Analytic Scaling Formulas for Crossed Laser Acceleration in Vacuum

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1 October 6, 4 ARDB Note Analytic Scaling Formulas for Crosse Laser Acceleration in Vacuum Robert J. Noble Stanfor Linear Accelerator Center, Stanfor University 575 San Hill Roa, Menlo Park, California 945 USA Abstract. Using an analytic result of Sprangle et al for the longituinal accelerating fiel experience by a relativistic electron from two crosse Gaussian lasers in vacuum, we erive a new approximation for the accelerating potential vali for any energy provie the electron is traveling near the spee of light. The expression is applicable when the laser crossing angle is small an the electron s istance from the focal point is small compare to the Rayleigh length of the focuse lasers. Both these conitions are the usual ones for high energy particle acceleration in a boune accelerating structure near a laser focus. We use our expression for the accelerating potential to obtain formulas for the maximum energy gain per slip length, the optimal crossing angle, an the corresponing optimal slip istance. In the high energy limit, the optimal crossing angle is times the iffraction angle an the graient is.65 times the prouct of the iniviual laser fiel strength an the laser iffraction angle. Near the optimal crossing angle, the graient increases approximately linearly with the ratio of crossing angle to iffraction angle.. Introuction The purpose of this note is to iscuss the optimiation of laser acceleration of electrons in vacuum with respect to key parameters etermine in experiments such as LEAP at Stanfor University an E-63 at SLAC []. In particular, we present new analytic scaling formulas for crosse laser acceleration vali for any energy provie the electron travels at nearly the spee of light. The prototypical vacuum laser acceleration configuration we consier was analye by Sprangle et al [,3] an is shown in Figure. In this scheme a pair of linearly polarie laser beams with Gaussian transverse profiles, having the same frequency an equal strength E, are focuse an intersecte in vacuum. The first laser propagates along the axis an the secon laser propagates along the axis, where the axis an the axis are rotate by the crossing angles of an, respectively, with respect to the axis. The phases of the lasers are such that the transverse electric fiels cancel along the axis while the axial fiels a. Properly phase electrons injecte along the axis can be accelerate by the net axial component of the laser fiel. Figure. Coorinate system an electric fiels for two intersecting laser beams.

The analysis by Sprangle et al was one assuming that the focuse Gaussian laser propagates as a nearly uniirectional wave of finite cross section escribe by a fiel E t = E(x,y, exp(ik-iωt, an E t >> E.

2 The analysis by Sprangle et al was one assuming that the focuse Gaussian laser propagates as a nearly uniirectional wave of finite cross section escribe by a fiel E t = E(x,y, exp(ik-iωt, an E t >> E. Uner this assumption the usual vector wave equation can be replace by the scalar Helmholt equation for the transverse fiel E(x,y,, an the longituinal fiel is etermine later by emaning iv E =. The Helmholt equation is solve in the so-calle paraxial approximation in which the fiel E(x,y, is assume to vary slowly along the irection on the scale of a wavelength. The paraxial approximation is vali when the laser spot sie (waist w is much greater than the wavelength λ. For small crossing angles <<, the axial accelerating fiel, as seen by an electron traveling with velocity v c along the axis, was calculate to be E ˆ ( / E = exp cosψ, t (a + ˆ ˆ + where ˆ = / R, R = π / λ is the Rayleigh length, = w / is the laser iffraction angle, w ψ ˆ ˆ + = tan ˆ φ ( γ t, (b + ˆ t = /v has been assume, v / ( γ, = ( / c / γ v, an φ = R φ is the arrival phase of the electron relative to the laser maximum fiel value (. In Fig. the normalie electric fiel E /E as a function of the position / R is plotte for the parameters of the E-63 experiment [] using nominal 6 MeV kinetic energy electrons an for the same parameters in the limit of high energy electrons (γ. E /E =.5 mra = 6.4 mra / R Figure. Normalie electric fiel for the E-63 experiment using nominal 6 MeV electrons (upper curve; re an in the high energy limit (lower curve; blue.

3 The phase velocity of the accelerating fiel is greater than the spee of light, an therefore the fiel slips ahea of the electron. An electron interacting with the laser fiels over the entire region = - to + will experience no net energy gain. But if the laser fiels are terminate, say by bounaries, then net energy gain can occur. An electron injecte π/ in phase relative the fiel crest will gain energy until the phase slip changes by π, at which point it begins to be ecelerate. From Eq. (b near the focal point (<< R, the istance require for the electron to slip by π is λγ + γ / γ, ( s c /( c / ( + where γ efines a critical energy. Whenγ < γ, slippage is ominate by the low c = velocity of the electron, an whenγ > γ, slippage is ominate by the phase shift ue to laser iffraction. c Experimentally s etermines the length of the accelerator cell, which is marke by bounaries to terminate the laser fiels. In Fig. these points correspon to the locations of the fiel ero crossings on either sie of the origin. The axial electric fiel can in principle be written as the graient of an effective potential, E = U / ˆ, provie one can ientify the potential from the integral U = E ζ. R As far as the author knows, the general integral of Eq. ( is not known in close form. However in the special limit where γ tan ˆ = tan ˆ / ˆ to c ˆ R, one can use the trigonometric ientity ( ( rewrite cosψ t as a prouct of sine an cosine functions an absorb the leaing ( ˆ + foun in E. This observation allowe Sprangle et al to ientify the potential in the high energy limit as 4E ˆ ˆ U ( ˆ = exp sin φ sinφ, (3 + + ˆ ˆ where k = π/λ, an the potential is efine to be ero at the origin. Note that high energy here actually means γ >>. The normalie potential U/(E /k is shown in Fig.3 along with the numerical integral of the electric fiel ( for the parameters of E-63. U/(E /k =.5 mra = 6.4 mra Figure 3. Normalie potential from the numerical integration of Eq. ( for 6 MeV electrons (smaller curve; re an the potential in the high energy limit from Eq. (3 (larger curve; blue. / R 3

