Abstract

A general formulation of optical second-harmonic generation in crystal-cored fibers is described. Although pump depletion of the fundamental wave and anisotropy of the core crystal are ignored, the theory and its numerical results provide a practical means of designing frequency doublers that make use of nonlinear-optical fibers. Comparison with previous results is also provided.

© 1992 Optical Society of America

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References

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  1. K. Chikuma and S. Umegaki, “Characteristics of optical second-harmonic generation due to Čerenkov-radiation-type phase matching,” J. Opt. Soc. Am. B 7, 768–775 (1990).
    [Crossref]
  2. K. I. White and B. K. Nayar, “Nonlinear fiber waveguide: the field overlap,” in Digest of Conference on Lasers and Electro-Optics (Optical Society of America, Washington, D.C., 1986), paper WK46, pp. 204–206;“Second-harmonic generation in nonlinear optical fiber waveguides: efficient designs using radiation modes,” J. Opt. Soc. Am. B 5, 317–324 (1988).
  3. N. A. Sanford and J. M. Connors, “Optimization of the Cerenkov sum-frequency generation in proton exchanged Mg:LiNbO3 channel waveguides,” J. Appl. Phys. 65, 1429–1437 (1989).
    [Crossref]
  4. D. Marcuse, “Coupled mode theory of round optical fibers,” Bell Syst. Tech. J. 52, 817–50842 (1973).
    [Crossref]
  5. D. S. Chemla and J. Zyss, eds., Nonlinear Optical Properties of Organic Molecules and Crystals (Academic, Orlando, Fla., 1987), Vols. 1 and 2.
  6. B. K. Nayar, “Nonlinear optical interactions in organic crystal-cored fibers,” ACS Symp. Ser. 233, 153–166 (1983).
    [Crossref]
  7. B. K. Nayar, “Optical fibers with organic crystalline cores,” in Nonlinear Optics: Materials and Devices, C. Flytzanis and J. L. Oudar, eds. (Springer-Verlag, Berlin, 1986), pp. 142–153.
    [Crossref]
  8. S. Umegaki, Y. Takahashi, A. Manabe, and S. Tanaka, “Optical second-harmonic generation in an organic crystal-core fiber,” in Extended Abstracts: Nonlinear Optical Materials, D. A. B. Miller, ed. (Materials Research Society, Pittsburgh, Pa., 1985), pp. 97–99.
  9. S. Umegaki, A. Hiramatsu, Y. Tsukikawa, and S. Tanaka, “Crystal growth of organic materials and optical second-harmonic generation in optical fiber,” in Molecular and Polymeric Optoelectronic Materials, G. Khanarian, ed., Proc. Soc. Photo-Opt. Instrum. Eng.682, 187–190 (1986).
    [Crossref]
  10. P. V. Vidakovic, M. Coquillay, and F. Salin, “N-(4-nitrophenyl)-N-methylamino-aceto-nitril: a new organic material for efficient second harmonic generation in bulk and waveguide configurations. I. Growth, crystal structure and characterization of organic crystal-cored fibers,” J. Opt. Soc. Am. B 4, 998–1012 (1987).
    [Crossref]
  11. K. I. White, B. K. Nayar, and R. Kashap, “Amplification and second-harmonic generation in non-linear fibre waveguides,” Opt. Quantum Electron. 20, 339–342 (1988).
    [Crossref]
  12. P. Kerkoc, Ch. Bosshard, H. Arend, and P. Gunter, “Growth and characterization of 4-(N,N-dimethylamino)-3-acetamidonitrobenzene single-crystal cored fibers,” Appl. Phys. Lett. 54, 487–489 (1989).
    [Crossref]
  13. T. Uemiya, N. Uenishi, Y. Shimizu, S. Okamoto, K. Chikuma, T. Tohma, and S. Umegaki, “Crystal-cored fiber using organic material and focusing properties of generated second-harmonic waves,” in Nonlinear Optical Properties of Materials, H. Schlossberg and R. Wick, eds., in Proc. Soc. Photo-Opt. Instrum. Eng.1148, 207–212 (1989).
    [Crossref]
  14. A. Harada, Y. Okazaki, K. Kamiyama, and S. Umegaki, “Generation of continuous wave blue coherent light from a semiconductor laser using nonlinear optical fiber with an organic core crystal,” in Digest of Conference on Lasers and Electro-Optics (Optical Society of America, Washington, D.C., 1990), paper CFE3, pp. 496–498.
  15. Equation (30b) of Ref. 1 has to be corrected by eliminating the factor K1(W)/J1(U).
  16. R. Morita, T. Kondo, Y. Kaneda, A. Sugihashi, N. Ogasawara, S. Umegaki, and R. Ito, “Dispersion of second-order nonlinear optical coefficient d11 of 2-methyl-4-nitroaniline (MNA),” Jpn. J. Appl. Phys. 27, L1131–L1133 (1988).
    [Crossref]
  17. M. Born and E. Wolf, eds., Principles of Optics (Pergamon, Oxford, 1980), App. III.

