>> The Birefringence of Single Mode Fiber
Circular core fibers whose axes are straight are not birefringent – that is, the two orthogonally polarized LP01 mode have the same effective indices. Bending such a fiber introduces stresses in the fiber and makes the fiber linearly birefringent with the fast and slow axes in the plane and perpendicular to the plane of the loop, respectively. The bending-induced birefringence of a single mode silica fiber is given by:
where nex and ney represent the effective indices of the LP01 modes polarized in the plane and perpendicular to the plane of the bend, respectively, b is the outer radius of the fiber, R is the radius of the loop, and C is a constant that depends on the fiber material and the elastooptic properties of the fiber. For silica fibers, C is about 0.133 at 633nm.
The above equation tells us that the smaller the loop radius R, the larger is the birefringence. Note that any bending will also introduce attenuation and, hence, very small bend radii are not very practical.
Let’s look at some examples.
> Example A
Let’s consider a silica fiber of outer radius b = 62.5 um bent into a circular loop of radius 30 mm. The birefringence of the fiber at 633nm is then
which is indeed very small compared with the core-cladding indices difference.
Although the induced birefringence is very small, by having the two polarizations propagate over a long fiber length, one can obtain large phase shifts. Thus, if the fiber is coiled around N loops of radius R, then the bend-induced phase difference between the two polarization is
Substituting for Δneff from the previous equation, we obtain
where we have disregarded an unimportant negative sign. For achieving phase differences of π (corresponding to a half-wave plate) or π/2 (corresponding to a quarter-wave plate), we must have
> Example B
For simulating a quarter-wave plate at λ = 633 nm, using bend-induced birefringence, if we have a single loop (N=1), then
Using the same loop radius of 2.1 cm, we can simulate a half-wave retardation plate using two loops (N=2). Bend-induced linear birefringence can be used to build in-line polarization controllers as shown in the following section.
>> In-line Fiber Polarization Controllers
The following figure shows an in-line fiber optic polarization controller that utilizes bend-induced birefringence. It consists of three fiber birefringence components; the first and the last are quarter-wave retarders and the central one is a half-wave retarder. The bent fibers are fixed at points marked A, B, C, and D. The three fiber loops are free to rotate as shown.
A rotation of each of the loops will rotate the principle axes of the birefringent fiber sections with respect to the input polarization state. This is analogous to rotation of a classical bulk half-wave or a quarter-wave plate with respect to the incident light. Thus, rotation of the three loops is equivalent to the rotations of a combination of a λ/4, λ/2, and λ/4 plate. One can show that with this combination, any input polarization state can be transformed to any other output polarization state.
The polarization controller described above is used in many applications such as in fiber optic sensors where control of the state of polarization of a the light propagating through the fiber is required.
Polarization controllers operating over a wavelength range of 1250 – 1600 nm with optical insertion loss variations of less than 0.004 dB over the band are commercially available. Such polarization controllers are extremely important components in the measurement of polarization dependence of optical devices such as optical isolators, EDFAs, etc..