Abstract

The propagation of a square beam and a Gaussian beam along the interface of two photorefractive crystals is studied theoretically. Both of the crystals have a diffusion nonlinearity of the gradient type (with a sign opposite that of the diffusion nonlinearity), which is able to balance the self-bending of the beams and hence to produce a photorefractive surface wave. A comparison of this wave with the symmetrical and antisymmetrical surface waves propagating along the photorefractive crystals’ interface is also given.

© 2001 Optical Society of America

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References

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  1. G. Duree, J. Shultz, G. J. Salamo, M. Segev, A. Yariv, B. Crosignani, P. Di Porto, E. J. Sharp, and R. R. Neurgaonkar, “Observation of self-trapping of an optical beam due to the photorefractive effect,” Phys. Rev. Lett. 71, 533–536 (1993).
    [CrossRef] [PubMed]
  2. M. D. Iturbe Castillo, P. A. Marquez Aguilar, J. J. Sánchez Mondragón, S. Stepanov, and V. Vysloukh, “Spatial solitons in photorefractive Bi12TiO20 with drift mechanism of non-linearity,” Appl. Phys. Lett. 64, 408–410 (1994).
    [CrossRef]
  3. M. Segev, B. Crosignani, A. Yariv, and B. Fisher, “Spatial solitons in photorefractive media,” Phys. Rev. Lett. 68, 923–926 (1992).
    [CrossRef] [PubMed]
  4. B. Crosignani, M. Segev, D. Engin, P. Di Porto, A. Yariv, and G. Salamo, “Self-trapping of optical beams in photorefractive media,” J. Opt. Soc. Am. B 10, 446–452 (1993).
    [CrossRef]
  5. G. S. Garcia Quirino, J. J. Sánchez Mondragón, and S. Stepanov, “Nonlinear surface optical waves in photorefractive crystals with a diffusion mechanism of nonlinearity,” Phys. Rev. A 51, 1571–1577 (1995).
    [CrossRef] [PubMed]
  6. G. S. Garcia Quirino, J. J. Sánchez Mondragón, S. Stepanov, and V. A. Vysloukh, “Guided modes in a dielectric slab with diffusion-type photorefractive nonlinearity,” J. Opt. Soc. Am. B 13, 2530–2535 (1996).
    [CrossRef]
  7. M. Cronin-Golomb, “Photorefractive surface waves,” Opt. Lett. 20, 2075–2077 (1995).
    [CrossRef] [PubMed]
  8. R. Torres-Cordoba, J. J. Sánchez-Mondragón, and V. A. Vysloukh, “Beam propagation and surface wave formation along the interface of photorefractive media,” submitted to Phys. Rev. E.
  9. A. V. Khomenko, E. Nippolainen, A. A. Kamshilin, A. Zuñiga Segundo, and T. Jaaskelainen, “Leaky photorefractive surface waves in Bi12TiO20 and Bi12SiO20 crystals,” Opt. Commun. 150, 175–179 (1998).
    [CrossRef]
  10. A. V. Khomenko, A. Garcia-Weidner, and A. A. Kamshilin, “Amplification of optical signals in Bi12TiO20 crystal by photorefractive surface waves,” Opt. Lett. 21, 1014–1016 (1996).
    [CrossRef] [PubMed]
  11. J. D. Jackson, “Plane electromagnetic waves and wave propagation,” in Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975), Chap. 7, Sec. 9, pp. 303–306.
  12. J. J. Sakurai, Modern Quantum Mechanics (Addison-Wesley, Reading, Mass., 1975), Chap. 2, Sec. 5.
  13. B. M. Abramowitz and I. A. Segun, eds., Handbook of Mathematical Functions (Dover, New York, 1968), p. 299, Eq. (7.1.2a).
  14. H. Kogelnik, “Waves in thick holograms,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
    [CrossRef]

1998 (1)

A. V. Khomenko, E. Nippolainen, A. A. Kamshilin, A. Zuñiga Segundo, and T. Jaaskelainen, “Leaky photorefractive surface waves in Bi12TiO20 and Bi12SiO20 crystals,” Opt. Commun. 150, 175–179 (1998).
[CrossRef]

1996 (2)

1995 (2)

