Page 1

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

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

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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Page 2

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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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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1

1

1

0.51

1

1

NM

= 32,= 2

NM

= 32, = 8

NM

= 32,= 32

NM

= 512,= 2

NM

= 512,= 8

0

-40

0

40

0

-10

0

10

0

-2.5

0

2.5

?norm(%)

?norm(%)

?norm(%)

NM

= 512,= 32

(a)

(b)

(c)

(d)

(e)

(f)

0.5

? / ?max

0.5

? / ?max

0.5

? / ?max

? / ?max

0.5

? / ?max

0.5

? / ?max

0

-40

0

40

0

-10

0

10

0

-2.5

0

2.5

?norm(%)

?norm(%)

?norm(%)

Fig. 4. Simulated normalized steering error εnormin the entire possible scanning range for

ϕ0=0, (a) N =32 and M = 2, (b) N =32 and M =8, (c) N =32 and M =32, (d) N =512

and M = 2, (e) N = 512 and M = 8, and (f) N = 512 and M = 32.

As a consequence, for an SLM whose pixels can take on any phase value, i.e., an analog

SLM, the phase errors inevitably induced by the staircase approximation due to the pixelation

of the SLM causes a decreased steering efficiency while the direction of the steered beam still

perfectlycoincides with the aimingdirection.However,as soon as the phase responseis limited

to a finite number of phase levels the induced phase errors generally affect the mean slope

of the approximate staircase grating, thus steering the beam in a slightly different direction

than intended. In Fig. 3, the realized steering angle using Eq. (4) with ϕ0= 0, is shown as a

function of the aiming angle when the beam is steered over the angular region [0,αspot]. Here,

the normalized steering error εnormhas been defined as the difference between realized and

aiming angle divided by the angular spot size. Also, the maximal normalized steering εnorm,max

is defined as the absolute maximum of εnormover the full steering range.

As seen in Fig. 3, the agreement between the aiming and realized steering angles can be

surprisingly poor, and in the aiming angle region [0.25,0.40]αspotthe beam even reverses its

scanning direction. For the worst aiming angle, indicated as Pos I in the figure, the maximal

normalized steering error εϕ0=0

is not primarily that the SLM can only provide certain steering angles but rather that the recipe

for realizing the wavefronts steers the beam in a direction sometimes quite far from the aiming

direction. For instance, in Pos I we aim for a steering angle of ∼ 0.2αspotbut obtain something

quite different,although it is obvious from the figure that the SLM is indeed physically capable

of producing a steering angle quite close to 0.2αspot.

One might argue that the simulation shown here was made for an unrealistically low value of

N = 32 and that the steering error would decrease for larger values. It is true that the steering

error, measured in absolute angle, does decrease as the number of pixels increases, keeping

everything else unchanged. However, when the number of SLM pixels increases, the angular

spot size αspot= λ/Np decreases as well and thus, as it turns out, the maximum angular steer-

ing error normalized to the angular spot size is in fact very similar for any choice of N. This

can be seen in Fig. 4 showing the results of steering the beam over the entire angular range

[0,αmax] achievable with the SLM. The figure shows the simulated normalized steering error

norm,maxis as large as 15%. However, it is evident that the problem

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181624

32

1

2

3

Pixel number

Phase level

(a)

181624

32

1

2

3

Pixel number

Phase level

(c)

18 1624

32

1

2

3

Pixel number

Phase level

(b)

18 1624

32

1

2

3

Pixel number

Phase level

(d)

Phase threshold levels

“ ” ?

“ ” ?

?0

?0

Realized mean phase tilt

Ideal phase

Realized phase

Fig.5. Idealphase (blue dash-dotted), realized staircase phase modulation (black solid), and

the mean phase tiltof the realized modulation (red dashed) for the aiming angles labeled (a)

Pos I and (b) Pos II in Fig. 3 for N = 32, M = 4, and ϕ0= 0. The tilt error, corresponding

to an error ε in steering angle is indicated. (c) and (d) show the two same cases as in (a)

and (b) but for the optimal choice of ϕ0. The black dotted lines indicate the threshold phase

levels used for rounding the phase to the closest allowed phase level; they are the same in

all figures.

εnormas a function of the aiming angle for N = 32 and 512 with M = 2, 8, and 32. Note that

while the error peaks are sharper for larger N-values the peak height is independentof N. Also,

Fig. 4 clearly shows that the maximal normalized steering error εnorm,maxdecreases with an in-

creasing numberof phase levels M, as it should.Furtheranalysis of the simulations showedthat

εϕ0=0

pixels. Hence, the steering error and its causes and remedies should be of interest also for the

large SLMs (N ≈ 1000) used today. Evidently, one has to be particularly careful when using a

binary SLM, for which M = 2.

Simulations also showed that if ϕ0is set such that ϕN/2=0, pinningthe phase modulationof

a pixel in the SLM center to a constant value for all aiming angles, a larger normalized steering

error of ε

less severe than the factor of 4 based on the simplified analysis of the numberof SLM changing

events in Section 2.2.

norm,max≈ 0.625M−1, which again underscores that the error is independent of the number of

ϕN/2=0

norm,max≈M−1is obtained. However,the deteriorationis thus a factor of 1.6, which is

3. Improved steering accuracy by optimizing ϕ0for each aiming angle

From Fig. 3, it is clear that the simple phase mapping with ϕ0= 0 used in the previous section

is not suitable for accurate beam steering. In Figs. 5(a) and (b) we illustrate why this method

fails for the aiming angles labeled Pos I and Pos II in Fig. 3, respectively. These figures show

the aimed ideal wavefront for beam steering to these angles, the realized staircase phase modu-

lation, and a wavefrontwith the same mean tilt as the realized modulation(for display purposes

the mean tilt wavefront was displaced vertically to coincide with the ideal wavefront at the left-

most pixel but it is only the wavefront slope that is of significance). As seen in Fig. 5(a) too

many pixels on the right side of the SLM are set to the higher phase level resulting in a too

large mean tilt of the grating, i.e., the steering error ε >0. Similarly, in Fig. 5(b),too few pixels

takes on the higher phase level resulting in a too small mean tilt, i.e., ε < 0.

Note that, accordingto the quantizationrule that the pixel phase takes on the permitted value

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0.2 0.4

?

0.6

?

0.81

0

1

2

3

(a)

0

10

20

30

(b)

0

0/ (2 / )

M

0.2 0.4

?

0.6

?

0.810

0/ (2 / )

M

?

??????

?norm

?

??????

?norm

Fig. 6. Absolute value of the normalized steering error εnormas a function of ϕ0for N =

128, M = 2, and (a) α = 0.47875αmaxand (b) α = 0.499αmax; the latter case is within the

difficult steering angle region around α = 0.5αmax.

0 0.20.4

Aiming angle / ?spot

0.60.81

0

0.2

0.4

0.6

0.8

1

0 0.20.4

Aiming angle / ?spot

0.60.81

0

Ideal

Realized, optimized ?0

Optimal ?0

Realized angle / ?

(a)

(b)

M

2?

spot

?opt ?0

norm,max

Fig. 7. (a) Simulations of 500 realized steering angles for an SLM (N = 32 and M = 4)

when ϕ0is optimized for each aiming angle. εopt ϕ0

of ϕ0corresponding to the aiming angle.

norm,max≈ −0.024. (b) The optimized value

that is closest to the ideal one, it is the intersections of the ideal phase setting and the imag-

inary phase threshold levels (positioned centered between any two neighboring phase levels)

that determine the shape of the resulting staircase phase modulation. The positions of these

intersections can be adjusted by changing ϕ0. Thus, a simple but powerful strategy to increase

the steering accuracy is to consider ϕ0as a free parameter and optimize its value for each steer-

ing angle α. Previous work has shown that a similar approach can improve the beam shaping

abilities of phase quantized diffractive optical elements (DOEs)[12].

The key to the success of this method is to choose ϕ0such that the obtained mean slope of

the phase staircase grating becomes as close to the aimed slope as possible. In general, εnorm

as a function of ϕ0shows an almost random behavior, see Fig. 6(a). However, in the vicinity

of angles that are particularly difficult to steer to, εnormshows a more slowly-varying behavior,

see Fig. 6(b). Thus, ϕ0is efficiently optimized in two steps: first the approximate optimum

is found using a discrete set (200 samples were used here) of values ϕ0∈ [0,2π/M]; then a

smaller region (20 samples were used here) close to the found approximate optimum is used to

obtain the final optimum. In Fig. 5(c) and (d), the effects of the described scheme are shown.

