Independent spatial intensity, phase and polarization distributions Erik H. Waller1,∗ and Georg von Freymann1,2 1 Physics
department and research center OPTIMAS, University of Kaiserslautern, 67663 Kaiserslautern, Germany 2 Fraunhofer-Institute for Physical Measurement Techniques (IPM), 67663 Kaiserslautern, Germany ∗ [email protected]
Abstract: Independent control of the spatial intensity, phase and polarization distribution has numerous applications in direct laser writing, microscopy and optical trapping. Especially, it is well known that the inversion of the Debye-Wolf diffraction integral usually leads to spatially varying intensity, phase and polarization maps. Here, we present a prism and grating free setup built around a single phase-only spatial-light-modulator for full control of spatial intensity, phase and polarization distributions. These distributions are not limited to non-diffractive beams and do not require any change of setup. We verify the versatility of the proposed method with wavefront and intensity measurements. © 2013 Optical Society of America OCIS codes: (050.1970) Diffractive optics; (070.6120) Spatial light modulators; (110.1080) Active or adaptive optics.
References and links 1. M. A. A. Neil, R. Juˇskaitis, M. J. Booth, T. Wilson, T. Tanaka, and S. Kawata, “Adaptive aberration correction in a two-photon microscope,” J. Microsc. 200, 105–108 (2001). 2. E. H. Waller, M. Renner, and G. von Freymann, “Active aberration- and point-spread-function control in direct laser writing,” Opt. Express 20, 24949–24956 (2012). 3. E. H. Waller and G. von Freymann, “Multi foci with diffraction limited resolution,” Opt. Express 21, 21708– 21713 (2013). 4. T. Satoh, Y. Terui, R. Moriya, B. A. Ivanov, K. Ando, E. Saitoh, T. Shimura, and K. Kuroda, “Directional control of spin-wave emission by spatially shaped light,” Nat. Photonics 6, 662–666 (2012). 5. I. Iglesias and J. J. S´aenz, “Scattering forces in the focal volume of high numerical aperture microscope objectives,” Opt. Commun. 284, 2430–2436 (2011). 6. M. R. Foreman, S. S. Sherif, P. R. T. Munro, and P. T¨or¨ok, “Inversion of the Debye-Wolf diffraction integral using an eigenfunction representation of the electric fields in the focal region,” Opt. Express 16, 4901–4917 (2008). 7. K. Jahn and N. Bokor, “Solving the inverse problem of high numerical aperture focusing using vector Slepian harmonics and vector Slepian multipole fields,” Opt. Commun. 288, 13–16 (2013). 8. E. G. van Putten, I. M. Vellekoop, and A. P. Mosk, “Spatial amplitude and phase modulation using commerical twisted nematic LCDs,” Appl. Opt. 47, 2076–2081 (2008). 9. J. A. Davis, D. M. Cottrell, J. Campos, M. J. Yzuel, and I. Moreno, “Encoding amplitude information onto phase-only filters,” Appl. Opt. 38, 5004–5013 (1999). 10. V. Bagnoud and J. D. Zuegel, “Independent phase and amplitude control of a laser beam by use of a single-phaseonly spatial light modulator,” Opt. Lett. 29, 295–297 (2004). 11. X.-L. Wang, J. Ding, W.-J. Ni, C.-S. Guo, and H.-T. Wang, “Generation of arbitrary vector beams with a spatial light modulator and a common path interferometric arrangement,” Opt. Lett. 32, 3549–3551 (2007). 12. D. Preece, S. Keen, E. Botvinick, R. Bowman, M. Padgett, and J. Leach, “Independent polarisation control of multiple optical traps,” Opt. Express 16, 15897–15901 (2008). 13. I. Moreno, C. Iemmi, J. Campos, and M. J. Yzuel, “Jones matrix treatment for optical Fourier processors with structured polarization,” Opt. Express 19, 4583–4594 (2011).
#197136 - $15.00 USD Received 4 Sep 2013; revised 24 Oct 2013; accepted 27 Oct 2013; published 8 Nov 2013 (C) 2013 OSA 18 November 2013 | Vol. 21, No. 23 | DOI:10.1364/OE.21.028167 | OPTICS EXPRESS 28167
14. J. H. Clegg and M. A. A. Neil, “Double pass, common path method for arbitrary polarization control using a ferroelectric liquid crystal spatial light modulator,” Opt. Lett. 38, 1043–1045 (2013). 15. M. A. A. Neil, F. Massoumian, R. Juˇskaitis, and T. Wilson, “Method for the generation of arbitrary complex vector wave fronts,” Opt. Lett. 27, 1929–1931 (2002). 16. H. Chen, J. Hao, B.-F. Zhang, J. Xu, J. Ding, and H.-T. Wang, “Generation of vector beam with space-variant distribution of both polarization and phase,” Opt. Lett. 36, 3179–3182 (2011). 17. F. Kenny, D. Lara, O. G. Rodr´ıguez-Herrera, and C. Dainty, “Complete polarization and phase control for focusshaping in high-NA microscopy,” Opt. Express 20, 14015–14029 (2012). 18. R. L. Eriksen, P. C. Mogensen, and J. Gl¨uckstad, “Elliptical polarisation encoding in two dimensions using phase-only spatial light modulators,” Opt. Commun. 187, 325–336 (2001). 19. I. Moreno, J. A. Davis, T. M. Hernandez, D. M. Cottrell, and D. Sand, “Complete polarization control of light from a liquied crystal spatial light modulator,” Opt. Express 20, 364–376 (2012). 20. D. Maluenda, I. Juvells, R. Mart´ınez-Herrero, and A. Carnicer, “Reconfigurable beams with arbitrary polarization and shape distributions at a given plane,” Opt. Express 21, 5424–5431 (2013). 21. C. Maurer, A. Jesacher, S. F¨urhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring arbitrary optical vector beams,” New J. Phys. 9, 1–20 (2007). 22. W. Han, Y. Yang, W. Cheng, and Q. Zhan, “Vectorial optical field generator for the creation of arbitrarily complex fields,” Opt. Express 21, 20692–20706 (2013).
