C. Roychoudhuri^1*, N. Prasad b², G. Fernandoa¹

¹ Research Professor, Physics Department, University of Connecticut, Storrs, CT 06269, USA.

² NASA Langley Research Center, MS 468, Hampton, VA 23681, USA.

*Corresponding Author:C Roychoudhuri, Research Professor, Physics Department, University of Connecticut, Storrs, CT 06269, USA.Tel: (860)486-2903; Fax: (860)486-3346

Citation: C Roychoudhuri, N Prasad b, G Fernandoa (2023)Where Lies the Quantumness behind Detecting Electromagnetic Waves for Frequencies from Infrared and Up. Nano Technol & Nano Sci J 6: 159.

Received: May 28, 2024; Accepted: June 25, 2024; Published: June 28, 2024.

Copyright: © 2024 C. Roychoudhuria, et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

The synthesis of Newtonian concept of corpuscles during emission with Huygens’ concept of secondary wavelets during propagation implies that all EM radiations from quantized atoms and molecules are released as discrete amount of energies. However, they propagate out as time finite Maxwellian light pulses. Huygens also underscored that his secondary wavelets keep propagating as independent pulses in the absence of any interacting medium, or until intercepted by an interface with a medium or a detector. Then we use the Superposition Principle and the coherence theory to derive Einstein’s photoelectric equation by summing innumerable random time-finite pulses. This process driven approach should yield the characteristic statistical variations of photoelectron current pulses, as generated by photodetectors for different kinds of light sources. Lamb & Scully originally proposed this semiclassical approach without assuming that light actually consists of time finite pulses. The quantumness remains confined within the excitation and de-excitations processes in the material particles.

Keywords: Physics of Photoelectric Effect; Photon Counting Statistics; Photoelectron Current Pulses (PCP), Generation of; Semiclassical Theory of PCP generation; Coherence Manipulation to Control PCP statistics; Photoelectric Equation; Superposition Principle.

Healthy debate to keep ourselves challenged

The modern debate behind the quantumness of light has re-emerged [1] because of the revival of the old “Wave-Particle Duality” (WPD) that was between Newton and Huygens. Several of the founders of the Quantum Mechanics had actively promoted this WPD and now it is considered as new science, even though it is really our ignorance about the realities behind EM waves and elementary particles. The fundamental problem is that all of our theories are necessarily incomplete because, even though, they have been formulated based upon postulates constructed by far-sighted contemporary geniuses, their knowledge about the workings rules of the universe were insufficient. This insufficiency will continue with us, probably, forever, even while our working knowledge of the universe keeps enhancing as we keep iteratively advancing newer theories. Therefore, the driving point of this article is to inspire people to keep challenging the older working theories to explore the possibilities of developing better theories that guides the visualization of nature’s working processes. Our engineering successes depend upon our capability to emulate nature allowed physical processes, in novel ways, even when the theory is not yet perfect. Newton understood and expressed that his debate with Huygens had remained un-resolved because neither of them wereable to figure out the proper and complete model of light, from generation to propagation to detection. Today, severalcenturies after Newton’s time, the founders of the Quantum Mechanics (QM) of early 1900have revived the old WPD because we have not yet figured out the detailed physical processes behind the generation, propagation and detectionprocesses of Electromagnetic radiation. Maxwell’s wave equation has given an excellent formalism to visualize theperpetual propagation EM waves through the free space and all material media leveraging the electromagnetic tension fields, both in the free space and in material media. Maxwell derived that the perpetual velocity ofl ightis empowered by the electromagnetic tension properties, c2= 1/ em , where  and m are the electromagnetic tension properties of the medium. Optical engineers always use refractive indices to determine the velocity of light in different media, which are derived using, e and m for the media. For the free space =1/ ; consequently and represent physical parameters of the free space, which we should not neglect [2-5]. We should note that the variable “c”, from medium tomedium, is a derived parameter from the electromagnetic tension properties; it is not a fundamental constant of nature. Maxwell’s waves are amplitude undulation of diverse electromagnetic media. WPD should not encourage us to neglect the critical and routine successes we have been deriving from Maxwell’s equations, before we can find an alternate mathematical theory that can explain how an electromagnetic“ energy bullet” (photon) can acquire perpetual velocity.

Flow of the paper

Let us briefly underscore that if we pay attention to the physical processes behind the emergence of our experimental data, we will find that the quantumness in detecting EM waves lies with the physical interaction properties of the detector. If the frequency-resonant classical detectors (dipole oscillator) can keep absorbing energy out of the stimulating EM wave continuously, there is no “quantumness” in the EM waves, as in radio and microwaves. For higher frequencies of EM waves, the frequency-resonant energy absorbing detecting dipoles are quantized atoms, molecules, or their assemblies. They have built-in finite quantum-cups. They can absorb energy out of the EM wave just to fill up their quantum cups. Once they fill up their quantum-cup with a discrete amount of energy, they cannot accept any more energy until they are completely recycled. It does not make logical sense to assign this “quantum cup” limitation of quantum detectors on to Maxwell’s EM waves. Figure 1 gives a brief summary of the EM wave spectrum and the suitable detectors for each range of frequencies.

