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A universal compensator for polarization changes induced by non-reciprocal circular birefringence on a retracing beam
Optics Communications 366 (2016) 119–121
Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
A universal compensator for polarization changes induced by non-reciprocal circular birefringence on a retracing beam Mario Martinelli, Paolo Martelli n, Annalaura Fasiello Politecnico di Milano, Dipartimento di Elettronica Informazione e Bioingegneria, PoliCom Lab, Via G. Ponzio 34/5, 20133 Milano, Italy
art ic l e i nf o
a b s t r a c t
Article history: Received 3 November 2015 Received in revised form 14 December 2015 Accepted 15 December 2015
In this communication we recognize that it is possible to cancel out the effects of the non-reciprocal circular birefringence on a retracing beam. The experimental results demonstrate that a linearly polarized beam is returned into an orthogonal state after retracing through a variable Faraday rotator, by exploiting the reflective action of a Porro prism with edge at 45° with respect to the initial polarization axis, for any amount of non-reciprocal Faraday rotation. & 2015 Elsevier B.V. All rights reserved.
Keywords: Polarization Birefringence Non-reciprocal circular birefringence Porro prism Faraday rotator mirror Retracing path
1. Introduction The very remarkable compensation property of the Faraday rotator mirror (FRM), i.e. a mirror with in front a Faraday rotator with 45° of rotation power, was illustrated for the first time in 1989 [1]. In particular, when the FRM is used, any reciprocal birefringence in the retracing path is canceled out. Although it was been initially conceived for the use in optical fiber sensor applications [2], the compensation property of the FRM was quickly exported in other contexts of optics, as quantum cryptography [3], measurements of quantum states of light [4] and optical communications [5]. Behind the success of the FRM there was a series of factors: – its compensation property holds at the single-photon level, so it can be also applied in quantum experiments; – it is passive, so it acts without limitation of response time in all its optical bandwidth of operation; – it allows a practical and zero-loss recovery of an orthogonally polarized returning beam, by means of a simple polarizing beam splitter (PBS). The aim of the present communication is to show that a n
Corresponding author. E-mail address: [email protected] (P. Martelli).
http://dx.doi.org/10.1016/j.optcom.2015.12.036 0030-4018/& 2015 Elsevier B.V. All rights reserved.
companion mirror of the FRM exists, based on a Porro prism set at 45° with respect to a reference polarization axis (e.g., identified by a PBS). We investigate the ability of this component, which we name Porro rotator mirror (PRM), when used as a mirror in a retracing path, in canceling out the non-reciprocal circular birefringence of the path. As for the FRM, this property holds at single-photon level, occurs in a passive way and produces an orthogonal state after retracing for a linearly polarized beam.
2. Optical retracing paths The field transformation properties of a Porro prism (also named right-angle prism or roof edge rotator) are known [6–8]. If the two right-angle reflective faces act as ideal metallic reflectors, then the Porro prism is a pure rotator for both the image and the polarization. In this case, there is not the phase shift between s and p components related to the total internal reflection condition and the linear states of polarization are not ellipticized. Moreover, the overall effect of the double reflection at the metallized faces is to preserve the circular polarization handedness. Coherently to the formalism introduced in [9] for describing retracing beams, we can write the following Jones matrix of an ideally-reflecting Porro prism with edge oriented at an angle ϕ with respect to the horizontal reference axis:
120
M. Martinelli et al. / Optics Communications 366 (2016) 119–121
⎡ cos (2ϕ) − sin (2ϕ)⎤ ⎥ = − R (2ϕ), JPP (ϕ) = − ⎢ ⎣ sin (2ϕ) cos (2ϕ) ⎦
(1)
which corresponds, apart from a phase shift of π, to a pure rotation matrix R (2ϕ) with rotation power 2ϕ. It is worthwhile to highlight that the above matrix summarizes both the effects of prism edge rotation and double reflection. If a non-reciprocal rotator with rotation power α is added to the Porro prism in a retracing path, then the Jones matrix of the whole path is given by
J = R ( − α )⋅JPP (ϕ)⋅R (α ) = − R ( − α )⋅R (2ϕ)⋅R (α ).
