Fiber characterization for application in heterodyne laser interferometry with nanometer uncertainty, part I: polarization state measurements

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1 Optical Engineering 44(2), (February 2005) Fiber characterization for application in heterodyne laser interferometry with nanometer uncertainty, part I: polarization state measurements B. A. W. H. Knarren S. J. A. G. Cosijns H. Haitjema P. H. J. Schellekens Eindhoven University of Technology Department of Mechanical Engineering Section Precision Engineering P.O. Box MB Eindhoven, The Netherlands Abstract. New measurement techniques and methods for characterizing the quality of optical fibers are presented. The fibers are to be used in a single-fiber-fed heterodyne laser interferometer with nanometer uncertainty. The polarization-mixing properties of several different polarizationmaintaining fiber types, as well as the effects of fiber lengths on highprecision length measurements, is investigated for the first time. Mixing ratio and polarization orthogonality were measured using three different methods. The measured mixing ratios varied between 1:100 and 1:1650 (intensity). Also the influence of the fiber assembly on the polarizationmixing properties is shown Society of Photo-Optical Instrumentation Engineers. [DOI: / ] Subject terms: fiber characterization; interferometry; metrology; lasers; fiber optic applications; polarization. Paper received Jan. 7, 2004; revised manuscript received Jul. 1, 2004; accepted for publication Jul. 23, 2004; published online Jan. 21, Introduction For precision displacement measurements, the laser interferometer is currently one of the most accurate instruments. Its noncontact nature, long range, low uncertainty, and high resolution make it an ideal displacement measurement system for use in high-precision machines, e.g., wafer scanners. Because of the increasing demand for accuracy in these machines, the laser sources themselves are becoming a limitation. The laser head is, besides being a light source, also one of the heat sources within the machine. This heat will affect the machine accuracy by increasing the measurement uncertainty directly, as well as indirectly by its thermomechanical interaction with the structure and with the artifact to be measured. To overcome these problems, the use of a flexible coupling between laser head and interferometer optics is investigated. In the measurement system to be developed we want to use only one optical fiber to transmit the light from the laser head to the interferometer. The introduction of an optical fiber must not influence the displacement measurement accuracy of the heterodyne two frequencies or polarizations interferometer system. The interferometer should be usable over a range of at least 300 mm, and the achievable uncertainty should be less than 1 nm when neglecting refractive effects. The use of optical fibers in precision-displacement laser interferometry has currently only been applied to homodyne interferometer systems such as the Renishaw, Heidenhain, and SIOS. Heterodyne systems have not until now been equipped with a single optical fiber; the heterodyne systems available nowadays apply two optical fibers /2005/$ SPIE to transmit the laser beam. For both the homodyne and the two-fiber heterodyne systems the output polarization can be filtered, e.g., by using polarizers. In the one-fiber heterodyne system the output emerging from the fiber cannot be filtered. Because every deviation from the ideal linear polarization state results in displacement measurement errors, high demands are put on the output with regard to its polarization state. Deviations from the ideal linear polarization state, emerging as displacement measurement errors also called nonlinear displacement errors are described by Cosijns et al. 1 in detail. For laser interferometric measurements at the nanometer level, imperfections of the polarization state have to be measured with high accuracy. In addition, the fibers are used with heterodyne laser light, of which both polarization states must be maintained. Therefore some of the methods developed can also be used to characterize the polarization states of the stabilized HeNe laser light sources used. Furthermore, the fibers we are using are much shorter than usual. The shorter fibers show characteristic results, thereby revealing the underlying problems of the fiber assemblies. This is briefly shown at the end of this paper. More details are presented in part II, dealing with the modeling of the fibers. 1.1 Optical Fibers The fibers used in this research are monomode optical glass fibers. To preserve polarization, polarization-maintaining PM fibers are used. Such a fiber maintains the polarization by an asymmetry in its cross section. Due to the birefringent mode, leakage is minimized and thus polarization mixing is minimized. For more information about highbirefringent fibers we refer to Refs. 2 and 3. The PM properties are investigated in this paper at the level required for Optical Engineering

