1. Introduction

Computer generated holography (CGH) enables projection of patterns in the focal plane of a lens. This ability to sculpt light has found diverse and rapidly developing applications ranging from optical trapping [1–3], laser fabrication (e.g., high throughput photopolymerization [4] and lithography [5]), and optical displays [6] to brain research (e.g., neuronal simulation [7–9] and functional imaging [10]). Phase-only Spatial Light Modulators (SLMs) enable arbitrary intensity pattern generation in the objective lens image plane. In a conventional computer generated holographic (CGH) setup [11], an expanded laser beam provides near-uniform illumination (typically Gaussian) of the entire surface of the SLM conjugated with, and in ideal cases covering, the entrance pupil at the objective back focal plane. Phase modulation at the entrance pupil plane results in intensity modulation at the objective image plane. Overfilling the objective back pupil exploits its full numerical aperture (NA), thus achieving optimal transverse resolution and axial confinement [12, 13].

To calculate the phase profile for a desired intensity pattern, an Iterative Fourier Transform Algorithm (IFTA) propagates the electric field back and forth between the back and front focal planes of the objective lens [14] in order to minimize the mean squared error in the intensity patterns [11]. Typically, the amplitude is constrained while the phase of the field is allowed to vary freely in both planes. The algorithm is initialized in the front focal plane with a random phase profile and the desired intensity pattern. Imposing a specific intensity pattern (Gaussian) in the back focal plane constrains the solution towards which the algorithm can converge and therefore resulting intensity patterns feature undesired inhomogeneities, or speckle (Fig. 1(a)). Specifically, destructive interferences arise at large phase mismatches between neighboring sample points [15], and especially at vortex phase singularities (or phase vortices) at which the optical field vanishes [16] as indicated by arrows in Fig. 1.

 

Fig. 1 Numerical simulation of a speckle patterns obtained by computer generated holography. a) intensity and b) phase. The arrow indicates an example of zero intensity associated with a vortex phase. The white circle in b) represents the boundary of the disk intensity in a).

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Non-homogeneous intensities are all the more critical in experiments involving non-linear light-matter interactions. For such systems exhibiting optical non-linearities (like two-photon absorption [8]), saturated optical responses (lasers [17], light-sentitive membrane channels [18]) or optical processes sensitive to temperature (photo-polymerization [19]), speckled patterns generated by regular CGH with phase-only modulation are thus detrimental and may result in experimental artifacts. Moreover, in applications using linear photo-excitation processes, the speckle pattern may also be problematic. For instance, when photo-stimulating biological cells with illumination patterns containing only a few speckle grains, statistical fluctuations may be large. In addition, fluorescence imaging with selective patterned excitation requires uniform illumination [20, 21]. Experimental approaches have been proposed to eliminate these inhomogeneities either by superposing statistically independent patterns [22], or by using a rotating diffuser [23], or through different techniques either based on a common path interferometer, the so-called generalized phase contrast (GPC) method [24], or using two SLMs to achieve complex amplitude modulation [25].

Alternatively, it is possible to modify (improve) the IFTA algorithm in order to limit the generation of speckle in the projected intensity pattern [26, 27]. However phase vortices are robust to the iterative algorithm [28] and speckle-free pattern generation necessitates careful engineering of the initial phase pattern. Pseudo-random diffusers were developed to avoid phase vortex creation [29] but, to our knowledge, such diffusers are not used for CGH mainly because the numerical solution would require to control also the amplitude of the impinging beam at the SLM plane, which is an experimental challenge. To match the required field amplitude, the SLM must thus impose amplitude modulation of the incident laser beam [25, 30–32] as well as phase modulation.

In addition, in order to retain the benefits associated with entire SLM illumination, the optimal field pattern - lacking phase singularities - should feature a power spectrum projection spanning the full SLM surface. Matching the power spectrum of the diffuser with a Gaussian beam profile impinging on the SLM also maximally exploits the laser energy budget. Ultimately, vortex-free CGH can only be performed by finding a tradeoff between intensity, uniformity, and diffraction efficiency.

