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Universal compensation of the non-reciprocal circular birefringence in a retracing path by a mirrored quarter-wave plate
Optics Communications 372 (2016) 123–125
Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Universal compensation of the non-reciprocal circular birefringence in a retracing path by a mirrored quarter-wave plate 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 2 March 2016 Accepted 28 March 2016
A quarter-wave plate combined with a mirror realizes a pure rotator on the reflected beam, hence it realizes the same polarization transformation of a Porro prism, which has been recently demonstrated as a universal compensator for the non-reciprocal circular birefringence present in a retracing path. In the present work, the mirrored quarter-wave plate has been experimentally proved to effectively compensate for the non-reciprocal circular birefringence introduced by a variable Faraday rotator. & 2016 Elsevier B.V. All rights reserved.
Keywords: Polarization Quarter-wave plate Non-reciprocal circular birefringence Faraday rotator mirror Retracing path
1. Introduction A recently published paper [1] demonstrated that a Porro prism used as a reflector compensates for whatever non-reciprocal circular birefringence present in a retracing path. In particular, a Porro prism set at 45° with respect to a reference horizontal axis, identified by the polarization axis of the light transmitted by a cube polarizing beam splitter (PBS), produces an orthogonallypolarized returning light (i.e., in a vertical state), after retracing any non-reciprocal circular birefringence. So, the Porro prism acts as a companion mirror of the Faraday rotator mirror, which compensates for whatever reciprocal birefringence in a retracing path [2]. A Porro prism with the two right-angle reflective faces coated with a metal, in order to avoid any birefringence related to the reflection, acts as a perfect rotator preserving the circular polarization handedness, because of the double reflection on the two right-angle faces, while reversing the propagation wave-vector. In fact, using the formalism introduced by [3] for retracing beams, the Jones matrix of a Porro prism with edge oriented at an angle ϕ is given by
⎡ cos(2ϕ) −sin(2ϕ)⎤ ⎥ = − R(2ϕ), JPP (ϕ) = − ⎢ ⎣ sin(2ϕ) cos(2ϕ) ⎦
(1)
representing a polarization rotator by an angle 2ϕ [1]. After the publication of [1] we realized that, if a retracing path is considered, the combination of a quarter-wave plate (QWP) n
Corresponding author. E-mail address: [email protected] (P. Martelli).
http://dx.doi.org/10.1016/j.optcom.2016.03.083 0030-4018/& 2016 Elsevier B.V. All rights reserved.
followed by a mirror is equivalent to the Porro prism, showing the same Jones matrix representation and operating the same polarization transformation. Hence, even the mirrored quarter-wave plate (MQWP), can be considered as a universal compensator for any non-reciprocal circular birefringence present in a retracing beam. In particular, we show in the present paper the experimental demonstration of this compensation for the most usual case of non-reciprocal circular birefringence, due to the Faraday effect.
2. Polarization action of the mirrored quarter-wave plate The action of a MQWP oriented at 45° with respect to a polarization axis was used for long time by the optical community as an isolator for blocking the returning light, as reported for example in [4]. In fact, the combination of the QWP oriented at 45° and a mirror allows for transforming a horizontally-polarized light into a returning vertically polarized light. It appears rather singular that no one before now has recognized the rotation property that the combination of a QWP and a mirror possesses. Indeed, let us consider a QWP oriented at a generic angle ϕ with respect to a horizontal reference and followed by a common mirror. The reflected beam retraces the QWP yielding the following Jones matrix for the MQWP, according to the formalism described by [3] for retracing paths, depending on the angle ϕ:
⎡ 1 0⎤ ⎡1 0 ⎤ ⎡ 1 0⎤ JMQWP = R(ϕ)⎢ R(−ϕ)⎢ R(−ϕ)⎢ R(ϕ), ⎣ 0 i ⎥⎦ ⎣ 0 −1⎥⎦ ⎣ 0 i ⎥⎦
(2)
where R(ϕ) is the Jones matrix of a polarization rotator by an angle
124
M. Martinelli et al. / Optics Communications 372 (2016) 123–125
ϕ, that is ⎡ cosϕ −sinϕ ⎤ ⎥. R(ϕ) = ⎢ ⎣ sinϕ cosϕ ⎦
(3)
Considering that
⎡1 0 ⎤ ⎡1 0 ⎤ R(−ϕ)⎢ R(−ϕ) = ⎢ ⎣ 0 −1⎥⎦ ⎣ 0 −1⎥⎦
(4)
and
