Adjustable refractive power from diffractive moiré elements

Generic setup (not to scale): two diffractive optical elements (DOEs) are placed directly behind each other. They can be mutually rotated around a central axis. This combined optical element manipulates the wavefront of an incident light wave in a predesigned way that changes with the mutual rotation angle.

A MDOE that acts as a Fresnel lens with a refractive power that depends on their mutual rotation angle. Gray values in the figure actually correspond to phase-shift values between 0 and $2\pi$ . Note that the two DOEs are identical if one of them is reversed (i.e., flipped upside down).

Superposition of the two DOEs in Fig. 2 at different mutual rotation angles of (upper row) $-75°$ , $-30°$ , and $-15°$ ; and (lower row) $+15°$ , $+30°$ , and $+70°$ . The results are perfect blazed Fresnel lenses with different refractive powers; however, they include a sector of an angular range that corresponds to the mutual rotation angle that comprises a Fresnel lens of a different focal length.

MDOEs forming an adjustable Fresnel lens. The MDOE is similar to the one in Fig. 2; however, it is calculated according to Eq. (12) such that there are no phase discontinuities (with the exception of $2\pi$ phase jumps) along circular paths around the center.

Result of the superposition of the two DOEs of Fig. 4 at different mutual rotation angles of (upper row:) $-75°$ , $-30°$ , and $-15°$ ; and (lower row:) $+15°$ , $+30°$ , and $+70°$ . A sector formation like that in Fig. 3 is now avoided.

Diffraction efficiency and refractive power of the Fresnel lenses in Fig. 5 as a function of the mutual rotation angle between the two DOEs. The joint DOE corresponds to a bifocal Fresnel lens with two refractive powers (red and blue curves), the relative efficiencies of which depend on the rotation angle.

Sensitivity of MDOEs to transverse misalignment: MDOEs generating a Fresnel lens (left two columns), and a Fresnel lens with avoided sector formation (right two columns), are plotted for two mutual rotation angles of $10°$ and $30°$ , and for three different decentering values of $1\mathrm{pixel}$ (upper row), $5\mathrm{pixels}$ (middle row), and $10\mathrm{pixels}$ (lowest row). The total size of each MDOE is $512\mathrm{pixels}\times 512\mathrm{pixels}$ . The misalignment introduces lens deformations but it does not completely destroy the performance of the element.

Two DOEs calculated according to Eq. (18) produce a combined DOE acting as a varifocal Fresnel lens with an offset refraction power; i.e., at a zero mutual rotation angle, the focal length is $f_{\text{offset}}$ .

Calculated DOEs for creating an axicon with a refractive power that depends on the mutual rotation angle. All DOEs in (a), (b), and (c) have to be combined with a second DOE that is identical to the first one, but flipped upside down. The DOE in (a) is calculated according to Eq. (19) and produces a perfect blazed axicon with, however, an undesired sector [see example (d)] for a mutual rotation angle of $25°$ ). The DOE in (b) is calculated according to Eq. (20) and produces axicons without the undesired sector formation [see example (e)]. The DOE in (c) is calculated according to Eq. (21) and produces an axicon with a variable refractive power that is superposed by a Fresnel lens with a constant focal length.

Example for a set of DOEs that generates a spiral phase element with a variable helical index that depends linearly on the mutual rotation angle between the two DOEs.

Combined DOE elements of Fig. 10 at rotation angles of $-25°$ , $-5°$ , $-1.5°$ , $+1.5°$ , $+5°$ , and $+25°$ . The corresponding transmission functions correspond to spiral phase elements with helical indices of approximately $-13$ , $-3$ , $-1$ , $+1$ , $+3$ , and $+13$ , respectively.