## Abstract

Two parallel combinative long-period fiber gratings (LPFGs) can convert the fundamental core mode LP_{01} in a single-mode fiber (SMF) into one desired higher order core mode LP_{0m} 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 LP_{0m} 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 LP_{02} 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. Furthermore, the conversion rate of mode power theoretically can approach as much as$98\%$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 LP_{01} 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 modes$L{P}_{lm}$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 [10]. For the modes$L{P}_{0m}$, composed of a single vector mode HE_{1m}, 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_{02} 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 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 LP_{01} 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 HE_{1m} 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 HE_{1m} (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 LPFG between LP_{01} 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 frequency$V$of SMF is 2.07, and that of FMF is 8.30; the eigenmodes
LP_{0m} supported in FMF include LP_{01}, LP_{02}, and LP_{03}
[12]. In order to reveal the coupling efficiency
between all cladding modes HE_{1m} and the core modes LP_{02} and
LP_{03}, the coupling coefficients between them are calculated by the
expression

The coupling coefficients calculated are demonstrated in Fig.
2, in which it is indicated that the coefficients between the core mode LP_{03}
and all cladding modes HE_{1m} are almost larger than those between the core mode
LP_{02} 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 FMF, compared to those for
LP_{02} and these cladding modes, which is roughly shown in Fig. 1. From the Fig. 2, the optimal
cladding mode selected as the medium is HE_{16} for both multiplexing LP_{02}
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
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 HE_{16} from SFM to FMF. The
cross-section of parallel SMF and FMF is shown in Fig.
3.When the cladding mode HE_{16} 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

_{16}in the core and cladding regions of FMF, respectively, and${E}_{su}$is 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 triangle${O}^{\prime}AO$shown in Fig. 4, a transformational relation can be found:

The value of ${C}_{16-16}^{F-S}$is dependent on the distance$d$and the refractive index of surroundings ${n}_{3}$that directly determines the value of the distribution of
${E}_{su}$. When two fibers touch each other, whilst simultaneously the
value of ${n}_{3}$approaches that of the cladding index${n}_{2}$, which is taken to 1.445 and can be obtained by immersing the
structure of LPFG pair into an index-matching medium,${C}_{16-16}^{F-S}$will 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 HE_{16} selected
in both SMF and FMF are very similar, i.e.${\beta}_{13}^{S}\approx {\beta}_{13}^{F}$, which implies that the two cladding modes HE_{16} are
fully phase-matched, and in this case, the coupling coefficient from FMF to SMF
${C}_{16-16}^{S-F}$is closed to${C}_{16-16}^{F-S}$ [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 radius${{a}^{\prime}}_{2}$of SMF to 54.375 μm, while the cladding
radius${a}_{2}$of 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 HE_{16} 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
HE_{16} 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 [14], so when the selected cladding mode HE_{16} is
coupled from SMF to FMF, the coupling may be dependent on polarization. The coupling
coefficients of the cladding mode HE_{16} from SMF to FMF 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 $100\%$coupling efficiency which is determined
by$C{L}_{c}=\pi /2$are calculated and listed in Table 2.It shows that the values of coupling coefficients for$x-x$and$y-y$polarization coupling are approximated, because the radial and
azimuthal field components of the cladding mode HE_{16} almost have the same values; in
other words, the cladding mode HE_{16} is almost completely linearly polarized [12]. The$x-y$ and$y-x$polarization coupling is zero due to the orthogonality of mode
field at the overlap region .

Actually, the coupling distance ${L}_{c}$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 [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 $x-x$ 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 [11]. Even in the case that the surrounding index is close to the index of fiber cladding, as in this article, the number of HE_{1m} 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 HE_{16} mode coupled from LP_{01} by the LPFG in SMF largely couples to the HE_{16} 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 HE_{16} 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 LPFG pair can be expressed as

_{01}in SMF, and ${A}_{i}$indicates the amplitude of the core mode LP

_{02 }$\left(i=2\right)$, or LP

_{03}$\left(i=3\right)$; ${B}_{1},{B}_{2},{B}_{3}$,and${B}_{4}$represent the cladding modes HE

_{16}in SMF, HE

_{16}, HE

_{56}, and HE

_{65}in FMF respectively; the symbol$\beta $indicates the propagation constant of corresponding modes; The eigenvalue equations and field distribution functions of each mode type are derived from [15]; $\kappa $and $C$indicate the coupling coefficients for LPFG’s coupling and the coupling between two fibers, defined in Eqs. (1) and (2), respectively; and $\Lambda $is the grating period of LPFG; the superscript $S$on these denotes SMF, and$F$denotes FMF, and the subscript corresponds to the mode order. Due to being closely phase-matched between two coupled modes, the coupling coefficients${\kappa}_{16-01}^{S}\approx {\kappa}_{01-16}^{S}$, ${C}_{16-16}^{F-S}\approx {C}_{16-16}^{S-F}$, 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
${\beta}_{01}^{S}-{\beta}_{16}^{S}=2\pi /{\Lambda}^{S}$ and ${\beta}_{0i}^{F}-{\beta}_{16}^{F}=2\pi /{\Lambda}_{i}^{F}$ [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 LP_{01} in SMF to LP_{02} and LP_{03} in FMF, the
conversion spectra of mode powers are plotted in Fig.
6.

The total conversion ratio can be expressed by$\eta ={\eta}_{S}{\eta}_{c}{\eta}_{F}$; here${\eta}_{S}$is the coupling efficiency from the LP_{01} mode to the cladding mode HE_{16} by LPFG in SMF,${\eta}_{F}$is that from this cladding mode HE_{16} to LP_{02} and LP_{03} modes by corresponding LPFG in FMF, and${\eta}_{c}$is that of the cladding mode HE_{16} 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 HE_{16} in SMF and the cladding mode HE_{16}, HE_{56}, and HE_{65} in FMF. It is obvious that the coupling crosstalk from the cladding mode HE_{16} in SMF to adjacent cladding modes HE_{56} and HE_{65} in FMF is very weak due to lack of full phase-matching. The final conversion ratios from LP_{01} in SMF to LP_{02} and LP_{03} in FMF shown in Fig. 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$86\%$ [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 LP_{02} 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_{01},
LP_{02}, and LP_{03} is drawn in Fig. 7,
where three fundamental modes LP_{01} 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 LP_{0m}
and cladding modes HE_{1m} supported in the FMF are listed in Table 4.Note that the coupling interferences here just may occur among this type of
HE_{1m} modes because of their circular symmetry of field intensity, similar to the
core modes LP_{02} and LP_{03.}

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 LP_{01} 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 LP_{01}. 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 LP_{01} and LP_{02}. 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 $L{P}_{lm}$ 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 LP_{02} and LP_{03} multiplexing, the cladding mode HE_{16} is selected as the medium coupled from SMF to FMF. LPFG in SMF converts mode from fundamental mode LP_{01} into this cladding mode and in turn converted into core modes LP_{02} or LP_{03} 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).

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