EE 5392/7392 – Fourier Optics Optics Tutorial

1. EE 5392/7392 – Fourier OpticsOptics Tutorial Manjunath Somayaji Spring 2010 Department of Electrical Engineering

2. Reflection 300 BCE: Euclid’s Catoptrics accurately describes that the angles of incidence and reflection of light at a surface equal each other Refraction 400 BCE: Plato’s Republic – First recorded mention 1621: Snell defines law of refraction Diffraction “Any deviation of light rays from rectilinear paths which cannot be interpreted by reflection or refraction” Sommerfeld, 1954 1665: Grimaldi – first accurate report and description of the diffractive behavior of light. 1678: Huygens proposes wave theory 1803: Young’s double-slit experiment Understanding Optical Phenomena – a Brief History

3. Interference and Young’s Double-Slit Experiment • One can add light to light and create darkness! • Ray optics cannot explain this phenomenon – one needs wave theory.

4. Historical Background of Wave Optics • Also known as Physical Optics • Proponents include Huygens, Young, Fresnel, among others. • Some Milestones: • 1803: Young’s double-slit experiment showing interference • 1818: Fresnel’s memoirs describing diffraction – leading to Poisson’s Spot

5. Fresnel’s Work on the Wave Nature of Light • Remarkably accurate predictions of diffraction patterns. • Presented to a prize committee at the French Academy of Scientists. • Poisson: it is absurd, it would predict a bright spot at the center of the shadow of an opaque disk! • Chair of committee, F. Arago, • does the experiment: ?

6. Coverage • Wavefunction, Wave equation • Wavevector, Wavenumber. • Spherical, paraboloidal, plane waves • Plane wave equation • Spherical wave equation • Paraxial approximation • Phase shift during propagation • Concepts – Monochromatic vs. Polychromatic light • Concepts – Coherent vs. Incoherent light

7. Wavefunction, Wave equation Refractive index and the speed of light: • Speed of light in vacuum = co = 3 x 108 m/s • Speed of light in a medium with refractive index n = c = co/n • Since |n| ≥ 1, we have c≤co The complex wavefunction: An optical wave is mathematically described by a real function of position r = (x, y, z) and time t. It is denoted by U(r, t). The wave equation: where

8. Complex amplitude Complex Amplitude, Intensity, Coherence Complex Amplitude: U(r,t) = a(r) exp[jφ(r)] exp(j2πνt) U(r,t) = U(r) exp(j2πνt) Optical Intensity: I(r) = |U(r)|2 Coherent and Incoherent Light: U(r) I(r)

9. Wavefront, Wavelength, Wavevector, Wavenumber, • Wavefront: surfaces of equal phase, φ(r) = constant. • Wavelength = λ = c/ν • Wavevector = k = (kx, ky, kz) – indicates direction of propagation and wavelength • Wavenumber = k = 2π/λ = |k| • Helmholtz equation: Optical waves must satisfy this relation

10. Plane Wave, Spherical Wave The Plane Wave:U(r) = A exp(–jk.r) = A exp[–j(kx x + ky y + kz z)] The Spherical Wave:U(r) = A/r exp(–jkr) U(r) = A/r exp[–jk(x2 + y2 + z2)1/2] The Paraboloidal Wave (Fresnel Approximation) Valid for small angles only! Spherical Paraboloidal Plane Wave wave wave

11. Refraction Relationship between rays and waves: Rays are normals to wavefronts!

12. d(x, y) x z y Transmission through Transparent Materials – Phase Shifts Complex amplitude transmittance: t(x, y) = U(x, y, d) / U(x, y, 0) t(x, y) = ho exp[–j(n–1)kod(x, y)] U(x, y, d) U(x, y, 0)

13. Complex Amplitude Transmittance do x t(x, y) = ho exp[–j(n–1)kox] Prism: Linear! i.e.: tilted plane wave!  z ho = exp[–jkodo] Quadratic! i.e.: A plane wave is converted to a spherical wave! Lens: t(x, y) = ho exp[jko{(x2+y2)/2f}]

14. The Phenomenon of Diffraction This effect becomes more pronounced as the dimensions of the aperture approach the wavelength of the propagating light!

15. The Huygens-Fresnel Principle Courtesy: Wikipedia Courtesy: Wikipedia Refraction Diffraction Watch MIT TechTV Video

16. The End