## Abstract

We present a theoretical method for analyzing radiation loss from surface roughness scattering in hollow-core photonic bandgap fibers (HC-PBGFs). We treat the scattering process as induced dipole radiation and combine statistical information about surface roughness, mode field distribution and fibre geometry to accurately describe the far-field scattering distribution and loss in fibers with an arbitrary cross-sectional distribution of air holes of any shape. The predicted angular scattering distribution, total scattering loss and the loss wavelength dependence are all shown to agree well with reported experimental data. Our method yields a simpler result than that obtained by more complex approaches and is to the best of our knowledge the first successful attempt to accurately describe roughness scattering in HC-PBGFs.

© 2012 Optical Society of America

## 1. Introduction

In their most common form, hollow-core photonic bandgap fibers (HC-PBGFs) consist of a two-dimensional triangular lattice of air holes embedded in a host material with the core formed by removing a number of adjacent capillaries, usually 3, 7 or 19 [1–4]. Light guidance in the air core is therefore achieved by virtue of the out-of plane photonic bandgap of the periodic cladding. When designed properly, HC-PBGFs can support fundamental modes with more than 99% of the guided power propagating in air at wavelengths within the bandgap. As a consequence, Rayleigh scattering and phonon or impurity absorption that dominate the attenuation in more conventional step or graded index fibres have a lower contribution to the overall loss in HC-PBGFs. This raises the possibility of producing HC-PBGFs with very low attenuation values which, together with the nearly three orders of magnitude decrease in effective nonlinearity and the much reduced latency make HC-PBGFs an interesting candidate for next generation long-haul data transmission systems.

However, the lowest loss value reported to date for HC-PBGFs stands at 1.7*dB/km* [5], nearly an order of magnitude higher than in typical single mode fibers. Indeed, HC-PBGFs suffer from different loss mechanisms. Though the finite extent of the periodic cladding implies that the guided modes are always leaky, the leakage loss can be reduced to negligible levels by incorporating an appropriate number of rings of air holes in the cladding. This has led to the identification of scattering from the many air-glass interfaces present within HC-PBGFs as the fundamental mechanism dominating their loss [6]. The intrinsic roughness on these surfaces is due to thermally excited surface capillary waves (SCWs) present at the surface of molten glass which freeze-in as the glass goes through phase transition during the fibre draw. The thermodynamic origin of this roughness makes the scattering loss mechanism as fundamental as Rayleigh scattering in solid fibers. However, unlike Rayleigh scattering which is solely determined by the material choice, the surface scattering also crucially depends on the specific fiber design. Enlarging the air core or moving the photonic bandgap to longer wavelengths for example are known ways to reduce the roughness scattering loss [4, 6].

Paramount to obtaining further loss reduction in HC-PBGFs is the development of an accurate mathematical description of the scattering process. The standard theoretical treatment of scattering from waveguide imperfections considers the scattering as coupling between the guided modes and all the radiation modes of the structure [7, 8]. Roberts *et al.* [9] have applied such an approach to solid-core photonic crystal fibers, using a method involving complicated Green tensor math which in practice limits its applicability to structures with circular air holes. Phan-Huy *et al* [10] have treated the same problem in solid-core PCFs by breaking down the process into coupling to higher order core modes and subsequent leakage, but their approach ignores the direct coupling to radiation modes. Radiation modes were also neglected when Dangui *et al.* [11] numerically implemented the coupled-mode theory for HC-PBGFs, with their result requiring the computation of an impractically large number of coupling coefficients for several thousand cladding modes.

Here we extend an approach pioneered by Rawson [12, 13] and later used by Mazumder *et al.* [14] for weakly guiding fibers and derive a simpler method in which the far-field distribution of the scattered light is calculated using dipole radiation theory. Our method is suitable for describing scattering from boundary imperfections in a broad range of optical waveguides and in particular HC-PBGFs with air holes of arbitrary shape. The method predicts both an angular distribution of scattered light and overall attenuation values that agree well with experiments. We also show that the predicted wavelength dependence of the scattering loss agrees well with reported experimental data. Finally, we highlight the strong dependence of the loss on the roughness power spectral density, by showing that surfaces with the same value of root mean square (*rms*) roughness but different power density spectra can yield very different attenuation values.

