Abstract

In this analytical paper, a formalismfor second-harmonic and sum-frequency generation in glass fibers is developed. It considers only the lowest-order nonlinearities in regular, step-index fibers. Plots showing the dependence of index-matching frequencies on the core radius and index difference across the core-cladding interface are given. Estimates are given for the conversion efficiencies due to the nonlinear polarization at the core-cladding interface and the bulk nonlinear polarization proportional to EE, which includes quadrupolar terms. It is concluded that the interface effects would be dominant. Calculations for the longitudinal-electric-field-induced nonlinear polarization proportional to E2 that involve the same mode coupling are also included. The calculated conversion efficiencies are small and cannot explain the recently observed high conversion efficiencies that were obtained under nonindex-matched conditions. A maximum conversion efficiency of around 10−5 is predicted under index-matching conditions.

© 1987 Optical Society of America

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  1. Y. Sasaki and Y. Ohmori, “Phase-matched sum-frequency light generation in optical fibers,” Appl. Phys. Lett. 39, 466–468 (1981).
    [CrossRef]
  2. Y. Ohmori and Y. Sasaki, “Two-wave sum-frequency light generation in optical fibers,” IEEE J. Quantum Electron. QE-18, 758–762 (1982).
    [CrossRef]
  3. J. M. Gabriagues and L. Fersing, “Second harmonic generation in optical fibers,” presented at the Thirteenth International Quantum Electronics Conference, Anaheim, California, June 18–21, 1984.
  4. U. Osterberg and W. Margulis, “Dye laser pumped by Nd:YAG laser pulses frequency doubled in a glass optical fiber,” Opt. Lett. 8, 516–518 (1986); “Experimental studies on efficient doubling in glass optical fibers,” Opt. Lett. 12, 57–59 (1987).
    [CrossRef]
  5. A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall, New York, 1983).
  6. R. H. Stolen and W. N. Leibert, “Optical fiber modes using stimulated four photon mixing,” Appl. Opt. 15, 239–243 (1976).
    [CrossRef] [PubMed]
  7. J. W. Fleming, “Dispersion in GeO2–SiO2glasses,” Appl. Opt. 23, 4486–4493 (1984).
    [CrossRef]
  8. N. Bloembergen, Nonlinear Optics (Benjamin, New York, 1965).
  9. Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984).
  10. R. H. Stolen, “Phase-matched-stimulated four-photon mixing in silica-fiber waveguides,” IEEE J. Quantum Electron. QE-l1, 100–103 (1975).
    [CrossRef]
  11. N. Bloembergen, R. K. Chang, S. S. Jha, and C. H. Lee, “Optical second-harmonic generation in reflection from media with inversion symmetry,” Phys. Rev. 174, 813–822 (1968).
    [CrossRef]
  12. C. C. Wang, “Second-harmonic generation of light at the boundary of an isotropic medium,” Phys. Rev. 178, 1457–1461 (1969).
    [CrossRef]
  13. R. W. Terhune, P. D. Maker, and C. M. Savage, “Optical harmonic generation in calcite,” Phys. Rev. Lett. 8, 404–406 (1962).
    [CrossRef]
  14. J. E. Bjorkholm and A. E. Siegman, “Accurate cw measurements of optical second-harmonic generation in ammonium dihydrogen phosphate and calcite,” Phys. Rev. 154, 851–860 (1967).
    [CrossRef]
  15. P. S. Pershan, “Nonlinear optical properties of solids: energy considerations,” Phys. Rev. 130, 919–929 (1963). In this reference Eqs. (23) and (24) are derived through defining separate coefficients for the magnetic-dipole and quadrupole susceptibilities.
    [CrossRef]
  16. L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, New York, 1960). Chapter 11, on the electrostatics of dielectrics, provides a starting point. We know of no reference that treats the problem fully, particularly when the material properties are a function of position. In the static case, the analysis can be outlined as follows:δ (total free energy)=∫Volume[F(E+δE)-F(E)]dr≡-∫Volume(P·δE)dr,where F is the free-energy density and P is the effective polarization. Using a near local model, we are concerned with a term of the form F= ∊0χEiEj∇kEl. The expression for F(E+ δE) will have three terms associated with the changes in Ei, Ej, and El. The δEl term will have the form χEiEj∇k(χEiEj). This term can be integrated by parts to give δEl∇k(χEiEj). The value of the term δEl(χEiEj) will be zero at all end points of the integration, as δE is arbitrary and can be chosen to be highly localized. With this change the desired identifications with P can be made. Beyond this, the derivation has to be extended to using the time-averaged free energy, 〈F〉, expressed in terms of time-dependent field amplitudes.
  17. P. Guyot-Sionnest, W. Chen, and Y. R. Shen, “General considerations on optical second-harmonic generation from surfaces and interfaces,” Phys. Rev. B 33, 8254–8263 (1986).
    [CrossRef]
  18. B. Dick, A. Gierulski, and G. Marowsky, “Determination of the nonlinear optical susceptibility χ(2) of surface layers by sum and difference frequency generation in reflection and transmission,” Appl. Phys. B 38, 107–116 (1985).
    [CrossRef]
  19. J. M. Hicks, K. Kemnitz, and K. B. Eisenthal, “Studies of liquid surfaces by second harmonic generation,” J. Phys. Chem. 90, 560–562 (1986).
    [CrossRef]
  20. T. F. Heinz, H. W. K. Tom, and Y. R. Shen, “Determination of molecular orientation of monolayer absorbates by optical second-harmonic generation,” Phys. Rev A 28, 1883–1885 (1983).
    [CrossRef]
  21. T. F. Heinz, C. K. Chen, D. Richard, and Y. R. Shen, “Spectroscopy of molecular monolayers by resonant second-harmonic generation,” Phys. Rev. Lett. 48, 478–481 (1982).
    [CrossRef]
  22. J. A. Litwin, J. E. Sipe, and H. M. Driel, “Picosecond and nanosecond laser-induced second-harmonic generation from centrosymmetric semiconductors,” Phys. Rev. B 31, 5543–5546 (1985).
    [CrossRef]
  23. D. A. Weinberger and R. W. Terhune, “Electric field induced harmonic generation in fibers,” to be presented at the 1987 Conference on Lasers and Electro-Optics, Baltimore, Md., April 27–May 1,1987.
  24. P. D. Maker and R. W. Terhune, “Study of optical effects due to an induced polarization third order in the electric field strength,” Phys. Rev. 137, A801–A818 (1985).
    [CrossRef]
  25. R. H. Stolen and Chinlon Lin, “Self-phase-modulation in silica optical fibers,” Phys. Rev. A 17, 1448–1453 (1978).
    [CrossRef]
  26. M. D. Levenson, “Feasibility of measuring the nonlinear index of refraction by third order frequency mixing,” IEEE J, Quantum Electron. QE-19, 110–115 (1974).
    [CrossRef]
  27. R. H. Stolen, E. P. Ippen, and A. R. Tynes, “Raman oscillations in optical waveguides,” Appl. Phys. Lett. 20, 62–64 (1972).
    [CrossRef]
  28. R. H. Stolen and E. P. Ippen, “Raman gain in glass optical waveguides,” Appl. Phys. Lett. 22, 276–278 (1973).
    [CrossRef]
  29. R. H. Stolen, AT&T Bell Laboratories, Holmdel, New Jersey 07733 (personal communication).

