December 15, 2013 / Vol. 38, No. 24 / OPTICS LETTERS
5401
Measuring mechanical strain and twist using helical photonic crystal fiber Xiaoming Xi,1,* Gordon K. L. Wong,1 Thomas Weiss,1 and Philip St.J. Russell1,2 1
2
Max Planck Institute for the Science of Light Guenther-Scharowsky Strasse 1, 91058 Erlangen, Germany Department of Physics, University of Erlangen-Nuremberg Guenther-Scharowsky Strasse 1, 91058 Erlangen, Germany *Corresponding author: [email protected] Received October 3, 2013; revised November 7, 2013; accepted November 12, 2013; posted November 13, 2013 (Doc. ID 198817); published December 10, 2013 Solid-core photonic crystal fiber (PCF) with a permanent helical twist exhibits dips in its transmission spectrum at certain wavelengths. These are associated with the formation of orbital angular momentum states in the cladding. Here we investigate the tuning of these states with mechanical torque and axial tension. The dip wavelengths are found to scale linearly with both axial strain and mechanical twist rate. Analysis shows that the tension-induced shift in resonance wavelength is determined both by the photoelastic effect and by the change in twist rate, while the torsion-induced wavelength shift depends only on the change in twist rate. Twisted PCF can act as an effective optically monitored torque-tension transducer, twist sensor, or strain gauge. © 2013 Optical Society of America OCIS codes: (060.2340) Fiber optics components; (060.5295) Photonic crystal fibers. http://dx.doi.org/10.1364/OL.38.005401
In recent years there has been growing interest in devices based on helically twisted optical fiber. Such fibers offer an alternative means of manipulating the loss and polarization state of guided light [1–6], and have found applications in current sensing [7–9] and suppression of higher-order modes in fiber lasers [10–12]. Some of these ideas have also been applied to photonic crystal fibers (PCFs) [13–18]. Because they offer properties that are unattainable in conventional optical fibers, PCFs are having an increasing impact in many fields of optics as well as in other scientific disciplines such as chemistry and biology [19,20]. Recently, we demonstrated that a continuously twisted solid-core PCF supports orbital angular momentum (OAM) states in the microstructured cladding [21]. These states couple topologically to the core mode, giving rise to a series of sharp dips in the transmission spectrum. In addition, in the high-transmission windows between the dips, twisted PCF exhibits optical activity via a non resonant geometrical effect that can only be understood if both spin and orbital angular momentum are considered [22,23]. It has previously been reported that twisted PCF can be used as a torsion sensor [16]. In this Letter, we show that the torsion and tension-induced shifts of resonant wavelength in a twisted PCF scale linearly with the applied twist rate and axial strain. Excellent agreement is obtained between analytical modeling, numerical simulations and experimental results. As pointed out previously, the Poynting vector of the fundamental “space-filling” mode (SM) in the cladding of an untwisted PCF points precisely along the fiber axis [21]. When the fiber is gently twisted [αΛ ≪ 1, where α is the twist rate (rad · mm−1 ) and Λ the inter-hole spacing], this SM is forced to follow a helical trajectory around the core, creating OAM and causing orbital resonances to form for certain combinations of radius, twist rate, and wavelength. Taking nSM as the refractive index vector of the SM mode along the helical path, the following orbital resonance condition can be derived by considering the round-trip phase along a closed azimuthal path [21]: 0146-9592/13/245401-04$15.00/0
λR 2πnSM ρ2 α∕jlj;
(1)
where ρ is the radius of the cladding resonance, l is an integer representing its order and λR is the resonant wavelength. This equation shows that λR depends linearly on α, shifting toward longer wavelength as α increases. Although it is difficult a priori to assign precise values to nSM and ρ, the data indicate that the product nSM ρ2 is constant for a given PCF. We are interested in how λR shifts when both torsion and tension are applied to the twisted fiber. Differentiation of Eq. (1) yields ΔλR λR0
ΔnSM Δα 2Δρ ; nSM0 α0 ρ0
(2)
where nSM0 , ρ0 , and α0 are the values in the unstrained PCF. Axial strain ε, caused by tension, will extend the fiber length and increase the twist period. Positive mechanical torque (resulting in a change in twist rate αM > 0), on the other hand, will reduce the twist period. The resulting net change in twist rate is Δα αM − α0 ε:
(3)
