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We describe the design, construction, calibration, and validation of a Stokes vector polarimeter for investigating the polarization characteristics of fiber optic gyroscope coils. The device measures the complete Stokes vector, and reports conventional polarization properties including the Degree of Polarization (DoP), the orientation and Degree of Linear Polarization (DoLP), and the handedness and Degree of Circular Polarization (DoCP). The sensor operates at 1550 nm and employs a division of aperture optical architecture to acquire full Stokes vectors at 8 kHz while calculating polarization properties at a rate of 200 Hz. Preliminary measurements performed on both traditionally and unconventionally wound gyroscope coils are also presented.
©2009 Optical Society of America
1. Introduction
Knowledge of the polarization properties of an optical fiber is important in fiber based applications where the emitted polarization state directly affects subsequent optical components, detection schemes, and data processing algorithms [1,2]. The fields of telecommunications, medicine, and many military applications all employ fiber based components and instruments which rely on polarization to perform accurately. Certain applications use relatively expensive polarization maintaining (PM) and, more recently, photonic crystal PM fiber to insure control over the emitted polarization state [3,4]. Recent investigations into the use of spirally coiled single mode (SM) fiber in a so-called depolarized fiber optic gyroscope (FOG) architecture [5] has shown that SM fiber can be an effective and inexpensive PM fiber surrogate. These investigations led to the development of a polarimeter to fully characterize these and more general fiber components [6].
The most thorough way to characterize the polarization properties of a fiber component is to measure the exiting light’s Stokes vector [7]. The Stokes vector has been well described and contains four elements obtained by making (at least) four measurements of the emitted light after passing it through unique polarizing filters. The data set may be acquired serially by changing the polarizing filter between measurements, or simultaneously using multiple filters and detectors. Regardless of the chosen technique, all of the light’s polarization properties may be extracted from the Stokes vector including the total Degree of Polarization (DOP), the magnitude and orientation of the linear component (DoLP and θ, respectively), and the magnitude and handedness of the circular component (DoCP and its sign, respectively). The described polarimeter measures the Stokes vector components and computes the derived polarization properties for any fiber based component operating at 1550 nm. These properties may then be analyzed to assess the applicability of the component under test for a polarization sensitive application. An example investigated here is the polarimetric characterization of spirally wound FOG coils designed to mimic the behavior of PM fiber based coils.
Section 2 presents a review of the design and construction of the fiber optic polarimeter. Section 3 describes the requisite radiometric and polarimetric calibration procedures. Section 4 highlights validation measurements aimed at ascertaining the device sensitivity to polarization metrics. Section 5 reports on preliminary FOG coil measurements and Section 6 presents a summary of the work.
2. Polarimeter Design and Construction
The polarimeter was designed to characterize fiber based components with standard FC termination operating at a wavelength of 1550 nm. The target application required a polarimetric sampling rate of 200 Hz which is well suited to a multiple channel, simultaneous acquisition design. The so called division of aperture (DoAp) optical architecture was chosen because it easily configurable to the known light input parameters such as the wavelength, mode field diameter, and numerical aperture [8,9]. In this approach, the optical pupil plane is sub-sectioned using an array of mini-lenses to create four discrete optical channels for processing. Three of the four channels contain linear polarizers oriented at 0°, 60°, and 120° with respect to the positive horizontal direction when looking toward the source. Channel four contains a ¼ wave retarder followed by a linear polarizer whose transmission axis is oriented 45° relative to the retarder’s fast axis. Taken together, these components form a circular polarizer although here they are operated in “reverse” with the retarder preceding the polarizer. In this way, a circular analyzer is formed which measures the circular polarization component of the incident light. Figure 1(a) shows a schematic of the sensor’s optical layout, which includes the light source and a (pseudo)collimating lens that produces a very slowly converging beam. This slow convergence permits the creation of an accessible pupil plane for the array lenses, assuring each channel detector “sees” the entire fiber output face. The figure only shows three of the four channels since, in this view, the middle channel actually lies “outside” the page toward the reader and obscures the fourth channel which lies “inside” the page. Figure 1(b) shows an isometric cross section of the back end of the system including the polarization filters, array lenses, and detectors along with their mechanical mounting components. The optical design parameters may be easily reconfigured for other wavelengths or fiber output specifications as needed.
Fig 1 Top: Optical ray trace of fiber optic polarimeter. Bottom: Opto-mechanical details of the polarization filters, array lenses, and detectors.
