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Optimized optical tweezers are of great importance for biological micromanipulation. In this paper, we present a detailed electromagnetic-based calculation of the spatial intensity distribution for a laser beam focused through a high numerical aperture objective when there are several discontinuities in the optical pathway of the system. For a common case of 3 interfaces we have shown that 0.01 increase in the refractive index of the immersion medium would shift the optimal trapping depth by 3–4μm (0.2–0.6μm) for aqueous (air) medium. For the first time, We have shown that the alteration of the refractive index of the immersion medium can be also used in aerosol trapping provided that larger increase in the refractive index is considered.
©2011 Optical Society of America
1. Introduction
Optical Tweezers are widely used as non-invasive micromanipulation tools in many scientific areas, from biology [1–4] to nanotechnology [5–8]. Typical Optical tweezers (OT) consist of a Gaussian laser beam tightly focused through a high Numerical Aperture (NA) objective lens producing a 3-D intensity gradient at the focus. An object with the refractive index greater than that of the surrounding medium experiences a Hookean restoring force toward the focus [1] for which the strength of the trap can be regarded as the spring constant. A micron (and nano)-sized sphere trapped by OT is widely used as a handle of a non-contact micromanipulator. Nanometer spatial resolution along with sub-Megahertz temporal resolution have turned OT to a widely desired tool in many scientific areas. OT are normally implemented into an optical microscope in order to visualize the specimen under manipulation. Oil immersion objective lenses are commonly used for OT-based micromanipulation due to their high NA which provides stronger trap along with more detailed visualization of the sample. A significant problem of using oil immersion objectives is the Spherical Abberation (SA) induced by the refractive index mismatch between the immersion (oil) and sample (water) media. It is well known that the SA dramatically increases as the trapping (and visualization) depth increases which limits the trapping depth range. For example, a 1μm polystyrene bead can only be trapped up to depth of ∼ 10μm. The situation becomes even worse when trapping of nanoparticles is on demand. Therefore, finding a method for optimized nanoparticle trapping deep inside the sample chamber would be of great interest for in-depth micromanipulation. Different methods are proposed [9–13] to compensate for the SA introduced by oil immersion objectives among which the changing the refractive index of the immersion medium [9] seems to be more feasible. Reihani et al. has shown that, first, for an immersion oil with a given refractive index, there would be a depth (so-called optimal depth) at which the stiffest trap occurs, second, by increasing the Refractive Index of the Immersion Medium (RIIM) the optimal depth shifts toward the deeper positions. In this letter, we present a detailed electromagnetic-based calculation of the intensity profile around the focus of the objective as well as the restoring force of the optical trap in presence of several refractive index discontinuities in the optical pathway of typical OT. Considering the case of 3 interfaces (very common case in OT applications), we have theoretically confirmed that for trapping inside water, 0.01 increase in RIIM would shift the optimal trapping depth by 3 – 4μm which is in very good agreement with the previously reported experimental results [9]. We also have shown that for the case of trapping in air, 0.01 increase in RIIM would shift the optimal trapping depth only by 0.2 – 0.6μm which implies that the alteration of the RIIM can also be used for aerosol trapping, provided that larger increase in the RIIM is considered. For example, we have shown that an immersion medium with refractive index of 2.11 would provide the optimal depth of ∼ 36μm which could be of great interest for aerosol trapping community.
