1. Introduction

Optical tracking of nano-objects and micro-objects represents nowadays an important tool to study elementary processes at the molecular level [1, 2], to perform variety of colloidal studies like colloidal crystallization, optical sorting of colloids, quantification of interactions between colloids [3, 4, 5, 6, 7, 8], to study single particle Brownian motion and diffusion under various conditions including optical ratchets [9, 10, 11]. Usually fluorescent objects are detected by a sensitive CCD camera in two dimensions because they can be tracked in complex environment [12, 13, 14] and various algorithms are used to extract the object position in “nanometer” resolution [15, 16]. Such CCD based techniques were extended to 3D by adding the axial direction using off-focus imaging [17, 18]. Other methods are especially connected with optical tweezers [19] and use of a quadrant photodiode to detect either the object lateral motion [20, 21] or even 3D motion of single micro-object with subnanometer spatial resolution and microsecond temporal resolution [22, 23, 24, 25]. The most precise method is based on the polarization changes of the light measured by a pair of detectors [26]. This method measures picometer displacements of transparent microscopic object but only in one direction. Unfortunately, these methods using photodiodes can only follow single object.

Recent analyses of object motion near an interface attracted great attention especially in connection with evanescent wave illumination. A new microscopic method has been developed - total internal reflection microscopy [27] – that employs exponential decay of the illuminating field with the increasing distance from the surface. Therefore, the intensity decay of detected scattered field or fluorescence signal is used to obtain the distance of the object from the surface [28, 29, 27, 30].

Here we present a method that uses a CCD camera to detect the field scattered by an object perpendicularly to the direction of the illuminating beam. The novelty of this method is based on a spatially periodic illuminating field – for example a standing wave or more complicated interference structure. If the size of the object is comparable or bigger than the spatial period of the field, the pattern of the object made by scattered light is very sensitive to the location of the object with respect to intensity maxima and minima of the illumination field. Therefore any classical particle tracking algorithm [15] gives faulty results and we were forced to develop a method providing the mutual position of the object and the illuminating field from the pattern shape. This principle can be extended into two dimensions if a sort of two dimensional spatially periodic illumination is used. It can be even extended to three dimensions if it is combined with evanescent field illumination and so the particle motion in the third direction can be estimated [31]. The presented method was tested experimentally on particle motion in standing wave strong enough that it served as a periodic system of optical traps [31]. Since the method reveals the mutual position of the object and an optical trap, properties of this trap can be experimentally studied and consequently used for precise micro-object delivery [32, 33] or sorting according to the object size or optical properties.

2. Principle of the method

This method is based on the light scattering by an object illuminated by a spatially periodic field. The scattered light is detected by CCD camera placed perpendicularly to the direction of the illuminating field. If the object size is comparable with the period of the field, the CCD camera detects a complex pattern of the object made by the scattered light (see Fig. 1).The position of the pattern in the plane of CCD (α, β) is given by the position of particle (x̄, z̄) with respect to the imaging system (see Fig. 1 for the system of coordinates). However, using the spatially periodic illumination the shape of the pattern changes if the mutual position of the object and the field is modified (see movie of Fig. 1 – part 1 and 2, Fig. 2). Due to the spatially periodic illumination, the pattern shape repeats if the mutual position of the object and the illumination field is shifted by one period of the illumination field.

If the particle stays static with respect to the imaging system (see movie of Fig 1 – part 3), the position of the pattern on CCD plane is fixed and in this case the optical intensity of the pattern measured at each point (pixel) α, β of the CCD has the form (see Appendix I for more details):

where k is the size of the wavevector component along the z̄ axis, z_Ψ is the position of the object with respect to the standingwave (position of the standing wave can be set by movable mirroras shown in Fig. 1, see movie-part 3), I _off(α, β), I _amp(α, β) and Ψ(α ,β) are real functions. Based on Eq. 1 we can expect that when the standing wave uniformly moves, the intensity follows the cosine function with the same frequency as the intensity maxima of the standing wave pass through the particle but with specific offset I _off, amplitude I _amp, and phase Ψ. Knowing these three parameters for each pixel (α, β) we can reconstruct the shape of CCD object pattern for any position z_Ψ of the object with respect to the standing wave.