1.
Introduction
Many areas in optics – e.g., aberration control – require an accurate control of the spatial phase profile of a laser beam [1]. Others, such as direct-laser-writing, profit from additionally having access to the spatial intensity distribution [2, 3]. Furthermore, especially when it comes to light-matter interaction, spatially shaping the polarization may be beneficial. For example, spin waves may be precisely triggered by circular polarized light [4]. Also, scattering forces in the focal volume heavily depend on the polarization while trapping forces are a result of the intensity distribution [5]. Controlling both usually requires the inversion of the Debye-Wolf diffraction integral. This, in general, leads to spatially varying intensity, phase and polarization distributions [6, 7]. A number of methods exist to spatially encode intensity and phase distributions in a phaseonly spatial-light-modulator (SLM) by using macro pixel or locally varying diffraction efficiencies [8–10]. Others achieve spatial polarization [11–14], spatial polarization and spatial phase [15–17] or spatial polarization and spatial intensity [18–20] control using a single SLM [12–16, 19] or two SLMs [17, 18, 20]. Two groups introduced a setup allowing for independent spatial polarization, phase and intensity modulation [21, 22]. Han et al. accomplish this by using two sections on two SLMs each [22]. However, due to the two SLMs, this setup is not very compact and cost effective. In addition to that, the power utilization efficiency is limited by the use of non-polarizing beam splitters and four reflections at the SLMs. We estimate the maximum power utilization efficiency to be below 5%. While they show control of independent spatial phase and polarization distributions as well as intensity and polarization distributions, no simultaneous measurements of independent phase, intensity and polarization distributions are presented. Maurer et al., for the first time, show that independent phase, intensity and polarization modulation is possible with a single phase-only SLM and a Wollaston-prism [21]. They use their setup to generate different Gauss-Laguerre and Hermite-Gauss modes. These modes, however, are non-diffractive and no phase measurements are shown in the paper, leaving it somewhat unclear if less error tolerant patterns are also obtainable. Additionally, they explain that they needed to change the wave retarder from λ /4 to λ /2 when switching from Laguerre-Gaussian to Hermite-Gaussian vector beams making it impractical for fast switching or remote controlled applications. Here, we introduce an easy to implement setup, similar to the setup which Preece et al. introduced for polarization-only control [12]. In contrast to several previously reported systems, #197136 - $15.00 USD Received 4 Sep 2013; revised 24 Oct 2013; accepted 27 Oct 2013; published 8 Nov 2013 (C) 2013 OSA 18 November 2013 | Vol. 21, No. 23 | DOI:10.1364/OE.21.028167 | OPTICS EXPRESS 28168
the proposed setup abstains from using special gratings or prisms (as used in [15, 16, 21]). This facilitates the integration in existing setups since it allows a more flexible choice of optics used. Our setup uses a single SLM (in opposition to [17, 22]) and still allows for full control of the spatial intensity, phase and polarization distribution. We show that accurate almost arbitrary non-diffractive beams (but not limited to those) are possible using spatial phase and intensity measurements. The various patterns are generated by electronically changing the patterns on the SLM without the need of changes in the experimental setup (such as in [21]). The maximum power utilization efficiency is limited by the diffraction efficiency of the SLM alone and may therefore be above 60 %. To the best of our knowledge, we show for the first time the combination of arbitrary polarization patterns with simultaneous phase and intensity profiles. 2.
Setup and principle
The setup is shown in Fig. 1. A linearly polarized helium-neon (HeNe) laser is expanded by the telescope consisting of L1 ( f = 50 mm) and L2 ( f = 500 mm) to overfill the aperture of a phase-only spatial-light-modulator (SLM, Hamamatsu X10468-01). Two different holograms are simultaneously displayed side by side on the SLM. The holograms are each overlayed with a blazed grating directing the reflected intensity into the first diffraction orders. The diffraction efficiency of the gratings may locally be reduced to allow for intensity modulation of the beams. Thus, independent phase and intensity modulation is possible. The two designed patterns pass through L2 which serves as the first lens in the 4-f-setup. The patterns are then separated and the spatial frequency of the grating is clipped in the focus of L2 . One branch passes through a halfwave plate leading to orthogonal polarizations (0◦ and 90◦ ) in the two arms. Note, that when working with pulsed lasers, mirrors M1 and M2 may optionally be adjusted to yield equal path lengths for the two branches. A polarizing beamsplitter combines the two beams which hereafter propagate in the same direction. L3 ( f = 400 mm) back transformes the combined pattern. We measure the obtained distribution with a wavefront camera that allows for simultaneous spatial phase and intensity measurements (Phaseview DWC 1000), with the sensor placed in the target plane. An analyzer is optionally placed in front of the camera to visualize the polarization distribution. Technically, this analyzer should be placed in the target plane, since it is there where the exact polarization is achieved. However, due to the rather paraxial setup we still achieve a good representation when placing the analyzer about 20 cm in front of this plane.
