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Anomalous absorption of bulk electromagnetic waves by an ultra-thin layer
1 June 1998
Optics Communications 151 Ž1998. 374–383
Full length article
Anomalous absorption of bulk electromagnetic waves by an ultra-thin layer D.K. Gramotnev
)
Centre for Medical and Health Physics, School of Physical Sciences, Queensland UniÕersity of Technology, GPO Box 2434, Brisbane QLD 4001, Australia Received 3 December 1997; revised 5 February 1998; accepted 6 February 1998
Abstract A strong maximum of absorption Žas much as 100%. of bulk TE electromagnetic waves by an ultra-thin film with large imaginary dielectric permittivity between two semi-infinite media is shown to exist at an optimal film thickness. This optimal thickness is demonstrated to be usually much smaller than the wavelength and the wave penetration depth in the film. A 100% absorption of TE electromagnetic waves is shown to occur in the middle infrared range in an extremely thin film of an ionic crystal of several tens of nanometers, or just several nanometers. Physical reasons for this anomalous absorption are discussed. Frictional contact approximation, in which the layer can be considered as an immediate contact between two half spaces with special boundary conditions, is introduced and justified. Similarity of these boundary conditions with mechanical boundary conditions at a contact with liquid friction is investigated. The anomalous absorption of electromagnetic waves in thin non-uniform layers with the dielectric permittivity varying across the layer is analysed in the frictional contact approximation. Significant differences in the anomalous absorption of TE and TM waves are pointed out and investigated. q 1998 Elsevier Science B.V. All rights reserved. PACS: 78.65; 42.78H; 42.85 Keywords: Optics of thin films; Absorption; Transmission; Reflection; Ionic crystals
1. Introduction The anomalous absorption of acoustic waves by a thin layer of viscous fluid enclosed between elastic solid media was analysed theoretically in our previous papers w1–6x. It is characterised by a strong maximum of absorption Žas much as 100%. occurring at some optimal layer thickness which is usually much smaller than the wavelength and the wave penetration depth in the fluid w1–6x. At not very high frequencies the absorptivity maximum and the corresponding layer thickness were shown to be independent of frequency w1–5x. In this case a frictional contact approximation is valid w2,4,5x, and tangential stresses at the contact are proportional to relative velocity of the solid sur-
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faces w2–6x. This effect is very important for development of new acoustic viscosity sensors, measurement and diagnostics techniques, for investigation of interfaces and extremely small amounts of fluids. The wave equation for shear Žtransverse. waves in a viscous Newtonian fluid has the same form as the wave equation for electromagnetic waves in a medium with purely imaginary dielectric permittivity. This suggests that anomalous absorption must also take place for bulk electromagnetic waves in a thin layer Žfilm. with purely imaginary, or almost imaginary dielectric permittivity. Anomalous absorption of bulk TM electromagnetic waves in a uniform thin layer with large, purely imaginary dielectric permittivity was analysed recently in Ref. w7x. Similarly to acoustics, the anomalous absorption of TM optical waves was shown to display a strong Žas much as 100%. maximum of absorptivity at some optimal layer thickness. This layer thickness is usually much smaller
0030-4018r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved. PII S 0 0 3 0 - 4 0 1 8 Ž 9 8 . 0 0 0 8 8 - 1
D.K. GramotneÕr Optics Communications 151 (1998) 374–383
than the wavelength and the wave penetration depth in the material of the layer. Therefore the anomalous absorption cannot be related with the frustrated total internal reflection of the incident wave w8x, or multiple reflections inside the layer. Optical analogue of mechanical linear Žliquid. friction was analysed. At the same time, noticeable differences between the anomalous absorption in acoustics and optics were demonstrated. For example, unlike acoustics, the anomalous absorption in optics is a highly frequencyselective effect. So-called coefficient of friction for TM electromagnetic waves was shown to depend on frequency explicitly and implicitly Žthrough a dispersion of the dielectric permittivity of the layer.. The anomalous absorption of bulk TM optical waves was mentioned to be important for development of new high-sensitive optical sensors, signal-processing devices, measurement and evaluation techniques. The anomalous absorption must also take place for bulk TE electromagnetic waves in ultra-thin films and coatings. However, the analysis of this type of waves is noticeably different from the analysis of TM electromagnetic waves. There is no direct analogy between bulk TE electromagnetic waves and bulk shear acoustic waves, because of different boundary conditions. Therefore the anomalous absorption of bulk TE electromagnetic waves is a separate important problem which will result in strongly different dependences of reflectivity, transmissivity and absorptivity on parameters of the film and surrounding media, frequency, and angle of incidence. Frictional contact approximation for TE electromagnetic waves will also be represented by boundary conditions which are noticeably different from those for acoustic waves w1–6x and TM optical waves w7x. Therefore the aim of this work is the theoretical analysis of the anomalous absorption of bulk TE electromagnetic waves by a thin layer with imaginary dielectric permittivity. The absorptivity maximum and the corresponding optimal layer thickness will be determined. Boundary conditions corresponding to the frictional contact approximation for TE electromagnetic waves will be derived and discussed. We will show that the anomalous absorption of TE electromagnetic waves is frequency-dependent even in the frictional contact approximation. Comparison with TM electromagnetic and shear acoustic waves will be carried out. The frictional contact approximation will be used for theoretical analysis of the anomalous absorption of TE and TM bulk electromagnetic waves by non-uniform thin layers with the dielectric permittivity varying across the layer.
e 4 e 1. Then the reflectivity of a bulk TE electromagnetic wave Žincident on the interface from medium 1 at an angle that is not too close to pr2. is very close to 100%, i.e. the absorptivity is nearly zero. This is because at e 4 e 1, the Maxwell equations and the boundary conditions give the electric field in the transmitted wave E2 ; E1Ž e 1re .1r2 < E1 Ž E1 is the electric field in the incident wave.. This means that the energy flow in the transmitted wave is about ; Ž ere 1 .1r2 times smaller than in the incident wave w9x. Therefore only a small part of energy of the incident wave penetrates into medium 2 and dissipates there Žsee also Refs. w2,7x.. Now consider a layer of thickness h with imaginary permittivity e 2 s i e between two media 1 and 3 with the permittivities e 1 and e 3 – Fig. 1 Ž e 4 e 1,e 3; in this section presenting a qualitative physical consideration we assume that e 3 is real and positive.. In this case the first feeling is that the absorption in the layer must decrease as compared to the bulk medium 2, because the volume in which the dissipation takes place is decreased. However this is not correct. Similar to acoustics w1,2x and TM optical waves w7x, the absorptivity of TE electromagnetic waves strongly increases with decreasing the layer thickness h, reaches its maximum Žas much as 100%. at some optimal value of h s h m , and then decreases to zero when h ™ 0. The optimal thickness h m is usually much smaller than the wavelength in the layer Žsee also Refs. w1–7x.. Therefore, the anomalous absorption is not related with multiple reflections of the wave inside the layer. The optimal thickness is also much smaller than the wave penetration depth in the material of the layer w1–7x. Therefore, the anomalous absorption is not related with the frustrated total internal reflection w8x of electromagnetic waves in layered structures. Moreover, the frustrated total internal reflection of TE waves in the considered structure is not possible, because linear TE surface waves do not exist w8x. The anomalous absorption is a new effect which can be explained by existence of two mechanisms governing the absorption in the layer. 1. If e 4 e 1, then due to continuity of the tangential component Ž x-component for TE waves and z-component for TM waves. of magnetic field across a boundary, this tangential magnetic field inside the layer at y s 0 is of the same order of magnitude as in the incident wave. If the angle of incidence u 1 is smaller than the critical angle u 0 for a interface between media 1 and 3, the amplitude of the
2. Physical reasons for the anomalous absorption Consider an interface between medium 1 with real positive dielectric permittivity e 1 and medium 2 with purely imaginary dielectric permittivity e 2 s i e, where
375
Fig. 1. Layered structure with anomalous absorption.
D.K. GramotneÕr Optics Communications 151 (1998) 374–383
376
magnetic field at the boundary y s h is reduced, because the transmitted wave carries energy away from this boundary into medium 3. The smaller the value of e 3 , the stronger this energy flow, i.e. the smaller the magnetic field at the boundary y s h. Therefore, if e 3 4 e, the amplitude of the magnetic field at the layer boundary y s h appears to be much smaller than at y s h in the case of semi-infinite medium 2. Thus the derivative of the tangential magnetic field inside the layer ŽEH2 xrE y for TE waves and EH2 zrE y for TM waves. strongly increases with decreasing layer thickness Žandror e 3 .. For example, typical value of this derivative for a TE electromagnetic wave increases from
E H2 xrE y ; H2 xrmin Ž d , l2 . ; e 11r2 E1rmin Ž d , l2 . ; v cy1 Ž e 1 e .
1r2
E1
Ž1.
for the infinite layer thickness to about
E H2 xrE y ; v cy1 eE1
Ž2.
for h ™ 0. Here, l 2 is the wavelength and d is the wave penetration depth in the material of the layer, E1 is the electric field in the incident wave, c is the speed of light in vacuum, v is the angular frequency. The amount of energy Q absorbed in a unit volume of the layer is given by the equation w9x: Q s v Ž 8p .
y1
e < E2 < 2 f c 2 Ž 8pv e .
y1
< E H 2 x rE y < 2 .
absorptivity on the incidence angle displays a very pronounced dip down to zero at u 1 s u 0 w7x. This is because at the critical angle the transmitted wave propagates parallel to the layer, and there is no any energy flow from the layer boundary y s h into medium 3. The magnetic field at the boundary y s h Žin the transmitted wave. has the same amplitude, phase and direction as at the boundary y s 0. Therefore, the derivative EH2 zrE y is next to zero in the layer; so is the electric field and the dissipation Žsee also Refs. w1,2x.. The situation is absolutely different for TE waves. In this case, if u 1 s u 0 , the tangential component H2 x of the magnetic field at the boundary y s 0 is non-zero. However, in the transmitted wave propagating parallel to the layer, the magnetic field is perpendicular to the boundary y s h. Thus the tangential component of magnetic field at this boundary is zero. Therefore, the derivative EH2 xrE y ; H1rh will strongly increase with decreasing layer thickness h. This increase of the derivative EH2 xrE y causes a very strong maximum of the absorptivity equal to 100% at the critical angle if h s h m Žsee below.. The anomalous absorption of electromagnetic waves in the case with e 3 - 0 can be explained similarly to the case with u 1 ) u 0 . If e 3 is complex, the qualitative physical consideration of the anomalous absorption will be more complicated, but the above methods and principles remain the same.
Ž3.