1986 (3)

P. Guyot-Sionnest, W. Chen, and Y. R. Shen, “General considerations on optical second-harmonic generation from surfaces and interfaces,” Phys. Rev. B 33, 8254–8263 (1986).
[CrossRef]

J. M. Hicks, K. Kemnitz, and K. B. Eisenthal, “Studies of liquid surfaces by second harmonic generation,” J. Phys. Chem. 90, 560–562 (1986).
[CrossRef]

U. Osterberg and W. Margulis, “Dye laser pumped by Nd:YAG laser pulses frequency doubled in a glass optical fiber,” Opt. Lett. 8, 516–518 (1986); “Experimental studies on efficient doubling in glass optical fibers,” Opt. Lett. 12, 57–59 (1987).
[CrossRef]

1985 (3)

J. A. Litwin, J. E. Sipe, and H. M. Driel, “Picosecond and nanosecond laser-induced second-harmonic generation from centrosymmetric semiconductors,” Phys. Rev. B 31, 5543–5546 (1985).
[CrossRef]

P. D. Maker and R. W. Terhune, “Study of optical effects due to an induced polarization third order in the electric field strength,” Phys. Rev. 137, A801–A818 (1985).
[CrossRef]

B. Dick, A. Gierulski, and G. Marowsky, “Determination of the nonlinear optical susceptibility χ(2) of surface layers by sum and difference frequency generation in reflection and transmission,” Appl. Phys. B 38, 107–116 (1985).
[CrossRef]

1984 (1)

1983 (1)

T. F. Heinz, H. W. K. Tom, and Y. R. Shen, “Determination of molecular orientation of monolayer absorbates by optical second-harmonic generation,” Phys. Rev A 28, 1883–1885 (1983).
[CrossRef]

1982 (2)

T. F. Heinz, C. K. Chen, D. Richard, and Y. R. Shen, “Spectroscopy of molecular monolayers by resonant second-harmonic generation,” Phys. Rev. Lett. 48, 478–481 (1982).
[CrossRef]

Y. Ohmori and Y. Sasaki, “Two-wave sum-frequency light generation in optical fibers,” IEEE J. Quantum Electron. QE-18, 758–762 (1982).
[CrossRef]

1981 (1)

Y. Sasaki and Y. Ohmori, “Phase-matched sum-frequency light generation in optical fibers,” Appl. Phys. Lett. 39, 466–468 (1981).
[CrossRef]

1978 (1)

R. H. Stolen and Chinlon Lin, “Self-phase-modulation in silica optical fibers,” Phys. Rev. A 17, 1448–1453 (1978).
[CrossRef]

1976 (1)

1975 (1)

R. H. Stolen, “Phase-matched-stimulated four-photon mixing in silica-fiber waveguides,” IEEE J. Quantum Electron. QE-l1, 100–103 (1975).
[CrossRef]

1974 (1)

M. D. Levenson, “Feasibility of measuring the nonlinear index of refraction by third order frequency mixing,” IEEE J, Quantum Electron. QE-19, 110–115 (1974).
[CrossRef]

1973 (1)

R. H. Stolen and E. P. Ippen, “Raman gain in glass optical waveguides,” Appl. Phys. Lett. 22, 276–278 (1973).
[CrossRef]

1972 (1)

R. H. Stolen, E. P. Ippen, and A. R. Tynes, “Raman oscillations in optical waveguides,” Appl. Phys. Lett. 20, 62–64 (1972).
[CrossRef]

1969 (1)

C. C. Wang, “Second-harmonic generation of light at the boundary of an isotropic medium,” Phys. Rev. 178, 1457–1461 (1969).
[CrossRef]

1968 (1)

N. Bloembergen, R. K. Chang, S. S. Jha, and C. H. Lee, “Optical second-harmonic generation in reflection from media with inversion symmetry,” Phys. Rev. 174, 813–822 (1968).
[CrossRef]

1967 (1)

J. E. Bjorkholm and A. E. Siegman, “Accurate cw measurements of optical second-harmonic generation in ammonium dihydrogen phosphate and calcite,” Phys. Rev. 154, 851–860 (1967).
[CrossRef]

1963 (1)

P. S. Pershan, “Nonlinear optical properties of solids: energy considerations,” Phys. Rev. 130, 919–929 (1963). In this reference Eqs. (23) and (24) are derived through defining separate coefficients for the magnetic-dipole and quadrupole susceptibilities.
[CrossRef]

1962 (1)

R. W. Terhune, P. D. Maker, and C. M. Savage, “Optical harmonic generation in calcite,” Phys. Rev. Lett. 8, 404–406 (1962).
[CrossRef]

Bjorkholm, J. E.

