## Abstract

Using an eigenmode decomposition technique, we numerically determine the
backreflection coefficient of the modes of air-core photonic bandgap fibers for
flat terminations. This coefficient is found to be very small for the
fundamental air-guided mode, of the order of 10^{-5} to 10^{-6},
in contrast with the surface and bulk modes, which exhibit significantly higher
reflections, by about three to four orders of magnitude. For the Crystal Fibre
HC-1550-2 fiber, we find a reflection coefficient of
1.9×10^{-6} for an air termination, and approximately 3.3%
for a silica termination. We also find that the Fresnel approximation is ill
suited for the determination of the modal reflection coefficient, and instead
propose a more accurate new formula based on an averaged modal index.

© 2007 Optical Society of America

## 1. Introduction

Air-core photonic bandgap fibers (PBFs) [1–3] offer the enticing theoretical prospect of very low nonlinearities, reduced sensitivity to environmental factors such as temperature [4], potentially low backscattering and hence very low propagation losses, and they are finding interesting novel applications in various photonics fields, including fiber sensors and high-power laser delivery. As is the case for conventional solid-core fibers, the performance of a fiber optic system based on PBFs is expected to be adversely affected by the reflection that takes place at the end of an air-core fiber, whether the fiber is terminated in air (free end) or butt-coupled or spliced to another air-core or solid-core fiber. Backreflection from such terminations are notoriously detrimental to the performance of fiber interferometers, communication systems, and lasers and amplifiers, as they result in increased noise, instabilities, and other deleterious effects. [5] While the reflection at the end of a photonic crystal fiber has been studied, [6] evaluating the magnitude of the reflection at the end of an air-core fiber, such as the HC-1550-2, [7] has never been reported before and it is therefore becoming a pressing issue.

The backreflection from the end of a conventional fiber terminated by a semi-infinite
medium of refractive index *n*
_{0} is characterized by a
power reflection coefficient that is usually calculated by the well-known
approximate Fresnel formula:

where *n _{eff}* is the effective index of the mode considered.
The Fresnel approximation is equivalent to replacing the waveguide with a medium of
uniform refractive index equal to the mode effective index, and to approximate the
mode by a plane wave at normal incidence. As a result, the Fresnel approximation
performs best for weakly-guiding waveguides with a small index contrast across the
waveguide cross-section, for modes possessing a small numerical aperture (lower
order modes), and for a strong refractive index contrast between the waveguide and
the termination medium. Conversely, this approximation has proved to be inaccurate
to evaluate the reflection coefficient of strongly guiding waveguides, [8] and it does not account for the spatial intensity
distribution of the mode. Because PBFs are strongly guided waveguides, and because
we are mostly concerned with the reflection of a leaky mode (the fundamental
air-guided core mode) into air (low refractive index contrast between waveguide and
termination medium), we expect the Fresnel approximation to be ineffective in this
context, and that a more elaborate calculation will be required.

In this paper, we detail for the first time a computation procedure for the derivation of the reflection coefficient of a PBF mode into a medium of arbitrary refractive index, which we then apply to calculate numerically the reflection coefficient of the fundamental core mode, as well as representative surface and bulk modes, of an air-core fiber, and their dependencies on wavelength and fiber parameters. We then study the influence of PBF design parameters on the reflection coefficient, in order to design PBF components with very low reflections for improved performance in fiber devices. We also derive a simple formula based on the Fresnel approximation that provides accurate predictions of the backreflection from the end of a PBF, especially for fibers with a high air-filling ratio (above 90 %).

## 2. Theory

The configuration studied is represented in Fig. 1. It consists of a semi-infinite PBF, extending over
the *z* < 0 region (region I). The *z*
> 0 region (region II) is a semi-infinite medium of uniform index
*n*
_{0}, such as air. The PBF parameters are as follows:
the cladding crystal periodicity is Λ, the core radius is
*R*, and the cladding air hole radius is
*ρ*. The transverse plane is hereafter defined as the
interface plane terminating the PBF. We consider the reflection at this interface of
a mode field propagating through the PBF in the +*z*
direction.

