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2 3 O the Eletrodiamis of Spae-Time Periodi Mediums i a Waveguide of Arbitrary Cross Setio Eduard A. Gevorya Dotor of physial-mathematial siee professor Mosow State uiversity of EoomisStatistis ad Iformatis 7 Nehisaya St. Mosow 95 Russia. Itrodutio Theoretial ivestigatios of parametri iteratio betwee the eletrodyamis waves ad spae-time periodi fillig of the waveguide of arbitrary ross setio are reviewed. The ases of dieletri aisotropi ad magetodieletri periodially modulated fillig are osidered. The aalytial method of solutio of the problems of eletrodyamis of spaetime periodi mediums i a waveguide is give. The propagatio of trasverse-eletri (TE) ad trasverse-mageti (TM) waves i the waveguide metioed above are ivestigated. Physial pheomea of eletrodyamis of spae-time periodi mediums i the regio of wea ad strog iteratios betwee the travellig wave i the waveguide ad the modulatio wave are studied. Propagatio of eletromageti waves i the medium whose permittivity ad permeability are modulated i spae ad time with help of pump waves of various ature (eletromageti wave ultrasoi wave et.) uder the harmoi law represets oe of the basi problem of the eletromageti theory. I the sietifi literature the most part of suh researhes oers to eletrodyamis of periodially o-statioary ad o-uiform mediums i the ulimited spae [-5] while the same problems i the limited modulated mediums for example i the waveguides of arbitrary ross setio remai still isuffiietly studied ad there is o strit aalytial theory of the propagatio of eletromageti waves i similar systems (although i the sietifi literature already appeared the artiles o the problems metioed above [6-5]. Meawhile the ivestigatio of the propagatio of eletromageti waves i the waveguides with spae-time periodially modulated fillig represets a great iterest ot oly from poit of view of developmet of theory but also from poit of view of possibility of pratie appliatio of similar waveguides i the ultrahigh frequey eletrois. For example the waveguides with periodially o-statioary ad o-uiform fillig a be applied for desigig of multifrequey distributig ba-ouplig lasers (DBS lasers) Bragg refletio lasers (DBR lasers) mode trasformers parametri amplifiers multifrequey geerators trasformers of low ad higher frequey Bragg resoators ad filters prismati polarier diffratio latties osillators mode overters wave-haelig devies with a fie periodi struture et [4] [6-3].

3 68 Wave Propagatio. Eletromageti waves i a waveguide with spae-time periodi fillig Let us osider the regular ideal waveguide of arbitrary ross setio whih axis oiides with the OZ axis of ertai Cartesia frame. Let the permittivity ad permeability of the fillig of the waveguide with help of pump wave are modulated i spae ad time uder the periodi law (Fig..) [3 5] ( ) ( ) = + m os ut μ = μ + mμ os ut (.) m и m μ are the modulatio idexes u is the modulatio wave veloity is the modulatio wave umber u is the modulatio wave frequey и μ are the permittivity ad permeability of the fillig i the absee of modulatio. The sigal wave with frequey ω Fig... Geometry of ross setio of a waveguide with harmoially modulated fillig. propagates i a similar waveguide alog their axis i the positive diretio. Suppose that the sigal wave does t hage the quatities of ad μ. It is mea that we have the approximatio of small sigals. The field i similar waveguide represets the superpositio of trasverse-eletri (TE) ad trasverse-mageti (TM) waves whih i this osideratio are desribed with help of logitudial ompoets of mageti ( ) H ad eletri ( E ) vetors. These ompoets satisfy to partial differetial equatios with variable oeffiiets whih are obtaied from the Maxwell equatios taig ito aout that the harge desity ad the urret desity are equal to ero. These wave equatios have a form [3-5] [3] TE field ( μh) ( μh) Δ H + = (.) μ t t

