Isotropic superresolution imaging for fluorescence emission difference microscopy Shangting You, Cuifang Kuang,* Zihao Rong, Xu Liu, and Zhihua Ding State Key Laboratory of Modern Optical Instrumentation, Department of Optical Engineering, Zhejiang University, Hangzhou 310027, China *Corresponding author: [email protected] Received 21 August 2014; revised 1 October 2014; accepted 16 October 2014; posted 23 October 2014 (Doc. ID 221431); published 12 November 2014
Fluorescence emission diffraction microscopy (FED) has proven to be an effective sub-diffraction-limited imaging method. In this paper, we theoretically propose a method to further enhance the resolving capability of FED. Using a coated mirror and only one objective lens, this method achieves not only the same axial resolution as 4Pi microscopy but also a higher lateral resolution. The point spread function (PSF) of our method is isotropic. According to calculations, the full width at half-maximum (FWHM) of the isotropic FED’s PSF is 0.17λ along all three spatial directions. Compared with confocal microscopy, the lateral resolution is improved 0.7-fold, and the axial resolution is improved 3.1-fold. Simulation tests also demonstrate this method’s advantage over traditional microscopy techniques. © 2014 Optical Society of America OCIS codes: (100.6640) Superresolution; (180.2520) Fluorescence microscopy. http://dx.doi.org/10.1364/AO.53.007838
1. Introduction
Far-field optical microscopy allows humans to see previously inaccessible microscosms, and it is widely used in the life sciences. However, this technique has a natural barrier: the diffraction limit. In recent decades, several methods have been adopted to overcome this restriction [1,2], such as stimulated emission depletion microscopy (STED) [3], structured illumination microscopy (SIM) [4], stochastic optical reconstruction microscopy (STORM) [5], photoactivated localization microscopy (PALM) [6], and superresolution optical fluctuation imaging (SOFI) [7]. Recently, new superresolution imaging methods such as fluorescence emission difference microscopy (FED) [8] and switching laser mode microscopy (SLAM) [9,10] have been proposed. Fluorescence emission difference microscopy is similar to STED, as both of them require a solid focal spot and a hollow 1559-128X/14/337838-07$15.00/0 © 2014 Optical Society of America 7838
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focal spot in the imaging process. They differ in that STED reduces the effective size of the excitation spot physically, whereas FED does this mathematically. Stimulated emission depletion microscopy uses a solid focal spot to stimulate fluorescence emission and a hollow spot to deplete the margin area of the fluorescence spot; hence, a high power is required for the depletion beam. Unlike STED, FED uses both the solid focal spot and the hollow focal spot to produce two images by means of the ordinary confocal method, and it then yields a superresolution image by subtraction. Hence, FED requires twice the time of confocal microscopy to obtain a superresolution image. Owing to the absence of the depletion process, FED does not require a complicated optical setup or high-power laser. Because of the absence of nonlinearity, FED has an intrinsic limitation for the resolution. Our previous experiments prove that the lateral resolution of FED is around 1/4 wavelength [8,11]. Among the superresolution microscopy methods mentioned above, most concentrate on the enhancement of the lateral resolution [8–10,12,13], although
the axial resolution is far lower than the lateral resolution. In contrast, 4Pi microscopy realizes the axial resolution enhancement by interference [14] but fails to improve the lateral resolution. Although using double hollow spots in STED can improve the resolution in all three directions [15], the axial resolution remains far lower than the lateral resolution. In this paper, we demonstrate that a solid focal spot (the spot size is similar to that of 4Pi illumination) and two hollow focal spots can be generated by phase modulation. Unlike 4Pi microscopy, only one objective lens is needed if a coated mirror is inserted at the focal plane [16,17]. Besides, the incoherent superposition of the two hollow focal spots is a very small, isotropic hollow focal spot [18]. Superresolution imaging with isotropic resolution is achieved by applying the solid spot and isotropic hollow spot to FED; we call this isotropic FED imaging. According to our calculations, the full width at halfmaximum (FWHM) of the point spread function (PSF) of this method in all spatial directions reaches 0.17λ.
The PSF for FED is given as follows [19–21]: PSFFED PSFe · PSFd ⊗ p − r · PSFe− · PSFd ⊗ p;
where PSFFED , PSFe , PSFe− , and PSFd are the PSFs for the FED imaging, solid fluorescence emission spot, hollow fluorescence emission spot, and detection spot, respectively. p is the transmission function of the pinhole. In our calculations, the wavelength difference between the excitation light and fluorescence light is neglected for simplicity. Also, the pinhole is considered as infinitely small. The fluorescence intensity is expected to be linear; thus, the saturation of the fluorescence molecules is not considered. B. Focal Spot Pattern
Vectorial diffraction theory [22] is used to calculate the focal field in this paper. The electric field vector near the focal spot is given by [23] 2
ZZ
2. Theory A.
Er2 ; φ2 ; z2 iC
Fluorescence Emission Difference Microscopy
The FED image is the difference between two confocal images [8]. One image is illuminated by a solid focal spot, and the other is illuminated by a hollow focal spot. Then, Eq. (1) is used to calculate the FED image. The subtraction process is given by I FED I c1 − r · I c2 ;
(1)
where I FED , I c1 , and I c2 are the spatial distributions of the normalized intensity of the FED image and the two confocal images, respectively. r is the subtractive factor. A larger r enhances the resolution but causes artifacts; thus, this value is chosen empirically from the range of 0.5–1.0. In practice, r is determined by comparing the FED image and the confocal image. As long as the FED image does not differ from confocal image dramatically (e.g., some structures are lost), the subtractive factor is suitable. In some positions, negative differences are inevitable, and these pixel values are set to zero.
