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Visualization of light polarization forms in the laser conoscopic method
Optik 158 (2018) 349–354
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Original research article
Visualization of light polarization forms in the laser conoscopic method Pikoul O. Yu Far Eastern State Transport University, 47 Seryshev Str., Khabarovsk, Russia
a r t i c l e
i n f o
Article history: Received 21 November 2017 Accepted 22 December 2017 Key words: Interference Polarization Conoscopic pattern Phase difference
a b s t r a c t The laser conoscopic method enables visual estimation of radiation polarization. Visually identifying the forms of light polarization is convenient at the stage of pre-characterizing the polarization properties of the light used. Using the conoscopic patterns of a crystal plate with its entry face perpendicular to its optical axis, it is possible to allocate linearly polarized, circularly polarized, and elliptically polarized radiation. There are two typical appearances of conoscopic patterns, which correspond to two different intervals of phase shift: 0 < ␦ < /2 and 3/2 < ␦ < 2, and /2 < ␦ < and < ␦ < 3/2 are characteristic of elliptically polarized radiation. © 2017 Elsevier GmbH. All rights reserved.
1. Introduction Knowledge of light polarization forms is important for conducting research using optical methods; at present, photometric and visual methods to determine the form of optical light polarization are known [1,2]. A photoeffect phenomenon underlies the photometric method to determine the form of optical light polarization. The light polarization form (elliptical, circular, or linear) is determined by the value of the intensity measured. The photometric method enables the determination of the form of optical light polarization with a high degree of confidence, but requires considerable time and expensive stationary equipment. Typically, an interference phenomenon underlies the visual method for determining the form of optical light polarization. In particular, a light polarization form is determined by the appearance of the interference (conoscopic) pattern. The visual method enables the determination of the form of optical light polarization with a high degree of confidence, and does not require significant time or expensive stationary equipment, because it is an express method to determine the form of optical light polarization. Conoscopic analysis of interference fringes is one of the most useful tools for investigating the properties of optical crystals and the properties of different types of radiation [3–11]. Moreover, conoscopic patterns are employed in singular optics to study topological and polarization properties of optical beams having a complex wave structure [9]. To determine the polarization form, an optical system is used, which consists of an analyzer and a screen. The investigated parallel beam of light is transmitted along the axis of the optical system through the analyzer onto the screen, and the changes in the intensity of the light spot on the screen while rotating the analyzer are compared [1]. A light polarization form is determined by the change in the intensity measured. A decrease in the intensity of the light spot on the screen to a minimal value equal to zero is indicative of linear polarization. A decrease in the intensity of the light spot on the screen to a minimal value not equal to zero suggests elliptical polarization. No change in the intensity of the light spot on
E-mail address: fi[email protected] https://doi.org/10.1016/j.ijleo.2017.12.117 0030-4026/© 2017 Elsevier GmbH. All rights reserved.
350
P.O. Yu / Optik 158 (2018) 349–354
Fig. 1. Diagram of the optical system: 1 – crystal plate, 2 – analyzer, 3 – screen. Conoscopic patterns (a), (d) – circularly polarized light; (b), (e) – elliptically polarized light; (c), (f) – linearly polarized light.
the screen implies either natural light or circularly polarized light. Thus, the given method enables only the determination of linearly and elliptically polarized light and does not enable discrimination between natural or circularly polarized light. Another method to determine light polarization employs a divergent beam of light, which is transmitted through an optical system consisting of a crystal plate, analyzer, and screen (Fig. 1). A conoscopic pattern is usually observed when the angle of light divergence is 70–100◦ . The sense of vectors E is the same for all beams emerging from the analyzer and coincides with the analyzer transmission direction. First, we choose a crystal plate with an optical axis positioned in the plane of the entry face [1]. With natural light, it remains natural while exiting the plate. An interference (conoscopic) pattern on the screen after the analyzer appears as a light spot of uniform intensity. A lack of change in the intensity of the light spot on the screen while rotating the analyzer is indicative of natural light. With any (elliptically, circularly, or linearly) polarized light, an interference (conoscopic) pattern on the screen appears as two hyperbola systems (Fig. 1). Upon the analyzer rotation, a conoscopic pattern appears as two hyperbola systems with less contrast, which, upon further rotation of the analyzer, decreases to zero, with the conoscopic pattern on the screen appearing as a light spot of uniform intensity. The change in the conoscopic pattern on the screen is evidence of polarization. Thus, the choice of the crystal plate with an optical axis in the plane of its entry face enables only the determination of the state of polarization and detection of natural and polarized light. 2. Experimental results 2.1. Influence of polarization on the conoscopic patterns of optical crystals If a crystal plate with its entry face perpendicular to its optical axis is chosen, its conoscopic pattern also appears on the screen. Then, the polarizer is rotated around the axis of the optical system until a certain conoscopic pattern appears,
P.O. Yu / Optik 158 (2018) 349–354
351
Fig. 2. Conoscopic patterns of (ɑ)–(d) an optically inactive LiNbO3 crystal plate, and (e)–((h)) an optically active TeO2 crystal plate.
