## Abstract

The effects of fiber structure on Rayleigh scattering were investigated in detail. Some step-index fibers such as GeO_{2-} and F-doped silica-based fibers and total-internal-reflection photonic crystal fiber are examined. The Rayleigh scattering loss (RSL) depends on the fiber materials and index profiles, and different types of fiber have different dependencies on those parameters because of the different optical power confinement factors in every layer. On the basis of these results, the RSL can be optimized by adjusting the fiber structure or by selecting different materials.

© 2002 Optical Society of America

## 1. Introduction

Silica-based optical fibers are used throughout the world for large-capacity and long-distance transmission systems with erbium-doped fiber amplifiers (EDFAs). Fiber loss reduction will accelerate the construction of various transmission systems with longer repeater spacing.

With the development of fiber fabrication techniques, it becomes possible to reduce optical losses, such as OH^{-} absorption loss, induced by impurities in optical fiber; an optical loss as low as 0.154dB/km @1560 nm has been obtained [1]. To reduce the additional imperfection loss, the viscosity-matching technique has been proposed, which utilized the dopants GeO_{2} and fluoride (F) to match the core and cladding viscosity [2–3]. Rayleigh scattering loss (RSL), which has also been investigated in detail, accounts for the majority of fiber loss in the 1550-nm wavelength region [4–9]. It was found that the Rayleigh scattering of glass/silica depends on the fictive temperature [9]; moreover, it has been reported that Rayleigh scattering in optical fibers can be reduced by lowering the fictive temperature [6,7,9], which can be achieved by annealing treatment or by lowering the fiber-drawing temperature [4,6,10].

In this paper, the effects of the fiber structure on Rayleigh scattering were investigated in detail, including some step-index fibers, such as GeO_{2-} and F-doped silica-based fibers and total-internal-reflection (TIR) photonic crystal fiber (PCF). The RSL depends on fiber materials and index profiles, and different fiber types have different dependencies on those parameters. From these results, we can optimize RSL by adjusting the optical power confinement in the fibers.

## 2. Fiber Loss

The spectral loss of an optical fiber can be expressed as

where α
_{R}
is the RSL, α
_{IM}
the imperfection loss, α
_{OH}
the OH^{-} absorption loss, α
_{IR}
the infrared absorption loss, α
_{UV}
the ultraviolet absorption loss, and α
_{im}
the absorption loss of other impurities. The infrared absorption loss is given by α
_{IR}
=*C* exp(-*D*/λ) [8], where coefficients *C* and *D* are dependent on materials. The OH^{-} absorption spectrum in optical fibers between 1200 and 1550 nm has been modeled. It can be fitted with high accuracy to the sum of four Lorentz components and one Gaussian component [11]. It has been reported that α
_{OH}
can be greatly reduced by many fabrication processes, such as all wave fiber fabricated by MCVD. Absorption by the UV tail of germanium (α
_{UV}
) can be ignored because it is very small in the low-loss window. α
_{IM}
and α
_{im}
can be also ignored because they can be reduced to almost zero by improving the fiber fabrication techniques.

The RSL is proportional to λ^{-4} and to the light intensity propagating in the fiber and is given by the light intensity *P(r)* and Rayleigh scattering coefficient *A(r)* (RSC) in the radial distance *r* as [8]

## 3. Rayleigh Scattering Coefficient of Fibers

From Eq. (2), we can introduce a parameter Ā as the average RSC of the fiber; this can be expressed as Eq. (3):

The RSCs of fiber preforms can be measured and then fitted as a function of the dopant concentration, *i.e.*, the relative refractive-index difference between the pure silica glass and the fiber preform induced by GeO_{2} and fluoride. The RSC of GeO_{2} and fluoride codoped silica glass is reported to be [5]

where *A*
_{0}, discussed in many articles [4,6–9], is the RSC of pure silica glass; in this paper, *A*
_{0}=0.8 dB/km.µm^{4}, and [GeO_{2}] and [F] are the dopant concentrations of GeO_{2} and fluoride, respectively.

According to Sellmeier’s law [12] and Flemming’s interpolation equation [13], at the radial distance *r*, the relationship between the relative refractive-index difference Δ(*r*) and the dopant concentration *D*(*r*) can be expressed as Eq. (5):

where *D*(*r*) is [GeO_{2}] or [F] and constants *C*
_{0} and *C*
_{1} are dependent on the dopant materials. Substituting Eq. (5) into Eq. (4), and neglecting the higher-order term, the RCS will be [8]

It is found that the RSC increases as the dopant increases and that the dependence is the same for F- and GeO_{2}-doped glass. This is because concentration increases with increasing dopant added to the silica. For GeO_{2}-doped fibers, the refractive index increases as the dopant of GeO_{2} increases. On the other hand, the refractive index of F-doped glass decreases as the F concentration increases.

