Tunable and fixed polarization systems have beenused in a variety of different fields, especially biologyandmedicine [1–6], material science [7,8], astronomy[9], and remote sensing [10–12]. They have been usedto convert any state of polarization of light from onearbitrary input state to any output state [13], to de-sign polarization-transforming elements equivalentto rotating quarter-wave plates and rotating half-wave plates [14], tunable linear polarization rotatorsand tunable polarizers [15], wavelength-independentcontinuous polarization rotators [16], and tunablefilters [17–19].
The matrix methods of Jones and Mueller [20–24]and the Poincaré sphere method [24,25] are the bestknown and widely used methods to describe theproperties of the polarization systems and polariza-tion transformations. The Poincaré sphere method isespecially useful for a geometrical visualization ofthe polarization state of a fully polarized electromag-netic wave in which the polarization is representedas a point on a sphere. The matrix method of Muellercan be used not only for fully polarized light but forpartially polarized light, as well. In the case of fully
polarized light, the 4 × 4Mueller matrix can be easilyderived from the 2 × 2 Jones matrix [23]. When theoptical system does not contain any partial polari-zers, but consists entirely of rotators and retardationplates, all of the matrices are unitary with unit de-terminant. In this case, the matrix method of Jonesis equivalent to the Poincaré sphere method becausethe group of unitary 2 × 2 matrices with unit deter-minant is isomorphic to the group of three-dimensional rotations.
It has been recently shown [26] that any homoge-neous nondepolarizing optical system described bythe Jones matrix can be presented as a combinationof four basic mechanisms: linear and circular phaseand linear and circular amplitude anisotropy in thefollowing order:
TGen ¼ TCPTLPTCATLA: ð1Þ
Here TLA is the Jones matrix of linear amplitude ani-sotropy, TCA is the Jones matrix of circular amplitudeanisotropy,TLP is the Jonesmatrix of linear phase an-isotropy, andTCP is the Jonesmatrix of circular phaseanisotropy. This generalized matrix equivalencetheorem is a direct generalization of the first andthe second Jones equivalence theorems [27].
The first Jones equivalence theorem states that anoptical system containing any number of retardationplates (polarization elements with linear phase
anisotropy) and rotators (polarization elements withcircular phase anisotropy) is optically equivalent to asystem containing only two elements—one a retarda-tion plate, and the other a rotator [27]. It can be ea-sily proved that any optical system containing onlyretardation plates is equivalent to an optical systemcontaining one retardation plate and one rotator [28]and the retardation plate is followed by the rotator.The Jones equivalence theorem allows to consider indetail the properties of the polarization system (op-tical activity, effective phase shift, orientation of thesystem axes), in contrast to other methods that firstof all consider the polarization transformationof light.
The first Jones theorem allows constructing polar-ization systems with desirable properties, namely,an adjustable half-wave plate [28], an adjustablequarter-wave plate [29,30], and an adjustable phaseretarder without optical activity [31]. The adjustablepolarization systems described in the papers [28–31]consist of retardation plates with almost arbitraryinvariable phase shifts; the adjustment of the sys-tems is achieved by tuning the angle between theslow axes of the phase plates.
It is possible to apply the first Jones theorem for de-scription of polarization systems containing polariza-tion elements with linear phase anisotropy controlledby applied voltage. For an example, the polarizationsystem allowing converting any state of polarizationof light from one arbitrary input state to any outputstate [13] contains three nematic liquid-crystal (LC)cells such that the relative orientations of these threecells are 45° with respect to one another, and theretardation of each cell is controlled by proper adjust-ment of the applied voltage. The polarization systemsequivalent to rotating quarter-wave plates and rotat-ing half-wave plates [14] consist of three and fournematic LC cells controlled by the applied voltage, re-spectively. These polarization systems can be easilydescribed in the frame of the equivalence theoremas the polarization systems containing one cell (retar-dation plate) and one rotator.
Although the equivalence theorems are widelydiscussed [32,33], they are not often used for real po-larization systems properties description.
