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Evanescent field characteristics of eccentric core optical fiber for distributed sensing Jianxia Liu and Libo Yuan* Key Laboratory of In-Fiber Integrated Optics, Ministry of Education, College of Science, Harbin Engineering University, Harbin 150001, China *Corresponding author: [email protected] Received November 1, 2013; revised December 21, 2013; accepted December 22, 2013; posted December 24, 2013 (Doc. ID 200564); published February 5, 2014 Fundamental core-mode cutoff and evanescent field are considered for an eccentric core optical fiber (ECOF). A method has been proposed to calculate the core-mode cutoff by solving the eigenvalue equations of an ECOF. Using conformal mapping, the asymmetric geometrical structure can be transformed into a simple, easily solved axisymmetric optical fiber with three layers. The variation of the fundamental core-mode cut-off frequency (V c ) is also calculated with different eccentric distances, wavelengths, core radii, and coating refractive indices. The fractional power of evanescent fields for ECOF is also calculated with the eccentric distances and coating refractive indices. These calculations are necessary to design the structural parameters of an ECOF for longdistance, single-mode distributed evanescent field absorption sensors. © 2014 Optical Society of America OCIS codes: (130.3120) Integrated optics devices; (260.2110) Electromagnetic optics; (280.4788) Optical sensing and sensors; (350.7420) Waves. http://dx.doi.org/10.1364/JOSAA.31.000475
1. INTRODUCTION Optical fibers can be used in evanescent wave absorption sensing applications, due to the fraction of modal power extended into the external environment as an evanescent wave field. This sensing technology has been widely applied in chemistry, biology, and environmental monitoring, food, and hygiene detection in recent years [1–3]. In order to obtain the evanescent fields, multimode fibers [4] and microfibers and nanofibers [5,6] have been proposed. Fiber tapering [7] and side polishing [8,9] techniques have been reported. But evanescent-field-absorption-based multimode fiber sensors have low sensitivity due to weak evanescent field energy. Although the microfiber and nanofiber sensors improved the sensing sensitivity, it is difficult in distributed sensing and harsh environment monitoring due to the small fiber diameter. Fiber tapering and side polishing techniques destroy the cladding and coating of the fiber to a certain extent, thus reducing the mechanical properties of the optical waveguide. Recently, there has been widespread activity based on microstructure optical fibers in which evanescent field sensing has been strongly exploited [10,11]. Eccentric core optical fiber (ECOF) is a kind of special fiber in which the fiber core is eccentrically positioned and has some special characteristics due to the asymmetry of the cross section. Some characteristics of ECOF without the coating [12,13] were analyzed, as the fundamental core mode must be operated below a certain wavelength for efficient transmission. The fundamental core-mode cutoff must be considered, due to the severely asymmetrical structure. In this paper, the asymmetric geometrical structure is transformed into a simple, easily solved three-layer axisymmetric optical waveguide using conformal mapping. By solving the eigenvalue equations of an ECOF, the variation 1084-7529/14/030475-05$15.00/0
of the fundamental core-mode cut-off frequency (V c ) is calculated with different eccentric distances, wavelengths, core radii, and coating refractive indices. The fractional power of the evanescent fields for the ECOF is also calculated with the eccentric distances and coating refractive indices. The result is important for designing the structural parameters of the ECOF for long-distance, single-mode distributed evanescent field absorption sensors. A strong evanescent field exits in the coating of the ECOF due to a small distance between the core and coating. ECOF is a promising candidate for distributed evanescent field sensors and can be applied in harsh environment monitoring.
