Breaking the diffraction barrier using coherent anti-Stokes Raman scattering difference microscopy

Dong Wang; Shuanglong Liu; Yue Chen; Jun Song; Wei Liu; Maozhen Xiong; Guangsheng Wang; Xiao Peng; Junle Qu

doi:10.1364/OE.25.010276

1. Introduction

Far-field optical microscopy has rapidly developed recently, such as two-photon excited luminescence (TPEL) [1], two-photon excited-fluorescence (TPEF) [2], second harmonic generation (SHG) [3,4], third harmonic generation (THG) [5–7], stimulated Raman scattering (SRS) [8–10], and coherent anti-Stokes Raman scattering (CARS) [11–15]. However, the diffraction barrier in the imaging system limits the lateral resolution of conventional optical microscopes to approximately half the illumination wavelength. Thus, breaking the diffraction barrier is an intense research topic. During the past two decades several methods of breaking the diffraction limit have been proposed, including stochastic optical reconstruction microscopy (STORM) [16], photoactivated localization microscopy (PALM) [17], stimulated emission depletion microscopy (STED) [18], structured illumination microscopy (SIM) [19,20], digital image processing [21,22], and total internal reflection microscopy based on a nanostructured substrate [23]. STORM, PALM, STED, and SIM are the most widely applied, and have all been demonstrated to achieve super-resolution in bioscience [24–27]. However, significant disadvantages of these super-resolution methods include: photobleaching, toxicity, and the effects of fluorophores on labeled molecules in live samples. Thus, nondestructive detection cannot be achieved for these methods, but nondestructive detection is important for in vivo experiments in life sciences.

CARS is one of the most important microscopy methods, and can realize nondestructive detection. However, it is difficult to improve spatial resolution for CARS systems, and CARS super-resolution has been an active research topic [28–35]. Fluorescence emission difference (FED) microscopy has been recently proposed for breaking the diffraction barrier, which is based on the intensity difference between two differently acquired images [36]. This paper employs a similar concept and presents a theoretical investigation of how this kind of difference method in CARS microscopy can break the diffraction limit. We have called the method coherent anti-Stokes Raman scattering difference (CARS-D) microscopy. We build a theoretical CARS-D model and analyze related parameters settings to obtain the best super-resolution effect. Comparing with existing methods, the present CARS-D method can achieve non-invasive super-resolution imaging in situ, in real-time and without labelling.

2. Principle

CARS microscopy is a third order nonlinear optical imaging method in which two laser pulses at different frequencies, pump light (ω_p) and Stokes light (ω_s), are focused onto a sample (the probe light is also ω_p). Raman resonance occurs when ω_p-ω_s matches a Raman active molecular vibrational level. CARS detects the output anti-Stokes signal (ω_as) at a new frequency, 2ω_p-ω_s, which is enhanced by the Raman resonance. The anti-Stokes signal intensity can be expressed as

I_{a s} \propto E_{a s}^{2} \propto {[ρ E_{p}^{2} (r) E_{s} (r)]}^{2},

where ρ is the density of the Raman active molecule, and E_as(r), E_p(r), E_s(r) are the electric fields, where r denotes the radial component of the cylindrical coordinate system and reflects the characteristic of the light spot of the anti-Stokes, pump, and Stokes lights, respectively.

Figure 1 illustrates the CARS-D method. Two different scanned images are processed: one being the conventional CARS image acquired when the sample is illuminated by a Gaussian shaped spot, and the other is obtained with the sample is illuminated by a doughnut shaped spot generated by modulating the illumination beam with a vortex 0–2π phase plate. The final super-resolution CARS-D image is constructed by intensity subtraction of these two images. The green lines and arrow in Fig. 1 represent the conventional CARS course, where Figs. 1(a) and 1(b) show the electric field distribution of Gaussian shaped Stokes (E_s₁) and pump (E_p) lights, respectively, and Fig. 1(d) shows the Gaussian shaped effective excitation area of anti-Stokes (I_as₁) light. The yellow lines and arrow represent the proposed CARS-D course, where Fig. 1(c) shows the electric field distribution of doughnut shaped Stokes (E_s₂), and Fig. 1(e) shows the doughnut shaped effective excitation area of anti-Stokes (I_as₂) light. The red lines and arrow represent the difference of the two courses, and Fig. 1(f) shows the effective excitation area of the CARS-D signal is smaller and sharper, although there is some negative area. Point by point scanning provides a higher resolution image by the excitation spot difference as shown in Fig. 1(f). In Section 3 we build a physical model to further analyze the impact of the relevant parameters on imaging results.

