ISSN 1054-660X, Laser Physics, 2009, Vol. 19, No. 2, pp. 214–221.
STRONG FIELD AND ATTOSECOND PHYSICS
© MAIK “Nauka /Interperiodica” (Russia), 2009. Original Text © Astro, Ltd., 2009.
Electron-Positron Pair Creation by a Strong Tightly Focused Laser Field1 A. M. Fedotov Moscow Engineering Physics Institute, Kashirskoe sh. 31, Moscow, 115409 Russia e-mail:
[email protected] Received August 29, 2008
Abstract—The analytical calculation of the electron-positron pair creation yield by an extremely strong focused laser pulse in the vacuum is presented. In particular, dependence of the total yield of the pair production on the focusing parameter and the polarization properties of the laser pulse are analyzed. Comparison with the previous consideration of N.B. Narozhny, V.S. Popov, V.D. Mur, and S.S. Bulanov is given and discussed. As a result, we confirm their conclusion that the electron-positron pair production can be observed in principle at the level of intensity of 1027 W/cm2. PACS numbers: 12.20.-m, 42.25.-p, 42.50.Gy DOI: 10.1134/S1054660X09020108 1
1. INTRODUCTION
The electron-positron pair creation from vacuum by electromagnetic fields is perhaps the most intriguing prediction of quantum electrodynamics. After its first theoretical recognition and calculations [1–3], a lot of work has been done for its better understanding as well as for calculation of the rate of this process for different field configurations. For example, let us mention a complete calculation of the S-matrix for the leptonic sector of the QED in the constant homogeneous external field [4], calculation of the pair creation yield in the time dependent homogeneous fields via the imaginary time technique [5, 6], etc. (for further references see the reviews [7–10]). According to the naive estimations, the effect could become observable if the field strength became comparable with the critical for QED value ES = m2c3/e = 1.32 × 1016 V/cm. The only experiment available by now [11] was confined to the multiphoton regime, since one of the photons participating in pair creation possessed energy of tens of gigaelectronvolts, whereas a more interesting tunnel regime related directly to confidence of the nonperturbative calculations in QFT has not been yet tested experimentally. The main motivation for revisiting this problem currently is the hope to observe the pair creation process directly in several years with the upcoming Extreme Light Infrastructure (ELI) facility which is going to enter readily into the research and development stage [12]. According to the preliminary claims, with this project it will be possible to achieve the unprecedentedly high intensities of the order of 1025–1026 W/cm2, which are by just several orders of magnitude smaller 2 than the characteristic intensity IS = (c/4π) E S = 4.65 × 1 The
article is published in the original.
1029 W/cm2. There are also more or less speculative proposals for further exhausting the available laser intensity, see, e.g., [13–15], which can be incorporated into the project with the course of time. Since the plane electromagnetic wave of arbitrary intensity does not produce pairs from vacuum, the most suitable and well approved tool of the intense field QED, the diagrammatic approach based on the Volkovdressed electron propagators, can not be applied to study pair creation by the laser field. The first attempt to study pair creation by field configurations which are more realistic from the point of view of possible experiments than the homogeneous time varying field, namely for a focused laser pulse and for a head-on collision of such pulses, was made quite recently in [16, 17]. The key conjecture of these papers was that since the smallest scale of variation of the field produced by the optical lasers, the carrier wavelength λ and the corresponding period λ/c, is much larger than the Compton scale, the approximation of the constant homogeneous field can be applied locally. The realistic structure and polarization properties of the field can be successively taken into account by incorporation of a realistic model of a focused field and by integration over the position of a teared vacuum loop in a focal region. The key conclusion of [16, 17] was that due to macroscopically large (~λ3) volume of a focal spot and time duration of a pulse (>λ/c) compared to the Compton scale, the intensities required for observation of pair creation may be several orders of magnitude less than those predicted by the naive estimations. In particular, the laser pulse focused in the vacuum can start to create pairs at the intensities of the order of 1028 W/cm2, although this prediction was very sensitive to the details of the field distribution and to the polarization properties of the pulse. For the case of the collided pulses, the
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time averaged threshold intensity was predicted to be about 1026–1027 W/cm2 (thus probably accessible for ELI) and was much less sensitive to the details of the pulses. In this paper we develop further the approach of papers [16, 17] by applying the steepest descent method for evaluation of the integral over the focal region instead of its numerical evaluation. Let us note that although this idea was already encoded implicitly in the discussion around the [17, Eq. (5.12)], its systematic implementation was lacking. As we will show, in this way the pair creation rate can be expressed analytically at least if the beam shape in the focus is controlled by not too much number of parameters. The primary goal of this improvement is the possibility of explicit and more clear formulation of the optimal design of the beam shape and its polarization properties. In addition, by comparing our analytical formulae with the numerical data provided by the paper [16], we observed that the latter suffers from a slight error, originated from usage of different normalization of the field than it was assumed by the authors. Although this error does not affect noticeably the main conclusions of that paper, nevertheless we decided to reexamine the confidence of calculations provided by the [17] by application of an independent model of the field at the focus.