(2)
Knowing that the commutative property holds for the product of two-dimension rotation matrices, we obtain
J = − R ( − α )⋅R (2ϕ)⋅R (α ) = − R (2ϕ)⋅R ( − α )⋅R (α ) = − R (2ϕ),
(3)
demonstrating the universal cancellation of any non-reciprocal circular birefringence operated by the Porro prism, for whatever value of both α and ϕ. In particular for ϕ = 45°, corresponding to the case of PRM, the Jones matrix becomes
⎡ 0 1⎤ JPRM = JPP (45°) = − R (90°) = ⎢ , ⎣ − 1 0⎥⎦
(4)
which is anti-diagonal and confirms that PRM transforms an initial beam with horizontal (or vertical) state of polarization into an orthogonally-polarized returning beam, independently of any presence of non-reciprocal circular birefringence on the retracing path. Hereafter, the birefringence compensation properties of FRM and PRM will be compared with the help of the Poincaré representation, as summarized in Fig. 1 considering the view from the top of the sphere of the states of polarization. Fig. 1a shows the polarization evolution on a retracing path through a reciprocal linearly birefringent retarder Wr[Δ,ψ], in presence of a FRM. An initial horizontal state H moves into S after the retarder, whose action is a rotation of Δ around the equatorial axis of azimuth ψ. Then the action of the FRM, which is equivalent to a half-wave plate with diagonal axes [10], is given by a rotation of π around the axis through Q and Q, moving S into S′. Finally, the beam retraces the reciprocal retarder, now represented by a rotation about the equatorial axis with azimuth ψ, obtaining an
exit vertical state V, orthogonal to the entrance state H. This property holds for any generic reciprocal retarder Wr[Δ,ψ], therefore is verified that FRM compensate for any reciprocal birefringence. On the other hand, the compensation property of FRM does not hold for non-reciprocal birefringence [10], as for instance due to the Faraday rotation effect. The behavior of the PRM in a retracing path including a nonreciprocal rotator Rnr[α] of rotation power α is depicted in Fig. 1b. The polarization evolution is given by a sequence of rotations around the polar axis of the Poincaré sphere. The initial state H is moved into P by a rotation of 2α, due to the effect of the nonreciprocal rotator. Then the PRM gives a rotation of 90°, moving P into P′. Finally, the action of the non-reciprocal rotator on the returning beam is a rotation of –2α, producing an exit vertical state V. This result is independent of α, hence the non-reciprocal circular birefringence is canceled out by the PRM.
3. Experimental results In order to experimentally verify that the PRM cancels out the non-reciprocal circular birefringence in a retracing path, the setup reported in Fig. 2 was realized onto an optical bench by means of bulk-optic components. A variable Faraday rotator (VFR), developed in a previous work [11], was placed between a 50/50 polarization-preserving cube beam splitter (BS) and a metallized Porro prism (produced by Lambda Research Optics) with edge at 45°, realizing the PRM. A horizontally polarized laser beam at the wavelength of 1555 nm, after transmission by the BS, went through the VFR and then was made to retrace by the reflective action of PRM. Finally, the returning beam was reflected by the BS, analysed in polarization by a rotatable polarizer and photodetected. The polarization azimuth of the beam after retracing was measured for different values of VFR rotation power, obtained by changing the electrical current injected in the VFR. It is evident from Fig. 3 that the polarization rotation after retracing is substantially constant, close to the value of 90° expected from the theory, independently of the VFR rotation power. Fig. 3 also shows that substituting the PRM with a simple mirror, the Faraday effect is no more canceled, obtaining a polarization rotation after retracing which is substantially twice of the VFR rotation power (that is the single-pass Faraday rotation), as expected.
4. Conclusions The property of the Porro prism of compensating for the nonreciprocal circular birefringence is pointed out, introducing the PRM as companion reflective component beside the well-known FRM. The PRM is based on the use of a 45°-oriented Porro prism and acts on the non-reciprocal circular birefringence as the FRM acts on the reciprocal one. The experimental results show the
Fig. 1. Representation of birefringence compensation schemes in retracing paths, with the help of the view from the top of the Poincaré sphere: (a) reciprocal retarder and Faraday rotator mirror (FRM); (b) non-reciprocal rotator and Porro rotator mirror (PRM).
Fig. 2. Experimental set-up for canceling out a non-reciprocal circular birefringence in a retracing path.
M. Martinelli et al. / Optics Communications 366 (2016) 119–121
121
a sort of duality between PRM and FRM: the reciprocal birefringence is compensated by the latter but not by the former, while the non-reciprocal circular birefringence is compensated by the former but not by the latter.
