Analytic solution of cut-off wavelength of bend-insensitive fibers based on frustrated total reflection Zhen Lin,1,2 Guobin Ren,1,2,* Siwen Zheng,1,2 Wei Jian,1,2 and Shuisheng Jian1,2 1
Key Laboratory of All Optical Network & Advanced Telecommunication Network of EMC, Beijing Jiaotong University, Beijing 100044, China 2
Institute of Lightwave Technology, Beijing Jiaotong University, Beijing 100044, China *Corresponding author: [email protected]
Received 24 October 2013; revised 28 December 2013; accepted 14 January 2014; posted 16 January 2014 (Doc. ID 200110); published 14 February 2014
An analytic method for solving the cut-off wavelength of single groove-assisted bend-insensitive fiber (BIF) was proposed. Combined with the concepts of frustrated total reflection and numerical aperture, the cut-off wavelength formula for the BIF was deduced. Experimental results showed that, compared to the traditional analytic and numeric methods, this method had higher accuracy in the calculation of cutoff wavelength for BIFs that had a large proportion of depressed inner-cladding layer, which significantly expanded the range of accurately predicting the cut-off wavelength. © 2014 Optical Society of America OCIS codes: (060.0060) Fiber optics and optical communications; (060.2310) Fiber optics; (060.2330) Fiber optics communications. http://dx.doi.org/10.1364/AO.53.001110
1. Introduction
With the vigorous development of fiber-to-the-home (FTTH) projects, the bend-insensitive fiber (BIF) has aroused widespread interest of researchers in recent years. In order to reduce the huge power loss caused by the small bending radius in the optical fiber, there has been a variety of BIF designs, including groove structure, hole-assisted structure, and nanohole structure [1–4], etc. Among them, the single-grooveassisted bend-insensitive fiber (SBIF) utilizes a depressed inner cladding (DIC) next to the core to help restrict the mode power, which guarantees excellent bend-insensitive characteristics [5–7]. It meets the ITU-T G.657 recommended standard and can be fully compatible with G.652.D generic optical fiber, which is widely used in today’s communication systems. In addition, simple DIC structure has advantages of convenient manufacture and low cost. Therefore 1559-128X/14/061110-08$15.00/0 © 2014 Optical Society of America 1110
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the SBIF is expected to realize the replacement of single-mode fiber (SMF) in the FTTH communication system and has attracted increasing attention. However, the International Telecommunication Union has not published any standard method for measuring cut-off wavelength of SBIF. Cut-off wavelength is one of the most important parameters for measuring the performance of optical fibers. The commonly used standard method for measuring cut-off wavelength of SMF is the bend reference method [8]. The spectral attenuation for the SMF with a loop of 60 mm diameter is measured in comparison with the straight fiber where the long wavelength edge of the bendinduced loss is larger than the long wavelength baseline by 0.1 dB is considered as the effective cut-off wavelength. Because the SBIF is not sensitive to bending, it’s difficult to measure the cut-off wavelength by existing standard methods. Several methods also have been reported to determine the cut-off wavelength of BIF, such as the multimode reference technique (MRT), the far-field MFD method, and the transverse splice loss technique (SLT) [9–11].
However, these methods are complex and need careful operation to carry out. Because of the difficulty for measurement, how to exactly predict the cutoff wavelength by simulating calculation is the key point for researching and designing the SBIF. Due to the setting of cladding region and boundary condition, the existing analytic method (e.g., transmission matrix method) and numeric method (e.g., finite element method) have limitations. If the DIC is thin, the calculation has a certain degree of accuracy. But once the DIC thickens, the deviation will be obvious. Consequently, existing methods need to be improved, or a new simulation method should be proposed to exactly predict the cutoff wavelength of the SBIF. This paper presented a new analytic method for calculating the cut-off wavelength of the SBIF. By using a simple three-layer planar model, and combined with the concept of frustrated total reflection (FTR) in the field of film and the traditional concept of numerical aperture of the optical fiber, we solved the equivalent total reflection critical angle. Then starting from the eigenvalue equation, we deduced the equivalent normalized cut-off frequency. The ultimate formula of cut-off wavelength, which was suitable for the SBIF, was figured out. This method greatly expanded the accurate prediction range of cut-off wavelength of the SBIF. It laid a theoretical foundation for future research and design of diversified and practical groove-assisted optical fibers. On the other hand, this groove-assisted structure could be transplanted into a variety of large-mode-area designs to improve the bending performance [7], which enabled this kind of fiber as a potential candidate for a high-power compact optical-fiber amplifier and laser. 2. Principle
For ordinary SMF with a step-index structure, there has been a perfect theory model to solve its cut-off wavelength. The analytic expression is [12] 2πa 2 n − n2clad 1∕2 : λc V c co
(1)
Given that the core radius is a, the refractive index of core and cladding are nco and nclad , respectively, the normalized cut-off frequency of second-order modes for the step-index fiber V c 2.4048, the cutoff wavelength λc can be calculated. The refractive index profile of the SBIF, shown in Fig. 1, is a dual-step-index structure. On the basis of the SMF, a DIC is added between the core and outer cladding, the thickness is d, the refractive index is nd , nd < nclad < nco . Δn nco − nd (a positive value) is the refractive index difference between core and DIC, Δn0 nd − nclad (a negative value) is the refractive index difference between DIC and outer cladding. Obviously Eq. (1) is no longer applicable for the optical fiber with multilayer structure. So we return to the light transmission theory. The transmission of light in optical fiber is essentially based on the total
Fig. 1. Refractive index profile of the SBIF.