215
grating over the azimuthal angle in the kx ky -plane we have 1 –ω ( it + b/c ) A ( r, t ) = – -------------e ( zˆ × ∇ ) ( C ⋅ ∇ ) 2 ( 2π ) ω
2 dk ⊥ k ⊥ 2 ω - J 0 ( k ⊥ r ⊥ ) cos ( z – ib ) -----2- – k ⊥ . × --------------------2 c 2 0 ω -----2- – k ⊥ c
∫
The remaining integral equals sin(ω/c)/, where
= r ⊥ + ( z – ib ) [18], so that we are left with 2
2
1 –ω ( it + b/c ) A ( r, t ) = – -------------e ( zˆ × ∇ ⊥ ) 2 ( 2π ) sin ( ω /c ) × ( C ⋅ ∇ ⊥ ) ---------------------------.
(3)
Note that the most natural approach to construct this kind of solutions is the so called “complex source method,” see [19] and references therein. In order to understand the physical meaning of the exact solution (3) let us first consider the limiting case b c/ω. In this case, in the region |z | b, r⊥ b we can exploit the expansion ≈ z – ib + r ⊥ /2(z – ib) (which is in fact just the conventional paraxial approximation) in the argument of sine and even a more rough approximation ≈ z – ib in the denominator. The contribution of the second exponent in the representation sin(ω/c) = (eiω/c – e–iω/c)/2i is suppressed by a factor exp(–2ωb/c) 1 and thus can be dropped. Finally, by evaluating the required derivatives we obtain 2
2. THE FIELD MODEL Let us start with consideration of some exact solution of the Maxwell equations in vacuum. In the transverse gauge, as is well known, the fields can be defined through the vector potential satisfying the equations ˙˙ – c2 ∆ A = 0 and ∇ · A = 0. Their general solution can A be represented in the form of the Fourier integral as follows A ( r, t ) =
3
d k
-a ( k ) exp ( ik ⋅ r – ckt ), ∫ -----------( 2π ) 3
(1)
k ⋅ a ( k ) = 0. Let us choose in particular a ( k ) = ( zˆ × k ⊥ ) ( C ⋅ k ⊥ )e
b ( k z – ω/c )
2
2 ω δ ⎛ k – -----2-⎞ , ⎝ c ⎠
(2)
and inspect the properties of the corresponding solution. Here zˆ is a unit vector directed along the z-axis, k⊥ and kz are the components of the wave vector orthogonal and parallel to z respectively, ω is the frequency, b is some parameter with dimension of length, and C is some constant vector orthogonal to z. We are going to clarify its meaning below. By plugging the Eq. (2) into the Eq. (1), evaluating the integral over kz by removing the δ-function and inteLASER PHYSICS
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⎧ ω ( C ⋅ r ⊥ ) ( zˆ × r ⊥ ) ⎫ ω A ( r, t ) = -------------------------------zˆ × C – -----------------------------------------⎬ 2 2⎨ c ( b + iz ) 8π c ( b + iz ) ⎩ ⎭ (4) 2 ωr ⊥ z⎞ ⎛ × exp –iω t – -- – ------------------------ . ⎝ c⎠ 2c ( b + iz ) As it follows from the expression (4), in the case of large b our solution represents nothing else but a focused Gaussian beam. The effective range of variation of the spatial coordinates in this case is |z| b, r⊥ bc/ω b, as it was assumed above. The center of the focal spot is located at the origin, where we have A(0, t) = (ω/8π2cb2) zˆ × Ce–iωt and E(0, t) = – A˙ (0, t)/c = i(ω2/8π2c2b2) zˆ × Ce–iωt. By equating the electric field strength at the origin to2 iE0e–iωt, where E0 is the field amplitude and is the complex polarization vector, we find that C = –(8π2c2b2E0 /ω2) zˆ × . 2 The
unconvential choice of a phase is not essential of course but is enforced by compatibility with the previous papers [16, 17].