Figure 1: EM waves, being harmonic oscillations, can transfer energy only to frequency-resonant oscillators (detecting dipoles). When the resonant oscillators can absorb energy continuously from the EM waves, as in the cases of radio to microwaves, the detector designers have no need to learn quantum mechanics. For higher frequency EM waves, the frequency-resonant detectors are usually atoms and molecules with quantum-cups discrete size for energy absorption. Therefore, these detectors can report only events of discrete energy absorption. Detectors enclosed within the dashed elliptical enclosures are discrete energy absorbers, while the ones within the dashed circle can keep on absorbing energy continuously. Therefore, we need to pay attention to the physical interaction processes that generate data in our instruments. Mathematical theory alone cannot explain the physics behind interaction processes. [This figure has been synthesized using borrowed cartoons from various freely available websites.]

Newton understood and expressed that his debate with Huygens had remained un-resolved because neither of them wereable to figure out the proper and complete model of light, from generation to propagation to detection. Today, severalcenturies after Newton’s time, the founders of the Quantum Mechanics (QM) of early 1900have revived the old WPDbecause we have not yet figured out the detailed physical processes behind the generation, propagation and detectionprocesses of Electromagnetic radiation. Maxwell’s wave equation has given an excellent formalism to visualize the perpetual propagation EM waves through the free space and all material media leveraging the electromagnetic tension fields, both in the free space and in material media. Maxwell derived that the perpetual velocity of light is empowered by the electromagnetic tension properties, c2= 1/ em, where e and m are the electromagnetic tension properties of the medium. Optical engineers always use refractive indices to determine the velocity of light in different media,which are derived using, e and m for the media. For the free space =1/ ; consequently and represent physical parameters of the free space, which we should not neglect [2-5]. We should note that the variable “c”, from medium tomedium, is a derived parameter from the electromagnetic tension properties; it is not a fundamental constant of nature. Maxwell’s waves are amplitude undulation of diverse electromagnetic media. WPD should not encourage us to neglect the critical and routine successes we have been deriving from Maxwell’s equations, before we can find an alternate mathematical theory that can explain how an electromagnetic“ energy bullet” (photon) can acquire perpetual velocity.

These three phenomenological physics equations have been helping us quantitatively understand and keep exploring many natural phenomena within each ones domain of applications for prolong periods. The first equation is directly a superposition equation, superposition of innumerable spherical wavelets. The second and third equations accept the linear combinations of all possible allowed solutions to their respective equations as new solutions, because they are linear differential equations. However, just the simultaneous presence of the allowed solutions (SP) within the same physical space does not automatically generate the emergence of the Superposition Effect (SE). There has to be a frequency-resonant detector that can simultaneously respond to all the simultaneously stimulating waves. Then the detector has to execute the square modulus operation, E* E or ????* ????, to absorb the necessary energy and create the measurable data. Fields, by themselves do not generate data. Unfortunately, we have developed a sustained cultural belief that the fields by themselves, or even a single component of the superposed field, can generate the Superposition Effect (SE). This defies the logics embedded within our successful mathematical theory (or prescription). This will be clearer from Eq.6 later, where we will discuss the importance of differentiating the mathematical SP (simultaneous presence of multiple amplitude signals in the same space) from SE (the energy exchange & data generation by some physical detector that executes the square modulus operation).

In Section 3, we propose a solution to the Wave-Particle Duality while synthesizing Newton’s “corpuscular” concept with Huygens’ “secondary wavelet” and, at the same time, maintain the conceptual continuity with the predictions of modern QM, both in discrete-energy-emission by atoms/molecules and discrete-energy-absorption by quantum detectors. We also propose potential analysis and experiments to derive the photoelectric current pulse (PCP) statistics using the fundamentals of emission, superposition and detection processes.

Incompleteness of Classical and Quantum Physics

Incompleteness in our overall knowledge is natural and pervasive. Incompleteness in our theories are deeply fundamental.

Gödel’s “Incompleteness Theorem” [6] has demonstrated that no mathematical theory can ever be complete, because all mathematical theories have to start with an axiom or a postulate, which are unproven but intelligent guesses only, and not some confirmed knowledge, due to our initial ignorance about the rules of nature. Therefore, we should subject all theories to continuous critical evaluation to advance our deeper knowledge on visualizing the physical interaction processes that nature executes. Evolution is a sustained story of continuous engineering successes by all species [7].

