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Basics of Optics -Maxwell’s equations -From Maxwell’s equations to the wave equation - Plane wave as the free-space solution to the wave equation - Plane wave at the boundary: reflection and refraction - Wave confinement and waveguide Maxwell’s Equations • Coulomb’s law: F e1e2 er 40 r122 1 E F 1 e1 1 Ze e er r e2 4 0 r122 4 0 r 2 • Feature 1: why inverse square law? Maxwell’s Equations • Total force flux must be conserved in 3D space and equal to the source (i.e., total “detected” = total “created”) • We immediately have Gaussian law: Ze E d s s 40 1 Ze 1 e d s s r 2 r 0 0 V dV E 0 • Coulomb force in spaces with other dimensions Maxwell’s Equations • Feature 2: centric force • We therefore know that the static electric field is curl-free: rB rA Ze rB 1 Ze rB 1 Ze rB 1 Ze 1 1 E dl e d l dr dr ( ) r 40 rA r 2 40 rA r 2 40 rA r 2 40 rA rB E dl 0 E 0 l • From Helmholtz’s law, a vector is uniquely specified by its divergence and curl, hence Gaussian law plus curlfree description is equivalent to Coulomb’s law • Why go from the direct (explicit) description to constraint (implicit) description? Maxwell’s Equations • Constraint description – Field is specified in a limited set of spatial points, not specified in the rest area (– not specified doesn’t necessarily mean that the field is zero) – Specification is given in the form of equations – implicit expressions – These equations must be in the differential or integral form, cannot be in simple algebraic form • Why constraint description – We have to deal the coupling between the electric and magnetic field later – it is easier to deal with coupling problems if we express source by field, rather than to express field by source. Maxwell’s Equations • Current definition: dr J v dt • Total charge must be conserved, the net current flow through a closed surface must be equal to the change of charge density enclosed by the surface, hence we have the charge conservation law: ( Ze) J d s dV s V t t J t Maxwell’s Equations • Another constant current flow will feel the Biot-Savert force, hence we have Biot-Savert’s law: 0 F 4 l1 l2 I1dl1 ( I 2 dl2 er ) r122 0 I 2 dl 2 er 0 B(r ) 2 l2 4 4 r I 1dl1 12 F l1 ' Idl er r ' l ' | r r ' |2 • Biot-Savert force is similar to Coulomb’s force, only the interactive scalar charges are replaced by the interactive unit current flow vectors • Coulomb’s force is originated by the point sources (and their superposition), whereas Biot-Savert’s force comes from the vortex sources (and their superposition) – a vortex cannot be reduced to a single point, and exists only in space with dimension higher than 3! • Biot-Savert’s force only acts on current flows, has no effect on static charges, hence is an independent force (cannot be included by Coulomb’s force) Maxwell’s Equations • Feature 1: close vector flow • Magnetic field is continuous in 3D space → divergence free • Vector potential can be introduced, and we have Gaussian law for magnetic field: B(r ) 0 4 l' ' Idl er r ' 0 4 | r r ' |2 B A J er r ' 0 dV ' V | r r ' |2 4 B 0 J V | r r ' |dV ' Maxwell’s Equations • Feature 2: non-centric field anymore, otherwise, divergence free plus curl free makes no magnetic field anywhere (following Helmholtz’s theorem); and the magnetic force, again, follows the inverse square law as a function of the interactive distance • Reason – magnetic flux must be conserved in 3D space • Also from which, we can derive Ampere’s law: 2 B A ( A) A 0 2 0 4 J V | r r ' |dV ' 0 J Maxwell’s Equations • Electric field interacts between two static charges, reflects a pure “adiabatic” effect • Magnetic field interacts between two moving charges, reflects a pure “derivative” effect • Electric + Magnetic field provides a complete description Maxwell’s Equations • Static charge distribution creates a curl free, divergence driven field; the field can be detected by another charge, hence “electric” field • Constant current flow not only creates an electric field, it also creates a divergence free, curl driven field; the