 

Fig. 1. Configuration of the particle tracking method using spatially periodic field illumination - here it is made by two counter-propagating evanescent waves. The movie shows 3 different cases of field and particle motion. The first part shows a case when the field is static and the particle moves along the surface alternatively in the direction of standing wave periodicity. This part demonstrates how the CCD pattern changes its position and shape. Second part of the movie shows also a particle moving in the static field but for the case of Brownian motion. Third part shows a case when the static particle is illuminated by the moving standing wave. In this case the pattern changes its shape but not the position. [Media 1]

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Fig. 2. Patterns of an object illuminated by a spatially periodic field (evanescent standing wave) and seen by a CCD camera. The presented five images were taken by high-speed CCD camera (IDT X-Stream XS3,4 GB, framerate 6120 fps, integration time 2 μs) during fast sweep of standing wave over the particle (polystyrene, diameter 1.070 μm). This set of images corresponds to the movement of the standing wave over its one period (200 nm).

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3. Calibration of the interference patterns

In this section we present how to get experimentally the functions I _off(α, β), I _amp(α, β) and Ψ(α, β) from Eq. (1) for position α, β of each CCD pixel. We assume here that the object is stationary while the standing wave uniformly moves (z_Ψ = const ∙ t). Experimentally the particle can be fixed using external optical tweezers and the motion of the standing wave is provided by movable mirror (see movie of Fig. 1 – part 3). We recorded the image pattern of the light scattered by the object during the wave movement and some frames from this record are shown in Fig. 2. We used fast CCD camera so that we had enough patterns within several standing wave periods to fit the dependence from Eq. (1) at each CCD pixel to get the desired parameters I _off(α, β), I _amp(α, β), and Ψ(α, β) at each pixel center α, β. Figure 3 presents an example of these parameters found by this procedure.

 

Fig. 3. Reconstructed functions I _off, I _amp and Ψ for CCD area of 100×100 pixels. These function were obtained using 30 frames of the record described in Fig. 2.

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Knowing the parameters at each CCD pixel, the intensity pattern I _CCD taken by the whole CCD can be reconstructed for any value of z_ψ using Eq. (1). Since z_Ψ describes the mutual position of the object and the standing wave structure, we are now able to calculate the pattern shape for any of these mutual positions - regardless if the wave or the object is stationary or motional.

4. Particle tracking

The classical measurement of the object motion is based on the time record of changeless object images. Consequent two-dimensional analyses of the object image motion gives the α, β object positions within the CCD. External calibration of the CCD camera is needed to get object positions x̄ and z̄ in the object plane.

As we demonstrated in Fig. 2, for the spatially periodic illumination the object motion along z̄ axis brings about not only image shift but also image deformation. Therefore, each frame recorded by the CCD camera has to be compared with the calculated one and the best coincidence between them has to be found by changing the parameter z_Ψ. At the same time the calculated image has to be shifted by x, z so that to get the best overlap of the measured and calculated image. This optimization results in the particle position in x̄ and z̄ axes with respect to the imaging system and in the particle position z w with respect to the periodic field (standing wave), respectively.

To shift the reconstructed pattern I _CCD(α, β,z_Ψ) to an arbitrary position α - x, β - z (non-integer multiples of CCD pixel size) we used the translational property of the Fourier transform [34]:

where FT means the Fourier transform and IFT is the inverse Fourier transform.

To decide which is the optimal reconstructed pattern shape and its optimal position we look for the minimal value of function

 

Fig. 4. Object trajectories along z̄ and x̄ directions obtained from analyzes of the CCD camera images for the same configuration and particle properties as in Fig. 2. During the recording the standing wave was stationary and so the curves show the Brownian motion of the particle in xz̄ plane.

