1. Introduction

Spatial light modulators (SLMs) are increasingly being used in coherence optics, especially for shaping the spatial or temporal profiles of laser beams in diverse applications ranging from the creation and control of optical tweezers [1] to coherent control of quantum dynamics [2]. The usefulness of an SLM depends on its ability to produce a specific amplitude and/or phase modulation independently in each addressable pixel. For SLMs based on liquid crystals the actual modulation that can be achieved depends on the polarizations of the incident and transmitted beams. Normally these dependences are not precisely known a priori, requiring the exacting user to perform a calibration procedure.

Twisted-nematic liquid crystal displays (TN-LCDs) are the commonest SLM but they suffer from a strong amplitude-phase coupling [3]. To produce a desired modulation for a given wavelength one typically needs to carefully optimize the input and output polarizations states. The calibration of an TN-LCD for a given wavelength is a two-step procedure: (1) a polarimetric characterization is done by fitting the measured intensity transmission, for specific polarizations, to an assumed Jones [4] or Mueller [5] matrix; (2) the polarimetric matrix is then used to predict the modulation for arbitrary polarizations and numerical optimization techniques are employed to achieve a desired modulation curve [6].

The optimized complex modulation curve has to be verified either by interferometric techniques, such as the Mach-Zender [3] and the Young double-slit interferometer [7], or by diffraction-based methods [8]. While diffractive techniques measure simultaneously the amplitude and phase modulation, most interferometric techniques are best suited for phase measurements, being the amplitude modulation measured separately with a photometer. Nevertheless, all commonly used techniques measure a single modulation level at time. A complete measure of a modulation curve thus requires a series of time consuming sequential measurements.

An interferometric technique based on an a shear plate was proposed [9] to measure phase modulation. Here we extend such technique by including an imaging system which enables one to simultaneously display phase and amplitude modulations for several modulation levels in the same interferogram. This ability allows one to directly optimize the performance of a TN-LCD, by visually monitoring the interferogram while manually varying the optical elements that determine the input and output polarization states. The capability to simultaneously observe a full range of amplitude and phase modulations allows one to avoid the time consuming process of polarimetric characterization together with the subsequent numerical optimization. In particular, it becomes possible to quickly identify and measure mostly-phase or -amplitude TN-LCD modes of operation.

2. Measuring setup

2.1 Shear imaging interferometer

We assume that the SLM is illuminated by a laser beam with a Gaussian spatial profile whose beam waist can be adjusted relative to the SLM exit plane. This beam is then reflected, as shown in Fig. 1 , at an incident angle θ by a parallel plate of thickness t and refractive index n. Two replica beams are formed, laterally sheared by the amount

 

Fig. 1 Shear interferometer with imaging. Two replica images of the SLM, laterally sheared s’, are superposed on the camera. If the SLM is illuminated by a Gaussian beam with radius of curvature R, interference fringes are formed in the image plane with a period given by Eq. (2).

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Consider first a laser beam that passes without modulation by the SLM. The imaging lens will produce an image of the interference of two virtual spherical wavefronts with radius of curvature R (determined by the position of the laser beam waist and its collimation) generated at two virtual sources deviated laterally by the shear s. This will result in interference fringes in the image plane, with period given by

The shear plate is aligned for shearing along the SLM horizontal direction while obtaining fringes in the vertical direction. The optical quality of the shear plate is essential to achieve straight and evenly spaced fringes. We used a glass parallel plate of interferometric quality (λ/10) with thickness t = 5 mm and at an incidence angle of ≈45°.

2.2 Amplitude-phase measurements

In order to display a full set of SLM modulation characteristics using the shear interferometer described in the preceding section a particular modulation pattern is written onto the SLM, as depicted in Fig. 2 . This pattern consists of a vertically graded modulation over a region in the left horizontal part x < 0, with complex transmission T(y); and a constant modulation in the right horizontal part x > 0, with uniform transmission T ₀. Furthermore, a vertical edge mask is inserted in front of the SLM exit pupil, as shown in the same figure.