1990 (1)

1989 (2)

P. Kerkoc, Ch. Bosshard, H. Arend, and P. Gunter, “Growth and characterization of 4-(N,N-dimethylamino)-3-acetamidonitrobenzene single-crystal cored fibers,” Appl. Phys. Lett. 54, 487–489 (1989).
[Crossref]

N. A. Sanford and J. M. Connors, “Optimization of the Cerenkov sum-frequency generation in proton exchanged Mg:LiNbO3 channel waveguides,” J. Appl. Phys. 65, 1429–1437 (1989).
[Crossref]

1988 (2)

K. I. White, B. K. Nayar, and R. Kashap, “Amplification and second-harmonic generation in non-linear fibre waveguides,” Opt. Quantum Electron. 20, 339–342 (1988).
[Crossref]

R. Morita, T. Kondo, Y. Kaneda, A. Sugihashi, N. Ogasawara, S. Umegaki, and R. Ito, “Dispersion of second-order nonlinear optical coefficient d11 of 2-methyl-4-nitroaniline (MNA),” Jpn. J. Appl. Phys. 27, L1131–L1133 (1988).
[Crossref]

1987 (1)

1983 (1)

B. K. Nayar, “Nonlinear optical interactions in organic crystal-cored fibers,” ACS Symp. Ser. 233, 153–166 (1983).
[Crossref]

1973 (1)

D. Marcuse, “Coupled mode theory of round optical fibers,” Bell Syst. Tech. J. 52, 817–50842 (1973).
[Crossref]

Arend, H.

P. Kerkoc, Ch. Bosshard, H. Arend, and P. Gunter, “Growth and characterization of 4-(N,N-dimethylamino)-3-acetamidonitrobenzene single-crystal cored fibers,” Appl. Phys. Lett. 54, 487–489 (1989).
[Crossref]

Bosshard, Ch.

P. Kerkoc, Ch. Bosshard, H. Arend, and P. Gunter, “Growth and characterization of 4-(N,N-dimethylamino)-3-acetamidonitrobenzene single-crystal cored fibers,” Appl. Phys. Lett. 54, 487–489 (1989).
[Crossref]

Chikuma, K.

K. Chikuma and S. Umegaki, “Characteristics of optical second-harmonic generation due to Čerenkov-radiation-type phase matching,” J. Opt. Soc. Am. B 7, 768–775 (1990).
[Crossref]

T. Uemiya, N. Uenishi, Y. Shimizu, S. Okamoto, K. Chikuma, T. Tohma, and S. Umegaki, “Crystal-cored fiber using organic material and focusing properties of generated second-harmonic waves,” in Nonlinear Optical Properties of Materials, H. Schlossberg and R. Wick, eds., in Proc. Soc. Photo-Opt. Instrum. Eng.1148, 207–212 (1989).
[Crossref]

Connors, J. M.

N. A. Sanford and J. M. Connors, “Optimization of the Cerenkov sum-frequency generation in proton exchanged Mg:LiNbO3 channel waveguides,” J. Appl. Phys. 65, 1429–1437 (1989).
[Crossref]

Coquillay, M.

Gunter, P.

P. Kerkoc, Ch. Bosshard, H. Arend, and P. Gunter, “Growth and characterization of 4-(N,N-dimethylamino)-3-acetamidonitrobenzene single-crystal cored fibers,” Appl. Phys. Lett. 54, 487–489 (1989).
[Crossref]

Harada, A.

A. Harada, Y. Okazaki, K. Kamiyama, and S. Umegaki, “Generation of continuous wave blue coherent light from a semiconductor laser using nonlinear optical fiber with an organic core crystal,” in Digest of Conference on Lasers and Electro-Optics (Optical Society of America, Washington, D.C., 1990), paper CFE3, pp. 496–498.

Hiramatsu, A.

S. Umegaki, A. Hiramatsu, Y. Tsukikawa, and S. Tanaka, “Crystal growth of organic materials and optical second-harmonic generation in optical fiber,” in Molecular and Polymeric Optoelectronic Materials, G. Khanarian, ed., Proc. Soc. Photo-Opt. Instrum. Eng.682, 187–190 (1986).
[Crossref]

Ito, R.

R. Morita, T. Kondo, Y. Kaneda, A. Sugihashi, N. Ogasawara, S. Umegaki, and R. Ito, “Dispersion of second-order nonlinear optical coefficient d11 of 2-methyl-4-nitroaniline (MNA),” Jpn. J. Appl. Phys. 27, L1131–L1133 (1988).
[Crossref]

Kamiyama, K.

A. Harada, Y. Okazaki, K. Kamiyama, and S. Umegaki, “Generation of continuous wave blue coherent light from a semiconductor laser using nonlinear optical fiber with an organic core crystal,” in Digest of Conference on Lasers and Electro-Optics (Optical Society of America, Washington, D.C., 1990), paper CFE3, pp. 496–498.

Kaneda, Y.

R. Morita, T. Kondo, Y. Kaneda, A. Sugihashi, N. Ogasawara, S. Umegaki, and R. Ito, “Dispersion of second-order nonlinear optical coefficient d11 of 2-methyl-4-nitroaniline (MNA),” Jpn. J. Appl. Phys. 27, L1131–L1133 (1988).
[Crossref]

Kashap, R.

K. I. White, B. K. Nayar, and R. Kashap, “Amplification and second-harmonic generation in non-linear fibre waveguides,” Opt. Quantum Electron. 20, 339–342 (1988).
[Crossref]

Kerkoc, P.