field can be detected by another current, enhance “magnetic” field • Up to now, the electric and magnetic field are static, without any coupling in between Maxwell’s Equations Static Electric Field Field Explicit Description Field Implicit Description E (r ) 1 4 0 Static Magnetic Field (r ')eˆrr ' | r r ' | 2 V' E (r ) (r ) 0 E (r ) 0 B(r ) dV ' 0 4 0 4 V' L' Idl ' eˆrr ' | r r ' |2 J (r ') eˆrr ' dV ' | r r ' |2 B(r ) 0 B(r ) 0 J Maxwell’s Equations • However, there is no absolute inertia system – moving or steady depends on the system where the observer stays, hence a static charge and a constant current can be exchanged • Therefore, the static electric field comes from a static charge and the static magnetic field comes from a constant current can be exchanged! • How can we have a consistent description then? Maxwell’s Equations • Unify the static electric and magnetic fields as a single 3D vector in a unified spatial - temporal 4D space, under the non-relativistic condition, we have: E' EvB B' B v E 2 c to make the E/M description consistent Maxwell’s Equations • Accelerated electron creates time-varying current, timevarying current creates time-varying magnetic field, timevarying magnetic field will create electric field vortex • Why? • Explanation – isolated natural system tend to eliminate any disturbance, with time evolution towards its eigen state: once a time-varying magnetic field is established, a electric field vortex will be induced, to establish yet another magnetic field with an opposite change towards a cancellation of the total field, hence Faraday’s law holds t B ds E dl 0 s l B E t Maxwell’s Equations • Accelerated charge causes a time-varying charge distribution, leading to a time-varying current; from Ampere’s law, time-varying magnetic field exists; from Faraday’s law, electric vortex exists • Therefore, not only electric field can be generated in its divergence form by the static charge distribution, it can also be generated in its curl form by the “temporary” charge distribution which • Indicating that the electric field is brought in by both its divergence and curl, following the Helmholtz theorem, the electric field is complete • Hence, time-varied magnetic field excites electric field, electric and magnetic fields become coupled; however, such coupling is still unidirectional Maxwell’s Equations Static charge Div. Time-varying charge distribution (time-varying current, temporary charge) Moving charge, static charge distribution (constant current) Div. Curl Curl Static Electric Field Time-varying electric field Static Magnetic Field Time-varying magnetic field Curl J t Charge conservation law Gauss’s law E 0 B 0 Ampere’s law B 0 J Faraday’s law B E t (Derived from Coulomb’s law) (Derived from Biot-Savert’s law) Maxwell’s Equations • Contradiction between Ampere’s law and the charge conservation law B 0 J 0 0 t • Maxwell mended Ampere’s law, solved the problem E B 0 ( J ) B 0 J 0 0 t t Gaussian law is called E 0 0 0 Maxwell’s Equations • Significance – likes a current, the time-varying electric field can generate magnetic field • Hence the time-varying rate of the electric displacement vector is equivalent to a current, named as the displacement current; the conventional current caused by the moving charge is then called the conduction current to make a difference • Not only time-varying magnetic field, equivalent to temporary charges, can excite electric field, time-varying electric field, equivalent to a “temporary” current (i.e., the displacement current), can also excite magnetic field • Finally, electric and magnetic fields are fully coupled through mutual excitation Maxwell’s Equations • Therefore – 1. accelerated charge creates time-varying magnetic field in its neighborhood (through the time-varying conduction current and following Ampere’s law); – 2. time-varying magnetic field excites electric field in its neighborhood (through Faraday’s law) – 3. time-varying electric field excites magnetic field in its neighborhood (through Ampere’s law) • Step 2 and 3 