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JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 27, NO. 22, NOVEMBER 15, 2009 4919 Guided Wave Analysis of Hollow Optical Fiber for Mode-Coupling Device Applications Sejin Lee, Jiyoung Park, Yoonseob Jeong, Hojoong Jung, and Kyunghwan Oh, Member, IEEE Abstract—In this paper, optical mode characteristics of hollow optical fibers are thoroughly analyzed using finite element method. The guided modes along the ring core and cladding are identified and their optical properties are investigated. For the core modes, we investigated intensity distribution, higher order mode cutoff, propagation constant, and chromatic dispersion. The mode coupling between the fundamental mode and the excited modes in both core modes and cladding modes are discussed for applications in mode-coupling devices. Index Terms—Finite element methods, optical fibers, waveguides. Fig. 1. (a) Waveguide structure and (b) cross section of a HOF. I. INTRODUCTION C ONVENTIONAL optical fibers based on step index structure have led the revolution of telecommunications in the last century. There have been intensive research and development efforts to modify the optical fiber waveguide structures to flexibly control propagation properties such as the chromatic dispersion and its slope, polarization of guided modes, bending mode cutoff within optical communication bands loss, and . These optical fibers extended their applications from optical communications to sensor , endoscope , and high-power laser . Recently, air–silica structures have been embedded in optical fibers to result in varieties of novel fibers: microstructured hole fibers , , omniguide fibers , , and hollow IR transmitting fibers . Another alternative structure of optical fibers called hollow optical fiber (HOF) has been recently proposed by the authors for various photonic device applications . There have been intensive research reports on hollow photonic bandgap fibers, where the light is guided along the central air hole region –. In contrast to those photonic bandgap fibers, the HOF in this study guides the light by the index difference between the ring core and other boundaries in the central air hole and the silica cladding. Therefore, the majority of the light is guided along the ring core region along with the evanescent wave in the air hole region. The guided modes in the HOF, therefore, can be flexibly controlled along with excellent longitudinal uniformity in optical characteristics. Schematic structure of HOF is shown in Fig. 1, HOF consists of three-layered structure. The central air hole of radius is Manuscript received March 03, 2009; revised June 01, 2009, July 09, 2009. First published July 21, 2009; current version published September 10, 2009. This work was supported in part by the KOSEF under Grant ROA-2008-00020054-0 and Grant R15-2004-024-00000-0, in part by the KICOS under Grant 2008-8-1893(GOSPEL project), in part by the ITEP under Grant 2008-8-1901 and Grant 2009-8-0809, and in part by the Brain Korea 21 Project of the KRF. The authors are with the Institute of Physics and Applied Physics, Yonsei University, Seoul 120-749, Korea (e-mail: [email protected]). Digital Object Identifier 10.1109/JLT.2009.2028162 surrounded by the germanosilicate glass ring core that carries most of the guided core modes, and the cladding of radius is made of fused silica. The ring core has the inner and outer radii . of and along with the thickness HOF has drawn attention due to its unique advantages in the following notions: 1) fiber devices can be fabricated by after filling novel materials such as gas, liquid, and solid colloid in the hollow region , 2) high form-birefringence could be induced by elliptical ring core , and 3) HOF provides inherently high compatibility with a standard single-mode fiber (SMF) by adiabatic mode transformation . In comparison to various experimental investigations, theoretical analysis on HOF has been rather scarce and limited. It is only recent that the guided core modes have been identified in highly multimode HOFs , . However, these multimode HOFs are not practical in device applications due to severe bending losses and modal interferences, which is detrimental to mode-coupling device applications. In mode-coupling devices such as long-period grating (LPG)  or acousto-optic tunable filters (AOTF) , the coupling between the fundamental core mode and a few excited modes either in the core or the cladding has been of primary concern. In the case of HOF, unique mode-coupling properties have not been fully exploited thus far. Especially, the phase matching conditions for “the fundamental core modes to excited core modes coupling” and “the fundamental core mode to the cladding modes” in HOF have not been investigated at all. In this paper, we systematically analyzed the guided modes both along the ring core and the cladding along with the phase matching conditions among them, for the first time to the best knowledge of the authors. Utilizing a finite element method (FEM) package, we report both the core and the cladding modes and their modal distributions along with the normalized propagation constants and chromatic dispersion. We also calcubetween lated the difference in the propagation constant the fundamental mode and excited modes both in the core and 0733-8724/$26.00 © 2009 IEEE Authorized licensed use limited to: Yonsei University. Downloaded on October 13, 2009 at 02:02 from IEEE Xplore. Restrictions apply. 