2 nanometer-displacement interferometry. This means that the extinction ratios and the orientation of the main axes must be measured at higher accuracies than normally done. 1.2 PM Fibers Used for the Experiments The tests described in this paper were subsequently carried out with: 1. normal monomode fiber, not PM Wave Optics 4 2. bowtie monomode PM fiber Wave Optics 4 3. oval inner-clad monomode PM fiber Wave Optics 4 4. panda-type monomode PM fiber Point Source 5 5. pure-mode panda monomode PM fiber OZ Optics 6 6. pure-mode panda monomode PM fiber length 5 m; OZ Optics 6 7. pure-mode panda monomode PM fiber length 15 m; OZ Optics 6 8. pure-mode panda monomode PM fiber length 50 m; OZ Optics 6. The first five fiber assemblies all have a nominal length of 3 m and were used to examine the differences between fiber types. The last four, only differing in length, were used to check for length dependence. All the fiber assemblies are commercially available. In addition to the mentioned suppliers, such fibers are made by other manufactures e.g., Corning Need for External Reference PM fibers exhibit large phase changes between the two orthogonal modes e.g., due to bending or temperature changes. A change in temperature changes the refractive indices of the modes differently, resulting in large phase shifts with the use of the standard laser interferometer setup, as shown in Ref. 8. To compensate for measurement errors due to phase shifts within the fiber, the reference phase needs to be measured at the fiber s output, just after the light has emerged from the fiber, but before it enters the interferometer. 9 2 Fiber Characterization Mixing Effects For describing an ideal displacement interferometer, orthogonal linearly polarized light is assumed. In the literature e.g., in Refs. 1, 10 the influence of various effects, such as beam imperfections and optical component misalignment, is described. In this section the polarization mixing of the output of an optical fiber, positioned after a nonideal source, is investigated. From measurements using the same methods presented in Ref. 11, it is shown that the beam emerging from a laser head typically shows an ellipticity of about 1:10,000, with a nonorthogonality angle of about 0.2 deg. 2.1 Definition of Axes In this section we define the axes and orientations that are used. A heterodyne laser source emits two frequencies ( f 1 and f 2 ), orthogonally polarized. In the models, the electric field with frequency f 1 is coincident with the vertical axis and is Fig. 1 The definition of the axes for the nonideal E vectors in a heterodyne laser interferometer. represented by E 1, while the other field, with frequency f 2, is coincident with the horizontal axis and is represented by E 2. As in a nonideal interferometer both frequencies are elliptically polarized, their 90 deg out-of-phase components are represented by E 01 and E 02. The nonorthogonality of the polarizations of E 1 and E 2 is represented by the angle. The polarizer in front of the detector is oriented at an angle P with respect to the reference coordinate system. All other optical components are assumed to be perfect and aligned perfectly with the x, y coordinate system. The axis and E-vector definitions are given graphically in Fig. 1. In an ideal interferometer 0 and P 45 deg. The two frequencies are linearly polarized in an ideal interferometer, so that E 01 E 02 0, and the amplitudes are equal, so that E 1 E Measurement Methods The polarization mixing is the most important characterization parameter of the optical fibers to be determined, because the measurement accuracy of the laser interferometer is limited by the nonlinearities resulting from this mixing. The mixing after the fiber is described by the extinction ratio, which is the ratio of the unwanted to the wanted output intensity, ER P unwanted P wanted, where P wanted is the intensity of the light in the wanted mode axis and P unwanted is the intensity in the unwanted mode. For the two main axes of the PM fiber the extinction ratios are ER 1 P 01 P 1 and ER 2 P 02 P 2, where P i are values of the wanted power, and P 0i are the unwanted leaked intensities. Assuming independent contributions from the uncertainties in P 1 and P 01, denoted by u(p 1 ) and u(p 01 ), respectively, the uncertainty u(er 1 )in ER 1 is given by u ER 1 ER 1 u P 1 P u P 2 1/2 01 P Optical Engineering