2. EIGENVALUE EQUATIONS OF ECCENTRIC CORE OPTICAL FIBER A. Eccentric Core Optical Fiber The optical fiber preform was fabricated using ultrasonic drilling or side-grooving technology. With ultrasonic drilling, for example, first a silica glass preform was drilled using a mechanical ultrasonic drill to form an eccentric hole, and then the hole was filled by a higher index glass rod. The optical fiber fabricated using ultrasonic drilling and side-grooving technology by our research group is shown in Fig. 1. The index difference Δn n1 − n2 is 0.005 at λ 0.65 μm. The radius of the fiber core is 5.75 μm, and the eccentric distance is 4.6 μm for this sample. The optical fiber sample contains multimode fibers at λ 1.55 μm due to the large core size. B. Eigenvalue Equations of Eccentric Core Optical Fiber The core of eccentric core fiber does not lie in the center but remains circular. Experience shows that the performance of this fiber does not cause it to be eccentric core dependent as long as the eccentric distance is reasonably large. But as a rule © 2014 Optical Society of America
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Fig. 1. Cross-sectional view of the ECOF.
of thumb, a few micrometers is often sufficient for evanescent field sensing. Although this concentric fiber model is different from the eccentric core fiber, the power of an electromagnetic wave should transmit mainly in the core. The problem now is to seek a simple solution for this asymmetric waveguide geometry. If we regard the eccentric core fiber (see Fig. 2) as a threelayer cylindrical optical fiber, using conformal mapping the asymmetric geometrical structure can be transformed into a simple, easily solved structure. The z plane represents the original coordinate z x; y, and the ζ plane represents the transformed new plane coordinate ζ η; ζ. The ill-fitted irregular geometry on the z plane—three eccentric circles— may be regulated to be three concentric circles on a new ζ plane through conformal mapping (see Fig. 3). Zero-order approximation is adequate to ensure the method is valid [14,15], and the Helmholtz wave equation is changed to the following: ∇2
1 −
y1 ∕y2 k20 n2
−
β2 ψ
0:
(1)
Here y1 and y2 are symmetric points and are defined by
y1 y2 R22 ; y1 − Ly2 − L R20
2
where L is the distance deviated from the coordinate origin. R0 and R2 are the radii of the core and cladding on the z plane, respectively. If all phase parameters are scaled by a factor of (1 − y1 ∕y2 ), the solution of Eq. (1) is written as [16,17]
Fig. 3. Three concentric circles mapped on the ζ plane.
ψ 1 AJ m u1 r∕R1 cosmϕ BI m w2 r∕R1 CK m w2 r∕R1 cosmϕ β > k0 n2 ; ψ2 BJ m u2 r∕R1 CY m u2 r∕R1 cosmϕ β < k0 n2 ψ 3 DK m w3 r∕R2 cosmϕ 3 where A, B, C, and D are constants and R1 and R2 are radii of inner and outer circles, respectively, on the ζ plane. R1 is defined by R1 R0 jy2 ∕L − y2 j, and u1 R1 k20 n21 − β2 1∕2 1 − y1 ∕y2 u2 R1 k20 n22 − β2 1∕2 1 − y1 ∕y2 w2 R1 β2 − k20 n22 1∕2 1 − y1 ∕y2
;
4
w3 R2 β2 − k20 n23 1∕2 1 − y1 ∕y2 where n1 , n2 , and n3 are the refractive indices of the core, cladding, and coating, respectively. So far, for any given eccentric core fiber design, it is always possible to work out the equivalent concentric structure using conformal mapping. Before the core-mode cutoff (β > k0 n2 ), the continuity of ψ and its derivative at the core–cladding and cladding– coating interfaces (boundary conditions) gives the eigenvalue equation J 0m u1 K 0m w3 w2 J 0m u1 pm − r R2 J m u1 K m w3 w3 R1 J m u1 m
w22 w2 K 0m w3 s ; qm − u1 R2 K m w3 u1 w3 R1 m
(5)
where qm , r m , and sm are Bessel cross-product functions [18] and are defined by pm I m w2 RK m w2 − I m w2 K m w2 R; qm I m w2 RK 0m w2 − I 0m w2 K m w2 R; r m I 0m w2 RK m w2 − I m w2 K 0m w2 R;
6
sm I 0m w2 RK 0m w2 − I 0m w2 K 0m w2 R;
Fig. 2.
Cross section of eccentric core fiber on the z plane.
where R is the cladding-to-core radius ratio (R2 ∕R1 ). By solving Eq. (5), the propagation constant β can be obtained. When n1 1.4510, n2 1.4468, n3 1air, m 0,
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R0 4.5 μm, R2 62.5 μm, k 1 μm, and β 5868076∕m, the neff corresponding to the solution is 1.4470.