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where I _exp(α, β) is the intensity at the CCD pixel position α, β obtained during the particle tracking (for example for free particle), I _CCD(α - x ,β - z, z_Ψ) is the calculated intensity for particular z_Ψ value and shifted by x, z from the original pixel position. Since we do not evaluate only single frame but a time record of many intensity patterns, the particle position found from one frame is used as the input parameter for the optimization procedure in the subsequent frame. This assures fast convergence to the correct particle positions. The z_Ψ values can be extended out of the single period of the illuminating field if the CCD camera framerate is so high that between two adjoining frames the particle shifts only by a fraction of the field period. An example of particle positions obtained by the described method from a record of free particle motion in the standing wave is presented in Fig. 4.

The above described method provides x and z positions in CCD pixels but the values of z_Ψ are in micrometers (the spatial period of the illuminating field has to be known). If the illuminating standing wave is stationary, the values z_Ψ and z must describe the same motion of the particle and therefore can be used to calibrate the CCD sensor. Therefore, the slope of the linear dependence of z_Ψ on z shown in Fig. 5 gives the calibration constant. Assuming the same properties of the imaging system in both x̄ and z̄ directions this pixel calibration can be extended also to the x̄ axis.

In the real experimental conditions set-up noise and vibrations are presented and so the experimental data do not exactly follow the linear dependence. Therefore, the difference Δz = z_Ψ - z informs us about the movement of the standing wave with respect to the imaging system and so it visualizes the instability of the system. An example of the Δz record is presented in Fig. 6 for the same dataset as above. One easily sees apparent oscillations of the imaging system with respect to the illuminating standing wave with amplitude lower than 50 nm. These oscillations show vibrations of our setup coming from acoustic noise in the laboratory. But due to the presented method they do not influence errors in the measured object position with respect to the standing wave. Therefore the Δz record can also serve for estimation of the method resolution (see Appendix II).

5. Discussion

In the case of one dimensional illumination the method provides information about the object position along x and z direction from the image shift but moreover it also provides object position with respect to the periodic illumination field along z axis. Therefore the information obtained from the image shifts is quite comparable to the classical methods using CCD camera [15, 16]. However the information about the position of the object with respect to the periodic illumination field is unique and can be used in many ways. For some types of experiments it is extremely useful to know the position of the object with respect to the spatially periodic field - for example optical trapping experiments in standing waves. It is not important if the periodic light structure is sinusoidal or not and if it moves or not, the method provides the same information about the object position with respect to this structure. Therefore, it will provide higher precision of particle tracking in studies dealing with Brownian motors, Brownian dynamics including hops between neighboring potential wells especially in travelling potentials or ratchet potentials [35]. The other area of applications contains measurement of vibrations of the imaging part of a system (microscope) with respect to the illumination part. The reason is that z position obtained from the image shift depends on the vibration of CCD camera and related path, but object z positions with respect to the illumination pattern is independent on these vibrations (see Fig. 6). In this paper we presented an evanescent version of 1D illumination field but in principle the method also works with other types of periodic illumination (using periodic masks etc.). It can be also extended to 2D periodic illumination field and so the position of the object with respect to the illumination field will be known in z and x axis. However, the presented algorithm should be extended to 2D.

The sensitivity of the method is influenced by the size of the object placed to the illumination field. If its size is smaller than the period of the illumination field, the image pattern of this object is not as sensitive on the position in the standing wave as we show in Fig. 2. The brightness of this pattern changes as the object moves over the standing wave and especially at the intensity minima, where the spot is darker, the signal to noise ratio is low. Therefore, the accuracy is much worse. In our experiments we proved that the method works well for particles of diameter in the range 300–1000 nm. In principle the method should work for larger particles as well.

 

Fig. 5. Dependence of particle position with respect to the periodic illumination field extended out of the interval of single period of the illuminating field (z_Ψ) on the position of the interference pattern on the CCD camera (z). The slope of the linear fit gives the calibration constant between the CCD pixels and micrometers if the standing wave is stationary.

