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1 DOI:.38/NPHOTON.2.85 This supplement has two parts. In part A, we provide the rigorous details of the wavefront correction algorithm and show numerical simulations and experimental data for the cases of very weak and very strong optical aberrations. In part B, we focus on the application of our method to optimize the performance of an optical trap. A - Wavefront correction algorithm: Numerical simulations and experimental data Interference conservation and mode orthogonality Our method is based on the orthogonality property of the modes generated by the different regions of the spatial light modulator (SLM). Here, we rigourously deduce this orthogonality relation within the paraxial approximation but we note that this property remains valid for Maxwell s equations [29]. Within this approximation, the general propagation of a light field is given by Huygens-Fresnel integral of the form [35] F (x, y, z) = ik 2πB e ikz F (x,y, )χ(x,y )e ik 2B (A(x2 +y 2 ) 2(xx +yy )+D(y 2 +y 2 )) dx dy, where F (x,y, ) is the field in the initial plane with x and y the local transversal coordinates, k the free space wavevector and F (x, y, z) the field in the final plane with (x, y, z) the coordinates in this plane where z is the axial distance of propagation. The ABCD coefficients correspond to the standard matrix describing the propagation through the paraxial system. The function χ(x, y) is a phase-front changing term only acting in the initial plane and chosen such that χ χ =. The intensity of the beam is conserved during propagation. Indeed, we have F (x, y, z)f (x, y, z)dx dy = F (x,y, )F (x,y, )dx dy. (S2) Next we consider the superposition between two initial fields F and G giving rise to the fields F and G after their propagation using integral (S). The energy conservation relation (S2) yields the two following relations: G F + GF dx dy = G F + G F dx dy ig F igf dx dy = ig F ig F dx dy, where we take into account the two independent superpositions F +G and F +ig. These two relations correspond respectively to the conservation of the real and imaginary part: G (x, y, z)f (x, y, z)dx dy = G (x,y, )F (x,y, )dx dy. (S3) This defines as the interference conservation relationship. Using the interference conservation relation, we can define the orthogonality of optical fields as being the case when the interference integral is equal to zero. Further, if two modes are orthogonal in the initial plane, these modes remain orthogonal during their propagation. Finally, we can easily ascertain that if two fields originate from two non-overlapping initial areas, then these fields are orthogonal. (S) Wavefront correction The wavefront correction method is based on the ability of the spatial light modulator (SLM) to split the beam into many orthogonal modes and to change both the phase and the amplitude of each of these modes independently. Here, we consider the optical field F (x, y) in the image plane (z = ) as a superposition of N fields F i (x, y), each resulting from one of the modes controlled by the SLM N F (x, y) = a i F i (x, y), i= NATURE PHOTONICS 2 Macmillan Publishers Limited. All rights reserved.

