Wave Propagation 2010 Part 8 pdf

Wave Propagation 202 At the boundaries of the considering regions (at θ πγ = − and 2 θ π = ) the mentioned above boundary conditions for tangential and normal components of electric field must be satisfied. Substituting into the boundary conditions the expressions for the field components and taking into account that ( ) cos cos π γγ −−= we have the following four equations: ( ) ( ) ( ) ( ) cos cos cos 0PAP CPD αα α γπγγ − −− =, ( ) ( ) ( ) ( ) cos cos cos 0 md d PAP CPD αα α εγε πγεγ ′′ ′ + −− =, 0BCD − −=, 0 pdd BCD ε εε − +−=, where ( ) ( ) ( ) () cos cosPdPd αα μ πγ πγ μ μ = − ′ −= . A nontrivial solution of the system exists when the determinant is equal to zero, () () ( ) () () () () () () () cos 0 cos cos cos 0 cos cos 0 0111 011 md pd PPP PPP ααα ααα γπγγ εε γ πγ γ εε −−− ′′′ −− = −− −− . By expansion the determinant we have () () () ( ) () ( ) () ( ) () { } () () () () () () () () {} cos cos cos cos cos cos cos cos cos cos 0. md pd pd PP P P P PP P P P αα α α α αα α α α εε γ πγ γ εε πγ γ γπγ γεε πγ γ ⎡⎤ ′ − ++ −− + ⎣⎦ ⎡⎤ ′′ ′′ +−−+−+= ⎣⎦ (2) Numerical calculations of the functions in (2) were carried out with the aid of the hypergeometric function. Taking into account the identity [Olver (1974)] ()( ) ,,, 1, 1, 1,Fabcz abFa b c z z ∂ =+++ ∂ , we find () () () cos sin 1 cos cos sin 1 1, 2 , 2 , 22 PdPd F αα μθ θ θ θθμμ αα αα θ = ∂− ⎛⎞ =− =− + − + + ⎜⎟ ∂ ⎝⎠ , () () () cos sin 1 cos cos sin 1 1, 2 ,2 , 22 PdPd F αα μθ θ θ θθμμ αα αα θ =− ∂+ ⎛⎞ −= =+ −++ ⎜⎟ ∂ ⎝⎠ , and therefore () ( ) cos 1 1cos 1, 2,2, 22 dP d F α μθ αα θ μμ α α = + − ⎛⎞ =−++ ⎜⎟ ⎝⎠ , Nanofocusing of Surface Plasmons at the Apex of Metallic Tips and at the Sharp Metallic Wedges. Importance of Electric Field Singularity 203 () ( ) cos 1 1cos 1, 2,2 , 22 dP d F α μθ αα θ μμ α α =− + + ⎛⎞ =−++ ⎜⎟ ⎝⎠ . By substituting these expressions into (2) we have () () 1cos 1, 2,2 , 2 1cos 1cos 1 , 1,1, 1 , 1,1, 22 1cos ,1,1, 2 1cos 11,2,2, 1 2 m d pp dd pp dd F FF F FF ε γ αα ε εε πγ γ αα αα εε γ αα εε πγ αα εε − ⎛⎞ −+ + × ⎜⎟ ⎝⎠ ⎧⎫ ⎛⎞ ⎛⎞ ⎛−−⎞ − ⎪⎪ ⎛⎞ +−+ +−−+ + ⎜⎟ ⎜⎟ ⎨⎜ ⎟ ⎬ ⎜⎟ ⎜⎟ ⎜⎟ ⎝⎠ ⎪⎪ ⎝⎠ ⎝⎠ ⎝⎠ ⎩⎭ − ⎛⎞ −+ × ⎜⎟ ⎝⎠ ⎛⎞ ⎛⎞ ⎛−−⎞ +−++ −−− ⎜⎟ ⎜⎟ ⎜⎟ ⎜⎟ ⎜⎟ ⎝⎠ ⎝⎠ ⎝⎠ 1cos 1, 2,2 , 0. 2 γ αα ⎧ ⎫ − ⎪ ⎪ ⎛⎞ + += ⎨ ⎬ ⎜⎟ ⎝⎠ ⎪ ⎪ ⎩ ⎭ If the particular case when 1 pd ε ε = is considered, this equation is transformed into ( ) ( ) ( ) ( ) cos cos cos cos 0 dm PP P P αα α α εγ γε γγ ′′ − +− =. Since () () 1 cos sin cos d PP d αα γ γγ γ − ′ =− and () () 1 cos sin cos d PP d αα γ γγ γ − ′ −= − we may write () () () () () cos cos cos cos 0 dm dd PP P P dd αα α α εγ γε πγ γ γγ − −− =. (3) Eq.(3) is identical to the corresponding Eq.(1) for the geometry without dielectric plate. In this case the cone tip with dielectric constant m ε is immersed into the uniform dielectric with constant d ε . This problem has been solved for example in [Petrin (2007)]. The minimal root α of Eq.(2) (which corresponds to the physically correct solution) defines the character of electric field singularity in the vicinity of the cone apex. From Eq.(2) it follows that α is a function of three independent variables: the angle γ and the ratios of dielectric constants md ε ε and p d ε ε , i.e. ( ) ,, md p d α αγε ε ε ε = . As it was shown below ( ) ,, md p d α αγε ε ε ε = is a complex function even for real arguments (it is important). Let use again Drude’s model for permittivity of metal without absorption 22 1 mp ε ωω =− , where p ω is the plasma frequency of the metal. Therefore, for fixed values of γ , d ε and p ε , we may find the dependence ( ) p α ωω . Taking into account that d r α Ψ ∼ , we have 1 ex Er α − ∼ . Note, that for fixed values of γ , d ε , p ε and p ω ω Eq.