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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 27 — Dec. 17, 2012
  • pp: 28839–28845
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Orthogonal polarization mode coupling for pure twisted polarization maintaining fiber Bragg gratings

Fei Yang, Zujie Fang, Zhengqing Pan, Qing Ye, Haiwen Cai, and Ronghui Qu  »View Author Affiliations


Optics Express, Vol. 20, Issue 27, pp. 28839-28845 (2012)
http://dx.doi.org/10.1364/OE.20.028839


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Abstract

Spectral characteristics of orthogonal polarization mode coupling for pure twisted polarization maintaining fiber Bragg gratings (PM-FBG) are proposed and analyzed experimentally and theoretically. Different from the polarization mode coupling in PM-FBG due to side pressure, a resonant peak at the middle of two orthogonal polarization modes is found when the PM-FBG is twisted purely which is attributed to the cross coupling of polarization modes. Its intensity increases with the twisting rate. A new coupled mode equation is built to describe the pure twist polarization mode coupling, in which both the normal strain induced by strain-applied parts and the tangential strain induced by twisting are taken into consideration and expressed in a unified coordinate. The novel phenomenon and its explanation are believed to be helpful for PM-FBG applications in fiber sensor and laser technologies.

© 2012 OSA

1. Introduction

In this paper, the spectral characteristics of orthogonal polarization mode coupling for pure twisted PM-FBG are illuminated and analyzed experimentally and theoretically. A new coupled mode equation is built to describe this pure twist polarization mode coupling, in which both the normal strain induced by strain-applied parts and the shear strain induced by twisting are taken into consideration and expressed in a unified coordinate. Some potential applications are also proposed using the novel mechanism of polarization mode coupling in the fields of sensors and fiber lasers.

2. Experiment

The PM-FBG used in the experiments was imprinted in PMF (YOFC PM1017-A) which beat length is LB = 2.8 mm by using a 248 nm excimer laser and phase masks. Its index modulation was apodized by sinc square function with its main lobe coincided with the grating length of 10 mm. The grating period is 531.2 nm. The reflection spectra of the PM-FBG were measured under externally induced strains in the experiment setup as shown in Fig. 1
Fig. 1 Schematic diagram of the experimental setup.
. One of the PM-FBG pigtails is held still on a stage at point A in Fig. 1, and the other was mounted on a fiber rotator, which is used to twist the PM-FBG. A Glan prism polarizer and two collimators are used to polarize the light from broad band source (BBS) and control its polarization orientation. A circulator is used to input the polarized optical beam to the grating, and an optical spectrum analyzer (OSA) with resolution of 0.02 nm is used to measure the reflection spectrum.

In the measurements, the reflection spectra without twisting were measured firstly by adjusting the polarizer orientation to get one of the two polarization modes separately, as shown in Fig. 2(a)
Fig. 2 Reflection (a) and transmission (b) spectra of PM-FBG measured under different polarizations without twisting.
by blue dash line curve and red dot-dash line curve for the fast and slow axes respectively. The central wavelengths of the two main reflective peaks are 1550.94 nm and 1551.52 nm respectively. The two polarization orientations were perpendicular with each other, corresponding to the two perpendicular principal axes of PMF. The black solid curve in the figure is the spectrum when the input polarization was set to 45° to the principal axis. For obtaining the grating strength, its transmission spectra were also measured as shown in Fig. 2(b). The reflectivity of both resonant peaks is 86.5%. Their effective index modulation and coupling coefficient are extracted to be Δneff = 4.1 × 10−5 and κ = 1.66 respectively.

The effect of fiber twisting on the spectrum of PM-FBG was investigated experimentally. Figure 3(a)
Fig. 3 Experimental reflective spectrum (a) and reflective intensity of the middle peak (b) of PM-FBG with different twisting rate.
shows the spectra of the PM-FBG with different twisting rate when the input polarization direction is set at 45°. It is seen that a middle reflective peak at 1551.23nm is observed at the midpoint between the two principal peaks, and the reflective intensity increases with the twisting rate τ, which is defined as the ratio of the rotation angle θ of the rotator and the length L between the two fixed points. Figure 3(b) gives the amplitude of the middle peak varied with the twisting rate, which can be fitted with function of –[11.9 + 12.2 × exp(-τ/45.0)] dB, where τ is the twisting rate in mrad/mm, shown as red curve in the figure. The fitting functions and parameters may be different for different FBGs.