Fig. 1. Scheme of the setup. A HeNe laser is expanded by lenses L1 and L2 . The beam impinges perpendicularly onto a phase-only SLM. Two separate holograms overlayed with diffraction gratings guide the modulated wave along path A and path B respectively. L2 and L3 compose a 4- f -system with a half-wave plate placed in path A. A digital wavefront camera is used to simultaneously measure the intensity and phase distributions. An analyzer is optionally placed before the DWC to visualize polarization distributions.
#197136 - $15.00 USD Received 4 Sep 2013; revised 24 Oct 2013; accepted 27 Oct 2013; published 8 Nov 2013 (C) 2013 OSA 18 November 2013 | Vol. 21, No. 23 | DOI:10.1364/OE.21.028167 | OPTICS EXPRESS 28169
3.
Methods
In the setup introduced above the two field distributions generated by the holograms may be treated independently. One hologram leads to the linear polarized wave: Et,1 (x, y) = At,1 (x, y) · exp[i · Pt,1 (x, y)] , (1) 0 while the other after passing through the half-wave plate produces: 0 Et,2 (x, y) = At,2 (x, y) · exp[i · Pt,2 (x, y)]
(2)
in the target plane. Let us suppose we want to create an intensity distribution It (x, y), a phase distribution Pt (x, y) and a polarization distribution Polt (x, y). In our target plane this would require the following field distribution (in analogy with the ordinary and extra-ordinary wave in retarders): Et (x, y) = Et,1 (x, y) + Et,2 (x, y) =
At,1 (x, y) · exp[i · Pt (x, y)] · exp[+i · Polt (x, y)/2] At,2 (x, y) · exp[i · Pt (x, y)] · exp[−i · Polt (x, y)/2]
. (3)
The common phase term Pt (x, y) sets the phase distribution while the local polarization is controlled by the phase shift between the two arms Polt (x, y) (0,π : linear, π /2 and 3π /2: circular, else: elliptical polarization). The direction α of the local polarization vector is adjusted by the 2 (x, y)+A2 (x, y) equals the desired intenratio of At,1 (x, y) and At,2 (x, y) while ensuring that At,1 t,2 sity distribution It (x, y). At,1 (x, y) and At,2 (x, y) are therefore calculated as follows (for circular polarization α is set to π /4): √
It cos(α ) , √ At,2 (x, y) = It sin(α ) .
At,1 (x, y) =
(4) (5)
For some α this leads to negative amplitudes. In this case the target amplitudes are modified to ensure that positve and negative amplitude values are out of phase: At,1 (x, y) = −At,1 exp(iπ ) ,
(6)
At,2 (x, y) = −At,2 exp(iπ ) .
(7)
The final function to inscribe is a complex function that needs to be encoded very accurately as a phase-only pattern. Commonly used encoding schemes show a comparable large dependence of the phase pattern on the intensity pattern. Since the phase pattern defines the resulting polarization and phase, we focus on a very accurate encoded phase at high fidelity of the encoded amplitudes. We experienced rather low cross-modulation between phase and intensity using the following encoding scheme for the two patterns (similar to the encoding scheme in [9]): P1 = [Pt + Polt /2 + angle(At,1 )] + blaze1 · sinc(1 − abs(At,1 )) ,
(8)
P2 = [Pt − Polt /2 + angle(At,2 )] + blaze2 · sinc(1 − abs(At,2 )) .
(9)
#197136 - $15.00 USD Received 4 Sep 2013; revised 24 Oct 2013; accepted 27 Oct 2013; published 8 Nov 2013 (C) 2013 OSA 18 November 2013 | Vol. 21, No. 23 | DOI:10.1364/OE.21.028167 | OPTICS EXPRESS 28170
The blazei are blazed phase gratings in the intervall −π to π . After summation P1 and P2 are each modulated to a 2π phase range. Some corrections to the above procedure may be neccessary. We account for differences in the intensity distribution reflected by P1 and P2 and ensure a linear target-amplitude to obtainedamplitude mapping by precompensation. Additionally, Polt is modified to Polt +Δ, with Δ being an offset allowing fine adjustment of the path lengths of the two branches. Fine adjustment of the positions and propagation directions of the beams is done by electronically shifting the displayed patterns against each other and adding a tilt to them respectively. 4.
Experimental results
The method explained above allows a very versatile definition of almost arbitrary intensity, phase and polarization distributions. Figure 2 shows measurements of different combinations of such distributions. In Figs. 2(a)-2(c) we set the polarization to circular polarization and in Figs. 2(d)-2(f) to radial polarization. The target intensity is displaying a uniform (first column), a shaded-ring-filter (second column) or a cross-shaped distribution (third column). Three target phases, a constant phase (Fig. 2(a)), astigmatism (Fig. 2(b)) and coma (Fig. 2(c)), are adressed for each combination of polarization and intensity distribution. The target Zernike coefficients of astigmatism and coma are set to λ and λ /2 respectively. The constant phase distribution leads to a phase that is determined by a combination of the aberrations in the two paths. Since these aberrations are slightly different for the two paths the measured phase is expected to change with the target polarization. Since precompensation of these aberrations is beyond the scope of our SLM – in consequence – the measured phase obtained when displaying the constant phase distribution is respectively taken as a reference for the other two target phase patterns. Figure 3 shows examples of more complex polarization patterns. The polarization patterns are visualized by the black arrows in Fig. 3(a). For the uniform intensity distribution and the shaded-ring-filter the polarization is set to circular in the inner ring, to radial in the middle ring and to azimuthal in the outer ring. For the cross intensity distribution we combine two orthogonal linear polarizations, one within and one outside the cross. The same intensity and phase patterns are targeted as in Fig. 2. Despite the complexity of the patterns the definition of the distributions is still very accurate. 5.
