4 This article presents a comprehensive study that furnishes compelling evidence bolstering 5 the notion of electromagnetic field continuity, accomplished through the implementation of a 6 self-heterodyne measurement technique. 7 The primary objective was to investigate the detection method of FMCW for extremely weak 8 signals, reaching energy levels akin to photons. Through thorough examination, significant con- 9 tradictionsemerged,favoringclassicalfieldtheoryandchallengingthenotionofelectromagnetic 10 fieldquantization. Tostrengthenthisclaim,acarefullydesignedexperimentwasconductedto 11 measure sub-photon energy levels. Thecomprehensive analysis of both theoretical insights and 12 experimental data presents compelling evidence, demonstrating the untenability of a quantized 13 modelfortheelectromagneticfield. Thesefindingshaveprofoundimplicationsforunderstanding 14 the continuity of the electromagnetic field, prompting further exploration in this field of research. 15 1. Introduction 16 A scientific manuscript that challenges the existence of photons is understandably poised to 17 encountersubstantialresistance. Overthespanofmorethanacentury, thepresenceofphotons 18 hasbeenestablishedthroughanarrayofexperiments. Orhasit? Thestraightforwardresponse 19 is an emphatic negation. The experiments often cited as validating the existence of photons 20 essentially affirm the discrete nature of interactions between fields and matter. The phenomenon 21 of the photoelectric effect serves to illustrate that the absorption of energy from the field is 22 inherentlyquantized. Additionally,MaxPlanck’sworkdemonstratesthatinteractionswiththe 23 electric field exclusively occur in quantized units, as to avert the "ultraviolet catastrophe." What 24 remainsunprovenbytheseexperiments,however,istheassertionthatthequantizedabsorption 25 or emissionof energyinto theelectromagnetic fieldinherently resultsin thesustained retention 26 of such quantized energy within said field. The presumption of a quantized electromagnetic 27 field has persistently rested as an unsupported postulation. Similarly, the Compton effect, 28 often attributed as the "proof" for the particle-like nature of photons, solely highlights the 29 discretizationofinteractions. Ininstanceswherematterassimilatesenergyfromthefieldand 30 thisassimilationtakesplaceindiscreteunitsof 𝐸=ℎ𝜈,acorrespondingexchangeofmomentum 31 𝑝=𝐸/𝑣=ℎ𝜈/𝑐=ℎ/𝜆emergesinadherencetomomentumconservationprinciples,owingto 32 the velocity of electromagnetic waves equating to the speed of light. This, however, does not 33 substantiatetheexistenceofdiscreteentities("photons"),butratherstemsfromthegoverning 34 law of momentumconservation. While this work doesnot challenge the provendiscrete nature 35 of interactions inthese experiments, it does questionthe existence of photons, which remains 36 unproven within the context of these experiments. 37 Frequency-ModulatedCoherentWave(FMCW)detectionisawell-establishedmeasurement 38 technique used in many fields, such as optical coherence tomography and LiDAR. While FMCW 39 is explained by classical field theory, understanding its operation at the quantum-mechanical 40 levelpresentschallenges. OnesignificantchallengeinFMCWLiDARisthatithasthecapability 41 ofdetectingsinglephotons,butthedetectionisbasedonfrequency,whichchangesovertime. 42 This raises the question of whether a photon would change its frequency over its course [1]. 43 Photons, unless subjected to redshift, blueshift, or reflection from a moving target, remain 44 impervious to alterations in their frequency. The work conducted by Barber et al., in its ultimate 45 reckoning, does not furnish an elucidation for the modus operandi of the measurement principle. 46 Instead, they make reference to a separate publication authored by Winzer et al. [2]. In this cited 47 work, the authors arrive at a pivotal conjecture: "Thus, in a heterodyne system, the classical 48 electromagnetic field quantities are added according to Maxwell’s equations and the sum is 49 detected, only then we can speak of a photon stream." Essentially, this embodies the solitary 50 avenue through which the principles underpinning FMCW measurements can be comprehended, 51 preservingvalidityevenwhendealingwithenergymagnitudesconsiderablylessthan ℎ𝜈. Inpoint 52 offact,thisdeclarationinherentlysignifiesthat"theelectromagneticfieldpreservescontinuity, 53 with the term ’photon’ finding relevance exclusively upon detection." According to the purview 54 advanced by this study, photons merely epitomize the dynamic interplay between the field 55 and matter, relinquishing their existential claims during the course of field propagation - a 56 phenomenon exclusively attributed to Maxwellian propagation. Regrettably, the authors veer 57 fromthisparadigmbyerroneouslygraftingtheconceptofuncertaintyontodomainswhereits 58 applicability is unwarranted. 