Traditional polariscopes are applied in optical crys-tallography to identify mineralogical specimens.In these polarizing microscopes a small region ofthe sample is illuminated with a collimated or astrongly convergent light beam so that, first, anorthoscopic evaluation �Fig. 1�a�� and, second, aconoscopic evaluation �Fig. 1�b��1,2 can be per-formed. In both cases, a homogeneous region of athin sample with plane-parallel faces orthogonal tothe optical axis is used. Polarimeters in which anorthoscopic illumination is used are also applied inthe measurement of the linear birefringence or thelinear retardation of polarizing optical elementssuch as wave-plate retarders. In this case the areaunder evaluation is not a small region of the spec-imen, and the homogeneous sample is thicker. Inany of these cases the borders of the sample do notcontribute to the interference pattern observed atthe output.
Gradient-index �GRIN� lenses are a different typeof specimen. First, they have the shape of a rightcircular cylinder with almost plane and parallel in-put and output faces, their diameters range from 1to 4 mm, and their thicknesses are larger than 2.5mm.3 Their cross sections are, in many cases,smaller than the cross section of a laser beam;hence the illumination beam must be tailored to
D. Tentori ([email protected]) and J. Camacho ([email protected]) are with the Centro de Investigacion y Centro deEnsenanza Superior de Ensenada, Fısica Aplicada, Ensenada,Baja California, Mexico.
avoid the contribution of the borders of these spec-imens. GRIN rods are not homogeneous; theypresent a strong refractive-index change betweenthe rod axis and the edge ��n � 0.3�. In this casethe intensity contribution introduced by Fresnel co-efficients must also be controlled. GRIN lenses arelenslike media with a parabolic refractive-indexprofile that introduces an on-axis aberration thatmust be taken into account when they are illumi-nated with either a collimated or a convergent beamof light. Furthermore, their birefringence has aradial orientation and a radial spatial variation.It is considered that their residual birefringence isproduced during the ion-exchange process used fortheir manufacture. The volume and polarizabilitydifferences between the original and the exchangedions introduce mechanical stresses that, becausethe free surface is cylindrical, present a radial ori-entation. As a result, the polarization modes areradial and tangential,4,5 and the birefringenceshows a radial variation.6
Though the birefringence evaluation of GRINlenses can also be performed with a polariscopicsetup, special care is required. The main difficultiescome from the fact that the edges of the sample andits lenslike performance introduce contributions withthe same radial symmetry predicted for the cono-scopic patterns.7 They can be produced by �1� theoverlapping of the geometric shadows of out-of-focusimages of the entrance and the exit faces of the GRINlens at the plane of observation, �2� total internalreflection at the lateral face, �3� the light wave frontfolding over in the neighborhood of a focusing point,and �4� diffraction at the input or output faces or bothof the GRIN lens.
In this paper we present conoscopic patterns ob-tained for one-quarter- and one-half-pitch GRIN
lenses using linear and circular polariscopes. In ad-dition to the birefringence information, the cono-scopic patterns we present contain contributionsintroduced by different phenomena. We also in-clude a practical optical configuration that can beused to investigate the birefringence performance ofGRIN lenses. Their design characteristics avoid theinfidelity sources that modify the conoscopic pat-terns.
2. Polarization Description of a Gradient-Index Lens
In the conoscopic evaluation of crystals, the differ-ence between the values of the transmission coeffi-cient of linearly polarized light parallel to theincidence plane and the transmission coefficient oflinearly polarized light perpendicular to the inci-dence plane is neglected. For GRIN lenses, owingto the strong radial change of the refractive index,this contribution is easily observed in a null linearpolariscope �Fig. 2�.7 Nevertheless, this transmit-tance variation can be negligible when a large ex-posure time is used to record the conoscopic
patterns from a null linear polariscope �Fig. 3� orwhen the evaluation is accomplished with a circularpolariscope. In Fig. 3 we can see how the Fresnelcontribution to optical transmission is diminishedas the exposure time is increased from 20 to 80 ms.We can also mention that for low exposure energiesthe intensity distribution produced by the Fresnelcontribution might be understood as an almost nullbirefringence along the whole cross section of theGRIN lens.
