Micromechanical Photonics - H. Ukita Part 6 pot

3.2 Theoretical Analysis 91 (b) w F g Fs -1 1 O Z f r f(r) f(r) n q1(r) s (a) r f(r) F m b b Z Y f O w w' X Y Z R m R m Fig. 3.11. Geometry for calculating axial trapping efficiency of polystyrene mi- croshere. The laser focus is on the optical axis which is parallel to the center line of the microsphere [3.4] (a) X Y Z r gb g a b O Y Z 1 -1 f W W' q 1 q 1 F m R m Fs Fg R m R m cotF m n A (b) W O f ' n A s' R Fig. 3.12. Geometry for calculating the transverse trapping efficiency of polystyrene microshere. The laser focus is located along the transverse center line of the sphere [3.4] Z 0.2 0.40 Trapping efficiency -0.4 -0.2 Focus point (a) (b) 0 0.2 0.4 0.6 0.8 1.0 0.1 0.2 0.3 0.4 0 Y Trapping efficiency Focus point Fig. 3.13. Total trapping efficiency Q t exerted on a polystyrene microsphere sus- pended in water by trap with a uniformly filled input aperture of NA = 1.25 for axial (a), and for transversal (b) directions 92 3 Optical Tweezers Table 3.4. Maximum trapping efficiency for axial trap with various laser beam profiles beam profile downward directed upward directed Gaussian (TEM 00 ) 0.21 0.33 uniform 0.25 0.39 donut (TEM 01 ∗ ) 0.26 0.41 the upward directed beam is more effective in trapping the microsphere than the downward-directed beam. Table 3.3 shows microsphere materials for the analysis in this book. The trapping efficiency dependence on the incident angle of a ray means that trapping efficiency is related to the profile of the laser beam. Table 3.4 shows the maximum trapping efficiency calculated for input beams with various mode intensity profiles: Gaussian, uniformly filled, and donut. The maximum Q increases as the outer part intensity increases. Good trapping is possible when the outer part of the aperture is filled by a high intensity to give a laser beam with a high convergence angle. Example 3.4. Calculate the axial trapping efficiency for a microsphere when the focus of the uniformly input laser beam is along the optical axis in the center line of the sphere. Solution. First, we find the incident angle θ 1 (r, β)ofarayenteringthein- put aperture of the objective lens at the arbitrary point (r, β), as shown in Fig. 3.11a [3.4]. Since axial trapping efficiency is independent on β due to axial symmetry, we consider r-dependence for the θ 1 (r, β). The angle φ(r) between the incidence ray and z-axis is r 0 sin θ 1 (r)=s sin φ(r) where r 0 is the radius of the microsphere (we take r 0 = 1 since the results in the ray optics model are independent on r), s is the distance between the center of the microsphere and the laser focus. From Fig. 3.11b, φ(r) = tan −1  r R m tan Φ m  , where R m is the lens radius and Φ m is the maximum convergence angle. Then the incident angle θ 1 (r) becomes θ 1 (r)=sin −1   sr tan Φ m R m   1+  r tan Φ m R m  2   . Next, the trapping efficiencies Q s (r)andQ g (r) are computed by the vector sum of the contributions of all rays within the convergence angle using (3.5) and (3.6). Here, the y-component is cancelled out due to the symmetry, only the z-component is calculated as Q sz (r)=Q s (r)cosφ(r), Q gz (r)=Q g (r) sin φ(r). 3.2 Theoretical Analysis 93 Finally, Q s and Q g are obtained by integrating all the rays using Q s = 1 πR 2 m 2π  0 R m  0 rQ sz (r)drdβ = 2 R 2 m R m  0 rQ sz (r)dr, Q g = 1 πR 2 m 2π  0 R m  0 rQ gz (r)drdβ = 2 R 2 m R m  0 rQ gz (r)dr. The total trapping efficiency is given by Q t =  Q 2 s + Q 2 g . 3.2.3 Effect of Beam Waist In the ray optics, a laser beam is decomposed into individual rays with appro- priate intensity, direction and polarization, which propagate in straight lines. In actual conditions, the focused light beam has a beam waist, which means that each ray varies its direction near the focus. Therefore, the incident angle θ 1 varies from that of the straight line, leading to the recalculation of the exact optical pressure force. We introduce a Gaussian beam profile (3.9) of a beam waist ω 0 and the depth of focus Z 0 instead of straight line ray optics as ω 0 = λ 2NA ,Z 0 = kω 2 0 , (3.9) where k is the wave number 2π/λ, λ is the wavelength, and NA is the numer- ical aperture of the objective. To determine the incident angle θ 1 (r) of a Gaussian