2 A2 θ; φ
1 cos θ − 1cos2 φ
p6 cos θ6 4 cos θ − 1 cos φ sin φ sin θ cos φ
(2)
Ω
iΔαθ;φ
·e
px
3
6 7 sinθ · A1 θ; φ · A2 θ; φ · 4 py 5 pz ·e
iknz2 cos θr2 sin θ cosφ−φ2
dθdφ: (3)
In the formula above, θ; φ are spherical surface coordinates describing a point on the exit pupil, where θ represents the polar angle and φ represents the azimuth angle. r2 ; φ2 ; z2 are cylindrical coordinates describing a point near the focal point where z2 , φ, and r2 represent the optical axis, azimuth angle, and radial distance, respectively. The origins of both sets of coordinates are the geometric focal point of the objective lens. Er2 ; φ2 ; z2 is the electric field vector near the focal point. C is a normalization constant. Ω is the exit pupil. A1 θ; φ is the amplitude function of the incident light. px; py; pz is a unit vector indicating the polarization of the incident light. Δαθ; φ is the phase delay function of the phase plate. k is the wavenumber in free space, and n is the refractive index. A2 θ; φ is the coordinate conversion coefficient:
cos θ − 1 cos φ sin φ 1 cos θ − 1sin2 φ sin θ sin φ
− sin θ cos φ
3
7 − sin θ sin φ 7 5: cos θ
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In this paper, the refractive index of the immersion oil, n, is 1.518, and the numerical aperture of the objective lens, NA, is 1.4. The incident beam is a Gaussian beam with a very large waist diameter; thus, the intensity distribution on the entrance pupil can be regarded as uniform. The traditional focal spot appears as an ellipsoid, extending along the direction of the optical axis. Hence, the axial resolution of fluorescence confocal microscopy is relatively low. The most common method to reduce the axial size is 4Pi illumination [14], wherein two lenses are placed at opposite sides with their focuses aligned and then the interferential focal field is separated into three segments. The segment in the middle, which is considered as the main lobe, is far smaller than the traditional focal field. Figures 1(a) and 1(b) show the focal spots for the traditional illumination and 4Pi illumination, respectively. Also, the linear profile of the normalized intensity along the optical axis is sketched in Fig. 1(c). In these figures, the x axis is assumed to be the optical axis. Recent research has revealed that an effect similar to 4Pi illumination can be achieved using one lens and a reflecting mirror [16,17,24]. By phase modulation, a single lens can yield two separated focal spots. If a reflection mirror is inserted on the vertical-midplane of the two focal spots, then one focal spot is reflected back and overlaps the other one. The two focal spots interfere and yield a similar pattern to that of 4Pi illumination. Both this method and 4Pi require a pair of well-aligned focal spots, yet it is simpler for this method to align. For this method, we only need to adjust a five-axis nanopositioning stage where the reflection mirror is on it.
Here, Δφ is as given in Eq. (5); and ϕ is the azimuthal angle. We set z0 2λ. The phase mask’s shape and the focal field’s pattern are shown in Fig. 2. The focal field shown in Fig. 2(a) is named “field A,” and that shown in Fig. 2(c) is named “field C”. We employ a coated mirror, which is designed for two specific operating wavelengths λ1 and λ2. It reflects with a half-wave loss at the operating wavelength of λ2 and without a half-wave loss at λ1. This mirror is inserted into both field A and field C. The operating wavelength can be either λ1 or λ2; hence, the coated mirror can be regarded as either a half-wave-loss mirror or a no-half-wave-loss mirror. Then, three types of interferential focal fields are obtained, which are shown in Fig. 3 [18]. The first, generated by field A together with the half-wave-loss
Fig. 1. (a) Normal focal spot. (b) Focal spot of 4pi illumination. Side length of the cubes is 2λ. (c) Linear profile of the intensity along the optical axis. The blue curve represents the normal profile, and the green curve represents the 4pi profile.
Fig. 2. (a) and (b) Focal spot “field A” and the phase mask described in Eq. (6). (c) and (d) Focal spot “field C” and the phase mask described in Eq. (7). The dimensions of the cubes are 4λ × 4λ × 8λ.
