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Modeling of a step index segmented core single mode optical fiber as a dispersion compensator
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Optik 116 (2005) 255–264 www.elsevier.de/ijleo
Modeling of a step index segmented core single mode optical fiber as a dispersion compensator M. Basu, D. Ghosh Department of Physics, Bengal Engineering & Science University, Shibpore, P.O. Botanic Garden, Howrah 711103, W. Bengal, India Received 26 August 2004; accepted 15 January 2005
Abstract A segmented core central step index dispersion compensating fiber (DCF) has been modeled by using spot size optimization technique. We have designed and optimized the DCF by suitably adjusting different fiber parameters, i.e. inner core radius (a), central relative refractive index difference (D), normalized outer core position (p), width (b) and height (h) at different values of given Petermann-II spot sizes ðW Þ: The studies on the propagation behavior as well as bendloss and other losses of the DCF have been presented here. The proposed design of the DCF can possess improved bend performances over single core DCFs. At the same time higher negative dispersion coefficient in the single mode region as well as considerably high value of figure of merit (FOM) for optimized fiber parameters can be achieved for smaller spot sizes ðW Þ of the DCF. Further, inclusion of length independent splice loss modified this conventional FOM to a slightly lesser value of modified FOM. r 2005 Elsevier GmbH. All rights reserved. Keywords: Spot size technique; Dispersion compensating fibers; Bend loss; Figure of merit; Modified figure of merit
1. Introduction The advent of Erbium-doped fiber amplifiers (EDFAs) operating at wavelength 1.55 mm, the lowest loss window of optical fibers, suggests the need of up gradation of existing 1.30 mm conventional single mode optical fiber (CSFs) links. When operated at 1.55 mm, these CSFs exhibit large positive chromatic dispersion (17 ps/km nm). Therefore in order to upgrade the present long haul fiber optic communication system, comprising of CSFs, a combination of EDFAs and dispersion compensating fibers (DCFs) would be the most viable choice to compensate dispersion as well as Corresponding author.
E-mail address: [email protected] (M. Basu). 0030-4026/$ - see front matter r 2005 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2005.01.019
loss of the link. There are several techniques [1–13] available for dispersion compensation. However out of these techniques, DCF has been accepted as the most prospective candidate and also widely developed in this connection to combat the chromatic dispersion of the fibers, as they are cascadable, commercially available and compatible with all optical network concepts. The primary characteristic of DCFs at present time is that when appropriately designed, it can provide high negative dispersion coefficient [1–13], which will be opposite in sign but equal to or greater than the positive chromatic dispersion (17 ps/km nm) of a CSF at 1.55 mm. Such a scheme could compensate the positive dispersion of the CSFs over a relatively short length of the DCF [5,6]. Since the DCF used in the link has to be spliced either at the input or the output end of a CSF, it
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would increase the total loss of the system. The transmission length of the DCF required for compensation can be reduced by suitably designing the fibers having very large negative dispersion coefficients [3–7,10,12,13]. To achieve a very high negative value of dispersion coefficient, DCFs are required to have a high value of relative refractive index difference D ¼ ðn21 n22 Þ=2n21 ; where n1 and n2 are the refractive indices of the core and cladding, respectively, and much lower value of core radius (a), resulting a V-value close to the cutoff of the fundamental mode. This gives rise to the modal field largely extended into the cladding region, resulting in a weakly guided mode, which exhibits large bend loss. Since DCFs suffer from high attenuation (0.6 dB/ km at 1.55 mm) [1], there will be additional losses introduced by them in the fiber optic link. By using Erbium doped DCFs the simultaneous compensation of loss and dispersion can be obtained [14]. Further, to achieve broadband dispersion compensation, different triple clad DCF designs [18] are also being studied theoretically as well as experimentally. The microbending resistance of DCF has also been found to be better in the triple clad DCFs [18] than that of a standard fiber. There is also a report for design optimization of a dual core dispersion slope compensating fiber (DSCF) for broadband dense wavelength division multiplexing transmission system [13] where a theoretical figure of merit (FOM) has been achieved to a value 900 ps/nm dB at 1.55 mm. To enhance the FOM of a DCF by enlarging the absolute value of its dispersion has been found to be an effective way to reduce the non-linear effects occurring in the DCF [10]. A study of many possible solutions to compensate the chromatic dispersion of optical links have been presented by different research groups in France and India [2], where a dispersion coefficient 1800 ps/km-nm has been achieved at 1.55 mm low loss window. Extrinsic losses like splice-loss as well as micro-bend and macro-bend-induced losses in single-mode fibers strongly depend on the fiber characteristic mode field spot size [17]. Only 1/e half-width of the Gaussian-like LP01 mode field distribution is not sufficient to characterize dimensionally a single-mode fiber. Petermann-II spot size W [3,15–17,19,21] and Petermann-III spot size WN [3,15–17,20,21] of the near field are used to design and optimize the fibers. To achieve low bend-loss and splice-loss simultaneously the ratio W 1 =W should be optimized in such a way to obtain its value as close to unity as possible. It has been reported that spot-size technique has been used so far for studying different types of DCFs, such as step index DCF [5], DCFs with a central dip in a parabolic index single core [6], different graded index fibers [7], etc. The present work is mainly concerned with the design and optimization of a single mode central step index segmented core dispersion
compensating optical fiber (DCF) by using the spot size technique. To characterize a segmented core step index DCF, its performance on the basis of variation of some fiber parameters such as the inner core radius (a), the relative refractive index difference (D), the ratio W 1 =W and cutoff wavelength (lc) of the first higher order mode LP11 with normalized outer core width ‘b’ and height ‘h’, have been studied and characterized for different dispersion coefficients (Dc) at three given spot sizes ðW Þ: Also the bending-loss sensitivity of this particular profile of the DCF has been studied. Other intrinsic losses like Rayleigh scattering loss, absorption losses, etc. are included along with bend loss to estimate an important parameter, named as the FOM of DCF. The performance characteristics of the DCF with bend and also without bend have been simulated here. From the variation of FOMwithbend and FOMwithoutbend with normalized parameters ‘b’ and ‘h’ for different dispersion values at three given spot sizes ðW Þ; the bend optimized values of ‘b’ and ‘h’ can be estimated. The study shows that these multiclad designs of DCF have much improved bend performances over standard single core step index DCFs. The result shows that higher FOM values can be achieved in the single mode operating zone of the proposed DCF for smaller value of given Petermann-II spot size ðW Þ: When length independent splice loss is added along with length dependent bend loss, another definition of FOM, called modified figure of merit (MFOM) can be introduced, which in general, is lesser than the conventional FOM of the fiber. Section 2 of the paper describes the refractive index profile of the proposed step index segmented core DCF. Section 3 deals with the basic theoretical framework to design the proposed fiber. The DCF have been stimulated and characterized using spot size optimization technique with respect to different fiber parameters in Section 4. Also in the same section the FOMs have been estimated for bent as well as straight DCFs. Finally all the results have been summarized in Section 5.
2. Refractive index profile parameters for a step index segmented core DCF The refractive-index profile of a dispersion compensated central step index segmented core fiber as shown in Fig. 1, can be described as, 8 n1 ð1 Dt0 Rq Þ; Rp1; > > > > < n1 ð1 Dt0 Þ ¼ n2 ; 1pRpP; (1) nðRÞ ¼ n1 ð1 Dt00 Þ ¼ n3 ; PpRpP; > > > > : n1 ð1 DÞ ¼ n4 ; RXC;
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257
near field, is defined as [3,15–17,19,21] 2
n1
2
W ¼
R1
c2 ðrÞ dr
0
R1 ðdc=drÞ2 r dr
,
(3)
0
where c(r) represents the transverse field distribution of the fundamental LP01 mode. W is related to the waveguide dispersion of a fiber and also to the loss due to transverse offset at a splice or joint between two fibers. The splice loss asl (in dB) due to transverse offset d is fully characterized by W and can be expressed as [3,15–17,19,21] 2 d asl ¼ 4:343 . (4) W
n3 n(R)
h
n2
1
0
n4
P
1
C b
R
p
Fig. 1. The refractive index profile of a segmented core central step index dispersion compensating optical fiber.
where R ¼ r/a, a being the core radius of the fiber, r is the radial co-ordinate from the center, n1 and n3, respectively, represent the maximum refractive index at the central core (0oRo1) and the side core (PpRpC) and n2 is the refractive index of the inner clad in between the central core and the side core (1pRpP); n4 is the refractive index of the cladding (RXC); q defines the core profile shape of the fiber, which is taken as N for step index central core and the relative refractive index differences are 4 D ¼ n1nn ¼ 1 nn41 ; 1 2 Dt0 ¼ n1nn ¼ 1 nn21 ; 1
(2)
3 Dt00 ¼ n1nn ¼ 1 nn31 : 1
For this particular proposed DCF n2 ¼ n4. The relative width (b) is defined as (CP) and relative height (h) of the side core is defined with respect to the central index difference (D) of central core. The position of the side core relative to the central core is defined by the parameter p.
3. Theoretical framework to design a step index segmented core DCF Several characteristics of optical fiber can be described in terms of Petermann-II spot size W [3,15–17,19,21] and Petermann-III spot size WN [3,15–17,20,21]. The Petermann-II spot size W of the
Therefore, from Eq. (4) it can be said that to achieve the low splice loss (asl), the spot size, W should be as large as possible. It should also be mentioned here that to suppress the non-linear effects in DCFs the effective core area (Aeff) should be maximum [12]. This Aeff is proportional to spot size W : The Petermann-III spot size WN of the near field, is defined as [3,15–17,20,21] W 21 ¼
l , pn1 ðb0 k0 n2 Þ
(5)
where b0 is the propagation constant of the fundamental mode, k0( ¼ 2p/l0) is the free space wave number, n1, n2 are the refractive indices on the axis of central core and in the cladding, respectively, and l0 is the operating wavelength. WN is related to the pure bend loss ab (dB/ km) of the fiber. The pure bending loss ab (dB/km) is given by [3,15–17,20,21] sffiffiffiffiffiffiffiffiffi ! W 31 4Rc l2 ab ¼ 4:343f ðnðrÞ; lÞ exp , (6) Rc 3p2 n21 W 31 where Rc is the radius of curvature and f(n(r),l) represents the function which depends on the refractive index profile of the fiber. From Eq. (6) we see that to achieve the low bending loss (ab), WN should be as small as possible. However, WN is always larger than W in the single mode region and we can write [3,15–17,19–21]: W 1 4W .