3. FUNDAMENTAL CORE-MODE CUTOFF The significance of the fundamental core-mode cutoff is to define a wavelength that one should operate well below for efficient transmission. The solution to the eigenvalue equation has been obtained for different eccentric distances, wavelengths, and fiber radii up to the core-mode cutoff. When there is no solution to the eigenvalue equation, the LP01 mode is cut off from the core. The value of V at the cutoff is denoted as V c 1.2616 when k 1 μm. Obviously, the fundamental mode has a nonzero cut-off frequency, because the light is not confined in the fiber core and completely leaks into the cladding when V is small [12]. But this is an approximate value, and it changes with eccentric distances, wavelengths, coating refractive indices, and the radius of the fiber core. So considering the condition of fundamental core-mode cutoff neff ncl n2 , Eq. (5) is rebuilt: J 00 V c K 00 w3 J 00 V c K 00 w3 ln R − 0: R2 J 0 V c K 0 w3 w3 R1 RJ 0 V c V c R2 K 0 w3 (7) The accurate value of V c can be calculated by solving Eq. (7) [19]. A plot of V c with eccentric distances, wavelengths, coating refractive indices, and core radii are shown in Figs. 4(a)–4(d), respectively. At the same time, the fundamental core-mode cutoff (V c ) of the axisymmetric optical fiber with three layers and the wavelengths and coating refractive indices are also considered and shown in Figs. 4(b) and 4(c). When the eccentric distance increases and the radius of the fiber core decreases, the fundamental core-mode cut-off frequency V c will decrease and then approach zero. The absorption wavelengths and normalized frequencies (V ) of some common gases are listed in Table 1. V is the normalized frequency given by
V
2π · R0 · λ
q n21 − n22 ;
(8)
where n1 1.4510, n2 1.4468, and R0 4.5 μm. The V corresponding to the absorption wavelength of these gases in Table 1 is between V c (see Fig. 4) and 2.405, except for oxygen (O2 ) and nitrogen dioxide (NO2 ). It is can be forecasted that the value of V will be smaller than V c with the increasing absorption wavelength of the gas. This means that the ECOF will contain multimode fibers when the absorption wavelength of the gas is short, and the fundamental core mode of the ECOF will be cut off when the absorption wavelength of the gas is long, in order to ensure the light energy can transmit efficiently in the ECOF for a single mode. The values of Y are calculated and listed in the Table 1. Y is denoted as q Y R0 · n21 − n22 :
(9)
Fig. 4. Plot of V c versus (a) eccentric distance, (b) wavelength, (c) refractive index of the coating, and (d) core radius for different eccentric distances.
4. EVANESCENT FIELD In this section, we consider only the coating region. Here the single-mode condition corresponds to a wavelength of
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Table 1. Normalized Frequency and Y-Values Corresponding to Absorption Wavelength of Gases [20,21]
Gas Acetylene (C2 H2 ) Hydrogen iodide (HI) Ammonia (NH3 ) Carbon dioxide (CO2 ) Hydrogen sulfide (H2 S) Methane (CH4 ) Hydrogen fluoride (HF) Hydrogen bromide (HBr) Water (H2 O) Oxygen (O2 ) Nitrogen dioxide (NO2 )
Absorption Wavelength (nm)
Normalized Frequency (V)
Y R0 δ
1533
2.033
0.37 < Y < 0.58
1541
2.023
0.37 < Y < 0.59
1544 1567
2.019 1.989
0.37 < Y < 0.59 0.38 < Y < 0.60
1578
1.975
0.38 < Y < 0.60
1667 1330
1.87 2.344
0.40 < Y < 0.63 0.32 < Y < 0.50
1341
2.324
0.32 < Y < 0.51
1365 761 800
2.284 4.096 3.897
0.32 < Y < 0.52 0.18 < Y < 0.29 0.19 < Y < 0.30
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1.55 μm, and the refractive indices (RIs) of the core, cladding, and coating are 1.4510, 1.4468, and 1, respectively. Considering these RIs and the wavelength, the radius of the fiber core for the single mode is calculated as 4.5 μm. Under these conditions electromagnetic energy is propagating mainly in the fiber core, and the evanescent field will leak into the coating. The fractional power of the evanescent field in the coating can be expressed as η
P coating n3 R22 K 21 w3 R2 − K 20 w3 R2 ; P total 2K 20 w3 R2 P total
(10)
where P coating is the power in the coating, P total is the total power, and w3 is the modal parameter. The fractional power of the evanescent field in the coating as a function of the eccentric distance and coating refractive index is shown in Figs. 5(a) and 5(b), respectively. It can be observed that if the eccentric distance is 1 μm at n3 1, the fractional power of evanescent field in the coating is 1.132%. Hence, if the eccentric distance is much larger, the fractional power of the evanescent field in the coating decreases. The fractional power of the evanescent field in the coating as a function of the coating refractive index shows obvious change, and it is possible to detect the ambient environment.