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-0.4

0.4

1

1

1

0.51

1

1

NM

= 32,= 2

NM

= 32,= 8

NM

= 32,= 32

NM

= 512,= 2

NM

= 512,= 8

0

0

0

0

0

0

?norm(%)

?norm(%)

?norm(%)

NM

= 512, = 32

(a)

(b)

(c)

(d)

(e)

(f)

0.5

? / ?max

0.5

? / ?max

0.5

? / ?max

? / ?max

0.5

? / ?max

0.5

? / ?max

0

0

0

0

0

0

?norm(%)

?norm(%)

?norm(%)

-5

5

-1.5

1.5

-0.5

0.5

-0.1

0.1

-0.025

0.025

Fig. 8. Simulated normalized steering error εnormusing the optimized value of ϕ0for each

aiming angle α, (a) N = 32 and M = 2, (b) N = 32 and M = 8, (c) N = 32 and M = 32, (d)

N = 512 and M = 2, (e) N = 512 and M = 8, and (f) N = 512 and M = 32.

Evidently, we can obtain a phase staircase with a mean slope that is very close to that of the

ideal wavefront, leading to a steering angle very close to the desired one.

In Fig. 7(a), the simulated beam steering is shown for the same task as in Fig. 3, but now

with the optimization of ϕ0for every aiming angle. The correspondingoptimized ϕ0values are

shown in Fig. 7(b). As can be seen, the agreement between aiming and realized angle is far

better than for the results shown in Fig. 3. For the worst aiming angle, the difference between

aiming and realized angle is only 2.4 % of the spot size (compared to 15% for the case when

ϕ0= 0). Thus, for this angular interval and choice of M, the maximum steering inaccuracy can

be reduced ∼6.3 times. The improvement in steering accuracy occurs not only for the specific

SLM (N =32 and M =4) and steering interval shown in Figs. 3 and 7, but overthe full steering

range of any SLM. This is seen in Fig. 8 which is for cases identical to those in Fig. 4, but with

ϕ0optimized for every aiming angle.

Comparing the left and right columns, it is also evident that with the ϕ0optimization, the ac-

curacy does improvewith a larger numberof pixels. Furtheranalysis of the simulations showed

that εopt ϕ0

value of ϕ0, which maximizes εnorm, in this case the normalized steering error is given by

εworst ϕ0

than the conventionalfixed ϕ0is given by εϕ0=0

Thus, the improvement is independent of M.

norm,max≈ 3N−1M−1. For each aiming angle, it is of course also possible to find a worst

norm,max≈1.5M−1. Further,theimprovementfactorforεnorm,maxusing theoptimizedϕ0rather

norm,max/εopt ϕ0

norm,max≈ 0.625M−1/3N−1M−1≈ 0.2N.

4.Measurements

The measurements were made with a Boulder Non-linear Systems SLM (BNS 512×512SLM

System) nematic liquid crystal (NLC) SLM, a schematic of the optical setup used is shown in

Fig. 9(a). To allow for accurate modulation, the phase and amplitude modulation of the SLM

was characterized using a diffraction-basedmethod [13]. During the characterization, the CCD

camera shown in the setup was replaced with a photo detector. The measured amplitude and

phase modulation of the SLM is shown in Fig. 9(b). Since the amplitude modulation was very

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185220255

0

0.2

0.4

0.6

0.8

1

SLM pixel setting

Amplitude (-) and phase (rad/2?)

Amplitude

Phase

Laser

L1

ND

CCD

SLM

L2

L4

L3

P1

P2

FP

(a)

(b)

(d)

(c)

MFP

Fig. 9. (a) Optical setup; The HeNe-laser beam (λ = 543.5 nm) is expanded (lenses L1

and L2), attenuated with a neutral density filter (ND) and polarized (P1) before it falls

on the SLM. Lens L3forms the Fourier plane (FP) which is then magnified by lens L4.

In the magnified FP (MFP), the steered diffraction spot is captured with a CCD camera.

Polarizer P2is used to block any non-modulated light. (b) Measured amplitude and phase

modulation of the SLM as functions of the pixel setting. (c) Typical SLM-frame in which

the central 128×128 pixels are used for the experiment. Outside the central pixels the SLM

is set to steer the light into directions not disturbing the measurements. (d) Typical image

captured by the CCD camera. The determined beam centroid and the intensity along its x-

and y-direction are shown.

small, the SLM was considered to be a purely phase modulating device.

Due to the SLM backplane being slightly curved, a typical feature of the used SLM [14], the

full area of the SLM was not used since the shape of the resulting spot became too deteriorated.