(7)
It can be said from Eqs. (4) and (6) that to reduce bend losses, WN should be as small as possible, while to reduce splice losses due to transverse offset, W should be as large as possible [3,15–17,19–21]. This implies that for a given value of W and dispersion, to achieve low bend loss as well as splice loss simultaneously, the design criteria is to achieve a ratio W 1 =W as close to unity as possible. For a long haul fiber optic link consisting of a CSF and a DCF, the total dispersion Dt due to the
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propagation along a length (ls) of a CSF and a length (lc) of a DCF, is given by [1,3] Dt ¼ Ds l s þ Dc l c .
Ds l s , jDc j
(10) (11)
where as and ac are the losses of a CSF and a DCF, respectively, in dB/km. From Eq. (10), it is seen that inserting DCF increases the total link attenuation. Hence the added attenuation must be compensated with additional gain of the amplifiers and also the DCFs should be optimized to minimize the total loss of the fiber. The compensating fiber loss ac (dB/km) can be expressed as sum of bend loss (ab) and other losses (ar) that include Rayleigh scattering loss and losses due to fiber manufacturing techniques. Thus ac can be written as ac ¼ ab þ ar .
(12)
Therefore to minimize the total loss of the link, it can be said from Eq. (11) that the DCF should have very high negative dispersion coefficient as well as very low attenuation loss. A measure of the dispersion compensation efficiency of the fiber can be expressed by an important parameter, named as figure of merit (FOM). It is the amount of dispersion per unit loss with practical units of ps/nm dB. FOM ¼ jDc j=ac . Thus, Eq. (11) can be re-written as Ds a ¼ as þ ls. FOM
or;
a ¼ as þ
Ds ls. MFOM
(15) (16)
Here, MFOM is the modified FOM, which is defined (9)
where Dc and Ds are the dispersion co-efficients of the DCF and CSF, respectively, in the units of ps/(km nm). The total attenuation of the link a (in dB) due to the length lc of DCF is given by [1,3] a ¼ as l s þ ac l c Ds or; a ¼ as þ ac ls, jDc j
a ¼ as l s þ ac l c þ asl
(8)
To minimize the total dispersion Dt, the length lc is so chosen that lc ¼
can now be written as
(13)
(14)
From Eq. (14), it is seen that for low loss a, a large value of FOM is desired. In fact, when FOM approaches infinity, the total link loss a approaches asls, implying the loss due to DCF is zero. Thus for the optimization of a DCF, we essentially require a smaller length (lc) of a DCF accompanied by a large FOM, that can be achieved at higher negative dispersion coefficient (Dc) with lower losses (ac) of a DCF. Since a DCF has to be spliced either at the input or the output end of a CSF and the spot sizes belonging to the two fibers may not be same, a length independent splice loss (asl) has to be added to the link. Therefore, the total link loss a (in dB)
as MFOM ¼ jDc jl c =a0c ,
(17)
where a0c ¼ ac l c þ as l.
(18)
Eq. (18) represents the sum of the losses due to bending and Rayleigh scattering and also the loss due to splices or joints (asl). It should be mentioned that in general MFOM is less than FOM except for the limiting cases when lc becomes too small or splice loss is insignificant.
4. Results and discussions In order to obtain the characteristics of the proposed DCF, the spot size optimization technique has been used here. For a given value of the Petermann-II spot size (W ) and negative dispersion co-efficient (Dc), the ratio ðW 1 =W Þ has been optimized to obtain its value as close to unity as possible by suitably adjusting different fiber parameters, i.e. inner core radius (a), central relative refractive index difference (D), outer core position (pa), outer core width (ba) and height (hD) as shown in Fig. 1. The so achieved values of W 1 =W have been plotted in Figs. 2(a) and (b) as a function of normalized outer core width (b) and height (h), respectively, for different negative dispersion values (Dc) with fixed normalized outer core position (p), at different spot sizes (W ¼ 2.5, 3.0 and 3.5 mm). As can be seen from Figs. 2(a) and (b) that at first W 1 =W increases with ‘b’, when ‘h’ is fixed at the value 0.555 as well as with ‘h’ when ‘b’ is fixed at the value 0.535, respectively, for different negative dispersion coefficients, and after reaching to a maximum value, it comes down with further increase in ‘b’ or ‘h’, respectively, in each case. It can also be seen from all these figures that for smaller spot size (i.e. W ¼ 2.5 mm), maximum negative dispersion coefficient Dc150 ps/ km nm in single mode regime can be achieved by this technique. Also it has been noticed that for a fixed spot size (W ), the curves shift left to achieve larger Dc values. The cutoff wavelength (lc) of the first higher order mode LP11 have also been plotted as a function of normalized side core width ‘b’ when ‘h’ is fixed at the value 0.555, as well as with ‘h’ when ‘b’ is fixed at the value 0.535 in Figs. 2(c) and (d), respectively, for different dispersion coefficients (Dc) as well as for three different spot sizes (W ¼ 2.5, 3.0, 3.5 mm). In general, Figs. 2(c) and (d)
ARTICLE IN PRESS M. Basu, D. Ghosh / Optik 116 (2005) 255–264 2.00
2.00 w =2.5 µm w =3.0 µm w =3.5 µm
1.95 1.90 1.85
(4)
(1)
1.90 1.85
h =0.555
1.80
1.75 1.70
1.70
(2) (3)
1.60
(6) (7)(2)
1.60 1.55
1.55
α
1.50
(5)
W /W
W /W
(1)
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(3)
1.50
(5)
1.45
α
1.45 1.40
1.40
1.35
1.35
1.30
1.30
1.25 0.40
b =0.535 (4)
1.75
(7) (6)
w =2.5 µm w =3.0 µm w =3.5 µm
1.95
1.80
0.45
0.50
0.55
0.60
0.65 0.70
0.75
0.80
0.85
1.25 0.40
0.45 0.50
0.55
0.60
(a) 1600 1590
1590
1580
1580
1570
1570
1560
1560
1540
(1)
1550
λ c (nm)
(6) (4)
(2) (7)
1530
h = 0.555
(3) (5)
1520 0.40 0.45 0.50 0.55
0.70
0.75
0.80
0.85
(b) 1600
1550
0.65
h
b
λ c (nm)
259
0.60 0.65
w =2.5 µm w =3.0 µm w =3.5 µm 0.70 0.75 0.80 0.85
1540
(6) (4)
(1)
(2) (7)
b = 0.535 (3)
w =2.5 µm w =3.0 µm w =3.5 µm
1530 (5) 1520 0.40 0.45
0.50
0.55
0.60
0.65
0.70 0.75
0.80
0.85
h
b
(c)
(d)
Fig. 2. Variation of W 1 =W and the cutoff wavelength lc of the first higher order mode LP11 as a function of ‘b’ when ‘h’ is fixed at 0.555 ((a) and (c), respectively) and as a function of ‘h’ when ‘b’ is fixed at 0.535 ((b) and (d), respectively) at three different spot sizes, W ¼ 2.5, 3.0, 3.5 mm with the following parameters: (1) Dc ¼ 150 ps/km nm, p ¼ 6.86; (2) Dc ¼ 120 ps/km nm, p ¼ 5.10; (3) Dc ¼ 100 ps/km nm, p ¼ 4.41; (4) Dc ¼ 100 ps/km nm, p ¼ 7.00; (5) Dc ¼ 50 ps/km nm, p ¼ 3.825; (6) Dc ¼ 60 ps/km nm, p ¼ 5.85; (7) Dc ¼ 50 ps/km nm, p ¼ 5.125.
show that lc (in nm) has a tendency to decrease at first with increase in ‘b’ value or ‘h’ value and after reaching to a minimum value it again starts to increase with further increase in ‘b’ or ‘h’ value, respectively. Unlike the solution for a single core step index profile, the profiles that include multiple claddings outside the central core have a finite cutoff wavelength for propagation of LP01 mode and consequently have a susceptibility of a design to bending loss. As long as the cutoff wavelength (lc) exists in the range of p1550 nm, the proposed fiber can be operated in single mode regime. For example it can be said from the above figures that for higher dispersion values like 150 ps/ km nm at W ¼ 2.5 mm, the normalized outer core width (b) can be varied from 0.455 to 0.535 with outer core height (h) fixed at 0.555 to operate in single mode zone and corresponding values of W 1 =W varies from 1.94 to 1.68. Similarly for the same dispersion value and spot size, the normalized outer core height (h) can be varied from 0.473 to 0.555 with outer core width (b) fixed at
0.535 and corresponding values of W 1 =W stays in the range of 1.94–1.69 when operated in single mode zone. This sharp decrease of W 1 =W reveals the fact that there should be improved bend performance of the proposed fiber for comparatively higher value of ‘b’ or ‘h’. It should also be mentioned that the larger negative dispersion coefficients (Dc) could be achieved when the value of normalized outer core position (p) is increased for a given spot size (W ). As the optimization has been done by adjusting inner core radius (a) and the central relative refractive index difference (D), it can be seen from Fig. 3(a) that the inner core radius (a) is decreasing and relative refractive index difference (D) is increasing continuously with increasing ‘b’ values, respectively, when the normalized outer core height (h) is fixed at 0.555, for different negative dispersion values and spot sizes (W ). At the same time the variation of the inner core radius (a) and central relative refractive index difference (D) as a function of ‘h’ have been studied in Fig. 3(b), when ‘b’ is kept fixed
ARTICLE IN PRESS M. Basu, D. Ghosh / Optik 116 (2005) 255–264 8
8
1.5 w = 2.5 µm w =3.0 µm w =3.5 µm
(5) (7)
7
(7)
(6)
1.2
b =0.535
6
6
0.9 0.8
(1)
0.7 3
(1)
0.6 (2) (3)
(4) 2
(1)
3
(1)
0.50
0.55
0.60
0.65
0.70
0.75
0.80
0.3 0. 0.85
b
(2)
0.8 0.7 (2)
2
(3)
0.6
(4)
1 0.40
0.45
0.5
(7)
(6)
(5)
(7) 0.45
4
0.4
(6) 1 0.40
0.5
1.0 (4)
0.9 ∆ in %
4
5 Inner core radius (a)µm
∆ in %
(2)
1.2
(3)
1.0
(4)
1.3
1.1
1.1 (3) 5
w = 2.5 µm w = 3.0 µm w = 3.5 µm
7 1.3
h = 0.555
(6)
1.4
(5)
1.4
Inner core radius (a)µm
260
(5) 0.50
0.55
0.60
0.65
0.70
0.75
0.80
0.4 0.85
h
(a)
(b)
Fig. 3. Variation of central relative refractive index difference (D) (left side) and inner core radius (a) (right side) as a function of ‘b’ when ‘h’ is fixed at 0.555 (a) and as a function of ‘h’ when ‘b’ is fixed at 0.535 (b) at three different spot sizes, W ¼ 2.5, 3.0, 3.5 mm with the following parameters: (1) Dc ¼ 150 ps/km nm, p ¼ 6.86; (2) Dc ¼ 120 ps/km nm, p ¼ 5.10; (3) Dc ¼ 100 ps/km nm, p ¼ 4.41; (4) Dc ¼ 100 ps/km nm, p ¼ 7.00; (5) Dc ¼ 50 ps/km nm, p ¼ 3.825; (6) Dc ¼ 60 ps/km nm, p ¼ 5.85; (7) Dc ¼ 50 ps/ km nm, p ¼ 5.125.
