Physics
Interference
Interference refers to the interaction of two or more waves, resulting in the combination of their amplitudes. In physics, interference can lead to the reinforcement or cancellation of waves, producing patterns of varying intensity. This phenomenon is commonly observed in optics, acoustics, and quantum mechanics, and plays a crucial role in understanding wave behavior.
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11 Key excerpts on "Interference"
- Swapan K Saha(Author)
- 2007(Publication Date)
- World Scientific(Publisher)
Chapter 3 Interference and diffraction 3.1 Fundamentals of Interference Principles of reflection and refraction are well explained using geometrical optics. Physical optics deals with light as a wave and the principle of linear superposition is particularly important. The most interesting cases of Interference usually involve identical waves, with the same amplitude and wavelength, coming together. Consider the case of just two waves, although one may generalize to more than two. When these waves are in phase and travel together are superposed, the intensity at the point of superposition varies from point to point between maxima which exceed the sum of the intensities in the beams, and minima, which may be zero. This is known as Interference. When the crest of one wave passes through the crest of another wave, it is referred as constructive Interference. It also occurs when the trough of one wave is superpositioned upon the trough of another wave. The other extreme case occurs when the trough of one corresponds with the crest of the other and tend to cancel each other out, resulting in a flat or no wave while interfering. This type of Interference is referred to as destructive Interference. Basic principles of optical Interference has wide range of applications ranging from on-line real-time wavefront control in astronomy to experi-ments in relativity. In what follows, the principle of Interference and diffrac-tion and the necessary conditions in physical applications are elucidated. 3.2 Interference of two monochromatic waves Let the two monochromatic waves E 1 and E 2 be superposed at the recom-bination point P . The correlator sums the instantaneous amplitudes of the 81 82 Diffraction-limited imaging with large and moderate telescopes fields.
eBook - PDFPhysical Optics
Principles and Practices
- Abdul Al-Azzawi(Author)
- 2018(Publication Date)
- CRC Press(Publisher)
2 Interference and Diffraction 2.1 INTRODUCTION The particle nature of light is well known and has important applications. There are two accepted behaviours of light to explain the dual nature of light. In some cases, light acts like a particle, and in others, it acts like a wave. The proof of the wave nature of light came with the discovery of Interference of light and diffraction. In this chapter, the wave nature of light will be studied with emphasis on two important wave phenomena, the diffraction and Interference of light. Also in this chapter are experiments designed to observe the diffraction patterns generated by objects, such as a blade, disk, washer, single-slit and double-slit holes, and grating, will be described. Students will practise light alignment techniques to generate diffraction patterns from the different geometrical objects. 2.2 Interference OF LIGHT When light waves from two light sources are mixed, the waves are said to interfere. This Interference can be explained by the principle of superposition. When two or more waves of the same phase and direction go past a point at the same time, the instantaneous amplitude at that point is the sum of the instantaneous amplitudes of the two waves. If the waves are in phase, then they add together, resulting in a larger amplitude. This is referred to constructive Interference, as shown in Figure 2.1(a). If the waves are out of phase with one another, then they cancel each other. This is referred to destructive Interference, as shown in Figure 2.1(b). If the waves differ in amplitude and are out of phase with one another, then they add to give a partial cancellation or elimination. This is referred to as partial cancellation or elimination Interference, as shown in Figure 2.1(c). Interference occurs with monochromatic light. This light has a single colour and, hence, a single frequency. In addition, if two or more sources of light are to show Interference, they must 23