which is due to the polarization form and the optical activity properties of the crystal plate. Types of conoscopic patterns corresponding to certain forms of optical light polarization are shown in Fig. 2. Conoscopic patterns of linearly polarized radiation obtained for a single uniaxial crystal plate are well known, interpreted, and described in the literature [1,2]. With a convergent bundle of rays traveling along the optical axis, the conoscopic pattern of an optically inactive crystal plate (Fig. 2(a)) consists of concentric isochromatic rings with a specific distribution of intensity, in terms of a bright or dark superimposed cross. The arms of the “Maltese cross,” consisting of two isogyres, intersect in the center of the field of vision and expand towards the ends. Optical activity in a crystal (gyrotropy) results in a certain change in conoscopic patterns when the linearly polarized light beam propagates along the optical axis. A specific feature of uniaxial optically active crystals consists of the absence of a black “Maltese cross” in the center of the pattern in the area of a certain angular diameter (Fig. 2(e)). When changing the polarization of radiation for each mentioned plate, variable conoscopic patterns arise. With circularly polarized radiation, the conoscopic pattern appears as shown in Fig. 2(b), (f). The two black dots in the center of the visual field on the conoscopic pattern of an optically inactive LiNbO3 crystal plate describe circularly polarized light (Fig. 2(b)). With circularly polarized radiation, the conoscopic pattern of an optically active TeO2 crystal plate forms a pattern consisting of two inlaid spirals (Fig. 2(f)). The “Maltese cross” is absent from the observed patterns. It is determined that the direction of spiral twisting from the periphery to the center of the pattern agrees with the rotation direction of the polarization plane [3]. When changing the radiation ellipticity, the conoscopic patterns change and stand between the patterns that correspond to linearly polarized radiation and circularly polarized radiation. The two types of conoscopic patterns that correspond to elliptically polarized radiation are shown in Fig. 2(c), (d), (g), and (h). Thus, it has been determined that the conoscopic method is highly sensitive to the polarization of light. The conoscopic method visually identifies the forms of light polarization (linearly, circularly, and elliptically polarized), which is of special convenience at a stage of pre-characterization of the polarization properties of the light under study. 2.2. Visualization of the phase shift interval of elliptically polarized radiation During the polarization visualization experiment, two typical varieties of conoscopic patterns of optical crystals (for both optically active and optically inactive crystals) were revealed. The phase shift interval of elliptically polarized radiation was found to influence the appearance of the conoscopic pattern of optical crystals. Elliptically polarized light was obtained by varying the azimuth of the entering light, using the polarizer rotation as well as /4 quartz plate rotation. The experiment setup is shown in Fig. 3. The laser beam axis coincides with the optical axis of crystal 5 and is perpendicular to the entrance face of the crystal. After polarizer 2, /4 phase plate 3 is positioned, which enables the changed light polarization. Because of the untraditional rotation of the /4 phase plate about its vertical axis by an angle of , the optical path length traveled by the light increases, resulting in a change in phase shift ␦ with constant plate thickness d. This enables one to obtain all forms of polarization required at the exit from the phase plate.
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P.O. Yu / Optik 158 (2018) 349–354
Fig. 3. System diagram: 1 – He-Ne laser (0 = 632.8 nm), 2 – polarizer, 3 – /4 phase plate to control light ellipticity, 4 – diffuser, 5 – crystal, 6 – analyzer, 7 – screen.