J. E. Bjorkholm and A. E. Siegman, “Accurate cw measurements of optical second-harmonic generation in ammonium dihydrogen phosphate and calcite,” Phys. Rev. 154, 851–860 (1967).
[CrossRef]

Bloembergen, N.

N. Bloembergen, R. K. Chang, S. S. Jha, and C. H. Lee, “Optical second-harmonic generation in reflection from media with inversion symmetry,” Phys. Rev. 174, 813–822 (1968).
[CrossRef]

N. Bloembergen, Nonlinear Optics (Benjamin, New York, 1965).

Chang, R. K.

N. Bloembergen, R. K. Chang, S. S. Jha, and C. H. Lee, “Optical second-harmonic generation in reflection from media with inversion symmetry,” Phys. Rev. 174, 813–822 (1968).
[CrossRef]

Chen, C. K.

T. F. Heinz, C. K. Chen, D. Richard, and Y. R. Shen, “Spectroscopy of molecular monolayers by resonant second-harmonic generation,” Phys. Rev. Lett. 48, 478–481 (1982).
[CrossRef]

Chen, W.

P. Guyot-Sionnest, W. Chen, and Y. R. Shen, “General considerations on optical second-harmonic generation from surfaces and interfaces,” Phys. Rev. B 33, 8254–8263 (1986).
[CrossRef]

Dick, B.

B. Dick, A. Gierulski, and G. Marowsky, “Determination of the nonlinear optical susceptibility χ(2) of surface layers by sum and difference frequency generation in reflection and transmission,” Appl. Phys. B 38, 107–116 (1985).
[CrossRef]

Driel, H. M.

J. A. Litwin, J. E. Sipe, and H. M. Driel, “Picosecond and nanosecond laser-induced second-harmonic generation from centrosymmetric semiconductors,” Phys. Rev. B 31, 5543–5546 (1985).
[CrossRef]

Eisenthal, K. B.

J. M. Hicks, K. Kemnitz, and K. B. Eisenthal, “Studies of liquid surfaces by second harmonic generation,” J. Phys. Chem. 90, 560–562 (1986).
[CrossRef]

Fersing, L.

J. M. Gabriagues and L. Fersing, “Second harmonic generation in optical fibers,” presented at the Thirteenth International Quantum Electronics Conference, Anaheim, California, June 18–21, 1984.

Fleming, J. W.

Gabriagues, J. M.

J. M. Gabriagues and L. Fersing, “Second harmonic generation in optical fibers,” presented at the Thirteenth International Quantum Electronics Conference, Anaheim, California, June 18–21, 1984.

Gierulski, A.

B. Dick, A. Gierulski, and G. Marowsky, “Determination of the nonlinear optical susceptibility χ(2) of surface layers by sum and difference frequency generation in reflection and transmission,” Appl. Phys. B 38, 107–116 (1985).
[CrossRef]

Guyot-Sionnest, P.

P. Guyot-Sionnest, W. Chen, and Y. R. Shen, “General considerations on optical second-harmonic generation from surfaces and interfaces,” Phys. Rev. B 33, 8254–8263 (1986).
[CrossRef]

Heinz, T. F.

T. F. Heinz, H. W. K. Tom, and Y. R. Shen, “Determination of molecular orientation of monolayer absorbates by optical second-harmonic generation,” Phys. Rev A 28, 1883–1885 (1983).
[CrossRef]

T. F. Heinz, C. K. Chen, D. Richard, and Y. R. Shen, “Spectroscopy of molecular monolayers by resonant second-harmonic generation,” Phys. Rev. Lett. 48, 478–481 (1982).
[CrossRef]

Hicks, J. M.

J. M. Hicks, K. Kemnitz, and K. B. Eisenthal, “Studies of liquid surfaces by second harmonic generation,” J. Phys. Chem. 90, 560–562 (1986).
[CrossRef]

Ippen, E. P.

R. H. Stolen and E. P. Ippen, “Raman gain in glass optical waveguides,” Appl. Phys. Lett. 22, 276–278 (1973).
[CrossRef]

R. H. Stolen, E. P. Ippen, and A. R. Tynes, “Raman oscillations in optical waveguides,” Appl. Phys. Lett. 20, 62–64 (1972).
[CrossRef]

Jha, S. S.

N. Bloembergen, R. K. Chang, S. S. Jha, and C. H. Lee, “Optical second-harmonic generation in reflection from media with inversion symmetry,” Phys. Rev. 174, 813–822 (1968).
[CrossRef]

Kemnitz, K.

J. M. Hicks, K. Kemnitz, and K. B. Eisenthal, “Studies of liquid surfaces by second harmonic generation,” J. Phys. Chem. 90, 560–562 (1986).
[CrossRef]

Landau, L. D.

