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

Liang Fang; Hongzhi Jia

doi:10.1364/OE.22.011488

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 modes $L P_{l m}$ 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 $L P_{0 m}$ , 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 LP₀₂ and LP₀₃ 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 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 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 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 frequency $V$ of SMF is 2.07, and that of FMF is 8.30; the eigenmodes LP_0m supported in FMF include LP₀₁, LP₀₂, and LP₀₃ [12]. In order to reveal the coupling efficiency between all cladding modes HE_1m and the core modes LP₀₂ and LP₀₃, the coupling coefficients between them are calculated by the expression

κ_{v - u} = \frac{1}{2} ω ε_{0} n_{1}^{2} δ (z) \int_{0}^{2 π} d ϕ \int_{0}^{a_{1}} E_{v} E_{u}^{*} r d r,

where

ω

and

ε_{0}

are the angular frequency and the dielectric constant in vacuum respectively; and

n_{1}

and

a_{1}

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 [11]; and

*

indicates the complex conjugate.

Table 1. Parameters of Two Fibers in MADM

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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 HE_1m 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 HE_1m.

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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

C_{16 - 16}^{F - S} = \frac{1}{2} ω ε_{0} [(n_{1}^{2} - n_{3}^{2}) \int_{0}^{2 π} d ϕ \int_{0}^{a_{1}} E_{s u} (r^{'}, ϕ^{'}) E_{c o}^{*} (r, ϕ) r d r + (n_{2}^{2} - n_{3}^{2}) \int_{0}^{2 π} d ϕ \int_{a_{1}}^{a_{2}} E_{s u} (r^{'}, ϕ^{'}) E_{c l}^{*} (r, ϕ) r d r],

where

E_{c o}, E_{c l}

are the functions of transverse electric field distribution of the cladding mode HE₁₆ in the core and cladding regions of FMF, respectively, and

E_{s u}

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^{'} A O

shown in Fig. 4, a transformational relation can be found:

\begin{array}{l} r^{'} = \sqrt{r^{2} + d^{2} + 2 r d \cos ϕ} \\ ϕ^{'} = \sin^{- 1} (\frac{r}{r^{'}} \sin ϕ), \end{array}

where

A

(r, ϕ)

for FMF and

(r^{'}, ϕ^{'})

for SMF are any points in the region of FMF, and

d

is 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 $C_{16 - 16}^{F - S}$ is dependent on the distance $d$ and the refractive index of surroundings $n_{3}$ that directly determines the value of the distribution of $E_{s u}$ . 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 HE₁₆ selected in both SMF and FMF are very similar, i.e. $β_{13}^{S} \approx β_{13}^{F}$ , which implies that the two cladding modes HE₁₆ 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 ${a^{'}}_{2}$ 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 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 with $n_{3} = 1.445$ in 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 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 $C L_{c} = π / 2$ 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 HE₁₆ almost have the same values; in other words, the cladding mode HE₁₆ 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 .

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

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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 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

\begin{array}{l} \frac{d A_{1}}{d z} = \frac{1}{2} j B_{1} κ_{16 - 01}^{S} \exp [- j (β_{01}^{S} - β_{16}^{S} - \frac{2 π}{Λ^{S}}) z] \\ \frac{d B_{1}}{d z} + j β_{16}^{S} B_{1} = j C_{16 - 16}^{F - S} B_{2} + j C_{56 - 16}^{F - S} B_{3} + j C_{65 - 16}^{F - S} B_{4} + \frac{1}{2} j A_{1} κ_{01 - 16}^{S} \exp [j (β_{01}^{S} - β_{16}^{S} - \frac{2 π}{Λ^{S}}) z], \\ \frac{d B_{3}}{d z} + j β_{56}^{F} B_{3} = j C_{16 - 56}^{S - F} B_{1} \\ \frac{d B_{4}}{d z} + j β_{65}^{F} B_{4} = j C_{16 - 65}^{S - F} B_{1} \\ \frac{d B_{2}}{d z} + j β_{16}^{F} B_{2} = j C_{16 - 16}^{S - F} B_{1} + \frac{1}{2} j A_{i} κ_{0 i - 16}^{F} \exp [j (β_{0 i}^{F} - β_{16}^{F} - \frac{2 π}{Λ_{i}^{F}}) z] \\ \frac{d A_{i}}{d z} = \frac{1}{2} j B_{2} κ_{16 - 0 i}^{F} \exp [- j (β_{0 i}^{F} - β_{16}^{F} - \frac{2 π}{Λ_{i}^{F}}) z] \end{array}

where

A_{1}

is the amplitude of core mode LP₀₁ in SMF, and

A_{i}

indicates the amplitude of the core mode LP₀₂

(i = 2)