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FEDOTOV
The expressions for the electric and the magnetic field strengths are deduced straightforwardly (compare to [16, 17, 20]), A˙ E = – ---c – iϕ
2 iE 0 e ⎧ ξ 2 ( x × zˆ ) ( x ∧ e ) ⎫ -2 ⎨ e + ------------------------------------- ⎬ exp ⎛ – -----------------⎞ , = -----------------------⎝ 1 + 2iχ⎠ 1 + 2iχ ( 1 + 2iχ ) ⎩ ⎭
H = ∇×A – iϕ 2 2 (5) iE 0 e ⎧ 4∆ ξ = -------------------------2 ⎨ zˆ × e 1 – ----------------- ⎛ 1 – --------------------------⎞ ⎝ ⎠ 1 + 2iχ 2 ( 1 + 2iχ ) ( 1 + 2iχ ) ⎩ 2
2x ( x ∧ e ) 6∆ ξ + ------------------------ 1 – ----------------- ⎛ 1 – --------------------------⎞ 1 + 2iχ 1 + 2iχ ⎝ 3 ( 1 + 2iχ )⎠ 2
2 2 ⎫ 8i∆zˆ ( x ∧ e ) ξ ξ + ----------------------------- ⎛ 1 – --------------------------⎞ ⎬ exp ⎛ – -----------------⎞ , ⎝ 1 + 2iχ⎠ 1 + 2iχ ⎝ 2 ( 1 + 2iχ )⎠ ⎭
where we have introduced the dimensionless transverse and longitudinal spatial variables x = r⊥/R and χ = z/L which are suitable for consideration of the Gaussian focused beam and ϕ = ω(t – z/c) is the fast phase. The parameters R = 2bc/ω = c/∆ω = λ/2π∆ and L = 2b = c/∆2ω = λ/2π∆2 are the focal radius and the diffractive length, respectively. The dimensionless parameter ∆ = c/2ωb = λ/2πR solely controls the degree of focusing for a given carrier frequency. Its direct physical meaning can be traced by applying the paraxial approximation directly in the k-space. Indeed, in virtue of the 2
2
2
E = 3iE 0 e H = 3E 0 e
– iϕ
– iϕ
[ e f + ( zˆ × z ⊥ ) ( e ∧ z ⊥ )g ],
{ [ e × zˆ g – z ⊥ ( e ∧ z ⊥ )h ] ( z ⋅ zˆ ) 2
+ zˆ ( e ∧ z ⊥ ) [ 4g + ζ ⊥ h ] }, sin ζ – ζ cos ζ -, f = ------------------------------3 ζ
2
( 3 – ζ ) sin ζ – 3ζ cos ζ -, g = ----------------------------------------------------5 ζ
2
(6)
2
3 ( 2ζ – 5 ) sin ζ – ζ ( ζ – 15 ) cos ζ -. h = ------------------------------------------------------------------------------7 ζ
2
expansion kz = ω /c – k ⊥ ≈ ω/c – ck ⊥ /2ω which is equivalent to the one applied above the exponent b ( k z – ω/c )
E H, H –E one can obtain a TH-mode corresponding to exactly the same intensity profile and the total power but with the magnetic field everywhere orthogonal to z. More general solutions can be constructed by taking a mixture of the TE- and TH-modes, as it was suggested in the [20]. Note that exactly the same classification regards our original exact solution as well. In the opposite limiting case b c/ω (or ∆ 1 with the notation introduced above) we have ≈ r, so that the field reduces to an equal mixture of convergent and divergent spherical waves, according to the representation sin(ωr/c) = (eiωr/c – e–iωr/c)/2i. Thus, our exact solution interpolates between a focused beam and a spherical standing wave (which is also a TE-mode in our case) as the focusing degree increases. This is very natural because the diffraction limit should be incorporated in the exact solution automatically. Let us examine it in more details below. In order to keep the meaning of the electric field amplitude at the origin for the parameter E0, let us choose for this time C = −12π2c4E0(ee × zˆ )/ω4. The electric and the magnetic fields in the limit b 0 are given explicitly by
2 2 2 2 –c k ⊥ /4ω ∆
e reduces e , so that in the framework of paraxial approximation the parameter ∆ 1 can be interpreted as the angular aperture of a focused beam. The total power of the beam averaged over the period is 2 2 2 2 given by P = (c/8π) d r ⊥ Re (E × H*) = cE 0 R e /16 .