reflection. The light should be totally reflected on the interface between core and cladding to continue propagating. For ordinary SMF, relative to the core, the cladding region can be considered as infinite. In this case, the total reflection occurs completely as long as the incident angle on the interface is equal to or greater than the total reflection critical angle. The cut-off wavelength can be expressed as λc
2πa sin θNA : Vc
(2)
θNA is the numerical aperture angle, which indicates the maximum reception capacity for the incident light of the optical fiber. Namely, in order to propagating in the fiber, the incident angle of the incident light on the interface between air and core should be less than or equal to θNA . As can be seen in Fig. 2, according to Snell’s law, the maximum incident angle (i.e., θNA ) on the fiber end meets that nair sin θNA nco cos θc . θc is the total refection critical angle on the interface between core and cladding, the refractive index of air nair 1. Equation (2) can be rewritten as λc
2πa n cos θc : V c co
(3)
We simplify the SBIF into three-layer planar structure, as illustrated in Fig. 3. In order to stimulate guide modes, the light that propagates in the core of the SBIF should be totally reflected on the interface between core and DIC. However, based on the theory of thin film optics, while the thickness of DIC is the order of wavelength, even if the incident angle is greater than the total reflection critical angle of the interface between core and DIC, the total reflection may not occur. From the energy point of view, the energy carried by light only partly returns
Fig. 2. Schematic of light propagation in the SMF. 20 February 2014 / Vol. 53, No. 6 / APPLIED OPTICS
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So the reflectivity R jrj2. Since light is incident at greater than the critical angle, sin θ1 > sin θc nd ∕ nco , δ is no longer a real number, but an imaginary number. We assume that δ iδ0 , a real number δ0 : δ0
to the core. This phenomenon is known as FTR [13]. FTR has had many applications in spectroscopy systems, such as thin-film device, interferometer, microporous membrane, and so on [14–16]. For SBIFs, the width of DIC is generally in the order of propagation wavelength. Therefore, FTR generates in the SBIF, resulting in the change of cut-off wavelength. Then in light of Eq. (3), the key for accurately predicting the cut-off wavelength of SBIF is to figure out the equivalent total reflection condition. Equation (3) should be rewritten as follows: 2πa n cos θe : V e co
(6)
Define reiθ Er ∕Ei , then reiθ r12 r23 ei2δ ∕1 r12 r23 ei2δ : 1112
r12 ∥
n2 cos θ1 − n1 cos θ2 ; n2 cos θ1 n1 cos θ2
(9)
r12 ⊥
n1 cos θ1 − n2 cos θ2 ; n1 cos θ1 n2 cos θ2
(10)
r23 ∥ −r32 ∥ −
n2 cos θ3 − n3 cos θ2 ; n2 cos θ3 n3 cos θ2
(11)
r23 ⊥ −r32 ⊥ −
n3 cos θ3 − n2 cos θ2 : n3 cos θ3 n2 cos θ2
(12)
The r∥ and r⊥ from Eq. (7) can be calculated by Eq. (8) to Eq. (12). R∥ jr∥ j2 and R⊥ jr⊥ j2 can be obtained:
(5)
Assuming Ei is the incident amplitude, at the interface between core and DIC the Stokes’ relations [17] connect the reflection and transmission coefficients r12 , t12 for light originating in the core and the coefficients r21 , t21 for the light originating in the DIC. There are r12 −r21 , t12 t21 r212 1. A similar set of relations applies to the mediums on the either sides of the interface between DIC and outer cladding. Then the reflected wave amplitudes back to the core are r12 Ei , t12 r23 t21 Ei ei2δ , t12 r23 r21 r23 t21 Ei ei4δ . Add these multiple reflection contributions, while making use of the Stokes’ relations to get the total reflected amplitude Er : Er Ei r12 r23 ei2δ ∕1 r12 r23 ei2δ :
(8)
(4)
V e and θe are the equivalent normalized cut-off frequency and equivalent total reflection critical angle of the dual-step-index structure, respectively. First, we use thin film interference theory to solve θe . As shown in Fig. 3, at the interface between core and DIC, θ1 is the incident angle, θ2 is the refracting angle, similarly at the interface between DIC and outer cladding, θ2 is the incident angle, and θ3 is the refracting angle. The phase thickness δ of DIC can be obtained easily. δ 2π∕λnd d cos θ2 . Using Snell’s law: nco sin θ1 nd sin θ2 nclad sin θ3 : q 2πd n2d − n2co sin2 θ1 : δ λ
q n2co sin2 θ1 − n2d :
In accordance with the Fresnel’s formula [17], the reflection coefficients for polarizations parallel with and perpendicular to the plane of incidence are
Fig. 3. Simple three-layer planar structure.
λc
2πd λ
(7)
APPLIED OPTICS / Vol. 53, No. 6 / 20 February 2014
R∥
A∥ sinh2 δ0 B∥ − 1 ; A∥ sinh2 δ0 B∥
(13)
R⊥
A⊥ sinh2 δ0 B⊥ − 1 ; A⊥ sinh2 δ0 B⊥
(14)
where A∥
i ih 2 n n2 sin2 θ1 −1 n32 1 sin2 θ1 −n32 1 2 1∕2 2 2 2 ; n3 n3 n1 2 2 4 n2 cosθ1 n2 −sin θ1 −1 n2 −sin θ1 2 2 1 h 2 1∕2 2 i2 n3 n −sin2 θ1 n32 cosθ1 n2 1 2 1∕2 ; B∥ n1 2 n3 2 3 4n2 cosθ1 n2 −sin θ1 1 1 2 2 n n1 −1 n32 −1 n22 2 2 1∕2 ; 2 A⊥ n2 n n1 2 1 4 n2 cosθ1 n2 −sin θ1 −1 n32 −sin2 θ1 2 2 1 h 2 1∕2 i n3 2 −sin θ1 cosθ1 n2 2 1∕2 : B⊥ 1 n 4cosθ1 n32 −sin2 θ1 n21 −1 n22
h
n23 −1 n22
1
n21 1 n22
Then the reflectivity R: R
R∥ R⊥ : 2
(15)
We consider two limiting cases. In the case of infinitesimal DIC, it’s equivalent to the structure with only core and outer cladding. The total reflection critical angle θ0 arcsinnclad ∕nco . In the case of infinite DIC, it’s equivalent to the structure with only core and DIC. The total reflection critical angle θ00 arcsinnd ∕nco . Accordingly, the equivalent total reflection critical angle should be located between θ0 and θ00 , θ00 < θe < θ0 . Assuming nco 1.448, nd 1.44, nclad 1.444, then 83.97° < θe < 85.74°. The reflectivity R changed by incident angles θ1 on the interface between core and DIC are shown in Fig. 4, with different thicknesses of DIC and core radius a 6.38 μm. One can see that, while d 0.5 μm, taking θ1 85.7°, which is considerably larger than the total reflection critical angle of the interface between core and DIC θ00 arcsinnd ∕nco ≈ 84°, a majority of power of the incident light still leak to the outer cladding and loss, the reflectivity is only about 60%. This phenomenon, namely, is FTR. Because of the role of FTR, the total reflection can’t occur. So we have to assume an equivalent total reflection condition for solving the equivalent total reflection critical angle. While the equivalent total reflection condition is satisfied, θe is equal to the corresponding incident angle. There are three ways to figure out the reflectivity, which meets the equivalent total reflection condition: (1) as the DIC layer is thin, calculations by the FE method are accurate, and the simulation results by FTR and FE methods should be consistent. Use the results for reverse solving the reflectivity. (2) From the confinement loss to determine the cut-off frequency; the confinement loss is related to the imaginary part of the effective refractive index of the mode [18]. The confinement loss of each mode determines whether the mode can be propagated. Then use the results to calibrate the reflectivity. (3) Use the experimental measurement results to calibrate the reflectivity. In these ways, the equivalent total reflection conditions for a class of
2
n
1 u2
wˆ12 Iˆ i
6 6 6 ρIˆ J 0 n2d Iˆ 0i 6 i uJ 2 ˆ ˆ nco wI i 6 det6 6 nIˆ j 1 − 1 6 1η w2 wˆ 2 6 4 2 0 n2d Iˆ 0j n K ρIˆ j nclad 2 wK − 2 ˆ ˆI n w co
co
j
n
1 u2
similar fiber structures can be solved. In this paper, we assume the equivalent total reflection condition is that the reflectivity reaches to 99.9% or more. This condition applies to the samples throughout this paper as well as similar fiber structures. For example, in Fig. 4, while d 9 μm, R 99.9% is corresponding to θ1 85.46°, so that θe θ1 85.46°. With the thickness of DIC increasing, θe gradually decreases within the interval from θ0 to θ00 . Here, 99.9% is not a completely fixed value, which can be freely adjusted to meet the need for precision. Consequently, the equivalent total reflection critical angle θe in Eq. (4) could be solved. Numerical aperture (NA) is the one of the most important indicators of the fiber. We introduce the concept of the equivalent numerical aperture NAe nco cos θe . NAe changes with the d and nd is shown in Fig. 5. ε Δn0 ∕Δn nd − nclad ∕nco − nd , absolute value of which reflects the depth of the DIC. Unless otherwise specified, the ε is consistent throughout the manuscript. The equivalent numerical aperture reflects the light gathering capacity of the fiber to a certain degree. For the SBIF with dual-step-index structure, the normalized cut-off frequency is also different from the step-index fiber. The eigenvalue equation takes the form [19]:
wˆ12 Kˆ i
n2 Kˆ 0 n2 dwˆ Kiˆ i co nKˆ j 1 1 1η w2 − w ˆ2 2 0 n2 Kˆ 0 n K d j ρKˆ j nclad 2 wK − 2 ˆ Kˆ n w ρKˆ i
Fig. 4. Reflectivity as a function of incident angles with various thicknesses of DIC and core radius a 6.38 μm.
