## Calibration of spatial light modulators suffering from spatially varying phase response |

Optics Express, Vol. 21, Issue 13, pp. 16086-16103 (2013)

http://dx.doi.org/10.1364/OE.21.016086

Acrobat PDF (1767 KB)

### Abstract

We present a method for converting the desired phase values of a hologram to the correct pixel addressing values of a spatial light modulator (SLM), taking into account detailed spatial variations in the phase response of the SLM. In addition to thickness variations in the liquid crystal layer of the SLM, we also show that these variations in phase response can be caused by a non-uniform electric drive scheme in the SLM or by local heating caused by the incident laser beam. We demonstrate that the use of a global look-up table (LUT), even in combination with a spatially varying scale factor, generally does not yield sufficiently accurate conversion for applications requiring highly controllable output fields, such as holographic optical trapping (HOT). We therefore propose a method where the pixel addressing values are given by a three-dimensional polynomial, with two of the variables being the (*x*, *y*)-positions of the pixels, and the third their desired phase values. The coefficients of the polynomial are determined by measuring the phase response in 8×8 sub-sections of the SLM surface; the degree of the polynomial is optimized so that the polynomial expression nearly replicates the measurement in the measurement points, while still showing a good interpolation behavior in between. The polynomial evaluation increases the total computation time for hologram generation by only a few percent. Compared to conventional phase conversion methods, for an SLM with varying phase response, we found that the proposed method increases the control of the trap intensities in HOT, and efficiently prevents the appearance of strong unwanted 0th order diffraction that commonly occurs in SLM systems.

© 2013 OSA

## 1. Introduction

1. E. Marom and N. Konforti, “Dynamic optical interconnections,” Opt. Lett. **12**, 539–541 (1987) [CrossRef] [PubMed] .

3. E. Hällstig, J. Öhgren, L. Allard, L. Sjöqvist, D. Engström, S. Hård, D. Ågren, S. Junique, Q. Wang, and B. Noharet, “Retrocommunication utilizing electroabsorption modulators and non-mechanical beam steering,” Opt.Eng. **44**, 045001 (2005) [CrossRef] .

4. M. Reicherter, T. Haist, E. U. Wagemann, and H. J. Tiziani, “Optical particle trapping with computer-generated holograms written on a liquid-crystal display,” Opt. Lett. **24**, 608–610 (1999) [CrossRef] .

5. E. R. Dufresne, G. C. Spalding, M. T. Dearing, S. A. Sheets, and D. G. Grier, “Computer-generated holographic optical tweezer arrays,” Rev. Sci. Instrum. **72**, 1810–1816 (2001) [CrossRef] .

6. M. A. Seldowitz, J. P. Allebach, and D. W. Sweeney, “Synthesis of digital holograms by direct binary search,” Appl. Opt. **26**, 2788–2798 (1987) [CrossRef] [PubMed] .

8. G. Milewski, D. Engström, and J. Bengtsson, “Diffractive optical elements designed for highly precise far-field generation in the presence of artifacts typical for pixelated spatial light modulators,” Appl. Opt. **46**, 95–105 (2007) [CrossRef] .

14. M. Persson, D. Engström, and M. Goksör, “Real-time generation of fully optimized holograms for optical trapping applications,” Proc. SPIE **8097**,80971H (2011) [CrossRef] .

10. M. W. Farn, “New iterative algorithm for the design of phase-only gratings,” Proc. SPIE **1555**, 34–42 (1991) [CrossRef] .

14. M. Persson, D. Engström, and M. Goksör, “Real-time generation of fully optimized holograms for optical trapping applications,” Proc. SPIE **8097**,80971H (2011) [CrossRef] .

15. X. D. Xun and R. W. Cohn, “Phase calibration of spatially nonuniform spatial light modulators,” Appl. Opt. **43**, 6400–6406 (2004) [CrossRef] [PubMed] .

16. J. Oton, P. Ambs, M. S. Millan, and E. Perez-Cabre, “Multipoint phase calibration for improved compensation of inherent wavefront distortion in parallel aligned liquid crystal on silicon displays,” Appl. Opt. **46**, 5667–5679 (2007) [CrossRef] [PubMed] .

