Polarization mode dispersion in single mode optical fibers due to core-ellipticity

Optics Communications 263 (2006) 36–41 www.elsevier.com/locate/optcom

Polarization mode dispersion in single mode optical fibers due to core-ellipticity Deepak Gupta 1, Arun Kumar *, K. Thyagarajan Department of Physics, Indian Institute of Technology Delhi, Hauz Khas, New Delhi 110 016, India Received 10 October 2005; received in revised form 27 December 2005; accepted 12 January 2006

Abstract The variation of polarization mode dispersion (PMD) with V-parameter in single mode optical fibers due to core-ellipticity is studied by performing numerical simulations taking into account both geometrical and thermal-stress-induced birefringences as well as the variation of fiber refractive indices with wavelength. Simple empirical relations are given for calculating the mean PMD for any value of core-ellipticity and V-parameter of a standard single mode fiber. It is observed that the mean PMD saturates for V J 1.8 leading to very small second order PMD.  2006 Elsevier B.V. All rights reserved. PACS: 42.81.Gs Keywords: Birefringence; Polarization; Polarization mode dispersion; Core-ellipticity; Thermal stress

1. Introduction Polarization mode dispersion (PMD) in optical fibers is one of the key factors limiting the performance of ultra high bit-rate long-haul fiber-optic communication systems (bit-rate > 10 Gbps) [1–3]. It is measured in terms of the differential group delay (DGD), which is the time delay acquired between the two orthogonally polarized components of the signal pulse [4], and is expressed in units of pffiffiffiffiffiffiffi ps/ km. The difference in group velocities arises due to the existence of birefringence in the fiber, which is mainly due to a small core-ellipticity and asymmetric thermalstress acquired during the fabrication process. The objective of this paper is to study the effect of core-ellipticity on PMD of standard single mode fibers (SSMF) corresponding to the ITU-G.652 specification which are typically employed in long-haul transmission systems. Through numerical simu-

*

Corresponding author. Tel.: +91 11 26591361; fax: +91 11 26581114. E-mail addresses: d_gupta10@rediffmail.com (D. Gupta), akumar@ physics.iitd.ac.in (A. Kumar). 1 Tel.: +91 11 26596579; fax: +91 11 26581114. 0030-4018/$ - see front matter  2006 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2006.01.017

lations, taking into account both geometrical and stress birefringences, a simple empirical relation is established for fiber PMD arising due to core-ellipticity. From this relation, one can estimate PMD due to core-ellipticity for any given practical value of core-ellipticity and V-parameter, which are the two governing quantities for determining fiber PMD. 2. Calculation of fiber birefringence The fiber birefringence due to core-ellipticity is comprised of two components – geometrical birefringence arising due to the elliptic cross-section of the core, and thermal-stress-induced birefringence arising due to an asymmetrical stress on the core owing to the difference in thermal expansion coefficients of core and cladding. Let a and b represent the semi-major and semi-minor axes of the elliptic core of the fiber which was desired to be circular of radius r0. Since the core area of the fiber does not change due to acquisition of ellipticity during fabrication, a, b and r0 are related as: a ¼ r0 ð1  e2 Þ1=4 ;

b ¼ r0 ð1  e2 Þ1=4 ;

ð1Þ

D. Gupta et al. / Optics Communications 263 (2006) 36–41

where e is the ellipticity, defined as e2 = 1  (b2/a2). The geometrical and thermal-stress-induced birefringences of this elliptic-core fiber are calculated separately as follows: 2.1. Geometrical birefringence Several methods [5–7] have been reported in the literature to calculate the geometrical birefringence in elliptic-core fibers. In the present paper we have used the equivalent waveguide model proposed by Kumar et al. [8] which is simple and shown to match with the accurate numerical results of Dyott et al. [7] extremely well. According to this method, the given elliptic-core fiber is approximated by a rectangular-core waveguide having the same aspect ratio, core area and core–cladding refractive indices. The dimensions of the equivalent rectangular-core waveguide are pffiffiffiffiffiffi pffiffiffiffiffiffi given by 2a0 ¼ pa and 2b0 ¼ pb. The propagation constants bx and by (of the x and y-polarized modes respectively) of this waveguide are in turn calculated by considering it to be a perturbed form of a pseudo rectangular-core waveguide of the same dimensions whose dielectric constant is separable in x and y. Thus we obtain the geometrical birefringence BG = bx  by as a function of the V-parameter, defined for the circular core fiber as V ¼ k 0 r0 ðn21  n22 Þ1=2 , where k0 is the free space wave vector and n1 and n2 are the core and cladding refractive indices of the step-indexed elliptic-core fiber. 2.2. Thermal-stress-induced birefringence The appearance of an elliptic core during fiber fabrication leads to an asymmetric stress distribution inside the fiber, owing to the difference in thermal expansion coefficients of the core and cladding materials [9–11]. This stress leads to a refractive index step which is superimposed on the already existing dopant-induced index step. For the x and y polarized modes, these index steps are given by [9] 1 ða1  a2 ÞDT 1 Dnx ¼ n3 ðp a þ p12 bÞ; 2 2 1m ða þ bÞ 11 1 ða1  a2 ÞDT 1 ðp b þ p12 aÞ; Dny ¼ n3 2 2 1m ða þ bÞ 11