59 In Barber et al.’s study, they managed to measure energy down to the scale of two photons. 60 Interestingly, they didn’t delve further into the concept of photons, and consequently, they didn’t 61 endeavorto measure energyin quantities smallerthan asingle photon. Thisappears perplexing 62 given the principles of FMCW operation. In FMCW, each measurement period is made up 63 of numerous smaller measurements. The measured energy is linked to the entirety of this 64 measurement period. To put it differently, if the researchers had focused on just a quarter of the 65 measurement period’s energy, they would have effectively measured what could be thought of as 66 half a "photon" (imagining very tiny units of energy), provided they averaged the measurements 67 accordingly. This missed opportunity could have prompted a deeper reexamination of our 68 understanding of photons and energy measurement. 69 This study aims to explore the possibility of measuring energy units smaller than a single 70 photon and seeks to elucidate the interpretation of such measurement outcomes. 71 2. Detection principle of FMCW 72 LOLaser collimator circulator spli er BDtarget electrical ampli erADC, FFT, … trigger signal Fig. 1. Typical setup for Frequency-Modulated Continuous Wave (FMCW) LiDAR detection Figure 1 illustrates a typical setup for FMCW LiDAR detection. The system consists of a 73 swept-source laser, which is divided into two paths: the local oscillator (LO) and the signal path. 74 Theopticalsignaltransmittedthroughthesignalpathinteractswithatargetandsubsequently 75 combines with the LO signal on a balanced detector (BD). The resulting signal is then amplified 76 using an electrical amplifier before being digitized and processed. Additionally, to mark the 77 initiation of the chirps generated by the laser system, a trigger signal is provided by the laser, 78 which facilitates the capture of the time series of measurements at the detector. 79 At the detector, the field strengths of the local oscillator (LO) and the signal reflected from the 80 targetcombine,resultingintheirclassicalobservation. Thedetectormeasurestheintensityof 81 the combined signals, which corresponds to the absolute square of the field strength. Due to 82 thesweepingnatureofthelaserandtheround-trippropagationtimeofthesignaltoandfrom 83 thetarget,thefrequenciesofthecombinedsignalsarenotidentical. Asaconsequence,abeat 84 signal is generated, characterized by a frequency significantly lower than the optical frequencies 85 of the laser signal, and it is sampled by the detector. Typical sampling frequencies utilized in 86 FMCW LiDAR systems fall within the range of 1GHz. The energy of the squared field strength 87 can be quantified in terms of photons and the resultant signal can be mathematically represented 88 as follows: 89 𝑖𝑏𝑑(𝑡)=2𝜂𝑞 𝑇√︁ 𝑁𝐿𝑂𝑁𝑠𝑖𝑔cos(2𝜋𝜅𝜏𝑡 +𝜙) +𝑖𝑠(𝑡) (1) InEquation1,thevariable 𝜅representsthelinearchirpofthelaserwithitsassociatedchirp 90 rate,𝜏corresponds to the round trip time, 𝜂denotes the quantum efficiency of the detector, 𝑞 91 signifiestheelementarycharge, 𝜙representsafixedphase,and 𝑁𝑠𝑖𝑔and𝑁𝐿𝑂denotethenumber 92 of photons in the signal and local oscillator, respectively. The signal is subject to noise 𝑖𝑠(𝑡), 93 primarily stemming from the shot noise of the local oscillator. 94 ForanLOphotonfluxmuchlargerthanthesignalflux, 𝑖𝑠(𝑡)isaGaussianrandomprocess 95 with zero mean and variance 96 𝜎2(𝑡)=2𝜂𝑞2 𝑇𝐵𝑁𝐿𝑂=𝜂𝑞2 𝑇2𝑙𝑒𝑛𝐹𝐹𝑇𝑁𝐿𝑂 (2) It’s important to note that in equation 2, the symbol 𝐵represents the bandwidth, not of the 97 integrationtime,butratherofasingledetectioninthetimedomain,denotedas 𝑇𝑇𝐷=𝑇/𝑙𝑒𝑛𝐹𝐹𝑇. 98 Equation 1 elucidates how the beat frequency is determined by a target located at a distance 𝑑, 99 𝑐represents the speed of light. 100 𝑓𝑏𝑒𝑎𝑡=𝜅𝜏=𝜅2𝑑 𝑐(3) The extraction of distance information from the detected signal in the time domain can be 101 achieved through the utilization of a Fourier transformation. To facilitate this process, the signal 102 ismultipliedbyawindowfunction,suchasHammingorHanning,beforeundergoingFourier 103 transformation. The resulting frequency spectrum reveals the detected beat frequency, which 104 directlycorresponds tothedistance information. It isimportanttonote thatfor comprehensive 105 coverage, the application of up- and down-chirps is employed to detect moving targets. This 106 approach allows for the calculation of the frequency shift resulting from the Doppler effect 107 in relation to the distance information. However, in the case of static targets located at fixed 108 positions,thedistinctionbetweenup-anddown-chirpsbecomesinconsequential,andasingle 109 chirp is sufficient for accurate measurements. 