The optical system design was first laid out paraxially and then optimized using Zemax. The collimating and mini-lenses are plano-convex and, due to the slow nature of the optical layout, are capable of producing near diffraction limited imagery at the detector plane. All four channel polarizers are near infrared (NIR) dichroic sheet and the retarder is a zero order quartz plate whose fast axis alignment to its companion polarizer was confirmed to within 3° during the calibration procedure described below. The detectors are low noise InGaAs photodiodes operating in photoconductive mode. Each detector current is transimpedence amplified, digitized (16 bits) at 100 kHz, and fed to a single board computer (SBC) containing four digital input channels. The SBC, housed within the polarimeter enclosure, averages 256 samples per channel and transmits the results via Ethernet to a control laptop computer running the polarization data reduction software. This software has been described [9] and, in brief, applies the polarimetric data reduction matrix obtained during calibration to the four measured averages to yield the current Stokes vector. From the Stokes vector, the DoP, DoLP and θ, and DoCP are reported at a user selectable rate of up to 200 Hz. The software also reports raw channel voltages (in counts) and the normalized Stokes vector elements which are helpful during alignment, testing, calibration, and validation of the sensor. All parameters are available numerically and graphically using a set of four configurable virtual strip charts. Additionally, polarization information may be viewed graphically via an available Poincarè sphere [10] animation, although this added rendering reduces the polarimetric refresh rate to a maximum of 70 Hz. Automatic file logging at user specified intervals is also available.
3. Calibration
Much has been published on the calibration of polarimeters, although most reports are quite specific to a particular sensor and or application. In all cases however, calibration consists of both radiometric and polarimetric correction for the inherent device signatures of each required measurement (i.e., each channel voltage in this case). Here, an unpolarized superluminescent diode operating at 1550 ±40 nm served as the source. The unpolarized nature of the source was verified by observing minimal (<1%) modulation as the output was passed through a rotating linear polarizer and monitored with a standard radiometer. For the radiometric portion, individual channel noise floors or offsets are first obtained with the source turned off. The source power is then adjusted until the detectors reach ≈90% of their peak values to obtain the individual channel gains. The radiometric offset and gain of each channel are used to normalize subsequent measurements for polarimetric processing. For the polarimetric calibration, the channel signatures are measured using a set of known polarization input states, from which the system data reduction matrix is calculated. Known inputs were obtained by inserting a piece of NIR sheet polarizer between the fiber and the collimating lens and rotating it with 0.01° accuracy over 360° in 5° increments. The radiometrically corrected (i.e., normalized) modulation observed on the four channels is shown in Fig. 2 . The three linear channels are expected to exhibit sinusoidal modulations of the same magnitude and to be shifted in relative phase by 60° (or 120°), corresponding to the angular difference of the channel polarizers’ transmission axes. Previous reports indicate the principal transmittances of the employed NIR sheet to be ≈86% and ≈0.006% at 1550 nm [11]. The concatenation of the calibration polarizer with any of the channels containing only a linear polarizer results in a maximum transmittance of (0.86)2 = 0.74 and a negligible minimum transmittance. Channels 1-3 in the figure exhibit this behavior, with any slight deviations being indicative of the actual polarizer alignment and transmission. For this set of linear input states, the circular channel transmittance should remain constant at half the peak values observed on the linear channels (i.e., ≈ 37%) but is seen to exhibit a weak modulation. The source of this modulation is a combination of the slight deviation of the angle between the polarizer and retarder axes from the ideal -45° position, coupled with the fact that the retarder is not exactly ¼ wave. It is shown below that the required values for these parameters to yield the exhibited modulation is -45.45° and 92.3° respectively. The modulation curves shown in Fig. 2, including their deviations from the ideal curves, represent the inherent polarimetric signature of the sensor.
