Complete snapshot Stokes polarimeter based on a single biaxial crystal IRENE ESTÉVEZ1, VICTOR SOPO1, ANGEL LIZANA 1,*, ALEX TURPIN1 AND JUAN CAMPOS1 1 Departament de física, Universitat Autònoma de Barcelona, Bellaterra E-08193, Spain *Corresponding author: angel.lizana@uab.es Received XX Month XXXX; revised XX Month, XXXX; accepted XX Month XXXX; posted XX Month XXXX (Doc. ID XXXXX); published XX Month XXXX We propose a new snapshot complete Stokes polarimeter Polarimeters based on the CR phenomenon are quite repeatable based on the conical refraction (CR) phenomenon. This is instruments because they provide very large data redundancy, this the first CR based polarimeter able to provide complete redundancy data allowing us to decrease experimental noise from raw polarimetric information by only using a single biaxial intensity measurements. Moreover, unlike most polarimeters, which crystal and a CCD camera. Compared with existing CR require time-sequential measures to decrease statistical noise, CR polarimeters, the proposed optical scheme is simpler and based polarimeters do that instantaneously because redundancy data reduces costs. In addition, all the benefits of CR based is spatially obtained from intensity images on a CCD camera [6, 9]. This feature was applied in Ref. [10], where the suitability of CR based polarimeters (snapshot, large data redundancy, etc.) are polarimeters under low-intensity conditions was highlighted. also present in the proposed architecture. The By contrast, biaxial crystals (BC) for themselves only provide linear polarimeter is experimentally implemented and information of input beams, so the polarimetric information is analyzed in terms of accuracy and sensibility. © 2015 incomplete [10]. If elliptically polarized light is of interest, a second arm Optical Society of America combining a BC with an optical retarder (OR) is required, this allowing OCIS codes: (140.3490) Lasers, distributed-feedback; (060.2420) Fibers, us to include elliptical polarization analyzers at the measuring polarization-maintaining; (060.3735) Fiber Bragg gratings. configuration. That is the reason why architectures of complete CR polarimeters present two optical arms (i.e. two BCs as well as two CCD http://dx.doi.org/10.1364/OL.99.099999 cameras), which raises the polarimeter price and imposes a proper synchronization between the two CCD cameras in the system. We propose a new complete CR based polarimeter that only Polarimetric information is used in a large number of applications, as requires one biaxial crystal and a single CCD camera. It is based on a for instance, in astronomy [1], materials characterization [2], bio- simple set-up (CCDs synchronization is avoided) that reduces costs medical applications [3], etc. To cover this need, different polarimeter when compared with existing prototypes [9, 10]. The optical scheme is architectures have been proposed in literature [4]. In the case of real- sketched in Fig. 1(a). A monochromatic light source is focused by the time applications, snap-shot polarimeters [5] are used to operate at convergent lens (L1). The focused beam enters into an Optical Module high velocity rates, they being commonly based on front-division or (OM, Fig. 1(b)) where it is divided in two sub-beams by means of a amplitude-division architectures. A new type of amplitude-division non-polarizing Beam-Splitter (BS1). At this point, the reflected beam polarimeter, based on the conical refraction (CR) phenomenon, was (RB) is totally reflected on a prism (P1) and then partially transmitted proposed in Ref. [6]. These polarimeters take benefit of the CR on a second beam-splitter (BS2), while the transmitted beam (TB) is phenomenon occurring in bi-axial crystals (BC) [7, 8]. In fact, when a totally reflected on a second prism (P2) and partially reflected on the focused Gaussian beam passes through one of the optical axes of a BC, BS2. Therefore, two convergent and parallel beams with the same at the focal plane we observe two concentric bright rings separated by intensity are obtained at the exit of the OM. In addition, the beams a dark (Poggendorff) ring under conditions of R0>>0, R0 being the separation can be controlled by performing linear displacements d of radius of the ring (it depending on the crystal length and on the the prism P2. Note that between the prism P1 and the BS2 elements, principal refractive indices n1, n2 and n3) and where 0 is the waist we have also set a linear retarder (LR), so the input polarization is radius of the focused input beam. Moreover, the polarization modified by this element. Once the two convergent light beams exit distribution observed at the ring plane is always linearly polarized, in a from the OM, they passes through one of the optical axis of a bi-axial way that each two light spots being diametrically placed on the ring crystal and thus, two rings-like