Thus, because of the increase of the derivative EH2 xrE y, the energy absorbed in a unit volume of the layer also strongly increases with decreasing layer thickness h. 2. On the other hand, as has already been mentioned, the absorptivity must decrease with decreasing volume in which the dissipation takes place, i.e., with decreasing layer thickness h. Competition of these two opposing mechanisms results in an optimum layer thickness h m at which the dissipation in the layer is maximal. If the angle of incidence is greater than the angle of total internal reflection from the interface between media 1 and 3, then the wave in medium 3 in a non-eigen, evanescent wave. Medium 3 resists such non-eigen oscillations of electric and magnetic fields. This also results in decrease of the magnetic field at the interface between the layer and medium 3, and the dissipation in a unit volume of the layer again increases with decreasing layer thickness h. As has already been mentioned, this consideration is also valid for bulk TM electromagnetic waves if the x-component of the magnetic field is replaced by its z-component. Similar physical explanation of the anomalous absorption by a thin fluid layer was presented in Ref. w2x for bulk shear acoustic waves. However, at the critical angle u 0 , the anomalous absorption of TE waves is strongly different from that of TM waves. In the case of TM waves, the dependence of the
3. Frictional contact approximation Let a bulk TE electromagnetic wave be incident on a layer with imaginary dielectric permittivity at the angle u 1 ŽFig. 1.. The dielectric permittivity of medium 1 is real and positive, while the dielectric permittivity of medium 3 is complex. From the Maxwell equations and boundary conditions, we obtain the energy coefficients of reflection R and transmission T of the incident electromagnetic wave:
Rs
Ts
Ž k 1 y k 3 y q a 2 . tanh Ž a h . q i a Ž k 1 y y k 3 y . Ž k 1 y k 3 y y a 2 . tanh Ž a h . q i a Ž k 1 y q k 3 y . 4 k 1 y a arcosh Ž a h .
2
,
Ž4.
,
Ž5.
2
Ž k 1 y k 3 y y a 2 . tanh Ž a h . q i a Ž k 1 y q k 3 y .
2
where k 1 y s k 0 e1r2 1 cos u 1 , k 32 y s k 02 e 3 y k 02 e 1 sin2u 1 ,
Ž6.
a 2 s k 02 e 1sin2u 1 y ik 02 e,
Ž7.
a s ReŽ k 3 y ., k 0 s vrc, v is the frequency and c is the speed of light in vacuum; ReŽ a . ) 0, and ReŽ k 3 y . ) 0 if ReŽ k 3 y . / 0, or ImŽ k 3 y . ) 0 if ReŽ k 3 y . s 0.
D.K. GramotneÕr Optics Communications 151 (1998) 374–383
The coefficient of absorption M of the incident wave by the layer is given by Ms1yRyT.
Ž8.
Eqs. Ž4., Ž5. and Ž8. are valid for a layer of arbitrary thickness h with the imaginary dielectric permittivity e 2 s i e. So far, no restrictions on magnitudes of e 1,2,3 have been imposed. However, as we will see below, the absorptivity M can be especially large when the layer thickness is small Žas compared with the wavelength and the wave penetration depth in material of the layer. and the layer permittivity is large Žas compared with the dielectric permittivities of media 1 and 3.. Therefore, we will analyse this case in more detail. If the inequality e 4 e 1sin2u 1
Ž9.
is satisfied, then Eq. Ž7. gives a 2 f yi k 02 e, and the wavelength and the wave penetration depth in the material Ž2re .1r2 and d f of the layer are l 2 f 2 p ky1 0 1r2 Ž . ky1 2re , respectively. If, in addition, 0 e 4 e 11r2
Ž < k 3 y
Ž 10.
and the layer thickness is small: k 0 h Ž er2 .
1r2
- 1,
377
waves, we need to recall that it is the magnetic field derivative with respect to the y-coordinate which determines the anomalous absorption in the layer Žsee Section 2.. Moreover, magnetic field in electromagnetic waves is analogous to displacements in shear Žtransverse. acoustic waves. Therefore, we should use the boundary conditions at the layer interfaces written in terms of the magnetic field and then apply the procedure similar to the derivation of the frictional contact approximation in acoustics w2x. The boundary conditions for TE waves, written in terms of magnetic field, are given by
ey1 1 Ž E H1 x rE y y E H1 y rE x . ys0 s yiey1 Ž E H2 xrE y y E H2 yrE x . ys0 , yiey1 Ž E H2 xrE y y E H2 yrE x . ysh s ey1 3 Ž E H 3 x rE y y E H 3 y rE x . ysh ,
Ž H1 x . ys0s Ž H2 x . ys0 , Ž H2 x . yshs Ž H3 x . ysh ,
Ž 15 .
where H1,2,3 are the vectors of the magnetic field in media 1, 2 and 3 respectively. Using inequalities Ž9. – Ž11. and the last two equations of conditions Ž15., we obtain
Ž E H2 xrE y . ys0 f Ž E H2 xrE y . ysh Ž 11.
f Ž H2 x . ysh y Ž H2 x . ys0 rh
then Eqs. Ž4. and Ž5. can be reduced as f Ž H3 x . ys0 y Ž H1 x . ys0 rh,
R f < k 0 y Wc Ž k 1 y y k 3 y . < 2r< k 0 q Wc Ž k 1 y q k 3 y . < 2 ,
T f 4 k 1 y a < W < 2 c 2r< k 0 q Wc Ž k 1 y q k 3 y . < 2 ,
Ž 12.
Ž E H2 yrE x . ys0 f Ž E H2 yrE x . ysh < Ž E H2 xrE y . ys0 ,
Ž 13.
and the first two of conditions Ž15. can be reduced as
ey1 1 Ž E H1 x rE y y E H1 y rE x . ys0
where W s Ž ev h.
y1
.
Ž 14.
The reason for taking modulus of W in the numerator of T will become clear below. Physically, inequality Ž9. means that the wave in the layer propagates almost normally to its boundaries. Inequality Ž10. means that the square of the y-component of the wave vector in the layer must be much larger than the product of the y-components of the wave vectors in the surrounding media 1 and 3. Note that condition Ž10. is always satisfied at the critical angle of incidence at which the absorptivity is maximal Žsee below.. Eqs. Ž12. and Ž13. can also be derived by considering transmission of bulk TE electromagnetic waves through an interface between media 1 and 3 Žwithout the intermediate layer. with some special boundary conditions. This approximation is called frictional contact approximation, because it is analogous to approximating a fluid layer between two solid surfaces by a contact with linear Žliquid. friction w1–6x. To derive boundary conditions corresponding to the frictional contact approximation for TE electromagnetic
s ey1 3 Ž E H 3 x rE y y E H 3 y rE x . ys0 ,
Ž 16a.
ey1 1 Ž E H1 x rE y y E H1 y rE x . ys0 s W Ž E H3 xrE t y E H1 xrE t . ys0 .
Ž 16b.
Comparing these boundary conditions with the boundary conditions in the frictional contact approximation for bulk acoustic waves w1–5x, we can see that Eq. Ž16a. is analogue to equality of mechanical stresses across the frictional contact w2x. Similarly, Eq. Ž16b. is analogue to equality of these stresses to the force of linear friction with the coefficient W, the derivatives EH3 xrEt and EH1 xrEt playing role of velocities of the surfaces in contact. We can also see that the frictional coefficient for TE electromagnetic waves is the same as for TM waves w7x. However, boundary conditions Ž16. are noticeably different from the boundary conditions for the frictional contact approximation for TM waves w7x. Conditions Ž16. at the interface between media 1 and 3 with the dielectric permittivities e 1 and e 3 result in Eqs.
D.K. GramotneÕr Optics Communications 151 (1998) 374–383
378
Ž12. and Ž13. for the reflectivity and transmissivity of TE bulk electromagnetic waves. Thus we can replace four boundary conditions Ž15. by only two conditions Ž16.. In terms of electric field E, conditions Ž16. can be re-written as
Ž E1 . ys0 s Ž E3 . ys0 ,
Ž 17a.
2 2 Ž E1 . ys0 s Wky2 0 Ž E E1 rE yE t y E E3 rE yE t . ys0 . Ž 17b .
In the frictional contact approximation the absorptivity M is determined by Eq. Ž8. with R and T given by Eqs. Ž12. and Ž13.. The maximum of the function M Ž h. is equal to Mm f 2 k 1 y a q k 1 y q < k 3 y < 2 q k 12 y q 2 k 1 y a
Ž
1r2 y1
.
,
Ž 18. and the optimum layer thickness, h m , at which this maximum is achieved, is given by h m s Ž ek 02 .
y1
1r2
< k 3 y < 2 q k 12 y q 2 k 1 y a
.
Ž 19.
If e 3 is real, then Eqs. Ž18. and Ž19. give: Mm f Ž 1 q k 3 yrk 1 y . h m f Ž ek 02 .
y1
y1
,
Ž 20.
Ž k 3 y q k1 y .
Ž 21.
if e 3 ) 0 and u 1 F u 0 , and Mm f 2 k 1 y k 0 Ž e 1 y e 3 . h m f Ž ek 0 .
y1
1r2
y1
q k1 y
,
Ž 22.
Ž e 1 y e 3 . 1r2 ,
Ž 23.
if e 3 - 0, or u 1 ) u 0 . The reflectivity and transmissivity of the wave at h s h m are R m f Ž 1 q k 1 yrk 3 y .
y2
,
Tm f Ž k 1 yrk 3 y . Ž 1 q k 1 yrk 3 y .
1r2
= Ž e1 y e3 .
Ž 24.
Tm s 0
y e 11r2 cos u 1
1r2
q e 11r2 cos u 1
y1
,
Ž 25 .
if e 3 - 0, or u 1 ) u 0 . If condition Ž9. is not satisfied, then a is determined by Eq. Ž7.. However this is not crucial for applicability of the frictional contact approximation Ž16. and Ž17. and Eqs. Ž12. and Ž13.. Indeed, if condition Ž9. is not satisfied, then Eqs. Ž12. and Ž13. can still be obtained using the boundary conditions Ž16. or Ž17. with the complex frictional coefficient W s yik 0r Ž c a 2 h .
< a < 2 4 k1 y < k 3 y <,
Ž 27.
h < a < - 1.
Ž 28.
