Abstract

A general microscopic theory that treats cascading of a wide range of nonlinear optical processes as a collective effect is developed. Important practical implications are discussed.

© 1997 Optical Society of America

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  1. G. R. Meredith, “Second-order cascading in third-order nonlinear oprical processes,” J. Chem. Phys. 77, 5863 (1982).
    [CrossRef]
  2. R. DeSalvo, D. J. Hagan, M. Sheik-Bahae, G. I. Stegeman, and E. W. Van Stryland, “Self-focusing and self-defocusing by cascaded second-order effects in KTP,” Opt. Lett. 17, 28 (1992).
    [CrossRef] [PubMed]
  3. G. I. Stegeman, M. Sheik-Bahae, E. Van Stryland, and G. Assanto, “Coherent interactions for all-optical signal processing via quadratic nonlinearities,” Opt. Lett. 18, 13 (1993).
    [CrossRef] [PubMed]
  4. R. Schiek, D. Y. Kim, M. L. Sundheimer, and G. I. Stegeman, “Second-order cascaded nonlinearity in lithium niobate channel waveguides,” in Nonlinear Optics: Materials, Fundamentals and Applications (Institute of Electrical and Electronics Engineers, New York, 1994), p. 36.
  5. R. Schiek, “Nonlinear refraction caused by cascaded second-order nonlinearity in optical waveguide structures,” J. Opt. Soc. Am. B 10, 1848 (1993).
    [CrossRef]
  6. C. N. Ironside, J. S. Aitchison, and J. M. Arnold, “An all-optical switch employing the cascaded second-order nonlinear effect,” IEEE J. Quantum Electron. 29, 2650 (1993).
    [CrossRef]
  7. D. C. Hutchings, J. S. Aitchison, and C. N. Ironside, “All-optical switching based on nondegenerate phase shifts from a cascaded second-order nonlinearity,” Opt. Lett. 18, 793 (1993).
    [CrossRef] [PubMed]
  8. Y. Baek, R. Schiek, G. I. Stegeman, G. Krijnen, I. Baumann, and W. Sohler, “All-optical integrated Mach–Zehnder switching due to cascaded nonlinearities,” Appl. Phys. Lett. 68, 2055 (1996).
    [CrossRef]
  9. Y. Baek, R. Schiek, and G. I. Stegeman, “All-optical switching in a hybrid Mach–Zehnder interferometer as a result of cascaded second-order nonlinearity,” Opt. Lett. 20, 2168 (1995).
    [CrossRef]
  10. L. Torner, C. R. Menyuk, and G. I. Stegeman, “Excitation of solitons with cascaded χ(2) nonlinearities,” Opt. Lett. 19, 1615 (1994).
    [CrossRef] [PubMed]
  11. J. B. Khurgin and Y. J. Ding, “Resonant cascaded surface-emitting second-harmonic generation: a strong third-order nonlinear process,” Opt. Lett. 19, 1016 (1994).
    [CrossRef] [PubMed]
  12. A. L. Lentine and D. A. B. Miller, “Evolution of the SEED technology: bistable logic gates to optoelectronic smart pixels,” IEEE J. Quantum Electron. 29, 655 (1993).
    [CrossRef]
  13. J. Khurgin and S. Li, “Coulomb enhancement of the third-order nonlinearities in the mesoscopic semiconductor structures,” Appl. Phys. A 53, 523 (1991).
    [CrossRef]
  14. J. Khurgin, “Coulomb enhancement of ultrafast nonlinearities in quantum-well structures,” J. Opt. Soc. Am. B 9, 157 (1992).
    [CrossRef]
  15. C. Flytzanis, in Quantum Electronics, H. Rabin and C. L. Tang, eds. (Academic, New York, 1975); Vol. 1A, p. 9-207.
  16. N. B. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetsky, “Holographic storage in electrooptic crystals. I. Steady state,” Ferroelectrics 22, 949 (1979).
    [CrossRef]
  17. M. Segev, Ming-Feng Shih, and G. C. Valley, “Photorefractive screening solitons of high and low intensity,” J. Opt. Soc. Am. B 13, 706 (1996).
    [CrossRef]
  18. G. H. Döhler, “Doping superlattices (‘n-i-p-i crystals’),” IEEE J. Quantum Electron. QE-22, 1683 (1986).
  19. G. R. Meredith, “Cascading in optical third-harmonic generation by crystalline quartz,” Phys. Rev. B 24, 5522 (1981).
    [CrossRef]