at the value 0.535. It can also be concluded from the above figures that higher values of negative dispersion coefficients can be achieved at the expense of very high relative refractive index difference (D) and very small inner core radius (a) for three different spot sizes. It has also been seen that larger negative dispersion values can be obtained for larger ‘p’ values, but the exact position of the side core determines the value of negative dispersion coefficient. To obtain large value of negative dispersion coefficient (Dc), at any optimized ‘b’ and ‘h’ value, which are taken as 0.535 and 0.555, respectively, the total core radius [(p+b)a] decreases at a given spot size which has been shown in Table 1. This means that to have a larger negative dispersion coefficient, the side core should come closer to the central core so that modal field spreading in cladding will be less and consequently the performance of this DCF will be better even at higher negative dispersion region. The already obtained values of W 1 =W (as shown in Figs. 2(a) and (b)) have been used to estimate the bend losses of DCF. In Fig. 4(a), the variations of bend loss in dB/km have been plotted as a function of outer core width (b), keeping ‘h’ value fixed at 0.555 for three different spot sizes (W ) and negative dispersion coefficients (Dc), with bend radii of curvature (Rc) equal to 2, 3 and 4 cm, respectively. It is seen that the nature of variation of bend loss with ‘b’ is same as expected with that of W 1 =W with ‘b’ at a fixed ‘h’. Bend loss decreases for increased values of bend radii of curvature (Rc) in all cases. At smaller spot size (W ¼ 2.5 mm), bend loss varies from the value of 3 101 dB/km to a very insignificant value of 3 10–6 dB/km when ‘b’ value changes from 0.455 to 0.535, keeping ‘h’ fixed at 0.555 in the single mode zone of the proposed fiber, even when the bend radius of curvature (Rc) is 2 cm. For RcX3 cm, bend loss is negligible (o106 dB/km) and this shows
that for smaller spot sizes and larger negative dispersion coefficients, we have an improved bend performance of the fiber. In Fig. 4(b), the variation of bend losses in dB/ km have been plotted as a function of normalized outer core height (h) keeping ‘b’ value fixed at 0.535, for the same set of spot sizes (W ) and dispersion values. It is observed that bend losses vary in the same fashion with ‘h’ as in Fig. 4(a). The so-obtained values of bend losses (ab in dB/km) for different bend radii of curvatures (Rc) and spot sizes (W ) along with Rayleigh scattering loss as well as losses due to fiber manufacturing technique such as absorption loss (ar), etc. [23] can be incorporated to calculate the FOM of this central step index segmented core dispersion compensated optical fiber. Fig. 5(a) shows the variation of the FOM as a function of normalized outer core width (b) when normalized outer core height (h) is fixed at 0.555, at spot sizes W ¼ 2.5, 3.0 and 3.5 mm, for different negative dispersion values (Dc) in ps/km nm. The losses due to absorption loss, Rayleigh scattering loss etc. solely depend on relative refractive index difference (D) and controls the value of FOM as shown in Fig. 5. Even when there is no bend, as the central relative refractive index difference (D) is increased with increasing ‘b’ value at ‘h’ ¼ 0.555, it is expected that Rayleigh scattering loss, absorption loss, etc. will increase and as a result FOMwithoutbend should decrease with ‘b’ for the three given spot sizes (W ) and negative dispersion coefficients (Dc). When the bending of DCF is considered the nature of variation of FOMwithbend with ‘b’ value is quite interesting. For W ¼ 3.5 mm and Dc ¼ 60 ps/km nm with outer core position ‘p’ ¼ 5.85, FOM is almost zero when bend radii of curvature (Rc) are 2 and 3 cm in the single mode operation region of the DCF. For larger bend radius, i.e. Rc ¼ 4 cm, FOM values are changing from 46.54 to 84.96 ps/nm dB with ‘b’ value from 0.515
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Table 1. Position of inner core and outer core at different negative dispersion coefficients and spot sizes when ‘b’ fixed at 0.535 and ‘h’ fixed at 0.555 W (mm)
Dc (ps/km nm)
p
D (%)
Inner core radius (a) (mm)
Outer core radius (p+b)a (mm)
2.5
100 120 150
4.41 5.10 6.86
2.376 2.865 5.095
0.928 0.786 0.499
4.588 4.432 3.690
3.0
50 100
3.825 7.0
1.449 4.006
1.288 0.566
5.617 4.261
3.5
50 60
5.125 5.85
1.535 1.975
1.079 0.894
6.110 5.706
10 10 10 10 10
10 10 10 10 10 10 10
w=2.5 µ m, p=6.86 D = -150(ps/km-nm) w=3.0 µ m, p=7.0 D = -100(ps/km-nm) w=3.5 µ m, p=5.85 D = -60(ps/km-nm)
5 4 3 2
h = 0.555
1 0
-1
Bend loss (dB/km)
Bend loss (dB/km)
10
6
-2 -3 -4 -5
6
10
5
10
4
10
3
10
2
10
1
10
0
w=2.5 µ m, p=6.86 D = -150(ps/km-nm) w=3.0 µ m, p=7.0 D = -100(ps/km-nm) w=3.5 µ m, p=5.85 D = -60(ps/km-nm) b = 0.535
-1
10
-2
10
-3
10
-4
10
-5
10
-6
0.42
10
-6
10 0.46
0.50
0.54
0.58
0.62
0.66
0.70
0.42
0.46
0.50
0.54
0.58
b
h
(a)
(b)
0.62
0.66
0.70
Fig. 4. Bend loss (dB/km) characteristics as a function of ‘b’ when ‘h’ is fixed at 0.555 (a) and as a function of ‘h’ when ‘b’ is fixed at 0.535 (b) for different bend radii of curvature (Rc) at three different spot sizes, W ¼ 2.5, 3.0, 3.5 mm.
to 0.540, respectively, at fixed ‘h’ ¼ 0.555 in the single mode region of the proposed fiber, where as at the same time FOMwithoutbend stays in the range of 156.09–139.73 ps/nm dB. The study shows that for larger spot sizes there is a sharp difference between FOMwithoutbend and FOMwithbend even when large bend radius (Rc) is considered. Further when we decrease the spot sizes (W ), we can see the bend performance of the DCF can be improved considerably. At W ¼ 3.0 mm, we can achieve comparatively higher value of negative dispersion co-efficient (Dc)100 ps/km nm in the single mode region of the fiber. Here FOMwithoutbend changes from 204 to 136 ps/nm dB with ‘b’ value ranging from 0.455 to 0.535. At the same time the FOM of a DCF having sharp bending (Rc ¼ 2.0 cm) is negligibly small. For Rc ¼ 3 cm, FOM changes from 8.28 to 135.92 ps/ nm dB in the same range of ‘b’. When Rc is comparatively larger, i.e. 4 cm, FOMwithbend merge with the curve of FOMwithoutbend. It is well known that minimum bend radius in fiber optic network that are widely accepted for long-term deployment of fibers in practical system installations, is 3.75 cm [22]. For this practically accepted standard bend radius (Rc ¼ 3.75 cm), FOMwithbend changes from 184.53 ps/nm dB to
135.92 ps/nm dB. These FOM values are much closer to the values of FOMwithoutbend in the above mentioned range of ‘b’ for single mode operation region, in comparison to the previous case of W ¼ 3.5 mm, where FOMRc ¼ 3.75 changes from 14.02 to 57.08 ps/nm dB which is far away from the corresponding FOM values for a straight fiber with ‘b’ value ranging from 0.515 to 0.540. Further decrease in spot sizes to a value of W ¼ 2.5 mm, shows that larger value of FOM can be achieved when Dc ¼ 150 ps/km nm at ‘p’ ¼ 6.86. In this case FOMwithoutbend changes from 237.04 to 163.51 ps/nm dB for ‘b’ varying from 0.455 to 0.535 in the single mode region when ‘h’ is fixed at 0.555. When bending is considered with very small radius of curvature, i.e. Rc ¼ 2 cm, FOM curve is showing a sharp change from 97.67 ps/nm dB at ‘b’ ¼ 0.455 and ultimately merges with the curve of FOMwithoutbend and attains a value of 163.51 ps/nm dB at ‘b’ ¼ 0.535. For RcX3 cm, it is seen that FOMwithbend curves completely merge with FOMwithoutbend as bend-loss (ab) is insignificant (o106 dB/km) in the above range of ‘b’ values corresponding to the single mode region of the proposed fiber. So maximum FOM of 237.04 ps/nm dB can be achieved here, even when bending (RcX3 cm) is
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280
w=2.5 µ m, p=6.86 D = -150(ps/km-nm) w=3.0 µ m, p=7.0 D = -100(ps/km-nm) w=3.5 µ m, p=5.85 D = -60(ps/km-nm)
h = 0.555
240
200
200
160
FOM (ps/nm-dB)
FOM (ps/nm-dB)
160
120
80
40
0 0.42
w=2.5 µ m, p=6.86 D = -150(ps/km-nm) w=3.0 µ m, p=7.0 D = -100(ps/km-nm) w=3.5 µ m, p=5.85 D = -60(ps/km-nm)
b = 0.535
240
0.46
0.50
0.54
0.58
0.62
0.66
120
80
40
0 0.42
0.46
0.50
0.54
0.58
0.62
0.66
h
b
(a)
(b)
Fig. 5. Variation of FOM (ps/nm dB) of the central step index segmented core straight DCF and also DCF with different bend radii of curvature (Rc) as a function of ‘b’ when ‘h’ is fixed at 0.555 (a) and as a function of ‘h’ when ‘b’ is fixed at 0.535 (b) at three different spot sizes, W ¼ 2.5, 3.0, 3.5 mm.