If we neglect the Fresnel coefficient contribution,the polarization description of the GRIN lens issimpler. In this paper we follow that approach.The polarization matrix we use describes the bire-fringence properties of the GRIN lens, slicing it intothin meridional planes �Fig. 2�.7 The azimuth ori-entation of each meridional plane is defined withrespect to the positive branch of the x axis.8 It hasbeen shown that along each one of these meridionalplanes the GRIN lens behaves as a linear retarderwhose retardation, for each point on the observationplane, depends on the optical path difference be-tween the ray coupled to the tangential polarizationmode and the ray coupled to the radial polarizationmode reaching that particular point. The polariza-tion matrix for any point �r, �� on the observationplane is
Fig. 1. In traditional polariscopes the sample is placed betweentwo orthogonal polarizers. �a� Orthoscopic setup. �b� Cono-scopic setup. In both cases the observation is made at the out-put, on a plane perpendicular to the optical axis located after theanalyzer.
Fig. 2. GRIN sample in a null linear polarizer. Along each thinmeridional plane, the GRIN lens behaves like a linear retarderwhose retardation value has a radial dependence.
Fig. 3. Conoscopic pattern obtained in a null linear polariscopewith a convergent beam of light �not from a spatial filter� focusedclose to the input face of a one-quarter-pitch GRIN lens. Theobservation plane is the output face of the GRIN sample. As theexposure time increases, the dark square produced by the Fresnelcontribution to intensity becomes negligible. �a� 20 ms, �b� 50 ms,�c� 80 ms.
where �r is the total retardation angle between thetangential and the radial modes for the point on theobservation plane with radial position r.
3. Ray-Tracing Model
We assume that the parabolic refractive-index profileof the two polarization modes �radial and tangential�satisfies
n2�r� � N02 � B2r2, (2)
where N0 is the refractive index on axis and B is aconstant for each polarization mode. Applying Fer-mat’s principle to an isotropic parabolic GRIN me-dium described by Eq. �2�, Marchand9 has shown thatthe trajectories followed by light rays can be de-scribed by use of a right Cartesian coordinate systemas
x � x0 cos z� �p0
Bsin z� ,
y � y0 cos z� �q0
Bsin z� , (3)
where
z� � �B�l0� z. (4)
The rod axis lies along the optical axis �z axis�, andthe input face is located at z 0 �Fig. 4�. The inci-dent ray is characterized by its input position on theentrance face of the GRIN lens r0 �x0, y0� and itspropagation direction. The propagation direction ofthe ray, inside the GRIN medium and in the imme-diate vicinity of the input face, is determined by thedirection cosines �p0, q0, l0� defined as
p0 � n�r0�cos ,
q0 � n�r0�cos �,
l0 � n�r0�cos �, (5)
where angles , �, and � in Eqs. �5� are the anglesformed by the ray and the x, y, and z axes, respec-tively, i.e.,
cos �drdx
, cos � �drdy
, cos � �drdz
. (6)
In what follows, these equations are used to deter-mine the light trajectories for a convergent lightbeam focused on the vertex of the entrance face anda collimated beam of light traveling parallel to theoptical axis. Because in this paper we present someimages in which the sample is slightly misaligned, we
start in both cases by describing the ray-tracing equa-tions for a misaligned beam of light.
A. Convergent Incident Beam
When our incident beam of light is a spherical con-vergent beam not focused on the optical axis, we canstill take advantage of the circular cylindrical sym-metry of the sample, selecting for the ray-tracingprocedure the rays traveling along the meridionalplane that contains the focusing point, i.e., a planethat contains the optical axis and the focusing point�Fig. 4�a��. For simplicity, we select this meridionalplane as the xz plane. In this case, for all rayswithin this plane, y0 0 and q0 0, and we candescribe the light paths of these meridional rays us-ing only the first equation in Eqs. �3�.
If we consider an incident meridional ray impingingon the entrance face of the GRIN lens at �x, y� �x0, 0�and forming with the normal an angle , we can show,using Fig. 4�a� and applying Snell’s law, that
x � x0 cos z� �sin
Bsin z� , (7)
z� �Bz
�N02 � B2r0
2 � sin2 �1�2 . (8)
Fig. 4. Ray tracing along a meridional plane. �a� Convergentincident beam focused close to the input face of the GRIN lens. �b�Collimated incident beam. The angles of the director cosines ofthe ray inside the GRIN medium are , 0, and �. The angle ofincidence at the input face is . The angle of incidence at theoutput face is �.
These equations describe both the ordinary and theextraordinary rays when � 0 or when the birefrin-gence is almost null.10
If r0 �x0, y0� �0, 0�, i.e., r0 0, the trajectoriesof the rays focused on the vertex of the input face ofthe GRIN lens are described by the equation
x �sin
Bsin z� , (9)
with
z� �Bz
�N02 � sin2 �1�2 . (10)
Equation �10� indicates that the length Lf covering acomplete cycle �z� 2��, i.e., the pitch length, varieswith the angle of incidence ,
Lf �2�
B�N0
2 � sin2 , (11)
and with the refractive-index-profile parameters N0and B.
