Nonlinear Optics Lab. Hanyang Univ. Nonlinear Optics ( 비선형 광학 ) 담당 교수 : 오 차 환 교 재 : A. Yariv, Optical Electronics in Modern Communications, 5 th Ed., Oxford university Press, 1997 부교재 : R. W. Boyd, Nonlinear Optics, Academic Press, 1992 A. Yariv, P. Yeh, Optical waves in Crystals, John Wiley & Sons, 봄학기

Nonlinear Optics Lab. Hanyang Univ. Chapter 1. Electromagnetic Theory 1.0 Introduction Propagation of plane, single-frequency electromagnetic waves in - Homogeneous isotropic media - Anisotropic crystal media 1.1 Complex-Function Formalism Expression for the sinusoidally varying time functions ; Typical expression ;

Nonlinear Optics Lab. Hanyang Univ. Distinction between the real and complex forms 1) 2) * Time averaging of sinusoidal products

Nonlinear Optics Lab. Hanyang Univ. 1.2 Considerations of Energy and Power in Electromagnetic Field Maxwell’s curl equations (in MKS units) ; [, ] Vector identity ;

Nonlinear Optics Lab. Hanyang Univ. Divergence theorem ; : Poynting theorem Total power flow into the volume bounded by s Power expended by the field on the moving charges Rate of increase of the vacuum electromagnetic stored energy Power per unit volume expended by the field on electric and magnetic dipoles

Nonlinear Optics Lab. Hanyang Univ. Dipolar dissipation in harmonic fields The average power per unit volume expended by the field on the medium electric polarization ; Assume, field and polarization are parallel to each other Put, : Isotropic media : Anisotropic media

Nonlinear Optics Lab. Hanyang Univ. Ex) single localized electric dipole, power DFDF Let, position of electron : electric field : power DFDF 1) power DFDF 2) power DFDF : The dipole(electron) continually loses power to the field : The field continually gives power to the dipole  Power exchange between the field and medium via dipole interaction

Nonlinear Optics Lab. Hanyang Univ. 1.3 Wave Propagation in Isotropic Media Electromagnetic plane wave propagating along the z-axis in homogeneous, isotropic, and lossless media Put, General solutions : * Phase velocity : * wavelength : * Relative amplitude :

Nonlinear Optics Lab. Hanyang Univ. Power flow in harmonic fields Intensity (average power per unit area carried in the propagation direction by a wave) : (1.3-17)  Electromagnetic energy density : (1.3-17)  For positive traveling wave :

Nonlinear Optics Lab. Hanyang Univ. 1.4 Wave Propagation in Crystals-The Index Ellipsoid In general, the induced polarization is related to the electric field as : electric susceptibility tensor If we choose the principal axes, (Diagonalization)

Nonlinear Optics Lab. Hanyang Univ. Secular equation For a monochromatic plane wave ; From Maxwell’s curl equations, In principal coordinate,

Nonlinear Optics Lab. Hanyang Univ. Simple example ( ) : wave propagating along the x-axis : transverse wave !! For nontrivial solution to exist, Det=0 ;

Nonlinear Optics Lab. Hanyang Univ. Normal surface Optic axis Simple example (, determinant equation  ) : circle : ellipse

Nonlinear Optics Lab. Hanyang Univ. Wave propagation in anisotropic media Maxwell equations  Define the unit vector along the propagation direction as ( : wave vector) Put,  =1, and Taking scalar product,on both sides : : propagation direction is perpendicular to the electric displacement vector not to the electric field vector (poynting vector)

Nonlinear Optics Lab. Hanyang Univ. Index ellipsoid The surface of constant energy density in D space : Energy density : or : Index ellipsoid

Nonlinear Optics Lab. Hanyang Univ. Classification of anisotropic media 1) Isotropic : ex) CdTe, NaCl, Diamond, GaAs, Glass, … 2) Uniaxial : (1) Positive uniaxial : ex) Ice, Quartz, ZnS, … (2) Negative uniaxial : ex) KDP, ADP, LiIO 3, LiNbO 3, BBO, … Fast/Slow axis 3) Biaxial : ex) LBO, Mica, NaNO 2, …