ð2Þ

37

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where U ¼ r0 k 20 n21  b20 and b0 is the propagation constant for the circular core fiber. The total birefringence B due to core-ellipticity is the sum of the geometrical and thermal-stress-induced birefringences: B = BG + BS. In all the above calculations, the variation of n1 and n2 with operating wavelength is taken into account by the Sellemeier equation for Ge-doped silica fibers given by [12] n2 ðk0 Þ ¼ 1 þ

3 X i¼1

Ai k20 ; k20  B2i

ð4Þ

where Ai and Bi are the Sellemeier coefficients corresponding to the GeO2-doping which produces the given core– cladding refractive index difference and k0 = 2p/k0 is the operating wavelength. 3. Calculation of PMD As mentioned in the beginning, random mode couplings along the fiber length impart a stochastic variation to the PMD due to core-ellipticity. To incorporate their effects, the transmission fiber of length L is considered to be a concatenation of N number of small segments each having a fixed set of birefringent axes, a section length which is randomly chosen around a certain mean length and oriented randomly with respect to each other [13,14]. Fig. 1 shows the schematic of the concatenation. The Jones matrix Ti for the ith section can be written down as !   0 expði Bh2 i Þ cos hi  sin hi Ti ¼ sin hi cos hi 0 expði Bh2 i Þ   sin hi cos hi  ; ð5Þ  sin hi cos hi where hi is the orientation of the fast axis of the ith segment with respect to a fixed laboratory axis (say x-axis), and hi is the length of the ith segment. Thus the Jones matrix corresponding to the entire length of fiber can be expressed as a multiplication over Ti for all the N sections: T ðV ; eÞ ¼

N Y

ð6Þ

T i.

i¼1

where n is the average refractive index of the fiber, a1 and a2 are the thermal expansion coefficients of the core and cladding materials respectively, DT is the difference of room temperature and the softening temperature of the fiber core material, m is the Poisson’s ratio of the core material, p11 and p12 are the strain-optic tensor coefficients, and a and b are the semi-major and semi-minor axes of the elliptic core. The resulting thermal-stress-induced birefringence is given by BS ¼ bx  by    U 2 1 3 ða1  a2 ÞDT ða  bÞ n ðp ¼ k0 1  2  p Þ 12 ; 2 1  m2 ða þ bÞ 11 V

To calculate DGD, the method of Jones matrix eigenanalysis is used [15]. The Jones matrix of the fiber is calculated for two nearby V-parameters, say V1 and V1 + dV and the DGD Ds is given by   argðq1 =q2 Þ  ; ð7Þ DsðV 1 ; eÞ ¼   dx where q1 and q2 are the eigenvalues of the matrix product T(V1 + dV)T1(V1), arg() is the argument function and dx is the change in angular frequency over the two V values and is given by dx ¼

ð3Þ

c dV r0 ðn21

 n22 Þ

1=2

;

ð8Þ

38

D. Gupta et al. / Optics Communications 263 (2006) 36–41

Fig. 1. Schematic showing the fiber as a concatenation of numerous small sections of random lengths and orientations.

where c is the velocity of light in free space. Here the value of dV is chosen such that the product Dsdx is sufficiently small to avoid any omission of full 2p rotations in the arg() function. To obtain the probability distribution of DGD, Ds is calculated a large number of times by treating hi to be a random quantity uniformly distributed in [0, p] and hi to be a random quantity chosen from a Gaussian distribution around the mean length hm = L/N with a standard deviation Dh [13,14]. These values of Ds are arranged in the form of a relative frequency distribution which is then fitted to a Maxwellian of the form ! 32 Ds2 4Ds2 P ðDsÞ ¼ 2 exp  ; ð9Þ 2 p hDsi3 phDsi where hDsi is the mean DGD or the most probable value of DGD. The mean PMD is then calculated as hDsi DPMD ¼ pffiffiffi . L