Using Eqs. �6� and �9�, as well as Fig. 4�a�, we cancalculate the angle of incidence of the ray on a planenormal to the axis, located in a position z on theoptical axis:
� � arcsin� �sin
�N02 � sin2 �1�2 cos� Bz
�N02 � sin2 �1�2�� .
(12)
We can see from Eq. �12� that at any position z thebeam shows a nonuniform divergence.
B. Collimated Incident Beam
For a collimated incident beam of light that travelsforming an angle with the optical axis, we can selectagain those rays contained in the meridional planeshown in Fig. 4�b�. In this case also, for these raysy0 0 and q0 0, p0 sin , and its trajectory withinthe GRIN lens is described by the first equation inEqs. �3� and by Eq. �4�. If the collimated beam oflight travels parallel to the optical axis � 0�, thetrajectories of the rays satisfy
x � x0 cos z� , (13)
z� �Bz
�N02 � B2x0
2�1�2 . (14)
Equation �14� indicates that the length Lc covering acomplete cycle �z� 2��, i.e., the pitch length for therays in a collimated beam of light traveling parallel tothe optical axis, varies with the radial incidence po-sition r0 x0,
Lc �2�
B�N0
2 � B2x02, (15)
and with the refractive-index-profile parameters N0and B.
Using Eqs. �6� and �13�, as well as Fig. 4�b�, we cancalculate the angle of incidence of the ray on a planenormal to the axis, located at a position z on theoptical axis:
� � arcsin� �x0 B�N0
2 � B2x02�1�2 cos� Bz
�N02 � B2x0
2�1�2�� .
(16)
Equation �16� tells us that at any position z the beamshows a nonuniform divergence.
C. On-Axis Aberration
If we send a collimated light beam � 0� through aone-quarter-pitch lens, the light beam will focus closeto the exit face. Because the pitch length varieswith the initial radial position, the position of thefocusing point also depends on the initial radial po-sition. This behavior is shown in Fig. 5�a� for a Sel-foc lens with N0 1.608, B 0.545 mm�1, and adiameter of 1.8 mm. We note that for marginal rays�r0 0.9 mm� the focusing point is located at z 4.41mm while for paraxial rays, z � 4.63 mm. There isa region along the axis of the GRIN lens in which rays
Fig. 5. Ray-tracing calculations for a collimated beam of lighttraveling parallel to the optical axis. The sample is a one-quarter-pitch GRIN lens for 633 nm. �a� The intersection of each incidentray with the optical axis varies with its input radial position. �b�Close to the focusing point, some of the output radial positionsoverlap an �0.8-mm region along the optical axis.
coming from different initial radial positions inter-sect with each other. This axial zone ��0.8 mm� isshown for the same lens in Fig. 5�b�.
GRIN lenses whose lengths are equal to any oddmultiple of one-quarter-pitch length, illuminatedwith a collimated beam of light traveling parallel tothe optical axis, or GRIN lenses whose lengths equalany multiple of one-half-pitch length, illuminatedwith a convergent beam of light focused on the vertexof the input face, focus the incident beam close to thevertex of the output face of the GRIN lens, showingbehavior similar to that presented in Fig. 5. We canobserve in Fig. 6 six conoscopic images obtained forthree different positions of the observation plane lo-cated close to a focusing point. The sample was aone-quarter-pitch lens in a linear polariscope. Fig-ures 6�b�, 6�d�, and 6�f � were obtained with a null
linear polariscope. Figures 6�a�, 6�c�, and 6�e� wererecorded at the same positions of the observationplane used for corresponding Figs. 6�b�, 6�d�, and 6�f �.For Figs. 6�a�, 6�c�, and 6�e� the axis of the analyzerwas aligned with the polarizer axis. The exposuretime was the same in all cases. Considering the twoimages obtained for each different position of the ob-servation plane, we can observe that there is a con-trast reversal between both images, and, for theimages obtained with a null polariscope, there is adiscontinuity in the fringe pattern that is due to the180° phase shift between the electric fields at bothsides of the isogyres �Ref. 7, Eq. �22��. Althoughthese conoscopic patterns present the phenomeno-logical behavior predicted by the theory, the addi-tional modulation added by the folding of eachpolarization-mode wave front is undesirable. In
Fig. 6. Conoscopic images obtained near the focusing point; the same exposure time was used. The sample was a one-quarter-pitchGRIN lens illuminated with a collimated beam of light parallel to the optical axis. The observation plane was behind the output face�inside the GRIN lens�. From left to right the distance from the observation plane to the output face of the GRIN lens was increased ��1to 3 mm�. For images �a�, �c�, and �e�, the analyzer and the polarizer were aligned. Images �b�, �d�, and �f � were obtained at the samepositions as the corresponding images �a�, �c�, �e�, with the analyzer orthogonal to the polarizer.
this case the complexity added by the longitudinalaberration obscures the interpretation of the cono-scopic patterns in the neighborhood of a focusingpoint.