intensity distributions are obtained at present orthogonal linear polarizations. In this situation, the effect of the focal plane of the lens L1. The intensity distribution of these two the BC on the input light at the focal plane can be described by the light rings corresponds to the projection of the input polarization on: input polarization projection on a basis of linear analyzers [6]. (i) a system of linear analyzers with different orientations due to the biaxial crystal (BC) effect (configuration related to the TB in Fig. 1(b)); and (ii) a system of elliptical analyzers with different ellipticity, they There are different possibilities to implement a complete being set due to the combination of the linear retarder LR and the BC polarimeter with the optical scheme proposed in Fig. 1. One choice to (configuration related to the RB in Fig. 1(b)). be done is to select the specific linear retarder to be placed into the optical module in Fig. 1(b). Two different situations are here analyzed: (a) a quarter-waveplate (QWP) oriented to a certain angle ; and (b) a variable retarder oriented at 0 degrees of the laboratory vertical. Each proposed configuration lead to a different polarimeter with different polarimetric performance. To estimate the quality of a polarimeter there are different parameters that can be revised, as its sensibility, Fig. 1. (a) Complete Stokes polarimeter scheme based on a single bi- accuracy, robustness or noise amplification, the last being a crucial axial crystal; and (b) detail of the Optical Module (OM) for the property of polarimeters. There are well-known quality indicators, as generation of two parallel and convergent beams. the condition number (CN) [12] or the equally weighted variance (EWV) [13] that provide the performance of polarimeters in terms of Following, the polarization analyzers (PAs) provided by our noise amplification from noise present at the intensity measurements polarimeter are determined. First, we calculate the PAs related to the to the final Stokes parameters measurements. By considering the form Transmitted Beam (TB) (see Fig. 1(b)). This TB is not affected by the of the CN and the EWV indicators, one realizes that such noise linear retarder and its corresponding PAs are those directly provided amplification is closely related with the specific arrangement of PAs by the CR phenomenon –i.e. a set of linear analyzers. By using the present in a polarimeter, or in other words, with the way in which the Mueller-Stokes (M-S) formalism, those analyzers are given by, matrix inversion present in Eq. (5) amplify noise in the radiometric 1 vector IRB-TB. aTB      1 cos  sin  0  , T 2 (1) Next, the performance of the optical configurations (a) and (b) where  is the angular position of the different linear analyzers at the described above, is analyzed in terms of the CN indicator. CN estimates ring and the subscript T denotes the vector transpose. The intensity how close a certain matrix is from singularity. In fact, the farther the distribution measured by the CCD camera for the TB is the projection matrix is from singularity (i.e. the smallest the CN), the lower the noise of the incident polarization over the different analyzers aTB in Eq. (1), amplification of the system. First, we study the configuration (a) ̶ i.e. a 1 quarter-waveplate (QWP) with an orientation  as linear retarder. ITB      S0  S1 cos   S2 sin   , Under this scenario, =90º and Eq. (3) becomes, 2 (2) � 1 � where S0, S1, S2 and S3 are the unknown Stokes parameters of the � � 1� cos  cos 2 2  sin  sin 2 cos 2 � input polarization. Note that Eq. (2) does not give information about aRB   ,    . 2 �cos  sin 2 cos 2  sin  sin 2 � 2 the S3 parameter, and thus, the Stokes information encrypted in Eq. (2) � � � � sin  2    � � is incomplete. (6) Regarding to the Reflected Beam (RB), it is affected by the Figure 2(a) shows the PAs related to the configuration (a) over the combination of the linear retarder and the linear analyzers set by the Poincaré sphere [11], which have been calculated according to Eq. (1) conical refraction phenomenon. By taking into account the general (red line) and Eq. (6) (blue line). The different blue curves correspond expression for the Mueller matrix of a linear retarder with an to different orientations  of the QWP (from 0º to 180º in steps of 10º). orientation  and a retardance  [11], the PAs related to the RB are, We see as a rotation of the QWP leads to a rotation of the aRB analyzers � 1 � curve around the S3 axis. For each angle  selected, we have calculated � � cos   cos 2 2  cos  sin 2 2   sin   1  cos   sin 2 cos 2 � 1� the corresponding CN, this being equal to 2.0 in all the cases (see inset aRB   ,  ,    � , � cos   1  cos   sin 2 cos 2  sin   sin 2  cos  cos 2  � 2� 2 2 in Fig. 2(a)). Therefore, the orientation of the QWP is not a relevant � cos  sin  sin 2  sin  sin  cos 2 � parameter in terms of system conditioning. In addition, the obtained � �(3) CN is very close to the theoretical minimum in polarimetric systems and the intensity distribution recorded by the CCD camera –this time (i.e. √3=1.73 [14]), this showing that the configuration (a) leads to related to the RB– is the projection of the incident polarization over the polarimeters with very low noise amplification. different analyzers aRB, �S0  S1 � �cos   cos 2 2  cos  sin 2 2   sin   1  cos   sin 2 cos 2 � � � � � 1� � � I RB     � S1 cos   1  cos   sin 2 cos 2  sin   sin 2  cos  cos 2  � � 2 2 � . 