The next task is to work out the effects of twist and strain on nSM . When a material is stretched uniaxially, Poisson’s ratio ν determines the degree to which the transverse dimensions shrink. This yields three orthogonal components of strain, each of which acts upon the transverse electric field of the light so as to change the polarizability of the glass. The starting relationship is Δ1∕n2T −νp11 p12 p12 ε;
(4)
where nT is the refractive index “seen” by the transversely polarized light and p11 and p12 are the photoelastic coefficients for fused silica [24]. Finally, the change in ρ can be estimated from Poisson’s ratio ν, yielding Δρ −ρ0 νε: © 2013 Optical Society of America
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Since the twist rate is slow and the hollow cladding channels represent a small fraction of the total cladding area, it seems reasonable to approximate nT by nSM , so that, from Eq. (4), the change in nSM due to axial strain may be written ΔnSM
n3SM0 νp11 p12 − p12 ε: 2
(6)
Substituting Eqs. (3), (5), and (6) into Eq. (2), the net wavelength shift can be written n2 ΔλR αM − ε − 2νε ε SM0 νp11 p12 − p12 : (7) λR0 α0 2 When the first term on the right-hand side is zero, i.e., for a special combination of twist and strain, the twist period will equal its value in the unstrained PCF. Neglecting the effect of the hollow channels, this will occur when the ratio of axial tension F to torque T has the magnitude F∕T 2Y ∕Y S a2 α, where Y is Young’s modulus (∼70 kN · mm−2 ), Y S the shear modulus (∼31 kN · mm−2 ) and a the outer radius of the fiber (62.5 μm). The inset in Fig. 1 shows a scanning electron micrograph of the microstructure of the fused silica PCF used in the experiments. The hollow channels in the hexagonal cladding array had a diameter ∼1 μm and spacing Λ ≈ 3 μm. Permanent twists were induced by focusing a CO2 laser beam on to the side of the fiber (so as to heat it up to the softening point of the glass) and scanning it along the fiber at a speed of ∼1.7 mm · s−1 while continuously rotating one fiber end, the other being fixed. The whole writing process was computer-controlled. A linearly polarized supercontinuum source and an optical spectrum analyzer were used to measure the transmission spectrum. The spectra were found to be independent of the launched polarization angle. In the torsion experiments the sample was fixed rigidly between a rotation stage and a fiber holder. –5
The mechanical twist rate αM was worked out from the angle of rotation and the distance between the fixed points. To maintain the strength of the fiber, a UV-cured polyimide coating with refractive index 1.376 was applied to the twisted section. It turned out that the refractive index of the coating material had a negligible effect on the spectral response, a conclusion that we further verified by submerging an uncoated sample in various liquids, including water (n 1.33) and index-matching fluid (n 1.640). This insensitivity to the external medium occurs because the OAM state (see inset of Fig. 1) has negligible field strength outside the microstructured cladding region—a key technical advantage for structural sensing applications because it avoids the need to develop special fiber coatings. We first explored the effect of pure torsion (ε 0) on a twisted section 5 cm long with twist period 437.3 μm, i.e., α0 14.36 rad · mm−1 . We focused on a transmission dip at 803.3 nm, with an extinction ratio of ∼30 dB, as shown in Fig. 1. The axial Poynting vector distribution of the right-circularly polarized mode for this resonance is shown in the inset of Fig. 1. It was calculated using a finite element method to solve Maxwell’s equations in a helicoidal coordinate frame. Almost identical numerical mode patterns were obtained for right- and left-circular polarization states. A series of spectral responses for different values of mechanical twist are plotted in Fig. 2; a wavelength tuning range of ∼16 nm was achieved. Higher twist rates were not applied to avoid damage to the fiber. The response was observed to be highly reversible and repeatable for 360° rotation over the 5 cm sample length. The shift in resonant wavelength is plotted against αM in Fig. 3, showing an almost perfect linear relationship with a twist sensitivity of 56 nm · mm · rad−1 . The results calculated both by finite element modeling (dashed line) and by fitting to the measurements using Eq. (7) with jnSM ρ2 ∕lj 8.9 μm2 , jlj 6, and ε 0 (solid line) are also shown. The agreement is excellent, showing that, as predicted by Eq. (7), the photoelastic effect has a negligible effect on the spectral tuning in the presence of pure torsion.