In this paper, we propose an original experimental and algorithmic scheme and demonstrate the possibility to project vortex-free patterns with a standard CGH system. With this system, we investigate the tradeoff between intensity, uniformity, and diffraction efficiency. Compared with regular CGH, the proposed experimental technique preserves a large and continuous field of excitation centered on the optical axis where diffraction efficiency reaches its maximum. In addition, the proposed technique offers any usual CGH systems, a uniform patterning illumination modality.

In section 2, we describe the iterative algorithm and the optical system used to project vortex-phase free shapes obtained by dual phase and amplitude control with a single phase-only modulator. In section 3, we characterize an experimentally obtained spot using the proposed scheme and compare it to a conventional speckled CGH spot. Finally, in section 4 we numerically characterize and discuss tradeoffs between the pattern’s intensity uniformity, diffraction efficiency and axial confinement. We also illustrate experimentally the tradeoff between uniformity and diffraction efficiency.

2. Modified IFTA and experimental complex amplitude modulation with a phase-only SLM

Here we used a modified IFTA initialized with a pseudo-random phase pattern. In contrast with previous pixel-by-pixel derived pseudo-random phase profiles [29], we generate a pseudo-random diffuser by applying a simple spatial filter to a random phase map. As detailed in Appendix A, filter width and gain are adjusted to achieve: 1. the absence of phase vortices and, 2. electric field power spectrum uniformity and width corresponding to that of the illumination pupil. The modified IFTA then reduces the mean squared error (with respect to the desired intensity pattern) while allowing the back focal plane amplitude within the pupil to vary, imposing only zero amplitude outside the pupil.

Our modified IFTA yields a complex amplitude pattern which requires complex amplitude modulation by the SLM. In order to do so experimentally, we decrease the resolution at the objective pupil plane by introducing a low-pass spatial filtering aperture block into an intermediate image plane, as shown in Fig. 2. Therefore the field at any given point in the back focal plane of the objective is the average of the field coming from two (or more) adjacent SLM pixels. This allows for an extra (or more) degree of freedom to modulate simultaneously amplitude and phase in the pupil plane. Modulating the amplitude of a one-dimensional grid (or a checkerboard) at the SLM allows intensity modulation at the back focal plane of the objective, by stopping ±1 diffraction orders by the beam block. Typically, the fraction f of the energy remaining unmodulated by a binary grid is f = cos²(Φ/2), where Φ is the grid phase modulation amplitude. Therefore, for an incident SLM field amplitude A_ill and the desired amplitude A, the phase at the SLM is modulated as:

 

Fig. 2 A typical CGH projection system. A Ti:Sapphire laser beam is expanded to illuminate the full SLM active area. A telescope (L1,L2) conjugates the SLM with the back pupil plane of a first objective (Obj. 1). For the measurements of the axial propagation we used second objective (Obj. 2) and a tube lens (TL) to conjugate the image plane with a CCD camera (Cam.). In order to achieve complex amplitude modulation with a single phase-only SLM, an aperture block (App. Block) is introduced into an intermediate image plane.

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The amplitude and the phase of the field can be independently modulated in the image plane because their control is performed in two different spectral regions of the phase pattern. The amplitude is coded in the high spatial frequency range (between 1/2p and 1/2a) and the phase of the pattern in the low spatial frequency range (between 0 and 1/2p). The spatial filter introduced in the standard CGH setup acts as a “don’t-care” region in the image plane [28, 33] that leaves a continuous and centered field of excitation at the sample. The usable field of excitation at the intermediate excitation plane is λ f₁/p. Considering that p must correspond to an even number of SLM pixel widths, it is maximal when p = 2a. In this case, the size of the clear aperture in the beam block, is only reduced by a factor two along the one-dimension grid axis compared to regular CGH. When modulating with a checkerboard, the field of excitation is reduced along the two orthogonal axes. It is noteworthy that the field of excitation along the third (axial) dimension is also reduced by the aperture block because it scales as the inverse of the pixel size of the SLM [34, 35]. Here, the aperture block filters high spatial frequencies of the field at the SLM – thus averaging the field on neighboring SLM pixels – which is equivalent to increasing the actual size of SLM phase actuators. The axial field of excitation is then expected to be reduced by a factor of 2 (if amplitude modulation is performed with a 2 by 2 pixel checkerboard) or $\sqrt{2}$ (with a one-dimensional 2-pixel periodic grid).

