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Improved beam steering accuracy of a

single beam with a 1D phase-only

spatial light modulator

David Engstr¨ om,1J¨ orgen Bengtsson,2Emma Eriksson,1

and Mattias Goks¨ or1

1Biophotonics Group, Department of Physics,

University of Gothenburg, SE-412 96 G¨ oteborg, Sweden

2Photonics Laboratory, Department of Microtechnology and Nanoscience,

Chalmers University of Technology, SE-412 96 G¨ oteborg, Sweden

david.engstrom@physics.gu.se

Abstract:

modulation values limit the positioning accuracy when a phase-only one

dimensional spatial light modulator (SLM) is used for beam steering.

Applying the straightforward recipe for finding the optimal setting of the

SLM pixels, based on individually optimizing the field contribution from

each pixel to the field in the steering position, the inaccuracy can be a

significant fraction of the diffraction limited spot size. This is especially

true in the vicinity of certain steering angles where precise positioning

is particularly difficult. However, by including in the optimization of

the SLM setting an extra degree of freedom, we show that the steering

accuracy can be drastically improved by a factor proportional to the

number of pixels in the SLM. The extra degree of freedom is a global

phase offset of all the SLM pixels which takes on a different value for

each steering angle. Beam steering experiments were performed with the

SLM being set both according to the conventional and the new recipe,

andtheresultswereinverygoodagreementwiththetheoreticalpredictions.

The limited number of pixels and their quantized phase

© 2008 Optical Society of America

OCIS codes: (090.1995) Digital holography; (090.2890) Holographic optical elements;

(090.1970) Diffractive optics; (230.6120) Spatial light modulators; (120.5060) Phase modula-

tion; (230.1950) Optical devices : Diffraction gratings; (350.4855) Optical tweezers or optical

manipulation

References and links

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2. D. C. O’Brien, R. J. Mears, T. D. Wilkinson, and W. A. Crossland, “Dynamic holographic interconnects that use

ferroelectric liquid-crystal spatial light modulators,” Appl. Opt. 33, 2795–2803 (1994).

3. D.-J. Lee, N. Kim, S.-H. Song, and H.-H. Suh, “Optical routing system consisting of spatial light modulator and

kinoform phase gratings,” Proc. SPIE 2689, 300–304 (1996).

4. M. Johansson, S. H˚ ard, B. Robertson, I. Manolis, T. Wilkinson, and W. Crossland, “Adaptive beam steering

implemented in a ferroelectric liquid-crystal spatial-light-modulator free-space, fiber-optic switch,” Appl. Opt.

41, 4904–4911 (2002).

5. E. H¨ allstig, J.¨Ohgren, L. Allard, L. Sj¨ oqvist, D. Engstr¨ om, S. H˚ ard, D. ˚ Agren, S. Junique, Q. Wang, and B.

Noharet, “Retrocommunication utilizing electroabsorption modulators and non-mechanical beam steering,” Opt.

Eng. 44, 45001–1–8 (2005).

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27 October 2008 / Vol. 16, No. 22 / OPTICS EXPRESS 18275

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6. M. Reicherter, T. Haist, E. Wagemann, and H. Tiziani, “Optical particle trapping with computer-generated holo-

grams written on a liquid-crystal display,” Opt. Lett. 24, 608–610 (1999).

7. J. E. Curtis, B. A. Koss, and D. G. Grier, “Dynamic holographic optical tweezers,” Opt. Commun. 207, 169–175

(2002).

8. K. L. Tan, S. T. Warr, I. G. Manolis, T. D. Wilkinson, M. M. Redmond, W. A. Crossland, R. J. Mears, and B.

Robertson, “Dynamic holography for optical interconnections. II. Routing holograms with predictable location

and intensity of each diffraction order,” J. Opt. Soc. Am. A 18, 205–215 (2001).

9. R. James, F. A. Fern´ andez, S. E. Day, M. Komarˇ cevi´ c, and W. A. Crossland, “Modeling of the diffraction effi-

ciency and polarization sensitivity for a liquid crystal 2D spatial light modulator for reconfigurable beam steer-

ing,” J. Opt. Soc. Am. A 24, 2464–2473 (2007).

10. C. H. J. Schmitz, J. P. Spatz, and J. E. Curtis, “High-precision steering of multiple holographic optical traps,”

Opt. Express 13, 8678–8685 (2005).

11. J. W. Goodman, “Introduction to Fourier optics,” (McGraw-Hill, 1996).

12. K. Ball¨ uder and M. R. Taghizadeh, “Optimized phase quantization for diffractive elements by use of a bias

phase,” Opt. Lett. 24, 1756–1758 (1999).

13. D. Engstr¨ om, 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).

14. J. Harriman, A. Linnenberger, and S. Serati, “Improving spatial light modulator performance through phase

compensation,” Proc. SPIE 5553, 58–67 (2004).

15. R. W. Gerchberg and W. O. Saxton,“A practical algorithm for the determination of phase from image and dif-

fraction plane pictures,” Optik 35, 237–246 (1972).

