INFRARED WAVE PROPAGATION IN A HELICAL WAVEGUIDE WITH INHOMOGENEOUS CROSS SECTION AND APPLICATION

Transcription

1 Progress In Electromgnetics Reserch, PIER 61, , 26 INFRARED WAVE PROPAGATION IN A HELICAL WAVEGUIDE WITH INHOMOGENEOUS CROSS SECTION AND APPLICATION Z. Menchem nd M. Mond Deprtment of Mechnicl Engineering Ben-Gurion University of the Negev Beer-Shev 8415, Isrel Abstrct This pper presents n improved pproch for the propgtion of electromgnetic (EM) fields long helicl dielectric wveguide with circulr cross section. The min objective is to develop mode model for infrred (IR) wve propgtion long helicl wveguide, in order to provide numericl tool for the clcultion of the output fields, output power density nd output power trnsmission for n rbitrry step s ngle of the helix. Another objective is to pply the inhomogeneous cross section for hollow wveguide. The derivtion is bsed on Mxwell s equtions. The longitudinl components of the fields re developed into the Fourier- Bessel series. The trnsverse components of the fields re expressed s functions of the longitudinl components in the Lplce plne nd re obtined by using the inverse Lplce trnsform by the residue method. The seprtion of vribles is obtined by using the orthogonlreltions. This model enbles us to understnd more precisely the influence of the step s ngle nd the rdius of the cylinder of the helix on the output results. The output power trnsmission nd output power density re improved by incresing the step s ngle or the rdius of the cylinder of the helix, especilly in the cses of spce curved wveguides. This mode model cn be useful tool to improve the output results in ll the cses of the hollow helicl wveguides (e.g., in medicl nd industril regimes).

2 16 Menchem nd Mond 1. INTRODUCTION Vrious methods for the nlysis of cylindricl hollow metllic or metllic with inner dielectric coting wveguide hve been studied in the literture [1 17]. A review of the hollow wveguide technology [1] nd review of IR trnsmitting, hollow wveguides, fibers nd integrted optics [2] were published. The first theoreticl nlysis of the problem of hollow cylindricl bent wveguides ws published by Mrctili nd Schmeltzer [3]. Their theory considers the bending s smll disturbnce nd uses cylindricl coordintes to solve Mxwell equtions. They derive the mode equtions of the disturbed wveguide using the rtio of the inner rdius r to the curvture rdius R s smll prmeter (r/r 1). Their theory predicts tht the bending hs little influence on the ttenution of hollow metllic wveguide. However, prcticl experiments hve shown lrge increse in the ttenution, even for rther lrge R. Mrhic [4] proposed mode-coupling nlysis of the bending losses of circulr metllic wveguide in the IR rnge for lrge bending rdii. His study of nonstedy-stte propgtion improves the understnding of experimentl procedures for the mesurement of bending losses nd provides prcticl limits to the rte of chnge of curvture consistent with single-mode propgtion. In the circulr guide it is found tht the preferred TE 1 mode cn couple very effectively to the lossier TM 11 mode when the guide undergoes circulr bend. The mode-coupling nlysis [4] developed to study bending losses in microwve guides hs been pplied to IR metllic wveguides t λ = 1.6 µm. For circulr wveguides, the microwve pproximtion hs been used for the index of refrction nd the stright guide losses, nd the results indicte very poor bending properties due to the ner degenercy of the TE 1 nd TM 11 modes, thereby offering n explntion for the high losses observed in prctice. Miygi et l. [5] suggested n improved solution, which provided greement with the experimentl results, but only for r/r 1. A different pproch [4, 6] trets the bending s perturbtion tht couples the modes of stright wveguide. Tht theory explins qulittively the lrge difference between the metllic nd metllicdielectric bent wveguide ttenution. The reson for this difference is tht in metllic wveguides the coupling between the TE nd TM modes cused by the bending mixes modes with very low ttenution nd modes with very high ttenution, wheres in metllic-dielectric wveguides, both the TE nd TM modes hve low ttenution. The EH nd HE modes hve similr properties nd cn be relted to modes tht hve lrge TM component.

