## Abstract

In this paper, for the first time, we report on systematic theoretical and experimental investigation of Phase Contrast Optical Tweezers (PCOT) which could be an indispensable tool for micromanipulation of the transparent micro and nano objects such as biological tissues and vesicles. The quadrant photodiode detection scheme and the power-spectrum calibration method is shown to be valid for this case. We have shown that the phase objective with new designed phase plates can provide nearly aberration-free condition at a desired depth. This could be a valuable advantage for simultaneous in-depth micro-manipulations and visualization of the sample.

©2010 Optical Society of America

## 1. Introduction

Optical Tweezers (OT) are indispensable micromanipulation tools in many scientific fields from biology [1–4] to nanotechnology [5–9]. OT are based on a Gaussian laser beam tightly focused through a high quality objective lens producing a 3-D Gaussian intensity profile at the focus. An object with the refractive index greater than that of the surrounding medium experiences a Hookean force towards the focus [10]. The strength of the trap can be regarded as the spring constant. Such an instrument is widely used for micromanipulation using micron (and nano)- sized objects without any mechanical contact with the sample. Nanometer spatial resolution along with sub-Megahertz temporal resolution have turned OT to a widely desired tool in many scientific communities. OT are normally implemented into a Bright Field Microscope (BFM) in order to visualize the specimen under manipulation. However, for phase specimens (specimens which differs only in the refractive index compared to the surrounding medium, *e.g.*, biological tissues) and tiny objects such as vesicles BFM fails to provide reasonable contrast. Therefore, combination of OT with different contrast enhancement techniques would be of great interest for manipulation of the phase specimens. There have been previous attempt in combination of OT with Differential Interference Contrast (DIC) techniques [11]. However, it is well known that Phase Contrast Microscopy (PCM) and DIC microscopy techniques should be considered as complementary (rather than competing) methods of viewing phase objects (such as biological tissues) depending on the size and the refractive index of the object [12,13]. PCM introduced by Zernike [14, 15] combined with OT would be of great interest especially for the biological micromanipulation community. Although this combination has been qualitatively used for visualization of the sample under manipulation [16], however, to our knowledge, there is no report of quantitative investigation available for this case. In this report we present the results of our theoretical and experimental systematic investigation on “Phase Contrast Optical Tweezers” (PCOT). The effect of the Objective Phase Plate (OPP) and the Condenser Annular Aperture (CAA), respectively, on the trap quality and the detection signal are widely investigated both by theory and experiment and the results are compared to that of the popular BFM based OT. We have shown that the phase plate implemented in the phase objective lens improves the axial trap to some extent which could be an advantage for in-depth micromanipulation applications. Our theoretical calculations promises design of new objective phase plates for specific applications such as spherical aberration compensation. We have also shown that the quadrant photodiode back focal plane detection scheme [17] and the power-spectrum calibration method [18] is still valid for PCOT. In fact, presence of the CAA could improve the sensitivity of the detection signal. Based on our theoretical calculations new designs of CAA could provide even better sensitivities.

## 2. Implementation of the Phase Contrast Optical Tweezers (PCOT)

The conventional BFM can be regarded as an amplitude contrast microscope. This microscopy technique is commonly used for visualization of *amplitude* microscopic objects (objects which only alter the intensity of the illuminating light). Though bright field microscope succeeds in many microscopic visualizations, however, it fails in observation of *phase* objects (objects which only alter the phase of the illuminating light, *e.g.*, biological tissues). Basically, PCM converts the contrast in the phase into the contrast in the intensity so that the sample could be visualized by human eye. As an example, Fig. 1 shows the images of a phospholipid vesicle with the thickness of ~ 5*nm* in the vicinity of a trapped polystyrene bead in both bright field (a) and phase contrast (b) modes. It is clear that in the bright field mode the vesicle can not be recognized while in the phase contrast mode one can easily see even some more details of the vesicle. PCM contains two additional components (compared to BFM) on its optical path, being: (a) Condenser Annular Aperture (CAA): an annular aperture which is located at the back focal plane of the condenser. The parallel light rays of the illumination source is cut by this aperture into a circular shape and then focused by means of a condenser forming a light cone illuminating the sample. (b) Objective Phase Plate (OPP): a phase plate containing a phase ring implemented in the front focal plane of the phase objective lens. The diameter of this phase ring is tuned so that the illuminating light cone (direct light) fully passes through this ring while diffracted light may pass through the whole area of the front focal plane of the objective. The phase plate introduces an extra phase difference (*ϕ _{p}*) between direct and diffracted rays. Depending on the imaging system (positive or negative) this phase could be