Discussion
As visible in Figs. 2 and 3, the addressed combinations follow the expected behaviour. For example, independent of the phase and intensity pattern, for circular target polarization no major intensity modulation is observed when rotating the analyzer. Meanwhile both the phase pattern and the intensity pattern are defined with high precision. To give a measure of the degree of polarization control, we calculate the average Stokes parameters for the uniform intensity distribution with circular polarization, shown in Fig. 2(a): ⎞ ⎛ ⎞ ⎛ 1 I0 + I90 ⎟ ⎜ −0.07 ⎟ ⎜ I0 − I90 ⎟=⎜ ⎟ S = ⎜ (10) ⎠ ⎝ −0.03 ⎠ . ⎝ √ I45 √− I135 2 I0 I90 sin(δ ) 0.99 Here, the Ii are the measured intensities with the numbers indicating the angle of the analyzer. The last parameter is calculated under the assumption that the relative phase δ between the two orthogonal waves is exactly π /2. All parameters are normalized to the first parameter.
#197136 - $15.00 USD Received 4 Sep 2013; revised 24 Oct 2013; accepted 27 Oct 2013; published 8 Nov 2013 (C) 2013 OSA 18 November 2013 | Vol. 21, No. 23 | DOI:10.1364/OE.21.028167 | OPTICS EXPRESS 28171
Fig. 2. Circular (a)-(c) and radial (d)-(f) target polarization for a uniform (first column), a shaded-ring (center column) and a cross (third column) intensity distribution. The reference phase (a) is set to a spatially constant phase value. Astigmatism (b) and coma (c) are set as target phases for each polarization-intensity combination. The white arrows indicate the analyzer setting to visualize the polarization. For the phase and intensity measurement the analyzer is removed from the setup.
#197136 - $15.00 USD Received 4 Sep 2013; revised 24 Oct 2013; accepted 27 Oct 2013; published 8 Nov 2013 (C) 2013 OSA 18 November 2013 | Vol. 21, No. 23 | DOI:10.1364/OE.21.028167 | OPTICS EXPRESS 28172
Fig. 3. Mixture of polarizations as visualized by the black arrows in the target intensity distributions. Uniform intensity distribution and shaded-ring-filter: Circular polarization in the inner ring, radial polarization in the middle ring and azimuthal polarization in the outer ring. Cross intensity distribution: 90◦ linear polarization inside the cross, 0◦ linear polarization outside the cross.
This compares very well with the Stokes vector for a perfectly circularly polarized wave: S = (1, 0, 0, 1)T . The mean-square error between the desired and obtained intensity distribution for this pattern is below 5 %. Exemplarily, after subtracting the reference, the obtained Zernike coefficient for the phase front of the uniform intensity distribution in Fig. 2(b) is 0.97λ (616 nm) - very close to the desired λ (633 nm). Since the setup is mostly common path, the distributions are very robust against vibrations, turbulences and drifts. However, while being cost-effective and compact, one drawback of the proposed method is its reduction of the bandwidth. As a result of the phase range being limited to 2π by the SLM, phase-wrapping leads to cross-modulation and phase jumps. The latter causes a sharp drop in intensity visible in all patterns with astigmatism and coma added. Cross-modulation occures for extreme phase values (when the pattern is distorted by the wrapping) and results in heightened or lowered intensity, for example, visible in the intensity in Figs. 2(b) or 2(d). In consequence this also leads to imperfections in the polarization distribution. Also, due to these limitations we cannot correct for the strong aberrations present in the system but take them as the reference wave front. The strong aberrations are introduced by the beamsplitter. The experimental distortions may be reduced using an SLM with a higher phase modulation range and using look-up tables to reduce cross-modulation.
#197136 - $15.00 USD Received 4 Sep 2013; revised 24 Oct 2013; accepted 27 Oct 2013; published 8 Nov 2013 (C) 2013 OSA 18 November 2013 | Vol. 21, No. 23 | DOI:10.1364/OE.21.028167 | OPTICS EXPRESS 28173
6.
Conclusions
We demonstrate a compact, flexible and cost-effective setup that allows for almost arbitrary independent phase, intensity and polarization distributions built around a single phase-only spatial-light-modulator. All combinations are created electronically without requiring a change in setup. We abstain from using special gratings or prisms and do not restrict ourselves to nondiffractive beams. We demonstrate circular and radial as well as combinations of circular and locally varying linear polarization distributions. Each polarization distribution is combined with different intensity and phase patterns. The obtained distributions are visualized by simultaneous wavefront and intensity measurements. Exemplary Stokes parameters and deviations from target distributions are calculated to quantifiy the quality of the obtained patterns. The patterns follow the expected behaviour very well.
