Abstract

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 as98%and the 3dB bandwidth can reach 10nm or more.

© 2014 Optical Society of America

1. Introduction

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 [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 de-multiplexed [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 (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 modesLPlmwherel>0, while propagating along the fiber for a long distance, will produce intermodal dispersion; as a result, the vector mode components may walk off [10]. For the modesLP0m, 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.

 

Fig. 1 Diagram of MADM and mode conversion from the fundamental mode LP01 to cladding mode HE16 and then to HOMs LP02 and LP03, with their transverse electric field distributions shown from left to right and up to down.

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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 [11]. 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 frequencyVof SMF is 2.07, and that of FMF is 8.30; the eigenmodes LP0m supported in FMF include LP01, LP02, and LP03 [12]. 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 expression

κvu=12ωε0n12δ(z)02πdϕ0a1EvEurdr,
whereωandε0are the angular frequency and the dielectric constant in vacuum respectively; andn1anda1are the refractive index and radius of the fiber core respectively;σ(z)is the modulation strength; EvandEuare the functions of electric field transverse distribution of mode v and u in the fiber core [11]; and indicates the complex conjugate.

Tables Icon

Table 1. Parameters of Two Fibers in MADM

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.

 

Fig. 2 Coupling coefficients for LP02 and LP03 to all cladding modes HE1m.

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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 by

C1616FS=12ωε0[(n12n32)02πdϕ0a1Esu(r,ϕ)Eco(r,ϕ)rdr+(n22n32)02πdϕa1a2Esu(r,ϕ)Ecl(r,ϕ)rdr],
whereEco,Eclare the functions of transverse electric field distribution of the cladding mode HE16 in the core and cladding regions of FMF, respectively, andEsuis that in the surroundings for SMF. 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 triangleOAOshown in Fig. 4, a transformational relation can be found:
r=r2+d2+2rdcosϕϕ=sin1(rrsinϕ),
where A(r,ϕ)for FMF and (r,ϕ)for SMF are any points in the region of FMF, and dis the distance between two fiber cores.

 

Fig. 3 Cross section of parallel SMF and FMF, and coordinate transformation in two fibers.

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Fig. 4 Effective indexes of the cladding mode HE16 in SMF and FMF with fiber parameters listed in Table 1.

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The value of C1616FSis dependent on the distancedand the refractive index of surroundings n3that directly determines the value of the distribution of Esu. When two fibers touch each other, whilst simultaneously the value of n3approaches that of the cladding indexn2, which is taken to 1.445 and can be obtained by immersing the structure of LPFG pair into an index-matching medium,C1616FSwill become large enough, and hence the periodic coupling length will be vastly shortened [13]. Furthermore, significant coupling only happens when the propagation constants of cladding mode HE16 selected in both SMF and FMF are very similar, i.e.β13Sβ13F, which implies that the two cladding modes HE16 are fully phase-matched, and in this case, the coupling coefficient from FMF to SMF C1616SFis closed toC1616FS [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 radiusa2of SMF to 54.375 μm, while the cladding radiusa2of 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.

 

Fig. 5 Radial electric field distributions of cladding mode HE16 withn3=1.445in FMF (solid line) and SMF (dotted line).

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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 [14], 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 thexpolarization (linearly polarized along thexaxis) and theypolarization (linearly polarized along theyaxis). The four types of coupling coefficients and the corresponding coupling lengthsLcwith 100%coupling efficiency which is determined byCLc=π/2are calculated and listed in Table 2.It shows that the values of coupling coefficients forxxandyypolarization 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 [12]. Thexy andyxpolarization coupling is zero due to the orthogonality of mode field at the overlap region .

Tables Icon

Table 2. Coupling coefficients and Coupling Lengths for Four Types of Polarization Coupling

Actually, the coupling distance Lcbetween 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 [13].

