Properties of elliptical-core two-mode fiber

Polarization and modal birefringence of elliptical-core two-mode fibers are investigated. Wavelengths corresponding to zero group delay difference (GDD) between the two spatial modes and between the orthogonal polarizations are computed when the fiber parameters, i.e., the relative core/cladding index difference and the ratio of major over minor axis, are varied. Simple relationships between the zero GDD wavelengths and fiber parameters are obtained. With proper fiber design, zero GDD between the two spatial modes and the two orthogonal polarizations can be achieved at the same wavelength. ©2005 Optical Society of America OCIS codes: (060.2270) Fiber characterization; (060.2280) Fiber design and fabrication; (060.2340) Fiber optics components References and links 1. S. Ramachandran, “Novel photonic devices in few-mode fibres,” IEE Proc.Circuits Devices Syst. 150, 473479 (2003). 2. A.M. Vengsarkar, W.C. Michie, L. Jankovic, B. Culshaw, and R.O. Claus, “Fiber-optic dual-technique sensor for simultaneous measurement of strain and temperature,” J. Lightwave Technol. 12, 170-177 (1994). 3. H.S. Park, K.Y. Song, S.H. Yun, and B. Y. Kim, “All-fiber wavelength-tunable acoustooptic switches based on intermodal coupling in fibers,” J. Lightwave Tech. 20, 1864-1868 (2002). 4. K. Y. Song, and B.Y. Kim, “Broad-band LP01 mode excitation using a fused-typed mode-selective coupler,” IEEE Photon. Technol. Lett. 15, 1734-1736 (2003). 5. H.S. Park, S.H. Yun, I.K. Hwang, S.B. Lee, and B.Y. Kim, “All-fiber add-drop wavelength-division multiplexer based on intermodal coupling,” IEEE Photon. Technol. Lett. 13, 460-462 (2001). 6. W.V. Sorin, B.Y. Kim, and H.J. Shaw, “Highly selective evanescent modal filter for two-mode optical fibers,” Opt. Lett. 11, 581-583 (1986). 7. S.H. Yun, I.K. Hwang, and B.Y. Kim, “All-fiber tunable filter and laser based on two-mode fiber,” Opt. Lett. 21, 27-29 (1996). 8. K.A. Murphy, M.S. Miller, A.M. Vengsarkar, and R.O. Claus, “Elliptical-core two-mode optical-fiber sensor implementation methods,” J. Lightwave Technol. 8, 1688-1696 (1990). 9. B.Y. Kim, J.N. Blake, S.Y. Huang, and H.J. Shaw, “Use of highly elliptical core fibers for two-mode fiber devices,” Opt. Lett. 12, 729-731 (1987). 10. J.N. Blake, S.Y. Huang, B.Y. Kim, and H.J. Shaw, “Strain effects on highly elliptical core two-mode fibers,” Opt. Lett. 12, 732-734 (1987). 11. S.Y. Huang, J.N. Blake, and B.Y. Kim, “Perturbation effects on mode propagation in highly elliptical core two-mode fibers,” J. Lightwave Technol. 8, 23-33 (1990). 12. K.Y. Song, I.K. Hwang, S.H. Yun, and B.Y. Kim, “High performance fused-type mode-selective coupler using elliptical core two-mode fiber at 1550nm,” IEEE Photon. Technol. Lett. 14, 501-503 (2002). 13. J. Ju, W. Jin, and M.S. Demokan, “Two-mode operation in highly birefringent photonic crystal fiber,” IEEE Photon. Technol. Lett. 16, 2472-2474 (2004). 14. A.W. Snyder and X.-H. Zheng, “Optical fibers of arbitrary cross sections,” J. Opt. Soc. Am. A 3, 600-609 (1986). 15. K. Thyagarajan, S.N. Sarkar, and B.P. Pal, “Equivalent step index (ESI) model for elliptic core Fibers,” J. Lightwave Technol. LT-5, 1041-1044 (1987). 16. Masashi Eguchi, Masanori Koshiba, “Accurate finite-element analysis of dual-mode highly elliptical-core fibers,” J. Lightwave Technol. 12, 607-613 (1994). 17. Robert L. Gallawa, I. C. Goyal, Yinggang Tu, and Ajoy K. Ghatak , “Optical waveguide modes: an approximate solution using Galerkin’s method with Hermite-Gauss basis functions,” IEEE J. Quan. Electron. 21, 518-522 (1991). 18. Tzong-Lin Wu, and Hung-chun Chang, “An efficient numerical approach for determining the dispersion characteristics of dual-mode elliptical-core optical fibers,” J. Lightwave Technol. 13, 1926-1934 (1995). 19. R.B. Dyott, Elliptical Fiber Waveguides, (Boston, Artech House, 1995). (C) 2005 OSA 30 May 2005 / Vol. 13, No. 11 / OPTICS EXPRESS 4350 #7093 $15.00 US Received 7 April 2005; revised 23 May 2005; accepted 24 May 2005 20. C.D. Poole, J.M. Wiesenfeld, and D.J. Digiovanni, “Elliptical-core dual-mode fiber dispersion compensator,” IEEE Photon. Technol. Lett. 5, 194-197 (1993). 21. C.D. Poole, J.M. Wiesenfeld, D.J. Digiovanni, and A.M. Vengsarkar, “Optical fiber-based dispersion compensation using higher order modes near cutoff,” J. Lightwave Technol. 12, 1746-1758 (1994). 22. Wang Zhi, Ren Guobin, Lou Shuqin, Liang Weijun, “Investigation of the supercell based orthonormal basis function method for different kinds of fibers,” Opt. Fiber Technol. 10, 296-311 (2004). 23. J.D. Joannopoulos, R.D. Meade, J.N. Winn, Photonic crystals: molding the flow of light, (New York, Princeton university press, 1995). 24. J. Blake, M.C. Pacitti, S.L.A. Carrara, “Splitting of the Second Order Mode Cutoff Wavelengths in Elliptical Core Fibers,” Optical Fiber Sensors Conference, 1992. 8th, Jan. 29-31, 125–128 (1992).