Our Evidence Based Science (EBS) has been thriving upon experimentally reproducible data to validate our theories. However, no apparatus can extract complete information about any unknown entity under study. We construct our apparatus to generate data through interaction between the unknown entity under study and a known referent entity. First, no apparatus we construct has 100% fidelity in transferring the ‘recorded’ data [8, See Ch.12]. Second, we never know the complete properties of any referent entity. We deliberately assume wide ranges of approximations and also assume all the forces are negligible except the one force of interaction under consideration. Further, we also neglect the influence of all the diverse “background fluctuations”. Thus, all of our data are approximate and hence our knowledge about the unknown entity can only be approximate, and not complete. That is why we do not have the choice but to accept that all of our theories will always be incomplete. We must accept this barrier of insurmountable incompleteness of all theories and develop a strategy to keep iteratively improving them. One of the proposed strategies is to incorporate the interaction process visualization, while taking guidance from the mathematical logics built into the working theory and iteratively keep pushing to develop better and better theories. We intend to apply this “interaction process visualization” strategy to enhance the theory of photoelectric emission [9].

Foundational background in framing an equation determines its future strengths

Let us briefly explore the foundational background of the three equations, Eq.1 to 3, presented above. Eq. 1 is a literal mathematical translation, made by Fresnel in 1817, of the postulate, “Secondary Wavelets”, presented by Huygens before 1690: Once a wave is triggered on a tension field by a suitable external force, every point on the wavefront keeps generating secondary wavelets moving in the same direction. The conceptual physical picture behind this postulate is as follows. A parent tension field cannot assimilate (absorb) the original external perturbed energy that has triggered the wave. Therefore, the tension field spontaneously tries to get rid of the perturbing energy by simply pushing it forward out of every point. Thus, the wave is perpetually regenerated as a superposition of innumerable secondary wavelets in the same forward direction. Therefore, the original wave packet keeps moving indefinitely. That is what happens for EM waves in the medium of the free space, or in the media of different materials. Thus, Huygens postulate, based upon his initial logical mental visualization, received a mathematical formulation by Fresnel. It became a great success story in physics. This wave theory guided the advancement of physics from many directions. Then, in 1876, Maxwell mathematically restructured the experimentally observed laws of Electrostatics and Magnetostatics, and unified them [10, 11]. Maxwell derived the wave equation for EM waves, Eq.2. Note that Eq.1 is functionally a solution of Maxwell’s wave equation since it is a linear superposition of innumerable spherical wavelets.

Thus, the evolutionary history of the Eq.1 and Eq.2 are grounded on centuries of experiments and understanding of the physical pictures of the natural processes involved in electromagnetism. This is why we see the continued successes in optical science and engineering including the successful emergence of the new applied fields of Plasmonic Photonics, Nanophotonic, and Metamaterials. However, Maxwell’s equation cannot model the physical processes behind the generation, and the absorption of light by atoms and molecules. Planck’s law of 1901 on the Blackbody Radiation underscored that the atomic and molecular radiation exchanges must take place with the emission and absorption of EM wave energy in discrete amounts. In 1913, Bohr proposed his “old quantum theory” and gave the “planetary” model of atoms having discrete energy levels. However, it was not extendable much beyond the Hydrogen atom. In 1914, Frank and Hertz experimentally demonstrated that the heavier atoms like Mercury also possess discrete quantized energy levels. Then, in 1925, Heisenberg presented his Matrix mechanics as one of the foundational approaches to modern Quantum Mechanics QM. It can precisely predict the quantum energies that we can measure. However, it was not helpful in visualizing the physical picture of atomic emission and absorption processes. In 1926, Schrodinger produced his “wave equation”, which appeared to be a better approach to QM from the standpoint of visualizing the physical pictures of “particles” as “plane waves”. Unfortunately, even the Schrodinger’s equation (Eq.3) has not succeeded in helping us visualize the physical processes behind the absorption or the emission processes taking place in atoms and molecules. It can predict only the final measurable discrete energy exchanges. The process remains hidden behind the postulate, “collapse of the wave function”! Therefore, the early founders of QM, while establishing the currently dominant Copenhagen Philosophy, promoted the philosophy that the purpose of physics theories is just to validate the experimental data, not to visualize the interaction processes that nature carries out everywhere. Schrodinger’s equation does give the precise position of an orbiting electron in an atom! Should we take this limitation as “forever”; or, should we take it as an entry point for further enquiry and further enhancement of the theory?