2. Calculation of optimal RIIM for OT
When a linearly polarized laser beam is focused through an aplanatic objective into a homogeneous medium (no refractive index mismatch), the electric field at a given point P around the focus (the origin located at the Gaussian focus center; defined by “O” in Fig. 1) can be written as [14]:
where k is wavenumber in the medium, ŝ = sxî + sy ĵ + szk̂ is a unit vector along a typical ray, Ω is the solid angle formed by all rays emerging the objective (defined by NA of the objective), r⃗P is the position vector of point P, Φ is the SA function introduced by the objective itself, and finally a⃗ is the electric strength vector at the entrance of the objective. A similar equation can be written for magnetic field. It is shown that in the presence of a planar interface located at z = −zI, with different refractive indices at two sides (n 1 and n 2), and assuming that the objective introduces no aberration (Φ = 0), the electric field at the interface (in the second medium) can be written as [15]: where k 1 is the wavenumber in the first medium, , and T 1→2 is an operator describing the change in the electric field of the ray passing through the interface. This operator would be a function of n 1, n 2, and incident (or refraction) angle at the interface [15]. Equation (2) can be extended to the general case where the system contains m different media (n 1 ,..., nm) with m-1 interfaces. For such a case, if the electric field at the entrance aperture of the objective considered to be as (w 0 being the beam waist and ), then the electric field inside the mth medium in the spherical coordinate system centered at the focus can be written as: where α is the convergence angle of the marginal rays defined by the NA of the objective (NA = nsinα, with n being the refractive index of the immersion medium), tk (k=2, 3, ..., m) is the thickness of kth medium, and ϕi (i=2, 3, ..., m) is the refraction angle in the equivalent medium. Esample is the electric field strength vector inside the sample medium given by , , and . Note that τp, and τs define the fresnel transmission coefficients for p and s polarization states, respectively. It is worth mentioning that the intensity distribution around the focus can be given by square of Eq. (3). On the other hand, it is well known that, the restoring force (gradient force) of OT is proportional to the intensity gradient [16–20]. Therefore one can use Eq. (3) to estimate the restoring (gradient) force of OT by calculation of the average intensity gradient over the extend of the trapped object [19, 20]. By maximizing the calculated restoring force, one can search for the parameters (such as refractive index of the immersion medium [9]) which provide the optimal trapping conditions.
Fig. 1 The optical pathway of a typical ray. The dotted line defines the optical path of the ray as if there is no refractive index mismatch in the optical pathway. The solid line represents the path of the same ray when nim > ng = nobjective > ns. Δz defines the shift of the ray in the axial direction.
In the most of the OT applications the sample medium (mainly water) is sandwiched between two coverglasses (refer to Fig. 1). In such a case there would be 3 planar discontinuities in the refractive indices (m=4), with media being the objective’s top lens (n 1 = nobj = 1.518), immersion medium (n 2 = nim), coverglass (n 3 = ng = 1.518, note that for a standard coverglass #1.5, t 3 = 170μm), and sample (n 4 = ns, t 4 = d =probe depth, refer to Fig. 1 for definition). Note that when the objective’s top lens, coverglass, and the immersion medium are index matched (nobj = nim = ng = 1.518) then there would be only one interface (coverslip-water) which very often happens in OT applications. In the following sections the results for the two most popular cases will be presented.
2.1. Trapping in water
Trapping inside an aqueous medium using an oil immersion objective is very common in OT applications for which ns = 1.33, and nim = 1.518. Figure 2 shows the resulted axial (Fig. (2a)) and typical lateral (Fig. (2c)) intensity profiles produced by an oil immersion objective (NA=1.3, working distance=200μm) through a coverglass of 170μm thick. Note that the lateral intensity profile varies at different depths. The calculated average intensity gradients acting on a 1μm polystyrene bead trapped in such intensity profiles are shown in Figs. 2(b) and 2(d), for the lateral and axial directions, respectively.
Fig. 2 Trapping inside water: intensity distribution in the axial (a) and lateral (c) directions for different immersion oils. Calculated axial (b) and lateral (d) Average Intensity Gradient (AIG) for a 1μm dielectric microsphere trapped using an objective with NA=1.3 and working distance of 200μm. The electric field beam waist of the incident laser beam was considered to be w 0 = 3mm.
Figures 2(a)–2(d) illustrate that: (1) for n = 1.518, where the system is considered to be abberation-free, the optimal trap occurs just in the vicinity of the coverglass inner surface (d = 0). (2) The trapping strength significantly decreases as the trapping depth is increased. This is very common for trapping using oil immersion objectives [9]. (3) By increasing the refractive index of the immersion medium, the optimal trapping depth (minimum spherical aberration) shifts toward the deeper positions which supports the previous experimental observasions [9]. (4) As it can be seen from Fig. 2(b), the maximum trapping strength decreases slightly by increasing the nim, e.g., the maximum for nim = 1.56 is 16.5% lower compared to nim = 1.518. For each case the total power at the focus was calculated by integrating the intensity over the focal plane. The results show that this effect is mainly due to the reduction (12.3%) in the power transmitted into the sample. In other words, the total transmission coefficient of the planar interfaces decreases as nim increases.
