|MadSci Network: Physics|
Greetings: My discussion of reflection is based on the the late Nobel Laureate, Professor Richard P. Feynman's lectures at the California Institute of Technology and I will try and paraphrase Professor Feynman's conclusions here without including the difficult mathematics. I have included some quotations from the lectures and you can also find the mathematical derivations in the following reference: Richard P. Feynman, Robert B. Leighton, Matthew Sands, The Feynman Lectures on Physics, Addison-Wesley 1964. Volume 2, Chapter 32, Refractive Index of Dense Materials Volume 2, Chapter 33, Reflection from Surfaces When an electromagnetic field ( e.g.. Radio, microwave, infrared or light) from a distant source (S) is incident on a thin sheet of glass we often use Snell's Law to describe the reflection and refraction that takes place as the waves pass through the glass. In these calculations we use the index of refraction (n) of the material at the frequency of the wave source. The refractive index is a measure of the velocity of light in the material (v) relative to the velocity of light in a vacuum (c) such that v = c/n. However, for completeness, n is in reality a complex number consisting of a real and an imaginary component. I'll label the real component of n, N(r) and I'll label the imaginary component of n, N(i). N(r) is the "apparent" velocity of the wave in the glass and N(i) is a measure of attenuation (absorption) in the glass and describes an exponential decay of the amplitude of the light wave as it passes through the glass. In many optically transparent materials such as glass N(i) is very small and we neglect it. Also, we tend to think of materials as opaque or transparent because they behave that way in the region of the spectrum we are using, be it microwaves or light; however, opaque materials can change into transparent materials in some regions of the spectrum and vice versa (e.g. color filters). For example we consider copper and other metals to be opaque at radio, microwave , infrared and optical wavelengths yet copper does become transparent to light in very thin layers (solar window coatings) and at X-ray wavelengths. Some metals such as Li, Na, K and Rb become transparent in the ultraviolet wavelengths. The point of this discussion is that general equations of matter must include the behavior of materials throughout the electromagnetic spectrum from radio waves through X-rays which in turn makes these equations very complex and difficult to comprehend in a physical sense. However, I feel Professor Feynman had a super ability to provide both a mathematical and a physical view of a problem. Professor Feynman writes: "We have said that light goes slower in water than in air, and slower, slightly , in air than in vacuum. This effect is described as the index of refraction (n). How does this come about? Let's use the following physical statements. (a) That the total electric field in any physical circumstance can always be represented by the sum of the fields from all of the charges in the universe. (b) That the field from a single charge is given by its acceleration evaluated with a retardation at the speed c, always (for the radiation field)." REFLECTION FROM GLASS If an external light source is incident on a thin sheet of glass we would say that the light in the glass is retarded at the speed c/N(r). Professor Feynman then writes: "That, however is not right, and we have to understand why it is not. It is approximately true that light or any electrical wave does appear to travel at a speed c/n through a material whose index of refraction is n, but the fields are still produced by the motions of all the charges - including the charges moving in the material - and with these basic contributions of the field traveling at the ultimate velocity c. Our problem is to understand how apparently slower velocity comes about. Why should there be charges moving in the glass? We know that all material consists of atoms which contain electrons. When the electric field of the source acts on these atoms it drives the electrons up and down, because it exerts a force on the electrons. And moving electrons generate a field - they constitute new radiators. These new radiators are related to the source S, because they are driven by the field of the source. This means that the field is not the same as the one that was there before the glass was there, but is modified in such a way that the field inside the glass appears to be moving at a different speed. These charges also radiate waves back toward the source. This backward going field is the one we see reflected from the surfaces of transparent materials. It does not come from just the surface. The backward radiation comes from every where in the interior , but it turns out that the total effect is equivalent to a reflection from the surface." REFLECTION FROM METAL "In metals some of the electrons have no binding force holding them to a particular atom; it is these free electrons which are responsible for the conductivity. There are other electrons which are bound - however; there influence is usually swamped by the effects of the conduction electrons." The motions of these free electrons travel with a velocity derived from the average field from the source but they follow a jagged path from collision to collision. One result of the free electrons is that : " The real and imaginary parts of n have the same magnitude". This is true for Feynman's low - frequency approximation , where the metal is opaque to light, which is derived in his lecture. "With such a large imaginary part to n, the wave is rapidly attenuated in the metal." The distance in which the amplitude of the wave decreases by a factor of 1/3 is called the skin depth and a calculation shows that for copper the penetration of the wave is a small fraction of a wavelength. While a wavelength of light is about 500 nanometers in length the atomic spacing in a metal is about one half of a nanometer. Thus a wavelength is 1000 atoms long and a plane area of one square wavelength has 1 million atoms. Thus even with less than a wavelength penetration, there are countless numbers of atoms and electrons to excite and reradiate energy near the metals surface! The rapid exponential decay near the metal's surface causes both a rapid decrease in the amplitude of the wave and a large phase shift in the excitation and reradiation from the free electrons. This 180 degree phase shift in a very small distance is required because a metallic surface should short out the voltage of a field hitting the surface. For this case Maxwell's wave equations require that the reflected field should be equal in magnitude to the incident wave only 180 degrees out of phase with the incident wave, so that the vector sum of the voltages of the incomming wave and the outgoing waveat the surface is zero (an effective short circuit). It is often difficult to realize that a fixed zero field plane can be composed of two large amplitude waves, traveling in opposite directions; yet this happens in many areas in physics including musical instruments, radio antennas as well as in optics. I have shown this interference effect graphically in my response to the following question: Physics : Re: When waves collide, do both kinds of interference occur? Adrian Popa, Staff Optical/Microwave Physics, Tue Jan 21 11:42:59 1997 Professor Feynman then concludes: "Metals do not reflect 100%, but many do reflect visible light very well. In other words the imaginary part of their indexes is very large. But we have seen that a large imaginary part of the index means a strong absorption. So there is a general rule that if any material gets to be a very good absorber at any frequency, the waves are strongly reflected at the surface and very little gets inside to be absorbed ." " We want to emphasize that the amplitude of a surface reflection is not a property of the material as in the index of refraction. It is a surface property, one that depends precisely on how the surface is made." Thus, polishing, coatings, films and surface corrosion are all part of the surface and must be considered in the calculations. Regards your Mad Scientist - Adrian E. Popa
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