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Abstract
Distinguishing chemically similar particles in a complex environment has been a challenging problem in spectroscopy, such as micro-Raman spectroscopy. Here, we show that it is possible to distinguish particles from their spectroscopic signals in a modulated optical trap, where the trapping field also acts as an excitation source. Using the overdamped Langevin equation, we report that spectroscopic signals averaged over a certain signal acquisition time exhibit several discrete minimas at unique modulating frequencies dependent on their drag coefficient, exemplified herein as a function of the particle size. In typical experimental conditions, such minimas can be as low as ∼50% of the signal of an unmodulated trap.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
With the merits of being a non-invasive, reagent-less and specific spectroscopic method suitable for in situ experiments, micro-Raman spectroscopy has become a powerful tool in cell identification, cell type discrimination and biomedical imaging [1,2]. The Raman spectrum of a cell provides rich information of its molecular constitution, allowing for quantitative and qualitative characterization of the cell at a sub-micron spatial resolution [3,4]. The combination of optical trapping with micro-Raman spectrometry, conveniently called a Raman tweezer, have been emerging as a powerful tool for highly specific molecular characterization of cells and micro-particles [5–10]. Various forms of this combination has recently been shown to increase the signal collection efficiency as well [11,12]. However, a major drawback of these methods is their inability to distinguish between chemically identical signal sources in a complex biochemical and optical environment. In this paper, we explore ways to circumvent this challenge through the access of hydrodynamic properties (e.g. size, shape) of the optically trapped particle in the trapping medium.
An optical trap can be approximated as a harmonic potential with a trap stiffness, $\kappa$. Therefore, several methods for calibrating $\kappa$ have been explored, such as by measuring the displacement produced by a known external force [13,14], or by analysing the force of thermal fluctuations by the power spectrum analysis [15]. A very practical method of calibrating $\kappa$ is by periodically modulating the trap stiffness achieved by modulating the optical power of the trapping source [16]. The effects of such modulated optical trap on Brownian motion and the position variance of the trapped particle were studied both theoretically and experimentally along with the report of errors it can produce in force measurements in Ref. [16]. This method has subsequently been applied to perform quantitative measurement with bio-molecules [17].
Because modulation of the trap directly modulates the harmonic potential, understanding the effects of trap modulation on the motion of Brownian particle is very important in optical traps. It has gained considerable interest in recent decades in context of calibrating $\kappa$. While there have been studies on the dynamics of Brownian particle in the modulated harmonic trap [16,18–22], the effects of such modulation on the spectroscopic signal such as Raman signal of the trapped particle has not yet been explored. The technical challenges associated with the single cell analysis and characterization [23,24] have called for alternate techniques like micro-Raman spectroscopy for immediate chemical identification and characterization. However, due to the diffraction limit of light, achievement of finer resolution to probe into the sub-cellular phenomenon to distinguish chemically similar signal sources requires integration of advanced imaging techniques such as scanning probe microscopy, which has its own limitations [25]. One alternate method would be through comparison of hydrodynamic properties. So, studying the response of trap modulation on the Raman signal would, in addition to chemical characterization, allow investigations of hydrodynamic and optical properties of the trapped particle, thus providing a distinction between chemically similar compounds. Based on the theoretical formulation of the position variance of optically trapped particle shown in Ref. [16], our work here will estimate the variations in Raman power of the trapped particle as a function of modulation frequency.
First we discuss the theoretical background and introduce an expression for the average Raman power, then we discuss the findings of our results along with the concluding remarks.
