EE 485, Winter 2004, Lih Y. Lin Chapter 5 Electromagnetic Optics - Introduce “vector” nature* of light Electrical field: E (r,t) * Magnetic field: H (r,t) 5.1 *Electromagnetic* Theory of Light E (r,t) and H (r,t) must satisfy Maxwell’s equations. In free space ∂E ∇ ×H = ε (5.1-1) 0 ∂t ∂H ∇ ×E = −µ (5.1-2) 0 ∂t ∇ ⋅E = 0 (5.1-3) ∇ ⋅H = 0 (5.1-4) 1 − Farad ε : Electric permittivity = ×10 9 ( ) in MKS units 0 36π Meter − Henry µ : Magnetic permeability = 4π×10 7 ( ) in MKS units 0 Meter The wave equation * * All components of E (r,t) and H (r,t) (Ex , Ey , Ez , Hx , Hy , Hz ) satisfy the wave equation: 1 ∂ 2u ∇2u − = 0 (5.1-5) 2 ∂ 2 c0 t 1 c = 0 ε µ 0 0 In free space, Maxwell equations and wave equations are linear, therefore principle of superposition applies. Maxwell’s equations in a medium Assume no free electric* charges or currents. Need two more vector fields: D (r,t) : Electric flux density, or electric displacement * B (r,t) : Magnetic flux density 1 EE 485, Winter 2004, Lih Y. Lin ∂D ∇ ×H = (5.1-7) ∂t ∂B ∇ ×E = (5.1-8) ∂t ∇ ⋅D = 0 (5.1-9) ∇ ⋅B = 0 (5.1-10) = ε + D 0E P (5.1-11) = µ + µ B 0H 0M (5.1-12) P : Polarization density (macroscopic sum of the electric dipole moments that the electric field induces) M : Magnetization density (macroscopic sum of the magnetic dipole moments that the magnetic field induces) In a non-magnetic medium (assumption of this course), M = 0 = µ → B 0H (5.1-13) Boundary conditions Assume no free charges or surface currents. In a homogeneous medium, E , H , D , B are continuous. At the boundary between two dielectric media, the tangential components of E and H, and the normal components of D and B are continuous. Intensity and power S =E ×H : Poynting vector (direction and magnitude of power flux) I = S : Power flow across a unit area normal to S 5.2 Dielectric Media* * - Linear: P (r,t)= constant ×E (r,t) - Non-dispersive: Response is instantaneous. P (t0 ) does not depend on E (t < t ) (An idealization. Any physical system has a finite response time). 0 * - Homogeneous: P (E ) does not depend on r . - Isotropic: P (E ) is independent of the direction of E . A. Linear, Nondispersive, Homogeneous, and Isotropic Media = ε χ P 0 E at any position and time (5.2-1) χ : Electric susceptibility (scalar constant) = ε + χ ≡ ε D 0 (1 )E E (5.2-2,3) 2 EE 485, Winter 2004, Lih Y. Lin ε: Electric permittivity of the medium ε ε ≡ = 1+ χ : Dielectric constant r ε 0 ∂E ∇ ×H = ε (5.2-4) ∂t ∂H ∇ ×E = −µ (5.2-5) 0 ∂t ∇ ⋅E = 0 (5.2-6) ∇ ⋅H = 0 (5.2-7) Wave equation: 1 ∂ 2u ∇2u − = 0 (5.2-8) c2 ∂t 2 1 c c = = 0 (5.2-9) εµ 0 n = ε = + χ n r 1 : Refractive index (5.2-10) B. Nonlinear, Dispersive, Inhomogeneous, or Anisotropic Media Inhomogeneous media * * * P = ε χ(r)E , D = ε(r )E , n = n(r) 0 * If the medium