3. Theoretical model

3.1 Dielectric constant in the PM-FBG

Fiber twisting not only induces shear strains inside the fiber, but also rotates the principal axis of the polarization maintaining fiber as shown in Fig. 4
Fig. 4 Schematic diagram of twisted PM-FBG and its four eigen modes.
. The torsion-induced shear strains are expressed as exz = τy and eyz = -τx [19

19. R. Ulrich and A. Simon, “Polarization optics of twisted single-mode fibers,” Appl. Opt. 18(13), 2241–2251 (1979). [CrossRef] [PubMed]

], where τ is the twisting rate. The principal axis of the birefringence is now rotating along with position z. It is necessary to convert the in-fiber coordinate to a unified coordinate, i.e. the laboratory coordinate to give out a composite effect of the strains. The coordinate conversion introduces off-diagonal terms of the strain state in x-y plane, expressed as exy = eyx = Δesin(2τz), where Δe = exey is the normal strain difference in the fiber core, induced by stress applying part (SAP). For the grating written in the fiber, the dielectric constant is modulated in z-direction by UV laser irradiation, expressed asδε=εuvcos2βBz, whereβB=π/Λ=2π/λB. The photo-elastic induced index increment and the index modulation of grating should be added as two perturbations on the original background index. Thus the complete dielectric constant is expressed as [20

20. Z. Fang, K. Chin, R. Qu, and H. Cai, Fundamentals of Optical Fiber Sensors, (John Wiley & Sons, 2012), Chap. 3.

]
ε=ε¯+ε¯2p44(ΔecosθΔesinθτyΔesinθΔecosθτxτyτxeΣ)+εuvcos2βBz.
(1)
whereθ=2τzand e = ex + ey are denoted for simplicity. ε¯is the homogeneous part, which includes the DC component of UV irradiation and the averaged photo-elastic effect. The second term is a 3 × 3 tensor, denoted as ε˜ hereinafter.

3.2 The coupled mode equations

It is shown that the twisted polarization maintaining fiber is an inhomogeneous and anisotropic medium. Helmholtz equation for such a medium is written as

2E+k2εE(E)=0.
(2)

The divergence of electric field is now different from that in isotropic homogeneous medium without free charge. It has to be deduced from the divergence of electric displacement:D=(εE)=ρf=0; the divergence of field is then expressed as

E=[(ε˜+δε)E]/ε¯.
(3)

(a1a2b1b2)=(iκbcosθκτ+iκbsinθiκgei2δz0κτ+iκbsinθiκbcosθ0iκgei2δziκgei2δz0iκbcosθκτiκbsinθ0iκgei2δzκτiκbsinθiκbcosθ)(a1a2b1b2).
(7)

In the equation, denotations are used: κb=n3kp44Δe/2=2π/LBwith birefringence beat lengthLB; κτ=n2p44τ/2 with elasto-optical coefficient p44and κg=Δnuvk0/2with UV-induced index incrementΔnuv.

The coupling mechanisms of PM-FBG are described in CME (7). The grating induces forward-backward coupling of the same polarization. It does not make coupling between two polarizations due to its essence of multiple-beam interference. Similar to FBG in single mode fiber, the grating determines the spectral characteristics, with the two resonant wavelengths due to birefringence. The birefringence is attributed to the unbalanced normal strainΔe. In the unified coordinate, however, its effect depends on the angle of its principal axis to the lab axis, which varies with the propagation in a twisted fiber. Torsion induces coupling between the two perpendicular polarization modes, as analyzed in reference [19

19. R. Ulrich and A. Simon, “Polarization optics of twisted single-mode fibers,” Appl. Opt. 18(13), 2241–2251 (1979). [CrossRef] [PubMed]

], which does not change the effective index itself. The effects of torsion and birefringence are described by the real coefficient and the imaginary coefficient respectively in (7), corresponding to variations of amplitudes and phases. For twisted fibers with and without birefringence, the equation is self-consistent.

