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A tunable aberration compensation device for rectangular micro-optical systems is proposed and demonstrated. This device, which is based in nematic liquid crystal and a micro-electrode structure, forms gradients in the index of refraction as a function of voltage. We have developed a fringe skeletonizing application in order to extract the 3D wavefront from an interference pattern. This software tool obtains the optical aberrations using Chebyshev polynomials. By using phase shifted electrical signals the aberrations can be controlled independently. A complete independent control over the spherical and coma aberration has been demonstrated. Also, an independent control over the astigmatism aberration has been demonstrated in a broad range. This device has promising applications where aberration compensation is required. The independent compensation achieved for some coefficients, such as astigmatism for example, is more than 2.4 waves.
© 2015 Optical Society of America
1. Introduction
Adaptive optics seeks to compensate optical aberrations of wavefronts via devices that transform the phase of the incoming light: in a first stage, measuring the aberrated wavefront compared to an ideal pattern; afterwards, correcting the optical aberration. Initially, phase and amplitude corrections using this technique were tackled in astronomy and military applications. However, the growing interest in many other fields has contributed to the maturing of adaptive optics with new materials, methods and procedures. Particularly, devices based on liquid crystal (LC) technology have been proposed for adaptive optics in many research fields. The relevance of LC technology arises from tuning capability without moving parts, small size, light weight, low driving voltage and low power consumption of the structures and devices filled with them.
A LC spatial light modulator (SLM) was included in the experimental arrangement used by G. D. Love, as one of the first proposals to implement a LC device as an aberration corrector by producing different Zernike modes [1]. A few years later, A. F. Naumov characterized the aberrations in spherical LC lenses [2], in terms of rms error of the parabolic approximation. This author in collaboration with G. D. Love proposed a method to reduce aberrations by using different harmonics in the control voltage [3]. Some years later these aberrations were measured by T. Takahashi et al. [4] using a commercial device based on Zernike coefficients. Among the applications to implement a LC spherical lens operating as an aberration compensator, Knittel et al. proposed the spherical aberration compensation in a blue-ray disc system [5]. And, most recently, C. Hsieh et. al have implemented a distortion aberration compensation device using a spherical lens array placed at the intermediate image plane of an optical system [6]. These aberration compensators are always based on spherical lenses that can be modeled by Zernike coefficients.
Monochromatic aberrations are caused by the geometry of the lens. Since P. L. Seidel defined the five main aberrations (Spherical Aberration, Coma, Astigmatism, Curvature of Field and Distortion), this classification has formed the basis of reference to the categorization of lens quality as defined by optical designers. Interferometrists often like to represent wavefronts as Zernike polynomials. This mathematical tool has been widely used because, unlike other polynomial set, they are orthogonal over a unit circle and represents balanced classical optical aberrations, yielding minimum variance over a circular pupil [7]. These characteristics are lost when non circular pupils are employed. Some schemes for square pupils have obtained square polynomials from the orthogonalization of circle polynomials [8], using a circle inside the square. This procedure has the disadvantage of losing the part of the wavefront that falls outside the circle. Some other studies overcome this problem by defining a unit square inside the circle [9]. In those studies, closed-form expressions for the low-order polynomials are used. These expressions are orthonormal over a determined pupil by using the recursive Gran-Smidt orthogonalization process [10]. Among the advantages of the last technique, it is the correspondence between polynomials’ coefficients and the peak value of each aberration. Besides, it can also be employed for different types of pupils [11]. Despite this, there are some drawbacks in order to evaluate optical systems with rectangular or square pupil. Zernike coefficients can be obtained from the orthonormal coefficients but these ones are really designed to use it on rotational symmetric optics. They do not represent balanced classical aberrations and do not have the physical significance as those in the aberrations of rectangular optics. To overcome the last drawback, Liu et al. have developed a complete study by introducing 2D Chebyshev polynomials (related to some Seidel aberrations), instead of Zernike polynomials to characterize optical rectangular apertures [12]. The main reason to employ this type of polynomial, is the rectangular correspondence between polar conformal mapping of Zernike and Chebyshev forms. In the referred work, a Schmidt corrector plate is characterized. Another interesting applications could be other types of lenses with non-rotational symmetry, the cylindrical ones. Functionality of the cylindrical lens is basically focused on beam focusing or expanding in one direction or pointing at a monochromator slit or a linear detector array. Current fields of application are: laser beam shaping with microlens arrays [13], uniform line pattern generation [14] and 3D imaging [15]. Similarly to spherical lenses, cylindrical lenses based on LC technology share their applications with conventional fixed lenses, so that some research works have being reported in optical interconnection, beam steering control [16] or imaging in autostereoscopic displays [17]. Despite all applications involved, a study concerned with the optical quality of LC cylindrical lenses cannot be found in the literature. Only one LC lens with square aperture has been reported in terms of Zernike coefficients [18], with the errors that the use of Mahajan coefficients brings.
