Holography has been described as capable of record-ing all the information carried by a wavefront. How-ever, in a normal holographic setup the informationabout the state of polarization of light is lost. Severalschemes to record the polarization state have beenproposed, such as the use of two separate referencebeams [1,2], or the use of a mosaic polarizer [3]. Aseparate two-reference-beam scheme for metal sur-faces was proposed by Garvanska [4]. These methodsuse standard silver halide holographic emulsionsas the recording media. A different approach is to de-velop a polarization-sensitive emulsion [5]. Themajority of available techniques employed in themeasurement of polarization rotation of light use in-tensity variation through an analyzing polarizer. Themain problem associated with this approach is therequirement for an absolute intensity measurement;therefore any unbalanced stray light does affect themeasurement. This is particularly difficult in situa-tions in which small-angle polarization rotation is
being considered. A method that to a large extentavoids some of the problems is given in [6], althoughit requires a succession of three images to obtain thepolarization information and therefore is not suita-ble for single-shot events. A technique that couldbe used for single-shot events is Savart plates [7],which consist of two polarizing beam splitting plateswith orthogonal optic axes. An incident beam propa-gating through plate 1 is resolved into ordinaryand extraordinary beams, which are displaced fromeach other in the first principal plane. Upon enteringplate 2, the ordinary beam becomes an extraordinarybeam, and vice versa. Splitting again takes place inthe second principal plane, which is orthogonal to thefirst. The result is two emerging beams displacedalong a diagonal. A set of interference fringes is ob-tained, in which the separation of the fringes is pro-portional to the plate thickness and the contrast is ameasure of the polarization angle. The main problemof this method is that stray light does affect thefringe contrast measurement.
In this paper we present a method to obtain thepolarization state of an arbitrary wavefront. Thisis achieved in a two-stage process. Two holograms
are recorded on the same holographic plate. The onlydifference between the two holograms is that theirreference beams are at a small angle with respectto each other and each has a different state ofpo>larization. The two reference beams are linearlypolarized, and their axes are at right angles to eachother. The dual hologram is reconstructed by using asingle beam, giving rise to two reconstructed wave-fronts bearing the same angle between them as thetwo reference beams. The reconstructed wavefrontsinterfere, generating a set of parallel fringes. Fromthe fringe contrast the polarization state of differentpoints of the wavefront are obtained. The methodis suitable for phase or reflective objects. It can beapplied to steady-state or transient situations.
2. Theoretical Background and Experimental Setup
The experimental configuration, shown in Fig. 1,consists of a Nd:YAG CW frequency-doubled laser(532nm, single longitudinal mode, 20mW). The laserbeam is separated into two beams; one is used asreference and the other as object beam. In the pathof each beam a half-wave plate (not shown) is locatedin order to control the polarization state of eachbeam. In the reference beam path a 0°400 polarizingwedge is located. The wedge generates two referencebeams with orthogonal polarizations and with anangular separation of 0:01 rad. One of the resultingreference beams polarization is parallel to the planeof the optical table, and the other’s is perpendicularto it. The object beam polarization is rotated by 10°relative to the polarization of one of the referencebeams to generate a contrast bias.The test object consists of a half-wave plate that
covers only half of the field of view; the other halfis left clear. Therefore the object consists of two ad-jacent areas of different polarizations; the polari-zation of the area in which the half-wave plate is
located can be modified by rotating the half waveplate. Image plane holograms, as well as Fresnelholograms, were made in this set of experiments.During the recording stage, two holograms, one foreach polarization component of the object beam,are formed on the same holographic emulsion. Afterdevelopment, a single beam is used to reconstruct thedual hologram, generating two reconstructed beams(one from each hologram obtained from the two refer-ence beams). The two reconstructed beams providetwo wavefronts from the same object, albeit at aslight angle given by the polarizing wedge. The re-constructed wavefronts correspond to the two polari-zation states of the object. The two reconstructedbeams have the same polarization state, as theyare obtained from the same illuminating beam.Therefore a set of parallel interference fringes isobtained. The fringe separation is controlled by theangle of the polarizing wedge.
We will use Jones matrix calculus to describe oursetup. The Jones vector for a 45° linearly polarizedbeam is given by
R�45° �1���2
p�10
�; �1�
and the Jones matrix for a polarizing wedge is givenby
JPW ��e�iω∥x 00 e�iω⊥x
�;
where ω∥ � �2π × sin θ∥�=λ and ω⊥ � �2π × sin θ⊥�=λ;θ∥ and θ⊥ are the beam angles of the parallel and per-pendicular polarization components relative to theholographic plate normal.
