The BMV experiment : a novel apparatus to study the propagation of light in a transverse magnetic field
Remy Battesti, Benoit Pinto Da Souza, Sebastien Batut, Cecile Robilliard, Gilles Bailly, Christophe Michel, Marc Nardone, Laurent Pinard, Oliver Portugall, Gerard Trenec, Jean-Marie Mackowski, Geert L.J.A. Rikken, Jacques Vigue, Carlo Rizzo
aa r X i v : . [ phy s i c s . op ti c s ] O c t EPJ manuscript No. (will be inserted by the editor)
The BMV experiment : a novel apparatus to study thepropagation of light in a transverse magnetic field
R´emy Battesti , Benoˆıt Pinto Da Souza , S´ebastien Batut , C´ecile Robilliard , Gilles Bailly , Christophe Michel ,Marc Nardone , Laurent Pinard , Oliver Portugall , Gerard Tr´enec , Jean-Marie Mackowski , Geert L.J.A. Rikken ,Jacques Vigu´e and Carlo Rizzo Laboratoire National des Champs Magn´etiques Puls´es (UMR 5147, CNRS - INSA - Universit´e Paul Sabatier Toulouse 3),31400 Toulouse cedex, France. Laboratoire Collisions Agr´egats R´eactivit´e (UMR 5589, CNRS - Universit´e Paul Sabatier Toulouse 3), IRSAMC, 31062Toulouse Cedex 9, France. Laboratoire des Mat´eriaux Avanc´es (LMA) (Universit´e Claude Bernard Lyon 1, CNRS, IN2P3), Villeurbanne Cedex, FranceReceived: date / Revised version: date
Abstract.
In this paper, we describe in detail the BMV (Bir´efringence Magn´etique du Vide) experiment,a novel apparatus to study the propagation of light in a transverse magnetic field. It is based on a veryhigh finesse Fabry-Perot cavity and on pulsed magnets specially designed for this purpose. We justify ourtechnical choices and we present the current status and perspectives.
PACS.
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Experiments on the propagation of light in a transversemagnetic field date from the beginning of the 20th cen-tury. In 1901 Kerr [1] and in 1902 Majorana [2] discoveredthat linearly polarized light, propagating in a medium inthe presence of a transverse magnetic field, acquires an el-lipticity. In the following years, this magnetic birefringencehas been studied in details by A. Cotton and H. Mouton[3] and it is known nowadays as the Cotton-Mouton ef-fect. The velocity of light propagating in the presence ofa transverse magnetic field B depends on the polariza-tion of light, i.e. the index of refraction n k for light po-larized parallel to the magnetic field is different from theindex of refraction n ⊥ for light polarized perpendicular tothe magnetic field. For symmetry reasons, the difference ∆n = ( n k − n ⊥ ) is proportional to B . Thus, in generalan incident linearly polarized light beam exits ellipticallypolarized from the magnetic field region. The ellipticity tobe measured Ψ can be written as Ψ = π Lλ ∆n sin 2 θ (1)where L is the optical path in the magnetic field region, λ the wavelength of the light, and θ the angle between lightpolarization and the magnetic field.In dilute matter like gases, such an effect is usually verysmall and it needs very sensitive ellipsometers to be mea-sured. Ab initio calculations can be performed using the
Send offprint requests to : [email protected] most advanced computational techniques and they stillremain very challenging [4].Propagation of light in vacuum in the presence of atransverse magnetic field has been experimentally studiedsince 1929 [5]. The first motivation was to look for a mag-netic moment of the photon. Only around 1970, thanks tothe effective Lagrangian established in 1935 and 1936 byKochel, Euler and Heisenberg [6] [7], it has been shownthat the Cotton-Mouton effect should also exist in a vac-uum [8] [9]. Quantum ElectroDynamics (QED) predictsthat a field of 1 T should induce an anisotropy of the in-dex of refraction of about 4 × − . This very fundamentalprediction has not yet been experimentally verified.Some of the earlier experiments were based on the useof an interferometer of the Michelson-Morley type. One ofthe two arms passed through a region where a transversemagnetic field was present inducing a difference in thelight velocity that should have been observed as a phaseshift [10] [11]. In 1979, Iacopini and Zavattini [12] proposedto measure the ellipticity induced on a linearly polarizedlaser beam by the presence of a transverse magnetic fieldusing an optical cavity in order to increase the optical pathin the field. The effect to be measured was modulated inorder to be able to use heterodyne technique to increasethe signal to noise ratio.In 1986, Maiani, Petronzio, and Zavattini [13] showedthat hypothetical low mass, neutral, spinless boson, scalaror pseudoscalar, that couples with two photons could in-duce an ellipticity signal in the Zavattini apparatus sim-ilar to the one predicted by QED. Moreover, an appar- Please give a shorter version with: \authorrunning and \titlerunning prior to \maketitle ent rotation of the polarization vector of the light couldbe observed because of conversion of photons into realbosons resulting in a vacuum magnetic dichroism which isabsent in the framework of standard QED. The measure-ments of ellipticity and dichroism, including their signs,can in principle completely characterize the hypotheticalboson, its mass m a , the inverse coupling constant M a , andthe pseudoscalar or scalar nature of the particle. Maiani,Petronzio, Zavattini’s paper was essentially motivated bythe search for Peccei and Quinn’s axions. These are pseu-doscalar, neutral, spinless bosons introduced to solve whatis called the strong CP problem [14]. However, it was soonclear that such an optical apparatus could hardly excludea range of axion parameters not already excluded by as-trophysical bounds [15].Following Zavattini’s proposal, after tests at CERN inGeneva, Switzerland, [16], an apparatus has been set upat the Brookhaven National Laboratory, USA [17]. No ev-idence for dichroism or for ellipticity induced by the mag-netic field was found. The sensitivity being insufficient todetect the QED effect, only limits on the axion parame-ters could be published in 1993 [17], M a > . × GeVat the 95% confidence level, provided m a < . × − eV.In 1991, a new attempt to measure the vacuum mag-netic birefringence has been started at the LNL in Leg-naro, Italy, by the PVLAS collaboration [18]. This exper-iment is again based on ref. [12]. A Fabry-Perot cavityis used to