Theoretical analysis of mode evolution in a tapered double clad fiber based on the coupled local mode theory

Xuanfeng Zhou

doi:10.1364/OPTCON.479690

1. Introduction

In the late 1980s, to solve the problems of insufficient pump power and unavoidable nonlinear effects in fiber lasers, Snitzer et al. proposed cladding pumping technology based on double clad fibers (DCFs) [1]. Nowadays, DCF has become the main choice of high-power fiber lasers [2,3]. DCF consists of three layers with core, inner cladding and outer cladding. In the application of fiber laser, the core and inner cladding can be used as the transmission channels of signal light and pump light respectively [4–6]. In addition, fibers with three-layer structure are widely used in many fields such as mode optimization and mode control [7,8].

A tapered fiber refers to the fiber whose size changes along the longitude direction. A tapered fiber can be used to change the distribution of optical field, so it is used in a variety of fiber devices such as mode filters, fiber sensors and pump couplers [9–12]. Strictly speaking, due to the longitudinal non-uniformity, tapered fibers cannot support any eigenmode, which brings many difficulties to the research. For this reason, most studies believe that the diameter of tapered fiber changes slowly, which means it satisfies the adiabaticity criteria [13,14]. In this case, the concept of local mode can be introduced to describe the optical field characteristics of tapered fiber at any longitudinal position. With this method, the mode characteristics in the tapered fiber can be converted into how to calculate the energy proportion of local modes [15–17].

A tapered double clad fiber (TDCF) has the characteristics of both DCF and tapered fiber, so it has a wider application prospect in mode conversion and other aspects [18–23]. However, mode characteristics of a TDCF should be analyzed based on three-layer waveguide model, so it is more complicated than that of a single clad fiber (SCF). If the beam propagation method (BPM) is used for numerical simulation, the structural characteristics of the tapered fiber are difficult to handle in the meshing process to determine a suitable grid size. This will lead to problems such as low calculation accuracy and slow calculation speed. Besides, the energy proportion of each local mode cannot be directly calculated, which is unfavorable for the mode analysis of a TDCF [16].

In this paper, mode evolution characteristics of TDCFs are analyzed based on the coupled local mode theory (CLMT). In the case of adiabatic taper, CLMT only needs to solve the coupled local mode equations (CLMEs) in the calculation process, so the calculation speed is much faster. At the same time, CLMT can directly calculate the energy proportion of each mode, so it can provide a clearer physical image. The paper is organized as follows. In Sec. 2, the theoretical model of a TDCF is established, and the mode characteristics of TDCF are analyzed by solving mode coupling coefficient. In Sec. 3, based on CLMT, TDCFs with different taper parameters are calculated by solving CLMEs and the simulation results are given. Sec. 4 gives an example of structure optimization process for a TDCF. Finally, a conclusion is given in Sec. 5.

2. Theoretical model and analysis

2.1 Basic structure of a TDCF

The basic structure of a TDCF is shown in Fig. 1. It consists of three parts: original fiber, taper region and taper waist. The cross section of the DCF is a three-layer structure. The corresponding refractive index is n₁ for the core, n₂ for the inner cladding and n₃ for the outer cladding. For the original fiber, the radial dimension is defined as ρ₀. Specifically, the core radius is a₀, the inner cladding radius is b₀, and the outer cladding radius is c₀. The ratio of inner cladding radius to core radius is defined as cladding core ratio s = b₀/a₀. For the taper waist, the radial dimension is defined as ρ_w. Specifically, the core radius is a_w, the inner cladding radius is b_w, and the outer cladding radius is c_w. The taper ratio of the taper region is defined as TR=ρ_w/ρ₀. The length of the taper region is defined as taper length L_t. The change rate of the radial dimension at position z is defined as taper angle θ(z).

Fig. 1. Basic structure diagram of a TDCF.

