Chapter 25 Wave Optics Chapter 34 Opener. The beautiful colors from the surface of this soap bubble can be nicely explained by the wave theory of light. A soap bubble is a very thin spherical film filled with air. Light reflected from the outer and inner surfaces of this thin film of soapy water interferes constructively to produce the bright colors. Which colors we see at any point depends on the thickness of the soapy water film at that point and also on the viewing angle. Near the top of the bubble, we see a small black area surrounded by a silver or white area. The bubble’s thickness is smallest at that black spot, perhaps only about 30 nm thick, and is fully transparent (we see the black background). We cover fundamental aspects of the wave nature of light, including two-slit interference and interference in thin films.

Wave Optics studies light depend on its wave nature properties that depend on its wave nature Originally, light was thought (by Newton & others) to be particles. That model successfully explained all of geometric optics.

Other experiments revealed properties of light that could only be explained with a wave theory. Maxwell’s theory of electromagnetism convinced physicists that light was a wave.

Wave vs. Geometric Optics The wavelength λ of light plays a key role in determining when geometric optics can or cannot be used. When discussing image characteristics over distances d much greater than the wavelength, geometric optics is extremely accurate. That is, when d >> λ geometric optics works very well.

When dealing with sizes d comparable to or smaller than the wavelength λ, wave optics is required. That is, wave optics is needed when d <  λ Examples include interference effects and propagation through small openings Later, more experiments led to the quantum theory of light (early 20th Century). In that theory, light has properties of both waves and particles

A Brief Side Note: Does light consist of waves or particles? Modern (quantum) theory of light asks the question: Does light consist of waves or particles?

(It depends on the experiment A Brief Side Note: Modern (quantum) theory of light asks the question: Does light consist of waves or particles? Answer: YES! (It depends on the experiment & on the property)

Interference A property unique to waves is interference. You might have noticed this for sound waves. For example, interference of sound waves can be produced by two speakers as in the figure. Interference. can be understood qualitatively by remembering that waves are sine or cosine functions of position & time. See figure.

Constructive Interference When the waves are in phase, their maxima occur at the same time at a given point in space The total amplitude at the listener’s location is the sum of the amplitudes of the two individual waves.

Constructive Interference occurs if 2 waves are in phase, so that the sum of their displacements is large This can produce a large amplitude wave with a large intensity!

Destructive Interference When the waves are out of phase, the maximum of one wave can coincide with the minimum of the other wave. In this case, the interference is destructive. If the waves are 180° out of phase, the sum of the displacements of the 2 waves is zero

Conditions for Interference Suppose that two or more interfering waves travel through different regions of space over at least part of their propagation from source to destination. Suppose that the waves are brought together at a common point. Suppose that the waves have the same frequency and also have a fixed phase relationship This means that over a given distance or time interval the phase difference between the waves remains constant Such waves are called coherent

Coherence The eye cannot follow variations of every cycle of the wave, so it averages the light intensity For waves to interfere constructively, they must stay in phase during the time the eye is averaging the intensity For waves to interfere destructively, they must stay out of phase during the averaging time Both of these possibilities involve the wave having precisely the same frequency

Coherence With slightly different frequencies, the interference changes from constructive to destructive and back as time goes on. Over a large number of cycles, the waves average no interference.

Some Brief History: Waves vs. Particles & Huygens’ Principle This principle states: “Every point on a wave front can be considered as a point source of tiny wavelets that spread out in the propagation direction at the wave speed”. Figure 34-1. Huygens’ principle, used to determine wave front CD when wave front AB is given. The overall wave front is the envelope for all of the wavelets. That is, the wave front is tangent to all of them. More on Huygens a little later!

Diffraction  The bending of a wave around an object. Huygens’ Principle is consistent with diffraction. Diffraction occurs for waves & not particles. The fact that light exhibits diffraction, is an indication that it consists of waves, & not particles. Figure 34-2. Huygens’ principle is consistent with diffraction (a) around the edge of an obstacle, (b) through a large hole, (c) through a small hole whose size is on the order of the wavelength of the wave.

Diffraction  The bending of a wave around an object. In Newton’s time, there were competing theories of light. Some were based on it being made up of particles. Others were based on it consisting of waves. The observation that light exhibits diffraction was considered a proof that it consists of waves & not particles. Figure 34-2. Huygens’ principle is consistent with diffraction (a) around the edge of an obstacle, (b) through a large hole, (c) through a small hole whose size is on the order of the wavelength of the wave.

Diffraction: Some Illustrations around an edge of an obstacle Figure 34-2. Huygens’ principle is consistent with diffraction (a) around the edge of an obstacle, (b) through a large hole, (c) through a small hole whose size is on the order of the wavelength of the wave.

Diffraction: Some Illustrations through a large hole, of size much larger than the wavelength. Diffraction around an edge of an obstacle Figure 34-2. Huygens’ principle is consistent with diffraction (a) around the edge of an obstacle, (b) through a large hole, (c) through a small hole whose size is on the order of the wavelength of the wave.

Diffraction: Some Illustrations through a large hole, of size much larger than the wavelength. Diffraction through a small hole, of similar size to the wavelength. Diffraction around an edge of an obstacle Figure 34-2. Huygens’ principle is consistent with diffraction (a) around the edge of an obstacle, (b) through a large hole, (c) through a small hole whose size is on the order of the wavelength of the wave.

Huygens’ Principle & the Law of Refraction can be used to derive The Law of Refraction (Snell’s Law): As the wavelets propagate from each point, they propagate more slowly in the medium of higher index of refraction. This leads to a bend in the wave front & therefore in the ray. The wave fronts are perpendicular to the rays.

sinθ1 = (v1t/AD) & sinθ2 = (v2t/AD) Snell’s Law can be derived using Huygens’ Principle plus geometry & the definition of the index of refraction n. v  (c/n). Speed of light in vacuum = c. Speed of light in a medium = v. From the figure: sinθ1 = (v1t/AD) & sinθ2 = (v2t/AD) or, (sinθ1/sinθ2) = (v1/v2) = (n2/n1) or, n1sinθ1 = n2sinθ2

n1sinθ1 = n2sinθ2 Figure: sinθ1 = (v1t/AD) & sinθ2 = (v2t/AD) or, (sinθ1/sinθ2) = (v1/v2) = (n2/n1) or, n1sinθ1 = n2sinθ2 This has made use of the fact that, when light travels from one medium to another, it’s frequency of does not change, but it’s wavelength λ does: