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We propose a method for eliminating the deformations in fluorescence emission difference microscopy (FED). Due to excessive subtraction, negative values are inevitable in the original FED method, giving rise to deformations. We propose modulating the beam to generate an extended solid focal spot and a hollow focal spot. Negative image values can be avoided by using these two types of excitation spots in FED imaging. Hence, deformations are eliminated, and the signal-to-noise ratio is improved. In deformation-free imaging, the resolution is higher than that of confocal imaging by 32%. Compared to standard FED imaging with the same level of deformations, our method provides superior resolution.
© 2014 Optical Society of America
1. Introduction
Confocal microscopy is widely used in biological imaging because of its high resolution and optical sectioning ability [1–3]. However, due to the diffraction limit, it is difficult to further enhance the resolution of confocal microscopy. Modern science and technology demand higher resolution; thus, several sub-diffraction-limited imaging methods have been invented, such as photoactivated localization microscopy (PALM) [4], stochastic optical reconstruction microscopy (STORM) [5], structured illumination microscopy (SIM) [6], stimulated emission depletion microscopy (STED) [7], and super-resolution optical fluctuation imaging (SOFI) [8]. Though these methods’ resolving capabilities have been proven, they are confronted with several problems, such as the massive data processing, the complexity of the optical system, the difficult sample preparation, and the high cost.
The image subtraction method can overcome these disadvantages. Since the original images are confocal images, the subtraction method does not increase the complexity of the optical system. This approach does not require massive data processing because subtraction is a very simple operation. Previous research has shown that the subtraction of confocal images acquired under different pinhole sizes can improve the resolution [9,10], but the signal-to-noise ratio is relatively low. Recently, better subtraction methods have been proposed, such as fluorescence emission difference microscopy (FED) [11–13] and switching laser mode microscopy (SLAM) [14,15].
However, a major drawback to FED is the associated distortions. FED employs a solid focal spot and a hollow focal spot to acquire two confocal images, and then the FED image is obtained by subtraction of these images. If we consider the point spread function (PSF), the FED PSF is the difference between the PSFs of the two confocal imaging systems. Unfortunately, the PSF profiles of the two confocal imaging systems are not matched; hence, the negative side-lobes in the FED PSF are inevitable under any subtractive factor. Therefore, FED images are always accompanied by negative pixel values. These negative pixel values are ultimately eliminated by setting those pixel values to zero.
The side-lobes of PSF may lead to distortions in the image. In this paper, distortion is grouped into two categories. One is named “artifact”, which is caused by the positive side-lobes. It appears to be some inexistent image by the side of the object’s image, but it does not influence the shape of the object’s image. The other is named “deformation”, which is caused by the negative side-lobes of the PSF. Since negative value is set to zero, it results in information loss. In some cases, useful but weak signals may be present among the negative values. When the negative values are removed, these signals are lost, leading to deformation of the image. The deformation influences the image’s shape and cause severe distortion, e.g., the discontinuity of lines, the disappearance of points, or the shrinkage of blocks. This may greatly degrade the image’s quality.
Although negative values do not necessarily result in deformations, it is convenient to use the minimum negative value in a FED image to evaluate the possibility of an deformation’s existence.
In order to eliminate the deformations, a pair of matched PSFs is needed. If the difference between the two confocal PSFs is everywhere non-negative, there will be no negative values in the FED image. In [16], a method was proposed to generate an illuminating spot with a flat-top profile to take the place of the conventional solid spot, which would result in the elimination of negative difference between the PSFs.
In this paper, we demonstrate a simpler approach to obtain an extended solid spot. By using a 0–2π counter-clockwise vortex phase plate to modulate a right-handed circularly polarized beam, an extended solid focal spot is obtained. By using a diaphragm to block the rays that are at the margin area of the entrance pupil, this spot can be further extended. The PSF of the further extended spot and the PSF of the hollow spot are well matched, avoiding negative difference between the PSFs. Thus, deformations in FED can be eliminated by replacing the normal solid spot with this further extended solid spot. We refer to this method as deformation-free FED imaging (dfFED).
Calculations indicate that if deformations are not allowed in the dfFED image, the full width at half maximum (FWHM) of the dfFED PSF is 0.21λ, which is a reduction of 32% compared with confocal imaging. In such conditions, dfFED achieves a higher resolution than confocal imaging.
When the subtractive factor is assigned a higher value, the resolution of dfFED is further enhanced, but deformations also occur, thus lowering the signal-to-noise ratio. Compared to conventional FED imaging with the same level of deformations, the dfFED method results in higher resolution than conventional FED.