1. Introduction

Phase-only spatial light modulators (SLMs) have been used for beam steering for various op-

tical tasks such as optical switching/routing [1–4], object tracking [5], and in optical tweez-

ers [6, 7]. For beam steering, two characteristics are of general importance; the steering effi-

ciency (fraction of total power in steered beam) and the steering (positioning) accuracy. The

steering efficiency has been studied in detail [8, 9], while only little work has been published

on the positioning accuracy of actual SLMs yielding a phase-only response [10].

In this paper we look into this issue for a parallel incident beam being steered/deflected by

a 1D phase-only SLM. First, in Section 2 we review the conventional method for setting the

phase of the SLM pixels and give a simple analytic expression for the approximate positioning

accuracy that can be expected with this method. We also point out that this method often yields

far from optimal results; for instance, during a continuous sweep there might even be two con-

secutive settings for which the beam movesin the reversedirection.To avoid the pooraccuracy,

which is caused by not accounting for the limited number of possible phase values in the SLM

modulation, in Section 3 we propose a new scheme to optimize the SLM for a higher preci-

sion in the realized steering angle. The simulation results are confirmed with measurements

presented in Section 4.

2. Beam steering with conventional setting of the SLM pixels

2.1.SLM pixel setting

Using a blazed grating, the diffractive counterpart of the prism, it is possible to steer a mono-

chromatic single beam into any given direction. However, when using a pixelated SLM, the

phase modulation of the blazed grating has to be approximated with a staircase grating with

phase resets. This limits the smallest realizable period of the grating to two pixel widths yield-

ing a maximal deflection angle of

αmax= sin−1λ

2p,

(1)

according to the grating equation, where λ is the wavelength of the incident light and p is the

pixel width. However, as will be shown in this work, while a pixelated SLM with an analog

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Received 18 Jul 2008; revised 17 Oct 2008; accepted 19 Oct 2008; published 23 Oct 2008

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012

xj/ p

345

1

2

3

4

y p

/

0

90

180

270

360

123456

Pixel number

Phase ( )

o

Phase (a.u.)

?j?

??

?

(a)

(c)

0

1

“”

??

0

(b)

123456

Pixel number

Fig. 1. (a) An ideal wavefront with a steering angle α (y = xtanα, xj= (j−1)p where p

is the pixel width) and the wave contributions from the pixels. (b) An ideal phase setting

(dashed) and the staircase approximations realizable with a pixelated SLM with an analog

(solid) and quantized (dotted), M = 4, phase modulation. (c) Two ideal phase settings, cor-

responding to two wavefronts with a difference in steering angle of δα, and the realizable

phase levels in between them.

phase response can still steer to any angle within the angular range (limited by ±αmax) a pixe-

lated SLM with a quantized phase response inevitably results in a lowered steering accuracy.

When steering a beam to an angle α, the wave leaving the SLM should ideally be a plane

wave with a wavefront tilted by the angle α, see Fig. 1(a). The phase modulation over the SLM

surface of the ideal tilted plane wave is

ϕideal(x,α) =2π

λxsinα +ϕ0,

(2)

where x is the spatial coordinate in the SLM plane and ϕ0is an arbitrary constant phase offset.

It is straightforward to find the optimal phase setting for a pixelated SLM with an analog phase

response; the phase values are chosen to coincide with the ideal phase in the pixel centers, see

the solid line in Fig. 1(b). The phase is given by

ϕideal

j

(α) =2π

λxjsinα +ϕ0,

(3)

where xjis the center x-coordinate for pixel j, j = 1,2,...,N, where N is the total number of

pixels. If the SLM only allows M equidistant phase levels between 0 and 2π, the phase values

in the pixel centers are set to the allowed phase level closest to the ideal values at the same

spatial position, see the dotted line in Fig. 1(b). This is achieved by the following expression

?

ϕj(α) = round

ϕideal

j

(α)M

2π

?

2π

M,

(4)

where “round” simply rounds the value to the closest integer value. Of course, in addressing

the SLM all the phase values are taken modulo 2π, yielding wrapped staircase gratings.

2.2. Coarse estimation of steering accuracy

There is a simple way to estimate the typical changeof the beam angle whenscanningthe beam

in as small steps as possible. The idea is simply to count the number of changing events S, i.e.,

the numberof times any of the pixels in the SLM has changedits setting - and therebyprobably

causing some motion of the beam – during during a scan from a nominal steering angle α to

α +δα. We can then estimate the typical smallest angular change, and thus typical positioning

error, simply as αerror= |δα|/S. Evidently, it is possible to have larger inaccuracies than this;

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1 2 3 4 5 6 7 8

log2( )

M

5678910

11

0

0.5

1

log2( )

N

Q

1 2 3 4 5 6 7 8

log2( )

M

0

0.5

1

Q

1 2 3 4 5 6 7 8

log2( )

M

0

0.5

1

Q

(a)(b) (c)

5678910

11

2( )

N

log

5678910

11

2( )

