Interference in Wedge Shaped Film (Reflected Rays)

Thin Film Interference

A film of thickness from 0.5 to 10 $\mu$ m is a transparent medium of glass, mica, air enclosed between glass, soap film, etc. When the light is made incident on this thin film partial reflection and partial refraction occur from the top surface of the film. The refracted beam travels in the medium and again suffers partial reflection and partial refraction at the bottom surface of the film. In this way several reflected and refracted rays are produces by a single incident ray. As they moves are superimposed on each other and produces interference pattern.

Interference in Parallel Film ( Reflected Rays)

Consider a thin film of uniform thickness ‘t’ and refractive index $\mu$ bounded between air. Let us consider monochromatic ray AB is made incident on the film, at B part of ray is reflected (R₁) and a part is refracted along BC.At C The beam BC again suffer partial reflection and partial refraction, the reflected beam CD moves again suffer partial reflection and partial refraction at D. The refracted beam R₂ moves in air. These two reflected rays R₁and R₂ interfere to produce interference pattern.

The optical path difference between the two reflected rays
$\Delta =\mu (BC+CD)-BN$

In ΔBDN, sin i = BN / BD and BC = CD as ΔBMC ≡ ΔMCD, therefore
$\Delta =2\mu BC-BDsin i$

In ΔBMC, cos r = t /BC, therefore
$\Delta =\frac{2\mu t}{cosr}-BDsin i$
$\Delta =\frac{2\mu t}{cosr}-2BMsin i$

In ΔBMC, tan r = BM / t , therefore
$\Delta =\frac{2\mu t}{cosr}-2t(tan r) sin i$
According to snell’s law $\mu =\frac{sin\mathbf{i}}{sin\mathbf{r}}$

$\Delta =\frac{2\mu t}{cosr}-2\mu t(tan \mathbf{r}) sin \mathbf{r}$
$\Delta =\frac{2\mu t}{cos\mathbf{r}}-\frac{2\mu tsin^{2} \mathbf{r}}{cos\mathbf{r}}$
$\Delta =\frac{2\mu t}{cos\mathbf{r}}(1-sin^{2}\boldsymbol{r})$
$\Delta =\frac{2\mu t}{cos\mathbf{r}}cos^{2}\boldsymbol{r}$
$\Delta =2\mu t cos\mathbf{r}$ ..........................(2.11)
Correction on account of phase change at reflection: when a beam is reflected from a denser medium (ray R₁ at B), a path change of $\lambda$ /2 occur for the ray.
Therefore the true path difference is $\Delta =2\mu t cos\mathbf{r}+\frac{\lambda }{2}$ ..........(2.12)

Condition of Maxima (Bright Fringe)
Maxima occur when path difference
$\Delta =n\lambda$
$2\mu t cos\mathbf{r}+\frac{\lambda }{2}=n\lambda$
$\boldsymbol{2\mu t cosr=\pm \frac{(2n-1)\lambda}{2}}$ ...............(2.13)

Condition for Minima (Dark Fringe)
Minima occur when path difference
$\Delta =\frac{(2n+1)\lambda}{2}$

$2\mu t cos\mathbf{r}+\frac{\lambda }{2}=\frac{(2n+1)\lambda }{2}$

$\boldsymbol{2\mu t cosr=\pm n\lambda}$ .....................(2.14)

Interference in Parallel Film (Transmitted Rays)

The optical path difference between transmitted rays T₁ and T₂ will be

This path difference is calculated in the same way as above to get
$\Delta =2\mu t cos\mathbf{r}$ ....................(2.15)

Condition of Maxima (Bright Fringe)
Maxima occur when path difference,
$\Delta =n\lambda$
$2\mu t cos\mathbf{r}=n\lambda$ ......................(2.16)

Condition for Minima (Dark Fringe)
Minima occur when path difference,
$\Delta =\frac{(2n+1)\lambda}{2}$
$2\mu t cos\mathbf{r}=\frac{(2n-1)\lambda}{2}$ ...........(2.17)

Interference in Wedge Shaped Film (Reflected Rays)

The wedge shaped film has a thin film of varying thickness, having thickness zero at one end and increases at the other. The angle of wedge is $\theta$ .

The optical path difference between the two reflected rays R₁ and R₂ will be

$\Delta =\mu (BC+CD)-BM$

From the geometry

$\Delta =\mu (BN+NC+CD)-BM$
As in ΔBMD;
$sin\mathbf{i}=\frac{BM}{BD}$

And in ΔBND
$sin\mathbf{r}=\frac{BN}{BD}$
According Snell’s Law,

$\mu =\frac{sin\mathbf{i}}{sin\mathbf{r}}=\frac{BM/BD}{BN/BD}=\frac{BM}{BN}$

Or $BM=\mu BN$

Thus $\Delta =\mu (NC+CD)$
$\Delta =\mu (NC+CL)$
$\Delta =\mu NL$
As in ΔNDL $cos(r+\theta)=\frac{NC}{2t}$
$\Delta =2\mu t cos(r+\theta)$

Correction on account of phase change at reflection: when a beam is reflected from a denser medium (ray R₁ at B), a path change of $\lambda$ /2 occur for the ray.

Therefore the true path difference is
$\Delta =2\mu t cos(r+\theta)+\frac{\lambda }{2}$

Condition of Maxima (Bright Fringe)
Maxima occur when path difference,
$\Delta =\pm n\lambda$
$2\mu t cos(r+\theta)+\frac{\lambda }{2}=\pm n\lambda$
$\boldsymbol{2\mu t cos(r+\theta)=\pm \frac{(2n-1)\lambda}{2}}$

Condition for Minima (Dark Fringe)
Minima occur when path difference
$\Delta =\pm \frac{(2n+1)\lambda}{2}$
$2\mu t cos(r+\theta)+\frac{\lambda }{2}=\pm \frac{(2n+1)\lambda}{2}$
$\boldsymbol{2\mu t cos(r+\theta)=\pm n\lambda}$

Lloyd's’ mirror experiment

Lloyd's mirror This is another method for finding the wavelength of light by the division of wavefront. Light from a slit So falls on a silvered surface at a very small grazing angle of incidence as shown in the diagram (Figure 1). A virtual image of So is formed at S1. Interference occurs between the direct beam from So to the observer (0) and the reflected beam The zeroth fringe will be black because of the phase change due to reflection at the surface. Application An interesting application of this effect may be observed when a helicopter flies above the sea near a radio transmitter. The helicopter will receive two signals: (a) one signal directly from the transmitter and (b) a second signal after reflection from the sea As the helicopter rises the phase difference between the two signals will alter and the helicopter will pass through regions of maxima and minima. Lloyd's mirror Experiment Lloyd’s Mirror is used to produce two-source interference

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Interference in Wedge Shaped Film (Reflected Rays)

Thin Film Interference

Interference in Wedge Shaped Film (Reflected Rays)

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