*ϕp*= ±

*π*/2 (one quarter of the wavelength at

*λ*= 550

*nm*) [13]. Along with another

*π*/2 phase difference due to the scattering from the specimen [12], constructive (

*ϕ*= 0) or destructive (

_{total}*ϕ*=

_{total}*π*) interference between direct and scattered lights produces contrast in the image. Since the intensity of the scattered light is considerably smaller than that of the direct light, an absorptive layer is normally coated on the phase ring which improves the visibility of the final image by reducing the intensity of the direct light. The CAA can be easily removed from the optical path if necessary but since the OPP is an integrated part of the phase objective, to remove the OPP from the optical path one has to replace the phase objective with similar bright field (normal) objective.

Our OT setup (Fig. 2) is based on a continuous wave laser beam (Nd:YAG, Compass *λ* = 1064*nm*, Coherent) introduced into an inverted microscope (Olympus, IX-71) through its fluorescence port. The sample chamber was assembled from a coverslip and a microscope slide spaced by two stripes of double sided Scotch tape as spacers. The sample chamber was mounted on a piezo stage (Physik Instrumente, PI-527.2cl) which along with a piezo equipped objective holder (Physik Instrumente, P-723.10) provides nanometer resolution for three dimensional positioning of the trap inside the chamber. The laser beam was focused through a high Numerical Aperture (NA) objective lens to the diffraction limit. The objectives used in this research were (Olympus, 100×, NA=1.3, oil) and (Olympus, 100×, NA=1.3, oil, Ph3) which hereafter will be called “normal” and “phase” objectives, respectively. The only difference between the objectives was the phase plate implemented in the phase objective (their transmission [19] is measured to be identical for our wavelength). The laser light passing through the sample was collected by means of a high numerical aperture (NA=1.4) condenser and directed towards the quadrant photodiode (Hamamatsu, S5981) positioned at the optical conjugate plane of the back-focal-plane of the condenser [17]. The polystyrene beads used in this research were purchased from Sigma (LB-11) with the mean diameter of 1.09*µm* and diluted in double distilled water. The immersion oil used in this research was from Olympus with the refractive index of *n* = 1.518.

#### 2.1. Power spectrum calibration

It is known that the motion of a particle inside a Hookean optical potential well can be given by Langevin equation, *γẋ* + *κ*Δ*x* = *ζ*(*t*), in which *γ*, *ζ*, *κ* and Δ*x* are drag coefficient, stochastic force, trap stiffness, and displacement of the particle from the center of the trap, respectively [18]. Fourier transform of the Langevin equation results in a Lorentzian form power spectrum,

where *k _{B}*,

*T*and ${f}_{c}=\frac{\kappa}{2\pi \gamma}$ are Boltzman constant, absolute temperature and corner frequency, respectively. Note that for a spherical particle, one can write

*γ*= 6

*πηr*with

*r*and

*η*being the radius of the particle and viscosity of the surrounding fluid (water).

One common method for calibration of optical tweezers is called power spectrum analysis method [18] in which the positional time series of the trapped bead is recorded by digitizing the signal of the quadrant photodiode at sampling frequency of *f _{s}*. Fourier transform of the recorded time series results in the power spectrum of the recorded data. The corner frequency and trap stiffness can be extracted by fitting the experimental power spectrum to Eq. (1) provided that

*T*,

*r*, and

*η*are known. It should be mentioned that if the trapped microsphere is very close to the glass surface (within couple of particle’s radius) the hydrodynamic effect of the glass wall should be considered [20].