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The phase modulation method to achieve bifocal focusing is given by the following equation: Δφ0
π sgnsinz0 k0 cos θ: 2
(5)
Here, θ is the angle between the optical axis and the ray, z0 is the half-distance between two focal spots, and k0 is the wavenumber in free space. We now introduce different types of focal spots, which will be distinguished in our imaging method. First, by using the phase masks described by the following equations, two types of focal spots are generated Δφ1 Δφ0 ;
(6)
Δφ2 Δφ0 ϕ:
(7)
Fig. 3. Intensity distribution of the interfered field. (a) Compressed solid spot, (b) laterally hollow spot, and (c) axially hollow spot. Side length of the cubes is 2λ.
mirror (i.e., using λ2), is called the “compressed solid spot.” The second, generated by field C together with the no-half-wave-loss mirror (i.e., using λ1), is called the “laterally hollow spot.” The third, generated by field A together with the no-half-wave-loss mirror, is called the “axially hollow spot.” 3. Method
The “compressed solid spot” can serve as the solid spot in FED, taking the place of the traditional confocal spot. The incoherent superposition of the “laterally hollow spot” and “axially hollow spot” is a 3D hollow spot, which serves as the hollow spot. Figure 4 indicates the optical setup of our design. The light sources are two lasers, λ1 and λ2. When using the solid spot for illumination, shutters S1 and S2 are closed; thus, the excitation beam λ2 passes the phase plate, PP0, and then a “compressed solid spot” is formed. Fluorescence light is emitted from the sample and focused onto the pinhole. The first confocal image is acquired by scanning. Subsequently, the hollow spot is used. The operating wavelength is chosen to be λ1. Shutter S0 is closed so that the excitation beam can pass through both PP1 and PP2; then, the “axially hollow spot” and “laterally hollow spot” are formed. Incoherent superposition is achieved by separating the two pulse trains by a slight temporal delay. The time delay should be larger than pulse width but smaller than the fluorescence lifetime. The second confocal image can also be acquired by scanning. Finally, the FED image is generated by subtraction.
Fig. 4. Optical setup to achieve isotropic FED imaging. L, converging lens; DM, dichroic mirror; BS, beam splitter; PBS, polarizing beam splitter; M, reflecting mirror; CM, coated mirror; BF, bandpass filter; PH, pinhole; OL, objective lens; PP0, PP1, and PP2, phase plates; S0, S1, and S2, shutters; λ∕4, quarter-wave plate; λ∕2, half-wave plate; D, detector; S, sample.
The quarter-wave plate near the objective lens turns the linear polarization into circular polarization; thus, the focal field is rotationally symmetrical with respect to the x–y plane. This avoids anisotropic lateral resolution. There is a half-wave plate near the laser λ1 that can tune the incident power proportion between PP1 and PP2. In order to optimize the FED imaging, we should determine a proper power proportion between the “laterally hollow spot” and “axially hollow spot.” Here, we introduce the “effective volume” in order to evaluate the PSF of the FED imaging. The PSF of the FED is calculated according to Eq. (2); then, the effective volume is the product of the FWHMs along three directions: V eff dx · dy · dz :
(8)
According to Fig. 5, as the subtractive factor increases, the effective volume shrinks. The power proportion also makes a difference. Intensities of 40% and 60% for the laterally hollow spot and axially hollow spot, respectively, were determined as the optimal proportions to minimize the effective volume of the PSF. 4. Results and Discussion A. Point Spread Function
The power proportion is set to the optimal value— 40% for the laterally hollow spot and 60% for the axially hollow spot—and the subtractive factor is set to 0.5. The PSFs of the isotropic FED imaging, 4Pi imaging (both illuminated and detected by two objective lenses), and confocal imaging are indicated in Fig. 6. It is obvious that the isotropic FED imaging achieves the same axial resolution as 4Pi microscopy, and its lateral resolution is far better than that of 4Pi. In order to compare the resolving capability among different methods, the FWHMs of the PSFs are listed in Table 1. The data for isotropic FED imaging are categorized per three different subtractive factors. Conservatively, we consider the subtractive factor
Fig. 5. Effective volume of PSFs. The ratio indicates the power proportion of the laterally hollow spot to the axially hollow spot. 20 November 2014 / Vol. 53, No. 33 / APPLIED OPTICS
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Fig. 6. Point spread functions for different imaging methods. (a) and (b) Perspective figure and sectional plane figure of the PSF for confocal imaging. (c) and (d) PSF for 4Pi imaging. (e) and (f) PSF for isotropic FED imaging. (g) Linear profiles of PSFs along the lateral direction. (h) Linear profiles of PSFs along the axial direction. Side length of the cubes is 2λ.
Table 1.
Full Width at Half-Maximum of the PSFs for Different Methods along All Directions
FWHM of the PSF along All Directions Method Confocal 4Pi Isotropic FED (r 0.5) Isotropic FED (r 0.7) Isotropic FED (r 1.0)
x
y
z
0.29λ 0.29λ 0.17λ 0.15λ 0.13λ
0.29λ 0.29λ 0.17λ 0.15λ 0.13λ
0.70λ 0.16λ 0.17λ 0.16λ 0.15λ
r 0.5; hence, the FWHM of the isotropic FED’s PSF is 0.17λ along all three directions, signifying an isotropic sphere.
