Abstract

In this work we applied a Hamiltonian formalism to simplify the equations of non-degenerate nonlinear four-wave mixing to the one-degree-of-freedom Hamiltonian equations with a three-parameter Hamiltonian. Thereby, a problem of signal amplification in a phase-sensitive double-pumped parametric fiber amplifier with pump depletion was reduced to a geometrical study of the phase portraits of the one-degree-of-freedom Hamiltonian system. For a symmetric case of equal pump powers and equal signal and idler powers at the fiber input, it has been shown that the theoretical maximum gain occurs on the extremal trajectories. However, to reduce the nonlinear interaction of waves, we proposed to choose the separatrix as the optimal trajectory on the phase plane. Analytical expressions were found for the maximum amplification, as well as the length of optical fiber and the relative phase of interacting waves allowing this amplification. Using the proposed approach, we optimized of the phase-sensitive parametric amplifier. As a result, the optimal parameters of the phase-sensitive amplifier were found and the maximum possible signal amplification was realized in a broad range of signal wavelengths.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

Full Article  |  PDF Article
OSA Recommended Articles
Simple geometric interpretation of signal evolution in phase-sensitive fibre optic parametric amplifier

A. A. Redyuk, A. E. Bednyakova, S. B. Medvedev, M. P. Fedoruk, and S. K. Turitsyn
Opt. Express 25(1) 223-231 (2017)

Gain characteristics of a frequency nondegenerate phase-sensitive fiber-optic parametric amplifier with phase self-stabilized input

Renyong Tang, Jacob Lasri, Preetpaul S. Devgan, Vladimir Grigoryan, Prem Kumar, and Michael Vasilyev
Opt. Express 13(26) 10483-10493 (2005)

Optimized design of six-wave fiber optical parametric amplifiers by using a genetic algorithm

Peipei Li, Hongna Zhu, Stefano Taccheo, Xiaorong Gao, and Zeyong Wang
Appl. Opt. 56(15) 4406-4411 (2017)

