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Generation of collinearly propagating orthogonally polarized beams
Optik 122 (2011) 1164–1168
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Generation of collinearly propagating orthogonally polarized beams N. Ghosh a,∗ , Y. Otani b , K. Bhattacharya a a b
Department of Applied Optics and Photonics, University of Calcutta, 92, Acharya Prafulla Chandra Road, Kolkata 700009, India Center for Optical Research and Education(CORE), Utsunomiya University, 7-1-2, Yoto, Utsunomiya, Tochigi, 3218585, Japan
a r t i c l e
i n f o
Article history: Received 22 February 2010 Accepted 22 July 2010
Keywords: Cube beam splitter Polarization phase shifting 3D-profilometry
a b s t r a c t A simple monolithic device for simultaneous generation of collinearly propagating orthogonally polarized light beams of equal intensity is presented. The common cube beam splitter masked by two linear polarizers is used to achieve the purpose. This is experimentally verified through the use of a Stoke’s polarimeter. It is also shown experimentally that the same setup behaves as a polarization phase shifting interferometer. © 2010 Elsevier GmbH. All rights reserved.
1. Introduction Orthogonally polarized light has a number of applications both in polarized light microscopy as well as in optical metrology. Several devices have been considered for generation of orthogonally polarized light beams. The Wollaston prism is one such device that splits the incident beam into two orthogonally plane-polarized components, the o-ray and the e-ray, which then emerge at an angle to each other. The Nomarski prism [1], which is a modification of the Wollaston prism, causes the light rays to come to a focal point outside the body of the prism. This allows for greater flexibility for the prism to be actively focused and is widely used in differential interference contrast microscopy where the two orthogonally plane-polarized components are focussed on the sample and produce coloured interference patterns with white light. A Zeeman-split laser produces a beam containing two circularly polarized components with opposite phase delays making the polarizations of the two components orthogonal but having two different frequencies with wavelengths a few MHz apart. In practice, however, the polarizations of the frequency components inside a laser cavity depart slightly from being perfectly circular, and the polarizations of the frequency components after the wave plates may not be perfectly linear or perfectly orthogonal. In particular, the frequency component intended to have the “P” polarization will have a small “S”-polarized component and vice-versa. Mansour and Habli [2] designed an orthogonal circular polarization axicon beamsplitter that generates orthogonal polarization states operating in the IR, Visible and UV spectrum. The beams are, however, generated
∗ Corresponding author. E-mail address: [email protected] (N. Ghosh). 0030-4026/$ – see front matter © 2010 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2010.07.021
through surface reflections of a thin-film substrate coated over the axicon and do not propagate collinear to each other. The proposed arrangement uses a polarizer-masked cube beam splitter, aligned in an unconventional orientation, for generation of orthogonally polarized components. The experimental results with a Stokes polarimeter are presented. It is also shown that in the proposed arrangement the device behaves as a compact polarization phase shifting interferometer suitable for small sample dimensions.
2. Theory and applications As shown in Fig. 1, consider an expanded and collimated circularly polarized laser beam from the He–Ne laser incident only on the upper half (A) of a cube beam splitter (CBS) with its dielectric coated partially reflecting interface parallel to the incident beam, as in Gates interferometer [3]. At the interface of the CBS, the incident beam is amplitude divided and emerges as two light beams propagating in parallel. If the incident light is circularly polarized, it appears that the beam that is reflected from the dielectric interface and emerges at A , reverses its sense of circular polarization on reflection whereas the component that is transmitted through it retains its initial state of circular polarization. Experimental results, however, are quite on the contrary. On oblique incidence at the reflecting interface within the beam splitter, the reflection coefficients for TE and TM components of the circularly polarized incident beam are unequal, resulting in arbitrary polarization states at the output of the CBS. Fig. 2 shows the Stokes parameters at 632 nm as well as the spectral variation of the Stokes parameters. Since the beam splitting layer is symmetrical about the interface, an identical situation occurs when the input beam is allowed to be incident on the lower half of the CBS.
N. Ghosh et al. / Optik 122 (2011) 1164–1168
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Fig. 1. (a) Setup showing right-circular incident beam. Results from the output beams for (b) beam 2 and (c) beam 1.
In order to circumvent this problem, the two input faces of the CBS are polarization masked so as to generate two strictly orthogonal but perfectly plane-polarized light beams on reflection and transmission at the partially reflecting interface. In this particular configuration, the reflected component remains strictly s-polarized (TE) and the transmitted component p-polarized (TM) on emerging from the CBS. The measured Stokes parameters of the emerging
Fig. 2. (a) Setup showing left-circular incident beam. Results from the output beams for (b) beam 2 and (c) beam 1.