To calculate the reflection coefficient, we use the formalism of eigenmode decomposition.[9,10] The incident and reflected fields (in region I) are
decomposed into a sum of PBF modes and the transmitted field (in region II) into a
sum of plane waves. The transverse incident electrical field is denoted * E^{I}*(

*) and can either be a core mode, a cladding mode, or a surface mode of the PBF, labeled*

**r***(*

**E**_{0}*). The reflected transverse electrical field, labeled*

**r***(*

**E**^{R}*), propagates in the -*

**r***z*direction and is written as a superposition of core and cladding mode fields of the PBF, which are collectively labeled

*(*

**E**_{m}*).*

**r**The modal decomposition of the reflected transverse electrical field is therefore written as:

where the summation is carried over (*N _{M}*+1)
modes, and

*a*is the amplitude of the signal reflected into mode

_{m}*m*. Furthermore, only modes of the same symmetry class as the incident mode will be present in the decomposition of the reflected field. For most application, we are concerned with the amount of light backreflected into the incident mode, which is labeled as

*m*= 0, since all other coefficients are much smaller (by many orders of magnitude). The intensity or power reflection coefficient into the fundamental mode is:

In the same fashion, the transverse transmitted electrical field propagating in
region II in the positive *z* direction is denoted * E^{T}*(

*) and decomposed in a sum of plane waves propagating in free space. The transverse magnetic fields are decomposed in the same fashion. Using the continuity of the transverse electric and magnetic fields, we obtain the following relationships at the PBF interface plane:*

**r**As described in the appendix, we can then use the orthogonality of the transverse
fields of the waveguide modes, and the orthogonality of the transverse fields of the
plane waves, to transform these equations into a system of linear equations, which
are then solved numerically to recover the reflection coefficients
*a _{m}*.

The PBFs were modeled using the Stanford Photonic Bandgap Fiber (SPBF) code, a finite-difference mode solver developed for the purpose of modeling the modes of waveguides with complex refractive index profiles such as air-core fibers. [11] In the simulations carried out for this paper, the SPBF code was used with periodic boundary conditions, with a simulation step of Λ/50 and a grid size of 10Λ×10Λ. We used typically 260 PBF modes and between 2000 and 5000 plane waves for the computation of the reflection coefficients.

To compute the projection of the PBF modes onto plane waves, we used a
two-dimensional discrete Fourier transform (2D DFT) of the computed fields. Figure 2 illustrates a typical matrix * A*. The non-zero coefficients are illustrated as blue dots, and the zero
coefficients are left out (shown as white spaces). The upper left corner indicates
the projection of PBF modes onto each other; the lower right corner shows the
projection of plane waves onto each other. The remaining portions of the matrix show
the projection of PBF modes onto plane waves propagating in air. The matrix size is
2082 × 2082 elements, of which nearly 950,000 are non-zero complex
elements. This sheer number of non-zero elements makes it impossible to invert such
a matrix accurately directly, so we resorted to an iterative method, described in
Ref [12].

## 3. Variation of the fundamental mode reflection coefficient across the PBF bandgap

In the first part of this work, we modeled the PBF whose refractive index profile is
illustrated in Fig. 3. The cladding holes' radius is
*ρ* = 0.47Λ, and the core radius
*R* = 0.8Λ. Figure 4 represents the calculated bandgap of this fiber, as
well as the dispersion curves of all of its core modes. The bandgap extends from
*λ* = 0.56Λ to
*λ* = 0.64Λ. Due to the choice of core radius, [13] this fiber does not support surface modes, and it has only
two core modes, the two degenerate, orthogonally polarized fundamental modes. Figure 5 shows the longitudinal Poynting vector profile of
one of the fundamental modes at *λ* = 0.60Λ.
The fundamental core modes have a very large fraction of their intensity (over 90 %)
concentrated in air, which is the reason why a weak reflection at an air interface
is expected.

The Fourier transform intensity of this same mode's transverse electric
field, plotted in logarithmic scale in the wavevector domain, is shown in Fig. 6. The black circle shows the boundary for all
wavevectors such that ${k}_{x}^{2}+{k}_{y}^{2}\le {k}_{0}^{2}={\left(\frac{2\pi}{\lambda}\right)}^{2}$, i.e., all wavevectors admissible for plane waves propagating in
air. Since the fundamental mode profile exhibits little variation across the
bandgap, as the wavelength varies across the bandgap, the most significant change
occurs in the value of *k*
_{0}, and thus in the radius of
this circle. As the wavelength increases, the radius of the circle decreases, and
thus fewer plane waves can be excited past the PBF termination plane, yielding a
decrease in transmitted field. We therefore expect the fundamental-mode reflection
to increase with wavelength. At the same time, since the circle boundary lies in a
region where the Fourier transform intensity of the electric field is very weak
(roughly 0.1% of its maximum value), we expect a weak dependence on wavelength.