3. Experimental characterization: setup and results

We used a typical CGH projection system (Fig. 2). A Ti:Sapphire laser (λ = 800 nm) illuminates a SLM (LCOS-SLM X10468-02, Hamamatsu Photonics) through a beam expander (BE). A telescope comprised of two lenses (f₁ = 1000 mm and f₂ = 500 mm) conjugates the SLM to the back focal plane of a first objective (Olympus LUMPLFLN 60XW, NA 0.9). The aforementioned aperture block was positioned at a Fourier plane relatively to the SLM (i.e. in an intermediate image plane). The intensity at the output of the first objective was imaged onto a CCD camera with a second microscope objective (Olympus UPLSAPO60XW, NA 1.0) and tube lens (f_TL = 180 mm). In order to evaluate the axial propagation of the holographic beams around the objective focal plane, we imaged the optical sections of the excitation volumes generated by different holographic beams by varying, with a piezo objective positioner (P-721.CDQ, Physik Instrument), the position of the first objective with respect to the collection objective kept at a fixed position. We achieve vortex-free phase holographic pattern projection with a phase only spatial light modulator, projecting a 10 µm disk to the objective focal plane with the system described in Fig. 2. First, we measured the Gaussian illumination profile A_ill on the SLM in order to achieve accurate amplitude control. We generated the initial diffuser by filtering a uniform random distribution phase map with a Gaussian kernel of waist w = 21 pixels, gain G₀ = 160 and oversampling s = 4 (see Appendix A for definition and discussion on w, G₀ and s). We then ran 100 iterations of our modified IFTA. Figure 3(b) shows the projected spot. For comparison, we generated a 10 µm spot with phase-only modulation (Fig. 3(a)), running a standard iterative algorithm, constraining amplitude in both focal planes. Figure 3(c) shows the intensity histograms fitted to Rician distributions [36] for both the holographic (dashed line, squares) and vortex-free spot (solid line, circles), the latter exhibiting a narrower peak centered on the average intensity. In the case of the holographic spot, the intensity histogram approached that of a speckle field integrated over the pixel size of the CCD [37]. Importantly, these histograms illustrate the suppression of high intensity speckle grains. We repeted these experiments for two photon fluorescence excitation, a configuration for which the presence of such high intensity fluctuations is all the more critical, and observed similar significant improvement as shown in Figs. 3(d)–3(f).

 

Fig. 3 Comparison of a standard 10 µm holographic disk (a) and a vortex-free holographic disk (b). The histogram (c) of vortex-free disk intensity (normalized by the average intensity < I >) exhibits a sharper, more pronounced peak (solid line, circles) than that of the speckled disk (dashed line, squares). Same comparison for two photon fluorescence excitation (d,e). Again the histogram (f) exhibits a sharper peak for the (solid line, circles) for the vortex-free disk compared to the that of the speckled disk (dashed line, squares).

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Next, we characterized the effect of speckle removal on axial confinement, comparing three approaches: the vortex-phase-free approach, the speckled phase-only approach and a flat-phased disk that can be obtained with interferometric configurations such as GPC [24, 38]. An intensity-stack (along the propagation axis) of the pattern shown in Fig. 3(b), can be seen in Visualization 1. Figures 4(a) and 4(b) show intensity profiles for experimentally obtained vortex-free and fully speckled shapes, respectively. Figure 4(c) (dashed green line) displays average intensity as a function of axial position for the vortex-free disk shown in b, which agrees well with the axial profile obtained through numerical simulation (solid green line). Predictably, the axial confinement for the vortex-free shape (∼ 70 µm) lies between the axial confinement of a speckled disk (dashed and solid blue lines) and the one of a flat-phase disk obtained with the GPC technique [38] (Fig. 4 red solid line, numerical simulation). For applications using two-photon excitation, the loss in axial confinement can be restored by combining CGH with temporal focusing which, as already shown both for conventional CGH and for GPC, enables to achieve an axial confinement of few microns independently on the spot shape an size [23, 38].