References
Fig. 3. Rotation of the polarization azimuth after retracing as a function of the rotation power of the variable Faraday rotator.
effectiveness of the PRM in canceling out the non-reciprocal Faraday rotation, transforming a horizontal state of polarization into an orthogonal vertical state after retracing. So the present work demonstrates, for the first time, a scheme for the universal compensation of any non-reciprocal circular birefringence. It is expected that applications of the PRM will emerge in all the contexts where is beneficial to cancel out the non-reciprocal circular birefringence, typically encountered in optics as polarization rotation induced by the Faraday effect. Finally, it is interesting to notice
[1] M. Martinelli, A universal compensator for polarization changes induced by birefringence on a retracing beam, Opt. Commun. 72 (1989) 341–344. [2] M.J. Marrone, A.D. Kersey, Visibility limits in fibre-optic Michelson interferometer with birefringence compensation, Electron. Lett. 27 (1991) 1422–1424. [3] D.S. Bethune, W.P. Risk, An autocompensating fiber-optic quantum cryptography system based on polarization splitting of light, IEEE J. Quantum Elect. 36 (2000) 340–347. [4] V. Secondi, F. Sciarrino, F. De Martini, Quantum spin-flipping by the Faraday mirror, Phys. Rev. A 70 (2004) 040301. [5] M. Martinelli, L. Marazzi, P. Parolari, M. Brunero, G. Gavioli, Polarization in retracing circuits for WDM-PON, IEEE Photonics Technol. Lett. 24 (2012) 1191–1193. [6] E.J. Galvez, M.R. Cheyne, J.B. Stewart, C.D. Holmes, H.I. Sztul, Variable geometric-phase polarization rotators for the visible, Opt. Commun. 171 (1999) 7–13. [7] R.M.A. Azzam, H.K. Khanfar, Polarization properties of retroreflecting rightangle prisms, Appl. Opt. 47 (2008) 359–364. [8] Y.-D. Liu, C. Gao, X. Qi, Field rotation and polarization properties of the Porro prism, 26, 2009, 1157. [9] R. Bhandari, Geometric phase in an arbitrary evolution of a light beam, Phys. Lett. A 135 (1989) 240–244. [10] R. Bhandari, A useful generalization of the Martinelli effect, 88, 1992, pp. 1–5. [11] P. Martelli, P. Boffi, M. Ferrario, L. Marazzi, P. Parolari, S. M. Pietralunga, R. Siano, A. Righetti, M. Martinelli, Polarization stabilizer for polarization-division multiplexed optical systems, Proceedings ECOC, 2007, paper 6.6.5.
Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
A universal compensator for polarization changes induced by non-reciprocal circular birefringence on a retracing beam Mario Martinelli, Paolo Martelli n, Annalaura Fasiello Politecnico di Milano, Dipartimento di Elettronica Informazione e Bioingegneria, PoliCom Lab, Via G. Ponzio 34/5, 20133 Milano, Italy
art ic l e i nf o
a b s t r a c t
Article history: Received 3 November 2015 Received in revised form 14 December 2015 Accepted 15 December 2015
In this communication we recognize that it is possible to cancel out the effects of the non-reciprocal circular birefringence on a retracing beam. The experimental results demonstrate that a linearly polarized beam is returned into an orthogonal state after retracing through a variable Faraday rotator, by exploiting the reflective action of a Porro prism with edge at 45° with respect to the initial polarization axis, for any amount of non-reciprocal Faraday rotation. & 2015 Elsevier B.V. All rights reserved.
Keywords: Polarization Birefringence Non-reciprocal circular birefringence Porro prism Faraday rotator mirror Retracing path
1. Introduction The very remarkable compensation property of the Faraday rotator mirror (FRM), i.e. a mirror with in front a Faraday rotator with 45° of rotation power, was illustrated for the first time in 1989 [1]. In particular, when the FRM is used, any reciprocal birefringence in the retracing path is canceled out. Although it was been initially conceived for the use in optical fiber sensor applications [2], the compensation property of the FRM was quickly exported in other contexts of optics, as quantum cryptography [3], measurements of quantum states of light [4] and optical communications [5]. Behind the success of the FRM there was a series of factors: – its compensation property holds at the single-photon level, so it can be also applied in quantum experiments; – it is passive, so it acts without limitation of response time in all its optical bandwidth of operation; – it allows a practical and zero-loss recovery of an orthogonally polarized returning beam, by means of a simple polarizing beam splitter (PBS). The aim of the present communication is to show that a n
Corresponding author. E-mail address: [email protected] (P. Martelli).
http://dx.doi.org/10.1016/j.optcom.2015.12.036 0030-4018/& 2015 Elsevier B.V. All rights reserved.
companion mirror of the FRM exists, based on a Porro prism set at 45° with respect to a reference polarization axis (e.g., identified by a PBS). We investigate the ability of this component, which we name Porro rotator mirror (PRM), when used as a mirror in a retracing path, in canceling out the non-reciprocal circular birefringence of the path. As for the FRM, this property holds at single-photon level, occurs in a passive way and produces an orthogonal state after retracing for a linearly polarized beam.
