Transfer matrix method for simulation of the fiber Bragg grating in polarization maintaining fiber

Optics Communications 452 (2019) 185–188

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Optics Communications journal homepage: www.elsevier.com/locate/optcom

Transfer matrix method for simulation of the fiber Bragg grating in polarization maintaining fiber Qiang Liu a , Qian Li a , Yudan Sun a,b , Quan Chai c , Bin Zhang c , Chao Liu a ,∗, TaoSun d , Wei Liu a , Jiudi Sun a , Zonghuan Ren a , Paul K Chu e a

School of Electronics Science, Northeast Petroleum University, Daqing 163318, PR China College of Mechanical and Electrical Engineering,Daqing Normal University, Daqing 163712, PR China c Key Lab of In-Fiber Integrated Optics of Ministry of Education, Harbin Engineering University, Harbin 150001, PR China d Media Lab, Massachusetts Institute of Technology, Cambridge, MA 02139, USA e Department of Physics, Department of Materials Science & Engineering, and Department of Biomedical Engineering, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong, China b

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Keywords: Transfer matrix method Fiber Bragg grating Polarization maintaining fiber Twisted fiber

ABSTRACT In this study, a new simulation method is proposed and verified for fiber Bragg grating patterned on polarization maintaining fiber(PM-FBG) using the transfer matrix approach. The method is designed to solve the twisted PM-FBG simulation. The four mode matrix element of a period grating is derived from the S matrix element and the method only involves the torsion-induced shear strain entry 𝑒𝑥𝑦 which can be considered as the rotation of the principal axis of the polarization maintaining fiber and ignores the torsion-induced shear strain entries 𝑒𝑥𝑧 and 𝑒𝑦𝑧 . The validity of the method is confirmed by numerical simulation of the twisted PM-FBG. The novel method provides the ability to analyze any type of PM-FBG.

1. Introduction Fiber Bragg grating (FBG) written in polarization maintaining fiber (PMFBG) has been investigated for sensing applications such as fiberbased sensors [1–6]. PMFBG can overcome the cross talking problem of FBG in measuring the temperature and axial strain using the Bragg peaks of the two orthogonal polarization modes [7–9]. In addition, the twist sensor [10,11], bending sensor [12], and inclinometer [13] has been developed recently based on PMFBG. In order to study the spectral characteristics of the PMFBG subjected by external factors, numerical analysis is widely adopted to simulate the reflection spectrum of PMFBG [14,15]. The basic approach of analyzing PMFBG is based on the four mode coupled-mode theory considering coupling between the two forward propagating polarization modes and two backward propagating modes. The amplitude evolution of the four modes is governed by the coupled mode equation [16,17] and the coupling coefficients are computed by approximation. The method can simulate the reflection spectrum of PMFBG subjected to shear strain induced by diametrical loading and twist [18]. However, the lower simulation efficiency affects practical application. The popular simulation approach of PMFBG is the transfer matrix method. Ref. [19] describes a transfer matrix method based on four mode coupling to analyze PMFBG subjected to the shear strain on transverse loading. Recently, a novel phenomenon was reported for the twisted PMFBG. A resonant peak at the middle ∗

region of the two principal Bragg peaks was discovered due to cross coupling between the two orthogonal polarization modes [10,16] but unfortunately, the existing transfer matrix method is not applicable to the twisted PMFBG. In this paper, a novel transfer matrix method is proposed to analyze the twisted PMFBG based on the four mode coupled-mode theory. The method only considers the torsion-induced shear strain entry 𝑒𝑥𝑦 and ignores the torsion-induced shear strain entries 𝑒𝑥𝑧 and 𝑒𝑦𝑧 resulting in a smaller coupling coefficient. The method is verified by simulating the reflection spectrum of the twisted PMFBG.