Let us briefly compare Maxwell’s Eq.2 with the Schrodinger’s Eq.3. Eq.2 is a proper wave equation since it balances the temporal “acceleration” with the spatial “acceleration” through the second derivative of time and space, respectively. This leads to the perpetual wave velocity of light, c² = 1/ ????????, where ???? and ???? are the nature’s action parameters, the electric and magnetic tension properties of the medium in which EM waves are traveling [5]. We should note that c is a medium-dependent variable and a derived parameter, and not a “fundamental constant” of nature. Schrodinger’s equation does not have the built in “temporal acceleration” (second derivative of time). That is why quantum particles cannot be “perpetual waves” like EM waves are. In fact, this is why there is a “potential gradient” term V (x, t) in Eq.3, which provides the “space limited” physical push/pull potential gradient for the particle to move; but it cannot move perpetually like the EM waves. However, the quantum mechanical beauty of this equation is that the allowed solutions are amplitude-harmonics and accommodates the Superposition Principle, just as Maxwell’s equation does.

 Thus, even though both the EM waves and the quantum particles “carry energies” to exchange during physical interactions to generate observable effects; form the standpoint of interactions, they are amplitude signals obeying the SP (Superposition Principle). Neither of these equations of propagation represent energy-bullets from the standpoint of triggering any energy exchange process directly. They trigger the interaction process as amplitude-amplitude stimulations. When the interactions are compatible, frequency-resonant, they will trigger linear amplitude stimulation, followed by the square-law energy transfer process (????* ????). This is a two-step process, built into our mathematical recipe. Therefore, EM waves neither propagate as energy bullets, nor can generate superposition effect (energy transfer) by itself without being facilitated by a detector to execute the square modulus operation. Huygens Principle categorically underscores that wavelets keep propagating without interacting with each other, or physically transforming each other’s characteristic wave properties. We call this important characteristic of EM waves as Non-Interaction of Waves (NIW) in the absence of interacting materials [8].

It is apparent that Nature is a marvelously creative system engineer. It is the diverse physical transformations through diverse physical interactions, from dust-to-dust, that nature has been maintaining a cyclically evolving universe, both in the Cosmosphere and in our Biosphere. All species, from viruses to humans are a product of three to four billion years of evolution where all the species have been constantly carrying out diverse routine and innovative engineering activities, by simply emulating nature-allowed processes, without knowing or deciphering the laws of nature. Humans have started succeeding in inventing approximate but serious mathematical theories starting less than one thousand years back. Our, key point is that successful biological evolution requires sustained and successful engineering innovations allowed by the rules of nature, irrespective of whether we can theorize the rules perfectly well or not. Therefore, it is more important for us to use the powerful logics of math to guide us to visualize the nature allowed interaction processes so that we can emulate them more efficiently, rather than seeking only esthetic beauty and harmony inside the elegant mathematical universe we can create.

Mathematical Superposition Principle (SP) is incomplete without recognizing the physical processes behind the recordable Superposition Effect (SE)

We have already mentioned that the three equations (Eq.1, 2, 3) have the common property of abiding by the Superposition Principle (SP). Eq.1 is a direct statement of the SP, as it represents linear summation of harmonic waves. The Eq.2 of Maxwell and Eq.3 of Schrodinger, both are linear differential equations. Therefore, mathematically, any linear combination of individual solutions of these equations will also satisfy their respective equations. We know this as the Superposition Principle (SP). Thus, we see that SP plays a very significant role both in the classical and in the quantum physics. This is a very important conceptual continuity in nature, which we should not neglect. Therefore, it is of critical importance for us to visualize the invisible operational processes, which generate the final measurable data in our instruments, after being “superposed” to interact with a detector, whether it is classical or quantum mechanical.

Differentiating between Superposition Principle (SP) and Superposition Effect (SE)

We need to recognize nature’s fundamental Interaction Principle. No observable and/or measurable physical transformation can happen in nature without some interaction between more than one physical entities, guided by some mutually compatible force of interaction. Even our mathematical theory defines superposition principle as summation of more than one physical entity containing amplitudes and phases carried by multiple entities. Therefore, a single particle, or a well-defined single pulse of EM wave, cannot generate by itself the measurable superposition effect. Eq.4 represents the mathematical expression for the generic amplitude  Superposition Principle (SP) where  can be considered as solutions of either the Maxwell or the Schrodinger equation. The two sets of n-parametric values cannot be carried by any single elementary entity:

We trust our mathematical equations when validated by diverse experiments. Then, we must respect and leverage the built-in logics behind working mathematical equations to explore and visualize the interaction processes it represents. However, Eq.4 does not generate data. We know that EM waves do not interact by themselves to generate data due to the NIW property of waves. Light beams from billions of galaxies and/or stars cross through each other during their long journey in space towards our earth. However, our telescopes, when record well resolved images, they preserve all the spectral and other characteristics of each cosmic entity under study. Without frequency sensitive light detector array, we cannot register the images of stars. The same is true for interferometers. We cannot register superposition fringes without the active participation of all the beams simultaneously stimulating the detector array [Ch.2 & 3 in 8]. However, it is a slightly different story for particles. Particles do interact with each other. However, one still needs a “particle” detector to register simultaneous presence of multiple particles bringing multiple amplitude and phase information, as is implied by the generic superposition Eq.4. Our mathematical recipe tells us that only the square modulus of the Eq.4 can generate measureable data. However, the “square modulus” has to be a physical operation executed by some material detector that is frequency resonant to be stimulated. Then it absorbs energy out of all the stimulating beams and undergoes physical transformation to report that the detector array has executed the square modulus operation. Therefore, for the registration of optical interference phenomenon, we must first recognize the generic electromagnetic polarizability of a suitable material dipole, as in Eq.5, consisting of both a linear term and a set of nonlinear terms, where c_n (n ) is the n-th polarizability interaction parameter of the detecting dipoles. If the detector is a quantum dipole,then c_n (n ) represents its quantum dipole property. For a radio receiver, it is the polarizability of the frequency tuned LCR circuit.

