Behaviour of Electromagnetic Waves in Different Media and Structures Part 14 doc

378 Behaviour of Electromagnetic Waves in Different Media and Structures We prefer to express physical quantities by laboratory time x0 rather than proper time τ To do so, we first calculate dt = dτ =  x0 (0) 2  2 x0 (0) − Z0 exp[ −2τ 0 ( B2 + 4Ω )τ ] (61)  x0 (0)exp[τ 0 ( B2 + 4Ω )τ ]  2 2 x0 (0)exp[2τ 0 ( B2 + 4Ω )τ ] − Z0 ] from Eq (54) Using integral formula  ( x 2 − 1)−1/2 dx = cosh −1 ( x ) ,we obtain τ t= 0  x0 (0)exp[τ 0 ( B2 + 4Ω )τ ]dτ  2 2 x0 (0)exp[2τ 0 ( B2 + 4Ω)τ ] − Z0 ]  x0 (0)   x0 (0) x0 (0) {cosh [ ]} = exp[τ 0 ( B2 + 4Ω )τ ]] − cosh −1 [ 2 0 0  τ 0 ( B + 4Ω ) Z Z Z0 , (62) −1 in getting this result, we have assumed that initial laboratory time t = 0 corresponds to initial proper time τ = 0 The distance of electron away from the origin varies with the laboratory time t can be written out according to Eqs (54) and (58) and the result is −  Z0 dr dr / dτ =− = exp[ −τ 0 ( B2 + 4Ω )τ ]  dt dt / dτ x0 (0) (63) If we can express the right hand side of this equation by the expression of t , then we get the equation describing r changing with laboratory time t From Eq (62), after some calculation we can obtain a quadratic equation of exp[τ 0 ( B2 + 4Ω)τ ] , which is exp[2τ 0 ( B2 + 4Ω )τ ] − 2 exp[τ 0 ( B2 + 4Ω )τ ]cosh[τ 0 ( B2 + 4Ω )t ]  2 Z0 1 + 2 + 2 cosh 2 [τ 0 ( B2 + 4Ω )t ] = 0   x0 (0) x0 (0) It is easy to solve this equation and the result is exp[τ 0 ( B2 + 4Ω)τ ] = cosh[τ 0 ( B2 + 4Ω)t ] ± 1 sinh[τ 0 ( B2 + 4Ω )t ]  x0 (0) (64)  We know that when the electron is stationary, x0 (0) = 1 and t = τ So we should take the “ + “ in Eq (64), and Eq (63) becomes −  Z0 dr = 2  dt x0 (0)cosh[τ 0 ( B + 4Ω)t ] + sinh[τ 0 ( B2 + 4Ω )t ] 379 The Influence of Vacuum Electromagnetic Fluctuations on the motion of Charged Particles This equation can be easily solved and the result is  Z0 dt t r (0) − r (t ) =  = = 0  2 Z0 τ 0 (B 2  x0 (0)cosh[τ 0 ( B2 + 4Ω )t ] + sinh[τ 0 ( B2 + 4Ω )t ] t dexp( y )   + 4Ω )  [ x (0) + 1]exp(2 y ) + [ x (0) − 1] 0 0 2 τ 0 ( B2 + 4 Ω ) 0   x0 (0) + 1 x0 (0) + 1 ]} exp[τ 0 ( B2 + 4Ω )t ]] − tan −1 [   x0 (0) − 1 x0 (0) − 1 {tan −1[ So the final expression of the electron’s distance away from the origin expressed by the laboratory time is r (t ) = r (0) −   2 x0 (0) + 1 x0 (0) + 1 exp[τ 0 ( B2 + 4Ω )t ]] − tan −1 ( {tan −1 [ )} ,   τ 0 ( B + 4Ω ) x0 (0) − 1 x0 (0) − 1 2 (65) which explicitly shows the inward spiral characteristic of the electron’s planar motion with a constant magnetic field along its normal direction Classically the final destiny of electrons performing such a motion is falling into the origin    For ultrarelativistic electrons, x0 (0) is very large and tan −1[ ( x0 (0) + 1) /( x0 (0) − 1) ]  π / 4 Using Eq (65), We can estimate the laboratory time needed for an ultrarelativistic electron to decrease its the distance from r (0) to r (0) / 2 , and the result is thalf = 1 1 + tan[τ 0 ( B2 + 4Ω)r (0) / 4] ln τ 0 ( B + 4Ω) 1 − tan[τ 0 ( B2 + 4Ω)r (0) / 4] 2 The initial value r (0) can be approximated by that of the situation without considering the  radiation reaction effects, namely r (0) ≈ x0 / B ，so thaff can be written as thalf =  1 1 + tan[τ 0 ( B2 + 4Ω )x0 (0) / 4 B] ln  τ 0 ( B + 4Ω) 1 − tan[τ 0 ( B2 + 4Ω)x0 (0) / 4 B] 2 (66) However, due to τ 0 being a very small quantity, we are justified to further simplify this expression and the result is thalf    x0 (0) mc x0 (0)  10 −3 2 eB B seconds for electrons which shows that thalf has little relationship with vacuum fluctuations For ultrarelativistic electrons, according to Eq (64), we have approximate expression exp[τ 0 ( B2 + 4Ω )τ ]  cosh[τ 0 ( B2 + 4Ω )t ] , so the energy of electrons can be simplified as  x0 (τ )   0 (0) x cosh[τ 0 ( B2 + 4Ω )t ]  2 x0 (0)cosh 2 [τ 0 ( B2 + 4Ω )t ] − Z0 2  cosh[τ 0 ( B2 + 4Ω )t ]  x0 (0)sinh[τ 0 ( B2 + 4Ω )t ] 380 Behaviour of Electromagnetic Waves in Different Media and Structures   The ratio of two x0 (τ ) / x0 (0) ’s respectively with and without the considerations of vacuum fluctuations is   [ x0 (τ ) / x0 (0)]with  1 − 4Ω /B2 ,   [ x0 (τ ) / x0 (0)]without (67) which shows that the effect of the vacuum fluctuations We look forward to seeing that this result would be tested in future experiments The nonzero effects of vacuum fluctuations had been recognized in microscopic world long time ago, such as the Lamb shift and the Casmir effect etc However, whether or not the vacuum fluctuations has a relationship with the radiation reaction for the motion of charges is still an open question The planar motion of high energy electrons with a constant magnetic field perpendicular to its moving plane provides a possible experimental scheme to test this viewpoint 7 Conclusion In this chapter, we presented a new reduction of order form of LDE, which coincides with that obtained by the method of Landau and Lifshitz in its Taylor series form Using the classical version of zero-point electromagnetic fluctuating fields of the vacuum, we obtained the contributions of vacuum fluctuations to radiation reaction of a radiating charge up to 2 the τ 0 term Then we use the obtained reduction of order equation of LDE including the 2 radiation reaction induced by external force and vacuum fluctuations up to the τ 0 term, which is accurate enough for any macroscopic motions of charges and even applicable to the electron’s motion of a hydrogen atom due to τ 0 being extremely small, to study the onedimensional uniformly accelerating motion produced by a constant electric field and the planar motion produced by a constant magnetic field Our calculations show that for any one-dimensional uniformly accelerating motion the velocity of charges has a limit value and almost all puzzles associated with this special motion disappear; while the planar motion of electrons provides an experimental scheme to