2. Optical retracing paths The field transformation properties of a Porro prism (also named right-angle prism or roof edge rotator) are known [6–8]. If the two right-angle reflective faces act as ideal metallic reflectors, then the Porro prism is a pure rotator for both the image and the polarization. In this case, there is not the phase shift between s and p components related to the total internal reflection condition and the linear states of polarization are not ellipticized. Moreover, the overall effect of the double reflection at the metallized faces is to preserve the circular polarization handedness. Coherently to the formalism introduced in [9] for describing retracing beams, we can write the following Jones matrix of an ideally-reflecting Porro prism with edge oriented at an angle ϕ with respect to the horizontal reference axis:
120
M. Martinelli et al. / Optics Communications 366 (2016) 119–121
⎡ cos (2ϕ) − sin (2ϕ)⎤ ⎥ = − R (2ϕ), JPP (ϕ) = − ⎢ ⎣ sin (2ϕ) cos (2ϕ) ⎦
(1)
which corresponds, apart from a phase shift of π, to a pure rotation matrix R (2ϕ) with rotation power 2ϕ. It is worthwhile to highlight that the above matrix summarizes both the effects of prism edge rotation and double reflection. If a non-reciprocal rotator with rotation power α is added to the Porro prism in a retracing path, then the Jones matrix of the whole path is given by
J = R ( − α )⋅JPP (ϕ)⋅R (α ) = − R ( − α )⋅R (2ϕ)⋅R (α ).
(2)
Knowing that the commutative property holds for the product of two-dimension rotation matrices, we obtain
J = − R ( − α )⋅R (2ϕ)⋅R (α ) = − R (2ϕ)⋅R ( − α )⋅R (α ) = − R (2ϕ),
(3)
demonstrating the universal cancellation of any non-reciprocal circular birefringence operated by the Porro prism, for whatever value of both α and ϕ. In particular for ϕ = 45°, corresponding to the case of PRM, the Jones matrix becomes
⎡ 0 1⎤ JPRM = JPP (45°) = − R (90°) = ⎢ , ⎣ − 1 0⎥⎦
(4)
which is anti-diagonal and confirms that PRM transforms an initial beam with horizontal (or vertical) state of polarization into an orthogonally-polarized returning beam, independently of any presence of non-reciprocal circular birefringence on the retracing path. Hereafter, the birefringence compensation properties of FRM and PRM will be compared with the help of the Poincaré representation, as summarized in Fig. 1 considering the view from the top of the sphere of the states of polarization. Fig. 1a shows the polarization evolution on a retracing path through a reciprocal linearly birefringent retarder Wr[Δ,ψ], in presence of a FRM. An initial horizontal state H moves into S after the retarder, whose action is a rotation of Δ around the equatorial axis of azimuth ψ. Then the action of the FRM, which is equivalent to a half-wave plate with diagonal axes [10], is given by a rotation of π around the axis through Q and Q, moving S into S′. Finally, the beam retraces the reciprocal retarder, now represented by a rotation about the equatorial axis with azimuth ψ, obtaining an
exit vertical state V, orthogonal to the entrance state H. This property holds for any generic reciprocal retarder Wr[Δ,ψ], therefore is verified that FRM compensate for any reciprocal birefringence. On the other hand, the compensation property of FRM does not hold for non-reciprocal birefringence [10], as for instance due to the Faraday rotation effect. The behavior of the PRM in a retracing path including a nonreciprocal rotator Rnr[α] of rotation power α is depicted in Fig. 1b. The polarization evolution is given by a sequence of rotations around the polar axis of the Poincaré sphere. The initial state H is moved into P by a rotation of 2α, due to the effect of the nonreciprocal rotator. Then the PRM gives a rotation of 90°, moving P into P′. Finally, the action of the non-reciprocal rotator on the returning beam is a rotation of –2α, producing an exit vertical state V. This result is independent of α, hence the non-reciprocal circular birefringence is canceled out by the PRM.
