1. Introduction

Photonic crystal fibers (PCFs) have attracted considerable attention because of their fine tunability of various guiding characteristics, such as single-mode operation, dispersion profiles, the magnitude of nonlinearity, and so on. Nonlinear optical effects in PCFs are perhaps the most interesting and important research area because of their ultrasmall transverse dimensions compared with conventional optical fibers. When the input power is large, nonlinear effects can have great impact even in large core PCFs. It is important to investigate the modal characteristics of PCFs under strong intensity light illumination for their nonlinear applications and some precedent theoretical works have been done with regard to the modal properties of nonlinear PCF [1]–[4]. Because PCF has very complex transverse geometry, the usage of rigorous numerical methods is almost indispensable to investigate nonlinear modal properties of PCFs and one of the promising candidates is the finite-element method (FEM). FEM has been widely used for investigating modal properties of general nonlinear optical waveguides and has been applied to nonlinear PCFs recently [2]. Especially, circular air hole geometries in PCFs and complex refractive index distributions due to the Kerr-type nonlinearity can be fully taken into account in FEM, therefore it can be considered the most suitable numerical method for nonlinear PCFs. However, it usually requires large computational resources and time due to the iteration for obtaining self-consistent solutions [2], while some tens or hundreds of iterations are required to obtain converged solutions around the critical power as will be shown later in this paper. Therefore, the development of simple and fast methods to obtain the nonlinear modal properties of PCFs is highly desired.

In this paper, approximate empirical relations for nonlinear PCFs are proposed. Replacing a PCF with a conventional step-index fiber (SIF) [5], closed form expressions for the effective refractive index and effective core area [6, 7] of nonlinear PCFs can be derived. To define the equivalent cladding index, the effective index of the so-called fundamental space-filling mode, which is calculated using empirical relations [8], for the effective normalized frequency, is introduced, and thus, nonlinear guided waves propagating in PCFs can be easily characterized without any need for numerical computations. Numerical results for nonlinear PCFs with nonsaturable and saturable nonlinearities are presented. The validity of the method proposed here is ensured by comparing the calculated results with those obtained by a full-vector FEM [2].

 

Fig. 1. (a) A geometry of PCF and (b) its equivalent classical optical fiber model.

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2. Approximate empirical relations for nonlinear PCFs

2.1 Replacing PCFs with SIFs

We consider a PCF as shown in Fig. 1(a), where d is the hole diameter, Λ is the hole pitch of triangular lattice structure, and the refractive index of the background material is given as n. To utilize the approximate analytical solutions, originally derived for axially symmetric nonlinear optical fibers, a complex PCF geometry as shown in Fig. 1(a) is replaced with the classical SIF equivalent model, as shown in Fig. 1(b). Here, n_co is the equivalent refractive index of the core and is given as n. a_eff is the effective core radius and is given as Λ/3 [5]. n_cl is the effective refractive index of the cladding, and is given as the effective index of the so-called fundamental space filling mode n_FSM. Usually, the use of numerical methods is mandatory to obtain n_FSM. However, it takes much time to calculate n_FSM if a pure numerical method is used. Therefore, recently proposed empirical relations for PCF designs [8] are utilized to obtain n_FSM. According to Ref. [8], the normalized frequency V of a linear PCF is given by

with λ being the free-space wavelength, while the values of the fitting coefficients, A_i (i=1 to 4) are summarized in Ref. [8]. As a result, n_co, a_eff, n_cl=n_FSM can be obtained without time-consuming numerical calculations.

2.2 Approximate empirical relations for nonlinear PCFs with nonsaturable nonlinearity

We consider the case that the background material of PCF has instantaneous Kerr-type nonlinearity whose nonlinear refractive index n is given by

where n_L stands for the linear part of the refractive index of the material, n ₂ [m²/W] is the nonlinear coefficient, Z ₀ is the free-space impedance, and E is the electric field. If the guided mode field distribution of nonlinear SIF shown in Fig. 1(b) is approximated as a Gaussian field ϕ given by

where A is the amplitude of field, and w is the spot size where the amplitude of the field drops as 1/e, the intensity-dependent effective refractive index n_eff is given as [6]:

Here, k ₀ is the free-space wavenumber, while the spot size w is given by

and V_NL is defined as

By taking into account the fact that the optical power P and the critical optical power P_c at which V_NL→∞ are given by

and

respectively, the effective index of nonlinear PCFs is given as

In addition, the effective core area of nonlinear PCFs A_eff is given as

In this case, V_NL defined in Eq. (6) can be rewritten as

By assigning the refractive index of the core n_co, the effective core radius a_eff, the normalized frequency V, and operating wavelength λ, empirical relation for the effective index of nonlinear PCF n_eff can be obtained as a function of optical power P with moderate accuracies by using Eq. (9). This approach was originally developed for nonlinear SIFs based on a scalar approximation, however, since the accurate analysis of the cladding effective index in PCF has to be based on a full-vectorial formalism, we use the approach proposed in Ref. [8] to calculate the cladding effective index.