(1) has many roots i α but not all of the roots have physical sense or represent the singular electric field at the cone apex. Obviously, that only roots smaller than unit ( 1 ex Er α − ∼ ) give the singular electric field. So, we will be interested by the solutions of Eq.(2) in the interval ( ) Re 1 α < . To define the lower boundary of the solution’s interval it is necessary to remind that in the vicinity of the apex the electric field density must be integrable. It means that the electric field and density must increase slower than 32 r − and 3 r − respectively when 0r → . So, the lower boundary of the roots interval is equal to 12 and the total roots interval of interest is ( ) 12 Re 1 α − << . Wave Propagation 204 Eq.(2) was solved numerically. For 15 γ = ° , 1 d ε = and 1 p ε = the results of calculations are the same as in Fig. 2 obtained from Eq.(1). The plots for other values of dielectric constant of the plate p ε are analogous to the plot of Fig. 2. Using the same approach as in the case of Fig. 3 it were calculated numerically (see Fig. 6) the dependences of the critical frequency cr ω (normalized on the plasma frequency of the metal) on the cone angle γ for 1 d ε = and several values of p ε . These dependences may be found analytically. The critical frequency cr ω corresponds to the root value 12 α =− . Substituting 12 α = − and 22 1 m p cr ε ωω =− into Eq.(2) we find the following expression which is valid for any values of d ε and p ε : 1cos 12,12,1, 2 1 1cos 32,32,2, 2 1 cos 1 cos 1 32,32,2, 1 32,32,2, 22 1cos 1cos 1 12,12,1, 1 12,12,1, 22 d cr pp p dd pp dd F F FF FF γ ε γ ω εε γγ ω εε εε γγ εε ⎧ − ⎛⎞ ⎜⎟ ⎪ ⎝⎠ ⎪ +× − ⎛⎞ ⎪ ⎜⎟ ⎪ ⎝⎠ = ⎨ ⎛⎞ ⎛⎞ +− ⎛⎞⎛⎞ +−− ⎜⎟ ⎜⎟ ⎜⎟⎜⎟ ⎝⎠⎝⎠ ⎝⎠ ⎝⎠ × ⎛⎞ ⎛⎞ +− ⎛⎞⎛⎞ ++− ⎜⎟ ⎜⎟ ⎜⎟⎜⎟ ⎝⎠⎝⎠ ⎝⎠ ⎝⎠ 1 2 − ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ Fig. 6. Normalized critical frequency cr p ω ω as a function of γ for 1 d ε = and several values of p ε . Curve 1 for 1 p ε = , 2 – 1.5 p ε = , 3 – 2.25 p ε = . It is shown the asymptotic of the curve when 90 γ →°. It was found that when 90 γ →° (the metal cone turns into metal plane and the free space between the cone and the dielectric plate disappears) the curves of the critical frequencies tends to the value 1 pp ω ε + – the utmost frequencies of SPP’s existence on the boundary metal-dielectric plate [Stern (1960)]. As in the previous part of the chapter we see that it is Nanofocusing of Surface Plasmons at the Apex of Metallic Tips and at the Sharp Metallic Wedges. Importance of Electric Field Singularity 205 absolutely unexpected that the utmost maximal frequency of SPP’s existence arises in the quastatic statement of the singularity existence problem. 3.2 Application of the theory to a silver tip So, as in the section 2.2, we see that if the working frequency is fixed, then there are two different types of singularity. In this case there is a critical angle cr γ which separates the regime with the first type of singularity from the regime with the second type of singularity. For 1 d ε = and several values of p ε the dependences ( ) 0cr cr γ γλ = of the critical angle on the wavelength of light in vacuum of the focused SPPs with frequency ω may be found from Fig. 6 by a recalculation as it was made (section 2.2) for microtip immersed into uniform medium. The plots ( ) 0cr cr γ γλ = for silver, 1 d ε = and for three values 1 p ε = , 1.5 p ε = and 2.25 p ε = are shown in Fig. 7. Calculating the plots ( ) 0cr cr γ γλ = we neglect by losses in silver. If the angle of the cone γ is more than cr γ , then the singularity at the apex is of the first type. If the angle γ is smaller than cr γ , then the singularity is of the second type. Fig. 7. Critical angle cr γ of silver cone as a function of wavelength in vacuum 0 λ of exciting laser for 1 d ε = and several values of p ε . Curve 1 for 1 