diffraction gratings and does - Breault Research Organization, Inc.

DIFFRACTION GRATINGS AND DOESFundamentals of Diffractive Optical Elements. . . . .The transmittance function of the prism looks similar to the blazed grating.Mathematically, the difference is manifest in the modulo operation. Physically, theblazed grating is the diffractive counterpart of the refractive prism. In general, amodulo 2 operation on the phase function of a refractive element yields itsdiffractive counterpart’s phase function.We are now in a position to examine a more complicated phase behavior; namely,that of a simple thin lens. A thin lens is one whose thickness is very smallcompared to its diameter. A ray intersecting one side of a thin lens essentially exitsat the same point on the other side of the lens.A thin lens is a phase transformer in the sense that it delays, in time, the phase ofthe wavefront in proportion to the thickness of the lens. A positive lens, such as abiconvex lens whose outer edge thickness is less than its center thickness, delayscollimated light, for example, more at the center of the lens than at the edge. Theedge of the wavefront has propagated further in the same amount of time than thecenter of the wavefront when it exits the center of the lens, because it is not“slowed” down by the additional refractive lens material. Since the wavefront iscontinuous and is a surface of constant phase, the wavefront must converge towarda point in space, which in this particular case, is the focal point of the lens.We can equivalently understand this phenomena by examining a ray of light. Sincethe wavefront is delayed more at the center of the lens than at its edge, the ray isalso delayed by the same amount because it is related to the wavefront and in factis perpendicular to the wavefront. The magnitude of the optical path length of theray at the edge of the lens is less than at the center. The optical path length of a rayis the distance that light travels in a vacuum in the same time it travels a givendistance in a material. Therefore, the edge ray is bent toward the focal point of thelens.The geometry for a radially symmetric convex-plano lens is illustrated in Equation10.Figure 5 The wavefront experiences a total phase delay due to the lens and theremaining region around the lens. Mathematically, this is,ASAP Technical Guide 19