2. Theory The reflection spectrum of the PM-FBG shows two resonant peaks corresponding to the slow axis and fast axis polarization mode transmission and are induced by forward–backward coupling of the same polarization modes. The twisted PMFBG generates shear strain entries 𝑒𝑥𝑦 , 𝑒𝑥𝑧 , 𝑒𝑦𝑧 . The shear strain entries 𝑒𝑥𝑧 and 𝑒𝑦𝑧 result in coupling between the longitudinal field component and transversal field of the orthogonally polarized modes. The computed coupling coefficients are too small to consider in weak guidance approximation [20]. The shear strain entry 𝑒𝑥𝑦 induced by the two orthogonally polarized modes

Corresponding author. E-mail address: [email protected] (C. Liu).

https://doi.org/10.1016/j.optcom.2019.07.034 Received 22 June 2019; Received in revised form 15 July 2019; Accepted 16 July 2019 Available online 18 July 2019 0030-4018/© 2019 Elsevier B.V. All rights reserved.

Q. Liu, Q. Li, Y. Sun et al.

Optics Communications 452 (2019) 185–188

Fig. 1. Diagram of the twisted PM-FBG.

The elements of the 4×4 scattering matrix S can be derived as follows:

𝑆11 = 𝑆13 = 𝑆21 = 𝑆23 = 𝑆31 =

Fig. 2. Periodic structure of the PMFBG.

𝑆33 = should not be ignored and can be regarded as the coordinate rotation of the principal axis of the polarization maintaining fiber. The schematic diagram of the twisted PMFBG is shown in Fig. 1 and the slow axis and fast axis of the PMFBG are represented by x and y, respectively. The corresponding rotated coordinate axis are x′ and y′ and the length and period of the PMFBG are 𝐿0 and 𝛬, respectively. The 𝜃0 stands for the twisted angle and the twisting rate is defined as the ratio of the rotated angle and length of the twisted fiber 𝜏 = 𝜃0 ∕𝐿0 . 𝑎1 and 𝑏1 refer to the amplitude of the incident and output polarization light along the fast axis of the PMFBG, respectively and similarly, 𝑎3 and 𝑏3 are the amplitude of the incident and output polarization light along the slow axis of the PMFBG, respectively. In order to deduce the transfer matrix formulation, the PMFBG can be approximately divided into 𝑁 segments with one grating period and 𝐿0 = 𝑁⋅𝛬, assuming that every grating period has a constant step refractive index. One periodic structure of the PMFBG is shown in Fig. 2. The refractive indexes of the slow axis and fast axis are 𝑛𝑥 and 𝑛𝑦 , respectively and 𝛥𝑛 denotes the UV-induced index increment. Hence, one period of the PMFBG includes interface 1, medium 2, interface 2, and medium 3 in sequence.

𝑆41 = 𝑆43 =

⎡ ⎢ ⎢ ⎢ ⎢ ⎣

𝑆12 =

=0

𝑆14 =

= 𝑡 cos 𝜃

𝑆22 =

= 𝑡 sin 𝜃

𝑆24 =

=0

𝑆32 =

=𝑟

𝑆34 =

= −𝑡 sin 𝜃

𝑆42 =

= 𝑡 cos 𝜃

𝑆44 =

𝑏1 𝑎2 𝑏1 𝑎4 𝑏2 𝑎2 𝑏2 𝑎4 𝑏3 𝑎2 𝑏3 𝑎4 𝑏4 𝑎2 𝑏4 𝑎4

= 𝑡 cos 𝜃 = −𝑡 sin 𝜃 = −𝑟 =0 (2) = 𝑡 sin 𝜃 = 𝑡 cos 𝜃 =0 = −𝑟

𝑏1 𝑏2 𝑏3 𝑏4

⎤ ⎡ ⎥ ⎢ ⎥=𝑆⎢ ⎥ ⎢ ⎥ ⎢ ⎦ ⎣

𝑎1 𝑎2 𝑎3 𝑎4

⎤ ⎡ ⎥ ⎢ ⎥=⎢ ⎥ ⎢ ⎥ ⎢ ⎦ ⎣

𝑆11 𝑆21 𝑆31 𝑆41

𝑆12 𝑆22 𝑆32 𝑆42

𝑆13 𝑆23 𝑆33 𝑆43

𝑆14 𝑆24 𝑆34 𝑆44

⎤⎡ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎦⎣

𝑎1 𝑎2 𝑎3 𝑎4

⎤ ⎥ ⎥ ⎥ ⎥ ⎦

(3)

