In the past decade, quantitative phase imaging gave a new dimension to optical microscopy, and the recent extension of digital holography techniques to nonlinear microscopy appears very promising, for the phase of nonlinear signal provides additional information, inaccessible to incoherent imaging schemes. In this work, we show that the position of second harmonic generation (SHG) emitters can be determined from their respective phase, at the nanometer scale, with single-shot off-axis digital holography, making possible real-time nanometric 3D-tracking of SHG emitters such as nanoparticles.
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With the downscaling of material sciences to the sub-micrometer range, nanomaterials of all sorts have attracted a lot of interest and are now being used for numerous applications, in both research and industrial activities. Nanoparticles (NPs), in general, are now commonly used in biology as labels or contrast agents for imaging  and for detection of specific pathogens or proteins . In addition, they appear very promising for bottom-up tissue engineering , for drug or gene delivery [4, 5] to specific cellular sites and even for tumor treatment . Gold nanoparticles, in particular, have been extensively investigated because of their excellent biocompatibility and chemically inert nature, and nowadays a vast knowledge on how to functionalize gold nanoparticles has been developed .
Be it for micro-fluidic or targeted biomedical applications, precise real-time 3D-tracking of nanoparticles is becoming an urgent need to fill. Over the years, many methods [8–14] were developed for determining the lateral position of NPs at presumably nanometer (or at least sub-pixel) resolution, although one must be very careful with such claims. These methods are mostly based on post-processing and analyses of optical images, by cross-correlation , sumabsolute difference , centroid , direct Gaussian fit [11, 12] or polynomial-fit Gaussianweight . A comparative study of these methods is proposed in Ref . The real challenge to 3D-tracking always has been and remains the determination of the axial position, which is no better than the micrometer range, unless the complex diffraction field can be accessed. Even in these cases, it does not reach the sub-micrometer without a priori knowledge of shape or size of the particle.
Given the challenge, it appears logical that digital holographic microscopy (DHM) be the ideal solution, since its single-shot acquisition provides real-time imaging, while its ability to recover the complex wavefront provides a very good axial resolution. It is with a total intern reflection illumination digital holographic setup that Atlan et al.  achieved video-rate 3D localization of gold nanoparticles. A year before, Lee and Grier  used in-line holographic methods to reconstruct the field diffracted by holographically trapped silica microspheres and to estimate their relative axial position. They could successfully differentiate four microspheres located 5 µm apart in the axial direction. For a unique colloidal microsphere in a backgroundfree environment, and using advanced fitting to the Lorenz-Mie theory, they estimated an axial resolution of 10 nm . However, in this last case, particle shape and refractive index must be known a priori, and fit is very sensitive to variations of the diffraction pattern. In any cases, these techniques are incompatible with biological media, such as cells or tissues, for the signal of interest will be drowned in a sea of scatterers.
This is one of the reasons why two groups from Switzerland decided to combine digital holography with nonlinear microscopy, more specifically second harmonic generation (SHG) [18, 19]. The main advantages (highly specific, background-free) over previously described digital holographic methods are consequences of the nature of SHG and overcome the problems posed by diffusive media. The two groups independently proposed a method, based on the same principle as in Ref. , for obtaining the axial position of SHG emitters from numerical reconstructions of digital holographic images. Although proposed, this method has never been fully characterized, and while its axial resolution was initially claimed to be of about 2 µm , we recently found out that it is in fact one order of magnitude better.
In this work, we describe how digital holography makes possible real-time, nanometric, 3D-tracking of SHG emitters and how this can be applied to tracking of nanoparticles. This paper mostly focuses on determination of the axial position, since the described holographic technique is compatible with previously-cited methods for determination of lateral position. First, we elaborate on the method proposed by Refs. [18, 19] and characterize it more thoroughly. Then, we propose and test a completely new, faster, more precise and accurate technique, based on direct measurement of the SHG phase recovered by DHM.