transmission (dB)
–10 –15 –20
0
= 14.4 rad·mm-1
0
0.2 0.4 0.6 0.8
1
–25 –30 –35 –40 5 µm
–45
760
780
800
820
840
860
wavelength (nm) Fig. 1. Measured transmission spectrum of a 5 cm long PCF with a static twist rate of 14.4 rad · mm−1 . Insets: scanning electron micrograph of the PCF structure (left) and calculated axial Poynting vector distribution of the right-circularly polarized OAM cladding resonance at 803.3 nm (right).
Fig. 2. Measured transmission spectra of the PCF in Fig. 1 for different values of mechanical twists (in steps of 0.0345 rad · mm−1 ). The original spectrum is shown in gray.
December 15, 2013 / Vol. 38, No. 24 / OPTICS LETTERS
Fig. 3. Measured shift in resonance wavelength as a function of mechanical twist rate αM (colored dots). The solid line was plotted using Eq. (7) with jnSM ρ2 ∕lj 8.9 μm2 and jlj 6. The dashed line shows the results of finite element modeling.
In addition to a linear response, which is important when a sensing device is to be used in real-world applications, twisted PCF can distinguish negative from positive twist without the need for pre-twisting and is capable of resolving very small twists. For the resonances treated in this Letter, finite element modeling indicates that the exponential amplitude decay rate of the cladding resonances (γ cl m−1 ) exceeds twice their rate of coupling (κ m−1 ) to the core mode. This has the interesting consequence that the transmission dips show an asymptotic length-dependence, becoming exponentially deeper following the relationship (from coupled-mode theory): q exp L −γ cl γ 2cl − 4κ2 ;
8
where L is the length of the twisted section. The decay rate approaches zero for γ cl ∕2κ ≫ 1 because coupling to the cladding resonance becomes insignificant. It is greatest when γ cl ∕2κ is close to 1, resulting in dip centers that can be more and more accurately located as L increases. Even higher accuracy is possible if several dips are monitored at the same time. We studied next the effects of superimposed tension and torsion. A second sample was prepared with a twist period of 434.8 μm (α0 14.45 rad · mm−1 ) and a 4 cm long twisted section, exhibiting a transmission dip at 719.5 nm. The protective coating of the fiber was removed to avoid disturbing the strain measurement. The set-up in the previous experiment was modified by adding a precision motorized linear stage and a commercial axial force sensor. Axial strain ε was applied to the twisted section by stretching the PCF using the linear stage until the desired tension (F πa2 Y ε) was reached. The sample was first subjected to increasing values of αM up to 0.045 rad · mm−1 , causing the measured dip wavelength to shift by 2.2 nm, to 721.7 nm [Fig. 4(a)]. This corresponds to a twist sensitivity of ∼49 nm · mm · rad−1 . The resonance exhibits a smaller twist sensitivity
5403
Fig. 4. Measured shift in resonance wavelength as a function of net change in twist rate Δα when the PCF was (a) subjected to mechanical twist at zero strain followed (b) by axial strain, keeping the twist angle between the fiber ends constant. The colored circles are experimental measurements and the solid lines are based on Eq. (7) with jnSM ρ2 ∕lj 7.9 μm2 and jlj 7. The blue dashed lines represent the net mechanical twist rate during this procedure.