4. Simulations and discussion

The spectral characteristics are investigated by simulation clearly. The new CME is now a differential equation with non-constant coefficients. We solved it using the finite difference method. The input linearly polarized wave is in 45° to the principal axis of PMF. The boundary conditions are thus expressed as: A(0)=(1/2,1/2,b1(0),b2(0))T andA(L)=(a1(LFBG),a2(LFBG),0,0)T, where LFBGis the length of the FBG.

Figure 5(a)
Fig. 5 Simulated reflective spectra (a) and reflective intensity of the middle peak (b) of PM-FBG with different twisting rate. The simulated fiber’s beat length is LB = 2.8 mm,its elasto-optical coeffient is p44 = −0.07. The simulated grating’s length is LFBG = 10 mm, its period is Λ = 531.2 nm, its index modulation is Δnuv = 1.8 × 10−4 × (sinc(2z/LFBG −1))2 with apodization.
shows the calculated reflection spectra expressed as
R1,2=|b1,2(0)|2|a1(0)|2+|a2(0)|2,
(8)
with input polarization setting at 45° to the principal axis. The peak wavelengths with no torsion are at 1550.94 and 1551.52 nm, as shown in the figure by solid line. The spectra for twisting rate of 41.9 and 125.7 mrad/mm are also shown in the figure by red dash-dot line and blue dash line respectively.

A reflective peak at 1551.23nm appears in Fig. 5(a); its amplitude rises with the torsion increasing, as shown in Fig. 5(b). The curve can be fitted by the function of –[11.64 + 27.45 × exp(–τ/42.8)] dB. Comparing with Fig. 3, the simulated spectral characteristic is coincident well with the experimental result qualitatively. The quantitative difference may be attributed to the fact that there is a background noise at the middle for untwisted PM-FBG, which brings about some uncertainty.

The evolutions of the four modes intensity at the wavelength of 1551.23 nm along the z axis of PM-FBG with twisting rate of 125.7 mrad/mm are calculated to show the coupling mechanism in detail. Figure 6(a)
Fig. 6 Intensity evolution along the z axis of PM-FBG subjected to a twisting rate of 125.7 mrad/mm with an incident light at a wavelength of 1551.23 nm. The polarization of the incident light is only along the slow (a) and fast (b) axis respectively. The other parameters are same with Fig. 5.
and 6(b) are the simulated results for input polarization directions set at slow axis and fast axis, respectively. Then only one mode a1 and a2 is excited at the input end of the PM-FBG. It is seen that energy of the input modes are transferred to the other polarization mode gradually by twisting, which is reflected partly by the grating and transmitted mostly. The reflected energy would generate the middle reflective peak. The other input polarization mode evolves in the same way. The reflected beam is thus composed by wave b1 and b2 which come from their corresponding orthogonal input polarization modes a2 and a1.

It is convinced that the middle peak is mainly attributed to the cross coupling between the two polarization modes induced by the torsion as described by the deduced CME (7). Such a phenomenon may be used as a twisting sensor. And in single frequency polarized fiber lasers incorporated with PM-FBG, the additive peak may affect their polarization performances.

5. Conclusion

The spectral properties of orthogonal polarization mode coupling for pure twisted PM-FBG are studied in this paper experimentally and theoretically. A resonant peak at the middle between two main polarization peaks is observed in the measured spectrum when the PM-FBG is twisted. And its amplitude increases with the twisting rate. A four mode coupled mode equation is deduced to give explanations of such phenomena. In the new CME, the rotating principal axis in twisted PMF is expressed in the lab coordinate, and the degenerate polarization modes are used as eigen modes, giving a self-consistent description of the birefringence and shear strains. The simulation based on the new CME shows good agreement with the experimental data. The effect may be used to develop some sensors, and helpful in understanding the characteristics of fiber lasers incorporated with fiber Bragg gratings.

