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Optics Express

Optics Express

  • Editor: Michael Duncan
  • Vol. 12, Iss. 8 — Apr. 19, 2004
  • pp: 1720–1726
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Form birefringence in UV-exposed photosensitive fibers computed using a higher order finite element method

N. Belhadj, S. LaRochelle, and K. Dossou  »View Author Affiliations


Optics Express, Vol. 12, Issue 8, pp. 1720-1726 (2004)
http://dx.doi.org/10.1364/OPEX.12.001720


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Abstract

The effective index change and form birefringence are calculated in UV-exposed fibers using a high-order vectorial finite element method. The birefringence is compared in optical fibers with and without photosensitive inner cladding.

© 2004 Optical Society of America

1. Introduction

2. The UV-induced refractive index profiles

Cladding-mode-suppression fibers (CMSFs) are manufactured with a photosensitive inner cladding to decrease the losses created by coupling to cladding modes. The parameters of the CMSF, considered in this paper, are as follows: before UV-illumination, the CMSF has a circularly symmetric step-index profile with a core radius ρ=3.05 µm and an index n1=1.4565, a photosensitive inner cladding with a radius of σ=6.25 µm and an index n2=1.444, and an infinite cladding with the same index n2=1.444. In this paper, it is assumed that the photosensitive inner cladding and the fiber core have the same UV-absorption and photosensitive response. More complex photosensitive responses or gratings, leading to radially and azimuthally asymmetric index change profile could be modeled using the same numerical technique. The form birefringence of a one-side illuminated CMSF will be compared to the results obtained for one-side illuminated step-index fiber (SF) having the same core-cladding parameters but without photosensitive cladding. The form birefringence of a one-side illuminated SF is also compared to the form birefringence obtained for a one-side illuminated Photosensitive Cladding Only Fiber (PCOF) with the same core-cladding parameters as the CMSF but without photosensitivity in the core. During exposure, the absorption of the UV-light incident on the side of the fiber results in an asymmetric index change profile in the photosensitive areas. For all fibers, considering illumination along the x-axis, and in the absence of saturation, the refractive index change is assumed to present an exponential decay across the photosensitive area of the fibers [3

3. D. Innis, Q. Zhong, A. M. Vengserkar, W. A. Reed, S. G. Kosinski, and P. J. Lemaire, “Atomic force microscopy study of uv-induced anisotropy in hydrogen-loaded germanosilicate fibers,” Appl. Phys. Lett. 65, 1528–1530 (1994). [CrossRef]

6

6. K. Dossou, S. LaRochelle, and M. Fontaine, “Numerical Analysis of the Contribution of the transverse asymmetry in the photo-induced index change profile to the birefringence of optical fiber,” J. Lightwave Technol. 20, 1463–1469 (2002). [CrossRef]

]:

{δn(x,y)=δnpexp[2α(x+η2y2)]forθrηδn(x,y)=0elsewhere
(1)

where δn p is the peak refractive index change on the side where the UV beam is incident, 2α the asymmetry coefficient, (x, y) the Cartesian coordinates in the fiber transverse plane with the origin on the fiber axis and r=(x 2+y2)1/2. The limits of the photosensitive region are: η=σ for CMSF and PCOF, η=ρ for SF, θ=0 for CMSF and SF, and θ=ρ for PCOF. In the non-photosensitive region, the index change is equal to zero. If the index change were proportional to the absorbed intensity, the parameter α would correspond to the UV-absorption coefficient of the photosensitive region. However, we prefer to refer to α as the asymmetry coefficient because of the modeling complexity of the photosensitive response, which depends nonlinearly on the exposure intensity and time [11

11. H. Patrick and S. L. Gilbert, “Growth of Bragg gratings produced by continuous-wave ultraviolet light in Optical fiber”, Opt. Lett. 18, 1484–1486 (1993). [CrossRef] [PubMed]

]. Figure 1 shows the index profiles in the SF, CMSF and PCOF for 2α=0.2 µm-1 and δnp=0.01. The UV exposure is incident from x<0. The index profile along the x-axis (y=0) is also shown in Fig. 2.

Fig. 1. The refractive index profiles for 2α=0.2 µm-1 and δnp=0.01 in the (a) SF (b) CMSF and (c) PCOF.

