1. The Study of Harmonics of the Heisenberg-Euler Lagrangian
Kiran Rao University of Western Ontario
S.R. Valluri University of Western Ontario
U. Jentschura Max Planck Institute for Nuclear Physics, Heidelberg, Germany
D. Lamm Georgia Institute of Technology, Atlanta, GA
In memory of Prof. Victor Elias

2. Outline The Heisenberg-Euler Lagrangian Nonlinear Maxwell’s equations
Fourier expansion of scattered fields
Harmonics in the scattered fields
Subharmonics

3. Introduction
Existence of higher harmonics of electromagnetic fields for the special case of strong fields has been demonstrated by Valluri and Bhartia (1980).
We wish to demonstrate the existence of higher harmonics in the scattered fields for arbitrary field strengths in the interaction of two monochromatic plane waves.
This is done by means of the nonlinear Heisenberg-Euler Lagrangian.
Ding and Kaplan (1989) showed that second-harmonic generation occurs in photon-photon scattering of intense laser radiation due to broken symmetry of the interaction.

4. Quantum Corrections to the Maxwell Lagrangian
Maxwell Lagrangian: L0 = ½(E2 – B2)
Nonlinear corrections
Born-Infeld Lagrangian
Heisenberg-Euler Lagrangian
- Heisenberg, Euler, Schwinger, Weisskopf
- valid for slowly varying fields
- imply a nonlinear behaviour of the electromagnetic field
- an escape from the “infinities” from the classical concept of a Maxwellian point singularity

5. Heisenberg-Euler Lagrangian
L = LMaxwell + LHE
Defining and , Maxwell’s equations are given by

6. Heisenberg-Euler Lagrangian
The general expression for the Heisenberg-Euler Lagrangian is
This is valid for slowly varying fields: ,
This form is not convenient for computations, so a series representation is used instead.

7. Heisenberg-Euler Lagrangian
In the special case of weak fields,
For strong fields (E and B are much larger than the critical field) and parallel polarization,
where
For strong fields and perpendicular polarization,
(Valluri and Bhartia, 1978)

8. Heisenberg-Euler Lagrangian
The general expression for the Heisenberg-Euler Lagrangian is given in series form by
where
(Jentschura et al., 2002)

9. Modified Maxwell’s Equations
Explicitly, the modified Maxwell’s equations become
where

10. Modified Maxwell’s Equations
and are defined as follows:
ie.

11. Solution of equations
These equations are solved through a series expansion:
and are the fields with zero sources (the classical fields).

12. Solution of equations
The scattered fields Ef and Bf satisfy wave equations:
Here, and
These equations are solved through Green’s functions.

13. Solution of equations
The additional scattered fields (non-classical) are given by
where

14. The Initial Fields
We assume the initial fields are two converging plane waves polarized parallel to each other, travelling in the +z and –z directions.
Perpendicular polarizations can also be considered.

15. The Initial Fields
We also assume the region of interaction of the waves is a small region of cross section A and length L.
The vector from the source to the field point is approximately the same as .

16. Fourier series expansion
Instead of using the expressions for Ei and Bi directly, we first expand these quantities in a 2-dimensional Fourier series (in 1 = ωt and 2 = kz).

17. Fourier series expansion
The Fourier coefficients Amn and Amn’ are plotted vs. m and n.

18. The Scattered Fields
Using these expansions, we calculate the scattered fields and :
where

19. The Scattered Fields
Intensity distribution in space of the first harmonic of the electric field:

20. The Scattered Fields
Intensity distribution in space of the second harmonic of the electric field:

21. The Scattered Fields
Intensity distribution in space of the third harmonic of the electric field:

22. Fourier coefficients
We have determined the contribution of each harmonic by calculating the Fourier coefficients numerically.
Expression of the Fourier coefficients in terms of the generic nonlinear expressions involving the electric and magnetic fields is an algebraically forbidding task.
The important question of using symbolic packages like MAPLE or Mathematica in this connection warrants further study.

23. Subharmonics?
The wave equation for a higher order term in the power series for E or B may contain a source term that is cubic in the field.
Consequently, there may be subharmonics in the scattered fields (waves of non-integer frequencies).
These subharmonics are expected to be very weak and difficult to detect. (This is also the case with subharmonics produced by musical instruments, as found by violinist Kimura.)

24. Conclusions
Nonlinear interaction of electromagnetic waves (scattering) may be verified experimentally in electron synchrotrons, storage rings and high intensity lasers due to the high intensity of electromagnetic fields generated.
They may also verify the existence of higher harmonics and/or subharmonics.
Other potential sources include pulsars and magnetars.

25. References
1. Dittrich, W. and Gies, H. Probing the quantum vacuum. Springer Tracts in Physics, Vol Springer, Berlin, Heidelberg, New York, 2000.
2. Jentschura, U.D., Gies, H., Valluri, S.R., Lamm, D.R. and Weniger, E.J. QED effective action revisited. Can J. Phys. 80: (2002).
3. Valluri, S.R. and Bhartia, P. An analytical proof for the generation of higher harmonics due to the interaction of plane electromagnetic waves. Can. J. Phys., 58:116 (1980).
4. Valluri, S.R., Lamm, D., and Mielniczuk, W.J. Applications of the representation of the Heisenberg-Euler Lagrangian by means of special functions. Can. J. Phys. 71: 389 (1993).

26. References
5. Bhartia, P. and Valluri, S.R. Non-linear scattering of light in the limit of ultra-strong fields. Can. J. Phys., 56:1122 (1978).
6. Wheeler, J. Craig, et al. (2000). Asymmetric supernovae, pulsars, magnetars, and gamma-ray bursts. The Astrophysical Journal. 537 (2), pp
7. Dall'Osso, Simone, et al. (2007). Newborn magnetars as sources of gravitational radiation: constraints from high energy observations of magnetar candidates. ArXiv Astrophysics e-prints [online]  Accessed: Feb. 18, 2007.
8. Baring, M.G. Astrophys. J. 440, L69 (1995).
9. Heyl, J.S. and Hernquist, L. Phys. Rev. D.: Part. Fields, 55:2449 (1997).