Transition of a particle between adjacent optical traps: A study using catastrophe theory

In spite of the widespread use of optical tweezers as a quantitative tool to measure small forces, there exists no unambiguous and simple experimental method for either validating its theoretically predicted form or empirically parameterizing it over the entire range. This problem is addressed by studying the transition of a colloidal particle between two spatially separated optical traps. The transition as a function of the relative intensity of the traps and the separation between them reveals a formal resemblance to the “butterfly catastrophe” which also maps onto to phase transitions observed, for example, in ferroelectrics on a phenomenological level. The method has been used to experimentally determine the force-displacement curve for an optical trap over its entire range.

Figure 1

Phase diagram showing the number of minima in the potential for various values of the relative intensity $p$ and separation $D$; circles correspond to one minimum, vertical lines correspond to two minima, and crosses correspond to more than two minima. The forms of the potential at different points in the $p$-$D$ plane marked by $A\cdots L$ are shown in the bottom panels. In the top panel $D$ is plotted in units of $\lambda$.

Figure 2

(a) The experimental setup. A 1064 nm laser beam is used to form optical traps. The incident wave front is phase modulated using an SLM to form multiple traps. The sample consists of 5 $\mu$m silica microspheres in an aqueous environment. (b) and (c) Holograms generated using the Gerchberg-Saxton algorithm to create single optical traps. (d) Hologram obtained by combining (b) and (c) at a pixel fraction, $p=0.5$, using the random masking algorithm. The holograms are grayscale images (false color in the online version has been used for clarity). (e), (f), and (g) The corresponding simulated intensity profiles obtained by taking the Fourier transform of unit amplitude complex numbers corresponding to the phase in the holograms.

Figure 3

Motion of the particle as the relative intensity between the two traps is changed for separations $D\lambda$$=$ 1.5, 3.6, and 5 $\mu$m is shown in (a), (b) and (c), respectively. The onward run is shown as squares (black), while the return run is shown by circles (gray). (d)–(f) The corresponding bottom panels are the curves obtained from numerical calculations for identical parameters. The inset in (a) shows the position as a function of time as $p$ is changed from 0.62 to 0.63. The corresponding change in position is 40 nm.

Figure 4

The force-displacement curve for optical tweezers obtained from $x$ vs $p$ curves shown in Fig. 3 using the method described in the text is shown by circles connected with straight line segments. The scatterplot in the inset shows the power spectrum of fluctuation in position in the configuration $p=0$. The solid line is a Lorentzian fit to the data, giving a corner frequency of 12 Hz. Using this, we calculate the small displacement limit (dotted line) of optical trap stiffness, which has been used to provide the physical scale to the experimentally obtained force-displacement curve. The curve obtained from numerical calculations employing scattering theory is shown as a solid line.

Figure 5

(a) Hysteresis loops done at three different time rates. The solid, dashed, dash-dotted lines correspond to 0.5, 5, and 50 mHz, respectively. The width of the hysteresis loop $\delta p$ increases with increasing rate, as shown in the inset, implying a finite role of temperature. (b) The trap stiffness $k_{\mathrm{opt}}$ obtained from thermal fluctuations in position as a function of $p$ for $D\lambda$$=$ 5 $\mu$m. Only the return part of the cycle is shown.

Figure 6

(a) Time series of the position in the presence of an oscillatory perturbation at 1 Hz in the configuration $D\lambda =4$ $\mu$m and $p=0.50$. (b) The spatial distribution of the residence time obtained from the data shown in (a) normalized with the total time of measurement. The bottom panel shows similarly obtained distributions of the residence time for $p=0.43,0.46,0.50,0.53,0.59$, respectively. The dumbbell shaped distribution in any one potential minimum is a consequence of the fact that in a sinusoidal oscillation a particle spends more time towards the extreme than in the center.