Abstract

The technical embodiment of the Huygens-Fresnel principle, an optical phased array (OPA) is an arrangement of optical emitters with relative phases controlled to create a desired beam profile after propagation. One important application of an OPA is coherent beam combining (CBC), which can be used to create beams of higher power than is possible with a single laser source, especially for narrow linewidth sources. Here we present an all-fiber architecture that stabilizes the relative output phase by inferring the relative path length differences between lasers using the small fraction of light that is back-reflected into the fiber at the OPA’s glass-air interface, without the need for any external sampling optics. This architecture is compatible with high power continuous wave laser sources (e.g., fiber amplifiers) up to 100 W per channel. The high-power compatible internally sensed OPA was implemented experimentally using commercial 15 W fiber amplifiers, demonstrating an output RMS phase stability of λ/194, and the ability to steer the beam at up to 10 kHz.

© 2016 Optical Society of America

Full Article  |  PDF Article
OSA Recommended Articles
Coherent beam combining using a 2D internally sensed optical phased array

Lyle E. Roberts, Robert L. Ward, Andrew J. Sutton, Roland Fleddermann, Glenn de Vine, Emmanuel A. Malikides, Danielle M. R. Wuchenich, David E. McClelland, and Daniel A. Shaddock
Appl. Opt. 53(22) 4881-4885 (2014)

High speed, high power one-dimensional beam steering from a 6-element optical phased array

W. Ronny Huang, Juan Montoya, Jan E. Kansky, Shawn M. Redmond, George W. Turner, and Antonio Sanchez-Rubio
Opt. Express 20(16) 17311-17318 (2012)

Internally sensed optical phased array

David J. Bowman, Malcolm J. King, Andrew J. Sutton, Danielle M. Wuchenich, Robert L. Ward, Emmanuel A. Malikides, David E. McClelland, and Daniel A. Shaddock
Opt. Lett. 38(7) 1137-1139 (2013)