## 2. Formulation

As shown in Fig. 1 the optical field of air-guided modes in HC-PBGFs always has some unavoidable overlap with the surrounding glass surfaces which generates scattered radiation and hence loss since these surfaces are rough. The intrinsic roughness causes all the air-glass interfaces to depart from those of an ideal fiber. If we consider a single air hole, its boundary imperfections can be described by a random function *f* (*s*, *z*) where *s* is the curvilinear coordinate along the hole perimeter.

Any local departure from the ideal boundary can be treated as a dipole of volume *dV* = |*f* (*s*, *z*)|*dsdz* which is excited by the incident modal field. This approximation is justified by the fact that the *rms* roughness resulting from frozen-in SCWs is of the order of 0.1*nm*, much smaller than the wavelength of light. Since the normal component of the electric displacement **D** and the parallel component of the electric field **E** are continuous at all interfaces, the induced dipole moment is generally given as:

*α*

_{||}and

*γ*

_{⊥}are the polarizability tensors [15]. In general, these quantities will depend on the shape of the dipole and the refractive index contrast. Johnson

*et al.*[15] have computed the polarizabilities

*α*and

*γ*for different dipole shapes and index contrasts. From their results, the effects of neglecting the exact shape of the dipole results in an error of less than 1% for silica-air interfaces at infrared wavelengths (index contrasts of ∼ 0.5 and ∼ 2). Further neglecting the effects of the discontinuity of the normal component of the electric field results in an additional error which is well below 5%. The polarizability additionally depends on the sign of the roughness defect. For example, when the perturbation is such that glass protrudes into air we speak of positive roughness, otherwise, the roughness is said to be negative. We take the polarizability

*α*

_{+}of a positive roughness “bump” as that of a glass sphere suspended in air and that of a negative one

*α*

_{−}as the polarizability of an air sphere in a glass background. The two differ in both sign and magnitude [15, 16], but since

*f*is random with zero average, we will neglect all the effects above and write the induced dipole moment in terms of the incident electric field at the interface as:

*α*

_{0}=

*ε*

_{0}(|

*α*

_{+}| + |

*α*

_{−}|)/2 ≈ 0.715

*ε*

_{0}for silica and air,

*ε*

_{0}being the free-space permittivity.

Our aim in this work is to calculate the far-field distribution of the light scattered by the collection of all the dipoles in a section of length 2*L* of the perturbed waveguide. This section is chosen so that it contains all the roughness spectral components of interest, but is also short enough for the incident field *E*_{0} to be approximately constant throughout. At point *P*(*ϑ*, *ϕ*) (see Fig. 2) on a distant sphere of radius *R*, the scattered field element from a single dipole located at the origin is [16]:

*k*

_{0}is the free-space propagation constant. The total field at

*P*is obtained by adding together all the scattered field elements while taking into account the relative phases. Taking as reference a ray that would be scattered from the origin, the phase difference for a ray scattered from position (

*s*,

*z*) can be calculated with the help of Fig. 2 as:

In this expression, *βz* represents the phase build up of the incident mode field between *z* = 0 and *z*, while *k*_{0}*zcosϕ* − *k*_{0}*y*′*sinϕ* is the phase accumulated by the reference ray in the same time. It has been assumed that all the scattered rays propagate through air, which is a reasonable approximation for the high air-filling fraction periodic cladding of HC-PBGFs. *y*′ = *ycosϑ* − *xsinϑ* is the new *y* coordinate of the radiating dipole in the frame obtained by rotating the original *xy* axes by an angle *ϑ* around the z-axis. Combining Eq. (3) and (4), the total scattered field at *P* from a single air hole is

*j*as an index to label each air hole, the total scattered field is therefore obtained by a summation over all surfaces, which with the help of Eq. (3) and (2) becomes:

Equation (6) contains a very general form of the boundary distortion. In simple structures, when the roughness can be separated into axial and azimuthal parts, the solution to Eq. (6) follows along the lines of the work by Mazumder *et al.* [14]. In the case of roughness due to frozen-in capillary waves at glass-air interfaces as in HC-PBGFs that we discuss in section 4.1, the most relevant features of the roughness and those that dictate the angular distribution of scattered light are those along the fiber axis. We therefore restrict *f* (*s _{i}*,

*z*) to

*f*(

*z*) only. The

*z*–dependent part of the above expression can be rewritten as:

*F*̃ is the spatial Fourier transform of the roughness distribution. Expanding the incident field at the scattering interface in terms of its components perpendicular (

*E*

_{0⊥}) and parallel(

*E*

_{0||}) to the

*Pz*plane and carrying out the vector product, Eq. (6) becomes:

**u**and

**v**are the unit vectors shown in Fig. 2. This result shows that the far-field distribution of scattered light is the product of a component depending on the roughness spectrum and a quantity that depends on the incident electric field at the interface, the geometry and optical properties of the fibre. Indeed, each spatial frequency component of the roughness scatters light in a specific direction

*ϕ*, a fact that conforms with the more complicated analysis of coupled-mode theory or volume-current methods [7, 8]. This also implies that only the spatial frequencies satisfying

*β*−

*k*

_{0}≤

*κ*≤

*β*+

*k*

_{0}contribute to radiation loss.

Equation (8) contains all the details of the scattered field but often the main quantity of interest is the average distribution of scattered power such as that typically measured in angularly resolved scattering (ARS) experiments:

*R*(

*u*) and the power spectral density (PSD) which are a Fourier transform pair need to be introduced. The power spectral density of the roughness is defined as [26]

*L*(

*ϕ*) the fraction of light scattered in the polar direction

*ϕ*that is actually lost. If

*P*

_{0}is the total optical power carried by the incident guided mode, the exponential loss coefficient is obtained by integration as:

Note that in deriving Eq. (11), we have assumed that the scattered light propagates in air, which is a reasonable approximation for within the microstructured area. In all HC-PBGFs however, the microstructure is surrounded by a silica outer cladding, and in ARS experiments an index-matched fluid is often used between the fiber and an external detector. In this case, the scattered light is refracted at the interface between the microstructured cladding and the glass outer jacket with refractive index *n _{gl}*. The measured scattering angle

*ϕ*is therefore obtained through Snell’s law

_{m}*n*cos

_{gl}*ϕ*= cos

_{m}*ϕ*.

## 3. Application to single-mode fibers

For a first verification of the accuracy of our model, we consider the classical problem of radiation loss due to scattering from a sinusoidal diameter variation in a standard single-mode fiber [17, 18]. The structure which is shown in the inset of Fig. 3 is a fibre with a core with radius *r* = *a* + *bsin*(2*πz*/Λ) and refractive index *n*_{1}, and a cladding with refractive index *n*_{2}. When the sinusoidal perturbation is small enough to satisfy the assumptions in our treatment (*b* ≪ *λ*), the scattering loss suffered by the *x*–polarized fundamental *HE*_{11} mode can be calculated using equation (11). Since the cladding is no longer air, the relative phase difference of equation (4) needs to be adjusted by replacing *k*_{0} with *k*_{0}*n*_{2}. For a pure sinusoidal perturbation over a long fiber section (*L* ≫ Λ), the power spectral density is

*n*

_{1}= 1.46,

*n*

_{2}= 1.458,

*a*= 5

*μm*and the wavelength of

*λ*= 1

*μm*is such that the waveguide is just below the cut-off of the first higher-order mode (

*V*= 2.4). To show the compatibility of our method with common numerical tools, the field components of the fundamental mode were calculated using a fully vectorial finite element method, and equation (8) was evaluated to obtain the scattered field. With

*L*(

*ϕ*) = 1 for all angles larger than the critical angle and

*L*(

*ϕ*) = 0 otherwise, the normalized scattering loss coefficient shown in Fig. 3 is identical to that obtained by Marcuse using a far more complex coupled mode theory [7]. This not only validates our simple approach but also shows that it can be implemented in combination with any arbitrary mode-solving numerical tool.