considered with ‘b’ ¼ 0.455 and ‘h’ ¼ 0.555. Similarly, in Fig. 5(b), the FOM characteristics have been studied as a function of normalized outer core height (h) when ‘b’ is fixed at 0.535 at W ¼ 2.5, 3.0 and 3.5 mm for the same set of ‘Dc’ and ‘p’ values as given in Fig. 5(a). In this case also for larger spot sizes, i.e. W ¼ 3.5 mm, FOM varies from 154.77 to 140 ps/nm dB with ‘h’ ranging from 0.530 to 560 and ‘b’ fixed at 0.535, even when no bending is considered. For CCITT recommended fiber with Rc ¼ 3.75 cm, the FOM has the range of values from 7.84 to 52.29 ps/nm dB in the single mode region of the proposed fiber, with ‘h’ varying from 0.530 to 0.560 at W ¼ 3.5 mm showing a large difference between FOMwithoutbend and FOMwithbend. When W is fixed at 3.0 mm for Dc ¼ 100 ps/km nm, it is seen that maximum FOM 204.3 ps/nm dB for a straight fiber can be obtained with ‘h’ ¼ 0.465 for single mode operation. When there is sharp bending, i.e. Rc ¼ 2 cm, the performance of the fiber is very poor as FOM is almost zero in this case at W ¼ 3.0 mm. For medium bending, i.e. Rc ¼ 3 cm, initially FOMwithbend becomes very small, i.e. 3.66 ps/nm dB and it approaches to comparatively larger FOM values 130 ps/nm dB for further increase in ‘h’ values in the fundamental operating zone of the DCF. When bend radius is considerably larger, FOMwithbend curve approaches very near to FOMwithoutbend curve. For a standard single mode CCITT recommended DCF with Rc ¼ 3.75 cm, initially FOM has a comparatively lower value of 156 ps/ nm dB and it merges with the curve of FOMwithoutbend afterwards. For the smallest spot size (i.e. W ¼ 2.5 mm) considered here with Dc ¼ 150 ps/km nm, the maximum FOM 238.43 ps/nm dB has been obtained at ‘h’ ¼ 0.473 with ‘b’ fixed at 0.535 for a straight fiber. Further for sharp bending, with Rc ¼ 2 cm, there is a considerable decrease in the value of FOMwithbend at ‘h’ ¼ 0.473, which is 93.2 ps/nm dB and approaches to the corresponding value of FOMwithoutbend when ‘h’
approaches towards the value of 0.555. The figure also shows that for medium and large bend radii of curvature (i.e. RcX3 cm) at W ¼ 2.5 mm, FOM is independent on bending at all. Therefore for a standard fiber in practical system network (Rc ¼ 3.75 cm), the maximum FOM 238.43 ps/ nm-dB can be achieved at normalized outer core height (h) 0.473 with width (b) fixed at 0.535, at W ¼ 2.5 mm and negative dispersion coefficient (Dc) ¼ 150 ps/km-nm. The above studies indicate that maximum negative dispersion coefficient (Dc) as well as FOM have been achieved for comparatively smaller spot size W ¼ 2.5 mm, where the effect of bend loss can be minimized in comparison to other spot sizes (W ¼ 3.0 and 3.5 mm). From the above figures it is also seen that even when there is sharp bending (Rc ¼ 2 cm) at W ¼ 2.5 mm, FOMwithbend is practically same as that of FOMwithoutbend when ‘b’ ranges from 0.520 to 0.535 at ‘h’ ¼ 0.555 and when ‘h’ ranges from 0.544 to 0.555 at ‘b’ ¼ 0.535, respectively. Thus we can also estimate the bend insensitive ‘b’ or ‘h’ values even when there is considerable bending, as the FOM curves for bent as well as straight fibers merge with each other in the above-mentioned range of ‘b’ and ‘h’. Thus Fig. 5 provides a quick reference for fiber optic designers to choose a best suitable DCF with the optimized value of fiber parameters, such as ‘p’, ‘b’ or ‘h’ having maximum possible value of FOM as well as large negative dispersion coefficient (Dc). The above studies have been reflected in Table 2 where we have optimized another important parameter named as MFOM, which includes the length independent splice losses (in dB) between standard fiber and compensating fiber. It is seen from the table that for smaller spot sizes, we can achieve larger negative dispersion coefficient (Dc) as well as smaller length (lc) of the DCF, which will be able to compensate the total dispersion accumulated by a standard fiber of length (ls) 60 km [8]. At W ¼ 2.5 mm, a smallest length 6.8 km of
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Table 2. Comparative study of FOM and MFOM for different fiber parameters at three different negative dispersion coefficients (Dc) and spot sizes (W ) W (mm)
Dc (ps/ km nm)
p
b
h
0.455
0.555 0.520 2.5
150
6.86 0.473 0.535 0.544
0.455 0.555 0.500 3.0
100
7.0 0.465 0.535 0.530
0.515 0.555 0.540 3.5
60
5.85 0.530 0.535 0.560
Rc (cm)
FOM with bend (ps/ dB nm)
2.0 3.0 3.75 2.0 3.0 3.75
97.67 237.04 237.04 175.73 175.73 175.73
2.0 3.0 3.75 2.0 3.0 3.75
93.20 238.82 238.82 171.90 171.90 171.90
2.0 3.0 3.75 2.0 3.0 3.75
0.0 8.28 184.53 0.0 91.88 165.50
2.0 3.0 3.75 2.0 3.0 3.75
0.0 3.92 156.73 0.0 107.06 151.11
2.0 3.0 3.75 2.0 3.0 3.75
0.0 0.1 14.02 0.0 3.87 57.08
2.0 3.0 3.75 2.0 3.0 3.75
0.0 0.0 7.84 0.0 8.76 52.29
DCF is sufficient to compensate the dispersion of the link. To estimate the splice loss the spot size (W ) of standard fiber has been considered of the order of 5.0 mm [18]. Inclusion of losses due to splices or joints reduces the value of FOM by a considerable amount. It has been seen that the maximum FOM 237–239 ps/ nm dB for the proposed DCF with RcX3 cm, can be obtained for two given sets of fiber parameters like ‘b’ ¼ 0.455, ‘h’ ¼ 0.555, ‘p’ ¼ 6.86 and ‘b’ ¼ 0.535, ‘h’ ¼ 0.473, ‘p’ ¼ 6.86, respectively, at W ¼ 2.5 mm with
FOM without bend (ps/ dB nm)
lc (km)
as (dB)
237.04
175.73 6.8
1.938
171.90
204.34
165.50 10.2
1.087
204.44
0.0 8.21 154.20 0.0 83.68 140.68 0.0 3.90 134.30 0.0 96.09 130.15
151.11
156.09
139.73 17.0
140.0
82.38 163.43 163.43 131.74 131.74 131.74 79.18 164.28 164.28 129.58 129.58 129.58
238.82
154.77
MFOM with bend (ps/ dB nm)
0.541
0.0 0.1 13.92 0.0 3.86 55.40 0.0 0.0 7.81 0.0 8.72 50.88
MFOM without bend (ps/ dB nm) 163.43
131.74
164.28
129.58
167.8
140.68
167.87
130.15
144.15
130.09
143.02
130.32
Dc ¼ 150 ps/km nm. When splice loss is included these FOM values have been modified to comparatively smaller value of MFOM 163–164 ps/nm dB. Again, another example shows that when ‘b’ ¼ 0.520, ‘h’ ¼ 0.555, ‘p’ ¼ 6.86 for W ¼ 2.5 mm, the proposed DCF with Dc ¼ 150 ps/km nm has FOM 175.73 ps/ nm dB irrespective of bending, where as it can be modified to a lesser value of MFOM 131.74 ps/nm dB. Similar studies have been reflected for other spot sizes of 3.0 and 3.5 mm with Dc ¼ 100 and 60 ps/km nm,
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respectively. Thus, it is found that all the above studies on FOM characteristics can estimate different fiber parameters for a segmented core central step index single mode DCF, operating at wavelength of 1550 nm.
5. Conclusion In this paper, we have designed a segmented core central step index single mode DCF by optimizing different important fiber parameters at the 1.55 mm window. Unlike the single core step or graded index DCFs, the proposed fiber design shows improved bend performances at comparatively smaller values of Petermann-II spot sizes (W ) of the DCF and considerably high value of FOM can be achieved. Moreover, the fiber parameters are optimized in such a way that bend loss insensitive DCFs can also be obtained. This kindles hope for fiber optic designers to develop a modern optical communication system comprising of CSFs and DCFs having smaller length (lc) and maximum FOM. However, it should be mentioned that inclusion of length independent splice loss controls the FOM to a comparatively lower limit. The above studies have a scope of future work for further improvement of DCFs both in linear and non-linear regime.
