3. Theory connecting acoustic, optical, microwave and radio regimes

In this section we summarize the development of a uniform view of force field tailoring, generating relations between frequency, wavelength, and particle dimensions.
 
3.1 Previous work

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.
 
 
 
Abstract 
Intro
Theory
Near-term: Acoustic
Mid-Term: L2 Habitat
Space Economy 
Far-Term: Radio-Wave Construction
Comments
Issues
Conclusions
Acknowledgements
References

 

3.2 Generalized Relations
From Maxwell’s Equations, the undamped electromagnetic wave equation in a non-dissipative medium is:
                                                                                                                               (1)

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.
 
 
 
 
 
Abstract 
Intro
Theory
Near-term: Acoustic
Mid-Term: L2 Habitat
Space Economy 
Far-Term: Radio-Wave Construction
Comments
Issues
Conclusions
Acknowledgements
References

Table 3.1. Important Parameters and Magnitudes: Comparison of Optics and Acoustic Force Fields.
 
Parameter 
Optics
Acoustics
Stress term 
Maxwell’s stress tensor 
Radiation stress tensor

Rayleigh regime size  Nanometers  Millimeters to centimeters 
Material parameter
Ratio of Refractive Indices of Solid to medium(or vacuum) 
Density ratio of particle to medium 
Intensity 
Optical intensity 
Sound pressure fluctuation intensity
Force order of magnitude 
Zemanek, Re.[10]: 514.5 nm laser; beam waist of  8 microns; 5nm glass sphere in water;  Force = 2.5 *10-22 N.
Wanis[24]: 156 dB at 800 Hz (1 0 0)  mode at 2mm radius rigid particles. Force = 3.3*10-6 N

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.
 
 
 
 
 
Text Box:  ACCELERATION / INTENSITY, m/Ws2
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. 
 
PARTICLE RADIUS a  (m)
 
 


 
Text Box:  ACCELERATION / INTENSITY, m/Ws2
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. 
 
PARTICLE RADIUS a  (m)
 



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. 
 
PARTICLE RADIUS a  (m)
 
 



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.

.
 
 
 
 
Abstract 
Intro
Theory
Near-term: Acoustic
Mid-Term: L2 Habitat
Space Economy 
Far-Term: Radio-Wave Construction
Comments
Issues
Conclusions
Acknowledgements
References