Hereweare going to demonstrate the advantages ofthe first Jones equivalence theorem for describing theproperties of a polarization system containing two re-tardation plates, onewaveplatewith a variable phaseshift, and one wave plate with an invariable phaseshift, and show that, if an invariable phase shift isequal to 90° and the relative orientation of these twoplates is 45° with respect to one other, this polariza-tion system will be equivalent to a quarter-wavesystem with an adjustable axes orientation or a quar-ter-wave system with adjustable optical activity de-pending on light propagation direction.
2. Theory. Properties of Systems
Let us consider polarized light propagation through apolarization system consisting of one wave plate with
a variable phase shift Γv and one wave plate with aninvariable phase shift Γf .
The angle between the slow axes of the plates φ canbe changed by means of the second plate rotationonly. We introduce the system of coordinates X, Y ,which, for the first plate, coincide with the slowand the fast axes, respectively. We can change theproperties of the system by changing the values ofΓv and φ. It is known that a polarization system con-sisting of two components with phase shifts Γ1, Γ2,and the angle between the slow axes of the platesφ is equivalent to a system consisting of one phaseplate and one optical rotator [28]. The Jones matrixTðφ;Γ1;Γ2Þ of our polarization system can be ex-pressed in the following matrix form:
Tðφ;Γ1;Γ2Þ ¼ TCPðθÞTLPðε;Γeff Þ: ð2Þ
Here thematrix TCPðθÞ is the Jonesmatrix of circularphase anisotropy:
TCPðθÞ ¼�
cos θ sin θ− sin θ cos θ
�; ð3Þ
where θ is a phase shift introduced for two orthogonalcircular components of the electric vector or an angleof optical activity. The matrix TLPðε;Γeff Þ is the Jonesmatrix of linear phase anisotropy:
where Γeff is a value (i.e., phase shift between twoorthogonal linear components of the electric vector)and ε is an azimuth of the anisotropy (i.e., the anglebetween the slow axis of the retarder and the X axisof our coordinate system). According to [28] the va-lues Γeff , ε, and θ can be obtained from the followingequations:
If φ ¼ 45°, Eqs. (9)–(11), can be represented in thefollowing form:
cosðΓeff Þ ¼ cosðΓ1Þ · cosðΓ2Þ; ð12Þ
tanð2εÞ ¼ tanΓ2
sinΓ1; ð13Þ
tanðθÞ ¼ tanðΓ1=2Þ · tanðΓ2=2Þ: ð14Þ
It follows from Eq. (12) that effective phase shift willbe constant if cosðΓ1Þ ¼ 0ðΓ1 ¼ 90°Þ or cosðΓ2Þ ¼ 0ðΓ2 ¼ 90°Þ, that means that, if the phase shift ofany plate is equal to 90°, the effective phase shift willbe equal to 90°, and will not depend on the phaseshift of the other plate.
Let us consider two systems. The first systemconsists of the first plate with invariable phase shiftequal to 90°, that is Γ1 ¼ Γf ¼ 90° and the secondplate with the variable phase shift Γ2 ¼ Γv. The sec-ond system consists of the first plate with variablephase shift Γ1 ¼ Γv, and the second plate with theinvariable phase shift equal to 90°, Γ2 ¼ Γf ¼ 90°.The system of coordinates X ;Y coincides with theslow and the fast axes of the first plate, respectively,for the first and the second systems.
Let us describe the properties of the first polariza-tion system. It follows from Eqs. (13) and (14) thatthe angle of the effective axes orientation ε and theangle of the optical activity θ depend on the variablephase shift as
ε ¼ Γv
2; θ ¼ Γv
2: ð15Þ
The equivalent optical system of the first polariza-tion system under consideration consists of the quar-ter-wave plate followed by the rotator; the angle ofthe quarter-wave plate axis orientation ε and the an-gle of the optical activity θ are controlled by the ad-justment voltage applied, namely, ε ¼ θ ¼ Γv=2. TheJones matrix of this polarization system can be ex-pressed in the following matrix form:
Tðφ;Γf ;ΓvÞ¼TCPðθ¼Γv=2ÞTLPðε¼Γv=2;Γeff ¼90°Þ;ð16Þ
Tðφ;Γf ;ΓvÞ ¼1ffiffiffi2
p�
cosðΓv=2Þ sinðΓv=2Þ− sinðΓv=2Þ cosðΓv=2Þ
�
×�1 − i cosΓv i sinΓv
−i sinΓv 1þ i cosΓv
�: ð17Þ
This means that the first system is a quarter-waveplate with adjustable angle of optical activity θ andadjustable angle of axes orientation ε.