4920 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 27, NO. 22, NOVEMBER 15, 2009 cladding to understand unique phase matching conditions in HOF for mode-coupling device applications. Sellmeier coefficients for SiO  and GeO in  were used for the calculation. The refractive index of the SiO cladding and the GeO -doped SiO ring core was derived from II. MODE ANALYSIS METHOD AND MODAL CHARACTERISTICS OF HOF (3) A. Mode Analysis Methodology The electric fields of guided modes in HOF could be obtained by solving the Maxwell’s equations for the given boundary conditions. For the propagating modes along -direction in HOF, the transverse components are derived from the longitudinal . component For a given waveguide structure as in Fig. 1(a), the longitudinal component of the electric field in a core mode is analytically given as (1) Here , and are Bessel functions of integer order . Characteristic equation can be derived for the given boundary conditions to satisfy continuity requirements for the fields and their derivatives. The propagation constant and subsequent propagation properties can be obtained from the characteristic equation. Although analytical solutions exist for this simple structure, we have chosen to use the FEM because it can easily be adapted to include the distortions in fabrication unavoidable in realistic structures. FEM has been one of the most accurate methods to investigate the propagation properties of guided modes providing highly reliable estimates on modal characteristics such as effective refractive index, field distribution, and cutoff wavelength –. In FEM analysis of an optical waveguide starts from the scalar wave equation where the mode field function satisfies equation (2) Subdividing the domain of interest into triangular meshes, a set of matrix equations is obtained applying the variational principle, and the eigenvalues of the equation could be solved for the given boundary conditions using FEM . In this study, we adopted FEM  to obtain modal characteristics for HOF such as effective refractive indices, electric field distribution, and modal intensity for both the core modes and the cladding modes. where SA, Sl, GA, and Gl are the Sellmeier coefficients for the SiO and GeO glasses, and is the mole fraction of GeO . C. Electric Field Distribution and Intensity Profiles of Guided Modes in HOF Using FEM analysis, the fundamental mode and the first five excited modes guided along the ring core were identified as in Fig. 2 at the wavelength of 800 nm. HOF parameters were the air m, thickness of the GeO -doped SiO hole radius of ring cores m, and relative refractive index difference between core and cladding %. The direction distribution of the electric fields in the core modes was similar to that of conventional SMFs. It is notemode is linearly polarworthy to find that the fundamental ized across the ring core. It is also highly notable that the excited , and are nondegenerate, as described modes in the following section, in contrast to the fourfold degenerate mode in conventional SMFs. Therefore, the guided modes in HOF were assigned following the hybrid mode nomenclature rather than weakly guiding LP mode assignment in conventional SMFs. , and have the Three symmetric modes twofold degeneracy for two orthogonal polarization directions. and in Fig. 2(d) and (e) The antisymmetric modes mode has the twofold degenare nondegenerate and the eracy. The intensity distributions of these core modes were very similar to one another in contrast to conventional SMFs, which can make the overlap integral between the modes in HOF significantly larger than conventional SMFs. We further analyzed the mode intensity profile and the results are shown in contour plots in Fig. 3. It is clearly observed that two high intensity lobes exist within the ring core for the funmode as in Fig. 3(a). This unique nonuniform damental mode in HOF is rather intensity profile of the fundamental unexpected and has not been reported yet, to the best knowledge of the authors. mode, uniform intensity within the In contrast to the and modes, where ring core was observed in the the electric field directions conform well to the ring geometry. In air–silica optical fibers, the amount of light in the air hole is one of the important parameters to estimate the evanescent wave interaction. In this HOF, the power fraction at the hollow mode was calculated as a channel of the fundamental function of wavelength, and the results are plotted in Fig. 4. The power fraction of light at the hollow channel for a certain guided mode is defined as B. Material Dispersion In order to include material dispersion effects in precise calculation of the effective indexes of the guided modes, we used Sellmeier equations of GeO doped SiO and SiO for the ring core and cladding, respectively. (4) The power fraction has a maximum value of about 0.035% near 1.1 m. This value is significantly lower than some of Authorized licensed