In our experiment the reference beam is obtainedby shining a 45° linearly polarized beam on a polar-izing wedge; the output will therefore be
Rr � JPW _R�45° �1���2
p�e�iω∥x
e�iω⊥x
�; �2�
The object beam goes through a half-wave plate toobtain a rotation of the original beam to providethe contrast bias, to go through the object. For thesake of simplicity, in what follows we will assumethat at any stage it is possible to normalize the beamamplitude, which does not affect the end result. TheJones matrix for an object that rotates the inputpolarization by an angle ϕ�x; y� is given by
JO ��e�iϕ�x;y�=2 0
0 eiϕ�x;y�=2
�: �3�
The input beam on the object has a fixed polarizationbias 2α, which for our case is 2α � 10°, and is given by
The last term of Eq. (9) shows that we have a set ofparallel fringes, corresponding to the last term inparenthesis, modulated by the polarization rotationof the object. The fringe separation is given by thedifference between ω∥ and ω⊥, which is providedby the polarizing wedge. The angle of the referencebeam to the object used in all our experiments is12°, and the angular separation of the two polariza-tions that the polarizing wedge provides is 0°400. Thelaser beam was filtered by using a spatial filter, re-sulting in a Gaussian beam of about 2 cm, in whichonly the central 1 cm was used for the experiment.The intensity variation across the beam was lessthan 20%.The relative amplitude of each of the two recon-
structed beams depends on the polarization angle be-tween the object beam and each of the referencebeams. For example, if at a particular point of the ob-ject the polarization is aligned with that of one of the
reference beams, the reconstruction of the corre-sponding hologram will have maximum amplitude,whereas the reconstruction from the other hologramwill have zero amplitude. Their amplitudes becomeequal for an angle of 45°. For an object in which thereis no rotation a set of parallel fringes of constant con-trast will be obtained, given that the object beam hasan arbitrary bias of 2α � 10°. The no-polarization-rotation area in the object provides a contrast biasfrom which it is possible to determine whether rota-tion is toward higher or lower polarization angles.The contrast bias is obtained by a 10° polarizationrotation of the object illumination beam. If the anglebetween the polarization of the object and one of thereference beams is θ, the amplitude of the recon-structed beams will be
A∥ � A sin θ; �10�
A⊥ � A cos θ; �11�
and the corresponding intensities will be
I∥ � A2 sin2θ; �12�
I⊥ � A2 cos2θ: �13�
As the two reconstructed beams have the same po-larization state, because they are obtained fromthe same illuminating reconstructing beam, theyinterfere. The interference between the two recon-structed beams generates a set of parallel interfer-ence fringes. The fringe separation is controlled bythe angle between the two beams. The fringe con-trast [8] is given by
V � Imax � Imin
Imax � Imin� 2
�������������������������sin2θ cos2θ
p: �14�
As it turns out, the fringe contrast is dependent onlyon the polarization state. From Eq. (9) it is apparentthat θ � ϕ�x; y� � 2α.
3. Experimental Procedure and Results
In general, holographic emulsions have different re-sponses that depend on the type of emulsion or thebatch in which they were made. To obtain absolutemeasurements it is essential to perform a calibra-tion. For calibration purposes an area of the objectwith a known polarization rotation is left; for allcases shown it is 10°. Figure 2 shows a reconstructedimage in which the right-hand side has a rotation of10° to be used as calibration. Reconstructed imagesare recorded by using a CCD camera with 1024 ×244 pixels and 16 bit resolution (SPH6 Apogee cam-era). An estimate of the minimum contrast from aCCD camera of a sine wave form can be obtained.In order to define a sine wave, at least six levels arerequired. For an 8 bit camera at mid-intensity the
visibility will be V8 bit � 6=128 � 0:047. It is appar-ent that measuring a 1% change in contrast requiresa camera with a minimum of 10 bits. From the recon-structed image, which is the interference patternbetween the two images formed with each of thereference beams, an intensity profile is obtained,as is shown in Fig. 3. The two profiles are along linesperpendicular to the fringes in both the known polar-ization bias region and the unknown polarizationregion. From the profile extreme values are obtained.The values thus obtained are used to calculatethe fringe contrast. The contrast is an averagefrom around ten successive fringes. The contrast ob-tained from the known bias region (10°) is V �0:342. From the values of contrast obtained in theknown contrast area a calibration factor is obtained.The calibration factor is the ratio of 0.342 (contrastfor the 10° area) to the measured value of contrast onthe 10° area for the case in study. All values obtainedfrom the image are subsequently multiplied by thissole calibration factor.The procedure has been applied to several types of
commercial holographic emulsion, which are de-tailed in Table 1. In the foregoing discussion itwas assumed that the emulsion has a linear responserange and that the holograms were recorded on thelinear range. We point out that the majority of testswere done with a solution physical developer [9],which has an extremely wide dynamic range, severalothers of magnitude of exposure energy. Furthertests are required on other types of developer; how-ever, as we apply a contrast bias, and the contrastvariations are around this bias value, any developerought to give comparable results.As can be seen from Table 1, there is a fairly broad
dispersion of values of the normalizing factor for thedifferent emulsions; however, after normalization
the data points fall within a few percent of the sametheoretical curve.
In Fig. 4 the theoretical curve for the fringe con-trast is shown in conjunction with the normalizedvalues for one of the emulsions tested. It is apparentthat the measured values are well within the errorbars of the measurement. Similar results wereobtained for all emulsions tested.
