The measurement of birefringence in optical fibers and photonic devices is of considerable interest. Applications range from measurements of polarization mode dispersion (PMD) for optical fiber telecommunication systems to fiber optic sensors. Commercial instrumentation has been developed by manufacturers for PMD measurements and the characterization of polarization (optical anisotropy) in fibers and fiber components [1
Polarization Measurements of Signals and Components, Product Note 8509-1, Agilent Technologies.
]. These measuring techniques are complex and can suffer from a lack of accuracy in cases in which the birefringence is very small and/or the length of the fiber or fiber component is short.
There is also a number of optical fiber sensors that are based upon measuring changes in refractive index, or in the relative change in refractive indices associated with two distinct modes of polarization. Strain sensors and polarimetric control of fiber laser sensors have been previously demonstrated [2–4
G. Ball, G. Meltz, and W. Morey, “Polarimetric heterodyning Bragg-grating fiber-laser sensor,” Opt. Lett.
18, 1976–1978 (1993) [CrossRef] [PubMed]
], and these techniques may be employed for a sensitive measurement of changes in lasing wavelength [5
J. Hernandez-Cordero, V.A. Kozlov, and T.F. Morse, “Highly accurate method for single-mode fiber laser
wavelength measurement,” Photon. Technol. Lett. , Vol. 14, 83–85 (2002) [CrossRef]
]. A further important application of PMB may be found in the field of biological and chemical sensing. By functionalizing the surface of a silica microsphere with an antigen and measuring the resonance frequency when an
antibody attaches to the antigen, the refractive index of the material coating of the microsphere changes along with the resonance frequency. Thus, an extremely small change in the refractive index of a whispering gallery sphere cavity [6
S. Arnold, M. Khoshsima, I. Teraoka, S. Holler, and F. Vollmer, “Shift of whispering-gallery modes in microspheres by protein adsorption,” Opt. Lett.
28, 272–274 (2003) [CrossRef] [PubMed]
] can be used to detect the presence of biologically active molecules that attach to a functionalized surface [7
F. Vollmer, S. Arnold, D. Braun, I. Teraoka, and A. Libchaber, “Multiplexed DNA Quantification by Spectroscopic Shift of Two Microsphere Cavities,” Biophys. J.
85, 1974–1979 (2003) [CrossRef] [PubMed]
In the following, we present a new technique for the measurement of optical anisotropy that is based on longitudinal mode beating initiated by the insertion of a birefringent element within the laser cavity. This can be applied to high accuracy birefringence measurements of optical fibers and photonic devices. The experimental results obtained with this technique are shown in good agreement when compared to measurements made using a polarization analyzer.
2. Intra-cavity technique for measuring polarization anisotropy
It is well known that interference between longitudinal modes in a laser cavity produces longitudinal mode beating (LMB) signals. The frequencies of these LMB signals can be readily observed in the frequency domain using a radio frequency spectrum analyzer (RFSA). If birefringence exists within the laser cavity, two orthogonal sets of longitudinal modes (x
polarization) can be produced in the laser system, [8
M.J.F. Digonnet, Rare-Earth-Doped Fiber Lasers and Amplifiers, 2nd ed. (Marcel Dekker, Inc, New York, 2001), p. 158.
] and, because they are orthogonal, these two sets of modes will not beat with each other. At the output of the laser, if the radiation passes through a linear polarizer oriented at 45° with respect to the birefringence axes, the independent x
polarization longitudinal modes will be projected onto a common axis, and they will “beat” against one another producing a new set of signals. This polarization mode beating (PMB), along with the self LMB in the x
directions, can be clearly observed in the frequency domain by using an RFSA. We will show how the x-y PMB frequencies may be related to the birefringence in the laser cavity.
Fig. 1. Experimental setup (WDM: wavelength division multiplexer, FBG: fiber Bragg grating, DUT: device under test, RFSA: radio frequency spectrum analyzer, IR: infrared, OSA: optical spectrum analyzer).
Referring to Fig. 1
, the principle of this measurement may be explained in the following manner. The resonance condition in the laser cavity presented above may be written as:
m is the laser resonance frequency, l
1 is the length of the gain section in the laser cavity, l
2 is the length of the device under test (DUT) inserted within the cavity, and l
1 + l
2 ≡ L the total cavity length. The refractive index of the gain section within the laser cavity is n1
, which in the present case is erbium doped fiber, and the refractive index of the DUT is n2
. The longitudinal mode index (or the mode order number) of the laser oscillation is given by m, and c is the speed of light in vacuum.
Within the laser cavity, we assume (for now) that the gain section of the cavity l
1 has no birefringence and the anisotropy of the entire cavity is due to the birefringence of the DUT. Thus, the refractive index of l
1 is n
1, and the refractive indices of the DUT can be characterized as n2x
, where x and y correspond to the two orthogonal polarization directions. Then, the laser resonance frequencies for the x and y polarizations are given as follows:
After the output light passes through a 45° polarizer, the x-y mode beating frequencies (PMB frequencies) can be expressed as
In general, there will be many longitudinal modes in the laser cavity. Indeed, the exact longitudinal mode indices for x and y polarizations (i.e., m and k) are unknown. However, this information is essential to relate the PMB to the refractive index anisotropy (Δn) that is responsible for the PMB. Therefore, even if we can accurately measure the PMB, we still do not know the relation between x-y beating frequency and the optical anisotropy.
