## Abstract

We investigated the modal properties of complex refractive-index core photonic crystal fibers (PCFs) with the supercell model. The validity of the approach is shown when we compare our results with those reported earlier on a step complex refractive-index profile. The imaginary part of the electric field results in wave-front distortion in the complex refractive-index profile PCFs, which means that the power flows out or into the doped region according to the sign of the imaginary part of the refractive index. A simple formula is proposed for calculating the gain or loss coefficients of these fibers. The numerical results obtained by the approximation formula agree well with the full-vectorial results.

©2004 Optical Society of America

## 1. Introduction

The critical characteristics in fiber lasers or amplifiers are the gain and loss along the length of the fiber. Except for models based on rate and propagation equations, an alternative method is to analyze optical fibers whose refractive-index profile is described in terms of a complex function. The gain or loss is decided by the imaginary component of the complex propagation constant, which is critically dependent on the imaginary component of the complex refractive-index profile.

The propagation characteristics of optical fibers with a circular complex refractive-index profile have been reported in the literature [1–5]. These analyses are based on scalar wave equations and thus are applicable only for weakly guiding fibers. With the advent of photonic crystal fibers (PCFs) [6] and Bragg fibers [7], the numerical method must be applicable for fibers with complicated refractive-index or high-index contrast profiles. PCFs used as active fibers were first reported by Wadsworth *et al*. [8], and they offer several advantages compared with standard fibers, such as mode control and high-dispersion tailoriability. Such PCFs have been used to design amplifiers with improved performance, such as low-threshold or high-power amplifiers and lasers [9,10].

In this paper, a full-vectorial method is used for modeling the complex refractive-index core PCF. The vectorial characteristics of modes are retained after recasting of the full vectorial wave equations into an eigenvalue system. We first introduce the algorithm briefly and then compare the numerical results with those reported earlier on a step complex refractive-index profile. The modal properties of complex refractive-index core PCFs are discussed, and a noticeable phenomenon of wave-front distortion is demonstrated. Finally, we propose a simple approximate formula for calculating the gain or loss coefficients of these fibers.

## 2. Simulation method

It is assumed that the PCF is uniform in the propagation (*z*) direction, so our main task here is to investigate the transverse modal field distribution *e⃗*_{t}
(*x y*) which can be divided into two polarization components along the *x* and *y* axes: *e⃗*_{t}
(*x*,*y*)=*x̂e*_{x}
(*x*,*y*)+*ŷe*_{y}
(*x*,*y*). For a dielectric waveguide, when its dielectric constant profile *ε*(*x*, *y*) has *x* and *y* axial symmetry, i.e., it is an even function of both *x* and *y*, it can be proved that *e*_{x}
(*x*,*y*) and *e*_{y}
(*x*,*y*) always have opposite parities in the *x* and *y* directions for each eigenmode [11].

For the case of compactness, two subscripts *m* and *n* are introduced to express the opposite parities of the mode electric field, which have the logical value 0 or 1, and are used to describe the symmetry of the *x* component *e*_{x}
(*x,y*) as *e*_{x}
(-*x,y*)=(-1)
^{m}*e*_{x}
(*x,y*) and *e*_{x}
(*x*,-*y*)=(-1)
^{n}*e*_{x}
(*x,y*). All the compositions of *mn* are [00, 01, 10, 11], which can completely express the symmetry of the mode electric field about both axes. To improve computational efficiency, the transverse electric field can be expanded with the localized orthonormal Hermite-Gaussian basis functions as follows:

where the bar over the subscript indicates the logical operator NOT and the subscript *mn* indicates that there are four sets of (*e*_{x}
, *e*_{y}
) with different parity. *F* is the number of expansion terms, *εab*^{s}
(*s*=*x*,*y*) are the expansion coefficients, and *ψ*_{i}
(*s*) is the ith-order orthonormal Hermite-Gaussian function [12,13,14].

For a dielectric constant structure with axially symmetric *x* and *y*, i.e., *ε*(-*x*,*y*)=*ε*(*x*,-*y*)=*ε*(*x*,*y*), the expression can be given as a sum of the cosine functions as

$$\mathrm{ln}\epsilon \left(\mathbf{r}\right)=\mathrm{ln}\epsilon (x,y)=\sum _{a,b=0}^{P-1}{P}_{\mathit{ab}}^{\mathrm{ln}}\mathrm{cos}\frac{2\pi \mathit{ax}}{{D}_{x}}\mathrm{cos}\frac{2\pi \mathit{by}}{{D}_{y}},$$

where *P* is the number of expansion items and *P*_{ab}
, ${{P}_{\mathit{\text{ab}}}}^{\text{ln}}$ are the expansion coefficients that can be analytically evaluated from the Fourier transform. *D*_{x}
(*D*_{y}
) is the characteristic period in the *x*(*y*) direction. When fibers with a complex refractive-index profile are examined, the expansion coefficients *P*_{ab}
, ${{P}_{\mathit{\text{ab}}}}^{\mathit{\text{ln}}}$ are complex. This approach is convenient especially when one is investigating PCFs or Bragg fibers [14,15].

