PHYC 3540 Summary I. II. Propagation of light a. History, wave vs. particle picture b. Fermats Principle – actual path is that for which the OPL is an extremum c. Huygen’s principle – every point on wavefront acts as a source i. Large aperture (>>) – rectilinear propagation (Geometric optics) ii. Small aperture ( ) – spreading of wavefront (Diffraction) d. Wavefronts, Rays – Laws of reflection and refraction (Snels law) Geometric Optics a. Imaging – Cartesian surfaces, Paraxial ray approximation b. Refraction at a Spherical interface n n' s s' R n n' n'n n n' s s' R f f' n n' P Power f f' c. Thin lens n n' n L n n L n' n n' P s s' R1 R2 f f' Magnificat ion M h' ns ' h n' s for n n' F’ F s 17/02/16 all s' Page 1 of 8 imaging d. Principal Planes (H,H’) i. Used to represent any optical element or system and are located such that the system images an object according to: n n' n n' P s s' f f' ii. The principal planes used to represent a spherical interface (or thin lens) must coincide at the interface (or centre of the thin lens). H H’ n' n f f' s' s e. Combination of two systems n n2 H1 H1’ H n' H’ H2’ H2 s s' f f' i. Power of combined system P P1 P2 where, P1 17/02/16 Page 2 of 8 dP1 P2 n2 n n n2 n' ' and P2 2 ' f1 f2 f1 f2 ii. The location of H, H’ are determined with respect to H1 and H2’ as f' P n' h' H 2 ' H ' d 1 d f1 ' P n2 follows: f P n h H 1 H d 2 d f2 P n2 f. Matrix methods in paraxial optics i. Matrices 1 L Translation matrix 0 1 1 Refraction matrix n n' Rn ' 1 Reflection matrix 2 R 0 n n' 0 1 0 1 Thin lens matrix 1 1 f ii. Cardinal point locations P n' 1 A h' d 1 P n2 C hd P2 P n n2 n D n' C g. Stops in optical systems i. Aperture stop Element that limits the bundle of rays (and hence light gathering power or speed) collected by optical system Is that element whose entrance pupil subtends the smallest angle at the object point in question Exit pupil is the image of the aperture stop in all elements following it Entrance pupil is the image of the aperture stop in all elements preceding it 17/02/16 Page 3 of 8 III. ii. Field stop Element of the optical system that limits the field of view That element whose entrance window subtends the smallest angle at the centre of the entrance pupil Exit window is the image of the field stop in all elements following it. Entrance window is the image of the field stop in all elements preceding it iii. Chief ray, marginal rays – definition of edge of field of view iv. Angular field of view Image space – angle formed by the entrance window as viewed from the centre of the entrance pupil Object space – angle formed by the exit window as viewed from the centre of the exit pupil h. Applications i. Cameras (aperture settings (f#), exposure, telephoto lens) ii. Human eye – myopia, hyperopia iii. Hand magnifiers, eyepieces, compound microscope iv. Telescope, binoculars Physical optics (wave properties are important) a. Wave equation – plane waves, spherical waves, cylindrical waves b. Amplitude reflection and transmission co-efficients 12 v2 v1 v1 v2 ; 12 2v2 v1 v2 c. Electromagnetic waves (E B; E, B in phase, speed c/n) i. Irradiance I 1 vEo 2 2 ; E o Amplitude ii. Transmittance T I2 n1n2 n2 2 4 12 12 21 I1 n1 (n1 n2 ) 2 iii. Reflectance (12) 2 n n1 I 2 12 2 21 R R 2 I 1 n2 n1 d. Interference i. Two source interference Irradiance I I 1 I 2 2 I 1 I 2 cos 2 1 where, 17/02/16 2 1 k (r2 r1 ) ( 2 1 ) Page 4 of 8 If 1 = 2 and I1=I2=Io there are maxima located at, sin m a Examples Double slit, Lloyds mirror, radio towers Dielectric layers, e.g. Haidinger bands, fringes of equal thickness observed in oil films, air wedges Antireflection coatings Newton’s rings ii. Multiple beam interference (coated dielectric layers) Transmitted irradiance 2 T Io 1 IT 2 4 R (1 R) sin 2 1 2 2 (1 R) where, 2 k o nd cos ' and n, ’ pertain to the dielectric layer. Irradiance maxima occur when 2nd cos ' mo Fabry-Perot Interferometer Finesse = separation between fringes R fringe width ( FWHM ) 1 R Free spectral range FSR 2 2nd ; v~FSR 1 2nd Fringe width free spectral range 2 finesse 2nd Spectral Resolution 2nd min v~ ~ 2ndv~ v 17/02/16 Page 5 of 8 e. Diffraction i. Fresnel–Kirchoff diffraction formula (Mathematical statement of Huygen’s principle) ikEo E 2 e ik ( r r ') aperture rr ' F dA where, F 1 cos(nˆ, r ) cos(nˆ, r ' ) obliquity 2 factor P r r' n ii. Fraunhofer limit (plane waves) diffraction integral reduces to I 0 when 2 where, sin m kb sin 2 b Circular aperture (diameter do = 2ao) 2 J I I o 1 2 where, = kaosin 1.22 ao Airy disc; Spatial resolution of optical devices; focussing limit N slits (width b, separation a) I 0 aperture Single slit (width b) sin I I o E C ' e ikr dA when sin 2 sin sin N I I o where, 2 Principal maxima when sin p 17/02/16 Page 6 of 8 a ka sin 2 Diffraction gratings Grating dispersion Spectral resolution mN f. Polarization i. Jones vector for polarization E ox General i E oy e 0 Vertical 1 Horizontal 1 0 cos Linearly polarized at an angle α, (1) m sin 1 1 Left Circularly polarized (i.e. counterclockwise) 2 i 1 1 (i.e. clockwise) 2 i A 1 Left Elliptically polarized 2 2 2 B iC A B C A 1 Right Elliptically polarized A 2 B 2 C 2 B iC Right Circularly polarized C B and the angle of inclination, α, of the ellipse is given by 2 Eox Eoy tan 2 2 Eox Eoy2 ii. Jones matrices for optical elements Linear polarizer (Transmission axis at an angle wrt xaxis) cos 2 sin cos sin 2 sin cos Phase retarders e i x 0 In general, where, x and y represent i y 0 e the advance in phase of the components Quarter wave plate (|y - x| = /2) 0 i 1 e 4 Slow axis vertical 0 i where, E ox A e 17/02/16 E oy B 2 C 2 i 4 tan 1 1 0 0 i Slow axis horizontal Page 7 of 8 Half wave plate (|y - x| = ) 0 i 1 e 2 Slow axis vertical 0 1 1 0 0 1 Slow axis horizontal Rotator (light polarized at an angle is rotated by an angle ) cos sin sin cos e i 2 g. Holography i. Hologram R( x, y, z ) r ( x, y, z )e i ( x , y , z ) re ikz O( x, y, z ) o( x, y, z )e i ( x , y , z ) oe ikr where r x2 y2 z2 I ( x, y ) O R OO * RR * OR * O * R 2 where, R – reference wave, O – object wave, I - intensity ii. Reconstructed wave 2 R( x, y )t p (tb BOO * ) R BR 2O* B R O where, t – transmission, first term – direct wave, second term – conjugate wave, third term – object wave 17/02/16 Page 8 of 8