## 4. Roughness scattering in HC-PBGFs

Since no analytical solution for wave propagation in HC-PBGFs can be obtained, our general approach is to solve for the field components at the interfaces of the guided modes using a fully vectorial finite element method [19]. A realistic model of the surface roughness inside the fiber is then incorporated and the far-field distribution of the scattered light obtained from Eq. (8). As no assumption has been made on the nature of the incident modal field, Eq. (8) can also be used to analyze the scattering loss of surface modes.

#### 4.1. Surface roughness due to frozen-in surface capillary waves

As already mentioned, thermally excited surface capillary waves impose an intrinsic roughness on glass surfaces. On a flat two-dimensional surface, they lead to a two-dimensional power spectral density given by the well-known expression [20]:

where*κ*is the two dimensional surface wavevector,

*ρ*the glass density and

*g*the gravity constant.

*k*represents Boltzmann’s constant,

_{B}*T*the glass transition temperature and

_{g}*γ*the surface tension of the glass, which for pure silica are ∼ 1500

*K*and ∼ 1

*J.m*

^{−2}respectively.

In the case of HC-PBGFs and holey fibers in which the glass surfaces are cylindrical rather than flat, the roughness PSD becomes periodic along the azimuthal direction and hence can be expanded into an infinite Fourier series. Neglecting the gravity contribution (since the fibers are made under an imposed vertical flow), integrating over all the azimuthal wavevectors leads to the 1-D form given by [6]:

The 2-D expression of the roughness PSD in Eq. (15) requires an upper wavevector cut-off |*κ _{u}*| for the surface roughness to be physically acceptable, and gravity imposes another lower wavevector cut-off determined by the capillary length, which is of the order of ∼ 4

*mm*[21]. Equation (16) implies a logarithmic roughness autocorrelation along

*z*which also requires the existence of low and high spatial frequency cut-offs to be physically acceptable. Since the high frequency cut-off is estimated to be of the order of a few molecular lengths (∼ 0.5

*nm*), it is of no particular concern here as the highest frequency

*β*+

*k*

_{0}contributing to scattering falls well below this value. In HC-PBGFs, the origins of a low-frequency cut-off are not yet known. Atomic Force microscopy (AFM) measurements on the inner surface of air-holes in HC-PBGFs and holey fibers revealed a roughness PSD consistent with frozen-in surface capillary waves of Eq. (16) between

*κ*= 0.2

*μm*

^{−1}and

*κ*= 30

*μm*

^{−1}but could not measure the low spatial frequency end of the spectrum [6,10]. The results presented in this paper are therefore obtained by imposing that spatial frequencies below the arbitrary cut-off point

*κ*= 0.1

_{c}*μm*

^{−1}have the same PSD value

*S*(

*κ*).

_{c}#### 4.2. Angular distribution of scattered power and loss

Using expression (16) for the PSD of surface capillary waves and the cut-off previously mentioned, we calculate the scattered field distribution according to Eq. (11). The fibre used in our example is an ideal HC-PBGF with a structure closely matching that of a low-loss fibre we recently reported [22], a 19c fiber with air holes similar to hexagons with rounded corners (such as shown in Fig. 1) and in which the pitch is Λ ∼ 4.4*μm* and *d*/Λ ∼ 0.975. Figure 4(a) shows plots of *P*(*ϕ _{m}*)/(2

*LP*

_{0}) as a function of scattering angle

*ϕ*for a number of guided modes within the fiber, calculated with the assumption that all scattered light is lost, or

_{m}*L*(

*ϕ*) = 1 for all

*ϕ*. The curve for the fundamental mode shows good qualitative agreement with the measured ARS data from Roberts et al, referring to a not too dissimilar fiber design [6]. Because the roughness described by Eq. (16) has high PSD at small spatial frequencies, scattering occurs primarily in the forward direction, as can be seen. Our model goes one step further in allowing us to fully evaluate the differential scattering loss between the guided modes of the fiber, which can be seen to be non-negligible for these fibers.