References

  • View by:
  • |
  • |
  • |

  1. R.G. Smith, “Optical power handling capacity of low loss optical fibers as determined by stimulated Raman and Brillouin scattering,” Appl. Opt. 11(11), 2489–2494 (1972).
    [Crossref] [PubMed]
  2. J. Mason, J. Stupl, W. Marshall, and C. Levit, “Orbital debris–debris collision avoidance,” Adv. Space Res. 48(10), 1643–1655 (2011).
    [Crossref]
  3. H. Bruesselbach, S. Wang, M. Minden, D.C. Jones, and M. Mangir, “Power-scalable phase-compensating fiber-array transceiver for laser communications through the atmosphere,” J. Opt. Soc. Am. B. 22(2), 347–353 (2005).
    [Crossref]
  4. P.F. McManamon, T.A. Dorschner, D.L. Corkum, L.J. Friedman, D.S. Hobbs, M. Holz, S. Liberman, H.Q. Nguyen, D.P. Resler, and R.C. Sharp, “Optical phased array technology,” Proc. IEEE 84(2), 268–298 (1996).
    [Crossref]
  5. T.M. Shay, V. Benham, J.T. Baker, B. Ward, A.D. Sanchez, M.A. Culpepper, D. Pilkington, J. Spring, D.J. Nelson, and C.A. Lu, “First experimental demonstration of self-synchronous phase locking of an optical array,” Opt. Express 14(25), 12015–12021 (2006).
    [Crossref] [PubMed]
  6. C.X. Yu, S.J. Augst, S.M. Redmond, K.C. Goldizen, D.V. Murphy, A. Sanchez, and T.Y. Fan, “Coherent combining of a 4 kw, eight-element fiber amplifier array,” Opt. Lett. 36(14), 2686–2688 (2011).
    [Crossref] [PubMed]
  7. G.S. Goodno, S.J. McNaught, J.E. Rothenberg, T.S. McComb, P.A. Thielen, M.G. Wickham, and M.E. Weber, “Active phase and polarization locking of a 1.4 kW fiber amplifier,” Opt. Lett. 35(10), 1542–1544 (2010).
    [Crossref] [PubMed]
  8. M.A. Vorontsov, S.L. Lachinova, L.A. Beresnev, and T. Weyrauch, “Obscuration-free pupil-plane phase locking of a coherent array of fiber collimators,” J. Opt. Soc. Am. A 27(11), A106–A121 (2010)
    [Crossref]
  9. D.J. Bowman, M.J. King, A.J. Sutton, D.M. Wuchenich, R.L. Ward, E.A. Malikides, D.E. McClelland, and D.A. Shaddock, “Internally sensed optical phased array,” Opt. Lett. 38(7), 1137–1139 (2013).
    [Crossref] [PubMed]
  10. L.E. Roberts, R.L. Ward, A.J. Sutton, R. Fleddermann, G. de Vine, D.M. Wuchenich, E.A. Malikides, D.E. McClelland, and D.A. Shaddock, “Coherent beam combining using a 2D internally sensed optical phased array,” Appl. Opt. 53(22), 4881–4885 (2014).
    [Crossref] [PubMed]
  11. D.A. Shaddock, “Digitally enhanced heterodyne interferometry,” Opt. Lett.,  32(22), 3355–3357 (2007).
    [Crossref] [PubMed]
  12. D. M. Wuchenich, T. Lam, J. H. Chow, D. E. McClelland, and D. A. Shaddock, “Laser frequency noise immunity in multiplexed displacement sensing,” Opt. Lett. 36(5), 672–674 (2011).
    [Crossref] [PubMed]
  13. G. de Vine, D. S. Rabeling, B. J. Slagmolen, T. Y. Lam, S. Chua, D. M. Wuchenich, D. E. McClelland, and D. A. Shaddock, “Picometer level displacement metrology with digitally enhanced heterodyne interferometry,” Opt. Exp. 17(2), 828–837 (2009).
    [Crossref]
  14. D.A. Shaddock, B. Ware, P. G. Halverson, R. E. Spero, and B. Klipstein, “Overview of the LISA phasemeter,” AIP Conf. Proc. 873, 654–660 (2006).
    [Crossref]
  15. A. Kobyakov, M. Sauer, and D. Chowdhury, “Stimulated Brillouin scattering in optical fibers,” Adv. Opt. Photonics 2(1), 1–59 (2010).
    [Crossref]
  16. B. Anderson, A. Flores, R. Holten, and I. Dajani, “Comparison of phase modulation schemes for coherently combined fiber amplifiers,” Opt. Express 23(21), 27046–27060 (2015).
    [Crossref] [PubMed]
  17. A. Flores, C. Robin, A. Lanari, and I. Dajani, “Pseudo-random binary sequence phase modulation for narrow linewidth, kilowatt, monolithic fiber amplifiers,” Opt. Express 22(15), 17735–17744 (2014).
    [Crossref] [PubMed]
  18. G.D. Goodno, C-C. Shih, and J.E. Rothenberg, “Perturbative analysis of coherent combining efficiency with mismatched lasers,” Opt. Express 18(24), 25403–25414 (2010).
    [Crossref] [PubMed]

2015 (1)

2014 (2)

2013 (1)

2011 (3)

2010 (4)

2009 (1)

G. de Vine, D. S. Rabeling, B. J. Slagmolen, T. Y. Lam, S. Chua, D. M. Wuchenich, D. E. McClelland, and D. A. Shaddock, “Picometer level displacement metrology with digitally enhanced heterodyne interferometry,” Opt. Exp. 17(2), 828–837 (2009).
[Crossref]

2007 (1)

2006 (2)

2005 (1)

H. Bruesselbach, S. Wang, M. Minden, D.C. Jones, and M. Mangir, “Power-scalable phase-compensating fiber-array transceiver for laser communications through the atmosphere,” J. Opt. Soc. Am. B. 22(2), 347–353 (2005).
[Crossref]

1996 (1)

P.F. McManamon, T.A. Dorschner, D.L. Corkum, L.J. Friedman, D.S. Hobbs, M. Holz, S. Liberman, H.Q. Nguyen, D.P. Resler, and R.C. Sharp, “Optical phased array technology,” Proc. IEEE 84(2), 268–298 (1996).
[Crossref]

1972 (1)

Anderson, B.

