On the observation of vacuum birefringence

Optics Communications 267 (2006) 318–321 www.elsevier.com/locate/optcom

On the observation of vacuum birefringence Thomas Heinzl

b

a,*

, Ben Liesfeld b, Kay-Uwe Amthor b, Heinrich Schwoerer b, Roland Sauerbrey c, Andreas Wipf d

a School of Mathematics and Statistics, University of Plymouth, Drake Circus, Plymouth PL4 8AA, UK Institut fu¨r Optik und Quantenelektronik, Friedrich-Schiller-Universita¨t Jena, Max-Wien-Platz 1, 07743 Jena, Germany c Forschungszentrum Rossendorf, Bautzner Landstraße 128, 01328 Dresden, Germany d Theoretisch-Physikalisches Institut, Friedrich-Schiller-Universita¨t Jena, Max-Wien-Platz 1, 07743 Jena, Germany

Received 3 May 2006; received in revised form 19 June 2006; accepted 20 June 2006

Abstract We suggest an experiment to observe vacuum birefringence induced by intense laser fields. A high-intensity laser pulse is focused to ultra-relativistic intensity and polarizes the vacuum which then acts like a birefringent medium. The latter is probed by a linearly polarized X-ray pulse. We calculate the resulting ellipticity signal within strong-field QED assuming Gaussian beams. The laser technology required for detecting the signal will be available within the next three years.  2006 Elsevier B.V. All rights reserved. PACS: 12.20.m; 42.50.Xa; 42.60.v

The interactions of light and matter are described by quantum electrodynamics (QED), at present the bestestablished theory in physics. The QED Lagrangian couples photons to charged Dirac particles in a gauge invariant way. At photon energies small compared to the electron mass, x  me, electrons (and positrons) will generically not be produced as real particles. Nevertheless, as already stated by Heisenberg and Euler, ‘‘. . . even in situations where the [photon] energy is not sufficient for matter production, its virtual possibility will result in a ‘polarization of the vacuum’ and hence in an alteration of Maxwell’s equations’’ [1]. These authors were the first to explicitly derive the nonlinear terms induced by QED for small photon energies but arbitrary intensities (see also [2]). The most spectacular process resulting from these modifications presumably is pair production in a constant electric field. This is an absorptive process as photons disappear

*

Corresponding author. Tel.: +44 1752 232754; fax: +44 1752 232780. E-mail addresses: [email protected] (T. Heinzl), [email protected] (R. Sauerbrey), [email protected] (A. Wipf). 0030-4018/$ - see front matter  2006 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2006.06.053

by disintegration into matter pairs. It can occur for field strengths larger than the critical one given by [3,4] Ec 

m2e ’ 1:3  1018 V=m: e

ð1Þ

In this electric field, an electron gains an energy me upon travelling a distance equal to its Compton wavelength, ke ¼ 1=me . The associated intensity is I c ¼ E2c ’ 4:4 2 1029 W=cm such that both field strength and intensity are way out of experimental reach for the time being – unless one can utilize huge relativistic gamma factors produced by large scale particle accelerators [5,6]. Alternatively, there are also dispersive effects that may be considered. These include many of the phenomena studied in nonlinear optics as well as ‘‘birefringence of the vacuum’’ first addressed by Klein and Nigam [7] in 1964, soon followed by more systematic studies [8–10]. In essence, the polarized QED vacuum acts like a birefringent medium (e.g. a calcite crystal) with two indices of refraction depending on the polarization of the incoming light. In a static magnetic field of 5 T a light polarization rotation has recently been observed [11]. The measured signal differs from the