References [1] A.J. Antos, D.K. Smith, Design and characterisation of dispersion compensating fibre based on LP01 mode, IEEE/OSA Journal of Lightwave Technology 12 (1994) 1739–1745. [2] J.L. Auguste, et al., Conception, realization, and characterization of a very high negative chromatic dispersion fiber, Optical Fiber Technology 8 (2002) 89–105. [3] M. Basu, R. Tewari, H.N. Acharya, Effect of grading on the characteristics of a dispersion compensated fibre, Optics Communications 174 (2000) 119–125. [4] K. Thyagarajan, et al., A novel design of a dispersion compensating fiber, IEEE Photonics Technology Letters 8 (1996) 1510–1512. [5] R. Tewari, M. Basu, H.N. Acharya, Spot size technique: a theoretical approach to optimise dispersion compensated optical fibres, Fiber and Integrated Optics 17 (1998) 221–230. [6] R. Tewari, M. Basu, H.N. Acharya, Modified figure of merit for dispersion compensated optical fibres, Optics Communications 155 (1998) 260–262.
[7] R. Tewari, M. Basu, H.N. Acharya, Studying the effect of a central dip on the performance of a dispersion compensated fibre, Optics Communications 174 (2000) 405–411. [8] H. Izadpanah, et al., Dispersion compensation in 1310 nm optimised SMFs using optical equalizer fibre, EDFAs and 1310/1550 nm WDM, Electronics Letters 28 (1992) 1469–1471. [9] H. Izadpanah, E. Goldstein, C. Lin, Broadband multiple wavelength simultaneous dispersion compensation near 1550 nm through single mode fibres optimised for 1310 nm, Electronics Letters 29 (1993) 364–365. [10] T. Kato, et al., Dispersion flattened transmission line consisting of wide-band non-zero dispersion shifted fiber and dispersion compensating fiber module, Optical Fiber Technology 8 (2002) 231–239. [11] M. Onishi, et al., Dispersion compensating fiber with a high figure of merit of 250 ps/nm-dB, Electronics Letters 30 (1994) 161–163. [12] M. Onishi, et al., High performance dispersion compensating fibers, Fiber and Integrated Optics 16 (1997) 277–285. [13] B.P. Pal, K. Pande, Optimization of a dual-core dispersion slope compensating fiber for DWDM transmission in the 1480–1610 nm band through G.652 single-mode fibers,, Optics Communications 201 (2002) 335–344. [14] P. Palai, K. Thyagarajan, B.P. Pal, Erbium-doped dispersion compensating fiber for simultaneous compensation of loss and dispersion, Optical Fiber Technology 3 (1997) 149–153. [15] A.K. Ghatak, K. Thyagarajan, Introduction to Fiber Optics, Cambridge University Press, UK, 1999. [16] A.K. Ghatak, A. Sharma, R. Tewari, Understanding Fibre Optics on a PC, VIVA Books, New Delhi, 1994. [17] E.G. Neumann, Single Mode Fibers: Fundamentals, Springer, Heldelberg, 1988. [18] L.G. Nielsen, et al., Invited paper: dispersion compensating fibre, Optical Fiber Technology 6 (2000) 164–180. [19] K. Petermann, Constraints for fundamental mode spot size for broad band dispersion compensated single mode fibres, Electronics Letters 19 (1983) 712–714. [20] K. Petermann, R. Ku¨hne, Upper and lower limits for the microbending loss in arbitrary single mode fibres, IEEE/ OSA Journal of Lightwave Technology 4 (1986) 2–7. [21] R. Tewari, B.P. Pal, U.K. Das, Dispersion shifted dual shape core fibres: optimization based on spot size definitions, IEEE/OSA Journal of Lightwave Technology 10 (1992) 1–5. [22] CCITT Recommendation G.652, 1986. [23] I. Garrett, C.J. Todd, Components and systems for longwavelength monomode fibre transmission, Optical and Quantum Electronics 14 (1982) 95–143.
Optik
Optics
Optik 116 (2005) 255–264 www.elsevier.de/ijleo
Modeling of a step index segmented core single mode optical fiber as a dispersion compensator M. Basu, D. Ghosh Department of Physics, Bengal Engineering & Science University, Shibpore, P.O. Botanic Garden, Howrah 711103, W. Bengal, India Received 26 August 2004; accepted 15 January 2005
Abstract A segmented core central step index dispersion compensating fiber (DCF) has been modeled by using spot size optimization technique. We have designed and optimized the DCF by suitably adjusting different fiber parameters, i.e. inner core radius (a), central relative refractive index difference (D), normalized outer core position (p), width (b) and height (h) at different values of given Petermann-II spot sizes ðW Þ: The studies on the propagation behavior as well as bendloss and other losses of the DCF have been presented here. The proposed design of the DCF can possess improved bend performances over single core DCFs. At the same time higher negative dispersion coefficient in the single mode region as well as considerably high value of figure of merit (FOM) for optimized fiber parameters can be achieved for smaller spot sizes ðW Þ of the DCF. Further, inclusion of length independent splice loss modified this conventional FOM to a slightly lesser value of modified FOM. r 2005 Elsevier GmbH. All rights reserved. Keywords: Spot size technique; Dispersion compensating fibers; Bend loss; Figure of merit; Modified figure of merit
1. Introduction The advent of Erbium-doped fiber amplifiers (EDFAs) operating at wavelength 1.55 mm, the lowest loss window of optical fibers, suggests the need of up gradation of existing 1.30 mm conventional single mode optical fiber (CSFs) links. When operated at 1.55 mm, these CSFs exhibit large positive chromatic dispersion (17 ps/km nm). Therefore in order to upgrade the present long haul fiber optic communication system, comprising of CSFs, a combination of EDFAs and dispersion compensating fibers (DCFs) would be the most viable choice to compensate dispersion as well as Corresponding author.
E-mail address: [email protected] (M. Basu). 0030-4026/$ - see front matter r 2005 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2005.01.019
loss of the link. There are several techniques [1–13] available for dispersion compensation. However out of these techniques, DCF has been accepted as the most prospective candidate and also widely developed in this connection to combat the chromatic dispersion of the fibers, as they are cascadable, commercially available and compatible with all optical network concepts. The primary characteristic of DCFs at present time is that when appropriately designed, it can provide high negative dispersion coefficient [1–13], which will be opposite in sign but equal to or greater than the positive chromatic dispersion (17 ps/km nm) of a CSF at 1.55 mm. Such a scheme could compensate the positive dispersion of the CSFs over a relatively short length of the DCF [5,6]. Since the DCF used in the link has to be spliced either at the input or the output end of a CSF, it
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would increase the total loss of the system. The transmission length of the DCF required for compensation can be reduced by suitably designing the fibers having very large negative dispersion coefficients [3–7,10,12,13]. To achieve a very high negative value of dispersion coefficient, DCFs are required to have a high value of relative refractive index difference D ¼ ðn21 n22 Þ=2n21 ; where n1 and n2 are the refractive indices of the core and cladding, respectively, and much lower value of core radius (a), resulting a V-value close to the cutoff of the fundamental mode. This gives rise to the modal field largely extended into the cladding region, resulting in a weakly guided mode, which exhibits large bend loss. Since DCFs suffer from high attenuation (0.6 dB/ km at 1.55 mm) [1], there will be additional losses introduced by them in the fiber optic link. By using Erbium doped DCFs the simultaneous compensation of loss and dispersion can be obtained [14]. Further, to achieve broadband dispersion compensation, different triple clad DCF designs [18] are also being studied theoretically as well as experimentally. The microbending resistance of DCF has also been found to be better in the triple clad DCFs [18] than that of a standard fiber. There is also a report for design optimization of a dual core dispersion slope compensating fiber (DSCF) for broadband dense wavelength division multiplexing transmission system [13] where a theoretical figure of merit (FOM) has been achieved to a value 900 ps/nm dB at 1.55 mm. To enhance the FOM of a DCF by enlarging the absolute value of its dispersion has been found to be an effective way to reduce the non-linear effects occurring in the DCF [10]. A study of many possible solutions to compensate the chromatic dispersion of optical links have been presented by different research groups in France and India [2], where a dispersion coefficient 1800 ps/km-nm has been achieved at 1.55 mm low loss window. Extrinsic losses like splice-loss as well as micro-bend and macro-bend-induced losses in single-mode fibers strongly depend on the fiber characteristic mode field spot size [17]. Only 1/e half-width of the Gaussian-like LP01 mode field distribution is not sufficient to characterize dimensionally a single-mode fiber. Petermann-II spot size W [3,15–17,19,21] and Petermann-III spot size WN [3,15–17,20,21] of the near field are used to design and optimize the fibers. To achieve low bend-loss and splice-loss simultaneously the ratio W 1 =W should be optimized in such a way to obtain its value as close to unity as possible. It has been reported that spot-size technique has been used so far for studying different types of DCFs, such as step index DCF [5], DCFs with a central dip in a parabolic index single core [6], different graded index fibers [7], etc. The present work is mainly concerned with the design and optimization of a single mode central step index segmented core dispersion
compensating optical fiber (DCF) by using the spot size technique. To characterize a segmented core step index DCF, its performance on the basis of variation of some fiber parameters such as the inner core radius (a), the relative refractive index difference (D), the ratio W 1 =W and cutoff wavelength (lc) of the first higher order mode LP11 with normalized outer core width ‘b’ and height ‘h’, have been studied and characterized for different dispersion coefficients (Dc) at three given spot sizes ðW Þ: Also the bending-loss sensitivity of this particular profile of the DCF has been studied. Other intrinsic losses like Rayleigh scattering loss, absorption losses, etc. are included along with bend loss to estimate an important parameter, named as the FOM of DCF. The performance characteristics of the DCF with bend and also without bend have been simulated here. From the variation of FOMwithbend and FOMwithoutbend with normalized parameters ‘b’ and ‘h’ for different dispersion values at three given spot sizes ðW Þ; the bend optimized values of ‘b’ and ‘h’ can be estimated. The study shows that these multiclad designs of DCF have much improved bend performances over standard single core step index DCFs. The result shows that higher FOM values can be achieved in the single mode operating zone of the proposed DCF for smaller value of given Petermann-II spot size ðW Þ: When length independent splice loss is added along with length dependent bend loss, another definition of FOM, called modified figure of merit (MFOM) can be introduced, which in general, is lesser than the conventional FOM of the fiber. Section 2 of the paper describes the refractive index profile of the proposed step index segmented core DCF. Section 3 deals with the basic theoretical framework to design the proposed fiber. The DCF have been stimulated and characterized using spot size optimization technique with respect to different fiber parameters in Section 4. Also in the same section the FOMs have been estimated for bent as well as straight DCFs. Finally all the results have been summarized in Section 5.