References

  • View by:
  • |
  • |
  • |

  1. J. Hansryd, P. A. Andrekson, M. Westlund, J. Li, and P. O. Hedekvist, “Fiber-based optical parametric amplifiers and their applications,” IEEE J. Sel. Top. Quantum Electron. 8(3), 506–520 (2002).
    [Crossref]
  2. M. E. Marhic, Fiber optical parametric amplifiers, oscillators and related devices (Cambridge University, 2008).
  3. C. McKinstrie and S. Radic, “Phase-sensitive amplification in a fiber,” Opt. Express 12(20), 4973–4979 (2004).
    [Crossref] [PubMed]
  4. C. McKinstrie, M. Raymer, S. Radic, and M. Vasilyev, “Quantum mechanics of phase-sensitive amplification in a fiber,” Opt. Commun. 257(1), 146–163 (2006).
    [Crossref]
  5. Z. Tong, C. Lundström, P. Andrekson, C. McKinstrie, M. Karlsson, D. Blessing, E. Tipsuwannakul, B. Puttnam, H. Toda, and L. Grüner-Nielsen, “Towards ultrasensitive optical links enabled by low-noise phase-sensitive amplifiers,” Nat. Photonics 5(7), 430–436 (2011).
    [Crossref]
  6. C. J. McKinstrie and G. G. Luther, “Solitary-wave solutions of the generalised three-wave and four-wave equations,” Phys. Lett. A 127(1), 14–18 (1988).
    [Crossref]
  7. G. Cappellini and S. Trillo, “Third-order three-wave mixing in single-mode fibers: exact solutions and spatial instability effects,” J. Opt. Soc. Am. B 8(4), 824–838 (1991).
    [Crossref]
  8. S. Trillo and S. Wabnitz, “Dynamics of the nonlinear modulational instability in optical fibers,” Opt. Lett. 16(13), 986–988 (1991).
    [Crossref] [PubMed]
  9. C. De Angelis, M. Santagiustina, and S. Trillo, “Four-photon homoclinic instabilities in nonlinear highly birefringent media,” Phys. Rev. A 51(1), 774–791 (1995).
    [Crossref] [PubMed]
  10. A. Bendahmane, A. Mussot, A. Kudlinski, P. Szriftgiser, M. Conforti, S. Wabnitz, and S. Trillo, “Optimal frequency conversion in the nonlinear stage of modulation instability,” Opt. Express 23(24), 30861–30871 (2015).
    [Crossref] [PubMed]
  11. C. J. McKinstrie, “Stokes-space formalism for Bragg scattering in a fiber,” Opt. Commun. 282(8), 1557–1562 (2009).
    [Crossref]
  12. H. Steffensen, J. R. Ott, K. Rottwitt, and C. J. McKinstrie, “Full and semi-analytic analyses of two-pump parametric amplification with pump depletion,” Opt. Express 19(7), 6648–6656 (2011).
    [Crossref] [PubMed]
  13. J. R. Ott, H. Steffensen, K. Rottwitt, and C. J. McKinstrie, “Geometric interpretation of four-wave mixing,” Phys. Rev. A 88(4), 043805 (2013).
    [Crossref]
  14. A. A. Redyuk, A. E. Bednyakova, S. B. Medvedev, M. P. Fedoruk, and S. K. Turitsyn, “Simple geometric interpretation of signal evolution in phase-sensitive fibre optic parametric amplifier,” Opt. Express 25(1), 223–231 (2017).
    [Crossref] [PubMed]
  15. J. M. Chavez Boggio, J. D. Marconi, S. R. Bickham, and H. L. Fragnito, “Spectrally flat and broadband double-pumped fiber optical parametric amplifiers,” Opt. Express 15(9), 5288–5309 (2007).
    [Crossref] [PubMed]
  16. P. F. Byrd and M. D. Friedman, Handbook of Elliptic Integrals for Engineers and Scientists (Springer, 1971).
    [Crossref]
  17. H. B. Dwight, Tables of Integrals and Other Mathematical Data (The Macmillan Company, 1961).
  18. M. E. Marhic, A. A. Rieznik, and H. H. Fragnito, “Investigation of the gain spectrum near the pumps of two-pump fiber-optic parametric amplifiers,” J. Opt. Soc. Am. B 25(1), 22–30 (2008).
    [Crossref]

2017 (1)

2015 (1)

2013 (1)

J. R. Ott, H. Steffensen, K. Rottwitt, and C. J. McKinstrie, “Geometric interpretation of four-wave mixing,” Phys. Rev. A 88(4), 043805 (2013).
[Crossref]

2011 (2)

H. Steffensen, J. R. Ott, K. Rottwitt, and C. J. McKinstrie, “Full and semi-analytic analyses of two-pump parametric amplification with pump depletion,” Opt. Express 19(7), 6648–6656 (2011).
[Crossref] [PubMed]

Z. Tong, C. Lundström, P. Andrekson, C. McKinstrie, M. Karlsson, D. Blessing, E. Tipsuwannakul, B. Puttnam, H. Toda, and L. Grüner-Nielsen, “Towards ultrasensitive optical links enabled by low-noise phase-sensitive amplifiers,” Nat. Photonics 5(7), 430–436 (2011).
[Crossref]

2009 (1)

C. J. McKinstrie, “Stokes-space formalism for Bragg scattering in a fiber,” Opt. Commun. 282(8), 1557–1562 (2009).
[Crossref]

2008 (1)

2007 (1)

2006 (1)

C. McKinstrie, M. Raymer, S. Radic, and M. Vasilyev, “Quantum mechanics of phase-sensitive amplification in a fiber,” Opt. Commun. 257(1), 146–163 (2006).
[Crossref]

2004 (1)

2002 (1)

J. Hansryd, P. A. Andrekson, M. Westlund, J. Li, and P. O. Hedekvist, “Fiber-based optical parametric amplifiers and their applications,” IEEE J. Sel. Top. Quantum Electron. 8(3), 506–520 (2002).
[Crossref]

1995 (1)

C. De Angelis, M. Santagiustina, and S. Trillo, “Four-photon homoclinic instabilities in nonlinear highly birefringent media,” Phys. Rev. A 51(1), 774–791 (1995).
[Crossref] [PubMed]

1991 (2)

1988 (1)

C. J. McKinstrie and G. G. Luther, “Solitary-wave solutions of the generalised three-wave and four-wave equations,” Phys. Lett. A 127(1), 14–18 (1988).
[Crossref]

Andrekson, P.