3 Progress In Electromgnetics Reserch, PIER 61, Hollow wveguides with both metllic nd dielectric internl lyers were proposed to reduce the trnsmission losses. Hollow-core wveguides hve two possibilities. The inner core mterils hve refrctive indices greter thn one (nmely, leky wveguides) or the inner wll mteril hs refrctive index of less thn one. A hollow wveguide cn be mde, in principle, from ny flexible or rigid tube (plstic, glss, metl, etc.) if its inner hollow surfce (the core) is covered by metllic lyer nd dielectric overlyer. This lyer structure enbles us to trnsmit both the TE nd TM polriztion with low ttenution [4, 6]. A method for the EM nlysis of bent wveguides [7] is bsed on the expnsion of the bend mode in modes of the stright wveguides, including the modes under the cutoff. A different pproch to clculte the bending losses in curved dielectric wveguides [8] is bsed on the well-known conforml trnsformtion of the index profile nd on vectoril eigenmode expnsion combined with perfectly mtched lyer boundry conditions to ccurtely model rdition losses. Light propgtion through inhomogeneous medi whose refrctive index hs wek chnges with cylindricl symmetry round the propgtion xis ws published in [9], nd two pproximtions bsed on geometric optics were exmined. The cylindricl TEM nd TEM 1 lser modes were considered, nd explicit nlyticl expressions for the light rys nd intensity profiles for short nd long times were derived. An improved ry model for simulting the trnsmission of lser rdition through metllic or metllic dielectric multibent hollow cylindricl wveguide ws proposed in [1, 11]. It ws shown theoreticlly nd proved experimentlly tht the trnsmission of CO 2 rdition is possible even through bent wveguide. The propgtion of EM wves in loss-free inhomogeneous hollow conducting wveguide with circulr cross section nd uniform plne curvture of the longitudinl xis ws considered in [12]. For smll curvture the field equtions cn, however, be solved by mens of n nlyticl pproximtion method. In this pproximtion the curvture of the xis of the wveguide ws considered s disturbnce of the stright circulr cylinder, nd the perturbed torus field ws expnded in eigenfunctions of the unperturbed problem. Using the Ryleigh-Schrodinger perturbtion theory, eigenvlues nd eigenfunctions contining first-order correction terms were derived. An extensive survey of the relted literture cn be found especilly in the book on EM wves nd curved structures [13]. The rdition from curved open structures is minly considered by using perturbtion pproch, tht is by treting the curvture s smll perturbtion of the stright configurtion. The perturbtive pproch is not entirely

4 162 Menchem nd Mond suitble for the nlysis of reltively shrp bends, such s those required in integrted optics nd especilly short millimeter wves. An nlyticl method to study generl helix-loded structure hs been published in [14]. The inhomogeneously-loded helix enclosed in cylindricl wveguide operting in the fst-wve regime. The tpe-helix model hs been used which tkes into ccount the effect of the spce-hrmonics, nd is used prticulrly in the cses tht the structure is operted t high voltges nd for high helix pitch ngles. The propgtion chrcteristics of n ellipticl step-index fiber with conducting helicl winding on the core-cldding boundry [15] re investigted nlyticlly where the coordinte systems re chosen for the circulr nd ellipticl fibers. In their wveguides the core nd the cldding regions re ssumed to hve constnt rel refrctive indices n 1 nd n 2, where n 1 >n 2. The fibers re referred to s the ellipticl heliclly cldded fiber nd the circulr heliclly cldded fiber. The models bsed on the perturbtion theory consider the bending s perturbtion (r/r 1), nd solve problems only for lrge rdius of curvture. An improved pproch hs been derived for the propgtion of EM field long toroidl dielectric wveguide with circulr cross-section [16]. The derivtion is bsed on Mxwell s equtions for the computtion of the EM field nd the rdition power density t ech point during propgtion long toroidl wveguide, with rdil dielectric profile. Tht method [16] employs toroidl coordintes (nd not cylindricl coordintes, such s in the methods tht considered the bending s perturbtion (r/r 1)). In ddition, terms up to fourth order in 1/R were considered in which the further orders re equl to zero. The min objective of this pper is to generlize the numericl mode model [16] from toroidl dielectric wveguide (pproximtely plne curve) with circulr cross-section to helicl wveguide ( spce curved wveguide for n rbitrry vlue of the step s ngle of the helix) with circulr cross-section. Another objective is to demonstrte the bility of the model to solve prcticl problems with inhomogeneous cross-section in the cse of hollow wveguide. The generlized mode model with the two bove objectives provides us numericl tool for the clcultion of the output fields nd output power trnsmission for n rbitrry step s ngle of the helix (δ p ). The results of this model re pplied to the study of hollow wveguides with spce curved shpes tht re suitble for trnsmitting IR rdition, especilly CO 2 lser rdition. In this pper we supposed tht the modes excited t the input of the wveguide by the conventionl CO 2 lser IR rdition (λ =1.6 µm) re closer to the TEM polriztion of the lser rdition. The TEM mode is the fundmentl nd the