In this work, we present the use of an adaptive LC phase-only array as an aberration corrector. The scheme proposed for the array is based on a striped pattern electrode, with hole-patterned manufacturing technology, where a gradual voltage across each micro-optical element is generated. In some cases, this voltage profile is capable of reproducing a parabolic refraction index gradient in the LC layer like a graded index, (GRIN) element, so mimicking the optical behavior of a conventional microlens. In others, a phase profile with capability to correct different aberrations is obtained. The tunable array has been designed and fabricated following a theory to avoid phase defects proposed by the authors [19]. They have previously reported the versatility of structures with comb electrodes for phase modulation in LC devices. Phase tunability [20], comparable performance of LC and conventional fixed microlenses [21] and capability of reproducing 3D vision [22], are some of the more distinctive features of this configuration. A key issue identified in the present work is how advantageous unique driving is for the implementation and feasibility of the aberration corrector. Traditionally, phase modulation for aberration compensation has been made point to point of the wavefront with multiple electrodes and driving signals (like in a SLM). Our lenticular arrays are addressed with two different electrical signals applied to the comb electrodes. This kind of striped electrode structures can be focused on aberration compensation of 2D wavefronts for lenticular cylindrical arrays in many different potential research works and industrial applications: Cylindrical lens arrays may be used as a device for measuring highly aberrated wavefronts where two orthogonal line patterns are detected on a CCD and are superimposed [24]; they are very useful to control the 3D distance in autostereoscopic displays [22]; or, their tunable focal length capability can be exploited to collimate effectively the beam emitted from a stacked laser diode [25]. The device could also be very useful for optical elements with micrometric size and rectangular aperture when each element of the array is used independently (e.g. light-sheet microscopy [26]).
A lenticular device has been designed and manufactured. Aberrations of the mico-optical elements, including all of the classical coefficients, have been modeled and measured emphasizing their voltage dependence. An important correction for the astigmatic aberration has been observed. This is very useful for some applications as autostereoscopic vision based on spatial multiplexing. For the first time of our knowledge, we report a compensation device for aberrations in an arrays of cylindrical lenses or micro-optical systems with square aperture, specifically manufactured with liquid crystal technology. We have developed specific software combining a fringe skeletonizing technique [27] to extract the 3D phase map and, also, an aberration estimation algorithm based on Chebyshev polynomials.
2. Liquid crystal cylindrical micro-optical array
A cylindrical micro-optical array has been proposed as a wavefront corrector. The device is a one dimensional array of adaptive LC phase-only passive elements where each micro-optical element of the array mimics the effect of a GRIN element (Fig. 1). The proposed device comprises two ITO coated substrates (20 Ω/sq). One of them has a patterned electrode. The other one is a continuous electrode. The patterned electrode consists of a pair of comb-type finger electrodes. These two electrodes are connected to different electrical signals (V1 and V2). The common ground is connected to the other substrate.
Fig. 1 Tunable liquid crystal one dimensional array of cylindrical micro-optical elements for aberration compensation. (a) Designed mask, (b) zoom over three electrodes and (c) 3D representation.
The device has been designed and implemented. The application of the aberration compensation device is focused on an autostereoscopic device with spatial multiplexing and a microlens sheet of 100lpi (lens per inch). The electrode configuration allow to generate also a 200lpi device. In the manufacturing process, a special chrome mask is required and the high resolution photolithographic process has been optimized. The control electrode of the array is based on a configuration hole-patterned: LC (MDA 98-1602 from Merck) is sandwiched between two glasses; the ITO electrode on the upper glass is designed with a pattern mask comb-shaped and the ITO electrode on the lower glass works like a continuous counter electrode (ground plane). To achieve a proper phase profile with minima defects, the device must meet a specific ratio d/t = diameter/thickness [19]. This requirement prevents the wavefront from shrinking effect of the active area near the edges and from the growing of a flat-like phase in the center. A ratio d/t = 2.4 is necessary to satisfy this condition. Considering that a 100lpi array has micro-optical elements of 254 µm in diameter, the device thickness has to be 105 µm. The available materials allowed us to manufacture devices with maximum thickness of 100 μm (using Mylar sheets). In the case of 200lpi (127 µm in diameter), the thickness is 50 µm. The resulting cavity is filled with the LC. The LC is used as an adaptive phase-only modulator so no twist of the molecules is necessary. For this reason, a planar alignment is used. A polyimide (PIA2000) is deposited over the two substrates. This layer acts as an alignment layer when it is rubbed in the direction of the electrodes for the upper substrate and in antiparallel direction for the other one. The LC molecules align parallel to the direction indicated in Fig. 1. A Cartesian coordinate system is defined, where the x-axis is pointing in the direction of the tooth of the comb and the y-axis in the transverse direction. The light propagates along the z-axis.
3. Theoretical background and software development
As commented above, there is not much literature about aberrations in rectangular apertures. The use of 2D Chebyshev polynomials (related to some Seidel aberrations), instead of Zernike polynomials to characterize optical rectangular apertures could be the solution for optical systems with non-rotational symmetry.