110 The determination of the number of detected photons in the signal path can be achieved using 111 thedetectedsignal[1]. Byaveragingalargenumberofpowerspectra,theaverageoftheresulting 112 power spectra associated with the signal is directly proportional to the quantity 𝑁𝐿𝑂𝑁𝑠𝑖𝑔, while 113 the averaged noise floor is proportional to 𝑁𝐿𝑂. Consequently, in a system limited by shot noise, 114 the normalized averaged power spectra exhibit a noise floor value of 1. The value corresponding 115 to the signal frequency in the normalized power spectra therefore represents: 116 𝑠𝑖𝑔𝑓𝑜𝑢𝑟𝑖𝑒𝑟 =𝜂𝑁𝑠𝑖𝑔 (4) InEquation4, 𝜂denotesthequantumefficiency,and 𝑁𝑠𝑖𝑔representsthenumberofphotonsin 117 the signal. The normalized signal strength in the shotnoise-limited system therefore corresponds 118 to the number of detected photoelectrons plus 1. 119 InrealFMCWsystemstherearefurthernoisesources. WhileforstrongLOsignalsthedetector 120 noiseislessimportant,thenoiseoftheelectricalamplifierisnotreallynegligible. Therefore, 121 whendetectingverysmallsignalstrengths,thefollowingprocedurecanbefollowed: First,the 122 average noise power of the amplifier is determined by disconnecting both LO and receive signal 123 fromthebalanceddetector. ThentheaveragenoisepoweroftheLOisdeterminedbyconnecting 124 only theLO signalto the balanceddetector. Thenoise power ofthe LOis determined fromthe 125 nowaveragednoisepower,minusthenoisepoweroftheelectricalamplifier. Todeterminethe 126 signalstrengthinunitsofphotonquanta,theobtainedpowerspectraareaveraged,fromwhichthe 127 averagednoiseoftheelectricalamplifierissubtracted,andthesignalthusobtainedisnormalized 128 bytheLOnoise. Iffurthernoisesourcesoccur,theyshowupasanoisefloorabovethevalue"1". 129 The signal value in photoelectrons is the signal value above that overall noise floor. 130 Materials and Methods 131 Theobjectiveofthisexperimentistoquantifyenergylevelswithineachmeasurementinterval 132 that fall below the magnitude of a single ’photon’. Initially, these measurements were carried out 133 usingstandardFMCWtargetmeasurements. However,theseFMCWmeasurementsweresubject 134 tothepresenceofspeckle,causedbythesurfaceroughnessofthetarget,resultinginasignificant 135 specklecontrast. Tomitigatetheimpactofspeckle,theexperimentalsetupwasstreamlinedto 136 the configuration illustrated in Figure 2. 137 LOLaser aenuator delay line spli er BD electrical ampli erADC, FFT, … trigger signal90 10 aenuator(2)Power meter50 3dB couplerspli er RX50 Fig. 2. Self-Heterodyne Measurement Configuration for Speckle Avoidance The experimental setup is based on Bridger’s swept source, which produces frequency sweeps 138 (power 3.2mW at 1550nm) and triggers a detection board. The laser source is utilized as the 139 swept source. For detection, a balanced detector (Thorlabs PDB570C) with a responsivity 140 of𝑅=1𝐴/𝑊and a quantum efficiency of 𝜂=0.8at 1550nm is employed, incorporating an 141 integratedelectricalamplifier. Giventhefeeblenatureofthesignal,anadditionalLow-Noise 142 Amplifier (Nooelec LaNA HF) is introduced into the setup. The augmented analog signal is 143 captured through a National Instruments DAQ system (NI PXIe-1082, NI PXIe-5763). All other 144 elements within the setup are standard. 145 The frequency transition rate of the sweep is specified at 100GHz/s, generating two sweeps in 146 both directions lasting approximately 35 µs each. The sampling rate is set to 500MSamples/s. 147 Tomitigateundesirednoise, theopticalgainwasminimized,and thedetectionwassettothe 148 "Balanced"mode. Thisconfigurationwaschoseninsteadofauto-balancing,asauto-balancing 149 introduced additional noise that could not be reduced through averaging. The laser source 150 generatedupanddownsweepswhiletriggeringateachstart. Theacquisitionsystemrecorded 151 44,000 samples for each up and down sweep. From the recorded data, 16,384 samples were 152 selected for each ramp, focusing on a region where the laser was clearly locked. 