This is indeed found to be the case. Figure 7 shows the reflection coefficient of the
*x*-polarized fundamental mode, defined as
|*a*
_{0}|^{2}, when the
fiber is terminated in air, calculated at several wavelengths across the bandgap.
The reflection coefficient remains weak across the entire bandgap, its coefficient
varying between ~0.6×10^{-5} at the short-wavelength
end of the bandgap and ~1.3×10^{-5} at the
long-wavelength end. We also observe a general trend towards an increase in the
backreflection as the wavelength increases, which was predicted earlier on the basis
of the smaller wavevectors associated with longer wavelengths: the mode energy is
transmitted in air over a smaller range of wavevectors, and fewer plane waves can be
excited, thus resulting in an increased reflection coefficient.

## 4. Reflection coefficient of various PBF modes

It was also interesting to determine the reflection coefficient for the other kinds
of modes supported by air-core fibers, in particular higher order core modes,
surface modes, and cladding modes. To do so, we retained the same cladding holes
radius (*ρ* = 0.47Λ) but increased the fiber
core radius to *R* = 2.0Λ, so that the fiber now supports
a large number of core modes, as well as some surface modes. As is well known, the
surface-mode energy is concentrated mostly in the silica at the core-cladding
boundary. The core region of this fiber is shown in Fig. 8, together with the intensity contour lines of an
exemplary surface mode.

Figure 9 shows the calculated reflection coefficients of
selected modes of this fiber, namely the six surface modes closest in effective
index to the fundamental modes, and the first six air-guided modes (the two
LP_{0,1}-like fundamental modes, labeled
HE_{1,1}
^{x,y}, and the
four LP_{1,1}-like first-order modes, labeled TE_{0,1},
HE_{2,1}
^{x,y}, and
TM_{0,1}, respectively). These coefficients were calculated at
*λ* = 0.6Λ, which is about at the center of
the bandgap, for a termination in air, and they represent reflection into the same
backward-traveling mode as the incident mode. The reflection coefficient for the
fundamental mode is 1.2×10^{-5}, which is close to the value
calculated for the previous (single-mode) fiber. The higher order core modes
(LP_{11}-like) have a significantly higher reflection coefficient,
averaging 7×10^{-5}. This is expected, since these modes have a
higher fraction of their energy in the silica portions of the PBF, and hence are
more strongly reflected. Note that the backreflection coefficients of the four
LP_{11}-like modes differ by as much as a factor of 3. This is due to
the vectorial nature of the PBF waveguide, in which the strong index contrast
between the air core and the silica cladding membranes lifts the near-degeneracy of
these modes normally present in weakly-guiding fibers. The surface modes, which have
an even higher fraction of their energy in silica, exhibit a much higher
backreflection coefficient, in the range of 8×10^{-3}. All of
these core and surface modes have very close effective indices, in the vicinity of
the refractive index of air, so the Fresnel approximation (Eq. 1) yields fairly similar reflection coefficients for all them
(see Fig. 9). Clearly, and expectedly, the Fresnel approximation
yields poor predictions.

The reflection coefficient was also calculated for multiple representative bulk modes (also for reflection into the backward-traveling fundamental mode of same polarization). All reflection coefficients were approximately equal to 1%. This is again entirely consistent with the field distributions of these bulk modes, which are strongly localized on the thin silica veins of the photonic-crystal cladding. This feature brings their reflection coefficient up to a value of the same magnitude as a reflection from silica into air (~3.37%).