To investigate the polarization transformation of abeam of light propagating through the first polariza-tion system, let us consider how two elementary po-larizing elements (the quarter-wave plate and therotator) influence the light state of polarization re-presented by a point on the Poincaré sphere (Fig. 1).
Let the direction of a slow axis of the plate coincidewith the Poincaré sphere axis OX.
Consider the polarization transformation of abeam of light with any azimuth α0 of linear polariza-tion relative to the plate slow axis. It is assumed thatthe azimuth of the linear polarized light measuredfrom the slow plate axis in the anticlockwise direc-tion is positive. The point A on the equator of thePoincaré sphere represents the state of linear polar-ized light at the moment when the phase shift Γv ofthe active element is equal to zero Γv ¼ 0. The in-crease of the phase shift Γv results in the change ofthe plate axis orientation at the angle ε ¼ Γv=2 andthe change of the azimuth of linear polarization atthe same angle ε ¼ Γv=2 relative to the quarter-waveaxis, namely, αΓv
¼ 2α0 � Γv=2. Point A will movealong the equator of the Poincaré sphere. If Γv ¼2α0 − 90°, point A will move to point A0, which corre-sponds to the azimuth αΓv
¼ 45° relative to thequarter-wave axis. The change of the light polariza-tion after the passage of light through the quarter-wave plate corresponds to a rotation of the Poincarésphere around axis OX through the angle Γeff ¼ 90°.The beam passage through the quarter-wave platetakes it to the state A00, which is circularly polarized.Circular polarization is eigenpolarization of the rota-tor, so it does not change passing through the follow-ing rotator and remains circularly polarized. IfαΓv
¼ 2α0 þ Γv=2 (point B0), the circular polarizationsense will be changed (point B00). So, it is possible totransform any linear polarization into a circular one
Fig. 1. Polarization transformation of a beam of light propagatingthrough the first polarization system (the quarter-wave plate withadjustable values of optical activity θ and axes orientation ε) repre-sented by a point on the Poincaré sphere.
by the proper adjustment of the voltage applied andto use the system as a circularly polarized lightgenerator.
If Γv is not equal to 2α0 þ 90° (point A moves alongthe equator of the Poincaré sphere to point C), thelinearly polarized light will be transformed into anelliptical one by the quarter-wave plate (point C0).The change of the light polarization after the passageof light through the rotator corresponds to a rotationof the Poincaré sphere around the axis OA00 throughthe angle θ ¼ Γv=2, and the passage through the ro-tator takes it to another elliptical state of polariza-tion (point C00), so that the ellipticity remainsconstant, but the orientation of the ellipse major axisvaries.
If the incident light is circular polarized and is re-presented by point B00 on the Poincaré sphere, thepassage of the beam through the quarter-wave platetakes it to state A0, which is linearly polarized withthe azimuth angle equal to 45° relative to the quar-ter-wave axis. The light passage through the rotatorresults in moving point A0 along the equator to pointD00; as a result, the beam remains linearly polarizedbut its azimuth varies.
Let us describe the properties of the second sys-tem. Equations (13) and (14) can be presented inthe following form:
tanðθÞ ¼ tan�Γv
2
�; tan2ε ¼ ∞: ð18Þ
It follows from Eq. (18) that the orientation of the ef-fective axes ε does not change under the change of thephase shift of the first element Γv and the angle ofoptical activity θ is equal to half of the variable phaseshift:
θ ¼ Γv
2; ε ¼ 45°: ð19Þ
The equivalent optical system of the second polar-ization system under consideration consists of theimmovable quarter-wave plate followed by the rota-tor, the angle of the optical activity θ is controlled byadjustment of the applied voltage. Then the Jonesmatrix Tðφ;Γv;Γf Þ of the second polarization systemcan be expressed in the following matrix form:
This polarization system can be regarded as a quar-ter-wave plate with adjustable optical activity.
