Mapping of intermodal beat length distribution in an elliptical-core two-mode fiber based on Brillouin dynamic grating

Yong Hyun Kim; Kwang Yong Song

doi:10.1364/OE.22.017292

1. Introduction

A two (spatial) mode fiber (TMF), which was researched for devices and sensors based on unique properties of LP₁₁ mode [1,2], has recently attracted a renewed interest as a potential platform of mode division multiplexing (MDM) for optical communication systems [3–5]. An elliptical-core (e-core) TMF was developed to stabilize the lobe orientation of the LP₁₁ mode [6–9] which is useful for several applications such as fiber sensors, tunable filters, and fiber lasers [10,11], and also applicable to a MDM transmission for the reduction of the modal cross-talk [12]. An intermodal beat length (IBL) is one of key parameters characterizing two- or few-mode fibers, determined by the difference between effective (phase) refractive indices (ERI’s) of two spatial modes. A few techniques have been developed for measuring the IBL such as an acousto-optic tunable filter (AOTF), far-field interference, prism coupler, and Fabry-Perot cavity [13–16], however these methods can provide only the average value within a test fiber.

In this paper we propose and experimentally demonstrate the mapping of the IBL distribution along an e-core TMF based on Brillouin dynamic grating (BDG) operated by different spatial and polarization modes. The BDG is generated by the stimulated Brillouin scattering (SBS) of the LP₀₁ mode (i.e. pump), and analyzed by the LP₁₁ mode (i.e. probe), where four different pairs of the spatial and polarization modes in the e-core TMF are applied for the pump and the probe (LP₀₁^x-LP₁₁^y, LP₀₁^y-LP₁₁^y, LP₀₁^x-LP₁₁^x, and LP₀₁^y-LP₁₁^x). We show that the BDG spectra can provide the information of not only the IBL between the pump and the probe modes but also the birefringence of each spatial mode, implementing the full characterization of the modal structure in the e-core TMF. A mode selective coupler (MSC) is used for selective launch and retrieval of the LP₀₁ and the LP₁₁ modes in the BDG operations. The local reflection spectra from the BDG are obtained by an optical time-domain analysis which is conducted for each pair of modes with a 1.5 m spatial resolution along a 75 m test fiber. The evaluation of temperature- and strain-dependence of the BDG spectra is also performed, and the results show that the spectral shift of the BDG reflection appears with a slope ranging from −0.16 MHz/°C to 4.9 MHz/°C for temperature and from −0.089 MHz/με to −0.012 MHz/με for strain, respectively, depending on the polarizations of the pump and the probe.

2. Principle

The BDG represents an acoustic wave generated in the process of SBS which plays a role of moving index grating. In general, the BDG is operated in a polarization maintaining fiber (PMF) utilizing the birefringence between orthogonally polarized pump and probe waves, where the BDG is generated by counter-propagating pump waves in one polarization and used to reflect the probe wave in the orthogonal polarization. The frequency offset between the pump and the probe (called BDG frequency, ν_D) in the BDG operation is determined by the ERI difference between the two polarization modes, which is several tens of GHz in PMF’s [17]. The BDG has proven to be useful for a variety of applications such as distributed sensing based on the birefringence variation, optical signal processing and microwave photonics [17–23]. Recently the operation of BDG in a circular-core few-mode fiber was reported by applying the ERI difference between the LP₀₁ and the LP₁₁ modes guided in the fiber [24].