test the conjecture that the interaction between charged particles and the vacuum electromagnetic fluctuations is anther mechnisim for the charge’s radiation reaction, which plays a dominant role only for onedimensional macroscopic motions of charged particles 8 Acknowledgment This work was supported by the postdoc foundation of Shanghai city, China The author would like to thank Dr Hui Li for reading the manuscript and proposing fruitful comment on the content of this chapter 9 References Aguirregabiria, J M., (1997), Solving Forward Lorentz-Dirac-like Equations, J Phys A: Math Gen, Vol 30, pp 2391-2402 Boulware, D., (1980), Radiation from a Uniformly Accelerated Charge, Ann Phys, Vol 124, pp 169-188 The Influence of Vacuum Electromagnetic Fluctuations on the motion of Charged Particles 381 Boyer, T H., (1980), Thermal Effects of Acceleration through Random Classical Radiation Phys Rev D, Vol 21, pp 2137-2148 Bell, J S., & Leinaas, J M., (1983), Electrons as Accelerated Thermometers, Nucl Phys B, Vol 212, pp.131-149; (1987), The Unruh Effect and Quantum Fluctuaions of Electrons in Storage Ring, Nucl Phys B, Vol 284, pp 488-508 Chen, P., & Tajima, T., (1999) Testing Unruh Radiation with Ultraintense Lasers, Phys Rev Lett, Vol 83, pp 256-259 Carati, A., (2001) An extension of Eliezer’s theorem on the Abraham-Lorentz-Dirac Equation, J Phys A Gen, Vol 34, pp 5937-5944 Cole, D C., & Zou, Y., (2003), Quantum mechanical Ground State of Hydrogen Obtained from Classical Electrodynamics, Phys Lett A, Vol 317, pp 14-20 Dirac, P A M., (1938) Classical Theory of Radiating Electrons, Proc R Soc London A, Vol 167, pp 148-168 Dewitt, B S., & Brehme, R., (1960), Radiation Damping in a Gravitational Field, Ann Phys, Vol 9, pp 220-259 Eliezer, C J., (1947) The Interaction of Electrons and an Electromagnetic Field, Rev Mod Phys, Vol 19, pp 147-184 Endres, D J., (1993), The Physical Solution to the Lorentz-Dirac Equation for Planar Motion in a Constant Magnetic Field, Nonlinearity, Vol 6, pp 953-971 Fulton, T., & Rohrlich, F., (1960), Classical Radiation from a Uniformly Accelerated charge, Ann Phys, Vol 9, pp 499-517 Ginzberg, V L., (1970), Radiation and Radiation Friction Force in Uniformly Accelerated Motion of a Charge, Sov Phys Usp, Vol 12, pp 569-585 Iso, S., Yamamoto, Y, & Zhang, S., (2010), Does an Accelerated Electron Radiate Unruh Radiation, available from, arXiv:1011.419lvl Johnson, P R., & Hu, B L., (2005), Uniformly Accelerated Charge in a Quantum Field: from Radiation Reaction to Unruh Effect, Foundations of Physics, Vol 35, pp 1117-1147 Jackson, J D., (1976), On Understanding Spin-Flip Synchrotron Radiation and the Transverse Polarization of Electrons in Storage Rings, Rev Mod Phys, Vol 48, pp 417-433 Kawaguchi, H., Tsubota, K., & Honma, T., Lorentz Group Lie Algebra Map of UltraRelativistic Radiating Electron, availabel from http://epaper.kek.jp/pac97/papers/pdf/8P086.PDF Landau, L D., & Lifshitz, E M., (1962), The Classical Theory of Fields, Pergamon, Oxford Lyle, S N., (2008), Uniformly Accelerating Charged Particles, Springer, ISBN 978-3-540-68469-5, Berlin, Heidelberg Lubart, N D., (1974), Solution of Radiation Reaction Problem for the Uniform Magnetic field, Phys Rev D, Vol 9, pp 2717-2722 Plass, G N., (1961) Classical Electrodynamic Equations of Motion with Radiative Reaction, Rev Mod Phys, Vol 33, pp 37-62 Spallicci, A., (2010), Free Fall and Self-Force: an Historical Perspective, available from, arXiv:1005.0611v1 Singal, A K., (1995), The Equivalence Principle and an Electric Charge in a Gravitational Field, Gen Rel Grav, Vol 27, pp 953-967; (1997), The Equivalence Principle and an Electric Charge in a Gravitational Field II A Uniformly Accelerated Charge does not Radiate, Gen Rel Grav, Vol 29, pp 1371-1390 382 Behaviour of Electromagnetic Waves in Different Media and Structures Thirolf, P G., Habs, D., Henig, A., Jung, D., Kiefer, D., Lang, C., Schrerber, J., Mais, C., Schaller, G., Schutzhold, R., & Tajima, T., (2009), Signatures of the Unruh Effect via High-Power, Short-Pulse Lasers, Eur Phys J D, Vol 55, pp 379-389 Teitelboim, C., (1970), Splitting of the Maxwell Tensor: Radiation Reaction without advanced Field, Phys Rev D, Vol 1, pp 1572-1582 Teitelboim, C., Villarroel, D., & Weert, C G., (1980), Classical Electrodynamics of Retarded Fields and Point Particles, Riv Nuovo Cimento, Vol 3, pp 1-64 Unruh, W G., (1998), Accleration Radiation for Orbiting Electrons, available from, arXiv:hep-th/9804158v1; Unruh, W G., (1976), Notes on Black Hole Evaporation, Phys Rev D, Vol 14, pp 870-892 Wang, Guozhong., Li, Hui., Shen, Yifeng., Yuan, Xianzhang., & Zi, Jian., (2010) AntiDampimg Effect of Radiation Reaction, Phys Scr, Vol 81, pp 1-10 Wheeler, J A., & Feynman, R P., (1945), Interaction with the Absorber as the Mechanism of Radiation, Rev Mod Phys, Vol 17, pp 157-181; (1949), Classical Electrodynamics in Terms of Diract Interparticle Action, Rev Mon Phys, Vol 21, pp 425-433 18 Observation of Cavity Interface and Mechanical Stress in Opaque Material by THz Wave Tsuguhiro Takahashi Central Research Institute of Electric Power Industry Japan 1 Introduction One of the recent topics of optical measurement techniques is the generation and utilization of the “Terahertz wave (THz wave)” In the long history of the research on electromagnetic waves, it remains as the last unrevealed area because its generation and detection techniques have not been sufficiently developed In recent years, some useful and convenient devices have become commercially available, and much related research is being carried out One important feature of THz wave is the high transmission probability through some opaque materials, many of which are dielectric materials through which a visible light beam cannot pass Solid insulating materials are usually opaque dielectric materials, and it is difficult to measure their interior