The birefringence properties of gradient-index�GRIN� lenses are important because of their wideapplication and special characteristics. In addition,the size and symmetry of GRIN lenses place them, asoptical elements, between bulk optics and fiber optics.Because the medium is not homogeneous, we canfollow a light beam before it reaches the lateral sur-face and separate the guiding effect associated withthe lateral surface from light propagation inside themedium. Optical methods and models developed forthe assessment of bulk nonisotropic optical media canbe applied to GRIN lenses, giving us information onlight propagation for each polarization mode. Wehave studied the propagation of the polarizationmodes inside the GRIN anisotropic medium, makinguse of the ray optics methods developed for anisotro-pic media1,2 and a polarization-mode description sup-ported by experimental work.3–9
Brandenburg10 showed that mechanical and ther-mal stresses, introduced during the ion-exchangeprocess used to generate the refractive-index profile,modify the isotropic structure of the original glassbeing the source of their intrinsic birefringence. Us-
D. Tentori �[email protected]� and J. Camacho �[email protected]� are with Centro de Investigacion y Centro de EnsenanzaSuperior de Ensenada, Fısica Aplicada, Ensenada, Baja California,Mexico.
Received 25 October 2002; revised manuscript received 23 April2003.
ing this result and taking into account the radialsymmetry of these elastic stresses, Su and Gilbert3
concluded that the principal birefringence directionsinduced by the fabrication process are radial and tan-gential and that the mechanical equilibrium at therod axis forces birefringence along the rod axis to becanceled. Using different optical techniques andGRIN rod samples, several authors found that theirexperimental results can be explained, assuming thatthe polarization modes are radial and tangential, andhave reported that the birefringence at the GRIN rodaxis is in fact null.3–9 Furthermore, when radial andtangential polarization modes were used, it has beenconcluded that the measured intrinsic birefringenceexhibits a circular cylindrical symmetry.7–9 Despiteadvances, there is a shortcoming. Until now the po-larization optics developed for these materials hasnot included the different refractive properties of or-dinary and extraordinary rays. In this work we pro-pose an electromagnetic representation for polarizedrays and relate the polarization modes of light withthe symmetry and physical properties of the medium.This model is a first step toward a better character-ization of the birefringence properties of materialswith a circular cylindrical symmetry.
Although it is accepted that the birefringence sym-metry of gradient-index lenses is different from thatof natural crystals, it is considered that the point-to-point refractive-index response of these optical me-dia,5,8 or any other optical medium, can be describedby using the index ellipsoid.1 In this work wepresent a theoretical investigation based on use of awave solution of Maxwell’s equations and a local in-dex ellipsoid that follows the circular cylindrical sym-
metry of the medium. In this study we link thephysical properties of the medium with the velocity ofpropagation of each polarization mode. The analy-sis here follows the procedure used by other authorsto analyze the propagation direction of plane wavesin crystals.1,2 In our case the analysis is restrictedto an arbitrary point on the GRIN medium, and weassume that the amplitude and the phase evolutioncan be analyzed separately. To verify the validity ofour description two experiments were performed:�1� The GRIN lens sample was located in a circularoptical polariscope and illuminated off axis with alow-divergence light source located on the entranceface of the GRIN lens. �2� The GRIN lens was lo-cated in a linear polariscope, and a second lightsource with a larger divergence was focused on theentrance face of the GRIN sample. The informationcontained within the images produced in each case bythe off-axis illumination is analyzed with the ray-tracing equations developed by Marchand11 for anoptical medium with a parabolic refractive-index pro-file.
Using an off-axis point source, we can distinguishbetween the images formed by ordinary and extraor-dinary rays. Use of a higher divergence allows us todemonstrate that the conclusions obtained for a pointobject and meridional rays can also be applied to anextended object and skew rays.
This paper is organized as follows: First wepresent the ray-tracing equations used to analyze theimages formed by ordinary rays and the mode de-scription proposed for GRIN lenses. Following Si-mon,1 we use a wave solution and Maxwell’sequations in Cartesian coordinates to obtain vectoridentities that allow us to describe the evolution oflight waves along the cylindrical GRIN medium.We assume that the evolution of the light path andthe change in the amplitude of the wave are indepen-dent. We work only with those relations that in-volve the light-path evolution and rewrite the vectorexpressions previously obtained, using circular cylin-drical coordinates. Taking into account their geom-etry and the polarization modes used to describe thepropagation of polarized rays in GRIN lenses,4–9 werelate the phase and the energy propagation direc-tions with the physical properties of the medium, i.e.,with the magnetic induction constant and principaldielectric constants in circular cylindrical coordi-nates. The images obtained in this work for an off-axis object, formed by a one-quarter-pitch GRIN lens,show that the electromagnetic description presentedhere is not restricted to meridional rays but holdsalso for skew rays.