2. Theory
2.1 Fluorescence emission difference microscopy
According to the principle of FED, super-resolution is obtained by taking the difference between two confocal fluorescence images. These two confocal images are obtained with different illumination patterns; thus, their associated PSFs are different. One PSF is a solid spot, and the other one is a donut-shaped pattern with a dark spot at its center. Therefore, the difference between these two PSFs is a smaller spot, which leads to a narrower FWHM and thus a higher resolution.
A typical FED system is shown in Fig. 1. The excitation beam is split into two beams. The beam in the upper path is modulated by the phase plate PP1, and the beam in the lower path is modulated by the phase plate PP2. In the original FED method [11], PP1 does not introduce a phase shift, whereas PP2 gives rise to a 0–2π vortex phase shift. Hence, the focal spot of the beam in the upper arm is a normal solid focal spot, and the focal spot of the beam in the lower arm is a donut-shaped hollow spot. During the scanning process, either shutter S1 or shutter S2 is open at any given time, and when one is opened, the other is shut. As a result, the illumination of the sample switches between these two different focal spots. Hence, two different confocal images are acquired. Finally, the FED image is generated by numerical subtraction:
Fig. 1 FED system setup: PBS, polarizing beam splitter; PP1 and PP2, phase plate; D, diaphragm; S1 and S2, shutter; S, sample; M, reflecting mirror; DM, dichroic mirror; QWP, quarter-wave plate; OL, objective lens; CL, converging lens; and PH, pinhole.
Here, and are the normalized intensities of confocal images illuminated by the solid spot and the hollow spot respectively and where is the subtractive factor. The quality of the FED image can be further enhanced with digital image processing such as the deconvolution method [17,18].
In our method, a diaphragm and a quarter-wave plate are added into this simple system because an apertured beam and a circularly polarized beam are needed. The aplanatic objective lens with a numerical aperture (NA) of 1.40 is immersed in oil with a refractive index of 1.518. The excitation beam is a Gaussian beam, the beam diameter of which is equal to the diameter of the objective lens’s entrance pupil. In our calculations, the pinhole is assumed to be infinitely small. Also, we assume that the wavelength difference between the excitation beam and the fluorescence beam is negligible.
2.2 Beam modulation
According to vectorial diffraction theory [19], various focal spot patterns can be created by using phase modulation and polarization modulation [20]. If the incident beam experiences no modulation, the focal spot is a normal solid focal spot, as shown in Fig. 2(a). When a 0–2π vortex phase modulation is applied to an optical beam, the polarization plays an important role. If a 0–2π counter-clockwise vortex phase shift is applied to a left-handed circularly polarized beam, the beam may form a donut-shaped hollow focal spot, as shown in Fig. 2(b). If a 0–2π counter-clockwise vortex phase shift is applied to a right-handed circularly polarized beam, the beam may form an extended solid spot, as shown in Fig. 2(c). The extended solid spot has a larger diameter than the normal solid spot, with a slight depression at the center of its intensity profile.
Fig. 2 Focal spot pattern under different modulations. (a) Normal focal spot. (b) Hollow focal spot. (c) Extended solid focal spot. (d) Further extended solid focal spot. The curve beneath each pattern is the linear profile of the intensity distribution along the central axis.
This extended solid spot can be further enlarged by using a diaphragm, leaving the polarization state and phase modulation unchanged. Since the diaphragm blocks the light in the periphery of the beam, the effective NA of the objective lens is manually reduced. A lower NA yields a larger focal spot. Thus the focal spot becomes larger. Figure 2(d) shows the focal spot produced by using a diaphragm that has a aperture diameter that is 0.6 times as large as the objective lens’s entrance pupil. Compared to the extended solid spot, its diameter is further extended, with a much more obvious depression at the center of its intensity profile. The size of this further extended solid spot can be controlled by tuning the aperture size of the diaphragm. A smaller aperture may result in a larger focal spot, and may give rise to a more obvious depression.
The original FED method uses the spots in Figs. 2(a) and 2(b) to acquire two confocal images. However, the solid illuminating spot can be replaced by the “extended solid spot” in Fig. 2(c) or the “further extended solid spot” in Fig. 2(d).
2.3 Point spread function
The PSF of confocal microscopy can be calculated using the relation [21]
Here, , , and are the point spread functions of confocal imaging, excitation light illumination, and fluorescence light detection, respectively. Further, is identical to the intensity distribution of the focal spot, hence it depends on the modulation. Here, is the transmission function of the pinhole, and represents the convolution operator. For the sake of simplicity, we assumed the relation: , where is the PSF of unmodulated excitation light illumination.The PSF of FED imaging is:
Here, and are the point spread functions of the two confocal images and is the subtractive factor.It is obvious that there may be negative values in the PSF of FED, which can cause negative values to the FED images. These negative values in FED images are ultimately eliminated by setting those pixel values to zero, which may lead to deformations. In order to eliminate the deformations in FED, the difference between the PSFs used to obtain the two confocal images must be non-negative.
