Ray Theory Transmission Theories of Optics • Light is an electromagnetic phenomenon described by the same theoretical principles that govern all forms of electromagnetic radiation • Electromagnetic optics provides the most complete treatment of light phenomena in the context of classical optics. • quantum theory provides the successful explanation for light-matter interaction(emission & absorption) • Electromagnetic radiation propagates in the form of two mutually coupled vector waves, an electric field- wave & magnetic field wave(classical optics) • Many optical phenomena such as diffraction is described by a single scalar wave function(scalar wave optics or wave optics) • Ray optics is the limit of wave optics when the wavelength is very short. Quantum Optics Electromagnetic Optics Wave Optics Ray Optics Engineering Model • Depending on the structure, we may use ray optics or electromagnetic optics Electromagnetic Optics • Electromagnetic radiation propagates in the form of two mutually coupled vector waves, an electric field wave & a magnetic field wave. Both are vector functions of position & time. • In a source-free, linear, homogeneous, isotropic & non-dispersive media, such as free space, these electric & magnetic fields satisfy the following partial differential equations, known as Maxwell’ equations: E H [1] t H E [2] t E 0 [3] H 0 [4] • In Maxwell’s equations, E is the electric field expressed in [V/m], H is the magnetic field expressed in [A/m]. [F/m] : Electric permittivi ty [H/m] : Magnetic permeabili ty : is divergence operation : is curl operation • The solution of Maxwell’s equations in free space, through the wave equation, can be easily obtained for monochromatic electromagnetic wave. All electric & magnetic fields are harmonic functions of time of the same frequency. Electric & magnetic fields are perpendicular to each other & both perpendicular to the direction of propagation, k, known as transverse wave (TEM). E, H & k form a set of orthogonal vectors. Electromagnetic Plane wave in Free space Ex Direction of Propagation k x z z y By An electromagnetic wave is a travelling wave which has time varying electric and magnetic fields which are perpendicular to each other and the direction of propagation, . Linearly Polarized Electromagnetic Plane wave E e E cos(ωt - kz) [5] x 0x H e H cos(ωt kz) y 0 y [6] where : ω 2f : Angular frequency [rad/m] [7] k : Wavenumber or wave propagation constant [1/m] 2 :Wavelength [m] k E0x [] : intrinsic (wave) impedance [8] H 0 y 1 v [m/s] : velocity of wave propagation [9] E and B have constant phase in this xy plane; a wavefront z E E k Propagation B E x E = E sin(wt–kz) x o z , has the same A plane EM wave travelling alongz Ex (or By) at any point in a given xy plane. All electric field vectors in a givenxy plane are therefore in phase. The xy planes are of infinite extent in thex and y directions. Wavelength & free space • Wavelength is the distance over which the phase changes by 2 . v [10] f • In vacuum (free space): 9 10 7 0 [F/m] 0 410 [H/m] 36 [11] 8 v c 310 m/s 0 120 [] EM wave in Media • Refractive index of a medium is defined as: c velocity of light (EM wave) in vacuum n r r v velocity of light (EM wave) in medium 0 0 [12] r : Relative magnetic permeability r : Relative electric permittivity (r 1) • For non-magnetic media : [13] n r Intensity & power flow of TEM wave 1 • The poynting vector S E H for TEM wave is parallel to the 2 wavevector k so that the power flows along in a direction normal to the wavefront or parallel to k. The magnitude of the poynting vector is the intensity of TEM wave as follows: E 2 I 0 [W/m 2 ] [14] 2 Connection between em wave optics & ray optics • The EM waves radiated by a small optical source can be represented by a train of spherical wave fronts with the source at the center • A wave front is the locus of all points in the wave train which exhibit the same phase • Large scale optical effects such as reflection & refraction can be analyzed by simple geometrical process called ray tracing(geometrical optics) Wave fronts (constant phase surfaces) Wave fronts Wave fronts k P E r rays k P O z A perfect plane wave A perfect spherical wave A divergent beam (a) (b) (c) Examples of possible EM waves Modes in an optical fiber Meridional and Skew rays • Meridional rays: The rays which