On elation lengths of thermal electromagnetic fields in uilibrium and out of uilibrium conditions Illarion Dorofeyev Institute for Physics of Microstructures Russian Academy of Sciences, GSP-5, Nizhny Novgorod, 6395, Russia Corresponding author: Illarion955@mail.ru Spatial coherence of thermal fields in far- and near-field zones generated by heated half-space into vacuum is studied at essentially different thermodynamical conditions. It is shown that elation lengths of fields in any field zone are different in uilibrium and out of uilibrium systems. In wide range of distances from sample surface elation functions are should be calculated using a total sum of evanescent and propagating contributions due to their mutual compensation at some conditions because of antielations. It is demonstrated that elation lengths as calculated with a proposed formula are in agreement with a behavior of elation functions of thermal fields in spectral range of surface excitations.introduction The study of the time and spatial coherence properties of spontaneous fields is an important part of modern science. Correlation properties of free thermal electromagnetic fields in uilibrium with surrounding matter are described in detail; see for example [-4]. Correlation tensors of the fields at any distances from bodies with arbitrary geometric shapes can be theoretically calculated both in uilibrium and in out of uilibrium when heated body radiates into a cold surrounding [5-8]. It should be emphasized that in accordance with the theoretic analysis thermal fields are formally divided into the propagating and evanescent fields with different spectral and elation properties, see for numerous examples [9-]. We recall that the elation radius of propagating waves of the black body field is on the order of the Wien wavelength in case of non-ual time elation functions, and is on the order of a wavelength of interest in case of ual time elation functions. The spatial elation scale of fields nearby heated bodies depends on electrodynamic properties of materials and other parameters of the problem under study. An analysis of properties of thermally stimulated fields in a vicinity of interfaces is extremely important for a description of dynamics of near surface processes. There has been a vast amount of scientific activity in the study of the coherence properties of thermal fluctuations at fruencies of the surface excitations and waveguide modes [-6]. The prominent result here is the increase of the spatial elation at the surface eigenfruencies. The increase can reach tens of wavelengths nearby an interface which is substantially larger than the elation scale of the uilibrium radiation in free space []. Furthermore, it was shown in [, 5] that the length of elation is determined by the dispersion characteristic for surface polaritons. The physical grounding is that the collective coherent excitation of surface charges or surface oscillations of the lattice at the interface becomes as the source of fields at esponding eigenfruencies transferring the spatial coherence to the fluctuating electromagnetic fields. Obviously, a study the coherence of spontaneous fields in the fruency range of resonance states or eigenmodes is especially important, because properties of eigenmodes are completely determined by whole set of characteristics of the system. Our paper addresses the following question: how different the coherence lengths of thermal electromagnetic fields in uilibrium and out of uilibrium systems. In sections we give general formulas for traces of spatial coherence tensors of the fluctuating electromagnetic fields generated by a half-space into cold space, and radiated by a halfspace into surroundings kept with the same temperature. Numerical results for coherence tensors of thermal fields and for the elation length within thermodynamically different systems and relating discussions are provided in section 3. Concluding remarks are given in section 4.. Spatial elations of thermal fields We consider a system of heated half-space radiating into a vacuum in different thermodynamic conditions. One condition esponds to the complete thermodynamic uilibrium and another condition esponds to the system in which the temperature of the half-space is much larger than the temperature of the ambient bodies at infinity. In both cases the general formulas for the crossspectral tensors of thermal fields at any distance from the half space with flat boundary can be found in [6, 9-]. All spatial elations are considered in vacuum over a heated half-space between points r (,, h) and
r ( CosΦ, SinΦ, h+ H), where h and h+ H are the distances along the normal to the boundary of a half-space, is the separation of the selected points along the interface. The angle Φ will be irrelevant after taking a trace over the cross-spectral matrices wij ( r, r; ), ( i, j x, y, z). It is remarkably that any statistical characteristics of thermal fields in the problems of our consideration can be divided into two separate parts esponding to the evanescent and propagating waves. The same is valid for traces of the cross-spectral matrices. Thus in case when the heated half-space radiates into cold surroundings the trace of the cross-spectral matrix for components of the electric field can be represented as follows n } n W (, H; ) Trace w (, H; ), () f (, H; ) + f (, H; ) where π / f (, H) πu dθsin θcos( k Hcos θ) () ( P S r )/ ( r )/ + J( ksin θ ), [ ] f (, H) πu dy exp k y( h+ H) P S y r r J k y ( + ) Im{ } + Im{ } ( + ), P, S 3 where r are the Fresnel coefficients, u Θ / π c is the spectral power density of the black-body radiation, Θ (, T) ( / ) cth( /k B T), J is the Bessel function, k / c. In uilibrium case the trace is expressed by the following formula W (, H; ) Trace{ wij (, H; ) }, (4) g (, H; ) + g (, H; ) + g (, H; ) where π / g (, H) π u dθsin θcos( k Hcos θ) (5) J ( k sin θ ), π / g H u d J k π θ θ θ {( S P r r θ ) exp ik h H θ } (, ) sin ( sin ) Re cos [ ( + )cos ], and g f form Eq.