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, New York, 1960). Chapter 11, on the electrostatics of dielectrics, provides a starting point. We know of no reference that treats the problem fully, particularly when the material properties are a function of position. In the static case, the analysis can be outlined as follows:δ (total free energy)=∫Volume[F(E+δE)-F(E)]dr≡-∫Volume(P·δE)dr,where F is the free-energy density and P is the effective polarization. Using a near local model, we are concerned with a term of the form F= ∊0χEiEj∇kEl. The expression for F(E+ δE) will have three terms associated with the changes in Ei, Ej, and El. The δEl term will have the form χEiEj∇k(χEiEj). This term can be integrated by parts to give δEl∇k(χEiEj). The value of the term δEl(χEiEj) will be zero at all end points of the integration, as δE is arbitrary and can be chosen to be highly localized. With this change the desired identifications with P can be made. Beyond this, the derivation has to be extended to using the time-averaged free energy, 〈F〉, expressed in terms of time-dependent field amplitudes.

Lee, C. H.

N. Bloembergen, R. K. Chang, S. S. Jha, and C. H. Lee, “Optical second-harmonic generation in reflection from media with inversion symmetry,” Phys. Rev. 174, 813–822 (1968).
[CrossRef]

Leibert, W. N.

Levenson, M. D.

M. D. Levenson, “Feasibility of measuring the nonlinear index of refraction by third order frequency mixing,” IEEE J, Quantum Electron. QE-19, 110–115 (1974).
[CrossRef]

Lifshitz, E. M.

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, New York, 1960). Chapter 11, on the electrostatics of dielectrics, provides a starting point. We know of no reference that treats the problem fully, particularly when the material properties are a function of position. In the static case, the analysis can be outlined as follows:δ (total free energy)=∫Volume[F(E+δE)-F(E)]dr≡-∫Volume(P·δE)dr,where F is the free-energy density and P is the effective polarization. Using a near local model, we are concerned with a term of the form F= ∊0χEiEj∇kEl. The expression for F(E+ δE) will have three terms associated with the changes in Ei, Ej, and El. The δEl term will have the form χEiEj∇k(χEiEj). This term can be integrated by parts to give δEl∇k(χEiEj). The value of the term δEl(χEiEj) will be zero at all end points of the integration, as δE is arbitrary and can be chosen to be highly localized. With this change the desired identifications with P can be made. Beyond this, the derivation has to be extended to using the time-averaged free energy, 〈F〉, expressed in terms of time-dependent field amplitudes.

Lin, Chinlon

R. H. Stolen and Chinlon Lin, “Self-phase-modulation in silica optical fibers,” Phys. Rev. A 17, 1448–1453 (1978).
[CrossRef]

Litwin, J. A.

J. A. Litwin, J. E. Sipe, and H. M. Driel, “Picosecond and nanosecond laser-induced second-harmonic generation from centrosymmetric semiconductors,” Phys. Rev. B 31, 5543–5546 (1985).
[CrossRef]

Love, J. D.

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall, New York, 1983).

Maker, P. D.

P. D. Maker and R. W. Terhune, “Study of optical effects due to an induced polarization third order in the electric field strength,” Phys. Rev. 137, A801–A818 (1985).
[CrossRef]

R. W. Terhune, P. D. Maker, and C. M. Savage, “Optical harmonic generation in calcite,” Phys. Rev. Lett. 8, 404–406 (1962).
[CrossRef]

Margulis, W.

U. Osterberg and W. Margulis, “Dye laser pumped by Nd:YAG laser pulses frequency doubled in a glass optical fiber,” Opt. Lett. 8, 516–518 (1986); “Experimental studies on efficient doubling in glass optical fibers,” Opt. Lett. 12, 57–59 (1987).
[CrossRef]

Marowsky, G.

B. Dick, A. Gierulski, and G. Marowsky, “Determination of the nonlinear optical susceptibility χ(2) of surface layers by sum and difference frequency generation in reflection and transmission,” Appl. Phys. B 38, 107–116 (1985).
[CrossRef]

Ohmori, Y.

Y. Ohmori and Y. Sasaki, “Two-wave sum-frequency light generation in optical fibers,” IEEE J. Quantum Electron. QE-18, 758–762 (1982).
[CrossRef]

Y. Sasaki and Y. Ohmori, “Phase-matched sum-frequency light generation in optical fibers,” Appl. Phys. Lett. 39, 466–468 (1981).
[CrossRef]

Osterberg, U.

U. Osterberg and W. Margulis, “Dye laser pumped by Nd:YAG laser pulses frequency doubled in a glass optical fiber,” Opt. Lett. 8, 516–518 (1986); “Experimental studies on efficient doubling in glass optical fibers,” Opt. Lett. 12, 57–59 (1987).
[CrossRef]

Pershan, P. S.

P. S. Pershan, “Nonlinear optical properties of solids: energy considerations,” Phys. Rev. 130, 919–929 (1963). In this reference Eqs. (23) and (24) are derived through defining separate coefficients for the magnetic-dipole and quadrupole susceptibilities.
[CrossRef]

Richard, D.

T. F. Heinz, C. K. Chen, D. Richard, and Y. R. Shen, “Spectroscopy of molecular monolayers by resonant second-harmonic generation,” Phys. Rev. Lett. 48, 478–481 (1982).
[CrossRef]

Sasaki, Y.

Y. Ohmori and Y. Sasaki, “Two-wave sum-frequency light generation in optical fibers,” IEEE J. Quantum Electron. QE-18, 758–762 (1982).
[CrossRef]

Y. Sasaki and Y. Ohmori, “Phase-matched sum-frequency light generation in optical fibers,” Appl. Phys. Lett. 39, 466–468 (1981).
[CrossRef]

Savage, C. M.

R. W. Terhune, P. D. Maker, and C. M. Savage, “Optical harmonic generation in calcite,” Phys. Rev. Lett. 8, 404–406 (1962).
[CrossRef]

Shen, Y. R.

