## Abstract

We investigate the evolution of the propagation mode and the group velocity dispersion in the taper region and analyze its contribution to the nonlinearity of tapered fibers, which is important for a comprehensive understanding of the light propagation characteristics and the mechanisms supporting the supercontinuum generation in tapered fibers.

©2004 Optical Society of America

## 1. Introduction

Supercontinuum generation in tapered fibers [1,2] typically takes place around the zero point of the group velocity dispersion (GVD). The most important mechanism during this process is the so-called “soliton splitting” [3–5]. The significant parameters supporting the supercontinuum generation in tapered fibers include a small diameter (1~4 µm), high intensity (on the order of TW/cm^{2}), and a correctly tailored GVD.

So far, most publications have concentrated on the *waist region* of the tapered fibers [1,2,6], where the strongest nonlinear effect is induced, and assume that the waist can be approximated by a constant diameter silica strand surrounded by air, having similar properties to a high air fill fraction PCF. However, a complete model of the propagation characteristics must take into account the transition or the taper region where the diameter is varying along the fiber.

Furthermore, all of the nonlinear processes engaged in the supercontinuum (SC) generation take place over the whole fiber, including the input and the output tapers. Comprehensive understanding of the nonlinear properties in the taper is of great importance for a complete picture of the generation of SC and its interaction with the pump. To determine the starting point of nonlinear processes such as self-phase modulation, four-wave mixing, and the soliton splitting, a full and precise analysis of the nonlinear parameter *γ* and the GVD in the input taper is required. Characterization of the pulse and dispersion properties of the SC after exiting the fiber should also include the contribution from the output taper. Especially when the tapered fiber is inserted into a feed-back scheme, acting as nonlinear element in an optical parametric amplifier or oscillator [7], the SC will have to pass through both the input and the output tapers several times and consequently experiences more nonlinearity and dispersion from the tapers. The nonlinear characteristics of both tapered regions will influence significantly the amplification or oscillation processes. Additionally, in experimental GVD measurements, due to the unknown GVD value in the tapers it is difficult to compare the directly measured GVD value along the whole tapered fiber and the theoretical GVD value of the waist. Therefore, the investigation of the mode and the GVD evolution in the taper region is significant for a comprehensive understanding of the light propagation and interaction in the fiber.

In this paper, we present theoretical studies of the evolution of the mode profile and the GVD in the taper region. To characterize the most important parameters as mentioned above, we first need to evaluate the propagation constant β.

There are generally two theoretical models to describe the propagation of light in tapered fibers: scalar wave equation and full vector Maxwell equation. In the scalar wave equation, the polarization of the field is assumed to be unchanged and the longitudinal component is ignored [8–10]. However, with the fiber tapered down, the polarization of the light varies with propagation, and the longitudinal component of the mode fields increases even to the same order of magnitude as the transverse components [11]. Therefore, the vector model can provide a more precise description of the light propagation.

Our calculation shows that the nonlinear parameter based on the field distribution is not sensitive to the difference between the scalar equation and the vector equation models when the diameter of the tapered fiber is larger than 1.0 µm. In this paper, we restrict our discussion to fiber tapers with a diameter larger than 1.8 µm, and we used the scalar equation to calculate β in order to simplify the calculation of the mode evolution. In contrast, the GVD, which is actually the second-order derivative of β, is extremely sensitive to any approximations in the model when the diameter of the taper becomes rather small. In this case, we used the full vector Maxwell equation to simulate the evolution characteristics of GVD in the tapered region.

Furthermore, when simulating the mode evolution, we also compared the results from the standard solution of the scalar wave equation with those obtained from the so-called “variational calculation” [8], which is in fact an approximation to the scalar equation.

## 2. Evolution of the radial distribution of the light intensity and the nonlinear parameter in the taper

#### 2.1 Standard solution of the scalar equation

For an adiabatic process [9] in a tapered fiber, the perturbation caused by the variation of the fiber radius is so small that the loss of power from the fundamental mode to the higher-order mode in the fiber is negligible [9, 12]. In this paper, we assume the fiber is an adiabatic single-mode tapered fiber.

In such a fiber, the light propagates always as a fundamental mode. For a given position *z* along the taper, we can build the fundamental-mode wave equation in a cylindrical system by making use of the geometry of the local profile (see Fig. 1).

To obtain the radial distribution of intensity, which is a function of the propagation constant *β*, we need to solve the scalar wave equation (1).

where *Ψ* is the transverse electric field, and *m* is equal to zero for the fundamental mode *LP _{01}*.