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There are two main shapes of taper region: linear taper shape and nonlinear taper shape. For the linear taper shape, the radial dimension changes linearly in the entire taper region. For the nonlinear taper shape, the following definition is adopted in this paper. The radial dimension of the taper region at the position z is:

(1)$$\rho (z) = \left\{ {\begin{array}{{cc}} {\begin{array}{{cc}} {\frac{{({\rho_w} - {\rho_0})}}{{{L_t}{L_k}}}{z^2} + {\rho_0}}&{(0 \le z \le {L_k})} \end{array}}\\ {\begin{array}{{cc}} {\frac{{{\rho_0} - {\rho_w}}}{{{L_t}({L_t} - {L_k})}}{z^2} + \frac{{2({\rho_w} - {\rho_0})}}{{{L_t} - {L_k}}}z + \frac{{{L_t}({\rho_0} - {\rho_w})}}{{{L_t} - {L_k}}} + {\rho_w}}&{(\; {L_k} \le z \le {L_t})} \end{array}} \end{array}} \right.$$

where z = 0 is the starting point of the taper region, and z = L_k is the knee point position for the taper angle.

The shape profiles of taper region for different taper shapes are shown in Fig. 2(a), where the knee point coefficient α_k is defined as α_k= L_k/L_t. Figure 2(b) shows the taper angles at different longitude positions.

Fig. 2. Shape profile and taper angle for different taper shapes.

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2.2 Mode characteristics of a DCF

Three-layer waveguide model is adopted to analyze mode characteristics of a DCF. When considering the weak guidance approximation, the mode field distribution of the fiber can be expressed in the form of linear polarization. The electric field corresponding to LP_mn mode (m is the azimuthal mode number, and n is the radial mode number) can be expressed as:

(2)$$\begin{array}{l} \begin{array}{{cccc}} {{\Psi _{mn}} = {A_{mn}}{J_m}({U_{mn}}\frac{r}{a})\left( {\begin{array}{{cc}} {\cos (m\theta )}\\ {\sin (m\theta )} \end{array}} \right)}&{\begin{array}{{cccc}} {\begin{array}{{cccc}} {\begin{array}{{cccc}} {}&{}&{} \end{array}}&{} \end{array}}&{\begin{array}{{cccc}} {}&{} \end{array}}&{}&{} \end{array}}&{r < a} \end{array}\\ \begin{array}{{cccc}} {{\Psi _{mn}} = \left[ {{B_{mn}}{I_m}({W_{mn}}\frac{r}{a}) + {C_{mn}}{K_m}({W_{mn}}\frac{r}{a})} \right]\left( {\begin{array}{{ccc}} {\cos (m\theta )}\\ {\sin (m\theta )} \end{array}} \right)}&{{n_{eff}} \ge {n_2}}&{a \le r \le b} \end{array}\\ \begin{array}{{cccc}} {{\Psi _{mn}} = \left[ {{B_{mn}}{J_m}({Q_{mn}}\frac{r}{b}) + {C_{mn}}{N_m}({Q_{mn}}\frac{r}{b})} \right]\left( {\begin{array}{{cccc}} {\cos (m\theta )}\\ {\sin (m\theta )} \end{array}} \right)}&{{n_{eff}} < {n_2}}&{a \le r \le b} \end{array}\\ \begin{array}{{cccc}} {{\Psi _{mn}} = {D_{mn}}{K_m}({T_{mn}}\frac{r}{b})\left( {\begin{array}{{cccc}} {\cos (m\theta )}\\ {\sin (m\theta )} \end{array}} \right)}&{\begin{array}{{cccc}} {\begin{array}{{cc}} {\begin{array}{{cc}} {\begin{array}{{cc}} {\begin{array}{{cc}} {}&{} \end{array}}&{} \end{array}}&{} \end{array}}&{\begin{array}{{ccc}} {}&{} \end{array}} \end{array}}&{}&{} \end{array}}&{r > b} \end{array} \end{array}$$

where A, B, C and D are coefficient constants, J_m and N_m are m-order Bessel functions of the first and second kind respectively, I_m and K_m are the m-order modified Bessel functions of the first and second kind respectively. $U = a{k_0}\sqrt {n_1^2 - n_{eff}^2}$, $W = a{k_0}\sqrt {n_{eff}^2 - n_2^2}$, $Q = b{k_0}\sqrt {n_2^2 - n_{eff}^2}$, and $T = b{k_0}\sqrt {n_{eff}^2 - n_3^2}$ are the normalized constants of the fiber, and n_eff is the effective refractive index.

The optical field of the local mode of the fiber can be solved by the scalar wave equation together with the solution on the continuity of ψ and dψ/dr at the boundary. Therefore, the boundary continuity conditions can be applied to the core mode and the cladding mode respectively.

