## Abstract

The harmonic diffractive lens is a diffractive imaging lens for which the optical path-length transition between adjacent facets is an integer multiple *m* of the design wavelength λ_{0}. The total lens thickness in air is *m*λ_{0}/(*n* − 1), which is *m* times thicker than the so-called modulo 2π diffractive lens. Lenses constructed in this way have hybrid properties of both refractive and diffractive lenses. Such a lens will have a diffraction-limited, common focus for a number of discrete wavelengths across the visible spectrum. A 34.75-diopter, 6-mm-diameter lens is diamond turned in aluminum and replicated in optical materials. The sag of the lens is 23 μm. Modulation transfer function measurements in both monochromatic and white light verify the performance of the lens. The lens approaches the diffraction limit for 10 discrete wavelengths across the visible spectrum.

© 1995 Optical Society of America

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### Equations (10)

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(1)
$$\begin{array}{cc}P\equiv \frac{1}{f}=\frac{k\lambda}{{{r}_{1}}^{2}}& \text{so}\phantom{\rule{0.2em}{0ex}}\text{that}\frac{\Delta P}{P}=\frac{\Delta \lambda}{\lambda}\end{array}\phantom{\rule{0.5em}{0ex}},$$
(2)
$$\begin{array}{ll}t\left(r,{\lambda}_{0}\right)=& \begin{array}{cc}\left(\frac{1}{\Delta n}\right){\text{MOD}}_{m{\lambda}_{0}}\left(\frac{{r}^{2}}{2f}\right),& m=1,2,3,4,\dots ,\end{array}\\ \phantom{\rule{1.6em}{0ex}}\Delta \mathrm{n}\equiv & \left(\mathrm{n}-1\right),\end{array}$$
(3)
$$\eta ={\text{sinc}}^{2}\left(\frac{m{\lambda}_{0}}{\lambda}-k\right).$$
(4)
$$P=\frac{k\left({\lambda}_{\text{eff}}/k\right)}{{{r}_{1}}^{2}}=\frac{{\lambda}_{\text{eff}}}{{{r}_{1}}^{2}}.$$
(5)
$$\begin{array}{cc}\frac{\Delta P}{P}=\frac{\Delta \lambda}{\lambda}=\frac{{\lambda}_{0}-\frac{m{\lambda}_{0}}{\left(m+1\right)}}{{\lambda}_{0}}\cong \frac{1}{m}& \text{and}\phantom{\rule{0.2em}{0ex}}\Delta {t}_{\text{max}}=\frac{m{\lambda}_{0}}{\Delta n}.\end{array}$$
(6)
$$\begin{array}{ll}t\left(r,{\lambda}_{0}\right)=& \left(\frac{1}{\Delta n}\right)\frac{{r}^{2}}{2f}-\left(\frac{1}{\Delta n}\right){\text{FLOOR}}_{m{\lambda}_{0}}\left|\frac{{r}^{2}}{2f}\right|,\\ \phantom{\rule{2em}{0ex}}m=& 1,2,3,\dots ,\end{array}$$
(7)
$${\text{FLOOR}}_{b}\left(a\right)\equiv b\phantom{\rule{0.2em}{0ex}}\text{INTEGER}\left(\frac{a}{b}\right),$$
(8)
$$U\left({P}_{0},{\lambda}_{i}\right)=\frac{-i}{{\lambda}_{i}}{\displaystyle \int {\displaystyle {\int}_{\sum}U\left({P}_{1},{\lambda}_{i}\right)\frac{\text{exp}\left(iks\right)}{s}r\mathrm{d}r\mathrm{d}\u04e8,}}$$
(9)
$$s\cong z+\frac{r\rho}{z}\text{cos}\left(\theta -\phi \right)+\frac{{r}^{2}}{2z},$$
(10)
$$\begin{array}{cc}\zeta ={r}^{2},& \mathrm{d}\zeta =2r\mathrm{d}r,\end{array}$$