## The Plane Wave Spectrum Representation of Electromagnetic Fields |

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Page 14

With A = (Ax, Ay, Az) the remaining condition (2.15) now reads Ax

sinh /3 = 0 , (2.20) which implies that A = (za sinh ft a

and fe are arbitrary complex parameters. Substitution from (2.19) and (2.21) into (

2.13), (2.14) shows that the field is formed by linear combination of the two plane

waves E,H = {(0,0, 1), y0(-*sinhft -

H, E = {(0, 0, 1), Z0 (/ sinh ft

former ...

With A = (Ax, Ay, Az) the remaining condition (2.15) now reads Ax

**cosh**j3 - iAysinh /3 = 0 , (2.20) which implies that A = (za sinh ft a

**cosh**ft 6) , (2.21) where aand fe are arbitrary complex parameters. Substitution from (2.19) and (2.21) into (

2.13), (2.14) shows that the field is formed by linear combination of the two plane

waves E,H = {(0,0, 1), y0(-*sinhft -

**cosh**/3,0)} X g-fc,>'slnh0e-ii«Jccosh/> (222) andH, E = {(0, 0, 1), Z0 (/ sinh ft

**cosh**ft 0)} x e-fc,>>slnh/3 g-jfcoxcosh/3 /2 23) Theformer ...

Page 141

flog/). = ,:,/». , ..,Lu.. = ».

objective can be achieved by expressing sin ft in the numerator as the sum of

functions even in n — ft and ft, respectively, through the formula n sin ft = (n -ft) sin

ft + ft sin ft. (6.77) For in this way (6.76) can be written -ft Cog/) = g(n - j8, y) + g(ft,

y) , (6.78) where sinhy i , „ . _ ~ ffP, y)~ cos y cos - 2

ysinhy ft sin ft -ysinhy\

flog/). = ,:,/». , ..,Lu.. = ».

**cosh**. y. dy ' sin ft + i sinh y * cosh2 y — cos2 ft ' (6.76) theobjective can be achieved by expressing sin ft in the numerator as the sum of

functions even in n — ft and ft, respectively, through the formula n sin ft = (n -ft) sin

ft + ft sin ft. (6.77) For in this way (6.76) can be written -ft Cog/) = g(n - j8, y) + g(ft,

y) , (6.78) where sinhy i , „ . _ ~ ffP, y)~ cos y cos - 2

**cosh**y + cos ft "5F ft sin ft +ysinhy ft sin ft -ysinhy\

**cosh**y — cos ft '**cosh**y + cos ft ...Page 143

r-r^- f ' r \zsin ft + smhy / Iv \ i 0 r r sinh r I — COS /S / TZ To- dr\ • (6'86) 71 J

COSh2 T - COS2 ft 0 J Such expressions for L2(cos ft) are quite complicated, but

it may be noted that, when cos ft is real, |L2(cos ft)\ is a simple function. If cos ft is

real and greater than — 1, /L2(cos ft, y)/2 =

follows by inspection of (6.85) for — 1 < cos ft < I (0 < ft < n) or of (6.86) for cos ft >

coshy(/S = -ift',ft' > y);and there can be no discontinuity at cos ft = 1 (ft = 0) or at

cos ft =

r-r^- f ' r \zsin ft + smhy / Iv \ i 0 r r sinh r I — COS /S / TZ To- dr\ • (6'86) 71 J

COSh2 T - COS2 ft 0 J Such expressions for L2(cos ft) are quite complicated, but

it may be noted that, when cos ft is real, |L2(cos ft)\ is a simple function. If cos ft is

real and greater than — 1, /L2(cos ft, y)/2 =

**cosh**y + cos ft . (6.87) This resultfollows by inspection of (6.85) for — 1 < cos ft < I (0 < ft < n) or of (6.86) for cos ft >

coshy(/S = -ift',ft' > y);and there can be no discontinuity at cos ft = 1 (ft = 0) or at

cos ft =

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### Contents

PRELIMINARIES | 3 |

PLANE WAVE REPRESENTATION | 11 |

SUPPLEMENTARY THEORY | 39 |

Copyright | |

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### Other editions - View all

The Plane Wave Spectrum Representation of Electromagnetic Fields: (Reissue ... P. C. Clemmow No preview available - 1996 |

### Common terms and phrases

angle angular frequency angular spectrum approximation asymptotic expansion Babinet's principle Bessel function boundary conditions branch-points Cerenkov radiation complex considered corresponding cos2 cosh current distribution dielectric tensor diffraction dipole direction of phase evaluated exponential expressed in terms factor finite formula Fourier Fresnel integral given gives half-plane half-space Hankel function homogeneous plane wave incident field integrand line-source magnetostatic field Maxwell's equations obtained P(cos path of integration perfectly conducting permittivity phase propagation plane wave spectrum point charge point of observation polarized field pole power radiated problem pure imaginary radiation field real axis reflection coefficient refractive index replaced respectively result scattered field screen simple sin2 sinh solution specified spectrum function steepest descents superposition surface current density surface wave theory time-averaged power flux time-harmonic tion total field transmitted two-dimensional uniaxial medium unity upper/lower sign vacuum field wave spectrum representation wavelength written z-axis zero