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Fig. 3. (a) Polarizer-masked beam splitter showing the propagation of TE-mode. Results for S-parameters for beam 1 and beam 2 shown in (b) and (c) respectively clearly shows that beam 1 does not have the desired state of polarization while that of beam 2 is perfectly circularly polarized.
Fig. 4. (a) Polarizer-masked beam splitter showing the propagation of TM-mode. Results for S-parameters for beam 1 and beam 2 shown in (b) and (c) respectively clearly shows that beam 1 does not have the desired state of polarization while that of beam 2 is perfectly circularly polarized.
beams, as shown in Figs. 3 and 4, substantiates this experimentally. When the polarizer-masked CBS is placed in the path of a parallel beam of light so as to simultaneously illuminate both its input faces, the beams that emerge from A (Figs. 3 and 4) is a
N. Ghosh et al. / Optik 122 (2011) 1164–1168
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superposition of two mutually orthogonal plane-polarized beams that are collinearly propagating. A quarter wave plate with one of its principal axis at /4 placed in the path of these output beam ensures the generation of orthogonally polarized circular beams of light at the output face A . The Stokes parameters for these output beams indicate the generation of two orthogonal and almost perfectly circularly polarized collinearly propagating light beams.
3. Applications In addition to the possible applications mentioned earlier, a simple but interesting application of the proposed scheme is the realization of a polarization phase shifting interferometer as schematically shown in Fig. 5. When the beam splitting interface of the CBS and the plane wavefront incident on it are not strictly normal to each other, the generated pair of orthogonally polarized light beams interfere with a small angle between them. A linear polarizer placed in the path of these beams then ensures that interference fringes are observed on the CCD placed immediately thereafter. Using the convenient methodology of Jones Calculus, the output amplitude ε1 for one of the circularly polarized components may be represented by
ε1 =
cos2 cos sin
cos sin sin2
1 i
= (cos + i sin )
cos sin
where is the orientation of the analyzer placed at the output end. The amplitude may therefore be expressed in the form ε1 = R1 exp(i1 ) where 1 = .
(1)
Similarly, the output amplitude ε2 for the orthogonally polarized circular component for which the Jones vector will be given as 1 −i
, is given as
ε2 = R2 exp(i2 ) where 2 = −.
(2)
It can be easily shown from the above equations that rotation through an angle of an output linear polarizer will introduce a phase difference of 2 between the two collinear but opposite circularly polarized beams intercepted by it. The two transmitted linearly polarized components emerging from the output polarizer have a phase shift of 2 between them and therefore interferes to produce a fringe pattern. This principle can effectively be used for phase shifting interferometry. It is interesting to note that the intensity ratio of these interfering beams are independent of the rotation angle of the polarizer, signifying that the fringe contrast remains constant irrespective of the polarization phase step introduced.
Fig. 5. Schematic for the generation of collinearly propagating orthogonal polarized light and its application to surface profilometry.
Fig. 6. Four /2 phase-shifted interferograms for orientations of 0◦ , 45◦ , 90◦ and 135◦ of the output linear polarizer.
The four phase-shifted fringe patterns produced at the output and recorded through a CCD camera are shown in Fig. 6. Phase shifting is achieved by rotating an output linear polarizer, P() and the interferograms are recorded for values of 0◦ , 45◦ , 90◦ and 135◦ respectively, yielding corresponding phase shifts of 0◦ , 90◦ , 180◦ and 270◦ respectively. The orthogonality of the circularly polarized states of the two beams generated at the output ensures that the phase shifting is linearly proportional to the rotation angle of the polarizer. This simple and convenient scheme of polarization phase shifting can be applied to 3D-profilometry by fringe projection and phase stepping. The reconstructed shape is shown in Fig. 7(a). For the sake of comparison, a photograph of the sample (a plastic die) is shown in Fig. 7(b). Acceptable reconstruction of the sample profile
Fig. 7. (a) Phase map of the object as seen in (b).
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N. Ghosh et al. / Optik 122 (2011) 1164–1168
bears proof of the /4 phase stepping and hence the generation of collinearly propagating orthogonally polarized light. Acknowledgement The authors gratefully acknowledge the Grants-in-aid no. 03(1134)/09/EMR-II from the Council for Scientific and Industrial Research, India, during the course of this study.
References [1] R. Allen, G. David Nomarski, The Zeiss-Nomarski differential interference equipment for transmitted-light microscopy, Z. Wiss. Mikrosk. 69 (4) (1969) 193–221. [2] M.K. Mansour, M.A. Habli, Orthogonal circular polarization axicon beam splitter, Microw. Opt. Technol. Lett. (2001) 260–265. [3] J.W. Gates, Reverse shearing interferometry, Nature 176 (1955) 359.