Again, we calculate the effective volume. The volumes are 0.0589λ3 , 0.0135λ3 , and 0.0049λ3 for the confocal imaging, 4Pi, and isotropic FED, respectively. The PSF’s volume for the isotropic FED imaging was reduced by 92% compared with that for the confocal imaging. B. Simulation Results for the Point Array Sample
We designed a 3 × 3 × 3 point array sample to verify the effect of the isotropic FED imaging. Figure 7(a) shows the sample; here, the distance between two adjacent points is 0.4λ. The figure indicates that neither laterally adjacent points nor axially adjacent points can be distinguished in the confocal imaging. In the 4Pi, only axially adjacent points can
Fig. 7. Simulation results for the point array sample. (a) Point array sample. (b) Confocal image. (c) 4Pi image. (d) Isotropic FED image. (e) Linear profile of the intensity distribution along the lateral direction. (f) Linear profile of the intensity distribution along the axial direction. Side length of the cubes is 2λ. 7842
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Natural Science Foundation of China (Nos. 61377013, 61205160, 61378051, 61335003, and 61427818). References
Fig. 8. Simulation results for the “ZJUOPT” pattern. (a) Sample of the “ZJUOPT” sample. (b) Confocal image. (c) 4Pi image. (d) Isotropic FED image. The box size is 2λ × 2λ × 6λ.
be distinguished. In the isotropic FED image, all 27 points can be distinguished. C.
Simulation Results for the “ZJUOPT” Pattern
A 3D sample of “ZJUOPT” pattern is also designed to compare the effects of different types of imaging. As shown in Fig. 8(a), the letters are located in different lateral planes. The confocal image shown in Fig. 8(b) suffers from a comparatively low axial resolution; hence, the six letters are not clearly separated. The 4Pi image shown in Fig. 8(c) overcomes this drawback, in which the letters are well distinguished. Each letter in the isotropic FED image shown in Fig. 8(d) can be discerned more clearly. Figure 8 confirms that the isotropic FED image exhibits the best resolution. However, artifacts occur in both the 4Pi image and the isotropic FED image, which are caused by the side lobes of the PSFs. These artifacts can be reduced by digital image processing, e.g., deconvolution methods [25,26].
5. Conclusion
In conclusion, this paper introduces a new method to improve the performance of FED microscopy. Bifocal focusing is achieved by phase modulation. A coated mirror is inserted on the focal plane; consequently, the two focal spots overlap and interfere with each other. We demonstrate how to generate a compressed solid spot, laterally hollow spot, and axially hollow spot with different types of phase masks and mirrors. The compressed solid spot illuminates the sample and yields a confocal image. The laterally hollow spot, combined with the axially hollow spot, illuminates the sample and yields another confocal image. Finally, we subtract the second confocal image from the first one and acquire an FED image. This method has an isotropic resolution in all spatial directions. Compared with 4Pi microscopy, it has the same level of axial resolution and a far higher lateral resolution. Compared with confocal microscopy, the lateral resolution is improved by 0.7-fold, and the axial resolution is improved by 3.1-fold. This work was financially supported by grants from the National Basic Research Program of China (973 Program) (No. 2015CB352003) and the National
1. S. W. Hell, “Microscopy and its focal switch,” Nat. Methods 6, 24–32 (2009). 2. S. W. Hell, “Far-field optical nanoscopy,” in Single Molecule Spectroscopy in Chemistry, Physics and Biology (Springer, 2010), pp. 365–398. 3. S. W. Hell and J. Wichmann, “Breaking the diffraction resolution limit by stimulated emission: stimulated-emissiondepletion fluorescence microscopy,” Opt. Lett. 19, 780–782 (1994). 4. M. G. Gustafsson, “Surpassing the lateral resolution limit by a factor of two using structured illumination microscopy,” J. Microsc. 198, 82–87 (2000). 5. M. J. Rust, M. Bates, and X. Zhuang, “Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy (STORM),” Nat. Methods 3, 793–796 (2006). 6. E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. LippincottSchwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science 313, 1642–1645 (2006). 7. T. Dertinger, R. Colyer, R. Vogel, M. Heilemann, M. Sauer, J. Enderlein, and S. Weiss, “Superresolution optical fluctuation imaging (SOFI),” in Nano-Biotechnology for Biomedical and Diagnostic Research (Springer, 2012), pp. 17–21. 8. C. Kuang, S. Li, W. Liu, X. Hao, Z. Gu, Y. Wang, J. Ge, H. Li, and X. Liu, “Breaking the diffraction barrier using fluorescence emission difference microscopy,” Sci. Rep. 3, 1441 (2013). 9. H. Dehez, M. Piché, and Y. De Koninck, “Resolution and contrast enhancement in laser scanning microscopy using dark beam imaging,” Opt. Express 21, 15912–15925 (2013). 10. H. Dehez, M. Piché, and Y. De Koninck, “High resolution imaging with TM01 laser beams,” Proc. SPIE 7386, 738606 (2009). 11. Z. Rong, S. Li, C. Kuang, Y. Xu, and X. Liu, “Real-time superresolution imaging by high-speed fluorescence emission difference microscopy,” J. Mod. Opt. 61, 1364–1371 (2014). 12. S. Li, C. Kuang, X. Hao, Y. Wang, J. Ge, and X. Liu, “Enhancing the performance of fluorescence emission difference microscopy using beam modulation,” J. Opt. 15, 125708 (2013). 13. A. Gasecka, A. Daradich, H. Dehez, M. Piché, and D. Côté, “Resolution and contrast enhancement in coherent anti-Stokes Raman-scattering microscopy,” Opt. Lett. 38, 4510–4513 (2013). 14. S. Hell and E. H. Stelzer, “Properties of a 4Pi confocal fluorescence microscope,” J. Opt. Soc. Am. A 9, 2159–2166 (1992). 15. B. Harke, C. K. Ullal, J. Keller, and S. W. Hell, “Threedimensional nanoscopy of colloidal crystals,” Nano Lett. 8, 1309–1313 (2008). 