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The wave-particle duality of light is a controversial topic in modern physics. In this context, this work highlights the ability of the wave-nature of light on its own to account for the conservation of energy in light-matter interaction. Two simple fundamental properties of light as wave are involved: its period and its power P. The power P depends only on the amplitude of the wave’s electric and magnetic fields (Poynting’s vector), and can easily be measured with a power sensor for visible and infrared lasers. The advantage of such a wave-based approach is that it unveils unexpected effects of light’s power P capable of explaining numerous results published in current scientific literature, of correlating phenomena otherwise considered as disjointed, and of making predictions on ways to employ the electromagnetic (EM) waves which so far are unexplored. In this framework, this work focuses on determining the magnitude of the time interval that, coupled with light’s power P, establishes the energy conserved in the exchange of energy between light and matter. To reach this goal, capacitors were excited with visible and IR lasers at variable average power P. As the result of combining experimental measurements and simulations based on the law of conservation of energy, it was found that the product of the period of the light by its power P fixes the magnitude of the energy conserved in light’s interaction with the capacitors. This finding highlights that the energy exchanged is defined in the time interval equal to the period of the light’s wave. The validity of the finding is shown to hold in light’s interaction with matter in general, e.g. in the photoelectric effect with x-rays, in the transfer of electrons between energy levels in semiconducting interfaces of field effect transistors, in the activation of photosynthetic reactions, and in the generation of action potentials in retinal ganglion cells to enable vision in vertebrates. Finally, the validity of the finding is investigated in the low frequency spectrum of the EM waves by exploring possible consequences in microwave technology, and in harvesting through capacitors the radio waves dispersed in the environment after being used in telecommunications as a source of usable electricity.

In the current debate on wave-particle duality of light and its role in light-matter interaction, two questions rise: 1) are photons necessary and sufficient participants? and 2) what decides the magnitude of the phenomena arising from light-matter interaction? We address these questions in our study of conservation of energy in light-matter interaction.

Photons are particles of light, which provide a beautiful example of conservation of energy through the photoelectric effect. Ultraviolet (UV) and x-ray photons with frequency ν p h and energy E p h = h ν p h , where h is Planck’s constant (6.63 × 10^{−34} Js), can eject an electron out of an atom such that h ν p h = E b + E k + φ , where E b is the binding energy of the electron in the atom, E k the kinetic energy of the electron after being ejected from the atom by the photon, and ϕ a work function. For example, an x-ray photon with energy E_{x} = 1.2536 keV produced by a Mg Ka source can extract an electron with E_{k} = 964.2 eV from a carbon atom to which the electron was bound with E_{b} = 285.4 eV and ϕ = 4 eV [_{2}/2L-WSe_{2} interface in heterostructure photocells between the conduction band (CB) of MoSe_{2} ( E C B M o ) and the CB of WSe_{2} ( E C B W ). Similarly, Adinolfi et al. [

The energy needed in photosynthetic processes also challenges the amount of energy that single photons can provide. Photosynthetic reactions are triggered by solar light at an average power per unit area P ≈ 136 mW cm 2 [_{2} exchange [_{2} oxidation [_{2} stimulated emission [_{x} state, the same amount of energy of ≈ 0.12 eV is exchanged when green light in at λ i n = 540 nm ( ν i n = 0.55 PHz ) releases yellow light at λ o u t = 580 nm ( ν o u t = 0.52 PHz ). We estimate that ~4, 6, and 14 couples of photons are required to provide the 0.5 eV, 0.72 eV, and 1.6 eV activation energies per molecule to initiate the photosynthetic reactions mentioned above. Here we raise the question whether the average power per unit area P ≈ 136 mW cm 2 of the solar light might play a role in providing a wider range and higher values of the energy exchanged between solar light and photosynthetic organisms.