Figure 3(a) plots the confocal imaging PSFs when using the normal solid spot and the hollow spot for illumination. It is obvious that these two curves are poorly matched. When , the shaded area indicates where the negative PSF differences occur. (If , should be scaled before marking the shaded area.) If we replace the normal solid spot with the “extended solid spot,” the PSF associated with the solid-spot-illuminated confocal imaging will become broader, as is shown in Fig. 3(b). The shaded area becomes smaller, but a negative PSF difference still exists. If we use the “further extended solid spot” in Fig. 2(d) for illumination, the resulting situation is shown in Fig. 3(c). By elaborately tuning the aperture size, the PSFs associated with the two confocal images may be well matched, thus achieving a non-negative difference. In our simulation conditions (NA = 1.4, n = 1.518), the aperture’s diameter should be 0.6 times as large as the objective lens’s entrance pupil.
Fig. 3 Point spread functions associated with the confocal imaging resulting from illumination with different focal spot. (a) PSFs resulting from illumination with a normal solid spot and with a hollow spot. (b) PSFs resulting from illumination with an extended solid spot and with a hollow spot. (c) PSFs resulting from illumination with a further extended solid spot and with a hollow spot. The shaded area shows where a negative PSF difference may occur.
The conventional FED employs the normal solid spot and the hollow spot for illumination; while our method, dfFED, employs the “further extended solid spot” and the hollow spot for illumination. The PSFs of FED and dfFED can be calculated with Eqs. (4) and (5).
If the subtractive factor is 0.4 for FED, the FWHM of its PSF is 0.21λ, and the minimum value of the PSF is −0.2. If the subtractive factor is 0.95 for dfFED, the FWHM of its PSF is also 0.21λ, and the minimum value of the PSF is 0. These PSFs are shown in Fig. 4. Thus, dfFED achieves a relatively high resolution and does not introduce negative values.
Fig. 4 Point spread functions for FED imaging. (a) PSF of the original FED method. The subtractive factor is 0.4. (b) PSF of deformation-free FED imaging. The subtractive factor is 0.95. (c) and (d) are respectively the linear profiles of (a) and (b) along the central axis.
Figure 5(a) shows the relation between the subtractive factor and the minimum value of the PSF. Under FED, negative values in the PSF are inevitable. In contrast, dfFED has a wide non-negative range, where the subtractive factor can be up to 0.95.
Fig. 5 (a) Minimum value of PSFs under different subtractive factors. (b) FWHMs of PSFs under different subtractive factors. (c) Relation between minimum value and PSF FWHM.
Figure 5(b) shows the relation between the subtractive factor and the FWHM of the PSF. The FWHM of the PSF decreases as the subtractive factor increases, which means that the resolving capability is enhanced. Note that the FWHM of the confocal imaging system’s PSF is about 0.31λ. Thus, the resolution of FED is always higher than that of confocal imaging, but dfFED’s resolution is higher than that of confocal imaging only when its subtractive factor exceeds 0.65.
The relation between the minimum value of the PSF and the FWHM of the PSF is plotted in Fig. 5(c). The plot indicates that dfFED has a PSF with a smaller size than FED under the same minimum value. In other words, dfFED has a smaller valley value than FED under the same PSF FWHM. Thus, dfFED has advantages over conventional FED.
3. Simulation and results
3.1 Deformation-free imaging
A sample pattern to be imaged, composed of an array of points, is designed for use in simulation tests. As shown in Fig. 6(a), there are nine points in the sample pattern. The four points at the four corners (white) have twice the intensity of the other points (orange). The spacing between adjacent points is 0.3λ. Side length of each square point is 0.04λ.
Fig. 6 Simulation results for imaging a sample array pattern consisting of nine points. (a) Actual array pattern, consisting of points of unequal intensities, (b) confocal image, (c) dfFED image, where the subtractive factor is 0.95, and (d)–(f) FED images, with subtractive factors of 0.4, 0.6, and 0.8 respectively.
The confocal image is shown in Fig. 6(b). It is obvious that only the four points at the corners are distinguishable. The other five points are obscured by the blurring of the bright spots at the corners. Therefore, confocal imaging is not capable of resolving this sample pattern.
The dfFED image is shown in Fig. 6(c). The subtractive factor is tuned to 0.95 so that it does not induce negative values. Hence, there are no deformations in this image. Although some very dim artifacts arise, they do not lead to the deformation of the object’s image. In practice, these dim artifacts tend to submerge in the background image. Hence, these artifacts are acceptable. Because of its higher resolution with respect to confocal imaging, all nine of the points are distinguishable.