Fortunately, for most EM wave detection under normal intensity levels, the approximation of keeping only the linear term is sufficiently accurate. Then, we can express the instantaneous energy available for absorption by a detecting dipole can be expressed as Eq.6, where  now represents n-different EM signals, simultaneously stimulating the material detecting dipoles:

Let us also note that we always design our superposition experimental apparatus such that more than one signal of similar class (mutually phase-steady) is generated and then superposed simultaneously on an interaction-compatible detector, or a detector array. That is why Michelson developed the technique of alignment of interferometers using white-light to ascertain zero relative path delays between the two arms of any two-beam interferometer [13]. With the advent of lasers, we now can generate wave trains that has phase-steady relation over a long-time duration and the alignment restrictions are relaxed. This is long coherence time (or length). The theory of optical coherence is a major subject developed by classical physics over a couple of centuries [14]. Glauber gave it a quantum mechanical formalism [15], which turns out to be mathematically equivalent to classical formalism [16].

We conclude this section by underscoring that the Superposition Principle (SP) is the correct mathematical starting point to start analyzing both the classical and the quantum mechanical superposition effects. However, the mathematical expression for SP does not represent any observables data. The mathematical square modulus does represent the energy transfer process, but only when the detectors’ polarizability parameter is incorporated to model the physical interaction process. Accordingly, the continued progress of Evidence Based Science along the right path critically depends upon the incorporation of visualizing the invisible interaction processes within our apparatus that generate the “evidence” (data) for us.

Proposed model for the Hybrid Photon wave packet to resolve the wave-particle duality conundrum

Conceptual models presented here can collectively eliminate the need for continuing with the postulate of Wave-Particle Duality (WPD). The WPD does not represent any new knowledge about “photons” and “particles” that can facilitate the continuous advance of our evidence based science. We need to keep exploring possible physical process models behind the absorption and emission of EM wave energy by atoms and molecules. Currently neither the Maxwell’s equation nor the Schrodinger equation provides explicit guidance in this direction. Yet, these are working theories; and hence definitely embed some of the nature’s actual working processes behind Maxwellian ether and Schrodinger’s “waving” particles. Therefore, further exploration is worthwhile.

Our approach relies upon preserving conceptual continuity and logical congruence between the working theories and diverse observations while introducing newer concepts. This paper does not suggest any new theories or any fundamental changes. Our key approach is to suggest newer views that can eventually be validated through new experiments within the bounds of the current theories, while opening up the gates for their future improvement.

The Hybrid photon wave packet

We are proposing that all EM energies, emitted by atoms and molecules, propagate out as time-finite Maxwellian wave packets [17] in the all-pervading electromagnetic tension field, whether cosmic vacuum or material media It tacitly accepts the Newtonian “corpuscles” and the Einsteinian “light quanta”, hn, in the sense that the total energy available out of the wave packet is hn. We chose the shape of the wave packet emergent out of atoms or molecules as dominantly an exponential to conform with the observation that spontaneous emissions show the spectral line width as Lorentzian [17], which is the mathematical Fourier transform of an exponential function. We have demonstrated elsewhere [8, See Ch.5] that the time-integrated fringe width function of classical spectrometers to any time finite wave pulse is the square modulus of the Fourier transform of the incident temporal wave envelope. We show this model of photon wave packet, consisting of quasi-exponential time envelope, in Fig.2.

According Eq.1, Huygens-Fresnel Diffraction Integral, light always propagates through the process of diffractive spreading. Therefore, individual light pulses are no longer capable of delivering all its energy to any remote atoms whose physical cross-section is approximately 1A. This is why a completely new separate field of micro-cavity QED has evolved for some time [18]. However, even for ordinary photodetectors, the detecting atoms and molecules have only Angstrom size physical cross-section. At very low light level, the so-called single-photon flux levels, the energy flux propagating through (1A)²– cross section in a 1mm diameter collimated 1mW laser beam could provide only~10^-18 photon-equivalent energy hn per second. Obviously, the effective energy absorption cross section of the detecting element has to be many orders of magnitude larger than (1A)2. From the classical dipole models, from radio to cell phone to atom, we already know that, at frequency resonance, the dipole projects itself as a much larger energy- harvesting cup.