2. Meridional Rays in Gradient-Index Lenses
To describe light propagation in a cylindrical rod witha parabolic refractive-index profile, Marchand11 con-sidered that this medium has only a transverse vari-ation in its optical properties; i.e., the longitudinal
variation along a line parallel to the axis of the GRINrod is zero:
dn�r, ��
dz� 0. (1)
In this case, when the Fermat principle and a para-bolic refractive-index profile are used, the ray-tracingequations are
x � x0 cos z� �p0
Bsin z� ,
y � y0 cos z� �q0
Bsin z� . (2)
Here z� � �B�l0�z and the refractive-index profile con-sidered is
n2 � N02 � B2r2; (3)
n � n�r�, N0 is the value of the refractive index forisotropic rays for a point on axis, and B is a constantparameter whose value �inverse of longitude units�defines the magnitude of the refractive-index gradi-ent. In Eq. �2� it is considered that inside the GRINmedium a light ray leaves the first surface of theGRIN lens at �x0, y0, 0� and its initial propagationdirection within the GRIN lens is given by the direc-tor cosines of the ray multiplied by the refractiveindex at that point, i.e., by its optical director cosines�p0, q0, l0�.
Because of the symmetry of revolution about theoptical axis, to consider the propagation of any me-ridional ray �a ray traveling through a plane contain-ing the optical axis�, we can rotate our coordinatesystem so that our meridional ray lies on the x–zplane. In this case y0 � 0, q0 � 0, and Eq. �3� showsus that meridional rays continue traveling, withinthe GRIN lens, along their initial meridional plane.For a skew ray q0 � 0 and the trajectory followed bylight is the out-of-plane curve.
In Fig. 1 we show, for a quarter-pitch GRIN lens,two types of rays that belong to beams formed only bymeridional rays. One of them belongs to a colli-mated beam of light traveling parallel to the opticalaxis and the other to a convergent beam of light fo-cused on the intersection of the optical axis with the
Fig. 1. Path followed by two different meridional rays. Whenlight travels through a GRIN medium, it presents a continuouschange in its propagation direction. For meridional rays the lighttrajectory evolves along the incidence meridional plane.
entrance face of the GRIN lens. In both cases thepoint-to-point propagation direction of light is modi-fied following the relations given in Eq. �2�.
3. Birefringence Symmetry of Gradient-Index Lensesand Polarization Modes
The experimental results obtained for GRIN lensesilluminated with polarized light have been explained,considering that for this medium the polarizationmodes have radial and tangential orientations.4–9 Itis assumed that any polarized ray incident on a pointP��, �� on the entrance face of the GRIN lens couplesto a radial polarization mode whose polarization vec-tor is parallel to the meridional plane that intersectsthe GRIN lens input face along the radius � at thatparticular point or to a tangential polarization modewhose polarization vector is tangential to the circle ofradius � at that point. In Fig. 2 we show a schematicrepresentation of the cross section of a GRIN sample.As we can see, both polarization modes are orthogo-nal to each other at any point P��, ��.
Although strong evidence supports the existence ofthese radial and tangential modes, we must point outthat the experimental work reported in the literaturefor GRIN lenses used with polarized light has beenperformed with light beams formed only by meridi-onal rays. Wozniak,4 Montoya-Hernandez andMalacara-Hernandez,5 and Rouke and Moore7,8
worked with collimated light, while Su and Gilbert3
and Camacho and Tentori6,9 worked with a conver-gent beam of light focused on the vertex of the en-trance face of the GRIN lens. It is still necessary tobe able to discuss the performance of GRIN lensesworking with polarized skew rays. In this work wetake advantage of ray equations based on parabolicrefractive-index profiles to describe the imaging prop-erties of GRIN lenses. We analyze, considering thepredictions obtained for the electromagnetic wave so-lution presented in this work and ray-tracing equa-tions, the images obtained with polarized skew rays.