J0 uJ
co
co
j
0 Iˆ 0 J ρIˆ i uJ wˆ Iˆi i 1 1 ˆ n u2 wˆ 2 I i 0 Iˆ 0 K ρIˆ j wK − wˆ Iˆj j nIˆ j 1 1 2 1η w2 − w ˆ
0 3 Kˆ 0 J ρKˆ i uJ wˆ Kˆi i 7 7 1 1 ˆ 7 n u2 wˆ 2 K i 7 7 0; 0 ˆ0 7 K j K ρKˆ j wK − wˆ Kˆ 7 7 j 7 5 nKˆ j 1 1 2 1η w2 − w ˆ
20 February 2014 / Vol. 53, No. 6 / APPLIED OPTICS
(16)
1113
Fig. 5. Equivalent numerical aperture as a function of innercladding thickness for various values of the inner-cladding refractive index.
Fig. 6. Equivalent normalized cut-off frequency as a function of inner-cladding thickness for various values of the inner-cladding refractive index.
where
where J J n u; ˆ Kˆ i K n w; ˆ Iˆ i I n w;
K K n 1 ηw;
ˆ c n2clad − n2d ∕n2co − n2clad 1∕2 vc : w
ˆ Kˆ j K n 1 ηw; ˆ Iˆ j I n 1 ηw;
and the mode parameters: u an2co k2 − β2 1∕2 ; ˆ aβ2 − n2d k2 1∕2 ; w w aβ2 − n2clad k2 1∕2 ; v u2 w2 1∕2 akn2co − n2clad 1∕2 : J n , I n , and K n are the Bessel and modified Bessel functions, n is the azimuthal mode number, ρ nco k∕β, k is the free-space wave number, β is the modal propagation constant, η d∕a. For the TE and TM modes, n 0. The cut-off frequency of the mode is defined to be the frequency at which the modal propagation constant is equal to the wave number in the outer cladding, i.e., β nclad k, so that w 0, uc vc , where the subscript “c” denotes the value at cutoff. For small w, using small variable approximation [20]: K 0 1 ηw K0 0 ∼ 1 ηw2 log w−1 : wK wK 0 1 ηw
(17)
Then the eigenvalue that is the equivalent normalized cut-off frequency V e vc could be found from the following equation by simplifying Eq. (16): 0 ˆ c ˆ c J 0 vc K 00 w K 0 w ˆ c K 0 w ˆ c vc J 0 vc w ˆ c K 0 1 ηw 0 0 ˆ c ˆ c J 0 vc I 0 w I 0 w ; ˆ c vc J 0 vc w ˆ c ˆ c I 0 w I 0 1 ηw 1114
The normalized cut-off frequency of the step-index fiber is only relevant to the refractive index of core and outer cladding. But the thickness of DIC of SBIF also affects the value of the equivalent normalized cut-off frequency. While the refractive index of core and outer cladding and diameter of core and outer cladding are fixed, the equivalent normalized cut-off frequency as a function of inner-cladding thickness for various values of the inner-cladding refractive index is shown in Fig. 6. As can be seen from the figure, V e increases by enlarging the width of the DIC and then tends to be a constant value vs . V e is proportional to the depth of the DIC. vs can be determined by ˆ c J 00 vs K 00 w − ; ˆ c ˆ c K 0 w vs J 0 vs w
(19)
where ˆ c n2clad − n2d ∕n2co − n2clad 1∕2 vs : w Equation (19) indicates that vs is only associated with the refractive index of each layer but is independent from the thickness of each layer. So as long as the structural parameters are determined, the equivalent normalized cut-off frequency V e in Eq. (4) could be calculated. Known nco , nd , nclad , a, d, the cut-off wavelength of SBIF can be obtained by solving the combined equation of Eqs. (4), (15), and (18). 3. Simulation Results and Experimental Verification
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APPLIED OPTICS / Vol. 53, No. 6 / 20 February 2014
For the case with the same structure, the simulation results of the cut-off wavelength for SBIF with FTR method and finite element (FE) method are compared in Fig. 7. d∕a reflects the thickness of DIC; ε
Fig. 7. Comparison of simulation results for the cut-off wavelength of SBIF with FTR method and FE method.
reflects the depth of DIC. The full vectorial finite element method is performed with the commercial software Comsol Multiphysics. The cut-off frequency of the mode is defined to be the frequency at which the effective index of the mode is equal to the refractive index of the outer cladding. A circular perfectly matched layer (PML) [21] in association with a perfect electric conductor is used to truncate the computational domain. The width of the PML is set at 20 μm. As d increases, λc sharp declines and then approximately remains unchanged. However, the result of the FTR method is different. The part of the decline completely coincides with the FE method. Then λc doesn’t remain constant, whereas λc gradually rises, which means that the core is weakly influenced by the outer cladding if the thickness of DIC is large enough. Meanwhile, the lower the refractive index of the DIC, the greater the amplitude of change of λc. We investigate the energy intensity distribution of the second-order mode of SBIF. Assuming the power factor Γ ∬d P∕∬all P, which is the ratio of energy integral on the area of DIC and energy integral on the whole mode field area. Γ changes with the inner-cladding thickness while ε is fixed (Fig. 8). One can see that the energy intensity distributed on the inner cladding increases with the inner-cladding thickness rapidly at first and then slowly declines, which means the loss of the second-order mode experiences a process of first increasing and then decreasing. The losses of second-order modes increase, which means second-order modes will be cut off more easily, resulting in single-mode transmission. Therefore the variation of the loss of second-order mode partly leads to the change of cut-off wavelength. The corresponding energy intensity distributions for three representative points are also simulated (displayed in the illustrations of Fig. 8). The same polarization direction (arrow surface) of the electric fields represents that they belong to the same mode. In addition, with the change of inner-cladding thickness, the energy intensity distributed on core and the effective areas also change. The combination of these factors leads to the movement of the cut-off wavelength.
Fig. 8. Power factor Γ as a function of inner-cladding thickness and corresponding energy intensity distributions of second-order mode.
