Phase quantization of diffractive optical elements for the formation of predetermined symmetric light distributions

A method for the effective formation of symmetric light fields with a given phase distribution by binary diffractive optical elements is proposed. The design method is based on two-level phase quantisation of multi-level diffractive optical elements that form a basic distribution of light field. Numerical simulation of the formation of predetermined light fields is performed. The possibility of forming such light fields by a spatial light modulator is experimentally demonstrated.

which is a distribution composed of light fields generated in the +1 and -1 diffraction orders, can theoretically lead to doubling of the efficiency value.

Design method
Let τ(x,y) be the phase transmission function multilevel DOE that forms a predetermined amplitudephase distribution. One of the many currently existing methods (amplitude encoding techniques [15,16], iterative methods [17]) can be used to calculate this function. Reducing the number of quantisation levels for the phase function in multilevel DOEs reduces the efficiency value. The DOE quantisation operation can be described by the expression: M is the number of quantisation levels, and [..] is an integer part of number.
Then, the complex transmission function of the DOE is written as In case of two-level quantisation (M = 2), N irregular diffraction grating is created. In this case, light fields that are symmetric about the origin are formed in symmetrical diffraction orders (for example, +1 and -1 diffraction orders). Under specific initial parameters of a multilevel binary DOE, quantisation may lead to the light distribution composed of the fields formed in these orders forming a symmetrical light field.
In the simulation, we used the integral transform, which describes the diffraction of a laser beam on a thin optical element supplemented with lens: where A(x,y) is the laser beam distribution, λ is the laser wavelength, g(x,y) is the complex function of the optical element, and f is the focal length. The simulation results for the case of calculating the binary DOE forming the light distribution in the form of a cross, the letter «S» and the contour of the rectangle are shown in Fig. 1. Fig. 1 also shows the light fields generated by a base multilevel DOE (Fig. 1, top row), which were taken as a basis, as part of a composite light field with symmetry.

Experimental results
To generate the calculated symmetric light fields experimentally, we utilised a SLM PLUTO VIS (1920 × 1080 pixel resolution, 8 μm pixel size). The optical setup is shown in Figure 2. The calculated binary phase displayed at the SLM is illuminated by a laser beam (wavelength of 532 nm). With the phase pattern of 1024 × 1024 pixels used in the experiment, the size of the phase element at the SLM output was approximately equal to 8.2 mm ×8.2 mm. A system composed of a lens L 1 (f 1 = 150 mm) and a lens L 2 (f = 250 mm) was utilised to expand the laser beam incident on the SLM

Conclusions
The method of calculation of DOEs that form a predetermined, symmetrical amplitude-phase distribution is proposed. This method is based on two-level phase quantisation of DOE. Due to the fact that the designed DOE has a binary relief, the technology for their production is significantly simplified. In addition, owing to the formation of the light field as a sum of the light fields formed in the +1 and -1 diffraction orders, the theoretical efficiency is the sum of the efficiency of diffraction orders.
The simulation of the formation of some symmetric fields in the thin optical element approximation is performed. Predetermined symmetrical fields are experimentally formed by a SLM. The experimental results are in qualitative agreement with the simulation results.