Probing vacuum birefringence by phase-contrast Fourier imaging under fields of high-intensity lasers
aa r X i v : . [ h e p - ph ] A p r myjournal manuscript No. (will be inserted by the editor) Probing vacuum birefringence by phase-contrast Fourier imaging underfields of high-intensity lasers
K. Homma , , D. Habs and T. Tajima Graduate School of Science, Hiroshima University, Kagamiyama, Higashi-Hiroshima 739-8526, Japan Fakult¨at f¨ur Physik, Ludwig-Maximilians Universit¨at M¨unchen, D-85748 Garching, GermanyReceived: date / Revised version: date
Abstract
In vacuum high-intensity lasers can causephoton-photon interaction via the process of virtual vac-uum polarization which may be measured by the phasevelocity shift of photons across intense fields. In theoptical frequency domain, the photon-photon interac-tion is polarization-mediated described by the Euler-Heisenberg effective action. This theory predicts the vac-uum birefringence or polarization dependence of the phasevelocity shift arising from nonlinear properties in quan-tum electrodynamics (QED). We suggest a method tomeasure the vacuum birefringence under intense opticallaser fields based on the absolute phase velocity shift byphase-contrast Fourier imaging. The method may servefor observing effects even beyond the QED vacuum po-larization.
To observe nonlinear responses of matter, the pump-probe technique is widely used: Matter is first excitedby an intense laser pulse and then probed by a delayedweaker laser pulse. When the vacuum is considered asa part of matter, the most natural approach to probeit is, hence, the pump-probe technique. Maxwell’s equa-tions in vacuum, however, allow only for linear superpo-sitions of laser fields. In quantum mechanics, a photoncan be resolved into a pair of virtual fermions over ashort time via the uncertainty principle in the higher fre-quency domain even below the fermion mass scale. Theloop of the virtual pair provides a coupling to photons,resulting in a photon-photon interaction. In the opticalfrequency domain, the electron-positron loop and pos-sibly the lightest quark-antiquark loop are expected togive rise to the photon-photon interaction with the massscale of the electron being 0 . c and of the lightest Send offprint requests to : Kensuke Homma
Fig. 1
Polarization directions of the target and probe lasers. quark ranging from ∼ −
100 MeV/ c , respectively. Be-low the electron mass scale, there is no known mass scalerelevant for photon-photon interactions in the standardmodel of particle physics. In this paper we focus on thephoton-photon interaction in the optical laser frequencyrange based on quantum electrodynamics (QED).In the low-frequency collision ¯ hω ≪ m e c , it is suf-ficient to describe the photon-photon interaction by theeffective one-loop Lagrangian [1,2,3] L − loop = 1360 α m e [4( F µν F µν ) + 7( F µν ˜ F µν ) ] , (1)where α = e ¯ hc is the fine structure constant, m e is theelectron mass, F µν = ∂A µ /∂x ν − ∂A ν /∂x µ is the an-tisymmetric field strength tensor and its dual tensor˜ F µν = 1 / ε µνιρ F ιρ with the Levi-Civita symbol ε µνιρ .Based on this Lagrangian, the dispersion relation forphotons in vacuum is expected to be modified by intenseelectromagnetic fields. This effect under a constant elec-tromagnetic field was first discussed by Toll [4]. At opti-cal frequencies, we may approximate the time-varying K. Homma , , D. Habs and T. Tajima electromagnetic field as a constant field, because therelevant time scale for the creation of virtual electron-positron pairs is much shorter than that of the inverseof optical frequencies. We can discuss the dispersion re-lation and the birefringent nature via measurements ofthe refractive index, i.e. , the inverse of the phase velocityas illustrated in Fig.1, where a linearly polarized probelaser beam crosses a linearly polarized target laser beam.The measurements of the phase velocity shift when theelectric fields of both lasers are either parallel or normalto each other are specified with subscriptions k or ⊥ , re-spectively. The theoretical derivation of these quantitiesin the linearly polarized electromagnetic field of the tar-get (the so-called crossed-field configuration, where theelectric field ˆ E and magnetic field ˆ B are normal at thesame strength) was originally studied in [5,6] and fur-ther derived from the generalized prescription based onthe polarization tensor, applicable to arbitrary externalfields, in [7]. This results in v k /c = 1 − α ρ t ρ c ,v ⊥ /c = 1 − α ρ t ρ c , (2)where v k /c and v ⊥ /c are the phase velocities when thecombination of linear polarizations of the probe and tar-get lasers is parallel and normal, respectively. The quan-tity ρ c ≡ m e c / ¯ h ∼ . × J/ µ m is the Comptonenergy density of an electron and ρ t is defined as z k /k where k is the wave number of the probe electromagneticfield with the unit vector of ˆ k . The Lorentz-invariantquantity z k is defined as z k = ( k α F ακ )( k β F βκ ) , (3)and the relation to the energy density ǫ in the crossedfield condition is z k k = ǫ (1 + (ˆ k · ˆ n )) , (4)with ǫ = E = cB and ˆ n = ˆ B × ˆ E with ˆ indicating theunit vector. Thus the second terms in Eq. (2) show thatthe deviation of the phase velocities of light v k and v ⊥ are proportional to the field energy density normalizedto the Compton energy density of an electron. The shiftof the refractive index from that of the normal vacuumis on the order of 10 − for the energy density ǫ of1 J/ µ m corresponding to the power density of a high-power laser beam focused to 10 W/cm at its waist.The refractive medium exhibits a polarization depen-dence, i.e. , it shows birefringence. The difference in v k and v ⊥ in Eq. (2) results from the first and second termsin the bracket of the effective one-loop Lagrangian inEq. (1).The dispersion relation and the birefringence under aconstant electromagnetic field in the UV limit ( ω → ∞ )may be evaluated via the Kramers-Kronig dispersion re-lation, as discussed in [8]. The phase velocity in both UV and IR is expected to be subluminal ( v phase < c ) underthe influence of the QED field [8,7]. The UV limit ofthe phase velocity is supposed to govern causality whichshould not exceed the velocity of light in vacuum. There-fore, it can be a fundamental test of a variety of effectivefield theories in the IR by testing whether the phase ve-locity in the UV limit, extrapolated from that of theIR, is superluminal ( v phase ( ∞ ) > c ) or not. Thus far thedispersion relation from IR to UV is theoretically knownonly in the QED field [8]. However, there is no data sofar even in the domain of IR frequencies. It is impor-tant, therefore, for experiments to quantitatively verifyor disprove the QED prediction. We note that the mea-surement of the refractive index in the domain of higherfrequencies may be sensitive to the part of the anomalousdispersion where the real part of the refractive index risesas discussed in [9], and, also, the measurement of theelectron-positron pair creation [10,11,12,13] in strongelectromagnetic fields may be directly sensitive to theabsorptive or imaginary