Thin film interference (exam tomorrow )

In summary, the conversation discusses the phenomenon of Newton's rings, which occur when monochromatic light is reflected from the air gap between a convex spherical piece of glass and a plane slab of glass. The dark fringes of the rings can be approximated by the equation r= \sqrt{mR\lambda}, where m is the order of the fringe, R is the radius of curvature of the spherical piece of glass, and lambda is the wavelength of the light. The conversation also explores different methods for deriving this formula, with one method involving equating the arc length of the circular cross section to the radius of the circles. However, another method suggests solving for r to first order in t (the thickness of the air gap) and
  • #1
jdstokes
523
1
Newton's rings are observed in monochromatic light reflected from the air gap between a convex spherical piece of glass (radius of curvature R) and a plane slap of glass. The two pieces are in contact in the centre.

(b)

Show that the dark fringes have radii given approximately by [itex]r= \sqrt{mR\lambda}[/itex] , lambda is the wavelength of the light and m = 0,1,2...

My solution:

Locally it can be regarded as an air gap of thickness t, separated by two parallel sheets of glass. The light reflected from the top layer of the air experiences no phase shift whereas the light reflected from the bottom suffers a pi rad phase shift since it encounters a more dense medium. Hence the condition from destructive interference is

[itex]2t = m \lambda[/itex] where m = 0,1,2,...

so

[itex]t = \frac{m \lambda}{2}[/itex].

Want to show that

[itex]r = \sqrt{2tR}[/itex]

WTS

[itex]t = \frac{r^2}{2R}[/itex].

The arc length along the spherical piece of glass is [itex]s = R\theta[/itex] which is approximately the distance along the flat piece of glass (r). So [itex]r \approx R\theta[/itex].

Then from the right-handed triangle I get [itex]\tan\theta = t/r[/itex] which for small theta is approximately theta. Hence,

[itex]r \approx R\theta \approx R\frac{t}{r}[/itex] so

[itex]t \approx \frac{r^2}{R}[/itex]

which is off by a factor of 1/2. Any ideas on this one?

Thanks in advance

James
 
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  • #2
I don't quite follow your equating the arc length with the radius of the circles. Instead, write the equation of the circular cross section:
[tex](y - R)^2 + x^2 = R^2[/tex]
(where the center of the circle/sphere is at (0,R))

Of course, y = t & x = r, so:
[tex](t - R)^2 + r^2 = R^2[/tex]
Solve for r to first order in t (ignore higher order terms) and you'll get your answer.
 
  • #3
Thanks Doc Al. That's a much better method.
 

1. What is thin film interference?

Thin film interference is a phenomenon that occurs when light passes through a thin transparent layer, such as a soap bubble or a thin film of oil on water. The light waves reflect off the top and bottom surfaces of the layer, causing interference and resulting in a colorful pattern.

2. How does thin film interference work?

Thin film interference occurs when light waves reflect off the top and bottom surfaces of a thin transparent layer. If the light waves are in phase when they reflect off the two surfaces, they will interfere constructively, resulting in bright colors. If they are out of phase, they will interfere destructively, resulting in dark colors or no color at all.

3. What factors affect thin film interference?

The thickness and refractive index of the thin film, as well as the angle of incidence of the light, can affect thin film interference. Thicker films and higher refractive index materials will produce more intense colors, while larger angles of incidence will result in a shift in the interference pattern.

4. What are some real-life examples of thin film interference?

Some examples of thin film interference in everyday life include the colorful patterns on soap bubbles, the colors on an oil slick, and the rainbow patterns on CDs and DVDs. It is also used in technology such as anti-reflective coatings on glasses and computer screens.

5. How is thin film interference related to the colors we see?

The colors we see in thin film interference are a result of the different wavelengths of light being reflected and interfering with each other. This creates a specific pattern of colors that we perceive with our eyes. The thickness and refractive index of the thin film determine which colors are reflected and which are canceled out, resulting in the colors we see.

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