## 2 Microfluidic Manipulation Techniques
### Optical Tweezers  
Optical tweezers (OTs) are a method to control the motion of a single or few particles using a highly focused laser beam [32; 33; 34]. The focused beam induces particle motion, exerting three types of optical forces (Figure 3) [97]: (1) scattering force, (2) absorption force, and (3) gradient force. The optical scattering and absorption forces push a particle along the direction of propagation of a focused laser, balancing with the Stokes' drag in a fluid. For a spherical particle in the laser field with a Gaussian beam profile, the forces are expressed as [34; 97; 98]  
Figure 2: Electrophoretic motion of a charged particle in a direct current (DC) electric field. Reproduced from [40], with the permission of AIP Publishing (2012).  
\[F_{scat}=\frac{8\pi I_{0}\alpha^{2}\varepsilon_{m}}{3C_{0}\lambda^{4}}, \tag{8}\]  
\[F_{abs}=\frac{2\pi\varepsilon_{m}I_{0}}{C_{0}\lambda}\text{Im}[\alpha], \tag{9}\]  
where \(I_{0}\), \(C_{0}\), and \(\lambda\) are the maximum intensity, speed, and wavelength of the incident laser light, respectively. The variable Im[ ] represents the imaginary part of the brackets. The parameter \(\alpha\) is given by  
\[\alpha=3V_{p}\big{(}\varepsilon_{p}-\varepsilon_{m}\big{)}\big{/}\big{(} \varepsilon_{p}+2\varepsilon_{m}\big{)}, \tag{10}\]  
where \(V_{p}\) is particle volume, and \(\varepsilon_{p}\) and \(\varepsilon_{m}\) are the electrical permittivities of the particle and suspending medium, respectively. The dielectric constants are a function of wavelength and consist of a real and an imaginary part, such that \(\varepsilon=\varepsilon_{1}+i\varepsilon_{2}\), where \(\varepsilon_{1}=n^{2}-k^{2}\) and \(\varepsilon_{2}=2 nk\). Here, \(n\) and \(k\) are the refractive index and absorption coefficient, respectively. In addition to the scattering and absorption forces, the optical gradient force is defined as follows [34; 97; 98]:  
\[F_{grad}=\frac{2\pi\
abla I_{0}\alpha}{C_{0}}, \tag{11}\]  
where \(\
abla I_{0}\) is the gradient of the incident laser intensity. It pulls a particle along the gradient or the reverse gradient of laser intensity, depending on the sign of the \(\alpha\). For a non-absorbing dielectric particle, \(\alpha\) is positive when the refractive index of the particle is larger than that of the surrounding medium. Then, the particle is attracted toward the region of highest laser intensity by the gradient force. On the contrary, when \(\alpha\) is negative, the particle is repelled from the highest intensity region. For \(\alpha>0\), the gradient force overwhelms the previously mentioned scattering and absorption forces in all directions. In this case, suspended particles in a fluid move along the gradient of laser intensity and stop moving at the laser beam waist, which is a stable trapping zone. This is the working principle of OTs.  
In a general implementation of OTs, a near-infrared (NIR) laser beam is chosen as a driving laser source to minimize the optically-induced damage on target particles and reduce the level of fluctuation in laser power [99]. The laser beam is tightly focused by a high-numerical aperture (N.A.) objective lens to create the trapping zone [32; 34; 100]. Then, the beam waist behaves as a linear spring for particles in the vicinity of the laser focus. Near the beam waist, particles experience forces that are proportional to displacement from the center of the trapping zone.  
OTs have been extensively used for transporting [34; 97], sorting [6; 34; 54], and characterizing [27; 54; 101] particles with high-resolution. However, they cannot achieve high-throughput manipulation of particles because the beam waist of a focused laser used for trapping is very small [33; 34]. Another problem is that excessive heating is produced during OT manipulation [33] and may cause thermal damage or change in the physical/chemical properties of the manipulated particles.  
Figure 3: Trapping of a single particle by optical tweezers. Reproduced from [97], with the permission of The Royal Society of Chemistry (2011).