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Interaction of Light with Solids

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Abstract

The interaction of light with solids is described by Maxwell’s equations, which treat the solid as a continuum and lead to its optical parameters as a function of the frequency of the electromagnetic radiation: the complex dielectric constant. The dielectric constant describes the ability of a solid to screen an electric field – with electronic and ionic contributions – and is one of the most important material parameters. This function is closely related to the index of refraction and the optical absorption (or extinction) coefficient. All these parameters are derived from measured quantities: the transmitted and reflected light as a function of wavelength, impinging angle, and polarization.

A periodic modulation of the dielectric constant along a spatial direction leads to a photonic bandgap for the propagation of specific modes along this direction, analogous to the electronic bandgap for electrons traveling in the periodic crystal potential. A complete bandgap for propagation along any direction can be created for three-dimensional periodicity; defects given by deviations from periodicity lead to localized states in such photonic crystals, similar to effects in the electronic counterpart, allowing for, e.g., waveguiding or suppresion of spontaneous emission.

At high field amplitudes, nonlinear optical effects occur due to the nonparabolicity of the lattice potential. These effects can be described by a field-dependent dielectric function. The resulting nonharmonic oscillations permit mixing of different signals with corresponding change in frequency and amplitude.

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Notes

  1. 1.

    This chapter contains quite a few equations; the most important relations are pointed out by a shading.

  2. 2.

    The operators \( \frac{\partial }{\partial \mathbf{r}}\times \) and \( \frac{\partial }{\partial \mathbf{r}}\cdot \) are also written as \( \nabla \times \) or rot and \( \nabla \cdot \) or div, respectively, with \( \nabla \) being the Nabla operator. The symbolic vector \( \frac{\partial }{\partial \mathbf{r}} \) has the components \( \left(\frac{\partial }{\partial x},\frac{\partial }{\partial y},\frac{\partial }{\partial z}\right) \), and the “×” and “\( \cdot \)” operations yield consequently a vector and a scalar, respectively.

  3. 3.

    An index r is added to the widely used symbol n to distinguish this quantity from the carrier density n.

  4. 4.

    We can understand this by equating damping with transfer of energy into heat and optical absorption with extraction of this energy from the radiation field. Such absorption occurs even outside a specific electronic or ionic resonance absorption – see chapters “Photon–Phonon Interaction” and “Photon–Free-Electron Interaction.”

  5. 5.

    The components of the electric field vector parallel and perpendicular to the plane of incidence, \( E_\| \mathrm{and}\, {E}_{\perp } \), are also referred to as p-polarized and s-polarized (s from German for senkrecht), respectively.

  6. 6.

    A Brewster angle Φ B is defined as the angle under which no component E || is reflected. This happens for Φ tΦ r, or Φ t + Φ r = 90°; from this, we obtain with Eq. 38 n r1 sin Φ i = n r2 cos Φ r, and eventually

    $$ \tan {\varPhi}_{\mathrm{B}}={n}_{\mathrm{r}2}/{n}_{\mathrm{r}1}. $$
    (39)
  7. 7.

    Reflectance (etc.) is used rather than reflectivity since it is not normalized to the unit area; this is similar to the use of the word resistance (not normalized) versus resistivity, distinguishing between the suffixes -ance and -ivity.

  8. 8.

    The reflected light is elliptically polarized if there is absorption in the sample. Without absorption, the reflected light remains linearly polarized, but with an altered angle of polarization with respect to the plane of incidence.

  9. 9.

    In semiconductors, mostly free electrons rather than ions (as in the Debye-Hückel theory) provide the screening. This characteristic length in semiconductor physics is more commonly called the Debye length.

  10. 10.

    An exception was the inclusion of an overlayer considered in ellipsometry (Eq. 84), leading to the description by an effective yet spatially not varying dielectric function 〈ε〉.

  11. 11.

    Equation 98 expressed in terms of E is correct but not hermitian, see Joannopoulos et al. (2008).

  12. 12.