The same calculations were repeated for the lateral direction, to search for the depth at which optimal lateral trap occurs (dopt,lat ). The results for both directions are summarized in Table 1.
Table 1 implies that 0.01 increment in nim results in 3 – 4μm shift for the depth at which the optimal axial trap occurs. This is in very good agreement with the previously reported experimental results [9]. For the lateral direction, our results suggest 3.5 – 4.2μm shift for the depth at which the optimal lateral trap occurs. Note that due to the refractive index mismatch, the real focus of the laser differs from the probe depth which is defined as the distance traveled by the objective. This calculation would be of great importance when the exact position of the trap inside the sample or the distance from the chamber wall is required.
2.2. Trapping in air
Optical tweezers have also been widely used for aerosol trapping [21]. For this case, same calculations can be repeated using m = 4 and nsample = 1 to find the optimal conditions for aerosol trapping. It is worth mentioning that the total internal reflection may limit the effective NA of the system. For example, in the case of trapping inside water, the upper limit for effective NA would be 1.33 while for the case of aerosol trapping it can not exceed 1 due to the total internal reflection at the glass-air interface. Figure 3 shows typical axial (Fig. (3a)) and lateral (Fig. (3c)) intensity distributions as well as the calculated average intensity gradient in both axial (Fig. (3b)) and lateral (Fig. (3d)) directions.
Fig. 3 Trapping in air: intensity distribution in the axial (a) and lateral (c) directions for different immersion oils. Calculated axial (b) and lateral (d) Average Intensity Gradient (AIG) for a 1μm dielectric microsphere trapped using an objective with effective numerical aperture of 1 and working distance of 200μm. The electric field beam waist of the incident laser beam was considered to be w 0 = 3mm.
Figure 3 shows that: (1) The maximum of the intensity graphs is considerably lower compared to the case of trapping in water. This is mainly due to the decreased transmission coefficient for the current case. (2) The intensity distributions are wider compared to the water case. These considerations explains why the restoring force of the trap both in the axial (Fig. (3b)) and lateral (Fig. (3d)) directions is considerably lower compared to the water case. Considering the larger refractive index contrast when the object is trapped in air (compared to water) and the fact that the trapped object would has larger wiggling due to the lower viscosity of the air (compared to water), it can be explained that why trapping in air is always harder than in water. Table 2 quantitatively summarizes the results for trapping in air using an objective with effective NA of 1.
From Table 2 it can be deduced that 0.01 increment in nim shifts the optimal depth for the axial (lateral) trap by 0.2 – 0.6μm (0.4 – 0.6μm). Note that the shift is very small compared to the case of trapping in water. Therefore, changing the refraction index of the immersion medium may not be very helpful for aerosol trapping unless a considerably larger increase in nim is considered. As an example, the axial intensity distribution as well as the axial AIG for the immersion medium with nim = 2.11 is shown in Fig. 4. Note that the optimal depth is shifted to d = 36μm using nim = 2.11 which could be of great importance for aerosol trapping applications.
Fig. 4 The axial intensity distribution (a) and AIG (b) for a 1μm dielectric sphere trapped using an objective with NA=1 and nim = 2.11. It is worth mentioning that the transmission coefficient for this case is reduced by %51 compared to the case of nim = 1.518.
3. Conclusion
We presented a detailed electromagnetic-based calculation of the intensity distribution around the focus of optical tweezers. These calculations can be used to search for the conditions which provide the ultimate functionality of OT. Considering the practical case of having 3 refractive index discontinuities in the optical pathway of the system, we have shown that for aqueous samples, 0.01 increase in the refractive index of the immersion medium would shift the optimal trapping depth by 3–4μm to deeper positions. This is in very good agreement with the previously reported experimental results. We also have shown that this method can be used for aerosol trapping applications provided that a larger increase in the refractive index of the immersion medium is considered. For example, we have shown that nim = 2.11 (which is commercially available) would shift the optimal depth to d = 36μm. These results could be of great importance for optical micromanipulation community.
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