Augst, S.J.

Baker, J.T.

Benham, V.

Beresnev, L.A.

Bowman, D.J.

Bruesselbach, H.

H. Bruesselbach, S. Wang, M. Minden, D.C. Jones, and M. Mangir, “Power-scalable phase-compensating fiber-array transceiver for laser communications through the atmosphere,” J. Opt. Soc. Am. B. 22(2), 347–353 (2005).
[Crossref]

Chow, J. H.

Chowdhury, D.

A. Kobyakov, M. Sauer, and D. Chowdhury, “Stimulated Brillouin scattering in optical fibers,” Adv. Opt. Photonics 2(1), 1–59 (2010).
[Crossref]

Chua, S.

G. de Vine, D. S. Rabeling, B. J. Slagmolen, T. Y. Lam, S. Chua, D. M. Wuchenich, D. E. McClelland, and D. A. Shaddock, “Picometer level displacement metrology with digitally enhanced heterodyne interferometry,” Opt. Exp. 17(2), 828–837 (2009).
[Crossref]

Corkum, D.L.

P.F. McManamon, T.A. Dorschner, D.L. Corkum, L.J. Friedman, D.S. Hobbs, M. Holz, S. Liberman, H.Q. Nguyen, D.P. Resler, and R.C. Sharp, “Optical phased array technology,” Proc. IEEE 84(2), 268–298 (1996).
[Crossref]

Culpepper, M.A.

Dajani, I.

de Vine, G.

L.E. Roberts, R.L. Ward, A.J. Sutton, R. Fleddermann, G. de Vine, D.M. Wuchenich, E.A. Malikides, D.E. McClelland, and D.A. Shaddock, “Coherent beam combining using a 2D internally sensed optical phased array,” Appl. Opt. 53(22), 4881–4885 (2014).
[Crossref] [PubMed]

G. de Vine, D. S. Rabeling, B. J. Slagmolen, T. Y. Lam, S. Chua, D. M. Wuchenich, D. E. McClelland, and D. A. Shaddock, “Picometer level displacement metrology with digitally enhanced heterodyne interferometry,” Opt. Exp. 17(2), 828–837 (2009).
[Crossref]

Dorschner, T.A.

P.F. McManamon, T.A. Dorschner, D.L. Corkum, L.J. Friedman, D.S. Hobbs, M. Holz, S. Liberman, H.Q. Nguyen, D.P. Resler, and R.C. Sharp, “Optical phased array technology,” Proc. IEEE 84(2), 268–298 (1996).
[Crossref]

Fan, T.Y.

Fleddermann, R.

Flores, A.

Friedman, L.J.

P.F. McManamon, T.A. Dorschner, D.L. Corkum, L.J. Friedman, D.S. Hobbs, M. Holz, S. Liberman, H.Q. Nguyen, D.P. Resler, and R.C. Sharp, “Optical phased array technology,” Proc. IEEE 84(2), 268–298 (1996).
[Crossref]

Goldizen, K.C.

Goodno, G.D.

Goodno, G.S.

Halverson, P. G.

D.A. Shaddock, B. Ware, P. G. Halverson, R. E. Spero, and B. Klipstein, “Overview of the LISA phasemeter,” AIP Conf. Proc. 873, 654–660 (2006).
[Crossref]

Hobbs, D.S.