The transmission matrix of interface 1 can be derived from the relation between the transmission matrix and scattering matrix, 𝑎1 , 𝑏1 , 𝑎3 , 𝑏3 and given by 𝑏2 , 𝑎2 , 𝑏4 , 𝑎4 : ⎡ ⎢ ⎢ ⎢ ⎢ ⎣

𝑎1 𝑏1 𝑎3 𝑏3

⎡ ⎢ ⎢ ⎢ =⎢ ⎢ ⎢ ⎢ ⎣

⎤ ⎡ ⎥ ⎢ ⎥ = 𝑇1 ⎢ ⎥ ⎢ ⎥ ⎢ ⎦ ⎣ cos 𝜃 𝑡 𝑟 cos 𝜃 𝑡 sin 𝜃 𝑡 𝑟 sin 𝜃 𝑡

⎤ ⎥ ⎥ ⎥ ⎥ ⎦ 𝑟 cos 𝜃 𝑡 𝑟2 cos 𝜃 + 𝑡 cos 𝜃 𝑡 𝑟 sin 𝜃 𝑡 𝑟2 sin 𝜃 + 𝑡 sin 𝜃 𝑡

𝑏2 𝑎2 𝑏4 𝑎4

sin 𝜃 𝑡 𝑟 sin 𝜃 − 𝑡 cos 𝜃 𝑡 𝑟 cos 𝜃 𝑡 −

𝑟 sin 𝜃 𝑡 𝑟2 sin 𝜃 − − 𝑡 sin 𝜃 𝑡 𝑟 cos 𝜃 𝑡 𝑟2 cos 𝜃 + 𝑡 cos 𝜃 𝑡 −

⎤ ⎥⎡ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎥ ⎦

𝑏2 𝑎2 𝑏4 𝑎4

⎤ ⎥ ⎥ ⎥ ⎥ ⎦ (4)

A phase shift will be generated as the two forward propagating polarization modes and two backward propagating modes travel through medium 2 and medium 3. The transfer matrixes of medium 2 and medium 3 are given by

As the twisted angle of the PMFBG is 𝜃0 , the coordinate axis of the adjacent grating rotates by 𝜃, as shown in Fig. 3(b). The coordinate axis of medium 1 and medium 2 are x, y and x′ , y′ respectively. Each element of the scattering matrix S for interface 1 can be expressed by the following equation: 𝑏𝑖 . 𝑎𝑗

=𝑟

The scattering matrix of interface 1 is given by

According to the scattering matrix principle, interface 1 can be considered as the scattering node with zero length, as shown in Fig. 3(a) [21]. 𝑎1 , 𝑎3 and 𝑏1 , 𝑏3 are the amplitudes of the forward and backward propagating light of the medium 1 for the two orthogonal polarization directions respectively, 𝑏2 , 𝑏4 and 𝑎2 , 𝑎4 are the amplitudes of the forward and backward propagating light of the medium 2 for the two orthogonal polarization directions respectively, and 𝑟 and 𝑡 are the reflection coefficient and transmission coefficient of interface 1, respectively. The reflection coefficient and transmission coefficient of the slow axis and fast axis are approximately equal due to the smaller 𝛥𝑛 and expressed as 𝑟 = 𝛥𝑛/(𝑛𝑥 + 𝑛𝑦 ). The transmission coefficient is 𝑡1 = 𝑡2 = 𝑡, and 𝑡2 = 1 − 𝑟2 is given by the Fresnel’s reflection law.

