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Research Article
Vol. 55, No. 4 / February 1 2016 / Applied Optics
Evaluation of a linear birefringence measurement method with increased sensitivity ´ MARZENA PRETKA ˛ ,* WŁADYSŁAW A. WOZNIAK , PIOTR KURZYNOWSKI,
AND
´ SŁAWOMIR DROBCZYNSKI
Department of Optics and Photonics, Faculty of Fundamental Problems of Technology, Wroclaw University of Technology, Wybrzeze Wyspianskiego 27, 50-370 Wroclaw, Poland *Corresponding author: [email protected] Received 17 July 2015; revised 2 December 2015; accepted 23 December 2015; posted 23 December 2015 (Doc. ID 246278); published 1 February 2016
The evaluation of a measurement method of linear birefringence with increased sensitivity is presented. The examined method is based on a substantial change of the geometrical phase caused by a small change of an examined medium’s birefringence. The measuring setup consists of a linear polarizer, Wollaston compensator, and circular analyzer. The measurement is performed by tracking the phase shift of a fringe pattern. Specific orientation of the elements modifies the setup’s response—the sensitivity of the setup can be controlled in a limited measuring range. The present paper concentrates on the factors affecting the setup’s sensitivity as well as the accuracy of obtained results. The validity of the proposed approach has been demonstrated by measuring a phase retardance introduced by the liquid crystal retarder. A thousandfold increase in sensitivity has been obtained in the presented experiments, which allows the measurement of retardance introduced by the linear birefringent medium with an accuracy of 0.003° within the limited measuring range. © 2016 Optical Society of America OCIS codes: (260.5430) Polarization; (260.1440) Birefringence. http://dx.doi.org/10.1364/AO.55.000868
1. INTRODUCTION Birefringence is one of the basic properties in the optics of anisotropic media. It carries information about the difference between refractive indices introduced by the material for its eigenwaves, i.e., waves with two special polarization states that can be transmitted unchanged by birefringent medium. The indices difference is related to another important quantity: a retardance describing the phase shift introduced by the birefringent medium between its eigenwaves. This phase shift depends on the properties of the sample (its birefringence or its thickness) that frequently change together with other physical properties like stress, temperature, pressure, electric, and magnetic fields. The change of the mentioned phase reveals the information about values of the properties. In recent years plenty of methods have been presented for measuring the linear birefringence of optical samples. Most of them are based on rotating elements [1–4], liquid crystal or electro-optic modulators [5,6], aperture splitting [7], or division of amplitude [8]. As a consequence they require several exposures or quite complex measurement systems. In our previous work [9] the phase retardance could be measured in polariscopic arrangements due to the interference realized by means of the analyzer. As indicated by Pancharatnam [10], the phase shift measured in a interferometric setup consists of two terms. The first one describes the dynamic phase 1559-128X/16/040868-05$15/0$15.00 © 2016 Optical Society of America
related only to the sample’s properties (in our case it is the phase retardance between eigenwaves described above). The second one is, so-called, the geometric phase resulting from the geometry of the measurement setup, i.e., mutual configuration of the setup elements. The presented method allows measuring the phase retardance with greatly increased sensitivity (as compared to conventional polariscope systems)—a very small change in the measured phase retardance triggers a large change in the geometric phase of the presented system. Furthermore, the sensitivity of the system can be controlled by slightly changing the mutual azimuthal orientation of the setup elements.
2. MEASURING SETUP The measuring setup consists of the following elements (see Fig. 1): (1) linear polarizer (P) with the azimuth angle αP 45°, (2) Wollaston compensator (W ) with the azimuth angle αW 0°, (3) sample (S) with the azimuth angle αS introducing phase shift γ S , (4) linear quarter-wave plate (Q) with the azimuth angle αQ 0°,
Research Article
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Fig. 1. The scheme of the measuring setup: P, polarizer; W , Wollaston compensator; S, examined sample; Q, quarter-wave plate; A, analyzer; CCD, camera.