3.1. Theoretical fundamentals of aberrations in rectangular optical systems
As mentioned above, the wave aberration is typically represented by using a decomposition in Zernike polynomials for square pupils in the analysis of aberrated systems. Particularly, the wave aberration W(x,y) for noncircular pupils (optical rectangular apertures) can be expanded in terms of Chebyshev polynomials that are orthonormal over the pupil [11],
where Fj are the 2D Chebyshev polynomials and Aj, are the aberration coefficients related each to a classical aberration of the wavefront. This expression is also named Chebyshev polynomial fit. Here x and y are the coordinates in the surface of the pupil.The aberration coefficients, Aj, following the property of orthogonality, are estimated by a double integral of the wave aberration multiplied by the Chebyshev functions and divided by a weighting function, see Eq. (2) [12]. The coefficients, Aj, can be determined directly when the wave aberration is a continuous function or, numerically, when the wave aberration is a data set. In this latter case, the most common approach is to use a numerical quadrature routine.
The 2D Chebyshev polynomial set, Fj, is obtained by multiplying 1D classical Chebyshev polynomials by one another; the complete set of 2D polynomials is defined in [12]. The polynomials that have a physical meaning similar to those used by Zernike polynomials, in classic optical systems, are listed in Table 1, in Cartesian representation up to 6th order. Like the Zernike polynomials, they range from −1 to + 1 over a defined interval and have a sinusoidal behavior as interferometric fringe maps. The normalized pupil, on the xy plane covers an area with square shape. For rectangular apertures the series have to be scaled. A normalization constant K, to maintain the orthogonality of the polynomials is also considered. The graphical representation of the polynomial terms Fj are shown in Fig. 2.
Table 1. 2D Chebyshev polynomials set related to classical aberrations [12].
In order to characterize the coefficients of the aberrated optical system, the measured wavefront is fitted to the Chebyshev polynomial set to subsequently derivate the coefficients Aj. The method to relate the measured wave aberration and the Chebyshev polynomial set is described below. Specific software has been programed to solve the problem. Finally, in order to check the measured wave aberration, a particular experimental arrangement has been implemented. It measures the optical wavefront and the aberrations coefficients, Aj, are extracted. In the following section the procedure is explained in detail. The aberration effects on the image can be estimated by studying the optical path length of the LC device.
3.2. Software development
To determine the solution of the system of linear equations, Eq. (2), where the unknown coefficients are Aj, a simple and fast method has been used. The proposed problem is solved using MATLAB software. The algorithm formulates a matrix equation and uses the matrix left division (mldivide) instruction. This operator employs different procedures depending on the coefficient matrix and computes the numerical solution in the least squares sense. The use of digitized interferograms could cause some errors in the final solution, but considering the number of sampling points produced by the CCD, these ones are negligible in comparison with the fitting error. The estimation of this error is described in a following section. In order to evaluate the proposed algorithm and demonstrate the use of a LC device as an aberration compensation device, in rectangular micro-optical systems, a specific device has manufactured and a characterization system has been set up.
4. Wavefront measurement and experimental arrangement
The wavefront surface, W(x,y), for the aberration measurement has been approached through the optical path length (OPL) that light experiences upon transmission through the LC layer. Throughout the process, LC device works in dynamic regime (unidirectional translation of the molecular director perpendicularly to substrates) and it is addressed by an external driving voltage. In a first step, the phase fringe patterns are recorded by means of a CCD sensor and the image intensity is enhanced. Next, 3D phase maps are extracted from the device electro-optic response. A simple method to do this has been carried out. Figure 3 shows a simplified scheme. Next, an imaginary horizontal line (placed in the x-axis direction) is swept in the y-axis direction, that is, from the top to the bottom of the active area. This line superimposed on the interference pattern defines an intensity profile for each position. The corresponding phase profile, ΔΦ(x,y), is generated from every intensity profile; an algorithm has been specifically developed for this task. The values of the phase profiles range from 0 to 255 (each point stored in one byte, 255 levels). Next, the set of phase profiles forms a 3D phase map. In this map, the optical phase shift, ΔΦ(x,y), is measured in units of 2π radians and is referred to a position in Cartesian coordinates (x,y) measured in microns.
The initial interference pattern is the result of placing the LC device between two crossed polarizers. This device produces a gradient in the index of refraction produced by a voltage variation across the surface. The result is a specific birefringence as a function of the position (x,y). The birefringence causes phase differences between ordinary and extraordinary components of light, changing the polarization state of the incoming light. For each position in the active area, there is a determined birefringence and thus a specific phase shift Φ(x,y). In this way, an interference pattern ranging from bright to dark intensities is observed, corresponding to 180° and 0° of phase shift, respectively. Because of its many applications, the studies of this type of interference patterns are numerous.
Until now several techniques have been proposed for phase reconstruction, among them, some of the most important are Fringe Skeletonizing [27], Fourier transform [28], use of wavelets [29], Phase Trackers [30] and Genetic Algorithm [31]. In our experiment, due to the proposed setup between crossed polarizer, the scattering produced by the LC, and the fact than only one light source is used, some algorithms are unfeasible or erroneous. For example, Phase Trackers or Fourier use an additional reference source. On the other hand, the image quality for other algorithms gives some errors. Checking different types of algorithms for the generated images of this specific setup, the Fringe Skeletonizing was found to be the most robust and fast. Although it is one of the oldest techniques, it is one of the most used, because it is faster and proven. This technique is based on the very definition of interference.