In a circular-core fiber, the LP₁₁ mode is an approximated mode composed of almost degenerate TM₀₁, TE₀₁, and HE₂₁ modes. Since the intensity lobes of the LP₁₁ mode is constructed by the interference of the TM₀₁ (or TE₀₁) mode and HE₂₁ mode with slightly different propagation constants, the orientation of the lobes is not maintained in the propagation along the fiber. In an e-core fiber, on the contrary, the LP₁₁ mode splits into two groups, i.e. LP₁₁^odd and LP₁₁^even modes, with considerably different ERI’s and well-defined and stable intensity patterns [6–9]. When the size and the ellipticity of the e-core are properly set, the LP₁₁^odd mode can be cut-off while the LP₁₁^even mode is still guided. This fiber is called an e-core TMF supporting two stable spatial modes, i.e. the LP₀₁ and the LP₁₁^even modes [8, 25]. Since the e-core also produces internal birefringence, overall four spatial and polarization modes – i.e. LP₀₁^x, LP₀₁^y, LP₁₁^even ^x (or simply LP₁₁^x), LP₁₁^even ^y (or LP₁₁^y) – are guided in the e-core TMF [8].

In our work the BDG is generated by the SBS of the LP₀₁ mode (pump1 and pump2) and analyzed by the LP₁₁ mode (probe) of an e-core TMF as schematically shown in Fig. 1.A MSC composed of a single-mode fiber (SMF) and an e-core TMF is used for high-efficiency and selective launch and retrieval of each mode [25,26]. The frequency offset between the counter-propagating pump1 and pump2 is set to the Brillouin frequency (ν_B) of the LP₀₁ mode in the TMF for the generation of the BDG. The probe is input to the SMF of the MSC in the LP₀₁ mode to be exclusively coupled to the LP₁₁ mode of the TMF, which is reflected by the BDG and coupled back to the LP₀₁ mode of the SMF as the BDG reflection. The inset ‘A’ of Fig. 1 depicts the spectral positions of relevant waves. It is notable that the frequency offset between the probe and the BDG reflection is expected to be equal to the ν_B of the LP₀₁ mode, the same as that between the pump1 and the pump2, determined by the frequency of the shared acoustic wave. Since the ERI of the LP₁₁ mode is smaller than that of the LP₀₁ mode, the optical frequency of the probe is set higher than that of the pump waves like the case of the BDG in a PMF [17, 24]. When n₀₁(ν) and n₁₁(ν) are the ERI’s of the LP₀₁ and the LP₁₁ modes, respectively, at an optical frequency ν, the BDG frequency ν_D is calculated as follows: The acoustic waves are shared in the SBS of the LP₀₁ mode and the BDG reflection of the LP₁₁ mode, so the phase matching condition for each process yields:

S B S : \frac{2 π}{Λ} = \frac{2 π}{c} (n_{01} (ν_{0}) \cdot ν_{0} + n_{01} (ν_{0} - ν_{B}) \cdot (ν_{0} - ν_{B}))

B D G : \frac{2 π}{Λ} = \frac{2 π}{c} (n_{11} (ν_{0} + ν_{D}) \cdot (ν_{0} + ν_{D}) + n_{11} (ν_{0} + ν_{D} - ν_{B}) \cdot (ν_{0} + ν_{D} - ν_{B}))

where c, ν₀ and Λ are the speed of light, the optical frequency of the pump1, and acoustic wavelength, respectively. Applying the 1st order Taylor expansion to n₀₁ and n₁₁ around ν₀ in Eqs. (1) and (2) we obtain the following equation:

\begin{array}{l} 2 n_{01} (ν_{0}) \cdot ν_{0} - n_{01} (ν_{0}) \cdot ν_{B} - ν_{0} ν_{B} {\frac{d n_{01}}{d ν} |}_{ν_{0}} + ν_{B}^{2} {\frac{d n_{01}}{d ν} |}_{ν_{0}} \\ = 2 n_{11} (ν_{0}) \cdot ν_{0} + 2 n_{11} (ν_{0}) \cdot ν_{D} - n_{11} (ν_{0}) \cdot ν_{B} + 2 ν_{0} ν_{D} {\frac{d n_{11}}{d ν} |}_{ν_{0}} - ν_{0} ν_{B} {\frac{d n_{11}}{d ν} |}_{ν_{0}} \\ + 2 ν_{D}^{2} {\frac{d n_{11}}{d ν} |}_{ν_{0}} - 2 ν_{B} ν_{D} {\frac{d n_{11}}{d ν} |}_{ν_{0}} + ν_{B}^{2} {\frac{d n_{11}}{d ν} |}_{ν_{0}} \end{array}