because it has order jlj 7 and a different PCF was used. Keeping the twist angle constant and using the linear stage, the fiber was then subjected to axial tension up to a value of ∼1700 μstrain, when the dip was observed to return linearly to its original value of 719.5 nm [Fig. 4(b)] with a strain sensitivity (obtained by fitting to the data) of 1.18 pm · μstrain−1 . Equation (7) predicts 1.094 pm · μstrain−1 for p11 0.113, p12 0.252, ν 0.16 [25], and nSM 1.44. This is in excellent agreement, considering uncertainties in the values of the stress-optical coefficients, ν and nSM . Also plotted in Fig. 4 is the net mechanical twist rate αM − α0 ε throughout this procedure. If the fiber is stretched so that αM α0 ε, the twist rate will again equal α0 , but the resonance will have shifted to a shorter wavelength as a result of stress-optical effects. Equation (7) predicts a wavelength shift of ΔλR −1.17 nm, very close to the experimental value [Fig. 4(b)]. Since the twist rate can be tuned by both axial strain and mechanical twist, the PCF can act as a transducer between tension and torque, offering an all-optical means of setting a precise twist angle simply by measuring the tension, or vice-versa. According to Eq. (7), the transducer could be operated as follows. If a certain twist angle ϕreq is required, one simply applies the correct axial tension to ensure that ε
2ϕreq ; α0 Ln2SM0 p12 − νp11 p12 4ν
(9)
where L is the length of the twisted section. One then twists the fiber until the dips in the spectrum have returned to their original positions, i.e., ΔλR 0. A similar procedure could be used to set the linear strain to a required value εreq . In this case, one would set the mechanical twist rate to the value n2 αM α0 εreq 1 2ν SM0 p12 − νp11 p12 2
(10)
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and then stretch the fiber until the dips have again returned to their original positions. We have previously reported that the twisted PCFs exhibit high-quality circular birefringence in the spectral regions between the dips [22]. This means that the plane of polarization of linearly polarized light rotates as the light propagates along the fiber. By adding a mechanical twist, the polarization state at the output can be finely adjusted, which could be useful in polarization control. In conclusion, twisted PCFs offer unique opportunities as angle and strain sensors as well as providing a promising platform for new developments in optical-fiber components. References 1. R. D. Birch, Electron. Lett. 23, 50 (1987). 2. C. D. Poole, C. D. Townsend, and K. T. Nelson, J. Lightwave Technol. 9, 598 (1991). 3. S. Oh, K. R. Lee, U.-C. Paek, and Y. Chung, Opt. Lett. 29, 1464 (2004). 4. V. I. Kopp, V. M. Churikov, J. Singer, N. Chao, D. Neugroschl, and A. Z. Genack, Science 305, 74 (2004). 5. C. Jáuregui and J. M. López-Higuera, Opt. Lett. 30, 14 (2005). 6. O. V. Ivanov, Opt. Lett. 30, 3290 (2005). 7. R. I. Laming and D. N. Payne, J. Lightwave Technol. 7, 2084 (1989). 8. J. R. Qian, Q. Guo, and L. Li, IEE Proc. Optoelectron. 141, 373 (1994). 9. V. P. Gubin, V. A. Isaev, S. K. Morshnev, A. I. Sazonov, N. I. Starostin, Y. K. Chamorovskii, and A. I. Usov, Quantum Electron. 36, 287 (2006). 10. G. T. Moore and J. R. Marciante, “Cladding-pumped fiber with helical rare-earth-doped core for fiber lasers and amplifiers,” U.S. patent 6,650,664 (November 18, 2003).
11. P. Wang, L. J. Cooper, W. A. Clarkson, J. Nilsson, R. B. Williams, J. Sahu, and A. K. Vogel, Electron. Lett. 40, 1325 (2004). 12. S. Lefrancois, T. S. Sosnowski, C.-H. Liu, A. Galvanauskas, and F. W. Wise, Opt. Express 19, 3464 (2011). 13. M. Fuochi, J. Hayes, K. Furusawa, W. Belardi, J. Baggett, T. Monro, and D. Richardson, Opt. Express 12, 1972 (2004). 14. A. Michie, J. Canning, I. Bassett, J. Haywood, K. Digweed, M. Åslund, B. Ashton, M. Stevenson, J. Digweed, A. Lau, and D. Scandurra, Opt. Express 15, 1811 (2007). 15. A. Argyros, J. Pla, F. Ladouceur, and L. Poladian, Opt. Express 17, 15983 (2009). 16. W. Shin, Y. L. Lee, B. A. Yu, Y. C. Noh, and K. Oh, Opt. Commun. 282, 3456 (2009). 17. Y. K. Chamorovsky, N. I. Starostin, S. K. Morshnev, V. P. Gubin, M. V. Ryabko, A. I. Sazonov, and I. L. Vorob’ev, Quantum Electron. 39, 1074 (2009). 18. V. M. Churikov, V. I. Kopp, and A. Z. Genack, Opt. Lett. 35, 342 (2010). 19. P. St.J. Russell, J. Lightwave Technol 24, 4729 (2006). 20. A. M. Cubillas, S. Unterkofler, T. G. Euser, B. J. M. Etzold, A. C. Jones, P. J. Sadler, P. Wasserscheid, and P. St.J. Russell, Chem. Soc. Rev. 42, 8629 (2013). 21. G. K. L. Wong, M. S. Kang, H. W. Lee, F. Biancalana, C. Conti, T. Weiss, and P. St.J. Russell, Science 337, 446 (2012). 22. X. M. Xi, T. Weiss, G. K. L. Wong, F. Biancalana, S. M. Barnett, M. J. Padgett, and P. St.J. Russell, Phys. Rev. Lett. 110, 143903 (2013). 23. T. Weiss, G. K. L. Wong, F. Biancalana, S. M. Barnett, X. M. Xi, and P. St.J. Russell, J. Opt. Soc. Am. B 30, 2921 (2013). 24. J. F. Nye, Physical Properties of Crystals (Oxford, 1985). 25. A. Bertholds and R. Dandliker, J. Lightwave Technol. 6, 17 (1988).
