1. LOW FREQUENCY LIMIT OF LASER FIELDS
LASER PHYSICS 2014
SOFIA, BULGARIA
15 July 2014
LOW FREQUENCY LIMIT
OF LASER FIELDS
H. R. Reiss
Max Born Institute, Berlin, Germany
American University, Washington, DC, USA

2. Preview →The tunneling model applies only to longitudinal fields.←
→ Laser fields are transverse. ←
Longitudinal and transverse fields are separate electromagnetic phenomena;
there can be no gauge transformation that connects them.
The Göppert-Mayer gauge transformation connects 2 longitudinal-field gauges.
At low frequencies, radiation pressure becomes increasingly important.
Radiation pressure exists only for transverse fields.
The often-used criterion that strong-field theories approach constant-electric-field results at low frequencies guarantees incorrect low-frequency answers.
Radiation pressure means that the dipole approximation fails at low frequencies.
The “tunneling limit” is actually a “tunneling-is-wrong” limit.
Dipole-approximation TDSE does not apply for low frequencies.
The 1980 HRR SFA adapts properly to low frequencies; other “SFAs” do not.

3. Longitudinal fields can oscillate with time,
Longitudinal fields are static or quasistatic electric (QSE) fields.
Lorentz invariant: E2- B2 > 0.
The electric field sets the sole preferred direction in space.
Longitudinal fields can oscillate with time,
but they do not propagate.

4. Typical diagram showing the addition of two longitudinal fields:
Coulomb potential + scalar potential for QSE field.
This is from “ETH Life”, but such diagrams abound in the literature for the unphysical explanation of ionization by lasers.

5. Transverse fields propagate with velocity c
We will adopt the convention that:
Transverse ↔ Plane Wave ↔ Propagating
Lorentz invariant: E2- B2 = 0; necessary for a field to be transverse.
That is: the magnetic field must always be present.
There are 3 preferred directions in space for: the electric field; the magnetic field; the propagation direction; all are mutually perpendicular.
Transverse fields propagate with velocity c

6. regardless of the gauge (LG or VG).
Application of the dipole approximation means replacing a transverse field with a longitudinal field,
regardless of the gauge (LG or VG).

7. THERE IS NO POSSIBLE GAUGE TRANSFORMATION THAT RELATES
LONGITUDINAL FIELDS AND TRANSVERSE FIELDS ARE
SEPARATE AND DISTINCT ELECTROMAGNETIC PHENOMENA
A change of gauge does not change the fields.
Longitudinal fields are identified by E2- B2 > 0.
Transverse fields are identified by E2- B2 = 0.
THERE IS NO POSSIBLE GAUGE TRANSFORMATION THAT RELATES
LONGITUDINAL AND TRANSVERSE FIELDS
… but what about the Göppert-Mayer gauge transformation?

8. LG and VG both describe QSE fields;
Göppert-Mayer (GM) Gauge Transformation
Length Gauge (LG) ↔ Velocity Gauge (VG)
LG and VG both describe QSE fields;
Neither LG nor VG describes a PW field.
There is NO DIRECT CONNECTION to PW fields.
Does this matter?
The dipole approximation should be valid.

9. STRONG FIELDS ARE AT THE TOP AND WEAK FIELDS ARE AT THE BOTTOM.
Extreme qualitative differences exist between longitudinal and transverse fields.
For QSE fields (i.e. longitudinal fields), the electric field is the sole descriptor of electromagnetic phenomena. In atomic physics,
E = 1 a.u.
defines the boundary between strong field and weak fields.
In the following intensity vs. frequency figure for QSE fields:
STRONG FIELDS ARE AT THE TOP
AND WEAK FIELDS ARE AT THE BOTTOM.

11. In the previous slide, there is an upper limit on the frequency due to the failure of the dipole approximation at high frequencies.
If a tunneling model is to be used to describe laser-induced ionization, there is a much more restrictive upper limit on frequency due to the fact that tunneling applies only if many “photons” are required for a single ionization event.