T. Heinzl et al. / Optics Communications 267 (2006) 318–321

QED expectations and may be caused by a new coupling of photons to an hitherto unobserved pseudoscalar. Detection of the tiny dispersive effects is an enormous challenge. In this paper, we point out that several orders of magnitude in field strength may be gained by employing high-power lasers. Distorting the vacuum with lasers has been suggested long ago [10] but was not considered experimentally for lack of sufficient laser power. However, recent progress in both laser technology and X-ray detection has lead to novel experimental capabilities. It is therefore due time to specifically address the feasibility of a strong-field laser experiment to measure vacuum birefringence. In the light of the results [11] such experiments are also necessary in order to test whether strong electromagnetic fields provide windows into new physics. We intend to utilize the high-repetition rate petawatt class laser system POLARIS which is currently under construction at the Jena high-intensity laser facility and which will be fully operational in 2007 [12]. POLARIS consists of a diode-pumped laser system based on chirped pulse amplification (CPA) which will be operating at K = 1032 nm (X = 1.2 eV) with a repetition rate of 0.1 Hz. A pulse duration of about 140 fs and a pulse energy of 150 J in principle allows to generate intensities in the focal region of I = 1022 W/cm2. This corresponds to a substantial electric field E ’ 2 · 1014 V/m, still about four orders of magnitude below Ec. The proposed experimental setup is shown in Fig. 1. A high-intensity laser pulse is focused by an off-axis parabolic mirror. A linearly polarized laser-generated ultrashort X-ray pulse is aligned collinearly with the focused optical laser pulse. After passing through the focus the laser induced vacuum birefringence will lead to a small ellipticity of the X-ray pulse which will be detected by a high contrast X-ray polarimeter [13]. The whole setup is located in an ultra-high vacuum chamber and is entirely computer controlled.

319

Shown in grey in Fig. 1 is an extension of the setup which enables us to accurately overlap two counter propagating high-intensity laser pulses. Accurate control over spatial and temporal overlap was convincingly demonstrated carrying out an autocorrelation of the laser pulses at full intensity [14] and generating Thomson backscattered X-rays from laser-accelerated electrons [15]. This counter propagating scheme, a table-top ‘‘photon collider’’, may also be employed for pair creation from the vacuum. For the probe pulse we have chosen an X-ray source of photon energy x J 1 keV, since the birefringence signal is proportional to x2 (see below). Our long-term plans are to replace the present source by an X-ray free electron laser (XFEL) or by a laser-based Thomson backscattering source [15] both of which deliver ultrashort and highly polarized X-rays. Refraction is a dispersive process based on modified propagation properties of the probe photons travelling through a region where a (strong) background field is present. The resulting corrections to pure Maxwell theory to leading order in the probe field al may be expressed in terms of an effective action [10] Z 1 d4 x d4 y al ðxÞPlm ðx; y; AÞam ðyÞ; dS  ð2Þ 2 where Al denotes the background field and Plm the polarization tensor. To lowest order in a loop (or  h) expansion the latter is given by the Feynman diagram ð3Þ with the heavy lines in the loop denoting the dressed propagator depending on the background field A. The dressed propagator is known exactly only for a few special background configurations (see [16] for an overview). Typically, one obtains rather unwieldy integral representations which have to be analyzed numerically. In our case, however, we can exploit the fact that we are working in the regime of both low energy and small intensities leading to two small parameters [17], namely

Laser pulse

m2  x2 =m2e ’ 4  106 ;

50/50 Beamsplitter

2

 E

x z Off-axis parabolic mirror

=E2c

ð4Þ 8

¼ I=I c ’ 2  10 :

ð5Þ

2

Off-axis parabolic mirror

X-ray laser pulse

Polarizer

2

Analyzer

Fig. 1. Proposed experimental setup for the demonstration of vacuum birefringence: A high-intensity laser pulse is focused by an F/2.5 off-axis parabolic mirror. A hole is drilled into the parabolic mirror in alignment with the z-axis (axes as indicated) in such a way that an X-ray pulse can propagate along the z-axis through the focal region of the high-intensity laser pulse. Using a polarizer-analyzer pair the ellipticity of the X-ray pulse may be detected. Shown in grey: Extension of the setup for the generation of counter propagating laser pulses and a high-intensity standing wave which may be used for pair creation.

