Abstract

Spatial solitons in quadratically nonlinear media result from the interplay of parametric gain, diffraction and cascading phase shift. Their main features are well understood in mathematical terms, and several experiments have been successfully carried out which demonstrate their observability and most important properties. Here we provide an intuitive interpretation of some of the underlying physics, outlining the processes that govern their excitation, propagation and interaction forces.

© 2002 Optical Society of America

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References

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  1. A. D. Boardman and A. P. Sukhorukov, Soliton Driven Photonics (Kluwer Acad. Publ., Dordrecht, 2001).
    [CrossRef]
  2. S. Trillo and W. E. Torruellas, Spatial Solitons (Springer-Verlag, Berlin, 2001).
  3. M. Segev and G. Stegeman, “Self-Trapping of Optical Beams: Spatial Solitons,” Phys. Today 51, 43–48 (1998).
    [CrossRef]
  4. M. Peccianti, A. De Rossi, G. Assanto, A. De Luca, C. Umeton, and I. C. Khoo, “Electrically Assisted Self-Confinement and Waveguiding in planar Nematic Liquid Crystal cells”, Appl. Phys. Lett. 77, 7–9 (2000).
    [CrossRef]
  5. A. W. Snyder, D. J. Mitchell, and Y. S. Kivshar, “Unification of Linear and Nonlinear Wave Optics,” Mod. Phys. Lett. B 9, 1479–1506 (1995).
    [CrossRef]
  6. G. I. Stegeman and M. Segev, “Optical Solitons and Their Interactions: Universality and Diversity,” Sci. 286, 1518–1523 (1999).
    [CrossRef]
  7. A. D. Boardman and K. Xie, “Theory of spatial solitons,” Radio Science 28, 891–899 (1993).
    [CrossRef]
  8. Y. N. Karamzin and A. P. Sukhorukov, “Mutual focusing of high-power light beams in media with quadratic nonlinearity”, Sov. Phys.-JETP,  41, 414–416 (1976).
  9. K. Hayata and M. Koshiba, “Multidimensional solitons in quadratic nonlinear media,” Phys. Rev. Lett. 71, 3275–3278 (1993).
    [CrossRef] [PubMed]
  10. A. V. Buryak and Y. S. Kivshar, “Spatial optical solitons governed by quadratic nonlinearity,” Opt. Lett. 19, 1612–1614 (1994).
    [CrossRef] [PubMed]
  11. C.R. Menyuk, R. Schiek, and L. Torner, “Solitary waves due to χ(2) :χ(2) cascading,” J. Opt. Soc. Am. B 11, 2434–2443 (1994).
    [CrossRef]
  12. A. V. Buryak, Y. S. Kivshar, and V. V. Steblina, “Self-trapping of light beams and parametric solitons in diffractive quadratic media,” Phys. Rev. A 52, 1670–1674 (1995).
    [CrossRef] [PubMed]
  13. G. I. Stegeman, M. Sheik-Bahae, E. VanStryland, and G. Assanto, “Large nonlinear phase shifts in second-order nonlinear optical processes,” Opt. Lett. 18, 13–15 (1993).
    [CrossRef] [PubMed]
  14. G. Leo, G. Assanto, and W. E. Torruellas, “Bidimensional spatial solitary waves in quadratically nonlinear bulk media,” J. Opt. Soc. Am. B 14, 3134–3142 (1997).
    [CrossRef]
  15. G. Assanto, “Diffraction with Second-Harmonic Generation for the formation of self-guided or ‘solitary’ waves,” in Diffractive optics and Optical Microsystems, A. N. Chester and S. Martellucci eds., 65–74 (Plenum Press, New York, 1997).
  16. M. J. Werner and P. D. Drummond, “Simulton solutions for the parametric amplifier,” J. Opt. Soc. Am. B 10, 2390–2393 (1993).
    [CrossRef]
  17. R. A. Fuerst, M. T. G. Canva, G. I. Stegeman, G. Leo, and G. Assanto, “Robust generation, properties, and potential applications of quadratic spatial solitons generated by optical parametric amplification,” Opt. & Quantum Electron. 30, 907–921 (1998).
    [CrossRef]
  18. A. V. Buryak, Y. S. Kivshar, and S. Trillo, “Stability of tree-wave parametric solitons in diffractive quadratic media,” Phys. Rev. Lett. 77, 5210–5213 (1996).
    [CrossRef] [PubMed]
  19. D. M. Baboiu and G. I. Stegeman, “Solitary-wave interactions in quadratic media near type I phase-matching conditions,” J. Opt. Soc. Am. 14, 3143–3150 (1997).
    [CrossRef]
  20. A. V. Buryak and V. V. Steblina, “Soliton collisions in bulk quadratic media: comprehensive analytical and numerical study,” J. Opt. Soc. Am. B 16, 245–255 (1999).
    [CrossRef]
  21. B. Costantini, C. De Angelis, A. Barthelemy, A. Laureti Palma, and G. Assanto, “Polarization multiplexed χ(2) solitary waves interactions,” Opt. Lett. 22, 1376–1378 (1997).
    [CrossRef]
  22. S. K. Johansen, O. Bang, and M. P. Soerensen, “Escape velocities in bulk χ(2) soliton interactions,” Phys. Rev. E 65, 026601–026604(2002).
    [CrossRef]
  23. V. V. Steblina, Y. S. Kivshar, and A. V. Buryak, “Scattering and spiraling of solitons in a bulk quadratic medium,” Opt. Lett. 23, 156–158 (1998).
    [CrossRef]