Another process that challenges the amount of energy that single photons can provide is the energy conversion in retinal ganglion cells (RGCs) to enable vision in vertebrates. The RGCs act as capacitors with baseline capacitance of the order of few pF [

In this work, we exploit the wave nature of light to address the mismatch between the low frequency photon’s energy and the energy conserved in the exchange of energy between light and devices or natural light harvesting (LH) systems. We start off by recalling that light is an EM wave whose power P per unit area is the modulus of Poynting’s vector, | S | = | E × H | = P A , where A is area, E the electric field, and H the magnetic field [

Capacitors. To test the interaction between visible and IR light and matter we used Custom Thermoelectric 07111-9L31-04B devices. These devices produce a voltage difference Δ V where there is a temperature difference Δ T across the device in agreement with the Seebeck effect Δ V = S Δ T , where S is the Seebeck coefficient. However, their multilayer structure consisting of AlO-Cu-Bi_{2}Te_{3}-Cu-AlO shown in Appendix 1 can also be viewed as a capacitive structure. The basic device has a height h = 3 mm and a capacitance C in the pF range. Adding insulating layers to the basic device, and varying the cross-sectional area with diameter D of the illuminating laser, enable us modifying the capacitance C. The configurations of the devices examined in our work, alongside their C and dielectric constant ε values, are summarized in Appendix 1. The ε values are calculated from C = ε ε 0 a r e a h , where ε 0 = 8.854 × 10 − 12 F m is the permittivity in vacuum.

Infrared sources. The characteristics of the visible and IR continuous wave (c.w.) and pulsed (PL) lasers used in our experiments are summarized in

Data acquisition and modeling. Using Keithley 2000 multi-meters, we acquired voltage Δ V ( t ) and temperature Δ T ( t ) differences as a function of time t. The temperatures of the illuminated ( T l i g h t ( t ) ) and non-illuminated ( T n o − l i g h t ( t ) ) faces of the capacitors were measured using OMEGA type E Ni-Cr/Cu-Ni thermocouple probes. We measured Δ V ( t ) and Δ T ( t ) using LabView 2012 and a National Instruments PXI-1042q communications chassis. We collected and analyzed a total of 116 data sets with different combinations of capacitor’s capacitance C, light’s average power P and/or period τ for c.w. and PL lasers. Each set consists of Δ V ( t ) , Δ T ( t ) , T l i g h t ( t ) and T n o − l i g h t ( t ) . We used Wolfram Mathematica 10.3 software for modeling.

Mode of operation | Period t [fs] | Maker and type | Average power P range [mW] | Beam diameter D [mm] |
---|---|---|---|---|

c.w. | 1.77 | Viasho solid state | 25 - 50 | 3 |

c.w. | 3.55 | ThorLabs solid state | 25 - 1000 | 3 |

c.w. | 6.67 | IPG Photonics pumped solid state | 50 - 200 | 5 |

Mode of operation | Period t [fs] | Maker and type | Repetition rate 1/T_{p} [1/s] | Pulse duration Dt_{p} [ns] | Average power P range [mW] | Peak power P_{peak} range | Beam diameter D [mm] |
---|---|---|---|---|---|---|---|

PL | 3.55 | Northrop Grumman solid state | 4*10^{3} | 0.7 | 5 - 50 | kW | 4 |

PL | 3.55 | New Wave Research Tempest | 10 | 7 | 150 - 450 | MW | 3 |

To estimate the magnitude of Δ t , and therefore of E c = P Δ t , we collected and analyzed 116 sets of voltage and temperature measurements resulting from the excitation of capacitors by lasers of average power P spanning from 10 to 1 × 10^{3} mW. In our experiments, we use visible and IR lasers with wavelength λ from 532 to 2 × 10^{3} nm (i.e. from frequency ν = 0.56 PHz or period τ = 1.77 fs , to ν = 0.15 PHz or τ = 6.70 fs ). We shine c.w. and PL lasers with beam diameter D onto capacitors with capacitance C in the ≈ 18 to ≈ 270 pF range (Section-2). We then monitor the voltage Δ V ( t ) and temperature Δ T ( t ) differences produced by the capacitors versus time t in the 400 seconds immediately following the start of the illumination with visible or IR light. The exponential time-dependent rise of Δ V ( t ) occurs in the time interval we call the exponential perturbation regime (EPR). The end of the EPR marks the beginning of the steady state laser illumination (ssli) where the voltage reaches a stable value. The two regimes are labelled in

we developed a model, based on the law of conservation of energy (derived in Appendix 2), to link the magnitude of Δ V and the evolution in time of Δ V ( t ) to E c , P ( t ) , Δ T ( t ) , and C. Here P ( t ) is the exponential function that describes the rise versus time of the laser’s power with time constant τ P such that:

P ( t ) = − P 0 e − t τ P + P s s l i , (1)

where P 0 ≈ P s s l i , and P s s l i is the average power in the ssli.

