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An in-plane liquid gradient index (L-GRIN) microlens is designed for dynamically adjusting the beam focusing. The ethylene glycol solution (core liquid) withde-ionized (DI) water (cladding liquid) is co-injected into the lens chamber to form a gradient refractive index profile. The influences of the diffusion coefficient, mass fraction of ethylene glycol and flow rate of liquids on the refractive index profile of L-GRIN microlens are analyzed, and the finite element method and ray tracing method are used to simulate the convection-diffusion process and beam focusing process, which is helpful for the prediction of focusing effects and manipulation of the device. It is found that not only the focal length but the focal spot of the output beam can be adjusted by the diffusion coefficient, mass fraction and flow rate of liquids. The focal length of the microlens varies from 942 to 11 μm when the mass fraction of the ethylene glycol solution varies from 0.05 to 0.4, and the focal length changes from 127.1 to 8 μm by varying the flow rate of the core liquid from 0.5 × 10^{3} to 5 × 10^{3} pL/s when there is no slip between the core and cladding inlet. The multiple adjustable microlens with a simple planar microfluidic structure can be used in integrated optics and lab-on-chip systems.

Tunable microlenses are widely used in microfluidic or lab-on-chip systems [

Several kinds of tunable microlenses are developed for providing adaptive focusing with adjustable curved refractive surfaces [

In this paper, an in-plane tunable L-GRIN microlens was designed for dynamically adjusting the beam focusing, which can be more readily integrated for lab-on-a-chip applications. The convection-diffusion process of liquids in the microfluidic chamber is firstly simulated with the finite element method (FEM) when a high-refractive-index solution is injected side-by-side into a low-refractive-index solution. During the convection and diffusing process, the diffusion profile of the liquids, and hence the refractive index profile within the L-GRIN microlens, varies with the hydrodynamic flow conditions. Thus, the refractive index profile of the L-GRIN microlens is calculated and discussed numerically under different flow conditions, and the beam transmission and focusing process in the L-GRIN microlens is simulated using the ray tracing method. The effects of the diffusion coefficient, mass fraction and flow rate of liquids on the input beam focusing effect, including the focal length and the size of the focal spot, were demonstrated. Therefore, an in-plane tunable beam focusing is achieved using a simple planar microfluidic structure; in addition, with the simulation of the convection and diffusing process, the ray tracing method gives us approaches to predict the focusing effect of an L-GRIN microlens.

The schematic of the L-GRIN microlens designed is shown in _{in_core}, _{in_core}_{in_clad}, _{in_clad}_{out}, _{out} represent the height and width of the core inlet, cladding inlet and outlet, respectively. The main part of the L-GRIN microlens is a micro-cylindrical chamber, where the diffusion and convection process of liquids occurs and the gradient refractive index profile appears. The _{core} which is designed to be 50 μm stands for the diameter of the core inlet. Similarly, _{clad} is the diameter of the cladding inlet and is designed to be 150 μm. In the chamber, ethylene glycolsolution (core liquid) is injected side-by-side into de-ionized (DI) water (cladding liquid) from the same direction as shown in

(_{in_core} = 50 μm, _{in_core} = 30 μm, _{in_clad} = 150 μm, _{in_clad} = 50 μm, _{out} = 150 μm, _{out} = 50 μm. The _{core} = 50 μm, _{clad} = 150 μm; (

To simulate and optimize the refractive index profile and light propagation in the L-GRIN microlens under different conditions, the FEM and optical ray tracing method are adopted. The refractive index profile can be calculated by simulating the diffusion and convection process of liquids in the microfluidic chamber. When the refractive index of liquids is mainly affected by diffusion, the process can be described by Fick’s second law:

In which

The concentration distribution for the full, developed, steady-state flow can be expressed as Equation (4) [

In the normalized coordinate system, _{0} is the normalized concentration, and _{0} stands for the initial concentration.

For the structure we designed, _{core} is the flow rate of the core liquid and _{core} is the flow rate of the cladding liquid; _{core} is the liquid viscosity of the core liquid and μ_{clad} is the liquid viscosity of the cladding liquid. The initial parameters are set as

Once the liquids for the core and the cladding liquids are determined, the liquid viscosity and diffusion coefficient are considered to be constant. The position of the focal point can be tuned along the

Because of the decisive effect of the convection-diffusion process on the refractive index profile of the L-GRIN microlens, the average velocity ^{−10} and 1.17 × 10^{−9} m^{2}/s with different mass fractions of 0.025 and 0.95 for ethylene glycol [^{−10} to 6.45 × 10^{−10} m^{2}/s with a fixed mass fraction of 0.8 for the ethylene glycol [