    Here, n is just a label not to be confused with the refractive index; the number of eigenvalues ω n is given by the size of the (generally truncated) matrix in the secular equation of the eigenvalue problem for a periodically perturbed ε. Considering more plane waves enlarges the secular equation and thereby the number of solutions ω n for a given k including increasingly more contributions of shorter waves with higher frequencies.

  13. 13.

    There are only two ways to center such standing waves: the nodes can either be centered in each ε > layer or in each ε < layer; other positions violate the symmetry of the unit cell about its center.

  14. 14.

    Instead of the notations M and K, also the labels X and J, respectively, are used in literature for critical points of a two-dimensional hexagonal Brillouin zone.

  15. 15.

    Light generated, e.g., by luminescence within a GaAs crystal (n r ≅ 3.5) is coupled out to air by only ~4% due to the small critical angle of total reflection (~17°, Eq. 40) and the consequential small fraction of the solid angle. The resulting poor external quantum efficiency is strongly increased if the light is emitted into the defect mode of a cavity cladding the light emitter. This is of particular importance for the performance of single-photon sources.

  16. 16.

    The emission of the quantum dot shifts to lower energy at increased temperature according to the change of the bandgap described by the Varshni dependence Eq. 44 of chapter “Bands and Bandgaps in Solids”; the energy of the cavity resonance also experiences a redshift by the T-dependence of the refractive index but at a much lower rate.

  17. 17.

    Material destruction is avoided by monochromatic irradiation in a wavelength range of little absorption and by the use of short pulses. Material destruction can occur by simple lattice heating, by dielectric breakdown (109 W/cm2 is equivalent to 106 V/cm oscillation amplitude and achieved by focusing a 103 W laser on a spot of 10 μm diameter), by stimulated Brillouin scattering with intense multiphonon absorption, or by self-focusing. Typical destruction thresholds are 100MW/cm2 for a 100 ns pulse in a low-absorbing range and 1 kW/cm2 for a cw laser beam (In comparison, sunlight on the earth’s surface transmits 100 mW/cm2). For more on damage, see Kildal and Iseler (1976).

  18. 18.

    Sometimes, an inverted relation between E and P is used; for instance, for the second-order term, one obtains

    $$ {E}_i\left({\omega}_3\right)=\frac{1}{\varepsilon_0}\sum \limits_{i,j,k}{\delta}_{ijk}\left(-{\omega}_3:{\omega}_1,{\omega}_2\right)\, {P}_j\left({\omega}_1\right)\;{P}_k\left({\omega}_2\right) $$
    (105)

    with

    $$ {\chi}_{ijk}^{(2)}\left(-{\omega}_3:{\omega}_1,{\omega}_2\right)={\varepsilon}_0\sum \limits_{l,m,n}{\chi}_{il}^{(1)}\left({\omega}_3\right)\;{\chi}_{jm}^{(1)}\left({\omega}_1\right)\;{\chi}_{kn}^{(1)}\left({\omega}_2\right){\delta}_{lmn}\left(-{\omega}_3:{\omega}_1,{\omega}_2\right). $$
    (106)
  19. 19.

    When the restoring potential is nonlinear but symmetrical with respect to the equilibrium position r 0, then the nonlinear oscillatory motion on both sides of r 0 is equal, and the Fourier analysis of the ensuing oscillation does not contain even coefficients. Still, such potential can produce odd harmonics (3rd, 5th, …). For producing even-order harmonics, the nonlinear medium must not have inversion symmetry.

  20. 20.

    Amplification is achieved when a small signal at ω s is mixed with a strong laser pump beam at ω p and results in the creation of an additional beam at ω i , the idler frequency, according to ω p = ω s + ω i . In this process, energy from ω p is pumped into ω s and ω i ; consequently, ω s is amplified.