(5) linear analyzer (A) with the azimuth angle αA −45°, and (6) CCD camera. The Wollaston compensator’s azimuth angle is αW 0°, which means that it introduces the phase shift γ W between its eigenwaves linearly depending on x coordinate (γ W kx, k const). This element determines the x-axis’ position of the system. If the polarizer is placed before the prism at the azimuth angle αP different from αW by 45° then these two elements modify the light into the light field with the ellipticity angle varying linearly with x coordinate, while the azimuth angle changes as a rectangular function from −45° to 45°. Such a field can be transformed into light fringes by properly positioning the circular analyzer (which in this setup consists of the quarter-wave plate Q and the linear analyzer A) and can finally be registered by the camera. The sample S inserted between the Wollaston compensator and the circular analyzer adds an additional phase difference γ S between its own eigenwaves. This light field change can be observed as a fringe shift φG at the output of the setup. The shift value depends on the phase difference γ S as well as the sample’s azimuthal orientation. Let us examine this relationship in more detail using the Stokes–Mueller formalism [11]. According to this formalism light can be described by a fourelement Stokes vector while a 4 × 4 element Muller matrix is required to characterize the birefringent medium. The Stokes vector describing the light emerging from an optical setup is given by the product of the Mueller matrices corresponding to the succeeding elements of a setup and the Stokes vector of the incident light. The formula for the intensity distribution at the output can be obtained as the first element of the resulting Stokes vector and it looks as follows [9]: I 1 cos 2αS · sin γ S · cos γ W − cos γ S · sin γ W :
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the sample and the Wollaston prism have the same azimuthal angles (i.e., αS αW 0°) the sample’s phase retardance γ S can simply be added to the phase difference γ W introduced by the prism. As a result the fringe pattern at the output is shifted with the shift value equal to the sample’s phase retardance γ S —the relationship between the phase shift on the sample and fringe shift is linear [Fig. 2(a)] with the maximum contrast of fringes [Fig. 2(b)]. This arrangement corresponds to the most common measurement configuration. Radically different behavior of the setup is observed for the azimuth angle of the sample αS equal to 45°. In this case the resulting fringe phase value is constant regardless the sample’s phase difference γ S , except for one point where the phase difference γ S is equal to 90° and the fringe phase is reversed [solid line in Fig. 2(c)]. At this point fringe contrast equals to zero [solid line in Fig. 2(d)]. A more interesting case occurs when the sample’s orientation is minimally different from 45°. As it was expected the fringe shift versus the sample’s phase retardance γ S has an intermediate form between the two cases described above. In this case a limited range exists in which very small changes of the sample’s phase retardance γ S cause significantly greater fringe shifts [dotted line in Fig. 2(c)]. Tracking the fringe shift allows measuring the of phase retardance γ S changes with greatly increased sensitivity. The setup’s increased sensitivity is accompanied by a decrease in the contrast of obtained fringes [dotted line in Fig. 2(d)], which makes the application of the Fourier analysis necessary in further calculations. The exemplary fringe patterns with contrast equal to one and with low contrast equal to 0.007 are presented in Fig. 3. This second fringe pattern corresponds to the minimum value of contrast from the dotted line in Fig. 2(d). The theoretical sensitivity C of the measurement method (understood as a fringe shift change due to changes in the sample’s phase retardance γ S ) can be described as
(1)
Upon revision this equation could take a more useful form of I x 1 V · sinγ W − φG 1 V · sink · x − φG : (2) It can be seen as describing a sine pattern, in which V serves as a contrast and φG describes the phase shift of the fringes: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi V cos2 γ S sin2 γ S · cos2 2αS ; (3) φG γ S arctantan γ S · cos 2αS :
(4)
The pattern contrast V and phase shift φG depend on the sample’s azimuthal orientation αS and its phase retardance γ S . If
Fig. 2. Fringe shift φG and contrast V as functions of phase retardance γ S for the sample azimuth equal to (a), (b) 0°; (c), (d) 45° (solid lines); and 45.2° (dotted lines).
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Fig. 3. Exemplary fringe patterns with contrast equal to (a) 1, (b) 0.007.
C
∂φG 1 tan γ S 2 cos2αS − : ∂γ S 1 tan γ S 2 cos 2αS 2
(5)
The sample placed at azimuth angle αS 0° results in constant sensitivity equal to 1. For all other orientations of the sample the sensitivity reaches its maximum for phase retardance value γ S 90°: C max lim C γ S →90°
1 : cos 2αS
Fig. 4. Theoretical fringe shift and sensitivity versus retardance relationships for sample and LCR azimuth angles equal to 45.02° (solid line) with corresponding sensitivity (dotted line).
(6)
The maximum of sensitivity tends to infinity when the angle between the orientation sample and the Wollaston prism tends to 45°.
3. MEASUREMENT WITH INCREASED SENSITIVITY As it was described above, the setup’s sensitivity varies quite significantly, reaching very high values for the sample’s retardance around 90°. If a required measuring range lies far from this value, the setup requires an additional compensating element like a liquid crystal retarder (LCR) placed at the same azimuthal angle as the sample. This element can complement the sample’s phase retardance to 90°. The LCR can also be used to measure the characteristics of the system [i.e., relationship between the fringe shift and the phase retardance of the sample with LCR (γ L γ S )]. The existence of an area with increased sensitivity allows measuring very small changes in the sample’s phase retardance. To do this, proceed as follows: 1. Set the polarizer P at the azimuth angle −45, and analyzer A perpendicularly to P by finding the minimum value of the mean intensity registered by the camera. 2. Insert a quarter-wave plate Q between the polarizers at an angle for which the maximum intensity is achieved, fix the value of this angle, and remove the quarter-wave plate from the setup. 3. Insert the Wollaston compensator W and adjust it to obtain the maximum amplitude of zero order of the Fourier transform calculated from the intensity distribution registered by the camera. 4. Insert the quarter-wave plate again at the earlier fixed orientation. 5. The liquid crystal retarder LCR with the phase shift of around 180° can now be inserted and its orientation adjusted to obtain high contrast of the fringe pattern. Spatial frequency of the fringes (carrier frequency) should be calculated.
