MICROLENS FOCAL LENGTH CONTROLING FOR OPTICAL TWEEZER ARRAYS BY ACOUSTO-OPTIC MODULATION IN GERMANIUM

The microlens arrays created by ultrasonic waves in acousto-optical material Germanium (Ge) is theoretically proposed. The simulated results show the proposed microlens arrays can be used for optical tweezers arrays to trap an assembly micro-beads. Moreover, with the control by changing of the ultrasonic intensity and frequency, the optical tweezers arrays will act as the dynamical one, which can sieve the beads in (X,Y) plane in the embedding fluid. The focal length of microlens in Germanium controling by frequency and intensity of ultrasonic wave is discussed. Indexing terms/


INTRODUCTION
Up to now, there are many methods as using diffractive elements [1,2,3], using microlens arrays fabricated by proton beam writing [4], using image processing techniques [5], and intelligent control techniques [6], proposed to create the optical tweezer arrays to trap an assembly of micro-particles.The aim of all methods to create an arrays of small laser spots, in which the intensity gradient is powerful [7].The arrays of microlens with high numerical aperture (NA) are appreciable tool for this aim.
As shown in work of Sow and colleagues [4], the microlens arrays must be fabricated through process from proton writing to thermal heating, unfortunately, the dimension of microlens and spatial period are fixed, related to the input fabrication parameters.So that, it should not be resourceful if they are used for the optical tweezer arrays, which should be changed in accordance with trapped objects.
As well known, the acousto-optical devices have been proposed to use for partial reflection of light (beamspliter), and the Bragg cell have numerous application in photonics [8].It is gread, McLeod and Arnold have discussed about the tunable acoustic gradient index lens in liquid [9][10][11].Moreover, in 2014, the acousto-optical deflectors for optical tweezer arrays is anounced in European Network of Excellence for Biophotonics [20].The results presented in those works give us the ideal to use two perpendicular ultrasonic waves for creating the 2D arrays of microlens in extra-dense flint glass (EDFG) [12] and first time show out the nanopartilce trapping capability of tweezers using created microlens [13,14].However, there are questions if microlens created by ultrasonis waves modulation in EDFG, which has small figure of merit can be used for bigger particle as bio-molecule and how to catch the big particles embedded sparsely in fluid.
In this work, firstly, we present the optical tweezer arrays created in Germanium by ultrasonic waves; secondly, the microlens length controling by frequency and intensity of ultrasonic wave for that the trapping positions fixed in 3D space is discussed.

THE SAMPLE OF OPTICAL TWEEZER ARRAYS
The optical tweezer arrays using microlens modulated by ultrasonic waves in Germanium is shown in Fig. 1a.The refractive index of a Square Germanium plate is modulated by two perpendicular ultrasonic waves from two LiNbO3transducers, which is supplied by the radio frequency signal generator (Fig. 1b).Then the Germanium plate become a microlens arrays.The original laser beam is expanded by two lenses, so that its intensity distribution approximately is uniform in entry plane of Germanium plate.Propagating through Germanium plate (microlens arrays) the laser beam is separated into sub-beams, which is focused in the fluid with microparticles.The foci of all laser sub-beams become the optical tweezer arrays.
The intensity and frequency of the radio signal are changing to control the frequency and intensity of ultrasonic waves.Consequently, the dimension of microlens and spatial period of microlens arrays will be changed.Moreover, with changing of the acoustic intensity, the gradient of refractive index, and consequently, the focal length of microlens and the numerical aperture of microlenses will be changed.So to control the intensity gradient of the laser in focus spot and the its position in Z-direction, the intensity of ultrasonic wave must be controled, while to control the focus spots of microlenses in the plane (X,Y), the initial phase and frequency of ultrasonic waves must be controled by the controling electric signal.The controling principle will be presented in the next section.