Instead, only the central N×N pixels of the SLM were used to emulate a 1D grating, while the

pixelsnotinvolvedin theexperimentwereset toabinarygratingdeflectingthelighttopositions

outside the measurement area in the beam steering experiment, see Fig. 9(c). Measurements

were performed for N = 128 and 256.

The far-field pattern of the steered beam was magnified and captured on a CCD camera, see

Fig. 9(d). The beam position was determined by calculating the center of mass of the thres-

holded (> 50% of the peak intensity) image. The region used in the center of mass calculation,

i.e., fulfilling the threshold condition, had a diameter of ∼150 and ∼100 pixels for N = 128

and 256, respectively.

Toperformthemeasurementsforthefullangularrangewouldhavebeentootimeconsuming.

Instead, measurements were performed for narrow angular regions centered at αc= 0.5αmax,

0.25αmax, 0.125αmax, and 0.47875αmax. In the listed order, these regions yield a decreasing

steering error since beam steering is an increasingly simple task in these regions. Note that

since the width of the error peaks decreases as the number of phase levels M increases, see

Fig. 4, smaller angular regions were used for larger values of M. For each angular region, 150

equally distant aiming angles (resulting in 150 SLM frames) were used.

In Fig. 10, measured and simulated normalized steering errors εnormare shown for the angle

αc= 0.5αmaxwith N = 128 and M = 2, 8, and 32. As seen, the agreement between measure-

ments and simulations is very good. Naturally, as an increasing M value yields a decreasing

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Simulated

(%)

?norm

Simulated

(%)

?norm

200

400600

-40

-20

0

20

40

Measurement number

(a)

-20

0

20

(c)(d)

-5

0

5

(e) (f)

Measured

(%)

?norm

Measured

(%)

?norm

Measured

(%)

?norm

0

200

400600

Measurement number

0

200

400600

Measurement number

norm

0

-40

0.492

-20

00

20

40

-20

0.496

0

20

-5

0

5

Simulated

(%)

?norm

norm

(b)

0.50.504

? ? ?max

0.50.508

? ? ?max

0.50.501

? ? ?max

0.499

Worst ?0

?0= 0

?N/2= 0

Best ?0

Fig. 10. Measured normalized steering error for N = 128, αc= 0.5αmax, and (a) M = 2,

(c) M = 8, and (e) M = 32. Simulations of the same cases are shown in (b), (d), and (f).

The error is shown for ϕ0taking on values corresponding to ϕ0=0 (red dashed), ϕN/2=0

(green dash-dotted), worst ϕ0for each aiming angle (black dotted), and optimal ϕ0for

each aiming angle (blue solid). For the used device, an εnormvalue of 1% corresponds to

2.12 μrad and αmaxis 13.6 mrad.

εnorm, the measurements for M = 32 are more sensitive to noise, which can also be seen in the

figure.

Similar results were obtained for all measurements (N = 128 and 256, M = 2, 8, and 32,

for the regions centered at the αcvalues listed above). For larger values of N, the agreement

between measurementsandsimulations was notquite as good,see Fig. 11.This is partly caused

by the limited beam stability which affects a smaller spotsize (for N = 256) more than a larger

one (for N = 128). However, measurements showed that the beam stability, i.e., the small fluc-

tuations of the beam position when the SLM was continuously updated with the same frame,

was ±1 % and ±3 % of the spot size (still defined as λ/Np) for N =128 and 256, respectively.

Thus, additional effects must contribute to the noise in the measurements. One effect is most

probablycaused by the backplaneof the SLM which is knownto be slightly curvedfor the used

model[14]. Also, for the larger N the incident intensity on the peripheral pixels is noticeably

lower as a result of the Gaussian laser beam, whereas a constant incident intensity was assumed

in the simulations.

5. Conclusions and discussion

We have shown that the commonlyused method to set the pixel values in a beam steering phase

modulatingSLM can lead to unexpectedlylargedeviations in the actual steering angle from the

desired. During a beam scan even reverse tracking can occur over a few consecutive settings of

the SLMs.