2.3 Approximate empirical relations for nonlinear PCF with saturable nonlinearity

We consider the case that the background material of PCF has saturable nonlinearity whose nonlinear refractive index n is given as:

where n_sat stands for the saturation coefficient. By assuming Gaussian field given in Eq. (3) for the electric field distributions of SIFs as shown in Fig. 1(b), the intensity-dependent effective refractive index n_eff is given as [7]:

where the normalized spot size R₀, V_sat, and Q are defined as

and

respectively, and n_FSM is calculated by using the empirical relations [8]. The normalized spot size R ₀ is calculated by the following equation:

Here, m is an integer and usually, a value of m=500 is enough to obtain convergence. Because R ₀ is not explicitly calculated from Eq. (18), numerical methods such as bisectional methods have to be used. However, the calculation time is almost negligible for obtaining R ₀. After calculating R ₀, n_eff can be easily obtained by using Eq. (13). In addition, the effective core area A_eff is given by

3. Guiding properties of nonlinear PCFs

3.1 PCF with nonsaturable nonlinearity

We consider a PCF as shown in Fig. 1(a), and we assume that the refractive index of the background material is given as in Eq. (2). The linear part of the refractive index of silica is taken as n_L=1.45. Solid curves in Figs. 2(a), (b), (c), and (d) show the effective refractive index of nonlinear PCFs as a function of P/P_c for λ/Λ=0.1, 0.2, 0.3, and 0.4, respectively. We can see that the value of the effective refractive index is increased for higher optical power. In these figures, the results obtained by full-vector FEM [2] are also depicted (dots) and are found to be in good agreement with those obtained by the approximate empirical relations proposed here. From these figures, we can observe that for smaller values of λ/Λ (relatively shorter wavelength), some discrepancies in the results can be seen for higher values of P/P_c (around 0.9 to 1.0). For smaller values of λ/Λ, the field is strongly confined in the core region, and the nonlinearity is enhanced. Therefore, if the optical power is increased to the value near P_c, the guided mode behaves like a Townes soliton [1, 9] which is a self-localized guided mode in the bulk nonlinear medium. The critical power for Townes soliton is given by P_Townes≈0.93P_c [10]. Dashed curves in Figs. 2(a), (b), (c), and (d) show the effective refractive index obtained by Eq. (9) when P_c is replaced with P_Townes. For λ/Λ=0.1 (Fig. 2 (a)), the results obtained by FEM agree well with dashed curves. On the other hand, for larger values of λ/Λ, the results obtained by FEM agree well with solid curves. These results indicate that for relatively stronger nonlinearity (smaller values of λ/Λ), the light becomes insensitive to the presence of the cladding air holes around the critical power of Townes soliton (P_Townes), while for relatively weak nonlinearity (larger values of λ/Λ), the light is confined by the nonlinearity effect and by the index-guiding effect. This is confirmed by the field distributions as shown in Fig. 3. Figs. 3(a) and (b) show the field distributions obtained by FEM with d/Λ=0.4, P/P_c=0.9, for λ/Λ=0.1 and 0.4, respectively. We can see that for λ/Λ=0.1, the field confinement in the core region is stronger than that for λ/Λ=0.4 and there are almost no overlaps with cladding air holes. This observation was also confirmed by our full-vector FEM in the regime of P=P_Townes, where a stable solution could not be obtained. Therefore the argument that the field in the regime around P=P_Townes is best described as a Townes soliton is physically correct.

 

Fig. 2. Effective refractive index with nonsaturable nonlinearity as a function of P/P_c for λ/Λ=(a) 0.1, (b) 0.2, (c) 0.3, and (d) 0.4.

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Fig. 3. Field distributions of nonlinear PCFs with d/Λ=0.4 and P/P_c=0.9, for λ/Λ=(a) 0.1 and (b) 0.4.

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Fig. 4. Errors of approximate empirical relations as a function of the normalized power for λ/Λ=(a) 0.1, (b) 0.4, and (c) 1.5.