To investigate the polarization transformation of abeam of light propagating through the second polar-
ization system, let us consider how two elementarypolarizing elements influence the light state of polar-ization represented by a point on the Poincaré sphere(Fig. 2). Let us assume that the coordinate system isconnected to the unmovable axes of the quarter-waveplate. Let the linearly polarized incident light withthe azimuth angle α be represented by point A onthe equator of the Poincaré sphere.
The change of the light polarization after the pas-sage of light through the quarter-wave plate corre-sponds to a rotation of the Poincaré sphere aboutthe axis OX through the angle Γeff ¼ 90°. The pas-sage of the beam through the quarter-wave platetakes it to state A0, which is elliptically polarized.The change of the light polarization after the passageof light through the rotator corresponds to a rotationof the Poincaré sphere about the axis OB00 throughthe angle θ ¼ Γv=2, so that point A0 moves alongthe parallel of latitude to point A00, so that ellipticityremains constant but the orientation of the ellipsemajor axis varies.
If the linearly polarized incident light is repre-sented by point B on the equator of the Poincarésphere (the azimuth angle α ¼ 45°), the passage ofthe beam through the quarter-wave plate takes itto state B00, which is circularly polarized. Circular po-larization is eigenpolarization of the rotator, so itdoes not change passing through the rotator and re-mains circularly polarized.
If the linearly polarized light is the eigenpolariza-tion of the quarter-wave plate (point C), the passageof the beam through the quarter-wave plate takes itto the same state C. Passage of the beam through therotator results in moving pointC along the equator topoint C00, so that the beam remains linearly polarizedbut the azimuth varies and can be controlled by ad-justment of the applied voltage.
Fig. 2. Polarization transformation of a beam of light propagatingthrough the second polarization system (the quarter-wave platewith adjustable values of optical activity θ and immovable axesorientation ε) represented by a point on the Poincaré sphere.
If the incident light is circularly polarized and isrepresented by point D on the Poincaré sphere, thepassage of the beam through the quarter-wave platetakes it to state B, which is linearly polarized withthe azimuth angle α ¼ 45°. Passage of light throughthe rotator results in moving point B along the equa-tor to point D00, so that the beam remains linearlypolarized but the azimuth varies.
This polarization system is commercially availableas a linear polarization rotator [34], but as far as weknow, this polarization system has not been de-scribed as a combination of a quarter-wave plate fol-lowed by a rotator with adjustable optical activityand its properties as quarter-wave plate are not pro-posed for beam polarization transformation.
3. Experiments
To investigate the polarization transformation of abeam of light by polarization systems, we used the ex-perimental setup shown inFig. 3.The light sourcewasan He–Ne laser operating at wavelength 632:8 nmwith power 35 mW. A quarter-wave plate and a Glanprism with extinction ratio 10−5 were used to changethe azimuth angle of linearly polarized light. The si-milar Glan prism mounted at a mobile holder wasused to analyze the state of light polarization at thesystem output. The output signal was registered byaphotodiodePDA100A (Thorlabs) andan oscilloscope(Velleman).
The system under investigation consisted of an LCcell and a compound quarter-wave plate. The phaseshift of the LC cell was controlled by voltage applica-tion, and the compound quarter-wave plate had in-ternal optical activity. The angle of the opticalactivity was equal to 17:5°, and this value has beentaken into consideration under investigation. The LCcell was a nematic sandwich cell. The gap of the cellwas 5 μm. The parallel aligned cell was filled by LCmixture MLC6003. The cell aperture was 1:5 cm ×1:5 cm. To change the phase shift of the LC cell,an electrical signal with the rectangular shape, fre-quency of 200 Hz, and amplitude in the range from 0to 20 V was used. The phase retardation of the cell,as a function of the applied voltage, is shown in Fig.