Fig. 1 Illustration of CARS-D method.

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3. Theoretical model

CARS is a third order nonlinear optical phenomenon, and based on Eq. (1) with some reasonable assumption we can obtain the following analytical model.

The Gaussian shaped pump (E_p) and Stokes (E_s₁) lights can be expressed as

E_{p} (r) = E_{p 0} e x p (- \frac{r^{2}}{r_{p}^{2}}),

and

E_{s 1} (r) = E_{s 10} e x p (- \frac{r^{2}}{r_{s 1}^{2}}),

where E_p₀ and E_s₁₀ are the amplitudes, and r_p and r_s₁ are the spot radii of the electric field of pump and Stokes lights, respectively. To simplify the model to more directly reflect the physical mechanism, the doughnut shaped Stokes (E_s₂) can be expressed as [18,37]

E_{s 2} (r) = E_{s 20} r e x p (- \frac{r^{2}}{r_{s 2}^{2}}),

where E_s₂₀ and r_s₂ are the amplitude and radius, respectively, of the electric field of the doughnut shaped Stokes light.

From Eqs. (1)–(3) the conventional and unconventional CARS courses with effective excitation areas of anti-Stokes (I_as₁ and I_as₂) can be expressed as

I_{a s 1} \propto {[ρ E_{p 0}^{2} \exp (- \frac{2 r^{2}}{r_{p}^{2}}) E_{s 1} \exp (- \frac{r^{2}}{r_{s 1}^{2}})]}^{2},

and

I_{a s 2} \propto {[ρ E_{p 0}^{2} \exp (- \frac{2 r^{2}}{r_{p}^{2}}) E_{s 2} r \exp (- \frac{r^{2}}{r_{s 2}^{2}})]}^{2} .

The difference of Eqs. (5) and (6) is the CARS-D course,

Δ I_{a s} = I_{a s 1} - k_{M} I_{a s 2},

where k_M is the subtractive factor. To simplify the model, we assume the radius of the pump light, r_p = 0.5λ (λ is the wavelength of the pump light), and the relationships between the radii r_p, r_s₁, and r_s₂ are assumed to be r_s₁ = k₁r_p and r_s₂ = k₂r_p (k₁ and k₂ are ratio coefficients). The amplitudes E_p₀ and E_s₁₀ are determined by the normalization condition and generally taken as 1, while the amplitude E_s₂₀ is replaced by M. Then, from Eqs. (5)–(7),

Δ I_{a s} \propto ρ^{2} \exp (- \frac{16 r^{2}}{λ^{2}}) {\exp [- \frac{8 r^{2}}{{(k_{1} λ)}^{2}}] - k_{M} M^{2} r^{2} \exp [- \frac{8 r^{2}}{{(k_{2} λ)}^{2}}]} .

We need to determine the factor M. There is some negative area using the CARS-D method, and the desired signal may be covered by the negative signal when the negative value is too large. Therefore, we define M as the specific value between the positive and negative signal when the subtractive factor k_M = 1 and ρ = 1 everywhere over the area. That is to say M equals the ratio between the volume integrals of I_as₁ and I_as₂. The volume integrals of I_as₁ and I_as₂ are, respectively,

