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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
1. Introduction
The current description of matter response to electro-magnetic wave rests on the foundational model of atomic oscillator proposed by H. A. Lorentz [1]. This model is highly regarded by the majority of the academic community as being exceptionally accurate, in spite of its simplicity, see for example [2]. However, in practice the optical properties of materials are computed using empirical equations, for example [3].
One of two foundational assumptions of the Lorentz model is that the electron returns to equilibrium due to the action of a force that is linearly proportional to the electron displacement. We were unable to find justification or in-depth discussion of this assumption. Usually, textbooks explain that since an electron in an atom is in a stable state, a returning force should act against any external forcing. Additionally, because the electron displacement is miniscule, this returning force could be approximated as a linear function of the displacement, for example [4–6]. The second foundational assumption is that the forced electron displacement is hindered by a “damping force” that is proportional to the electron displacement velocity [1]. Similarly, Lorentz stated this assumption without explanation or justification. The reader can find more in-depth discussion of these assumptions in our Arxiv submission [7].
It is interesting to note, and it is tempting to suggest, that both foundational assumptions of Lorentz’s model are consistent with J.J. Thompson’s “plum pudding” model of atomic structure, rather than Bohr’s planetary model, and with the aether hypotheses, rather than with the current model of the Universe. Both Thompson’s atom structure and the aether hypotheses were dominant during the time when H.A. Lorentz developed his atomic oscillator model. The negative electrons (“plums”) move inside of the positively charged cloud (“pudding”) and this motion is subject to friction against aether; hence, the approximation of the damped linear harmonic oscillator.
The approximation of the linear harmonic oscillator assumes that the potential in which the electron moves (commonly referred to as effective potential) is a quadratic function of electron displacement from its equilibrium location. If the depth of this effective potential is U0 then the natural oscillation frequency of the electron is [7]
A linear harmonic oscillator by definition is incapable of generating neither higher nor sub-harmonics. Additionally, if the electron in an atom, exposed to an electro-magnetic wave, behaves as a liner harmonic oscillator, then the material’s refractive index would be independent on the field amplitude.
From practical experience, we know that some materials generate harmonics and sub-harmonics and that the refractive index could be dependent on the intensity of the laser light. This suggests that the electron’s effective potential differs from quadratic dependence and the returning force is nonlinear. The current approach to handle such nonlinearity is to represent the dependence of the returning force on the electron displacement from equilibrium as a Taylor series:
Retaining the first two terms of the series (2) leads to the solution of the electron motion equation that combines harmonic functions at the frequency of the oscillating force, ${{\boldsymbol \omega }_{\boldsymbol L}}$ (fundamental) and the frequency 2${{\boldsymbol \omega }_{\boldsymbol L}}$ (second harmonic). Retaining three terms leads to the solution that has the fundamental, second harmonic and third harmonic at the frequency 3${{\boldsymbol \omega }_{\boldsymbol L}}$. Note that such mathematical manipulation limits the number of harmonics by the number of the retained terms of the Taylor series. Additionally, as mentioned above, the amplitude of each higher harmonic is chosen to be significantly smaller than the previous one. However, in general, the accurate representation of the dependence of the returning force on the electron displacement in the form of a Taylor series could require the retention of a larger number of terms. Then the number of generated harmonics should exceed the typically considered two or three terms. In general, the amplitude of the higher harmonics could be larger or smaller than the amplitudes of the lower harmonics; however, the Taylor series will still be convergent.
Obviously, the preferred approach is in finding an exact solution of the electron motion equation forced by the electromagnetic wave in the exact (known, not approximated) effective potential of the electron. The major difficulty of such an approach is in determining the shape of the electron effective potential via either measurements or theoretical modeling and computation using some fundamental material properties. In the particular case of s-orbitals of hydrogen or a hydrogen-like free atom, the effective electron potential can be theoretically determined using a classical approach. Equating the Coulomb attraction force to the centrifugal force and using the angular momentum conservation principle, the “returning” force acting on electron expressed as the derivative of the electron effective potential over the dimensionless displacement is [7]:
2. General description of forced electron oscillations in a hydrogen-like atom
The effective potential of a hydrogen-like unbound atom computed using classical approach [7] and used in the Eq. (3) is shown in the Fig. 1. The effective potential increases approaching zero as the displacement from the stable orbit increases. Thus, the electron displaced by a sufficiently large external force can leave the atom, i.e. ionization takes place. Obviously, the depth of the electron effective potential is the ionization energy. In the particular example of calculations described below we have chosen the ionization energy of 1 eV. This is significantly lower than the ionization energy of typical atoms in ground state. However, excited metastable atoms that, for example, are produced in plasma discharges, could have such low ionization energy.