17. D. Engström, M. Persson, and M. Goksör, “Spatial phase calibration used to improve holographic optical trapping,” in *Biomedical Optics and 3-D Imaging*, OSA Technical Digest (Optical Society of America, 2012), paper DSu2C.3 [CrossRef] .

18. G. Thalhammer, R. W. Bowman, G. D. Love, M. J. Padgett, and M. Ritsch-Marte, “Speeding up liquid crystal SLMs using overdrive with phase change reduction,” Opt. Express **21**, 1779–1797 (2013) [CrossRef] [PubMed] .

19. S. Reichelt, “Spatially resolved phase-response calibration of liquid-crystal-based spatial light modulators,” Appl. Opt. **52**, 2610–2618 (2013) [CrossRef] [PubMed] .

20. Z. Zhang, H. Yang, B. Robertson, M. Redmond, M. Pivnenko, N. Collings, W. A. Crossland, and D. Chu, “Diffraction based phase compensation method for phase-only liquid crystal on silicon devices in operation,” Appl. Opt. **51**, 3837–3846 (2012) [CrossRef] [PubMed] .

## 2. Impact of a spatially varying phase response on ideal holograms

*P*

_{0}, is very low, 0.2% and 0.1% of the total power for the circle and array, respectively.

*P*

_{0}are hardly affected at all, see Figs. 1(e) and 1(h). By compensating for such aberrations it has been shown that it is possible to restore the ideal shape of the spots [21

21. T. Cizmar, M. Mazilu, and K. Dholakia, “In situ wavefront correction and its application to micromanipulation,” Nat. Photonics **4**, 388–394 (2010) [CrossRef] .

22. R. W. Bowman, A. J. Wright, and M. J. Padgett, “An SLM-based ShackHartmann wavefront sensor for aberration correction in optical tweezers,” J. Opt. **12**, 124004 (2010) [CrossRef] .

## 3. Spatial variations in phase response of LC SLMs

*φ*of the polarized light exiting such a reflective SLM (thus passing the LC layer twice) is given by Here,

*λ*is the used wavelength, Δ

*n*=

*n*

_{eff}−

*n*

_{o}is the difference between the effective and the ordinary refractive index of the LC material, and

*d*is the thickness of the LC layer. The effective refractive index, and consequently the phase, for each pixel can be varied (

*n*

_{o}≤

*n*

_{eff}≤

*n*

_{e};

*n*

_{e}being the extraordinary refractive index) by rotating the rod-shaped LC molecules in a plane normal to the polarization of the incident beam. This is achieved by controlling the electric field over the pixel, in turn accomplished by applying a voltage over the pixel electrode and the common electrode. The SLM is addressed with an 8 or 16-bit number for each pixel, hereafter referred to as the pixel value (PV), which is converted into voltage by the SLM driving hardware.

*φ*

_{desired}(

*x*,

*y*) is the phase of the ideal hologram in position (

*x*,

*y*). The LUT can either be provided by the SLM manufacturer or determined by measuring the phase response of the SLM [23

23. T. H. Barnes, K. Matsumoto, T. Eijo, K. Matsuda, and N. Ooyama, “Grating interferometer with extremely high stability, suitable for measuring small refractive index changes,” Appl. Opt. **30**, 745–751 (1991) [CrossRef] [PubMed] .

26. D. Engström, G. Milewski, J. Bengtsson, and S. Galt, “Diffraction-based determination of the phase modulation for general spatial light modulators,” Appl. Opt. **45**, 7195–7204 (2006) [CrossRef] [PubMed] .

### 3.1. Thickness variations in liquid crystal SLMs

*d*(

*x*,

*y*). Such a thickness variation also affects the electric field applied across the pixels, resulting in a spatially dependent effective refractive index

*n*

_{eff}(

*x*,

*y*). Thus, Eq. (2) becomes dependent on

*x*and

*y*and therefore a non-flat backplane results in a spatially varying phase response.

*f*(

*x*,

*y*), hereafter referred to as the “scaling matrix method” [15

15. X. D. Xun and R. W. Cohn, “Phase calibration of spatially nonuniform spatial light modulators,” Appl. Opt. **43**, 6400–6406 (2004) [CrossRef] [PubMed] .

16. J. Oton, P. Ambs, M. S. Millan, and E. Perez-Cabre, “Multipoint phase calibration for improved compensation of inherent wavefront distortion in parallel aligned liquid crystal on silicon displays,” Appl. Opt. **46**, 5667–5679 (2007) [CrossRef] [PubMed] .

