Two parallel combinative long-period fiber gratings (LPFGs) can convert the fundamental core mode LP01 in a single-mode fiber (SMF) into one desired higher order core mode LP0m in a few-mode fiber (FMF), in the process of which one specific cladding mode acts as a medium coupled from one fiber to another. Different LP0m modes can be obtained by controlling the grating period of LPFG in FMF to meet the phase matching condition. In this article we focus on the design and analyses of LP02 and LP03 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. Furthermore, the conversion rate of mode power theoretically can approach as much asand the 3dB bandwidth can reach 10nm or more.
© 2014 Optical Society of America
Single-mode fiber (SMF), 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 [1, 2]. In these technologies, mode selective couplers or mode multiplexers / de-multiplexers (MUXs/DEMUXs) are key devices that convert the fundamental mode LP01 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 mode-selective couplers (MSCs) [5, 6], fused fiber mode couplers , and tapered mode-selective couplers . Due to being simple, lower-loss and more compact waveguide-based solutions, the mode couplers are regarded as promising MUXs/DENUXs in MDM transmission . 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 de-multiplexed . The greater the number of multiplexed modes is, the worse the interferences will occur, which may cause design difficulties . 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 (LPFG), 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 LPFG written in FMF.
This MADM has a structure of two parallel combinative LPFGs; one LPFG converts the fundamental core mode into one cladding mode in SMF touched with FMF, then this cladding mode is coupled from SMF to FMF; finally, the other LPFG written in FMF converts this cladding mode into one desired core HOM. Theoretically, each LP mode can be multiplexed in one FMF as independent data channel. However, scalar modeswhere, while propagating along the fiber for a long distance, will produce intermodal dispersion; as a result, the vector mode components may walk off . For the modes, composed of a single vector mode HE1m, 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 LP02 and LP03 modes in this article; other LP0m modes multiplexing can be extended by the basic principle discussed here. Compared with proposed mode-selective couplers based on multi-core fiber and tapered structure, the structure and fabrication of this MADM on the basis of conventional fiber are simpler and more facile.
2. Analysis of mode coupling
The diagram of MADM is shown in Fig. 1, where the parallel SMF and FMF are placed close together, and two LPFGs are written in SMF and FMF, respectively. Fundamental core mode LP01 transmitting in SMF is coupled into one specific cladding mode through one LPFG, and then this cladding mode is coupled to FMF, and finally converted into a desired core HOM by another LPFG in FMF.
2.1 Couping between cladding mode and HOMs
Since one of the cladding modes HE1m is selected as the medium, through LPFG written in FMF with uniform modulation this cladding mode can be strongly coupled to nothing but the core modes HE1m (if the fiber is weakly guiding, i.e., scalar modes LP0m), because of the complete circular symmetry of their field intensities in the fiber core region. The mode coupling of LPFG between LP01 and cladding modes in SMF is well known . In this section, our work lays emphasis on the characteristics of mode conversion between the cladding modes and core HOMs through LPFG in FMF.
The parameters of SMF and FMF as components of MADM is shown in Table 1.The normalized waveguide frequencyof SMF is 2.07, and that of FMF is 8.30; the eigenmodes LP0m supported in FMF include LP01, LP02, and LP03 . In order to reveal the coupling efficiency between all cladding modes HE1m and the core modes LP02 and LP03, the coupling coefficients between them are calculated by the expression11]; and indicates the complex conjugate.
The coupling coefficients calculated are demonstrated in Fig. 2, in which it is indicated that the coefficients between the core mode LP03 and all cladding modes HE1m are almost larger than those between the core mode LP02 and these cladding modes. This is due to the more similarity of electric field distributions between LP03 mode and these cladding modes in the core region of FMF, compared to those for LP02 and these cladding modes, which is roughly shown in Fig. 1. From the Fig. 2, the optimal cladding mode selected as the medium is HE16 for both multiplexing LP02 and LP03. 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 LPFG in FMF 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 LPFG.
2.2 Coupling between two fibers
In this section, we analyze the coupling of cladding mode HE16 from SFM to FMF. The cross-section of parallel SMF and FMF is shown in Fig. 3.When the cladding mode HE16 is coupled from SMF to FMF, its electric field distributed in the surroundings for SMF will be perturbed by the refractive index of both cladding and core areas in FMF. So the complete coupling coefficient is defined byFig. 4, a transformational relation can be found:
The value of is dependent on the distanceand the refractive index of surroundings that directly determines the value of the distribution of . When two fibers touch each other, whilst simultaneously the value of approaches that of the cladding index, which is taken to 1.445 and can be obtained by immersing the structure of LPFG pair into an index-matching medium,will become large enough, and hence the periodic coupling length will be vastly shortened . Furthermore, significant coupling only happens when the propagation constants of cladding mode HE16 selected in both SMF and FMF are very similar, i.e., which implies that the two cladding modes HE16 are fully phase-matched, and in this case, the coupling coefficient from FMF to SMF is closed to [5, 13]. 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 radiusof SMF to 54.375 μm, while the cladding radiusof FMF is maintained to 62.5 μm.The index matching relation is illustrated in Fig. 4. It shows the effective indexes of the cladding mode HE16 in SMF and FMF are equal in the wavelength of 1550 nm, which means the full phase match with the designed fiber parameters. Figure 5 shows the radial electric field distributions of cladding mode HE16 in FMF and SMF when the LPFG pair is immersed in the index-matching 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.