#197136 - $15.00 USD Received 4 Sep 2013; revised 24 Oct 2013; accepted 27 Oct 2013; published 8 Nov 2013 (C) 2013 OSA 18 November 2013 | Vol. 21, No. 23 | DOI:10.1364/OE.21.028167 | OPTICS EXPRESS 28174
department and research center OPTIMAS, University of Kaiserslautern, 67663 Kaiserslautern, Germany 2 Fraunhofer-Institute for Physical Measurement Techniques (IPM), 67663 Kaiserslautern, Germany ∗ [email protected]
Abstract: Independent control of the spatial intensity, phase and polarization distribution has numerous applications in direct laser writing, microscopy and optical trapping. Especially, it is well known that the inversion of the Debye-Wolf diffraction integral usually leads to spatially varying intensity, phase and polarization maps. Here, we present a prism and grating free setup built around a single phase-only spatial-light-modulator for full control of spatial intensity, phase and polarization distributions. These distributions are not limited to non-diffractive beams and do not require any change of setup. We verify the versatility of the proposed method with wavefront and intensity measurements. © 2013 Optical Society of America OCIS codes: (050.1970) Diffractive optics; (070.6120) Spatial light modulators; (110.1080) Active or adaptive optics.
References and links 1. M. A. A. Neil, R. Juˇskaitis, M. J. Booth, T. Wilson, T. Tanaka, and S. Kawata, “Adaptive aberration correction in a two-photon microscope,” J. Microsc. 200, 105–108 (2001). 2. E. H. Waller, M. Renner, and G. von Freymann, “Active aberration- and point-spread-function control in direct laser writing,” Opt. Express 20, 24949–24956 (2012). 3. E. H. Waller and G. von Freymann, “Multi foci with diffraction limited resolution,” Opt. Express 21, 21708– 21713 (2013). 4. T. Satoh, Y. Terui, R. Moriya, B. A. Ivanov, K. Ando, E. Saitoh, T. Shimura, and K. Kuroda, “Directional control of spin-wave emission by spatially shaped light,” Nat. Photonics 6, 662–666 (2012). 5. I. Iglesias and J. J. S´aenz, “Scattering forces in the focal volume of high numerical aperture microscope objectives,” Opt. Commun. 284, 2430–2436 (2011). 6. M. R. Foreman, S. S. Sherif, P. R. T. Munro, and P. T¨or¨ok, “Inversion of the Debye-Wolf diffraction integral using an eigenfunction representation of the electric fields in the focal region,” Opt. Express 16, 4901–4917 (2008). 7. K. Jahn and N. Bokor, “Solving the inverse problem of high numerical aperture focusing using vector Slepian harmonics and vector Slepian multipole fields,” Opt. Commun. 288, 13–16 (2013). 8. E. G. van Putten, I. M. Vellekoop, and A. P. Mosk, “Spatial amplitude and phase modulation using commerical twisted nematic LCDs,” Appl. Opt. 47, 2076–2081 (2008). 9. J. A. Davis, D. M. Cottrell, J. Campos, M. J. Yzuel, and I. Moreno, “Encoding amplitude information onto phase-only filters,” Appl. Opt. 38, 5004–5013 (1999). 10. V. Bagnoud and J. D. Zuegel, “Independent phase and amplitude control of a laser beam by use of a single-phaseonly spatial light modulator,” Opt. Lett. 29, 295–297 (2004). 11. X.-L. Wang, J. Ding, W.-J. Ni, C.-S. Guo, and H.-T. Wang, “Generation of arbitrary vector beams with a spatial light modulator and a common path interferometric arrangement,” Opt. Lett. 32, 3549–3551 (2007). 12. D. Preece, S. Keen, E. Botvinick, R. Bowman, M. Padgett, and J. Leach, “Independent polarisation control of multiple optical traps,” Opt. Express 16, 15897–15901 (2008). 13. I. Moreno, C. Iemmi, J. Campos, and M. J. Yzuel, “Jones matrix treatment for optical Fourier processors with structured polarization,” Opt. Express 19, 4583–4594 (2011).
#197136 - $15.00 USD Received 4 Sep 2013; revised 24 Oct 2013; accepted 27 Oct 2013; published 8 Nov 2013 (C) 2013 OSA 18 November 2013 | Vol. 21, No. 23 | DOI:10.1364/OE.21.028167 | OPTICS EXPRESS 28167
14. J. H. Clegg and M. A. A. Neil, “Double pass, common path method for arbitrary polarization control using a ferroelectric liquid crystal spatial light modulator,” Opt. Lett. 38, 1043–1045 (2013). 15. M. A. A. Neil, F. Massoumian, R. Juˇskaitis, and T. Wilson, “Method for the generation of arbitrary complex vector wave fronts,” Opt. Lett. 27, 1929–1931 (2002). 16. H. Chen, J. Hao, B.-F. Zhang, J. Xu, J. Ding, and H.-T. Wang, “Generation of vector beam with space-variant distribution of both polarization and phase,” Opt. Lett. 36, 3179–3182 (2011). 17. F. Kenny, D. Lara, O. G. Rodr´ıguez-Herrera, and C. Dainty, “Complete polarization and phase control for focusshaping in high-NA microscopy,” Opt. Express 20, 14015–14029 (2012). 18. R. L. Eriksen, P. C. Mogensen, and J. Gl¨uckstad, “Elliptical polarisation encoding in two dimensions using phase-only spatial light modulators,” Opt. Commun. 187, 325–336 (2001). 19. I. Moreno, J. A. Davis, T. M. Hernandez, D. M. Cottrell, and D. Sand, “Complete polarization control of light from a liquied crystal spatial light modulator,” Opt. Express 20, 364–376 (2012). 20. D. Maluenda, I. Juvells, R. Mart´ınez-Herrero, and A. Carnicer, “Reconfigurable beams with arbitrary polarization and shape distributions at a given plane,” Opt. Express 21, 5424–5431 (2013). 21. C. Maurer, A. Jesacher, S. F¨urhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring arbitrary optical vector beams,” New J. Phys. 9, 1–20 (2007). 22. W. Han, Y. Yang, W. Cheng, and Q. Zhan, “Vectorial optical field generator for the creation of arbitrarily complex fields,” Opt. Express 21, 20692–20706 (2013).