form a sustainable loop, and make electricmagnetic fields propagation! Such an electric-magnetic non-local oscillation is named as the electromagnetic wave Maxwell’s Equations in Vacuum Faraday’s law B E t Maxwell modified Ampere’s law, E B 0 J 0 0 derived from Biot-Savert’s law t E / 0 Gaussian law, derived from Coulomb’s law B 0 Gaussian law, derived from Biot-Savert’s law Maxwell’s Equations in General Form B E t D H J t D B 0 D E B H J E 17 equations, yet 16 unknowns, one equation is redundant, e.g., the 4th equation is embedded in the 1st equation Why independent sources, i.e., the free charge and conduction current, have to be taken as unknowns as well? Wave Equation • From Maxwell’s equations to the wave equation sourceless, harmonic B t H J D t E homogeneous medium E j B j H E 2 E H j D j E H 2 H ( E ) 2 E 2 E coupling term independent term Wave in Free Space • Simplified wave equation sourceless, homogenous D 0 E E E E 0 E 0 simplified wave equation 2 E 2 E 0 from B 0 H 0 we have 2 H 2 H 0 • Obviously, to manipulate the polarization through coupling among different field components, the introduction of medium inhomogeneity and/or structure is necessary • Free space solution of the wave equation – plane harmonic wave E0 e j ( k r t ) j ( kr t ) H 0e where 2 | k | 2 or | k | The Wave Concept • Solution to wave equation – traveling wave – 1D f ( z vt ) or f ( z t ) propagation along – 3D f (k r t ) propagation along k t z • The element of traveling wave and its complex expression – plane harmonic wave sin(k r t ), cos(k r t ) e j ( k r t ) The Wave Concept • Standing wave – formed by the superposition of two contradirectionally propagating (traveling) waves cos(k r ) cos(t ) cos(k r )e jt • Evanescent wave – spatially decayed wave e ki r cos(kr r t ) e ki r e j ( kr r t ) • Wave coupling – phase matching condition cos(k1 r 1t ), cos(k2 r 1t ) kˆ11 kˆ22 | k1 | | k2 | • Phase velocity and group velocity vp c d d (c | k | / n ) | k | dn dn , vg v p (1 ) v p (1 ) |k| n d |k| d |k| n d |k| n d Wave in Free Space • Dispersion in free space 1/ 1/ |k | • Plane harmonic wave characteristics E 0 [ E0e j ( k r t ) ] e j ( k r t ) E0 ( jk E0 )e j ( k r t ) 0 k E0 0 for the same reason k H 0 0 we have k0 E0 / H0 also from E j H H jE k0 H0 / E0 where k0 k / | k | Wave in Free Space • Therefore, the plane harmonic wave in free space has its electric field, magnetic field, and propagation direction all orthogonal – it is a condition forced by the electric and magnetic divergence free requirement | E0 | / | H 0 | E [V/m] k free space impedance H [A/m] Wave in Free Space • Significance of the plane harmonic wave – eigen solution of the wave equation in free space, i.e., for whatever excitation, after waiting for infinitely long, the solution at infinitely far distance from the source in free space can only be the plane harmonic wave • Therefore, regardless of the source distribution, the electromagnetic wave in free space will become the plane harmonic wave after infinitely long at infinity, the evolution process is the spatial diffraction that eventually smears out any initial nonuniformity Plane Harmonic Wave at Boundary z ki i r [ E0i e j ( ki r t ) E0 r e j ( kr r t ) ]t [ E0t e j ( kt r t ) ]t kr z=0 t kt must hold for any point (x, y) at plane z=0, hence ki r |z 0 kr r |z 0 kt r |z 0 or kix krx ktx , kiy k ry kty the three wave vectors have identical projection on the boundary plane Feature 1:The incident wave vector can always be selected in a plane (y=0) perpendicular to the boundary plane (z=0). As such, the reflected and refracted wave vectors must be in the same plane (y=0), since kiy=0 requires kry= kty=0 as well. This plane (y=0) is called the incident plane. Feature 2: ki sin i kr sin r kt sin t or 11 sin i 11 sin r 2 2 sin t i.e., i r , 11 sin i 2 2 sin t Snell’s law Plane Harmonic Wave at Boundary z E-field parallel to the boundary (s-wave): i r ki E0i E0 r E0t kr z=0 t or kt E H Hence E0 r E0i H 0i cos i H 0 r cos r H 0t cos t 1 cos i 1 1 cos i 1 E0i E0 r E0t 1 E0i cos i 1 E0 r cos r 2 E0t cos