153 When dealing with measurements of weak signals, it becomes necessary to fine-tune the delay 154 linetoalignthesignalpreciselywithasinglebin. Thisadjustmenthastheaddedadvantageof 155 enabling deliberate avoidance of a window function, which has the potential to modify the signal 156 characteristics. Moreover, to facilitate examination of shorter measurement intervals, the bin 157 numbermustbedivisiblebyafactorrepresentingapowerof2. Inthiscontext,theinitialpeak 158 was aligned to a bin that is evenly divisible by 8. 159 Results 160 Acquired data, along with the corresponding signal processing, can be found in Dataset 1, [3]. 161 Aside from the contribution of electrical amplifier noise, the performance of the setup is 162 primarily constrained by shot noise. A residual, minor reflection might stem from the initial 163 attenuator,whichcouldn’tbeentirelynullifiedevenwiththeinclusionofanadditionaloptical 164 isolator. The noise baseline exclusively exhibits shot noise characteristics. This is achieved 165 byeliminatingtheaveragedelectricalamplifiernoisefromboththesignalandlocaloscillator, 166 and subsequently normalizing the signal with the local oscillator noise. It’s worth noting that 167 there could be a slight remaining offset due to variations introduced by multiple measurements - 168 reconnectingfibersmayleadtomarginalfluctuationsineffectivelaserpower. However,since 169 this offset remains uniform, it can be rectified by intentionally scaling the noise baseline to a 170 value of 1. 171 Given that the detector employed is an Avalanche Photodiode (APD), which inherently 172 manifestsexcessnoise,thenoisebaselineisnotunitybutratherrepresentedas 𝐹=𝑀𝑖,where 173 𝑀signifiesthegain(M=2.5forgivendetectoratlowestgain)and 𝑖representstheexcessnoise 174 index, which is specified as 0.3 for this particular sensor. Consequently, the count of photons 175 within a bin is derived by normalizing the signal, subtracting 1, and subsequently scaling it with 176 𝑀𝑖/𝜂=2.50.3/0.8=1.65. 177 Upon significant averaging, static noise stemming from factors such as reflections and Analog- 178 to-Digital Converter (ADC) noise may manifest. This static noise introduces an unvarying offset 179 acrossallaveragedmeasurements. Insituationswheredistinctsetsofaveragedmeasurements 180 are available, they can reciprocally function as local oscillator baselines for one another’s signals. 181 By subtracting the noise baseline from a measurement lacking any signal (pure noise) from 182 another measurement conducted at a point where a signal is present, the outcome is the effective 183 eradicationofstaticnoise. Thisoperationleadstoanamplificationofshotnoisebyafactorof 184√ 2, while concurrently mitigating the presence of static noise. 185 Figure 3 illustrates that through ample averaging, the noise floor can be sufficiently smoothed, 186 revealing signals with energy levels below that of a single photon. 187 Given that the displayed signal is a result of averaging, the current inquiry revolves around 188 determiningwhethertheinputconstitutesaquantizedquantity,withtheoutcomereflectingan 189 averaging of discrete values. Alternatively, it prompts us to consider whether the input signal is 190 inherently continuous and causally aligns with the already-averaged value. 191 Discussion 192 TheFouriertransformis alinearfunction. Therefore, forthe Fouriertransform ofequation1, the 193 following holds true: 194 200 220 240 260 280 300 FFT bin0.000.050.100.150.20N PhSignal Up-Chirp (averaged, n = 100.000), T = 32. 8 /uni03BCμFig. 3. Detection of Energy Levels Below One Photon per Measurement Interval F (𝑖𝑏𝑑(𝑡))=F (2𝜂𝑞 𝑇√︁ 𝑁𝐿𝑂𝑁𝑠𝑖𝑔cos(2𝜋𝜅𝜏𝑡 +𝜙)) + F (𝑖𝑠(𝑡)) (5) Hence, in the discourse concerning signal generation, it becomes feasible to delineate the 195 signal itself from the superimposed shot noise originating from the local oscillator. 196 The signal emerges when we perform a mathematical operation called the Fourier transform. 197 This operation takes the data collected from the signal over a period of time and processes 198 it using specific intervals. In this case, we’re using 16,384 data points. Before applying this 199 transformation, we already have 16,384 individual data points gathered in the time domain. 200 As per the current scientific understanding, when energy is released into the electromagnetic 201 field, it stays contained within that field until it’s detected. It doesn’t get divided and spread 202 acrossdifferentmeasurements. Thinkofitlikesmallpacketsofenergy. Thesepacketsmustbe 203 completely detected in a single measurement of time; they can’t be split into parts. 