2. Theoretical formulation
Since an optically trapped micron sized particle in a viscous medium exhibits Brownian motion, the particle has a natural tendency to diffuse away from the equilibrium position. The average time the particle will stay in the equilibrium position is determined by the the inverse of its roll-off frequency, $\omega _o$; defined as the ratio of trap-stiffness, $\kappa$ and drag coefficient, $\gamma$. As $\kappa$ is directly proportional to the laser power, a sinusoidally modulated laser power results in the modulation of $\kappa$. So, the equation of motion given by the Langevin equation is
Here, $x(t)$ is the trajectory of the Brownian particle, $\omega$ is the modulation frequency, $\epsilon$ is the modulation depth, $-\kappa x(t)$ is the harmonic force from the trap, and $D = k_BT/\gamma$, is the Einstein’s equation relating diffusion constant with the Boltzmann energy and drag coefficient. For a spherical particle of radius $r$ trapped in a medium of density $\rho$ and kinematic viscosity $\nu$, the drag coefficient $\gamma$ is approximated by the Stokes’ law as $\gamma = 6\pi \rho \nu r$. $\sqrt {2D}\eta (t)$, is a stochastic Gaussian process that represents Brownian forces at absolute temperature $T$ such that for all $t$ and $t'$:
Following the derivation of Itô’s lemma [26,27], the position variance of the trapped particle satisfies the equation (see Appendix A for the derivation)
From the equipartition theorem the initial position variance in the $x$ direction takes the equilibrium value of
The solution of Eq. (3) with the initial position variance $\sigma _{xx}^2(0)$ is the position variance of the trapped particle. Writing the ratio of $\sigma _{xx}^2(t) / \sigma _{xx}^2(0)$ as $V(t)$ we have
This equation gives insights as to how the modulation frequency affects the position variance of the trapped particle. From the simulation of Eq. (5) in Fig. 1, we see that the position variance approaches the equilibrium value, $\sigma _{xx}^2(0)$ in the high modulation frequency region, where as in the low frequency region, the particle will have enough time to diffuse away from the trapping potential, showing significant position variance.
Fig. 1. Position variance as a function of modulation frequency, $\omega$ at two different modulation depths, $\epsilon = 1$ and $0.5$ for an integration time of $300$ oscillation cycles.
2.1 Raman power of the trapped particle
We are interested to know the nature of Raman power that can be collected from the optically trapped particle as a function of modulation frequency. For a trapped particle at position, $x(t)$ the Raman signal emitted from the particle with the Raman cross-section, $\sigma _R$ is
where $I_o$ is the intensity of the laser. Assuming the same excitation and collection coefficient, the normalized Raman power is where $w(z)$ is the beam waist radius at which the field value falls to $1/e$ of its axial value and $z$ is the propagation direction. Since both the excitation and collection signal is a Gaussian beam, using the Gaussian distribution function, we can evaluate the average Raman power collected to be,Writing $w(z)$ in terms of the position variance at the equilibrium $\sigma _{xx}^2(0)$
where $\xi$ is a conversion factor relating the confocal area of the trapping Gaussian field with the equilibrium position variance of the trapped particle, the average Raman power of modulated Raman signal will be (see Appendix B for an alternate derivation using Itô’s lemma [26,27] for stochastic processes).For unmodulated case, $V(t) = 1$
which is the change in Raman signal due to diffusion. Therefore,Using Eq. (5) and choosing the value of the conversion factor $\xi$ according to the source Gaussian beam and the optical components, we can analyse Eq. (12) to gain some insights into the nature of Raman signal with respect to the modulating frequency.
3. Results and discussion
For an optical trap with a source wavelength, $\lambda = 785nm$, a high Numerical Aperture objective lens of $NA = 1$, the Gaussian beam waist radius at the focus $w(z = 0)$ is $250$ $nm$. From the simulation of Eq. (12) for a $1$ $\mu m$ particle trapped in an aqueous medium with $\rho = 10^3$ $kg/m^3$ as the density of water at room temperature and $\nu = 10^{-6}$ $m^2/s$ as the kinematic friction coefficient of water shown in Fig. 2, we see that the maximum Raman power variance, ${(R_{mod}/R_{unmod})}_{min}$, is seen at a particular modulation frequency, $\omega _R$ in the low frequency region ($\omega < \omega _o$) due to the increased position variance of the particle in the low-frequency region, while in the high frequency region the Raman power approaches the value for an unmodulated case given by Eq. (11). Since from Eq. (12) we know that the Raman power variance depends on the trap-stiffness $\kappa$ and drag coefficient $\gamma$ (determined by the friction coefficient of the medium and particle size), and also since the conversion factor $\xi$ is dependent on the trap stiffness, it is necessary to first calibrate the trap stiffness of the optical trap. In the present study, we first analyze Eq. (12) over a range of trap stiffness to optimize $\kappa$ that show maximum average Raman power variance for a $1$ $\mu m$ trapped particle with the measurement time $t$ as that for 300 cycles of oscillation shown in Fig. 3(a). Then for the optimised $\kappa$, we show the dependence of $\omega _R$ with the diameter, $d$ of the trapped particle.