increase the effect to be measured, while a su-perconductive 5 T magnet rotates around its own axis tomodulate it. Eventually, the collaboration has publishedthe observation of a magnetically induced dichroism invacuum [19], and also of magnetically induced elliptic-ity in excess of what is expected according to QED [20].These results have triggered a lot of interest in the field,in particular because the existence of axions could be theexplanation for this unexpected signal [21]. The proposedrange is 2 × GeV < M a < . × GeV, provided that1 × − eV < m a < . × − eV. Recently, the PVLAScollaboration has disclaimed their previous observations.An ellipticity signal is still present at 5.5 T, while no ellip-ticity signal is observed at 2.3 T [22]. No clear explanationis given as to why the original signal has disappeared.As soon as the original PVLAS results have becomeknown, they have created big expectations, and the re-search field has gained a large momentum since this couldbe the first positive sighting of axions or axionlike parti-cles [21]. Indeed, the PVLAS result is in contradiction withthe solar axion search, and in particular with the resultsof the CERN axion telescope CAST [23], which excludeaxionlike particles with a mass smaller than 10 − eV anda inverse coupling constant smaller than 10 GeV. To ac-commodate both results one should change the nature ofaxionlike particle interaction to justify their confinementto the interior of the sun (see e.g. [24]).From the experimental point of view, different propos-als have been put forward (see e.g. [25]), most of themplanning photoregeneration experiments. Such a kind ofexperiment is based on the idea that once an axionlikeparticle is created by photon conversion in a magnetic field, the particle escapes from the magnet region whilethe light beam can be easily confined. Now, if the createdparticle passes through another magnet region, it can beconverted back into a photon that one can detect in aregion where no photon should exist [26]. QED vacuumbirefringence is also put forward in the proposal OSQAR[27]. Most of these projects are expected to take data in2007. First results of a photoregeneration experiment spe-cially designed to test the PVLAS claim have appearedvery recently [28] excluding the axionlike particle inter-pretation of the PVLAS original result with a confidencelevel greater than 99.9%.In the meantime new results have also been postedon the arXiv web site by the Q&A (Quantum electrody-namics test & search for Axion) project based at NationalTsing Hua University, Taiwan. This project has startedaround 1996. The experimental set-up is similar to thePVLAS one. No effect has been detected, in contradictionwith the original result from the PVLAS collaboration. Asfar as we understand, the confidence level is about 68%[29].In this paper, we describe a novel apparatus to studythe propagation of light in a transverse magnetic field : theBMV (Bir´efringence Magn´etique du Vide) experiment. Itis based on a very high finesse Fabry-Perot cavity andon a pulsed transverse magnet specially designed for thispurpose. We justify our technical choices and we presentthe current status and perspectives. We show that our ap-paratus has the sensitivity to test the PVLAS result, andeventually to measure the vacuum magnetic birefringence.A very important QED prediction that has been waitingto be tested since about 75 years.
In fig. 1, we present a sketch of our optical apparatus. Theoptical cavity and the laser source are put on two differentoptical tables, linked by a polarization maintaining opticalfiber (PM fiber).On the first table, light emitted by a mono-mode fre-quency controlled Nd:YAG laser at a wavelenght λ = 1064nm passes through a Faraday isolator FI.A telescope T1 adapts the beam waist to be injectedinto the fiber. Mirrors M1 and M2 transport the beamto a λ/ lease give a shorter version with: \authorrunning and \titlerunning prior to \maketitle mirrors M3 and M4. These two mirrors are also used toalign the optical beam with the optical axis of the Fabry-Perot cavity made by two very high reflectivity mirrorsCM1 and CM2.A system of high precision translations and rotationsfor the Fabry-Perot cavity mirrors and also for the polar-izers has been designed and assembled. The piezoelectricstacks of the mirror alignment system as well as the mech-anism allowing their rotation are adapted to ultra-highvacuum.Before entering the Fabry-Perot cavity light is polar-ized by the polarizer P. Polarization of the light trans-mitted by the cavity is analyzed by the analyzer A. Theextraordinary ray is detected by photodiode PdT, whilethe ordinary ray after being reflected by a mirror M5 isdetected by photodiode PdI0. Both signals are used in thedata analysis. Mirror M5 turns with prism A and extrac-tion of the ordinary ray can be done at any position ofA. All the optical components from the polarizer P to theanalyzer A are placed in a ultra high vacuum chamber notshown in the figure.Light reflected back by the Fabry-Perot cavity arrivesat the beamsplitter BS and it is sent to the photodiodePdR, the signal of which is used to drive the phase lockingcircuit PhLC that locks the laser frequency to the Fabry-Perot cavity. LASER FI λ /2 T1 M2M1P2
FoutEOBSP3
PdP
PolLC PM fiber
T2M3M4
PhLC
PdR
Table 1
Source optical table
P ACM1 PdI0PdT
Table 2
Cavity optical table
CM2 M5
Fin
Laser
PhLCPM fiber
Fig. 1.
Scheme of the optical apparatus.
Our Fabry-Perot cavity is made by two identical mirrorsof 25.4 mm (1 inch) diameter. The length of the cavityD in the final version of our setup should not exceed 2.5meters. The radius of curvature C of the mirrors has beenchosen equal to 8 m. The resonant frequencies of the cavitymodesis given by the following formula ν qlm = " q + ( l + m + 1) arccos (cid:0) − DC (cid:1) π c D (2)We want that only one mode resonates at the fre-quency of the T EM mode, thusarccos (cid:0) − DC (cid:1) π = pk + 1 (3)where p and k are integers. Choosing C = 8 m guar-antees that only modes with l + m ≫ w is situated at the center of thecavity and it is given by eq. 4. w = s(cid:18) λ π (cid:19) p D (2 C − D ) (4)On the mirror the waist w m is given by the followingformula : w m = s(cid:18) λ π (cid:19) r C D C − D (5)For C = 8 m and D = 2 . w ≃ w m ≃ .