For the reliability of the experimental results, we applied three measurement methods for BIFs to measure the cut-off wavelengths of four laboratory made SBIF samples, and compared experimental results with simulation results. The microscope photograph of one of the sample fiber is shown in Fig. 9. Three measurement methods: (1) the improved bend reference technique (BRT) used Photon Kinetics 2210 optical fiber analysis systems; the length of measured fiber is 2 m. Because the bending loss for SBIFs at the standard loop diameter of 60 mm is almost negligible, the higher-order modes cannot be removed. With regard to different samples (different bending losses), 3–8 small loops of 20 mm diameter were used to replace with the big loop of 60 mm diameter in the traditional method, which increases the loss of higher-order modes and forms a reference light path involved to determine the cut-off wavelength. (2) The MRT also used Photon Kinetics 2210 optical fiber analysis systems,
Fig. 9. Microscope photograph of the sample. 20 February 2014 / Vol. 53, No. 6 / APPLIED OPTICS
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compared with BRT that winding a small loop to obtain a reference light path, the MRT replaced the measured fiber with a short piece of multimode fiber. The power spectrum of the same wavelength range was measured as the reference transmission power spectrum, which was involved in determining the cut-off wavelength. The length of the multimode fiber (Yangtze Optical Fibre & Cable) is 1 m. (3) The transverse SLT [11]: two of same fiber sample (about 0.5 m long) were fixed and aligned in fiber fusion splicer, and their facets were separated less than 5 μm. Through the electronic screen of fiber fusion splicer, it’s easy to transversely move one of the fibers (parallel to the end face) for the distance of radius of core. A broadband light source was used as the input, and the output power was measured by an optical spectrum analyzer. The splice loss was measured with reference to the perfectly aligned fibers. At the vicinity of the cut-off wavelength, the transverse splice loss was the lowest and jumped to the high value when only LP01 mode propagated in the fiber. The wavelength where we got the lowest splice loss was the cut-off wavelength. The sample structure parameters, simulation results, and experiment results are listed in Table 1 and shown in Fig. 10. In Fig. 10, the simulation data uses the same core radius and epsilon as four samples and contains the simulation results (FTR method and FE method) and experimental measurement results (BRT method and SLT method) of accurate structure parameters of four samples. They are marked by special symbols, e.g., hollow triangle, hollow square, solid circular, and hollow star. It can be observed that the structure parameters of Sample 1 and Sample 2 belong to or are near the overlapped part for the FTR method and FE method. The simulation results with FTR method and FE method are similar and slightly differ from each other. The experimental results are not far off the simulation results. However, the structure parameters of Sample 3 and Sample 4 obviously belong to the separate part. The simulation results of the FE method have a large deviation from the results of the FTR method, Table 1.
Structure Parameters, Simulation Results, and Experiment Results
SBIFs Samples nco nd nclad a (μm) d (μm) Cut-off wavelength (nm)
1
FTR FE BRT MRT SLT
Sample 4 at 1550 nm Bending radius (mm) 5 Macro-bend loss 0.3464 (dB/loop)
1116
2
3
4
1.4472 1.4472 1.4481 1.4468 1.441 1.4408 1.4401 1.4405 1.444 1.444 1.444 1.444 5.46 6.3 6.62 7.18 8.54 8.7 11.2 12.6 1087 1301 1636 1565 1090 1250 1495 1310 1035 1247 1579 1507 1035 1252 1573 1506 1030 1250 1590 1520 7.5 0.1157
10 0.0684
APPLIED OPTICS / Vol. 53, No. 6 / 20 February 2014
Fig. 10. Simulation results and experimental results of samples.
and the actual experimental results are significantly closer to the results of the FTR method. Overall, the theoretical and experimental values well match the used FTR method for the DIC profile. However, the theoretical values are generally larger than the experimental values about 50 nm [3], which are same as well-known phenomenon for SMF. There are inevitable error factors in manufacturing process and from the external experiments, such as bending leakage losses, torsion losses, stress losses, and so on, that make it easier for higher-order modes to be lost and easier to be cut off. Experiments show that compared to the traditional analytic and numeric methods, the FTR method has higher accuracy in the calculation of the cut-off wavelength for SBIFs that have a large proportion of the DIC layer. The FTR method significantly expands the range of accurately predicting the cut-off wavelength of SBIFs. 4. Conclusion
An analytic method for solving the cut-off wavelength of the single groove-assisted BIF with DIC was presented. We used a simple three-layer planar model and the eigenvalue equation, which were combined with the concept of FTR in the field of film and the traditional concept of numerical aperture of the optical fiber to figure out the equivalent total reflection critical angle and the equivalent normalized cutoff frequency. Then the ultimate calculation formula of the cut-off wavelength was deduced. It turned out that the simulation result of the FTR method proposed in this paper was different from the result of the traditional finite element method if the DIC had a certain thickness and depth. Experimental results indicated that compared to the traditional analytic and numeric methods, the FTR method had higher accuracy in the calculation of the cut-off wavelength for BIFs, which had a large proportion of DIC layer. This method greatly expanded the accurate prediction range of cut-off wavelength of the SBIF. It laid a theoretical foundation for future research and design of diversified and practical grooveassisted optical fibers.
This work is supported in part by the Major State Basic Research Development Program of China (grant no. 2010CB328206), the National Natural Science Foundation of China (NSFC) (grant nos. 61178008, 61275092), and the Fundamental Research Funds for the Central Universities, China. References 1. M.-J. Li, P. Tandon, D. C. Bookbinder, S. R. Bickham, M. A. McDermott, R. B. Desorcie, D. A. Nolan, J. J. Johnson, K. A. Lewis, and J. J. Englebert, “Ultra-low bending loss singlemode fiber for FTTH,” J. Lightwave Technol. 27, 376–382 (2009). 2. H. Kuniharu, M. Shoichiro, G. Ning, and W. Akira, “Lowbending-loss single-mode fibers for fiber-to-the-home,” J. Lightwave Technol. 23, 3494–3499 (2005). 3. P. R. Watekar, S. Ju, and W.-T. Han, “Optimized design of trenched optical fiber for ultralow bending loss at 5 mm of bending diameter,” Appl. Opt. 50, E97–E101 (2011). 4. M. S. Oskouei, S. Makouei, A. Rostami, and Z. D. K. Kannani, “Proposal for optical fiber designs with ultrahigh effective area and small bending loss applicable to long haul communications,” Appl. Opt. 46, 6330–6339 (2007). 5. F. Wu, D. Peckham, R. Smith, J. Pedder, P. Weimann, S. Jost, T. Goddard, P. Wang, J. Hartpence, J. Kim, K. Bradley, R. Lingle, Jr., W. Hatton, E. Barish, and D. Mazzarese, “A new G.652D, zero water peak fiber optimized for low bend sensitivity in access networks,” in Proceedings of the 55th International Wire & Cable Symposium, Providence, Rhode Island (2006), pp. 348–356. 6. J. M. Fini, P. I. Borel, P. A. Weimann, P. Kristensen, J. Bjerregaard, K. Carlson, M. F. Yan, P. W. Wisk, S. Ramachandran, A. D. Yablon, D. J. DiGiovanni, D. Trevor, C. J. Martin, and A. McCurdy, “Bend-insensitive fibers for FTTX applications,” in Optical Fiber Communication Conference (Optical Society of America, 2009), paper OTuL4. 7. Z. Lin, G. B. Ren, S. W. Zheng, W. Jian, J. J. Zheng, and S. S. Jian, “Analysis and characterization of Er-Yb codopeddepressed inner cladding fiber,” Appl. Opt. 52, 5856–5861 (2013).