M. Cronin-Golomb, “Photorefractive surface waves,” Opt. Lett. 20, 2075–2077 (1995).
[CrossRef] [PubMed]

G. S. Garcia Quirino, J. J. Sánchez Mondragón, and S. Stepanov, “Nonlinear surface optical waves in photorefractive crystals with a diffusion mechanism of nonlinearity,” Phys. Rev. A 51, 1571–1577 (1995).
[CrossRef] [PubMed]

1994 (1)

M. D. Iturbe Castillo, P. A. Marquez Aguilar, J. J. Sánchez Mondragón, S. Stepanov, and V. Vysloukh, “Spatial solitons in photorefractive Bi12TiO20 with drift mechanism of non-linearity,” Appl. Phys. Lett. 64, 408–410 (1994).
[CrossRef]

1993 (2)

G. Duree, J. Shultz, G. J. Salamo, M. Segev, A. Yariv, B. Crosignani, P. Di Porto, E. J. Sharp, and R. R. Neurgaonkar, “Observation of self-trapping of an optical beam due to the photorefractive effect,” Phys. Rev. Lett. 71, 533–536 (1993).
[CrossRef] [PubMed]

B. Crosignani, M. Segev, D. Engin, P. Di Porto, A. Yariv, and G. Salamo, “Self-trapping of optical beams in photorefractive media,” J. Opt. Soc. Am. B 10, 446–452 (1993).
[CrossRef]

1992 (1)

M. Segev, B. Crosignani, A. Yariv, and B. Fisher, “Spatial solitons in photorefractive media,” Phys. Rev. Lett. 68, 923–926 (1992).
[CrossRef] [PubMed]

1969 (1)

H. Kogelnik, “Waves in thick holograms,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
[CrossRef]

Cronin-Golomb, M.

Crosignani, B.

B. Crosignani, M. Segev, D. Engin, P. Di Porto, A. Yariv, and G. Salamo, “Self-trapping of optical beams in photorefractive media,” J. Opt. Soc. Am. B 10, 446–452 (1993).
[CrossRef]

G. Duree, J. Shultz, G. J. Salamo, M. Segev, A. Yariv, B. Crosignani, P. Di Porto, E. J. Sharp, and R. R. Neurgaonkar, “Observation of self-trapping of an optical beam due to the photorefractive effect,” Phys. Rev. Lett. 71, 533–536 (1993).
[CrossRef] [PubMed]

M. Segev, B. Crosignani, A. Yariv, and B. Fisher, “Spatial solitons in photorefractive media,” Phys. Rev. Lett. 68, 923–926 (1992).
[CrossRef] [PubMed]

Di Porto, P.

G. Duree, J. Shultz, G. J. Salamo, M. Segev, A. Yariv, B. Crosignani, P. Di Porto, E. J. Sharp, and R. R. Neurgaonkar, “Observation of self-trapping of an optical beam due to the photorefractive effect,” Phys. Rev. Lett. 71, 533–536 (1993).
[CrossRef] [PubMed]

B. Crosignani, M. Segev, D. Engin, P. Di Porto, A. Yariv, and G. Salamo, “Self-trapping of optical beams in photorefractive media,” J. Opt. Soc. Am. B 10, 446–452 (1993).
[CrossRef]

Duree, G.

G. Duree, J. Shultz, G. J. Salamo, M. Segev, A. Yariv, B. Crosignani, P. Di Porto, E. J. Sharp, and R. R. Neurgaonkar, “Observation of self-trapping of an optical beam due to the photorefractive effect,” Phys. Rev. Lett. 71, 533–536 (1993).
[CrossRef] [PubMed]

Engin, D.

Fisher, B.

M. Segev, B. Crosignani, A. Yariv, and B. Fisher, “Spatial solitons in photorefractive media,” Phys. Rev. Lett. 68, 923–926 (1992).
[CrossRef] [PubMed]

Garcia Quirino, G. S.

G. S. Garcia Quirino, J. J. Sánchez Mondragón, S. Stepanov, and V. A. Vysloukh, “Guided modes in a dielectric slab with diffusion-type photorefractive nonlinearity,” J. Opt. Soc. Am. B 13, 2530–2535 (1996).
[CrossRef]

G. S. Garcia Quirino, J. J. Sánchez Mondragón, and S. Stepanov, “Nonlinear surface optical waves in photorefractive crystals with a diffusion mechanism of nonlinearity,” Phys. Rev. A 51, 1571–1577 (1995).
[CrossRef] [PubMed]

Garcia-Weidner, A.