For the core mode, its eigenvalue equation can be expressed as:

(3)$$\left|{\begin{array}{{cccc}} {{J_m}(U)}&{ - {I_m}(W)}&{ - {K_m}(W)}&0\\ {\frac{U}{a}{J_m}^\prime (U)}&{ - \frac{W}{a}{I_m}^\prime (W)}&{ - \frac{W}{a}{K_m}^\prime (W)}&0\\ 0&{{I_m}(Ws)}&{{K_m}(Ws)}&{ - {K_m}(T)}\\ 0&{\frac{W}{a}{I_m}^\prime (Ws)}&{\frac{W}{a}{K_m}^\prime (Ws)}&{ - \frac{T}{b}{K_m}^\prime (T)} \end{array}} \right|= 0$$

For the cladding mode, its eigenvalue equation can be expressed as:

(4)$$\left|{\begin{array}{{cccc}} {{J_m}(U)}&{ - {J_m}({Q / s})}&{ - {N_m}({Q / s})}&0\\ {\frac{U}{a}{J_m}^\prime (U)}&{ - \frac{Q}{b}{J_m}^\prime ({Q / s})}&{ - \frac{Q}{b}{N_m}^\prime ({Q / s})}&0\\ 0&{{J_m}(Q)}&{{N_m}(Q)}&{ - {K_m}(T)}\\ 0&{\frac{Q}{b}{J_m}^\prime (Q)}&{\frac{Q}{b}{N_m}^\prime (Q)}&{ - \frac{T}{b}{K_m}^\prime (T)} \end{array}} \right|= 0$$

According to Eq. (2), the electric field expression on the boundary is:

(5)$$\begin{array}{l} {\Psi _{mn}}(a) = {A_{mn}}{J_m}({U_{mn}})\left( {\begin{array}{{cc}} {\cos (m\theta )}\\ {\sin (m\theta )} \end{array}} \right)\\ {\Psi _{mn}}(b) = {D_{mn}}{K_{mn}}({T_{mn}})\left( {\begin{array}{{cc}} {\cos (m\theta )}\\ {\sin (m\theta )} \end{array}} \right) \end{array}$$

Mode field distribution of the original DCF is calculated as shown in Fig. 3. The basic parameters are a₀= 7.5 µm, b₀= 22.5 µm, n₁= 1.4595, n₂= 1.4518, n₃= 1.444. The cladding core ratio is s = 3. The numerical apertures of the core and inner cladding are NA₁= 0.15 and NA₂= 0.15 respectively. The wavelength of light is set to be λ= 1550 nm. It should be noted that LP₀₁, LP₁₁, LP₂₁ and LP₀₂ modes are core modes, and other modes are cladding modes. Considering that both LP₁₁ and LP₂₁ modes have odd mode and even mode, it can be concluded that the original fiber with this parameter is a six-mode fiber when only core modes are considered.

Fig. 3. Mode field distribution of low-order modes in a DCF.

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2.3 Coupled local mode equations

According to CLMT, the CLME in a tapered fiber can be expressed as a differential equation about the mode complex amplitude [24]:

(6)$$\frac{{d{b_k}}}{{dz}} - i{\beta _k}{b_k} = \sum\limits_{j \ne k} {{C_{jk}}{b_j}}$$

where b is the mode complex amplitude, and C_jk the mode coupling coefficient from the j-th local mode to the k-th local mode.

Mode coupling coefficient is a key parameter of a tapered fiber, which is directly related to the mode evolution characteristics. Under the condition of weak guidance approximation, the mode coupling coefficient can be expressed as:

(7)$${C_{jk}} ={-} \frac{{{k_0}}}{{2{Z_0}({\beta _j} - {\beta _k})}}\int_{S\infty } {\frac{{\partial {n^2}}}{{\partial z}}{{\hat{\psi }}_j}{{\hat{\psi }}_k}} dS$$

where β_j and β_k are the propagation constants of the j-th and k-th local modes respectively. ${\hat{\psi }_j}$ and ${\hat{\psi }_k}$ are the normalized mode field distributions of the j-th and k-th local modes respectively. Z₀ is wave impedance in vacuum, and n is the refractive index distribution of the fiber.

