Optical tweezers based on nonlinear focusing

Abstract

Recently, some authors have proposed to add an optical Kerr effect (OKE) while focusing the Gaussian beam (GB) enlightening an optical trap. These authors conclude that the introduction of a nonlinear lensing (NL) is benefit to the optical trapping capacity. The proposed modelling was based on the Gaussian approximation (GA) which assimilates the Kerr lensing effect to a pure lensing effect free from any aberration. In this paper, we evaluate the longitudinal and radial figures of merit of the optical trap based on NL using a diffraction integral taking into account the aberration associated with the OKE. The conclusion is that the GA modelling underestimates (overestimates) the improving of the longitudinal (radial) trapping ability of the optical trap. In summary, what is gained in the longitudinal efficiency of the nonlinear trap is lost in the radial force which decreases, thereby reducing the possibility to keep trapped the particle.

Appendices

Appendix

1. Calculation of the diffracted field $E_{d} (r,z)$ by a series:

The first step for determining an analytical form for the integral given by Eq. (12) is to expand the phase term $\exp [ - i\Delta \varphi (\rho )]$ where $\Delta \varphi (\rho ) = \varphi_{0} \exp [ - 2\rho^{2} /W_{1}^{2} ]$

$$\exp [ - i\Delta \varphi (\rho )] = \sum\limits_{m = 0}^{\infty } {\frac{{[ - i\varphi_{0} ]^{m} \exp [ - 2m\rho^{2} /W_{1}^{2} ]}}{m!}} .$$

(A1)

The diffraction integral takes the following form:

$$E_{d} (r,z) = \frac{{2\pi E_{0} }}{\lambda z}\sum\limits_{m = 0}^{\infty } {\frac{{[ - i\varphi_{0} ]^{m} }}{m!}\int\limits_{0}^{\infty } {\exp \left[ { - \rho^{2} \left( {\frac{2m + 1}{{W_{1}^{2} }} + i\gamma } \right)} \right]} } J_{0} (\beta \rho )\rho d\rho .$$

(A2)

With $\beta = 2\pi r/(\lambda f_{L} )$ and $\gamma = \frac{\pi }{\lambda }\left( {\frac{1}{z} - \frac{1}{{f_{L} }}} \right).$

In the above equation we can recognise the following HANKEL transform

$$F(\beta ) = \int\limits_{0}^{\infty } {f(\rho )J_{0} (\beta \rho )\rho d\rho } .$$

(A3)

In the case where the function f takes the form $f(\rho ) = \exp [ - a^{2} \rho^{2} /2]$, the Hankel transform $F(\beta )$ is given by

$$F(\beta ) = \frac{1}{{a^{2} }}\exp \left[ { - \frac{{\beta^{2} }}{{2a^{2} }}} \right].$$

(A4)

Finally the diffracted field is written as follows:

$$E_{d} (r,z) = \frac{{2\pi E_{0} }}{\lambda z}\sum\limits_{m = 0}^{\infty } {\frac{{[ - i\varphi_{0} ]^{m} }}{m!}} \frac{1}{{2\left[ {\frac{2m + 1}{{W_{1}^{2} }} + i\gamma } \right]}}\exp \left[ { - \frac{{r^{2} }}{{W_{m}^{2} }}} \right]\exp \left[ {i\frac{{r^{2} }}{{2R_{m} }}} \right],$$

(A5)

where

$$W_{m}^{2} = W_{F}^{2} \left[ {(2m + 1) + \frac{{\gamma^{2} W_{1}^{4} }}{(2m + 1)}} \right]\;{\text{and}}\;R_{m} = \frac{{W_{F}^{2} }}{2\gamma }\left[ {\left( {\frac{2m + 1}{{W_{0} }}} \right)^{2} + W_{1}^{2} \gamma^{2} } \right],$$

(A6)

where $W_{F} = \frac{\lambda z}{{\pi W_{1} }}.$

The summation in Eq. (A5) is truncated to a finite number of terms noted $m_{\sup }$. Its value, for an accurate evaluation of $E_{d} (r,z)$, increases with the value of $\varphi_{0}$. For $\varphi_{0} = 10rad$, only 25 terms in the sum are needed to be accurate within 1%.

2. The trap figure of merit $[Z_{L} ]_{GA}$ for Gaussian approximation

In this modelling, the Kerr lens is assimilated to a linear lens having a focal length $f_{NL}$ given by Eq. (15).

3.1 Focusing without OKE

We start with the Gaussian intensity distribution given by Eq. (3) where $W_{0} = \lambda f_{L} /(\pi W_{1} )$ is the beam-waist radius (best focus), and z = 0 the position of the best focus. The gradient of the on-axis intensity $I(0,z)$ is written as follows:

$$\hat{z}\vec{\nabla }_{z} I(0,z) = \frac{\partial I(0,z)}{{\partial z}} = \frac{{ - I_{0} 2zz_{0} }}{{(z^{2} + z_{0}^{2} )^{2} }},$$

(A7)

where $z_{0} = \pi W_{0}^{2} /\lambda$ is the Rayleigh distance of the focused Gaussian beam. It is easy to establish that the longitudinal gradient given above is minimum at position $z_{\min } = z_{0} /\sqrt 3$. At this position, the gradient is minimum and its value is given by

$$\hat{z}\left[ {\vec{\nabla }_{z} I(0,z)} \right]_{\min } = - \frac{{\sqrt {27} I_{0} }}{{8z_{0} }} = - \frac{{\sqrt {27} }}{8\pi }\frac{{\lambda I_{0} }}{{W_{0}^{2} }}.$$

(A8)

The intensity at position $z = z_{\min }$ is then given by

$$I(0,z_{\min } ) = \frac{{3I_{0} }}{4}.$$

(A9)

By applying Eq. (20), we obtain

$$[R]_{\varphi = 0} = \frac{{\sqrt {27} }}{6\pi }\frac{\lambda }{{W_{0}^{2} }}.$$

(A10)

3.2 Focusing with OKE

The previous calculation remains valid when $\varphi_{0} > 0$ provided to replace $W_{0}$ by $W_{0}^{^{\prime}}$ given by

$$\frac{1}{{W_{0}^{^{\prime}} }} = \frac{{\pi W_{1} }}{\lambda }\left[ {\frac{1}{{f_{L} }} + \frac{1}{{f_{NL} }}} \right] = \frac{{f_{L} }}{{W_{0} }}\left[ {\frac{1}{{f_{L} }} + \frac{{2\lambda \varphi_{0} }}{{3\pi W_{1}^{2} }}} \right].$$

(A11)

In presence of nonlinear lensing, Eq. (A10) is modified as follows:

$$[R]_{\varphi = 0} = \frac{{\sqrt {27} }}{6\pi }\frac{\lambda }{{W_{0}^{2} }}\left[ {1 + \frac{{2\lambda f_{L} \varphi_{0} }}{{3\pi W_{1}^{2} }}} \right]^{2} .$$

(A12)

Finally, we get the dimensionless figure of merit $[Z_{L} ]_{GA}$ obtained in the framework of the Gaussian Approximation

$$[Z_{L} ]_{GA} = \frac{{[R]_{{\varphi_{0} > 0}} }}{{[R]_{{\varphi_{0} = 0}} }} = \left[ {1 + \frac{{2\lambda f_{L} \varphi_{0} }}{{3\pi W_{1}^{2} }}} \right]^{2} .$$

(A13)