Abstract

A novel technique is proposed for measuring the longitudinal fiber parameters of multi-core fiber (MCF). The mode field diameter (MFD)of a fiber link composed of MCF is successfully estimated with a modified optical time domain reflectometer (OTDR). The measurement accuracy of the MFD distribution is revealed by simulation as a function of the mode coupling coefficient. It is also shown that the relative-index difference and chromatic dispersion of MCF can be estimated with the present technique.

© 2014 Optical Society of America

1. Introduction

As the power launched into communications fiber has increased to cope with the rapid increase in transmission capacity, optical nonlinear effects and the fiber fuse phenomenon have become a problem [1, 2]. Because of these effects, the transmission capacity of conventional fiber is approaching its fundamental limit [3]. Recently, the research and development of novel optical fibers have been undertaken throughout the world with the goal of overcoming the above limit [4, 5].

One innovative optical fiber is multi-core fiber (MCF) which accommodates a number of single-mode fiber cores, each of which carries a different signal. In MCFs, the mode field diameter (MFD), cutoff wavelength and optical loss are important parameters as well as the crosstalk, namely the maximum power transferred between the cores. There have been many reports on techniques for measuring the MFD, cutoff wavelength, optical loss and chromatic dispersion. The MFD and cutoff wavelength are especially important and there are many techniques for their measurements including the far-field pattern technique [6], the near field pattern technique [6], and the optical time reflectometer (OTDR) technique [710].

In particular, the OTDR technique provides a very simple way of measuring the MFD and it can measure its longitudinal distribution [9, 10]. Moreover, there have been many reports on the use of OTDR techniques to estimate the relative-index difference [11], chromatic dispersion [810], and Raman gain coefficients [1214] in a fiber link. However, there have been few reports on the use of OTDR to measure the fiber parameters of MCFs.

In this paper, we propose a simple technique for measuring the longitudinal fiber parameters and the crosstalk of MCF based on an improvement we made to conventional OTDR. We reveal the measurement accuracy of the MFD distribution as a function of the mode coupling coefficient by simulation. We successfully estimate the MFD, relative-index difference and chromatic dispersion distributions of a fiber link composed of MCF with our proposed technique.

2. Principle of our proposed technique

2.1 Bi-directional OTDR technique

The experimental setup for measuring the longitudinal fiber parameters of MCF is shown in Fig. 1.

 

Fig. 1 Experimental setup for measuring MFD.

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We consider a homogeneous seven-core MCF as shown in Fig. 1. Thus, we assume that the mode coupling coefficients h between center and outer cores are the same and the mode coupling coefficients between outer cores are also as the same. Here, we consider the longitudinal fiber parameter of the center core in MCF. When the input power P0(0) is launched into the center core of MCF, the powers of center and outer cores can be obtained by solving the coupled power equations with regards to seven cores. P1(z) and P2(z) denote the powers of center core and one of outer cores with respect to fiber length z, respectively. Both powers P1(z)and P2(z) can be obtained as [15]

P1(z)=P0(0)[1+6exp(7hz)7]
P2(z)=P0(0)[1exp(7hz)7]

Two different kinds of reference fiber with a length of L0 as shown in Fig. 1 are employed to determine the absolute value of the longitudinal fiber parameter before an MCF of length L. By using the Eqs. (1) and (2) and reference [16], OTDR signals of the center core S1(z) and S2(z) (in dB) launched from the opposite ends (subscripts 1 and 2) of a fiber of length L + L0 can be expressed as

S1(z)={10logP1+10log(αs(z))+10log(B(z))2αz10loge(0zL0)10logP1+10log(αs(z))+10log(B(z))2αz10loge+10log[1+6exp[14h(zL0)]7](L0+LzL0),
S2(z)={10logP2+10log(αs(z))+10log(B(z))2α(L+L0z)10loge+10log[1+6exp[14hL]7](0zL0)10logP2+10log(αs(z))+10log(B(z))2α(L+L0z)10loge+10log[1+6exp[14h(L+L0z)]7](L0+LzL0),
where P1 and P2 are the input powers, αs the local scattering coefficient, B the backscattered capture fraction, and α the local attenuation coefficient. h denotes the mode coupling coefficient between center core and outer cores in an MCF.

Equations (3) and (4) include the optical losses caused by optical power decay, imperfection and mode coupling. When we estimate the longitudinal fiber parameter of the core in an MCF using the bidirectional OTDR technique, we need to know the loss induced by mode coupling between center core and outer cores. The imperfection loss contribution U(z) [7] excluding the mode coupling loss can be obtained as

U(z)=5logP1P2+10log(αs(z)B(z))2α(L+L0)(10loge)={S1(z)+S2(z)25log[1+6exp[14hL]7](0zL0)S1(z)+S2(z)25log{[1+6exp[14h(zL0)]][1+6exp[14h(L+L0z)]]49}(L0zL0+L).

The backscatter capture fraction B is expressed using the refractive index of the core n and the MFD 2w(λ,z) as [7]

B(λ,z)=32log[λ2πnw(λ,z)]2.

The imperfection loss contribution Un(z) normalized by the value at the first reference point z = z0 can be expressed as

Un(z)=U(z)U(z0)=10log(αs(z)n2(z0)αs(z0)n2(z))+20log(2w(λ,z0)2w(λ,z)).

With conventional single-mode fibers, the variation in the local scattering coefficient αs is negligible compared with that in the MFD. The second reference point z = z1 satisfies Eq. (7). Therefore, the longitudinal MFD distribution of the core in the MCF can be estimated by using the normalized imperfection loss and MFDs at the two reference points z0 and z1 as [10]

2w(λ,z)=2w(λ,z0)(2w(λ,z0)2w(λ,z1))U(z)U(z0)U(z1)U(z0).

To obtain the imperfection loss contribution U(z), we have to estimate the crosstalk between cores. When the OTDR pulse is launched into center core of an MCF with a length L, the backscattered powers S3(z) ( = 10logP3(z)) of center core and S4(z) ( = 10logP4(z)) of the outer core in dB from a given position z in an MCF are expressed as

S3(z)=10logP3+10log(αs1B1)2α1z(10loge)+10log[1+6exp(14hz)7],
S4(z)=10logP3+10log(αs2B2)2α2z(10loge)+10log[1exp(14hz)7],
where P3 is the input power, αsi (i = 1,2) the local scattering coefficient, and Bi (i = 1,2) the backscattered capture fraction, and αi(i = 1,2) the local attenuation coefficient.