2002 (1)

S. K. Johansen, O. Bang, and M. P. Soerensen, “Escape velocities in bulk χ(2) soliton interactions,” Phys. Rev. E 65, 026601–026604(2002).
[CrossRef]

2000 (1)

M. Peccianti, A. De Rossi, G. Assanto, A. De Luca, C. Umeton, and I. C. Khoo, “Electrically Assisted Self-Confinement and Waveguiding in planar Nematic Liquid Crystal cells”, Appl. Phys. Lett. 77, 7–9 (2000).
[CrossRef]

1999 (2)

G. I. Stegeman and M. Segev, “Optical Solitons and Their Interactions: Universality and Diversity,” Sci. 286, 1518–1523 (1999).
[CrossRef]

A. V. Buryak and V. V. Steblina, “Soliton collisions in bulk quadratic media: comprehensive analytical and numerical study,” J. Opt. Soc. Am. B 16, 245–255 (1999).
[CrossRef]

1998 (3)

R. A. Fuerst, M. T. G. Canva, G. I. Stegeman, G. Leo, and G. Assanto, “Robust generation, properties, and potential applications of quadratic spatial solitons generated by optical parametric amplification,” Opt. & Quantum Electron. 30, 907–921 (1998).
[CrossRef]

M. Segev and G. Stegeman, “Self-Trapping of Optical Beams: Spatial Solitons,” Phys. Today 51, 43–48 (1998).
[CrossRef]

V. V. Steblina, Y. S. Kivshar, and A. V. Buryak, “Scattering and spiraling of solitons in a bulk quadratic medium,” Opt. Lett. 23, 156–158 (1998).
[CrossRef]

1997 (3)

1996 (1)

A. V. Buryak, Y. S. Kivshar, and S. Trillo, “Stability of tree-wave parametric solitons in diffractive quadratic media,” Phys. Rev. Lett. 77, 5210–5213 (1996).
[CrossRef] [PubMed]

1995 (2)

A. V. Buryak, Y. S. Kivshar, and V. V. Steblina, “Self-trapping of light beams and parametric solitons in diffractive quadratic media,” Phys. Rev. A 52, 1670–1674 (1995).
[CrossRef] [PubMed]

A. W. Snyder, D. J. Mitchell, and Y. S. Kivshar, “Unification of Linear and Nonlinear Wave Optics,” Mod. Phys. Lett. B 9, 1479–1506 (1995).
[CrossRef]

1994 (2)

1993 (4)

1976 (1)

Y. N. Karamzin and A. P. Sukhorukov, “Mutual focusing of high-power light beams in media with quadratic nonlinearity”, Sov. Phys.-JETP,  41, 414–416 (1976).

Assanto, G.

M. Peccianti, A. De Rossi, G. Assanto, A. De Luca, C. Umeton, and I. C. Khoo, “Electrically Assisted Self-Confinement and Waveguiding in planar Nematic Liquid Crystal cells”, Appl. Phys. Lett. 77, 7–9 (2000).
[CrossRef]

R. A. Fuerst, M. T. G. Canva, G. I. Stegeman, G. Leo, and G. Assanto, “Robust generation, properties, and potential applications of quadratic spatial solitons generated by optical parametric amplification,” Opt. & Quantum Electron. 30, 907–921 (1998).
[CrossRef]

G. Leo, G. Assanto, and W. E. Torruellas, “Bidimensional spatial solitary waves in quadratically nonlinear bulk media,” J. Opt. Soc. Am. B 14, 3134–3142 (1997).
[CrossRef]