Since ν₀ is over 500 times larger than ν_D (and also ν_B) [17, 24], the derivative terms multiplied by ν₀ are dominant among the derivatives in Eq. (3). When the group index n_g₀₁ (n_g₁₁) of the LP₀₁ (LP₁₁) mode at ν₀ is introduced and the derivative terms without ν₀ are ignored, Eq. (3) is reduced to the following equation:

2 (n_{01} (ν_{0}) - n_{11} (ν_{0})) \cdot ν_{0} - (n_{g 01} - n_{g 11}) \cdot ν_{B} = 2 n_{g 11} ν_{D}

Considering the similar order of magnitude (10⁻⁴ ~10⁻³) of the phase and the group index differences between the LP₀₁ and LP₁₁ modes guided in silica fibers without strong absorption, the second term on the left hand side is negligible compared to the first term in Eq. (4), which leads to a simple relation between IBL (L_B) and ν_D as follows:

ν_{D} = \frac{n_{01} (ν_{0}) - n_{11} (ν_{0})}{n_{g}_{11}} \cdot ν_{0} \equiv \frac{Δ n}{n_{g}_{11}} \cdot ν_{0} = \frac{c}{L_{B} \cdot n_{g}_{11}}

Therefore, the distribution map of the IBL can be acquired from that of local ν_D’s.

Fig. 1 Schematic of the operation of the BDG based on an e-core TMF. Inset ‘A’ shows the spectral configuration of optical waves: MSC, Mode selective coupler; ν_B, Brillouin frequency; ν_D, BDG frequency.

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Owing to the internal birefringence, the BDG operation in the e-core TMF can take place in four different pairs of the pump-probe modes according to the polarization (LP₀₁^x-LP₁₁^y, LP₀₁^x-LP₁₁^x, LP₀₁^y-LP₁₁^y, and LP₀₁^y-LP₁₁^x) as depicted in Fig. 2.The polarization-dependent splits of the ERI’s result in corresponding splits of the ν_D’s. Therefore one can find the ERI difference between any pair of optical modes from the difference of ν_D’s by Eq. (4), so investigate the modal structure along the fiber.

Fig. 2 The ERI structure of the guided modes in an e-core TMF and the possible BDG operation according to the polarizations of the LP₀₁ and the LP₁₁ modes.

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3. Experiments

In experiments, we employed a measurement setup as shown in Fig. 3. Two distributed feedback laser diodes (DFB-LD’s) were used as light sources with the operation wavelength of 1550 nm for the pump and 1547 nm for the probe, respectively. The output of the pump LD was divided by a 50/50 coupler, and one of the output waves was modulated by an electro-optic modulator (EOM) and a pulse generator to generate a 50 ns rectangular pulse for pump1. The other output wave was modulated by a single sideband modulator (SSBM) and a microwave generator to prepare a continuous Stokes wave for pump2 with a frequency offset of ν_B from the pump1. The output from the probe LD was modulated by another EOM and the pulse generator to generate a 15 ns bell-shaped pulse for the probe. A polarization controller (PC) was used for each of the pump and the probe waves. Er-doped fiber amplifiers (EDFA’s) were used to control the power of the pump1 and the pump2 which counter-propagated in a fiber under test (FUT) to build the BDG by SBS. The peak power of the pump1 and the probe was about 24 dBm and 27 dBm, respectively, and the continuous power of the pump2 was 15 dBm. The probe pulse was propagated in the same direction as the pump1, and the time trace of the BDG reflection was recorded by a 125 MHz photo receiver and an oscilloscope through a fiber Bragg grating (FBG). The pulse propagation of the pump1 and the probe was synchronized with a 30 ns delay as depicted in the inset ‘A’ of Fig. 3 to maximize the signal amplitude, and the pulse repetition rate was 500 kHz. The pump1 and the pump2 were launched to the FUT in the LP₀₁ mode through mode strippers (MS’s, by bending) removing the unwanted LP₁₁ mode from the pump waves which is coupled at the splice point between the lead SMF and the FUT. The probe was launched in the LP₁₁ mode through a MSC, following the schematic of Fig. 1. The coupling efficiency of the MSC was about 80%, and the mode extinction ratio of the LP₁₁ mode to the LP₀₁ mode in the probe wave of the TMF was about 23 dB. The measurement of the BDG spectrum was carried out by sweeping the frequency offset Δν between the pump1 and the probe with about 4 MHz step using the current control of the pump LD.