Here, if we assume that αs1 = αs2 = αs, α1 = α2 = α, and B1 = B2 = B, the crosstalk (XT) between two cores in an MCF estimated using the OTDR is obtained with Eqs. (9) and (10) as the following equation.

XT(z)=10log[P4(z)P3(z)]=10log[1exp(14hz)1+6exp(14hz)].

Note that the crosstalk given by Eq. (11) estimated using the OTDR is that for a fiber length of 2z. The average mode coupling coefficient h(z) over the fiber length z can be estimated from Eq. (11). Thus, the imperfection loss contribution U(z) can be estimated by taking the mode coupling coefficient h(z) into account. The MFD distribution of the core in an MCF can be measured with our present technique.

Once the longitudinal MFD distribution is known, the relative-index difference Δ(z) (in %) can be estimated as [11]

Δ(z)=1k[(1+kΔ(z0))×10U(λ,z)U(λ,z0)20log(2w(λ,z0)2w(λ,z))101],
where k is a proportional constant of the Rayleigh scattering coefficient to the dopant concentration of GeO2-doped core fiber.

Moreover, we can estimate the longitudinal chromatic dispersion D(z) by using the MFDs and the relative-index difference Δ. The chromatic dispersion D is given as the sum of the waveguide dispersion Dw and the material dispersion Dm. The material dispersion Dm and waveguide dispersion Dw are expressed as [16]

D=Dm+D,w
Dm=λcd2ndλ2,
Dw=λ2π2cnddλ(λw2),
where c is the light velocity and n is the refractive index of the core.

The material dispersion Dm can be estimated from the dopant concentration in an optical ðber by using Sellmeier’s relation [17]. By contrast, the waveguide dispersion Dw can be estimated by determining the wavelength dependence of the MFD [18]. We approximated the wavelength dependence of the MFD [19] as

w(λ)=g0+g1λ1.5.

Substituting Eq. (16) into Eq. (15), the waveguide dispersion can be expressed as

Dw=λ2π2cnw2(13g1λ1.5w).

Here, when the MFDs are given at two wavelengths, the coefficient g1 is determined by solving Eq. (16) with regards to the two wavelengths.

2.2 Measurement accuracy of MFD

We discuss the effect of the mode coupling between cores on the MFD estimation. The imperfection loss I(z) including the mode coupling loss contribution can be obtained from Eq. (3) as

I(z)={U(z)+5log[1+6exp[14hL]7](0zL0)U(z)+5log{[1+6exp[14h(zl0)]][1+6exp[14h(L+l0z)]]49}(L0zL0+L).
Note that I(z) corresponds to U(z) when there is no mode coupling in the MCF.

We examined the MFD estimation with and without mode coupling as a function of the mode coupling coefficient h for the fiber link. 2w*(λ, z) denotes the MFD that we estimated from the imperfection loss I(z). Here, we assumed that the MFD can be estimated using Eq. (8) accurately. Therefore, we consider the principle relative error Δε of the MFD estimation caused by the mode coupling between cores as

Δε=2w(λ,z)2w*(λ,z)2w(λ,z)=12w*(λ,z)2w(λ,z)=1(2w(λ,z0)2w(λ,z1))F(λ,z,h)U(z1)U(z0),F(λ,z,h)=5log[1+6exp(14h(zL0)][1+6exp(14h(L+L0z)]7[1+6exp(4hL)].

The imperfection loss difference Un(z1) = U(z1)-U(z0) between two reference points can be expressed in terms of relative-index difference Δ and MFD 2w as

Un(z1)=U(z1)U(z0)=10log[{1+kΔ(z1)}{12Δ(z0)}{1+kΔ(z0)}{12Δ(z1)}]+20log[2w(λ,z0)2w(λ,z1)]

We calculated the relative error using Eqs. (19) and (20). In calculations, we used the parameters of the two reference fibers in Table 1.

Tables Icon

Table 1. Parameters of Test MCF and Reference Fibers used in calculations

Figure 2 shows the relationship between fiber length and relative error Δε(in %) of MFD as a function of mode coupling coefficient h. Here, the length of MCF was assumed to be 4km.

 

Fig. 2 Relationship between fiber length and relative error Δεof the MFD as a function of mode coupling coefficient h.

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It is seen that the relative error is maximized at z = 2 km, which corresponds to the center position of the MCF. This is because the function of F(λ,z,h) becomes minimum in the center position of the MCF. It is also seen that the principle relative error decreases as h decreases.

The maximum relative errors for the fiber lengths L of 4km, 20km, and 40km were calculated as a function of mode coupling coefficient as shown in Fig. 3.

 

Fig. 3 Relationship between the mode coupling coefficient h and maximum relative error Δεof the MFD as a function of fiber length.

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It is seen that the maximum relative error increases as the mode coupling coefficient h (1/km) increases. It is also seen that the relative error increases with fiber length of MCF when the h is given. As a result, it is clarified from Fig. 3 that MFD of 4km long MCF with a mode coupling coefficient h of less than 0.02 (1/km) can be estimated with an accuracy of better than 1% without taking the mode coupling between cores into account. Therefore, in this case, we can estimate the longitudinal fiber parameters of an MCF by using the conventional bidirectional OTDR technique [9, 10].

Next, we examine validity of the approximation when deriving Eq. (12) regarding crosstalk. The crosstalk between center and outer cores is obtained from Eqs. (10) and (11) as

XT(z)=S4(z)S3(z)=10log(αs2B2αs1B1)+2(α1α2)z(10loge)+10log[1exp(14hz)1+6exp(14hz)].

Equation (21) can be rewritten in terms of refractive index, relative-index difference Δ and MFD as

XT(z)=g(z,λ,Δ1,Δ2,w1,w2)+10log[1exp(14hz)1+6exp(14hz)]g(z,λ,Δ1,Δ2,w1,w2)=10log[(1+kΔ2)(1+kΔ1)]+10log[n12w12n22w22]+R0kλ4(Δ1Δ2)2z(10loge).
where the function g (in dB) represents the ratio of Rayleigh scattering coefficient αs, the backscatter capture fraction B and optical loss in the outer core to that in the center core. n1and n2 denote the refractive indexes of the center and outer cores, respectively.

The relative error Δε between the exact hth value in Eq. (22) and approximated hap value in Eq. (11) can be derived as

Δε=hexhaphex==ln[110XTg10110XT101+610XT101+610XTg10]ln[110XTg101+610XTg10].

Figure 4 shows the relationship between the relative error Δε (in %) and g (in dB) as a function of XT (in dB).

 

Fig. 4 Relationship between g value and relative error Δεof the h as a function of crosstalk between center and outer cores.