3 Fig. 4 Schematic diagram of the dc measurement setup for measuring the change in polarization state due to fiber effects. Fig. 2 Setup used in the first ac method. The beam to be measured is mixed with a circularly polarizing reference source, and for various polarizer angles the beat signals are measured using a spectrum analyzer and an avalanche photodetector. Using the orientations of the analyzer for both extinction ratios, the nonorthogonality is determined as is described in detail in Sec and in Ref. 11. To measure the extinction ratio, three different measurement methods are compared and discussed. 2.3 AC Methods In order to measure the polarization state of a heterodyne laser beam, two ac measurement methods have been developed, called the carrier frequency method and the direct beat method. 11,12 In Fig. 2 the setup used with the carrier frequency method is shown. In this method the laser beam under investigation here the fiber output is mixed with the circularly polarized reference. The beat signals between the reference beam and the beam under investigation are measured with an avalanche photodetector and a spectrum analyzer. The polarizer in front of the detector is used to let the two beams interfere. The measured beat signals are used to calculate the extinction ratio. The orientation of the polarizer at minimum beat signal is used to calculate the nonorthogonality. Contrary to the results presented for the laser head output in Ref. 11, the output of the fiber is, due to the birefringence, by no means stable for the time the measurements would last. Therefore the same warming up and cooling down procedure as is described in Sec is applied in order to see the complete change in polarization. In this way the minimal achievable extinction ratio can still be determined accurately. In Fig. 3 the setup used with the direct beat method is shown. In this method no circularly polarized reference is used. The beat signal is obtained from the two frequencies of the heterodyne laser itself. The polarizer in front of the detector is used to let the two polarizations interfere. As Fig. 3 Setup used in the second ac method. The beat signals are measured for various polarizer angles, using a spectrum analyzer and an avalanche photodetector. with the first method, the orientation of the polarizer and the signal strength of the spectrum analyzer are used to calculate the extinction ratio and nonorthogonality. Also, the cooling-down procedure must be used in combination with the birefringent fibers. A disadvantage of these ac methods in characterizing the fiber output is that with them we can never measure the fiber without the influence of the laser head used. However, we can indeed measure the two axes independently, as is done with the dc method. 2.4 DC Method For the measurement of a polarization state change in the fiber e.g., due to bending or temperature change, the measurement setup schematically given in Fig. 4 was used. In this setup a frequency-stabilized, circularly polarized HeNe laser was used as the light source. For unambiguous definition of the state of polarization a Glan-Thompson 13 polarizer was used, which can be rotated around its optical axis in order to make a linearly polarized beam at any desired angle. To prevent amplitude change during rotation, the output of the laser source must be circularly polarized. Because the light is circularly polarized, its intensity is independent of the polarizer orientation. Because only small deviations around the optimal alignment are used, deviations due to imperfection in the circular polarization may be neglected. After the fiber, a second, pivoted Glan-Thompson polarizer was placed as an analyzer. By setting the orientations of the polarizers perpendicular ( a p /2), ideally all light should be blocked. To detect the light parallelpolarized along the analyzer transmission axis an intensity detector is used. A photograph of the setup is shown in Fig. 5. Using the Jones formalism, 14 the resulting output electric field of the dc measurement setup can be described by E out A R a H fiber R p P R p E crs P R a /2 H fiber R p P R p E crs, where P is the Jones matrix for the polarizer, A is the Jones matrix for the analyzer, R is the rotation matrix, and E crs is the Jones vector of the circularly polarized light source. For our situation the Jones matrix of the fiber is just that of a linear birefringent retarder: H fiber ei x e i y. 6 Optical Engineering

4 Fig. 5 Photograph of the dc measurement setup. Linearly polarized light is obtained by placing a polarizer (2) in front of the circularly polarizing source (1). The light is transmitted by a fiber (5) through another polarizer, called the analyzer (7), to the detector (8). After the first of the collimator lenses (3,6), an extra fixation (4) is used to prevent intensity changes due to bending near the connector. Before measurements were done, the system itself was carefully examined. The complete system, without a fiber, had an extinction ratio of 1:200,000. This means that the measurement system is able to measure changes in the intensity ratio down to 5 ppm. With the use of fiber keying with a positioning key used to restrict the connector to mating with other components at a single angular orientation 3 deg, the fiber is nominally by eye aligned with