The typical trends of Δ V ( t ) and Δ T ( t ) are displayed in

Next, in

In

The experimental data indicate that Δ V increases when the EM wave’s average power P increases. Moreover, the trends of π Δ V versus C and τ suggest that the increase in Δ V with P is enhanced when C decreases and τ increases. To model these trends we derive Equation (2a) and Equation (2b), which stem from the law of conservation of energy applied to the interaction between visible or IR light and a capacitor. We present the derivation of Equation (2a) and Equation (2b) in Appendix 2. A similar approach is adopted in Ref. [

E ( t ) = 1 2 C Δ V ( t ) 2 + 1 2 q ( t ) 2 C − Σ 0 Δ T ( t ) , (2a)

where Σ 0 is the entropy in a closed system, as discussed in Appendix 2. In each instant of time t, the variables Δ V ( t ) , q ( t ) , and Δ T ( t ) adjust their values to that of E ( t ) = P ( t ) Δ t . Likewise, in the ssli the energy E c conserved in the interaction between light and matter is:

E c = 1 2 C Δ V 2 + 1 2 q s s l i 2 C − Σ 0 Δ T , (2b)

where q s s l i is the surface charge in the ssli.

From Equation (2a) we then derive the voltage difference Δ V ( t ) in the EPR as follows:

Δ V ( t ) = 2 C E ( t ) − 1 C 2 q ( t ) 2 + 2 Σ 0 C Δ T ( t ) . (3a)

The voltage in the EPR is V ( t ) E P R = Δ V ( t ) + ϕ V where ϕ V is an offset with magnitude of ~ m V labelled in

Δ V = 2 C E c − 1 C 2 q s s l i 2 + 2 Σ 0 C Δ T . (3b)

The voltage in the ssli is V s s l i = Δ V + ϕ V . Equation (3a) and Equation (3b) indicate that Δ V ( t ) and Δ V increase when C decreases, in agreement with the experimental findings in

We now combine Equation (2) and Equation (3) to (i) determine the magnitude of Δ t , (ii) unveil the value of E c when the experimental value of the average power P is known, and (iii) clarify the dependence of Δ V ( t ) , Δ V and π Δ V on the period τ .

To illustrate the general method enabling us to determine the magnitude of Δ t we exploit the data in

Step 1: rough estimate of E c and Δ t . The data in

Step 2: modeling of Δ V ( t ) and search for the estimated parameters q s s l i and Σ 0 . With the rough hypotheses that Δ t ~ τ and E ( t ) ~ P ( t ) τ we model the voltage difference Δ V ( t ) in the EPR from Equation (3a). To reach this goal we first construct an expression for the evolution with time of the surface charge q ( t ) (see Appendix 2). Then, we superimpose Equation (3a) to the function that fits the data in

Step 3: refined expression for E c and value for Δ t . By plugging into Equation (2b) for E c the experimental parameters and the estimated parameters found through Step 2, we extract a refined value for E c ≈ 2.4 fJ and calculate Δ t = E c P = 3.4 fs ≈ τ = 3.55 fs for the data in

The same method illustrated above leads us to the three additional examples of good match between experimental and modeled Δ V ( t ) shown in

Δ V ( t ) = 2 C P ( t ) τ − 1 C 2 q ( t ) 2 + 2 Σ 0 C Δ T ( t ) (4a)

and

Δ V = 2 C P τ − 1 C 2 q s s l i 2 + 2 Σ 0 C Δ T . (4b)

Equation (4a) and Equation (4b) align with the experimental finding that π Δ V increases when τ does.

