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Abstract
In order to stabilize the extinction cross section measurement of a single nanoparticle, we propose to analyze the blurriness parameter of aperture edge images in real time, which provides a feedback to lock the sample position. Unlike the conventional spatial modulation spectroscopy (SMS) technique, a probe beam experiences both the spatial modulation by a piezo stage and the temporal modulation by a chopper. We experimentally demonstrate that the measurement uncertainty is one order magnitude less than that in the previous report. The proposed method can be readily implemented in conventional SMS systems and can help to achieve high stability for sensing based on light extinction by a single nanoparticle, which alleviate the impact from laboratory environment and increase the experimental sensitivity.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Single nanoparticles (NPs) exhibit significantly different optical responses in comparison with the cluster assemblies [1–4]. For example, the scattering cross section of a single subwavelength NP can be arbitrarily large by creating the degeneracy of local resonances, which is referred to as super-scatterings [5,6]. To study and take advantages of these local resonance characters, the techniques to measure the scattering or absorption properties of a single NP have drawn much attention in the past decades, ranging from near-field to far-field optical techniques, including near-field optical microscope technique [7], dark-field microscopy [8–10], photothermal measurements [11–14], transient absorption [15–17], direct absorption [18–20], and spatial modulation spectroscopy [21,22].
In particular, the spatial modulation spectroscopy (SMS) technique provides a solution to measure the optical spectra of a single NP with size down to nanometer scale [22–28]. The SMS technique is based on modulation of the particle position located in a tightly focused beam by a piezo stage, and a lock-in amplifier is used to measure the amplitude of light modulation by particle displacement. This method has been experimentally demonstrated to measure the extinction cross section of a single NP with background suppression and high signal-to-noise ratio, for both metallic and dielectric NPs [23,24,29–32]. However, the measurement stability is significantly affected by the field distribution variation of focused spot [29]. Conventionally, since the probe beam has a narrow waist under tight focus, measurement instability occurs when the sample tends to drift from the focal plane due to environmental disturbances. In particular, temperature fluctuation can induce microscope stage movement along the optical axis with a distance larger than the Rayleigh range under tight focus [33]. Also it is important to maintain stability of sample position in the case of a small NP with extremely weak signal, because the time constant of the lock-in amplifier needs increase and the measurement period of extinction cross section spectrum becomes longer.
In this paper, we design a feedback-lock system to stabilize the sample plane during the measurement of extinction cross section. In recent years, there have been different approaches for stabilizing sample surfaces by nanoscale distance detection, such as image-based correlation method [34], video-imaging analysis [35,36], reflection interference contrast microscopy [37,38], autofocusing in microscopy [39,40], laser-based back-focal-plane detections [41,42], drift tracking using fluorescent or scattering fiducial markers [43–45]. These methods are designed for super-resolution microscopy techniques, single-molecule experiments, and other nanoscale imaging and measurement systems [34,43], where wavelength filters can be used to separate the beam for stabilizing the sample from the one for specific measurement. However, since the SMS technique usually measures the cross section over a broad wavelength range from visible to infrared region, such a filter scheme is not compatible with the SMS method.
In our proposed feedback-lock technique, a probe beam and a monitoring one illuminate samples simultaneously. For the feedback, during the measurement, we analyze the blurriness of aperture edge images with the monitoring beam to lock the sample plane and stabilize the measurement. The probe beam is spatially modulated with vibration by a piezo stage and temporally modulated by a chopper, while the monitoring one only experiences the spatial modulation. In order to extract the probe signal, two different approaches are proposed to measure the extinction cross section by lock-in detecting the signals at the temporal modulation frequency and at the differential frequency between the spatial and temporal modulations, respectively. The proposed method can be readily implemented in current SMS systems and has no limitation on the measured operation wavelength range.
Indeed, we can reduce the impact of environmental disturbance and realize a highly stable measurement of the extinction cross section of a single NP. As an experimental demonstration, we measure the extinction cross section of a single 150-nm-diameter gold nanocube at the wavelength $0.6328 ~\mu \mathrm {m}$. By the proposed technique with the feedback-lock monitoring, the drift of the sample is down to $17 ~ \mathrm {nm}$, and the uncertainty of the extinction cross section is only $0.92 \%$, one order magnitude less than the measurement uncertainty of $3.77\%$ in the previous report [21]. Such experimental results also agree well with the numerical simulation for uncertainty analysis.
