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Physical interpretation of intercore crosstalk in multicore fiber: effects of macrobend, structure fluctuation, and microbend: comment

Masanori Koshiba

Open Access Open Access

Abstract

To estimate the intercore crosstalk in uncoupled multicore fibers with cores’ propagation constants perturbed by bending, twist, and structure fluctuations, in the paper [Opt. Express 21, 5401 (2013) [CrossRef]  ] entitled “Physical interpretation of intercore crosstalk in multicore fiber: effects of macrobend, structure fluctuation, and microbend,” Hayashi et al. derived an intuitively interpretable expression of the average power-coupling coefficient as the convolution of the arcsine and Lorentzian distributions, and calculated the derived expression numerically by performing the fast Fourier transform. In the present paper, we derive a closed-form solution of the convolution of the arcsine and Lorentzian distributions, and point out that the closed-form analytical expression derived here is exactly equivalent to the previously reported closed-form expression.

© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

Hayashi et al. [1] derived an intuitively interpretable expression of the average power-coupling coefficient (PCC) for estimating intercore crosstalk (XT) between two cores (core m and core $n$) in uncoupled multicore fibers (UC-MCFs) with cores’ propagation constants perturbed by bending, twist, and structure fluctuations. Prior to Ref. [1], Koshiba et al. [2] derived a closed-form expression of the average PCC ${h_{mn}}$ as (Eq. (12) in Ref. [2])

$${h_{mn}} = \sqrt 2 {K}_{mn}^2{l_\textrm{c}}\left[ {1/\sqrt {a\left( {b + \sqrt {ac} } \right)} + 1/\sqrt {c\left( {b + \sqrt {ac} } \right)} } \right],$$
$$a = 1 + {({q - p} )^2},\; \; \; \; \; b = 1 + {q^2} - {p^2},\; \; \; \; \; c = 1 + {({q + p} )^2},$$
$$p = \frac{{{\beta _m}{\varLambda _{mn}}{l_\textrm{c}}}}{{{R_\textrm{b}}}},\; \; \; \; \; q = \Delta {\beta _{mn}}{l_\textrm{c}},$$
where ${{K}_{mn}}$ is the redefined mode-coupling coefficient from core n to core m that is symmetrical (${{K}_{mn}} = {{K}_{nm}}$) [3], ${l_\textrm{c}}$ is the correlation length, ${\beta _m}$ is the propagation constant in core m (${\beta _m} \approx {\beta _n}$), ${\varLambda _{mn}}$ is the core-to-core distance (${\varLambda _{mn}} = {\varLambda _{nm}}$), ${R_\textrm{b}}$ is the bending radius, and $\varDelta {\beta _{mn}}$ is the propagation constant difference ($\varDelta {\beta _{mn}} = {\beta _m} - {\beta _n} ={-} \varDelta {\beta _{nm}}$).

The closed-form expression in Eq. (1) is very powerful and has been widely used to estimate the intercore XT in various UC-MCFs; however, as pointed out in Ref. [1], it is difficult to interpret physical meaning of Eq. (1) intuitively. Hayashi et al. [1] found that the average PCC can be expressed as the convolution of the arcsine and Lorentzian distributions. The arcsine and Lorentzian distributions represent the spectra of the perturbations induced by macrobends and structure fluctuations, respectively. Since the inverse Fourier transforms of the arcsine and Lorentzian distributions are the Bessel function of the first kind of order zero and the two-sided exponential function, respectively, the average PCC can be expressed with the Fourier transform as (Eq. (32) in Ref. [1])

$$\begin{gathered} {h_{mn}} = {K}_{mn}^2\mathop \int \nolimits_{ - \infty }^\infty {J_0}\left( {\frac{{{\beta _m}{\varLambda _{mn}}\left| \zeta \right|}}{{{R_{\text{b}}}}}} \right)\operatorname{exp} \left( { - \frac{{\left| \zeta \right|}}{{{l_{\text{c}}}}}} \right)\operatorname{exp} \left( {j\Delta {\beta _{mn}}\zeta } \right)d\zeta \hfill \\ = 2{K}_{mn}^2\mathop \int \nolimits_0^\infty {J_0}\left( {\frac{{{\beta _m}{\varLambda _{mn}}\zeta }}{{{R_{\text{b}}}}}} \right){\text{exp}}\left( { - \frac{\zeta }{{{l_{\text{c}}}}}} \right)\cos \left( {\Delta {\beta _{mn}}\zeta } \right)d\zeta . \hfill \\ \end{gathered}$$