In support to the equation E c = P τ , we point out that Har-Shemesh & Di Piazza [

Photoelectric effect. An x-ray photon with energy E x = 1.2536 keV has a frequency ν p h = 0.3 × 10 18 Hz (or τ x = 3.3 as ) derived from the equation E x = h ν p h . Assuming E x = E = c P x τ x we estimate P x ≈ 67 W , which is in good agreement with the value P x ≈ 100 W for commercial x-ray sources [

Electronic devices. Barati et al. [_{2}/2L-WSe_{2} heterostructure photocells [

Photosynthesis. Let us assume that the average solar light power per unit area P ≈ 136 mW cm 2 [_{2} stimulated emission [_{x} state at τ i n = 1.82 fs and consequent release of yellow light at τ o u t = 1.93 fs , the exchanged energy is | P ( τ o u t − τ i n ) | ≈ 85 eV . These values of exchanged energies enable in a small time interval the activation of multiple photosynthetic reactions requiring different activation energies per molecule (e.g. 0.5 eV for the oxidation of cytochrome [_{2} exchange [_{2} oxidation [

Vision in vertebrates. Let us assume that solar green light at τ = 1.82 fs ( ν p h = 0.55 PHz ) and P ≈ 136 mW cm 2 [

We explore now other consequences of E c = P τ being the energy conserved in the interaction between matter and light in the low frequency range of the EM spectrum:

Microwave technology. Electromagnetic waves can generate, or be generated by, mechanical motion [

Radio wave energy harvesting. We collected on a capacitor with capacitance C = 37 pF the radio waves emitted and received by a cellular phone tuned to the Global System for Mobile communications (GSM) frequency of ν G S M = 900 MHz ( τ G S M = 1.1 ns ). We observed the production of a Δ V G S M = 0.095 mV with a time constant τ Δ V G S M = 95 s and Δ T G S M = 0.02 ° C . Using the approximate expression E c = P τ G S M G S M ≈ 1 2 C Δ V G S M 2 derived from Equation (2b) we estimate P G S M = 0.15 nW , E c = P G S M τ G S M = 0.165 aJ , and expect a voltage responsivity Δ V G S M P G S M = π Δ V ≈ 0.6 × 10 6 V W , about six orders of magnitude larger than that obtained with near IR light in

By revisiting the wave nature of light, we investigate the magnitude of the energy E c conserved in light’s interaction with matter. Upon combing experimental measurements and simulations based on the law of conservation of energy, we find that E c = P τ = n τ h ν p h , where P and τ are the power and the period of the electromagnetic (EM) wave, n τ is the number of photons in a time interval equal to τ , h is Planck’s constant, and ν p h is the photon’s frequency. We highlight that the magnitude of the energy conserved E c is defined in a time interval equal to τ both for light as wave, for which E c = P τ , and for light as particle, for which E c = n τ h ν p h . By addressing the interaction of light with matter in the photoelectric effect, in photo-excited field effect transistors, in photosynthesis and in vertebrate’s vision, we show that E c = P τ = n τ h ν p h is valid independently of the employed photo-detecting device and in the whole EM spectrum.

In addition, as a consequence of the law of conservation, we notice that the energy E c constraints the magnitude of the variables involved in the energy exchange between light and a specific photo-detecting device, and imposes a reciprocal self-limitation among those variables. In our experiment, for example, the magnitude of E c , in cooperation with capacitor’s capacitance C and light’s period τ , dictates the magnitude of the voltages produced by the capacitors and provides a strategy to estimate the magnitude of entropy. In photo-excited field effect transistors, E c constraints the magnitude of the energy supplied to electrons thus promoting or suppressing photocurrent production. Similarly, the magnitude of the capacitance C in retinal ganglion cells evolved in time in harmony with the power P and period τ of the solar light such that E c can elicit action potentials with the proper value to enable vision in vertebrates.