V_{1} = \int_{0}^{\infty} r d r \int_{0}^{2 π} d θ \exp (- \frac{16 r^{2}}{λ^{2}}) \exp [- \frac{8 r^{2}}{{(k_{1} λ)}^{2}}] = \frac{π}{a_{1}},

a_{1} = \frac{16}{λ^{2}} + \frac{8}{{(k_{1} λ)}^{2}},

and

V_{2} = \int_{0}^{\infty} r d r \int_{0}^{2 π} d θ \exp (- \frac{16 r^{2}}{λ^{2}}) (r^{2}) \exp [- \frac{8 r^{2}}{{(k_{2} λ)}^{2}}] = \frac{π}{a_{2}^{2}},

a_{2} = \frac{16}{λ^{2}} + \frac{8}{{(k_{2} λ)}^{2}},

and, hence,

M = \frac{V_{1}}{V_{2}} = \frac{a_{2}^{2}}{a_{1}} .

Note that all results given in Eqs. (9)-(13) have been transformed to no-dimensional form. The integration of the formula (8) will be larger than zero when there exists noise in the process. Instead, if the volume integral is smaller than zero, it means that some useful information in these images is lost. Therefore, the optimal integral should be equal to zero where ρ = k_M = 1 to achieve a perfect super-resolution image without noise or information loss. In order to obtain nice images around the optimal point, we need to adjust the factor k_M around 1 in the following studies, and define M (shown in Eq. (13) for facilitate analysis.

4. Numerical results

In this section, we present numerical results for images based on conventional CARS and the present CARS-D method by Matlab simulation according to above-mentioned analytical equations (see Eqs. (1)-(13)). To verify our numerical model we have carried out imaging experiments based on a commercial CARS imaging system. The experimental samples are polystyrene beads with the 110 nm diameter. Figure 2(a) is the TEM (Transmission Electron Microscope) image of the polystyrene beads with the size 6.25 μm × 6.25μm. Based on the CARS imaging system, Fig. 2(b) gives the image of one polystyrene bead marked with the green dotted circle in Fig. 2(a), where the brighter color corresponds to the stronger CARS signal (I_as). The wavelengths of pump and Stokes lights are 655 nm and 820 nm, respectively, while the Raman shift of the polystyrene bead is 1600 cm⁻¹. Figure 2(c) is the intensity distribution of I_as along the green dotted line in the Fig. 2(b), where the position along the line is the abscissa. We can obtain the value of the spatial resolution as 484 nm for the current CARS system.

Fig. 2 (a) TEM image of the polystyrene beads with the size 6.25 μm $\times$ 6.25 μm; (b) Experimental CARS image of one polystyrene bead marked with the green circle in Fig. 2 (a); (c) Intensity distribution of I_as along the green line in Fig. 2 (b); (d) Theoretical CARS image of Fig. 2 (a); (e) Intensity distribution of I_as along the green line in Fig. 2 (d); (f) Local enlarged curve marked with the green circle in Fig. 2 (e).

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Then we apply our numerical model to image the same sample given in Fig. 2(a), and all parameters in the simulation results from above experimental data. The diameter of the pump light spot is 655 nm, the coefficient k₁ is 820/655, and k_M is 1. The corresponding calculated CARS image is shown in Fig. 2(d). The intensity distribution of I_as along the line in Fig. 2(d) is given in Fig. 2(e), and a local enlarged curve marked with the green circle in Fig. 2 (e) can be found in Fig. 2(f). From the numerical result, we can learn the theoretical value of the spatial resolution is 480nm, which agrees well with the previous experimental value (see Fig. 2(c)).

Furthermore, we also obtain the theoretical images based on the doughnut CARS image (Fig. 3(a)) and the CARS-D image (Fig. 3(b)), respectively, with the coefficient k₂ = 820/655. Figure 3(c) is the intensity distribution of I_as along the green dotted line in Fig. 3(b), and Fig. 3(d) is the local enlarged curve marked with the green circle in Fig. 3 (c). We can find that CARS-D method has better spatial resolution (338 nm) than conventional CARS method.

Fig. 3 (a) Doughnut CARS image of the polystyrene beads; (b) CARS-D image of the polystyrene beads; (c) Intensity distribution of I_as along the green line in Fig. 3(b); (d) Local enlarged curve marked with the green circle in Fig. 3 (e).

