1. Introduction

Optical microscopy, since its invention 400 years ago, has become an essential tool in many disciplines. When considering optical imaging, there are some important factors, including contrast, resolution, magnification, imaging speed, penetration depth, noninvasiveness, etc. Among them, contrast is arguably the most important one. Without contrast, nothing can be observed no matter how good other factors are. During the last century, most of the major developments in optical microscopy are related to contrast, such as dark field, phase contrast, differential interference contrast, fluorescence, etc [1].

During the last decade, there were significant breakthroughs in resolution of optical microscopy, relying on the nonlinearities of fluorescence [2–7], such as switching and saturation. To avoid the issue of photobleaching with fluorescence, we recently discovered saturable scattering in an isolated plasmonic particle. Combined with saturated excitation (SAX) microscopy technique [7], our discovery provides a new contrast agent for superresolution microscopy, with full-width-at-half-maximum (FWHM) of point spread function (PSF) reduced to λ/8 [8]. In addition, we also found that scattering from a single plasmonic particle exhibits reverse saturation behavior at higher excitation intensities [9], where very narrow side lobes and significantly reduced width of main lobe in the PSF are observed, on a single particle basis. The nonlinear behavior of scattering is suggested to be similar to saturable and reverse saturable absorption, which is one of the most studied nonlinearities in plasmonics [10–13].

It will be very interesting to extract the reversed saturated part and to further reduce PSF width by separating the side lobes from the main lobe. However, such separation is not possible with conventional confocal microscopy scheme. Here we explore the combination of reverse saturable scattering and SAX microscopy, which provides the capability to extract nonlinear parts of the detected signals, and thus to enhance resolution with scattering contrast. In our results, we found that the intensity dependencies of SAX signals based on saturable and reverse saturable scattering exhibit multiple dips, showing very different features from SAX signals based on saturation of fluorescence. To explain the origin of these dips, high-order nonlinearities are introduced into our theoretical treatment. Excellent correspondence was found between theory and experiment, including slope variation in the intensity dependence, and phase flip when SAX signals crossing the dips. By quantitative amplitude and phase analyses of different SAX frequency components, exceptional PSF width reduction is found after each dip, showing the potential of this technique.

2. Experimental setup

At the focal point of a laser scanning confocal microscope, a three-dimensional PSF is formed by constructive interference. The intensity at the center of the PSF is higher than that at the peripheral, so if the excited signals exhibit saturation, the saturation starts from the center. Therefore, spatial resolution can be effectively enhanced by extracting the saturated part. That’s the main concept behind SAX microscopy, in which the saturated part is extracted by adding a temporal sinusoidal modulation onto the excitation, and by performing Fourier transform to the emitted signals. When no saturation occurs, there is only one peak with fundamental modulation frequency in the Fourier domain. However, when signals start to saturation, higher harmonics in the Fourier domain appear. By collecting these higher harmonics, the non-saturated region is filtered out, and thus resolution is enhanced.

The experimental setup, which is shown in Fig. 1, is basically the same as that in the previous work [8]. Here the excitation wavelength is 532 nm, and 80-nm gold nanospheres (GNSs) are used. Two acousto-opitc modulators (AOMs) (AOM-402AF1, IntraAction Corp.) are used to produce sinusoidal modulation. The modulation frequencies of the two AOMs are 40.00 and 40.01 MHz, respectively. By splitting the laser to pass through each AOM and recombining them afterwards, modulation at beating frequency (10kHz) is obtained. This frequency is sent as a reference input to a lock-in amplifier (HF2LI, Zurich Instrument) via a photodiode. The modulated excitation goes through a laser-scanning confocal microscope, and the backscattered signals from a GNS are collected by a photomultiplier tube (PMT). The signals are then sent to the lock-in amplifier to extract signals at different harmonics of modulation frequencies. By synchronizing laser-scanning system with the lock-in amplifier, two-dimensional images are formed with different harmonic signals.

 

Fig. 1 Experimental setup of the SAX microscope. The blue square is the part for generating temporal modulation, and the red square shows the part of laser-scanning confocal microscope.