8. A. Ghatak and K. Thyagarajan, Introduction to Fiber Optics (Cambridge University, 1998). 9. International Standard IEC 60793–1-42, 2007–04 (2007). 10. K. Nakajima, J. Zhou, K. Tajima, C. Fukai, K. Kurokawa, and I. Sankawa, “Cutoff wavelength measurement in a fiber with improved bending loss,” IEEE Photon. Technol. Lett. 16, 1918–1920 (2004). 11. P. R. Watekar, S. Ju, L. Htein, and W.-T. Han, “A simple and reliable method to determine LP11 cutoff wavelength of bend insensitive fiber,” Opt. Express 18, 13761–13771 (2010). 12. N. S. Kapany, Fiber Optics: Principles and Applications (Academic, 1967). 13. S. Zhu, A. W. Yu, D. Hawley, and R. Roy, “Frustrated total internal refection: a demonstration and review,” Am. J. Phys. 54, 601–607 (1986). 14. N. Petrov, “Frustrated-total-internal-reflection-based thinfilm color separator,” Opt. Lett. 32, 2744–2746 (2007). 15. M. Daehler and P. A. R. Ade, “Michelson interferometer with frustrated-total-internal-reflection beam splitter,” J. Opt. Soc. Am. 65, 124–130 (1975). 16. B. Z. Volchek, S. V. Kononova, E. N. Vlasova, R. K. Mamedov, and K. A. Mikhalev, “Study of microporous membranes using frustrated total internal reflection spectroscopy,” J. Opt. Technol. 70, 22–24 (2003). 17. M. Born and E. Wolf, Principles of Optics (Pergamon, 1975). 18. P. Viale, S. Fevrier, F. Gerome, and H. Vilard, “Confinement loss computations in photonic crystal fibers using a novel perfectly matched layer design,” in Proceedings of the COMSOL Multiphysics User’s Conference, Paris (2005), http://www .comsol.com/papers/1083/. 19. S. Kawakami and S. Nishida, “Characteristics of a doubly clad optical fiber with a low-index inner cladding,” IEEE J. Quantum Electron. 10, 879–887 (1974). 20. G. N. Watson, Theory of Bessel Functions (Cambridge University, 1944). 21. J.-P. Berenge, “Three-dimensional perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 127, 363–379 (1996).
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Key Laboratory of All Optical Network & Advanced Telecommunication Network of EMC, Beijing Jiaotong University, Beijing 100044, China 2
Institute of Lightwave Technology, Beijing Jiaotong University, Beijing 100044, China *Corresponding author: [email protected]
Received 24 October 2013; revised 28 December 2013; accepted 14 January 2014; posted 16 January 2014 (Doc. ID 200110); published 14 February 2014
An analytic method for solving the cut-off wavelength of single groove-assisted bend-insensitive fiber (BIF) was proposed. Combined with the concepts of frustrated total reflection and numerical aperture, the cut-off wavelength formula for the BIF was deduced. Experimental results showed that, compared to the traditional analytic and numeric methods, this method had higher accuracy in the calculation of cutoff wavelength for BIFs that had a large proportion of depressed inner-cladding layer, which significantly expanded the range of accurately predicting the cut-off wavelength. © 2014 Optical Society of America OCIS codes: (060.0060) Fiber optics and optical communications; (060.2310) Fiber optics; (060.2330) Fiber optics communications. http://dx.doi.org/10.1364/AO.53.001110
1. Introduction
With the vigorous development of fiber-to-the-home (FTTH) projects, the bend-insensitive fiber (BIF) has aroused widespread interest of researchers in recent years. In order to reduce the huge power loss caused by the small bending radius in the optical fiber, there has been a variety of BIF designs, including groove structure, hole-assisted structure, and nanohole structure [1–4], etc. Among them, the single-grooveassisted bend-insensitive fiber (SBIF) utilizes a depressed inner cladding (DIC) next to the core to help restrict the mode power, which guarantees excellent bend-insensitive characteristics [5–7]. It meets the ITU-T G.657 recommended standard and can be fully compatible with G.652.D generic optical fiber, which is widely used in today’s communication systems. In addition, simple DIC structure has advantages of convenient manufacture and low cost. Therefore 1559-128X/14/061110-08$15.00/0 © 2014 Optical Society of America 1110
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the SBIF is expected to realize the replacement of single-mode fiber (SMF) in the FTTH communication system and has attracted increasing attention. However, the International Telecommunication Union has not published any standard method for measuring cut-off wavelength of SBIF. Cut-off wavelength is one of the most important parameters for measuring the performance of optical fibers. The commonly used standard method for measuring cut-off wavelength of SMF is the bend reference method [8]. The spectral attenuation for the SMF with a loop of 60 mm diameter is measured in comparison with the straight fiber where the long wavelength edge of the bendinduced loss is larger than the long wavelength baseline by 0.1 dB is considered as the effective cut-off wavelength. Because the SBIF is not sensitive to bending, it’s difficult to measure the cut-off wavelength by existing standard methods. Several methods also have been reported to determine the cut-off wavelength of BIF, such as the multimode reference technique (MRT), the far-field MFD method, and the transverse splice loss technique (SLT) [9–11].
However, these methods are complex and need careful operation to carry out. Because of the difficulty for measurement, how to exactly predict the cutoff wavelength by simulating calculation is the key point for researching and designing the SBIF. Due to the setting of cladding region and boundary condition, the existing analytic method (e.g., transmission matrix method) and numeric method (e.g., finite element method) have limitations. If the DIC is thin, the calculation has a certain degree of accuracy. But once the DIC thickens, the deviation will be obvious. Consequently, existing methods need to be improved, or a new simulation method should be proposed to exactly predict the cutoff wavelength of the SBIF. This paper presented a new analytic method for calculating the cut-off wavelength of the SBIF. By using a simple three-layer planar model, and combined with the concept of frustrated total reflection (FTR) in the field of film and the traditional concept of numerical aperture of the optical fiber, we solved the equivalent total reflection critical angle. Then starting from the eigenvalue equation, we deduced the equivalent normalized cut-off frequency. The ultimate formula of cut-off wavelength, which was suitable for the SBIF, was figured out. This method greatly expanded the accurate prediction range of cut-off wavelength of the SBIF. It laid a theoretical foundation for future research and design of diversified and practical groove-assisted optical fibers. On the other hand, this groove-assisted structure could be transplanted into a variety of large-mode-area designs to improve the bending performance [7], which enabled this kind of fiber as a potential candidate for a high-power compact optical-fiber amplifier and laser. 2. Principle
For ordinary SMF with a step-index structure, there has been a perfect theory model to solve its cut-off wavelength. The analytic expression is [12] 2πa 2 n − n2clad 1∕2 : λc V c co
(1)
Given that the core radius is a, the refractive index of core and cladding are nco and nclad , respectively, the normalized cut-off frequency of second-order modes for the step-index fiber V c 2.4048, the cutoff wavelength λc can be calculated. The refractive index profile of the SBIF, shown in Fig. 1, is a dual-step-index structure. On the basis of the SMF, a DIC is added between the core and outer cladding, the thickness is d, the refractive index is nd , nd < nclad < nco . Δn nco − nd (a positive value) is the refractive index difference between core and DIC, Δn0 nd − nclad (a negative value) is the refractive index difference between DIC and outer cladding. Obviously Eq. (1) is no longer applicable for the optical fiber with multilayer structure. So we return to the light transmission theory. The transmission of light in optical fiber is essentially based on the total
Fig. 1. Refractive index profile of the SBIF.
