Multi-lobe superoscillation and its application to structured illumination microscopy

Niv Shapira; Niv Shapira; Zhigui Deng; Zhigui Deng; Roei Remez; Danveer Singh; Eytan Katzav; Ady Arie

doi:10.1364/OE.27.034530

1. Introduction

Superoscillations (SO), can be described as the superposition of Fourier components of a band-limited function, that generates locally oscillations that are faster than the function's highest Fourier component. The seemingly paradox or violation of the Fourier theorem can be solved by noting the fast local oscillations can be cancelled by other parts of the function when the function is integrated from –infinity to infinity. Nevertheless, it has been shown, that moderate SOs in waves can persist over long distances and time farther than the commonly known evanescent-waves [1]. Early research on SOs includes the studies of super-directive antenna designs by Alexander Schelkunoff [2] and of super resolving pupils for optical lenses by Toraldo di Francia [3]. In 1988, the phenomenon was explored by Yakir Aharonov et al., in quantum physics, and is associated to what is now known as quantum weak measurement [4]. In recent years, the research on the topic accelerated and various applications were studied extensively [5], including imaging [6–8], nonlinear optics [9], particle trapping [10], temporal shaping of ultrafast pulses [11,12], nanometric displacements detection [13], free-space SO and plasmonics analogies [14], and spatial shaping of matter waves [15]. Most of these works concentrated on generating a single SO lobe. Here, instead, we study methods to generate a multi-lobe function with nearly constant intensity and constant local frequency. Such functions open interesting new possibilities that utilize the region that contains high local frequencies. Only in the last decade it became clear that designing the superoscillations to more desirable shapes was possible, e.g. via interpolating superoscillations [16,17]. Some insight into the basic energetic limitations involved in exceeding the Nyquist rate also developed [18,19]. The multi lobe superoscillating functions that we realize in this paper are then tested for one potential application – the possibility of improving the resolution of structured illumination microscope.

The structure of this paper is the following: In Section 2 we analyze two different methods to design SO beams and in Section 3 we present their experimental realization in an optical setup. To test the potential of these beams for structured illumination microscopy, we conduct in Section 4 a Moiré imaging experiment of a fluorescent grating. Full reconstruction of images is theoretically analyzed in Section 5, and the results are then summarized in Section 6.

2. Design of multi lobe superoscillating functions

Recent two works [9,16], suggested two different possibilities for the creation of these multi-lobe SO. The first multi-lobe SO we consider is based on a shifted cosine function [9]

y_{1} (x) = {[\cos (2 π f_{0} x) - s]}^{2}

Where

0 < s < 1

defines the cosine vertical offset. This periodic function consists of a single central SO lobe in every period and its width defined by the zero-crossings

\cos^{- 1} (s) / π f_{0}

. The function is said to be SO around

x = 0

, because as the parameter s gets closer to 1 the central lobe width gets arbitrarily smaller, independently of the function’s maximal Fourier frequency component. In order to achieve multi-lobe SO, all is needed is to iteratively repeat the initial process (i.e. shifting the previous function and squaring), and by that increase the number of SOs and their spatial extent. Given the initial frequency,

f_{0}

, we define

y_{0} = \cos (2 π f_{0} x)

. The iterative process is then carried out by:

y_{k} = {(\frac{y_{k - 1} - s_{k - 1}}{1 - s_{k - 1}})}^{2}

where

0 < s_{0} < 1

sets how narrow the SOs will be and

s_{k} = 0.5

for every

k > 0

. The generated number of zero crossings in each superoscillating region corresponds to

2^{k}

. This method shares some similarity with a recently proposed method [20] as it involves multiplication of band-limited functions. However, the iterative scheme used here allows for translations too, hence allowing a richer family of functions.

