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Studying the effect of a central dip on the performance of a dispersion compensated fibre
1 February 2000
Optics Communications 174 Ž2000. 405–411 www.elsevier.comrlocateroptcom
Studying the effect of a central dip on the performance of a dispersion compensated fibre R. Tewari ) , M. Basu, H.N. Acharya Department of Physics, Indian Institute of Technology, Kharagpur 721302, India Received 2 June 1999; received in revised form 26 October 1999; accepted 27 October 1999
Abstract The spot size technique has been used to study the effect of a central dip on the performance of a dispersion compensated fibre ŽDCF.. A Gaussian dip around the axis of the fibre core has been considered. Such a type of dip in the refractive index profile is obtained during the fibre fabrication by modified chemical vapour deposition ŽMCVD. technique. The percentage change in figure of merit Ž dFOMrFOM Ž%.. of a DCF with a central dip with respect to a DCF having no dip has been estimated for different dip depths and dip widths, respectively. It is seen that for larger values of bend radii with large negative dispersion values the magnitude of dFOMrFOM Ž%. changes rapidly with change in other fibre losses, d a r , than in case of smaller bend radii with small dispersion values. q 2000 Elsevier Science B.V. All rights reserved. Keywords: Dispersion compensated fibre; Spot size technique; Figure of merit; Modified chemical vapour deposition technique
1. Introduction One of the methods of choice for dispersion compensation is to use the single mode negative dispersion fibres. With the aim of obtaining large negative dispersion coefficient Ž Dc . and figure of merit Ž FOM ., a large number of DCFs have been investigated having different types of designs, supporting single mode or higher order modes w1–7x. In this paper, we have studied the effect of a central Gaussian refractive index dip w8–10x on the performance of a single mode DCF. The central dip in the refractive index profile is obtained during the fibre fabrication by modified chemical vapour deposition
) Corresponding author. Fax: q91-3222-55303; e-mail: [email protected]
ŽMCVD. technique w8–10,18x. In this paper, a DCF having a central Gaussian refractive index dip w8–10x in the core is designed and optimized by using spot size optimization technique w6,7,11–14,16x. The spot size technique used here has already been shown to work for dispersion shifted single clad w12x and multiclad fibres with segmented core w11x and dual shape core w12x profiles. In recent papers w6,7x, the widely used step index DCF has been designed and optimised, using this technique w6,7,11–14,16x with the aim of attaining maximum negative dispersion Ž Dc . and large value of FOM. According to the spot size technique w6,7,11–14,16x for a given value of W w6,7,11–15x and dispersion, to achieve the lowest bending loss w6,7,11–14,16,17x in the single mode region, the ratio W` rW should be optimised by changing the core radius, a, and the relative refractive index difference, D. At a given value of W, the
0030-4018r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 Ž 9 9 . 0 0 6 5 8 - 6
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R. Tewari et al.r Optics Communications 174 (2000) 405–411
so achieved value of W` w6,7,11–14,16,17x can be used to calculate the optimum bend loss Ž a b . w6,7,11–14,16,17x and thereafter the bend loss–dispersion diagram can be obtained theoretically. In this paper, dispersion compensated fibres having a central dip and without dip have been designed and optimised using spot size technique w6,7,11–14,16x. It is seen that for a given W and bend radius Ž R ., bend loss increases with negative dispersion values for both the fibres i.e., with and without dip, respectively. It is also seen that at a given dispersion value, bend loss is more for larger values of dip width d˜Žs dra. as compared to the smaller dip widths of a DCF. Thus, at a given value of dispersion, FOM can be calculated, which requires inclusion of bending loss Ž a b . as well as other losses Ž a r . which include Rayleigh scattering loss and losses associated with fibre manufacturing techniques such as absorption losses etc. w1,3x. Instead of estimating absolute value of FOM for the fibres having dip, the percentage change in FOM Ži.e., dFOMrFOM Ž%.. of the fibres has been estimated by taking into account the corresponding bending losses Ž a b . and other losses Ž a r . associated with the fibre having central dip in the core and that of the fibre having no central dip, respectively. Calculations show that for larger bend radii of curvature Ž R . and larger values of dispersion Ž Dc ., the dFOMrFOM Ž%. is positive for smaller value of d a r Žs a rŽwithout dip. y a rŽdip. . but becomes negative for larger values of d a r . This implies that initially FOM of a fibre with dip is smaller as compared to FOM of a fibre without dip but for larger values of d a r the FOM of a fibre with dip increases as compared to FOM without dip. On the other hand, for smaller bend radii of curvature dFOMrFOM Ž%. is always positive with change in d a r implying that FOM of a fibre with dip is always smaller than the FOM of a fibre without dip. Section 2 of this paper describes the refractive index profile of a DCF with a central dip on the axis. The next section deals with the theoretical background for designing such types of fibres with dip and without dip. The percentage change in FOM, i.e., dFOMr FOM Ž%. of a fibre with a central dip in the core with respect to a fibre having no dip is estimated in Section 3. Finally, all the results obtained from this technique are discussed and summarized in Section 4 and Section 5, respectively.
2. Refractive index profie of a DCF with and without a central dip in the core In the presence of a Gaussian central dip at the axis of the fibre, as shown in Fig. 1, the index profile can be described as w8–10x: n2 Ž R .
s n2o Ž R . y Ž n12 y n22 . p˜ Ž exp Ž y R 2 r d˜2 yexp Ž y1r d˜2 .
R-1
s n 22
Ž 1.