part. The Kramers-Kronig rela-tion connects the real and imaginary parts of the forwardscattering amplitude or the refractive index. Therefore,the systematic measurements of real and imaginary partsover a wide frequency range may provide a test groundof QED and the Kramers-Kronig relation itself, when itis applied to the vacuum.The key issue is how to detect the extremely small re-fractive index change, resulting from the photon-photoninteraction between the target and probe lasers. The con-ventional way in the X-ray frequency range is based on ameasurement of the ellipticity caused by the target field-induced birefringence with respect to the linearly polar-ized incident probe photons [9,23]. Since nowadays high-precision X-ray polarimetery technique is available [14],we may reach the sensitivity to QED-induced birefrin-gence, if high-intensity lasers such as those attainable inELI [15] are provided. As explained above, the probe fre-quency dependence of the birefringence is important tocomplete the QED-induced dispersion relation. There-fore, we need measurements in the optical frequencyrange as well. The conventional ways in the range ofoptical frequencies that were performed [21] and pro-posed [22] are again based on a measurement of the el-lipsoid caused by the birefringence and a measurement ofthe rotation angle of a linearly polarized probe laser bymaking it propagate for a long distance under the influ-ence of a weak magnetic [21] or electromagnetic field [22].This method has the advantage to enhance the phaseshift by a long optical path without introducing costlystrong target electromagnetic fields. In the case of a con-stant magnetic field on the order of 1 T, one encountersthe limits of physical sensitivity to the QED nonlineareffects. In the case of an electromagnetic field, we maybe sensitive to the QED-induced birefringence within afew days with a 1J CW laser according to the claim in[22]. However, if one aims at the sensitivity even beyondQED-induced birefringence as we discuss in section 3, robing vacuum birefringence by phase-contrast Fourier imaging under fields of high-intensity lasers 3 it is essential to introduce a high-intensity pulse evenbeyond the capability of the ELI facility [15]. In suchcircumstances the storage of a high-intensity laser pulsein a cavity is limited by the damage threshold of theoptical elements needed to store the target field over along time.On the other hand, if we could localize the field-induced refractive index change by tightly focusing ahigh-intensity target laser pulse and measuring the spa-tially inhomogeneous phase effect of the vacuum on apulse-by-pulse basis, there will be no physical limit inincreasing the intensity of the laser pulse until the vac-uum itself breaks down. In order to increase the shift ofthe refractive index, corresponding to the inverse of thephase velocities in Eq. (2), i.e. , the intensity of the tar-get laser pulse as expected from Eq. (3) and Eq. (4), itis necessary to use a focused laser pulse by confining thelarge laser energy into a small space-time volume. Thiscauses a locally varying refractive index along the tra-jectory of the target laser pulse in vacuum. A variationof the refractive index arises over the high-intensity partand the remaining vacuum. If the probe laser penetratesinto both parts simultaneously, the corresponding phasecontrast should be embedded in the transverse profileof the same probe laser. Our suggestion is to directlymeasure the phase contrast and to determine the abso-lute value of the refractive index change by controllingthe combination of polarizations of the probe and tar-get laser pulses. This should result in the birefringenceas expected in Eq. (2). The birefringence measurementbased on the measurement of absolute phase velocitieswe suggest here should be contrasted to any existingtechniques to measure the ellipticity where only relativephase differences can be discussed.In the following sections we introduce the basic con-cept of the phase-contrast Fourier imaging by crossingtarget and probe lasers and discuss a way to extracta physically induced phase in the presence of a phasebackground. We then discuss also other physical con-tributions beyond QED, to which this imaging methodmay be applied. We now consider an experimental setup, where we cre-ate a high-intensity spot by focusing a laser pulse in vac-uum and probe its refractive index shift by a second laserpulse. We call the first laser pulse the target laser pulse,while from hereon the second laser pulse will be denotedas the probe laser pulse. We need to detect the extremelysmall shift of the phase velocity by the target-probe in-teraction. For this we also need an intense probe laserin order to enhance the visibility. However, if we utilizeconventional interferometer techniques, providing a ho-mogeneous phase contrast over the probe laser profile,such small refractive index changes are hard to detect.
Fig. 2
Far-field diffraction patterns from a thin-wire tar-get, when a Gaussian laser pulse is shot onto the wire. Theleft figure describes the situation without wire. The right fig-ure shows the case when the thin-wire target is horizontallyarranged.
This is because the resulting intensity modulation alwaysappears on top of a huge pedestal intensity, with an ex-tremely small contrast between the modulation and thepedestal. Any photo-detection device will not be sensi-tive to the small number of photons spatially distributedover the pedestal intensity beyond 1 J ( ∼ visi-ble photons), due to the limited dynamic range of thephoton intensity measurable by a camera pixel withoutcausing saturation of the intensity measurement. On theother hand, broadening the dynamic range by loweringthe gain of the electric amplification of photo-electronsdegrades the sensitivity to the small number of the spa-tially distributed photons or the sensitivity to the smallphase shift. Therefore, we need to invent a method thatcan spatially separate the weakly modulated character-istic intensity pattern from the strong pedestal.In order to overcome this difficulty, we suggest utiliz-ing the inhomogeneous phase-contrast Fourier imagingin the focal plane by focusing the probe laser. The phys-ically embedded phase contrast on the transverse pro-file of the probe laser amplitude is Fourier transformedonto the focal plane due to the effect of the added phaseby the lens. Actually, a parabolic mirror is necessary toavoid dispersion and damage by high-intensity irradia-tion. This will be considered later. The intensity pat-tern in the focal plane exhibits the preferable feature,that the characteristic phase boundary causes outer re-gions of the intensity profile far from the focal point toexpand, whereas a Gaussian laser beam with a homo-geneous phase converges into a small focal spot at itswaist. It is instructive to illustrate the characteristic na-ture of the diffraction pattern from a wire-like targetas shown in Fig. 2. Here the far-field pattern, knownas Fraunhofer diffraction, is shown in the case when aGaussian laser beam irradiates a thin-wire target. This K. Homma , , D. Habs and T. Tajima can be understood as the Fourier transform of the wireshape, approximated as a rectangle of 2 µ × ν . It is wellknown that a lens produces a far-field diffraction pattern,corresponding to the exact Fourier-transformed image ofthe object in the front focal plane (e.g. see [25,27]). Inorder to understand the diffraction image, we may qual-itatively refer to Babinet’s principle, which states thatthe diffraction pattern from an opaque wire plus thatof a slit of the same size and shape form an amplitudedistribution identical to that of the incident wave. There-fore, the characteristic diffraction patterns from the wireand the slit are similar, but deviate from each other suchthat they interfere to reconstruct the incident wave. Theintensity pattern after Fourier transform of such a rect-angular slit is expressed as (cid:18) sin( µω x ) µω x (cid:19) (cid:18) sin( νω y ) νω y (cid:19) , (5)where ω x = πλf x and ω y = πλf y are the spatial frequen-cies for the given position ( x, y ) in the focal plane of thelens/mirror with the focal length f at the wavelength λ , respectively. In the case of a slit with µ ≫ ν , therectangular profile in the focal plane emerges as a pat-tern of dark and bright fringes perpendicular to the slit(see Fig. 2 (right)). The narrower the slit size is, thefurther the fringes move apart. On the other hand, aGaussian beam without wire or slit remains unchanged,because the Fourier transform of a Gaussian beam re-mains a Gaussian beam (see Fig. 2 (left)). This is thekey feature that drastically improves the detectability ofsmall phase shifts by sampling only outer parts of thediffraction pattern. This may also be interpreted as thecounter-concept to the conventional spatial filter, whereouter parts are eliminated to maintain a smooth phaseon the transverse profile of the Gaussian distribution.Given the intuitive picture above, a quantitative for-mulation of our proposed method is presented as follows.In order to discuss the amount of the phase shift, weneed a distinct geometry of both the target and probelasers. Let us first consider the laser profile assumingGaussian beams. The solution of the electromagneticfield propagation along z in vacuum is well-known [27].The electric field component corresponding to the trans-verse mode l, m and e.g. polarized along y is expressedas E ( x, y, z, t ) = Re { e y ψ l,m ( x, y, z ) e iωt } with ψ l,m ( x, y, z ) = A w w ( z ) H l √ w ( z ) x ! H m √ w ( z ) y ! × exp ( − i [ kz − ( l + m + 1) η ( z )] − r w ( z ) + ik R ( z ) !) , (6)where the H l are l -th order Hermite polynomials, k =2 π/λ , r = p x + y , w is the waist, which cannot besmaller than λ due to the diffraction limit, and other definitions are as follows: w ( z ) = w (cid:18) z z R (cid:19) , (7) R = z (cid:18) z R z (cid:19) , (8) η ( z ) = tan − (cid:18) zz R (cid:19) , (9) z R ≡ πw λ . (10)In order to determine the normalization factor A , weuse the orthonormal condition of the n th Hermite poly-nomial as follows Z + ∞−∞ H n ( ξ ) e − ξ dξ = 2 n n ! √ π. (11)With the replacement ξ = √ w x we obtain Z + ∞−∞ H l √ w x ! e − x w dx Z + ∞−∞ H m √ w y ! e − y w dy = 12 w π l + m l ! m ! . (12)At z = 0 we can then relate the amplitude A with thebeam power P by P = A Z + ∞−∞ Z + ∞−∞ ψ l,m ( x, y ) ψ ∗ l,m ( x, y ) dxdy (13)yielding A ≡ P πσ l + m l ! m ! , (14)where w is replaced by 2 σ and A is the on-axis intensityat the waist.Figure 3 illustrates the conceptual experimentalsetup for the phase-contrast Fourier imaging. In whatfollows, subscripts p and t always denote the probe and target quantities, respectively. Both the target and probelaser beams are focused with different waist sizes w t and w p with the incident beam diameters d t and d p , respec-tively. Both laser beams cross each other at their waists,where wave fronts are close to flat with R = ∞ in Eq. (8).We assume that the target waist w t is smaller than theprobe waist w p , which embeds the phase contrast δ at z within the amplitude on the transverse profile of theprobe laser. The probe laser then propagates to a lens offocal length f p and the inverse Fourier imaging is per-formed in the back focal plane of that lens.We then define the geometry of the laser intersec-tion. Figure 4 illustrates geometrical relations where thetightly focused target pulse with time duration τ t , beamwaist diameter 2 w t , and Rayleigh length z Rt propagatesalong the Z -axis, and the probe pulse with the larger robing vacuum birefringence by phase-contrast Fourier imaging under fields of high-intensity lasers 5 Fig. 3
Conceptual experimental setup for the suggestedphase-contrast Fourier imaging. At the crossing point be-tween the probe and target laser beams, the target pulsecauses a shift in the index of refraction, and a correspondingphase shift δ embedded into the probe pulse as explained inFig. 4. beam waist diameter 2 w p and longer time duration τ p propagates along the z -axis tilted by θ with respect tothe Z -axis. In Fig.4 a) and b) the pulses are assumed tohave a rectangular intensity profile along the propaga-tion direction. The dashed rectangular pulses occupy thepositions at time t = 0 and the solid ones are those at t = t . The probe and target pulses overlap each otherat the position marked by ∗ . At this moment in time, t is taken as zero. We need to express δl to estimate thepass length where an additional phase is embedded. Thepath length δl is defined as the distance where the frontof the probe pulse meets the edges of the target laserat t = t , beyond which the target laser is no longerpresent. In Fig.4 a) because A , B , and δl form a righttriangle, we hence obtain the following relation( δl ) = A + B , (15)where δl = ct , A = c ( τ t − t ) and B = tan( π − θ ) c ( τ t − t )with the velocity of light c , resulting in t = τ t π − θ ) . (16)Depending on the relation between the target beamwaist 2 w t and the pulse length cτ t , namely, whethera) A ≤ w t or b) where the probe wavefront meets theside of the target laser before reaching the tail as shownin Fig.4, the optical path length δl is expressed asa) δl = ct = cτ t − cos θ for cτ t < w t − cos θ sin θ , (17)b) δl = 2 w t sin θ for cτ t ≥ w t − cos θ sin θ , (18) Fig. 4
Geometry of the embedded phase contrast in theprobe pulse. where the equations should not be applied to the cases θ = 0 or ϑ = π . In the case a) the path lengths withphase shift are not constant below or above the starpoint along the wavefront of the probe laser. In the caseb) the path lengths are constant over the part of theprobe wavefront which penetrates both sides of the tar-get laser. The residual part along the wavefront, how-ever, meets the head or tail of the target laser pulse andcauses deviations from the constant path length. An ex-actly equal path length over the probe wavefront duringthe propagation time of the target laser ∼ z Rt /c is re-alized only in the case of θ = π/