Figure 4(b) shows a plot of the calculated roughness scattering loss and confinement loss across the photonic bandgap for the fundamental mode of the same fibre, along with the measured attenuation spectrum. As can be seen, our model suggests that the total loss is dominated by the roughness scattering contribution near the center of the bandgap, while the bandgap edges are determined by confinement loss. The calculated scattering loss values are within a factor of 2 of the measured ones, which is an excellent agreement considering the simplicity of our method.

#### 4.3. Wavelength dependence of scattering loss

A crucial factor in determining the ultimate loss performance of HC-PBGFs is its wavelength dependence. If the effects of material dispersion are rigidly ignored, the wavelength *λ _{c}* at which the lowest loss value occurs is proportionally scaled when an HC-PBGF with a given cross-section is drawn to different dimensions [23]. It has been shown experimentally that when a given fiber cross-section is drawn to different diameters, its minimum loss decreases as
${\lambda}_{c}^{-3}$ until a wavelength around 2

*μm*where infrared absorption mechanisms begin to dominate. However, a theoretical understanding of this dependence has not hitherto been provided [6,24]. Despite the explicit ${k}_{0}^{4}$ in Eq. (11), our model confirms the experiments by predicting a scattering loss that indeed follows a ${\lambda}_{c}^{-a}$ trend, with

*a*= 3.06 in the case of the low-loss HC-PBGF, as shown in Fig. 5. In general,

*a*depends on both the specific design of the fibre cross-section and the roughness PSD, and it has been found to assume values between 2.5 and 3.5 for a range of fibers we studied. This confirms that the scattering loss and its wavelength dependence are very much dependent on the design parameters.

On the same figure, we also show how the normalized interface field intensity

*F*was calibrated using measurements and simulations for the fibre in [22] at a wavelength of 1.5

*μm*so that

*F*= 0.0174

*μm*

^{−1}corresponds to 3.5

*dB/km*. As can be seen, this scalar quantity does not explain the observed wavelength dependence of the loss and is thus inaccurate in quantifying it. This can be understood by realizing that in addition to the fibre cross-section determining the field at the interfaces and how it changes with wavelength, when the operational wavelength is increased, the range of roughness spatial frequencies that contribute to the scattering also changes and becomes narrower. For example, at a wavelength of 1

*μm*the range of spatial frequencies required to calculate the scattering in air at angles

*ϕ*from 0 to 180° ranges from 0 to 12.55

*μm*

^{−1}while at a wavelength of 2

*μm*, it goes from 0 to 6.275

*μm*

^{−1}. Therefore only models that fully account for the roughness PSD like our method (and do not just use its

*rms*value like the

*F*–parameter) can predict the wavelength dependence of the loss.

## 5. The impact of roughness PSD

In order to accurately predict roughness scattering loss in HC-PBGFs, accurate knowledge of the roughness statistics and its PSD in particular is of utmost importance. However, the practical limitations of AFM measurements make it impossible to obtain information about the very important low spatial frequency region of the spectrum and alternative techniques are yet to be implemented. In the previous sections, we have assumed that the PSD for surface capillary waves is cut-off at an arbitrary spatial frequency *κ _{c}* = 0.1

*μm*

^{−1}. In Fig. 6(a), we plot the computed scattering loss for our HC-PBGF as a function of the cut-off frequency at the wavelength of

*λ*= 1.55

_{c}*μm*. As

*κ*increases, the PSD

_{c}*S*(

*κ*) is lower for a broad range of frequencies, leading to the observed decrease in the scattering loss.

