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four level approximation. As the number of quantized steps, N, increases, the
lens more closely approximates the continuous kinoform phase profile and the
diffraction efficiency of the element will approach 100%.
As will be discussed in more detail later, a multilevel diffractive lens is
fabricated using a set of binary masks. Each mask produces two phase levels.
Thus, M masks are needed to produce 2 M = N phase levels. A two step (N=2)
binary lens therefore only requires a single mask, a four level element (N=4)
requires two masks and similarly for N=M.
The transition points for the boundaries of each zone in a multilevel
element are calculated by examining the phase function given by Equation 3.4.
For the case of the binary lens already discussed, the zone boundaries were
determined by setting Equation 3.4 equal to multiples of n to satisfy the zone
theory of Fresnel. It can be seen from Figure 3.7 that the size of these zone
boundaries decreases as the number of levels increases. With each
successive mask, the zone widths decrease by a factor of two, and two more
phase levels are added. Since every mask produces 2 phase levels, the
boundary of each zone occurs when the phase function in Equation 3.4 equals
7c/M. Equivalently this may be expressed as
3.8
Solving Equation 3.8 for r m and m as before gives the boundary for each zone
and the maximum number of zones created by M masks,