Mode add/drop multiplexers of LP 02 and LP 03 modes with two parallel combinative long-period fiber gratings

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1 Mode add/drop multiplexers of LP 0 and LP 03 modes with two parallel combinative long-period fiber gratings Liang ang and Hongzhi Jia* Engineering Research Center of Optical Instruments and Systems, Ministry of Education. Shanghai Key Laboratory of Modem Optical System. School of Optical-electrical and Computer Engineering, University of Shanghai for Science and Technology, No. 56 JunGong Road, Shanghai 00093, China *hzjia@usst.edu.cn Abstract: Two parallel combinative long-period fiber gratings (LPGs) can convert the fundamental core mode LP 0 in a single-mode fiber (SM) into one desired higher order core mode LP 0m in a few-mode fiber (M), in the process of which one specific cladding mode acts as a medium coupled from one fiber to another. Different LP 0m modes can be obtained by controlling the grating period of LPG in M to meet the phase matching condition. In this article we focus on the design and analyses of LP 0 and LP 03 mode add / drop multiplexers (MADMs). This device has some advantages of facile and good scalability, and particularly, of eliminating coupling interferences for the ahead multiplexed modes by the posterior MADMs or couplers. urthermore, the conversion rate of mode power theoretically can approach as much as 98% and the 3dB bandwidth can reach 0nm or more. 04 Optical Society of America OCIS codes: ( ) iber optics communications; (060.80) Buffers, couplers, routers, switches, and multiplexers; ( ) Multiplexing; ( ) iber Bragg gratings. References and links. Y. Kokubun and M. Koshiba, Novel multi-core fibers for mode division multiplexing: proposal and design principle, IEICE Electron. Express 6(8), 5 58 (009).. N. Bozinovic, Y. Yue, Y. X. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, Terabit-scale orbital angular momentum mode division multiplexing in fibers, Science 340(640), (03). 3. A. Li, A. Al Amin, X. Chen, and W. Shieh, Transmission of 07-Gb/s mode and polarization multiplexed CO- ODM signal over a two-mode fiber, Opt. Express 9(9), (0). 4. C. Koebele, M. Salsi, D. Sperti, P. Tran, P. Brindel, H. Mardoyan, S. Bigo, A. Boutin,. Verluise, P. Sillard, M. Astruc, L. Provost,. Cerou, and G. Charlet, Two mode transmission at 00 Gb/s, over 40 km-long prototype few-mode fiber, using LCOS-based programmable mode multiplexer and demultiplexer, Opt. Express 9(7), (0). 5. N. Riesen and J. D. Love, Weakly-guiding mode-selective fiber couplers, IEEE J. Quantum Electron. 48(7), (0). 6. J. D. Love and N. Riesen, Mode-selective couplers for few-mode optical fiber networks, Opt. Lett. 37(9), (0). 7. A. Li, X. Chen, A. A. Amin, and W. Shieh, used fiber mode couplers for few-mode transmission, IEEE Photon. Technol. Lett. 4(), (0). 8. N. Riesen and J. D. Love, Ultra-broadband tapered mode-selective couplers for few-mode optical fiber networks, IEEE Photon. Technol. Lett. 5(4), (03). 9. Y. Xie, S. u, H. Liu, H. Zhang, M. Tang, P. Shum, and D. Liu, Design and numerical optimization of a mode multiplexer based on few-mode fiber couplers, J. Opt. 5(), 5404 (03). 0. Y. Yue, Y. Yan, N. Ahmed, N. Ahmed, J. Yang, L. Zhang, Y. Ren, H. Huang, K. M. Birnbaum, B. I. Erkmen, S. Doliner, M. Tur, and A. E. Willner, Mode properties and propagation effects of optical orbital angular momentum (OAM) modes in a ring fiber, IEEE Photon. J. 4, (0).. T. Erdogan, Cladding-mode resonances in short- and long period fiber grating filters, J. Opt. Soc. Am. A 4(8), (997).. K. Okamoto, undamentals of Optical Waveguides (Elsevier Academic Press, 006), Chap. 3. (C) 04 OSA 9 May 04 Vol., No. 0 DOI:0.364/OE OPTICS EXPRESS 488