1996 (2)

Y. Baek, R. Schiek, G. I. Stegeman, G. Krijnen, I. Baumann, and W. Sohler, “All-optical integrated Mach–Zehnder switching due to cascaded nonlinearities,” Appl. Phys. Lett. 68, 2055 (1996).
[CrossRef]

M. Segev, Ming-Feng Shih, and G. C. Valley, “Photorefractive screening solitons of high and low intensity,” J. Opt. Soc. Am. B 13, 706 (1996).
[CrossRef]

1995 (1)

1994 (2)

1993 (5)

1992 (2)

1991 (1)

J. Khurgin and S. Li, “Coulomb enhancement of the third-order nonlinearities in the mesoscopic semiconductor structures,” Appl. Phys. A 53, 523 (1991).
[CrossRef]

1986 (1)

G. H. Döhler, “Doping superlattices (‘n-i-p-i crystals’),” IEEE J. Quantum Electron. QE-22, 1683 (1986).

1982 (1)

G. R. Meredith, “Second-order cascading in third-order nonlinear oprical processes,” J. Chem. Phys. 77, 5863 (1982).
[CrossRef]

1981 (1)

G. R. Meredith, “Cascading in optical third-harmonic generation by crystalline quartz,” Phys. Rev. B 24, 5522 (1981).
[CrossRef]

1979 (1)

N. B. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetsky, “Holographic storage in electrooptic crystals. I. Steady state,” Ferroelectrics 22, 949 (1979).
[CrossRef]

Aitchison, J. S.

C. N. Ironside, J. S. Aitchison, and J. M. Arnold, “An all-optical switch employing the cascaded second-order nonlinear effect,” IEEE J. Quantum Electron. 29, 2650 (1993).
[CrossRef]

D. C. Hutchings, J. S. Aitchison, and C. N. Ironside, “All-optical switching based on nondegenerate phase shifts from a cascaded second-order nonlinearity,” Opt. Lett. 18, 793 (1993).
[CrossRef] [PubMed]

Arnold, J. M.

C. N. Ironside, J. S. Aitchison, and J. M. Arnold, “An all-optical switch employing the cascaded second-order nonlinear effect,” IEEE J. Quantum Electron. 29, 2650 (1993).
[CrossRef]

Assanto, G.

Baek, Y.

Y. Baek, R. Schiek, G. I. Stegeman, G. Krijnen, I. Baumann, and W. Sohler, “All-optical integrated Mach–Zehnder switching due to cascaded nonlinearities,” Appl. Phys. Lett. 68, 2055 (1996).
[CrossRef]

Y. Baek, R. Schiek, and G. I. Stegeman, “All-optical switching in a hybrid Mach–Zehnder interferometer as a result of cascaded second-order nonlinearity,” Opt. Lett. 20, 2168 (1995).
[CrossRef]

Baumann, I.

Y. Baek, R. Schiek, G. I. Stegeman, G. Krijnen, I. Baumann, and W. Sohler, “All-optical integrated Mach–Zehnder switching due to cascaded nonlinearities,” Appl. Phys. Lett. 68, 2055 (1996).
[CrossRef]

DeSalvo, R.

Ding, Y. J.

Döhler, G. H.

G. H. Döhler, “Doping superlattices (‘n-i-p-i crystals’),” IEEE J. Quantum Electron. QE-22, 1683 (1986).

Flytzanis, C.

C. Flytzanis, in Quantum Electronics, H. Rabin and C. L. Tang, eds. (Academic, New York, 1975); Vol. 1A, p. 9-207.

Hagan, D. J.

Hutchings, D. C.

Ironside, C. N.

C. N. Ironside, J. S. Aitchison, and J. M. Arnold, “An all-optical switch employing the cascaded second-order nonlinear effect,” IEEE J. Quantum Electron. 29, 2650 (1993).
[CrossRef]

D. C. Hutchings, J. S. Aitchison, and C. N. Ironside, “All-optical switching based on nondegenerate phase shifts from a cascaded second-order nonlinearity,” Opt. Lett. 18, 793 (1993).
[CrossRef] [PubMed]

Khurgin, J.