the horizontal and vertical axes. Now the main axes of the laser, the polarizer axes, and the fiber s main axes are roughly in the same plane, within a few degrees. First the orientation of these axes with respect to each other must be determined with an accuracy better than 0.1 deg, for the assembly to be used in the interferometer. Due to mechanical tolerances, the rotational accuracy 1 deg of the fiber keying is well below what is required for alignment. In addition, the extinction ratios as generally specified for our wavelength are well below what is required to obtain nanometer precision. Therefore the methods described are also used for selection measurement to accept or reject fibers to a quality at which they are not guaranteed. 2.5 Axis Alignment Procedure First the analyzer axis is set visually to an arbitrary position near the nominal position, and the polarizer angular resolution 0.01 deg is rotated 5 deg around its nominal position. For each polarizer orientation, the fiber is repeatedly heated and cooled as is also done by others, e.g., Ref. 15, causing phase changes between the two orthogonal principal axes of the fiber. Thus at a given polarizer and analyzer angle, the state of polarization changes repeatedly from linear to elliptical and back to linear, due to the phase shifts introduced by thermal changes of the fiber. When the output polarization is linear, the intensity at the detector is minimal with the analyzer s transmission axis perpendicular to this linear axis. On the other hand, the elliptical state of polarization results in a maximal intensity minimum. In this polarization state one component will always strike the detector. If the intensity signal versus temperature or time is recorded, a sinusoidal relation is found, as plotted in Fig Alignment of the polarizer The minimum intensity marked with in Fig. 6 and the maximum intensity marked with are recorded for each polarizer angle and plotted versus the polarizer s orientation. An example of such a measurement result is given in Fig. 7. The minimum of the maximal intensity identifies the correct alignment between fiber input and polarizer. This orientation defines the fibers main axis at the input side. Fig. 6 The intensity after the analyzer, while the fiber is cooling down. The minimal ( ) and the maximal ( ) intensity are recorded. The polarizer and analyzer angles are fixed during this measurement. Fig. 7 The minimal ( ) and maximal ( ) intensities, recorded after the analyzer while the fiber is cooling down, to be used for alignment of the polarizer s main axes. The minimum of the maximal intensity defines the fiber s main axes at the input side of the fiber. These results are from a different experiment from that shown in Fig. 6. Optical Engineering

5 Fig. 8 The minimum ( ) and maximum ( ) intensity after the analyzer, while the fiber is cooling down, to be used for alignment of the analyzer s main axes. The three characteristic lines indicated by a, b, and c signify orientations that are discussed in detail in Ref Analyzer alignment Now the polarizer is moved to this orientation and fixed, and the orientation of the analyzer is changed, following the same procedure as for the polarizer. An example of a result for the rotation of the analyzer is given in Fig. 8. The minimum of the maximal intensity is the optimal alignment between fiber output and analyzer. The typical behavior of the minimal intensity is seen clearly. In Ref. 16 this typical intensity profile is discussed in detail. Both polarizer and analyzer are now aligned optimally. The intensity measured at this optimal alignment gives the minimal polarization mixing or polarization leakage attainable for this fiber. The fiber s main axes, both for input and for output, are known with respect to the analyser s and polarizer s reference coordinate systems. The same procedure can be repeated for the other main axis of the fiber. In this way any nonorthogonality angle between the axes is measured. Table 1 Results from measurements of the extinction ratio of a standard panda type PM fiber for various measurement methods. It can be seen that these are all in good agreement. Method Extinction ratio u(er) (worst) Orthogonality error (deg) Dc (0 deg) 1:(93 7) 0 Dc (90 deg) 1:(120 9) Ac 1, f 1 1:(97 18) 1 Ac 1, f 2 1:(121 22) Ac 2, f 1 1:(98 13) 0 Ac 2, f 2 1:(110 13) Fig. 9 Intensity after analyzer, while the fiber is (a) held still, (b) substantially bent, (c) heated, and (d) cooling down. 2.6 AC and DC Measurement Results Compared To verify the dc method, the measurement results of the three methods are compared. The results from the measurements for a standard panda PM fiber by both ac methods, as well as the dc method, are given in Table 1. From these results it can be seen that all measurements are in the same range; the only difference is in the nonorthogonality measured with the first ac method, which is probably due to