2. Experimental details
Holography consists in exploiting the interference resulting from mixing of two coherent waves: one reference wave and one object wave. The general idea is to use the reference wave to encode the complex diffraction pattern of an object into an interference pattern called hologram. Classically, holograms were recorded on a photo-sensitive plate and had to be photo-chemically developed and re-illuminated to produce 3D images. With digital holography, holograms are recorded by a digital camera. In opposition to the classical holography, no photo-chemical development step is required and the image reconstruction is performed numerically . In our off-axis configuration, the detector is located some distance away from the system image plane and image reconstruction is performed according to Ref. . For more details on digital holographic microscopy applied to second harmonic generation, see Ref. .
2.1. Experimental setup
Schematics of the optical setup used for experiments can be found in Fig 1(a). The laser source is a 800 nm wavelength Ti:Sapphire laser delivering 250 fs pulses at approximately 250 kHz repetition rate. In the object arm O of our Mach-Zehnder type interferometric setup, light is focused in the specimen plane by a f/4 aperture condenser lens and then collected by a 100× microscope objective. A β-barium borate (BBO) frequency doubler crystal (FDC) is inserted in the reference arm R to generate the second harmonic reference wave. Holograms were recorded at a frame rate of 8.3 Hz and an exposure time of 120 ms, using a -20C-cooled Hamamatsu Orca ER CCD camera (12-bits), designed for fluorescence imaging. A 400 ± 20 nm bandpass filter eliminates the fundamental-wavelength component of light. The studied specimen consists of a solution of BaTiO3 nanoparticles of tetragonal crystalline structure and of relatively large mean size (200 nm) deposited on a glass cover slip. Comparison with recent work by Pu et al.  on similar BaTiO3 particles allows to estimate the SHG cross-sections of our particles to be of the order of 1 × 105 GM, with 1 GM corresponding to 10−50 cm4 ·s ·photon−1. The laser peak power in the specimen plane is of approximately 30 GW·cm−2·pulse−1, which is about one order of magnitude below the threshold for biological cell damage , and could potentially be reduced by the use of a more sensitive camera, e.g. EMCCD-type, or more efficient NPs.
To test and compare the two methods for determination of axial position of SHG emitters, we have mounted the specimen on a feedback-looped piezoelectric stage, with a sub-nanometer resolution. During the experiment, the specimen was axially scanned over a range of 2 micrometers, with a sampling step of 25 nm, and eight holograms were recorded at each position for statistic analysis of the efficiency of each technique.
2.2. Phase fluctuations and offset adjustment
It is very important to emphasize that digital holography, as proposed here, is a single-shot technique. This means that a single hologram acquisition leads to complex wavefront retrieval over the entire field of view. As a consequence, the technique is said to be vibration-insensitive, in opposition to scanning microscopy techniques. However, this is no longer the case when comparing the reconstructed fields from different holograms, as the absolute phase may have been arbitrarily shifted between the two acquisitions. In classical, i.e. linear, digital holography, this problem is easily overcome by referencing the phase variations of the specimen (e.g. a cell) to the phase offset in a region of the background where the phase is expected to remain unchanged (e.g. the cover slip) . Unfortunately, this is not possible with a background-free technique, such as the one presented here, where the only collected SHG signal comes from the specimen. To produce a uniform, unvarying SHG background on which to reference the phase variations and eliminate those attributed to system vibrations, we have introduced a lithium triborate (LBO) crystal — indicated by FDC in Fig. 1(a) — in the focal plane of the beam expander located in the object arm of the interferometer. Referencing the SHG phase from the specimen to that of the LBO-generated background cancels out almost all vibrations-induced phase variations. Even so, phase variations caused by vibrations between the LBO crystal and the specimen remain, because they introduce a different phase shift for the fundamental (that will generate the SHG at the specimen) and the second harmonic (generated by the LBO crystal) wavelengths. The remaining phase variations account for approximately 5% of the overall phase noise.