J. Khurgin, “Coulomb enhancement of ultrafast nonlinearities in quantum-well structures,” J. Opt. Soc. Am. B 9, 157 (1992).
[CrossRef]

J. Khurgin and S. Li, “Coulomb enhancement of the third-order nonlinearities in the mesoscopic semiconductor structures,” Appl. Phys. A 53, 523 (1991).
[CrossRef]

Khurgin, J. B.

Kim, D. Y.

R. Schiek, D. Y. Kim, M. L. Sundheimer, and G. I. Stegeman, “Second-order cascaded nonlinearity in lithium niobate channel waveguides,” in Nonlinear Optics: Materials, Fundamentals and Applications (Institute of Electrical and Electronics Engineers, New York, 1994), p. 36.

Krijnen, G.

Y. Baek, R. Schiek, G. I. Stegeman, G. Krijnen, I. Baumann, and W. Sohler, “All-optical integrated Mach–Zehnder switching due to cascaded nonlinearities,” Appl. Phys. Lett. 68, 2055 (1996).
[CrossRef]

Kukhtarev, N. B.

N. B. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetsky, “Holographic storage in electrooptic crystals. I. Steady state,” Ferroelectrics 22, 949 (1979).
[CrossRef]

Lentine, A. L.

A. L. Lentine and D. A. B. Miller, “Evolution of the SEED technology: bistable logic gates to optoelectronic smart pixels,” IEEE J. Quantum Electron. 29, 655 (1993).
[CrossRef]

Li, S.

J. Khurgin and S. Li, “Coulomb enhancement of the third-order nonlinearities in the mesoscopic semiconductor structures,” Appl. Phys. A 53, 523 (1991).
[CrossRef]

Markov, V. B.

N. B. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetsky, “Holographic storage in electrooptic crystals. I. Steady state,” Ferroelectrics 22, 949 (1979).
[CrossRef]

Menyuk, C. R.

Meredith, G. R.

G. R. Meredith, “Second-order cascading in third-order nonlinear oprical processes,” J. Chem. Phys. 77, 5863 (1982).
[CrossRef]

G. R. Meredith, “Cascading in optical third-harmonic generation by crystalline quartz,” Phys. Rev. B 24, 5522 (1981).
[CrossRef]

Miller, D. A. B.

A. L. Lentine and D. A. B. Miller, “Evolution of the SEED technology: bistable logic gates to optoelectronic smart pixels,” IEEE J. Quantum Electron. 29, 655 (1993).
[CrossRef]

Odulov, S. G.

N. B. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetsky, “Holographic storage in electrooptic crystals. I. Steady state,” Ferroelectrics 22, 949 (1979).
[CrossRef]

Schiek, R.

Y. Baek, R. Schiek, G. I. Stegeman, G. Krijnen, I. Baumann, and W. Sohler, “All-optical integrated Mach–Zehnder switching due to cascaded nonlinearities,” Appl. Phys. Lett. 68, 2055 (1996).
[CrossRef]

Y. Baek, R. Schiek, and G. I. Stegeman, “All-optical switching in a hybrid Mach–Zehnder interferometer as a result of cascaded second-order nonlinearity,” Opt. Lett. 20, 2168 (1995).
[CrossRef]

R. Schiek, “Nonlinear refraction caused by cascaded second-order nonlinearity in optical waveguide structures,” J. Opt. Soc. Am. B 10, 1848 (1993).
[CrossRef]

R. Schiek, D. Y. Kim, M. L. Sundheimer, and G. I. Stegeman, “Second-order cascaded nonlinearity in lithium niobate channel waveguides,” in Nonlinear Optics: Materials, Fundamentals and Applications (Institute of Electrical and Electronics Engineers, New York, 1994), p. 36.

Segev, M.

Sheik-Bahae, M.

Shih, Ming-Feng

Sohler, W.

Y. Baek, R. Schiek, G. I. Stegeman, G. Krijnen, I. Baumann, and W. Sohler, “All-optical integrated Mach–Zehnder switching due to cascaded nonlinearities,” Appl. Phys. Lett. 68, 2055 (1996).
[CrossRef]

Soskin, M. S.

N. B. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetsky, “Holographic storage in electrooptic crystals. I. Steady state,” Ferroelectrics 22, 949 (1979).
[CrossRef]

Stegeman, G. I.

Sundheimer, M. L.

R. Schiek, D. Y. Kim, M. L. Sundheimer, and G. I. Stegeman, “Second-order cascaded nonlinearity in lithium niobate channel waveguides,” in Nonlinear Optics: Materials, Fundamentals and Applications (Institute of Electrical and Electronics Engineers, New York, 1994), p. 36.

