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Optics Express

Optics Express

  • Editor: Michael Duncan
  • Vol. 13, Iss. 12 — Jun. 13, 2005
  • pp: 4476–4484
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Theoretical and experimental investigations of macro-bend Losses for standard single mode fibers

Qian Wang, Gerald Farrell, and Thomas Freir  »View Author Affiliations


Optics Express, Vol. 13, Issue 12, pp. 4476-4484 (2005)
http://dx.doi.org/10.1364/OPEX.13.004476


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Abstract

Modeling of macro-bend losses for single mode fibers with multiple cladding or coating layers is presented. Macro-bend losses for standard single mode fibers (SMF28) are investigated theoretically and experimentally, showing that the inner primary coating layer of SMF28 has a significant impact on the bend losses and most of the radiation field is absorbed in the inner primary coating layer of SMF28. The agreement between theoretical calculations and experimental measurements suggests that the so-called elastooptical correction in modeling is not required for SMF28.

© 2005 Optical Society of America

1. Introduction

It is well known that a radiation loss occurs when a single mode fiber is bent. Accurate modeling of this bend loss is essential for the design of fibers employed in optical communications or optical devices based on a bent fiber, such as some forms of optical sensor [1

1. R. C. Gauthier and C. Ross, “Theoretical and experimental considerations for a single-mode fiber-optic bend-type sensor,” Appl. Opt. 36, 6264–6273 (1997). [CrossRef]

]. The simplest model treats a bent fiber as a core-infinite cladding structure [2

2. D. Marcuse, “Curvature loss formula for optical fibers,” J. Opt. Soc. Am. 66, 216–220 (1976). [CrossRef]

,3

3. D. Marcuse, “Bend loss of slab and fiber modes computed with diffraction theory,” IEEE J. Quantum Electron. 29, 2957–2961 (1993). [CrossRef]

]. In fact, a practical fiber with coating layer(s) offering mechanical protection shows quite different bend loss characteristics to those predicted by the simplest model. Existing theoretical calculations of fiber bend losses treat a fiber as a core-cladding-infinite coating structure, when considering the impact of coating layer [4

4. C. Vassallo, “Perturbation of an LP mode of an optical fiber by a quasi-degenerate field: a simple formula,” Opt. & Quantum Electron. 17, 201–205 (1985). [CrossRef]

7

7. L. Faustini and G. Martini, “Bend loss in single-mode fibers, ” J. Lightwave Technol. 15, 671–679 (1997). [CrossRef]

]. As we know most fibers have double coating layers or some fibers themselves have more than one cladding layer, such as depressed-cladding fibers but no existing formulas have been presented for modeling bend losses of these fibers except the prediction of maximum or minimum bend losses conditions in relation to fiber parameters and input wavelength [8

8. A. J. Harris and P.F. Castle, “Bend loss measurement on high numerical aperture single-mode fibers as function of wavelength and bend radius,” J. Lightwave Technol. 4, 34–40 (1986). [CrossRef]

,9

9. R. Morgan, J.S. Barton, P.G. Harper, and J.D.C. Jones, “wavelength dependence of bending loss in mononmode optical fibers:effect of the fiber buffer coating, ” Opt. Lett. 15, 947–949 (1990). [CrossRef] [PubMed]

]. Therefore, calculation of bend losses for a single mode fiber with multiple cladding layers or coating layers based on perturbation theory is firstly presented in Section 2, which can be used for simulation and design of fiber devices with multiple cladding layers or multiple coating layers when macro-bend losses are involved.

2. Theoretical calculations of fiber bend loss

Total loss of a bent fiber includes the pure bend loss in the bent section and the transition loss caused by the mismatch of propagation mode between the bent and the straight sections. For a single mode bent fiber of length L, the pure bend loss can be calculated by [8

8. A. J. Harris and P.F. Castle, “Bend loss measurement on high numerical aperture single-mode fibers as function of wavelength and bend radius,” J. Lightwave Technol. 4, 34–40 (1986). [CrossRef]

]

Ls=10log10(exp(2αL))=8.686αL
(1)

where α is the so-called bend loss coefficient, which is determined by the fiber structure, bending radius and wavelength of the light. Most theoretical investigations on fiber bend losses are focused on calculations of this bend loss coefficient.