204 The path where this detection occurs is labeled as ’RX’ in Figure 1. At a certain point, it 205 dividesintotwopartsandthencombineswithanothersignalknownasthelocaloscillator,or 206 ’LO’. When there’s only a tiny amount of energy, like a few ’photons’, half of these ’photons’ go 207 toonedetector,andtheotherhalfgotoadifferentone. Ifweweretocountthese’photons’as 208 whole units, the count would follow a specific pattern described by Poisson statistics, with an 209 averagenumberdeterminedby’RX/2’. Onlyatspecificmomentswhena’photon’arrives,we 210 observeanelectronbeingdetected. Inallotherinstances,thecountofdetectedelectronsremains 211 at zero, and there is no interaction between the signals. 212 Figure4illustratesasimulatedscenarioinwhichanaverageof20photoelectrons( 𝜂=1)is 213 detected. In this case, the signal conforms to a continuous behavior. 214 In Figure 5, we observe a simulated scenario in which only 100 photoelectrons are dispersed 215 acrosstheentiretyofanFMCWmeasurementintervalencompassing200samples. Consequently, 216 themeancountperdetectionintervalstandsat0.5. Theresultantinterferencesignalpredominantly 217 comprises a multitude of zeros. 218 Both figures also depict the signal in a non-quantized scenario. It’s important to note that 219 what’s illustrated is solely the signal component of the comprehensive detected signal, which is 220 still susceptible to the addition of noise from the local oscillator. 221 0 25 50 75 100 125 150 175 200 Sample No.460048005000520054005600Photo electronsTime Domain Signal, NPE=20 quantized continuousFig. 4. Simulated Signal with Quantized Field: Average of 20 Photoelectrons per Sample 0 25 50 75 100 125 150 175 200 Sample No.480048504900495050005050510051505200Photo electronsTime Domain Signal, NPE=0.5 quantized continuous Fig. 5. Simulated Signal with Quantized Field: Average of 0.5 Photoelectrons per Sample Asthesignalstrengthofthereceivepathdiminishes,anincreaseintheoccurrenceofzeros 222 becomesevident withinthe time-domainsignal. Given thenon-uniformnature ofa sinesignal, 223 thisphenomenonleadstoareductioninthecountofdetectedphotoelectronsforeachreceived 224 photoelectron. This simulation outcome for decreasing quantities of received photoelectrons is 225 presentedinFigure6. They-axisemploysasquare-rootscale,revealingthathalvingtheinput 226 energy corresponds to a roughly fourfold decrease in the detected number of photoelectrons. 227 Consequently, the apparent count of photoelectrons would consistently fall short of the actual 228 /uni00000013 /uni00000015/uni00000018/uni00000013/uni00000013 /uni00000018/uni00000013/uni00000013/uni00000013 /uni0000001a/uni00000018/uni00000013/uni00000013 /uni00000014/uni00000013/uni00000013/uni00000013/uni00000013 /uni00000014/uni00000015/uni00000018/uni00000013/uni00000013 /uni00000014/uni00000018/uni00000013/uni00000013/uni00000013 /uni00000014/uni0000001a/uni00000018/uni00000013/uni00000013 /uni00000015/uni00000013/uni00000013/uni00000013/uni00000013 /uni0000002c/uni00000051/uni00000053/uni00000058/uni00000057/uni00000003/uni00000033/uni0000004b/uni00000052/uni00000057/uni00000052/uni00000048/uni0000004f/uni00000048/uni00000046/uni00000057/uni00000055/uni00000052/uni00000051/uni00000056/uni00000003 N PE/uni00000014/uni00000013/uni00000013/uni0000001c/uni00000013/uni00000013/uni00000015/uni00000018/uni00000013/uni00000013/uni00000017/uni0000001c/uni00000013/uni00000013/uni0000001b/uni00000014/uni00000013/uni00000013N PE , 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. 6. Correlation Between Detected and Received Photoelectrons for Simulated Quantized Field Energy at Low Energy Levels (Note the Square Root Scale) count within this detection scheme, particularly when the average photoelectron count per 229 individual time domain detection dips into the single digits or even drops below one. Illustrating 230 this effect, Figure 7 presents the attenuation of detected photoelectrons in relation to input 231 photoelectrons,expressedindecibels(dB),acrossacomprehensivedetectionwindowcomprising 232 16,384 values. Notably, as photon counts per detection interval decrease, the attenuation grows 233 notably pronounced, ranging between 20 and 25 dB. 234 /uni00000013 /uni00000015/uni00000018/uni00000013/uni00000013 /uni00000018/uni00000013/uni00000013/uni00000013 /uni0000001a/uni00000018/uni00000013/uni00000013 /uni00000014/uni00000013/uni00000013/uni00000013/uni00000013 /uni00000014/uni00000015/uni00000018/uni00000013/uni00000013 /uni00000014/uni00000018/uni00000013/uni00000013/uni00000013 /uni00000014/uni0000001a/uni00000018/uni00000013/uni00000013 /uni00000015/uni00000013/uni00000013/uni00000013/uni00000013 /uni0000002c/uni00000051/uni00000053/uni00000058/uni00000057/uni00000003/uni00000033/uni0000004b/uni00000052/uni00000057/uni00000052/uni00000048/uni0000004f/uni00000048/uni00000046/uni00000057/uni00000055/uni00000052/uni00000051/uni00000056/uni00000003 