Investigations on asymmetric transmittivity of optical devices and different diode-like behaviors

•Proof that reciprocal optical devices cannot guarantee asymmetric transmittance•Proof nonreciprocity is necessary for being diodes•Discussions on thermodynamics of both and nonreciprocal devices•Comparisons the diode-like behaviors between electronic diodes This study theoretically proved although can show transmittivity (AT) under controlled incident modes (i.e., conditional AT), they AT with arbitrary light modes, whereas only possibly AT. Besides, were discussed to second law valid anyway. Furthermore, compared. Electrons are identical devices, so could have conductance regardless electrons. In contrast, electromagnetic waves different from as be different, showing when arbitrary. The mathematical proof characteristic comparisons diodes, which firstly presented here, should help clarifying required diodes. Many demonstrated in two opposite directions experiments or numerical calculations.1Yao Y. Liu H. Wang Li Song B. R.P. Povinelli M.L. Wu W. 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Therefore, study, it mathematically proven whose described matrices modes. To best our knowledge, first time argument mathematically. clarification requirements avoid further futile efforts developing related discussed. discussion prevent improper use function comparing awareness intrinsic differences raised suggest accurate “diode”. Flowing circuits make no difference electrons) would consistently. characterized may transmittivities even same waveguide. fail exhibit change. It argued AT, intrinsically terms transferring physical quantities, hence, As mentioned previously, gratings. feature structurally directions. shown Figure 1A, structure taken example represent spatially (SARDG). Assuming SARDG (the adds light), outgoing Lout (ai′, bi′) incoming (ai, bi) Lout=SLin, S matrix. corresponding expanded form Lout=SLin given Equation 1, on-diagonal Sa′i,ai Sb′i,bi coefficients ai bi ai′ bi′, respectively, off-diagonal coefficients, Sa′i,aj (i≠j) Sa′i,bj aj bj ai′, respectively.[a1′a2′a3′⋮b1′b2′b3′⋮]=[Sa′1,a1Sa′1,a2Sa′1,a3⋯Sa′1,b1Sa′1,b2Sa′1,b3⋯Sa′2,a1Sa′2,a2Sa′2,a3⋯Sa′2,b1Sa′2,b2Sa′2,b3⋯Sa′3,a1Sa′3,a2Sa′3,a3⋯Sa′3,b1Sa′3,b2Sa′3,b3⋯⋮⋮⋮⋱⋮⋮⋮⋱Sb′1,a1Sb′1,a2Sb′1,a3⋯Sb′1,b1Sb′1,b2Sb′1,b3⋯Sb′2,a1Sb′2,a2Sb′2,a3⋯Sb′2,b1Sb′2,b2Sb′2,b3⋯Sb′3,a1Sb′3,a2Sb′3,a3⋯Sb′3,b1Sb′3,b2Sb′3,b3⋯⋮⋮⋮⋱⋮⋮⋮⋱][a1a2a3⋮b1b2b3⋮],(Equation 1) Consider Case 1 2 Figures 1B 1C, respectively. According transmitted power side calculated Equations 3, respectively.PCase1=[b1′b2′b3′⋮]+[b1′b2′b3′⋮]=|b1′|2+|b2′|2+|b3′|2+…=(|Sb′1,a1|2+|Sb′2,a1|2+|Sb′3,a1|2+⋯)|a1|2,(Equation 2) PCase2=[a1′a2′a3′⋮]+[a1′a2′a3′⋮]=|a1′|2+|a2′|2+|a3′|2+…=(|Sa′1,b1|2+|Sa′2,b1|2+|Sa′3,b1|2+⋯)|b1|2,(Equation 3) superscript ‘+’ denotes conjugate transpose compare 2, 3 marked red blue line box matrix, 2. reciprocal, must according theorem. So Sb′1,a1 equals Sa′1,b1 while Sb′2,a1 does necessarily equal Sa′2,b1, Sb′3,a1 Sa′3,b1, on. explains attributed nonzero-order Clearly, amplitude a1 b1, |a1|2=|b1|2, Actually, theorem, satisfy Sb′j,ai=Sa′i,bj. equality model Sb′j,ai Sa′j,bi. inequality explained 3.Figure 3Schematic coefficientsShow full caption(A) coefficient bj′. ai′. Sb′j,ai=Sa′i,bj means always trace way back original devices.(B) Sa′j,bi aj′. mode, devices.View Large Image ViewerDownload Hi-res image Download (PPT) (A) devices. (B) realized previously conditions upper lower sides plane wave. Under condition, will longer exist. follows. For convenience, rewritten as:[A′B′]=[SA′ASA′BSB′ASB′B][AB],(Equation 4) B stand column vector [a1a2a3…]T [b1b2b3…]T, respectively ‘T’ transpose); A′ B′ [a1′a2′a3′…]T [b1′b2′b3′…]T, respectively; SA′A stands left quadrant SA′B right quadrant, SB′A SB′B quadrant. Considering only, [A0]T, 4 [0B]T, relationships Cases 5 6, respectively.