Interaction in the Steady State between Electromagnetic Waves and Matter

It is common experience that our eyes do not perceive significant changes in color when we observe for long time an object continuously exposed to light. We always see plants to be green in summer until in autumn factors external to our vision, such as changes in the length of daylight and temperature, cause the break-down of chlorophyll and, in turn, spectacular changes in plant’s colors. Likewise, the photocurrent produced in solar panels or field effect transistors achieves a steady state magnitude shortly after the start of the illumination. The steady state photocurrent lasts until the illumination stops. Understanding the origin of the steady state response of a device or light harvesting (LH) system to illumination with electromagnetic (EM) waves motivates the research presented in this work. In our experiments, we used capacitors as LH systems and illuminated them with infrared (IR) light over an 80 hours time period. We investigated the interaction between light and matter by monitoring versus time the voltage output of the capacitors. By combining modeling and experimental observations, we concluded that the steady state voltage is established soon after the start of the illumination as the consequence of the law of conservation of energy. We also found that the magnitude of the voltage in the steady state depends on the power and period of the illuminating IR light, and on the capacitance of the capacitor. When light’s power undergoes fluctuations, also the voltage produced by the capacitor and the surface charge density on the capacitors do so. These findings suggest that the law of conservation of energy has a significant repercussion when light is absorbed by matter in the steady state, for example in the mechanism of vision in vertebrates. Likewise, these findings are true when light is emitted from matter, for example in the mechanism of formation of the Cosmic Microwave Background (CMB).


Introduction
Is exposure time important in light-matter interaction?The question arises naturally when considering light as a wave.In this case, the energy transferred to matter is P t ∆ , where t ∆ is a time interval [1].The average power 2 P ∝ E , where E is the wave's electric field, can easily be measured with a power sensor.Therefore, because of the dependence on t ∆ of the transferred energy, when matter is exposed to light, two scenarios might occur: in the first scenario, the energy transferred from light to matter increases with time.In the second scenario, the energy transferred from light to matter reaches a steady state.By exploring the first scenario, we unveil outstanding consequences.One is that a prolonged and continuous exposure of a field effect transistor (FET) to light would increase with time the magnitude of the produced photocurrent.Another consequence is that the passing of time would change all year long the color of plants exposed to light, not just in autumn when chlorophyll breaks down due to the shortening of daylight and to temperature decrease.An additional consequence is that prolonged exposure to solar light of the photoreceptor cells (rods, cones and photosentitive retinal ganglion cells) in vertebrate's eyes would increase the magnitude of the action potential V ap produced in the process enabling vision.This increase would be such that 2 ap P t V ∆ ∝ [1].A further consequence would be the increase with exposure time t ∆ of a rotation rate rot ρ such that 2 1 2 , where exp I is a moment of inertia.Such a situation could occur to a support plate in a microwave oven irradiated by microwaves [1].
As a final consequence, we mention that the frequency CMB ν of the cosmic microwave background (CMB) radiation would increase with time after its initial emission since , as we will discuss elsewhere.Considering experimental results, however, leads to conclusions different than those just described.For example, Sarker et al. [2] suggest that photocurrent production in FETs reaches a steady state rather than indefinitely increasing with time.Our own experience indicates that no significant changes in color occur in plants in the three-month summer period, until in autumn the significant and steady variations in temperature and daylight change this trend.In general, from our own experience of vision, we know that colors remain steady unless something happens in the environment.Thus, we must admit that a drift in the magnitude of the action potential V ap with time of exposure to light of the photoreceptor cells does not occur.We can explain this fact either by assuming that V ap reaches a steady state upon hyperpolarization of the photoreceptor cells, or via other phenomena, such as thermal energy dissipation or mechanical motion, arising to keep V ap constant.We will show elsewhere that the steady state value of the action potential V ap upon hyperpolarization is the result of the evolutionary adaptation of the vertebrate's vision to the average power of solar light (136 mW/cm 2 [3] [4]).Moreover, everyday experience shows us that the rotation rate rot ρ of the support plate in a microwave is not accelerated by a prolonged exposure to the microwaves.Finally, accurate measurements of the frequency of World Journal of Condensed Matter Physics the CMB radiation convey that it peaks at CMB 160.23 GHz ν = , which makes the CMB radiation uniquely identifiable [5].These experimental evidences strongly suggest that with time the energy transferred from light to matter reaches a steady state in agreement with the second scenario.
In this work, we embrace the second scenario and show that achieving a steady state illumination (ssi) follows from the description of light-matter interaction through the law of conservation of energy.To achieve this conclusion, we study experimentally and through modeling the magnitude of the voltage ( ) versus time produced by the interaction of infrared (IR) light with a capacitor.We observe that, upon illuminating with IR light the capacitor for hours or days, the voltage ( ) V t levels off in the ssi regime at a value ( ) . We assume that in the ssi the voltage ( ) produced by the capacitor, the temperature difference ( ) T t ∆ and the surface charge ( ) q t are constant.However, fluctuations can occur in the average power ( ) P t of the IR light due to the operation of the Q301 globar source as described in Ref. [6].These fluctuations trigger perturbations of ( )

