TwoBeam Interference Constructive or destructive superposition of two

Two-Beam Interference Constructive or destructive superposition of two light waves: Ē 1(r, t) = Ē 01 e i(k 1·r – w 1 t + f 1) Ē 2(r, t) = Ē 02 e i(k 2·r – w 2 t + f 2) Ē = Ē 1 + Ē 2 Intensity measured at a detector: I = Єoc < Ē 2> < > denotes the time average I = Єoc <Ē·Ē* > I = Єoc < (Ē 1 + Ē 2)·(Ē 1* + Ē 2*) > I = Єoc < |Ē 1|2 + |Ē 2|2 + Ē 1·Ē 2* + Ē 2·Ē 1* > I = Єoc <|Ē 1|2> + Єoc <|Ē 2|2> + 2Єoc <Re{Ē 1·Ē 2*}> I 1 I 2 I 12 interference term

Fizeau Fringes • Fizeau fringes produced by a wedge-shaped film • Difference in film thickness between adjacent fringes is lo/2 n 1 Dd = lo/2 n 1 Fringes of equal thickness (FET) real fringes lo S lo virtual fringes no n 1 n 2

Twyman-Green Interferometer • Twyman-Green interferometer used to observe Fizeau fringes • Equivalent to Michelson interferometer but using collimated light viewing microscope collimated light lo M ~ normal incidence

Measurement of Optical Flatness • Difference in film thickness between adjacent fringes is lo/2 n 1 Dd = lo/2 n 1 lo no n 1 n 2

Interferometric Microscopy Dd = lo/2 n 1 lo

Measurement of Film Thickness Dd´/Dd = 2 t/lo t = (lo/2) Dd´/Dd Dd´ = t Dd = lo/2 tilted mirror, M 2 lo M 1 film thickness, t M 2´

Measurement of Film Thickness Dd´/Dd = 2 t/lo t = (lo/2) Dd´/Dd • Resolution ~ (550 nm / 2) (1/200) ~ 1. 4 nm

Measurement of Film Thickness • Fringe locations move with wavelength d = (l 2/2) Dl / (l 1 – l 2) Fringes of equal chromatic order (FECO) Dl Monochromator l 1 tilted mirror, M 2 M 1 film thickness, t l 2 white light source M 2´

Transparent Films P S lens no n 1 d n 2 constructive interference: OPD = 2 n 1 dcosqt = mlo destructive interference: OPD = 2 n 1 dcosqt = (m + ½)lo If reflection coefficients (r, r´) are not small then multiple reflections must be added

Multiple-Beam Interference (Etalons) 1 2 3 4 N E o ) . . . r, t. . . qt lens tt´r´ r´ 2(N-1 qi 0 r. E o tt´r´ E tt´r´ 3 o E tt´r´ 5 o E tt´r´ 7 o Eo no d r´, t´ NEo 2 3 4 ´ 2 1 no tt´r ´ 8 E o tt´r ´ 6 E o tt´r ´ 4 E o tt´r ´ 2 E o tt´r tt´E o 0 . . . n 1 N lens

Multiple-Beam Interference OPD between adjacent rays, D = n 1(AB + BC) – no(AD) = 2 n 1 d cosqt Phase difference between adjacent rays, d = k. D = (4 pn 1 d / lo) cos qt A D no C d n 1 qt B ER = r. Eoeiwt + tt´r´Eoei(wt-d) + tt´r´ 3 Eoei(wt-2 d) +. . . no (ray 0) (ray 1) (ray 2) (rays 3 to N)

Coefficient of Finesse Define coefficient of Finesse, F = 4 r 2/(1 - r 2)2 1 T = IT/Io = 1 + Fsin 2(d/2) R = IR/Io = Fsin 2(d/2) 1 + Fsin 2(d/2) Note: R + T = 1 (conservation of energy) Phase difference between adjacent rays, d = (4 pn 1 d / lo) cos qt

Reflectance from a thin film Single layer thin film (n 1) on glass (n 2=1. 5) R(%) d = (4 pn 1 d / lo) cos qt

Thin Film Thickness Monitoring • Variation in R with d can be used to monitor film thickness (d) during deposition R(%) d = (4 pn 1 d / lo) cos qt

Transparent Films • 2 methods to produce interference in transparent films • Vary the angle of incidence with wavelength fixed VAMFO (variable angle monochromatic fringe observation) • Vary the wavelength of light with a fixed angle of incidence CARIS (constant-angle reflection interference spectroscopy) From Ohring, Fig. 6 -3, p. 257

Comparison of Film Thickness Measurement Techniques From Ohring, Fig. 6 -2, p. 253

Microscopy Conventional microscopy is not sensitive to phase specimens undisturbed amplitude phase light specimen Df

Phase Specimens e. g. , a cell (5 mm thick) in aqueous medium R = [(1. 335 -1. 36)/(1. 335 -1. 36)]2 = 0. 0086% T = 99. 9914% OPD = (1. 36 – 1. 335)(5 mm) = 0. 125 mm = lo/4 Df = 90° t ~ 5 mm n = 1. 335 n = 1. 36 lo = 500 nm

Differential Interference Contrast (DIC) Microscopy -45° polarizer Wollaston prism objective lens sample condenser lens Wollaston prism 45° polarizer Df 2 mm separation between beams

DIC Microscopy Contrast produced by phase gradients light is blocked some light transmitted polarizer phase specimens

Phase Contrast Microscopy Phase object E(t) transform plane superposition, Eo + Ed undiffracted wave, Eo diffracted wave, Ed t I(t) particle surround t

Phase Contrast Microscopy Fritz Zernike: Nobel Prize for Physics, 1953 transform plane phase ring Phase object E(t) superposition, Eo + Ed undiffracted wave, Eo diffracted wave, Ed t I(t) surround particle t