Redirection of a light path and change of momentum as it passes through a microsphere or "particle" with a high refractive index associated with the medium (left). The momentum of equal magnitude and opposite direction is transferred from the photons to the sphere according to momentum conservation (right).---16. Because of the light gradient, the path originating from the center of the beam carries more photons than the light path starting at the edges of the beam, resulting in a greater force pulling the particle toward the focal point.-- -16.
Polarized laser beam propagates towards the SLM window as a plane wave with constant phase throughout its wavefront and with uniform intensity. The desired field is Fourier transformed in the Labview program and coded in the SLM window. In the "phase calibration part", the phase values are converted to the range [0.255], which corresponds to the input value of the SLM.
The allowable error between the values of the processed desired array and origin desired array can be set to "Error". In "select monitor" part, by adjusting the value, phase mask is loaded on the window of SLM or by changing the value, the phase mask is displayed on the computer monitor so that the phase mask can be checked. It can change the size of the circle, center position of the circle and phase delay value.
The black solid line is a parabolic fit of the red dots and corresponds to the potential energy curve of the trap.---42.
Introduction
Optical Trapping
Holographic Optical Tweezers
Theory
Optical Trapping principle and regime
In the light of beam optics, the optical principle is explained by momentum change of light before and after particle. Momentum of equal magnitude and opposite direction is transferred from the photons to the sphere according to momentum conservation (right). Because of the light gradient, the path originating from the center of the beam carries more photons than the light path starting from the outline of the beam, resulting in a greater force pulling the particle toward the focal point.
2-4 Theoretical stiffness of optical traps calculated using Rayleigh scattering (point dipole approximation), Lorentz-Mie theory, and the ray optics model. There are three types of theoretical developments of the force applied to the trapped spherical particle depending on the particle diameter, a, and the laser wavelength, λ, ratio. The ray optics approach describes for a >> 10λ,[6] and a Rayleigh distribution or dipole point approximation for a case << λ/10.[7, 8] Most physical objects that are useful or interesting to capture, in practice, tend to fall in this intermediate size range (0.1 – 10 λ) which is considered as a ~ λ case.[9, 10] In this case, the generalized Lorenz–Mie theory (GLMT) is applied .
Particle motion in a fluid
Fourier Transform in the system
Method
- Equipartition Theorem
- Hologram generating program
- Particle tracking program
- Fourier Transform by lens
For 2D signals, the "FFT vi" provided in Labview calculates the Discrete Fourier Transform (DFT) of the input matrix. The Gerchberg-Saxton algorithm[4] is an iterative algorithm for obtaining the phase of a pair of light distributions related via a propagation function, such as the Fourier transform, if their intensities in their respective optical plane are known. If the intensity difference between the Fourier-transformed field and the desired field is above the specified error, the Fourier-transformed and phase-only information of the field is taken to be multiplied by the reference intensity.
After this process, the field is Fourier-transformed and again compared with the desired intensity. In "phase calibration part" the phase values are converted to the range of [0,255] corresponding to the input of SLM. In this section, we will see how the above logic is realized in Labview by looking at the front panel of the program.
So the program only changes the phase information with an intensity predetermined by the input beam profile. The allowed error between the values of the processed desired array and the origin desired array can be set in the "Error" part. Assume that electromagnetic wave propagates from the left side of the lens to the right side.
It is assumed that the beam propagates through a virtual rectangle whose thickness (Δ0) covers the thickest part of the lens. We considered the biconvex lens, where R1 is the radius of curvature of the convex part and R2 is the radius of curvature of the concave part. The above equation, however, can be applied to any type of optical lens by appropriately changing the signs of R1 and R2.
In general, the diameter of the lens is much smaller than the radius of curvature, and we can safely use binomial approximation. Assume that there is an input field, EO, that propagates a distance d, passes through a lens, and then propagates a lens focal length f. The same relationship is applied for Ef and El' for the distance f, the focal length of the lens.
Experiment
- Sample preparation
- Trapping laser
- Objective lens
- SLM
- System scheme
- SLM calibration
- Stiffness calibration
- Results: Trapped image
The transmissive type passes the beam through the liquid crystal and the reflective type passes the beam through the liquid crystal and bounces back so that the beam passes through the liquid crystal again. The reflective type has a thinner liquid crystal layer and its switching speed is faster compared to the transmissive type. The 4-4 liquid crystal pixels in the top 3 rows are aligned parallel to the substrate, while the bottom 2 rows are tilted.
This experiment uses a spatial light modulator with liquid crystals on silicon (Hamamatsu Photonics, LCOS-SLM 13267-03). In Figure 4-4, the top three layers of liquid crystal are aligned parallel to the silicon substrate, while the bottom two layers of liquid crystal are at a certain angle to the silicon substrate. At this point, only the phase-modulated polarization direction of the linearly polarized incident beam is the same as the alignment of the liquid crystal layer direction.
A half wave plate (HWP) and a polarizing beam splitter (PBS) are used to reduce the power of the laser. SLM is placed at the focal plane of L1 while the angle between input beam and output beam is 7.7°. It is therefore necessary to polarize the input beam before it reaches the window of SLM, to make its polarization parallel to the alignment direction of the liquid crystals.
To make the most of the SLM window, the input beam should be centered on the beam and in the center of the SLM window. And a focusing lens is placed so that the SLM is in the focal plane of the lens. To get the conversion factor, I set the input field polarization to 45˚ in the x-direction by rotating the half-wave plate.
11 Intensity graph of the field by SLM and PBS while input value of SLM varies from 0 to 255. As we expected, the intensity changes by changing the SLM input, and we can get the SLM calibration from the period and the offset values of the the above function. Because the Brownian particle is in thermal equilibrium with the heat bath consisting of the water molecules, its probability distribution follows the Maxwell-Boltzmann distribution [13].
The black solid line is a parabolic fit of the red dots and corresponds to the potential energy curve of the trap. The images of the grid optical trap, trap position changes over time and the sum of arbitrarily positioned optical traps are shown.
Conclusion & discussion
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Liska, "Optical collection of Rayleigh particles using a Gaussian standing wave", (in English), Optics Communications, vol. Schaub, “Theoretical determination of the net radiant force and torque for a spherical particle illuminated by a focused laser beam,” (in English), Journal of Applied Physics, vol. Stelzer, “Optical trapping of dielectric particles in random fields”. in English), Journal of the Optical Society of America a-Optics Image Science and Vision, vol.
Nieminen et al., "Optical tweezers computational toolbox," Journal of Optics A: Pure and Applied Optics, vol. Martin-Badosa, "Fast generation of holographic optical tweezers by random mask coding of Fourier components," (in English), Optics Express, vol. Wang, "Elimination of a zero-order beam induced by a pixelated spatial light modulator for holographic projection," (in English), Applied Optics, vol.
Vohnsen, "A direct comparison between a MEMS deformable mirror and a liquid crystal spatial light modulator in signal-based wavefront sensing," (in English), Journal of the European Optical Society-Rapid Publications, vol.