Substituting the decomposition equations of the dielectric constant (3) and the modal field equations (1) and (2) into the full-vectorial coupling wave equations [16], we obtain four sets of eigenequations as shown in Eq. (4).

where *β* is the propagation constant corresponding to the mode field distribution (*e*_{x}
, *e*_{y}
). If the complex dielectric structure *ε*(*x*, *y*) is considered, the corresponding propagation constant is thus complex and can be written as *β*=*β*_{r}
+*iβ*_{i}
, *β*_{r}
, *β*_{i}
are the phase item and the gain (loss) item, respectively. The gain or loss of the modal field is hence 8.686*β*_{i}
in decibels per meter.

*L*_{mn}
is a four-dimensional matrix, and *M*
_{1}, *M*
_{2}, *M*
_{3}, and *M*
_{4} are the overlapping integrals, which are a four-dimensional *F*×*F*×*F*×*F* matrix. These overlapping integrals can be calculated analytically. All the overlapping integrals are not shown here because of their complicated form, which we have discussed previously [15]. Through the subscript transform, *L*_{mn}
and *ε*^{s}
can be transferred into a [2×*F*
^{2}]×[2×*F*
^{2}] two-dimensional complex matrix and a vector with 2×*F*
^{2} elements, with which the eigensystem Eq. (4) can be solved by a complex solver. When the real parts of the eigenvalues at the wavelength *λ* are labeled in decreasing order, the modal electric fields from the fundamental to higher order can be obtained. If the vector items *M*
_{3x}, *M*
_{3y}, *M*
_{4x}, and *M*
_{4y} are neglected, the eigenvalue problems transform into a scalar approximation.

The complex refractive index of the fiber can generally be written as *n*(*x*, *y*)=*n*_{r}
(*x*, *y*)+*i*
*n*_{i}
(*x*, *y*). The positive (negative) sign of the imaginary part indicates a gain (lossy) media. If we take the complex index as *n*
^{*}, where the asterisk represents a complex conjugate operation, it can be proved that the matrix *L*_{mn}
will be ${L}_{\mathit{\text{mn}}}^{*}$
. According to Eq. (4), We have

The imaginary part of the eigenvector determines the imaginary part of the electric field [Eq. (2)]; therefore, we can conclude that the complex conjugate operation of *n* results in the complex conjugate of the propagation constant and the modal field.

## 3. Propagation properties of complex index PCFs

#### 3.1 Validity of the method

To illustrate the use of the method, we first choose a simple refractive-index profile (step index fiber) for comparing with Ref. [5]. The fiber parameters used in the calculation are *a*=2.2 µm (core radius), *n*_{core}
=1.475+*in*_{i}
, *n*_{clad}
=1.458, at *λ*=1.55 µm; *n*_{i}
is the imaginary part of the refractive index in fiber core. The numbers of the expansion items in Eqs. (1)–(3) are taken as *F*=18, *P*=600. The values of real and imaginary parts of mode index *n*_{e}
=*n*_{er}
+*in*_{ei}
obtained by our method are listed in Table 1. As seen from the table, the scalar method gives quite an accurate modal index. A larger *F* gives more-accurate results, but there is a trade-off between the accuracy and computation time. The mode indices obtained by the vectorial method are also shown in table. We note that the vectorial results are a little smaller than the scalar results; this is because the vector items *M*
_{3x}, *M*
_{3y} and the coupling items *M*
_{4x}, *M*
_{4y} are taken into account in Eq. (4).

#### 3.2 Propagation properties of complex index core PCF

After validating the full-vectorial method, we apply this method to analyze PCFs with a complex index core. One of the most important PCF configurations consists of a silica fiber with a solid core surrounded by a silica cladding pierced by rings of air holes, which are typically hexagonally packed. The parameter hole spacing *Λ* and relative hole size *d/Λ* are used to define the structure of the PCF (shown in Fig. 1). We assume that the fiber core is doped with core radius *R* and that its refractive index has an imaginary part. We take the fiber parameters as *Λ*=4.4 µm, *R*=2.2 µm and the relative hole size to be *d/Λ*=0.3. The refractive index of the fiber core is taken as *n*_{core}
=1.475+i*n*_{i}
, where *n*_{i}
=10^{-3}, the index of the cladding matrix is *n*_{matrix}
=1.458, and the index of air is *n*_{air}
=1. The complex refractive-index profile can be considered as a function of radial distance, pump and signal wavelength, and dopant profiles for active medium [4]. For simplicity, we assume that the complex refractive index is a constant in fiber core throughout this paper.