4. Light Waves in a Heterogeneous Medium
GRIN cylindrical rods are nonconducting, nonmag-netic media that do not absorb light. We assumethat the Maxwell equations that describe the electro-
magnetic properties of such media at any point P�x, y,z� are
� � H � �D��t, (4)
� � E � �B��t, (5)
� � D � 0, (6)
� � H � 0, (7)
and, since the medium is nonmagnetic,
B � 0H. (8)
This GRIN medium is nonhomogeneous, so the solu-tion to the Maxwell equations is not a plane wave.To overcome this problem, in this work we use amicroscopic point of view and assume that the wavesolution to the Maxwell equations that describes lightpropagation along the GRIN medium is given in Car-tesian coordinates by
E � E3 � x, y, z�exp i2�
� �� x, y, z� � u� x, y, z�t�,
(9)
H � H3 � x, y, z�exp i2�
� �� x, y, z� � u� x, y, z��,
(10)
D � D3 � x, y, z�exp i2�
� �� x, y, z� � u� x, y, z�t�,
(11)
where the amplitudes of the electric-field intensity,the magnetic-field intensity, and the electric displace-ment, E3, H3, and D3, are functions of the space coordi-nates, and ��x, y, z� and u�x, y, z� describe, respec-tively, the light wavefront and the phase velocity atpoint P�x, y, z�.
Substitution of the wave solutions given in Eqs.�9�–�11� into the Maxwell equations can be simplified,defining
M �2�
� �� x, y, z� � u� x, y, z�t�
and making use of
� exp�iM�
� x� i
2�
� ���
� x� t
�u� x�exp�iM�,
� exp�iM�
� y� i
2�
� ���
� y� t
�u� y�exp�iM�, (12)
� exp�iM�
� z� i
2�
� ���
� z� t
�u� z�exp�iM�,
d exp�iM�
dt� i
2�
�u exp�iM�. (13)
Fig. 2. Vector polarization modes �radial and tangential� for pointP with polar coordinates ��, ��. For GRIN lenses �or rods� therefractive-index profile and the birefringence present a circularcylindrical symmetry.
The equations obtained when the solutions given inEqs. �10� and �11� are substituted into Eq. �4� can beseparated into two independent equation systems,considering the point of view of geometrical optics,i.e., assuming that the phase evolution and ampli-tude evolution are independent. Under this scopethe first system is
Hz���
� y� � ���
� z�Hy � uDx,
Hx���
� z� � Hz���
� x� � uDy,
Hy���
� x� � Hx���
� y� � uDz. (14)
Defining
n � ���
� x,
��
� y,
��
� z� , (15)
we can write Eqs. �14� as
n � H � uD. (16)
Hence, for any light ray, at any point P�x, y, z� withinthe GRIN medium, vectors n and H are orthogonal tovector D. Note that from the vector calculus Eq. �15�indicates that n is normal to the light wavefront de-fined by ��x, y, z�. In our case n depends on theposition.
The second system of equations is
i2�
�t� H3 � �u� � � � H3 . (17)
Using Eq. �8� and substituting Eqs. �9� and �10� intoEq. �5�, we obtain again two equation systems. Thefirst can be written by using vector n, defined in Eq.�15�, as
n � E � u0H. (18)
Equation �19� indicates that E � H.The second equation system is
i2�
�t� E3 � �u� � � � E3 . (19)
We introduce now the solution given in Eq. �11� aswell as Eqs. �12� and �13� in Eq. �6�. We obtain twosystems of linearly independent equations. Thefirst system can be written as
� � D � n � D � 0. (20)
This result, like Eq. �16�, shows us that n � D.The second system of equations can be written as
� � D3 � 0. (21)
Substituting the solution given by Eq. �10� as well asEq. �12� in Eq. �7�, we obtain two systems of linearlyindependent equations.
The first system is
� � H � n � H � 0. (22)
The result is in agreement with Eqs. �16� and �18�,indicating that vectors n and H are orthogonal.
The second system of equations can be written as
� � H3 � 0. (23)
Resuming, Eqs. �16�, �18�, �20�, and �22� indicate thatvectors D, E, and n are coplanar, since they are or-thogonal to H, D and E are in the same plane, but ingeneral E is not normal to vector n �perpendicular tothe wavefront�. This result is common to lightwaves traveling through an optically anisotropic me-dium.1,2 In this case, as usual, it is convenient toremember that energy flows in the direction of thePoynting vector,
S � E � H, (24)
and to define a unit vector R, in the direction of thePoynting vector,
R �S
�S�. (25)
Following the vector analysis based on geometry, de-veloped by Simon,1 we represent all vectors lying ona plane normal to vector H in Fig. 3. In our case adifferential surface of the wavefront at point P�x, y, z�is normal to the plane of the figure. Using Fig. 3, wesee that E and D must form the same angle with Rand n, since R � E and n � D.