R)1
where d˜Žs dra. and p˜ are the dip width and the dip depth, respectively and n2o Ž R . represents the refractive index profile, in the absence of the axial dip, of a parabolic index fibre which is given by w8–10x: n 2o Ž R .
s n12 w 1 y 2 D R 2 x
R-1
s n 22
R)1
Ž 2.
where R s rra, a is the radius of the core, n1 and n 2 represent maximum refractive index of the central core Ž 0 - r - a. and cladding Ž r ) a., respectively and D s Ž n1 y n 2 .rn1 is the relative index difference.
3. Theoretical background to design a DCF with and without a central dip The characteristics of a DCF with a central dip in the core can be described in terms of spot sizes W w6,7,11–15x and W` w6,7,11–14,16,17x. We consider the following two definitions for the mode spot sizes.
Fig. 1. Refractive index profile of a DCF having a Gaussian dip at the centre of the core.
R. Tewari et al.r Optics Communications 174 (2000) 405–411
The first spot size W of the near field, known as Petermann y2 spot size, is defined as w6,7,11–15x: `
H0 c
2 2
W s
2
Ž r .dr
`
H0
Ž 3.
2
Ž d crd r . rd r
d
2
ž / W
d
Ž 4.
We see from Eq. Ž4. that to achieve low splice loss a d , W should be as large as possible. It should also be mentioned that in order to suppress the non-linear effects in DCFs the effective core area Ž A eff . should be large which is directly proportional to the spot size w5,19x. The second spot size considered is W` w6,7,11–14,16,17x of the near field which can be defined as: W`2 s
l
Ž 5.
p n1 Ž bo y k o n 2 .
where bo is the propagation constant of the fundamental mode, k o Žs 2prl. is the free space wave number, n1 and n 2 are the refractive indices on the central axis and in the cladding level, respectively ŽFig. 1. and l is the operating wavelength. W` w6,7,11–14,16,17x is also called the Petermann y3 spot size which is related to the pure bending loss of the fibre. The pure bend loss coefficient g b Žper unit length., in a fibre with a radius of curvature R, is given by w6,7,11–14,16,17x:
g b s f Ž n Ž r . , l.
W`3
( ½ R
exp
We see from Eqs. Ž6. and Ž7. that to achieve the low bending loss a b , W` should be as small as possible. However it is seen that W` is always larger than W in the single mode region and we can write w6,11– 14,16x: W` ) W .
where c Ž r . represents the transverse field distribution of the LP01 mode. The spot size W is inversely proportional to the rms spot size that can be measured from the angular dependence of the far field intensity. The spot size W is related to the transverse offset at a splice between the two fibres. The splice loss a d Žin decibels., due to transverse offset Ž d . at a splice or joint between two fibres, is fully characterized by the Petermann y2 spot size W and can be expressed as w6,7,11–15x:
a d s 4.343
407
y4R l2 3p
2
n12 W`3
5
Ž 8.
Thus, we see from Eqs. Ž3. – Ž8. that for small bending loss, W` should be as small as possible w6,7,11–14,16,17x, while for small splice loss due to transverse offset, W should be as large as possible w6,7,11–15x. Thus for a given W and dispersion, to achieve low bending loss and splice losses along with the low non-linear effects, the design criteria requires to achieve a ratio W` rW as close to unity as possible w6,7,11–14,16,17x. An important parameter, known as figure of merit Ž FOM ., which is the amount of dispersion per unit loss with practical units of psrnm dB, can be used to characterise the DCFs w2,3,6,7x. The DCFs should be optimized to minimize the total loss of the fibre. The compensating fibre loss a c Žin dBrkm. can be described as the sum of bend loss Ž a b . and the other losses Ž a r . which include Rayleigh scattering loss and losses associated with the fibre manufacturing techniques such as absorption losses, etc. Thus, a c can be written as:
ac s a b q ar .
Ž 9.
Thus, the figure of merit Ž FOM . of a DCF can be written as: FOM s
Dc
a b q ar
.
Ž 10 .
The percentage change in figure of merit Ž dFOMrFOM Ž%.. of a fibre with a central dip in the core with respect to the FOM of a fibre without any central dip can be estimated from the above Eqs. Ž9. and Ž10. in terms of a b and a r and is given by: dFOM FOM
Ž %. s
yŽ d ar q d a b .
a rŽ with dip. q a bŽ with dip.
= 100
Ž 11 .
where .
Ž 6.
Thus the pure bend loss a b Žin dBrkm. is given by:
d a b s a bŽ without dip. y a bŽ with dip.
a b s 4.343g b .
d a r s a rŽ without dip. y a rŽ with dip.
Ž 7.
¶ • ß
dFOMs FOMŽ without dip . y FOMŽ with dip .
Ž 12 .
