Abstract

The asymmetry of the UV-induced refractive-index profile in photosensitive optical fibers causes a deformation of the modal fields, resulting in transition losses between UV-exposed and unexposed fiber sections up to 0.1 dB for UV-induced index changes of the order of ≤10-2. A numerical analysis of the transition losses is developed. It is verified experimentally that the loss is caused by coupling of power of the fundamental mode to cladding modes at such a transition.

© 2000 Optical Society of America

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References

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  1. For example, Bragg Gratings, Photosensitivity, and Poling in Glass Fibers and Waveguides: Applications and Fundamentals, Vol. 17 of 1997 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1997).
  2. M. Janos, J. Canning, M. G. Sceats, “Incoherent scattering losses in optical fiber Bragg gratings,” Opt. Lett. 21, 1827–1829 (1996).
    [Crossref] [PubMed]
  3. D. Johlen, P. Klose, H. Renner, E. Brinkmeyer, “Strong LP11-mode splitting in UV side-written tilted fiber gratings,” in Bragg Gratings, Photosensitivity, and Poling in Glass Fibers and Waveguides: Applications and Fundamentals, Vol. 17 of 1997 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1997), paper BMG12, pp. 219–221.
  4. C. K. Madsen, J. J. DeMarco, C. H. Henry, E. J. Laskowsky, R. E. Scotti, T. A. Strasser, “Apodized UV-induced gratings in planar waveguides for compact add–drop filters,” in Bragg Gratings, Photosensitivity, and Poling in Glass Fibers and Waveguides: Applications and Fundamentals, Vol. 17 of 1997 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1997), paper BTuA4, pp. 262–264.
  5. T. T. Vu, R. J. Orazi, M. N. McLandrich, “Four channel fused fibre mux/demux couplers and add/drop filters in the 1300 and 1550 nm regions,” Electron. Lett. 34, 583–585 (1998).
    [Crossref]
  6. D. Johlen, P. Klose, H. Renner, E. Brinkmeyer, “Narrow-band-mode converting Fabry–Perot output coupler for fiber lasers,” in Optical Fiber Communication Conference, Vol. 2 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), paper FA4, pp. 372–373.
  7. J. Canning, A. L. G. Carter, M. G. Sceats, “Correlation between photodarkening and index change during 193 nm irradiation of germanosilicate and phosphosilicate fibers,” J. Lightwave Technol. 15, 1349–1356 (1997).
    [Crossref]
  8. H. Renner, “Modes of UV-written planar waveguides,” Opt. Lett. 23, 111–113 (1998).
    [Crossref]
  9. I. S. Gradshteyn, I. M. Ryzhik, Tables of Integrals, Series, and Products (Academic, London, 1980).
  10. A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983).
  11. P. H. Moon, D. E. Spencer, Field Theory Handbook (Springer-Verlag, Berlin, 1988).
  12. T. A. Birks, Y. W. Li, C. D. Hussey, “Waveguides with delta function layers,” Opt. Commun. 83, 203–209 (1991).
    [Crossref]
  13. Fiber CD 120.02 (Optical Fiber Technology Center, Sydney, Australia).
  14. D. Johlen, H. Renner, A. Ewald, E. Brinkmeyer, “Fiber Bragg grating Fabry–Perot interferometer for a precise measurement of the UV-induced index change,” in Proceedings of 24th European Conference on Optical Communication 1998 (Lerko Print, Madrid, 1998), paper WdA06, pp. 393–394.
  15. D. Johlen, F. Knappe, H. Renner, E. Brinkmeyer, “UV-induced absorption, scattering and transition losses in UV side-written fibers,” in Optical Fiber Communication Conference and the International Conference on Integrated Optics and Optical Fiber Communication, 1999 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1999), paper ThD1, pp. 50–52.
  16. Radiation modes of the uncoated fiber, which could contribute to radiation of power out of the cladding into air, oscillate rapidly over the region of the fundamental-mode fields. Thus, they are excited to a negligible amount only and are not considered here. The actual transition losses of the fundamental mode would even be increased by that small amount of power possibly coupled to radiation modes.
  17. E. M. Dianov, S. A. Vasiliev, A. A. Frolov, O. I. Medvedkov, “Germanosilicate glass refractive-index change under singlet and triplet excitation of germanium oxygen deficient centers,” in Bragg Gratings, Photosensitivity, and Poling in Glass Fibers and Waveguides: Applications and Fundamentals, Vol. 17 of 1997 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1997), paper JMF4, pp. 175–177.