In an anisotropic medium the propagation direc-tion of light rays can be different from the propaga-tion direction of the wavefront, so it is useful to definevelocity � in the direction of energy flux in order tocompare this direction with that associated with thephase velocity u. From Fig. 3 we see that
�� � u cos �, � � u�cos �; (26)
�� holds for a material with positive anisotropy.�The propagation velocity of ordinary rays is higherthan the propagation velocity of extraordinaryrays.12� For material with a negative anisotropy thevelocity of extraordinary rays is labeled �.
Fig. 3. Relative orientation of those vectors located in a planeorthogonal to vector H.
It is convenient to define a unit vector n in the di-rection of phase change, since in general �n� � n � 1,
n �n
�n�. (27)
Substituting Eq. �18� into Eq. �16�, we have
E � �0 u2�n2�D � �n � E�n. (28)
The vectors in Eq. �28� are represented in Fig. 3.From Fig. 3 we can obtain a similar mathematicaldescription for vector R if we project vector 0u2D inthe direction of vector R. The result is
�0 u2�n2�D � E cos2 � � �0 u2�n2��R � D�R. (29)
Using Fig. 3, we can also show that
�0 u2�n2��R � D� � �n � E�cos �. (30)
Substituting Eqs. �26� and �30� into Eq. �29�, we ob-tain different relations for positive and negativeanisotropic materials:
�0 u2�n2�D ���
2
u2 E ���
u�n � E�R,
�0 u2�n2�D �u2
�2 E �
u�
�n � E�R. (31)
5. Light Waves in a Circularly Cylindrical BirefringentMedium
The nondiagonal components of the dielectric con-stant tensor of a GRIN lens, written in Cartesiancoordinates, are not null. Owing to its circular cy-lindrical anisotropy, for a GRIN lens
Dj � εjEj j � 1, 2, 3 (32)
only when we use a circular cylindrical coordinatesystem. Below we use a circular cylindrical coordi-nate system to deduce, taking advantage of the pre-vious analysis, the value of angle � and theexpressions relating the velocities u and � with theparameters of the medium �dielectric constant andpermittivity� for each polarization mode.
We make use of the standard orthogonal base ofunit vectors, e�, e�, ez, utilized to generate a coordi-nate system with a circular cylindrical symmetry.These base vectors are a linear combination of theCartesian unit vectors ex, ey, ez:
e� � cos �ex � sin �ey,
e� � sin �ex � cos �ey,
ez � ez. (33)
Using this base, we introduce Eq. �32� into the vectorequations that we obtained working with the wavesolution to Maxwell equations in Cartesian coordi-nates. To do so, we rewrite the vector equations thatwe use.
The component relations obtained when Eq. �16� isrewritten in circular cylindrical coordinates are
n�Hz � nz H� � uD�,
nz H� � n�Hz � uD�,
n�H� � n�H� � uDz. (34)
For Eq. �18� we obtain
n�Ez � nz E� � 0 uH�,
nz E� � n�Ez � 0 uH�,
n�E� � n�E� � 0 uHz. (35)
Equation �20� becomes
n�D� � n�D� � nz Dz � 0, (36)
and Eq. �22� is
n�H� � n�H� � nz Hz � 0. (37)
Using also circular cylindrical coordinates, we re-write Eq. �24� and obtain
S � e��E�Hz � H�Ez� � e��Ez H� � E�Hz�
� ez�E�H� � E�H��. (38)
Below we transform the relations obtained with Fig.3. The relations in Eq. �28� become
E� � �0 u2�n2�ε�E� � �n�E� � n�E� � nz Ez�n�,
E� � �0 u2�n2�ε�E� � �n�E� � n�E� � nz Ez�n�,
Ez � �0 u2�n2�εz Ez � �n�E� � n�E� � nz Ez�nz, (39)
and Eq. �31�, for a positive anisotropic medium, is
Assuming that the polarization modes are radial andtangential, any light wave with an electric displace-ment vector
D � �D�, D�, Dz� (41)
can be written in terms of two orthogonal polarizationmodes as
D � Dt � Dr. (42)
In order to use Eqs. �34�–�40� to determine at pointP��, �, z� the propagation direction of light for eachpolarization mode, we assume that the incident beamof light is completely polarized and has only one po-larization mode.
