- Email: [email protected]
Recursive wavefront aberration correction method for LCoS spatial light modulators
Optics and Lasers in Engineering 49 (2011) 743–748
Contents lists available at ScienceDirect
Optics and Lasers in Engineering journal homepage: www.elsevier.com/locate/optlaseng
Recursive wavefront aberration correction method for LCoS spatial light modulators J. Garcı´a-Ma´rquez n, J.E.A. Landgrave, N. Alcala´-Ochoa, C. Pe´rez-Santos ´ ptica A.C., Loma del Bosque 115, CP 37150, Leo ´n, Mexico Centro de Investigaciones en O
a r t i c l e i n f o
abstract
Available online 5 March 2011
We present two accurate and relatively simple interferometric methods for the correction of wavefront aberrations of about 3 wavelengths (3l) in spatial light modulators (SLMs) of the liquid crystal on silicon (LCoS) type. The first is based on a recursive use of a wavefront fitting algorithm in a WykoTM interferometer, in which Zernike polynomials are employed as the basis functions. We show here that the successive use of only three measurements is required to obtain a peak-to-valley (PV) error as low as l/10, with an uncertainty of l/30, in the compensated wavefront. The second method makes use of the actual optical path difference (OPD) computed by the interferometer at each pixel of the image of the interferogram of the LCoS modulator (LCoS-M). From numerical interpolation of these OPD values we were able to assign the required OPD compensation at each pixel of the LCoS-M. With this method, PV errors of the compensated wavefront as low as l/16, with an uncertainty of l/30, were obtained for the entire LCoS-M, or of l/33 for the disk that we used as the domain of the Zernike polynomials in the first method. & 2011 Elsevier Ltd. All rights reserved.
Keywords: Spatial light modulators Aberration compensation Phase measurement Pupil engineering
1. Introduction Spatial light modulators (SLM) have been used in a broad variety of applications, such as corneal topography [1], saturatedpatterned excitation microscopy [2,3], apodization [4], focus tracking [5], astronomic instrumentation [6], and high-resolution wavefront sensing and correction [7]. In most of these applications, the wavefront aberration introduced by the SLM has to be taken into consideration. Given that SLMs with aberrations larger than 2l are common [8], and that this would have a marked effect on the image quality of the optical system in which they work, the need for compensating their aberrations is amply justified. To achieve this, some authors have first measured interferometrically the out coming wavefront of an SLM, and then compensated its aberrations by inducing a phase with the opposite sign in the modulator [6,9]. They have shown that an SLM with an aberration comparable to its maximum modulation depth, typically around 1l, can be reduced to roughly l/10 in small matrix arrays. It has also been shown that peak-to-valley (PV) aberrations as large as 19l in a Liquid Crystal on Silicon Modulator (LCoS-M) can be reduced to nearly l/10, by wrapping the compensating phase, modulo 2p, in the modulator [10]. This was achieved after careful measurements of the birefringence induced by successive gray level screens on the LCoS-M, and interferometric tests of the aberrated wavefront emerging from the inactive modulator.
n
Corresponding author. E-mail address: [email protected] (J. Garcı´a-Ma´rquez).
0143-8166/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.optlaseng.2011.01.024
Finally, a third, non-interferometric method [11] can be used to compensate the aberration, not just of the modulator, but of the entire optical system of which this forms a part. By generating an optical vortex with the SLM (Fig. 1), a light intensity distribution in the form of a doughnut is projected on a CCD. Due to the aberrations of the optical elements in the system, including those of the LCoS-M, this doughnut will appear distorted in the image that we obtain from the digital camera. By means of the Gerchberg–Saxton algorithm [12], the doughnut distortion is iteratively removed through successive phase modifications in the SLM, achieving thus the desired correction of aberrations in the whole system. An SLM that is corrected from aberrations, or programmed to correct those of a complete optical system, can in principle contribute to form diffraction limited images. This, of course, would be essential in applications like those mentioned at the beginning of this section. In this paper we show how to correct an aberrated SLM up to l/16 PV error, yielding nearly perfect tilt fringes in an interferometric test, or equivalently, a Strehl ratio which is fairly close to 1, in the point spread function (PSF) computed with the corrected wavefront. In our results in this article l ¼633 nm.
2. Aberration correction of the LCoS-M by means of a phase-modulated grating The SLM that we used for this work was a liquid crystal on silicon (LCoS) bi-dimensional array of 1024 768 pixels,
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´rquez et al. / Optics and Lasers in Engineering 49 (2011) 743–748 J. Garcı´a-Ma
Holograma ( I )
Fig. 1. 3-D OPD graph of an optical vortex of topological charge 1 obtained with an aberration-free LCoS-M in a WykoTM interferometer.
Fig. 3. Phase-modulated grating used to compensate aberrations in an LCoS-M. It has an average period of 12 pixels and 85 line-pairs (lp). Its contrast can be readily varied to control the intensity of the diffracted beams.
CCD Camera Focused at the LCoS-M
Point Source
Beam Splitter
Collimator
Aberrated Wavefront
Reference LCoS -M Flat
Fig. 4. Testing configuration to measure the aberration introduced after reflection at an LCoS-M. The LCoS is slightly tilted, in such a way that the beam diffracted in the first order is aligned to be inspected by the interferometer. The tilt adjustment would be clearly dependent on the frequency of the phase grating that we generate in the LCoS.