P.F. McManamon, T.A. Dorschner, D.L. Corkum, L.J. Friedman, D.S. Hobbs, M. Holz, S. Liberman, H.Q. Nguyen, D.P. Resler, and R.C. Sharp, “Optical phased array technology,” Proc. IEEE 84(2), 268–298 (1996).
[Crossref]

Holten, R.

Holz, M.

P.F. McManamon, T.A. Dorschner, D.L. Corkum, L.J. Friedman, D.S. Hobbs, M. Holz, S. Liberman, H.Q. Nguyen, D.P. Resler, and R.C. Sharp, “Optical phased array technology,” Proc. IEEE 84(2), 268–298 (1996).
[Crossref]

Jones, D.C.

H. Bruesselbach, S. Wang, M. Minden, D.C. Jones, and M. Mangir, “Power-scalable phase-compensating fiber-array transceiver for laser communications through the atmosphere,” J. Opt. Soc. Am. B. 22(2), 347–353 (2005).
[Crossref]

King, M.J.

Klipstein, B.

D.A. Shaddock, B. Ware, P. G. Halverson, R. E. Spero, and B. Klipstein, “Overview of the LISA phasemeter,” AIP Conf. Proc. 873, 654–660 (2006).
[Crossref]

Kobyakov, A.

A. Kobyakov, M. Sauer, and D. Chowdhury, “Stimulated Brillouin scattering in optical fibers,” Adv. Opt. Photonics 2(1), 1–59 (2010).
[Crossref]

Lachinova, S.L.

Lam, T.

Lam, T. Y.

G. de Vine, D. S. Rabeling, B. J. Slagmolen, T. Y. Lam, S. Chua, D. M. Wuchenich, D. E. McClelland, and D. A. Shaddock, “Picometer level displacement metrology with digitally enhanced heterodyne interferometry,” Opt. Exp. 17(2), 828–837 (2009).
[Crossref]

Lanari, A.

Levit, C.

J. Mason, J. Stupl, W. Marshall, and C. Levit, “Orbital debris–debris collision avoidance,” Adv. Space Res. 48(10), 1643–1655 (2011).
[Crossref]

Liberman, S.

P.F. McManamon, T.A. Dorschner, D.L. Corkum, L.J. Friedman, D.S. Hobbs, M. Holz, S. Liberman, H.Q. Nguyen, D.P. Resler, and R.C. Sharp, “Optical phased array technology,” Proc. IEEE 84(2), 268–298 (1996).
[Crossref]

Lu, C.A.

Malikides, E.A.

Mangir, M.

H. Bruesselbach, S. Wang, M. Minden, D.C. Jones, and M. Mangir, “Power-scalable phase-compensating fiber-array transceiver for laser communications through the atmosphere,” J. Opt. Soc. Am. B. 22(2), 347–353 (2005).
[Crossref]

Marshall, W.

J. Mason, J. Stupl, W. Marshall, and C. Levit, “Orbital debris–debris collision avoidance,” Adv. Space Res. 48(10), 1643–1655 (2011).
[Crossref]

Mason, J.

J. Mason, J. Stupl, W. Marshall, and C. Levit, “Orbital debris–debris collision avoidance,” Adv. Space Res. 48(10), 1643–1655 (2011).
[Crossref]

McClelland, D. E.

D. M. Wuchenich, T. Lam, J. H. Chow, D. E. McClelland, and D. A. Shaddock, “Laser frequency noise immunity in multiplexed displacement sensing,” Opt. Lett. 36(5), 672–674 (2011).
[Crossref] [PubMed]

G. de Vine, D. S. Rabeling, B. J. Slagmolen, T. Y. Lam, S. Chua, D. M. Wuchenich, D. E. McClelland, and D. A. Shaddock, “Picometer level displacement metrology with digitally enhanced heterodyne interferometry,” Opt. Exp. 17(2), 828–837 (2009).
[Crossref]

McClelland, D.E.

McComb, T.S.

McManamon, P.F.

