Controlling the transverse momentum distribution of a
light field via azimuth division of a hologram in
holographic optical tweezers
Sheng-Yang Tseng* and Long Hsu
Department of Electrophysics, National Chiao Tung University, 1001 University Road, Hsinchu, Taiwan 300
*Corresponding author: [email protected]
Received 6 June 2011; revised 3 September 2011; accepted 9 September 2011; posted 14 September 2011 (Doc. ID 148755); published 10 November 2011
This study proposes a method for creating a light field with controlled distribution of transverse momen-tum (TM) by displaying a hologram only in an azimuth region that centers atθ0and has a range ofΔθ of a spatial light modulator in holographic optical tweezers. This study utilized ray optics to analyze the TM of the resultant field, revealing that the direction of the TM is determined by the center angle of the azimuth region and that the magnitude of the TM is proportional to sinðΔθ=2Þ, without regarding the intensity. The relationship was verified experimentally. In addition, this study demonstrated moving particles along a designed path and depleting particles by the fields. © 2011 Optical Society of America
OCIS codes: 140.7010, 090.1760, 230.6120.
1. Introduction
Using holographic optical tweezers (HOTs) [1–4] to create a light field with momentum transverse to the optical axis provides a simpler approach for control-ling the motions of microparticles. Without any light intensity change or external force, particles in the field can move along the region of highest intensity because of the transverse momentum (TM). A large number of applications have thus utilized special light modes with the TM, such as optical vortices. [5–7]. Studies of creating a light field with a trolled distribution of momentum have been con-ducted as well [8–12].
In electromagnetic theory, the TM of a field is pro-portional to the phase gradient and intensity of the field [13]. Applying only phase modulation to a light field is insufficient to control the TM distribution. Certain studies adjust both the amplitude and phase of a light field to produce a desired TM distribution. For instance, the shape-phase algorithm [10] by
Roichman and Grier uses a phase-only hologram to direct partial light with assigned phase shifts, ac-cording to a shape function that produces a required amplitude distribution. Another method, proposed by Jesacher et al. [11], is based on controlling ampli-tude and phase distribution with two cascade phase-only holograms.
In ray optics, models of light fields are composed of rays. Because the momentum of a ray rests along the ray’s direction, inclined rays contribute the TM [14]. In conventional optical tweezers [15], numerous inclined rays are created when incident rays are brought into focus. However, because of the azimuth symmetry of incident light intensity, the TM contrib-uted by the incident rays is cancelled out completely at the focus. This implies that breaking the azimuth symmetry can realize alternative approaches toward creating a light field with TM distribution.
Based on this notion, this study proposes a method for creating a light field with a controlled TM distri-bution by displaying a hologram only in an azimuth region of a spatial light modulator (SLM) in HOTs. This paper uses ray optics to analyze the TM dis-tribution of the resultant field. By tracing all the
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incident rays from the SLM to the focal plane of an objective in HOTs and summing the TM contributed by all the rays passing through the same points of the focal plane, the TM distribution of the field can be obtained. Implementation of this approach reveals a criterion for creating a light field with a controlled TM distribution. To verify the proposed method, electromagnetic theory is used to calculate the TM distributions of resultant fields numerically. In addi-tion, the ability of moving particles of these fields is experimentally demonstrated.
2. Theory
This section presents the usage of ray optics to ana-lyze the TM flux density, the TM per unit time per unit area, of an optical pattern produced by a holo-gram that is displayed only in an azimuth region of an SLM in HOTs. The azimuth region centers atθ0 and has an angle range ofΔθ. A normally incident, collimated ray bundle of power ptotal that extends over the entire input aperture of an objective models the incident light. Before considering the diffraction effect of the hologram, the TM flux at focus is analyzed, to which all of the incident rays within the azimuth region converge.
In HOTs, a hologram displayed on an SLM is imaged onto the back focal plane of an objective. The resultant light field is consequently created on the front focal plane of the objective. Thus, the setup of HOTs can be simplified to an objective between the front and back focal planes. A typical objective used in HOTs is an infinite-corrected objective, which conforms to the Abbe sine condition, and can be mod-eled by the first principal plane (PP), the second PP, and a spherical surfaceS [16]. The spherical surface S is centered at the focus, and is a tangent to the second PP.