T t ∆
and ( ) q t .The perturbations in ( ) ( ) T t ∆ can experimentally be observed [1] [6] [7], but those in ( ) q t and in the surface charge density ( ) t σ cannot be detected.Nevertheless, literature results support evidences for the existence of such perturbations.For instance, light with wavelength from 1600 nm to 700 nm was found to trigger changes in the surface charge density ( ) t σ in polyethylene [8] with magnitude dependent upon the IR light's power over area, i.e. the intensity, and with a charge responsivity of about 5.3 pC/W [8].The fluctuations in the charge density were found not to be related to heating effects, in agreement with results of Ref. [1].Thus, supported by experimental observations and by modeling, we incorporate into the energy conservation equation describing the interaction of IR light with matter [1] [6] the experimentally observed sinusoidal fluctuations of the average power ( )

( )
V t derived from the energy conservation equation and the experimentally observed ( ) V t .This result extends to the ssi regime the va- lidity of the law conservation of energy for the transfer of energy from light to matter.The role of this law was initially established in the exponential perturbation regime (EPR) immediately following the start of the illumination with IR light [1] [7].The importance of these findings is that they help understanding several natural phenomena such as the generation of the CMB and the mechanism of vision in vertebrates that we will discuss elsewhere.

Experimental Set-Up
Capacitors: We studied the interaction of IR light with matter through the vol-World Journal of Condensed Matter Physics , and two capacitors without IT as in Figure 1(c) enabling 526.9 pF C = .Infrared light: We used the continuous broadband IR light in the 350 -7500 cm −1 wavenumber (28,600 nm -1300 nm wavelength or 95.33 fs -4.33 fs period) range produced by a Q301 globar source in a N 2 -purged Bruker Vertex 70 spectrometer [6] [7].The beam diameter is 10 mm D = . The average power is 25 mW P = between 700 -7500 cm −1 (14,286 nm -1300 nm, or 47.62 fs -4.33 fs period, the middle IR (MIR) range) and about 21.2 μW between 350 -700 cm −1 (28,600 nm -14,286 nm or 95.33 fs -47.62 fs, the far IR (FIR) range).We placed a polyethylene polarizer (Pike Technologies) with a 500 -10 cm −1 spectral range between the IR light and the capacitor to achieve the 21.2 μW average power in the FIR range.We monitored versus time t the average power ( ) P t of the IR light in the MIR range using a power-meter sensor Coherent Power-Max RS PS19, sensitive to the 300 -11,000 nm wavelength range and to the 100 μW to 1 W power range.
Temperature measurements: We measured the temperatures of the illumi-

Results
To investigate the magnitude of the voltage versus time ( ) V t in the ssi, we collected for ~80 hours the voltage ( ) V t generated by a capacitor interacting with IR light.This amount of time assures that fluctuations can be observed.Without fluctuations of the average power ( ) P t we expect the voltage in the ssi to be , where 0 V is the voltage before the start of the illumina- tion with IR light and V ∆ the jump in voltage in the EPR with a time constant World Journal of Condensed Matter Physics of tens of seconds [1].With sinusoidal fluctuations arising in ( ) P t , in time we expect perturbations to occur in the voltage ( ) V t , the temperature difference ( )