[A′B′]=[SA′ASA′BSB′ASB′B][A0],(Equation 5) [A′B′]=[SA′ASA′BSB′ASB′B][0B],(Equation 6) Then, 7 8, respectively.B′+B′=(SB′AA)+SB′AA=A+SB′A+SB′AA,(Equation 7) A′+A′=(SA′BB)+SA′BB=B+SA′B+SA′BB,(Equation 8) And 9 10, respectively.τCase3=A+SB′A+SB′AAA+A,(Equation 9) τCase4=B+SA′B+SA′BBB+B,(Equation 10) SARDG, SB′AT. τCase4 10 as:τCase4=B+SB′ASB′A+BB+B,(Equation 11) Using M denote SB′A+SB′A 9, N SB′ASB′A+ 11, since Hermitian matrices, eigenvalues λi. [A0]T [0B]T represented A=∑i=1nxiui B=∑i=1nyivi, ui vi orthogonal eigenvectors N, λi, xi yi coefficients. Subsequently, τCase3 11 12 13, respectively.τCase3=∑i=1nλi|xi|2∑i=1n|xi|2,(Equation 12) τCase4=∑i=1nλi|yi|2∑i=1n|yi|2,(Equation 13) λmin≤τCase3≤λmax λmin≤τCase4≤λmax, λmin λmax minimal maximal values limit values. Since any exist combination equal. Consequently, guaranteed. nonreciprocal, asymmetric, hence value range τCase4, smaller than larger τCase4. way, (λmin,Case4−λmin,Case3 λmax,Case3−λmax,Case4). support argument, 3D printer (BMF NanoArch S130) sputtering machine, 4. printed UV-curing resin then sputtered silver. composed isosceles triangles placed equidistantly, height, bottom span, space 50 μm, geometry optimized finite-difference time-domain (FDTD) simulation results has high contrast ratio indicated 5A. FDTD simulation, linearly forward backward same. transmittance about times direction. However, transmittances FTIR (Fourier Transform Infrared Spectroscopy, SHIMADZU IRaffinity-1S) almost same, 5B. conditional. condition experimental measurement nonpolarized contain shows nearly AT.Figure 5Transmittance directionsShow Numerical simulation.(B) FTIR.View simulation. FTIR. transfer closely radiative heat transfer, confusion thermodynamics,44Allahverdyan Could cuticle beetles serve thermoregulation?.Opt. Photon. 17-22Google Clausius statement colder warmer body change.45Clausius HIRST T.A. TYNDALL Mechanical Theory Heat, Applications Steam-Engine Physical Properties Bodies.in: Archer Hirst Translated John Tyndall. 1867Google assumed bodies temperatures exchange radiatively isolated system, Body temperature (T1) (T2) spontaneously if reverse direction, 6. discussions separated cases system isolated, body’s radiations reach only. ideally. Even managed emit direction) flow spontaneously. because reflected continuously do find ways go ideal release holds realize explaining why fails explain anyway violated seemingly. fact themselves absorptions cases. absorbed eventually reradiate 1. Nonetheless, protect emitting reflections, systems. like asymmetry. Hence, suggested Note widely nowadays property, 7. easily pass hardly rectification. Electronic circuit (excluding above voltage). There another circuit. so-called (reciprocal AT) light, view we believe inappropriate “optical diode” insufficient quantity From perspective mathematics, scalar (some tensor form, here), though (nonlinear current-voltage diodes), section reveals physics. traveling considered waves, Based aforementioned observations, understood light. manipulated, kind circumstance happen practice naming demonstrating checking natural idea actually accurate. Reciprocal Instead, asymmetry conclusion, symmetry, limiting Nonreciprocity (breaking reciprocity) validity thermodynamics. Additionally, calling inaccurate comparison key When But Only asymmetrical unawareness electrons, might cause incautious mistaken Also, settle dispute over