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Next, we further accurately estimate the performance of the proposed CARS-D method and optimize its performance to achieve super-resolution images. For the particular case k₁ = k₂ = k_M = 1, we obtained the effective excitation areas of conventional CARS signal (I_as₁), doughnut CARS signal (I_as₂), and CARS-D signal (ΔI_as), as shown in Fig. 4(a), corresponding to the red, blue, and green dotted lines, respectively. To estimate the attainable spatial resolution, we performed the simulation test on a spoke-like sample [36], as shown in Fig. 4(b) with size 18λ/π $\times$ 18λ/π. In the process of our theory analysis, results are closely related to several key parameters: the wavelength of the pump light λ, and the radii of the Stokes light spot determined by the factors k₁ and k₂. Therefore, both the size of the test sample and the pixels of the sample image are set to be related to λ. Then, in the following analysis the value of wavelength λ will be not especially emphasized anymore.

Fig. 4 (a) Illustration of the effective excitation areas of conventional CARS signal (red dotted line), doughnut CARS signal (blue dotted line) and CARS-D signal (green dotted line). (b) A spoke-like test sample.

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The spoke-like sample is widely used because the resolution of an imaging technique can be assessed very simply by measuring the radius of a circle delimiting the ‘unresolved’ central area and the ‘resolved’ peripheral area in the corresponding image. Figure 5 shows the imaging results using an excitation spot to scan the sample point by point. Figures 5(a), (b), and (c) correspond to conventional CARS, doughnut CARS, and CARS-D images, respectively. From the Fig. 5(a), one also sees that the patterns beyond the black circle are the same as those shown in Fig. 4(b), but the patterns have such obvious fuzzy characteristic that the spoke-like lines cannot be distinguished from Fig. 5(a) obtained by the conventional CARS method due to the effect of diffraction limit. Similar patterns can also be seen in Fig. 3(b) with the only difference of the irradiation light (i.e., a doughnut shaped spot). Figure 5(c) shows the CARS-D signal (ΔI_as) with a constraint that a point is set to zero when the difference is negative. Comparing Figs. 5(a) and 5(c), while features inside the region indicated by the black circle cannot be clearly discerned in the conventional CARS image, some features in the same region of the CARS-D image can be distinguished, which indicates enhanced resolution.

Fig. 5 (a) Conventional CARS image. (b) Doughnut CARS image. (c) CARS-D image.

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Figures 6 and 7 (a)–(f) show the effect of k_M from 0.8 to 1.3, step 0.1, for CARS-D images, respectively. The images are colored following the parula colormap. Figure 5 shows that when k_M = 1, the features in the middle region can be distinguished clearly. However, as k_M decreases, the features in the middle region cannot be as well discerned, due to increased noise in the middle region with smaller k_M. When k_M>1, features in the middle region are also eliminated, and this effect increases with increasing k_M. To display this problem more intuitively, Fig. 7 shows the same images colored following the lines colormap. There are two regions in Figs. 7(a)–5(f), indicated by the inner and outer black circles. Within the inner black circle, where there is full noise, features cannot be completely discerned. Outside the outer black circle, features are discernible with little or no noise. Tuning k_M shows that features between the inner and outer black circles are most clearly distinguished for k_M = 1. Combined with Eqs. (8)–(13), Fig. 7 shows that to obtain the best CARS-D resolution, the volume integral of Eq. (8) must be zero for ρ = 1. For k_M>1, the volume integral of Eq. (8) is negative, which means that some of the information will be lost in the process of the difference. However for k_M<1, the volume integral of Eq. (8) is positive, which means that there exists a certain amount of noise between inner and outer black circles in the CARS-D method. Therefore, only for k_M = 1, we can expect that neither the noise or information loss can be found, i.e., a perfect super-resolution image can be achieved.

Fig. 6 CARS-D images with different subtractive factors using the parula color-map.

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Fig. 7 CARS-D images with different subtractive factors using the lines color-map.