T t ∆
, the surface charge ( ) q t and the surface charge density ( ) In particular, the experimentally observable ( ) , where ( ) F t is the perturbation transferred by the fluctuations to ( ) V t .This perturbation justifies the non-constant trends of the voltage ( ) V t in the ssi, as we observe in Figure 2. Here, all the panels show that ( ) V t achieves a mini- mum or various minima over a time interval several hours long.Specifically, Figure 2     V t , the first of which with min 2.51 mV V = appears 8 hours after the start of the illumination.
In each panel of Figure 2,

( )
V t can be fitted using a hyperbolic secant func- tion [6]: ( ) where osc V is the oscillation in the voltage due to the quasi-sinusoidal fluctuations of ( )

Model
To determine 1) the magnitude of ssi V , 2) the surface charge density ( ) t σ , and 3) the mechanism that transfers to ( ) V t the of ( ) P t , we use the law of conservation of energy [1]: where ( ) E t is the energy transferred from the IR light to the capacitor, and 0 Σ is the entropy in a closed system derived as in Ref. [1].
1) Determining the magnitude of V ssi By manipulating Equation ( 2) we obtain the voltage ( ) In Figure 3, we display the voltage ( ) V t modeled from Equation (3) along- side the fitting functions obtained with Equation (1) of the experimental voltage ( ) V t of the four cases reported in Figure 2. The values of ssi V , viewed as ( ) , are derived from the comparison between the panels in Figure 3 and the plateau level achieved by the jump in voltage V ∆ at the beginning of the illumination with IR light (the EPR) for the each of the capacitors under various illumination conditions.These jumps in voltage V ∆ are pictured in good match between the values of ssi V extracted from Figure 3 and Figure 4 suggests that the law of conservation of energy in Equation ( 2) describes in a satisfactory manner the light-matter interaction in the EPR and in the ssi.In both the EPR and the ssi, the unknown variables are 0 Σ , ( ) ( ) t σ .The en- tropy 0 Σ can be derived as in Ref. [1] for the whole time interval that covers the EPR and the ssi.Whereas, the surface charge ( ) q t and surface charge den- sity ( ) t σ in the ssi requires a more dedicated discussion which we present in the next Section.
2) Determining the surface charge density variation in time,

( ) t σ
To determine the surface charge ( ) q t and the surface charge density ( ) t σ in the ssi we proceed as in Ref. [1], which addressed the same issue in the EPR.
In the ssi, as in the EPR, the average power ( ) P t of the IR light acts on the World Journal of Condensed Matter Physics surface charges ( ) q t on the capacitor with a force ( ) ( ) ( ) force displaces the charges away from the location in which the IR waves illuminate the capacitor, locally decreasing their surface density as ( ) ( )  V t is produced as described in Equation (3).In this process, ( ) varies in time t as well as in space.The space variable r is a complex variable in the 2D space represented as , where i is the imaginary unit [1].In the EPR the average power ( ) P t increases exponentially until it achieves the value ssi P selected for the measurement [1].In the ssi, however, ( ) P t sinu- soidally oscillates around ssi P due to the operation of the Q301 globar source affected by the temperature fluctuations of the closed sample compartment, as described in Ref. [6].We expect the oscillations of ( ) P t to be reflected on the Equation ( 6) suggests that the sinusoidal instability in the average power ( ) P t produces a hyperbolic instability in ( ) 3) Determining the mechanism that transfers to the voltage ( ) , in Equation ( 6) we write the spatial variables as follows: To solve the integrals in Equation ( 7) we make a substitution of the variables such that To know the value of ( ) q t in the ssi requires estimating ssi σ , osc σ , x v , y v , x t σ , and y t σ .We assume that ssi σ coincides with the values found in the in time t by a 07111-9L31-04B device manufactured by Custom Thermolectric Inc.The basic device, illustrated in Figure1, is a thermoelectric device consisting of a sequence of layers: 1) an insulating alumina (AlO) plate on the face exposed to the IR light, 2) a metallic Cu plate, 3) a layer of pillars made of a doped semiconducting Bi 2 Te 3 -based alloy, 4) another Cu plate, and 5) another AlO plate.This multi-layer structure resembles that of a capacitor, as shown in previous research[1] [6][7].To change the capacitance C of the capacitor we combined the capacitors in series or added a layer of insulating tape (IT) on the face exposed to the IR light.The IT consists of heavy cotton cloth pressure sensitive tape with strong adhesive and tensile properties.In this research we used three specific structures: two capacitors in series with IT as in Figure1(a) featuring 148.5 pF C = , one capacitor with IT as in Figure 1(b) with 298.1 pF C = of the capacitors using OMEGA type E Ni-Cr/Cu-Ni thermocouple probes[1] [6][7].Data collection: We exposed the capacitors to IR light for ~80 hours after starting the illumination, and collected the voltage ( ) V t , the temperature dif- time t for the entire duration of the measurements in the ssi.The data were collected using LabView 2012 and a National Instruments PXI-1042q communications chassis[1] [6][7].