6. Now, the phase on the LCR can be changed to 90° leading to low contrast. The orientation of the LCR can now be adjusted by monitoring the amplitude of the Fourier transform calculated from the intensity distribution at the spatial frequency carried out previously—it should be minimal for the desired 45° orientation. The minimum amplitude value for the spatial frequency corresponds to the minimum contrast of fringes. 7. Insert the sample behind the LCR at an azimuth angle for which the minimum of the fringes contrast is achieved. 8. Measure the characteristics of the system (dots in Fig. 4) by changing the voltage applied to the LCR and fit the dependency function to the measured points (solid line in Fig. 4). 9. Set the voltage at the LCR to obtain the fringes’ minimum contrast and, thus its maximum sensitivity (which corresponds to the sum of γ L γ S equal to 90°). 10. Now, tracking fringes will allow the measurement of a very small change of the sample’s phase retardance γ S . 4. MEASURING THE SETUP’S SENSITIVITY AND ACCURACY OF OBTAINED RESULTS As can be seen from Eq. (6) the theoretical maximum of sensitivity tends to infinity when the angle between the sample and the Wollaston compensator is precisely 45°. In the practical situation the accuracy of the both elements’ orientation is limited and the maximum sensitivity of the setup is high but finite. Moreover, equations introduced in Section 2 were derived assuming that other elements of the measuring system were also set precisely to the required azimuthal orientations. Some inaccuracies in the setup’s adjustment as well as quality of the elements used in experiments can affect the obtained sensitivity. To investigate that issue more detailed numerical calculations were carried out: the influence of small azimuthal orientation changes on one of the elements was examined while the other elements were adjusted perfectly (αP 45°, αW 0°, αS 45°, αQ 0°, αA −45°). All elements were considered consecutively except for the Wollaston compensator, which was treated as a reference element.
Research Article To verify the reliability of the numerical calculations several measurements were performed. The LCR (LRC-200, Meadowlark Optics) was used as both the sample and the compensating element at the same time. Before using the LCR the dependence between its retardation and applied voltage has been carefully examined by measuring the intensity of light passing through the LCR inserted between crossed polarizers. Results are consistent with specification from the manufacturer. A He–Ne tunable laser set to 632.8 nm wavelength (HTPS-EC, Thorlabs) with a collimator (expanding the beam to the width of 4.5 in.) were used as the light source. The experimental setup comprised also two linear polarizers (LPVIS100-MP, Thorlabs), a Wollaston prism with separation angle 11’, and a seventh-order quarter-wave plate made in the workshop at our faculty and also the CCD camera (Guppy F-503B, Allied Vision, pixel size 2.2 μm, resolution 1944 pixels × 2592 pixels). The LCR was mounted in the motorized precision rotation mount (PRM1Z8, Thorlabs, resolution 0.1°). The other mounts, allowing continuous rotation with an accuracy of about 0.25°, were made in the workshop at our faculty. The measurement results (asterisk in Fig. 5) are in accordance with the calculations (solid lines in Fig. 5). The obtained results show that the proper mutual orientation of the setup’s elements affect the relationship between the sample’s phase retardance and the fringe shift in two ways: the sensitivity can be reduced (i.e., the steep part of the curve is flattened) or the range with increased sensitivity can be displaced (the sensitivity reaches its maximum for the value of the phase retardance different from 90°). Clearly, the azimuthal alignment of a sample (in this case the LCR) has the greatest influence on the sensitivity value. It does not affect the position of the increased sensitivity range. The sensitivity value is, to a smaller degree than in the case of LCR changes, modified by the quarter-wave plate azimuth changes. The azimuthal alignment of the quarter-wave plate also affects the position of the
Vol. 55, No. 4 / February 1 2016 / Applied Optics
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increased sensitivity range. Inaccurate azimuth angle of the analyzer shifts the range without affecting sensitivity. The change of the azimuth angle of the polarizer does not affect the obtained results at all; hence they were not shown in Fig. 5. The two parameters a and b were added to the mathematical model to account for these two effects: a for the steep part of the curve flattening and b for the change of sensitivity maximum position. The additional parameter c describes the initial fringe shift resulting from the position of the camera relative to the fringe pattern. Now the practical formula for the fringe shift versus retardance dependence based on theoretical dependence (4) with amendments derived from the calculation and measurement results looks as follows: φG γ S arctantanγ S − b · cos 2αS − a c:
(7)
The a, b, and c values have to be determined empirically. Several series of measurements with different a parameters were performed to demonstrate the ability to control the sensitivity. Figure 6 shows the fringe shift versus the phase retardance measurements (dots) with fitted dependency function (solid lines). The adjusted R-square coefficient (R 2 ) values very close to 1 indicate the accurateness of the fit. The characteristics of sensitivity for several series of measurements (obtained for the different azimuth angles of LCR) have been shown in Fig. 7. It is clear that the sensitivity can vary significantly. From Eq. (7) the final formula for the phase retardance measurements can be taken:
Fig. 6. Measured fringe shift versus phase retardance (dots) with fitted dependency function (solid lines).