To deal with the medium characteristics associatedwith the polarization modes, we start our discussionby considering polarized meridional rays. In Figs. 4and 5 we consider how a ray of each polarizationmode couples to the GRIN medium. To be able tocompare the results obtained for both modes, for anylight trajectory, we use in both cases the same refer-ence system to define the state of polarization oflight13 and the spatial coordinates of the light path.In these circumstances, to get an input ray polarizedonly in the tangential or the radial mode, we need toconsider an input light ray that travels parallel to theoptical axis. We consider a collimated beam of light
reaching the flat entrance face of a GRIN sample, asshown in Figs. 4 and 5.
A. Tangential Mode
As we see from Fig. 4, if the light beam is initiallytangentially polarized,
Dt � �0, D�, 0�, (43)
the electric displacement vector before and after thechange in direction can be described by using Eq.�43�. Thus from Eq. �32�
Et � �0, E�, 0�. (44)
The tangential mode corresponds to the transverseelectric mode with respect to axis z. SubstitutingEq. �43� into Eqs. �35� and making use of Eq. �32�, wehave
n�ε�E� � 0, (45)
i.e.,
n� � 0. (46)
Equation �46� indicates that the wavefront is normaland is contained in a meridional plane. Substitut-ing Eqs. �42� and �45� into the first and third equa-tions of �34�, we have
nz H� � 0, n�H� � 0. (47)
Furthermore, since H � E, we must have, using Eq.�43�,
H�E� � H�E� � Hz Ez � H�E� � 0. (48)
For the tangential mode Eqs. �47� and �48� hold if
H� � 0. (49)
Thus, for this mode Ht � �H�, 0, Hz�, a vector orthog-onal to Dt and Et.
Now we substitute Eqs. �44�, �46�, and �49� intoEqs. �39� and �40�, obtaining
ut � �t� � �t � n� 10ε�
�1�2
. (50)
As we can see, for the tangential mode, rays travel inthe direction of propagation of the wavefront �� � 0�.This result is consistent with the fact that, in thiscase, Dt and Et are parallel. Note also from Eq. �50�that the relationship between the physical constantsof the medium and the velocity of light has a formvery similar to the usual form. Equation �50� indi-cates that for any point P��, �, z� in the GRIN mediumthe light propagation velocity depends on its initialpropagation direction.
B. Radial Mode
We see that for an incident ray of the radial mode,shown in Fig. 5, the change in the propagation direc-tion produces coupling of the radial polarization modeto the longitudinal polarization mode. To write
Fig. 4. Collimated beam of light reaching the flat entrance face ofa GRIN sample. Change of direction does not modify the polar-ization state of a meridional ray of the tangential polarizationmode.
Fig. 5. Collimated beam of light reaching the flat entrance face ofa GRIN sample. Part of the energy of the meridional incident ray,radially polarized, couples to the longitudinal polarization modewhen it reaches the entrance face of the GRIN sample.
properly the components of a meridional ray travel-ing through a GRIN rod, we must consider the un-avoidable presence of this coupling, so that
Dr � �D�, 0, Dz�. (51)
Since E � H and, from Eqs. �51� and �32� we knowthat Er lies on the plane ��, z�,
Hr � �0, H�, 0�. (52)
From Eq. �52� we see that the radial mode corre-sponds to the transverse magnetic mode, since themagnetic field is transverse to the z direction.
Substituting Eqs. �51� and �52� into Eqs. �35�,
n�Ez � 0, n�E� � 0. (53)
Equations �53� corroborate that the normal to thewavefront is contained within a meridional plane,i.e.,
n� � 0. (54)
Substituting Eqs. �51�, �52�, and �54� in Eqs. �34�, weobtain
n�ε�E� � nzεz Ez. (55)
Equation �55� can also be obtained from Eq. �36�.The first and third equations of �39� can be rewrit-
ten in cylindrical coordinates by using again Eqs.�51�, �52�, and �54�:
n2E� � 0 ur2ε�E� � �n�E� � nz Ez�n�n
2,
n2Ez � 0 ur2εz Ez � �n�E� � nz Ez�nz n2. (56)
Substituting Eq. �55� into Eqs. �56� and simplifying,we obtain
ur � � 10ε�
�1�2�nz2 � n�
2 ε�
εz�1�2
. (57)
To obtain an expression for the propagation veloc-ity of energy ��, we calculate the components of thePoynting vector for the radial mode, substituting Eqs.�51�, �52�, and �54� into Eq. �38�:
Sr � e��H�Ez� � e��0� � ez�E�H��, (58)
and in Eq. �25�,
R� � Ez
�E�2 � Ez
2�1�2 , R� � 0, Rz �E�
�E�2 � Ez
2�1�2 .