References

  • Armstrong JA, Bloembergen N, Ducuing J, Pershan PS (1962) Interactions between light waves in a nonlinear dielectric. Phys Rev 127:1918

    Article  ADS  Google Scholar 

  • Aspnes DE (1976) Spectroscopic ellipsometry of solids. In: Seraphin BO (ed) Optical properties of solids: new developments. North-Holland, Amsterdam, p 799

    Google Scholar 

  • Aspnes DE (1980) Modulation spectroscopy/electric field effects on the dielectric function of semiconductors. In: Moss TS, Balkanski M (eds) Optical properties of solids, vol 2, North-Holland, Amsterdam, p 109

    Google Scholar 

  • Aspnes DE, Studna AA (1983) Dielectric functions and optical parameters of Si, Ge, GaP, GaAs, GaSb, InP, InAs, and InSb from 1.5 to 6.0 eV. Phys Rev B 27:985

    Article  ADS  Google Scholar 

  • Azzam RMA, Bashara NM (1987) Ellipsometry and polarized light. Elsevier Science BV, Amsterdam

    Google Scholar 

  • Balkanski M, Moss TS (1994) Optical properties of semiconductors. North-Holland, Amsterdam

    Google Scholar 

  • Basu PK (1998) Theory of optical processes in semiconductors. Clarendon Press, Oxford

    Google Scholar 

  • Bayer M, Reinecke TL, Weidner F, Larionov A, McDonald A, Forchel A (2001) Inhibition and enhancement of the spontaneous emission of quantum dots in structured microresonators. Phys Rev Lett 86:3168

    Article  ADS  Google Scholar 

  • Birner A, Busch K, Müller F (1999) Photonische Kristalle. Phys Bl 55(4):27 (Photonic crystals, in German)

    Article  Google Scholar 

  • Blanco A, Chomski E, Grabtchak S, Ibisate M, John S, Leonard SW, Lopez C, Meseguer F, Miguez H, Mondia JP, Ozin GA, Toader O, van Driel HM (2000) Large-scale synthesis of a silicon photonic crystal with a complete three-dimensional bandgap near 1.5 micrometres. Nature 405:437

    Article  ADS  Google Scholar 

  • Bloembergen N (1982) Nonlinear optics and spectroscopy. Rev Mod Phys 54:685

    Article  ADS  Google Scholar 

  • Born M, Wolf E (2002) Principles of optics, 7th edn. Cambridge University Press, Cambridge

    Google Scholar 

  • Boyd RW (2008) Nonlinear optics, 3rd edn. Academic Press, London

    Google Scholar 

  • Bruggemann DAG (1935) Berechnung verschiedener physikalischer Konstanten von heterogenen Substanzen. Ann Phys (Leipzig) 24:636 (Calculation of various physical constants of heterogeneous substances, in German)

    Article  ADS  Google Scholar 

  • Brust D (1972) Model calculation of the q-dependent dielectric function of some zinc-blende semiconductors. Phys Rev B 5:435

    Article  ADS  Google Scholar 

  • Burge DK, Bennett HE (1964) Effect of a thin surface film on the ellipsometric determination of optical constants. J Opt Soc Am 54:1428

    Article  ADS  Google Scholar 

  • Busch K, Wehrspohn R (2003) Special issue on Photonic Crystals: optical materials for the 21th century. Phys Stat Sol A 197(3):593–702

    Article  ADS  Google Scholar 

  • Busch K, von Freymann G, Linden S, Mingaleev SF, Tkeshelashvili L, Wegener M (2007) Periodic nanostructures for photonics. Phys Rep 444:101

    Article  ADS  Google Scholar 

  • Byer RL, Herbst RL (1977) Parametric oscillation and mixing. Top Appl Phys 16:81

    Article  Google Scholar 

  • Cardona M (1969) Modulation spectroscopy. In: Seitz F, Turnbull D, Ehrenreich H (eds) Solid state physics, suppl 11. Academic Press, New York

    Google Scholar 

  • Chemla DS, Jerphagnon J (1980) Nonlinear optical properties. In: Moss TS, Balkanski M (eds) Handbook on semiconductors, vol 2. North-Holland, Amsterdam, p 545