4 O the Eletrodiamis of Spae-Time Periodi Mediums i a Waveguide of Arbitrary Cross Setio 69 TM field ( E ) ( E ) Δ E + μ = t t (.3) Δ = x + y is the two-dimesioal Laplae operator is the veloity of light i vauum. The solutio of wave equatios (.) ad (.3) we loo for the form of deompositio by orthoormal eigefutios of the seod ad first boudary-value problems for the ross setio of the waveguide ( ψ( xy ) ψ( xy )). These futios satisfy to the followig Helmholt equatios with orrespodig boudary oditios o the surfae of the waveguide [3] [3]: seod boudary-value problem first boudary-value problem Ψ Δ Ψ ( xy ) + λψ ( xy ) = = (.4) λ Σ ( ) Δ Ψ ( xy ) + Ψ ( xy ) = Ψ xy = (.5) λ ad λ are the eigevalues of the seod ad first boudary-value problems for the trasverse ross setio of the waveguide Σ is the otour of the waveguide s ross setio is the ormal to Σ. From Maxwell equatios the trasverse ompoets of trasverse-eletri (TE) ad trasverse-mageti (TM) fields a be represeted i terms of as follows [3]: TE field H ( x y t) = H ( t) Ψ ( x E ( x y t) = E ( t) Ψ ( x (.6) ( t ) ( t ) H ( t ) μ Hτ ( x y t) λ = Ψ( x μ Σ (.7) TE field ( t ) H ( t ) μ Eτ ( x y t) λ = Ψ ( x (.8) t ( t. ) E ( t) Hτ ( x y t) λ = Ψ ( x t (.9) ( t ) ( t ) E ( t ) Eτ ( x y t) = λ Ψ x y ( ) (.)

5 7 Wave Propagatio = i( / x) + j( / is the two-dimesioal abla operator. If ito the wave equatios (.) ad (.3) of variables ad t to itrodue the ew quatities by the formulas H = μ H E = E (.) ad to pass to the ew variables ξ ad η aordig to the formulas [] ξ dξ ξ = ut η = u u (.) ( ξ) μ( ξ) β μ β = u μ ad whe u the ξ η t ad the solutios of reeived partial differetial equatios to loo for the form [] i H e γη H x y = ( ξ ) Ψ( ) E ( ) ( ) = E ξ Ψ x y i e γη (.3) taig ito aout the orthoormaliatio of the eigefutios ψ ( x y ) ad ψ ( x y ) the we reeive for oeffiiets: H ad E the followig ordiary differetial equatios with variable d dh μ dξ μ μ ξ β μ μ χ β + H = d μ (.4) d de dξ μ ξ β μ χ β + E = d μ μ (.5) γ μ χ μ λ = β μ γ μ χ μ λ = β. (.6) μ I this ivestigatio we are limited of small quatities of modulatio idexes of the waveguide fillig. It is explaied that i real experimet the modulatio idexes are very 4 small ad they a hage from to 4 (the quatity 4 is fixed i the hrome gelati). Note that if the veloity of modulatio wave satisfies the oditio u 8 υph υph = μ is the phase veloity i the o-disturbae medium the side by side of modulatio idexes is small the parameter = ( m + mμ) β b ( b = β ) too that is l <<. The with help of haged of variables

6 O the Eletrodiamis of Spae-Time Periodi Mediums i a Waveguide of Arbitrary Cross Setio 7 s = ( ) β ξ μ d ξ μ μ β μ s = ( ) β ξ d ξ (.7) μ β μ ad taig ito aout that permittivity ad permeability of the fillig hage by the harmoi law (.) the above reeived differetial equatios (.4) ad (.5) o variables ξ ad η are trasformed to the differetial equatios with periodi oeffiiets of Mathie-Hill type [3]. I the first approximatio o small modulatio idexes they have a form [3] d H ( s) is + θ H = (.8) ds = d E θ ( s) ds = 4 = b e ( s) ( s) is + θ E = (.9) e ( χ ) 4 θ ( ) χ = (.) b γ 4 θ± = ( χ ) ( χ ) mμ b u (.) b γ 4 θ± = ( χ ) ( χ ) m b u (.) b ( ) γ = b χ μ λ ( ) The solutios of the equatios (.8) ad (.9) we loo for the form H s C iμ s is ( ) = ( ) e = e γ χ = μ λ b. (.3) e e iμ s is = E s = C. (.4) Substitutig these expressios ito Mathie-Hill equatios (.8) ad (.9) for determiatio of harateristi idexes μ ad μ we reeive the followig dispersio equatios: μ ( θ ) = θ + + ( θ ) ( μ ) θ ( μ ) θ (.5) μ ( θ ) ( θ ) ( μ ) θ ( μ ) θ = θ + +. (.6)

7 7 Wave Propagatio The aalysis of these dispersio equatios show that uder the followig oditios [33] θ δ θ δ (.7) we beome to the regio of wea iteratio betwee the sigal wave ad the modulatio wave the harateristi idexes μ ad μ are real ad the the Mathie-Hill equatios have the stable solutios. With help of obtaied solutios of dispersio equatios ad the expressios for the oeffiiets C ( μ ) = θ μ = θ ( ) θ C θ C ± = ± = C ( ± θ ) ( ± θ ) (.8) (.9) 4 4 C ad C are defied from the oditios of ormaliig we obtaied the aalytial expressios for the H ad E of TE ad TM fields i the waveguide i the regio of wea iteratio. They have a form [33] i P t i ut = Ψ( ) μ = ( ω ) ( ) H x y C V e e (.3) e i ( ) ( P ωt) i ( ut) (.3) E = Ψ x y C V = e V Δ C m = + C μ V Δ C m = + C (.3) μ ω μ Δ = mμ + ω Δ = m + (.33) u u 4 θ χ λ χ m (.34) ( ) ( ) ± = β b + b ( ) ( ) 4 θ χ λ χ m (.35) ( P ) ± = β b + b ( ω ) ( ω ) = μ λ ( P) = μ λ μ. (.37) As is see from the expressios (.3) ad (.3) TE ad TM fields i the waveguide with modulated fillig are represeted as the set of spae-time harmois with differet

8 O the Eletrodiamis of Spae-Time Periodi Mediums i a Waveguide of Arbitrary Cross Setio 73 amplitudes. At that time the amplitude of the ero (fudametal) harmoi are idepedet of small modulatio idexes while the amplitudes of the plus ad mius first harmois (side harmois) are proportioal to the small modulatio idexes i the first degree. At the realiatio the followig oditio [3] [33] θ δ (.38) η δ = 4 β β 4λ η = + b m β = (.39) b ( ω ) 4 u λ β ω C u = β u b b β θ = = (.4) (here are show the results for the TE field whe μ = ) the strog (resoae) iteratio betwee the sigal wave ad the modulatio wave taes plae whe ours the osiderable eergy exhage betwee them. The aalytial expressio for the frequey of strog iteratio is foud i the form Δ +Δ ω = ( η + β ) ω ω ω ω ω s s s u β (.4) ad is show that the width of strog iteratio is small ad proportioal to the modulatio idex i the first degree [3] [33] Δ ω = 8 β ( + ηβ )( η β ) u. (.4) η I the regio of strog iteratio the dispersio equatio (.5) has omplex solutios i the followig form ( ) θ δ μ = ± i θ = δ. (.43) (.43) allow to reeive the aalytial expressios for the amplitudes of differet harmois i the form [3] [33] V ( η + β )( η + 3β ) V. 6 β (.44) The aalysis of these expressios shows that i the ase of forward modulatio whe the diretios of propagatio of the sigal wave ad the modulatio wave oiide the