P. Kerkoc, Ch. Bosshard, H. Arend, and P. Gunter, “Growth and characterization of 4-(N,N-dimethylamino)-3-acetamidonitrobenzene single-crystal cored fibers,” Appl. Phys. Lett. 54, 487–489 (1989).
[Crossref]

Kondo, T.

R. Morita, T. Kondo, Y. Kaneda, A. Sugihashi, N. Ogasawara, S. Umegaki, and R. Ito, “Dispersion of second-order nonlinear optical coefficient d11 of 2-methyl-4-nitroaniline (MNA),” Jpn. J. Appl. Phys. 27, L1131–L1133 (1988).
[Crossref]

Manabe, A.

S. Umegaki, Y. Takahashi, A. Manabe, and S. Tanaka, “Optical second-harmonic generation in an organic crystal-core fiber,” in Extended Abstracts: Nonlinear Optical Materials, D. A. B. Miller, ed. (Materials Research Society, Pittsburgh, Pa., 1985), pp. 97–99.

Marcuse, D.

D. Marcuse, “Coupled mode theory of round optical fibers,” Bell Syst. Tech. J. 52, 817–50842 (1973).
[Crossref]

Morita, R.

R. Morita, T. Kondo, Y. Kaneda, A. Sugihashi, N. Ogasawara, S. Umegaki, and R. Ito, “Dispersion of second-order nonlinear optical coefficient d11 of 2-methyl-4-nitroaniline (MNA),” Jpn. J. Appl. Phys. 27, L1131–L1133 (1988).
[Crossref]

Nayar, B. K.

K. I. White, B. K. Nayar, and R. Kashap, “Amplification and second-harmonic generation in non-linear fibre waveguides,” Opt. Quantum Electron. 20, 339–342 (1988).
[Crossref]

B. K. Nayar, “Nonlinear optical interactions in organic crystal-cored fibers,” ACS Symp. Ser. 233, 153–166 (1983).
[Crossref]

B. K. Nayar, “Optical fibers with organic crystalline cores,” in Nonlinear Optics: Materials and Devices, C. Flytzanis and J. L. Oudar, eds. (Springer-Verlag, Berlin, 1986), pp. 142–153.
[Crossref]

K. I. White and B. K. Nayar, “Nonlinear fiber waveguide: the field overlap,” in Digest of Conference on Lasers and Electro-Optics (Optical Society of America, Washington, D.C., 1986), paper WK46, pp. 204–206;“Second-harmonic generation in nonlinear optical fiber waveguides: efficient designs using radiation modes,” J. Opt. Soc. Am. B 5, 317–324 (1988).

Ogasawara, N.

R. Morita, T. Kondo, Y. Kaneda, A. Sugihashi, N. Ogasawara, S. Umegaki, and R. Ito, “Dispersion of second-order nonlinear optical coefficient d11 of 2-methyl-4-nitroaniline (MNA),” Jpn. J. Appl. Phys. 27, L1131–L1133 (1988).
[Crossref]

Okamoto, S.

T. Uemiya, N. Uenishi, Y. Shimizu, S. Okamoto, K. Chikuma, T. Tohma, and S. Umegaki, “Crystal-cored fiber using organic material and focusing properties of generated second-harmonic waves,” in Nonlinear Optical Properties of Materials, H. Schlossberg and R. Wick, eds., in Proc. Soc. Photo-Opt. Instrum. Eng.1148, 207–212 (1989).
[Crossref]

Okazaki, Y.

A. Harada, Y. Okazaki, K. Kamiyama, and S. Umegaki, “Generation of continuous wave blue coherent light from a semiconductor laser using nonlinear optical fiber with an organic core crystal,” in Digest of Conference on Lasers and Electro-Optics (Optical Society of America, Washington, D.C., 1990), paper CFE3, pp. 496–498.

Salin, F.

Sanford, N. A.

N. A. Sanford and J. M. Connors, “Optimization of the Cerenkov sum-frequency generation in proton exchanged Mg:LiNbO3 channel waveguides,” J. Appl. Phys. 65, 1429–1437 (1989).
[Crossref]

Shimizu, Y.

T. Uemiya, N. Uenishi, Y. Shimizu, S. Okamoto, K. Chikuma, T. Tohma, and S. Umegaki, “Crystal-cored fiber using organic material and focusing properties of generated second-harmonic waves,” in Nonlinear Optical Properties of Materials, H. Schlossberg and R. Wick, eds., in Proc. Soc. Photo-Opt. Instrum. Eng.1148, 207–212 (1989).
[Crossref]

Sugihashi, A.

R. Morita, T. Kondo, Y. Kaneda, A. Sugihashi, N. Ogasawara, S. Umegaki, and R. Ito, “Dispersion of second-order nonlinear optical coefficient d11 of 2-methyl-4-nitroaniline (MNA),” Jpn. J. Appl. Phys. 27, L1131–L1133 (1988).
[Crossref]

Takahashi, Y.

S. Umegaki, Y. Takahashi, A. Manabe, and S. Tanaka, “Optical second-harmonic generation in an organic crystal-core fiber,” in Extended Abstracts: Nonlinear Optical Materials, D. A. B. Miller, ed. (Materials Research Society, Pittsburgh, Pa., 1985), pp. 97–99.

Tanaka, S.