by conventional optical measurement techniques If THz wave can pass through an insulating material sufficiently, the same measurement technique as with a visible light beam, already developed and practically applied, such as the technique of utilizing polarization, will become applicable In this chapter, the research work on the introduction of THz wave techniques to the internal measurement of solid insulating materials is reviewed [1,2] 2 Applicability of THz wave to internal measurement of solid insulating materials The dielectric strength of insulating materials generally increases in the order of gas < liquid < solid, but from the viewpoints of the selfrecovery of insulation after discharge, the cooling method and so on, SF6 gas and insulating oil are generally adopted in high voltage electric power equipment On the other hand, polyethylene is successfully utilized in high voltage XLPE cables with well controlled manufacturing techniques In the research work on the “All Solid Insulated Substation” [3], high voltage moulded transformer, compact connector and bus system are being developed Such equipment is expected to show high insulation performance, if there is no fault in the manufacturing process and it is operated with the monitoring of aging Cost reduction will be realized by compactification and long time use, if solid insulating materials can be sufficiently utilized Practical problems of solid insulating materials are, internal cavities (voids) occurrence, electrical and mechanical distortions and so on In order to utilize solid insulating materials effectively, these faults should be monitored during production and operation, but there is no established non-destructive measurement method for opaque insulating materials 384 Behaviour of Electromagnetic Waves in Different Media and Structures In order to obtain internal information from the outside, the detecting “probe” must be inserted For a solid material, it is not possible to insert some objects, but sonic or electromagnetic waves can be utilized These waves should have high transmission probability and sufficient interaction with the target to detect physical quantities An example of the classifications of an electromagnetic waves is shown in Fig 1 Fig 1 Names and wave lengths of electromagnetic waves “THz wave” means an electromagnetic wave with a “terahertz” frequency band Generally, research work on electromagnetic waves of several100 GHz – several10 THz in frequency is being carried out recently This frequency band has remained an “unrevealed area”, but because of the recent development of several related techniques, such as the femtosecond laser, the generation and detection techniques of THz wave have been developed (Table 1 and Table 2), and several related applications are being examined THz wave techniques are expected to be applied in a wide range of areas in the future 2.1 Transmission probability of THz wave for solid insulating materials Most solid insulating materials are opaque dielectric materials “Opaque” means, visible light of about 400 nm – 750 nm in wavelength and electro magnetic waves of around this band cannot be sufficiently transmitted It is said that electromagnetic waves of longer wavelengths (far red) can be transmitted, and it is known that microwaves and millimeter waves, which have much longer wave lengths, can also be transmitted In order to utilize THz waves for transmission measurement of solid insulating materials, it should be clarified, which frequency is suitable for target materials But there is not yet a systematic database of such data Examples of measurement results of transmission probability for PE (polyethylene), Epoxy, silicone rubber, and EPDM (ethylene propylene diene Monomer) rubber, which are typical insulating materials, are shown in Fig 2, 3, 4 and 5 Except for PE, there are large and almost flat absorptions for the indicated frequency range Other transmission probability measurements for several kinds of solid materials have been made, as shown in Fig 6 as an example1 It was obtained with the BWO2 spectroscope (Fig 7), which is commercially available In Fig 6, adjacent points have been averaged, but there 1 These characteristics are those of author’s samples There is a possibility the same material with other filler and manufacturing process will show other characteristics 2 Back-ward Wave Oscillator Tube; it generates single frequency electromagnetic waves, which can be adjusted in the range of millimeter wave to sub-millimeter waves by changing the applied voltage The changeable range depends on the model number of the tube Observation of Cavity Interface and Mechanical Stress in Opaque Material by THz Wave 385 still remains oscillation, which is called the “etalon effect”, which is decided by the thickness of samples (5mm) and frequency According to this data, the transmission probability of THz waves is not high for many solid materials From the next section, actual measurement made for polyethylene, through which THz waves can transmit well, are described, but the same measurement can be applied for other materials with lower frequency waves (, in such a case, the spatial resolution of measurement should be worse) Method frequency power source material several 10nW(CW) 11mW(PW) 0.1μW(PW) GaAs organic DAST crystal InGaP/InGaAs/GaAs Parametric oscillation [6,7] 0.65 2.6THz 200mW(PW) LiNbO3 LiTaO3 photo conduction switch [8] 3THz 1μW(PW) GaAs super conductive optical switch [9,10] 2THz 0.5μW(PW) YBCO semiconductor (by surface electric field) [11,12] 37THz (PW) InP, GaAs semiconductor (by coherent phonon) [13] 4THz (PW) Te, PbTe, CdTe semiconductor in high magnetic field [14] 2THz 100mW(PW) InAs, GaAs single quantum well [13,15] 1.4—2.6THz (PW) GaAs/ Al0.3Ga0.7As (quantum well structure) double quantum well [13,16] 1.5THz (PW) Al0.2Ga0.8As gas laser 0.3 7THz 