Nonlinear Optics Lab. Hanyang Univ. Example of index ellipsoid (positive uniaxial) propagation direction

Nonlinear Optics Lab. Hanyang Univ. Intersection of the index ellipsoid Birefringence :

Nonlinear Optics Lab. Hanyang Univ. Normal index surface : The surface in which the distance of a given point from the origin is equal to the index of refraction of a wave propagating along this direction. 1) Positive uniaxial (n e >n o ) 2) negative uniaxial (n e <n o ) 3) biaxial ( )

Nonlinear Optics Lab. Hanyang Univ. 1.5 Jones Calculus and Its Application in Optical Systems with Birefringence Crystals Jones Calculus (1940, R.C. Jones) : - The state of polarization is represented by a two-component vector - Each optical element is represented by a 2 x 2 matrix. - The overall transfer matrix for the whole system is obtained by multiplying all the individual element matrices. - The polarization state of the transmitted light is computed by multiplying the vector representing the input beam by the overall matrix. Examples) - Polarization state : - Linear polarizer (horizontal) : - Relative phase changer : Report) matrix expressions - Linear polarizers (horizontal, vertical) - Phase retarder - Quarter wave plate (fast horizontel, vertical) - Half wave plate

Nonlinear Optics Lab. Hanyang Univ. Retardation plate (wave plate) : Polarization-state converter (transformer) Polarization state of incident beam : where, : complex field amplitudes along x and y s, f axes components : Polarization state of the emerging beam :

Nonlinear Optics Lab. Hanyang Univ. Define, - Difference of the phase delays : - Mean absolute phase change : Polarization state of the emerging beam in the xy coordinate system : where,

Nonlinear Optics Lab. Hanyang Univ. Transfer matrix for a retardation plate (wave plate) Transfer matrix is a unitary ( ) : Physical properties are invariant under unitary transformation => If the polarization states of two beams are mutually orthogonal, they will remain orthogonal after passing through an arbitrary wave plate.

Nonlinear Optics Lab. Hanyang Univ. Ex) Half wave plate : x-polarized beam Report : Problem 1.7

Nonlinear Optics Lab. Hanyang Univ. Ex) Quarter wave plate : left circularly polarized beam : y-pol. : right circularly polarized beam : x-pol.

Nonlinear Optics Lab. Hanyang Univ. Intensity transmission In many cases, we need to determine the transmitted intensity, since the combination of retardation plates and polarizers is often used to control or modulate the transmitted optical intensity. Incident beam intensity : Output beam intensity : Transmissivity :

Nonlinear Optics Lab. Hanyang Univ. Ex) A birefringent plate sandwiched between parallel polarizers : fn. of d and Ex) A birefringent plate sandwiched between a pair of crossed polarizers

Nonlinear Optics Lab. Hanyang Univ. Circular polarization representation It is often more convenient to express the field in terms of “basis” vectors that are circularly polarized ; : constitute a complete set that can be used to describe a field of arbitrary polarization. Right circularly polarized Left circularly polarized Rectangular representation : Circular representation :

Nonlinear Optics Lab. Hanyang Univ. Transformation examples) Report :

Nonlinear Optics Lab. Hanyang Univ. Faraday rotation In certain optical materials containing magnetic atoms or ions, the two counter-rotating, circularly-polarized modes have different indices of refraction when an external magnetic field is applied along the beam propagation direction. This difference is due to the fact that the individual atomic magnetic moments process in a unique sense about the z-axis (magnetic field direction) and thus interact differently with the two counter-rotating modes.

Nonlinear Optics Lab. Hanyang Univ. Ignoring the prefactor, exp[-(i/2)(  + +  - ], where, Why (Faraday) rotation angle ? In rectangular representation,

Nonlinear Optics Lab. Hanyang Univ. Basic difference between propagation in a magnetic medium and in a dielectric birefringent medium : CW for +z CW for -z CW for +z CCW for -z B Report : proof by calculating Jones matrix.

Nonlinear Optics Lab. Hanyang Univ. Optical isolator