If we use a convergent beam of light focused on thevertex of the input face of a one-quarter-pitch-lengthGRIN lens �or any odd multiple of one-quarter pitch�,the output beam is nearly collimated; hence, for anypoint along a plane normal to the optical axis, in theneighborhood of the exit face of the GRIN lens, thereis a single angle of incidence at the input face of theGRIN sample, for each output position on the obser-vation plane, for each polarization mode. Neverthe-less, we must remember that the emerging beam isnot completely collimated but shows a small diver-gence �Eq. �12��. In addition, for a GRIN lens whoselength is equal to a multiple of one-half-pitch length,illuminated with a collimated beam of light, there isa single input position for each output position on theobservation plane �close to the output face of theGRIN lens�, for each polarization mode. In this casea small divergence is also present �Eq. �16��.
4. Polariscopic Evaluation of Gradient-Index Lenses
To cancel out the angular intensity contribution in-troduced by isogires, we work with circularly polar-ized light. A simplified diagram of the null circularpolariscope we used is shown in Fig. 7. The inputcircular polarizer and the circular analyzer are builtwith quarter-wave plates and linear polarizers. Thecombination of a linear polarizer with an azimuthangle of �45° and a quarter-wave plate with an azi-muth angle of 0° is used as the input right circularpolarizer. The combination of a quarter-wave platewith an azimuth angle of 0° and a linear polarizerwith an azimuth angle equal to �45° forms the cir-cular analyzer. The sample is placed between thecircular polarizer and the circular analyzer.
The input circularly polarized beam of light is righthanded; it can be either collimated or focused on thevertex of the entrance face of the GRIN lens. Theequations we present can be applied to an observa-tion plane located inside the GRIN lens, viewedthrough the circular analyzer. The polarized field atthe output is
14 � 1 �i
�1 �i�MG�r, ��� 1�i�
� 12
sin�r
2�sin 2� � i cos 2���1
1�12
cos�r
2 � 1�1�
. (17)
From Eq. �17� we find that the output intensity isgiven by
I � sin2 �r
2
cos2 �r
2
. (18)
The intensity distribution associated with relation�18� shows us that the conoscopic pattern is an inter-ference pattern formed by continuous circular con-centric fringes. Bright and dark fringes will beobtained, for a circular analyzer built with a linearpolarizer with an azimuth angle of �45°, for thoseradial positions for which
�r � �2m � 1��,
�r � 2m�, (19)
respectively, where m is an integer. If the azimuthangle of the linear polarizer forming part of the cir-cular analyzer is �45°, a contrast reversal is ob-tained.
The optical path difference OPD�r, z� between theradial and the tangential polarization modes at theobservation point on the output plane of the GRINlens is related to �r by
�r �2�
�OPD�r, z�, (20)
where � is the wavelength of the monochromaticbeam of light used for the birefringence evaluation.
A. Optical Path for a Convergent Beam of Light
To determine the optical path difference we use theequation9
OP�r, z� �1l0 �
0
z
n2�r�dz, (21)
developed by Marchand for a transverse GRIN. Theoptical path OPR,T�r, z� of the radial and the tangen-tial modes, at the radial position r x on the obser-vation plane located at z, can be calculated by thesubstitution of Eqs. �2�, �9�, and �10� into Eq. �21�.When r0 x0 0, the optical path is given by
OPR,T�r, z� � � N02
�N02 � sin2 �1�2 �
sin2
2�N02 � sin2 ��z
�sin2
4BR,Tsin
2BR,Tz�N0
2 � sin2 �1�2 , (22)
where R and T are used to denote the radial and thetangential polarization modes.
Fig. 7. GRIN sample in the circular polariscope used in this pa-per. A linear polarizer with an azimuth angle equal to 45° and aquarter-wave plate �azimuth angle of 0°� formed the input circularpolarizer. A quarter-wave plate �azimuth angle of 0°� and a linearpolarizer whose azimuth angle can be 45° �null circular polari-scope� or �45° formed the circular analyzer.