2. In this case, after thepenetration of the probe laser pulse, the profile of theprobe laser in the x − y plane contains a trajectory witha constant phase shift δ along the projection of the pathof the target laser on the probe wavefront, as shown inFig. 4 c).We express the phase shift δ in the vicinity of thewaist | z | = cτ t / ≤ z Rt , where we assume that the wave-front is flat as indicated by Eq. (7) and Eq. (8) δ = 2 πλ p δnδlϕ t ( x p , y p ) , (19)where δn is the refractive index shift, δl is the pathlength with an effectively constant phase shift over thecrossing time and ϕ t ( x p , y p ) is a weighting function toreflect the path length difference depending on the inci-dent position with respect to the target profile expressedas a function of the position ( x p , y p ) in the transverseplane of the probe laser. If we limit the origin of thelaser-induced refractive index change to QED, based onEq. (2), (3), and (4), we parametrize the refractive indexshift as δn qed = ζN (1 − cos θ ) E t πw t cτ t , (20)where ζ is 4 or 7 for the polarization combinations k or ⊥ , respectively. N is the coefficient to convert from K. Homma , , D. Habs and T. Tajima energy density to the refractive index shift defined as N ≡ α ¯ h m e c = 1 . × − [ µ m / J]. The incident angle θ varies from 0 to π which is measured from the propa-gation direction of the target pulse to that of the probepulse as depicted in Fig. 4. E t is the energy of the targetpulse given in [J], and πw t cτ t is the volume in [ µ m ] forthe given target profile with the waist w t from Eq. (7).By respecting the constant path length over ∼ z Rt /c for simplicity, we consider only the case of (18) with θ = π/
2. By substituting Eq. (20) and (18) into Eq. (19),we obtain the simplest expression for δ qed δ qed ∼ ζN E t λ p w t cτ t , (21)where cτ t ≥ w t must be satisfied from Eq.(18) andwe take the approximation ϕ t ( x p , y p ) ∼ ϕ t ( x p , y p ) based on the precise profile ofthe target laser reflecting actual experimental setups).In this limit we approximate the target profile as a rect-angular of the size 2 µ × ν , inside which the phase shiftis assigned to be constant. The effective slit sizes are de-fined by the transverse sizes of the focused laser beamsthrough the relation µ ∼ z Rt and ν ∼ w t . (22)We then explicitly define the window functions rec and rec as rec ( µ, ν ) = (cid:26) | x | ≤ µ and | y | ≤ ν | x | > µ or | y | > ν (cid:27) ,rec ( µ, ν ) = (cid:26) | x | ≤ µ and | y | ≤ ν | x | > µ or | y | > ν (cid:27) . (23)This window provides a unit region of a constant phase,which may be applied even to arbitrary phase maps com-posed of a collection of the unit window cells.We now discuss how the probe laser including thephase δ embedded at the focal plane propagates intothe image plane via the lens system and evaluate theexpected intensity distribution in the image plane. Sinceour discussion is based on local phases with rectangularshape and both Fourier and inverse Fourier transformsof a rectangular function give identical sinc functions, werepresent the lens effect as Fourier transform. For eachpropagation from the object plane ( x , y ) at z to theimage plane ( x, y ) at z (see Fig.3), we always take theFresnel diffraction. Based on (6) and (13), the probe fieldprofile in the plane where the phase δ is embedded canbe defined as T ( x , y ) = A p H l (cid:18) x √ σ (cid:19) H m (cid:18) y √ σ (cid:19) e − x y σ )2 (24)where A p ≡ q I πσ l + m l ! m ! is the on-axis waist ampli-tude of the probe laser. The linearly synthesized ampli-tude at z is then expressed as Ψ ( x , y , z ) = α ( z ) rec ( µ, ν ) T ( x , y ) + β ( z ) rec ( µ, ν ) T ( x , y ) , (25)where α ( z ) and β ( z ) are propagation factors of the probewaves at the point z after probe-target crossing. Thefunctions α containing the phase shift δ caused by thelocal refractive index shift and β are defined as α ( z ) = e i ( kz + δ ) ,β ( z ) = e ikz . (26)The Fourier transform F of the synthesized amplitude Ψ in the image plane ( x, y ) at z after the lens [28] isexpressed as F { Ψ ( x , y ) } = α ( z ) F { rec ( µ, ν ) T ( x , y ) } + β ( z ) F { rec ( µ, ν ) T ( x , y ) } = ( α ( z ) − β ( z )) Z µ − µ Z ν − ν dx dy T ( x , y ) e − i ( ω x x + ω y y ) + β ( z ) Z ∞−∞ Z ∞−∞ dx dy T ( x , y ) e − i ( ω x x + ω y y ) , (27)where we define ( ω x , ω y ) ≡ ( πf p λ p x, πf p λ p y ) at z . We in-troduce the coefficient C s for the first term in Eq. (27),containing the information on how much the phase shift,representing the signal , is localized, resulting in thephoton-photon interaction. We decompose C s into itsreal and imaginary parts, because Hermite polynomialscontain even and odd functions and the non-zero valuesof these integrals appear even in the imaginary part. Wedenote them asRe C s ( ω x , ω y ) ≡ C sR = Z µ − µ dx Z ν − ν dy { cos( ω x x ) cos( ω y y ) − sin( ω x x ) sin( ω y y ) } T ( x , y )andIm C s ( ω x , ω y ) ≡ C sI = − Z µ − µ dx Z ν − ν dy { cos( ω x x ) sin( ω y y ) +sin( ω x x ) cos( ω y y ) } T ( x , y ) . (28)We also define the coefficient C b for the second term ofEq. (27), which corresponds to the background pedestalas C b ( ω x , ω y ) ≡ ( − i ) l + m (2 σ ) πH l ( √ σω x ) H m ( √ σω y ) e − ω x + ω y (2 σ )2 , (29)where the fact is used that the n th -order Hermite func-tion is the eigen-function of the Fourier transform,namely, (2 π ) − / F { H n ( ξ ) e − ξ / } = ( − i ) n H n ( ω ) e − ω / .We note that the coefficient (2 π ) − / arises due to ourdefinition of the Fourier transform with the prefactor of robing vacuum birefringence by phase-contrast Fourier imaging under fields of high-intensity lasers 7 unity applied to the lens system. The Fourier transformof the amplitude is then expressed as F { Ψ } = ( α − β )( C sR + iC sI ) + β ( C bR + iC bI ) . (30)When no confusions are expected, we will omit( x , y , z ) and ( ω x , ω y , z ) for all relations below (30).By substituting Eq. (26), (28) and (29) into Eq. (30),the intensity pattern at the image plane is expressed as | F { Ψ }| = ψ l,m ψ ∗ l,m = (cid:18) A p f p λ p (cid:19) ×{ − cos δ ) ( C s C ∗ s − ( C sR C bR + C sI C bI )) − δ ( C sI C bR − C sR C bI ) + C b C ∗ b } , (31)where ( f p λ p ) − arises from the Fresnel diffraction. Ac-cording to (29), when l + m is even or odd, C bI or C bR becomes zero, respectively.Equation (31) indicates that this method works asan interferometer via the cross terms with coefficients1 − cos δ and sin δ . The second term with sin δ vanishesfor any combinations of l and m , as long as the symmet-ric rectangular ranges around ( x , y ) = 0 are assumedin the definitions of Eq.(28). This interferometer differsfrom a conventional one in that the modulating part dueto the phase shift δ can be spatially separated from theconfined strong part C b C ∗ b , due to the characteristic pat-tern of C s C ∗ s which creates images in the region of higherspatial frequencies. In Eq.(31) the third term C b C ∗ b cor-responds to the intense pedestal pattern insensitive tothe phase δ which keeps the same shape as that at z with a different transverse scale based on Eq.(29), be-cause Hermite functions are the eigen-functions of theFourier transform. The second term shows proportion-ality to δ for δ ≪