1998 (2)

T. T. Vu, R. J. Orazi, M. N. McLandrich, “Four channel fused fibre mux/demux couplers and add/drop filters in the 1300 and 1550 nm regions,” Electron. Lett. 34, 583–585 (1998).
[Crossref]

H. Renner, “Modes of UV-written planar waveguides,” Opt. Lett. 23, 111–113 (1998).
[Crossref]

1997 (1)

J. Canning, A. L. G. Carter, M. G. Sceats, “Correlation between photodarkening and index change during 193 nm irradiation of germanosilicate and phosphosilicate fibers,” J. Lightwave Technol. 15, 1349–1356 (1997).
[Crossref]

1996 (1)

1991 (1)

T. A. Birks, Y. W. Li, C. D. Hussey, “Waveguides with delta function layers,” Opt. Commun. 83, 203–209 (1991).
[Crossref]

Birks, T. A.

T. A. Birks, Y. W. Li, C. D. Hussey, “Waveguides with delta function layers,” Opt. Commun. 83, 203–209 (1991).
[Crossref]

Brinkmeyer, E.

D. Johlen, H. Renner, A. Ewald, E. Brinkmeyer, “Fiber Bragg grating Fabry–Perot interferometer for a precise measurement of the UV-induced index change,” in Proceedings of 24th European Conference on Optical Communication 1998 (Lerko Print, Madrid, 1998), paper WdA06, pp. 393–394.

D. Johlen, F. Knappe, H. Renner, E. Brinkmeyer, “UV-induced absorption, scattering and transition losses in UV side-written fibers,” in Optical Fiber Communication Conference and the International Conference on Integrated Optics and Optical Fiber Communication, 1999 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1999), paper ThD1, pp. 50–52.

D. Johlen, P. Klose, H. Renner, E. Brinkmeyer, “Strong LP11-mode splitting in UV side-written tilted fiber gratings,” in Bragg Gratings, Photosensitivity, and Poling in Glass Fibers and Waveguides: Applications and Fundamentals, Vol. 17 of 1997 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1997), paper BMG12, pp. 219–221.

D. Johlen, P. Klose, H. Renner, E. Brinkmeyer, “Narrow-band-mode converting Fabry–Perot output coupler for fiber lasers,” in Optical Fiber Communication Conference, Vol. 2 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), paper FA4, pp. 372–373.

Canning, J.

J. Canning, A. L. G. Carter, M. G. Sceats, “Correlation between photodarkening and index change during 193 nm irradiation of germanosilicate and phosphosilicate fibers,” J. Lightwave Technol. 15, 1349–1356 (1997).
[Crossref]

M. Janos, J. Canning, M. G. Sceats, “Incoherent scattering losses in optical fiber Bragg gratings,” Opt. Lett. 21, 1827–1829 (1996).
[Crossref] [PubMed]

Carter, A. L. G.

J. Canning, A. L. G. Carter, M. G. Sceats, “Correlation between photodarkening and index change during 193 nm irradiation of germanosilicate and phosphosilicate fibers,” J. Lightwave Technol. 15, 1349–1356 (1997).
[Crossref]

DeMarco, J. J.

C. K. Madsen, J. J. DeMarco, C. H. Henry, E. J. Laskowsky, R. E. Scotti, T. A. Strasser, “Apodized UV-induced gratings in planar waveguides for compact add–drop filters,” in Bragg Gratings, Photosensitivity, and Poling in Glass Fibers and Waveguides: Applications and Fundamentals, Vol. 17 of 1997 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1997), paper BTuA4, pp. 262–264.

Dianov, E. M.

