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Chapter 34 The Wave Nature of Light;

Interference

34 - 5 Interference in Thin Films

34 - 6 Michelson Interferometer

34 - 7 Luminosity Intensity

Units of Chapter 34

Double-­‐Slit  Experiment  destructive interference

constructive interference

Two sources of light are coherent if they have the same frequency and maintain the same phase relationship.

Intensity as a function of angle

34-4 Intensity in the Double-Slit Interference Pattern

34-4 Intensity in the Double-Slit Interference Pattern

34-3 Coherence

Two sources of waves are perfectly coherent

if they emit waves with the same frequency and maintain the same phase relationship at all times.

If the waves from the two sources have random phase with respect to each other over time the sources are incoherent.

If there is a very thin film of material – a few wavelengths thick – light will reflect from both the bottom and the top of the layer, causing interference. This can be seen in soap bubbles and oil slicks.

34-5 Interference in Thin Films

If the path difference ABC equals one or a whole number of wavelengths in the film (λ/n), the two waves will reach the eye in phase and interfere constructively.

34-5 Interference in Thin Films

A similar effect takes place when a shallowly curved piece of glass is placed on a flat one. When viewed from above, concentric circles appear that are called Newton’s rings.

34-5 Interference in Thin Films

A beam of light reflected by a material with index of refraction greater than that of the material in which it is t ravel ing, changes phase by 180° or ½ cycle.

34-5 Interference in Thin Films

Example 34-6: Thin film of air, wedge-shaped.

A very fine wire 7.35 x 10-3 mm in diameter is placed between two flat glass plates. Light whose wavelength in air is 600 nm falls (and is viewed) perpendicular to the plates and a series of bright and dark bands is seen. How many light and dark bands will there be in this case? Will the area next to the wire be bright or dark?

34-5 Interference in Thin Films

34-5 Interference in Thin Films

Solution: The path lengths are different for the rays reflected from the upper and lower surfaces; in addition, the ray reflected from the lower surface undergoes a 180° phase change. Dark bands will occur when 2t = mλ and bright bands when 2t = (m + ½)λ. At the position of the wire, 2t is 24.5 wavelengths. This is a half-integer; the area next to the wire will be bright, and there will be 25 dark bands between it and the other edge. Including the band next to the wire, there will also be 25 light bands.

k1r1 � k2r2 = 2t2⇡

�; '1 � '2 = ⇡; 2t

2⇡

�+ ⇡ = n⇡ n = 1, 3, 5...

k1r1 � k2r2 = 2t2⇡

�; '1 � '2 = ⇡; 2t

2⇡

�+ ⇡ = n⇡ n = 1, 3, 5...

Example 34-7: Thickness of soap bubble skin.

A soap bubble appears green (λ = 540 nm) at the point on its front surface nearest the viewer. What is the smallest thickness the soap bubble film could have?

Assume n = 1.35.

34-5 Interference in Thin Films

34-5 Interference in Thin Films

The incident and reflected rays are assumed to be perpendicular to the bubble’s surface Solution: The path length difference is twice the thickness of the film; the ray reflecting from the first surface undergoes a 180° phase change, while the other ray does not. The smallest thickness where the green light will be bright is when 2t = λ/2n, or t = 100 nm. Constructive interference then occurs for every additional 200 nm in thickness.

k1r1 � k2r2 = 2t 2⇡�/n ; '1 � '2 = ⇡;

2t 2⇡�/n + ⇡ = m2⇡ m = 1, 2, 3...

Problem Solving: Interference

1.  Interference occurs when two or more waves arrive simultaneously at the same point in space.

2.  Constructive interference occurs when the waves are in phase.

3.  Destructive interference occurs when the waves are out of phase.

4.  An extra half-wavelength shift occurs when light reflects from a medium with higher refractive index.

34-5 Interference in Thin Films

Example 34-8: Nonreflective coating.

What is the thickness of an optical coating of MgF2 (mangnessioum difluride) whose index of refraction is n = 1.38 and which is designed to eliminate reflected light at wavelengths (in air) around 550 nm when incident normally on glass for which n = 1.50?

34-5 Interference in Thin Films

The Michelson interferometer is centered around a beam splitter, which transmits about half the light hitting it and reflects the rest. It can be a very sensitive measure of length.

34-6 Michelson Interferometer

If the movable mirror is moved a distance λ/4 and no π phase shift occurs at the beam splitter, one beam will travel an extra distance equal to λ/2, and the two beams will destructively interfere.

34-6 Michelson Interferometer

34-7 Luminous Intensity

The intensity of light as perceived depends not only on the actual irradiance but also on the sensitivity of the eye at different wavelengths.

Luminous flux:

1 lumen = 1/683 W of 555-nm light

Luminous intensity:

1 candela = 1 lumen/steradian

Illuminance: luminous flux per unit area