Figure 2: A hybrid photon mode that harmoniously accommodates the concepts of Newton, Huygens, Maxwell, Planck, Einstein, Lorentz and Schrödinger. All lights emitted by atoms and molecules are Newtonian pulses, but propagate as Huygens “secondary wavelets”, following Maxwell’s wave equation, and remaining as independent pulses following Huygens’ postulate of Non-Interaction of Waves (NIW). So far, neither Maxwell’s equations, nor that of Schrodinger, provide explicit model for “the transition domain”, as to how the discrete packet of energy hn evolves into a classical wave packet. For radio and microwaves, the radiating dipoles physically oscillate to generate the radiation. Does nature have a very different model for the atomic world; or are we missing something?

We would call this behavior of frequency-resonant atoms as projecting an enlarged quantum cup (Fig.3). The computed energy converging field lines are shown in Fig.3b [19, 20]. We would call this as a push-pull phenomenon, jointly generated by the electromagnetic tension field and the frequency resonant dipole. The tension field, with imposed wave generated on it by some earlier dipole, is perpetually seeking out some energy sink to get rid of the perturbation energy so it can come back to its original unperturbed quiescent state. A ground-state dipole, after stimulation in the presence of a frequency-resonant signal, collaborates with the tension field to pull in (suck in) the amount energy it can. A cell phone or a radio oscillator can keep pulling in energy as long as their circuit is in “on” state.In contrast, quantum atoms or molecules can pull in only the allowed quantum cupful of energy until it is recycled to its original ground state again.

Figure 3: There is a field-dipole “Push-Pull” interaction process. (a) The quantum cup concept: multiple photon wave packets simultaneously stimulate a detecting dipole. (b) Computed model for converging Poynting vector lines justifying the quantum cup concept [19]. (c) Hertz’s symmetric dipole radiation model [20].

Limits of Einstein’s photoelectric equation

Einstein’s photoelectric equation (Eq.7) represents an energy-balancing equation based on the measurement of the kinetic energy of electrons already free out of its quantum mechanical bound state. The Eq.7 is correct within its limited domain of energy balancing book keeping.

It is not a phenomenological equation that can help us explore the physical processes behind light-matter interactions, which triggers the release of quantum mechanically bound electrons inside materials and its ejection process. It does not model the initial light-dipole amplitude-amplitude quantum mechanical stimulation process. Unfortunately, Einstein modeled his equation in 1905, eight years before Bohr’s “old quantum theory” and twenty years before the “modern” quantum theory became known. We should not try to extract phenomenological interpretations about electromagnetic radiation out of Eq.7, overriding unusually successful phenomenological equations of Huygens- Fresnel, Maxwell and Schrodinger. Eq.1 and Eq.2 clearly imply light propagate as wave amplitudes, and not as energy bullets. Even Schrodinger equation is successful because it treats the excited states of quantum entities as harmonically oscillating amplitude entities. Measured energy exchange always happens after a quadratic square modulus operation ???? *???? is executed by a frequency-resonant amplitude oscillation that has been successfully triggered.

Semiclassical model of photon wave-packet

We are now combining the Hybrid Photon Model (Fig.2) with the push-pull postulate of absorption of a quantum cupful of energy (Fig.3), to build the semiclassical model for photoelectron emission. Given the overwhelming successes of Eq.1 ad Eq.2, it is now obvious that only way a sufficient amount of quantum cup-full of energy can be gathered, would be from a large number of propagating wave packets simultaneously stimulating the detecting dipole. We show this in the two cartoons of Fig.4. A proper theory need to use the Superposition Principle of simultaneous stimulation by many, many time-finite classical photon wave packets, E(n_q , t) , stimulating a quantum detector, whose interaction parameter is embedded in its polarizability factor c(n_q ) for the photo detecting atoms/molecules.Eq.8 shows the multi-step interaction processes-- first the frequency-resonant amplitude-amplitude stimulation a(t )due to multiple wave packets, E(n_q , t), followed by the quadratic step of perceived available energy flow per unit time, or the intensity. I(t).

Figure 4: Two samples of random photon wave packets traveling towards a photodetector array. In both representations, the total number of photon wave packets are the same; but for (b), the wave packets are distributed over a longer time stretch than for (a). Therefore, for case (b), the effective number of pulses available for detectors to absorb energy over the same period dt is lower, hence, the rate of photon counts will also be lower with more statistical fluctuations. Further, if the relative phases of the pulses are same (laser source), or random (thermal source), the resultant effective intensity fluctuation will be different, generating different photon counting statistics during the same time interval.