(59)
Then, substituting Eqs. �58� and �59� into Eqs. �40�,we obtain
0 ur2
n2 ε�E� ��r�
2
ur2 E� �
�r�
ur�n�E� � nz Ez�
�Ez
�E�2 � Ez
2�1�2 ,
0 ur2
n2 εz Ez ��r�
2
ur2 Ez �
�r�
ur�n�E� � nz Ez�
�E�
�E�2 � Ez
2�1�2 , (60a)
0 ur2
n2 ε�E� �ur
2
�r2 E� �
ur
�r
�n�E� � nz Ez�
�Ez
�E�2 � Ez
2�1�2 ,
0 ur2
n2 εz Ez �ur
2
�2 Ez �ur
�r
�n�E� � nz Ez�
�E�
�E�2 � Ez
2�1�2 . (60b)
The velocity of propagation of energy can be calcu-lated by substitution of Eq. �55� into Eqs. �60�.
�r� �1
n�0�1�2
�nz2 � n�
2ε��εz�3�2
�nz2 � n�
2ε�2�εz
2�1�2 ,
�r �n
�0�1�2 �nz
2 � n�2ε�
2�εz2
nz2 � n�
2ε��εz�1�2
. (61)
If
ε� � εz, (62)
substituting Eq. �62� into Eqs. �57� and �61�, we ob-tain
ur � �r� � �r � n� 10ε�
�1�2
; (63)
i.e., we obtain, for the rays of the radial mode, anequation similar to the typical expression for the ve-locity of light in terms of the physical properties of themedium, 0 and ε�. As expected, in this case thedirection of propagation of energy coincides with thedirection of propagation of the wavefront since Dr andEr are parallel.
We observe that for ε� � εz Eqs. �57� and �61� pre-dict that the direction of propagation of energy andthe direction of propagation of the wavefront are dif-ferent; i.e., in this case Snell’s law is not satisfied bythose rays coupled to the radial mode. The angulardeviation �� between those rays satisfying Snell’srefraction law and extraordinary rays satisfies
Part of the amplitude of a practical linearly polarizedincident skew ray is polarized along the tangentialdirection, while the other part is polarized in theradial direction. So, from the point of view of elec-tromagnetic theory, the analysis presented in Section6 could also be used to describe incident skew rays.
7. Conoscopic Setups
Two conoscopic setups were used. The first experi-ment was performed with a null circular polariscope,and for the second experiment a linear polariscopewas utilized. A detailed diagram of the null circularpolariscope that we used is shown in Fig. 6. In thistype of polariscope we avoid the presence of zerotransmittance zones �isogyres� obtained for null lin-ear polariscopes. The light source used was a polar-ized He–Ne laser �633 nm, TEM00, 50 mW�. A rightcircular polarizer, formed by a calcite prism polarizer�Glan Thompson; azimuth, 45°� and a quartz quarter-wave plate �azimuth � 0°� were used to produce ourinput polarized beam of light. Circularly polarizedlight was focused on the GRIN lens input face by aspatial filter �Newport Model 915� formed by a micro-scope objective �10�, N.A. � 0.25� and a pinhole �15m� mounted on a micropositioning stage.
This combination generates a small-sized objectformed by the Airy disk and one ring of the Fraun-hofer diffraction pattern of the lens aperture. Thesample, a Selfoc GRIN lens �1�4 pitch for 633 nm;N0 � 1.608; N.A., 0.46; length, 4.63 mm�, was fixed toa yz micropositioning stage formed by two lineartranslation mounts �Aerotech Model ATS 303,0.5-m resolution�. For full control of the angle be-tween the optical axis of the polariscope and theGRIN rod axis ��, ��, on top of these micropositioningstages was placed a rotation mount �Aerotech ModelATS301R� used to modify angle �, and the samplewas mounted on a metal strip that could be tilted asshown in Fig. 6 to modify angle �. This support alsoallowed us to place the GRIN lens in close contactwith the pinhole. At the output we used a circular
analyzer aligned with the input circular polarizer,formed by a quartz quarter-wave plate �azimuth �0°� and a plate polarizer �azimuth, 45°; clear aper-ture, 7 mm�. Since the lenslike contribution of theGRIN samples is produced by a parabolic refractive-index profile, there is a nonzero on-axis aberrationthat cannot be neglected. Hence this optical setuprequires the selection of a fixed observation plane.The observation plane that we used is located on theoutput face of the GRIN lens.14 To obtain an ampli-fied image of the GRIN lens output face, a wide-fieldmicroscope objective of a dissection polarizing micro-scope was used as a projection lens �Olympus ModelDF PLAN 1X� supported by an xyz mount. Theworking distance of this microscope objective, whenused with an inverted orientation, is large enough toinsert, between the GRIN sample and the detector,the quarter-wave plate, and the linear polarizermounted on their rotating mechanical mounts. Theimage of the selected object plane was projected onthe detector array of a CCD camera �Electrim Corpo-ration, EDC-1000U�. To locate the image of the ob-ject plane, the output face of the GRIN sample waslaterally illuminated with white light by using twomultimode optical fibers at the right and left sides ofthe output face of the GRIN lens.