In the following sections the effect of each of the CAA and OPP on the OT system will be examined by both theory and experiment.

#### 2.2. The effect of the objective phase plate (Experiment)

In order to have a well defined OT it is crucial to have a linear dependency between the laser power and the trap stiffness. To check it out for PCOT, first CAA was removed from the optical path and a polystyrene bead was trapped using the phase objective and then it was moved to the depth of 4*µm* where the positional signals were recorded at different laser powers with sampling rate of 22KHz. Power spectrum of each time series was fitted to Eq. (1) using a freely available Matlab program [18] to extract the corner frequency (*f _{c}*) and trap stiffness (

*κ*) by anti-aliasing the data and setting the sampling and Niquist frequencies to 22KHz and 11KHz, respectively.

*T*,

*r*, and

*η*were set to 298

*K*, 0.545

*µm*and 10

^{−3}

*N.S.m*

^{−2}, respectively. Considering the temperature increase rate of 7

*K*/

*W*[21], the increase in the temperature at the focus would be less than 0.7

*K*. Therefore, viscosity was considered constant throughout the experiments. Figure 3 shows the resulted graphs for the lateral (a) and axial (b) directions which clearly confirms the linear behavior. The immediate question would be, “

*How the quality of the trap compares for the PCOT and bright field OT at a given laser power?*” To answer it we performed a series of trapping experiments at a fixed laser power (40 mW at the sample) using the phase and normal objectives. For the measurements of the lateral direction the condenser iris was fully open while for the measurements of the axial direction the condenser iris was closed to one third [22]. Figure 4 shows the lateral as well as the axial trapping stiffness at different depths for the normal and phase objectives. Figure 4 illustrates that: (1) Both normal and phase objectives provide a maximal stiffness at a certain depth (optimal depth [23]) in the lateral and axial directions which is due to the minimization of the spherical aberration [23]. Note that for axial direction the immersion oil with refractive index of 1.518 along with an objective from Leica provided the optimal depth somewhere inside the coverglass [23] while in the present work the optimal depth using an Olympus objective occurs at certain depth inside the chamber [Fig. 4(b)]. We tested two objectives from each brand and found out that the optimal depth is almost same for each brand. In short one can conclude that objectives from different brands enter different amount of SA into the system. (2) The optimal depth differs considerably for the axial (~ 4

*µm*) and lateral (~ 17

*µm*) directions. To our knowledge, this is the first time such an effect is reported. In fact, we noticed that this difference in the optimal depth is size dependant which would makes sense considering the fact that beads with different sizes would cover different portions of the focal spot. Note that at depth of ~ 17

*µm*the axial stiffness reduces to about one third of the maximal value, and at depth of ~ 4

*µm*the lateral stiffness drops by ~ 25%. Therefore, it is important to choose the right depth for the manipulation experiment in order to use the ultimate efficiency of the trap. (3) In the axial direction both traps has almost equal stiffness at the optimal depth, however, as the depth is increased first the phase graph drops slightly faster but later it decreases slower so that at depths above 30

*µm*it approaches to a plateau shape. As it can be seen from the graphs the bead was stably trapped in 3-D up to the depth of 40

*µm*(26

*µm*) using the phase (normal) objective. This implies ~ %54 improvement in the accessible depths using the phase objective (compared to the normal objective) which could be valuable when the experiment has to be conducted far away from the chamber walls. It is worth mentioning that for submicron beads which can only be trapped in a very narrow depth band (say few microns) using oil immersion objectives, such a cheap method (compared to other methods such as using SLM [24]) of extension in the accessible depth would be highly desired. Note that the objective phase plate is not designed for our wavelength and new designs could provide even better improvements.