p ε = , 2 – 1.5 p ε = , 3 – 2.25 p ε = . The left boundaries of the plots ( ) 21 s ppp cr c λ πεω =+ are the limits of the spectrums of SPP on plane surface. If (as in section 2.2) the wavelength of the laser 0 λ , the dielectric constant of the working medium d ε and the dielectric constant of the dielectric plate p ε are given, then the cone angle γ at the apex of focusing SPPs microtip defines the type of singularity. From Fig. 7 it may be seen that the more p ε the more cr γ under the other things being equal. As in the previous section of the chapter, consider the setup of the work [De Angelis (2010)] on local Raman’s microscopy. The waveength of the laser excited the focused SPPs is equal Wave Propagation 206 to 0 532 λ = nm. The SPPs travel along the surface of the microtip cone and focus on its apex. From Fig. 7 it may be seen that the critical angle for 1 d ε = and 1 p ε = is equal to 24.7 cr γ ≈° (as in the previous section). If the dielectric constant p ε is equal to 1.5 or 2.25 then the critical angles are 26.7 cr γ ≈ ° and 28.9 cr γ ≈ ° , respectively. 4. Nanofocusing of surface plasmons at the edge of metallic wedge. Conditions for electric field singularity existence at the edge immersed into a uniform dielectric medium. In this part of the chapter we focus our attention on finding the condition for electric field singularity of focused SPP electric field at the edge of a metal wedge immersed into a uniform dielectric medium. SPP nanofocusing at the apex of microtip (considered in the previous sections) corresponds (based on the analogy with conventional optics) to the focusing by spherical lens. Thus, SPP nanofocusing at the edge of microwedge corresponds to the focusing by cylindrical lens at the edge [Gramotnev (2007)]. The main advantage of the wedge SPP waveguide in nanoscale is the localization of plasmon wave energy in substantially smaller volume [Moreno (2008)] due to the electric field singularity at the edge of the microwedge. This advantage is fundamentally important for miniaturization of optical computing devices which have principally greater data processing rates in comparison with today state of the art electronic components [Ogawa (2008), Bozhevolnyi (2006)]. As it will be shown below the electric field singularity at the edge of the microwedge may be of two types due to frequency dependence of dielectric constant of metal in optical frequency range. This phenomenon is analogues to the same phenomenon for microtips which was considered in the previous parts of this chapter. The investigation of these types of electric field singularities at the edge of metal microwedge is the goal of this chapter section. 4.1 Condition for electric field singularity at the edge of metallic wedge Let consider the metal microwedge (see Fig. 8) with dielectric constant of the metal m ε . The frequency of the SPP is ω . The wedge is immersed into a medium with dielectric costant d ε . Fig. 8. Geometry of the wedge. Nanofocusing of Surface Plasmons at the Apex of Metallic Tips and at the Sharp Metallic Wedges. Importance of Electric Field Singularity 207 Let calculate the electric field distribution near the edge. In cylindrical system of coordinates with origin O and angle θ (see Fig. 8), a symmetric quasistatic potential ϕ obeys Laplace’s equation. The two independent solutions of the Laplace’s equation are the functions ( ) sinr α α θ and ( ) cosr α α θ [Landau, Lifshitz (1982)] where α is a constant; θ is the angle from the axis OX; r is the radial coordinate from the origin O. Taking into account that the electric potential in the metal and dielectric depends on r as the same power we may write the following expressions for potential in metal and dielectric respectively cos( ) m