Reduction of modal evolution fluctuation in 2-LP mode optical time domain reflectometry

Atsushi Nakamura; Yusuke Koshikiya; Tetsuya Manabe

doi:10.1364/OE.25.020727

1. Introduction

Optical time domain reflectometry (OTDR) is a widely used technique for diagnosing and characterizing optical fibers. Recently, several studies have reported an OTDR that conducts measurements in the 2-LP mode region of the fiber under test (FUT) for the purpose of diagnosing single-mode fibers (SMFs) [1–3] and characterizing the properties of few-mode fibers (FMFs) [4–8]. The techniques individually observe the fundamental (LP₀₁) and second-order (LP₁₁) mode components of the backscattered light by employing a mode selective coupler (MSC). Moreover, the planar lightwave circuit (PLC) based or phase-plate based MSCs used in the technique separately extract the two orthogonal LP₁₁ spatial mode (so called LP_11a and LP_11b) components, i.e. the transmission efficiencies of the LP_11a and LP_11b mode components are sensitive to the cross-sectional intensity distribution of the LP₁₁ mode. This characteristic causes amplitude fluctuations in the OTDR traces yielded by the LP_11a and LP_11b mode components of the backscattered light, since the intensity distribution of the LP₁₁ mode evolves along the FUT owing to modal birefringence [9] and internal or external factors such as structural imperfections, bends, distortion and lateral pressure which are randomly distributed along the FUT. We term the amplitude fluctuation “modal evolution fluctuation.” Although the modal evolution fluctuation may useful to get information about the mode coupling period [4], the fluctuation also obscures the local attenuation information. Therefore, the modal evolution fluctuation must be eliminated in the loss distribution measurements.

Averaging backscattered signals with independent intensity distribution of the LP₁₁ mode is expected to be one simple solution for reducing the modal evolution fluctuation. It can be achieved by scrambling the state of polarization (SOP) of four degenerate higher-order vector modes (TM₀₁, TE₀₁, and even/odd HE₂₁ modes) that actually constitute the LP₁₁ mode, since the intensity distribution is determined by the relative phase relationships between the vector modes. Therefore, an understanding how the SOP of the higher-order vector modes evolves in the backscattering process is important to decrease the modal evolution fluctuation. To the best of our knowledge, although some recent investigations have described the Rayleigh backscattering in few- or multi-mode fibers [2,10–12], the SOP evolution of the backscattered higher-order vector modes in the backscattering process is still incompletely understood.

This paper first analyzes Rayleigh backscattering amplitude coefficients from the fundamental vector mode (HE₁₁ mode), launched into the FUT as a probe pulse, to higher-order vector modes recaptured as the backscattered light. We then clarify the SOP evolution of the higher-order vector modes with respect to that of the probe HE₁₁ mode in the backscattering process and propose a modal evolution fluctuation reduction technique. Finally, proof-of-concept experiments are carried out on optical fibers in the laboratory and field environments to verify the feasibility of our proposed technique.

2. Theoretical discussion

2.1 Rayleigh backscattering amplitude coefficients

This section analyzes the Rayleigh backscattering amplitude coefficients of the higher-order vector modes generated from the HE₁₁ mode traversing the FUT as a probe pulse. Several studies have derived the backscattering coefficient of SMF by treating the backscattering analysis as the problem of surface wave excitation by electric dipoles [13,14]. We extend the analysis to derive the backscattering coefficients from the LP₀₁ mode to the vector modes that constitute the LP₁₁ mode. Throughout this paper, the analysis is formulated in the frequency domain with the assumption of the time-varying factor exp(jωt).

