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Two-Beam Interference

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 = Є o c < Ē 2 > < > denotes the time average.

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Two-Beam Interference

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  1. Two-Beam Interference Constructive or destructive superposition of two light waves: Ē1(r, t) = Ē01 e i(k1·r – w1t + f1) Ē2(r, t) = Ē02 e i(k2·r – w2t + f2) Ē = Ē1 + Ē2 Intensity measured at a detector: I = Єoc < Ē2> < > denotes the time average I = Єoc <Ē·Ē* > I = Єoc < (Ē1 + Ē2)·(Ē1* + Ē2*) > I = Єoc < |Ē1|2 + |Ē2|2 + Ē1·Ē2* + Ē2·Ē1* > I = Єoc <|Ē1|2> + Єoc <|Ē2|2> + 2Єoc <Re{Ē1·Ē2*}> I12 I1 I2 interference term

  2. Fizeau Fringes • Fizeau fringes produced by a wedge-shaped film • Difference in film thickness between adjacent • fringes is lo/2n1 Dd = lo/2n1 Fringes of equal thickness (FET) real fringes lo lo S no n1 n2 virtual fringes

  3. 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

  4. Measurement of Optical Flatness • Difference in film thickness between adjacent • fringes is lo/2n1 Dd = lo/2n1 Dd = lo/2n1 lo lo no no n1 n1 n2 n2

  5. Interferometric Microscopy Dd = lo/2n1 lo

  6. Measurement of Film Thickness Dd´/Dd = 2t/lo t = (lo/2) Dd´/Dd Dd´ = t Dd = lo/2 tilted mirror, M2 lo M1 M2´ film thickness, t

  7. Measurement of Film Thickness Dd´/Dd = 2t/lo t = (lo/2) Dd´/Dd • Resolution ~ (550 nm / 2) (1/200) ~ 1.4 nm

  8. Measurement of Film Thickness • Fringe locations move with wavelength • d = (l2/2) Dl / (l1 – l2) Fringes of equal chromatic order (FECO) Dl Monochromator  l2 l1 tilted mirror, M2 white light source M1 M2´ film thickness, t

  9. Transparent Films P S lens no n1 d n2 constructive interference: OPD = 2n1dcosqt = mlo destructive interference: OPD = 2n1dcosqt = (m + ½)lo If reflection coefficients (r, r´) are not small then multiple reflections must be added

  10. Multiple-Beam Interference (Etalons) 0 Eo 1 2 3 4 N lens qi tt´r´Eo rEo tt´r´3Eo ... tt´r´5Eo tt´r´7Eo tt´r´r´2(N-1)Eo no r, t ... d n1 qt r´, t´ no ... tt´r´4Eo tt´r´8Eo tt´Eo tt´r´6Eo tt´r´2NEo tt´r´2Eo lens 1 2 3 4 N 0

  11. Multiple-Beam Interference OPD between adjacent rays, D = n1(AB + BC) – no(AD) = 2n1d cosqt Phase difference between adjacent rays, d = kD = (4pn1d/ lo) cos qt D A no C d n1 qt B no ER = rEoeiwt (ray 0) + tt´r´Eoei(wt-d) (ray 1) + tt´r´3Eoei(wt-2d) (ray 2) + ... (rays 3 to N)

  12. Coefficient of Finesse Define coefficient of Finesse, F = 4r2/(1 - r2)2 1 1 + Fsin2(d/2)  T = IT/Io = Fsin2(d/2) 1 + Fsin2(d/2) R = IR/Io = Note: R + T = 1 (conservation of energy) Phase difference between adjacent rays, d = (4pn1d / lo) cos qt

  13. Reflectance from a thin film Single layer thin film (n1) on glass (n2=1.5) R(%) d = (4pn1d / lo) cos qt

  14. Thin Film Thickness Monitoring • Variation in R with d can be used to monitor • film thickness (d) during deposition R(%) d = (4pn1d / lo) cos qt

  15. 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

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

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

  18. 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

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

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

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

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

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