B. Costantini, C. De Angelis, A. Barthelemy, A. Laureti Palma, and G. Assanto, “Polarization multiplexed χ(2) solitary waves interactions,” Opt. Lett. 22, 1376–1378 (1997).
[CrossRef]

G. I. Stegeman, M. Sheik-Bahae, E. VanStryland, and G. Assanto, “Large nonlinear phase shifts in second-order nonlinear optical processes,” Opt. Lett. 18, 13–15 (1993).
[CrossRef] [PubMed]

G. Assanto, “Diffraction with Second-Harmonic Generation for the formation of self-guided or ‘solitary’ waves,” in Diffractive optics and Optical Microsystems, A. N. Chester and S. Martellucci eds., 65–74 (Plenum Press, New York, 1997).

Baboiu, D. M.

D. M. Baboiu and G. I. Stegeman, “Solitary-wave interactions in quadratic media near type I phase-matching conditions,” J. Opt. Soc. Am. 14, 3143–3150 (1997).
[CrossRef]

Bang, O.

S. K. Johansen, O. Bang, and M. P. Soerensen, “Escape velocities in bulk χ(2) soliton interactions,” Phys. Rev. E 65, 026601–026604(2002).
[CrossRef]

Barthelemy, A.

Boardman, A. D.

A. D. Boardman and K. Xie, “Theory of spatial solitons,” Radio Science 28, 891–899 (1993).
[CrossRef]

A. D. Boardman and A. P. Sukhorukov, Soliton Driven Photonics (Kluwer Acad. Publ., Dordrecht, 2001).
[CrossRef]

Buryak, A. V.

Canva, M. T. G.

R. A. Fuerst, M. T. G. Canva, G. I. Stegeman, G. Leo, and G. Assanto, “Robust generation, properties, and potential applications of quadratic spatial solitons generated by optical parametric amplification,” Opt. & Quantum Electron. 30, 907–921 (1998).
[CrossRef]

Costantini, B.

De Angelis, C.

De Luca, A.

M. Peccianti, A. De Rossi, G. Assanto, A. De Luca, C. Umeton, and I. C. Khoo, “Electrically Assisted Self-Confinement and Waveguiding in planar Nematic Liquid Crystal cells”, Appl. Phys. Lett. 77, 7–9 (2000).
[CrossRef]

De Rossi, A.

M. Peccianti, A. De Rossi, G. Assanto, A. De Luca, C. Umeton, and I. C. Khoo, “Electrically Assisted Self-Confinement and Waveguiding in planar Nematic Liquid Crystal cells”, Appl. Phys. Lett. 77, 7–9 (2000).
[CrossRef]

Drummond, P. D.

Fuerst, R. A.

R. A. Fuerst, M. T. G. Canva, G. I. Stegeman, G. Leo, and G. Assanto, “Robust generation, properties, and potential applications of quadratic spatial solitons generated by optical parametric amplification,” Opt. & Quantum Electron. 30, 907–921 (1998).
[CrossRef]

Hayata, K.

K. Hayata and M. Koshiba, “Multidimensional solitons in quadratic nonlinear media,” Phys. Rev. Lett. 71, 3275–3278 (1993).
[CrossRef] [PubMed]

Johansen, S. K.

S. K. Johansen, O. Bang, and M. P. Soerensen, “Escape velocities in bulk χ(2) soliton interactions,” Phys. Rev. E 65, 026601–026604(2002).
[CrossRef]

Karamzin, Y. N.

Y. N. Karamzin and A. P. Sukhorukov, “Mutual focusing of high-power light beams in media with quadratic nonlinearity”, Sov. Phys.-JETP,  41, 414–416 (1976).

Khoo, I. C.

M. Peccianti, A. De Rossi, G. Assanto, A. De Luca, C. Umeton, and I. C. Khoo, “Electrically Assisted Self-Confinement and Waveguiding in planar Nematic Liquid Crystal cells”, Appl. Phys. Lett. 77, 7–9 (2000).
[CrossRef]

Kivshar, Y. S.