The optical path OPR,T�r, z� of the radial and thetangential modes, at the radial position r x on theobservation plane located at z, can be calculated foran incident collimated beam by the substitution ofEqs. �2�, �13�, and �14� into Eq. �21�. When the col-limated beam of light travels parallel to the opticalaxis,
OPR,T�r, z� � z�N02 � �BR,T
2x02�2�
�N02 � BR,T
2x02�1�2� �
BR,Tx02
4
� sin2BR,Tz
�N02 � BR,T
2x02�1�2 . (23)
C. Optical Path Difference
The optical path difference between the rays of theradial polarization mode and the tangential polariza-tion mode reaching the same radial position r at aplane located at z is
OPD�r, z� � OPR�r, z� � OPT�r, z�. (24)
For a circular analyzer built with a linear polarizerwith an azimuth angle of �45°, bright and dark in-terference fringes will be observed for
OPD�r, z� � �m �12��,
OPD�r, z� � m�, (25)
respectively, where m is an integer. When the azi-muth angle of the linear polarizer forming part of thecircular analyzer is �45°, a contrast reversal is ob-tained.
The equations presented in this paper were ob-tained assuming that Snell’s refraction law holds atthe input face of the GRIN lens. Hence they can beapplied off axis to both polarization states only if theinput beam of light is collimated or nearly collimatedbecause, as we know, it is only in that case thatSnell’s law is valid for both polarization modes.10
On axis the birefringence of GRIN samples is null,4–6
so Snell’s law can be applied to both polarizationmodes for any incidence angle. We must also re-member that the polarization matrix we use �Eq. �1��was obtained assuming that all the rays are in phaseat the input face of the GRIN lens. Hence this ma-trix description is valid only for a convergent lightbeam focused on the vertex of the input face or for acollimated beam of light traveling parallel to the op-tical axis.
5. Location of the Observation Plane
In traditional polariscopes, when an orthoscopic ob-servation is used the sample is illuminated with acollimated linearly polarized beam of light �Fig. 1�a��,and, because the sample is not a lenslike medium, theoutput beam of light is also collimated. For cono-scopic observations a convergent beam of linearly po-larized light is focused on the sample, and an outputlens is used to collimate the emerging beam �Fig.
1�b��. The observation plane in both cases is locatedat any position along the collimated output beam.1,2
In the case of GRIN lenses, owing to its lenslikeperformance, when its length is equal to any oddmultiple of one-quarter-pitch length, only one lens isrequired. In the case of a sample whose length isequal to any multiple of one-half-pitch length, theoptical setup shown in Fig. 1�a� can be used to studythe birefringence properties of these specimens.But, because of the nonzero on-axis aberration ofGRIN lenses, the optical polariscope used for the bi-refringence analysis of GRIN rods requires the selec-tion of a fixed observation plane because the form ofthe interference pattern changes with the position ofthe observation plane. When our observation planeis inside or outside the GRIN sample and not close tothe input or output faces of the GRIN lens, we observethe out-of-focus images of the input and the outputfaces of the GRIN sample. The type of result we getis shown in Fig. 8. We obtained Fig. 8�a� by focusingthe convergent beam inside the sample, as is shownin Fig. 1�b�. Figure 8�b� was obtained with an ob-servation plane located near the output face of theGRIN sample. As we can see, the Fresnel contribu-tion and a radial light intensity distribution are ob-served; the symmetry of these patterns is similar tothat predicted for a conoscopic pattern, but we arejust looking at the geometrical shadows of the inputand output faces of the GRIN sample.
If we select an observation plane located inside theGRIN lens, owing to the on-axis aberration, the cono-scopic pattern will be distorted. The origin of thisdistortion can be understood when the radial varia-tion of the optical path is calculated. To do so, weuse a GRIN lens with N0 1.608, B 0.545 mm�1,a diameter of 1.8 mm, and a length of 4.63 mm.Figure 9�a� shows the optical path variation of a con-vergent light beam focused on the vertex of the inputface of the GRIN lens �numerical aperture 0.46�.Figure 9�b� shows the optical path variation for thesame lens illuminated with a collimated beam of light� 0�. We can see from both graphs that the opticalpath is shorter on axis, so when we focus our viewingsystem on a plane inside the GRIN lens, the image ofthe intersection of the lateral face of the GRIN sam-ple with the selected plane does not contain the imageof the vertex of this geometrical plane.
To calculate the optical path difference using Eqs.�22� and �23�, we must be able to focus our viewingsystem on a geometrical plane �position z�. This ispossible if the plane is located on the output face ofthe GRIN lens or outside it, in front of the GRIN lens.If we select an observation plane located outside theGRIN lens, in an arbitrary position, we need to takeinto account light refraction at the output face of theGRIN sample because light is not collimated butrather shows a small divergence �Eqs. �12� and �16��.The determination of the additional optical path dif-ference can be avoided if the plane of observation islocated on the exit face of the GRIN lens.