    Google Scholar 

  • Choy MM, Byer RL (1976) Accurate second-order susceptibility measurements of visible and infrared nonlinear crystals. Phys Rev B 14:1693

    Article  ADS  Google Scholar 

  • Cohen MH (1963) Generalized self-consistent field theory and the dielectric formulation of the many-body problem. Phys Rev 130:1301

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Cui TJ, Smith DR, Liu R (eds) (2010) Metamaterials: theory, design, and applications. Springer, New York

    Google Scholar 

  • Ejder E (1971) Refractive index of GaN. Phys Stat Sol A 6:445

    Article  ADS  Google Scholar 

  • Engheta N, Ziolkowski RW (eds) (2006) Metamaterials: physics and engineering explorations. Wiley, Hoboken

    Google Scholar 

  • Franken PA, Ward JF (1963) Optical harmonics and nonlinear phenomena. Rev Mod Phys 35:23

    Article  ADS  MATH  Google Scholar 

  • Gérard JM, Sermage B, Gayral B, Legrand B, Costard E, Thierry-Mieg V (1998) Enhanced spontaneous emission by quantum boxes in a monolithic optical microcavity. Phys Rev Lett 81:1110

    Article  ADS  Google Scholar 

  • Haken H (1963) Theory of excitons II. In: Kuper CG, Whitfield GD (eds) Polarons and excitons. Oliver & Boyd, Edinburgh, p 294

    Google Scholar 

  • Ho K-M, Chan CT, Soukoulis CM, Biswas R, Sigalas M (1994) Photonic band gaps in three dimensions: new layer-by-layer periodic structures. Sol State Commun 89:413

    Article  ADS  Google Scholar 

  • Hobden MV (1967) Phase-matched second-harmonic generation in biaxial crystals. J Appl Phys 38:4365

    Article  ADS  Google Scholar 

  • Inoue K, Ohtaka K (eds) (2004) Photonic crystals. Springer, Berlin

    Google Scholar 

  • Joannopoulos JD, Johnson SG, Winn JN, Meade RD (2008) Photonic crystals – molding the flow of light, 2nd edn. Princeton University Press, Princeton

    MATH  Google Scholar 

  • John S (1987) Strong localization of photons in certain disordered dielectric superlattices. Phys Rev Lett 58:2486

    Article  ADS  Google Scholar 

  • Kildal H, Iseler GW (1976) Laser-induced surface damage of infrared nonlinear materials. Appl Optics 15:3062

    Article  ADS  Google Scholar 

  • Kildal H, Mikkelsen JC (1973) The nonlinear optical coefficient, phasematching, and optical damage in the chalcopyrite AgGaSe2. Opt Commun 9:315

    Article  ADS  Google Scholar 

  • Kirchner A, Busch K, Soukoulis CM (1998) Transport properties of random arrays of dielectric cylinders. Phys Rev B 57:277

    Article  ADS  Google Scholar 

  • Kurtz SK, Jerphagnon J, Choi MM (1979) Nonlinear dielectric susceptibilities. In: Hellwege KH, Hellwege AM (eds) Landolt-Börnstein new series III/11. Springer, Berlin, p 671

    Google Scholar 

  • Lee KF, Ahmad HB (1996) Fractional photon model of three-photon mixing in the non-linear interaction process. Opt Laser Technol 28:35

    Article  ADS  Google Scholar 

  • Mandel P (2010) Nonlinear optics. Wiley-VCH, Weinheim

    Google Scholar 

  • Manley JM, Rowe HE (1959) General energy relations in nonlinear reactances. Proc IRE 47:2115

    Google Scholar 

  • Marple DTF (1964) Refractive index of GaAs. J Appl Phys 35:1241

    Article  ADS  Google Scholar 

  • Maruani A, Oudar JL, Batifol E, Chemla DS (1978) Nonlinear spectroscopy of biexcitons in CuCl by resonant coherent scattering. Phys Rev Lett 41:1372

    Article  ADS  Google Scholar 

  • Meintjes EM, Raab RE (1999) A new theory of Pockels birefringence in non-magnetic crystals. J Opt A 1:146

