Recent Optical and Photonic Technologies 2012 Part 9 potx

Recent Optical and Photonic Technologies 228 Hu, B. B.; Zhang, X C. & Auston, D. H. (1990). Free-space radiation from electro-optical crystals. Appl. Phys. Lett., Vol. 56, No. 6, (February 1990) 506-508 Kapteyn, H. C. (1992). Photoionization-pumped x-ray lasers using ultrashort-pulse excitation. Appl. Opt., Vol. 31, No. 24, (August 1992) 4931-4939 Kawachi, T.; Kado, M.; Tanaka, M.; Sasaki, A.; Hasegawa, N.; Kilpio, A. V.; Namba, S.; Nagashima, K.; Lu, P.; Takahashi, K.; Tang, H.; Tai, R.; Kishimoto, M.; Koike, M.; Daido, H. & Kato, Y. (2002). Gain saturation of nickel-like silver and tin x-ray lasers by use of a tabletop pumping laser system. Phys. Rev. A, Vol. 66, No. 3, (September 2002) 033815 Klebniczki, J.; Hebling, J. & Kuhl, J. (1990). Generation of tunable femtosecond pulses in a traveling-wave amplifier. Opt. Lett., Vol. 15, No. 23, (December 1990) 1368- 1370 Kobayashi, T. & Shirakawa, A. (2000). Tunable visible and near-infrared pulse generation in a 5 fs regime. Appl. Phys. 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Introduction In recent years, the investigation of composite material systems in the terahertz (THz) regime has drawn a considerable attention from a wide spectrum of scientific areas, for instance the fields of nano-science (Beard et al., 2002), (Hendry et al., 2006) and metamaterials (Levy et al., 2007). The interaction of terahertz waves with a composite system consisting of particles embedded in a host material as illustrated in Fig. 1 can be described by effective material properties and effective medium theories (EMTs) enabling the calculation of the resulting macroscopic permittivity ε R . ε R E H Fig. 1. The interaction between an electromagnetic wave and a composite system can be described by an effective permittivity ε R . If the particle size is much smaller than the wavelength of interest, as visualized in Fig. 2, scattering effects are negligible and quasi-static models suffice. Otherwise, scattering effects have to be taken into account. In this book chapter we will review common quasi-static EMTs and their application to various composite material systems. The selection of theoretical models comprises the Landau-Lifshitz-Looyenga model, which is applicable to mixtures of arbitrarily shaped particles, the Polder-van-Santen theory, which explicitly considers the influence of the inclusions shape and orientation, the differential Bruggeman theory and a recent extension to the latter proposed by the authors. Recent Optical and Photonic Technologies 232 λ<<r λ>>r r r ε R Fig. 2. In the case of small particles compared to the wavelength, quasi-static models suffices, otherwise scattering effects as to be taken into account. The first application scenario that we will study is the characterisation of polymeric compounds. By adding microscopic particles to a polymeric host material, the resulting properties of the plastic like colour, material strength and flammability can be optimized. Moreover, the additives induce a change of the optical parameters of the mixture that can be studied with terahertz time domain spectroscopy (THz TDS). The resulting refractive index depends on the volumetric content and the dielectric constant of the additives as well as the particle shape. Due to the variety of commonly used additives, ranging from rod like glass fibres, over cellulose based fillers to spherical nanoparticles, polymeric compound systems are ideal to illuminate the applicability and limitations of the different EMTs. Apart from the polymeric compounds, we will also discuss the usability of the EMTs to describe biological systems. As one example, the water content of plant leaves considerably effects their dielectric properties. Utilizing the EMTs allows for the determination of the water content of the plants with terahertz radiation. In summary, the chapter will review a selection of effective medium theories and outline their applicability to various scientific problems in the terahertz regime. Additionally, a short overview on the THz time domain spectroscopy (TDS) which is employed to experimentally validate the models' predictions is presented. 