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Fig. 6. Time record of the quantity Δz = z_Ψ - z. Low frequency vibration of the CCD camera with respect to the illumination field is clearly visible.

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6. Conclusions

We have presented in details a new method for particle tracking which uses a spatially periodic illumination - like a standing wave or any kind of two-beam interference field. If the period of this field is known, the position sensor (CCD camera) can be calibrated without any extra experimental procedure. The unique feature of this method is that it provides not only the object position with respect to imaging system but also position of the object with respect to the illuminating field regardless if this field is static or in motion. This could be particularly useful to study particle motion in optical Brownian ratchets [9] or traveling periodic potentials [35, 31, 33]. Using the frequency analysis we demonstrated that resolution of the method is better than 1 nm. The method, as it is presented in this paper, is not applicable for the case when more proximate particles should be tracked at the same time because their coherent images on CCD influence each other.

Appendix I:

Scattered light interference pattern

In this section we are not going to derive the shape of the scattered interference pattern at the plane of CCD but rather to show that this pattern changes periodically with respect to the position of the object in the periodic illumination field. Let us assume that a scatterer (particle) stays static in the laboratory system of Cartesian coordinates x̄, ȳ, z̄ (optical system and CCD) and let us illuminate the particle by a standing wave moving in the direction of z̄ axis (see Fig. 1). The optical system images the scattered field into the plane of the CCD camera.

• 1). The scatterer is placed into the traveling standing wave having electric field E(x̄,ȳ,z̄,t) parallel to the axis z̄. Even vectorial description of the field could be used but for simplicity and without lack of generality we assume here this field orientation. Further we assume that the traveling standing wave is created by interference of two counter-propagating beams with time varying phase in one of them. To simplify the explanation but to keep the key features we assume that the complex amplitudes of these beams are z̄-independent. This assumption is valid for plane waves, non-diffracting beams, or evanescent waves and can be used for approximative description of real spatially limited beams assuming the geometrical optics approximation. The motional standing wave has the form:

(4)$$E\left(\overline{x},\overline{y},\overline{z},t\right)=E_1\left(\overline{x},\overline{y}\right)\exp \left[ik\overline{z}-i\omega t\right]+E_2\left(\overline{x},\overline{y}\right)\exp \left[-ik\overline{z}+{2iz}_{\psi }-i\omega t\right]$$
$$=\exp \left(-i\omega t\right)\left[E_1\left(\overline{x},\overline{y}\right)\exp \left(ik\overline{z}\right)+\exp \left({2ikz}_{\psi }\right)E_2\left(\overline{x},\overline{y}\right)\exp \left(-ik\overline{z}\right)\right]$$

where E ₁(x̄,ȳ) and E ₂(x̄,ȳ) are the complex amplitudes of the electric field. Changing the value of z_Ψ causes the movement of the standing wave structure. We expect here that z_Ψ corresponds to the shift of the movable mirror that retro-reflects one of the incident waves [32].

The light scattered by the object placed into the field E(x̄,ȳ,z̄,t) creates an electromagnetic field E _CCD(α,β,t) in the plane of the CCD (see Fig. 1). Fields E(x̄,ȳ,z̄,t) and E _CCD(α,β,t) are linked by an impulse response function f(x̄,ȳ,z̄,α,β) of the imaging system and the particle. This function describes how the unity volume of the particle submerged into the field E(x̄,ȳ,z̄,t) at x̄,ȳ,z̄ forms the field E _CCD(α,β,t) at the position of α,β in the CCD plane. If we expect linear, homogeneous, and isotropic medium in between the sample and CCD camera, the following contribution dE _CCD(α,β,t) to the final field E _CCD(α,β,t) on the camera from the volume dx̄dȳdz̄ placed around x̄,ȳ,z̄ of the sample can be written:

(5)$${dE}_{\mathrm{CCD}}(\alpha ,\beta ,t)=f\left(\overline{x},\overline{y},\overline{z},\alpha ,\beta \right)E\left(\overline{x},\overline{y},\overline{z},t\right)d\overline{x}d\overline{y}d\overline{z}.$$