S. Umegaki, A. Hiramatsu, Y. Tsukikawa, and S. Tanaka, “Crystal growth of organic materials and optical second-harmonic generation in optical fiber,” in Molecular and Polymeric Optoelectronic Materials, G. Khanarian, ed., Proc. Soc. Photo-Opt. Instrum. Eng.682, 187–190 (1986).
[Crossref]

S. Umegaki, Y. Takahashi, A. Manabe, and S. Tanaka, “Optical second-harmonic generation in an organic crystal-core fiber,” in Extended Abstracts: Nonlinear Optical Materials, D. A. B. Miller, ed. (Materials Research Society, Pittsburgh, Pa., 1985), pp. 97–99.

Tohma, T.

T. Uemiya, N. Uenishi, Y. Shimizu, S. Okamoto, K. Chikuma, T. Tohma, and S. Umegaki, “Crystal-cored fiber using organic material and focusing properties of generated second-harmonic waves,” in Nonlinear Optical Properties of Materials, H. Schlossberg and R. Wick, eds., in Proc. Soc. Photo-Opt. Instrum. Eng.1148, 207–212 (1989).
[Crossref]

Tsukikawa, Y.

S. Umegaki, A. Hiramatsu, Y. Tsukikawa, and S. Tanaka, “Crystal growth of organic materials and optical second-harmonic generation in optical fiber,” in Molecular and Polymeric Optoelectronic Materials, G. Khanarian, ed., Proc. Soc. Photo-Opt. Instrum. Eng.682, 187–190 (1986).
[Crossref]

Uemiya, T.

T. Uemiya, N. Uenishi, Y. Shimizu, S. Okamoto, K. Chikuma, T. Tohma, and S. Umegaki, “Crystal-cored fiber using organic material and focusing properties of generated second-harmonic waves,” in Nonlinear Optical Properties of Materials, H. Schlossberg and R. Wick, eds., in Proc. Soc. Photo-Opt. Instrum. Eng.1148, 207–212 (1989).
[Crossref]

Uenishi, N.

T. Uemiya, N. Uenishi, Y. Shimizu, S. Okamoto, K. Chikuma, T. Tohma, and S. Umegaki, “Crystal-cored fiber using organic material and focusing properties of generated second-harmonic waves,” in Nonlinear Optical Properties of Materials, H. Schlossberg and R. Wick, eds., in Proc. Soc. Photo-Opt. Instrum. Eng.1148, 207–212 (1989).
[Crossref]

Umegaki, S.

K. Chikuma and S. Umegaki, “Characteristics of optical second-harmonic generation due to Čerenkov-radiation-type phase matching,” J. Opt. Soc. Am. B 7, 768–775 (1990).
[Crossref]

R. Morita, T. Kondo, Y. Kaneda, A. Sugihashi, N. Ogasawara, S. Umegaki, and R. Ito, “Dispersion of second-order nonlinear optical coefficient d11 of 2-methyl-4-nitroaniline (MNA),” Jpn. J. Appl. Phys. 27, L1131–L1133 (1988).
[Crossref]

S. Umegaki, Y. Takahashi, A. Manabe, and S. Tanaka, “Optical second-harmonic generation in an organic crystal-core fiber,” in Extended Abstracts: Nonlinear Optical Materials, D. A. B. Miller, ed. (Materials Research Society, Pittsburgh, Pa., 1985), pp. 97–99.

S. Umegaki, A. Hiramatsu, Y. Tsukikawa, and S. Tanaka, “Crystal growth of organic materials and optical second-harmonic generation in optical fiber,” in Molecular and Polymeric Optoelectronic Materials, G. Khanarian, ed., Proc. Soc. Photo-Opt. Instrum. Eng.682, 187–190 (1986).
[Crossref]

T. Uemiya, N. Uenishi, Y. Shimizu, S. Okamoto, K. Chikuma, T. Tohma, and S. Umegaki, “Crystal-cored fiber using organic material and focusing properties of generated second-harmonic waves,” in Nonlinear Optical Properties of Materials, H. Schlossberg and R. Wick, eds., in Proc. Soc. Photo-Opt. Instrum. Eng.1148, 207–212 (1989).
[Crossref]

A. Harada, Y. Okazaki, K. Kamiyama, and S. Umegaki, “Generation of continuous wave blue coherent light from a semiconductor laser using nonlinear optical fiber with an organic core crystal,” in Digest of Conference on Lasers and Electro-Optics (Optical Society of America, Washington, D.C., 1990), paper CFE3, pp. 496–498.

Vidakovic, P. V.

White, K. I.

K. I. White, B. K. Nayar, and R. Kashap, “Amplification and second-harmonic generation in non-linear fibre waveguides,” Opt. Quantum Electron. 20, 339–342 (1988).
[Crossref]

K. I. White and B. K. Nayar, “Nonlinear fiber waveguide: the field overlap,” in Digest of Conference on Lasers and Electro-Optics (Optical Society of America, Washington, D.C., 1986), paper WK46, pp. 204–206;“Second-harmonic generation in nonlinear optical fiber waveguides: efficient designs using radiation modes,” J. Opt. Soc. Am. B 5, 317–324 (1988).

