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Divided-aperture confocal Brillouin microscopy for simultaneous high-precision topographic and mechanical mapping

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Abstract

Confocal Brillouin microscopy (CBM) is a novel and powerful technique for providing non-contact and direct readout of the micro-mechanical properties of a material, and thus used in a broad range of applications, including biological tissue detection, cell imaging, and material characterization in manufacturing. However, conventional CBMs have not enabled high precision mechanical mapping owing to the limited depth of focus and are subject to system drift during long-term measurements. In this paper, a divided-aperture confocal Brillouin microscopy (DCBM) is proposed to improve the axial focusing capability, stability, and extinction ratio of CBM. We exploit high-sensitivity divided-aperture confocal technology to achieve an unprecedented 100-fold enhancement in the axial focusing sensitivity of the existing CBMs, reaching 5 nm, and to enhance system stability. In addition, the dark-field setup improves the extinction ratio by 20 dB. To the best of our knowledge, our method achieves the first in situ topographic imaging and mechanical mapping of the sample and provides a new approach for Brillouin scattering applications in material characterization.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Brillouin scattering, an inelastic light scattering caused by local thermal vibration of the matter, can provide non-invasive detection of mechanical properties of materials, such as stiffness, temperature, strain, and elasticity constants, and has been widely used in material characterization and optical sensing [13]. Confocal Brillouin microscopy (CBM) refers to the combination of Brillouin spectroscopy with confocal microscopy, which can achieve three-dimensional micro-mechanical properties mapping of samples [47]. In the field of materials science, CBM has been used to characterize the elastic constants of spider silks and nanostructured hydrogel networks, and has successfully provided the stiffness changes in supercontraction and helped understand the complex biomaterial interactions [8,9]. In the field of biomedical science, mechanical mapping of cells through CBM has unprecedentedly revealed the cell biomechanics-function relationships in the process of gene expression and environmental changes [1017]. The applications of CBM to the stiffness detection of tissues can be used for early diagnosis and treatment [5,1824]. The development of technology has led to a constant increase in the demand for high-precision mechanical mapping of micro-sample in the fields mentioned. Nevertheless, the shortcomings of traditional CBMs in axial focusing capability, system stability, and extinction ratio hinder their further extension and application.

During the mapping process, the focused spot size of CBM varies with sample morphology at each scanning position for the lack of rapid and high-precision axial focusing capability. As a result, traditional CBMs cannot collect the Brillouin scattered light exactly at the smallest excitation spot on the sample surface and the actual spatial resolution is lower than the theoretical spatial resolution. Several methodologies based on optical path reconstruction, such as dual-axis configuration [5] and annular pupil [25], have significantly reduced the effective excitation area of the focusing spot and thus achieving micron or even sub-micron spatial resolution. Confocal Brillouin microscopy based on adaptive optics configuration has been developed to engineer the incident wavefront and correct for aberrations, achieving 1.4-fold improvement in axial resolution [26]. The above methods can all achieve high spatial resolution CBM. However, they are not suitable for mechanical mapping of non-flat samples.

Regarding system stability, the CBM imaging mechanism of point-excited and point-detecting is bound to cause time-consuming imaging, even reaching tens of hours. During the long-term imaging, the defocus caused by environmental fluctuations reduces the spectral intensity of CBM. For this reason, improving system stability is critical for high quality image. Although some studies have used image pattern recognition of a marked area to detect the system drift, and achieved automatic axial focusing via a feedback loop to ensure long-term stability in the imaging process, the complexity of the algorithm in such methods highly increases the imaging time [21,27,28]. Besides, the samples have to be marked. Thus, how to ensure a rapid and accurate focus tracking in the imaging process remains a technical challenge for the application of CBM.

Moreover, Brillouin scattering is extremely sensitive to the strong elastic background light. In particular, specular reflections arising near the surface or interface can easily overwhelm the Brillouin peaks, significantly reducing the extinction ratio and spectral measurement accuracy. Previously, spatial filtering [29,30] and molecular absorption [31] techniques have been proposed, in which a mask and molecular absorption cell are used to filter out the elastic scattered light in the spatial or frequency domains, respectively. Recently, the destructive interference [32,33] and equalization techniques [34], respectively based on multi-beam interference and phase matching, have been developed to eliminate the elastic scattered light. Especially, background-deflection technique based on diffraction masks has been proposed to achieve an unprecedented 10,000-fold enhancement in the spectral contrast [14].