Figure 1 shows the path of a normally incident ray in HOTs from the SLM plane through an objec-tive to the focal plane. The optical axis is in thez di-rection. The ray originates from the azimuth region at the positionrr¼ rrcosðθÞx þ rrsinðθÞy and crosses the first PP at PointA1, which is at the same position rr. According to the Abbe sine condition [16], the ray emerges from Surface S at Point As, of which the
transverse component of the position vector is also rr, before converging to the focus. At the focus, the TM flux contributed by the ray Pt;ray can be shown as follows: Pt;ray¼ − npray cfobj rr ¼ − np ray cfobj rr½cosðθÞx þ sinðθÞy; ð1Þ wheren is the refractive index of the surroundings, prayis the power transported by the ray,c is the speed of light in vacuum,fobj is the focal length of the ob-jective, andx and y are the unit vectors in the x and y direction, respectively. Since all the normal incident rays within the azimuth region converge to the focus, after summing, or integrating, the TM contributed by all the rays within the region, the total TM at the focus Pptotal
t;total can be obtained by Pptotal t;total ¼ Z R 0 Z θ 0þΔθ=2 θ0−Δθ=2 NrPt;rayrrdθdrr ¼ − Z R 0 Z θ 0þΔθ=2 θ0−Δθ=2 Nr n cfobj ptotal πR2N r ×½cosðθÞx þ sinðθÞyr2 rdθdrr ¼ − 2 3πc n R fobj ptotalsinðΔθ=2Þ
×½cosðθ0Þx þ sinðθ0Þy; ð2Þ whereR is the radius of the objective’s back aperture, Nr is number density of the incident rays on the SLM, andpray is substituted byptotal=ðπR2N
rÞ. When considering the diffraction effect of the holo-gram, the incident ray bundle is assumed to be split into a large number of collimated ray bundles bear-ing different headbear-ings and power after the hologram. This is similar to the concept of the angular spectrum in Fourier optics [17], which decomposes a field into plane waves, though these ray bundles originate only from the azimuth region. Once these ray bundles pass through the objective, they converge to points
on the focal plane and form an intensity distribu-tionIðx; yÞ.
Usually, the size of the optical pattern is signifi-cantly smaller compared to the focal length of the ob-jective. The inclined angles between the ray bundles and the optical axis are close to zero. This study thus assumes that regardless of the inclined angles, these ray bundles contribute the same TM flux at the focal plane as long as they have the same power. In other words, the total TM flux of the entire optical pattern is equal to the TM flux at the focus to which all the incident rays converge. Consequently, the TM flux density of the optical pattern Jptotal
totalðx; yÞ can be ob-tained by redistributing the total TM at the focus to the entire optical pattern according to the inten-sity distribution, as follows:
Jptotal
totalðx; yÞ ¼ Ppt;totaltotalInðx; yÞ; ð3Þ where
Inðx; yÞ ¼R∞ Iðx; yÞ −∞
R∞
−∞Iðx; yÞdxdy; ð4Þ which is a normalized intensity distribution of the optical pattern. After substituting Eq. (2) into Eq. (3), the TM flux density of the optical pattern becomes Jptotal t;totalðx; yÞ ¼ − 2 3πc nfR obj ptotalsinðΔθ=2Þ
×½cosðθ0Þx þ sinðθ0ÞyInðx; yÞ: ð5Þ From Eq. (5), the direction of the TM flux density is along −½cosðθ0Þx þ sinðθ0Þy, which is determined only by θ0. The magnitude of the TM flux density is proportional to the normalized intensity Inðx; yÞ and the total incident power ptotal. In addition, the magnitude of the TM flux density is also proportional to sinðΔθ=2Þ. When Δθ is equal to 0°, the TM flux density is zero due to an absence of incident rays. AsΔθ increases, the magnitude of the TM flux den-sity increases in conjunction with the extension of the azimuth region, reaching its maximum value at Δθ ¼ 180°. However, when Δθ is larger than 180°, more rays arrive at the focal plane, though part of the TM flux density is cancelled by the newly added rays. The magnitude of the TM flux density thus decreases and experiences a complete cancellation whenΔθ ¼ 360°. This is the reason why the TM flux, or the transverse scattering force, is not observed in optical tweezers.