13. STRONG FIELDS ARE AT THE LEFT AND WEAK FIELDS ARE AT THE RIGHT.
For PW fields (i.e. transverse fields), many factors play a role:
In the relativistic domain, the dipole approximation is certainly not valid. This sets a low-frequency limit.
Radiation pressure exists, and will introduce momentum in the direction of field propagation. This becomes significant when the momentum due to this cause introduces a kinetic energy approaching the binding energy of an electron.
Radiation pressure produces a displacement of the electron in the propagation direction. This is important when it approaches 1 a.u.
These radiation pressure effects impose more stringent constraints than just generic relativistic effects.
In the following intensity vs. frequency figure for PW fields:
STRONG FIELDS ARE AT THE LEFT
AND WEAK FIELDS ARE AT THE RIGHT.

15. There is a domain in which QSE fields have behavior similar to PW fields, as long as the dipole approximation is valid.
THIS IS CALLED THE
“DIPOLE OASIS”.
A tunneling model can be used to approximate PW field effects only within a limited subset of the Dipole Oasis.
“TUNNEL OASIS”.

17. Point A: Point B: Points labeled A, B in the figure:
Magnetic field effects already exist at Points A and B.

18. Longitudinal fields at low frequencies
As ω → 0, an ac field goes to a dc field.
Processes become slower and go to an adiabatic limit.
The Keldysh parameter goes to 0:
The Keldysh parameter is called the “adiabaticity parameter” for this reason.
Transverse fields behave completely differently.

19. Transverse fields at low frequencies
As ω ↓ :
x-ray → UV → visible → IR → microwave → RF → ELF
ELF (extreme low frequency) fields have absolutely no relationship to constant electric fields.
Instead of representing an adiabatic limit, ELF fields require enormous power to produce them.
For transverse fields, ω ↓ represents an approach to an extreme relativistic limit.

20. K → 0 is an extreme relativistic domain where B = E and
tunneling has no possible applicability.

22. Two examples of an ELF field are shown next:
One example is a man-made ELF field (76 Hz);
the other example is an ELF field in Nature (7.8Hz).

23. From Wikipedia

24. From Wikipedia

25. Note:
At the Schumann resonant frequency of 7.83 Hz, the wavelength is 6 Earth radii.
This would seem to be an adiabatic process since the frequency is low even for sound waves.
It is not adiabatic: The entire wavelength of 3 Earth diameters acts coherently.

26. Summary to this point So … what can be done?
Laser fields are transverse.
Tunneling methods apply only to longitudinal fields.
The E2 - B2 Lorentz invariant is different for transverse and longitudinal fields.
Therefore, there is no possible gauge transformation that connects them.
There is a limited domain of field parameters (Tunnel Oasis) where transverse and longitudinal fields have similar effects.
The Tunnel Oasis is bounded at low frequencies because of radiation pressure, a magnetic field effect.
Therefore, the dipole approximation fails at low frequencies.
Longitudinal fields at low frequencies tend to constant electric fields.
Transverse fields at low frequencies tend to ELF behavior, not to constant-field behavior
Failure of the dipole approximation means that imposition of the requirement that constant-electric-field results should obtain as ω→0 is a guarantee of wrong results.
Failure of the dipole approximation at low frequencies means that TDSE is not applicable.
So … what can be done?

27. ONE SOLUTION: Solve the Dirac equation numerically in 3 spatial dimensions.
This can be done in principle, but it is extremely difficult.
ANOTHER SOLUTION: Use relativistic Volkov techniques.
This has been done for initial ground-state hydrogen, but it can be applied to any hydrogenic state and, with relativistic Hartree-Fock, it can be extended to any initial state.
See: HRR, JOSA B 7, 574 (1990) for Dirac-Volkov
See: HRR, PRA 42, 1476 (1990) for Klein-Gordon-Volkov
THERE ARE MANY SITUATIONS WHERE TUNNELING IS NOT VALID, BUT WHERE CONDITIONS ARE NOT REALLY RELATIVISTIC; WHAT THEN?
The problem can be formulated relativistically (so that the magnetic field is always present to make propagation possible), and then a nonrelativistic limit can be employed when justified by specific local conditions.
This is the nature of the 1980 SFA: HRR, PRA 22, 1680 (1980).
The 1980 SFA predicts both the LES (low-energy structure) and the high energy plateau; it is especially accurate for circular polarization and higher photoelectron energies in general; it is accurate for frequencies much higher and much lower than the limits of the Tunnel Oasis.

28. Questions are welcome, now or later.
END