1; however, the intensity pattern isconstrained by C b . This implies the phase informationis attainable only in the vicinity of the background pat-tern C b C ∗ b , though the signal is strong. Therefore, thesignal-to-pedestal ratio is not expected to be large. Onthe other hand, the first term indicates proportionalityto δ for δ ≪ C s C ∗ s term is not affected by C b and it producesa pattern characterized by spatial frequencies. Hence thesignal-to-pedestal ratio is expected to be drastically im-proved circumventing the most intense spot. Therefore,depending on the value of δ and the allowed dynamicrange of the photo-detection device used, we have choiceson which term we focus.We consider a Gaussian beam with l = m = 0. There-fore, we do not expect the terms proportional to phase δ due to the imaginary parts originating from Hermitepolynomials with odd orders. Nevertheless, if we needto stick to the proportionality to δ , we may add a localoffset phase π/ Fig. 5
Conceptual experimental setup for the suggestedphase-contrast Fourier imaging. At the crossing point be-tween the probe and target lasers, the target laser causesa shift in the index of refraction, which amounts to the re-fractive phase shift δ vac embedded into the probe laser asexplained in Fig. 4. The offset phase is embedded by a holo-graphic phase plate in advance which provides the offsetphase + π/ rec ( µ , ν ) at z via Fourier transform by the first lens. − cos( δ + π/
2) in Eq. (31) becomes ∼ δ . It is im-portant to be able to discuss whether the phase shift isincreased or decreased, since it directly reflects the dy-namics of the local interaction. From a technical pointof view, more importantly, this has a definite advantageof enhancing the signal due to the proportionality to δ compared to the δ sensitivity in 1 − cos δ in case of anextremely small δ . However, in turn, one must accept thesituation that the local offset phase contrast produces anintrinsic diffraction pattern as a new kind of pedestal,which now has an equal diffraction pattern comparedto the one caused by the photon-photon interaction.Thanks to the proportionality to δ , we can reduce theintensity of the probe laser pulse. On the other hand, thenew pedestal pattern would occupy the dynamic rangeof the camera device. In order to reduce the amount ofthe pedestal intensity, we may add more intelligent char-acteristic offset patterns by mixing δ ≡ δ vac ± π/ δ vac and the offsetphase ± π/ δ vac ± π/ rec ( µ , ν ) with µ = N µ and ν = N ν and N ≥ ± π/ K. Homma , , D. Habs and T. Tajima Fig. 6
An example how to implement the offset phase by aholographic device.
For this region we introduce the coefficient C ( ω x , ω y )by replacing ( µ, ν ) with ( µ , ν ) in Eq.(28) as well. Theintensity profile in the focal plane is then re-expressedas | F { Ψ }| = ψ l,m ψ ∗ l,m ∼ (cid:18) A p f p λ p (cid:19) ×{ C − C b )( C ± δ vac C s ) + C b C ∗ b } , (32)where ± are cases when the offset phase ± π/ rec ( µ , ν ) and rec ( µ , ν ), respectively. We note thatthis relation is applicable to both real and imaginary co-efficients as long as either all real or all imaginary coeffi-cients are simultaneously zero. Actually we can confirmthat Eq.(31) becomes identical with Eq.(32) under thiscondition, when N = 1 and δ vac = 0, namely, C s = C and δ = ± π/ π/ rec ( µ , ν ) at z via Fourier transform bythe first lens in Fig.5. If such a long distance is not avail-able, we may use a holographic device as illustrated inFig.6 where the phase plate is placed at the focal planeand a laser produces a proper Fourier image which isstored in the holographic device by mixing with a refer-ence laser. If we replace the reference laser by the probelaser in Fig.5, we can produce the Fourier image in frontof the first lens in Fig.5. In a practical case as shown inFig.9, the holographic device may be located before thefocal point when it records the local phases in advance,in order to supply an offset distance before the propersinc distribution is reconstructed by probe laser pulsesat the exact point where we need it. We performed numerical calculations with the rectangu-lar offset phase + π/ l = m = 0 (TEM ). The param-eters used for Fig. 7 are summarized in Tab. 1, wherethe parameters of the target and probe lasers, the em-bedded offset and the physical phase shifts due to thenonlinear QED effect used to obtain the Fourier trans-formed intensity distributions are specified. Figure 7 top-left illustrates the intensity pattern due to Eq.(32) as afunction ( x, y ) in the image plane when δ vac = 0 andthe offset phase + π/ x and y -axes. Figure 7 top-right shows theexpected number of pedestal photons, N ped integratedover a 1cm x 1cm cell along the y-axis at x = 0. Fig-ure 7 bottom-left shows N sig − N ped per 1cm x 1cm cellalong the y-axis at x = 0, where N sig is the integratednumber of photons per 1cm x 1cm cell for signal, namely,with δ = δ vac + π/
2. In the actual experimental setupthe subtraction should be performed on the shot-by-shotbasis as illustrated in Fig.9 where a probe laser pulse isequally split into the signal path with the target laserpulse and the calibration path without it so that we cancompare the two cases. The solid-red, dashed-blue anddotted-green histograms show the case when the probewavelengths of 800nm, 840nm, and 760nm are assumed,respectively, and the yellow band shows the statisticalfluctuations p N ped due to the quantum efficiency of thephoton detection. Figure 7 bottom-right shows the sta-tistical significance of the signal photons with respectto the statistical fluctuations of the pedestal photons;( N sig − N ped ) / p N ped per 1cm x 1cm cell along the y-axis at x = 0. The colors have the same meaning as thosein Fig. 7 bottom-left .Figure 7 bottom-right indicates that we can expectseveral cells in which the number of signal photons is ei-ther increased or decreased by more than two standarddeviations from the pedestal fluctuations. If we couldcount the number of photons per 1cm x 1cm cell in theside band around the pedestal peak without detector sat-uration, we can measure the phase velocity shift due tothe QED effect even by one probe-target laser crossing.This side band structure appears owing to interferencebetween C ( ω x , ω y ) and C s ( ω x , ω y ) in Eq.(32).Although the spectral width of the probe laser showsfaint effects as shown in Fig.7 bottom-left and bottom-right , the characteristic pattern along the y-axis is simi-lar. As long as the wavelength distribution can be mea-sured at the same time, we can reconstruct δ vac based onthe measured wavelength distribution and the intensitypattern along the y-axis.The most difficult issue is the dynamic range of exist-ing cameras used in research which typically have 16-bitresolution and at most 28-bit per pixel. In order to solve robing vacuum birefringence by phase-contrast Fourier imaging under fields of high-intensity lasers 9 x[cm]-20 -15 -10 -5 0 5 10 15 20 y [ c m ] -20-15-10-505101520 110 π Pedestal distribution with phase offset + y[cm]-20 -15 -10 -5 0 5 10 15 20 The number of pedestal photons per 1cm x 1cm cell along y-axis y[cm]-20 -15 -10 -5 0 5 10 15 20-800-600-400-2000200400600800 × per 1cm x 1cm cell pedestal -N signal N y[cm]-20 -15 -10 -5 0 5 10 15 20-4-3-2-101234 per 1cm x 1cm cell pedestal N) / pedestal -N signal (N Fig. 7
