Abstract

A theoretical model of the nonlinear electron oscillations in an atom exposed to electro-magnetic wave is described. This model considers electron motion as a nonharmonic periodic function, unlike the current approach based on a perturbation solution of the linear harmonic oscillator. Our work demonstrates the use of realistic electron potential and correct radiation damping in the modeling of optical electron response. One of the main benefits of our approach is that it provides a way of computing the efficiencies of the up- and down-conversion of laser frequency.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. H. A. Lorentz, The theory of electrons and its applications to the phenomena of light and radiant heat, second edition (B. G. Teubner, 1916).
  2. R.P. Feynman, R.B. Leighton, and M. Sands, The Feynman Lectures on Physics, Vol. 13 (Addison Wesley; Later Printing edition 1971).
  3. Handbook of Optics Third Edition, 5 Volume Set 3rd Edition (McGraw-Hill Education; 3 edition December 2009).
  4. S.A. Akhmanov and A.Yu. Nikitin, Physical Optics (Clarendon Press, 1997), chapter 8.
  5. N. Bloembergen, Nonlinear optics. A lecture note (Harvard University, W. A. Benjamin, Inc., 1965).
  6. R. W. Boyd, Nonlinear Optics (Academic Press, 2003).
  7. V.V. Semak and M.N. Shneider, Invicem Lorentz Oscillator Model (ILOM), arXiv:1709.02466.
  8. H. Helmholtz, Sensation of Tone (London, 1895).
  9. A. Blaguire, Nonlinear System Analysis (Acad. Press, 1966).
  10. C. Hayashi, Nonlinear Oscillations in Physical Systems (McGraw-Hill Book Co, 1964).
  11. B. Santra, M. N. Shneider, and R. Car, “In situ characterization of nanoparticles using rayleigh scattering,” Sci. Rep. 7(1), 40230 (2017).
    [Crossref]
  12. N. G. Khlebtsov, “Anisotropic properties of plasmonic nanoparticles: depolarized light scattering, dichroism, and birefringence,” J. Nanophotonics 4(1), 041587 (2010).
    [Crossref]
  13. Y. P. Raizer and M. N. Shneider, “Longitudional structure of the cathode portions of glow discharges,” High Temp. 29(6), 833–843 (1991).
  14. Y. P. Raizer, M. N. Shneider, and N. A. Yatsenko, Radio-Frequency Capacitive Discharges (CRC Press, 1995).
  15. M. N. Shneider, “Ponderomotive perturbations of low density low-temperature plasma under laser Thomson scattering diagnostics,” Phys. Plasmas 24(10), 100701 (2017).
    [Crossref]

2017 (2)

B. Santra, M. N. Shneider, and R. Car, “In situ characterization of nanoparticles using rayleigh scattering,” Sci. Rep. 7(1), 40230 (2017).
[Crossref]

M. N. Shneider, “Ponderomotive perturbations of low density low-temperature plasma under laser Thomson scattering diagnostics,” Phys. Plasmas 24(10), 100701 (2017).
[Crossref]

2010 (1)

N. G. Khlebtsov, “Anisotropic properties of plasmonic nanoparticles: depolarized light scattering, dichroism, and birefringence,” J. Nanophotonics 4(1), 041587 (2010).
[Crossref]

1991 (1)

Y. P. Raizer and M. N. Shneider, “Longitudional structure of the cathode portions of glow discharges,” High Temp. 29(6), 833–843 (1991).

Akhmanov, S.A.

S.A. Akhmanov and A.Yu. Nikitin, Physical Optics (Clarendon Press, 1997), chapter 8.

Blaguire, A.

A. Blaguire, Nonlinear System Analysis (Acad. Press, 1966).

Bloembergen, N.

N. Bloembergen, Nonlinear optics. A lecture note (Harvard University, W. A. Benjamin, Inc., 1965).

Boyd, R. W.

R. W. Boyd, Nonlinear Optics (Academic Press, 2003).

Car, R.

B. Santra, M. N. Shneider, and R. Car, “In situ characterization of nanoparticles using rayleigh scattering,” Sci. Rep. 7(1), 40230 (2017).
[Crossref]

Feynman, R.P.

R.P. Feynman, R.B. Leighton, and M. Sands, The Feynman Lectures on Physics, Vol. 13 (Addison Wesley; Later Printing edition 1971).

Hayashi, C.