This difficulty can be circumvented in the following manner. As seen in Fig. 1
, laser tuning is easily achieved upon adjusting the resonance wavelength of the fiber Bragg grating (FBG). During tuning, the observed PMB will shift due to the change of the mode order numbers. For example, at wavelength λ1
, the PMB frequency (Δv
) can be expressed as
and at wavelength λ2 , the PMB frequency will be
The difference in PMB frequencies at λ1 and λ2 is Δv
Because tuning of the laser wavelength can be made to be very small, we assume that the difference of the change in mode index integer in the x polarization is the same as the change in the y polarization mode, i.e.,
We thus define an integer h that is approximated as
Therefore, the equation for the PMB frequency difference that occurs at two wavelengths can be simply written as
The refractive index anisotropy can therefore be obtained as
Thus, the birefringence in the fiber cavity is directly proportional to the magnitude of the slope of the PMB with respect to wavelength. As will be described below, the absolute value in Eq. (11
) is a consequence of the fact that the slope can be positive or negative. Notice that only linear birefringence has been considered for the analysis. In general, the fiber cavity will have elliptical birefringence and matrix analysis would be therefore required. However, unless special devices are used within the fiber cavity, linear birefringence typically dominates over other types of birefringence. Thus, we have obtained a first approximation to calculate approximately the anisotropy within a fiber laser cavity.
As shown above, by measuring the PMB frequencies as a function of wavelength the
anisotropy in the fiber cavity may be immediately obtained. In the measuring process, the
fiber or fiber component whose birefringence is to be measured may be spliced or butt-coupled to an active fiber. Ideally, this active fiber will have birefringence much lower than that of the birefringence of the DUT. The active fiber plus the DUT will then compose a single laser cavity.
3. Experimental results
In our experimental arrangement, a 978 nm LD was used as the pump source, which was
spliced into a WDM coupler. The center wavelength of the FBG was at 1540.84nm, with a
peak reflectivity of ~97.5%, and a FWHM of ~0.1nm. The FBG was glued at the midpoint of an aluminum bar: length ~26.5cm, width ~5.5cm, thickness ~0.2cm. The bar was supported on both ends and weights in increments of 94g were placed at the center of the plate to cause a deformation of the plate. The FBG glued to the bar was stretched by the addition of these weights, and tuning was approximately 3.14 × 10-4 nm/g. Erbium doped fiber was used as the gain section which was spliced with the DUT.
Experiments were carried out using a Hi-Bi polarization maintaining fiber and a one
meter long chirped fiber grating. The other output of the WDM (see Fig. 1
) was sent to an optical spectrum analyzer (OSA) to monitor the lasing wavelength. A dichroic filter was used to eliminate the residual pump at 978 nm prior to the laser emission being detected by a 1GHz bandwidth InGaAs photodetector. Signals from the detector were sent to an radio frequency spectrum analyzer that displayed the LMB and PMB frequencies. A typical radio frequency spectrum is shown in Fig. 2
, where the first two peaks correspond to the PMB signals (PMBA
respectively) and the third peak is the first LMB frequency.
Fig. 2. Radio frequency spectrum of the fiber laser showing the longitudinal mode beating and
the polarization mode beating (PMB) frequencies.
Fig. 3. Theoretical laser mode arrangement as a function of frequency (v). As the fiber laser
is tuned from Lambda 1 to Lambda 2 different modes with different modal indices will generate the beat frequencies.
To further illustrate how tuning affects PMB signals, we consider a laser with orthogonal longitudinal modes arranged as shown in Fig. 3
, where the Bragg grating determines the lasing wavelength. There can be many longitudinal modes that can simultaneously oscillate as a consequence of the modal separation and the width of the Bragg grating. The dotted lines in Fig. 3
represent the bandwidth to which the Bragg grating has been stretched, so that a different set of longitudinal modes are permitted to oscillate at different wavelengths. As an example, PMBA Fig. 2
could originate from the beating between Xm+1
polarization longitudinal mode with mode number m
+1 and y
polarization longitudinal mode with mode number n
+1), and PMBB
could arise from beating between Xm+2
. LMB results from the beating between Xm+1
3.1 Measurements of refractive index anisotropy in polarization maintaining Fiber
We first examine three different lengths of Hi-Bi polarization maintaining (PM) fiber. The lengths were 0.45m, 2.60m, and 4.06m. The PM fiber measurements are presented in Fig. 4
and Eq. 11
was used to obtain the refractive index anisotropy (Δn) from the slope of the PMB changes as a function of wavelength. Using a straight line fit for the measured data, we obtain a slope of 5.027MHz/nm for the 0.45m section, 13.165MHz/nm for the 2.60m section, and 13.322MHz/nm for the 4.06m section, which results in an overall average value of Δn = 3.55±0.97x10-4
. The difference in the sign of the slopes in Fig. 4
may appear to be an anomaly. However, we must realize that tuning from λ1
in Fig. 3
results in a positive PMB vs wavelength slope, whereas the tuning from λ3
gives a negative PMB vs wavelength slope.