 

Fig. 4 Axial intensity profiles of the speckled (a) and singularity-free disk (b). Light energy axial confinement is plotted (c) for a speckled holographic disk (blue), a speckle free disk (green) and a disk with flat phase (red). Dashed lines show experimental data and solid lines numerical simulations. For the GPC beams the axial resolution was obtained from the Rayleigh range of a Gaussian beam defined as: $z_R=n\pi w_0^2/\lambda$ where n is the index of refraction and w₀ is the waist of the Gaussian beam. Experimental curves closely follow theoretical predictions.

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Regarding diffraction efficiency, we measured experimentally that the energy in the vortex-free disk was only 4% of the energy in the phase-only speckled disk. One of the cmm-FWHM) overfills the entrance pupil of the microscope objective (5.4 mm in diameter) which can in principle optimize axial confinement. Conversely, the intensity distribution required for speckle-free patterning is narrower than the entrance pupil diameter as illustrated in Appendix A. In the following section we aim at optimizing the diffraction efficiency by introducing a saturation of the complex amplitude while keeping the same illumination condition. Doing so will lead us to characterize the trade-off between shape uniformity, diffraction efficiency, and axial confinement.

4. Shape uniformity, diffraction efficiency, and axial confinement trade-offs

Our phase-vortex suppression method enables improved pattern uniformity at the cost of light efficiency and axial confinement. Specifically, complex amplitude modulation reduces efficiency by discarding a fraction of laser light incident on the SLM. Furthermore, our approach reduces amplitude uniformity at the objective back pupil and thus reduces the effective NA with which the wavefront propagates, degrading pattern axial confinement in the output plane. In order to mitigate these effects, we introduced a second modification to the IFTA. We clipped intensity peaks from the amplitude profiles at the SLM plane during the iterative algorithm. The ratio A/A_ill (as defined in section 2) was normalized causing energy loss for almost every pixel (where A/A_ill < 1). At the SLM, intensity peaks in the required intensity profile thus account for most of energy loss although their contribution to the total power may be negligible. Clipping intensity peaks may then improve efficiency. With each iteration, the normalized amplitude ratio A/A_ill is multiplied by a saturation factor > 1. Then, a threshold ceiling is applied to clip values surpassing 1. The saturation factor increases the similarity between uniform SLM illumination and the desired complex amplitude distribution, thus improving efficiency. As shown in Fig. 5(a), saturating by a factor of 1.5 for 50 iterations increased efficiency from less than 7% to more than 60%, while still producing an illumination distribution more uniform than generated with standard phase-only holography. Moreover, intensity clipping distributed more light over the objective back pupil periphery, improving the exploited system NA and axial confinement, see Fig. 5(b). For the saturation factor 1.5 high uniformity is expected (standard deviation < 22%) as well as improved axial confinement. Here, numerical simulations were performed using a Gaussian illumination profile whose full width at half maximum equals the pupil diameter (as in the experiment of section 3) and the initial pseudo-random phase mask was generated using the following filtering parameters (see Appendix A) : s = 4, w = 6, G₀ = 18.

 

Fig. 5 (Numerical simulations) Trade-offs between shape uniformity, diffraction efficiency and axial confinement. Diffraction efficiency increased and uniformity decreased through the course of 50 iterations of the IFTA(a). Algorithm convergence is represented by connected dots where the final 50^th iteration corresponds to the highest efficiency. The amplitude at the SLM was clipped at every iteration after saturating by factors indicated next to each line (a) or point (b). Red “x” points correspond to values obtained with phase-only holography. Higher saturation ratios produced more efficient, less uniform, and more axially confined patterns. Illustrations of shapes obtained for saturation factors of 1.0, 1.5 and 2.1 are shown in figures (c), (d) and (e) respectively as well as a shape with phase vortices generated with standard phase-only holography (f).

