We point out that photon regeneration-experiments that search for the axion,
or axion-like particles, may be resonantly enhanced by employing matched
Fabry-Perot optical cavities encompassing both the axion production and
conversion magnetic field regions. Compared to a simple photon regeneration
experiment, which uses the laser in a single-pass geometry, this technique can
result in a gain in rate of order ${\cal F}^2$, where ${\cal F}$ is the finesse
of the cavities. This gain could feasibly be $10^{(10-12)}$, corresponding to
an improvement in sensitivity in the axion-photon coupling, $g_{a\gamma\gamma}$
, of order ${\cal F}^{1/2} \sim 10^{(2.5-3)}$, permitting a practical purely
laboratory search to probe axion-photon couplings not previously excluded by
stellar evolution limits, or solar axion searches.Comment: 4 pages, 2 figure

A. A. Ansel’m

D. B. Tanner

G. G. Raffelt

Karl van Bibber

P. Sikivie

English

arXiv

We point out that photon regeneration-experiments that search for the axion, or axion-like particles, may be resonantly enhanced by employing matched Fabry-Perot optical cavities encompassing both the axion production and conversion magnetic field regions. Compared to a simple photon regeneration experiment, which uses the laser in a single-pass geometry, this technique can result in a gain in rate of order ${\cal F}^2$, where ${\cal F}$ is the finesse of the cavities. This gain could feasibly be $10^{(10-12)}$, corresponding to an improvement in sensitivity in the axion-photon coupling, $g_{a\gamma\gamma}$ , of order ${\cal F}^{1/2} \sim 10^{(2.5-3)}$, permitting a practical purely laboratory search to probe axion-photon couplings not previously excluded by stellar evolution limits, or solar axion searches.We point out that photon regeneration-experiments that search for the axion, or axion-like particles, may be resonantly enhanced by employing matched Fabry-Perot optical cavities encompassing both the axion production and conversion magnetic field regions. Compared to a simple photon regeneration experiment, which uses the laser in a single-pass geometry, this technique can result in a gain in rate of order ${\cal F}^2$, where ${\cal F}$ is the finesse of the cavities. This gain could feasibly be $10^{(10-12)}$, corresponding to an improvement in sensitivity in the axion-photon coupling, $g_{a\gamma\gamma}$ , of order ${\cal F}^{1/2} \sim 10^{(2.5-3)}$, permitting a practical purely laboratory search to probe axion-photon couplings not previously excluded by stellar evolution limits, or solar axion searches

Sikivie, P.

Tanner, D.B.