16. E. Mudry, E. Le Moal, P. Ferrand, P. C. Chaumet, and A. Sentenac, “Isotropic diffraction-limited focusing using a single objective lens,” Phys. Rev. Lett. 105, 203903 (2010). 17. E. Le Moal, E. Mudry, P. C. Chaumet, P. Ferrand, and A. Sentenac, “Two-photon fluorescence isotropic-single-objective microscopy,” Opt. Lett. 37, 85–87 (2012). 18. S. Li, C. Kuang, X. Hao, Z. Gu, and X. Liu, “Generation of a 3D isotropic hollow focal spot for single-objective stimulated emission depletion microscopy,” J. Opt. 14, 085704 (2012). 19. M. Muller, Introduction to Confocal Fluorescence Microscopy (SPIE, 2006), Vol. 69. 20. M. Martinez-Corral, “Point spread function engineering in confocal scanning microscopy,” Proc. SPIE 5182, 112–122 (2004). 21. M. Gu, Principles of Three Dimensional Imaging in Confocal Microscopes (World Scientific, 1996). 22. B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. Royal Soc. A 253, 358–379 (1959). 20 November 2014 / Vol. 53, No. 33 / APPLIED OPTICS
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23. X. Hao, C. Kuang, T. Wang, and X. Liu, “Effects of polarization on the de-excitation dark focal spot in STED microscopy,” J. Opt. 12, 115707 (2010). 24. E. Le Moal, E. Mudry, P. C. Chaumet, P. Ferrand, and A. Sentenac, “Isotropic single-objective microscopy: theory and experiment,” J. Opt. Soc. Am. A 28, 1586–1594 (2011).
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25. G. Vicidomini, R. Schmidt, A. Egner, and S. Hell, “Automatic deconvolution in 4Pi-microscopy with variable phase,” Opt. Express 18, 10154–10167 (2010). 26. J. Bewersdorf, R. Schmidt, and S. W. Hell, “Comparison of I5M and 4Pi-microscopy,” J. Microsc. 222, 105–117 (2006).
Fluorescence emission diffraction microscopy (FED) has proven to be an effective sub-diffraction-limited imaging method. In this paper, we theoretically propose a method to further enhance the resolving capability of FED. Using a coated mirror and only one objective lens, this method achieves not only the same axial resolution as 4Pi microscopy but also a higher lateral resolution. The point spread function (PSF) of our method is isotropic. According to calculations, the full width at half-maximum (FWHM) of the isotropic FED’s PSF is 0.17λ along all three spatial directions. Compared with confocal microscopy, the lateral resolution is improved 0.7-fold, and the axial resolution is improved 3.1-fold. Simulation tests also demonstrate this method’s advantage over traditional microscopy techniques. © 2014 Optical Society of America OCIS codes: (100.6640) Superresolution; (180.2520) Fluorescence microscopy. http://dx.doi.org/10.1364/AO.53.007838
1. Introduction
Far-field optical microscopy allows humans to see previously inaccessible microscosms, and it is widely used in the life sciences. However, this technique has a natural barrier: the diffraction limit. In recent decades, several methods have been adopted to overcome this restriction [1,2], such as stimulated emission depletion microscopy (STED) [3], structured illumination microscopy (SIM) [4], stochastic optical reconstruction microscopy (STORM) [5], photoactivated localization microscopy (PALM) [6], and superresolution optical fluctuation imaging (SOFI) [7]. Recently, new superresolution imaging methods such as fluorescence emission difference microscopy (FED) [8] and switching laser mode microscopy (SLAM) [9,10] have been proposed. Fluorescence emission difference microscopy is similar to STED, as both of them require a solid focal spot and a hollow 1559-128X/14/337838-07$15.00/0 © 2014 Optical Society of America 7838
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focal spot in the imaging process. They differ in that STED reduces the effective size of the excitation spot physically, whereas FED does this mathematically. Stimulated emission depletion microscopy uses a solid focal spot to stimulate fluorescence emission and a hollow spot to deplete the margin area of the fluorescence spot; hence, a high power is required for the depletion beam. Unlike STED, FED uses both the solid focal spot and the hollow focal spot to produce two images by means of the ordinary confocal method, and it then yields a superresolution image by subtraction. Hence, FED requires twice the time of confocal microscopy to obtain a superresolution image. Owing to the absence of the depletion process, FED does not require a complicated optical setup or high-power laser. Because of the absence of nonlinearity, FED has an intrinsic limitation for the resolution. Our previous experiments prove that the lateral resolution of FED is around 1/4 wavelength [8,11]. Among the superresolution microscopy methods mentioned above, most concentrate on the enhancement of the lateral resolution [8–10,12,13], although
the axial resolution is far lower than the lateral resolution. In contrast, 4Pi microscopy realizes the axial resolution enhancement by interference [14] but fails to improve the lateral resolution. Although using double hollow spots in STED can improve the resolution in all three directions [15], the axial resolution remains far lower than the lateral resolution. In this paper, we demonstrate that a solid focal spot (the spot size is similar to that of 4Pi illumination) and two hollow focal spots can be generated by phase modulation. Unlike 4Pi microscopy, only one objective lens is needed if a coated mirror is inserted at the focal plane [16,17]. Besides, the incoherent superposition of the two hollow focal spots is a very small, isotropic hollow focal spot [18]. Superresolution imaging with isotropic resolution is achieved by applying the solid spot and isotropic hollow spot to FED; we call this isotropic FED imaging. According to our calculations, the full width at halfmaximum (FWHM) of the point spread function (PSF) of this method in all spatial directions reaches 0.17λ.