Finally, the experimental wavefront surface, W(x,y), can be easily determined from the phase shift measurements, ΔΦ(x,y). For this, the phase shift measurements are converted to optical path length, Δnd(x,y), using Fresnel approximation [Eq. (3)] and the dimensions are normalized,
where Δn(x,y) is the effective birefringence at some specific point (x,y) of the active area, d is the thickness of the LC layer, ΔΦ(x,y) is the phase shift measurement and λ is the wavelength of the incoming light.4.1. Experimental arrangement
Based on the proposed software, the lenticular array is analyzed in terms of the individual elements. For this purpose, a couple of electrodes for each micro-optical element is located and the area of interest is selected (254μm × 254μm). This image of this area is processed and the phase is unwrapped. With the 3D phase maps, aberration coefficients, Aj, are extracted by using Chebyshev polynomials. Also, some of the classical Seidel aberrations are evaluated by changing the applied voltage. Finally, the fit error is also estimated.
The system is placed in an optical table as shown in Fig. 4. The experimental set-up consists of: a He-Ne laser source (632.8nm), neutral density filters, a linear polarizer at 45°, a LC sample with the alignment direction at 90°, a × 10 microscope objective, another polarizer at −45° and a B/W CCD digital camera (effective no. of pixels 1344 × 1024). All the angles are referred to the horizontal direction of the table plane.
Fig. 4 Experimental set-up for characterizing a tunable LC cylindrical micro-optical array for aberration compensation.
Given the micrometric size of each optical element, LC arrays are placed onto a special micropositioner with micrometer control of the step via computer. Furthermore, function of the microscope objective is to be used like an optical zoom that allows visualizing each element in the CCD (x10 and x20, in the case of 100lpi and 200lpi respectively). In order to generate three phase shifted electrical signals, a driving module has been designed and built. It consists of a custom electrical phase shift waveform generator that uses an NI sbRIO-9633 module. As an example, an inset in Fig. 4 shows two signals with 180° of electrical phase shift. This module is an embedded control and acquisition device that integrates a real-time processor and a Xilinx Spartan-6 LX25 FPGA with four 12-bit analog output channels ( ± 10 V). The LC arrays are connected to the FPGA and are switched by 1 kHz square signals. The characterization software is based on a Graphic User Interface (GUI) programmed with MATLAB. This one receives the fringe captures through the CCD. Some examples of the captured images are shown in Fig. 5. Several fringe patterns are shown in order to demonstrate the homogeneity of the sample.
Besides, it has several options for image processing, for example: contrast improvements on fringe images, automatic rotations (when the device is at a specific angle), electrodes detection, area of interest selection, etc. Once the images from the CCD are processed, the program is capable of unwrapping the phase in two or three axis by Fringe Skeletonizing technique [Fig. 6].
Fig. 6 Fringe Skeletonizing technique (a) interference pattern and Skeletons (b) Skeletons taken the mean value of the peaks (c) comparison between captured skeletons and mean value.
There is certain ambiguity in the spatial position of the fringe position. After a fringe skeletonizing by peak detection [Fig. 6(a)] the red lines show the result. The software takes an average across all peaks detected [Fig. 6(b)], this mean value is drawn as a blue line. As can be seen in Fig. 6(c), the differences between the mean lines (blues) and the measured lines (reds) are unappreciable. In the worst case (fringes in the center of the active area),the maximum deviation is 0.5µm.
5. Results and discussion
The results of this work are presented and described in three steps. In the first stage, the result of the 3D phase map extraction, in addition to some important aberrations, are shown. Then we focus the result in the compensation of the astigmatism aberration. Finally, the error of the measurements is compared and discussed.
5.1. Estimation of aberration coefficients
In this section the process of the aberration estimation is shown. In this case, the 100lpi device is studied. The first step executed by the program is the 3D phase map extraction. Phase maps connect the phase shifts in radians with the position (x,y) on the active area. The area between two electrodes is first located by means of the computer program. The x and y axis, which range from 0 to 254 μm, are converted to normalized units (−1 to 1). Resulting phase shift diagrams are extracted from interference patterns of Fig. 7(a); 3D (x, y, z) and 2D (x, y) maps are shown in Fig. 7(a) and 7(b) respectively. These graphs indicate the maximum phase shift as well as the different shapes of the phase profiles for different voltages.
Fig. 7 Experimental wavefront for 100lpi device (normalized), W(x,y). (a) Interference fringes (b) 3D phase maps. (c) 2D (x,y) phase maps.