For step index DCF, its refractive index distribution can be expressed as:

(8)$${n^2}(r) = n_1^2 - (n_1^2 - n_2^2)H({r - a} )- (n_2^2 - n_3^2)H({r - b} )$$

Therefore, the mode coupling coefficient can be expressed as:

(9)$$\begin{aligned} {C_{jk}} &= \theta {\kappa _{jk}}\\ {\kappa _{jk}} &= \frac{{{k_0}}}{{2{Z_0}({\beta _j} - {\beta _k})}}({NA_1^2\oint_{r = a} {{{\hat{\psi }}_j}{{\hat{\psi }}_k}dl + NA_2^2\oint_{r = b} {{{\hat{\psi }}_j}{{\hat{\psi }}_k}dl} } } )\end{aligned}$$

where ${\kappa _{jk}}$ is defined as mode overlap coefficient between two local modes.

The beat length between mode j and mode k is defined as:

(10)$$L{p_{jk}} = \frac{{2\pi }}{{|{{\beta_j} - {\beta_k}} |}}$$

The mode overlap factor between mode j and mode k is defined as:

(11)$${\delta _{jk}} = NA_1^2\oint_{r = a} {{{\hat{\psi }}_j}{{\hat{\psi }}_k}dl + NA_2^2\oint_{r = b} {{{\hat{\psi }}_j}{{\hat{\psi }}_k}dl} }$$

Substituting Eq. (5) into Eq. (11), it can be found that mode coupling can only occur in two modes with equal m values, and they must be either two odd modes or two even modes. (These modes are called family modes in this paper.) The mode overlap factor between mode LP_mn and mode LP_mn’ is:

(12)$${\sigma _{mn - mn^{\prime}}} = \left\{ \begin{array}{l} \begin{array}{{cc}} {2\pi aNA_1^2{A_{0n}}{J_0}({U_{0n}}){A_{0n^{\prime}}}{J_0}({U_{0n^{\prime}}}) + 2\pi bNA_2^2{D_{0n}}{K_0}({T_{0n}}){D_{0n^{\prime}}}{K_0}({T_{0n^{\prime}}})}&{m = 0} \end{array}\\ \pi aNA_1^2{A_{mn}}{J_m}({U_{mn}}){A_{mn^{\prime}}}{J_m}({U_{mn^{\prime}}}) + \begin{array}{{cc}} {\pi bNA_2^2{D_{mn}}{K_m}({T_{mn}}){D_{mn^{\prime}}}{K_m}({T_{mn^{\prime}}})}&{m \ne 0} \end{array} \end{array} \right.$$

2.4 Mode characteristics of a TDCF

For a TDCF, its eigenmode can be calculated based on size parameters and refractive index parameters at any longitude position, which is called the local mode of tapered fiber at that position. According to Eq. (3) and Eq. (4), the effective refractive indices of different modes of TDCF are calculated as shown in Fig. 4. Here the taper ratio and cladding core ratio are set to be equal with TR = s.

Fig. 4. Effective indices of low-order modes for a TDCF.

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Modes with effective refractive index larger than inner cladding are called core modes, while modes with effective refractive index smaller than inner cladding are called cladding modes [25]. It can be seen from Fig. 4 that LP₀₁ mode is core mode in the whole taper range. LP₁₁, LP₂₁ and LP₀₂ modes gradually evolve from core mode to cladding mode with the decrease of core size. Other modes are cladding mode in the whole taper range. It should be mentioned that the effective refractive indices of family modes do not overlap.

According to the effective refractive index and eigenvalue equation, the normalized coefficient constants can be further calculated. According to Eq. (3), it can be obtained that the relationship between the coefficient constants of the core mode is:

(13)$$\begin{aligned} A &= B\frac{{{I_m}(W)}}{{{J_m}(U)}} + C\frac{{{K_m}(W)}}{{{J_m}(U)}}\\ D &= B\frac{{{I_m}(Ws)}}{{{K_m}(T)}} + C\frac{{{K_m}(Ws)}}{{{K_m}(T)}}\\ \frac{C}{B} &= \frac{{W{J_m}(U){{I^{\prime}}_m}(W) - U{I_m}(W){{J^{\prime}}_m}(U)}}{{U{K_m}(W){{J^{\prime}}_m}(U) - W{J_m}(U){{K^{\prime}}_m}(W)}} \end{aligned}$$

According to Eq. (4), it can be obtained that the relationship between the coefficient constants of the cladding mode is:

