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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

U_{p}={e^{2}E^{2} \over 4m\omega _{0}^{2}}

,

where $e$ is the electron charge, $E$ is the linearly polarised electric field amplitude, $\omega _{0}$ is the laser carrier frequency and $m$ is the electron mass.

In terms of the laser intensity $I$ , using $I=c\epsilon _{0}E^{2}/2$ , it reads less simply:

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 $\epsilon _{0}$ is the vacuum permittivity.

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

U_{p}(\mathrm {eV} )=9.33\cdot I(10^{14}\ \mathrm {W/cm} ^{2})\cdot \lambda ^{2}(\mathrm {\mu m} ^{2})

,^[2]

where the laser wavelength is $\lambda =2\pi c/\omega _{0}$ , and $c$ is the speed of light. The units are electronvolts (eV), watts (W), centimeters (cm) and micrometers (μm).

Atomic units

In atomic units, $e=m=1$ , $\epsilon _{0}=1/4\pi$ , $\alpha c=1$ where $\alpha \approx 1/137$ . If one uses the atomic unit of electric field,^[3] then the ponderomotive energy is just

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 $q$ interacts with an electric field $E\,\cos(\omega t)$ . The force on the charged particle is

F=qE\,\cos(\omega t)

.

The acceleration of the particle is

a_{m}={F \over m}={qE \over m}\cos(\omega t)

.

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

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

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 $U_{p}$ .

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

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[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]

@@ Line 1: / Line 1: @@
+In strong-field [[laser physics]], '''ponderomotive energy''' is the cycle-averaged quiver energy of a free electron in an [[electromagnetic field]].<ref>''Highly Excited Atoms''. By J. P. Connerade. p. 339</ref>
-{{Unreferenced|date=January 2007}}
-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,
+The ponderomotive energy is given by
+:<math>U_p = {e^2 E^2 \over 4m \omega_0^2}</math>,
+where <math>e</math> is the [[electron charge]], <math>E</math> is the linearly [[Polarization (waves)|polarised]] electric field amplitude, <math>\omega_0</math> is the laser [[carrier frequency]] and <math>m</math> is the [[electron mass]].
-<math>U_p=e^2E_a^2/4m\omega_0^2</math>
+In terms of the laser [[Intensity (physics)|intensity]] <math>I</math>, using <math>I=c\epsilon_0 E^2/2</math>, it reads less simply:
-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]].
+:<math>U_p={e^2 I \over 2 c \epsilon_0 m \omega_0^2}={2e^2 \over c \epsilon_0 m} \cdot {I \over 4\omega_0^2}</math>,
+where  <math>\epsilon_0</math> is the vacuum permittivity.
-===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>.
+For typical orders of magnitudes involved in laser physics, this becomes:
-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>.
+:<math> U_p (\mathrm{eV}) = 9.33 \cdot I(10^{14}\ \mathrm{W/cm}^2) \cdot \lambda^2(\mathrm{\mu m}^2) </math>,<ref>https://www.phys.ksu.edu/personal/cdlin/class/class11a-amo2/atomic_units.pdf {{Bare URL PDF|date=March 2022}}</ref>
-The acceleration of the electron is
-:<math>a_{m} = F/m = (eE/m) \, \exp(-i\omega t)</math>.
+where the laser wavelength is <math>\lambda= 2\pi c/\omega_0</math>, and <math>c</math> is the speed of light. The units are electronvolts (eV), watts (W), centimeters (cm) and micrometers (μm).
-Because the electron executes harmonic motion, the electron's position is
-:<math>x = -a /\omega^2 = -(eE/m\omega^2) \, \exp(-i\omega t)</math>.
+===Atomic units===
-For a particle experiencing harmonic motion, the time-averaged energy is
+In [[atomic units]], <math>e=m=1</math>, <math>\epsilon_0=1/4\pi</math>, <math>\alpha c=1</math> where <math>\alpha \approx 1/137</math>. 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
-:<math>U = \textstyle{\frac{1}{2}}m\omega^2 \langle x^2\rangle = e^2 E^2/ 4 m \omega^2 </math>.
+:<math>U_p = \frac{E^2}{4\omega_0^2}.</math>
+==Derivation==
-In laser physics, this is called the ponderomotive energy <math>U_p</math>.
+The formula for the ponderomotive energy can be easily derived. A free particle of charge
+<math>q</math> interacts with an electric field <math>E \, \cos(\omega t)</math>. The force on the charged particle is
+:<math>F = qE \, \cos(\omega t)</math>.
+The acceleration of the particle is
-===Atomic units===
+:<math>a_{m} = {F \over m} = {q E \over m} \cos(\omega t)</math>.
-{{also|Atomic units}}
+Because the electron executes harmonic motion, the particle's position is
-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>[http://physics.nist.gov/cgi-bin/cuu/Value?auefld%20CODATA%20Value:%20atomic%20unit%20of%20electric%20field http://physics.nist.gov/cgi-bin/cuu/Value?auefld CODATA Value: atomic unit of electric field] </ref> then the ponderomotive energy is just
-:[[Atomic units|<math>U_p = </math>]]<math>\frac{I}{4\omega^2}.</math>
+:<math>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)</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 = {q^2 E^2 \over 4 m \omega^2}</math>.
+In laser physics, this is called the ponderomotive energy <math>U_p</math>.
 ==See also==
 *[[Ponderomotive force]]
+*[[Electric constant]]
 *[[Harmonic generation]]
 *[[List of laser articles]]
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 [[Category:Laser science]]
 [[Category:Energy (physics)]]
 {{Physics-stub}}