## Abstract

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 3–4*μm* (0.2–0.6*μm*) 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 OSA

## 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 1*μ*m polystyrene bead can only be trapped up to depth of ∼ 10*μm*. 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 – 4*μm* 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 – 0.6*μm* 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 ∼ 36*μm* 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]:

*ŝ*=

*s*+

_{x}î*s*

_{y}*ĵ*+

*s*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),

_{z}k̂*r⃗*is the position vector of point

_{P}*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*= −

*z*, with different refractive indices at two sides (

_{I}*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]:

*k*

_{1}is the wavenumber in the first medium, ${\mathbf{\text{W}}}_{e}\hspace{0.17em}=\hspace{0.17em}\frac{\overrightarrow{a}({s}_{1x},{s}_{1y})}{{s}_{1z}}$, 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}

*,..., n*) with m-1 interfaces. For such a case, if the electric field at the entrance aperture of the objective considered to be as $\overrightarrow{E}\hspace{0.17em}=\hspace{0.17em}{E}_{0}{e}^{-{\rho}^{2}/{w}_{0}^{2}}\widehat{i}$ (

_{m}*w*

_{0}being the beam waist and $\rho \hspace{0.17em}=\hspace{0.17em}\sqrt{{x}^{2}\hspace{0.17em}+\hspace{0.17em}{y}^{2}}$), then the electric field inside the

*m*medium in the spherical coordinate system centered at the focus can be written as:

_{th}*α*is the convergence angle of the marginal rays defined by the NA of the objective (

*NA*=

*n*sin

*α*, with

*n*being the refractive index of the immersion medium),

*t*(

_{k}*k*=2, 3, ...,

*m*) is the thickness of

*k*medium, and

_{th}*ϕ*(

_{i}*i*=2, 3, ...,

*m*) is the refraction angle in the equivalent medium.

*E*is the electric field strength vector inside the sample medium given by ${E}_{\mathit{\text{sample}},x}\hspace{0.17em}=\hspace{0.17em}{\Pi}_{l=1}^{l=m-1}\hspace{0.17em}{\tau}_{\mathit{\text{pl}}}\hspace{0.17em}{\text{cos}}^{2}\hspace{0.17em}\theta \hspace{0.17em}\text{cos}\hspace{0.17em}{\varphi}_{m}\hspace{0.17em}+\hspace{0.17em}{\Pi}_{h=1}^{h=m-1}\hspace{0.17em}{\tau}_{\mathit{\text{sh}}}\hspace{0.17em}{\text{sin}}^{2}\hspace{0.17em}\theta $, ${E}_{\mathit{\text{sample}},y}\hspace{0.17em}=\hspace{0.17em}{\Pi}_{l=1}^{l=m-1}\hspace{0.17em}{\tau}_{\mathit{\text{pl}}}\hspace{0.17em}\text{cos}\hspace{0.17em}\theta \hspace{0.17em}\text{sin}\hspace{0.17em}\theta \hspace{0.17em}\text{cos}\hspace{0.17em}{\varphi}_{m}\hspace{0.17em}-\hspace{0.17em}{\Pi}_{h=1}^{h=m-1}\hspace{0.17em}{\tau}_{\mathit{\text{sh}}}\hspace{0.17em}\text{cos}\hspace{0.17em}\theta \hspace{0.17em}\text{sin}\hspace{0.17em}\theta $, and ${E}_{\mathit{\text{sample}},z}\hspace{0.17em}=\hspace{0.17em}-{\Pi}_{l=1}^{l=m-1}\hspace{0.17em}{\tau}_{\mathit{\text{pl}}}\hspace{0.17em}\text{cos}\hspace{0.17em}\theta \hspace{0.17em}\text{sin}\hspace{0.17em}{\varphi}_{m}$. Note that

_{sample}*τ*, and

_{p}*τ*define the fresnel transmission coefficients for

_{s}*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.

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} = *n _{obj}* = 1.518), immersion medium (

*n*

_{2}=

*n*), coverglass (

_{im}*n*

_{3}=

*n*= 1.518, note that for a standard coverglass #1.5,

_{g}*t*

_{3}= 170

*μm*), and sample (

*n*

_{4}=

*n*,

_{s}*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 (

*n*=

_{obj}*n*=

_{im}*n*= 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.

_{g}#### 2.1. Trapping in water

Trapping inside an aqueous medium using an oil immersion objective is very common in OT applications for which *n _{s}* = 1.33, and

*n*= 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=200

_{im}*μm*) through a coverglass of 170

*μm*thick. Note that the lateral intensity profile varies at different depths. The calculated average intensity gradients acting on a 1

*μm*polystyrene bead trapped in such intensity profiles are shown in Figs. 2(b) and 2(d), for the lateral and axial directions, respectively.

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 *n _{im}*,

*e.g.*, the maximum for

*n*= 1.56 is 16.5% lower compared to

_{im}*n*= 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

_{im}*n*increases.

_{im}The same calculations were repeated for the lateral direction, to search for the depth at which optimal lateral trap occurs (*d _{opt,lat}* ). The results for both directions are summarized in Table 1.

Table 1 implies that 0.01 increment in *n _{im}* results in 3 – 4

*μm*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 – 4.2

*μm*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 *n _{sample}* = 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.

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 *n _{im}* shifts the optimal depth for the axial (lateral) trap by 0.2 – 0.6

*μm*(0.4 – 0.6

*μm*). 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

*n*is considered. As an example, the axial intensity distribution as well as the axial AIG for the immersion medium with

_{im}*n*= 2.11 is shown in Fig. 4. Note that the optimal depth is shifted to

_{im}*d*= 36

*μm*using

*n*= 2.11 which could be of great importance for aerosol trapping applications.

_{im}## 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 3–4*μm* 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 *n _{im}* = 2.11 (which is commercially available) would shift the optimal depth to

*d*= 36

*μm*. These results could be of great importance for optical micromanipulation community.

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