eBook - PDF- David Halliday, Robert Resnick, Kenneth S. Krane(Authors)
- 2019(Publication Date)
- Wiley(Publisher)
941 41-1 TWO-SOURCE Interference When otherwise identical waves from two sources overlap at a point in space, the combined wave intensity at that point can be greater or less than the intensity of either of the two waves. We call this effect Interference. The interfer- ence can be either constructive, when the net intensity is greater than the individual intensities, or destructive, when the net intensity is less than the individual intensities. Whether the Interference is constructive or destructive de- pends on the relative phase of the two waves. Although any number of waves can in principle inter- fere, we consider here the Interference of only two waves. We assume that the sources of the waves each emit at only a single wavelength. (The case of sources that emit waves of several wavelengths can be handled by considering the separate Interferences of the individual component wave- lengths.) Figure 41-1 represents the time dependence of two iden- tical waves that arrive at the same point P in space. In Fig. 41-1a, the two waves arrive in phase; that is, they line up crest to crest and valley to valley. The net effect at point P is due to the combination of the two waves, and the resultant wave would have twice the amplitude of the component waves. This is the condition of maximal constructive interfer- ence. The pattern of Fig. 41-1a would look the same if we shifted one of the waves by a full cycle (2 radians) or by any whole number of full cycles. We can therefore say that maximal constructive Interference of two waves occurs when their phase difference (in radians) is 0, 2, 4, . . . . In Fig. 41-1b, the waves line up crest to valley and valley to crest. This is the condition for complete destructive inter- ference, in which the two waves cancel one another’s effect at point P. To go from a situation of constructive Interference CHAPTER 41 CHAPTER 41 Interference I n this chapter and the next one, we discuss interfer- ence and diffraction of light waves.
eBook - PDFOptics
Principles and Applications
- Kailash K. Sharma(Author)
- 2006(Publication Date)
- Academic Press(Publisher)
C H A P T E R 6 Interference of Light Waves 6.1 Interference When two or more light waves cross each other, the resultant field, given by the superposition principle, is the vector sum of the fields associated with the individual waves, i.e., E r t = i E i r t (6.1) where E i r t is the electric field associated with the i th wave. Here, we have considered superposition of the electric fields since, as was first shown by Wiener, most optical detectors including the human eye are influenced primarily by the electric field of the light waves. Needless to state that the superposition principle applies to magnetic fields as well: B r t = i B i r t The superposition principle permits the resultant irradiance I r t at a given point to differ from the sum of the irradiances of the individual waves when present alone, i.e., superposition principle for the intensities may not always hold. When I r t = i I i r t (6.2) the light waves are said to have interfered with each other. The waves, after interfering, continue to move forward unaltered, except for a phase retardation in a lossless medium which bears no relationship, whatsoever, to the fact that Interference among the waves has taken place in the region of their overlap. As has already been mentioned in Chapter 2, only mutually coherent waves can interfere but the lack of Interference does not necessarily imply incoherence of the interfering waves. Arago and Fresnel had concluded that orthogonally polarized coherent light waves do not interfere. 255 256 Chapter 6: Interference OF LIGHT WAVES 6.2 TWO-WAVE Interference Interference between two monochromatic waves of the same frequency gives rise to a spatially stationary distribution of time averaged intensities. Interference among monochromatic waves with widely different frequencies can hardly be observed.