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 xx polarization and yy polarization is not distinct, therefore we just simulate the xx polarized mode coupling in this section. It is well known that there are a large number of cladding eignemodes supported in ordinary fiber [11]. 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 as

dA1dz=12jB1κ1601Sexp[j(β01Sβ16S2πΛS)z]dB1dz+jβ16SB1=jC1616FSB2+jC5616FSB3+jC6516FSB4+12jA1κ0116Sexp[j(β01Sβ16S2πΛS)z],dB3dz+jβ56FB3=jC1656SFB1dB4dz+jβ65FB4=jC1665SFB1dB2dz+jβ16FB2=jC1616SFB1+12jAiκ0i16Fexp[j(β0iFβ16F2πΛiF)z]dAidz=12jB2κ160iFexp[j(β0iFβ16F2πΛiF)z]
where A1is the amplitude of core mode LP01 in SMF, and Aiindicates the amplitude of the core mode LP02 (i=2), or LP03(i=3); B1,B2,B3,andB4represent the cladding modes HE16 in SMF, HE16, HE56, and HE65 in FMF respectively; the symbolβindicates the propagation constant of corresponding modes; The eigenvalue equations and field distribution functions of each mode type are derived from [15]; κand Cindicate the coupling coefficients for LPFG’s coupling and the coupling between two fibers, defined in Eqs. (1) and (2), respectively; and Λis the grating period of LPFG; the superscript Son these denotes SMF, andFdenotes FMF, and the subscript corresponds to the mode order. Due to being closely phase-matched between two coupled modes, the coupling coefficientsκ1601Sκ0116S, C1616FSC1616SF, 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 [16]. For the LPFG, the mode resonances occur in the phase matching conditions β01Sβ16S=2π/ΛS and β0iFβ16F=2π/ΛiF [11]. 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.

Tables Icon

Table 3. Parameters of Device Components of MADMs

 

Fig. 6 (a) Exchange relationship of mode power with coupling crosstalk from SMF to FMF. (b) Coupling efficiency of respective LPFG in SMF and FMF. (c) Total conversion ratio of MADMs for multiplexing LP02 and LP03 .

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The total conversion ratio can be expressed byη=ηSηcηF; hereηSis the coupling efficiency from the LP01 mode to the cladding mode HE16 by LPFG in SMF,ηFis that from this cladding mode HE16 to LP02 and LP03 modes by corresponding LPFG in FMF, andηcis 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) approach98%. 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 as65%and86% [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 [19].

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.

 

Fig. 7 Connection diagram of multiplexing modes LP01, LP02 and LP03.

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Tables Icon

Table 4. Effective Indexes of the Core Modes LP0m and Cladding Modes HE1m

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 LPlm modes where l>1 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].

5. Conclusion

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 98%and the 3 dB bandwidth of these devices can reach 10nm or more.

Acknowledgment

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]  

6. J. D. Love and N. Riesen, “Mode-selective couplers for few-mode optical fiber networks,” Opt. Lett. 37(19), 3990–3992 (2012). [CrossRef]   [PubMed]  

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]  

20. K. S. Lee and T. Erdogan, “Fiber mode conversion with tilted gratings in an optical fiber,” J. Opt. Soc. Am. A 18(5), 1176–1185 (2001). [CrossRef]   [PubMed]  

References

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  • |

  1. Y. Kokubun, 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, 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, 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, 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, J. D. Love, “Weakly-guiding mode-selective fiber couplers,” IEEE J. Quantum Electron. 48(7), 941–945 (2012).
    [CrossRef]
  6. J. D. Love, N. Riesen, “Mode-selective couplers for few-mode optical fiber networks,” Opt. Lett. 37(19), 3990–3992 (2012).
    [CrossRef] [PubMed]
  7. A. Li, X. Chen, A. A. Amin, W. Shieh, “Fused fiber mode couplers for few-mode transmission,” IEEE Photon. Technol. Lett. 24(21), 1953–1956 (2012).
    [CrossRef]
  8. N. Riesen, 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, 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, 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, M. N. Ng, “Analysis of two parallel long-period fiber gratings,” J. Lightwave Technol. 22(5), 1358–1366 (2004).
    [CrossRef]
  14. M. Z. Alam, 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, 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, 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, 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]
  20. K. S. Lee, T. Erdogan, “Fiber mode conversion with tilted gratings in an optical fiber,” J. Opt. Soc. Am. A 18(5), 1176–1185 (2001).
    [CrossRef] [PubMed]

2013 (4)

N. Bozinovic, Y. Yue, Y. X. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340(6140), 1545–1548 (2013).
[CrossRef] [PubMed]

N. Riesen, 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]

Y. Xie, S. Fu, H. Liu, H. Zhang, M. Tang, P. Shum, D. Liu, “Design and numerical optimization of a mode multiplexer based on few-mode fiber couplers,” J. Opt. 15(12), 125404 (2013).
[CrossRef]

M. Z. Alam, J. Albert, “Selective excitation of radially and azimuthally polarized optical fiber cladding modes,” J. Lightwave Technol. 31(19), 3167–3175 (2013).
[CrossRef]

2012 (4)

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, 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).