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 , so when the selected cladding mode HE16 is coupled from SMF to FMF, the coupling may be dependent on polarization. The coupling coefficients of the cladding mode HE16 from SMF to FMF may differ in thepolarization (linearly polarized along theaxis) and thepolarization (linearly polarized along theaxis). The four types of coupling coefficients and the corresponding coupling lengthswith coupling efficiency which is determined byare calculated and listed in Table 2.It shows that the values of coupling coefficients forandpolarization coupling are approximated, because the radial and azimuthal field components of the cladding mode HE16 almost have the same values; in other words, the cladding mode HE16 is almost completely linearly polarized . The andpolarization coupling is zero due to the orthogonality of mode field at the overlap region .
Actually, the coupling distance between two fibers shown in Fig. 1 can be overlapped more or less on the grating extents of two LPFGs in SMF and FMF; however, in this design it will increase the whole coupling length of the parallel combative LPFG pair .
3. Discussion and simulation
Because of the polarization independence of the LPFG coupling in SMF and FMF due to the complete circular symmetry of the LPFG structure, the polarization mode coupling in the whole process in the LPFG 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 polarization and polarization is not distinct, therefore we just simulate the 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 HE1m modes is as much as seven, which are listed in Fig. 2, exclusive of other HE, EH, TM and TE modes. The power of the HE16 mode coupled from LP01 by the LPFG in SMF largely couples to the HE16 cladding mode in FMF 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 HE16 mode. Among all cladding modes, we discover that the modes HE56 and HE65 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 LPFG pair can be expressed as15]; and indicate the coupling coefficients for LPFG’s coupling and the coupling between two fibers, defined in Eqs. (1) and (2), respectively; and is the grating period of LPFG; the superscript on these denotes SMF, anddenotes FMF, and the subscript corresponds to the mode order. Due to being closely phase-matched between two coupled modes, the coupling coefficients, , 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 . For the LPFG, the mode resonances occur in the phase matching conditions and . The parameters of device components of MADM are listed in Table 3.In order to reveal the efficiency of coupling crosstalk from SMF and FMF, and the coupling efficiencies of respective LPFG in SMF and FMF, as well as to exhibit the final mode conversion ratios from LP01 in SMF to LP02 and LP03 in FMF, the conversion spectra of mode powers are plotted in Fig. 6.
The total conversion ratio can be expressed by; hereis the coupling efficiency from the LP01 mode to the cladding mode HE16 by LPFG in SMF,is that from this cladding mode HE16 to LP02 and LP03 modes by corresponding LPFG in FMF, andis that of the cladding mode HE16 from SMF to FMF, which is shown in Fig. 6(b) and 6(a) respectively. Note that in Fig. 6(a), we give the exchange relationship of mode power between the cladding mode HE16 in SMF and the cladding mode HE16, HE56, and HE65 in FMF. It is obvious that the coupling crosstalk from the cladding mode HE16 in SMF to adjacent cladding modes HE56 and HE65 in FMF is very weak due to lack of full phase-matching. The final conversion ratios from LP01 in SMF to LP02 and LP03 in FMF shown in Fig. 6(c) approach. 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 asand [17, 18]. Furthermore, the 3dB bandwidths of mode conversion are nearly 10 μm, and it indicates that this device has lower wavelength dependence. Because of the large bandwidth of the cladding mode coupling from SMF to FMF, the total conversion bandwidth is mainly dependent on the spectra of LPFG shown in Fig. 6(b); in other words, the number of grating periods of each LPFG, the greater of the number, the narrower of the bandwidth .
4. Analysis of coupling interferences in MDM transmission
Single MADM for multiplexing modes LP02 or LP03 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 LP01, LP02, and LP03 is drawn in Fig. 7, where three fundamental modes LP01 are multiplexed in one FMF 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 LP0m and cladding modes HE1m supported in the FMF are listed in Table 4.Note that the coupling interferences here just may occur among this type of HE1m modes because of their circular symmetry of field intensity, similar to the core modes LP02 and LP03.
