The capability of a twisted nematic liquid crystal display to generate a set of equi-azimuth polarization states is used to achieve a phaseonly modulation regime. For this purpose, a liquid crystal display followed by a quarter-wave plate is launched between two polarizers. Theoretical support is provided by means of the Jones matrix calculus and the Poincaré sphere representation. Laboratory results for a commercial liquid crystal display are presented. A phase modulation deep of 270° is obtained at 514 nm with a residual intensity variation which is lower than 2.5 %.
©2006 Optical Society of America
Twisted nematic liquid crystal displays (TNLCDs) are programmable pixelated devices with an intrinsic capability of modulating spatially a light beam. They have been widely used in many optical applications such as the implementation of active lenses , the design of programmable axicons  or the generation of optical tweezers [3, 4].
It is known that, in general, the twisted nematic structure changes the polarization state of input light . As a result, a TNLCD inserted between two polarizers produces a coupled intensity and phase modulation. However, many applications require a phase-only response. Although parallel aligned nematic displays are suitable for this application , extensive efforts have been devoted to optimize the phase-modulation capability of TNLCDs due to their ease of availability and low cost.
The first thick TNLCDs could achieve pure phase modulation by simply sandwiching them between two polarizers properly oriented and keeping the applied voltage below the optical threshold (at which disruption of the twisted structure occurs) . This technique has to be revised because of the change of modulation capabilities of progressively thinner commercial TNLCDs. Several authors have proposed the use of elliptically polarized rotated eigenvectors, which propagates unchanged, except for a rotation, through the liquid crystal cell [8, 9]. This allows one to overcome the inconvenience of polarization modulation which is inherent to the twisted alignment. The eigenvalues associated with these polarization states are purely phase factors and the total phase shift is a function of the voltage addressed to the cell. In order to generate and detect the eigenvectors, the classical polarizer-TNLCD-analyzer configuration is substituted by a polarimetric arrangement with two polarizers and two quarter wave plates. The main drawback of this procedure is the voltage dependence of the rotated eigenvectors, since it forces to define an average eigenvector over the entire range of applied voltages.
An alternative approach to achieve a phase-only TNLCD modulation response is the use of numerical simulations to find the optimal input and output polarization configurations. For this purpose a reliable model to describe the optical behavior of the liquid crystal display in the on-state is required. Lu and Saleh proposed a simple mathematical model based on the Jones matrix calculus . However, it is founded on several assumptions that can not be applied to the current flat TNLCDs. One solution is to perform an experimental calibration of the TNLCD by means of intensity measurements . In this approach the Jones matrix of a TNLCD is determined for each value of applied voltage. In this context, we have recently presented a new description of an on-state TNLCD. It is based on the equivalence between a twisted nematic liquid crystal cell and a system consisting of one retardation plate and one rotator [12, 13]. The equivalent system is completely characterized by two parameters, namely, the equivalent phase retardation and the equivalent rotation angle, which can be determined experimentally for each addressed voltage . In contrast to Jones matrix elements, the equivalent parameters have a simple physical meaning. Moreover, the action of a TNLCD over an arbitrary input polarization state can be interpreted as two successive rotations on the Poincaré sphere, so that modulation properties of a liquid crystal cell can be easily predicted. In particular, we have shown that a TNLCD launched between a polarizer and a quarter-wave plate can generate a set of equi-azimuth polarization states (EAPSs) whose ellipticity depends on the applied voltage .
In this paper, we demonstrate that from a set of EAPSs it is possible to achieve phase-only modulation with the aid of a properly oriented analyzer. Note that the present method does not try to minimize the change in the polarization ellipse inherent to the twisted nematic alignment. Instead, polarization state modulation is just used to obtain a phase-only response. Our approach provides two clear advantages. First, the residual intensity variations in the optimal configuration, which deteriorate the display performance as phase-only modulator, are significantly reduced. Second, and in contrast to other previously reported techniques, the design of an EAPS generator is carried out by a simple numerical simulation based on the equivalent reatarder-rotator approach .
This article has been organized as follows. In Sec. 2 we demonstrate that phase-only modulation can be achieved with a set of EAPSs passing through a properly oriented polarizer. In Sec. 3 we show that a commercial TNLCD can act as an EAPS generator when it is sandwiched between a polarizer and a quarter-wave plate. In order to describe the optical behavior of the liquid crystal cells we use the equivalent retarder-rotator model. In Sec. 4 we complete the optical system presented in Sec. 3 with the introduction of an analyzer to achieve the phase-only modulation regime. In Sec. 5 a simple method based on the Talbot effect is used to measure the phase response. The operation curve corresponding to a commercial display is presented. Finally, conclusions are summarized in Sec. 6.