We find that only fundamental mode (HE_{11}) and second-order modes (HE_{21}, TM_{01} and TE_{01}) exist at wavelength *λ*=1550 nm. The expression *E*(*x*,*y*)=*E*_{r}
(*x*,*y*)+*iE*_{i}
(*x*,*y*) is used to represent the modal field, where *E*_{r}
(*x*,*y*) and *E*_{i}
(*x*,*y*) are the real and imaginary parts of the modal field, respectively. Figures 2 and 3 show |*E*_{r}
|^{2} and |*E*_{i}
|^{2} for the HE_{11} and TM_{01} modes. It is shown that both the real and the imaginary parts of modal field reflect the symmetry of the dielectric structure; the imaginary part of the modal field is much smaller than the real part because of the small *n*_{i}
. An apparent characteristic of the imaginary part of the modal field is that the modal field reaches the minimum near the edge of the fiber core.

The imaginary part of the modal electric field is a result of the imaginary part of the refractive index introduced into the fiber core. Figure 4 shows the phase distribution in the fiber’s cross section for mode HE_{11x}. This phase can be considered to be the wave front of the fundamental mode. It is well known that the wave front is a plane for ideal fibers (no loss or gain), but when the imaginary part of the refractive index is introduced into the fiber core, the wave front is distorted. The phase distortion revealed in Fig. 4 shows that there is a power outflow from the doped region to cladding, which is induced by the axial gain. Sharma [3] has proved that the power orthogonality is not valid for optical waveguides or fibers with a complex refractive-index profile. We can also draw this conclusion by affirming that the matrix *L*_{mn}
is not Hermitian in Eq. (4). This is an important characteristic for fibers with an imaginary index profile. It implies that the modal fields are linearly correlated and are not independent from each other. It can be assumed that the modal field is a combination of some normal modes of a corresponding fiber without gain or loss and hence induce the wave-front distortion.

We find that the phase distortion is principally dominated by *n*_{i}
—the imaginary part of the refractive index in the fiber core. Figure 5 shows the phase distribution along the *x*=0 axis with different *n*_{i}
for *d/Λ*=0.3, *λ*=1550nm. In Fig. 5, the phase is multiplied by 10 for *n*_{i}
=10^{-4} and 100 for *n*_{i}
=10^{-5}. It is obvious that these curves are overlapped; therefore it can be concluded that the phase distribution *P*(*x*,*y*) satisfies *P*(*x*,*y*)∝*n*_{i}
. For different *n*_{i}
, the line types of these phase distributions are the same except for the amplitude.

The relative air hole size *d/Λ* is an important parameter for determining the modal characteristics of PCF. Figure 6 shows the dependence of mode index *n*_{e}
=*β*/*k*
_{0}=*n*_{er}
+*in*_{ei}
of the fundamental mode on *d/Λ* at wavelengths of 980, 1310, and 1550 nm. *k*
_{0}=2*π/λ* is the wave number of the vacuum. At a certain wavelength, the real part of mode index *n*_{er}
decreases with increasing air hole; whereas the imaginary part of mode index *n*_{ei}
increases as *d/Λ* increases. On the other hand, the mode index (including *n*_{er}
and *n*_{ei}
) increases as the wavelength decreases because more field energy is confined in the high-index region (core). The introduction of air hole into the fiber cladding will enhance the gain of the fiber efficiently. It is shown that the use of erbium- or ytterbinm-doped PCF as fiber lasers or amplifiers allows for improved features of amplification properties with respect to standard step-index fibers [9].

For practical implementation, single-mode fibers are generally required. It is shown that the PCF will be endlessly single mode when *d/Λ* is less than 0.406 [17] for standard PCF (the fiber core is undoped). The active PCF discussed in Figs. 2 and 3 (*d/Λ*=0.3) is multimode (including the fundamental mode and second-order modes) at wavelength *λ*=1550 nm. Because the fiber core is up-doped, it is reasonable that the critical value of *d/Λ* for endlessly single-mode operation will decrease and depend on the refractive index of the core. To extend the available parameter space, the refractive index of the fiber core should be depressed, for example, by codoping with fluorine, to compensate the refractive-index increase.

## 4. Approximate formula for calculation gain or loss coefficient

The power loss (gain) coefficient for a waveguide composed of arbitrary absorbing (amplifying) media can be expressed in terms of modal fields [16]

where *β*_{i}
is the imaginary part of propagation constant and *n*_{r}
(*x*,*y*), *n*_{i}
(*x*,*y*) denote the real and imaginary parts of the refractive-index profile. When the waveguide is only slightly absorbing (amplifying), as is normally the case in practice, we can replace the modal fields with their corresponding modal fields of nonabsorbing waveguide (the imaginary part of the refractive-index profile is neglected).