Fig. 5. Fringe shift as a function of LCR phase retardance for the setup’s configuration with a small azimuthal change Δα on one of the elements while the other elements are adjusted according to the maximum sensitivity conditions (αA 45°, αW 0°, αLCM 45°, αP −45°): results of measurements (dots, asterisks, and crosses) and numerical calculations (solid lines).
Fig. 7. Sensitivities for the series of measurements from Fig. 6.
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Fig. 8. Sensitivity and uncertainty of retardation determination for the C series of measurements from Fig. 6.
Table 1. Sensitivity and Uncertainty of Retardation Determination for the Series of Measurements from Fig. 6 Series of Measurements
Width of Measuring Range [°]
The Maximum Inaccuracy [°]
3,077 1,044 0,355 0,032
0,088 0,032 0,013 0,003
A B C D
γ S arctan
tanφG − c b: cos 2αS − a
(8)
This formula also allows for calculation of measurement uncertainty of obtained phase retardance using the exact differential method. Its value depends on the uncertainty of a, b, c coefficients as well as on the accuracy of fringe shift measurements. The accuracy of the fringe shift measurement was estimated at 1° on the basis of scatter of results over time and the impact of the accuracy of the carrier frequency determination in the Fourier transform. Figure 8 illustrates the uncertainty of retardation determination obtained for the C series of measurements in comparison with the sensitivity values as in Fig. 7. For every measuring series there is a point with maximum sensitivity, which decreases significantly with the distance from the point. High sensitivity is accompanied by a small measurement uncertainty and vice versa. And both these quantities change their values in a very wide range. If we confine the area of our interest to the range, for example, where the sensitivity is reduced by half, we get convenient criteria for comparing the results: the width of this range and the maximum inaccuracy of the obtained phase retardance. The data for the measured series were collected in Table 1. 5. CONCLUSION It has been shown that the proposed method enables the measurement of linear birefringence changes with increased sensitivity. The measuring setup as well as the measurement
procedure have been described. The ability to modify the sensitivity by changing the mutual orientations of the setup’s elements has been demonstrated with LCR as a sample. The highest sensitivity obtained in experiments is nearly 6000, which allows the measurement of retardance introduced by the linear birefringent medium with an accuracy of 0.003° within the limited measuring range. The factors affecting the setup’s sensitivity have been examined. The most important are the azimuthal orientations of the sample and a compensation element; however, the possible influence of the other components cannot be neglected either. The great advantages of this method are the setup’s simplicity and lack of mechanical movement of its elements while performing measurements (with the exception of the calibration process). The disadvantage is the fact that the increased sensitivity can only be obtained in a limited measuring range. This method requires only one exposure so it is suitable for measuring the dynamic changes of birefringent media properties (e.g., birefringence versus temperature change, electro-optic or magneto-optic effects, etc.). Moreover, the time between the successive measurements depends only on the camera frame rate, which enables performing real-time measurements. The principal axis angle of the sample does not need to be known prior to the birefringence measurement as the sample is aligned in the calibration process. It is worth considering the applicability of this calibration process as a method for measuring the principal axis angle of the sample. In future work we would like to measure the spatial distribution of birefringence with increased sensitivity. Funding. Polish National Science Centre (DEC-2013/11/ B/ST7/01155). REFERENCES 1. W. Pin and A. Asundi, “Full-field retardation measurement of a liquid crystal cell with a phase shift polariscope,” Appl. Opt. 47, 4391–4395 (2008). 2. A. Mori and R. Tomita, “Semi-automated Sènarmont method for measurement of small retardation,” Instrum. Sci. Technol. 43, 379–389 (2015). 3. N. N. Nagib, “New formulas for phase retardance measurements of birefringent plates,” Opt. Laser Technol. 31, 309–313 (1999). 4. J. F. Lin, “Measurement of linear birefringence using a rotating-waveplate Stokes polarimeter,” Optik 121, 2144–2148 (2010). 5. J. F. Lin and Y. L. Lo, “The new circular heterodyne interferometer with electro-optic modulation for measurement of the optical linear birefringence,” Opt. Commun. 260, 486–492 (2006). 6. M. Shribak, “Complete polarization state generator with one variable retarder and its application for fast and sensitive measuring of twodimensional birefringence distribution,” J. Opt. Soc. Am. A 28, 410–419 (2011). 7. S. Shibata, T. Onuma, and Y. Otani, “Realtime birefringence mapping by polarization camera,” in Proceedings of International Symposium on Optomechatronic Technologies (ISOT) (2012), pp. 1–2. 8. R. Fang, A. Zeng, L. Zhu, L. Liu, and H. Huang, “Simultaneous measurement of retardation and fast axis angle of eighth-wave plate in real time,” Opt. Commun. 285, 4884–4886 (2012). ´ 9. M. Borwinska and W. A. Wo´zniak, “Method of residual birefringence measurements in interferometer with increased sensitivity,” Proc. SPIE 8697, 869706 (2012). 10. S. Pancharatnam, Collected Works (Oxford University, 1975). 11. R. A. Chipman, Handbook of Optics (McGraw-Hill, 1995), Vol. 2, Chap. 22.