Thus inequalities Ž27. and Ž28. are the conditions of the frictional contact approximation for TE waves if condition Ž9. is not satisfied. However, for large imaginary dielectric permittivities of the layer Žwhich we consider in this paper and which are also required by conditions Ž10. and Ž27.. it is difficult to imagine a real situation in which condition Ž9. is frustrated. That is why, throughout this paper, we mainly use the frictional coefficient determined by Eq. Ž14.. Note that in the case of TM waves, if condition Ž9. is not satisfied, the coefficient of friction is still determined by Eq. Ž14.. This is another difference in the anomalous absorption of TE and TM electromagnetic waves. If condition Ž10. Žor Ž27.. is not satisfied, we are not able to present equations for R and T in the form of Ž12. and Ž13.. Thus condition Ž10. is vital for the frictional contact approximation. If we inverse it, then the absorptivity is near zero, i.e. the anomalous absorption disappears. Thus condition Ž10. is important not only for the frictional contact approximation, but also for significant absorption in the layer. However, this is the case only for TE waves, whereas a very strong anomalous absorption of TM waves can occur when the condition similar to Ž10. is inversed w10x.
4. Analysis of the solutions y2
if e 3 ) 0 and u 1 F u 0 , and Rm f Ž e1 y e3 .
modulus of W. This was done to make Eq. Ž13. valid also for the complex frictional coefficient Ž26.. In this case other two applicability conditions of the frictional contact approximation Ž10. and Ž11. should be written as
Ž 26.
Žwhich is easily reduced to Eq. Ž14. if condition Ž9. holds.. Now we can see the reason why in Eq. Ž13. we used
Unlike acoustics, where the anomalous absorption does not depend on frequency if conditions similar to Ž9. – Ž11. are satisfied w1–6x, the anomalous absorption in optics is frequency-dependent Žsee also Ref. w7x.. This dependence is represented not only by the explicit dependence of k 1,3 and W on v , but also by dispersion of the dielectric permittivity of the layer. Moreover, large imaginary dielectric permittivities of the layer can be achieved only in the vicinity of a sharp line of absorption, where the dielectric permittivity very strongly Žresonantly. depends on frequency Žsee, for example, an optical resonance related with generation of transverse optical phonons in ionic crystals w11x.. Thus the anomalous absorption in optics appears to be a highly frequency-selective resonant effect, and this is its very important and strong distinction from the anomalous absorption in acoustics w1–6x. From Eqs. Ž20., Ž21. and Ž24. we see that if k 1 y s k 3 y Žhalf spaces with equal dielectric permittivities., then Mm s 0.5, R m s Tm s 0.25, h m s 2 k 1 yrŽ ek 02 ., i.e. the coefficients of absorption, reflection and transmission, corre-
D.K. GramotneÕr Optics Communications 151 (1998) 374–383
sponding to the maximum of absorption, are independent of frequency, angle of incidence, and parameters of the media in contact. In addition, Eqs. Ž13. and Ž24. give TmrT Ž h s 0. s 0.25, i.e. this ratio is also independent of any material parameters. The same invariants are valid for TM electromagnetic waves and for shear acoustic waves polarised perpendicular to the plane of incidence w2x. If the angle of incidence u 1 G u 0 Žor e 3 - 0., then the optimum layer thickness h m is given by Eq. Ž23. which is independent of the angle of incidence. This does not mean that the anomalous absorption is independent of the angle of incidence, but that the maximum of the absorptivity Ždifferent for different values of u 1 – see Eq. Ž22.. is achieved at the same optimum layer thickness for all u 1 G u 0. If the angle of incidence is equal to the critical angle u 0 , then k 3 y s 0 and Eqs. Ž20., Ž22. and Ž24. give Mm s 1, R m s Tm s 0, and h m is determined by Eq. Ž23.. Thus we have total absorption of the TE electromagnetic wave at the critical angle. From Eqs. Ž8., Ž12. and Ž13. we can see that for any value of the layer thickness h satisfying condition Ž11., the dependence of the absorptivity on the incidence angle reaches its maximum at the critical angle. For h s h m , this maximum is equal to 100%. This is a strong distinction of the anomalous absorption of TE electromagnetic waves from the anomalous absorption of TM waves Žfor TM waves we had Mm s 0 at the critical angle w7x.. This difference was explained in Section 2. For example, consider a CaF2 film on a transparent substrate with e 1 s 3 Ž e 3 s 1, i.e. medium 3 is vacuum.. A TE electromagnetic wave with the length 2 prk 0 s 38.905 mm in vacuum Žat this wavelength the dielectric permittivity of the layer e 2 f i233 w11x. is incident on the layer from the substrate at the angle u 1 s 0. Then Eqs. Ž20., Ž21. and Ž24. give Mm f 0.64, R m f 0.13, Tm f 0.23, h m f 73
Fig. 2. The dependences of the absorptivity M of TE electromagnetic waves on the thickness h of a CaF2 film with the permittivity e 2 s i233 Ž2 p r k 0 s 38.905 mm.; e 1 s 3, e 3 s1. Ž1. u 1 s 08, Ž2. u 1 s u 0 f 35.278, Ž3. u 1 s608. Curve 4 presents the dependence of M on h for TM waves at 608 angle of incidence.
379
Fig. 3. The dependences of the absorptivity M, reflectivity R and transmissivity T of TE waves on the angle of incidence u 1 for a CaF2 film of thickness hs h m Ž u 1 G u 0 . f 38 nm with the permittivity e 2 s i233 Ž2 p r k 0 s 38.905 mm.; e 1 s 3, e 3 s1 Ž u 0 f 35.278.. The Mi curve represents the absorptivity in the bulk CaF2 , i.e. at hsq`. The Mc curve represents the absorptivity in the bulk CaF2 within the area of 38 nm thickness near the interface.
nm Žif h ™ q`, then M ™ 0.27.. If u 1 s u 0 f 35.27 0 , then Mm s 1 and h m f 38 nm Žif h s q`, then M f 0.23.. The dependences of the absorptivity M on the thickness h of a CaF2 film with the permittivity e 2 s i233 Ž2 prk 0 s 38.905 mm. on a transparent substrate of e 1 s 3 Žmedium 3 is vacuum: e 3 s 1. are presented in Fig. 2 for three different angles of incidence: u1 s 0 Žcurve 1., u 1 s u 0 f 35.278 Žcurve 2., and u 1 s 608 Žcurve 3.. Curve 4 presents the dependence of M on the layer thickness in the same structure but for TM waves at 608 angle of incidence Žto calculate this dependence, Eqs. Ž1. – Ž3. of Ref. w7x were used.. It is obvious that curve 1 for the normal incidence for TE waves is exactly the same as for TM waves, because at u 1 s 0 there is no difference between these two types of waves. It can be seen that typical widths of the absorptivity maximums for TE waves are smaller than for TM waves, so are typical optimum layer thicknesses corresponding to the absorptivity maximums ŽFig. 2.. The dependences of the absorptivity M, reflectivity R and transmissivity T on the angle of incidence for the CaF2 film of thickness h s 38 nm with the permittivity e 2 s i233 Ž2 prk 0 s 38.905 mm. on a transparent substrate of e 1 s 3 Žmedium 3 is vacuum: e 3 s 1. are presented in Fig. 3. The value of the layer thickness of 38 nm is equal to the optimum thickness h m at the angles of incidence u 1 G u 0 . Therefore, the absorptivity at the critical angle of incidence is equal to 100% ŽFig. 3.. The Mi curve represents the absorptivity in bulk CaF2 , i.e. at h s q`. Thus we can see that the absorptivity characterising the anomalous absorption in a thin layer Žthe M curve., is substantially larger than the conventional absorptivity in the bulk
380
D.K. GramotneÕr Optics Communications 151 (1998) 374–383
medium Žthe Mi curve.. Moreover, the anomalous absorption takes place in a very thin layer Žin our example h s 38 nm., whereas the conventional absorption takes place in a fairly thick area Žwith the thickness of about d ; 570 nm. near the interface. Therefore the Mc curve in Fig. 3 represents the conventional absorption in the area of 38 nm thick near the interface in the semi-infinite medium 2. Comparison of the M and Mc curves in Fig. 3 is more correct and interesting than that of the M and Mi curves, because the Mc and M curves correspond to the conventional and the anomalous absorption, respectively, in regions of the same thickness equal to 38 nm. From the comparison of these two curves, we can see that at the critical angle, the absorptivity due to the anomalous absorption is about 33 times greater than the absorptivity due to the conventional absorption within the same thickness of material 2. In other words, the dissipation in a unit volume of medium 2 is ; 33 times greater for the anomalous absorption than for the conventional one. Fig. 4 is similar to Fig. 3, but the layer thickness h s 9 nm and the dielectric permittivity of the third medium e 3 s 2.9, i.e. very close to e 1 s 3. In this case the critical angle u 0 f 79.488. The layer thickness of 9 nm is equal to the optimum layer thickness h m at u 1 G u 0 . Therefore the absorptivity is equal to 100% at the critical angle. From the comparison of the M and Mc curves in Fig. 4, corresponding to the anomalous and conventional absorption in CaF2 of the same thickness, we can see that the anomalous absorption is ; 500 times greater than the conventional one Žthe absorptivity given by the Mc curve at u 1 s 79.488 is about 2 = 10y3 .. Even if we reduce the layer thickness to 0.9 nm Ži.e. to a couple of atomic layers., we will still get a substantial absorption of about
Fig. 4. The dependences of the absorptivity M, reflectivity R and transmissivity T of TE waves on the angle of incidence u 1 for a CaF2 film of thickness hs h m Ž u 1 G u 0 . f9 nm with the permittivity e 2 s i233 Ž2 p r k 0 s 38.905 mm.; e 1 s 3, e 3 s 2.9 Ž u 0 f 79.488.. The Mi curve represents the absorptivity in the bulk CaF2 , i.e. at hsq`. The Mc curve represents the absorptivity in the bulk CaF2 within the area of 9 nm thickness near the interface.
Fig. 5. The dependences of the optimal thickness h m of a CaF2 layer on the angle of incidence u 1 for TE waves Ž2 p r k 0 s 38.905 mm.; e 1 s 3. Ž1. e 3 s1, Ž2. e 3 s 2, Ž3. e 3 s 2.9, Ž4. e 3 s 3.