*x*,

*y*),

*f*(

*x*,

*y*) is chosen such that Eq. (4) is exactly fulfilled for some fixed value of

*φ*

_{desired}(

*x*,

*y*), e.g.,

*π*or 2

*π*.

### 3.2. Non-uniform electric drive scheme

*U*

_{pixel}V and 2.5 +

*U*

_{pixel}V at a frequency of 1 kHz. This results in a maximum usable voltage

*U*

_{pixel}of 2.5 V, for backplane electrode voltage switching between 0 and 5 V.

28. D. Preece, R. Bowman, A. Linnenberger, G. Gibson, S. Serati, and M. Padgett, “Increasing trap stiffness with position clamping in holographic optical tweezers,” Opt. Express **17**, 22718–22725 (2009) [CrossRef] .

*τ*is proportional to

*d*

^{2}/(

*E*

_{LC}+

*C*

_{LC}) where

*E*

_{LC}is the electric field applied over the LC and

*C*

_{LC}is a material constant [29

29. M. Schadt and W. Helfrich, “Voltage-dependent optical activity of a twisted nematic liquid crystal,” Appl. Phys. Lett. **18**, 127–128 (1971) [CrossRef] .

*U*

_{pixel}and 5 −

*U*

_{pixel}, the full backplane voltage of 5 V can be utilized while the LC is still DC-balanced [30]. However, since the pixel electrodes are switched sequentially (row by row, or similarly), i.e., at slightly different times relative to the switching of the common electrode, the latter approach often induces a spatially varying phase behavior.

### 3.3. LC heating induced by high laser power

*π*/2 but a relatively low value to reach 2

*π*. It is also evident that the LUT (or a more advanced addressing scheme) must be determined using the same power of the incident beam as in subsequent experiments.

## 4. Method

### 4.1. Phase modulation characterization

*in-situ*with a minimum of modifications of the setup, a diffraction-based method was used. This avoids the problems with an interferometric approach or mapping of the phase as a grayscale intensity [19

19. S. Reichelt, “Spatially resolved phase-response calibration of liquid-crystal-based spatial light modulators,” Appl. Opt. **52**, 2610–2618 (2013) [CrossRef] [PubMed] .

25. Z. Zhang, G. Lu, and F. T. S. Yu, “Simple method for measuring phase modulation in liquid crystal televisions,” Opt. Eng. **33**, 3018–3022 (1994) [CrossRef] .

26. D. Engström, G. Milewski, J. Bengtsson, and S. Galt, “Diffraction-based determination of the phase modulation for general spatial light modulators,” Appl. Opt. **45**, 7195–7204 (2006) [CrossRef] [PubMed] .

*π*equal one; a typical result is shown in Fig. 4(c). Finally, the phase is calculated according to

*i*is the region index and

*i*.

### 4.2. Fitting a 3D polynomial to the measured data

*f*(

*φ*

_{desired},

*x*,

*y*) that gives the optimal PV for a desired phase value

*φ*

_{desired}at the position on the SLM given by

*x*and

*y*. To do this, the determined relations between

*φ*

_{desired}and PV for all measured SLM regions are arranged in a linear equation system. Exemplified using a polynomial of the seventh order we end up with

*N*is the number of SLM subregions used for the fit,

*K*is the number of measured phase levels,

**c**= [

*c*

_{0}

*c*

_{1}···

*c*]

_{M}*contains the coefficients that determine the polynomial, and*

^{T}*M*is the number of terms in the polynomial. In the example given above, with a polynomial of the seventh order,

*M*= 120. Each row in the matrix on the left side contains the polynomial terms for a set (

*φ*,

*x*,

_{i}*y*) and on the same row in the vector on the right side is the corresponding PV that yielded the phase

_{i}*φ*. Here,

*x*and

_{i}*y*correspond to the coordinates defining the center of SLM subregion

_{i}*i*. Finally, the coefficients are obtained by solving the, generally overdetermined, linear equation system in Eq. (6).

### 4.3. Phase compensation using the 3D polynomial

*φ*

_{desired}(

*x*,

*y*), to the suitable PV matrix; for each SLM pixel the PV is given by the polynomial

14. M. Persson, D. Engström, and M. Goksör, “Real-time generation of fully optimized holograms for optical trapping applications,” Proc. SPIE **8097**,80971H (2011) [CrossRef] .