9 74 Wave Propagatio amplitude of mius first harmoi does t deped from the modulatio idex while the amplitude of the plus first harmoi is proportioal to the modulatio idex i the first degree. I other words i the regio of strog iteratio besides the fudametal harmoi the substatial role plays the mius first harmoi refleted from the periodi struture of the fillig o the frequey u ω = ω = η β ( ) s s u β η > β. (.45) I the baward modulatio ase whe the diretios of propagatio of the sigal wave ad the modulatio wave do t oiide the mius first ad plus first harmois hage their roles. The results reeived above admit the visual physial explaatio of the effet of strog iteratio betwee the sigal wave ad the modulatio wave. Below the physial explaatio we show by example of TE field i the ase of forward modulatio. The ero harmoi i the modulated fillig of the waveguide is iidet o the desity maxima of the fillig at the agle ϕ ad is refleted from them at the agle ϕ (Fig.). These agles are defied from the followig orrelatios [33] ω ω ( ) os ϕ = λ + β os ϕ β os ϕ =. (.46) + β β osϕ At that time the iidet ad refletio agles are differet beause of the movig of the modulatio wave of the fillig ad the frequeies of iidet ad refleted waves satisfy to the followig orrelatio [33] ω siϕ ω si ϕ. = (.47) Fig.. The physial explaatio of the effet of strog iteratio.

10 O the Eletrodiamis of Spae-Time Periodi Mediums i a Waveguide of Arbitrary Cross Setio 75 If ow we apply the first-order Wolf-Bragg oditio whe the waves refleted from highdesity poits of the iterferee patter are amplified we obtai the followig equatio ( ) ( ) β ω osϕ β =. (.48) It is ot diffiult to ote taig ito aout (.46) that the solutio of the equatio (.48) preisely oiides with the expressio of the frequey of strog iteratio (see (.4)) reeived above. 3. Propagatio of eletromageti waves i a waveguide with a periodially modulated aisotropi isert Cosider a waveguide of arbitrary ross setio with a aisotropi omageti ( μ = ) modulated isert (modulated uiaxial rystal) the permittivity tesor of whih has the form ( t ) = t ( ) ( t ) ompoets ( t ) ad ( t ) are modulated by the pumpig wave i spae ad time aordig to the harmoi law (.) ( t ) = [ + m os ( ut)] ( t ) = [+ m os ( ut)]. (.) Here ad are the permittivities i the absee of a modulatig wave; m ad m are the modulatio idies; ad ad и are respetively the waveumber ad veloity of the modulatig wave. Cosider the propagatio of a sigal eletromageti wave at frequey ω i this waveguide uder the assumptio that the modulatio idies are small ( m m m m ) << <<. Note that whe the oditio β.8 is satisfied β = u / ot oly the modulatio idies but also parameter l = mβ / ( β ) are small ( l <<. ) As i my earlier wors (see e.g. [3] [3] [34-37]) trasverse eletri (ТЕ) ad trasverse mageti (TM) waves i the waveguide will be desribed through the logitudial ompoets of the mageti ( H ) ad eletri ( E ) field. The bearig i mid that Dx = ( t) Ex Dy = ( t) Ey D = ( t) E ad В = H ad usig the Maxwell equatios we obtai equatios for H (x y f) ad E(x y ; amely for the TE wave H H Δ H + = t t (.3)

11 76 Wave Propagatio for the TM wave E E Δ E + = (.4) t E = E. Δ is the two-dimesioal Laplaia ad It is easy to he i this ase that the trasverse ompoets of the ТЕ ad TM fields a be expressed i terms of (.6) as: for TE wave ( t) H Hτ = λ Ψ ( x (.5) for TM wave ( t) H Eτ = λ Ψ( x t (.6) ( t) E Hτ λ = Ψ ( x t (.7) E Let us itrodue the ew variables τ ( t) E = λ Ψ ξ ( x. (.8) dξ ξ = ut η = u u β ξ / (.9) ( ) ito equatios (.3) ad (.4) ad see for solutios to the above equatios i the form ( ω) ipu η = Ψ H e H x y ( ω) ipu η = Ψ ( ξ ) ( ) ( ξ ) ( ) Taig ito aout that futios ψ ( x y ) ad ( x y ) (.) E e E x y. (.) ψ satisfy the Helmholt equatios (.4) ad (.5) we get ordiary seod-order differetial equatios i variable ξ to fid E ξ H ( ξ ) ad ( ) d dh χ β H dξ + = (.) dξ β