Torner, L.

Valley, G. C.

Van Stryland, E.

Van Stryland, E. W.

Vinetsky, V. L.

N. B. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetsky, “Holographic storage in electrooptic crystals. I. Steady state,” Ferroelectrics 22, 949 (1979).
[CrossRef]

Appl. Phys. A (1)

J. Khurgin and S. Li, “Coulomb enhancement of the third-order nonlinearities in the mesoscopic semiconductor structures,” Appl. Phys. A 53, 523 (1991).
[CrossRef]

Appl. Phys. Lett. (1)

Y. Baek, R. Schiek, G. I. Stegeman, G. Krijnen, I. Baumann, and W. Sohler, “All-optical integrated Mach–Zehnder switching due to cascaded nonlinearities,” Appl. Phys. Lett. 68, 2055 (1996).
[CrossRef]

Ferroelectrics (1)

N. B. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetsky, “Holographic storage in electrooptic crystals. I. Steady state,” Ferroelectrics 22, 949 (1979).
[CrossRef]

IEEE J. Quantum Electron. (3)

C. N. Ironside, J. S. Aitchison, and J. M. Arnold, “An all-optical switch employing the cascaded second-order nonlinear effect,” IEEE J. Quantum Electron. 29, 2650 (1993).
[CrossRef]

A. L. Lentine and D. A. B. Miller, “Evolution of the SEED technology: bistable logic gates to optoelectronic smart pixels,” IEEE J. Quantum Electron. 29, 655 (1993).
[CrossRef]

G. H. Döhler, “Doping superlattices (‘n-i-p-i crystals’),” IEEE J. Quantum Electron. QE-22, 1683 (1986).

J. Chem. Phys. (1)

G. R. Meredith, “Second-order cascading in third-order nonlinear oprical processes,” J. Chem. Phys. 77, 5863 (1982).
[CrossRef]

J. Opt. Soc. Am. B (3)

Opt. Lett. (6)

Phys. Rev. B (1)

G. R. Meredith, “Cascading in optical third-harmonic generation by crystalline quartz,” Phys. Rev. B 24, 5522 (1981).
[CrossRef]

Other (2)

R. Schiek, D. Y. Kim, M. L. Sundheimer, and G. I. Stegeman, “Second-order cascaded nonlinearity in lithium niobate channel waveguides,” in Nonlinear Optics: Materials, Fundamentals and Applications (Institute of Electrical and Electronics Engineers, New York, 1994), p. 36.

C. Flytzanis, in Quantum Electronics, H. Rabin and C. L. Tang, eds. (Academic, New York, 1975); Vol. 1A, p. 9-207.

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Figures (4)

Fig. 1
Fig. 1

General four-wave mixing in (a) χ(3) medium, (b) χ(2) medium.

Fig. 2
Fig. 2

Geometry of cascading: (a) Long–intermediate-wavelength limit, (b) collinear, and (c) transverse with the cavity for the intermediate wave.

Fig. 3
Fig. 3

Microscopic picture of cascading.

Fig. 4
Fig. 4

Space-charge nonlinearity as a special case of cascading.

Equations (50)

