Theoretical Analysis of Cladding-Mode Waveguide Dispersion and Its Effects on the Spectra of Long-Period Fiber Grating

1838 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 21, NO. 8, AUGUST 2003 Theoretical Analysis of Cladding-Mode Waveguide Dispersion and Its Effects on the Spectra of Long-Period Fiber Grating H. Jeong and K. Oh, Member, IEEE, Member, OSA Abstract A new method to control the free spectral range (FSR) of a long-period fiber grating (LPFG) is proposed and theoretically analyzed. As the refractive index decreases radially outward in the silica cladding by graded doping of fluorine, waveguide dispersion in the cladding modes was modified to result in the effective indexes change and subsequently the phase-matching conditions for coupling with the core mode in a LPFG. Enlargement of the FSR in a LPFG was theoretically confirmed. Index Terms Cladding mode, free spectral range (FSR), LPG (LPG), optical communication, optical fiber, waveguide dispersion. I. INTRODUCTION LONG period gratings (LPGs) have been intensively studied for applications in optical communications and sensor systems due to benefits such as all-fiber configuration, low insertion loss, high flexibility in spectral design, and low cost. LPG devices have been developed in recent years as gain equalizers in erbium-doped fiber amplifiers (EDFAs) for wavelength-division-multiplexing (WDM) systems [1], bandpass filters, polarizers [2], demodulators [3], and sensing elements. In an LPG, a periodic index structure with a pitch of hundreds of microns imposes the phase-matching condition between the fundamental core mode and forward-propagating cladding modes in an optical fiber, as follows: Here, is the grating pitch, and and are the effective index of the core mode and that of the cladding modes, respectively. Recently the gain bandwidth of optical amplifier is rapidly expanding to fully exploit the bandwidth allowed in silica optical fibers. Examples are thulium-doped fiber amplifier (TDFA) Manuscript received October 14, 2002; revised March 31, 2003. This work was supoorted in part by the Korea Science and Engineering Foundation (KOSEF) through the Ultra Fast Optical Network (UFON) Engineering Research Center (ERC) program in Kwangju Institute of Science and Technology (K-JIST), the Brain Korea21 (BK21) program supported by the Ministry of Education (MOE), and the Center for Hybrid Optical Access Network (CHOAN) Information Technology Research Center (ITRC) program supported by the Ministry of Information and Communication (MIC). H. Jeong was with the Kwangju Institute of Science and Technology, Gwangju 500-712, Korea. He is now with the Micro-Mechatronics team, Korea Institute of Industrial Technology, ChonAn 330-825, Korea. K. Oh is with the Department of Information and Communications, Kwangju Institute of Science and Technology, Gwangju 500-712, Korea (e-mail: koh@ kjist.ac.kr). Digital Object Identifier 10.1109/JLT.2003.815641 (1) for the -band (1450 1490 nm) [4], fiber Raman amplifier (FRA) for the -band (1490 1530 nm) [5], as well as a combination of the conventional EDFA for the -band (1530 1570 nm) [6] and the -band (1570 1610 nm) [7]. In spite of continuous development of ultrawide-band amplifiers, an LPG optimized for certain gain equalization filter could induce a high insertion loss in the neighboring gain bands because of the relatively close-spaced resonance peaks of cladding modes. Fig. 1 shows this circumstance: Fig. 1 shows a typical LPG s transmission spectrum used as a gain equalization filter in -band EDFA, and Fig. 1 shows the same LPGs transmission spectrum in a wider wavelength range. Note that the gain equalization filter optimized for the band incurs additional insertion loss over 10 db in 1.4 m and 5 db in 1.3 m due to the closely spaced resonance cladding modes. Thus, the ability to control the free spectral range (FSR) of the resonance peaks in LPGs is in need for novel fiber filters in ultrawide-band optical communication systems. Recently, attempts to control the FSR of the fiber-based periodic coupling devices have been reported. Locally etching the fiber cladding with an Hydroflouric (HF) acid solution in longperiod fiber gratings (LPFGs) has shown the ability to tune continuously the resonance wavelengths, using the dependence of the propagation constants of the cladding modes on the cladding diameter [8]. Wide resonance peak spacing was also demonstrated by Li et al. in an acoustooptic tunable filter (AOTF) based on the cladding-etched fiber [9]. The etching technique is using conventional