Z. Tong, C. Lundström, P. Andrekson, C. McKinstrie, M. Karlsson, D. Blessing, E. Tipsuwannakul, B. Puttnam, H. Toda, and L. Grüner-Nielsen, “Towards ultrasensitive optical links enabled by low-noise phase-sensitive amplifiers,” Nat. Photonics 5(7), 430–436 (2011).
[Crossref]

Andrekson, P. A.

J. Hansryd, P. A. Andrekson, M. Westlund, J. Li, and P. O. Hedekvist, “Fiber-based optical parametric amplifiers and their applications,” IEEE J. Sel. Top. Quantum Electron. 8(3), 506–520 (2002).
[Crossref]

Angelis, C. De

C. De Angelis, M. Santagiustina, and S. Trillo, “Four-photon homoclinic instabilities in nonlinear highly birefringent media,” Phys. Rev. A 51(1), 774–791 (1995).
[Crossref] [PubMed]

Bednyakova, A. E.

Bendahmane, A.

Bickham, S. R.

Blessing, D.

Z. Tong, C. Lundström, P. Andrekson, C. McKinstrie, M. Karlsson, D. Blessing, E. Tipsuwannakul, B. Puttnam, H. Toda, and L. Grüner-Nielsen, “Towards ultrasensitive optical links enabled by low-noise phase-sensitive amplifiers,” Nat. Photonics 5(7), 430–436 (2011).
[Crossref]

Byrd, P. F.

P. F. Byrd and M. D. Friedman, Handbook of Elliptic Integrals for Engineers and Scientists (Springer, 1971).
[Crossref]

Cappellini, G.

Chavez Boggio, J. M.

Conforti, M.

Dwight, H. B.

H. B. Dwight, Tables of Integrals and Other Mathematical Data (The Macmillan Company, 1961).

Fedoruk, M. P.

Fragnito, H. H.

Fragnito, H. L.

Friedman, M. D.

P. F. Byrd and M. D. Friedman, Handbook of Elliptic Integrals for Engineers and Scientists (Springer, 1971).
[Crossref]

Grüner-Nielsen, L.

Z. Tong, C. Lundström, P. Andrekson, C. McKinstrie, M. Karlsson, D. Blessing, E. Tipsuwannakul, B. Puttnam, H. Toda, and L. Grüner-Nielsen, “Towards ultrasensitive optical links enabled by low-noise phase-sensitive amplifiers,” Nat. Photonics 5(7), 430–436 (2011).
[Crossref]

Hansryd, J.

J. Hansryd, P. A. Andrekson, M. Westlund, J. Li, and P. O. Hedekvist, “Fiber-based optical parametric amplifiers and their applications,” IEEE J. Sel. Top. Quantum Electron. 8(3), 506–520 (2002).
[Crossref]

Hedekvist, P. O.

J. Hansryd, P. A. Andrekson, M. Westlund, J. Li, and P. O. Hedekvist, “Fiber-based optical parametric amplifiers and their applications,” IEEE J. Sel. Top. Quantum Electron. 8(3), 506–520 (2002).
[Crossref]

Karlsson, M.

Z. Tong, C. Lundström, P. Andrekson, C. McKinstrie, M. Karlsson, D. Blessing, E. Tipsuwannakul, B. Puttnam, H. Toda, and L. Grüner-Nielsen, “Towards ultrasensitive optical links enabled by low-noise phase-sensitive amplifiers,” Nat. Photonics 5(7), 430–436 (2011).
[Crossref]

Kudlinski, A.

Li, J.

