Superresolution imaging with optical fluctuation using speckle patterns illumination

Superresolution fluorescence microscopy possesses an important role for the study of processes in biological cells with subdiffraction resolution. Recently, superresolution methods employing the emission properties of fluorophores have rapidly evolved due to their technical simplicity and direct applicability to existing microscopes. However, the application of these methods has been limited to samples labeled with fluorophores that can exhibit intrinsic emission properties at a restricted timescale, especially stochastic blinking. Here, we present a superresolution method that can be performed using general fluorophores, regardless of this intrinsic property. Utilizing speckle patterns illumination, temporal emission fluctuation of fluorophores is induced and controlled, from which a superresolution image can be obtained exploiting its statistical property. Using this method, we demonstrate, theoretically and experimentally, the capability to produce subdiffraction resolution images. A spatial resolution of 500 nm, 300 nm and 140 nm with 0.4, 0.5 and 1.4 NA objective lenses respectively was achieved in various samples with an enhancement factor of 1.6 compared to conventional fluorescence microscopy.

(demagnification) system. To compare this result with the experiment result, we measured speckle patterns using the experimental setup combined with disordered medium and motorize stage, to generate speckle patterns and their correlation, as shown in Fig. S2a. To prevent loss of speckle pattern spatial frequency information, an objective lens with a greater numerical aperture (NA = 0.8) compared to that of the illumination objective lens (NA = 0.5), was used as the collection objective lens. The speckle pattern correlation was calculated and averaged from 100 frames of the measured speckle patterns shown in Fig. S2b. As a result, the line profile (red dot) from the correlation calculated using the measured speckle pattern is well matched with the line profile (black line) calculated using the diffraction-limited point spread function shown in Fig. S2c. This result shows that the speckle correlation modified by the linear optics system is well described by equation (S7).
However, the original SOFI equation needs two types of correlation: cross-correlation ( ≠ ) and autocorrelation ( = ). Regarding the original SOFI, since the blinking signals of a fluorophore has no crosscorrelation with the blinking signals of other fluorophores, this is not considered, and amounts to a major assumption of SOFI. In contrast, for S-SOFI, since the fluctuation spot of a speckle pattern has limited size, we have to consider cross-correlation of fluctuation signals from fluorophores in the same speckle spot. To do this, we need to consider an equation including both types of correlation in the analytic model used for calculation of S-SOFI. In order to simplify this problem, we expand the spot size of the incident laser toward a diffuser. Then, because a speckle pattern made from the large scale spot of an incident laser can ignore small dephasing fluctuations, the speckle pattern just shifts laterally as the diffuser is moved incrementally by the motorized stage (Fig. S3). Therefore, since the speckle pattern shifts laterally with small dephasing, the correlation function in equation (S8) can be used not only for auto-correlation, but also for cross-correlation. Substituting equation (S8) into equation (S10), we need to consider the step size of the motorized stage since the speckle pattern shifts laterally following the motorized stage; then equation (S10) becomes where ( ) is step size of the motorized stage and ( ) is rescaled ( ) according to the optical magnification. Because ( ) is a negligible distance compared with the FWHM of the intensity correlation function , we can ignore ( ), and get the equation (3) in the main text from equation (S11).

b. Simulation of virtual sample containing two fluorophores
To define a resolution enhancement of S-SOFI, we consider two fluorophores with molecular brightness , = 1 apart from each other with distance , as shown in Fig. S4a. To calculate using the analytic model, we define a fluorophore position as an expressed vector in the form − 1 =̂+̂ and − 2 = ( − )̂+ ̂, such that equation (S12) becomes Since the measured image of a fluorophore is an incoherent image, we can use an incoherent PSF defined where is the full width at half maximum of the Gaussian function, which corresponds to 2 in conventional microscopy, is the wavelength of the detection signal. As shown in equation (S14), the width of Gaussian form PSF is reduced by factor √2 since PSF is squared according to the SOFI algorithm. Notice that the final term in equation (S14) is regarded as the contribution of a virtual fluorophore made by crosscorrelation of the original two fluorophores at their center. This term affects the resolution enhancement of S-SOFI. Using this analytic model equation (S14), we can get the analytically simulated image in Fig. S4b, where = 532 , = 0.5, and = 532 . We can check the resolution enhancement using the analytic model, and the intensity peak ratio of the Abbe limit. Therefore, the resolution enhancement of S-SOFI is 1.3× that of the resolution of conventional microscopy when = 532 , and = 0.5.