ð10Þ

4. Results and discussion Using the simulation model discussed above, we analyze the PMD distribution of a standard single mode fiber (SSMF) corresponding to the ITU-G.652 specification for a range of core-ellipticities e from 0.1 to 0.4. This range covers nearly all typical core-ellipticity values found in already installed fiber links. Also, the V-parameter is restricted to the range 1.0 < V < 2.4, which is the typical useful single mode range. The physical parameters and material constants of the fiber and the simulation parameters are given in Table 1. The physical parameters are taken from the Corning SMF-28 fiber datasheet, which is a typical SSMF employed in fiber links. The simulation parameters are taken from Refs. [13,14]. Material constants of the fiber are calculated using the following relations [16]: (i) Poisson’s ratio is calculated as m ¼ ð1  mÞmSiO2 þ mmGeO2 , where m is the GeO2-dopant concentration in the core (in mol.%), and mSiO2 ¼ 0:165 and mGeO2 ¼ 0:197 are the Poisson’s ratios for pure SiO2 and pure GeO2. (ii) The softening temperature TS of the fiber core is taken to decrease from 1100 C (pure SiO2) by 10 C per mol.% of GeO2 concentration. Room temperature TR is taken as 25 C to calculate the temperature difference DT = TR  TS.

Table 1 Material constants [16], physical parameters (Corning SMF-28 fiber datasheet) and simulation parameters [13,14] of the SSMF Value Material constants of SSMF Thermal expansion coefficient a1 (core) Thermal expansion coefficient a2 (clad) Temperature difference DT Poisson’s ratio m Strain-optic tensor coefficient p11 Strain-optic tensor coefficient p12

9.19 · 107 C1 5.6 · 107 C1 1040 C 0.166 0.126 0.260

Physical parameters of SSMF Core radius r0 Cladding refractive index n2 (at 1550 nm) Core–cladding refractive index difference D

4.1 lm 1.44402 0.36%

Simulation parameters Fiber length L No. of sections N Mean section length hm = L/N Standard deviation Dh

80 km 800 100 m 20 m

(iii) The thermal expansion coefficient of the core is taken to increase from that of the cladding by 1.04 · 107 C1 per mol.% of GeO2 concentration. Fig. 2 shows the variations of geometrical birefringence BG (dotted curves), thermal-stress-induced birefringence BS (dashed curves) and the total birefringence B (solid curves) in each section as functions of V for varying ellipticity e. For all the curves, thermal-stress-induced birefringence dominates the geometrical one and the two add up to increase the net birefringence, which is expected for GeO2-doped low index difference (D < 1–2%) silica fibers [17]. Fig. 3 shows the corresponding contributions of these birefringences to the DGD per unit length in the various sections of the fiber, for varying ellipticity e. The DGD per unit length is simply the derivative of the birefringence with respect to the optical angular frequency. The dotted, dashed and solid curves correspond to contributions of geometrical, stress and total birefringence respectively. It shows that the contribution of stress-induced birefringence to the DGD is more than that of geometrical birefringence for all ellipticities, and highlights the fact that merely considering geometrical effects for computation of PMD of elliptic-core fibers is not sufficient. Stress-induced effects are indeed significant and need to be taken into account for a correct description of PMD. It should also be noted that in principle a compensation of PMD due to geometri-

D. Gupta et al. / Optics Communications 263 (2006) 36–41

Fig. 2. Variations of fiber birefringences with V for different core-ellipticities of the SSMF: geometrical (dotted curves), stress-induced (dashed curves) and total birefringence (solid curves).

39

Fig. 4. Plot of the DGD probability distribution for V = 1.8 and varying e showing simulated curves (solid) and their Maxwellian fits (dotted).

Fig. 5. Variation of mean PMD DPMD with V-parameter for varying coreellipticity e. Fig. 3. Contribution of fiber birefringences to DGD per unit length with V for different core-ellipticities of the SSMF: geometrical (dotted curves), stress-induced (dashed curves) and total birefringence (solid curves).

cal and thermal-stress-induced effects can lead to low intrinsic PMD in dispersion shifted fibers which have a refractive index profile that is different from the step-index profile of SSMFs considered here [18]. The curves of Fig. 2 are used to generate the DGD distribution using Eqs. (5)– (10), where dV is chosen to be 104 and 10 000 DGD points are generated for each set of V and e. Fig. 4 shows the generated distributions (solid curves) and their corresponding Maxwellian fits (dotted curves) for V = 1.8 and different values of e. The mean DGD hDsi is found out from these curves and the corresponding mean PMD is calculated using Eq. (10). Fig. 5 shows the variation of the mean PMD, DPMD as a function of V with different values of e.