eBook - PDF- Md Nazoor Khan, Simanchala Panigrahi(Authors)
- 2017(Publication Date)
- Cambridge University Press(Publisher)
2 Interference 2.1 Introduction In Section 1.16 of the previous chapter we learned that two beams of light waves can cross each other without either one producing any modification on the other after passing beyond the region of crossing. However, from the concepts explained in Section 1.16.2, we expect some modifications in the amplitudes or intensity (since intensity ∝ amplitude 2 ) of the two waves inside the region of crossing. The intensity of the resultant wave becomes a function of the position of the point. At certain points intensity is maximum and at other points it is minimum. In other words, we say that the two waves interfere with each other inside the region of crossing. This modification of intensity obtained by the superposition of two or more beams of light waves is called Interference of light. The phenomenon of Interference of light complements the validity of the concept that light is a wave. As a result of the short wavelength and disordered phase relationships of the interfering light waves, the Interference pattern is not visible to the naked eye without special arrangements. It was in the year 1801 that Thomas Young for the first time demonstrated the Interference of sunlight experimentally. Before discussing the Interference phenomenon, let us discuss Huygens’ principle, a helpful tool and an early concept in favour of the wave theory of light when the scientific world was mesmerized by Newton’s corpuscular theory of light. 2.2 Huygens’ Principle Huygens, a Dutch mathematician, in 1678, propounded a theory regarding the propagation of light wave in any medium. According to this theory, light is a sort of disturbance in the medium in which it propagates in all direction from a point source. To explain the propagation of light in vacuum, he postulated an all-pervading medium called ‘ether’ (Later on, in the year 1881, Michelson and Morley, American scientists, performed a
eBook - PDF- Charles A. Bennett(Author)
- 2022(Publication Date)
- Wiley(Publisher)
As discussed in Section 1.4, we may plot a harmonic traveling wave as a function of either position or time, so the horizontal axes of the plots of Figure 7.1 are not labeled to allow for either possibility. When waves are combined in phase, they combine to give a larger amplitude wave, and when they are 𝜋 rad out of phase, they tend to cancel. The general term for this is Interference, with the situation of Figure 7.1(a) referred to as constructive Interference and that of Figure 7.1(e) referred to as destructive Interference. Principles of Physical Optics, Second Edition. Charles A. Bennett. © 2022 John Wiley & Sons, Inc. Published 2022 by John Wiley & Sons, Inc. 214 7 Superposition and Interference (a) (b) (c) (d) (e) Figure 7.1 In each column, the top wave (wave 1) sums algebraically with the middle wave (wave 2) to give the bottom wave. In each case, waves 1 and 2 have the same amplitude and wavelength. (a) Waves 1 and 2 are in phase, giving constructive Interference. (b)–(d) Wave 2 is shifted in phase by 𝜋∕4, 𝜋∕2, and 3𝜋∕4 relative to wave 1. (e) Wave 2 is shifted in phase by 𝜋 relative to wave 1, giving destructive Interference. Since the amplitudes of waves 1 and 2 are equal, the destructive Interference in (e) is complete. 7.3 Interference Between Two Monochromatic Electromagnetic Waves We begin our discussion of Interference by examining the combination of two travel- ing monochromatic electromagnetic waves. As we will see later, no source is perfectly monochromatic; nevertheless, this assumption allows us to utilize the harmonic wave solutions to Maxwell’s equations determined in Chapter 2 and will provide an intuitive starting point that can be refined as we proceed. Figure 7.2 shows two sources S 1 and S 2 , located by position vectors ⃗ s 1 and ⃗ s 2 relative to an arbitrary coordinate origin labeled O. Also shown is an observation point P (located by posi- tion vector ⃗ r relative to O), where the superposition will be observed.
eBook - PDF- John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
- 2015(Publication Date)
- Wiley(Publisher)
In Figure 27.1, these distances are , 1 5 2 1 4 wavelengths and , 2 5 3 1 4 wave- lengths. In general, when the waves start out in phase, constructive Interference will result at P whenever the distances are the same or differ by any integer number of wavelengths—in other words, assuming that , 2 is the larger distance, whenever , 2 2 , 1 5 ml, where m 5 0, 1, 2, 3, . . . . Figure 27.2 shows what occurs when two identical waves arrive at the point P out of phase with one another, or crest-to-trough. Now the waves mutually cancel, according to the principle of linear superposition, and destructive Interference results. With light waves this would mean that there is no brightness. The waves begin with the same phase but are out of phase at P because the distances through which they travel in reaching this spot differ by one-half of a wavelength (, 1 5 2 3 4 l and , 2 5 3 1 4 l in the drawing). In general, for waves that start out in phase, destructive Interference will take place at P whenever the distances differ by any odd integer number of half-wavelengths—that is, whenever , 2 2, 1 5 1 2 l , 3 2 l, 5 2 l, . . . , where , 2 is the larger distance. This is equivalent to , 2 2, 1 5(m 1 1 2 )l, where m 5 0, 1, 2, 3, . . . . Wave Interference occurs when two or more waves exist simultaneously at the same place. Light is an electromagnetic wave and, therefore, can exhibit Interference effects. The Interference of light waves is responsible for the lovely irides- cent colors of the feathers on this purple-crested turaco from South Africa. 27 | Interference and the Wave Nature of Light 766 Chapter | 27 LEARNING OBJECTIVES After reading this module, you should be able to... 27.1 | Apply the principle of linear superposition to light waves. 27.2 | Analyze double-slit Interference. 27.3 | Analyze thin-film Interference. 27.4 | Understand the operation of the Michelson interferometer. 27.5 | Analyze single-slit diffraction. 27.6 | Determine the resolving power of lenses.