van Bibber, Karl

CERN Document Server

arXiv:hep-ph/0701198v1   23 Jan 2007Resonantly Enhanced Axion-Photon RegenerationP. Sikivie,a,b D.B. Tanner,a and Karl van Bibberca Department of Physics, University of Florida, Gainesville, FL 32611, USAb Theoretical Physics Division, CERN, CH-1211 Gene`ve 23, Switzerlandc Lawrence Livermore National Laboratory, Livermore, CA 94550, USAWe point out that photon regeneration-experiments that search for the axion, or axion-like parti-cles, may be resonantly enhanced by employing matched Fabry-Perot optical cavities encompassingboth the axion production and conversion magnetic field regions. Compared to a simple photonregeneration experiment, which uses the laser in a single-pass geometry, this technique can resultin a gain in rate of order F2, where F is the finesse of the cavities. This gain could feasiblybe 10(10−12) , corresponding to an improvement in sensitivity in the axion-photon coupling, gaγγ ,of order F1/2 ∼ 10(2.5−3), permitting a practical purely laboratory search to probe axion-photoncouplings not previously excluded by stellar evolution limits, or solar axion searches.PACS numbers: PACS numbers: 12.38.-t, 12.38.Qk, 14.80.Mz, 29.90.+r, 95.35.+dThe axion remains the most attractive solution to thestrong-CP problem and is one of two leading dark-mattercandidates [1]. Recently, it has been realized that the ax-ion represents a fundamental underlying feature of stringtheories; there could be several or even a great number ofaxions or axion-like particles within any particular stringtheory [2]. From the experimental viewpoint, there arenow several photon regeneration experiments in variousstages of preparation [3], motivated in part by the reportof the PVLAS collaboration of a non-zero magnetically-induced dichroism of the vacuum [4], which may be in-terpreted as the production of a light boson with a two-photon coupling [5]. Although the particle interpretationof the PVLAS experiment is in principle excluded by themuch more stringent limit set by the CAST solar ax-ion search [6], it is important to check this result by apurely laboratory experiment, particularly if one couldultimately improve such a measurement to reach previ-ously unexplored regions of (ma, gaγγ).The simplest and most unambiguous purely laboratoryexperiment to look for light pseudoscalars is photon re-generation (“shining light through walls”) [7]. A laserbeam traverses a dipole magnet, wherein a small frac-tion of the photons are converted into axions with thesame energy. An optical barrier blocks the primary laserbeam, whereas the axion component of the beam travelsthrough the wall unimpeded and enters a second dipolemagnet, where it is reconverted to photons with the sameprobability (we will assume magnets of identical lengthL and field strength B0 without loss of generality). How-ever, because the photon-regeneration rate goes as g4aγγ,the sensitivity of the experiment to small values of gaγγis poor in its basic form, improved appreciably only byincreasing B0 or L.In this Letter, we point out that matched Fabry-Perotcavities incorporated into the production and detectionmagnets can improve the sensitivity in the axion-photoncoupling, gaγγ by the square root of the cavities’ finesse,F1/2. This factor may be 10(2.5−3).The action density for the dynamics of photons andaxions isL = 12(ǫE2−B2)+ 12(∂ta)2− 12(~∇a)2− 12m2aa2−ga ~E · ~B(1)where ~E, ~B and a are respectively the electric, magneticand axion fields. The electromagnetic fields are givenin terms of scalar and vector potentials, ~E = −~∇Φ −∂t ~A, ~B = ~∇ × ~A, as usual. The coupling g ≡ gaγγis written without subscripts here, for simplicity. ǫ isassumed constant in both space and time. In the presenceof a large static magnetic field ~B0(~x), the equations ofmotion areǫ~∇ · ~E = g ~B0 · ~∇a~∇× ~B − ǫ∂t ~E = −g ~B0∂ta∂2t a− ~∇2a+m2aa = −g ~E · ~B0 . (2)~B now represents the magnetic field minus ~B0, and termsof order gE and gB are neglected. Eqs. (2) describe theconversion of axions to photons and vice-versa.Using these equations, it can be shown [5, 7, 8] thatthe axion-to-photon conversion probability in a regionof length L, permeated by a constant magnetic field B0transverse to the direction of propagation and a dielectricconstant ǫ, is given by (h¯ = c = 1)p =14βa√ǫ(gB0L)2(2qLsinqL2)2, (3)with βa the axion speed and q = ka − kγ the momentumtransfer. In terms of the energy ω, which is the same forthe axion and the photon, ka =√ω2 −m2a, βa = kaω andkγ =√ǫω. The photon to axion conversion probabilityin this same region is also equal to p. Everything elsebeing the same, the conversion probability is largest whenq ≈ 0. For ma << ω, and propagation in a vacuumq = −m2a2ω+ (1 −√ǫ)ω . (4)2(a)(b)WallPhotonDetectora!LaserLB0MagnetB0MagnetLMatched Fabry-PerotsIOLaserMagnetMagnetPhotonDetectorsFIG. 