∫
The field model (5) is a generalization to the case of arbitrary polarization of the model suggested originally in the [20] from quite a different line of reasoning and then successively applied in the papers [16, 17]. The main goal of this section is pointing out explicitly one of the exact solutions of Maxwell equations expressed in terms of simple analytical functions which is reduced in the paraxial limit to those proposed by Narozhny et al. Note that since the electric field E is everywhere orthogonal to the focal axis z, the solution under consideration corresponds to the so-called TE-mode. In contrast, the magnetic field possesses in general a longitudinal component. By a duality transformation
Here we used the notations ϕ = ωt and z = ωr/c which are more natural for the spherical geometry than those used above for the case of weak focusing. Note that the magnetic field vanishes at the origin, exactly as it should be expected for a standing wave. In order to calculate the incoming power, in this case only one of the waves, e.g. the incoming wave, should be considered. By direct evaluation, the incoming power averaged 3 2 2 2 over a period equals P = (9/15) c E 0 e /ω . This last formula determines the maximal strength of the electric field achievable in our model for the prescribed value of the incoming power. The typical size of the focal spot is given by c/ω, as it is forced by the diffraction limit. 3. CALCULATION OF THE PAIR CREATION YIELD The length scale l typical for the pair creation process can be estimated naively by equating the work eEl produced by the field and the proper energy 2mc2 required to create a pair. Basing on this reasoning one LASER PHYSICS
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can obtain l ~ mc2/eE. However, as it was proposed in [7], the actual length of formation of the process besides the classically forbidden region should also include a wider region where the systematical semiclassical approach breaks down. For the case of pair creation, we have l ~ (mc2/eE) E S /E = lC (IS /I)3/4, where lC = /mc = 3.86 × 10–11 cm. Therefore the electromagnetic field in vacuum can be considered as constant and homogeneous as long as its carrier wavelength satisfies the condition λ l ~ lC (IS /I)3/4. As it was pointed out in the [17] and will be confirmed below, the threshold intensity for pair production in the vacuum is greater than about 10–3IS. Assuming that the pair creation is substantial it is clear that the fields can be considered as locally constant for all λ 0.1 nm, i.e. starting from the soft X-ray range. For optical frequencies this condition is satisfied with a huge margin. Therefore, the average number of created pairs per unit time and volume can be calculated with the formula derived for the constant homogeneous fields [2– 4], whereas their actual variation can be taken into account only on the final stage of integration of the rate density of pair creation over the available space and time [16, 17], 2 πE 3 πη e - η coth ⎛ -------⎞ exp ⎛ – ---------S⎞ d r dt, (7) N e + e – = ---------------2 2 ⎝ ⎠ ⎝ ⎠ 4π c
∫
where the field invariants , η =
2
2 2
2
2
E –H E –H ⎞ 2 ⎛ ----------------- + ( E ⋅ H ) ± ------------------ . ⎝ 2 ⎠ 2
(8)
Let us assume for the time being that the temporal integral is taken over a period, thus defining the number of pairs created per one period. As for the spatial integral, it can be taken over the whole space since the region where the approximation of a constant homogeneous field is invalid due to the weakness of the field does not contribute anyway. Our strategy is to calculate the 4fold integral in the Eq. (7) with the fields given in paraxial approximation (see Eqs. (5)) and for the extremely focused field (see Eqs. (6)) by application of the steepest descent technique. As it is clear from the Eqs. (5) in the framework of paraxial approximation (∆ 1) the fields are given by the dimensional parameter E0 multiplied by the dimensionless functions of the dimensionless variables x, χ, ϕ and the dimensionless parameter ∆. As a consequence, the field invariants and η can be represented in the same form. Since ∆ is small we can expand the corresponding dimensionless functions in power series with respect to ∆, and, because these invariants vanish identically for the case of a plane wave (∆ = 0), they are proportional to ∆. Of course, this property can be verified directly. Therefore, we have (r, t) = ESϑ (x, χ, ϕ) LASER PHYSICS