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).
Thus, for an electron on s-orbital, the equation for the dimensionless displacement induced by the incident electro-magnetic wave would be as follows:
Solving the Eq. (4) will give dimensionless displacement of the electron y(t). Unlike all previous theories that assume the electron motion as a perturbed harmonic function, we consider the most unrestricted case that, in general, would describe forced electron displacement as a non-harmonic periodic function. Taking the second time derivative of this displacement will give the value proportional to the electric field of the re-radiated electro-magnetic wave, $\ddot{y} = \Psi (\textrm{t} )$. The Fourier analysis of this function will give the spectrum of the re-radiated electric field, and the second power of the spectrum of the re-radiated electric field gives the spectrum of re-radiated power.
In the most general case this spectrum will have an infinite number of harmonics. This is opposite to the common approximation that forces the solution to be limited to only two or three harmonics. As mentioned above, the amplitudes of the harmonics could have various possible values, also contrary to the current belief that these amplitudes are rapidly decreasing as a function of harmonic number.
Our general theory of the forced atomic oscillator opens an avenue to the formulation of tangible (quantitative) criteria for “linearity” of the optical response of matter to the irradiation. It is indisputable that the effective electron potential is a non-quadratic function of electron displacement. Therefore, optics is always nonlinear; however, as known from practice, a linear approximation is acceptably accurate for low light intensities. What are the intensities at which the linear approximation becomes inaccurate? The commonly used model of the atomic oscillator is inherently incapable of providing tangible criteria for linear to non-linear transition. However, our theoretical model of the atomic oscillator that uses forced electron undulations in a real effective potential allows for the formulation of such criteria.
Another useful application of our theoretical model could be in the development and synthesis of materials with high efficiencies of nonlinear frequency conversion. The computation of the re-radiation spectrum, using the fundamental properties of material, provides information that determines the efficiencies of the laser frequency up-conversion to higher harmonics. Additionally, our theoretical approach enables the computation of the efficiency of laser frequency down conversion, currently assumed to be generated as optical parametric oscillation (OPO) via mixing of the laser frequency with the ill-defined frequency of thermal fluctuations in crystal. Same as in the case of higher harmonics generation, the generation of sub-harmonics in our theory naturally follows from the specific shapes of the effective electron potential.
Below we present possible criteria of optical nonlinearity for the cases of long pulses and short pulses, as well as the examples of computed spectra, that could be used for first principles based computation of the conversion efficiencies. Also, below we will present examples of spectral analysis of the re-radiated field that could be used for calculation of the conversion efficiencies.
3. Criteria of optical nonlinearity that follows from our theoretical model
If the duration of the pulse of incident electro-magnetic wave significantly exceeds its period (for example, CW or long pulse laser), the time dependent electric field re-radiated by the electron, $\Psi (\textrm{t} )= \ddot{y}$, can be represented as discrete frequency spectrum, or as Fourier series:
The average power of the dipole radiation for the period is given by the Parseval theorem:
4. Analysis of higher harmonic and sub-harmonic generation by a hydrogen-like atom
For s-orbital, it is a straightforward procedure to compose effective electron potential consisting of the Coulomb attraction potential, centrifugal potential energy, and a repulsion component that simulates action from the neighboring atom located at the distance ra. This procedure leads to the equation of electron motion (4).
Solving this equation gives the time dependence of the displacement of the electron from the equilibrium orbit. Then, the time dependent acceleration determines the dynamics of the electric field of the re-radiation produced by the electron that undulates, forced by the incident electro-magnetic wave, $E(t )\sim \Psi (t )= \; \ddot{y}(t )$
It is easy to anticipate from the equation of electron motion (4) that the natural oscillation of the electron will look as a spectral line centered at the frequency ${\omega _0}\; $ given by the Eq. (1). It is important to note that the spectral line of natural oscillations of the electron will have asymmetric shape, unlike the Lorentz spectra line shape that follows from the common perturbed harmonic oscillation approximation.
From the theory of nonlinear mechanical and electrical oscillators [7–9], it is known that if the frequency of harmonic force, ${\omega _L}$, is lower than the center natural oscillation frequency, ${\omega _0}$, then the oscillator generates higher harmonics with the center frequencies that are multiples of the frequency of harmonic force. It is reasonable to suggest that the same is valid for the optically driven atomic oscillators. The numerical solution of atomic oscillator equation with a realistic effective electron potential confirms this (Figs. 1,2)
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.