Fig. 4. (a), (d) Polarization ellipses of the light obtained at the exit from the /4 plate upon rotation of the polarizer; Ein – direction of electric vector of the light entering the /4 plate; z – optical axis of the /4 plate. (b), (e) Conoscopic patterns of a LiNbO3 crystal plate. (c), (f) Conoscopic patterns of a TeO2 crystal plate corresponding to the incidence light polarization ellipses. Phase difference ␦ = /2.
When the /4 phase plate rotation angle = 0◦ , the light is found to be elliptically polarized. A beam from laser 1 passes through polarizer 2; phase plate 3, which provides for the degree of light ellipticity required; then to diffuser 4, which produces a divergent beam. This beam of light passes through plane-parallel crystal plate 5 and produces a conoscopic pattern on screen 7. The light polarization ellipse, whose axes do not coincide with the axes of a crystal plate, is given by [1]: Ex2 a2o
+
Ey2 a2e
−
2Ex Ey cos ı = sin2 ı, ao ae
(1)
where EX and Ey are the components of the intensity vector of the crystal-exiting light, and ␦ is the phase difference between the ordinary and extraordinary rays on the exit from the /4 phase plate. Polarization ellipse axes are always positioned in the plane perpendicular to the light propagation direction. Rotation of the polarizer about the beam axis [4] results in the elliptically polarized light at the exit from the /4 phase plate. The phase difference introduced by the crystal plate with normal orientation makes ␦ = /2, which corresponds to the optical length difference of /4 phase plate. The orientation of the axes of the exit light polarization ellipse coincides with the main directions of the /4 phase plate with any azimuth of the entry light. Conoscopic patterns of LiNbO3 and TeO2 crystals with the elliptically polarized light produced by this method are shown in Fig. 4. The experiment results show that with ␦ = /2, the azimuth of the incident light, and the direction of the E-vector’s traces, the light polarization ellipse influences the appearance of the conoscopic patterns. With the same polarization ellipse azimuth but a direction opposite to the tracing direction (Fig. 4(a), (d)), the axis of symmetry of the conoscopic pattern rotates approximately 90◦ (Fig. 4). In this case, the general appearance of the conoscopic pattern remains the same. This conclusion may be drawn for both optically active and optically inactive crystals. A more frequent event in practical measurements is the case where the azimuth of light entering the crystal plate remains constant and equal to ␣ = 45◦ (diagonal position of the plate) while the value of ␦ varies. This situation is likely if the ellipticity
P.O. Yu / Optik 158 (2018) 349–354
353
Fig. 5. Conoscopic patterns of a (a)–(d) LiNbO3 crystal plate and (e)–((h)) TeO2 crystal plate with elliptical light obtained at the exit from the /4 phase plate upon its rotation about the vertical axis, and the corresponding polarization ellipses. Ein – direction of the light vector at the entry to the /4 plate; z – optical axis of the /4 quartz plate. Phase difference interval: (a), (b), (e), (f) – 0 < ␦ < /2 and 3/2 < ␦ < 2; (c), (d), (g), ((h)) – /2 < ␦ < and < ␦ < 3/2.
of light varies, for instance, when the /4 phase plate is untraditionally rotated about the vertical axis lying in the plane of its entry face. It is evident that in this case the ellipse of polarization will continuously change its shape and orientation. It is clear from Eq.(1) that with ␦ = 0 and ␦ = , the light will preserve its linear polarization, whereas in the second case the E-vector will rotate by 90◦ upon exiting the plate. With ␦ = /2 and ␦ = 3/2, the light will acquire circular polarization with the opposite tracing direction. With phase difference values of 0 < ␦ < /2 and /2 < ␦ < , < ␦ < 3/2 and 3/2 < ␦ < 2, the light will become elliptical, and the ellipsis of light polarization for any value of ␦ in these intervals is always oriented at an angle of 45◦ relative to the principal directions of the /4 phase plate, but the major axis of the ellipse is located in opposite quadrants (Fig. 5). The next test determines the effect the polarization ellipse orientation produces on the appearance of the crystal’s cono/ /2, there scopic pattern (Fig. 5). An essential distinction is that for elliptically polarized light with a phase difference of ␦ = are already two typical appearances of conoscopic patterns.