We consider an optical fiber with a local fluctuating permittivity. The permittivity can be written as

ε (x, y, z) = ε \cdot [1 + ε_{r} (x, y, z)],

where ε is the average permittivity and ε_r(x,y,z) is the local fluctuation. The refractive index can be written as follows

n (x, y, z) = \sqrt{ε (x, y, z) / ε_{0}} = n \cdot [1 + n_{r} (x, y, z],

where ε₀ is permittivity in vacuum. n and n_r(x,y,z) are the average refractive index and the local fluctuation, respectively. Assuming that n_r << 1, we obtain ε_r = 2n_r. When light passes through the FUT, the fluctuation components behave as sources of scattered light [15–17]. The incident electric field E_in induces an oscillating dipole, and the fluctuation part of the dipole moment P is given by

P = Δ ε (x, y, z) E_{i n},

where

Δ ε = ε \cdot ε_{r} (x, y, z) = 2 ε \cdot n_{r} (x, y, z) .

Given the weakly guiding approximation [18], the transverse electric field of a particular guided mode can be given by

E_{m} = E_{m} (x, y) \exp (- j β_{m} z) \approx E_{m} (x, y) \exp (- j k n z),

where E_n and β_n are the transverse function and the propagation constant, respectively. n is the mode number index, respectively. k is the wavenumber. We disregard the longitudinal component of the field, since it is negligible compared to the transverse component in weakly guiding fibers. With the help of the Lorentz reciprocity theorem [19,20], the field amplitude of the backscattered light excited by the dipole can be written as

a_{m} = \frac{- j ω \int_{V} P \cdot E_{m} d V}{2 \int_{S} [{| E_{m} (x, y) |}^{2} / Z] d S},

where Z, the characteristic impedance, is given by Z = (μ/ε)^1/2. V and S are the scattering volume and the cross section, respectively. dS is the cross section element, and dV is the volume element of dSdz. Length element dz is assumed to be small enough that the incident field is in phase within the length element.

We consider first 6 vector modes, i.e. even/odd HE₁₁ (eHE₁₁/oHE₁₁), TM₀₁, TE₀₁, and even/odd HE₂₁ (eHE₂₁/oHE_21b) modes. The intensity and polarization patterns of the modes are shown in Fig. 1.

Fig. 1 Intensity and polarization patterns of first 6 vector modes.

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The transverse function can be expressed as follows:

\begin{array}{l} E_{x} (x, y) = F_{1} \hat{x} \\ E_{y} (x, y) = F_{1} \hat{y} \\ E_{e}^{+} (x, y) = F_{2} (\hat{x} \cos θ + \hat{y} \sin θ) \\ E_{o}^{+} (x, y) = F_{2} (- \hat{x} \sin θ + \hat{y} \cos θ) \\ E_{e}^{-} (x, y) = F_{2} (\hat{x} \cos θ - \hat{y} \sin θ) \\ E_{o}^{-} (x, y) = F_{2} (\hat{x} \sin θ + \hat{y} \cos θ) \end{array}},

where

E_{x}

,

E_{y}

,

E_{e}^{+}

,

E_{o}^{+}

,

E_{e}^{-}

and

E_{o}^{-}

represent the eHE₁₁, oHE₁₁, TM₀₁, TE₀₁, eHE₂₁ and oHE_21b modes, respectively. F₁ and F₂ are the radial field functions for the modes.

\hat{x}

and

\hat{y}

are unit vectors that represent the polarization directions. θ is the polar angle relative to the x-axis.

Using Eqs. (3)-(7), the field amplitude of the backscattered TM₀₁, TE₀₁, eHE₂₁ and oHE₂₁ modes generated from eHE₁₁ and oHE₁₁ modes can be obtained as follows:

\begin{matrix} a_{x e}^{\pm} = - j G_{1} \\ a_{x o}^{\pm} = \pm j G_{2} \\ a_{y e}^{\pm} = \mp j G_{2} \\ a_{y o}^{\pm} = - j G_{1} \end{matrix}},

where

a_{x e}^{+}

,

a_{x o}^{+}

,

a_{x e}^{-}

and

a_{x o}^{-}

represent the field amplitude of the backscattered TM₀₁, TE₀₁, eHE₂₁ and oHE₂₁ modes generated from eHE₁₁ mode, respectively.