E. M. Dianov, S. A. Vasiliev, A. A. Frolov, O. I. Medvedkov, “Germanosilicate glass refractive-index change under singlet and triplet excitation of germanium oxygen deficient centers,” in Bragg Gratings, Photosensitivity, and Poling in Glass Fibers and Waveguides: Applications and Fundamentals, Vol. 17 of 1997 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1997), paper JMF4, pp. 175–177.

Ewald, A.

D. Johlen, H. Renner, A. Ewald, E. Brinkmeyer, “Fiber Bragg grating Fabry–Perot interferometer for a precise measurement of the UV-induced index change,” in Proceedings of 24th European Conference on Optical Communication 1998 (Lerko Print, Madrid, 1998), paper WdA06, pp. 393–394.

Frolov, A. A.

E. M. Dianov, S. A. Vasiliev, A. A. Frolov, O. I. Medvedkov, “Germanosilicate glass refractive-index change under singlet and triplet excitation of germanium oxygen deficient centers,” in Bragg Gratings, Photosensitivity, and Poling in Glass Fibers and Waveguides: Applications and Fundamentals, Vol. 17 of 1997 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1997), paper JMF4, pp. 175–177.

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, Tables of Integrals, Series, and Products (Academic, London, 1980).

Henry, C. H.

C. K. Madsen, J. J. DeMarco, C. H. Henry, E. J. Laskowsky, R. E. Scotti, T. A. Strasser, “Apodized UV-induced gratings in planar waveguides for compact add–drop filters,” in Bragg Gratings, Photosensitivity, and Poling in Glass Fibers and Waveguides: Applications and Fundamentals, Vol. 17 of 1997 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1997), paper BTuA4, pp. 262–264.

Hussey, C. D.

T. A. Birks, Y. W. Li, C. D. Hussey, “Waveguides with delta function layers,” Opt. Commun. 83, 203–209 (1991).
[Crossref]

Janos, M.

Johlen, D.

D. Johlen, H. Renner, A. Ewald, E. Brinkmeyer, “Fiber Bragg grating Fabry–Perot interferometer for a precise measurement of the UV-induced index change,” in Proceedings of 24th European Conference on Optical Communication 1998 (Lerko Print, Madrid, 1998), paper WdA06, pp. 393–394.

D. Johlen, F. Knappe, H. Renner, E. Brinkmeyer, “UV-induced absorption, scattering and transition losses in UV side-written fibers,” in Optical Fiber Communication Conference and the International Conference on Integrated Optics and Optical Fiber Communication, 1999 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1999), paper ThD1, pp. 50–52.

D. Johlen, P. Klose, H. Renner, E. Brinkmeyer, “Strong LP11-mode splitting in UV side-written tilted fiber gratings,” in Bragg Gratings, Photosensitivity, and Poling in Glass Fibers and Waveguides: Applications and Fundamentals, Vol. 17 of 1997 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1997), paper BMG12, pp. 219–221.

D. Johlen, P. Klose, H. Renner, E. Brinkmeyer, “Narrow-band-mode converting Fabry–Perot output coupler for fiber lasers,” in Optical Fiber Communication Conference, Vol. 2 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), paper FA4, pp. 372–373.

Klose, P.

D. Johlen, P. Klose, H. Renner, E. Brinkmeyer, “Narrow-band-mode converting Fabry–Perot output coupler for fiber lasers,” in Optical Fiber Communication Conference, Vol. 2 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), paper FA4, pp. 372–373.

D. Johlen, P. Klose, H. Renner, E. Brinkmeyer, “Strong LP11-mode splitting in UV side-written tilted fiber gratings,” in Bragg Gratings, Photosensitivity, and Poling in Glass Fibers and Waveguides: Applications and Fundamentals, Vol. 17 of 1997 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1997), paper BMG12, pp. 219–221.

Knappe, F.

D. Johlen, F. Knappe, H. Renner, E. Brinkmeyer, “UV-induced absorption, scattering and transition losses in UV side-written fibers,” in Optical Fiber Communication Conference and the International Conference on Integrated Optics and Optical Fiber Communication, 1999 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1999), paper ThD1, pp. 50–52.

Laskowsky, E. J.

