Arbitrary polarization control by magnetic field variation Elena Stoyanova,1 Svetoslav Ivanov,1 and Andon Rangelov1 1Department of Physics, Sofia University, James Bourchier 5 blvd, 1164 Sofia, Bulgaria We propose a universal scheme for the construction of a device for arbitrary to arbitrary polar- ization transformation, which consists of two quarter-wave plates and two Faraday rotators. Using this device, one can continuously change the retardance and the rotation angle simply by changing the magnetic fields in each Faraday rotator. I. INTRODUCTION polarization manipulation. Indeed, every reversible po- larization transformation, (a reversible change in the po- In many practical applications the ability to observe larization vector from any initial state to any final state) and control polarization is critical, as polarization is one can be achieved using a composition of a retarder and of the basic characteristics of transverse light waves [1– a rotator [21]. An arbitrary transformation requires one 5]. Two prominent optical devices for controlling light’s half-wave plate and two quarter-wave plates [22], or just polarization state are optical polarization rotators and two quarter-wave plates [7, 23, 24] if the apparatus it- optical polarization retarders [1–5]. A polarization ro- self is rotated. Arbitrary polarization transformations, tator rotates the plane of linear light polarization at a however, require one to perform rotations on individual specified angle [6–10], while a retarder (or a waveplate) plates which may turn out to be very impractical in par- introduces a phase difference between two orthogonal po- ticular applications where one needs to change the angles larization components of a light wave [6, 7, 11–13]. with certain frequency and speed. Retarders are usually made from birefringent materi- In this paper we attempt to solve this problem by sub- als. Fresnel rhombs [14, 15] are also widely used and, stituting mechanical rotations of the plates with variation based on total internal reflections, they achieve retar- of magnetic fields. The device we propose is a modified dation at a broader range of wavelengths. Two common version of Simon-Mukunda’s controller and consists of types of retarders are the half-wave plate and the quarter- two quarter-wave plates and two rotators. In this setting wave plate. By introducing a phase shift of π between the we can perform fast and continuous variation of the po- two orthogonal polarization components for a particular larization vector simply by changing the magnetic fields wavelength, the half-wave plate effectively rotates the po- in each Faraday rotator. larization vector to a predefined angle. The quarter-wave plate introduces a shift of π/2, and thereby converts lin- II. PREFACES early polarized light into circularly polarized light and vice versa [1–5]. Although half-wave and quarter-wave plates clearly The Jones matrix that describes a rotator with rota- dominate in practice, retarders of any desired retardation tion angle θ is can be designed and successfully applied. Tuning the re- cos θ sin θ tardance is a valuable feature because in some practical R(θ)= , (1) − sin θ cos θ settings one may need a half-wave plate, while in others a quarter-wave plate is required. Tunable retardance can while the Jones matrix that represents a retarder is be achieved by liquid-crystals [16–18]. Alternatively, one can use the technique recently suggested by Messaadi et eiϕ/2 0 J(ϕ)= − . (2) al. [19], which is based on two half-wave plates cascad- 0 e iϕ/2 ing between two quarter-wave plates. This basic optical system functions as an adjustable retarder that can be Here ϕ is the phase shift between the two orthogonal arXiv:2011.07834v1 [physics.optics] 16 Nov 2020 controlled by spinning one of the half-wave plates. polarization components of the light wave. The most A polarization rotator may employ Faraday rotation or widely-used retarders are the half-wave plate (ϕ = π) birefringence. The Faraday rotator consists of a magne- and the quarter-wave plate (ϕ = π/2) [6, 7]. toactive material that is put inside a powerful magnet [1– Consider now a single polarizing birefringent plate of 5]. The magnetic field causes a circular anisotropy (Fara- phase shift ϕ, whose fast axis is rotated to an angle of day effect), which makes left- and right-circular polarized θ relative to the vertical axis (azimuth angle). In a 2- waves “feel” different refraction indices. As a result, the dimensional rectangular coordinate system whose axes