2 3. K. S. Chiang,. Y. M. Chan, and M. N. Ng, Analysis of two parallel long-period fiber gratings, J. Lightwave Technol. (5), (004). 4. M. Z. Alam and J. Albert, Selective excitation of radially and azimuthally polarized optical fiber cladding modes, J. Lightwave Technol. 3(9), (03). 5. C. Tsao, Optical ibre Waveguide Analysis (Oxford University, 99), Part Abrishamian, S. Sato, and M. Imai, A new method of solving multimode coupled equations for analysis of uniform and non-uniform fiber Bragg gratings and its application to acoustically induced superstructure modulation, Opt. Rev. (6), (005). 7. M. J. Kim, Y. M. Jung, B. H. Kim, W. T. Han, and B. H. Lee, Ultra-wide bandpass filter based on long-period fiber gratings and the evanescent field coupling between two fibers, Opt. Express 5(7), (007). 8. Y. Liu, K. S. Chiang, Y. J. Rao, Z. L. Ran, and T. Zhu, Light coupling between two parallel CO-laser written long-period fiber gratings, Opt. Express 5(6), (007). 9. T. Erdogan, iber Grating Spectra, J. Lightwave Technol. 5(8), (997). 0. K. S. Lee and T. Erdogan, iber mode conversion with tilted gratings in an optical fiber, J. Opt. Soc. Am. A 8(5), (00).. Introduction Single-mode fiber (SM), a finite bandwidth capacity, is insufficient to satisfy current increasing bandwidth requirements in global information. In order to expand the bandwidth capacity, mode-division multiplexing (MDM), and modal orbital angular momentum (OAM) multiplexing as new technologies have recently been proposed [, ]. In these technologies, mode selective couplers or mode multiplexers / de-multiplexers (MUXs/DEMUXs) are key devices that convert the fundamental mode LP 0 to different higher order modes (HOMs) or OAMs which are then multiplexed as independent data channels transmitting in one fiber. So far several kinds of mode MUXs/DEMUXs or couplers have been proposed [3, 4], particularly, based on the principle of mode coupling, such as two or three-core modeselective couplers (MSCs) [5, 6], fused fiber mode couplers [7], and tapered mode-selective couplers [8]. Due to being simple, lower-loss and more compact waveguide-based solutions, the mode couplers are regarded as promising MUXs/DENUXs in MDM transmission [8]. However, the mode couplers proposed for different multiplexed modes mainly differ in the coupling lengths or angular offset. When several modes are multiplexed simultaneously and orderly in one fiber by corresponding couplers, the posterior couplers will cause coupling interferences, even de-multiplexing for ahead multiplexed modes and then result in power loss of these modes. The interferences also occur when these multiplexed modes are demultiplexed [8]. The greater the number of multiplexed modes is, the worse the interferences will occur, which may cause design difficulties [9]. In this article, a novel mode add / drop multiplexer (MADM) is proposed, based on the principle of coupling between the core HOM and the cladding mode by long period fiber grating (LPG), and the cladding mode coupling from one fiber to another. This design can effectively eliminate the coupling interferences for the multiplexed core modes transmitting through the posterior multiplexer, for the resonance condition of coupling from the selected cladding mode to desired core modes is strictly dependent on the grating period of LPG written in M. This MADM has a structure of two parallel combinative LPGs; one LPG converts the fundamental core mode into one cladding mode in SM touched with M, then this cladding mode is coupled from SM to M; finally, the other LPG written in M converts this cladding mode into one desired core HOM. Theoretically, each LP mode can be multiplexed in one M as independent data channel. However, scalar modes LPlm where l > 0, while propagating along the fiber for a long distance, will produce intermodal dispersion; as a result, the vector mode components may walk off [0]. or the modes LP 0m, composed of a single vector mode HE m, the dispersion will not occur, so this type of mode is of more practical significance to MDM transmission. Therefore we focus on the design of MADMs of LP 0 and LP 03 modes in this article; other LP 0m modes multiplexing can be extended by the basic principle discussed here. Compared with proposed mode-selective couplers based on (C) 04 OSA 9 May 04 Vol., No. 0 DOI:0.364/OE OPTICS EXPRESS 489