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Finally, we have experimentally shown that this modified algorithm to validate the former numerical results, using the same parameters (see Fig. 6). To do so, we used a simplified experimental setup by directly placing a camera at the focal plane of lens L1 in Fig. 2 (replacing the mirror M by the camera) and using visible light (λ = 635nm). In Fig. 6(a), the plot of Fig. 5(a) is reproduced experimentally and exhibits a similar trend: intensity uniformity globally degrades when increasing the diffraction efficiency. Interestingly, experimentally, the best shape uniformity was obtained for saturation parameters slightly larger than one, or more precisely, for diffraction efficiencies larger than a few percents. This difference is due to the low intensity of the spot under this condition, making interference with parasitic light (such as due to imperfect anti-reflexion coating of optics for instance) all the more critical. Using higher saturation factors improves significantly diffraction efficiencies at the expanse of intensity uniformity. Experimentally, we observed that the uniformity was critically sensitive to geometrical aberrations in the optical system, especially at high diffraction efficiencies, which explains that the larger inhomogeneities measured experimentally compared with numerical values. An illustration of so-generated patterns is shown in Fig. 6(b) for various saturation parameters. A qualitative optimal tradeoff between uniformity and diffraction efficiency can be identified for a 1.3 saturation parameter. Remarkably, up to a saturation parameter of 2.1 reaching efficiencies higher than 95%, uniformity could be improved compared with regular phase-only modulation.

 

Fig. 6 Experimental measurement of the trade-off between shape uniformity and diffraction efficiency. As predicted numerically, diffraction efficiency increases at the expense of intensity uniformity as illustrated in (a) (measurements performed for saturation parameters ranging from 1.0 to 2.1 as well as for phase-only modulation (Ph)). The patterns obtained for saturation parameters from 1.2 to 1.7, as well as phase-only modulation, are illustrated in (b).

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5. Conclusion

The use of a phase-only spatial light modulator in a 4-f configuration efficiently redistributes laser energy while exploiting the full NA of the lens, thus optimizing axial confinement of the projected pattern. However, phase-only modulation induces speckle in the projected patterns. In this paper, we proposed solutions to perform computer generated holography with reduced speckle contrast. We have suggested a novel technique to generate diffusers avoiding the creation of phase vortices and have experimentally demonstrated the generation of vortex-free spot featuring greater amplitude uniformity compared to a standard holographic spot. This improvement in uniformity comes at the expense of axial confinement, and we analyzed the achievable trade-offs between uniformity and axial confinement. Such trade-offs are fundamentally related to spatial modulation of a laser beam in the Fourier plane and thus are not specific to the new technique we proposed and exist regardless of the algorithm used [33]. For application using 2P excitation the loss in axial resolution can be recuperated by combining vortex-free CGH with temporal focusing. Compared to GPC methods the advantage of this approach lies in the possibility to generate multi-scale pattern (from a single diffraction limited spot to a large area) on a larger field of view without affecting the diffraction efficiency (Supp Fig. 7 in [23]). The improvement of intensity uniformity has the potential to widen the range of R&D applications of phase-only computer-generated holography’s utility.

 

Fig. 7 Probability P(Δφ = +2π) of generating a vortex as a function of the standard deviation for the phase difference between two adjacent corners of a s-wide square after filter: σ_Δ. Results from numerical simulation (dots) is the ratio of positive vortices to the number of diffraction limited spots in the field of view. For low values of P (see inset), we find that $P=\sqrt{2}{P}_k$.

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Appendix A: Pseudo-random diffuser generation

In this appendix we detail the mathematical basis for generating diffusers without phase vortices. The processes involves setting two parameters of a Gaussian filter (width and gain) which are thus partially constrained by this objective.