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and η(r, t) = ESϑ η (x, χ, ϕ), where ϑ = ∆E0 /ES is the dimensionless intensity parameter of the focused field and , η are the normalized dimensionless functions depending exclusively on the structure and polarization properties of the laser beam and not containing small or large parameters. With this notation after passage to integration over the dimensionless variables x, χ, ϕ instead of r⊥, z, t we obtain 4 λ 1 2 N e + e – = -------------2 ⎛ ---------------⎞ ϑ I ( ϑ ), ⎝ ⎠ 2π∆l C ( 2π )
I(ϑ) =
∫
(9)
πη π 2 η coth ⎛ -------⎞ exp ⎛ – ------⎞ d ξ dχ dϕ. ⎝ ⎠ ⎝ ϑ⎠ 2
The dimensionless factor (e2/2c)(R2L/ω) E S = (λ/2π∆lC)4 in front of the integral is macroscopically large, for instance for λ = 1 µm it is of the order of 1027. This quantity can be interpreted visually as a number of virtual loops settled in the focal region during a laser period. For a prescribed polarization of the focused beam, the remaining integral I(ϑ) depends just on the unique intensity parameter ϑ, which hereafter is assumed to be small, ϑ 1. Since the normalized invariant which enters the exponent in the Eq. (9) is everywhere positive and is fast decreasing outside of the focal region due to the whole decreasing of the field, it should possess one or more maxima in the focal region. In the case ϑ 1 the Laplace method which is a version of the steepest descent technique [21] can be applied to evaluation of the integral. Only the global (highest) among the maxima contribute to the integral with exponential accuracy. In order to illustrate the technique, let us first consider in more details the case of the TE-mode described explicitly in the preceding section and assume circular polarization (ee = xˆ + iyˆ ). In this case the invariants depend on the phase variable ϕ and the azimuthal angle φ of the vector x in the focal plane only through the combination ϕ – φ, so that their spatial distributions rotate as a whole around the focal axis with variation of the phase. Therefore integration over the phase variable is idle bringing just a factor 2π. Also, in this case there is a unique well pronounced global maximum for located at the origin ξ = χ = 0. Near the origin up to the cubic terms we have 3 ( 6 – cos [ 2 ( ϕ – φ ) ] ) 2 2 = 2 2 – ---------------------------------------------------ξ – 11 2χ + …, 2 sin [ 2 ( ϕ – φ ) ] 2 η = 2 2χ – ---------------------------------ξ + …. 2
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FEDOTOV
Thus, at the origin η coth( πη/ ) = ( 2 2 )2/π, and consequently 2 2π
(2 2) 2 I ( ϑ ) ≈ ----------------- dϕ d ξ π
∫ ∫ 0
+∞
×
∫ dχe
2 11 2 π 3 – -------------- 1 + --- ( 6 – cos 2 ( ϕ – φ ) )ξ + ------ χ 4 2 2 2ϑ
–∞
3/2
512ϑ π = -------------------- exp ⎛ – --------------⎞ . ⎝ 2 2ϑ⎠ 385 2 The corrections up to an arbitrary order can be found straightforwardly in principle. In order to do this, we pass to the new variables ξ˜ = ξ/ ϑ , χ˜ = χ/ ϑ , bring – π/ max ϑ
the exponent e in front of the integral, expand the integrand in powers of ϑ and evaluate the resulting Gaussian integrals. Practically, this procedure is very tedious so that we were forced to apply the computer algebra. Proceeding in this way, we have
Unlike the case of circular polarization (or, more generally, large ellipticity), in the case of linear polarization or small ellipticity the spatial distributions of both field invariants do not rotate around the focal axis, but “breath,” actually in an opposite phase to each other, successively reaching their maxima twice during the laser period. Therefore, integration over the phase variable in this case is not idle any more and should be also performed by the Laplace method. Therefore the number of created pairs acquires an additional smallness by approximately a factor 1.5 ϑ compared to the case of large ellipticity. We have also examined the case of a circularly polarized TH-mode which is obtained by a duality transformation from the TE-mode. This transformation permutes the invariants and η, so that the main contribution to the integral comes from the maxima of the invariant η obtained above for the case of the TE-mode. This invariant possesses two equal global maxima located in the focal plane (χ = 0) symmetrically with respect to the focal axis. As it follows from the numerical calculation, their positions are given by ξmax = 0.6585, φ = ϕ ± π/2 whereas their normalized height is η max = 2.126. The analogue of formula (11) in this case takes the form
3/2 – π/2 2ϑ
I ( ϑ ) = 7.314ϑ e
2
3
3
× [ 1 – 4.283ϑ + 20.32ϑ + O ( ϑ ) ].