In the second experiment we used a null linearpolariscope. In that case the input and the outputquarter-wave plates were removed. The calciteprism polarizer was used as input linear polarizers,and the linear analyzer was a plate polarizer fixed toa rotating mount. The divergence of the object beamwas also different; in addition to the Airy disk weincluded four diffraction rings. To generate this ob-ject beam, we used with the 10� microscope objectivea 25-m-diameter pinhole.
8. Experiments
As we mentioned above, in the first experiment thesize of the pinhole produces an object beam in whichin addition to the Airy disk we also have one ring ofthe Fraunhofer diffraction pattern of the microscopeobjective clear aperture. When this light beam isfocused on the input face of the GRIN lens at a radialposition located �5 m from the vertex �the intersec-tion between the optical axis and the entrance face ofthe GRIN lens�, we obtain on the output face of theGRIN lens the intensity distribution shown in Fig. 7.In Fig. 7 we can observe two intensity distributionsformed by the tangential and radial polarizationmodes. The additional rings are produced by dif-fraction at the pinhole edge. Between both light dis-tributions we can observe the interference patternproduced by the superposition of both beams, coupledat the analyzer. When we move the input objectfarther from the vertex, the brighter intensity distri-bution keeps its circular symmetry, while the eccen-tricity of the elliptical one is increased. Using thisconoscopic setup, we could observe these output in-tensity distributions only in the neighborhood of theoptical axis, since for an input off-axis position of �15m the propagation direction of the distorted inten-
Fig. 6. Experimental arrangement used to study the anisotropicresponse of quarter-pitch GRIN samples. The incident conver-gent beam is produced with a spatial filter, SF. The projectinglens L is a dissection microscope objective �inverted orientation�.The conoscopic pattern on the output face of the GRIN lens isprojected on a CCD arrangement.
sity distribution exceeded the clear aperture of thequarter-wave plate �7 mm�.
The second experiment was performed in a nulllinear polariscope, including the Airy disk and fourdiffraction rings in the illumination beam. The in-put convergent beam was focused on the input face ofthe GRIN lens, in the neighborhood of its optical axis,within the zero birefringence zone �off-axis position,�4 m�. The intensity distribution obtained whenthe analyzer is orthogonal to the input polarizer isshown in Fig. 8. In Fig. 8 there is a distorted inten-sity distribution that presents a higher angular vari-ation, and a lower intensity pattern, verticallydisplaced. In this second pattern the intensity dis-tribution associated with the Airy disk keeps its cir-cular shape. The intensity distribution of thediffraction rings surrounding the Airy disk of thesecond pattern is very low. When the linear ana-lyzer was rotated 70° we obtained the image shown inFig. 9. We can observe that in this case the intensitydistribution that we observe presents uniform lateralamplification �the number of rings is higher becauseof light diffraction at the pinhole edge�, and the in-tensity of the distorted distribution becomes negligi-ble. To obtain this image it was necessary to use aneutral filter between the light source and the inputpolarizer, since the intensity of the intensity distri-bution formed by ordinary rays is higher. In Fig. 10we present the output intensity distribution obtainedfor an intermediate angular orientation �27°� of theanalyzer �before the attenuator was inserted�. Aswe see, for this orientation of the linear analyzer bothmodes carry a similar intensity.
