Variable Thickness (Wedge shaped) films Film with zero thickness at one end, progressively increasing to a particular thickness at the other end. Wedge angle very small, fraction of a degree.

When a parallel beam of monochromatic light illuminates the wedge from above, the reflected rays interfere with each other. Path difference between the reflected rays from the lower and upper surfaces of the air film varies along its length due to variation in film thickness. Alternate bright and dark fringes are formed on its top surface.

BC and DE reflected from the top and bottom of the air film are coherent, derived from the same ray AB (division of amplitude) The rays are very close if the thickness of the film is of the order of wavelength of light. Interference of rays --- fringes depending on the phase difference for small air film thickness

Thickness of the glass plate is large as compared to the wavelength of incident light so that entire pattern is due to air film only. Optical path difference between BC and DE. is the abrupt phase change of on reflection from the boundary of air to glass interface. (Point F) where

For Maxima Path difference For bright fringe For minima For dark fringe

Assuming normal incidence cos r = 1, Let thickness of film at A = t1 And a dark fringe is formed (say) then Next dark fringe occurs at C where, the thickness CL = t2 , then at C,

Subtracting (1) from (2) AB is the distance between successive dark fringes (bright fringes also)→ fringe width (β)

Then For small values of θ , tan θ ≈ θ Therefore, From the above equation fringe width  is constant for given wedge angle. As θ increases,  decreases. ( fringes move closer). As the value of θ, approaches 900, the interference pattern vanishes.

Fringe at Apex is Dark At Apex, the two glass slides are in contact with each other. Therefore, the thickness t ≈ 0 The optical path difference then becomes It implies that the path difference of or a phase difference of π occurs between the reflected waves at the edge. The two waves interfere destructively. Hence the fringe at the apex is dark

Straight and parallel fringes Each fringe pattern is produced by the interference of rays reflected from the section of the wedge having same thickness. Locus of points having same thickness lie along lines parallel to the contact edge →Straight and parallel edges. Also called the fringes of equal thickness.

Equidistant fringes since λ and θ are constants,  is a constant for a given wedge angle (hence equidistant fringes) and hence parallel. Localized fringes Fringes are formed very close to the top surface of wedge.

Determination of Wedge Angle Using microscopes note the positions of dark fringes at two distant points Q and R , Let OQ = x1, and OR = x2 Thickness of the wedge is t1 at Q, t2 at R For dark fringes at Q For small θ, Therefore, For dark fringe at R, where N = number of dark fringes lying between Q and R. Subtracting gives

Therefore For air So Determination of the thickness of the spacer Spacer forms the wedge shaped air film between the glass slides For thickness ‘t’ of the spacer where l = length of the air wedge

Since Therefore Fizeau Fringes Equal optical thickness of fringes localized at the top of the film Fizeau Fringes.

Anti reflection films (APPLICATION) [Anti reflection coatings (AR)] One of the most important applications of thin film interference is in producing antireflection coatings. Optical instruments such as cameras and telescopes use multi-component glass lenses. It is noted that:

A part of the light incident on a glass surface is reflected and that amount is subtracted from the transmitted light. For a large number of reflections, the quality of image produced by a device will be poor. Alexander Smakula (1935) discovered that reflection from a surface can be reduced by coating the surface with a thin transparent dielectric film.

A transparent thin film coated on a surface to suppress the surface reflections is called an antireflection coating (AR) or non reflecting film. Conditions: Phase condition -wave reflected from the top and bottom surfaces of the thin films are in opposite phase. Overlapping leads to destructive interference.   Amplitude Condition – Reflected waves have equal amplitude.

Antireflection coating

a b t 1 2 air(na) film(nf) glass(ng) ng > nf > na

Thickness- t, Refractive index of film material = Phase of beams 1 and 2 reflected from the top and bottom surfaces of the thin films should be 180º out of phase. Δ between 1 and 2 = , n = 0, 1, 2… First   change at the top surface of film (air to film boundary) Second   change at the lower surface of AR film (film to glass boundary) Assuming normal incidence of light i.e. cos r = 1

Addition or subtraction of a full wave ( ) does not affect the phase. Rays 1 and 2 interface destructively if Therefore For the film to be transparent, the thickness of the film should be minimum which is possible for n = 0. Therefore 2nf tmin = λ/2 or tmin = λ/4nf

Optical thickness of Antireflection coating should be λ/4 Optical thickness of Antireflection coating should be λ/4. Such quarter wavelength components suppress the reflections and allow the light to pass into the transmitted component. Amplitude condition:

Amplitude of ray 1 = ar Amplitude of ray 2 = ar’tt’ For complete destructive interference ray 1 and 2 must have the same amplitude, i.e.

Now na = 1 for air Therefore, It implies that the refractive index of the thin film should be less than substrate. i.e glass plate. In the case of glass., if ng is 1.5 , then nf will be 1.22 . The materials having this refractive index are magnesium fluoride etc. .

Anti-Glare deals with external sources of reflection off a surface – like bright sunlight or high ambient lighting conditions – and its impact on the readability of the image To deal with the external sources of reflection, Anti-Glare uses diffusion mechanisms to breakup the reflected light off the surface. Diffusion works by reducing the coherence of the reflected image, making these unwanted images unfocused to the eye, thereby reducing their interference with viewing of the intended image contained in the display.