2. Effective medium theories The analysis of dielectric mixture systems, for instance particles embedded in a host material, is a problem of enormous complexity if every single particle is considered individually. Alternatively, the resulting macroscopic material parameter of the mixture can be derived which characterize the interaction between the material system and electromagnetic waves. To calculate this effective material parameter, effective medium theories (EMTs) can be employed. In this chapter, we will exemplarily present a selection of the most common quasi static EMTs which can directly be applied to the description of heterogeneous dielectrics in the THz range. Table 1 provides a basic overview of the characteristics of these models, which will be further described below. 2.1 Maxwell-Garnett One of the first and probable the most well known EMT is the Maxwell-Garnett (MG) model (Maxwell-Garnett, 1904) which is based on analyzing the effective polarizability of spherical inclusions with the permittivity ε p embedded in a vacuum environment as illustrated in Fig. 3. Applications of Effective Medium Theories in the Terahertz Regime 233 Model Volumetric content Particle shape Area of Application Maxwell- Garnett Low spheres Very low concentrations Polder and van Santen High ellipsoidal Ellipsoidal particles, anisotropic systems Extended Bruggeman High ellipsoidal High permittivity contrast, ellipsoidal particles, anisotropic systems Landau, Lifshitz, Looyenga Middle arbitrary Mixtures of irregular, unknown shaped particles Complex Refractiv Index middle arbitrary Mixtures with small permittivity contrast Table 1. Overview of the EMTs mentioned in the text. ε p a ε h =1 EE Fig. 3. To derive the MG model, the resulting polarizability of a single spherical particle is derived. Following the basics of electrostatics the resulting polarizability α p of a single spherical particle is given by (Jackson, 1999) 3 0 1 4 2 p p p a ε απε ε − = + (1) where ε 0 is the permittivity of the vacuum and a is the radius of the particle. Now it is assumed, that the polarizability remains constant if multiple particles are present. Consequently the Clausius Mossoti relation (Kittel, 1995) that connects the relative permittivity ε r of a material with the polarizability of a number of N microscopic particles 0 1 23 j j j r r N α ε εε − = + ∑ (2) can be exploited to calculate the effective permittivity ε R of this inhomogeneous medium, where f p is the volumetric content of the particles: 1 1 22 p R p Rp f ε ε εε − − = + + (3) Recent Optical and Photonic Technologies 234 If the particles are embedded in a host material with given permittivity ε h , Eq. 3 changes into the MG equation: 22 p h Rh p R hph f ε ε εε ε εεε − − = ++ (4) As can be seen from these deductions, the assumption is violated if a larger volumetric fraction of the medium is formed by the inclusions, since in this case the effective background permittivity changes. Thus, the model can be applied to very low concentrations only. 2.2 Polder and van Santen Another approach with extended validity was derived by Polder and van Santen: Instead of employing the host ε h in the calculation to derive Eq. 4, the effective dielectric constant ε R is utilized. That way, the effect of the slightly increasing effective background permittivity can be taken into account. The equation 32 p h Rh p R pR f ε ε εε ε εε − − = + (5) results, which is known as the Böttcher equation (Böttcher, 1942). Despite this extension, the model is still restricted to spherical shaped inclusions. By including depolarization factors N in the deductions, it is possible to expand the validity to ellipsoidal particles. These factors can be calculated by the following equations (Kittel, 1995): () 2222 0 2 ()()() lll x llll xyz du N x uxuyuzu ∞ = + +++ ∫ (6) 1 xyz NNN + += (7) The Fig. 4 shows the numerically calculated N x values for different aspect ratios between the axis x and y in a) and the axis x and z in b). Fig. 4. Values of the depolarisation factor N x as a function of the aspect ratio between the axis x and y in a) and the axis x and z in b). Applications of Effective Medium Theories in the Terahertz Regime 235 In the case of ideal disc-like particles the aspect ratio x/y converges toward zero while the N x value tends