Simulated distributions of ψ , ψ ∗ , (TEM ) in image plane z based on Eq.(32) with parameters given in Tab. 1. top-left : Patterns of Eq.(32) with the offset phase + π/ top-right : The number of pedestal photons, N ped integratedover a 1cm x 1cm cell along the y-axis at x = 0 of the top-left distribution. bottom-left : N sig − N ped per 1cm x 1cm cellalong the y-axis at x = 0, where N sig is the integrated number of photons per 1cm x 1cm cell with δ = δ vac + π/
2. Thesolid-red, dashed-blue, and dotted-green histograms in Fig.7 show the case when probe wavelengths of 800nm, 840nm, and760nm are assumed, respectively, and the yellow band shows statistical fluctuations, p N ped , due to the quantum efficiencyof the photon detector. bottom-right : The statistical significance of signal photons with respect to the statistical fluctuationsof the background photons; ( N sig − N ped ) / p N ped per 1cm x 1cm cell along the y-axis at x = 0. The colors have the samemeaning as those in the bottom-left chart.0 K. Homma , , D. Habs and T. Tajima Target laser parameters Probe laser parameters τ t = 15 fs τ p = 2 z Rt /c = 24 fs E t = 250J E p = 25J λ t = 800 ±
40 nm λ p = 800 ±
40 nm d t = 39 . d p = 7 . f t = 75 cm f p = 25 cm w t ∼ f t λ t πd t = 0 . µ m w p ∼ f p λ p πd p = 1 . µ m z Rt = πw t /λ t = 3 . µ m z Rp = πw p /λ p = 18 . µ m Embedded physical phase by assuming only QED effect δ qed = 1 . × − from Eq.(21) with ζ = 4 and θ = π/ Shape of physical and offset phases µ = z Rt and ν = w t µ = 5 × µ and ν = 5 × ν with offset phase + π/ Table 1
Laser parameters used to produce Fig. 7 based on the conceptual experimental setup shown Figs. 4 and 5. Thesubscripts t and p refer to the target and probe lasers, respectively. This choice of parameters is explained in the text insections 2 and 3. the limited dynamic range, let us suppose that we sam-ple photons per 1cm x 1cm cell by ∼ pixels. In sucha case the number of photons per pixel is ∼ with re-spect to ∼ photons at around 5cm from the pedestalpeak (see Fig.7 top-right ). Even if we use 10-bit reso-lution, the number of photons per resolution becomes10 / ∼ photons. Compared to the N sig − N ped of ∼ at around 5cm from the pedestal peak (seeFig.7 bottom-left ), the sensitivity of 10 photons per res-olution is sufficient to observe the intensity modulationsbeyond two standard deviations from the pedestal fluc-tuations without intensity saturation (see Fig.7 bottom-right ). This suggests that in principle it is possible todetect the laser-induced QED effect by a single shot onlyif the conditions listed in Tab.1 are realized. Therefore,by assigning camera devices for individual 1cm x 1cmcell with ∼ pixel readout, we can overcome the lim-ited dynamic range of cameras even with the currentlyexisting technology.In order to study the laser-induced vacuum birefrin-gence, we inject a linearly polarized probe pulse whoseelectric field vector is turned by 45 deg with respect tothat of the target pulse so that its electric field along the x and y axes are equal. We then put two polarization fil-ters at the image plane symmetrically with respect to y = 0 as illustrated in Fig.9 to cover the regions + y and − y along the y -axis, respectively, which select orthog-onal polarizations at the image plane. The asymmetrybetween the number of modulated photons from that ofthe pedestal pattern between the regions ± y providesdirect information of the birefringence on the pulse-by-pulse basis.We note that this method bears similarity to thatin [24], where two intense target laser pulses are treatedas a matterless double slit and the interference betweenspherical waves from these slits is discussed as a signa-ture of the photon-photon interaction. In [24] the oc-currence of diffraction is caused by the laser-laser inter- action itself. In our method the target laser causes therefractive phase shift experienced by the probe laser, asindicated in Fig. 5. This phase shift is embedded in arefracted, nearly-plane wave in the forward direction ofthe probe laser, as explicitly formulated in Eq. (25) andEq. (26). We then set a lens to the right of the inter-action between the target and probe lasers as shownin Fig. 5. The diffraction or Fourier transform in ourmethod is incurred by the added phase of the lens andthe spherical wave propagation from the lens to the fo-cal plane. The advantage of our method is an enhancedsensitivity to a small phase shift on the pulse-by-pulsebasis, as it is demonstrated due to a more efficient col-lection of photons by the lens using the much simplertarget geometry. On the other hand, the disadvantageis the deviation from the ideal phases included in thepath of the probe laser except the laser-induced vacuumphase. The ways to correct for this kind of backgroundphase aberrations and the other background source forthe phase-contrast Fourier imaging will be discussed inthe following subsections. In actual experiments it is unavoidable that the probepulse includes local phase fluctuations on a pulse-by-pulse basis even in the absence of the laser-induced sig-nal δ as illustrated in Fig.8. The figure corresponds tothe case when the phase contrast δ in Fig.5 c) is embed-ded in the presence of background phase fluctuations φ i ≡ φ ( X ) as a function of the position X ≡ ( x , y ) atthe object plane where i denotes a corresponding regionwith the constant phase φ i in the transverse plane of theprobe pulse. Compared to δ , the φ i ’s are expected to bemuch larger. However, if the values of the local phase set { φ i } on each probe pulse is suppressed below the offsetphase π/ a priori measured, we are inprinciple able to correct for the effect of the background robing vacuum birefringence by phase-contrast Fourier imaging under fields of high-intensity lasers 11 Fig. 8
The phase δ induced by the target laser pulse in thepresence of local phase fluctuations in the transverse planeof the probe laser pulse at the object plane where the targetand the probe pulses cross each other. fluctuations. In the next subsection 3.2 we discuss howto measure the phase set on a pulse-by-pulse basis indetail. In this subsection, however, we focus on how todetermine δ on a pulse-by-pulse basis, if the measuredphase set is given in advance.Let us extend the expressions from Eq. (23) through(27). In general, the integration limits defined byEq. (23) and used in the first equation of (27) can takeany shape and size. We replace the rectangular region rec with the region R i ≡ R ( X ), where a constant phase ismapped within R i . By denoting the spatial frequency as W = ( ω x , ω y ) = (2 πx/ ( f p λ p ) , πy/ ( f p λ p )), for the po-sition ( x, y ) at the image plane with the integral kernel f ( W, X ) ≡ T ( x , y ) e − i ( ω x x + ω y y ) , the Fourier trans-form including the local phase fluctuations φ i is ex-pressed as Ψ ( W ; φ ) = F { Ψ ( X ; φ ) } = N X X i { α ( φ i ) − β } Z R i dXf ( X, W )= N X X i { α ( φ i ) − β }F i ( W ) + β F ∞ ( W ) , (33)where N X is the number of regions in the transverseplane at z , α ( φ i ) = e i ( kz + φ i ) , β = e ikz , F i ( W ) = R R i dXf ( X, W ), and F ∞ ( W ) = R ∞−∞ dXf ( X, W ). Wenote that this expression corresponds to the regional cutand paste on T ( x , y ); i.e. , cutting a region with a phasedetermined from β at z and paste the same region byadding φ i in α ( φ i ).Given φ i on a pulse-by-pulse basis, we can numeri-cally calculate the real and imaginary parts of Ψ ( W ; φ ).The estimated background intensity pattern I bg ( φ ) inthe image plane with the phase fluctuations φ without the laser-induced phase is given by I bg ( W ; φ ) = { Re Ψ ( W ; φ ) } + { Im Ψ ( W ; φ ) } . (34)We now include as well a template of the laser-induced phase by the target laser pulse δ ≡ δ ( X ). Thephase shift δ can be evaluated from the geometry of theenergy density profile of the target laser pulse. Based onEq.