A theoretical study of diffraction limit breaking via coherent control of the relative phase in coherent anti-Stokes Raman scattering microscopy

Dong Wang; Wenyong Fu; Yunfei Lei; Houzhi Cai; Jinyuan Liu

doi:10.1364/OE.27.005005

1. Introduction

Far-field optical microscopy is a basic and useful technique for sample analysis in bioscience. In addition to classical transmission and reflection microscopy, fluorescence microscopy and nonlinear microscopy techniques have developed rapidly in the recent past, 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]. With proper manipulation of the illumination field or sample properties, it is possible to extract spatial features that are much smaller than the diffraction limit, leading to super-resolution optical microscopy, which has become a hot research topic in recent years. In fluorescence microscopy, several super-resolution techniques have been successfully demonstrated in experiments, such as photoactivated localization microscopy (PLAM) [16], stochastic optical reconstruction microscopy (STORM) [17], and stimulated emission depletion (STED) microscopy [18,19], which rely on the ability to control the emissive properties of fluorophores with optical radiation. However, significant disadvantages of fluorescence include: photobleaching, toxicity, and the effects of fluorophores on labeled molecules in live samples.

For nonlinear microscopy that does not include exogenous labeling agents of chemical species, several super-resolution methods have been proposed theoretically, such as super-resolution CARS microscopy [20–28] and super-resolution SRS microscopy [29,30]. These methods require new and powerful light sources, the optical systems are quite complicated to build, and the high-energy photons used in their methods can damage biological tissue. In this study, with the intention of avoiding these defects, we propose a scheme to suppress the CARS signal by controlling the relative phase between the pump light and the Stokes light to realize super-resolution CARS microscopy.

2. Principle

CARS microscopy is a third-order nonlinear optical imaging method in which two laser pulses at different frequencies, denoted as pump light ( $ω_{p}$ ) and Stokes light ( $ω_{s}$ ), are focused onto a sample. Raman resonance occurs when $ω_{p} - ω_{s}$ matches a Raman-active molecular vibrational level. CARS system detects the outputted anti-Stokes signal ( $ω_{a s}$ ) at a new frequency $2 ω_{p} - ω_{s}$ which is enhanced by the Raman resonance, and should satisfy the frequency-matching $Δ ω = 2 ω_{p} - ω_{s} - ω_{a s} = 0$ . In addition, the pump, Stokes, and anti-Stokes lights should also satisfy the wave-vector-matching $Δ \vec{K} = 2 {\vec{K}}_{p} - {\vec{K}}_{s} - {\vec{K}}_{a s} = 0$ , where ${\vec{K}}_{p}$ , ${\vec{K}}_{s}$ , ${\vec{K}}_{a s}$ are the wave vectors of the pump light, the Stokes light, and the anti-Stokes signal, respectively [13,31–33]. CARS is a parametric phenomenon in which the input and output photons exchange energy, while the quantum state of the molecule is left unchanged. As a vibrationally enhanced four-wave mixing process, the anti-Stokes signal in CARS is generated at a new frequency that differs to the input beams, which is accompanied by a non-resonant signal caused by electronic contributions to the four-wave mixing process. So, for these photons, frequency-matching means energy conservation, and wave-vector-matching means momentum conservation. Moreover, for CARS microscopy, the wave-vector-matching conditions are relaxed because of the large cone of wave vectors and the short interaction length [11].

The phases of pump light, Stokes light and anti-Stokes signal are $ω_{p} t + {\vec{K}}_{p} \cdot \vec{r} + ϕ_{p}$ , $ω_{s} t + {\vec{K}}_{s} \cdot \vec{r} + ϕ_{s}$ , and $ω_{a s} t + {\vec{K}}_{a s} \cdot \vec{r} + ϕ_{a s}$ , where $ϕ_{p}$ , $ϕ_{s}$ , $ϕ_{a s}$ are the initial phases of the pump light, the Stokes light, and the anti-Stokes signal, respectively (t is the time coordinate, and $\vec{r}$ is the spatial vector). The relative phase $Δ ϕ$ between the pump light, Stokes light and anti-Stokes signal is

