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PHILIPS RESEARCH REPORTS a Bimonthly Publication Containing Papers on the Research Work Carried out ~ the Various Philips Laboratories . " , 1:. 1 PHILIPS RESEARCH REPORTS, . A Journalof Theoretical and Experimental Research in Physics, Chemistry and Allièd'Fi'e.lds PHI LIP S,R ESEA R CH lAB 0 RA'T 0 R lES , I . Philips Res. Repts Vol. 30 pp. 1*-380* 1975 Printed In lbo Netherlands .~.. ~ • t., \ ! I ;' r Ó, t ;,' • , .. PHILIPS RESEARCH REPORTS A bimonthly publication containing papers on the research work carried out ~ the various Philips laboratories. Published in annual volumes of about 600 pages. © N.V. Philips' Gloeilampenfabrieken, Eindhoven, Netherlands, 1975. Articles or illustrations reproduced, in whole or in part, must be accompanied by full acknowledgement of the source: PHILIPS RESEARCH REPORTS OTHER PHILIPS PUBLICATIONS ( I PHILIPS TECHNICAL REVIEW A monthly publication dealing with technical problems relating to the products, processes, and investigations of the Philips Industries . { About 32 pages per issue. - . ELECTRONIC APPLICATIONS , ( A quarterly publication containing articles on electronics. 40-60 pages per issue. PHILIPS TELECOMMUNICATION REVIEW A quarterly publication containing articles on all subjects in the field of telecommunications. 40 pages per issue. For particulars regarding these publications apply to the distributor mentioned on the back-cover of this issue. N.V. P"'UPS' GlO£llAMrtNFABR\lKl~ . - SPECIAL ISSUE IN HONOUR OF C. J. BOUWKAMP ON THE OCCASION OF HIS SIXTIETH BIRTHDAY AND HIS RETIREMENT AS SCIENTIFIC ADVISER OF PHILlPS RESEARCH LABORATORIES AND EDITOR-IN-CHIEF OF PHILlPS RESEARCH REPORTS Christoffel Jacob Bouwkamp was bom at Hoogkerk, The Netherlands, on June 26, 1915. He studied at the University of Groningen, The Netherlands, where he graduated in theoretical physics, mechanics, and mathematics in 1938. After his military service he completed his doctoral thesis (under F. Zernike) on diffraction theory, and gave courses on mechanics and quantum mechanics at the University of Groningen. In 1941 he joined the staff of Philips Research Laboratories, N.V. Philips' Gloeilampenfabrieken, Eindhoven, The Netherlands. From J 946 he also was an Editor, and since 1952 the Editor-in-Chief of Philips Research Reports. In 1969 he was appointed Scientific Adviser for Applied Mathematics at Philips Research Laboratories. He has been Professor Extraordinary of Applied Mathematics since 1955, first at the University of Utrecht, The Netherlands, until 1958, and since then at the Technological University Eindhoven. On several occasions he was Visiting Lecturer, Research Associate, and Research Mathematician at various institutes in Europe and the United States. He is a Member ofthe "Koninklijke Nederlandse Akademie van Wetenschappen" (Royal Dutch Academy of Sciences). Any centre of human activity - such as, for example, a research laboratory _ is necessarily dependent on the form of its organization, on its budget, on the atmosphere, on its technical facilities, and - in short - on many matters over which the individual has little influence. When, therefore, an individual is lauded, praised or eulogized he is likely to reply in terms of: "I was only one of a team", or "I stood on the shoulders of others", or "I grasped the opportunity afforded me by others". Yet we all know that such remarks express only a part of the truth, for the individual is indeed unique and he moulds the course of events. No one is irreplaceable but no individual can take the place of another, ten good math- ematicians do not make one great one, and the new vintage, however good, will not have the taste of the old. With effect from July 1, 1975 Bouwkamp the mathematician will no longer be in the ranks of the Philips Research Laboratories. With effect from that date Philips Research Reports has to be edited and guided by another hand. With effect from that date Bouwkamp the man will be absent from our corridors. An individualleaves us; we lose a voice, a talent, an adviser, a hobbyist and a master of his craft. Mathematics in general, this laboratory and this journal are greatly in his debt. We all wish to express our thanks but in no standardized manner and in no blaze of publicity. We were of the opinion that a special issue of Philips Research Reports to which he devoted so much of his enthousiasm, would be a good synthesis between the honour to an individual and the manifestation of what may be achieved by joint activity. We hope that the scientific world will appreciate this collective publication inspired by and dedicated to a colleague and friend. Many have contributed to this issue; Bouwkamp was not a contributor but without him it could not have been written. The Directors, Philips Research Laboratories CONTENTS Page H. B. G. Casimir Laser modes; an imperfect tribute to C. J. 1· Bouwkamp O.Bottema A lesson in elementary geometry 