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It is seen that the absolute value of relative error increases as g value move away from zero. It is also seen that h is underestimated as the g value is positive and h is overestimated as the g value is negative. Moreover, it is also seen that when the g value of 3dB, the relative error becomes 100% and the approximated hap value corresponds to twice exact hex value. For example, we consider the 4km long MCF, which is composed of a center core of conventional single-mode fiber (SMF) and six outer cores of dispersion-shifted fibers (DSFs). By using the parameter values in Table 1, the g value was obtained to be 0.92 dB. Here, a k value of 0.62 was used. The relative error of h is about −23% from Fig. 4. Therefore, the assumption in Eq. (11) approximately holds taking the relative error of MFD in Fig. 3 into account if the average h is assumed to be 10−2 (1/km) for the fiber length of 4km.

3. Experiments

3.1 MFD distribution

To confirm the effectiveness of our present technique, we measured the MFD distribution along a fiber link composed of two single core fibers (Ref #1, Ref #2) and a test MCF with seven cores. The parameters of these fibers are listed in Table 2. Ref #2 was spliced with the center core in the test MCF as shown in Fig. 1.

Tables Icon

Table 2. Parameters of Test MCF and Reference Fibers

Fig. 5 shows the experimental setup for measuring the crosstalk in an MCF using a modified OTDR [16, 20, 21]. The modified OTDR consists of a conventional OTDR including a light source and a detector, a circulator, and a switch. The backscattered power of the adjacent core is measured using the switch. The coupling between the conventional single-mode fiber and each core of the multi-core fiber was realized with a space coupling device as shown in Fig. 5.

 

Fig. 5 Experimental setup for measuring crosstalk using modified OTDR.

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Figures 6(a) and 6(b) show the bidirectional backscattered powers of the fiber link at λ = 1.31 and 1.55 μm, respectively. The modified OTDR at wavelengths of both 1.31 and 1.55 μm was used to measure the backscattered signal powers for the link. In our measurements, the OTDR pulse width was set at 1μs and the averaging time was 3 min.

 

Fig. 6 Bidirectional backscattered powers of fiber link at (a) 1.31 μm and (b) 1.55 μm.

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Figure 7(a) and 7(b) show the backscattered powers of the center and outer cores of the test MCF at λ = 1.31 and 1.55 μm obtained using the modified OTDR when the optical pulse from the OTDR was launched into the center core of the MCF. The backscattered power from the outer core was measured by switching the output port to the outer core as shown in Fig. 4. In the OTDR measurements, the pulse width of the OTDR was 1 μs and the averaging time was 3 min.

 

Fig. 7 Backscattered powers of center and outer cores in MCF when an optical pulse was launched into the center core. (a) 1.31 μm, and (b) 1.55 μm.

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Figure 8 shows the crosstalk between the cores for the test MCF at λ = 1.31 and 1.55 μm. The crosstalk was estimated from the backscattered powers of the center and outer cores as shown in Fig. 7. We found that the estimated crosstalk of L = 4km at 1.55 μm was in good agreement with the value measured with the conventional power method. We also found that the estimated crosstalk at 1.31 μm was about −55 dB at a distance of 4km. The difference between the crosstalk values at 1.31 and 1.55 μm was about 30dB at L = 4 km.

 

Fig. 8 Crosstalk between cores at λ = 1.31 and 1.55 μm.

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Figure 9 shows the average mode coupling coefficient h at λ = 1.55 μm as a function of fiber length. The average mode coupling coefficient h over the fiber length can be estimated by using Fig. 8 and Eq. (12).

 

Fig. 9 Relationship between fiber length and average h at 1.55 μm.

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The average mode coupling coefficient over entire fiber length of 4km was obtained to be 6.2x10−4 (1/km) from Fig. 9. The crosstalk of the MCF at 4km was about −26dB at 1.55 μm. This is because the seven cores were step-index profiles and the core pitch Λ was about 40 μm.

Figure 10 shows the imperfection loss contribution U(z) of the fiber link at 1.55 μm. The imperfection loss was obtained from the bidirectional OTDR in Fig. 5 and the crosstalk between the cores in Fig. 7.

 

Fig. 10 Imperfection loss contribution U(z)of fiber link at 1.55 μm.

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Figure 11 shows the MFD distribution 2w(z) of the fiber link at λ = 1.55 μm. It is seen that the MFD of MCF decreases as the fiber length increase. We found that the experimental result for the MFD of the test MCF agreed well with the value measured with the far field pattern (FFP) technique as shown in Table 1. As a result, we confirmed that the MFD distribution was successfully estimated by our proposed technique.

 

Fig. 11 MFD distribution of test fiber link at λ = 1.55 μm.

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3.2 Other longitudinal distributions of MCF

The longitudinal properties of the MCF were measured with the OTDR technique as mentioned in Section 2.

Figure 12 shows the relative-index difference distribution Δ(z) of the fiber link. The relative-index difference Δ was estimated by using Eq. (12). It is seen that the relative-index difference of the test MCF is almost the same as that of the conventional single-mode fiber, Ref #2. The Δ of Ref #2 was used to estimate the relative-index difference of the MCF. Therefore, the Δ value of Ref #1 is slightly smaller than that shown in Table 2 [11]. The Δ of MCF becomes large as the fiber length increase. This property of the MCF has an effect on the MFD distribution as shown in Fig. 11.

 

Fig. 12 Relative-index difference Δ (%) distribution of fiber link.

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Figure 13 shows the chromatic dispersion distribution D(z) of the fiber link at a wavelength of 1.55 μm. The chromatic dispersion D can be estimated by using the estimated relative-index difference Δ and the wavelength dependence of the MFD. We used the experimental MFD results that we obtained at 1.31 and 1.55 μm. Here, a k value of 0.62 was used.

 

Fig. 13 Chromatic dispersion distribution of fiber link at 1.55 μm.

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The longitudinal parameters such as MFD 2w, relative-index difference Δ and chromatic dispersion D can be successfully estimated with the present OTDR technique. This technique will provide a powerful way of measuring the longitudinal fiber parameter of MCF.

4. Conclusions

We proposed a novel technique for obtaining the longitudinal fiber parameter of MCF based on the bidirectional OTDR technique. By employing a simulation, we clarified that the MFD distribution of the core in MCF with a mode coupling coefficient h of less than 0.02 (1/km) can be estimated within an accuracy of 1% without taking the mode coupling between cores into account. We clarified experimentally that the technique can be applied to a link consisting of MCF. The MFD distribution of the fiber link was successfully estimated by using the crosstalk measured with the modified OTDR. We also showed that the relative-index difference Δ and chromatic dispersion distributions of the fiber link can be estimated with the present technique.