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One appearance of the conoscopic pattern corresponds to the case where the azimuths of the entry light and polarization ellipse coincide and the Ein -vector of the entry light and the major axis of the ellipse are located in the same quadrant, which is true for the phase difference interval of 0 < ␦ < /2 and 3/2 < ␦ < 2 (Fig. 5(a), (b), (e), (f)). The second appearance of the pattern corresponds to the case when the Ein -vector of the entry light and the major axis of the polarization ellipse are located in adjacent quadrants and make an angle of 90◦ , which is true for the phase difference interval of /2 < ␦ < and < ␦ < 3/2 (Fig. 5(c), (d), (g), (h)). In both the first and second cases, a change in the Ein -vector tracing direction for the opposite, with the orientation of the ellipse remaining the same, results in an approximately 90◦ rotation of the symmetry axis of the conoscopic pattern. Two varieties of conoscopic patterns of optical crystals with elliptically polarized light greatly differ from each other in appearance. The difference in the conoscopic patterns is due to the different interval of change in the elliptical light phase difference, which, in turn, results in the coincidence or non-coincidence of the azimuth of light entering the /4 plate and the azimuth of polarization ellipse upon exiting therefrom. It has been established that a change in the E-vector tracing direction for the opposite, with the orientation of the ellipse remaining the same, results in an approximately 90◦ rotation of the symmetry axis of the conoscopic pattern. 3. Conclusion The laser conoscopic method is applied to visualization of polarizing properties of optical radiation. For these purposes, the investigated radiation travels to a crystal plate with an optical axis perpendicular to the entrance side. Correlations between forms of polarization of radiation (i.e., linear, circular, or elliptic) and types of conoscopic pattern of the crystal plate (optical active and inactive) observed on the screen were experimentally established. There are two different intervals of phase shift: 0 < ␦ < /2 and 3/2 < ␦ < 2, and /2 < ␦ < and < ␦ < 3/2 are characteristic of elliptically polarized radiation, which correspond to two typical appearances of conoscopic (polarization) patterns. Conoscopic patterns enable the visualization of optical radiation polarization properties with great confidence and do not require any expensive stationary equipment or significant time; they constitute a rapid method for determining the polarization forms of optical radiation. Funding This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors. References [1] R. Stoiber, S. Morse, Microscopic Identification of Crystals, The Ronald Press Company, New York, 1972. [2] M. Born, E. Wolf, Principles of Optics, sixth ed., Pergamon, London, 1986. [3] O.Yu Pikul, K.A. Rudoy, A.I. Livashvili, V.I. Doronin, V.I. Stroganov, Spiral structure in conoscopic figures of optically active crystals, J. Opt. Technol. 72 (2005) 69–70. [4] J.A. Haigh, Y. Kinebas, A.J. Ramsay, Inverse conoscopy: a method to measure polarization using patterns generated by a single birefringent crystal, Appl. Opt. 53 (2) (2017) 184–188. [5] O. Pikoul, Determination of the optical sign of a crystal by a conoscopic method, J. Appl. Cryst. 43 (2010) 949–954. [6] L. Dumitrascu, I. Dumitrascu, D.O. Dorohoi, E.C. Subbarao, G.S. Hirane, F. Jona, Conoscopic method for determination of main refractive indices and thickness of a uniaxial crystal cut out parallel to its optical axis, J. Appl. Cryst. 42 (2009) 878–884. [7] F.E. Veiras, M.T. Garea, L.I. Perez, Wide angle conoscopic interference patterns in uniaxial crystals, Appl. Opt. 51 (2012) 3081–3090. [8] I.O. Buinyi, V.G. Denisenko, M.S. Soskin, Topological structure in polarization resolved conoscopic patterns for nematic liquid crystal cells, Opt. Comm. 282 (2009) 143–155. [9] A.F. Konstantinova, B.V. Nabatov, E.A. Evdishchenko, K.A. Rudoy, V.I. Stroganov, O.Yu. Pikul, The influence of optical activity on the intensity and polarization parameters of transmitted light in crystals, Cryst. Rep. 48 (2003) 884–892. [10] S.K. Pal, P.S. Ruchi, Polarization singularity index sign inversion by a half-wave plate, Appl. Opt. 56 (2017) 6181–6190. [11] N.V. Sidorov, A.A. Kruk, O.Y. Pikoul, M.N. Palatnikov, N.A. Teplyakova, A.A. Yanichev, O.V. Makarova, Integrated research of structural and optical homogeneities of the lithium niobate crystal with low photorefractive effect, Optik 126 (2015) 1081–1089.