V. V. Steblina, Y. S. Kivshar, and A. V. Buryak, “Scattering and spiraling of solitons in a bulk quadratic medium,” Opt. Lett. 23, 156–158 (1998).
[CrossRef]

A. V. Buryak, Y. S. Kivshar, and S. Trillo, “Stability of tree-wave parametric solitons in diffractive quadratic media,” Phys. Rev. Lett. 77, 5210–5213 (1996).
[CrossRef] [PubMed]

A. V. Buryak, Y. S. Kivshar, and V. V. Steblina, “Self-trapping of light beams and parametric solitons in diffractive quadratic media,” Phys. Rev. A 52, 1670–1674 (1995).
[CrossRef] [PubMed]

A. W. Snyder, D. J. Mitchell, and Y. S. Kivshar, “Unification of Linear and Nonlinear Wave Optics,” Mod. Phys. Lett. B 9, 1479–1506 (1995).
[CrossRef]

A. V. Buryak and Y. S. Kivshar, “Spatial optical solitons governed by quadratic nonlinearity,” Opt. Lett. 19, 1612–1614 (1994).
[CrossRef] [PubMed]

Koshiba, M.

K. Hayata and M. Koshiba, “Multidimensional solitons in quadratic nonlinear media,” Phys. Rev. Lett. 71, 3275–3278 (1993).
[CrossRef] [PubMed]

Laureti Palma, A.

Leo, G.

R. A. Fuerst, M. T. G. Canva, G. I. Stegeman, G. Leo, and G. Assanto, “Robust generation, properties, and potential applications of quadratic spatial solitons generated by optical parametric amplification,” Opt. & Quantum Electron. 30, 907–921 (1998).
[CrossRef]

G. Leo, G. Assanto, and W. E. Torruellas, “Bidimensional spatial solitary waves in quadratically nonlinear bulk media,” J. Opt. Soc. Am. B 14, 3134–3142 (1997).
[CrossRef]

Menyuk, C.R.

Mitchell, D. J.

A. W. Snyder, D. J. Mitchell, and Y. S. Kivshar, “Unification of Linear and Nonlinear Wave Optics,” Mod. Phys. Lett. B 9, 1479–1506 (1995).
[CrossRef]

Peccianti, M.

M. Peccianti, A. De Rossi, G. Assanto, A. De Luca, C. Umeton, and I. C. Khoo, “Electrically Assisted Self-Confinement and Waveguiding in planar Nematic Liquid Crystal cells”, Appl. Phys. Lett. 77, 7–9 (2000).
[CrossRef]

Schiek, R.

Segev, M.

G. I. Stegeman and M. Segev, “Optical Solitons and Their Interactions: Universality and Diversity,” Sci. 286, 1518–1523 (1999).
[CrossRef]

M. Segev and G. Stegeman, “Self-Trapping of Optical Beams: Spatial Solitons,” Phys. Today 51, 43–48 (1998).
[CrossRef]

Sheik-Bahae, M.

Snyder, A. W.

A. W. Snyder, D. J. Mitchell, and Y. S. Kivshar, “Unification of Linear and Nonlinear Wave Optics,” Mod. Phys. Lett. B 9, 1479–1506 (1995).
[CrossRef]

Soerensen, M. P.

S. K. Johansen, O. Bang, and M. P. Soerensen, “Escape velocities in bulk χ(2) soliton interactions,” Phys. Rev. E 65, 026601–026604(2002).
[CrossRef]

Steblina, V. V.

Stegeman, G.

M. Segev and G. Stegeman, “Self-Trapping of Optical Beams: Spatial Solitons,” Phys. Today 51, 43–48 (1998).
[CrossRef]

Stegeman, G. I.

G. I. Stegeman and M. Segev, “Optical Solitons and Their Interactions: Universality and Diversity,” Sci. 286, 1518–1523 (1999).
[CrossRef]

R. A. Fuerst, M. T. G. Canva, G. I. Stegeman, G. Leo, and G. Assanto, “Robust generation, properties, and potential applications of quadratic spatial solitons generated by optical parametric amplification,” Opt. & Quantum Electron. 30, 907–921 (1998).
[CrossRef]

D. M. Baboiu and G. I. Stegeman, “Solitary-wave interactions in quadratic media near type I phase-matching conditions,” J. Opt. Soc. Am. 14, 3143–3150 (1997).
[CrossRef]

G. I. Stegeman, M. Sheik-Bahae, E. VanStryland, and G. Assanto, “Large nonlinear phase shifts in second-order nonlinear optical processes,” Opt. Lett. 18, 13–15 (1993).
[CrossRef] [PubMed]

Sukhorukov, A. P.

Y. N. Karamzin and A. P. Sukhorukov, “Mutual focusing of high-power light beams in media with quadratic nonlinearity”, Sov. Phys.-JETP,  41, 414–416 (1976).