a_{y e}^{+}

,

a_{y o}^{+}

,

a_{y e}^{-}

and

a_{y o}^{-}

represent the field amplitude of the backscattered TM₀₁, TE₀₁, eHE₂₁ and oHE₂₁ modes generated from oHE₁₁ mode, respectively. ± j denotes the phase difference of ± π/2 between the incident and backscattered light. G₁ and G₂ are given by:

\begin{array}{l} G_{1} = k n \int_{V} n_{r} (x, y, z) \cdot F_{1} \cdot F_{2} \cos θ \cdot \exp (- 2 k n z) d V \\ G_{2} = k n \int_{V} n_{r} (x, y, z) \cdot F_{1} \cdot F_{2} \sin θ \cdot \exp (- 2 k n z) d V \end{array}} .

Assuming that the local refractive index fluctuation n_r(x,y,z) is uniform in the radial direction yields the following relation

G = G_{1} = G_{2} .

From Eqs. (8)-(10), it is found that the relative phase relationships between the higher-order vector modes generated from the incident HE₁₁ mode depend on the incident polarization state, while the moduli of their complex amplitudes are independent.

Since the main scope of this section is to clarify the amplitude coupling coefficients in the backscattering process, we do not mention the power recapture factor for each mode, which is very important in predicting the backscattered power for each mode component, in the main text. We refer in the Appendix to the power recapture factor.

2.2 SOP evolution of higher-order vector modes and modal evolution fluctuation reduction

This section clarifies the SOP evolution of the backscattered higher-order vector modes with respect to that of the probe HE₁₁ mode. We then propose a technique for reducing the modal evolution fluctuation.

The complex amplitude of the probe light with HE₁₁ mode at the scattering position can be expressed as

[\begin{array}{l} A_{x} \\ A_{y} \end{array}] = A_{0} [\begin{matrix} \cos φ \\ e^{j δ} \sin φ \end{matrix}],

where A₀ is amplitude of the probe light, and φ is angle which determines the ratio of x and y polarizations. δ is the relative phase difference between the x and y polarizations. The backscattered light with TM₀₁, TE₀₁, eHE₂₁ and oHE₂₁ modes generated from the HE₁₁ mode can be given by

[\begin{matrix} A_{e}^{\pm} \\ A_{o}^{\pm} \end{matrix}] = [\begin{matrix} a_{x e}^{\pm} & a_{y e}^{\pm} \\ a_{x o}^{\pm} & a_{y o}^{\pm} \end{matrix}] [\begin{matrix} A_{x} \\ A_{y} \end{matrix}] = - j G [\begin{matrix} \cos φ \pm e^{j δ} \sin φ \\ \mp \cos φ + e^{j δ} \sin φ \end{matrix}],

where

A_{e}^{+}

,

A_{o}^{+}

,

A_{e}^{-}

and

A_{o}^{-}

represent TM₀₁, TE₀₁, eHE₂₁ and oHE₂₁ modes, respectively.

To describe the SOP evolution of the probe HE₁₁ mode and the backscattered higher-order vector modes, we use Stokes parameters and the Poincaré sphere representation for HE₁₁ mode and higher-order Stokes parameters and higher-order Poincaré spheres for TM₀₁, TE₀₁, eHE₂₁ and oHE₂₁ modes [21]. The Stokes parameters for the probe light with HE₁₁ mode are:

\begin{array}{l} S_{0} = {| A_{x} |}^{2} + {| A_{y} |}^{2} \\ S_{1} = {| A_{x} |}^{2} - {| A_{y} |}^{2} \\ S_{2} = 2 Re (A_{x}^{*} A_{y}) \\ S_{3} = 2 Im (A_{x}^{*} A_{y}) \end{array}},

The higher-order Stokes parameters can be given by:

\begin{array}{l} S_{0}^{\pm} = {| A_{e}^{\pm} |}^{2} + {| A_{o}^{\pm} |}^{2} \\ S_{1}^{\pm} = {| A_{e}^{\pm} |}^{2} - {| A_{o}^{\pm} |}^{2} \\ S_{2}^{\pm} = 2 Re (A_{e}^{\pm *} A_{o}^{\pm}) \\ S_{3}^{\pm} = 2 Im (A_{e}^{\pm *} A_{o}^{\pm}) \end{array}} .