Finally, the finding that the voltage responsivity of capacitors increases with τ and is about six orders of magnitude larger with radio waves than with visible and infrared light has strong implications in the search for alternative sources of renewable and sustainable energy: it suggests that it would be very efficient to harvest the radio waves dispersed in the environment after being used in telecommunications. This effort requires a systematic exploration of the behavior of temperature, entropy and time constants in the process of energy transfer, an effort that soon we plan on undertaking.

In conclusion, our results suggest that the wave-nature of light is necessary and sufficient to describe the energy conserved in light-matter interaction. The number of photons involved can be determined as a simple consequence, and the magnitude of the phenomena arising from light-matter interaction and their connection can be established once the wave properties of light are known.

This work was supported by the U.S. Office of Naval Research (ONR) awards # N000141410378 and N000141512158, the JMU 4-VA Consortium (2016-2017), the JMU-Madison Trust-Fostering Innovation and Strategic Philanthropy-Innovation Grant-2015, the Thomas F. Jeffress and Kate Miller Jeffress Memorial Trust, Grant # J-1053, the JMU Center for Materials Science, the JMU Department of Physics and Astronomy, and the JMU College of Science and Mathematics. A portion of this research was conducted at the Center for Nanophase Materials Sciences, which is a Department of Energy (DOE), Office of Science User Facility at the Oak Ridge National Laboratory under the Grant CNSM2014-R16. The authors thank M. Currie and V. D. Wheeler (Naval Research Laboratory-NRL) for their help in the experiments, and B. C. Utter (Bucknell University), J. C. Zimmerman (JMU), W. W. Shiflet (JMU), G. N. Parsons (NCSU), and F. Segatta (University of Bologna, Italy) for fruitful discussions.

Boone, D.E., Jackson, C.H., Swecker, A.T., Hergenrather, J.S., Wenger, K.S., Kokhan, O., Terzić, B., Melnikov, I., Ivanov, I.N., Stevens, E.C. and Scarel, G. (2018) Probing the Wave Nature of Light-Matter Interaction. World Journal of Condensed Matter Physics, 8, 62-89. https://doi.org/10.4236/wjcmp.2018.82005

In this section we illustrate the structure of the Custom Thermoelectric 07111-9L31-04B devices used in our experiments. _{2}Te_{3}-Cu-AlO. The capacitor has a thickness h = 3 mm. The cross-sectional area A c s of the visible or IR laser has diameter D and corresponds to the illuminated area of the capacitor.

two capacitors respectively. The ε values are calculated from C = ε ε 0 a r e a h , where ε 0 = 8.854 × 10 − 12 F m is the permittivity in vacuum.

Device description | Beam diameter D [mm] | Capacitance C [pF] | Dielectric constant e |
---|---|---|---|

1 capacitor | 3 | 270.5 | 17312 |

1 capacitor | 4 | 481.0 | 17316 |

1 capacitor | 5 | 751.5 | 17315 |

1 capacitor + IT | 3 | 43.4 | 2995.4 |

1 capacitor + IT | 4 | 77.3 | 3001.0 |

1 capacitor + IT | 5 | 120.6 | 2996.4 |

Device description | Beam diameter D [mm] | Capacitance C [pF] | Dielectric constant e |
---|---|---|---|

2 capacitors | 3 | 245.7 | 30832.8 |

2 capacitors | 4 | 436.7 | 30825.8 |

2 capacitors | 5 | 682.4 | 30828.3 |

2 capacitors + IT | 3 | 18.5 | 2785.9 |

2 capacitors + IT | 4 | 32.95 | 2791.0 |

2 capacitors + IT | 5 | 51.4 | 2791.9 |

A2-A. Conservation of energy

To justify the observed magnitudes and trends of the voltage responsivity π Δ V of capacitors with capacitance C illuminated by visible and infrared (IR) lasers with average power P and period τ, we exploit the law of conservation of energy [^{3} mW range, light interacts with matter by transferring energy which subsequently is distributed into an electrical and a thermal component. Thus, we describe as follows the total differential dE of the energy transferred from the visible and IR lasers to the capacitor on an area A c s with diameter D corresponding to the cross-section of the laser beam:

d E = q d Δ V + Δ V d q − Σ d Δ T − Δ T d Σ , (A1)

where q = σ A c s is charge, σ = q A c s is surface charge density, and Σ is entropy.