Iturbe Castillo, M. D.

M. D. Iturbe Castillo, P. A. Marquez Aguilar, J. J. Sánchez Mondragón, S. Stepanov, and V. Vysloukh, “Spatial solitons in photorefractive Bi12TiO20 with drift mechanism of non-linearity,” Appl. Phys. Lett. 64, 408–410 (1994).
[CrossRef]

Jaaskelainen, T.

A. V. Khomenko, E. Nippolainen, A. A. Kamshilin, A. Zuñiga Segundo, and T. Jaaskelainen, “Leaky photorefractive surface waves in Bi12TiO20 and Bi12SiO20 crystals,” Opt. Commun. 150, 175–179 (1998).
[CrossRef]

Kamshilin, A. A.

A. V. Khomenko, E. Nippolainen, A. A. Kamshilin, A. Zuñiga Segundo, and T. Jaaskelainen, “Leaky photorefractive surface waves in Bi12TiO20 and Bi12SiO20 crystals,” Opt. Commun. 150, 175–179 (1998).
[CrossRef]

A. V. Khomenko, A. Garcia-Weidner, and A. A. Kamshilin, “Amplification of optical signals in Bi12TiO20 crystal by photorefractive surface waves,” Opt. Lett. 21, 1014–1016 (1996).
[CrossRef] [PubMed]

Khomenko, A. V.

A. V. Khomenko, E. Nippolainen, A. A. Kamshilin, A. Zuñiga Segundo, and T. Jaaskelainen, “Leaky photorefractive surface waves in Bi12TiO20 and Bi12SiO20 crystals,” Opt. Commun. 150, 175–179 (1998).
[CrossRef]

A. V. Khomenko, A. Garcia-Weidner, and A. A. Kamshilin, “Amplification of optical signals in Bi12TiO20 crystal by photorefractive surface waves,” Opt. Lett. 21, 1014–1016 (1996).
[CrossRef] [PubMed]

Kogelnik, H.

H. Kogelnik, “Waves in thick holograms,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
[CrossRef]

Marquez Aguilar, P. A.

M. D. Iturbe Castillo, P. A. Marquez Aguilar, J. J. Sánchez Mondragón, S. Stepanov, and V. Vysloukh, “Spatial solitons in photorefractive Bi12TiO20 with drift mechanism of non-linearity,” Appl. Phys. Lett. 64, 408–410 (1994).
[CrossRef]

Neurgaonkar, R. R.

G. Duree, J. Shultz, G. J. Salamo, M. Segev, A. Yariv, B. Crosignani, P. Di Porto, E. J. Sharp, and R. R. Neurgaonkar, “Observation of self-trapping of an optical beam due to the photorefractive effect,” Phys. Rev. Lett. 71, 533–536 (1993).
[CrossRef] [PubMed]

Nippolainen, E.

A. V. Khomenko, E. Nippolainen, A. A. Kamshilin, A. Zuñiga Segundo, and T. Jaaskelainen, “Leaky photorefractive surface waves in Bi12TiO20 and Bi12SiO20 crystals,” Opt. Commun. 150, 175–179 (1998).
[CrossRef]

Salamo, G.

Salamo, G. J.

G. Duree, J. Shultz, G. J. Salamo, M. Segev, A. Yariv, B. Crosignani, P. Di Porto, E. J. Sharp, and R. R. Neurgaonkar, “Observation of self-trapping of an optical beam due to the photorefractive effect,” Phys. Rev. Lett. 71, 533–536 (1993).
[CrossRef] [PubMed]

Sánchez Mondragón, J. J.

G. S. Garcia Quirino, J. J. Sánchez Mondragón, S. Stepanov, and V. A. Vysloukh, “Guided modes in a dielectric slab with diffusion-type photorefractive nonlinearity,” J. Opt. Soc. Am. B 13, 2530–2535 (1996).
[CrossRef]

G. S. Garcia Quirino, J. J. Sánchez Mondragón, and S. Stepanov, “Nonlinear surface optical waves in photorefractive crystals with a diffusion mechanism of nonlinearity,” Phys. Rev. A 51, 1571–1577 (1995).
[CrossRef] [PubMed]

M. D. Iturbe Castillo, P. A. Marquez Aguilar, J. J. Sánchez Mondragón, S. Stepanov, and V. Vysloukh, “Spatial solitons in photorefractive Bi12TiO20 with drift mechanism of non-linearity,” Appl. Phys. Lett. 64, 408–410 (1994).
[CrossRef]

Segev, M.