5. CONCLUSION In conclusion, the ECOF is transformed into a concentric three-layered optical fiber using conformal mapping. The eigenvalue equation at the fundamental core-mode cutoff is built. The fundamental core-mode cutoff of an ECOF has been obtained by solving the eigenvalue equations graphically. The V c at the fundamental core-mode cutoff is calculated for different eccentric distances, wavelengths, coating refractive indices, and radii of the fiber core. The fractional power of the evanescent field for the ECOF is also calculated for different eccentric distances and coating refractive indices. The above solutions will help to design ECOF parameters, such as eccentric distance, radius of the fiber core, working wavelength, power in the fiber, and also the selection of an external medium (chemicals that can permeate into the coating) for evanescent field absorption. It is also convenient to control all these parameters during the fabrication process and helpful to know their influence on chemical sensor properties.
ACKNOWLEDGMENTS This work was supported by the National Natural Science Foundation of China, under grant numbers 11274077 and 61290314, and Heilongjiang Provence Natural Science Foundation, under grant number F201328, and partially supported by the 111 project (B13015) and Sino-Japan S & T cooperation project (2010DFA-2770 and 2011DFB11520), to Harbin Engineering University.
REFERENCES
Fig. 5. Plot of evanescent field versus (a) eccentric distance and (b) coating refractive index.
1. P. K. Choudhury and T. Yoshino, “On the pH response of fiber optic evanescent field absorption sensor having a U-shaped probe: an experimental analysis,” Optik 114, 13–18 (2003). 2. M. D. DeGrandpre and L. W. Burgess, “Long path fiber-optic sensor for evanescent field absorbance measurements,” Anal. Chem. 60, 2582–2586 (1988). 3. M. Marazuela and M. Moreno-Bondi, “Fiber-optic biosensors— an overview,” Anal. Bioanal. Chem. 372, 664–682 (2002).
J. Liu and L. Yuan 4.
W. Wojcik, A. Kotyra, S. Przylucki, A. Smolarz, and J. T. Szymczak, “Evanescent field absorption sensor,” Proc. SPIE 3054, 132–136 (1997). 5. L. Tong, J. Lou, and E. Mazur, “Single-mode guiding properties of subwavelength-diameter silica and silicon wire waveguides,” Opt. Express 12, 1025–1035 (2004). 6. L. Tong, J. Lou, and E. Mazur, “Modeling of subwavelengthdiameter optical wire waveguides for optical sensing applications,” Proc. SPIE 5634, 416–423 (2005). 7. L. Su, T. H. Lee, and S. R. Elliott, “Evanescent-wave excitation of surface-enhanced Raman scattering substrates by an opticalfiber taper,” Opt. Lett. 34, 2685–2687 (2009). 8. J. Senosiain, I. Diaz, A. Gaston, and J. Sevilla, “High sensitivity temperature sensor based on side-polished optical fiber,” IEEE Trans. Instrum. Meas. 50, 1656–1660 (2001). 9. S. T. Huntington, K. A. Nugent, A. Roberts, P. Mulvaney, and K. M. Lo, “Field characterization of a D-shaped optical fiber using scanning near-field optical microscopy,” J. Appl. Phys. 82, 510–513 (1997). 10. A. S. Webb, F. Poletti, D. J. Richardson, and J. K. Sahu, “Suspended-core holey fiber for evanescent-field sensing,” Opt. Eng. 46, 010503 (2007). 11. A. Candiani, A. Bertucci, S. Giannetti, A. Cucinotta, R. Corradini, and S. Selleri, “Label-free DNA biosensor based on a peptide nucleic acid-functionalized microstructure optical fiber-Bragg grating,” J. Biomed. Opt. 18, 057004 (2013).