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Original research article
Visualization of light polarization forms in the laser conoscopic method Pikoul O. Yu Far Eastern State Transport University, 47 Seryshev Str., Khabarovsk, Russia
a r t i c l e
i n f o
Article history: Received 21 November 2017 Accepted 22 December 2017 Key words: Interference Polarization Conoscopic pattern Phase difference
a b s t r a c t The laser conoscopic method enables visual estimation of radiation polarization. Visually identifying the forms of light polarization is convenient at the stage of pre-characterizing the polarization properties of the light used. Using the conoscopic patterns of a crystal plate with its entry face perpendicular to its optical axis, it is possible to allocate linearly polarized, circularly polarized, and elliptically polarized radiation. There are two typical appearances of conoscopic patterns, which correspond to two different intervals of phase shift: 0 < ␦ < /2 and 3/2 < ␦ < 2, and /2 < ␦ < and < ␦ < 3/2 are characteristic of elliptically polarized radiation. © 2017 Elsevier GmbH. All rights reserved.
1. Introduction Knowledge of light polarization forms is important for conducting research using optical methods; at present, photometric and visual methods to determine the form of optical light polarization are known [1,2]. A photoeffect phenomenon underlies the photometric method to determine the form of optical light polarization. The light polarization form (elliptical, circular, or linear) is determined by the value of the intensity measured. The photometric method enables the determination of the form of optical light polarization with a high degree of confidence, but requires considerable time and expensive stationary equipment. Typically, an interference phenomenon underlies the visual method for determining the form of optical light polarization. In particular, a light polarization form is determined by the appearance of the interference (conoscopic) pattern. The visual method enables the determination of the form of optical light polarization with a high degree of confidence, and does not require significant time or expensive stationary equipment, because it is an express method to determine the form of optical light polarization. Conoscopic analysis of interference fringes is one of the most useful tools for investigating the properties of optical crystals and the properties of different types of radiation [3–11]. Moreover, conoscopic patterns are employed in singular optics to study topological and polarization properties of optical beams having a complex wave structure [9]. To determine the polarization form, an optical system is used, which consists of an analyzer and a screen. The investigated parallel beam of light is transmitted along the axis of the optical system through the analyzer onto the screen, and the changes in the intensity of the light spot on the screen while rotating the analyzer are compared [1]. A light polarization form is determined by the change in the intensity measured. A decrease in the intensity of the light spot on the screen to a minimal value equal to zero is indicative of linear polarization. A decrease in the intensity of the light spot on the screen to a minimal value not equal to zero suggests elliptical polarization. No change in the intensity of the light spot on
E-mail address: fi[email protected] https://doi.org/10.1016/j.ijleo.2017.12.117 0030-4026/© 2017 Elsevier GmbH. All rights reserved.
350
P.O. Yu / Optik 158 (2018) 349–354
Fig. 1. Diagram of the optical system: 1 – crystal plate, 2 – analyzer, 3 – screen. Conoscopic patterns (a), (d) – circularly polarized light; (b), (e) – elliptically polarized light; (c), (f) – linearly polarized light.
the screen implies either natural light or circularly polarized light. Thus, the given method enables only the determination of linearly and elliptically polarized light and does not enable discrimination between natural or circularly polarized light. Another method to determine light polarization employs a divergent beam of light, which is transmitted through an optical system consisting of a crystal plate, analyzer, and screen (Fig. 1). A conoscopic pattern is usually observed when the angle of light divergence is 70–100◦ . The sense of vectors E is the same for all beams emerging from the analyzer and coincides with the analyzer transmission direction. First, we choose a crystal plate with an optical axis positioned in the plane of the entry face [1]. With natural light, it remains natural while exiting the plate. An interference (conoscopic) pattern on the screen after the analyzer appears as a light spot of uniform intensity. A lack of change in the intensity of the light spot on the screen while rotating the analyzer is indicative of natural light. With any (elliptically, circularly, or linearly) polarized light, an interference (conoscopic) pattern on the screen appears as two hyperbola systems (Fig. 1). Upon the analyzer rotation, a conoscopic pattern appears as two hyperbola systems with less contrast, which, upon further rotation of the analyzer, decreases to zero, with the conoscopic pattern on the screen appearing as a light spot of uniform intensity. The change in the conoscopic pattern on the screen is evidence of polarization. Thus, the choice of the crystal plate with an optical axis in the plane of its entry face enables only the determination of the state of polarization and detection of natural and polarized light. 2. Experimental results 2.1. Influence of polarization on the conoscopic patterns of optical crystals If a crystal plate with its entry face perpendicular to its optical axis is chosen, its conoscopic pattern also appears on the screen. Then, the polarizer is rotated around the axis of the optical system until a certain conoscopic pattern appears,
P.O. Yu / Optik 158 (2018) 349–354
351
Fig. 2. Conoscopic patterns of (ɑ)–(d) an optically inactive LiNbO3 crystal plate, and (e)–((h)) an optically active TeO2 crystal plate.