C. Fourier reweighting method
The Fourier reweighting (FRW) method was introduced by Dertinger et al. to improve the resolution enhancement of SOFI 7 . According to the FRW method, use of a simple reweighting of the optical transfer function (OTF), the resolution enhancement of SOFI can be improved even more (e.g., √2 → 2). Specifically, the FRW method can improve the resolution of SOFI by replacing the 2-fold PSF with 4-fold PSF using the reweighting factor in Fourier space. In order to calculate the resolution enhancement of S-SOFI with FRW, we also apply FRW method to the two fluorophores analytic model as in the following step. First, Fourier transform is applied equation (14) as follows, where ̃ denotes the Fourier transform, ⊗ denotes convolution, 2 represents the Fourier transform of 2 , ̃ represents the OTF (Fourier transformed ), and denotes the spatial frequency vector and =2 ( −3 2 2 2 ).
Next, equation (S15) is multiplied using a Fourier reweighting factor defined as =̃( α ≪ 1 denotes a damping factor to prevent division by close-to-zero numbers, and then equation (S15) And is substituted by ′ = 2 . Then equation (S16) can be rewritten as Note that ̃( 2 ⁄ ) is the Fourier transform of (2 ) = (− applied to equation (S17), and then equation (S17) is given by Because ′ is 2 , ′ can be considered2 . Finally, using this relation, the FRW analytic model is gotten as Using this equation, we can get simulated images from the two fluorophores analytic model as shown in Fig.   S4c. The resolution enhancement of S-SOFI with FRW was then evaluated using modeling and the criterion of the Abbe limit, to determine that the resolution enhancement was 1.6× compared with the resolution of conventional microscopy when = 532 , and = 0.5.

D. Resolution enhancement of S-SOFI with step size of the motorize stage
In manuscript of experiment, ( ) is relatively large value and thus cannot be neglected (e.g., when the step size of motorized stage = 6 m, then the ( ) = 120nm in the nanopattern; then the step size of the motorized stage = 2 m and 7 m, then the ( ) = 200 nm and 70 nm in a biological sample). Thus, we should consider effect of ( ) in S-SOFI processing. In order to estimate the ( ) effect on the S-SOFI resolution, we needed to revisit the two fluorophore model used in section B-b. Using the two fluorophore model, the S-SOFI equation with ( ) is mathematically expressed as Here, the cross correlation term affected by ( ) should be considered to calculate resolution enhancement of S-SOFI. Additionally, it is main factor that prevents resolution enhancement of S-SOFI approach to the root of cumulant order. Using 2 ) = 2 (−( 2 + 2 )) ℎ ( 2 2 ) relation, equation (S20) can be rewritten as Comparing with equation (S14), equation (S21) has additional two terms which are ( − 2 2 ), ℎ ( 2   2 ).
The first term,