C. K. Madsen, J. J. DeMarco, C. H. Henry, E. J. Laskowsky, R. E. Scotti, T. A. Strasser, “Apodized UV-induced gratings in planar waveguides for compact add–drop filters,” in Bragg Gratings, Photosensitivity, and Poling in Glass Fibers and Waveguides: Applications and Fundamentals, Vol. 17 of 1997 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1997), paper BTuA4, pp. 262–264.

Li, Y. W.

T. A. Birks, Y. W. Li, C. D. Hussey, “Waveguides with delta function layers,” Opt. Commun. 83, 203–209 (1991).
[Crossref]

Love, J. D.

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983).

Madsen, C. K.

C. K. Madsen, J. J. DeMarco, C. H. Henry, E. J. Laskowsky, R. E. Scotti, T. A. Strasser, “Apodized UV-induced gratings in planar waveguides for compact add–drop filters,” in Bragg Gratings, Photosensitivity, and Poling in Glass Fibers and Waveguides: Applications and Fundamentals, Vol. 17 of 1997 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1997), paper BTuA4, pp. 262–264.

McLandrich, M. N.

T. T. Vu, R. J. Orazi, M. N. McLandrich, “Four channel fused fibre mux/demux couplers and add/drop filters in the 1300 and 1550 nm regions,” Electron. Lett. 34, 583–585 (1998).
[Crossref]

Medvedkov, O. I.

E. M. Dianov, S. A. Vasiliev, A. A. Frolov, O. I. Medvedkov, “Germanosilicate glass refractive-index change under singlet and triplet excitation of germanium oxygen deficient centers,” in Bragg Gratings, Photosensitivity, and Poling in Glass Fibers and Waveguides: Applications and Fundamentals, Vol. 17 of 1997 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1997), paper JMF4, pp. 175–177.

Moon, P. H.

P. H. Moon, D. E. Spencer, Field Theory Handbook (Springer-Verlag, Berlin, 1988).

Orazi, R. J.

T. T. Vu, R. J. Orazi, M. N. McLandrich, “Four channel fused fibre mux/demux couplers and add/drop filters in the 1300 and 1550 nm regions,” Electron. Lett. 34, 583–585 (1998).
[Crossref]

Renner, H.

H. Renner, “Modes of UV-written planar waveguides,” Opt. Lett. 23, 111–113 (1998).
[Crossref]

D. Johlen, F. Knappe, H. Renner, E. Brinkmeyer, “UV-induced absorption, scattering and transition losses in UV side-written fibers,” in Optical Fiber Communication Conference and the International Conference on Integrated Optics and Optical Fiber Communication, 1999 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1999), paper ThD1, pp. 50–52.

D. Johlen, H. Renner, A. Ewald, E. Brinkmeyer, “Fiber Bragg grating Fabry–Perot interferometer for a precise measurement of the UV-induced index change,” in Proceedings of 24th European Conference on Optical Communication 1998 (Lerko Print, Madrid, 1998), paper WdA06, pp. 393–394.

D. Johlen, P. Klose, H. Renner, E. Brinkmeyer, “Narrow-band-mode converting Fabry–Perot output coupler for fiber lasers,” in Optical Fiber Communication Conference, Vol. 2 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), paper FA4, pp. 372–373.

D. Johlen, P. Klose, H. Renner, E. Brinkmeyer, “Strong LP11-mode splitting in UV side-written tilted fiber gratings,” in Bragg Gratings, Photosensitivity, and Poling in Glass Fibers and Waveguides: Applications and Fundamentals, Vol. 17 of 1997 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1997), paper BMG12, pp. 219–221.

Ryzhik, I. M.

I. S. Gradshteyn, I. M. Ryzhik, Tables of Integrals, Series, and Products (Academic, London, 1980).

Sceats, M. G.

J. Canning, A. L. G. Carter, M. G. Sceats, “Correlation between photodarkening and index change during 193 nm irradiation of germanosilicate and phosphosilicate fibers,” J. Lightwave Technol. 15, 1349–1356 (1997).
[Crossref]

M. Janos, J. Canning, M. G. Sceats, “Incoherent scattering losses in optical fiber Bragg gratings,” Opt. Lett. 21, 1827–1829 (1996).
[Crossref] [PubMed]

Scotti, R. E.