5· C.H.Papas On the equation of motion in electrodynamics 14· M.Kac An example of "counting without counting" 20· A. Erdé/yi Fourier transforms of integrable generalized 23· functions Murray S. Klamkin Asymptotic heat conduction in arbitrary bodies 31· J. H. vall Lint and An asymmetrie contest for properties of ar- 40· H.O.Pollak bitrary value Irvin Kay Near and far field HF radar ground wave return 56· from the sea N. G. van Kampen The collapse of the wave function 65· B. R. A. Nijboer On a relation between the scattering cross- 74· section in dense media and the energy of a dilute electron gas Joseplt B. Keiler Effective conductivity, dielectric constant and 83'" permeability of a dilute suspension P. Delsarte, J. M. Goethals Bounds for systems of lines, and Jacobi poly- 91· and J. J. Seidel nomials V. Belevitch and Y. Genin Reciprocity invariants in equivalent networks 106· J. B. Alblas Relaxation phenomena in electro-magneto- 122· elasticity John W. Miles Asymptotic approximations for oblate spher- 140· oida! wave functions J. Boersma Analysis of Weinstein's diffraction function 161'" J. A. Geurst Continuum theory for type-A smectic liquid 171'" crystals Leopold B. Felsen Complex rays 187'" H. Freudentha/ On the cardinality offinite Tits geometries 196'" R. L. Brooks, C. A. B. Smith, Leaky electricity and triangulated triangles 205'" A. H. Stone and W. T. Tutte P. J. Federico The number of polyhedra 220'" Jose! Meixnèr and Some remarks on the treatment of the diffrac- 232'" Schiu Sche tion through a circular aperture Harold Levine Acoustical diffraction radiation 240'" K. M. Adams The non-amplification property of networks of 277. II-terminal resistive devices F. E. J. Kruseman Aretz alld FFT algorithms 288* J. A. Zonneveld A. T. de Hoop The N-port receiving antenna and its equivalent 302* electrical network H. J. Butterweck Noise voltages of bulk resistors due to random 316* fluctuations of conductivity A. J. Dekkers N-omino enumeration 322* A. J. W. Duijvestijn Fast calculation of inverse matrices occurring in 329* squared-rectangle calculation N. G. de Bruijn and A finite basis theorem for packing boxes with. 337* D. A. Klarner bricks F. L. H. M. Stumpers Some notes on the correspondence between Sir 344* Edward Appleton and Balth. van der Pol N. Marcuvitz Eigenmodes, quasimodes, and quasiparticles 357* Publications by C. J. Bouwkamp 376* , VOL. 30 ISSUE IN HO~OUR OF: C. J. BOUWKAMP 1975 Philips, ,R~sea~ch Rep()rts • EDiTED BY Till! RESEARCH LABORATORY OF N.V. PHILIPS' GLOEILAMPENFABRIEKEN, EINDHOVEN, NETHERLANDS R883 Philips Res. Repts 30, 1*-4*, 1975 LASER MODES; AN IMP;ERFECT TRIBUTE TO C. J. BOUWKAMP by H. B. G. CASIMIR " v-. ~ '. , ~.. ) I feel rather ashamed that in an issue of the Philips Research Reports dedicated to C. J. Bouwkamp I have nothing better to offer than some sketchy considerations. ',' The truth of the matter is that for several years already 1 have from time to time been thinking on a subject that, properly worked out, might have made a suitable contribution. I have even occasionally discussed the matter with Bouwkamp and he has taken me to task for not yet having filled in the analytical details. But somehow I have been too busy (or' too làzy?) to do .this with a precision and completeness that would meet Bouwkamp's stringent standards. Let me all the same state the problem. ,; In 1928 Gamow published his theory of IX-decay 1). The essential idea is that o-particles are escaping from a nucleus by tunnelling through apotential barrier. Gamow treated the problem by introducing the notion of complex eigenvalues. They correspond to wave functions that are outgoing waves .for large r, are exponential in the region where It'..:... V < 0, a~d are to a highdegree of approx- imation standing waves inside the barrier. To my knowledge such complex eigenvalues have not been used very exten- sively, and I am not aware - but this rrî~y be. my lack of familiarity with the relevant literature - of the existence of à body of general theory describing the relation of such eigenstates to .the complete set of orthogonal eigenfunctions. But it struck me that the concept, of complex éigenvalues might conveniently be used to calculate electromagnetic mode's in certain arrangements in gas lasers. Let us consider the following simple arrangement. The space between two perfectly conducting mirrors at z = ° and z = L is subdivided in three regions: It ~ . (I) x <-d;" (II) =d-«; x <d; - (Ill) x> d. .' ," ::'t In (I) and (Ill) the dielectric constant is 8 > 1; in (ll) it is unity. , We look for an' electrömagnetic Wave öf the following form: 2* H. B. G. CASIMIR in (I) and (Ill) E" = B sin (koz) exp (-i"lxl) exp (iwt), (1) in (Il) E" = A sin (koz) cos (k,.x) exp (iwt). (2) These expressions are solutions of Maxwell's equations, if (3) (4) The boundary condition at x = d is (5) The boundary condition at x = -d is then automatically fulfilled.
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