Low intensity,   1, means that we can work to lowest nontrivial order in the external field i.e. O(2). Low energy, m  1, implies that we may safely expand Plm in derivatives or, after Fourier transformation, in powers of the probe 4momentum k = x(1, nk) where k2 = 1 and n P 1 is the index of refraction. Thus, the derivative expansion is in powers of x2 or, equivalently, of m2. For the physics we are interested in it is sufficient to work at leading order in frequency which turns out to be O(m2). As a result of this discussion we may straightforwardly employ the Heisenberg–Euler Lagrangian [1] to leading order in intensity, 2, 1 1 dLðS; PÞ ¼ c S2 þ cþ P2 : 2 2

ð6Þ

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T. Heinzl et al. / Optics Communications 267 (2006) 318–321

The basic building blocks in Eq. (6) are the scalar and pseudoscalar invariants [4,9] 1 1 S   F lm F lm ¼ ðE 2  B 2 Þ; 4 2 1 P   F lm Fe lm ¼ E  B; 4

ð7Þ ð8Þ

where Flm denotes the electromagnetic field-strength tensor (comprising both background and probe photon field) and Fe lm its dual. The nonlinear couplings in Eq. (6) are given by cþ  7q;

c  4q;

q

a 1 ; 45p E2c

ð9Þ

with a = 1/137 being the fine-structure constant. To proceed we split the fields into an intense but slowlyvarying background and a weak probe field according to the replacement Flm ! Flm + flm with upper (lower) case letters for electromagnetic quantities henceforth referring to the background (probe). The latter may be modelled as a plane or standing wave or more realistic variants thereof like Gaussian beams (see discussion R below). Writing the Heisenberg–Euler action, dS ¼ d4 xdL, in the form of Eq. (2) yields the polarization tensor Plm ¼ c k 2 SPlm þ c bl bm þ cþ ~ bl ~ bm ; lm

l m

ð10Þ

2

where P ¼ glm  k k =k is the standard projection orthogonal to k and S denotes the background invariant. In addition we have introduced the new 4-vectors [9], bl  F lm k m ;

~ bl  Fe lm k m :

ð11Þ

Note that we have b  k ¼ 0 ¼ ~ b  k and hence Plmkm = 0 as required by gauge invariance. The polarization tensor from Eq. (10) becomes particularly simple for a plane wave background which has S ¼ 0. In what follows we want to show that this simplification does actually occur for any background at leading order in frequency m and intensity 2. If we write the index of refraction as n = 1 + D we expect D = O(2) the deviation of n from unity being due to the external fields. Hence k2 is no longer zero but rather k2 = O(2m2) implying k 2 S ¼ Oð4 m2 Þ. For generic geometrical settings the invariant b2 = O(2m2). The upshot of this power counting exercise is the important inequality jk 2 Sj  jb2 j;

ð12Þ

by means of which we may neglect k 2 S. This justifies the statement in [16] that to leading order in  and m the eigenvalues of Plm do not depend on the invariants S and P. Hence, under the assertion Eq. (12), slowly-varying but otherwise arbitrary background fields behave as crossed fields (E and B orthogonal and of the same magnitude) for which strictly S ¼ P ¼ 0. In addition, one has achieved orthogonality, b  ~ b ¼ 0, so that Eq. (10) turns into a spectral representation, Plm ¼ c bl bm þ cþ ~ bl ~ bm :

ð13Þ

This is indeed the polarization tensor for crossed fields [18,19] evaluated at leading order in the derivative-intensity expansion. Hence, we can simply quote the (coinciding) results for the index of refraction for either constant or crossed-field background ([8–10,20] or [18,19], respectively) which is given by n ¼ 1 þ ð11  3ÞqI ¼ 1 þ

a I ð11  3Þ : 45p Ic

ð14Þ

As is well known, the two different values imply birefringence with a relative phase shift between the two rays proportional to Dn  n+  n, d 4a d I 4a d 2 : D/ ¼ 2p Dn ¼ ¼ k 15 k I c 15 k