3. Numerical method

In the full-vectorial approach, polarization effects are computed by solving the vectorial wave equation ∇×(∇×ψ⃗)-k 2 n 2(x,y)ψ⃗=0 where ψ⃗ is the electric field [12

12. M. Koshiba, S. Maruyama, and K. Hirayama, “A Vector Finite Element Method with the high-order mixed-interpolation-type triangular elements for optical waveguiding problems,” J. Lightwave Technol. 12, 495–502(1994). [CrossRef]

]. In the Vectorial Finite Element Methods (VFEM) used in [6

6. K. Dossou, S. LaRochelle, and M. Fontaine, “Numerical Analysis of the Contribution of the transverse asymmetry in the photo-induced index change profile to the birefringence of optical fiber,” J. Lightwave Technol. 20, 1463–1469 (2002). [CrossRef]

], the transverse field E t is approximated by linear P 1 edge finite elements while the longitudinal component E z is approximated by standard nodal quadratic P 2 finite elements. To increase the accuracy of the results, we use in this paper a Higher Order Vectorial Finite Element Methods (HO-VFEM) [10

10. K. Dossou, Département de mathématique et de statistique, Université Laval, (Québec) Canada G1K 7P4, and M. Fontaine are preparing a manuscript to be called “A high order isoparametric finite element method for computation of waveguide modes.”

], where the component E t is interpolated by P 3 edge finite elements and E z by continuous P 4 finite elements. In order to have a high accuracy in the geometric representation, curvilinear triangles were used to approximate the core-cladding interface. The vector shape functions, especially on the curvilinear triangles, were selected in such a way that the compatibility condition is respected [10

10. K. Dossou, Département de mathématique et de statistique, Université Laval, (Québec) Canada G1K 7P4, and M. Fontaine are preparing a manuscript to be called “A high order isoparametric finite element method for computation of waveguide modes.”

, 13

13. L. Demkowicz and L. Vardapetyan, “Modeling of electromagnetic absorption/scattering problems using hp-adaptive finite elements,” Comput. Methods Appl. Mech. Engrg. 152, 103–124 (1998). [CrossRef]

]. For the unexposed SF fiber, the exact calculation of the effective index gave n eff=1.450469714 at 1550 nm while the VFEM method resulted in n eff=1.450463617, which represents an error of 0.6×10-5, and the HO-VFEM method resulted in n eff=1.450469727, which represents an error of 1.3×10-8. In UV-exposed fibers, the birefringence calculations reported in [7

7. N. Belhadj, K. Dossou, X. Daxhelet, S. LaRochelle, S. Lacroix, and M. Fontaine, “A comparative study of numerical methods for the calculation of the birefringence of UV-illuminated fibers”, OSA Technical Digest : Conference on Bragg Gratings, Photosensitivity and Poling in Glass Waveguides, Monterey, California, USA, September 1–3, (paper MD20) 112–114 (2003).

] show a remarkable agreement between the HO-VFEM and the Scalar Finite Difference method with Polarization Correction. The difference between the calculated birefringence values is smaller than 1.4×10-7 for an effective index change, δneff, lowers than 5×10-3. The results presented in [7

7. N. Belhadj, K. Dossou, X. Daxhelet, S. LaRochelle, S. Lacroix, and M. Fontaine, “A comparative study of numerical methods for the calculation of the birefringence of UV-illuminated fibers”, OSA Technical Digest : Conference on Bragg Gratings, Photosensitivity and Poling in Glass Waveguides, Monterey, California, USA, September 1–3, (paper MD20) 112–114 (2003).

] also show that the VFEM underestimates the form birefringence. For δneff=5×10-3, the difference between the form birefringence values calculated by using the VFEM and the HO-VFEM is of the order of 1×10-6 for 2α=0.2µm-1 and 2.5×10-6 for 2α=0.4µm-1. To improve the accuracy of the calculations, the results presented in this paper are therefore computed using HO-VFEM.

4. Results

For the index profiles shown in Fig. 1, we calculated the normalized electric fields of the modes of the one-side exposed fibers. The modes are shown in Fig. 2 where the field profiles are superposed on their respective refractive index profiles. In all three cases, the electric field has an asymmetric profile shifted towards the side of the fiber directly exposed to the UV. For the PCOF, the overlap between index perturbation and the guided mode is smaller because there is no index change in the fiber core. That explains the smaller effective index change and the smaller shift of the electric field profile observed in this type of fiber compared to the SF and CMSF.