ray passing at r = r in the aperture of the objective enters at the point (α, β) on the sphere surface as shown in Fig. 3.14. The coordinates (α, β) are expressed α = 2sZ 2 0 −  4s 2 Z 2 0 − 4Z 2 0  s 2 − r 2 0 +  r R m  2 ω 2 0  Z 2 0 +  r R m  2 ω 2 0  2  Z 2 0 +  r R m  2 ω 2 0  , (3.10) β =  r 2 0 − (s −α) 2 . (3.11) Then the incident angle θ 1 (r) is calculated as the angle between the tan- gent vector a of the Gaussian ray at (α, β) and the direction vector b pointing to the center of the sphere. After the incident angle θ 1 (r) is defined, the trap- ping efficiency along the optical axis can be computed. Figure 3.15 show the result for a polystyrene sphere suspended in water. Considering the beam 94 3 Optical Tweezers y Focus point 0 -1 1 S o Z (ab) Fig. 3.14. Geometry for calculating exact axial trapping efficiency for microsphere considering beam waist s 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0 0.4 0.60.2 0.8 1 Normalized distance between particle center and focus point Trapping efficiency Ray optics Diameter 20 mm 10 5 2 1 Beam waist Fig. 3.15. Axial trapping efficiency of polystyrene microsphere suspended in water by converging ray approximations of straight line (ray optics) and parabolic line (beam waist) with beam waist ω 0 waist, it is seen from the figure that the axial trapping efficiency decreases to 50% that of the straight lines. This is caused by the fact that focused rays are almost parallel to the optical axis near the focus, as shown in the upper left sketch in the figure. Figure 3.16 shows the transverse trapping efficiency along the axis perpen- dicular to the optical axis. It is seen from the figure that both straight and parabolic Gaussian beam rays have almost the same numerical results. This is based on the fact that the incident angles at the surface of the sphere are almost the same for both approximations because the laser focus is located near the surface edge, maximum trapping efficiency, on the center line of the sphere (see the upper left sketch in the figure). Example 3.5. Compute the trapping efficiency of a microsphere suspended in water along the propagation axis by the laser beam emitted from the tapered 3.2 Theoretical Analysis 95 Normarized distance between microsphere center and focus point Trapping efficiency Diameter 40 mm 10 2 20 4 Ray optics Beam waist 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05 0 0 0.2 0.4 0.6 0.8 1.0 s Fig. 3.16. Transverse trapping efficiency of polystyrene microsphere by two con- verging ray approximations d 2 d 1 w 1 w 2 n 2 n 1 R r(z) Fig. 3.17. Geometry for calculating trapping efficiency for microsphere along prop- agation axis by laser beam emitted from tapered lensed optical fiber lensed optical fiber of curvature R =10µm, beam waist radius ω 1 =5.0 µm, core refractive index n 1 =1.462, as shown in Fig. 3.17. The focus distance from the tapered lensed fiber end d 2 and the beam radius r(z) with the beam waist ω 2 are given as d 2 = − n 2 R(n 2 − n 1 ) (n 2 − n 1 ) 2 + R 2  λ πω 2 1  2 ,r(z)=ω 2  1+  z kω 2 2  2 . Solution. An equation of a ray going along the z-direction is expressed by the variable parameter t(0 ≤ t ≤ 1) as y = tω 2  1+  z Z 0  2 ,Z 0 = kω 2 2 , where t = r/R m and ω 2 = ω 1   πω 2 1 λ  2  n 2 −n 1 n 1 R  2 +1 . 96 3 Optical Tweezers The equation of the microsphere located on the z-axisis(z −s) 2 + y 2 = r 2 0 where r 0 is the radius of the microsphere and s is the distance between the center of the microsphere and the beam waist. From the two equations given carlier, the intersection point α between the ray and the sphere surface is α = 2sZ 2 0 −  4s 2 Z 2 0 − 4Z 2 0 (s 2 − r 2 0 + t 2 ω 2 2 )(Z 2 0 + t 2 ω 2 2 ) 2(Z 2 0 + t 2 ω 2 2 ) . According to the Pythagoras theorem β =  r 2 0 − (s −α) 2 . The incident angle θ 1 of a Gaussian ray entering the sphere at the inter- section point (α, β) is the angle between the tangential vector a of the ray and the vector b pointing from the point (α, β) to the center of the sphere is θ 1 = arccos ab |a |·|b| , where a =(1,f(t, α)),f is the derivative function of y, that is f(t, α)= tω 2 α Z 2 0  1+ α 2 Z 2 0 , b =(s −α, −β). Here θ 2 =arcsin{(n 1 /n 2 )sinθ 1 }, R(t, s)= 1 2   tan(θ 2 − θ 1 ) tan(θ 2 + θ 1 )  2 +  sin(θ 2 − θ 1 ) sin(θ 2 + θ 1 )  2  , and T =1−R. The trapping efficiencies Q s and Q g are given from (3.5) and (3.6) as Q s =1+R(t, s) cos(2θ 1 ) − T 2 {cos(2θ 1 − 2θ 2 )+R(t, s) cos(2θ 1 )} 1+R(t, s) 2 +2R(t, s) cos(2θ 2 ) , Q g = R(t, s) sin(2θ 1 ) − T 2 {sin(2θ 1 − 2θ 2 )+R(t, s) sin(2θ 1 )} 1+R(t, s) 2 +2R(t, s) cos(2θ 2 ) . Considering the z-component, Q  s =Q s cos φ, cosφ = 1  1+f(t, s) 2 , Q  g = Q g sin τ, sin τ = f(t, s)  1+f(t, s) 2 . The trapping efficiency along the z-axis due to a ray is given as Q z = Q  s +Q  g . 