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Figures 4(a), (b), and (c) show the errors of the approximate solutions Δn_eff as a function of the normalized power for λ/Λ=0.1, 0.4 and 1.5, respectively. Here, Δn_eff is defined as

where n_eff,FEM is the effective refractive index of nonlinear PCFs obtained by using FEM [2]. Solid and dashed curves in the figures are the results obtained for P_c=λ²/(2πn_con ₂) and P_c=P_Townes, respectively. We can see that the errors become smaller for smaller values of λ/Λ. For λ/Λ=0.4, |Δn_eff| lies within 0.5%. For larger values of λ/Λ, the accuracy of the solution obtained by the present method becomes worse. For λ/Λ=1.5, |Δn_eff| is 5 to 15% around the critical power and 2 to 3% around the half of critical power. From Eq. (9), it is clear that n_eff is independent to d/Λ when P=0.5P_c (P=0.465P_c for Townes soliton) and given by

It is interesting to note that the results obtained by FEM also show this feature.

 

Fig. 5. Normalized effective core area with nonsaturable nonlinearity as a function of P/P_c for λ/Λ=(a) 0.1, (b) 0.2, (c) 0.3, and (d) 0.4.

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To obtain the converged solutions by using FEM around the critical power, it takes some tens or hundreds of iterations. On the other hand, by using the present method, solutions can be obtained almost instantaneously with moderate accuracy. Therefore, it is very useful to use the present method for initial design predictions or grasping general tendencies of the characteristics of nonlinear PCF.

Solid curves in Figs. 5(a), (b), (c), and (d) show the normalized effective core area as a function of P/P_c for λ/Λ=0.1, 0.2, 0.3, and 0.4, respectively. Dashed curves in Fig. 4 represent the normalized effective core area obtained by using the Eq. (10), by replacing P_c with P_Townes. The results obtained by the present method agree well with those obtained by FEM (dots) for larger values of d/Λ and smaller values of λ/Λ. This is because for smaller values of d/Λ, field is leaked into the cladding region because of the weak confinement and the Gaussian field assumption used to derive Eq. (9) is not satisfied. For larger values of λ/Λ, the situation is the same. Because the wavelength of light is relatively long compared with Λ, the field is leaked into the cladding region, and therefore the Gaussian field assumption is not valid any more.

 

Fig. 6. Effective refractive index with saturable nonlinearity as a function of P/P_c for d/Λ (a)=0.4 and (b)=0.8, where the normalized wavelength is λ/Λ=0.1.

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Fig. 7. Effective refractive index with saturable nonlinearity as a function of P/P_c for d/Λ (a)=0.4 and (b)=0.8, where the normalized wavelength is λ/Λ=0.4

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3.2 PCF with saturable nonlinearity

We consider a PCF as shown in Fig. 1(a), and we assume that the refractive index of silica is given as in Eq. (12). Solid curves in Figs. 6(a) and (b) show the effective refractive index of the nonlinear PCF as a function of P/P_c for λ/Λ=0.1 with d/Λ=0.4 and 0.8, respectively. Solid curves in Figs. 7(a) and (b) show the same thing as in Fig. 6 except for λ/Λ=0.4. By decreasing the value of V_sat/V (smaller values of n_sat), we can see strong saturation of the effective refractive index at high optical power. The results obtained by the present method agree well with those obtained by FEM and the general tendencies are well described. For smaller values of d/Λ, some discrepancies in the results can be seen. This is due to the fact that in the linear regime (P/P_c=0), the results are not so accurate because of the inappropriateness of the Gaussian field assumption. Solid curves in Figs. 8(a) and (b) represent the normalized effective core area of the nonlinear PCF as a function of P/P_c for λ/Λ=0.1 with d/Λ=0.4 and 0.8, respectively. Solid curves in Figs. 9(a) and (b) show the same thing as in Fig. 8 except for λ/Λ=0.4. Again, the general behavior of effective core area is well described by the approximate empirical relations.

 

Fig. 8. Normalized effective core area with saturable nonlinearity as a function of P/P_c for d/Λ (a)=0.4 and (b)=0.8, where the normalized wavelength is λ/Λ=0.1.

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Fig. 9. Normalized effective core area with saturable nonlinearity as a function of P/P_c for d/Λ(a)=0.4 and (b)=0.8, where the normalized wavelength is λ/Λ=0.4.

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4. Conclusion

We have proposed approximate empirical relations for nonlinear PCFs. Replacing a PCF with a conventional step-index fiber, closed form expressions for the effective refractive index and effective core area of nonlinear PCFs have been derived. The validity of the proposed method was confirmed by comparing the results with those obtained by FEM. We have confirmed that, for the intervals of λ/Λ<0.4 and 0.3<d/Λ<0.8, a standard error for Eqs. (9) and (13) is expected to be less than 1%. Although the accuracy of solutions obtained by the present approach is inferior to those obtained by general numerical methods due to its approximation, solutions are almost instantaneously obtained. Therefore, the proposed approach is very useful for the initial design or grasping general characteristics of nonlinear PCFs. We believe that the present methodology can applied to any PCF configurations by carefully define all the critical parameters like the effective core radius of the equivalent model and the effective cladding index.