Note that for instance in the case ∆ = 0.1 and E0 = 0.1ES, so that ϑ = 0.01, the first and the second order corrections contribute by 4.0 and 0.2% respectively, which means that the applied method of evaluation of the integral is very reliable. Since large laser intensities are currently available only in the femtosecond pulses, the finite duration of a pulse should be taken into account. Although we could consider the exact pulse-like solutions basing on the technique of the preceding section, let us assume that the pulses contain at least several periods and use the conventional approach involving the envelope function. Following [16, 17], we choose the envelope function of 2
2 2
– ϕ /4ω τ
the form g(ϕ) = e , where τ 2π/ω is the pulse duration. The only distinction with the previous calculation is that now we should make a replacement ϑ ϑg(ϕ) everywhere in the integrand and integrate over ϕ from –∞ to +∞. After the passage to a new integration variable ϕ˜ = ϕ/ωτ, which produces an additional factor ωτ, the integral over ϕ˜ can be evaluated by the Laplace method as well. In this way for the total number of pairs created per one shot we obtain 3
4
( shot ) – 4 λ ϑ cτ – π/2 N e + e – = 3.998 × 10 -----------------e 4 4 lC ∆ 2
3
2ϑ
× [ 1 – 6.196ϑ + 37.24ϑ + O ( ϑ ) ].
(11)
4
( shot ) – 4 λ ϑ cτ N e + e – = 5.262 × 10 ----------------4 4 lC ∆
(10) ×e
– 1.4778/ϑ
–1
2
(12) 3
[ 1 – 4.806ϑ + 22.35ϑ + O ( ϑ ) ]. –1
Since η max – max 0.12, the pair creation by a THmode is exponentially suppressed in comparison to pair creation by a TE-mode for all ϑ 0.1. The total number of pairs created by a circularly polarized 10 fs pulse with carrier wavelength λ = 1 µm as a function of the ratio E0 /ES for two values of ∆ is depicted at the Figs. 1a and 1b according to the formulas (11) and (12), respectively. We have also provided comparison with the data obtained by numerical computation in the [17] (star scatters), see the Table 1 of that paper. An impressive discrepancy up to 11 orders in magnitude between the two independent calculations is explained by a slight error in the [16, 17] made in the calculation of the field invariants. Namely, in these papers the expression for the quantity (E2 – H2)/2 contains an extra factor 2, so that the maximal values of the invariants and η, which enter the exponent, are overestimated by 2 . This error can be easily corrected by stretching their data by a factor 2 along the independent variable (see the circle scatters). After this operation, we come to a quite good agreement for the case of TE-mode, which is almost perfect for ∆ = 0.1. Fortunately, the mentioned corrections do not modify essentially the most important conclusions of Narozhny et al. because the field and the intensity required for observation of pair production are very LASER PHYSICS
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219
1015 1016
(a)
1011
1012
107
Ne +e –(shot)
108
103
104
10–1
100 10–4
10–5
10–8
10–9
10–12
10–13
10–16
10–17
10–20 0.1
(b)
0.2
0.3
0.4
0.5
0.6
The total number of pairs created per one shot N
0.7 ( shot ) + –
e e
0.8 E0 /Es
10–21 0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8 E0 /Es
vs. the ratio E0 /ES for ∆ = 0.1 (solid line) and for ∆ = 0.05 (dashed line)
compared to the results of numerical computation from [17] (filled stars for ∆ = 0.1 and empty stars for ∆ = 0.05) for the cases of the TE-mode [panel (a)] and the TH-mode [panel (b)]. The filled and empty circles correspond to the transformation of this numerical data by stretching E0 2E 0 (λ = 1 µm, τ = 10 fs, circular polarization).
insensitive to a particular definition of the effective threshold. It should be noted in addition that the intensity averaged over the pulse duration rather than the peak intensity was mentioned in the [16, 17]. In the most interesting range of the field strength near the threshold our corrections can be reduced roughly to replacement of the average intensity by the peak intensity in their conclusions. Finally, let us turn to consideration of the pair creation in the case of the extremely tight focusing. Since so far the maximal pair production was predicted for a circularly polarized TE-mode, let us assume in the following just this case. Basing on the Eqs. (6), the invariant e again possesses a global maximum at the origin, but this time max = E0 due to-vanishing of the magnetic field. Near the origin we have ⎧ 5 + ( 13 – 3 cos 2(φ – ϕ) ) sin2 θ 2 3 ⎫ = E 0 ⎨1 – ----------------------------------------------------------------------- ζ + O(ζ )⎬, 50 ⎩ ⎭ and η = O(ζ). Therefore, proceeding in the same way as above we have 3 25 2λ cτ ⎛ E 0⎞ 4 –πE/ES ( shot ) - ----- e . N e + e – = -----------------------5 4 24 7π l C ⎝ E S⎠
(13)
Note that simple extrapolation of the formula (11) to the value ∆ = ( 2 2 )–1 0.354 underestimates the exact result (13) just by approximately 4.55 times, which is almost perfect coincidence for a purpose of estimation of the required field strength. Let us now estimate the number of created pairs for the most optimistic scenario, which will be very probaLASER PHYSICS