Once we have selected the output face of the GRINsample as the observation plane, we must remember
that light should not be focused on this plane becausethe wave front folds over, and for some radial posi-tions on the output face we will have more than oneray from the same polarization mode reaching thatpoint. The emerging light beam must be nearly col-limated, so we must use a convergent input beamfocused on the vertex of the input face of the GRINlens for samples whose length is equal to an oddmultiple of one-quarter-pitch length and a collimatedinput beam, traveling parallel to the optical axis, forsamples whose length is equal to a multiple of one-half-pitch length.
6. Influence of the Borders and the Lateral Face of theCylindrical Samples
When the lateral size of the input beam is larger thanthe optical diameter of the GRIN sample, diffraction
effects are easily observed �Fig. 10�. The pattern ofcircular concentric rings introduced by diffraction issuperimposed onto the conoscopic pattern, addingcomplexity and obscuring the evaluation procedure.In particular, the image in Fig. 10 was obtained whenFig. 8. Conoscopic patterns obtained for a one-quarter-pitch
GRIN lens illuminated with a convergent light beam focused insidethe sample. �a� Observation plane located inside the sample. �b�Observation plane located outside the sample.
Fig. 9. Optical path calculated for a one-quarter-pitch GRIN lensfor 633 nm. �a� The incident beam is a convergent beam focusedon the vertex of the GRIN lens’s input face. �b� The incident beamis a collimated light beam traveling parallel to the optical axis.
Fig. 10. Comparison between two conoscopic patterns obtainedwith a linear polariscope. The sample �a half-pitch GRIN lens�was illuminated with a collimated beam of light. �top� The ana-lyzer was orthogonal to the input polarizer. �bottom� The ana-lyzer was aligned with the input polarizer.
a one-half-pitch GRIN lens was illuminated with acollimated beam of light traveling parallel to the op-tical axis. Thus the emerging beam is nearly colli-mated at the output face. The conoscopic patternshown here was observed at the output face of theGRIN lens. The concentric fringes close to the axisof the GRIN rod can be understood as a nonzerobirefringence for that region, a result opposite to thereported measurements and observations.4–6
If the input beam is not diffracted at the input face,but its numerical aperture is larger than the numer-ical aperture of the GRIN lens, total internal reflec-tion at the lateral face adds several reflected beams.The light beam coupled to the tangential polarizationmode can be reflected polarized as a tangential or aradial mode, and the light beam coupled to the radialmode can be reflected with a radial or a tangentialpolarization. In each case the angle for which thesereflections �total internal reflection� are obtained isdifferent because the index of refraction for each po-larization mode is different. The complicated pat-tern resulting from these reflections is shown in Fig.11. The random pattern of the outer ring is theimage of the metallic region surrounding the pinhole,illuminated by light reflected at the input face of theGRIN sample �amplified speckle pattern�. The in-ner rings are conoscopic patterns produced by lightreflected, at different angles, from the lateral face ofthe GRIN lens. As we can see, these contributionsintroduce strong changes, so they must be avoided.
7. Circular Polariscope
A detailed diagram of the circular polariscope, withone-quarter-pitch GRIN lenses �or odd multiples ofone-quarter-pitch length�, that we propose to workwith is shown in Fig. 12. The light source must be
spatially coherent because the interference is pro-duced between rays coming from different parts ofthe beam. It should also be monochromatic becausethe optical path difference shows a fast variation outof the paraxial region. We used a He–Ne laser �633nm, TEM00, 50 mW�. To be able to follow the align-ment procedure covering the whole range of illumi-nation and introducing a minimum disturbance tothe alignment of the optical setup, we have found thatit is convenient to use a plate polarizer P0 locatedbetween the laser source and the input polarizer.By changing the relative orientation of the axis of thisadditional polarizer with respect to the polarizationaxis of the linear polarizer P1 in the input circularpolarizer, we can easily control the intensity of theinput light beam. A right circular polarizer, formedby a calcite prism polarizer �Glan–Thompson, azi-muth 45°� and a quartz quarter-wave plate �azi-muth 0°�, was used to produce our input polarizedbeam of light.