From Eqs. (11)-(14), the normalized Stokes parameters for the probe light with HE₁₁ mode can be given by:

\begin{array}{l} s_{1} = S_{1} / S_{0} = \cos 2 φ \\ s_{2} = S_{2} / S_{0} = \sin 2 φ \cos δ \\ s_{3} = S_{3} / S_{0} = \sin 2 φ \sin δ \end{array}} .

The normalized higher-order Stokes parameters can be written as follows:

\begin{array}{l} s_{1}^{\pm} = S_{1}^{\pm} / S_{0}^{\pm} = \pm \sin 2 φ \cos δ \\ s_{2}^{\pm} = S_{2}^{\pm} / S_{0}^{\pm} = \mp \cos 2 φ \\ s_{3}^{\pm} = S_{3}^{\pm} / S_{0}^{\pm} = \mp \sin 2 φ \sin δ \end{array}} .

From Eqs. (15) and (16), we find that the SOP of the backscattered higher-order vector modes is determined uniquely by the SOP of the scattered HE₁₁ mode. Figure 2 shows examples of the SOP evolution for the scattered HE₁₁ mode and backscattered higher-order vector modes. When the SOP of the scattered HE₁₁ mode evolves from [s₁, s₂, s₃]^T = [1, 0, 0]^T through [0, 1, 0]^T to [-1, 0, 0]^T, that of the backscattered TM₀₁/TE₀₁ mode evolves from [s₁⁺, s₂⁺, s₃⁺]^T = [0, −1, 0]^T through [1, 0, 0]^T to [0, 1, 0]^T while that of the backscattered HE₂₁ mode evolves from [s₁^-, s₂^-, s₃^-]^T = [0, 1, 0]^T through [-1, 0, 0]^T to [0, −1, 0]^T. We find from these results that the SOP of the backscattered higher-order vector modes uniformly covers the higher-order Poincaré spheres by changing the SOP of the scattered HE₁₁ mode uniformly on a Poincaré sphere.

Fig. 2 Examples of SOP evolution: (a) Poincaré sphere for the scattered HE₁₁ mode, (b) higher-order Poincaré sphere for TM₀₁ and TE₀₁ modes generated from the HE₁₁ mode, (c) higher-order Poincaré sphere for HE₂₁ mode generated from the HE₁₁ mode.

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When the backscattered higher-order vector modes travel back to the MSC, their relative phase relationships vary along the propagation owing to modal birefringence. We exclude the transmission loss from the formula to simplify the discussion, so the complex amplitude at the MSC is given by

{[\begin{matrix} A_{e}^{\pm} \\ A_{o}^{\pm} \end{matrix}]}_{M S C} = [\begin{matrix} \exp (j β_{e}^{\pm} z_{s}) & 0 \\ 0 & \exp (j β_{o}^{\pm} z_{s}) \end{matrix}] [\begin{matrix} A_{e}^{\pm} \\ A_{o}^{\pm} \end{matrix}],

where

β_{e}^{+}

,

β_{o}^{+}

,

β_{e}^{-}

and

β_{o}^{-}

represent the propagation constants for the TM₀₁, TE₀₁, eHE₂₁ and oHE₂₁ modes, respectively. z_s is the distance from the scattering position to the MSC. The MSC transforms the backscattered higher-order vector modes into the LP₁₁ modes at the MSC. The relation between the higher-order vector modes and the LP₁₁ modes is given by