ð15Þ

A realistic laser field will lead to an intensity distribution along the z-axis (choosing k = ez). If z0 measures the typical extension of the distribution we may set s  z/z0 and write the intensity as I(s) = I0 g(s) with peak intensity I0 and a dimensionless distribution function g(s). The phase shift Eq. (15) is then replaced by the expression [21] D/ ¼

4a z0 I 0 j; 15 k I c

where the correction factor j is the integral Z s0 ds gðsÞ ¼ Oð1Þ: j ¼ jðs0 Þ 

ð16Þ

ð17Þ

s0

Here, s0 denotes the half-width of the intensity distribution in units of z0. In general it is a reasonable approximation to let s0 ! 1. For a single Gaussian beam, z0 is the Rayleigh length and the intensity I1 follows a Lorentz curve, hence g1(s) = 1/(1 + s2) implying j1(1) = p. Identifying d = 2z0 this differs from Eq. (15) by a factor of p/2 = O(1). For two counter propagating Gaussian beams (‘standing wave’) obtained from splitting a beam of intensity I1 one gains a factor of two in peak intensity but the distribution gets thinned out due to the usual cos2 modulation, which cancels the gain in intensity leading to the same correction factor j2 = p = j1. A linearly polarized electromagnetic wave undergoing vacuum birefringence with a polarization vector oriented under an angle of 45 with respect to both background fields E and B will be rendered elliptically polarized with ellipticity d (ratio of the field vectors). In the experiment, intensities will be measured  and 2 the experimental quantity to be determined is d2 ’ 12 D/ . In Table 1 expected ellipticity values for given experimental parameters are listed. These results clearly show the challenging nature but also the feasibility of the proposed experiment. Presently, a petawatt class laser facility such as POLARIS is expected to reach about 1022 W/cm2 at unprecedented repetition rates of 0.1 Hz [12]. The values of d2 obtained for such lasers (Table 1) are at the limit of the accuracy that can now be obtained with high-contrast X-ray polarimeters using multiple Bragg reflections from channel-cut perfect

T. Heinzl et al. / Optics Communications 267 (2006) 318–321 Table 1 Numerical values for the phase shift Eq. (16) and ellipticity signal d2 using the Jena laser facility POLARIS x/keV 12 15

k/nm 0.1 0.08

z0/lm 10 25

d2

D//rad 5

1.5 · 10 4.4 · 105

5.6 · 1011 4.8 · 1010

Its peak intensity is taken to be I0 = 1022 W/cm2. First line: laser-generated Thomson backscattered X-ray probe and present typical Rayleigh length. Second line: XFEL probe and large Rayleigh length.

crystals [13,22,23]. These instruments are in principle capable of a sensitivity of d2 ’ 1011 [23]. Since the expected signal is proportional to both I2 and k2 it may be greatly enhanced by increasing the laser intensity or choosing a smaller probe pulse wave length. For example, with the proposed ELI laser facility reaching 1025 W/cm2 [24] a sensitivity of the polarimeter of only 107, . . . , 104 is required which is within presently demonstrated values of sensitivity [13]. The required X-ray probe pulse may be generated either with an XFEL synchronized to a petawatt laser or by the use of Thomson scattered laser photons from monochromatic laser accelerated electron beams [15,25]. It seems worthwhile to point out that although a standing wave for the background (which may be created in the ‘‘photon collider’’ setup as shown in Fig. 1) does not lead to an increase in integrated intensity and hence of the birefringence signal, it does yield double peak intensity. This is important for the observation of effects sensitive to localized intensity like Cherenkov radiation and pair production. Acknowledgements This work was supported by the DFG Project TR 18. The authors gratefully acknowledge stimulating discus-

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