THEORICAL BACKGROUNDS 3.1 Distribution of refractive index
Considering in the X-and Y-directions of Germanium plate (see Fig. 1) travel two identical acoustic plane waves, then the strain at positions x and y and time t are given as follows [8,17]: (2), the strain at position ( , ) The strain x y t creates a propotional perturbation of the refractive index in Ge, analogous to the Kerr effect [8]: where  is a phenomennological coefficient known as the photoelastic constant, n is the refractive index of Ge in the absence of sound.Because the acoustic frequency is typical much smaller than the optical frequency, an adiabatic appoach for studying light-sound interaction may be adopted.Considering that two waves are phase matching, i.e.


 is a figure of merit for the strength of the acousto-optic effect in the material,  is the mass density of the medium, and s I is the intensity of sound.The acousto-optic material with refractive index distributing as shown in Eq. ( 5) became a microlens arrays.

Focal length of microlens
Now, we pay attention on the circle of acousto-optic material centered at point   where xy   is defined as the radial radius.From Eq. ( 8), the refractive index distribution in the circle of radius /2  can be rewritten as follows: which describes the refractive index distribution of the GRIN microlens [8].From Eq.( 7), the focal length of microlens is given by: where d is the thickness of acousto-optic material.

SIMULATION AND DISCUSSION FOR GERMANIUM
The Germanium (Ge) is one of the effective acousto-optic crystals with figure merit is about 3   10 times larger than one of the extra-dense flint glass [17,18]. .With considering the initial phase of two waves is same of 0   , the refractive index distribution in area 33   of crystal Ge is simulated and illustrated in Fig. 2a.The period appearance of graded- index areas of   (see Fig. 2b) leads the acousto-modulated crystal Ge became 2D microlens arrays.
There are advantages in comparision to the microlens arrays fabricated by proton beam writing [4], that the acoustomodulated microlens arrays have the controlable spatial period,  , by acoustic frequency and the controlable position of microlenses in Ge crystal,   , ii xy by the acoustic frequency or initial phase (Fig. 2b-d).If the acoustic modulated microlens arrays used to design an optical tweezers, the trapping positions,   , ii xy, in the plane (X,Y) at a deepth in the fluid, d s zf  can be controled by the frequency and initial phase of acoustic waves.By this ways the microparticles randomly located in the plane (X,Y) of fluid will be sieved.
As shown in Eq.8, the focal length of microlens will be changed when the acoustic frequency changing, that means the deepth of plane (X,Y), d z , will be changed on the Z-direction in the sieving process (see Fig. 3a).To fix the plane (X,Y) at chosen deepth in fluid when changing frequency, i.e. the focal length is unchanged, it is necessary to choice the acoustic intensity suitable to the controling frequency.With parameters given as Up to now, the LiNbO3 -transducer generating the ultrasonic wave with intensity upto

CONCLUSION
The microlens arrays in the Germanium crystal modulated by two perpendicular ultrasonic (acoustic frequency in range of MHz) waves is proposed for the optical tweezers.The simulated results show that by those microlens arrays, the trapping positions (x,y,z) in fluid embedding the microparticles can be controled the changing the acoustic frequency, consequently, the randomly locating microparticles in fluid can be siewed to tweezers centers.Moreover, the microparticles located in fixed plane (X,Y) of the fluid will be trapped if the acoustic intensity is chosen suitable to the controling frequency.With the focal of milimeters and the dimenssion of micrometers, the arrays of the microlens Ge modulated by ultrasonic waves with frequency in range (25÷100) MHz generated from LiNbO3-transducer are useful for trapping the biological molecules.
are the initial phase in x- and y-direction, respectively; s F is frequency; and s V is the velocity.Since interference of two waves, from Eq. (1) and Eq.
of refractive-index wave, i.e. a material parameter representing the effectiveness

,
ii xyand limited in the circle of radius  .Considering the refractive index distribution reaches the maximum value at point   1 , i xy, and the initial phase to be zero 0   , i.e.
With considerations and using trigonemetry relations, from Eq.(5) the refractrive index distribution in the area of   -optic circle can be approximately rewritten as follows:

Fig. 2 .
Fig. 2 Distribution of refraction index in Ge (a); Position of microlens in Ge (x i ,y i ) with different initial phases of acoustic waves: b-0   , c-

Fig. 3
Fig.3 Focal length (a) and Acoustic intensity (b and c) vs Acoustic frequency.
and manufactured[19], so to fix the trapping plane (X,Y) at16 d z mm for example, the frequency can be reduced from 100MHz downto 25MHz as shown in Fig.3c.