J. Hansryd, P. A. Andrekson, M. Westlund, J. Li, and P. O. Hedekvist, “Fiber-based optical parametric amplifiers and their applications,” IEEE J. Sel. Top. Quantum Electron. 8(3), 506–520 (2002).
[Crossref]

Lundström, C.

Z. Tong, C. Lundström, P. Andrekson, C. McKinstrie, M. Karlsson, D. Blessing, E. Tipsuwannakul, B. Puttnam, H. Toda, and L. Grüner-Nielsen, “Towards ultrasensitive optical links enabled by low-noise phase-sensitive amplifiers,” Nat. Photonics 5(7), 430–436 (2011).
[Crossref]

Luther, G. G.

C. J. McKinstrie and G. G. Luther, “Solitary-wave solutions of the generalised three-wave and four-wave equations,” Phys. Lett. A 127(1), 14–18 (1988).
[Crossref]

Marconi, J. D.

Marhic, M. E.

McKinstrie, C.

Z. Tong, C. Lundström, P. Andrekson, C. McKinstrie, M. Karlsson, D. Blessing, E. Tipsuwannakul, B. Puttnam, H. Toda, and L. Grüner-Nielsen, “Towards ultrasensitive optical links enabled by low-noise phase-sensitive amplifiers,” Nat. Photonics 5(7), 430–436 (2011).
[Crossref]

C. McKinstrie, M. Raymer, S. Radic, and M. Vasilyev, “Quantum mechanics of phase-sensitive amplification in a fiber,” Opt. Commun. 257(1), 146–163 (2006).
[Crossref]

C. McKinstrie and S. Radic, “Phase-sensitive amplification in a fiber,” Opt. Express 12(20), 4973–4979 (2004).
[Crossref] [PubMed]

McKinstrie, C. J.

J. R. Ott, H. Steffensen, K. Rottwitt, and C. J. McKinstrie, “Geometric interpretation of four-wave mixing,” Phys. Rev. A 88(4), 043805 (2013).
[Crossref]

H. Steffensen, J. R. Ott, K. Rottwitt, and C. J. McKinstrie, “Full and semi-analytic analyses of two-pump parametric amplification with pump depletion,” Opt. Express 19(7), 6648–6656 (2011).
[Crossref] [PubMed]

C. J. McKinstrie, “Stokes-space formalism for Bragg scattering in a fiber,” Opt. Commun. 282(8), 1557–1562 (2009).
[Crossref]

C. J. McKinstrie and G. G. Luther, “Solitary-wave solutions of the generalised three-wave and four-wave equations,” Phys. Lett. A 127(1), 14–18 (1988).
[Crossref]

Medvedev, S. B.

Mussot, A.

Ott, J. R.

Puttnam, B.

Z. Tong, C. Lundström, P. Andrekson, C. McKinstrie, M. Karlsson, D. Blessing, E. Tipsuwannakul, B. Puttnam, H. Toda, and L. Grüner-Nielsen, “Towards ultrasensitive optical links enabled by low-noise phase-sensitive amplifiers,” Nat. Photonics 5(7), 430–436 (2011).
[Crossref]

Radic, S.

C. McKinstrie, M. Raymer, S. Radic, and M. Vasilyev, “Quantum mechanics of phase-sensitive amplification in a fiber,” Opt. Commun. 257(1), 146–163 (2006).
[Crossref]

C. McKinstrie and S. Radic, “Phase-sensitive amplification in a fiber,” Opt. Express 12(20), 4973–4979 (2004).
[Crossref] [PubMed]

Raymer, M.

C. McKinstrie, M. Raymer, S. Radic, and M. Vasilyev, “Quantum mechanics of phase-sensitive amplification in a fiber,” Opt. Commun. 257(1), 146–163 (2006).
[Crossref]

Redyuk, A. A.

Rieznik, A. A.

Rottwitt, K.

Santagiustina, M.

C. De Angelis, M. Santagiustina, and S. Trillo, “Four-photon homoclinic instabilities in nonlinear highly birefringent media,” Phys. Rev. A 51(1), 774–791 (1995).
[Crossref] [PubMed]

Steffensen, H.