P.F. McManamon, T.A. Dorschner, D.L. Corkum, L.J. Friedman, D.S. Hobbs, M. Holz, S. Liberman, H.Q. Nguyen, D.P. Resler, and R.C. Sharp, “Optical phased array technology,” Proc. IEEE 84(2), 268–298 (1996).
[Crossref]

McNaught, S.J.

Minden, M.

H. Bruesselbach, S. Wang, M. Minden, D.C. Jones, and M. Mangir, “Power-scalable phase-compensating fiber-array transceiver for laser communications through the atmosphere,” J. Opt. Soc. Am. B. 22(2), 347–353 (2005).
[Crossref]

Murphy, D.V.

Nelson, D.J.

Nguyen, H.Q.

P.F. McManamon, T.A. Dorschner, D.L. Corkum, L.J. Friedman, D.S. Hobbs, M. Holz, S. Liberman, H.Q. Nguyen, D.P. Resler, and R.C. Sharp, “Optical phased array technology,” Proc. IEEE 84(2), 268–298 (1996).
[Crossref]

Pilkington, D.

Rabeling, D. S.

G. de Vine, D. S. Rabeling, B. J. Slagmolen, T. Y. Lam, S. Chua, D. M. Wuchenich, D. E. McClelland, and D. A. Shaddock, “Picometer level displacement metrology with digitally enhanced heterodyne interferometry,” Opt. Exp. 17(2), 828–837 (2009).
[Crossref]

Redmond, S.M.

Resler, D.P.

P.F. McManamon, T.A. Dorschner, D.L. Corkum, L.J. Friedman, D.S. Hobbs, M. Holz, S. Liberman, H.Q. Nguyen, D.P. Resler, and R.C. Sharp, “Optical phased array technology,” Proc. IEEE 84(2), 268–298 (1996).
[Crossref]

Roberts, L.E.

Robin, C.

Rothenberg, J.E.

Sanchez, A.

Sanchez, A.D.

Sauer, M.

A. Kobyakov, M. Sauer, and D. Chowdhury, “Stimulated Brillouin scattering in optical fibers,” Adv. Opt. Photonics 2(1), 1–59 (2010).
[Crossref]

Shaddock, D. A.

D. M. Wuchenich, T. Lam, J. H. Chow, D. E. McClelland, and D. A. Shaddock, “Laser frequency noise immunity in multiplexed displacement sensing,” Opt. Lett. 36(5), 672–674 (2011).
[Crossref] [PubMed]

G. de Vine, D. S. Rabeling, B. J. Slagmolen, T. Y. Lam, S. Chua, D. M. Wuchenich, D. E. McClelland, and D. A. Shaddock, “Picometer level displacement metrology with digitally enhanced heterodyne interferometry,” Opt. Exp. 17(2), 828–837 (2009).
[Crossref]

Shaddock, D.A.

Sharp, R.C.

P.F. McManamon, T.A. Dorschner, D.L. Corkum, L.J. Friedman, D.S. Hobbs, M. Holz, S. Liberman, H.Q. Nguyen, D.P. Resler, and R.C. Sharp, “Optical phased array technology,” Proc. IEEE 84(2), 268–298 (1996).
[Crossref]

Shay, T.M.

Shih, C-C.

Slagmolen, B. J.

G. de Vine, D. S. Rabeling, B. J. Slagmolen, T. Y. Lam, S. Chua, D. M. Wuchenich, D. E. McClelland, and D. A. Shaddock, “Picometer level displacement metrology with digitally enhanced heterodyne interferometry,” Opt. Exp. 17(2), 828–837 (2009).
[Crossref]

Smith, R.G.

Spero, R. E.

D.A. Shaddock, B. Ware, P. G. Halverson, R. E. Spero, and B. Klipstein, “Overview of the LISA phasemeter,” AIP Conf. Proc. 873, 654–660 (2006).
[Crossref]

Spring, J.

Stupl, J.