always pass through the axis of fiber giving high optical intensity at the center of the core of the fiber. • Skew Rays : The rays which never intersect the axis of the fiber, giving low optical intensity at the center and high intensity Meridional and Skew rays Number of fiber modes vs. the V number Limitations of the Ray-model • (1) The ray model gives an impression that during total internal reflection the energy is confined to the core only. However, it is not so. In reality the optical energy spreads in cladding also. • (2) The ray model does not speak of the discrete field patterns for propagation inside a fiber. • (3) The ray model breaks down when the core size becomes comparable to the wavelength of light. The ray model therefore is not quite justified for a SM fiber. • The limitations of the Ray model are overcome in the wave model General form of linearly polarized plane waves Any two orthogonal plane waves Can be combined into a linearly Polarized wave. Conversely, any arbitrary linearly polarized wave can be resolved into two independent Orthogonal plane waves that are in phase. E e x E0x cos(ωt kz) e y E0 y cos(ωt kz) E E E 2 E 2 0x 0 y [15] E tan 1 ( 0 y ) E0x Elliptically Polarized plane waves E e x Ex e y E y e x E0x cos(ωt kz) e y cos(ωt kz ) 2 2 E E E E x y 2 x y cos sin 2 E0x E0 y E0x E0 y [16] 2E0x E0 y cos tan(2) 2 2 E0x E0 y Circularly polarized waves Circular polarizati on : E E E & [17] 0x 0 y 0 2 : right circularly polarized, - : left circularly polarized Laws of Reflection & Refraction Reflection law: angle of incidence=angle of reflection Snell’s law of refraction: n1 sin 1 n2 sin 2 [18] Total internal reflection, Critical angle Transmitted (refracted) light k t 2 90 2 n 2 Evanescent wave n 1 > n2 k k 1 i r 1 c Critical angle 1 c TIR Incident Reflected light light n2 (a) sin c (b) (c) n1 Light wave travelling in a more dense medium strikes a less dense medium. Depending on the incidence anglewrt to c , which is determined by the ratio of the refractive indices, the wave may be transmitted (refracted) or reflected. (a) 1 c and total internal reflection (TIR). (c) (b) 1 c 1 c n2 [19] sin c n1 Phase shift due to TIR • The totally reflected wave experiences a phase shift however which is given by: n 2 cos 2 1 n n2 cos 2 1 tan N 1 ; tan p 1 [20] 2 nsin 1 2 sin 1 n n 1 n2 • Where (p,N) refer to the electric field components parallel or normal to the plane of incidence respectively. Optical wave guiding by TIR: Dielectric Slab Waveguide Propagation mechanism in an ideal step-index optical waveguide. Launching optical rays to slab waveguide n2 sin min ; minimum angle that supports TIR n1 [21] Maximum entrance angle, 0 max is found from the Snell’s relation written at the fiber end face. 2 2 [22] nsin 0max n1 sin c n1 n2 Numerical aperture: 2 2 NA nsin 0 max n1 n2 n1 2 [23] n1 n2 [24] n1 Optical rays transmission through dielectric slab waveguide n n ; 1 2 c 2 c O For TE-case, when electric waves are normal to the plane of incidence must be satisfied with following relationship: 2 2 n d sin m n cos 2 n tan 1 1 2 [25] 2 n sin 1 EM analysis of Slab waveguide • For each particular angle, in which light ray can be faithfully transmitted along slab waveguide, we can obtain one possible propagating wave solution from a Maxwell’s equations or mode. • The modes with electric field perpendicular to the plane of incidence (page) are called TE (Transverse Electric) and numbered as: TE 0 ,TE 1,TE 2 ,... Electric field distribution of these modes for 2D slab waveguide can be expressed as: Em (x, y, z,t) e x f m (y)cos(ωt m z) [26] m 0,1,2,3 (mode number) wave transmission along slab waveguides, fibers & other type of optical waveguides can be fully described by time & z dependency of the mode: j(wtm z) cos(ωt m z) or e TE modes in slab waveguide y z Em (x, y, z,t) e x f m (y)cos(ωt m z) m 0,1,2,3 (mode number) Modes in slab waveguide • The order of the mode is equal to the # of field zeros across the guide. The order of the mode is also related to the angle in which the ray congruence corresponding to this mode makes with the plane of the waveguide (or axis of the fiber). The steeper the angle, the higher the order of the mode. • For higher order modes the fields are distributed more toward the edges of the guide and penetrate further into the cladding region.