Probing the Wave Nature of Light-Matter Interaction Probing the Wave Nature of Light-Matter Interaction

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Introduction
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 ph [1] [2]. Thus, in the photoelectric effect, the conserved energy is Once the photon interacts with matter, this energy distributes itself into various components: one component of magnitude b E unbinds the electron from the atom, another one of magnitude φ unbinds the electron from the material, and the last component transfers into the kinetic energy k E of the free electron. This concept is successfully exploited in x-ray photoelectron spectroscopy (XPS) [3] [4]. However, as the photon's frequency ph ν decreases below that of the UV light, the energy of one single photon becomes small and less than the magnitude conserved in the interaction between light and matter. Such a mismatch is observed, for example, in electronic devices where photons are used to generate a photocurrent by providing energy to electrons to jump over an energy barrier. To overcome a 0.32 eV barrier, a photon with frequency 77. et al. [5] found that near IR light at 0.3 PHz ν = and average power can excite the electrons over the 0.32 eV gap at the MoSe 2 /2L-WSe 2 interface in heterostructure photocells between the conduction band (CB) of MoSe 2 ( CBMo E ) and the CB of WSe 2 ( CBW E ). Similarly, Adinolfi et al. [6], report that near IR light at 0.23 PHz ν = with P in the nW range, or below, maximizes the photocurrent production in silicon-based photovoltage field effect transistors (FETs).
Sarker et al. [7] confirm the trend by showing that green light at 0.56 PHz ν = with 5 μW P = generates photoexcited carriers by triggering electrons to jump from in-gap impurity levels to the CB in 4H-SiC used as substrate in graphene FETs. These results suggest that the average power of the electromagnetic (EM) wave might play a role in photocurrent generation.
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 2 mW 136 cm P ≈ [8] [9]. Typical activation energies per molecule are, e.g., 0.5 eV for the oxidation of cytochrome [10], or 0.72 eV for the CO 2 exchange [11], or 1.6 eV for the CO 2 oxidation [12].
A careful analysis of the light's wavelengths in and out of a photosynthetic organism shows that, e.g., in the carotenoid S 2 stimulated emission [13] with green light in at 520 nm in λ = 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 [14]. When illuminated by solar light, the RGCs elicit an action potential 70 mV 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, area, E the electric field, and H the magnetic field [15]. Experimentally, the average power P of visible and IR lasers is easily measured with power sensors. We then hypothesize that the energy conserved in the interaction between light and matter is c