35% at the critical angle of incidence. Of course, the presented macroscopic approach is not applicable in this case and, to obtain correct equations for the energy coefficients, we must use a microscopic approach for analysis of the anomalous absorption by just several atomic layers. However, this consideration is not the aim of this paper. Thus we have seen that the optimum layer thickness strongly depends on the dielectric permittivities of the surrounding media and the angle of incidence. In particular, Eq. Ž23. gives that if e 1 ) e 3 ) 0, then the optimum layer thickness at u 1 G u 0 decreases when e 1 ™ e 3 Žcompare with Fig. 3 and 4 plotted for h s 38 nm and 9 nm respectively.. The dependences of h m on the angle of incidence u 1 are presented in Fig. 5 for TE waves Ž2 prk 0 s 38.905 mm. in the structure with a CaF2 layer; e 1 s 3 is the permittivity of medium 1, and the permittivity of medium 3 takes four different values: e 3 s 1 Žcurve 1.; e 3 s 2 Žcurve 2.; e 3 s 2.9 Žcurve 3.; e 3 s 3 Žcurve 4.. As has already been mentioned, the optimum layer thickness is angular independent at incidence angles that are greater than the critical angle Žsee Eq. Ž23. and Fig. 5.. To compare the anomalous absorption of TE and TM waves, the dependences of the absorptivities of both these types of electromagnetic waves in a CaF2 layer Ž2 prk 0 s 38.905, e 2 s i233, e 1 s 3, e 3 s 1. on the angle of incidence are presented in Figs. 6a, 6b for different values of the layer thickness. As expected, at normal incidence both these types of waves give the same absorptivity ŽFigs. 6a, 6b.. The main difference at small layer thicknesses is that the anomalous absorption of TE waves has maximum at the critical angle, while the anomalous absorption of TM waves at this angle is characterised by a sharp pronounced minimum down to zero ŽFig. 6a.. At the thicknesses comparable with the wave penetration depth in the layer, the pronounced dip at the critical angle for TM waves becomes smaller and disappears for the infinite layer thick-
D.K. GramotneÕr Optics Communications 151 (1998) 374–383
Fig. 6. The dependences of the absorptivities of TE and TM electromagnetic waves in a CaF2 layer Ž2 p r k 0 s 38.905, e 2 s i233, e 1 s 3, e 3 s1. on the angle of incidence. Ža. Ž1. TE wave for hs 38 nm, Ž2. TE wave for hs95 nm, Ž3. TM wave for hs 38 nm, Ž4. TM wave for hs95 nm. Žb. Ž1. TE wave for hs800 nm, Ž2. TE wave for hsq`, Ž3. TM wave for hs800 nm, Ž4. TM wave for hsq`.
381
ultra-thin layers. However, because typical layer thicknesses, which are required for the anomalous absorption, can be very small Ža few tens of nanometers, or even a few nanometers., optical parameters of the layer may be noticeably affected by the interfaces. In particular, the dielectric permittivity can vary across the layer. An isolated interface between two semi-infinite media can also result in modifying properties of near-interface regions of media in contact. Thus we obtain very thin layers with varying dielectric permittivity near the interface. Anomalous absorption of electromagnetic waves can be used for investigation of such layers and interfaces. Therefore, anomalous absorption in non-uniform layers with the permittivity varying across the layer is a very important practical problem. Frictional contact approximation appears to be a powerful technique for simple theoretical analysis of the anomalous absorption in such non-uniform thin films. This is because optical properties of the film in this approximation are represented only by the frictional coefficient Žsee Eqs. Ž16. and Ž17... Therefore, if we are able to determine the frictional coefficient W for a thin non-uniform layer, we can easily find the coefficients of reflection, absorption and transmission of the incident wave by means of Eqs. Ž8., Ž12. and Ž13.. Consider a non-uniform layer with imaginary dielectric permittivity varying across the layer, i.e. e is a function of the y-coordinate ŽFig. 1.. If inequalities Ž9. – Ž11. and condition Ž4b. from Ref. w7x are satisfied for all values of e, then the value of yey1EH2rE y is approximately constant across the layer. Here, H2 is the x- or y-component Žfor TE waves., or z-component Žfor TM waves. of the magnetic field in the layer. Then Žin accordance with Eqs. Ž15., Ž16.., F s w Ž y . dÕ,
Ž 29.
where ness ŽFig. 6b.. In addition, a sharp maximum of absorption appears near the critical angle just in front of the dip ŽFig. 6b.. This maximum can be observed only for TM waves. It is probably related with the wave interference inside the layer. At smaller thicknesses this maximum is less sharp and strong. Therefore, it is overwhelmingly suppressed by the strong minimum related with the anomalous absorption ŽFig. 6a.. From Fig. 6b, we can also see that as the layer thickness decreases from a large value, the absorption slightly decreases for almost all angles of incidence, and only then increases dramatically due to the anomalous absorption.
5. Non-uniform layers In the previous sections, we analysed only the anomalous absorption of electromagnetic waves in uniform
F s ey1 2 E H 2 z rE y, dÕ s Ž E H2 zrE t . yqd y y Ž E H2 zrE t . y
for TM waves,
F s ey1 2 Ž E H 2 y rE x y E H 2 x rE y . , dÕ s Ž E H2 xrE t . yqd y y Ž E H2 xrE t . y
for TE waves,
w Ž y . s Ž e v d y .y1 is the coordinate dependent frictional coefficient between two planes y and y q d y inside the layer. The notations are chosen from the view point of analogy with mechanical friction Ž F is analogous to the force of friction, and dÕ to the relative velocity of the planes y and y q d y .. Substituting w Ž y . into Eq. Ž29., rearranging it and integrating over the layer thickness gives Õ 3 y Õ1 s Fv
h
H0 e Ž y . d y s FrW ,
D.K. GramotneÕr Optics Communications 151 (1998) 374–383
382
where Õ1,3 s EH1,3rEt, H1,3 are the x- Žor z-. component of the magnetic field at the interfaces of media 1 and 3, respectively, for TE Žor TM. electromagnetic waves, and W is the total frictional coefficient for the non-uniform layer Žcompare with Eq. Ž16b... Thus y1
ž
h
Ws v
H0 e Ž y . d y
/
.
Ž 30.
If condition Ž9. is not satisfied for the minimal value of the dielectric permittivity in the layer, then in Eq. Ž29. we have to use w Ž y . s yik 0rŽ c a 2 d y ., and the total frictional coefficient is given by Wsy
ik 0 c
y1
ž
h
H0 a
2
Ž y .d y
/
.
Ž 31.
If condition Ž9. is satisfied for all values of the permittivity in the layer, then Eq. Ž31. is reduced to Eq. Ž30.. Calculating the frictional coefficient for a non-uniform layer from Eq. Ž30. or Ž31. and substituting it into Eqs. Ž12., Ž13. and Ž8., we can easily determine the energy coefficients of reflection, transmission and absorption of TE electromagnetic waves. Similarly, substituting the frictional coefficient Ž30. into Eqs. Ž6. and Ž7. of Ref. w7x, we can find the energy coefficients characterising the anomalous absorption of TM waves in a non-uniform layer. Thus using the frictional contact approximation, we can easily analyse the anomalous absorption of bulk electromagnetic waves in non-uniform layers, which would be very difficult to do without this approximation.
6. Conclusions This paper has analysed a new optical effect – the anomalous absorption of bulk TE and TM electromagnetic waves by an ultra-thin film with purely imaginary dielectric permittivity. Detailed analysis of the anomalous absorption was carried out in the frictional contact approximation. The absorptivity maximum and the corresponding optimal layer thickness were determined for TE waves as functions of frequency, angle of incidence, film thickness, and dielectric permittivities of the media in contact. Comparison of the anomalous absorption of TE and TM electromagnetic waves was carried out. The main difference between these two types of waves was discovered near the critical angle, where in the frictional contact approximation the absorptivity of TE waves reaches its maximum of 100%, and the absorptivity of TM waves drops to zero. The anomalous absorption was also shown to display as much as several hundred times greater dissipation in a unit volume of the film than the conventional absorption in the bulk medium Ži.e. at infinite thickness of the film.. As an example, the anomalous absorption of TE and TM bulk electromagnetic waves was analysed in thin films of CaF2 in the infrared range, where a large imaginary
dielectric permittivity of the film can be obtained at the frequency of transverse optical phonons. Typical film thicknesses, which are required for the anomalous absorption in this case, were found to be about several tens of nanometers, or just several nanometers. The frictional contact approximation was demonstrated to be a powerful tool for substantial simplification of theoretical analysis of the anomalous absorption of electromagnetic waves in thin films. This is not only because it allows the reduction of the number of boundary conditions by two, but also because the properties of the film are taken into account only through the frictional coefficient. Therefore the frictional contact approximation allowed us to analyse analytically the anomalous absorption in nonuniform films with varying permittivity. In addition, the frictional contact approximation will be very important for theoretical analysis of electromagnetic waves in complex multi-layered structures, anisotropic media, etc. The anomalous absorption of electromagnetic waves is not restricted to infrared radiation in ionic crystals. It can be realised in the vicinity of any sufficiently sharp absorption line for electromagnetic waves of arbitrary frequency from radio waves to ultraviolet radiation. The only principle requirement is that the dielectric permittivity of the film must have large imaginary part. Current paper was restricted only to large purely imaginary dielectric permittivities of the film. However, such permittivities can be obtained only at some special frequencies, for example, at the frequency of transverse optical phonons in ionic crystals. In this case, even small variations of the frequency result in large Žnegative or positive. real part of the film permittivity w11x. This large real part will definitely affect the anomalous absorption in the film. Therefore, analysis of the anomalous absorption of electromagnetic waves in thin films with complex dielectric permittivities, as well as investigation of frequency response of the anomalous absorption, are important practical problems which will be analysed in our next paper w10x. The anomalous absorption can be realised not only for bulk, but also for guided and surfaces electromagnetic waves. Thus it will be important for development of new signal-processing devices, sensors, measurement and evaluation techniques using bulk and guided optical modes.
Acknowledgements The author is grateful to J.M. Bell and G. Voevodkin for stimulating and fruitful discussions.
References w1x D.K. Gramotnev, S.N. Ermoshin, Sov. Tech. Phys. Lett. ŽUSA. 15 Ž1989. 856 wPis’ma Zh. Tekh. Fiz. 15 no. 21 Ž1989. 62x.
D.K. GramotneÕr Optics Communications 151 (1998) 374–383 w2x D.K. Gramotnev, S.N. Ermoshin, L.A. Chernozatonskii, Sov. Phys. Acoust. ŽUSA. 37 Ž1991. 343 wAkust. Zh. 37 Ž1991. 660x. w3x M.L. Vyukov, D.K. Gramotnev, L.A. Chernozatonskii, Sov. Phys. Acoust. ŽUSA. 37 Ž1991. 229 wAkust. Zh. 37 Ž1991. 448x. w4x L.A. Chernozatonskii, D.K. Gramotnev, M.L. Vyukov, J. Phys. D 25 Ž1992. 226. w5x L.A. Chernozatonskii, D.K. Gramotnev, M.L. Vyukov, Phys. Lett. A 164 Ž1992. 126. w6x D.K. Gramotnev, M.L. Vyukov, Int. J. Mod. Phys. B 8 Ž1994. 1741.