2. Stable measurement technique for the extinction cross section
Figure 1 schematically shows a proposed system, where a probe beam from a laser source experiences both spatial and temporal modulations. We note that here we only consider a reflection scheme, and it is readily extended to the transmission scheme. A sample is mounted on a piezo stage, which is fixed on an inverted microscope frame, and the probe beam is tightly focused on the sample surface by an objective. A laboratory coordinate system $(x, y, z)$ is defined and shown in Fig. 1, where $z$ is the optical axis perpendicular to the sample surface. Since the probe beam has a narrow waist under the tight focus, measurement instability occurs when the sample tends to drift under environmental disturbances. We note that by observing the center position of the probe beam, the piezo stage is adjusted to horizontal and ensure that the drift is just along z axis.
Fig. 1. Experimental setup: The sample is mounted on a piezo stage, which is fixed on an inverted microscope. The yellow and the red beams represent the monitoring and the probe beams, respectively. The monitoring beam illuminates an aperture (A) and images on the sample plane. The probe beam is modulated by a chopper (CH) and a piezo stage. The beams illuminate on the sample plane by an objective (Obj), where the aperture is imaged on a camera, and the probe beam is detected by a photodiode and sent to a lock-in amplifier (Lock-in). The lock-in reference frequency $f_{\textrm{vib}}$ is generated internally and is synchronized with a digital piezo controller (DPC), while the reference frequency $f_{\textrm{chop}}$ is generated by a chopper controller (CHC). Here $\delta _x$ is the vibration amplitude of spatial modulation, and $(x, y, z)$ denotes the laboratory coordinate, where $z$ is the optical axis perpendicular to the sample surface.
2.1 Feedback of sample plane position by analyzing edge blurriness of the aperture image
When the sample plane drifts from the focal plane, we realize the feedback-lock of the sample plane by analyzing the varied blurriness of aperture images. Here we suppose that the probe beam tightly focuses on the plane of $z=0$ [the red profile in Fig. 2(a)], and $z_{\mathrm {s}}$ is the drift distance of the sample plane. We note that when the sample plane drifts from the $z=0$ plane to either sides, the blurriness of the aperture image would change [see Figs. 2(b)-(d)]. Therefore, we can detect the drift and control the piezo stage to move the sample toward the reverse direction until it is backed to the focus of the probe beam.
Fig. 2. (a) Schematic of varied blurriness of aperture images, where the probe beam tightly focuses on the plane of $z=0$, and $z_{\mathrm {s}}$ is the drift distance of the sample plane. As the sample plane drifts from the focus plane, the edges of octagonal aperture images change from sharp (c) to blur (b) or (d), respectively.
In order to quantitatively analyze the image blurriness, we consider the image of a one-dimension edge under the illumination of an incoherent light source. In this case, the intensity distribution around the aperture edge can be given as
where $\operatorname {erf}$ denotes the error function, $a$ is the center position of the edge, and $k$, $h$ are the parameters related to the intensity of light source and the background, respectively. The parameter $B$ is defined as the blurriness, which is reversely proportional to the spread parameter of a point spread function [46–48]. More importantly, $B$ is independent of the light source intensity. Therefore, $B$ can be retrieved by normalizing the intensity of the aperture image near the edge and fitting with Eq. (1). For example, Figs. 3(a)-(e) are the left edges of normalized octagonal aperture images, which are experimentally captured by the camera when the piezo stage moves to five equally spaced planes (see the details in Sec. 4.1). Figures 3(f)-(j) show the averaged experimental data along $y$ of (a)-(e) (red dotted lines) and the fitting results (blue solid ones) with Eq. (1), respectively, where the corresponding $B$ values are shown in the corners. The good agreement between the experimental and the fitting results indicates that $B$ can be used as a quantitative parameter for feedback-lock monitoring.Fig. 3. Feedback of the sample plane positions by analyzing edge blurriness of aperture images. (a-e) Five edge images of an octagonal aperture captured by the experimental setup (see the details in Sec. 4.1), when the piezo stage moves to five different positions. (f-j) The averaged experimental data along $y$ of (a-e) (red dotted lines) and the fitting results (blue solid ones) with Eq. (1) with the corresponding $B$ values shown in the corners. (k) The experimental results of the parameter $B$ versus $z_{\mathrm {s}}$ (blue dotted line) where the five red circles indicate the $B$ results corresponding to (f-j), respectively, and the fitting results with Eq. (2) (blue solid line). The green solid line corresponds to the theoretical sensitivity by computing $d B / d z_{\mathrm {s}}$, while the green dotted ones are the experimental results.
