Abstract

The limited number of pixels and their quantized phase 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, and the results were in very good agreement with the theoretical predictions.

© 2008 Optical Society of America

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References

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2007

2006

2005

2004

J. Harriman, A. Linnenberger, and S. Serati, "Improving spatial light modulator performance through phase compensation," Proc. SPIE 5553, 58-67 (2004).
[CrossRef]

2002

2001

1999

1996

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).
[CrossRef]

1994

1987

1972

R. W. Gerchberg and W. O. Saxton, "A practical algorithm for the determination of phase from image and diffraction plane pictures," Optik 35, 237-246 (1972).

Ballüder, K.

Bengtsson, J.

Crossland, W. A.

Crossland,, W.

Curtis, J. E.

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).
[CrossRef] [PubMed]

J. E. Curtis, B. A. Koss, and D. G. Grier, "Dynamic holographic optical tweezers," Opt. Commun. 207, 169-175 (2002).
[CrossRef]

Day, S. E.

Engström, D.

Fernández, F. A.

Galt, S.

Gerchberg, R. W.

R. W. Gerchberg and W. O. Saxton, "A practical algorithm for the determination of phase from image and diffraction plane pictures," Optik 35, 237-246 (1972).

Grier, D. G.

J. E. Curtis, B. A. Koss, and D. G. Grier, "Dynamic holographic optical tweezers," Opt. Commun. 207, 169-175 (2002).
[CrossRef]

Haist, T.

Hård, S.

Harriman, J.

J. Harriman, A. Linnenberger, and S. Serati, "Improving spatial light modulator performance through phase compensation," Proc. SPIE 5553, 58-67 (2004).
[CrossRef]

James, R.

Johansson, M.

Kim, N.

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).
[CrossRef]

Komarcevic, M.

Konforti, N.

Koss, B. A.

J. E. Curtis, B. A. Koss, and D. G. Grier, "Dynamic holographic optical tweezers," Opt. Commun. 207, 169-175 (2002).
[CrossRef]

Lee, D.-J.

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).
[CrossRef]

Linnenberger, A.

J. Harriman, A. Linnenberger, and S. Serati, "Improving spatial light modulator performance through phase compensation," Proc. SPIE 5553, 58-67 (2004).
[CrossRef]

Manolis, I.

Manolis, I. G.

Marom, E.

Mears, R. J.

Milewski, G.

O’Brien, D. C.

Redmond, M. M.

Reicherter, M.

Robertson, B.

Saxton, W. O.

R. W. Gerchberg and W. O. Saxton, "A practical algorithm for the determination of phase from image and diffraction plane pictures," Optik 35, 237-246 (1972).

Schmitz, C. H. J.

Serati, S.

J. Harriman, A. Linnenberger, and S. Serati, "Improving spatial light modulator performance through phase compensation," Proc. SPIE 5553, 58-67 (2004).
[CrossRef]

Song, S.-H.

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).
[CrossRef]

Spatz, J. P.

Suh, H.-H.

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).
[CrossRef]

Taghizadeh, M. R.

Tan, K. L.

Tiziani, H.

Wagemann, E.

Warr, S. T.

Wilkinson, T.

Wilkinson, T. D.

Appl. Opt.

J. Opt. Soc. Am. A

Opt. Commun.

J. E. Curtis, B. A. Koss, and D. G. Grier, "Dynamic holographic optical tweezers," Opt. Commun. 207, 169-175 (2002).
[CrossRef]

Opt. Express

Opt. Lett.

Optik

R. W. Gerchberg and W. O. Saxton, "A practical algorithm for the determination of phase from image and diffraction plane pictures," Optik 35, 237-246 (1972).

Proc. SPIE

J. Harriman, A. Linnenberger, and S. Serati, "Improving spatial light modulator performance through phase compensation," Proc. SPIE 5553, 58-67 (2004).
[CrossRef]

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).
[CrossRef]

Other

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, 45001-1-8 (2005).
[CrossRef]

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

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Figures (11)

Fig. 1.
Fig. 1.

(a) An ideal wavefront with a steering angle α (y=x tanα, 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, corresponding to two wavefronts with a difference in steering angle of δα, and the realizable phase levels in between them.

Fig. 2.
Fig. 2.

Ratio Q between the actual number of changes S actual in the SLM setting and the number S given by Eq. (6) when the beam is being steered over an angular interval [αstartstartspot ] with (a) α start=0, (b) α start=0.5 ×α max, and (c) α start=0.625×α max. The number of SLM pixels N=32, 64, 128,…, 2048. The number of phase levels M=2, 4, 8,…, 256.

Fig. 3.
Fig. 3.

(a) Simulations of 500 realized steering angles for an SLM (N=32 and M=4) with φ 0=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, ε norm , max φ 0 = 0 0 . 15 .

Fig. 4.
Fig. 4.

Simulated normalized steering error εnorm in the entire possible scanning range for φ 0=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.

Fig. 5.
Fig. 5.

Ideal phase (blue dash-dotted), realized staircase phase modulation (black solid), and the mean phase tilt of 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 φ 0=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 φ 0. 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.

Fig. 6.
Fig. 6.

Absolute value of the normalized steering error ε norm as a function of ϕ 0 for N=128, M=2, and (a) α=0.47875α max and (b) α=0.499α max; the latter case is within the difficult steering angle region around α=0.5α max.

Fig. 7.
Fig. 7.

(a) Simulations of 500 realized steering angles for an SLM (N=32 and M=4) when φ 0 is optimized for each aiming angle. ε norm , max opt φ 0 0 . 024 . (b) The optimized value of φ 0 corresponding to the aiming angle.

Fig. 8.
Fig. 8.

Simulated normalized steering error ε norm using the optimized value of φ 0 for 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.

Fig. 9.
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 L3 forms 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 P2 is 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.

Fig. 10.
Fig. 10.

Measured normalized steering error for N=128, α c=0.5α max, 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 φ 0 taking on values corresponding to φ 0=0 (red dashed), φN /2=0 (green dash-dotted), worst φ 0 for each aiming angle (black dotted), and optimal φ 0 for each aiming angle (blue solid). For the used device, an ε norm value of 1% corresponds to 2.12 µrad and α max is 13.6 mrad.

Fig. 11.
Fig. 11.

Measured normalized steering error for N=256, α c=0.125α max, 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 φ 0 taking on values corresponding to φ 0=0 (red dashed), φ N/2=0 (green dash-dotted), and optimal φ 0 for each aiming angle (blue solid). For the used device, an ε norm value of 1% corresponds to 1.06 µrad and α max is 13.6 mrad.

Equations (9)

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α max = sin 1 λ 2 p ,
φ ideal ( x , α ) = 2 π λ x sin α + φ 0 ,
φ j ideal ( α ) = 2 π λ x j sin α + φ 0 ,
φ j ( α ) = round ( φ j ideal ( α ) M 2 π ) 2 π M ,
S j φ j ideal ( α + δ α ) φ j ideal ( α ) 2 π M 2 π λ x j sin ( α + δ α ) sin α 2 π M M x j δ α λ ,
S = j = 1 N S j = { x j = ( j 1 ) p } = M δ α p λ j = 1 N ( j 1 ) = M δ α p λ ( N 1 ) N 2 δ α p M N 2 2 λ .
α error = 2 λ p 1 M N 2 .
α error 2 M N α spot ,
α error 1 0.8 2 M N α spot .

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