By use of the Kubelka-Munk theory, the Mie theory and the independent scattering approximation, we obtain the explicit expression of the emittance of an infrared coating attached to a radar absorber with a high emittance, in the 3~5μm window. Taking aluminum particles with spherical shape as the pigments within the coating, we give the dependence of the coating emittance with respect to the particle radius, the thickness of the coating. At a volume fraction of 0.05, we propose the optimum particle radius range of the pigment particles is around 0.35~0.6μm. When the thickness of the coating exceeds 300μm, the decrease of emittance at 4μm wavelength becomes negligible. Too much thickness of IR layer wouldn’t contribute to the decrease of emittance. We study the influence of the infrared coating on the performance of the radar absorber, and believe that not too much thick infrared coating consisting of spherical Al particles wouldn’t result in a remarkable deterioration of the absorbing ability of the radar absorber.
© 2005 Optical Society of America
Radar absorbing materials (RAM) are widely used in defence technology. In order to decrease the radar scattering cross section of an aircraft, researchers always coat a RAM layer on the metal surface of the aircraft. But in general, the RAM layer is highly emissive. The high emittance of the exterior surface of the aircraft caused by the RAM greatly increases the probability of being detected by an infrared detector. Infrared guided weapons have been the main threats for aircrafts. For an object with a high temperature, which is the order of hundreds of degrees centigrade, such as the exterior surface of the tailpipe of an aircraft, the peak radiation is distributed in 3~5μm band. In order to decrease the high emittance from the hot RAM layer, it is possible to add an additional thin layer with low infrared emittance to the RAM layer. In a previous paper, Liu et al. discussed  the emissive properties of a highly emissive substrate coated with a monolayer in the 8~14μm window. Here we extend the study to a double-layer structure. We study this composite structure, discuss its emissive properties, and the influence of the infrared layer on the performance of a radar absorber.
Consider a typical coated exterior structure of an aircraft. It can be simulated by a metal substrate coated with a RAM Layer and an infrared layer, as illustrated in Fig. 1. We think that the width of the plate is much larger than the thickness of the coatings, so in most case this structure can be considered as an infinite plane plate coated with two thin layers. In general case the thermal radiation can be approximately viewed as isotropic , so the radiative transfer within the IR layer can be described by the Kubelka-Munk theory(KMT)[3,4]. The explicit expression of the emittance of a coated substrate is given by [1, 3, 4]:
where Rv and y are defined as
Re denotes the external diffuse reflectance at the front interface, and Ri is the internal diffuse reflectance at the same location, namely z = 0, and RRAM denotes the diffuse reflectance at the interface between IR layer and RAM layer when illuminated from the IR layer. Re is determined by the following expression [1, 5, 6],
where m1 is the effective refractive index of the IR layer in the 3~5μm waveband. When the characteristic size of the particle is comparable with the incident wavelength, the higher order terms of Mie [7, 8] series cannot be ignored. Niklasson et al.  conceived a random unit cell (RUC) being a coated sphere consisting of a core with dielectric permittivity εp surrounded by a shell with dielectric permittivity εh, with εp and εh being the dielectric permittivity of the particles and the host medium. The volume ratio of the core to the coated sphere is equal to the volume fraction of pigment particles f. When the RUC is embedded in an effective medium with a dielectric permittivity εeff, the exact forward scattering vanishes, i.e.,
where δMG is the revision term which can be figured out with Eq. (4). For small spherical particles, only the dipole term of Mie series is retained, Eq. (5) reduces to the original Maxwell-Garnett Theory (MGT)  as follows:
In the computation for m 1, we adopt the EMGT, which is general for all the radius range we consider. Although for the small particle such as radius=0.1μm, incident by 3~5μm, EMGT reduces to MGT, we still make use of EMGT for consistency, just as we take advantage of Mie theory to compute the optical efficiency in all the radius range, in stead of using Rayleigh model to calculate in the small particle radius range and using Mie theory in the rest big radius range.
2.2 Determination of K-M coefficients in terms of Mie theory
Although KMT has been implemented widely in paint and papermaking, the physical meaning of the coefficients K, S is not clear . The average pathlength parameter (APP) has to be introduced [11, 12] to describe the propagation of each diffuse flux as a whole. Maheu, Letoulouzan, and Gouesbet set up a four-flux model to describe the light propagation in a particulate medium. In their model the absorption coefficient and scattering coefficient are not phenomenological and can be expressed in terms of single-scattering property:
where ξ, n and Cabs denote APP, the number density of the pigment particles, and the absorption cross section of single particle respectively. σc is the forward scattering ratio for collimated radiation and Csca is the scattering cross section of single particle. Considering that the incident thermal radiation is nearly perfectly diffuse, we can think that the radiation in the coating is completely diffuse as well provided that sufficient scattering effect exists in the coating, although the internal illumination is not perfectly diffuse due to refraction. This is true for the most particle radius range that we shall discuss below. APP can be readily obtained equaling 2 in the case of perfectly diffuse radiation. For the small particles which act as a group of Rayleigh scatterers in the coating, the scattering is too weak to guarantee that the internal radiation is completely diffuse. However, we treat this case by use of perfectly diffuse radiation approximation (PDRA) as well, and will discuss the errors caused by this approximation in the following computation.