\begin{array}{l} Δ ϕ = 2 (ω_{p} t + {\vec{K}}_{p} \cdot \vec{r} + ϕ_{p}) - (ω_{s} t + {\vec{K}}_{s} \cdot \vec{r} + ϕ_{s}) - (ω_{a s} t + {\vec{K}}_{a s} \cdot \vec{r} + ϕ_{a s}) \\ = Δ ω t + Δ \vec{K} \cdot \vec{r} + 2 ϕ_{p} - ϕ_{s} - ϕ_{a s} \\ = 2 ϕ_{p} - ϕ_{s}, \end{array}

where we ignore the initial phases (

ϕ_{a s} = 0

) of the anti-Stokes light. Because in CARS, the phase of the anti-Stokes field is only important when it is detected in an interferometric fashion. If the anti-Stokes field is applied as a local oscillator field, its phase can be tuned, and the transition between the ground state and the first vibrationally excited state can be controlled. This was explained elegantly by S. Rahav et al. in [34], where the CARS process with a phase-controlled local oscillator field was presented as an interference between two quantum pathways in the material. In the absence of the local oscillator field, the initial phase of anti-Stokes field is a constant value, which cannot be controlled by the pump and Stokes light (in order to simplify the physical model, the initial phase of the anti-Stokes light can be taken as zero). The relative phase of a circular level system is very important as it can impact some physical quantities of the system [35], such as the nonlinear susceptibility. Coincidentally, there are experiment and quantum theory studies about that THz wave intensity generated from a two-color laser-induced gas plasma can also be coherent controlled by the relative phase [36–39], which is the same as four-wave mixing phenomenon. Then by the same theory we can think that the relative phase of the CARS system can also influence the intensity of the CARS signal, and the relationship between the intensity

I_{a s}

and the relative phase

Δ φ

is [40]

I_{a s} \propto \sin^{2} Δ ϕ .

This formula demonstrates that the intensity of the anti-Stokes light $I_{a s}$ can be coherently controlled by the relative phase between the pump light and Stokes light $Δ ϕ$ , which satisfies the sine curve relationship. It is noteworthy that the anti-Stokes light $I_{a s}$ can even be zero for the condition $Δ ϕ = N π$ .

3. Analysis and discussion

Next, we present an experimental method to regulate and control the relative phase. A pair of wedge crystals are put along the light path, as shown in Fig. 1(a). When the pump light and Stokes light pass through the pair of crystals, the direction of the two-colored lights remains the same. Due to the angle of the wedge, we can also control the length of the vertical direction of the crystals to change the optical path length. However, the relative phase of the two-colored lights changes when the optical path length changes. Thus, we can control the length of the vertical direction of the crystals to alter the relative phase. Thenc, with the correct angle of the wedge and the appropriate dispersion we can obtain the relative phase $Δ ϕ$

Δ ϕ = 2 ϕ_{p} - ϕ_{s} = 2 π (2 \frac{n_{p} Δ l}{λ_{p}} - \frac{n_{s} Δ l}{λ_{s}}) = \frac{2 ω_{p} n_{p} - ω_{s} n_{s}}{c} Δ l = \tan θ \frac{2 ω_{p} n_{p} - ω_{s} n_{s}}{c} Δ h,

where

θ

is the angle of the wedge, c is the velocity of light in a vacuum,

Δ h

and

Δ l

are, respectively, the length changes in the vertical and horizontal directions of the crystals,

λ_{p}

and

λ_{s}

are, respectively, the wavelengths of the pump and Stokes lights in a vacuum, and

n_{p}

,

n_{s}

are the refractive indexes of the pump and Stokes lights in the crystal, respectively. If we assume that Eq. (3) is equal to

2 π

, the rate of the regulation

ρ

can be obtained as

Fig. 1 The sketch of the pair of wedge crystals (a) and the phase plate (b).

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ρ = \frac{n_{s} Δ l}{λ_{s}} = \frac{λ_{p} / λ_{s}}{2 n_{p} / n_{s} - λ_{p} / λ_{s}} .