 

Fig. 2 The SLM is modulated with a vertically graded modulation pattern T, surrounded by a reference uniform modulation T ₀. An edge barrier is inserted laterally leaving a modulated area with width equal to the shear s. The shear interferometer (Fig. 1) superposes the T and T ₀ regions, producing interference fringes whose relative shift measure phase modulation. The edge barrier blocks interference fringes in region T, revealing amplitude modulation.

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The shear interferometer system (Fig. 1) superposes the images of the SLM pattern and edge with replica images shifted horizontally by s’. The resulting interferogram will be divided in two regions: (i) the region -s’ < x’ < 0, where the sheared image of the edge superposes the graded modulated area T, shows intensity modulation ∝|T|²; (ii) the region 0 < x’ < s’ were the sheared image of the modulated area is superimposed with the reference area. This area contains interference fringes, with spacing d_f given by Eq. (2) and horizontal shifts Δx’, which are a direct measure of phase modulation ΔΦ/2π = Δx’/d _f. Use of the edge barrier enables amplitude modulation to be seen directly and also hides ghost images from multiple reflections between the shear plate front and rear surfaces.

2.3 Spatial filtering

The pixelated structure of the SLM may produce Moiré patterns when two sheared images are superposed, or when the imaged SLM pixel pitch is close to the camera pixel pitch: such effects obscure the interferograms. To depixelate the images we performed a spatial filtering, by placing a pinhole at the center of the back focal plane of the imaging lens. The size of the pinhole is chosen to cut diffraction angles larger than the width of the 0th order diffraction pattern. For a pixel pitch Δ and an imaging lens with focal length f at a wavelength λ, the pinhole diameter is λf / Δ. Notice that after spatial filtering we are actually measuring the modulation of entire pixels, including the passive part around the active area that may contribute to the total modulation with a nonzero constant value.

A finite thickness of the parallel plate normally imposes a different focus for the two reflected images; this relative defocus is tolerable if it is of the order of the depth of focus of the imaging system. After spatial filtering the imaging resolution equals the pixel pitch and the depth of focus equals the corresponding Rayleigh range, Δ ²/λ, which is typically in the millimeter range.

3. Optimization of a TN-LCD

3.1 Modulator setup

Figure 3 shows a diagram of a spatial light modulator implemented with a TN-LCD (Sony LCX016AL-6). The TN-LCD is placed between a polarization state generator and a polarization state detector, both consisting of an association between a linear polarizer (an association between a polarizing cube and half-wave plate) and a quarter-wave plate. This configuration allows the selection of arbitrary elliptical polarization states and gives total freedom to calibrate the TN-LCD modulator [10].

 

Fig. 3 The SLM consists of a TN-LC display between elliptical polarization generator and detector. By adjusting the collimating lens of the beam expander it is possible to vary the radius of curvature of the incoming incident beam, therefore varying the fringe spacing displayed in the shearing interferometer (Fig. 1).

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The entrance polarizing cube makes the TN-LCD input power independent of the generated polarization state whereas the exit polarizing cube makes the reflection of the shear plate surfaces independent of the detected polarization state. The beam exiting the modulator is picked off by a shear plate used for the interferometer depicted in Fig. 1. The shear plate could be positioned on a flip mount allowing an in situ use of the interferometer. The laser beam (from an Argon ion laser at 514 nm) was spatially filtered and expanded to a diameter larger then the shearing s = 3.8 mm, determined by Eq. (1). The collimator was then adjusted for a fringe period of d _f ≈0.4 mm, corresponding, by Eq. (2), to |R| ≈3m (imaging magnification m ≈1). To ensure the LCD is being imaged, instead of one of its Talbot self images, focusing was done with an irregular pattern applied to the modulator.

Particular care must be taken to avoid aberrations of the expanded beam entering the SLM which distort the interference fringes. It is especially important to minimize spherical aberration that will cause barrel or pincushion distortions, depending on whether the beam is convergent or divergent (sign of R). For that reason we used aberration corrected doublet lenses in the beam expander.

3.2 Interferograms

Figure 4 shows an illustrative interferogram obtained with the shearing interferometer (Fig. 1) for the SLM based on the TN-LCD (Fig. 3) and using a modulation pattern such as described in Fig. 2. The various levels of gray in the left part are a direct measure of amplitude modulation whereas the right part displays interference fringes whose relative shifts are a direct measure of phase modulation. After spatial filtering, LCD pixel resolution (32 micron) was nearly achieved and the imaging deep of focus was ≈2 mm, about the distance between the edge mask (a metallic sheet) and the LCD panel.