B. Crosignani, M. Segev, D. Engin, P. Di Porto, A. Yariv, and G. Salamo, “Self-trapping of optical beams in photorefractive media,” J. Opt. Soc. Am. B 10, 446–452 (1993).
[CrossRef]

G. Duree, J. Shultz, G. J. Salamo, M. Segev, A. Yariv, B. Crosignani, P. Di Porto, E. J. Sharp, and R. R. Neurgaonkar, “Observation of self-trapping of an optical beam due to the photorefractive effect,” Phys. Rev. Lett. 71, 533–536 (1993).
[CrossRef] [PubMed]

M. Segev, B. Crosignani, A. Yariv, and B. Fisher, “Spatial solitons in photorefractive media,” Phys. Rev. Lett. 68, 923–926 (1992).
[CrossRef] [PubMed]

Sharp, E. J.

G. Duree, J. Shultz, G. J. Salamo, M. Segev, A. Yariv, B. Crosignani, P. Di Porto, E. J. Sharp, and R. R. Neurgaonkar, “Observation of self-trapping of an optical beam due to the photorefractive effect,” Phys. Rev. Lett. 71, 533–536 (1993).
[CrossRef] [PubMed]

Shultz, J.

G. Duree, J. Shultz, G. J. Salamo, M. Segev, A. Yariv, B. Crosignani, P. Di Porto, E. J. Sharp, and R. R. Neurgaonkar, “Observation of self-trapping of an optical beam due to the photorefractive effect,” Phys. Rev. Lett. 71, 533–536 (1993).
[CrossRef] [PubMed]

Stepanov, S.

G. S. Garcia Quirino, J. J. Sánchez Mondragón, S. Stepanov, and V. A. Vysloukh, “Guided modes in a dielectric slab with diffusion-type photorefractive nonlinearity,” J. Opt. Soc. Am. B 13, 2530–2535 (1996).
[CrossRef]

G. S. Garcia Quirino, J. J. Sánchez Mondragón, and S. Stepanov, “Nonlinear surface optical waves in photorefractive crystals with a diffusion mechanism of nonlinearity,” Phys. Rev. A 51, 1571–1577 (1995).
[CrossRef] [PubMed]

M. D. Iturbe Castillo, P. A. Marquez Aguilar, J. J. Sánchez Mondragón, S. Stepanov, and V. Vysloukh, “Spatial solitons in photorefractive Bi12TiO20 with drift mechanism of non-linearity,” Appl. Phys. Lett. 64, 408–410 (1994).
[CrossRef]

Vysloukh, V.

M. D. Iturbe Castillo, P. A. Marquez Aguilar, J. J. Sánchez Mondragón, S. Stepanov, and V. Vysloukh, “Spatial solitons in photorefractive Bi12TiO20 with drift mechanism of non-linearity,” Appl. Phys. Lett. 64, 408–410 (1994).
[CrossRef]

Vysloukh, V. A.

Yariv, A.

B. Crosignani, M. Segev, D. Engin, P. Di Porto, A. Yariv, and G. Salamo, “Self-trapping of optical beams in photorefractive media,” J. Opt. Soc. Am. B 10, 446–452 (1993).
[CrossRef]

G. Duree, J. Shultz, G. J. Salamo, M. Segev, A. Yariv, B. Crosignani, P. Di Porto, E. J. Sharp, and R. R. Neurgaonkar, “Observation of self-trapping of an optical beam due to the photorefractive effect,” Phys. Rev. Lett. 71, 533–536 (1993).
[CrossRef] [PubMed]

M. Segev, B. Crosignani, A. Yariv, and B. Fisher, “Spatial solitons in photorefractive media,” Phys. Rev. Lett. 68, 923–926 (1992).
[CrossRef] [PubMed]

Zuñiga Segundo, A.