Then the individual quantum entity, or quantum cup, takes the time interval dt (see the last integral of Eq.8) to fill up their quantum cups. If the flowing light beam carries (nhn+ x) amount of energy over the integration interval of dt, then n bound electrons will be released and the leftover x amount of fractional energy will continue to flow through the photodetector array, unused. This is with the assumption that the quantum efficiency of the detector is ideally 100%, which can never be realistically realized in practice. The physical process steps are: (i) Joint amplitude stimulation or the application of the Superposition Principle; (ii) Nonlinear square modulus operation; (iii) Time integrated energy collection over a finite time interval. These three steps are literally built into our mathematical prescriptions. The last two steps constitute the Superposition Effect, which the detector registers. Defying these logically self-consistent steps in favor of accepting “single photon interferes”, deprives us from advancing mathematically self-consistent theories   relying upon visualizing the invisible interaction processes that are built into working mathematical theories.

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Eq.8 provides us with the multi-step visualizable picture behind the light and material dipole stimulation, which remains missing in Einstein’s direct data modeling postulate behind Eq.7. The period dt for collecting the energy hn before a quantum mechanically bound electron can be released is still an unsettled number, but continuing measurements imply that it is most likely around 1ps or may be less [21].

We know that the temporal and spatial coherence characteristics play critical roles in determining the visibility of interference (superposition) fringes in all interferometry [22]. The same is also true for the statistical distribution in generating the photoelectric current pulses (PCP). We have presented this modelling guideline in Eq.8. We also know through consistent measurements that the statistical distribution of PCP’s (i) for thermal sources is super-Poissonian, (ii) for lasers, it is Poissonian and (iii) for special sources like nonlinear down conversion, it is sub-Poissonian [23]. QM has also established that the de-excitation of atoms and molecular quantum levels are statistically random with characteristic lifetimes. Then the statistical distribution of “bullet” photons as they propagate towards a detector cannot be the only explanation behind the emergence of PCP statistics. Of course, laser sources, due to very fast-stimulated emission process should produce more “lumpy” wave packets compared to thermal sources. However, we believe that the superposition model of photon wave packet, presented here, is a more realistic approach to derive PCP statistics from the fundamental interaction process dictated by the coherence and amplitudes of the light pulses. The detecting dipole’s quantum cup must need a finite time interval to harvest the quantum cupful of energy it needs, however short time it may be. The statistical fluctuation arises due fluctuation in the available intensity variation, hence in the energy, dictated by the coherence properties of the flowing photon wave packets (amplitude and phase).

The time-fluctuating intensity curves for I (t) in the bottom of Fig.4 underscore this point. Fig.4a and 4b show the same number of photon wave packets. However, their temporal density is different and correspondingly, the potential intensity variation with time is different. Hence, the availability of nhn amount of energy over any dt period, or the fluctuations in the number n of photoelectrons, will be different for the two cases shown in Fig.4a and b. By incorporating the fractional energy term x in (nhn+ x) in Eq.8, we are underscoring that the classical light flux can be continuously reduced to any value, even below hn . Therefore, the time intervals during non-emission of photoelectrons for extremely low levels of light flux would not necessarily mean complete absence EM wave energy.

The “push-pull” dipole model clears up the misleading belief that “Granularity proves the discreteness of photon

Fig, 4(i) is a sample photograph of systematic buildup of spatial granularity with prolonged exposure to a very weak beam-flux passing through a double slit [borrowed from the web]. Such photographs have been consistently used to justify the quantumness of Maxwell’s EM waves. Photographic plates have small randomly distributed Ag-Halide crystallites and CCD arrays have regular array of small detecting pixels. Therefore, irrespective of the incident beam intensity and the exposure time, under sufficient enlargement of the “pictures”, they will always show the spatial granularity. It is the buildup of the temporal granularity under low light exposures, which we need to explain using our photon wave packet and quantum cup models.

Both the photographic film and the CCD detector are built out of assemblies of quantum mechanical detecting dipoles. Under the influence of frequency resonant EM waves, the push-pull interaction process starts, unfurling the dipolar “quantum cups” [see Fig.3(b) and Fig.5(ii)].

When the detecting dipoles are densely packed, the arrays of opened up quantum cup fields overlap with each other (Fig.5(ii)) and the detecting entities compete with each other to fill up their individual quantum cup with the necessary hn quantity of energy out of all the photon wave packets flowing through them and stimulating them simultaneously [24]. The number of photoelectron generation has been defined by the last integral in Eq.8, reproduced here as Eq.9:

At high levels of propagating energy-flux, when n in Eq.9 is larger than the number of detector elements intercepted by the light beam, all quantum cups succeed in filling up their energy requirement and get exposed (or, release photoelectron). At low levels of flowing energy density, when n in Eq.9 is less than the number of detecting elements, statistically some quantum cups will win over their neighbors during the first propagation interval dt. Since the exposed detecting element can no longer compete for further energy, during the follow-on intervals of dt, the unexposed detecting elements will have the opportunity to pull in the available energy to fill up their quantum cups. This is the physical explanation behind the “slow” buildup of temporal granularity, pictorially explained in Fig.5(ii), where only two dt exposure intervals have been presented.