Acknowledgments

The authors want to acknowledge the support of the National Natural Science Foundation of China (Grant No. 60871067), NSAF (Grant No. 11076028) and the projects of STCSM (Grant No. 2012AA041203).

References and links

1.

L. A. Ferreira, F. M. Araujo, J. L. Santos, and F. Farahi, “Simultaneous measurement of strain and temperature using interferometrically interrogated fiber Bragg grating sensors,” Opt. Eng. 39(8), 2226–2234 (2000). [CrossRef]

2.

G. H. Chen, L. Y. Liu, H. Z. Jia, J. M. Yu, L. Xu, and W. C. Wang, “Simultaneous strain and temperature measurements with fiber Bragg grating written in novel Hi-Bi optical fiber,” IEEE Photon. Technol. Lett. 16(1), 221–223 (2004). [CrossRef]

3.

C. C. Ye, S. E. Staines, S. W. James, and R. P. Tatam, “A polarization-maintaining fibre Bragg grating interrogation system for multi-axis strain sensing,” Meas. Sci. Technol. 13(9), 1446–1449 (2002). [CrossRef]

4.

T. Mawatari and D. Nelson, “A multi-parameter Bragg grating fiber optic sensor and triaxial strain measurement,” Smart Mater. Struct. 17(3), 035033 (2008). [CrossRef]

5.

C. M. Lawrence, D. V. Nelson, E. Udd, and T. Bennett, “A fiber optic sensor for transverse strain measurement,” Exp. Mech. 39(3), 202–209 (1999). [CrossRef]

6.

C. L. Zhao, X. F. Yang, C. Lu, N. J. Hong, X. Guo, P. R. Chaudhuri, and X. Y. Dong, “Switchable multi-wavelength erbium-doped fiber lasers by using cascaded fiber Bragg gratings written in high birefringence fiber,” Opt. Commun. 230(4-6), 313–317 (2004). [CrossRef]

7.

Y. G. Liu, X. H. Feng, S. Z. Yuan, G. Y. Kai, and X. Y. Dong, “Simultaneous four-wavelength lasing oscillations in an erbium-doped fiber laser with two high birefringence fiber Bragg gratings,” Opt. Express 12(10), 2056–2061 (2004). [CrossRef] [PubMed]

8.

C. Spiegelberg, J. H. Geng, Y. D. Hu, Y. Kaneda, S. B. Jiang, and N. Peyghambarian, “Low-noise narrow-linewidth fiber laser at 1550 nm (June 2003),” J. Lightwave Technol. 22(1), 57–62 (2004). [CrossRef]

9.

F. Bosia, P. Giaccari, J. Botsis, M. Facchini, H. G. Limberger, and R. P. Salathe, “Characterization of the response of fibre Bragg grating sensors subjected to a two-dimensional strain field,” Smart Mater. Struct. 12(6), 925–934 (2003). [CrossRef]

10.

Y. P. Wang, B. F. Yun, N. Chen, and T. P. Cui, “Characterization of a high birefringence fibre Bragg grating sensor subjected to non-homogeneous transverse strain fields,” Meas. Sci. Technol. 17(4), 939–942 (2006). [CrossRef]

11.

K. O. Hill, F. Bilodeau, B. Malo, and D. C. Johnson, “Birefringent photosensitivity in monomode optical fibre: application to external writing of rocking filters,” Electron. Lett. 27(17), 1548–1550 (1991). [CrossRef]

12.

K. S. Lee and J. Y. Cho, “Polarization-mode coupling in birefringent fiber gratings,” J. Opt. Soc. Am. A 19(8), 1621–1631 (2002). [CrossRef] [PubMed]

13.