3. Method 1: Determination of axial position from reconstruction distances
A first method to determine the axial position of SHG emitters has been independently proposed last year in Refs. [18,19]. Since at the time it was only proposed and not very well characterized, we have investigated it in more details, in order to determine its strengths and limitations.
The method is based on numerical propagation of the SHG field recovered from hologram processing [20, 22]. The idea is that the intensity of the second harmonic field generated by a NP will reach a maximum at an axial position corresponding to the image plane. Then, the exact position of the image plane (in the image space) can be related to the axial position of that NP (in the object space), as depicted in Fig. 2. In other words, the reconstruction distance d between the hologram plane and the image plane depends on the axial position of the SHG emitter, and it is possible to deduce the relative axial positions (in the object space) of different SHG emitters from the reconstruction distances that maximize their respective SHG field intensity (in the image space). The efficiency of this method strongly depends on the precision at which the reconstruction distance can be determined. Ultimately, it depends on the longitudinal magnification (ML) of the microscope objective, which, under some approximation, can be related to its transverse magnification (MT) by
For our 100× objective, moving the specimen by 25 nm causes the reconstruction distance to change by 250 µm. Over such distance, the diffraction algorithm used for numerical propagation only spreads the field to immediate neighbor pixels. At such small scale, it is therefore not the field of the entire image that contributes to finding the maximum intensity along the optical axis, but only the field of a few pixels in the vicinity of the particles center. Consequently, while the reconstruction distance can, in principle, be incremented by infinitely small values, the real sensitivity can be limited by the wavelength, the pixel pitch and the camera bit depth. Furthermore, as the specimen moves too far away from the working distance, the relation of Eq. (1) on which is based this method is no longer valid.
We have tested this method on the stack of holograms we have recorded according to the protocol described in Section 2. For each of the 640 holograms (i.e. 8 holograms at each of the eighty 25 nm steps) the phase of the constant SHG background generated by the BBO crystal in the object arm (see Fig. 1(a)) was set to zero, to eliminate the nefast effects of vibrations in the system (see Section 2.2). Each hologram was processed independently. An automatic procedure was used for hologram reconstruction and optimization of the reconstruction distance was based on maximization of the SHG intensity.
Typical reconstructed SHG field intensity in the hologram plane and after numerical propagation to the image plane are respectively displayed in Fig. 3(a) and 3(b). Fig. 3(c) maps the intensity along the white line present in the two previous images, for reconstruction distances varying between −15 and −8 cm. For every specimen position and reconstruction distance, the intensity of the brightest pixel is color-coded in the background of the graph of Fig. 3(d), where the optimized reconstruction distance is plotted against the axial position returned by the piezoelectric stage. The graph shows an overall linear relation between axial position of the specimen in the object space and reconstruction distance (or axial position of the image plane) in the image space. It should be noted that the objective working distance more or less corresponds to the 0 µm axial position in the graph, where the slope is more regular and, as expected from Eq. (1), corresponds to M2 T — plotted in black line. From the graph of Fig. 3(d), we evaluate the axial precision of this method to be a few hundreds of nanometers, for positions close to the working distance.
We estimate the axial resolution of this method to be roughly 200–400 nm, and limited by typical point-spread-function, pixel pitch and camera bit depth. That value is one order of magnitude better than the resolution initially estimated by Hsieh et al. . It is comparable to that of Ref.  and is much better than many other optical tracking techniques, but still not as high as one would expect from an interferometric technique. One limitation is that this method is based on an imaging scheme and is only correct as long as the rough approximation of Eq. (1) remains valid, which might no longer be the case for non-ideal imaging conditions. Another limitation is that this method is extremely time-consuming, therefore incompatible with realtime imaging, since after the complex field is recovered from each hologram, an optimization routine is needed for determination of the reconstruction distance that maximizes the intensity of the SHG field. In digital holography, numerical field propagation is the step requiring the most processing time and optimization routines roughly require to perform between 10 and 100 of these numerical propagation steps. The major and possibly only advantage of this method is its large axial range, for it does not suffer from 2π phase indetermination (see Section 4). For this reason, it could be advantageously used to complete the other method we will describe is this work.
