3. Theory connecting acoustic, optical, microwave and radio regimes
3. Theory connecting acoustic, optical, microwave and radio regimes
It is known that optical radiation exerts pressure on solid objects. Solar radiation pressure is significant enough at the orbit of Earth to be included [14] as a low-order effect in trajectory calculations for dust in the vicinity of Earth. When solid particle radius is less than 5% of the incident wavelength, the force exerted on a particle by radiation can be modeled using Maxwell’s relations, simplified for the Rayleigh regime [9]. Here the phase differences between radiation falling on different parts of the particle are negligible, simplifying the interference between incident and scattered radiation. In this regime, particles experience a net force in the direction of the incident beam, where there is only one beam, but in the direction of increasing intensity (towards the beam waist) if the beam is focused. Ref.[10] discusses the trapping force experienced on such particles, and shows that radiation forces can be increased by 3 orders of magnitude, and the “trap stiffness” increased by seven orders of magnitude, when a standing wave pattern is created. Positioning is improved when the reflected beam from a mirror interferes with the incident beam. Particles move towards nodes or antinodes of the standing wave field depending on their relative refractive index. Ref. [13] presents a method for computing forces on neutral particles in an electromagnetic waveguide.
Similar phenomena occur in the field of ultrasonics [15,16]. Beissner [17] discussed models for the radiation pressure in ultrasonic fields from the points of view adopted by Langevin and Brillouin, and compared them in the context of measurements of the radiation force on an absorber at oblique incidence. Collas [18] showed results on acoustic levitation in ground experiments. Yarin et al [19] calculated acoustic radiation pressure using a boundary element method and predicted shapes of levitated droplets, which showed good agreement with experimental measurements. They showed that displacement of the droplet center relative to the pressure node due to the presence of gravity (or other steady force) was significant and could be computed. Wang and Lee [20] reviewed the subject of radiation pressure and acoustic levitation, keeping in view the applications to containerless processing in microgravity. In these applications, ultrasonic frequencies were used, with extremely high amplitudes achieved in the resonator. Zhuyou et al [21] report levels of 183dB inside their ultrasonic levitator used to levitate steel spheres. Refs. [20,22] discuss the issues of acoustic streaming inside these chambers, and their influence on the levitated particles. With ultrasonics, the practical size range of levitated particles goes beyond the Rayleigh regime, and the streaming flow around the particles has a profound influence on thermal gradients, spinning motion, vibration and the ability to retain a coherent trapping force.
We [23] recognized that high sound levels are not necessary, and that acoustic manipulation of objects in reduced gravity would work with audible sound frequencies. In experiments aboard the NASA KC-135 flight laboratory, we showed that positioning worked better when the sound levels were low enough so that the streaming effects were small. Refs. [24-25] extended the flight test results to ground experiments with liquids and powder suspensions. These were the first demonstrations that a multitude of particles inside a resonant chamber would form single-particle-thick walls parallel to the nodal surfaces, and not agglomerate around points of minimum potential.
Our approach in this
chapter starts from the observation that the equations describing the generation
of radiation forces and trapping stiffness in optics and acoustics are
similar. We confirmed this similarity through results from flight and ground
experiments using audible sound, comparing them with results from optics
and ultrasonics in other wavelength and size regimes. Our results on acoustics
show that complex surface shapes can be generated by suitably tailoring
frequency and resonator geometry. Predictions for cylindrical and rectangular
resonators show that various surface shapes of practical interest can be
generated. We generalize these observations to explore the use of long-wave
electromagnetic fields to move and position construction raw material in
microgravity along desired wall shapes, automatically and gradually, using
a continuous input of solar-derived energy. A comparison is developed where
particles of the same material are used with optical, acoustic and microwave
fields, exploring the power requirements in different wavelength regimes
to achieve the acceleration level needed to overcome noise. The comparison
is confined to the case of transparent materials in standing wave fields.
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A critical parameter for determining the interaction of radiation with solid particles is the refractive index. In the Rayleigh regime where target particle radius is much smaller than the wavelength (a < 0.05l) the particle experiences a uniform instantaneous field due to the electromagnetic wave. Only the electric field need be considered, which makes the problem equivalent to that of an isotropic, homogeneous, dielectric sphere in a uniform field. A spherical target in this size range acts as an oscillating electric dipole. For an incident wave of unit intensity, and transparent particles (refractive index mostly real) the scattered intensity [10] is:
(2)
The main feature of Rayleigh scattering
in the above is the dependence of the scattered intensity upon the inverse
fourth power of the wavelength. Total scattered energy can be obtained
by integration over the sphere surface. When a wave is reflected off a
mirror and a standing wave pattern is formed, there are sharp intensity
gradients in the beam. Under these conditions, the two main contributions
to the electromagnetic forces acting on the particle in a standing wave
field are the Gradient force and the Scattering force. Following [10]:
(3)
and
(4)
The force expressions in the electromagnetic
field are similar in form to those in the acoustic field, for the moderate-intensity,
Rayleigh regime of acoustics where acoustic streaming and the generation
of harmonics by nonlinearity are secondary. The electromagnetic fields
do not offer mechanisms for the generation of such nonlinearities in their
simple form, though such effects cannot be ruled out when interaction with
large numbers of particles is considered in detail. Parameters may
be compared roughly as shown in Table 3.1. While small, the acoustic
field numbers in Table 3.1 are seen to be adequate [24] to rapidly form
walls with various types of particles, even in the presence of g-jitter
of the order of 1m/s2.