31. Software available at http://www.physics.gu.se/forskning/komplexa-system/biophotonics/download/hotlab/

## 5. Experiments and results

### 5.1. Optical setup

*μ*m×10.6

*μ*m) was used for capturing bright field images except for the trapping experiments described in Section 5.4.3.

*in-situ*within the HOT setup. Only minor adjustments of the HOT setup were made, see Fig. 5(b). First, the IR filter positioned in front of the bright field camera was removed in order to image the reflection of the trapping laser. Second, the 0th order spot was blocked outside the microscope. For measurements with high optical power incident on the SLM,

*P*

_{SLM}≥ 0.5 W, a reflective ND filter (optical density 2–3) was also placed in the beam path outside the microscope. The latter modifications were done in order to reduce the amount of light incident on the camera. Since only 1/64 of the SLM area is used to diffract light to the 1st order in each sequence, the optical power in the 0th order is very high and might even damage the camera sensor.

### 5.2. Phase characterization

8. G. Milewski, D. Engström, and J. Bengtsson, “Diffractive optical elements designed for highly precise far-field generation in the presence of artifacts typical for pixelated spatial light modulators,” Appl. Opt. **46**, 95–105 (2007) [CrossRef] .

32. M. Persson, D. Engström, and M. Goksör, “Reducing the effect of pixel crosstalk in phase only spatial light modulators,” Opt. Express **20**, 22334–22343 (2012) [CrossRef] [PubMed] .

*φ*=

_{i}*π*, 2

*π*, etc. This raw data curve was then normalized in segments between these PV values, by subtracting a constant “dark intensity” and multiplying by an appropriate constant, such that the value is either zero or one at the beginning and end of each segment, depending on whether

*φ*is an even or odd integer of

_{i}*π*in that position. The phase

*φ*(PV) was then extracted from the normalized curve according to Eq. (5) and is shown in Figs. 6(b) and 6(d) for all subregions. Table 1 shows the ranges of PV that yield a phase of

_{i}*π*, 2

*π*, and 3

*π*somewhere on the SLM. From these measurements, it is evident that the phase response varies drastically across the surface of the SLM. In some cases, the obtained phase modulation differs by

*π*for the same PV in different positions on the SLM. As a remark, this spatial variation in the phase response is larger than for the SLM used in Ref. 19

19. S. Reichelt, “Spatially resolved phase-response calibration of liquid-crystal-based spatial light modulators,” Appl. Opt. **52**, 2610–2618 (2013) [CrossRef] [PubMed] .

### 5.3. Polynomial fitting and accuracy

*π*; Fig. 7(a) shows the error in PV between the measurements (averaged for each sub-region) and the polynomial of 7th order for four of these phase values. By averaging the PVs obtained from the polynomial for each subregion and phase value, the phase response could be determined from the measurements. For each subregion the mean and maximal phase error (among the 20 phase values) were determined. The largest mean and maximal phase error from all 52 subregions and 32 most central subregions are shown in Figs. 7(b) and 7(c), respectively.

**52**, 2610–2618 (2013) [CrossRef] [PubMed] .

20. Z. Zhang, H. Yang, B. Robertson, M. Redmond, M. Pivnenko, N. Collings, W. A. Crossland, and D. Chu, “Diffraction based phase compensation method for phase-only liquid crystal on silicon devices in operation,” Appl. Opt. **51**, 3837–3846 (2012) [CrossRef] [PubMed] .

**52**, 2610–2618 (2013) [CrossRef] [PubMed] .

### 5.4. Method evaluation

*f*(

*x*,

*y*) is chosen such that Eq. (4) is fulfilled for

*φ*

_{desired}=

*π*. In the optical trapping experiment, the method was compared to the use of a global LUT.

#### 5.4.1. Binary gratings covering a subregion of the SLM

*φ*

_{desired}was stepped from 0 to 2

*π*. The three tested methods for converting desired phase to PV were then used to convert each grating to the corresponding PV matrix. Finally, the power in one of the 1st diffraction orders was measured and normalized to the maximum value and the realized phase was calculated using Eq. (5).