5401
Measuring mechanical strain and twist using helical photonic crystal fiber Xiaoming Xi,1,* Gordon K. L. Wong,1 Thomas Weiss,1 and Philip St.J. Russell1,2 1
2
Max Planck Institute for the Science of Light Guenther-Scharowsky Strasse 1, 91058 Erlangen, Germany Department of Physics, University of Erlangen-Nuremberg Guenther-Scharowsky Strasse 1, 91058 Erlangen, Germany *Corresponding author: [email protected] Received October 3, 2013; revised November 7, 2013; accepted November 12, 2013; posted November 13, 2013 (Doc. ID 198817); published December 10, 2013 Solid-core photonic crystal fiber (PCF) with a permanent helical twist exhibits dips in its transmission spectrum at certain wavelengths. These are associated with the formation of orbital angular momentum states in the cladding. Here we investigate the tuning of these states with mechanical torque and axial tension. The dip wavelengths are found to scale linearly with both axial strain and mechanical twist rate. Analysis shows that the tension-induced shift in resonance wavelength is determined both by the photoelastic effect and by the change in twist rate, while the torsion-induced wavelength shift depends only on the change in twist rate. Twisted PCF can act as an effective optically monitored torque-tension transducer, twist sensor, or strain gauge. © 2013 Optical Society of America OCIS codes: (060.2340) Fiber optics components; (060.5295) Photonic crystal fibers. http://dx.doi.org/10.1364/OL.38.005401
In recent years there has been growing interest in devices based on helically twisted optical fiber. Such fibers offer an alternative means of manipulating the loss and polarization state of guided light [1–6], and have found applications in current sensing [7–9] and suppression of higher-order modes in fiber lasers [10–12]. Some of these ideas have also been applied to photonic crystal fibers (PCFs) [13–18]. Because they offer properties that are unattainable in conventional optical fibers, PCFs are having an increasing impact in many fields of optics as well as in other scientific disciplines such as chemistry and biology [19,20]. Recently, we demonstrated that a continuously twisted solid-core PCF supports orbital angular momentum (OAM) states in the microstructured cladding [21]. These states couple topologically to the core mode, giving rise to a series of sharp dips in the transmission spectrum. In addition, in the high-transmission windows between the dips, twisted PCF exhibits optical activity via a non resonant geometrical effect that can only be understood if both spin and orbital angular momentum are considered [22,23]. It has previously been reported that twisted PCF can be used as a torsion sensor [16]. In this Letter, we show that the torsion and tension-induced shifts of resonant wavelength in a twisted PCF scale linearly with the applied twist rate and axial strain. Excellent agreement is obtained between analytical modeling, numerical simulations and experimental results. As pointed out previously, the Poynting vector of the fundamental “space-filling” mode (SM) in the cladding of an untwisted PCF points precisely along the fiber axis [21]. When the fiber is gently twisted [αΛ ≪ 1, where α is the twist rate (rad · mm−1 ) and Λ the inter-hole spacing], this SM is forced to follow a helical trajectory around the core, creating OAM and causing orbital resonances to form for certain combinations of radius, twist rate, and wavelength. Taking nSM as the refractive index vector of the SM mode along the helical path, the following orbital resonance condition can be derived by considering the round-trip phase along a closed azimuthal path [21]: 0146-9592/13/245401-04$15.00/0
λR 2πnSM ρ2 α∕jlj;
(1)
where ρ is the radius of the cladding resonance, l is an integer representing its order and λR is the resonant wavelength. This equation shows that λR depends linearly on α, shifting toward longer wavelength as α increases. Although it is difficult a priori to assign precise values to nSM and ρ, the data indicate that the product nSM ρ2 is constant for a given PCF. We are interested in how λR shifts when both torsion and tension are applied to the twisted fiber. Differentiation of Eq. (1) yields ΔλR λR0