#### 2.3. The effect of the objective phase plate (Theory)

For an objective satisfying the sine condition and illuminated with a linearly polarized Gaussian laser beam, the distribution of the electric field’s amplitude around the focus of the objective can be written as [25]:

$$\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\times {\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}J}_{0}\left({k}_{0}{n}_{1}r\phantom{\rule{.2em}{0ex}}\mathrm{sin}{\theta}_{1}\right)\mathrm{exp}\left({\mathrm{ik}}_{0}{n}_{2}z\mathrm{cos}{\theta}_{2}\right)d{\theta}_{1}$$

Where *α*
_{0} is the convergence angle of the objective; *θ*
_{1} and *θ*
_{2} are the incident and refraction angles in the glass-water interface; f, *w*, and *k*
_{0} are the focal length, beam waist of the Gaussian beam before the objective, and the wavenumber in the vacuum, respectively. *τ _{s}* and

*τ*are the Fresnel transmission coefficients for

_{p}*s*and

*p*polarization states.

*z*and

*r*define the axial and lateral components of the cylindrical coordinate system positioned at the focus.

*J*

_{0}denotes the Bessel function of first kind zero order. The phase

*Ψ*=

*−d*(

*n*

_{1}cos

*θ*

_{1}−

*n*

_{2}cos

*θ*

_{2}) denotes the spherical aberration due to the mismatch between the refractive indices of the sample and immersion media in which

*d*defines the depth of the trap [23] when there is no focus shift or simply the distance traveled by the objective (refer to the inset of Fig. 2). It has been previously shown that by increasing the refractive index of the immersion medium the optimal depth (minimum aberration) shifts to deeper positions [23]. In such a case

*d*denotes depth of trap with respect to the optimal depth. Typical axial and lateral intensity distributions near the focus of the objective for different depths are shown in Figs. 5(a) and 5(c) (black curves). It is known that the restoring force of OT is proportional to the intensity gradient, therefore, the Averaged Intensity Gradient (AIG) over the part of the focus which is covered by the particle could be regarded as the restoring force [26]. Figures 5(b) and 5(d), respectively, show the results of such calculations on the axial and lateral intensity distributions at different depths.

The effect of OPP on the electric field distribution can be deduced from Eq. (2) provided that the phase introduced by OPP is properly included [27]. In order to do that, the integration rang of the Eq. (1) was divided into 3 sub-ranges: (a) from zero to *α*
_{1}, (b) from *α*
_{1} to *α*
_{2} (for this range the phase introduced by OPP should be included), and (c) from *α*
_{2} to *α*.

The definitions of *α*
_{1}, *α*
_{2} and *α* are shown in Fig. 6(a). Figure 6(b) illustrates a photo of the OPP implemented into the phase objective in which *r*
_{1}, *r*
_{2}, and *R* are measured to be 1.60*mm*, 2.05*mm*, and 3.30*mm*, respectively. Note that the values of *α*
_{1} and *α*
_{2} can be simply defined as
${\alpha}_{1}=\mathrm{arctan}\left\{\frac{{r}_{1}}{R}\mathrm{tan}\left[\mathrm{arcsin}\left(\frac{\mathrm{NA}}{n}\right)\right]\right\}$
and
${\alpha}_{2}=\mathrm{arctan}\left\{\frac{{r}_{2}}{R}\mathrm{tan}\left[\mathrm{arcsin}\left(\frac{\mathrm{NA}}{n}\right)\right]\right\}$
in which *NA* and *n* are the numerical aperture of the objective, and the refractive index of the immersion medium, respectively. For our case *α*
_{1} and *α*
_{2} were 38.8° and 45.8°, respectively. Since the refractive index of the phase plate (normally made of *MgF*
_{2}) does not change over the visible and near infra-red wavelengths [28], therefore, one can write
${\varphi}_{p,1064}=\frac{\pi}{2}\times \frac{550}{1064}\simeq \frac{\pi}{4}$
. Considering the values of *α*
_{1}, *α*
_{2}, and *ϕ _{p}* the calculations were repeated for the phase objective and the results are shown as gray curves in Fig. 5.