In order to validate our numerical simulation, the comparison between the observed light propagation in Reference [_{clad} = 50 μL·min^{−1} (_{clad} = 0.5 μL·min^{−1}, the light exhibits curved-ray trajectories and converges repeatedly for different core widths,

(_{clad} = 50 μL·min^{−1}, the light is confined in the core due to the step-index distribution. At _{clad} = 0.5 μL·min^{−1}, the diffusion-induced gradient of the refractive index causes the light to repeatedly merge. With increasing _{1/6} = 270 μm, _{1/3} = 300 μm and _{1/2} = 340 μm. The focusing period also increases; for example, the first period for _{1/3} is 300 μm, the second is extended to 490 μm and the third is 590 μm. (Scale bar equals 300 μm.)(_{1/6} = 300 μm, _{1/3} = 360 μm and _{1/2} = 410 μm. The focusing period also increases; for example, the first period for _{1/3} is 360 μm, the second is extended to 540 μm and the third is 680 μm.

Similarly, the length of the first section increases with the core width and decreases almost linearly with the flow rate of the core fluid in both figures. However, when examining the light propagation pattern in detail, it is found that the self-focusing period is chirped such that the focusing period increases. For instance, as shown in _{1/3} is 300 μm, the second is extended to 490 μm, and the third is 590 μm, owing to the diminishing bidirectional gradient contrast downstream as a result of diffusion [_{1/3} is 360 μm, the second is extended to 540 μm, and the third is 680 μm in

In order to form the gradient refractive index profile in the lens chamber, the ethylene glycolsolution (_{core} = 1.432) and DI water (_{clad} = 1.332) are co-injected into the lens chamber from the same direction.Once contacting DI water, the ethylene glycol starts to diffuse from the ethylene glycol solution into the DI water. The average velocity ^{−9} m^{2}/s and ^{−10} m^{2}/s) as shown in

The refractive index profiles in the _{clad} = 8, _{core} = 8 × 10^{3} pL/s). (^{−9} m^{2}/s; (^{−10} m^{2}/s.

It is known that a relatively lower average velocity of liquids offers remarkable diffusion effect, and the focus with an adjustable focal length of the L-GRIN microlens forms by changing the diffusion process. The diffusion process of the liquid is influenced by the average velocity

Because the diffusion coefficient changes with the concentration of the core liquid, the refractive index profile of the L-GRIN microlens varies with the mass fraction of the ethylene glycol solution. As the diffusion progresses, the solution flowing into different regions has a different concentration, and the diffusion coefficient is, therefore, also different. However, according to Equation (4), compared with the concentrated ethylene glycol solution, the diluted ethylene glycol solution will have an obvious gradient refractive index profile whose diffusion coefficient is larger and can be considered approximately constant during the diffusion process. To facilitate the analysis, smaller mass fractions of ethylene glycol ranging from 0.05 to 0.4 were selected and the focal length of the L-GRIN microlens with different mass fractions was calculated while the flow rate of the liquids was kept the same (_{core} = 1 × 10^{3} pL/s, _{clad} = 8 × 10^{3} pL/s). Firstly, we simulated the refractive index profiles of the L-GRIN microlens with the mass fraction increasing from 0.05 to 0.4 by 0.05, and simulation results showed that the sharper peak profile of the refractive index could be achieved with the increase of the mass fraction.

(_{core} = 1 × 10^{3} pL/s, _{clad} = 8 × 10^{3} pL/s; (

The refractive index profile of the L-GRIN microlens can also be tuned by changing the flow rate of the liquids. In order to analyze the influence of the flow rate of the liquids on the focal length of the L-GRIN microlens, the ethylene glycol solution is co-injected with DI water into the lens chamber at the same flow rate ranging from 0.5 × 10^{3} to 5 × 10^{3} pL/s. In the analysis, we suppose there is no relative slip between the ethylene glycol solution and the DI water. Additionally, for simplifying the calculation, simulations were started by setting ^{−3} Pa·s with a fixed mass fraction of 0.3.

We simulated the refractive index profiles with the flow rate increasing from 0.5 × 10^{3} to 5 × 10^{3} pL/s by 0.5 × 10^{3} pL/s, and the results demonstrated that the diffusion was no longer dominant with the increase of the flow rate. ^{3} pL/s, and the cross-sectional refractive index profiles at five different locations (^{3} to 5 × 10^{3} pL/s. However, when the flow rate is larger than 5 × 10^{3} pL/s, the converging of the light beam caused by focusing becomes more and more unobvious. Therefore, continuously tuning the flow rate of the liquids in a certain range provides a tunable focal length when the mass fraction of the ethylene glycol is kept constant.

