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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 16, Iss. 17 — Aug. 18, 2008
  • pp: 13354–13363
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Excitation of s-polarized surface electromagnetic waves in inhomogeneous dielectric media

Kihong Kim  »View Author Affiliations


Optics Express, Vol. 16, Issue 17, pp. 13354-13363 (2008)
http://dx.doi.org/10.1364/OE.16.013354


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Abstract

We consider a model of an inhomogeneous dielectric slab first studied by Shvartzburg, Petite and Auby [J. Opt. Soc. Am. B 16, 966 (1999)] and several variations of that model and study the excitation of s-polarized surface electromagnetic waves on the surface of inhomogeneous dielectric media. Using the invariant imbedding theory of wave propagation in stratified media, we calculate the reflectance and the absorptance of an s wave incident obliquely on a dielectric slab in the Otto configuration, as a function of incident angle and frequency. We also calculate the spatial distribution of the electric field intensity in the inhomogeneous region. We find that in all cases we have considered, s-polarized surface waves are excited at certain incident angles and frequencies. We discuss the physical mechanism of the surface wave generation and the possibility of experimental observations of these effects.

© 2008 Optical Society of America

1. Introduction

Various kinds of surface waves, which propagate along the two-dimensional boundaries of the systems under consideration, have been studied extensively in many branches of science. Well-known examples include surface plasma waves and surface acoustic waves. Surface plasma waves, often referred to as surface plasmons, can be excited on the surface of metals by external electromagnetic radiation and have been the subject of intense research in recent years due to their strong potential for applications in nanophotonic devices and sensors [1–7

1. H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings (Springer-Verlag, 1988).

].

The basic theory of surface plasmons based on Maxwell’s equations indicates that in order to excite them, the incident electromagnetic wave needs to be p-polarized. In addition, because of the large difference in momentum between external electromagnetic waves and surface plasmons, there has to be a way to provide the momentum difference. For this purpose, one uses the Kretschmann or Otto configuration, or breaks the translational symmetry by etching gratings or drilling holes into metal [1

1. H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings (Springer-Verlag, 1988).

, 8

8. T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667–669 (1998). [CrossRef]

].

It has been pointed out by many researchers that in some complex media including nonlinear media, magnetic media and highly inhomogeneous media, s-polarized incident waves can also excite surface electromagnetic waves [9–15

9. G. I. Stegeman, J. D. Valera, C. T. Seaton, J. Sipe, and A. A. Maradudin, “Nonlinear s-polarized surface plasmon polaritons,” Solid State Commun. 52, 293–297 (1984). [CrossRef]

]. Most notably, Shvartzburg, Petite and Auby derived an exact analytical expression for the dispersion relation of s-polarized surface electromagnetic waves which can be excited on the surface of inhomogeneous semiconductors [14

14. A. Shvartzburg, G. Petite, and N. Auby, “S-polarized surface electromagnetic waves in inhomogeneous media: exactly solvable models,” J. Opt. Soc. Am. B 16, 966–970 (1999). [CrossRef]

]. They assumed a special kind of one-dimensional inhomogeneity, where the dielectric permittivity of an inhomogeneous slab takes a maximum value at the surface and decreases monotonically as a specific power law until it reaches a constant value, as the depth from the surface increases.

In the present paper, we first revisit the model studied in [14

14. A. Shvartzburg, G. Petite, and N. Auby, “S-polarized surface electromagnetic waves in inhomogeneous media: exactly solvable models,” J. Opt. Soc. Am. B 16, 966–970 (1999). [CrossRef]

] and point out some crucial misstatement and numerical errors made in that paper. We also study several nontrivial variations of that model for the first time. Our treatment is not limited to obtaining the dispersion relation. Using the invariant imbedding theory of wave propagation in stratified media [16–25

16. R. Bellman and G. M. Wing, An Introduction to Invariant Imbedding (Wiley, 1976).

], we calculate the reflectance and the absorptance of incident s waves in the Otto configuration, which can be directly measured experimentally, in a numerically exact manner. We also calculate the spatial distribution of the electric field intensity inside the inhomogeneous region exactly. We find that s-polarized surface electromagnetic waves are excited in all cases we have studied. This suggests that they can exist in a fairly broad range of inhomogeneous dielectrics.