In this paper, we present divided-aperture confocal Brillouin microscopy (DCBM) as a precise and accurate mechanical mapping method with rapid and high-sensitivity focusing capability. The system stability and extinction ratio are enhanced during the long-time spectral detection and imaging. We also demonstrate that DCBM can achieve in situ mechanical mapping, topographical imaging, and dark-field imaging under the same excitation.

2. Methods

2.1 Divided-aperture confocal Brillouin microscopy method

A schematic of the DCBM system is shown in Fig. 1, consisting of a divided-aperture confocal system and a confocal Brillouin spectroscopy system, which are symmetrical about the optical axis of the objective. The reflected light can be spatially separated from the scattered light by lateral illumination and lateral detection, in which the backscattering light is collected by the confocal Brillouin spectroscopy system of DCBM, thus effectively suppressing the crosstalk with the reflected light and improving the extinction ratio through the dark-field setup. Additionally, DCBM can realize dark field imaging through Rayleigh scattered light detected by FP interferometer. Meanwhile, the reflected light is collected by the divided-aperture confocal system to enable focus tracking and in situ topographical imaging.

 figure: Fig. 1.

Fig. 1. Schematic of the DCBM system. The beam emitted from a laser source passes through a polarizer, a beam splitter (BS), an aperture, and is focused onto the sample by an objective. The scattered light (S) is transmitted along the original light path, focused on a pinhole (PH1) by a lens, and finally detected by a Fabry–Perot interferometer. Simultaneously the reflected light (R) is collected in the symmetrical direction of the illumination light path through the aperture and is focused on a second pinhole (PH2) through an attenuator (ATT) and a lens, and then detected by a photomultiplier tube (PMT). IC is the confocal axial response curve obtained by PMT, IS is the curve shifted by IC, ID is the curve after subtraction of IS and IC, LD is the fitting line of ID, us is the shift offset of curve IS relative to curve IC, and u is the normalized axial coordinate.

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In the divided-aperture confocal system, the confocal axial response curve IC (as shown in Fig. 1) can be obtained by PMT after axial scanning of the sample. Since the curve IC is symmetrical about its peak, and the two segments of data near the half-maximum positions on both sides of IC are very sensitive to variations of the axial position. We axially shift one segment of IC by us to get the curve IS which is intersected with the other segment, and then subtract the corresponding data of the two segments, according to Eq. (1), in which, (vx, vy, u) are the normalized optical coordinates of (x, y, z). We select the best linearity data of the subtracted curve ID to fit a line LD, according to Eq. (2). By calculating the zero position of the fitting line LD and offsetting it by us/2, the optical defocus of the sample can be precisely obtained [35].

$${I_D}({v_x},{v_y},u,{u_s}) = {I_S}({v_x},{v_y},u,{u_s}) - {I_C}({v_x},{v_y},u)$$
$${L_D}({v_x},{v_y},u,{u_s}) = f({{I_D}({v_x},{v_y},u,{u_s})} )$$

We make a closed loop mechanical adjustment of focus with the measured optical defocus, thereby eliminating the system drift and improving system stability during the long-time measurement. High-precision axial focusing also ensures the smallest size of the focused spot on the sample surface at each scanning position. Moreover, the in situ topography of the sample can be obtained by the optical defocus.

The constant shift us directly determines the slope of the subtracted curve ID at u = us/2. Therefore, the value of the constant shift us is evaluated to ensure that the curve ID has the maximum slope at u = us/2, that is, the axial focusing sensitivity of DCBM is the best at u = us/2. The curve ID can be expressed as:

$$\begin{array}{c} { {{I_D}({v_x},{v_y},u,u{}_s)} |_{{v_x}\textrm{ = }{v_y}\textrm{ = 0}}}\textrm{ = }{\left|{\int_{ - \frac{\pi }{2}}^{\frac{\pi }{2}} {\int_0^{\cos \theta } {P({\rho ,\theta } )\exp \left[ {\frac{{i({u - {u_s}} ){\rho^2}}}{2}} \right]\rho d\rho d\theta } } \times \int_{\frac{\pi }{2}}^{\frac{{3\pi }}{2}} {\int_0^{ - \cos \theta } {P({\rho ,\theta } )\exp \left[ {\frac{{i({u - {u_s}} ){\rho^2}}}{2}} \right]\rho d\rho d\theta } } } \right|^2} - \\ {\left|{\int_{ - \frac{\pi }{2}}^{\frac{\pi }{2}} {\int_0^{\cos \theta } {P({\rho ,\theta } )\exp \left( {\frac{{iu{\rho^2}}}{2}} \right)\rho d\rho d\theta } } \times \int_{\frac{\pi }{2}}^{\frac{{3\pi }}{2}} {\int_0^{ - \cos \theta } {P({\rho ,\theta } )\exp \left( {\frac{{iu{\rho^2}}}{2}} \right)\rho d\rho d\theta } } } \right|^2} \end{array}$$