Introduction
Two-mode optical fibers have been studied for a number of devices and sensors applications [1][2][3][4][5][6][7][8][9][10][11][12][13].Two-mode operation can be achieved by operating a conventional circular-core fiber below cut-off and hence it supports the fundamental LP 01 and the second order hybrid LP 11 modes.The performance of these devices is however not stable because the hybrid LP 11 is actually consists of four true modes that have slightly different propagation constants and coherent mixing of these modes results in variation of the hybrid mode intensity pattern along the fiber length and with environmental disturbance.The same applies to the polarization states of the fundamental mode.These problems can be overcome by elliptical core fibers [5][6][7][8][9][10].In an elliptical core fiber, the hybrid LP 11 mode split into two groups, i.e., LP 11 odd and LP 11 even modes, with well defined mode intensity patterns.The LP 11 odd and LP 11 even modes have significantly different cut off wavelengths, which allows the existence of a wavelength range within which only LP 01 and LP 11 even modes are supported by the fiber.
The elliptical-core fibers that support two stable spatial modes, i.e., the LP 01 and LP 11 even modes, are named elliptical-core two-mode fibers.The applications of these fibers include interferometric modal/polarimetric sensors that can be used to measure strain, temperature or both at the same time [2,[8][9][10].The performances of the interferometric systems are affected by two important parameters, i.e., the group delay difference (GDD) and the phase delay difference (PDD) between the two spatial modes or between the two orthogonal polarizations of the same spatial mode.For example, in a two-mode fiber interferometer in which the GDD between two modes can be designed to approximately zero, a light source with a short coherence length can be used without losing interference signals even though large PDD between the two modes may be employed [9].There are some analyses about the ellipticalcore two-mode fibers, which paid attention to modeling the dispersion properties [14][15][16][17][18][19][20][21][22] of the x-and y-components of the fundamental mode (LP 01 x , LP 01 y ) and the dispersion properties of the second order even mode (LP 11 even) near cutoff.However, to the best of our knowledge, little or no effort has been made to study the group delay difference between the two well-confined (away from cutoff) spatial modes as functions of fiber parameters, which are important for optimizing the performance of two mode interferometers.
In this paper, we study the modal behavior of step-index elliptical-core two-mode fibers, with emphases on the group delay difference between the two confined spatial modes, and the dependence of the characteristic wavelengths on the fiber structural parameters, (i.e., the cutoff wavelengths of the LP 11 even and odd modes, the wavelength corresponding to zero-GDD between two spatial modes and between two orthogonal polarizations of the same spatial mode), which is believed to be useful for designing fiber for interferometer sensor applications.The relationship between zero-GDD wavelengths of the two orthogonal polarizations of both LP 01 and LP 11 even modes and the fiber parameters are presented in Section 2; the properties of the two-mode elliptical core fibers, including the cutoff wavelength of the even and odd LP 11 modes, the zero-GDD wavelength between the two spatial modes and the modal beat length at this wavelength are presented in Section 3; a particular fiber, which has the same zero-GDD wavelength for the two orthogonal polarizations of the fundamental mode and for the two spatial modes, is presented in Section 4.