In order to enable highly sensitive feedback, we further analyze the sensitivity of the blurriness parameter $B$ as a function of sample position $z_{\mathrm {s}}$. Based on the incoherent image theory [46,47], our analysis shows that the blurriness parameter $B$ is given as
2.2 Two approaches to measure the extinction cross section
We note that since the monitoring beam has a wavelength range which covers the wavelength of the probe beam. In order to extract the probe signal, in addition to the spatial modulation by the piezo stage, we use a temporal one only on the probe beam with a chopper. The spatial and temporal modulation frequencies $f_{\textrm{vib}}$ and $f_{\textrm{chop}}$ are generated by the lock-in amplifier and the chopper controller, respectively.
Let us first consider the case where the probe beam is only modulated by the chopper. Since the particle size is small enough in comparison with the probe beam waist, according to the local approximation of extinction [21,31,49], the output power for detection at the frequency $f_{\textrm{chop}}$ is
Here $\sigma _{\textrm {ext}}$ is the extinction cross section under linear optical response, and $P_{\mathrm {r}}$ is the output power for the background case where the particle moves out of the probe beam, i.e. when ${\sigma _{\textrm {ext}}}=0$. $(x, y)$ is the swept measuring position. $I_{N}$ is the normalized intensity profile of the probe beam such that $\iint I_{N}(x, y) d x d y=1$. Conventionally, $I_{N}$ can be directly obtained by measuring the intensity of the probe beam with a camera [21]. In an alternative way, when $\sigma _{\textrm {ext}}$ is relatively large and induces an observable reduction of the output power in comparison to the background fluctuations, $I_{N}$ can be obtained by measuring $P_{f_{\textrm {chop}}}$ through scanning the sample plane and normalizing the reduction of the output power, because such a reduction is proportional to $I_{N}$ according to Eq. (3) [29]. Therefore, in this case, $\sigma _{\textrm {ext}}$ is evaluated from the ratio of the reduction of the output power $P_{f_{\textrm {chop}}}$ to $I_{N}$.
If the reduction of the output power $P_{f_{\textrm {chop}}}$ is too weak to measure due to low signal-to-noise ratio, the second approach is proposed that the probe beam is modulated simultaneously with both spatial and temporal modulations. Considering only the first order harmonic temporal modulation, the signal is written as
From Eq. (4), $\sigma _{\textrm {ext}}$ is obtained from the signal $P_{f_{\textrm {chop}}-f_{\textrm {vib}}}$ at each position. We note that in the conventional SMS method the lock-in frequency is $f_{\textrm {vib}}$ only related to the spatial modulation [21,22,24–26,29]. Unlike the conventional SMS, here the two-beam temporal and spatial modulation technique is proposed to detect at the differential frequency between spatial and temporal modulations, such that we can extract the probe signal from the monitoring beam.
3. Theoretical analysis of measurement uncertainty
To estimate the performance of feedback-lock procedure, we theoretically analyze the measurement uncertainty of extinction cross section $\sigma _{\textrm {ext}}$ due to the fluctuation of $z_{\mathrm {s}}$. Here we define the measurement uncertainty as $\eta =\Delta \sigma _{\textrm {ext}} / \sigma _{\textrm {ext0}}$, where $\sigma _{\textrm {ext0}}$ is the accurate result, and $\Delta \sigma _{\textrm {ext}}$ is the absolute uncertainty estimated by standard deviation. $\eta$ are simulated for different Gaussian beam waists $w_{\textrm {probe}}$ and initial sample positions $z_{\mathrm {s} 0}$, assuming that $z_{\mathrm {s}}$ varies with the normal random probability distribution $N\left (z_{\mathrm {s} 0}, \Delta z_{\mathrm {s}}\right )$ around the initial sample plane position, where $\Delta z_{\mathrm {s}}$ is the uncertainty of sample position. The details for the simulation and the code are given in Supplement 1.