which is due to the polarization form and the optical activity properties of the crystal plate. Types of conoscopic patterns corresponding to certain forms of optical light polarization are shown in Fig. 2. Conoscopic patterns of linearly polarized radiation obtained for a single uniaxial crystal plate are well known, interpreted, and described in the literature [1,2]. With a convergent bundle of rays traveling along the optical axis, the conoscopic pattern of an optically inactive crystal plate (Fig. 2(a)) consists of concentric isochromatic rings with a specific distribution of intensity, in terms of a bright or dark superimposed cross. The arms of the “Maltese cross,” consisting of two isogyres, intersect in the center of the field of vision and expand towards the ends. Optical activity in a crystal (gyrotropy) results in a certain change in conoscopic patterns when the linearly polarized light beam propagates along the optical axis. A specific feature of uniaxial optically active crystals consists of the absence of a black “Maltese cross” in the center of the pattern in the area of a certain angular diameter (Fig. 2(e)). When changing the polarization of radiation for each mentioned plate, variable conoscopic patterns arise. With circularly polarized radiation, the conoscopic pattern appears as shown in Fig. 2(b), (f). The two black dots in the center of the visual field on the conoscopic pattern of an optically inactive LiNbO3 crystal plate describe circularly polarized light (Fig. 2(b)). With circularly polarized radiation, the conoscopic pattern of an optically active TeO2 crystal plate forms a pattern consisting of two inlaid spirals (Fig. 2(f)). The “Maltese cross” is absent from the observed patterns. It is determined that the direction of spiral twisting from the periphery to the center of the pattern agrees with the rotation direction of the polarization plane [3]. When changing the radiation ellipticity, the conoscopic patterns change and stand between the patterns that correspond to linearly polarized radiation and circularly polarized radiation. The two types of conoscopic patterns that correspond to elliptically polarized radiation are shown in Fig. 2(c), (d), (g), and (h). Thus, it has been determined that the conoscopic method is highly sensitive to the polarization of light. The conoscopic method visually identifies the forms of light polarization (linearly, circularly, and elliptically polarized), which is of special convenience at a stage of pre-characterization of the polarization properties of the light under study. 2.2. Visualization of the phase shift interval of elliptically polarized radiation During the polarization visualization experiment, two typical varieties of conoscopic patterns of optical crystals (for both optically active and optically inactive crystals) were revealed. The phase shift interval of elliptically polarized radiation was found to influence the appearance of the conoscopic pattern of optical crystals. Elliptically polarized light was obtained by varying the azimuth of the entering light, using the polarizer rotation as well as /4 quartz plate rotation. The experiment setup is shown in Fig. 3. The laser beam axis coincides with the optical axis of crystal 5 and is perpendicular to the entrance face of the crystal. After polarizer 2, /4 phase plate 3 is positioned, which enables the changed light polarization. Because of the untraditional rotation of the /4 phase plate about its vertical axis by an angle of , the optical path length traveled by the light increases, resulting in a change in phase shift ␦ with constant plate thickness d. This enables one to obtain all forms of polarization required at the exit from the phase plate.
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P.O. Yu / Optik 158 (2018) 349–354
Fig. 3. System diagram: 1 – He-Ne laser (0 = 632.8 nm), 2 – polarizer, 3 – /4 phase plate to control light ellipticity, 4 – diffuser, 5 – crystal, 6 – analyzer, 7 – screen.