02 mm.Assuming a good mode matching, cavity transmissiondepends on the mirror transmissivity T and mirror losses P . The ratio r I between the intensity of the transmittedlight I t and the intensity of the incident light can be writ-ten as r I = I t I i = (cid:18) TT + P (cid:19) (6)The value of the sum T + P is fixed by the value ofthe requested finesse since F = π − R where the mirrorreflectivity R is equal to 1 − ( T + P ). For having a finessebetween 500 000 and 1 000 000, let’s assume that P =3 × − and T = 2 × − . In this case r I ≃ . Ψ s of the order of 10 − rad Hz − / . Aswe show in the following (see eq. 31), assuming a quantumefficiency of the photodiode q = 0 .
7, one finds that I t hasto be about 40 mW. If r I ≃ .
12, the incidence intensity I i has to be about 250 mW.Now, the intensity I m on the surface of the exit mirrorof the cavity is I t /T i.e. about 20 kW. This correspondsto a power density of the order of 600 kW/cm . Please give a shorter version with: \authorrunning and \titlerunning prior to \maketitle
Mirrors are always unintentionally slightly birefringent(see e.g. [34]) and therefore the Fabry-Perot cavity reso-nance line will be separated in two. Light polarized asthe fast axis of birefringence will see a different opticalpath with respect to the light polarized as the slow axis ofbirefringence. In ref. [35] the issue of a birefringent Fabry-Perot cavity in the framework of ellipticity measurementshas been discussed in detail. We just recall here that thefrequency separation of the two polarizations is given by ∆ν δ = δ π c D (7)where δ is the mirror birefringence. The line width is ∆ν = 1 F c D (8)To avoid that the cavity acts as polarizer prism, it isnecessary that ∆ν δ < ∆ν , thus δ < πF (9)Thus for a 1 000 000 finesse cavity δ < . × − ,which is demanding but which has already been observedfor high finesse mirrors [34]. In our project, we plan to use a very high finesse cavityto increase the ellipticity signal to be measured. For 1 000000 cavity finesse and 2 meters cavity length, the photonlifetime in the cavity τ = F D/ ( πc ) is about 2 msec. Thisduration is not negligible with respect to the magneticpulse duration.In the following, we calculate the output field of aFabry-Perot interferometer in the case where the opticalpath is a slowly varying function of time. More precisely,the phase acquired by the light wave while going forwardand backward in the Fabry-Perot cavity is given by: ψ = 2 kD + φ ( t ) (10)We assume that φ ( t ) has negligible variations over thetime t D = D/c taken by the light wave to go from onemirror to the other. However, because the cavity has avery high finesse F , the variations of φ ( t ) over the photonlifetime in the cavity τ are not necessarily negligible andthe goal of the calculation is to express how the photonlifetime averages the phase φ ( t ). It seems that there is nogeneral solution for the field exiting from a Fabry-Perotinterferometer which is not stationary in time. The presentcalculation is approximate and assumes that the phase φ ( t ) is very small ( φ ( t ) could be large but the varyingpart must be small and we assume that there is no timeindependent part).We calculate the light field exiting from the Fabry-Perot at the instant t , by summing the contributions ofthe rays which have been transmitted with n return paths: E out ( t ) T E in exp [ iφ ( t − ( t D ))] = (11) ∞ X j =0 R j exp " i jkD + j X p =1 φ ( t − pt D ) ! We assume that the amplitude reflection (respectively trans-mission) coefficients of the two mirrors are real and wenote their product as R (respectively T ). The incidentfield E in is taken as perfectly monochromatic. The lightwhich has been reflected n times at both ends of the Fabry-Perot interferometer has sampled the phase shift φ ( t ) atthe times ( t − pt D ) with p varying from 1 to j . For thegeneral case, this expression cannot be evaluated analyti-cally.We assume that the Fabry-Perot cavity is at resonance(2 kD = 0 [2 π ]) and that the phase φ ( t ) is very smallso that we can replace the exponential by its first-orderexpansion:exp " i j X p =1 φ ( t − pt D ) ≈ i j X p =1 φ ( t − pt D ) (12)The condition is not only φ ( t ) ≪ F , we must have F φ ( t ) ≪
1. We must calculate two sums: ∞ X j =0 R j = 11 − R (13) ∞ X j =1 R j j X p =1 φ ( t − pt D ) = ∞ X p =1 φ ( t − pt D ) ∞ X j = p R j = (14) ∞ X p =1 φ ( t − pt D ) R p − R We can rewrite R p : R p = exp [ − ( Γ/ p (2 t D )] (15)where Γ = 1 /τ ≈ (1 − R ) /t D and we get: ∞ X j =1 R j j X p =1 φ ( t − pt D ) = ∞ X p =1 φ ( t − pt D ) exp [ − ( Γ/ p (2 t D )] 11 − R ≈ (16)1(1 − R ) Z + ∞ ( Γ/ dt ′ φ ( t − t ′ ) exp [ − ( Γ/ t ′ ]The final result is that the output field is given by: lease give a shorter version with: \authorrunning and \titlerunning prior to \maketitle E out ( t ) T E in exp [ iφ ( t − t D )] ≈ − R (cid:20) i h φ i − R ) (cid:21) (17)where h φ i = Z + ∞ ( Γ/ dt ′ φ ( t − t ′ ) exp [ − ( Γ/ t ′ ] (18)It is easy to make φ ( t ) = 0 in the result and then wefind the traditional result of the Fabry Perot cavity atresonance: E out ( t ) T E in = 11 − R (19)For the sake of simplicity, let’s assume that B ( t ′ ) = B sin( ωt ′ ). B ( t ′ ) can be written as B ( t ′ ) = B h − e (2 iωt ′ ) − e ( − iωt ′ ) i (20)and h φ i ∝ Z ∞ B h − e (2 iω ( t − θ )) − e ( − iω ( t − θ )) i e ( − ( Γ/ θ ) dθ (21)where θ = t − t ′ . Finally, we obtain that h φ i ∝ B − q ωΓ/ ) cos(2 ωt + ϕ ) (22)with tan ϕ = − ωΓ Thus when the magnetic field varies during the lifetimeof the photons in the cavity, the ellipticity acquired bythe light depends on an attenuated averaged value of thesquare of the magnetic field and moreover the ellipticityis not in phase with the square of the field. To avoid suchan effect, it is clear that one needs2 ωΓ/ ≪ To increase as much as possible our signal to noise ratio,we need a cavity with a very high finesse. As far as weknow, the highest finesse ever published is about 2 × [36], while the highest quality factor Q is the one of thePVLAS cavity [19], Q ≃ × corresponding to a cavitylinewidth of about 200 Hz and a storage time of 0.5 ms.Our cavity mirrors are made by the Laboratoire desMat´eriaux Avanc´es (LMA). Thanks to their know-how wehave currently at our disposal 4 mirrors with losses rang-ing from 3 to 9 ppm and transmission ranging from 1.5 to 1.9 ppm. This corresponds to finesses ranging from 300000 to 700 000. The main problem is to handle these mir-rors in an adequate way because they are very sensitiveto pollution. Therefore we have built a special clean roomwith laminar flow cabinets for their manipulation. Theaccess to this room is limited to fully equipped personnel(see fig 2). Fig. 2.