[\begin{matrix} A_{11 a x} \\ A_{11 a y} \\ A_{11 b x} \\ A_{11 b y} \end{matrix}] = \frac{1}{\sqrt{2}} [\begin{matrix} 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 0 & - 1 & 0 & 1 \\ 1 & 0 & - 1 & 0 \end{matrix}] {[\begin{matrix} A_{e}^{+} \\ A_{o}^{+} \\ A_{e}^{-} \\ A_{o}^{-} \end{matrix}]}_{M S C},

where A₁₁_ax, A₁₁_ay, A₁₁_bx and A₁₁_by represent the complex amplitudes for the LP_11ax, LP_11ay, LP_11bx and LP_11by modes, respectively. From Eqs. (12), (17) and (18), the powers of the LP_11a and LP_11b mode components can be written as follows:

\begin{matrix} \begin{matrix} P_{11 a} \propto {| A_{11 a x} |}^{2} + {| A_{11 a y} |}^{2} \\ = [\cos^{2} (\frac{β_{e}^{+} - β_{e}^{-}}{2} z_{s}) + \sin^{2} (\frac{β_{o}^{-} - β_{o}^{+}}{2} z_{s})] \cos^{2} φ \\ + [\sin^{2} (\frac{β_{e}^{+} - β_{e}^{-}}{2} z_{s}) + \cos^{2} (\frac{β_{o}^{-} - β_{o}^{+}}{2} z_{s})] \sin^{2} φ \\ - \frac{1}{2} {\sin [(β_{e}^{+} - β_{e}^{-}) z_{s}] - \sin [(β_{o}^{-} - β_{o}^{+}) z_{s}]} \sin (2 φ) \sin δ \end{matrix} \\ \begin{matrix} P_{11 b} \propto {| A_{11 b x} |}^{2} + {| A_{11 b y} |}^{2} \\ = [\sin^{2} (\frac{β_{e}^{+} - β_{e}^{-}}{2} z_{s}) + \cos^{2} (\frac{β_{o}^{-} - β_{o}^{+}}{2} z_{s})] \cos^{2} φ \\ + [\cos^{2} (\frac{β_{e}^{+} - β_{e}^{-}}{2} z_{s}) + \sin^{2} (\frac{β_{o}^{-} - β_{o}^{+}}{2} z_{s})] \sin^{2} φ \\ + \frac{1}{2} {\sin [(β_{e}^{+} - β_{e}^{-}) z_{s}] - \sin [(β_{o}^{-} - β_{o}^{+}) z_{s}]} \sin (2 φ) \sin δ \end{matrix} \end{matrix}},

where P₁₁_a and P₁₁_b represent the powers of the LP_11a and LP_11b mode components. We find from Eq. (19) that P₁₁_a and P₁₁_b fluctuate with distance z_s owing to the difference between the propagation constants of the higher-order vector modes if only a specific SOP of the probe HE₁₁ mode is given. We also find that the fluctuation can be reduced either by scrambling the SOP of the probe HE₁₁ mode or by using two orthogonal SOPs while averaging the backscattered signals. Note, however, that even if one injects two orthogonal SOPs at different times, orthogonality is not always maintained along the fiber when the birefringence distribution varies over the averaging time due to external factors such as vibration. In practical use, scrambling the SOP is therefore preferred over using two orthogonal SOPs.

3. Proof-of-concept experiments

We carried out experiments on two kinds of optical fibers that comply with ITU-T G.652: one is wound around a fiber bobbin (laboratory environment), and the other is installed in the field. To confirm the effectiveness of our proposed technique, we undertook measurements under the following conditions: a) without using the proposed technique, and b) with the proposed technique. The amplitude fluctuations in the LP₀₁, LP_11a and LP_11b mode components of the backscattered light were evaluated by the root mean square of the difference between the linear regression line and the measured data, and comparisons were made between the results obtained without and with the proposed technique.