N PE/uni000000ed/uni00000015/uni00000018/uni000000ed/uni00000015/uni00000013/uni000000ed/uni00000014/uni00000018/uni000000ed/uni00000014/uni00000013/uni000000ed/uni00000018N PE , detected/dB/uni00000027/uni00000048/uni00000057/uni00000048/uni00000046/uni00000057/uni00000048/uni00000047/uni00000003/uni00000036/uni0000004c/uni0000004a/uni00000051/uni00000044/uni0000004f/uni00000003/uni0000002f/uni00000048/uni00000059/uni00000048/uni0000004f/uni00000003/uni00000049/uni00000052/uni00000055/uni00000003/uni00000034/uni00000058/uni00000044/uni00000051/uni00000057/uni0000004c/uni0000005d/uni00000044/uni00000057/uni0000004c/uni00000052/uni00000051/uni00000003/uni00000052/uni00000049/uni00000003/uni00000029/uni0000004c/uni00000048/uni0000004f/uni00000047/uni00000003/uni00000028/uni00000051/uni00000048/uni00000055/uni0000004a/uni0000005c/uni00000003/uni0000000b/uni00000051/uni00000020/uni00000014/uni00000019/uni00000016/uni0000001b/uni00000017/uni0000000c Fig. 7. Correlation between Detected and Received Photoelectrons for Simulated Quantized Field Energy at Low Energy Levels (Logarithmic Scale) Nonetheless,suchbehaviorremainsabsentinactualdataobservations,asdepictedinFigure 235 8 (please note the linear scale). In instances where the signal energy in the receiving path is 236 reduced, thedetected photon countaligns withthe diminishment insignal energy. Thereis just 237 a constant factor of about 2 between the expected and measured values. Further exploration 238 revealedthatthisisexplainable: themeasuredlaserpowerisanaverageofthelaseroutput,while 239 thepowerduringthesweepsislower. Additionally,alongsidethecouplinglossesbetweenthe 240 usedattenuators, thereiscoupling losstothe detector,andanother possibilityisthat theexcess 241 noisefactoroftheAPDcouldbeslightlyhigher. Thedetectedenergiesclearlyexhibitalinear 242 relationshipwithinputpower,andtheynotablysurpassthelevelsanticipatedforthequantized 243 scenario (roughly -25 dB relative to input power, as illustrated in Figure 7). 244 /uni00000013 /uni00000015/uni00000013/uni00000013 /uni00000017/uni00000013/uni00000013 /uni00000019/uni00000013/uni00000013 /uni0000001b/uni00000013/uni00000013 /uni0000002c/uni00000051/uni00000053/uni00000058/uni00000057/uni00000003/uni00000033/uni0000004b/uni00000052/uni00000057/uni00000052/uni00000051/uni00000003/uni00000028/uni00000054/uni00000058/uni0000004c/uni00000059/uni00000044/uni0000004f/uni00000048/uni00000051/uni00000057/uni00000056/uni00000003 N/uni00000018/uni00000013/uni00000014/uni00000013/uni00000013/uni00000014/uni00000018/uni00000013/uni00000015/uni00000013/uni00000013/uni00000015/uni00000018/uni00000013/uni00000016/uni00000013/uni00000013/uni00000016/uni00000018/uni00000013N 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 Fig.8. CorrelationbetweenDetectedandReceivedPhotonEnergyEquivalentsatLow Energy Levels: Experimental Data (Linear Scale) The linear relationship can be comprehended through the premise that the energy is uniformly 245 spread across the measurement interval, aligning with the "continuous" curves in Figures 4 246 and5. Furthermore,theenergiesdetectedinindividualinstances(i.e.,persingletimedomain 247 measurement) are significantly smaller than that of a single photon. It’s also important to 248 emphasize that the energies constituting the signal cannot stem from a solitary photon. This 249 phenomenonisascribedtothefactthatthesignalemergesfromtheinterplaybetweentwodistinct 250 frequencyramps-onefromthelocaloscillatorandtheotherfromthesignalitself. Crucially,the 251 specific frequenciesof both thelocal oscillatorand the signalalter witheachinstanceof time 252 domain measurement. 253 Moreover, a coherent and uninterrupted signal would open avenues for scrutinizing energy 254 distribution within narrower measurement intervals. If the measurement interval were divided in 255 half,eachmeasurementshouldtheoreticallyencapsulatehalftheenergy,providedsignalcontinuity. 256 Theinvestigationofshortenedmeasurementintervalscanbestreamlinedbyscrutinizingareduced 257 number of samples while preserving the same dataset. The ensuing outcome is illustrated in 258 Figure 9, thereby affirming the existence of a uniformly distributed energy profile. 259 Table 1 presents vital measurements along with their associated confidence levels. The value 260 of𝜎was computed as the square root of the variance present within the averaged noise floor. 