In which, ρ and θ are the normalized polar coordinates in the objective pupil plane, and P (ρ,θ) is the pupil function of the objective. The slope k(u,us) of the curve ID at u = us/2 can be expressed as:

$$k({u,u{}_s} )\textrm{ = }{\left. {\frac{{\partial {I_D}(0,0,u,u{}_s)}}{{\partial u}}} \right|_{u = \frac{{{u_s}}}{2}}}$$

According to Eqs. (3) - (4), the relationship between the constant shift us and the slope k(u,us) shown in Fig. 2 can be obtained, and it can be seen from Fig. 2 that the optimal constant shift is us= 4.

 figure: Fig. 2.

Fig. 2. The relationship between the constant shift us and the slope of the curve ID.

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2.2 Laser divided-aperture confocal Brillouin microscope

A laser divided-aperture confocal Brillouin microscope was constructed based on the principle of DCBM, as shown in Fig. 1. A single longitudinal mode laser (COHERENT Verdi G2) with a wavelength of 532 nm was used as the light source, and a microscope objective (OLYMPUS LMPlanFLN 50× NA=0.5) was used as the system objective. The Brillouin scattering was acquired by a tandem multi-pass Fabry–Perot interferometer (JRS Scientific Instruments TFP-1) and a 100 µm pinhole (Newport, PH-100). A two-dimensional translation stage (Physik Instrumente P-542.2CD) and a piezoelectric transducer (PZT, Physik Instrumente P-725.4CD) were used to scan the test sample in the xy plane and along the z-axis, respectively. A 10 µm pinhole (Newport, PH-10) and a photomultiplier tube (Hamamatsu, H10723-01) were used to construct the divided-aperture confocal system.

The details of the imaging process of DCBM are as follows. For each scanning position, PZT drives the objective to scan the sample axially. During the axial scanning process, the reflected light intensity signal detected by PMT and the axial position of PZT are continuously acquired to obtain the axial response curve IC. The axial position of the sample surface deviating from the focus is obtained by shift, subtraction and fitting of the axial curves. Then PZT drives the objective to coincide the system focus with the sample surface, and the Brillouin spectrum is precisely collected by FP interferometer. After the single-point spectrum acquisition is completed, the two-dimensional translation stage moves the sample to the next scanning position, and the above spectrum acquisition process is repeated to achieve Brillouin imaging finally.

In DCBM, the single point axial focusing time is equal to twice the response time of PZT, and the response time of the PZT (P-725.4CD) is about 42.4 ms. Therefore, in our system, the single-point axial focusing time is less than 100 ms. Compare with the single acquisition time of Brillouin spectrum which is between several seconds to tens of seconds, the single point axial focusing time is negligible for the imaging time of the one tested position. Therefore, the axial focusing process of DCBM hardly affects the image acquisition time.

3. Results

3.1 Axial focusing capability

The axial focusing sensitivity is a measure of the system capability to detect the lowest intensity difference along the axis, which directly determines the axial measurement capability of 3D imaging. Therefore, a flat mirror and a single-crystal silicon wafer were used to verify the axial focusing sensitivity of DCBM and CBM. The objective was driven by the calibrated objective scanner to move axially in steps of 5 nm, and the reflected light intensity near the zero position of the curve ID measured by PMT is shown in Fig. 3(a). As the mirror was defocused, a clear step profile formed, which indicated that the axial focusing sensitivity of DCBM could reach 5 nm. Similarly, the axial focusing sensitivity of CBM was tested using the silicon wafer. Figure 3(b) shows the spectral intensity of silicon obtained by CBM, indicating an axial focusing sensitivity of 500 nm.

 figure: Fig. 3.

Fig. 3. Axial focusing sensitivity measurement results. (a) Divided-aperture confocal Brillouin microscopy. (b) Confocal Brillouin microscopy.