Szriftgiser, P.

Tipsuwannakul, E.

Z. Tong, C. Lundström, P. Andrekson, C. McKinstrie, M. Karlsson, D. Blessing, E. Tipsuwannakul, B. Puttnam, H. Toda, and L. Grüner-Nielsen, “Towards ultrasensitive optical links enabled by low-noise phase-sensitive amplifiers,” Nat. Photonics 5(7), 430–436 (2011).
[Crossref]

Toda, H.

Z. Tong, C. Lundström, P. Andrekson, C. McKinstrie, M. Karlsson, D. Blessing, E. Tipsuwannakul, B. Puttnam, H. Toda, and L. Grüner-Nielsen, “Towards ultrasensitive optical links enabled by low-noise phase-sensitive amplifiers,” Nat. Photonics 5(7), 430–436 (2011).
[Crossref]

Tong, Z.

Z. Tong, C. Lundström, P. Andrekson, C. McKinstrie, M. Karlsson, D. Blessing, E. Tipsuwannakul, B. Puttnam, H. Toda, and L. Grüner-Nielsen, “Towards ultrasensitive optical links enabled by low-noise phase-sensitive amplifiers,” Nat. Photonics 5(7), 430–436 (2011).
[Crossref]

Trillo, S.

Turitsyn, S. K.

Vasilyev, M.

C. McKinstrie, M. Raymer, S. Radic, and M. Vasilyev, “Quantum mechanics of phase-sensitive amplification in a fiber,” Opt. Commun. 257(1), 146–163 (2006).
[Crossref]

Wabnitz, S.

Westlund, M.

J. Hansryd, P. A. Andrekson, M. Westlund, J. Li, and P. O. Hedekvist, “Fiber-based optical parametric amplifiers and their applications,” IEEE J. Sel. Top. Quantum Electron. 8(3), 506–520 (2002).
[Crossref]

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

J. Hansryd, P. A. Andrekson, M. Westlund, J. Li, and P. O. Hedekvist, “Fiber-based optical parametric amplifiers and their applications,” IEEE J. Sel. Top. Quantum Electron. 8(3), 506–520 (2002).
[Crossref]

J. Opt. Soc. Am. B (2)

Nat. Photonics (1)

Z. Tong, C. Lundström, P. Andrekson, C. McKinstrie, M. Karlsson, D. Blessing, E. Tipsuwannakul, B. Puttnam, H. Toda, and L. Grüner-Nielsen, “Towards ultrasensitive optical links enabled by low-noise phase-sensitive amplifiers,” Nat. Photonics 5(7), 430–436 (2011).
[Crossref]

Opt. Commun. (2)

C. McKinstrie, M. Raymer, S. Radic, and M. Vasilyev, “Quantum mechanics of phase-sensitive amplification in a fiber,” Opt. Commun. 257(1), 146–163 (2006).
[Crossref]

C. J. McKinstrie, “Stokes-space formalism for Bragg scattering in a fiber,” Opt. Commun. 282(8), 1557–1562 (2009).
[Crossref]

Opt. Express (5)

Opt. Lett. (1)

Phys. Lett. A (1)

C. J. McKinstrie and G. G. Luther, “Solitary-wave solutions of the generalised three-wave and four-wave equations,” Phys. Lett. A 127(1), 14–18 (1988).
[Crossref]

Phys. Rev. A (2)

C. De Angelis, M. Santagiustina, and S. Trillo, “Four-photon homoclinic instabilities in nonlinear highly birefringent media,” Phys. Rev. A 51(1), 774–791 (1995).
[Crossref] [PubMed]

J. R. Ott, H. Steffensen, K. Rottwitt, and C. J. McKinstrie, “Geometric interpretation of four-wave mixing,” Phys. Rev. A 88(4), 043805 (2013).
[Crossref]

Other (3)