Fig. 2. Interferogram of an inactive LCoS-M obtained with a WykoTM interferometer.
commercialized by HoloeyeTM as model LC-R 2500. The optical quality of this display was first tested in a Twyman–Green configuration and it was established that a relatively strong astigmatism was present (Fig. 2). But correction of a mixture of aberrations exceeding 2l is not an easy task in a device, which can only attain a maximum phase excursion of 2p. Although this direct approach has been used for aberration compensation [6,9,10], as we mention above, the experimentalist should be aware of its limitations. Perhaps the most important of them is the need to use the whole phase displacement range, or phase stroke, of the device to attain the compensation of its aberration. This would prevent him, for example, from reducing the depth of modulation that is required in order to lessen the flickering in the LCoS, as suggested in Ref. [13]. An alternative approach to correct the aberrations of the LCoS-M is the use of a phase-modulated grating, where the phase distortion due to the aberrations of the modulator is encoded in the phase of the grating (Fig. 3). This grating may be thought of as a hologram, in which the conjugate of the aberrated wave is
recorded, using a plane wave as the reference beam. This method requires, of course, an accurate measurement of the aberrated wavefront of the LCoS-M. A Fizeau digital interferometer (WykoTM) was used for this purpose, allowing us to represent the aberration in terms of Zernike polynomials, or as a matrix with the values of the optical path differences (OPDs) sensed by the digital camera of the interferometer (Fig. 4). The advantage of the first description is that we only need a few dozens of numbers, typically 36 polynomial coefficients, to have a good account of the wavefront aberration over a circular domain in the LCoS-M surface. In the second case we need thousands, but they offer a thorough description of the aberration through the entire surface of the LCoS-M. It is to be expected that the matrix representation of the aberration would lead to a better aberration compensation of the LCoS-M. We shall next present a simple theory to discuss some details common to both methods. 2.1. Theory Let r0 ¼ a exp½ij0 ðx,yÞ
ð1Þ
´rquez et al. / Optics and Lasers in Engineering 49 (2011) 743–748 J. Garcı´a-Ma
745
be the complex amplitude reflectance of the LCoS-M in the absence of any induced phase modulation. For simplicity we assume a is a constant approaching 1. j0(x,y), on the other hand, is the phase distortion introduced by reflection on the inactive LCoS-M, related to the wavefront aberration W(x,y) through the equation j0(x,y) ¼(2p/l)W(x,y), where l is the wavelength of the light, and (x,y) are the coordinates of the pixels of the modulator. When the LCoS-M is activated with a video signal, its complex amplitude reflectance becomes
and g is the function that relates the intensity of a pixel in the screen pattern with the phase induced at the corresponding pixel in the LCoS-M. Here Imax and Imin are, respectively, the maximum and the minimum intensities of the screen pattern, u0 is the spatial frequency assigned to the grating, and j0(x,y) the phase distortion introduced by the passive LCoS-M. When we illuminate the modulator with a unit amplitude plane wave at normal incidence, from Eqs. (2) and (3) follows that the complex amplitude U(x,y) after reflection will be
r ¼ r0 exp½iji ðx,yÞ ¼ a expfi½j0 ðx,yÞ þ ji ðx,yÞg ¼ a exp½ijðx,yÞ
Uðx,yÞ ¼ rðx,yÞU1 ¼ a expfi½j0 ðx,yÞ þ ji ðx,yÞg ¼ a exp ij0 ðx,yÞ þ ig½Iðx,yÞ
ð2Þ
where r0 is given by Eq. (1), ji(x,y) is the induced phase modulation and j(x,y) is the actual phase modulation that we obtain from the LCoS-M. Obviously, ji(x,y)¼ j(x,y) j0(x,y). Therefore, if we wish to obtain a given phase modulation j(x,y) from the modulator, we must induce a phase modulation ji(x,y) in it which is the difference between the desired phase modulation, j(x,y), and the phase distortion introduced by the passive LCoS-M, j0(x,y). One form of achieving this is by encoding the phase distortion j0(x,y) in a phase grating. This method has several advantages. The main one is that phase distortions larger than the maximum modulation depth attained by the LCoS-M could be compensated; the larger the frequency of the grating, the larger the amount of potential compensation. Therefore, the limit now would be set by the pixel size of the LCoS-M. An additional advantage of this method is that there is no need to elaborate a look up table (LUT) with the induced phases in the LCoS-M from 256 gray levels at the monitor screen. What we need now is an accurate method of finding the phase distortion function j0(x,y). A non-linear phase response of the LCoS has, therefore, no effect on the aberration compensation. It is the shape of the lines of the grating that matter (Fig. 3). Their profile, which would be defined by phase non-linearities, would only affect the amount of light that is sent into different diffracted orders. Finally, a further, but not minor, advantage of this method is that there is no need of using polarizers, which would introduce their own aberrations, unless they are made of high optical quality. Ordinary polarizers are of poor optical quality and mechanically unstable. Their aberrations, therefore, would be difficult to compensate with any phase modulator. When working on Point Spread Function (PSF) engineering, therefore, if possible, we should avoid the use of polarizing elements in the setup. The main disadvantages of the method are that diffracted beams overlap each other, and that these beams often have small intensities. In our case, where our interest lays on apodization of the focused beams, these do not represent a serious drawback. We may even benefit by another feature of this technique, namely, the ability to change the intensity of the compensated beam by the simple expedient of varying the modulation depth of the phase grating. This could not have been achieved if we had compensated the LCoS-M in the usual form. We believe there are instances where this feature represents a significant advantage. Let I(x,y) be the intensity distribution of the screen pattern that we shall use to generate the grating that will encode the aberration compensation of the LCoS-M, where I assumes integer values between 0 and 255 (the gray levels), and (x,y) are the coordinates associated to the 1024 768 pixels of the modulator. What we want is to induce in it a phase modulation of the form