, or LP₀₃

(i = 3)

;

B_{1}, B_{2}, B_{3}

,and

B_{4}

represent the cladding modes HE₁₆ in SMF, HE₁₆, HE₅₆, and HE₆₅ in FMF 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 [15];

κ

and

C

indicate the coupling coefficients for LPFG’s coupling and the coupling between two fibers, defined in Eqs. (1) and (2), respectively; and

Λ

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

κ_{16 - 01}^{S} \approx κ_{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 $β_{01}^{S} - β_{16}^{S} = 2 π / Λ^{S}$ and $β_{0 i}^{F} - β_{16}^{F} = 2 π / Λ_{i}^{F}$ [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 LP₀₁ in SMF to LP₀₂ and LP₀₃ in FMF, the conversion spectra of mode powers are plotted in Fig. 6.

Table 3. Parameters of Device Components of MADMs

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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 by $η = η_{S} η_{c} η_{F}$ ; here $η_{S}$ is the coupling efficiency from the LP₀₁ mode to the cladding mode HE₁₆ by LPFG in SMF, $η_{F}$ is that from this cladding mode HE₁₆ to LP₀₂ and LP₀₃ modes by corresponding LPFG in FMF, and $η_{c}$ is 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 $98 %$ . 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 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 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 LP₀₂ and LP_03.

Fig. 7 Connection diagram of multiplexing modes LP₀₁, LP₀₂ and LP_03.

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Table 4. Effective Indexes of the Core Modes LP_0m and Cladding Modes HE_1m

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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 $L P_{l m}$ 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

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	Core radius (μm)	Cladding radius (μm)	Core index $n_{1}$	Cladding index $n_{2}$	Surrounding index $n_{3}$
SMF	3.000	54.375	1.460	1.450	1.445
FMF	12.000	62.500	1.460	1.450	1.445

Polarization coupling	$x - x$	$x - y$	$y - x$	$y - y$
Coupling coefficients (nm⁻¹)	$6.53 \times 10^{- 8}$	0	0	$6.48 \times 10^{- 8}$
Coupling length $L_{c}$ (cm)	$2.41$	-	-	$2.42$

	LPFG in SMF	Coupling of HE₁₆	LPFG in FMF
	LPFG in SMF	Coupling of HE₁₆	LP₀₂	LP₀₃
Modulation strength $δ (z)$	$5.00 \times 10^{- 4}$	_	$5.00 \times 10^{- 4}$	$5.00 \times 10^{- 4}$
Grating or coupling length $L$ (cm)	$1.63$	$2.41$	$3.02$	$1.20$
Grating period $Λ$ (μm)	$222.50$	_	$170.20$	$354.80$

Core modes LP_0m	LP₀₁			LP₀₂			LP₀₃
Effective indexes	1.45933			1.45653			1.45179
Cladding modes HE_1m	HE₁₁	HE₁₂	HE₁₃		HE₁₄	HE₁₅		HE₁₆	HE₁₇
Effective indexes	1.44993	1.44970	1.44933		1.44882	1.44818		1.44742	1.44658

	Core radius (μm)	Cladding radius (μm)	Core index $n_{1}$	Cladding index $n_{2}$	Surrounding index $n_{3}$
SMF	3.000	54.375	1.460	1.450	1.445
FMF	12.000	62.500	1.460	1.450	1.445

Mode add/drop multiplexers of LP₀₂ and LP₀₃ modes with two parallel combinative long-period fiber gratings

Abstract

1. Introduction

2. Analysis of mode coupling

2.1 Couping between cladding mode and HOMs

2.2 Coupling between two fibers

3. Discussion and simulation

4. Analysis of coupling interferences in MDM transmission

5. Conclusion

Acknowledgment

References and links

Cited By

Figures (7)

Tables (4)

Equations (4)

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