use limited to: Yonsei University. Downloaded on October 13, 2009 at 02:02 from IEEE Xplore. Restrictions apply. LEE et al.: MODE-COUPLING DEVICE APPLICATIONS 4921 Fig. 2. Electric field direction distributions in HOF cladding modes. Here (a)–(c) are symmetric and (d)–(f) are antisymmetric core modes of HOF. The wavelength of the light is at 800 nm. The intensity distributions are also shown below the electric field direction distribution as a function of radial position. (a) HE , (b) EH , (c) HE , (d) TE , (e) TM , and (f) HE . Fig. 3. Contour plot of intensity profile of the core modes. (a) HE , (b) EH , (c) HE , (d) TE , (e) TM , and (f) HE . photonic crystal fibers, which showed maximum fraction up to 50% . The low power fraction in HOF directly indicates strong confinement of the guided modes along the ring core and is attributed to the large refractive index at the boundary between the air hole and the ring core region. In contrast to the air hole and ring core boundary, the evanescent wave significantly extends into silica cladding, as shown in the intensity profiles in Fig. 2. In recent fiber optic devices, the coupling between the fundamental core mode and cladding modes has been widely adopted especially in LPG  and AOTF . In LPGs, the uniform intensity grating in the core provides phase matching conditions between the fundamental core mode and symmetric cladding modes. On the contrary AOTF provides phase matching conditions for the antisymmetric cladding mode coupling. Authorized licensed use limited to: Yonsei University. Downloaded on October 13, 2009 at 02:02 from IEEE Xplore. Restrictions apply. 4922 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 27, NO. 22, NOVEMBER 15, 2009 Fig. 5. Electric field direction distribution of cladding modes in HOF. (a)–(f) are antisymmetric and (g)–(i) symmetric cladding modes. (a) TE , (b) HE , (c) TM , (d) TE , (e) HE , (f) TM , (g) HE , (h) EH , and (i) HE . D. Propagation Properties of the Core Modes Fig. 4. Power fraction at the hollow channel of the HE of wavelength. mode as a function We analyzed first few cladding modes and categorized into symmetric and antisymmetric modes as in Fig. 6. Antisymmetric cladding modes are (a)–(f), and (g)–(i) are symmetric and modes are nondegenerate, modes. Note that and HE and EH are twofold degenerate modes. In the calculation, we assumed the cladding diameter of 125 m, surrounded by air of refractive index 1. We also included the material dispersion in both cladding and ring core. Due to presence of the central air hole, the cladding modes of HOF also showed significantly different intensity distribution compared with conventional SMFs, where the symmetric cladding modes have intensity peaks at the center while the antisymmetric modes have nodes. HOF both symmetric and antisymmetric modes have the annular structures and modal overlap integrals between the fundamental core mode would be significantly different from those in conventional fibers, which could provide a new opportunity in LPG and AOTF device design. Among the modes in Fig. 5, it is expected that the overlap symmetric cladding modes integral of core mode would be significantly larger than the with the mode rest of modes. In the case of antisymmetric modes, is expected to provide the largest overlap integral. Further investigation on the modal overlap and mode-coupling strength in HOF is being pursued by the authors. The propagation properties of the guided modes in HOF were of the analyzed by studying the effective refractive index guided modes as a function of wavelength. The plots of for the core and cladding modes are shown in Fig. 6(a) and (b). , and , are nondeThe first excited modes, generate with distinctive cutoff wavelength as in Fig. 6(a). The HOF cladding modes in Fig. 6(b) also showed distinctive nondegenerate effective index dispersion due to the presence of the central air hole. Chromatic dispersion is one of the key factors to determine the transmission capacity of an optical communication link. The chromatic dispersion that includes waveguide and material dispersion of each mode is obtained from the following relation. (5) and first excited For the fundamental core mode, modes, we calculated the chromatic dispersion, and the results are shown in Fig. 7. The mode showed the zero dispersion near 1600 nm, which is quite different from 1310 nm of conventional SMFs. Anomalous dispersion range is significantly reduced and in the U band the dispersion is less than 10 ps/nm km. The dispersion values in 1310 and 1550 nm are and ps/nm km, respectively. The sign of dispersion slope near 1550 nm is also opposite to conventional SMF. The excited modes have relatively large negative dispersion near the cutoff as expected. We further investigated the impact of the air hole radius on chromatic dispersion and compared with the conventional SMF, where the hole radius is equal to zero. The results are summarized in Fig. 8 for the range of hole radius from 0 to 4 m. It is noted that as soon as the central air hole is introduced, the chromatic dispersion is in general downshifted to result in a lower dispersion value. Authorized licensed use limited to: Yonsei University. Downloaded on October 13, 