We assume that the pigment particles embeded inside the host are all spherical and in identical size (monodisperse system). According to Mie theory, the extinction and scattering efficiencies of a uniform sphere are written as [7, 8]
where x = 2πmha/λ is the size parameter of the particle (a is the radius of the particle, mh is the refractive index of the host medium of the IR layer, λ is the wavelength in free space).
The absorption coefficient K and the backscattering coefficient S are given by 
and the phase function with S 1(μ) and S 2(μ) being the scattering amplitudes .
2.3 Radar Reflectivity of substrate coated with double layer
The IR layer shall certainly influence the performance of RAM layer. We can make use of the method which treats with the double-layer RAM  to study the effect of IR layer on RAM layer. For RAM layer, the normalization input impedance is given by
where ε 2r, μ 2r are the relative dielectric permittivity and magnetic permeability of the RAM layer and are measurable quantities. And for the double layer, the normalization input impedance is as follows
where is the impedance of the IR layer, and can be estimated in terms of the effective medium theory. Considering the facts that the pigment particle size is much less than radar wavelength, and the volume fraction of pigment is much less than 1, and the host medium is continuous, we choose MGT to figure out the effective dielectric permittivity of the IR layer in the radar waveband. The intrinsic dielectric function of metallic particles εp in the radar waveband can be obtained by use of the Drude formula [14, 15] as will be discussed in detail in section 3. The effective magnetic permeability can be considered as unity since we choose the nonmagnetic materials as the pigment particles in general case.
So the radar reflectivity of the substrate coated with a RAM monolayer is given by
And similarly the radar reflectivity of the substrate coated with RAM and IR double-layer is
It should be pointed out that Ref.13 used the time-harmonic factor exp(iωt), whereas exp(-iωt) was adopted in Ref. 8 and our work. When we use Eq. (12)–(15) to compute the radar reflectivity, we have to adopt the corresponding conjugates of the complex dielectric function and magnetic permeability.
3. Computation and discussion
We assume for the sake of simplicity that the host media is not absorptive in 3~5μm band and the refractive index is equal to 1.514, which is representative of an acrylic resin . We choose spherical Al particles as the pigments and sample the wavelengths in the 3~5μm window. The complex refractive indexes of Al at room temperature are shown in Table 1. The optical constants of Al at hundreds of Celsius degree are evaluated by use of Drude equation. Considering that the dielectric permittivity of Al in IR band is mainly due to intraband transition of conduction electrons, we can evaluate the dielectric constants in 3~5μm window in terms of Drude equation as follows:
where ε 0 denotes the contribution to the dielectric function from the interband transition, with a typical value of 1.03. Ωp is the plasma frequency and τ is the intraband relaxation time. Mathewson gave the values  of Ωp and τ at 552K: ħΩp =133eV and τ = 0.52×10-14s. The refractive index of Al at 552K derived from Drude formula and the optical constants at room temperature coming from Ref.  are compared in Table 1. We assume that the temperature of the metal substrate and the pigments is 552K, and take the evaluated data in the following computations.
3.1 Optimum range of pigment particle radius
With an appropriate program  we can compute the values of Q ext, Q sca, Q abs, and (1-σc)Q sca for each sampled wavelength. The computation results are plotted in Fig. 2(a). We can readily figure out the emittance at different wavelength by the method mentioned above. As illustrated in Fig. 2(b), if we choose spherical Al particles as the pigments to decrease the emittance of a radar absorber, the optimum range of particle radius is about 0.2~0.6μm. For the particle with radius less than 0.2μm, Rayleigh scattering dominates and the scattering is weak. So the IR layer cannot curtain off enough infrared radiation. With the radius of particle reaching 0.2~0.6μm, resonant regime prevails gradually and the scattering and backscattering become markedly strong, and the coating curtains off most radiation in this radius range. When particle radius increases (>0.6μm), the backscattering efficiency of particle (1-σc)Q sca decreases gradually, and approaches a saturation value. Therefore the ability of coating layer to curtain off infrared radiation declines gradually. When the particle radius is larger than 3 μm, the layer almost cannot curtain off any infrared radiation since the scattering of large particle is peaked mainly in the forward hemisphere .