*π*, and the phase response curves, see Figs. 8(d)–8(f), should have a constant slope of 1 and no offset. While the scaling matrix method brings the phase response curves closer to the ideal line – the maximum error is reduced from 0.8

*π*to 0.6

*π*– it fails to compensate for their varying shapes. With the 3D polynomial method, the response curves are brought much closer to the ideal line and the maximum error is reduced to 0.3

*π*.

#### 5.4.2. Holograms covering the full SLM

*π*. Again, a grating period of 16 SLM pixels were used. The powers in the zeroth and the two first diffraction orders were measured using a camera. The power in the two first diffraction orders should then ideally vary as

*P*

_{tot}(2/

*π*)

^{2}sin

^{2}(

*φ*/2) and the zeroth order should vary as

*P*

_{tot}cos

^{2}(

*φ*/2), where

*P*

_{tot}is the total power in the trapping plane. Thus, the zeroth order should completely vanish at

*φ*=

*π*and equal

*P*

_{tot}at

*φ*= 2

*π*and the first diffraction orders should completely vanish at

*φ*= 2

*π*. In Fig. 9, the powers for the three measured diffraction orders are plotted for each of the three methods.

*P*

_{tot}, 0.26

*P*

_{tot}, and 0.074

*P*

_{tot}for the three methods, respectively. At

*φ*= 2

*π*the zeroth order equals 0.91

*P*

_{tot}, 0.96

*P*

_{tot}, and 0.97

*P*

_{tot}and the highest of the two 1st order powers equals 0.16

*P*

_{tot}, 0.069

*P*

_{tot}, and 0.028

*P*

_{tot}, respectively. The phase for which the 0th order minimum and ±1st order maxima is found equals 1.05

*π*, 0.88

*π*, and 0.93

*π*radians for the three methods, respectively. Also, an analysis of the shapes of the curves shows that the global LUT and the 3D polynomial give an equally decent fit to the ideal sine-squared shapes while the scaling matrix method degrades the curve shapes. As seen, none of the methods removes the 0th order completely for a phase step of

*π*radians, nor do the first orders completely vanish when the phase step reaches 2

*π*. The reason for this might be that the spatial phase response is still not perfectly corrected for. However, a more pronounced effect is likely that the realized phase gratings are smeared out due to pixel crosstalk resulting in non-ideal “binary” gratings [8

8. G. Milewski, D. Engström, and J. Bengtsson, “Diffractive optical elements designed for highly precise far-field generation in the presence of artifacts typical for pixelated spatial light modulators,” Appl. Opt. **46**, 95–105 (2007) [CrossRef] .

32. M. Persson, D. Engström, and M. Goksör, “Reducing the effect of pixel crosstalk in phase only spatial light modulators,” Opt. Express **20**, 22334–22343 (2012) [CrossRef] [PubMed] .

*P*

_{SLM}= 50 mW, 0.5 W, and 1.5 W.

*α*

_{max}and one producing 24 spots forming a regular 5×5 grid with a spacing of 0.125

*α*

_{max}; the center position excluded. Here,

*α*

_{max}= sin

^{−1}(

*λ*/2

*p*) is the maximal steering angle allowed by the pixelated SLM;

*p*is the pixel pitch. Both spot arrangements were centered on the optical axis. The two holograms were optimized using a modified Gerchberg-Saxton algorithm to obtain nearly perfect theoretical uniformity [14

**8097**,80971H (2011) [CrossRef] .

#### 5.4.3. Holographic optical trapping

*μ*m×14

*μ*m).

*μ*m) suspended in water were trapped and positioned with a spacing of 7.6

*μ*m along the

*x*-axis; the central bead coinciding with the optical axis. To obtain bright field images suitable for bead position determination the traps were positioned in a plane 2.4

*μ*m in front of the imaging/trapping plane of the microscope objective. The latter plane coincides with the imaged Fourier plane of the SLM, which is the plane where the 0th order spot is located. The 2.4

*μ*m longitudinal displacement of the traps was accomplished by including a spherical phase curvature when optimizing the hologram. Finally, the sample was placed such that the beads were positioned 10

*μ*m above the glass substrate.