3 multi-core fiber and tapered structure, the structure and fabrication of this MADM on the basis of conventional fiber are simpler and more facile.. Analysis of mode coupling The diagram of MADM is shown in ig., where the parallel SM and M are placed close together, and two LPGs are written in SM and M, respectively. undamental core mode LP 0 transmitting in SM is coupled into one specific cladding mode through one LPG, and then this cladding mode is coupled to M, and finally converted into a desired core HOM by another LPG in M.. Couping between cladding mode and HOMs Since one of the cladding modes HE m is selected as the medium, through LPG written in M with uniform modulation this cladding mode can be strongly coupled to nothing but the core modes HE m (if the fiber is weakly guiding, i.e., scalar modes LP 0m ), because of the complete circular symmetry of their field intensities in the fiber core region. The mode coupling of LPG between LP 0 and cladding modes in SM is well known []. In this section, our work lays emphasis on the characteristics of mode conversion between the cladding modes and core HOMs through LPG in M. x y z L S L c L ig.. Diagram of MADM and mode conversion from the fundamental mode LP 0 to cladding mode HE 6 and then to HOMs LP 0 and LP 03, with their transverse electric field distributions shown from left to right and up to down. Table. Parameters of Two ibers in MADM Core radius (μm) Cladding radius (μm) Core index n Cladding index n SM M Surrounding n index The parameters of SM and M as components of MADM is shown in Table. The normalized waveguide frequencyv of SM is.07, and that of M is 8.30; the eigenmodes LP 0m supported in M include LP 0, LP 0, and LP 03 []. In order to reveal the coupling efficiency between all cladding modes HE m and the core modes LP 0 and LP 03, the coupling coefficients between them are calculated by the expression π a v u 0 ( ) v u 0 0 κ ωε n δ z dφ E E rdr, = () (C) 04 OSA 9 May 04 Vol., No. 0 DOI:0.364/OE OPTICS EXPRESS 490

4 whereω and ε 0 are the angular frequency and the dielectric constant in vacuum respectively; and n anda are the refractive index and radius of the fiber core respectively; σ ( z) is the modulation strength; E v and E u are the functions of electric field transverse distribution of mode v and u in the fiber core []; and indicates the complex conjugate. 6 x 0-4 Coupling Coefficients/δ(z) (nm - ) LP 0 -HE m LP 03 -HE m Radial Number m ig.. Coupling coefficients for LP 0 and LP 03 to all cladding modes HE m. The coupling coefficients calculated are demonstrated in ig., in which it is indicated that the coefficients between the core mode LP 03 and all cladding modes HE m are almost larger than those between the core mode LP 0 and these cladding modes. This is due to the more similarity of electric field distributions between LP 03 mode and these cladding modes in the core region of M, compared to those for LP 0 and these cladding modes, which is roughly shown in ig.. rom the ig., the optimal cladding mode selected as the medium is HE 6 for both multiplexing LP 0 and LP 03. Actually, when the number of MADMs, i.e. multiplexed HOMs is increased to three or more, the cladding modes can be selected differently in practice, in view of avoiding the coupling interferences from the ahead multiplexed modes to the other cladding modes by the LPG in M of the posterior MADMs; in other words, to break the phase matching conditions of these modes that are likely to cause coupling interferences when transmitted through the LPG.. Coupling between two fibers In this section, we analyze the coupling of cladding mode HE 6 from SM to M. The crosssection of parallel SM and M is shown in ig. 3. When the cladding mode HE 6 is coupled from SM to M, its electric field distributed in the surroundings for SM will be perturbed by the refractive index of both cladding and core areas in M. So the complete coupling coefficient is defined by π a π a S C = ωε ( ) (, ) (, ) + ( ) (, ) (, ), n n 3 dφe r φ E r φ rdr n n su co 3 dφ E r φ E r φ rdr su cl () a where E, E are the functions of transverse electric field distribution of the cladding mode co cl HE 6 in the core and cladding regions of M, respectively, and E su is that in the surroundings (C) 04 OSA 9 May 04 Vol., No. 0 DOI:0.364/OE OPTICS EXPRESS 49

5 for SM. In order to unify the polar coordinates of the two field functions and facilitate the numerical calculation, according to the laws of cosines and sines in triangle OAO shown in ig. 4, a transformational relation can be found: where A ( r, φ ) for M and ( r, φ ) distance between two fiber cores. = + + cosφ r r d rd r φ = φ r sin sin, for SM are any points in the region of M, and d is the (3) M n O n φ r A a a n 3 n φ r d SM a a n O ig. 3. Cross section of parallel SM and M, and coordinate transformation in two fibers. Effective Refractive Index of Cladding Mode HE SM M Wavelength (nm) ig. 4. Effective indexes of the cladding mode HE 6 in SM and M with fiber parameters listed in Table. (C) 04 OSA 9 May 04 Vol., No. 0 DOI:0.364/OE OPTICS EXPRESS 49