C. K. Madsen, J. J. DeMarco, C. H. Henry, E. J. Laskowsky, R. E. Scotti, T. A. Strasser, “Apodized UV-induced gratings in planar waveguides for compact add–drop filters,” in Bragg Gratings, Photosensitivity, and Poling in Glass Fibers and Waveguides: Applications and Fundamentals, Vol. 17 of 1997 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1997), paper BTuA4, pp. 262–264.

Snyder, A. W.

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983).

Spencer, D. E.

P. H. Moon, D. E. Spencer, Field Theory Handbook (Springer-Verlag, Berlin, 1988).

Strasser, T. A.

C. K. Madsen, J. J. DeMarco, C. H. Henry, E. J. Laskowsky, R. E. Scotti, T. A. Strasser, “Apodized UV-induced gratings in planar waveguides for compact add–drop filters,” in Bragg Gratings, Photosensitivity, and Poling in Glass Fibers and Waveguides: Applications and Fundamentals, Vol. 17 of 1997 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1997), paper BTuA4, pp. 262–264.

Vasiliev, S. A.

E. M. Dianov, S. A. Vasiliev, A. A. Frolov, O. I. Medvedkov, “Germanosilicate glass refractive-index change under singlet and triplet excitation of germanium oxygen deficient centers,” in Bragg Gratings, Photosensitivity, and Poling in Glass Fibers and Waveguides: Applications and Fundamentals, Vol. 17 of 1997 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1997), paper JMF4, pp. 175–177.

Vu, T. T.

T. T. Vu, R. J. Orazi, M. N. McLandrich, “Four channel fused fibre mux/demux couplers and add/drop filters in the 1300 and 1550 nm regions,” Electron. Lett. 34, 583–585 (1998).
[Crossref]

Electron. Lett. (1)

T. T. Vu, R. J. Orazi, M. N. McLandrich, “Four channel fused fibre mux/demux couplers and add/drop filters in the 1300 and 1550 nm regions,” Electron. Lett. 34, 583–585 (1998).
[Crossref]

J. Lightwave Technol. (1)

J. Canning, A. L. G. Carter, M. G. Sceats, “Correlation between photodarkening and index change during 193 nm irradiation of germanosilicate and phosphosilicate fibers,” J. Lightwave Technol. 15, 1349–1356 (1997).
[Crossref]

Opt. Commun. (1)

T. A. Birks, Y. W. Li, C. D. Hussey, “Waveguides with delta function layers,” Opt. Commun. 83, 203–209 (1991).
[Crossref]

Opt. Lett. (2)

Other (12)

D. Johlen, P. Klose, H. Renner, E. Brinkmeyer, “Strong LP11-mode splitting in UV side-written tilted fiber gratings,” in Bragg Gratings, Photosensitivity, and Poling in Glass Fibers and Waveguides: Applications and Fundamentals, Vol. 17 of 1997 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1997), paper BMG12, pp. 219–221.

C. K. Madsen, J. J. DeMarco, C. H. Henry, E. J. Laskowsky, R. E. Scotti, T. A. Strasser, “Apodized UV-induced gratings in planar waveguides for compact add–drop filters,” in Bragg Gratings, Photosensitivity, and Poling in Glass Fibers and Waveguides: Applications and Fundamentals, Vol. 17 of 1997 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1997), paper BTuA4, pp. 262–264.

D. Johlen, P. Klose, H. Renner, E. Brinkmeyer, “Narrow-band-mode converting Fabry–Perot output coupler for fiber lasers,” in Optical Fiber Communication Conference, Vol. 2 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), paper FA4, pp. 372–373.

I. S. Gradshteyn, I. M. Ryzhik, Tables of Integrals, Series, and Products (Academic, London, 1980).

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983).

P. H. Moon, D. E. Spencer, Field Theory Handbook (Springer-Verlag, Berlin, 1988).

Fiber CD 120.02 (Optical Fiber Technology Center, Sydney, Australia).

D. Johlen, H. Renner, A. Ewald, E. Brinkmeyer, “Fiber Bragg grating Fabry–Perot interferometer for a precise measurement of the UV-induced index change,” in Proceedings of 24th European Conference on Optical Communication 1998 (Lerko Print, Madrid, 1998), paper WdA06, pp. 393–394.