2. Model

We consider the propagation of an s wave of vacuum wave number k 0=ω/c incident on a stratified medium where the dielectric permittivity ε varies only in the z direction. We assume that the medium lies in 0≤zL and the wave propagates in the xz plane. Then the complex amplitude of the electric filed, E=E(z), satisfies

d2Edz2+[k02ε(z)q2]E=0,
(1)

where q is the x component of the wave vector. We assume that the wave is incident from the region where z>L and ε=ε 1 and is transmitted to the region where z<0 and ε=ε 1. When θ is defined as the angle of incidence, q is equal to ε1k0sinθ .

We notice that the wave equation, Eq. (1), has the same form as the Schrödinger equation for a particle with energy 𝓔(= 2 k 2/2m) incident on a potential V(z) in one dimension, if we identify V(z)/𝓔 with [1-ε(z)/ε 1]/cos2 θ and k with ε1k0cosθ [24

24. K. Kim, F. Rotermund, and H. Lim, “Disorder-enhanced transmission of a quantum mechanical particle through a disordered tunneling barrier in one dimension: Exact calculation based on the invariant imbedding method,” Phys. Rev. B 77, 024203 (2008). [CrossRef]

]. An important point we want to make here is that the quantity [1-ε(z)/ε 1]/cos2 θ plays the role of a potential in quantum mechanics. This leads us to claim that in order to excite s-polarized surface electromagnetic waves in linear dielectric media, it is necessary to have strong inhomogeneity in the z dependence of ε so that it can provide a confining potential for surface waves, similarly to the cases where quantum mechanical particles are confined in a potential well.

In the present paper, we consider a generalized version of the model studied in [14

14. A. Shvartzburg, G. Petite, and N. Auby, “S-polarized surface electromagnetic waves in inhomogeneous media: exactly solvable models,” J. Opt. Soc. Am. B 16, 966–970 (1999). [CrossRef]

]. We assume that the dielectric permittivity of a semiconductor slab of thickness D, which lies in 0≤zD, is given by

ε(z)=εL{1ωp2ω(ω+iγ)[11b+F2(z)b]},
F(z)=(1+Dza)α,ωp2=4πe2N0εLmeff,
(2)

where the function F(z) describes the spatial dependence of ε near the surface at z=D and a is the length scale representing the thickness of the inhomogeneous region. The dimensionless parameter b controls the strength of inhomogeneity. The spatial variation of ε(z) is supposed to describe approximately the carrier depletion layer formed on the surface of a degenerate semiconductor [26–28

26. I. P. Ipatova, A. Yu. Maslov, L. V. Udod, G. Benedek, and G. Panzarini, “The enhancement factor of hyper-Raman scattering from an inhomogeneous semiconductor surface,” Surf. Sci. 377–379, 436–439 (1997). [CrossRef]

]. εL is the frequency-independent part of ε and γ is the damping parameter. The plasma frequency ωp is expressed in terms of N 0, which is the density of charge carriers at z=D, and the carrier effective mass m eff. The dispersion relation for surface waves when α=1 was derived and analyzed in detail in [14

14. A. Shvartzburg, G. Petite, and N. Auby, “S-polarized surface electromagnetic waves in inhomogeneous media: exactly solvable models,” J. Opt. Soc. Am. B 16, 966–970 (1999). [CrossRef]

]. In this paper, we reanalyze this problem and point out some misstatement and numerical errors made in that paper. Our theory is not limited to obtaining the dispersion relation and to the α=1 case. We consider the cases where α=0.5, 0.75, 1, 1.25 and 1.5 and calculate the reflectance, the absorptance and the electric field distribution inside the inhomogeneous region.

