- Email: [email protected]
Second harmonic generation in a polarized Dirac vacuum
2 November 1998
PHYSICS
ELSEZVIER
LETTERS
A
Physics Letters A 248 (1998) 16-18
Second harrnonic generation in a polarized Dirac vacuum S.G. Oganesyan ’ Lmerain Tekhnika R&D Company, 21 Shopron
str., Yerevan 375090, Armenia
Received 28 November 1997; revised rn~~sc~~& received 20 April 1998; accepted for publication I1 August 1998 Communicated by J.P. Vigier
Abstract The Dirac vacuum, polarized by a strong electric field Eo, is discussed as nonlinear medium for laser radiation (LR). It is shown that such a medium leads to (a) a LR refractive index appearing, (b) LR polarization plane rotation, (c) LR second harmonic generation. It is proposed to use these effects for EO field diagnostics. @ 1998 Elsevier Science B.V.
Studies of laser radiation-free electron interaction have demonstrated a number of interesting effects, such as polarization and modulation of electron beams by LR, rotation of the LR polarization plane, and LR harmonic generation [ I]. All of these phenomena can be used both for electron beam and laser beam structure investigations. Note that the latter problem is actual since in recent years there has been great interest in formation of ultrashort intense pulses and in nonlinear quantum electrodynamics in those pulse fields [ 21. It should be emphasized that the experimental study of strong fields is a very complicated problem because they destroy an ordinary molecular medium. Since electron-beam densities are not too high, they also seem to be not suitable. In the present work we have attempted to apply the Dirac vacuum (DV) [ 31 to this problem. Note two principal advantages of this medium: (a) DV is the densest medium of Fermi particles, f b) the critical field at which the DV bre~down occurs (via e-e+ pair production) is strong enough, E,, = nz2c3/lelfi = 1.3 x lOi V/cm.
’ E-mail: [email protected].
The nonlinear theory of the DV was developed by Heisenberg and Euler [4]. They obtained a Lagrangian density expression in arbitrary strong constant electric and magnetic fields. If these field strengths are not too high (E, B << EC,) then (to fourth order accuracy [5] ) the Lagrangian density is 15= & + L’, where L _ B2 - B= 0 --
87r
L’=R[(E2
’
-B2)‘+7(E*B)*].
(1)
Here the factor R = ~/36~*~~~, and cy = e2/fic is the fine structure constant. Further analysis [ 61 showed that one may use Lagrangian (1) in alternating fields as well, if their frequencies are not too large (o < c/A,, where A, = ri/mc is the electron Compton wavelength). In particular, the second-quantized Lagrangian ( 1) describes correctly the well-known effect of light-by-light scattering [7]. In the present work we consider the simple case when a strong electric field EO (Eo < E,,) polarizes the DV in a region of space dimension of order 1. We study the influence of this medium on linear polarized LR,
037%9601/98/$ - see front matter @ 1998 Elsevier Science B.V. AII rights reserved. PIISO375-9601(98)00591-X
S.G. Oganesyan/Physics Letters A 248 (1998) 16-18
E=Eisin(wt-kiz), B = Br sin(ot
- kiz) .
(2)
Further we show that LR-DV interaction leads to (a) a LR refractive index appearing, (b) a changing LR polarization state, (c) LR second harmonic generation. Consider effects (a) and (b)first. Let us neglect the DV influence on the EO field and calculate ki , El, Bt values on the basis of a common technique [ _5,7]. Namely, we substitute electric (E = Ea + Ei ) and magnetic (Br ) fields in ( 1) and calculate the polarization (P = aL’/JE) and the magnetization (I = JL’/c?B) vectors. As a result, in the linear LR field approximation the vectors of the electric and magnetic induction are Di = EijEj* Bi = pi,iHj, where the electric and magnetic constant tensors are, respectively, cij = Sij( 1 + 16~REg)
+ 32~REo~Eoj 3
p;,i = 6i.i ( 1 - 16rREi)
+ 56TREoiEoj
.
(3)
Using Z and P expressions one may write the component j, of the current j = crotl + dP,/Jt, which oscillates at frequency 0, j,x = 2RoI (4E& + 7E&)E,,
- 3Eo.~Eo,E1,1
17
also that if LR is elliptic potarized from the outset (at z = O), then interaction leads to its polarization plane rotation by
(p = iarctg
26sin[(w/c)(nll
- nl)l] (6)
l-62
angle (here 6 = EI,/EI,, El, and El, are the initial ellipse principal axes) [ S] . Consider now the problem of LR h~monic generation Here we restrict our analysis only to the problem of the second harmonic production. We write its field in the form E = E2 sin( 2ut - kzz) ,
B = B2 sin(2ot
- k.g) ,
(7)
where the wave vector kl = 2on(2w)/c. On substituting (7), (2) and Ea values into ( 1) and using Eqs. (4)) we get the component j, of current j, which oscillates at frequency 20~. The analysis proceeds as before showing that the second harmonic electric field is directed along Ea, i.e. E2 I[ Eo. If Ea Ij El 11p, then n(w) = nil, n( 20) = ~11. Since in this case the LR and the second harmonic phase synchronization occurs, the conversion coefficient
xcos(ot-kiz), jiy = ~R~[-~Eo~EoJEI~
+
(4E& +
7E&)E1yl
x cos(wt - k,z). In order to simplify the further analysis, we assume that the EO vector is directed along the x axis (i.e., Ea /I i). On solving the Maxwell equations 47T. rotB=Tj+;z,
1 JE
rOtE=: -.! !!!