In the phase maps a complete tunability of the wavefront is observed. As stated previously, there are no shrinking or plateau defects both at the sides and center of the active area, respectively, due to the application of the theory to avoid phase defects proposed by the authors. On the next step, the experimental wavefront surfaces, W(x,y), are estimated from the 3D phase maps, see Eq. (3). For this, the process commented on section 4 is used. The result is the different experimental wavefront surfaces as a function of voltage. After that, these wavefronts are related to the Chebyshev polynomials by Eq. (1). The resultant Chebyshev aberration coefficients, Aj, have been displayed in Fig. 8. Astigmatism coefficient has been suppressed in this figure (because it is very high compared to the others) and is studied individually in the next section.
Fig. 8 Amplitude of 36 Chebyshev aberration coefficients, Aj, for different voltages applied to the LC tunable micro-optical array of 100lpi.
Chebysev coefficient can be both positive and negative, depending on the orientation of the deformation. As can be seen, there are some coefficients whose amplitude stands out from the others. These are the ones corresponding to spherical (A10, A21) and coma (A6, A15) aberrations in the x axis for the primary and secondary terms. Also, the aberrations can be controlled by means of the voltage applied to the LC array. For a better comprehension, each coefficient is shown as a function of voltage on independent graphs (Fig. 9 and Fig. 10).
Fig. 9 Amplitude of spherical aberration coefficients as a function of voltage. (a) Primary spherical X (A10). (b) Secondary spherical X (A21).
Fig. 10 Amplitude of coma aberration coefficients as a function of voltage. (a) Primary coma X (A6). (b) Secondary coma X (A15).
The spherical aberration causes the rays of light coming from different apertures (of different microlenses in the LC array) to focus at different planes, resulting in resolution losses. Unlike spherical lenses, which are symmetric in both axis, cylindrical lenses are symmetric either in x plane or y plane. For this reason, this aberration receives a new type of denomination, spherical X or spherical Y. In this case, due to the position of the LC micro-optical element along Y direction (see Fig. 3), spherical aberration X is considered. There is a tunable range from −0.6 to 0.8 waves. For some specific voltage, this aberration can even be suppressed, achieving an ideal lens.
The coma aberration coefficients, A6 and A15, are shown in Fig. 10. These two coefficients are related to coma; it causes an effect of different focusing of raylights in off-axis directions in an extended lens, causing a wedge shape at the image of a point source. In this case, the coma aberration can be compensated in the amplitude range of −0.4 to 0.6 waves, for voltages from 2 to 2.6Vrms respectively.
The resulting amplitudes of the different aberration coefficients, for the LC cylindrical micro-optical elements, extend from positive to negative values as a function of voltage applied to the LC device. This versatile tunability of the coefficients suggests, at least in principle, the very high potential of these devices to be used as aberration compensators particularly for cylindrical micrometric optical systems.
Particular attention has been paid in this study to the astigmatic aberration coefficient (A3), due to its utility in correcting the astigmatic deviations, for example, for autostereoscopic applications for 3D vision, of growing interest in research at present. In fact, there is a direct proportionality between the astigmatic aberration coefficient [Fig. 11(a)] and the focal length of each cylindrical micro-optical element of the array. In Fig. 11(b) the focal length obtained by the Fresnel approximation (2π·|A3|/λ) [Fig. 11(b) “A3”] is compared to the focal length obtained from the captured interference patterns [Fig. 11(b) “FP”]. Also the associated RMS error is shown in Fig. 11(c). As can be observed the higher the voltage the lower the difference. As the astigmatism is predominating for higher voltages the effect of the other aberration in the wave amplitude is depreciable. For this reason, by using only the astigmatism coefficient in the Fresnel approximation equation, the result is similar to the focal length obtained from the fringe patterns. The equations of the polynomial un-normalized form for X and Y astigmatic aberrations are proportional to a cylinder equation, as described in Table 1.
Fig. 11 (a) Amplitude of astigmatism X aberration, A3 (b) focal distance as a function of voltage (FP from interference patterns and A3 from astigmatism coefficient), (c) RMS error.
The results of the aberration characterization, in terms of this parameter, determine that the astigmatic aberration coefficient, A3, can be tuned in a broad range from −0.5 to −6.5 waves, as shown in Fig. 11(a). Therefore, this outcome indicates that the device could be used as a tunable astigmatic device for removing the astigmatism intrinsic to the lenticular sheets of the devices in the application for which it is intended. This is the reason for the normalized size of the lenticular array manufactured.
A very linear and high sensitivity is obtained. With a small range of voltage (1.4Vrms-3Vrms) a huge tunability is obtained (6 waves). The result is a sensitivity of 3.75 λ/Vrms. This indicates that the individual elements of the array could be used in microscopy systems. This device can help to reduce the system aberration and the complexity of the set-up.
5.2. Independent control of aberrations
In the previous section, a broad control over astigmatism has been demonstrated. Other aberrations are maintained between −0.6 to 0.8 waves. Manipulation with any of the selected aberrations is accompanied by an appearance of other aberrations. In this section, selectivity in formation of aberrations is demonstrated. For this, four hypothesis are investigated. The use of different harmonics, phase shifted electrical signals, over-voltage control and a combination of the previous two.
In this case, the 200lpi device is used. With this configuration, two different electrical signals can be applied at the sides of the active area. Firstly, the same experiment as in the previous section is carried out (in phase electrical signals, V1 = V2). The results are shown in Fig. 12.