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Furthermore, a sample prepared by electron beam lithography was imaged by CBM and DCBM, which is a Poly methyl methacrylate (PMMA) sub-micronic pattern coated on silicon wafer. The imaging results formed by the PMMA spectral intensity are shown in Fig. 4(a) and Fig. 4(b), respectively. The image size was 120 × 60 pixels and the scanning step was 60 nm. The single spectrum acquisition time was 5 s. The total imaging time of CBM and DCBM were 36000 s and 36720 s, respectively. The additional imaging time of DCBM relative to CBM was caused by the axial focusing process. During the imaging, DCBM constantly used the confocal curve obtained by reflected light for focus tracking on the sample surface. The objective changed the axial position with the height of the sample, so that the scanning point was always located in the focal plane of the system. As DCBM ensured mechanical imaging with the smallest excitation spot size at each scanning position by high-sensitivity axial focusing, the 400 nm periodic region on the sample can be clearly imaged. In contrast, CBM could only use the microscopic image to perform axial focusing once before imaging. During the imaging process, the objective could not move dynamically with the height of the sample, that is, the focal plane of the system was unchanged. The excitation spot size continuously varied with the sample morphology and only the 600 nm periodic region could be resolved. Thus it can be seen from Fig. 4 that, the better imaging effect of DCBM is a benefit of the improved focusing capability.

 figure: Fig. 4.

Fig. 4. Imaging measurement results. (a) Divided-aperture confocal Brillouin microscopy. (b) Confocal Brillouin microscopy.

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3.2 Anti-drift capability

The anti-drift ability is directly related to the system stability. To verify the DCBM anti-drift capability, we used CBM and DCBM to perform Brillouin imaging on an Su-8 micro-pillar array, which is a common micro-structure used in microfluidics and MEMS. Figure 5(a) shows the white light image of the Su-8 micro-pillar, and the red square indicates the scanning area. Initially, the sample was moved onto the focal plane of the system for spectral scanning. During the spectral scanning process, we gradually and uniformly moved the Z stage to defocus the sample. At the end of the scanning, the Z stage had been moved axially by 5 µm. The Su-8 Brillouin scattering intensity imaging results of CBM and DCBM are shown in Fig. 5(b) and Fig. 5(c), respectively. The image size was 240 × 240 pixels, and the scanning step was 1 µm. The acquisition time of single spectrum was 10 s. The mirror spacing and the mirror scanning range of FP interferometer were 2 mm and 250 nm, respectively.

 figure: Fig. 5.

Fig. 5. Su-8 micro-pillar images. (a) White light image. (b)-(c) Brillouin intensity images by CBM and DCBM, respectively.

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As CBM could not perform focus tracking after the system was disturbed, the sample was out of focus, and the image was degraded by defocus, resulting in a Brillouin intensity image inconsistent with the sample appearance. Moreover, the spectral intensity eventually decreased by approximately 63%. The reduction of spectral intensity shows that the Brillouin scattering may not be able to excite after sample defocusing. Since DCBM used the constructed divided-aperture confocal system for focus tracking, the sample surface remained in focus regardless of the defocus caused by the axial movement. The decrease in spectral intensity present in CBM did not occur in the experimental result obtained by DCBM. DCBM performed axial focusing through reflected light to ensure excitation intensity of Brillouin scattering. And DCBM could eliminate the drift caused by environmental disturbances and temperature changes during the long-term spectral imaging, thus demonstrating that it has superior system stability compared to CBM.

3.3 Extinction ratio performance

Figure 6(a) shows the Brillouin spectrum of chicken breast tissue detected by DCBM and CBM. The laser was focused near the tissue-glass interface at a power of 10 mW through the objective and the single acquisition time of the FP interferometer was 10 s. The mirror spacing and the mirror scanning range were 2 mm and 100 nm, respectively. The average results after ten acquisitions are shown in Fig. 6(a). As CBM simultaneously detected both reflected light and Brillouin scattered light, the detected Brillouin spectrum was affected by the reflected light near the sample interface. On the contrary, DCBM effectively eliminated the reflected light component in the spectrum through the dark-field setup, and the collected spectral center peak intensity was much lower than CBM, which greatly reduced the background light of the Brillouin spectrum.

 figure: Fig. 6.

Fig. 6. (a) Brillouin spectra of chicken breast tissue measured by CBM and DCBM. (b) Extinction ratios of CBM and DCBM.