P. F. Byrd and M. D. Friedman, Handbook of Elliptic Integrals for Engineers and Scientists (Springer, 1971).
[Crossref]

H. B. Dwight, Tables of Integrals and Other Mathematical Data (The Macmillan Company, 1961).

M. E. Marhic, Fiber optical parametric amplifiers, oscillators and related devices (Cambridge University, 2008).

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 (5)

Fig. 1
Fig. 1 Phase portraits: k = −1.5 (a), k = 0 (b), k = 1.5 (c) and k = 4 (d).
Fig. 2
Fig. 2 (a) Dependence of parameter k on signal wavelength. (b)–(d) Phase portraits for k = −0.5 (b), −0.96 (c) and −2 (d). Separatrices are depicted by black dots. (b) Schematic depiction of the of the power transfer process in PS-FOPA with a fixed gain.
Fig. 3
Fig. 3 (a) Schematic representation of closed separatrices of two types for k = −0.5 and k = 0.5. (b) The maximum attainable power of the signal p = P3/PT for motion along separatrices on the parameter k.
Fig. 4
Fig. 4 Evolution of the pump power (solid red), signal (idler) power (solid green) and relative phase θ (dashed blue) during two full periods of closed trajectories, shown in Fig. 2.
Fig. 5
Fig. 5 Dependence of maximum signal gain Gmax, optimal fiber length and relative phase parameter on signal wavelength. P1 = P2 = 2.1 W, P3 = P4 = 0.01 mW, λ1 = 1495.9 nm and λ2 = 1611.9 nm. Black horizontal line indicates the gain (G = 53.2 dB) that would be obtained when depleting completely the pumps. Red circles depict signal amplification, found from the solution of the system of Eqs. (35)(36) at optimum values of the relative phase and the fiber length. Dashed line depicts signal amplification, found from simulations of the NLSE at the optimum values of the relative phase and the fiber length.

Tables (1)

Tables Icon

Table 1 Maximum powers for extremal trajectories

Equations (65)