2� � � � S3  cos  sin  sin 2  sin  sin  cos 2  � � (4) Thus, complete polarimetric information of the input beam is obtained by considering the intensity distributions of the RB and TB beams. At this point, different strategies could be applied to extract the Fig. 2. Polarization analyzers as a function of: (a) the QWP orientation encrypted input polarization from Eqs. (2) and (4), as applying the  (configuration (a)); and (b) the variable retarder retardance  synchronous detection method [10] or some inversion method. We use (configuration (b)). In both cases, the corresponding CN values are this last approach by applying the following matrix system, shown as an inset image. Sinput  A%1 I RB TB , (5) Second, we study the configuration (b) ̶ i.e. a variable linear retarder where Sinput is the Stokes vector of the input polarization and A% is the 1 oriented to an angle =0º. In such situation Eq. (3) becomes, pseudo-inverse of the matrix A (i.e. the so-called instrumental matrix), the A matrix being a nx4 matrix whose rows contain the n polarization 1 aRB   ,     1 cos  sin  cos   sin  sin   , T analyzers of the system –both aTB and aRB analyzers in Eqs. (1) and (3). 2 (7) Likewise, IRB-TB is the radiometric vector, which is constructed from the Figure 2(b) shows the PAs related to the configuration (b). Whereas projections of the input polarization on the different polarization the red curve shows the linear analyzers (aTB), calculated according to analyzers (i.e. ring intensities at different angles ). Eq. (1), the green curve shows the polarization analyzers (aRB), calculated according to Eq. (7). The different green curves correspond (a) (i.e. 0.0028). For its part, S3 shows a minimum value (0.0056) for to different retardances  of the variable retarder (from -90º to 90º in =±90º, when the variable retarder acts as a QWP and the configuration (b) steps of 20º). Note that this element could be experimentally turns into configuration (a). By contrast, as lower the value for , as larger implemented, for instance, by using a parallel aligned liquid crystal S3, it tending to ∞ when  tends to 0 (see as well CN() in the inset of panel. A change of the retardance  leads to a rotation of the aRB Fig. 2(b)). It is interesting to note that S2 remains constant (0.0056, analyzers curve over the S1 axis. Therefore, unlike the configuration (a) this being the maximum value reached in configuration (a)) for all the above discussed (Fig. 2(a)), this configuration (b) leads to different CN  values, except for =0, where the variances S1 and S2 values depending of the retardance  selected (see inset in Fig. 2(b)). simultaneously show a minimum (0.0028, see Fig. 3(b)). Thus, Thus, the noise amplification of the system strongly depends on this configuration (b) with =0º is the best configuration for linear parameter. Two limit situations are observed for =0º and  =±90º. If polarization metrology in terms of noise amplification –this is because =0º, both aRB and aTB analyzers are at the sphere equator and there is both aRB and aTB PAs are on the Poincaré sphere equator. This situation no S3 information (incomplete polarimeter; CN=∞). By contrast, when arises very interesting as PAs curves can be rotated in a controlled way =±90º, the configuration (b) turns into the configuration (a) (for the by using linear retarders properly placed at the optical module (Fig. 1). case =90º and =0º), and the CN reaches the minimum possible However, the aim of this work is to perform complete polarimetry value (CN=2.0), so the noise propagation is minimized. under optimal conditions, and thus, the configuration (a) is chosen for It is also interesting to analyze the configurations (a) and (b) in the subsequent experimental implementation. terms of the specific Stokes variances (SV) [13, 15], they providing The optical arrangement in Fig. 1 was implemented for the information of the noise amplification at each particular Stokes configuration (a). A Gaussian beam (He–Ne laser; 633 nm) channel. The SV components can be calculated by error propagating was used as light source. The beam reaches a Polarization State Eq. (5), this leading to [13, 15], Generator (PSG) formed by a linear polarizer and two Parallel n Aligned (PA) Liquid Crystals panels (LCC1223T-A provided by Si  I 2 �qik2 , Thorlabs). This