1 General overview

Above-mentioned techniques are limited to the control of the irradiance at the output plane. However, for some applications, it would be interesting to control not just the amplitude, but also the phase of the light wave. As reviewed on the previous chapter to reconstruct an irradiance pattern, one of the most popular solutions relies on the encoding of complex holograms. Nevertheless, complex modulation in a single phase-only SLM is not trivial. Therefore, it is necessary to use some method to modulate both the phase and the amplitude of the light wave. The first idea is to use two devices, one that modulates amplitude, while the second is used for the phase. For instance, in [92], two liquid-crystal televisions were combined with an afocal imaging system and aligned for a pixel-to-pixel matching. Tudela et al. reported a similar technique employing two ferroelectric LC SLMs to encode a complex Fresnel hologram [93]. Subsequently, other more sophisticated methods that also employ two modulators to shape arbitrary complex fields appeared. For example, in [94] two nematic SLMs are employed in a lensless configuration to shape simultaneously amplitude (using optical polarizing elements) and phase. In [95], two DOEs are encoded on conjugated Fourier planes. The first one is employed to redistribute the intensity of an incoming light beam to an arbitrary pattern by on-axis diffraction (using the GSA). This pattern is projected to a second SLM, placed at the Fourier plane of a 4f imaging system, which is employed to perform the phase modulation of the Fourier components. In the same manuscript [95] the authors propose an alternative arrangement using just one SLM. They suggest encoding the two masks side by side in the same SLM. For this, they substitute the first lens by a slightly tilted concave mirror that reflects the light back to the SLM; the mirror sends the beam to the area of the SLM that encodes the phase. Zhu et al. reported another example of complex encoding with two SLMs in 2014 [96]. Again, one SLM is conjugated with a second one by an imaging system. By using a set of

polarizers, they are able to control the polarization direction of the input beam throughout the entire system. In that way, the first SLM controls the amplitude of the light at the output plane of the system, while the second one only affects the phase. In all these cases, two identical devices have been used.

Single-phase SLMs are perfect for easily encoding phase patterns, while DMDs are basically aimed to manipulate the amplitude. The combination of these two elements in the same optical setup can create a fully complex modulated wavefront [97]. Several disadvantages are implicit with these strategies. First, the economic cost of two optical devices makes it advisable to use another technique that only requires a single device. Moreover, when one device should be imaged onto a second one, there is a high dependence between the alignment precision and the quality of the reconstruction. In particular, the corresponding pixels of both devices must be coupled to avoid unwanted problems.

Complex modulation with a single DMD has also been probed successfully [98]. A method called superpixel employs this kind of devices combined with a spatial filter to independently modulate amplitude and phase at a very high frame rate, since DMDs are comparatively much faster than SLMs. The main idea is to merge groups of micromirrors into superpixels and use a 4f imaging system made up of two slightly off axis lenses. The spatial filter, placed at the Fourier plane of this 4f system, is employed to block the high spatial. In that way, the phase response of each micromirror on a superpixel is dependent on its position within the DMD, and the response of a superpixel at the output plane can be obtained from the sum of the micromirrors. The efficiency is the main drawback of this kind of methods, since reported values are around 5%. Compared to DMDs, SLMs are slower devices; however, multiple techniques developed to encode complex fields in phase-only devices are more efficient than their counterparts for DMDs. Some of the iterative techniques discussed in the previous chapter are able to have certain control over the phase of the complex field. This is the case of the aforementioned WGS-PC, an IFTA with phase control. In a similar way, another iterative method, but this time based on a conjugate gradient minimization technique has also been reported as alternative to generate patterns with independent control over the phase and the amplitude of light [99]. Basically, this algorithm defines a cost function based on the difference between the calculated electric field and the desired pattern. Unlike IFTA based algorithms, which are based on Fourier transforms, this is a minimization algorithm that calculates the error for each pixel. A point

in favor to this technique is that the minimization approach facilitates convergence. In general iterative algorithms, present some notable disadvantages. Specifically, obtained results are only an approximation of the target complex field. In addition, compared to other techniques more computing time is necessary for reliable results.

Another approach proposed a spatial cross-modulation method (SCMM) [100]. SCMM uses a phase-only SLM and a random phase diffuser to encode a complex object as a scattered phase image by letting the object to transmit through the random diffuser. Basically the technique is executed in two steps: digital encode step and optical decode step. The first one is developed completely by computer algorithms and its objective is to obtain a phase pattern to be encoded into the SLM. To do this, they arbitrarily define both the amplitude and phase of a complex field, which is Fourier transformed. Then the result is multiplied by the spatial phase distribution of a random phase diffuser. Through an inverse Fourier transform of the above result, a diffusion image is obtained at the output plane of the system. Finally, the amplitude of the diffusion image is ignored whereas the phase term is conjugated to obtain an image referred to as cross-modulated image. This cross-modulated image is the final element encoded into the SLM. The second step is intended to be experimental. The SLM is placed at the input plane of a 4f imaging system with the random diffuser placed at the Fourier plane. In that way, the desired complex field is retrieved at the output plane of the 4f imaging system. As one can notice, this method is quite similar to phase conjugated techniques. Some advantages of this method are the suppression of speckle noise and a high diffraction efficiency. As the encoded pattern lacks of the abrupt phase discontinuities typical of other methods, most of the light is confined to a single diffraction order and there is no need to filter out other orders. There is a compromise between the quality of the reconstruction and the pixel size of the object. The higher the ratio between the spatial resolution of the diffuser and the desired complex object, the greater the quality of the reconstruction. In order to increase that ratio, it is possible to reduce the resolution of the target object, despite this is generally not desirable.

A theoretical exact solution was proposed by Bolduc et al. [101]. However, computational cost was so intensive that, in practice, they needed an approximation for experimental realization. This finally results in an adjustment of [78], with an additional phase term, allowing the method to control both phase and amplitude. Nevertheless, this adjustment no longer provides an exact solution. More recently, a random technique was employed

to sample two functions [102]. The first function contains the phase of the desired complex field, whereas the second one is a diverging optical element aimed to control the amplitude by redirecting undesired light out of the optical axis. Two complementary random binary patterns are generated to choose, for each pixel, which function is encoded. This technique does not require iterative algorithms or computational costly calculus making it suitable for certain applications, such as visual optics [103].