6 The value of S C 6 6 is dependent on the distance d and the refractive index of surroundings n3 that directly determines the value of the distribution of E su. When two fibers touch each other, whilst simultaneously the value of n3 approaches that of the cladding index n, which is taken to.445 and can be obtained by immersing the structure of LPG pair into an indexmatching medium, S C 6 6 will become large enough, and hence the periodic coupling length will be vastly shortened [3]. urthermore, significant coupling only happens when the propagation constants of cladding mode HE 6 selected in both SM and M are very similar, S i.e. β3 β3, which implies that the two cladding modes HE 6 are fully phase-matched, and S in this case, the coupling coefficient from M to SM C S 6 6 is closed to C 6 6 [5, 3]. If the two fibers are identical, the two cladding mode will naturally reach the phase matching condition. In our design work here, it can be achieved by reducing the cladding radius a of SM to μm, while the cladding radius a of M is maintained to 6.5 μm.the index matching relation is illustrated in ig. 4. It shows the effective indexes of the cladding mode HE 6 in SM and M are equal in the wavelength of 550 nm, which means the full phase match with the designed fiber parameters. igure 5 shows the radial electric field distributions of cladding mode HE 6 in M and SM when the LPG pair is immersed in the indexmatching medium. It reveals that the evanescent field spread into surroundings increases, which enlarges the overlap region between two cladding modes in one side of two fibers, thereby makes the coupling coefficient large enough. x 0-3 Radial Electric ield of Cladding Mode HE 6 (V/m) Core Cladding M SM Surroundings Radial Position (nm) x 0 4 ig. 5. Radial electric field distributions of cladding mode HE 6 with n 3 =.445 in M (solid line) and SM (dotted line). It should be noticed that the cladding modes are not identical in radial and azimuthal field components, and the higher order of the mode, the more distinct of the two components [4], so when the selected cladding mode HE 6 is coupled from SM to M, the coupling may be dependent on polarization. The coupling coefficients of the cladding mode HE 6 from SM to M may differ in the x polarization (linearly polarized along the x axis) and the y polarization (linearly polarized along the y axis). The four types of coupling coefficients and the corresponding coupling lengths L c with 00% coupling efficiency which is determined by CLc = π are calculated and listed in Table. It shows that the values of coupling coefficients for x x and y y polarization coupling are approximated, because the (C) 04 OSA 9 May 04 Vol., No. 0 DOI:0.364/OE OPTICS EXPRESS 493

7 radial and azimuthal field components of the cladding mode HE 6 almost have the same values; in other words, the cladding mode HE 6 is almost completely linearly polarized []. The x y and y xpolarization coupling is zero due to the orthogonality of mode field at the overlap region. Table. Coupling coefficients and Coupling Lengths for our Types of Polarization Coupling Polarization coupling x x x y y x y y Coupling coefficients (nm ) Coupling length L (cm) c Actually, the coupling distance Lc between two fibers shown in ig. can be overlapped more or less on the grating extents of two LPGs in SM and M; however, in this design it will increase the whole coupling length of the parallel combative LPG pair [3]. 