According to the integral expression for the propagation constant of nonabsorbing waveguides [16]

the subscript *r* denote the real values. In the weakly guiding approximation, we have

where *n*_{er}
=*β*_{r}*/k*. Substituting Eq. (8) into Eq. (6), we have

Referring to the optical fiber with a circular step-index profile, we obtain the imaginary part of the mode index

where *k* denotes the *k*th layer from the fiber core to the out-cladding, *n*_{kr}
(*n*_{ki}
) is the real (imaginary) part of refractive index in the *k*th layer, and *Γ*_{k}
is the optical power confinement factor in the *k*th layer. For the complex index core PCF discussed in this paper, we have

where *n*_{corer}
(*n*_{corei}
) is the real (imaginary) part of the refractive index in the fiber core and *Γ* is the fraction of modal power in the core. The imaginary part of the propagation constant (i.e., gain or loss coefficient) is *β*_{i}
=*k*
_{0}
*n*_{ei}
. According to Eq. (11), *n*_{ei}
is determined by the confinement factor and the mode index *n*_{er}
when the refractive-index profile of the fiber is fixed. The configuration of PCF can provide more flexibility to control modal properties and then the gain (loss) coefficient.

The approximation expression of the imaginary part of the mode index in Eq. (10) is especially useful when a weakly absorbing or amplifying waveguide is discussed. Only the mode index and power confinement factor of the corresponding waveguide, in which the imaginary part of the refractive index is neglected, are needed for calculating the imaginary part of the propagation constant (the gain or loss). Hence the approximate formulas provide a simple approach for evaluating the gain or loss coefficients, and the complicated process for solving the complex propagation constants of bound modes of the eigenvalue equation can be avoided.

We first compare the mode index *n*_{e}
and the power confinement factor *Γ* between the complex refractive-index core PCF and the corresponding fibers, in which the imaginary part of the imaginary refractive index is neglected. For clarity, fiber A represents PCF with *n*_{core}
=1.475+i10^{-3} and the refractive index of the cladding matrix is *n*_{matrix}
=1.458. Fiber B represents PCF with *n*_{core}
=1.475 and *n*_{matrix}
=1.458. Figure 7 shows the mode indices (for fiber A, only the real part of mode index are demonstrated) and the confinement factors of fiber A and B as a function of *d/Λ*. The differences of mode index and confinement factor are also given in the figure. We note that as the relative air hole size *d/Λ* increases, the differences of two types of fiber decrease. The restraining assumption of the stand method of perturbation is that the gain or loss exhibited by the waveguide does not alter the field significantly [1]. The assumption is supported by Fig. 7, in which the difference of fiber A and B is of an extremely small magnitude: 10^{-5} for mode index, and 10^{-4} for power confinement factor. Hence for practical fibers, the perturbation method is also applicable.

To verify Eq. (11), we illustrate both the full-vectorial results and the approximate results obtained with Eq. (11) in Fig. 8. In Fig. 8(a) the imaginary parts of the effective mode indexes are shown as a function of relative air hole size *d/Λ* at wavelengths of 980, 1310, and 1550 nm; in Fig. 8(b), the imaginary parts of the effective mode indices are shown as a function of wavelength for different relative air hole size *d/Λ* (0.1, 0.3 0.5 0.7). As shown in Fig. 8, Eq. (11) agrees well with the full-vectorial method. We find that the difference of the two groups of curves is less than 10^{-6}. It is observed that the difference between the approximate formula and the full-vectorial method is small at short wavelengths but increases for long wavelengths. Further, the difference increases as the air hole size *d/Λ* increases. This is understood to occur because the weakly guiding approximation used in Eq. (11) brings more error for the large air hole and long wavelength.

## 5. Conclusion

We investigated the propagation properties of complex refractive-index core PCF with the supercell method. The validity of the approach is shown by comparison of the numerical results with those reported earlier on a step complex refractive-index profile. The imaginary part of the electric field results in wave-front distortion in the complex refractive-index profile PCF, which means that there is a power flow out or into the doped region according to the sign of the imaginary part of the refractive index. A simple formula is proposed for calculating gain or loss coefficient; with this formula the complicated process for solving the complex eigenvalue equation can be avoided. It is shown that the gain or loss coefficients are critically dependent on the power confinement factor and the mode index when the refractive-index profile is fixed. The fact that the numerical results obtained by the approximation formula are in good agreement with the full-vectorial results shows the validity of the formula.

## Acknowledgments

This research has been supported by the Foundation of Beijing Jiaotong University (grant NJTU PD238, PD241), China.

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