    Article  ADS  Google Scholar 

  • Midwinter JE, Warner J (1965) The effects of phase matching method and of uniaxial crystal symmetry on the polar distribution of second-order non-linear optical polarization. Brit J Appl Phys 16:1135

    Article  ADS  Google Scholar 

  • Moss TS, Burrell GJ, Ellis B (1973) Semiconductor Optoelectronics. Wiley, New York

    Google Scholar 

  • Nara H (1965) Screened impurity potential in Si. J Phys Soc Jpn 20:778

    Article  ADS  Google Scholar 

  • Nara H, Morita A (1966) Shallow donor potential in silicon. J Phys Soc Jpn 21:1852

    Article  ADS  Google Scholar 

  • Nazarewicz P, Rolland P, da Silva E, Balkanski M (1962) Abac chart for fast calculation of the absorption and reflection coefficients. Appl Optics 1:369

    Article  ADS  Google Scholar 

  • Noda S, Tomoda K, Yamamoto N, Chutinan A (2000) Full three-dimensional photonic bandgap crystals at near-infrared wavelengths. Science 289:604

    Article  ADS  Google Scholar 

  • Palik ED (ed) (1985) Handbook of optical constants of solids I. Academic Press, New York

    Google Scholar 

  • Palik ED (ed) (1991) Handbook of optical constants of solids II. Academic Press, San Diego

    Google Scholar 

  • Penn DR (1962) Wave-number-dependent dielectric function of semiconductors. Phys Rev 128:2093

    Article  ADS  MATH  Google Scholar 

  • Petroff Y (1980) Optical properties of semiconductors above the band edge. In: Moss TS, Balkanski M (eds) Optical properties of solids, vol 2, Handbook on semiconductors. North-Holland, Amsterdam, p 1

    Google Scholar 

  • Philipp HR, Taft EA (1960) Optical constants of silicon in the region 1 to 10 eV. Phys Rev 120:37

    Article  ADS  Google Scholar 

  • Pollak FH, Shen H (1993) Modulation spectroscopy of semiconductors: bulk/thin film, microstructures, surfaces/interfaces and devices. Mater Sci Eng R10:275

    Google Scholar 

  • Purcell EM, Torrey HC, Pound RV (1946) Resonance absorption by nuclear magnetic moments in a solid. Phys Rev 69:37

    Article  ADS  Google Scholar 

  • Rabi II, Millman S, Kusch P, Zacharias JR (1939) The molecular beam resonance method for measuring nuclear magnetic moments; the magnetic moments of 3Li6, 3Li7 and 9 F19. Phys Rev 55:526

    Article  ADS  Google Scholar 

  • Rabin H, Tang CL (eds) (1975) Quantum electronics: a treatise, vol 1, Nonlinear optics. Academic Press, New York

    Google Scholar 

  • Reithmaier JP, Sek G, Löffler A, Hofmann C, Kuhn S, Reitzenstein S, Keldysh LV, Kulakovskii VD, Reinecke TL, Forchel A (2004) Strong coupling in a single quantum dot-semiconductor microcavity system. Nature 432:197

    Article  ADS  Google Scholar 

  • Rice A, Jin Y, Ma XF, Zhang X-C, Bliss D, Larkin J, Alexander M (1994) Terahertz optical rectification from <110> zinc-blende crystals. Appl Phys Lett 64:1324

    Article  ADS  Google Scholar 

  • Russell PSJ (1986) Optics of Floquet-Bloch waves in dielectric gratings. Appl Phys B 39(4):231

    Article  ADS  Google Scholar 

  • Rustagi KC (1970) Effect of carrier scattering on nonlinear optical susceptibility due to mobile carriers in InSb, InAs, and GaAs. Phys Rev B 2:4053

    Article  ADS  Google Scholar 

  • Sakoda K (2001) Optical properties of photonic crystals. Springer, Berlin

    Book  Google Scholar 

  • Seraphin BO (1973) Proceedings of the first international conference on modulation spectroscopy. Surf Sci 37:1–1011