The third box shows that all the detecting elements have been exposed. The temporal granularity does not require “bullet photon” model. The concept of “bullet photon” actually defies the physical logics built into the three historically successful phenomenological equations, the Huygens-Fresnel diffraction integral, Maxwell’s wave equation and Schrodinger’s harmonic equation for quantum particles, shown in Eq.1, Eq.2 and Eq.3, respectively. They all propagate amplitudes, not energy bullets.

Figure 5: (i) Sequential buildup of higher contrast fringes with prolonged exposure for very weak diffracting signals. Detecting surfaces consist of discrete and spatially separated detecting pixels that are quantum cups. (ii) Application of quantum-cup concept along with the joint “push-pull” property of electromagnetic tension field, which pushes energy to the frequency–resonant dipole and the dipole, pulls in one quantum cupful of energy out of the propagating EM waves. Under extremely low flux density, densely packed quantum cups compete with each other and there are statistical winners and losers. However, over prolonged exposure all of them can fill up their quantum cup dring successive exposure periods. The first box, “Exposure at t₁”: the dipoles 2, 4 & 5 are the winners and 1, 3 and 6 are the losers in the competition. The second box, Exposure at t₂”: Dipoles 1, 3 and 6 are free now to absorb the available energy, since dipoles 2, 4 and 5 can no longer absorb any more energy. The last box shows that all quantum-cup detectors are now “exposed”.

We started with the objective of eliminating the need for continuing with the concept of wave-particle duality (WPD), which actually represents our ignorance about the realistic physical models for waves and particles. Interested readers may consult the literature [25-29], where a large number of authors have been trying to develop the semiclassical treatment of the quantum world for decades. We took the photoelectric effect as a case example to show that it is possible to eliminate the WPD with the semiclassical approach when we assume that all atomic and molecular emissions consist of semi-exponential classical light pulses. We have provided the rationales behind this model that are self- consistent.

It is the field of elementary particles that are still in some controversy [30-32], besides the Copenhagen Interpretation of the Schrodinger’s equation. We understand that Huygens-Fresnel Diffraction Integral and Maxwell’s Wave Equation do not represent the ultimate and complete knowledge about electromagnetism. In spite of that, these phenomenological equations of electromagnetism have been guiding us for over several centuries through uninterrupted evolution of optical science and engineering without any serious controversies. Newer applied fields are successfully emerging and maturing– Nanophotonics, Plasmonic Photonics, Metamaterials and a wide variety of Biophotonics. In all these fields, the electromagnetic tension properties, ???? & ????, in regular materials have been playing the key roles through the parameter of refractive index [5].In none of these successfully evolving fields, the leading scientists are propagating EM waves as energy bullets; they are using Maxwell’s equation set. QM has not developed any systematic theory that can explicitly provide the perpetual velocity to “indivisible bullet photons”. We have purposefully underscored that all the three equations of significance in fundamental physics (Eq.1, 2, and 3) deal with amplitudes-amplitude superposition, or interaction. Next comes the square modulus operation, E * (t)E(t) or ???? * (t)???? (t), to derive the “intensity” and then the energy exchange happens only after an integration of the “intensity” over a finite period, however short it could be. The postulate of “Wave function collapse” only suppresses this necessary enquiry to keep advancing physics. The concepts we are promoting are built-into the mathematical logics of our working equations. As good engineers, we must leverage these mathematical logics to maximize the visualization of the interaction processes that nature has been carrying out. These are successful phenomenological equations. None of them indicate the existence and propagation of EM waves as discrete “energy bullets”. That is why our efforts to improve upon these successful phenomenological equations would be productive and useful [5, 25-29]. One of our suggestions is to re-develop the equations electromagnetism to find an electromagnetic model for the elementary particles as localized harmonic oscillators [5], because Schrodinger’s equation represent localized harmonic oscillators, not perpetually propagating “plane waves” like Maxwellian EM waves; Schrodinger’s particles require a separate potential gradient around it to move in space, obeying Newtonian inertia.

1. Hentschel K (2018) [Photons: The History and Mental Models of Light Quanta], Springer.
2. Navarro J Ed (2018) [Ether and Modernity: The Recalcitrance of an Epistemic Object in the Early Twentieth Century], Oxford University Press.
3. Wikipedia (2022) Luminiferous aether.
4.  Wilson RJ (2010) [The Ether Dispute].
5. Roychoudhuri C (2021) Cosmic Ether, Possessing Electric-Tension and Magnetic-Resistance, Is the Unified Field for Physics”, Journal of Modern Physics 12: 671-699.