12 O the Eletrodiamis of Spae-Time Periodi Mediums i a Waveguide of Arbitrary Cross Setio 77 d de χ + E ξ =. (.3) dξ β β d Here ( up ) ω ( ) χ λ β up ω = χ = λ β (.4) ( p ) ω λ = I terms of the ew variables ( p ) β ω = λ b β b β = u β =. (.5) ξ b s = dξ b dξ s = β (.6) β ξ Equatios (.) ad (.3) tae the form of the Mathieu-Hill equatios dh + H = 4χ ds b de ds ( ) 4χ b + E =. (.7) Note that the frequey domai desribed by the oditios θ θ >> δ θ θ δ 4 ωβ 4 ωβ θ = p b u θ = p b u >> (.8) (.9) ( up ) ω b θ θ θ λ = = + b u 4 θ θ = θ = m m (.) b is the domai of wea iteratio betwee the sigal wave ad the wave that modulates the isert. Solvig (.7) by the method developed i [3] [3] [34-37] ad disardig the terms proportioal to the modulatio idies i the first power we obtai the followig expressios for the ТЕ ad TM field i the frequey domai defied by formulas (.8): [38]

13 78 Wave Propagatio for the TE wave H = Ψ x y e V e ip ( ) ( ) ω t i ( ) ut = (.) V ω = + u for the TM wave θ ± = 4± ( θ ) (.) ip ( ) ( ) ω t i ( ) ut E = Ψ x y e V e = (.3) ω V = + ( m + m) u θ ± = 4± ( θ ). (.4) Note that quatities ad i (.) ad (.3) are foud from the ormaliatio oditio. As follows from (.) ad (.3) whe a eletromageti wave propagates i a waveguide with a isert harmoially modulated i spae ad time the ТЕ ad TM fields represet a superpositio of spae-time harmois of differet amplitudes. I the domai of wea iteratio betwee the sigal ad modulatio waves the amplitudes of harmois + ad - prove to be small (they are liearly related to the modulatio idies) ompared with the amplitude of the fudametal harmoi (whih is idepedet of modulatio idies). It is ow [] that whe θ ad θ ted to uity i.e. whe the oditios θ δ θ δ (.5) are satisfied the sigal wave ad the wave that modulates the isert strogly iterat (the first-order Bragg oditio for waves refleted from a high-desity area is met) ad vigorously exhage eergy. Coditio (.5) a be reast (for the TM field) as ωs give by ω Δω ω ω +Δ ω (.6) s s u 4λ s ( ) β b ω = β + η η = + λ = λ (.7) is the frequey ear whih the strog iteratio taes plae ad

14 O the Eletrodiamis of Spae-Time Periodi Mediums i a Waveguide of Arbitrary Cross Setio 79 ( + βη ) u Δ ω = (.8) θ 8 βη is the width of the domai of strog iteratio. Calulatios show that ~ V V ~ m m (.9) i frequey domai (.6). From relatioships (.9) it follows that the amplitude of refleted harmoi - is idepedet of modulatio idies i the domai the sigal wave ad the wave that modulates the aisotropi isert strogly iterat. I other words ot oly the eroth harmoi of the sigal but also refleted harmoi - of frequey u ω = ( η β )( η > β ) (.3) s β plays a sigifiat role i this domai. Note i olusio that the results obtaied here tur ito those reported i [37] i the limit m ; i the limit u oe arrives at results for a waveguide with a ihomogeeous but statioary aisotropi isert. 4. Iteratio of eletromageti waves with spae-time periodi aisotropi mageto-dieletri fillig of a waveguide Let the axis of a regular waveguide of a arbitrary ross setio oiides with the OZ axis of a Certai Cartesia oordiate frame. Assume that the waveguide is filled with a periodially modulated aisotropi mageto- dieletri fillig whose tesor permittivity ad permeability are speified by the formulas I (3.) ost μ ost futios i spae ad time: = ( t) μ μ = μ ( t) μ = = ad the ( t) ad ( t) ( t ) = + m os( ut). (3.) μ ompoets are harmoi (3.) ( t ) = + m os( ut) μ μ μ (3.3) m << ad m μ << are small modulatio idexes = ost ad μ = ost are respetively the permittivity ad permeability of the fillig i the absee of a modulatio wave. Let a sigal wave uit amplitude with frequey ω propagates i suh a waveguide i a positive Diretio of a axis OZ. After some algebra the wave equatios for the logitudial ompoets H ( x y t) ad E ( x y t) of TE ad TM fields a be obtaied from Maxwell equatios