ji ðx,yÞ ¼ g½Iðx,yÞ
ð3Þ
where Iðx,yÞ ¼
1 Imax þImin þ ðImax Imin Þcos 2pu0 xj0 ðx,yÞ 2
ð4Þ
ð5Þ
The function g is highly non-linear, and its actual form can only be known from an LUT. For the purpose of simplifying our demonstration, however, we shall assume that g is linear, so that we can write
ji ðx,yÞ ¼ d þ bIðx,yÞ
ð6Þ
where d and b are appropriate constants. From Eqs. (5) and (6) it follows that 1 Uðx,yÞ ¼ a exp d þ bðImax þImin Þ 2 1 ð7Þ exp j0 ðx,yÞ þ bðImax Imin Þcos 2pu0 xj0 ðx,yÞ 2 Making use of the Jacobi–Anger expansion [14]: eiz cos y ¼
1 X
Jn ðzÞeiny
ð8Þ
n ¼ 1
where Jn(z) is the Bessel function of the first kind and nth order, after dropping a constant factor we obtain: 1 X 1 Uðx,yÞ ¼ exp ij0 ðx,yÞ Jn bðImax Imin Þ exp in 2pu0 xj0 ðx,yÞ 2 n ¼ 1
ð9Þ Therefore, the amplitude of the wave corresponding to the first diffracted order (n ¼1) is 1 Uðx,yÞ1 ¼ exp ij0 ðx,yÞ J1 bðImax Imin Þ exp i 2pu0 xj0 ðx,yÞ 2 1 ð10Þ ¼ J1 bðImax Imin Þ expði2pu0 xÞ 2 The phase distortion j0(x,y) has then been removed from this wave, leaving an undistorted plane wave of amplitude J1 ½ð1=2Þ bðImax Imin Þ. Notice that this amplitude depends on the contrast of the fringe pattern I(x,y), as we mentioned above. In conclusion, we can compensate the aberration introduced by an LCoS-M if we induce a phase grating whose phase itself is modulated with the phase distortion introduced by the LCoS-M, but with the opposite sign. 2.2. Results Fig. 5 shows four successive OPD maps with increasing degree of aberration correction. They were obtained for a circular zone of 14 mm diameter of the LCoS-M, the maximum that could be used within the rectangular aperture of the modulator. Initially, the wavefront reflected from the inactive LCoS-M had a PV aberration of approximately 212l. The corresponding phase distortion j0(x,y) was then represented in terms of Zernike polynomials, and subsequently used to modulate the phase of the grating that will be induced in the LCoS-M (Eq. (4)). After repeating this process 3 times, the PV aberration of the compensated LCoS-M over the circular domain was only of l/10. This can be clearly seen in Fig. 6, that shows the three-dimensional OPD graph before and after
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correction of the wavefront reflected from the LCoS-M. Notice the scale range in each map. Instead of using the Zernike coefficients provided by the interferometer to represent the wavefront aberration over a circular zone on the LCoS-M, we can read out the matrix with the values of the OPDs corresponding to each pixel of the interferogram obtained from the entire LCoS-M. Fig. 7 shows a sequence of interferograms of increasingly corrected, slightly tilted, wavefronts. All of them correspond to the first diffraction order of the phase-modulated grating induced at the LCoS-M (Actually, the first of them is obtained in the absence of modulation in the grating, corresponding thus to the aberration of the inactive LCoS-M.) Notice that the sequence ends up as a set of virtually straight equidistant fringes. The corresponding OPD maps of this sequence are shown in Fig. 8. From them we can see an aberration reduction from nearly 3l to only l/16. If these interferograms are masked, so that the inspected area of the LCoS-M is a disk of 14 mm diameter (the maximum beam diameter that can be admitted by the LCoS-M), the residual wavefront aberration would be l/33, which is beyond the uncertainty limit of the measurements performed with the interferometer.
It is worth mentioning here some technical details related to this alternative approach. The camera in the interferometer had a CCD with 736 480 pixels, but we only needed a fraction of them to record the interferograms of the LCoS-M. The entire compensation of the LCoS-M, on the other hand, required phase-related integer numbers between 0 and 255 at each of its 1024 768 pixels. To obtain these numbers, we needed first to build a rectangular matrix with the OPDs that were obtained after phase-stepping the interferogram of the LCoS-M. This matrix has significantly lower dimensions than the matrix of the LCoS-M, particularly when we realize that some of its marginal rows and columns are discarded because not all their elements were filled with data. Typical values in our case were in the order of 190 120. With them we had to address the 1024 768 pixels of the LCoS-M; in other words, we needed to interpolate the matrix with OPD values of the interferometer at each pixel of the LCoS-M. This task can be readily done with standard numerical routines, like the function imresize of MATLAB. From OPD data of the corrected wavefronts over a circular domain we can obtain the corresponding PSFs through Fourier transformation. Graphs of these are also provided by the software of the interferometer, together with an estimation of the Strehl ratio. Since our motivation for using LCoS-M was PSF engineering, this was very relevant information. When data from Figs. 5 and 8 were used, the graphs of the PSFs resembled very closely an Airy pattern (which is, of course, the PSF of a clear circular pupil), with Strehl ratios estimations greater than 0.989. Finally, a word about the precision of our measurements. The interferometer that was used for the compensation of two LCoS-Ms is the instrument that we use for the calibration of flat surfaces in our research center. It is a commercial instrument, but as a certified center that provides flatness calibration services, its uncertainty had to be traceable. This was done and established at l/ 30. The details can be found in Ref. [15].