The rate of regulation is defined as $ρ = n_{s} Δ l / λ_{s}$ , as shown in Eq. (4), which reflects the tenability of the relative phase $Δ ϕ$ by a length change $Δ l$ . The quantity $ρ$ is larger, and this method is easier to be realized in experiments. If the rate of the regulation $ρ$ is too small, the length change $Δ l$ maybe below the nanometer order of magnitude, which is hard to realize in experiments by stepping motor. As shown in Fig. 2(a), we have plotted a 3-dimensional image of Eq. (4) where the three quantities $ρ$ , $n_{p} / n_{s}$ and $λ_{p} / λ_{s}$ are the coordinate variables. From Fig. 2(a) it is found that the regulation rate $ρ$ is infinite on the 2-dimensional hyperplane $2 n_{p} / n_{s} - λ_{p} / λ_{s} = 0$ . In order to illustrate this more clearly, the top panel of Fig. 2(a) is replotted in Fig. 2(b), where the hyperplane instead appears as a line. In the CARS system, the wavelength of the Stokes light is longer than the pump light. In other words, we obtain the inequality condition $λ_{p} / λ_{s} < 1$ . For the normal dispersion condition, which has the relationship $n (λ) = a + b / λ^{2} + c / λ^{4}$ , we also obtain another inequality condition, $n_{p} / n_{s} > 1$ , implying we only need to study the quantities in the rectangular region, (i.e. the black dashed box shown in the top left corner of Fig. 2(b), where Fig. 2(c) is a 3-dimensional image of this region). The results show that if we want to obtain larger values of $ρ$ , the quantities $n_{p} / n_{s}$ and $λ_{p} / λ_{s}$ must respectively be smaller and larger. Furthermore, we obtained more intuitive curves of the variables $ρ$ and $λ_{p} / λ_{s}$ with different values of $n_{p} / n_{s}$ , as shown in Fig. 2(d), where the parameter $n_{p} / n_{s}$ was chose as 1, 1.1, 1.2, 1.3, 1.4, shown respectively as the colors red, yellow, green, blue, and pink dotted.

Fig. 2 2-dimensional and 3-dimensional images of the three physical quantities $ρ$ , $n_{p} / n_{s}$ and $λ_{p} / λ_{s}$ .

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For many kinds of media, the small difference between $n_{p}$ and $n_{s}$ can be ignored because the frequency difference $2 ω_{p} - ω_{s}$ in the CARS system is not zero. However, the optical delay $Δ τ$ of the pump and Stokes lights is written as

Δ τ = (n_{p} - n_{s}) \frac{Δ l}{c} = \tan θ (n_{p} - n_{s}) \frac{Δ h}{c} .

If we compare Eqs. (3) and (5), it is found that we can control the relative phase $Δ ϕ$ without altering the optical delay $Δ τ$ , even in the air condition. This means that the intensity of the anti-Stokes light also depends on the light path length in air. This dependence may explain why it is not easy to obtain anti-Stokes signals in experiment.

Thus, in order to break the diffraction limit, we must first demonstrate an appropriate theoretical phase plate. The proposed phase plate has a phase shift of $π / 2$ in the central area, as shown in Fig. 1(b). $D_{o}$ and $D_{i}$ are the outer and inner diameters of the phase plate, respectively. Through proper adjustment to the pair of wedge crystals, the relative phase in the outer area can be $N π$ , and the inner area can be $(N + 0.5) π$ . Therefore, the intensity of the anti-Stokes light in the outer area is zero because of Eq. (2), and the effective area needed to generate an anti-Stokes signal can be reduced to just the inner area. An illustration of this physical mechanism is shown in Fig. 3, where Fig. 3(a) is the normal CARS system, and Fig. 3(b) is the phase plate system that requires a narrower area to generate an anti-Stokes signal. Figures 3(a) and 3(b) are just sketches, where the actual values of the x-axis and y-axis don't make sense. Hence, the units in these graphs are missing.

Fig. 3 Sketches of the normal CARS system (a) and the super-resolution CARS system (b).

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Finally, in order to verify that our scheme is suitable for realizing super-resolution CARS microscopy, we considered a numerically generated test sample and simulated super-resolution images, as shown in Fig. 4. The sample consists of a dendritic line structure similar to structures found in, for example, brain tissue, as shown in Fig. 4(a), where the size is 1000 × 1000nm, and the line thickness is 6 nm. CARS is a third-order nonlinear optical phenomenon, and through Eq. (2) we can obtain the following equation

I_{a s} \propto E_{a s}^{2} \propto ρ [\sin^{2} (Δ ϕ) + b] {[E_{p}^{2} (r) E_{s} (r)]}^{2},

where

ρ

is the density of the Raman-active molecule, b is the signal offset,

E_{a s}

,

E_{p}

,

E_{s}

are the electric field, 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. Here the Gaussian function is used to simplify the physical model and describe the spot shape, and the amplitude is assumed to be 1. It is noteworthy that they are not Gaussian beams, in which specific phase is needed and depends on spatial coordinates. Thus, the electric fields