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3. SAX signal extraction of saturable and reverse saturable scattering

Figure 2(a) shows the intensity dependency of scattering from a single gold nanoparticle. When the excitation intensity is less than 1 MW/cm², the scattering intensity is linearly dependent on the excitation. However, when the excitation increases over 1 MW/cm², the scattering intensity deviates from linear trend, showing saturation behavior. Furthermore, when the excitation intensity reaches 2.2 MW/cm², the scattering increases again, showing reverse saturable scattering [9].

 

Fig. 2 (a) Intensity dependency of scattering versus excitation from a 80-nm GNS. (b) Intensity dependencies of 1f_m, 2f_m, and 3f_m SAX signals. Each data point is averaged from four 80-nm GNS. S means local slope, which is sketched by blue dotted lines. The purple Ds indicate the location of dips.

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To extract the nonlinear parts of scattering, SAX technique is applied, and the SAX curves are shown in Fig. 2(b). From Fig. 2(b), there are two interesting observations. First, the curves of 2f_m and 3f_m are not smooth, with several dips (D_2f and D_3f) at specific intensities along the curves. This result is not expected since there is no dip in the SAX curves from fluorescence in our previous report [7]. Second, the slopes change with excitation intensity. When excitation intensity is less than 0.5 MW/cm², the slopes of 1f_m, 2f_m, and 3f_m are 1, 2, and 3, respectively. However, after each dip, the slopes become larger. It is known that the slopes of signals in log-log scale reflects nonlinearity orders, and the higher the nonlinearity, the better is the imaging resolution, so the results indicate that the resolution of SAX images will increase after every dip.

4. Theoretical analysis

To find out the origin and the properties of dips in the SAX signals, we have to establish the relationship between nonlinear coefficients of intensity dependence and the amplitudes of harmonic frequencies in Fourier domain. Quantitatively, the nonlinear dependency of scattering signal S(I) can be expanded in a polynomial series, which is similar to the definitions in nonlinear absorption [12],

The nonlinear coefficients α, β, γ can be obtained by fitting the intensity dependency in Fig. 2(a), as shown by the inset of Fig. 3(a). Then the amplitude of each higher frequency components can be deduced, as presented in Fig. 3(a). Obviously, although both saturation and reverse saturation of scattering can be obtained with the third-order equation, it shows only one dip in 2f_m signal, and no dip in 3f_m signal. However, there are at least two dips in both 2f_m and 3f_m signals in Fig. 2(b).

 

Fig. 3 Calculated intensity dependence of 1f_m, 2f_m, and 3f_m SAX signals based on (a) third-order polynomial and (b) fifth-order polynomial nonlinear models. The insets show the corresponding intensity dependency fitting of scattering signals.

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The reason is that here we only considered third-order nonlinearity, following previous χ⁽³⁾ treatment in saturable absorption [10]. Apparently, our results require additional nonlinearities. Due to the centrosymmetry of GNS, the next higher nonlinearity should be χ⁽⁵⁾. Therefore, additional terms should be considered, i.e. S(I) = αI + βI² + γI³ + δI⁴ + εI⁵ + …, where δ, and ε are fourth- and fifth-order nonlinear coefficients.

Figure 3(b) shows both the scattering intensity dependence fitting (inset) and the amplitude of each frequency components when considering fifth-order nonlinearity. Not only the inset curve fits better to experimental results, but now there are also three dips in 2f_m and two dips in 3f_m signals, agreeing well to experiments in both Figs. 2(a) and 2(b).

In addition, slope change after each dip is also observable in Fig. 3(b). Based on the high-order polynomial nonlinear model, the dip represents the excitation intensity when next higher nonlinearity starts to take dominance. For example, the first dip in the 2f_m curve originates from the A₂ term when 2γ = −3βI, i.e. the contributions of I² and I³ terms are comparable. Before the dip, I² term is dominant, but after the dip, the I³ term takes over governance. Therefore, after each dip, the slope should increase by one, and furthermore, the sign of SAX signal should exhibit sign flip.