optical fiber and the spectral response is modified by the amount of local etching of the cladding. The technique, however, inevitably suffers from mechanical reliability since HF-water solution etched fiber is very vulnerable to micro-crack, which can degrade the fiber tensile strength by orders of magnitude [10]. The etching technique, therefore, requires significantly more stringent and expensive packaging standards than pristine fiber devices. Furthermore, the outer diameter of the fiber should be carefully monitored to meet the desired spectral response and very tight tolerance in diameter variation should be maintained, which hinders mass production of fiber devices. It is, therefore, highly desirable to provide a fundamental solution to synthesize the spectral response of LPG and AOTF using a pristine fiber. In this study, we propose a new technique to control the FSR of periodically coupling fiber devices using a pristine fiber with a refractive-index modulation in the cladding. Refractive-index modulation could be effectively achieved by synthesis of F-doped sol gel tubes [11], cold isostatic pressing (CIP) tubes 0733-8724/03$17.00 2003 IEEE

JEONG AND OH: THEORETICAL ANALYSIS OF CLADDING-MODE WAVEGUIDE DISPERSION 1839 the LPG spectrum and proposed a new technique to enhance the FSR ofthe LPG, for the first time to the best knowledge of authors. II. NUMERICAL ANALYSIS The dispersive characteristics of binary silica glass doped with dopants such as GeO and F have been precisely calculated using the Sellmeier equation for given ranges of concentrations [14]. The Sellmeier equation for the refractive index is (2) Fig. 1. Transmission spectrum for a typical long-period fiber grating (LPFG) for a C-band EDFA gain equalization filter in 1500 nm 1600 nm wavelength range and 1200 nm 1600 nm wavelength range. [12] or commercially available Heraeus F500 series tubes. Note that the proposed pristine fiber technique aims at complete obviation of the secondary etching process in the prior arts and removal of deteriorations in device reliability, packaging costs, and mass production caused by HF etching. The newly proposed method to control the LPG spectrum is based on the waveguide dispersion into cladding modes. In previous studies on LPGs, direct modification of cladding-mode waveguide dispersion and its subsequent effects on the spectra of LPGs have not been fully investigated so far. Recently the material dispersion change of cladding modes by doping transition metals and its impacts on the FSRs in LPG have been reported by the authors [13]. By a proper design of the refractive-index profile, the magnitude of change in the effective indexes of the cladding modes in the proposed fiber could be made large enough to affect the phase-matching conditions for the coupling with the fundamental core mode. The phase-matching conditions will subsequently affect LPG transmission spectra and the FSR. In this study the authors theoretically analyzed the effect of the refractive-index profile of the cladding region on Sellmeier coefficients, denoted by and, for binary silica doped with GeO and F are summarized in Table I. In fiber devices such as LPG and fiber acoustooptic filter, the main mechanism to generate a spectral response is the coupling between the fundamental mode ( ) guided through the core and the cladding modes guided through an air silica multimode waveguide. In the analysis the core was assumed to be germanosilicate glass as in conventional single-mode fibers (SMFs). The refractive index of cladding was assumed to be depressed by gradual doping of fluorine (F) into silica glass to construct triangular and trapezoidal refractive-index profile. Various technologies have been recently developed for silica tubes aiming for large and economic optical fiber preform fabrications. Sol gel techniques have been successfully introduced to mass production, and optical fibers have been manufactured using the sol-gel derived tubes [11]. The sol gel tubes are very versatile in doping additives such as Ge, B, F to vary the index of refraction. Gradient of doping concentration of F have been achieved with diffusion of F [15], along with stepwise sintering processing. As an alternative, CIP has been reported where silica particle are mechanically shaped in a very high hydrostatic pressure [13]. The technique starts from forming a porous silica body and,similar to the sol process, doping of index modifying additives are easily adopted,sintering the porous body in the precursor gas environment. Other than these techniques there are