As can be observed, the astigmatism behavior is very similar to the 100lpi case. The only difference is that in this case only half of the amplitude can be achieved (half of the phase retardation due to having half of the thickness). As in the previous section, the control over the astigmatism is accompanied by the appearance of a decreasing spherical aberration and an oscillation coma aberration.
Among the four hypothesis suggested below, the use of harmonics has not had satisfactory results. The LC permittivity has a small dispersion of the permittivity (at electrical frequencies) so very high frequencies (>100 kHz) have to be applied in order to observe changes in the optical phase profile.
The second experiment is based on the observation of a saturation regime for some specific coefficients (for voltages higher than 2.8 Vrms). The phase shift reaches a maximum at the sides (related to the saturation voltage). Despite this, there is a curve before the birefringence is saturated (in a small voltage range, 0.6Vrms in this case). This curve affects the shape of the phase profile near to the electrode region while the amplitude is almost constant. For this reason, a saturation in some coefficients is observed (astigmatism and coma) while spherical aberration is modified. As can be seen in Fig. 13, astigmatism is maintained in −2.4 waves while coma is −0.4 waves. These aberrations could be compensated by other optical elements while at the same time the spherical coefficient can be corrected in 1 wave by using only an increment of 0.6 Vrms.
Fig. 13 Independent control of spherical aberration. Amplitude of (a) astigmatism X aberration and (b) spherical X and coma X.
The third experiment uses phase shifted electrical signals to control the phase profile. When the upper comb-type finger electrodes (V1 and V2) have an electrical phase shift of 180° with respect to each other, the signals are cancelled at the center of the electrodes. From 0° phase shift to 180° phase shift the voltage at the center is controlled while the voltage at the sides is the same. With this effect, the phase amplitude in the center of the active area is modified. When the electrical phase shift is increased the voltage gradient (between the center and the sides) is increased, as well as the phase amplitude. For this reason, the astigmatism is also increased (keeping the other aberrations in a reasonable margin).. In this specific case, V1 = 1.8Vrms and V2 = 1.8Vrms but phase-shifted from 45° to 180°.
As is demonstrated in Fig. 14, the astigmatism can be controlled in more than 2 waves for an electrical phase shift between 45° and 180°. The other aberrations are maintained in a range of ± 0.2 waves.
Fig. 14 Independent control of astigmatism aberration. Amplitude of (a) astigmatism X aberration and (b) spherical X and coma X.
Finally, the combination of the two previous experiments has been studied. For this, a phase-shifted electrical signal (180°) is carried out at an over-voltage regime (from 1.8 to 4.5 Vrms). The results are very interesting, the spherical aberration is almost suppressed while the aberration astigmatism is maintained at a constant value (Fig. 15). The physical explanation is in the voltage profile produced by the phase shifted electrical signals. As in the previous experiment, the phase shifted electrical signals produce a zero voltage at the center of the active area. When the voltage is increased in the electrodes, the birefringence experiences the typical curve preceding the saturation region. In this case, the curve of the threshold voltage is maintained by the phase shifted electrical signals while the electrode region suffers from the curve of the saturation region. For this reason, the shape of the phase profile is curved at the sides and the center, while the phase amplitude is almost constant. Some aberrations as spherical and astigmatism are maintained constant while the amplitude of the coma is increased (because a phase profile with a coma type curve is produced).
Fig. 15 Independent control of coma aberration. Amplitude of (a) astigmatism X aberration and (b) spherical X and coma X.
The combination of both over-voltage and phase-shifted electrical signals open up endless possibilities. Unlike astigmatism coefficient, which always has a decreasing tendency, other aberrations have several zero crossings. This can be used to achieve an astigmatism with zero aberrations. The use of a simulation model (taking into account the Chebysev polynomials and the Frank-Oseen equations) could lead us to design the necessary electrical signals to obtain specific aberration coefficients. This idea has to be further investigated.
5.3 Manufacturing and software errors
Errors can be caused by several sources. They can be divided in the errors produced by the manufacturing process and the error caused by the software. In the first case, the errors can be caused either by the voltage distribution in the electrodes or non-uniformity of LC thickness. We have evaluated these parameters. The proposed structure has been simulated by Finite Element Method (FEM). Despite the ITO layer has a sheet resistance of 20Ω/sq, the voltage is properly distributed even to the end of the electrode fingers (it has also been checked in the interference patterns). The non-uniformity of the LC cell has been evaluated previous to filling. There is a tolerance of 0.5% in the total thickness. This can produce little variations in the phase shift when the whole array is used. Finally, meshing aberrations are not observed for low voltages. These have to be taken into account when shrinking effect begins to appear or the whole array is going to be used.