P. Guyot-Sionnest, W. Chen, and Y. R. Shen, “General considerations on optical second-harmonic generation from surfaces and interfaces,” Phys. Rev. B 33, 8254–8263 (1986).
[CrossRef]

T. F. Heinz, H. W. K. Tom, and Y. R. Shen, “Determination of molecular orientation of monolayer absorbates by optical second-harmonic generation,” Phys. Rev A 28, 1883–1885 (1983).
[CrossRef]

T. F. Heinz, C. K. Chen, D. Richard, and Y. R. Shen, “Spectroscopy of molecular monolayers by resonant second-harmonic generation,” Phys. Rev. Lett. 48, 478–481 (1982).
[CrossRef]

Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984).

Siegman, A. E.

J. E. Bjorkholm and A. E. Siegman, “Accurate cw measurements of optical second-harmonic generation in ammonium dihydrogen phosphate and calcite,” Phys. Rev. 154, 851–860 (1967).
[CrossRef]

Sipe, J. E.

J. A. Litwin, J. E. Sipe, and H. M. Driel, “Picosecond and nanosecond laser-induced second-harmonic generation from centrosymmetric semiconductors,” Phys. Rev. B 31, 5543–5546 (1985).
[CrossRef]

Snyder, A. W.

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall, New York, 1983).

Stolen, R. H.

R. H. Stolen and Chinlon Lin, “Self-phase-modulation in silica optical fibers,” Phys. Rev. A 17, 1448–1453 (1978).
[CrossRef]

R. H. Stolen and W. N. Leibert, “Optical fiber modes using stimulated four photon mixing,” Appl. Opt. 15, 239–243 (1976).
[CrossRef] [PubMed]

R. H. Stolen, “Phase-matched-stimulated four-photon mixing in silica-fiber waveguides,” IEEE J. Quantum Electron. QE-l1, 100–103 (1975).
[CrossRef]

R. H. Stolen and E. P. Ippen, “Raman gain in glass optical waveguides,” Appl. Phys. Lett. 22, 276–278 (1973).
[CrossRef]

R. H. Stolen, E. P. Ippen, and A. R. Tynes, “Raman oscillations in optical waveguides,” Appl. Phys. Lett. 20, 62–64 (1972).
[CrossRef]

R. H. Stolen, AT&T Bell Laboratories, Holmdel, New Jersey 07733 (personal communication).

Terhune, R. W.

P. D. Maker and R. W. Terhune, “Study of optical effects due to an induced polarization third order in the electric field strength,” Phys. Rev. 137, A801–A818 (1985).
[CrossRef]

R. W. Terhune, P. D. Maker, and C. M. Savage, “Optical harmonic generation in calcite,” Phys. Rev. Lett. 8, 404–406 (1962).
[CrossRef]

D. A. Weinberger and R. W. Terhune, “Electric field induced harmonic generation in fibers,” to be presented at the 1987 Conference on Lasers and Electro-Optics, Baltimore, Md., April 27–May 1,1987.

Tom, H. W. K.

T. F. Heinz, H. W. K. Tom, and Y. R. Shen, “Determination of molecular orientation of monolayer absorbates by optical second-harmonic generation,” Phys. Rev A 28, 1883–1885 (1983).
[CrossRef]

Tynes, A. R.

R. H. Stolen, E. P. Ippen, and A. R. Tynes, “Raman oscillations in optical waveguides,” Appl. Phys. Lett. 20, 62–64 (1972).
[CrossRef]

Wang, C. C.

C. C. Wang, “Second-harmonic generation of light at the boundary of an isotropic medium,” Phys. Rev. 178, 1457–1461 (1969).
[CrossRef]

Weinberger, D. A.

D. A. Weinberger and R. W. Terhune, “Electric field induced harmonic generation in fibers,” to be presented at the 1987 Conference on Lasers and Electro-Optics, Baltimore, Md., April 27–May 1,1987.

Appl. Opt. (2)

Appl. Phys. B (1)

B. Dick, A. Gierulski, and G. Marowsky, “Determination of the nonlinear optical susceptibility χ(2) of surface layers by sum and difference frequency generation in reflection and transmission,” Appl. Phys. B 38, 107–116 (1985).
[CrossRef]

Appl. Phys. Lett. (3)

Y. Sasaki and Y. Ohmori, “Phase-matched sum-frequency light generation in optical fibers,” Appl. Phys. Lett. 39, 466–468 (1981).
[CrossRef]

R. H. Stolen, E. P. Ippen, and A. R. Tynes, “Raman oscillations in optical waveguides,” Appl. Phys. Lett. 20, 62–64 (1972).
[CrossRef]

R. H. Stolen and E. P. Ippen, “Raman gain in glass optical waveguides,” Appl. Phys. Lett. 22, 276–278 (1973).
[CrossRef]

IEEE J, Quantum Electron. (1)

M. D. Levenson, “Feasibility of measuring the nonlinear index of refraction by third order frequency mixing,” IEEE J, Quantum Electron. QE-19, 110–115 (1974).
[CrossRef]

IEEE J. Quantum Electron. (2)

Y. Ohmori and Y. Sasaki, “Two-wave sum-frequency light generation in optical fibers,” IEEE J. Quantum Electron. QE-18, 758–762 (1982).
[CrossRef]

R. H. Stolen, “Phase-matched-stimulated four-photon mixing in silica-fiber waveguides,” IEEE J. Quantum Electron. QE-l1, 100–103 (1975).
[CrossRef]

J. Phys. Chem. (1)

J. M. Hicks, K. Kemnitz, and K. B. Eisenthal, “Studies of liquid surfaces by second harmonic generation,” J. Phys. Chem. 90, 560–562 (1986).
[CrossRef]