The thermal component of Equation (A1) is preceeded by a negative sign to signify that part of the energy transferred to the capacitor as electrical energy is thermally dissipated. Initially we hypothesize that capacitance C is such that

C = q Δ V : we will refine this assumption in Sub-section A2-D. Hypothesizing that C = q Δ V enables us to separate the terms containing q and Δ V , thus reducing Equation (A1) to:

d E = C Δ V d Δ V + q C d q − Σ d Δ T , (A2)

where we assume d Σ to be negligible. In a generalized grand-canonical ensemble [

canonical ensemble, such that Σ = Σ 0 − k B β q Δ V [

d E = C Δ V d Δ V + q C d q − Σ 0 d Δ T + k B β q Δ V d Δ T . (A3)

Assuming k B β q Δ V to give just a slight correction to Σ 0 , d E is further reduced to:

d E ≈ C Δ V d Δ V + q C d q − Σ 0 d Δ T . (A4)

In the exponential perturbation regime (EPR), at the start of the illumination of the capacitor with visible or IR light, the energy conserved in each instant of time t is E ( t ) = P ( t ) Δ t . Here P ( t ) is the exponential function that describes the rise in time of the laser’s power with a time constant τ P such that:

P ( t ) = − P 0 e − t τ P + P s s l i , (A5)

where P 0 ≈ P s s l i ≈ P and P is the average power in the steady state laser illumination (ssli) regime [

∫ Δ E min Δ E M A X d E = C ∫ Δ V min Δ V M A X Δ V d Δ V + 1 C ∫ q min q M A X q d q − Σ 0 ∫ Δ T min Δ T M A X d Δ T . (A6)

We redefine the variables such that their minima are set at zero in their respective units and their maxima correspond to the values of the variables at an arbitrary instant of time t such that Δ V max = Δ V ( t ) , Δ E max = Δ E ( t ) , q max = q ( t ) , and Δ T max = Δ T ( t ) . In the EPR the integration leds to:

E ( t ) = P ( t ) Δ t = 1 2 C Δ V ( t ) 2 + 1 2 q ( t ) 2 C − Σ 0 Δ T ( t ) . (A7a)

In the ssli the integration gives:

E c = P Δ t = 1 2 C Δ V 2 + 1 2 q s s l i 2 C − Σ 0 Δ T , (A7b)

where q s s l i is the surface charge in the ssli on the area A c s .

To model in the EPR the measured voltage difference Δ V ( t ) , we rearrange Equation (A7a) such that:

Δ V ( t ) = 2 C P ( t ) Δ t − 1 C 2 q ( t ) 2 + 2 Σ 0 C Δ T ( t ) (A8a)

The voltage in the EPR is V ( t ) E P R = Δ V ( t ) + ϕ V where ϕ V is an offset with magnitude of ~ m V . The offset ϕ V is due to contributions from the environment and from the capacitor. In the ssli instead, Δ V is:

Δ V = 2 C P Δ t − 1 C 2 q s s l i 2 + 2 Σ 0 C Δ T . (A8b)

The voltage in the ssli is V s s l i = Δ V + ϕ V . In the main text we use Equation (A8a) and Equation (A8b) to model the experimental data obtained with both continuous wave (c.w.) and pulsed (PL) lasers.

A2-B. Model for the charge density variation in time, σ ( t ) .

