# Interference in Thin Films

The film of transparent material like a drop of oil spread on the surface of water, show brilliant colours when exposed to an extended source of light. This phenomenon can be explained on interference basis. Here interference takes place between rays reflected from the upper and from the lower surface of the film.

Case I: Thin film of uniform· thickness:

(a) Reflection Pattern: Let us consider a thin film of thickness t, refractive index µ and a ray AB of monochromatic light of λ is falling on it at an angle i. This ray is partly reflected and partly refracted as shown in Figure 4.4.

Path difference           = (BD + DE)film– (BH)air

= [µ (BD +DE)- BH]air

= [µ(BD + DG + GE) – BH]

= µ (BD + DG) As                                                        µGE = BH

BH = EB sin i

GE = EB sin r

µ GE = EB = sin i

µ = sin i / sin r

Extend BL and ED to meet at K we have

LKD = LNDE = r

Triangles BLD and LKD are congruent

BD=DK

Then path difference   = µ(BD + DG) = µ(KD + DG)

= µ KG = µ KB cos r

∆ = 2 µt cos r

Now, here since reflection from denser medium is taking place by one ray so additional path difference of λ/2 is taken into account. Thus the total path difference is

T = 2 µt cos r – λ/2                                                     … (5)

Condition for maxima

T = (2 µ cos r- λ /2) = n λ

2 µt cos r = (2n + 1) λ /2         n = 0, 1, 2, 3, ….. .

Condition for minima

T = (2 µt cos r- λ /2} = (2n + 1) λ /2

2 µt cos r = (n + 1) λ

n is the integer so (n + 1) is also integer and can be taken as n

2 µt cos r = nλ             n = 0, 1, 2

when cos r is kept constant and thickness increases gradually, the path difference becomes λ /2, λ, 3 λ /2, 2 λ, 5 λ /2, . . . and as a result the film will appear dark (t = 0), bright, dark and so on. On the other hand if t is constant and r is varied we again get a series of maxima and minima.

(b) Transmitted Pattern: Here the path difference two rays OEMS and PR will be

∆= µ (DE + EM) – DH

= 2 µt cos r … (9)

In this case there will be no additional path difference so the total path difference

T = 2 µt cos r

Condition for maxima

∆ = 2 µt cos r = nλ                              n = 0, 1, 2, 3, ….. .

Condition for minima

∆= 2 µt cos r = (2n + 1) λ/2                            n = 0, 1, 2, 3, …… . .. (12)

We find that the conditions for maxima and minima are found in case of transmitted pattern are opposite to those found in case of reflected pattern. Under the same conditions of the film looks dark in reflected pattern it will look bright in transmitted pattern.

Colours of thin films: When white light is incident on a thin film only few wavelengths will satisfy the condition of maxima and therefore corresponding colours will seen in the pattern. For other wavelengths condition of minima is satisfied, and so corresponding colours will be missing in the pattern. The coloration of film vary with t and r. Therefore if one vary either t or r a different set of colours will be observed.

Since the condition for maxima and minima are opposite in case of reflected and transmitted pattern. So the colours found in two patterns will be complimentary to each other.

Necessity of broad source: When point source is used only a small portion of the film can be seen through eye and as a result the whole interference pattern cannot be seen. But when a broad source is used rays of light are incident at different angles and reflected parallel beam reach the eye and whole beam and complete pattern is visible.

Case II: The film is of varying thickness (Wedge shaped thin film): Consider the wedge shaped film as shown in Figure 4.5. Let a ray from S is falling on the film and after deflections produce interference pattern. ∆= [(PF + FE)film + (PK)air]

= µ (PF + FE) – (PK)

= µ (PN + NF + FE) – PK

= µ (PN + NF + FE – PN)

= µ (NF +FE)

= µ (NF + FL)

= µ (NL) = 2 µ t cos (r +θ)

Then total path difference considering refraction from denser medium is taking place

∆T = 2 µt cos (r+θ) – λ/2

Condition for maxima

2µcos (r + θ) = (2n + 1) λ/2                                         n = 0, 1, 2, ….. . ..15)

Condition for minima

2µcos (r + θ) = nλ                                                       n = 0, 1, 2, 3, …… . .. (16)

Thus the film will appear bright if the thickness t satisfies the condition of maxima

t = (2n + 1) λ / 4µcos (r + θ)                                                   .. . (17)

and it will appear dark when

t = n λ / 2µ tanθ cos (r + θ)                                                     … (18)

Hence, we move along the direction of increasing thickness we observe dark, bright, dark λ. .. … fringes. For t= 0 i.e., at the edge of film ∆ = λ/2 so the film will appear dark. Then width of the fringes so observed can be found

β = n λ / 2µ tanθ cos (r + θ)

In case of normal incidence r = 0

2 µt= (2n + 1) λ/2 (maxima)

2 µt = n λ (minima)

β = λ/ 2µθ = fringe width 