In fact, it is the LPFG written in the FMF of MADM that possibly influence the foregoing multiplexed modes. For the MADM1, according to the resonance condition of its LPFG in the FMF and its grating period listed in Table 3, if there is one mode that the transmitted LP01 can be coupled to, the effective index of this mode should be 1.45022. However, there are no modes corresponding to this index; therefore MADM1 will not produce coupling interferences for the transmitted LP01. Similarly, for the MADM2, 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 LP01 and LP02. Therefore, this kind of MADM can effectively eliminate the coupling interferences for ahead multiplexed modes by back MADMs.
When three or more modes are multiplexed in FMF or MMF, these coupling interferences can be intentionally avoided by means of selecting properly different cladding modes as the mediums coupled from SMF to FMF or MMF to break the resonance conditions of possible coupling interferences. Furthermore, the structure of proposed MADM can be extended to multiplexing modes where and their spatial-orientation modes, provided that the grating profile of LPFG written in FMF 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 [14, 20].
A novel MADM with two parallel combinative LPFGs has been proposed and theoretically analyzed and discussed in this article. For modes LP02 and LP03 multiplexing, the cladding mode HE16 is selected as the medium coupled from SMF to FMF. LPFG in SMF converts mode from fundamental mode LP01 into this cladding mode and in turn converted into core modes LP02 or LP03 by LPFG in FMF. 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. Furthermore, the conversion rate of mode power can theoretically approach and the 3 dB bandwidth of these devices can reach 10nm or more.
This work is supported by the National Basic Research Program of China (2011CB707500), the Innovation Fund Project for Graduate Student of Shanghai (JWCXSL1302), the cultivating fund for national projects of University of Shanghai for Science and Technology (USST).
References and links
1. Y. Kokubun and M. Koshiba, “Novel multi-core fibers for mode division multiplexing: proposal and design principle,” IEICE Electron. Express 6(8), 522–528 (2009). [CrossRef]
2. 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(6140), 1545–1548 (2013). [CrossRef] [PubMed]
3. A. Li, A. Al Amin, X. Chen, and W. Shieh, “Transmission of 107-Gb/s mode and polarization multiplexed CO-OFDM signal over a two-mode fiber,” Opt. Express 19(9), 8808–8814 (2011). [CrossRef] [PubMed]
4. C. Koebele, M. Salsi, D. Sperti, P. Tran, P. Brindel, H. Mardoyan, S. Bigo, A. Boutin, F. Verluise, P. Sillard, M. Astruc, L. Provost, F. Cerou, and G. Charlet, “Two mode transmission at 2×100 Gb/s, over 40 km-long prototype few-mode fiber, using LCOS-based programmable mode multiplexer and demultiplexer,” Opt. Express 19(17), 16593–16600 (2011). [CrossRef] [PubMed]
5. N. Riesen and J. D. Love, “Weakly-guiding mode-selective fiber couplers,” IEEE J. Quantum Electron. 48(7), 941–945 (2012). [CrossRef]
7. A. Li, X. Chen, A. A. Amin, and W. Shieh, “Fused fiber mode couplers for few-mode transmission,” IEEE Photon. Technol. Lett. 24(21), 1953–1956 (2012). [CrossRef]
8. N. Riesen and J. D. Love, “Ultra-broadband tapered mode-selective couplers for few-mode optical fiber networks,” IEEE Photon. Technol. Lett. 25(24), 2501–2504 (2013). [CrossRef]
9. Y. Xie, S. Fu, 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. 15(12), 125404 (2013). [CrossRef]
10. 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, 535–543 (2012).
11. T. Erdogan, “Cladding-mode resonances in short- and long period fiber grating filters,” J. Opt. Soc. Am. A 14(8), 1760–1773 (1997). [CrossRef]
12. K. Okamoto, Fundamentals of Optical Waveguides (Elsevier Academic Press, 2006), Chap. 3.
13. K. S. Chiang, F. Y. M. Chan, and M. N. Ng, “Analysis of two parallel long-period fiber gratings,” J. Lightwave Technol. 22(5), 1358–1366 (2004). [CrossRef]
14. M. Z. Alam and J. Albert, “Selective excitation of radially and azimuthally polarized optical fiber cladding modes,” J. Lightwave Technol. 31(19), 3167–3175 (2013). [CrossRef]
15. C. Tsao, Optical Fibre Waveguide Analysis (Oxford University, 1992), Part 3.
16. F. 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. 12(6), 467–471 (2005). [CrossRef]
17. 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 15(17), 10855–10862 (2007). [CrossRef] [PubMed]
18. Y. Liu, K. S. Chiang, Y. J. Rao, Z. L. Ran, and T. Zhu, “Light coupling between two parallel CO2-laser written long-period fiber gratings,” Opt. Express 15(26), 17645–17651 (2007). [CrossRef] [PubMed]
19. T. Erdogan, “Fiber Grating Spectra,” J. Lightwave Technol. 15(8), 1277–1294 (1997). [CrossRef]