Experimental Methods
Capacitors.  Table 1 and

Results and Discussion
To estimate the magnitude of t ∆ , and therefore of c E P 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.
where 0 ssli P P ≈ , and ssli P is the average power in the ssli.
The typical trends of ( ) Figure 1(a) and Figure 1(b). Similar trends with current appear in Ref. [7]. We fit the experimental data in Figure 1(a) with a sum of two exponential functions, and those in Figure 1 Figure 1(c) and Figure 1(d). We measure the slope of V ∆ versus P in Figure 1(c) to obtain the voltage responsivity V π ∆ [16], and the slope of T ∆ versus P in Figure 1(d) to obtain the temperature responsivity T ξ ∆ . The V π ∆ and T ξ ∆ responsivities indicate the amount of voltage and temperature produced by the capacitor at a certain average power P of the visible or IR lasers.
Next, in Figure 2(a) and Figure 2(b) we plot the voltage V π ∆ and temperature T ξ ∆ responsivities versus the capacitance C of the capacitors. We exhibit data obtained with c.w. and PL lasers. The specific PL lasers used have peak power peak P in the kW and MW ranges (Section 2). Clearly, V π ∆ increases when C decreases, while, within the uncertainties, T ξ ∆ is not sensitive to C. We show additional data in Appendix 3 confirming these trends with C. The small uncertainties on the responsivities signify negligible defects in our devices [7].
There are no noticeable differences between the trends of the V π ∆ and T ξ ∆ responsivities obtained with c.w. and PL lasers. This result aligns with the findings of Geiregat et al. [17] inferred from the study of amplified spontaneous emission spectra from thin films.
In Figure 2(c) and Figure  The experimental data indicate that V ∆ increases when the EM wave's average power P increases. Moreover, the trends of V π ∆ versus C and τ suggest that the increase in 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. [18].
According to our model, in each instant of time t in the EPR the energy where 0 Σ is the entropy in a closed system, as discussed in Appendix 2.  We also display data obtained upon illumination of the same capacitors with pulsed (PL) lasers with τ = 3.55 fs and peak P in the kW and MW ranges. Trends of (c) where ssli q is the surface charge in the ssli.
as follows: The voltage in the EPR is ( ) φ is an offset with magnitude of ~mV labelled in Figure 1(a). The offset V φ is due to contributions from the environment and from the capacitor. Likewise, from Equation (2b) we obtain V ∆ in the ssli as: The voltage in the ssli is ssli increase when C decreases, in agreement with the experimental findings in Figure 2(a).
We now combine Equation (2) and Equation (3) to (i) determine the magnitude of t ∆ , (ii) unveil the value of c E when the experimental value of the average power P is known, and (iii) clarify the dependence of ( ) To illustrate the general method enabling us to determine the magnitude of t ∆ we exploit the data in Figure 1(a) and Figure 1(b). The same method is equally effective when applied to all the 116 sets of data we measured and analyzed, whether collected with c.w. or PL lasers, in the whole average power and wavelength ranges explored, and for all capacitances considered. The method requires three steps: Step 1: rough estimate of c E and t ∆ . The data in Figure 1 increase with τ in agreement with the trends observed in Figure 2(c). Despite the agreement, the hypothesis that t τ ∆ needs further support.
Step 2: modeling of ( )  Figure 3(a). The agreement is very good, despite the slight mismatch at the inflection point, which we attribute to the uncertainties in P τ and T τ ∆ .
Step 3: refined expression for c E and value for t ∆ . By plugging into Equation (2b) for c E the experimental parameters and the estimated parameters  ( ) PL, experimental parameters: . Figure 3(c) and Figure 3(d) and obtained with c.w. (Figure 3(b) and Figure   3(c)) and PL lasers (Figure 3(d)  Similarly, Sarker et al. [7] shone green light at 1.77 fs τ =  The mechanism of energy transfer from light into a natural LH system and subsequent distribution into various chemical reaction pathways requiring different activation energies is called down-conversion [23]. We observe that, assuming We explore now other consequences of c E Pτ = being the energy conserved in the interaction between matter and light in the low frequency range of the EM

Summary and Conclusions
By revisiting the wave nature of light, we investigate the magnitude of the energy

Appendix1. Schematics of the Capacitors and Summary Their Characteristics
In this section we illustrate the structure of the Custom Thermoelectric 07111-9L31-04B devices used in our experiments. Figure A1 shows the schematics of a single capacitor and that of a single capacitor with additional insulating tape (IT) on the illuminated face. The IT consists of heavy cotton cloth pressure sensitive tape with strong adhesive and tensile properties. We use the IT to modify the capacitance C of the capacitors. We highlight the multy-layer structure consisting of AlO-Cu-Bi 2 Te 3 -Cu-AlO. The capacitor has a thickness h = 3 mm. The cross-sectional area cs A of the visible or IR laser has diameter D and corresponds to the illuminated area of the capacitor. Figure A2 depicts the schematics of two capacitors in series, one with and one without the additional IT. Placing two capacitors in series is an alternative way to modify the capacitance C. The characteristics of the capacitors, their C and dielectric constant ε values, are summarized in Tables A1 and Table A2 Figure A2. Schematics of (a) the two capacitors, and (b) two capacitors with additional insulating tape (IT). The other symbols are defined as in Figure A1. Table A1. Layer structure, diameter D of the illuminated area, capacitance C, and dielectric constant ε of the single capacitors used for the data presented in the main text. We modify the C values using insulating tape (IT), a pressure-sensitive tape made of heavy cotton cloth with strong adhesive and tensile properties.  Table A2. Layer structure, diameter D of the illuminated area, capacitance C, and dielectric constant ε of the two capacitors used to produce the results presented in the Figure  A7. Also for two capacitors, we modify the C values using insulating tape (IT), a pressure-sensitive tape made of heavy cotton cloth with strong adhesive and tensile properties.