Zero GDD wavelength for two orthogonal polarizations
Figure 1(a) shows the cross-section of a step index elliptical-core fiber (ECF), which is characterized by three parameters: a, the semi major axis; b, the semi minor axis; and Δ, the core-cladding relative refractive index difference.The polarization modal birefringence (PMB) Δβ, which is defined as the difference between the propagation constants of the orthogonal polarization components of the mode (i.e., Δβ PMB1 between LP 01 x and LP 01 y , Δβ PMB2 between LP 11 x even and LP 11 y even), tends to zero at both short and long wavelengths and reaches its maximum value at an intermediate wavelength [19].Take the elliptical-core fiber with the structural parameters of a=3μm, Δ=0.5%, and the aspect ratio η=a/b=2.5, as an example, Δβ and the GDD between both polarization components of both LP 01 and LP 11 even modes are plotted in Fig. 1(b).At λ MPMB1 , Δβ PMB1 reaches its maximum, the GDD between the two orthogonal polarizations of LP 01 mode, i.e., equals to zero, where c is the light velocity in vacuum.Hence λ MPMB1 is also the wavelength (λ 0p1 ) at which the GDD between the two orthogonal polarizations is zero.   .The four lines are plotted by using Eq.(2a), which is fitted from the numerical results.
From Fig. 1(b), it can also be seen that Δβ PMB2 reaches its maximum at an intermediate wavelength λ MPMB2 which is also the wavelength λ 0P2 at which the GDD between the two orthogonal polarizations of the LP 11 even mode is zero.The relationship between the wavelength λ 0P2 and fiber structure parameters can also be fitted to a simple form as given by Eq. (2b).The numerical and the fitted results are plotted in Fig. 1(d) as data points and lines.
Based on the scaling property of the Maxwell's equations [23], the wavelength in Fig. 1(c), (d) and in Eq.( 2) are normalized to the semi major axis (a) of the elliptical core and should be applicable to any value of a. λ 0p1 (λ MPMB1 ) and λ 0p2 (λ MPMB2 ) are proportional to Δ 1/2 .The results obtained from Eq. ( 2) have good agreement with those reported by others [16,19], which is helpful for designing an ECF to achieve the max-PMB at a particular wavelength.The splitting of the cutoff wavelengths of LP 11 even and odd modes, shown as λ c1 and λ c2 in Fig. 2(a), has been reported by J. Blake et al. [24].The beat length between the first and second order spatial modes, and the wavelength at which the first and second order spatial modes propagated with the same group velocity have however not been accurately predicted to our knowledge.These calculations are much more difficult than predicting the general modal behavior because the results depend on the difference in propagation constants between two spatial modes.

properties of two-mode fibers
3.1 cutoff wavelengths of LP 11 even and LP 11 odd modes λ c1 and λ c2 are the cutoff wavelength of LP 11 even and LP 11 odd modes, respectively.The ECF will support two modes (LP 01 and LP 11 even) within the wavelength range of from λ c2 to λ c1 .Figure 2(b) and 2(c) shows the dependences of λ c1 and λ c2 on the fiber parameters (Δ and η), respectively.Again, the data points are numerically computed values and the lines are the fitted curves of according to Eq. (3a) and (3b), respectively.
For comparison, the ratio of λ c1 to λ c2 calculated from Eq. ( 3) and that obtained from ref.
[24] are plotted in Fig. 2(d), where the numerical results of Δ=0.2% and Δ=1% are also plotted.It shows that the fitted expressions have a good agreement with Ref. [24].

zero-GDD wavelength
In a two-mode fiber, GDD (  In general, Δτ gs , Δτ ps and Δβ SMB are all polarization dependent.The GDD, PDD and SMB between the y polarization components of the two spatial modes for the ECF with parameters given in Fig. 1(b) are plotted in Fig. 3(a).As expected, the wavelength λ 0s where the GDD is zero equals to the wavelength λ MSMB where Δβ SMB is maximized, and the GDD equals to the PDD at the wavelength λ Pmax where PDD is maximized.Figure 3(b) shows the relationship between λ 0sy (subscript 'y' represents the y polarization component) and Δ for various values of η.Again, the data points are obtained from numerical calculation and are fitted into a simple formula given in Eq. (5a).The four lines are obtained from Eq. (5a) and agree well with the numerical results.Eq. (5b) shows the fitted form of the zero-GDD wavelength of the x-components between LP 01 and LP 11 even modes.It is obvious that there is only a little difference between the results for the orthogonal polarization components.3) and ( 5), it can be found that the zero-GDD wavelength is outside the twomode range, i.e., λ 0s < λ c2 , when η is less than ~1.90.Fig. 4, in which λ/(a•sqrt(Δ)) are plotted from Eq. (2), Eq. (3) and Eq. ( 5), also demonstrates the critical value of the aspect ratio η c ≈ 1.90.This means that the first and second order modes can not propagate with the same group velocity in a two-mode ECF with η< 1.90.From Fig. 4, it can also be seen that the zero GDD between the two spatial modes and the zero GDD (or maximum birefringence) between the orthogonal polarizations of the fundamental modes can be achieved at the same wavelength (λ 0s = λ 0P1 ) at η ≈ 2.4.This will be further discussed in section 4.