ACS Symp. Ser. (1)

B. K. Nayar, “Nonlinear optical interactions in organic crystal-cored fibers,” ACS Symp. Ser. 233, 153–166 (1983).
[Crossref]

Appl. Phys. Lett. (1)

P. Kerkoc, Ch. Bosshard, H. Arend, and P. Gunter, “Growth and characterization of 4-(N,N-dimethylamino)-3-acetamidonitrobenzene single-crystal cored fibers,” Appl. Phys. Lett. 54, 487–489 (1989).
[Crossref]

Bell Syst. Tech. J. (1)

D. Marcuse, “Coupled mode theory of round optical fibers,” Bell Syst. Tech. J. 52, 817–50842 (1973).
[Crossref]

J. Appl. Phys. (1)

N. A. Sanford and J. M. Connors, “Optimization of the Cerenkov sum-frequency generation in proton exchanged Mg:LiNbO3 channel waveguides,” J. Appl. Phys. 65, 1429–1437 (1989).
[Crossref]

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

Jpn. J. Appl. Phys. (1)

R. Morita, T. Kondo, Y. Kaneda, A. Sugihashi, N. Ogasawara, S. Umegaki, and R. Ito, “Dispersion of second-order nonlinear optical coefficient d11 of 2-methyl-4-nitroaniline (MNA),” Jpn. J. Appl. Phys. 27, L1131–L1133 (1988).
[Crossref]

Opt. Quantum Electron. (1)

K. I. White, B. K. Nayar, and R. Kashap, “Amplification and second-harmonic generation in non-linear fibre waveguides,” Opt. Quantum Electron. 20, 339–342 (1988).
[Crossref]

Other (9)

M. Born and E. Wolf, eds., Principles of Optics (Pergamon, Oxford, 1980), App. III.

T. Uemiya, N. Uenishi, Y. Shimizu, S. Okamoto, K. Chikuma, T. Tohma, and S. Umegaki, “Crystal-cored fiber using organic material and focusing properties of generated second-harmonic waves,” in Nonlinear Optical Properties of Materials, H. Schlossberg and R. Wick, eds., in Proc. Soc. Photo-Opt. Instrum. Eng.1148, 207–212 (1989).
[Crossref]

A. Harada, Y. Okazaki, K. Kamiyama, and S. Umegaki, “Generation of continuous wave blue coherent light from a semiconductor laser using nonlinear optical fiber with an organic core crystal,” in Digest of Conference on Lasers and Electro-Optics (Optical Society of America, Washington, D.C., 1990), paper CFE3, pp. 496–498.

Equation (30b) of Ref. 1 has to be corrected by eliminating the factor K1(W)/J1(U).

K. I. White and B. K. Nayar, “Nonlinear fiber waveguide: the field overlap,” in Digest of Conference on Lasers and Electro-Optics (Optical Society of America, Washington, D.C., 1986), paper WK46, pp. 204–206;“Second-harmonic generation in nonlinear optical fiber waveguides: efficient designs using radiation modes,” J. Opt. Soc. Am. B 5, 317–324 (1988).

D. S. Chemla and J. Zyss, eds., Nonlinear Optical Properties of Organic Molecules and Crystals (Academic, Orlando, Fla., 1987), Vols. 1 and 2.

B. K. Nayar, “Optical fibers with organic crystalline cores,” in Nonlinear Optics: Materials and Devices, C. Flytzanis and J. L. Oudar, eds. (Springer-Verlag, Berlin, 1986), pp. 142–153.
[Crossref]

S. Umegaki, Y. Takahashi, A. Manabe, and S. Tanaka, “Optical second-harmonic generation in an organic crystal-core fiber,” in Extended Abstracts: Nonlinear Optical Materials, D. A. B. Miller, ed. (Materials Research Society, Pittsburgh, Pa., 1985), pp. 97–99.

S. Umegaki, A. Hiramatsu, Y. Tsukikawa, and S. Tanaka, “Crystal growth of organic materials and optical second-harmonic generation in optical fiber,” in Molecular and Polymeric Optoelectronic Materials, G. Khanarian, ed., Proc. Soc. Photo-Opt. Instrum. Eng.682, 187–190 (1986).
[Crossref]

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Figures (9)

Fig. 1
Fig. 1

Configuration of the crystal-cored fiber and radiation angles of the second-harmonic wave generated by phase matching of Čerenkov-type radiation.

Fig. 2
Fig. 2

(a) Conversion efficiency P2ω/Pω and (b) Čerenkov radiation angle θA outside the fiber as a function of core radius a for several kinds of cladding glass for a DAN crystal and a fundamental wavelength 1064 nm.

Fig. 3
Fig. 3

(a) Conversion efficiency P2ω/Pω and (b) Čerenkov radiation angle θA outside the fiber as a function of core radius a for several kinds of cladding glass for a MNA crystal and a fundamental wavelength 1064 nm.

Fig. 4
Fig. 4

Conversion efficiency P2ω/Pω as a function of the fundamental wavelength taking core radius a as a parameter for the DAN crystal-cored fiber with SF4 glass cladding.

Fig. 5
Fig. 5

(a) Mode dispersion curves for the fundamental and its second-harmonic wavelength and (b) conversion efficiency P2ω/Pω and Čerenkov radiation angle θA as a function of core radius a for a DAN crystal-cored fiber with SF1 glass cladding.

Fig. 6
Fig. 6

Dependence of the conversion efficiency P2ω/Pω on fiber length L for the two cladding radii b for a DAN crystal-cored fiber with SF1 glass cladding.