A. D. Boardman and A. P. Sukhorukov, Soliton Driven Photonics (Kluwer Acad. Publ., Dordrecht, 2001).
[CrossRef]

Torner, L.

Torruellas, W. E.

Trillo, S.

A. V. Buryak, Y. S. Kivshar, and S. Trillo, “Stability of tree-wave parametric solitons in diffractive quadratic media,” Phys. Rev. Lett. 77, 5210–5213 (1996).
[CrossRef] [PubMed]

S. Trillo and W. E. Torruellas, Spatial Solitons (Springer-Verlag, Berlin, 2001).

Umeton, C.

M. Peccianti, A. De Rossi, G. Assanto, A. De Luca, C. Umeton, and I. C. Khoo, “Electrically Assisted Self-Confinement and Waveguiding in planar Nematic Liquid Crystal cells”, Appl. Phys. Lett. 77, 7–9 (2000).
[CrossRef]

VanStryland, E.

Werner, M. J.

Xie, K.

A. D. Boardman and K. Xie, “Theory of spatial solitons,” Radio Science 28, 891–899 (1993).
[CrossRef]

Appl. Phys. Lett. (1)

M. Peccianti, A. De Rossi, G. Assanto, A. De Luca, C. Umeton, and I. C. Khoo, “Electrically Assisted Self-Confinement and Waveguiding in planar Nematic Liquid Crystal cells”, Appl. Phys. Lett. 77, 7–9 (2000).
[CrossRef]

J. Opt. Soc. Am. (1)

D. M. Baboiu and G. I. Stegeman, “Solitary-wave interactions in quadratic media near type I phase-matching conditions,” J. Opt. Soc. Am. 14, 3143–3150 (1997).
[CrossRef]

J. Opt. Soc. Am. B (4)

Mod. Phys. Lett. B (1)

A. W. Snyder, D. J. Mitchell, and Y. S. Kivshar, “Unification of Linear and Nonlinear Wave Optics,” Mod. Phys. Lett. B 9, 1479–1506 (1995).
[CrossRef]

Opt. & Quantum Electron. (1)

R. A. Fuerst, M. T. G. Canva, G. I. Stegeman, G. Leo, and G. Assanto, “Robust generation, properties, and potential applications of quadratic spatial solitons generated by optical parametric amplification,” Opt. & Quantum Electron. 30, 907–921 (1998).
[CrossRef]

Opt. Lett. (4)

Phys. Rev. A (1)

A. V. Buryak, Y. S. Kivshar, and V. V. Steblina, “Self-trapping of light beams and parametric solitons in diffractive quadratic media,” Phys. Rev. A 52, 1670–1674 (1995).
[CrossRef] [PubMed]

Phys. Rev. E (1)

S. K. Johansen, O. Bang, and M. P. Soerensen, “Escape velocities in bulk χ(2) soliton interactions,” Phys. Rev. E 65, 026601–026604(2002).
[CrossRef]

Phys. Rev. Lett. (2)

A. V. Buryak, Y. S. Kivshar, and S. Trillo, “Stability of tree-wave parametric solitons in diffractive quadratic media,” Phys. Rev. Lett. 77, 5210–5213 (1996).
[CrossRef] [PubMed]

K. Hayata and M. Koshiba, “Multidimensional solitons in quadratic nonlinear media,” Phys. Rev. Lett. 71, 3275–3278 (1993).
[CrossRef] [PubMed]

Phys. Today (1)

M. Segev and G. Stegeman, “Self-Trapping of Optical Beams: Spatial Solitons,” Phys. Today 51, 43–48 (1998).
[CrossRef]

Radio Science (1)

A. D. Boardman and K. Xie, “Theory of spatial solitons,” Radio Science 28, 891–899 (1993).
[CrossRef]

Sci. (1)

G. I. Stegeman and M. Segev, “Optical Solitons and Their Interactions: Universality and Diversity,” Sci. 286, 1518–1523 (1999).
[CrossRef]

Sov. Phys.-JETP (1)

Y. N. Karamzin and A. P. Sukhorukov, “Mutual focusing of high-power light beams in media with quadratic nonlinearity”, Sov. Phys.-JETP,  41, 414–416 (1976).

Other (3)

A. D. Boardman and A. P. Sukhorukov, Soliton Driven Photonics (Kluwer Acad. Publ., Dordrecht, 2001).
[CrossRef]

S. Trillo and W. E. Torruellas, Spatial Solitons (Springer-Verlag, Berlin, 2001).

G. Assanto, “Diffraction with Second-Harmonic Generation for the formation of self-guided or ‘solitary’ waves,” in Diffractive optics and Optical Microsystems, A. N. Chester and S. Martellucci eds., 65–74 (Plenum Press, New York, 1997).