The unexpected high-order nonlinearity might be explained by thermal mechanism. Under our experimental condition, the temperature rise of a single GNS is estimated to be 100 °C at 1 MW/cm² excitation intensity. Further studies are required to quantify the effect of thermal nonlinearity. At this stage, it is important to note that all nonlinear behaviors reported in this work are fully reversible and reproducible, since the temperature rise is far below particle destruction level. One more note is that although we are working with planar imaging result here, SAX is known to provide resolution enhancement in all three dimensions [7], so our nonlinear analysis should be directly applicable to three-dimensional imaging.

5. SAX imaging: excellent correspondence of theory and experiment

With the highly nonlinear dependence of scattering, enhancement of optical resolution is expected, with the aid of the SAX microscopy. Figures 4(a)-4(c) show SAX images with different excitation intensities. 1f_m images exist for all intensities, with similar FWHM throughout our measurement. In Fig. 4(a), where the excitation intensity is enough to induce saturation, but not enough to induce dips, FWHM reduction is observed for both 2f_m and 3f_m signals, similar to our previous observation [8].

 

Fig. 4 SAX images with different excitation intensities under the condition of (a) slight saturation, (b) first dip of the 2f_m signal, and (c) first dip of the 3f_m signal. The image size is 750 × 750 nm², with 20-nm pixel size. (d) and (e) are theoretical comparison of image profiles corresponding to 2f_m in (b) and 3f_m in (c), respectively.

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In Fig. 4(b), by increasing the excitation intensity to induce the first dip of 2f_m curve, the 2f_m image shows donut geometry, as we expected, while the 3f_m image remains a Gaussian profile, with greatly reduced FWHM. Similarly, when the excitation intensity increases up to the first dip of 3f_m, the 2f_m image recovers Gaussian profile with reduced FWHM, while the 3f_m image becomes donut in shape, as shown in Fig. 4(c).

Based on Fig. 3(b), the theoretical profiles of 2f_m and 3f_m donut images can be derived, as shown in Figs. 4(d) and 4(e), respectively. It is interesting to notice that the line widths of the donuts are 110 and 65 nm for 2f_m and 3f_m, respectively. The experimental data are included in the two figures, showing reasonable agreement, and thus confirming the correctness of introducing fifth-order nonlinearity. Comparing to fluorescence, no such high-order nonlinearity is found, and the width of 2f_m and 3f_m donut here is much smaller than imaging resolution in fluorescence-based SAX imaging [7].

From the prediction of Eq. (3), the sign of SAX signal should flip before and after each dip. Figure 5 confirms the theory by taking 2f_m signals with the in-phase component (“X”) output of the lock-in amplifier, instead of the magnitude (“R”) output. The excitation intensity is 8 MW/cm², which is high enough to induce multiple dips. From the “R” channel, five peaks are observed, corresponding to the last two dips of the 2f_m curve in Figs. 1(b) and 3(b). From the “X” channel, it is clear that the sign of peaks flips after each dip. By collecting only the negative part from the “X” channel, we can selectively highlight the central peak, whose FWHM is less than 90 nm. It provides another possibility to further enhance spatial resolution of the 2f_m signals.

 

Fig. 5 The 2f_m SAX images in “R” and “X” channels of a lock-in amplifier, respectively.

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6. Summary

In this study, we combined SAX microscopy with saturation and reverse saturation of scattering. Based on our analysis, fifth order of nonlinearity is required to explain the intensity dependence of SAX signal. Such high order of nonlinearity is unexpected, especially when compared to previous fluorescence or nonlinear absorption reports. As a result, very interesting PSF shapes with different harmonics are observed, such as donuts, and the width of the SAX imaging PSF can be significantly reduced. Excellent agreements between experiment and theory are found, in terms of intensity dependence, imaging profile, and SAX signal phase. The results can be applied for superresolution imaging in not only plasmonic materials, but also biological tissues, where the distorted PSF will provide a sensitive probe to local environment through careful shape examination. Our analysis model can be further extended to analyze versatile nonlinear behaviors of other materials.