The data of Fig. 2 can be used to obtain the so-called polarimetric data reduction matrix, which is applied to all subsequent measurements to calculate the input light’s Stokes vector. A number of ways to do so have been reported and a technique based on nonlinear regression has been used here [12,13]. In this approach, the calibration modulation curves are fit using the Levenberg-Marquardt technique with the Stokes-Muller algebra providing the model function for fitting [14]. Concentrating first on the linear channels, the measured power on any particular channel is a function of the Stokes vector of the incident light and the employed polarizers’ principle transmittances and orientations and may be written as
where M(q, r, θi) is the Mueller matrix of a linear polarizer with principal transmittances q and r, oriented at θi. M cal(q, r, θcal)) is the calibration polarizer’s Mueller matrix and, since the same material was used for the calibration polarizer as the channel polarizers, q and r are the same as for M. S in is the unpolarized input Stokes vector and the [[1]] notation indicates the first element of the vector obtained on the right hand side. The Mueller matrices and Stokes vector of Eq. (1) are well known and have been described elsewhere [15]. A similar approach can be performed on the circular channel in which case the model function takes the formHere, M ret(δ, θret) is the Mueller matrix of a linear retarder with retardance δ whose fast axis is oriented along θret. Performing the algebra leads to the following model equations for the 3 linear channels;
where i = {1,2,3} and {θ1 = 0°, θ2 = 60°, θ3 = 120°}. The model function for the circular channel is given by;The data of Fig. 2 are fit to the models of Eqs. (3) and (4) using q, r, θi, δ, and θret as the fit parameters. The numerical nature of the fitting procedure requires seed values for the parameters, which were obtained from component specification sheets and/or the measured component properties (e.g., principal transmittances and orientations). For the linear channels, seed values of q = 0.86, r = 0.006, and θi = 0°, 60°, 120°, were used and the extra circular channel parameters were seeded using δ = 90° and θret = -45° (i.e., 135°). Table 1 contains the regressed parameter values, obtained separately for each channel. Ideally, the regressed q and r values would be constant across all channels and consecutive linear channel orientations would be spaced by 60°. As well, the regressed retardance and fast axis orientation in channel 4 should be 90° and 135° respectively, based on manufacturer’s specifications. Table 1 shows that all of the regressed values are quite close to the seed values and it was found that the fitting procedure was relatively insensitive to the seed values, consistently converging to the same set of regressed parameter values. Importantly, q and r are practically constant over the linear channels and the slight deviations of the regressed values from the seed values represent the inherent polarimetric signature which the calibration procedure strives to correct.
Table 1. Regressed parameter values after data fitting
Note that the fitting procedure is performed across the entire 60 element data set so that regressing only 5 parameters results in an overspecified system of equations. Indeed only 5 power measurements are sufficient to obtain the regression values but the overspecification technique has been found to improve the general accuracy of the sensor. Analysis has shown that using the minimum number of required calibration states to obtain the data reduction matrix subsequently yields very accurate Stokes vector measurements of input states corresponding to those used during the calibration but measurements of intermediate (i.e., non-calibration) states exhibited reduced accuracy. Employing the entire calibration data set in the regression has the effect of only slightly reducing the measurement accuracy of the states used during calibration, while greatly improving it for arbitrary inputs.
Once the regressed values are known, they are plugged into the model equations and θcal is allowed to range from 0° to 360°. These modulation curves are shown in Fig. 3 along with the calibration data of Fig. 2. It is seen that the fit curves quite accurately match the data and thus the regressed parameter values can be relied upon to generate an accurate data reduction matrix (DRM) for the sensor.
Fig. 3 The data of Fig. 2 (symbols) overlain with the fits of Eq. (3) using the regressed polarization parameters of Table 1 (curves).
As an interim step to obtaining the system data reduction matrix, the so-called calibration matrix is obtained by modeling each channel using only the Mueller matrices of the polarization element(s) contained therein, with the regressed parameter values plugged in as required. The first rows of each of the 4 resulting matrices, called the analyzer vectors, are extracted and concatenated into the 4x4 calibration matrix. It can be shown that each channel’s analyzer vector is simply the Stokes vector of the eigenpolarization of that channel. Figure 4 shows the measured normalized calibration matrix as well as the ideal normalized calibration matrix (assuming perfect polarizing elements) for the channel arrangements described. There is an overall good agreement between the two with slight deviations that are attributable to the slight rotational misalignments of the channel polarization elements, measurement noise, and/or numerical round off errors that propagate through the polarimetric calculations.
As a final step, the calibration matrix is inverted to yield the DRM which will inherently correct for the channel polarization signatures. The DRM is subsequently multiplied by the radiometrically normalized four element measurement vector (P1, P2, P3, P4) † to yield the input light’s Stokes vector, where † indicates transpose.