Fig. 3 Experimental setup. Inset ‘A’ shows the time traces of the pump1 and the probe pulses, and inset ‘B’ is the BDG spectrum observed by an optical spectrum analyzer in front of the FBG: EDFA, Er-doped fiber amplifier; EOM, electro-optic modulator; SSBM, single-sideband modulator; FBG, fiber Bragg grating; PC, polarization controller; DAQ, data acquisition.

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The FUT used in the experiment is a 75 m e-core TMF with the index difference Δ of 0.6%, the core radius of 5.5 μm x 3.5 μm, and the cladding diameter of 100 μm. The cut-off wavelength of the LP₁₁^odd mode is about 1500 nm. The ν_B of the LP₀₁ mode was 10.614 GHz, and the ERI difference between the LP₀₁ and the LP₁₁ modes was about 3.6 x 10⁻³ estimated by an AOTF [13], which corresponds to about 480 GHz by Eq. (5) at a wavelength of 1550 nm with n_g₁₁ = 1.45.

First we roughly measured the spectral position of ν_D using an optical spectral analyzer (OSA) in front of the FBG. The BDG reflection was observed at Δν ~450 GHz as depicted in the inset ‘B’ of Fig. 3 in which the leakage of the continuous pump2 (i.e. the output from the SSBM) is seen together. In order to determine the sweep range of Δν in the acquisition of time traces we carried out a test measurement by sweeping Δν in the vicinity of 450 GHz (from 430 GHz to 470 GHz) while recording the time traces of the BDG reflection with the oscilloscope. We found out that the BDG reflection appeared in three different frequency regions of Δν centered nearly at 445 GHz (low), 449 GHz (middle), and 453 GHz (high) with the reflection amplitude strongly dependent on the polarization states of the pump and the probe waves. According to the theoretical expectation in Fig. 2, we thought that the low, the middle, and the high frequency regions correspond to the pump-probe combinations of LP₀₁^y-LP₁₁^x, LP₀₁^x-LP₁₁^x (and LP₀₁^y-LP₁₁^y), and LP₀₁^x-LP₁₁^y, respectively. In order to acquire proper spectra of the BDG reflection for each pair of the pump-probe, our time-domain measurements were carried out with following steps:

All the PC’s (pump1, 2 and probe) are controlled to maximize the BDG reflection (i.e. time trace) at the high frequency region of Δν and the BDG spectrum of LP₀₁^x-LP₁₁^y is measured.
The probe PC is controlled (while the pump PC’s fixed) to minimize the BDG reflection at the high frequency region of Δν, and the BDG spectrum of LP₀₁^x-LP₁₁^x is measured at the middle frequency region of Δν.
The pump PC’s are controlled (while the probe PC fixed) to maximize the BDG reflection at the low frequency region of Δν, and the BDG spectrum of LP₀₁^y-LP₁₁^x is measured.
The probe PC is controlled (while the pump PC’s fixed) to minimize the BDG reflection at the low frequency region of Δν and the BDG spectrum of LP₀₁^y-LP₁₁^y is measured at the middle frequency region of Δν.

In the above steps the sweep ranges of Δν were 442.76 – 446.96 GHz (for the low), 447.16 – 451.36 GHz (for the middle), and 451.08 – 455.48 GHz (for the high), respectively. The accuracy of the optical frequency was about 1 GHz for the absolute value of Δν limited by the resolution of the OSA, and about 10 MHz for the relative value between the sweep ranges by the frequency drift of the LDs.