(14)$$\begin{aligned} A &= B\frac{{{J_m}({Q / s})}}{{{J_m}(U)}} + C\frac{{{N_m}({Q / s})}}{{{J_m}(U)}}\\ D &= B\frac{{{J_m}(Q)}}{{{K_m}(T)}} + C\frac{{{N_m}(Q)}}{{{K_m}(T)}}\\ \frac{C}{B} &= \frac{{T{J_m}(Q){{K^{\prime}}_m}(T) - Q{K_m}(T){{J^{\prime}}_m}(Q)}}{{Q{K_m}(T){{N^{\prime}}_m}(Q) - T{N_m}(Q){{K^{\prime}}_m}(T)}} \end{aligned}$$

Solving Eq. (13) and Eq. (14), the normalized coefficient constants for different core sizes can be obtained. Figure 5 shows the change curve of normalized coefficient constants for LP_0n modes. Along with the taper region, the coefficients change continuously. It should be noted that during the evolution of LP₀₂ mode, both B and C changes suddenly at the junction point of core mode and cladding mode.

Fig. 5. Normalized coefficient constants of low-order LP_0n modes in a TDCF.

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Mode field distribution characteristics with different core sizes are calculated, and their one-dimensional electric field evolution patterns are shown in Fig. 6. It can be seen that the shape and size of the mode field are constantly changing during the evolution process. This is the reason why the calculated mode is called the local mode.

Fig. 6. One dimensional electric field evolution patterns of low-order modes in a TDCF.

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According to Eqs. (9)–(11), the beat length, mode overlap factor and mode overlap coefficient between low-order modes of TDCF with different core sizes are calculated, as shown in Fig. 7. It can be seen from Fig. 7(a) that the beat length between modes does not change monotonously with the change of core radius. Among them, the beat length between LP₀₁ and LP₀₂ is the shortest, which is less than 0.3 mm in the whole range. The beat length between LP₀₂ and LP₀₃ is the longest, which exceeds 1 mm at the peak position. From Fig. 7(b), we can see that change of the overlap factor between modes is also non-monotonic. In Fig. 7(c), the mode overlap coefficients of mode coupling for LP₂₁-LP₂₂ and LP₀₂-LP₀₃ are larger, which means that such mode couplings are likely to occur in the six-mode TDCF.

Fig. 7. Mode coupling parameters between low-order modes in a TDCF.

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3. Simulation results

3.1 Mode evolution of TDCFs with different taper shapes

In this section, mode evolution characteristics in TDCFs with different parameters are calculated. The fourth-order Runge-Kutta method is used to solve CLME. In the calculation process, coupling among five modes are considered. According to the analysis above, the beat lengths corresponding to the four LP modes are basically within 1 mm. Therefore, the taper length is set with L_t= 1 mm firstly. The calculation results are shown in Fig. 8, where α-LP_mn indicates the energy proportion of LP_mn mode among all transmission modes. Figure 8(a1)–(d1) show the evolution process of injection mode. Figure 8(a2)–(d2) show the evolution process of the adjacent family mode (Δn = 1). Figure 8(a3)–(d3) show that the evolution process of the non-adjacent family mode (Δn ≠1).

Fig. 8. Mode characteristics of a TDCF with different taper shapes.

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It can be seen from Fig. 8 that energy of different modes show oscillation characteristics along with the propagation direction. The oscillation period is basically consistent with the beat length. Mode coupling mainly occurs between adjacent family modes (Δn = 1). In comparison, mode coupling can be ignored when LP₀₁ or LP₁₁ mode is injected, and the energy proportion at the output end is more than 98%. However, mode coupling cannot be ignored when LP₂₁ or LP₀₂ mode is injected, and the energy proportion at the output end is less than 90% for most cases. The taper shape has an important influence on the mode coupling process. Taking the injection of LP₀₂ mode as an example, the energy proportions at the output end for the four kinds of taper shapes are 81.7%, 62.2%, 70.1% and 97.2%, respectively.

3.2 Mode evolution of TDCFs with different taper lengths

In order to reduce the mode coupling in tapered fiber, it is necessary to satisfy the adiabaticity criteria better. A simple method is increasing the taper length. Mode evolution characteristics of a TDCF with different taper lengths are simulated with results shown in Fig. 9. Only energy proportion of the injection mode is shown in the figures, where the taper length in Fig. 9(a)–(d) is L_t= 5 mm, and the taper length in Fig. 9(e)–(h) is L_t= 10 mm.

Fig. 9. Mode characteristics of a TDCF with different taper lengths.