This technique will constitute a powerful approach for measuring the longitudinal fiber parameter of MCF.

Acknowledgments

The authors would like to thank Mr. K. Kawazu for his cooperation in experiments. Part of this research uses results from a study entitled ‘Research and development of innovative optical fiber for practical use’, commissioned by the National Institute of Information and Communications Technology (NICT).

References and links

1. T. Morioka, “New generation optical infrastructure technologies: EXAT initiative: toward 2020 and beyond,” in Proc. OECC2009, paper FT4(2009). [CrossRef]  

2. R. Kashyap and K. J. Blow, “Observation of catastrophic self-propelled self-focusing in optical fibres,” Electron. Lett. 24(1), 47–49 (1988). [CrossRef]  

3. R.-J. Essiambre, G. Kramer, P. J. Winzer, G. J. Foschini, and B. Goebel, “Capacity limits of optical fiber networks,” J. Lightwave Technol. 28(4), 662–701 (2010). [CrossRef]  

4. P. J. Winzer and G. J. Foschini, “MIMO capacities and outage probabilities in spatially multiplexed optical transport systems,” Opt. Express 19(17), 16680–16696 (2011). [CrossRef]   [PubMed]  

5. M. Koshiba, K. Saitoh, and Y. Kokubun, “Heterogeneous multi-core fibers: proposal and design principle,” IEICE Electron. Express 6(2), 98–103 (2009). [CrossRef]  

6. ITU-T Recommendation G.650.1.

7. M. S. O’Sullivan and J. Ferner, “Interpretation of SM fiber OTDR signatures,” Proc. SPIE 86(661), 171–176 (1986). [CrossRef]  

8. M. Ohashi and M. Tateda, “Novel technique for measuring longitudinal chromatic dispersion distribution in single-mode fibres,” Electron. Lett. 29(5), 426–427 (1993). [CrossRef]  

9. K. Nakajima, M. Ohashi, and M. Tateda, “Chromatic dispersion distribution measurement along a single-mode optical fiber,” J. Lightwave Technol. 15(7), 1095–1101 (1997). [CrossRef]  

10. A. Rossaro, M. Schiano, T. Tambosso, and D. D’Alessandro, “Spatially resolved chromatic dispersion measurement by a bidirectional OTDR technique,” IEEE J. Sel. Top. Quantum Electron. 7(3), 475–483 (2001). [CrossRef]  

11. M. Ohashi, “Novel technique for measuring relative-index difference of fiber links,” IEEE Photon. Technol. Lett. 18(24), 2584–2586 (2006). [CrossRef]  

12. M. Wuilpart, G. Ravet, P. Megret, and M. Blondel, “Distributed measurement of Raman gain spectrum in concatenations of optical fibres with OTDR,” Electron. Lett. 39(1), 88–89 (2003). [CrossRef]  

13. K. Toge, K. Hogari, and T. Horiguchi, “Raman gain efficiency distribution measurement in single-mode optical fibers by using backscattering technique,” IEEE Photon. Technol. Lett. 17(8), 1704–1706 (2005). [CrossRef]  

14. Y. Tsutsumi and M. Ohashi, “Indirect technique for measuring Raman gain efficiency spectrum using OTDR,” IEICEJ95-B(2), 146–154 (2012) (in Japanese).

15. K. Takenaga, Y. Arakawa, S. Tanigawa, N. Guan, S. Matsuo, K. Saitoh, and M. Koshiba, “An investigation on crosstalk in multi-core fibers by introducing random fluctuation along longitudinal direction,” IEICE Trans. Commun., E-94-B(2), 409–416 (2011).

16. M. Ohashi, K. Kawazu, A. Nakamura, and Y. Miyoshi, “Simple backscattered power technique for measuring crosstalk of multi-core fibers,” in Proc. OECC2012, 357–358 (2012). [CrossRef]  

17. N. Shibata, M. Kawachi, and T. Edahiro, “Optical loss characteristics of high-GeO2 content silica fibers,” IEICE Trans. E63(12), 837–841 (1980).

18. C. Pask, “Physical interpretation of Petermann’s strange spot size for single-mode fibres,” Electron. Lett. 20(3), 144–145 (1984). [CrossRef]  

19. D. Marcuse, “Loss analysis of single-mode fiber splices,” Bell Syst. Tech. J. 56(5), 703–718 (1977). [CrossRef]  

20. M. Nakazawa, M. Yoshida, and T. Hirooka, “Nondestructive measurement of mode couplings along a multi-core fiber using a synchronous multi-channel OTDR,” Opt. Express 20(11), 12530–12540 (2012). [CrossRef]   [PubMed]  