3. Conclusions
Fig. 5. Results of the recursive aberration correction of the LCoS-M by means of Zernike polynomials. They were obtained in a circular zone of 14 mm diameter of the modulator.
We have demonstrated that a high degree of aberration compensation can be achieved on an LCoS-M by inducing a phase-modulated grating on it. The correction is attained recursively, and three iterations are enough to have a residual error of l=16. If, as it is commonly the case, the two-dimensional, phase modulation of the grating is required over a circular domain, this can be prescribed with enough accuracy and economy in terms of Zernike polynomials. If the phase modulation must be provided for the entire surface of the LCoS-M, then this can be given after interpolating the raw data (OPDs) from the interferometer at each pixel of the modulator. In the first case, we had a residual aberration of l=10 over a 14 mm circular domain (Fig. 5).
Fig. 6. 3-D OPD graph of the reflected wavefront from the LCoS-M: (a) before correction and (b) after correction. Note the range of the scales.
´rquez et al. / Optics and Lasers in Engineering 49 (2011) 743–748 J. Garcı´a-Ma
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Fig. 7. Interferograms of the wavefront reflected from the LCoS-M, with a slight tilt added: (a) before correction, (b) after the first aberration correction of the modulator, (9c) after the correction of the first residual aberration, and (d) after the correction of the second residual aberration. Notice the straightness of the fringes in (d). The dashed circle is of 14 mm diameter. This is the maximum beam diameter accepted by the modulator.
Fig. 8. Results of the recursive aberration correction of the LCoS-M by means of an OPD matrix.
In the second, the residual aberration was of l/33 within the same domain, shown with a solid circle in Fig. 8d. The main advantages of the method of aberration compensation that we proposed are three: (1) there is no need to spend the whole phase modulation depth (phase stroke) to correct the LCoS, (2) there is no need of high optical quality polarizers to use with the LCoS, and (3) it is easy to adjust the amplitude of the corrected wavefront. The main disadvantages, on the other hand, are two: (1) the intensity of the beam that has been corrected from
aberrations is small (only a small percentage of the intensity of the beam reflected by the inactive LCoS-M) and (2) all the beams diffracted by the grating overlap each other over long distances of propagation. For some applications, however, these drawbacks do not represent a serious impediment. The compensation of an LCoS-M is not a long term affair. After several weeks of operation, we observed a loss of aberration correction in the modulators that we had compensated. Possible causes for this drift are the handling and repeated operation of
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the modulators, and their long term exposure to ambient temperature fluctuations. External mechanical forces, like the ones we exert on the modulators each time we fix them to their mounts, can affect the aberration compensation by several tenths of a wavelength. Because of these facts, it is important to check periodically the aberration status of the LCoS-M. In pupil engineering the requirement of an aberration-free wavefront is very stringent. The added possibility of correcting the residual aberrations of the whole optical system, apart from generating the required pupil filter, makes the LCoS technology very attractive.
Acknowledgment The authors would like to express their gratitude to Maximi˜ anza Te´cnica Industrial (CETI), in liano Gala´n, of Centro de Ensen Guadalajara, Mexico, for joining them to compensate the aberrations of his own LCoS-M, of the same model and brand than the authors’ modulator, enriching thus the experimental validation of the method of aberration compensation presented here. N.A-O. wants to acknowledge CONACYT for grant 133495. References [1] Prieto PM, Ferna´ndez EJ, Manzanera S, Artal P. Adaptive optics with a programmable phase modulator: applications in the human eye. Opt Express 2004;12:4059–71. [2] Heintzmann R, Jovin ThM, Cremer C. Saturated patterned excitation microscopy – a concept for optical resolution improvement. J Opt Soc Am A 2002;19:1599–609.