2. Refractive index profile parameters for a step index segmented core DCF The refractive-index profile of a dispersion compensated central step index segmented core fiber as shown in Fig. 1, can be described as, 8 n1 ð1 Dt0 Rq Þ; Rp1; > > > > < n1 ð1 Dt0 Þ ¼ n2 ; 1pRpP; (1) nðRÞ ¼ n1 ð1 Dt00 Þ ¼ n3 ; PpRpP; > > > > : n1 ð1 DÞ ¼ n4 ; RXC;
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near field, is defined as [3,15–17,19,21] 2
n1
2
W ¼
R1
c2 ðrÞ dr
0
R1 ðdc=drÞ2 r dr
,
(3)
0
where c(r) represents the transverse field distribution of the fundamental LP01 mode. W is related to the waveguide dispersion of a fiber and also to the loss due to transverse offset at a splice or joint between two fibers. The splice loss asl (in dB) due to transverse offset d is fully characterized by W and can be expressed as [3,15–17,19,21] 2 d asl ¼ 4:343 . (4) W
n3 n(R)
h
n2
1
0
n4
P
1
C b
R
p
Fig. 1. The refractive index profile of a segmented core central step index dispersion compensating optical fiber.
where R ¼ r/a, a being the core radius of the fiber, r is the radial co-ordinate from the center, n1 and n3, respectively, represent the maximum refractive index at the central core (0oRo1) and the side core (PpRpC) and n2 is the refractive index of the inner clad in between the central core and the side core (1pRpP); n4 is the refractive index of the cladding (RXC); q defines the core profile shape of the fiber, which is taken as N for step index central core and the relative refractive index differences are 4 D ¼ n1nn ¼ 1 nn41 ; 1 2 Dt0 ¼ n1nn ¼ 1 nn21 ; 1
(2)
3 Dt00 ¼ n1nn ¼ 1 nn31 : 1
For this particular proposed DCF n2 ¼ n4. The relative width (b) is defined as (CP) and relative height (h) of the side core is defined with respect to the central index difference (D) of central core. The position of the side core relative to the central core is defined by the parameter p.
3. Theoretical framework to design a step index segmented core DCF Several characteristics of optical fiber can be described in terms of Petermann-II spot size W [3,15–17,19,21] and Petermann-III spot size WN [3,15–17,20,21]. The Petermann-II spot size W of the
Therefore, from Eq. (4) it can be said that to achieve the low splice loss (asl), the spot size, W should be as large as possible. It should also be mentioned here that to suppress the non-linear effects in DCFs the effective core area (Aeff) should be maximum [12]. This Aeff is proportional to spot size W : The Petermann-III spot size WN of the near field, is defined as [3,15–17,20,21] W 21 ¼
l , pn1 ðb0 k0 n2 Þ
(5)
where b0 is the propagation constant of the fundamental mode, k0( ¼ 2p/l0) is the free space wave number, n1, n2 are the refractive indices on the axis of central core and in the cladding, respectively, and l0 is the operating wavelength. WN is related to the pure bend loss ab (dB/ km) of the fiber. The pure bending loss ab (dB/km) is given by [3,15–17,20,21] sffiffiffiffiffiffiffiffiffi ! W 31 4Rc l2 ab ¼ 4:343f ðnðrÞ; lÞ exp , (6) Rc 3p2 n21 W 31 where Rc is the radius of curvature and f(n(r),l) represents the function which depends on the refractive index profile of the fiber. From Eq. (6) we see that to achieve the low bending loss (ab), WN should be as small as possible. However, WN is always larger than W in the single mode region and we can write [3,15–17,19–21]: W 1 4W .
(7)
It can be said from Eqs. (4) and (6) that to reduce bend losses, WN should be as small as possible, while to reduce splice losses due to transverse offset, W should be as large as possible [3,15–17,19–21]. This implies that for a given value of W and dispersion, to achieve low bend loss as well as splice loss simultaneously, the design criteria is to achieve a ratio W 1 =W as close to unity as possible. For a long haul fiber optic link consisting of a CSF and a DCF, the total dispersion Dt due to the
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propagation along a length (ls) of a CSF and a length (lc) of a DCF, is given by [1,3] Dt ¼ Ds l s þ Dc l c .
Ds l s , jDc j
(10) (11)
where as and ac are the losses of a CSF and a DCF, respectively, in dB/km. From Eq. (10), it is seen that inserting DCF increases the total link attenuation. Hence the added attenuation must be compensated with additional gain of the amplifiers and also the DCFs should be optimized to minimize the total loss of the fiber. The compensating fiber loss ac (dB/km) can be expressed as sum of bend loss (ab) and other losses (ar) that include Rayleigh scattering loss and losses due to fiber manufacturing techniques. Thus ac can be written as ac ¼ ab þ ar .
(12)
Therefore to minimize the total loss of the link, it can be said from Eq. (11) that the DCF should have very high negative dispersion coefficient as well as very low attenuation loss. A measure of the dispersion compensation efficiency of the fiber can be expressed by an important parameter, named as figure of merit (FOM). It is the amount of dispersion per unit loss with practical units of ps/nm dB. FOM ¼ jDc j=ac . Thus, Eq. (11) can be re-written as Ds a ¼ as þ ls. FOM
or;
a ¼ as þ
Ds ls. MFOM
(15) (16)
Here, MFOM is the modified FOM, which is defined (9)
where Dc and Ds are the dispersion co-efficients of the DCF and CSF, respectively, in the units of ps/(km nm). The total attenuation of the link a (in dB) due to the length lc of DCF is given by [1,3] a ¼ as l s þ ac l c Ds or; a ¼ as þ ac ls, jDc j
a ¼ as l s þ ac l c þ asl
(8)
To minimize the total dispersion Dt, the length lc is so chosen that lc ¼
can now be written as
(13)
(14)
From Eq. (14), it is seen that for low loss a, a large value of FOM is desired. In fact, when FOM approaches infinity, the total link loss a approaches asls, implying the loss due to DCF is zero. Thus for the optimization of a DCF, we essentially require a smaller length (lc) of a DCF accompanied by a large FOM, that can be achieved at higher negative dispersion coefficient (Dc) with lower losses (ac) of a DCF. Since a DCF has to be spliced either at the input or the output end of a CSF and the spot sizes belonging to the two fibers may not be same, a length independent splice loss (asl) has to be added to the link. Therefore, the total link loss a (in dB)
as MFOM ¼ jDc jl c =a0c ,
(17)
where a0c ¼ ac l c þ as l.
(18)
Eq. (18) represents the sum of the losses due to bending and Rayleigh scattering and also the loss due to splices or joints (asl). It should be mentioned that in general MFOM is less than FOM except for the limiting cases when lc becomes too small or splice loss is insignificant.
4. Results and discussions In order to obtain the characteristics of the proposed DCF, the spot size optimization technique has been used here. For a given value of the Petermann-II spot size (W ) and negative dispersion co-efficient (Dc), the ratio ðW 1 =W Þ has been optimized to obtain its value as close to unity as possible by suitably adjusting different fiber parameters, i.e. inner core radius (a), central relative refractive index difference (D), outer core position (pa), outer core width (ba) and height (hD) as shown in Fig. 1. The so achieved values of W 1 =W have been plotted in Figs. 2(a) and (b) as a function of normalized outer core width (b) and height (h), respectively, for different negative dispersion values (Dc) with fixed normalized outer core position (p), at different spot sizes (W ¼ 2.5, 3.0 and 3.5 mm). As can be seen from Figs. 2(a) and (b) that at first W 1 =W increases with ‘b’, when ‘h’ is fixed at the value 0.555 as well as with ‘h’ when ‘b’ is fixed at the value 0.535, respectively, for different negative dispersion coefficients, and after reaching to a maximum value, it comes down with further increase in ‘b’ or ‘h’, respectively, in each case. It can also be seen from all these figures that for smaller spot size (i.e. W ¼ 2.5 mm), maximum negative dispersion coefficient Dc150 ps/ km nm in single mode regime can be achieved by this technique. Also it has been noticed that for a fixed spot size (W ), the curves shift left to achieve larger Dc values. The cutoff wavelength (lc) of the first higher order mode LP11 have also been plotted as a function of normalized side core width ‘b’ when ‘h’ is fixed at the value 0.555, as well as with ‘h’ when ‘b’ is fixed at the value 0.535 in Figs. 2(c) and (d), respectively, for different dispersion coefficients (Dc) as well as for three different spot sizes (W ¼ 2.5, 3.0, 3.5 mm). In general, Figs. 2(c) and (d)
ARTICLE IN PRESS M. Basu, D. Ghosh / Optik 116 (2005) 255–264 2.00
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w =2.5 µm w =3.0 µm w =3.5 µm
1530 (5) 1520 0.40 0.45
0.50
0.55
0.60
0.65
0.70 0.75
0.80
0.85
h
b
(c)
(d)
Fig. 2. Variation of W 1 =W and the cutoff wavelength lc of the first higher order mode LP11 as a function of ‘b’ when ‘h’ is fixed at 0.555 ((a) and (c), respectively) and as a function of ‘h’ when ‘b’ is fixed at 0.535 ((b) and (d), respectively) at three different spot sizes, W ¼ 2.5, 3.0, 3.5 mm with the following parameters: (1) Dc ¼ 150 ps/km nm, p ¼ 6.86; (2) Dc ¼ 120 ps/km nm, p ¼ 5.10; (3) Dc ¼ 100 ps/km nm, p ¼ 4.41; (4) Dc ¼ 100 ps/km nm, p ¼ 7.00; (5) Dc ¼ 50 ps/km nm, p ¼ 3.825; (6) Dc ¼ 60 ps/km nm, p ¼ 5.85; (7) Dc ¼ 50 ps/km nm, p ¼ 5.125.