Creating an optical pattern with controlled TM distribution thus becomes two parts: (1) controlling the direction of the TM by adjustingθ0of an azimuth region withΔθ < 360°; and (2) calculating a required hologram for producing a desired intensity distribu-tion, which can be accomplished by most algorithms used in HOTs, such as the Gerchberg–Saxton (GS) algorithm [18], the weighted GS algorithm (GSW)
[19], or the generalized adaptive add algorithm (GAA) [3].
3. Setup
The schematic illustration of the experimental setup is shown in Fig.2. The trapping laser is a fiber laser (YLR-10-1064-LP; IPG photonics) with an output power of up to 10 W at 1064 nm. The laser beam is first expanded by a beam expander to fulfill the ac-tive area of the SLM (X10468-03; Hamamatsu). A half-wave plate and a polarized beam splitter are utilized to control incident laser power on the SLM. The laser beam immediately leaving the SLM is consequently imaged by a telescope onto the back focal plane of a 100× water-immersion objective, with a numerical aperture (NA) that equals 1.1 (Plan; Nikon). Once the laser beam passes through the ob-jective, a designed optical field is formed on a sample held on a stage. The sample is illuminated by a light-emitting diode (LED) and is imaged on a CCD camera (PL-B955G; PixeLink) by the objective and a tube lens.
4. Results
Figure3(a)shows a phase-only hologram for produ-cing two trap arrays with opposite directions of TM. The hologram is divided into two azimuth regions, separated by a horizontal dash line. The hologram in each region produces a point trap array labeled with the same number as that of the azimuth region, as shown in Fig.3(b). The trap arrays extend in they direction. The traps in each trap array are arranged to set the distances between the neighboring traps at 2 μm. According to Eq. (5), the central angle of Region 1 is 90°; the TM of the resultant trap array is there-fore along−y. Similarly, the central angle of Region 2 is 270°; the TM of the resultant trap array is hence along y.
Figures 3(c) and 3(d) show the normalized simu-lated TM distributions of the resultant field in the x and y directions, respectively. The field produced by the hologram was calculated by performing a fast Fourier transform (FFT) on the transmittance of the
hologram. The TM distribution of the field was sub-sequently calculated by multiplying the intensity distribution by the gradient of the phase distribution of the field [13] before normalizing by the maximum magnitude of the result. These figures show that the TM of the trap array produced by Region 1 is along −y, indicated by a black color, and that of the trap array produced by Region 2 is along y, indicated by a white color. These results are consistent with the results predicted by Eq. (5).
These trap arrays were projected in a solution of 9 μm diameter polystyrene beads, which was placed between a cover slip and a slide. Figure 3(e) shows the sequential snapshots of beads moving along the trap arrays. The beads were pushed against the slide surface and were confined in the trap arrays. Within the trap arrays, the beads experienced only light gra-dient force due to cancellation between the gragra-dient forces from different traps. As the beads scattered the photons of the trap array, they obtained the TM and thus moved along they direction. Equation (5) shows that the TM is proportional to sinðΔθ=2Þ and ptotal; the average moving speeds of the beads should thus also be proportional to sinðΔθ=2Þ and ptotal.
Figure4(a)shows the average moving speed in the y direction of the beads in the trap arrays at different Δθ and ptotal. The solid lines are a result of the fits to the functionVmaxsinðΔθ=2Þ, where Vmaxis a free parameter. The data show the same dependence
Fig. 4. (Color online) Average moving speed of a 9 μm diameter bead in a trap array at differentΔθ and ptotal: (a) is the average
moving speed at different Δθ from 60° to 300°. The solid lines fit the functionVmaxSinðΔθ=2Þ; (b) is Vmax at different incident
powerptotal. The straight line is a linear fit, which does not pass
through the origin, indicating that a minimum incident power of approximately 63:2 mW is required to overcome the resistance between the particle and slide surface.