Research Article
Vol. 55, No. 4 / February 1 2016 / Applied Optics
Evaluation of a linear birefringence measurement method with increased sensitivity ´ MARZENA PRETKA ˛ ,* WŁADYSŁAW A. WOZNIAK , PIOTR KURZYNOWSKI,
AND
´ SŁAWOMIR DROBCZYNSKI
Department of Optics and Photonics, Faculty of Fundamental Problems of Technology, Wroclaw University of Technology, Wybrzeze Wyspianskiego 27, 50-370 Wroclaw, Poland *Corresponding author: [email protected] Received 17 July 2015; revised 2 December 2015; accepted 23 December 2015; posted 23 December 2015 (Doc. ID 246278); published 1 February 2016
The evaluation of a measurement method of linear birefringence with increased sensitivity is presented. The examined method is based on a substantial change of the geometrical phase caused by a small change of an examined medium’s birefringence. The measuring setup consists of a linear polarizer, Wollaston compensator, and circular analyzer. The measurement is performed by tracking the phase shift of a fringe pattern. Specific orientation of the elements modifies the setup’s response—the sensitivity of the setup can be controlled in a limited measuring range. The present paper concentrates on the factors affecting the setup’s sensitivity as well as the accuracy of obtained results. The validity of the proposed approach has been demonstrated by measuring a phase retardance introduced by the liquid crystal retarder. A thousandfold increase in sensitivity has been obtained in the presented experiments, which allows the measurement of retardance introduced by the linear birefringent medium with an accuracy of 0.003° within the limited measuring range. © 2016 Optical Society of America OCIS codes: (260.5430) Polarization; (260.1440) Birefringence. http://dx.doi.org/10.1364/AO.55.000868
1. INTRODUCTION Birefringence is one of the basic properties in the optics of anisotropic media. It carries information about the difference between refractive indices introduced by the material for its eigenwaves, i.e., waves with two special polarization states that can be transmitted unchanged by birefringent medium. The indices difference is related to another important quantity: a retardance describing the phase shift introduced by the birefringent medium between its eigenwaves. This phase shift depends on the properties of the sample (its birefringence or its thickness) that frequently change together with other physical properties like stress, temperature, pressure, electric, and magnetic fields. The change of the mentioned phase reveals the information about values of the properties. In recent years plenty of methods have been presented for measuring the linear birefringence of optical samples. Most of them are based on rotating elements [1–4], liquid crystal or electro-optic modulators [5,6], aperture splitting [7], or division of amplitude [8]. As a consequence they require several exposures or quite complex measurement systems. In our previous work [9] the phase retardance could be measured in polariscopic arrangements due to the interference realized by means of the analyzer. As indicated by Pancharatnam [10], the phase shift measured in a interferometric setup consists of two terms. The first one describes the dynamic phase 1559-128X/16/040868-05$15/0$15.00 © 2016 Optical Society of America
related only to the sample’s properties (in our case it is the phase retardance between eigenwaves described above). The second one is, so-called, the geometric phase resulting from the geometry of the measurement setup, i.e., mutual configuration of the setup elements. The presented method allows measuring the phase retardance with greatly increased sensitivity (as compared to conventional polariscope systems)—a very small change in the measured phase retardance triggers a large change in the geometric phase of the presented system. Furthermore, the sensitivity of the system can be controlled by slightly changing the mutual azimuthal orientation of the setup elements.
2. MEASURING SETUP The measuring setup consists of the following elements (see Fig. 1): (1) linear polarizer (P) with the azimuth angle αP 45°, (2) Wollaston compensator (W ) with the azimuth angle αW 0°, (3) sample (S) with the azimuth angle αS introducing phase shift γ S , (4) linear quarter-wave plate (Q) with the azimuth angle αQ 0°,
Research Article
Vol. 55, No. 4 / February 1 2016 / Applied Optics
Fig. 1. The scheme of the measuring setup: P, polarizer; W , Wollaston compensator; S, examined sample; Q, quarter-wave plate; A, analyzer; CCD, camera.