commercially available F-doped silica tubings, Heraeus F500 series, with various index differences so that stepwise index control in the cladding is readily available. The proposed fiber structure is based on refractive-index modulation of cladding by F-doping and current technologies developed for optical fiber preform claddings could be applied to suffice the desired index structure and make the propose device mass-producible. In the calculations, the refractive index of air was set 1.0 independent of wavelengths. The mode-field profiles of core and cladding modes could be calculated by finite-difference analysis with inverse power iteration method [16]. With these assumptions, we could calculate the mode-field profiles and intensity distributions of both the core and the cladding modes defined by the proposed fiber structure. The coupling coefficients between them could be calculated by the overlap integral for a given index perturbation. If we limit the calculation to gratings that consist of perfect circular-symmetric index perturbation in the transverse plane of the fiber, the only nonzero coupling coefficients between the

1840 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 21, NO. 8, AUGUST 2003 TABLE I SELLMEIER COEFFICIENTS OF BINARY SILICA GLASS DOPED WITH GeO OR F [14]; COEFFICIENTS ARE VALID FOR SPECTRAL RANGE OF 0.4 1.7 m core mode and the cladding modes involve cladding modes of azimuthal number [17]. We will confine the calculation for only 1 cladding modes. The coupled-mode equation that describes interactions in an LPG is given by [16] Here, stands for the induced-index fringe modulation, where. And the coupling constant for core mode and cladding modes and the phase shift, can be expressed as Here, is the normalized induced-index change in the core. and are the amplitudes of the core mode and th cladding modes, respectively. Here, we assumed that the induced-index change was uniform across the core such that was taken out of the overlap integral. By the boundary conditions that are, and, where is the grating length, the transmission through the grating is simply given by.for propagation parameters, the electric fields of core and cladding modes for each boundary could be obtained by numerically solving Maxwell s equation as in [18]. (3) (4) (5) (6) III. CLADDING-MODE ANLAYSIS The effective indexes of the first ten cladding modes were calculated for step, triangular, and trapezoidal refractive-index profiles. The waveguide dispersion as well as material dispersion calculated using the Sellmeier equation were taken into account in the numerical analysis. In this study, we modify the refractive-index profile of cladding region from the step waveguide to a triangular or a trapezoidal waveguide, as shown in Fig. 2, in order to change waveguide dispersion of cladding modes. Note that the core parameters were set similar to those of conventional SMF. In triangular profiles, Fig. 2, the refractive indexes in the cladding linearly decrease to the minimum values 1.454, 1.451, and 1.448. The profiles are designated by the index difference relative to the pure silica clad in conventional step SMF. The refractive-index difference was set by the experimental results where conventional chemical vapor deposition process can achieve with Fluorine containing precursors such as, [19]. In trapezoidal profiles, the refractive index decreases linearly after a certain radial position, the upper pedestal of the trapezoid, in the cladding and the minimum index was set as 1.448, with the index difference of 0.009 relative to the pure silica clad. By solving the characteristic equations for the guided modes [16] [18] for different index profiles in Fig. 2, effective indexes of the first ten cladding modes were numerically evaluated at 1.5 m, and the calculated results are plotted in Fig. 3. Two significant modifications were observed. First, the effective indexes of the cladding modes decrease by as much as or by a factor of 0.12 referenced to conventional step SMF, for the 0.009 triangular waveguide at the cladding-mode number of 1, as in Fig. 3. And the difference in effective indexes referenced to the step SMF gradually increases for higher order cladding modes. Similar effects were also observed in the trapezoidal waveguides as shown in Fig. 3. Induced change in effective refractive index for both structures is significant enough to change the phase-matching