Figure 1 gives the schematic cross-section view of a bent fiber with multiple cladding or coating layers. The bending radius is denoted by R. For the q-th cladding layer, the refractive index is nq and the thickness is x q+1-xq. Based on the approximations made in Refs. [4

4. C. Vassallo, “Perturbation of an LP mode of an optical fiber by a quasi-degenerate field: a simple formula,” Opt. & Quantum Electron. 17, 201–205 (1985). [CrossRef]

7

7. L. Faustini and G. Martini, “Bend loss in single-mode fibers, ” J. Lightwave Technol. 15, 671–679 (1997). [CrossRef]

], the field in the cladding layers of the bent fiber is

ψq(x,y)=12π+[Dq(ζ)Bi(Xq)+Hq(ζ)Ai(Xq)]exp(iζy)dζ
(2)

where the notation in Refs. [6

6. H. Renner, “Bending losses of coated single-mode fibers: a simple approach,” J. Lightwave Technol. 10, 544–551 (1992). [CrossRef]

] and [7

7. L. Faustini and G. Martini, “Bend loss in single-mode fibers, ” J. Lightwave Technol. 15, 671–679 (1997). [CrossRef]

] is used and X(x,ζ)=(R2k2nq2)23[β2+ζ2k2nq2(1+2xR)]. Bi and Ai are Airy functions, respectively. For the outermost infinite layer, there is a relationship between DN(ζ) and HN(ζ), namely, HN(ζ)=-jDN(ζ). For any two adjacent layers, according to field-continuous boundary conditions, we have

{Dq(ζ)Bi[Xq(xq,ζ)]+Hq(ζ)Ai[Xq(xq,ζ)]=Dq+1(ζ)Bi[Xq+1(zq,ζ)]+Hq+1(ζ)Ai[Xq+1(zq,ζ)]Dq(ζ)B'i[Xq(zq,ζ)]+Hq(ζ)A'i[Xq(xq,ζ)]=Dq+1(ζ)B'i[Xq+1(xq,ζ)]+Hq+1(ζ)A'i[Xq+1(Xq,ζ)].
(3)

Therefore [D1(ζ)H1(ζ)]=[M11M12M21M22][DN(ζ)HN(ζ)],, considering all the cladding layers, or in a short form D 1(ζ)=GH 1(ζ) consequently.

Fig. 1. ross-section of a bent fiber with multiple cladding layers.

Based on the boundary condition between the first cladding layer and the core layer [7

7. L. Faustini and G. Martini, “Bend loss in single-mode fibers, ” J. Lightwave Technol. 15, 671–679 (1997). [CrossRef]

], we have H1(ζ)=π{GBi[X1(x1,ζ)]+Ai[X1(x1,ζ)]}(γ2+ζ2)12exp[a(γ2+ζ2)12] and with the perturbation theory, the bend loss coefficient can be calculated by

2α=2κ22πβV2K12(aγ)Im(H1(ζ)Ai[X2(0,ζ)]dζ)
(4)

The advantage of this model is that it can not only be used to calculate bend losses of fibers containing only one coating layer as presented in Ref. [5

5. I. Valiente and C. Vassallo, “New formalism for bending losses in coated single-mode optical fibers,” Electron. Lett. 25, 1544–1545 (1989). [CrossRef]

7

7. L. Faustini and G. Martini, “Bend loss in single-mode fibers, ” J. Lightwave Technol. 15, 671–679 (1997). [CrossRef]

], but also it is suitable for fibers with multiple cladding layers (depressed-cladding fibers) or coating layers. The fiber used in Ref. [9

9. R. Morgan, J.S. Barton, P.G. Harper, and J.D.C. Jones, “wavelength dependence of bending loss in mononmode optical fibers:effect of the fiber buffer coating, ” Opt. Lett. 15, 947–949 (1990). [CrossRef] [PubMed]

] has two coating layers and the presented experimental results and theoretical investigations (predicting the maximum or minimum bend losses conditions in relation to fiber parameters and input wavelength) in Ref. [9

9. R. Morgan, J.S. Barton, P.G. Harper, and J.D.C. Jones, “wavelength dependence of bending loss in mononmode optical fibers:effect of the fiber buffer coating, ” Opt. Lett. 15, 947–949 (1990). [CrossRef] [PubMed]

] show that for that fiber, the radiated field penetrates through both the inner and the outer primary coating layers. With the above formulas, theoretical modeling, including the outer primary coating, shows a better agreement between the experimental and theoretical results, by comparison with the case where only the inner layer is considered.

3. Experimental investigations about bend losses for SMF28

Figure 2 gives the experimental setup used for our measurement of fiber bend losses. The optical spectrum analyzer is used instead of an optical power meter because it can measure the bend loss at the peak output wavelength of the tunable laser rather then over a range of wavelengths. The fiber used in the experiment is SMF28, which is a very common fiber and widely used in optical communication systems. It has core, cladding, inner and outer primary coating layers. Corresponding parameters are presented in Table 1.