21. E. P. Patent, 0982577–A1, “ Device for measuring crosstalk between multicore optical fibres,” (1999).

References

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  • |

  1. T. Morioka, “New generation optical infrastructure technologies: EXAT initiative: toward 2020 and beyond,” in Proc. OECC2009, paper FT4(2009).
    [Crossref]
  2. R. Kashyap and K. J. Blow, “Observation of catastrophic self-propelled self-focusing in optical fibres,” Electron. Lett. 24(1), 47–49 (1988).
    [Crossref]
  3. R.-J. Essiambre, G. Kramer, P. J. Winzer, G. J. Foschini, and B. Goebel, “Capacity limits of optical fiber networks,” J. Lightwave Technol. 28(4), 662–701 (2010).
    [Crossref]
  4. P. J. Winzer and G. J. Foschini, “MIMO capacities and outage probabilities in spatially multiplexed optical transport systems,” Opt. Express 19(17), 16680–16696 (2011).
    [Crossref] [PubMed]
  5. M. Koshiba, K. Saitoh, and Y. Kokubun, “Heterogeneous multi-core fibers: proposal and design principle,” IEICE Electron. Express 6(2), 98–103 (2009).
    [Crossref]
  6. ITU-T Recommendation G.650.1.
  7. M. S. O’Sullivan and J. Ferner, “Interpretation of SM fiber OTDR signatures,” Proc. SPIE 86(661), 171–176 (1986).
    [Crossref]
  8. M. Ohashi and M. Tateda, “Novel technique for measuring longitudinal chromatic dispersion distribution in single-mode fibres,” Electron. Lett. 29(5), 426–427 (1993).
    [Crossref]
  9. K. Nakajima, M. Ohashi, and M. Tateda, “Chromatic dispersion distribution measurement along a single-mode optical fiber,” J. Lightwave Technol. 15(7), 1095–1101 (1997).
    [Crossref]
  10. A. Rossaro, M. Schiano, T. Tambosso, and D. D’Alessandro, “Spatially resolved chromatic dispersion measurement by a bidirectional OTDR technique,” IEEE J. Sel. Top. Quantum Electron. 7(3), 475–483 (2001).
    [Crossref]
  11. M. Ohashi, “Novel technique for measuring relative-index difference of fiber links,” IEEE Photon. Technol. Lett. 18(24), 2584–2586 (2006).
    [Crossref]
  12. M. Wuilpart, G. Ravet, P. Megret, and M. Blondel, “Distributed measurement of Raman gain spectrum in concatenations of optical fibres with OTDR,” Electron. Lett. 39(1), 88–89 (2003).
    [Crossref]
  13. K. Toge, K. Hogari, and T. Horiguchi, “Raman gain efficiency distribution measurement in single-mode optical fibers by using backscattering technique,” IEEE Photon. Technol. Lett. 17(8), 1704–1706 (2005).
    [Crossref]
  14. Y. Tsutsumi and M. Ohashi, “Indirect technique for measuring Raman gain efficiency spectrum using OTDR,” IEICEJ95-B(2), 146–154 (2012) (in Japanese).
  15. K. Takenaga, Y. Arakawa, S. Tanigawa, N. Guan, S. Matsuo, K. Saitoh, and M. Koshiba, “An investigation on crosstalk in multi-core fibers by introducing random fluctuation along longitudinal direction,” IEICE Trans. Commun., E-94-B(2), 409–416 (2011).
  16. M. Ohashi, K. Kawazu, A. Nakamura, and Y. Miyoshi, “Simple backscattered power technique for measuring crosstalk of multi-core fibers,” in Proc. OECC2012, 357–358 (2012).
    [Crossref]
  17. N. Shibata, M. Kawachi, and T. Edahiro, “Optical loss characteristics of high-GeO2 content silica fibers,” IEICE Trans. E63(12), 837–841 (1980).
  18. C. Pask, “Physical interpretation of Petermann’s strange spot size for single-mode fibres,” Electron. Lett. 20(3), 144–145 (1984).
    [Crossref]
  19. D. Marcuse, “Loss analysis of single-mode fiber splices,” Bell Syst. Tech. J. 56(5), 703–718 (1977).
    [Crossref]
  20. M. Nakazawa, M. Yoshida, and T. Hirooka, “Nondestructive measurement of mode couplings along a multi-core fiber using a synchronous multi-channel OTDR,” Opt. Express 20(11), 12530–12540 (2012).
    [Crossref] [PubMed]
  21. E. P. Patent, 0982577–A1, “ Device for measuring crosstalk between multicore optical fibres,” (1999).

2012 (1)

2011 (2)

K. Takenaga, Y. Arakawa, S. Tanigawa, N. Guan, S. Matsuo, K. Saitoh, and M. Koshiba, “An investigation on crosstalk in multi-core fibers by introducing random fluctuation along longitudinal direction,” IEICE Trans. Commun., E-94-B(2), 409–416 (2011).

P. J. Winzer and G. J. Foschini, “MIMO capacities and outage probabilities in spatially multiplexed optical transport systems,” Opt. Express 19(17), 16680–16696 (2011).
[Crossref] [PubMed]

2010 (1)

2009 (1)

M. Koshiba, K. Saitoh, and Y. Kokubun, “Heterogeneous multi-core fibers: proposal and design principle,” IEICE Electron. Express 6(2), 98–103 (2009).
[Crossref]

2006 (1)

M. Ohashi, “Novel technique for measuring relative-index difference of fiber links,” IEEE Photon. Technol. Lett. 18(24), 2584–2586 (2006).
[Crossref]

2005 (1)

K. Toge, K. Hogari, and T. Horiguchi, “Raman gain efficiency distribution measurement in single-mode optical fibers by using backscattering technique,” IEEE Photon. Technol. Lett. 17(8), 1704–1706 (2005).
[Crossref]

2003 (1)

M. Wuilpart, G. Ravet, P. Megret, and M. Blondel, “Distributed measurement of Raman gain spectrum in concatenations of optical fibres with OTDR,” Electron. Lett. 39(1), 88–89 (2003).
[Crossref]

2001 (1)

A. Rossaro, M. Schiano, T. Tambosso, and D. D’Alessandro, “Spatially resolved chromatic dispersion measurement by a bidirectional OTDR technique,” IEEE J. Sel. Top. Quantum Electron. 7(3), 475–483 (2001).
[Crossref]

1997 (1)

K. Nakajima, M. Ohashi, and M. Tateda, “Chromatic dispersion distribution measurement along a single-mode optical fiber,” J. Lightwave Technol. 15(7), 1095–1101 (1997).
[Crossref]

1993 (1)

M. Ohashi and M. Tateda, “Novel technique for measuring longitudinal chromatic dispersion distribution in single-mode fibres,” Electron. Lett. 29(5), 426–427 (1993).
[Crossref]

1988 (1)

R. Kashyap and K. J. Blow, “Observation of catastrophic self-propelled self-focusing in optical fibres,” Electron. Lett. 24(1), 47–49 (1988).
[Crossref]

1986 (1)

M. S. O’Sullivan and J. Ferner, “Interpretation of SM fiber OTDR signatures,” Proc. SPIE 86(661), 171–176 (1986).
[Crossref]

1984 (1)

C. Pask, “Physical interpretation of Petermann’s strange spot size for single-mode fibres,” Electron. Lett. 20(3), 144–145 (1984).
[Crossref]

1980 (1)

N. Shibata, M. Kawachi, and T. Edahiro, “Optical loss characteristics of high-GeO2 content silica fibers,” IEICE Trans. E63(12), 837–841 (1980).

1977 (1)

D. Marcuse, “Loss analysis of single-mode fiber splices,” Bell Syst. Tech. J. 56(5), 703–718 (1977).
[Crossref]

Arakawa, Y.

K. Takenaga, Y. Arakawa, S. Tanigawa, N. Guan, S. Matsuo, K. Saitoh, and M. Koshiba, “An investigation on crosstalk in multi-core fibers by introducing random fluctuation along longitudinal direction,” IEICE Trans. Commun., E-94-B(2), 409–416 (2011).

Blondel, M.

M. Wuilpart, G. Ravet, P. Megret, and M. Blondel, “Distributed measurement of Raman gain spectrum in concatenations of optical fibres with OTDR,” Electron. Lett. 39(1), 88–89 (2003).
[Crossref]

Blow, K. J.

