5 Diffraction-based phase calibration of SLM with binary phase Fresnel

We proposed a phase calibration diffractive technique. Interferometric techniques involve a very precise alignment and suffer from sensibility to vibrations. Moreover, the methods shown above require a large number of optical elements. In [58], polarizing beam splitters, linear polarizers and two- lens Kepler type are included, apart from other common optical elements; while in [57] two polarizers, a beam splitter, a piezo mirror, a shutter, and a camera are required. On the other hand, diffractive-based methods may be affected by non-diffracted light, which introduces a harmful effect at the zero diffraction order and by residual intensity modulation. In our proposal just a single beam-splitter and an intensity detector is enough to get the calibration. As it does not use an interferometric arrangement, it is very insensitive to vibrations or other environmental fluctuations unlike two-arm interferometers.

The main idea of this technique lies in the encoding of a set of binary phase Fresnel lenses (BPFLs) into the phase-only SLM and measure the resulting irradiance foci with the detector. In this case, a charge-coupled device (CCD) camera is employed to measure the foci but another intensity-recording device (i.e. a bucket detector) can be used. A schematic of the setup is shown in Fig. II.7.

Fig. II.7 – Experimental setup employed for phase calibration of the phase-only SLM. Only a camera and a beam-splitter cube are needed in

addition to the SLM.

The BPFL is a lens made of a set of concentric rings that alternate their phase between two values (ϑ1 and ϑ2). The diffraction efficiency ηm can be expressed

in mathematical terms as 2 2 2 1 2 sin 2 m m

 





where m =1 represents the selected focus. Each lens generates several foci located at different axial positions, and m = 1 denotes the main focus or the one with the highest intensity. According to Eq. (I.1), it is evident that the diffraction efficiency of the lens depends on the difference

 



Fig. II.8 – BPFL example. The concentric rings alternate their phase values between ϑ1 and ϑ2

To perform the calibration, two curves must be compared: an experimental one, which relates the level of gray sent to the device with the irradiance of the foci measured by the CCD, and another theoretical one, which relates that same irradiance to the phase, according to the Eq. (I.1). To obtain the experimental calibration curve, a series of steps, which are detailed below must be carried out. The objective is to obtain a curve that allows to know precisely ϑ2 in the generation of BPFL affects to the intensity of the measured

focus.

The first step consists in the generation of the BPFLs that have to be sent to the modulator. In all of them, ϑ1 will take the value 0, while ϑ2 will take a

different value for each BPFL. To optimize the final result and obtain a calibration curve as accurate as possible, it is advisable to generate 255 BPFLs from 1 to 255; however, a faster calibration can be obtained by generating a reduced number of lenses. In this way, fewer points will be enough to obtain the gray level - radiance curve and less time is needed to make the measurements.

The second step is to synchronize the modulator with the camera (or the device used to measure the irradiance). Then, it is possible to send the BPFLs generated in the previous step to the modulator. As the spot widths remains the same for all BPFLs, the criteria used to measure diffraction efficiency is unique for all measurements. In our case, the 1/e2 _{criterion was employed to}

Once the experimental curve has already been achieved (Fig. II.9.a), the theoretical one can be obtained from Eq. (I.1). This curve is represented in Fig. II.9.b. The main difference between both figures is that the first relates the irradiance and the gray level while the second one links the irradiance with the phase. The comparison between both curves allows the correlation of the gray levels with the phase. To fit both curves, it must be taken into account that for each gray level sent to the SLM, a phase value associated to that gray level is encoded at the LC layer. In that sense, and thanks to the theoretical basis explained before, the point of maximum irradiance can be related to a phase difference between ϑ1 and ϑ2 of π, and the minimum with a phase difference

of 0 or a multiple of 2π. Please, note that depending on the modulator model, the phase shift that the modulator can encode may be greater than 2π. One should also keep in mind that the experimental curve should be normalized to improve the result as the irradiance values are measured in arbitrary units.

Fig. II.9 – Experimental results. a) experimental curve obtained by sending 255 BPFLs. b) theoretical curve obtained. c) calibration curve

The next step consists of separating each of the minimum-maximum or maximum-minimum sections, in the experimental curve, as they are related to phase shifts of π (from 0 to π, from π to 2π, and so on). The irradiance values of those sections are substituted into Eq. (I.1) linking the gray level used to obtain a certain irradiance value with the phase. The reason to split in sections is because a real function must be obtained. At this point, the resulting function of joining the different sections accurately indicates the phase encoded in the modulator according to the level of gray sent to it, which is precisely the curve that we want to obtain as a result of the calibration, and can be employed as a LUT function (Fig. II.9.c). From Fig. II.9.b one can realize that the phase-only SLM calibrated here is able to encode a phase shift of 3π. To test the validity of the presented method, a comparison (Fig. II.10) with one of the best-known techniques [63] has been made. In this technique, 255 Ronchi grating patterns, designed in a similar way to the BPFLs, are sent to the modulator; the setup is almost identical to the one employed in our experiment but including an extra refractive lens. The irradiance is also measured analogously to our experiment, using the same 1/e2 _criterion.

Fig. II.10 – Comparison between the proposed technique using BPFLs (red dots) with a well-stablished calibration method using Ronchi gratings

(blue line).

At a glance, one can realize that both results agree satisfactorily. In detail, the calculated root mean squared error between the two methods is approximately 6%. Differences are mainly produced due to laser fluctuations

in both position and energy. These results provide a clear proof that the method offers reliable data for phase calibration purposes.

In order to prove the utility of the technique in other experimental conditions, other tests has been performed with different specifications: further periods for the BPFLs, a different criterion for the beam width (instead of the 1/e2_),

and selecting a different focal plane (m ≠ 1) for the focal measurements. The results obtained, in all cases, yield root mean squared errors less than 5%. Moreover, to predict how different amplitude distributions affects the technique, simulations have been made. The result of the simulations confirms that this parameter does not influence the calibrations function, which is something expected as relative irradiance measurements are performed. This method has some restrictions: the pixel size of the SLM limits the minimum size of the rings that make up the BPFLs. This prevents the generation of a main focus very close to the SLM, since rings smaller than the pixel size of the SLM are needed. This technique requires at least two pixels to properly encode a ring, leading to a closest distance of, approximately, 150 mm. On the other hand, in this technique all the pixels of the modulator are involved in the calibration. That means that the LUT obtained is global and is used equally in all pixels, regardless of their spatial position on the screen; however, as discussed before, it is possible that not all pixels respond in the same way. For most applications and devices this is not a problem, since the decrease in diffraction efficiency is not significant. More information about the results obtained with this experiment can be found in [64].