5 Progress In Electromgnetics Reserch, PIER 61, Figure 1. The circulr helicl wveguide. most importnt mode. This mens tht cross-section of the bem hs Gussin intensity distribution. The output power trnsmission is improved by incresing the step s ngle or the rdius of the cylinder of the helix, especilly in the cses of spce curved wveguides. 2. FORMULATION OF THE PROBLEM The method presented in [16] is generlized to provide numericl tool for the clcultion of the output trnsverse fields nd power density in helicl wveguide (see Fig. 1), for n rbitrry vlue of the step s ngle of the helix (δ p ). Fig. 1 shows the geometry of the helicl wveguide with circulr cross section. The direction of the IR wve propgtion is long the xis of the helicl wveguide. The xis of the helicl wveguide is shown in Fig. 2. The deployment of the helix nd the step s ngle δ p re shown in Fig. 3. We strt by finding the metric coefficients from the helicl trnsformtion of the coordintes. The ltters will be used in the wve equtions s will be outlined in the next section. The helicl trnsformtion of the coordintes is chieved by two

6 164 Menchem nd Mond rottions nd one trnsltion, nd is given in the form: X cos(φ c) sin(φ c ) 1 Y = sin(φ c ) cos(φ c ) cos(δ p ) sin(δ p ) Z 1 sin(δ p ) cos(δ p ) r sin θ R cos(φ c) + R sin(φ c ), (1, b, c) r cos θ ζ sin(δ p ) where ζ is the coordinte long the helix xis, R is the rdius of the cylinder, δ p is the step s ngle of the helix (see Figs. 2 3), nd φ c = (ζ cos(δ p ))/R. Likewise, r + δ m, where 2 is the internl dimeter of the cross-section of the helicl wveguide, δ m is the thickness of the metllic lyer, nd d is the thickness of the dielectric lyer (see Fig. 4). Z Z A K ζ Y X X Figure 2. Rottions nd trnsltion of the orthogonl system (X,ζ,Z) from point A to the orthogonl system (X, Y, Z) tpoint K. Figure 2 shows the rottions nd trnsltion of the orthogonl system (X, ζ, Z) from point A to the orthogonl system (X, Y, Z) t point K. In the first rottion, the ζ nd Z xes rotte round the X xis of the orthogonl system (X, ζ, Z) t the point A until the Z xis becomes prllel to the Z xis (Z Z), nd the ζ xis becomes prllel

7 Progress In Electromgnetics Reserch, PIER 61, _ Z δ p Z _ ζ ζ A δ p 2 ζ δ p Rφ c 2 π R Figure 3. Deployment of the helix. substrte metl lyer dielectric lyer ir d δm Figure 4. A cross-section of the wveguide (r, θ). to the X, Y plne (ζ (X, Y )) of the orthogonl system (X, Y, Z) t the point K. In the second rottion, the X nd ζ xes rotte round the Z xis (Z Z) of the orthogonl system (X, ζ, Z) until X X nd ζ Y. After the two bove rottions, we hve one trnsltion from the orthogonl system (X,ζ,Z) t point A to the orthogonl system (X, Y, Z) t the point K. Figure 3 shows the deployment of the helix depicted in Fig. 2. The condition for the step s ngle δ p is given ccording to tn(δ p ) 2( + δ m) 2πR, (2)