## 5. Effect of PBF design parameters on the fundamental core mode reflection coefficient

Unlike in a conventional fiber, the magnitude of the backreflection at the end of a PBF is strongly influenced by the fiber parameters. These parameters can therefore be adjusted to minimize backreflection. One such parameter is the core radius. A proper choice of core radius can prevent the PBF from supporting surface modes. [13] At the same time, the absence of surface modes has been linked to a reduction in the fraction of fundamental mode power localized in the silica regions of the PBF. [4] We thus expect that the reflection of the fundamental mode will be higher for core radii that support surface modes. Since larger cores support fundamental modes that are more strongly confined in the air core (i.e., the fundamental mode intensity at the core-cladding interface is weaker), we also expect that as the core radius increases, the fundamental mode reflection should generally decrease.

To confirm these trends, we plot in Fig. 10 the calculated dependence of the fundamental mode
reflection coefficient on the core radius, for the same cladding holes radius as
used before (*ρ* = 0.47Λ) and at about the
center of the bandgap (wavelength *λ* = 0.6Λ).
As expected, as the core radius increases, the reflection tends to decrease, except
for radii where the PBF supports surface modes, such as for radii between
1.05Λ and 1.3Λ, and between 1.45Λ and 1.7Λ.[13] For those radii, the fundamental mode reflection
coefficient increases very sharply and significantly, again because of the
substantial increase in the fraction of fundamental mode energy in silica.[4] This result points to another reason for avoiding surface
modes: they increase backreflection of the fundamental (and other) core modes.

The minimum backreflection predicted for a single-mode PBF is determined to occur for
a radius of *R* = 0.95 Λ, and is approximately equal to
2.9×10^{-6}. Further reduction in the backreflection can be
achieved by selecting a radius *R* = 1.9Λ, where the
backreflection is predicted to be 4.2×10^{-7}, but at the expense
of the PBF no longer being single-mode.

## 6. Effect of the termination medium refractive index on the reflection coefficient

While the reflections of the core modes of a PBF terminated in air are found to be low, much stronger reflections are expected for a solid or liquid termination medium (such as silica or lithium niobate, for example), because the core modes then propagate into a medium with a much greater index than their own. Since the Fresnel approximation is more accurate for a large index contrast between waveguide and termination medium, it is also expected to give better estimates of the reflection coefficients.

To investigate the influence of the termination medium index, we show in Fig. 11 the reflection of the fundamental mode for a PBF
with *R* = 0.8Λ and *ρ* =
0.47Λ, and a wavelength *λ* = 0.6Λ,
as a function of the termination refractive index. The reflections predicted by the
Fresnel approximation are shown as a red curve. While both methods are in agreement
for large termination medium indices, there are as expected significant differences
for a termination medium index *n*
_{0} close to the
refractive index of air. When *n*
_{0} is close to 1, the
Fresnel approximation ceases to be valid and yields inaccurate answers, as we have
seen before, by as much as 3 orders of magnitude. One interesting feature of these
curves is the location of their minima. The Fresnel approximation predicts a minimum
reflection for a termination medium of refractive index equal to
*n _{eff}* < 1, while the calculated reflection
exhibits a minimum for a termination medium index of approximately 1.005.

For larger values of the termination medium index, the agreement between both approaches is much improved. Notably, the calculated reflection coefficient for a silica termination is found to be 3.13%, differing by 8% with the value of 3.37% expected for a plane wave at normal incidence to an air-to-silica interface. The Fresnel approximation predicts a reflection of 3.67% for the fundamental mode at the same air-silica interface, within 17% of the computed value. This reflection coefficient is also approximately the reflection we expect from the butt-coupling of this PBF to a conventional index-guiding silica fiber, such as an SMF28 fiber.

## 7. Simplified formulation

Since the Fresnel reflection coefficient is a poor approximation for the modal backreflection coefficient, we developed a new approximate formula for evaluating that quantity. This approximation is based on a Fresnel coefficient in which the mode effective index is replaced by an averaged modal index. We derived the formula for the averaged modal index in a heuristic fashion, to obtain the best match with our calculation of the modal reflection coefficient.

The averaged modal index is defined as the normalized overlap of the transverse electric field intensity with the refractive index distribution:

By definition, 1 ≤ *n _{modal}* ≤

*n*, and the larger the fraction of energy the mode carries in air, the closer the averaged modal index is to 1. We show that the modal reflection coefficient can be well approximated by:

_{silica}The physical interpretation of this approximation is that, from a reflection
standpoint, the light effectively propagates as a plane wave under normal incidence,
in a medium of refractive index *n _{modal}*. The use of the
averaged modal index also allows to account for the occurrence of a minimum in
reflection for a termination medium with a refractive index larger than 1.