N. Riesen, J. D. Love, “Weakly-guiding mode-selective fiber couplers,” IEEE J. Quantum Electron. 48(7), 941–945 (2012).
[CrossRef]

J. D. Love, N. Riesen, “Mode-selective couplers for few-mode optical fiber networks,” Opt. Lett. 37(19), 3990–3992 (2012).
[CrossRef] [PubMed]

A. Li, X. Chen, A. A. Amin, W. Shieh, “Fused fiber mode couplers for few-mode transmission,” IEEE Photon. Technol. Lett. 24(21), 1953–1956 (2012).
[CrossRef]

2011 (2)

2009 (1)

Y. Kokubun, M. Koshiba, “Novel multi-core fibers for mode division multiplexing: proposal and design principle,” IEICE Electron. Express 6(8), 522–528 (2009).
[CrossRef]

2007 (2)

2005 (1)

F. Abrishamian, S. Sato, 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]

2004 (1)

2001 (1)

1997 (2)

Abrishamian, F.

F. Abrishamian, S. Sato, 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]

Ahmed, N.

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, 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).

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, 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).

Al Amin, A.

Alam, M. Z.

Albert, J.

Amin, A. A.

A. Li, X. Chen, A. A. Amin, W. Shieh, “Fused fiber mode couplers for few-mode transmission,” IEEE Photon. Technol. Lett. 24(21), 1953–1956 (2012).
[CrossRef]

Astruc, M.

Bigo, S.

Birnbaum, K. M.

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, 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).

Boutin, A.

Bozinovic, N.

N. Bozinovic, Y. Yue, Y. X. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340(6140), 1545–1548 (2013).
[CrossRef] [PubMed]

Brindel, P.

Cerou, F.

Chan, F. Y. M.

Charlet, G.

Chen, X.

A. Li, X. Chen, A. A. Amin, W. Shieh, “Fused fiber mode couplers for few-mode transmission,” IEEE Photon. Technol. Lett. 24(21), 1953–1956 (2012).
[CrossRef]

A. Li, A. Al Amin, X. Chen, 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]

Chiang, K. S.

Doliner, S.

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, 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).

Erdogan, T.

Erkmen, B. I.

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, 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).

Fu, S.

Y. Xie, S. Fu, H. Liu, H. Zhang, M. Tang, P. Shum, D. Liu, “Design and numerical optimization of a mode multiplexer based on few-mode fiber couplers,” J. Opt. 15(12), 125404 (2013).
[CrossRef]

Han, W. T.

Huang, H.

N. Bozinovic, Y. Yue, Y. X. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340(6140), 1545–1548 (2013).
[CrossRef] [PubMed]

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, 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).

Imai, M.

F. Abrishamian, S. Sato, 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]

Jung, Y. M.

Kim, B. H.

Kim, M. J.

Koebele, C.

Kokubun, Y.

Y. Kokubun, M. Koshiba, “Novel multi-core fibers for mode division multiplexing: proposal and design principle,” IEICE Electron. Express 6(8), 522–528 (2009).
[CrossRef]

Koshiba, M.

Y. Kokubun, M. Koshiba, “Novel multi-core fibers for mode division multiplexing: proposal and design principle,” IEICE Electron. Express 6(8), 522–528 (2009).
[CrossRef]

Kristensen, P.

N. Bozinovic, Y. Yue, Y. X. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340(6140), 1545–1548 (2013).
[CrossRef] [PubMed]

Lee, B. H.

Lee, K. S.

Li, A.

A. Li, X. Chen, A. A. Amin, W. Shieh, “Fused fiber mode couplers for few-mode transmission,” IEEE Photon. Technol. Lett. 24(21), 1953–1956 (2012).
[CrossRef]

A. Li, A. Al Amin, X. Chen, 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]

Liu, D.

Y. Xie, S. Fu, H. Liu, H. Zhang, M. Tang, P. Shum, D. Liu, “Design and numerical optimization of a mode multiplexer based on few-mode fiber couplers,” J. Opt. 15(12), 125404 (2013).
[CrossRef]

Liu, H.