[3] Heintzmann R. Saturated patterned excitation microscopy with two-dimensional excitation patterns. Micron 2003;34:283–91. [4] Davis JA, Escalera JC, Campos J, Ma´rquez A, Yzuel MJ, Iemmi C. Programmable axial apodizing and hyperresolving amplitude filtres with a liquid-crystal spatial light modulator. Opt Lett 1999;24:628–30. [5] Wright AJ, Patterson BA, Poland SP, Girkin JM, Gibson GM, Padgett MJ. Dynamic closed-loop system for focus tracking using a spatial light modulator and a deformable membrane mirror. Opt Express 2006;14: 222–8. [6] Love GD. Wave-front correction and production of Zernike modes with a liquid-crystal spatial light modulator. Appl Opt 1997;36:1517–24. [7] Vorontsov MA, Justh EW, Beresnev LA. Adaptive optics with advanced phasecontrast techniques. I. High-resolution wave-front sensing. J Opt Soc Am A 2001;18:1289–99. ¨ ¨ [8] Wernicke G, Kruger S, Kamps J, Gruber H, Demoli N, Durr M, et al. Application of a liquid crystal display spatial light modulator system as dynamic diffractive element and in optical image processing. J Opt Commun 2004;25:141–8. [9] Hu L, Xuan L, Liu Y, Cao Z, Li D, Mu QQ. Phase-only liquid-crystal spatial light modulator for wave-front correction with high precision. Opt Express 2004;12:6403–9. [10] Wang X, Wang B, Pouch J, Miranda F, Anderson JE, Bos PJ. Performance of a liquid-crystal on silicon spatial light modulator. Opt Eng 2004 2769–74. ¨ [11] Jesacher A, Schwaighofer A, Furhapter S, Maurer C, Bernet S, Ritsch-Marte M. Wavefront correction of spatial light modulators using an optical vortex image. Opt Express 2007;15:5801–8. [12] Gerchberg RW, Saxton WO. A practical algorithm for the determination of phase from image and diffraction plane pictures. Optik 1972;35:237–46. [13] Jesacher A. Applications of spatial light modulators for optical trapping ¨ Innsbruck; and image processing. D. Thesis Leopold-Franzens Universitat, 2009. [14] Abramowitz M, Stegun IA. Handbook of mathematical functions. New York: Dover; 1970. [15] Garcı´a-Ma´rquez J, Alcala´ N, Pe´rez C, Moya J, Montoya M. Calibracio´n de un interfero´metro Fizeau: ca´lculo de incertidumbres. Rev Mex Fis 2005;50: 114–20.
Contents lists available at ScienceDirect
Optics and Lasers in Engineering journal homepage: www.elsevier.com/locate/optlaseng
Recursive wavefront aberration correction method for LCoS spatial light modulators J. Garcı´a-Ma´rquez n, J.E.A. Landgrave, N. Alcala´-Ochoa, C. Pe´rez-Santos ´ ptica A.C., Loma del Bosque 115, CP 37150, Leo ´n, Mexico Centro de Investigaciones en O
a r t i c l e i n f o
abstract
Available online 5 March 2011
We present two accurate and relatively simple interferometric methods for the correction of wavefront aberrations of about 3 wavelengths (3l) in spatial light modulators (SLMs) of the liquid crystal on silicon (LCoS) type. The first is based on a recursive use of a wavefront fitting algorithm in a WykoTM interferometer, in which Zernike polynomials are employed as the basis functions. We show here that the successive use of only three measurements is required to obtain a peak-to-valley (PV) error as low as l/10, with an uncertainty of l/30, in the compensated wavefront. The second method makes use of the actual optical path difference (OPD) computed by the interferometer at each pixel of the image of the interferogram of the LCoS modulator (LCoS-M). From numerical interpolation of these OPD values we were able to assign the required OPD compensation at each pixel of the LCoS-M. With this method, PV errors of the compensated wavefront as low as l/16, with an uncertainty of l/30, were obtained for the entire LCoS-M, or of l/33 for the disk that we used as the domain of the Zernike polynomials in the first method. & 2011 Elsevier Ltd. All rights reserved.
Keywords: Spatial light modulators Aberration compensation Phase measurement Pupil engineering
1. Introduction Spatial light modulators (SLM) have been used in a broad variety of applications, such as corneal topography [1], saturatedpatterned excitation microscopy [2,3], apodization [4], focus tracking [5], astronomic instrumentation [6], and high-resolution wavefront sensing and correction [7]. In most of these applications, the wavefront aberration introduced by the SLM has to be taken into consideration. Given that SLMs with aberrations larger than 2l are common [8], and that this would have a marked effect on the image quality of the optical system in which they work, the need for compensating their aberrations is amply justified. To achieve this, some authors have first measured interferometrically the out coming wavefront of an SLM, and then compensated its aberrations by inducing a phase with the opposite sign in the modulator [6,9]. They have shown that an SLM with an aberration comparable to its maximum modulation depth, typically around 1l, can be reduced to roughly l/10 in small matrix arrays. It has also been shown that peak-to-valley (PV) aberrations as large as 19l in a Liquid Crystal on Silicon Modulator (LCoS-M) can be reduced to nearly l/10, by wrapping the compensating phase, modulo 2p, in the modulator [10]. This was achieved after careful measurements of the birefringence induced by successive gray level screens on the LCoS-M, and interferometric tests of the aberrated wavefront emerging from the inactive modulator.
n
Corresponding author. E-mail address: [email protected] (J. Garcı´a-Ma´rquez).
0143-8166/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.optlaseng.2011.01.024
Finally, a third, non-interferometric method [11] can be used to compensate the aberration, not just of the modulator, but of the entire optical system of which this forms a part. By generating an optical vortex with the SLM (Fig. 1), a light intensity distribution in the form of a doughnut is projected on a CCD. Due to the aberrations of the optical elements in the system, including those of the LCoS-M, this doughnut will appear distorted in the image that we obtain from the digital camera. By means of the Gerchberg–Saxton algorithm [12], the doughnut distortion is iteratively removed through successive phase modifications in the SLM, achieving thus the desired correction of aberrations in the whole system. An SLM that is corrected from aberrations, or programmed to correct those of a complete optical system, can in principle contribute to form diffraction limited images. This, of course, would be essential in applications like those mentioned at the beginning of this section. In this paper we show how to correct an aberrated SLM up to l/16 PV error, yielding nearly perfect tilt fringes in an interferometric test, or equivalently, a Strehl ratio which is fairly close to 1, in the point spread function (PSF) computed with the corrected wavefront. In our results in this article l ¼633 nm.