Opt. Lett. (1)

U. Osterberg and W. Margulis, “Dye laser pumped by Nd:YAG laser pulses frequency doubled in a glass optical fiber,” Opt. Lett. 8, 516–518 (1986); “Experimental studies on efficient doubling in glass optical fibers,” Opt. Lett. 12, 57–59 (1987).
[CrossRef]

Phys. Rev A (1)

T. F. Heinz, H. W. K. Tom, and Y. R. Shen, “Determination of molecular orientation of monolayer absorbates by optical second-harmonic generation,” Phys. Rev A 28, 1883–1885 (1983).
[CrossRef]

Phys. Rev. (5)

N. Bloembergen, R. K. Chang, S. S. Jha, and C. H. Lee, “Optical second-harmonic generation in reflection from media with inversion symmetry,” Phys. Rev. 174, 813–822 (1968).
[CrossRef]

C. C. Wang, “Second-harmonic generation of light at the boundary of an isotropic medium,” Phys. Rev. 178, 1457–1461 (1969).
[CrossRef]

P. D. Maker and R. W. Terhune, “Study of optical effects due to an induced polarization third order in the electric field strength,” Phys. Rev. 137, A801–A818 (1985).
[CrossRef]

J. E. Bjorkholm and A. E. Siegman, “Accurate cw measurements of optical second-harmonic generation in ammonium dihydrogen phosphate and calcite,” Phys. Rev. 154, 851–860 (1967).
[CrossRef]

P. S. Pershan, “Nonlinear optical properties of solids: energy considerations,” Phys. Rev. 130, 919–929 (1963). In this reference Eqs. (23) and (24) are derived through defining separate coefficients for the magnetic-dipole and quadrupole susceptibilities.
[CrossRef]

Phys. Rev. A (1)

R. H. Stolen and Chinlon Lin, “Self-phase-modulation in silica optical fibers,” Phys. Rev. A 17, 1448–1453 (1978).
[CrossRef]

Phys. Rev. B (2)

P. Guyot-Sionnest, W. Chen, and Y. R. Shen, “General considerations on optical second-harmonic generation from surfaces and interfaces,” Phys. Rev. B 33, 8254–8263 (1986).
[CrossRef]

J. A. Litwin, J. E. Sipe, and H. M. Driel, “Picosecond and nanosecond laser-induced second-harmonic generation from centrosymmetric semiconductors,” Phys. Rev. B 31, 5543–5546 (1985).
[CrossRef]

Phys. Rev. Lett. (2)

R. W. Terhune, P. D. Maker, and C. M. Savage, “Optical harmonic generation in calcite,” Phys. Rev. Lett. 8, 404–406 (1962).
[CrossRef]

T. F. Heinz, C. K. Chen, D. Richard, and Y. R. Shen, “Spectroscopy of molecular monolayers by resonant second-harmonic generation,” Phys. Rev. Lett. 48, 478–481 (1982).
[CrossRef]

Other (7)

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall, New York, 1983).

J. M. Gabriagues and L. Fersing, “Second harmonic generation in optical fibers,” presented at the Thirteenth International Quantum Electronics Conference, Anaheim, California, June 18–21, 1984.

N. Bloembergen, Nonlinear Optics (Benjamin, New York, 1965).

Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984).

D. A. Weinberger and R. W. Terhune, “Electric field induced harmonic generation in fibers,” to be presented at the 1987 Conference on Lasers and Electro-Optics, Baltimore, Md., April 27–May 1,1987.

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, New York, 1960). Chapter 11, on the electrostatics of dielectrics, provides a starting point. We know of no reference that treats the problem fully, particularly when the material properties are a function of position. In the static case, the analysis can be outlined as follows:δ (total free energy)=∫Volume[F(E+δE)-F(E)]dr≡-∫Volume(P·δE)dr,where F is the free-energy density and P is the effective polarization. Using a near local model, we are concerned with a term of the form F= ∊0χEiEj∇kEl. The expression for F(E+ δE) will have three terms associated with the changes in Ei, Ej, and El. The δEl term will have the form χEiEj∇k(χEiEj). This term can be integrated by parts to give δEl∇k(χEiEj). The value of the term δEl(χEiEj) will be zero at all end points of the integration, as δE is arbitrary and can be chosen to be highly localized. With this change the desired identifications with P can be made. Beyond this, the derivation has to be extended to using the time-averaged free energy, 〈F〉, expressed in terms of time-dependent field amplitudes.

R. H. Stolen, AT&T Bell Laboratories, Holmdel, New Jersey 07733 (personal communication).

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Figures (4)

Fig. 1
Fig. 1

Dependence on frequency of the effective index of refraction, βk/k, of fibers with a 1.5-μm-radiuscore containing different mole fractions of GeO2. The dashed curves are the LP01 mode data replotted with the frequency scale doubled. The intersections with the curves for other modes provide the second-harmonic frequencies for which one can realize index matching with the input radiation in the LP01 mode. Of the intersections shown, the effects discussed in this paper can provide coupling only to the LP12 and LP31 modes.

Fig. 2
Fig. 2

The same data as for Fig. 1 but for fibers with 3-μm-radius cores.

Fig. 3
Fig. 3

Dependence of the frequency for index-matched doubling of radiation in the LP01 mode on the mole fraction GeO2 in the fiber core, for different radius cores. A minimum mole fraction near 0.1 (δn = 0.15) is required for fibers with larger cores. Higher values are needed for fibers with small-radius cores. The dependence of the matching frequency on the mole fraction of GeO2 can be seen to be small.