(5) linear analyzer (A) with the azimuth angle αA −45°, and (6) CCD camera. The Wollaston compensator’s azimuth angle is αW 0°, which means that it introduces the phase shift γ W between its eigenwaves linearly depending on x coordinate (γ W kx, k const). This element determines the x-axis’ position of the system. If the polarizer is placed before the prism at the azimuth angle αP different from αW by 45° then these two elements modify the light into the light field with the ellipticity angle varying linearly with x coordinate, while the azimuth angle changes as a rectangular function from −45° to 45°. Such a field can be transformed into light fringes by properly positioning the circular analyzer (which in this setup consists of the quarter-wave plate Q and the linear analyzer A) and can finally be registered by the camera. The sample S inserted between the Wollaston compensator and the circular analyzer adds an additional phase difference γ S between its own eigenwaves. This light field change can be observed as a fringe shift φG at the output of the setup. The shift value depends on the phase difference γ S as well as the sample’s azimuthal orientation. Let us examine this relationship in more detail using the Stokes–Mueller formalism [11]. According to this formalism light can be described by a fourelement Stokes vector while a 4 × 4 element Muller matrix is required to characterize the birefringent medium. The Stokes vector describing the light emerging from an optical setup is given by the product of the Mueller matrices corresponding to the succeeding elements of a setup and the Stokes vector of the incident light. The formula for the intensity distribution at the output can be obtained as the first element of the resulting Stokes vector and it looks as follows [9]: I 1 cos 2αS · sin γ S · cos γ W − cos γ S · sin γ W :
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the sample and the Wollaston prism have the same azimuthal angles (i.e., αS αW 0°) the sample’s phase retardance γ S can simply be added to the phase difference γ W introduced by the prism. As a result the fringe pattern at the output is shifted with the shift value equal to the sample’s phase retardance γ S —the relationship between the phase shift on the sample and fringe shift is linear [Fig. 2(a)] with the maximum contrast of fringes [Fig. 2(b)]. This arrangement corresponds to the most common measurement configuration. Radically different behavior of the setup is observed for the azimuth angle of the sample αS equal to 45°. In this case the resulting fringe phase value is constant regardless the sample’s phase difference γ S , except for one point where the phase difference γ S is equal to 90° and the fringe phase is reversed [solid line in Fig. 2(c)]. At this point fringe contrast equals to zero [solid line in Fig. 2(d)]. A more interesting case occurs when the sample’s orientation is minimally different from 45°. As it was expected the fringe shift versus the sample’s phase retardance γ S has an intermediate form between the two cases described above. In this case a limited range exists in which very small changes of the sample’s phase retardance γ S cause significantly greater fringe shifts [dotted line in Fig. 2(c)]. Tracking the fringe shift allows measuring the of phase retardance γ S changes with greatly increased sensitivity. The setup’s increased sensitivity is accompanied by a decrease in the contrast of obtained fringes [dotted line in Fig. 2(d)], which makes the application of the Fourier analysis necessary in further calculations. The exemplary fringe patterns with contrast equal to one and with low contrast equal to 0.007 are presented in Fig. 3. This second fringe pattern corresponds to the minimum value of contrast from the dotted line in Fig. 2(d). The theoretical sensitivity C of the measurement method (understood as a fringe shift change due to changes in the sample’s phase retardance γ S ) can be described as
(1)
Upon revision this equation could take a more useful form of I x 1 V · sinγ W − φG 1 V · sink · x − φG : (2) It can be seen as describing a sine pattern, in which V serves as a contrast and φG describes the phase shift of the fringes: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi V cos2 γ S sin2 γ S · cos2 2αS ; (3) φG γ S arctantan γ S · cos 2αS :
(4)
The pattern contrast V and phase shift φG depend on the sample’s azimuthal orientation αS and its phase retardance γ S . If
Fig. 2. Fringe shift φG and contrast V as functions of phase retardance γ S for the sample azimuth equal to (a), (b) 0°; (c), (d) 45° (solid lines); and 45.2° (dotted lines).
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Fig. 3. Exemplary fringe patterns with contrast equal to (a) 1, (b) 0.007.
C
∂φG 1 tan γ S 2 cos2αS − : ∂γ S 1 tan γ S 2 cos 2αS 2
(5)
The sample placed at azimuth angle αS 0° results in constant sensitivity equal to 1. For all other orientations of the sample the sensitivity reaches its maximum for phase retardance value γ S 90°: C max lim C γ S →90°
1 : cos 2αS
Fig. 4. Theoretical fringe shift and sensitivity versus retardance relationships for sample and LCR azimuth angles equal to 45.02° (solid line) with corresponding sensitivity (dotted line).
(6)
The maximum of sensitivity tends to infinity when the angle between the orientation sample and the Wollaston prism tends to 45°.
3. MEASUREMENT WITH INCREASED SENSITIVITY As it was described above, the setup’s sensitivity varies quite significantly, reaching very high values for the sample’s retardance around 90°. If a required measuring range lies far from this value, the setup requires an additional compensating element like a liquid crystal retarder (LCR) placed at the same azimuthal angle as the sample. This element can complement the sample’s phase retardance to 90°. The LCR can also be used to measure the characteristics of the system [i.e., relationship between the fringe shift and the phase retardance of the sample with LCR (γ L γ S )]. The existence of an area with increased sensitivity allows measuring very small changes in the sample’s phase retardance. To do this, proceed as follows: 1. Set the polarizer P at the azimuth angle −45, and analyzer A perpendicularly to P by finding the minimum value of the mean intensity registered by the camera. 2. Insert a quarter-wave plate Q between the polarizers at an angle for which the maximum intensity is achieved, fix the value of this angle, and remove the quarter-wave plate from the setup. 3. Insert the Wollaston compensator W and adjust it to obtain the maximum amplitude of zero order of the Fourier transform calculated from the intensity distribution registered by the camera. 4. Insert the quarter-wave plate again at the earlier fixed orientation. 5. The liquid crystal retarder LCR with the phase shift of around 180° can now be inserted and its orientation adjusted to obtain high contrast of the fringe pattern. Spatial frequency of the fringes (carrier frequency) should be calculated.