Fig. 7
Fig. 7

Dependence of the conversion efficiency P2ω/Pω on the cladding radius b for core radius a of 1 μm for a DAN crystal-cored fiber with SF1 glass cladding.

Fig. 8
Fig. 8

Comparison of the present results with the previous ones for the DAN crystal-cored fiber with the SF1 glass cladding and MNA crystal-cored fiber with the SF4 glass cladding.

Fig. 9
Fig. 9

Functions T(θ′) and S(θ′) for (a) a DAN and (b) a MNA crystal-cored fiber. The core radius and the cladding glass are 1 μm and SF1, respectively, in (a) and 0.5 μm and SF4, respectively, in (b).

Tables (1)

Tables Icon

Table 1 Types of Generated Second-Harmonic Waves and Corresponding Phase-Matching Conditions

Equations (53)

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

P 2 ω = ω V Im ( E 2 ω P NL * ) d V ,
2 E 2 ω G + 4 ω 2 μ 0 2 ω G E 2 ω G = 4 ω 2 μ 0 P NL eff ,
2 E 2 ω S + 4 ω 2 μ 0 2 ω S E 2 ω S = 0 ,
2 E 2 ω A + 4 ω 2 μ 0 2 ω A E 2 ω A = 0 ,
G ( r , r ) = exp ( i k 2 ω G | r r | ) 4 π | r r | = { i 4 π m = 0 m cos m ( ϕ ϕ ) 0 H m ( 1 ) ( r γ 2 ω ) J m ( r γ 2 ω ) cos K 2 ω ( z z ) d K 2 ω r > r i 4 π m = 0 m cos m ( ϕ ϕ ) 0 H m ( 1 ) ( r γ 2 ω ) J m ( r γ 2 ω ) cos K 2 ω ( z z ) d K 2 ω r > r ,
g m B ( r , r ) = g m ( r , r ) + A m ( K 2 ω ) J m ( r γ 2 ω ) J m ( r γ 2 ω ) ,
g m ( r , r ) = { H m ( 1 ) ( r γ 2 ω ) J m ( r γ 2 ω ) r > r H m ( 1 ) ( r γ 2 ω ) J m ( r γ 2 ω ) r < r .
G ( r , r ) = i 4 π m = 0 m cos m ( ϕ ϕ ) × 0 g m B ( r , r ) cos K 2 ω ( z z ) d K 2 ω .
E 2 ω G ( r ) = 4 ω 2 μ 0 P NL e ff ( r ) G ( r , r ) d r .
P NL e ff ( r ) = 0 d e ff C J 0 ( U ω r ) 2 exp ( 2 i β ω z ) ,
E 2 ω G ( r ) = i ω 2 μ 0 0 d eff C π 0 a 0 2 π L / 2 L / 2 0 r d r d ϕ d z d K 2 ω × m = 0 m cos m ( ϕ ϕ ) g m B ( r , r ) J 0 ( U ω r ) 2 × exp ( 2 i β ω z ) cos K 2 ω ( z z ) ,
E 2 ω G ( r ) = i ω 2 μ 0 0 d eff C 0 a g 0 B ( r , r ) J 0 ( U ω r ) 2 r d r × sin [ ( K 2 ω / 2 β ω ) L ] ( K 2 ω / 2 β ω ) exp ( i K 2 ω z ) d K 2 ω ,
g 0 B ( r , r ) = { H 0 ( 1 ) ( r γ 2 ω ) J 0 ( r γ 2 ω ) + A ( K 2 ω ) J 0 ( r γ 2 ω ) J 0 ( r γ 2 ω ) r > r H 0 ( 1 ) ( r γ 2 ω ) J 0 ( r γ 2 ω ) + A ( K 2 ω ) J 0 ( r γ 2 ω ) J 0 ( r γ 2 ω ) r < r .
E 2 ω S ( r ) = i 4 π B ( K 2 ω ) H 0 ( 1 ) ( r η 2 ω ) exp ( i K 2 ω z ) d K 2 ω ,
A ( K 2 ω ) = η 2 ω H 0 ( 1 ) ( a γ 2 ω ) H 1 ( 1 ) ( a η 2 ω ) γ 2 ω H 1 ( 1 ) ( a γ 2 ω ) H 0 ( 1 ) ( a η 2 ω ) η 2 ω J 0 ( a γ 2 ω ) H 1 ( 1 ) ( a η 2 ω ) γ 2 ω J 1 ( a γ 2 ω ) H 0 ( 1 ) ( a η 2 ω ) ,