The PSF for FED is given as follows [19–21]: PSFFED PSFe · PSFd ⊗ p − r · PSFe− · PSFd ⊗ p;
where PSFFED , PSFe , PSFe− , and PSFd are the PSFs for the FED imaging, solid fluorescence emission spot, hollow fluorescence emission spot, and detection spot, respectively. p is the transmission function of the pinhole. In our calculations, the wavelength difference between the excitation light and fluorescence light is neglected for simplicity. Also, the pinhole is considered as infinitely small. The fluorescence intensity is expected to be linear; thus, the saturation of the fluorescence molecules is not considered. B. Focal Spot Pattern
Vectorial diffraction theory [22] is used to calculate the focal field in this paper. The electric field vector near the focal spot is given by [23] 2
ZZ
2. Theory A.
Er2 ; φ2 ; z2 iC
Fluorescence Emission Difference Microscopy
The FED image is the difference between two confocal images [8]. One image is illuminated by a solid focal spot, and the other is illuminated by a hollow focal spot. Then, Eq. (1) is used to calculate the FED image. The subtraction process is given by I FED I c1 − r · I c2 ;
(1)
where I FED , I c1 , and I c2 are the spatial distributions of the normalized intensity of the FED image and the two confocal images, respectively. r is the subtractive factor. A larger r enhances the resolution but causes artifacts; thus, this value is chosen empirically from the range of 0.5–1.0. In practice, r is determined by comparing the FED image and the confocal image. As long as the FED image does not differ from confocal image dramatically (e.g., some structures are lost), the subtractive factor is suitable. In some positions, negative differences are inevitable, and these pixel values are set to zero.
2 A2 θ; φ
1 cos θ − 1cos2 φ
p6 cos θ6 4 cos θ − 1 cos φ sin φ sin θ cos φ
(2)
Ω
iΔαθ;φ
·e
px
3
6 7 sinθ · A1 θ; φ · A2 θ; φ · 4 py 5 pz ·e
iknz2 cos θr2 sin θ cosφ−φ2
dθdφ: (3)
In the formula above, θ; φ are spherical surface coordinates describing a point on the exit pupil, where θ represents the polar angle and φ represents the azimuth angle. r2 ; φ2 ; z2 are cylindrical coordinates describing a point near the focal point where z2 , φ, and r2 represent the optical axis, azimuth angle, and radial distance, respectively. The origins of both sets of coordinates are the geometric focal point of the objective lens. Er2 ; φ2 ; z2 is the electric field vector near the focal point. C is a normalization constant. Ω is the exit pupil. A1 θ; φ is the amplitude function of the incident light. px; py; pz is a unit vector indicating the polarization of the incident light. Δαθ; φ is the phase delay function of the phase plate. k is the wavenumber in free space, and n is the refractive index. A2 θ; φ is the coordinate conversion coefficient:
cos θ − 1 cos φ sin φ 1 cos θ − 1sin2 φ sin θ sin φ
− sin θ cos φ
3
7 − sin θ sin φ 7 5: cos θ
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In this paper, the refractive index of the immersion oil, n, is 1.518, and the numerical aperture of the objective lens, NA, is 1.4. The incident beam is a Gaussian beam with a very large waist diameter; thus, the intensity distribution on the entrance pupil can be regarded as uniform. The traditional focal spot appears as an ellipsoid, extending along the direction of the optical axis. Hence, the axial resolution of fluorescence confocal microscopy is relatively low. The most common method to reduce the axial size is 4Pi illumination [14], wherein two lenses are placed at opposite sides with their focuses aligned and then the interferential focal field is separated into three segments. The segment in the middle, which is considered as the main lobe, is far smaller than the traditional focal field. Figures 1(a) and 1(b) show the focal spots for the traditional illumination and 4Pi illumination, respectively. Also, the linear profile of the normalized intensity along the optical axis is sketched in Fig. 1(c). In these figures, the x axis is assumed to be the optical axis. Recent research has revealed that an effect similar to 4Pi illumination can be achieved using one lens and a reflecting mirror [16,17,24]. By phase modulation, a single lens can yield two separated focal spots. If a reflection mirror is inserted on the vertical-midplane of the two focal spots, then one focal spot is reflected back and overlaps the other one. The two focal spots interfere and yield a similar pattern to that of 4Pi illumination. Both this method and 4Pi require a pair of well-aligned focal spots, yet it is simpler for this method to align. For this method, we only need to adjust a five-axis nanopositioning stage where the reflection mirror is on it.