reflection. The light should be totally reflected on the interface between core and cladding to continue propagating. For ordinary SMF, relative to the core, the cladding region can be considered as infinite. In this case, the total reflection occurs completely as long as the incident angle on the interface is equal to or greater than the total reflection critical angle. The cut-off wavelength can be expressed as λc
2πa sin θNA : Vc
(2)
θNA is the numerical aperture angle, which indicates the maximum reception capacity for the incident light of the optical fiber. Namely, in order to propagating in the fiber, the incident angle of the incident light on the interface between air and core should be less than or equal to θNA . As can be seen in Fig. 2, according to Snell’s law, the maximum incident angle (i.e., θNA ) on the fiber end meets that nair sin θNA nco cos θc . θc is the total refection critical angle on the interface between core and cladding, the refractive index of air nair 1. Equation (2) can be rewritten as λc
2πa n cos θc : V c co
(3)
We simplify the SBIF into three-layer planar structure, as illustrated in Fig. 3. In order to stimulate guide modes, the light that propagates in the core of the SBIF should be totally reflected on the interface between core and DIC. However, based on the theory of thin film optics, while the thickness of DIC is the order of wavelength, even if the incident angle is greater than the total reflection critical angle of the interface between core and DIC, the total reflection may not occur. From the energy point of view, the energy carried by light only partly returns
Fig. 2. Schematic of light propagation in the SMF. 20 February 2014 / Vol. 53, No. 6 / APPLIED OPTICS
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So the reflectivity R jrj2. Since light is incident at greater than the critical angle, sin θ1 > sin θc nd ∕ nco , δ is no longer a real number, but an imaginary number. We assume that δ iδ0 , a real number δ0 : δ0
to the core. This phenomenon is known as FTR [13]. FTR has had many applications in spectroscopy systems, such as thin-film device, interferometer, microporous membrane, and so on [14–16]. For SBIFs, the width of DIC is generally in the order of propagation wavelength. Therefore, FTR generates in the SBIF, resulting in the change of cut-off wavelength. Then in light of Eq. (3), the key for accurately predicting the cut-off wavelength of SBIF is to figure out the equivalent total reflection condition. Equation (3) should be rewritten as follows: 2πa n cos θe : V e co
(6)
Define reiθ Er ∕Ei , then reiθ r12 r23 ei2δ ∕1 r12 r23 ei2δ : 1112
r12 ∥
n2 cos θ1 − n1 cos θ2 ; n2 cos θ1 n1 cos θ2
(9)
r12 ⊥
n1 cos θ1 − n2 cos θ2 ; n1 cos θ1 n2 cos θ2
(10)
r23 ∥ −r32 ∥ −
n2 cos θ3 − n3 cos θ2 ; n2 cos θ3 n3 cos θ2
(11)
r23 ⊥ −r32 ⊥ −
n3 cos θ3 − n2 cos θ2 : n3 cos θ3 n2 cos θ2
(12)
The r∥ and r⊥ from Eq. (7) can be calculated by Eq. (8) to Eq. (12). R∥ jr∥ j2 and R⊥ jr⊥ j2 can be obtained:
(5)
Assuming Ei is the incident amplitude, at the interface between core and DIC the Stokes’ relations [17] connect the reflection and transmission coefficients r12 , t12 for light originating in the core and the coefficients r21 , t21 for the light originating in the DIC. There are r12 −r21 , t12 t21 r212 1. A similar set of relations applies to the mediums on the either sides of the interface between DIC and outer cladding. Then the reflected wave amplitudes back to the core are r12 Ei , t12 r23 t21 Ei ei2δ , t12 r23 r21 r23 t21 Ei ei4δ . Add these multiple reflection contributions, while making use of the Stokes’ relations to get the total reflected amplitude Er : Er Ei r12 r23 ei2δ ∕1 r12 r23 ei2δ :
(8)
(4)
V e and θe are the equivalent normalized cut-off frequency and equivalent total reflection critical angle of the dual-step-index structure, respectively. First, we use thin film interference theory to solve θe . As shown in Fig. 3, at the interface between core and DIC, θ1 is the incident angle, θ2 is the refracting angle, similarly at the interface between DIC and outer cladding, θ2 is the incident angle, and θ3 is the refracting angle. The phase thickness δ of DIC can be obtained easily. δ 2π∕λnd d cos θ2 . Using Snell’s law: nco sin θ1 nd sin θ2 nclad sin θ3 : q 2πd n2d − n2co sin2 θ1 : δ λ
q n2co sin2 θ1 − n2d :
In accordance with the Fresnel’s formula [17], the reflection coefficients for polarizations parallel with and perpendicular to the plane of incidence are
Fig. 3. Simple three-layer planar structure.