1.
Introduction
Many areas in optics – e.g., aberration control – require an accurate control of the spatial phase profile of a laser beam [1]. Others, such as direct-laser-writing, profit from additionally having access to the spatial intensity distribution [2, 3]. Furthermore, especially when it comes to light-matter interaction, spatially shaping the polarization may be beneficial. For example, spin waves may be precisely triggered by circular polarized light [4]. Also, scattering forces in the focal volume heavily depend on the polarization while trapping forces are a result of the intensity distribution [5]. Controlling both usually requires the inversion of the Debye-Wolf diffraction integral. This, in general, leads to spatially varying intensity, phase and polarization distributions [6, 7]. A number of methods exist to spatially encode intensity and phase distributions in a phaseonly spatial-light-modulator (SLM) by using macro pixel or locally varying diffraction efficiencies [8–10]. Others achieve spatial polarization [11–14], spatial polarization and spatial phase [15–17] or spatial polarization and spatial intensity [18–20] control using a single SLM [12–16, 19] or two SLMs [17, 18, 20]. Two groups introduced a setup allowing for independent spatial polarization, phase and intensity modulation [21, 22]. Han et al. accomplish this by using two sections on two SLMs each [22]. However, due to the two SLMs, this setup is not very compact and cost effective. In addition to that, the power utilization efficiency is limited by the use of non-polarizing beam splitters and four reflections at the SLMs. We estimate the maximum power utilization efficiency to be below 5%. While they show control of independent spatial phase and polarization distributions as well as intensity and polarization distributions, no simultaneous measurements of independent phase, intensity and polarization distributions are presented. Maurer et al., for the first time, show that independent phase, intensity and polarization modulation is possible with a single phase-only SLM and a Wollaston-prism [21]. They use their setup to generate different Gauss-Laguerre and Hermite-Gauss modes. These modes, however, are non-diffractive and no phase measurements are shown in the paper, leaving it somewhat unclear if less error tolerant patterns are also obtainable. Additionally, they explain that they needed to change the wave retarder from λ /4 to λ /2 when switching from Laguerre-Gaussian to Hermite-Gaussian vector beams making it impractical for fast switching or remote controlled applications. Here, we introduce an easy to implement setup, similar to the setup which Preece et al. introduced for polarization-only control [12]. In contrast to several previously reported systems, #197136 - $15.00 USD Received 4 Sep 2013; revised 24 Oct 2013; accepted 27 Oct 2013; published 8 Nov 2013 (C) 2013 OSA 18 November 2013 | Vol. 21, No. 23 | DOI:10.1364/OE.21.028167 | OPTICS EXPRESS 28168
the proposed setup abstains from using special gratings or prisms (as used in [15, 16, 21]). This facilitates the integration in existing setups since it allows a more flexible choice of optics used. Our setup uses a single SLM (in opposition to [17, 22]) and still allows for full control of the spatial intensity, phase and polarization distribution. We show that accurate almost arbitrary non-diffractive beams (but not limited to those) are possible using spatial phase and intensity measurements. The various patterns are generated by electronically changing the patterns on the SLM without the need of changes in the experimental setup (such as in [21]). The maximum power utilization efficiency is limited by the diffraction efficiency of the SLM alone and may therefore be above 60 %. To the best of our knowledge, we show for the first time the combination of arbitrary polarization patterns with simultaneous phase and intensity profiles. 2.
Setup and principle
The setup is shown in Fig. 1. A linearly polarized helium-neon (HeNe) laser is expanded by the telescope consisting of L1 ( f = 50 mm) and L2 ( f = 500 mm) to overfill the aperture of a phase-only spatial-light-modulator (SLM, Hamamatsu X10468-01). Two different holograms are simultaneously displayed side by side on the SLM. The holograms are each overlayed with a blazed grating directing the reflected intensity into the first diffraction orders. The diffraction efficiency of the gratings may locally be reduced to allow for intensity modulation of the beams. Thus, independent phase and intensity modulation is possible. The two designed patterns pass through L2 which serves as the first lens in the 4-f-setup. The patterns are then separated and the spatial frequency of the grating is clipped in the focus of L2 . One branch passes through a halfwave plate leading to orthogonal polarizations (0◦ and 90◦ ) in the two arms. Note, that when working with pulsed lasers, mirrors M1 and M2 may optionally be adjusted to yield equal path lengths for the two branches. A polarizing beamsplitter combines the two beams which hereafter propagate in the same direction. L3 ( f = 400 mm) back transformes the combined pattern. We measure the obtained distribution with a wavefront camera that allows for simultaneous spatial phase and intensity measurements (Phaseview DWC 1000), with the sensor placed in the target plane. An analyzer is optionally placed in front of the camera to visualize the polarization distribution. Technically, this analyzer should be placed in the target plane, since it is there where the exact polarization is achieved. However, due to the rather paraxial setup we still achieve a good representation when placing the analyzer about 20 cm in front of this plane.