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w7x D.K. Gramotnev, Optics Lett. 23 Ž1998. 91. w8x V.M. Agranovich, D.L. Mills, ŽEds.., Surface Polaritons. Electromagnetic Waves at Surfaces and Interfaces, North Holland, Amsterdam, 1982. w9x L.D. Landau, E.M. Lifshitz, Electrodynamics of Continuous Media, Pergamon, New York Ž1984.. w10x D.K. Gramotnev, J.A. Ross, Frequency response of anomalous absorption of electromagnetic waves in a layered structure, Optical and Quantum Electronics, submitted for publication. w11x E.D. Palik. Handbook of Optical Constants of Solids II, Boston, Academic Press, 1991.
Optics Communications 151 Ž1998. 374–383
Full length article
Anomalous absorption of bulk electromagnetic waves by an ultra-thin layer D.K. Gramotnev
)
Centre for Medical and Health Physics, School of Physical Sciences, Queensland UniÕersity of Technology, GPO Box 2434, Brisbane QLD 4001, Australia Received 3 December 1997; revised 5 February 1998; accepted 6 February 1998
Abstract A strong maximum of absorption Žas much as 100%. of bulk TE electromagnetic waves by an ultra-thin film with large imaginary dielectric permittivity between two semi-infinite media is shown to exist at an optimal film thickness. This optimal thickness is demonstrated to be usually much smaller than the wavelength and the wave penetration depth in the film. A 100% absorption of TE electromagnetic waves is shown to occur in the middle infrared range in an extremely thin film of an ionic crystal of several tens of nanometers, or just several nanometers. Physical reasons for this anomalous absorption are discussed. Frictional contact approximation, in which the layer can be considered as an immediate contact between two half spaces with special boundary conditions, is introduced and justified. Similarity of these boundary conditions with mechanical boundary conditions at a contact with liquid friction is investigated. The anomalous absorption of electromagnetic waves in thin non-uniform layers with the dielectric permittivity varying across the layer is analysed in the frictional contact approximation. Significant differences in the anomalous absorption of TE and TM waves are pointed out and investigated. q 1998 Elsevier Science B.V. All rights reserved. PACS: 78.65; 42.78H; 42.85 Keywords: Optics of thin films; Absorption; Transmission; Reflection; Ionic crystals
1. Introduction The anomalous absorption of acoustic waves by a thin layer of viscous fluid enclosed between elastic solid media was analysed theoretically in our previous papers w1–6x. It is characterised by a strong maximum of absorption Žas much as 100%. occurring at some optimal layer thickness which is usually much smaller than the wavelength and the wave penetration depth in the fluid w1–6x. At not very high frequencies the absorptivity maximum and the corresponding layer thickness were shown to be independent of frequency w1–5x. In this case a frictional contact approximation is valid w2,4,5x, and tangential stresses at the contact are proportional to relative velocity of the solid sur-
)
E-mail: [email protected]
faces w2–6x. This effect is very important for development of new acoustic viscosity sensors, measurement and diagnostics techniques, for investigation of interfaces and extremely small amounts of fluids. The wave equation for shear Žtransverse. waves in a viscous Newtonian fluid has the same form as the wave equation for electromagnetic waves in a medium with purely imaginary dielectric permittivity. This suggests that anomalous absorption must also take place for bulk electromagnetic waves in a thin layer Žfilm. with purely imaginary, or almost imaginary dielectric permittivity. Anomalous absorption of bulk TM electromagnetic waves in a uniform thin layer with large, purely imaginary dielectric permittivity was analysed recently in Ref. w7x. Similarly to acoustics, the anomalous absorption of TM optical waves was shown to display a strong Žas much as 100%. maximum of absorptivity at some optimal layer thickness. This layer thickness is usually much smaller
0030-4018r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved. PII S 0 0 3 0 - 4 0 1 8 Ž 9 8 . 0 0 0 8 8 - 1
D.K. GramotneÕr Optics Communications 151 (1998) 374–383
than the wavelength and the wave penetration depth in the material of the layer. Therefore the anomalous absorption cannot be related with the frustrated total internal reflection of the incident wave w8x, or multiple reflections inside the layer. Optical analogue of mechanical linear Žliquid. friction was analysed. At the same time, noticeable differences between the anomalous absorption in acoustics and optics were demonstrated. For example, unlike acoustics, the anomalous absorption in optics is a highly frequencyselective effect. So-called coefficient of friction for TM electromagnetic waves was shown to depend on frequency explicitly and implicitly Žthrough a dispersion of the dielectric permittivity of the layer.. The anomalous absorption of bulk TM optical waves was mentioned to be important for development of new high-sensitive optical sensors, signal-processing devices, measurement and evaluation techniques. The anomalous absorption must also take place for bulk TE electromagnetic waves in ultra-thin films and coatings. However, the analysis of this type of waves is noticeably different from the analysis of TM electromagnetic waves. There is no direct analogy between bulk TE electromagnetic waves and bulk shear acoustic waves, because of different boundary conditions. Therefore the anomalous absorption of bulk TE electromagnetic waves is a separate important problem which will result in strongly different dependences of reflectivity, transmissivity and absorptivity on parameters of the film and surrounding media, frequency, and angle of incidence. Frictional contact approximation for TE electromagnetic waves will also be represented by boundary conditions which are noticeably different from those for acoustic waves w1–6x and TM optical waves w7x. Therefore the aim of this work is the theoretical analysis of the anomalous absorption of bulk TE electromagnetic waves by a thin layer with imaginary dielectric permittivity. The absorptivity maximum and the corresponding optimal layer thickness will be determined. Boundary conditions corresponding to the frictional contact approximation for TE electromagnetic waves will be derived and discussed. We will show that the anomalous absorption of TE electromagnetic waves is frequency-dependent even in the frictional contact approximation. Comparison with TM electromagnetic and shear acoustic waves will be carried out. The frictional contact approximation will be used for theoretical analysis of the anomalous absorption of TE and TM bulk electromagnetic waves by non-uniform thin layers with the dielectric permittivity varying across the layer.
e 4 e 1. Then the reflectivity of a bulk TE electromagnetic wave Žincident on the interface from medium 1 at an angle that is not too close to pr2. is very close to 100%, i.e. the absorptivity is nearly zero. This is because at e 4 e 1, the Maxwell equations and the boundary conditions give the electric field in the transmitted wave E2 ; E1Ž e 1re .1r2 < E1 Ž E1 is the electric field in the incident wave.. This means that the energy flow in the transmitted wave is about ; Ž ere 1 .1r2 times smaller than in the incident wave w9x. Therefore only a small part of energy of the incident wave penetrates into medium 2 and dissipates there Žsee also Refs. w2,7x.. Now consider a layer of thickness h with imaginary permittivity e 2 s i e between two media 1 and 3 with the permittivities e 1 and e 3 – Fig. 1 Ž e 4 e 1,e 3; in this section presenting a qualitative physical consideration we assume that e 3 is real and positive.. In this case the first feeling is that the absorption in the layer must decrease as compared to the bulk medium 2, because the volume in which the dissipation takes place is decreased. However this is not correct. Similar to acoustics w1,2x and TM optical waves w7x, the absorptivity of TE electromagnetic waves strongly increases with decreasing the layer thickness h, reaches its maximum Žas much as 100%. at some optimal value of h s h m , and then decreases to zero when h ™ 0. The optimal thickness h m is usually much smaller than the wavelength in the layer Žsee also Refs. w1–7x.. Therefore, the anomalous absorption is not related with multiple reflections of the wave inside the layer. The optimal thickness is also much smaller than the wave penetration depth in the material of the layer w1–7x. Therefore, the anomalous absorption is not related with the frustrated total internal reflection w8x of electromagnetic waves in layered structures. Moreover, the frustrated total internal reflection of TE waves in the considered structure is not possible, because linear TE surface waves do not exist w8x. The anomalous absorption is a new effect which can be explained by existence of two mechanisms governing the absorption in the layer. 1. If e 4 e 1, then due to continuity of the tangential component Ž x-component for TE waves and z-component for TM waves. of magnetic field across a boundary, this tangential magnetic field inside the layer at y s 0 is of the same order of magnitude as in the incident wave. If the angle of incidence u 1 is smaller than the critical angle u 0 for a interface between media 1 and 3, the amplitude of the
2. Physical reasons for the anomalous absorption Consider an interface between medium 1 with real positive dielectric permittivity e 1 and medium 2 with purely imaginary dielectric permittivity e 2 s i e, where
375
Fig. 1. Layered structure with anomalous absorption.
D.K. GramotneÕr Optics Communications 151 (1998) 374–383
376
magnetic field at the boundary y s h is reduced, because the transmitted wave carries energy away from this boundary into medium 3. The smaller the value of e 3 , the stronger this energy flow, i.e. the smaller the magnetic field at the boundary y s h. Therefore, if e 3 4 e, the amplitude of the magnetic field at the layer boundary y s h appears to be much smaller than at y s h in the case of semi-infinite medium 2. Thus the derivative of the tangential magnetic field inside the layer ŽEH2 xrE y for TE waves and EH2 zrE y for TM waves. strongly increases with decreasing layer thickness Žandror e 3 .. For example, typical value of this derivative for a TE electromagnetic wave increases from
E H2 xrE y ; H2 xrmin Ž d , l2 . ; e 11r2 E1rmin Ž d , l2 . ; v cy1 Ž e 1 e .
1r2
E1
Ž1.
for the infinite layer thickness to about
E H2 xrE y ; v cy1 eE1
Ž2.
for h ™ 0. Here, l 2 is the wavelength and d is the wave penetration depth in the material of the layer, E1 is the electric field in the incident wave, c is the speed of light in vacuum, v is the angular frequency. The amount of energy Q absorbed in a unit volume of the layer is given by the equation w9x: Q s v Ž 8p .
y1
e < E2 < 2 f c 2 Ž 8pv e .
y1
< E H 2 x rE y < 2 .
absorptivity on the incidence angle displays a very pronounced dip down to zero at u 1 s u 0 w7x. This is because at the critical angle the transmitted wave propagates parallel to the layer, and there is no any energy flow from the layer boundary y s h into medium 3. The magnetic field at the boundary y s h Žin the transmitted wave. has the same amplitude, phase and direction as at the boundary y s 0. Therefore, the derivative EH2 zrE y is next to zero in the layer; so is the electric field and the dissipation Žsee also Refs. w1,2x.. The situation is absolutely different for TE waves. In this case, if u 1 s u 0 , the tangential component H2 x of the magnetic field at the boundary y s 0 is non-zero. However, in the transmitted wave propagating parallel to the layer, the magnetic field is perpendicular to the boundary y s h. Thus the tangential component of magnetic field at this boundary is zero. Therefore, the derivative EH2 xrE y ; H1rh will strongly increase with decreasing layer thickness h. This increase of the derivative EH2 xrE y causes a very strong maximum of the absorptivity equal to 100% at the critical angle if h s h m Žsee below.. The anomalous absorption of electromagnetic waves in the case with e 3 - 0 can be explained similarly to the case with u 1 ) u 0 . If e 3 is complex, the qualitative physical consideration of the anomalous absorption will be more complicated, but the above methods and principles remain the same.