Fig. 4. (a), (d) Polarization ellipses of the light obtained at the exit from the /4 plate upon rotation of the polarizer; Ein – direction of electric vector of the light entering the /4 plate; z – optical axis of the /4 plate. (b), (e) Conoscopic patterns of a LiNbO3 crystal plate. (c), (f) Conoscopic patterns of a TeO2 crystal plate corresponding to the incidence light polarization ellipses. Phase difference ␦ = /2.
When the /4 phase plate rotation angle = 0◦ , the light is found to be elliptically polarized. A beam from laser 1 passes through polarizer 2; phase plate 3, which provides for the degree of light ellipticity required; then to diffuser 4, which produces a divergent beam. This beam of light passes through plane-parallel crystal plate 5 and produces a conoscopic pattern on screen 7. The light polarization ellipse, whose axes do not coincide with the axes of a crystal plate, is given by [1]: Ex2 a2o
+
Ey2 a2e
−
2Ex Ey cos ı = sin2 ı, ao ae
(1)
where EX and Ey are the components of the intensity vector of the crystal-exiting light, and ␦ is the phase difference between the ordinary and extraordinary rays on the exit from the /4 phase plate. Polarization ellipse axes are always positioned in the plane perpendicular to the light propagation direction. Rotation of the polarizer about the beam axis [4] results in the elliptically polarized light at the exit from the /4 phase plate. The phase difference introduced by the crystal plate with normal orientation makes ␦ = /2, which corresponds to the optical length difference of /4 phase plate. The orientation of the axes of the exit light polarization ellipse coincides with the main directions of the /4 phase plate with any azimuth of the entry light. Conoscopic patterns of LiNbO3 and TeO2 crystals with the elliptically polarized light produced by this method are shown in Fig. 4. The experiment results show that with ␦ = /2, the azimuth of the incident light, and the direction of the E-vector’s traces, the light polarization ellipse influences the appearance of the conoscopic patterns. With the same polarization ellipse azimuth but a direction opposite to the tracing direction (Fig. 4(a), (d)), the axis of symmetry of the conoscopic pattern rotates approximately 90◦ (Fig. 4). In this case, the general appearance of the conoscopic pattern remains the same. This conclusion may be drawn for both optically active and optically inactive crystals. A more frequent event in practical measurements is the case where the azimuth of light entering the crystal plate remains constant and equal to ␣ = 45◦ (diagonal position of the plate) while the value of ␦ varies. This situation is likely if the ellipticity
P.O. Yu / Optik 158 (2018) 349–354
353
Fig. 5. Conoscopic patterns of a (a)–(d) LiNbO3 crystal plate and (e)–((h)) TeO2 crystal plate with elliptical light obtained at the exit from the /4 phase plate upon its rotation about the vertical axis, and the corresponding polarization ellipses. Ein – direction of the light vector at the entry to the /4 plate; z – optical axis of the /4 quartz plate. Phase difference interval: (a), (b), (e), (f) – 0 < ␦ < /2 and 3/2 < ␦ < 2; (c), (d), (g), ((h)) – /2 < ␦ < and < ␦ < 3/2.
of light varies, for instance, when the /4 phase plate is untraditionally rotated about the vertical axis lying in the plane of its entry face. It is evident that in this case the ellipse of polarization will continuously change its shape and orientation. It is clear from Eq.(1) that with ␦ = 0 and ␦ = , the light will preserve its linear polarization, whereas in the second case the E-vector will rotate by 90◦ upon exiting the plate. With ␦ = /2 and ␦ = 3/2, the light will acquire circular polarization with the opposite tracing direction. With phase difference values of 0 < ␦ < /2 and /2 < ␦ < , < ␦ < 3/2 and 3/2 < ␦ < 2, the light will become elliptical, and the ellipsis of light polarization for any value of ␦ in these intervals is always oriented at an angle of 45◦ relative to the principal directions of the /4 phase plate, but the major axis of the ellipse is located in opposite quadrants (Fig. 5). The next test determines the effect the polarization ellipse orientation produces on the appearance of the crystal’s cono/ /2, there scopic pattern (Fig. 5). An essential distinction is that for elliptically polarized light with a phase difference of ␦ = are already two typical appearances of conoscopic patterns.