3. Experimental results In order to experimentally verify that the PRM cancels out the non-reciprocal circular birefringence in a retracing path, the setup reported in Fig. 2 was realized onto an optical bench by means of bulk-optic components. A variable Faraday rotator (VFR), developed in a previous work [11], was placed between a 50/50 polarization-preserving cube beam splitter (BS) and a metallized Porro prism (produced by Lambda Research Optics) with edge at 45°, realizing the PRM. A horizontally polarized laser beam at the wavelength of 1555 nm, after transmission by the BS, went through the VFR and then was made to retrace by the reflective action of PRM. Finally, the returning beam was reflected by the BS, analysed in polarization by a rotatable polarizer and photodetected. The polarization azimuth of the beam after retracing was measured for different values of VFR rotation power, obtained by changing the electrical current injected in the VFR. It is evident from Fig. 3 that the polarization rotation after retracing is substantially constant, close to the value of 90° expected from the theory, independently of the VFR rotation power. Fig. 3 also shows that substituting the PRM with a simple mirror, the Faraday effect is no more canceled, obtaining a polarization rotation after retracing which is substantially twice of the VFR rotation power (that is the single-pass Faraday rotation), as expected.
4. Conclusions The property of the Porro prism of compensating for the nonreciprocal circular birefringence is pointed out, introducing the PRM as companion reflective component beside the well-known FRM. The PRM is based on the use of a 45°-oriented Porro prism and acts on the non-reciprocal circular birefringence as the FRM acts on the reciprocal one. The experimental results show the
Fig. 1. Representation of birefringence compensation schemes in retracing paths, with the help of the view from the top of the Poincaré sphere: (a) reciprocal retarder and Faraday rotator mirror (FRM); (b) non-reciprocal rotator and Porro rotator mirror (PRM).
Fig. 2. Experimental set-up for canceling out a non-reciprocal circular birefringence in a retracing path.
M. Martinelli et al. / Optics Communications 366 (2016) 119–121
121
a sort of duality between PRM and FRM: the reciprocal birefringence is compensated by the latter but not by the former, while the non-reciprocal circular birefringence is compensated by the former but not by the latter.
References
Fig. 3. Rotation of the polarization azimuth after retracing as a function of the rotation power of the variable Faraday rotator.
effectiveness of the PRM in canceling out the non-reciprocal Faraday rotation, transforming a horizontal state of polarization into an orthogonal vertical state after retracing. So the present work demonstrates, for the first time, a scheme for the universal compensation of any non-reciprocal circular birefringence. It is expected that applications of the PRM will emerge in all the contexts where is beneficial to cancel out the non-reciprocal circular birefringence, typically encountered in optics as polarization rotation induced by the Faraday effect. Finally, it is interesting to notice
[1] M. Martinelli, A universal compensator for polarization changes induced by birefringence on a retracing beam, Opt. Commun. 72 (1989) 341–344. [2] M.J. Marrone, A.D. Kersey, Visibility limits in fibre-optic Michelson interferometer with birefringence compensation, Electron. Lett. 27 (1991) 1422–1424. [3] D.S. Bethune, W.P. Risk, An autocompensating fiber-optic quantum cryptography system based on polarization splitting of light, IEEE J. Quantum Elect. 36 (2000) 340–347. [4] V. Secondi, F. Sciarrino, F. De Martini, Quantum spin-flipping by the Faraday mirror, Phys. Rev. A 70 (2004) 040301. [5] M. Martinelli, L. Marazzi, P. Parolari, M. Brunero, G. Gavioli, Polarization in retracing circuits for WDM-PON, IEEE Photonics Technol. Lett. 24 (2012) 1191–1193. [6] E.J. Galvez, M.R. Cheyne, J.B. Stewart, C.D. Holmes, H.I. Sztul, Variable geometric-phase polarization rotators for the visible, Opt. Commun. 171 (1999) 7–13. [7] R.M.A. Azzam, H.K. Khanfar, Polarization properties of retroreflecting rightangle prisms, Appl. Opt. 47 (2008) 359–364. [8] Y.-D. Liu, C. Gao, X. Qi, Field rotation and polarization properties of the Porro prism, 26, 2009, 1157. [9] R. Bhandari, Geometric phase in an arbitrary evolution of a light beam, Phys. Lett. A 135 (1989) 240–244. [10] R. Bhandari, A useful generalization of the Martinelli effect, 88, 1992, pp. 1–5. [11] P. Martelli, P. Boffi, M. Ferrario, L. Marazzi, P. Parolari, S. M. Pietralunga, R. Siano, A. Righetti, M. Martinelli, Polarization stabilizer for polarization-division multiplexed optical systems, Proceedings ECOC, 2007, paper 6.6.5.