Figure 3 shows the experimental setup. The fiber laser (FL) emitted a linearly polarized continuous wave with a center wavelength of 1050 nm as the probe light. Although we did not consider the interference between the Rayleigh backscattered lights in Section II, this interference results in fading noise, which is a well-known problem with coherent detection OTDR as well as direct detection OTDR. To reduce the fading noise, the optical frequency was shifted by over 100 GHz during the measurements [22]. The SOP of the probe light was varied by a polarization scrambler (PS). The probe light was pulsed through an acousto-optic modulator (AOM) driven by a pulse generator (PG). The AOM provided 1 μs long probe pulses, which yields the spatial resolution of 100 m. The probe pulse was passed through an optical circulator (OC), followed by a MSC. The MSC was composed of two PLC which are orthogonally connected to each other and pigtail fibers; the structure and operation are shown in Fig. 4. By choosing the arbitrary pigtail fiber, the MSC not only selected the mode of the excited probe pulse but also individually extracted an arbitrary mode component of the backscattered light. The LP₀₁ mode of the probe pulse was then launched into the FUT. The LP₀₁, LP_11a and LP_11b mode components of the backscattered light that returned to the input end were divided into the individual modes. Note that the LP₀₁, LP_11a and LP_11b mode components contain two orthogonal polarizations, i.e. LP_01x/LP_01y, LP_11ax/LP_11ay and LP_11bx/LP_11by modes, respectively. Each component was received by avalanche photodetectors (APDs) before the analogue-to-digital converter (ADC). The OTDR traces were acquired after the measured signals were averaged 2¹⁶ times.

Fig. 3 Experimental setup. FL: fiber laser. PS: polarization scrambler. AOM: acousto-optic modulator. PG: pulse generator. OC: optical circulator. MSC: mode selective coupler. FUT: fiber under test. APD: avalanche photodetector. ADC: analogue-to-digital converter.

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Fig. 4 Structure and operation of the MSC.

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Figure 5 shows the OTDR traces for the fiber in laboratory environment. The blue, red and green traces represent the optical intensity for the LP₀₁, LP_11a and LP_11b mode components of the backscattered light, respectively. We first confirmed that the amplitude fluctuations in the LP₀₁ mode component of Figs. 5(a) and 5(b) were almost equal; about 0.01 dB. This is expected result, since we used polarization independent devices in this experiment.

Fig. 5 OTDR traces for the fiber in the laboratory environment: (a) without the proposed technique, (b) with the proposed technique. The blue, red and green traces represent the optical intensity for the LP₀₁, LP_11a and LP_11b mode components of the backscattered light, respectively.

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On the other hand, we also confirmed that although the amplitude fluctuations in the LP_11a and LP_11b mode components of Fig. 5(a) were 0.03 dB and 0.02 dB, those of Fig. 5(b) were both 0.01 dB. The large fluctuations in Fig. 5(a) relative to those of Fig. 5(b) are interpreted as the effect of the modal evolution fluctuation, and the effect was reduced by scrambling the SOP of the probe light.

Figure 6 shows the OTDR traces for the fiber installed in the field environment; the insets show enlarged views. The blue, red and green traces have the same meaning as in Fig. 5. We confirmed the amplitude fluctuations in the LP₀₁ mode component of Fig. 6(a) and 6(b) were both 0.01 dB. We also confirmed the amplitude fluctuations in the LP_11a and LP_11b mode components of Fig. 6(a) and 6(b) were 0.06 dB and 0.05 dB, and those of Fig. 6(b) were 0.01 dB and 0.01 dB, respectively. Moreover, a loss event near 10 km can be found in the inset of Fig. 6(b), while the event in the inset of Fig. 6(a) was obscured by the modal evolution fluctuation. From these results, it can be conclude that the proposed technique is effective in reducing the modal evolution fluctuations in the LP_11a and LP_11b mode components of the backscattered light.

Fig. 6 OTDR traces for the fiber installed in the field environment: (a) without the proposed technique, (b) with the proposed technique. The blue, red and green traces represent the optical intensity for the LP₀₁, LP_11a and LP_11b mode components of the backscattered light, respectively. The insets show enlarged views.