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Given the black circles in Fig. 7, we can define inner and outer spatial resolutions (SR_in and SR_out, respectively) to estimate CARS-D imaging effects. Each spatial resolution (SR) can be calculated by the diameter, D, of the black circle, i.e., SR = πD/36 (there are 36 lines in the spoke-like sample as shown in Fig. 4(b)). Then we may analyze the impact of the factors k₁ and k₂ on CARS-D spatial resolution. Figure 8 shows SR_in and SR_out as functions of k₁ and k₂, keeping one factor constant while varying the other. The spatial resolution reduces as the factor reduces, and tends toward a constant when the factor is large enough. The results suggested as long as the radius of the pump light spot is a determined value, the radii of the Gaussian shaped and doughnut shaped Stokes spots have little influence on the spatial resolution, unless the radii are negligible values. However k₂ has a greater effect on SR than k₁. Maximum SR_out≈0.35λ and SR_in≈0.065λ. However, it is unreasonable to take the spatial resolution of CARS-D as 0.065λ, because that not all the features between the inner and outer black circles can be distinguished. Therefore, we can only regard the spatial resolution SR of CARS-D as between 0.35λ and 0.065λ, and probably closer to 0.065λ.

Fig. 8 Spatial resolution as function of factors k₁ and k₂: (a) inner spatial resolution, SR_in, as a function of k₁ with k₂ = k_M = 1; (b) SR_in as a function of k₂ with k₁ = k_M = 1; (c) outer spatial resolution, SR_out, as a function of k₁ with k₂ = k_M = 1; (d) SR_out as a function of k₂ with k₁ = k_M = 1.

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As above all simulations are based on the spoke-like test sample as shown in Fig. 4(b), we will further evaluate the reliability of the CARS-D method by using an irregular sample. To illustrate that the CARS-D method provides superior imaging than conventional CARS based on irregular sample, we used the Golgi complex image as an irregular test sample, as shown in Fig. 9. Figure 9(a) shows a visual image of the Golgi complex, and the size is 9λ × 9λ. Note that the size is not the real size of the Golgi complex in bioscience, but an irregular test sample to highlight details. Figures 9(b), (c) and (d) show conventional CARS doughnut CARS and CARS-D images, respectively, which are numerical simulations for the special case with k₁ = k₂ = k_M = 1. There is considerable resolution improvement when this kind of irregular sample was imaged by CARS-D method. Many features disclosed by CARS-D imaging are completely hidden under conventional CARS and doughnut CARS observation. Therefore, the present CARS-D method can effectively be used for super-resolution imaging not only based on the special regular samples (e.g., spoke-like samples) but also on any irregular samples.

Fig. 9 (a) Golgi complex visual image (as an irregular test sample), (b) conventional CARS image, (c) doughnut CARS image, and (d) CARS-D image.

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5. Conclusion

We proposed a CARS-D method to improve the resolution of CARS microscopy. On the basis of intensity differences between two differently acquired images, CARS-D can achieve spatial resolution between 0.35λ and 0.065λ, which is beyond the conventional diffraction barrier that conventional CARS microscopy cannot break. In the proposed CARS-D theoretical model, the best resolution is produced when the volume integral of Eq. (8) is zero for ρ = 1. Based on numerical simulation, we verify the CARS-D can be effectively applied for super-resolution microscopy.

Funding

National Natural Science Foundation of China (NSFC) (61235012, 61620106016, 61525503, 61378091, 61405123); National Basic Research Program of China (Program 973) (2015CB352005); Guangdong Natural Science Foundation Innovation Team (2014A030312008); Hong Kong, Macao and Taiwan cooperation innovation platform & major projects of international cooperation in Colleges and Universities in Guangdong Province (2015KGJHZ002); and Shenzhen Basic Research Project (JCYJ20160328144746940, JCYJ20160308093035903, JCYJ20150930104948169).

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Breaking the diffraction barrier using coherent anti-Stokes Raman scattering difference microscopy

Abstract

1. Introduction

2. Principle

3. Theoretical model

4. Numerical results

5. Conclusion

Funding

References and links

Cited By

Figures (9)

Equations (13)

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