_{c}Figure 6(b) shows the total loss across the photonic bandgap for three values of *κ _{c}*. A decrease in

*κ*corresponds to a higher PSD at the short spatial frequencies responsible for stronger scattering at longer wavelengths. As a result, the loss at longer wavelengths becomes higher than at shorter wavelengths, as can be readily seen in the figure. Additionally, we note that the scattering loss is directly proportional to the ratio

_{c}*T*/

_{g}*γ*which we have assumed is 1500

*K*/(

*J.m*

^{−2}), an increase in this ratio will therefore shift the scattering loss curve upwards.

SCWs may not be the only mechanism generating surface roughness in a HC-PBGF. While they have been found to dominate the roughness spectrum in HC-PBGFs between the spatial frequencies of 0.2 and 30*μm*^{−1}, roughness from other sources may be present at frequencies below those practically measurable. It is well known that most realistic surfaces have roughness statistics that present one or more of three basic components: a long-range waviness, short-range random roughness and periodicity [26]. These additional roughness components, corresponding to long range waviness and possibly originating from drawing parameter fluctuations during fiber fabrication, may ultimately have an effect on the overall loss [11].

To investigate this, we study the loss that would be produced in our HC-PBGF by roughness components with an exponential or Gaussian auto-correlation. The analytical expressions for these are *R*(*u*) = *σ*^{2}*exp*(−|*u*|/*L _{c}*) and
$R(u)={\sigma}^{2}\mathit{exp}\left(-{u}^{2}/{L}_{c}^{2}\right)$ respectively, with

*L*being the correlation length and

_{c}*σ*the corresponding

*rms*roughness. The PSD arising from these auto-correlation functions are a Lorentzian $S(\kappa )=2{\sigma}^{2}{L}_{c}/\left(1+{L}_{c}^{2}{\kappa}^{2}\right)$ and a Gaussian $S(\kappa )={\sigma}^{2}{L}_{c}/\sqrt{\pi}\mathit{exp}\left(-{\kappa}^{2}{L}_{c}^{2}/4\right)$ respectively.

Figure 7(a) shows the scattering loss contribution arising from such roughness components as a function of the correlation length *L _{c}* at wavelength

*λ*= 1.55

*μm*, assuming

*σ*of 0.1

*nm*. In both cases, the loss increases with

*L*, peaks near

_{c}*L*4

_{c}*mm*and then decreases quickly, as typically observed with exponential or Gaussian roughness [27]. In a broad range of correlation lengths (70

*μm*− 3.7

*cm*) the scattering loss from such roughness components alone is higher than the measured value, which would suggest discarding this range. If however a long-range exponential roughness component with

*L*of the order of a few centimeters is present in our HC-PBGF in addition to SCWs, this leads to a predicted scattering loss value very close to the measured 3.5

_{c}*dB/km*. The contribution of this additional roughness component would be many orders of magnitude below that of SCWs over AFM-measurable spatial frequencies and would therefore be compatible with reported measurements, as can be seen in Fig. 7(b). Although clearly speculative, this confirms the importance of accurate roughness information in the low spatial frequency region to allow us making even more accurate loss predictions.

## 6. Conclusion

We have presented a simple theoretical method based on dipole radiation to describe light scattering from surface roughness in HC-PBGFs. Our method shows excellent agreement with the more standard but considerably more complex approach of coupled-mode theory. Our model not only returns an accurate value of loss, but also gives detailed information about the far-field distribution of scattered light, including the angular distribution of scattered power typically measured in angularly resolved scattering experiments. Using our model and a number of reasonable assumptions about the surface roughness, the predicted loss from frozen-in SCWs alone is within a factor of 2 for a state-of-the-art HC-PBGF. Realistic considerations on other possible origins of roughness such as a long-range waviness, especially in the spectral region not accessible by current roughness measurement methods, can lead to more accurate loss predictions. Our simple approach allows calculation of scattering losses from any microstructured fiber of arbitrary geometry for which the power spectral density of the roughness is known and it can be used in combination with any numerical method to calculate the field intensity of the guided mode at the interfaces.

## Acknowledgments

This work was supported by the EU 7th Framework Programme under grant agreement 228033 ( MODE-GAP).

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