Photo of the experimental room.
The mirrors are put in a home made kinematic mountwhich is activated by two piezoelectric wedges. They allowus to align the mirror reflecting surface perpendicular tothe cavity axis with a precision better than 4 µrad . Thisaligning system operates in ultra high vacuum. Finally,we use a bronze wheel for rotating the mirror mount tominimize the cavity birefringence. It allows us a rotationsmaller than 1 mrad.
The laser source is locked to the Fabry-Perot cavity tomaximize the injection of the light into the magnetic fieldregion using the usual Pound-Drever-Hall technique [37].The laser crystal itself is used to phase modulate the laserlight [38].The light source is a tuneable non planar ring oscil-lator Nd:YAG laser emitting ∼
200 mW of power at awavelength λ = 1064 nm ( ν = 2 . × Hz). The fre-quency of the laser can be changed by two methods: a fastone (bandwidth >
10 kHz) based on a piezoelectric actua-tor acting on the laser crystal and a slow one (bandwidth < × Hz, whereas the latter yields changes ofabout 100 × Hz.In fig. 3 we show a simplified scheme of our lockingcircuit.The novelty, compared for example to ref. [39], is thatwe control the laser frequency using three different feed-back signals instead of two. As usual, a very low band-
Please give a shorter version with: \authorrunning and \titlerunning prior to \maketitle "Pound-Drever" errorFast PZT amplifier High Volt. PZT amplifier Fast PZT amplifier High Volt. PZT amplifier Thermo-electric cooler 0 0 0
Electro-optical processAnalog feedback controllers C on t r o ll ed i npu t s M ea s u r ed ou t pu t s Fig. 3.
Conceptual design of our locking circuit. width one acts on the crystal temperature, a second onewith an important dynamical range acts on one of the twoends of the piezoelectric actuator, and a third one acts onthe other end of the piezoelectric actuator allowing a finetuning of the laser frequency on the cavity resonance fre-quency.
In fig. 4 we show the basic principle of the apparatus fol-lowing ref. [12]. B (t)P A η Γ LaserLight
Fig. 4.
Basic principle of the detection technique.
A laser beam is polarized by the polarizer P. Let’s as-sume that the light intensity after P is I t . Let’s also assumethat the light intensity after analyzer A is I e . Going fromP to A, light acquires an ellipticity η ( t ) thanks to an ellip-ticity modulator, a static ellipticity Γ , and an ellipticityto be measured Ψ ( t ). In our case Ψ ( t ) ∝ B ( t ) . The de-tection technique suggested in ref. [12] is the heterodyneone. The same technique has been used in the experiencesof ref. [17] and [19].At the extinction σ , when A is crossed with respectto P, I e can be written as I e ( t ) = I t σ + I t [ η ( t ) + Γ + Ψ ( t )] (24)For the sake of simplicity, let us now assume that η ( t ) = η cos (2 πΩ m t + θ m ) and Ψ ( t ) = Ψ cos (2 πΩ e t + θ e ),with Ω m ≫ Ω e . It is straightforward to show that I e isconstituted by the frequency components given in table 1: frequency component amplitude phaseDC I DC I t (cid:0) σ + η / Ψ / Γ (cid:1) Ω e I Ω e I t (2 Γ Ψ ) θ e Ω e I Ω e I t (cid:0) Ψ / (cid:1) θ e Ω m I Ω m I t (2 Γ η ) θ m Ω m ± Ω e I Ω m ± Ω e I t ( η Ψ ) θ m ± θ e Ω m I Ω m I t (cid:0) η / (cid:1) θ m Table 1.