⎡ 1 0⎤⎡ 1 0 ⎤⎡ 1 0⎤ ⎡ 1 0⎤ ⎢⎣ ⎥⎢ ⎥⎢ ⎥=⎢ ⎥, 0 i ⎦⎣ 0 −1⎦⎣ 0 i ⎦ ⎣ 0 1⎦
(5)
the Jones matrix expressed by (2) becomes
⎡ cos(2ϕ) −sin(2ϕ)⎤ ⎡ 1 0⎤ ⎥ = R(2ϕ). JMQWP = R(ϕ)⎢ R(ϕ) = ⎢ ⎥ ⎣ 0 1⎦ ⎣ sin(2ϕ) cos(2ϕ) ⎦
(6)
This Jones matrix describes a polarization rotator by an angle 2ϕ, exactly as a Porro prism with edge rotated by an angle ϕ, hence the MQWP shows the same non-reciprocal circular birefringence compensation property demonstrated in [1] for the Porro prism. In fact, when the MQWP is placed after a non-reciprocal device with circular birefringence, like a Faraday rotator of power α, we obtain for the resulting Jones matrix
J = R(−α )JMQWP R(α ) = R(−α )R(2ϕ)R(α ) = R(2ϕ) = JMQWP .
(7)
It is thus confirmed that also the MQWP acts as universal compensator for the non-reciprocal birefringence like the Porro prism. In particular, in case of a MQWP set at an angle of 45°, the corresponding Jones matrix becomes
JMQWP
45°
⎡ 0 1⎤ , =⎢ ⎣ −1 0⎥⎦
(8)
which is anti-diagonal and gives an exchange between horizontal and vertical states of polarization after retracing, for any non-reciprocal circular birefringence on the retracing path. Moreover, the transformation of an initial horizontal state (point H) into a returning vertical state (point V) is shown in Fig. 1, by means of the Poincaré sphere representation of the action of a 45°-oriented MQWP on a retracing path including a generic non-reciprocal rotator Rnr(α) of rotation power α. Indeed, considering the top view of the Poincaré sphere on the right of Fig. 1, the point H is moved at first in P1 by a counter-clockwise rotation of 2α along the equator, representing the action of the non-reciprocal rotator. Then the QWP oriented at 45° moves P1 in P2, by a rotation of 90° around the axis passing through the points Q and Q, representing the diagonal linear states. According to [3], the mirror is equivalent to a half-wave plate with horizontal and vertical
Fig. 1. Polarization evolution from a horizontal state (H) to a vertical state (V) in the retracing path obtained cascading a non-reciprocal rotator Rnr(α) of power α, a 45°-oriented QWP and a mirror.
eigen states, hence is represented on the Poincaré sphere by a rotation of 180° around the axis passing through H and V (vertical state), moving P2 in P3. After the mirror, the beam retraces the QWP, with state of polarization moved from P3 to P4, and finally retraces the non-reciprocal rotator, with state moved from P4 to V.
3. Experimental results In order to experimentally prove the above theoretical results, the setup schematized by Fig. 2 was realized. A 1555-nm horizontally-polarized laser beam is transmitted through a PBS and retraces itself after crossing a low-order QWP and a metallic mirror. A variable Faraday rotator (VFR), developed in a previous work [5], is either absent or present in the retracing path. A first experiment, carried out without VFR in the setup of Fig. 2, permits to verify the 2ϕ-rotator behavior of the mirrored QWP oriented at an angle ϕ: the optical intensity, photodetected after the reflection of the returning beam by the PBS, is measured as a function of the QWP orientation angle. Considering the Jones matrix of Eq. (6) for MQWP, the intensity of the vertical polarization component reflected by the PBS is expected to depend on the angle ϕ according to the function sin2(2ϕ). A very good agreement between the measured values, represented in case of no VFR by asterisks, and the expected values, represented by the dashed curve, is evidenced by Fig. 3, confirming the polarization rotation effect of the MQWP on the retracing beam. In order to experimentally check the effectiveness of the MQWP in compensating for any non-reciprocal circular birefringence, we have inserted the VFR in the setup of Fig. 2 and measured the optical intensity reflected by PBS for different values of polarization rotation given by the VFR. The measure results are reported in Fig. 3 and show a substantial cancellation of the Faraday effect in the retracing path, confirming the compensation property of the MWQP, as predicted by Eq. (7). In particular, the measured optical intensity is maximized in the case of MQWP oriented at 45°, for any value of Faraday rotation. In fact, in this condition a horizontal state of polarization is transformed into a vertical state by the action of the mirrored 45°-oriented QWP, according to the Jones matrix of Eq. (8), which also cancels the non-reciprocal circular birefringence present in the retracing path.