408
R. Tewari et al.r Optics Communications 174 (2000) 405–411
where the bend loss Ž a b . of the DCF with dip and without dip have been obtained using the spot size technique w6,7,11–14,16x. The other losses Ž a rŽw ith dip. . of the DCFs have been obtained by decreasing an amount d a r from the value of the other losses Ž a rŽw ithout dip. . of the DCF w1,3x as explained in Eq. Ž12.. It should be mentioned that the fabricated DCFs have the other losses, a r which include Rayleigh scattering loss and losses associated with fibre manufacturing techniques such as absorption losses etc., lying between 0.3 to 0.5 dBrkm w1,3x and the corresponding DCF with central dip are supposed to have less value of a r due to less amount of GeO 2 concentration at and around the centre of the core. The percentage change in FOM which is an important factor to study the effect of central dip on the characteristics of a DCF has been estimated from Eqs. Ž11. and Ž12.. 4. Results and discussions Refractive index profiles obtained from MCVD technique have been found to have a central dip at the axis of the core of the fibre ŽFig. 1.. In order to design and optimize the fibres in the presence of a central dip in the core, the scalar wave equation was solved numerically using the spot size optimization technique w6,7,11–14,16x. For a given value of dispersion, Dc , and spot size, W, and for different values of dip widths Ž d˜. and dip depths Ž p˜ ., the ratio W` rW has been optimized, by adjusting the core radius Ž a. and the relative refractive index difference Ž D .. This yielded the optimized value of W` which has been used to calculate the bend loss Žcf Eqs. Ž6. and Ž7.. at different dispersion values for typical value of spot size W s 3.0 mm. Fig. 2 shows the plot of bend loss in dBrkm against negative dispersion value Žpsrkm nm., for DCFs having central dip in the core along with the DCFs without any dip in the core. The curves have been plotted for two different dip widths d˜s 0.3 and 0.5, respectively, with dip depths p˜ remaining constant and equal to 0.5. It is seen during calculation that the variation of dip depths, p˜ does not make any significant change in bend loss when dip width is kept constant. It should be mentioned that the measured dip width Ž d˜. and dip depth Ž p˜ . in fabricated fibres are typically in the range of 0.28–0.33 and 0.34–0.72, re-
Fig. 2. Bend loss characteristics as a function of negative disper˜ 0.3, ps sion of a DCF with central dip in the core Ž ds ˜ 0.5 and ˜ 0.5, ps ds ˜ 0.5. as well as without any dip in the core, for spot size W s 3.0 mm and bend radii Rs 2, 3 and 4 cm, respectively.
spectively w18x. The values of dip width Ž d˜. and dip depth Ž p˜ . of the DCF considered here are of the same order with the typical values of dip width Ž d˜. and dip depth Ž p˜ . reported earlier w18x. In Fig. 2, the curves have been drawn for the typical value of the spot size W s 3.0 mm w6,7x and bend radii of curvature Ž R s 2, 3 and 4 cm. w6,7x. In order to have the quantitative values we have given a table ŽTable 1. to show the values of the optimised ratio W` rW along with the bend losses Ž a b . of the DCFs with a central dip in the core as well as without any dip in the core for given values of dispersion and bend radii Ž R . at spot size W s 3.0 mm. It is seen from Fig. 2 that at smaller negative dispersion values, the bend loss Ž a b . is negligible for larger bend radius Ž R s 4 cm. and on the other hand for smaller bend radii Ž R s 2 cm and 3 cm. bend loss is more dominant. In this figure it is also seen that for a particular bend radius Ž R ., the bend loss is more for larger values of dip widths Ž d˜. than in case of smaller values of d.˜ Thus, from the view point of bend loss only, it can be said the presence of dip has degraded the performance of a DCF.
R. Tewari et al.r Optics Communications 174 (2000) 405–411
409
Table 1 Difference in bend losses Ž d a b . of DCFs with a central dip and without dip at different values of negative dispersion Ž Dc . when the spot size W is fixed at 3.0 mm Dip parameters
Dc Žpsrkm nm.
W` rW
R Žcm.
a b ŽdBrkm.
without dip
y30
1.888
2
1.2
–
d˜s 0.3, p˜ s 0.5
y30
1.899
2
2.0
y0.8
d˜s 0.5, p˜ s 0.5 without dip
y30 y50
1.916 2.142
2 3
2.8 0.35
y1.6 –
d˜s 0.3, p˜ s 0.5
y50
2.145
3
0.63
y0.28
d˜s 0.5, p˜ s 0.5 without dip
y50 y60
2.150 2.254
3 4
0.90 0.075
y0.55 –
d˜s 0.3, p˜ s 0.5
y60
2.261
4
0.1
y0.025
d˜s 0.5, p˜ s 0.5
y60
2.264
4
0.175
y0.1
At a given value of dispersion and bend radius, FOM of the fibres without a central dip has been calculated which requires inclusion of bending loss Ž a b . as well as other losses Ž a r . which include Rayleigh scattering loss and losses associated with fibre manufacturing techniques such as absorption losses, etc. w1,3x. Table 2 shows the FOM values of the DCFs without central dip in the core for different dispersion values Ž Dc s y30, y50 and y60 psrkm nm. and bend radii Ž R s 2, 3 and 4 cm. at spot size W s 3.0 mm. This is needed to estimate the percentage change in figure of merit Ž FOM . due to the presence of a dip at the central axis of a fibre with respect to a fibre having no dip at the centre. In Fig. 3Ža. and Fig. 3Žb., the percentage changes in FOM, i.e., dFOMrFOM Ž%. is plotted against the change
Table 2 Figure of merit Ž FOM . of a DCF without any dip in the core at different values of negative dispersion Ž Dc . and bend radii Ž R . when the spot size W is fixed at 3.0 mm Dc R Žcm. a bŽwithout dip. a rŽwithout dip. FOMŽwithout dip. Žpsrkm nm. ŽdBrkm. ŽdBrkm. Žpsrnm dB. y30 y30 y30 y50 y50 y50 y60 y60 y60
2 2 2 3 3 3 4 4 4
1.2 1.2 1.2 0.35 0.35 0.35 0.075 0.075 0.075
0.5 0.4 0.3 0.5 0.4 0.3 0.5 0.4 0.3
17.6 18.8 20.0 58.8 66.7 76.9 104.3 126.3 160.0
d a b ŽdBrkm.