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En(r, t)=Ene˜n exp[i(±knz-ωnt)]ψn(x, y)+c.c.;
n=1, 2, 3, 4,
P4(3)(ω4=ω1+ω2+ω3)=0χ(3)(ω1+ω2+ω3; ω1, ω2, ω3)E1E2E3×exp(-iω4t)ψ1(x, y)ψ2(x, y)ψ3(x, y)×exp[i(k1+k2+k3)z],
χ(3)(ω1+ω2+ω3; ω1, ω2, ω3)=jkl Ne4rijrjkrklrli0(Eji-ω1)[Eki-(ω1+ω2)][Eli-(ω1+ω2+ω3)],
rij=i|r|j.
2E4-n2c22E4t2=10c22P4(3)t2,
dE4dz=i ω42cnκ1234χ(3)(ω4; ω1, ω2, ω3)E1E2E3×exp[(k1+k2+k3-k4)z],
κ1234=ψ1ψ2ψ3ψ4dxdyψ42dxdy.
Pi(2)(ωi=ω1+ω2)=0χ(2)(ω1+ω2; ω1, ω2)E1E2 
×exp(-iωit)ψ1(x, y)ψ2(x, y)
×exp[i(k1+k2)z],
χ(2)(ω1+ω2; ω1, ω2)
=jk Ne3rijrjkrki0(Eji-ω1)[Eki-(ω1+ω2)].
2Ei-n2c22Eit2-n2c2τcEit=10c22Pit2,
ψi(x, y)=ψ1(x, y)ψ2(x, y),
Ei=-1r(ωi)χ(2)(ω1+ω2; ω1, ω2)×E1E2 exp[i(k1+k2)z].
Ei(z)=ωi2cnχ(2)(ω1+ω2; ω1, ω2)×E1E2κ12i 1-exp(-iΔkz)Δk×exp[i(k1+k2-ki)z],
Δk=k1+k2-ki,
κ12i=ψ1ψ2ψidxdyψi2dxdy.
ωi2cn1-exp(-iΔkz)Δki ωi2cnL2=i πL2λir(ωi)=i ωiτp2r(ωi),
ωi2cn1-exp(-iΔkz)Δkωi2cn1Δk=lcoh2λir(ωi)=ωiτcoh2πr(ωi),
Ei(z)=i ωiτcn2χ(2)(ω1+ω2; ω1, ω2)E1E2κ12i×exp[i(k1+k2)z].
Ei(z)=χ(2)(ω1+ω2; ω1, ω2)×βκi34-1E1E2 exp[i(k1+k2-ki)z],
κi34=ψiψ3ψ4dxdyψi4dxdy.
β=Mr(ωi)Qiκ12iκi34,
P4(2+2)(ω4=ω1+ω2+ω3)
 =0χ(2)(ωi+ω3; ωi, ω3)EiE3×exp(-iω4t)ψi(x, y)ψ3(x, y)×exp[i(ki+k3)z]
=β0χ(2)(ωi+ω3; ωi, ω3)×χ(2)(ω1+ω2; ω1, ω2)×κi34-1E1E2E3ψi(x, y)ψ3(x, y)×exp[i(k1+k2+k3)z],
χ(2)(ωi+ω3; ωi, ω3)
=kl Ne3rijrjkrki0[Eli-(ω1+ω2+ω3)][Eki-(ω1+ω2)].
dE4dz=i ω42cnβχ(2)(ωi+ω3, ωi, ω3)×χ(2)(ω1+ω2; ω1, ω2)E1E2E3×exp[(k1+k2+k3-k4)z].
χ(2+2)(ω1+ω2+ω3; ω1, ω2, ω3)
=χ(2)(ω1+ω2; ω1, ω2)
×χ(2)(ωi+ω3; ωi, ω3) Mr(ωi)Qi κ12iκi34κ1234.
ηcasc=χ(2+2)χ(3)=ηmatηgeo,
ηmat=χ(2)(ωi; ω1, ω2)χ(2)(ω4; ωi, ω3)r(ωi)χ(3)(ω4; ω1, ω2, ω3)
ηgeo=MQi κ12iκi34κ1234,
ηmat=Ne2d¯20r(ωi)δE¯.
ηmat=Ne2d¯20δE¯r(ωi)χ(1)(ωi)r(ωi)r(ωi)-1r(ωi);
pi(ωi)=erkiρki exp(-iωit).
Ei(ωi)=QiNpi(ωi)/0r(ωi)=QiNerkiρki exp(-iωit)/0r(ωi).
ρki exp(-iωit)=erikEi(ωi)/(Eki-ωi)=Qi Ne2rkirik0r(ωi)(Eki-ωi)ρki exp(-iωit)Qi r(ωi)-1r(ωi)ρki exp(-iωit).
I(ωi=ω1-ω2)=n2η0E1E2ψ1ψ2×exp[i(k1-k2)z-ωit],
dρdt=eα(ω1)I(ωi)/ωi-ρτr,
α(ω1)α(ω2)=ω1ncj Ne2rijrji0Γ.
ρ(ωi)=Ne3d¯2E1E22Γ(τr-1-jωi).
Eiρ(ωi)dscr(ωi)0=Ne3rifrfidsc2r(ωi)0Γτr-1E1E2,
χsc(2)=Ne3rifrfidsc20Γτr-1,
Ei=1r(ωi)χsc(2)E1E2 exp[i(k1-k2)z].
ηsc=χsc(2)χ(2)τr(δω¯)-1dscd¯,

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