Of all the terms in Equation (A8a), q ( t ) C is the least constrained by experimental parameters. Thus, we exploit the surface charge q ( t ) to achieve the best

fit between modeled and experimental data. To do so, we carefully develop a model for q ( t ) , which is related to the evolution with time in the EPR of the surface charge density, σ ( t ) [

to the area A c s with diameter D is the Poynting vector | S | = | E × H | = P A c s ,

where E and H are the electric and magnetic fields, respectively [

charge density, which can be described as σ ( r , t ) = q ( r , t ) A c s . To mathematically

sketch such perturbation we hypothesize that, while hitting the surface of the capacitor, light affects the charges through a force f ( t ) = q ( t ) E ( t ) . Such force displaces the surface charges away from the location in which the visible or IR light impinges on the capacitor. Such a phenomenon is pictured for a broad range of low frequency electromagnetic (EM) waves in review articles (Ref. [

average power P ( t ) = A c s | S ( t ) | = A c s | E ( t ) × H ( t ) | , we can further picture that the surface charge density locally decreases such that σ ( r , t ) ∝ 1 P ( t ) without

ejecting charges away of the capacitor’s surface. To depict reference system, orientation and phase of the spatial variables defining σ ( r , t ) , we represent the 2D space variable r as a complex variable z = r x + i r y , where i is the imaginary unit. This choice resembles that adopted to describe light polarization through Jones matrices [

z = [ r x r y ] = [ r 0 x e i ϕ x r 0 y e i ϕ y ] = [ r a + i r b r c + i r d ] . (A9)

All possible rotations of the reference system, phases, and positions in the 2D plane can be obtained by selecting magnitude and sign of r a , r b , r c , and r d . With this premise we can model σ ( z , t ) as exponentially decreasing with the increase of P ( t ) according to:

σ ( z , t ) = R E ( | σ 0 | e − t τ σ + r x k x + φ e i r y k y − | σ f | ) = | σ 0 | e − t τ σ + f x L x k x + ϕ cos ( f y L y k y ) − | σ f | . (A10)

The parameters σ 0 and σ f ≈ σ s s l i are surface charge densities, where σ s s l i is the value in the ssli. They are considered as absolute values because the density of the charge is independent of the charge’s sign. The time constant τ σ of the surface charge density’s decay is effective with a slight delay compared to the time constant τ P of the power P ( t ) . k , with components k x and k y , is a vector with units of inverse length. To simplify the determination of the magnitude of the phase φ , of k x and k y , we assume that | k x | = | k y | = κ . The quantities L x and L y are such that | L x | = | L y | = D . We note that the exponential behavior of σ ( z , t ) is modulated by the oscillatory function e i r y κ . In Equation (A10) we write the spatial variables as follows: r x = f x L x and r y = f y L y , where

f x = r x L x and f y = r y L y are dimensionless fractions varying from 0 to 1 and

enabling us to locate any position on the surface of the capacitor. For example, by choosing the origin of the reference systems on the lower left corner as in

density according to Equation (A10) and pertinent to the specific case of

A2-C. The integral of σ ( r , t ) over space.

While σ ( z , t ) is defined by a spatiotemporal set of variables, the functions Δ V ( t ) and Δ T ( t ) depend only on time t. To decouple the spatial variable z from σ ( z , t ) , we integrate σ ( z , t ) in Equation (A10) over the illuminated surface area of the capacitor approximated as a square with side D. We thus obtain a time-dependent surface charge density σ ( t ) such that:

σ ( t ) = − | σ f | + | σ 0 | ∫ A c s d A c s ( e − t τ σ + f x L x κ + φ cos ( f y L y κ ) ) = − | σ f | + | σ 0 | e − t τ σ + φ ∫ f x 0 f x f d f x e f x L x κ ∫ f y 0 f y f d f y cos ( f y L y κ ) . (A11)

To solve the integrals, we change the variables from f x to Ξ = f x L x κ , and from f y to Υ = f y L y κ . We obtain d f x = d Ξ L x κ , Ξ 0 = f x 0 L x κ , and Ξ f = f x f L x κ . Similarly, we also have d f y = d Υ L y κ , Υ 0 = f y 0 L κ y , and Υ f = f y f L y κ . Consequently, σ ( t ) becomes:

σ ( t ) = − | σ f | + | σ 0 | ( e − t τ σ + φ L x L y κ 2 ) ∫ Ξ 0 Ξ f d Ξ e Ξ ∫ Υ 0 Υ f d Υ cos ( Υ ) = − | σ f | + | σ 0 | ( e − t τ σ + φ L x L y κ 2 ) ( e Ξ f − e Ξ 0 ) ( sin ( Υ f ) − sin ( Υ 0 ) ) = − | σ f | + | σ 0 | ( e − t τ σ + φ L x L y κ 2 ) ( e f x f L x κ − e f L x 0 x κ ) ( sin ( f y f L y κ ) − sin ( f y 0 L y κ ) ) . (A12)