4. Validation and Testing
Two measurements were performed to assess the instrument’s validity, one aimed at quantifying its accuracy while the other was designed to determine its repeatability. In the first case, a measurement that mimicked the calibration measurement was performed. Again, 60 linear states oriented in 5° increments were introduced and the DRM obtained from the calibration procedure was applied to the measurement vectors. The resulting Stokes vectors were then processed into the polarization products including the degree of polarization (DoP), degree of linear polarization (DoLP), degree of circular polarization (DoCP), and the orientation of the linear component (θ). Using the conventional Stokes vector notation, S = (s0, s1, s2, s3)†, these polarization products have been described extensively [12] and are given by
Based on these definitions, the measured polarization products as the input polarizer is rotated 360° are shown in Fig. 5 . For such a measurement, the DoP and DoLP are ideally 100% while the DoCP should be 0% and the orientation of the linear component should vary linearly throughout the 360° rotation. The first three properties are found in Fig. 5 to behave as expected to within 2.5% with the largest error observed in the DoCP. Averaging over the entire data set yields mean DoP, DoLP, and DoCP values of 99.6%, 99.59%, and 0.7% respectively. To better reveal the orientation accuracy, the figure shows the orientation residuals, defined as the measured orientation minus the known orientation and represented as Δθ. The polarizer was mounted in a high resolution rotation stage (specified accuracy of 0.001°) which provided the known value. The figure shows orientation residuals within an envelope of ±0.325° across the entire measurement with the average residual of 0.18°. The slight deviations between the polarization products and the ideal results are primarily attributable to measurement noise, since this is the only signal contribution that differs from the calibration measurement used to derive the DRM.
Fig. 5 Measured polarization properties for a rotating linearly polarized input. Top: DoP and DoLP. Bottom: DoCP and orientation residuals.
Repeatability testing consisted of performing multiple measurements of the same input which is again due to a rotating linear polarizer in the beam. Three measurements were made consecutively within a 20 minute time period and the Stokes vector and polarization product means and standard deviations (SDs) calculated. Figure 6 shows the polarization products’ mean values across the measurements, with error bars defined as ± 1 SD. The figure shows accuracies similar to those of Fig. 5 with very small error bars, indicating high repeatability.
Fig. 6 Polarization products' accuracy and repeatability. Accuracy over 3 identical measurements is similar to that of Fig. 5, with excellent repeatability (typically < ±0.15%).
As a final evaluation of the sensor, a validation measurement was performed where the input beam was first passed through a fixed linear polarizer and then through a rotating wave plate of approximately ¼ wave retardance. The wave plate was specified as ½ wave of retardance at 860 nm which, accounting for the spectral dispersion [16], corresponds to 95.3° of retardance at 1550 nm. The list plots shown in Fig. 7 display the measured DoP, DoLP, and DoCP as the retarder is rotated through one full cycle. The polarizer-retarder pair is nominally nondepolarizing so that the DoP should ideally be 100% across the range. Slight variations are observed, although most of the data lie well within ±2% of this ideal value. The DoLP and DoCP plots of Fig. 7 also include solid curves displaying the theoretical values for the nominal arrangement and they are seen to correlate very well to the observed modulations. Taken together, these two plots show how the polarization state evolves from purely linear, through elliptical, to almost purely circular during the first 45° of the revolution, after which it returns to the orthogonal linear state during the next 45°. The pattern is repeated 4 times with good symmetry throughout the 360° revolution of the retarder.
Fig. 7 Polarization products of a linear polarizer followed by a rotating 95.3° waveplate. Note the sign of the DoCP indicates polarization handedness.
5. Fiber optic gyroscope coil measurements
The polarimeter was next applied in the diagnosis of the polarization properties of a number of FOG coils. The studied components included a traditional FOG coil containing 350 m of SM fiber wound on a 1.2” diameter by 2” high mandrel, and two crossover-free (CF) coils [17] containing 120 m and 360 m of SM fiber wound into 8 and 24 layers respectively. Each layer of the CF coils consists of a tightly spaced Archimedean spiral of fiber on a thin but rigid substrate, with the layers stacked to prevent so-called crossovers. Crossovers are defined here as points where a section of wound fiber passes over a previously wound section such that a localized stress point (i.e., retardance) is created in both sections. Traditionally wound coils contain innumerable crossovers and the combined effect of the stress points is an unpredictable overall coil birefringence. The CF coil is designed to minimize this effect while also imparting consistently oriented birefringence axes aligned normal to and parallel with the coil substrate. A 2 m length of polarization maintaining (PM) fiber, whose polarization properties are readily calculated [18], was also examined as a reference.