a measurement error, as both other methods measured the expected orthogonality. A small rotational misalignment of the nonpolarizing beamsplitter, which influences only one of the two polarization directions, could be responsible for this difference. As shown in Table 1, the measurement results for all three methods are in good agreement. Because the dc measurements are the easiest and are capable of measuring the effects of the fiber without the influence of the laser head used, in the following sections this method is used. Also, the dc method is less complicated and the achievable measurement uncertainties with this method are also lower. The ac methods are, however, useful to validate the results of the dc-method and to validate the assumption that the two polarizations are being measured independently. 2.7 Effects Influencing the Polarization Mixing As the fiber-fed heterodyne laser interferometer must be used in servo systems with changeable environmental conditions, both validations just mentioned are essential. In addition, the effect of changing the type of PM fiber and the length dependence are investigated. The effect of mixing on the achievable interferometric displacement uncertainty is calculated in Ref Bending versus temperature To test if the leakage between the two modes in the fiber is altered by bending or temperature change, the following experiment is conducted. For fixed angles of the polarizer and analyzer, the fiber s minimum and maximum mixing intensity in various conditions is determined. This is done by first placing the fiber in standard laboratory conditions, and then moving, twisting, and bending it substantially. Finally the fiber is heated, and cooled down again. In all cases the phase change between the modes is 2 ; thus for all cases all possible polarization states are present. A change in mixing would result in a change of maximal amplitude, observable by recording the intensity after the analyzer. Results of this experiment are found in Fig. 9. A total of about 45 s is recorded. We observe the following stages: the Optical Engineering

6 fiber at rest, till 10 s section a in Fig. 9, the bending of the fiber between 10 and 15 s section b, and the cooling down of the fiber from 25 s on section d. The heating of the fiber between 17 and 22 s section c of Fig. 9 is not clearly visible. From these results it can be concluded that the minimum and maximum intensities for both bending and temperature change are the same. This means that only the phase difference between the main axes of the fiber, and not the leakage from one mode to the other, changes. From this it is clear that the temperature change gives a good approximation for characterizing the fiber. This is very useful, as it is difficult or impossible to test the effects of bending repeatably. That is so because the effects causing mode coupling like bending are most effective if applied at an angle of 45 deg with respect to the fiber s main axis. 3 Because the angle at which the perturbation is applied is not known, one cannot tell what disturbance of the fiber is actually made. The temperature change, however, can be made controllably, easily, and repeatably. Because the mixing values found for temperature changes are a good approximation for characterizing the complete fiber, no other tests disturbing the fiber e.g., by bending are needed. A comment should be made about the first part of the figure, where the initial mixing is seen. This initial mixing can have any value between the maximum and the minimum intensity. This is due to the arbitrary phase difference between the two main axes. The radius of curvature of the bent fiber must not be too small; otherwise the output intensity decreases due to leakage into the cladding. The radii of curvature used at least 100 mm are adequate to render the effects of these losses due to bending negligible Orthogonality of axes after the fiber To prevent mixing in the interferometer and consequent reduction of its displacement measurement accuracy, the output polarizations of the fiber must be orthogonal. Otherwise the two frequencies polarizations cannot be split within the beamsplitter and will mix. Therefore the orthogonality of the polarization output was measured. This was done by determining both of the fiber s main axes and their extinction ratios with respect to the reference coordinate system. Results from measurements done both for 0 and 90 deg are shown in Fig. 10. From our experiments no deviation from 90 deg could be measured; the orthogonality was confirmed within the measurement resolution of 0.1 deg. From results after fitting see Ref. 16 a nonorthogonality less than 0.06 deg was obtained. Both results are in good agreement with the expected and the previously reported nonorthogonalities. In Ref. 17 the reported orthogonality of the output was within the measurement uncertainty of 0.5 deg. The measured nonorthogonality would result in a displacement measurement uncertainty of 0.1 nm. Therefore it can be concluded that this orthogonality will be adequate for the development of a fiber-fed laser interferometer with nanometer uncertainty. It is mentioned that combined effects of mixing and orthogonality must be taken into account for determining the achievable displacement uncertainty. 