Therefore, using Eqs. (5) and (4) the total field in the plane of CCD can be expressed in the following form:

(6)$$E_{\mathrm{CCD}}(\alpha ,\beta ,t)=\underset{\mathrm{scatterer}}{\iiint }f\left(\bar{x},\bar{y},\bar{z},\alpha ,\beta \right)E\left(\bar{x},\bar{y},\bar{z},t\right)d\bar{x}d\bar{y}d\bar{z}$$
$$=\exp \left(-\mathrm{i\omega t}\right)[\underset{\mathrm{scatterer}}{\iiint }f\left(\bar{x},\bar{y},\bar{z},\alpha ,\beta \right)E_1\left(\bar{x},\bar{y}\right)\exp \left(\mathit{ikz}\right)d\bar{x}d\bar{y}d\bar{z}$$
$$+\exp \left({2\mathrm{ikz}}_{\psi }\right)\underset{\mathrm{scatterer}}{\iiint }f\left(\bar{x},\bar{y},\bar{z},\alpha ,\beta \right)E_2\left(\bar{x},\bar{y}\right)\exp \left(-\mathit{ik}\bar{z}\right)d\bar{x}d\bar{y}d\bar{z}$$

Even though the function f(x̄,ȳ,z̄,α,β) is not known, we can threat the integrals in Eq. (6) as two unknown complex functions F ₁(α, β) and F ₂(α, β):

(7)$$F_1(\alpha ,\beta )=\mid F_1(\alpha ,\beta )\mid \exp \left[{i\psi }_1(\alpha ,\beta )\right],$$

(8)$$F_2(\alpha ,\beta )=\mid F_2(\alpha ,\beta )\mid \exp \left[{i\psi }_2(\alpha ,\beta )\right],$$

giving

(9)$$E_{\mathrm{CCD}}\left(\alpha ,\beta ,t\right)=exp\left(\mathit{-}\mathit{i\omega t}\right)\left[F_1\left(\alpha ,\beta \right)+\exp \left({2\mathit{ikz}}_{\psi }\right)F_2\left(\alpha ,\beta \right)\right].$$

The optical intensity I _CCD measured at each point α, β (pixel) of the CCD has the form:

(10)$$I_{\mathrm{CCD}}\left({\alpha ,\beta ,z}_{\psi }\right)=\frac{1}{2}{\mathit{c\epsilon }}_0{E}_{\mathrm{CCD}}(\alpha ,\beta ,t)E_{\mathrm{CCD}}(\alpha ,\beta ,t)*,$$
$$\equiv I_{\mathrm{off}}(\alpha ,\beta )+I_{\mathrm{amp}}(\alpha ,\beta )\cos \left[{2\mathrm{kz}}_{\psi }+\psi (\alpha ,\beta )\right].$$

where I _off(α, β), I _amp(α, β) and Ψ(α, β) are real functions:

(11)$$I_{\mathrm{off}}(\alpha ,\beta )=\frac{1}{2}{\mathit{c\epsilon }}_0\left[{\mid F_1(\alpha ,\beta )\mid }^2+{\mid F_2(\alpha ,\beta )\mid }^2\right],$$
$$I_{\mathrm{amp}}\left(\alpha ,\beta \right)={\mathit{c\epsilon }}_0\mid F_1\left(\alpha ,\beta \right)\mid \mid F_2\left(\alpha ,\beta \right)\mid ,$$
$$\psi \left(\alpha ,\beta \right)={\psi }_2\left(\alpha ,\beta \right)-{\psi }_1\left(\alpha ,\beta \right).$$

If position of the sphere is changed with respect to the standing wave (z_Ψ), the intensity at each point (pixel) of the CCD changes with the period of the illumination standing wave (π/k) = (λ/2).

 

Fig. 7. The power spectral density for Δz record. The blue-plotted data corresponds to the range which is influenced by the setup instability, the red part shows the range which can be associated with the Gaussian white noise caused by the method inaccuracy. The reason why the boundary is chosen at the frequency of 2000 Hz is discussed in the text below.

