621.317.382.029.6 Monograph No. 23 MEASUREMENTS SECTION ABSOLUTE POWER MEASUREMENT AT MICROWAVE FREQUENCIES By A. L. CULLEN, Ph.D., B.Sc.(Eng.), Associate Member. (The paper was first received 13/// June, and in revised form 9 th November, 1951. // was published as an INSTITUTION MONOGRAPH 15th February, 1952.) SUMMARY In 1876, Bartoli2 derived Maxwell's result from the second law The radiation pressure of electromagnetic waves on a reflecting of thermodynamics in a very general way, showing that it was surface may be used as the basis of a method of power measurement at applicable to all streams of energy. He also attempted to verify microwave frequencies. The method is absolute, depending only on the result experimentally, but was unable to overcome disturbing measurements of mass, length and time; no secondary electrical effects due to the heating of the reflector, which gives rise to con- standards are involved in the experimental procedure. Moreover, no vection currents of air and to the radiometer effect, collectively appreciable power is absorbed by the apparatus. described as "gas action." This method has been compared with a balanced calorimeter watt- 3 meter, and agreement to within 1-2 watts has been obtained in the Lebedew, using light waves, succeeded in 1900 in eliminating range 10-50 watts. The major difficulty encountered in obtaining these unwanted effects from the final result, and he was the first greater accuracy with the radiation-pressure apparatus described is to give a reliable demonstration of the reality of radiation the disturbing effect of convection currents of air, caused by the heating pressure. By 1902, Nichols and Hull4 had made a thorough of the reflector by the small fraction of the incident microwave power investigation of the "gas action," and had established the which is not reflected. quantitative accuracy of Maxwell's result. Recently, Carrara and Lombardini7 have demonstrated the LIST OF PRINCIPAL SYMBOLS existence of radiation pressure at microwave frequencies, but u,b Dimensions of rectangular waveguide. apart from an indication that the pressure was of the correct A Normalized characteristic admittance. order of magnitude, no quantitative results were obtained. B Normalized susceptance. Independently of this work, a quantitative method of measur- c Velocity of light in vacua. ing power at microwave frequencies by making use of the F Force due to radiation pressure. radiation pressure effect has been put forward.8 Preliminary h Height of ripple. experimental results have already been reported;9 it is the purpose H Magnetic field strength. of the present paper to describe the techniques adopted and to rms R.M.S. magnetic field in incident wave. explain their theoretical basis. I = Moment of inertia. in, a • Coupling coefficients. (1.2) Elementary Theory P Instantaneous value of radiation pressure. Maxwell's derivation of an expression for the pressure of P = Incident power. electromagnetic radiation was based upon energy considerations r Radial co-ordinate. and gave no indication of the mechanism of the effect. Sir J. J. 'l = Radius of a circular guide. Thomson later obtained the same result by considering the '2 Ratio of amplitudes in successive swings. mechanical force exerted by the magnetic field of the wave on the Characteristic wave impedance. currents induced by it in the reflecting surface. Phase constant. This more physical approach will be followed here to prove t Damping coefficient. the desired result. Consider a simple harmonic plane wave A Penetration depth. normally incident on a highly conducting surface. The wave will Free-space permittivity. be almost completely reflected at the surface with approximate Guide wavelength. doubling of the magnetic field and partial cancellation of the Angular position of movement. electric field. Some energy will flow into the conductor, but will P - Field reflection coefficient. penetrate only slightly, in accordance with the well-known skin- PP - Power reflection coefficient. effect. Inside the conductor, displacement current will be X - Radial propagation coefficient. swamped by conduction current; if the instantaneous magnetic = Free-space permeability. field in the conductor is H, and the instantaneous current density - Angular frequency. is t, curl H = 6 at all points. (1) BASIC PRINCIPLES With a Cartesian co-ordinate system, and OX as the direction (1.1) Introduction of propagation of the plane wave, OY as the direction of the Tn 1873, Maxwell1 showed from his electromagnetic theory of electric vector, and OZ as the direction of the magnetic vector, light that a beam of radiation falling normally on to a flat surface and the conductor surface perpendicular to the OX axis, this would give rise to a pressure equal in magnitude to the electro- reduces