The second multi-lobe SO we consider was developed by Katzav and Schwartz [16] for “Yield” optimization of an interpolated SO functions. Here too, a periodic signal is considered and imposed by a number of M constraints which force it to oscillate in a pre-defined interval $[- a, a]$ with $a < π$ , and between the values $\pm 1$ . The function can be written using its Fourier series as

f (x) = \frac{a_{0}}{\sqrt{2 π}} + \sum_{n = 1}^{N} \frac{a_{n}}{\sqrt{π}} \cos (n x)

where

{a_{n}}

are the Fourier coefficients. This function is then optimized with respect to the energy available in its superoscillatory interval compared to the total energy of the signal. Formally, this boils down to optimizing the superoscillatory yield defined by:

Y = \frac{{\int_{- a}^{a} | f (x) |}^{2} d x}{{\int_{- π}^{π} | f (x) |}^{2} d x}

subject to the aforementioned constraints. The closely related method for constructing interpolating superoscillations by Ferreira and Kempf [18,19] differs from this method mainly by the fact that it does not maximize the superoscillatory yield, but rather minimizes the total energy of the signal, subject to the constraints. Another minor difference, is that it considers a continuous spectrum, rather than a discrete one used here (see [17] for a more detailed comparison). Examples of signals generated by both methods are presented in Fig. 1. The Fourier coefficients for the Yield optimized functions examples are given in Table 1. Next important measure to define is the SO_power, which is the relative increase of the SO frequency

(f_{S O})

over the highest Fourier component,

f_{m a x}^{F T}

:

S O_{p o w e r} = \frac{f_{S O}}{f_{\max}^{F T}} - 1

We note that there isn't an exact value for the SO frequency. This is due to the fact that both functions exhibit inherent quadratic chirp that becomes more significant as the number of oscillations grows. The chirp can be estimated by calculating the initial to final frequency difference over the SO interval. However, ignoring the chirp, some measure for the effective frequency is still desired and may vary corresponding to the specific application. Such a measure can be devised in the spatial domain, by counting the number of oscillations in the maximal interval for which the function has SOs. Alternatively, it can be done in the frequency domain, by Fourier transforming the function only in the SO interval. Obviously, the result will deviate from the perfect delta functions of a cosine transform, depending on the amount of existing chirp and the number of periods considered.

Fig. 1 Examples for signals generated by the Shifted-Cosine and the Yield-Optimized functions (first and second rows respectively). Each profile contains a single period. In the first row, K is the number of iterations. In the second row, N is the number of frequency components (except the zero component), M the number of constraints and [-a, a] the pre-defined SO interval. A scale of the length inverse to the bandwidth (BW) is attached to each profile. The figure presents an energy comparison between the two methods (in the second and third columns), considering the SO region and measured in terms of the intensity ratio of the SO over the side-lobes. Growing energetic superiority is clearly seen for the Yield-optimized function over the Shifted Cosine function as the number of oscillations and the SO_power increases.

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Table 1. Yield-optimized Fourier coefficients

View Table

Regarding the practicality of the functions as illumination patterns, one would like to compare the two realizations. The Yield-optimized interpolated function can generate arbitrarily desired number of SO lobes, while the Shifted-Cosine version is limited to only a specific discrete set of lobes (integer powers of 2, minus 1). Furthermore, the Yield-optimization method provides much higher relative energy in the SO region (i.e., a higher superoscillatory-yield) with respect to the surroundings compared to the shifted cosine method, in particular for a large number of SOs. However, as expected by [17], this mainly comes at the expense of the function’s regularity of frequency or amplitude, while the Shifted-Cosine exhibits high similarity to a perfect cosine. Examples can be seen in Fig. 1.

Lastly, the bottom line depends on the specific application under consideration. If structured illumination is considered, one may choose the Shifted-Cosine to reduce image restoration artifacts or the Yield optimized interpolating functions to enable higher energy in the SO frequencies.

3. Optical generation and characterization of multi lobe beams

For the purpose of implementing the SO as a light pattern, a beam modulation experimental setup was established (Fig. 2) based on a reflective phase-only spatial light modulator (SLM), HOLOEYE-Pluto-II, which consists of 1920x1080 pixels of 8µm size. The laser source is a laser diode with a wavelength of 532nm and a power of 4.5mW. A telescope was placed between the SLM and illumination objective to relay the hologram to the back pupil of the illumination objective. The focal length of L1 is 300mm and the focal length of L2 is 150mm. Two objectives Olympus UPlanSApo 40x/0.95 and Nikon LU Plan Flour 100x/0.9 were used for the illumination and detection, with tube lens of 1-inch plano-convex 200mm focal length.

Fig. 2 The experimental setup: a piece of cover glass was used for the measurement of various SO patterns, and a grating was used for the measurement of Moiré patterns later. (a) Hologram of the SO generated by computer. The period of the blazed grating determines the angle between the zero-order and first-order. (b) Experimental image of the 1st diffraction order reflected by the SLM; (c) Simulation of SO pattern created by a SLM hologram; (d) SEM image of the fluorescent grating with a period of 190nm; (e) Simulation of Moiré pattern formed after the SO pattern through the fluorescent grating.