The solution of this equation can be expressed by a linear composition of Bessel functions and modified Bessel functions. Using the boundary conditions that the wave function and its first derivative are continuous at the core-cladding and cladding-air interfaces, we can obtain the propagation constant of the fundamental mode at each position z. In Fig. 2, the color-scaled contour shows the evolution of the normalized radial distribution of intensity along the fiber at a wavelength of 800 nm.

The red lines in the graph describe the core-cladding and cladding-air interfaces. The variation of the radius of the cladding can be described by:

which is a fit to measured data (with *z* in *mm*) [13]. We have assumed that the ratio between the radius of the cladding and the core remains constant (around 15.2) during the pulling process. The Sellmeier equation for the cladding is ${n}_{\mathit{cladding}}\left(\lambda \right)=\sqrt{3.0+0.009\u2044\left({\lambda}^{2}-0.01\right)+84.1\u2044\left({\lambda}^{2}-96.0\right)}$ with λ in µm. The refractive index difference between the cladding and the core is 0.36%. In the simulation of the mode evolution, we assume that power is conserved: ${\int}_{0}^{2\pi}{\int}_{0}^{\infty}\frac{1}{2}\mathrm{Re}\left(\overrightarrow{E}\times {\overrightarrow{H}}^{*}\right)r\mathit{dr}d\varphi =\mathit{const}.$, and the ratio between the transverse electric field and transverse magnetic field is approximated as *µ*
_{0}/*ε*.

At the beginning of the taper region, the light propagates as a core mode and most of the energy is confined within the core. As the fiber is tapered down, the difference between the refractive indices of the core and the cladding is not large enough to confine the mode in the core. Therefore, the light begins to spread out into the cladding and propagates as a cladding mode that is guided by the boundary between the cladding and the air. As a result, the energy redistributes into the cladding and the intensity of the light decreases due to the relatively large diameter of the cladding. As the fiber is tapered down further, the intensity of the mode, which is now confined by the cladding-air interface, increases again due to the rather small radius of the cladding and reaches its highest magnitude at the end of the taper. The position where the propagation mode transfers from the core mode to the cladding refers to as the transition point, or the so-called ‘core-mode cut off’ [14]. Furthermore, the evolution of the mode and the process of the light propagation in the output taper region are mirror symmetric to Fig. 2 about the waist.

The evolution described above depends significantly on the pump wavelength. Figure 3 (a) and (b) demonstrate the evolution processes for wavelengths at 500 nm and 1064 nm, respectively.

At shorter wavelength, the mode can be confined to a smaller core and propagates as a core mode over a longer distance compared to the mode at longer wavelength. For a given tapered fiber, the value of core-cladding transition parameter *V _{cc}*, which is given by [14],

${V}_{\mathit{cc}}\approx \left(\frac{2}{\mathrm{ln}\phantom{\rule{.2em}{0ex}}s}\right){\left(1+\frac{0.26}{\mathrm{ln}\phantom{\rule{.2em}{0ex}}s}\right)}^{-\frac{1}{2}},$,

where *s* is the ratio between the radius of the cladding and the core, does not depend on the wavelength. The local V-value of the core at position z is a function of wavelength and can be expressed as ${V}_{\mathit{core}}\left(z\right)=\frac{2\pi \xb7{r}_{\mathit{core}}\left(z\right)}{\lambda}\xb7\sqrt{\left({n}_{\mathit{core}}^{2}-{n}_{\mathit{cladding}}^{2}\right)}$. When *V _{core}*(

*z*) is larger than the critical parameter

*V*, the light propagates as a core mode. At the transition point, where

_{cc}*V*(

_{core}*z*) is equal to

*V*, the local V-value of the core

_{cc}*V*(

_{core}*z*) becomes too small to confine the core mode any more, therefore, when

*V*(

_{core}*z*) is smaller than

*V*, the cladding becomes the new guiding medium and the light propagates as a cladding mode. Therefore, for a smaller wavelength, to match the core-cladding transition point, the radius of the core should be smaller than that for a longer wavelength.

_{cc}Knowing the transverse distribution of the mode along the taper, we can evaluate the nonlinear parameter *γ* that is important for the understanding of the nonlinear properties in the tapered region. *γ* is defined as *n*
_{2}
*ω*
_{0}/*c A _{eff}* [15], inversely proportional to the effective area of the mode

*A*that is related to the modal distribution function

_{eff}*E*(

*r,ϕ*) as:

${A}_{\mathit{eff}}=\frac{{\left({\int}_{0}^{2\pi}{\int}_{0}^{\infty}{\mid E(r,\varphi )\mid}^{2}r\phantom{\rule{.2em}{0ex}}\mathit{dr}\phantom{\rule{.2em}{0ex}}d\varphi \right)}^{2}}{{\int}_{0}^{2\mathit{\pi}}{\int}_{0}^{\infty}{\mid E(r,\varphi )\mid}^{4}r\phantom{\rule{.2em}{0ex}}\mathit{dr}\phantom{\rule{.2em}{0ex}}d\varphi},$,

where *n _{2}* is the nonlinear refractive index of the fiber material,

*ω*and

_{0}*c*are the given frequency and the speed of light in vacuum, respectively. Figures 4(a) and (b) illustrate the evolution of the effective area

*A*and the nonlinear parameter

_{eff}*γ*along the SMF 28 fused silica tapered fiber at different pump frequencies.

