 6 γ 4 ) * ) ) sin ( θ ) F 01 ( r ) F 11 ( r ) .
Ξ 1 = | γ 1 | 2 ; Ξ 2 = | γ 2 | 2 ; Ξ 3 + 5 = | γ 3 | 2 + | γ 5 | 2 ; Ξ 4 + 6 = | γ 4 | 2 + | γ 6 | 2 ; Ξ 35 = γ 3 γ 5 * ; Ξ 46 = γ 4 γ 6 * ; Ξ 13 + 5 = γ 1 ( γ 3 + γ 5 ) * ; Ξ 26 + 4 = γ 2 ( γ 6 + γ 4 ) * ; Ξ 13 5 = γ 1 ( γ 3 γ 5 ) * ; Ξ 26 4 = γ 2 ( γ 6 γ 4 ) * .
Γ 35 ( min ) = min ( | γ 3 | , | γ 5 | ) ; Γ 46 ( min ) = min ( | γ 4 | , | γ 6 | ) ; Γ 35 ( max ) = max ( | γ 3 | , | γ 5 | ) Γ 46 ( max ) = max ( | γ 4 | , | γ 6 | ) .
δ a b = ϕ a ϕ b .
2 | P x ^ E t | 2 = | γ 1 + γ 2 | 2 F 01 ( r ) 2 + ( | γ 3 + γ 6 | 2 + | γ 5 + γ 4 | 2 ) F 11 ( r ) 2 + 2 cos ( 2 θ ) ( γ 3 + γ 6 ) ( γ 5 + γ 4 ) * F 11 ( r ) 2 2 sin ( 2 θ ) ( γ 3 + γ 6 ) ( γ 5 + γ 4 ) * F 11 ( r ) 2 + 2 cos ( θ ) ( γ 1 + γ 2 ) ( γ 3 + γ 6 + γ 5 + γ 4 ) * F 01 ( r ) F 11 ( r ) + 2 sin ( θ ) ( γ 1 + γ 2 ) ( γ 3 + γ 6 γ 5 γ 4 ) * F 01 ( r ) F 11 ( r )
2 | P y ^ E t | 2 = | γ 1 γ 2 | 2 F 01 ( r ) 2 + ( | γ 3 γ 6 | 2 + | γ 5 γ 4 | 2 ) F 11 ( r ) 2 + 2 cos ( 2 θ ) ( γ 3 γ 6 ) ( γ 5 γ 4 ) * F 11 ( r ) 2 2 sin ( 2 θ ) ( γ 3 γ 6 ) ( γ 5 γ 4 ) * F 11 ( r ) 2 + 2 cos ( θ ) ( γ 1 γ 2 ) ( γ 3 γ 6 + γ 5 γ 4 ) * F 01 ( r ) F 11 ( r ) + 2 sin ( θ ) ( γ 1 γ 2 ) ( γ 3 γ 6 γ 5 + γ 4 ) * F 01 ( r ) F 11 ( r ) .
Ξ 12 ( x ) = | γ 1 + γ 2 | 2 ; Ξ 12 ( y ) = | γ 1 γ 2 | 2 ; Ξ 36 + 54 ( x ) = | γ 3 + γ 6 | 2 + | γ 5 + γ 4 | 2 ; Ξ 36 + 54 ( y ) = | γ 3 γ 6 | 2 + | γ 5 γ 4 | 2 ; Ξ 3654 ( x ) = ( γ 3 + γ 6 ) ( γ 5 + γ 4 ) * ; Ξ 3654 ( y ) ( γ 3 γ 6 ) ( γ 5 γ 4 ) * ; Ξ 1236 + 54 ( x ) = ( γ 1 + γ 2 ) ( γ 3 + γ 6 + γ 5 + γ 4 ) * ; Ξ 1236 + 54 ( y ) = ( γ 1 γ 2 ) ( γ 3 γ 6 + γ 5 γ 4 ) * ; Ξ 1236 54 ( x ) = ( γ 1 + γ 2 ) ( γ 3 + γ 6 γ 5 γ 4 ) * ; Ξ 1236 54 ( y ) = ( γ 1 γ 2 ) ( γ 3 γ 6 γ 5 + γ 4 ) * .
z = Ξ 3645 ( x ) Ξ 35 Ξ 46 * = | γ 3 | | γ 4 | e i δ 43 + | γ 5 | | γ 6 | e i ( δ 64 + δ 35 ) e i δ 43 .
( z z ) = ( | γ 5 | | γ 6 | cos ( δ 64 + δ 35 ) = | γ 3 | | γ 4 | | γ 5 | | γ 6 | sin ( δ 64 + δ 35 ) | γ 5 | | γ 6 | sin ( δ 64 + δ 35 ) | γ 5 | | γ 6 | cos ( δ 64 + δ 35 ) | γ 3 | | γ 4 | ) ( cos δ 43 sin δ 43 ) .
( Ξ 13 + 5 / | γ 1 | Ξ 13 5 / | γ 1 | ) = ( | γ 5 | cos δ 53 + | γ 3 | | γ 5 | sin δ 53 | γ 5 | sin δ 53 | γ 5 | cos δ 53 | γ 3 | ) ( cos ϕ 3 sin ϕ 3 ) ,
( Ξ 26 + 4 / | γ 2 | Ξ 26 4 / | γ 2 | ) = ( | γ 6 | cos δ 64 + | γ 4 | | γ 6 | sin δ 64 | γ 6 | sin δ 64 | γ 6 | cos δ 64 | γ 4 | ) ( cos δ 24 sin δ 24 ) ,
γ ˜ 2 = γ 2 * ; γ ˜ 3 = γ 5 * ; γ ˜ 5 = γ 3 * ; γ ˜ 4 = γ 6 * ; γ ˜ 6 = γ 4 * .
γ ^ = argmin γ p , x | I p ( x ) P p E t ( x , γ ) | 2 ,

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