𝑆𝑖𝑗 =

𝑏1 𝑎1 𝑏1 𝑎3 𝑏2 𝑎1 𝑏2 𝑎3 𝑏3 𝑎1 𝑏3 𝑎3 𝑏4 𝑎1 𝑏4 𝑎3

(1) 186

⎡ ⎢ 𝑇2 = ⎢ ⎢ ⎢ ⎣

𝑒𝑖𝜑1 0 0 0

0 𝑒−𝑖𝜑1 0 0

0 0 𝑒𝑖𝜑3 0

0 0 0 𝑒−𝑖𝜑.3

⎤ ⎥ ⎥ ⎥ ⎥ ⎦

⎡ ⎢ 𝑇4 = ⎢ ⎢ ⎢ ⎣

𝑒𝑖𝜑2 0 0 0

0 𝑒−𝑖𝜑2 0 0

0 0 𝑒𝑖𝜑4 0

0 0 0 𝑒−𝑖𝜑.4

⎤ ⎥ ⎥ ⎥ ⎥ ⎦

(5)

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Optics Communications 452 (2019) 185–188

Fig. 3. Interface 1 of one period grating.

Fig. 4. Reflection spectra of the PMFBG with different twisting rates: (a) Apodization functions of the grating, (b) Reflection spectra of the twisted PMFBG, and (c) Intensity of the intersection region for different twisting rates..

In the equation, 𝜑1 and 𝜑3 are the phase shifts of the 𝑥-axis and 𝑦axis polarization modes of medium 2, respectively and 𝜑2 and 𝜑4 are the phase shifts of the 𝑥-axis and 𝑦-axis polarization modes of medium 3, respectively. The transmission matrix of interface 2 can be deduced by Eq. (4) with 𝜃 = 0 and negative reflection coefficient due to the inverse medium with interface 1. The transmission matrix is deduced to be 1

⎡ 𝑡 ⎢ −𝑟 𝑇3 = ⎢ 𝑡 ⎢ 0 ⎢ ⎣ 0

− 𝑟𝑡

0 0

0 0

1 𝑡

1 𝑡

− 𝑟𝑡

0 0 − 𝑟𝑡 1 𝑡

⎤ ⎥ ⎥ ⎥ ⎥ ⎦

Therefore, the transmission matrix 𝑇 of one period grating is 𝑇 = 𝑇 1 ⋅𝑇2 ⋅𝑇3 ⋅𝑇4 which is given in Box I. 3. Simulation and discussion The transmission matrix method is deduced under the weak guidance approximation. To validate the method, the spectral characteristics of the twisted PMFBG are calculated using analogous parameters. The effective refractive indexes of the fast and slow axes are 1.45984 and 1.46041, respectively. The grating length is 𝐿0 = 10 mm and its period 𝛬 is 631.2 nm. In addition to the modulation depth of refractive index (𝛥𝑛) with the value of 1.8×10−4 , the apodization function] is

(6)

187

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( ⎡ 𝑒𝑖𝜑+ ⎢ ( ⎢ 𝑒𝑖𝜑+ 𝑇 =⎢ ( ⎢ 𝑒𝑖𝜑+ ⎢ ( ⎢ 𝑒𝑖𝜑+ ⎣

) − 𝑟2 𝑡12 cos 𝜃 ) 𝑟 − 1 𝑡2 cos 𝜃 ) − 𝑟2 𝑡12 sin 𝜃 ) − 1 𝑡𝑟2 sin 𝜃

Optics Communications 452 (2019) 185–188

( −𝑖𝜑 𝑒 + ( −𝑖𝜑 𝑒 + ( −𝑖𝜑 𝑒 + ( −𝑖𝜑 𝑒 +

)

𝑟 cos 𝜃 𝑡2 ) 1 2 − 𝑟 𝑡2 cos 𝜃 ) − 1 𝑡𝑟2 sin 𝜃 ) − 𝑟2 𝑡12 sin 𝜃

−1

(

′

−𝑒𝑖𝜑+ + 𝑟2

)