15 8 Wave Propagatio D= E D B urlh = urle = t t = μ μh B with allowae for the equalities 9 ( π ) divd = divb = (3.4) = /4 9 F/ m μ = 4π H / m (3.5) 7 D x E = x Dy = E D = ( t) E (3.6) y B μμh x = x By μμ We arrive at the followig equatios: for TE waves for the TM waves ( ) Δ + = = H B = μμ ( t) H. (3.7) y t H H H μ μμ ( t) (3.8) μ t ( ) Δ + = t E E E μμ ( t) (3.9) t = μ ( t) H E = ( ) H t E. (3.) With the use of Maxwell equatios (3.4) ad (3.5) the trasverse ompoets of ТЕ ad TM fields a be represeted i terms of as follows: for TE waves for TM waves H = H t Ψ x y ( ) ( ) E = E ( t) Ψ( x ( t ) H ( t ) μ Hτ λ = Ψ μ ( t ) H ( t ) λ t (3.) ( x (3.) μ Eτ μ = Ψ ( x (3.3) ( t ) E ( t ) Hτ = Ψ ( x (3.4) λ t

16 O the Eletrodiamis of Spae-Time Periodi Mediums i a Waveguide of Arbitrary Cross Setio 8 With the ew variables E τ ( t ) E ( t ) = λ Ψ ( x. (3.5) ut ξ = ut η = u u β (3.6) β = u μμ wave equatios (3.8) ad (3.9) a be modified to obtai ( t ) ( β ) ( t) H μ Δ + μ μ = (3.7) H H μ ξ β η ( t ) ( β ) ( t) Δ E + =. (3.8) E μμ E ξ β η Let us see solutios of equatios (3.7) ad (3.8) i the form (.3). The taig ito H ξ E ξ the followig seod-order aout (.4) ad (.5) we obtai for ( ) ad ( ) ordiary differetial equatios with the periodi Mathieu-Hill oeffiiets: dh ( ξ) ( t )( β ) ( t ) μ β dξ μ μ μ + γ λ H ( ξ) = (3.9) de ( ξ) ( t )( β ) ( t ) μ μ + γ λ E ( ξ) =. (3.) ξ β d With the ew variable ζ = ξ / equatios (3.9) ad (3.) a be modified ito the form d H ( ζ ) dζ = + θ exp ζ ζ = (3.) ( i ) H ( ) d E ( ζ ) dζ = ( i ) E ( ) + θ exp ζ ζ = (3.) quatities θ ad θ are the oeffiiets of the Fourier deompositios of the expressios that appear before futios H ( ζ ) ad E ( ) ζ eterig equatios (3.9) ad (3.). I the first approximatio for small parameters m ad m μ these oeffiiets are expressed aordig to the formulas 4μ θ = μ μ γ λ ( b ) b μ μλ θ± = bμ mμ (3.3)