Figure 4 (a) demonstrates that the effective area increases with the wavelength around the transition point and this transition point appears earlier at longer wavelengths. Thus, the nonlinear parameter *γ* becomes larger at shorter wavelengths and the peak value of *γ* shifts to the waist direction with shortening the wavelength, as demonstrated in Fig. 4(b). Figure 4 (b) shows nicely that the value of the nonlinear parameter early in the taper becomes comparable with that in the waist, for example at a wavelength of 500 nm, which implies that nonlinear processes such as self phase modulation and four-wave mixing can already take place before the transition point! This effect leads to a wavelength dispersion of the nonlinear parameter *γ* and the transition points. It will consequently influence the propagating supercontinuum in the output taper region, so that different wavelengths will experience different nonlinear interactions and different time-dependent propagation characteristics.

#### 2.2 Variational calculation

We also performed “variational calculation” [8], where we assumed a Gaussian distribution for the fundamental mode $\psi \left(r\right)=\frac{\sqrt{2}}{w}{e}^{-{(r\u2044w)}^{2}}$, which proved to be a good approximation to the true ground state radial distribution if we were not too close to the transition point. The effective width of the Gaussian mode is obtained by maximizing $\u3008{\beta}^{2}\u3009\equiv \u3008{{n}_{\mathit{eff}}}^{2}{{k}_{0}}^{2}\u3009={\int}_{0}^{\infty}\mathit{dr}\phantom{\rule{.2em}{0ex}}r\psi \left(r\right)\left(\frac{1}{r}\frac{\partial}{\partial r}r\frac{\partial}{\partial r}+{n}^{2}\left(r\right){k}_{0}^{2}\right)\psi \left(r\right)$.

Using $\mathrm{Using}\frac{d\u3008{\beta}^{2}\u3009}{\mathit{dw}}=0$ and solving the subsequently derived equation with respect to *w*,

we calculated the value of w directly.

Figures 5(a) and (b) show the evolution of the effective mode area and the nonlinear parameter *γ* along the tapered fiber at 800 nm obtained by the variational calculation and the standard Bessel-function solution, respectively.

Qualitatively, these two methods give approximately similar results considering their overall distribution and the shapes of the curves in Fig. 5(a), especially at the edges of the taper. The largest discrepancy between them is around the transition point due to the Gaussian-distribution ansatz along the whole taper region.

Nevertheless, the variational theory is an effective method for approximately solving the scalar wave equation and a useful tool for analysing the transverse mode evolution in tapered fibers.

## 3. Evolution of the GVD in the taper

In order to fully characterize the propagation in a tapered fiber, we have to include the GVD in the taper region, which is defined as:

where β is the propagation constant at a given frequency *ω*, λ is the corresponding wavelength, and c is the speed of light in vacuum. β can be derived by solving the full vector Maxwell equation [8].

Following the standard approach of Snyder and Love [8], we obtain the solutions of the longitudinal components (*E _{z}* and

*H*) in the taper region.

_{z}In the core, where *r*<*r*
_{1} (*r*
_{1} is the radius of the core), we get:

where *A _{1}* and

*A*are constants,

_{2}*k*=2

_{0}*π*/

*λ*is the wave number,

_{0}*n*is the refractive index of the core,

_{core}*n*=

_{eff}*β/k*is the effective index, m=1 for the fundamental mode

_{0}*HE*, and

_{11}*J*is the mth order Bessel function of the first kind.

_{m}*E*and

_{z1}*H*are the longitudinal components of the electric and the magnetic field in the core, respectively.

_{z1}In the cladding, where *r*
_{1}≤*r*≤*r*
_{2} (*r*
_{2} is the radius of the cladding), the light can propagate either as a core mode or a cladding mode. This is determined by the local fiber profile. For a core mode, the effective index *n _{eff}* has a value between the refractive indices of the core and the cladding. For a cladding mode,

*n*is smaller than the refractive index of the cladding. The position where

_{eff}*n*is exactly equal to

_{eff}*n*refers to as the transition point. Therefore, when the light propagates along the taper,

_{cladding}*n*undergoes a transition from a value larger than

_{eff}*n*to a value smaller than it.