R. Kashyap and K. J. Blow, “Observation of catastrophic self-propelled self-focusing in optical fibres,” Electron. Lett. 24(1), 47–49 (1988).
[Crossref]

D’Alessandro, D.

A. Rossaro, M. Schiano, T. Tambosso, and D. D’Alessandro, “Spatially resolved chromatic dispersion measurement by a bidirectional OTDR technique,” IEEE J. Sel. Top. Quantum Electron. 7(3), 475–483 (2001).
[Crossref]

Edahiro, T.

N. Shibata, M. Kawachi, and T. Edahiro, “Optical loss characteristics of high-GeO2 content silica fibers,” IEICE Trans. E63(12), 837–841 (1980).

Essiambre, R.-J.

Ferner, J.

M. S. O’Sullivan and J. Ferner, “Interpretation of SM fiber OTDR signatures,” Proc. SPIE 86(661), 171–176 (1986).
[Crossref]

Foschini, G. J.

Goebel, B.

Guan, N.

K. Takenaga, Y. Arakawa, S. Tanigawa, N. Guan, S. Matsuo, K. Saitoh, and M. Koshiba, “An investigation on crosstalk in multi-core fibers by introducing random fluctuation along longitudinal direction,” IEICE Trans. Commun., E-94-B(2), 409–416 (2011).

Hirooka, T.

Hogari, K.

K. Toge, K. Hogari, and T. Horiguchi, “Raman gain efficiency distribution measurement in single-mode optical fibers by using backscattering technique,” IEEE Photon. Technol. Lett. 17(8), 1704–1706 (2005).
[Crossref]

Horiguchi, T.

K. Toge, K. Hogari, and T. Horiguchi, “Raman gain efficiency distribution measurement in single-mode optical fibers by using backscattering technique,” IEEE Photon. Technol. Lett. 17(8), 1704–1706 (2005).
[Crossref]

Kashyap, R.

R. Kashyap and K. J. Blow, “Observation of catastrophic self-propelled self-focusing in optical fibres,” Electron. Lett. 24(1), 47–49 (1988).
[Crossref]

Kawachi, M.

N. Shibata, M. Kawachi, and T. Edahiro, “Optical loss characteristics of high-GeO2 content silica fibers,” IEICE Trans. E63(12), 837–841 (1980).

Kokubun, Y.

M. Koshiba, K. Saitoh, and Y. Kokubun, “Heterogeneous multi-core fibers: proposal and design principle,” IEICE Electron. Express 6(2), 98–103 (2009).
[Crossref]

Koshiba, M.

K. Takenaga, Y. Arakawa, S. Tanigawa, N. Guan, S. Matsuo, K. Saitoh, and M. Koshiba, “An investigation on crosstalk in multi-core fibers by introducing random fluctuation along longitudinal direction,” IEICE Trans. Commun., E-94-B(2), 409–416 (2011).

M. Koshiba, K. Saitoh, and Y. Kokubun, “Heterogeneous multi-core fibers: proposal and design principle,” IEICE Electron. Express 6(2), 98–103 (2009).
[Crossref]

Kramer, G.

Marcuse, D.

D. Marcuse, “Loss analysis of single-mode fiber splices,” Bell Syst. Tech. J. 56(5), 703–718 (1977).
[Crossref]

Matsuo, S.

K. Takenaga, Y. Arakawa, S. Tanigawa, N. Guan, S. Matsuo, K. Saitoh, and M. Koshiba, “An investigation on crosstalk in multi-core fibers by introducing random fluctuation along longitudinal direction,” IEICE Trans. Commun., E-94-B(2), 409–416 (2011).

Megret, P.

M. Wuilpart, G. Ravet, P. Megret, and M. Blondel, “Distributed measurement of Raman gain spectrum in concatenations of optical fibres with OTDR,” Electron. Lett. 39(1), 88–89 (2003).
[Crossref]

Nakajima, K.

K. Nakajima, M. Ohashi, and M. Tateda, “Chromatic dispersion distribution measurement along a single-mode optical fiber,” J. Lightwave Technol. 15(7), 1095–1101 (1997).
[Crossref]

Nakazawa, M.

O’Sullivan, M. S.

M. S. O’Sullivan and J. Ferner, “Interpretation of SM fiber OTDR signatures,” Proc. SPIE 86(661), 171–176 (1986).
[Crossref]

Ohashi, M.

M. Ohashi, “Novel technique for measuring relative-index difference of fiber links,” IEEE Photon. Technol. Lett. 18(24), 2584–2586 (2006).
[Crossref]

K. Nakajima, M. Ohashi, and M. Tateda, “Chromatic dispersion distribution measurement along a single-mode optical fiber,” J. Lightwave Technol. 15(7), 1095–1101 (1997).
[Crossref]

M. Ohashi and M. Tateda, “Novel technique for measuring longitudinal chromatic dispersion distribution in single-mode fibres,” Electron. Lett. 29(5), 426–427 (1993).
[Crossref]

Y. Tsutsumi and M. Ohashi, “Indirect technique for measuring Raman gain efficiency spectrum using OTDR,” IEICEJ95-B(2), 146–154 (2012) (in Japanese).

Pask, C.

C. Pask, “Physical interpretation of Petermann’s strange spot size for single-mode fibres,” Electron. Lett. 20(3), 144–145 (1984).
[Crossref]

Ravet, G.

M. Wuilpart, G. Ravet, P. Megret, and M. Blondel, “Distributed measurement of Raman gain spectrum in concatenations of optical fibres with OTDR,” Electron. Lett. 39(1), 88–89 (2003).
[Crossref]

Rossaro, A.

A. Rossaro, M. Schiano, T. Tambosso, and D. D’Alessandro, “Spatially resolved chromatic dispersion measurement by a bidirectional OTDR technique,” IEEE J. Sel. Top. Quantum Electron. 7(3), 475–483 (2001).
[Crossref]

Saitoh, K.

K. Takenaga, Y. Arakawa, S. Tanigawa, N. Guan, S. Matsuo, K. Saitoh, and M. Koshiba, “An investigation on crosstalk in multi-core fibers by introducing random fluctuation along longitudinal direction,” IEICE Trans. Commun., E-94-B(2), 409–416 (2011).

M. Koshiba, K. Saitoh, and Y. Kokubun, “Heterogeneous multi-core fibers: proposal and design principle,” IEICE Electron. Express 6(2), 98–103 (2009).
[Crossref]

Schiano, M.

A. Rossaro, M. Schiano, T. Tambosso, and D. D’Alessandro, “Spatially resolved chromatic dispersion measurement by a bidirectional OTDR technique,” IEEE J. Sel. Top. Quantum Electron. 7(3), 475–483 (2001).
[Crossref]

Shibata, N.

