Light is part of the electromagnetic spectrum and therefore is governed by Maxwell's equations. In this chapter we introduce these equations to describe the propagation of light. In later chapters we will use them as a basis for the design of a wide range of diffractive optical elements. Although we begin with the vector form of Maxwell's equations, this representation is difficult to use for most calculations, but they can be simplified by a series of assumptions to their scalar forms, where they are most useful. However, when the wavelength of the incident radiation is comparable to or smaller than the size of the diffractive features, it is necessary to retain the vector forms, and the computations are more difficult. They are discussed in Chapter 3. While several modeling techniques are used in this text, it is important to remember that they are all derived from Maxwell's equations.
When designing an optical element using any mathematical model, it is critical to know that the model is accurate for your design. By understanding the assumptions that are made in the derivation of that model, you can assure yourself that the modeling technique is valid for that element. In Sec. 2.1 of this chapter, the assumptions that are made to reduce Maxwell's equations to the scalar representation are outlined. Then in Secs. 2.2 and 2.3, the use of Fourier analysis to model the performance of a diffractive optical element performance is described. In Sec. 2.4, first-order scalar theory is used to calculate the efficiency of a diffractive structure; Sec. 2.5 discusses the extension of scalar theory to better analyze the efficiency of diffractive structures.
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