Figures 4(a)-(c) are the simulation results of the measurement uncertainty $\eta$ for $\Delta z_{s}=0.32 \lambda , 0.16 \lambda , 0.027 \lambda$, respectively. As expected, they show that $\eta$ is reduced for large beam waist and small sample drift uncertainty. Given the beam waist $w_{\textrm {probe}}=0.525 \lambda$, Fig. 4(d) compares the three cases, which shows that the measurement uncertainty can be dramatically reduced by stabilizing the sample position. For example, when $\Delta z_{s}=0.027 \lambda$, the measurement uncertainty is less than $1.6 {\%}$, when the initial position is within $(-0.3 \lambda , 0.3 \lambda )$.
Fig. 4. (a-c) Measurement uncertainties of extinction cross section $\eta$ as a function of the initial sample positions $z_{\mathrm {s} 0}$ and Gaussian beam waists $w_{\textrm {probe}}$, under three different drift uncertainties $\Delta z_{s}=0.32 \lambda , 0.16 \lambda , 0.027 \lambda$ respectively. (d) Given the Gaussian beam waist $w_{\textrm {probe}}=0.525 \lambda$, the red, green, and blue lines correspond to the results in (a)-(c), respectively.
4. Experimental demonstration
We experimentally demonstrate the proposed technique by measuring the extinction cross section of a single gold NP. In order to show the measurement stability, we specifically choose a relatively large nanoparticle, so that we can use both two approaches through Eq. (3) and Eq. (4) to evaluate $\sigma _{\textrm {ext}}$. In our experimental demonstration, for the reflection scheme, a silica substrate is coated with 200-nm-thick gold, and 150-nm-diameter gold nanocubes are sparsely distributed on the surface. We randomly choose a nanocube and the inset of Fig. 5(c) shows the image of the measured single nanocube by scanning electron microscope (SEM). Here the monitoring beam is from an LED lamp, and the probe light source is a collimated super continuous laser (YSL SC-5) through a 1-nm-bandwidth filter at $\lambda =0.6328 ~\mu \mathrm {m}$, whose each pulse has $100~\mathrm {ps}$ width and $2~\mathrm {pJ}$ energy. Both the monitoring beam and the probe beam have a rather low power density, and therefore the nonlinear response of the gold NP is negligible [50]. The sample is fixed on a piezo stage (PI P-733.3 CD, controlled by a digital piezo controller, PI E-727). The beams pass through an objective (100x, numerical aperture 0.90) and illuminate the sample.
Fig. 5. Experimental results for detection at the frequency (a) $f_{\textrm {chop}}$ and (b) $f_{\textrm {chop}}-f_{\textrm {vib}}$, respectively. (c) Experimental normalized intensity profile $I_{N}$ at ${y}=0~\mu \mathrm {m}$ (blue dotted line) and the fitting result with an ideal Gaussian (blue solid one). Experimental result of $P_{f_{\textrm {chop}}-f_{\textrm {vib}}} / P_{\mathrm {r}}$ (green dotted one) and the fitting result with Eq. (4) (green solid one). The inset shows the SEM image of the measured nanocube on the gold-coated substrate.