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

d A 1 d z = i β 1 A 1 + i γ { [ | A 1 | 2 + 2 ( | A 2 | 2 + | A 3 | 2 + | A 4 | 2 ) ] A 1 + 2 A 2 * A 3 A 4 } ,
d A 2 d z = i β 2 A 2 + i γ { [ | A 2 | 2 + 2 ( | A 1 | 2 + | A 3 | 2 + | A 4 | 2 ) ] A 2 + 2 A 1 * A 3 A 4 } ,
d A 3 d z = i β 3 A 3 + i γ { [ | A 3 | 2 + 2 ( | A 1 | 2 + | A 2 | 2 + | A 4 | 2 ) ] A 3 + 2 A 1 A 2 A 4 * } ,
d A 4 d z = i β 4 A 4 + i γ { [ | A 4 | 2 + 2 ( | A 1 | 2 + | A 2 | 2 + | A 3 | 2 ) ] A 4 + 2 A 1 A 2 A 3 * } ,
d A l d z = i H A l * , d A l * d z = i H A l
H = l = 1 4 β l | A l | 2 + γ ( l = 1 4 | A l | 2 ) 2 γ 2 l = 1 4 | A l | 4 + 2 γ ( A 1 * A 2 * A 3 A 4 + A 1 A 2 A 3 * A 4 * ) .
d P l d z = H θ l = 4 γ P 1 P 2 P 3 P 4 sin θ d θ d θ l ,
d θ l d z = H P l = β l + 2 γ r = 1 4 P r γ P l + 2 γ P l 1 P 1 P 2 P 3 P 4 cos θ
H = l = r 4 β r P r + γ ( r = 1 4 P r ) 2 γ 2 r = 1 4 P r 2 + 4 γ P 1 P 2 P 3 P 4 cos θ ,
( p 3 = P 3 , θ ) , ( p 1 = P 1 + P 3 , θ 1 ) , ( p 2 = P 2 + P 3 , θ 2 ) , ( p 4 = P 4 P 3 , θ 4 ) .
P 1 = p 1 p 3 , P 2 = p 2 p 3 , P 3 = p 3 , P 4 = p 4 + p 3
H = β l p l + β 2 p 2 + Δ β p 3 + β 4 p 4 + γ ( p 1 + p 2 + p 4 ) 2 γ 2 [ ( p 1 p 3 ) 2 + ( p 2 p 3 ) 2 + p 3 2 + ( p 4 + p 3 ) 2 ] + 4 γ ( p 1 p 3 ) ( p 2 p 3 ) p 3 ( p 4 + p 3 ) cos θ ,
d p 3 d z = H θ , d θ d z = H p 3
H = Δ β p 3 γ 2 [ ( p 1 p 3 ) 2 + ( p 2 p 3 ) 2 + p 3 2 + ( p 3 p 4 ) 2 ] + 4 γ ( p 1 p 3 ) ( p 2 p 3 ) p 3 ( p 3 + p 4 ) cos θ ,
P l | z = 0 = P l 0 , θ l | z = 0 = θ l 0 .
P 1 = P 10 x , P 2 = P 20 x , P 3 = P 30 + x , P 4 = P 40 + x .
H ( x , θ ) = ( Δ β γ + Δ P 0 ) x 2 x 2 + 4 P 1 P 2 P 3 P 4 cos θ ,
d x d Z = H θ = 4 P 1 P 2 P 3 P 4 sin θ ,
d θ d Z = H x = Δ β γ + Δ P 0 4 x 2 P 1 P 2 P 3 P 4 ( P 1 1 + P 2 1 P 3 1 P 4 1 ) cos θ ,
P 10 x , P 20 x , x P 30 , x P 40 .
H ( 0 , θ 0 ) = 4 P 10 P 20 P 30 P 40 cos θ 0 .
H ( P 20 , θ 1 ) = ( Δ β γ + Δ P 0 ) P 20 2 P 20 2 ,
Δ β = γ ( P 30 + P 20 P 10 ) .
4 P 10 P 20 P 30 P 40 cos θ 0 = ( Δ β γ + Δ P 0 ) P 20 2 P 20 2 ,
G m a x = P 30 + P 20 P 30 = 1 + P 20 P 30 .
p = P 40 + x 2 R , R = P 20 + P 40 ,
H = ( k + 1 + δ 1 δ 2 ) p 2 p 2 + 4 Q 1 Q 2 Q 3 Q 4 cos θ , k = Δ β 2 γ R ,
Q 1 = P 10 + P 40 2 R p = 1 / 2 + δ 1 p , Q 2 = 1 / 2 p , Q 3 = P 30 P 40 2 R + p = δ 2 + p , Q 4 = p ,
d p d l = H θ = 4 Q 1 Q 2 Q 3 Q 4 sin θ ,
d θ d l = H p = ( k + 1 + δ 1 δ 2 ) 4 p 2 Q 1 Q 2 Q 3 Q 4 ( Q 1 1 + Q 2 1 Q 3 1 Q 4 1 ) cos θ ,
θ A = 2 π n , θ B = 2 π n + π , n Z , p C = 0 , p D = 1 / 2.
