Electromagnetic optics!

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1 1 EM theory Electromagnetic optics! EM waves Monochromatic light

2 2 Electromagnetic optics! Electromagnetic theory of light Electromagnetic waves in dielectric media Monochromatic light References: Fundamentals of Photonics, Ch. 5

3 3 From ray optics to quantum optics! Ray optics Wave optics Electromagnetic optics! Quantum optics

4 E-M fields are described by two related vector fields functions of ( r,t ) 4 Electric field E(r,t) [V/m] Magnetic field H(r,t) [A/m] Both fields are related by a set of PDE s: E H = ε 0 t H E = µ 0 t H = 0 E = 0 Maxwell s equations! (in free space)! Both E(r,t) and H(r,t) are real functions ε 0 1/36π 10-9 F/m is the electric permittivity μ 0 4π 10-7 H/m is the magnetic permeability

5 5 After 155 years, Maxwell s equations are still famous!

6 6 The wave equation connects E-M optics and wave optics! Each of the components (E x,e y,e z ) and (H x,h y,h z ) must satisfy 2 u 1 c u t 2 = 0 The wave equation! (in free space)! c 0 = 1 ε 0 µ m/s is the speed of light in vacuum (that also appears in wave optics). The (wave)function u(r,t) is any of the components.

7 7 Inside a medium two additional vector fields appear! Electric flux density D(r,t) [C/m 2 ] Magnetic flux density! B(r,t) [Wb/m 2 ] The 4 fields are related by Maxwell s equations in a source-free medium: H = D t E = B t B = 0 D = 0 Maxwell s equations! (source-free medium)! Polarization density P(r,t) Magnetization densitym(r,t) D = ε 0 E + P B = µ 0 H + µ 0 M (in free space P = M = 0)

8 8 Boundary conditions! Homogeneous medium: all components of E, H, D and B are continuous functions of position. Boundary between two different media: Dielectric Dielectric: E //, H //, D and B are continuous Dielectric Metal: E // = 0

9 9 The Poynting vector governs The direction of power flow is perpendicular to both E and H: the flow of power! The optical intensity I is the time-averaged magnitude of the Poynting vector. Calculating the divergence of S we obtain Poynting s theorem: the power flowing from a region equals the change in stored energy ( ) S = E H S = E H [E] = N/C = N/A.s [H] = A/m [S] = N/m.s = W/m 2 = t ( 1 ε 2 0E µ 2 0H 2 ) + E P t + µ 0 H M t energy density of electric and magnetic fields energy on the electric and magnetic dipoles

10 10 An electromagnetic wave also carries linear momentum! This results in radiation pressure on objects from which the wave reflects or scatters. In free space, the linear momentum density / unit volume is a vector ε 0 E B = 1 c 2 S Remember: EM waves carry Power Intensity Energy Momentum

11 11 Orbital angular momentum! Light beams with an azimuthal phase dependence carry orbital angular momentum. Such beams have: Laguerre-Gaussian amplitude! helical phase front!

12 12 A medium is characterized by the constitutive relations! These relate P, M to E, H but in most practical cases we only need to consider the dielectric properties, i.e. P to E. Type of dielectric medium! linear nondispersive Properties! P(r,t) is proportional to E(r,t) P(t) only depends on E(t) homogeneous independent of r! isotropic independent of direction of E(r,t) (so we must have P // E ) E(r,t) medium P(r,t)

13 13 Examples of different media! Optical glass BK7 Calcite GRIN lens KDP crystal

14 A) Linear, nondispersive, homogeneous, and isotropic media! 14 E χ P The relation at every (r,t ) is simply (χ is the electric susceptibility) D and E are also parallel (ε is the electric permittivity) H = ε E t E = µ H t P = ε 0 χe D = ε 0 E + P = ε 0 (1+ χ)e = εe B = µh H = 0 E = 0

15 15 The properties of the medium define the speed of light! Each component of E and H satisfies the wave equation 2 u 1 2 u c 2 t = 0 where c = 1 2 εµ The speed of light inside the medium is related to the refractive index and the susceptibility: n = c 0 c = εµ ε 0 µ 0 = 1+ χ Poynting s theorem has the form of a continuity equation, where W = energy density stored in the medium S = W t W = 1 2 εe µh2