The effective index change, δneff,x, was calculated as a function of the peak refractive index change, δnp, for various asymmetry coefficients. The results for the CMSF and PCOF are compared to that of the SF in Fig. 3. The effective index change varies almost linearly with a weak quadratic contribution for the three illuminated fiber types. In the case of the PCOF, the curvature is larger. Furthermore, the comparison of the CMSF and SF (Fig. 3(a)) also shows that a higher peak index change is required in the CMSF to reach the same δneff,x than in the SF. This is expected because in CMSF the peak index change first occurs in the photosensitive cladding where the overlap with the guided mode is smaller. This also explains the fact that an even higher peak index change is required in a PCOF to reach the same δneff (Fig. 3(b)), because in this case, there is no refractive index increase in the core. Similar results are obtained for the effective index changes of the mode polarized along the y-axis. We also notice that, as the asymmetry coefficient increases, higher peak index change are required to obtain the same δneff,x. In [6

6. K. Dossou, S. LaRochelle, and M. Fontaine, “Numerical Analysis of the Contribution of the transverse asymmetry in the photo-induced index change profile to the birefringence of optical fiber,” J. Lightwave Technol. 20, 1463–1469 (2002). [CrossRef]

] it was observed that, for a given peak refractive index change, there is a value of the asymmetry coefficient (2α=0.3 to 0.4 µm-1) that will maximize the photo-induced birefringence in SF.

The UV-induced birefringence is determined using B=n eff,x-n eff,y where n eff,x and n eff,y are the effective indices of the fundamental mode polarized along the x- and y-axis. Notice that, in the SF case, the slow axis corresponds to the x-axis and the fast axis to the y-axis. However, in the case of PCOF and CMSF, this notion depends on the value of the index change. As is shown below, for small changes of δnp, the x-axis is associated to the slow axis in the case of the CMSF and to the fast axis in the case of the PCOF. For higher index changes, y becomes the slow axis for the CMSF and the fast axis for the PCOF.

Fig. 2. The normalized electric field (solid line) superposed to the asymmetric refractive index profile (dashed line) for δnp=0.01 and 2α=0.2 µm-1. We show in (a) the SF, in (b) the CMSF and in (c) the PCOF fibers.
Fig. 3. The effective index change as a function of the peak refractive index change for (a) CMSF (solid lines) and SF (dashed lines) and (b) PCOF (solid lines) and SF (dashed lines).

In Fig. 4, we present the birefringence calculations for various values of 2α. The absolute value of the form birefringence of the SF shows the expected quasi-quadratic dependence on δneff [5

5. H. Renner, “Effective index increase, form birefringence and transmission losses in UV-illuminated photosensitive fiber,” Opt. Express 9, 546–560 (2001) http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-11-546. [CrossRef] [PubMed]

7

7. N. Belhadj, K. Dossou, X. Daxhelet, S. LaRochelle, S. Lacroix, and M. Fontaine, “A comparative study of numerical methods for the calculation of the birefringence of UV-illuminated fibers”, OSA Technical Digest : Conference on Bragg Gratings, Photosensitivity and Poling in Glass Waveguides, Monterey, California, USA, September 1–3, (paper MD20) 112–114 (2003).

]. For small values of the asymmetry coefficient (2α≤0.1 µm-1), the birefringence of CMSF displays a similar quasi-quadratic dependence. For higher asymmetry coefficients, 0.3 µm-1≤2α≤0.4 µm-1, the birefringence reaches a maximum for 1×10-3 < δneff < 3×10-3. After this maximum, the birefringence decreases and goes through zero. The change in the birefringence sign indicates that there is a cross-over between the effective indices of the two orthogonally polarized modes: the mode polarized along the y-axis becomes associated to the slow axis and the mode polarized along the x-axis becomes associated to the fast axis. This cross-over occurs when the refractive index in the cladding reaches a value close to that of the core (Fig. 5), a situation not likely to occur in experiments because of the saturation of the photosensitive response.

In the PCOF case, the form birefringence has a quasi-quadratic shape for an asymmetry coefficient smaller than 2α=0.1µm-1 and a parabolic shape for larger asymmetry coefficients. The form birefringence reaches an extremum for 0.5×10-3<δneff<1.×10-3. After this extremum, the form birefringence increases quickly and goes through zero. That reflects the same cross-over phenomenon observed in the case of the CMSF.

Fig. 4. The form birefringence as a function of the effective index of δneff,x for (a) the CMSF (solid lines) and SF (dashed lines) and (b) the PCOF (solid lines) and SF (dashed lines).