 

Fig. 4 Interferogram obtained with the shear interferometer for a TN-LCD. Left half part displays amplitude modulation whereas right part displays phase modulation. For illustration only four levels of modulation are used. The lateral shear imposed by the parallel plate is illustrated by the twin image of the computer mouse pointer. (Media 1 for a detailed view.)

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For a quantitative measurement of the modulation curves the interferograms were digitally analyzed. A region of interest in the digital image, corresponding to a known set of modulation levels, was scanned line by line. For the amplitude part of the image, integration directly yielded the transmission of the modulator, square of the amplitude modulation. For phase measurements, fringes were analyzed by the Fourier phase shifting determination method [11].

In our method we assume the modulation characteristics are spatially invariant, an implicit assumption also made in other characterization methods, e.g [3,7,8]. Since TN-LCDs are mostly used in video displays, they are expected to have high modulation uniformity. Nevertheless non flatness of the device component layers may cause localized aberrations that invalidate the spatial invariance property. To compensate the effect of such aberrations, as well as possible aberrations or nonuniformity of the illumination beam, a reference interferogram, i.e. one obtained with no modulation applied to the TN-LCD, was used. Reference phase values were subtracted from the measured ones to correct distortion of interference fringes. Amplitude measurements were corrected by division with the corresponding region in the reference interferogram. A dark image was subtracted from both measuring and reference interferograms to remove any background that would limit the contrast ratio.

3.3 Modulation modes

The majority of SLM applications require a mostly-phase (minimum amplitude coupling) or mostly-amplitude (minimum phase coupling) modes of operation. These modes of operation can be readily identified by visual inspection of the interferograms while manually varying the input and output polarizations [see Fig. 5(a) ]. Even with four degrees of freedom (four wave plates’ rotations) we found that it was possible to obtain either mostly-amplitude or -phase modes in a few minutes. Later analysis of the interferograms gave the corresponding modulation curves [Fig. 5(b)].

 

Fig. 5 Mostly-phase modulation obtained for the TN-LCD at a wavelength of 514 nm. (a) Interferogram for 64 levels of modulation [uniformly covering 0-255 gray levels] optimized visually for maximum phase variation and minimum amplitude variation; (b) corresponding modulation curve obtained afterwards by analysis of the optimized interferogram.

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Our optimized modulation curve for mostly-phase operation at 514 nm has a phase depth of 258° and an amplitude coupling of nearly 10%. Others before [12] have optimized the same device at a near wavelength (532 nm) by performing a Jones matrix characterization followed by a numerical search for a mostly-phase operation. Using a diffraction-based modulation measuring technique they found a phase depth of 207° and an amplitude coupling of 5%.

4. Conclusion

We have demonstrated a new technique to measure the complex modulation of a spatial light modulator based on a shearing interferometer with imaging. The interferometer superposes adjacent areas of the modulator with variable modulation levels, enabling the simultaneous observation of amplitude and phase modulation for several different modulation levels to be observed in a single interferogram. We applied the shear interferometer to the calibration of a TN-LCD device. Direct observation of amplitude and phase modulation curves permitted a quick manual optimization of the elliptical input and output polarization states leading to a mostly–phase operating mode that is characteristic for this device. For a quantitative characterization of the optimized modulation, we used a standard Fourier fringe analysis method.

The distinctive feature of the proposed interferometer technique is its ability to visualize the complete operation curves instantaneously. This feature allows a heuristic optimization of operating modes, without the need for a polarimetric characterization and numerical optimization of input and output polarizations. This reduces tremendously the effort needed to test and optimize the performance of a modulator under alternative configurations, particularly of the TN-LCD type. Independently of the use we have made of our proposed interferometer to optimize an TN-LCD device, the same interferometer could be used for a full polarimetric characterization of any type of modulator, reducing drastically the number of interferograms needed with other interferometric or diffraction measuring techniques. In addition, using a flip mount for the shearing plate, the proposed interferometer can be added to any application setup allowing an in situ calibration system.