B ( K 2 ω ) = 8 i ω 2 μ 0 0 d eff C a [ η 2 ω J 0 ( a γ 2 ω ) H 1 ( 1 ) ( a η 2 ω ) γ 2 ω J 1 ( a γ 2 ω ) H 0 ( 1 ) ( a η 2 ω ) ] × sin [ ( K 2 ω / 2 β ω ) / L ] ( K 2 ω / 2 β ω ) 0 a J 0 ( r γ 2 ω ) J 0 ( U w r ) 2 r d r .
P 2 ω = 2 π ω 3 μ 0 0 2 d eff 2 C 2 × F ( K 2 ω ) sin 2 [ ( K 2 ω / 2 β ω ) / L ] ( K 2 ω / 2 β ω ) 2 d K 2 ω ,
F ( K 2 ω ) = { 1 + Re [ A ( K 2 ω ) ] } | 0 a J 0 ( r γ 2 ω ) J 0 ( U ω r ) 2 r d r | 2 = Re [ i G ( K 2 ω ) ] | 0 a J 0 ( r γ 2 ω ) J 0 ( U ω r ) 2 r d r | 2 ,
G ( K 2 ω ) = η 2 ω N 0 ( a γ 2 ω ) H 1 ( 1 ) ( a η 2 ω ) γ 2 ω N 1 ( a γ 2 ω ) H 0 ( 1 ) ( a η 2 ω ) η 2 ω J 0 ( a γ 2 ω ) H 1 ( 1 ) ( a η 2 ω ) γ 2 ω J 1 ( a γ 2 ω ) H 0 ( 1 ) ( a η 2 ω ) .
E 2 ω S ( r ) = i 4 π [ C ( K 2 ω ) J 0 ( r η 2 ω ) × D ( K 2 ω ) N 0 ( r η 2 ω ) ] exp ( i K 2 ω z ) d K 2 ω .
E 2 ω A ( r ) = i 4 π E ( K 2 ω ) K 0 ( r ξ 2 ω ) exp ( i K 2 ω z ) d K 2 ω ,
A ( K 2 ω ) = O ( K 2 ω ) P ( K 2 ω ) i Q ( K 2 ω ) R ( K 2 ω ) O ( K 2 ω ) P ( K 2 ω ) + Q ( K 2 ω ) R ( K 2 ω ) ,
O ( K 2 ω ) = η 2 ω J 0 ( a γ 2 ω ) J 1 ( a η 2 ω ) γ 2 ω J 1 ( a γ 2 ω ) J 0 ( a η 2 ω ) , O ( K 2 ω ) = η 2 ω H 0 ( 1 ) ( a γ 2 ω ) J 1 ( a η 2 ω ) γ 2 ω H 1 ( 1 ) ( a γ 2 ω ) J 0 ( a η 2 ω ) , P ( K 2 ω ) = ξ 2 ω N 0 ( b η 2 ω ) K 1 ( b ξ 2 ω ) η 2 ω N 1 ( b η 2 ω ) K 0 ( b ξ 2 ω ) , Q ( K 2 ω ) = γ 2 ω N 0 ( a η 2 ω ) N 1 ( a γ 2 ω ) η 2 ω N 1 ( a η 2 ω ) N 0 ( a γ 2 ω ) , Q ( K 2 ω ) = η 2 ω H 0 ( 2 ) ( a γ 2 ω ) N 1 ( a η 2 ω ) γ 2 ω H 1 ( 2 ) ( a γ 2 ω ) N 0 ( a η 2 ω ) , R ( K 2 ω ) = ξ 2 ω J 0 ( b η 2 ω ) K 1 ( b ξ 2 ω ) η 2 ω J 1 ( b η 2 ω ) K 0 ( b ξ 2 ω ) .
G ( K 2 ω ) = O ( K 2 ω ) P ( K 2 ω ) + Q ( K 2 ω ) R ( K 2 ω ) O ( K 2 ω ) P ( K 2 ω ) + Q ( K 2 ω ) R ( K 2 ω ) ,
Q ( K 2 ω ) = η 2 ω N 0 ( a γ 2 ω ) J 1 ( a η 2 ω ) γ 2 ω N 1 ( a γ 2 ω ) J 0 ( a η 2 ω ) , Q ( K 2 ω ) = γ 2 ω N 0 ( a η 2 ω ) J 1 ( a γ 2 ω ) η 2 ω N 1 ( a η 2 ω ) J 0 ( a γ 2 ω ) .
J 0 ( a γ 2 ω ) γ 2 ω J 1 ( a γ 2 ω ) = K 0 ( a τ 2 ω ) τ 2 ω K 1 ( a τ 2 ω ) ,
P 2 ω = 2 π 2 ω 3 μ 0 0 2 d eff 2 C 2 × j Res [ G ( K 2 ω , j ) ] sin 2 [ ( K 2 ω , j / 2 β ω ) / L ] ( K 2 ω , j / 2 β ω ) 2 × | 0 a J 0 ( r γ 2 ω , j ) J 0 ( U ω r ) 2 r d r | 2 ,
K 2 ω = 2 β ω = 2 ( N ω ω / c ) ,
N ω = n 2 ω S cos θ ( = n 2 ω G cos θ ) ,
N ω = N 2 ω .
P 2 ω = 4 π 2 ω 3 μ 0 0 2 d eff 2 C 2 L F ( 2 β ω ) = 4 π 2 ω 3 μ 0 0 2 d eff 2 C 2 L Re [ i G ( K 2 ω ) ] × | 0 e J 0 ( r γ 2 ω ) J 0 ( U ω r ) 2 r d r | 2 | K 2 ω = 2 β ω .
sin 2 [ ( K 2 ω / 2 β ω ) / L ] ( K 2 ω / 2 β ω ) 2 L π δ ( K 2 ω / 2 β ω ) .
P 2 ω = 2 π 2 ω 3 μ 0 0 2 d eff 2 C 2 L 2 Res [ G ( K 2 ω , m ) ] × | 0 a J 0 ( r γ 2 ω , m ) J 0 ( U ω r ) 2 r d r | 2 .