(19) we parametrize δ as δ = κϕ t ( X ) , (35)where κ is a constant parameter that considers the abso-lute value of the phase shift induced by the target laser.The profile can be a priori determined by the experi-mental design of the focal spot. We can monitor if thecenter of the spot is in fact stable and further correct forits deviation from the fixed geometry of the target laser.Given δ , we only have to replace the phase by φ i → φ i + δ with a constant parameter κ as follows I bg + sig ( W ; φ + κϕ t ) = { Re Ψ ( W ; φ + κϕ t ) } + { Im Ψ ( W ; φ + κϕ t ) } , (36)where bg + sig refers to the fact that the laser-inducedphase is embedded in the background phase fluctuations.Given the measured intensity pattern I meas in theimage plane per probe pulse, we define χ with Eq. (36)as a function of κχ ( κ ) ≡ N W − N W X j | I meas ( W j ) − I bg + sig ( W j ; φ + κϕ t ) | I meas ( W j ) + I bg + sig ( W j ; φ + κϕ t ) , (37)where N W is the number of sampling points in the im-age plane and j runs over all regions in this plane. Theparameter κ can be determined by minimizing χ on apulse-by-pulse basis within the required accuracy. Figure 9 illustrates a schematic view of the entire systemfor the phase-contrast Fourier imaging including parts tocorrect all phase aberrations in the system. The targetlaser pulse moves perpendicular to the drawing plane.Its focus or waist lies in this plane. The signal path(SP) consists of the inverse Fourier transform part asdiscussed in Fig.5 and an array of mega-pixel camerasensors at the end to sample the intensity profile byindividual 1cm x 1cm cells as discussed with Fig.7. Inthe SP the probe laser pulses are injected with the po-larization tilted by 45 deg with respect to that of thetarget laser pulses. In front of the sensors, two polar-izers (P) selecting photons with the orthogonal combi-nation of polarizations so that the birefringence can bemeasured in a shot, which allows the statistical integra-tion of the measurement over many shots by minimiz-ing the systematic error due to shot-by-shot fluctuations , , D. Habs and T. Tajima of the probe pulse energy. After the implementation ofthe holographic plate (HP) which produces the offsetphase contrast in the laser interaction zone, we intro-duce a beam splitter (BS2) followed by the identical im-age transferring system as that in the SP. We refer to thisleg as the calibration path (CP). We classify the originsof local phase fluctuations into the static component bythe optical elements in the paths and the pulse-by-pulsecomponent such as wavefront fluctuations included inthe probe pulse coming from the upstream laser system.Since the repetition rate of the target laser is limited,we may inject a single-mode CW laser with the samewavelength as the dominant part of the probe pulse spec-trum into both the SP and the CP, while the target laserpulses are not injected (Wavefront aberrations resultingfrom reflection by BS mode at a single longitu-dinal mode, we expect to be able to accurately deter-mine the static phase component as the average valueby using huge photon statistics accumulated over a longtime period for an experiment while probe pulses are notinjected. For the pulse-by-pulse component we use theintensity profile observed at the end of the CP to recon-struct a set of local phases caused by wavefront fluctu-ations included in the probe pulse on the pulse-by-pulsebasis.The measurable four types of local phase sets aredenoted as φ CWSP , φ CWCP , φ P LSSP , and φ P LSCP where super-scripts specify cases of CW and pulse laser injections,respectively, and subscripts refer to the different pathsthe beams take. In the following, all phase sets are in-terpreted as those defined on the focal plane, even if thelocal phases are actually embedded in different prop-agation points. The two phase sets in the SP are ex-pressed by phases ϕ ’s with subscripts corresponding tothe names of the optical elements along the path in Fig.9as follows: φ CWSP = ϕ HP + ϕ BS + ϕ P M SP + ϕ P M SP + ϕ P SP , (38)and φ P LSSP = ϕ P LS + φ CWSP , (39)where BS ϕ P LS is the pure phase set caused by only the pulse-by-pulse component which is not correctable by the CWlaser. The two phase sets in the CP are expressed as well φ CWCP = ϕ HP + ϕ BS + ϕ P M CP + ϕ P M CP + ϕ P CP , (40)and φ P LSCP = ϕ P LS + φ CWCP . (41)Combining Eq.(39) and (41), we can restore the offsetphase for the probe pulse injection in the SP, φ P LSSP bythe other measured sets of phases as φ P LSSP = φ P LSCP − φ CWCP + φ CWSP . (42) This implies that φ P LSSP can be restored by other measur-able quantities, which is a necessary condition to allowthe correction within the same probe pulse injection inthe SP in the presence of the laser-induced vacuum phaseshift. We finally describe the entire phase set in the focalplane in the SP when a target laser pulse exists as φ = φ P LSSP + δ vac , (43)where δ vac = κϕ t as parametrized in Eq.(35). By sub-stituting Eq.(43) into Eq.(37), we can, in principle, de-termine κ for the physical template based on the targetlaser profile ϕ t .The template analysis discussed in Sect. 3.1 can alsobe applied to determine the individual set of phases inthe right hand side of Eq.(42). By assigning a squareshape to the region R i in Eq.(33), representing a cell in-stead of physical template ϕ t in Eq. (37), we estimate κ i for each R i ( X ). The number of photons at a point W i inthe image plane contains the convoluted phase informa-tion of the amplitude from all points in the transverseplane of the probe N X as seen from Eq. (33). Therefore,as long as the number of sampling points in the imageplane N W is larger than that in the transverse probeprofile N X , we can, in principle, determine a phase setfrom Eq. (37) by scanning κ i over the expected dynamicrange of the phase variation. The achievable resolutionof the phase reconstruction depends on the scanning stepon κ i in the χ -test. As discussed with Fig.7 the phase-contrast Fourier imaging achieves at least the sensitiv-ity of ∼ − for the physical phase shift by samplingthe side band of the intensity distribution on the imageplane. Therefore, we can introduce the same resolutionstep to determine κ i . We may measure the initial coarsephase sets a priori by a commercially available wavefrontsensor. From the phase measurement we can extract theset of phases at the focal plane by performing Fouriertransform from the image plane back to the focal plane.Starting from this initial phase set at the focal plane,we perform the χ -test to determine κ i more accuratelyby comparing the computed Fourier image to the mea-sured intensity at the image plane. If the resolution ofcommercially available wavefront sensors is limited to ∼ λ/ times for scanning κ i , in orderto reach the same phase resolution as ∼ − . Accord-ingly, a proper computing power is necessary to restorethe sets of the offset phases in Eq.