E_{p}

and

E_{s}

have the expressions

E_{p} (r) \propto \exp (- r^{2} / r_{p}^{2})

and

E_{s} (r) \propto \exp (- r^{2} / r_{s}^{2})

, where

r_{p}

and

r_{s}

are the spot widths. Then, Eq. (6) can be expressed as

I_{a s} \propto ρ [\sin^{2} (Δ ϕ) + b] \exp (- \frac{r^{2}}{r_{e f f}^{2}}),

r_{e f f} = {(\frac{4}{r_{p}^{2}} + \frac{2}{r_{s}^{2}})}^{- \frac{1}{2}},

where

r_{e f f}

is the effective width of the CARS course. However, areas that are larger than the effective width

r_{e f f}

can also generate a CARS signal, and so we assumed the radius of the area, which can effectively generate a CARS signal, to be

r_{o} = r_{e f f} / 0.8 = 160 n m

(i.e.

r_{p} = r_{s} = 313.53 n m

). In other words, no CARS signal is generated in an area whose radius is larger than

r_{o}

. The outer diameter of the phase plate

D_{o}

is larger than

2 r_{o} = 320 n m

.

Fig. 4 (a) Test sample simulating a fine structure (e.g. brain tissue). By simulating the super-resolution CARS imaging course of the test sample, the CARS images are obtained for phase plate inner diameters $D_{i}$ of 320nm (b), 160nm (c), 40nm (d) and 20nm (e), where the signal offset b is zero. The remaining images are for signal offsets b of 1 (f), 0.8 (g), 0.4 (h) and 0.1 (i), where the inner diameter $D_{i}$ is 20nm.

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The sub-micrometric spot widths are obtained by focusing the beams with a high numerical aperture (and consequently low working distance) objective. If the phase plate is placed before the objective, we should consider the amplitude and phase modification induced by the objective [41,42]. Therefore, the phase plate may be some nano-structure films after the objective. However, this is a theoretical study, and then we simplify the physical model to better reflect the physical connotation without concerning about the objective. With these arguments, we simulated CARS super-resolution images with Matlab, as shown in Figs. 4(b)-4(i), where the colors from blue to red correspond to the intensity of the anti-Stokes light from weak to strong. The Figs. 4(b)-4(e) correspond to phase plate inner diameters $D_{i}$ of 320 nm, 160 nm, 40 nm and 20 nm, where the signal offset b was zero. When comparing these imaging results, it is found that images with smaller inner diameters display more detail. However, it is more difficult to fabricate smaller inner diameter phase plates.

Next, we studied the influence of the signal offset b on the simulated images, as shown in Figs. 4(f)-4(i), which correspond to signal offsets of b = 1, 0.8, 0.4 and 0.1, where the inner diameter $D_{i}$ was chosen as 20 nm. Signal offset b is the noise signal offset, which depends on precision and stability of the experiment system. From these results, it is found that with a small noise signal offset we can still obtain super-resolution images.

4. Conclusions

In summary, we presented a theoretical study on the suppression of CARS by the relative phase between pump light and Stokes light and its application to super-resolution imaging. Here the relative phase of the pump and Stokes lights was defined as $Δ ϕ = 2 ϕ_{p} - ϕ_{s}$ . The relationship between the anti-Stokes signal and the relative phase was derived as $I_{a s} \propto \sin^{2} Δ ϕ$ . Subsequently, we studied the possibility of regulating and controlling the relative phase by a pair of wedge crystals. In this approach, we proposed a special phase plate to reduce the effective region of the generation of the CARS signal and simulated super-resolution images. We note however, that in practical terms, this kind of phase plate may not be fabricated very easily, though it may be realized via metamaterials or the other nano-optics technology.

Funding

National Natural Science Foundation of China (NSFC) (11775147); Science and Technology Program of Shenzhen (STPS) (JCYJ20170302153912966, JCYJ20160608173121055); Scientific Research Start-up Project for Newly Introduced Teacher of Shenzhen University (2017015).

Acknowledgments

Thanks for the help of professor Junle Qu and professor Jun Song.

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A theoretical study of diffraction limit breaking via coherent control of the relative phase in coherent anti-Stokes Raman scattering microscopy

Abstract

1. Introduction

2. Principle

3. Analysis and discussion

4. Conclusions

Funding

Acknowledgments

References

Cited By

Figures (4)

Equations (8)

Optics Express