1. Phys102 Lecture 25 The Wave Nature of Light; Interference
Key Points
Huygens’ Principle
Interference – Young’s Double-Slit Experiment
Intensity in the Double-Slit Interference Pattern
References
24-1,2,3,4,8.

2. The Wave Nature of Light
Figure Huygens’ principle, used to determine wave front CD when wave front AB is given.

3. Wave Interference
These graphs show the interference of two waves. (a) Constructive interference; (b) Destructive interference; (c) Partially destructive interference.
Figure Graphs showing two identical waves, and their sum, as a function of time at three locations. In (a) the two waves interfere constructively, in (b) destructively, and in (c) partially destructively.
(c) Partially destructive interference
(a) Constructive interference
(b) Destructive interference

4. Interference – Young’s Double-Slit Experiment
If light is a wave, there should be an interference pattern.
Figure If light is a wave, light passing through one of two slits should interfere with light passing through the other slit.

5. Interference – The Double Slit Experiment
Screen

6. Huygens’ Principle
Huygens’ principle: every point on a wave front acts as a point source; the wave front as it develops is tangent to all the wavelets.
Figure Huygens’ principle, used to determine wave front CD when wave front AB is given.

7. Young’s Double-Slit Experiment
The interference occurs because each point on the screen is not the same distance from both slits. Depending on the path length difference, the wave can interfere constructively (bright spot) or destructively (dark spot).
Figure How the wave theory explains the pattern of lines seen in the double-slit experiment. (a) At the center of the screen the waves from each slit travel the same distance and are in phase. (b) At this angle θ, the lower wave travels an extra distance of one whole wavelength, and the waves are in phase; note from the shaded triangle that the path difference equals d sin θ. (c) For this angle θ, the lower wave travels an extra distance equal to one-half wavelength, so the two waves arrive at the screen fully out of phase. (d) A more detailed diagram showing the geometry for parts (b) and (c).
Constructive interference:
Destructive interference:

8. Example: Line spacing for double-slit interference.
A screen containing two slits mm apart is 1.20 m from the viewing screen. Light of wavelength λ = 500 nm falls on the slits from a distant source. Approximately how far apart will adjacent bright interference fringes be on the screen?
Figure Examples 34–2 and 34–3. For small angles θ (give θ in radians), the interference fringes occur at distance x = θl above the center fringe (m = 0); θ1 and x1 are for the first-order fringe (m = 1), θ2 and x2 are for m = 2.
Solution: Using the geometry in the figure, x ≈ lθ for small θ, so the spacing is 6.0 mm.

9. Conceptual Example: Changing the wavelength.
i-clicker: What happens to the interference pattern in the previous example if the incident light (500 nm) is replaced by light of wavelength 700 nm?
A) The fringe spacing increases.
B) The fringe spacing decreases.
C) The fringe spacing remains the same.
b) i-clicker: What happens instead if the wavelength stays at 500 nm but the slits are moved farther apart?
Solution: a. As the wavelength increases, the fringes move farther apart.
b. Increasing the slit spacing causes the fringes to move closer together.

10. Interference in Thin Films
Another way path lengths can differ, and waves interfere, is if they travel through different media. 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.
Figure Thin film interference patterns seen in (a) a soap bubble, (b) a thin film of soapy water, and (c) a thin layer of oil on wet pavement.

11. Interference in Thin Films
Wavelength in medium = λ/n
Figure Light reflected from the upper and lower surfaces of a thin film of oil lying on water. This analysis assumes the light strikes the surface nearly perpendicularly, but is shown here at an angle so we can display each ray.

12. 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.
Figure Newton’s rings. (a) Light rays reflected from upper and lower surfaces of the thin air gap can interfere. (b) Photograph of interference patterns using white light.

13. When n2 > n1, the phase changes by 180° (λ/2) upon reflection.

14. Example: 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.
Compare the two reflected beams:
There is a 180° phase shift at the first surface. Then, constructive interference occurs when path difference is λ/2 rather than λ.
Figure 34-21: Example 34–7.The incident and reflected rays are assumed to be perpendicular to the bubble’s surface. They are shown at a slight angle so we can distinguish them.
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.