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12. C. Y. Guan, L. B. Yuan, F. J. Tian, and Q. Dai, “Characteristics of near-surface-core optical fibers,” J. Lightwave Technol. 29, 3004–3008 (2011). 13. J. X. Liu, H. C. Deng, and L. B. Yuan, “Characteristics analysis of coating layer power distribution of eccentric core optical fiber,” Proc. SPIE 8561, 85611R (2012). 14. H. Kober, Dictionary of Conformal Representations (Dover, 1952). 15. C. Y. Guan, L. B. Yuan, F. J. Tian, Q. Dai, and X. Z. Tian, “Mode field analysis of eccentric optical fibers by conformal mapping,” Proc. SPIE 7753, 77535W (2011). 16. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, 1983). 17. C. Y. H. Tsao, D. N. Payne, and L. Li, “Modal propagation characteristics of radially stratified and D-shaped metallic optical fibers,” Appl. Opt. 28, 588–594 (1989). 18. E. T. Goodwin, “Recurrence relations for cross-products of Bessel functions,” Q. J. Mech. Appl. Math. 11, 72–74 (1949). 19. R. J. Black and R. Bourbonnais, “Core-mode cutoff for finite cladding light guides,” IEE Proc-J 133, 377–384 (1986). 20. B. Culshaw, G. Stewart, F. Dong, C. Tandy, and D. Moodie, “Fibre optic techniques for remote spectroscopic methane detection—from concept to system realization,” Sens. Actuators B 51, 25–37 (1998). 21. G. Whitenett, G. Stewart, H. B. Yu, and B. Culshaw, “Investigation of a tunable mode-locked fiber laser for application to multipoint gas spectroscopy,” J. Lightwave Technol. 22, 813–819 (2004).
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Evanescent field characteristics of eccentric core optical fiber for distributed sensing Jianxia Liu and Libo Yuan* Key Laboratory of In-Fiber Integrated Optics, Ministry of Education, College of Science, Harbin Engineering University, Harbin 150001, China *Corresponding author: [email protected] Received November 1, 2013; revised December 21, 2013; accepted December 22, 2013; posted December 24, 2013 (Doc. ID 200564); published February 5, 2014 Fundamental core-mode cutoff and evanescent field are considered for an eccentric core optical fiber (ECOF). A method has been proposed to calculate the core-mode cutoff by solving the eigenvalue equations of an ECOF. Using conformal mapping, the asymmetric geometrical structure can be transformed into a simple, easily solved axisymmetric optical fiber with three layers. The variation of the fundamental core-mode cut-off frequency (V c ) is also calculated with different eccentric distances, wavelengths, core radii, and coating refractive indices. The fractional power of evanescent fields for ECOF is also calculated with the eccentric distances and coating refractive indices. These calculations are necessary to design the structural parameters of an ECOF for longdistance, single-mode distributed evanescent field absorption sensors. © 2014 Optical Society of America OCIS codes: (130.3120) Integrated optics devices; (260.2110) Electromagnetic optics; (280.4788) Optical sensing and sensors; (350.7420) Waves. http://dx.doi.org/10.1364/JOSAA.31.000475
1. INTRODUCTION Optical fibers can be used in evanescent wave absorption sensing applications, due to the fraction of modal power extended into the external environment as an evanescent wave field. This sensing technology has been widely applied in chemistry, biology, and environmental monitoring, food, and hygiene detection in recent years [1–3]. In order to obtain the evanescent fields, multimode fibers [4] and microfibers and nanofibers [5,6] have been proposed. Fiber tapering [7] and side polishing [8,9] techniques have been reported. But evanescent-field-absorption-based multimode fiber sensors have low sensitivity due to weak evanescent field energy. Although the microfiber and nanofiber sensors improved the sensing sensitivity, it is difficult in distributed sensing and harsh environment monitoring due to the small fiber diameter. Fiber tapering and side polishing techniques destroy the cladding and coating of the fiber to a certain extent, thus reducing the mechanical properties of the optical waveguide. Recently, there has been widespread