in other losses Ž d a r . for different bend radii of curvature Ž R s 2, 3 and 4 cm. and different values of dispersion coefficients Ž Dc s y30, y50 and y60 psrkm nm.. The corresponding values of a b and Ž d a b s a bŽwithout dip. y a bŽwith dip. . for the two set of parameters d˜s 0.3, p˜ s 0.5 and d˜s 0.5, p˜ s 0.5, respectively have been taken from Table 1. The figures show that dFOMrFOM Ž%. values are changing in a range of q50 to y90 with the value of d a r when the other losses Ž a rŽwithout dip. . are fixed at 0.5, 0.4 and 0.3 dBrkm, respectively w1,3x. It should be mentioned that the fabricated DCFs have the other losses, a r , which include Rayleigh scattering loss and losses associated with fibre manufacturing techniques such as absorption losses, etc., lying between 0.3 dBrkm and 0.5 dBrkm w1,3x and the corresponding DCF with central dip, are supposed to have a lower value of a r due to the lower amount of GeO 2 concentration at the centre of the core. Comparison of Fig. 3Ža. and Fig. 3Žb. shows that at a given value of a rŽw ithout dip. , d a r and R, when we consider smaller dip width Ž d˜s 0.3., the dFOMr FOM Ž%. is less compared to the case of a DCF with larger dip width Ž d˜s 0.5.. This is due to the fact that the bend loss is smaller in case of d˜s 0.3 than in the case of d˜s 0.5 for a given set of dispersion and bend radius Žcf. Fig. 2 and Table 1.. The above figures also show that for larger bend radii as well as for larger dispersion coefficients, there is more rapid change in dFOMrFOM Ž%. with change in the magnitude of d a r than in case of smaller bend radii and smaller dispersion values of
410
R. Tewari et al.r Optics Communications 174 (2000) 405–411
and Fig. 3Žb. also show that for bend radii of curvature Ž R s 2 cm and 3 cm. dFOMrFOM Ž%. is always positive with the change in other fibre losses, d a r , implying that FOM of a fibre with dip is always smaller than the FOM of a fibre without dip, while for larger bend radius i.e., R s 4 cm and dispersion Ž Dc s y60 psrkm nm., the dFOMrFOM Ž%. is initially positive for smaller value of d a r but becomes negative for larger values of d a r . This implies that initially FOM of a fibre with dip is smaller as compared to FOM of a fibre without dip but for larger values of d a r Ž) 0.02 dBrkm in Fig. 3Ža. and ) 0.10 dBrkm in Fig. 3Žb.., the FOM of a fibre with dip increases as compared to FOM without dip. It should be mentioned that in order to obtain negative value of dFOMrFOM Ž%. throughout d a r values i.e., d a r G 0, d a b has to be zero at d a r s 0 Žcf. Eqs. Ž11. and Ž12... According to Fig. 2 Žand Table 1. this is possible only either for a straight fibre or for a fibre having no dip.
5. Conclusion
Fig. 3. Ža. The variation of the percentage change in figure of merit Ž dFOMr FOM in Ž%.. against change in other losses Ž d a r . for different bend radii Ž R . and negative dispersion values Ž Dc . of ˜ 0.3, ps a DCF with a central dip Ž ds ˜ 0.5. in the core. Žb. The variation of the percentage change in figure of merit Ž dFOMr FOM in Ž%.. against change in other losses Ž d a r . for different bend radii Ž R . and negative dispersion values Ž Dc . of a ˜ 0.5, ps DCF with a central dip Ž ds ˜ 0.5. in the core.
the DCF. This is because at larger bend radii Ž R ., the bend loss Ž a b . is negligible Žcf. Fig. 2 and Table 1. in comparison to other losses Ž a rŽw ith dip. . and the percentage change factor Ž dFOMrFOM Ž%.. will be dominated only by d a r as d a b is also comparatively small, whereas for smaller bend radii and smaller dispersion values, the bend loss Ž a b . is already large Žcf. Fig. 2 and Table 1. and therefore the increase in d a r has less effect on dFOMrFOM Ž%. as d a b is also comparable or more. Fig. 3Ža.
Using spot size based optimization technique, the bend loss–dispersion diagram for a DCF with central dip as well as without central dip in the core has been obtained. Using this diagram, the effect of a central dip in the core, on the performance of a DCF has been estimated in terms of percentage change in figure of merit Ži.e., dFOMr FOM Ž%... It is seen that for larger bend radii of curvature initially dFOMrFOM Ž%. is positive for smaller value of d a r but becomes negative for larger values of d a r whereas dFOMrFOM Ž%. is always positive in case of smaller values of bend radii of curvature. The calculations also show that for larger values of bend radii with large negative dispersion values the magnitude of dFOMrFOM Ž%. changes rapidly with d a r than in the case of smaller bend radii with small dispersion values.
Acknowledgements This work was partially supported by Indo-French Centre for the promotion of Advanced Research
R. Tewari et al.r Optics Communications 174 (2000) 405–411
ŽIFCPAR. under the project title, ‘‘R & D in Dispersion Compensated Optical Fibres and Amplifiers’’.
References w1x H. Izadpanah, C. Lin, J.L. Gimlett, A.J. Antos, D.W. Hall, D.K. Smith, Electron. Lett. 28 Ž1992. 1469. w2x A.J. Antos, D.W. Hall, D.K. Smith, OFCr100C, 1993, p. 204. w3x A.J. Antos, D.K. Smith, IEEE OSA J. Lightwave Technol. 12 Ž1994. 1739. w4x C.D. Poole, J.M. Wiesenfeld, A.R. McDomick, K.T. Nelson, Opt. Lett. 17 Ž1992. 985. w5x M. Onishi, H. Ishikawa, Y. Ishiguro, Y. Koyano, M. Nishimura, H. Kanamori, Fiber Integr. Opt. 16 Ž1997. 277. w6x R. Tewari, M. Basu, H.N. Acharya, Fiber Integr. Opt. 17 Ž1998. 221. w7x R. Tewari, M. Basu, H.N. Acharya, Opt. Commun. 155 Ž1998. 260.