Although it is possible to obtain two beams ofperpendicular polarization with different angulardirections using a combination of polarizing beamsplitters and mirrors, the proposed scheme using apolarizing wedge is significantly simpler to align.Several of these schemes were tried, and alwaysthe presence of spurious fringes would render the re-construction unusable. The optical alignment of thepolarizing wedge setup is nearly as simple as a stan-dard Fresnel hologram.
To assess the method described for measuring po-larization rotation, a set of measurements of knownsamples was carried out. Two different solutionswere used in a 1 cm × 1 cm cross-section cuvette. A25mm diameter object beam was used, leaving anarea around the cuvette that was used to obtainthe calibration factor. Tartaric acid and saccharosewere used as solute to make a solution in water.For both cases we used 41 g of solute for 100 g ofsolution. The results are given in Table 2.
One of the advantages of the method proposed forpolarization rotation is that it does have spatialresolution. To test this aspect of the method, inthe same cuvette a solution with a gradient was ob-tained by slowly pouring two different concentra-tions. A solution of the same concentration used inthe previous cases was poured into the cuvette, withpure water slowly poured on top of it. Initially somemixing does occur at the interface. It is necessary towait around 20 min to obtain a useful gradient. The
Fig. 2. Image of the reconstruction of a hologram. The reconstruc-tion is divided into two zones; the right-hand side has a bias of 10°.
Fig. 3. (Color online) Typical line profile obtained from the imagein Fig. 2.
Table 1. Tested Emulsions and Corresponding Typical Normalizing Factors Obtained
Emulsion Beam Ratio Exposure Energy μJ=cm2 Process Normalizing Factor
results obtained are shown in Fig. 5, which demon-strate the capability of the method to obtain spatialinformation. Each division is approximately 1mm onthe actual object.
4. Conclusions
A method suitable for measuring the polarizationrotation of coherent light has been proposed, whichis a two-stage process. Initially two holograms arerecorded simultaneously on the same holographicemulsion, using reference beams that have mutuallyperpendicular polarizations. The two reconstructedimages from the hologram interfere to produce aset of parallel fringes; the rotation measurementsare obtained from the fringe contrast. This methodis well suited for performing two-dimensional mea-surements with spatial resolution. This method isparticularly well suited to transient phenomena,as in dense plasma studies. For this type of study,a pulsed laser can obtain the polarogram, to be ana-lyzed at a later stage. This type of study is notfeasible using [6], as it requires the acquisition of sev-eral images to obtain a measurement. To provide a
calibrating factor for the holographic emulsion usedit is necessary to have an area with known rotation.The values obtained from themeasurements have er-rors that are less than 5%, and spatial variation canalso be obtained. One of the interesting advantagesof the present measurement technique is that nonco-herent light does not affect the results obtained. Thisis because only the coherent light from the laser con-tributes to the hologram formation; any incoherentlight will not show up in the reconstruction of theholograms.
The authors gratefully acknowledge funding underFondo Nacional de Desarrollo Científico y Tecnológi-co (FONDECYT) grant 1030958.
References
1. A. W. Lohmann, “Reconstruction of vectorial wavefronts,”Appl. Opt. 4, 1667–1668 (1965).
2. K. Gasvik, “Holographic reconstruction of the state of polari-zation,” J. Mod. Opt. 22, 189–206 (1975).
3. H. Kubo and R. Nagata, “Further consideration of photoelas-ticity using polarization holography,” J. Mod. Opt. 23, 519–528 (1976).
4. D. Garvanska, “Polarization holography applied to detectionof shape deviations of metal surfaces,” J. Opt. 12, 201–206 (1971).
5. T. Todorov, L. Nikolova, and N. Tomova, “Polarization hologra-phy. 1: new high-efficiency organic material with reversiblephotoinduced birefringence,” Appl. Opt. 23 4309–4312 (1984).
6. R. J. Wijngaarden, K. Heeck, M. Welling, R. Limburg, M. Pan-netier, K. van Zetten, V. L. G. Roorda, and A. R. Voorwinden,“Fast imaging polarimeter for magneto-optical investiga-tions”, Rev. Sci. Instrum. 72, 2661–2664 (2001).
7. M. Born and E. Wolf, Principles of Optics, 4th ed. (Perga-mon, 1970).
8. W. T. Welford, Optics, Vol. 14 of Oxford Physics Series (OxfordUniversity Press, 1976).
9. R. Aliaga, H. Chuaqui, and P. Pedraza, “Achival and wide ex-posure latitude process for holography,” Appl. Opt. 29, 2861–2863 (1990).
Fig. 5. (Color online) Comparison of normalized measured valuesfor a typical case against the theoretical curve.
Fig. 4. (Color online) Polarization rotation along the length of thecell in which a solution with a gradient concentration of tartaricacid was established.
Table 2. Rotation Values for Solutions (41 g Solute, 100 g Solution)