Frequency components of the signal I e (see eq. 24) The existence of the two I Ω m ± Ω e components is typ-ical of the heterodyne technique in which one beats theeffect to be measured with a carrier effect, η ( t ) in ourcase. I Ω m ± Ω e components have a linear dependence on theeffect for the detected signal. Let us note that the com-ponent I Ω e linear in Ψ exists even if η ( t ) = 0 because ofthe existence of the spurious static ellipticity Γ . Detectingthe signal only by modulating the effect is an example ofwhat is called homodyne technique. Homodyne techniquehas the advantage of demanding a simpler optical appara-tus compared to the heterodyne technique because of theabsence of the ellipticity modulator. Homodyne detectionis the technique we have chosen for the BMV experiment.As we show in the following paragraph, homodyne detec-tion is particularly interesting when Γ ≫ σ . This is thecase for our BMV experiment. Actually, σ ≈ − while Γ ≈ − . Γ is due to the birefringence of the mirrorsthat constitute our Fabry-Perot cavity. In the following we will justify our choice by comparingthe heterodyne technique and the homodyne one. In par-ticular we calculate the sensitivity expected using the twotechniques. We prove that they are completely equivalentwith respect to this very important parameter.In the case of the heterodyne technique one has Ψ = s I Ω m ± Ω e I t I Ω m (25)The homodyne technique, corresponding to η = 0,becomes interesting when Γ ≫ σ , Ψ /
2. In this case Ψ = s I Ω e I t I DC (26)The limiting noise of both techniques is due to thecorpuscular nature of light (shot noise). Shot noise is pro-portional to the square root of the number of photonsdetected by the photodiode after the prism A. This is es-sentially proportional to I DC . i shotnoise = r e qI t ( σ + η / Ψ / Γ ) hν (27) lease give a shorter version with: \authorrunning and \titlerunning prior to \maketitle where e is the absolute value of the electron charge, q isthe quantum efficiency of the photodiode, h is the Planck’sconstant, and ν is the frequency of light. The rate of pho-tons corresponding to the signal is proportional to I Ω m ± Ω e in the case of heterodyne technique or to I Ω e in the caseof the homodyne technique. The signal-to-noise ratio canbe written as r hetero = i signal i shotnoise = s qI t Ψ η ( σ + η / Ψ / Γ ) hν (28)(where the factor 2 in the square root takes into accountthe existence of two sidebands) and r homo = i signal i shotnoise = s qI t Ψ Γ ( σ + Ψ / Γ ) hν (29)respectively.It is therefore clear that optimal working conditionsthus imply that η dominates in the I DC component forthe heterodyne technique and that Γ dominates in the I DC component for the homodyne technique. In this case,let us finally obtain an expression for the sensitivity Ψ shete and Ψ shomo respectively. For this we impose the conditionof a signal-to-noise ratio equal to one. We get Ψ shetero = s hν qI t (30) Ψ shomo = s hν qI t (31)The shot noise limit is hardly ever reached. To get amore general expression we note that at optimal workingconditions equations 25 and 26 can also be written as Ψ = η I Ω m ± Ω e I DC (32) Ψ = Γ I Ω e I DC (33)respectively. In the following, we introduce the residualellipticity noise γ ( Ω ) that is the noise signal I noise ( Ω )due to the existence of Γ divided by I DC . This quantitycan be measured by looking at the Fourier spectrum ofthe signal detected by the photodiode after the analyzerA set at maximum extinction when η = 0 and no effectis present. We assume that η ( t ) does not add any extraellipticity noise. Now, by imposing a signal-to-noise ratioequal to one, equations 32 and 33 can be written as Ψ shete = η I Ω m γ ( Ω e )2 I DC = Γ I DC γ ( Ω e )2 I DC = Γ γ ( Ω e ) (34) Ψ shomo = Γ I DC γ ( Ω e ) I DC = Γ γ ( Ω e ) (35) respectively, and we find that the heterodyne and homo-dyne techniques are completely equivalent. It is also clearthat a sensitivity limited by the shot noise can only bereached if γ ( Ω e ) is small enough. In the case of the homo-dyne technique γ ( Ω e ) < s hν qI t Γ (36)To obtain a Ψ shomo ≃ − rad Hz − / , if Γ ≃ − rad, γ ( Ω e ) < − Hz − / .The heterodyne technique needs an external modula-tion of the magnetic field and of the ellipticity which ismore demanding. In our experiment we have chosen thehomodyne technique ( η ( t ) = 0) thanks to our pulsed fieldwhich gives us a intrinsic modulation. A modified version of the apparatus shown in fig. 4 hasalso been proposed to measure small ellipticities and inparticular vacuum magnetic birefringence [30]. A quarter-wave plate is inserted between the polarizer and the ana-lyzer. The wave plate is set in such a way that the inten-sity of the ordinary and of the extraordinary ray exitingthe analyzer are equal when no birefringence exists alongthe light path. The presence of an ellipticity in the beamreaching the analyzer prism unbalances the intensities ofthese two rays. The two ray intensities are measured bytwo identical photodiodes. The difference of these two in-tensities gives the final signal for the analysis. This detec-tion method is usually called balanced polarimetry.One can show that this method is in principle totallyequivalent to the one proposed in ref. [12]. The electronicextinction given by the substraction of the electric signalscorresponding to the intensities of the two rays exitingthe analyzer prism play the role of the optical extinction σ . The presence of a static uncompensated ellipticity Γ gives a DC component to the signal, as any modulatedellipticity gives a corresponding modulation to the signal[30].In practice all the results of the previous paragraphapply also for the balanced polarimetry. As we have shown in the previous paragraphs, the ellip-ticity can be written as Ψ ( t ) = αB ( t ) or equivalently : Ψ ( t ) = I e ( t ) − I DC I t Γ , (37)where I DC has been defined in sec. 3, and we assume thatthe average effect given in formula 22 can be neglected.The signal analysis based on the Fourier spectrum dis-cussed in the previous paragraph is the most appropriateonly when the ellipticity Ψ ( t ) is a harmonic function of Please give a shorter version with: \authorrunning and \titlerunning prior to \maketitle time. As we use a pulsed field, we will use a different tech-nique.To recover the value of the constant α one can use thefollowing formula α = R T Ψ ( t ) B ( t ) dt R T B ( t ) dt (38)with B (0) = B ( T ) = 0.An interesting property of formula 38 is that if Ψ ( t )is proportional to the derivative of B ( t ) or B ( t ) , α = 0.This means that this technique allows an easy rejection ofthis kind of spurious signals.On the other hand, formula 38 gives a value of α = 0even if Ψ ( t ) is not proportional to B ( t ) . A slightly morecomplicated formula can be used α ′ ( τ ) = R T Ψ ( t ′ + τ ) B ( t ′ ) dt ′ R T B ( t ′ ) dt ′ (39)It is evident that α = α ′ (0). Now one can compare thefunction α ′ ( τ ) with α ′ B ( τ ) = R T B ( t ′ + τ ) B ( t ′ ) dt ′ R T B ( t ′ ) dt ′ (40)Only if Ψ ( t ) is proportional to B ( t ) , α ′ ( τ ) is propor-tional to α ′ B ( τ ) for any value of τ .It is important to notice that I e ( t ) will usually be mea-sured at θ = 45 ◦ which gives the maximum value for Ψ ( t )(see formula 1). Since at θ = 0 ◦ , Ψ ( t ) = 0, even if B ( t ) = 0,the measurement of I e ( t ) at θ = 0 ◦ and the comparison ofit with I DC is a crucial test. Any difference from zero ofthe quantity I e ( t, θ = 0 ◦ ) − I DC is an indication of noiseinduced by the magnetic field on the apparatus during thepulse. A first analytical calculation of a model pulsed coil ge-ometry has been presented in ref. [31]. A field as high as25 T over a meter length seemed to be achievable. Thismodel has been considered as the starting point of theactual coil design in the framework of the BMV project.[32]. The basic idea was to get the current creating themagnetic field as close as possible of the light path over alength as long as possible. For the magnetic birefringenceapplication, one has to maximize the integral of the squareof the field over the magnet length L : B L = B L eq = Z L/ − L/ B y ( z ) dz (41)Here, if B is the field maximum, we define L eq as theequivalent length of a magnet giving an uniform field value B on the axis. The ratio L eq /L gives us a measure of thefield’s uniformity. In fig. 5 we show the configuration wehave studied and realized. Fig. 5.