N

log

Fig. 2. Ratio Q between the actual number of changes Sactualin the SLM setting and

the number S given by Eq. (6) when the beam is being steered over an angular in-

terval [αstart,αstart+αspot] with (a) αstart= 0, (b) αstart= 0.5 ×αmax, and (c) αstart=

0.625 ×αmax. The number of SLM pixels N = 32, 64, 128, ..., 2048. The number of

phase levels M = 2, 4, 8, ..., 256.

the estimation does not account for the actual movement of the beam and in general the pixel

changing events are not evenly distributed over the nominal steering angles. For instance, sev-

eral pixel changing events might even coincide completely, which in effect reduces the actual

number of changing events. To calculate S, we calculate the number of times pixel number j

changes its setting when the nominal steering angle changes from α to α +δα, which is given

by the total phase change divided by the step size of the phase as

Sj≈|ϕideal

j

(α +δα)−ϕideal

2π/M

j

(α)|

≈

2π

λxj|sin(α +δα)−sinα|

2π/M

≈Mxj|δα|

λ

,

(5)

since α and δα are small angles. The total number of changes of any of the N pixels in the

SLM is obtained by summing Sjover all the pixels

S =

N

∑

j=1

Sj= {xj= (j−1)p} =M|δα|p

λ

N

∑

j=1

(j−1)=M|δα|p

λ

(N −1)N

2

≈|δα|pMN2

2λ

. (6)

Thus

αerror=2λ

p

1

MN2.

(7)

Finally, we relate this to the diffractionlimited spot size of the beam, which we conveniently

define as αspot= λ/Np, so the angular inaccuracy is approximately

αerror≈

2

MNαspot,

(8)

which is in accordance with the results from the analysis in Ref. [10].

As mentioned this value is approximate, partly because we count as multiple events a single

event where simultaneouslymore than one pixel changes. We have simulated a large numberof

situations in which the beam was steered over the nominal angular interval [αstart,αstart+αspot]

in a continuous scan, i.e., the desired steering angle was changed in very small steps. Thus for

most steps there was no change at all in the settings of the realized SLM, but those steps where

the SLM setting did change were counted to get the true number of changing events, Sactual.

Apart from situations when the SLM setting was nearly periodic, we found as a rule of thumb

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00.20.4

Aiming angle / ?spot

0.60.81

0

0.2

0.4

0.6

0.8

1

Realized angle / ?spot

Pos IPos II

Ideal

Realized, Eq. 4, ? ?

0

?

?norm,max

?0= 0

Fig. 3. (a) Simulations of 500 realized steering angles for an SLM (N =32 and M =4) with

ϕ0= 0. Pos I and II indicate an angle for which the steering error is positive and negative,

respectively. The maximum deviation from the aiming angle, relative to the beam spot size,

εϕ0=0

norm,max≈ 0.15.

that Sactual≈0.8S, as illustrated in Fig. 2, thereby slightly increasing the positioninginaccuracy

measure to

αerror≈

0.8

In our analysis we used x ∈ [0,p,...,(N −1)p] for the pixel center positions, and Eq. (4) for

finding the realizable phase setting of each pixel. Thus we see that the phase of the leftmost

pixel, at x = 0, is always kept fixed, while the others are trying to approximate the wavefront

at the nominal angle α. Of course, we could choose any other pixel to be the one that is al-

ways fixed; however this increases the number of events when two or more pixels are changed

simultaneously and thus lowers Sactualleading to higher inaccuracy. This situation is worst if

the phase is fixed in the most central pixel in the SLM, which unfortunately is a likely choice

for anyone who is not considering inaccuracy effects in their beam steering SLM. Then our

simulations show that Sactualis only ∼ 0.2S, and hence our simple estimation would suggest a

fourfold increase in positioning inaccuracy. Thus, we have already obtained a very simple de-

sign rule: choose the pixel whose phase you keep fixed to be in the periphery, not in the center,

of the SLM.

1

2

MNαspot.

(9)

2.3. Actual steering accuracy

To this point we considered the number of different SLM settings during a scan of the beam.

This gives a crude measure of the positioning accuracy,but what in fact determines the steering

angle is the mean slope of the phase modulation ”staircase” generated by the SLM pixels. The

mean slope is the slope of the line obtainedas the best linear fit to the staircase. In fact there is a

virtually exact correspondence, such that the position of the beam from an SLM with a certain

value of the mean slope of the phase staircase, measured in radians per unit length along the

SLM, is exactly (within a very small fraction of αspot) the same as that of a beam produced by

a perfect wavefront with the same slope. Although this might seem quite reasonable, we also

checked this claim thoroughly by comparing the mean-slope method for obtaining the steering

angle with rigorous calculations of the spot position using the Fresnel diffraction integral[11]

(data not shown), and both yielded in all essence the same results. Thus, all beam positioning

simulations that follow are directly based on the calculation of the mean slope of the phase

modulation staircase.

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Received 18 Jul 2008; revised 17 Oct 2008; accepted 19 Oct 2008; published 23 Oct 2008

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