A novel polarisation modulation scheme for polarimeters based on Fabry-Perot
cavities is presented. The application to the proposed HERA-X experiment aiming
to measuring the magnetic birefringence of vacuum with the HERA superconducting
magnets is discussed

Della Valle, F.

Ejlli, A.

Ruoso, G.

Zavattini, G.

English

arXiv

A novel polarisation modulation scheme for polarimeters based on Fabry–Perot cavities is presented. The application to the measurement of the magnetic birefringence of vacuum with the HERA superconducting magnets in the ALPS-II configuration is discussed

Archivio della Ricerca - Università degli Studi di Siena

A polarisation modulation scheme for measuring vacuum magnetic birefringence with static fields

ZAVATTINI, Guido

DELLA VALLE, Federico

EJLLI, Aldo

Archivio istituzionale della ricerca - Università di Ferrara

G. Zavattini

F. Della Valle

A. Ejlli

G. Ruoso

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The European Physical Journal C

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DELLA VALLE, FEDERICO

Archivio istituzionale della ricerca - Università di Trieste

Eur. Phys. J. C  (2016) 76:294 DOI 10.1140/epjc/s10052-016-4139-0Special Article - Tools for Experiment and TheoryA polarisation modulation scheme for measuring vacuummagnetic birefringence with static fieldsG. Zavattini1,2,a, F. Della Valle3,4, A. Ejlli1,2, G. Ruoso51 Dip. di Fisica e Scienze della Terra, Università di Ferrara, via G. Saragat 1, Edificio C, 44122 Ferrara, FE, Italy2 INFN, Sez. di Ferrara, via G. Saragat 1, Edificio C, 44122 Ferrara, FE, Italy3 Dip. di Fisica, Università di Trieste, via A. Valerio 2, 34127 Trieste, TS, Italy4 INFN, Sez. di Trieste, via A. Valerio 2, 34127 Trieste, TS, Italy5 INFN, Lab. Naz. di Legnaro, viale dell’Università 2, 35020 Legnaro, PD, ItalyReceived: 18 January 2016 / Accepted: 12 May 2016© The Author(s) 2016. This article is published with open access at Springerlink.comAbstract A novel polarisation modulation scheme forpolarimeters based on Fabry–Perot cavities is presented. Theapplication to the measurement of the magnetic birefringenceof vacuum with the HERA superconducting magnets in theALPS-II configuration is discussed.1 IntroductionVacuum magnetic birefringence is a non linear electrody-namic effect in vacuum closely related to light-by-light elas-tic scattering. Predicted by the Euler–Heisenberg–Weisskopfeffective Lagrangian density see also: [1–11] written in 1936,LEHW = 12μ0(E2c2− B2)+ Aeμ0[(E2c2− B2)2+ 7(Ec· B)2],it takes into account the vacuum fluctuations of electron-positron pairs. As of today, LEHW still needs experimentalconfirmation. HereAe = 245μ0α2λ¯3emec2= 1.32 × 10−24 T−2.The ellipticity ψ induced on a linearly polarised beam of lightwith wavelength λ passing through a medium with birefrin-gence n and length L , and whose axes are defined by theexternal magnetic field, isψ = π Lλn sin 2φa e-mail: guido.zavattini@unife.itwhere φ is the angle between the magnetic field and the polar-isation direction. The birefringence predicted by LEHW is[7–11]n = 3AeB2  4 × 10−24B2.Several experiments are underway, of which the most sen-sitive at present are based on polarimeters with very highfinesse Fabry–Perot cavities and variable magnetic fields[12–14]. The Fabry–Perot cavity is necessary to increase theoptical path L within the magnetic field region, whereas thevariable magnetic field is necessary to induce a time depen-dent effect. Both of these aspects significantly increase thesensitivity of the polarimeters.Ideas to use high field superconducting magnets, such asthose used in the LHC and HERA accelerators, have alsobeen suggested but their use is limited by the difficulty inmodulating, in one way or another, their magnetic fields. Towork around this problem, proposals of rotating the polar-isation have been considered [15], but the presence of theFabry–Perot cavity, whose mirrors always present an intrin-sic birefringence whose induced ellipticity is orders of mag-nitude larger than the ellipticity due to vacuum magneticbirefringence, have made this idea unfeasible.In this note, a novel modulation scheme is presented thatmight profitably be employed with large superconding mag-nets.2 Preliminary considerationsIn a recent workshop in Hamburg [16], a new scheme, pre-sented in this paper, has been suggested to measure themagnetic birefringence of vacuum predicted on the basis ofthe 1936 effective Lagrangian LEHW. One of the presenta-123 294 Page 2 of 5 Eur. Phys. J. C   (2016) 76:294 tions [17], proposed the idea (called HERA-X: Heisenberg-Euler-biRefringent-ALPS-eXperiment) of making use