4. One can see from Fig. 4 that the cell phase retar-dation was changed from 126° to 343° and the Fre-dericks threshold voltage is 2:2 V. The angle φbetween axes of the LC cell and the quarter-waveplate was equal to 45°.
Let us consider the polarization system presentedin Fig. 3(a). The method, described in [35] was usedto determine the absolute value of the effective phaseretardation deviation from 90°, namely, ΔΓeff ¼j90° − Γeff j. The deviation of the effective phase retar-dation of the first polarization system ΔΓeff as afunction of the phase retardation of the LC cell isshown in Fig. 5.
To determine the effective axes angle and the angleof optical activity, the azimuth angle of linearly polar-ized light at the system input was changed by the po-larizer rotation. Along with the polarizer rotation,the analyzer was rotated to achieve full light extin-guishing at the system output. The angles of the po-larizer position γ1 and the analyzer position γ2 weredetermined under light extinguishing at the systemoutput. The angles allow us to determine the angle ofoptical activity. If the system under considerationdoes not have optical activity, the difference betweentwo angles γ1 and γ2 will be equal to 90°. If the systemhas optical activity, then the angle of optical activitywill be determined as follows: θ ¼ jγ1 � γ2 � 90°j.
To investigate the dependence of the system effec-tive axes and optical activity on the phase retarda-tion of the LC cell, the applied voltage was changedfrom 0 to 20 V and the angle of the axes position andthe angle of optical activity were determined. The an-gle of the axes position ε and the angle of optical ac-tivity θ as a function of the LC cell retardation areshown in Fig. 6. One can see from Fig. 6 that the de-pendence of the angle of effective axes ε on the phaseshift Γv and the dependence of the angle of opticalactivity θ on the phase shift Γv are linear and the pro-portionality factor is equal to 0.5, in conformity withEq. (15).
The method described in [36] was used to convertlinearly polarized light into circularly polarized light
Fig. 3. Experimental setup: a, a quarter-wave plate with adjus-table axes; b, a quarter-wave plate with adjustable optical activity.
Fig. 4. Phase retardation of the LC cell as a function of theapplied voltage.
under different values of the LC cell phase retarda-tion. The azimuth of linearly polarized lightconverted into circularly polarized light was deter-mined for each value of the phase retardation. Thedependence of the azimuth angle α on the phase re-tardation value Γv is presented in Fig. 7. Figure 7shows a linear dependence between the azimuth αof linearly polarized light converted into circularlypolarized and the phase retardation Γv.
The results presented in Figs. 6 and 7 show thatthe first polarization system can be used for thefabrication of a quarter-wave plate with adjustableposition of the axes. The quarter-wave plate allowsconverting linearly polarized light with any azimuthinto circularly polarized light by adjusting the effec-tive axes position.
To investigate the polarization transformation of alight beam by the second polarization system, weused the experimental setup shown in Fig. 3b. The
method described above was used to determine theangle of effective optical activity and the angle ofaxes position as a function of the LC cell phase shiftΓv. It has been revealed that the angle of effectiveaxes position was equal to 45° and did not change un-der the increase of the phase shift Γv.
The experimentally determined deviation of the ef-fective phase retardation of the second polarizationsystemΔΓeff as a function of the phase shift Γv is pre-sented in Fig. 8. One can see from Fig. 8 that theabsolute value of the effective phase retardation de-viation from 90° Γeff does not exceed 3°� 2:5° anddoes not depend on the value of Γv.
Linearly polarized light with different azimuths(α ¼ 15°, 30°, 45°) was propagated through the polar-ization system in order to investigate the polariza-tion state transformation. The angle of the ellipsemajor axis at the system output was determined
Fig. 5. Deviation of the effective phase retardationΔΓeff ¼ j90° −Γeff j of the first polarization system with adjustable effective axisorientation as a function of the phase retardation of the LC cell.
Fig. 6. Angle of optical activity (a) and the angle of effective angleposition (b) as a function of the LC cell phase retardation for thefirst polarization system under investigation. The solid curve isthe function θ ¼ Γv=2.