Figure 12 shows a plot of the averaged modal index calculated
from Eq. (9) for the fundamental mode as a function of core radius for a
PBF with *ρ* = 0.47Λ, and at a wavelength of
*λ* = 0.6Λ. The averaged modal index also
shows sharp increases whenever the PBF supports surface modes, due to an enhanced
overlap of the mode with the silica regions, [4] as does the fundamental mode reflection coefficient (see Fig. 10). This is in particular visible for core radii
between 1.05Λ and 1.3Λ, and core radii between
1.45Λ and 1.7Λ. As the core radius increases, the averaged
modal index outside of these bands decreases, which is again consistent with the
predicted behavior of the reflection coefficient.

In Fig. 13, we have plotted a comparison between the fundamental
mode reflection coefficient obtained by our complete eigenmode decomposition
calculation (blue curve), and by the averaged modal index approximation (red curve).
We note that while the trends of the reflection coefficient are well predicted by
this new approximation, the actual values predicted differ by a factor of 2 to 6,
depending of the core radius, and that the fit has a somewhat better quality when
the PBF supports surface modes. We attribute this difference mainly to the angular
spread of the energy of the fundamental mode, which is quite significant in these
PBFs (nearly 10° for a core radius *R* = 0.8Λ).

It is therefore of interest to consider PBFs where the angular spread of the fundamental mode is much reduced, so as to have a mode behaving nearly like a plane wave under normal incidence at a PBF termination. A larger air-filling ratio than those considered heretofore (around 85 %) is required for that purpose, and we elected to model the HC-1550-2 fiber by Crystal Fibre, [7] which has an air-filling ratio of over 90%. Figure 14 shows a scanning electron microscope picture of the PBF structure, with the entire photonic crystal cladding visible (8 layers of holes). We modeled the PBF using the refractive index structure shown in part in Fig. 15, using 5 layers of holes and periodic boundary conditions.

The longitudinal Poynting vector component of the fundamental mode at a wavelength of
*λ* = 0.4Λ is shown in Fig. 16. There is very strong confinement of the mode field
into the air core, with a field intensity overlap with silica now less than 1%. The
Fourier transform intensity of this same mode's transverse electric
field, plotted in logarithmic scale in the wavevector domain, is shown in Fig. 17. The black circle shows the boundary for all
wavevectors such that ${k}_{x}^{2}+{k}_{y}^{2}\le {k}_{0}^{2}={\left(\frac{2\pi}{\lambda}\right)}^{2}$, i.e. all wavevectors admissible for plane waves propagating in
air. The energy of the fundamental mode shows a much smaller angular spread, of the
order of 3°. The calculated value for the fundamental mode reflection
coefficient is 1.9×10^{-6} for an air-termination.

The averaged modal index approximation is compared to the calculations of the
reflection coefficient in Fig. 17, for the PBF HC-1550-2 at a wavelength of
*λ* = 0.4Λ, as a function of the
termination medium index. The agreement is now excellent, the disagreement between
the two approaches being at most 50% for all the points shown in the figure. Also,
the averaged modal index model predicts that the minimum in the reflection occurs
for a termination index of 1.0036, in good agreement with the prediction of the
exact model (black dots), which falls somewhere between 1.003 and 1.004. The
conclusion is that the averaged modal index approximation can be used for
qualitative predictions when modeling a fiber with a low air-filling ratio (around
85%), and for quantitative predictions when modeling a fiber with an air-filling
ratio of 90% or higher.

## 8. Conclusion

We have used an eigenmode decomposition method to compute the backreflection
coefficient of various modes in PBFs, for terminations in a uniform medium, since
the assumptions on which the commonly used Fresnel approximation relies cease to be
valid for air-core fibers. The predicted values range from 10^{-5} to
10^{-6} for the fundamental mode, a few 10^{-3} for surface
modes, and around 10^{-2} for bulk modes. Furthermore we observed strong
variations of the fundamental mode reflection as a function of core radius, due to
the presence of surface modes, which enhance the overlap of the fundamental mode
with the silica. For the Crystal Fibre HC-1550-2 PBF we predicted a reflection
coefficient of 1.9×10^{-6} for an air termination, and 3.3% for a
silica termination, which would correspond to the splice reflection of a PBF with an
index-guiding conventional silica optical fiber.