A. V. Khomenko, E. Nippolainen, A. A. Kamshilin, A. Zuñiga Segundo, and T. Jaaskelainen, “Leaky photorefractive surface waves in Bi12TiO20 and Bi12SiO20 crystals,” Opt. Commun. 150, 175–179 (1998).
[CrossRef]

Appl. Phys. Lett. (1)

M. D. Iturbe Castillo, P. A. Marquez Aguilar, J. J. Sánchez Mondragón, S. Stepanov, and V. Vysloukh, “Spatial solitons in photorefractive Bi12TiO20 with drift mechanism of non-linearity,” Appl. Phys. Lett. 64, 408–410 (1994).
[CrossRef]

Bell Syst. Tech. J. (1)

H. Kogelnik, “Waves in thick holograms,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
[CrossRef]

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

Opt. Commun. (1)

A. V. Khomenko, E. Nippolainen, A. A. Kamshilin, A. Zuñiga Segundo, and T. Jaaskelainen, “Leaky photorefractive surface waves in Bi12TiO20 and Bi12SiO20 crystals,” Opt. Commun. 150, 175–179 (1998).
[CrossRef]

Opt. Lett. (2)

Phys. Rev. A (1)

G. S. Garcia Quirino, J. J. Sánchez Mondragón, and S. Stepanov, “Nonlinear surface optical waves in photorefractive crystals with a diffusion mechanism of nonlinearity,” Phys. Rev. A 51, 1571–1577 (1995).
[CrossRef] [PubMed]

Phys. Rev. Lett. (2)

M. Segev, B. Crosignani, A. Yariv, and B. Fisher, “Spatial solitons in photorefractive media,” Phys. Rev. Lett. 68, 923–926 (1992).
[CrossRef] [PubMed]

G. Duree, J. Shultz, G. J. Salamo, M. Segev, A. Yariv, B. Crosignani, P. Di Porto, E. J. Sharp, and R. R. Neurgaonkar, “Observation of self-trapping of an optical beam due to the photorefractive effect,” Phys. Rev. Lett. 71, 533–536 (1993).
[CrossRef] [PubMed]

Other (4)

R. Torres-Cordoba, J. J. Sánchez-Mondragón, and V. A. Vysloukh, “Beam propagation and surface wave formation along the interface of photorefractive media,” submitted to Phys. Rev. E.

J. D. Jackson, “Plane electromagnetic waves and wave propagation,” in Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975), Chap. 7, Sec. 9, pp. 303–306.

J. J. Sakurai, Modern Quantum Mechanics (Addison-Wesley, Reading, Mass., 1975), Chap. 2, Sec. 5.

B. M. Abramowitz and I. A. Segun, eds., Handbook of Mathematical Functions (Dover, New York, 1968), p. 299, Eq. (7.1.2a).

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Figures (5)

Fig. 1
Fig. 1

Field continuity, where the Maxwell equations are applied in z=0, for PRC–PRC.

Fig. 2
Fig. 2

Approximation of solitonlike and surface-wave formation D=0.5 for x0 and D=-0.5 for x<0.

Fig. 3
Fig. 3

Propagation and diffraction of the beam between two media, PRC and PRC, in the Fresnel region, with D=0.5 for x>0 and D=-0.5 for x<0.

Fig. 4
Fig. 4

Gaussian beam propagation along the interface of the PRC and the PRC media, with D=0.5 for x0 and D=-0.5 for x<0.

Fig. 5
Fig. 5

Quasi stability of Gaussian beam propagation along the interface of the PRC and the PRC media, with D=0.5 for x0 and D=-0.5 for x<0.

Equations (57)