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Table 3.1. Important Parameters and Magnitudes:
Comparison of Optics and Acoustic Force Fields.
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Parameter
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Optics
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Acoustics
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| Radiation
stress tensor
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| Rayleigh regime size | Nanometers | Millimeters to centimeters |
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Material
parameter
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Ratio
of Refractive Indices of Solid to medium(or vacuum)
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Density
ratio of particle to medium
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Intensity
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Optical
intensity
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Sound
pressure fluctuation intensity
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Force
order of magnitude
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Zemanek, Re.[10]: 514.5 nm laser; beam waist
of 8 microns; 5nm glass sphere in water; Force = 2.5 *10-22
N.
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Wanis[24]: 156 dB at 800 Hz (1 0 0) mode
at 2mm radius rigid particles. Force = 3.3*10-6 N
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3.3 Development of a common basis for comparison across wavelength and particle size
In Figures 3.1 – 3.3, this comparison is
extended to a standing microwave field to get a different range of wavelength
and particle size. In this first consideration of the generalized problem,
we used the following logic to enable a direct comparison of different
types of waves and particles, drawing upon each application area. Optical
tweezers usually use visible wavelengths and the theoretical expressions
are simpler for transparent particles (glass, which is mostly silicon dioxide).
Microwaves transmit through silicon dioxide, and acoustic shaping works
on most materials. This enables us to choose material of the same density
(roughly 2000 kg/m3), and assume the refractive index of glass
relative to vacuum for both the optical and microwave cases. In Figures
3.1 – 3.3, the force per unit incident radiation intensity is divided by
particle mass to obtain the acceleration per unit intensity. In the case
of gradient forces, the gradient is approximated by dividing the intensity
by a quarter-wavelength (Chapter 7 includes a more refined calculation
which justifies this). The abcissa is the particle radius. For each particle
radius, the wavelength used is 20 times the particle radius to stay within
the Rayleigh regime definition and remove some of the wavelength dependence.
The acceleration in each case depends inversely on particle radius. This
poses a drawback in dealing with raw material until powerful long-wave
resonators can be developed, or we learn to generate adequate coherent
forces in the Mie scattering regime.
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Figure 3.1 Estimate of the acceleration
per unit intensity, experienced by glass spheres in a standing wave field
of optical radiation in vacuum, with the radiation wavelength being 20
times the particle radius.
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Figure 3.2 Estimate of the acceleration per unit intensity, experienced by silicon dioxide spheres in a standing wave field of acoustic radiation in air, with the radiation wavelength being 20 times the particle radius. | ||
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Figure 3.3 Estimate of the acceleration per unit intensity, experienced by silicon dioxide spheres in a standing wave field of microwave radiation in vacuum, with the radiation wavelength being 20 times the particle radius. | ||
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To-date, as seen from Table 3.1, theRayleigh-domain experimental data are in the acoustic regime with millimeter-scale objects [23-35] and the optical regime with nanometer-scale objects [10]. The above results indicate that high microwave intensity would be required to move particles. It is a good rule of thumb that intensities achievable inside resonators can reach 3 orders of magnitude higher than source beam intensity. Our experiments on acoustic shaping (below) show that 40kW/m2 corresponding to the 156dB resonant field shown in [24] is adequate for forming walls from ceramic materials in acoustic resonators.
In the optical regime, the values of acceleration per unit intensity are 1 to 2 orders of magnitude lower than those in the acoustic case. As the wavelength (and hence the maximum particle size considered) increase, the acceleration per unit intensity decreases in inverse proportion. However, the feasibility of generating high power improves rapidly, and the cost of power generation at the desired wavelength decreases. For example, infrared lasers achieve 1kW routinely for far less cost per watt than, say, a visible-range laser.
Going into the microwave regime, we see that the values of acceleration per unit intensity are 6 orders of magnitude below those in acoustics. We have no experimental evidence so far of particles being positioned using microwaves; however, JPL’s web pages speak of a microwave sail being developed, as an extension to solar sail technology. Clearly, microwave intensities needed to produce significant acceleration will be quite large. Microwave beam intensities up to 8MW /m2 have been demonstrated in ground-based laboratory experiments [26]. With a resonator Q-factor of 1000 for short-duration operation in a wall-formation application, we may thus expect to achieve microwave resonator intensity in space experiments of 8GW/m2. It thus appears reasonable that microwave-induced electromagnetic shaping using raw materials such as silicon dioxide (primary component of lunar regolith) is feasible in prototype experiments where we can use closed, metal-cased enclosures.
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