1 𝑡2

sin 𝜃

𝑖𝜑′+

(−𝑒 + 1) 𝑡𝑟2 sin 𝜃 ( ′ ) 𝑒𝑖𝜑+ − 𝑟2 𝑡12 cos 𝜃 ( ′ ) 𝑒𝑖𝜑+ − 1 𝑡𝑟2 cos 𝜃

′ ⎤ (−𝑒−𝑖𝜑+ + 1) 𝑡𝑟2 sin 𝜃 ⎥ ( ) 1 −𝑖𝜑′+ 2 −𝑒 + 𝑟 𝑡2 sin 𝜃 ⎥ ⎥ ′ ⎥ (𝑒−𝑖𝜑+ − 1) 𝑡𝑟2 cos 𝜃 ⎥ ( ) 1 −𝑖𝜑′+ 2 𝑒 − 𝑟 𝑡2 cos 𝜃 ⎥⎦

(7)

𝜙+ = 𝜙1 + 𝜙2 = 𝜋 + (𝛽𝑓 𝑎𝑠𝑡 − 𝛽𝑓 )𝛬

(8)

𝜙′+

(9)

= 𝜙3 + 𝜙4 = 𝜋 + (𝛽𝑠𝑙𝑜𝑤 − 𝛽𝑠 )𝛬

where 𝛽𝑓 𝑎𝑠𝑡 and 𝛽𝑠𝑙𝑜𝑤 are the propagation constants of the fast axis and slow axis, 𝛽𝑓 = 𝛽𝑠 = 𝜋∕𝛬. The matrix of PMFBG can be calculated by multiplication of each period matrix.

Box I.

(sinc(2z/L−1))2 , as shown in Fig. 4(a). The incident light wave is linearly polarized and set to be 45◦ . The boundary conditions are √ √ 𝐵(0) = (1∕ 2, 𝑏1 (0), 1∕ 2, 𝑏3 (0))𝑇 (10) 𝐵(𝐿) = (𝑏2 (𝐿), 0, 𝑏4 (𝐿), 0)𝑇

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The simulated reflection spectra are shown in Fig. 4(b). The solid line is the original spectrum of the PMFBG and the two dominant reflection peaks at 1551.54 nm and 1550.93 nm correspond to the slow axis and fast axis polarization modes transmission. As the PMFBG is twisted, a reflection peak emerges from the intersection region (1551.25 nm) of the two principal Bragg peaks. The reflection spectra with the twisting rates of 41.9, 83.8, and 125.7 mrad/mm are shown in Fig. 4(b). The peak varies with increasing twisting rate, as shown in Fig. 4(c). The simulated results are consistent with previous studies [16] and so the proposed method can be employed to simulate twisted PMFBG. 4. Conclusion A novel transfer matrix method for four mode coupling is proposed and validated. The method ignores the torsion-induced shear strain entries 𝑒𝑥𝑧 and 𝑒𝑦𝑧 and only considers the shear strain entry 𝑒𝑥𝑦 . To evaluate the developed matrix method, the reflection spectra of the twisted PMFBG is simulated. The results from the proposed method show good agreement with those calculated by the coupling mode equation. The method provides a more concise and efficient approach to simulate the performance of any kind of PMFBG. Acknowledgments This work was jointly supported by the Natural Science Foundation of Heilongjiang Province, China [grant number E2017001], Youth Science Foundation of Northeast Petroleum University, China [grant number 15071120517], Provincial talent project, China [grant number ts26180221], Hong Kong Research Grants Council (RGC) General Research Funds (GRF), China [grant number City U 11205617], as well as City University of Hong Kong, China Strategic Research Grant (SRG) [grant number 7005105]. References [1] T. Guo, L. Shang, F. Liu, C. Wu, B. Guan, H. Tam, J. Albert, Polarizationmaintaining fiber-optic-grating vector vibroscope, Opt. Lett. 38 (2013) 531–533. [2] B. Hofp, B. Fischer, T. Bosselmann, A. Koch, J. Roths, Strain-independent temperature measurements with surface-glued polarization-maintaining fiber Bragg grating sensor elements, Sensors 19 (2019) 144. [3] C. Liu, L. Yang, X. Lu, Q. Liu, F. Wang, J. Lv, T. Sun, H. Mu, Paul K. Chu, Mid-infrared surface plasmon resonance sensor based on photonic crystal fibers, Opt. Express 15 (2017) 14227.

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