J. Mason, J. Stupl, W. Marshall, and C. Levit, “Orbital debris–debris collision avoidance,” Adv. Space Res. 48(10), 1643–1655 (2011).
[Crossref]

Sutton, A.J.

Thielen, P.A.

Vorontsov, M.A.

Wang, S.

H. Bruesselbach, S. Wang, M. Minden, D.C. Jones, and M. Mangir, “Power-scalable phase-compensating fiber-array transceiver for laser communications through the atmosphere,” J. Opt. Soc. Am. B. 22(2), 347–353 (2005).
[Crossref]

Ward, B.

Ward, R.L.

Ware, B.

D.A. Shaddock, B. Ware, P. G. Halverson, R. E. Spero, and B. Klipstein, “Overview of the LISA phasemeter,” AIP Conf. Proc. 873, 654–660 (2006).
[Crossref]

Weber, M.E.

Weyrauch, T.

Wickham, M.G.

Wuchenich, D. M.

D. M. Wuchenich, T. Lam, J. H. Chow, D. E. McClelland, and D. A. Shaddock, “Laser frequency noise immunity in multiplexed displacement sensing,” Opt. Lett. 36(5), 672–674 (2011).
[Crossref] [PubMed]

G. de Vine, D. S. Rabeling, B. J. Slagmolen, T. Y. Lam, S. Chua, D. M. Wuchenich, D. E. McClelland, and D. A. Shaddock, “Picometer level displacement metrology with digitally enhanced heterodyne interferometry,” Opt. Exp. 17(2), 828–837 (2009).
[Crossref]

Wuchenich, D.M.

Yu, C.X.

Adv. Opt. Photonics (1)

A. Kobyakov, M. Sauer, and D. Chowdhury, “Stimulated Brillouin scattering in optical fibers,” Adv. Opt. Photonics 2(1), 1–59 (2010).
[Crossref]

Adv. Space Res. (1)

J. Mason, J. Stupl, W. Marshall, and C. Levit, “Orbital debris–debris collision avoidance,” Adv. Space Res. 48(10), 1643–1655 (2011).
[Crossref]

AIP Conf. Proc. (1)

D.A. Shaddock, B. Ware, P. G. Halverson, R. E. Spero, and B. Klipstein, “Overview of the LISA phasemeter,” AIP Conf. Proc. 873, 654–660 (2006).
[Crossref]

Appl. Opt. (2)

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

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

H. Bruesselbach, S. Wang, M. Minden, D.C. Jones, and M. Mangir, “Power-scalable phase-compensating fiber-array transceiver for laser communications through the atmosphere,” J. Opt. Soc. Am. B. 22(2), 347–353 (2005).
[Crossref]

Opt. Exp. (1)

G. de Vine, D. S. Rabeling, B. J. Slagmolen, T. Y. Lam, S. Chua, D. M. Wuchenich, D. E. McClelland, and D. A. Shaddock, “Picometer level displacement metrology with digitally enhanced heterodyne interferometry,” Opt. Exp. 17(2), 828–837 (2009).
[Crossref]

Opt. Express (4)

Opt. Lett. (5)

Proc. IEEE (1)

P.F. McManamon, T.A. Dorschner, D.L. Corkum, L.J. Friedman, D.S. Hobbs, M. Holz, S. Liberman, H.Q. Nguyen, D.P. Resler, and R.C. Sharp, “Optical phased array technology,” Proc. IEEE 84(2), 268–298 (1996).
[Crossref]

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (8)