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w8x E. Sharma, A. Sharma, I.C. Goyal, IEEE J. Quantum Electron. 18 Ž1982. 1484. w9x E. Khular, A. Kumar, A.K. Ghatak, B.P. Pal, Opt. Commun. 23 Ž1977. 263. w10x B.P. Pal, A. Kumar, A.K. Ghatak, Electron. Lett. 16 Ž1980. 505. w11x R. Tewari, K. Petermann, European Conference on Optical Communication ŽECOC., 1987, p. 215. w12x R. Tewari, B.P. Pal, U.K. Das, IEEE OSA J. Lightwave Technol. 4 Ž1992. 1. w13x E.G. Neumann, Single Mode Fibers: Fundamentals, Springer Verlag, Heidelberg, 1988. w14x A.K. Ghatak, A. Sharma, R. Tewari, Understanding Fibre Optics on a PC, VIVA Books, New Delhi, 1994. w15x K. Petermann, Electron. Lett. 19 Ž1983. 712. w16x K. Petermann, R. Kuhne, IEEE OSA J. Lightwave Technol. ¨ 4 Ž1986. 2. w17x W.A. Snyder, J.D. Love, Optical Waveguide Theory, Chapmann and Hall, New York, 1983. w18x H.M. Presby, D. Marcuse, W.G. French, Appl. Opt. 18 Ž1979. 4006. w19x G.P. Agrawal, Nonlinear Fibre Optics, Academic Press, New York, 1989.
Optics Communications 174 Ž2000. 405–411 www.elsevier.comrlocateroptcom
Studying the effect of a central dip on the performance of a dispersion compensated fibre R. Tewari ) , M. Basu, H.N. Acharya Department of Physics, Indian Institute of Technology, Kharagpur 721302, India Received 2 June 1999; received in revised form 26 October 1999; accepted 27 October 1999
Abstract The spot size technique has been used to study the effect of a central dip on the performance of a dispersion compensated fibre ŽDCF.. A Gaussian dip around the axis of the fibre core has been considered. Such a type of dip in the refractive index profile is obtained during the fibre fabrication by modified chemical vapour deposition ŽMCVD. technique. The percentage change in figure of merit Ž dFOMrFOM Ž%.. of a DCF with a central dip with respect to a DCF having no dip has been estimated for different dip depths and dip widths, respectively. It is seen that for larger values of bend radii with large negative dispersion values the magnitude of dFOMrFOM Ž%. changes rapidly with change in other fibre losses, d a r , than in case of smaller bend radii with small dispersion values. q 2000 Elsevier Science B.V. All rights reserved. Keywords: Dispersion compensated fibre; Spot size technique; Figure of merit; Modified chemical vapour deposition technique
1. Introduction One of the methods of choice for dispersion compensation is to use the single mode negative dispersion fibres. With the aim of obtaining large negative dispersion coefficient Ž Dc . and figure of merit Ž FOM ., a large number of DCFs have been investigated having different types of designs, supporting single mode or higher order modes w1–7x. In this paper, we have studied the effect of a central Gaussian refractive index dip w8–10x on the performance of a single mode DCF. The central dip in the refractive index profile is obtained during the fibre fabrication by modified chemical vapour deposition
) Corresponding author. Fax: q91-3222-55303; e-mail: [email protected]
ŽMCVD. technique w8–10,18x. In this paper, a DCF having a central Gaussian refractive index dip w8–10x in the core is designed and optimized by using spot size optimization technique w6,7,11–14,16x. The spot size technique used here has already been shown to work for dispersion shifted single clad w12x and multiclad fibres with segmented core w11x and dual shape core w12x profiles. In recent papers w6,7x, the widely used step index DCF has been designed and optimised, using this technique w6,7,11–14,16x with the aim of attaining maximum negative dispersion Ž Dc . and large value of FOM. According to the spot size technique w6,7,11–14,16x for a given value of W w6,7,11–15x and dispersion, to achieve the lowest bending loss w6,7,11–14,16,17x in the single mode region, the ratio W` rW should be optimised by changing the core radius, a, and the relative refractive index difference, D. At a given value of W, the
0030-4018r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 Ž 9 9 . 0 0 6 5 8 - 6
406
R. Tewari et al.r Optics Communications 174 (2000) 405–411
so achieved value of W` w6,7,11–14,16,17x can be used to calculate the optimum bend loss Ž a b . w6,7,11–14,16,17x and thereafter the bend loss–dispersion diagram can be obtained theoretically. In this paper, dispersion compensated fibres having a central dip and without dip have been designed and optimised using spot size technique w6,7,11–14,16x. It is seen that for a given W and bend radius Ž R ., bend loss increases with negative dispersion values for both the fibres i.e., with and without dip, respectively. It is also seen that at a given dispersion value, bend loss is more for larger values of dip width d˜Žs dra. as compared to the smaller dip widths of a DCF. Thus, at a given value of dispersion, FOM can be calculated, which requires inclusion of bending loss Ž a b . as well as other losses Ž a r . which include Rayleigh scattering loss and losses associated with fibre manufacturing techniques such as absorption losses etc. w1,3x. Instead of estimating absolute value of FOM for the fibres having dip, the percentage change in FOM Ži.e., dFOMrFOM Ž%.. of the fibres has been estimated by taking into account the corresponding bending losses Ž a b . and other losses Ž a r . associated with the fibre having central dip in the core and that of the fibre having no central dip, respectively. Calculations show that for larger bend radii of curvature Ž R . and larger values of dispersion Ž Dc ., the dFOMrFOM Ž%. is positive for smaller value of d a r Žs a rŽwithout dip. y a rŽdip. . but becomes negative for larger values of d a r . This implies that initially FOM of a fibre with dip is smaller as compared to FOM of a fibre without dip but for larger values of d a r the FOM of a fibre with dip increases as compared to FOM without dip. On the other hand, for smaller bend radii of curvature dFOMrFOM Ž%. is always positive with change in d a r implying that FOM of a fibre with dip is always smaller than the FOM of a fibre without dip. Section 2 of this paper describes the refractive index profile of a DCF with a central dip on the axis. The next section deals with the theoretical background for designing such types of fibres with dip and without dip. The percentage change in FOM, i.e., dFOMr FOM Ž%. of a fibre with a central dip in the core with respect to a fibre having no dip is estimated in Section 3. Finally, all the results obtained from this technique are discussed and summarized in Section 4 and Section 5, respectively.