Sr α ϕ αθ = , where 22 ψ θψ − ≤≤ , ( ) cos( ) d Sr α ϕ αθ π =−, where ( ) 222 ψθπψ ≤≤ − , where m S and d S are constants, ψ is the total angle of the metallic wedge. The boundary conditions for tangential and normal components of electric field may be written as ,,md EE τ τ = and ,,mmn ddn EE ε ε = . Using the above expressions for electric potential in the two media the boundary conditions may be rewritten in the following form () 1 , cos mm ErSθ r α τ ϕ α α − ∂ =− =− ∂ , () () 1 , cos dd ErS r α τ ϕ α αθ π − ∂ =− =− − ∂ , () 1 , 1 sin mn m ErSθ r θ α ϕ α α − ∂ =− = ∂ , () () 1 , 1 sin dn d ErS r θ α ϕ α αθ π − ∂ =− = − ∂ . At the first boundary of the wedge (where 2 θψ = ) the boundary conditions give two equations () ( ) ( ) 11 2 cos cos md rS θ rS αα θψ ααααθπ −− = −=− −, () ( ) ( ) 11 2 sin sin mm dd θ rS θ rS αα ψ εα α εα αθ π −− = =−, or ( ) ( ) ( ) cos 2 cos 2 md SS α ψαψπ =−, ( ) ( ) ( ) sin 2 sin 2 mm dd SS ε αψ ε α ψ π =−. Wave Propagation 208 Note, that at the second boundary of the wedge (where 2 θ ψ = − ) the boundary conditions give absolutely identical equations due to symmetry of the problem. A nontrivial solution of the system exists when the determinant is equal to zero, ( ) ( ) ( ) ( ) ( ) ( ) cos 2 sin 2 sin 2 cos 2 0 md ε α ψ π αψ ε α ψ π αψ − −− = . (4) From this equation it follows that the index of singularity α is a function of two variables: angle ψ and the ratio md ε ε , i.e. ( ) min , md α αψεε = . Note, that in electrostatic field m ε →∞ and, therefore, in the limit we have ( ) ( ) ( ) cos 2 sin 2 0 αψ π αψ − = . The minimal root of this equation will be when ( ) ( ) cos 2 0 αψ π − = (we are interested in the interval ψ π < ). Therefore, the minimal value of α is defined by equation ( ) 22 α ψπ π −=− or ( ) 2 α ππψ =− (it is well-known result [Landau, Lifshitz (1982)]). When 0 ψ → , we have 12 α → . Using Drude’s model for permittivity of metal without absorption 22 1 mp ε ωω =− and considering that the wedge is immersed into vacuum ( 1 d ε = ) we have 22 1 md p ε εωω =− . Therefore, for fixed value of we may find the dependence ( ) min p α αωω = . Taking into account that r α ϕ ∼ , we have 1 ex Er α − ∼ . Note, that from the physical sense of the electric potential ϕ it follows that allways ( ) Re 0 α ≥ . Therefore, the interval α of singularity existence is ( ) 0Re 1 α ≤<. Fig. 9 shows the dependences ( ) min p α αωω = obtained from Eq.(4) for the wedge angle 30 ψ =°. We can see that as in the case of cone tip in the case of wedge there are two types of electric field singularity at the edge of metallic wedges. The first type of electric field singularity takes place when cr ω ω < . Here, the index α has a pure real value. The second type of electric field singularity takes place when cr ω ω > and the index α has a pure image value. Fig. 9. Real (curve 1) and image (curve 2) part of the index α as a function of normalized frequency p ω ω for the wedge angle 30 ψ = ° . Nanofocusing of Surface Plasmons at the Apex of Metallic Tips and at the Sharp Metallic Wedges. Importance of Electric Field Singularity 209 Fig. 10 shows the plots of ( ) Re α and ( ) Im α as functions of p ω ω (the same functions depicted in Fig.9) in the vicinity of the critical frequency cr ω without losses. For comparison in Fig. 10 the plots of ( ) Re α and ( ) Im α as functions of p ω ω for silver (the metal with losses) are shown. Fig. 10. The vicinity of the critical frequency cr ω . Real (curve 1) and image (curve 2) parts of the index α as a function of normalized frequency p ω ω for the wedge angle 30 ψ =° and no losses in the metal. For comparison, the analogous curves 3 and 4 for silver wedge (metal with losses). From the dependences like of Fig. 9 it was numerically found cr ω (normalized on the plasma frequency p ω ) as a function of the wedge angle ψ (see Fig. 11). The obtained function is excellently approximated by the elementary function [] 0.05255 de g . cr p ωω ψ ≈ . This is not a coincidence. Indeed, the condition cr ω ω = implies that 0 α = and therefore from Eq.