In Fig. 1, we show typical spatial variations of ε(z) considered here. We use the parameter values N 0=3×1016 cm-3, εL=12, m eff=0.01me, γ=0 and b=-0.53, where me is the electron mass. The value of the plasma frequency ωp is about 2.82×1013/sec. We choose ω=5×1013/sec and α=0.5,1,1.5 and plot (1-ε), which plays the role of an effective potential when θ=0, as a function of the normalized distance from the surface of a semiconductor in contact with the air. We note that there is a deep potential well near the surface. Since the effective potential is inversely proportional to cos2 θ, the potential well becomes infinitely deeper as θπ/2.

Fig. 1. Spatial variations of (1-ε), when an inhomogeneous dielectric slab located in 0≤zD is in contact with the air in the region z>D. The dielectric permittivity of the inhomogeneous slab is given by Eq. (2) with the parameter values N 0=3×1016 cm-3, εL=12, m eff=0.01me, γ=0, b=-0.53, ω=5×1013/sec and α=0.5,1,1.5.

Similarly to the case of the excitation of surface plasmons by incident p waves in layered metallic structures, it is necessary to use the Kretschmann or Otto configuration to compensate the large difference in momentum between the incident electromagnetic waves and the surface waves. For this purpose, we place a uniform dielectric slab with ε=εd (<ε 1) in D<z<L.

3. Dispersion relation in a special case

In the special case where the exponent α is equal to 1 in Eq. (2), it is possible to transform Eq. (1) into the Bessel equation

d2Qdu2+1udQdu+(s2ρ2u2)Q=0,
(3)

where

Q(u)=E(z)F(z),u=1+Dza,
ρ2=εLb(ωpac)2+14,s2=(qa)2εL(ωac)2+εL(ωpac)2(11b).
(4)

In order to have a solution that decays exponentially as u increases to infinity, we need the condition that s 2 should be always positive. Then the solution of Eq. (3) is given by the modified Bessel function of the second kind, K ρ(su). In [14

14. A. Shvartzburg, G. Petite, and N. Auby, “S-polarized surface electromagnetic waves in inhomogeneous media: exactly solvable models,” J. Opt. Soc. Am. B 16, 966–970 (1999). [CrossRef]

], it has been claimed that ρ2 should also be nonnegative in order to have a decaying solution. This latter condition, however, is unnecessary, since K ρ(x) with a pure imaginary ρ decays exponentially as x→∞, too [29

29. T. M. Dunster, “Bessel functions of purely imaginary order, with an application to second-order linear differential equations having a large parameter,” SIAM J. Math. Anal. 21, 995–1018 (1990). [CrossRef]

].

We assume that the inhomogeneous medium described by the Bessel equation is in contact with the air. By expressing the electric field E and the transverse component of the magnetic field, Hx, in the medium in terms of the Bessel function K ρ(su) and using the continuity of these fields across the surface, we obtain the dispersion relation for surface waves propagating in the x direction along the surface:

(qa)2=(ωac)2+[12+1Kρ(s)Kρ(su)uu=1]2,
(5)

which relates q and ω in an implicit manner.

Fig. 2. Dispersion relations of s-polarized surface waves, q(ω), for the model with α=1 in Eq. (2). The parameters used in the calculation are N 0=3×1016 cm-3, εL=12, m eff=0.01me, a=10-4 cm, γ=0 and b=-0.4,-0.53,-0.72,-1.1,-2.1. The dashed line represents the dispersion of light in a vacuum.

Fig. 3. Incident angle at which an s-polarized surface wave is excited plotted versus the frequency of the incident wave, when α=1, N 0=3×1016 cm-3, εL=12, m eff=0.01me, a=10-4 cm, γ=0, b=-0.53 and ε 1=2.25. The point P corresponds to ω=5.2×1013/sec and θ=64.88°.