2009 at 02:02 from IEEE Xplore. Restrictions apply. LEE et al.: MODE-COUPLING DEVICE APPLICATIONS 4923 Fig. 6. Effective refractive index of (a) the core modes (b) the cladding modes. TABLE I CUTOFF WAVELENGTH OF GUIDED CORE MODES IN HOF Fig. 7. Chromatic dispersion of core modes of the HOF. index air hole would result in a lower net material dispersion. In fact, the chromatic dispersion is addition of material dispersion and waveguide dispersion, and material dispersion has positive value contribution while the waveguide dispersion has negative value in the range of 1.2–1.6 m. Inclusion of air in the core region will effectively lower the material dispersion to reduce the chromatic dispersion. Further increase of the air hole radius beyond 3 m did not change the dispersion value itself but the slope was affected significantly, which is directly related to the mode cutoff at longer wavelength. One of unique propagation characteristics of HOF is the existence of the fundamental mode cutoff. In Fig. 7(a), it is noted of mode cross zero near . The cutoff that the wavelength of the core modes in a HOF with the given waveguide parameters are listed in Table I. Single mode window is and mode cutoff. bounded by the E. Mode-Coupling Characteristics Fig. 8. Impacts of the central air hole radius (m) on chromatic dispersion. Fiber mode-coupling devices have been developed for various applications such as tunable filter, bandpass filter, band rejection filter, and frequency shifter. In Fig. 9, schematic diagrams for the mode-coupling mechanism are shown for the cases of the core-to-core mode and the core-to-cladding mode coupling. The fundamental mode of HOF is converted to higher order core modes or the cladding modes by satisfying the phase matching conditions imposed by the periodic perturbations. The primary principle behind the mode-coupling devices lies in the phase matching condition. This lower chromatic dispersion is attributed to a speculation that replacing the germanosilicate glass by the lower refractive Authorized licensed use limited to: Yonsei University. Downloaded on October 13, 2009 at 02:02 from IEEE Xplore. Restrictions apply. (6) 4924 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 27, NO. 22, NOVEMBER 15, 2009 Fig. 9. Schematic diagram for the principle of mode-coupling devices. Fig. 10. Propagation constant difference between the HE core mode and (a) higher order core modes, (b) cladding modes. (c) Magnified image 1.2–1.65 m of (b). Fig. 11. Phase matching pitches for the mode coupling between (a) HE mode and higher order core modes (b) HE core mode and cladding. where is the propagation constant of the fundamental mode and is that of an excited mode where the fundamental mode is coupled. is the pitch of the periodic perturbation caused by either acoustic wave modulations or permanent index gratings. In this study, we investigated the phase matching conditions for both core-to-core mode and core-to-cladding mode coupling and their behavior as a function of wavelength in HOF, for the first time. is the fundamental basis for mode-couInformation on in HOF: pling device design. We calculated two types of core mode to the higher order core mode coupling and 1) 2) core mode to cladding mode coupling, which can be readily applicable for fiber grating and AOTF devices. The results are summarized in Fig. 10. It is well known that the bandwidth of the mode coupling or equivalently would directly depend on the slope of such that if it is close to zero at a certain wavelength, , then the coupling results in a broadband spectra near . If increase or decrease monotonically with a nonzero the slope, then the coupling results in a narrowband spectra . core mode to antisymmetric core mode coupling In the plot in Fig. 10(a) shows a plateau with a near-zero in HOF, slope in the range from 0.9 to 1.0 m. Therefore, broadband spectra in the above wavelength range can be provided in a HOF. to However, the symmetric coupling of the modes shows a monotonic increase with a positive slope, and they are expected to result in narrow line spectra. core mode to cladding mode coupling, both symIn the metric and antisymmetric coupling showed a similar pattern: 1) monotonic decrease with a negative slope up to 1.4 m and 2) a plateau extending from 1.4 to 1.6 m, which corresponds to narrowband and broadband spectra, respectively. Broadband coupling in HOF has been observed in the 1.5 m region with bandwidth exceeding 100 nm , , which agrees well with theoretical prediction in Fig. 11(b) and (c). We obtained the modulation pitches for the phase matching core mode and conditions between the fundamental higher order core modes or the cladding modes. The results are summarized in Fig. 11. As in Fig. 11(a), the phase matching mode to higher order core modes in HOF pitch for the shows unique parabolic shapes in contrast to monotonically increasing behaviors in conventional SMFs . These parabolic shapes in the phase matching condition will provide very wide coupling bandwidth. For example, a wide band coupling is expected between 0.9 to 1.1 m with the bandwidth over to coupling for the periodic pitch 200 nm for the Authorized licensed use limited to: Yonsei University. Downloaded