**8097**,80971H (2011) [CrossRef] .

^{−3}pN/nm and 0.36·10

^{−3}pN/nm for the global LUT and the 3D polynomial, respectively; the trap stiffness uniformity, calculated with the maximum and minimum stiffness values similarly to Eq. (1), is 90% and 96%, respectively. For a desired trap power of 20% of the total power the mean trap stiffness is 0.65·10

^{−3}pN/nm and 0.68

*·*10

^{−3}pN/nm for the global LUT and the 3D polynomial, respectively. Furthermore, the trap stiffness uniformity is 81% and 94%, respectively. Thus, the 3D polynomial method increases both the trap stiffness of the traps and the uniformity thereof.

*μ*m behind it. However, when a hologram optimized to yield five traps with a desired power in each trap equal to only 2% of the total power, for the global LUT method the 0th diffraction order spot is strong enough to capture the bead from the middle trap, see Fig. 11(e). When the 3D polynomial method was used, the 0th diffraction order was still too weak to affect the bead in the middle trap, see Fig. 11(f). Thus, with this method very weak traps can be efficiently used also in close vicinity to the 0th diffraction order; this is important, e.g., if sensitive living cells are to be trapped.

## 6. Conclusions

*x*,

*y*)-coordinates and the desired phase of each pixel. Experimental evaluations of holographically generated configurations of intensity spots and optically trapped beads confirm that the SLM behaves more ideally than when previously proposed conversion methods are used. The main advantages are that the unwanted 0th diffraction order, i.e., optical power on the optical axis of the system, is strongly suppressed, and that the optical power is more accurately distributed among the desired spots/traps. In HOT, the suppression of the 0th diffraction order is a major improvement as it means that unwanted particles being drawn into the optical axis, typically in the center of the measurement region, is no longer such a severe problem. Thus, there is no need to block the 0th diffraction order outside the microscope. Instead, traps close to – or even coinciding with – the optical axis behave as any other trap. Furthermore, the 3D polynomial method has shown to increase both the trap stiffness and the trap uniformity.

32. M. Persson, D. Engström, and M. Goksör, “Reducing the effect of pixel crosstalk in phase only spatial light modulators,” Opt. Express **20**, 22334–22343 (2012) [CrossRef] [PubMed] .

## Acknowledgments

## References and links

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2. | P. F. McManamon, T. A. Dorschner, D. L. Corkum, L. J. Friedman, D. S. Hobbs, M. Holtz, S. Liberman, H. Q. Nguyen, D. P. Resler, R. C. Sharp, and E. A. Watson, “Optical phased array technology,” Proc. SPIE |

3. | E. Hällstig, J. Öhgren, L. Allard, L. Sjöqvist, D. Engström, S. Hård, D. Ågren, S. Junique, Q. Wang, and B. Noharet, “Retrocommunication utilizing electroabsorption modulators and non-mechanical beam steering,” Opt.Eng. |

4. | M. Reicherter, T. Haist, E. U. Wagemann, and H. J. Tiziani, “Optical particle trapping with computer-generated holograms written on a liquid-crystal display,” Opt. Lett. |

5. | E. R. Dufresne, G. C. Spalding, M. T. Dearing, S. A. Sheets, and D. G. Grier, “Computer-generated holographic optical tweezer arrays,” Rev. Sci. Instrum. |

6. | M. A. Seldowitz, J. P. Allebach, and D. W. Sweeney, “Synthesis of digital holograms by direct binary search,” Appl. Opt. |

7. | B. K. Jennison, J. P. Allebach, and D. W. Sweeney, “Efficient design of direct-binary-search computer-generated holograms,” J. Opt. Soc. Am. A |

8. | G. Milewski, D. Engström, and J. Bengtsson, “Diffractive optical elements designed for highly precise far-field generation in the presence of artifacts typical for pixelated spatial light modulators,” Appl. Opt. |

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12. | D. Engström, A. Frank, J. Backsten, M. Goksör, and Jörgen Bengtsson, “Grid-free 3D multiple spot generation with an efficient single-plane FFT-based algorithm,” Opt. Express |

13. | S. Bianchi and R. Di Leonardo, “Real-time optical micro-manipulation using optimized holograms generated on the GPU,” Comput. Phys. Commun. |

14. | M. Persson, D. Engström, and M. Goksör, “Real-time generation of fully optimized holograms for optical trapping applications,” Proc. SPIE |