2. Equi-azimuth polarization states
It is known that any elliptical polarization state can be characterized by the azimuth angle, α, and the ellipticity angle, ε, which are the parameters that describe the orientation and shape of the polarization ellipse. The azimuth α is the angle between the long axis of the polarization ellipse and the Cartesian x-axis. The ellipticity angle ε is defined so that tanε=b a, where a and b are, respectively, the length of the semi-major and semi-minor axes of the ellipse. The normalized Jones vector of an elliptical polarization state in terms of α and ε is given by 
where δ is the absolute phase determining the angle between the electric vector and the major axis of the ellipse at the initial time. Note that the Jones representation in terms of α and ε, shown in Eq. (1), is directly connected with the geometrical representation of the polarization states in the Poincaré sphere. On this surface, the spherical coordinates of any point can be written as (1, 2α, 2ε). This fact becomes an efficient tool to understand by simple geometrical arguments the results obtained with the Jones matrix calculus.
Now, let us consider a set of polarization states with the same azimuth 0 α and variable ellipticity angle ε, i.e., a family of polarization ellipses with a fixed orientation and variable shape. In the Poincaré sphere these states are located along a meridian line (equi-azimuth contour) . If a light beam in one of such polarization states passes through a polarizer with its axis oriented at an angle ξ with respect to the x-direction, the output Jones vector E out, expressed in the polarizer framework, can be calculated by the matrix product
where P(ξ) is the matrix of an ideal linear polarizer in the above framework. The transmitted intensity of the output light beam is given by
Inspection of Eq. (2) shows that if
then E out is found to be
Moreover, the transmittance T becomes independent of ε and equal to 1/2. The constant value of T under the condition given by Eq. (4) can be easily interpreted in geometrical terms with the aid of the Poincaré sphere. In this representation, the intensity transmitted by a polarizer is simply given by cos2(γ/2), where γ is the angle between the point that represents the incident state of polarization and the polarizer axis subtended from the centre of the sphere . Therefore, all points that are located on a meridian circle that is normal to the polarizer axis yield to the same intensity, since for all of them γ=π 2. This is precisely the situation achieved when Eq. (4) is verified (we recall that the angles of the polarization ellipse are doubled on the Poincaré sphere). Concerning Eq. (5), note that the global phase factor exp[j(±ε+δ)] produces different phase shifts in the states distributed along the meridian line defined by the azimuth α 0. This fact can only be recognized by means of the Jones calculus, since polarization states that differ in a global phase factor are represented by the same point in the Poincaré sphere.
3. Generation of equi-azimuth polarization states with a liquid crystal display
The optical behavior of a twisted nematic liquid crystal cell is entirely equivalent to that of a system consisting of a retardation plate followed by a rotator. The properties of the equivalent system are fully characterized by the equivalent retardation angle δeq and the equivalent rotator angle φeq.. With this approach, the Jones matrix of the liquid crystal cell, written in the frame with the x-axis in the direction of the input molecular director, is 
where R is the rotation matrix, WP is the Jones matrix of a retarder, φ is the molecular twist, and β is the birefringence of the material. In the on-state, the equivalent parameters δeq and φeq (and of course the birefringence β) become functions of the addressed gray-level, g. An easily implemented polarimetric procedure can be used to measure experimentally the equivalent parameters δeq and φeq for each value of g. Figure l shows the calibration curves δeq(g) and φeq(g) that where obtained in Ref.  for a wavelenght of 514 nm corresponding to a commercial display, Sony LCX016AL. The twist angle of this display is φ=-1.594±0.03 rad. This value was measured with the single-wavelength polarimetric method presented in Ref. .
We note that for low gray-levels both equivalent parameters δeq and φeq tend to zero. This means that M TNLCD is close to the identity matrix. Therefore, a zero gray-level corresponds to the maximum operating voltage (in this situation, the bulk liquid crystal directors are aligned along the electric field direction). Concerning high gray-levels, Fig. 1 shows that δeq progressively decreases for low operating voltages. A zero value of δeq for g=255 indicates that the Gooch-Tarry minimum-condition have been reached . Therefore, the TNLCD should become a pure optical rotator.
The equivalent retarder-rotator approach is an efficient tool for the design of LC-based devices. From Eq. (6), the action of a liquid crystal cell over an input polarization state can be visualized on the Poincaré sphere as two successive rotations, one corresponding to the equivalent retardation plate and the other to the equivalent rotator. The final polarization state depends on the values of the equivalent parameters δeq and φeq and, therefore, on the addressed gray-level. In this way, the input state is mapped into a set of polarized states distributed on the sphere surface. We have shown in a previous paper  that it is possible to find an input linearly polarized state that is projected by the LCD onto a set of states located on a circle on the Poincaré sphere. Then, a properly oriented quarter wave plate transforms these polarizations states on a circle onto outgoing states on a meridian line. The complete optical setup is shown in Fig. 2.
When the axis of the polarizer and the slow axis of the quarter-wave plate are oriented at angles θ 1=-28° and θ 2=16°, respectively, the Sony LCX016AL display generates a set of quasi EAPSs, as was demonstrated in Ref. . The final points are located around the meridian α 0=-2° as shown in Fig. 3.