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It can be seen from Fig. 9 that mode coupling for injection of LP₀₁ and LP₁₁ modes can also be ignored. For LP₂₁ and LP₀₂ modes, the energy proportion are larger than 99.5% for most cases. However, it can be seen from Fig. 9 (c) and Fig. 9 (d) that when the taper length is not large enough, the mode coupling is still greatly affected by the taper shapes. Therefore, optimization of a TDCF should at least include the following two aspects: taper shape and taper length.

3.3 Comparison between CLMT and BPM

The optical field transmission characteristics of the above TDCFs can also be simulated by finite-difference beam propagation method (FD-BPM). The simulation results for injection of LP₀₁ mode are shown in Fig. 10. The horizontal grid size in the simulation is set with 0.1 µm, and the longitudinal grid size is 0.5 µm. Since BPM cannot directly give the energy proportion of the mode, the calculation results are obtained by mode decomposition of optical fields at different longitude positions.

Fig. 10. Comparison of simulation results by CLMT and BPM.

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It can be seen from Fig. 10 that the mode evolution trend calculated by these two methods are basically the same, which can verify the accuracy of CLMT in calculating tapered fibers. The deviation of calculation results mainly appears in the region close to the taper waist. The main reason is the grid size is relatively large in the taper waist since the size of the fiber core is very small. In contrast, the calculation error of CLMT mainly comes from the calculation of mode coupling coefficient and the solution of CLMEs. The mode coupling coefficient is calculated based on the analytical method, so the calculation accuracy can be improved through setting the computational grid adaptively. The fourth-order Runge-Kutta method is used to calculate CLMEs, which can also achieve high accuracy only if the slowly varying envelope approximation is satisfied.

4. Optimization design of a TDCF

In this section, we introduce an example of optimization design of a TDCF. The TDCF is required to connect with a SCF, and all modes in the SCF must be transmitted with high efficiency. (This may be used in the manufacture of fiber devices in application areas such as fiber laser and fiber communication [26].) As can be seen from Fig. 1, we can intuitively determine parameters of the SCF as follows: core radius a₀, core refractive index n₂, and cladding refractive index n₃. On the other hand, core radius of the original DCF is set to be equal with the cladding radius at taper waist with a₀= b_w. The numerical aperture of core and inner cladding are equal with NA₁= NA₂. Hence, light in the core of the input SCF firstly transmit to the core of the original DCF, then it evolves into the inner cladding at taper waist, and finally it couples into the core of the output SCF [27]. Next, the most important work is to optimize the parameters of the TDCF.

4.1 Optimization of the cladding core ratio

Since structures of SCF and TDCF are different, it is necessary to calculate the mode coupling efficiency at the end face firstly. The calculation results for DCF with different cladding core ratios are shown in Fig. 11. Figure 11 (a) gives the mode coupling efficiency between the SCF and the TDCF at the input end. Based on the mode decomposition method [28], coupling efficiency can bebdetermined from:

(15)$$C = \frac{{\int\!\!\!\int_S {{E_{in}}\cdot E_{out}^ \ast dxdy} }}{{\sqrt {\left( {\int\!\!\!\int_S {{{|{{E_{in}}} |}^2}dxdy} } \right)\left( {\int\!\!\!\int_S {{{|{{E_{out}}} |}^2}dxdy} } \right)} }}$$

where E_in is electric field of the input fiber mode, E_out is electric field of the output fiber mode.

Fig. 11. Mode coupling efficiency between SCF and TDCF.

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It can be seen that as long as the cladding core ratio satisfies s ≥ 3, mode coupling efficiency can be larger than 99.9%. Figure 11(b) gives the mode coupling efficiency between the SCF and the TDCF at the output end. For most cases, the mode coupling efficiency is increasing along with the cladding core ratio.

The difference of mode fields between the TDCF and the output SCF can also be analyzed by the effective indices. Figure 12 shows the effective refractive index curves with different cladding core ratios. It can be seen from the figures that the effective refractive index of the mode decreases along with the taper direction. When the cladding core ratio is small, the effective refractive index curve is smooth. The curve becomes more distorted with the gradual increase of cladding core ratio. At the taper waist, the effective refractive index of the TDCF is closer to that of the SCF when the cladding core ratio is larger, which means that the mode field distributions are more similar. Therefore, in order to ensure that all modes have high coupling efficiency at the output end, we choose the cladding core ratio as s = 8.

Fig. 12. Effective refractive index of TDCF with different cladding core ratios.