Scheme of the Xcoil.
Because of the particular geometry, in the following wewill call it Xcoil [33].In the approximation that the length of the coil is muchbigger that its width, one can calculate the field in the cen-ters of the Xcoil by using the expression of the field createdby four infinite wires. Following these analytical results,we have realized an actual coil based on the Xcoil scheme.In fig. 6 we can see the comparison between experimentaland theoretical field computed by finite element analysis.Here, the field is measured by a pick-up coil and a lock-inamplifier for an oscillating current of 1 A at 230 Hz.
Fig. 6.
Computed (line) and measured (line and markers) fieldprofile.
Our parameters leads to a field factor of 1.68 T/kA,an equivalent length L eq of 13.2 cm : almost half of thetotal magnet length. The relevant electrical parametersare L=400 µH and R @77 K = 52 mΩ .The higher experimental field in fig. 6 is explained bythe shape of the wire used that does not exactly corre-spond to the infinitesimal filaments in the simulation.The coil support is manufactured using G10, a com-posite material commonly used to deal with high stressesand cryogenic conditions. External reinforcements fromthe same material are added after winding to contain the lease give a shorter version with: \authorrunning and \titlerunning prior to \maketitle magnetic pressure that can be as high as 500 MPa at thefield maximum.We have produced several Xcoils, having outer dimen-sions of 250 mm x 100 x 100 mm. The internal hole avail-able for optical measurements is of 17 mm. This value ismainly given by the dimension of the vacuum tube andthickness of the tubes for double-walled cryostat.Tests at high field have been performed using a pulsedpower supply, that consists of six capacitors switched bythyristors. This capacitor bank is directly down-scaledfrom the LNCMP’s 14 MJ capacitor bank and it containesa maximum energy of 200 kJ.Due to the increase of resistance by the Joule heatingduring the shot, the pulse duration is voltage-dependant: it takes 18 ms at low fields (4T) and 6 ms at 14 T. Likefor conventional pulsed magnets, the coil is placed in a liq-uid nitrogen cryostat to limit heating consequences. Thewhole cryostat is double-walled with a vacuum thermal in-sulation, including for the inner tube. As shown in fig. 7,the cryostat provides two additional functions. It housesthe cavity which passes through an opening at room tem-perature arranged through the cryostat and it allows thecavity to be mechanically disconnected from the coil. Dryand warm nitrogen gas coming from the main bath is rein-serted between the cavity and the internal bore of thecryostat in order to prevent any trapping of air moisture.Finally, the cryostat allows for fast disassembly and re-placement in the event of degradation of the coil. Fig. 7.
Drawing of the liquid nitrogen cryostat. The coil isalso shown.
In fig. 8 we show a series of magnet pulses obtained byincreasing the power supply voltage. The maximum fieldobtained without generating significant resistance changesin the coil, that would indicate the onset of conductorageing, has been 14.3 T.The long-term effect of aging at lower field has beenstudied by pulsing a coil for 100 times at 11.5 T and 100times at 12.5 T. No significant change in its resistance hasbeen detected.
Fig. 8.
Series of magnet pulses.
Another crucial point for optical application is that themechanical noise created by the coil during the pulse hasto be as low as possible. We measured for one of our coils,the mechanical noise on the floor of the test area duringthe pulse. The pulse was 7.5 T high and its duration wasabout 5 ms. In fig. 9 we show such a measurement. Aninteresting feature is that the shock wave arrives after thepulse. The mechanical path was less than one meter.
Fig. 9.
Mechanical noise during and after a pulse.
During normal operation, by measuring the resistanceafter each shot, a pick-up signal to monitor the field valueand the frequency in vibration spectrum, we can monitorthe coil behavior.Fig. 10 shows a drawing of the experimental set upcorresponding to the Fabry-Perot cavity and the field re- \authorrunning and \titlerunning prior to \maketitle gion. Two cryostats are shown. The ultra high vacuumchambers for the optics sit on an optical table 3.6 meterlong. The length of the cavity is 2.2 meters. The cryostatsare supported above the optical table by a structure me-chanically decoupled from the optical cavity and the vi-brations’s path to the mirrors is more than 2 meters long.
Fig. 10.
Drawing of our experimental set up with the twocryostats in place.