Comparing Eqs. (4) and (6), there exists divergence for F-doped fiber. *A*(*r*) increases linearly with [GeO_{2}] and parabolically with [F] in Eq. (4), but linearly with both Δ in Eq (6). We do not further pursue this question or use of Eq. (6) in this paper.

Referring to the optical fiber, which has the step-index profile, the RSCs are different in every layer because of different dopants. The average RSC of the fiber can be derived from Eq. (3) and expanded as the sum of Eq. (7):

$$={A}_{1}\frac{{\int}_{0}^{{r}_{1}}P\left(r\right)rdr}{{\int}_{0}^{+\infty}P\left(r\right)rdr}+{A}_{2}\frac{{\int}_{{r}_{1}}^{{r}_{2}}P\left(r\right)rdr}{{\int}_{0}^{+\infty}P\left(r\right)rdr}+\cdots +{A}_{m}\frac{{\int}_{{r}_{m-1}}^{+\infty}P\left(r\right)rdr}{{\int}_{0}^{+\infty}P\left(r\right)rdr}\equiv \sum _{i=1}^{m}{A}_{i}{\Gamma}_{i},$$

where *i* denotes the *i*th layer from the fiber core (*i* = 1) to the out-cladding (*i=m*); *A*_{i}
is the RSC in the *i*th layer, which can be obtained from Eq. (7) and is assumed to be constant for every layer; and Γ
_{i}
is the optical power confinement factor in the *i*th layer.

## 4. Estimation of Rayleigh Scattering Loss

On the basis of the RSC of preform prepared for fibers, the loss property due to dopant, drawing temperature, and drawing tension has been investigated in detail [4,6,7,10,11]. In this section, we discuss the RSL caused by the optical property of different fibers with different dopants and different index profiles.

#### 4.1 GeO_{2}-Doped Silica-Based SMF (G.652)

ITU-T G.652 fiber is a GeO_{2}-doped core, pure silica cladding fiber; its Rayleigh scattering loss index profile is shown as Fig. 1. Its RSC can be expressed as Eq. (8), referring to Eq. (6) and Eq. (7):

For single-cladding step-index fiber, Γ_{1}, the power confinement factor in the core of the fundamental mode LP_{01} depends on the fiber structure parameters (*a* and Δ) and on the propagation constant β. According to the solution of the scalar-wave equations of step-index weakly guided fiber, as discussed in many books on optical waveguides, Γ_{1} can be expressed as Eq. (9):

where **E** and **H** are the electric field and the magnetic field, respectively; *z*̂ denotes unit vector along the longitudinal direction of the fiber; *k*
_{0}=2π/λ is the wave number in vacuum; *U*
^{2}=*a*
^{2}[${{k}_{0}}^{2}$
${{n}_{0}}^{2}$/(1–2Δ)-β^{2}]; the relative refractive-index difference is defined as Δ=(${{n}_{1}}^{2}$-${{n}_{0}}^{2}$)/2${{n}_{1}}^{2}$; and *J*
_{i}(*U*) (*i*=0,1) is the *i*th-order Bessel function. Then the RSC can be expressed as Eq. (10) after substituting Eq. (9) into Eq. (8):

where *W*
^{2}=*a*
^{2}(β^{2}-${{k}_{0}}^{2}$
${{n}_{0}}^{2}$).

From Eq. (10), the dependence of the relative-refractive index difference Δ and the fiber core radius *a* can be obtained through numerical computation. Figure 2 shows the relationship between *a* and RSL at 1550 nm for various Δ. Figure 3 shows the relationship between Δ and RSL at 1550 nm for various *a*. It is found from Figs. 2 and 3 that the loss limitation due to Rayleigh scattering is ~0.153 dB/km for conventional fibers with Δ of 0.3%, *a* of 4.5 µm. It is also shown that the RSL increases as Δ increases or *a* increases, *i.e.*, the fiber parameters greatly influence the optical loss, but the dependence of Δ is larger than that of the core radius for GeO_{2}-doped fibers. This is because the RSC of the core increases as Δ increases, and because Γ_{1} increases as Δ or *a* increases. These are discussed in detail in Ref. [8].