(4) it follows () ( ) ( ) ( ) () () () 0 sin 2 cos 2 2 lim 1 cos 2 sin 2 m cr α αψ π αψ π ε ψ αψ π αψ → − ==− − . Thus, 2 cr p ω ωψπ = or [] [] de g . /360 0.0527046 de g . cr p ωω ψ ψ =≈ exactly (analytical solution)! We can see that the accuracy of the numerical results is substantially high. Fig. 11 also shows the frequency 2 p ω , above which SPP can not exist. Note, that when 180 ψ →° (the wedge turns into a plane) the critical frequency tends to 2 p ω – the utmost frequency of SPP existence on plane surface. Again it is absolutely unexpectable that the utmost frequency of SPP existence arises in the quasistatic problem of singularity existence. Wave Propagation 210 Fig. 11. Normalized critical frequency of singularity existence cr p ω ω as a function of the wedge angle ψ . Solid line is the approximating function [] 0.05255 deg. cr p ωω ψ ≈ . 4.2 Results of calculation for a silver microwegde From the results of the previous section 4.1 we see that the problem of finding of the SPP nanofocusing properties of microwedge is the following. On the one hand the wavelength of SPP must be possibly smaller. So, the SPP frequency must be close (but smaller) to the critical frequency of SPP existence 2 p ω . On the other hand it is necessary to use the effect of additional increasing of the SPP electric field at the edge of the microwedge due to electric field singularity at the edge. As we have seen the electric field singularity at the edge exists for any frequency of SPP, but there two different types of electric field singularity. The choice of the singularity depends on the particular technical problem (in this work this problems do not discuss). Consider the following problem. Let there is a microwedge on the edge of which SPPs with frequency ω are focused (the wavelength in vacuum of a laser exciting the SPP is equal to 0 λ ). What is the value of wedge angle cr ψ which separates regimes of nanofocusing with different types of singularities? Consider a microwedge made from silver (plasma frequency of silver is equal to 16 1.36 10 p ω =× 1 s − [Fox (2003)] and no losses). Based on the dependences of Fig. 11 it is elementary to find the function ( ) 0cr cr ψ ψλ = for silver (see Fig. 12) which is the solution of the considering problem. Indeed, 2 cr p ω ωψπ = ⇒ 0 22 pcr c π λωψ π = ⇒ ( ) 32 2 2 0 8 cr p c ψ πωλ = or for the angle in degrees [] ( ) 22 2 2 0 deg. 1440 cr p c ψ πωλ = . Thus, we have obtained for this problem the exact analytical solution [...]... correlation approximation employed earlier (73) 18 230 Electromagnetic Waves Wave Propagation The above is an equation for the second moment of the wave function E, which can be ¯ ˜ decomposed into a coherent part E and a diffuse part E Therefore, ˜ ˜ E ⊗ E∗ = E ⊗ E∗ + E ⊗ E∗ (74) The coherent part is indeed known for our problem Our primary interest is in the diffuse part Therefore, we write (73) in terms... H., et al (1997) Phys Rev Lett Vol 78, p 1667 Mehtani D., Lee N., Hartschuh R D., et al (2006) J Opt A Vol 8, p S 183 Moreno E., Rodrigo S G., Bozhevolnyi S I., et al (20 08) Phys Rev Lett Vol 100, p 023901 Landau L D., Lifshitz E M (1 982 ), Electrodynamics of continuos media, Nauka, Moscow Naber A., Molenda D., Fischer U C., et al (2002) Phys Rev Lett Vol 89 , p 21 080 1 Nerkararyan K V., Abrahamyan T.,... function On the other hand, the RT 8 220 Electromagnetic Waves Wave Propagation equation, as we saw earlier, is an equation for the specific