f ± ( p ) = ( k + 1 + δ 1 δ 2 ) 4 p 2 Q 1 Q 2 Q 3 Q 4 ( Q 1 1 + Q 2 1 Q 3 1 Q 4 1 )
f ± ( p ) = ± ( 1 + 2 δ 1 ) δ 2 p 1 / 2 , p + 0 ,
f ± ( p ) = ( 1 + 2 δ 2 ) δ 1 q 1 / 2 , ( 1 / 2 p ) = p + 0.
g ( H 0 , p ) = ( k + 1 + δ 1 δ 2 ) 4 p 1 2 ( 2 p 2 ( k + 1 + δ 1 δ 2 ) p + H 0 ) ( Q 1 1 + Q 2 1 Q 3 1 Q 4 1 ) ,
H = p 1 / 2 [ ( k + 1 + δ 1 δ 2 ) p 1 / 2 2 p 3 / 2 + 4 ( q + δ 1 ) q ( p + δ 2 ) cos θ ] = 0
H 1 2 ( k + δ 1 δ 2 ) = q 1 / 2 [ ( k + 1 δ 1 + δ 2 ) q 1 / 2 2 q 3 / 2 + 4 ( q + δ 1 ) ( p + δ 2 ) p cos θ ] .
H ( p , θ ) = ( k + 1 + 2 cos θ ) p 2 ( 1 + 2 cos θ ) p 2 ,
d p d l = H θ = 4 ( 1 2 p ) p sin θ ,
d θ d l = H p = ( k + 1 + 2 cos θ ) 4 ( 1 + 2 cos θ ) p .
θ A = 2 π n , θ B = 2 π n + π , p C = 0 , p D = 1 2 , n Z .
p A = k + 3 12 , p B = 1 k 4 , cos θ C = k + 1 2 , cos θ D = k 1 2 .
A = ( k + 3 12 , 2 π n ) , B = ( 1 k 4 , 2 π n + π ) , C = ( 0 , θ C + 2 π n ) , D = ( 1 2 , θ D + 2 π n ) .
d θ d p = k + 1 + 2 cos θ 4 ( 1 2 p ) p sin θ ( 1 + 2 cos θ ) ( 1 2 p ) sin θ .
k + 1 + 2 cos θ = 0.
H = p [ 4 ( 1 2 p ) ( cos θ + 1 2 ) + k ] , H = ( 1 2 p ) [ 4 p ( cos θ + 1 2 ) k ] + k 2 .
H = 4 p ( 1 2 p ) ( cos θ + 1 2 )
p = 0 , p = 1 2 + k 4 cos θ + 2 = f ( θ ) , p = 1 2 , p = k 4 cos θ + 2 = f ( θ ) 1 2 .
G m a x = P 40 + P 20 P 40 = R P 40 = 1 2 p 0 .
G m a x = p ( k ) p 0 = 3 + k 6 p 0 ,
G m a x = 1 / ( 2 p 0 ) ,
( d p d l ) 2 = 16 Q 1 Q 2 Q 3 Q 4 sin 2 θ = 16 Q 1 Q 2 Q 3 Q 4 [ H ( k + 1 + δ 1 δ 2 ) p + 2 p 2 ] 2 .
( d p d l ) 2 = 16 p 2 ( 1 2 p ) 2 [ H ( k + 1 ) p + 2 p 2 ] 2 = ( H 0 H ( p , 0 ) ) ( H ( p , π ) H 0 ) ,
p A ± = p A ± 1 6 ( H A H 0 ) = p A ( 1 ± 1 H 0 6 p A 2 ) ,
p B ± = p B ± 1 2 ( H 0 H B ) = p B ( 1 ± 1 + H 0 2 p B 2 ) ,
H ( p , θ ) = H ( p , 0 ) + 4 p ( 1 2 p ) ( cos θ 1 ) = H ( p , π ) + 4 p ( 1 2 p ) ( cos θ + 1 ) .
p B + > p A + p 0 p A > p B ,
12 l ( p ) = p A p d p ( p B + p ) ( p A + p ) ( p p A ) ( p p B ) = g F ( φ , m ) ,
g = 2 ( p B + p A ) ( p A + p B ) , sin 2 φ = ( p A + p B ) ( p p A ) ( p A + p A ) ( p p B ) , m 2 = ( p B + p B ) ( p A + p A ) ( p B + p A ) ( p A + p B ) .
12 l ( p ) = p 2 p A d p p ( 2 p B p ) ( 2 p A p ) = L ( 2 p A ) L ( p ) ,
L ( p ) = 1 4 p A p B ln ( 2 4 p A p B ( 2 p A p ) ( 2 p B p ) + 4 p A p B p 2 ( p A + p B ) ) .
p = p 0 , 2 p A p 0 ,
p B ± = 1 k 4 ± [ ( 1 k 4 ) 2 + 1 2 p 0 ( k + 3 6 p 0 ) ] 1 / 2 .
p A ± = k + 3 12 ± [ ( k + 3 12 ) 2 1 6 p 0 ( k 1 + 2 p 0 ) ] 1 / 2 .
p = p 0 , 2 p B p 0 .

Metrics