To evaluate the results using our technique as described above, we used a polarization
analyzer to characterize two sections of the PM fiber. The polarization analyzer gives
information about the differential group delay (DGD, in ps), which is related to birefringence as
The lengths of the fibers measured were 0.96m and 2.15m, and the results of the DGD as
measured with the polarization analyzer are presented in Fig. 4d
. These measurements result in an average value of birefringence of Δn = 3.86±0.27x10-4
. The average birefringence values obtained with the intra-cavity technique and the polarization analyzer show good agreement; Table 1
and Table 2
summarize and compare the data obtained with both methods.
Table1. Intra-cavity fiber laser measurement on three sections of PM fiber.
1 (m)||slope (MHz/nm)||Δn|
Table 2. Polarization analyzer measurement on two sections of the PM fiber.
L (m)||Δτ (ps)||Δn|
Fig. 4. Polarization mode beating (PMB) frequencies as a function of wavelength for different lengths of Newport PM fiber ((a) 45cm, (b) 2.6m, (c) 4.06m). The differential group delay (DGD) for two lengths of the same fiber measured with the HP 8509B is shown in (d).
3.2 Measurement of refractive index anisotropy in a chirped fiber grating
The second set of measurements were made on a 1m-long chirped Bragg grating (the total
length of the device is 3.5 m with pigtails). An optical anisotropy in these devices is generally created during the “writing” process with UV laser radiation, and for a typical Bragg grating this is of the order of Δn ≈ 10-5
. Measurements on this fiber device were repeated several times in order to obtain an average value and evaluate the divergence of the results. Figure 5
shows the change in PMB frequencies as a function of wavelength for the chirped grating. A linear fit to the data yields an average slope of 0.6663±0.06 MHz/nm, which, at a wavelength of 1.54 μm, corresponds to a birefringence of Δn =(1.7±0.1)x10-5
, including the birefringence from the gain fiber. In order to take into account the birefringence associated with the gain fiber itself, we made a measurement of the laser cavity using only the gain fiber, and without the chirped grating. These results are presented in Fig. 6
. From this we calculated the value of Δn for the gain fiber to be approximately 1.5x10-5
. Thus, the difference in birefringence between the data with the chirped grating and without the chirped grating is approximately 2x10-6
. The accuracy of this measurement was limited by the fact that the orientation of the axes of birefringence of the gain fiber and the DUT was unknown; hence the birefringence axes for the two fibers were not aligned. Furthermore, as explained in the previous section, our analysis does not consider elliptical birefringence, which in this case can affect the accuracy as well.
Fig. 5. Polarization mode beating (PMB) frequency vs. wavelength for chirped fiber grating
Fig. 6. Polarization mode beating (PMB) frequency vs. wavelength for laser cavity without a
device under test (DUT)
A comparable measurement of differential group delay (DGD) can be obtained by using
the polarization analyzer. These results are shown in Fig. 7
. The noise in this measurement is significant, because the polarization anisotropy (Δn) is two orders of magnitude smaller than in the preceding example. The average DGD is 0.013 ± 0.005ps, which, for a length of 1m corresponds to Δn = 3.9±1.5x10-6
. Thus, the measurements obtained from the two methods are in relatively good agreement, and, with further improvement in the knowledge of the overall birefringence of the fiber cavity, they could be improved.
As a first approximation, the proposed method seems to yield good results for measuring
the anisotropy of a fiber device placed within a fiber laser cavity. However, the accuracy is limited by the knowledge of the intra-cavity birefringence. In order to improve the
performance of our method, a more detailed analysis of the laser cavity is required. In
particular, matrix analysis could include the effects of elliptical birefringence within the laser cavity. More general and realistic cases such as misalignments between the birefringence ellipses of both, the gain fiber and the DUT, could be analyzed. Due to the birefringence ellipses, variations on the alignment of the birefringent axes will yield a range for the anisotropy as measured with our method. As an example, the measured anisotropy when the fast axes of the DUT and the gain fiber coincide with each other should be different from that obtained if both axes are orthogonal to each other. Thus, matrix analysis could provide a way of relating the birefringence of the gain fiber, the DUT and the two together yielding more accurate results.
Fig. 7. Differential group delay (DGD, ps) vs. wavelength for chirped fiber grating
Although there are commercial devices yielding better accuracy than our method, most of
them are based on measurement techniques requiring widely tunable sources. As demonstrated in our experiments, the intra-cavity technique requires small tuning ranges in order to yield information regarding anisotropy of a DUT. While further and more detailed analysis is required in order to achieve better accuracy, the fact that small optical bandwidths are required with the intra-cavity technique could be of interest for developing measurement systems based on this method.