Fig. 2. Raman power variance of a one micron trapped particle plotted as a function of modulation frequency, $\omega$ from $10$ $rad/s$ to $5000$ $rad/s$ for trap-stiffness, $\kappa = 1.32 \times 10^{-5}$ $N/m$ and conversion factor, $\xi = 200$. The vertical line denote the roll-off frequency at $\omega _o = 1400.6$ $rad/s$.
Fig. 3. (a) The dependence of maximum Raman power variance, ${(R_{mod}/R_{unmod})}_{min}$ and the frequency corresponding to the maximum Raman power variance, $\omega _R$ on the trap stiffness, $\kappa$ for a one micron diameter particle in the modulation frequency range of $10$ $rad/s$ to $1500$ $rad/s$. The average optimum trap-stiffness is $8.17 \times 10^{-6}$ $N/m$, where about 35% Raman power variation is seen. The non-linearity seen below $\kappa = 1.32 \times$ $10^{-5}$ $N/m$ is due to the theoretical limits of the approximation presented in Appendix B. (b) The profile of modulation frequencies showing maximum Raman power variance with respect to the diameter of the particle along with the corresponding measurement time (top axis) for the optimized trap-stiffness.
From Fig. 3(b) we see that for a given particle size, the maximum Raman power variance occurs at a unique modulation frequency, $\omega _R$ at the measurement time of a 300 oscillation cycles. Hence, it is possible to calibrate the size of the trapped particle according to the modulating frequency at which the maximum Raman power variance is seen. The dependence of Raman power variance on the diameter of the particle for a given $\kappa$ was insignificant, as the discrepancy in the average Raman signal variance was less than 1% (data not shown).
The range ($1-100$ $pN/\mu m$) from which the trap-stiffness was optimized for a one micron particle in our studies applies for typical polystyrene beads used for trap-stiffness calibrations [15] [16]. However, for studies of larger particles such as biological cells for instance, higher values of trap-stiffness is required through higher laser power, as the trap-stiffness has linear dependence on laser power [28]. Nevertheless, the same calculations can be performed to optimize the trap stiffness that shows significant Raman power variance.
The particle size calibration can also be done by solely studying the Raman power. If the measurement is done at a constant measurement time for particles of different sizes in a given trap-stiffness, then the Raman power variances for different particles will be seen at the same modulation frequency, but the values of such fluctuations will be different according to the size of the particle as shown in Fig. 4. In such a case, the size of the trapped particle can be calibrated by studying the Raman signal fluctuations in the modulated optical trap.
Fig. 4. Raman power variances for three different diameters $1$ $\mu m$, $5$ $\mu m$ and $10$ $\mu m$ for $\kappa = 8.17 \times 10^{-6}$ $N/m$, showing maximum fluctuations of approximately $60\%$, $80\%$ and $15\%$ respectively.
We performed our calculations for spherical particles for mathematical convenience, as it allows for approximating the drag coefficient from Stoke’s law. However, our method does not necessitate a spherical particle. The particle’s roll-off frequency, which is dependent on drag coefficient (Eq. (1)) irrespective of shapes and sizes, can be obtained by studying the Raman power variance. In this regard, the present theoretical model can be extended to particles of arbitrary shapes and sizes to study their hydrodynamic properties from spectroscopic signals (or, vice-versa). Though the results are based on analysis of the average Raman power, the formalism translates equally for any spectroscopic study done in modulated trapping beam.