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Generation of collinearly propagating orthogonally polarized beams N. Ghosh a,∗ , Y. Otani b , K. Bhattacharya a a b
Department of Applied Optics and Photonics, University of Calcutta, 92, Acharya Prafulla Chandra Road, Kolkata 700009, India Center for Optical Research and Education(CORE), Utsunomiya University, 7-1-2, Yoto, Utsunomiya, Tochigi, 3218585, Japan
a r t i c l e
i n f o
Article history: Received 22 February 2010 Accepted 22 July 2010
Keywords: Cube beam splitter Polarization phase shifting 3D-profilometry
a b s t r a c t A simple monolithic device for simultaneous generation of collinearly propagating orthogonally polarized light beams of equal intensity is presented. The common cube beam splitter masked by two linear polarizers is used to achieve the purpose. This is experimentally verified through the use of a Stoke’s polarimeter. It is also shown experimentally that the same setup behaves as a polarization phase shifting interferometer. © 2010 Elsevier GmbH. All rights reserved.
1. Introduction Orthogonally polarized light has a number of applications both in polarized light microscopy as well as in optical metrology. Several devices have been considered for generation of orthogonally polarized light beams. The Wollaston prism is one such device that splits the incident beam into two orthogonally plane-polarized components, the o-ray and the e-ray, which then emerge at an angle to each other. The Nomarski prism [1], which is a modification of the Wollaston prism, causes the light rays to come to a focal point outside the body of the prism. This allows for greater flexibility for the prism to be actively focused and is widely used in differential interference contrast microscopy where the two orthogonally plane-polarized components are focussed on the sample and produce coloured interference patterns with white light. A Zeeman-split laser produces a beam containing two circularly polarized components with opposite phase delays making the polarizations of the two components orthogonal but having two different frequencies with wavelengths a few MHz apart. In practice, however, the polarizations of the frequency components inside a laser cavity depart slightly from being perfectly circular, and the polarizations of the frequency components after the wave plates may not be perfectly linear or perfectly orthogonal. In particular, the frequency component intended to have the “P” polarization will have a small “S”-polarized component and vice-versa. Mansour and Habli [2] designed an orthogonal circular polarization axicon beamsplitter that generates orthogonal polarization states operating in the IR, Visible and UV spectrum. The beams are, however, generated
∗ Corresponding author. E-mail address: [email protected] (N. Ghosh). 0030-4026/$ – see front matter © 2010 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2010.07.021
through surface reflections of a thin-film substrate coated over the axicon and do not propagate collinear to each other. The proposed arrangement uses a polarizer-masked cube beam splitter, aligned in an unconventional orientation, for generation of orthogonally polarized components. The experimental results with a Stokes polarimeter are presented. It is also shown that in the proposed arrangement the device behaves as a compact polarization phase shifting interferometer suitable for small sample dimensions.
2. Theory and applications As shown in Fig. 1, consider an expanded and collimated circularly polarized laser beam from the He–Ne laser incident only on the upper half (A) of a cube beam splitter (CBS) with its dielectric coated partially reflecting interface parallel to the incident beam, as in Gates interferometer [3]. At the interface of the CBS, the incident beam is amplitude divided and emerges as two light beams propagating in parallel. If the incident light is circularly polarized, it appears that the beam that is reflected from the dielectric interface and emerges at A , reverses its sense of circular polarization on reflection whereas the component that is transmitted through it retains its initial state of circular polarization. Experimental results, however, are quite on the contrary. On oblique incidence at the reflecting interface within the beam splitter, the reflection coefficients for TE and TM components of the circularly polarized incident beam are unequal, resulting in arbitrary polarization states at the output of the CBS. Fig. 2 shows the Stokes parameters at 632 nm as well as the spectral variation of the Stokes parameters. Since the beam splitting layer is symmetrical about the interface, an identical situation occurs when the input beam is allowed to be incident on the lower half of the CBS.
N. Ghosh et al. / Optik 122 (2011) 1164–1168
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Fig. 1. (a) Setup showing right-circular incident beam. Results from the output beams for (b) beam 2 and (c) beam 1.
In order to circumvent this problem, the two input faces of the CBS are polarization masked so as to generate two strictly orthogonal but perfectly plane-polarized light beams on reflection and transmission at the partially reflecting interface. In this particular configuration, the reflected component remains strictly s-polarized (TE) and the transmitted component p-polarized (TM) on emerging from the CBS. The measured Stokes parameters of the emerging
Fig. 2. (a) Setup showing left-circular incident beam. Results from the output beams for (b) beam 2 and (c) beam 1.
