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The design of solar power satellites ( SPS) or a lunar power system (LPS) involves beaming energy to Earth, perhaps in the form of microwaves. Unfortunately, the magnitude of such projects has discouraged the planning of demonstrations. In order to understand the physical requirements that tend to push the size of a space power system upward, it is necessary to consider the physics of wireless power transmission, or power beaming. As most space advocates know, radio waves, microwaves, infrared rays,visible light, ultraviolet rays, and xrays are all forms of electromagnetic radiation. They differ merely in their wavelength; that is, the distance from the crest (or trough) of one wave and the crest (or trough) of the next wave. Regardless of wavelength, all electromagnetic radiation travels at the same velocity, namely, the speed of light (300,000 kilometers per second or 186,000 miles per second). Therefore, the shorter the distance between the waves, the more of them will pass a given point in one second. In other words, the shorter the wavelength, the higher the frequency. The relationshipbetween frequency (f) and wavelength (L) is given by the following equation: c f =  L (Equation 1) where c is the speed of light. Note that the units must be consistent. Typically, c is given as 300 million meters per second and L is given in meters. Thus, f is given in "inverse seconds" or cycles (waves) per second, also known as hertz (Hz). The unit hertz is too small to conveniently describe commonlyused frequencies, so prefixes are added. Radio waves are measured in thousands of hertz (kilohertz or kHz) or millions of hertz (megahertz or MHz). Microwaves are measured in billions of hertz (gigahertz or GHz). Transmitting power is conceptually simpler than conventional radio and television broadcasting, since information need not be encoded into the waves. Radio waves or microwaves are typically produced by using a transmitter, which is simply a device which pushes electric charges (electrons) back and forth in a piece of metal onehalf wavelength long (an antenna). This will cause a varying electric and magnetic field (that is, an electromagnetic wave) to radiate outward. The receiving antenna must likewise be onehalf wavelength long. The reason for this is that antennas are analogous to strings on a musical instrument and work by the electromagnetic equivalent of resonance. A typical halfwavelength antenna, known as a dipole, can transmit or receive in all directions, except straight off the ends. In order to focus the energy in one direction, additional antenna elements are added. These are known as reflectors and directors, and may be slightly longer or shorter than the main ("driven") element. If an antenna needs to receive a wide variety of frequencies (such as television channels), it may need several driven elements, reflectors, and directors of various sizes. This is why television antennas typically have an arrowshaped appearance. Since microwaves have wavelengths of only a few centimeters, or even millimeters, the dipole antennas used are correspondingly small. This makes another type of beam focusing apparatus feasible; one known as a phasedarray antenna. This is a large number of dipoles attached to the same flat surface. Each one is fed energy from the transmitter. If they are spaced no more than onehalf wavelength apart, then the waves from each one will merge together. The beam can be steered electronically by slightly varying the timing, or phase, from one dipole to the next, so that the wave crests connect up at an angle. In any case, the waves emanating from the array act as though they are oversized light waves shining through a giant hole. Many of us recall high school or college physics demonstrations in which a laser beam shines through a narrow opening and spreads out or diffracts. There is typically a bright spot seen on a far wall, with fainter spots on either side of it. This is because different parts of the wall are at varying distances from different parts of the hole. Waves will therefore reinforce each other at some locations and cancel each other out at others. Diffraction will be most apparent if the hole is not that much larger than a wavelength of light. However, even for small wavelength spassing through large openings, the central spot will spread out wider than the opening, if projected a long distance away. (Diffraction from a grating is a separate, but related, phenomenon in which light waves shine through many closely spaced holes, or reflect off of grooves.) The situation with microwaves is simply a scaledup version of diffraction, although the geometry of a circular "hole" or phased array makes the light and dark pattern slightly different from that of a narrow slit typically used in optics demonstrations. A typical microwave diffraction pattern consists of a circular central maximum (or "main lobe," in the language of antenna designers), which contains 84% of the energy. Surrounding this central maximum will be ringshaped secondary maxima, also called sidelobes. Between these secondary maxima are regions of little or no energy, known as minima. A microwave power beam is captured by an array of dipole antennas known as a rectifying antenna, or rectenna. A rectenna is generally a bit smaller than the main lobe, since the mainlobe edge and the sidelobes contain energy which is very dilute. It is thus the main lobe that is of interest to us in designing a power transmission system. The width of the main lobe increases with increasing distance from the transmitter. In addition, the main lobe spreads out more if the transmitting phased array is smaller. The relevant quantity