16 B) Linear, nondispersive, inhomogeneous, isotropic media! 16 E(r) χ(r) P(r) The relations become (χ, ε and n become functions of position) P = ε 0 χ(r)e D = ε(r)e The wave equation gains a new term but for slowly varying ε(r) we may write 2 E 2 E 1 ε E µ ε 0 ε 2 E t = 0 2 c(r) = 1/ 1 c 2 (r) 2 E t 2 0 µ 0 ε(r) = c 0 / n(r)

17 C) Linear, nondispersive, homogeneous, anisotropic media! 17 E xyz χ jj P xyz The relation becomes dependent on the direction of the vector E! The dielectric properties of the medium are described by an array of 3 3 constants: the electric susceptibility tensor {χ ij } and the electric permittivity tensor {ε ij } P i = D = j j ε 0 χ ij E j ε ij E j

18 D) Linear, dispersive, homogeneous, isotropic media! 18 E(t ) P(t ) χ(t ) The response depends on the values of E(t ) for all t t : Since a correlation in the time domain is a product in the frequency domain: impulse response function (time domain) transfer function (frequency domain) P(t) = ε 0 χ(t t )E( t )d t P(ν) = ε 0 χ(ν)e(ν) ε 0 χ(t) ε 0 χ(ν)

19 E) Nonlinear, nondispersive, homogeneous, and isotropic media! 19 The relation is nonlinear; the previous Maxwell s equation in a linear medium and the wave equation are not valid.! P χ (1) E + χ (2) E 2 + χ (3) E 3 + Starting from the general Maxwell s eqns. in a medium, and considering a homogeneous and isotropic medium: E E = µ H t ( ) = µ H t general wave equation for a homogeneous, isotropic medium 2 E 1 c E t 2 = µ 0 2 P t 2 Most dielectric media are approximately linear, except when using very powerful, focused laser beams.

20 20 Nonlinear optics was born with the discovery of the laser! When the electric field exceeds the interatomic electric fields (~10 8 V/m) strange phenomena start happening

21 21 Monochromatic electromagnetic waves! Just as we did for wave optics, let s see some examples of solutions of the wave equation. Things are much simpler when we consider monochromatic waves. Introducing a complex representation: It s easy to calculate the time derivative of these waves: Maxwell s equations in a medium become: E(r,t) = Re{ E(r )e iωt } H(r,t) = Re{ H(r )e iωt } t t ( E(r )e iωt ) = iωe(r )e iωt ( H(r )e iωt ) = iωh(r )e iωt H = jωd B = 0 E = jωb D = 0 (D = ε 0 E + P)

22 22 Intensity and Helmholtz equation for monochromatic E-M waves! Writing the complex Poynting vector: The optical intensity is then the magnitude of the vector Re{S} S = Re{ Ee iωt } Re{ He iωt } S = = Re{S} 1 2 E H In a linear, nondispersive, homogeneous, isotropic medium we have Like in wave optics, a very simple equation can be now derived for U = (E x,y,z ) or (H x,y,z ): the well-known Helmholtz equation! D = εe B = µh 2 U + k 2 U = 0 ( k = nk 0 = ω εµ )

23 23 The Transverse E-M plane wave! Let s consider a monochromatic E-M wave whose E and H fields are plane waves: From wave optics, we already know that plane waves are solutions of the Helmholtz equation, with Replacing H, E in Maxwell s equations: It follows that: E is perpendicular to both H and k H is perpendicular to both E and k!! This is then a transverse electromagnetic (TEM) wave! H(r) = H o exp( ik r) E(r) = E o exp( ik r) k = k = nk 0 k H o = ωεe o k E o = ωµh 0

24 24 Intensity of a TEM wave! We have for the complex Poynting vector: Where we have introduced the parameter called impedance of the medium: (impedance of vacuum 377 Ω) The intensity is then: S = S = 1 2 E H η = E 0 H 0 = 1 2 E 0H 0 = E 0 2 / 2η I = E 0 2 / 2η µ ε The most powerful laser in the world has a peak intensity of ~10 22 W/cm 2. What is the corresponding magnitude of the electric field? Compare with the electric fields used at a particle accelerator (~10 7 V/m)

25 25 Electron acceleration in plasmas waves using lasers!

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