2 DOI:.38/NPHOTON.2.85 where the coefficients a i correspond to their complex amplitudes. The intensity at the point of reference (i.e. probe position) (x,y ) is given by I(a i ) = F (x,y )F (x,y ) N N = a i Fi a j F j, i= j= (S4) where the fields F i are measured at the point (x,y ). To determine the maximum intensity we can treat equation (S4) as an action integral that needs to be maximised with respect to the complex coefficients a i. Taking into account that the different fields considered originate from different regions of the SLM, we need to maximise this integral with an additional constraint a i a i = for each amplitude a i. This can be determined by introducing the Lagrange multipliers μ i and defining S S(a i,μ i ) = N N N a i a j Fi F j μ i (a i a i ), i= j= i= (S5) which is the modified action integral including the constraints. Maximising S, we find the two Euler equations: =a i a i i =..N (S6) N μ i a i = a j Fi F j i =..N. (S7) j= The second equation can be simplified considering that N j= a jf j = F a is a scalar product, that the Lagrangian multipliers μ i are real, and the complex coefficients a i are of the form exp(iψ i ). Combining these two equations we find: μ i exp(iφ i )=(F a)f i. This means that the phase factors ψ i, defining the complex coefficients a i, can be determined up to a global phase factor (same for all modes) as: ψ i = arg(fi ) + arg(f a) = arg(fi )+const., (S8) while the Lagrangian multipliers correspond to μ i = F i (F a). This approach corresponds to the phase-only correction of the wave front. At this point, we remark that the Lagrangian minimisation approach, using the constraint relations a i a i =, for each complex amplitude a i, delivers the same optimum as the method developed in [25]. The advantage of our approach is the possibility to modify the intensity constraints or the degrees of freedom of the system making a further optimisation possible. Here, we consider changing the amplitude/intensity of the individual mode to achieve the additional optimisation. Indeed, using solution (S8) together with (S4), we can determine the maximal intensity achievable within the considered constraints: I(a i ) = = N N a i Fi a j F j (S9) i= j= ( N 2 F i ). (S) i= We remark that the maximal intensity depends on the intensities of the individual modes and is maximum when all the modes have the same intensity. Equilibrating the intensities can be achieved either on the back aperture of the objective (ex situ) or at the focal point (in situ). This can be achieved through an amplitude modulation of the diffraction efficiency of each mode on the SLM [39]. This approach corresponds to implementing a complex modulation in our method in contrast to the phase-only approach presented by I.M. Vellekoop and colleagues [22-25]. 2 NATURE PHOTONICS 2 Macmillan Publishers Limited. All rights reserved.

3 DOI:.38/NPHOTON.2.85 SUPPLEMENTARY INFORMATION In practical terms, using the complex modulation, we reduce the intensity of the higher efficiency modes such that all modes have the same intensity in the back aperture implying an uniform illumination. The final amplitude mask on the SLM gives direct insight into the alignment and optical quality of the system. In the case of very high optical distortions, the intensity of many modes can be very low at the focal point which may make it inefficient to equilibrate the intensity of each mode in situ. Numerical implementation We consider a Gaussian beam incident on the SLM (waist 3μm) and focussed through a microscope objective (see figure S). The propagation of the beam is modelled through the water in the sample (index n =.33) using soft aperture Beam Diffuser Probe Particle SLM plane Objective FIG. S: Diagram showing the different elements considered in the numerical simulation. We consider a beam incident upon the SLM which acts as an aperture. This is then focused upon a probe particle with an intermediately placed diffusing medium. the ABCD matrix approach including laterally offset optical elements. The effect of the SLM on the optical beam is modelled by approximating each independent region of the SLM as a soft Gaussian aperture with a waist of 3μm and positioned upon a square grid with a lattice constant of 6μm. In this manner, the propagation of the beamlet after the soft aperture can still be expressed in a closed analytical form. The resulting beamlets are then focussed by the objective which is modelled by a thin short focal distance lens (f = 2mm). The optical distortion is modelled by including a random phase plane placed before the focus. The phase front correction algorithm is implemented by introducing a variable phase delay on each of the modes and, in case of the amplitude modulation, a variable transmittance of each of the soft Gaussian apertures. Figure S2b shows the distorted field in the focal plane of the objective while Fig. S2a shows, for comparison, the beam in the absence of distortion. Figures S2c-d exemplify the effects of the two different phase front correction methods; phase-only and complex wavefront correction. For both approaches, equation () is used to retrieve the phase and amplitude needed for the correction. Using the same model, it is possible to identify the wavefront flatness as given by equation (2). In Figure S3 we observe the characteristic wavefront flatness for turbid media. Experimental implementation In this paragraph we present an experimental study showing the performance of the optimization algorithm implemented by the SLM. As the method requires a single point intensity detection it is important to discuss its sensitivity. In the original zero-order approach [22-25], the test mode interferes with the superposition of all the other modes. With an increasing number of total modes, the influence of the tested mode on the overall measured intensity thus becomes miniscule. This saturates the dynamic range of the intensity detector and, at the same time, accelerates photobleaching when using a fluorescent particle as the intensity probe. In contrast to the zero order technique, our method employs a reference mode in the first diffractive order that interferes with the tested modes one by one while NATURE PHOTONICS Macmillan Publishers Limited. All rights reserved.