The images acquired by conventional FED are shown in Figs. 6(d)–6(f), under subtractive factors of 0.4, 0.6, and 0.8, respectively. Negative pixel values occur in these three figures, but they are set to zero. For the subtractive factor r = 0.4, the FED image looks like the confocal image. Only the four points at the corners and the central point are distinguishable in Fig. 6(d). As the subtractive factor increases, the resolution is enhanced, but the deformations are also amplified. Thus, the four strong points seem to be clearer, but the five weak points tend to disappear. No matter what the subtractive factor is, the FED image cannot resolve all nine of those points.
3.2 Superior resolution of dfFED imaging
In the previous section, we avoided deformations in the dfFED image by careful selection of the subtractive factor; if instead we allow negative values to appear in the dfFED figure, dfFED demonstrates clearly better performance than FED.
We first note that if FED and dfFED are of the same resolution, dfFED suffers less from negative value than FED does. A sample pattern consisting of an array of parallel lines is designed for use in simulation tests. The line width is related to the resolution since it will be broaden by different extent when imaging with different system. An image with a greater line width represents a worse resolution. The sample pattern is shown in Fig. 7(a), where the spacing between two adjacent lines is 0.6λ. The line width is 0.02λ. The corresponding dfFED image at a subtractive factor of 1.0 is shown in Fig. 7(b), and the FED figure at a subtractive factor of 0.60 is shown in Fig. 7(c). In order to consider the deformations, negative values are not set to zero here. A linear profile is shown in Fig. 7(d); as can be seen, the positive parts of the results of the two methods coincide. Thus, these two images are of the same resolution. However, their negative parts differ. The valley value of the FED image’s profile is below −0.6, while the valley value of the dfFED image’s profile is not below −0.25. Therefore, FED suffers from more severe deformations than dfFED.
Fig. 7 Simulation results for imaging a sample array pattern of parallel lines. (a) Original sample pattern, (b) dfFED image, where the subtractive factor is 1.00, (c) FED image, where the subtractive factor is 0.60, and (d) linear intensity profiles, taken along horizontal lines through the centers of (b) and (c).
We also note that if FED and dfFED are restricted to the same level of deformations, namely, the same minimum negative values, dfFED has a higher resolution than FED. Figure 8(a) shows a spoke-like sample pattern that is widely used in resolution tests. For FED imaging, the subtractive factor is set to 0.51, so the negative values of the FED image are no smaller than −0.2. For dfFED imaging, the subtractive factor is set to 1.029, so the negative values of dfFED image are also no smaller than −0.2. Hence, the two images are restricted to the same level of deformations. The FED image and dfFED image are shown in Figs. 8(b) and 8(c) respectively. It is obvious that the diameter of the unresolved circular area in Fig. 8(b) is larger than that in Fig. 8(c). Linear profiles along the green lines in Figs. 8(b) and 8(c) are plotted in Fig. 8(d). The FED profile is a flat line while the dfFED profile displays obvious peaks and troughs. Thus, in this case, dfFED has a higher resolution than conventional FED.
Fig. 8 Simulation results for imaging a spoke-like sample pattern. (a) Spoke-like sample pattern, (b) FED image, where the subtractive factor is 0.51, (c) dfFED image, where the subtractive factor is 1.029, and (d) linear profiles of intensity along the green lines in (a), (b), and (c). The minimum values in (b) and (c) are both −0.2.
4. Conclusion
In conclusion, we proposed a method for eliminating the deformations in FED. The main idea is to find a pair of matched PSFs, where the matching is based on whether the difference between the PSFs is always non-negative. We used a diaphragm and a 0–2π counter-clockwise vortex phase plate to modulate a right-handed circularly polarized beam, which yielded an extended solid focal spot. We also used a 0–2π counter-clockwise vortex phase plate to modulate a left-handed circularly polarized beam, which yielded a donut-shaped hollow focal spot. These two spots were used in confocal imaging. The corresponding PSFs were well matched. Thus, after subtraction, there were almost no negative values in the dfFED PSF. Also, there were no deformations in the dfFED image.
If deformations are not allowed in the image, the subtractive factor of dfFED should be set to 0.95. In this case, the FWHM of the dfFED PSF is 0.21λ. The FWHM of the PSF for traditional confocal imaging is 0.31λ. Therefore, dfFED achieves high-resolution deformation-free imaging.
If deformations are allowed, the subtractive factor can exceed 0.95, and dfFED performs better than the original FED method. If FED and dfFED are of the same resolution, dfFED suffers from less deformation than FED, and if FED and dfFED are restricted to the same level of deformation, dfFED has a higher resolution than FED.
Because of the simple optical system and better imaging performance, we believe that dfFED is a useful and promising super-resolution imaging method.
Acknowledgments
This work was financially supported by grants from the National Basic Research Program of China (973 Program) (No.2015CB352003) and the National Natural Science Foundation of China (No. 61377013, 61205160, 61378051, 61335003 and 61427818).
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