Here, Δφ is as given in Eq. (5); and ϕ is the azimuthal angle. We set z0 2λ. The phase mask’s shape and the focal field’s pattern are shown in Fig. 2. The focal field shown in Fig. 2(a) is named “field A,” and that shown in Fig. 2(c) is named “field C”. We employ a coated mirror, which is designed for two specific operating wavelengths λ1 and λ2. It reflects with a half-wave loss at the operating wavelength of λ2 and without a half-wave loss at λ1. This mirror is inserted into both field A and field C. The operating wavelength can be either λ1 or λ2; hence, the coated mirror can be regarded as either a half-wave-loss mirror or a no-half-wave-loss mirror. Then, three types of interferential focal fields are obtained, which are shown in Fig. 3 [18]. The first, generated by field A together with the half-wave-loss
Fig. 1. (a) Normal focal spot. (b) Focal spot of 4pi illumination. Side length of the cubes is 2λ. (c) Linear profile of the intensity along the optical axis. The blue curve represents the normal profile, and the green curve represents the 4pi profile.
Fig. 2. (a) and (b) Focal spot “field A” and the phase mask described in Eq. (6). (c) and (d) Focal spot “field C” and the phase mask described in Eq. (7). The dimensions of the cubes are 4λ × 4λ × 8λ.
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The phase modulation method to achieve bifocal focusing is given by the following equation: Δφ0
π sgnsinz0 k0 cos θ: 2
(5)
Here, θ is the angle between the optical axis and the ray, z0 is the half-distance between two focal spots, and k0 is the wavenumber in free space. We now introduce different types of focal spots, which will be distinguished in our imaging method. First, by using the phase masks described by the following equations, two types of focal spots are generated Δφ1 Δφ0 ;
(6)
Δφ2 Δφ0 ϕ:
(7)
Fig. 3. Intensity distribution of the interfered field. (a) Compressed solid spot, (b) laterally hollow spot, and (c) axially hollow spot. Side length of the cubes is 2λ.
mirror (i.e., using λ2), is called the “compressed solid spot.” The second, generated by field C together with the no-half-wave-loss mirror (i.e., using λ1), is called the “laterally hollow spot.” The third, generated by field A together with the no-half-wave-loss mirror, is called the “axially hollow spot.” 3. Method
The “compressed solid spot” can serve as the solid spot in FED, taking the place of the traditional confocal spot. The incoherent superposition of the “laterally hollow spot” and “axially hollow spot” is a 3D hollow spot, which serves as the hollow spot. Figure 4 indicates the optical setup of our design. The light sources are two lasers, λ1 and λ2. When using the solid spot for illumination, shutters S1 and S2 are closed; thus, the excitation beam λ2 passes the phase plate, PP0, and then a “compressed solid spot” is formed. Fluorescence light is emitted from the sample and focused onto the pinhole. The first confocal image is acquired by scanning. Subsequently, the hollow spot is used. The operating wavelength is chosen to be λ1. Shutter S0 is closed so that the excitation beam can pass through both PP1 and PP2; then, the “axially hollow spot” and “laterally hollow spot” are formed. Incoherent superposition is achieved by separating the two pulse trains by a slight temporal delay. The time delay should be larger than pulse width but smaller than the fluorescence lifetime. The second confocal image can also be acquired by scanning. Finally, the FED image is generated by subtraction.
Fig. 4. Optical setup to achieve isotropic FED imaging. L, converging lens; DM, dichroic mirror; BS, beam splitter; PBS, polarizing beam splitter; M, reflecting mirror; CM, coated mirror; BF, bandpass filter; PH, pinhole; OL, objective lens; PP0, PP1, and PP2, phase plates; S0, S1, and S2, shutters; λ∕4, quarter-wave plate; λ∕2, half-wave plate; D, detector; S, sample.