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One appearance of the conoscopic pattern corresponds to the case where the azimuths of the entry light and polarization ellipse coincide and the Ein -vector of the entry light and the major axis of the ellipse are located in the same quadrant, which is true for the phase difference interval of 0 < ␦ < /2 and 3/2 < ␦ < 2 (Fig. 5(a), (b), (e), (f)). The second appearance of the pattern corresponds to the case when the Ein -vector of the entry light and the major axis of the polarization ellipse are located in adjacent quadrants and make an angle of 90◦ , which is true for the phase difference interval of /2 < ␦ < and < ␦ < 3/2 (Fig. 5(c), (d), (g), (h)). In both the first and second cases, a change in the Ein -vector tracing direction for the opposite, with the orientation of the ellipse remaining the same, results in an approximately 90◦ rotation of the symmetry axis of the conoscopic pattern. Two varieties of conoscopic patterns of optical crystals with elliptically polarized light greatly differ from each other in appearance. The difference in the conoscopic patterns is due to the different interval of change in the elliptical light phase difference, which, in turn, results in the coincidence or non-coincidence of the azimuth of light entering the /4 plate and the azimuth of polarization ellipse upon exiting therefrom. It has been established that a change in the E-vector tracing direction for the opposite, with the orientation of the ellipse remaining the same, results in an approximately 90◦ rotation of the symmetry axis of the conoscopic pattern. 3. Conclusion The laser conoscopic method is applied to visualization of polarizing properties of optical radiation. For these purposes, the investigated radiation travels to a crystal plate with an optical axis perpendicular to the entrance side. Correlations between forms of polarization of radiation (i.e., linear, circular, or elliptic) and types of conoscopic pattern of the crystal plate (optical active and inactive) observed on the screen were experimentally established. There are two different intervals of phase shift: 0 < ␦ < /2 and 3/2 < ␦ < 2, and /2 < ␦ < and < ␦ < 3/2 are characteristic of elliptically polarized radiation, which correspond to two typical appearances of conoscopic (polarization) patterns. Conoscopic patterns enable the visualization of optical radiation polarization properties with great confidence and do not require any expensive stationary equipment or significant time; they constitute a rapid method for determining the polarization forms of optical radiation. Funding This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors. References [1] R. Stoiber, S. Morse, Microscopic Identification of Crystals, The Ronald Press Company, New York, 1972. [2] M. Born, E. Wolf, Principles of Optics, sixth ed., Pergamon, London, 1986. [3] O.Yu Pikul, K.A. Rudoy, A.I. Livashvili, V.I. Doronin, V.I. Stroganov, Spiral structure in conoscopic figures of optically active crystals, J. Opt. Technol. 72 (2005) 69–70. [4] J.A. Haigh, Y. Kinebas, A.J. Ramsay, Inverse conoscopy: a method to measure polarization using patterns generated by a single birefringent crystal, Appl. Opt. 53 (2) (2017) 184–188. [5] O. Pikoul, Determination of the optical sign of a crystal by a conoscopic method, J. Appl. Cryst. 43 (2010) 949–954. [6] L. Dumitrascu, I. Dumitrascu, D.O. Dorohoi, E.C. Subbarao, G.S. Hirane, F. Jona, Conoscopic method for determination of main refractive indices and thickness of a uniaxial crystal cut out parallel to its optical axis, J. Appl. Cryst. 42 (2009) 878–884. [7] F.E. Veiras, M.T. Garea, L.I. Perez, Wide angle conoscopic interference patterns in uniaxial crystals, Appl. Opt. 51 (2012) 3081–3090. [8] I.O. Buinyi, V.G. Denisenko, M.S. Soskin, Topological structure in polarization resolved conoscopic patterns for nematic liquid crystal cells, Opt. Comm. 282 (2009) 143–155. [9] A.F. Konstantinova, B.V. Nabatov, E.A. Evdishchenko, K.A. Rudoy, V.I. Stroganov, O.Yu. Pikul, The influence of optical activity on the intensity and polarization parameters of transmitted light in crystals, Cryst. Rep. 48 (2003) 884–892. [10] S.K. Pal, P.S. Ruchi, Polarization singularity index sign inversion by a half-wave plate, Appl. Opt. 56 (2017) 6181–6190. [11] N.V. Sidorov, A.A. Kruk, O.Y. Pikoul, M.N. Palatnikov, N.A. Teplyakova, A.A. Yanichev, O.V. Makarova, Integrated research of structural and optical homogeneities of the lithium niobate crystal with low photorefractive effect, Optik 126 (2015) 1081–1089.