4. Method 2: Determination of axial position from direct SHG phase value
A more precise way of determining the relative z-position of different SHG point-source emitters relies simply on the phase value of the complex SHG field recovered at the camera plane. Figure 4(b) illustrates how this phase directly depends on the axial position of the emitter, here a NP lying on a glass cover slip. This representation includes the background SHG field generated by the LBO crystal, indicated by . This figure is relatively complex and can be much simplified. To provide a clear understanding of why the SHG phase depends on the NP position, let’s consider the two cases depicted by the cartoon of Fig. 4(a). The only difference between these two cases is a ∆z shift of the SHG emitter along the optical axis. A first observation is that as long as ∆z occurs in a non-dispersive medium (e.g. air), the total optical path length is independent of the axial position of the SHG emitter. However, the phase detected at the camera plane will vary with respect to the exact position where second harmonic is generated and the phase difference between the two cases will be
For displacements occurring in a dispersive medium, the exact refractive indices at the fundamental and SHG wavelengths must be known in order to retrieve ∆z. But for displacements occurring in a uniform, non-dispersive medium characterized by a refractive index n, Eq. 2 simplifies to
The previous equation relates the observed phase at the hologram plane to the relative axial position of the SHG emitters. Interestingly, the relation of Eq. (3) corresponds to the phase advance term of the fundamental field at wavelength λ 0. As a consequence, each 25 nm displacement steps correspond to an 11.25° phase shift and emitters separated by ∆z > λ 0 are subject to a 2π phase indetermination. Fortunately, if needed, this indetermination can be lifted by using the less sensitive method based on digital propagation that we first described for axial position determination.
On a more general note, this development gives the correct interpretation of the information contained in the phase of nonlinear signals, that is not simply an OPL difference, as it is the case for linear holography, but rather a more subtle difference of the OPL traveled at the different wavelengths.
We have tested this method on the stack of holograms we have recorded according to the protocol described in Section 2.1. Typical images of the intensity and the phase of the second harmonic field generated by the nanoparticle are presented in Fig. 5(a). Because this method, unlike the first one, is not based on imaging principle but only on direct phase difference, there is no need to numerically propagate the SHG field to form in focus images, which reduces the required processing steps and considerably shortens the processing time. For this reason, the images of Fig.5(a) represent the SHG field in the camera plane, i.e. without numerical propagation to the image plane.
For each hologram, the phase of the SHG field generated by the nanoparticle was extracted from the pixel coordinates corresponding to the maximum intensity and referenced to the phase of the constant SHG background generated by the LBO crystal in the object arm (see Fig. 1(a)), as it was the case with the first method, to eliminate the nefast effects of vibrations in the system (see Section 2.2). The result is plotted in Fig. 5(b) against the nanoparticle position returned by the piezoelectric stage, for the entire stack of holograms. The graph in Fig. 5(b) shows that a very clear, linear relation exists between the SHG phase and the nanoparticle position, as predicted by Eq. (3), for the case of second harmonic generation in a non-dispersive medium—the nanoparticle is located in air environment, and thus the dispersion between the fundamental and second harmonic fields is negligible. As expected for a wavelength of 800 nm, the phase gradually changes by 5π over the 2 µm scan range. A first observation is that this method is more accurate than the first one, as the data points do not deviate from the expected linear behavior. This is because the method is not based on imaging considerations and therefore does not suffer as much from aberrations when the specimen is located far away from the working distance of the objective. Another observation is that the method is also more precise, for the data points are more closely packed on the graph. The standard deviation of the phase, for the different holograms recorded at the same position provides a reasonable estimation of the axial precision that ranges between 10 to 50 nm over the entire scanning range, for hologram to hologram comparison. Of course, the axial precision for comparison of different nanoparticles within the same hologram will be much less, possibly smaller than 10 nm, since such a measurement will no longer suffer from system vibrations.