15. | X. D. Xun and R. W. Cohn, “Phase calibration of spatially nonuniform spatial light modulators,” Appl. Opt. |

16. | J. Oton, P. Ambs, M. S. Millan, and E. Perez-Cabre, “Multipoint phase calibration for improved compensation of inherent wavefront distortion in parallel aligned liquid crystal on silicon displays,” Appl. Opt. |

17. | D. Engström, M. Persson, and M. Goksör, “Spatial phase calibration used to improve holographic optical trapping,” in |

18. | G. Thalhammer, R. W. Bowman, G. D. Love, M. J. Padgett, and M. Ritsch-Marte, “Speeding up liquid crystal SLMs using overdrive with phase change reduction,” Opt. Express |

19. | S. Reichelt, “Spatially resolved phase-response calibration of liquid-crystal-based spatial light modulators,” Appl. Opt. |

20. | Z. Zhang, H. Yang, B. Robertson, M. Redmond, M. Pivnenko, N. Collings, W. A. Crossland, and D. Chu, “Diffraction based phase compensation method for phase-only liquid crystal on silicon devices in operation,” Appl. Opt. |

21. | T. Cizmar, M. Mazilu, and K. Dholakia, “In situ wavefront correction and its application to micromanipulation,” Nat. Photonics |

22. | R. W. Bowman, A. J. Wright, and M. J. Padgett, “An SLM-based ShackHartmann wavefront sensor for aberration correction in optical tweezers,” J. Opt. |

23. | T. H. Barnes, K. Matsumoto, T. Eijo, K. Matsuda, and N. Ooyama, “Grating interferometer with extremely high stability, suitable for measuring small refractive index changes,” Appl. Opt. |

24. | A. Bergeron, J. Gauvin, F. Gagnon, D. Gingras, H. H. Arsenault, and M. Doucet, “Phase calibration and applications of a liquid-crystal spatial light modulator,” Appl. Opt. |

25. | Z. Zhang, G. Lu, and F. T. S. Yu, “Simple method for measuring phase modulation in liquid crystal televisions,” Opt. Eng. |

26. | D. Engström, G. Milewski, J. Bengtsson, and S. Galt, “Diffraction-based determination of the phase modulation for general spatial light modulators,” Appl. Opt. |

27. | A. Linnenberger, S. Serati, and J. Stockley, “Advances in Optical Phased Array Technology,” Proc. SPIE |

28. | D. Preece, R. Bowman, A. Linnenberger, G. Gibson, S. Serati, and M. Padgett, “Increasing trap stiffness with position clamping in holographic optical tweezers,” Opt. Express |

29. | M. Schadt and W. Helfrich, “Voltage-dependent optical activity of a twisted nematic liquid crystal,” Appl. Phys. Lett. |

30. | A. Linnenberger and Teresa Ewing, Boulder Nonlinear Systems, 450 Courtney Way, #107 Lafayette, CO 80026, USA (personal communication, February 2013). |

31. | Software available at http://www.physics.gu.se/forskning/komplexa-system/biophotonics/download/hotlab/ |

32. | M. Persson, D. Engström, and M. Goksör, “Reducing the effect of pixel crosstalk in phase only spatial light modulators,” Opt. Express |

33. | C. Runge, “Über empirische Funktionen und die Interpolation zwischen äquidistanten Ordinaten,” in |

**OCIS Codes**

(090.1760) Holography : Computer holography

(090.1970) Holography : Diffractive optics

(090.2890) Holography : Holographic optical elements

(120.5060) Instrumentation, measurement, and metrology : Phase modulation

(140.7010) Lasers and laser optics : Laser trapping

(230.6120) Optical devices : Spatial light modulators

(090.1995) Holography : Digital holography

(350.4855) Other areas of optics : Optical tweezers or optical manipulation

**ToC Category:**

Optical Devices

**History**

Original Manuscript: April 29, 2013

Revised Manuscript: June 10, 2013

Manuscript Accepted: June 13, 2013

Published: June 28, 2013

**Virtual Issues**

Vol. 8, Iss. 8 *Virtual Journal for Biomedical Optics*

**Citation**

David Engström, Martin Persson, Jörgen Bengtsson, and Mattias Goksör, "Calibration of spatial light modulators suffering from spatially varying phase response," Opt. Express **21**, 16086-16103 (2013)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-21-13-16086

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