Since the time of Helmholtz’s consideration of nonlinear acoustic oscillator [8], it is known that if the natural oscillation frequency, ${\omega _0}$, of a nonlinear oscillator is smaller than the forcing frequency, ${\omega _L}$, the generation of subharmonics occurs. In this case the solution of the nonlinear oscillator equation , assuming approximation of the nonlinear returning force as a Taylor series with only three terms, has a spectrum that contains frequencies ${\omega _n} = {\omega _L}/n$, where ${\omega _L}$ is the frequency of driving force, and $n = 2,\; 3,\; 4 \ldots $, [9,10].
The calculations using our model show that, indeed, in the case where the laser frequency is larger than the frequency of the natural atomic oscillation, ${\omega _L}$ > ${\omega _0}$, the fractional harmonics are generated in combination with the higher harmonics. However, unlike the perturbation approximations, our unrestricted numeric solution of atomic oscillator equation demonstrates that the frequency of the sub-harmonic is continuously changing with the change of the natural oscillation frequency, ${\omega _0}$. (Figs. 3,4). This is a noticeable result and, possibly, could provide an alternate explanation to the experimentally observed down-conversion of laser frequency knows as OPO. In some crystals, the natural oscillation frequency of the electrons on the outer shell could be lower than the laser frequency. Then, in such crystals, the frequency down-conversion takes place. The ability of tuning the down-converted frequency could be due to the anisotropy of the effective electron potential of the atoms in such crystals. It is worth noting that the down-conversion of the laser frequency is an inherent property of nonlinear oscillator, unlike the theory of OPO which is based on frequency mixing with a frequency of mysterious thermal noise.
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. a,b Same as in Fig. 1, except for r1=5r0. ${\omega _0}/2\pi = 3.56 \cdot {10^{14}}\; \textrm{Hz}$, $\zeta \approx 1.97 \cdot {10^{ - 3}}$ .
Using our theoretical model of the atomic oscillator with the effective potential formulated for a spherical orbit (s-orbital) of an atom, described previously, [7], we performed numerical study of the optical response. The calculation results presented below provide illustration of the proposed criteria for transition from linear approximation to nonlinear approximation. Our approach demonstrates the proposed mechanisms of higher harmonics and tunable down-conversion of laser frequency. It is worth mentioning that our approach allows prediction of the efficiencies of up- and down-conversion of laser frequency since the computation of the emission spectra uses the fundamental atomic property. This is unlike the current model that is based on the empirically determined coefficients of the abbreviated Taylor series representation of the electron effective potential.
As was mentioned, the results presented below were obtained for the effective potential of an electron on the spherical orbit in an atom. The computation of the effective potential of the electron on any orbit for atoms bound in a material is currently in progress.
The results demonstrating transition between generation of higher harmonics to generation of fractional harmonics are shown in Figs. 2–8. The figures show the normalized spectrum of E-field of re-radiated light (“a” component of Figs. 2–8) and the corresponding phase diagram of electron velocity vs position (“b” component of Figs. 2–8). The computations were performed for the case of infinitely monochromatic laser light centered at the frequency corresponding to the wavelength of ${\lambda _L} = $ 532 nm with laser intensity of ${I_L}$ = 7.95 1014 W/m2 and long pulse duration exceeding 10000 cycles of laser radiation. The depth of effective electron potential was chosen to be 1 eV, and is the same for all calculations. The values of the radius of the electron orbit were chosen to be r0 (Fig. 2), 2.5r0 (Fig. 3), 5r0 (Fig. 4), and 10r0 (Fig. 5), 15r0 (Fig. 6), 20r0 (Fig. 7), 30r0 (Fig. 8), where r0 is the Bohr radius.
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. 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. 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. a, b Same as in Fig. 1, except for r1=30r0. ${\omega _0}/2\pi = 5.94 \cdot {10^{13}}\; \textrm{Hz}$, $\zeta \approx 1.52 \cdot {10^{ - 8}}$.
The calculations verify our hypothesis that when the laser frequency, ${\omega _L} = 2\pi c/{\lambda _L}$, is smaller than the electron natural oscillation frequency, ${\omega _0}$, the higher harmonics are generated (Fig. 2). When it is larger, the sub-harmonics are produced (Fig. 2–7), and the frequency of this sub-harmonics can be continuously changed by changing the natural oscillation frequency.