1: (color) (a) Simple photon regeneration. (b) Resonantphoton regeneration, employing matched Fabry-Perot cavi-ties. The overall envelope schematically shown by the thindashed lines indicates the important condition that the axionwave, and thus the Fabry-Perot mode, in the conversion mag-net must follow that of the hypothetically unimpeded photonwave from the Fabry-Perot mode in the production magnet.Between the laser and the cavity is the injection optics (IO)which manages mode matching of the laser to the cavity, im-poses RF sidebands for reflection locking of the laser to thecavity, and provides isolation for the laser. The photon detec-tors are also preceded by matching and beam-steering optics.Not shown at all is the electro-optical system required to lockthe two cavities together in frequency.Fig. 1a shows the axion-photon regeneration exper-iment as usually conceived. If P0 is the power of thelaser, the power of the axion beam traversing the wall isp P0 where p is the conversion probability in the magneton the LHS of Fig. 1a. Let p′ be the conversion probabil-ity in the magnet on the RHS. The power in regeneratedphotons is P = p′ p P0.Fig. 1b shows the two improvements we propose forthe experiment. The first improvement is to build up thepower on the photon-to-axion conversion side of the ex-periment using a Fabry-Perot cavity, as illustrated. Pho-tons in the production cavity will then convert to ax-ions with probability p for each pass through the cavity.The standing wave in the production cavity is the sum ofleft-moving and right-moving components of equal am-plitude. If the reflectivity of the cavity mirrors is givenbyR = 1− η (5)and the power of the laser is P0, the power of the right-moving wave in the production cavity is 1η P0. Thereforethe axion power through the wall in the setup of Fig. 1bis 1η p P0. Assuming the lasers in Fig. 1a and Fig. 1bhave the same power, the axion flux is increased by thefactor 1η .Increasing the axion production rate, and thus the pho-ton regeneration rate, by building up the optical power inthe first magnet is not a new idea. In fact, the only pho-ton regeneration experiment performed and published todate, by Ruoso et al. utilized an “optical delay line,” i.e.,an incoherent cavity encompassing the production mag-net, causing the laser beam to traverse the magnet 200times before exiting. With relatively modest magnets(4.4 m, 3.7 T each), a limit of gaγγ < 7.7× 10−7 GeV−1was set [9].There is substantial gain from building up the laserpower in the axion production magnet; however, it isimmaterial whether one “recycles” the photons incoher-ently, as in an optical delay line, or coherently, as in aFabry-Perot (FP) cavity. In contrast, the coherent casealone can provide a large additional gain in sensitivity forphoton regeneration. Thus, the second improvement is toalso install a Fabry-Perot cavity on the regeneration sideof the experiment, making a symmetric arrangement, asillustrated in Fig. 1b. When the second FP cavity islocked to the first, the probability of axion to photonconversion in the second FP cavity is 2η′ p′ = 2π F ′ p′where F ′ is the finesse of the cavity, and p′ is the axion-to-photon conversion probability in the absence of thecavity. The calculation which yields this result is out-lined in the next paragraph.The cavity modes are described by~An = An(t)yˆ sin(nπLz) (6)where zˆ is in the direction of light propagation and yˆ is atransverse direction. The dependence of the mode func-tion on the transverse coordinates (x and y) is neglectedhere, but will be discussed later. Using Eqs (2) one canshow that, in the presence of an axion beam travellingthrough the cavity in the z-directiona(z, t) = A sin(kaz − ωt) , (7)the coefficients An(t) satisfy(d2dt2+ γddt+ ω2n)An(t) = C sin(ωt−qL2) , (8)where ωn =nπ√ǫLandC =1ǫgωB0A 2Lqsin(qL2) . (9)As before, q = ka− kn =√ω2 −m2a− nπL is the momen-tum transfer. When the production cavity and the regen-eration cavity are tuned to the same frequency, ωn = ωfor some n. Then~An = yˆCωγsin(nπLz) sin(ωt− qL2− π2) , (10)up to transients. The energy stored in the cavity isE = 14SLA2ǫω2 where A = Cωγ and S is the cross-sectional area of the cavity mode. The power emittedby the cavity is P = γE, assuming that there are nolosses other than by transmission through the mirrors.3The power of the axion beam is Pa =12SA2ωka, assum-ing it has the same cross-sectional area S as the cavitymode. The axion to photon conversion probability in theFP cavity is thereforepFP =PPa=12g2B20ǫβaωQL(2qLsin(qL2))2, (11)where Q = ωγ is the quality factor of the cavity. In termsof the conversion probability p′ in the same region (lengthL, magnetic field B0, and dielectric constant ǫ) withoutthe cavity, we have pFP =2Q√ǫLωp′ = 2F′π p′ as announced.Including both improvements, the regenerated photonpower emitted from the cavity isP =2η′ηp′ p P0 . (12)Half of the power P is right-moving and half is left-moving. To detect all the regenerated photons, detec-tors are installed