N. Shibata, M. Kawachi, and T. Edahiro, “Optical loss characteristics of high-GeO2 content silica fibers,” IEICE Trans. E63(12), 837–841 (1980).

Takenaga, K.

K. Takenaga, Y. Arakawa, S. Tanigawa, N. Guan, S. Matsuo, K. Saitoh, and M. Koshiba, “An investigation on crosstalk in multi-core fibers by introducing random fluctuation along longitudinal direction,” IEICE Trans. Commun., E-94-B(2), 409–416 (2011).

Tambosso, T.

A. Rossaro, M. Schiano, T. Tambosso, and D. D’Alessandro, “Spatially resolved chromatic dispersion measurement by a bidirectional OTDR technique,” IEEE J. Sel. Top. Quantum Electron. 7(3), 475–483 (2001).
[Crossref]

Tanigawa, S.

K. Takenaga, Y. Arakawa, S. Tanigawa, N. Guan, S. Matsuo, K. Saitoh, and M. Koshiba, “An investigation on crosstalk in multi-core fibers by introducing random fluctuation along longitudinal direction,” IEICE Trans. Commun., E-94-B(2), 409–416 (2011).

Tateda, M.

K. Nakajima, M. Ohashi, and M. Tateda, “Chromatic dispersion distribution measurement along a single-mode optical fiber,” J. Lightwave Technol. 15(7), 1095–1101 (1997).
[Crossref]

M. Ohashi and M. Tateda, “Novel technique for measuring longitudinal chromatic dispersion distribution in single-mode fibres,” Electron. Lett. 29(5), 426–427 (1993).
[Crossref]

Toge, K.

K. Toge, K. Hogari, and T. Horiguchi, “Raman gain efficiency distribution measurement in single-mode optical fibers by using backscattering technique,” IEEE Photon. Technol. Lett. 17(8), 1704–1706 (2005).
[Crossref]

Tsutsumi, Y.

Y. Tsutsumi and M. Ohashi, “Indirect technique for measuring Raman gain efficiency spectrum using OTDR,” IEICEJ95-B(2), 146–154 (2012) (in Japanese).

Winzer, P. J.

Wuilpart, M.

M. Wuilpart, G. Ravet, P. Megret, and M. Blondel, “Distributed measurement of Raman gain spectrum in concatenations of optical fibres with OTDR,” Electron. Lett. 39(1), 88–89 (2003).
[Crossref]

Yoshida, M.

Bell Syst. Tech. J. (1)

D. Marcuse, “Loss analysis of single-mode fiber splices,” Bell Syst. Tech. J. 56(5), 703–718 (1977).
[Crossref]

Electron. Lett. (4)

R. Kashyap and K. J. Blow, “Observation of catastrophic self-propelled self-focusing in optical fibres,” Electron. Lett. 24(1), 47–49 (1988).
[Crossref]

M. Ohashi and M. Tateda, “Novel technique for measuring longitudinal chromatic dispersion distribution in single-mode fibres,” Electron. Lett. 29(5), 426–427 (1993).
[Crossref]

M. Wuilpart, G. Ravet, P. Megret, and M. Blondel, “Distributed measurement of Raman gain spectrum in concatenations of optical fibres with OTDR,” Electron. Lett. 39(1), 88–89 (2003).
[Crossref]

C. Pask, “Physical interpretation of Petermann’s strange spot size for single-mode fibres,” Electron. Lett. 20(3), 144–145 (1984).
[Crossref]

IEEE J. Sel. Top. Quantum Electron. (1)

A. Rossaro, M. Schiano, T. Tambosso, and D. D’Alessandro, “Spatially resolved chromatic dispersion measurement by a bidirectional OTDR technique,” IEEE J. Sel. Top. Quantum Electron. 7(3), 475–483 (2001).
[Crossref]

IEEE Photon. Technol. Lett. (2)

M. Ohashi, “Novel technique for measuring relative-index difference of fiber links,” IEEE Photon. Technol. Lett. 18(24), 2584–2586 (2006).
[Crossref]

K. Toge, K. Hogari, and T. Horiguchi, “Raman gain efficiency distribution measurement in single-mode optical fibers by using backscattering technique,” IEEE Photon. Technol. Lett. 17(8), 1704–1706 (2005).
[Crossref]

IEICE Electron. Express (1)

M. Koshiba, K. Saitoh, and Y. Kokubun, “Heterogeneous multi-core fibers: proposal and design principle,” IEICE Electron. Express 6(2), 98–103 (2009).
[Crossref]

IEICE Trans. (1)

N. Shibata, M. Kawachi, and T. Edahiro, “Optical loss characteristics of high-GeO2 content silica fibers,” IEICE Trans. E63(12), 837–841 (1980).

IEICE Trans. Commun., (1)

K. Takenaga, Y. Arakawa, S. Tanigawa, N. Guan, S. Matsuo, K. Saitoh, and M. Koshiba, “An investigation on crosstalk in multi-core fibers by introducing random fluctuation along longitudinal direction,” IEICE Trans. Commun., E-94-B(2), 409–416 (2011).

J. Lightwave Technol. (2)

R.-J. Essiambre, G. Kramer, P. J. Winzer, G. J. Foschini, and B. Goebel, “Capacity limits of optical fiber networks,” J. Lightwave Technol. 28(4), 662–701 (2010).
[Crossref]

K. Nakajima, M. Ohashi, and M. Tateda, “Chromatic dispersion distribution measurement along a single-mode optical fiber,” J. Lightwave Technol. 15(7), 1095–1101 (1997).
[Crossref]

Opt. Express (2)

Proc. SPIE (1)

M. S. O’Sullivan and J. Ferner, “Interpretation of SM fiber OTDR signatures,” Proc. SPIE 86(661), 171–176 (1986).
[Crossref]

Other (5)

T. Morioka, “New generation optical infrastructure technologies: EXAT initiative: toward 2020 and beyond,” in Proc. OECC2009, paper FT4(2009).
[Crossref]

ITU-T Recommendation G.650.1.

M. Ohashi, K. Kawazu, A. Nakamura, and Y. Miyoshi, “Simple backscattered power technique for measuring crosstalk of multi-core fibers,” in Proc. OECC2012, 357–358 (2012).
[Crossref]

Y. Tsutsumi and M. Ohashi, “Indirect technique for measuring Raman gain efficiency spectrum using OTDR,” IEICEJ95-B(2), 146–154 (2012) (in Japanese).