As seen from the figure, the mean PMD increases with V and saturates beyond V J 1.8 leading to a very small variation of mean PMD with V-parameter. Since for a given fiber V could only be changed by changing wavelength, it implies that the mean PMD varies very little with wavelength in this region. This leads us to an important conclusion that the second order PMD, which depends on the variation of PMD with wavelength, is very small for V J 1.8. To verify the correctness of these results, we compare them with the experimental results on PMD due to coreellipticity reported by Mabrouki et al. [19]. Fig. 6 shows this comparison in detail by plotting the mean PMD as a function of ellipticity e. The triangles and crosses correspond to measured PMD values at 1310 nm (Fig. 6(a)) and 1550 nm (Fig. 6(b)) by interferometric and Jones

40

D. Gupta et al. / Optics Communications 263 (2006) 36–41

this case, the variation can be described by the following relation: c1 c2 ð11Þ DPMD ¼ c0 þ þ 2 . V V Fig. 7 shows the variation of DPMD with V1 (solid curves) and the quadratic fit as given by Eq. (11) (dotted curves). As can be seen clearly, the fitted curves and the simulated curves match very well in the entire range of V. To complete the empirical relation, we examine the variation of the best-fit coefficients ci with ellipticity e and observe that it can be described by the following quadratic relation: ci ¼ d 0i þ d 1i e þ d 2i e2 .

Fig. 6. Variation of mean PMD with ellipticity e showing experimental results of Ref. [19] and the present simulated results at (a) 1310 nm and (b) 1550 nm wavelength. Experimental points (crosses and triangles) correspond to Jones matrix eigenanalysis and interferometric method respectively.

matrix methods respectively of Ref. [19]. The solid curves correspond to the present results for V = 2.4 and 2.0, corresponding to wavelengths of 1310 nm and 1550 nm respectively. Since the values of various fiber parameters are not mentioned in Ref. [19] an exact match between the two is not expected. However, it can be seen that for some values of e the two results match very well. The large difference between our results and the experimental results at other ellipticities can be attributed to some random birefringences present in those fibers which is evident from the erratic behaviour of measured PMD with core-ellipticity in some tested fibers. Given the curves of Fig. 5, we proceed to establish an empirical relation for DPMD for the given SSMF, for which we plot this quantity versus inverse of V-parameter (V1) and note that the variation follows a parabola. In

ð12Þ

Fig. 7. Variation of mean PMD DPMD with inverse of V-parameter for different core-ellipticities: simulated (solid curves) and quadratic fit (dotted curves).

Fig. 8. Variations of the best-fit coefficients ci with e (solid curves) and their corresponding quadratic fits as given by Eq. (12) (dotted curves).

D. Gupta et al. / Optics Communications 263 (2006) 36–41 Table 2 Coefficients for the empirical relation given by Eq. (12) ci c0 c1 c2

41

Acknowledgements

dji (ps km1/2) d0i

d1i

d2i

0.51 2.79 2.26

3.49 24.13 20.11

11.90 111.64 110.73

D. Gupta is thankful to the Industrial Research and Development unit at Indian Institute of Technology Delhi for granting a High Value Research Assistantship. The authors are thankful to the reviewer for useful suggestions. References

Here i takes the values 0, 1 and 2. Fig. 8 shows the variation of the coefficients ci with e (solid curves) and their corresponding quadratic fits as given by Eq. (12) (dotted curves). Again the fitted curves match quite well with the original one. The values of d0i, d1i and d2i are given in Table 2. Eqs. (11) and (12) represent the complete empirical relations for PMD due to core-ellipticity. Using these relations, the mean PMD can be calculated for any given fiber length and any typical value of V and e, provided the fiber corresponds to the G.652 specification. 5. Conclusions We have analyzed the effect of V-parameter on PMD arising due to core-ellipticity in a standard single mode fiber considering both geometrical and stress-induced birefringences, and have observed that the latter has a significant contribution to PMD and cannot be neglected for a correct description of PMD due to core-ellipticity. It has been shown that the mean PMD varies as a quadratic function of V1 for the typical useful single mode region, and the second order contribution to PMD is very small for V J 1.8. Simple empirical relations are given which can be used to easily calculate the PMD characteristics of a standard single mode fiber for any given values of V-parameter and core-ellipticity.

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