Radiometry And The Friis Transmission Equation

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Radiometry and the Friis transmission equationJoseph A. ShawCitation: Am. J. Phys. 81, 33 (2013); doi: 10.1119/1.4755780View online: http://dx.doi.org/10.1119/1.4755780View Table of Contents: hed by the American Association of Physics TeachersRelated ArticlesThe reciprocal relation of mutual inductance in a coupled circuit systemAm. J. Phys. 80, 840 (2012)Teaching solar cell I-V characteristics using SPICEAm. J. Phys. 79, 1232 (2011)A digital oscilloscope setup for the measurement of a transistor’s characteristic curvesAm. J. Phys. 78, 1425 (2010)A low cost, modular, and physiologically inspired electronic neuronAm. J. Phys. 78, 1297 (2010)Spreadsheet lock-in amplifierAm. J. Phys. 78, 1227 (2010)Additional information on Am. J. Phys.Journal Homepage: http://ajp.aapt.org/Journal Information: http://ajp.aapt.org/about/about the journalTop downloads: http://ajp.aapt.org/most downloadedInformation for Authors: htmlDownloaded 07 Jan 2013 to 153.90.120.11. Redistribution subject to AAPT license or copyright; see http://ajp.aapt.org/authors/copyright permission

Radiometry and the Friis transmission equationJoseph A. ShawDepartment of Electrical & Computer Engineering, Montana State University, Bozeman, Montana 59717(Received 1 July 2011; accepted 13 September 2012)To more effectively tailor courses involving antennas, wireless communications, optics, and appliedelectromagnetics to a mixed audience of engineering and physics students, the Friis transmissionequation—which quantifies the power received in a free-space communication link—is developedfrom principles of optical radiometry and scalar diffraction. This approach places more emphasis onthe physics and conceptual understanding of the Friis equation than is provided by the traditionalderivation based on antenna impedance. Specifically, it shows that the wavelength-squareddependence can be attributed to diffraction at the antenna aperture and illustrates the importantdifference between the throughput (product of area and solid angle) of a single antenna or telescopeand the throughput of a transmitter-receiver pair. VC 2013 American Association of Physics Teachers.[http://dx.doi.org/10.1119/1.4755780]I. INTRODUCTIONIt is becoming increasingly common for physics and electrical engineering students to take courses together in optics,photonics, and topics in applied electromagnetics such aswireless communications, antennas, or remote sensing. Foreffective learning with students from different disciplines, itis helpful to identify examples where principles from onesubject area can be applied to another. Exactly such an opportunity arises when students encounter principles of radiometry in optics courses or the free-space Friis transmissionequation in antennas or wireless communications courses.The Friis transmission equation relates the received powerto the transmitted power, antenna-separation distance, andantenna gains in a free-space communication link.1 However, students first encountering the usual textbook derivation, based on equivalent dipole antenna impedance, tend togain little physical insight into the Friis equation. Even forstudents familiar with impedance concepts, the physical basis of the Friis equation becomes more apparent with a derivation based on radiometry2–4 and scalar diffractiontheory.5–8 Radiometry is the field of science devoted toquantifying the amount of power carried by a beam of electromagnetic radiation using geometric optics, areas, andsolid angles. Scalar diffraction theory allows us to use simpleFourier transform relationships to account for the beamspreading effects induced by the finite size of antenna or optical apertures.5–8 (Vector diffraction theory does the samething more accurately by using Maxwell’s equations withoutthe simplifying scalar-field assumptions.9)Textbooks on antennas and communications10–13 oftenwrite the Friis equation as Prk 2k 2¼ e r etDr Dt ¼Gor Got ;Pt4pR4pR(1)where P is power in Watts, e is the antenna efficiency, k isthe electromagnetic wavelength, R is the far-field lineof-sight distance between the antennas, D is the antenna directivity relative to isotropic, G is the antenna gain (G ¼ eD),and the subscripts r and t denote the receiver and transmitter,respectively. Friis actually developed the transmission equation in terms of effective areas Ar and At, of receiving andtransmitting antennas, respectively, resulting in33Am. J. Phys. 81 (1), January 2013http://aapt.org/ajpPr Ar At¼:Pt k2 R2(2)Following his derivation of the transmission equationfrom geometry and antenna impedance,1 Friis proposed thathenceforth antennas should be described in terms of theireffective area rather than antenna gain or radiation resistance, and that antenna radiation should be expressed aspower flow per unit area rather than field strength in Voltsper meter. In doing so, he effectively proposed moving froma traditional circuit viewpoint, which deals with current andvoltage, to a radiometric one, which deals with power, powerdensity, etc. However, in textbooks and classroom presentations, the Friis equation is normally derived or explained interms of antenna impedance, leading to Eq. (1). Unfortunately, without a deeper understanding of how the antennagain depends on wavelength, Eq. (1) can be mistakenly interpreted to mean that the received power increases with wavelength, whereas in fact the opposite occurs. Additionalconfusion often arises because the antenna impedance derivation is based on the effective area of an infinitesimaldipole, but applied broadly to antennas of any shape.10,11Several authors have discussed alternate derivations ofthe Friis equation. Friis himself presented an argument thatresembles a simpler form of the antenna derivation that iscommon in modern antenna textbooks, but he left the resultin terms of antenna areas as seen in Eq. (2). Hogg14 derivedthe Friis transmission equation using three differentapproaches—Fresnel zones, Gaussian beams, and modes—all of which lead to Eq. (2). Bush15 outlined a derivationbased on Fraunhofer diffraction integrals but did not combine diffraction with radiometry. Shortly thereafter, Heald16pointed out that the most significant benefit of Bush’s paperwas that it removed the oft-confusing antenna impedancefrom the derivation.The development presented here builds on this background to provide an even simpler path to the Friis equationbased on optical radiometry and a simple result of scalar diffraction theory. Many undergraduate and graduate studentshave been exposed to enough basic optical principles thatthis development is intuitively appealing, particularly in amixed class of physics and engineering students. This paperpresents a combined discussion of the Friis transmissionequation and optical radiometry, relying on the geometricC 2013 American Association of Physics TeachersVDownloaded 07 Jan 2013 to 153.90.120.11. Redistribution subject to AAPT license or copyright; see http://ajp.aapt.org/authors/copyright permission33