JEONG AND OH: THEORETICAL ANALYSIS OF CLADDING-MODE WAVEGUIDE DISPERSION 1841 Fig. 2. Refractive-index profile of the proposed fiber for triangular structure and trapezoidal structure, evaluated at 0.6328 m. Note that 0.003, 0.006, and 0.009 in indicated the maximum clad-refractive-index difference referenced to the step-index profile. The minimum refractive index in the trapezoidal profiles was set to 1.448, with the index difference of 0.009. Fig. 3. Effective indexes of cladding modes for triangular refractive-index profile, as shown in Fig. 2, and trapezoidal refractive-index profile, as shown in Fig. 2. Both material dispersion and waveguide dispersion are included in calculations. conditions with the core mode. Second, the spacing in the effective indexes between the adjacent modes does increases, which will directly affect the FSR. For example, the effective index spacing between the third and fourth modes increase by as much as a factor over 2, for the 0.009-triangular profile and trapezoidal profile-2. These modifications are attributed to the effects of waveguide dispersion as well as the material dispersion due to fluorine doping. It is, therefore, expected from those observations that the waveguide modification in the cladding indeed modifies the phase-matching conditions, and subsequently the free-spectral range can be controlled. Detailed analysis on the phase matching and spectral response will be discussed in the following section. The other important aspect in LPG spectrum analysis is the overlap integral between the core mode and the cladding mode, which will determine the strength of the resonant coupling as shown in (5). After finding mode fields of the fundamental core mode and the first ten cladding modes, the overlap integrals were evaluated at 1.5 m. The results are presented for various index profiles in Fig. 4. In conventional step-index SMF, the overlap integral, or equivalently the coupling constant, increases in higher order cladding modes and reaches the maximum around the eighth mode, which has been reported in the [16]. In the case of the triangular and the trapezoidal waveguides, however, the overlap integral shows contrasting characteristics. Especially in triangular waveguide, the overlap integral has its maximum at the first cladding mode and decreases in higher order modes. Similarly in trapezoidal profiles, the overlap integral reaches its maximum in a lower order mode than step-index SMF. To explain this feature, mode-field distribution was calculated for step- and triangular-index profile. Fig. 5 shows the calculated mode profiles of the first three cladding modes for the step waveguide, and Fig. 5 shows the 0.009-triangular waveguide. In the step waveguide, the cladding modes exhibit an oscillatory behavior in the entire radial direction and the intensity near the core increases for higher order modes, as shown in Fig. 5. In the case of the triangular index profile, however, the field is much more

1842 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 21, NO. 8, AUGUST 2003 Fig. 5. Plots of the optical field as radial position for the first three cladding modes in a fiber having step-index profile and 0.009-triangular index profile. waveguide, which directly indicates a wider FSR. The results in Fig. 3 are consistent with the observations in Fig. 6. From the phase-matching condition, (1), the resonance wavelength of th order and ( )th-order cladding modes can be expressed as Fig. 4. Overlap integral between the fundamental core mode and cladding modes for triangular refractive-index profile and for trapezoidal refractive-index profile. The overlap integrals were evaluated at 1.5 m. confined to the core region for the lower order modes, which contributes to the larger overlap integral in (5). IV. LPG SPECTRAL ANALYSIS From (1), the phase-matching grating pitch was calculated as a function of the coupling wavelength and results are shown in Fig. 6 for the first nine cladding modes. The results for the step-index profile are shown in Fig. 6, those for the 0.009-triangular index profile in Fig. 6, and those for the trapezoidal (Trap-2) index profile in Fig. 6(c), respectively. Compared with the results of the step waveguide, the phase-matching conditions for the triangular and trapezoidal index profiles show significant differences in terms of the slope in the plot of the grating pitch versus the resonance wavelength. The slope is shallower in the triangular and trapezoidal waveguide compared with the step waveguide, which is attributed to the modification of dispersion of cladding modes as indicated in Fig. 3. Another notable feature is that the spacing between the spectral resonances is significantly wider in