In fig. 11 we show a scheme of our vacuum system. Themost critical parts are the two tubes that allow light topass through the magnetic field region. Their length isabout 50 cm and the inner diameter is 10 mm.The main point is that the presence of residual gas inthe vacuum pipe induces an ellipticity on the light beambecause of the magnetic birefringence of gases (Cotton-Mouton effect) [4]. To keep such an ellipticity small com-pared to the one to be measured, the maximum pressure P g of a residual gas constituent has to satisfy the followingformula P g ( atm ) ≪ λΨ ( t )2 F L∆n u B ( t ) (42)where following [4] ∆n u is the anisotropy of the indexof refraction for the residual gas component for a magneticfield of 1 T and a pressure of 1 atm at a temperature of0 ◦ C. For example, if Ψ ( t ) ≃ − , F ≃ × , L ≃ ∆n u ≃ − as for O gas, B ( t ) ≃
10 T, P g ( atm ) ≪ . × − torr.The vacuum system set up allows a dry roughing bya spiral pump and a turbo molecular pump, then a per-manent vibration free pumping thanks to two ion pumps.As tests have shown, in the vacuum pipe a pressure betterthan 10 − mbar is expected, while a slightly better pres-sure should be reached in the vacuum chambers wherethe mirrors are inserted. Finally, a gas analyzer will be put in between the two vacuum pipe passing through themagnets to check the nature of the residual gas, and mon-itoring the Cotton-Mouton of the residual gas. Fig. 11.
Scheme of our vacuum system.
The stray magnetic field of the magnet induces systematiceffects on the optics and especially on the mirrors whichare the elements closest to the magnets. Mirror Faraday ef-fect, i.e. the rotation of the polarization of a linearly polar-ized light induced by a magnetic field perpendicular to themirror surface, has been reported in ref. [40]. Faraday ro-tation was measured being of the order of K F = 3 . × − rad T − per reflection. Mirror Cotton-Mouton effect, i.e.the ellipticity induced on a linearly polarized light by amagnetic field parallel to the mirror surface, has been re-ported in ref. [41] and it amounted to about K CM = 10 − T − per reflection. The mirrors tested in ref. [41] were ofdifferent quality than the ones used in ref. [40]. Both re-sults depend on the number of reflecting layers and on thelayer materials. However they can be used to estimate themaximum stray field tolerable at the mirror location. Forexample, the ellipticity induced by the mirrors because oftheir own Cotton-Mouton effect is negligible compared tothe ellipticity to be measured, when the stray magneticfield parallel to the mirror surface obeys the following re-lation. B k stray ≪ r π F Ψ K CM (43)For example, if Ψ ≃ − and F ≃ B k stray ≪ × − T.An equivalent formula can be found for the componentof the stray magnetic field perpendicular of the mirrorsurface. B ⊥ stray ≪ π F ρ K F (44) lease give a shorter version with: \authorrunning and \titlerunning prior to \maketitle where ρ is the rotation in which one would be inter-ested, and we have assumed as usual that the number ofreflections on the mirrors in a Fabry-Perot cavity is equalto 2 F /π .Using one of our Xcoils, powered for a central field of7 T, we have measured the stray field on the axis of thecoil. At 70 cm from the coil’s center the field is of theorder of 1 mT / kA, which corresponds to a reduction ofa factor 500 with respect to the field at the centre of themagnet. We have also measured that the shielding factorgiven by a 4 mm copper plate is almost 5 at this field.In our experiment, the current is around 8000 A, whatgives us a field on the nearest mirror of 0,1 mT. With ashielding of at least 80% (we certainly can expect morethan 90 %), it gives a field on the mirror of 0,02 mT. Ifwe need a better shielding, we only need to put a secondcopper plate. This is a major advantage of pulsed fieldsover static fields. The whole optical system is operational, and the laser hasbeen locked to different cavities. In particular, the laserhas been locked to linear cavities (up to 2 m length) andring cavities (round trip of 2.4 m) of finesses up to 50 000.The measuring technique has been tested by measuring abirefringence induced by an electric field in a gas (Kerreffect) of about 10 − [42].After years of developments and tests, we have finallyput optics and magnet together and started debuggingand studying sensitivity. First with just one magnet inplace, then, when test runs will be completed, with twomagnets in place. This configuration corresponds to 40T m. Finesse greater than 300 000 is expected. Sensitivityshould also be at least 10 − rad Hz − / , thanks to thehigh central frequency of the modulated effect. All theseexperimental parameters exceed the corresponding valuesfor the PVLAS experiment. Therefore, we will be ableto test the PVLAS results using this first version of ourapparatus.Once this first step is accomplished, we will continuetowards the QED vacuum magnetic birefringence mea-surement. The critical points will be to reduce laser noiseto reach a sensitivity