activity based on microstructure optical fibers in which evanescent field sensing has been strongly exploited [10,11]. Eccentric core optical fiber (ECOF) is a kind of special fiber in which the fiber core is eccentrically positioned and has some special characteristics due to the asymmetry of the cross section. Some characteristics of ECOF without the coating [12,13] were analyzed, as the fundamental core mode must be operated below a certain wavelength for efficient transmission. The fundamental core-mode cutoff must be considered, due to the severely asymmetrical structure. In this paper, the asymmetric geometrical structure is transformed into a simple, easily solved three-layer axisymmetric optical waveguide using conformal mapping. By solving the eigenvalue equations of an ECOF, the variation 1084-7529/14/030475-05$15.00/0
of the fundamental core-mode cut-off frequency (V c ) is calculated with different eccentric distances, wavelengths, core radii, and coating refractive indices. The fractional power of the evanescent fields for the ECOF is also calculated with the eccentric distances and coating refractive indices. The result is important for designing the structural parameters of the ECOF for long-distance, single-mode distributed evanescent field absorption sensors. A strong evanescent field exits in the coating of the ECOF due to a small distance between the core and coating. ECOF is a promising candidate for distributed evanescent field sensors and can be applied in harsh environment monitoring.
2. EIGENVALUE EQUATIONS OF ECCENTRIC CORE OPTICAL FIBER A. Eccentric Core Optical Fiber The optical fiber preform was fabricated using ultrasonic drilling or side-grooving technology. With ultrasonic drilling, for example, first a silica glass preform was drilled using a mechanical ultrasonic drill to form an eccentric hole, and then the hole was filled by a higher index glass rod. The optical fiber fabricated using ultrasonic drilling and side-grooving technology by our research group is shown in Fig. 1. The index difference Δn n1 − n2 is 0.005 at λ 0.65 μm. The radius of the fiber core is 5.75 μm, and the eccentric distance is 4.6 μm for this sample. The optical fiber sample contains multimode fibers at λ 1.55 μm due to the large core size. B. Eigenvalue Equations of Eccentric Core Optical Fiber The core of eccentric core fiber does not lie in the center but remains circular. Experience shows that the performance of this fiber does not cause it to be eccentric core dependent as long as the eccentric distance is reasonably large. But as a rule © 2014 Optical Society of America
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Fig. 1. Cross-sectional view of the ECOF.
of thumb, a few micrometers is often sufficient for evanescent field sensing. Although this concentric fiber model is different from the eccentric core fiber, the power of an electromagnetic wave should transmit mainly in the core. The problem now is to seek a simple solution for this asymmetric waveguide geometry. If we regard the eccentric core fiber (see Fig. 2) as a threelayer cylindrical optical fiber, using conformal mapping the asymmetric geometrical structure can be transformed into a simple, easily solved structure. The z plane represents the original coordinate z x; y, and the ζ plane represents the transformed new plane coordinate ζ η; ζ. The ill-fitted irregular geometry on the z plane—three eccentric circles— may be regulated to be three concentric circles on a new ζ plane through conformal mapping (see Fig. 3). Zero-order approximation is adequate to ensure the method is valid [14,15], and the Helmholtz wave equation is changed to the following: ∇2
1 −
y1 ∕y2 k20 n2
−
β2 ψ
0:
(1)
Here y1 and y2 are symmetric points and are defined by
y1 y2 R22 ; y1 − Ly2 − L R20
2
where L is the distance deviated from the coordinate origin. R0 and R2 are the radii of the core and cladding on the z plane, respectively. If all phase parameters are scaled by a factor of (1 − y1 ∕y2 ), the solution of Eq. (1) is written as [16,17]
Fig. 3. Three concentric circles mapped on the ζ plane.