4 DOI:.38/NPHOTON.2.85 μ μ μ μ μ μ μ μ FIG. S2: Wavefront correction in the case of an optical diffuser. a, Intensity cross-section of the perfectly focussed beam without any diffuser and b, including a diffuser. The effect of the two different wavefront correction methods is seen when applying c, phase only modulation and d, complex modulation. wavefront flatness < g > separation of modes [mm] FIG. S3: Wavefront flatness. Analogous to figure 5 in the main manuscript but with parameters given in the supplementary text. all the other inactive modes are diffracted to the zeroth order. As the individual modes carry the same intensity, the ratio between the oscillation amplitude and the constant background term in equation () is maximal. This gives the largest dynamic range possible for the detection of the relative phase and amplitude of the modes, regardless of the number of the modes involved. This trend is illustrated in Fig. S4. The harmonic signal in equation () can be fully described by three real parameters: a constant term, an oscillation amplitude and its phase. Therefore one needs at least three intensity measurements for different phase values of the tested mode to extract the optimal phase for each mode. Naturally, the more data-points that are recorded, the more precise the results obtained. This is especially true if one scans through a number of periods (phase values from to 2πN) where the optimal phase extraction is analogous to an extremely sensitive lock-in amplification technique. Nevertheless, since our method offers a very high signal sensitivity, we wish to present its performance in the extreme case when only three measurements are taken, representing the shortest possible time-intervals for the optimization procedure. In our test system, introduced in the main article, we studied the influence of the number of modes upon the intensity enhancement in the case of 4 NATURE PHOTONICS 2 Macmillan Publishers Limited. All rights reserved.

5 DOI:.38/NPHOTON.2.85 SUPPLEMENTARY INFORMATION FIG. S4: Saturation of the detector dynamic range. The saturation is defined by the ratio between the amplitude of the periodic term (2 E t E r ) and the constant term ( E t 2 + E r 2 ) from Equation (). low aberrations (caused mainly by the SLM curvature). Even though our SLM features a refresh rate of 6Hz, we operate at 2Hz due to systematic delays between the computer, SLM and camera. Figure S5 shows that the system aberrations can be quantified well even in this case. FIG. S5: Optimization procedure at low-aberrations. a-f, Retrieved phase aberrations for different numbers of modes. i, Corresponding relative intensity enhancement with respect to the original peak intensity. In contrast to the case of the entirely randomized light, with an increasing number of modes, the enhanced intensity saturates as the modes become more correlated. Analogously to the study in [22], we employed our method to optimize intensity of light randomized by a diffuser (see Fig. S6). To enhance the resulting quality we employed 44 modes and increased the number of intensity measurements for each of them to 6. After a period of 3.2 hours we obtained an intensity enhancement (ratio between the optimized intensity and the original averaged speckled background) of 76. B - Aberration free trapping Here, we present our quantitative analysis of optical trapping performance in the system with the correction method implemented. Since most of the relevant data available in the literature is related to measurements at wavelengths NATURE PHOTONICS Macmillan Publishers Limited. All rights reserved.