A small detail concerning the value of the phase of the SHG field was left unmentioned in the previous paragraph: the phase retrieved from the hologram does not directly correspond to the phase of the nanoparticle. Actually, because the condenser introduces dispersion of the laser pulse, the incident fundamental wavelength field arrives at the specimen slightly before the background SHG field generated by the LBO crystal. We have selected our condenser lens so that the temporal lag is enough to shift the two fields out of temporal coherence. The hologram therefore consists of two independent, mutually incoherent object waves interfering with an off-axis reference wave at different region on the CCD, along the off-axis trajectory: the first one being the SHG field generated by the NP and the second being the SHG background generated by the LBO crystal. This is illustrated in Fig. 1(b). For such holograms recorded with multiple, mutually incoherent object waves, the retrieved phase is a mean value of the phase of each object wave, weighted by their respective coherence coefficient with the reference wave. If these coefficients are equal and if no spatial overlap exists between the retrieved phase maps of the two object waves, as it is the case here, then the phase of one object wave is obtained by doubling the phase retrieved from the hologram. While using such a SHG background wave allows to correct the phase fluctuations between different holograms, it also reduces the sensitivity of the technique by half.
The precision and accuracy of this method are both much higher those of the first method we presented. We evaluate the axial precision of this method to be roughly 10–50 nm, for hologram to hologram comparison and ≲ 10 nm for comparison within a same hologram. Because this method is not based on an imaging scheme, it works very well in extreme conditions, e.g. for a specimen located far away from the working distance of the microscope objective. As a consequence, the technique can be used over a very large axial range, hence the high accuracy over the entire scan range. Additionally, since the method does not require numerical propagation, it basically only consists of a couple of Fourier transforms that can be performed at video rate, even for relatively large images. It is therefore intrinsically suited for real-time applications. The major important limitation is the 2π phase ambiguity for steps larger than the fundamental wavelength, but can be overcome by the first method we presented, that used numerical field propagation.
This method will work for multiple nanoparticles dispersed in the field of view, as long as their respective phase patterns can be spatially (laterally or axially) resolved. At high nanoparticle densities, this could require numerical field propagation. Furthermore, assuming that each particle introduces a phase shift that depends on its size (i.e. that there is no phase matching inside the particles), one would have to use nanoparticles with a very narrow size distribution if using this method to determine the relative axial position of multiple particles dispersed in the field of view. Otherwise, the precision will be limited by the width of the particle size distribution
Finally, this method can be used with any other frequency mixing nonlinear light-matter interaction (e.g. higher harmonic generation, coherent anti-Stokes Raman scattering, sum- or difference-frequency, etc.). For instance, third harmonic generation would be even more sensitive, as the phase-position relation would vary twice as much as for the case of SHG here.
In conclusion, we have proposed and experimentally tested a new, real-time technique for nanometric 3D-tracking of SHG emitters. This technique directly relates the SHG phase to the relative axial position of the emitter(s) and presents hologram to hologram axial precision of 10–50 nm and an even better in-hologram axial precision of ≲ 10 nm. Furthermore, the technique can be straightforwardly adapted to other nonlinear fields, such as third harmonic generation, sum- or difference-frequency, or coherent anti-Stokes Raman scattering. Finally, we demonstrated the application of this technique to 3D-tracking of BaTiO3 nanoparticles and believe that it has a bright future in biomedical imaging, for monitoring of targeted receptors, with use of bio-conjugated nanoparticles.
Authors would like to thank Paul Bowen of the Powder Technology Laboratory at EPFL for supplying BaTiO3 nanocrystals. This work was financially supported in part by the Swiss National Competence Center in Biomedical Imaging (NCCBI) and by the Swiss National Science Foundation (SNSF), grant #205320-120118.
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