2. Aberration correction of the LCoS-M by means of a phase-modulated grating The SLM that we used for this work was a liquid crystal on silicon (LCoS) bi-dimensional array of 1024 768 pixels,
744
´rquez et al. / Optics and Lasers in Engineering 49 (2011) 743–748 J. Garcı´a-Ma
Holograma ( I )
Fig. 1. 3-D OPD graph of an optical vortex of topological charge 1 obtained with an aberration-free LCoS-M in a WykoTM interferometer.
Fig. 3. Phase-modulated grating used to compensate aberrations in an LCoS-M. It has an average period of 12 pixels and 85 line-pairs (lp). Its contrast can be readily varied to control the intensity of the diffracted beams.
CCD Camera Focused at the LCoS-M
Point Source
Beam Splitter
Collimator
Aberrated Wavefront
Reference LCoS -M Flat
Fig. 4. Testing configuration to measure the aberration introduced after reflection at an LCoS-M. The LCoS is slightly tilted, in such a way that the beam diffracted in the first order is aligned to be inspected by the interferometer. The tilt adjustment would be clearly dependent on the frequency of the phase grating that we generate in the LCoS.
Fig. 2. Interferogram of an inactive LCoS-M obtained with a WykoTM interferometer.
commercialized by HoloeyeTM as model LC-R 2500. The optical quality of this display was first tested in a Twyman–Green configuration and it was established that a relatively strong astigmatism was present (Fig. 2). But correction of a mixture of aberrations exceeding 2l is not an easy task in a device, which can only attain a maximum phase excursion of 2p. Although this direct approach has been used for aberration compensation [6,9,10], as we mention above, the experimentalist should be aware of its limitations. Perhaps the most important of them is the need to use the whole phase displacement range, or phase stroke, of the device to attain the compensation of its aberration. This would prevent him, for example, from reducing the depth of modulation that is required in order to lessen the flickering in the LCoS, as suggested in Ref. [13]. An alternative approach to correct the aberrations of the LCoS-M is the use of a phase-modulated grating, where the phase distortion due to the aberrations of the modulator is encoded in the phase of the grating (Fig. 3). This grating may be thought of as a hologram, in which the conjugate of the aberrated wave is
recorded, using a plane wave as the reference beam. This method requires, of course, an accurate measurement of the aberrated wavefront of the LCoS-M. A Fizeau digital interferometer (WykoTM) was used for this purpose, allowing us to represent the aberration in terms of Zernike polynomials, or as a matrix with the values of the optical path differences (OPDs) sensed by the digital camera of the interferometer (Fig. 4). The advantage of the first description is that we only need a few dozens of numbers, typically 36 polynomial coefficients, to have a good account of the wavefront aberration over a circular domain in the LCoS-M surface. In the second case we need thousands, but they offer a thorough description of the aberration through the entire surface of the LCoS-M. It is to be expected that the matrix representation of the aberration would lead to a better aberration compensation of the LCoS-M. We shall next present a simple theory to discuss some details common to both methods. 2.1. Theory Let r0 ¼ a exp½ij0 ðx,yÞ
ð1Þ
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be the complex amplitude reflectance of the LCoS-M in the absence of any induced phase modulation. For simplicity we assume a is a constant approaching 1. j0(x,y), on the other hand, is the phase distortion introduced by reflection on the inactive LCoS-M, related to the wavefront aberration W(x,y) through the equation j0(x,y) ¼(2p/l)W(x,y), where l is the wavelength of the light, and (x,y) are the coordinates of the pixels of the modulator. When the LCoS-M is activated with a video signal, its complex amplitude reflectance becomes
and g is the function that relates the intensity of a pixel in the screen pattern with the phase induced at the corresponding pixel in the LCoS-M. Here Imax and Imin are, respectively, the maximum and the minimum intensities of the screen pattern, u0 is the spatial frequency assigned to the grating, and j0(x,y) the phase distortion introduced by the passive LCoS-M. When we illuminate the modulator with a unit amplitude plane wave at normal incidence, from Eqs. (2) and (3) follows that the complex amplitude U(x,y) after reflection will be
r ¼ r0 exp½iji ðx,yÞ ¼ a expfi½j0 ðx,yÞ þ ji ðx,yÞg ¼ a exp½ijðx,yÞ
Uðx,yÞ ¼ rðx,yÞU1 ¼ a expfi½j0 ðx,yÞ þ ji ðx,yÞg ¼ a exp ij0 ðx,yÞ þ ig½Iðx,yÞ
ð2Þ
where r0 is given by Eq. (1), ji(x,y) is the induced phase modulation and j(x,y) is the actual phase modulation that we obtain from the LCoS-M. Obviously, ji(x,y)¼ j(x,y) j0(x,y). Therefore, if we wish to obtain a given phase modulation j(x,y) from the modulator, we must induce a phase modulation ji(x,y) in it which is the difference between the desired phase modulation, j(x,y), and the phase distortion introduced by the passive LCoS-M, j0(x,y). One form of achieving this is by encoding the phase distortion j0(x,y) in a phase grating. This method has several advantages. The main one is that phase distortions larger than the maximum modulation depth attained by the LCoS-M could be compensated; the larger the frequency of the grating, the larger the amount of potential compensation. Therefore, the limit now would be set by the pixel size of the LCoS-M. An additional advantage of this