2. Refractive index profie of a DCF with and without a central dip in the core In the presence of a Gaussian central dip at the axis of the fibre, as shown in Fig. 1, the index profile can be described as w8–10x: n2 Ž R .
s n2o Ž R . y Ž n12 y n22 . p˜ Ž exp Ž y R 2 r d˜2 yexp Ž y1r d˜2 .
R-1
s n 22
Ž 1.
R)1
where d˜Žs dra. and p˜ are the dip width and the dip depth, respectively and n2o Ž R . represents the refractive index profile, in the absence of the axial dip, of a parabolic index fibre which is given by w8–10x: n 2o Ž R .
s n12 w 1 y 2 D R 2 x
R-1
s n 22
R)1
Ž 2.
where R s rra, a is the radius of the core, n1 and n 2 represent maximum refractive index of the central core Ž 0 - r - a. and cladding Ž r ) a., respectively and D s Ž n1 y n 2 .rn1 is the relative index difference.
3. Theoretical background to design a DCF with and without a central dip The characteristics of a DCF with a central dip in the core can be described in terms of spot sizes W w6,7,11–15x and W` w6,7,11–14,16,17x. We consider the following two definitions for the mode spot sizes.
Fig. 1. Refractive index profile of a DCF having a Gaussian dip at the centre of the core.
R. Tewari et al.r Optics Communications 174 (2000) 405–411
The first spot size W of the near field, known as Petermann y2 spot size, is defined as w6,7,11–15x: `
H0 c
2 2
W s
2
Ž r .dr
`
H0
Ž 3.
2
Ž d crd r . rd r
d
2
ž / W
d
Ž 4.
We see from Eq. Ž4. that to achieve low splice loss a d , W should be as large as possible. It should also be mentioned that in order to suppress the non-linear effects in DCFs the effective core area Ž A eff . should be large which is directly proportional to the spot size w5,19x. The second spot size considered is W` w6,7,11–14,16,17x of the near field which can be defined as: W`2 s
l
Ž 5.
p n1 Ž bo y k o n 2 .
where bo is the propagation constant of the fundamental mode, k o Žs 2prl. is the free space wave number, n1 and n 2 are the refractive indices on the central axis and in the cladding level, respectively ŽFig. 1. and l is the operating wavelength. W` w6,7,11–14,16,17x is also called the Petermann y3 spot size which is related to the pure bending loss of the fibre. The pure bend loss coefficient g b Žper unit length., in a fibre with a radius of curvature R, is given by w6,7,11–14,16,17x:
g b s f Ž n Ž r . , l.
W`3
( ½ R
exp
We see from Eqs. Ž6. and Ž7. that to achieve the low bending loss a b , W` should be as small as possible. However it is seen that W` is always larger than W in the single mode region and we can write w6,11– 14,16x: W` ) W .
where c Ž r . represents the transverse field distribution of the LP01 mode. The spot size W is inversely proportional to the rms spot size that can be measured from the angular dependence of the far field intensity. The spot size W is related to the transverse offset at a splice between the two fibres. The splice loss a d Žin decibels., due to transverse offset Ž d . at a splice or joint between two fibres, is fully characterized by the Petermann y2 spot size W and can be expressed as w6,7,11–15x:
a d s 4.343
407
y4R l2 3p
2
n12 W`3
5
Ž 8.
Thus, we see from Eqs. Ž3. – Ž8. that for small bending loss, W` should be as small as possible w6,7,11–14,16,17x, while for small splice loss due to transverse offset, W should be as large as possible w6,7,11–15x. Thus for a given W and dispersion, to achieve low bending loss and splice losses along with the low non-linear effects, the design criteria requires to achieve a ratio W` rW as close to unity as possible w6,7,11–14,16,17x. An important parameter, known as figure of merit Ž FOM ., which is the amount of dispersion per unit loss with practical units of psrnm dB, can be used to characterise the DCFs w2,3,6,7x. The DCFs should be optimized to minimize the total loss of the fibre. The compensating fibre loss a c Žin dBrkm. can be described as the sum of bend loss Ž a b . and the other losses Ž a r . which include Rayleigh scattering loss and losses associated with the fibre manufacturing techniques such as absorption losses, etc. Thus, a c can be written as:
ac s a b q ar .
Ž 9.
Thus, the figure of merit Ž FOM . of a DCF can be written as: FOM s
Dc
a b q ar
.
Ž 10 .
The percentage change in figure of merit Ž dFOMrFOM Ž%.. of a fibre with a central dip in the core with respect to the FOM of a fibre without any central dip can be estimated from the above Eqs. Ž9. and Ž10. in terms of a b and a r and is given by: dFOM FOM
Ž %. s
yŽ d ar q d a b .
a rŽ with dip. q a bŽ with dip.
= 100
Ž 11 .
where .
Ž 6.
Thus the pure bend loss a b Žin dBrkm. is given by:
d a b s a bŽ without dip. y a bŽ with dip.
a b s 4.343g b .
d a r s a rŽ without dip. y a rŽ with dip.
Ž 7.
¶ • ß
dFOMs FOMŽ without dip . y FOMŽ with dip .
Ž 12 .