9. Discussion
The experimental results presented in Section 8 wereobtained by using an incident off-axis illuminationthat includes meridional and skew rays. We canobserve that in both experiments we obtained only
Fig. 7. Images of the intensity distribution at the output face of aGRIN lens �quarter pitch�, formed by the radial and the tangentialpolarization modes. The circularly polarized input object �locatedon the entrance face of the GRIN lens� was formed by the Airy diskand the first diffraction ring of the Fraunhofer diffraction patternof the microscope objective clear aperture. The tangential modeintensity distribution �ordinary rays� is not distorted while theradial mode intensity distribution �extraordinary rays� becomesdistorted.
Fig. 8. Dominant intensity distribution corresponding to the ex-traordinary rays of a small-sized object �linearly polarized� locatedat the entrance face of a quarter-pitch GRIN lens. The inputillumination is formed by the Airy disk and the first four diffractionrings of the Fraunhofer diffraction pattern of the microscope ob-jective clear aperture. The output intensity distribution associ-ated with the ordinary rays �vertically displaced� keeps thesymmetry of the input light beam. This image was obtained in anull linear polariscope.
Fig. 9. Dominant intensity distribution corresponding to the or-dinary rays of a small-sized object located at the entrance face of aquarter-pitch GRIN lens �with a linear polariscope�. The inputillumination is formed by the Airy disk and the first four diffractionrings of the Fraunhofer diffraction pattern of the microscope ob-jective clear aperture. The output intensity distribution associ-ated with the extraordinary rays is very low; only the zones withhigher brightness can be observed. This image was obtained byrotating the analyzer 70°.
two intensity distributions. They are associatedwith two off-axis light beams polarized in orthogonaldirections. The continuous intensity distributionsobtained for each diffraction ring in Figs. 7 and 10show that the light beams associated with both po-larization modes contain meridional as well as skewrays.
Going back to the theoretical analysis presentedhere, we can see that we have ordinary rays �raystraveling in the direction predicted by Fermat prin-ciple� when the light is tangentially polarized Eq.�43�� or when it is radially polarized, and the dielec-tric constant components along the radial and thelongitudinal directions have the same value Eqs.�62� and �63��. In this case we should observe onlyone ring pattern when ε� � ε� � εz, i.e., when there isno birefringence. When ε� � εz � ε� the differentpropagation velocity associated with each polariza-tion mode Eqs. �50� and �63�� can be described interms of a different refractive-index profile for eachpolarization mode. In this case we should observe,at the output face of the quarter-pitch GRIN sample,two ring patterns with different sizes.
If we assume that ε� � εz we must observe a radi-ally polarized pattern formed by rays traveling in thedirection of the Poynting vector; i.e., rays associatedwith the radial mode are extraordinary rays. Wesee from Eqs. �61� that for radially polarized rays thepropagation velocity at the same position ��, z� of theGRIN sample varies with the shape of the wave front;i.e., they cannot be described in terms of a parabolicrefractive-index profile. �This assumption is validonly within the paraxial region.9� Since the refrac-tive index that these rays “see” depends, for eachpoint P��, �� within the sample, on the angle of prop-agation � �measured with respect to the z axis�, as
previously proposed by Wozniak4 and Rouke andMoore,8 the intensity distribution of the ring patternproduced by these rays must be distorted.
We observe in Figs. 7, 8, and 10 that one of the lightdistributions does not keep the original shape, whilethe other does. From our discussion above we canassociate the undistorted ring pattern with ordinaryrays �the tangential polarization mode� and the dis-torted ring pattern with extraordinary rays �the ra-dial polarization mode�.
Note that in both cases the light distributions mustbe formed by meridional and skew rays. So we con-clude that the experimental results indicate that forthese samples ε� � εz, or the projection rules used todetermine the magnitude of the components of theelectric-field vector cannot be applied when light hascoupled to the medium.
10. Conclusions
We have presented a polarization-mode descriptionfor an anisotropic medium with a circular cylindricalsymmetry. Taking advantage of previous experi-mental results obtained for light beams formed onlyby meridional rays, we assume that there are twoorthogonal polarization modes oriented radially andtangentially. The images that we obtained for theintensity distributions obtained with a one-quarter-pitch GRIN lens, using an off-axis input object andthe model presented here, show that rays associatedwith the tangential mode behave as ordinary rayswhile rays associated with the radial mode are ex-traordinary rays. The experimental results pre-sented here also show that the electromagneticdescription proposed is not restricted to meridionalrays but can also be applied to skew rays.
This work was supported by Consejo Nacional deCiencia y Tecnologia CONACYT, project G-37000,and by a scholarship granted by Asociacion Nacionalde Universidades e Institutos de Educacion Superiorto Javier Camacho.
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