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4. Conclusion

This paper theoretically analyzed Rayleigh backscattering amplitude coefficients from the HE₁₁ mode to higher-order vector modes. Based on the analysis, we clarified the SOP evolution of the backscattered higher-order vector modes with respect to that of the probe HE₁₁ mode, and then proposed a technique for reducing modal evolution fluctuation in the OTDR traces obtained from the LP_11a and LP_11b mode components of the backscattered light. The technique is simply realized by scrambling the SOP of the launched probe HE₁₁ mode during the measurement. Proof-of-concept demonstrations were carried out on optical fibers in the laboratory and field environment. The measured results showed that the technique clearly reduced the modal evolution fluctuation and discovered a loss event that was otherwise obscured by the modal evolution fluctuation.

Although the experiments were performed in the 2-LP mode region of widely used optical fibers that comply with ITU-T G.652, the modal evolution fluctuation will also occur when assessing FMFs optimized for operation over the C band by using OTDRs that individually detect the LP_11a and LP_11b mode components such as those proposed in [4–8]. This is because the modal evolution fluctuation is caused by the intensity distribution evolution of the LP₁₁ mode resulting from the modal birefringence between the four degenerate higher-order vector modes, and the modal birefringence occurs in circular symmetric optical fibers regardless of operating wavelength. Therefore, the proposed technique will also be useful in the OTDR assessment of FMFs.

Appendix Power recapture factor in backscattering process

This appendix examines to the power recapture factor from the HE₁₁ mode to each higher-order vector mode in the backscattering process. We approximate the radial field functions of the HE₁₁ mode and the higher-order vector modes by Gaussian and Laguerre-Gaussian functions [23,24], which can be written as:

\begin{array}{l} F_{1} = \sqrt{\frac{2}{π w_{1}^{2}}} \cdot \exp (- \frac{r^{2}}{w_{1}^{2}}) \\ F_{2} = \sqrt{\frac{4}{π w_{2}^{4}}} \cdot r \exp (- \frac{r^{2}}{w_{2}^{2}}) \end{array}},

where w₁ and w₂ are the mode field radius for the HE₁₁ and the higher-order vector modes, respectively.

We consider the Rayleigh backscattering from the eHE₁₁ mode to the TM₀₁ mode. Since the refractive index fluctuation that causes scattering is random, the backscattered power is given by the ensemble average of the backscattered light:

P_{x e}^{+} = \frac{〈 {| a_{x e}^{+} E_{e}^{+} |}^{2} 〉}{Z},

where the angle bracket represents ensemble averaging. The power recapture factor in the backscattering process can be obtained as the ratio of the incident power to the backscattered power. Equations (7)-(10), (20) and (21) yield the power recapture factor

B_{x e}^{+}

from the eHE₁₁ mode to the TM₀₁ mode as follows:

B_{x e}^{+} = \frac{1}{4} \frac{3}{2} {(\frac{λ}{2 π n})}^{2} {(\frac{2 w_{1}}{w_{1}^{2} + w_{2}^{2}})}^{2} .

Since the moduli of the complex amplitude of each higher-order vector mode are independent on the incident polarization state as described in Section 2.1, the power recapture factors from the eHE₁₁ or oHE₁₁ mode to the TE₀₁, eHE₂₁ and oHE₂₁ modes are equal to the result of Eq. (22). Therefore, the power recapture factor from the HE₁₁ mode group (eHE₁₁ and oHE₁₁) to the higher-order vector mode group (TM₀₁, TE₀₁, eHE₂₁ and oHE₂₁) is quadruple the result of Eq. (22), and is given by:

B = \frac{3}{2} {(\frac{λ}{2 π n})}^{2} {(\frac{2 w_{1}}{w_{1}^{2} + w_{2}^{2}})}^{2} .

The recapture factor from the HE₁₁ mode group to the higher-order vector mode group agrees with the result of previous work [7].

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Reduction of modal evolution fluctuation in 2-LP mode optical time domain reflectometry

Abstract

1. Introduction

2. Theoretical discussion

2.1 Rayleigh backscattering amplitude coefficients

2.2 SOP evolution of higher-order vector modes and modal evolution fluctuation reduction

3. Proof-of-concept experiments

4. Conclusion

Appendix Power recapture factor in backscattering process

References and links

Cited By

Figures (6)

Equations (23)

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