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
tilted
mirror, M2 l1
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)Eo 1 2 3 4 N
lens qi
rEo
tt´r´Eo
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

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 = [( )/( )]2 = %
T = %
OPD = (1.36 – 1.335)(5 mm) = 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
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