8 166 Menchem nd Mond where the internl dimeter is denoted s 2, the thickness of the metllic lyer is denoted s δ m, nd the rdius of the cylinder is denoted s R. According to Eqs. (1) (1c), the helicl trnsformtion of the coordintes becomes X =(R + r sin θ) cos(φ c )+rsin(δ p ) cos θ sin(φ c ), Y =(R + r sin θ) sin(φ c ) r sin(δ p ) cos θ cos(φ c ), Z = r cos θ cos(δ p )+ζ sin(δ p ), (3) (3b) (3c) where φ c =(ζ/r) cos(δ p ),Ris the rdius of the cylinder, nd (r, θ) re the prmeters of the cross-section. Note tht ζ sin(δ p )=Rφ c tn(δ p ). The metric coefficients in the cse of the helicl wveguide ccording to Eqs. (3) (3c) re: h r =1, h θ = r, (4) (4b) h ζ = (1 + r R sin θ)2 cos 2 (δ p ) + sin 2 (δ p )(1 + r2 R 2 cos2 θcos 2 (δ p )) = 1+ 2r R sin θcos2 (δ p )+ r2 R 2 sin2 θcos 2 (δ p )+ r2 R 2 cos2 θcos 2 (δ p )sin 2 (δ p ) 1+ r R sin θcos2 (δ p ). (4c) Furthermore, the third nd the fourth terms in the root of the metric coefficient h ζ re negligible in comprison to the first nd the second terms when (r/r) 2 1. Nonetheless, the metric coefficient h ζ still depends on δ p, the step s ngle of the helix (Fig. 3). Note tht the metric coefficient h ζ is function of r nd θ, which cuses difficulty in the seprtion of vribles. Thus, the nlyticl methods re not suitble for the helicl or the curved wveguide. In this method, the seprtion of vribles is performed by employing the orthogonlreltions. The cross-section of the helicl wveguide in the region r + δ m is shown in Fig. 4, where δ m is the thickness of the metllic lyer, nd d is the thickness of the dielectric lyer. For smll vlues of the step s ngle δ p (sin(δ p ) tn(δ p ) δ p, cos(δ p ) 1), condition (2) becomes δ p 2(+δ m )/(2πR). For smll vlues of the step s ngle, the helicl wveguide becomes toroidl wveguide, where the rdius of the curvture of the helix cn then be pproximtely by the rdius of the cylinder (R). In this cse, the toroidl system (r, θ, ζ) in conjunction with the curved wveguide is

9 Progress In Electromgnetics Reserch, PIER 61, Figure 5. A generl scheme of the toroidl system (r, θ, ζ) nd the curved wveguide. shown in Fig. 5, nd the trnsformtion of the coordintes (3) (3c) is given s specil cse of the toroidl trnsformtion of the coordintes, s follows ( ) ( ) ζ ζ X =(R + r sin θ) cos, Y =(R + r sin θ) sin, Z = r cos θ, R R (5, b, c) nd the metric coefficients re given by h r =1, h θ = r, h ζ =1+ r sin θ. (6, b, c) R By using the Serret-Frenet reltions for sptil curve, we cn find the curvture (κ) nd the torsion (τ) for ech sptil curve tht is chrcterized by θ = const nd r = const for ech pir (r, θ) in the rnge. This is chieved by using the helicl trnsformtion introduced in equtions (3) (3c). The curvture nd the torsion (see Appendix A) re constnts for constnt vlues of the rdius of the cylinder (R), the step s ngle (δ p ) nd the prmeters (r, θ) of the cross-section. The curvture nd the torsion re given by κ = 1+C t R(1 + tn 2 (δ p )+C t ), τ = tn(δ p ) R(1 + tn 2, (7, b) (δ p )+C t ) where C t = r2 R 2 sin2 θ +2 r r2 sin θ + R R 2 sin2 (δ p )cos 2 θ. The rdius of curvture nd the rdius of torsion re given by ρ =1/κ, nd σ =1/τ, respectively. For smll vlues of the step s ngle (δ p 1), the helicl wveguide becomes toroidl wveguide (Fig. 5), where