4. Invariant imbedding method

We consider a plane s wave of unit magnitude (x, z)=E(z)exp(iqx)=exp[ip(L-z)+iqx], where p=ε1k0cosθ , incident on the medium from the region where z>L. The quantities of main interest are the reflection and transmission coefficients, r=r(L) and t=t(L), defined by the wave functions outside the medium:

E˜(x,z)={[eip(Lz)+r(L)eip(zL)]eiqx,z>Lt(L)eipz+iqx,z<0.
(6)

Using the invariant imbedding theory of wave propagation, we have derived exact ordinary differential equations satisfied by r and t:

1iε1k0drdl=2cosθr(l)+ε˜(l)12cosθ[1+r(l)]2,
1iε1k0dtdl=cosθt(l)+ε˜(l)12cosθ[1+r(l)]t(l),
(7)

where ε˜=ε/ε 1 [19

19. K. Kim, H. Lim, and D.-H. Lee, “Invariant imbedding equations for electromagnetic waves in stratified magnetic media: Applications to one-dimensional photonic crystals,” J. Korean Phys. Soc. 39, L956–L960 (2001).

]. These are supplemented with the initial conditions r(0)=0 and t(0)=1. For given values of k 0 and θ and for an arbitrary function ε(l), we integrate Eq. (7) from l=0 to l=L and obtain the reflection and transmission coefficients r(L) and t(L).

We have also derived the invariant imbedding equation for the electric field amplitude inside the inhomogeneous medium, which has the form

1iε1k0El=cosθE(z,l)+ε˜(l)12cosθ[1+r(l)]E(z,l),
(8)

where E is considered to be a function of z and l. For a given z(0<z<L), E(z,L) is obtained by integrating this equation, together with the equation for r(l) in Eq. (7), from l=z to l=L using the initial condition E(z, z)=1+r(z).

5. Results

We first consider the case where the exponentα is equal to 1. In Fig. 4, we show the reflectance R (=|r|2) and the absorptance A (=1-|r|2-|t|2) of an s wave of frequency ω=5.2×1013/sec as a function of incident angle. In all results shown in this section, we use the parameter values N 0=3×1016 cm-3, εL=12, m eff=0.01me, a=10-4 cm, ε 1=2.25, εd=1, D=100a and b=-0.53. In Fig. 4, the thickness of the gap between the inhomogeneous dielectric slab and the prism is 15a and the damping parameter γ is 0.001 in (a) and (b) and 0.01 in (c) and (d). We observe that there appears a very sharp absorption peak at θ=64.88°, which is due to the excitation of a surface wave. This angle agrees perfectly with the value obtained from the dispersion relation shown in Fig. 3 and is well above the critical angle θc. We find that as γ increases, the absorption peak gets broadened and the maximum value of A decreases.

Fig. 4. (a), (c) Reflectance and (b), (d) absorptance of an s wave of frequency ω=5.2×1013/sec incident obliquely on an inhomogeneous dielectric slab, the permittivity of which is given by Eq. (2) with α=1, in the Otto configuration, plotted versus incident angle. In Figs. 4–7, we use the common parameter values N0=3×1016 cm-3, εL=12, m eff=0.01me, a=10-4 cm, ε 1=2.25, εd=1, D=100a and b=-0.53. The damping parameter γ is 0.001 in (a) and (b) and 0.01 in (c) and (d). The thickness of the air gap, L-D, is equal to 15a. Note that there appear sharp reflection dips and absorption peaks at θ=64.88°, which corresponds precisely to the point P in Fig. 3.

In Fig. 5, we plot the reflectance and the absorptance of an s wave incident at θ=65° versus frequency, obtained for the same parameters as in Fig. 4. When γ=0.001, we find an extremely sharp and narrow absorption peak at ω=5.2×1013/sec, which is again due to the excitation of a surface wave. Similarly to the behavior shown in Fig. 4, as γ increases, the absorption peak gets broadened and the maximum value of A decreases.

