INSA de Rouen - STPI2 - Année 2011-2012

P4-2 SIB 1

Wave Basics

Wave function : Ψ(x,t) At a point M : ΨM (x,t) = Ψ(xM ,t) = Ψ(0;t − τ) with τ =

xM the delay v

Harmonic wave : sinusoidal oscillation Caracteristics :

• Amplitude A

• Angular Frequency ω =

• Period T • Frequency ν =

1 T

• Wavelength λ = vT =

2π = 2πν T

v ν

• Velocity of Propagation v =

λ T

ω 2π u~0 = u~0 where u~0 points in the direction of propagation, k~ u0 k = 1 v λ −−→ Wave Function : Ψ(x, y, z,t) = A cos(ωt − k~0 ·~r) where k~0 = (kx , ky , kz ) and~r = OM = (x, y, z) Wave Vector : k~0 = k~ u0 =

Transverse wave : displacement perpendicular to k~0 Longitudinal wave : displacement parallel to k~0 Wave Equation :

2

∂ 2Ψ ∂ 2Ψ ∂ 2Ψ ∂ 2Ψ = v2 ∆Ψ where ∆Ψ = ∇2 Ψ = + 2 + 2 is the Laplacian. 2 ∂t ∂ x2 ∂y ∂z

Electromagnetic Waves

Light = Electromagnetic Wave = Propagation of Magnetic Field ~B and Electric Field ~E Monochromatic Planar Wave (MPW) :

• Monochromatic : One single angular frequency ω • Planar : Perpendicular plane to the direction of propagation • Polarization of the wave : direction of oscillation of the wave

Wave Speed : In Matter, v = Material Wavelength : λ = λn0 Material Wave Vector : ~k = nk~0

c n

where n is the refraction index (n = 1 for vacuum).

v0 where v~0 points in the direction of oscillation. (unit : V.m−1 = N.C−1 ) Electric Field : ~E = E0 cos(ωt − k~0 ·~r)~ 1 E0 Magnetic Field : ~B = ~k ∧ ~E ⇒ ~B = cos(ωt − k~0 ·~r)(~ u0 ∧ v~0 ) = B0 cos(ωt − k~0 ·~r)w ~ 0 (unit : Tesla T ) ω c Then, (~ u0 , v~0 , w ~ 0 ) is a "‘right-handed"’ system. 1 ~E ∧ ~B (density vector of energy flux, unit : W.m−2 ) µ0 ~k and ~R have the same direction : ~R = k~Rk~ u0 Poynting Vector : ~R =

1

1 RT E2 k~Rk dt = · · · = 0 T0 2µ0 c ae a e or I = E·~E ∗ = · · · = E02 in complex representation, with ~E and ~E ∗ conjugate complex vectors 2 2

Irradiance : I = hk~Rkit =

3

Polarization of Light

3.1

Description

k= Ex = 0 0 ~ ~ ~ We set E = Ey = E0 cos(ωt − k0 ·~r + ϕ) and k = 0 0 E = E 00 cos(ωt − k~ ·~r) z 0 0

2π λ

=

ω c

Linear Polarization : E00 6= E000 and ϕ ≡ 0[π] E 00 tan α = 00 E0 Circular Polarization : E00 = E000 and ϕ ≡ π2 [π] Elliptical Polarization : E00 6= E000 or ϕ 6= 0[π] ϕ > 0 : clockwise direction, right polarization ϕ < 0 : anticlockwise direction, left polarization Unpolarized Wave : Ex = 0 ~E = Ey = E0 cos(ωt − k~0 ·~r + ϕy (t)) where ϕy (t) ans ϕz (t) are random functions of time. E = E cos(ωt − k~ ·~r + ϕ (t)) z z 0 0 0 ae ~0 ·~r) jϕy (t) j(ωt− k Irradiance : I = ~E·~E ∗ = · · · = ae E02 Complex Representation : E = E0 e e 2 e jϕz (t)

3.2

Changing polarization

Polarizers : Transmitting one single orientation of ~E along its transmission axis ~P : E~P = (~E · ~P)

~P k~Pk

0 a a I e e 0 ∗ 2 2 ~ ~ ~ On unpolarized lightwave : I0 = ae E0 =⇒ IP = h EP ·EP it = · · · = E0 = where P = cos α 2 2 2 sin α 0 ae Malus’s Law : After polarizer ~P, new polarizer ~A = cos β =⇒ IA = · · · = E02 cos2 (α − β ) = IP cos2 (α − β ) 2 sin β

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Optical Wave Interference

Superposition of two planar lightwaves (Same wavelength λ andsame initial axes) :  ~ ~ ~ ~ Let E1 = A1 e j(ωt−k1 ·~r)~z0 and E2 = A2 e j(ωt−k2 ·~r)~z0 =⇒ E=E1 +E2 = A1 e j(ωt−k1 ·~r) + A2 e j(ωt−k2 ·~r) ~z0      √ 2π =⇒ I = · · · = I1 + I2 + 2 I1 I2 cos k~2 − k~1 ·~r where k~2 − k~1 ·~r = δ (M) = ϕ (M) λ {z } | term of interference

• Bright Fringes (Constructive Interference) : I = Imax ⇐⇒ cos (ϕ (M)) = +1 : Interference Fringes : • Dark Fringes (Destructive Interference) : I = Imin ⇐⇒ cos (ϕ (M)) = −1 Imin + Imax Mean Value of I : Imean = = · · · = I1 + I2 2 2

Fringe Visibility : V =

Imax − Imin Imax + Imin

• V = 0 =⇒ Imax = Imin ⇐⇒ no interference • V = 1 =⇒ Imin = 0 ⇐⇒ perfect contrast

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