10 168 Menchem nd Mond the rdius of the curvture of the helix cn then be pproximtely by the rdius of the cylinder (ρ R). The generliztion of the method from toroidl dielectric wveguide [16] (pproximtely plne curve) to helicl wveguide ( spce curved wveguide for n rbitrry vlue of the step s ngle of the helix) is presented in the following derivtion. The derivtion is bsed on Mxwell s equtions for the computtion of the EM field nd the rdition power density t ech point during propgtion long helicl wveguide, with rdil dielectric profile. The longitudinl components of the fields re developed into the Fourier-Bessel series. The trnsverse components of the fields re expressed s function of the longitudinl components in the Lplce trnsform domin. Finlly, the trnsverse components of the fields re obtined by using the inverse Lplce trnsform by the residue method, for n rbitrry vlue of the step s ngle of the helix (δ p ). 3. SOLUTION OF THE WAVE EQUATIONS The wve equtions for the electric nd mgnetic field components in the inhomogeneous dielectric medium ɛ(r) re derived in this section for lossy dielectric medi in metllic boundries of the wveguide. The cross-section of the helicl wveguide is shown in Fig. 4 for the ppliction of the hollow wveguide, in the region r + δ m, where δ m is the thickness of the metllic lyer, nd d is the thickness of the dielectric lyer. The derivtion is given for the lossless cse to simplify the mthemticl expressions. In liner lossy medium, the solution is obtined by replcing the permitivity ɛ by ɛ c = ɛ j(σ/ω) in the solutions for the lossless cse, where ɛ c is the complex dielectric constnt, nd σ is the conductivity of the medium. The boundry conditions for lossy medium re given fter the derivtion. For most mterils, the permebility µ is equl to tht of free spce (µ = µ ). The wve equtions for the electric nd mgnetic field components in the inhomogeneous dielectric medium ɛ(r) re given by ( 2 E + ω 2 µɛe + E ɛ ) =, (8) ɛ nd 2 H + ω 2 µɛh + ɛ ( H) =, ɛ (8b) respectively. The trnsverse dielectric profile (ɛ(r)) is defined s ɛ (1 + g(r)), where ɛ represents the vcuum dielectric constnt, nd

11 Progress In Electromgnetics Reserch, PIER 61, g(r) is its profile function in the wveguide. The normlized trnsverse derivtive of the dielectric profile (g r ) is defined s (1/ɛ(r))( ɛ(r)/ r). From the trnsformtion of Eqs. (3) (3c) we cn derive the Lplcin of the vector E (i.e., 2 E), nd obtin the wve equtions for the electric nd mgnetic fields in the inhomogeneous dielectric medium. It is necessry to find the vlues of E, ( E), E, nd ( E) in order to obtin the vlue of 2 E, where 2 E = ( E) ( E). All these vlues re dependent on the metric coefficients (4) (4c). The ζ component of 2 E is given by ( 2 E) ζ = 2 E ζ + 2 Rh 2 ζ [ sin θ ζ E r + cos θ ζ E θ ] 1 R 2 h 2 E ζ, (9) ζ where 2 E ζ = 2 r 2 E ζ + 1 r h ζ [ sin θ R θ 2 E ζ + 1 r r E ζ + cos θ rr r E ζ θ E ζ h ζ ζ 2 E ζ ], (1) nd in the cse of h ζ =1+(r/R) sin θcos 2 (δ p ). The longitudinl components of the wve equtions (8) nd (8b) re obtined by deriving the following terms [ (E ɛ ɛ ) ] ζ = 1 h ζ ζ [E r g r ], (11) nd [ ] [ ] ɛ ɛ ɛ ( H) = jωɛ ɛ E = jωɛg r E θ. (12) ζ ζ The longitudinl components of the wve equtions (8) nd (8b) re then written in the form ( ) 2 E + k 2 E ζ + 1 ) (E r g r =, (13) h ζ ζ ζ ( ) 2 H ζ + k 2 H ζ + jωɛg r E θ =, (14) where ( 2 E) ζ, for instnce, is given in eq. (9). The locl wve number prmeter is k = ω µɛ(r) =k 1+g(r), where the free-spce wve number is k = ω µ ɛ.