optics propagation of radiance (power per unit area per unitsolid angle) through free space. Beam spreading is introduced via scalar diffraction at the antenna aperture. Thisapproach still relies on aperture areas, but in a much moredirect manner and without any reference to antenna impedance. In this development, it becomes clear that the source ofthe wavelength-squared dependence is diffraction at theantenna aperture. This discussion also conveniently mergesseveral topics from antenna theory and radiometry. Therefore, for students familiar with radiometry this approachbuilds on familiar principles to enhance understanding of theFriis equation, and for students not familiar with radiometrythis approach teaches radiometry in the practical context of afree-space communication link.II. BASIC CONCEPTS OF OPTICAL RADIOMETRYThe radiometric development of the Friis transmissionequation relies on the concepts of radiance and throughput,both central to radiometry.2,3 As indicated in Table I, radiance is the power incident on (emitted from) a surfaceper unit area, from (into) a given solid angle, with units ofW/(m2 sr). Suggestive of its physical meaning, radiance often is called brightness in the microwave radiometry and radio astronomy communities.17,18 The concepts of radianceand related radiometric quantities can be introduced andused in courses ranging from antennas to photonics and optical design because they provide a consistent framework forcalculating power transmitted from or received by an opticaldetector or antenna. While the five quantities in Table I aresufficient for power flow calculations, radiance is the mostfundamental because it is invariant in a lossless system. Similarly, conservation of energy requires that the product ofarea and solid angle is invariant, a quantity referred to by optical scientists and engineers as throughput (sometimescalled e tendue). Antenna texts10,11 and optical texts discussing coherent receivers19 provide an added quantification ofthis concept by showing that the area-solid-angle product fora single-spatial-mode, diffraction-limited system is equal tothe wavelength squared. This provides a theoretical limit tothe field-of-view solid angle achievable with such systemsfor a given aperture size and wavelength.Whether or not the classroom discussion extends to a fullcoverage of all the radiometric quantities in Table I, the mostbasic of radiometric principles can be taught and used toderive the Friis transmission equation,1 the radar range equation,20 and the lidar equation.21 One of the most important ofthese radiometric principles is that the throughput is theproduct of an area and a solid angle that always opens awayfrom the area, as shown in Fig. 1. Students often need to bereminded that the steradian is a dimensionless unit of solidangle and can be thought of in a manner similar to the morecommon planar angle. As shown in Fig. 2, a planar angle inFig. 1. Geometry for radiometric calculations, showing that a solid anglealways opens away from the area (A) in which incident power is calculated.radians is defined as the ratio of (circular) arc length to radius (s/r), whereas a solid angle in steradians is defined asthe ratio of (spherical) surface area to the square of the radius(Asph/r2). In a more complicated situation, we can find a solidangle x by integrating in spherical coordinates over theappropriate range of polar and azimuthal anglesx ¼ 兼 sin h dh d/:(3)Although many antenna and physics textbooks use a capital omega (X) for all solid angles, in radiometry a careful distinction is made between solid angle x and projected solidangle X. For a projected solid angle, we replace the sphericalsurface area with a flat cross-sectional area, such as the projected aperture area of a lens, mirror, or antenna. The integraldefinition of the projected solid angle X is the same asEq. (3) but includes a factor of cos(b) in the integrand, whereb is the angle between the area’s surface normal and theviewing direction. The small-angle approximation allows usto see that the projected solid angle subtended by an objectcan be estimated simply as the ratio of the object’s projectedcross-sectional area divided by the square of the distance Rfrom the observer to the objectX¼A:R2(4)From dimensional analysis, the power collected by a receiver with area Ar from a transmitter subtending solid angleTable I. Radiometric quantities and antenna equivalents.Quantity SymbolUnitsAntenna PEMILWPower, PW/m2 incidentPower density, WW/m2 leaving surfacePower density, WW/srRadiation intensity, UW/(m2 sr)Brightness, B (in radio astronomy)34Am. J. Phys., Vol. 81, No. 1, January 2013Fig. 2. In plane geometry an angle h is defined as the ratio of circular arclength s to radius r (left); in spherical geometry a solid angle x is the ratioof spherical surface area A to the square of the radius r2. The projected solidangle X results from considering the projected area instead of the actualspherical area.Joseph A. ShawDownloaded 07 Jan 2013 to 153.90.120.11. Redistribution subject to AAPT license or copyright; see http://ajp.aapt.org/authors/copyright permission34