Numerical generation of phase profiles are performed over pixelated maps. To resolve the speckle pattern, the sampling period in the image plane must be smaller than the Point Spread Function (PSF) width λ/(2NA), where λ is the illumination wavelength. In order to probe and constrain intensity at intermediate positions during the numerical iterative process, the image plane must be oversampled by a factor s > 1. The distance between two sampling points is then λ/(2sNA).

A vortex occurs if the summed phase shift on a closed loop around a point is greater than 2π. In the case of a pixelated phase map, this summed phase shift is expressed as:

(3)$$\mathrm{\Delta }\phi =\mathrm{\Delta }{\phi }_1+\mathrm{\Delta }{\phi }_2+\mathrm{\Delta }{\phi }_3+\mathrm{\Delta }{\phi }_4$$

where Δφ_k is the phase difference between two adjacent corners of a s-wide square defined (wrapped) in the range ]− π, π], thus yielding Δφ = ±2π in the presence of a vortex. Singularities can therefore be prevented by reducing phase differences between neighboring pixels. Previous methods based on iterative pixel-by-pixel construction achieve smoothly varying phase [29]. Here, we propose a novel method for generating two-dimensionally phase-smooth diffuser maps by spatially filtering a random distribution of phases. Spatial filtering of an initially random phase map smooths the steep phase steps, increasing spatial phase correlation.

A compromise has to be found since an excessively smooth phase map deteriorates axial confinement of the projected pattern. Indeed the axial confinement is optimal when the power spectrum of the electric field uniformly spans the full NA of the lens. In order to achieve: 1. the absence of phase singularities and, 2. electric field power spectrum uniformity and width corresponding to that of the illumination pupil, we adjusted two parameters of the filtered phase mask: the filter width and its gain.

If the initial random phases are uniformly distributed between −π and +π, the obtained distribution has expected value µ₀ = 0 and variance ${\sigma }_0^2={\pi }^2/3$. After filtering by a Gaussian filter

(4)$$\frac{G_0}{2\pi w^2}\exp \left(-\frac{x^2+y^2}{2w^2}\right)$$

of width w and static gain G₀ (and considering that the width is sufficient for the central-limit theorem to be valid), the filter phases will follow a Gaussian distribution with expected value µ_w = µ₀ = 0 and variance

(5)$${\sigma }_w^2={\sigma }_0^2\frac{G_0^2}{2\pi w^2}=\frac{G_0^2\pi }{6w^2}$$

Finally, the variance for the phase difference Δφ_k between two adjacent corners of the above mentioned s-wide square will be twice this value, reduced by the factor accounting for the correlation introduced by the filtering:

(6)$${\sigma }_{\mathrm{\Delta }}^2=2{\sigma }_w^2\left[1-\exp (-s^2/(4w^2))\right]$$

and the probability that one of the phase difference Δφ_k will be wrapped is:

(7)$$P_k=\frac{1}{2}\text{erfc}\left(\frac{\pi }{\sqrt{2{\sigma }_{\mathrm{\Delta }}^2}}\right)$$

Expressing the probability P(Δφ = +2π) of generating a positive vortex depends on the individual probabilities P_k, Eq. 7, for each phase difference Δφ_k, which are also correlated. In the case where the probability P is small we assume that it is simply proportional to the probability P_k. We validated this assumption through numerical simulations, shown in Fig. 7, and found that in the cases where the total number of positive vortices (Δφ = +2π) is less than 1% (P < 0.01) then $P=\sqrt{2}{P}_k$.

After eliminating phase singularities from the image plane, we need to optimize power spectrum uniformity in the Fourier (or pupil) plane in order to achieve a situation as close as similar as possible to uniform SLM illumination. In order to do this we need to adjust the filter parameters G₀ and w, which are linked through Eqs. (5–7) for a give a chosen probability P_k. We will vary w and adjust G₀ in order to keep the same vortex probability. Figure 8 illustrates power spectra in the pupil plane and the phase distribution in the image plane with filters of various width w.