261 QuantummechanicsattributesthePoissondistributionofphotoneventstothefundamental 262 uncertainty principle. Consider a scenario where the target remains static. In such cases, the 263 100 110 120 130 140 150 FFT bin0.000.020.040.060.080.10N PhSignal Up-Chirp (averaged, n = 200.000), T = 16.4 /uni03BCμ 50 55 60 65 70 75 FFT bin0.000.010.020.030.040.05N PhSignal Up-Chirp (averaged, n= 400.000), T= 8.2 /uni03BCμFig. 9. Reduction in Measurement Interval Duration frequency of a photon emitted toward the target remains unchanged until it reaches the detector. 264 When the number of photons reaching the detector decreases, according to the uncertainty 265 principle, the arrival time uncertainty should increase, leading to statistically different arrival 266 times for photons of a specific frequency. 267 However, for accurately deriving the sine function, which eventually generates a distinct 268 peakatthebeatfrequency,precisearrivaltimesareimperative. Thisnecessityarisesfromthe 269 dependence of the beat frequency on both the local oscillator’s frequency and the signal photons’ 270 frequency at a specific moment. While an increased uncertainty in arrival times would typically 271 result in a broadening of the frequency peak, empirical observations challenge this expectation. 272 Notably, in real-world measurements, such broadening remains absent. 273 These observations suggest that we may not be dealing with discrete photons but instead 274 withpureelectromagneticwavescharacterizedbyprecisearrivaltimes. It’sworthnotingthat 275 theuncertaintyprincipleprimarilyapplieswhenadetectorrecordsindividualdetections. The 276 Table 1. Signal Levels and Associated Confidence Intervals For Bin 240, Up-Chirp Measurement Interval 𝑇=32.768𝜇𝑠𝑇=16.384𝜇𝑠𝑇=8.192𝜇𝑠 Number of Photons 0.21 0.11 0.053 𝜎 0.0091 0.0064 0.0047 confidence / 𝜎 23.1 16.8 11.9 apparentreasonforthisuncertaintymaybelinkedtothestatisticalprocessofenergyaccumulation 277 until a sufficient amount of energy is available for a quantized detection from the field. 278 However,evenifwesetasidetheaforementionedargumentsandmaintainacommitmentto 279 someformofaveraging,accordingtoaquantizedmodeltheincoming’photons’permeasurement 280 intervalalsowouldneedtoadheretoPoissonstatistics(additionallyignoringthefacttheythen 281 wouldhavetobedetectedalsoduringseveraltimedomaindetections,whatisnotpossiblefor 282 quantized entities per definition). Granted, the precise identification of the signal component 283 withinindividualmeasurementintervalsisconfoundedbytheshotnoiseoriginatingfromthelocal 284 oscillator. However,whatcanbeestablishedisastatisticaloverviewencompassingnumerous 285 measurements. This statistical overview can then be contrasted with a simulation, both for 286 the scenario involving Poisson statistics and that of a continuous, unchanging distribution of 287 energy. To execute this comparison, a sequence of measurements is conducted to determine 288 the average energy value. Subsequently, the distribution corresponding to this average value 289 is simulated for both statistical scenarios. Given the substantial discrepancy between the two 290 statistics, particularly when the averaged value of 𝜂𝑁/𝐹falls within the range of 1...4, dedicated 291 measurement series are devised to attain the requisite values. 292 The results of simulations conducted for specific values of 𝜂𝑁/𝐹are illustrated in Figure 293 10. Thesignalstatisticsstarklyalignwiththescenarioofcontinuousenergydistribution,thus 294 contradicting the assumption of Poisson statistics. 295 This underscores that even in scenarios with a limited count of photons per measurement 296 interval (again, disregarding the ’impossible’ detection scheme spanning multiple time domain 297 measurements),thesignalleveldeviatesfromPoissonstatistics. Instead,itexhibitsauniform 298 energy distribution. 299 Conclusion 300 Through an objective exploration of Frequency-Modulated Continuous Wave (FMCW) measure- 301 ments,particularlyfocusingontherealmofself-heterodynemeasurements,itbecomesevident 302 thatthesemethodologiesarebettersuitedtoelucidatethenatureofacontinuousfieldasopposed 303 to one that is discretized. This conclusion emerges from an unbiased analysis of the evidence at 304 hand. 305 In situations involving exceedingly weak signals, a discretized field would undergo severe 306 attenuationandfailtoyieldadetectablesignal. However,measurementdataclearlydisprovesthis 307 expectationbyshowinganabsenceofsuchattenuation. Instead,itstronglysupportsacontinuous 308 energydistribution,effectivelyrefutingtheconceptofaquantizedfield. Uponcloserexamination 309 of smaller measurement intervals, a notable trend emerges: the detected energy maintains a 310 directproportionalitytothelengthofthedetectioninterval. Thisobservationprovidesadditional 311 compellingevidenceinsupportofthecontinuousnatureoftheelectromagneticfield. Lastbutnot 312 least, the statistical behavior of the signals aligns closely with a continuous energy distribution. 