C. J. S. de Matos, P. Torres, L. C. G. Valente, W. Margulis, and R. Stubbe, “Fiber Bragg grating (FBG) characterization and shaping by local pressure,” J. Lightwave Technol. 19(8), 1206–1211 (2001). [CrossRef]

14.

J. F. Botero-Cadavid, J. D. Causado-Buelvas, and P. Torres, “Spectral properties of locally pressed fiber Bragg gratings written in polarization maintaining fibers,” J. Lightwave Technol. 28(9), 1291–1297 (2010). [CrossRef]

15.

M. S. Muller, T. C. Buck, H. J. El-Khozondar, and A. W. Koch, “Shear strain influence on fiber Bragg grating measurement systems,” J. Lightwave Technol. 27(23), 5223–5229 (2009). [CrossRef]

16.

M. S. Muller, L. Hoffmann, A. Sandmair, and A. W. Koch, “Full strain tensor treatment of fiber Bragg grating sensors,” IEEE J. Quantum Electron. 45(5), 547–553 (2009). [CrossRef]

17.

M. S. Müller and C. D. A. Schnarr, “Analytical coherency matrix treatment of shear strained fiber Bragg gratings,” Opt. Express 17(25), 22624–22631 (2009). [CrossRef] [PubMed]

18.

M. S. Müller, H. J. El-Khozondar, T. C. Buck, and A. W. Koch, “Analytical solution of four-mode coupling in shear strain loaded fiber Bragg grating sensors,” Opt. Lett. 34(17), 2622–2624 (2009). [CrossRef] [PubMed]

19.

R. Ulrich and A. Simon, “Polarization optics of twisted single-mode fibers,” Appl. Opt. 18(13), 2241–2251 (1979). [CrossRef] [PubMed]

20.

Z. Fang, K. Chin, R. Qu, and H. Cai, Fundamentals of Optical Fiber Sensors, (John Wiley & Sons, 2012), Chap. 3.

OCIS Codes
(060.0060) Fiber optics and optical communications : Fiber optics and optical communications
(060.2420) Fiber optics and optical communications : Fibers, polarization-maintaining
(060.3735) Fiber optics and optical communications : Fiber Bragg gratings

ToC Category:
Fiber Optics and Optical Communications

History
Original Manuscript: September 25, 2012
Revised Manuscript: November 8, 2012
Manuscript Accepted: December 9, 2012
Published: December 12, 2012

Citation
Fei Yang, Zujie Fang, Zhengqing Pan, Qing Ye, Haiwen Cai, and Ronghui Qu, "Orthogonal polarization mode coupling for pure twisted polarization maintaining fiber Bragg gratings," Opt. Express 20, 28839-28845 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-27-28839