λc
2πd λ
(7)
APPLIED OPTICS / Vol. 53, No. 6 / 20 February 2014
R∥
A∥ sinh2 δ0 B∥ − 1 ; A∥ sinh2 δ0 B∥
(13)
R⊥
A⊥ sinh2 δ0 B⊥ − 1 ; A⊥ sinh2 δ0 B⊥
(14)
where A∥
i ih 2 n n2 sin2 θ1 −1 n32 1 sin2 θ1 −n32 1 2 1∕2 2 2 2 ; n3 n3 n1 2 2 4 n2 cosθ1 n2 −sin θ1 −1 n2 −sin θ1 2 2 1 h 2 1∕2 2 i2 n3 n −sin2 θ1 n32 cosθ1 n2 1 2 1∕2 ; B∥ n1 2 n3 2 3 4n2 cosθ1 n2 −sin θ1 1 1 2 2 n n1 −1 n32 −1 n22 2 2 1∕2 ; 2 A⊥ n2 n n1 2 1 4 n2 cosθ1 n2 −sin θ1 −1 n32 −sin2 θ1 2 2 1 h 2 1∕2 i n3 2 −sin θ1 cosθ1 n2 2 1∕2 : B⊥ 1 n 4cosθ1 n32 −sin2 θ1 n21 −1 n22
h
n23 −1 n22
1
n21 1 n22
Then the reflectivity R: R
R∥ R⊥ : 2
(15)
We consider two limiting cases. In the case of infinitesimal DIC, it’s equivalent to the structure with only core and outer cladding. The total reflection critical angle θ0 arcsinnclad ∕nco . In the case of infinite DIC, it’s equivalent to the structure with only core and DIC. The total reflection critical angle θ00 arcsinnd ∕nco . Accordingly, the equivalent total reflection critical angle should be located between θ0 and θ00 , θ00 < θe < θ0 . Assuming nco 1.448, nd 1.44, nclad 1.444, then 83.97° < θe < 85.74°. The reflectivity R changed by incident angles θ1 on the interface between core and DIC are shown in Fig. 4, with different thicknesses of DIC and core radius a 6.38 μm. One can see that, while d 0.5 μm, taking θ1 85.7°, which is considerably larger than the total reflection critical angle of the interface between core and DIC θ00 arcsinnd ∕nco ≈ 84°, a majority of power of the incident light still leak to the outer cladding and loss, the reflectivity is only about 60%. This phenomenon, namely, is FTR. Because of the role of FTR, the total reflection can’t occur. So we have to assume an equivalent total reflection condition for solving the equivalent total reflection critical angle. While the equivalent total reflection condition is satisfied, θe is equal to the corresponding incident angle. There are three ways to figure out the reflectivity, which meets the equivalent total reflection condition: (1) as the DIC layer is thin, calculations by the FE method are accurate, and the simulation results by FTR and FE methods should be consistent. Use the results for reverse solving the reflectivity. (2) From the confinement loss to determine the cut-off frequency; the confinement loss is related to the imaginary part of the effective refractive index of the mode [18]. The confinement loss of each mode determines whether the mode can be propagated. Then use the results to calibrate the reflectivity. (3) Use the experimental measurement results to calibrate the reflectivity. In these ways, the equivalent total reflection conditions for a class of
2
n
1 u2
wˆ12 Iˆ i
6 6 6 ρIˆ J 0 n2d Iˆ 0i 6 i uJ 2 ˆ ˆ nco wI i 6 det6 6 nIˆ j 1 − 1 6 1η w2 wˆ 2 6 4 2 0 n2d Iˆ 0j n K ρIˆ j nclad 2 wK − 2 ˆ ˆI n w co
co
j
n
1 u2
similar fiber structures can be solved. In this paper, we assume the equivalent total reflection condition is that the reflectivity reaches to 99.9% or more. This condition applies to the samples throughout this paper as well as similar fiber structures. For example, in Fig. 4, while d 9 μm, R 99.9% is corresponding to θ1 85.46°, so that θe θ1 85.46°. With the thickness of DIC increasing, θe gradually decreases within the interval from θ0 to θ00 . Here, 99.9% is not a completely fixed value, which can be freely adjusted to meet the need for precision. Consequently, the equivalent total reflection critical angle θe in Eq. (4) could be solved. Numerical aperture (NA) is the one of the most important indicators of the fiber. We introduce the concept of the equivalent numerical aperture NAe nco cos θe . NAe changes with the d and nd is shown in Fig. 5. ε Δn0 ∕Δn nd − nclad ∕nco − nd , absolute value of which reflects the depth of the DIC. Unless otherwise specified, the ε is consistent throughout the manuscript. The equivalent numerical aperture reflects the light gathering capacity of the fiber to a certain degree. For the SBIF with dual-step-index structure, the normalized cut-off frequency is also different from the step-index fiber. The eigenvalue equation takes the form [19]:
wˆ12 Kˆ i
n2 Kˆ 0 n2 dwˆ Kiˆ i co nKˆ j 1 1 1η w2 − w ˆ2 2 0 n2 Kˆ 0 n K d j ρKˆ j nclad 2 wK − 2 ˆ Kˆ n w ρKˆ i
Fig. 4. Reflectivity as a function of incident angles with various thicknesses of DIC and core radius a 6.38 μm.
J0 uJ
co
co
j
0 Iˆ 0 J ρIˆ i uJ wˆ Iˆi i 1 1 ˆ n u2 wˆ 2 I i 0 Iˆ 0 K ρIˆ j wK − wˆ Iˆj j nIˆ j 1 1 2 1η w2 − w ˆ
0 3 Kˆ 0 J ρKˆ i uJ wˆ Kˆi i 7 7 1 1 ˆ 7 n u2 wˆ 2 K i 7 7 0; 0 ˆ0 7 K j K ρKˆ j wK − wˆ Kˆ 7 7 j 7 5 nKˆ j 1 1 2 1η w2 − w ˆ
20 February 2014 / Vol. 53, No. 6 / APPLIED OPTICS
(16)
1113
Fig. 5. Equivalent numerical aperture as a function of innercladding thickness for various values of the inner-cladding refractive index.
Fig. 6. Equivalent normalized cut-off frequency as a function of inner-cladding thickness for various values of the inner-cladding refractive index.
where
where J J n u; ˆ Kˆ i K n w; ˆ Iˆ i I n w;
K K n 1 ηw;
ˆ c n2clad − n2d ∕n2co − n2clad 1∕2 vc : w
ˆ Kˆ j K n 1 ηw; ˆ Iˆ j I n 1 ηw;
and the mode parameters: u an2co k2 − β2 1∕2 ; ˆ aβ2 − n2d k2 1∕2 ; w w aβ2 − n2clad k2 1∕2 ; v u2 w2 1∕2 akn2co − n2clad 1∕2 : J n , I n , and K n are the Bessel and modified Bessel functions, n is the azimuthal mode number, ρ nco k∕β, k is the free-space wave number, β is the modal propagation constant, η d∕a. For the TE and TM modes, n 0. The cut-off frequency of the mode is defined to be the frequency at which the modal propagation constant is equal to the wave number in the outer cladding, i.e., β nclad k, so that w 0, uc vc , where the subscript “c” denotes the value at cutoff. For small w, using small variable approximation [20]: K 0 1 ηw K0 0 ∼ 1 ηw2 log w−1 : wK wK 0 1 ηw
(17)
Then the eigenvalue that is the equivalent normalized cut-off frequency V e vc could be found from the following equation by simplifying Eq. (16): 0 ˆ c ˆ c J 0 vc K 00 w K 0 w ˆ c K 0 w ˆ c vc J 0 vc w ˆ c K 0 1 ηw 0 0 ˆ c ˆ c J 0 vc I 0 w I 0 w ; ˆ c vc J 0 vc w ˆ c ˆ c I 0 w I 0 1 ηw 1114
The normalized cut-off frequency of the step-index fiber is only relevant to the refractive index of core and outer cladding. But the thickness of DIC of SBIF also affects the value of the equivalent normalized cut-off frequency. While the refractive index of core and outer cladding and diameter of core and outer cladding are fixed, the equivalent normalized cut-off frequency as a function of inner-cladding thickness for various values of the inner-cladding refractive index is shown in Fig. 6. As can be seen from the figure, V e increases by enlarging the width of the DIC and then tends to be a constant value vs . V e is proportional to the depth of the DIC. vs can be determined by ˆ c J 00 vs K 00 w − ; ˆ c ˆ c K 0 w vs J 0 vs w
(19)
where ˆ c n2clad − n2d ∕n2co − n2clad 1∕2 vs : w Equation (19) indicates that vs is only associated with the refractive index of each layer but is independent from the thickness of each layer. So as long as the structural parameters are determined, the equivalent normalized cut-off frequency V e in Eq. (4) could be calculated. Known nco , nd , nclad , a, d, the cut-off wavelength of SBIF can be obtained by solving the combined equation of Eqs. (4), (15), and (18). 3. Simulation Results and Experimental Verification
18
APPLIED OPTICS / Vol. 53, No. 6 / 20 February 2014
For the case with the same structure, the simulation results of the cut-off wavelength for SBIF with FTR method and finite element (FE) method are compared in Fig. 7. d∕a reflects the thickness of DIC; ε
Fig. 7. Comparison of simulation results for the cut-off wavelength of SBIF with FTR method and FE method.