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Appendix II:

The estimation of the method resolution

The quantity Δz = z_Ψ - z mentioned in Fig. 6 can also assist to determine the resolution of the method because this quantity incorporates the error from the image analyses and also the noise coming from the mutual motion of the CCD and the periodic field illumination caused by the set-up. To analyze the record quantitatively we calculated the power spectral density by

(12)$${\mathit{PSD}}_{\Delta z}=\frac{{\mid \mathit{DFT}\left[\Delta z\right]\mid }^2}{N},$$

where DFT[Δz] is discrete Fourier transform of the record Δz and N is the number of points in the Δz record. PSDΔz plot is shown in Fig. 7 and the constant high-frequency part demonstrates the dominance of the white noise. This white noise comes from the CCD noise, rounding of numbers during the particle tracking procedure, deviations caused by the Brownian motion during the calibration of interference patterns, and other frequency independent effects. If the setup oscillations were eliminated, the white noise component would be the only disturbing contribution to the signal and therefore it limits the resolution of the method.

The criterion determining the method resolution is the magnitude (standard deviation)σ_Δz of the white noise component of Δz record that can be seen in its power spectral density (red part of Fig. 7). In the case of Gaussian white noise the probability density distribution p(PSD _Δz) of the values of PSD _Δz is exponential [36] and depends on the standard deviation σ_Δz of the positions Δ_z:

(13)$$p\left({\mathit{PSD}}_{\Delta z}\right)=\frac{1}{{\sigma }_{\mathit{\Delta }z}^2}\left(\frac{{\mathit{PSD}}_{\Delta z}}{{\sigma }_{\Delta z}^2}\right).$$

Therefore, if PSD _Δz data are taken only for frequencies higher than f_b(f_b is the boundary separating the data influenced and uninfluenced by the setup oscillations), a histogram from this part has the form of an exponentially decaying function (see an example in Fig. 8).

The probability distribution (13) is fitted to the histogram obtained for several boundary frequencies f_b and σ² _Δ2(f_b) is found together with 95% confidence level (see Fig. 9).

It is seen that for f_b ≤ 2000 Hz this dependency monotonically decreases. It is caused by the influence of setup oscillations. For higher boundary frequencies f_b ≥ 2000 Hz the value of σ_Δz(f_b) does not change significantly. However the uncertainty of its estimation grows due to decreasing amount of data points. Since there is no sharp boundary, the frequency f_b = 2000 Hz is a good compromise to set the beginning of the region where the white noise dominates.

For the record presented in the Fig. 4 we obtained the standard deviation σ_Δz(f_b = 2000 Hz) = 1.02 ± 0.01 nm. We generated random white noise sample with normal distribution width done by this value. Its power spectral density is plotted in Fig. 10 (green color) and compared with the experimental record (blue). Both records overlap very well for frequencies higher than 2000 Hz.

 

Fig. 8. The normalized histogram of PSD _Δz corresponding to the red-plotted part of Fig. 7.

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Fig. 9. The value of σ_Δz(f_b) determined from the fit to the power spectral density histogram presented in Fig. 7 (for 50 bins) for each boundary frequency f_b.

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Fig. 10. The comparison of power spectral density PSDσ_Δz and the power spectral density of the generated Gaussian-distributed white noise for the determined value of σ_Δz

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Since the white noise component in Δz is caused by the deviations of both z_Ψ and z values and since the x values are obtained in the same way as the z values, we can conclude that the standard deviations caused due to the method inaccuracy do not exceed the limit of 1 nm for all x, z and z_Ψ values obtained by this method.

Acknowledgments

This work was partially supported by the Institutional Research Plan of the Institute of Scientific Instruments (AV0Z20650511), Ministry of Education, Youth, and Sports of the Czech Republic (LC06007), European Commission via 6FP NEST ADVENTURE Activity (ATOM3D, project No. 508952).

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