t 1 1 2 2 cos t 2 2 cos t 2 1 cos i 1 E0t E0i 1 cos i 2 cos t 1 2 2 Plane Harmonic Wave at Boundary H-field parallel to the boundary (p-wave): z i r ki z=0 t H Hence H 0i H 0 r H 0t kr kt E0i cos i E0 r cos r E0t cos t or H 0i H 0 r H 0t 1 1 2 H 0i cos i H 0 r cos r H 0t cos t 1 1 2 E H 0r H 0i 1 cos i 1 1 cos i 1 2 1 cos t cos i 2 1 E0 r E0i 2 1 cos t cos i 2 1 1 cos i H 0t 1 H 0i 1 2 cos i cos t 1 2 2 E0t E0i 2 cos t 2 2 cos t 2 2 cos i 2 1 2 cos i cos t 1 2 2 Dielectric-Dielectric Boundary For non-magnetic materials 1 2 0 E0 r E0i s-wave E0t E0i E0 r E0i 1 cos i 2 cos t k ktz iz kiz ktz 1 cos i 2 cos t 2 1 cos i 1 cos i 2 cos t 2kiz kiz ktz 2 cos i 1 cos t k / ktz / 2 iz 1 kiz / 1 ktz / 2 2 cos i 1 cos t p-wave E0t E0i 2 1 cos i 2 cos i 1 cos t 2kiz / 1 1 2 kiz / 1 ktz / 2 Dielectric-Dielectric Boundary 1 E0 r / E0i s B 0 p 1 E0 r / E0i 1 2 1 2 p C 90 0 90 s -1 -1 B Total Internal Reflection Under the internal sin t 1 sin i sin i reflection scheme 2 t i for i C sin 1 2 1 2 total reflection happens. Since cos t 1 sin t 1 1 2 sin i j | 1 sin 2 i 1| 2 2 is purely imaginary, the refracted wave vector becomes: ktz 2 0 cos t j 2 0 | ktx 2 0 sin t 2 0 1 sin 2 i 1| 2 1 sin i 10 sin i kix 2 i.e., the refracted wave is propagating along the boundary, decaying in the direction perpendicular to the boundary. Therefore, the refracted wave under TIR is reduced to a surface wave propagating along the boundary only, formed by the projection of the incident and reflected wave vectors on the boundary plane. TIR as an All-Pass Filter Once i C we find s-wave p-wave E0 r kiz j | ktz | |k | e2 js , s tan 1 tz tan 1 E0i kiz j | ktz | kiz 2 | E0 r kiz / 1 j | ktz | / 2 |k | 2 j e p , p tan 1 1 tz tan 1 E0i kiz / 1 j | ktz | / 2 2 kiz 1 sin 2 i 1| 2 1 cos i 1 | 1 2 sin i 1| 2 2 cos i Applications of TIR • APF – for polarization splitting and conversion (example: wave plate) • Stop the traveling wave – for waveguide • Under TIR, refracted wave is traveling along the surface – for surface wave excitation • TIR is not wavelength sensitive, however, its combination (multiple TIR) is wavelength sensitive! – for construction of BPF Total Refraction If i t 90 we have Hence 1 sin i 2 sin(90 i ) 2 cos i or i tan 1 2 cos i 1 cos t 2 cos B 1 sin B 2 cos B (1 2 B 1 1 tan B ) 0 2 For p-wave, if the incident angle hits the Brewster angle, total refraction happens. For s-wave, total refraction is impossible, since for 1 2 we always have 1 cosi 2 cost 1 1 sin 2 i 2 2 sin 2 t 1 1 sin 2 i 2 1 sin 2 i 0 Total Reflection and Total Refraction • Total reflection happens to both s- and p- waves for internal reflection (when incident light from high refractive index medium); it doesn’t happen to external reflection (when incident light from low refractive index medium); hence the name total internal reflection. • Total refraction, however, only happens to p-wave, regardless of internal or external incidence; it doesn’t happen to s-wave. • Why total refraction can happen? • What will happen at the boundary with identical permittivity but different permeability? Applications of Total Refraction • Polarization splitting • Filtering (Once the medium is dispersive, the Brewster angle becomes wavelength dependent. Otherwise, a front stage, e.g., a prism or a diffractional grating, is needed to convert the wavelength change into the angle change) Free Space and Plane Wave • Propagating wave in free-space – for any source, it will gradually turn itself into a plane wave as it propagates in free-space, and becomes more difficult to be collected in a distance away, for a limited receiver surface • How can we force the EM wave to propagate along a specific direction without any spreading in the 3D world? – That comes to the waveguide concept The Concept of Waveguide • The wave has to localized in certain directions • How to localize the wave? – Convert the traveling wave into the standing wave • Introduce the transverse resonance kx2 | k2 | 2 k x1 x | k2 | k 2 2 2 x2 | k1 | 1 s-wave reflection at the boundary: z | k1 |2 2 k x21 2 1 R k x1 k x 2 k x1 