(^{3} pL/s; (_{clad} = 8×10^{3} pL/s, _{core} = 1×10^{3} pL/s; (

In general, the focal length of the lens is related to the lens radius, the refractive index difference between the lens center and border, and the lens thickness; in addition, it is dependent on the wavelength of the incident light which will change the refractive index profile of the lens. Thus, a shorter wavelength of incident light (when other parameters are constant) results in larger refractive index contrast, which causes light to bend toward the lens axis more significantly and leads to the decreased focal length.

The focus becomes a tunable focal spot at a relatively higher average velocity. The adjustment of the focal spot of the L-GRIN microlens can also be achieved by changing the flow rate of the liquids when the average velocity is higher. The case of no relative slipbetween the core and cladding liquids with a lower average velocity has been analyzed above. Analyzing a more sophisticated case will be helpful to better understand the influence of flow rate on the refractive index profile. For instance, the ethylene glycol solution is injected in different flow rates with DI water, andthere is a relative slip between the core and cladding liquids. For analyzing the influence of the relative slip between the core and cladding liquids on the refractive index profile, the cladding flow rate was kept constant (40 × 10^{3} pL/s) and the flow rate of the core inlet varied from 2 × 10^{3} pL/s to 50 × 10^{3} pL/s by 5 × 10^{3} pL/s. ^{3} pL/s, and the cross-sectional refractive index profiles at five different locations (_{core} = 25×10^{3} pL/s, the core width of the refractive index is 24 μm, as shown in _{core}. When the core inlet flow rate remainsconstant, with a relatively low core inlet flow rate (lower than 10 × 10^{3} pL/s),the core width approachs zero. In that case, the L-GRIN microlens can theoretically focus the beam onto one point when the ratio of the core inlet flow rate to the cladding inlet flow rate is lower than 0.25. However, the core width increases along with the increaseof the core inlet flow rate (higher than 10 × 10^{3} pL/s); that is, when increasing the ratio of the flow rates of the ethylene glycol solution to the DI water, the size of the output spot may continue to grow. Thus, controlling the ratio of the core inlet flow rate to the cladding inlet flow rate is a useful way to dynamically adjust the size of the light beam in microscale.

(_{core} = 25 × 10^{3} pL/s; (_{core} = 25 × 10^{3} pL/s; (_{core}.

_{core} = 2.5 × 10^{4} pL/s and _{core} = 4 × 10^{4} pL/s can be expressed as:

(_{core} = 2.5 × 10^{4} and _{core} = 4 × 10^{4} pL/s; (

In the L-GRIN microlens, the diameter of the focal spot is affected by the focal length and the wavelength of incident light. For a given incident light, one can conclude that the focal spot is the smallest when the focal length is the shortest. As the wavelength increases, the focal length is gradually increased, and the focal spot is increased.

These discussions above are all for a steady-state flow. However, the response time of the focal length and the focal spot change are important parameters for the stability of an adaptive lens operation, which is mainly dependent on the diffusion speed of two inlet flows. According to Equation (1), the response time varies with the concentration

We designed an in-plane L-GRIN microlens that can realize two-dimensional light beam focusing dynamically. The diffusion coefficient, mass fraction and flow rate of the core inlet and the cladding inlet are demonstrated to be the main influencing factors for changing the refractive index profile of the L-GRIN microlens, and the influences of these factors on the focusing process of the L-GRIN microlens are studied. It is found that adjusting the mass fraction and flow rate of the ethylene glycol is an efficient way to change the focal length of the output beam. The focal length varies from 942 to 11 μm at the mass fraction of ethylene glycol varying from 0.05 to 0.4. In addition, by varying the core and cladding inlet flow rate from 0.5 × 10^{3} to 5 × 10^{3} pL/s, the focal length of the microlens changes from 127.1 to 8 μm. In addition, the adjustable size of the output beam spot can be provided by controlling the relative slip between the core inlet and the cladding inlet flow rate with a relatively high average velocity. The flexible properties of manipulating the light beam in microscale have advantages for integrated devices in lab-on-a-chip applications.

This work is supported by the National Natural Science Foundation of China (Grant No. 61172081) and the Natural Science Foundation of Zhejiang Province, China (Grant No. LZ13F010001).

Zichun Le and Yunli Sun planned and performed the reported designs and their analysis. Ying Du has written the main manuscript and prepared the figures and tables. All authors reviewed the manuscript.

The authors declare no conflict of interest.