## Abstract

We show how to design a round optical fiber so that it is effectively single moded, with no polarization degeneracy. Such fibers would be free from the consequences of polarization degeneracy or near degeneracy – phenomena such as polarization fading in interferometry, and polarization mode dispersion – and so may offer an alternative to polarization maintaining fibers for the avoidance of these phenomena. The design presented builds on an earlier observation of polarization selective refection in Bragg fibers.

© 2002 Optical Society of America

We are used to polarization of light as it occurs in free plane waves – formed by a superposition of two orthogonal linear polarizations with any relative amplitude and phase, giving the familiar linear, circular and elliptical polarization states. Polarization of light guided in an ordinary single mode fiber is similar. It is perhaps easy to forget that polarization in this sense is not an inherent property of light. Just as the radiation field is modified in photonic crystal structures [1,2] the accessible polarization states of the field in a waveguide may differ from those of free radiation. In a sense this is obvious – ordinary step index fibers have TE and TM modes without polarization degeneracy. This is easily overlooked as in ordinary single mode fiber these modes are unguided, and do not commonly intrude on observations, and in multimode fiber, in the ray optics regime, polarization reappears in its 2-fold form.

Bragg fibers [3] guide light through coherent
radial reflection off concentric rings of alternating high-low index, rather than by
the ordinary mechanism of radial evanescence in the outer or cladding layer (Fig. 1). We show how to design a Bragg fiber
so that there is only a single guided mode, with no polarization degeneracy. The
electric field in this mode is strictly transverse and azimuthally directed (i.e.
parallel to the rings) and shares these properties with the TE_{0n} modes of
an ordinary optical fiber (Fig. 2, and see
for example [4]). We call fibers with this
property “TE Bragg fibers” for short. The TE modes in a Bragg fiber, by contrast
with ordinary fibers, tend to be better guided than other modes [5], and the singular feature of our TE Bragg
fibers is the clear promotion of a single TE mode into the best guided position. To
achieve this result the Brewster angle phenomenon is exploited in the design so as
to reduce the reflection (and thus the guidance) of TM and hybrid modes. In a
network of TE Bragg fibers, the polarization control is provided by the fabric of
the medium – rather than by a superimposed birefringence, as in polarization
maintaining fibers – and may more easily accommodate forming fiber couplers and
other devices, because both mode and fiber are round, thus overcoming the
orientation problem presented by ordinary polarizing fiber, or by the recently
proposed “supersquare” photonic crystal fiber [6]. In a network of TE Bragg fibers, the light is in a setting where the
familiar polarization states and related phenomena do not exist.

Bragg fibers of finite radius support no strictly bound modes, only leaky modes with
varying loss. However, as a practical matter, a sufficiently low loss rate is as
good as binding by radial evanescence. Whether a mode with a given loss coefficient
(fractional loss per m) should be regarded as in practice guided or not, depends on
the length of the fiber. Whether a fiber should be regarded as single moded or not,
depends on its length and on the relative lossiness of the least lossy mode and all
the other modes. For example, consider a fiber having three modes with fractional
power losses of 10^{-6}, 10^{-4} and 10^{-2} per m
respectively, and with all other modes much lossier. For a fiber length of 1 m, all
three modes are guided, at 100 m, the first two are guided and the third is not,
while at 10 km, the fiber is effectively single moded.

Since all modes in a Bragg fiber are lossy, it is clear that single modedness depends on there being one mode which has a low enough loss, while all other modes are sufficiently lossy – it is the contrast which counts. Our design strategy makes use of the Brewster angle phenomenon to help to achieve such contrast.

The calculations were done essentially using Chew’s method [7], and constitute accurate modal solutions of Maxwell’s
equations. Each mode is characterized by a complex propagation constant β or
effective refractive index *n*
_{eff} = *β/k* (*k* = free space wave number).
We do not attempt to take the calculation of loss rates further by including the
effects of perturbations, material absorption or scattering. We contend that it is
reasonable to assume provisionally that the effects of such perturbations are small
in a wide range of practical circumstances, as is the case for ordinary single mode
fibers. We display key modal properties of two similar TE Bragg fiber designs in
Tables 1a and 1b, assuming a free space wavelength of 1 μm. The motivation
for our particular choice of cross sections is discussed below. The fiber A of Table 1a has an air core (n=1) of radius
1.3278 μm, followed by 16 pairs of alternating high-low index (n=1.49, thickness
0.2133 μm and n=1.17, thickness 0.346 μm), followed by an infinite medium of the
higher index (1.49). Fiber B is identical, except that the core radius is 1.8278μm.
Although the index contrast used is larger than conventionally obtainable, such ring
structures have been approximated by “holey” techniques where the low index rings
were approximated by rings of holes [8],
giving an average refractive index in the low index rings of about the desired
value.

