![]() Largely in parallel to these efforts on disordered media, it was recently realised that materials and devices can get entirely new functionalities when adding to them a suitably arranged combination of gain and loss. More recent studies focused instead on controlling the medium itself, for example, through the material fabrication process 22 or through a spatially modulated pumping 23, leading, for example, to a versatile control of random and micro-cavity lasers 24, 25, 26, 27, 28. In a first generation of corresponding experiments the focus was laid on shaping the input wave front impinging on an immutable disordered sample such as to achieve a desired output, like a spatial or temporal focus behind the medium 18, 19, 20, 21. Remarkable progress in these endeavours has recently been made in the optical domain, largely due to the availability of spatial light modulators and new concepts for how to apply them on turbid media 16, 17. While much work has been invested into understanding the ‘statistical’ properties of the corresponding wave phenomena 13 there has recently been a surge of interest in controlling the scattering of waves through ‘individual’ systems for specific purposes such as detection, imaging, and efficient transmission across disordered materials 14, 15. Among the many important physical aspects of wave propagation that have been studied the phenomenon of Anderson localisation has received particular attention 4, 5, 6, 7, 8, 9, 10, 11, 12. The scattering of waves through disordered media is a paradigmatic phenomenon that has captured the interest of various communities for quite some time now 1, 2, 3. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution License. ![]() Use the information below to generate a citation. Then you must include on every digital page view the following attribution: If you are redistributing all or part of this book in a digital format, Then you must include on every physical page the following attribution: If you are redistributing all or part of this book in a print format, Want to cite, share, or modify this book? This book uses the This book may not be used in the training of large language models or otherwise be ingested into large language models or generative AI offerings without OpenStax's permission. Since the arc subtends an angle ϕ ϕ at the center of the circle, To calculate the intensity at an arbitrary point P on the screen, we return to the phasor diagram of Figure 4.7. In solving that problem, you will find that they are less than, but very close to, ϕ = 3 π, 5 π, 7 π, … rad. The exact values of ϕ ϕ for the maxima are investigated in Exercise 4.120. As a result, E 1 E 1 and E 2 E 2 turn out to be slightly larger for arcs that have not quite curled through 3 π 3 π rad and 5 π 5 π rad, respectively. Since the total length of the arc of the phasor diagram is always N Δ E 0, N Δ E 0, the radius of the arc decreases as ϕ ϕ increases. These two maxima actually correspond to values of ϕ ϕ slightly less than 3 π 3 π rad and 5 π 5 π rad. ![]() The proof is left as an exercise for the student ( Exercise 4.119). In part (e), the phasors have rotated through ϕ = 5 π ϕ = 5 π rad, corresponding to 2.5 rotations around a circle of diameter E 2 E 2 and arc length N Δ E 0. The amplitude of the phasor for each Huygens wavelet is Δ E 0, Δ E 0, the amplitude of the resultant phasor is E, and the phase difference between the wavelets from the first and the last sources is The phasor diagram for the waves arriving at the point whose angular position is θ θ is shown in Figure 4.7. ![]() This distance is equivalent to a phase difference of ( 2 π a / λ N ) sin θ. If we consider that there are N Huygens sources across the slit shown in Figure 4.4, with each source separated by a distance a/N from its adjacent neighbors, the path difference between waves from adjacent sources reaching the arbitrary point P on the screen is ( a / N ) sin θ. To calculate the intensity of the diffraction pattern, we follow the phasor method used for calculations with ac circuits in Alternating-Current Circuits. Calculate the intensity relative to the central maximum of an arbitrary point on the screen.Calculate the intensity relative to the central maximum of the single-slit diffraction peaks.By the end of this section, you will be able to:
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