50mW(CW) CH3OH, CH2F2, CH3Cl quantum cascade laser [17] 1.5 THz 数10mW(PW) GaAs/AlGaAs, InGaAs/InAlAs millimeter wave + frequency multiplier [18] 2.7THz (CW) GaAs schottky diode backward wave oscillation from electron beam in high magnetic field [19] 1.4THz several mW(CW) backward wave oscillator tube femto second plus + optical mixing [4,5] 3THz 1.5 7THz 1—10THz Table 1 Generation methods of THz wave (CW: continuous wave, PW: pulsed wave) 386 Behaviour of Electromagnetic Waves in Different Media and Structures method notes photo conduction switch [13,16] The same switch as in the generation method is utilized Frequency spectrum is obtained from a waveform in the time domain by Fourier transform thermal absorbtion Temperature change of materials (gas cell) absorbing THz wave is detected (, applicable for wide frequency range) EO sampling [13,16] THz wave intensity is converted to visible light beam intensity by the Pockels effect Imaging with a CCD camera is realized photon counting [20] Single electron transistor of quantum dot structure is utilized Intensity is measured directly as numbers of photons Superconductive tunnel junction device [21] Photon-assisted tunneling effect is detected There is the possibility of being arrayed Table 2 Detection methods of THz wave Fig 2 Transmission probability measurement for PE (thickness: 2.7mm; dot colours represent measurement ID numbers) 392 Behaviour of Electromagnetic Waves in Different Media and Structures Fig 9 Scanning image of PE panel with 1.5mm hole (scanning step: 0.54mm, scanning area: 22.1×21.1mm) When two orthogonally crossed coordinates and their normal stresses σ1 and σ2 are considered, the photoelestic effect is expressed by phase difference θ between two electric field components of the incident light beam along those coordinates as θ= 2π ⋅ h λ ⋅ C (σ 1 − σ 2 ) (1) where λ is the wave length of the incident light beam, h is the thickness of the sample, C is the photoelastic constant A phase difference such as that in Eq (1) appears in materials with dielectric anisotropy It can be detected by the “probe” light beam, as shown in Fig 11 The dielectric anisotropy (refractive index anisotropy) affects the linearly polarized incident light beam, and the polarization of the output light beam becomes elliptical, the ellipticity of which is governed by the phase difference θ Generally a polarizer is utilized to make the polarization of the incident light beam linear, and an analyzer (, which is the same optical parts as a polarizer) is utilized to detect the ellipticity of the output light beam As in the Observation of Cavity Interface and Mechanical Stress in Opaque Material by THz Wave 393 experiment corresponding to Fig 10, when a mechanically stressed sample is set in a crossNicol system and the direction of normal stresses are set to the angle of 45 degrees to the transmitting direction of the polarizer and analyzer, the output light beam intensity from the analyzer I is expressed as I= I0 {1 + cos(Δθ + θ 0 )} 2 (2) where I0 is the incident light intensity, and θ0 is the phase difference cased by the natural birefringence (, birefringence when σ1=σ2) The relationship between I and θ is shown in Fig.12 without polarizer/analyzer Fig 10 Scanning result of stressed PE panel (detected stress direction: horizontal, scanning step: 0.54mm, scanning area: 23.2×33.1mm) In order to confirm the positive relationship between applied mechanical stress and the difference in the refractive index, and to estimate the photoelastic constant of PE in the THz frequency range, the output THz wave strength from the analyzer has been measured by changing the applied mechanical stress The stress was applied with a hydraulic jack, which was replaced with the sample stage in Fig 7 The transmitted THz wave was 1.0 THz in frequency, and the dimensions of the measured PE sample block were 70.4 mm in width, 30.5 mm in thickness (direction of mechanical stress application) and 30.9 mm in length (direction of THz wave transmission) The area of applying mechanical stress with the hydraulic jack was 70.4mm x 30.9mm Under these conditions, 394 Behaviour of Electromagnetic Waves in Different Media and Structures the relationship between the applied mechanical stress and the phase difference θ in Eq (1) was obtained, as shown in Fig 13 In this measurement, very large stress was applied to the PE sample; therefore, some distortion was observed7 By measuring the sample size during this experiment, the relationship between applied mechanical stress and distortion was obtained, as shown in Fig 14 The range in which mechanical stress is almost proportional to distortion can be regarded as the elastic region, and when stress is larger than the compressive yield strength of PE, it can be said to be the plastic region and the distortion increases rapidly8 In Fig 13, a linear approximation was made in the elastic region, and the photoelastic constant under these experimental conditions was estimated C = 4 × 10 −11 (3) The estimation of the photoelastic constant in this frequency range seems to be unique Fig 11 Polarization change of “probe” light beam in material with dielectric anisotropy 7 The distortion of PE changed gradually after mechanical stress was applied In this experiment, values of applied mechanical stress were recorded after the change was almost saturated (, about 30 – 60 minutes after), from the stress value indicator of the hydraulic jack 8 In this experiment, the compressive yield strength was obtained to be about 18 – 23 MPa, which is in agreement with some references (e.g., The Society of Polymer Science Japan, “Polymer Data Handbook” (1986)) Observation of Cavity Interface and Mechanical Stress in Opaque Material by THz Wave 395 Fig 12 Relationship between output light intensity from analyzer and phase difference θ Fig 13 Relationship between phase difference θ and