eBook - PDF- John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
- 2021(Publication Date)
- Wiley(Publisher)
Interference effects can also be detected if the two speakers are fixed in position and the listener moves about the room. Consider Figure 17.6, where the sound waves spread outward from each of two in-phase speakers, as indicated by the concentric circular arcs. Each solid arc represents the middle of a condensation, and each dashed arc represents the middle of a rarefaction. Where the two waves overlap, there are places of construc- tive Interference and places of destructive Interference. Constructive Interference occurs wherever two condensations or two rarefactions intersect, and the drawing shows four such places as solid dots. A listener stationed at any one of these locations hears a loud sound. On the other hand, destructive Interference occurs wherever a condensation and a rarefaction intersect, such as the two open dots in the picture. A listener situated at a point of destructive Interference hears no sound. At locations where neither constructive nor destructive Interference occurs, the two waves partially reinforce or partially cancel, depending on the position relative to the speakers. Thus, it is possible for a listener to walk about the overlap region and hear marked variations in loudness. The individual sound waves from the speakers in Figure 17.6 carry energy, and the energy delivered to the overlap region is the sum of the energies of the individual waves. This fact is consistent with the principle of conservation of energy, which we first encountered in Section 6.8. This principle states that energy can neither be created nor destroyed, but can only be converted from one form to another. One of the interesting consequences of Interference is that the energy is redistributed, so there are places within the overlap region where the sound is loud and other places where there is no sound at all. Interference, so to speak, “robs Peter to pay Paul,” but energy is always conserved in the process.
eBook - PDF- John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
- 2018(Publication Date)
- Wiley(Publisher)
Constructive Interference occurs wherever two condensations or two rarefactions intersect, and the drawing shows four such places as solid dots. A listener stationed at any one of these locations hears a loud sound. On the other hand, destructive Interference occurs wherever a condensation and a rarefaction intersect, such as the two open dots in the picture. A listener situated at a point of destructive Interference hears no sound. At locations where neither constructive nor destructive Interference occurs, the two waves partially reinforce or partially cancel, depending on the position relative to the speakers. Thus, it is possible for a listener to walk about the overlap region and hear marked variations in loudness. The individual sound waves from the speakers in Figure 17.6 carry energy, and the energy delivered to the overlap region is the sum of the energies of the individual waves. This fact is consistent with the principle of conservation of energy, which we first encountered in Section 6.8. This principle states that energy can neither be created nor destroyed, but can only be converted from one form to another. One of the interesting consequences of Interference is that the energy is redistributed, so there are places within the overlap region where the sound is loud and other places where there is no sound at all. Interference, so to speak, “robs Peter to pay Paul,” but energy is always conserved in the process. Example 1 illustrates how to decide what a listener hears. FIGURE 17.5 Noise-canceling headphones utilize destructive Interference. Noise Noise Speaker Microphone Electronic circuitry Out-of-phase noise Reduced noise level 1 wavelength R C R C R C FIGURE 17.6 Two sound waves overlap in the shaded region. The solid lines denote the middle of the condensations (C), and the dashed lines denote the middle of the rarefactions (R). Constructive Interference occurs at each solid dot (●) and destructive Interference at each open dot ( ).
