4.3 Characterization methods of active tapered fibers
4.3.3 Birefringence measurements
Polarization mode beat measurement
The most reliable and simplest way to measure the magnitude of birefringence in optical fiber is by measuring the polarization mode beating[72, 73].
Slow mode
Fast mode
Beat length
Figure 4.4 Evolution of state of polarization along PM-fiber, if a linearly polarized light was launched between fast and slow axes.
First of all, measurements require a broadband source, the spectrum of which does not overlap with the absorption spectrum of the active fiber. In the case of the ytterbium amplifier, the spontaneous emission spectrum of an erbium amplifier at 1.5μm is used.
Next, linearly polarized light must be introduced at a 45◦angle between the fast and slow axes of the fiber to excite both polarization modes of equal amplitude. The easiest way to do this is to load the fibers into a splicer that allows PM splicing.
After the excitation of two polarization modes, they propagate at different velocities
along the fast and slow axes. This means that the phase difference between the two polarization modes changes along with the fiber, and therefore the state of polarization also changes, as shown in Figure 4.4. At a distance equal to the so-called polarization beat length, the phase relationship between the two waves is restored, and the polarization state turns out to be the same as at the input of the fiber. Polarization beat length is
Lb= λ
Δn, (4.4)
where λ is the vacuum wavelength, and Δn is birefringence. The spectrum, meanwhile, remains unchanged at the fiber output (Fig. 4.5a).
Since different spectral components have different velocities in fiber, they acquire different phase shifts between the two polarization modes and have a different state of polarization. If a polarizer is installed at the fiber output, it transmits those wavelengths that have accumulated such a phase delay that the polarization state is linear and coincides with the polarizer axis. If other wavelengths have linear polarization orthogonal to the polarizer axis, they are completely blocked. The rest of the spectral components pass partially through the polarizer. As a result, the spectrum at the exit from the polarizer is modulated. By rotating the polarizer, the position can be found at which the polarization mode beating in the spectrum reaches its maximum amplitude (Fig. 4.5b).
Figure 4.5 a) Spontaneous emission spectrum of an erbium amplifier, b) spectral oscillations.
As can be seen, the envelope curve of the beat repeats the shape of the spontaneous emission spectrum. To normalize the beat spectrum, one can subtract the spontaneous emission spectrum from it. The resulting graph (Fig. 4.6) gives a
more accurate estimation of the polarization extinction ratio (PER) and oscillation periodΔλ:
Δλ= Lbλ
L , (4.5)
whereLis fiber length.
,QWHQVLW\G%
:DYHOHQJWKQP Figure 4.6 Spectrum of polarization mode beating after processing.
Finally, since fiber lengthLand oscillation periodΔλare known, birefringence Δncan be calculated as follows:
Δn= λ2
ΔλL, (4.6)
Typically, PM T-DCF has a birefringence of the order of 10-4[P1]. This method works well with fibers several meters long, which have a large birefringence greater than 10-5. If the birefringence is less, then the spectrum is not wide enough to observe the oscillations. Generally speaking, measuring low birefringence in short fibers is not a trivial task, because many other techniques (e.g., twist method[74], the Lyot-Sagnac interferometer[75], the polarization-sensitive optical time domain reflectometry (OTDR)[76]and the Brillouin OTDR[77]) require long fibers.
Birefringence calculation using polarization eigenstates
Taking certain assumptions, the birefringence calculation can be performed if the
polarization eigenstates of the fiber are known. These calculations assume that linear birefringence exists only due to the macro bending of the fiber. Therefore, it is necessary to arrange the fiber so that it is coiled with a large diameter to reduce birefringence caused by macro bending. However, since the length of the tapered fiber is usually several meters, the final bending and twisting always exist in the actual implementation of the experimental setup. Therefore, both linear and circular birefringence of small magnitude exist. Thus, theJe x p matrix should include both linear and circular birefringence components and have the following form:
J=exp(iφ)
⎡
⎣cosεL2 +iβεsinεL2 −α+iγε sinεL2
α−iγε sinεL2 cosεL2 −iβε sinεL2
⎤
⎦, (4.7)
Hereε2=α2+β2+γ2is elliptical birefringence,αis circular birefringence,β is linear birefringence, andγis related to the linear part whose eigenaxes are parallel to the bisectors of x and y.γcan be neglected if it is much smaller than other matrix elements. Knowing the real Jones matrix of the fiber (Eq. 3) and assuming that it has a form of Eq. 4.7, it is possible to find theαandβpairs (Fig. 4.7). Except for the coefficient written in front of the matrix, which represents the absolute phase, the matrixJ is periodic[78]. This means that the number of pairsαandβis infinite.
In other words, the Jones matrix only determines the relationship between different types of birefringence, but not their absolute values.
Figure 4.7 Poincare sphere. Orange, green, and blue arrows indicate birefringence and its components.
Since the ratio betweenαandβis calculated, it is sufficient to find the absolute value of at least one type of birefringence. If the cross-sectional radius r of the fiber and the bend radiusRare known, the linear birefringenceβcan be found as follows:
β=0.5Cs r2
R2 (4.8)
whereCs≈2.02×106m-1for fused silica[79]. The radius of the tapered fiber changes with length, so the linear birefringence can be calculated as a function of fiber length.