ψ 1 AJ m u1 r∕R1 cosmϕ BI m w2 r∕R1 CK m w2 r∕R1 cosmϕ β > k0 n2 ; ψ2 BJ m u2 r∕R1 CY m u2 r∕R1 cosmϕ β < k0 n2 ψ 3 DK m w3 r∕R2 cosmϕ 3 where A, B, C, and D are constants and R1 and R2 are radii of inner and outer circles, respectively, on the ζ plane. R1 is defined by R1 R0 jy2 ∕L − y2 j, and u1 R1 k20 n21 − β2 1∕2 1 − y1 ∕y2 u2 R1 k20 n22 − β2 1∕2 1 − y1 ∕y2 w2 R1 β2 − k20 n22 1∕2 1 − y1 ∕y2
;
4
w3 R2 β2 − k20 n23 1∕2 1 − y1 ∕y2 where n1 , n2 , and n3 are the refractive indices of the core, cladding, and coating, respectively. So far, for any given eccentric core fiber design, it is always possible to work out the equivalent concentric structure using conformal mapping. Before the core-mode cutoff (β > k0 n2 ), the continuity of ψ and its derivative at the core–cladding and cladding– coating interfaces (boundary conditions) gives the eigenvalue equation J 0m u1 K 0m w3 w2 J 0m u1 pm − r R2 J m u1 K m w3 w3 R1 J m u1 m
w22 w2 K 0m w3 s ; qm − u1 R2 K m w3 u1 w3 R1 m
(5)
where qm , r m , and sm are Bessel cross-product functions [18] and are defined by pm I m w2 RK m w2 − I m w2 K m w2 R; qm I m w2 RK 0m w2 − I 0m w2 K m w2 R; r m I 0m w2 RK m w2 − I m w2 K 0m w2 R;
6
sm I 0m w2 RK 0m w2 − I 0m w2 K 0m w2 R;
Fig. 2.
Cross section of eccentric core fiber on the z plane.
where R is the cladding-to-core radius ratio (R2 ∕R1 ). By solving Eq. (5), the propagation constant β can be obtained. When n1 1.4510, n2 1.4468, n3 1air, m 0,
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R0 4.5 μm, R2 62.5 μm, k 1 μm, and β 5868076∕m, the neff corresponding to the solution is 1.4470.
3. FUNDAMENTAL CORE-MODE CUTOFF The significance of the fundamental core-mode cutoff is to define a wavelength that one should operate well below for efficient transmission. The solution to the eigenvalue equation has been obtained for different eccentric distances, wavelengths, and fiber radii up to the core-mode cutoff. When there is no solution to the eigenvalue equation, the LP01 mode is cut off from the core. The value of V at the cutoff is denoted as V c 1.2616 when k 1 μm. Obviously, the fundamental mode has a nonzero cut-off frequency, because the light is not confined in the fiber core and completely leaks into the cladding when V is small [12]. But this is an approximate value, and it changes with eccentric distances, wavelengths, coating refractive indices, and the radius of the fiber core. So considering the condition of fundamental core-mode cutoff neff ncl n2 , Eq. (5) is rebuilt: J 00 V c K 00 w3 J 00 V c K 00 w3 ln R − 0: R2 J 0 V c K 0 w3 w3 R1 RJ 0 V c V c R2 K 0 w3 (7) The accurate value of V c can be calculated by solving Eq. (7) [19]. A plot of V c with eccentric distances, wavelengths, coating refractive indices, and core radii are shown in Figs. 4(a)–4(d), respectively. At the same time, the fundamental core-mode cutoff (V c ) of the axisymmetric optical fiber with three layers and the wavelengths and coating refractive indices are also considered and shown in Figs. 4(b) and 4(c). When the eccentric distance increases and the radius of the fiber core decreases, the fundamental core-mode cut-off frequency V c will decrease and then approach zero. The absorption wavelengths and normalized frequencies (V ) of some common gases are listed in Table 1. V is the normalized frequency given by
V
2π · R0 · λ
q n21 − n22 ;
(8)
where n1 1.4510, n2 1.4468, and R0 4.5 μm. The V corresponding to the absorption wavelength of these gases in Table 1 is between V c (see Fig. 4) and 2.405, except for oxygen (O2 ) and nitrogen dioxide (NO2 ). It is can be forecasted that the value of V will be smaller than V c with the increasing absorption wavelength of the gas. This means that the ECOF will contain multimode fibers when the absorption wavelength of the gas is short, and the fundamental core mode of the ECOF will be cut off when the absorption wavelength of the gas is long, in order to ensure the light energy can transmit efficiently in the ECOF for a single mode. The values of Y are calculated and listed in the Table 1. Y is denoted as q Y R0 · n21 − n22 :
(9)
Fig. 4. Plot of V c versus (a) eccentric distance, (b) wavelength, (c) refractive index of the coating, and (d) core radius for different eccentric distances.