Ž3.
Thus, because of the increase of the derivative EH2 xrE y, the energy absorbed in a unit volume of the layer also strongly increases with decreasing layer thickness h. 2. On the other hand, as has already been mentioned, the absorptivity must decrease with decreasing volume in which the dissipation takes place, i.e., with decreasing layer thickness h. Competition of these two opposing mechanisms results in an optimum layer thickness h m at which the dissipation in the layer is maximal. If the angle of incidence is greater than the angle of total internal reflection from the interface between media 1 and 3, then the wave in medium 3 in a non-eigen, evanescent wave. Medium 3 resists such non-eigen oscillations of electric and magnetic fields. This also results in decrease of the magnetic field at the interface between the layer and medium 3, and the dissipation in a unit volume of the layer again increases with decreasing layer thickness h. As has already been mentioned, this consideration is also valid for bulk TM electromagnetic waves if the x-component of the magnetic field is replaced by its z-component. Similar physical explanation of the anomalous absorption by a thin fluid layer was presented in Ref. w2x for bulk shear acoustic waves. However, at the critical angle u 0 , the anomalous absorption of TE waves is strongly different from that of TM waves. In the case of TM waves, the dependence of the
3. Frictional contact approximation Let a bulk TE electromagnetic wave be incident on a layer with imaginary dielectric permittivity at the angle u 1 ŽFig. 1.. The dielectric permittivity of medium 1 is real and positive, while the dielectric permittivity of medium 3 is complex. From the Maxwell equations and boundary conditions, we obtain the energy coefficients of reflection R and transmission T of the incident electromagnetic wave:
Rs
Ts
Ž k 1 y k 3 y q a 2 . tanh Ž a h . q i a Ž k 1 y y k 3 y . Ž k 1 y k 3 y y a 2 . tanh Ž a h . q i a Ž k 1 y q k 3 y . 4 k 1 y a arcosh Ž a h .
2
,
Ž4.
,
Ž5.
2
Ž k 1 y k 3 y y a 2 . tanh Ž a h . q i a Ž k 1 y q k 3 y .
2
where k 1 y s k 0 e1r2 1 cos u 1 , k 32 y s k 02 e 3 y k 02 e 1 sin2u 1 ,
Ž6.
a 2 s k 02 e 1sin2u 1 y ik 02 e,
Ž7.
a s ReŽ k 3 y ., k 0 s vrc, v is the frequency and c is the speed of light in vacuum; ReŽ a . ) 0, and ReŽ k 3 y . ) 0 if ReŽ k 3 y . / 0, or ImŽ k 3 y . ) 0 if ReŽ k 3 y . s 0.
D.K. GramotneÕr Optics Communications 151 (1998) 374–383
The coefficient of absorption M of the incident wave by the layer is given by Ms1yRyT.
Ž8.
Eqs. Ž4., Ž5. and Ž8. are valid for a layer of arbitrary thickness h with the imaginary dielectric permittivity e 2 s i e. So far, no restrictions on magnitudes of e 1,2,3 have been imposed. However, as we will see below, the absorptivity M can be especially large when the layer thickness is small Žas compared with the wavelength and the wave penetration depth in material of the layer. and the layer permittivity is large Žas compared with the dielectric permittivities of media 1 and 3.. Therefore, we will analyse this case in more detail. If the inequality e 4 e 1sin2u 1
Ž9.
is satisfied, then Eq. Ž7. gives a 2 f yi k 02 e, and the wavelength and the wave penetration depth in the material Ž2re .1r2 and d f of the layer are l 2 f 2 p ky1 0 1r2 Ž . ky1 2re , respectively. If, in addition, 0 e 4 e 11r2
Ž < k 3 y
Ž 10.
and the layer thickness is small: k 0 h Ž er2 .
1r2
- 1,
377
waves, we need to recall that it is the magnetic field derivative with respect to the y-coordinate which determines the anomalous absorption in the layer Žsee Section 2.. Moreover, magnetic field in electromagnetic waves is analogous to displacements in shear Žtransverse. acoustic waves. Therefore, we should use the boundary conditions at the layer interfaces written in terms of the magnetic field and then apply the procedure similar to the derivation of the frictional contact approximation in acoustics w2x. The boundary conditions for TE waves, written in terms of magnetic field, are given by
ey1 1 Ž E H1 x rE y y E H1 y rE x . ys0 s yiey1 Ž E H2 xrE y y E H2 yrE x . ys0 , yiey1 Ž E H2 xrE y y E H2 yrE x . ysh s ey1 3 Ž E H 3 x rE y y E H 3 y rE x . ysh ,
Ž H1 x . ys0s Ž H2 x . ys0 , Ž H2 x . yshs Ž H3 x . ysh ,
Ž 15 .
where H1,2,3 are the vectors of the magnetic field in media 1, 2 and 3 respectively. Using inequalities Ž9. – Ž11. and the last two equations of conditions Ž15., we obtain
Ž E H2 xrE y . ys0 f Ž E H2 xrE y . ysh Ž 11.
f Ž H2 x . ysh y Ž H2 x . ys0 rh
then Eqs. Ž4. and Ž5. can be reduced as f Ž H3 x . ys0 y Ž H1 x . ys0 rh,
R f < k 0 y Wc Ž k 1 y y k 3 y . < 2r< k 0 q Wc Ž k 1 y q k 3 y . < 2 ,
T f 4 k 1 y a < W < 2 c 2r< k 0 q Wc Ž k 1 y q k 3 y . < 2 ,
Ž 12.
Ž E H2 yrE x . ys0 f Ž E H2 yrE x . ysh < Ž E H2 xrE y . ys0 ,
Ž 13.
and the first two of conditions Ž15. can be reduced as
ey1 1 Ž E H1 x rE y y E H1 y rE x . ys0
where W s Ž ev h.
y1
.
Ž 14.
The reason for taking modulus of W in the numerator of T will become clear below. Physically, inequality Ž9. means that the wave in the layer propagates almost normally to its boundaries. Inequality Ž10. means that the square of the y-component of the wave vector in the layer must be much larger than the product of the y-components of the wave vectors in the surrounding media 1 and 3. Note that condition Ž10. is always satisfied at the critical angle of incidence at which the absorptivity is maximal Žsee below.. Eqs. Ž12. and Ž13. can also be derived by considering transmission of bulk TE electromagnetic waves through an interface between media 1 and 3 Žwithout the intermediate layer. with some special boundary conditions. This approximation is called frictional contact approximation, because it is analogous to approximating a fluid layer between two solid surfaces by a contact with linear Žliquid. friction w1–6x. To derive boundary conditions corresponding to the frictional contact approximation for TE electromagnetic
s ey1 3 Ž E H 3 x rE y y E H 3 y rE x . ys0 ,
Ž 16a.
ey1 1 Ž E H1 x rE y y E H1 y rE x . ys0 s W Ž E H3 xrE t y E H1 xrE t . ys0 .
Ž 16b.
Comparing these boundary conditions with the boundary conditions in the frictional contact approximation for bulk acoustic waves w1–5x, we can see that Eq. Ž16a. is analogue to equality of mechanical stresses across the frictional contact w2x. Similarly, Eq. Ž16b. is analogue to equality of these stresses to the force of linear friction with the coefficient W, the derivatives EH3 xrEt and EH1 xrEt playing role of velocities of the surfaces in contact. We can also see that the frictional coefficient for TE electromagnetic waves is the same as for TM waves w7x. However, boundary conditions Ž16. are noticeably different from the boundary conditions for the frictional contact approximation for TM waves w7x. Conditions Ž16. at the interface between media 1 and 3 with the dielectric permittivities e 1 and e 3 result in Eqs.
D.K. GramotneÕr Optics Communications 151 (1998) 374–383
378
Ž12. and Ž13. for the reflectivity and transmissivity of TE bulk electromagnetic waves. Thus we can replace four boundary conditions Ž15. by only two conditions Ž16.. In terms of electric field E, conditions Ž16. can be re-written as
Ž E1 . ys0 s Ž E3 . ys0 ,
Ž 17a.
2 2 Ž E1 . ys0 s Wky2 0 Ž E E1 rE yE t y E E3 rE yE t . ys0 . Ž 17b .
In the frictional contact approximation the absorptivity M is determined by Eq. Ž8. with R and T given by Eqs. Ž12. and Ž13.. The maximum of the function M Ž h. is equal to Mm f 2 k 1 y a q k 1 y q < k 3 y < 2 q k 12 y q 2 k 1 y a
Ž
1r2 y1
.
,
Ž 18. and the optimum layer thickness, h m , at which this maximum is achieved, is given by h m s Ž ek 02 .
y1
1r2
< k 3 y < 2 q k 12 y q 2 k 1 y a
.
Ž 19.
If e 3 is real, then Eqs. Ž18. and Ž19. give: Mm f Ž 1 q k 3 yrk 1 y . h m f Ž ek 02 .
y1
y1
,
Ž 20.
Ž k 3 y q k1 y .
Ž 21.
if e 3 ) 0 and u 1 F u 0 , and Mm f 2 k 1 y k 0 Ž e 1 y e 3 . h m f Ž ek 0 .
y1
1r2
y1
q k1 y
,
Ž 22.
Ž e 1 y e 3 . 1r2 ,
Ž 23.
if e 3 - 0, or u 1 ) u 0 . The reflectivity and transmissivity of the wave at h s h m are R m f Ž 1 q k 1 yrk 3 y .
y2
,
Tm f Ž k 1 yrk 3 y . Ž 1 q k 1 yrk 3 y .
1r2
= Ž e1 y e3 .
Ž 24.
Tm s 0
y e 11r2 cos u 1
1r2
q e 11r2 cos u 1
y1
,
Ž 25 .
if e 3 - 0, or u 1 ) u 0 . If condition Ž9. is not satisfied, then a is determined by Eq. Ž7.. However this is not crucial for applicability of the frictional contact approximation Ž16. and Ž17. and Eqs. Ž12. and Ž13.. Indeed, if condition Ž9. is not satisfied, then Eqs. Ž12. and Ž13. can still be obtained using the boundary conditions Ž16. or Ž17. with the complex frictional coefficient W s yik 0r Ž c a 2 h .