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References

  1. L. A. Ferreira, F. M. Araujo, J. L. Santos, and F. Farahi, “Simultaneous measurement of strain and temperature using interferometrically interrogated fiber Bragg grating sensors,” Opt. Eng.39(8), 2226–2234 (2000). [CrossRef]
  2. G. H. Chen, L. Y. Liu, H. Z. Jia, J. M. Yu, L. Xu, and W. C. Wang, “Simultaneous strain and temperature measurements with fiber Bragg grating written in novel Hi-Bi optical fiber,” IEEE Photon. Technol. Lett.16(1), 221–223 (2004). [CrossRef]
  3. C. C. Ye, S. E. Staines, S. W. James, and R. P. Tatam, “A polarization-maintaining fibre Bragg grating interrogation system for multi-axis strain sensing,” Meas. Sci. Technol.13(9), 1446–1449 (2002). [CrossRef]
  4. T. Mawatari and D. Nelson, “A multi-parameter Bragg grating fiber optic sensor and triaxial strain measurement,” Smart Mater. Struct.17(3), 035033 (2008). [CrossRef]
  5. C. M. Lawrence, D. V. Nelson, E. Udd, and T. Bennett, “A fiber optic sensor for transverse strain measurement,” Exp. Mech.39(3), 202–209 (1999). [CrossRef]
  6. C. L. Zhao, X. F. Yang, C. Lu, N. J. Hong, X. Guo, P. R. Chaudhuri, and X. Y. Dong, “Switchable multi-wavelength erbium-doped fiber lasers by using cascaded fiber Bragg gratings written in high birefringence fiber,” Opt. Commun.230(4-6), 313–317 (2004). [CrossRef]
  7. Y. G. Liu, X. H. Feng, S. Z. Yuan, G. Y. Kai, and X. Y. Dong, “Simultaneous four-wavelength lasing oscillations in an erbium-doped fiber laser with two high birefringence fiber Bragg gratings,” Opt. Express12(10), 2056–2061 (2004). [CrossRef] [PubMed]
  8. C. Spiegelberg, J. H. Geng, Y. D. Hu, Y. Kaneda, S. B. Jiang, and N. Peyghambarian, “Low-noise narrow-linewidth fiber laser at 1550 nm (June 2003),” J. Lightwave Technol.22(1), 57–62 (2004). [CrossRef]
  9. F. Bosia, P. Giaccari, J. Botsis, M. Facchini, H. G. Limberger, and R. P. Salathe, “Characterization of the response of fibre Bragg grating sensors subjected to a two-dimensional strain field,” Smart Mater. Struct.12(6), 925–934 (2003). [CrossRef]
  10. Y. P. Wang, B. F. Yun, N. Chen, and T. P. Cui, “Characterization of a high birefringence fibre Bragg grating sensor subjected to non-homogeneous transverse strain fields,” Meas. Sci. Technol.17(4), 939–942 (2006). [CrossRef]
  11. K. O. Hill, F. Bilodeau, B. Malo, and D. C. Johnson, “Birefringent photosensitivity in monomode optical fibre: application to external writing of rocking filters,” Electron. Lett.27(17), 1548–1550 (1991). [CrossRef]
  12. K. S. Lee and J. Y. Cho, “Polarization-mode coupling in birefringent fiber gratings,” J. Opt. Soc. Am. A19(8), 1621–1631 (2002). [CrossRef] [PubMed]
  13. C. J. S. de Matos, P. Torres, L. C. G. Valente, W. Margulis, and R. Stubbe, “Fiber Bragg grating (FBG) characterization and shaping by local pressure,” J. Lightwave Technol.19(8), 1206–1211 (2001). [CrossRef]
  14. J. F. Botero-Cadavid, J. D. Causado-Buelvas, and P. Torres, “Spectral properties of locally pressed fiber Bragg gratings written in polarization maintaining fibers,” J. Lightwave Technol.28(9), 1291–1297 (2010). [CrossRef]
  15. M. S. Muller, T. C. Buck, H. J. El-Khozondar, and A. W. Koch, “Shear strain influence on fiber Bragg grating measurement systems,” J. Lightwave Technol.27(23), 5223–5229 (2009). [CrossRef]
  16. M. S. Muller, L. Hoffmann, A. Sandmair, and A. W. Koch, “Full strain tensor treatment of fiber Bragg grating sensors,” IEEE J. Quantum Electron.45(5), 547–553 (2009). [CrossRef]
  17. M. S. Müller and C. D. A. Schnarr, “Analytical coherency matrix treatment of shear strained fiber Bragg gratings,” Opt. Express17(25), 22624–22631 (2009). [CrossRef] [PubMed]
  18. M. S. Müller, H. J. El-Khozondar, T. C. Buck, and A. W. Koch, “Analytical solution of four-mode coupling in shear strain loaded fiber Bragg grating sensors,” Opt. Lett.34(17), 2622–2624 (2009). [CrossRef] [PubMed]
  19. R. Ulrich and A. Simon, “Polarization optics of twisted single-mode fibers,” Appl. Opt.18(13), 2241–2251 (1979). [CrossRef] [PubMed]
  20. Z. Fang, K. Chin, R. Qu, and H. Cai, Fundamentals of Optical Fiber Sensors, (John Wiley & Sons, 2012), Chap. 3.

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