very close to the quantum limit andto continue the magnet R&D to reach 25 T. A 3-D com-plete computer modelling of our coils will be implementedto study the behavior of our coils under constraints.In any case, we will need more powerful transversepulsed magnets. Xcoils with a field region of 25 cm havebeen successfully tested, we are confident that final ap-paratus will consist of magnets capable to deliver a fieldover a length such that B L = 600 T m. QED elliptic-ity to be measured will be of the order of 4 × − rad,which should be reached in few hundreds of magnet pulsescorresponding to a few weeks of data acquisition. We thank J. Billette, M. Fouch´e, D. Forest, J-P. Lau-rent, J. Mauchain, J-L. Montorio, L. Polizzi, and A. Zi-touni. We also acknowledge the strong support of theengineering and technical staff of LCAR, LNCMP andLMA. This work is supported by the
ANR-Programmenon th´ematique (ANR-BLAN06-3-139634), and by the
CNRS-Programme National Astroparticules . References
1. J. Kerr, Br. Assoc. Rep. (1901) 568.2. Q. Majorana, Rendic. Accad. Lincei (1902) 374; Ct. r.hebd. Sanc Acad. Sci. Paris (1902) 159, 235.3. A. Cotton et H. Mouton, Ct. r. hebd. S´eanc Acad. Sci. Paris (1905) 317, 349; Ibid. (1906) 203; Ibid. (1907)229; Ann. Chem. Phys. (1907) 145, 289.4. C. Rizzo, A. Rizzo et D. M. Bishop, Int. Rev. Phys. Chem. (1997) 81.5. W. H. Watson, Proc. Roy. Soc. London A (1929), 345.6. H. Euler et K. Kochel, Naturwiss. (1935), 246.7. W. Heisenberg et H. Euler, Z. Phys. (1936), 714.8. Z. Bialynicka-Birula et I. Bialynicki-Birula, Phys. Rev. D (1970), 2341.9. S. L. Adler, Ann. Phys. (N. Y.) (1971), 599.10. C. C. Farr et C. J. Banwell, Proc. Roy. Soc. London A (1932), 275.11. C. J. Banwell et C. C. Farr, Proc. Roy. Soc. London A (1940), 1.12. E. Iacopini et E. Zavattini, Phys. Lett. B (1979) 151.13. L. Maiani, R. Petronzio et E. Zavattini, Phys. Lett. B (1986), 359.14. R. D. Peccei and H. Quinn, Phys. Rev. Lett. , 1440(1977).15. G. Raffelt and L. Stodolsky, Phys. Rev. D , 1237 (1988).16. E. Iacopini, B. Smith, G. Stefanini and E. Zavattini, NuovoCimento B (1981) 21.17. R. Cameron, G. Cantatore, A. C. Melissinos, G. Ruoso, Y.Semertzidis, H. Halama, D. Lazarus, A. Prodell, F. Nezrich,C. Rizzo and E. Zavattini, Phys. Rev. D (1993) 3707.18. D. Bakalov, F. Brandi, G. Cantatore, G. Carugno, S. Caru-sotto, F. Della Valle, A.M. De Riva, U. Gastaldi, E. Iacopini,P. Micossi, E. Milotti, R. Onofrio, R. Pengo, F. Perrone, G.Petrucci, E. Polacco, C. Rizzo, G. Ruoso, E. Zavattini, G.Zavattini, Quantum Semiclass. Opt. (1998) 239.19. E. Zavattini, G. Zavattini, G. Ruoso, E. Polacco, E.Milotti, M. Karuza, U. Gastaldi, G. Di Domenico, F. DellaValle, R. Cimino, S. Carusotto, G. Cantatore, and M. Bre-gant, Phys. Rev. Lett. ∼ axions/talks/Giovanni Cantatore.pdf21. S. Lamoreaux, Nature,
31 (2006).22. E. Zavattini et al., arXiv:0706.3419 [hep-ex].23. K.Zioutas et al. (CAST Collaboration), Phys. Rev. Lett. , 121301 (2005).24. E. Mass´o, J. Redondo, J. of Cosmology and AstroparticlePhysics ∼ axions/axions.shtml26. K. Van Bibber, N.R. Dagderiven, S.E. Koonin, A.K. Ker-man, H.N. Nelson, Phys. Rev. Lett. , 759 (1987).2 Please give a shorter version with: \authorrunning and \titlerunning prior to \maketitle ∼ axions/talks/Pierre Pugnat.pdf28. C. Robilliard, R. Battesti, M. Fouch´e, J. Mauchain, A-M.Sautivet, F. Amiranoff, and C. Rizzo, Phys. Rev. Lett. inpress (arXiv:0707.1296 [hep-ex]).29. Sheng-Jui Chen, Hsien-Hao Mei, Wei-Tou Ni,arXiv:hep-ex/0611050 [hep-ex].30. C.Rizzo Europhys. Lett. , 483 (1998).31. S. Askenazy, C. Rizzo, and O. Portugall, Physica B , (2001) 5.32. S. Askenazy, J. Billette, P. Dupr´e, P. Ganau, J. Mackowski,J. Marquez, L. Pinard, O. Portugall, D. Ricard, G.L.J.A.Rikken, C. Rizzo, G. Tr´enec, J. Vigu´e, AIP Conf. Proc. ,(2001) 115.33. S. Batut et al., submitted to IEEE Trans. Applied Super-conductivity.34. F. Brandi, F. Della Valle, A.M. De Riva, P. Micossi, F.Perrone, C. Rizzo, G. Ruoso, G. Zavattini, Appl. Phys. B , 351 (1997).35. G. Zavattini, G. Cantatore, R. Cimino, G. Di Domenico,F. Della Valle, M. Karuza, E. Milotti, G. Ruoso, Appl. Phys.B , 571 (2006).36. G. Rempe, R.J. Thompson, H.J. Kimble, R. Lalezari, Opt.Lett. , 363 (1992).37. R.V. Pound, Rev. Sci. Instrum. , 460 (1946); R.W.P.Drever, J.L. Hall, F.B. Kowalsky, J. Hough, G.M. Ford, A.J.Munley and H. Ward, Appl. Phys. B , 97 (1983).38. G. Cantatore, F.D. Valle, E. Milotti, P. Pace, E. Zavattini,E. Polacco, F. Perrone, C. Rizzo, G. Zavattini, and G. Ruoso,Rev. Sci. Instrum. , 2785 (1995).39. A.M. De Riva, G. Zavattini, S. Marigo, C. Rizzo, G. Ru-oso, G. Carugno, R. Onofrio, S. Carusotto, M. Papa, F. Per-rone, E. Polacco, G. Cantatore, F. Della Valle, P. Micossi,E. Milotti, P. Pace, E. Zavattini, Rev. Sci. Instrum. , 2680(1996).40. E. Iacopini, G. Stefanini and E. Zavattini, Appl. Phys. A , 63 (1983).41. G. Bialolenker, E. Polacco, C.Rizzo, G.Ruoso, Appl. Phys.B , 703 (1999).42. F. Bielsa, R. Battesti, C. Robilliard, G. Bialolenker, G.Bailly, G. Trenec, A. Rizzo, C. Rizzo, Eur. Phys. J. D36