of thepowerful infrastructure of the ALPSIIc set-up [18]: about5000 T2m, which could go up to about 7700 T2m if the peakfield of 6.6 T is employed. In this configuration the magneticbirefringence would ben(HERA-X) ≈ 10−22for the 5.3 T magnetic field. With this birefringence, the max-imum ellipticityψ (HERA-X) = π Lλn(HERA-X)is ψ (HERA-X) = 5×10−14 for λ = 1064 nm and L = 177 m.In the usual setups, the magnetic field is modulated to gainsensitivity. In the particular case of the HERA superconduct-ing magnets the electric current can be modulated at about amillihertz frequency [19].Let’s analyse the measurement scheme of Fig. 1, featur-ing two crossed polarisers, a variable magnetic field (fixeddirection) at a frequency νB, and an ellipticity modulatorat a frequency νm. The purpose of the ellipticity modula-tor is twofold: it allows heterodyne detection for improvingsensitivity by linearising the signal and shifting it to a highfrequency; it permits the distinction between an ellipticitysignal and a rotation signal [12]. In this scheme the inten-sity collected at the photodiode PDE is, at the lowest usefulorder,I⊥(t)  I0[η2(t) + 2η(t)ψ(t)]+ O[ψ(t)2].The interesting signal is found, in a Fourier transform ofthe signal from the photodiode, at the two sidebands ±νBfrom the carrier frequency νm of the ellipticity modula-tor.The resulting peak shot-noise sensitivity in such a schemeisSshot =√2eI0q,where e is the electron charge, I0 is the intensity reachingthe analyser, and q is the quantum efficiency of the photodi-ode. With I0  100 mW and q = 0.7 A/W, the shot-noisepeak sensitivity is Sshot  2 × 10−9 1/√Hz. Despite theexceptional parameters of the magnetic field of HERA-X,Fig. 1 A simple heterodyne ellipsometer. PDE extinction photodiode,PDT transmission photodiodethe integration time T to achieve a unitary signal-to-noiseratio remains too long, even supposing to work at shot-noisesensitivity:T ∼(Sshotψ (HERA-X))2∼ 109 s.As mentioned above, further amplification is required. Thiscan be achieved with a Fabry–Perot cavity, which can bethought of as a lengthening of the optical path by a factorN = 2F/π , where F is the finesse of the cavity. The pro-posed finesse for HERA-X is F = 60,000. With such afinesse, the ellipticity ψ increases by a factor N = 38,000and the integration time therefore diminishes by a factor N 2.Assuming shot-noise sensitivity, on paper, this device shouldeasily allow the measurement.A problem remains, however, regarding the actual sensi-tivity that one may reasonably think to achieve at low fre-quencies with such a long cavity. Let us consider the exper-iments on this subject realised so far with a scheme similarto the one proposed with HERA-X [12,20–23]. In Fig. 2 weshow the noise densities in birefringenceSn = Sψ λπ(2Fπ)dmeasured in these apparatuses as a function of the frequencyof the effect. In this formula Sψ is the ellipticity sensitivity ofeach experiment, λ is the wavelength, F is the finesse and dthe cavity length. Note that the cavity length d has been usedinstead of the length L of the magnetic region; what is plottedis therefore the best sensitivity in birefringence that could beobtained by the experiments. In the figure we did not report aFig. 2 Birefringence noise densities measured in polarimeters set upto measure the magnetic vacuum birefringence plotted as function of thefrequency. Data from the experiments BFRT [20], PVLAS-LNL [21,22], PVLAS-2013 [23], PVLAS-FE [12] are normalised to the length ofthe optical cavities, to the number of passes and to the wavelength. Theleftmost point has been measured during the 2015 data taking campaignof the PVLAS experiment. The two almost equivalent points from BFRTare measured with two different cavities, one having 34 passes and theother 578 passes. The error bars are an estimated 50 %123Eur. Phys. J. C   (2016) 76:294 Page 3 of 5  294 much worse sensitivity value of the Q &A experiment [13].The data are fitted with a power function.The message put forward by Fig. 2 is that increasing theeffective length (finesse and magnetic field length) does notguarantee the shortening of the necessary integration timeto reach a unitary signal-to-noise ratio; seeking the high-est finesse possible is not necessarily the optimal choice.Increasing the birefringence modulation frequency seems tobe more effective. Furthermore, with lower finesses, the cav-ity will have a shorter decay time and therefore a higher cutofffrequency allowing higher modulation frequencies. Figure 2suggests, therefore, that the finesse of the cavity should be thehighest for which the polarimeter is still limited by intrinsicnoises (shot-noise, Johnson-noise, etc.).The figure suggests that it is unlikely that, at 1 mHz, a sen-sitivity better than S(1 mHz)n ≈ 10−16/√Hz can be reached.As a matter of fact, the sensitivity of a giant 200 m cav-ity, necessarily built with the end mirrors sitting on separatebenches, can hardly be predicted. Even assuming for HERA-X the sensitivity of Fig. 2 at 1 mHz, a SNR = 1 could onlybe reached in aboutT =(S(1 mHz)nn(HERA-X))2≈ 1012 s.3 MethodIn this note, we present a novel modulation scheme that wouldbring in several advantages. This idea has never been testedin a laboratory, but is likely to be more effective than theone described above. In this way one can work at higher fre-quencies for the best sensitivity. In this scheme the magneticfield does not need to be modulated. The scheme consistsin introducing a pair of co-rotating half-wave-plates L1 andL2 inside the Fabry–Perot cavity, as schematically shownin Fig. 3. The polarisation within the magnetic field wouldrotate at twice the frequency of the wave-plates and shouldallow to increase substantially the modulation frequency ofthe effect. An important feature of this scheme is that thepolarisation direction of the light on the Fabry–Perot mirrorswould remain fixed, thereby eliminating the contribution ofthe ellipticity due to the intrinsic birefringence of the mir-rors. Furthermore, the polarisation direction on each mirrorFig. 3 Proposed modulation scheme. L1,2 rotating half-wave-plates,PDE extinction photodiode, PDT transmission photodiodecould be chosen; the input polariser defines the polarisationdirection on the first mirror whereas on the second mirror thepolarisation direction is defined by the relative angle betweenthe axes of the two wave-plates.Let us indicate with νL the rotation frequency of the wave-plates, that we suppose to rotate synchronously but not nec-essarily aligned one to the other. The Jones representation ofthe electric field at the exit of the cavity isEout(δ) =(Eout,‖Eout,⊥)= E0∞∑n=0[Reiδ L2 · X · L21 · X · L2]n·T eiδ/2 L2 · X · L1 ·(10)= E0[I − Reiδ L2 · X · L21 · X · L2]−1·T eiδ/2 L2 · X · L1 ·(10),where δ is the round-trip phase acquired by the light betweenthe two cavity mirrors, R and T are the reflectivity and trans-missivity of the mirrors, I is the identity matrix,X =(eiψ 00 e−iψ)is the magnetic birefringence of vacuum generating an ellip-ticity ψ in the polarisation of the light, andL1,2 = R(−φ − φ1,2) · L0(π + α1,2) · R(φ + φ1,2)are the rotating wave-plates. HereL0(α) =(eiα/2 00 e−iα/2)represents the wave-plate andR(φ) =(cos φ sin φ− sin φ cos φ),the rotation matrix, with φ the variable azimuthal angle ofthe wave-plates: φ(t) = 2πνLt . The angle φ2 − φ1 is theconstant relative phase between the slow axes of the tworotating wave-plates, andα1,2 allow for small deviations fromπ of the retardation of the two imperfect wave-plates. Theelectric field after the analyser is thenE(δ) = A · H · R(2φ2 − 2φ1) · Eout(δ),whereH =(1 iη(t)iη(t) 1)and A =(0 00 1)are the ellipticity modulator, placed at 45◦ with respect tothe output polarisation, and the analyser set to maximumextinction, respectively. In the expression for H, η(t) =123 294 Page 4 of 5 Eur. Phys. J. C   (2016) 76:294 η0 cos 2πνmt . The rotation matrix between the cavity andthe ellipticity modulator ensures that the modulator and theanalyser are correctly oriented. To first order in α1, α2, andψ , the intensity detected by the photodiode PDE is given byI (δ) ≈ I0 T21 − R cos δ + R2×{η(t)2+ 2 η(t) (1−R)1−R cos δ+R2[ψ sin(4φ(t)+4φ1)+ α1 sin(2φ(t)+2φ1)+α2 sin(2φ(t)+4φ1−2φ2)]}.The interesting result from this formula is that the signal ofthe magnetic birefringence of vacuum is found at the frequen-cies νm ± 4νL deriving from the product η(t) ψ sin(4φ(t) +4φ1), while the signals due to a small retardance differencefrom λ/2 of the wave-plates appear at νm ± 2νL. Higherorder imperfections in the phase delay of the rotating wave-plates may be present and may be described by the followingexpansion:α1,2(t) = α(0)1,2 + α(1)1,2 cos φ(t) + α(2)1,2 cos 2φ(t) + · · ·The various orders of imperfection can be estimated fromthe specifications of the producers. A typical value for α(0)1,2is α(0)1,2  10−2. The main contribution for α(1)1,2 may comefrom the parallelism of the surfaces of the wave-plate (wedge)coupled to the distance of the beam from the center of rota-tion. The typical parallelism of a wave-plate is 2 × 10−6 rad.With an off-center rotation of ≈1 mm this gives an estimatedvalue of α(1)1,2  2×10−9. The effect of such a defect, though,would generate signals at νm ± 3νL and νm ± νL and notat νm ± 4νL. To generate