1 Fig. 10 Results from the fiber axis orthogonality measurement. The intensity after the analyzer for both the fiber s main axes is plotted respectively in the upper and the lower graph. The fiber s main axes are perpendicular within our measurement resolution. 2.8 Extinction Ratio Measurement Results Behavior of commercially available PM fiber types As a high extinction ratio would result in high displacement measurement accuracy of the fiber-fed laser interferometer, different types of PM fiber types were tested first to see if there are differences among them. For each fiber the polarization mixing extinction ratio is determined by dividing the leakage intensity this is the maximum intensity when the variation is minimal, e.g., 0.7 W at 89.5 deg in Fig. 8 by the maximum intensity. The maximum intensity is found by rotating the analyzer 90 deg away from the orientation of the leak intensity. The extinction ratio is then the ratio between the intensity transmitted and the maximum intensity. Results from several different tested fibers, all nominally the same length, are presented in Table 2. From these results it can be seen that the polarization mixing is less than 1%, as is standard for all types, which are guaranteed to have an extinction ratio of at least 20 db. The relatively large difference between the two axes in the panda fiber is probably due to effects within the glued fiber collimator system, which was used with this fiber only. Using these results, the maximal achievable uncer- Table 2 Results of extinction measurements for various fiber types. The ratio is given for the maximum intensity of the coupled mode. The effects of the extinction ratio on the displacement uncertainty can be calculated. 16 Fiber type Extinction ratio u(er) (worst) Orthogonality error (deg) Panda (0 deg) 1:(94 7) 0 Panda (90 deg) 1:(120 9) Bowtie (0 deg) 1:(129 9) 0 Bowtie (90 deg) 1:(125 9) Elliptical (0 deg) 1:(103 7) 0 Inner clad (90 deg) 1:(104 7) Optical Engineering

7 Table 3 Extinction ratio measured for different fiber lengths. From the measurements no length dependence is found. As is shown in the text, the connector is probably the dominating factor. Fiber length (m) Extinction ratio u(er) (worst) 3 1:(900 27) 5 1:(844 17) 15 1:(563 16) 50 1:( ) tainty in displacement measurement is discussed in detail in Ref. 16; measured displacement errors are presented in Ref Specially selected fiber The fiber assemblies in the previous section show extinction ratios of around 1:100. From measurements it turned out that the maximal achievable displacement uncertainty with heterodyne laser interferometry with such fibers is not at the 1-nm level. The uncertainty was up to 5 nm. In order to make a fiber-fed laser interferometer with 1-nm uncertainty, a better PM fiber is needed. After intensive search, a company was found that supplied a PM fiber with better specifications. Its performance is now examined, to see if it is better. If the measured fiber is more suitable, that will appear not only in a higher measured extinction ratio, but also in a change in the measured typical intensity profile as shown in Fig. 8. For a fiber with a higher extinction ratio the intensity variation at optimal alignment should be smaller. In addition, the angle between the global minimum of the minimum intensity and the optimal alignment should be smaller this is the angle marked between the dashed lines a and b in this figure. For a detailed analysis of these effects we refer to part II of this paper. As shown in Table 3, the extinction ratio of a fiber with the same length as the fibers measured in Sec is substantially better. This shows that our measurement methods are capable of measuring a broad range of extinction ratios as well as that improvement in the output polarization state of the fibers is possible. The measurements of the resulting nonlinearities in a heterodyne laser interferometer are described in Ref Effect of fiber lengths For most applications the main goal is to position the laser head outside the machine to reduce thermal effects. In this section therefore the length dependence of the fiber properties is investigated. To analyze the influence of fiber length on the extinction ratio, the ratio was measured for several different lengths of the specially selected quality fiber, between 3 and 50 m. Again the same experiments as described with the dc method were done for all lengths. The measurement result for the 50-m fiber is given in Fig. 11. Clearly the two global minima are absent. As is shown in Ref. 16, this is one