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bly realizable with the ELI facility [12]. Namely, let us assume that λ = 800 nm, τ = 10 fs and that the peak incoming laser power is Ppeak = 7EW, so that E0 /ES = 2
( 5π/6 )P peak /I S λ = 7.847 × 10–2. Although the Poynting vector identically vanishes at the origin for a standing wave, however its maximal value at the focal 2 plane is approximately Smax = 0.740( cE 0 /4π ) 2 × 1027 W/cm2. With all these assumptions we obtain about 20 created pairs per laser shot, which corresponds to approximately 3 × 104 pair creation events per second provided that the repetition rate is 1.5 kHz. Note that our estimation lies in an unexpectedly perfect quantitative agreement with the estimation of [17], where it was predicted that several pairs should be created in a head-on collision of two circularly polarized tightly focused pulses with λ = 1 µm, τ = 10 fs and ∆ = 0.1, each of the time averaged intensity Iav = 2.5 × 1026 W/cm2 and thus of the peak intensity Ipeak ≈ 8 × 1026 W/cm2 at the focus, independently on were they TE- or TH-modes. Since there is just a qualitative similarity between a spherical standing wave and collided tightly focused beams, such an almost exact coincidence seems to be accidental of course. But it nevertheless confirms the main conclusion that pair creation can be observed in principle at intensities 1027 W/cm2 insensitively to particular shapes or polarization states of the 6olliding laser pulses. 4. DISCUSSION There are no doubts that the approximation of slowly varying fields adopted in the [16, 17] as well as in the current paper is valid and suitable for applica-
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FEDOTOV
tions to study different QED processes in the intense XUV or optical pulses. In this paper we have demonstrated how the asymptotically exact analytical formulas for the total pair creation yield in vacuum can be derived for a prescribed model of a laser pulse. In particular, we report such formulas for the most simple realistic models of a laser pulse, namely for circularly polarized TE- and TH-modes containing just the zeroth and the second cylindrical harmonics. We have also estimated quantitatively the suppression of pair production in the case of small ellipticity of a laser beam near the focal axis. Namely, the number of pairs created by a linearly polarized pulse is just by an order of magnitude less than for the case of circular polarization. Due to a very sharp dependence of the number of created pairs on the peak intensity, this implies that although circular polarization is preferred to observe the pair production, the difference between the required intensities for both cases is in fact very small. We have compared our analytical formulas with the numerical data obtained previously for the same problem in the papers [16, 17]. To the best of our knowledge, the main predictions of these papers concerning the intensities required for experimental observation of pair production in vacuum by the individual (1028 W/cm2) and collided focused laser pulses (1026– 1027 W/cm2) has been not confirmed yet by any independent study. At the same time, such a confirmation seems to be very important due to a recent announcement that the level of 1026 W/cm2 will be very likely accessible with the upcoming ELI facility. Although we have discovered and explained some discrepancy with the papers [16, 17] in the part concerning pair production by individual focused pulses, nevertheless we confirm all the most general conclusions claimed by Narozhny et al. In addition, by consideration of pair production by a spherical standing wave (or converging spherical pulse), we obtained an extra confidence that the minimal level of the required peak intensity and the peak laser power are 1027 W/cm2 and 5 EW almost insensitively to the shapes of the colliding pulses. Since posterior overcoming the ELI intensity level of 1026 W/cm2 seems to be very expensive, further search of the possibilities to optimize the pair creation yield by controlling the shape of a laser beam at the focus should be done, in our opinion. An important lesson from our calculations is that boost of pair production requires most of all further increasing of the maximal value of the invariant e at the focus for a prescribed incoming power (or peak intensity). However, up to now our knowledge of the relation between the beam shape and the value of max is very poor, so that it is very probable that the circularly polarized TE-mode can be not the best candidate for the experiments with individual focused pulses. Possible affection of the higher order cylindrical harmonics on the value of max also remains unclear yet.