a spurious signal at νm ± 4νLthe term α(2)1,2 is necessary. Assuming an ellipticity valueto be measured due to vacuum magnetic birefringence ofψ = 2.5 × 10−11 [formula (2)], it is not unreasonable toimagine that α(2)1,2  ψ ≈ α(1)1,2/100. In all cases, though,there will not be a contribution from these terms if the rota-tion axis of each wave-plate coincides with the beam position,condition which can be obtained with a careful alignment ofthe optics. The reduction of these systematic effects is to beperformed with the magnets turned off.In the above formulas we have not considered the intrinsicbirefringence of the mirrors [24]. In this scheme, by choosingappropriately φ1 and φ2 it should be possible to minimisethe effect of this birefringence by independently aligning, oneach mirror, the polarisation of the light to the birefringenceaxes of the mirrors [12].Clearly the presence of the two half-wave-plates inside thecavity introduces some losses. Therefore there is an upperlimit to the finesse one can obtain due to the absorption ofthe wave-plates. With a correct antireflective coating, wave-plates can be obtained with a total absorption/reflection of0.1 % each. Unwanted effects due to the reflected lightfrom the wave-plates inside the cavity can be eliminated bymisaligning the wave-plates very slightly so as to send thereflected light against the baffles which would be presentinside the vacuum tube. Considering that the finesse F isF = π1 − R =πT + P ,where R+T + P = 1, and assuming that the transmission ofthe mirrors T are such that T  P = 4×10−3 (four passagesthrough the waveplates), the absorption of the wave-plateslimits the finesse toFmax  πP 800. (1)In this case the predicted QED ellipticity signal would beψ(WavePlates) =(2Fmaxπ)ψ (HERA-X) ≈ 2.5 × 10−11. (2)Assuming for HERA-X the best birefringence sensitivity asshown in Fig. 2, which is independent from the finesse, thiswould mean a value of S(100 Hz)n  2.5 × 10−20 1/√Hz @100 Hz (with νL = 25 Hz). Using F = 800 the sensitivityin ellipticity isSψ = 2F Sn Lλ≈ 7.5 × 10−9 1/√Hz.The corresponding integration time to reach S/N = 1 wouldtherefore beT =(Sψψ(WavePlates))2 105 s. (3)Such a sensitivity remains to be demonstrated in the excep-tional conditions of the proposed HERA-X experiment, butwith such a low finesse, near shot-noise ellipticity sensitiv-ities have been demonstrated. The minimum output powerfrom the cavity to avoid being limited by shot-noise is≈10 mW. With a finesse of F = 800 the circulating powerinside the cavity is about 5 W distributed over a surfaceof about 0.1 cm2 determined by the beam radius. Thisis well below the damage threshold of the wave-plates of1 kW/cm2 which therefore guarantees a correct and stableoperation of the wave-plates. Furthermore very long Fabry–Perot cavities have been shown to be stable at frequencies ofa few tens of hertz by LIGO and VIRGO reaching shot-noiseperformances [25]. The numbers seem to be within reach andwe believe that this scheme could be a viable solution whenusing high field static magnetic fields generated by supercon-ducting magnets.4 ConclusionIn this note we have proposed a new scheme for a sensitivepolarimeter dedicated to measuring vacuum magnetic bire-123Eur. Phys. J. C   (2016) 76:294 Page 5 of 5  294 fringence based on a Fabry–Perot cavity which would allowthe use of static magnetic fields generated by superconduct-ing magnets. The modulation of the birefringence, necessaryto reach high sensitivities, is performed by two co-rotatinghalf-wave-plates inside the cavity, thus satisfying two con-ditions: rotating polarisation of the light inside the magneticfield; fixed polarisation direction on the Fabry–Perot mirrors.Furthermore the polarisation direction on the two mirrors canbe controlled independently.Open Access This article is distributed under the terms of the CreativeCommons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution,and reproduction in any medium, provided you give appropriate creditto the original author(s) and the source, provide a link to the CreativeCommons license, and indicate if changes were made.Funded by SCOAP3.References1. H. Euler, B. Kockel, Naturwiss. 23, 246 (1935)2. H. Euler, Ann. Phys. (Leipzig) 26, 398 (1936)3. W. Heisenberg, H. Euler, Z. Phys. 98, 714 (1936)4. V.S. Weisskopf, K. Dan, Vidensk. Selsk. 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A polarisation modulation scheme for measuring vacuum magnetic
  birefringence with static fields

A polarisation modulation scheme for measuring vacuum magnetic birefringence with static fields

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