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Shaping the desired light field profile is done by means of an off-axis computer-generated hologram. The chosen method enables the modulation of both amplitude and phase using a single phase-only hologram. The SO patterns were created at the focal plane of a high numerical aperture microscope objective lens.

The hologram coding method [21,22] is based on a blazed grating pattern. It spatially modulates the depth-of-phase of the blazing in the grating pattern and by that modifies the spatial diffraction efficiency. Light that is not diffracted into the first diffraction order is mainly sent into the zero order, effectively allowing for amplitude modulation at the first diffraction order. Concretely, the transmission function of the SLM is given by:

T (m, n) = \exp [i M (m, n) M o d (F (m, n) + \frac{2 π m}{Λ}, 2 π)]

Where m and n are the pixels of the SLM and

Λ

is the period of the blazed grating. The modulation functions

M

and

F

are defined by:

M = 1 + \frac{1}{π} \sin c^{-1} (A (x, y, 0)); F = Φ (x, y, 0) - π M

Where

A (x, y, 0)

and

Φ (x, y, 0)

represent the desired spatial field amplitude and phase in the hologram plane.

Being periodic functions, SO signals generated by both methods can be written in terms of their Fourier series. It is then directly seen that while the hologram spans the diffraction orders in the vertical dimension, from which only the first is selected, the rest of the function modulation is performed on the horizontal dimension, and thus every Fourier frequency component eventually transforms into a vertical stripe with a certain amplitude and phase. Various SO intensity patterns and the corresponding system PSF intensities were experimentally measured and compared to theoretical simulations (Fig. 3).

Fig. 3 Experimental measurements of various SO patterns and its corresponding system PSF intensity. The first row presents horizontal cross-section and the second row the corresponding camera image along a zoomed view to a single period. (a) Shifted-cosine SOs; (c) Yield-optimized SOs; (e) System PSF; (b,d) Camera image (top) with zoomed view (bottom); (f) PSF experimental image (top) compared to the simulated image (bottom). The notation $Λ_{a v g}$ refers to the measured averaged SO period and the $f_{c o}$ the cut-off frequency of the system.

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For aberration correction, a piece of cover glass was inserted between the objectives. Later, it will be replaced by sample under test. The system Point Spread Function (PSF) intensity measurement is crucially required in order to assure a diffraction limited system with a known numerical aperture. In our experiment, the PSF size was $0.98 μ m$ and its side lobes (which are twice as small) were $0.49 μ m$ . Note that we did not fill the entire aperture of the Fourier lens, in order to avoid aberrations, hence the effective NA is 0.544. In comparison, the two SO beams we show in Figs. 3(a) and 3(c), having 8 and 12 oscillations respectively, had average lobe size of $0.37 μ m$ and $0.418 μ m$ , with chirp of about $0.0845 μ m^{- 2}$ and $4 \cdot 10^{- 4} μ m^{- 2}$ . They represent local oscillations that are 32% and 17% faster than the system cutoff frequency. Utilizing these multi-lobe SOs as illumination patterns to an optical apparatus could enable various interesting applications which require their nearly periodic form. An interesting example for such application is the Structured Illumination microscopy (SIM), and will be discussed in the next two sections.

4. Moiré imaging of fluorescent grating with SO beam

SIM was originally suggested as a super-resolution method for incoherent fluorescence wide-field microscope [23,24]. It enables resolution improvement of up to twice the classical diffraction limit. In the original linear-SIM, instead of uniform wide field illumination, the sample under test is illuminated by a series of sinusoidal striped patterns of high spatial frequency of the form:

I_{i n} (r) = I_{0} [1 + m \cos (2 π f_{θ} \cdot r + ϕ)]

Where

I_{0}

is a constant intensity factor,

m

the grating modulation,

f_{θ}

the spatial frequency vector in the direction

θ

,

r

the spatial coordinate and

ϕ

the phase shift. The purpose of SIM is the enlargement of the imaging bandwidth, beyond the original diffraction limited optical transfer function (OTF) cut-off by utilizing the concept of Moiré fringes. Essentially, information from outside the classical diffraction limit is shifted into the detectable bandwidth. However, assuming the illumination microscope lens has the same NA as that of the imaging lens, the same diffraction limited OTF also limits the frequency of the modulation, and thus only a factor of two in the resolution can be originally achieved. For a more general case in which the illumination and imaging objectives are different, the highest frequency is

2 N A_{i} / λ_{i} + 2 N A_{e} / λ_{e}

, where

i

and

e

stand for illumination and emission respectively.