on October 13, 2009 at 02:02 from IEEE Xplore. Restrictions apply. core LEE et al.: MODE-COUPLING DEVICE APPLICATIONS of 1.2 mm. Broadband coupling in HOF has been reported in AOTF by the authors , and this tendency is in good agreement with the experimental results in the reference. Another notable point in the HOF mode coupling is the fact that the phase matching pitches are several times longer core mode than conventional SMF. For example, the to cladding mode coupling in conventional SMFs requires MHz for the case of the pitch length of about 500 m or index grating and AOTF, respectively. In contrast to SMF, the modulation pitches in HOF are in the order of a few millimeter, which is longer by several factors. From this HOF phase matching conditions, we could expect that HOF-AOTF requires RF frequency in tens of kHz range, which is also lower than the conventional SMF by several factors. These changes in the modulation pitch would be of significant benefit to reduce tolerance in the fabrication and facilitate mass production. Furthermore, longer modulation pitch would also indicate robust optical characteristics against environmental perturbation. Fabrication and characterization of mode-coupling devices based on HOF are being pursued by the authors for sensor applications. III. CONCLUSION Guided modes in HOF have been theoretically investigated using FEM for three-layered boundary conditions: central air hole, germanosilicate glass ring core, and silica cladding. Even with core–cladding relative index difference of 0.0045, which is similar to weakly guiding conventional solid core fibers, the , and were found to be nondeexcited modes mode was generate. Intensity profile of the fundamental not uniform along the ring core, yet showed two peaks at oppomode showed cutoff site sides. In HOF, the fundamental at longer wavelength and single-mode window is bounded by cutoff. By inclusion of low refracthe short wavelength tive index air within the core resulted in reduction of chromatic dispersion in 1.2–1.7 m range with zero dispersion wavelength near 1.6 m. 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QE-9, no. 9, pp. 919–933, Sep. 1973. Authorized licensed use limited to: Yonsei University. Downloaded on October 13, 2009 at 02:02 from IEEE Xplore. Restrictions apply. 4926 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 27, NO. 22, NOVEMBER 15, 2009  Y. Jung, S. B. Lee, J. W. Lee, and K. Oh, “Bandwidth control in a hybrid fiber acousto-optic filter,” Opt. Lett., vol. 30, no. 1, pp. 84–86, 1984.  W. Ha, W. Jung, J. Kim, W. Shin, I. Sohn, D. Ko, J. Lee, and K. Oh, “Fabrication and characterization of a broadband long-period-grating on a hollow optical fiber with femtosecond laser pulses,” J Kor. Phys. Soc., vol. 53, no. 6, pp. 3814–3817, 2008. Sejin Lee was born in 1983, Gyung-ki prefecture, Korea. He received the B.S. degree in physics (summa cum laude) from Yonsei University, Seoul, Korea, in 2007, where currently he is working toward the Ph.D. degree in applied physics since 2007. His recent research interests include specialty optical fiber and waveguide for photonic devices and components. Jiyoung Park was born in 1985. She received the B.S. degree in physics from Yonsei University, Seoul, Korea, in 2009, where she is currently working toward the Ph.D. degree in physics and applied physics. Her research interests include photonic device and fiber sensors. Yoonseob Jeong was born in 1983. He received the B.S. degree in physics from Yonsei University, Seoul, Korea, in 2008, where he is currently working toward the M.S. degree in physics and applied physics. His major is optical tapers and coupling devices and will be working toward optical trapping and optical communication. Hojoong Jung was born in 1982. He received the B.S. degree in physics from Yonsei University, Seoul, Korea, in 2008, where he is currently working toward the M.S. degree in physics and applied physics. His research interests include photonic device and fiber optics. Kyunghwan Oh (M’96) was born in 1963, Seoul, Korea. He received the B.S. degree in physics from Seoul National University, Seoul, Korea, and the M.S. degree in engineering and the Ph.D. degree in physics from Brown University, Providence, RI, in 1991 and 1994, respectively. He was with Gwangju institute of Science and Technology (GIST) from 1996 to 2006. He was a Visiting Scientist at Murray Bell laboratory, optical fiber research division, from September 2000 to February 2002, and a Visiting Scientist at the Institute of Physical High Technology (IPHT), Jena, Germany, in the summer and winter of 2004. Currently, he is a Professor at the Department of Physics, Yonsei University, Seoul. His research interests include specialty optical fiber and waveguide for photonic devices and components. Dr. Oh is a Topical Editor of Optics Letters (OSA), Optical Fiber Technology (Elsevier Science) and Editorial Board Member of Optics Communications (Elsevier Science). He was awarded with Alexander von Humboldt Research Fellow (Germany), Chevening Scholar, (UK) and JSPS Invitation Fellow (Japan). Authorized licensed use limited to: Yonsei University. Downloaded on October 13, 2009 at 02:02 from IEEE Xplore. Restrictions apply.