< a < 2 4 k1 y < k 3 y <,
Ž 27.
h < a < - 1.
Ž 28.
Thus inequalities Ž27. and Ž28. are the conditions of the frictional contact approximation for TE waves if condition Ž9. is not satisfied. However, for large imaginary dielectric permittivities of the layer Žwhich we consider in this paper and which are also required by conditions Ž10. and Ž27.. it is difficult to imagine a real situation in which condition Ž9. is frustrated. That is why, throughout this paper, we mainly use the frictional coefficient determined by Eq. Ž14.. Note that in the case of TM waves, if condition Ž9. is not satisfied, the coefficient of friction is still determined by Eq. Ž14.. This is another difference in the anomalous absorption of TE and TM electromagnetic waves. If condition Ž10. Žor Ž27.. is not satisfied, we are not able to present equations for R and T in the form of Ž12. and Ž13.. Thus condition Ž10. is vital for the frictional contact approximation. If we inverse it, then the absorptivity is near zero, i.e. the anomalous absorption disappears. Thus condition Ž10. is important not only for the frictional contact approximation, but also for significant absorption in the layer. However, this is the case only for TE waves, whereas a very strong anomalous absorption of TM waves can occur when the condition similar to Ž10. is inversed w10x.
4. Analysis of the solutions y2
if e 3 ) 0 and u 1 F u 0 , and Rm f Ž e1 y e3 .
modulus of W. This was done to make Eq. Ž13. valid also for the complex frictional coefficient Ž26.. In this case other two applicability conditions of the frictional contact approximation Ž10. and Ž11. should be written as
Ž 26.
Žwhich is easily reduced to Eq. Ž14. if condition Ž9. holds.. Now we can see the reason why in Eq. Ž13. we used
Unlike acoustics, where the anomalous absorption does not depend on frequency if conditions similar to Ž9. – Ž11. are satisfied w1–6x, the anomalous absorption in optics is frequency-dependent Žsee also Ref. w7x.. This dependence is represented not only by the explicit dependence of k 1,3 and W on v , but also by dispersion of the dielectric permittivity of the layer. Moreover, large imaginary dielectric permittivities of the layer can be achieved only in the vicinity of a sharp line of absorption, where the dielectric permittivity very strongly Žresonantly. depends on frequency Žsee, for example, an optical resonance related with generation of transverse optical phonons in ionic crystals w11x.. Thus the anomalous absorption in optics appears to be a highly frequency-selective resonant effect, and this is its very important and strong distinction from the anomalous absorption in acoustics w1–6x. From Eqs. Ž20., Ž21. and Ž24. we see that if k 1 y s k 3 y Žhalf spaces with equal dielectric permittivities., then Mm s 0.5, R m s Tm s 0.25, h m s 2 k 1 yrŽ ek 02 ., i.e. the coefficients of absorption, reflection and transmission, corre-
D.K. GramotneÕr Optics Communications 151 (1998) 374–383
sponding to the maximum of absorption, are independent of frequency, angle of incidence, and parameters of the media in contact. In addition, Eqs. Ž13. and Ž24. give TmrT Ž h s 0. s 0.25, i.e. this ratio is also independent of any material parameters. The same invariants are valid for TM electromagnetic waves and for shear acoustic waves polarised perpendicular to the plane of incidence w2x. If the angle of incidence u 1 G u 0 Žor e 3 - 0., then the optimum layer thickness h m is given by Eq. Ž23. which is independent of the angle of incidence. This does not mean that the anomalous absorption is independent of the angle of incidence, but that the maximum of the absorptivity Ždifferent for different values of u 1 – see Eq. Ž22.. is achieved at the same optimum layer thickness for all u 1 G u 0. If the angle of incidence is equal to the critical angle u 0 , then k 3 y s 0 and Eqs. Ž20., Ž22. and Ž24. give Mm s 1, R m s Tm s 0, and h m is determined by Eq. Ž23.. Thus we have total absorption of the TE electromagnetic wave at the critical angle. From Eqs. Ž8., Ž12. and Ž13. we can see that for any value of the layer thickness h satisfying condition Ž11., the dependence of the absorptivity on the incidence angle reaches its maximum at the critical angle. For h s h m , this maximum is equal to 100%. This is a strong distinction of the anomalous absorption of TE electromagnetic waves from the anomalous absorption of TM waves Žfor TM waves we had Mm s 0 at the critical angle w7x.. This difference was explained in Section 2. For example, consider a CaF2 film on a transparent substrate with e 1 s 3 Ž e 3 s 1, i.e. medium 3 is vacuum.. A TE electromagnetic wave with the length 2 prk 0 s 38.905 mm in vacuum Žat this wavelength the dielectric permittivity of the layer e 2 f i233 w11x. is incident on the layer from the substrate at the angle u 1 s 0. Then Eqs. Ž20., Ž21. and Ž24. give Mm f 0.64, R m f 0.13, Tm f 0.23, h m f 73
Fig. 2. The dependences of the absorptivity M of TE electromagnetic waves on the thickness h of a CaF2 film with the permittivity e 2 s i233 Ž2 p r k 0 s 38.905 mm.; e 1 s 3, e 3 s1. Ž1. u 1 s 08, Ž2. u 1 s u 0 f 35.278, Ž3. u 1 s608. Curve 4 presents the dependence of M on h for TM waves at 608 angle of incidence.
379
Fig. 3. The dependences of the absorptivity M, reflectivity R and transmissivity T of TE waves on the angle of incidence u 1 for a CaF2 film of thickness hs h m Ž u 1 G u 0 . f 38 nm with the permittivity e 2 s i233 Ž2 p r k 0 s 38.905 mm.; e 1 s 3, e 3 s1 Ž u 0 f 35.278.. The Mi curve represents the absorptivity in the bulk CaF2 , i.e. at hsq`. The Mc curve represents the absorptivity in the bulk CaF2 within the area of 38 nm thickness near the interface.
nm Žif h ™ q`, then M ™ 0.27.. If u 1 s u 0 f 35.27 0 , then Mm s 1 and h m f 38 nm Žif h s q`, then M f 0.23.. The dependences of the absorptivity M on the thickness h of a CaF2 film with the permittivity e 2 s i233 Ž2 prk 0 s 38.905 mm. on a transparent substrate of e 1 s 3 Žmedium 3 is vacuum: e 3 s 1. are presented in Fig. 2 for three different angles of incidence: u1 s 0 Žcurve 1., u 1 s u 0 f 35.278 Žcurve 2., and u 1 s 608 Žcurve 3.. Curve 4 presents the dependence of M on the layer thickness in the same structure but for TM waves at 608 angle of incidence Žto calculate this dependence, Eqs. Ž1. – Ž3. of Ref. w7x were used.. It is obvious that curve 1 for the normal incidence for TE waves is exactly the same as for TM waves, because at u 1 s 0 there is no difference between these two types of waves. It can be seen that typical widths of the absorptivity maximums for TE waves are smaller than for TM waves, so are typical optimum layer thicknesses corresponding to the absorptivity maximums ŽFig. 2.. The dependences of the absorptivity M, reflectivity R and transmissivity T on the angle of incidence for the CaF2 film of thickness h s 38 nm with the permittivity e 2 s i233 Ž2 prk 0 s 38.905 mm. on a transparent substrate of e 1 s 3 Žmedium 3 is vacuum: e 3 s 1. are presented in Fig. 3. The value of the layer thickness of 38 nm is equal to the optimum thickness h m at the angles of incidence u 1 G u 0 . Therefore, the absorptivity at the critical angle of incidence is equal to 100% ŽFig. 3.. The Mi curve represents the absorptivity in bulk CaF2 , i.e. at h s q`. Thus we can see that the absorptivity characterising the anomalous absorption in a thin layer Žthe M curve., is substantially larger than the conventional absorptivity in the bulk
380
D.K. GramotneÕr Optics Communications 151 (1998) 374–383
medium Žthe Mi curve.. Moreover, the anomalous absorption takes place in a very thin layer Žin our example h s 38 nm., whereas the conventional absorption takes place in a fairly thick area Žwith the thickness of about d ; 570 nm. near the interface. Therefore the Mc curve in Fig. 3 represents the conventional absorption in the area of 38 nm thick near the interface in the semi-infinite medium 2. Comparison of the M and Mc curves in Fig. 3 is more correct and interesting than that of the M and Mi curves, because the Mc and M curves correspond to the conventional and the anomalous absorption, respectively, in regions of the same thickness equal to 38 nm. From the comparison of these two curves, we can see that at the critical angle, the absorptivity due to the anomalous absorption is about 33 times greater than the absorptivity due to the conventional absorption within the same thickness of material 2. In other words, the dissipation in a unit volume of medium 2 is ; 33 times greater for the anomalous absorption than for the conventional one. Fig. 4 is similar to Fig. 3, but the layer thickness h s 9 nm and the dielectric permittivity of the third medium e 3 s 2.9, i.e. very close to e 1 s 3. In this case the critical angle u 0 f 79.488. The layer thickness of 9 nm is equal to the optimum layer thickness h m at u 1 G u 0 . Therefore the absorptivity is equal to 100% at the critical angle. From the comparison of the M and Mc curves in Fig. 4, corresponding to the anomalous and conventional absorption in CaF2 of the same thickness, we can see that the anomalous absorption is ; 500 times greater than the conventional one Žthe absorptivity given by the Mc curve at u 1 s 79.488 is about 2 = 10y3 .. Even if we reduce the layer thickness to 0.9 nm Ži.e. to a couple of atomic layers., we will still get a substantial absorption of about
Fig. 4. The dependences of the absorptivity M, reflectivity R and transmissivity T of TE waves on the angle of incidence u 1 for a CaF2 film of thickness hs h m Ž u 1 G u 0 . f9 nm with the permittivity e 2 s i233 Ž2 p r k 0 s 38.905 mm.; e 1 s 3, e 3 s 2.9 Ž u 0 f 79.488.. The Mi curve represents the absorptivity in the bulk CaF2 , i.e. at hsq`. The Mc curve represents the absorptivity in the bulk CaF2 within the area of 9 nm thickness near the interface.
Fig. 5. The dependences of the optimal thickness h m of a CaF2 layer on the angle of incidence u 1 for TE waves Ž2 p r k 0 s 38.905 mm.; e 1 s 3. Ž1. e 3 s1, Ž2. e 3 s 2, Ž3. e 3 s 2.9, Ž4. e 3 s 3.