Figures 4(a)-4(d) show the measured distributions of the BDG spectra along the FUT for different pump-probe pairs. As seen in the examples of local BDG spectra (at z = 9.1, 37.2, 67.2 m in the case of LP₀₁^y-LP₁₁^x) of the insets the BDG reflection appeared in broad spectral range (~2 GHz) with several side peaks. The BDG can be viewed as a weak Bragg grating [18], so the reflection spectrum is sensitive to the uniformity of the BDG. Considering that the BDG frequency of current scheme is determined by the local ERI difference of two modes in Eq. (5) and that the spectral widths (< 100 MHz) of the pump1 and the probe pulses are much narrower than 2 GHz, we think it is reasonable to attribute the broad spectrum by several side peaks to the non-uniformity of the fiber within the spatial resolution (~1.5 m) of the measurement. The width of the BDG spectrum is measured to be ~60 MHz in Gaussian fitting when side peaks are suppressed (such as the position z = 37.2 m in Fig. 4). It is also notable that similar spectral shapes and widths of the BDG signal are obtained for different pair of pump-probe modes at the same position of the FUT. In addition, a gradual increase (~3 GHz) of the center frequency is observed along the FUT which was loosely wound without intentional strain or temperature gradient. Since the variation of 3 GHz in the ν_D corresponds to the change of only about 0.67% in Δn or L_B, we think the gradual change shows the long-range non-uniformity of the FUT occurred in the fabrication process of the preform or the drawing process of the fiber. It is also notable that the overall distributions of the BDG spectra of different pump-probe pairs appear to be quite similar to one another except difference center frequencies.

Fig. 4 Distributions of the BDG spectra in the e-core TMF for the pump-probe pair of (a) LP₀₁^x-LP₁₁^y, (b) LP₀₁^y-LP₁₁^y, (c) LP₀₁^x-LP₁₁^x, and (d) LP₀₁^y-LP₁₁^x. The insets are the examples of local BDG spectra in the case of LP₀₁^y-LP₁₁^x at different positions.

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Distributed maps of ν_D and corresponding L_B obtained by the peak search (spectral centroid) of the measured BDG spectra are plotted in Fig. 5(a) for the pump-probe pairs of different polarizations. The L_B’s are calculated from Eq. (4) assuming n_g₁₁ = 1.45. The polarization dependence of n_g₁₁ is thought to be negligible considering the amount of birefringence (~3 x 10⁻⁵) of the FUT. The ν_D ranges from 443.3 GHz to 454.5 GHz in the FUT according to the polarization states of the pump-probe pairs and the positions, and local fluctuations with a gradual increase of ν_D are commonly seen for all the cases. It is also notable that the v_B is almost uniform within 2 MHz along the FUT while the BDG frequency changes as much as ~3 GHz. Figure 5(b) depicts the frequency difference (Δν_D) between the ν_D’s of different pump-probe pairs with respect to the position acquired by the cross-correlation of the local BDG spectra which is useful in calculating the spectral shift of multi-peak spectrum [27]. It is remarkable that the differences are quite uniform along the fiber compared to the strong fluctuations seen in the map of ν_D. This feature confirms that the variations of the BDG spectra are almost uniform along the fiber for different pump-probe pairs, and their origin is not the noise of the measurement but the structural variation of the FUT itself.

Fig. 5 (a) Distribution maps of ν_D and L_B for different pump-probe pairs. Note that L_B is calculated with n_g₁₁ = 1.45. (b) Frequency difference (Δν_D) between the ν_D’s of different pump-probe pairs along the position.