C. Hayashi, Nonlinear Oscillations in Physical Systems (McGraw-Hill Book Co, 1964).

Helmholtz, H.

H. Helmholtz, Sensation of Tone (London, 1895).

Khlebtsov, N. G.

N. G. Khlebtsov, “Anisotropic properties of plasmonic nanoparticles: depolarized light scattering, dichroism, and birefringence,” J. Nanophotonics 4(1), 041587 (2010).
[Crossref]

Leighton, R.B.

R.P. Feynman, R.B. Leighton, and M. Sands, The Feynman Lectures on Physics, Vol. 13 (Addison Wesley; Later Printing edition 1971).

Lorentz, H. A.

H. A. Lorentz, The theory of electrons and its applications to the phenomena of light and radiant heat, second edition (B. G. Teubner, 1916).

Nikitin, A.Yu.

S.A. Akhmanov and A.Yu. Nikitin, Physical Optics (Clarendon Press, 1997), chapter 8.

Raizer, Y. P.

Y. P. Raizer and M. N. Shneider, “Longitudional structure of the cathode portions of glow discharges,” High Temp. 29(6), 833–843 (1991).

Y. P. Raizer, M. N. Shneider, and N. A. Yatsenko, Radio-Frequency Capacitive Discharges (CRC Press, 1995).

Sands, M.

R.P. Feynman, R.B. Leighton, and M. Sands, The Feynman Lectures on Physics, Vol. 13 (Addison Wesley; Later Printing edition 1971).

Santra, B.

B. Santra, M. N. Shneider, and R. Car, “In situ characterization of nanoparticles using rayleigh scattering,” Sci. Rep. 7(1), 40230 (2017).
[Crossref]

Semak, V.V.

V.V. Semak and M.N. Shneider, Invicem Lorentz Oscillator Model (ILOM), arXiv:1709.02466.

Shneider, M. N.

B. Santra, M. N. Shneider, and R. Car, “In situ characterization of nanoparticles using rayleigh scattering,” Sci. Rep. 7(1), 40230 (2017).
[Crossref]

M. N. Shneider, “Ponderomotive perturbations of low density low-temperature plasma under laser Thomson scattering diagnostics,” Phys. Plasmas 24(10), 100701 (2017).
[Crossref]

Y. P. Raizer and M. N. Shneider, “Longitudional structure of the cathode portions of glow discharges,” High Temp. 29(6), 833–843 (1991).

Y. P. Raizer, M. N. Shneider, and N. A. Yatsenko, Radio-Frequency Capacitive Discharges (CRC Press, 1995).

Shneider, M.N.

V.V. Semak and M.N. Shneider, Invicem Lorentz Oscillator Model (ILOM), arXiv:1709.02466.

Yatsenko, N. A.

Y. P. Raizer, M. N. Shneider, and N. A. Yatsenko, Radio-Frequency Capacitive Discharges (CRC Press, 1995).

High Temp. (1)

Y. P. Raizer and M. N. Shneider, “Longitudional structure of the cathode portions of glow discharges,” High Temp. 29(6), 833–843 (1991).

J. Nanophotonics (1)

N. G. Khlebtsov, “Anisotropic properties of plasmonic nanoparticles: depolarized light scattering, dichroism, and birefringence,” J. Nanophotonics 4(1), 041587 (2010).
[Crossref]

Phys. Plasmas (1)

M. N. Shneider, “Ponderomotive perturbations of low density low-temperature plasma under laser Thomson scattering diagnostics,” Phys. Plasmas 24(10), 100701 (2017).
[Crossref]

Sci. Rep. (1)

B. Santra, M. N. Shneider, and R. Car, “In situ characterization of nanoparticles using rayleigh scattering,” Sci. Rep. 7(1), 40230 (2017).
[Crossref]

Other (11)

Y. P. Raizer, M. N. Shneider, and N. A. Yatsenko, Radio-Frequency Capacitive Discharges (CRC Press, 1995).

H. A. Lorentz, The theory of electrons and its applications to the phenomena of light and radiant heat, second edition (B. G. Teubner, 1916).

R.P. Feynman, R.B. Leighton, and M. Sands, The Feynman Lectures on Physics, Vol. 13 (Addison Wesley; Later Printing edition 1971).

Handbook of Optics Third Edition, 5 Volume Set 3rd Edition (McGraw-Hill Education; 3 edition December 2009).