The quarter-wave plate near the objective lens turns the linear polarization into circular polarization; thus, the focal field is rotationally symmetrical with respect to the x–y plane. This avoids anisotropic lateral resolution. There is a half-wave plate near the laser λ1 that can tune the incident power proportion between PP1 and PP2. In order to optimize the FED imaging, we should determine a proper power proportion between the “laterally hollow spot” and “axially hollow spot.” Here, we introduce the “effective volume” in order to evaluate the PSF of the FED imaging. The PSF of the FED is calculated according to Eq. (2); then, the effective volume is the product of the FWHMs along three directions: V eff dx · dy · dz :
(8)
According to Fig. 5, as the subtractive factor increases, the effective volume shrinks. The power proportion also makes a difference. Intensities of 40% and 60% for the laterally hollow spot and axially hollow spot, respectively, were determined as the optimal proportions to minimize the effective volume of the PSF. 4. Results and Discussion A. Point Spread Function
The power proportion is set to the optimal value— 40% for the laterally hollow spot and 60% for the axially hollow spot—and the subtractive factor is set to 0.5. The PSFs of the isotropic FED imaging, 4Pi imaging (both illuminated and detected by two objective lenses), and confocal imaging are indicated in Fig. 6. It is obvious that the isotropic FED imaging achieves the same axial resolution as 4Pi microscopy, and its lateral resolution is far better than that of 4Pi. In order to compare the resolving capability among different methods, the FWHMs of the PSFs are listed in Table 1. The data for isotropic FED imaging are categorized per three different subtractive factors. Conservatively, we consider the subtractive factor
Fig. 5. Effective volume of PSFs. The ratio indicates the power proportion of the laterally hollow spot to the axially hollow spot. 20 November 2014 / Vol. 53, No. 33 / APPLIED OPTICS
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Fig. 6. Point spread functions for different imaging methods. (a) and (b) Perspective figure and sectional plane figure of the PSF for confocal imaging. (c) and (d) PSF for 4Pi imaging. (e) and (f) PSF for isotropic FED imaging. (g) Linear profiles of PSFs along the lateral direction. (h) Linear profiles of PSFs along the axial direction. Side length of the cubes is 2λ.
Table 1.
Full Width at Half-Maximum of the PSFs for Different Methods along All Directions
FWHM of the PSF along All Directions Method Confocal 4Pi Isotropic FED (r 0.5) Isotropic FED (r 0.7) Isotropic FED (r 1.0)
x
y
z
0.29λ 0.29λ 0.17λ 0.15λ 0.13λ
0.29λ 0.29λ 0.17λ 0.15λ 0.13λ
0.70λ 0.16λ 0.17λ 0.16λ 0.15λ
r 0.5; hence, the FWHM of the isotropic FED’s PSF is 0.17λ along all three directions, signifying an isotropic sphere.
Again, we calculate the effective volume. The volumes are 0.0589λ3 , 0.0135λ3 , and 0.0049λ3 for the confocal imaging, 4Pi, and isotropic FED, respectively. The PSF’s volume for the isotropic FED imaging was reduced by 92% compared with that for the confocal imaging. B. Simulation Results for the Point Array Sample
We designed a 3 × 3 × 3 point array sample to verify the effect of the isotropic FED imaging. Figure 7(a) shows the sample; here, the distance between two adjacent points is 0.4λ. The figure indicates that neither laterally adjacent points nor axially adjacent points can be distinguished in the confocal imaging. In the 4Pi, only axially adjacent points can
Fig. 7. Simulation results for the point array sample. (a) Point array sample. (b) Confocal image. (c) 4Pi image. (d) Isotropic FED image. (e) Linear profile of the intensity distribution along the lateral direction. (f) Linear profile of the intensity distribution along the axial direction. Side length of the cubes is 2λ. 7842
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Natural Science Foundation of China (Nos. 61377013, 61205160, 61378051, 61335003, and 61427818). References
Fig. 8. Simulation results for the “ZJUOPT” pattern. (a) Sample of the “ZJUOPT” sample. (b) Confocal image. (c) 4Pi image. (d) Isotropic FED image. The box size is 2λ × 2λ × 6λ.
be distinguished. In the isotropic FED image, all 27 points can be distinguished. C.
Simulation Results for the “ZJUOPT” Pattern
A 3D sample of “ZJUOPT” pattern is also designed to compare the effects of different types of imaging. As shown in Fig. 8(a), the letters are located in different lateral planes. The confocal image shown in Fig. 8(b) suffers from a comparatively low axial resolution; hence, the six letters are not clearly separated. The 4Pi image shown in Fig. 8(c) overcomes this drawback, in which the letters are well distinguished. Each letter in the isotropic FED image shown in Fig. 8(d) can be discerned more clearly. Figure 8 confirms that the isotropic FED image exhibits the best resolution. However, artifacts occur in both the 4Pi image and the isotropic FED image, which are caused by the side lobes of the PSFs. These artifacts can be reduced by digital image processing, e.g., deconvolution methods [25,26].
5. Conclusion
In conclusion, this paper introduces a new method to improve the performance of FED microscopy. Bifocal focusing is achieved by phase modulation. A coated mirror is inserted on the focal plane; consequently, the two focal spots overlap and interfere with each other. We demonstrate how to generate a compressed solid spot, laterally hollow spot, and axially hollow spot with different types of phase masks and mirrors. The compressed solid spot illuminates the sample and yields a confocal image. The laterally hollow spot, combined with the axially hollow spot, illuminates the sample and yields another confocal image. Finally, we subtract the second confocal image from the first one and acquire an FED image. This method has an isotropic resolution in all spatial directions. Compared with 4Pi microscopy, it has the same level of axial resolution and a far higher lateral resolution. Compared with confocal microscopy, the lateral resolution is improved by 0.7-fold, and the axial resolution is improved by 3.1-fold. This work was financially supported by grants from the National Basic Research Program of China (973 Program) (No. 2015CB352003) and the National
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