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

2E1(X, Y, Z, t)-με0ε(X, Z) 2E1(X, Y, Z, t)t2=0,
x0.
ε(X, Z)=ε±δε(X, Z),
E1(X, Z, t)=E1(X, Z)exp(-iωt)
2E1(X, Z)X2+2E1(X, Z)Z2+ω2με0ε(X, Z)E1(X, Z)=0.
E1(X, Z)=E(X, Z)exp(-iβZ),
2E(X, Z)X2+2E(X, Z)Z2-2iβ E(X, Z)Z
-[β2-ω2με(X, Z)ε0]E(X, Z)=0.
δε(X, Z)=2n4r KBTe  ln|E(X, Z)|2X,
2E(X, Z)Z2β E(X, Z)Z
k0-2 2E(X, Z)X2+D E(X, Z)X
-ibk0-1 E(X, Z)Z-CE(X, Z)=0,
2E(x, z)x2+D E(x, z)x-ib E(x, z)z=0
2E(x, z)x2+sign(x)D E(x, z)x-ib E(x, z)z=0,
sign(x)=1x0-1x<0 .
E1(x, z)|x=0=E2(x, z)|x=0=G(z),z>0,
E1(x, z)xx=0=E2(x, z)xx=0=F(z)
E1(x, z)|z=0=E2(x, z)|z=0=K1(x);
d2E(x, s)dx2+D dE(x, s)dx-ibsE(x, s)=-ibK1(x)
E(x, s)=exp-Dx2G(s)cosβ1(s)x2+H1(s)sinβ1(s)x2+g1(x, s),
H1(s)=[DG(s)+2F(s)]β1-1(s)
β1=-i2 D2+4ibs=-iγ2+ibs,
g1(x, z)=2ib0xK1(t)×exp-D2(x-t) sin[β1(s)(x-t)]iβ1(s) dt.
Ls-1g1(x, s)=g1(x, z)
g1(x, z)=ib4πz exp(-ηz) 0xK1(t)×exp-ib(x-t)24z-D(x-t)2dt,
G(x, z, t, 0)=[b(4iπz)-1]exp[-½ib(t-x)2(2z)-1]×exp[½D(t-x)]
g1(x, z)=exp(-ηz)0xK1(t)G(x, z, t, 0)dt.
K1(x)=g0-h  x  h0otherwise,
K1(x)=g0 exp-x24,
E(x, z)=exp-Dx2[EI(x, z)+EII(x, z)+EIII(x, z)]+g1(x, z),
EI(x, z)=τx2π 0zG(z-u)expτ2x24u-ηuu-3/2du,
EII(x, z)=D2π 0zG(z-u)expr2x24u-ηuu-1/2du,
EIII(x, z)=1π 0zF(z-u)expτ2x24u-ηuu-1/2du.
E(x, z)=f1g0(f2+f3-f4f5-f6f7+f8+f9);
g1(x, z)=g02
exp{-¼x2+¼(D-x)2[1+(ib)-1z]-1}[1+(ib)-1z]1/2g2(x, z),
g2(x, z)=erf(D+z-1ibx)2(1+z-1ib)1/2-erf(D-x)2(1+z-1ib).
E01(x, z)
=exp{-½Dx+¼D2+¼(D-x)2[1+(ib)-1z]-1}[1+(ib)-1z]1/2
E02(x, z)=g02E03(x, z)exp{ηz-¼[x+(ib)-1Dz]2[1+(ib)-1z]-1}[1+(ib)-1z]1/2.
E03(x, z)=erf[x+(ib)-1Dz]2{(ib)-1z[1+(ib)-1z]}1/2,
E04(x, z)=14D -10[v-(ib)-1z]-5(v+1)-5×exp[x+(ib)-1zD]24[v-(ib)-1z]dv,
E04(x, z)=14Dg0ibz B(½, 1)1+(ib)-1zΦ1×exp-[x+(ib)-1zD]24[1+(ib)-1z],
Φ1=Φ1{0.5, 1, 1.5:-(ib)-1z, ¼ib×[x+(ib)-1zD]2z-1[1+(ib)-1z]-1},
Φ1(α, β, γ, u, q)=m,n=0 (α)m+n(β)n(γ)m+nm!n!umqn,
E(x, z)=E01(x, z)+E02(x, z)+g1(x, z).
limz E(x, z)=E(x).
limz E(x, z)=E(x)0,
f1=¼ exp-Dx2,
f2=1-xx2expDx22erfD2 zib+12 ibx2z,
f3=1+xx2exp-Dx22×erfD2 zib-12 ibx2z,
f4=1-x-(h/2)(x-h/2)2expDh4,
f5=expD(x-h/2)22×erfD2 zib+12 ib(x-h/2)2z,
f6=1+x-h/2(x-h/2)2expDh4,
f7=exp-D(x-h/2)22×erfD2 zbi-12 bi(x-h/2)2z,
f8=2xx2 coshD2 x2-2 sinhD2 x2,
f9=expDh4-2(x-h/2)(x-h/2)2 coshD2 x-h22+2 sinhD2 x-h22.

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