on both sides of the Fabry-Perot cavity,as illustrated. The combined improvements yield an in-crease by a factor 2/ηη′ in signal power. With presenttechnology, this factor may be as large as 1012.In general, the loss of power from the cavity will haveother contributions, η′ = η′trans + η′abs + η′scatt, where thelatter two terms represent absorption and scattering (in-cluding diffraction) losses. If transmission is not entirelydominant, then Eq. (12) should be modified by a mul-tiplicative factor f = η′trans/η′. There is no such factorcorresponding to the production side, as the transmittedpower is irrelevant, and the laser power can always bebrought up to the limit established by optical damage.There are practical limitations to the optics which es-tablish the achievable sensitivity of a resonant photonregeneration experiment. Here, we present a plausibleexperimental realization, utilizing a 10 W CW Nd:YAG(λ = 1.064 µm) laser, similar to the LIGO laser [10], anda total of eight LHC dipole magnets (8.75 T, 14.3 m, 50mm ⊘), which are well-suited to the experiment.With the magnets set end-to-end, four for the pro-duction leg and four for the regeneration leg, the overalllength of each cavity will exceed 60 m. We use 66 m inthis estimate, giving a cavity free spectral range (FSR)of 2.3 MHz. We will refer to this as a 4 + 4 configurationof LHC dipole magnets. The magnet diameter deter-mines the maximum size of the TEM00 Gaussian modeof the cavities. Furthermore, the spatio-temporal profilesof the axion and photon modes are identical, i.e., the ax-ion mode follows that of a hypothetically unobstructedphoton beam. Thus the cavities should be symmetric,with the beam waist at the optical barrier, such that thetwo end mirrors would support the optical mode if thebarrier and inboard mirrors were removed (Figure 1b).There is no constraint that the length of the two cavitiesbe exactly equal, nor that their relative separation equala multiple number of wavelengths. For the case of 1.064µm light, and for 130 m between end mirrors, the beamis everywhere smaller than 36 mm, and is not clipped bythe sagitta of the LHC dipoles [11].Another limit is that the intracavity power density atthe mirrors be below the damage threshold, which for thebest multilayer dielectric mirrors approaches 1 GW/cm2.Putting the waist of the mode at the barrier implies thatthe limitation on power density will be first encounteredfor the slightly convex inboard mirrors. The total circu-lating power in the production magnet is ∼1 MW.The best Fabry-Perot resonators have achieved a fi-nesse of a few million; here, we take for the finesse,F ∼ 3.1 × 105, an order of magnitude lower, which isstill ambitious, but feasible. For the line-width impliedby this finesse, 7.3 Hz, the vibration tolerance for thecavity mirrors is of order 10−3 nm, well within the expe-rience of LIGO detectors [10]. Also for this finesse, wecan design the cavity to ensure that transmission domi-nates loss (scattering and diffraction), and this has beenassumed.Intrinsic to resonantly enhanced photon regenerationis the requirement that the production and regenerationcavities remained locked in frequency together, withintheir bandwidth, ∆ν = 2ν/Q, where the quality factor,Q = nF of the cavity, and n its mode index. The 10 Wlaser is reflection locked to one of the cavity modes ofthe production cavity using the Pound-Dreever-Hall re-flection locking scheme [12]. Then, the challenge is toimplement the locking of the regeneration cavity in away that does not introduce any spurious photons intothe detectors. The scheme we envision is to use a lowpower Nd:YAG laser, offset locked by (integer)×FSR ofthe production cavity (say, 50 MHz) from the main laser,and use the same Pound-Dreever-Hall reflection lockingmethod to control the length of the regeneration cavity.For additional rejection, the locking could be done in theorthogonal polarization state, i.e. perpendicular to thedipole magnetic field ~B0.There are several possible schemes for detection of theweak signal from the regeneration cavity. The simplestis to focus the cavity output on a cooled InGaAs charge-coupled devices (CCDs), with pixels of the order of 15µm. Modern CCDs developed for astronomy have verylow dark current rates, of order 1-2 emin−1pixel−1. How-ever, the best scheme seems to be heterodyne detection,mixing the generation cavity output with the lockinglaser at an RF photodiode and detecting both in-phaseand quadrature signals at the difference frequency be-tween the laser in the generating cavity and the lockingbeam, especially as the frequencies of both lasers (andtheir difference) are well known.Defining a discovery to be 5σ, where σ is assumed to bedominated by the square-root counting statistics of thedark-current background, we find that the experimentis sensitive to axions or generalized pseudoscalars withgaγγ = 2.7×10−11 GeV−1, after 10 days cumulative run-4PVLASLaser experimentsSolar searchHB StarsResonantly enhancedphoton regenerationma (eV)10-6 10-5 10-4 10-3 10-210-1610-810-1010-1210-1410-410-6ga!!