system, analogous to that described in Ref. [15], k 1 (8) allows us to calibrate and to test the CR based polarimeter. In with i=(0,1,2,3), where n is the number of elements of the radiometric addition, as the PSG is static (i.e. different polarizations are vector IRB-TB , qik are the coefficients of the pseudo-inverse matrix A% 1 digitally addressed), it also minimize beam wander effects [16] and where I is an statistical noise that has been considered uniformly and permits us to arbitrarily generate any fully polarized State distributed on the components of the vector IRB-TB. of Polarization (SoP). After that, the Optical Module is placed (Fig. Figure 3 shows the SV values, calculated according to Eq. (8), as a 1(b)). The beam is focalized by a convergent lens (L1) (focal length of function of the orientation  of the QWP (configuration (a); Fig. 3(a)) f1=16 cm) and it is divided in two sub-beams by using 10 mm beam and as a function of the selected retardance  for the variable retarder splitters and prisms. The two exiting beams reaches a biaxial crystal (configuration (b); Fig. 3(b)). In both cases, we set an arbitrary value (made of KGd(WO4)2, fabricated by CROptics) with a length of for the statistical noise I in Eq. (8) (i.e., I=1). L=23.38 mm and conicity c=16.9 mrad. As a consequence, each light beam form a ring-like structure at the focal plane of the L1 lens, both with a radius of R0=397 m. Finally, a second convergent lens (L2) images the obtained rings to a CCD camera (6-bits with a resolution of 1000x1000 pixels distributed by Basler) with certain magnification. Before testing the polarimeter performance, it was experimentally calibrated. The instrumental matrix A can be experimentally determined by setting different known input polarizations and measuring the corresponding ring intensity distributions (see Eq. 5). Fig. 3. SV as a function of: (a) the QWP orientation  (configuration To this end, six controlled SoPs were generated with our PSG: four (a)); and (b) the retardance  for the variable retarder (configuration equally spaced linear states (horizontal, vertical, 45º and 135º linear (b)). polarizations) and two circular states (right-hand and left-hand circular polarizations). In addition, to link the calibration ring-like For the configuration (a) (Fig. 3(a)), the four Stokes variances intensities with the corresponding pixels at the CCD camera, the ring- remain very small (>6·10-3) as a function of the orientation of the centering and ring-correlation algorithm detailed in Ref. [10] was QWP. This is logical because in all the cases the system is optimized applied. Under this scenario, we calibrated the polarization analyzers (see Fig. 2(a)) and thus, the noise amplification must be always small. (PAs) of our system, with the QWP oriented to an arbitrary angle. The However, whereas S0 and S3 remain constant (0.0014 and 0.0056 obtained PAs are represented in Fig. 4(a) over the Poincaré sphere, as respectively), S1 and S2 follow a sinusoidal behavior as a function of blue and red curves for the aRB and the aTB analyzers respectively. The . Therefore, users interested in measuring some specific differences observed between the theoretical prediction (Fig. 2(a)) and polarimeteric information can take profit of the data in Fig. 3(a). In this the experimental implementation (Fig. 4(a)) are mainly related with sense, if variances for the linear content want to be equally balanced, extra retardations introduced by total internal reflection occurring in we have to select the orientations  {22.5º, 67.5º, 112.5º or 157.5º}, prisms as well as by reflections at metal thin film in beam splitters (see these values being related with the S1 and S2 crossings in Fig. 3(a) Fig. 1). For a better understanding of these differences, the aRB case (variances of 0.0041). By contrast, if noise amplification in S1 or S2 (blue curve) is following analyzed. The ideal situation is represented in wants to be minimized, the QWP orientations to be selected are {0º, Fig. 4(b). The linear PAs set by the biaxial crystal (red empty spots in 90º or 180º} and {45º or 135º}, leading to variances of 0.0028 for S1 Fig. 4(b)) are transformed by the QWP, which performs a rotation and S2, respectively. In the case of the configuration (b) (Fig. 3(b)), (with a value of 1=/2) around its fast axis (FA), the FA being set in the plane S1-S2. In the case represented in Fig. 4(b), the FA is oriented S0 and S1 remain constant as a function of the retardance  – at 0 degrees, so the QWP fast axis coincides with the S1 axis. However, whereas S0 shows the same value than that in configuration (a), S1 due to the effect of prisms and beam-splitters, which act as an remain constant to the minimum value reached for the configuration equivalent retarder, the resultant PAs are rotated once again, this time represented by red circles. In all the cases, the SD