linear polarization plane is rotated. A birefringent rota- are aligned with the horizontal and the vertical direc- tor can be achieved as a combination of two half-wave tions, the Jones matrix is given by plates. Such a rotator would have an angle of rotation Jθ(ϕ)= R(−θ)J(ϕ)R(θ). (3) equal to twice the angle between the optical axes of the two half-wave plates [20]. If behind this plate one places another plate with azimuth Wave plates and rotators are basic building blocks for angle θ + α/2, the resulting Jones matrix is given by the 2 following arrangements QW-HW-QW, QW-QW-HW, or D T T HW-QW-QW. The first one is the most popular (Fig. 1 a) and is also used as a fiber polarization controller [25]. T The Simon–Mukunda polarization controller [22] oper- ates in the following way: the first quarter-wave plate turns the input elliptical polarization into a linear polar- 4: ization. Then the half-wave plate rotates the obtained +: S linear polarization vector, which is finally transformed into the required elliptical polarization output by the sec- 4: S ond quarter-wave plate. The Jones matrix for the Simon– Mukunda polarization controller is J J J J E = θ3 (π/2) θ2 (π) θ1 (π/2). (5) 4: Because the unit matrix can be written as &ĂƌĂĚĂLJZŽƚĂƚŽƌ 1ˆ = J (π/2)J (−π/2), (6) 4: θ1 θ1 &ĂƌĂĚĂLJZŽƚĂƚŽƌ we obtain J J J J J J − = θ3 (π/2) θ2 (π) θ1 (π/2) θ1 (π/2) θ1 ( π/2). (7) % Next we use that F ɴ J 1 (π)= J 1 (π/2)J 1 (π/2) (8) & θ θ θ $ ɲ to simplify Eq. (7): J J J J J − = θ3 (π/2) θ2 (π) θ1 (π) θ1 ( π/2). (9) Next we make use of Eq. (4) for the combination of two half-wave plates R − −J J (2(θ1 θ2)) = θ2 (π) θ1 (π), (10) FIG. 1. (Color online)(a) The Simon–Mukunda polarization to get the final expression of the Jones matrix: controller in a configuration of the form QW-HW-QW (b) The J −J R J − scheme of arbitrary to arbitrary polarization transformation = θ3 (π/2) (α) θ1 ( π/2), (11) device, composed by two quarter-wave plates and two Faraday rotators. The orientation of the quarter-wave plates is fixed. where the rotator angle is α = 2(θ1 − θ2). Therefore (c) Polarization evolution on the Poincare sphere. The initial the Simon–Mukunda polarization controller can be con- polarization is at point A and the final polarization is at point structed as combination of two quarter-wave plates along B. As can be seen from Eq. (14) the first part of the evolution with a rotator between them. This device would operate is rotation at angle β between points A and C, followed by a in a similar way as the one before: the first quarter-wave α retardation at angle from point C to point B. plate turns the input elliptical polarization into a linear polarization vector, which is then rotated by the rotator element, and is finally transformed into the required el- product liptical output polarization by the second quarter-wave plate. cos α − sin α J + 2(π)J (π)= − , (4) θ α/ θ sin α cos α IV. ARBITRARY RETARDER AS A SPECIAL that represents a Jones rotator matrix (up to an unim- CASE OF THE MODIFIED SIMON–MUKUNDA portant phase of π) [8, 20]. POLARIZATION CONTROLLER In the special case when the two quarter-wave plates III. MODIFIED SIMON–MUKUNDA POLARIZATION CONTROLLER are oriented such that their fast optical axes are perpen- dicular to each other (cf. Eq. 11) (θ1 = θ3 = π/4) we obtain a retarder with retardation 2α [19]: A general scheme of a device capable of arbitrary po- larization transformations is the Simon–Mukunda polar- eiα 0 J0(2α)= J 4(π/2)R(α)J 4(−π/2) = − − . ization controller [22]. It consists of one half-wave plate π/ π/ 0 e iα (HW) and two quarter-wave plates (QW) in one of the (12) 3 · 7 rad · nm2 If the two quarter-wave plates are achromatic (for exam- where E = 4.45 10 T · m and λ0 = 257.5 nm is ple as in [11–13] or if Fresnel rhombs are used as quarter- the wavelength, often close to the Terbium ion’s 4f-5d wave plates) then one can achieve a wavelength tunable transition wavelength. In the range of 400 — 1100 nm, half-wave or quarter-wave plate that operates differently excluding 470 — 500 nm (absorption window [33]), the compared to previously suggested tunable wave plates TGG crystal has optimal material properties for a Fara- [26, 27]. However, in contrast to previously used tun- day rotator. The Verdet constant decreases with increas- able wave plates, here we do not need to rotate the wave ing wavelength for most materials (in absolute value): rad plates, but rather change the magnetic field in the Fara- for the TGG crystal it is equal to 475 T · m at 400 nm day rotator.