4 . Effective Accelerating Potential for Arbitrary Energy Although useful as a limit, Eq. (3 oes not allow one to unerstan the optimiation of parameters for a real experiment at finite energy or to erive simple scaling results for arbitrary energy. The analytic integration of Eq. ( to obtain the potential for arbitrary γ is hampere by the fact that the phase ψ t is nonlinear in ẑ. To make progress we note that one is only intereste in E an the potential U over a slip length aroun the origin, i.e. in the range [- s /, + s /]. Diviing the slip length s in Eq. ( by R one fins ˆ s π / =, (4 + + γ which is always less than. This suggests the following approximation for E vali for a relativistic electron when ˆ <<, which is essentially all of the accelerating region, ( ˆ E E exp ˆ cos ˆ ˆ + φ. (5 γ Here we have replace + ˆ by an linearie the arctangent function in Eq. ( within this approximation. In Fig. 4 the normalie electric fiel (5 for the E-63 parameters is plotte along with the fiels shown earlier in Fig.. The approximation is in excellent agreement with Eq. ( in the central region of π phase slip where the accelerator cell is locate. E /E =.5 mra = 6.4 mra Figure 4. Normalie electric fiel from the approximation in Eq. (5 for the E-63 parameters (ashe; brown compare with the exact fiel from Eq. ( (mile curve; re an high energy limit (lower; blue. The integral of E in Eq. (5 over the range [, ˆ] can be expresse in terms of the real part of the error function erf(w with complex argument w = u+iv [4]. The approximate effective potential is then / R 4

5 π E i ( = φ U ˆ exp( v R[ e ( erf ( u + iv erf ( iv], (6a where the quantities u an v are efine by u( ˆ = ˆ an v = + +. (6b γ Note that u ( ẑ is proportional to the longituinal coorinate, an v is proportional to the phase shift per unit length in Eq. (5. From Eq. (4, we can write v = / π / ˆ. ( s The error function is convenient to work with because so many approximations exist for it epening on the magnitues of u, v, an w. We anticipate that the optimal value of for the maximum energy gain will be of orer because if the crossing angle is much greater than the iffraction angle the slip istance is too short for the electron to gain much energy, while if the crossing angle is too small the projecte longituinal electric fiel will not prouce much energy gain. The analytic results in the next section will be consistent with this assumption. The error function in Eq. (6 can be greatly simplifie uner the conitions that u << an w >>. The first conition is always true within the slip region since ˆ << an is of orer, an the secon conition is a statement that v >>, which is vali in practice since v 3/ for all. Performing an asymptotic approximation of the error function uner these conitions yiels E ( ( u U ˆ exp u + sin( uv + φ + cos( uv + φ + sinφ, (7 v v v v where terms up to orer /v 3 an to first orer in u/v have been retaine. The /v 3 terms are essential for a goo approximation. The error function potential (6 an the asymptotic approximation (7 are plotte in Fig. (5 along with the two potentials shown earlier in Fig. 3. These potentials are excellent approximations to the numerical integral in the central region of π phase slip where the accelerator cell is locate. U/(E /k =.5 mra = 6.4 mra Figure 5. Approximate potentials from Eqs. (6 (ashe; brown an (7 (ash-ot; magenta compare with the potentials plotte in Fig. 3. / R 5

6 3. Analytic Scaling Formulas Because there are only exponential an trigonometric functions in the potential of Eq. (7, it is very useful for eriving simple scaling formulas for crosse laser acceleration at arbitrary energy. The energy gain per unit charge is efine by W = U ˆ U ˆ. (8 ( ( ( final initial For energy gain over a slip istance, we have ˆ = ˆ /, an ˆ = ˆ /. At these points, uv = ± π /. Using Eq. (7, the energy gain per unit charge in a slip istance is 8 E 3 W = cosφ ( exp( r + r π r / 4, (9a where / r =. (9b + + γ The optimal crossing angle which maximies the energy gain is etermine by the conition W / ( / =, or using the chain rule ( W / r ( r / ( / =. Solving ( W / r = yiels one real root for r, but when substitute into Eq. (9b, the corresponing solution for / is complex. The other partial erivative yiels ( / r / ( / = =, ( + ( / + ( / γ ( + ( / + ( / γ which yiels the optimal crossing angle to iffraction angle ratio ( / = +. ( opt γ The maximum energy gain per unit charge over a slip length is 4E π Wmax cosφ + exp 3 / opt / 6 / opt initial = ( ( ( opt s final s. ( In the limit γ, the optimum crossing angle becomes, an the maximum energy gain is 5 E cosφ E cosφ Wmax ( γ = exp( π / 3.6. (3 The optimal slip length (= accelerator cell length corresponing to the optimal crossing angle is λ λ 4 s ( opt = =. (4 + ( / opt γ The optimal slip length ivie by the Rayleigh length ( π R = λ is π / 4 ˆ s ( opt =, (5 + γ which is less than one for all γ. 6

