arXiv:2011.07834v1 [physics.] 16 Nov 2020 qa otieteagebtenteotclae fthe of axes rotation optical of the [20]. angle between plates half-wave an half-wave angle two two have the would twice of to rotator combination equal a a Such as rota- achieved plates. birefringent be A can the rotated. result, tor is a plane As indices. linear refraction polarized different right-circular “feel” and waves (Fara- left- anisotropy makes circular which a effect), causes day [1– field magnetic powerful The a magne- inside 5]. a put is of that consists material toactive rotator The . be plates. half-wave can the that of retarder one spinning adjustable by optical an cascad- controlled basic as plates This half-wave functions two plates. system quarter-wave on two based between et is ing Messaadi which by [19], suggested recently al. technique the one Alternatively, use [16–18]. can liquid-crystals can others by retardance in Tunable achieved while be required. plate, is practical half-wave plate quarter-wave some a a need in may because one re- feature settings the valuable Tuning a applied. is successfully tardance and designed be can oiaei rcie eadr of retarders practice, in dominate and light polarized circularly [1–5]. versa into vice light polarized early lt nrdcsasitof shift quarter-wave a The introduces angle. plate predefined a to po- vector the larization rotates particular effectively plate a half-wave the for wavelength, components polarization orthogonal two aepae yitouigapaesitof shift phase a quarter- introducing the By common and plate plate. Two half-wave wave retar- the wavelengths. and, are achieve of retarders used of range they types widely broader reflections, a also at internal are dation total 15] on [14, rhombs based Fresnel 11–13]. als. 7, [6, wave light a waveplate) of po- a a orthogonal components two (or at larization between retarder difference polarization phase a light a while introduces linear ro- [6–10], of angle polarization plane specified A the and rotates [1–5]. rotators tator retarders polarization polarization light’s optical optical controlling are for [1– state devices waves optical polarization light prominent transverse Two of one 5]. characteristics is polarization basic as the critical, of is polarization control and aepae n oaosaebscbidn lcsfor blocks building basic are rotators and plates Wave or rotation Faraday employ may rotator polarization A lhuhhl-aeadqatrwv ltsclearly plates quarter-wave and half-wave Although eadr r sal aefo iernetmateri- birefringent from made usually are Retarders observe to ability the applications practical many In h antcfilsi ahFrdyrotator. Faraday and each retardance in the fields change magnetic continuously the can quarter-wav one two device, of this consists which transformation, ization epooeauieslshm o h osrcino devic a of construction the for scheme universal a propose We .INTRODUCTION I. 1 eateto hsc,Sfi nvriy ae orhe b 5 Bourchier James University, Sofia Physics, of Department rirr oaiaincnrlb antcfil variation field magnetic by control polarization Arbitrary π/ ln Stoyanova, Elena ,adteeycnet lin- converts thereby and 2, any eie retardation desired π ewe the between 1 vtsa Ivanov, Svetoslav fbhn hspaeoepae nte lt ihazimuth with plate another places angle one plate this behind If r lge ihtehrzna n h etcldirec- vertical by the given is and matrix horizontal Jones the the axes tions, with whose aligned system are coordinate rectangular dimensional θ hs shift phase n h ure-aepae( plate quarter-wave the and hl h oe arxta ersnsartre is retarder a represents that matrix Jones the while inangle tion aiainvco ipyb hnigtemgei fields magnetic rotator. po- Faraday the the each changing of in by variation simply continuous vector and larization setting fast this perform of In can rotators. consists modified we two a and and is plates controller quarter-wave propose two Simon-Mukunda’s we device of The version fields. variation magnetic with of plates the of rotations mechanical stituting speed. and angles the par- frequency change in certain to impractical with needs very one be where to individual applications out ticular on turn rotations may it- which perform transformations, plates apparatus to polarization one the Arbitrary require if however, rotated. 