ΔnSM Δα 2Δρ ; nSM0 α0 ρ0
(2)
where nSM0 , ρ0 , and α0 are the values in the unstrained PCF. Axial strain ε, caused by tension, will extend the fiber length and increase the twist period. Positive mechanical torque (resulting in a change in twist rate αM > 0), on the other hand, will reduce the twist period. The resulting net change in twist rate is Δα αM − α0 ε:
(3)
The next task is to work out the effects of twist and strain on nSM . When a material is stretched uniaxially, Poisson’s ratio ν determines the degree to which the transverse dimensions shrink. This yields three orthogonal components of strain, each of which acts upon the transverse electric field of the light so as to change the polarizability of the glass. The starting relationship is Δ1∕n2T −νp11 p12 p12 ε;
(4)
where nT is the refractive index “seen” by the transversely polarized light and p11 and p12 are the photoelastic coefficients for fused silica [24]. Finally, the change in ρ can be estimated from Poisson’s ratio ν, yielding Δρ −ρ0 νε: © 2013 Optical Society of America
(5)
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OPTICS LETTERS / Vol. 38, No. 24 / December 15, 2013
Since the twist rate is slow and the hollow cladding channels represent a small fraction of the total cladding area, it seems reasonable to approximate nT by nSM , so that, from Eq. (4), the change in nSM due to axial strain may be written ΔnSM
n3SM0 νp11 p12 − p12 ε: 2
(6)
Substituting Eqs. (3), (5), and (6) into Eq. (2), the net wavelength shift can be written n2 ΔλR αM − ε − 2νε ε SM0 νp11 p12 − p12 : (7) λR0 α0 2 When the first term on the right-hand side is zero, i.e., for a special combination of twist and strain, the twist period will equal its value in the unstrained PCF. Neglecting the effect of the hollow channels, this will occur when the ratio of axial tension F to torque T has the magnitude F∕T 2Y ∕Y S a2 α, where Y is Young’s modulus (∼70 kN · mm−2 ), Y S the shear modulus (∼31 kN · mm−2 ) and a the outer radius of the fiber (62.5 μm). The inset in Fig. 1 shows a scanning electron micrograph of the microstructure of the fused silica PCF used in the experiments. The hollow channels in the hexagonal cladding array had a diameter ∼1 μm and spacing Λ ≈ 3 μm. Permanent twists were induced by focusing a CO2 laser beam on to the side of the fiber (so as to heat it up to the softening point of the glass) and scanning it along the fiber at a speed of ∼1.7 mm · s−1 while continuously rotating one fiber end, the other being fixed. The whole writing process was computer-controlled. A linearly polarized supercontinuum source and an optical spectrum analyzer were used to measure the transmission spectrum. The spectra were found to be independent of the launched polarization angle. In the torsion experiments the sample was fixed rigidly between a rotation stage and a fiber holder. –5
The mechanical twist rate αM was worked out from the angle of rotation and the distance between the fixed points. To maintain the strength of the fiber, a UV-cured polyimide coating with refractive index 1.376 was applied to the twisted section. It turned out that the refractive index of the coating material had a negligible effect on the spectral response, a conclusion that we further verified by submerging an uncoated sample in various liquids, including water (n 1.33) and index-matching fluid (n 1.640). This insensitivity to the external medium occurs because the OAM state (see inset of Fig. 1) has negligible field strength outside the microstructured cladding region—a key technical advantage for structural sensing applications because it avoids the need to develop special fiber coatings. We first explored the effect of pure torsion (ε 0) on a twisted section 5 cm long with twist period 437.3 μm, i.e., α0 14.36 rad · mm−1 . We focused on a transmission dip at 803.3 nm, with an extinction ratio of ∼30 dB, as shown in Fig. 1. The axial Poynting vector distribution of the right-circularly polarized mode for this resonance is shown in the inset of Fig. 1. It was calculated using a finite element method to solve Maxwell’s equations in a helicoidal coordinate frame. Almost identical numerical mode