show that lc (in nm) has a tendency to decrease at first with increase in ‘b’ value or ‘h’ value and after reaching to a minimum value it again starts to increase with further increase in ‘b’ or ‘h’ value, respectively. Unlike the solution for a single core step index profile, the profiles that include multiple claddings outside the central core have a finite cutoff wavelength for propagation of LP01 mode and consequently have a susceptibility of a design to bending loss. As long as the cutoff wavelength (lc) exists in the range of p1550 nm, the proposed fiber can be operated in single mode regime. For example it can be said from the above figures that for higher dispersion values like 150 ps/ km nm at W ¼ 2.5 mm, the normalized outer core width (b) can be varied from 0.455 to 0.535 with outer core height (h) fixed at 0.555 to operate in single mode zone and corresponding values of W 1 =W varies from 1.94 to 1.68. Similarly for the same dispersion value and spot size, the normalized outer core height (h) can be varied from 0.473 to 0.555 with outer core width (b) fixed at
0.535 and corresponding values of W 1 =W stays in the range of 1.94–1.69 when operated in single mode zone. This sharp decrease of W 1 =W reveals the fact that there should be improved bend performance of the proposed fiber for comparatively higher value of ‘b’ or ‘h’. It should also be mentioned that the larger negative dispersion coefficients (Dc) could be achieved when the value of normalized outer core position (p) is increased for a given spot size (W ). As the optimization has been done by adjusting inner core radius (a) and the central relative refractive index difference (D), it can be seen from Fig. 3(a) that the inner core radius (a) is decreasing and relative refractive index difference (D) is increasing continuously with increasing ‘b’ values, respectively, when the normalized outer core height (h) is fixed at 0.555, for different negative dispersion values and spot sizes (W ). At the same time the variation of the inner core radius (a) and central relative refractive index difference (D) as a function of ‘h’ have been studied in Fig. 3(b), when ‘b’ is kept fixed
ARTICLE IN PRESS M. Basu, D. Ghosh / Optik 116 (2005) 255–264 8
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260
(5) 0.50
0.55
0.60
0.65
0.70
0.75
0.80
0.4 0.85
h
(a)
(b)
Fig. 3. Variation of central relative refractive index difference (D) (left side) and inner core radius (a) (right side) as a function of ‘b’ when ‘h’ is fixed at 0.555 (a) and as a function of ‘h’ when ‘b’ is fixed at 0.535 (b) at three different spot sizes, W ¼ 2.5, 3.0, 3.5 mm with the following parameters: (1) Dc ¼ 150 ps/km nm, p ¼ 6.86; (2) Dc ¼ 120 ps/km nm, p ¼ 5.10; (3) Dc ¼ 100 ps/km nm, p ¼ 4.41; (4) Dc ¼ 100 ps/km nm, p ¼ 7.00; (5) Dc ¼ 50 ps/km nm, p ¼ 3.825; (6) Dc ¼ 60 ps/km nm, p ¼ 5.85; (7) Dc ¼ 50 ps/ km nm, p ¼ 5.125.
at the value 0.535. It can also be concluded from the above figures that higher values of negative dispersion coefficients can be achieved at the expense of very high relative refractive index difference (D) and very small inner core radius (a) for three different spot sizes. It has also been seen that larger negative dispersion values can be obtained for larger ‘p’ values, but the exact position of the side core determines the value of negative dispersion coefficient. To obtain large value of negative dispersion coefficient (Dc), at any optimized ‘b’ and ‘h’ value, which are taken as 0.535 and 0.555, respectively, the total core radius [(p+b)a] decreases at a given spot size which has been shown in Table 1. This means that to have a larger negative dispersion coefficient, the side core should come closer to the central core so that modal field spreading in cladding will be less and consequently the performance of this DCF will be better even at higher negative dispersion region. The already obtained values of W 1 =W (as shown in Figs. 2(a) and (b)) have been used to estimate the bend losses of DCF. In Fig. 4(a), the variations of bend loss in dB/km have been plotted as a function of outer core width (b), keeping ‘h’ value fixed at 0.555 for three different spot sizes (W ) and negative dispersion coefficients (Dc), with bend radii of curvature (Rc) equal to 2, 3 and 4 cm, respectively. It is seen that the nature of variation of bend loss with ‘b’ is same as expected with that of W 1 =W with ‘b’ at a fixed ‘h’. Bend loss decreases for increased values of bend radii of curvature (Rc) in all cases. At smaller spot size (W ¼ 2.5 mm), bend loss varies from the value of 3 101 dB/km to a very insignificant value of 3 10–6 dB/km when ‘b’ value changes from 0.455 to 0.535, keeping ‘h’ fixed at 0.555 in the single mode zone of the proposed fiber, even when the bend radius of curvature (Rc) is 2 cm. For RcX3 cm, bend loss is negligible (o106 dB/km) and this shows
that for smaller spot sizes and larger negative dispersion coefficients, we have an improved bend performance of the fiber. In Fig. 4(b), the variation of bend losses in dB/ km have been plotted as a function of normalized outer core height (h) keeping ‘b’ value fixed at 0.535, for the same set of spot sizes (W ) and dispersion values. It is observed that bend losses vary in the same fashion with ‘h’ as in Fig. 4(a). The so-obtained values of bend losses (ab in dB/km) for different bend radii of curvatures (Rc) and spot sizes (W ) along with Rayleigh scattering loss as well as losses due to fiber manufacturing technique such as absorption loss (ar), etc. [23] can be incorporated to calculate the FOM of this central step index segmented core dispersion compensated optical fiber. Fig. 5(a) shows the variation of the FOM as a function of normalized outer core width (b) when normalized outer core height (h) is fixed at 0.555, at spot sizes W ¼ 2.5, 3.0 and 3.5 mm, for different negative dispersion values (Dc) in ps/km nm. The losses due to absorption loss, Rayleigh scattering loss etc. solely depend on relative refractive index difference (D) and controls the value of FOM as shown in Fig. 5. Even when there is no bend, as the central relative refractive index difference (D) is increased with increasing ‘b’ value at ‘h’ ¼ 0.555, it is expected that Rayleigh scattering loss, absorption loss, etc. will increase and as a result FOMwithoutbend should decrease with ‘b’ for the three given spot sizes (W ) and negative dispersion coefficients (Dc). When the bending of DCF is considered the nature of variation of FOMwithbend with ‘b’ value is quite interesting. For W ¼ 3.5 mm and Dc ¼ 60 ps/km nm with outer core position ‘p’ ¼ 5.85, FOM is almost zero when bend radii of curvature (Rc) are 2 and 3 cm in the single mode operation region of the DCF. For larger bend radius, i.e. Rc ¼ 4 cm, FOM values are changing from 46.54 to 84.96 ps/nm dB with ‘b’ value from 0.515
ARTICLE IN PRESS M. Basu, D. Ghosh / Optik 116 (2005) 255–264
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Table 1. Position of inner core and outer core at different negative dispersion coefficients and spot sizes when ‘b’ fixed at 0.535 and ‘h’ fixed at 0.555 W (mm)
Dc (ps/km nm)
p
D (%)
Inner core radius (a) (mm)
Outer core radius (p+b)a (mm)
2.5
100 120 150
4.41 5.10 6.86
2.376 2.865 5.095
0.928 0.786 0.499
4.588 4.432 3.690
3.0
50 100
3.825 7.0
1.449 4.006
1.288 0.566
5.617 4.261
3.5
50 60
5.125 5.85
1.535 1.975
1.079 0.894
6.110 5.706
10 10 10 10 10
10 10 10 10 10 10 10
w=2.5 µ m, p=6.86 D = -150(ps/km-nm) w=3.0 µ m, p=7.0 D = -100(ps/km-nm) w=3.5 µ m, p=5.85 D = -60(ps/km-nm)
5 4 3 2
h = 0.555
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Bend loss (dB/km)
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6
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6
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5
10
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3
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0.70
0.42
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0.50
0.54
0.58
b
h
(a)
(b)
0.62
0.66
0.70
Fig. 4. Bend loss (dB/km) characteristics as a function of ‘b’ when ‘h’ is fixed at 0.555 (a) and as a function of ‘h’ when ‘b’ is fixed at 0.535 (b) for different bend radii of curvature (Rc) at three different spot sizes, W ¼ 2.5, 3.0, 3.5 mm.
to 0.540, respectively, at fixed ‘h’ ¼ 0.555 in the single mode region of the proposed fiber, where as at the same time FOMwithoutbend stays in the range of 156.09–139.73 ps/nm dB. The study shows that for larger spot sizes there is a sharp difference between FOMwithoutbend and FOMwithbend even when large bend radius (Rc) is considered. Further when we decrease the spot sizes (W ), we can see the bend performance of the DCF can be improved considerably. At W ¼ 3.0 mm, we can achieve comparatively higher value of negative dispersion co-efficient (Dc)100 ps/km nm in the single mode region of the fiber. Here FOMwithoutbend changes from 204 to 136 ps/nm dB with ‘b’ value ranging from 0.455 to 0.535. At the same time the FOM of a DCF having sharp bending (Rc ¼ 2.0 cm) is negligibly small. For Rc ¼ 3 cm, FOM changes from 8.28 to 135.92 ps/ nm dB in the same range of ‘b’. When Rc is comparatively larger, i.e. 4 cm, FOMwithbend merge with the curve of FOMwithoutbend. It is well known that minimum bend radius in fiber optic network that are widely accepted for long-term deployment of fibers in practical system installations, is 3.75 cm [22]. For this practically accepted standard bend radius (Rc ¼ 3.75 cm), FOMwithbend changes from 184.53 ps/nm dB to
135.92 ps/nm dB. These FOM values are much closer to the values of FOMwithoutbend in the above mentioned range of ‘b’ for single mode operation region, in comparison to the previous case of W ¼ 3.5 mm, where FOMRc ¼ 3.75 changes from 14.02 to 57.08 ps/nm dB which is far away from the corresponding FOM values for a straight fiber with ‘b’ value ranging from 0.515 to 0.540. Further decrease in spot sizes to a value of W ¼ 2.5 mm, shows that larger value of FOM can be achieved when Dc ¼ 150 ps/km nm at ‘p’ ¼ 6.86. In this case FOMwithoutbend changes from 237.04 to 163.51 ps/nm dB for ‘b’ varying from 0.455 to 0.535 in the single mode region when ‘h’ is fixed at 0.555. When bending is considered with very small radius of curvature, i.e. Rc ¼ 2 cm, FOM curve is showing a sharp change from 97.67 ps/nm dB at ‘b’ ¼ 0.455 and ultimately merges with the curve of FOMwithoutbend and attains a value of 163.51 ps/nm dB at ‘b’ ¼ 0.535. For RcX3 cm, it is seen that FOMwithbend curves completely merge with FOMwithoutbend as bend-loss (ab) is insignificant (o106 dB/km) in the above range of ‘b’ values corresponding to the single mode region of the proposed fiber. So maximum FOM of 237.04 ps/nm dB can be achieved here, even when bending (RcX3 cm) is
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w=2.5 µ m, p=6.86 D = -150(ps/km-nm) w=3.0 µ m, p=7.0 D = -100(ps/km-nm) w=3.5 µ m, p=5.85 D = -60(ps/km-nm)
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80
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0 0.42
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b
(a)
(b)
Fig. 5. Variation of FOM (ps/nm dB) of the central step index segmented core straight DCF and also DCF with different bend radii of curvature (Rc) as a function of ‘b’ when ‘h’ is fixed at 0.555 (a) and as a function of ‘h’ when ‘b’ is fixed at 0.535 (b) at three different spot sizes, W ¼ 2.5, 3.0, 3.5 mm.
