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2012-06-02T19:49:59Z

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In strong field [[laser]] [[physics]], the term '''Ponderomotive Energy'''<ref>Highly Excited Atoms. By J. P. Connerade. p339</ref> refers to the cycle averaged quiver energy of a free electron in an E-field.

==Equation ==
The Ponderomotive Energy equation is given by,
:[[Atomic units|<math>U_p = </math>]][[electron charge|<math>e^2</math>]][[Fresnel_equations#Amplitude_equations|<math>E_a^2</math>]][[electron mass|<math>/4m</math>]][[Frequency_comb#Carrier-envelope_offset_control|<math>\omega_0^2</math>]]

Where <math>e</math> is the [[electron charge]], <math>E_a</math> is the linearly [[polarised]] electric field amplitude, <math>\omega_0^2</math> is the laser carrier frequency and <math>m</math> is the [[electron mass]].

===Description===
In terms of the laser intensity <math>I</math>, using <math>I=c\epsilon_0 E_a^2/2</math>, it reads less simply <math>U_p=e^2 I/2 c \epsilon_0 m \omega_0^2=2e^2/c \epsilon_0 m \times I/4\omega_0^2</math>. Now, [[atomic units]] provide <math>e=m=1</math>, <math>\epsilon_0=1/4\pi</math>, <math>\alpha c=1</math> where <math>\alpha \approx 1/137</math>. Thus, <math>2e^2/c \epsilon_0 m=8\pi/137</math>.

The formula for the ponderomotive energy can be easily derived. A free electron of charge
<math>e</math> interacts with an electric field <math>E \, \exp(-i\omega t)</math>. The force on the electron is
:<math>F = eE \, \exp(-i\omega t)</math>.

The acceleration of the electron is
:<math>a_{m} = F/m = (eE/m) \, \exp(-i\omega t)</math>.

Because the electron executes harmonic motion, the electron's position is
:<math>x = -a /\omega^2 = -\frac{eE}{m\omega^2} \, \exp(-i\omega t) = -\frac{e}{m\omega^2} \sqrt{\frac{2I_0}{c\epsilon_0}} \, \exp(-i\omega t)</math>.

For a particle experiencing harmonic motion, the time-averaged energy is
:<math>U = \textstyle{\frac{1}{2}}m\omega^2 \langle x^2\rangle = e^2 E^2/ 4 m \omega^2 </math>.

In laser physics, this is called the ponderomotive energy <math>U_p</math>.

===Atomic units===
{{also|Atomic units}}

Converting between SI units and atomic units is more subtle than the introduction suggests. As presented, the Ponderomotive energy in atomic units appears to have some issues. If one uses the [[Atomic units|atomic unit of electric field]],<ref>CODATA Value: [http://physics.nist.gov/cgi-bin/cuu/Value?auefld atomic unit of electric field] </ref> then the ponderomotive energy is just
:[[Atomic units|<math>U_p = </math>]][[Coulomb's law|<math>\frac{I}{4\omega^2}.</math>]]

==See also==
*[[Ponderomotive force]]
*[[Electric constant]]
*[[Harmonic generation]]
*[[List of laser articles]]

==References and notes==
{{reflist}}

[[Category:Laser science]]
[[Category:Energy (physics)]]
{{Physics-stub}}

Subspace theorem - Revision history

Deltahedron: more on consequences