patterns were obtained for right- and left-circular polarization states. A series of spectral responses for different values of mechanical twist are plotted in Fig. 2; a wavelength tuning range of ∼16 nm was achieved. Higher twist rates were not applied to avoid damage to the fiber. The response was observed to be highly reversible and repeatable for 360° rotation over the 5 cm sample length. The shift in resonant wavelength is plotted against αM in Fig. 3, showing an almost perfect linear relationship with a twist sensitivity of 56 nm · mm · rad−1 . The results calculated both by finite element modeling (dashed line) and by fitting to the measurements using Eq. (7) with jnSM ρ2 ∕lj 8.9 μm2 , jlj 6, and ε 0 (solid line) are also shown. The agreement is excellent, showing that, as predicted by Eq. (7), the photoelastic effect has a negligible effect on the spectral tuning in the presence of pure torsion.
transmission (dB)
–10 –15 –20
0
= 14.4 rad·mm-1
0
0.2 0.4 0.6 0.8
1
–25 –30 –35 –40 5 µm
–45
760
780
800
820
840
860
wavelength (nm) Fig. 1. Measured transmission spectrum of a 5 cm long PCF with a static twist rate of 14.4 rad · mm−1 . Insets: scanning electron micrograph of the PCF structure (left) and calculated axial Poynting vector distribution of the right-circularly polarized OAM cladding resonance at 803.3 nm (right).
Fig. 2. Measured transmission spectra of the PCF in Fig. 1 for different values of mechanical twists (in steps of 0.0345 rad · mm−1 ). The original spectrum is shown in gray.
December 15, 2013 / Vol. 38, No. 24 / OPTICS LETTERS
Fig. 3. Measured shift in resonance wavelength as a function of mechanical twist rate αM (colored dots). The solid line was plotted using Eq. (7) with jnSM ρ2 ∕lj 8.9 μm2 and jlj 6. The dashed line shows the results of finite element modeling.
In addition to a linear response, which is important when a sensing device is to be used in real-world applications, twisted PCF can distinguish negative from positive twist without the need for pre-twisting and is capable of resolving very small twists. For the resonances treated in this Letter, finite element modeling indicates that the exponential amplitude decay rate of the cladding resonances (γ cl m−1 ) exceeds twice their rate of coupling (κ m−1 ) to the core mode. This has the interesting consequence that the transmission dips show an asymptotic length-dependence, becoming exponentially deeper following the relationship (from coupled-mode theory): q exp L −γ cl γ 2cl − 4κ2 ;
8
where L is the length of the twisted section. The decay rate approaches zero for γ cl ∕2κ ≫ 1 because coupling to the cladding resonance becomes insignificant. It is greatest when γ cl ∕2κ is close to 1, resulting in dip centers that can be more and more accurately located as L increases. Even higher accuracy is possible if several dips are monitored at the same time. We studied next the effects of superimposed tension and torsion. A second sample was prepared with a twist period of 434.8 μm (α0 14.45 rad · mm−1 ) and a 4 cm long twisted section, exhibiting a transmission dip at 719.5 nm. The protective coating of the fiber was removed to avoid disturbing the strain measurement. The set-up in the previous experiment was modified by adding a precision motorized linear stage and a commercial axial force sensor. Axial strain ε was applied to the twisted section by stretching the PCF using the linear stage until the desired tension (F πa2 Y ε) was reached. The sample was first subjected to increasing values of αM up to 0.045 rad · mm−1 , causing the measured dip wavelength to shift by 2.2 nm, to 721.7 nm [Fig. 4(a)]. This corresponds to a twist sensitivity of ∼49 nm · mm · rad−1 . The resonance exhibits a smaller twist sensitivity
5403
Fig. 4. Measured shift in resonance wavelength as a function of net change in twist rate Δα when the PCF was (a) subjected to mechanical twist at zero strain followed (b) by axial strain, keeping the twist angle between the fiber ends constant. The colored circles are experimental measurements and the solid lines are based on Eq. (7) with jnSM ρ2 ∕lj 7.9 μm2 and jlj 7. The blue dashed lines represent the net mechanical twist rate during this procedure.