The individual modes are characterized by two lengths, L_{1%}, at which the
transmitted power in that mode is reduced to 1%, and
L_{.01%}=2L_{1%}, at which the power is reduced to 0.01%. We
characterize each fiber as a whole by two lengths, L_{max} = L_{1%}
for the best guided mode, and L_{sm} = L_{.01%} for the second best
guided mode. We consider the fiber to be usefully single moded for lengths between
L_{sm} and L_{max}. Thus fiber A is usefully single moded for
lengths between about 400 μm and 26 m and fiber B is single moded for lengths
between about 2 cm and 400 m. We take into account only the second best guided mode
in calculating the onset of single modedness, as the remaining higher loss modes do
not contribute significantly. Modes omitted from Tables 1 are lossier than those shown.

In reference [5], the polarization dependence
of Fresnel reflection at a plane interface, which includes the Brewster angle
phenomenon [9], is presented as an
explanation for the tendency in Bragg fibers for the TE modes to be the best guided.
In this paper we go further by incorporating a Brewster angle condition into the
fiber design, and with higher refractive index contrasts than used in reference
[5], achieve a best guided mode that is
many orders of magnitude less lossy than any other mode. This use of the Brewster
angle condition appears to be favoured by having a low index core, and in line with
this we have chosen an air core, which lends itself to fabrication in the
rings-of-holes approximation to a Bragg fiber. Apart from these constraints related
to the Brewster angle condition (and apart from being physically reasonable) the
fiber designs are fairly arbitrary, with many remaining degrees of freedom for
further optimization. The *precise* values of the design parameters
used in this paper have no significance.

Consider a ray incident on a plane boundary between two transparent media. If the
angle of incidence is at the Brewster angle, i.e. if it is such that the reflected
ray is at right angles to the transmitted ray, the component of the incident light
with polarization parallel to the plane of incidence, analogous to TM radiation, is
*not reflected*. If in a planar multilayer stack with alternating
indices n_{1}, n_{2}, the Brewster angle condition holds at one
interface, it will hold at all interfaces. The condition that a TM wave be not
reflected at the interfaces of a curved concentric stack approaches the Brewster
condition for a plane stack at large radius. If *θ*
_{i}(i = 1, 2) is the angle of incidence in the layer with refractive index
*n*
_{i}, the Brewster condition is *θ*
_{1}+*θ*
_{2} = π/2. Snell’s law then gives *n*
_{1}sin*θ*
_{1} = *n*
_{2}sin*θ*
_{2} = *n*
_{2}cos*θ*
_{1} and then from the geometry of Fig.
3, ${n}_{1}^{2}$
*β*
^{2} = ${n}_{2}^{2}$(${n}_{1}^{2}$
*k*
^{2} – *β*
^{2}) or

For a given wavelength and a Bragg fiber with indices n_{1}, n_{2}
the n_{eff} of the best guided mode can be adjusted quite freely by varying
the thickness of the layers (and so the Bragg condition) and the size of the core.
For values of n_{eff} close to the Brewster condition (equation 1) Bragg reflection is undermined for
TM modes, and for hybrid modes, which have a component of TM polarization. In the
language of band gaps, the superposition of the Brewster and Bragg effects results
in a polarization dependent band gap; a band gap for TE but not for TM polarization.
The modes must satisfy a cavity resonance condition (a resonance with a width
corresponding to the loss). A single TE mode can be isolated by reducing the core
size so that all TE modes but one lie outside the range of *n*
_{eff} that corresponds to the Bragg condition. Increasing the core size
allows other TE modes to be better guided, leading to an increase in the length
required for single modedness. This is illustrated by comparison of fibers A and B,
which differ only in the core size.

A constraint on the value of n_{eff} is that it must be lower than all the
refractive indices that make up the fiber. This ensures that the fields are
oscillatory (and not evanescent) in the core and the surrounding layers, which is
necessary for Bragg reflection. Equation 1 ensures that n_{eff} is less than n_{1} and
n_{2}, so the remaining constraint is that it be less than the core
index. For a depressed core of index n_{co} the condition $\frac{{n}_{1}{n}_{2}}{\sqrt{{n}_{1}^{2}+{n}_{2}^{2}}}<{n}_{\mathit{co}}$ is equivalent to

which imposes the necessary condition

We present briefly, for an instructive contrast with the fibers of Table 1, another air core Bragg fiber [10]. This fiber has extremely low structural
loss for the best guided modes, but does not satisfy the Brewster condition, and has
poor contrast of lossiness, with L_{max} only about twice L_{sm}
(Table 2). While nominally single
moded for lengths greater than L_{sm} = 5330 km, non-structural losses would
supervene at much shorter lengths. By our criteria, the fiber is effectively
multimoded.

While this paper offers a very limited exploration of the Bragg profiles which take into account the Brewster condition, the examples show that the required high modal loss contrasts are attainable with its aid, and open up the prospect of truly effectively single mode optical fiber and optical fiber networks from which the problems associated with polarization have been eliminated. Optimization of the designs presented, including decreasing the overall loss, will be the subject of future publications.

## Acknowledgements

The authors thank Martijn de Sterke, Maryanne Large, Martijn van Eijkelenborg, John Canning and Simon Fleming for helpful discussions, especially concerning the meaning of singlemodedness for waveguides with no bound modes.

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