3. Discussion and simulation Because of the polarization independence of the LPG coupling in SM and M due to the complete circular symmetry of the LPG structure, the polarization mode coupling in the whole process in the LPG pair is only determined by the cladding mode coupling between two fibers. However, according to the analysis of polarization dependence in above section, the mode coupling in x x polarization and y y polarization is not distinct, therefore we just simulate the x x polarized mode coupling in this section. It is well known that there are a large number of cladding eignemodes supported in ordinary fiber []. Even in the case that the surrounding index is close to the index of fiber cladding, as in this article, the number of HE m modes is as much as seven, which are listed in ig., exclusive of other HE, EH, TM and TE modes. The power of the HE 6 mode coupled from LP 0 by the LPG in SM largely couples to the HE 6 cladding mode in M due to the full phase-matching, while simultaneously coupling with crosstalk to other cladding modes, of which have effective indexes close to that of the HE 6 mode. Among all cladding modes, we discover that the modes HE 56 and HE 65 meet the crosstalk condition, so they need to be involved in the analysis of the coupling crosstalk. The coupled mode equations describing the whole coupling process in the parallel combative LPG pair can be expressed as da db π jb j z S S S = κ exp 6 0 β β 0 6 Λ S π jβ B jc B jc B jc B jaκ exp j β β z, S S S S S S S + = Λ S db3 S + jβ B = jc B db4 S + jβ B = jc B db da π j B jc B ja j z S + β = + κ exp β β i i 0i 6 Λ i π jb κ exp j β β z i = 6 0i 0i 6 Λ i (4) (C) 04 OSA 9 May 04 Vol., No. 0 DOI:0.364/OE OPTICS EXPRESS 494

8 where A is the amplitude of core mode LP 0 in SM, and core mode LP 0 ( i = ), or LP 03 ( 3) Ai indicates the amplitude of the i = ; B, B, B 3,and B 4 represent the cladding modes HE 6 in SM, HE 6, HE 56, and HE 65 in M respectively; the symbol β indicates the propagation constant of corresponding modes; The eigenvalue equations and field distribution functions of each mode type are derived from [5]; κ and C indicate the coupling coefficients for LPG s coupling and the coupling between two fibers, defined in Eqs. () and (), respectively; and Λ is the grating period of LPG; the superscript S on these denotes SM, and denotes M, and the subscript corresponds to the mode order. Due to being closely phase-matched S S S S between two coupled modes, the coupling coefficientsκ 6 0 κ 0 6, C6 6 C6 6, and other coefficient pairs are the same as these. The numerical calculation and simulation of the respective coupling efficiency for device components and the whole propagating interactions for the MADM can be achieved by solving the coupled mode equations with the transfer matrix method [6]. or the LPG, the S S S mode resonances occur in the phase matching conditions β0 β6 = π Λ and β0i β6 = π Λ i []. The parameters of device components of MADM are listed in Table 3. In order to reveal the efficiency of coupling crosstalk from SM and M, and the coupling efficiencies of respective LPG in SM and M, as well as to exhibit the final mode conversion ratios from LP 0 in SM to LP 0 and LP 03 in M, the conversion spectra of mode powers are plotted in ig. 6. Table 3. Parameters of Device Components of MADMs Modulation strength δ ( z) Grating or coupling length L (cm) Grating period Λ (μm) LPG in SM Coupling of HE _ LPG in M LP 0 LP _ The total conversion ratio can be expressed byη = ηηη S c ; hereη S is the coupling efficiency from the LP 0 mode to the cladding mode HE 6 by LPG in SM, η is that from this cladding mode HE 6 to LP 0 and LP 03 modes by corresponding LPG in M, andη c is that of the cladding mode HE 6 from SM to M, which is shown in ig. 6(b) and 6(a) respectively. Note that in ig. 6(a), we give the exchange relationship of mode power between the cladding mode HE 6 in SM and the cladding mode HE 6, HE 56, and HE 65 in M. It is obvious that the coupling crosstalk from the cladding mode HE 6 in SM to adjacent cladding modes HE 56 and HE 65 in M is very weak due to lack of full phasematching. The final conversion ratios from LP 0 in SM to LP 0 and LP 03 in M shown in ig. 6(c) approach 98%. The similar structures with the two parallel gratings through the evanescent-field coupling between their cladding modes used to wavelength add / drop multiplexing have been experimentally reported with coupling efficiencies as much as 65% and 86% [7, 8]. urthermore, the 3dB bandwidths of mode conversion are nearly 0 μm, and it indicates that this device has lower wavelength dependence. Because of the large bandwidth of the cladding mode coupling from SM to M, the total conversion bandwidth is mainly dependent on the spectra of LPG shown in ig. 6(b); in other words, (C) 04 OSA 9 May 04 Vol., No. 0 DOI:0.364/OE OPTICS EXPRESS 495