17 8 ( b ) 4 θ = μ μ γ λ λ θ± = m b b We see solutios to equatios (3.) ad (3.) i the form Wave Propagatio b = β. (3.4) H = e Ce iμζ iζ ( ζ ) ( ζ ) = iμζ iζ = E = e C e. (3.5) It is ow [33] that uder the oditios (.7) whih provide for wea iteratio betwee the sigal wave ad the wave of the waveguide-fillig modulatio quatities μ μ C ± ad C ± have the form (.8) ad (.9) (aurate to withi small parameters m ad m μ ilusivel. Taig ito aout (3.5) (.8) (.9) ad hagig to variables ad t we obtai from (3.) aalyti expressios for H ad E of ТЕ ad TM waves. These expressios orrespod to the first approximatio for m ad m μ are valid i the regio of wea iteratio betwee the sigal wave ad the wave of the waveguide-fillig modulatio ad have the form [39] ip t i ut = Ψ( ) μ = ( ω ) ( ) H x y e C V e (3.5) ip ( ) ( ωt) i ( ut) (3.6) E = Ψ x y e C V e = C m μ C m 4 β μ β ω V = V θ ω λ C = = C b u μ u θ ω λ 4 β β ω = b u u β μ P = ω λ β P ω λ u μ u (3.7) =. (3.8) Note that for the frequey ad frequey width of the strog iteratio regio (see [3] [33]) the followig expressios a easily be obtaied from (.5): for TE waves u ω с = ( β + η ) 4μλ η = + β μ b for TM waves u ( + βη ) Δ ω = с δ (3.9) 4βη u ω с = ( β + η ) 4λ η = + β b ( + βη ) u Δ ω с = δ. (3.3) 4βη

18 O the Eletrodiamis of Spae-Time Periodi Mediums i a Waveguide of Arbitrary Cross Setio 83 For the quatities V ± ad V ± i this ase we obtai V V μλ bμ 4bμ m μ (3.3) V V λ b 4b m. (3.3) Aordig to (3.3) ad (3.3) i the strog- iteratio regio a substatial role is played ot oly by the fudametal harmoi but also by the refleted mius-first harmoi that exists at the frequey: for TE waves for TM waves u ω = η β β ( ) u ω = η β β ( ) η > β (3.33) > β. (3.34) Note that i limitig ase u the above obtaied relatioships yield results for the statioary ihomogeeous aisotropi mageto-dieletri fillig of a waveguide. 5. Refrees [] Brilloui L. Parodi M. Rasprostraeie Vol v Periodihesih Struturah. Perevod s Fratsusovo M.: IL 959. [] Bor M. Volf E. Osovi Optii. Perevod s Agliysovo M.: Naua 973. [3] Cassedy E. S. Olier A. A. TIIER 5 No 33 (963). [4] Barsuov K. A. Bolotovsiy B. M. Ivestiya Vuov. Seriya Radiofiia 7 No 9 (964). [5] Barsuov K. A. Bolotovsiy B. M. Ivestiya Vuov. Seriya Radiofiia 8 No 4 76 (965). [6] Cassedy E. S. TIIER 55 No 7 37 (967). [7] Peg S. T. Cassedy E. S. Proeedigs of the Symposium o Moder Optis. Brooly N. Y.: Politei Press MRI-7 99 (967). [8] Barsuov K. A. Gevorya E. A. Zvoiov N. A. Radiotehia i Eletroia No 5 98 (975). [9] Tamir T. Wag H. C. Olier A. A. IEEE Trasatios o Mirowave Theory ad Tehiques MTT- 34 (964). [] Averov S. I. Boldi V. P. Ivestiya Vuov. Seriya Radiofiia 3 No 9 6 (98). [] Ase J. TIIER 59 No 9 44 (968). [] Rao. TIIER 65 No 9 44 (968). [3] Yeh C. Cassey K. F. Kaprielia Z. A. IEEE Trasatios o Mirowave Theory ad Tehiques MTT-3 97 (965). η