4.1 Feedback-lock monitoring
We first estimate the dynamic range of the sample plane with the proposed feedback-lock monitoring. The monitoring beam from an LED lamp illuminates the aperture, which is detected by a CMOS camera (Thorlabs DCC-1545M). By moving the objective, the probe beam is initially focused on the sample, and the initial sample position is estimated as $\left |z_{\mathrm {s} 0}\right | \leq 0.05~ \mu \mathrm {m}$. Then the aperture position is adjusted for the highest sensitivity. For each fixed aperture position, by moving the sample at different $z_{\mathrm {s}}$ positions, the edge of the aperture image is fitted with Eq. (1) to obtain the parameter $B$. Figures 3(a)-(e) and (f)-(j) show different aperture-edge images and fitting results at five different $z_{\mathrm {s}}$. The fitting parameters $B$ as a function of $z_{\mathrm {s}}$ are shown as the blue dotted line in Fig. 3(k). By fitting with Eq. (2), shown as the blue solid line in Fig. 3(k), the result exhibits good fitting agreement with $z_{\max }=0.280~ \mu \mathrm {m}$, $R=2.53~ \mu \mathrm {m}^{-1}$ and $\sigma _{\mathrm {g}}=10.456 ~\mu \mathrm {m}$. Meanwhile, the green dotted and green solid lines are the sensitivity $d B / d z_{\mathrm {s}}$ from the experimental results and the theoretical ones with the fitting parameters, respectively. The good agreement indicate the validation of Eqs. (1) and (2). More importantly, according to the sensitivity analysis in Sec. 2.1, it is observed that here $z_{\max } \approx \sqrt {2} / 2 R$, which means that we almost achieved the optimization case. In this case, the maximal sensitivity is $d B / d z_{\mathrm {s}}=0.1008~(\mu \mathrm {m} \cdot \mathrm {pix})^{-1}$. A step-by-step guide for feedback-lock process is given in Supplement 1.
4.2 Measurement of the extinction cross section
We measure the extinction cross section with our feedback-lock monitoring. Here the time constant of the lock-in amplifier (Stanford Research, SR865) is set as $300 ~\mathrm {ms}$. The temporal modulation frequencies of the chopper (Stanford Research, SR540) and the spatial modulation of the piezo stage are $f_{\mathrm {chop}}=320 ~\mathrm {Hz}$ and $f_{\mathrm {vib}}=40 ~\mathrm {Hz}$, respectively. In order to show the measurement stability, we evaluate $\sigma _{\textrm {ext}}$ using the two approaches through Eqs. (3) and (4), respectively. The photodetector is homemade with a silicon photodiode (S5973).
With the first approach, the waist of the probe beam is measured by sweeping the sample in the x-y plane. With the reference frequency $f_{\textrm {chop}}$, $P_{f_{\textrm {chop}}} / P_{\mathrm {r}}$ is measured and shown in Fig. 5(a). According to Eq. (3), the beam waist $w_{\textrm {probe}}$ can be deduced by fitting with a Gaussian profile from the reduction of the output power $P_{f_{\textrm {chop}}} / P_{\mathrm {r}}$. The blue dotted and solid lines in Fig. 5(c) correspond to the normalized intensity profile $I_{N}$ at ${y}=0~\mu \mathrm {m}$ and the fitting result of an ideal Gaussian profile with the beam width $w_{\textrm {probe}}=0.332 ~\mu \mathrm {m}=0.525 \lambda$ (the Rayleigh range is $0.547~\mu \mathrm {m}$), respectively. According to Eq. (3), the result is evaluated as $\sigma _{\mathrm {ext}}=0.0816~ \mu \mathrm {m}^{2}$.
With the second approach, the spatial modulated signal is detected with the reference frequency $f_{\textrm {chop}}-f_{\textrm {vib}}$. The spatial modulation is set as $\delta _{x}=0.273 ~\mu \mathrm {m}$ (about $0.8 w_{\textrm {probe}}$) for a high signal-to-noise ratio [21]. Figure 5(b) shows the experimental result $P_{f_{\textrm {chop}}-f_{\textrm {vib}}} / P_{\mathrm {r}}$ and exhibits a one-peak and one-dip profile, which approximately corresponds to the first-order derivative of the Gaussian profile. In order to evaluate the extinction cross section, the experimental result at ${y}=0~\mu \mathrm {m}$ is plotted as the green dotted line in Fig. 5(c). By substituting the normalized intensity profile $I_{N}$ to Eq. (4), the experimental result is fitted with the extinction cross section $\sigma _{\mathrm {ext}}=0.0821~ \mu \mathrm {m}^{2}$ (green sold line).