6 DOI:.38/NPHOTON.2.85 FIG. S6: Optimizing light throughput through a diffuser. a, The original speckled intensity. b, The intensity after the optimization procedure. around 64 nm we have assembled our system for this wavelength and performed quantitative measurements. Let us briefly describe the procedure for the in situ correction for this case. The laser light reflected from the SLM (Hamamatsu LCOS X468-3) was imaged onto the back aperture of an objective (Zeiss C-Apochromat 63x/.2 W), similar to that seen in Figure 3. For the intensity probe, we used a 2nm gold particle adhered to the bottom of the sample chamber. The scattered light from this particle was collected by a CCD camera and used as a probe signal. To minimize reflections from the surface where the particle was adhered, the sample was filled with glass index matching immersion oil. To minimize any possible thermal drift, we have introduced a 532nm probe beam focused on the same particle. The light from this was collected by the condenser and imaged onto a position sensor (Pacific Silicon Sensor Inc. DL-7PCBA3). Its signal was used to measure the position of the gold particle with respect to the objective. Positional data were utilized in a feedback loop employing a nano-positioning stage (PI P-733.3DD) to keep the position fixed with an accuracy of ± nm. This allows access to longer time intervals for performing our optimization procedure. To evaluate the importance of in situ correction, we performed an ex situ correction as a reference experiment. In this instance, the objective was replaced with a lens of focal distance of 4 mm that focused the laser light onto a CCD camera. A single CCD pixel was used as the intensity probe. At the same time, we were able to demonstrate the advantage of our complex correction method. In the case of this setup, the SLM chip was not significantly overfilled. Therefore we obtained a non-uniform amplitude signal seen in the results for the ex situ correction (Fig. S7 a). In the optimal case, the objective back aperture was illuminated uniformly, which can still be achieved in our setup using an amplitude modulation on the SLM together with the phase modulation. We followed Cizmar et al. [39] for this purpose. We determined the trap stiffness for a μm polystyrene particle from the power spectra obtained using a back focal plane interferometry technique [7]. The data are summarized in Figure S7. It is seen that both in situ correction, as well as complex correction, bring a significant enhancement of the optical trapping quality. While the lateral stiffnesses typically improves up to 2%, the axial stiffness increased by a factor of three. This results support the well known fact that the axial stiffness is the most sensitive to the trap quality. We also observed a notable reduction of the lateral stiffnesses between the phase only and complex in situ corrections (Fig. S7 d and f). This is most likely due to the imperfect implementation of the amplitude modulation. Indeed, the design of the holographic mask for this case is an iterative process that does not converge to offer a perfect solution. This results in a very low speckled intensity background surrounding the optical trap in the focal plane. This background does not influence the optical trap significantly, but effectively leads to an over-estimate of the optical power measured in the back focal plane. The work of Rohrbach [4], where the same selection of water immersion objective NA and optical wavelength were used, showed near-perfect trapping performance. Our data agrees very well with his results. In particular, the ratio between the axial and the lateral stiffnesses gives the same value as recorded in his work. 6 NATURE PHOTONICS 2 Macmillan Publishers Limited. All rights reserved.

7 DOI:.38/NPHOTON.2.85 SUPPLEMENTARY INFORMATION a) 6 Field amplitude Correction ex situ 6 Field phase 3 b) 6 Field amplitude Correction in situ 6 Field phase y [pixels] y [pixels] c) Phase only correction ex situ kx = 2.38 pn/μm/mw ky = 2.29 pn/μm/mw kz =.8 pn/μm/mw d) Phase only correction in situ kx = 2.82 pn/μm/mw ky = 2.5 pn/μm/mw kz =.424 pn/μm/mw e) Complex correction ex situ kx = 2.54 pn/μm/mw ky = 2.44 pn/μm/mw kz =.467 pn/μm/mw f) Complex correction in situ kx = 2.57 pn/μm/mw ky = 2.3 pn/μm/mw kz =.539 pn/μm/mw FIG. S7: Quantitative analysis of optical trapping performance. a, and b, Results obtained from the optimization procedure ex situ and in situ respectively. c, - f, Trap stiffnesses for a μm polymer particle (n=.59) vs. the optical power in the focal plane. The values of optical power were obtained from the power measured at the back focal plane multiplied by the objective transmittance (experimentally evaluated by a dual objective method). Measurements are taken for the cases of: c, phase only, ex situ correction; d, phase only, in situ correction; e, complex, ex situ correction and f, complex in situ correction. NATURE PHOTONICS Macmillan Publishers Limited. All rights reserved.

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