Y. Xie, S. Fu, H. Liu, H. Zhang, M. Tang, P. Shum, D. Liu, “Design and numerical optimization of a mode multiplexer based on few-mode fiber couplers,” J. Opt. 15(12), 125404 (2013).
[CrossRef]

Liu, Y.

Love, J. D.

N. Riesen, 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]

N. Riesen, J. D. Love, “Weakly-guiding mode-selective fiber couplers,” IEEE J. Quantum Electron. 48(7), 941–945 (2012).
[CrossRef]

J. D. Love, N. Riesen, “Mode-selective couplers for few-mode optical fiber networks,” Opt. Lett. 37(19), 3990–3992 (2012).
[CrossRef] [PubMed]

Mardoyan, H.

Ng, M. N.

Provost, L.

Ramachandran, S.

N. Bozinovic, Y. Yue, Y. X. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340(6140), 1545–1548 (2013).
[CrossRef] [PubMed]

Ran, Z. L.

Rao, Y. J.

Ren, Y.

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, 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).

Ren, Y. X.

N. Bozinovic, Y. Yue, Y. X. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340(6140), 1545–1548 (2013).
[CrossRef] [PubMed]

Riesen, N.

N. Riesen, 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]

J. D. Love, N. Riesen, “Mode-selective couplers for few-mode optical fiber networks,” Opt. Lett. 37(19), 3990–3992 (2012).
[CrossRef] [PubMed]

N. Riesen, J. D. Love, “Weakly-guiding mode-selective fiber couplers,” IEEE J. Quantum Electron. 48(7), 941–945 (2012).
[CrossRef]

Salsi, M.

Sato, S.

F. Abrishamian, S. Sato, 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]

Shieh, W.

A. Li, X. Chen, A. A. Amin, W. Shieh, “Fused fiber mode couplers for few-mode transmission,” IEEE Photon. Technol. Lett. 24(21), 1953–1956 (2012).
[CrossRef]

A. Li, A. Al Amin, X. Chen, 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]

Shum, P.

Y. Xie, S. Fu, H. Liu, H. Zhang, M. Tang, P. Shum, D. Liu, “Design and numerical optimization of a mode multiplexer based on few-mode fiber couplers,” J. Opt. 15(12), 125404 (2013).
[CrossRef]

Sillard, P.

Sperti, D.

Tang, M.

Y. Xie, S. Fu, H. Liu, H. Zhang, M. Tang, P. Shum, D. Liu, “Design and numerical optimization of a mode multiplexer based on few-mode fiber couplers,” J. Opt. 15(12), 125404 (2013).
[CrossRef]

Tran, P.

Tur, M.

N. Bozinovic, Y. Yue, Y. X. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340(6140), 1545–1548 (2013).
[CrossRef] [PubMed]

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, 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).

Verluise, F.

Willner, A. E.

N. Bozinovic, Y. Yue, Y. X. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340(6140), 1545–1548 (2013).
[CrossRef] [PubMed]

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, 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).

Xie, Y.

Y. Xie, S. Fu, H. Liu, H. Zhang, M. Tang, P. Shum, D. Liu, “Design and numerical optimization of a mode multiplexer based on few-mode fiber couplers,” J. Opt. 15(12), 125404 (2013).
[CrossRef]

Yan, Y.

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, 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).

Yang, J.

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, 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).

Yue, Y.

N. Bozinovic, Y. Yue, Y. X. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340(6140), 1545–1548 (2013).
[CrossRef] [PubMed]

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, 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).

Zhang, H.

Y. Xie, S. Fu, H. Liu, H. Zhang, M. Tang, P. Shum, D. Liu, “Design and numerical optimization of a mode multiplexer based on few-mode fiber couplers,” J. Opt. 15(12), 125404 (2013).
[CrossRef]

Zhang, L.

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, 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).

Zhu, T.

IEEE J. Quantum Electron. (1)

N. Riesen, J. D. Love, “Weakly-guiding mode-selective fiber couplers,” IEEE J. Quantum Electron. 48(7), 941–945 (2012).
[CrossRef]

IEEE Photon. J. (1)

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, 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).