408
R. Tewari et al.r Optics Communications 174 (2000) 405–411
where the bend loss Ž a b . of the DCF with dip and without dip have been obtained using the spot size technique w6,7,11–14,16x. The other losses Ž a rŽw ith dip. . of the DCFs have been obtained by decreasing an amount d a r from the value of the other losses Ž a rŽw ithout dip. . of the DCF w1,3x as explained in Eq. Ž12.. It should be mentioned that the fabricated DCFs have the other losses, a r which include Rayleigh scattering loss and losses associated with fibre manufacturing techniques such as absorption losses etc., lying between 0.3 to 0.5 dBrkm w1,3x and the corresponding DCF with central dip are supposed to have less value of a r due to less amount of GeO 2 concentration at and around the centre of the core. The percentage change in FOM which is an important factor to study the effect of central dip on the characteristics of a DCF has been estimated from Eqs. Ž11. and Ž12.. 4. Results and discussions Refractive index profiles obtained from MCVD technique have been found to have a central dip at the axis of the core of the fibre ŽFig. 1.. In order to design and optimize the fibres in the presence of a central dip in the core, the scalar wave equation was solved numerically using the spot size optimization technique w6,7,11–14,16x. For a given value of dispersion, Dc , and spot size, W, and for different values of dip widths Ž d˜. and dip depths Ž p˜ ., the ratio W` rW has been optimized, by adjusting the core radius Ž a. and the relative refractive index difference Ž D .. This yielded the optimized value of W` which has been used to calculate the bend loss Žcf Eqs. Ž6. and Ž7.. at different dispersion values for typical value of spot size W s 3.0 mm. Fig. 2 shows the plot of bend loss in dBrkm against negative dispersion value Žpsrkm nm., for DCFs having central dip in the core along with the DCFs without any dip in the core. The curves have been plotted for two different dip widths d˜s 0.3 and 0.5, respectively, with dip depths p˜ remaining constant and equal to 0.5. It is seen during calculation that the variation of dip depths, p˜ does not make any significant change in bend loss when dip width is kept constant. It should be mentioned that the measured dip width Ž d˜. and dip depth Ž p˜ . in fabricated fibres are typically in the range of 0.28–0.33 and 0.34–0.72, re-
Fig. 2. Bend loss characteristics as a function of negative disper˜ 0.3, ps sion of a DCF with central dip in the core Ž ds ˜ 0.5 and ˜ 0.5, ps ds ˜ 0.5. as well as without any dip in the core, for spot size W s 3.0 mm and bend radii Rs 2, 3 and 4 cm, respectively.
spectively w18x. The values of dip width Ž d˜. and dip depth Ž p˜ . of the DCF considered here are of the same order with the typical values of dip width Ž d˜. and dip depth Ž p˜ . reported earlier w18x. In Fig. 2, the curves have been drawn for the typical value of the spot size W s 3.0 mm w6,7x and bend radii of curvature Ž R s 2, 3 and 4 cm. w6,7x. In order to have the quantitative values we have given a table ŽTable 1. to show the values of the optimised ratio W` rW along with the bend losses Ž a b . of the DCFs with a central dip in the core as well as without any dip in the core for given values of dispersion and bend radii Ž R . at spot size W s 3.0 mm. It is seen from Fig. 2 that at smaller negative dispersion values, the bend loss Ž a b . is negligible for larger bend radius Ž R s 4 cm. and on the other hand for smaller bend radii Ž R s 2 cm and 3 cm. bend loss is more dominant. In this figure it is also seen that for a particular bend radius Ž R ., the bend loss is more for larger values of dip widths Ž d˜. than in case of smaller values of d.˜ Thus, from the view point of bend loss only, it can be said the presence of dip has degraded the performance of a DCF.
R. Tewari et al.r Optics Communications 174 (2000) 405–411
409
Table 1 Difference in bend losses Ž d a b . of DCFs with a central dip and without dip at different values of negative dispersion Ž Dc . when the spot size W is fixed at 3.0 mm Dip parameters
Dc Žpsrkm nm.
W` rW
R Žcm.
a b ŽdBrkm.
without dip
y30
1.888
2
1.2
–
d˜s 0.3, p˜ s 0.5
y30
1.899
2
2.0
y0.8
d˜s 0.5, p˜ s 0.5 without dip
y30 y50
1.916 2.142
2 3
2.8 0.35
y1.6 –
d˜s 0.3, p˜ s 0.5
y50
2.145
3
0.63
y0.28
d˜s 0.5, p˜ s 0.5 without dip
y50 y60
2.150 2.254
3 4
0.90 0.075
y0.55 –
d˜s 0.3, p˜ s 0.5
y60
2.261
4
0.1
y0.025
d˜s 0.5, p˜ s 0.5
y60
2.264
4
0.175
y0.1
At a given value of dispersion and bend radius, FOM of the fibres without a central dip has been calculated which requires inclusion of bending loss Ž a b . as well as other losses Ž a r . which include Rayleigh scattering loss and losses associated with fibre manufacturing techniques such as absorption losses, etc. w1,3x. Table 2 shows the FOM values of the DCFs without central dip in the core for different dispersion values Ž Dc s y30, y50 and y60 psrkm nm. and bend radii Ž R s 2, 3 and 4 cm. at spot size W s 3.0 mm. This is needed to estimate the percentage change in figure of merit Ž FOM . due to the presence of a dip at the central axis of a fibre with respect to a fibre having no dip at the centre. In Fig. 3Ža. and Fig. 3Žb., the percentage changes in FOM, i.e., dFOMrFOM Ž%. is plotted against the change
Table 2 Figure of merit Ž FOM . of a DCF without any dip in the core at different values of negative dispersion Ž Dc . and bend radii Ž R . when the spot size W is fixed at 3.0 mm Dc R Žcm. a bŽwithout dip. a rŽwithout dip. FOMŽwithout dip. Žpsrkm nm. ŽdBrkm. ŽdBrkm. Žpsrnm dB. y30 y30 y30 y50 y50 y50 y60 y60 y60
2 2 2 3 3 3 4 4 4
1.2 1.2 1.2 0.35 0.35 0.35 0.075 0.075 0.075
0.5 0.4 0.3 0.5 0.4 0.3 0.5 0.4 0.3
17.6 18.8 20.0 58.8 66.7 76.9 104.3 126.3 160.0
d a b ŽdBrkm.