Fig. 2. Experimental setup for measuring fiber bend loss

Table 1. Parameters of SMF28 at wavelength 1550nm

table-icon
View This Table

Using a bending fiber of length 1~2 m, we measured the bend loss in the bend radius range of 8.5 mm to 12 mm inclusive, in increments of 0.5 mm and in the wavelength range from 1500 nm to 1600 nm. For bend radius smaller than 8.5 mm, the bend fiber is easily broken while for a bending radius larger than 12 mm, the bend loss is too low for reliable and repeatable measurement.

Figure 3 presents typical measured bend losses for SMF28 with or without an absorbing layer applied to the outside of the fiber. The curves for bend losses for a fiber without an absorbing layer outside have random variations that are small relative to the absolute bend loss at a given wavelength. The maximal variation in Fig. 3 is 3dB when the bend loss is 18dB. Further the measured results are not exactly repeatable and differ each time (Fig. 3 gives bend losses for ten measurements). After we coated the fiber with an absorbing layer, these random variations disappear and the measured bend losses also become invariant. This indicates that these random variations are caused by the reflection that occurs at the interface between the outer primary coating layer and air. It also shows that most of the radiated field is absorbed in the coating layers. Only a small amount of radiated field reaches the fiber surface and is reflected back resulting in interference with the propagation mode. Otherwise, according to the measured results and analysis method presented in Ref. [9

9. R. Morgan, J.S. Barton, P.G. Harper, and J.D.C. Jones, “wavelength dependence of bending loss in mononmode optical fibers:effect of the fiber buffer coating, ” Opt. Lett. 15, 947–949 (1990). [CrossRef] [PubMed]

], the curve for bend losses as a function of wavelength should have a periodical oscillatory nature comparable in amplitude to the bend losses themselves due to the reflection of the interface between the outer primary coating layer and air rather than these small random variations in our experiments.

Fig. 3. Measured bend losses for bending radius R=10.5 mm and bent length of 0.66 m.

4. Comparisons between theoretical and experimental results

Detailed comparisons between theoretical and experimental results have been carried out in this section to investigate the accuracy of different models; the impact of the coating layers on bend losses and the so-called elastooptical corrections for modeling in the previously published investigations.

Fig. 4. Measured and calculated bend loss for different bending radii at wavelength a) 1500 nm and b) 1600 nm.

The agreements between experimental measured bend losses and calculated results in Fig. 4 also suggest that the inner primary coating layer absorbs most of the radiated field and the outer primary coating layer has little impact on the bend loss characteristics. In order to verify it, we carry out the calculation of bend losses considering the outer primary coating layer with the formula presented in Section 2, which has the advantage that it can calculate bend losses for more than one coating layer. The calculation treats that the inner primary coating layer as ideally transparent and the outer primary coating layer to be infinite. Figures 5(a) and 5(b) present the corresponding bend losses in wavelength range from 1500 nm to 1600 nm for bending radii R=9 mm and R=10 mm, respectively. The measured bend losses and the calculated results considering only the inner coating layer are also presented in Fig. 5 for comparison. From Fig. 5(a) and 5(b), one can see, that the calculated bend losses considering only the inner coating layer are much closer to the experimentally measured results. The existence for a two coating layer structure of a wave-like variation in the bend loss of the fiber with wavelength is mainly caused by the reflections at the interface between the inner and outer coating layers. The fact that the measured results do not display this wave-like variation support a conclusion that the inner coating layer absorbs most of the radiated field from the cladding.

Fig. 5. Calculated (with one-coating layer and with two-coating layers, respectively) and measured bend losses in wavelength range from 1500 nm to 1600 nm for a) for R=10 mm and b) R=9 mm.

Fig. 6. Calculated (with and without elastooptical correction) and measured bend losses in wavelength range from 1500 nm to 1600 nm for a) for R=10 mm and b) R=9 mm.

5. Conclusion

References

1.

R. C. Gauthier and C. Ross, “Theoretical and experimental considerations for a single-mode fiber-optic bend-type sensor,” Appl. Opt. 36, 6264–6273 (1997). [CrossRef]

2.

D. Marcuse, “Curvature loss formula for optical fibers,” J. Opt. Soc. Am. 66, 216–220 (1976). [CrossRef]

3.