4. Phase-mostly modulation
Let us consider an EAPS generator followed by an analyzer. The electric field transmitted through the complete optical system shown in Fig. 4, expressed in the analyzer framework, is given by the Jones matrix product
In Eqs. (7–9), θ 3 denotes the angle between the transmission axis of the analyzer and the input molecular director. The Jones vector in Eq. (7), which corresponds to a light beam polarized in the direction of the transmission axis of the analyzer, can be rewritten as
where T is the normalized transmitted intensity,
and σ is the total phase shift,
When the values of θ 1, θ 2 and θ 3 are fixed, the transmitted intensity T and the second term of the total phase-shift σ become functions of the addressed gray-level g through the set of values taken by the equivalent parameters δeq and φeq, see Eqs. (8) and (9). Concerning Eq. (12), note that β is an increasing function of the gray-level that can range from a zero value at g=0 to a maximum value, β max, corresponding to the off-state .
Now, let us face the problem of achieving a phase-mostly modulation. In Sec. 3, we have shown that the TNLCD behaves as an equi-azimuth generator when θ 1=-28° and θ=16°. In such configuration, the polarization states emerging from the quarter-wave plate have an azimuth of about -2°. Taking into account Eq. (4), the angle of the analyzer has to be set at θ 3=43° or θ 3=-47° for obtaining a pure phase response. For our analysis, the best choice is θ=-47° since for this value the two terms of Eq. (12) add with the same sign yielding to a maximum phase shift.
5. Experimental verification
In order to measure the phase modulation of the TNLCD we have used a method based on the fractional-Talbot effect [17–20]. Let us consider a binary phase grating of period d and phase step height Δφ illuminated with a parallel light beam of wavelength λ. It can be proved that the phase modulation of the binary phase grating is converted into a binary intensity pattern at fractional-Talbot planes located at distances
from the phase grating, where D=2d 2/λ is the Talbot distance and m is an integer. The visibility of such intensity patterns is closely related to the phase step height through the equation 
This property of Fresnel images (which reveals their intrinsically interferometric character) can be used to measure the phase modulation of a TNLCD. In this case, gratings with two gray levels, one fixed (typically g=0) and the other variable, are successively implemented with the liquid crystal display. By measuring the contrast of the Fresnel images at a quarter of the Talbot distance D, the relative phase shift Δφ(g)=φ(g)-φ(0) can be determined. Provided that in the phase-mostly configuration there is a residual amplitude transmittance,, Eq. (14) must be conveniently rewritten to take into account this fact, giving 
With the theoretical foundation just given, experimental results have been obtained in the optimal configuration (θ 1=-28°, θ 2=16°, θ 3=-47°) with our commercial display. The gray-level g was varied in 8-level steps, from 0 to 255. The display was illuminated by a collimated Ar/Kr laser beam with λ=514 nm. The period of the binary grating addressed to the TNLCD was of 16 pixels (i.e., d=0.512 mm), for which the Talbot distance is D=1.02 m. The intensity patterns located at a quarter of the Talbot distance were recorded by a CCD camera. To achieve the maximum phase shift, the contrast of the display controller was set at its maximum value. Figure 5 shows the Fresnel images for several values of g. With regard to the residual amplitude modulation, the normalized transmitted intensity T(g) was obtained by displaying uniform images for each gray-level on the TNLCD and measuring the transmitted intensity I 1(g) with a photometer. Normalization was performed by measuring the intensity I 2(g) corresponding to θ 3=43° (perpendicular configuration) and evaluating the ratio
Figure 6a shows a plot of the phase-shift, Δφ, versus the gray-level, g, while Fig. 6b shows the operation curve, i.e., the transmitted complex amplitude √T exp(-iΔφ of our polarimetric arrangement. We can see that the results are close to a pure phase modulation regime. Note that the dependence of Δφ with the gray-level is clearly non-linear. The maximum phase-modulation depth is slightly greater than 3π/2 and the coupled intensity
modulation is minor than 2.5%. As is expected, the values of T(g) oscillates around 0.5, accordingly with Eqs. (3) and (4). Concerning the tolerance of the optimal configuration to variations in the orientations of polarizing elements, we found experimentally that a misalignment less than ±0.5° does not yield to a sensitive alteration of the operation curve. A detailed study of the robustness of the modulation regimes of a TNLCD inserted between two polarizers and two retarders can be found in Ref .
In summary, we have presented a method for achieving a phase-only modulation with a TNLCD. It is based on the generation of equi-azimuth polarization states by a TNLCD sandwiched between a polarizer and a quarter-wave plate. We have shown that this polarimetric arrangement must be completed with a properly oriented analyzer to obtain a phase-mostly modulation regime. This fact is demonstrated by a simple reasoning based on the Jones calculus. Experimental results have been presented for a commercial display. In order to measure the phase-modulation we have used the fractional-Talbot method. The operation curve provided by our TNLCD evidence the validity of the presented approach.
This research was funded by the Dirección General de Investigación Científica y Técnica, Spain, under the project FIS2004-02404 and FEDER. We also acknowledge financial support from the Generalitat Valenciana (grant GV05/110).
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