7 From Eqs. ( an (4, one obtains the average graient when the energy gain is maximie, 4E π G = Wmax / s( opt = cosφ ( / + exp opt π ( / ( opt 6 / opt. (6 In the limit γ, this graient becomes 5 Ginf = exp( π / 3 E cosφ.65e cosφ. (7 π It shoul be stresse that G in Eq. (6 is not the maximum graient that can be achieve in the accelerator cell, but rather is the particular value when the energy gain is maximie. In fact W tens to be nearly constant for / near the optimal value, while the slip length s tens to fall off rapily with increasing /. Hence one can increase the graient by increasing / somewhat above the optimum an still achieve about the same energy gain in a shorter accelerator cell. For a given laser strength E, which may be limite by surface amage threshols, the accelerating graient can be increase in this way. One fins that the graient increases approximately linearly with / within the approximations of our analysis. In Fig. 6 the normalie energy gain from Eq. (9, the slip length an graient are plotte as a function of / to illustrate this relation for the E-63 parameters. We have not plotte the exact energy gain calculate from the numerical integration of Eq. ( for comparison in Fig. 6 since the curve woul lie irectly over the approximation. Normalie graient s / R W/(-4Ecosφ/ / Figure 6. Normalie energy gain W/(-4E cosφ /, slip length s / R, an normalie graient (ratio of these quantities as a function of the laser crossing angle ratio / for the E-63 parameters. The optimal value of / for maximal energy gain in a cell is.94, but larger graients are possible by increasing the crossing angle with little reuction in the energy gain. In Table we summarie the E-63 experimental parameters from the proposal [] an the optimal values for the key parameters calculate from our analysis above. It shoul be note that the esign crossing angle of.5 mra originally chosen for E-63 was etermine by using the Sprangle high energy limit for the potential (3 an the slip length ( for a 6 MeV electron, an then varying the crossing angle to achieve a maximum energy gain. This was the only metho available without an analytic formula vali for arbitrary energy. The preicte energy gain using the previous metho overestimates the energy gain by % 7

8 accoring to our analysis. This can be seen graphically in Fig. 5 where the potential in the high energy limit is significantly larger than the finite energy potential near the turning points. Table : Parameters for the E-63 laser acceleration experiment. 4. Conclusion Parameter Value Laser E 5.9x 9 V/m λ.8 µm w 4 µm R Design 6.8 mm 6.4 mra.5 mra γ c 68.5 Electron KE 6 MeV γ 8.4 -ew (using Eq.(3 9 kev (/ opt.94 opt.5 mra -ew max 34 kev s(opt.63 mm W max / s(opt 89 MV/m 6 MV/m G inf Using an analytic result of Sprangle et al for the longituinal accelerating fiel experience by a relativistic electron from two crosse Gaussian lasers in vacuum, we have erive a new approximation for the accelerating potential vali for any energy provie the electron is traveling near the spee of light. The optimal laser crossing angle for maximum energy gain in a slip length, an the corresponing slip length were etermine. It was shown that in the high energy limit, the optimal crossing angle is times the iffraction angle, an the graient is approximately.65 times the prouct of the iniviual laser fiel strength an the laser iffraction angle. The energy gain function has a broa maximum aroun the optimal crossing angle, while the slip length falls off rapily with angle. As a result in the neighborhoo of the optimal value, the graient increases almost linearly with crossing angle. The calculations escribe in this paper have been written into a Mathca program, available from the author, which allows the user to vary the key parameters an plot the corresponing accelerating fiels an potentials. ACKNOWLEDGMENTS This work was supporte by the U.S. Department of Energy uner contract DE-AC-76SF55. REFERENCES. E. Colby et al, Laser Acceleration of Electrons in Vacuum, SLAC Proposal E-63, August 4,.. E. Esarey, P. Sprangle, an J. Krall, Laser acceleration of electrons in vacuum, Phys. Rev. E, Vol. 5, 5443 ( P. Sprangle, E. Esarey, J. Krall, an A. Ting, Vacuum laser acceleration, Optics Comm., Vol. 4, 69 ( M. Abramowit an I.A. Stegun, Hanbook of Mathematical Functions, p. 95, Ninth eition, November 97. 8

Table of Common Derivatives By David Abraham

Prouct an Quotient Rules: Table of Common Derivatives By Davi Abraham [ f ( g( ] = [ f ( ] g( + f ( [ g( ] f ( = g( [ f ( ] g( g( f ( [ g( ] Trigonometric Functions: sin( = cos( cos( = sin( tan( = sec