6. Now, the phase on the LCR can be changed to 90° leading to low contrast. The orientation of the LCR can now be adjusted by monitoring the amplitude of the Fourier transform calculated from the intensity distribution at the spatial frequency carried out previously—it should be minimal for the desired 45° orientation. The minimum amplitude value for the spatial frequency corresponds to the minimum contrast of fringes. 7. Insert the sample behind the LCR at an azimuth angle for which the minimum of the fringes contrast is achieved. 8. Measure the characteristics of the system (dots in Fig. 4) by changing the voltage applied to the LCR and fit the dependency function to the measured points (solid line in Fig. 4). 9. Set the voltage at the LCR to obtain the fringes’ minimum contrast and, thus its maximum sensitivity (which corresponds to the sum of γ L γ S equal to 90°). 10. Now, tracking fringes will allow the measurement of a very small change of the sample’s phase retardance γ S . 4. MEASURING THE SETUP’S SENSITIVITY AND ACCURACY OF OBTAINED RESULTS As can be seen from Eq. (6) the theoretical maximum of sensitivity tends to infinity when the angle between the sample and the Wollaston compensator is precisely 45°. In the practical situation the accuracy of the both elements’ orientation is limited and the maximum sensitivity of the setup is high but finite. Moreover, equations introduced in Section 2 were derived assuming that other elements of the measuring system were also set precisely to the required azimuthal orientations. Some inaccuracies in the setup’s adjustment as well as quality of the elements used in experiments can affect the obtained sensitivity. To investigate that issue more detailed numerical calculations were carried out: the influence of small azimuthal orientation changes on one of the elements was examined while the other elements were adjusted perfectly (αP 45°, αW 0°, αS 45°, αQ 0°, αA −45°). All elements were considered consecutively except for the Wollaston compensator, which was treated as a reference element.
Research Article To verify the reliability of the numerical calculations several measurements were performed. The LCR (LRC-200, Meadowlark Optics) was used as both the sample and the compensating element at the same time. Before using the LCR the dependence between its retardation and applied voltage has been carefully examined by measuring the intensity of light passing through the LCR inserted between crossed polarizers. Results are consistent with specification from the manufacturer. A He–Ne tunable laser set to 632.8 nm wavelength (HTPS-EC, Thorlabs) with a collimator (expanding the beam to the width of 4.5 in.) were used as the light source. The experimental setup comprised also two linear polarizers (LPVIS100-MP, Thorlabs), a Wollaston prism with separation angle 11’, and a seventh-order quarter-wave plate made in the workshop at our faculty and also the CCD camera (Guppy F-503B, Allied Vision, pixel size 2.2 μm, resolution 1944 pixels × 2592 pixels). The LCR was mounted in the motorized precision rotation mount (PRM1Z8, Thorlabs, resolution 0.1°). The other mounts, allowing continuous rotation with an accuracy of about 0.25°, were made in the workshop at our faculty. The measurement results (asterisk in Fig. 5) are in accordance with the calculations (solid lines in Fig. 5). The obtained results show that the proper mutual orientation of the setup’s elements affect the relationship between the sample’s phase retardance and the fringe shift in two ways: the sensitivity can be reduced (i.e., the steep part of the curve is flattened) or the range with increased sensitivity can be displaced (the sensitivity reaches its maximum for the value of the phase retardance different from 90°). Clearly, the azimuthal alignment of a sample (in this case the LCR) has the greatest influence on the sensitivity value. It does not affect the position of the increased sensitivity range. The sensitivity value is, to a smaller degree than in the case of LCR changes, modified by the quarter-wave plate azimuth changes. The azimuthal alignment of the quarter-wave plate also affects the position of the
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increased sensitivity range. Inaccurate azimuth angle of the analyzer shifts the range without affecting sensitivity. The change of the azimuth angle of the polarizer does not affect the obtained results at all; hence they were not shown in Fig. 5. The two parameters a and b were added to the mathematical model to account for these two effects: a for the steep part of the curve flattening and b for the change of sensitivity maximum position. The additional parameter c describes the initial fringe shift resulting from the position of the camera relative to the fringe pattern. Now the practical formula for the fringe shift versus retardance dependence based on theoretical dependence (4) with amendments derived from the calculation and measurement results looks as follows: φG γ S arctantanγ S − b · cos 2αS − a c:
(7)
The a, b, and c values have to be determined empirically. Several series of measurements with different a parameters were performed to demonstrate the ability to control the sensitivity. Figure 6 shows the fringe shift versus the phase retardance measurements (dots) with fitted dependency function (solid lines). The adjusted R-square coefficient (R 2 ) values very close to 1 indicate the accurateness of the fit. The characteristics of sensitivity for several series of measurements (obtained for the different azimuth angles of LCR) have been shown in Fig. 7. It is clear that the sensitivity can vary significantly. From Eq. (7) the final formula for the phase retardance measurements can be taken:
Fig. 6. Measured fringe shift versus phase retardance (dots) with fitted dependency function (solid lines).