D. Marcuse, “Bend loss of slab and fiber modes computed with diffraction theory,” IEEE J. Quantum Electron. 29, 2957–2961 (1993). [CrossRef]

4.

C. Vassallo, “Perturbation of an LP mode of an optical fiber by a quasi-degenerate field: a simple formula,” Opt. & Quantum Electron. 17, 201–205 (1985). [CrossRef]

5.

I. Valiente and C. Vassallo, “New formalism for bending losses in coated single-mode optical fibers,” Electron. Lett. 25, 1544–1545 (1989). [CrossRef]

6.

H. Renner, “Bending losses of coated single-mode fibers: a simple approach,” J. Lightwave Technol. 10, 544–551 (1992). [CrossRef]

7.

L. Faustini and G. Martini, “Bend loss in single-mode fibers, ” J. Lightwave Technol. 15, 671–679 (1997). [CrossRef]

8.

A. J. Harris and P.F. Castle, “Bend loss measurement on high numerical aperture single-mode fibers as function of wavelength and bend radius,” J. Lightwave Technol. 4, 34–40 (1986). [CrossRef]

9.

R. Morgan, J.S. Barton, P.G. Harper, and J.D.C. Jones, “wavelength dependence of bending loss in mononmode optical fibers:effect of the fiber buffer coating, ” Opt. Lett. 15, 947–949 (1990). [CrossRef] [PubMed]

10.

A. B. Shama, A. H. Al-Ani, and S. J. Halme, “Constant-curvature loss in monomode fibers: an experimental investigation,” Appl. Opt. 23, 3297–3301 (1984). [CrossRef]

11.

K. Nagano, S. Kawakami, and S. Nishida, “ Change of the refractive index in an optical fiber due to external forces,” Appl. Opt. 17, 2080–2085 (1978). [CrossRef] [PubMed]

OCIS Codes
(060.2310) Fiber optics and optical communications : Fiber optics
(060.2430) Fiber optics and optical communications : Fibers, single-mode

ToC Category:
Research Papers

History
Original Manuscript: April 8, 2005
Revised Manuscript: May 19, 2005
Published: June 13, 2005

Citation
Qian Wang, Gerald Farrell, and Thomas Freir, "Theoretical and experimental investigations of macro-bend losses for standard single mode fibers," Opt. Express 13, 4476-4484 (2005)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-12-4476


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References

  1. R. C. Gauthier and C. Ross, �??Theoretical and experimental considerations for a single-mode fiber-optic bend-type sensor,�?? Appl. Opt. 36, 6264-6273 (1997). [CrossRef]
  2. D. Marcuse, �??Curvature loss formula for optical fibers,�?? J. Opt. Soc. Am. 66, 216-220 (1976). [CrossRef]
  3. D. Marcuse, �??Bend loss of slab and fiber modes computed with diffraction theory,�?? IEEE J. Quantum Electron. 29, 2957-2961 (1993). [CrossRef]
  4. C. Vassallo, �??Perturbation of an LP mode of an optical fiber by a quasi-degenerate field: a simple formula,�?? Opt. & Quantum Electron. 17, 201-205 (1985). [CrossRef]
  5. I. Valiente and C. Vassallo, �??New formalism for bending losses in coated single-mode optical fibers,�?? Electron. Lett. 25, 1544-1545 (1989). [CrossRef]
  6. H. Renner, �??Bending losses of coated single-mode fibers: a simple approach,�?? J. Lightwave Technol. 10, 544-551 (1992). [CrossRef]
  7. L. Faustini and G. Martini, �??Bend loss in single-mode fibers, �?? J. Lightwave Technol. 15, 671-679 (1997). [CrossRef]
  8. A. J. Harris and P.F. Castle, �??Bend loss measurement on high numerical aperture single-mode fibers as function of wavelength and bend radius,�?? J. Lightwave Technol. 4, 34-40 (1986). [CrossRef]
  9. R. Morgan, J.S.Barton, P.G. Harper and J.D.C. Jones, �??Wavelength dependence of bending loss in mononmode optical fibers:effect of the fiber buffer coating, �?? Opt. Lett. 15, 947-949 (1990). [CrossRef] [PubMed]
  10. A. B. Shama, A. H. Al-Ani and S. J. Halme, �??Constant-curvature loss in monomode fibers: an experimental investigation, �?? Appl. Opt. 23, 3297-3301 (1984). [CrossRef]
  11. K. Nagano, S. Kawakami and S. Nishida, �??Change of the refractive index in an optical fiber due to external forces, �?? Appl. Opt. 17, 2080-2085 (1978). [CrossRef] [PubMed]

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