4. Conclusion In this work it has been shown that a MQWP, that is the combination of a quarter-wave plate and a mirror, can be used to compensate for any non-reciprocal circular birefringence in a retracing path, showing the same property recently recognized for a Porro prism. The MQWP realizes a pure polarization rotator, which can be also used in other applications. In particular a horizontal state of polarization transmitted by a PBS can be transformed by a
Fig. 2. Scheme of the experimental setup.
M. Martinelli et al. / Optics Communications 372 (2016) 123–125
125
from the initial beam without splitting loss, for any non-reciprocal circular birefringence present in the retracing path.
References [1] M. Martinelli, P. Martelli, A. Fasiello, A universal compensator for polarization changes induced by non-reciprocal circular birefringence on a retracing beam, Opt. Commun. 366 (2016) 119–121. [2] M. Martinelli, A universal compensator for polarization change induced by birefringence on a retracing beam, Opt. Commun. 72 (1989) 341–344. [3] R. Bhandari, Geometric phase in an arbitrary evolution of a light beam, Phys. Lett. A 135 (1989) 240–244. [4] D.K. Mansfield, A. Semet, L.C. Johnson, A lossless passive isolator for optically pumped far-infrared lasers, Appl. Phys. Lett. 37 (1980) 688–690. [5] 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, Proc. ECOC (2007), paper 6.6.5. Fig. 3. Optical intensity reflected by the PBS as a function of the QWP orientation for different amounts of non-reciprocal polarization rotation in the retracing path: (asterisks) experiments without VFR; (circles) experiments with VFR of power α ¼20°; (squares) experiments with VFR of power α ¼35°; (triangles) experiments with VFR of power α ¼60°; (dashed line) theoretical prediction.
MQWP set at 45° into a returning vertical state of polarization, which is then completely reflected by the PBS and hence separated
Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Universal compensation of the non-reciprocal circular birefringence in a retracing path by a mirrored quarter-wave plate 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 2 March 2016 Accepted 28 March 2016
A quarter-wave plate combined with a mirror realizes a pure rotator on the reflected beam, hence it realizes the same polarization transformation of a Porro prism, which has been recently demonstrated as a universal compensator for the non-reciprocal circular birefringence present in a retracing path. In the present work, the mirrored quarter-wave plate has been experimentally proved to effectively compensate for the non-reciprocal circular birefringence introduced by a variable Faraday rotator. & 2016 Elsevier B.V. All rights reserved.
Keywords: Polarization Quarter-wave plate Non-reciprocal circular birefringence Faraday rotator mirror Retracing path
1. Introduction A recently published paper [1] demonstrated that a Porro prism used as a reflector compensates for whatever non-reciprocal circular birefringence present in a retracing path. In particular, a Porro prism set at 45° with respect to a reference horizontal axis, identified by the polarization axis of the light transmitted by a cube polarizing beam splitter (PBS), produces an orthogonallypolarized returning light (i.e., in a vertical state), after retracing any non-reciprocal circular birefringence. So, the Porro prism acts as a companion mirror of the Faraday rotator mirror, which compensates for whatever reciprocal birefringence in a retracing path [2]. A Porro prism with the two right-angle reflective faces coated with a metal, in order to avoid any birefringence related to the reflection, acts as a perfect rotator preserving the circular polarization handedness, because of the double reflection on the two right-angle faces, while reversing the propagation wave-vector. In fact, using the formalism introduced by [3] for retracing beams, the Jones matrix of a Porro prism with edge oriented at an angle ϕ is given by
⎡ cos(2ϕ) −sin(2ϕ)⎤ ⎥ = − R(2ϕ), JPP (ϕ) = − ⎢ ⎣ sin(2ϕ) cos(2ϕ) ⎦
(1)
representing a polarization rotator by an angle 2ϕ [1]. After the publication of [1] we realized that, if a retracing path is considered, the combination of a quarter-wave plate (QWP) n
Corresponding author. E-mail address: [email protected] (P. Martelli).
http://dx.doi.org/10.1016/j.optcom.2016.03.083 0030-4018/& 2016 Elsevier B.V. All rights reserved.
followed by a mirror is equivalent to the Porro prism, showing the same Jones matrix representation and operating the same polarization transformation. Hence, even the mirrored quarter-wave plate (MQWP), can be considered as a universal compensator for any non-reciprocal circular birefringence present in a retracing beam. In particular, we show in the present paper the experimental demonstration of this compensation for the most usual case of non-reciprocal circular birefringence, due to the Faraday effect.