#### 4.2 Pure Silica Core Fiber

Figure 4 shows the refractive-index profile of the pure silica core fiber (PSCF) with depressed cladding; the core is pure silica, and the inner cladding and the outer cladding are F-doped. It is well known that the depressed cladding is introduced to a large effective area (*A*_{eff}
) to improve the bending characteristics by confining the optical power into the core [14].

According to Eq. (7), the average RSC for the PSCF can be expressed as

$$\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\approx {A}_{0}\left[1+41\mid {\Delta}^{+}-{\Delta}^{-}\mid \left(1-{\Gamma}_{1}\right)\right].$$

The third term on the right-hand side of Eq. (11) is ignored because the power confinement factor in the out-cladding is very small. Thus the PSCF fiber can be treated as a step-index fiber with parameter *a* and Δ ≡ |Δ^{+}-Δ^{-}|, and Γ_{1} can be obtained from Eq. (9) as well. As a result, Eq. (11) will have the form

where *U*
^{2}=*a*
^{2}[${{k}_{0}}^{2}$
${{n}_{0}}^{2}$-β^{2}], *W*
^{2}=*a*
^{2}[β^{2}-${{k}_{0}}^{2}$
${{n}_{0}}^{2}$(1–2Δ)], Δ=(${{n}_{0}}^{2}$-${n}_{2}^{2}$)/2${{n}_{0}}^{2}$.

Figure 5 shows the relation between *a* and RSL at 1550 nm for various Δ, and Fig. 6 shows the relation between Δ and RSL at 1550 nm for various *a*. It is found that the RSL decreases because Γ_{1} increases with increase of *a*. As shown in Figs. 5 and 6, when the fiber core is narrower, for example, *a* < 3 µm, RSL increases as Δ increases, but when *a* > 3 µm, RSL decreases as Δ increases. For clarity and simplicity, the derivation of the RSC of PSCF to the relative-index difference, *i.e.*, ∂Ā/∂Δ, at 1550 nm, will be derived in the following progression.

In Eq. (12), all parameters, except *U, W*, and β, are independent of Δ; thus we can write ∂Ā/∂*U* as follows:

where ∂Ā/∂*U* and ∂Ā/∂*W* can be obtained from Eq. (12) and expanded as

$$\frac{\partial \overline{A}}{\partial W}=-\frac{41{A}_{0}}{{k}_{0}^{2}{n}_{0}^{2}{a}^{2}}\frac{{J}_{0}^{2}\left(U\right)}{{J}_{1}^{2}\left(U\right)}W.$$

*U* and *W* are the structural parameters of optical fibers, and they depend on Δ. The derivation of *U* or *W* to Δ can be calculated from their definitions and expressed as

$$\frac{\partial W}{\partial \Delta}=\frac{1}{2W}\frac{\partial {W}^{2}}{\partial \Delta}=\frac{{a}^{2}\beta}{W}\frac{\partial \beta}{\partial \Delta}+\frac{{k}_{0}^{2}{n}_{0}^{2}{a}^{2}}{W},$$

where ∂β/∂Δ is the derivation of β to Δ; this can be obtained by solving the scalar wave equation of step-index fiber.

From Eq. (13) to Eq. (15), ∂Ā/∂Δ is calculated and plotted in Fig. 7 for various fiber core radii. The figure shows that for greater *a* (4 µm, 7 µm in Fig.7), ∂Ā/∂Δ is negative, *i.e.*, Ā decreases as Δ increases, and for thinner fiber core [2 µm, 3µm in Fig.4(c)], ∂Ā/∂Δ is positive when Δ is not too high, *i.e.*, Ā increases as Δ increases. From these findings we can determine that there is an optimized Δ for certain *a*. For example, when *a*=2 µm, the first zero point of ∂Ā/∂Δ is ~Δ=0.36%, *i.e.*, PSCF fiber with parameters (*a*=2 µm, Δ ≈ 0.36%) has the lowest RSL. Another example is *a*=3 µm, Δ ≈ 0.24%.

In general, both parameters a and Δ influence the optical loss, but the core radius dependence is larger than that of the refractive-index difference for F-doped fibers.

#### 4.3 Total-Internal-Reflection Guided Photonic Crystal Fiber

A promising new field in optical fiber technology has appeared within the past few years through the realization of so-called PCF. The recently fabricated PCF has been design following the model of a triangular photonic crystal cladding structure and centrally a single, missing air hole forming the core, as shown in Fig. 8. Remarkable properties for this type of PCF have been reported, for example, single-mode operation over an unusually wide wavelength range (at least 337–1550 nm) [15], and many other properties significantly different from standard optical fibers are expected [16,17]. However, it is important to note that no photonic bandgap (PBG) effects occur in these PCFs. This is because the fundamental mode confined to the high-index core region experiences the surrounding photonic crystal cladding as a medium with an effectively lower index, thereby allowing total internal reflection (TIR) to take place. An effective-index model [15,18] for the TIR PCF was presented a few years ago, and we will use this model to investigate the RCL property of the TIR PCFs.