here is the ratio of the phased array diameter to the wavelength. Thus, there are two ways to keep the main lobe from spreading. One is to increase the physical diameter of the phased array. The other is to increase its effective diameter by decreasing the wavelength (that is, by increasing the frequency). The tradeoffs involved in designing a power beaming system are now apparent. If a narrow beam is desired (allowing for a small rectenna), then a larger phased array transmitting antenna must be deployed, thereby increasing launch costs of a solar power satellite. This can be circumvented by turning up the frequency. However, too high a frequency will result in a beam that may be dangerously intense. In addition, frequencies in the tens of gigahertz are subject to absorption by rain and air. Frequencies less than about two and a half gigahertz will cause the beam to spread out, and will interfere with communications. A frequency of 2.45 GHz is commonly considered. The relationship between transmitting phased array diameter (Dt), received main beam lobe ("spot") diameter (Dr), wavelength (L) and transmittertoreceiver distance ("separation" S) is given by (Dt x Dr)  = 2.44 (L x S) (Equation 2) If the lefthand side of Equation 2 is greater than or equal to 2.44, then at least 84% of the energy of the beam will be captured. However, this percentage is not linear; in other words, if we wish to capture only 42% of the energy, this does not mean that we can use 1.22 on the righthand side of Equation 2. Nevertheless, for a given energy percentage, transmittertoreceiver separation, and wavelength, the product of Dt and Dr must have a certain minimum value. If one antenna is small, then the other must be large. Equation 2 represents a bestcase scenario. Practical antenna sizes may have to be larger if most of the beam is to be captured. If the transmitting antenna array is many wavelengths wide, then the beam will not spread out much over a short distance, but it will start out with width Dt. The rectenna will have to be at least as large as Dt, even if Equation 2 says Dr is smaller. There are ways in which such a wide beam can be focused, but it cannot be focused to a point. It can be focused only to a minimum size Dr, as given by Equation 2. For low Earth orbit power beaming demonstrations, it is easier to put the smaller antenna in space and the larger antenna on Earth. Furthermore, early demonstrations may capture only a small percentage of the total power, in order to keep antenna sizes small. This means that to light up a 60 watt bulb, thousands of watts may have to be transmitted. Since it is likely to be costly to launch such a power generating apparatus, the most feasible demonstration project may be Earthtospace transmission from a large transmitting antenna (such as the Arecibo dish) to a smaller rectenna in space. The peak beam intensity (I) at the rectenna (in, say, watts per square meter) is given by P Dt 2I = (pi x ) x () 4 L x S (Equation 3) where P is the total power transmitted (and pi = 3.14...). For example, consider a typical solar power satellite transmitting 5 GW (5 billion watts) of power from geostationary orbit (35,800 kilometers high) to the equator at a frequency of 2.45 GHz. We wish to estimate the size of a practical rectenna if the transmitting antenna array is 1 kilometer in diameter. First, we must find the wavelength, using a rearranged version of Equation 1: meters 300,000,000  c sec L =  =  = 0.1224 meters. f 2,450,000,000  sec Again, note that we must be consistent in the units we use. For the purpose of estimation, we can consider the rectenna to be roughly equal in size to the main lobe. Thus, 2.44 x L x S 2.44 x 0.1224 m x 35,800,000 m Dr =  =  Dt 1000 m Dr = 10,700 meters = 10.7 kilometers which is about what we expect for an SPS rectenna "farm." Equation 3 gives a peak beam intensity of 205 watts per square meter, or 20.5 milliwatts per square centimeter. The average intensity is about a third of this. The peak is only about twice the US industrial standard for human exposure. However, it is enough to provide usable power, since the rectenna converts it to electricity at about 90% efficiency. For a rectangular transmitting antenna array, the mathematics are slightly different, but the same general principles apply. The central maximum of the beam will contain 82% of the transmitted energy. It will be rectangular in shape, but it will spread out more along the transmitter array's short direction, than its long direction. For example, Canada's Radarsat has a 1.5 meter by 15 meter rectangular transmitting antenna, resulting in a "footprint" on the ground of 50 kilometers by 7 kilometers (at a frequency of 5.3 GHz and an altitude of 800 kilometers). With an output of 5000 watts, the power is too spread out at the ground to use in a practical demonstration project. Practical microwave transmission systems often use certain transmitting methods which can reduce the level of the sidelobes, and put some of the sidelobe energy into the main lobe. The price paid for this is that the main lobe spreads out, necessitating a larger rectenna. Thus, Equation 2 may be optimistic. Nevertheless, it is a useful rule of thumb in determining the feasibility of power beaming systems. In addition, the principle of diffraction also limits the resolution of optical systems such as lenses and telescopes. It is therefore an important concept in the design of space systems.
Seth D Potter was a Research Scientist in Physics at New York University when this article was written. He is currently an engineer at The Boeing Company in Seal Beach, California, USA, and serves on the Board of Directors of the National Space Society Education Chapter.