Hayashi et al. [1] calculated Eq. (4) numerically by performing the fast Fourier transform (FFT) and confirmed that the results of the FFT agree well with those of the closed-form expression in Eq. (1). So far, however, there has been no report on the closed-form solution of Eq. (4). Here, applying the integral formula (Eq. (46) on p. 11 in Ref. [4]) to Eq. (4), we derive another closed-form expression of the average PCC as

$${h_{mn}} = \frac{{\sqrt 2 {K}_{mn}^2{l_\textrm{c}}\sqrt {u + v} }}{v},$$
$$u = 1 + {p^2} - {q^2},\; \; \; \; \; v = \sqrt {{u^2} + 4{q^2}} .$$

Very recently, based on the coupled-mode theory, Ng et al. [5] have derived an analytical expression of the average intercore XT as (Eqs. (15) and (23) in Ref. [5])

$$\frac{{XT}}{L} = {h_{mn}} = \frac{{\sqrt 2 {K}_{mn}^2{l_\textrm{c}}\sqrt {\textrm{Re}(Z )+ |Z |} }}{{A|Z |}},$$
$$Z = {A^2} + {p^2} - {q^2} + j2Aq,\; \; \; \; \; A = \sqrt {1 + 4{{({{{K}_{mn}}{l_\textrm{c}}} )}^2}} ,$$
where L is the fiber length, $\textrm{Re}(Z )$ and $|Z |$ denote the real part and absolute value of the complex number Z, respectively, and j is the imaginary unit. Although Eq. (7) is not the closed-form solution of Eq. (4), when ${{K}_{mn}}{l_\textrm{c}} \ll 1$ (${{K}_{mn}}{l_\textrm{c}}$ is usually small compared to other terms [5]), i.e., $A \approx 1$, $\textrm{Re}(Z )$ and $|Z |$ are reduced to u and v in Eq. (6), respectively; therefore, we can see that Eq. (7) is coincident with Eq. (5). Furthermore, Ng et al. [5] have shown that when ${{K}_{mn}}{l_\textrm{c}} = 0$ in the parameter A, Eq. (7) (i.e., Eq. (5)) is equivalent to the closed-form expression in Eq. (1) (Appendix D in Ref. [5]).

In conclusion, we can say that Eq. (1) is exactly equivalent to Eq. (5) derived here and that not only Eq. (5) but also Eq. (1) is the closed-form solution of the convolution of the arcsine and Lorentzian distributions in Eq. (4).

Disclosures

The author declares no conflicts of interest.

Data availability

No data were generated or analyzed in the presented research.

References

1. T. Hayashi, T. Sasaki, E. Sasaoka, K. Saitoh, and M. Koshiba, “Physical interpretation of intercore crosstalk in multicore fiber: effects of macrobend, structure fluctuation, and microbend,” Opt. Express 21(5), 5401–5412 (2013). [CrossRef]  

2. M. Koshiba, K. Saitoh, K. Takenaga, and S. Matsuo, “Analytical expression of average power-coupling coefficients for estimating intercore crosstalk in multicore fibers,” IEEE Photonics J. 4(5), 1987–1995 (2012). [CrossRef]  

3. M. Koshiba, K. Saitoh, K. Takenaga, and S. Matsuo, “Multi-core fiber design and analysis: coupled-mode theory and coupled-power theory,” Opt. Express 19(26), B102–B111 (2011). [CrossRef]  

4. A. Erdélyi ed, Tables of Integral Transforms, vol. II (McGraw-Hill, New York, 1954).

5. K. Ng, V. Nazarov, S. Kuchinsky, A. Zakharian, and M.-J. Li, “Analysis of crosstalk in multicore fibers: statistical distributions and analytical expressions,” Photonics 10(2), 174 (2023). [CrossRef]  

Data availability

No data were generated or analyzed in the presented research.

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Equations (8)

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h m n = 2 K m n 2 l c [ 1 / a ( b + a c ) + 1 / c ( b + a c ) ] ,
a = 1 + ( q − p ) 2 , b = 1 + q 2 − p 2 , c = 1 + ( q + p ) 2 ,
p = β m Λ m n l c R b , q = Δ β m n l c ,
h m n = K m n 2 ∫ − ∞ ∞ ⁡ J 0 ( β m Λ m n | ζ | R b ) exp ⁡ ( − | ζ | l c ) exp ⁡ ( j Δ β m n ζ ) d ζ = 2 K m n 2 ∫ 0 ∞ ⁡ J 0 ( β m Λ m n ζ R b ) exp ( − ζ l c ) cos ⁡ ( Δ β m n ζ ) d ζ .
h m n = 2 K m n 2 l c u + v v ,
u = 1 + p 2 − q 2 , v = u 2 + 4 q 2 .
X T L = h m n = 2 K m n 2 l c Re ( Z ) + | Z | A | Z | ,
Z = A 2 + p 2 − q 2 + j 2 A q , A = 1 + 4 ( K m n l c ) 2 ,
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