To eliminate the f x and f y variables in Equation (A12) we choose the reference system for f x and f y as in

σ ( t ) = − | σ f | + | σ 0 | ( e − t τ σ + φ L x L y κ 2 ) ( e L x κ − 1 ) sin ( L y κ ) . (A13)

From Equation (A13) we can infer that to evaluate q ( t ) = σ ( t ) * A c s requires knowing the values of σ f , σ 0 , τ σ , φ , and κ . A rough value for σ f can be estimated from Equation (A8b) such that:

σ f ≈ C Δ V A c s . (A14)

We evaluate the parameters τ σ , φ , and κ from the best fit of the model to the experimental data for the voltage difference Δ V ( t ) . We found the magnitude of q ( t ) = σ ( t ) * A c s in the p C range in agreement with Ref. [

A2-D. Calculation of the correction factor ζ .

In the EPR the visible and IR lasers significantly disturb the surface charge density σ ( t ) on the capacitor through a sudden transition from a quiescent state to a perturbed one. A realistic view of the phenomenon requires going

beyond the simple assumption that σ ( t ) = Δ V ( t ) C A c s adopted in Sub-section

A2-A. Indeed, since both electrical and thermal phenomena are activated on the capacitor by the illumination through visible and IR lasers, we can assume that σ ( t ) = σ ( t ) e l + σ ( t ) t h . The electrical phenomena induce a voltage difference Δ V ( t ) e l linked to the capacitive behavior of the capacitors with capacitance C through the surface charge density σ ( t ) e l . We depict σ ( t ) e l as:

σ ( t ) e l ≈ Δ V ( t ) C A c s . (A15)

On the other hand, we assume that the thermal phenomena induce a voltage difference Δ V ( t ) t h linked to the raise in temperature difference Δ T ( t ) associated with a surface charge density σ ( t ) t h . Since it is challenging to directly relate the temperature difference Δ T ( t ) to σ ( t ) t h , we assume σ ( t ) t h to be:

σ ( t ) t h = ζ Δ V ( t ) 2 A c s , (A16)

where ζ is a correction factor related to Δ T ( t ) .

With these observations, the overall surface charge density σ ( t ) becomes:

σ ( t ) = σ ( t ) e l + σ ( t ) t h = Δ V ( t ) C + ζ Δ V ( t ) 2 A c s (A17)

and the correction factor ζ can be evaluated as:

ζ = q ( t ) Δ V ( t ) 2 − C Δ V ( t ) . (A18)

The magnitude of ζ has units of nC V 2 and can be extracted from the experimental data. For each set of data obtained with the same type of laser, τ , and

C, we display ζ versus average power P in

vary between about − 100 and 100 nC V 2 . As expected, we note that ζ e q significantly deviates from the zero line for large Δ T as τ moves toward

the visible region of the EM spectrum (see

In

ξ Δ T responsivities versus the capacitance C of devices consisting of two capacitors in series. These devices are described in

Mode of operation | Period t [fs] | Maker and type | Repetition rate 1/T_{p} [1/s] | Pulse duration Dt_{p} [ns] | Average power P range [mW] | Peak power P_{peak} range | Beam diameter D [mm] |
---|---|---|---|---|---|---|---|

PL | 6.67 | ThorLabs-OPO* | 20 | 4 | 20-70 | MW | 5 |

*Continuum surelite OPO laser tuned to specified wavelengths and pumped with frequency-tripled continuum surelite-II laser.

using devices consisting of two capacitors in series with C = 18.5 pF and C = 245.7 pF . We observe an increase in π Δ V when τ increases, i.e. toward the far IR and microwave regions of the EM spectrum. The temperature responsivity ξ Δ T exhibits a slight increase toward the visible region of the EM spectrum. The trends in