Fig. 1. Scheme of the setup. A HeNe laser is expanded by lenses L1 and L2 . The beam impinges perpendicularly onto a phase-only SLM. Two separate holograms overlayed with diffraction gratings guide the modulated wave along path A and path B respectively. L2 and L3 compose a 4- f -system with a half-wave plate placed in path A. A digital wavefront camera is used to simultaneously measure the intensity and phase distributions. An analyzer is optionally placed before the DWC to visualize polarization distributions.
#197136 - $15.00 USD Received 4 Sep 2013; revised 24 Oct 2013; accepted 27 Oct 2013; published 8 Nov 2013 (C) 2013 OSA 18 November 2013 | Vol. 21, No. 23 | DOI:10.1364/OE.21.028167 | OPTICS EXPRESS 28169
3.
Methods
In the setup introduced above the two field distributions generated by the holograms may be treated independently. One hologram leads to the linear polarized wave: Et,1 (x, y) = At,1 (x, y) · exp[i · Pt,1 (x, y)] , (1) 0 while the other after passing through the half-wave plate produces: 0 Et,2 (x, y) = At,2 (x, y) · exp[i · Pt,2 (x, y)]
(2)
in the target plane. Let us suppose we want to create an intensity distribution It (x, y), a phase distribution Pt (x, y) and a polarization distribution Polt (x, y). In our target plane this would require the following field distribution (in analogy with the ordinary and extra-ordinary wave in retarders): Et (x, y) = Et,1 (x, y) + Et,2 (x, y) =
At,1 (x, y) · exp[i · Pt (x, y)] · exp[+i · Polt (x, y)/2] At,2 (x, y) · exp[i · Pt (x, y)] · exp[−i · Polt (x, y)/2]
. (3)
The common phase term Pt (x, y) sets the phase distribution while the local polarization is controlled by the phase shift between the two arms Polt (x, y) (0,π : linear, π /2 and 3π /2: circular, else: elliptical polarization). The direction α of the local polarization vector is adjusted by the 2 (x, y)+A2 (x, y) equals the desired intenratio of At,1 (x, y) and At,2 (x, y) while ensuring that At,1 t,2 sity distribution It (x, y). At,1 (x, y) and At,2 (x, y) are therefore calculated as follows (for circular polarization α is set to π /4): √
It cos(α ) , √ At,2 (x, y) = It sin(α ) .
At,1 (x, y) =
(4) (5)
For some α this leads to negative amplitudes. In this case the target amplitudes are modified to ensure that positve and negative amplitude values are out of phase: At,1 (x, y) = −At,1 exp(iπ ) ,
(6)
At,2 (x, y) = −At,2 exp(iπ ) .
(7)
The final function to inscribe is a complex function that needs to be encoded very accurately as a phase-only pattern. Commonly used encoding schemes show a comparable large dependence of the phase pattern on the intensity pattern. Since the phase pattern defines the resulting polarization and phase, we focus on a very accurate encoded phase at high fidelity of the encoded amplitudes. We experienced rather low cross-modulation between phase and intensity using the following encoding scheme for the two patterns (similar to the encoding scheme in [9]): P1 = [Pt + Polt /2 + angle(At,1 )] + blaze1 · sinc(1 − abs(At,1 )) ,
(8)
P2 = [Pt − Polt /2 + angle(At,2 )] + blaze2 · sinc(1 − abs(At,2 )) .
(9)
#197136 - $15.00 USD Received 4 Sep 2013; revised 24 Oct 2013; accepted 27 Oct 2013; published 8 Nov 2013 (C) 2013 OSA 18 November 2013 | Vol. 21, No. 23 | DOI:10.1364/OE.21.028167 | OPTICS EXPRESS 28170
The blazei are blazed phase gratings in the intervall −π to π . After summation P1 and P2 are each modulated to a 2π phase range. Some corrections to the above procedure may be neccessary. We account for differences in the intensity distribution reflected by P1 and P2 and ensure a linear target-amplitude to obtainedamplitude mapping by precompensation. Additionally, Polt is modified to Polt +Δ, with Δ being an offset allowing fine adjustment of the path lengths of the two branches. Fine adjustment of the positions and propagation directions of the beams is done by electronically shifting the displayed patterns against each other and adding a tilt to them respectively. 4.
Experimental results
The method explained above allows a very versatile definition of almost arbitrary intensity, phase and polarization distributions. Figure 2 shows measurements of different combinations of such distributions. In Figs. 2(a)-2(c) we set the polarization to circular polarization and in Figs. 2(d)-2(f) to radial polarization. The target intensity is displaying a uniform (first column), a shaded-ring-filter (second column) or a cross-shaped distribution (third column). Three target phases, a constant phase (Fig. 2(a)), astigmatism (Fig. 2(b)) and coma (Fig. 2(c)), are adressed for each combination of polarization and intensity distribution. The target Zernike coefficients of astigmatism and coma are set to λ and λ /2 respectively. The constant phase distribution leads to a phase that is determined by a combination of the aberrations in the two paths. Since these aberrations are slightly different for the two paths the measured phase is expected to change with the target polarization. Since precompensation of these aberrations is beyond the scope of our SLM – in consequence – the measured phase obtained when displaying the constant phase distribution is respectively taken as a reference for the other two target phase patterns. Figure 3 shows examples of more complex polarization patterns. The polarization patterns are visualized by the black arrows in Fig. 3(a). For the uniform intensity distribution and the shaded-ring-filter the polarization is set to circular in the inner ring, to radial in the middle ring and to azimuthal in the outer ring. For the cross intensity distribution we combine two orthogonal linear polarizations, one within and one outside the cross. The same intensity and phase patterns are targeted as in Fig. 2. Despite the complexity of the patterns the definition of the distributions is still very accurate. 5.