Fig. 4
Fig. 4

Dependence of the frequency for index-matched doubling of radiation in the LP01 mode on the core radius, for different mole fractions of GeO2. The amount of harmonic generation in fibers would most likely be limited by variations in the core radius. In this case, the frequency range over which harmonics would be observed, typically several inverse centimeters, is inversely proportional to the slope of the above curves.

Tables (3)

Tables Icon

Table 1 Mode Parameters for Index-Matched Doubling

Tables Icon

Table 2 Conversion-Efficiency Calculations for Index-Matched Doubling

Tables Icon

Table 3 Mode Parameters for Non-Index-Matched Doublinga

Equations (49)

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E m ( ω ) = a m ( ω ) e m ( ω , x , y ) exp [ i β m ( ω ) z ] exp ( - i ω z )
H m ( ω ) = a m ( ω ) h m ( ω , x , y ) exp [ i β m ( ω ) z ] exp ( - i ω t ) .
N m ( ω ) = ½ ( e m × h m * ) · z ^ d x d y ,
N m ( ω ) ½ 0 n core 2 ( v g ) m - - ( n / n core ) 2 × e m ( ω , x , y ) · e * m ( ω , x , y ) d x d y ,
- - ( n / n core ) 2 [ c m e m ( ω , x , y ) · f m e m * ( ω , x , y ) ] d x d y 1 ,
f m [ ½ 0 n core 2 ( v g ) m ] 1 / 2 .
d a m ( ω , z ) / d z = - ¼ - - e m * · J ( x , y , z ) × exp [ i ω t - i β m ( ω ) z ] d x d y .
J ( x , y , z ) = P NL ( x , y , z ) / t = - i ( ω 1 + ω 2 ) P NL
P NL = a 1 a 2 p NL ( ω 1 + ω 2 , x , y ) exp [ i ( β 1 + β 2 ) z ] × exp [ - i ( ω 1 + ω 2 ) t ] ,
d a 3 ( ω 3 , z ) / d z = i a 1 a 2 l NL exp ( i Δ β z ) ,
1 / l NL ω 3 4 - - e 3 * · p NL d x d y .
a 3 = a 1 a 2 Δ β l NL [ exp ( i Δ β z ) - 1 ]
a 3 a 3 * = a 1 a 1 * a 2 a 2 * z 2 l NL l NL * sin 2 ( Δ β z / 2 ) ( Δ β z / 2 ) 2 .
a 3 a 3 * ( Δ β z = m π ) = a 1 a 1 * a 2 a 2 * m 2 l NL l NL * 4 ( Δ β ) 2 .
a 3 a 3 * ( Δ β z 1 ) = a 1 a 1 * a 2 a 2 * z 2 l NL l NL * .
δ Δ β = Δ β ρ δ ρ + Δ β M δ M + Δ β ω δ ( ω - ω match ) .
a 3 a 3 * = a 1 a 1 * a 2 a 2 * 2 Δ β 2 l NL l NL * L l seg .
I ( ω L ) seg 0 a 3 a 3 * d ω L = a 1 a 1 * a 2 a 2 * l seg l NL l NL * 2 π Δ β / ω L .
I ( ω L ) = a 1 a 1 * a 2 a 2 * L l NL l NL * 2 π Δ β / ω L .
δ ω match = ( Δ β / ln ρ ) δ ln ρ + ( Δ β / ln M ) δ ln M Δ β / ω .
P k ( 2 ω ) / 0 = ( δ - β - 2 γ ) E i ( ω ) i E k ( ω ) + β E k ( ω ) i E i ( ω ) + 2 γ E i ( ω ) k E i ( ω ) .
J = P / t + c ( × M ) - ( · Q ) / t ,
- F / 0 = 2 χ ( 2 ) i j k l ( - ω 3 , ω 2 , ω 1 ) E * i ( ω 3 ) E j ( ω 2 ) k E l ( ω 1 ) + 2 χ ( 2 ) i l k j ( - ω 3 , ω 1 , ω 2 ) E * i ( ω 3 ) E l ( ω 1 ) k E j ( ω 2 ) + 2 χ ( 2 ) j l k i ( ω 2 , ω 1 , - ω 3 ) E * j ( ω 2 ) E l ( ω 1 ) k E * i ( ω 3 ) + c . c . ,
P i ( ω 3 ) / 0 = 2 [ χ ( 2 ) i j k l ( - ω 3 , ω 2 , ω 1 ) - χ ( 2 ) j l k i ( ω 2 , ω 1 , - ω 3 ) ] × E j ( ω 2 ) k E l ( ω 1 ) + { if ω 1 ω 2 2 [ χ ( 2 ) i l k j ( - ω 3 , ω 1 , ω 2 ) - χ ( 2 ) j l k i ( ω 2 , ω 1 , - ω 3 ) ] × E l ( ω 1 ) k E j ( ω 2 ) } .
δ = 2 [ χ ( 2 ) x x x x ( - 2 ω , ω , ω ) - χ ( 2 ) x x x x ( ω , ω , - 2 ω ) ] , β = 2 [ χ ( 2 ) x x y y ( - 2 ω , ω , ω ) - χ ( 2 ) x y y x ( ω , ω , - 2 ω ) ] , 2 γ = 2 [ χ ( 2 ) y x y x ( - 2 ω , ω , ω ) - χ ( 2 ) x x y y ( ω , ω , - 2 ω ) ] .