(3). It should be noted that the evanescent contributions of the traces f and g are identical in uilibrium and in nonuilibrium problems. Using the formulas for traces we numerically studied the elation length of thermally stimulated fields generated by the material half space whose dielectric functions are given by the Drude or by the oscillatory models. (3) (6) 3. The elation length of thermal fields It should be emphasized that no striclty defined notion of the elation length following from some basic physical principles. The experimentally meausred value is the elation function. The elation length itself is the measure of the essentiality of elations of some process. It may be measured, for example from 5 per cent up to per cent level. For instance, from dimensionality reasons the elation length along some X direction can be defined X } z ( X ) } dx Tr w ( ) X ( z), (7) Tr w where z is any real numbers, in the simplest case they are natural numbers z n,,3,.... Here we take the often-used expression for elation length along, for example direction as follows d Tr{ wij } ( ) } ( ), (8) Tr w and similar expression for H direction. 3.. The elation length in free space in uilibrium and out of uilibrium cases In uilibrium in free space, or sufficiently far from any boundaries of a cavity the thermal electromagnetic field is the black-body radiation. In this case all chracteristics are known and we obtain a numerical value of the elation length using different approaches. For example, the normalized cross-spectral tensor in Eq. (3.5) from [3] is as follows Tr w ( R) 3sin R/π, (9) } where R represents the separation in units of the wavelength λ of two arbitrary spatial points. Taking into account Eqs.(8) and (9) we have the ellation length R λ /along the arbitrary direction R. From other side, in uilibrium and far ( kh ) from the heated half-space we obtain from Eq.(4) that W (, H) g (, H), where g ( H, ) is given by Eq.(5). In this case using Eq.(8) we have the identical H elation lengths λ / both along the direction H () and (H). We note that the espondence between these distances is + H R. Thus, we recall the well known results that in uilibrium in free space the ellation length is in order of the wavelength of the black body radiation. Besides, other characteristics of the black-body field are independent on optical properties of surrounding bodies. z
In its turn in the out of uilibrium case at kh we n have from Eq.() that W (, H) f (, H), where f ( H, ) is given by Eq.(). The spatial elations become the fruency dependent characteristics. For numerical calulations we have chosen the following parameters of matter: the plasma fruency P cm and damping ν. P for the Drude model, the fruencies of transversal and longitudinal phonons T 793cm, 969cm, the damping γ 4.76 cm and ε 6.7 for the oscillatory model esponding SiC. Figure exemplifies the normalized elation length H, / λ in units of the wavelength λ of thermal fields in free space versus a fruency in case when a heated halfspace radiates into cold surroundings. Fig.a)- for metal, Fig.b) for SiC. Upper curves in both figures espond to the elation length along of the H direction, and low curves along of the direction. 3.. The elation length nearby surface sample in uilibrium and out of uilibrium cases operties of fluctuating electromagnetic fields are especially important in the nearest vicinity of interfaces both in uilibrium and out of uilibrium problems. A possible way to descibe the fields just at interfaces was proposed in [7]. As we have already mentioned the spectral and elation properties of thermal fields can be computed based on the classical electrodynamics and using the fluctuation dissipation theorem [5-]. This approach is valid at distances from interfaces which are much larger than the interatomic distances, at least. Different contributions in traces appear additively and independently of each other in Eqs.() and (4). Figure repersents normalized values g ( ) g (,)/ π u - (a), g ( ) g (,) / π u -(b)and g ( ) g (,)/ π u - (c) of different contributions to the total trace as the functions of normalazed lateral separations / λ in accordance with Eqs.(4)-(6). The functions g ( ) and g ( ) are computed at the fixed distance h / λ.3. H, Fig.. Normalized elation length / λ of electrical part of thermal fields in free space versus a fruency in case when a heated half-space radiates into cold surroundings. Fig.a)- for metal, Fig.b) for SiC. It is clearly seen from the figure that the elation lengths are increased at the eigen fruencies of the systems. In case of a metal the elation length peaks at the plasma fruency P cm, and in case of SiC at the fruencies of the transversal T 793cm and longitudinal 969cm optical phonons. It should be emphasized that the elation length of thermal fields within a nonuilibrium system is larger than the elation length within a system in uilibrium. Fig.. Normalized values g ( ) - (a), g ( ) - (b) and g ( ) - (c) of different contributions to the total trace as the functions of normalized lateral separations / λ in accordance with Eqs.(4)- (6).