to magnetic energy density at the surface. He suggested the WJlx = - i. (1) possibility of experimental verification in the following words: "Such rays falling on a thin metallic disc, delicately sus- Consider a volume element dxdydz inside the conductor. The pended in a vacuum, might perhaps produce an observable current passing through this element is iydxdydz, and its length mechanical effect." in the direction of the current is dy. Thus the force on the element (Bli) is Bziydxdydz or fi0H2Lvdxdz. This may be Dr. Cullen is at University College, London. regarded as a pressure acting on the side dydz and equal to The paper is based on part of a thesis approved for the degree of Doctor of Philosophy in the University of London. Written contributions on Monographs are invited for consideration with a view to publication. The total instantaneous pressure, p, is found by integrating the [100] CULLEN: ABSOLUTE POWER MEASUREMENT AT MICROWAVE FREQUENCIES 101 pressures on the sides of all such elements; thus, if the conductor If the incident r.m.s. magnetic field strength is H£tx and is semi-infinite the total r.m.s. magnetic field strength at the surface of the con- ductor is Hrms, then Using eqn. (1) p= -H\HzdHz o Therefore H}ms ~ (7) The incident power in the beam is and since // 0 (2) Substituting from eqns. (7) and (8) in eqn. (4), and remem- This equation gives the instantaneous pressure in terms of the bering that c = lf-\/(jjLQK ) where c is the velocity of light instantaneous magnetic field at the conductor surface. 0 Jt is more useful in practice to know the average pressure over a long period of time, so that if Hrms is the r.m.s. value of the magnetic field, the average pressure p is This is the equation derived by Maxwell and verified by * = WO» (3) Nichols and Hull. If the reflector were perfect, pp would be unity. Thus, the upper limit for the force is The total force F exerted on reflecting surface by a beam of cross-sectional area a' is given by F=2P+/c (9a) 2 F = l{J>0Hr msa' (4) At optical wavelengths, p will be appreciably less than unity; e.g. for one of the silver reflectors used by Nichols and Hull the The experimental verification of this equation presents some value of p was 0-92. At microwave frequencies, however, difficulty, because H cannot be measured directly. In the p rmi almost complete reflection is easily obtained; e.g. in copper, the experiments of Nichols and Hull, the following procedure was 4 penetration depth is 11 x 10 cm at a wavelength of 10 cm, adopted: so that, using eqn. (6), the reflection coefficient differs from unity (a) First, the pressure of a given beam of radiation on a reflecting by about 1 part in 104. surface was determined. The order of magnitude involved is such that, for an inci- (b) Secondly, the beam of radiation was allowed to fall on a dent power of 1 watt, the force is 0-677 x 10~8 newton or wholly absorbing surface, and the resulting temperature rise was 3 employed to calculate the incident power P+ in the beam. 0-667 x 10~ dyne. Clearly, a sensitive apparatus is necessary (c) Finally, the power-reflection coefficient pp of the reflecting to detect such a force, and one must expect to find experimental surface was determined. difficulties in its accurate determination. The details of experimental technique are not relevant at It is instructive to derive eqn. (9) from a simple quantal treatment in which the beam of radiation is treated as a stream present, but it is necessary to consider the calculation of Hrms of photons. Let there be n photons per unit volume in the inci- from p and pp. If ^displacement currents in the conductor are neglected, the reflection coefficient p for the magnetic field can dent beam. Each photon has energy hv and momentum hv/c be written as and travels with velocity c. The power P+ in the incident beam is then P+ - nhvca' (10) a J If the power-reflection coefficient of the reflector is pp, a fraction pp of the incident photons will be reflected* and a fraction (1 — pp) will be absorbed. Equating force and rate of change of momentum gives This expression can be put into a more convenient form by F -- 2nhva'p + nhva'(\ — p ) introducing the free-space wavelength A and the penetration p p depth in the metal, A. or F— nhva'(\ r pp) • (ID Since 2TT/A — a>-\/(/v<x>) an£* ^ = VX^/WJUOCT) it follows that Substitution from eqn. (10) yields 2TTA P+ 0 The previous neglect of displacement currents in the metal is now seen to correspond to neglecting (A/A)2 in comparison with which is in agreement with eqn. (9). unity. It is therefore appropriate to work to the first order in A/A in the following calculation, when eqn. (5) becomes (1.3) Previous Experimental Investigations The experiments of Lebedew and of Nichols and Hull were p~l -(1 +J)2TTA/\ similar in principle.