As mentioned above, the PDRA shall lose its validity if the radius of particle is much small. We assume that the illumination external to the coating is perfectly diffuse and unpolarized, so the solid angle distribution density of the specific intensity external is 1/4π, and the illumination internal is distributed within a cone with an apex angle equal to arcsin(1/m 1) and outside the cone the illumination vanishes. Within the cone, the angular distribution of the specific intensity is given  as
with T∥ and T ⊥ being the Fresnel transmission coefficients.
Taking the 4μm wavelength, which is in the middle of the 3~5μm window, for example, we study the validity of this approximation. By use of Eq. (4)–(5), we can obtain m 1= 1.63 + 0.05i when a=0.2μm With Eq. (17), we can see that the internal illumination is somewhat peaked in the forward direction as shown in Fig. 3(a). And the Forward APP at the interface can be obtained with the definition of APP [1, 11], which is 1.09. Analogously for a=0.6μm, the illumination get not so anisotropic as for a=0.2μm, and the initial APP=1.11. As indicated by Vargas and Niklasson , APP will approach rapidly a saturation value even for a collimated incidence provided that the scattering is sufficiently strong (usually get saturated when the optical depth>3~5). For a=0.6μm, which is in the resonant regime and with a higher initial APP, the condition is met, and APP can approach quickly the saturation value. We can ignore the z-dependence of APP and treat the radiation as an ideal completely diffuse. On the other hand, the validity of PDRA will be enhanced by increasing the optical depth. The thicker the layer, the higher ratio the depth in which PDRA is valid occupies. Fig. 3(b) shows the dependence of optical depth (τ = nCextz = 3fQextz/4a) of thickness on particle radius, which corresponds to the case in Fig. 2(b). In the big radius region of 0.2~0.6μm range, the optical depth >>3~5, so the invalidity of PFRA caused by the anisotropy of internal illumination can be ignored to some extent.
In the small particle radius region of the range 0.2~0.6μm, the PDRA might lose its validity during the initial path of the incident photons, and will cause an over evaluation on APP. The over evaluated APP will result in a lowered emittance compared with the practical value, so the decrease of emittance in the small radius region might be not as low as shown in Fig. 2(b). Considering the approximation conditions we adopted, it is more appropriate to redefine the optimum radius range as around 0.35~0.6μm.
3.2 Dependence of emittance on thickness of IR layer
Taking again the 4μm wavelength for example, we discuss the relation of the emittance to the thickness of the IR layer. As illustrated in Fig. 4, the emittance decreases rapidly with an increasing thickness, but approaches a saturation value when d 1>300μm. It can be explained easily from equation (1). When d 1 → ∞, and M so ε = 1 - R → const. So too much thickness of IR layer doesn’t contribute to the decrease of the emittance. There is similar saturation thickness for different wavelength in 3~5μm band. This is consistent with our experiments and is helpful to construct low emittance layer which wouldn’t harm obviously the performance of the radar absorber.
3.3 On the validity of independent scattering approximation
We adopt independent scattering approximation (ISA) in this work. ISA  means that the clearances among particles are so large that the interactions among particles can be ignored, and the exciting field of every particle is equal to the incident field. In fact, when the concentration of particles increases to some value, the interactions among particles cannot be ignored, and the dependent scattering regime comes into action. How to treat with dependent scattering effect is a formidable task since the interactions among the random particles have to be taken into account. Peoples used to take the independent/dependent scattering regime into account in terms of δ/d, the ratio of center to center particles spacing to diameter or c/λ, the ratio of particle clearance to wavelength. Hottel et al. found  experimentally that dependent scattering effects were better correlated by c/λ than δ/d. Furthermore, c/λ=0.3 or 1/3 was always considered as the demarcation of independent and dependent scattering. In rhombohedral arrangement of particles, the correlation of f to c/λ is given  by At the same volume fraction, the less the particle radius, the larger the probability that dependent scattering dominates. According to Hottel criterion and the computation result, for the incident wavelength being 3μm, 4μm and 5μm, when the volume fraction f =0.05, the lower limit of particle radius in which ISA is valid, are 0.206μm, 0.274μm and 0.343μm respectively. That is to say, when the volume fraction f=0.05 and particle radius is larger than these values, ISA is an appropriate approximation to treat with the scattering of pigment particles in coating. The optimum particle radius range we set previously exceeds the lower limit, so we believe that the optimum particle radius range is valid.
3.4 Influence of IR layer on performance of radar absorber
We take the influence of IR layer in 2~18GHz into account. First of all, we extrapolate the dielectric functions of Al in 2~18GHz band by means of Drude formula. The results are shown in Fig. 5(a) and (b). We figure out the effective dielectric permittivity of the IR layer in terms of MGT, which is shown in Fig. 5(c) and (d).