method is that there is no need to elaborate a look up table (LUT) with the induced phases in the LCoS-M from 256 gray levels at the monitor screen. What we need now is an accurate method of finding the phase distortion function j0(x,y). A non-linear phase response of the LCoS has, therefore, no effect on the aberration compensation. It is the shape of the lines of the grating that matter (Fig. 3). Their profile, which would be defined by phase non-linearities, would only affect the amount of light that is sent into different diffracted orders. Finally, a further, but not minor, advantage of this method is that there is no need of using polarizers, which would introduce their own aberrations, unless they are made of high optical quality. Ordinary polarizers are of poor optical quality and mechanically unstable. Their aberrations, therefore, would be difficult to compensate with any phase modulator. When working on Point Spread Function (PSF) engineering, therefore, if possible, we should avoid the use of polarizing elements in the setup. The main disadvantages of the method are that diffracted beams overlap each other, and that these beams often have small intensities. In our case, where our interest lays on apodization of the focused beams, these do not represent a serious drawback. We may even benefit by another feature of this technique, namely, the ability to change the intensity of the compensated beam by the simple expedient of varying the modulation depth of the phase grating. This could not have been achieved if we had compensated the LCoS-M in the usual form. We believe there are instances where this feature represents a significant advantage. Let I(x,y) be the intensity distribution of the screen pattern that we shall use to generate the grating that will encode the aberration compensation of the LCoS-M, where I assumes integer values between 0 and 255 (the gray levels), and (x,y) are the coordinates associated to the 1024 768 pixels of the modulator. What we want is to induce in it a phase modulation of the form
ji ðx,yÞ ¼ g½Iðx,yÞ
ð3Þ
where Iðx,yÞ ¼
1 Imax þImin þ ðImax Imin Þcos 2pu0 xj0 ðx,yÞ 2
ð4Þ
ð5Þ
The function g is highly non-linear, and its actual form can only be known from an LUT. For the purpose of simplifying our demonstration, however, we shall assume that g is linear, so that we can write
ji ðx,yÞ ¼ d þ bIðx,yÞ
ð6Þ
where d and b are appropriate constants. From Eqs. (5) and (6) it follows that 1 Uðx,yÞ ¼ a exp d þ bðImax þImin Þ 2 1 ð7Þ exp j0 ðx,yÞ þ bðImax Imin Þcos 2pu0 xj0 ðx,yÞ 2 Making use of the Jacobi–Anger expansion [14]: eiz cos y ¼
1 X
Jn ðzÞeiny
ð8Þ
n ¼ 1
where Jn(z) is the Bessel function of the first kind and nth order, after dropping a constant factor we obtain: 1 X 1 Uðx,yÞ ¼ exp ij0 ðx,yÞ Jn bðImax Imin Þ exp in 2pu0 xj0 ðx,yÞ 2 n ¼ 1
ð9Þ Therefore, the amplitude of the wave corresponding to the first diffracted order (n ¼1) is 1 Uðx,yÞ1 ¼ exp ij0 ðx,yÞ J1 bðImax Imin Þ exp i 2pu0 xj0 ðx,yÞ 2 1 ð10Þ ¼ J1 bðImax Imin Þ expði2pu0 xÞ 2 The phase distortion j0(x,y) has then been removed from this wave, leaving an undistorted plane wave of amplitude J1 ½ð1=2Þ bðImax Imin Þ. Notice that this amplitude depends on the contrast of the fringe pattern I(x,y), as we mentioned above. In conclusion, we can compensate the aberration introduced by an LCoS-M if we induce a phase grating whose phase itself is modulated with the phase distortion introduced by the LCoS-M, but with the opposite sign. 2.2. Results Fig. 5 shows four successive OPD maps with increasing degree of aberration correction. They were obtained for a circular zone of 14 mm diameter of the LCoS-M, the maximum that could be used within the rectangular aperture of the modulator. Initially, the wavefront reflected from the inactive LCoS-M had a PV aberration of approximately 212l. The corresponding phase distortion j0(x,y) was then represented in terms of Zernike polynomials, and subsequently used to modulate the phase of the grating that will be induced in the LCoS-M (Eq. (4)). After repeating this process 3 times, the PV aberration of the compensated LCoS-M over the circular domain was only of l/10. This can be clearly seen in Fig. 6, that shows the three-dimensional OPD graph before and after
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correction of the wavefront reflected from the LCoS-M. Notice the scale range in each map. Instead of using the Zernike coefficients provided by the interferometer to represent the wavefront aberration over a circular zone on the LCoS-M, we can read out the matrix with the values of the OPDs corresponding to each pixel of the interferogram obtained from the entire LCoS-M. Fig. 7 shows a sequence of interferograms of increasingly corrected, slightly tilted, wavefronts. All of them correspond to the first diffraction order of the phase-modulated grating induced at the LCoS-M (Actually, the first of them is obtained in the absence of modulation in the grating, corresponding thus to the aberration of the inactive LCoS-M.) Notice that the sequence ends up as a set of virtually straight equidistant fringes. The corresponding OPD maps of this sequence are shown in Fig. 8. From them we can see an aberration reduction from nearly 3l to only l/16. If these interferograms are masked, so that the inspected area of the LCoS-M is a disk of 14 mm diameter (the maximum beam diameter that can be admitted by the LCoS-M), the residual wavefront aberration would be l/33, which is beyond the uncertainty limit of the measurements performed with the interferometer.