is directly proportional to the interaction region length E,This is not the case if EO -I_2. In this device n(w) = nl, whereas ~(20) = nil; and the conversion coefficient rll =
c at ’
l(jTR@;
_
l)EoE;
1sin(w/c’(nL - nll)‘l . nl
-
1111
we obtain ki = 3 nll._~ 9
BI = q,iG
1
(4)
nl = 1 + 28rREg
(5)
where nil = 1 + 16rREi,
are the refractive indices of the x and y projections of LR (2). Apparently, LR-DV interaction transforms the linear polarized LR into the elliptic polarized one. Note
Nevertheless, if the parameter (o/c) (rtl- 1~11) 1 << 1, then ~1 = 7~1i/12, i.e. ~1 N I as well. Since all of the obtained effects are determined quantitatively by the EO field, one may use them for field diagnostics. Moreover, an alternating field EO = E sin coot may be studied in the same way if its frequency 00 is not too large (wa < min{o, c/l}). This work was supported by International and Technology Center Grant No. A-87.
Science
18
S.G. Oganesyan/Physics Letters A 248 (1998) 16-18
References [i J V.M. H~utunian, S.C. Og~esy~, Phys. Rep. 270 (19%) 217. [ 21 K.T. McDonald, Proposal for experimental studies of nonlinear quantum electrodynamics, DOEfER/3072-38, Sept. 2, 1986. draft. [3] S.S. Schweber, An Introduction to Relativistic Quantum Field Theory (New York, 1962) Section 4g.
141 W. Heisenberg, H. Euler. 2. Phys. 98 (3936) 714. IS] E.M. Lifshitz, L.P. Pitaevsky, Relativistic Qu~tum Theory, Vol. 2 (Moscow, 1971) Section 126. f6] A.A. Grib. S.G. Mameyev, V.M. Mostep~enko, Vacuum Quantum Effects in Strong Fields (Moscow, 1988) Section 6. [ 71 A.I. Ahiezer. V.B. Berestetzky, Quantum Electrodynamics (Moscow, 1969) Section 41.2. [8] S.G. Oganesyan, JTF Lett., to be published.
PHYSICS
ELSEZVIER
LETTERS
A
Physics Letters A 248 (1998) 16-18
Second harrnonic generation in a polarized Dirac vacuum S.G. Oganesyan ’ Lmerain Tekhnika R&D Company, 21 Shopron
str., Yerevan 375090, Armenia
Received 28 November 1997; revised rn~~sc~~& received 20 April 1998; accepted for publication I1 August 1998 Communicated by J.P. Vigier
Abstract The Dirac vacuum, polarized by a strong electric field Eo, is discussed as nonlinear medium for laser radiation (LR). It is shown that such a medium leads to (a) a LR refractive index appearing, (b) LR polarization plane rotation, (c) LR second harmonic generation. It is proposed to use these effects for EO field diagnostics. @ 1998 Elsevier Science B.V.
Studies of laser radiation-free electron interaction have demonstrated a number of interesting effects, such as polarization and modulation of electron beams by LR, rotation of the LR polarization plane, and LR harmonic generation [ I]. All of these phenomena can be used both for electron beam and laser beam structure investigations. Note that the latter problem is actual since in recent years there has been great interest in formation of ultrashort intense pulses and in nonlinear quantum electrodynamics in those pulse fields [ 21. It should be emphasized that the experimental study of strong fields is a very complicated problem because they destroy an ordinary molecular medium. Since electron-beam densities are not too high, they also seem to be not suitable. In the present work we have attempted to apply the Dirac vacuum (DV) [ 31 to this problem. Note two principal advantages of this medium: (a) DV is the densest medium of Fermi particles, f b) the critical field at which the DV bre~down occurs (via e-e+ pair production) is strong enough, E,, = nz2c3/lelfi = 1.3 x lOi V/cm.
’ E-mail: [email protected].
The nonlinear theory of the DV was developed by Heisenberg and Euler [4]. They obtained a Lagrangian density expression in arbitrary strong constant electric and magnetic fields. If these field strengths are not too high (E, B << EC,) then (to fourth order accuracy [5] ) the Lagrangian density is 15= & + L’, where L _ B2 - B= 0 --
87r
L’=R[(E2
’
-B2)‘+7(E*B)*].