In order to overcome this barrier, we suggest to utilize the multi-lobe SO as the structured illumination (SO-SIM). This will enable a larger pattern frequency, larger spectrum shift and thus a resolution improvement larger than the two-fold improvement of SIM. In order to use the SO pattern within SIM, only the SO interval should interact with the sample under examination. Thus, the SO interval should be isolated from the surroundings by some sort of interaction block (e.g., an opaque mask). In the following example we assume that the SO function locally oscillates at a frequency which is 19% above the cutoff of $2 N A_{i} / λ_{i}$ . Note that the more oscillations being used the larger the field of view. However, there is a tradeoff between using more oscillations and an energetic cost, leading to a lower superoscillatory-Yield.

A simplified one-dimensional experiment was designed, based on the Moiré concept, to test the resolution improvement. A Rhodamine 6G-based fluorescence grating with a period of $0.19 μ m$ was inserted into the experimental setup (see Fig. 2) and illuminated, at wavelength $0.532 μ m$ , by a reference cosine pattern and a SO pattern, separately. Due to the fluorescent grating, the cut-off frequency of the optical system depends on the imaging objective with $N A = 0.9$ and the fluorescence wavelength $λ = 0.57 μ m$ , as illustrated by the red lines in Fig. 4. For the sake of simplicity, it is assumed that the amplitude grating can be described by $\frac{1}{2} [1 + \cos (2 π f_{g} x)]$ and the intensity of the illumination pattern described by $\frac{1}{2} [1 + c o s (2 π f x)]$ , where $f_{g}$ and $f$ are the grating and illumination frequencies respectively. Therefore, the intensity distribution just after the grating can be written as:

I (x) = \frac{1}{4} [1 + \cos (2 π f_{g} x)] [1 + \cos (2 π f x)]

Imaging the above distribution onto the CCD and transforming to the frequency domain, will result in four different frequencies, as can be seen in Fig. 4. The highest frequency component of the SO function is

f_{c o s i n e}

, while the frequency

f_{s o}

generated is 119% larger. Thus, while the frequency difference

f_{g} - f_{c o s i n e}

lies at the edge of the OTF support and attenuated dramatically, the frequency difference

f_{g} - f_{s o}

will be at a lower frequency that is observable and thus the unknown grating frequency,

f_{g}

, could be extracted.

Fig. 4 Simulation of the illuminated imaged grating spectra (a) Cosine pattern illumination; (b) SO pattern illumination. The red and green lines represent the cut-off frequency and the frequency difference respectively. The bright area is the diffraction-limited region.

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The grating itself was fabricated by focused ion beam (FIB) milling of PMMA mixed with Rhodamine 6G dye, see [27] for details. A SEM image of the grating is shown in Fig. 2(c) The grating period is much shorter than the incoherent resolution limit of $λ_{e} / (2 N A_{e}) ~ 0.316 μ m$ , and therefore a direct optical imaging cannot resolve the grating structure.

The cosine illumination frequency (Fig. 5(b1)) was chosen to have the largest possible value of $(N A_{i} / N A_{e}) \times (λ_{e} / λ_{i})$ . Since the illumination NA (0.6) is smaller than the emission NA (0.9), this frequency is 71.6% of the system resolution limit, i.e. $2.26 μ m^{- 1}$ (or $1 / 442 n m$ ). The SO illumination (Fig. 5(d1)) was generated such that this same frequency would be its largest Fourier component, and thus the local SO frequency 119% larger, i.e. $2.68 μ m^{- 1}$ (or 1/ $372.5 n m$ ). The grating was illuminated by the patterns where the surrounding silver region served as the desired aperture (Figs. 5(a3)–5(d3)). The sample was imaged onto the CCD and a Fourier transform was performed on the obtained image. Since the fluorescent grating has a rectangular response rather than a cosine, the spectrum acts accordingly and more significant side lobes are apparent. Finally, we compare the resolution improvement obtained by using the SIM and SO-SIM. The maximal resolved frequency is obtained in the limit in which $f_{g, m a x} - f_{s o} = f_{c o}$ . Thus, the maximal resolved frequencies are $f_{g, m a x} = f_{c o} + f_{c o s i n e}$ for SIM and $f_{g, m a x} = f_{c o} + f_{s o}$ for SO-SIM.