Figure 8 is a schematic of the test arrangement showing a fiber pigtailed 1550 nm amplified spontaneous emission source with an 80 nm wide bandwidth whose output goes through a free space coupler containing a linear polarizer that could be rotated about the beam axis. The polarized output is injected into the coil under test and the output is monitored as the polarizer is rotated at a constant angular velocity.
Figure 9 displays the polarization properties of the light exiting the traditional coil. The coil is seen to marginally retain the DoP, reaching maxima of about 52% and 48% when its eigenpolarizations are injected. The DoCP oscillates about 0% with a range of approximately ± 40% while the DoLP ranges from 30% - 50%. These behaviors imply the presence of both retardance and depolarization in the coil, presumably arising from the multiple crossover points. These points create localized but uncorrelated stress induced retardance and scattering centers. The former have arbitrarily aligned fast axes resulting in an overall arbitrary retardance magnitude while the latter tend to reduce the DoP. The orientation of the linear component is expected to vary linearly with the input polarizer’s orientation except in the special case when the retardance is one quarter wave (± n 2π for integral n) as described below. A pseudo-linear variation is observed in the bottom panel of the Fig. 9 implying that the overall induced coil retardance is not close to λ/4 ± n 2π.
Fig. 9 Time evolution of the polarization properties of a traditionally wound FOG coil. From top to bottom are the DoLP, DoCP, DoP, and θ measured as the input polarizer was rotated 500° (≈1.5 cycles).
Figure 10 shows the 8 and 24 layer CF coil performances. These coils display much better polarization maintenance, reaching DoP maxima close to 100% for input eigenpolarizations. This is presumably due to the absence of crossover points with their associated scattering and depolarization. The 8 layer coil exhibits reduced symmetry in its response, an offset in the DoCP of 22.5% and the linear orientation is seen to follow the injected orientation. It can be shown that this behavior, coupled with the appearance of a nonzero circular component ranging from -25°-60° and a DoLP range of about 65%- 95%, implies that the coil is acting to first order as a linear wave plate with retardance approximately equal to 132° [12]. The 24 layer coil response was more symmetric and shows almost complete conversion of the output state from linear to circular as the input state is rotated. This time however, the orientation is seen to exhibit a binary type behavior, rapidly transitioning from one orientation to another (offset by 90°) as the output passes through the completely circular state. This behavior is indicative of a ¼ wave linear retarder [12]. Using the Mueller algebra, these data have been fit to a first order model of the system comprising a single linear retarder and yield an overall coil retardance magnitude of ≈ 96°.
For comparison sake, Fig. 11 shows the measured response of a 2 m long PM patch cord. The patch cord behaves somewhat like the CF coils (particularly the 24 layer coil) in that high DoP and DoLP values are obtained for input eigenpolarizations. This is to be expected since again there are no crossovers to confound the polarization degrees. An offset in the DoCP is observed as with the 8 layer CF coil although the overall magnitude and modulation envelope are relatively small. The orientation evolution is similar to that of the 24 layer coil showing a binary response with a 90° envelope, this time centered at about 100°. In this case, the DoCP never approaches 100% so the fiber is not acting simply as a ¼ wave plate even though the orientation behavior is binary. This behavior is typical of a number of tested PM fibers and is currently being further investigated. In short however, the polarimetric properties of the CF coils are found to better approximate the behavior of a PM device than the traditionally wound coil, supporting the premise that the CF design can potentially serve as a FOG coil. These results are preliminary and more thorough testing is planned to better understand the details of coil winding parameters and the measured results.
6. Summary
In summary, a fiber optic polarimeter capable of analyzing the polarimetric properties of light emitted from optical fibers and its application to the measurement of FOG coils has been presented. The sensor design, construction, and calibration approaches have been detailed along with validation measurements to assess system accuracy and repeatability. The polarization evolution, as the input to a number of FOG coils is varied, is given as an example of the sensor utility for characterizing polarization sensitive fiber components. Initial measurements indicate that SM coils wound in the crossover-free configuration behave similarly to PM fiber and may serve as an inexpensive alternative to PM based coils.
Acknowledgements
This work was supported by the Defense Advanced Research Projects Agency under contract number W31P4Q-07-C-0191. Information contained herein has been approved for public release, distribution unlimited.
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