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4.2 Optimization of taper length

The beat lengths between the above four LP modes and their adjacent modes with different cladding core ratios are calculated respectively, with the results shown in Fig. 13. It can be noted that the peak positions of the beat length for different modes are also different. The peak position of LP₀₁ mode is close to the taper waist, while the peak positions of LP₂₁ mode and LP₀₂ mode are close to the original DCF. Obviously, the larger the cladding core ratio, the longer the beat length is. For s = 8, the longest beat length is LP₂₁-LP₂₂ mode, which exceeds 15 mm. It can be seen from the previous calculation, to achieve high transmission efficiency, the taper length needs to be longer than the beat length. Combined with the package requirements of most fiber devices, the taper length is set with L_t= 30 mm.

Fig. 13. Beat length of TDCFs with different cladding core ratios.

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4.3 Optimization of taper shape

If the linear taper shape is adopted, the mode energy proportions at different positions are calculated based on the CLMT, as shown in Fig. 14. Among them, the energy proportions of LP₀₁ and LP₁₁ modes at the output end are more than 99.9%, as shown in Fig. 14(a). The energy proportions of LP₂₁ and LP₀₂ modes at the output end are less than 75%, as shown in Fig. 14(b). Therefore, it is necessary to further optimize the taper shape to increase transmission efficiency of LP₂₁ and LP₀₂ modes.

Fig. 14. Mode characteristics of a TDCF with linear taper shape.

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For nonlinear taper shapes, the mode energy proportions at the output end with different knee point coefficients are shown in Fig. 15. It can also be seen from Fig. 15(a) that LP₀₁ and LP₁₁ modes are not sensitive to the change of knee point coefficient, and the energy proportions at the output end are larger than 99.9%. However, the knee point coefficient has a great impact on LP₂₁ and LP₀₂ modes. As shown in Fig. 15(b), when the knee point position is close to the beat length peak position, the mode energy proportions at the output end reach their minimum values, with 61.8% for LP₂₁ mode and 47.7% for LP₀₂ mode. When the knee point position is far away from the beat length peak position, the mode energy proportion at the output end gradually increases. Energy proportions of LP₂₁ and LP₀₂ modes reach their maximum values when the knee point position is at the output end, which are 93.7% and 99.1% respectively.

Fig. 15. Mode energy proportions at the output end of nonlinear TDCF.

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Therefore, when the knee point coefficient of the nonlinear taper shape is selected with α_k= 1, the largest mode transmission efficiency can be obtained. Combined with the transmission efficiency along the TDCF and coupling efficiency at the end face, the total transmission efficiencies of LP₀₁, LP₁₁, LP₂₁ and LP₀₂ modes are 99.0%, 100.0%, 93.6% and 96.5%, respectively. In other words, the transmission losses of connection with the six-mode SCF are all less than 0.3 dB.

5. Conclusion

In this paper, the simulation method based on CLMT is adopted to investigate the mode evolution characteristics of a TDCF. The calculation process is introduced comprehensively, which mainly includes solving the mode parameters of DCF with the three-layer waveguide model, and solving the CLME with the fourth-order Runge-Kutta method. The TDCF with different parameters is simulated by this method, and their influences on mode coupling characteristics are analyzed. On this basis, the optimization design process of a TDCF is introduced, and the theoretical transmission losses with a six-mode fiber are all less than 0.3 dB. This work further proves the advantages of CLMT in analyzing the mode evolution characteristics of a tapered fiber, which is of great value to the optimization design of tapered fibers.

Funding

National Natural Science Foundation of China (11904398).

Disclosures

The author declare no conflicts of interest.

Data availability

The data that support the findings of this study are available on request from the corresponding author, upon reasonable request.

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Theoretical analysis of mode evolution in a tapered double clad fiber based on the coupled local mode theory

Abstract

1. Introduction

2. Theoretical model and analysis

2.1 Basic structure of a TDCF

2.2 Mode characteristics of a DCF

2.3 Coupled local mode equations

2.4 Mode characteristics of a TDCF

3. Simulation results

3.1 Mode evolution of TDCFs with different taper shapes

3.2 Mode evolution of TDCFs with different taper lengths

3.3 Comparison between CLMT and BPM

4. Optimization design of a TDCF

4.1 Optimization of the cladding core ratio

4.2 Optimization of taper length

4.3 Optimization of taper shape

5. Conclusion

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (15)

Equations (15)

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