We also proposed a heuristic simple reflection formula, based on an averaged modal index, and allowing the reflection of a PBF mode to be predicted accurately, within less than an order of magnitude for PBFs with a low air-filling ratio (around 85%) and more accurately for an air-filling ratio of 90% and over.

## Appendix: Eigenmode decomposition method

The transverse incident electrical field is denoted * E^{I}*(

*) and is assumed to be a mode (core or cladding mode) of the PBF,*

**r***(*

**E**_{0}*). The reflected transverse electrical field propagates in the negative*

**r***z*direction and is a superposition of PBF modes (core and cladding modes), and is denoted

*(*

**E**^{R}*). The modal decomposition of the reflected transverse electrical field is of the form:*

**r**where *a*
_{0} is the amplitude reflection coefficient
of the incident mode considered into itself. The modes present in the
decomposition of the reflected field are of the same symmetry class as the
fundamental mode. The intensity reflection coefficient is:

In the same fashion, the transverse transmitted electrical field propagating
in region II in the positive *z* direction is denoted * E^{T}*(

*) and decomposed in a sum of plane waves propagating in free space:*

**r**where Ω is the set of plane waves used to decompose * E^{T}*,

*s*is a normalization constant, and

*and*

**E**^{1}_{W,k}*denote the two amplitudes along orthogonal polarizations of a plane wave propagating in air, with a wavevector*

**E**^{2}_{W,k}*=[*

**k***k*

_{x}*k*

_{y}*k*] satisfying:

_{z}The transverse magnetic fields are decomposed in the same fashion:

Using the continuity of the transverse electric and magnetic fields, we obtain the following relationships:

$$\Rightarrow {E}_{0}\left(r\right)+\sum _{m=0}^{{N}_{M}}{a}_{m}{E}_{m}\left(r\right)=\sum _{k\in \Omega}\left({b}_{k}^{1}{E}_{W,k}^{1}+{b}_{k}^{2}{E}_{W,k}^{2}\right)\frac{{e}^{ik\bullet r}}{S},$$

$$\Rightarrow {H}_{0}\left(r\right)+\sum _{m=0}^{{N}_{M}}{a}_{m}{H}_{m}\left(r\right)=\sum _{k\in \Omega}\left({b}_{k}^{1}{H}_{W,k}^{1}+{b}_{k}^{2}{H}_{W,k}^{2}\right)\frac{{e}^{ik\bullet r}}{S},$$

and we define a hermitian product over the function space by:

where the integral is calculated across the transverse plane.

We can then exploit the orthogonality of the transverse fields of the
waveguide modes, and the orthogonality of the transverse fields of the plane
waves, to derive the reflection coefficients *a _{m}*
and transmission coefficients

*b*:

_{k}yielding (1+N_{M}+2N_{W}) linear
equations to compute the (N_{M}+1)
*a _{m}* and 2N

_{W}

*b*coefficients. Equations (A11) and (A12) are derived from Eq. (A8), and Eq. (A13) and (A14) are derived from Eq. (A9).

_{k}These linear equations are encoded in a sparse matrix * A* of size
(1+N

_{M}+2N

_{W})×(1+N

_{M}+2N

_{W}), containing 1+N

_{M}+2N

_{W}+4(1+N

_{M})N

_{W}nonzero terms. The reflection coefficients

*are determined by solving an equation of the form:*

**ν**which is solved by performing an incomplete LU decomposition of * A*, which is used as a preconditioner to lower the conditioning
number of

*, and we thus solve:*

**A**which allows for faster and more accurate solving of the reflection
coefficient vector through the numerical method of the stabilized
biconjugate gradients method BICGSTAB,[11] with an approximate solution * ν_{0}* with an error measured by:

thus ensuring the very high accuracy of the solution within a few iterations of BICGSTAB (fewer than 10 usually), in a few seconds of calculation.

## Acknowledgments

This work was supported by Litton Systems, Inc., a wholly owned subsidiary of Northrop Grumman Corporation.

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