3.2 Theoretical Analysis 97 Next, the trapping efficiency due to a circular element of radius β is given as Q c =2πβQ z . Finally, this trapping efficiency is integrated over the entire cross-section of the sphere for all individual rays using the Shimpson formula under the conditions in Table 3.5. Figure 3.18 shows the axial trapping efficiency dependence on the distance from the optical fiber end for a polystyrene sphere of radii 2.0 and 2.5 µm. The laser beam profile is Gaussian and the wavelength is 1.3 µm. It is seen from the figure that trapping force increases as axial distance increases from zero to a beam waist of 40 µm, i.e., it increases over the region in which the fiber lens is focusing, and then begins to decrease monotonically as the beam diverges beyond the focus. Therefore, we can expect that the optimum dual fiber lens spacing will exists at a point where axial trapping efficiency is changing rapidly (see Sect. 3.3.4). 3.2.4 Off-axial Trapping by Solitary Optical Fiber In recent years, studies of optical tweezers have been conducted on optical- fiber tweezers [3.12] to improve their operation in the fields of life science and Table 3.5. Conditions for analysis of tapered lensed optical fiber trapping efficiency refractive index water 1.33 particle 1.59 fiber core 1.446 beam waist in the core (µm) 5.0 beam waist distance (µm) 49.24 radius of curvature (µm) 10 wavelength (µm) 1.31 particle radius (µm) 2–10 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0 50 100 150 200 250 300 350 400 Distance from fiber end (mm) Trapping efficiency Diameter 5 mm 4 mm Fig. 3.18. Axial trapping efficiency dependence on distance from optical fiber end of polystyrene sphere 98 3 Optical Tweezers micromachines. The optical fiber implementation of such tweezers is simple and inexpensive. The apparatus that uses a laser diode and an optical fiber is particularly simple since no external optics such as a dichromatic mirror, a beam splitter, and filters are required. Trapping forces can be resolved into two components: the gradient force F g , which pulls microspheres in the direction of the strong light intensity, and te scattering force F s , which pushes microspheres in the direction of light propagation. If a microsphere is located on the light propagation axis, the gradient forces cancel out, thereby resulting in pushing the sphere. Therefore, two counterpropagating coaxially aligned optical fibers are used to trap the sphere suspended in water [3.13]. Although the sphere is stabilized axially at a location where the scattering forces of the two beams balance each other, the trapping in the transverse direction is weak. The freedom of operation for the counterpropagating coaxially aligned optical fibers is poor. In this section, we theoretically analyze an off-axial microsphere trapping force [3.14] in three dimensions in order to trap it with a solitary optical fiber. Analysis of Off-axial Trapping Trapping efficiency for a microsphere on an optical axis can be calculated, from axial symmetry, as shown in Fig. 3.19a, by integrating the optical pres- sure force due to an individual ray in two dimensions. On the other hand, calculation in three dimensions is necessary for the off-axial trapping effi- ciency because of axial dissymmetry. Figure 3.19b shows that a ray enters at (a) Y Z F s F g F g F s Total trapping force (b) Intersection(x,y,z) Incident angle q 1 Y Z Beam profile Sphere center (0,B,A) Axial distance A Off-axial distance B F s F g Fig. 3.19. Geometry for calculating trapping efficiency for a microshere when focus is located on optical axis (a), and at off-axis (b) 3.2 Theoretical Analysis 99 the incident angle θ 1 on the arbitrary