the proposed waveguides than that in step Here, is the effective index of the fundamental core mode( ), and are the effective indexes of th-order ( ), and ( )th-order ( ) cladding modes in step-index cladding fiber, and and are the effective indexes change of th-order and ( )th-order cladding modes due to refractive-index change in the cladding. Using (7) and (8), the wavelength separation between th and ( )th cladding mode,, or equivalently the FSR of an LPG, can be expressed as In order to solve this transcendental equation for the resonance wavelengths and, effective indexes of the first two cladding modes were numerically obtained as shown in Fig. 7. In this figure, we focused on the first cladding mode where the coupling was set to take place at 1300 nm along with its nearest neighboring second mode for the step and triangular cladding (7) (8) (9)

JEONG AND OH: THEORETICAL ANALYSIS OF CLADDING-MODE WAVEGUIDE DISPERSION 1843 (c) Fig. 6. Plot of the phase-matching grating pitch 3 versus the coupling wavelength for the first nine cladding modes. Results for the step waveguide. Results for the 0.009 triangular waveguide. (c) Results for the trapezoidal (Trap-2) waveguide. index profiles. Note that in Fig. 7, both the slopes and the magnitude of the effective indexes did change as a function of degree of the refractive-index suppression or the index differences given from 0.003 to 0.009. Keeping the resonance of first mode at 1300 nm by adjusting grating pitches, the transcendental (9) was numerically solved to obtain the FSR, as a function of the index difference in the triangular profile. The grating pitch for the step-index cladding was set to 470 m. As the cladding index further suppressed from 0.003, 0.006, to 0.009 in the cladding, the grating pitches were varied from 369, 329, and 316 m, respectively (see Fig. 6 for phase-matching conditions). The numerical results on FSRs are shown in Fig. 8. As Fig. 7. Effective indexes of first and second cladding modes for step and triangular waveguides. Both material dispersion, calculated from (2), and waveguide dispersion are included in calculations. Absolute effective index slopes of first and second cladding modes increased by as much as 0:1082 2 10 ( 8.7%) and 0:1457 2 10 ( 11.4%) reference to conventional step waveguide for the 0.009 triangular waveguide. the refractive-index difference in the triangular profile, the FSR monotonically increases from 37 to 322 nm, by a factor of 9. Transmission spectra of LPGs were numerically simulated using the overlap integral equations (5) (6) for various cladding index profiles. Induced index change was assumed to be uniform over the cross section of fiber core. A uniform step function was assumed for the periodic index grating structure along the core of the fibers in order to simplify numerical analysis. In the grating structurem, radial and angular dependence were not considered. Neither chirping nor apodization were assumed to focus on the waveguide dispersion of cladding modes. A periodic index modulation of a step function with the amplitude of, with the duty cycle of, was assumed along the length of in the core, and numerical values for these parameters are,, and 25.4 mm, similar to previously reported paper [15]. The

1844 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 21, NO. 8, AUGUST 2003 be confined only to the wavelength region of interests with negligible crosstalks among adjacent bands using this graded-index cladding technique. Fig. 8. FSR as a function of refractive-index difference in the triangular waveguides. V. CONCLUSION A new optical fiber structure that modifies the cladding-mode waveguide dispersion was proposed for fiber filters based on coupling between the core and cladding modes. The effects of modification of the refractive-index profile in cladding region on the waveguide dispersion of cladding modes were theoretically analyzed using finite-difference analysis and coupled-mode theory. An effective index decrease of more than 1.7, or by a factor of 0.12 referenced to step waveguide was theoretically estimated for a triangular waveguide. This change in waveguide dispersion of cladding mode induced significant modification in the phase-matching conditions. By coupling to first symmetric cladding modes, enhancement of the FSR over 300 nm was theoretically predicted, which could be applied for noble fiber filters in ultrawide-band optical communication systems. Fig. 9. LPG transmission spectra for various cladding index profiles. grating pitch was adjusted as described in the previous