Res [ G ( K 2 ω , m ) ] = τ 2 ω , m 2 π a 2 β ω ( γ 2 ω , m 2 + τ 2 ω , m 2 ) J 1 ( a γ 2 ω , m ) 2 .
P ω = ( P ω ) core + ( P ω ) clad = C π a 2 4 β ω μ 0 ω { ( ω 2 μ 0 ω G + β ω 2 ) [ J 0 ( U ω a ) 2 + J 1 ( U ω a ) 2 ] + ( ω 2 μ 0 ω S + β ω 2 ) J 1 ( U ω a ) 2 U ω 2 K 1 ( W ω a ) 2 W ω 2 × [ K 1 ( W ω a ) 2 K 0 ( W ω a ) 2 ] } ,
T ( θ ) = | 0 a J 0 ( r k 2 ω G sin θ ) J 0 ( U ω r ) 2 r d r | 2 ,
S ( θ ) = sin 2 [ ( k 2 ω G cos θ / 2 β ω ) / L ] [ ( k 2 ω G cos θ / 2 β ω ) / L ] 2 ,
T ( θ ) = ( a 4 / 4 ) [ 2 J 1 ( a k 2 ω G sin θ ) / ( a k 2 ω G sin θ ) ] 2 .
( n 2 ω G ) 2 N ω 2 < 0.093 ( λ ω / a ) 2 ,
H 0 ( 1 ) ( r η 2 ω ) ( 2 / π r η 2 ω ) 1 / 2 exp [ i ( r η 2 ω π / 4 ) ] for r
E 2 ω S ( r ) = i ω 2 μ 0 0 d eff C g ( K 2 ω ) exp [ i f ( K 2 ω ) ] d K 2 ω ,
f ( K 2 ω ) = η 2 ω r + K 2 ω Z ,
g ( K 2 ω ) = ( 2 / π r η 2 ω ) 1 / 2 exp ( i π / 4 ) × sin [ ( K 2 ω / 2 β ω ) / L ] ( K 2 ω / 2 β ω ) 0 a J 0 ( r η 2 ω ) J 0 ( U ω r ) 2 r d r .
E S 2 ω ( r ) = 2 ω 2 μ 0 0 d eff C exp ( i k 2 ω S R ) R × sin [ ( k 2 ω S cos θ / 2 β ω ) / L ] ( k 2 ω S cos θ / 2 β ω ) × 0 a J 0 ( k 2 ω S r sin θ ) J 0 ( U ω r ) 2 r d r ,
P 2 ω = 2 π ω 3 μ 0 0 2 d eff 2 C 2 sin 2 [ ( K 2 ω / 2 β ω ) / L ] ( K 2 ω / 2 β ω ) 2 × | 0 a J 0 ( r η 2 ω ) J 0 ( U ω r ) 2 r d r | 2 d K 2 ω ,
{ Res [ G ( K 2 ω , m ) ] } 1 = d d K 2 ω [ 1 G ( K 2 ω ) ] | K 2 ω = K 2 ω , m .
1 G ( K 2 ω ) = τ 2 ω J 0 ( a γ 2 ω ) K 1 ( a τ 2 ω ) γ 2 ω J 1 ( a γ 2 ω ) K 0 ( a τ 2 ω ) τ 2 ω N 0 ( a γ 2 ω ) K 1 ( a τ 2 ω ) γ 2 ω N 1 ( a γ 2 ω ) K 0 ( a τ 2 ω ) .
τ 2 ω , m J 0 ( a γ 2 ω , m ) K 1 ( a τ 2 ω , m ) = γ 2 ω , m J 1 ( a γ 2 ω , m ) K 0 ( a τ 2 ω , m ) ,
d d K 2 ω [ 1 G ( K 2 ω ) ] | K 2 ω = K 2 ω , m = a K 2 ω , m ( γ 2 ω , m 2 + τ 2 ω , m 2 ) J 1 ( a γ 2 ω , m ) K 1 ( a τ 2 ω , m ) [ τ 2 ω , m N 0 ( a γ 2 ω , m ) K 1 ( a τ 2 ω , m ) γ 2 ω , m N 1 ( a γ 2 ω , m ) K 0 ( a τ 2 ω , m ) ] .
J n ( b η 2 ω ) ( 2 π b η 2 ω ) 1 / 2 cos ( b η 2 ω π 4 n π 2 ) , N n ( b η 2 ω ) ( 2 π b η 2 ω ) 1 / 2 sin ( b η 2 ω π 4 n π 2 ) , K n ( b ξ 2 ω ) ( 2 π b ξ 2 ω ) 1 / 2 exp ( b ξ 2 ω ) .
P ( K 2 ω ) = exp ( b ξ 2 ω ) b ( ξ 2 ω η 2 ω ) 1 / 2 × [ ξ 2 ω sin ( b η 2 ω π 4 ) η 2 ω sin ( b η 2 ω 3 π 4 ) ] , R ( K 2 ω ) = exp ( b ξ 2 ω ) b ( ξ 2 ω η 2 ω ) 1 / 2 × [ ξ 2 ω cos ( b η 2 ω π 4 ) η 2 ω cos ( b η 2 ω 3 π 4 ) ] .
P ( K 2 ω ) = i b , R ( K 2 ω ) = 1 b .
G F = i O ( K 2 ω ) + Q ( K 2 ω ) i O ( K 2 ω ) + Q ( K 2 ω ) .

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