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We placed a reflector at the objective focal plane of DCBM and CBM, respectively. The laser was focused on the reflector through the attenuator and the objective, and the reflected light and scattered light were simultaneously excited. In the two systems, we used the FP interferometer to record the spectral intensity at different spectral frequencies. In DCBM, the FP interferometer only collected the scattered light through the dark field setup, while the FP interferometer collected both the reflected light and the scattered light in CBM. The spectral intensity was divided by the corresponding laser power and multiplied by the corresponding attenuator optical densities. The results of the spectra reconstructed by rescaling the spectral intensity are shown in Fig. 6(b), which are the extinction ratios of DCBM and CBM. The mirror spacing and the mirror scanning range were 2 mm and 230 nm, respectively. Obviously, the extinction ratio of DCBM was increased by more than 20 dB compared to CBM.

3.4 Topographic imaging, mechanical mapping, and dark-field imaging performance

In the fields of semiconductor device and micro-nanoscale measurement, the distributions of the three-dimensional profile, elastic properties, and the thermodynamic parameter of a device directly determine its performance and reliability. The in situ topographic imaging and mechanical mapping of a sample are thus of significance for analyzing interface failures and the working mechanisms of micro devices. Therefore, by way of illustration, we used DCBM to perform imaging on a SG3524 chip to evaluate the DCBM imaging characteristics, which is a commonly used dual-channel adjustable pulse width modulation controller.

Figure 7(a) shows the microscopic image of the chip and indicates the scanning area of the topographic image in Fig. 7(b). The scanning range was 64 × 64 µm2 and the scanning step was 0.64 µm. Figure 7(c) shows the Brillouin spectrum of the point A shown in Fig. 7(a), and the spectral fitting results by considering the aperture broadening and the system convolution [36,37]. Hence, by fitting the spectrum at each scanning point, the in-situ Brillouin frequency shift mapping shown in Fig. 7(d) can be obtained.

 figure: Fig. 7.

Fig. 7. Images of SG3524 chip obtained by DCBM. (a) Microscopic image. (b) Topographic image. (c) Brillouin spectrum of point A. (d) Brillouin frequency shift mapping. (e) Dark-field image. (f) Fusion image of topographic image and Brillouin mapping.

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Here, the Brillouin frequency shifts corresponding to the three regions of red, green, and blue in Fig. 7(d) are 33.70 ± 0.13 GHz, 18.44 ± 0.07 GHz, and 15.28 ± 0.09 GHz, respectively. Through the Brillouin shift, density, and refractive index, the mechanical parameter information such as the elastic modulus of the sample can be directly obtained. These parameters are closely related to the mechanical and thermodynamic properties of the sample. Accurate measurement can reveal the mechanism of the chip, and is of great significance for microstructure design and analysis.

Figure 7(e) shows the dark-field image obtained by DCBM through the detected Rayleigh scattered light intensity. Since the crosstalk of the reflected light was eliminated, the Rayleigh scattered light changed significantly at the edges and the miniscule particles, providing more details regarding the sample.

As DCBM uses the same excitation spot to perform confocal imaging and Brillouin mapping simultaneously, the topographic information in Fig. 7(b) and the spectral information in Fig. 7(d) correspond by pixel to pixel. By fusing the information of the two images, we could obtain a fusion image as shown in Fig. 7(f), which contains topographic information and mechanical information. The value of the coordinate axis represents the three-dimensional structure of the sample, and the color distinguishes Brillouin frequency shifts in different regions. Besides, we can also refocus the excitation spot onto the sample surface by the measured optical defocus and closed loop feedback, thereby ensuring that the DCBM performs high-precision Brillouin mapping on the complex topography sample and in the long-time imaging process.

4. Discussion

Currently, in the research fields such as materials science and biomedical science, a confocal Brillouin microscope with high performance is urgently needed to enable the high-precision mechanical mapping for biological detection and material analysis. However, current CBMs could not achieve rapid and high-sensitivity focus tracking on the sample surface during the imaging process. The defocus caused by the three-dimensional topography of the sample would therefore reduce the imaging quality and spectral intensity. Meanwhile, current CBMs also lack rapid focusing capability to improve system stability during the long imaging process. Besides, conventional CBMs improve the extinction ratio by eliminating the reflected light. As a result, only the viscoelastic information of the sample can be obtained.

We demonstrate that in situ reflected light and the Brillouin scattered light are spatially separated by divided aperture technology. In contrast to conventional CBMs, the separated reflected light achieves nanometer-level axial focusing sensitivity by shifting the confocal axial curve. The axial focusing sensitivity and lateral resolution could reach 5 nm and 400 nm, respectively, as shown in Fig. 3 and Fig. 4. We also use the obtained optical defocus of the sample to perform mechanical feedback and suppress the axial drift of CBM during the long time-consuming imaging, as demonstrated in Fig. 5. Additionally, the extraction of the Brillouin scattered light from the reflected light also significantly increases the extinction ratio, making our system more suitable for mechanical mapping of biological tissue samples with multiple layers, which is verified in Fig. 6.