    Article  ADS  Google Scholar 

  • Seraphin BO, Bottka N (1965) Franz-Keldysh effect of the refractive index in semiconductors. Phys Rev 139:A560

    Article  ADS  Google Scholar 

  • Sharma AC, Auluck S (1981) Transverse dielectric function for a model semiconductor. Phys Rev B 24:4729

    Article  ADS  Google Scholar 

  • Sheik-Bahae M, Wang J, Van Stryland EW (1994) Nondegenerate optical Kerr effect in semiconductors. IEEE J Quantum Electron 30:249

    Article  ADS  Google Scholar 

  • Shelby RA, Smith DR, Schultz S (2001) Experimental verification of a negative index of refraction. Science 292:77

    Article  ADS  Google Scholar 

  • Soukoulis CM, Wegener M (2010) Optical metamaterials – more bulky and less lossy. Science 330:1633

    Article  ADS  Google Scholar 

  • Stegeman GI, Stegeman RA (2012) Nonlinear optics: phenomena, materials and devices. Wiley, Hoboken

    MATH  Google Scholar 

  • Stern F (1963) Elementary theory of the optical properties of solids. In: Seitz F, Turnbull D (eds) Sol state physics, vol 15. Academic Press, New York, p 299

    Google Scholar 

  • Tosatti E, Pastori Parravicini G (1971) Model semiconductor dielectric function. J Phys Chem Sol 32:623

    Article  ADS  Google Scholar 

  • van Vechten JA (1980) A simple man’s view of the thermochemistry of semiconductors. In: Moss TS, Keller SP (eds) Handbook on semiconductors, vol 3. North-Holland, Amsterdam, p 1

    Google Scholar 

  • Veselago VG (1968) The electrodynamics of substances with simultaneously negative values of ε and μ. Sov PhysUspekhi 10:509

    Article  ADS  Google Scholar 

  • Villeneuve PR, Piché M (1994) Photonic bandgaps in periodic dielectric structures. Progr Quantum Electr 18:153

    Article  ADS  Google Scholar 

  • Walter JP, Cohen ML (1970) Wave-vector-dependent dielectric function for Si, Ge, GaAs, and ZnSe. Phys Rev B 2:1821

    Article  ADS  Google Scholar 

  • Wang CC, Ressler NW (1969) Nonlinear optical effects of conduction electrons in semiconductors. Phys Rev 188:1291

    Article  ADS  Google Scholar 

  • Wang CC, Ressler NW (1970) Observation of optical mixing due to conduction electrons in n-type germanium. Phys Rev B 2:1827

    Article  ADS  Google Scholar 

  • Ward L (1994) The optical constants of bulk materials and films, 2nd edn. Institute of Physics Publishing, Bristol

    Google Scholar 

  • Warner J (1975) Quantum electronics: a treatise, vol 1, Nonlinear optics. Academic Press, New York, p 103

    Google Scholar 

  • Wolf PA, Pearson GA (1966) Theory of optical mixing by mobile carriers in semiconductors. Phys Rev Lett 17:1015

    Article  ADS  Google Scholar 

  • Yablonovitch E (1987) Inhibited spontaneous emission in solid-state physics and electronics. Phys Rev Lett 58:2059

    Article  ADS  Google Scholar 

  • Yablonovitch E, Gmitter TJ, Leung KM (1991) Photonic band structure: the face-centered-cubic case employing nonspherical atoms. Phys Rev Lett 67:2295

    Article  ADS  Google Scholar 

  • Zernike F, Midwinter JE (1973) Applied non-linear optics. Wiley, New York

    Google Scholar 

  • Zouhdi S, Sihvola A, Vinogradov AP (2009) Metamaterials and plasmonics: fundamentals, modelling, applications. Springer, New York

    Book  Google Scholar 

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Böer, K.W., Pohl, U.W. (2018). Interaction of Light with Solids. In: Semiconductor Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-69150-3_10

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