Xt,r as seen from the receiver, is given by the product of thesource radiance L (W m 2 sr 1) and the geometric throughputArXt,r (m2 sr)P ¼ LAr Xt;r ffi LAr At:R2(5)For example, in a Friis transmission scenario the area Arrepresents the effective area of a receiving antenna and Xt,ris the projected solid angle subtended by the transmittingantenna with effective area At located a distance R from thereceiver (Xt,r ¼ At/R2 is a common approximation wherethe flat cross-sectional area At is used instead of integratingover the projected spherical area). Because the resultingthroughput is equal to the product of the two areas dividedby the square of the distance between them, it is an invariant of the system regardless of which area is the receiverand which is the transmitter. This invariance means that asthe beam is compressed into a smaller area, its solid angleexpands proportionally. An important consequence of thisis that light focused onto a tiny single-mode fiber, for example, often has a solid angle that exceeds the maximum anglethat can propagate in the fiber, resulting in a very low lightcoupling efficiency. A similar concern exists when oneattempts to focus any electromagnetic energy into a smallarea.When discussing antennas it is important to emphasizethat the receiver solid angle used in the Friis transmissionequation is not the “beam solid angle” of antenna theory,except in the unlikely case of the transmit antenna beamexactly filling the receiver field of view at distance R; rather,we simply use the solid angle subtended by one antenna asseen from the other, a solid angle that usually is smaller thanthe antenna beam solid angle. One final note is that all solidangles discussed in the remainder of this paper are projectedsolid angles (the difference between solid angles and projected solid angles is tiny for small angles).III. FRIIS EQUATION DEVELOPED FROMRADIOMETRY AND SCALAR DIFFRACTIONTo develop the Friis transmission equation from opticalradiometry and scalar diffraction theory, we begin with acommon antenna calculation to incorporate the concept ofdirectivity into radiometry. The directivity D is an importantmeasure of how an antenna radiates preferentially in a givendirection relative to an isotropic antenna that radiates uniformly in all directions (see Fig. 3). Directivity can bedefined as the ratio of an antenna’s radiation intensityU (W sr 1) in a given direction, usually the direction ofmaximum radiation, to the isotropic radiation intensity U0(W sr 1). For total radiated power P, the radiation intensitycan be written as U ¼ P/X, while the isotropic radiation intensity is U0 ¼ P/(4p). Therefore, the directivity can be written as either a ratio of antenna radiation intensity U toisotropic radiation intensity U0 or a ratio of isotropic solidangle 4p to the antenna’s beam solid angle X UP4p4pD¼¼ :¼U0XPX(6)Notice that a narrower beam solid angle X results in alarger directivity D. This concept of solid angle ratio is used35Am. J. Phys., Vol. 81, No. 1, January 2013Fig. 3. Directivity is a concept that describes how much of an antenna’sradiated power is transmitted into a given direction (per unit solid angle X)relative to the amount radiated by an isotropic antenna radiating uniformlyinto solid angle Xo ¼ 4p sr.often in optical radiometry but is not usually referred to asdirectivity, a useful term that probably should be adopted bythe optics community. We can incorporate directivity intothe present discussion by writing the solid angle of the transmitted beam as the cross-sectional beam area Ab divided bythe distance squaredXt ¼Ab:R2(7)Using this result in Eq. (6) gives a transmitter directivityofDt ¼4pR2:Ab(8)It is here that diffraction must be invoked to bring thewavelength into the equation. Even for students who havenot studied scalar diffraction theory,5–8 an explanation canbe offered that diffraction causes a beam of electromagneticenergy to spread into an angle hd that is proportional to theratio of wavelength k over the size d of the aperture orobstruction causing the diffractionkhd ffi :d(9)This result is exactly true for a square aperture, but if d isthe radius of a circular aperture, Eq. (9) must be multipliedby 1.22 to obtain the Airy disk radius.5–8 This is a detail thatcan be discussed or not, depending on student background,available time, and instructor interest. At distance R sufficiently large that the beam dimension is much greater thanthe original aperture area, Eq. (9) and a small-angle approximation results in an estimate for the transverse dimension ofthe beamy ffi Rhd ffiRk:d(10)Now the beam area can be approximated asA b ffi y2 ¼R2 k2 R2 k2¼;d2At(11)Joseph A. ShawDownloaded 07 Jan 2013 to 153.90.120.11. Redistribution subject to AAPT license or copyright; see http://ajp.aapt.org/authors/copyright permission35