of the effects of increasing the fiber length, due to the unpredictable phase of the mode leakage. Results from experiments on different length of fibers are summarized in Table 3. Fig. 11 The minimal and maximal intensity after the analyzer: results from the 50-m fiber. Due to the large attenuation around 12 db/km of the PM fibers, increasing their length will reduce the output intensity. However, this amounts only to about 3% for a 10-m fiber. As no significant differences between fibers of various lengths are found, other aspects must be responsible for limiting the fiber assembly quality. In part II these aspects are discussed in more detail. There were clear indications that the quality extinction ratio of the fiber is determined by the connector and/or the connector assembly. So more attention was paid to the connector Connector replacement To verify our assumptions that, especially for short fibers, it is not the fiber but the fiber connectors that determine the overall measured quality extinction ratio, we had the connectors of the 3-m fiber used in the previous paragraph replaced by the manufacturer. The fiber s extinction ratio was measured again with the procedure described in Sec. 2.5, and was found to have changed from 1:900 to 1:300, clearly showing the influence of the connectors and/or the assembly process. This indicates that the connectors are of major importance. If the measured extinction ratio is low, it might be improved by replacing the connectors and then performing an inspection using our methods to verify if the demanded specifications for a fiber-fed interferometer with nanometer uncertainty are met. If still not within specification, the procedure might be repeated until the required specification is met. The influence of the connectors may be explained by considering the production process. In production the cleaved fiber is inserted into a ceramic tube and the two are glued together. The diameter tolerances between fiber and ceramics allow the glue to vary in thickness around the fiber. While hardening, the fiber may be deformed. This deformation can reduce the internal birefringence, especially if applied at or near 45 deg. To improve output quality the thickness of the glue should be made as uniform as possible. This might be done during assembly by aligning the not perfectly circular ceramic ferrules with the optical axes of the fiber. By doing so one may be avoid Optical Engineering

8 stresses induced by the connector assembly process that reduce the birefringence. 3 Conclusions To develop a fiber-fed heterodyne laser interferometer with nanometer uncertainty, the polarization state of the fiber output is crucial. To characterize the fiber output we defined the extinction ratio, describing the fiber quality in one parameter. The extinction ratios needed for the development of a fiber-fed interferometer are much higher than normally used at this wavelength. Therefore three different measurement techniques were developed to characterize the fiber. Among them, the ac methods can be used in addition to characterize the heterodyne laser beam used. They can thus be used for characterizing the laser beam before and after the fiber. The three fiber characterization measurement techniques show consistent results. Because the dc method is able to measure the properties of the fiber without influence of the laser source used, this method is preferred. To characterize the quality of a PM fiber, the phase between the two modes has to be changed. It was shown that changing the fiber temperature is a easy and repeatable tool for achieving this. Using the dc measurement technique, first several different types of PM optical fibers were examined, and no significant differences in their extinction ratio were found. All three types of PM fibers show an extinction ratio of about 1:100, which is also the specification generally offered by suppliers. Finally, a supplier was found who could offer a fiber specially selected out of a very good bulk material. The extinction ratio of that specially selected fiber was measured to be 1:900. In addition, length dependence was investigated by measuring the extinction ratio of these specially selected fibers with lengths ranging from 3 to 50 m. Measured extinction ratios varied between 1:550 and 1:1650, and showed no significant length dependence. Not only polarization mixing but also rotational misalignment and axis nonorthogonality would reduce measurement accuracy. Our measurement methods allowed precise determination of the orientation of the main axes, with much higher accuracy than with fiber keying. With our measurement methods the orientation of the main axes can be determined with a measurement uncertainty of 0.03 deg. Using these methods, the fine adjustment of the main axes with respect to the polarization optics and laser head was done. Thereby the orientation of the main axes was maintained within 