As DCBM fully utilizes the reflected light abandoned by traditional CBMs, not only the imaging performance is improved, but also the sample information obtained is highly increased. We also show that DCBM can simultaneously obtain topography and Brillouin data, as shown in Fig. 7. Such an optical setup enables accurate and precise in situ three-dimensional topography imaging, high extinction ratio mechanical mapping, and dark-field imaging, which is of great importance to the imaging of nanoscale mechanical features, especially for non-flat samples in the field of thin-film coating and chip manufacturing and testing process. For example, the in situ topography and elasticity information of the designing materials for biology and medicine can be simultaneously obtained by DCBM, which will contribute to a better understanding of the biochemical functions of materials.

5. Conclusion

In conclusion, we have demonstrated a divided-aperture confocal Brillouin microscopy featuring high axial focusing capability and high stability. Specifically, the axial focusing sensitivity has been improved to 5 nm. In addition, the specular reflections have effectively been reduced by the dark-field setup, which has significantly enhanced the extinction ratio by 20 dB. Furthermore, DCBM has achieved in situ topographical imaging, Brillouin mapping, and dark-field imaging simultaneously, which can provide accurate and information-rich images, thereby elucidating new pathways for applying Brillouin microscopy to the fields of materials engineering and bio-imaging.

Funding

National Natural Science Foundation of China (51535002, 61635003).

Disclosures

The authors declare no conflicts of interest.

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Figures (7)

Fig. 1.
Fig. 1. Schematic of the DCBM system. The beam emitted from a laser source passes through a polarizer, a beam splitter (BS), an aperture, and is focused onto the sample by an objective. The scattered light (S) is transmitted along the original light path, focused on a pinhole (PH1) by a lens, and finally detected by a Fabry–Perot interferometer. Simultaneously the reflected light (R) is collected in the symmetrical direction of the illumination light path through the aperture and is focused on a second pinhole (PH2) through an attenuator (ATT) and a lens, and then detected by a photomultiplier tube (PMT). IC is the confocal axial response curve obtained by PMT, IS is the curve shifted by IC, ID is the curve after subtraction of IS and IC, LD is the fitting line of ID, us is the shift offset of curve IS relative to curve IC, and u is the normalized axial coordinate.
Fig. 2.
Fig. 2. The relationship between the constant shift us and the slope of the curve ID.
Fig. 3.
Fig. 3. Axial focusing sensitivity measurement results. (a) Divided-aperture confocal Brillouin microscopy. (b) Confocal Brillouin microscopy.
Fig. 4.
Fig. 4. Imaging measurement results. (a) Divided-aperture confocal Brillouin microscopy. (b) Confocal Brillouin microscopy.
Fig. 5.
Fig. 5. Su-8 micro-pillar images. (a) White light image. (b)-(c) Brillouin intensity images by CBM and DCBM, respectively.
Fig. 6.
Fig. 6. (a) Brillouin spectra of chicken breast tissue measured by CBM and DCBM. (b) Extinction ratios of CBM and DCBM.
Fig. 7.
Fig. 7. Images of SG3524 chip obtained by DCBM. (a) Microscopic image. (b) Topographic image. (c) Brillouin spectrum of point A. (d) Brillouin frequency shift mapping. (e) Dark-field image. (f) Fusion image of topographic image and Brillouin mapping.

Equations (4)

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I D ( v x , v y , u , u s ) = I S ( v x , v y , u , u s ) I C ( v x , v y , u )
L D ( v x , v y , u , u s ) = f ( I D ( v x , v y , u , u s ) )
I D ( v x , v y , u , u s ) | v x  =  v y  = 0  =  | π 2 π 2 0 cos θ P ( ρ , θ ) exp [ i ( u u s ) ρ 2 2 ] ρ d ρ d θ × π 2 3 π 2 0 cos θ P ( ρ , θ ) exp [ i ( u u s ) ρ 2 2 ] ρ d ρ d θ | 2 | π 2 π 2 0 cos θ P ( ρ , θ ) exp ( i u ρ 2 2 ) ρ d ρ d θ × π 2 3 π 2 0 cos θ P ( ρ , θ ) exp ( i u ρ 2 2 ) ρ d ρ d θ | 2
k ( u , u s )  =  I D ( 0 , 0 , u , u s ) u | u = u s 2
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