Informatics Educational Institutions & Programs

In strong-field laser physics, ponderomotive energy is the cycle-averaged quiver energy of a free electron in an electromagnetic field.^[1]

Equation

The ponderomotive energy is given by

<img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d10941f82bde0669197c0df9a1edd38d9ddcdad" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:12.284ex; height:6.843ex;" alt="{\displaystyle U_{p}={e^{2}E^{2} \over 4m\omega _{0}^{2}}}">

,

where $<img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd253103f0876afc68ebead27a5aa9867d927467" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle e}">$ is the electron charge, $<img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}">$ is the linearly polarised electric field amplitude, $<img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a713d16c489051d4f515e12b1f86061c6be799b" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.5ex; height:2.009ex;" alt="{\displaystyle \omega _{0}}">$ is the laser carrier frequency and $<img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}">$ is the electron mass.

In terms of the laser intensity $<img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/535ea7fc4134a31cbe2251d9d3511374bc41be9f" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}">$ , using $<img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1858105bffc887fcf4dc224b5f2754f986324840" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.448ex; height:3.176ex;" alt="{\displaystyle I=c\epsilon _{0}E^{2}/2}">$ , it reads less simply:

<img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a837f778f63c4f612315b778badb7c73be62886f" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:30.447ex; height:6.843ex;" alt="{\displaystyle U_{p}={e^{2}I \over 2c\epsilon _{0}m\omega _{0}^{2}}={2e^{2} \over c\epsilon _{0}m}\cdot {I \over 4\omega _{0}^{2}}}">

,

where $<img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d2cae6289b0fe626d1f9472a3416ac73e87bc5a3" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.998ex; height:2.009ex;" alt="{\displaystyle \epsilon _{0}}">$ is the vacuum permittivity.

For typical orders of magnitudes involved in laser physics, this becomes:

<img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04adb02a20d9113af3b044f65953c1718b577963" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:40.135ex; height:3.509ex;" alt="{\displaystyle U_{p}(\mathrm {eV} )=9.33\cdot I(10^{14}\mathrm {W/cm} ^{2})\cdot \lambda (\mathrm {\mu m} )^{2}}">

,^[2]

where the laser wavelength is $<img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d56ccc5bd596dee259717e827d575a7e8a77706" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.618ex; height:2.843ex;" alt="{\displaystyle \lambda =2\pi c/\omega _{0}}">$ , and $<img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}">$ is the speed of light. The units are electronvolts (eV), watts (W), centimeters (cm) and micrometers (μm).

Atomic units

In atomic units, $<img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9cffd46f3e645175b2a3558847239d67a44d12fb" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.483ex; height:2.176ex;" alt="{\displaystyle e=m=1}">$ , $<img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e7a7a9db6c803ea8c683c2254d4a55e462cae9b" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.916ex; height:2.843ex;" alt="{\displaystyle \epsilon _{0}=1/4\pi }">$ , $<img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/26168dc553362e2e0100804eba8e7d785f100082" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.755ex; height:2.176ex;" alt="{\displaystyle \alpha c=1}">$ where $<img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55eb6d9bcf4654c0fa682001756e9fa66582d3e1" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.398ex; height:2.843ex;" alt="{\displaystyle \alpha \approx 1/137}">$ . If one uses the atomic unit of electric field,^[3] then the ponderomotive energy is just

<img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07469efd44e6a68796bb595c4cc47789f9eaf3d2" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:10.891ex; height:6.843ex;" alt="{\displaystyle U_{p}={\frac {E^{2}}{4\omega _{0}^{2}}}.}">

Derivation

The formula for the ponderomotive energy can be easily derived. A free particle of charge $<img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}">$ interacts with an electric field $<img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c911a25876bd54c825c912a2bc66dc58bfd23d6c" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.756ex; height:2.843ex;" alt="{\displaystyle E\,\cos(\omega t)}">$ . The force on the charged particle is

<img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b1fb4abd38bc33ad0af136af8c87beca47223b8" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.664ex; height:2.843ex;" alt="{\displaystyle F=qE\,\cos(\omega t)}">

.

The acceleration of the particle is

<img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e60708a135df53b893e576fb3680f73bf8420878" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:23.253ex; height:5.343ex;" alt="{\displaystyle a_{m}={F \over m}={qE \over m}\cos(\omega t)}">

.

Because the electron executes harmonic motion, the particle's position is

<img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7486359b26d36aebca7103080442b4878008d9f8" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:51.228ex; height:7.676ex;" alt="{\displaystyle x={-a \over \omega ^{2}}=-{\frac {qE}{m\omega ^{2}}}\,\cos(\omega t)=-{\frac {q}{m\omega ^{2}}}{\sqrt {\frac {2I_{0}}{c\epsilon _{0}}}}\,\cos(\omega t)}">

.

For a particle experiencing harmonic motion, the time-averaged energy is

<img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0153bfbe4d013607ee644d5027ae12109a8cc673" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:23.326ex; height:4.676ex;" alt="{\displaystyle U=\textstyle {\frac {1}{2}}m\omega ^{2}\langle x^{2}\rangle ={q^{2}E^{2} \over 4m\omega ^{2}}}">

.

In laser physics, this is called the ponderomotive energy $<img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5fdf01ea5c43d1e338d8b847556322c42a8c65a9" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.647ex; height:2.843ex;" alt="{\displaystyle U_{p}}">$ .

References and notes

^ Highly Excited Atoms. By J. P. Connerade. p. 339
^ https://www.phys.ksu.edu/personal/cdlin/class/class11a-amo2/atomic_units.pdf ^{[bare URL PDF]}
^ CODATA Value: atomic unit of electric field

[1] Highly Excited Atoms. By J. P. Connerade. p. 339

[2] ttps://www.phys.ksu.edu/personal/cdlin/class/class11a-amo2/atomic_units.pdf ^{[bare URL PDF]}

[3] CODATA Value: atomic unit of electric field

[1]

[2]

[3]

Informatics Educational Institutions & Programs

Contents

Equation

Atomic units

Derivation

See also

References and notes