By comparing the independent experimental results from the two approaches above, the feedback-lock monitoring indeed realizes the stable measurement. To further show the measurement stability, we then experimentally verify the uncertainty $\eta$ by multiple detections of $P_{f_{\textrm {chop}}-f_{\textrm {vib}}}$ at the left peak position with total measurement times ${n}=100$. The extinction cross section of each measurement is shown as the blue dotted line in Fig. 6(a), where the average value of $\sigma _{\mathrm {ext}}$ is $0.0829~ \mu \mathrm {m}^{2}$ shown as the red solid one. The standard deviation is $7.63 \times 10^{-4} ~\mu \mathrm {m}^{2}$, and $\eta$ is $0.92\%$ from the 100 times of measurements. The feedback is shown as the blurriness parameter $B=760.2 \times 10^{-4} ~\mathrm {pix}^{-1}$ (green dotted line in Fig. 6(a)) with an uncertainty $\Delta B=17.0 \times 10^{-4} ~\mathrm {pix}^{-1}$ due to the noise of our camera. According to the sensitivity measurement as Fig. 3(k), the residual sample drift can be estimated with an uncertainty of $\Delta {z_{\mathrm {s}}} \approx \Delta B/(d B / d z)=0.017~ \mu \mathrm {m}=0.027 \lambda$. From the simulation result as blue line in Fig. 4(d), the theoretical uncertainty $\eta$ is about from $0.12 \%$ to $0.53 \%$. Therefore, the experimental uncertainty is in a good agreement with the theoretical result.
Fig. 6. (a) The 100 times of experimental results with the feedback-lock monitoring. The experimental results of extinction cross section and the average value are shown as the blue dotted and red solid lines, respectively. The blurriness parameters are the green dots. (b) The 100 times of experimental results without the feedback-lock monitoring.
We also record the results without the feedback-lock monitoring, where the feedback is turned off and the piezo stage is backed to the initial z position, without any intervention of the sample drift. The 100 times of the experimental results of extinction cross section are shown as the blue dotted line in Fig. 6(b), with a standard deviation of $0.0148~\mu \mathrm {m}^{2}$ and a measurement uncertainty $\eta$ of $17.90 \%$. During the measurement, the blurriness parameters $B$ are also recorded and shown as the green dotted line in Fig. 6(b) with a high uncertainty as large as $\Delta B=199.2 \times 10^{-4}~\mathrm {pix}^{-1}$. The corresponding uncertainty of sample drift can be estimated as $\Delta {z_{\mathrm {s}}} \approx 0.202~ \mu \mathrm {m}=0.32 \lambda$. Correspondingly, the red line in Fig. 4(d) shows that the theoretical $\eta$ is about $11.67 \%$. Consistently, the experimental uncertainty is on the same order with the theoretical one. Comparing the cases with and without the feedback-lock monitoring, Fig. 6 shows that our proposed technique indeed reduces the impact of environmental disturbances and realizes a highly stable measurement of the extinction cross section of a single NP.
5. Discussion and conclusions
We further discuss the trade-off between the measurement stability and the feedback time. With the proposed feedback-lock monitoring, the residual measurement uncertainty caused by the residual sample drift of z-axis is mainly affected by the detection noise of the CMOS camera. The noise can be reduced by averaging multiple frames, while it takes rather long feedback time.
Since the proposed method can stabilize the measurement of extinction cross section, it is greatly helpful for the sensing applications of single NPs. For examples, the extinction cross section spectrum of a single NP can be exploited to identify biomolecules [51–54] or organic groups [55] and characterize chemistry reactions [56]. Our method provides stable measurements to alleviate the impact from laboratory environment and increase the experimental sensitivity.
In conclusion, we propose a highly stable method for measuring the extinction cross section of a single nanoparticle. We analyze the blurriness parameter of aperture edge images for a feedback-lock monitoring to stabilize the sample plane. Based on the spatial and temporal modulations and the lock-in measurement, the monitoring beam and the probe beam can be separated. Therefore, such a method can be readily implemented in current SMS systems.
Funding
National Natural Science Foundation of China (91850108, 61675179); National Key Research and Development Program of China (2017YFA0205700); Open Research Program of Key Laboratory of 3D Micro/Nano Fabrication and Characterization of Zhejiang Province; State Key Laboratory of Modern Optical Instrumentation; Vetenskapsrådet (2016-03911).
Acknowledgement
M.Y. would like to acknowledge support from Vetenskapsrådet through project No. 2016-03911.
Disclosures
Z.R., Q.Y., and J.Z. are named inventors on a number of patent applications related to this work.
Data availability
Data underlying the results presented in this paper are available in Ref. [57].
Supplemental document
See Supplement 1 for supporting content.
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