S.A. Akhmanov and A.Yu. Nikitin, Physical Optics (Clarendon Press, 1997), chapter 8.

N. Bloembergen, Nonlinear optics. A lecture note (Harvard University, W. A. Benjamin, Inc., 1965).

R. W. Boyd, Nonlinear Optics (Academic Press, 2003).

V.V. Semak and M.N. Shneider, Invicem Lorentz Oscillator Model (ILOM), arXiv:1709.02466.

H. Helmholtz, Sensation of Tone (London, 1895).

A. Blaguire, Nonlinear System Analysis (Acad. Press, 1966).

C. Hayashi, Nonlinear Oscillations in Physical Systems (McGraw-Hill Book Co, 1964).

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Figures (8)

Fig. 1.
Fig. 1. Schematic of the effective electron potential as function of the dimensionless displacement of electron from the equilibrium orbit (x-axis). It is assumed that the effective potential depth, U0, is 1 eV (y-axis in eV).
Fig. 2.
Fig. 2. a – Normalized spectrum of the re-radiation produced by the forced atomic oscillator with x-axis in units of 1015 Hz; b – corresponding phase diagram of forced electron displacement from the equilibrium, where y is dimensionless displacement of the electron from the equilibrium orbit. The natural oscillation frequency ${\omega _0}/2\pi = 1.78 \cdot {10^{15}}\; \textrm{Hz}$, and the computed non-linearity parameter $\zeta \approx 0.139$ , that corresponds to the chosen electron orbit radius r1, in the particular case r1=r0.
Fig. 3.
Fig. 3. a,b Same as in Fig. 2, except for r1=2.5r0. ${\omega _0}/2\pi = 7.13 \cdot {10^{14}}\; \textrm{Hz}$, $\zeta \approx 0.347$ .
Fig. 4.
Fig. 4. a,b Same as in Fig. 1, except for r1=5r0. ${\omega _0}/2\pi = 3.56 \cdot {10^{14}}\; \textrm{Hz}$, $\zeta \approx 1.97 \cdot {10^{ - 3}}$ .
Fig. 5.
Fig. 5. a, b Same as in Fig. 1, except for r1=10r0. ${\omega _0}/2\pi = 1.78 \cdot {10^{14}}\; \textrm{Hz}$, $\zeta \approx 1.22 \cdot {10^{ - 5}}$ .
Fig. 6.
Fig. 6. a, b Same as in Fig. 1, except for r1=15r0. ${\omega _0}/2\pi = 1.18 \cdot {10^{14}}\; \textrm{Hz}$, $\zeta \approx 9.5 \cdot {10^{ - 7}}$.
Fig. 7.
Fig. 7. a, b Same as in Fig. 1, except for r1=20r0. ${\omega _0}/2\pi = 8.9 \cdot {10^{13}}\; \textrm{Hz}$, $\zeta \approx 1.57 \cdot {10^{ - 7}}$.
Fig. 8.
Fig. 8. a, b Same as in Fig. 1, except for r1=30r0. ${\omega _0}/2\pi = 5.94 \cdot {10^{13}}\; \textrm{Hz}$, $\zeta \approx 1.52 \cdot {10^{ - 8}}$.

Equations (9)

Equations on this page are rendered with MathJax. Learn more.

ω 0 = ( 2 U 0 m e r 0 2 ) 1 / 2 ,
F ( x ) = a 1 x + a 2 x 2 + a 3 x 3 + ,
U e f f ( y ) y = 2 U 0 ( 1 ( y + 1 ) 2 1 ( y + 1 ) 3 ) ,
y ¨ + 2 U 0 m e r 0 2 ( 1 ( y + 1 ) 2 1 ( y + 1 ) 3 ) ξ m e y = e m e r 0 E ( t ) .
Ψ ( t ) = A 0 + k = 1 A k cos ( k ω L t θ k ) ,
P = 1 T 0 T Ψ 2 d t = A 0 2 + 1 2 k = 1 A k 2 .
ζ 1 2 k = 2 A k 2 / ( A 0 2 + 1 2 A 1 2 ) << 1.
Ψ (t) 2 dt = 1 2 π | F( ω ) | 2 d ω .
ζ 1 2 ω L + δ | F ( ω ) | 2 d ω / 0 ω L + δ | F ( ω ) | 2 d ω << 1.

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