Interference and diffraction 131
FIGURE 7.5: Explaining fringes of equal inclination.
The phase difference associated with this path difference isk0Λ. Noting that there will be an additional phase accumulation in reflection from at least one interface by ±π irrespective of the fact that the refractive index of the film is lower or higher than that of its environment, the phase difference can be written as
δ=4πnt
λ dcosθt±π (7.38)
or
δ=4πd
λ (n2t−n2isin2θi)1/2±π. (7.39) We choose the negative sign in Eq. (7.38) to make it look simpler. Interference maxima will occur atδ= 2mπ, which can be rewritten as
dcosθt= (2m+ 1)λt
4 , (7.40)
whereλt=λ/nt. The same conditions correspond to minima in the transmit- ted light. Interference minima in reflection (maxima in transmission) occur forδ= (2m±1)πand for such cases
dcosθt= 2mλt
4 . (7.41)
A comment regarding the refractive indices is in order. If the refractive indices are in increasing or decreasing order, the additional phase shift of±πwould not be there and the formulas for the maxima and minima would be inter- changed. Since the phase difference is mainly controlled byθ, such fringes are generally referred to as fringes of equal inclination.
7.3.3 Fringes of equal width
In contrast to the dominating role ofθ, a whole class of fringes exists for which the optical thickness of the film plays the most important part. These
Interference and diffraction 133
S
P
nt d α x
FIGURE 7.6: Explaining fringes of equal thickness.
are referred to as the fringes of equal width. Each fringe is the locus of all points in the film for which the optical thickness is a constant. These fringes are quite useful to determine the surface features. The surface under study can be put in contact with an optical flat (not having a deviation of more thanλ/4). The air gap between the two generates a thin film interference pattern. For flat test surfaces, the fringe will be a series of straight, equally spaced bands. This indicates a wedge-shaped air film. When viewed at nearly normal incidence as in Fig. 7.6, the contours from a nonuniform film are known as Fizeau fringes. For a thin wedge of small angle α, the path difference between the two reflected rays can be given by Λ = 2ntdcosθt, anddcan be approximated byd≈xα. For small values ofθi, the condition for interference maximum can be written as
(m+ 1/2)λ= 2ntdm= 2ntαxm. (7.42) This yields
xm=
m+ 1/2 2α
λt, (7.43)
whereλt=λ/ntis the wavelength in the film. Thus maxima occur at distances from the apex atλt/4α,3λt/4α, etc. The separation between the bright fringes is given by
∆x=λt/2α. (7.44)
Note that the difference in film thickness for adjacent maxima is given byλt/2.
Since the beam reflected by the lower surface traverses the thickness twice, adjacent maxima differ in the optical path byλt. In terms of the thickness, the location of maxima is given by
dm= (m+ 1/2)λt
2. (7.45)
Traversing the film twice gives a phase shift of π, which when added with additional π phase shift under reflection puts the two rays back in phase.
Hence the interference maxima result.
7.3.4 Newton’s rings
Two pieces of glass slides, when pressed at a point illuminated by normally incident light, can exhibit concentric fringe patterns. These patterns are known as Newton’s rings. These can be studied systematically by the arrangement shown in Fig. 7.7. On top of a glass optical flat, we place a lens. The system is illuminated with normally incident quasi-monochromatic light. The amount of uniformity of the circular pattern is a measure of how perfect the lens is.
LetRbe the radius of curvature of the convex lens. The relation between the distancexand the film thicknessdis given by
x2=R2−(R−d)2= 2Rd−d2. (7.46) Sinced≪R, Eq. (7.46) can be rewritten as
x2= 2Rd. (7.47)
Considering only the first two reflected beams, the m-th order interference maximum occurs at
2ntdm= (m+ 1/2)λ. (7.48)
The radius of them-th bright ring is then given by
xm= [(m+ 1/2)λtR]1/2. (7.49)
S
L F P
BS
R x d
FIGURE 7.7: Setup for observing Newton’s rings.
Interference and diffraction 135 The radius of them-th dark ring will be
xm= [mλtR]1/2. (7.50)
If there are no dust particles at the center between the lens and the optical flat and the contact is good, then we will have a minimum of intensity at the center (atx= 0) sinced= 0 at that point.
7.3.5 Mirrored interferometers: Michelson interferometer A variety of amplitude-splitting interferometers are based on multiple mir- rors and beam-splitters. Perhaps the best known is the Michelson interferom- eter. The arrangement of the interferometer is shown in Fig. 7.8. An incident beam is split into two by means of a beam-splitterBS. Both the transmitted and the reflected beams are reflected back onto the beam-splitter by mirrors M′ andM′′. A compensator C is placed in the path of the transmitted beam in order to compensate for the additional path traversed by the reflected beam in passing through theBS (since it gets reflected by the bottom surface). In order to understand how the fringes are formed, it is better to refer to the equivalent diagram shown inFig. 7.9. An observer at locationD will simul- taneously see the two mirrors and the source. Let the mirror separation bed.
Thus interference will be observed from light coming from two virtual sources S′ and S′′ separated by 2d. Optical path difference between the two rays coming fromS′ andS′′is given by 2dcosθand the condition for interference