Fig. 5. The refractive index profile of the CMSF before and after illumination for (a) (2α=0.4 µm-1, δneff,x=2.29×10-3) and, (b) (2α=0.4 µm-1, δneff,x=2.88×10-3).

Conclusion

We presented effective index and birefringence calculations using a high order vectorial finite element method in UV side-illuminated specialty fibers with a photosensitive region in the cladding. The calculations show that, in cladding mode suppression fibers and in photosensitive cladding fibers, the birefringence has a quasi-quadratic form for small asymmetry coefficients and a parabolic shape for higher values. For more usual values of effective index change, δneff<1.×10-3, the form birefringence can be significantly higher in the photosensitive cladding fiber (without photosensitivity in the core) than in other fiber types. Further information about photo-induced birefringence could be obtained by developing a more complete physical model of the refractive index change that would consider saturation and anisotropy. Both these effects could be taken into account in the calculations performed using the high-order VFEM.

References

1.

R. Gafsi and M. A. El-Sherif, “Analysis of Induced-Birefringence Effects on Fiber Bragg Gratings,” Opt. Fiber Technol. 6, 299–323 (2000). [CrossRef]

2.

T. Erdogan and V. Mizrahi, “Characterization of UV-induced birefringence in photosensitive Ge-doped silica optical fibers,” J. Opt. Soc. Am. B 11, 2100–2105 (1994). [CrossRef]

3.

D. Innis, Q. Zhong, A. M. Vengserkar, W. A. Reed, S. G. Kosinski, and P. J. Lemaire, “Atomic force microscopy study of uv-induced anisotropy in hydrogen-loaded germanosilicate fibers,” Appl. Phys. Lett. 65, 1528–1530 (1994). [CrossRef]

4.

A. M. Vengsarkar, Q. Zhong, D. Innis, W. A. Reed, P. J. Lemaire, and S. G. Kosinski “Birefringence reduction in side-written photoinduced fiber devices by a dual-exposure method,” Opt. Lett. 19, 1260–1262 (1994). [CrossRef] [PubMed]

5.

H. Renner, “Effective index increase, form birefringence and transmission losses in UV-illuminated photosensitive fiber,” Opt. Express 9, 546–560 (2001) http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-11-546. [CrossRef] [PubMed]

6.

K. Dossou, S. LaRochelle, and M. Fontaine, “Numerical Analysis of the Contribution of the transverse asymmetry in the photo-induced index change profile to the birefringence of optical fiber,” J. Lightwave Technol. 20, 1463–1469 (2002). [CrossRef]

7.

N. Belhadj, K. Dossou, X. Daxhelet, S. LaRochelle, S. Lacroix, and M. Fontaine, “A comparative study of numerical methods for the calculation of the birefringence of UV-illuminated fibers”, OSA Technical Digest : Conference on Bragg Gratings, Photosensitivity and Poling in Glass Waveguides, Monterey, California, USA, September 1–3, (paper MD20) 112–114 (2003).

8.

L. Dong, G. Qi, M. Marro, V. Bhatia, L. L. Hepburn, M. Swan, A. Collier, and D. L. Weidman “Suppression of Cladding Mode Coupling Loss in Fiber Bragg Gratings,” J. Lightwave Technol. 18, 1583–1590 (2000). [CrossRef]

9.

L. Dong, W. H. Loh, J.E. Caplen, and J.D. Minelly, “Efficient single-frequency fiber lasers with novel photosensitive Er/Yb optical fibers,” Opt. Lett. 22, 694–696 (1997). [CrossRef] [PubMed]

10.

K. Dossou, Département de mathématique et de statistique, Université Laval, (Québec) Canada G1K 7P4, and M. Fontaine are preparing a manuscript to be called “A high order isoparametric finite element method for computation of waveguide modes.”

11.

H. Patrick and S. L. Gilbert, “Growth of Bragg gratings produced by continuous-wave ultraviolet light in Optical fiber”, Opt. Lett. 18, 1484–1486 (1993). [CrossRef] [PubMed]

12.

M. Koshiba, S. Maruyama, and K. Hirayama, “A Vector Finite Element Method with the high-order mixed-interpolation-type triangular elements for optical waveguiding problems,” J. Lightwave Technol. 12, 495–502(1994). [CrossRef]

13.