intersection (x, y,z)ofthesurfaceofa sphere, whose center is located at (0,B,A). The y-coordinate is expressed as y (x,z) = B +  r 2 − x 2 − (z − A) 2 (3.12) The beam profiles for the x-andy-directions are given as ω y = tω 0  1+  z Z 0  2 ,ω x = uω 0  1+  z Z 0  2 , (3.13) where ω 0 is the radius at the beam waist, Z 0 is the depth of focus, and t(0 ≤ t ≤ 1) and u(0 ≤ u ≤ 1) are variable parameters. Next, the incident angle θ 1 of a ray entering the sphere at the inter- section point (x, y, z) is defined as the angle between the tangential vector a =  ω  x ,ω  y , 1  of the ray and the vector b =  x, B − y (x,z) ,A−z  pointing from the intersection (x, y, z) to the center (0,B,A) of the sphere θ 1 = arccos ab |a |·|b| . (3.14) As a result, the trapping efficiencies Q s(x,z) and Q g(x,z) owing to a ray hits the intersection (x, y, z) can be obtained using (3.5) and (3.6). The entire trapping efficiency due to the entire surface of the microsphere is given later. Figure 3.20 shows the sectional view of the off-axial trapping (a), indicating how to integrate Q s(x,z) and Q g(x,z) along the z-axis (b). Calculate the incident angle at the arbitrary point z in the circle in the yz plane and compute the optical trapping efficiency for the ray. Then integrate Q s(x,z) and Q g(x,z) along the z-direction leading to Q s z (x) and Q g z (x) in the yz plane. The integration is carried out for the upper and lower hemispheres individually because of the dissymmetry due to off-axial trapping. The integration starts from the Beam profile (a) A-r z upper (x) (t max ) z lower (x) (t max ) z (x) (t min ) dx dz Beam profile (b) x=0 x(u max ) zz Y X Fig. 3.20. Method of optical pressure integration when a sphere is located at an off-axis, side view (a), and top view (b) 100 3 Optical Tweezers left side z (x) (t min )=A − √ r 2 − x 2 in Fig. 3.20a for both the upper and lower hemispheres. The integration ends at the tangential point between the ray and the surface profiles of the upper and the lower hemispheres. The integration end points z upper (x) (t max ) for the upper hemisphere and z lower (x) (t max ) for the lower hemisphere are given by the solution between two equations shown as r 2 − x 2 =  y (x,z) − B  2 +(z − A) 2 ω y = tω 0  1+  z Z 0  2      . (3.15) Then, Q z s(x) and Q z g(x) are given as Q z s(x) =  z upper (x) (t max ) z (x) (t min ) Q s(x,z) dz +  z lower (x) (t max ) z (x) (t min ) Q s (x,z) dz, (3.16) Q z g(x) =  z upper (x) (t max ) z (x) (t min ) Q g(x,z) dz +  z lower (x) (t max ) z (x) (t min ) Q g (x,z) dz. (3.17) Next, our integration goes along the x-axis. Figure 3.20b shows the top view, indicating how to integrate along the x-axis. The trapping efficiencies Q z s(x) and Q z g(x) in the yz plane are summed along the x-axis in the xz plane. In this case, the integration starts from x = 0 and ends at x = x(u max ), which is the tangential point between the ray profile (3.13) and the sphere circle (3.18) in the xz plane x 2 +(z − A) 2 = r 2 ω x = uω 0  1+  z Z 0  2      . (3.18) Then, Q all s and Q all g are given as Q all s =2  x(u max ) 0 Q z s(x) dx, (3.19) Q all g =2  x(u max ) 0 Q z g(x) dx. (3.20) As a result, the total trapping efficiency comes from (3.7). Followings are the numerical results for the off-axial trapping in three dimensions. Off-axial Distance and Microsphere Radius Dependence In the analysis a circularly polarized laser beam by a laser diode with a 1.3 µm wavelength, a tapered lensed optical fiber with a curvature of 10 µm, and mi- crospheres 2–10 µm in radius are used under the conditions listed in Table 3.6. First, transverse trapping efficiency on the off-axial distance (transverse offset) is analyzed for a polystyrene sphere of 2.5 µm radius located at different [...]... 101 Table 3 .6 Microspheres for analysis of solitary fiber trapping density (g cm−3 ) 1. 06 2.54 refractive index 1 .6 1.51 Optical trapping efficiency material polystyrene glass 0. 06 Fiber end At beam waist 0.04 0.02 -1 0 radius (µm) 2–10 2–10 -5 2 fold beam waist 3 fold beam waist 0 5 10 -0 .02 -0 .04 -0 . 06 Off-axial distance (mm) Fig 3.21 Variation in transverse trapping efficiency for a 2. 