because it has order jlj 7 and a different PCF was used. Keeping the twist angle constant and using the linear stage, the fiber was then subjected to axial tension up to a value of ∼1700 μstrain, when the dip was observed to return linearly to its original value of 719.5 nm [Fig. 4(b)] with a strain sensitivity (obtained by fitting to the data) of 1.18 pm · μstrain−1 . Equation (7) predicts 1.094 pm · μstrain−1 for p11 0.113, p12 0.252, ν 0.16 [25], and nSM 1.44. This is in excellent agreement, considering uncertainties in the values of the stress-optical coefficients, ν and nSM . Also plotted in Fig. 4 is the net mechanical twist rate αM − α0 ε throughout this procedure. If the fiber is stretched so that αM α0 ε, the twist rate will again equal α0 , but the resonance will have shifted to a shorter wavelength as a result of stress-optical effects. Equation (7) predicts a wavelength shift of ΔλR −1.17 nm, very close to the experimental value [Fig. 4(b)]. Since the twist rate can be tuned by both axial strain and mechanical twist, the PCF can act as a transducer between tension and torque, offering an all-optical means of setting a precise twist angle simply by measuring the tension, or vice-versa. According to Eq. (7), the transducer could be operated as follows. If a certain twist angle ϕreq is required, one simply applies the correct axial tension to ensure that ε
2ϕreq ; α0 Ln2SM0 p12 − νp11 p12 4ν
(9)
where L is the length of the twisted section. One then twists the fiber until the dips in the spectrum have returned to their original positions, i.e., ΔλR 0. A similar procedure could be used to set the linear strain to a required value εreq . In this case, one would set the mechanical twist rate to the value n2 αM α0 εreq 1 2ν SM0 p12 − νp11 p12 2
(10)
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OPTICS LETTERS / Vol. 38, No. 24 / December 15, 2013
and then stretch the fiber until the dips have again returned to their original positions. We have previously reported that the twisted PCFs exhibit high-quality circular birefringence in the spectral regions between the dips [22]. This means that the plane of polarization of linearly polarized light rotates as the light propagates along the fiber. By adding a mechanical twist, the polarization state at the output can be finely adjusted, which could be useful in polarization control. In conclusion, twisted PCFs offer unique opportunities as angle and strain sensors as well as providing a promising platform for new developments in optical-fiber components. References 1. R. D. Birch, Electron. Lett. 23, 50 (1987). 2. C. D. Poole, C. D. Townsend, and K. T. Nelson, J. Lightwave Technol. 9, 598 (1991). 3. S. Oh, K. R. Lee, U.-C. Paek, and Y. Chung, Opt. Lett. 29, 1464 (2004). 4. V. I. Kopp, V. M. Churikov, J. Singer, N. Chao, D. Neugroschl, and A. Z. Genack, Science 305, 74 (2004). 5. C. Jáuregui and J. M. López-Higuera, Opt. Lett. 30, 14 (2005). 6. O. V. Ivanov, Opt. Lett. 30, 3290 (2005). 7. R. I. Laming and D. N. Payne, J. Lightwave Technol. 7, 2084 (1989). 8. J. R. Qian, Q. Guo, and L. Li, IEE Proc. Optoelectron. 141, 373 (1994). 9. V. P. Gubin, V. A. Isaev, S. K. Morshnev, A. I. Sazonov, N. I. Starostin, Y. K. Chamorovskii, and A. I. Usov, Quantum Electron. 36, 287 (2006). 10. G. T. Moore and J. R. Marciante, “Cladding-pumped fiber with helical rare-earth-doped core for fiber lasers and amplifiers,” U.S. patent 6,650,664 (November 18, 2003).
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