12 17 Menchem nd Mond The trnsverse Lplcin opertor is defined s The Lplce trnsform h 2 ζ ã(s) =L{(ζ)} = ζ= 2 ζ 2. (15) (ζ)e sζ dζ (16) is pplied on the ζ-dimension, where (ζ) represents ny ζ-dependent vribles, where ζ =(Rφ c )/ cos(δ p ). The next steps re given in detil in Ref. [16], s prt of our derivtion. Let us repet these steps, in brief. 1). By substituting Eq. (9) into Eq. (13) nd by using the Lplce trnsform (16), the longitudinl components of the wve equtions (Eqs. (13) (14)) re described in the Lplce trnsform domin, s coupled wve equtions. 2). The trnsverse fields re obtined directly from Mxwell s equtions, nd by using the Lplce trnsform (16), nd re given by { 1 Ẽ r (s) = s 2 + k 2 h 2 jωµ [ ] r ζ r R cos θcos2 (δ p ) H ζ + h ζ θ H ζ h ζ [ } sin θ +s ]+se r jωµ H θ h ζ, (17) Ẽ θ (s) = 1 s 2 + k 2 h 2 ζ R cos2 (δ p )Ẽζ + h ζ { s r [ rẽζ r R cos θcos2 (δ p )Ẽζ + h ζ +jωµ h ζ [ sin θ R cos2 (δ p ) H ζ +h ζ r H ζ ] ]+se θẽζ } θ +jωµ H r h ζ (17b) { [ ] 1 jωɛ r H r (s) = s 2 + k 2 h 2 ζ r R cos θcos2 (δ p )Ẽζ + h ζ h ζ [ ] θẽζ } sin θ +s R cos2 (δ p ) H ζ + h ζ r H ζ + sh r + jωɛe θ h ζ, (17c) { [ ] 1 s r H θ (s) = s 2 + k 2 h 2 ζ r R cos θcos2 (δ p ) H ζ + h ζ θ H ζ [ } sin θ jωɛh ζ R cos2 (δ p )Ẽζ +h ζ ]+sh rẽζ θ jωɛe r h ζ. (17d),

13 Progress In Electromgnetics Reserch, PIER 61, Note tht the trnsverse fields re dependent only on the longitudinl components of the fields nd s function of the step s ngle (δ p ) of the helix. 3). The trnsverse fields re substituted into the coupled wve equtions. 4). The longitudinl components of the fields re developed into Fourier-Bessel series, in order to stisfy the metllic boundry conditions of the circulr cross-section. The condition is tht we hve only idel boundry conditions for r =. Thus, the electric nd mgnetic fields will be zero in the metl. 5). Two sets of equtions re obtined by substitution the longitudinl components of the fields into the wve equtions. The first set of the equtions is multiplied by cos(nθ)j n (P nm r/), nd fter tht by sin(nθ)j n (P nm r/), for n. Similrly, the second set of the equtions is multiplied by cos(nθ)j n (P nmr/), nd fter tht by sin(nθ)j n (P nmr/), for n. 6). In order to find n lgebric system of four equtions with four unknowns, it is necessry to integrte over the re (r, θ), where r =[,], nd θ =[, 2π], by using the orthogonl-reltions of the trigonometric functions. 7). The propgtion constnts β nm nd β nm of the TM nd TE modes of the hollow wveguide [17] re given, respectively, by β nm = ko 2 (P nm /) 2 nd β nm = ko 2 (P nm/) 2, where the trnsverse Lplcin opertor ( 2 ) is given by (P nm/) 2 nd (P nm/) 2 for the TM nd TE modes of the hollow wveguide, respectively. The seprtion of vribles is obtined by using the preceding orthogonl-reltions. Thus the lgebric equtions (n ) re given by α (1) n A n + β (1) n D n = 1 (BC1) π n, α (2) n B n + β (2) n C n = 1 (BC2) π n, β (3) n B n + α (3) n C n = 1 (BC3) π n, β (4) n A n + α (4) n D n = 1 (BC4) π n. (18) (18b) (18c) (18d) Further we ssume n = n = 1. The elements (α (1) n, β (1) n, etc.), on the left side of (18) for n = 1 re given for n rbitrry vlue of the