The computed spectrum of the re-radiated field shown in the Fig. 2 demonstrates that the intensities of the harmonics do not necessarily decrease rapidly with the increase of the harmonic number. Under certain conditions, the power of the third harmonic can be higher than the power of the second harmonic (Fig. 2) and in other cases it can be lower (Figs. 3–8).
The computations also demonstrated the value of our criterion for nonlinearity of the atomic response. For large values of the parameter, $\zeta $, such as in the Figs. 2 and 3, a large number of higher harmonics with relatively higher power is produced. With the decrease of the parameter $\zeta $ the power of higher harmonics is rapidly decreasing and the phase diagram appears to look more like it should for a linear oscillator.
When the electron natural oscillation frequency, ${\omega _0}$, is smaller than the laser oscillation frequency, ${\omega _L}$, the subharmonics are generated as a natural result of nonlinearity of the electron oscillations, see Figs. (4–8). The frequency of subharmonics can smoothly change in correspondence to varying of the natural electron oscillation frequency. Such an intrinsic property of the nonlinear oscillator could provide an explanation of the known and widely used phenomenon of down-conversion of laser frequency in certain nonlinear crystals. Current explanation for this down-conversion involves mixing of frequencies of the laser and some mysterious frequency produced by thermal fluctuations in the crystal. Our results demonstrate that if the shape of the effective electron potential in the atomic or molecular system is anisotropic, the natural oscillation frequency of optical electrons will depend on the direction of the wave vector of the incident electro-magnetic wave. Then, if the material design is such that the electron oscillation frequency is lower than the laser frequency, the down-conversion with smoothly variable output frequency will take place as a result of the nonlinearity of electron motion.
The described above contribution to area of laser frequency up- and down-conversion demonstrate that our model could serve as a stage for development of numerical software that would be used for calculation of the efficiencies of harmonic and sub-harmonic generation based on the fundamental properties of materials or engineered structures.
In near future we plan to use the developed theoretical model to explain properties of several physical systems that exhibit or potentially could exhibit intriguing nonlinear optical properties. In the current stage of our research the plan is to conduct simulation of nonlinear optical response of two physical platforms: 1. metal and carbon nanoparticles [11,12]; and 2. plasmas with high density of metastable atoms, like region of the negative glow in glow discharges [13], or RF discharges in inert gases [14]. Also, we plan to investigate scattering on artificial periodic structures created by the action of ponderomotive forces in plasma [15].
5. Conclusion
A theoretical model that describes the atomic oscillator using realistic effective potential of the electron has significant practical value. Among the most obvious examples of its usefulness are the tangible (numerical) criterion for transition from “linear” to “nonlinear” optics, as well as the ability for first principles computation of efficiencies up-conversion and down-conversion of laser frequency. The former is, possibly, of mostly academic and educational value; however, the latter has large practical application. With further development, this model could facilitate the design of materials and structures (meta-materials) with specific nonlinear optical properties.
The developed approach is demonstrated for the case when the electron potential U(r) is simple to deduce analytically, i.e. for s-orbital of external (optical) electron. However, this approach can be applied without limitations to obtain the optical response of electron residing on non-spherical orbitals, providing the 3D electron potential is obtained from more sophisticated theoretical models.
The simulations using our model demonstrate, in particular, that if the laser frequency is below the natural oscillation frequency of the optical electron, then laser interaction produces higher harmonics. In the opposite case, in addition to higher harmonics, the sub-harmonics may occur. It is notable that our theoretical model provides first principles prediction of the generation and the calculation of the efficiencies of the laser frequency up- and down-conversion.
In addition to the immediate practical value of predicting of the efficiencies of harmonic and subharmonic generation, we envision that our theoretical model could provide valuable theoretical predictions of the dynamic transient spectrum produced during ultrashort laser interaction with media in highly nonlinear regime. Such spectral characterization could provide model verification and refinement and, also, lead to possibly novel spectral detection of species.
Funding
Defense Advanced Research Projects Agency (DARPA) (HR011-18-C-0109).
Acknowledgments
This research was possible thanks to the funding from the DSO, DARPA. The authors would like to thank DSO Program Manager, Dr. Michael Fiddy and the team of SETA members for insightful discussions leading to important improvements of the research plans. The authors are thankful to Brian Schimmoller from Signature Science, LLC for support and frequent discussions of the research results leading to substantial deepening of understanding of how an electron moves in an atomic system.
References
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]