Discussion
As visible in Figs. 2 and 3, the addressed combinations follow the expected behaviour. For example, independent of the phase and intensity pattern, for circular target polarization no major intensity modulation is observed when rotating the analyzer. Meanwhile both the phase pattern and the intensity pattern are defined with high precision. To give a measure of the degree of polarization control, we calculate the average Stokes parameters for the uniform intensity distribution with circular polarization, shown in Fig. 2(a): ⎞ ⎛ ⎞ ⎛ 1 I0 + I90 ⎟ ⎜ −0.07 ⎟ ⎜ I0 − I90 ⎟=⎜ ⎟ S = ⎜ (10) ⎠ ⎝ −0.03 ⎠ . ⎝ √ I45 √− I135 2 I0 I90 sin(δ ) 0.99 Here, the Ii are the measured intensities with the numbers indicating the angle of the analyzer. The last parameter is calculated under the assumption that the relative phase δ between the two orthogonal waves is exactly π /2. All parameters are normalized to the first parameter.
#197136 - $15.00 USD Received 4 Sep 2013; revised 24 Oct 2013; accepted 27 Oct 2013; published 8 Nov 2013 (C) 2013 OSA 18 November 2013 | Vol. 21, No. 23 | DOI:10.1364/OE.21.028167 | OPTICS EXPRESS 28171
Fig. 2. Circular (a)-(c) and radial (d)-(f) target polarization for a uniform (first column), a shaded-ring (center column) and a cross (third column) intensity distribution. The reference phase (a) is set to a spatially constant phase value. Astigmatism (b) and coma (c) are set as target phases for each polarization-intensity combination. The white arrows indicate the analyzer setting to visualize the polarization. For the phase and intensity measurement the analyzer is removed from the setup.
#197136 - $15.00 USD Received 4 Sep 2013; revised 24 Oct 2013; accepted 27 Oct 2013; published 8 Nov 2013 (C) 2013 OSA 18 November 2013 | Vol. 21, No. 23 | DOI:10.1364/OE.21.028167 | OPTICS EXPRESS 28172
Fig. 3. Mixture of polarizations as visualized by the black arrows in the target intensity distributions. Uniform intensity distribution and shaded-ring-filter: Circular polarization in the inner ring, radial polarization in the middle ring and azimuthal polarization in the outer ring. Cross intensity distribution: 90◦ linear polarization inside the cross, 0◦ linear polarization outside the cross.
This compares very well with the Stokes vector for a perfectly circularly polarized wave: S = (1, 0, 0, 1)T . The mean-square error between the desired and obtained intensity distribution for this pattern is below 5 %. Exemplarily, after subtracting the reference, the obtained Zernike coefficient for the phase front of the uniform intensity distribution in Fig. 2(b) is 0.97λ (616 nm) - very close to the desired λ (633 nm). Since the setup is mostly common path, the distributions are very robust against vibrations, turbulences and drifts. However, while being cost-effective and compact, one drawback of the proposed method is its reduction of the bandwidth. As a result of the phase range being limited to 2π by the SLM, phase-wrapping leads to cross-modulation and phase jumps. The latter causes a sharp drop in intensity visible in all patterns with astigmatism and coma added. Cross-modulation occures for extreme phase values (when the pattern is distorted by the wrapping) and results in heightened or lowered intensity, for example, visible in the intensity in Figs. 2(b) or 2(d). In consequence this also leads to imperfections in the polarization distribution. Also, due to these limitations we cannot correct for the strong aberrations present in the system but take them as the reference wave front. The strong aberrations are introduced by the beamsplitter. The experimental distortions may be reduced using an SLM with a higher phase modulation range and using look-up tables to reduce cross-modulation.
#197136 - $15.00 USD Received 4 Sep 2013; revised 24 Oct 2013; accepted 27 Oct 2013; published 8 Nov 2013 (C) 2013 OSA 18 November 2013 | Vol. 21, No. 23 | DOI:10.1364/OE.21.028167 | OPTICS EXPRESS 28173
6.
Conclusions
We demonstrate a compact, flexible and cost-effective setup that allows for almost arbitrary independent phase, intensity and polarization distributions built around a single phase-only spatial-light-modulator. All combinations are created electronically without requiring a change in setup. We abstain from using special gratings or prisms and do not restrict ourselves to nondiffractive beams. We demonstrate circular and radial as well as combinations of circular and locally varying linear polarization distributions. Each polarization distribution is combined with different intensity and phase patterns. The obtained distributions are visualized by simultaneous wavefront and intensity measurements. Exemplary Stokes parameters and deviations from target distributions are calculated to quantifiy the quality of the obtained patterns. The patterns follow the expected behaviour very well.
#197136 - $15.00 USD Received 4 Sep 2013; revised 24 Oct 2013; accepted 27 Oct 2013; published 8 Nov 2013 (C) 2013 OSA 18 November 2013 | Vol. 21, No. 23 | DOI:10.1364/OE.21.028167 | OPTICS EXPRESS 28174