(42) on the pulse-by-pulse basis. A background source of the current measurement is therefractive index shift due to the plasma creation from theresidual gas along the path of the focused target laserpulse. The refractive index of the static plasma in the robing vacuum birefringence by phase-contrast Fourier imaging under fields of high-intensity lasers 13
Fig. 9
Setup to correct local phase fluctuations. limit of negligible collisions between charged particles isexpressed as N = s − ω p γω , (44)where ω is the angular frequency of the target laser, ω p is the plasma angular frequency defined as p πe n e /m e and γ is the relativistic Lorentz factor given as p a with a = 0 . × − λ [ µm ] p I [ W/cm ]. In the low-pressure limit of the residual gas, the amount of refrac-tive index shift ∆N ≡ N − ω p / γω .Although the refractive index in the plasma becomessmaller than that of the peripheral area with neutralatoms, the inverted phase contrast of the phase shift in-side the probe pulse still maintains a rectangular shapealong the trajectory of the target laser. Therefore, itshould produce the characteristic diffraction pattern atsimilar locations to the nonlinear QED case as expectedfrom the Babinet’s principle, which requires that thediffraction pattern from an opaque slit plus the invertedslit of the same size and shape form an amplitude dis-tribution identical to that of the incident wave as wediscussed in section 2. In order to reduce this effect, weneed to reduce the electron density n e in the residual gas.If we take γ ∼ ∆N estimate,the refractive index shift ∼ − due to the nonlinearQED effect for a reference energy density ∼ µ m , cor-responding to a residual gas pressure of ∼ − Pa. Thecollisional frequency due to interactions between elec-trons and ions is expected to be 10 − s − at the crit-ical electron density n cr [ cm − ] = 1 . × /λ [ µm ],where ω p equals ω . For a duration time of ∼ fs of thetarget laser pulse, the inverse bremsstrahlung radiationdue to collisional processes in the residual gas is negligi-ble at ∼ − Pa.Plasma formation is also caused by the probe pulsealong its waist over a distance of ∼ ∼ − Pa the associated plasma induced phase shift is one order of magnitude smaller than that due to QED.Hence the pressure of the residual gas in the interactionchamber has to be kept at this level.We note that the actual processes will be more dy-namical, due to the pondermotive force executed by thehigh-intensity laser field. In such a case the refractiveindex shift based on static plasma gives only the upperbound on the amount of the local refractive and phaseshift.
In the previous sections we discussed the design ofthe phase-contrast Fourier imaging by aiming at prob-ing the vacuum birefringence through the QED effect,namely, the electron-positron loop to which photons cou-ple. However, if the quark mass in vacuum is of the sameorder as the electron mass, we should expect that quarksalso contribute to the vacuum birefringence by replac-ing the electron-positron loop with the quark-antiquarkloop. Whether this effect has a sizable contribution ornot is, however, difficult to quantify with presently ex-isting field theoretical approaches, because of the strongcoupling of quantum-chromodynamics (QCD) in vac-uum, where the coupling is too large to allow for a per-turbative treatment. Moreover, bare quark masses notconfined in hadrons are not precisely known. In addi-tion to calculations based on the QCD field theory [16,17], there is another possibility that the duality betweenstring theory with higher dimensions and field theoryin 3+1 dimensions (holography) [18,19] could be di-rectly applicable to this birefringence problem [20]. TheQCD and holographic approaches may give different pre-dictions for the balance of coefficients between the twoterms of the Euler-Heisenberg Lagrangian. Therefore, wemay be able to pin down such theoretical issues by ac-cumulating statistics more than a single shot and alsoexpecting further increase of the laser intensity in thefuture.Moreover, we note that because the photon-photonscattering cross section of QED interaction in the per-turbative regime is so small, 10 − b at optical frequency(see [29,30]), we experience little ’noise’, providing apristine experimental environment to search for some-thing beyond QED. Suppose then the detected disper-sion and birefringence quantitatively deviate from theexpectation of QED, including potential QCD correc-tions. This should indicate that undiscovered fields maybe mediating photons beyond QED and QCD. Scalarand pseudoscalar types of fields in vacuum may con-tribute via the first and second products in the bracketsof Eq. (1), respectively. They may be candidates of colddark matter, if the coupling to photons and the massare reasonably small [31]. Therefore, the measurementof the absolute value of the phase shift depending on thepolarization combinations and the comparison to the ex-pectations from nonlinear QED including potential QCD , , D. Habs and T. Tajima corrections may be a general test of unknown nature invacuum. We suggest an approach to probe the vacuum birefrin-gence under the influence of intense lasers. The phase-contrast Fourier imaging technique can provide a sensi-tive method to measure the absolute phase shift of lightcrossing intense laser fields. With this method nonlinearQED effects of the Euler-Heisenberg Lagrangian may bedetected requiring no more than lasers of the hundredPW-class. The method provides a window for scopingthe vacuum via the dynamics of the electron mass scaleand possibly the lightest quark mass. Such a detectionhas never been made to date, and it heralds the researchin the physics of the vacuum with a high-field approach.Given the high-intense optical lasers available in the ELIproject [15] in the near future, the realization of thissuggestion may become an exciting challenge for futureexperiments exploring vacuum physics.
Acknowledgment
This research has been supported by the DFG Cluster ofExcellence MAP (Munich-Center for Advanced Photon-ics). K. Homma appreciates the support by the Grant-in-Aid for Scientific Research no.21654035 from MEXTof Japan. T. Tajima is Blaise Pascal Chair Laureate atthe ´Ecole Normale Sup´erieure. We thank H. Gies andS. Sakabe for their advices and P. Thirolf for his carefulreading of our manuscript.
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