1 / l 2 γ = [ ¼ 0 ( ω 1 + ω 2 ) ] [ 2 γ core / ( f 1 f 2 f 3 ) ] O 2 γ ,
O 2 γ = f 1 f 2 f 3 - - ( γ / γ core ) ( e 3 * ) k ( e 2 ) i k ( e 1 ) d x d y [ + the same term with ( 1 2 ) if ω 1 ω 2 or mode 1 mode 2 ]
1 / l δ - β - 2 γ = [ ¼ 0 ( ω 1 + ω 2 ) ] [ ( δ - β - 2 γ core ) / ( f 1 f 2 f 3 ) ] O δ - β - 2 γ ,
O δ - β - 2 γ = f 1 f 2 f 3 - - ( δ - β - 2 γ ) / ( δ - β - 2 γ ) core × ( e 3 * ) k ( e 2 ) i ( e 1 ) k d x d y [ + the same term with ( 1 2 ) if ω 1 ω 2 or mode 1 mode 2 ] .
O 2 γ = f 1 f 2 f 3 - - ½ ( γ / γ core ) ( 2 · e 3 * ) ( e 1 ) i ( e 1 ) i d x d y + 2 i β 1 f 1 f 2 f 3 - - ½ ( γ / γ core ) ( e 3 * ) z ( e 1 ) i ( e 1 ) i d x d y ,
O 2 γ = f 1 f 2 f 3 0 2 π ½ ( γ / γ core ) ( e 3 * ) r ( e 1 ) i ( e 1 ) i r = ρ - η r = ρ + η ρ d ϕ - f 1 f 2 f 3 - - ½ ( γ / γ core ) ( 2 · e 3 * ) ( e 1 ) i ( e 1 ) i d x d y + 2 i β 1 f 1 f 2 f 3 - - ½ ( γ / γ core ) ( e 3 * ) z ( e 1 ) i ( e 1 ) i d x d y .
O 2 γ = K 1 O A + K 2 O D + ( Δ β / β 3 ) O z ,
Δ β = 2 β 1 - β 3 , K 1 = [ 1 - ( γ clad / γ core ) ( n core 4 / n clad 4 ) 1 ( n core 2 / n clad 2 ) 3 ] / 2 ,
K 2 = [ 1 - ( γ clad / γ core ) ( n core 2 / n clad 2 ) 1 ] / 2.
O A f 1 f 2 f 3 0 2 π [ ( e 3 * ) r ( e 1 ) r 2 ] r = ρ - η ρ d ϕ , O D f 1 f 2 f 3 0 2 π { ( e 3 * ) r [ ( e 1 ) z 2 + ( e 1 ) ϕ 2 ] r = ρ - η ρ d ϕ , O s f 1 f 2 f 3 - - ½ ( γ / γ core ) ( e 3 * ) z ( e 1 ) i ( e 1 ) i d x d y .
P r ( 2 ω ) / 0 = A E r ( ω ) 2 + C [ E r 2 ( ω ) + E ϕ 2 ( ω ) + E z 2 ( ω ) ] , P ϕ ( 2 ω ) / 0 = B E r ( ω ) E ϕ ( ω ) , P z ( 2 ω ) / 0 = B E r ( ω ) E z ( ω ) .
1 / l interface = ( 0 ω / 2 f 1 f 2 f 3 ) ( A O A + B O B + C O C ) ,
O B f 1 f 2 f 3 0 2 π { ( e 1 ) r [ ( e 1 ) ϕ ( e 3 * ) ϕ + ( e 1 ) z ( e 3 * ) z ] } r = ρ - η ρ d ϕ .
[ E * ( 2 ω ) ] ρ - η · P ( 2 ω ) = ρ - η ρ + η E * ( 2 ω ) · P ( 2 ) NL d r + γ E * r ( 2 ω ) E i ( ω ) E i ( ω ) r ρ - η r ρ + η ,
A + C = 1 [ E * r ( 2 ω ) E r ( ω ) E r ( ω ) ] r = ρ - η × { ρ - η ρ + η δ E * r ( 2 ω ) E r ( ω ) E r ( ω ) r d r - ρ - η ρ + η E * r ( 2 ω ) E r ( ω ) E r ( ω ) χ ( 2 ) r r r r ( ω , ω , - 2 ω ) r d r + γ E * r ( 2 ω ) E r ( ω ) E r ( ω ) r = ρ - η r = ρ + η } ,
B = 1 [ E * r ( ω ) ] r = ρ - η ρ - η ρ + η β E r ( ω ) r d r ,
C = 1 [ E * r ( ω ) ] r = ρ - η ( - ρ - η ρ + η E * r ( 2 ω ) { [ χ ( 2 ) z z r r ( ω , ω , - 2 ω ) ] / r } d r ) + γ E * r ( 2 ω ) r = ρ - η r = ρ + η } .
P i ( 2 ω ) / 0 = 12 c 1221 ( - 2 ω , 0 , ω , ω ) E j ( 0 ) E j ( ω ) E i ( ω ) + 6 c 1122 ( - 2 ω , 0 , ω , ω ) E i ( 0 ) E j ( ω ) E j ( ω ) .
1 / l E ( 0 ω / 2 f 1 f 2 f 3 ) E dc ( 12 c 1221 O 1221 + 6 c 1122 O 1122 ) ,
O 1221 f 1 f 2 f 3 0 2 π - ( e 3 * ) n ( u ^ dc ) l ( e 1 ) l ( e 1 ) n r d r d ϕ ,
O 1122 f 1 f 2 f 3 0 2 π - ( e 3 * ) n ( u ^ dc ) n ( e 1 ) l ( e 1 ) l r d r d ϕ .
A i B j j C i = A · ( B j j ) C = A i B n n C l + ( 1 / r ) ( B ϕ ) ( A ϕ C r - A r C ϕ ) ,
ν 1 ± ν 2 + ν 3 = 0 ,
δ(totalfreeenergy)=Volume[F(E+δE)-F(E)]dr-Volume(P·δE)dr,

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