In Fig. 6, we show the spatial distributions of the electric field intensity inside the inhomogeneous medium, when an s wave of frequency ω=5.2×1013/sec is incident at θ=64.88° and 64°, obtained for the same parameters as in Figs. 4 and 5. The vertical dashed line indicates the boundary between the inhomogeneous medium located in 0≤z≤100a and the air gap located in 100az≤115a. The wave is assumed to be incident from the right and transmitted to the left. We find that the electric field is very strong near the boundary in the θ=64.88° case, which clearly demonstrates the excitation of a surface wave. The fact that the electric field becomes much weaker when the incident angle is slightly different from 64.88° shows that the surface wave excitation is a resonant phenomenon.

Fig. 5. (a), (c) Reflectance and (b), (d) absorptance of an s wave incident at θ=65° on an inhomogeneous dielectric slab, the permittivity of which is given by Eq. (2) with α=1, in the Otto configuration, plotted versus frequency. L is 115a and γ is equal to 0.001 in (a) and (b) and 0.01 in (c) and (d).
Fig. 6. Spatial distributions of the electric field intensity inside the inhomogeneous medium, when an s wave of frequency ω=5.2×1013/sec is incident at θ=64.88° and 64° on an inhomogeneous dielectric slab, the permittivity of which is given by Eq. (2) with α=1, in the Otto configuration. L is 115a and γ is 0.001. The vertical dashed line indicates the boundary between the inhomogeneous medium located in 0≤z≤100a and the air gap located in 100az≤115a. The wave is assumed to be incident from the right side.

So far we have discussed the excitation of surface waves in the special case where the exponent α is equal to 1. We have done similar calculations for several other values of the exponent α and found that the surface waves are excited in those cases, too. In Figs. 7(a–d), we show the absorptance A of an s wave incident obliquely on an inhomogeneous dielectric slab, the permittivity of which is given by Eq. (2) with α=0.5, 0.75, 1.25 and 1.5, in the Otto configuration as a function of incident angle. The thickness of the air gap is 20a and the damping parameter γ is 0.001. The frequency of the incident wave is 5×1013/sec in (a) and (b) and 5.05×1013/sec in (c) and (d). All other parameters are the same as in Fig. 4. We find sharp absorption peaks in all cases, at θ=68.82°, 53.01°, 48.12° and 47°, when α is 0.5, 0.75, 1.25 and 1.5 respectively. In Figs. 7(e–h), we show the spatial distributions of the electric field intensity corresponding to the absorption peaks listed above. We find that the surface waves are indeed excited near the boundary in all of these cases. It appears that they are more strongly and narrowly confined to the surface when α is smaller.

Fig. 7. (a–d) Absorptance A of an s wave incident obliquely on an inhomogeneous dielectric slab, the permittivity of which is given by Eq. (2) with α=0.5, 0.75, 1.25 and 1.5, in the Otto configuration, plotted versus incident angle. L is 120a and γ is 0.001. The frequency ω is 5×1013/sec in (a) and (b) and 5.05×1013/sec in (c) and (d). (e–h) Spatial distributions of the electric field intensity, |E|2, corresponding to θ=68.82° in (a), θ=53.01° in (b), θ=48.12° in (c) and θ=47° in (d). The vertical dashed line indicates the boundary between the inhomogeneous medium and the air gap. The wave is assumed to be incident from the region where z≥120a.

6. Discussion and conclusion

The s-polarized surface waves we have considered so far are similar in nature to the waveguide modes propagating in an inhomogeneous dielectric slab. They are also similar to the bound states of a quantum mechanical particle in a potential well. In our case, the effective potential well has a sharp minimum at the surface so that the bound states are strongly localized near there. The fact that we found s-polarized surface waves for all cases we considered suggests that they exist for a fairly broad range of functional forms of the dielectric permittivity. As long as ε decreases sufficiently rapidly near the surface of a dielectric and a deep effective potential well is formed there, there is a possibility to observe s-polarized surface waves. Since this condition is expected to be satisfied in heavily doped degenerate semiconductors due to the formation of a carrier depletion layer [26–28