35% at the critical angle of incidence. Of course, the presented macroscopic approach is not applicable in this case and, to obtain correct equations for the energy coefficients, we must use a microscopic approach for analysis of the anomalous absorption by just several atomic layers. However, this consideration is not the aim of this paper. Thus we have seen that the optimum layer thickness strongly depends on the dielectric permittivities of the surrounding media and the angle of incidence. In particular, Eq. Ž23. gives that if e 1 ) e 3 ) 0, then the optimum layer thickness at u 1 G u 0 decreases when e 1 ™ e 3 Žcompare with Fig. 3 and 4 plotted for h s 38 nm and 9 nm respectively.. The dependences of h m on the angle of incidence u 1 are presented in Fig. 5 for TE waves Ž2 prk 0 s 38.905 mm. in the structure with a CaF2 layer; e 1 s 3 is the permittivity of medium 1, and the permittivity of medium 3 takes four different values: e 3 s 1 Žcurve 1.; e 3 s 2 Žcurve 2.; e 3 s 2.9 Žcurve 3.; e 3 s 3 Žcurve 4.. As has already been mentioned, the optimum layer thickness is angular independent at incidence angles that are greater than the critical angle Žsee Eq. Ž23. and Fig. 5.. To compare the anomalous absorption of TE and TM waves, the dependences of the absorptivities of both these types of electromagnetic waves in a CaF2 layer Ž2 prk 0 s 38.905, e 2 s i233, e 1 s 3, e 3 s 1. on the angle of incidence are presented in Figs. 6a, 6b for different values of the layer thickness. As expected, at normal incidence both these types of waves give the same absorptivity ŽFigs. 6a, 6b.. The main difference at small layer thicknesses is that the anomalous absorption of TE waves has maximum at the critical angle, while the anomalous absorption of TM waves at this angle is characterised by a sharp pronounced minimum down to zero ŽFig. 6a.. At the thicknesses comparable with the wave penetration depth in the layer, the pronounced dip at the critical angle for TM waves becomes smaller and disappears for the infinite layer thick-
D.K. GramotneÕr Optics Communications 151 (1998) 374–383
Fig. 6. The dependences of the absorptivities of TE and TM electromagnetic waves in a CaF2 layer Ž2 p r k 0 s 38.905, e 2 s i233, e 1 s 3, e 3 s1. on the angle of incidence. Ža. Ž1. TE wave for hs 38 nm, Ž2. TE wave for hs95 nm, Ž3. TM wave for hs 38 nm, Ž4. TM wave for hs95 nm. Žb. Ž1. TE wave for hs800 nm, Ž2. TE wave for hsq`, Ž3. TM wave for hs800 nm, Ž4. TM wave for hsq`.
381
ultra-thin layers. However, because typical layer thicknesses, which are required for the anomalous absorption, can be very small Ža few tens of nanometers, or even a few nanometers., optical parameters of the layer may be noticeably affected by the interfaces. In particular, the dielectric permittivity can vary across the layer. An isolated interface between two semi-infinite media can also result in modifying properties of near-interface regions of media in contact. Thus we obtain very thin layers with varying dielectric permittivity near the interface. Anomalous absorption of electromagnetic waves can be used for investigation of such layers and interfaces. Therefore, anomalous absorption in non-uniform layers with the permittivity varying across the layer is a very important practical problem. Frictional contact approximation appears to be a powerful technique for simple theoretical analysis of the anomalous absorption in such non-uniform thin films. This is because optical properties of the film in this approximation are represented only by the frictional coefficient Žsee Eqs. Ž16. and Ž17... Therefore, if we are able to determine the frictional coefficient W for a thin non-uniform layer, we can easily find the coefficients of reflection, absorption and transmission of the incident wave by means of Eqs. Ž8., Ž12. and Ž13.. Consider a non-uniform layer with imaginary dielectric permittivity varying across the layer, i.e. e is a function of the y-coordinate ŽFig. 1.. If inequalities Ž9. – Ž11. and condition Ž4b. from Ref. w7x are satisfied for all values of e, then the value of yey1EH2rE y is approximately constant across the layer. Here, H2 is the x- or y-component Žfor TE waves., or z-component Žfor TM waves. of the magnetic field in the layer. Then Žin accordance with Eqs. Ž15., Ž16.., F s w Ž y . dÕ,
Ž 29.
where ness ŽFig. 6b.. In addition, a sharp maximum of absorption appears near the critical angle just in front of the dip ŽFig. 6b.. This maximum can be observed only for TM waves. It is probably related with the wave interference inside the layer. At smaller thicknesses this maximum is less sharp and strong. Therefore, it is overwhelmingly suppressed by the strong minimum related with the anomalous absorption ŽFig. 6a.. From Fig. 6b, we can also see that as the layer thickness decreases from a large value, the absorption slightly decreases for almost all angles of incidence, and only then increases dramatically due to the anomalous absorption.
5. Non-uniform layers In the previous sections, we analysed only the anomalous absorption of electromagnetic waves in uniform
F s ey1 2 E H 2 z rE y, dÕ s Ž E H2 zrE t . yqd y y Ž E H2 zrE t . y
for TM waves,
F s ey1 2 Ž E H 2 y rE x y E H 2 x rE y . , dÕ s Ž E H2 xrE t . yqd y y Ž E H2 xrE t . y
for TE waves,
w Ž y . s Ž e v d y .y1 is the coordinate dependent frictional coefficient between two planes y and y q d y inside the layer. The notations are chosen from the view point of analogy with mechanical friction Ž F is analogous to the force of friction, and dÕ to the relative velocity of the planes y and y q d y .. Substituting w Ž y . into Eq. Ž29., rearranging it and integrating over the layer thickness gives Õ 3 y Õ1 s Fv
h
H0 e Ž y . d y s FrW ,
D.K. GramotneÕr Optics Communications 151 (1998) 374–383
382
where Õ1,3 s EH1,3rEt, H1,3 are the x- Žor z-. component of the magnetic field at the interfaces of media 1 and 3, respectively, for TE Žor TM. electromagnetic waves, and W is the total frictional coefficient for the non-uniform layer Žcompare with Eq. Ž16b... Thus y1
ž
h
Ws v
H0 e Ž y . d y
/
.
Ž 30.
If condition Ž9. is not satisfied for the minimal value of the dielectric permittivity in the layer, then in Eq. Ž29. we have to use w Ž y . s yik 0rŽ c a 2 d y ., and the total frictional coefficient is given by Wsy
ik 0 c
y1
ž
h
H0 a
2
Ž y .d y
/
.
Ž 31.
If condition Ž9. is satisfied for all values of the permittivity in the layer, then Eq. Ž31. is reduced to Eq. Ž30.. Calculating the frictional coefficient for a non-uniform layer from Eq. Ž30. or Ž31. and substituting it into Eqs. Ž12., Ž13. and Ž8., we can easily determine the energy coefficients of reflection, transmission and absorption of TE electromagnetic waves. Similarly, substituting the frictional coefficient Ž30. into Eqs. Ž6. and Ž7. of Ref. w7x, we can find the energy coefficients characterising the anomalous absorption of TM waves in a non-uniform layer. Thus using the frictional contact approximation, we can easily analyse the anomalous absorption of bulk electromagnetic waves in non-uniform layers, which would be very difficult to do without this approximation.
6. Conclusions This paper has analysed a new optical effect – the anomalous absorption of bulk TE and TM electromagnetic waves by an ultra-thin film with purely imaginary dielectric permittivity. Detailed analysis of the anomalous absorption was carried out in the frictional contact approximation. The absorptivity maximum and the corresponding optimal layer thickness were determined for TE waves as functions of frequency, angle of incidence, film thickness, and dielectric permittivities of the media in contact. Comparison of the anomalous absorption of TE and TM electromagnetic waves was carried out. The main difference between these two types of waves was discovered near the critical angle, where in the frictional contact approximation the absorptivity of TE waves reaches its maximum of 100%, and the absorptivity of TM waves drops to zero. The anomalous absorption was also shown to display as much as several hundred times greater dissipation in a unit volume of the film than the conventional absorption in the bulk medium Ži.e. at infinite thickness of the film.. As an example, the anomalous absorption of TE and TM bulk electromagnetic waves was analysed in thin films of CaF2 in the infrared range, where a large imaginary
dielectric permittivity of the film can be obtained at the frequency of transverse optical phonons. Typical film thicknesses, which are required for the anomalous absorption in this case, were found to be about several tens of nanometers, or just several nanometers. The frictional contact approximation was demonstrated to be a powerful tool for substantial simplification of theoretical analysis of the anomalous absorption of electromagnetic waves in thin films. This is not only because it allows the reduction of the number of boundary conditions by two, but also because the properties of the film are taken into account only through the frictional coefficient. Therefore the frictional contact approximation allowed us to analyse analytically the anomalous absorption in nonuniform films with varying permittivity. In addition, the frictional contact approximation will be very important for theoretical analysis of electromagnetic waves in complex multi-layered structures, anisotropic media, etc. The anomalous absorption of electromagnetic waves is not restricted to infrared radiation in ionic crystals. It can be realised in the vicinity of any sufficiently sharp absorption line for electromagnetic waves of arbitrary frequency from radio waves to ultraviolet radiation. The only principle requirement is that the dielectric permittivity of the film must have large imaginary part. Current paper was restricted only to large purely imaginary dielectric permittivities of the film. However, such permittivities can be obtained only at some special frequencies, for example, at the frequency of transverse optical phonons in ionic crystals. In this case, even small variations of the frequency result in large Žnegative or positive. real part of the film permittivity w11x. This large real part will definitely affect the anomalous absorption in the film. Therefore, analysis of the anomalous absorption of electromagnetic waves in thin films with complex dielectric permittivities, as well as investigation of frequency response of the anomalous absorption, are important practical problems which will be analysed in our next paper w10x. The anomalous absorption can be realised not only for bulk, but also for guided and surfaces electromagnetic waves. Thus it will be important for development of new signal-processing devices, sensors, measurement and evaluation techniques using bulk and guided optical modes.
Acknowledgements The author is grateful to J.M. Bell and G. Voevodkin for stimulating and fruitful discussions.
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w7x D.K. Gramotnev, Optics Lett. 23 Ž1998. 91. w8x V.M. Agranovich, D.L. Mills, ŽEds.., Surface Polaritons. Electromagnetic Waves at Surfaces and Interfaces, North Holland, Amsterdam, 1982. w9x L.D. Landau, E.M. Lifshitz, Electrodynamics of Continuous Media, Pergamon, New York Ž1984.. w10x D.K. Gramotnev, J.A. Ross, Frequency response of anomalous absorption of electromagnetic waves in a layered structure, Optical and Quantum Electronics, submitted for publication. w11x E.D. Palik. Handbook of Optical Constants of Solids II, Boston, Academic Press, 1991.