5- m-radius microsphere... incident θ for an optical fiber [3.15] 3 Optical Tweezers Optical trapping efficiency 102 0.5 Sphere 10 radius 9 (mm) 8 7 6 0.4 0.3 0.2 0.1 -2 0 20 10 0 -1 0 2 -0 .1 4 5 -0 .2 -0 .3 3 -0 .4 -0 .5 Off-axial distance (mm) Fig 3.22 Variation in transverse trapping efficiency at beam waist as function of off-axial distance, with microspore radius as a parameter Y Beam axis Incident angle Fgz Fsz Z Fs Fg Fig 3.23 Balancing... (deg) 80 70 Sphere Radius (mm) 2 3 60 4 50 5 40 30 20 10 6 7 8 9 5 0 10 10 Off-axial distance (mm) Fig 3.24 Relationship between optimum fiber incident angle and radius of microsphere for Fsz = Fgz at maximum Fg on off-axial position Incident angle (deg) 90 80 70 60 50 40 30 20 0 2 4 6 8 10 Sphere radius (mm) Fig 3.25 Variation in fiber incident angle as function of off-axial distance at Fs = Fg , with... coverslip ax Pmin with an upward-directed (lower) laser beam is less than that with the downward-directed (upper) laser beam because the scattering force is added to the gradient force to trap the particle with an upward-directed beam The discrepancy between the predicted and the measured forces is found to be smaller for a heavy particle (glass) than that for a light particle (polystyrene) This may... trapping apparatus Upper and lower objective lenses are seen around the microscope stage lower left in (a) Two objective lenses used in trapping particles in water with downward-directed and upward-directed laser beams (b) 4 Upward directed 3 2 Downward 1 0 0 2 4 6 8 Particle diameter (mm) Fig 3.28 Dependence of minimum axial trapping power on diameter for polystyrene spheres (a), and glass spheres (b) beam... manipulating particles using upwarddirected and downward-directed YAG laser beams at a wavelength of 1. 06 µm 104 3 Optical Tweezers Monitor Cover glass Liquid Slide glass CCD YAG laser Dichroic mirror Spacer l/4 plate Objective lens Enlarged view Upper objective Stage Lower objective ND filter Beam expander Fig 3. 26 Experimental setup for trapping and manipulating microobjects using upward-directed and... (friction force at the surface due to the 2-D trapping) is smaller than that of the glass In summary, we measured the optical-trapping force on polystyrene and glass microspheres of different diameters in two orthogonal directions with upward-directed and downward-directed laser beams and optical fibers Following are our experimental results: 1 We confirmed that the upward-directed beam has a higher trapping... transverse trapping efficiency for a 2. 5- m-radius microsphere as function of off-axial distance, with beam axial distance as a parameter axial distances of zero (fiber end), beam waist, two-fold beam waist, and three-fold beam waist The axial distance of the sphere is measured along the symmetry axis of the laser beam and the off-axial distance of the sphere is measured as a relative distance to the symmetry... setup for trapping and manipulating microobjects using upward-directed and downward-directed YAG laser beams with wavelength of 1. 06 µm having a TEM00 mode structure is shown in Fig 3. 26 The laser beam diameter is increased from 0.7 to 8.2 mm by a beam expander to fill the entire aperture of the objective uniformly A quarter-wave plate is placed to generate a circularly polarized beam The intensity of the... transverse trapping power on the velocity for d = 10 µm polystyrene particle 100 80 Polystyrene 60 40 20 0 0 10 20 30 40 Particle diameter (mm) 50 Fig 3.32 Dependence of minimum transverse trapping power on diameter of polystyrene microspheres trans Figure 3.31 shows the dependence of Pmin on sphere velocity for d = trans 10 µm polystyrene particles (small gravity) Pmin increases as bead velocity increases . of the gradient force F g , which always pulls a sphere to the beam axis and the scattering force F s , which always pushes a sphere along the beam axis. Figure 3.23 shows the concept of the horizontal. Tweezers The equation of the microsphere located on the z-axisis(z −s) 2 + y 2 = r 2 0 where r 0 is the radius of the microsphere and s is the distance between the center of the microsphere and the. of the suspend- ing medium (water), respectively; c is the speed of light, and His the height of the specimen chamber (150 µm). A transversely moving sphere should stably remain near the sphere