According to the effective-index model, the effective refractive index *n*_{eff}
in the photonic crystal cladding can be obtained by solving the scalar-wave equations within a unit cell centered on one of the holes. We can establish an effective step-index profile for the PCFs as shown in Fig. 9. The effective core radius is *a*=0.625*D* [19], and the effective relative refractive-index difference is

Then, through the same processing as with GeO_{2}-doped or F-doped fibers, the average RSC of TIR PCFs can be obtained and expressed as

where *A*
_{2} is the effective RSC in the photonic crystal cladding. *A*
_{2} is usually much larger than *A*
_{0} [20,21]. It is a reasonable hypothesis that *A*
_{2} depends on the background material and the air filling ratio *d/D*. For the case of simplicity, it is assumed to be a constant of 4.0 dB/km.um^{4} in this paper.

Figure 10 shows the relationship between the nearest-neighbor hole spacing *D* and RSL at 1550 nm for various *d/D*, and Fig. 11 shows the relationship between *d/D* and RSL at 1550 nm for various *D*. From Fig. 10 and Fig. 11, the RSL decreases as *D* increases or *d/D* increases. This is because Γ_{1} increases as D increases or d/D increases.

## 5. Discussion

#### 5.1 Rayleigh Scattering Loss of Doped Silica-Based Fibers

The RSL of GeO_{2}-doped silica-based fiber is greater than that of F-doped silica-based fiber, because there is major power confined to the pure silica core of the PSCF where the RSC is the lowest. We have estimated a RSL of 0.153 dB/km for GeO_{2}-doped fiber and 0.142 dB/km for PSCF when the fiber structure parameters are the same as *a*=4.5 µm, Δ=0.3% (shown in Figs. 2, 3, 5–7). Many different records have been recorded for the lowest RSL for F-doped fibers, such as 0.128 dB/km [22], 0.142 dB/km [4], and 0.11 dB/km [23]. The reason for such variation is that the researchers used different *A*
_{0}, the RSC of pure silica as it varies with the fabrication process.

#### 5.2 Rayleigh Scattering Loss of Total-Internal-Reflection Photonic Crystal Fiber

In Ref. [20], two PCFs with RSLs of 0.36 dB/km and 0.18 dB/km were fabricated. They have different *D* (3.2 and 4.2 µm, respectively) with the same *d/D*=0.44.In this paper, it can be understood qualitatively that the RSL decreases as *D* increases. There are some errors between our estimation and the experiments, because the average RSC in the photonic crystal cladding *A*
_{2} cannot be set accurately.

#### 5.3 Power Confinement Factor of G.652 Fiber and Total-Internal-Reflection Photonic Crystal Fiber

As discussed in above, the power confinement factor will be predominant when we consider the RSL of a fiber, and each layer of the fiber will contribute its own characteristic to the RSL. Figure 12 illustrates the power confinement factor in the core region of G.652 fiber (SMF) and TIR PCF, with structural parameters (*a*=4.5 µm, Δ=0.3%) and (*D*=2.3 µm, *d/D*=0.6), respectively. It can be found that Γ_{1} of the TIR PCF is greater than that of the single step-index G.652 fiber (SMF). As we know, when the fiber core is pure silica, the RSL decreases as Γ_{1} increases, so the RSL of TIR PCF will possibly be less than that of the PSCF and much less than that of the G.652 fiber. Hence we can design TIR PCF for low-loss transmission systems based on consideration of the power confinement factor.

## 6. Conclusion

In this paper the effect of the fiber structure on Rayleigh scattering has been investigated in detail. Three different fibers, GeO_{2-} and F-doped silica-based fibers and total-internal-reflection photonic crystal fiber (PCF), are included here. The RSL depends on fiber materials and on index profiles, and different types of fiber have different dependencies on those parameters because of different optical power confinement factors in every layer. For GeO_{2}-doped fiber, RSL increases as the fiber core radius increases or as the relative index difference increases. For F-doped PSCF, RSL decreases as the fiber core radius increases, and it varies in a complicated manner with the dopant concentration. For TIR PCF, RSL decreases as the hole spacing increases or as the air filling ratio increases. From these results, we can design and fabricate the optimized fiber with the best loss property.

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