applied mechanical stress 396 Behaviour of Electromagnetic Waves in Different Media and Structures Fig 14 Relationship between distortion of PE sample block and applied mechanical stress 4 Conclusion THz wave technology is outstanding in recent optical engineering Much development and application research is being carried out In this chapter, one such research, the study of its application in the internal measurement of insulating materials, has been reviewed Some measurements have been carried out with polyethylene, in which THz waves show high transmission probability, and the internal cavity interface and mechanical stress were detected The applicability of such internal measurement in polyethylene was presented A calibration (and estimation) method should be investigated for it Other internal problems, such as locally high electric field and temperature rise, are also expected to be detected by the same transmitting measurement of THz wave in the future Moreover, surface (and near surface) conditions of materials are expected to be detected by the measurement of reflected THz wave 5 Acknowledgment The author would like to thank Prof Hidaka and his laboratory members of the University of Tokyo for their instruction and collaboration 6 References [1] T Takahashi et al., Application of Measurement Technique for Polyethylene with Mechanical Stress/a Cavity Interface using THz Wave, Trans Of IEE Japan-A, Vol 127-A, No 10, pp 593-598 (2007) [in Japanese] Observation of Cavity Interface and Mechanical Stress in Opaque Material by THz Wave 397 [2] T Takahashi, T Takahashi and T Okamoto, `Study on measurement technique for polyethylene by using THz beam detection of vacant gap interface of mm order and mechanichal stress ', CRIEPI Report W03014 (2004) [in Japanese] [3] M Shibuya et al., Proposition of all solid insulated substation', CRIEPI Report W00047 (2000) (in Japanese) [4] S Matsuura et al., A compact terahertz radiation source for high-resolution spectroscopy, IEICE Technical Report, No LQE97-5 (1997) [in Japanese] [5] T Taniuchi et al., Widely Tunable Terahertz Electromagnetic Radiation by Nonlinear Optical Effect, The Review of Laser Engineering (The Laser Society of Japan), Vol 30, No 7, pp 365-369 (2002) [in Japanese] [6] K Kawase et al., Widely Tunable Coherent THz-Wave Generation Using Nonlinear Optical Effect, Trans Of IEICE, Vol J81-C-I, No 2, pp 66-73 (1998) [in Japanese] [7] K Kawase et al., “Injection-seeded terahertz-wave parametric generator with wide tenability”, Appl Phys Lett., Vol 80, No 2, pp 195-197 (2002) [8] J T Darrow et al., Saturation Properties of Large-Aperture Photoconducting Antennas, IEEE J Quantum Electron., 28, pp 1607-1616 (1992) [9] M Hangyo et al., Terahertz Radiation from High-Temperature Superconductors, The Review of Laser Engineering (The Laser Society of Japan), Vol 26, No 7, pp 536-540 (1998) [in Japanese] [10] M Tonouchi ,THz radiation from high-Tc superconductor, OYOBUTURI(JSAP), Vol 66, No 9, pp 988-989 (1997) [in Japanese] [11] X -C Zhang et al., Generation of femtosecond electromagnetic pulses from semiconductor surfaces, Appl Phys Lett., Vol 56(11), pp 1011-1013 (1990) [12] X -C Zhang et al., Optoelectronic measurement of semiconductor surfaces and interfaces with femtosecond optics, J Appl Phys., Vol 71(1), pp 326-338 (1992) [13] K Sakai et al., Terahertz Electromagnetic Waves: Generation and Applications, The Review of Laser Engineering (The Laser Society of Japan), Vol 26, No 7, pp 515-521 (1998) [14] X -C Zhang et al., Influence of electric and magnetic fields on THz radiation, Appl Phys Lett., Vol 62(20), pp 2477-2479 (1993) [15] H Ohtake et at., Intense THz Radiation from Semiconductors in a Magnetic Field and Its ApplicationA33, The Review of Laser Engineering (The Laser Society of Japan), Vol 30, No 7, pp 360-364 (2002) [in Japanese] [16] K Sakai et al., Terahertz optoelectronics, OYOBUTURI(JSAP), Vol 70, No 2, pp 149155 (2001) [in Japanese] [17] Optoelectronic Industry and Technology Development Association, Optoelectronics Technology Report (2007) [in Japanese] [18] F Maiwald et al., Design and performance of a 2.7 THz waveguide tripler, Proc of 12th Int Symp on Space terahertz Technology, pp 320-329 (2001) [19] A Volkov, Submillimeter BWO spectroscopy of solids, Int J Infrared Millim Waves , Vol 8, No 1, pp 55-61 (1987) [20] S Komiyama et al., Ultrahigh Sensitivity Detection of Terahertz Waves Using Quantum Dots, The Review of Laser Engineering (The Laser Society of Japan), Vol 30, No 7, pp 385-390 (2002) [in Japanese] 398 Behaviour of Electromagnetic Waves in Different Media and Structures [21] S Ariyoshi et al., Terahertz Imaging with a Superconducting Detector Array, IEICE Technical Report, Vol 106, No 403(ED2006 184-199), pp 59-63 (2006) [in Japanese] 19 Reciprocity in Nonlocal Optics and Spectroscopy Huai-Yi Xie Research Center for Applied Sciences, Academia Sinica, Taipei Taiwan 1 Introduction In any wave propagation, the wave can go through various scattering processes through interaction with target in the environment from a source to a detector In such a process, reciprocity refers to the equality in the signal received when the source and the detector are reversed, that is; their respective positions switched (Potton, 2004) We can find many interesting applications which based on either its validity or its breakdown, in the large number of areas involving transmission of signals ranging of classical optical problems In classical optics, reciprocity is a powerful result which finds applications in many problems in optics (Potton, 2004) and spectroscopy (Hill et al., 1997) For example, we can establish relations between far fields and near fields from different sources as well as spectroscopic analysis of surface enhanced Raman scattering (SERS) at metallic structure (Kahl & Voges 2000; Ru & Etchegoin, 2006) However, in the