k x 2 standing wave is formed underneath the boundary The Concept of Waveguide 2 d E0 1 x E0 R 2 e jk x1 (2 d ) z The resonance condition for standing wave in transverse direction (x): E0 E0 R 2e jkx1 (2 d ) R 2e jk x1 (2 d ) 1 A necessary condition is: | R | 1 How to make it possible? TIR – dielectric waveguide Conductor reflection – metallic waveguide Photonic crystal – Bragg waveguide Plasma reflection – plasmonic polariton waveguide The Dielectric Waveguide If k x 2 becomes purely imaginary, or: 2 | k2 |2 2 2 kx 2 | k2 |2 2 j 2 2 2 k k x 2 k x1 j | k x 2 | |k | R x1 e2 j , tan 1 x 2 tan 1 k x1 k x 2 k x1 j | k x 2 | k x1 2 2 2 21 2 The resonance condition becomes: e4 j e2 jk d 1 2k x1d 4 2m x1 d 1 2 tan 2 2 1 2 2 2 2 2 2 d m 2 2 m tan( 1 ) 2 2 21 2 21 2 Even mode tan 1 2 2 2 2 2 1 2 2 Odd mode Obviously, we have: 2 1 we find: r 2 n2 neff n1 r1 1 tan 21 2 or: Dispersion relation for the dielectric slab waveguide 2 2 2 21 2 With definition 0 neff neff - waveguide effective index The Dielectric Waveguide Dispersion relation E-field E0 ( x )e j ( z t ) (y-component) Symmetric (even mode): 1/ 2 c / n2 1/ neff 0 c / neff 1/ 1 c / n1 Ae k x 2 ( x d / 2 ) , x d / 2 E0 ( x) B cos( k x1 x), d / 2 x d / 2 k x 2 ( x d / 2) Ce , x d / 2 Anti-symmetric (odd mode): Ae k x 2 ( x d / 2 ) , x d / 2 E0 ( x ) B sin( k x1 x ), d / 2 x d / 2 Ce k x 2 ( x d / 2 ) , x d / 2 A, B, C – given by the tangential boundary condition H-field is given by the Faraday’s law, with x and z components only – that’s the TE wave Metallic Waveguide By letting k x 2 j in previous derivations, we will be able to obtain the EM wave solution in metallic waveguide. 1D (slab) or 2D dielectric waveguide – support TE and TM waves, not TEM wave 1D (slab) metallic waveguide – support all TEM, TE, and TM waves 2D (hollow) metallic waveguide – support TE and TM waves, not TEM wave Transmission Line – TEM Wave 2 2 E0 x ( x, y) For TEM wave: k Hence: ( 2 2 ) 0 E ( x , y ) x y 0 y TEM solution is the same as the static electric and magnetic field solution. E I I E H parallel lines or H twisted pair coaxial cable TEM Wave - Characteristics • TEM wave – “localized” plane wave with k, E, H mutually orthogonal: k is along the direction in which the waveguide (transmission line) is extended; Eand H- fields are restricted in the 2D cross-section, with their longitudinal dependence identical to the plane wave, and transverse dependence identical to the static E- and H- fields with the same boundary condition. • Propagation of the TEM wave relies on the free charge and conduction current on the metal (conductor) – dielectric surface. Namely, the TEM wave is a resonance between the EM fields and the free charge distribution. • For the TEM wave, we can readily introduce the voltage and current concept to turn a field problem into a circuit problem. • The TEM wave can be supported by the dual conductor transmission line: – Parallel lines or twisted pair – Coaxial cable – Printed metal stripe lines (on PCB or other substrates) • The TEM wave has no cut-off frequency, it can send DC power through. Various Waveguides for EM Waves WG Structure Guiding Pattern Applications Remarks DC Paired metal wires TEM Power transmission Transmission in freespace impossible VLF Paired metal wires or free-space TEM or plane wave Power transmission, submarine communication Huge antenna size required for broadcasting LF-MF Paired metal strips on PCB or free-space TEM or plane wave Circuits, (AM) broadcasting MF for radio broadcasting HF-VHF Coaxial cables, microstrips on PCB, or freespace TEM or plane wave Circuits, (FM) broadcasting, wireless communication, remote control, blue tooth, etc. TV broadcasting MW Coaxial cables, hollow metallic waveguides, or free-space TEM, TE/TM, HE/EH, or plane wave Circuits, radars, space (satellite) communication Long-haul telecommunication through antenna relay LW Optical fibres, dielectric waveguides, or freespace TE/TM, HE/EH, or plane wave Optical communication, sensor systems, space (satellite) communication Li-Fi (Lightening LED for Wi-Fi)