9 the number of grating periods of each LPG, the greater of the number, the narrower of the bandwidth [9]. Mode Power Exchange S HE 6 HE 6 HE 56 HE 65 Conversion Ratios of Mode Power Coupling Efficiencies (a) S S LP -HE6 0 HE -LP0 6 HE -LP (b) S LP -LP0 0 S LP -LP (c) ig. 6. (a) Exchange relationship of mode power with coupling crosstalk from SM to M. (b) Coupling efficiency of respective LPG in SM and M. (c) Total conversion ratio of MADMs for multiplexing LP 0 and LP Analysis of coupling interferences in MDM transmission Single MADM for multiplexing modes LP 0 or LP 03 has been successfully designed and analyzed above. In this section we focus on the coupling interferences for ahead multiplexed modes by back MADMs. The connection diagram of multiplexing modes LP 0, LP 0, and LP 03 is drawn in ig. 7, where three fundamental modes LP 0 are multiplexed in one M through two MADMs. The structure of de-multiplexing modes in the receiver can be designed to a symmetric structure relative to the transmitter here. The effective indexes of core modes LP 0m and cladding modes HE m supported in the M are listed in Table 4. Note that the coupling interferences here just may occur among this type of HE m modes because of their circular symmetry of field intensity, similar to the core modes LP 0 and LP 03. In fact, it is the LPG written in the M of MADM that possibly influence the foregoing multiplexed modes. or the MADM, according to the resonance condition of its LPG in the M and its grating period listed in Table 3, if there is one mode that the transmitted LP 0 can (C) 04 OSA 9 May 04 Vol., No. 0 DOI:0.364/OE OPTICS EXPRESS 496

10 be coupled to, the effective index of this mode should be.450. However, there are no modes corresponding to this index; therefore MADM will not produce coupling interferences for the transmitted LP 0. Similarly, for the MADM, there are also not any modes that can be found among all these modes listed in Table 4 to involve in the coupling interferences for the transmitted modes LP 0 and LP 0. Therefore, this kind of MADM can effectively eliminate the coupling interferences for ahead multiplexed modes by back MADMs. MADM MADM M SM SM ig. 7. Connection diagram of multiplexing modes LP 0, LP 0 and LP 03. Table 4. Effective Indexes of the Core Modes LP 0m and Cladding Modes HE m Core modes LP 0m LP 0 LP 0 LP 03 Effective indexes Cladding modes HE m HE HE HE 3 HE 4 HE 5 HE 6 HE 7 Effective indexes When three or more modes are multiplexed in M or MM, these coupling interferences can be intentionally avoided by means of selecting properly different cladding modes as the mediums coupled from SM to M or MM to break the resonance conditions of possible coupling interferences. urthermore, the structure of proposed MADM can be extended to multiplexing LP lm modes where l > and their spatial-orientation modes, provided that the grating profile of LPG written in M of MADM are properly tilted and the tilt direction of grating profile is changed according to the spatial-orientation of multiplexed modes, similar to the principle presented in the [4, 0]. 5. Conclusion A novel MADM with two parallel combinative LPGs has been proposed and theoretically analyzed and discussed in this article. or modes LP 0 and LP 03 multiplexing, the cladding mode HE 6 is selected as the medium coupled from SM to M. LPG in SM converts mode from fundamental mode LP 0 into this cladding mode and in turn converted into core modes LP 0 or LP 03 by LPG in M. The coupling crosstalk and possible coupling interferences have been analyzed with detail. This MADM has advantages of facile and good scalability, and of eliminating coupling interferences for ahead multiplexed mode by back MADMs or couplers. urthermore, the conversion rate of mode power can theoretically approach 98% and the 3 db bandwidth of these devices can reach 0nm or more. Acknowledgment This work is supported by the National Basic Research Program of China (0CB707500), the Innovation und Project for Graduate Student of Shanghai (JWCXSL30), the cultivating fund for national projects of University of Shanghai for Science and Technology (USST). (C) 04 OSA 9 May 04 Vol., No. 0 DOI:0.364/OE OPTICS EXPRESS 497