in other losses Ž d a r . for different bend radii of curvature Ž R s 2, 3 and 4 cm. and different values of dispersion coefficients Ž Dc s y30, y50 and y60 psrkm nm.. The corresponding values of a b and Ž d a b s a bŽwithout dip. y a bŽwith dip. . for the two set of parameters d˜s 0.3, p˜ s 0.5 and d˜s 0.5, p˜ s 0.5, respectively have been taken from Table 1. The figures show that dFOMrFOM Ž%. values are changing in a range of q50 to y90 with the value of d a r when the other losses Ž a rŽwithout dip. . are fixed at 0.5, 0.4 and 0.3 dBrkm, respectively w1,3x. It should be mentioned that the fabricated DCFs have the other losses, a r , which include Rayleigh scattering loss and losses associated with fibre manufacturing techniques such as absorption losses, etc., lying between 0.3 dBrkm and 0.5 dBrkm w1,3x and the corresponding DCF with central dip, are supposed to have a lower value of a r due to the lower amount of GeO 2 concentration at the centre of the core. Comparison of Fig. 3Ža. and Fig. 3Žb. shows that at a given value of a rŽw ithout dip. , d a r and R, when we consider smaller dip width Ž d˜s 0.3., the dFOMr FOM Ž%. is less compared to the case of a DCF with larger dip width Ž d˜s 0.5.. This is due to the fact that the bend loss is smaller in case of d˜s 0.3 than in the case of d˜s 0.5 for a given set of dispersion and bend radius Žcf. Fig. 2 and Table 1.. The above figures also show that for larger bend radii as well as for larger dispersion coefficients, there is more rapid change in dFOMrFOM Ž%. with change in the magnitude of d a r than in case of smaller bend radii and smaller dispersion values of
410
R. Tewari et al.r Optics Communications 174 (2000) 405–411
and Fig. 3Žb. also show that for bend radii of curvature Ž R s 2 cm and 3 cm. dFOMrFOM Ž%. is always positive with the change in other fibre losses, d a r , implying that FOM of a fibre with dip is always smaller than the FOM of a fibre without dip, while for larger bend radius i.e., R s 4 cm and dispersion Ž Dc s y60 psrkm nm., the dFOMrFOM Ž%. is initially positive for smaller value of d a r but becomes negative for larger values of d a r . This implies that initially FOM of a fibre with dip is smaller as compared to FOM of a fibre without dip but for larger values of d a r Ž) 0.02 dBrkm in Fig. 3Ža. and ) 0.10 dBrkm in Fig. 3Žb.., the FOM of a fibre with dip increases as compared to FOM without dip. It should be mentioned that in order to obtain negative value of dFOMrFOM Ž%. throughout d a r values i.e., d a r G 0, d a b has to be zero at d a r s 0 Žcf. Eqs. Ž11. and Ž12... According to Fig. 2 Žand Table 1. this is possible only either for a straight fibre or for a fibre having no dip.
5. Conclusion
Fig. 3. Ža. The variation of the percentage change in figure of merit Ž dFOMr FOM in Ž%.. against change in other losses Ž d a r . for different bend radii Ž R . and negative dispersion values Ž Dc . of ˜ 0.3, ps a DCF with a central dip Ž ds ˜ 0.5. in the core. Žb. The variation of the percentage change in figure of merit Ž dFOMr FOM in Ž%.. against change in other losses Ž d a r . for different bend radii Ž R . and negative dispersion values Ž Dc . of a ˜ 0.5, ps DCF with a central dip Ž ds ˜ 0.5. in the core.
the DCF. This is because at larger bend radii Ž R ., the bend loss Ž a b . is negligible Žcf. Fig. 2 and Table 1. in comparison to other losses Ž a rŽw ith dip. . and the percentage change factor Ž dFOMrFOM Ž%.. will be dominated only by d a r as d a b is also comparatively small, whereas for smaller bend radii and smaller dispersion values, the bend loss Ž a b . is already large Žcf. Fig. 2 and Table 1. and therefore the increase in d a r has less effect on dFOMrFOM Ž%. as d a b is also comparable or more. Fig. 3Ža.
Using spot size based optimization technique, the bend loss–dispersion diagram for a DCF with central dip as well as without central dip in the core has been obtained. Using this diagram, the effect of a central dip in the core, on the performance of a DCF has been estimated in terms of percentage change in figure of merit Ži.e., dFOMr FOM Ž%... It is seen that for larger bend radii of curvature initially dFOMrFOM Ž%. is positive for smaller value of d a r but becomes negative for larger values of d a r whereas dFOMrFOM Ž%. is always positive in case of smaller values of bend radii of curvature. The calculations also show that for larger values of bend radii with large negative dispersion values the magnitude of dFOMrFOM Ž%. changes rapidly with d a r than in the case of smaller bend radii with small dispersion values.
Acknowledgements This work was partially supported by Indo-French Centre for the promotion of Advanced Research
R. Tewari et al.r Optics Communications 174 (2000) 405–411
ŽIFCPAR. under the project title, ‘‘R & D in Dispersion Compensated Optical Fibres and Amplifiers’’.
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