Fig. 5 Comparison between SIM and SO-SIM. (a1)-(b1) Cosine pattern; (c1)-(d1) SO pattern (white dotted frame area); (a2)-(d2) The frequency domain of illumination; (a3)-(b3) The CCD image of grating illuminating by cosine pattern. The axes are given in units of the grating plane (normalized by the setup's magnification); (c3)-(d3) The CCD image of grating illuminating by SO pattern; (a4)-(d4) The frequency domain of CCD image; The green peak is the frequency difference between grating and SO pattern.

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For the same $N A_{i}$ and $N A_{e}$ , the SO-SIM provides 119% improvement in resolution with respect to the incoherent resolution limit, assuming that the local frequency is 19% above the cutoff. With respect to SIM, which is already twice this limit, it therefore represents a 9.5% improvement. For the experimental parameters that we use, in which the $N A_{i} / N A_{e} = 0.6 / 0.9$ , the improvements of SIM and SO-SIM with respect to the incoherent resolution limit are 71.4% and 85.0%. and the relative improvement of SO-SIM with respect to SIM is 7.9%. The theoretical spatial resolution limits of SIM and SO-SIM in our system are therefore $0.185 μ m$ and $0.171 μ m$ respectively. In both cases the resolution improvement is less than a factor of 2 because the effective $N A_{i}$ of the illuminating microscope objective is smaller than the $N A_{e}$ of the imaging lens. The period of the grating that we measured, $0.19 μ m$ , is just slightly bigger than the SIM resolution, but it is attenuated dramatically by the OTF and cannot be detected by the conventional SIM, as shown in Figs. 5(a4) and 5(b4). The spectrum of the image shows mainly the peak at the illumination frequency, and some other smaller peaks owing to the rectangular aperture. However, this grating is resolvable using the SO illumination, as shown in Figs. 5(c4) and 5(d4), where the signal that depends on the grating frequency is marked in green. These results also indicate that using illumination $N A_{i}$ which is identical to the imaging $N A_{e}$ , will enable the SO-SIM method to achieve more than two-fold improvement in resolution.

5. Simulation of structured illumination microscopy with SO beams

Within the more general 2D framework of SIM, the concept we present suggests to gather the conventional SIM images and, on that basis, exploit the added information of the SO illuminated images. Considering the super-resolved image reconstruction and using a multi-frame weighted Wiener filter [25], the new SO high-resolution spectra are accounted with respect to their noise variances. In a conventional SIM, a total of 9 images (representing three images with different phase shifts in three different directions) are the minimum required to compute the super resolution image. In the case of SO-SIM, more orientations should be applied to sufficiently cover the effective observable region and to avoid large gaps between the spectrum components. Theoretical image reconstruction simulations can theoretically be done using a large number of oscillations which exhibits very uniform, properly defined frequency and a perfect cosine shape. Therefore, simulation results could in principle suggest dramatic resolution improvement with no image quality loss. However, practically, structured illumination SO patterns which were measured in the lab consists of 4-12 oscillations and 20-40% SO_power, and therefore the simulation was done using similar parameters. Figure 6 presents simulations of super-resolved image obtained with an yield-optimized superoscillating function [16,17] having only 15 oscillations and 34.5% SO_power. Note that the image reconstruction is obtained only in the region where all the illumination patterns overlap. The region of interest is thus central to the illumination and marked by a white circle. The outer regions will experience deformations due to partial overlap of the patterns. In the central region, the SO-SIM shows clear improvement in resolution or wider bandwidth with respect to the standard SIM.

Fig. 6 (a)-(c) Imaging simulation comparison of the “USAF 1951” resolution table using (a) deconvolved wide-field image (b) Conventional SIM with 80% super-resolution and (c) SO-SIM as described. (d)-(f) similar comparison for the “Spikes” image resolution target. (g)-(i) are the “USAF 1951” corresponding reconstructed spectra in the frequency domain.