19 84 Wave Propagatio [4] Elahi Ch. TIIER 64 No (976). [5] Karpov S. Yu. Stolyarov S. N. Uspehi Fiihesih Nau 63 No 63 (993). [6] Elahi Ch. Yeh C. Joural of Applied Physis (973). [7] Elahi Ch. Yeh C. Joural of Applied Physis (974). [8] Peg S. T. Tamir T. Bertoi H. L. IEEE Trasatios o Mirowave Theory ad Tehiques MTT-3 3 (975). [9] Seshadri S. R. Applied Physis 5 (98). [] Krehtuov V. M. Tyuli V. A. Radiotehia i Eletroia 8 9 (983). [] Simo J. C. IRE Trasatios o Mirowave Theory ad Tehiques MTT-8 No 8 (96). [] Barsuov K. A. Radiotehia i Eletroia 9 No 7 73 (964). [3] Gevorya E. A. Proeedigs of Iteratioal Symposium o Eletromageti Theory Thessaloii Greee May (998). [4] Gevorya E. A. Mehduvedomstveiy Tematihesiy Nauhiy Sbori. Rasseyaie Eletromagitih Vol. Tagaro TRTU No 55 (). [5] Gevorya E. A. Boo of Abstrats of the Fifth Iteratioal Cogress o Mathematial Modellig Duba Russia September 3 Otober 6 99 (). [6] Gaydu V. I. Palatov K. I. Petrov D. M. Fiihesie Osovi Eletroii SVCH Mosow Sovetsoe Radio (97). [7] Yariv A. Kvatovaya Eletroia I Nelieyaya Optia. Perevod s Agliysovo M.: Sovetsoe Radio (973). [8] Volovodaya Optoeletroia. Pod Redatsiey T. Tamir. Perevod s Agliysovo M.: Mir (974). [9] Marue D. Optihesie Volovodi. Perevod s Agliysovo M.: Mir (974). [3] Yariv A. Yuh P. Optihesie Voli v Kristallah. Perevod s Agliysovo M.: Mir (987). [3] Barsuov K. A. Gevorya E. A. Radiotehia i Eletroia 8 No 37 (983). [3] Ma-Lahla N. V. Teoriya i Priloheiya Futsiy Mathe. Perevod s Agliysovo M.: Fimatgi (963). [33] Gevorya E. A. Uspehi Sovremeoy Radioeletroii No 3 (6). [34] Barsuov K. A. Gevorya E. A. Radiotehia i Eletroia (986). [35] Barsuov K. A. Gevorya E. A. Radiotehia i Eletroia 39 7 (994). [36] Gevorya E. A. Proeedigs of Iteratioal Symposium o Eletromageti Theory Kiev Uraie September (). [37] Gevorya E. A. Proeedigs of Iteratioal Symposium o Eletromageti Theory Depropetrovs Uraie September (4). [38] Gevorya E.A. Zhural Tehihesoy Fiii 76 No 5 34 (6) (Tehial Physis (6)). [39] Gevorya E.A. Radiotehia i Eletroia 53 No (8) (Joural of Commuiatios Tehology ad Eletrois 53 No (8)).

20 Wave Propagatio Edited by Dr. Adrey Petri ISBN Hard over 57 pages Publisher ITeh Published olie 6 Marh Published i prit editio Marh The boo ollets origial ad iovative researh studies of the experieed ad atively worig sietists i the field of wave propagatio whih produed ew methods i this area of researh ad obtaied ew ad importat results. Every hapter of this boo is the result of the authors ahieved i the partiular field of researh. The themes of the studies vary from ivestigatio o moder appliatios suh as metamaterials photoi rystals ad aofousig of light to the traditioal egieerig appliatios of eletrodyamis suh as ateas waveguides ad radar ivestigatios. How to referee I order to orretly referee this sholarly wor feel free to opy ad paste the followig: Eduard A. Gevorya (). O the Eletrodiamis of Spae-Time Periodi Mediums i a Waveguide of Arbitrary Cross Setio Wave Propagatio Dr. Adrey Petri (Ed.) ISBN: ITeh Available from: ITeh Europe Uiversity Campus STeP Ri Slava Krautea 83/A 5 Rijea Croatia Phoe: +385 (5) Fax: +385 (5) ITeh Chia Uit 45 Offie Blo Hotel Equatorial Shaghai No.65 Ya A Road (West) Shaghai 4 Chia Phoe: Fax:

21 The Author(s). Liesee ItehOpe. This hapter is distributed uder the terms of the Creative Commos Attributio-NoCommerial- ShareAlie-3. Liese whih permits use distributio ad reprodutio for o-ommerial purposes provided the origial is properly ited ad derivative wors buildig o this otet are distributed uder the same liese.