is locally homogeneous, that is, ε(r) varies sufficiently slowly, 1 ∂ 2E ∇2E − * = 0 (5.2-12) c2 (r) ∂t 2 Anisotropic media P and E are not necessarily parallel. If the medium is linear, non-dispersive, and homogeneous, = ε χ Pi ∑ 0 ijEj i, j = 1, 2, 3 denote the x, y, and z components j {χ } ij : Susceptibility tensor (3 × 3 matrix) = ε Di ∑ ijEj j {ε } ij : Electric permittivity tensor Dispersive media t = ε − P (t) 0 ∫ x(t t')E (t')dt' (5.2-17) −∞ 3 EE 485, Winter 2004, Lih Y. Lin χ(ν) is the Fourier transform of x(t) → χ = χ(ω), ε = ε(ω), n = n(ω) Nonlinear media If the medium is homogeneous, isotropic, and nondispersive, P = Ψ(E ) for all position and time, Ψ is a nonlinear function. Wave equations: 1 ∂ 2E ∂ 2Ψ(E ) ∇2E − = µ (5.2-20) 2 ∂ 2 0 ∂ 2 c0 t t → Principle of superposition no longer applicable 5.3 Monochromatic Electromagnetic Waves Define complex amplitude: E (r,t) = Re{}E(r) exp( jωt) (5.3-1) H (r,t) = Re{}H(r) exp( jωt) Likewise, for P, D, B . Maxwell’s equations ∇ × H = jωD (5.3-2) ∇ × E = − jωB (5.3-3) ∇ ⋅ D = 0 (5.3-4) ∇ ⋅B = 0 (5.3-5) = ε + D 0E P (5.3-6) = µ B 0H (5.3-7) Optical intensity and power 1 S ≡ E× H *: Complex Poynting vector (5.3-9) 2 Optical intensity S = Re{}S (5.3-8) Linear, nondispersive, homogeneous, and isotropic media ∇ × H = jωεE (5.3-11) ∇ × = − ωµ E j 0H (5.3-12) ∇ ⋅E = 0 (5.3-13) ∇ ⋅ H = 0 (5.3-14) Components of E and H must satisfy the Helmholtz equation: 4 EE 485, Winter 2004, Lih Y. Lin ∇2U + k 2U = 0 (5.3-15) = k nk0 Inhomogeneous media Eqs. (5.3-11) ~ (5.3-14) remain applicable, ε → ε(r) . For locally homogeneous, ε(r) varies slowly with respect to wavelength ε(r) → k = n(r)k , n(r) = 0 ε 0 Dispersive media = ε χ ν P 0 ( )E (5.3-16) D = ε(ν)E (5.3-18) ε ν = ε []+ χ ν ( ) 0 1 ( ) (5.3-19) → The only difference between non-dispersive medium and dispersive medium is that ε and χ are frequency-dependent. The Helmholtz equation applicable with ε(ν) k = n(ν)k , n(ν) = (5.3-20) 0 ε 0 5.4 Elementary Electromagnetic Waves Consider monochromatic waves. Assume the medium is linear, homogeneous, and isotropic. The transverse electromagnetic (TEM) plane wave = − ⋅ E(r) E0 exp( jk r) (5.4-1) = − ⋅ H(r) H0 exp( jk r) (5.4-2) = k nk 0 Substituting into Maxwell’s equations, we obtain: k × H = −ωεE 0 0 × = ωµ k E0 0H0 → E, H, and k are mutually orthogonal. E 1 µ η 0 = 0 ≡ η = 0 : Impedance of the medium (5.4-5,6) ε H0 n 0 n µ η = 0 = 120π = 377Ω: Impedance of free space (5.4-7) 0 ε 0 5 EE 485, Winter 2004, Lih Y. Lin E 2 Magnitude of Poynting vector S = 0 = I : Intensity (5.4-8) 2η 5.5 Absorption and Dispersion A. Absorption Light-absorbing dielectric materials have complex susceptibility, χ = χ'+ jχ" (5.5-1) and comlex permittivity, ε = ε + χ 0 (1 ) The Helmholtz equation is still valid, but = + χ + χ k 1 ' j "k0 (5.5-2) is now complex-valued. 