0.1 deg, compared to 1.9 deg when using fiber keying. Measurements showed no deviation from axis orthogonality to within 0.1 deg for the fibers, which is smaller than that in the laser head used 0.2 deg. The assumption that the connectors are responsible for the measured overall fiber quality was validated by the replacement of the connectors of a fiber assembly. This experiment showed also that our measurement methods can be used for selection. In an separate paper 16 the modeling of the fiber is described, giving a theoretical explanation of the measurement results presented in this paper. Also, the influence of the connectors and other results described in this article is analyzed by examining local and global disturbances and their locations. The effects of the fiber quality on the displacement measurement uncertainty and the elimination of phase changes are presented in Ref. 8. Acknowledgment This research was financially supported by ASML Holding N.V. and Agilent Technologies Netherlands B.V. References 1. S. J. A. G. Cosijns, H. Haitjema, and P. H. J. Schellekens, Modeling and verifying nonlinearities in heterodyne displacement interferometry, Precision Eng. 26 4, I. P. Kaminow, Polarization in optical fibers, IEEE J. Quantum Electron. 17 1, D. N. Payne, A. J. Barlow, and J. J. R. Hansen, Development of lowand high-birefringent optical fibers, IEEE J. Quantum Electron. 18 4, Wave Optics, Catalog Sep Point Source, product catalog Sep OZ Optics, Standard Tables, catalog Sep Corning Inc., Corning PureMode PM Photonic Fibers highperforming polarization maintaining fibers, catalog Sep B. A. W. H. Knarren, S. J. A. G. Cosijns, H. Haitjema, and P. H. J. Schellekens, Validation of a fiber fed heterodyne laser interferometer with nanometer uncertainty Precision Engineering to be published. 9. P. H. J. Schellekens and H. Haitjema, Heterodyne laser interferometer met fiber, Netherlands Patent NL W. Hou and G. Wilkening, Investigation and compensation of the nonlinearity of heterodyne interferometers, Precision Eng. 14, D. J. Lorier, B. A. W. H. Knarren, S. J. A. G. Cosijns, H. Haitjema, and P. H. J. Schellekens, Laser polarization state measurement in heterodyne interferometry, CIRP Ann. 52 1, B. A. W. H. Knarren, Application of optical fibers in precision heterodyne laser interferometry, PhD Thesis, Eindhoven University of Technology F. L. Pedrotti and L. S. Pedrotti, Introduction to Optics, 2nd ed., Prentice-Hall, Englewood Cliffs, NJ R. C. Jones, A new calculus for the treatment of optical systems. I. Description and discussion of the calculus, J. Opt. Soc. Am. 31, T. T. Aalto, M. Harjanne, and M. Kapulainen, A frequency domain method for the measurement of nonlinearity in heterodyne interferometry, Opt. Eng , B. A. W. H. Knarren, S. J. A. G. Cosijns, H. Haitjema, and P. H. J. Schellekens, Fiber characterization for application in heterodyne laser interferometry, part II: modeling and analysis, Opt. Eng G. D. Van Wiggeren and R. Roy, Transmission of linearly polarized light through a single-mode fiber with random fluctuations of birefringence, Appl. Opt , B. A. W. H. Knarren obtained his MSc in mechanical engineering at the Eindhoven University of Technology in He finished his PhD study on the subject of Applications of optical fibers in precision heterodyne laser interferometry in Currently he is a product development engineer at IPSE Optical Engineering, Best, The Netherlands. His research interests include (fiber) optical displacement metrology and optical system development. S. J. A. G. Cosijns obtained her MSc in physics at the Eindhoven University of Technology in From 1999 to 2004 she carried out her PhD work in the Precision Engineering Section on the subject of displacement interferometry with subnanometer uncertainty. Since September 2004 she has been employed by at ASM- Litography B. V. in Veldhoven, The Netherlands. Optical Engineering

9 H. Haitjema received his MSc in physics at the Utrecht University in 1985 and his PhD at the Delft University in From 1989 to 1997 he worked at the NMi van Swinden laboratory in Delft in the field of dimensional metrology. Since 1997 he has been an assistant professor of precision metrology in the Precision Engineering Section of the Eindhoven University of Technology, The Netherlands. His research interests include surface and displacement interferometry down to the nanometer level and theoretical aspects of metrology and uncertainty estimations. manufacturing. P. H. J. Schellekens obtained his MSc in technical physics at the Eindhoven University of Technology in In 1986 he obtained his PhD in the Dimensional Metrology Group on the subject of absolute accuracy of laser interferometer systems. Since 1990 he has been a full professor of precision engineering in the Faculty of Mechanical Engineering of the Eindhoven University. His interests are in precision metrology, precision design, and precision Optical Engineering