values are lower than an angle 2 around the equivalent retarder FA (back dashed line in Fig. 0.3º, but redundancy decreases this value by following the statistical 4(c)), which will also be in the S1-S2 plane, but in general, to a certain law 1 / N . Note that with our proposed polarimeter the SD values angle of the QWP fast axis. This new transformation defines the final decrease without a measuring time increase, because data redundancy PAs (orange spots in Fig. 4(c)), they being displaced from the desired is achieved spatially, not from time-sequential measurements. ones (blue spots). Contrary to expectations from data in Fig. 2(a), the In conclusion, we provided a new optical scheme for a snapshot QWP orientation is not negligible in terms of CN and it must be and complete CR based polarimeter. To perform complete polarimetric optimized in the experimental implementation. To achieve the optimal measurements, such polarimeters are nowadays based on bulky configuration (Fig. 4(d)), the FA of the equivalent retarder has to be systems (they require two biaxial crystals and two CCD cameras coincident with the S2 axis, so PAs rotations produced by the properly synchronized) [6, 9]. Other proposal based on more faceable equivalent retarder do not affect the system in terms of CN, optical arrangements are restricted to perform linear polarization independently of the 2 value (Fig. 4(d)). This is achieved when the metrology [10, 17]. Unlike this, the proposed optical set-up leads to a QWP and the equivalent retarder FAs form a relative angle of 45º much cheaper (one biaxial crystal and one CCD camera) and (angles on the Poincaré sphere are doubled). simpler (in-line) set-up. The polarimeter was optimized by following the condition number criteria (CN=2 and CN=2.21 for the theoretical and experimental configurations, respectively). Finally, the CR polarimeter was experimentally implemented and tested by measuring different controlled input SoPs. Regarding to the accuracy, the maximum absolute errors, in terms of the polarimetric angles azimuth and ellipticity, are of 1.56% and 1.30% respectively. By contrast, the repeatability was tested by calculating the standard deviation (SV) for a linear polarization measured 20 times, this leading (in the case of N=720) to a SV of 0.04% both for the azimuth and ellipticity. These parameters provide the optimized performance of the proposed polarimeter. Fig. 4. (a) Experimental analyzers over the Poincaré sphere; (b) Scheme of the theoretical analyzers for the QWP-biaxial crystal system; (c) Effect of the extra retardance introduced by prisms and BS; and (d) Analyzers obtained when the relative angle between the QWP and the equivalent retarder fast axes is optimized. We performed a new experimental analyzers calibration but this time the QWP and the equivalent retarder were oriented to 45º one to Fig. 5. (a) Azimuth (left-axis) and ellipticity (right-axis) values for the each other. By following the same geometrical argument discussed CR polarimeter (black crosses) and for the commercial one (red above, an extra QWP was also set in the transmitted beam to properly circles) related to different SoPs; (b) Azimuth (green squares) and obtain the desired linear analyzers aTB. Experimental results are given ellipticity (red circles) absolute errors for the same SoPs; and (c) over the Poincare sphere in Fig. 4(e) (blue line for aRB and red line for Azimuth (green squares) and ellipticity (red circles) SV as a function of aTB). The calibrated PAs are now very close to the theoretical ones (Fig. the analyzers number (N), for the measure of a linear polarization at 2(a)), they leading to an experimental CN equal to 2.21 (the optimal CN 25º. for CR based polarimeters is 2.0). The implemented polarimeter quality was tested by analyzing its Funding. Spanish MINECO (FIS2015-66328-C3-1-R, BES-2013- accuracy and repeatability. First, the accuracy was studied. We 064046 and fondos FEDER); Catalan Government (SGR 2014-1639). generated different input SoPs with our PSG system. In particular, we set a fixed voltage (2.25 V) to the second PA panel in the PSG, while the References first PA voltage was varied from 0.9-8.5 V. By doing this, the generated 1. N. Uribe-Patarroyo, A. Alvarez-Herrero, R. L. Heredero, J. C. del Toro polarizations describe a cut over the Poincaré sphere and they were Iniesta, A. C. López Jiménez, V. Domingo, J. L. Gasent, L. Jochum, and V. measured both by using our proposed polarimeter (for the Martínez Pillet, Phys. Status Solidi C 5, 1041 (2008) configuration shown in Fig. 4(e); CN = 2.21) and by means of a 2. E. Garcia-Caurel, R. Ossikovski, M. Foldyna, A. Pierangelo, A. De Martino, commercial polarimeter (PAN 5710VIS, S/N: M60217605; distributed B. Drévillon, in Ellipsometry at the Nanoscale, Ch. 2, (Springer, 2013), by Thorlabs). 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