Figure 1 .
Figure 1.The cross-section of the capacitors used in this research consists of a layer of Bi 2 Te 3 semiconducting elements embedded between alumina (AlO) and Cu plates.Configuration (a) illustrates two capacitors in series with insulating tape (IT) on the illuminated face.This combination features a capacitance 148.5 pF C = .Configuration (b) depicts one capacitor with IT on the face exposed to the IR light which enables 298.1 pF C = .Configuration (c) pictures two capacitors in series without IT on the face exposed to the IR light allowing 526.9 pF C = .The IT consists of heavy cotton cloth pressure sensitive tape with strong adhesive and tensile properties.
(a)  shows the case of a capacitor with 148.In this case, 0.25 mV ssi V = and the sinusoidal fluctuations of ( ) P t generate a minimum voltage min 0.20 mV V = occurring in ( )V t 56 hours after the start of the illumination.

Figure 2 (
b) illustrates the case of World Journal of Condensed Matter Physics

Figure 2 .
Figure 2. The variation of voltage versus time ( ) V t in the time interval of ~80 hours after starting the illumination with IR light.The experimental data are presented along with the fitting line (continuous black line) derived from Equation (1).We selected four cases with different angle of incidence 0 θ , capacitance C, period τ , and average power P. (a) 0 0 θ =  , 148.5 pF C = , 83 fs τ = , 21.2 W ssi P ≅ µ , average voltage in the steady

Figure 4 .
Figure 4. Using the results from Figure 3 and Figure 4 we obtain the following values for ssi V in the four examined cases: a) capacitance 148.5 pF C = and period 83 fs τ =

Figure 3 .
Figure 3. Experimental fitting line from Equation (1) (red line) and modeled line (black line) for different capacitors under various illumination conditions.The experimental fitting lines (red lines) correspond to those in Figure 2. The fluctuations in voltage occurring in the time interval of ~80 hours after starting the illumination with IR light are all of the same size, despite different capacitor and illumination conditions.We modeled the four cases reported in the panels using Equation (3) and the following parameters: (a) Angle of incidence 0 0 θ = 

Figure 4 .
Figure 4. Experimental fitting line (red line) and modeled line (black line) for the jump in voltage V ∆ occurring for different capacitors under various illumination conditions at the beginning of the illumination with IR light.It is noticeable that V ∆ depends on capacitance C, period τ , and average power P. We performed the modeling for the cases reported in the panels using the following parameters: (a) Angle of incidence 0 0 θ =  , capacitance y axes, and x t σ and y t σ are the critical times for their relaxation.The quantities x L and y L is the IR beam's diameter[1].Considering ssi σ and osc σ the average surface charge density in the ssi and the amplitude of its deviation from equilibrium, respectively, we obtain: 0 to 1 and enabling us to locate any position on the surface of area A of the capacitor[1].For example, by choosing the origin of the reference systems on the lower left corner of a square we have that, when 0 With these variables, in the ssi we integrate Equation (6)over A and obtain:

t of the IR light, and, consequently, of the voltage
t , cV t is critical time of the voltage, and