reflects the depth of DIC. The full vectorial finite element method is performed with the commercial software Comsol Multiphysics. The cut-off frequency of the mode is defined to be the frequency at which the effective index of the mode is equal to the refractive index of the outer cladding. A circular perfectly matched layer (PML) [21] in association with a perfect electric conductor is used to truncate the computational domain. The width of the PML is set at 20 μm. As d increases, λc sharp declines and then approximately remains unchanged. However, the result of the FTR method is different. The part of the decline completely coincides with the FE method. Then λc doesn’t remain constant, whereas λc gradually rises, which means that the core is weakly influenced by the outer cladding if the thickness of DIC is large enough. Meanwhile, the lower the refractive index of the DIC, the greater the amplitude of change of λc. We investigate the energy intensity distribution of the second-order mode of SBIF. Assuming the power factor Γ ∬d P∕∬all P, which is the ratio of energy integral on the area of DIC and energy integral on the whole mode field area. Γ changes with the inner-cladding thickness while ε is fixed (Fig. 8). One can see that the energy intensity distributed on the inner cladding increases with the inner-cladding thickness rapidly at first and then slowly declines, which means the loss of the second-order mode experiences a process of first increasing and then decreasing. The losses of second-order modes increase, which means second-order modes will be cut off more easily, resulting in single-mode transmission. Therefore the variation of the loss of second-order mode partly leads to the change of cut-off wavelength. The corresponding energy intensity distributions for three representative points are also simulated (displayed in the illustrations of Fig. 8). The same polarization direction (arrow surface) of the electric fields represents that they belong to the same mode. In addition, with the change of inner-cladding thickness, the energy intensity distributed on core and the effective areas also change. The combination of these factors leads to the movement of the cut-off wavelength.
Fig. 8. Power factor Γ as a function of inner-cladding thickness and corresponding energy intensity distributions of second-order mode.
For the reliability of the experimental results, we applied three measurement methods for BIFs to measure the cut-off wavelengths of four laboratory made SBIF samples, and compared experimental results with simulation results. The microscope photograph of one of the sample fiber is shown in Fig. 9. Three measurement methods: (1) the improved bend reference technique (BRT) used Photon Kinetics 2210 optical fiber analysis systems; the length of measured fiber is 2 m. Because the bending loss for SBIFs at the standard loop diameter of 60 mm is almost negligible, the higher-order modes cannot be removed. With regard to different samples (different bending losses), 3–8 small loops of 20 mm diameter were used to replace with the big loop of 60 mm diameter in the traditional method, which increases the loss of higher-order modes and forms a reference light path involved to determine the cut-off wavelength. (2) The MRT also used Photon Kinetics 2210 optical fiber analysis systems,
Fig. 9. Microscope photograph of the sample. 20 February 2014 / Vol. 53, No. 6 / APPLIED OPTICS
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compared with BRT that winding a small loop to obtain a reference light path, the MRT replaced the measured fiber with a short piece of multimode fiber. The power spectrum of the same wavelength range was measured as the reference transmission power spectrum, which was involved in determining the cut-off wavelength. The length of the multimode fiber (Yangtze Optical Fibre & Cable) is 1 m. (3) The transverse SLT [11]: two of same fiber sample (about 0.5 m long) were fixed and aligned in fiber fusion splicer, and their facets were separated less than 5 μm. Through the electronic screen of fiber fusion splicer, it’s easy to transversely move one of the fibers (parallel to the end face) for the distance of radius of core. A broadband light source was used as the input, and the output power was measured by an optical spectrum analyzer. The splice loss was measured with reference to the perfectly aligned fibers. At the vicinity of the cut-off wavelength, the transverse splice loss was the lowest and jumped to the high value when only LP01 mode propagated in the fiber. The wavelength where we got the lowest splice loss was the cut-off wavelength. The sample structure parameters, simulation results, and experiment results are listed in Table 1 and shown in Fig. 10. In Fig. 10, the simulation data uses the same core radius and epsilon as four samples and contains the simulation results (FTR method and FE method) and experimental measurement results (BRT method and SLT method) of accurate structure parameters of four samples. They are marked by special symbols, e.g., hollow triangle, hollow square, solid circular, and hollow star. It can be observed that the structure parameters of Sample 1 and Sample 2 belong to or are near the overlapped part for the FTR method and FE method. The simulation results with FTR method and FE method are similar and slightly differ from each other. The experimental results are not far off the simulation results. However, the structure parameters of Sample 3 and Sample 4 obviously belong to the separate part. The simulation results of the FE method have a large deviation from the results of the FTR method, Table 1.
Structure Parameters, Simulation Results, and Experiment Results
SBIFs Samples nco nd nclad a (μm) d (μm) Cut-off wavelength (nm)
1
FTR FE BRT MRT SLT
Sample 4 at 1550 nm Bending radius (mm) 5 Macro-bend loss 0.3464 (dB/loop)
1116
2
3
4
1.4472 1.4472 1.4481 1.4468 1.441 1.4408 1.4401 1.4405 1.444 1.444 1.444 1.444 5.46 6.3 6.62 7.18 8.54 8.7 11.2 12.6 1087 1301 1636 1565 1090 1250 1495 1310 1035 1247 1579 1507 1035 1252 1573 1506 1030 1250 1590 1520 7.5 0.1157
10 0.0684
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Fig. 10. Simulation results and experimental results of samples.
and the actual experimental results are significantly closer to the results of the FTR method. Overall, the theoretical and experimental values well match the used FTR method for the DIC profile. However, the theoretical values are generally larger than the experimental values about 50 nm [3], which are same as well-known phenomenon for SMF. There are inevitable error factors in manufacturing process and from the external experiments, such as bending leakage losses, torsion losses, stress losses, and so on, that make it easier for higher-order modes to be lost and easier to be cut off. Experiments show that compared to the traditional analytic and numeric methods, the FTR method has higher accuracy in the calculation of the cut-off wavelength for SBIFs that have a large proportion of the DIC layer. The FTR method significantly expands the range of accurately predicting the cut-off wavelength of SBIFs. 4. Conclusion
An analytic method for solving the cut-off wavelength of the single groove-assisted BIF with DIC was presented. We used a simple three-layer planar model and the eigenvalue equation, which were combined with the concept of FTR in the field of film and the traditional concept of numerical aperture of the optical fiber to figure out the equivalent total reflection critical angle and the equivalent normalized cutoff frequency. Then the ultimate calculation formula of the cut-off wavelength was deduced. It turned out that the simulation result of the FTR method proposed in this paper was different from the result of the traditional finite element method if the DIC had a certain thickness and depth. Experimental results indicated that compared to the traditional analytic and numeric methods, the FTR method had higher accuracy in the calculation of the cut-off wavelength for BIFs, which had a large proportion of DIC layer. This method greatly expanded the accurate prediction range of cut-off wavelength of the SBIF. It laid a theoretical foundation for future research and design of diversified and practical grooveassisted optical fibers.
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