Circularly polarized light was focused on theGRIN lens’s input face by use of a spatial filter SF�Newport, Model 915� formed by a low-numerical-aperture microscope objective and a pinholemounted on a micropositioning stage. We selectedthe spatial filter by taking into account the wide flatarea that surrounds the pinhole; it allows us toplace the GRIN sample in contact with the pinhole.The numerical aperture of the input beam must belower than the numerical aperture of the GRIN rod�0.46 or 0.60�3 to avoid reflections at the cylindricalsurface of the GRIN lens and large enough to illu-minate a wide area of the GRIN sample’s crosssection. The use of this illumination also preventsnoise introduced by diffraction at the edges of theinput and output lens faces.
In an ideal circular polariscope, the input and out-put faces of each birefringent element �linear polar-izers, retarders� are flat and perpendicular to theoptical axis. In practice, fabrication tolerances forpolarization optics are large enough to contribute tothe deformation of the conoscopic patterns. These
Fig. 11. Conoscopic pattern obtained in a linear null polariscopefor a one-quarter-pitch GRIN lens illuminated by a convergentlight beam focused close to the vertex of the input face. Thenumerical aperture of the incident beam is larger than the numer-ical aperture of the GRIN lens. The observation plane is locatedinside the GRIN lens.
Fig. 12. Diagram of the experimental arrangement used for theconoscopic evaluation of the birefringence of one-quarter-pitchGRIN samples. In combination with the circular polariscope, weuse a plate polarizer P0 at the input to control the light intensity.The incident convergent beam is produced with a spatial filter.The projecting lens L is a dissection microscope objective �invertedorientation�. The conoscopic pattern on the output face of theGRIN lens is projected on a CCD arrangement.
optical elements are wedges. The optical path dif-ference they introduce varies �3� over a 1-cm2 diam-eter area. There is a compromise among opticalalignment, the performance of the input circular po-larizer, and light reflected from the faces of the opti-cal elements behind the microscope objective.Unwanted reflections imaged on the pinhole may de-teriorate the intensity distribution of our input beamof light �Fig. 13�.
To place the GRIN lens in close contact with thepinhole, we must mount it on a thin and narrow rigidsupport; we used a 1�32-in. �1 in. 2.54 cm� stainlesssteel wire. To align the sample, we must fix thisrigid support to a five-degree-of-freedom microposi-tioning mount. The xyz micropositioning stage weused to hold up the fiber is an optical fiber coupler�Melles Griot, Model 17AMB003�MD�. This is acompact and reliable option for controlling the lineartranslation of the specimen. To have full control ofthe angles between the optical axis of the polariscopeand the GRIN rod axis ��, ��, we must place thismicropositioning stage on top of a rotation and aninclination mount. These rotary stages are used tomodify angles � and � �Fig. 12�.
At the output we used a circular analyzer formedby a quartz quarter-wave plate �azimuth 0°� and aplate polarizer P2 �azimuth �45°�. The observa-tion plane we used is in the output face of the GRINlens. To obtain an amplified image of the GRINlens’s output face, we used a wide-field microscopeobjective of a dissection polarizing microscope as pro-jection lens L �Olympus, Model DF PLAN 1X�. Theworking distance of this microscope objective, whenused with an inverted orientation, is large enough sothat the objective can be inserted between the GRINsample, which is followed by the quarter-wave plateand the linear polarizer mounted on their rotatingmechanical mounts, and the detector. The image ofthe selected object plane was projected on the detec-tor array of a CCD camera �Electrim Corporation,EDC-1000U�. To locate the image of the object
Fig. 13. When the directly transmitted beam of light is combinedwith beams reflected and redirected along the optical axis in theforward direction, we obtain at the pinhole the interference pro-duced by their superposition. The circle at the right is the outputface of the GRIN lens.
Fig. 14. When the convergent beam of light is focused at thevertex of the GRIN sample, a 90° rotation of the linear polarizer inthe circular analyzer produces a contrast inversion, as is predictedby the theory. �a� Conoscopic pattern obtained with a null circu-lar polariscope. �b� Conoscopic pattern obtained when the inputcircular polarizer and the circular analyzer were aligned. �c� Ifthe input beam is not focused at the vertex of the input face, thereis an additional optical phase that varies with the angle of inci-dence, modifying the typical result obtained for plane-incidentwave fronts. This image was obtained with the same sample inthe same null circular polariscope.
plane, we laterally illuminated the output face of theGRIN sample with white light by use of two multi-mode optical fibers at the right and left sides of theoutput face of the GRIN lens.