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It is interesting to note that the suggested SO-SIM can also be beneficial in coherent imaging systems, in addition to the incoherent imaging systems that were discussed till now. Currently, resolution enhancement in coherent imaging is unavailable in conventional SIM, because in such an arrangement the structured light could always be replaced with a sequential oblique plane wave illumination resulting with resolution not better than the Abbe diffraction limit [26]. While this is true for sinusoidal illumination it is not the case with SO beam illumination. As already been shown in the experiment above, the SO effective local frequency can be higher than the optical cut-off frequency. Such illumination is equivalent to illumination of the object by oblique plane waves in a larger angle than the maximal cut-off. Hence it cannot be replaced by a sequential plane waves illumination.

6. Summary and conclusions

In conclusion, we have studied methods to generate a multi-lobe SO and demonstrated experimentally the generation of SO patterns having 4 to 12 sub-wavelength oscillations, with local frequency of 20% to 40% above the band-limit. Multi-lobe SO beams were then used to illuminate a fluorescent grating, in order to demonstrate the possible resolution improvement given by utilizing the SO function instead of a standard sinusoidal structured light. Lastly, a complete 2D SO-SIM simulation suggested that implementing the SO pattern to the SIM approach enables resolution better than the two-fold improvement achieved by the SIM approach. Disadvantages of the method we presented include its sensitivity to the mask accuracy and artifacts in the image induced by the chirp or amplitude variation of the SO. However, since these deviations from a perfect sine wave are known, it may be possible to overcome them by suitable processing algorithm. Another disadvantage, is the current limitation on the practical number of SO periods within the pattern, and the requirement for a physical aperture that selects only the SO region of the beam. These impose limits on the field of view and requires to use an aperture near the object. Moreover, these may increase the edges artifacts, although they are not very dominant in the 2D simulations above. A possible way to overcome these limitations, is to eliminate the aperture and illuminate with the entire pattern, which consists of periodic repetitions of relatively weak SO regions, separated by strong lobes (see for example Fig. 4(b)). In this case, we expect that with proper tuning of the illumination power, the fluorescent signal from the strong lobes will be bleached, enabling to obtain a super-resolved signal only from the periodic SO intervals. Finally, we note that the parameters we used for the multilobe SO beams are feasible with fairly standard spatial light modulators and imaging optics, but with carefully fabricated phase masks it is possible to generate SO beams with even higher local frequencies and with a larger number of oscillations, that can provide further improvements in resolution.

Funding

Israel Science Foundation (ISF) (1415/17).

Disclosures

The authors declare that there are no conflicts of interest related to this article.

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27. The fluorescence grating was formed with the following protocol. The solution was prepared using 0.3mM of Rhodamine 6G (Rh6G) in 15mL of Poly (methyl methacrylate) (PMMA) A4. The concentration of Rh6G was chosen to ensure sufficient fluorescence light emission from low power illumination. The solution was stirred for 3 hours to dissolve the Rh6G completely in the PMMA A4. The prepared solution was spin coated at the 3000 RPM on the Si substrate, generating a uniform 0.25μm thick layer. The substrate was then softly baked at 90oC for 90 sec durations. 0.12μm of silver (Ag) was deposited using e-beam evaporation. The Raith ion line focused beam was then used to mill the grating in the silver layer.

$a_{n}$	N = 5, M = 4 a = 1.9	N = 5, M = 5 a = 2.3	N = 13, M = 9, a = 1.8
$a_{0}$	−0.269	0.284	0.1659	$a_{7}$	0.2577
$a_{1}$	0.117	−0.286	−0.0629	$a_{8}$	−0.1749
$a_{2}$	0.332	−0.008	−0.2033	$a_{9}$	−0.1863
$a_{3}$	−0.358	0.346	0.1750	$a_{10}$	0.3322
$a_{4}$	−0.190	−0.580	0.1142	$a_{11}$	−0.0027
$a_{5}$	0.800	0.617	−0.2473	$a_{12}$	−0.4760
$a_{6}$			0.0187	$a_{13}$	0.5944

Multi-lobe superoscillation and its application to structured illumination microscopy

Abstract

1. Introduction

2. Design of multi lobe superoscillating functions

3. Optical generation and characterization of multi lobe beams

4. Moiré imaging of fluorescent grating with SO beam

5. Simulation of structured illumination microscopy with SO beams

6. Summary and conclusions

Funding

Disclosures

References

Cited By

Figures (6)

Tables (1)

Equations (9)

Optics Express