D. Johlen, F. Knappe, H. Renner, E. Brinkmeyer, “UV-induced absorption, scattering and transition losses in UV side-written fibers,” in Optical Fiber Communication Conference and the International Conference on Integrated Optics and Optical Fiber Communication, 1999 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1999), paper ThD1, pp. 50–52.

Radiation modes of the uncoated fiber, which could contribute to radiation of power out of the cladding into air, oscillate rapidly over the region of the fundamental-mode fields. Thus, they are excited to a negligible amount only and are not considered here. The actual transition losses of the fundamental mode would even be increased by that small amount of power possibly coupled to radiation modes.

E. M. Dianov, S. A. Vasiliev, A. A. Frolov, O. I. Medvedkov, “Germanosilicate glass refractive-index change under singlet and triplet excitation of germanium oxygen deficient centers,” in Bragg Gratings, Photosensitivity, and Poling in Glass Fibers and Waveguides: Applications and Fundamentals, Vol. 17 of 1997 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1997), paper JMF4, pp. 175–177.

For example, Bragg Gratings, Photosensitivity, and Poling in Glass Fibers and Waveguides: Applications and Fundamentals, Vol. 17 of 1997 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1997).

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Figures (6)

Fig. 1
Fig. 1

Normalized propagation constants B of the first three modes versus maximum relative UV-induced refractive-index increase H for V = 1.86 and A = 6.

Fig. 2
Fig. 2

Calculated mode intensities: (a) fundamental mode of the unexposed fiber, (b) fundamental mode, (c) odd LP11 mode, (d) even LP11 mode of the UV-written fiber with H = 5 and A = 5. The circles indicate the core–cladding boundary.

Fig. 3
Fig. 3

(a) Squared LP01–LP01 excitation coefficient C 00 2 and LP01–LP11 excitation coefficient C 01 2 at a single transition between UV-exposed and unexposed fiber sections for A = 0 (where C 01 2 ≡ 0) and A = 5. (b) Transition loss versus normalized effective-index increase δB = B - .

Fig. 4
Fig. 4

Measured transmission loss that is due to UV exposure of short fiber sections.

Fig. 5
Fig. 5

Length dependence of transmission losses in UV side-written fiber sections.

Fig. 6
Fig. 6

Measurement of power coupled to cladding modes. (i) Stripping of unwanted cladding modes in front of the UV-written fiber section (transmission decreases by approximately 0.01 dB). (ii) Power coupled to cladding modes by the pair of transitions of the UV-written fiber section stripped by immersion oil, leading to a decrease in transmission by approximately 0.18 dB.

Equations (60)