IEEE Photon. Technol. Lett. (2)

A. Li, X. Chen, A. A. Amin, W. Shieh, “Fused fiber mode couplers for few-mode transmission,” IEEE Photon. Technol. Lett. 24(21), 1953–1956 (2012).
[CrossRef]

N. Riesen, 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]

IEICE Electron. Express (1)

Y. Kokubun, M. Koshiba, “Novel multi-core fibers for mode division multiplexing: proposal and design principle,” IEICE Electron. Express 6(8), 522–528 (2009).
[CrossRef]

J. Lightwave Technol. (3)

J. Opt. (1)

Y. Xie, S. Fu, H. Liu, H. Zhang, M. Tang, P. Shum, D. Liu, “Design and numerical optimization of a mode multiplexer based on few-mode fiber couplers,” J. Opt. 15(12), 125404 (2013).
[CrossRef]

J. Opt. Soc. Am. A (2)

Opt. Express (4)

Opt. Lett. (1)

Opt. Rev. (1)

F. Abrishamian, S. Sato, 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]

Science (1)

N. Bozinovic, Y. Yue, Y. X. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340(6140), 1545–1548 (2013).
[CrossRef] [PubMed]

Other (2)

K. Okamoto, Fundamentals of Optical Waveguides (Elsevier Academic Press, 2006), Chap. 3.

C. Tsao, Optical Fibre Waveguide Analysis (Oxford University, 1992), Part 3.

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Figures (7)

Fig. 1
Fig. 1

Diagram of MADM and mode conversion from the fundamental mode LP01 to cladding mode HE16 and then to HOMs LP02 and LP03, with their transverse electric field distributions shown from left to right and up to down.

Fig. 2
Fig. 2

Coupling coefficients for LP02 and LP03 to all cladding modes HE1m.

Fig. 3
Fig. 3

Cross section of parallel SMF and FMF, and coordinate transformation in two fibers.

Fig. 4
Fig. 4

Effective indexes of the cladding mode HE16 in SMF and FMF with fiber parameters listed in Table 1.

Fig. 5
Fig. 5

Radial electric field distributions of cladding mode HE16 with n 3 = 1.445 in FMF (solid line) and SMF (dotted line).

Fig. 6
Fig. 6

(a) Exchange relationship of mode power with coupling crosstalk from SMF to FMF. (b) Coupling efficiency of respective LPFG in SMF and FMF. (c) Total conversion ratio of MADMs for multiplexing LP02 and LP03 .

Fig. 7
Fig. 7

Connection diagram of multiplexing modes LP01, LP02 and LP03.

Tables (4)

Tables Icon

Table 1 Parameters of Two Fibers in MADM

Tables Icon

Table 2 Coupling coefficients and Coupling Lengths for Four Types of Polarization Coupling

Tables Icon

Table 3 Parameters of Device Components of MADMs

Tables Icon

Table 4 Effective Indexes of the Core Modes LP0m and Cladding Modes HE1m

Equations (4)

Equations on this page are rendered with MathJax. Learn more.

κ v u = 1 2 ω ε 0 n 1 2 δ ( z ) 0 2 π d ϕ 0 a 1 E v E u r d r ,
C 16 16 F S = 1 2 ω ε 0 [ ( n 1 2 n 3 2 ) 0 2 π d ϕ 0 a 1 E s u ( r , ϕ ) E c o ( r , ϕ ) r d r + ( n 2 2 n 3 2 ) 0 2 π d ϕ a 1 a 2 E s u ( r , ϕ ) E c l ( r , ϕ ) r d r ] ,
r = r 2 + d 2 + 2 r d cos ϕ ϕ = sin 1 ( r r sin ϕ ) ,
d A 1 dz = 1 2 j B 1 κ 1601 S exp[ j( β 01 S β 16 S 2π Λ S )z ] d B 1 dz +j β 16 S B 1 =j C 1616 FS B 2 +j C 5616 FS B 3 +j C 6516 FS B 4 + 1 2 j A 1 κ 0116 S exp[ j( β 01 S β 16 S 2π Λ S )z ], d B 3 dz +j β 56 F B 3 =j C 1656 SF B 1 d B 4 dz +j β 65 F B 4 =j C 1665 SF B 1 d B 2 dz +j β 16 F B 2 =j C 1616 SF B 1 + 1 2 j A i κ 0i16 F exp[ j( β 0i F β 16 F 2π Λ i F )z ] d A i dz = 1 2 j B 2 κ 160i F exp[ j( β 0i F β 16 F 2π Λ i F )z ]

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