Although we assume that we are observing theconoscopic pattern at the output face of the GRINlens, owing to the different divergence of each polar-ized beam, the real interference pattern we are re-cording is formed on the plane of the CCD.Although for the paraxial region the optical path con-tribution of the circular analyzer is negligible, theconoscopic pattern off axis is quite sensitive to thereal optical path traveled by each ray before reachingthe plane of the detector. The alignment and opticaltolerances of the linear retarder and the linear polar-izer used to build up the circular analyzer may intro-duce important modifications in the conoscopicpattern outside of the paraxial region. In tradi-tional polariscopes these optical elements are locatedbehind the collimating lens. In our case the finallens is an amplification lens, so it is better to insertthe circular analyzer before the projection lens be-cause the beam divergence is lower before this lens.
During the alignment procedure it is more conve-nient to work with a circular analyzer aligned withthe input circular polarizer. Using this configura-tion, we have more light, as we can observe all imagesin which the input polarizer and the analyzer arealigned. As the manufacturer states, the lightpower coupled to the orthogonal mode is lower thanthe light power that preserves its polarization state�Ref. 3, page 6�. When the sample is almost alignedwith the optical setup, we can rotate the polarizer P2so that the circular analyzer is orthogonal to the in-put circular polarizer and finish the alignment pro-cedure.
In Fig. 14 we observe three images of the GRINlens’s output face illuminated with white light andwith the laser light beam; the same exposure timewas used to record these images. The numericalaperture of the microscope objective used to focus theinput light beam at the vertex of the GRIN samplewas 0.25 �10��. The image in Fig. 14�b� shows thatthe illuminated region on the output face is small; thelight beam does not reach the lens edges. Figure14�a� was obtained with a circular analyzer orthogo-nal to the input circular polarizer. In Fig. 14�b� thecircular analyzer is aligned with the input circularpolarizer �second part in relation �18��. When theinput beam of light is not focused on the vertex of theinput face of the GRIN lens, there is an additionalcontribution to phase that is not included in relation�18�. Figure 14�c� shows us that in this case theinitial phase distribution modifies the intensity dis-tribution of the conoscopic pattern. From Figs. 14�a�and 14�b� we can clearly observe that birefringence is
almost null in a wide region in the neighborhood ofthe GRIN rod axis. The information contained inthese conoscopic patterns is in agreement with otherreported results4–6 and is opposite to that containedin the images of Figs. 10 and 11, where severalfringes are observed close to the GRIN rod axis.
8. Conclusion
Traditional conoscopic methods can be applied to thebirefringence evaluation of GRIN lenses if the specialcharacteristics of these samples are taken into ac-count. The polariscope design must �1� minimizethe contribution of Fresnel coefficients at the inputand output faces, �2� avoid the contribution of theedges of the entrance and output faces, �3� avoid thecontribution of the lateral face of these cylindricalsamples, and �4� take into account the on-axis aber-ration produced by the parabolic refractive-index pro-file.
A polariscope that satisfies these conditions andthe results obtained for a one-quarter-pitch GRINlens have been presented.
This research was supported by Consejo Nacionalde Ciencia y Tecnologı, project G-37000 and by thescholarship granted by Associacion Nacional de Uni-versidades e Instituciones de Educacion Superior toJavier Camacho Gonzalez. The authors appreciatethe valuable technical assistance of Miguel Farfan inthe preparation of some experimental setups.
References1. F. D. Bloss, Introduccion a Los Metodos de la Cristalografia
Optica, 5th ed. �Ediciones Omega, Barcelona, 1994�, pp. 124–128 �Translated from An Introduction to the Methods of OpticalCrystallography �Holt, Rinehart & Winston, New York, 1961��.
2. S. Tolansky, An Introduction to Interferometry, 2nd ed. �Long-man, London, 1973�, p. 203.
4. W. A. Wozniak, “Residual birefringence in gradient index lens-es,” Opt. Appl. 19, 429–437 �1989�.
5. W. Su and J. A. Gilbert, “Birefringent properties of diametri-cally loaded gradient-index lenses,” Appl. Opt. 35, 4772–4781�1986�.
6. D. Tentori and J. Camacho, “Birefringence characterization ofone-quarter-pitch GRIN lenses,” Opt. Eng. 41, 2468–2475(2002).
7. J. Camacho and D. Tentori, “Polarization optics of GRIN lens-es,” J. Opt. A: Pure Appl. Opt. 3, 89–95 �2001�.
8. R. C. Jones, “A new calculus for the treatment of optical sys-tems. I. Description and discussion of the calculus,” J. Opt.Soc. Am. 31, 488–493 �1941�.
9. E. W. Marchand, Gradient Index Optics �Academic, New York,1978�, pp. 53–60.
10. M. Francon and S. Mallick, Polarization Interferometers: Ap-plications in Microscopy and Macroscopy �Wiley Interscience,London, 1971�, pp. 140.