4. EVANESCENT FIELD In this section, we consider only the coating region. Here the single-mode condition corresponds to a wavelength of
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Table 1. Normalized Frequency and Y-Values Corresponding to Absorption Wavelength of Gases [20,21]
Gas Acetylene (C2 H2 ) Hydrogen iodide (HI) Ammonia (NH3 ) Carbon dioxide (CO2 ) Hydrogen sulfide (H2 S) Methane (CH4 ) Hydrogen fluoride (HF) Hydrogen bromide (HBr) Water (H2 O) Oxygen (O2 ) Nitrogen dioxide (NO2 )
Absorption Wavelength (nm)
Normalized Frequency (V)
Y R0 δ
1533
2.033
0.37 < Y < 0.58
1541
2.023
0.37 < Y < 0.59
1544 1567
2.019 1.989
0.37 < Y < 0.59 0.38 < Y < 0.60
1578
1.975
0.38 < Y < 0.60
1667 1330
1.87 2.344
0.40 < Y < 0.63 0.32 < Y < 0.50
1341
2.324
0.32 < Y < 0.51
1365 761 800
2.284 4.096 3.897
0.32 < Y < 0.52 0.18 < Y < 0.29 0.19 < Y < 0.30
J. Liu and L. Yuan
1.55 μm, and the refractive indices (RIs) of the core, cladding, and coating are 1.4510, 1.4468, and 1, respectively. Considering these RIs and the wavelength, the radius of the fiber core for the single mode is calculated as 4.5 μm. Under these conditions electromagnetic energy is propagating mainly in the fiber core, and the evanescent field will leak into the coating. The fractional power of the evanescent field in the coating can be expressed as η
P coating n3 R22 K 21 w3 R2 − K 20 w3 R2 ; P total 2K 20 w3 R2 P total
(10)
where P coating is the power in the coating, P total is the total power, and w3 is the modal parameter. The fractional power of the evanescent field in the coating as a function of the eccentric distance and coating refractive index is shown in Figs. 5(a) and 5(b), respectively. It can be observed that if the eccentric distance is 1 μm at n3 1, the fractional power of evanescent field in the coating is 1.132%. Hence, if the eccentric distance is much larger, the fractional power of the evanescent field in the coating decreases. The fractional power of the evanescent field in the coating as a function of the coating refractive index shows obvious change, and it is possible to detect the ambient environment.
5. CONCLUSION In conclusion, the ECOF is transformed into a concentric three-layered optical fiber using conformal mapping. The eigenvalue equation at the fundamental core-mode cutoff is built. The fundamental core-mode cutoff of an ECOF has been obtained by solving the eigenvalue equations graphically. The V c at the fundamental core-mode cutoff is calculated for different eccentric distances, wavelengths, coating refractive indices, and radii of the fiber core. The fractional power of the evanescent field for the ECOF is also calculated for different eccentric distances and coating refractive indices. The above solutions will help to design ECOF parameters, such as eccentric distance, radius of the fiber core, working wavelength, power in the fiber, and also the selection of an external medium (chemicals that can permeate into the coating) for evanescent field absorption. It is also convenient to control all these parameters during the fabrication process and helpful to know their influence on chemical sensor properties.
ACKNOWLEDGMENTS This work was supported by the National Natural Science Foundation of China, under grant numbers 11274077 and 61290314, and Heilongjiang Provence Natural Science Foundation, under grant number F201328, and partially supported by the 111 project (B13015) and Sino-Japan S & T cooperation project (2010DFA-2770 and 2011DFB11520), to Harbin Engineering University.
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