_{cladding}If *n _{eff}*>

*n*, the solutions can be written as:

_{cladding}If *n _{eff}*<

*n*, the solutions become

_{cladding}where *B _{1}*,

*B*,

_{2}*C*,

_{1}*C*are constants,

_{2}*n*is the refractive index of the cladding,

_{cladding}*I*and

_{m}*K*are the modified Bessel functions of the first kind and the second kind, respectively, and

_{m}*Y*is the Bessel function of the second kind.

_{m}*E*and

_{z2}*H*are the longitudinal components of the electric and the magnetic field in the cladding, respectively.

_{z2}In the air, where *r*>*r2*, *n _{eff}*>

*n*, we get:

_{air}where *D _{1}, D_{2}* are constants, and

*n*is the refractive index of the air.

_{air}*E*and

_{z3}*H*are the longitudinal intensities of the electric and the magnetic field in the air, respectively.

_{z3}By substituting the expressions of *E _{z}* and

*H*into the equations of the relationships between field components [8], we can obtain all the transverse components of the field. Using the boundary conditions,

_{z}*E*
_{z1}=*E*
_{z2}, *H*
_{z1}=*H*
_{z2}, *E*
_{φ1}=*E*
_{φ2}, *H*
_{φ1}=*H*
_{φ2} at *r*=*r*
_{1},

*E*
_{z2}=*E*
_{z3}, *H*
_{z2}=*H*
_{z3}, *E*
_{φ2}=*E*
_{φ3}, *H*
_{φ2}=*H*
_{φ3} at *r*=*r*
_{2},

we can build up an 8×8 homogeneous matrix equation where those constants are the variables. In order to get a set of non-zero solutions for the variables, the determinant of this matrix should be zero. Finally, we can obtain the result of the propagation constant by finding the roots of the determinant numerically. The GVD of a tapered fiber can then be calculated according to the formula given in Eq. (4).

Figure 6 shows the evolution of the GVD as a function of position along the taper for different wavelengths. The GVD values range between -150 and +200 ps/nm/km. We assume an SMF 28 fiber (Corning) to be tapered down to a waist diameter of 1.8 µm. The variation of the radius of the cladding follows Eq. (2) and the other parameters of the fiber profile are the same as discussed in Section 2.1.

The theoretical calculation also shows that when the cladding diameter of the fiber is larger than 6.3 µm, the GVD value of the taper obtained from the full vector Maxwell equation is almost equal to that from the scalar equation, and the difference between them is less than 1%, as marked by the dashed line in Fig. 7(a). Therefore, for a fiber with a cladding diameter larger than 6.3 µm, we can simply use the scalar equation to evaluate the GVD with sufficient precision. However, when the outer diameter becomes smaller than 6.3 µm, the difference increases significantly and cannot be neglected anymore. In Fig. 7(a), we compare the evolution of the GVD values that are calculated by the vector Maxwell equation and the scalar equation, respectively, where the pump wavelength is 800 nm and the cladding diameter varies from 9.3 µm to 1.8 µm.

When the core of the fiber becomes so thin that its influence on the intensity distribution can be neglected, the fiber can be considered to consist of only the cladding, and we define the corresponding model as cladding-air model. The full vector Maxwell equation, which is named as the core-cladding-air vector equation here, can be replaced by the cladding-air vector equation. Calculations show that when the diameter of the fiber is smaller than 6.9 µm, the difference between the GVD values calculated from the two vector equations does not exceed 1%, as marked by the dashed line in Fig. 7(b). The two curves in Fig. 7(b) show the evolution of the GVD along the tapered fiber, which are calculated by the cladding-air vector Maxwell equation and the core-cladding-air equation, respectively. The pump wavelength is 800 nm, and the diameter of the taper changes from 25.5 µm to 4.7 µm.

We conclude for the fiber SMF 28 that when the diameter of the tapered fiber is larger than 6.3 µm, the scalar equation can be used as a reasonable approximation to replace the full vector Maxwell equation, and when the diameter decreases below 6.9 µm, the cladding-air vector equation can simplify the full vector equations with sufficient precision.

## 4. Conclusion

Precise calculations have shown the evolution of the transverse intensity distribution, the nonlinear parameter, and the GVD of the fiber mode in the taper region of a tapered fiber. We have demonstrated a complete characterization of light propagation and nonlinear interaction processes in tapered fibers, which should be taken into account in both experimental analysis and device design. We have especially pointed out that the dispersion of the GVD and the nonlinear interaction within the taper region has significant influence on the propagation of spectrally broad supercontinuum pulses.

## Acknowledgments

We sincerely thank Prof. John Dudley for many valuable discussions. The work has been supported by DFG (FOR557, SPP1113) and BMBF (13N8340).

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