E. Temporal correlation length control
The temporal correlation length using intrinsic blinking of fluorophores cannot be controlled using external methods in other superresolution methods since it is already determined by intrinsic properties of the fluorophores. To control this correlation, chemical treatment 8,9 of the fluorophores is required. However, temporal correlation in S-SOFI can be changed by controlling random patterns of illumination that can induce blinking of fluorophores. In this paper, speckle patterns were employed as random patterns, and temporal correlation made by converting spatial correlation of the speckle patterns into temporal correlation using the motorized stage. Therefore, temporal correlation could be controlled by adjusting the step size of the motorized stage. As shown in Fig. S5a, the temporal correlation length is reduced with increasing step size of the motorized stage. Since the number of image frames needed for superresolution imaging depends on the temporal correlation length, it could also be controlled by adjusting the step size of the motorized stage. However, as the step size increases, the rescaled step size ( ) is no longer small enough to be ignored, as shown in equation (S11). Thus, ( ) influences the resolution of S-SOFI and distortion occurs during resolution enhancement in the superresolution image. To remove this effect, zero is chosen as time lag in equation (S10) when images are analyzed using the SOFI algorithm. A similar method already used in previous work to reduce the number of image frames needed for SOFI 10,11 . Then, equation (S10) can be rewritten as We can get the same equation shown in equation (S12) since ( = 0) is zero. Therefore, the number of image frames needed for S-SOFI can be reduced using control of the temporal correlation length, and by applying equation (S20) without distortion of resolution enhancement. To demonstrate this, 350 images (exposure time = 0.1 s) from the biological sample (motorized stage step = 4 m) illuminated by speckle patterns, and analyzed using SOFI and the Fourier reweighting method. As a result, we were able to get similar results using fewer image frames, as shown in Fig. S7 b-c. Moreover, this control of temporal correlation and use of equation (S20) provides not only that a reduced image frames with increasing step size of the motorized stage, but also that the random speckle patterns can be used for S-SOFI without strict control of the motorized stage. However, the number of image frames for S-SOFI cannot be reduced infinitely with increasing step size because the signal to noise ratio (SNR) decreases with the number of image frames as shown in Fig. S6. Thus, the number of image frames can be reduced by adjusting step size within the range needed to acquire enough image frames to satisfy the target SNR. Another, possible factor in the reduction of the number of image frames is the distortion effect of the speckle pattern may be used to increase randomness of the speckle pattern, unlike as shown in Fig. S3.
When we apply this distortion effect to S-SOFI, it is expected that the number of image frames will be reduced even more.

F. Uniform illumination using speckle pattern
Each frame of the speckle pattern shows nonuniform illumination characteristics, as shown in Fig. S3. To achieve a uniform illumination using the average of the speckle patterns, a sufficient number of frames of the speckle patterns are required. To quantify this, we used the concept of the intensity contrast of the speckle pattern. Theoretically, when a diffuser is continuously moving, the speckle pattern contrast is determined by following equation 12 : where is the contrast of speckle pattern, is the full with half maximum of the point spread function, is the velocity of the diffuser, is the total exposure time and is the correlation time of the speckle pattern.
Similarly, because S-SOFI uses varying speckle patterns over time, the averaging process of the speckle patterns can be regarded as a continuously moving diffuser. To apply the moving diffuser equation for S-SOFI, the equation (S23) is modified as follows: = 0.52√ , when < 4 (Circular pupil case and continuous regime), = √ 1 , when < 4 (General case and discontinuous regime), Where D denotes the demagnified step size of the motorized stage and M denotes the number of frames. Using these equations, contrast of the averaged speckle pattern can be calculated as shown in Fig. S9. To distinguish two objects from one another with half the wavelength, the variation in image contrast should not exceed 10% based on Abbe's criterion. From this result, a sufficient number of frames of the speckle patterns can be estimated. Moreover, the results of the averaged image from biological sample exhibit similar contrast over estimated number of frames using contrast equation, as shown in Fig. S10. However, some results from the biological sample show clearer image than the predicted one although they used images that had less than the estimated number of frames. This can be attributed to the overestimation of the speckle contrast equation because we only took into account simple parameters of the speckle patterns for the calculation. Nevertheless, the equations used well describe the relationship between the uniformity and the speckle contrast.

Figure legends
Figure S1| Modeling of 4f system for analytic calculation of speckle correlation. Here, is the speckle field that exists on a surface, at coordinate ( , ), made from the disordered medium. This speckle field propagates and is changed to speckle field on the object plane at coordinate ( , ) after passing through the 4f system made of lens with focal length , lens with focal length , and the pupil of the second lens with coordinate ( , ).