considered with ‘b’ ¼ 0.455 and ‘h’ ¼ 0.555. Similarly, in Fig. 5(b), the FOM characteristics have been studied as a function of normalized outer core height (h) when ‘b’ is fixed at 0.535 at W ¼ 2.5, 3.0 and 3.5 mm for the same set of ‘Dc’ and ‘p’ values as given in Fig. 5(a). In this case also for larger spot sizes, i.e. W ¼ 3.5 mm, FOM varies from 154.77 to 140 ps/nm dB with ‘h’ ranging from 0.530 to 560 and ‘b’ fixed at 0.535, even when no bending is considered. For CCITT recommended fiber with Rc ¼ 3.75 cm, the FOM has the range of values from 7.84 to 52.29 ps/nm dB in the single mode region of the proposed fiber, with ‘h’ varying from 0.530 to 0.560 at W ¼ 3.5 mm showing a large difference between FOMwithoutbend and FOMwithbend. When W is fixed at 3.0 mm for Dc ¼ 100 ps/km nm, it is seen that maximum FOM 204.3 ps/nm dB for a straight fiber can be obtained with ‘h’ ¼ 0.465 for single mode operation. When there is sharp bending, i.e. Rc ¼ 2 cm, the performance of the fiber is very poor as FOM is almost zero in this case at W ¼ 3.0 mm. For medium bending, i.e. Rc ¼ 3 cm, initially FOMwithbend becomes very small, i.e. 3.66 ps/nm dB and it approaches to comparatively larger FOM values 130 ps/nm dB for further increase in ‘h’ values in the fundamental operating zone of the DCF. When bend radius is considerably larger, FOMwithbend curve approaches very near to FOMwithoutbend curve. For a standard single mode CCITT recommended DCF with Rc ¼ 3.75 cm, initially FOM has a comparatively lower value of 156 ps/ nm dB and it merges with the curve of FOMwithoutbend afterwards. For the smallest spot size (i.e. W ¼ 2.5 mm) considered here with Dc ¼ 150 ps/km nm, the maximum FOM 238.43 ps/nm dB has been obtained at ‘h’ ¼ 0.473 with ‘b’ fixed at 0.535 for a straight fiber. Further for sharp bending, with Rc ¼ 2 cm, there is a considerable decrease in the value of FOMwithbend at ‘h’ ¼ 0.473, which is 93.2 ps/nm dB and approaches to the corresponding value of FOMwithoutbend when ‘h’
approaches towards the value of 0.555. The figure also shows that for medium and large bend radii of curvature (i.e. RcX3 cm) at W ¼ 2.5 mm, FOM is independent on bending at all. Therefore for a standard fiber in practical system network (Rc ¼ 3.75 cm), the maximum FOM 238.43 ps/ nm-dB can be achieved at normalized outer core height (h) 0.473 with width (b) fixed at 0.535, at W ¼ 2.5 mm and negative dispersion coefficient (Dc) ¼ 150 ps/km-nm. The above studies indicate that maximum negative dispersion coefficient (Dc) as well as FOM have been achieved for comparatively smaller spot size W ¼ 2.5 mm, where the effect of bend loss can be minimized in comparison to other spot sizes (W ¼ 3.0 and 3.5 mm). From the above figures it is also seen that even when there is sharp bending (Rc ¼ 2 cm) at W ¼ 2.5 mm, FOMwithbend is practically same as that of FOMwithoutbend when ‘b’ ranges from 0.520 to 0.535 at ‘h’ ¼ 0.555 and when ‘h’ ranges from 0.544 to 0.555 at ‘b’ ¼ 0.535, respectively. Thus we can also estimate the bend insensitive ‘b’ or ‘h’ values even when there is considerable bending, as the FOM curves for bent as well as straight fibers merge with each other in the above-mentioned range of ‘b’ and ‘h’. Thus Fig. 5 provides a quick reference for fiber optic designers to choose a best suitable DCF with the optimized value of fiber parameters, such as ‘p’, ‘b’ or ‘h’ having maximum possible value of FOM as well as large negative dispersion coefficient (Dc). The above studies have been reflected in Table 2 where we have optimized another important parameter named as MFOM, which includes the length independent splice losses (in dB) between standard fiber and compensating fiber. It is seen from the table that for smaller spot sizes, we can achieve larger negative dispersion coefficient (Dc) as well as smaller length (lc) of the DCF, which will be able to compensate the total dispersion accumulated by a standard fiber of length (ls) 60 km [8]. At W ¼ 2.5 mm, a smallest length 6.8 km of
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Table 2. Comparative study of FOM and MFOM for different fiber parameters at three different negative dispersion coefficients (Dc) and spot sizes (W ) W (mm)
Dc (ps/ km nm)
p
b
h
0.455
0.555 0.520 2.5
150
6.86 0.473 0.535 0.544
0.455 0.555 0.500 3.0
100
7.0 0.465 0.535 0.530
0.515 0.555 0.540 3.5
60
5.85 0.530 0.535 0.560
Rc (cm)
FOM with bend (ps/ dB nm)
2.0 3.0 3.75 2.0 3.0 3.75
97.67 237.04 237.04 175.73 175.73 175.73
2.0 3.0 3.75 2.0 3.0 3.75
93.20 238.82 238.82 171.90 171.90 171.90
2.0 3.0 3.75 2.0 3.0 3.75
0.0 8.28 184.53 0.0 91.88 165.50
2.0 3.0 3.75 2.0 3.0 3.75
0.0 3.92 156.73 0.0 107.06 151.11
2.0 3.0 3.75 2.0 3.0 3.75
0.0 0.1 14.02 0.0 3.87 57.08
2.0 3.0 3.75 2.0 3.0 3.75
0.0 0.0 7.84 0.0 8.76 52.29
DCF is sufficient to compensate the dispersion of the link. To estimate the splice loss the spot size (W ) of standard fiber has been considered of the order of 5.0 mm [18]. Inclusion of losses due to splices or joints reduces the value of FOM by a considerable amount. It has been seen that the maximum FOM 237–239 ps/ nm dB for the proposed DCF with RcX3 cm, can be obtained for two given sets of fiber parameters like ‘b’ ¼ 0.455, ‘h’ ¼ 0.555, ‘p’ ¼ 6.86 and ‘b’ ¼ 0.535, ‘h’ ¼ 0.473, ‘p’ ¼ 6.86, respectively, at W ¼ 2.5 mm with
FOM without bend (ps/ dB nm)
lc (km)
as (dB)
237.04
175.73 6.8
1.938
171.90
204.34
165.50 10.2
1.087
204.44
0.0 8.21 154.20 0.0 83.68 140.68 0.0 3.90 134.30 0.0 96.09 130.15
151.11
156.09
139.73 17.0
140.0
82.38 163.43 163.43 131.74 131.74 131.74 79.18 164.28 164.28 129.58 129.58 129.58
238.82
154.77
MFOM with bend (ps/ dB nm)
0.541
0.0 0.1 13.92 0.0 3.86 55.40 0.0 0.0 7.81 0.0 8.72 50.88
MFOM without bend (ps/ dB nm) 163.43
131.74
164.28
129.58
167.8
140.68
167.87
130.15
144.15
130.09
143.02
130.32
Dc ¼ 150 ps/km nm. When splice loss is included these FOM values have been modified to comparatively smaller value of MFOM 163–164 ps/nm dB. Again, another example shows that when ‘b’ ¼ 0.520, ‘h’ ¼ 0.555, ‘p’ ¼ 6.86 for W ¼ 2.5 mm, the proposed DCF with Dc ¼ 150 ps/km nm has FOM 175.73 ps/ nm dB irrespective of bending, where as it can be modified to a lesser value of MFOM 131.74 ps/nm dB. Similar studies have been reflected for other spot sizes of 3.0 and 3.5 mm with Dc ¼ 100 and 60 ps/km nm,
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respectively. Thus, it is found that all the above studies on FOM characteristics can estimate different fiber parameters for a segmented core central step index single mode DCF, operating at wavelength of 1550 nm.
5. Conclusion In this paper, we have designed a segmented core central step index single mode DCF by optimizing different important fiber parameters at the 1.55 mm window. Unlike the single core step or graded index DCFs, the proposed fiber design shows improved bend performances at comparatively smaller values of Petermann-II spot sizes (W ) of the DCF and considerably high value of FOM can be achieved. Moreover, the fiber parameters are optimized in such a way that bend loss insensitive DCFs can also be obtained. This kindles hope for fiber optic designers to develop a modern optical communication system comprising of CSFs and DCFs having smaller length (lc) and maximum FOM. However, it should be mentioned that inclusion of length independent splice loss controls the FOM to a comparatively lower limit. The above studies have a scope of future work for further improvement of DCFs both in linear and non-linear regime.
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