The image processing can cause also some errors. Sometimes, Fringe Skeletonizing can generate ambiguous or disconnected skeletons. In this case and after an image processing, we do not obtain undesired defects, such as broken fringe centerlines, cross-connected fringe centerlines or fringe centerlines with short branches. As commented in section 4.1, there is certain ambiguity in the spatial position of the fringe position. The software takes an average across all peaks detected. In the worst case (fringes in the center of the active area), the maximum deviation is 0.5µm. Another important error is the RMS fit error. In this case, the 100lpi device is studied. Due to the high resolution of the CCD camera, the error caused by using a digitized interferogram can be neglected. Despite this, there is an error caused by the fitting process between the measured and the reconstructed data. There are little differences in some parts of the phase maps. Despite this, they are too small to be appreciated on the graphs. For a quantification of the quality of the fit we have done a Root Mean Square (RMS) of the Fit Error following Eq. (4).
where NxM is the total number of data points, W(i,j) is the measured wavefront and W'(i, j) is the Chebyshev fit at x = i and y = j. The result is represented in Fig. 11,As can be seen, in Fig. 16 the maximum fit error is 0.0126 waves with a maximum fluctuaction of 0.006 waves. This error is dependent on the tolerance of the least squares method employed, but the main reason is the discretized phase map obtained by the proposed method. This error could be minimized by using another type of setup, for example, phase shifting interferometry in a double pass configuration.
6. Conclusions
In summary, we have described a novel method to extract aberrations in adaptive LC phase-only passive elements, designed and fabricated an aberration compensation device for rectangular micro-optical systems and presented experimental results of its electro-optic behavior. A complete independent control over the spherical and coma aberration has been demonstrated. Also, an independent control over the astigmatism aberration has been demonstrated in a broad range. For some specific voltage the aberrations can be reduced to minimum values, achieving an ideal lens. The use of a simulation model (taking into account the Chebysev polynomials and the Frank-Oseen equations) to design the necessary electrical signals to obtain specific aberration coefficients has to be further investigated. Moreover, as with other related cases, the use of arrays could improve the image quality in some types of optical systems. The device could also be very useful for optical elements with micrometric size and rectangular aperture when each element of the array is used independently. The resulting fitting error has been measured, resulting in less than 0.013 λ.
Acknowledgments
Authors acknowledge funding support from the Spanish Ministerio de Economía y Competitividad (grant no. TEC2013-47342-C2-2-R) and the R&D Program SINFOTON S2013/MIT-2790 of the Comunidad de Madrid. N. Bennis acknowledges financial support through Project Number POIG.01.03.01-14-016/08 (Poland).
References and links
1. G. D. Love, “Wave-front correction and production of Zernike modes with a liquid-crystal spatial light modulator,” Appl. Opt. 36(7), 1517–1520 (1997). [CrossRef] [PubMed]
2. A. F. Naumov, M. Y. Loktev, I. R. Guralnik, and G. Vdovin, “Liquid-crystal adaptive lenses with modal control,” Opt. Lett. 23(13), 992–994 (1998). [CrossRef] [PubMed]
3. A. Naumov, G. Love, M. Y. Loktev, and F. Vladimirov, “Control optimization of spherical modal liquid crystal lenses,” Opt. Express 4(9), 344–352 (1999). [CrossRef] [PubMed]
4. T. Takahashi, M. Ye, and S. Sato, “Aberrations of liquid crystal lens,” Electron. Lett. 42(25), 1476–1477 (2006). [CrossRef]
5. J. Knittel, H. Richter, M. Hain, S. Somalingam, and T. Tschudi, “Liquid crystal lens for spherical aberration compensation in a Blu-ray disc system,” IEE Proc. Sci. Meas. Technol. 152(1), 15–18 (2005). [CrossRef]
6. C. T. Hsieh, Y. F. Hsu, C. W. Chung, M. F. Chen, W. C. Su, and C. Y. Huang, “Distortion aberration correction device fabricated with liquid crystal lens array,” Opt. Express 21(2), 1937–1943 (2013). [CrossRef] [PubMed]
7. M. Born and E. Wolf, Principles of Optics, 7th ed. (Oxford, 1999).
8. M. Bray, “Orthogonal polynomials: a set for square areas,” Proc. SPIE 5252, 314–321 (2004). [CrossRef]
9. V. N. Mahajan and G. M. Dai, “Orthonormal polynomials in wavefront analysis: analytical solution,” J. Opt. Soc. Am. A 24(9), 2994–3016 (2007). [CrossRef] [PubMed]
10. W. Swantner and W. W. Chow, “Gram-Schmidt orthonormalization of Zernike polynomials for general aperture shapes,” Appl. Opt. 33(10), 1832–1837 (1994). [CrossRef] [PubMed]
11. G. M. Dai and V. N. Mahajan, “Orthonormal polynomials in wavefront analysis: error analysis,” Appl. Opt. 47(19), 3433–3445 (2008). [CrossRef] [PubMed]
12. F. Liu, B. M. Robinson, P. J. Reardon, and J. M. Geary, “Analyzing optics test data on rectangular apertures using 2-D Chebyshev polynomials,” Opt. Eng. 50(4), 043609(2011). [CrossRef]
13. A. Bich, J. Rieck, C. Dumouchel, S. Roth, K. J. Weible, M. Eisner, R. Voelkel, M. Zimmermann, M. Rank, M. Schmidt, R. Bitterli, N. Ramanan, P. Ruffieux, T. Scharf, W. Noell, H.-P. Herzig, and N. De Rooij, “Multifunctional micro-optical elements for laser beam homogenizing and beam shaping,” Proc. SPIE 6879, Photon Processing in Microelectronics and Photonics VII, 68790Q (2008).