 

Fig. 8 Example phase patterns with histogram of the distribution (a–c) obtained by filtering an initial random phase map and their corresponding power spectra (e–g) for a uniform intensity for the disked-shape in the image plane. The filter widths used are w = 1 pixels (a, d), w = 4 pixels (b, d), w = 10 pixels (c, f). The oversampling factor is s = 4. The filter gain is ajusted so the vortex probability is 10⁻⁴: G₀ ≈ 1, 2 (a–d), G₀ ≈ 0 9 (b–e), G₀ ≈ 58 (c–f). In (g) we show the distibution of the intensity across the pupil, averaged over all directions.

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We now discuss the two theoretical limit cases, the shortest and longest possible correlation lengths w. For w = 1 pixel, illustrated in Fig. 8(a): as w is small, the variance of the random phase must also be small to avoid the creation of phase vortices. In this case, the field at the sample plane A = A₀ exp(iϕ) may be simplified by the weak phase approximation: A ≃ A₀(1 + iϕ) whose Fourier transform is: $\tilde{A}\tilde{A}{\tilde{A}}_0+i{\tilde{A}}_0\ast \tilde{\phi }$. If A₀ is a large shape of the uniform square in Figs. 8(a)–8(c), the power spectrum therefore exhibits a bright cross shaped diffraction pattern on the optical axis, see Fig. 8(d). Here, the phase power spectrum determines the pupil aperture energy distribution. The pupil plane intensity is thus comprised of a centered bright spot added to the diffuser power spectrum. In Fig. 8(e), the contrast is dominated by the central bright spot and does not allow for the visualization of the power spectrum spreading due to the diffuser. The central hotspot precludes this solution’s implementation with SLMs.

For w larger than the target pattern, the in-shape phase correlation is high and the phase profile may be approximated by the first few Zernike polynomial orders (offset, tip/tilt, defocus, etc.). Tip and tilt displace the hot spot within the pupil and defocus increases the power spectrum, tightly focusing a spot in a plane above or below the pattern. Hence, large w values generate suboptimal solutions since the plane featuring the highest intensity and axial resolution does not coincide with that of the pattern.

Optimal w widths thus correspond to fractions of the target pattern size as illustrated in Fig. 8(b) and 8(c) where w is 4 and 10 pixels, respectively with oversampling s = 4. Fig. 8(e) and 8(f) show the corresponding power spectra. The relative intensity of the central peak splits into several peaks as w increases, leading to more uniform distribution of the intensity in the pupil as seen in Fig. 8(g).

Subsequently, we apply an iterative Fourier transform algorithm to the diffuser created in the outlined way, in order to reduce mean squared error with respect to the desired intensity pattern. We constrain the intensity of the desired shape in the sample plane, impose zero intensity outside the objective pupil at the back focal plane, and allow the phases to vary freely. After a few iterations the algorithm converges towards a pattern with a mean square inhomogeneity (of the intensity) of the order of 10%. Figure 9(a) shows an example initial diffuser and Fig. 9(c), the diffuser after 15 iterations. Their corresponding power spectra are shown respectively in Fig. 9(b) and Fig. 9(d). Figure 9(e) shows the final intensity of the pattern of the generated disk, and Fig. 9(f) its intensity histogram. As illustrated by Fig. 8 and Fig. 9, the required intensity profile at the back focal plane is not uniform, necessitating complex amplitude modulation.

 

Fig. 9 Example output from the iterative Fourier algorithm. An initial diffuser (a) is produced with s = 4, w = 20 pixels in a 90 × λ/(2NA)-wide disk, generating the power spectrum shown in (b). Final diffuser (c) and intensity distribution at the pupil (d) after 15 iterations between image and Fourier planes, imposing uniform intensity throughout the image disk and zero intensity outside of the pupil. The final disk (e) exhibit improved uniformity as shown by its intensity histogram (f).

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Funding

Europe Research Council (ERC) (247024); Agence Nationale de la Recherche (ANR) (14-CE17-0006); National Institutes of Health (NIH) (1-U01-NS090501-01); Getty Lab.

Acknowledgments

The authors thank Oscar Hernandez for helping with the alignment of the optical system and Alexander Jesacher and Gregor Thalhammer for helpful discussions.

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