Lec21 - 21 Lecture 21 21.1 Diffraction and optical instruments In the last lecture we saw that parallel rays after going through a slit are not

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Unformatted text preview: 21. Lecture 21 21.1 Diffraction and optical instruments In the last lecture we saw that parallel rays after going through a slit are not parallel anymore but have a range in angles from zero to θm = λ . Either experiment or a more a involved calculation show that, if the hole is round instead of a slit, then the minima occurs at an angle λ θm ￿ 1.22 , circular hole (21.1) a Since any optical instrument, for example a telescope has a hole through which light comes in, even perfect parallel rays produce an image blurred as if they were not actually parallel. For example in the simple case of a convergent lens shown in fig.118, horizontal parallel rays produce an image of size r given by λ r = 1.22 f a (21.2) where f is the focal distance and a is the diameter of the lens. Namely, although rays arrive parallel to the lens axis, after going through the lens diffraction spreads them over a cone. In fact the image is a solid circle of radius r surrounded by diffraction rings. These rings would make clear that the blurring is from diffraction and not from the other aberrations we have already encountered. Since the wave-length λ is small this blurring does not seem to be very important but we should remember that a telescope further enlarges the image produced by the initial lens or mirror. Diffraction puts a limit to how much this image can be enlarged because once we reach a resolution where we can see the diffraction pattern, further enlargement would be useless. This is sometimes called a theoretical limit, as opposed to other aberrations, nothing can be done to improve it (other than making the telescope wider). Notice also that spherical aberration requires using mostly the region of the lens close to its center. Restricting the opening would accomplish that at the expense of increased diffraction. One simple but very important device to consider is our own eye. Since the pupil has a diameter of approximately 4mm then the image projected on the retina cannot have more angular resolution than θm ￿ 1.22 λ 400nm ￿ 1.22 ￿ 10−4 4mm 4mm (21.3) in radians. This is quite small. It also implies that, if Nature was kind to us, we would have enough cells in the retina to reach such resolution. A simple check is to consider, at night, how far we would be able to see the two headlight of an incoming car as – 138 – separate sources. If the headlights are separated by 1m the distance at which we can resolve them is 1m D= = 104 m = 10Km ￿ 6miles (21.4) θm which is the right order of magnitude as we know from our driving experience. This means that the human eye indeed reaches close to its theoretical resolution limit. To improve we would need bigger eyes which would make focusing more complicated. Figure 118: Diffraction through the lens imply that perfectly parallel rays will give rise to a blurred image even for a perfect lens. The effect is very small but detectable when we enlarge the image or look at it with enough resolution. This gives a theoretical limit to the magnification of any optical device (in terms of its width). 21.2 Light-matter interaction: Photoelectric effect Up to know we have studied the properties of light when it propagates through a medium. However, light interacts with matter in many different ways, for example it is important to study how light is emitted an absorbed by matter. We start this study by considering the photoelectric effect. As we discussed at the beginning of the course, electrons are free to move inside a metal, which explains why they are conductors. When light shines on a metal these electrons can be ripped off the metal. This is called the photoelectric effect. Notice that electrons normally do not leave a metal mainly because if they do so, the metal would be positively charged and would attract them back. Light can kicked them out. In fact certain night-vision systems work in this way by accelerating the electrons and amplifying the resulting current. From a physical point of view we will be mainly concerned with the energy of the electrons which are ripped off the metal. It turns out that the number of electrons coming out – 139 – is proportional to the intensity of light but their energy is proportional to frequency of the light. This is rather surprising since the intensity of light describes precisely how much energy reaches a certain area of the metal. It make sense that the total energy transmitted to the electrons is proportional to the intensity. In fact it is because the larger the intensity the larger the number of electrons. It is surprising however that the energy of each individual electron depends only on the frequency of light and not the intensity. The experimental result is described in fig.119. No electrons emerge for frequencies smaller than a cut-off fc . For larger frequencies f the energy of the electrons is given by Eel. = h(f − fc ) (21.5) The explanation of this fact was given by Einstein. He proposed that light is made out of quanta which behave similarly as particles. Each quantum is called a photon and has an energy given by h Eph. = hf = ω = ￿ω (21.6) 2π where h is a universal constant known as Planck’s constant. We also used the relation between frequency and angular frequency ω = 2π f and defined ￿= h 2π (21.7) This is an entirely new physical description. Planck had already observed that light is emitted and absorbed in discrete amounts but Einstein took the photons as the real picture of what light is made of. Now the explanation of the photoelectric effect is very simple. To extract an electron of the metal a minimal energy W of needed to overcome the Coulomb attraction. This energy W is called the work function and depends on the substance. If the frequency of the light is smaller that fc = W h (21.8) then a photon has not enough energy to kick out an electron. Two or more photons would be needed but the probability of two photons hitting the same electron at the same time turns out to be very small and can be ignored. If the photon has larger energy than W the excess energy is transfered to the electron as kinetic energy: Eph. = hf = W + Eel. = hfc + Eel. from where eq.21.5 follows. – 140 – (21.9) Figure 119: Photoelectric effect. Electrons are ejected from the metal by incident light. For frequencies smaller that fc no electrons are ejected, for larger frequencies f , the energy of the electrons is given by the formula Eel. = h(f − fc ) – 141 – ...
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This note was uploaded on 12/07/2011 for the course PHY 219 taught by Professor Na during the Fall '11 term at Purdue University-West Lafayette.

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