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Another important result obtained from the relative difference of ν_D is the amount of birefringence Δn^xy ( = n_x – n_y) of each spatial mode. For example, Δn^xy of the LP₀₁ mode is calculated from Eq. (5) as follows:

ν_{D}^{x x} - ν_{D}^{y x} = \frac{(n_{01}^{x} - n_{11}^{x}) - (n_{01}^{y} - n_{11}^{x})}{n_{g 11}} \cdot ν_{0} = \frac{Δ n_{01}^{x y}}{n_{g 11}} \cdot ν_{0}

The distribution maps of Δn_xy for the LP₀₁ and the LP₁₁ modes are plotted in Fig. 6 which are calculated from the result of Fig. 5(b). It is seen that the birefringence of the LP₁₁ mode is slightly (~4%) larger than that of the LP₀₁ mode in this e-core TMF, with a similar shape of local fluctuations.

Fig. 6 Birefringence distribution of each spatial mode calculated from the results of Fig. 5(b).

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Temperature (C_T) and strain (C_ε) coefficients of the ν_D are useful parameters for high-sensitivity and discriminative sensing based on a PMF [22, 23]. We measured the C_T and the C_ε of the ν_D’s for different pump-probe pairs as a feasibility study of distributed sensing by the BDG in the e-core TMF. The measurements were conducted by applying temperature and strain variations to short sections of the FUT (5 m for temperature, 2 m for strain). The results are shown in Figs. 7 and 8 for the temperature and the strain, respectively. As seen in Fig. 7 the C_T’s are measured to be −0.16, 2.3, 2.9, and 4.9 (MHz/°C) for ν_D^xy, ν_D^yy, ν_D^xx, and ν_D^yx, respectively. It is remarkable that not only the amplitude but also the sign of C_T’s are different according to the polarization states of the pump and the probe, and the C_T appears to be smaller for the cases with larger ν_D. Such different C_T’s for different polarization pairs of modes indicate that the temperature dependence of the birefringence is different for each mode. We think further investigation is needed on this point for clear understanding. It is also interesting that the maximum value of 4.9 MHz/°C is about 5 times larger than that of the ν_B of ordinary SMF’s (~1 MHz/°C), and about 10 times smaller than that of the BDG based on a PANDA PMF [23].

Fig. 7 Temperature dependence of ν_D for the pump-probe pair of (a) LP₀₁^x-LP₁₁^y, (b) LP₀₁^y-LP₁₁^y, (c) LP₀₁^x-LP₁₁^x, and (d) LP₀₁^y-LP₁₁^x.

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Fig. 8 Strain dependence of ν_D for the pump-probe pair of (a) LP₀₁^x-LP₁₁^y, (b) LP₀₁^y-LP₁₁^y, (c) LP₀₁^x-LP₁₁^x, and (d) LP₀₁^y-LP₁₁^x.

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The strain coefficient C_ε’s are measured to be −0.018, −0.012, −0.089, and −0.081 (MHz/με) for ν_D^xy, ν_D^yy, ν_D^xx, and ν_D^yx, respectively, as depicted in Fig. 8. The maximum strain coefficient is observed in the case of ν_D^xx, not ν_D^yx like the case of C_T, and the maximum value is about twice that of the ν_B of ordinary SMF’s (~0.05 MHz/με) with opposite sign. We think the large differences shown in the C_T and the C_ε of the ν_D in the e-core TMF could be applied to discriminative sensing of temperature and strain.

In conclusion we have demonstrated the optical time domain measurement and the characterization of the intermodal beat length in an e-core TMF based on the analysis of the BDG spectra. It was shown that the BDG operations are possible with four different pairs of the spatial and polarization modes and that the modal structure in the e-core TMF can be investigated using the distribution maps of the BDG frequencies. Additionally, the BDG frequencies of different polarization pairs show large differences in the temperature and the strain coefficients, which could be applied for distributed fiber sensors.

Acknowledgments

This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2012R1A1A2009103).

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Mapping of intermodal beat length distribution in an elliptical-core two-mode fiber based on Brillouin dynamic grating

Abstract

1. Introduction

2. Principle

3. Experiments

Acknowledgments

References and links

Cited By

Figures (8)

Equations (6)

Optics Express