towards unity. For ideal rod like particles, the aspect ratio increase to infinity and N x descends to zero. These shapes are illustrated together with the resulting depolarization factors in Fig. 5. N=0x N =1/3x N=1x x y z Fig. 5. Values of the deplarisation factor N x for a) a rod b) a sphere and c) a disc 32 p h Rh p R pR f ε ε εε ε εε − − = + (5) Analogously to Eq. 5 the effective material parameter can be calculated by employing these factors which results in the Polder and van Santen (PvS) model (Polder & van Santen, 1946): () 3 11 1 3() h R pp h i R pRi f N ε ε εε εεε = −− +− ∑ (8) The special forms of the PvS model for ideal shapes, which are orientated isotropically in the mixture, are the following (Hale, 1976): Spheres: 32 p h Rh p R pR f ε ε εε ε εε − − = + (9) Discs: 23 p h Rh p Rp p f ε ε εε εε ε − − = + (10) Rods: 53() ph Rh p R RpR f ε ε εε ε εεε − − = ++ (11) As the Böttcher model is a special case of the PvS model, the Eq. 9 for spherical shaped particles equals the Böttcher equation Eq. 5. Due to the consideration of the influence of the particles shape and the increasing background permittivity, the PvS model is widely applicable. Especially anisotropic mixture systems like orientated glass fibres can be described by this approach. Recent Optical and Photonic Technologies 236 2.3 Bruggeman While the PvS model well describes a variety of mixtures, a strong contrast in the permittivity between the mixture components still affects its validity. Here, a differential approach (Bruggeman, 1935) can be utilized. The Bruggeman theory makes use of a differential formulation of Eq. 4. After integration, the equation results: 3 1 h R pR p ph f ε ε εε εε − −= − (12) which is the basic form of the Bruggeman model. This basic form describes spherical particles embedded in a host where a large contrast in permittivity occurs. By combining the two approaches (Bruggeman and PvS), more general forms of this model can be derived (Banhegyi 1986), (Scheller et al., 2009, a). The equation for this extended Bruggeman [EB] model in the general case, where one polarization factor is given by N, the other two by 1-N/2 is: ()() ()() 2 2 2 12 18 2 33 9125 31 13 53 1 13 53 NN NN NN N pR p h h p RphpR NN f NN εε ε ε ε εεεεε ⎡ ⎤ −− ⎡⎤ −+ ⎢ ⎥ ⎢⎥ −− ⎢ ⎥ + ⎣ ⎦ ⎢⎥ ⎣⎦ ⎛⎞⎛ ⎞ −++− ⎛⎞ =− ⎜⎟⎜ ⎟ ⎜⎟ ⎜⎟⎜ ⎟ −++− ⎝⎠ ⎝⎠⎝ ⎠ (13) For the case of isotopically orientated particles with ideal shapes the following set of equations results: Spheres: 32 p h Rh p R pR f ε ε εε ε εε − − = + (14) Discs: 2 1 2 pR ph p ph pR f ε εεε ε εεε ⎛⎞⎛ ⎞ −+ −= ⎜⎟⎜ ⎟ ⎜⎟⎜ ⎟ −+ ⎝⎠⎝ ⎠ (15) Rods: 2 5 5 1 5 pR hp p ph R p f εε εε εε εε ⎡ ⎤ ⎢ ⎥ ⎣ ⎦ ⎛⎞⎛ ⎞ −+ −= ⎜⎟⎜ ⎟ ⎜⎟⎜ ⎟ −+ ⎝⎠⎝ ⎠ (16) 2.4 Landau, Lifshitz, Looyenga Additionally, the Landau, Lifshitz, Looyenga (LLL) model (Looyenga 1965) makes use of a different assumption: Instead of taking the shape of the particles into account a virtual sphere is considered, which includes a given volumetric fraction of particles with unknown shape as illustrated in Fig. 3. By successively adding an infinitismal amount of particles, the effective permittivity increases slightly which can be described by a Taylor approximation. This procedure leads to the equation. ( ) 3 3 3 1 R pp p h ff ε εε =+− (17) [...]... M.; Salin, F ( 199 3) Self-mode locking of solid-state lasers without apertures, Opt Lett., OSA, 199 3, 18, 1041 Polder, D.; van Santen, J H ( 194 6) The effective permeability of mixtures of solids Physica,Vol No 12, 5, 194 6, 257-271 250 Recent Optical and Photonic Technologies Ralph, S E.; Grischkowsky, D ( 197 4) Trap-enhanced electric fields in semi-insulators: The role of electrical and optical carrier... 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No. 6, (February 199 0) 506-508 Kapteyn, H. C. ( 199 2). Photoionization-pumped x-ray lasers using ultrashort-pulse excitation. Appl. Opt., Vol. 31, No. 24, (August 199 2) 493 1- 493 9 Kawachi, T.;. A. ( 199 8). Efficient broadband second-harmonic generation by dispersive achromatic nonlinear conversion using only prisms. Opt. Lett., Vol. 23, No. 7, (April 199 8) 497 - 499 Saikan, S. ( 197 6) Recent Optical and Photonic Technologies 228 Hu, B. B.; Zhang, X C. & Auston, D. H. ( 199 0). Free-space radiation from electro -optical crystals. Appl. Phys.