A2-A. Conservation of energy
To justify the observed magnitudes and trends of the voltage responsivity V π ∆ 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 [18]. With P in the 10 to 1 × 10 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 cs A with diameter D corresponding to the cross-section of the laser beam: where cs q A σ = is charge, cs q A σ = 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 : we will refine this assumption in Sub-section A2-D. Hypothesizing enables us to separate the terms containing q and V ∆ , thus reducing Equation (A1) to: where we assume dΣ to be negligible. In a generalized grand-canonical ensemble [31], the entropy Σ is the Legendre transformation of 0 Σ , the entropy of the canonical ensemble, such that is Boltzmann's constant, and 1 B k T β = . Therefore: Assuming B k q V β ∆ to give just a slight correction to 0 Σ , dE is further reduced to: 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 t = ∆ . Here ( ) P t is the exponential function that describes the rise in time of the laser's power with a time constant P τ such that: where 0 ssli P P P ≈ ≈ and P is the average power in the steady state laser illumination (ssli) regime [32]. In the ssli the energy conserved is c E P t = ∆ . To obtain 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 In the EPR the integration leds to: In the ssli the integration gives: where ssli q is the surface charge in the ssli on the area cs A .
To model in the EPR the measured voltage difference The voltage in the EPR is φ is an offset with magnitude of ~mV . The offset V φ is due to contributions from the environment and from the capacitor. In the ssli instead, V ∆ is: The voltage in the ssli is ssli All possible rotations of the reference system, phases, and positions in the 2D plane can be obtained by selecting magnitude and sign of a r , b r , c r , and d r .
With this premise we can model ( ) To solve the integrals, we change the variables from x f to where ζ is a correction factor related to With these observations, the overall surface charge density and the correction factor ζ can be evaluated as: The magnitude of ζ has units of 2 nC V 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 Figure A5 and observe that ζ levels-off at a value which we call the equilibrium correction factor eq ζ . In  Table A1 and Table A2. The values of eq ζ vary between about 100 − and 2 nC 100 V . As expected, we note that eq ζ significantly deviates from the zero line for large T ∆ as τ moves toward the visible region of the EM spectrum (see Figure 2(d) and Figure A7(d)).
However, we also discover that large capacitances C contribute to large deviations of eq ζ from the zero line.
Appendix-3. Additional Trends of the Voltage π ∆ V and Temperature ζ ∆ T Responsivities with Capacitance C and Laser's Period τ In Figure A7(a) and Figure A7 Table A1 and Table A2. The PL lasers we used to obtain the results with τ = 6.67 fs are deascribed in Table A3.
T ξ ∆ responsivities versus the capacitance C of devices consisting of two capacitors in series. These devices are described in Figure A2 and their characteristics are summarized in Table A2. In Figure A7(a) and Figure A7(b) we observe that V π ∆ increases when C decreases, while, within the uncertainties, T ξ ∆ is not sensitive to C. In Figure A7(c) and Figure A7  lasers. The data were collected with c.w. lasers with periodτ = 1.77 fs, 3.55 fs, and 6.67 fs illuminating a cross-sectional area cs A with radius 1.5 mm 2 D = on capacitors with C = 18.51 pF and C = 245.7 pF. These capacitors are described in Figure A2 and their characteristics are summarized in Table A2. Table A3. Mode of operation (monochromatic), period τ , maker and type, repetition rate 1 p T , pulse duration p t ∆ , average power P range, peak power peak P range, and beam diameter D of the IR pulsed laser (PL) sources used to collect the data with τ = 6.67 fs discussed in Figure A6. We used continuum surelite optical parametric oscillator (OPO) lasers to tune the desired beams to specified wavelengths. These lasers were pumped with a frequency-tripled continuum surelite-II laser.