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N. Ghosh et al. / Optik 122 (2011) 1164–1168
Fig. 3. (a) Polarizer-masked beam splitter showing the propagation of TE-mode. Results for S-parameters for beam 1 and beam 2 shown in (b) and (c) respectively clearly shows that beam 1 does not have the desired state of polarization while that of beam 2 is perfectly circularly polarized.
Fig. 4. (a) Polarizer-masked beam splitter showing the propagation of TM-mode. Results for S-parameters for beam 1 and beam 2 shown in (b) and (c) respectively clearly shows that beam 1 does not have the desired state of polarization while that of beam 2 is perfectly circularly polarized.
beams, as shown in Figs. 3 and 4, substantiates this experimentally. When the polarizer-masked CBS is placed in the path of a parallel beam of light so as to simultaneously illuminate both its input faces, the beams that emerge from A (Figs. 3 and 4) is a
N. Ghosh et al. / Optik 122 (2011) 1164–1168
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superposition of two mutually orthogonal plane-polarized beams that are collinearly propagating. A quarter wave plate with one of its principal axis at /4 placed in the path of these output beam ensures the generation of orthogonally polarized circular beams of light at the output face A . The Stokes parameters for these output beams indicate the generation of two orthogonal and almost perfectly circularly polarized collinearly propagating light beams.
3. Applications In addition to the possible applications mentioned earlier, a simple but interesting application of the proposed scheme is the realization of a polarization phase shifting interferometer as schematically shown in Fig. 5. When the beam splitting interface of the CBS and the plane wavefront incident on it are not strictly normal to each other, the generated pair of orthogonally polarized light beams interfere with a small angle between them. A linear polarizer placed in the path of these beams then ensures that interference fringes are observed on the CCD placed immediately thereafter. Using the convenient methodology of Jones Calculus, the output amplitude ε1 for one of the circularly polarized components may be represented by
ε1 =
cos2 cos sin
cos sin sin2
1 i
= (cos + i sin )
cos sin
where is the orientation of the analyzer placed at the output end. The amplitude may therefore be expressed in the form ε1 = R1 exp(i1 ) where 1 = .
(1)
Similarly, the output amplitude ε2 for the orthogonally polarized circular component for which the Jones vector will be given as 1 −i
, is given as
ε2 = R2 exp(i2 ) where 2 = −.
(2)
It can be easily shown from the above equations that rotation through an angle of an output linear polarizer will introduce a phase difference of 2 between the two collinear but opposite circularly polarized beams intercepted by it. The two transmitted linearly polarized components emerging from the output polarizer have a phase shift of 2 between them and therefore interferes to produce a fringe pattern. This principle can effectively be used for phase shifting interferometry. It is interesting to note that the intensity ratio of these interfering beams are independent of the rotation angle of the polarizer, signifying that the fringe contrast remains constant irrespective of the polarization phase step introduced.
Fig. 5. Schematic for the generation of collinearly propagating orthogonal polarized light and its application to surface profilometry.
Fig. 6. Four /2 phase-shifted interferograms for orientations of 0◦ , 45◦ , 90◦ and 135◦ of the output linear polarizer.
The four phase-shifted fringe patterns produced at the output and recorded through a CCD camera are shown in Fig. 6. Phase shifting is achieved by rotating an output linear polarizer, P() and the interferograms are recorded for values of 0◦ , 45◦ , 90◦ and 135◦ respectively, yielding corresponding phase shifts of 0◦ , 90◦ , 180◦ and 270◦ respectively. The orthogonality of the circularly polarized states of the two beams generated at the output ensures that the phase shifting is linearly proportional to the rotation angle of the polarizer. This simple and convenient scheme of polarization phase shifting can be applied to 3D-profilometry by fringe projection and phase stepping. The reconstructed shape is shown in Fig. 7(a). For the sake of comparison, a photograph of the sample (a plastic die) is shown in Fig. 7(b). Acceptable reconstruction of the sample profile
Fig. 7. (a) Phase map of the object as seen in (b).
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N. Ghosh et al. / Optik 122 (2011) 1164–1168
bears proof of the /4 phase stepping and hence the generation of collinearly propagating orthogonally polarized light. Acknowledgement The authors gratefully acknowledge the Grants-in-aid no. 03(1134)/09/EMR-II from the Council for Scientific and Industrial Research, India, during the course of this study.
References [1] R. Allen, G. David Nomarski, The Zeiss-Nomarski differential interference equipment for transmitted-light microscopy, Z. Wiss. Mikrosk. 69 (4) (1969) 193–221. [2] M.K. Mansour, M.A. Habli, Orthogonal circular polarization axicon beam splitter, Microw. Opt. Technol. Lett. (2001) 260–265. [3] J.W. Gates, Reverse shearing interferometry, Nature 176 (1955) 359.