4. Conclusion
We show that in addition to chemical identification and characterization offered by Raman or florescence spectroscopy, rich hydrodynamic information are also accessible through modulation of the trapping beam and measurement of the average spectroscopic signal as a function of modulation frequency. In an experiment, by fitting the experimental data of the mean square displacement with the predicted power spectrum of [15], a robust estimation of the trap-stiffness can be obtained which has been useful for quantitative measurements of the optical forces on biomolecules [17]. But since our method is based on measurements of spectroscopic signals specific to the particle, this offers a favorable alternative that overcomes the difficulties associated with the study of hydrodynamic properties in a complex environment consisting of several particles of similar optical characteristics in the observation volume. The key advantage of this method is that spectroscopic signals can be used for relative estimation of a particle’s drag coefficient, and conversely, the originating particle from which the spectroscopy signal is emitted can be distinguished according to the relative drag coefficients.
In this study we have shown the influence of size of the trapped particle on the spectroscopic signal in a modulated optical trap to offer a method of comparison of drag coefficients which is useful in distinguishing sub-micron particles in a complex environment. Furthermore, features like practically low modulation frequency and practical integration time, and not necessitating the need for accurate calibration of the trap stiffness for relative signal analysis, adds to the relevance of this method from an experimental point of view. This method serves as a spectroscopic method to study hydrodynamic properties of particles in a complex environment like that in a cellular matrix in situ in a single experiment, circumventing the difficulties and limitations associated with the present micro-Raman spectroscopy and time consuming in vitro imaging techniques [23–25,29] and introducing a new paradigm of particle analysis.
Appendix A
The overdamped equation of motion of the trapped particle is
Here $\langle \frac {\partial f}{\partial t}\rangle = 0$, $\langle \mu _t \frac {\partial f}{\partial x}\rangle = -2 \omega _o (1 + \epsilon cos\omega t) \sigma ^2_{xx}(t)$ and $\langle \frac {\sigma _t^2}{2} \frac {\partial ^2 f}{\partial x^2} \rangle = 2D$ and $\langle \sigma _t \frac {\partial f}{\partial x} dB_t \rangle = 0$ as $\langle xdB_t \rangle = 0$. So the position variance is
Appendix B
Here we present an alternate derivation of the Raman signal for a modulated Gaussian beam using the stochastic differential equation.
The Taylor’s expansion of the normalized Raman power Eq. (7) is:
Taking the first two terms, we write $f = 1 - \frac {4x^2}{w^2(z)}$. Since $\mu _t = -\omega _o(1 + \epsilon cos \omega t)x_t$ and $\sigma _t = \sqrt {2D}$, $\langle \mu _t \frac {\partial f}{\partial x} \rangle = 8\omega _o(1 + \epsilon cos \omega t) \frac {\sigma ^2_{xx}(t)}{w^2(t)}$, $\langle \frac {\sigma ^2_t \partial ^2 f}{2\partial x^2} \rangle = -8\frac {D}{w^2(z)}$ and $\langle \sigma _t \frac {\partial f}{\partial x} dB_t \rangle = 0$ as $\langle xdB_t \rangle = 0$. Using Eq. (15), we have
Using Eqs. (5) and (9) to express in terms of $V(t)$ and $\xi$ we have,
Now to compare this expression with our equation of the normalized Raman power for the modulated case, we take the time derivative of Eq. (10).
Since Eq. (19) and Eq. (20) differs by a factor $\left (\frac {\xi }{\xi + 8V(t)}\right )^{\frac {3}{2}}$, for an optical trap system in the experiments such as [15,16] where $\xi$ is such that, the factor $\left (\frac {\xi }{\xi + 8V(t)}\right )^{\frac {3}{2}} = 1$, Eq. (12) would be a good approximation to estimate the fluctuations of the average Raman signal for the modulated case.
Funding
The World Academy of Sciences (18-013RG/Phys/AS_I, 21-334 RG/PHYS/AS_G, 22-244RG/PHYS/AS_G).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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