26. I. P. Ipatova, A. Yu. Maslov, L. V. Udod, G. Benedek, and G. Panzarini, “The enhancement factor of hyper-Raman scattering from an inhomogeneous semiconductor surface,” Surf. Sci. 377–379, 436–439 (1997). [CrossRef]

], we believe that the phenomenon discussed in this paper can be observed in those materials. We also point out that it is possible to artificially fabricate thin layers with designed refractive index profiles by various methods [15

15. A. B. Shvartsburg, V. Kuzmiak, and G. Petite, “Optics of subwavelength gradient nanofilms,” Phys. Rep. 452, 33–88 (2007). [CrossRef]

, 30–32

30. L. A. A. Pettersson, L. Hultman, and H. Arwin, “Porosity depth profiling of thin porous silicon layers by use of variable-range spectroscopic ellipsometry: a porosity graded-layer model,” Appl. Opt. 37, 4130–4136 (1998). [CrossRef]

].

In summary, we have studied the possibility of the excitation of s-polarized surface electromagnetic waves for several models of an inhomogeneous dielectric slab. We have calculated the reflectance, the absorptance and the electric field distribution associated with an s wave incident obliquely on an inhomogeneous dielectric slab in the Otto configuration. In all models we have considered, s-polarized surface waves are found to be excited at certain angles and frequencies. We have discussed the physical mechanism of the surface wave generation and argued that these effects can be observed experimentally in degenerate semiconductors and artificially fabricated materials.

Acknowledgments

This work has been supported by the Korea Science and Engineering Foundation grant (No. R0A-2007-000-20113-0) funded by the Korean Government.

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G. I. Stegeman, J. D. Valera, C. T. Seaton, J. Sipe, and A. A. Maradudin, “Nonlinear s-polarized surface plasmon polaritons,” Solid State Commun. 52, 293–297 (1984). [CrossRef]

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A. Shvartzburg, G. Petite, and N. Auby, “S-polarized surface electromagnetic waves in inhomogeneous media: exactly solvable models,” J. Opt. Soc. Am. B 16, 966–970 (1999). [CrossRef]

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A. B. Shvartsburg, V. Kuzmiak, and G. Petite, “Optics of subwavelength gradient nanofilms,” Phys. Rep. 452, 33–88 (2007). [CrossRef]

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R. Bellman and G. M. Wing, An Introduction to Invariant Imbedding (Wiley, 1976).

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K. Kim, F. Rotermund, and H. Lim, “Disorder-enhanced transmission of a quantum mechanical particle through a disordered tunneling barrier in one dimension: Exact calculation based on the invariant imbedding method,” Phys. Rev. B 77, 024203 (2008). [CrossRef]

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I. P. Ipatova, A. Yu. Maslov, L. V. Udod, G. Benedek, and G. Panzarini, “The enhancement factor of hyper-Raman scattering from an inhomogeneous semiconductor surface,” Surf. Sci. 377–379, 436–439 (1997). [CrossRef]

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T. M. Dunster, “Bessel functions of purely imaginary order, with an application to second-order linear differential equations having a large parameter,” SIAM J. Math. Anal. 21, 995–1018 (1990). [CrossRef]

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L. A. A. Pettersson, L. Hultman, and H. Arwin, “Porosity depth profiling of thin porous silicon layers by use of variable-range spectroscopic ellipsometry: a porosity graded-layer model,” Appl. Opt. 37, 4130–4136 (1998). [CrossRef]

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OCIS Codes
(240.6690) Optics at surfaces : Surface waves
(260.2710) Physical optics : Inhomogeneous optical media

ToC Category:
Optics at Surfaces

History
Original Manuscript: July 9, 2008
Manuscript Accepted: August 11, 2008
Published: August 14, 2008

Citation
Kihong Kim, "Excitation of s-polarized surface electromagnetic waves in inhomogeneous dielectric media," Opt. Express 16, 13354-13363 (2008)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-17-13354


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References

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