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0.125

? ? ?max

0.1260.124

Simulated

(%)

?norm

Simulated

(%)

?norm

200

400600

-10

0

10

Measurement number

(a)

-10

0

10

(c) (d)

-5

0

5

(e)(f)

Measured

(%)

?norm

Measured

(%)

?norm

Measured

(%)

?norm

0

200

400600

Measurement number

0

200

400600

Measurement number

norm

0

Simulated

(%)

?norm

norm

(b)

-10

0

10

-10

0

10

-5

0.1245

0

5

0.125

? ? ?max

0.1260.124

0.125

? ? ?max

0.1255

?0= 0

?N/2= 0

Best ?0

Fig. 11. Measured normalized steering error for N = 256, αc= 0.125αmax, and (a) M = 2,

(c) M = 8, and (e) M = 32. Simulations of the same cases are shown in (b), (d), and (f).

The error is shown for ϕ0taking on values corresponding to ϕ0=0 (red dashed), ϕN/2=0

(green dash-dotted), and optimal ϕ0for each aiming angle (blue solid). For the used device,

an εnormvalue of 1% corresponds to 1.06 μrad and αmaxis 13.6 mrad.

To overcome this problem and obtain a higher steering accuracy, it is crucial to optimize ϕ0

foreachaiminganglesuchthat agoodagreementis obtainedbetweentheidealwavefrontphase

slope and the mean slope of the realized staircase phase modulation. The maximal normalized

steering error over the full angular range of the SLM εnorm,maxis ∼ 3N−1M−1or ∼ 0.625M−1,

if the optimized or commonly used scheme is used to obtain the phase grating, respectively.

Thus, an improvement in εnorm,maxof ∼ 0.2N can be obtained.

We also found that even if one keeps to the commonly used method, an extremely simple

measure can improve the steering accuracy significantly, namely, setting the peripheral, rather

than the central, pixel to a fixed phase modulation value for all steering angles. Anyway, in the

light of the ϕ0-optimizationmethodboth conventionalmethodsperformpoorly;they yield only

∼ 2.4 and ∼ 1.5 times better accuracy, respectively, than if ϕ0is actively chosen to take on the

worst possible value for each steering angle.

The simulations have been verified with experiments in a free-space optics setup. The agree-

ment between experimentsand simulations was excellent exceptfor the cases when the theoret-

ical angular error was so small that non-ideal properties of the SLM limited the performance,

such as the backplane curvature and errors in the phase modulation characteristics.

It should be noted that the procedure of optimizing ϕ0does not only affect the peak position

of the spot but also have a slight influence on the spot shape and power. Therefore, the exact

improvement of the positioning accuracy should depend on the metric used to define the spot

position. We investigated the positioning accuracy both with the spot position defined as the

position of the peak intensity, and the position defined as the center-of-mass of the power dis-

tributionaroundthe spot abovea certaincut-offlevel.Evenfor the worst case, M =2, therewas

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no significant difference, and so the reported improvement of the accuracy was in all essence

identical for either definition. This was also confirmed by the good agreement between sim-

ulations, based on peak position, and measurements, based on center-of-mass positions. Only

for an application which is particularly sensitive to the detailed features of the side-lobes or a

slight asymmetry in the tails of the spot intensity distribution, should there be a reason to do a

more detailed analysis, with a spot quality measure including more features, to find the optimal

phase offset. Similarly, there might be cases for which some accuracy can be traded for a slight

increase in spot power. In such a situation, it should be possible to use a similar approach as

presented here to optimize ϕ0such that the highest spot power close to an aiming position is

obtained.

It is important to understand that the problem causing the steering error is a phase mapping

problem.Thereforethe same type of steering error,with the same typical magnitude,will occur

for any design methodthat yields a continuousphase modulationas its primaryoutput, such as,

e.g., the commonly used Gerchberg-Saxtonalgorithm[15].

Another possible approach to obtain high steering accuracy would be to allow phase slope

irregularities, i.e., remove the requirement for a monotonically varying phase modulation, so

that we no longer try to approximate an ideal wavefront. With such an approach the theoretical

steering accuracy could likely be increased even further, but at the expense of a slower algo-

rithm for finding the optimal setting of the SLM pixels since that would be a multidimensional

optimization, whereas the method presented in this paper has only one optimization variable –

the phase offset ϕ0.

Acknowledgments

This work was conducted at the Center for Biophysical Imaging (University of Gothenburg)

and financially supported by the Swedish Research Council, the SSF Bio-X program, and the

Carl Trygger Foundation for Scientific Research. The SLM was kindly provided by Prof. M.

Padgett, Dept of Physics and Astronomy, University of Glasgow.

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27 October 2008 / Vol. 16, No. 22 / OPTICS EXPRESS 18287