previous literature, the optical reciprocity always has been discussed under local optics (Potton, 2004) Now we try to describe the optical reciprocity from electrostatics to electrodynamics under nonlocal optics in order to consider some quantum effects of the particles Our goal is that the general conditions to determine that the optical reciprocity remains or breaks down will be constructed under nonlocal optics Some examples and applications will also be discussed 2 Reciprocity in electrostatics (Green reciprocity) If we consider an object whose size is much smaller than the wavelength of the incident light, then the effect of retardation can be neglected Hence we can simply use electrostatics to discuss the interaction between the light and the material In mathematics, we usually use two popular forms to describe optical reciprocity One is the Lorentz lemma in electrostatics and the other is the symmetry of the scalar Green function 2.1 Lorentz lemma in electrostatics Lorentz lemma in electrostatics form is well-known with local optical response of the medium in the literature We will extend to consider nonlocal optical response of the medium, since it is known that such response is rather significant with metallic nano structures due to the large surface-to-volume ratio of these systems First we write the mathematical form of the Lorentz lemma in electrostatics as follows (Griffiths 1999; Jackson, 1999): 400 Behaviour of Electromagnetic Waves in Different Media and Structures        ρ (r ) Φ (r ) d r =  ρ (r ) Φ (r ) d r , 3 1 3 2 2 (1) 1 where Φ 1 ( Φ 2 ) is the electric potential resulting from the total charge density ρ 1 ( ρ 2 ) Here we will derive this lemma in two different kinds of circumstances 2.1.1 Anisotropic local response In the beginning, we start from the Poisson equations with two different distributions of charge density ρ 1 and ρ 2 :     ∇ ⋅ ε ( r ) ⋅ ∇Φ 1  = −4πρ 1         , ∇ ⋅ ε ( r ) ⋅ ∇Φ 2  = −4πρ 2     (2)     where ε ( r ) is a dielectric tenser Next we use Eq (A1) and put the tensor λ = ε , the value Φ = Φ 1 and Ψ = Φ 2 Thus we have the following equality:             ˆ   Φ ∇ ⋅ (ε ⋅ ∇Φ ) − Φ ∇ ⋅ (ε ⋅ ∇Φ ) d r =  n ⋅ Φ ε ⋅ ∇Φ 1 2 3 2 1 1 S 2   − Φ 2ε ⋅ ∇Φ 1  da ,  (3) under the symmetric condition of a dielectric tensor ε ij = ε ji Combining with Eq (2) and extending the finite volume to all space (  3 ), we can remove the surface integral in Eq (3) and obtain Eq (1) Hence we prove the Lorentz lemma in electrostatics under the symmetry condition of the dielectric tensor, that is; the optical reciprocity does not break down under the symmetry condition of the dielectric tensor ( ε ij = ε ji ) in the case of anisotropic local response of the medium 2.1.2 Anisotropic nonlocal response In this case, we will extend to consider the nonlocal response Here we write the Poisson equations with two different charge densities ρ 1 and ρ 2 :         ∇ ⋅   ε ( r , r ′ ) ⋅ ∇′Φ 1 ( r ′ ) d 3r ′ = −4πρ 1 ( r )     (4)         ,  ε ( r , r ′ ) ⋅ ∇′Φ 2 ( r ′ ) d 3r ′ = −4πρ 2 ( r )  ∇ ⋅          and we use Eq (A5) with λ ( r , r1 ) = ε ( r , r1 ) , the value Φ = Φ 1 and Ψ = Φ 2 Thus we get the following identity:                  d r  d r {Φ ( r ) ∇ ⋅ ε ( r , r ) ⋅ ∇ Φ ( r ) − Φ ( r ) ∇ ⋅ ε ( r , r ) ⋅ ∇ Φ ( r )} ,                  ˆ  =  da  d r {n ⋅ Φ ( r ) ε ( r , r ) ⋅ ∇ Φ ( r ) − Φ ( r ) ε ( r , r ) ⋅ ∇ Φ ( r ) }   3 3 1 1 1 1 2 1 2 1 1 1 3 S 1 1 1 1 2 1 2 1 1 1 1 (5) 1     under the condition ε ij ( r , r ′ ) = ε ji ( r ′, r ) Next we combine Eq (4) and extend the finite volume to  3 ; thus we can remove the surface integral in Eq (5) and get Eq (1) again Hence we prove the Lorentz lemma in electrostatics under the symmetry condition of the     dielectric tensor ε ij ( r , r ′ ) = ε ji ( r ′, r ) Thus the optical reciprocity does not break down under the symmetry condition of the dielectric tensor in the case of anisotropic nonlocal response of the medium 401 Reciprocity in Nonlocal Optics and Spectroscopy 2.2 Scalar Green function Another method to describe the optical reciprocity is the symmetry of the scalar Green function The mathematical form is (Jackson, 1999):     G ( r ′′, r ′ ) = G ( r ′, r ′′ ) (6) We divide into two cases, and consider two kinds of boundary conditions to discuss the symmetry of the scalar Green function One is the Dirichlet boundary condition and the other is the Neumann boundary condition 2.2.1 Anisotropic local response Referring to Eq (2), the corresponding two equations of scalar Green function are in the following forms:         ∇ ⋅ ε ( r ) ⋅ ∇G ( r , r ′ )  = −4πδ ( r − r ′ )             , ∇ ⋅ ε ( r ) ⋅ ∇G ( r , r ′′ )  = −4πδ ( r − r ′′ )     (7)    where r and r ′ ( r ′′ ) are the positions of the field and source, respectively δ denotes the        Dirac delta function Let us apply Eq (A1) and put λ = ε ( r ) , Φ = G ( r , r ′ ) and Ψ = G ( r , r ′′ ) Hence we have the following equality:                   {G ( r , r′) ∇ ⋅ ε ( r ) ⋅ ∇G ( r , r′′) − G ( r , r′′) ∇ ⋅ ε ( r ) ⋅ ∇G ( r , r′)} d r ,                   ˆ ⋅ G ( r , r ′ ) ε ( r ) ⋅ ∇G ( r , r ′′ ) − G ( r , r ′′ ) ε ( r ) ⋅ ∇G ( r , r ′ )  da = n    3 (8) S   under the condition ε ij ( r ) = ε ji ( r ) and we also combine Eq (6) to get the following result:     −4π G ( r ′′, r ′ ) + 4π G ( r ′, r ′′ )               =  n ⋅ G ( r , r ′ ) ε ( r ) ⋅ ∇G ( r , r ′′ ) − G ( r , r ′′ ) ε ( r ) ⋅ ∇G ( r , r ′ )  da S ˆ   (9) Next we will divide into two different boundary conditions to discuss In the case of the Dirichlet boundary condition, we have:     G ( r , r ′ ) = G ( r , r ′′ ) = 0 , (10)  with r ∈ S Substitute this into Eq (9) and we obtain Eq (6), establishing the symmetry of the scalar Green function with the Dirichlet boundary condition under the symmetry condition of the dielectric tensor In the case of the Neumann boundary condition, let us generalize the results in Kim et al (Kim, 1993) to introduce the following Neumann boundary conditions (Xie, 2010):     ˆ   n ⋅ ε ( r ) ⋅ ∇GN ( r , r ′ )    r ∈S      ˆ  n ⋅ ε ( r ) ⋅ ∇GN ( r , r ′′ )    r ∈S 4π A , 4π =− A =− where A is the area of the closed boundary S Eq (9) then becomes: (11) 402 Behaviour of Electromagnetic Waves in Different Media and Structures 4π     −4π GN ( r ′′, r ′ ) + 4π GN ( r ′, r ′′ ) = − A 4π      G ( r , r′) da + A  G ( r , r′′) da   S N (12) N S We can then follow our pervious work (Xie, 2010) to define the following symmetrized Green function: 1     S   GN ( r ′′, r ′ ) = GN ( r ′′, r ′ ) −  GN ( r , r ′ ) da , A S (13) which can be shown explicitly to lead to the same solution for the potential with no contributions from the additional surface term 2.2.2 Anisotropic nonlocal response In this case, we will consider the anisotropic nonlocal response in the material The Poisson equations with two different distributions of charge density as given in Eq (4) will have the corresponding scalar Green functions satisfying:            r r r r r    d  ∇ ⋅ ε ( , ) ⋅ ∇ G ( , ′) = −4πδ ( r − r′)       d r ∇ ⋅ ε ( r , r ) ⋅ ∇ G ( r , r ′′ )  = −4πδ ( r − r ′′ )      3 1 1 1 1 1 1 1 1 (14) 3         Next we use Eq (A5) and let λ = ε , Φ ( r ) = G ( r , r ′ ) and Ψ ( r ) = G ( r , r ′′ ) , leading to:                      d r  d r {G ( r , r′) ∇ ⋅ ε ( r , r ) ⋅ ∇ G ( r , r′′) − G ( r , r′′) ∇ ⋅ ε ( r , r ) ⋅ ∇ G ( r , r′)} ,                      ˆ  =  da  d r {n ⋅ G ( r , r ′ ) ε ( r , r ) ⋅ ∇ G ( r , r ′′ ) − G ( r , r ′′ ) ε ( r , r ) ⋅ ∇ G ( r , r ′ ) }   3 3 1 1 1 1 1 1 1 3 S 1 1 1 1 1 1 (15) 1     under the condition ε ij ( r , r ′ ) = ε ji ( r ′, r ) , combining Eqs (14) and (15) leads to:     −4π G ( r ′′, r ′ ) + 4π G ( r ′, r ′′ )  ˆ                 =  da  d 3r1 n ⋅ G ( r , r ′ ) ε ( r , r1 ) ⋅ ∇ 1G ( r1 , r ′′ ) − G ( r , r ′′ ) ε ( r , r1 ) ⋅ ∇ 1G ( r1 , r ′ )  S   { } (16) Now we will divide into two different kinds of the boundary conditions First we consider the Dirichlet boundary condition (Eq (10)) The RHS of Eq (16) becomes zero and thus we obtain the optical reciprocity under the symmetry of the dielectric tensor Next we consider the Neumann boundary condition GN Again, introducing the following generalized Neumann conditions (Xie, 2010) :        ˆ n ⋅  d 3r1 ε ( r , r1 ) ⋅ ∇ 1GN ( r1 , r ′ )     r ∈S        ˆ n ⋅  d 3r1 ε ( r , r1 ) ⋅ ∇ 1GN ( r1 , r ′′ )     r ∈S 4π A , 4π =− A =− (17) the Green function can be symmetrized in the form as in Eq (13) under the following     symmetric condition for the dielectric tensor: ε ij ( r , r ′ ) = ε ji ( r ′, r ) Note further that just like the local case, this symmetrized Green function can be shown to lead to the same solution for the potential Now we demonstrate explicitly that the newly-constructed symmeterized 403 Reciprocity in Nonlocal Optics and Spectroscopy Green functions in Eq (13) yield the same solution for the potential with no contribution  from the additional surface term To achieve this, we start with Eq (A5) and set Φ = Φ ( r ) ,        S   Ψ ( r ) = GN ( r , r ′ ) , and λ ( r , r1 ) = ε ( r , r1 ) to obtain       1 1       ˆ   da d r {n ⋅ Φ ( r ) ε ( r , r ) ⋅ ∇ G ( r , r′) − G ( r , r ′) ε ( r , r ) ⋅ ∇ Φ ( r )}                       =  d r  d r {Φ ( r ) ∇ ⋅ ε ( r , r ) ⋅ ∇ G ( r , r ′ )  − G ( r , r ′ ) ∇ ⋅ ε ( r , r ) ⋅ ∇ Φ ( r ) }   3 S 3 1 3 1 1   S N 1 S N 1 S N 1 S N 1  1 1 1 1 (18) 1 S With Eq (13) into the GN of Eq (18) and using Eq (14) together with the Poisson equation for Φ , we obtain the following result:  1 1       3 ˆ    Φ ( r ′ ) =  Φ ( r ) da + S da d r1GN ( r , r′) n ⋅ ε ( r , r1 ) ⋅ ∇1Φ ( r1 ) S   A 4π           1   3 ˆ    +  ρ ( r ) GN ( r , r ′ ) d 3 r + F ( r ′ )   ρ ( r ) d 3 r +  S da d r1n ⋅ ε ( r , r1 ) ⋅ ∇1Φ ( r1 )  , (19)  4π       1 1        3 3 ˆ  =  Φ ( r ) da +  da d r1GN ( r , r′) n ⋅ ε ( r , r1 ) ⋅ ∇1Φ ( r1 ) +  ρ ( r ) GN ( r ,r′) d r  A S 4π S   where the surface term F ( r ′ ) has no contribution since the term { } in Eq (18) vanishes based on the nonlocal version of the Gauss law for Φ [See Eq (4)] Thus the symmetrized S GN leads to the same potential as the one obtained from using GN 2.3 Some examples for scalar Green functions We will give some examples to display the explicit forms of the Green function in both Dirichlet and Neumann boundary conditions (Chang 2008; Xie, 2010) Consider a metal sphere (radius a ) with an isotropic (for simplicity) but nonlocal dielectric response ε ( k , ω ) For the case of Dirichlet condition, we have previously applied the model of the nonlocal polarizability by Fuchs and Claro (Fuchs & Claro, 1987) to obtain the following symmetric Green function: ∞   G ( r , r ′ ) = 4π  1   r <  2 + 1  r    = 0 m =−  +1 > − α NL  ( rr′) +1  Y*m (θ ′,ϕ ′ ) Ym (θ ,ϕ ) ,   (20) where ( r< , r> ) denote the smaller or greater of ( r , r ′ ) , and α NL = ξ − 1 a2  + 1 ,   + 1 ξ +      (21) with the “effective dielectric function” given by:  2 ( 2 + 1) a ξ (ω ) =    π  ∞ 0 j2 ( ka ) −1  dk  ε ( k ,ω )   (22) For the same problem under the Neumann condition, first we get the following Green function by solving the corresponding boundary value problem: 404 Behaviour of Electromagnetic Waves in Different Media and Structures ∞  1 1 1  r r  + 1 ( rr ′ ) + 1   = 1 m =−  2  + 1  r>   (23) which does not satisfy Eq (6) However, we can use the pervious methods to symmetrize the above asymmetric Green function Applying Eq (13) to calculate the surface term using only the first two terms in Eq (23) and obtain the following: 1 1 1    GN ( r , r′) da = r′ − a A S (24) We thus obtain the final symmetrized Neumann Green function for the region outside a nonlocal metal sphere in the following form:  1  1 1  1 S   GN ( r , r ′ ) =  −  +  +   r>  r r ′  a  α NL  * 1  r   ∞ (25)  In our examples, we can find that the optical reciprocity does not break down since the     dielectric function