It is clear that in nature only the sum of these contribution is relevant at any distance from a sample surface. We note here that in case of spectral power density it is reasonably valid to neglect by the propagating or evanescent contributions in the near- or far-field zones. But, it should be careful with neglections in studying of elation functions at different distances from an interface. The cross-spectral tensors have alternating signs esponding to the elations or antielations. The first term in Eq.(4) is the distance independent term relevant to all distances including the near-field zone. Moreover, the second term in Eq.(4) esponding to Fig.b) starts with antielations. This term is comparable with other two at distances h / λ >.and effectively compensates them due to the antielations. That is why we must taking into account all contributions to descibe elations ectly. In it s turn, Fig.3 demonstrates normalised values f ( ) f (,)/ π u - (a) and f ( ) f (,)/ π u - (b) in case of the system out of uilibrium with use of Eqs.()-(3). The function f ( ) is computed at the fixed distance h / λ.3. Fig.4. Total normalized traces W ( ) and n W ( ) versus for uilibrium-(a), (b) and out of uilibrium- (c), (d) problems calculated with use of Eqs.(4) and (), espondingly in the case of metal sample. Fig.3. Normalised values f ( ) - (a) and f ( ) - (b) in case of the system out of uilibrium as the functions of normalized lateral separations / λ in accordance with Eqs.()-(3). It can be seen from comparing of figures and 3 that the total traces in uilibrium and out of uilibrium problems are different despite of identical evanescent terms. These total normalized traces W ( ) W ( )/ W () and W n ( ) W n ( )/ W n () versus / λ are demonstrated in Fig.4 for uilibrium-(a), (b) and out of uilibrium- (c), (d) problems with use of Eqs.(4) and (), espondingly in the case of metal sample. Herewith, the distance dependent terms were computed at h / λ.3. Figures 4a) and 4c) are calculated at the fruency 4cm relating to the spectral range of surface plasmon excitations ( Re{ ε( )} < ). Figures 4b) and 4d) are obtained at the fruency cm which is out of the spectral range of surface excitations. These examples show evidently that elation functions are different in uilibrium and out of uilibrium problems. Corresponding elation lengths are anticipated to be different due to obvious difference in the elation functions. For the system in uilibrium using Eqs.(4) and (8) the relative elation length along the direction can be formally represented as follows
d g ( ) + g ( ) + g ( ). () g () + g () + g () For the system out of uilibrium we use Eqs.() and (8) to obtain the elation length in this case d f ( ) + f ( ). () f () + f () following expressions in order to take into account the structural independence of each term from all other terms d g ( ) g ( ) g ( ) + +, () g () + g () + g () d f ( ) f ( ) +. (3) f () + f () Figure 5 exemplifies normalized elation lengths / λ versus h / λ in uilibrium a) and in out of uilibrium b) cases calculated with use of Eqs.() and (). In both figures thick curves espond to the fruency 4cm and thin curves- to the fruency cm. Figure 6 shows normalized elation lengths / λ versus h / λ in uilibrium a) and in out of uilibrium b) cases calculated with use of Eqs.() and (3). In both figures thick curves espond to the fruency 4cm and thin curves- to the fruencycm. It is seen that these curves are better matched to elation functions from Fig.5. h λ in Fig.5. Normalized elation lengths / λ versus / uilibrium a) and in out of uilibrium system b) cases calculated with use of Eqs.() and (). Obviously, the juxtaposition of figures, for instance 5b) with 4c), 4d) does not result in a conclusion on good espondence between them, especially at the fruency of surface excitations. That is why we make an attempt to modify formulas such as in Eqs. () and () in order to take an increase of elations, at least in Fig.4c). The expressions in Eqs.() and (4) for spectral traces mean that esponding contributions for propagating and evanescent parts are independent. But, the elation lengths as exprressed by Eq.() and () allow for their cross-product terms. Hence, we suggest omitting these cross-product terms in Eq.() and () and using the Fig.6. Normalized elation lengths / λ versus h / λ in uilibrium a) and in out of uilibrium system b) cases in accordance with Eqs.() and (3). We would like especially to emphasize that the known increase of elations of thermal fields at fruencies of surface polaritons, see figures 4a) and 4c), is more aduately described by the modified Eqs.() and (3) as it is exemplified in figure 6. 4. Conclusions Spatial coherence of thermally stimulated fields in farand near-field zones generated by heated half-space radiating into vacuum is studied in uilibrium and out of uilibrium conditions. It is demonstrated that elation lengths of fields are different at different thermodynamical conditions. Correlation lengths of thermal fields in vacuum
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