Comparison of the black and gray curves of Fig. 5 reveals that: (1) At the depth of 1*µm* the restoring force produced by the phase objective is 7% (13%) lower in the axial (lateral) direction which is in good agreement with the experimental results considering the experimental error bars. (2) At deeper positions the phase objective provides stronger restoring force both in the lateral and axial directions. This means that the phase objective can provide stable trap for deeper positions compared to the normal objective which confirms the experimental results.

So far we have examined the effect of the phase ring implemented in the phase objective and found out that it can improve the trap by sharpening the intensity distribution at the focus. However, as it is mentioned earlier, the specification of the phase ring may not be optimum for trapping purposes. We believe that using Eq. (2) one can design new phase rings which optimizes the trap for specific purposes such as spherical aberration compensation. As an example we searched for the conditions at which the spherical aberration at depth of 10*µm* is minimized. The variables were the number of the rings, their radii and the phase induced by each ring. We considered one simple case (with 2 rings one of which is exactly same as the phase ring of our phase objective) and one complicated case with 5 separate rings. Typical results are shown in Fig. 7. Considering that the intensities are normalized to the maximum intensity produced by the normal objective at zero depth (zero spherical aberration), one can conclude that the 5 ring case significantly sharpen the distribution by compensating the spherical aberration. It is worth mentioning that this phase plates do not have to be implemented in the objective and instead they can be positioned where is optical conjugate to the objective’s front focal plane (somewhere outside the objective).

#### 2.4. The effect of the condenser annular aperture on the detection system (Experiment)

As explained earlier the second optical element of a typical PCM would be the CAA at the back focal plane of the condenser. It is clear that such an element would cut out a reasonable portion of the laser light illuminating the photodiode, hence, reducing the signal level. However, it is essential to know the effect of such reduction on the detection system and the parameters deduced from the signal such as the stiffness and the conversion factor. Basically, what one should do is to repeat some measurements with and without this optical element in the optical path. Like previous section, first the linearity of the stiffness with the laser power was examined by repeating the measurements of section 2.2 with keeping the CAA in the optical path. Apart from the CAA the rest of the experimental details were kept similar to those in the section 2.2. The resulted graphs are shown in Fig. 8. Comparison of Fig. 8 and Fig. 3 reveals that the resulted linear fits are in very good agreement. Therefore, one can conclude that the CAA does not alter the resulted stiffness though it reduces the signal level. Figure 9 shows the results of measured stiffness produced by the phase objective as a function of the trapping depth with and without the CAA in the optical path. It can be seen that the resulted graphs (black squares and red circles) are very similar, both in the lateral and axial directions. In other words, the presence of the CAA in the optical path does not change the results of measured stiffness even in the presence of the spherical aberration. The last thing to check would be the effect of the CAA on the positional response (voltage-position) curve of the photodiode. Since the output of the photodiode is in voltage, it is crucial to have a linear relationship between the photodiode signal (in volts) and the real displacement of the bead in the trap (in nanometer). For the linear region of the positional response curve one can write *x* = *β _{x}V_{x}* where

*x*,

*β*, and

_{x}*V*are the displacement of the bead in the trap, a constant so called conversion factor, and the voltage change in the photodiode signal, respectively. To extract the positional response curve, a stuck bead was scanned through the focus of the laser with nanometer resolution while recording the output signal of the photodiode. To make sure that the beads were firmly stuck to the glass chamber wall, the bead solution was diluted in 300mM NaCl buffer before it is loaded into the chamber. To make sure the position of the bead with respect to the focus is as if it is trapped, zoomed images of trapped and stuck beads were compared. Figure 10 shows typical resulted curves with and without having CAA in the optical path. It should be mentioned that due to the difference in the signal levels of the two cases, the signals were normalized before they are plotted together in one graph. Figure 10 illustrates that: (1) there is a linear region in the central part of the both graphs which means the quadrant photodiode back focal plane detection scheme [17] is applicable in the presence of the CAA. (2) The slope of the linear part is greater in the presence of the CAA by ~ 28% which means this system potentially is capable of detecting finer displacements provided that the signal covers the entire voltage range (more amplification is required). (3) The linear range seems to be slightly shorter for the case with CAA which might limit the range at which position detection can be done.