Fig. 7. Dependence of the azimuth of linearly polarized light con-verted into circularly polarized light by the first system under in-vestigation on the phase retardation value of the LC cell. Thedashed curve (A) is the best fit approximation to the experimentalpoints, the solid curve (B) is the function α ¼ α0 � Γv=2.
Fig. 8. Deviation of the effective phase retardationΔΓeff ¼ j90° −Γeff j of the second polarization system with adjustable opticalactivity as a function of the LC cell phase shift.
by means of determining the angle of polarizer trans-mission axis γ3. The ellipticity of transmitted light ewas calculated after determining theminimum valueof the light intensity Imin and the maximum value ofthe light intensity Imax:
e ¼ Imin=Imax:
The experimentally obtained ellipticity of lighttransmitted through the second polarization systemas a function of the phase shift Γv is presented inFig. 9. One can see from Fig. 9 that the light ellipti-city at the system output depends on the linear po-larization azimuth α and does not depend on the LCcell phase shift Γv.
The angle of the polarization ellipse major axis θ0at the frame of reference of the first plate can becalculated as γ3 � 45° and its value is equal to the
optical activity angle of the system under considera-tion. Figure 10 shows the angle of the polarizationellipse major axis θ0 as a function of the LC cell phaseshift Γv. With reference to Fig. 10, it can be seen thatthe dependence θ0ðΓvÞ is linear and the angle θ0 is ap-proximately equal to Γv=2 for any linear polarizationazimuth at the system input. Hence, it is experimen-tally shown that the second system under considera-tion is a quarter-wave system with adjustable opticalactivity.
The system allows converting linearly polarizedlight into elliptically polarized light with the re-quired parameters. The value of ellipticity is definedby the azimuth of linearly polarized light α at the sys-tem input, and the rotation of the elliptically or lin-early polarized light can be performed by the LC cellphase shift Γv change.
Thus, it was demonstrated experimentally andtheoretically that a system consisting of one elementwith invariable quarter-wave shift and the secondone with variable phase shift oriented at the angleφ ¼ 45° relative to each other can be used as aquarter-wave polarization system with additionaladjustable parameters.
4. Conclusion
We have demonstrated the advantages of the firstJones equivalence theorem for constructing polariza-tion systems with desirable properties describing po-larization systems containing two retardation plates,one wave plate with a variable phase shift and onewave plate with an invariable phase shift. It has beendemonstrated, that if an invariable phase shift isequal to 90° and the relative orientation of thesetwo plates is 45° with respect to one another, this po-larization system can be regarded as an optical sys-tem consisting of one quarter-wave plate followed byone rotator. The properties of the system depend onthe plates’ order or the light direction propagation.
If the plate with the invariable retardation is thefirst, it is possible to change the quarter-wave plateaxis orientation and the optical activity angle of therotator by adjusting the voltage applied; this config-uration is a quarter-wave system with adjustableaxes orientation. This system can be used for thetransformation of linearly polarized light with anyazimuth into circularly polarized light and the trans-formation of circularly polarized light into linearlypolarized with any azimuth.
If the plate with the variable retardation is thefirst, it is possible to change the optical activity angleof the rotator by adjusting the voltage applied, butthe quarter-wave plate axis orientation remains un-changed; this configuration is a quarter-wave systemwith adjustable optical activity. The system can beused for the transformation of linearly polarizedlight into polarized light at any point on the Poincarésphere and for the rotation of the polarization ellipseat the required angle.
The properties of the quarter-wave system havebeen investigated experimentally and the reasonable
Fig. 9. Ellipticity of light transmitted through the second polar-ization system as a function of the phase shift for different valuesof linearly polarized light azimuth: a, α ¼ 15°; b, α ¼ 30°; c,α ¼ 45°.
Fig. 10. Angle of the ellipse major axis at the system output as afunction of the LC cell phase shift: a, α ¼ 15°; b, α ¼ 30°; c, α ¼ 45°.The solid curve is the function θ ¼ Γv=2.
coincidence between experimental and theoreticalresults has been observed.
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