For the radar absorber with an IR layer and that without, we compute the radar reflectivity in 2~18GHz as shown in Fig. 6(c). Fig. 6(a) and (b) show the dielectric permittivity and magnetic permeability of the RAM layer which are measured from experiment. It is obvious that the decrease of performance of radar absorber is slight. The IR coating strengthens the absorbing ability of the radar absorber in the lower frequency, and deteriorates a little the performance at the higher frequency (beyond the absorption peak frequency).
If we select spherical aluminium particles as the pigments to construct the low emittance coating layer of a radar absorber in 3~5μm window, at a volume fraction 5% (within ISA), the optimum range of Al particle radius is around 0.35~0.6μm. In this optimum radius range, the emittance can decrease from 0.9 to below 0.5 with a sufficient thickness of infrared layer. When the thickness of the infrared layer exceeds 300μm, the emittance at 4μm wavelength will not decrease obviously. There is similar saturation thickness for different wavelength in 3~5μm band. Too much thickness of IR layer wouldn’t contribute to the decrease of the emittance. A little deterioration of the performance of radar absorber due to infrared layer with a saturation thickness takes place when the frequency exceeds the peak absorption frequency, whereas the absorbing ability strengthens in the low frequency before the peak frequency.
Many thanks to Dr. M. I. Mishchenko and Dr. W.E. Vargas for the monograph and dissertation generously presented by them respectively.
References and Links
1 . L. Liu , R. Gong , D. Huang , Y. Nie , and C. Liu , “ Calculation of emittance of a coating layer with the Kubelka-Munk theory and the Mie-scattering model ,” J. Opt. Soc. Am. A 22 , 2424 – 2429 ( 2005 ). [CrossRef]
2 . M. F. Modest , Radiative heat transfer , ( New York: McGraw-Hill , 1993 ).
3 . A. Ishimaru , Wave Propagation and Scattering in Random Media , ( Academic , 1978 ).
4 . P. S. Kubelka and F. Munk , “ Ein Beitrag zur Optik des Farbanstriche ,” Z. Tech. Phys. 12 , 593 – 601 ( 1931 ).
5 . J. C. Richmond , “ Relation of emittance to other optical properties ,” J. Res. National Bureau Standards-C 67C , 217 – 226 ( 1963 ).
6 . R. Molenaar , J. J. ten Bosch , and J. R. Zijp , “ Determination of Kubelka-Munk scattering and absorption coefficients by diffuse illumination ,” Appl. Opt. 38 , 2068 – 2077 ( 1999 ). [CrossRef]
7 . G. Mie , “ Beitrâge zur optik truber Medien, speziell kolloidaler Metallosungen ,” Ann. Phys. 25 , 377 – 452 ( 1908 ). [CrossRef]
8 . C. F. Bohren and D.F. Huffman , Absorption and Scattering of Light by Small Particles , ( New York: Willey , 1983 ).
10 . S. E. Orchard , “ Reflection and transmission of light by diffusing suspension ,” J. Opt. Soc. Am. 59 , 1584 – 1597 ( 1969 ). [CrossRef]
11 . B. Maheu , J.N. Letoulouzan , and G. Gouesbet , “ Four-flux models to solve the scattering transfer equation in terms of Mie parameters ,” Appl. Opt. 23 , 3353 – 3362 ( 1984 ). [CrossRef] [PubMed]
13 . K. Kim , W. Kim , and S. Hong , “ A study on the behavior of laminated electromagnetic wave absorber ,” IEEE T. Magn. 29 , 2134 – 2138 ( 1993 ). [CrossRef]
14 . E. D. Palik , Handbook of Optical Constants of Solids , ( New York: Academic , 1985 ).
15 . A.G. Mathewson and H.P. Myers , “ Optical absorption in aluminium and the effect of temperature ,” J. Phys. F: Metal Phys. 2 , 403 – 415 ( 1972 ). [CrossRef]
16 . E. S. Thiele and R. H. French , “ Computation of light scattering by anisotropic spheres of rutile titania ,” Adv. Mater. 10 , 1271 – 1276 ( 1998 ). [CrossRef]
17 . W. E. Vargas and G. A. Niklasson , “ Generalized method for evaluating scattering parameters used in radiative transfer models ,” J. Opt. Soc. Am. A 14 , 2243 – 2252 ( 1997 ). [CrossRef]
18 . M. I. Mishchenko et al. Scattering, Absorption, and Emission of Light by Small Particles , ( Cambridge , 2002 ) Chapter 3.
19 . H. C. Hottel , A. F. Sarofim , W. H. Dalzell , and I. A. Vasalos , “ Optical properties of coatings. Effect of pigment concentration ,” AIAA J. 9 , 1895 – 1898 ( 1971 ). [CrossRef]
20 . M. Q. Brewster and C. L. Tien , “ Radiative transfer in packed fluidized beds: dependent versus independent scattering ,” J. Heat Transfer 104 , 573 – 579 ( 1982 ). [CrossRef]