1 k ≡ β − j α (5.5-3) 2 1 exp(− jkz) = exp(− αz) exp(− jβz) 2 → Intensity attenuated by exp(-αz) after propagating a distance z. α: Absorption coefficient (attenuation coefficient, or extinction coefficient) β : Propagation constant = nk0 (5.5-4) = The wave travels with a phase velocity c c0 n . α n − j = 1+ χ'+χ" (5.5-5) 2k0 → Relating the refractive index and the absorption coefficient to the real and imaginary parts of the susceptibility. Weekly absorbing media χ'<< 1, χ"<<1 1 → n ≈ 1+ χ' (5.5-6) 2 α ≈ − χ k0 " (5.5-7) B. Dispersion χ = χ ν = ν = ν = ν ( ), n n( ), c c( ) c0 / n( ) 6 EE 485, Winter 2004, Lih Y. Lin Pulse broadening (example: chromatic dispersion in optical fibers): Measures of dispersion Examples: 1) For glass optical components used with white light, n −1 V number ≡ D − nF nC nF, nD, nC: Refractive indices at blue (486.1 nm), yellow (589.2 nm), and red (656.3 nm) dn 2) dλ λ=λ 0 dθ dθ dn Prism: d = d dλ dn dλ C. The Resonant Medium d 2P dP + σ + ω2P = ω2ε χ E (5.5-12) dt 2 dt 0 0 0 0 ↔ Classical harmonic oscillator for bound charges in the medium d 2 x dx F + σ + ω2 x = (5.5-13) dt 2 dt 0 m 7 EE 485, Winter 2004, Lih Y. Lin m: Mass of the bound charge κ ω = : Resonant angular frequency 0 m κ: Elastic constant σ: Damping coefficient F = eE : Force Polarization density P = Nex , N: Number of charges per volume e2 N → χ = 0 ε ω2 m 0 0 Substituting E (t) = Re{}E exp( jωt) and P (t) = Re{}P exp( jωt) into Eq. (5.5- 12), we obtain: χ ω2 P = ε 0 0 E = ε χ(ν)E (5.5-14) 0 ω2 − ω2 + σω 0 0 j ν2 χ(ν) = χ 0 (5.5-15) 0 ν2 − ν2 + ν∆ν 0 j σ ∆ν = 2π ν2 ()ν2 − ν2 χ'(ν) = χ 0 0 (5.5-16) 0 ()ν2 − ν2 2 + ()ν∆ν 2 0 ν2ν∆ν χ"(ν) = −χ 0 (5.5-17) 0 ()ν2 − ν2 2 + ()ν∆ν 2 0 ν << ν → χ'(ν) ≈ χ : Low - frequency susceptibility 0 0 χ"(ν) ≈ 0 ν >> ν → χ ν ≈ χ ν ≈ 0 '( ) 0, "( ) 0, the medium acts like free space 8 EE 485, Winter 2004, Lih Y. Lin ν ν = ν → χ'(ν ) = 0, - χ"(ν) = 0 χ 0 0 ∆ν 0 Near resonance: ν / 2 χ(ν) = χ 0 (5.5-18) 0 ν − ν + ∆ν ( 0 ) j / 2 ν ∆ν 1 χ"(ν) = −χ 0 (5.5-19) 0 ()()ν − ν 2 + ∆ν 2 4 0 / 2 ν − ν χ'(ν) = 2 0 χ"(ν) (5.5-20) ∆ν ∆ν : FWHM of χ"(ν) 5.6 Pulse Propagation in Dispersive Media Assume the medium is linear, homogeneous, and isotropic, α = α ν = ν β ν = πν ν ( ), n n( ), ( ) 2 n( ) / c0 Pulsed plane wave in z-direction: = []ω − β U (z,t) A (z,t) exp j( 0t 0 z) (5.6-1) β = β ν 0 ( 0 ) A (z,t) : Complex envelope of the pulse, slow varying in ν comparison with 0 Knowing A (0,t) → Need to determine A (z,t) Linear-system description Suppose A (0,t) = A(0, f ) exp( j2πft) A(z, f ) = A(0, f )H ( f ) (5.6-2) = − 1 α − ()β − β H ( f ) exp f +ν z j f +ν ν z (5.6-3) 2 0 0 0 Determine A (z,t) from A (0,t) : Fourier transform 9 EE 485, Winter 2004, Lih Y.