Chapter 24 Wave optics

24.1 Conditions for Interference

• When two or more waves are present simultaneously in the same region of space, the resultant wave is the sum of the individual waves (Superposition Principle).

• Two sources are coherent if they emit waves that have a constant phase relationship.

• Conditions for Interference

1. The sources must be coherent.

2. The sources must have identical wavelengths.

3. The superposition principle must apply.

• When two identical waves (same wavelength and same amplitude) arrive at the point P in phase, the waves reinforce each other and constructive interference occurs.

• When two identical waves arrive at the point P out of phase with each other, the waves cancel each other and destructive interference occurs.

• Examples

24.2 Young’s Double-Slit Experiment

• An experiment demonstrated the wave nature of light.

• Schematic diagram (Figure 24.1)

• Light passes through a pair of closely spaced narrow slits and produces a pattern of alternating bright and dark fringes on a viewing screen. The fringes arise because of constructive and destructive interfaces.

• The path difference, = ₁ - ₂ = d sin , creates the phase difference between the two waves. (Figure 24.3)

• Constructive interference occurs when the path difference is an integer number m of wavelengths, m (m = 0, 1, 2, …).

(Bright fringes) = d sin = m

• Destructive interference occurs when the path difference is an half integer number of wavelengths, m (m = 0, 1, 2, …).

(Dark fringes) = d sin = (m + )

• Examples

24.3 Change of Phase Due to Reflection

• Figure 24.6

• A ray reflection from a medium with higher index of refraction undergoes a 180° phase change equivalent of one-half of a wavelength ( / 2).

• A ray reflection from a medium with lower index of refraction undergoes a no phase change.

24.4 Interference in Thin-Film

• Constructive and destructive interference of light wave can occur with thin films of transparent materials, where the light waves that reflect from the top and bottom surfaces of the film interfere.

• Schematic diagram (Figure 24.7)

• Important facts

1. Change of phase due to reflection:

When n₂ > n₁, the reflected ray has 180° phase change.

When n₂ < n₁, the reflected ray has no phase change.

2. The wavelength within the film is

_n = /n

where is the wavelength of light in vacuum.

• The difference in path between two waves, = ₁ - ₂ = 2 t (t is the film thickness), occurs inside the thin film. Hence, the important wavelength in creating the interference pattern is the wavelength within the film.

_n = /n

where n is the index of refraction for the film.

• When light travels through a material with a smaller index of refraction toward a material with larger index of refraction, the reflected at the boundary has a phase change of one-half of a wavelength (_film / 2). On the other hand, light reflected from a smaller index of refraction surface has no phase change.

• When the index of the film is the largest or smallest compared to the surrounding two media (n₁ < n_f > n₂ or n₁ > n_f < n₂),

1. The constructive interference condition is

2 t = (m + )_n ; 2 n t = (m + )

2. The destructive interference condition is

2 n t = m

� Problem Solving Strategy �

1. Identify the thin film causing the interference.

2. The type of interference that occurs determined by the phase relationship between the portion of the wave reflected at the upper surface of the film and the portion reflected at the lower surface.

3. Phase differences between two portions of the wave have two causes: (a) differences in the distances traveled by the two portions and (b) phase changes occurring on the reflection.

4. When distance and phase changes on reflection are both taken into account, the interference is constructive if the path difference is an integral multiple of .

<Newton’s Rings>

• Examples

24.5 Diffraction

• When light passes through small openings, around obstacles, or by sharp edges, it deviates from a straight-line path and enters the region that would otherwise be shadowed. This divergence of light is called diffraction.

• Figure 24.11, 12, and 13

• Central maximum; secondary maxima; minima

• Fraunhofer diffraction is one type of diffraction which occurs when the observing screen is far from the slit.

24.6 Single-Slit Diffraction

• Huygen’s principle: Every point on a wave front acts as a source of tiny wavelets that move forward with the same speed as the wave; the wave front at a later instant is the surface that is tangent to the wavelets. (Figure 24.15)

• Schematic diagram (Figure 24.14 and 16)

• When the path difference between ray 1 and ray 3, = sin , is half of wavelength, the destructive interference occurs (Dark fringes).

(The first Dark fringe) sin = ; sin =

(The second Dark fringe) sin = 2

(Destructive interference) sin = m with m = 1, 2, 3,…

• Examples

24.6 Polarization of Light Waves

<Polarized Electromagnetic Waves>

• A linearly polarized electromagnetic wave is one in which all oscillations of the electric field occur along one direction, which is taken to be the direction of polarization. (Figure 24-18)

• In unpolarized light the direction of polarization does not remain fixed, but fluctuate randomly in time. (most of the natural light)

• Polarizing materials allow only the component of the wave’s electric field along one direction to pass through them. The preferred transmission direction for the electric field is called the transmission axis of the material. (Figure 24-19)

<Marius’ Law >

• Figure 24-19

• The average intensity of the light leaving the analyzer is

= _o cos²

• Examples

< Polarization by Reflection >

• When light is incident on a nonmetallic surface at the Brewster angle (_B), the reflected light is completely polarized parallel to the surface.

= tan _p = (24.13)

• Examples

< Polarization by Scattering >

< Optical Activity >

< Application: Liquid Crystals >