where At is the effective area of the transmit antenna.Substituting this beam area into Eq. (8), we obtain awavelength-dependent expression for the transmit antennadirectivityDt ¼4pR2 At 4pAt¼ 2 ;k2 R2k(12)which can be solved to obtain the standard relationshipexpressing the antenna effective area in terms of transmitterdirectivity At ¼ k2Dt :4p(13)In this development students see directly that thewavelength-squared term expresses the diffraction-inducedbeam spreading of an electromagnetic wave launched from afinite aperture. In fact, Eqs. (6) and (13) together reproducethe traditional antenna resultAX ¼ k2 ;PtPt R2ffi:At Xb At Ab(15)Using Eq. (11) for the beam area Ab, we find thatLt ¼Pt R2 AtPt¼ :At R2 k2 k2(16)Here we see that the transmitted radiance can be writtensimply as the ratio of transmitted power to the wavelengthsquared, again a result of the throughput equality expressedin Eq. (14). Equation (16) is another relatively commonantenna concept that would be useful in discussions ofdiffraction-limited optical systems.Radiance propagates unchanged in a lossless medium,so all that is required to find the received power is to multiply the radiance by the appropriate receiver throughput.This is where students sometimes make the mistake of multiplying by the effective receiver area (correct) and the36Pr ¼ Lt Ar Xtr ¼Pt Ar At:k2 R2(17)The final step to obtain the Friis equation in the form ofEq. (1) is to use Eq. (13) to convert the effective receiver andtransmitter areas into quantities involving directivitiesPr1 k2 Dt k2 Dr 1¼ 2¼Pt k 4p 4p R2 k4pR 2Dr Dt :(18)(14)which says that the beam throughput is equal to the wavelength squared.10,11,19 In both the antenna and radiometricdiscussions of the Friis equation, this expression plays a central role because it demonstrates that as the wavelengthincreases, either the antenna area must increase even faster(as the square of the wavelength) or the beam will spreadinto a proportionally larger solid angle. We arrive at thisresult using only simple geometric radiometry and a basicresult of scalar diffraction theory, without employing radiation resistances, equivalent circuits, power-transfer relations,or anything unique to an infinitesimal dipole. The derivationusing those quantities can be useful, but the radiometricmethod is an alternative that enhances understanding of thephysics behind the equation and makes the material more accessible to a wider range of students.The Friis equation derivation continues with the recognition that, in radiometric terminology, the transmit antennaemits a radiance given by the ratio of transmitted power tothe transmitter throughputLt ¼receive-antenna beam solid angle (incorrect). This mistakeis equivalent to multiplying Eq. (16) by k2, which wouldresult in 100% of the transmitted power being collected bythe receiv

II. BASIC CONCEPTS OF OPTICAL RADIOMETRY The radiometric development of the Friis transmission equation relies on the concepts of radiance and throughput, both central to radiometry.2,3 As indicated in Table I, radi-ance is the power incident on (emitted from) a surface per unit area, from (into) a given solid angle, with units of W/(m2 sr .

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