313 Thephenomenonofquantizationoccursbothattheemissionanddetectionstagesofenergy 314 /uni000000ed/uni00000015 /uni00000013 /uni00000015 /uni00000017 /uni00000019 /uni0000001b /uni00000014/uni00000013 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 η * N /F/uni0000000c/uni00000020/uni00000014/uni00000011/uni0000001c/uni00000016 /uni00000036/uni0000004c/uni00000050/uni00000058/uni0000004f/uni00000044/uni00000057/uni00000048/uni00000047/uni00000003/uni00000034/uni00000058/uni00000044/uni00000051/uni00000057/uni0000004c/uni0000005d/uni00000044/uni00000057/uni0000004c/uni00000052/uni00000051/uni00000003/uni0000000b/uni00000033/uni00000052/uni0000004c/uni00000056/uni00000056/uni00000052/uni00000051/uni0000000c /uni00000036/uni0000004c/uni00000050/uni00000058/uni0000004f/uni00000044/uni00000057/uni00000048/uni00000047/uni00000003/uni00000026/uni00000052/uni00000051/uni00000057/uni0000004c/uni00000051/uni00000058/uni0000004c/uni00000057/uni0000005c/uni00000003/uni0000000b/uni00000046/uni00000052/uni00000051/uni00000056/uni00000057/uni00000044/uni00000051/uni00000057/uni00000003/uni00000031/uni0000000c /uni00000030/uni00000048/uni00000044/uni00000056/uni00000058/uni00000055/uni00000048/uni00000050/uni00000048/uni00000051/uni00000057/uni00000003/uni00000027/uni00000044/uni00000057/uni00000044(a) Fig. 10. Signal Statistics: Poisson and Continuous Signal Distribution, Along with Measured Data Statistics /uni000000ed/uni00000015 /uni00000013 /uni00000015 /uni00000017 /uni00000019 /uni0000001b /uni00000014/uni00000013 /uni00000039 /uni00000044/uni0000004f/uni00000058/uni00000048/uni00000013/uni00000011/uni00000013/uni00000013/uni00000013/uni00000011/uni00000013/uni00000018/uni00000013/uni00000011/uni00000014/uni00000013/uni00000013/uni00000011/uni00000014/uni00000018/uni00000013/uni00000011/uni00000015/uni00000013/uni00000013/uni00000011/uni00000015/uni00000018/uni00000013/uni00000011/uni00000016/uni00000013/uni00000013/uni00000011/uni00000016/uni00000018/uni00000013/uni00000011/uni00000017/uni00000013/uni00000033/uni00000055/uni00000052/uni00000045/uni00000044/uni00000045/uni0000004c/uni0000004f/uni0000004c/uni00000057/uni0000005c/uni00000036/uni0000004c/uni0000004a/uni00000051/uni00000044/uni0000004f/uni00000003/uni00000027/uni0000004c/uni00000056/uni00000057/uni00000055/uni0000004c/uni00000045/uni00000058/uni00000057/uni0000004c/uni00000052/uni00000051/uni0000000f/uni00000003/uni00000050/uni00000048/uni00000044/uni00000051/uni0000000b η * N /F/uni0000000c/uni00000020/uni00000014/uni00000011/uni00000016/uni0000001b /uni00000036/uni0000004c/uni00000050/uni00000058/uni0000004f/uni00000044/uni00000057/uni00000048/uni00000047/uni00000003/uni00000034/uni00000058/uni00000044/uni00000051/uni00000057/uni0000004c/uni0000005d/uni00000044/uni00000057/uni0000004c/uni00000052/uni00000051/uni00000003/uni0000000b/uni00000033/uni00000052/uni0000004c/uni00000056/uni00000056/uni00000052/uni00000051/uni0000000c /uni00000036/uni0000004c/uni00000050/uni00000058/uni0000004f/uni00000044/uni00000057/uni00000048/uni00000047/uni00000003/uni00000026/uni00000052/uni00000051/uni00000057/uni0000004c/uni00000051/uni00000058/uni0000004c/uni00000057/uni0000005c/uni00000003/uni0000000b/uni00000046/uni00000052/uni00000051/uni00000056/uni00000057/uni00000044/uni00000051/uni00000057/uni00000003/uni00000031/uni0000000c /uni00000030/uni00000048/uni00000044/uni00000056/uni00000058/uni00000055/uni00000048/uni00000050/uni00000048/uni00000051/uni00000057/uni00000003/uni00000027/uni00000044/uni00000057/uni00000044 (b) Fig. 10. (continued) within the electromagnetic field. The electromagnetic field itself maintains continuity, adhering 315 strictly to Maxwell’s equations and the principles of superposition, even at low energy levels. 316 Quantizationis distinctivelymanifested duringboth emissionanddetectionprocesses. When 317 electromagnetic fields are coherently added, the sum of the fields is subject to quantization. 318 However,thereisnoquantizationintheelectromagneticfieldandtherearecertainlyno’photons’. 319 Theprecisemechanismgoverningthequantizationprocessduringbothemissionanddetection 320 withinanotherwisecontinuousfieldstandsasacompellingavenuewarrantingfurtherexploration 321 in research. 322 3. Disclosure 323 The author declares no conflicts of interest. 324