Supplementary Material (2)

» Media 1: MPG (206 KB)     
» Media 2: MPG (205 KB)     

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Figures (7)

Fig. 1
Fig. 1

Intuitive sketch on the origin of the “cascading” phase shift in Type I SHG with plane waves.

Fig. 2
Fig. 2

Simulated evolution of the transverse intensity of (a) a linearly diffracting 1D beam and (b) a quadratic spatial soliton excited by a gaussian FF beam. (c) The FF phase-front evolution for the latter case. Propagation distances are in units of the diffraction length LD, whereas the transverse coordinate is in units of the input waist w0. Here ΔkLD=2.

Fig. 3
Fig. 3

Beam narrowing through SHG. The input is an FF beam. From top to bottom: Linear diffraction at low powers, weak SHG below threshold for soliton generation, and spatial soliton formation.

Fig. 4
Fig. 4

(205 + 205KB) Animations showing the phase rotation of FF and SH field vectors in standard SHG (left) and quadratic soliton propagation (right).

Fig. 5
Fig. 5

Soliton collisions: schematic illustration of the interaction terms (in green) due to overlapping envelopes at FF and SH for solitons “a” and “b”.

Fig. 6
Fig. 6

Soliton collisions: as in Figure 5, but taking into account the spatial overlap with the eigen-distributions.

Fig. 7
Fig. 7

BPM propagation (z versus y) 3D-graphs for the various cases of interactions mentioned above. Resulting FF intensity for equi-power Gaussian beams launched parallel to z. Units are real (distances and intensity) but should be viewed as arbitrary.

Equations (23)

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

2 i k FF z a FF ( y ) 2 y 2 a FF ( y ) = 2 k F F Γ [ a S H ( y ) a F F * ( y ) ] e i Δ k z
2 i k SH z a SH ( y ) 2 y 2 a SH ( y ) = 2 k SH Γ [ a FF 2 ( y ) ] e i Δ k z
Δ a FF ( y ) = i Γ a SH ( y ) a FF * ( y ) Δ z Δ a SH ( y ) = i Γ a FF 2 ( y ) Δ z
P FF = ε o χ ( 2 ) ( a SH a FF * + b SH b FF * + a SH b FF * + b SH a FF * )
P SH = ε o χ ( 2 ) ( a FF a FF + b FF b FF + 2 a FF b FF )
2 i k FF z a FF 2 y 2 a FF = 2 k F F Γ [ a S H a F F * + δ ( a F F ) ]
2 i k FF z b FF 2 y 2 b FF = 2 k F F Γ [ b S H b F F * + δ ( b F F ) ]
2 i k SH z a SH 2 y 2 a SH = 2 k S H Γ [ a F F 2 + δ ( a S H ) ]
2 i k SH z b SH 2 y 2 b SH = 2 k S H Γ [ b F F 2 + δ ( b S H ) ]
δ ( a F F ) = a SH b FF * + b SH a FF * + b SH b FF *
δ ( b F F ) = a SH b FF * + b SH a FF * + a SH a FF *
δ ( a SH ) = b FF 2 + 2 a FF b FF
δ ( b SH ) = a FF 2 + 2 a FF b FF
y d y ( a FF * / a FF x ) a SH b FF * y d y ( a FF * / a FF x ) b SH a FF * y d y ( b FF * / b FF x ) a SH b FF * y d y ( b FF * / b FF x ) b SH a FF *
y d y ( a SH * / a SH x ) a FF b F F y d y ( a SH * / a SH x ) b F F 2 y d y ( b SH * / b SH x ) a FF b F F y d y ( b SH * / b SH x ) a F F 2
δ ( a F F ) = a SH b FF * b SH a FF * + b SH b FF *
δ ( b F F ) = a SH b FF * b SH a FF * + a SH a FF *
δ ( a SH ) = b FF 2 2 a F F b F F
δ ( b SH ) = a FF 2 2 a F F b F F
δ ( a FF ) = i a SH b FF * + i b SH a FF * + b SH b FF *
δ ( b FF ) = i a SH b FF * + i b SH a FF * + a SH a FF *
δ ( a SH ) = b FF 2 + 2 i a FF b FF
δ ( b SH ) = a FF 2 2 i a FF b FF

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