It is worth mentioning here some technical details related to this alternative approach. The camera in the interferometer had a CCD with 736 480 pixels, but we only needed a fraction of them to record the interferograms of the LCoS-M. The entire compensation of the LCoS-M, on the other hand, required phase-related integer numbers between 0 and 255 at each of its 1024 768 pixels. To obtain these numbers, we needed first to build a rectangular matrix with the OPDs that were obtained after phase-stepping the interferogram of the LCoS-M. This matrix has significantly lower dimensions than the matrix of the LCoS-M, particularly when we realize that some of its marginal rows and columns are discarded because not all their elements were filled with data. Typical values in our case were in the order of 190 120. With them we had to address the 1024 768 pixels of the LCoS-M; in other words, we needed to interpolate the matrix with OPD values of the interferometer at each pixel of the LCoS-M. This task can be readily done with standard numerical routines, like the function imresize of MATLAB. From OPD data of the corrected wavefronts over a circular domain we can obtain the corresponding PSFs through Fourier transformation. Graphs of these are also provided by the software of the interferometer, together with an estimation of the Strehl ratio. Since our motivation for using LCoS-M was PSF engineering, this was very relevant information. When data from Figs. 5 and 8 were used, the graphs of the PSFs resembled very closely an Airy pattern (which is, of course, the PSF of a clear circular pupil), with Strehl ratios estimations greater than 0.989. Finally, a word about the precision of our measurements. The interferometer that was used for the compensation of two LCoS-Ms is the instrument that we use for the calibration of flat surfaces in our research center. It is a commercial instrument, but as a certified center that provides flatness calibration services, its uncertainty had to be traceable. This was done and established at l/ 30. The details can be found in Ref. [15].
3. Conclusions
Fig. 5. Results of the recursive aberration correction of the LCoS-M by means of Zernike polynomials. They were obtained in a circular zone of 14 mm diameter of the modulator.
We have demonstrated that a high degree of aberration compensation can be achieved on an LCoS-M by inducing a phase-modulated grating on it. The correction is attained recursively, and three iterations are enough to have a residual error of l=16. If, as it is commonly the case, the two-dimensional, phase modulation of the grating is required over a circular domain, this can be prescribed with enough accuracy and economy in terms of Zernike polynomials. If the phase modulation must be provided for the entire surface of the LCoS-M, then this can be given after interpolating the raw data (OPDs) from the interferometer at each pixel of the modulator. In the first case, we had a residual aberration of l=10 over a 14 mm circular domain (Fig. 5).
Fig. 6. 3-D OPD graph of the reflected wavefront from the LCoS-M: (a) before correction and (b) after correction. Note the range of the scales.
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Fig. 7. Interferograms of the wavefront reflected from the LCoS-M, with a slight tilt added: (a) before correction, (b) after the first aberration correction of the modulator, (9c) after the correction of the first residual aberration, and (d) after the correction of the second residual aberration. Notice the straightness of the fringes in (d). The dashed circle is of 14 mm diameter. This is the maximum beam diameter accepted by the modulator.
Fig. 8. Results of the recursive aberration correction of the LCoS-M by means of an OPD matrix.
In the second, the residual aberration was of l/33 within the same domain, shown with a solid circle in Fig. 8d. The main advantages of the method of aberration compensation that we proposed are three: (1) there is no need to spend the whole phase modulation depth (phase stroke) to correct the LCoS, (2) there is no need of high optical quality polarizers to use with the LCoS, and (3) it is easy to adjust the amplitude of the corrected wavefront. The main disadvantages, on the other hand, are two: (1) the intensity of the beam that has been corrected from
aberrations is small (only a small percentage of the intensity of the beam reflected by the inactive LCoS-M) and (2) all the beams diffracted by the grating overlap each other over long distances of propagation. For some applications, however, these drawbacks do not represent a serious impediment. The compensation of an LCoS-M is not a long term affair. After several weeks of operation, we observed a loss of aberration correction in the modulators that we had compensated. Possible causes for this drift are the handling and repeated operation of
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the modulators, and their long term exposure to ambient temperature fluctuations. External mechanical forces, like the ones we exert on the modulators each time we fix them to their mounts, can affect the aberration compensation by several tenths of a wavelength. Because of these facts, it is important to check periodically the aberration status of the LCoS-M. In pupil engineering the requirement of an aberration-free wavefront is very stringent. The added possibility of correcting the residual aberrations of the whole optical system, apart from generating the required pupil filter, makes the LCoS technology very attractive.
Acknowledgment The authors would like to express their gratitude to Maximi˜ anza Te´cnica Industrial (CETI), in liano Gala´n, of Centro de Ensen Guadalajara, Mexico, for joining them to compensate the aberrations of his own LCoS-M, of the same model and brand than the authors’ modulator, enriching thus the experimental validation of the method of aberration compensation presented here. N.A-O. wants to acknowledge CONACYT for grant 133495. References [1] Prieto PM, Ferna´ndez EJ, Manzanera S, Artal P. Adaptive optics with a programmable phase modulator: applications in the human eye. Opt Express 2004;12:4059–71. [2] Heintzmann R, Jovin ThM, Cremer C. Saturated patterned excitation microscopy – a concept for optical resolution improvement. J Opt Soc Am A 2002;19:1599–609.
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