_{x}#### 2.5. The effect of the condenser annular aperture on the detection system (Theory)

Consider a particle which is initially placed at the focus of the laser. In the spherical coordinate system positioned at the focus, the change in the intensity pattern at the back focal plane of the condenser due to the displacement *x*′ of the particle from the optical axis in the direction defined by *θ* and *ϕ* (refer to Fig. 11) can be written as [17]:

where *k*, *α*, *w*
_{0} and *I _{tot}* are the wave number inside the solution, polarizability of the particle, the beam waist of the laser at the focus, and the total intensity on the photodiode, respectively.

*r*denotes the radial coordinate of different positions on the photodiode (Fig. 11).

Integration of Eq. (3) for two half planes (right and left) followed by subtraction of the resulted values (*I*
_{+} − *I*
_{−}) gives rise to:

where *θ _{min}* and

*θ*denote the marginal angles at which light rays reach the photodiode. Note that in Eq. (4),

_{max}*I*

_{−}=

*−I*

_{+}[17]. For a condenser with a given NA one can write

*θ*= 0 and

_{min}*θ*= arcsin(

_{max}*NA*/

_{r}*n*) in which

_{cond}*n*is the refractive index of the immersion medium of the condenser, and

_{cond}*NA*represents the effective NA at which the light rays are collected by the condenser.

_{r}*NA*is given by the NA of the restricting element (could be the condenser or objective). In our case, for the bright field case (without CAA) the objective was the restricting element (lower NA), hence,

_{r}*NA*= 1.3 which implies

_{r}*θ*= 0° and

_{min}*θ*= 58.9°. For the phase contrast case (with CAA) the CAA cut some parts of the rays, hence, the CAA of the condenser would be the restricting element. For this case

_{max}*θ*and

_{min}*θ*are measured to be 55° and 58°, respectively. Fig. 12 represents the plots of the Eq. (4) for the bright field (black squares) and phase contrast (red circles) cases. Figure 12 shows that: (1) There is a linear region for both graphs which supports the experimental results. (2) Like the experimental results, the slope of the phase contrast case is greater than that of the bright field, however, theoretical increment of the slope (~ 54%) is somewhat bigger than that of the experimental results (~ 28%).

_{max}## 3. Conclusion

Optical tweezers are indispensable tool for micromanipulation. Phase contrast microscope is proven to be a favorable choice for visualization of the biological tissues. Combination of the optical tweezers and phase contrast microscope could be of great interest for micromanipulation of the biological tissues as well as other transparent nanostructures such as vesicles. We have widely examined such a combination by both theory and experiment. In fact, we found out that: (1) The back focal plane detection scheme and the power spectrum calibration method is valid for PCOT. (2) The presence of the objective phase plate and the condenser annular aperture not only do not alter the results but also, to some extent, improves the optical trap and the detection system, respectively. The OPP increases the 3-D stable trapping depth for a 1.09*µm* polystyrene bead by 54% which would be valuable for trapping submicrone particles which can only be trapped in a very narrow depth window. The CAA improves the sensitivity of the detection system by 28%. (3) Our theoretical results show that wise design of the OPP can result in nearly aberration-free condition deep inside the sample chamber. This could be of great interest for in-depth micromanipulation experiment using oil immersion objectives. The combination of phase contrast microscope and optimal optical trap could provide a valuable tool for biological micromanipulation community.

## Acknowledgements

We acknowledge F. Hajizadeh for stimulating discussion and her assistance for the numerical part. We also thanks P. M. Bendix for providing us with the vesicle sample.

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