beat length at zero-GDD wavelength
Polarization and modal beat lengths are important parameters for polarimeters and modal interferometers.In a two-mode elliptical core fiber, four possible interferometers may be implemented, i.e., polarimeters formed between the orthogonal polarizations of the LP 01 and LP 11 even modes; and modal interferometers formed between the LP 01 and LP 11 even modes for each of the orthogonal polarizations.Hence, four beat lengths, i.e., polarization beat length for LP 01 mode (L PMB1 =2π/Δβ PMB1 ), the polarization beat length for LP 11 even mode (L PMB2 =2π/Δβ PMB2 ), the spatial modal beat length (L SMBy =2π/Δβ SMBy ) for the y-polarization component, and the spatial modal beat length (L SMBx =2π/Δβ SMBx ) for the x-polarization component, need to be considered for the corresponding interferometers.(6c) The beat length L SMBx is not shown because it is not an independent quantity and can be obtained from the other three beat lengths through Eq. (6d).The polarization modal beat lengths (L PMB1 and L PMB2 ) have Δ -3/2 dependence, and the spatial modal beat lengths (L SMBy and L SMBx ) have a Δ -1/2 dependence.
In design two-mode interferometers based on elliptical-core fibers, the properties of the fiber, including the two-mode wavelength range, the zero-GDD wavelengths and the beat lengths at the these wavelengths, can be estimated from Eqs. (3), ( 5) and ( 6), respectively, and hence the performance of the interferometer.

A particular elliptical core fiber
From Eqs. ( 2) and ( 5), or Fig. 4, it can be seen that λ MPMB1 (i.e., λ 0P1 ) equals to λ MSMB (i.e., λ 0s ) when η≈ 2.4.Fibers satisfy this condition can have zero GDD between the two orthogonal polarizations and the two spatial modes at the same wavelength of λ 0 ≈ 3.11a•sqrt(Δ).Fig. 6 shows the Δβ PMB1 , Δβ SMBy and GDD as function of wavelength for a particular elliptical-core fiber with η=2.4,a=5μm, Δ=1%.This fiber has a zero-GDD wavelength for both polarizations and spatial modes at around 1.55μm.The zero-GDD wavelengths (λ 0sy and λ MPMB1 ), the spatial modal beat length and the polarization modal beat length at λ 0sy , are presented in Table 1.The difference between the results obtained numerically and those obtained from the fitted formulas compared well with errors given in the last column of the table.The value of λ 0sx /a, which is not listed in the table, equals to 0.310434 and is very close to λ 0sy.

Conclusion
In conclusion, the modal properties of the step-index elliptical-core two-mode fibers are investigated.It is found that the wavelengths corresponding to zero GDD between the two spatial modes and between the orthogonal polarization components, and the polarization and modal beat lengths at the zero GDD wavelengths have simple relationships with the aspect ratio η and relative core/cladding refractive index Δ.These relationships can be used to design two-mode fibers and two-mode fiber interferometers with optimized performance.

Figure 1 (
Figure 1(c) shows the relationship between λ 0P1 and Δ for different aspect ratio η.The data points '+', 'o', '*' and '□' are the results from numerical calculations using a super-cell

Fig. 2 .
Fig. 2. (a) The effective mode index of an ECF with the parameters same as given in Fig. 1(b); (b) and (c) the numerical and fitted results of the cutoff wavelengths of LP 11 even and LP 11 odd modes; (d) the ratio of the even to odd LP 11 mode cutoff wavelengths as a function of aspect ratio.

Figure 2 (
Figure2(a) shows the effective mode index (n eff ) of the y-polarization of three lowest-order modes (LP 01 , LP 11 even, LP 11 odd) of an ECF with the same parameters as given in Fig.1(b).λ c1 and λ c2 are the cutoff wavelength of LP 11 even and LP 11 odd modes, respectively.The ECF will support two modes (LP 01 and LP 11 even) within the wavelength range of from λ c2 to λ c1 .Figure2(b) and 2(c) shows the dependences of λ c1 and λ c2 on the fiber parameters (Δ and η), respectively.Again, the data points are numerically computed values and the lines are the fitted curves of according to Eq. (3a) and (3b), respectively.

Fig. 3 .
Fig. 3. (a) The GDD, PDD and the SMB of an ECF with the same parameters as given in Fig.1(b), (b)The relationship between zero-GDD wavelength λ 0s and the fiber parameters η and Δ.

Fig. 6 .
Fig.6.The PMB, SMB and GDD of a two-mode ECF with the structural parameters a=5μm, η=2.4,Δ=0.01.The PMB is amplified by 100 times in order to be plotted with the SMB.

Table 1 .
The fitted and the numerical results of a two-mode ECF with a=5μm, η=2.4,Δ=0.01.