24] 23, is just [7, self or plates [22], and quarter-wave one plates retarder quarter-wave two requires two a transformation and arbitrary of plate An half-wave composition [21]. a state) rotator using final a any achieved to be state initial can po- any the from in vector change po- larization reversible reversible (a every transformation, larization Indeed, manipulation. polarization ieyue eadr r h afwv lt ( most plate The half-wave the wave. are light retarders the widely-used of components polarization Here eaiet h etclai aiuhage.I 2- a In angle). (azimuth axis vertical the to relative osdrnwasnl oaiigbrfign lt of plate birefringent polarizing single a now Consider h oe arxta ecie oao ihrota- with rotator a describes that matrix Jones The nti ae eatmtt ov hspolmb sub- by problem this solve to attempt we paper this In 1 ltsadtoFrdyrttr.Using rotators. Faraday two and plates e ϕ θ n no Rangelov Andon and + stepaesitbtentetoorthogonal two the between shift phase the is h oainagesml ychanging by simply angle rotation the α/ θ ϕ ,tersligJnsmti sgvnb the by given is matrix Jones resulting the 2, o rirr oabtaypolar- arbitrary to arbitrary for e is hs atai srttdt nageof angle an to rotated is axis fast whose , J R J v,16 oa Bulgaria Sofia, 1164 lvd, θ ( ( ( ϕ θ ϕ I PREFACES II. = ) = ) = )   R e − ( iϕ/ cos − 0 sin 1 ϕ θ 2 θ ) θ J = e ( − ϕ π/ cos sin iϕ/ 0 ) R )[,7]. [6, 2) θ θ 2 ( θ   ) . , . ϕ = (3) (1) (2) π ) 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 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 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. Therefore, the suggested tunable retarder rad and 41 T · m at 1064 nm [33]. Our simulations were car- is not mechanical and can be used as a fast switcher, ried out for three different values for the magnetic field, where the switching on/off time of the optical activity B1 = 0.5 T, B2 = 1 T and B3 = 2 T, at a fixed length is in the order of microseconds with the state of the art L = 0.05 m of the TGG crystal. As can be seen from approaching subnanosecond [28, 29]. Fig. 2, any pair of rotation angles α and β in the in- terval [0, 2π] can be achieved with magnetic field smaller than 1T for the visible spectrum. Therefore, with com- V. ARBITRARY TO ARBITRARY mercial Faraday rotators available on the market, the POLARIZATION CONVERTER practical realization of the proposed polarization control device should be straightforward. Based on the fact that combining an arbitrary rotator with an arbitrary retarder allows to achieve any polar- 12 p ization transformation [21], we can combine the tunable retarder from Eq. (12) with an additional rotator to get 10 p an arbitrary-to-arbitrary polarization manipulation de- vice. Its Jones matrix is 8p J = J0(2α)R(β), (13) 6p [rad] or q − 4p J = Jπ/4(π/2)R(α)Jπ/4( π/2)R(β). (14) The proposed optical device, illustrated schematically 2p in Fig. 1 b, has potential advantages over the Simon– Mukunda polarization controller [7, 22], where one has 0 to adjust the spatial orientation of all three wave plates. 400 500 600 700 800 900 1000 1100 The proposed device (cf. Eq. (14)) is more convenient to l [nm] use in the sense that the rotator angle and the retardation are obtained by changing the magnetic field of the first and the second , respectively. Further- FIG. 2. (Color online) The Faraday rotation angle θ vs the light wavelength λ for three different magnetic fields B1 = 0.5 more, our scheme is fast switchable on and off, because T (red dotted), B2 = 1 T (green solid) and B3 = 2 T (blue in the absence of a magnetic field the polarization is not dashed). changed. Finally, we investigate the possibility of achieving any pair of rotation angles α and β (cf. (14)) for the pro- posed arbitrary-to-arbitrary polarization conversion de- VI. CONCLUSION vice. The rotation angle for the Faraday rotator is given by In conclusion, we have suggested two useful polariza- θ(λ)= V (λ)BL, tion manipulation devices. The first device is the mod- where B is the external magnetic field, L is the magneto- ified Simon–Mukunda polarization controller, which in optical element length, and V (λ) is Verdet’s constant. contrast to the traditional controller is constructed as a We do the most common calculation involving the Ter- combination of two quarter-wave plates and a Faraday bium Gallium Garnet (TGG) crystal, as this crystal rotator between them. Our second device for arbitrary yields a high Verdet constant. So far, the dispersion to arbitrary polarization transformation is composed of of Verdet’s constant for the TGG crystal has been ex- two Faraday rotators and two quarter-wave plates, where tensively studied [30–33] and the wavelength dependence the retardance and the rotation can be continuously mod- has been shown to be described by the formula ified merely by changing the magnetic fields of the two Faraday rotators. Because they use Faraday rotation, E the suggested schemes are non-reciprocal. We hope the V (λ)= 2 2 , (15) λ0 − λ proposed methods for polarization control would be cost- 4 effective and useful in any scientific laboratory.

ACKNOWLEDGMENT

This work was supported by Sofia University Grant 80-10-191/2020.

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