Summary of Fourier Optics

Transcription

1 Summary of Fourier Optics Diffraction of the paraxial wave is described by Fresnel diffraction integral, u(x, y, z) = j λz dx 0 dy 0 u 0 (x 0, y 0 )e j(k/2z)[(x x 0) 2 +(y y 0 ) 2 )], Fraunhofer diffraction is the limit of Fresnel diffraction for large distances between the input plane and the observation plane via the Fourier transform, the radius of the central Airy disk, diffraction pattern of a circular disk, define the limit of diffraction, θ = 1.22λ/D, with a 4f-system, one can modify the image by using a spatial filter in the mask plane. Optoelectronic, 2007 p.1/21

2 Polarization polarization of light is determined by the direction of x,y,z components of E(r, t) vary in time with different amplitudes and phases, for TEM waves z-component is zero, E(z, t) = Re{Aexp(i2πν(t z c )}, where A = A x ê x + A y ê y = a x exp(iφ x )ê x + a y exp(iφ y )ê y, complex electric field vector, E(z, t) = E x ê x + E y ê y, = a x cos[2πν(t z c ) + φ x] + a y cos[2πν(t z c ) + φ y], Optoelectronic, 2007 p.2/21

3 Linear polarized light E x = a x cos[2πν(t z c ) + φ x], E y = a y cos[2πν(t z c ) + φ y], linear polarized light, a y = 0, field points in x direction, linear polarized light, a x = 0, field points in y direction, Optoelectronic, 2007 p.3/21

4 Circular polarized light φ = φ y φ x = ±π/2 and a x = a y = a 0, E x = a 0 cos[2πν(t z c ) + φ x], E y = a 0 sin[2πν(t z c ) + φ x], Ex 2 + E2 y = a2 0, which is a circle, right-circular polarized light, φ = +π/2; and left-circular polarized light, φ = π/2, Optoelectronic, 2007 p.4/21

5 Polarization circulation Optoelectronic, 2007 p.5/21

6 Polarization ellipse E 2 x a 2 x + E2 y a 2 y 2 cos φ E xe y a x a y = sin 2 φ, Optoelectronic, 2007 p.6/21

7 Optics of anisotropic media anisotropy - optical properties depend on the orientation of the medium, P i = j ǫ 0 χ ij E j, each component of P is a linear combination of the components of E, χ is now a tensor (susceptibility tensor), P are not necessarily parallel to E anymore, D = ǫe + P = ǫ(1 + χ)e = ǫe Optoelectronic, 2007 p.7/21

8 the permeability tensor choosing coordinate system that way that ǫ is diagonal, ǫ = ǫ ǫ 22 0 = ǫ x ǫ y ǫ ǫ z the components of ǫ are then the principal axes, principle refractive indices, n x = ǫ11 ǫ 0, n y = optically isotropic, n x = n y = n z, uniaxial, n x = n y = n o (ordinary index), and n z = n e (extraordinary index, optical axis), biaxial, n x n y n z, ǫ22 ǫ 0, and n z = ǫ33 ǫ 0, Optoelectronic, 2007 p.8/21

9 optical anisotropy optical anisotropy is related crystal structure Optoelectronic, 2007 p.9/21

10 vectors of electromagnetic waves general considerations for biaxial crystals S = 1 µ 0 E B, energy flow is not in the direction of k, Optoelectronic, 2007 p.10/21

11 Propagation along a principle axis a linear polarized light propagates along principle axis, k = 0 0 k z, D = ǫ ǫ ǫ 33 E x 0 0, phase velocity c 0 n x, and the wave vector k = n x k 0, D = ǫ 11 E, D parallel to E, equivalent for polarization in y-direction, these special cases are called "normal modes" of the crystal for propagation along a principle axis Optoelectronic, 2007 p.11/21

12 Index ellipsoid Optoelectronic, 2007 p.12/21

13 Uniaxial crystals Optoelectronic, 2007 p.13/21

14 Arbitrary polarization in the x-y plane D = ǫ ǫ ǫ 33 E x E y 0, both field components travel with different phase velocity, for a travelled distance d, the phase differnce between x- and y-components, E x = a x cos[2πν(t z c ) + φ x], E y = a y cos[2πν(t z c ) + φ y], φ = φ y φ x = (n y n x )k 0 d, Optoelectronic, 2007 p.14/21

15 Normal modes Optoelectronic, 2007 p.15/21

16 Polarizers Optoelectronic, 2007 p.16/21

17 Half-wavelength-plate Optoelectronic, 2007 p.17/21

18 Pockels cell Optoelectronic, 2007 p.18/21

19 Liquid Crystals fluid materials, which have orientational order but typically positional disorder, phases of liquid crystal, Optoelectronic, 2007 p.19/21

20 Molecules for liquid crystals Optoelectronic, 2007 p.20/21

21 Liquid Crystal Displays - LCD Optoelectronic, 2007 p.21/21