2. Polarization action of the mirrored quarter-wave plate The action of a MQWP oriented at 45° with respect to a polarization axis was used for long time by the optical community as an isolator for blocking the returning light, as reported for example in [4]. In fact, the combination of the QWP oriented at 45° and a mirror allows for transforming a horizontally-polarized light into a returning vertically polarized light. It appears rather singular that no one before now has recognized the rotation property that the combination of a QWP and a mirror possesses. Indeed, let us consider a QWP oriented at a generic angle ϕ with respect to a horizontal reference and followed by a common mirror. The reflected beam retraces the QWP yielding the following Jones matrix for the MQWP, according to the formalism described by [3] for retracing paths, depending on the angle ϕ:
⎡ 1 0⎤ ⎡1 0 ⎤ ⎡ 1 0⎤ JMQWP = R(ϕ)⎢ R(−ϕ)⎢ R(−ϕ)⎢ R(ϕ), ⎣ 0 i ⎥⎦ ⎣ 0 −1⎥⎦ ⎣ 0 i ⎥⎦
(2)
where R(ϕ) is the Jones matrix of a polarization rotator by an angle
124
M. Martinelli et al. / Optics Communications 372 (2016) 123–125
ϕ, that is ⎡ cosϕ −sinϕ ⎤ ⎥. R(ϕ) = ⎢ ⎣ sinϕ cosϕ ⎦
(3)
Considering that
⎡1 0 ⎤ ⎡1 0 ⎤ R(−ϕ)⎢ R(−ϕ) = ⎢ ⎣ 0 −1⎥⎦ ⎣ 0 −1⎥⎦
(4)
and
⎡ 1 0⎤⎡ 1 0 ⎤⎡ 1 0⎤ ⎡ 1 0⎤ ⎢⎣ ⎥⎢ ⎥⎢ ⎥=⎢ ⎥, 0 i ⎦⎣ 0 −1⎦⎣ 0 i ⎦ ⎣ 0 1⎦
(5)
the Jones matrix expressed by (2) becomes
⎡ cos(2ϕ) −sin(2ϕ)⎤ ⎡ 1 0⎤ ⎥ = R(2ϕ). JMQWP = R(ϕ)⎢ R(ϕ) = ⎢ ⎥ ⎣ 0 1⎦ ⎣ sin(2ϕ) cos(2ϕ) ⎦
(6)
This Jones matrix describes a polarization rotator by an angle 2ϕ, exactly as a Porro prism with edge rotated by an angle ϕ, hence the MQWP shows the same non-reciprocal circular birefringence compensation property demonstrated in [1] for the Porro prism. In fact, when the MQWP is placed after a non-reciprocal device with circular birefringence, like a Faraday rotator of power α, we obtain for the resulting Jones matrix
J = R(−α )JMQWP R(α ) = R(−α )R(2ϕ)R(α ) = R(2ϕ) = JMQWP .
(7)
It is thus confirmed that also the MQWP acts as universal compensator for the non-reciprocal birefringence like the Porro prism. In particular, in case of a MQWP set at an angle of 45°, the corresponding Jones matrix becomes
JMQWP
45°
⎡ 0 1⎤ , =⎢ ⎣ −1 0⎥⎦
(8)
which is anti-diagonal and gives an exchange between horizontal and vertical states of polarization after retracing, for any non-reciprocal circular birefringence on the retracing path. Moreover, the transformation of an initial horizontal state (point H) into a returning vertical state (point V) is shown in Fig. 1, by means of the Poincaré sphere representation of the action of a 45°-oriented MQWP on a retracing path including a generic non-reciprocal rotator Rnr(α) of rotation power α. Indeed, considering the top view of the Poincaré sphere on the right of Fig. 1, the point H is moved at first in P1 by a counter-clockwise rotation of 2α along the equator, representing the action of the non-reciprocal rotator. Then the QWP oriented at 45° moves P1 in P2, by a rotation of 90° around the axis passing through the points Q and Q, representing the diagonal linear states. According to [3], the mirror is equivalent to a half-wave plate with horizontal and vertical
Fig. 1. Polarization evolution from a horizontal state (H) to a vertical state (V) in the retracing path obtained cascading a non-reciprocal rotator Rnr(α) of power α, a 45°-oriented QWP and a mirror.