E. P. Patent, 0982577–A1, “ Device for measuring crosstalk between multicore optical fibres,” (1999).

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Figures (13)

Fig. 1
Fig. 1 Experimental setup for measuring MFD.
Fig. 2
Fig. 2 Relationship between fiber length and relative error Δε of the MFD as a function of mode coupling coefficient h.
Fig. 3
Fig. 3 Relationship between the mode coupling coefficient h and maximum relative error Δε of the MFD as a function of fiber length.
Fig. 4
Fig. 4 Relationship between g value and relative error Δε of the h as a function of crosstalk between center and outer cores.
Fig. 5
Fig. 5 Experimental setup for measuring crosstalk using modified OTDR.
Fig. 6
Fig. 6 Bidirectional backscattered powers of fiber link at (a) 1.31 μm and (b) 1.55 μm.
Fig. 7
Fig. 7 Backscattered powers of center and outer cores in MCF when an optical pulse was launched into the center core. (a) 1.31 μm, and (b) 1.55 μm.
Fig. 8
Fig. 8 Crosstalk between cores at λ = 1.31 and 1.55 μm.
Fig. 9
Fig. 9 Relationship between fiber length and average h at 1.55 μm.
Fig. 10
Fig. 10 Imperfection loss contribution U(z)of fiber link at 1.55 μm.
Fig. 11
Fig. 11 MFD distribution of test fiber link at λ = 1.55 μm.
Fig. 12
Fig. 12 Relative-index difference Δ (%) distribution of fiber link.
Fig. 13
Fig. 13 Chromatic dispersion distribution of fiber link at 1.55 μm.

Tables (2)

Tables Icon

Table 1 Parameters of Test MCF and Reference Fibers used in calculations

Tables Icon

Table 2 Parameters of Test MCF and Reference Fibers

Equations (23)

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P 1 (z)= P 0 (0)[ 1+6exp(7hz) 7 ]
P 2 (z)= P 0 (0)[ 1exp(7hz) 7 ]
S 1 (z)={ 10log P 1 +10log( α s (z))+10log(B(z))2αz10loge(0z L 0 ) 10log P 1 +10log( α s (z))+10log(B(z))2αz10loge +10log[ 1+6exp[ 14h(z L 0 ) ] 7 ]( L 0 +Lz L 0 ) ,
S 2 (z)={ 10log P 2 +10log( α s (z))+10log(B(z))2α( L+ L 0 z )10loge +10log[ 1+6exp[ 14hL ] 7 ](0z L 0 ) 10log P 2 +10log( α s (z))+10log(B(z))2α( L+ L 0 z )10loge +10log[ 1+6exp[ 14h(L+ L 0 z) ] 7 ]( L 0 +Lz L 0 ) ,
U(z)=5log P 1 P 2 +10log( α s (z)B(z))2α(L+ L 0 )(10loge) ={ S 1 (z)+ S 2 (z) 2 5log[ 1+6exp[ 14hL ] 7 ](0z L 0 ) S 1 (z)+ S 2 (z) 2 5log{ [ 1+6exp[ 14h(z L 0 ) ] ][ 1+6exp[ 14h(L+ L 0 z) ] ] 49 } ( L 0 z L 0 +L) .
B(λ,z)= 3 2 log [ λ 2πnw(λ,z) ] 2 .
U n (z)=U(z)U( z 0 )=10log( α s (z) n 2 ( z 0 ) α s ( z 0 ) n 2 (z) )+20log( 2w(λ, z 0 ) 2w(λ,z) ).
2w(λ,z)=2w(λ, z 0 ) ( 2w(λ, z 0 ) 2w(λ, z 1 ) ) U(z)U( z 0 ) U( z 1 )U( z 0 ) .
S 3 (z)=10log P 3 +10log( α s1 B 1 )2 α 1 z(10loge)+10log[ 1+6exp(14hz) 7 ],
S 4 (z)=10log P 3 +10log( α s2 B 2 )2 α 2 z(10loge)+10log[ 1exp(14hz) 7 ],
XT(z)=10log[ P 4 (z) P 3 (z) ]=10log[ 1exp(14hz) 1+6exp(14hz) ].
Δ(z)= 1 k [ ( 1+kΔ( z 0 ) )× 10 U(λ,z)U(λ, z 0 )20log( 2w(λ, z 0 ) 2w(λ,z) ) 10 1 ],
D= D m +D , w
D m = λ c d 2 n d λ 2 ,
D w = λ 2 π 2 cn d dλ ( λ w 2 ),
w(λ)= g 0 + g 1 λ 1.5 .
D w = λ 2 π 2 cn w 2 ( 1 3 g 1 λ 1.5 w ).
I(z)={ U(z)+5log[ 1+6exp[ 14hL ] 7 ](0z L 0 ) U(z)+5log{ [ 1+6exp[ 14h(z l 0 ) ] ][ 1+6exp[ 14h(L+ l 0 z) ] ] 49 } ( L 0 z L 0 +L) .
Δε= 2w(λ,z)2 w * (λ,z) 2w(λ,z) =1 2 w * (λ,z) 2w(λ,z) =1 ( 2w(λ, z 0 ) 2w(λ, z 1 ) ) F(λ,z,h) U( z 1 )U( z 0 ) , F(λ,z,h)=5log [ 1+6exp(14h( z L 0 ) ][ 1+6exp(14h( L+ L 0 z ) ] 7[ 1+6exp(4hL) ] .
U n ( z 1 )=U( z 1 )U( z 0 )=10log[ { 1+kΔ( z 1 ) }{ 12Δ( z 0 ) } { 1+kΔ( z 0 ) }{ 12Δ( z 1 ) } ]+20log[ 2w(λ, z 0 ) 2w(λ, z 1 ) ]
XT(z)= S 4 (z) S 3 (z) =10log( α s2 B 2 α s1 B 1 )+2( α 1 α 2 )z(10loge)+10log[ 1exp(14hz) 1+6exp(14hz) ].
XT(z)=g(z,λ, Δ 1 , Δ 2 , w 1 , w 2 )+10log[ 1exp(14hz) 1+6exp(14hz) ] g(z,λ, Δ 1 , Δ 2 , w 1 , w 2 )=10log[ (1+k Δ 2 ) (1+k Δ 1 ) ]+10log[ n 1 2 w 1 2 n 2 2 w 2 2 ]+ R 0 k λ 4 ( Δ 1 Δ 2 )2z(10loge).
Δε= h ex h ap h ex == ln[ 1 10 XTg 10 1 10 XT 10 1+6 10 XT 10 1+6 10 XTg 10 ] ln[ 1 10 XTg 10 1+6 10 XTg 10 ] .

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