satisfies ε ( r , r ′ ) = ε ( r − r ′ ) 3 Reciprocity in electrodynamics In the pervious discussion, we have restricted our problem to the “long wavelength approximation” in which electrostatics has been applied (Chang, 2008) In problems with very high frequency source (e.g scattering between X-rays and a nanoparticle (Ruppin, 1975)) in which electrostatics breaks down and nonlocal effects can become even more significant due to the large value of the wavevector, the previous formulation (Chang, 2008) becomes inadequate Here we use the method of exact electrodynamics to study the optical reciprocity Again, we shall refer to both the Lorentz lemma in electrodynamics and the symmetry of dyadic Green function 3.1 Lorentz lemma in electrodynamics The mathematical form of the Lorentz lemma in electrodynamics is as follows (Landau et al., 1984):            J (r ) ⋅ E (r ) d r =  J (r ) ⋅ E (r ) d r , 1 3 2 2 3 1 (26)     where E1 ( E2 ) is the electric field resulting from the current density J 1 ( J 2 ) Next we will derive this formula in two different cases 3.1.1 Anisotropic local response In the beginning, let us consider time harmonic Maxwell’s equations (  e − iωt ): Reciprocity in Nonlocal Optics and Spectroscopy 405   ∇ ⋅ D = 4πρ   iω   ∇×E = B   c (27)     4π  iω   ∇⋅B = 0 ∇×H = J − D c c     Now we consider two different sources J 1 and J 2 which correspond to two electric fields     E1 and E2 and two magnetic fields H 1 and H 2 , respectively We obtain from Eq (27) the following result:             H 2 ⋅ ∇ × E1 − E1 ⋅ ∇ × H 2 + E2 ⋅ ∇ × H 1 − H 1 ⋅ ∇ × E2       iω   4π     , = ( B1 ⋅ H 2 − H 1 ⋅ B2 ) + ( E1 ⋅ D2 − D1 ⋅ E2 )  +  c ( J 1 ⋅ E2 − J 2 ⋅ E1 ) c  (28) which can be simplified to the following form:      4π     ∇ ⋅ ( H 1 × E2 − E1 × H 2 ) = ( J1 ⋅ E2 − J 2 ⋅ E1 ) , c (29) if the dielectric tensor satisfies the symmetry condition ε ij = ε ji and the permeability satisfies the symmetry condition μij = μ ji Furthermore, let us integrate over  3 in Eq (29) and use the divergence theorem to convert the left side of Eq (29) into a surface integral which can be removed Hence we obtain the Lorentz lemma in electrodynamics (i.e Eq (26)) 3.1.2 Anisotropic nonlocal response Next we will extend the local response to the nonlocal response where the relation between the auxiliary field and the magnetic field is described by        H ( r ) =  d 3r ′μ −1 ( r , r ′ ) ⋅ B ( r ′ ) (30) Thus Eq (28) will become:      ∇ ⋅ ( H 1 × E2 − E1 × H 2 ) iω 3     −1            d r ′{B1 ( r ) ⋅ μ ( r , r ′ ) ⋅ B2 ( r ′ ) − μ −1 ( r , r ′ ) ⋅ B1 ( r ′ ) ⋅ B2 ( r )} c  4π            iω         +  d 3r ′{E1 ( r ) ⋅ ε ( r , r ′ ) ⋅ E2 ( r ′ ) − ε ( r , r ′ ) ⋅ E1 ( r ′ ) ⋅ E2 ( r )} + ( J1 ⋅ E2 − J 2 ⋅ E1 ) c c = (31) We integrate over  3 and use the divergence theorem to remove the surface term again Thus Eq (31) becomes: 4π     3  ( J1 ⋅ E2 − J 2 ⋅ E1 ) d r c  iω                 =  d 3r  d 3r ′{B1 ( r ) ⋅ μ −1 ( r , r ′ ) ⋅ B2 ( r ′ ) − μ −1 ( r , r ′ ) ⋅ B1 ( r ′ ) ⋅ B2 ( r )} , c iω                 +  d 3r  d 3r ′{E1 ( r ) ⋅ ε ( r , r ′ ) ⋅ E2 ( r ′ ) − ε ( r , r ′ ) ⋅ E1 ( r ′ ) ⋅ E2 ( r )} c (32) 406 Behaviour of Electromagnetic Waves in Different Media and Structures which can be simplified to get the Lorentz lemma in electrodynamics under the symmetry     condition of the dielectric tensor ε ij ( r , r ′ ) = ε ji ( r ′, r ) and the permeability tensor     μij ( r , r ′ ) = μ ji ( r ′, r ) 3.2 Dyadic Green function The other method to describe the optical reciprocity is the symmetry of dyadic Green function and the mathematical form is the following form (Tai, 1993; Xie, 2009a, 2009b):    T    Ge ( r ′, r ′′ )  = Ge ( r ′′, r ′ ) ,   (33) where T is the transport We shall establish our results in two steps and restrict ourselves to the case of boundary conditions for the dyadic Green function 3.2.1 Anisotropic local response First we consider only local response which is simpler and sets the framework for the treatment of the more complicated nonlocal case Thus we assume the following constitutive             relations: D ( r ) = ε ( r ) ⋅ E ( r ) and B ( r ) = μ ( r ) ⋅ H ( r ) For fields with time harmonic dependence, we have the following vector identity:       ω2     4π   ∇ × μ −1 ( r ) ⋅ ∇ × E ( r ) − 2 ε ( r ) ⋅ E ( r ) = iω 2 J ( r ) , c c (34) which implies the following differential equation for the electric dyadic Green function of the problem:        ω2      4π    (35) I δ ( r − r′) ∇ × μ −1 ( r ) ⋅ ∇ × Ge ( r , r ′ ) − 2 ε ( r ) ⋅ Ge ( r , r ′ ) = c c            Using Eq (A9), we set A = Ge ( r , r ′ ) , B = Ge ( r , r ′′ ) and λ = μ −1 ( r ) to obtain the following identity:   { ∇ × μ  }    T        −1     T        ⋅ ∇ × Ge ( r , r ′′ )  ⋅ Ge ( r , r ′ ) − Ge ( r , r ′′ )  ⋅ ∇ × μ −1 ⋅ ∇ × Ge ( r , r ′ ) d 3r    T              T ˆ =  n × Ge ( r , r ′′ )  ⋅ μ −1 ⋅ ∇ × Ge ( r , r ′ ) −  μ −1 ⋅ ∇ × Ge ( r , r ′′ )  ⋅ n × Ge ( r , r ′ ) da S  ˆ    { ( ) } (36) Hence from either the dyadic Dirichlet condition:    ˆ n × Ge ( r , r ′ ) = 0 , (37)      ˆ n ×  μ −1 ⋅ ∇ × Ge ( r , r ′ )  = 0 ,   (38) or the dyadic Neumann condition1: 1     ˆ  Note that the condition introduced in Eq (38) is a generalization of n × ∇ × G e ( r , r ′ ) = 0  explicit inclusion of the magnetic permeability tensor with the ... direction in Fig.6(b) is detected 392 Behaviour of Electromagnetic Waves in Different Media and Structures Fig Scanning image of PE panel with 1.5mm hole (scanning step: 0.54mm, scanning area:... Detection of Terahertz Waves Using Quantum Dots, The Review of Laser Engineering (The Laser Society of Japan), Vol 30, No 7, pp 385-390 (2002) [in Japanese] 398 Behaviour of Electromagnetic Waves in Different. .. during production and operation, but there is no established non-destructive measurement method for opaque insulating materials 384 Behaviour of Electromagnetic Waves in Different Media and Structures