Fig. 5. Fringe shift as a function of LCR phase retardance for the setup’s configuration with a small azimuthal change Δα on one of the elements while the other elements are adjusted according to the maximum sensitivity conditions (αA 45°, αW 0°, αLCM 45°, αP −45°): results of measurements (dots, asterisks, and crosses) and numerical calculations (solid lines).
Fig. 7. Sensitivities for the series of measurements from Fig. 6.
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Fig. 8. Sensitivity and uncertainty of retardation determination for the C series of measurements from Fig. 6.
Table 1. Sensitivity and Uncertainty of Retardation Determination for the Series of Measurements from Fig. 6 Series of Measurements
Width of Measuring Range [°]
The Maximum Inaccuracy [°]
3,077 1,044 0,355 0,032
0,088 0,032 0,013 0,003
A B C D
γ S arctan
tanφG − c b: cos 2αS − a
(8)
This formula also allows for calculation of measurement uncertainty of obtained phase retardance using the exact differential method. Its value depends on the uncertainty of a, b, c coefficients as well as on the accuracy of fringe shift measurements. The accuracy of the fringe shift measurement was estimated at 1° on the basis of scatter of results over time and the impact of the accuracy of the carrier frequency determination in the Fourier transform. Figure 8 illustrates the uncertainty of retardation determination obtained for the C series of measurements in comparison with the sensitivity values as in Fig. 7. For every measuring series there is a point with maximum sensitivity, which decreases significantly with the distance from the point. High sensitivity is accompanied by a small measurement uncertainty and vice versa. And both these quantities change their values in a very wide range. If we confine the area of our interest to the range, for example, where the sensitivity is reduced by half, we get convenient criteria for comparing the results: the width of this range and the maximum inaccuracy of the obtained phase retardance. The data for the measured series were collected in Table 1. 5. CONCLUSION It has been shown that the proposed method enables the measurement of linear birefringence changes with increased sensitivity. The measuring setup as well as the measurement
procedure have been described. The ability to modify the sensitivity by changing the mutual orientations of the setup’s elements has been demonstrated with LCR as a sample. The highest sensitivity obtained in experiments is nearly 6000, which allows the measurement of retardance introduced by the linear birefringent medium with an accuracy of 0.003° within the limited measuring range. The factors affecting the setup’s sensitivity have been examined. The most important are the azimuthal orientations of the sample and a compensation element; however, the possible influence of the other components cannot be neglected either. The great advantages of this method are the setup’s simplicity and lack of mechanical movement of its elements while performing measurements (with the exception of the calibration process). The disadvantage is the fact that the increased sensitivity can only be obtained in a limited measuring range. This method requires only one exposure so it is suitable for measuring the dynamic changes of birefringent media properties (e.g., birefringence versus temperature change, electro-optic or magneto-optic effects, etc.). Moreover, the time between the successive measurements depends only on the camera frame rate, which enables performing real-time measurements. The principal axis angle of the sample does not need to be known prior to the birefringence measurement as the sample is aligned in the calibration process. It is worth considering the applicability of this calibration process as a method for measuring the principal axis angle of the sample. In future work we would like to measure the spatial distribution of birefringence with increased sensitivity. Funding. Polish National Science Centre (DEC-2013/11/ B/ST7/01155). REFERENCES 1. W. Pin and A. Asundi, “Full-field retardation measurement of a liquid crystal cell with a phase shift polariscope,” Appl. Opt. 47, 4391–4395 (2008). 2. A. Mori and R. Tomita, “Semi-automated Sènarmont method for measurement of small retardation,” Instrum. Sci. Technol. 43, 379–389 (2015). 3. N. N. Nagib, “New formulas for phase retardance measurements of birefringent plates,” Opt. Laser Technol. 31, 309–313 (1999). 4. J. F. Lin, “Measurement of linear birefringence using a rotating-waveplate Stokes polarimeter,” Optik 121, 2144–2148 (2010). 5. J. F. Lin and Y. L. Lo, “The new circular heterodyne interferometer with electro-optic modulation for measurement of the optical linear birefringence,” Opt. Commun. 260, 486–492 (2006). 6. M. Shribak, “Complete polarization state generator with one variable retarder and its application for fast and sensitive measuring of twodimensional birefringence distribution,” J. Opt. Soc. Am. A 28, 410–419 (2011). 7. S. Shibata, T. Onuma, and Y. Otani, “Realtime birefringence mapping by polarization camera,” in Proceedings of International Symposium on Optomechatronic Technologies (ISOT) (2012), pp. 1–2. 8. R. Fang, A. Zeng, L. Zhu, L. Liu, and H. Huang, “Simultaneous measurement of retardation and fast axis angle of eighth-wave plate in real time,” Opt. Commun. 285, 4884–4886 (2012). ´ 9. M. Borwinska and W. A. Wo´zniak, “Method of residual birefringence measurements in interferometer with increased sensitivity,” Proc. SPIE 8697, 869706 (2012). 10. S. Pancharatnam, Collected Works (Oxford University, 1975). 11. R. A. Chipman, Handbook of Optics (McGraw-Hill, 1995), Vol. 2, Chap. 22.