(GeV-1)Microwave cavitydark matter searchesAxion modelsFIG. 2: (color) Current exclusion plot of mass and photoncoupling (ma, gaγγ) for the axion, and the 5 σ discovery po-tential for the resonantly enhanced photon regeneration ex-periment calculated for a configuration of 4 + 4 LHC dipolemagnets. The existing exclusion limits indicated on the plotinclude the cavity microwave experiments assuming axionssaturate the dark matter halo density [1], the best direct solaraxion search (CAST collaboration) [6], the Horizontal BranchStar limit [13], and previous laser experiments [9]. The rederror ellipse indicates the positive result of the PVLAS collab-oration, if interpreted as a light pseudoscalar, based on mea-surements of magnetically-induced dichroism of the vacuum[4]. For the estimated limits of resonantly enhanced photonregeneration presented here, the solid curve corresponds tothe ↑↑↑↑ configuration of the individual LHC dipole magnetsin both the production and regeneration strings; the dottedcurve indicates the extension of the mass reach by addition-ally running in the ↑↑↓↓, and ↑↓↑↓ configurations.ning, up to an axion mass of ma ∼ 2 × 10−4 eV (Figure2). Note the experiment’s reach in gaγγ degrades sharplywith increasing mass beyond that value. This upper limitin mass for which one still has effectively maximum sensi-tivity is dictated by the length of the dipole magnetic fieldL, as the momentum mismatch between a massless pho-ton and a massive axion q ∼ L−1 defines the oscillationlength of the problem. As pointed out in ref. [7] however,there is a practical strategy to extend the mass range up-wards, if the total magnetic length L is comprised of astring of N individual identical dipoles of length l. Inthis case, one may configure the magnet string as a “wig-gler” to cover higher regions of mass, up to values corre-sponding to the oscillation length determined by a singledipole, i.e. q ∼ l−1. Figure 2 shows that the combinationof magnet configurations ↑↑↑↑, ↑↑↓↓, and ↑↓↑↓ extendthe mass reach up to ∼ 4× 10−4 eV.While resonant photon regeneration marks a signifi-cant improvement over the simple experiment, in fact thesensitivity in gaγγ still only gains (or loses) as F1/2. Thusin the example above, the experiment would still reacha limit of gaγγ = 8.5 × 10−11 GeV−1 even if the Fabry-Perot only achieved a finesse of F ≃ 30,000. As it willlikely be easier to attain higher values of F for shorterbaselines, we further note that as gaγγ ∝ (BL)−1, a limitof gaγγ ∼ 10−10 GeV−1, i.e. equal to the CAST and theHorizontal Branch Star limit, should still be achievableeven with only one LHC dipole in each leg.A fully-detailed derivation of the central result Eq.(12), and a detailed discussion of the experimental de-sign worked out as an example in this Letter will appearin a future publication.We thank Frank Nezrick, Stan Whitcomb, Tom Car-ruthers, Kem Cook, Bruce Macintosh, Georg Raffelt andGuido Mueller for useful conversations during the de-velopment of the concept. This work was supported inpart by the U.S. Department of Energy under contractsW-7405-ENG-48 and DE-FG02-97ER41029. P.S. grate-fully acknowledges the hospitality of the Aspen Center ofPhysics while working on this project.[1] R. Bradley et al., Rev. Mod. Phys. 75, 777 (2003).[2] P. Svrcˇek and E. Witten, hep-th/0605206[3] See, for example, R. Rabada´n, A. Ringwald and K. Sig-urdson, Phys. Rev. Lett. 96, 110407 (2006); A. Afanasev,Jefferson Laboratory proposal, March 25, 2006; L. Duvil-laret et al., CERN-SPSC-2005-034[4] E. Zavattini et al. (PVLAS Collaboration), Phys. Rev.Lett. 96, 110406 (2006).[5] P. Sikivie, Phys. Rev. Lett. 51, 1415 (1983); Phys. RevD32, 2988 (1985).[6] K. Zioutas et al. (CAST Collaboration), Phys. Rev. Lett.94, 121301 (2005).[7] K. van Bibber, N.R. Dagdeviren, S.E. Koonin, A.K. Ker-man and H.N. Nelson, Phys. Rev. Lett. 59, 759 (1987).[8] G. Raffelt and L. Stodolsky, Phys. Rev. D37, 1237(1988).[9] G. Ruoso et al., Z. Phys. C56, 505 (1992).[10] B. Abbott et al. (LIGO Scientific Collaboration), Nucl.Instrum. Methods A, 517, 154–179 (2004).[11] P. Pugnat, private communication (2006). The LHCdipoles were built with a 9-mm sagitta to accommodatethe orbit of 2804-m mean radius; the cold mass could beflexed to take the sagitta out if necessary.[12] R.W.P. Drever, J.L. Hall, F.V. Kowalski, J. Hough, G.M.Ford, A.J. Munley, and H. Ward, Appl. Phys. B 31, 97(1983); Eric D. Black, Am. J. Phys. 69, 79 (2001).[13] G.G. Raffelt, Stars as Laboratories for Fundamen-tal Physics, University of Chicago Press 1996; andhep-ph/0611350.

Resonantly Enhanced Axion-Photon Regeneration

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Resonantly Enhanced Axion-Photon Regeneration

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