paragraph for different refractive-index profile to keep the resonance at 1300 nm. The results of numerical simulation on the LPG transmission spectra are shown in Fig. 9. By inducing the triangular profile into the clad region, and consequently changing the waveguide dispersion of cladding modes, we could isolate only one cladding mode resonance peak of LPG in the spectral range from 1200 to 1600 nm at the 0.009-triangular waveguide. The FSR was estimated at about 330 nm as shown in Fig. 8. We, therefore, confirmed that the proposed technique to modify the cladding index profile does control the FSR. Selective flattening of a gain band without incurring insertion loss in the neighboring bands can be achieved using LPGs utilizing the proposed fiber structures. It is noted that the trapezoidal profile provided multiple resonance couplings in the range 1100 1600 nm with an FSR of less than 200 nm. While in the triangular profile, singular coupling was achieved in the same wavelength range with the maximum FSR of 320 nm. In comparison with prior reports based on etched cladding [8], [9], the FSR increased by more than a factor of two in the proposed fiber profile. A novel filtering devices, therefore, could be further designed whose characteristics will REFERENCES [1] A. M. Vengsarkar, P. J. Lemaire, J. B. Judkins, V. Bhatia, and T. Erdogan, Long-period fiber gratings as band-rejection filters, J. Lightwave Technol., vol. 14, pp. 58 65, Jan. 1996. [2] A. S. Kurkov, M. Douay, O. Duhem, B. Leleu, J. F. Henninot, J. F. Bayon, and L. 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JEONG AND OH: THEORETICAL ANALYSIS OF CLADDING-MODE WAVEGUIDE DISPERSION 1845 [16] M. Stern, Finite-difference analysis of planar optical waveguides, in PIER 10 (Progress in Electromagnetic Research 10), W. P. Huang, Ed. Cambridge, MA: EMW Publishing, 1995, ch. 4. [17] T. Erdogan, Cladding-mode resonances in short- and long-period fiber grating fibers, J. Opt. Soc. Amer. A, vol. 14, pp. 1760 1773, 1997. [18] C. Tsao, Optical Fiber Waveguide Analysis. New York: Oxford Univ. Press, 1992, ch. 10. [19] J. W. Fleming and D. L. Wood, Refractive index dispersion and related properties influorine doped silica, Appl. Opt., vol. 22, pp. 3102 3104, 1983. H. Jeong was born in Korea in 1973. He received the M.S. degree in material science from Pohang University of Science and Technology, Pohang, Korea, in 1998 and the Ph.D degree in information and communications from Kwangju Institute of Science and Technology, Gwangju, Korea, in 2003. Since his Ph.D. work, he has been engaged in research and development of optical fiber lasers, amplified spontaneous emission light sources, and long-period fiber grating. From March to September 2002, he was with the High Power Fiber Laser Group, Optoelectronics Research Centre, University of Southampton, Southampton, U.K., where he was engaged in research on high-power fiber lasers. Currently, he is with the Micro-Mechatronics Team, Korea Institute of Industrial Technology, ChonAn, Korea, where he has been engaged in research of gas sensors for fuel cell application, planar waveguide devices, and sensors using nano-optics. K. Oh (M 96) received the B.S. and M.S. degrees in physics from Seoul National University, Seoul, Korea, in 1986 and 1988, respectively, and the M.S. degree in engineering and the Ph.D. degree in physics from Brown University, Providence, RI, in 1991 and 1994, respectively. Subsequently, he was appointed as a Postdoctoral Research Associate in the Laboratory for Lightwave Technology. Returning to Korea, he was involved in specialty fiber development as a Senior Researcher in the fiber optics and telecommunication laboratory in Lucky Goldstar (LG) cable in 1995. From 1996 to 2000, he was an Assistant Professor in the Department of Information and Communications, Kwangju Institute of Science Technology (K-JIST), Gwangju, Korea. He became an Associate Professor in 2000. In the summer of 1998, he was appointed as a Visiting Professor of research at the Division of Engineering, Brown University. From September 2000 to March 2002, he was a Visiting Scientist with Bell Laboratories, Lucent Technologies, Murray Hill, NJ. In the summer of 2002, he was a Visiting Scientist in the Fiber Optic Material Research Program, Rutgers University, Piscataway, NJ. His research interests are in the areas of specialty fiber design and fabrication for active and passive fiber devices in optical communications. Dr. Oh is a Member of the IEEE Lasers and Electro-Optics Society (LEOS) and the Optical Society of America (OSA).