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

n2r, ϕ=n¯2r+δnUV2r, ϕ,
n¯2r=nco2,0r<ρcorencl2,ρ<r<cladding.
δnUV2r, ϕ=δnˆUV2 exp-2αx+ρ2-y2,0r<ρ,
δnUV2r, ϕδnˆUV2h0+hδρgϕδr-ρ,0rρ0,ρ<r<,
gϕ=0,-π/2ϕπ/2-cosϕ,π/2ϕ3π/2.
ξm=1πρ2+mδnˆUV2-- xmδnUV2x, ydxdy, m=0, 1.
ξm=1πρ2+my=-ρ+ρ exp-2αρ2-y2×x=-ρ2-y2+ρ2-y2 xm exp-2αxdxdy,
ξ0=12παρ2y=-ρ+ρ1-exp-4αρ2-y2dy,
=1παρ2ρ-Jα.
Jα=y=0ρ exp-4αρ2-y2dy,
Jα=t=0ρtρ2-t2 exp-4αtdt.
Jα=πρ2L14αρ-I14αρ+ρ,
ξ0=I14αρ-L14αρ2αρ.
ξ1=14πα2ρ3y=-ρ+ρ1-2αρ2-y2-1+2αρ2-y2exp-4αρ2-y2dy.
ξ1=14πα2ρ32ρ-παρ2-2Jα+α dJαdα.
dIνzdz=Iν-1z-νz Iνz,
dLνzdz=Lν-1z-νzLνz,
ξ1=3I14αρ-L14αρ-4αρI04αρ-L04αρ-2αρ8α2ρ2.
A=2αρ
ξ0=I12A-L12AA,
ξ1=3I12A-L12A-2AI02A-L02A-A2A2=3ξ0-2I02A+2L02A-12A.
ξ0=h0+2hδπ,  ξ1=-hδ2.
h0=ξ0+4π ξ1,  hδ=-2ξ1,
t2+k2n2r, ϕ-β2ψr, ϕ=0,
2r2+1rr+1r22ϕ2+κ2ψr, ϕ=0,
κ=k2nco2+h0δnˆUV2-β21/2.
2r2+1rr+1r22ϕ2-γ2ψr, ϕ=0,
γ=β2-k2ncl21/2.
ψr, ϕ=ψco=m=1-η amJmκrJmUcosmϕsinmϕ,0r<ρψcl=m=1-η bmKmγrKmWcosmϕsinmϕ,ρ<r<,
bm=am.
1rrr ψr+1r22ψϕ2+k2n2r, ϕ-β2ψ=0.
r ψr, ϕrr=ρ-r=ρ+=-r=ρ-r=ρ+1r2ψr, ϕϕ2dr-r=ρ-r=ρ+k2n2r, ϕ-β2ψr, ϕrdr.
r ψr, ϕrr=ρ-r=ρ+=- k2δnˆUV2hδρ2gϕψρ, ϕ- r=ρ-r=ρ+1r2ψr, ϕϕ2dr-r=ρ-r=ρ+k2n¯2r+k2δnˆUV2h0sr-β2ψr, ϕrdr
sr=1,0r<ρ0,ρ<r<
lim0ψr, ϕrr=ρ+-ψr, ϕrr=ρ-=-k2δnˆUV2hδρgϕψρ, ϕ.
ψcor, ϕrr=ρ=ψclr, ϕrr=ρ+Sρ gϕψρ, ϕ.
S=hδV2H
V=kρnco2-ncl2
H=δnˆUV2nco2-ncl2
amU JmUJmU-W KmWKmW1+δm0π-Sρl=1-η alϕ=02π gϕcoslϕcosmϕsinlϕsinmϕdϕ=0
M · a=0
a=am
M=Mm,l
Mm,m=1+ηδm0πW Km+1WKmW-U Jm+1UJmU-S1+2η-1-1m1-4m2
Mm,l=S×-1l-m/21-l-m2+2η-1-1l+m/21-l+m2,l+m even, lmπ41+ηδm0,l=m+1π41+ηδl0,m=l+10,l+m odd, |l-m|3
δm0=1for m=00for m0
detM=0,
B=β2-k2ncl2k2nco2-ncl2β-knclknco-ncl,
C=ϕ=02πr=0 ψψ¯rdrdϕϕ=02πr=0 ψ2rdrdϕ ϕ=02πr=0 ψ¯2rdrdϕ1/2,
ψ¯r=ψ¯co=J0κ¯rJ0U¯,0r<ρψ¯cl=K0γ¯rK0W¯,ρ<r<.
U¯=ρκ¯, κ¯=k2nco2-β¯21/2,
W¯=ργ¯, γ¯=β¯2-k2ncl21/2,
U¯J1U¯J0U¯=W¯K1W¯K0W¯,
C=22a0W¯J0U¯TVδU2δW2J1U¯m=01+δm0am2Qm1/2,
T=δW2UJ1UJ0U+δU2WK1WK0W-h0ShδU¯J1U¯J0U¯,
Qm=Km+1WKm-1WKm2W-Jm+1UJm-1UJm2U,
1ρ20ρ Jν2κrrdr=12Jν2U-Jν-1UJν+1U,
1ρ20ρ JνκrJνκ¯rrdr=U¯JνUJν-1U¯-UJν-1UJνU¯U2-U¯2,
1ρ2ρ Kν2γrrdr=12Kν-1WKν+1W-Kν2W,
1ρ2ρ KνγrKνγ¯rrdr=W¯KνWKν-1W¯-WKν-1WKνW¯W¯2-W2.

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