eigen states, hence is represented on the Poincaré sphere by a rotation of 180° around the axis passing through H and V (vertical state), moving P2 in P3. After the mirror, the beam retraces the QWP, with state of polarization moved from P3 to P4, and finally retraces the non-reciprocal rotator, with state moved from P4 to V.
3. Experimental results In order to experimentally prove the above theoretical results, the setup schematized by Fig. 2 was realized. A 1555-nm horizontally-polarized laser beam is transmitted through a PBS and retraces itself after crossing a low-order QWP and a metallic mirror. A variable Faraday rotator (VFR), developed in a previous work [5], is either absent or present in the retracing path. A first experiment, carried out without VFR in the setup of Fig. 2, permits to verify the 2ϕ-rotator behavior of the mirrored QWP oriented at an angle ϕ: the optical intensity, photodetected after the reflection of the returning beam by the PBS, is measured as a function of the QWP orientation angle. Considering the Jones matrix of Eq. (6) for MQWP, the intensity of the vertical polarization component reflected by the PBS is expected to depend on the angle ϕ according to the function sin2(2ϕ). A very good agreement between the measured values, represented in case of no VFR by asterisks, and the expected values, represented by the dashed curve, is evidenced by Fig. 3, confirming the polarization rotation effect of the MQWP on the retracing beam. In order to experimentally check the effectiveness of the MQWP in compensating for any non-reciprocal circular birefringence, we have inserted the VFR in the setup of Fig. 2 and measured the optical intensity reflected by PBS for different values of polarization rotation given by the VFR. The measure results are reported in Fig. 3 and show a substantial cancellation of the Faraday effect in the retracing path, confirming the compensation property of the MWQP, as predicted by Eq. (7). In particular, the measured optical intensity is maximized in the case of MQWP oriented at 45°, for any value of Faraday rotation. In fact, in this condition a horizontal state of polarization is transformed into a vertical state by the action of the mirrored 45°-oriented QWP, according to the Jones matrix of Eq. (8), which also cancels the non-reciprocal circular birefringence present in the retracing path.
4. Conclusion In this work it has been shown that a MQWP, that is the combination of a quarter-wave plate and a mirror, can be used to compensate for any non-reciprocal circular birefringence in a retracing path, showing the same property recently recognized for a Porro prism. The MQWP realizes a pure polarization rotator, which can be also used in other applications. In particular a horizontal state of polarization transmitted by a PBS can be transformed by a
Fig. 2. Scheme of the experimental setup.
M. Martinelli et al. / Optics Communications 372 (2016) 123–125
125
from the initial beam without splitting loss, for any non-reciprocal circular birefringence present in the retracing path.
References [1] M. Martinelli, P. Martelli, A. Fasiello, A universal compensator for polarization changes induced by non-reciprocal circular birefringence on a retracing beam, Opt. Commun. 366 (2016) 119–121. [2] M. Martinelli, A universal compensator for polarization change induced by birefringence on a retracing beam, Opt. Commun. 72 (1989) 341–344. [3] R. Bhandari, Geometric phase in an arbitrary evolution of a light beam, Phys. Lett. A 135 (1989) 240–244. [4] D.K. Mansfield, A. Semet, L.C. Johnson, A lossless passive isolator for optically pumped far-infrared lasers, Appl. Phys. Lett. 37 (1980) 688–690. [5] 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, Proc. ECOC (2007), paper 6.6.5. Fig. 3. Optical intensity reflected by the PBS as a function of the QWP orientation for different amounts of non-reciprocal polarization rotation in the retracing path: (asterisks) experiments without VFR; (circles) experiments with VFR of power α ¼20°; (squares) experiments with VFR of power α ¼35°; (triangles) experiments with VFR of power α ¼60°; (dashed line) theoretical prediction.
MQWP set at 45° into a returning vertical state of polarization, which is then completely reflected by the PBS and hence separated