intensity, which is a ‘phase-space’ quantity Wigner transforms serve as a bridge to link these two quantities (Yoshimori, 19 98; Friberg, 1 986 ; Marchand & Wolf, 1974; Pederson & Stamnes, 2000) We introduce the Wigner transforms of waves and Green’s functions as r+r... Huffman D R (1 983 ), Absorption and Scattering of Light by Small Particles, John Wiley & Sons, New York Bouhelier A., Renger J., Beversluis M R., Novotny L (2003) J Microsc Vol 210, p 220 212 Wave Propagation Bozhevolnyi S I., Volkov V S., Devaux E., et al (2006) Nature, Vol 440, p 5 08 De Angelis F., Das G., Candeloro P., et al (2010) Nature Nanotech Vol 5, p 67 Economou E N (1969) Phys Rev Vol 182 , p.539... ) = Iδjk δ(r − r ), (58a) j ˇ ˇ ¯ ¯ n j × G jk (r⊥ , ζ j ; r ) = n j × G( j+1) k (r⊥ , ζ j ; r ), ˆ ˆ (58b ) ˇ ˇ ¯ ¯ n j × ∇ × G jk (r⊥ , ζ j ; r ) = n j × ∇ × G( j+1) k(r⊥ , ζ j ; r ) ˆ ˆ (58c) Another pair of equations similar to (58b) and (58c) corresponding to the ( j − 1)-th interface must be added to this list Using these Green’s functions and the radiation conditions the wave functions can be... Novotny L (2006) J Opt A Vol 8, p S227 Angot A (1962), Complements de mathematiques a l’usage des ingenieurs de l’electrotechnique et des telecommunications, La Revue d’Optique, Paris, 2nd ed Babadjanyan A J., Margaryan N L., Nerkararyan K V (2000) J Appl Phys Vol 87 , p 8 Barnes W L (2006) J Opt A: Pure Appl Opt Vol 8, p S87 Bezus E A., Doskolovich L L., Kazanskiy N L et al (2010) J Opt Vol 12, p 015001... Guckenberger R (2002) Appl Phys Lett Vol 81 , p 5030 Gay G., Alloschery O., Weiner J., et al (2007) Phys Rev E, Vol 75, p 016612 Gramotnev D K (2005) J Appl Phys Vol 98, p 104302 Gramotnev D K., Vernon K C (2007) Appl Phys B Vol 86 , p 7 Hillenbrand R., Taubner T., Kellmann F (2002) Nature Vol 4 18, p 159 Ichimura T., Hayazawa N., Hashimoto M., et al (2004) Phys Rev Lett Vol 92, p 22 080 1 Kawata S (2001), Near-Field... μ (k ⊥ ) + q ∗ (k ⊥ ) − a q μ (α⊥ ) + q ∗ (α⊥ ) (μ · μ )(ν · ν )Eμ ν (z, α⊥ ), ν ν 2 2 (35b ) 10 222 Electromagnetic Waves Wave Propagation ¯a where Eμν represents scattering due to the coherent part of E , whereas the integral term in (35) ¯a represents scattering due to the diffuse part of E We may also regard Eμν as the source to our transport equations and calculate it to obtain 1 1 ¯a Eμν = |k... currently used, perhaps the most widely used approach is the radiative transfer (RT) approach (Ulaby et al., 1 986 ; Lam & Ishimaru, 1993; Karam & Fung, 1 982 ; Shin & Kong, 1 989 ; Caron et al., 2004; Ulaby et al., 1990; Liang et al., 2005; Fung & Chen, 1 981 ) Here, one formulates the scattering and propagation in each layer by using the radiative transfer equation, which involves only the parameters of the... Furthermore, there are well-established techniques for numerical analysis of the radiative transfer equations (Clough et al., 2005; Stamnes et al., 1 988 ; Berk et al., 19 98; Lenoble, 1 985 ) However,the RTT is heuristic and lacks the rigour of the statistical wave theory The fundamental quantity in the RTT is the specific intensity, which is a measure of energy flux density per unit area, per unit steradian . V. (2000) J. Appl. Phys. Vol. 87 , p. 8. Barnes W. L. (2006) J. Opt. A: Pure Appl. Opt. Vol. 8, p. S87. Bezus E. A., Doskolovich L. L., Kazanskiy N. L. et al. (2010) J. Opt. Vol. 12, p. 015001 equations (Clough et al., 2005; Stamnes et al., 1 988 ; Berk et al., 19 98; Lenoble, 1 985 ). However,the RTT is heuristic and lacks the rigour of the statistical wave theory. The fundamental quantity in the. Phys. Rev. Lett. Vol. 78, p. 1667. Mehtani D., Lee N., Hartschuh R. D., et al. (2006) J. Opt. A Vol. 8, p. S 183 . Moreno E., Rodrigo S. G., Bozhevolnyi S. I., et al. (20 08) Phys. Rev. Lett. Vol.