14. C.-H. Chen, C.-C. Chen, and W.-C. Liang, “Light pipe line beam shaper,” Opt. Rev. 14(4), 231–235 (2007). [CrossRef]
15. N. S. Holliman, N. A. Dodgson, G. E. Favalora, and L. Pockett, “Three-Dimensional Displays: A Review and Applications Analysis,” IEEE Trans. Broadcast 57(2), 362–371 (2011). [CrossRef]
16. E. Otón, A. Carrasco, R. Vergaz, J. M. Otón, J. M. Sanchez-Pena, X. Quintana, and M. Geday, “2D tunable beam steering - lens device based on high birefringence liquid crystals,” International Conference on Space Optical Systems and Applications, ICSOS'11, art. no. 5783671, 218–219, (2011). [CrossRef]
17. Y. P. Huang, C. W. Chen, T. C. Shen, and J. F. Huang, “Autostereoscopic 3D display with scanning Multi-Electrode driven liquid crystal (MeD-LC) Lens,” J. 3D Res. 1(1), 39–42 (2010). [CrossRef]
18. J. F. Algorri, G. D. Love, and V. Urruchi, “Modal liquid crystal array of optical elements,” Opt. Express 21(21), 24809–24818 (2013). [CrossRef] [PubMed]
19. J. F. Algorri, V. Urruchi, N. Bennis, and J. M. Sánchez-Pena, “Using an Analytical Model to Design Liquid Crystal Microlenses,” IEEE Photon. Technol. Lett. 26(8), 793–796 (2014). [CrossRef]
20. V. Urruchi, J. F. Algorri, J. M. Sánchez-Pena, N. Bennis, and J. M. Otón, “Electrooptic Characterization of Tunable Cylindrical Liquid Crystal Lenses,” Mol. Cryst. Liq. Cryst. (Phila. Pa.) 553(1), 211–219 (2012). [CrossRef]
21. V. Urruchi, J. F. Algorri, J. M. Sánchez-Pena, M. A. Geday, X. Quintana, and N. Bennis, “Lenticular Arrays Based on Liquid Crystals,” Opto-Electron. Rev. 20(3), 260–266 (2012).
22. J. F. Algorri, V. Urruchi, J. M. Sánchez-Pena, and J. M. Otón, “An Autostereoscopic Device for Mobile Applications Based on a Liquid Crystal Microlens Array and an OLED Display,” J. Disp. Technol. 10(9), 713–720 (2014). [CrossRef]
3. Patent title “Patterned glass cylindrical lens arrays for concentrated photovoltaic systems, and/or methods of making the same,” Inventors: Alexey Krasnov, Willem den Boer. Pub. No.: US20140069143 A1 International Application No.: PCT/US2011/000644. Publication Date: 13.03.2014.
24. M. Ares, S. Royo, and J. Caum, “Shack-Hartmann sensor based on a cylindrical microlens array,” Opt. Lett. 32(7), 769–771 (2007). [CrossRef] [PubMed]
25. Q. Deng, C. Du, C. Wang, C. Zhou, X. Dong, Y. Liu, and T. Zhou, “Microlens array for stacked laser diode beam collimation,” Proc. SPIE 5636, 666–670 (2005). [CrossRef]
26. C. Bourgenot, C. D. Saunter, J. M. Taylor, J. M. Girkin, and G. D. Love, “3D adaptive optics in a light sheet microscope,” Opt. Express 20(12), 13252–13261 (2012). [CrossRef] [PubMed]
27. G. Wang, Y. J. Li, and H. C. Zhou, “Application of the radial basis function interpolation to phase extraction from a single electronic speckle pattern interferometric fringe,” Appl. Opt. 50(19), 3110–3117 (2011). [CrossRef] [PubMed]
28. M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. 72(1), 156–160 (1982). [CrossRef]
29. M. A. Gdeisat, D. R. Burton, and M. J. Lalor, “Spatial carrier fringe pattern demodulation by use of a two-dimensional continuous wavelet transform,” Appl. Opt. 45(34), 8722–8732 (2006). [CrossRef] [PubMed]
30. H. Wang and Q. Kemao, “Frequency guided methods for demodulation of a single fringe pattern,” Opt. Express 17(17), 15118–15127 (2009). [CrossRef] [PubMed]
31. S. K. Ghorai, D. Kumar, and B. K. Hura, “Strain measurement in a Mach–Zehnder fiber interferometer using genetic algorithm,” Sens. Actuators A Phys. 122(2), 215–221 (2005). [CrossRef]
32. P. Yeh and C. Gu, Optics of Liquid Crystal Displays, 2nd ed. (John Wiley & Sons, 2008), New Jersey, NJ, USA.