There is another problem related to controlling pair production by the beam shaping, although for the time being of mostly theoretical interest. As it was first recognized in the [16], since currently all the ways of obtaining high intensities require focusing, the rapid growth of pair production due to extracting the energy from a laser pulse can impose insuperable limits for accessible laser intensities lying very closely to the socalled Schwinger Limit I = IS. Although a systematic study of this phenomenon is lacking (except for the case of time varying homogeneous field, [22]), there are some clear indications that the electron-positron plasma, once created, would speed up even more the exhausting of the laser pulse [23]. Therefore, in order to establish actual limitations on the peak intensity of a focused beam, the beam shapes corresponding to maximally reduced values of max should be identified as well. ACKNOWLEDGMENTS The author is very grateful to V.S. Popov, G. Mourou, A.M. Sergeev, V. T. Tikhonchuk, V.Yu. Bychenkov, and especially to N.B. Narozhny and V.D. Mur, for valuable discussions and comments. This work was supported by the Russian Foundation for Basic Research (grant 06-02-17370-a), the Ministry of Science and Education of Russian Federation and the Russian Federation President grant. MK-2364.2007.2. REFERENCES 1. F. Sauter, Z. Phys. 69, 742 (1931); Z. Phys. 73, 547 (1931). 2. W. Heisenberg and H. Euler, Z. Phys. 98, 714 (1936). 3. J. Schwinger, Phys. Rev. 82, 664 (1951). 4. A. I. Nikishov, Zh. Eksp. Teor. Fiz. 77, 1210 (1969) [Sov. Phys. JETP 30, 660 (1969)]; N. B. Narozhny and A. I. Nikishov, Sov. J. Nucl. Phys. 11, 1072 (1970). 5. E. Brezin and C. Itzykson, Phys. Rev. D 2, 1191 (1970). 6. V. S. Popov, Pis’ma Zh. Eksp. Teor. Fiz. 13, 261 (1971) [JETP Lett. 13, 185 (1971)]; Zh. Eksp. Teor. Fiz. 61, 1334 (1971) [Sov. Phys. JETP 34, 709 (1972)]. 7. V. I. Ritus and A. I. Nikishov, “Quantum Electrodynamics of Phenomena in a Strong Field,” Trudy Fiz. Inst. Akad. Nauk SSSR, No. 111 (Moscow, 1979). 8. W. Greiner, B. Müller, and J. Rafelski, Quantum Electrodynamics of Strong Fields (Springer, Berlin, 1985). 9. A. A. Grib, S. G. Mamaev, and V. M. Mostepanenko, Vacuum Quantum Effects in Strong Fields (Atomizdat, Moscow, 1988; Friedmann Labor. Publ., St. Petersburg, 1994). 10. E. S. Fradkin, D. M. Gitman, and Sh. M. Shvartsman, Quantum Electrodynamics with Unstable Vacuum (Springer, Berlin, 1991). 11. D. L. Burke, R. C. Field, G. Horton-Smith, et al., Phys. Rev. Lett. 79, 1626 (1997). 12. http://www.extreme-light-infrastructure.eu/pictures/ELIscientific-case-id17.pdf. LASER PHYSICS
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ELECTRON-POSITRON PAIR CREATION 13. B. Shen and M. Yu, Phys. Rev. Lett. 89, 275004 (2002). 14. S. V. Bulanov, T. Esirkepov, and T. Tajima, Phys. Rev. Lett. 91, 085001 (2003). 15. S. Gordienko, A. Pukhov, O. Shorokhov, and T. Baeva, Phys. Rev. Lett. 94, 103903 (2005). 16. N. B. Narozhny, S. S. Bulanov, V. D. Mur, and V. S. Popov, Phys. Lett. A 330, 1 (2004); Pis’ma Zh. Eksp. Teor. Fiz. 80, 434 (2004) [JETP Lett. 80, 382 (2004)]. 17. N. B. Narozhny, S. S. Bulanov, V. D. Mur, and V. S. Popov, Zh. Eksp. Teor. Fiz. 129, 14 (2006) [JETP 102, 9 (2006)]. 18. I. S. Gradshtein and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic Press, Boston, 1994).
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19. A. M. Fedotov, K. Yu. Korolev, and M. V. Legkov, SPIE Proc. 6726, 672613 (2007). 20. N. B. Narozhny, and M. S. Fofanov, JETP 90, 753 (2000); Phys. Lett. A 295, 87 (2002). 21. A. Erdelyi, Asymptotic Expansions (Dover, New York, 1956). 22. S. Schmidt, D. Blaschke, G. Röpke, S. A. Smolyansky, A. V. Prozorkevich, and V. D. Toneev, Int. J. Mod. Phys. E 7, 709 (1998); V. N. Pervushin, V. V. Skokov, A. V. Reichel, S. A. Smolyansky, and A. V. Prozorkevich, Int. J. Mod. Phys. A 20, 5689 (2005). 23. S. S. Bulanov, A. M. Fedotov, and F. Pegoraro, Phys. Rev. E 71, 016404 (2005).