- David Halliday, Jearl Walker, Patrick Keleher, Paul Lasky, John Long, Judith Dawes, Julius Orwa, Ajay Mahato, Peter Huf, Warren Stannard, Amanda Edgar, Liam Lyons, Dipesh Bhattarai(Authors)
- 2020(Publication Date)
- Wiley(Publisher)
The key for the waves (sound, light, whatever their type might be) is the path length difference ΔL, or more to the point, how ΔL compares to the wavelength of the waves. Consider two waves that are initially in phase but then travel along different paths with a length differ- ence of ΔL. From module 17.3, we know that fully constructive Interference (here, maximum brightness) occurs when ΔL = integer, (35.17) which we can write as ΔL = m, for m = 0, 1, 2, …. (35.18) In words, if ΔL/ is an integer, the phase relation between the waves does not change. For example, if the ratio is 6.0, then the phase difference between the waves is 6.0 wavelengths, which is equivalent to 0.0 wavelengths. They are still in phase and produce fully constructive Interference. We also know that fully destructive Interference (here, complete darkness) occurs when ΔL = odd number 2 , (35.19) Pdf_Folio:832 832 Fundamentals of physics which we can write as ΔL = m + 1 2 , for m = 0, 1, 2, …. (35.20) For example, if the ratio is 6.5, then the phase difference between the waves is 6.5 wavelengths, which is equivalent to 0.5 wavelength. The waves are exactly out of phase and produce fully destructive Interference. Intermediate values of ΔL/ correspond to intermediate Interference and thus also illumination. Rainbows and optical Interference In module 33.5, we discussed how the colours of sunlight are separated into a rainbow when sunlight travels through falling raindrops. We dealt with a simplified situation in which a single ray of white light entered a drop. Actually, light waves pass into a drop along the entire side that faces the Sun. Here we cannot discuss the details of how these waves travel through the drop and then emerge, but we can see that different parts of an incoming wave will travel different paths within the drop. That means waves will emerge from the drop with different phases.
eBook - PDF- David Halliday, Robert Resnick, Jearl Walker(Authors)
- 2021(Publication Date)
- Wiley(Publisher)
C H A P T E R 3 5 1112 CHAPTER 35 Interference New position of wavefront at time t = Δt a e b d Wavefront at t = 0 c Δt Figure 35.1.2 The propagation of a plane wave in vacuum, as portrayed by Huygens’ principle. Nature has long used optical Interference for coloring. For exam- ple, the wings of a Morpho butterfly are a dull, uninspiring brown, as can be seen on the bottom wing surface, but the brown is hidden on the top surface by an arresting blue due to the Interference of light reflecting from that surface (Fig. 35.1.1). Moreover, the top surface is color-shifting; if you change your perspective or if the wing moves, the tint of the color changes. Similar color shifting is used in the inks on many currencies to thwart counterfeiters, whose copy machines can duplicate color from only one perspective and therefore cannot duplicate any shift in color caused by a change in perspective. FCP To understand the basic physics of optical Interference, we must largely abandon the simplicity of geometrical optics (in which we describe light as rays) and return to the wave nature of light. Light as a Wave The first convincing wave theory for light was in 1678 by Dutch physicist Christian Huygens. Mathematically simpler than the electromagnetic theory of Maxwell, it nicely explained reflection and refraction in terms of waves and gave physical meaning to the index of refraction. Huygens’ wave theory is based on a geometrical construction that allows us to tell where a given wavefront will be at any time in the future if we know its present position. Huygens’ principle is: Here is a simple example. At the left in Fig. 35.1.2, the present location of a wavefront of a plane wave traveling to the right in vacuum is represented by plane ab, perpen- dicular to the page. Where will the wavefront be at time Δt later? We let several points on plane ab (the dots) serve as sources of spherical secondary wavelets that are emitted at t = 0.
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