L. Demkowicz and L. Vardapetyan, “Modeling of electromagnetic absorption/scattering problems using hp-adaptive finite elements,” Comput. Methods Appl. Mech. Engrg. 152, 103–124 (1998). [CrossRef]

OCIS Codes
(060.0060) Fiber optics and optical communications : Fiber optics and optical communications
(060.2340) Fiber optics and optical communications : Fiber optics components
(060.2430) Fiber optics and optical communications : Fibers, single-mode

ToC Category:
Research Papers

History
Original Manuscript: March 16, 2004
Revised Manuscript: April 6, 2004
Published: April 19, 2004

Citation
N. Belhadj, S. LaRochelle, and K. Dossou, "Form birefringence in UV-exposed photosensitive fibers computed using a higher order finite element method," Opt. Express 12, 1720-1726 (2004)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-8-1720


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References

  1. R. Gafsi, and M. A. El-Sherif, �??�??Analysis of Induced-Birefringence Effects on Fiber Bragg Gratings,�??�?? Opt. Fiber Technol. 6, 299-323 (2000). [CrossRef]
  2. T. Erdogan, and V. Mizrahi, �??�??Characterization of UV-induced birefringence in photosensitive Ge-doped silica optical fibers,�??�?? J. Opt. Soc. Am. B 11, 2100-2105 (1994) [CrossRef]
  3. D. Innis, Q. Zhong, A. M. Vengserkar, W. A. Reed, S. G. Kosinski, and P. J. Lemaire, �??�??Atomic force microscopy study of uv-induced anisotropy in hydrogen-loaded germanosilicate fibers,�??�?? Appl. Phys. Lett. 65, 1528-1530 (1994) [CrossRef]
  4. A. M. Vengsarkar, Q. Zhong, D. Innis, W. A. Reed, P. J. Lemaire, and S. G. Kosinski �??�??Birefringence reduction in side-written photoinduced fiber devices by a dual-exposure method,�??�?? Opt. Lett. 19, 1260-1262 (1994). [CrossRef] [PubMed]
  5. H. Renner, �??�??Effective index increase, form birefringence and transmission losses in UV-illuminated photosensitive fiber,�??�?? Opt. Express 9, 546-560 (2001) <a href= "http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-11-546">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-11-546</a> [CrossRef] [PubMed]
  6. K. Dossou, S. LaRochelle, and M. Fontaine, �??�??Numerical Analysis of the Contribution of the transverse asymmetry in the photo-induced index change profile to the birefringence of optical fiber,�??�?? J. Lightwave Technol. 20, 1463-1469 (2002). [CrossRef]
  7. N. Belhadj, K. Dossou, X. Daxhelet, S. LaRochelle, S. Lacroix, and M. Fontaine, �??�??A comparative study of numerical methods for the calculation of the birefringence of UV-illuminated fibers�??�??, OSA Technical Digest : Conference on Bragg Gratings, Photosensitivity and Poling in Glass Waveguides, Monterey, California, USA, September 1-3, (paper MD20) 112-114 (2003).
  8. L. Dong, G. Qi, M. Marro, V. Bhatia, L. L. Hepburn, M. Swan, A. Collier, and D. L. Weidman �??�??Suppression of Cladding Mode Coupling Loss in Fiber Bragg Gratings,�??�?? J. Lightwave Technol. 18, 1583-1590 (2000). [CrossRef]
  9. L. Dong, W. H. Loh, J.E. Caplen, and J.D. Minelly, �??�??Efficient single-frequency fiber lasers with novel photosensitive Er/Yb optical fibers,�??�?? Opt. Lett. 22, 694-696 (1997). [CrossRef] [PubMed]
  10. K. Dossou, Département de mathématique et de statistique, Université Laval, (Québec) Canada G1K 7P4, and M. Fontaine are preparing a manuscript to be called "A high order isoparametric finite element method for computation of waveguide modes."
  11. H. Patrick and S. L. Gilbert, "Growth of Bragg gratings produced by continuous-wave ultraviolet light in Optical fiber", Opt. Lett. 18, 1484-1486 (1993). [CrossRef] [PubMed]
  12. M. Koshiba, S. Maruyama, and K. Hirayama, �??�??A Vector Finite Element Method with the high-order mixed-interpolation-type triangular elements for optical waveguiding problems,�??�?? J. Lightwave Technol. 12, 495-502(1994) [CrossRef]
  13. L. Demkowicz, and L. Vardapetyan, �??�??Modeling of electromagnetic absorption/scattering problems using hp-adaptive finite elements,�??�?? Comput. Methods Appl. Mech. Engrg. 152, 103-124 (1998). [CrossRef]

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