Fig. 1
Fig. 1 Idealized fiber amplifier compatible internally sensed optical phased array.
Fig. 2
Fig. 2 Forward path phase contributions from each unique length of fiber in the optical system. For clarity only two fibers are shown.
Fig. 3
Fig. 3 Return path phase contributions from each unique length of fiber in the optical system. For clarity only two fibres are shown.
Fig. 4
Fig. 4 Phase wrapping algorithm used to exploit the 2π ambiguity of phase. At ∼14 seconds the feedback signal encounters the lower threshold at −0.6 cycles. This prompts the wrapping algorithm to add 1 cycle to the feedback signal, which immediately jumps to +0.4 cycles.
Fig. 5
Fig. 5 Experimental configuration of the optical system used to characterize the high-power compatible internally sensed OPA.
Fig. 6
Fig. 6 (a) Time series measurements of ΦRMS when the OPA is unlocked (green), forward-path locked (blue), and fully locked (magenta) without DEHI. The inset shows the zoomedin behavior of ΦRMS. (b) RPSD of the measurements shown in (a); the noise-shelf at frequencies below 10 Hz is typical of cyclic phase noise introduced by parasitic interference somewhere in the optical system.
Fig. 7
Fig. 7 (a) Time series measurements of ΦRMS when the OPA is unlocked (green), forward-path locked (blue), and fully locked (magenta) with DEHI. The inset shows the zoomed-in behavior of ΦRMS. (b) RPSD of the measurements shown in (a). The harmonic distortion visible on the forward and return path locked RPSD in (b) is caused by residual PRN noise introduced by the demodulator.
Fig. 8
Fig. 8 (a) Time series measurement of 5 kHz sinusoidal beam-steering with a high controller bandwidth. (b) Measured magnitude response of beam-steering for different controller bandwidths. The 5 kHz tone shown in (a) is identified by the blue circle in (b).

Equations (15)

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

s Fn ( t ) = A n sin ( ω h t + Φ Fn + β c ( t τ n ) )
s Fn ( t ) = p ( t τ n ) A n sin ( ω h t + Φ Fn )
s Rk ( t ) = k r Rk ( t ) = k p ( t τ k ) B k sin ( ω h t + Φ Rk )
Φ F 0 = ( ϕ 0 + ϕ A ) ( ϕ 1 + ϕ 2 + ϕ a ) Φ F 1 = ( ϕ 0 + ϕ B ) ( ϕ 1 + ϕ 2 + ϕ b )
Φ R 0 = ( ϕ 0 + ϕ A + 2 ϕ c + ϕ a + ϕ 2 + ϕ R ) ( ϕ 1 + ϕ LO )
Φ R 1 = ( ϕ 0 + ϕ B + 2 ϕ d + ϕ b + ϕ 2 + ϕ R ) ( ϕ 1 + ϕ LO )
Φ ^ F 0 = ( ϕ 0 + ϕ A ) ( ϕ 1 + ϕ 2 + ϕ a ) = 0 ( ϕ 0 + ϕ A ) = ( ϕ 1 + ϕ 2 + ϕ a )
Φ ^ F 1 = ( ϕ 0 + ϕ B ) ( ϕ 1 + ϕ 2 + ϕ b ) = 0 ( ϕ 0 + ϕ B ) = ( ϕ 1 + ϕ 2 + ϕ b )
Φ ^ R 0 = 2 ϕ a + 2 ϕ c + 2 ϕ 2 + ϕ R ϕ LO Φ ^ R 1 = 2 ϕ b + 2 ϕ d + 2 ϕ 2 + ϕ R ϕ LO
Φ ^ error = ( 2 ϕ b + 2 ϕ d ) ( 2 ϕ a + 2 ϕ c ) = 2 ϕ Y 2 ϕ X
ϕ X = ϕ a + ϕ c ϕ Y = ϕ b + ϕ d
Φ ^ error , Y X = 2 ϕ Y 2 ϕ X = 0 ϕ Y = ϕ X
Φ ^ error , Z X = 2 ϕ Z 2 ϕ X = 0 ϕ Z = ϕ Y = ϕ X
P th 21 b A e g B L e ( 1 + Δ ν L Δ ν B )
A ( τ ) = { 2 N 1 for τ = 0 , L , 2 L 1 for any other τ

Metrics