(1)
Here the factor R = ~/36~*~~~, and cy = e2/fic is the fine structure constant. Further analysis [ 61 showed that one may use Lagrangian (1) in alternating fields as well, if their frequencies are not too large (o < c/A,, where A, = ri/mc is the electron Compton wavelength). In particular, the second-quantized Lagrangian ( 1) describes correctly the well-known effect of light-by-light scattering [7]. In the present work we consider the simple case when a strong electric field EO (Eo < E,,) polarizes the DV in a region of space dimension of order 1. We study the influence of this medium on linear polarized LR,
037%9601/98/$ - see front matter @ 1998 Elsevier Science B.V. AII rights reserved. PIISO375-9601(98)00591-X
S.G. Oganesyan/Physics Letters A 248 (1998) 16-18
E=Eisin(wt-kiz), B = Br sin(ot
- kiz) .
(2)
Further we show that LR-DV interaction leads to (a) a LR refractive index appearing, (b) a changing LR polarization state, (c) LR second harmonic generation. Consider effects (a) and (b)first. Let us neglect the DV influence on the EO field and calculate ki , El, Bt values on the basis of a common technique [ _5,7]. Namely, we substitute electric (E = Ea + Ei ) and magnetic (Br ) fields in ( 1) and calculate the polarization (P = aL’/JE) and the magnetization (I = JL’/c?B) vectors. As a result, in the linear LR field approximation the vectors of the electric and magnetic induction are Di = EijEj* Bi = pi,iHj, where the electric and magnetic constant tensors are, respectively, cij = Sij( 1 + 16~REg)
+ 32~REo~Eoj 3
p;,i = 6i.i ( 1 - 16rREi)
+ 56TREoiEoj
.
(3)
Using Z and P expressions one may write the component j, of the current j = crotl + dP,/Jt, which oscillates at frequency 0, j,x = 2RoI (4E& + 7E&)E,,
- 3Eo.~Eo,E1,1
17
also that if LR is elliptic potarized from the outset (at z = O), then interaction leads to its polarization plane rotation by
(p = iarctg
26sin[(w/c)(nll
- nl)l] (6)
l-62
angle (here 6 = EI,/EI,, El, and El, are the initial ellipse principal axes) [ S] . Consider now the problem of LR h~monic generation Here we restrict our analysis only to the problem of the second harmonic production. We write its field in the form E = E2 sin( 2ut - kzz) ,
B = B2 sin(2ot
- k.g) ,
(7)
where the wave vector kl = 2on(2w)/c. On substituting (7), (2) and Ea values into ( 1) and using Eqs. (4)) we get the component j, of current j, which oscillates at frequency 20~. The analysis proceeds as before showing that the second harmonic electric field is directed along Ea, i.e. E2 I[ Eo. If Ea Ij El 11p, then n(w) = nil, n( 20) = ~11. Since in this case the LR and the second harmonic phase synchronization occurs, the conversion coefficient
xcos(ot-kiz), jiy = ~R~[-~Eo~EoJEI~
+
(4E& +
7E&)E1yl
x cos(wt - k,z). In order to simplify the further analysis, we assume that the EO vector is directed along the x axis (i.e., Ea /I i). On solving the Maxwell equations 47T. rotB=Tj+;z,
1 JE
rOtE=: -.! !!!
is directly proportional to the interaction region length E,This is not the case if EO -I_2. In this device n(w) = nl, whereas ~(20) = nil; and the conversion coefficient rll =
c at ’
l(jTR@;
_
l)EoE;
1sin(w/c’(nL - nll)‘l . nl
-
1111
we obtain ki = 3 nll._~ 9
BI = q,iG
1
(4)
nl = 1 + 28rREg
(5)
where nil = 1 + 16rREi,
are the refractive indices of the x and y projections of LR (2). Apparently, LR-DV interaction transforms the linear polarized LR into the elliptic polarized one. Note
Nevertheless, if the parameter (o/c) (rtl- 1~11) 1 << 1, then ~1 = 7~1i/12, i.e. ~1 N I as well. Since all of the obtained effects are determined quantitatively by the EO field, one may use them for field diagnostics. Moreover, an alternating field EO = E sin coot may be studied in the same way if its frequency 00 is not too large (wa < min{o, c/l}). This work was supported by International and Technology Center Grant No. A-87.
Science
18
S.G. Oganesyan/Physics Letters A 248 (1998) 16-18
References [i J V.M. H~utunian, S.C. Og~esy~, Phys. Rep. 270 (19%) 217. [ 21 K.T. McDonald, Proposal for experimental studies of nonlinear quantum electrodynamics, DOEfER/3072-38, Sept. 2, 1986. draft. [3] S.S. Schweber, An Introduction to Relativistic Quantum Field Theory (New York, 1962) Section 4g.
141 W. Heisenberg, H. Euler. 2. Phys. 98 (3936) 714. IS] E.M. Lifshitz, L.P. Pitaevsky, Relativistic Qu~tum Theory, Vol. 2 (Moscow, 1971) Section 126. f6] A.A. Grib. S.G. Mameyev, V.M. Mostep~enko, Vacuum Quantum Effects in Strong Fields (Moscow, 1988) Section 6. [ 71 A.I. Ahiezer. V.B. Berestetzky, Quantum Electrodynamics (Moscow, 1969) Section 41.2. [8] S.G. Oganesyan, JTF Lett., to be published.