Fig. 3. (Color online) Moving particles along two trap arrays (a number near the trap array indicates the region from which the trap array was created): (a) is the phase pattern for creating two trap arrays; (b) is the corresponding intensity distribution on the focal plane; (c) and (d) are the simulated TM distributions in thex andy directions, respectively; and (e) represents the sequential snapshots of two 9 μm diameter beads moving in the trap arrays.
Fig. 5. (Color online) Particles moving along a heart-shaped path: (a) is the phase pattern for producing a heart-shaped path; (b) is the corresponding intensity distribution on the focal plane; (c) and (d) are the simulated TM distributions in thex and y directions, respectively; and (e) represents the sequential snapshots of beads moving along the path (Media 1).
on Δθ as the TM flux density predicted by Eq. (5). Figure4(b) shows Vmax at different ptotal. The solid line is a linear fit and does not pass through the ori-gin, indicating that a minimum input power of approximately 63:2 mW is required to move the beads. This may be caused by the resistance between the surface and the beads.
A more complex path can also be created. A phase pattern which produces point traps that form a heart-shaped path is shown in Fig. 5(a). The traps are arranged to set the direction of TM along the path. The corresponding intensity and the simulated TM distributions in thex and y directions are shown in Figs. 5(b)–5(d). Figure 5(e) (Media 1) represents the sequential snapshots of moving 9 μm diameter beads along the path. The time interval between snapshots is 1 s. The white triangle in each snapshot indicates the same bead at different times. These fig-ures show that the beads move along the path in a direction, as predicted by the simulated results.
In addition to moving particles along closed loops, trap arrays can also be used to deplete particles. Figure6(a)shows a phase pattern for producing an equally-spaced trap array with 2 μm spacing, which forms a hexagon. The phase pattern is divided into six regions. The phase pattern in each region pro-duces a triangular trap array, which is labeled with
the same number as the corresponding region, shown in Fig.6(b). The TM of these triangular trap arrays directs outward from the center of the hexagon, as shown in Figs. 6(c) and 6(d). Figure 6(e) (Media 2) represents the sequential snapshots when the trap array was projected in a solution containing 0:5 μm diameter beads. The time interval between snap-shots is 1 s. At the beginning of the experiment, a large number of beads were located in the center of the screen. Over time, the beads jumped outward because of the TM and help from Brownian motions and particle collisions. When most particles were outside the trap array, particle collisions became rare. The Brownian motions of particles became the only aid for the particles to overcome the gradient forces. The depletion rate thus decreased and some particles remained trapped at the end of the experi-ment. This process can be reversed with the reversal of the direction of the TM in each subtrap array. In this manner, particles can be concentrated at the central region and local particle concentration can thus be controlled.
5. Conclusion
This paper proposes an alternative method for con-trolling the TM distribution of a light field, other than by modulating its phase and amplitude distri-bution. This study used ray optics to analyze the TM distribution of a field produced by a hologram. The results reveal that a field with controlled TM dis-tribution can be created by displaying a hologram only in an azimuth region with an angle range of Δθ < 360°. The direction of the resultant TM distri-bution is determined by the center angleθ0 and lies along the direction−½cosðθ0Þx þ sinðθ0Þy. The magni-tude of the TM is proportional to sinðΔθ=2Þ, without regarding the intensity distribution of the field. The control of the TM distribution of a light field is thus divided into two parts: (1) controlling the direc-tion of the TM by adjustingθ0of an azimuth region withΔθ < 360°; and (2) calculating a required holo-gram to produce a desired intensity distribution.
The relationship is experimentally verified. In addition, this study created point trap arrays with controlled TM distributions to demonstrate moving particles along predetermined paths and the deple-tion of particles.
The authors thank Sien Chi and Yi-Ren Chang for their thought-provoking discussions and suggestions.
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