14 172 Menchem nd Mond step s ngle (δ p )by: )[( (s 2 + β 21m α 1 (1)mm β 1 (1)mm = π s 2 + k 2 ) + k 2 1 +π 1 ( 1 R 4 k 2 s 2 4 cos4 (δ p ) cos4 (δ p ) 3 { ( +πk 2 s R 2 ( + 3 2R 2 β2 1m cos4 (δ p ) + 1 ( 4R 4 cos4 (δ p ) + 1 ( 8R 4 cos8 (δ p ) [ +πs ) ] ) + 1 ) )} 2R 2 cos2 (δ p )G (1)mm + 1 ( 4R 2 cos 4 (δ p )β 2 1m G(1)mm 2 + cos 2 (δ p ) 9 ( + 1 P 1m 2R 2 cos2 (δ p ) )] 2 cos2 (δ p ) 11 [ ( ) 3 +πk 4 cos 4 (δ p ) 2R ( )] 8R 4 cos8 (δ p ) , (19) { ( 1 = jωµ πs cos2 (δ p )+ 3 ) 1 4 cos4 (δ p ) R 2 G(1)mm 14 ( ) cos2 (δ p ) R 2 G(1)mm R 2 G(1)mm cos 2 (δ p ) 1 } P 1m R 2 16, (19b) where the elements of the mtrices (, etc.) re given in Appendix B. Similrly, the rest of the elements on the left side in ) )

15 Progress In Electromgnetics Reserch, PIER 61, Eqs. (18) (18d) re obtined. We estblish n lgebric system of four equtions with four unknowns. All the elements of the mtrices in the Lplce trnsform domin re dependent on the step s ngle of the helix (δ p ), the Bessel functions; the dielectric profile g(r); the trnsverse derivtive g r (r); nd (r, θ). The elements of the boundry conditions (e.g., (BC2)1 )tζ = + on the right side in (18b) re dependent on the step s ngle δ p s follows: where (BC2) = 2π (BC2) 1 = (BC2) sin θj 1 (P 1m r/)rdrdθ, [( )(se ζ + E ζ )] s 2 + k 2 h 2 ζ + jωµ H θ sg r h 2 ζ ) (jωµ H θ s + k 2 E r h ζ + 2 R h ζ sin θ + 2 R h ζ cos θ ( jωµ H r s + k 2 E θ h ζ ) + k 2 h 3 ζe r g r, nd for h ζ =1+ r R sin θcos2 (δ p ). The boundry conditions t ζ = + for TEM mode in excittion become to: { } (BC2) 1 = 2π Q(r)(k(r)+js)J 1m (P 1m r/)rdr δ 1n + 4jsπ { R 2 cos2 (δ p ) + 9π 2R 2 cos4 (δ p ) + 3jsπ 2R 2 cos4 (δ p ) + 8π R 2 cos2 (δ p ) } Q(r)k(r)J 1m (P 1m r/)r 2 dr δ 1n { } Q(r)k 2 (r)j 1m (P 1m r/)r 3 dr δ 1n { } Q(r)k(r)J 1m (P 1m r/)r 3 dr δ 1n { } Q(r)k 2 (r)j 1m (P 1m r/)r 2 dr δ 1n (2) where : Q(r) = E n c (r)+1 g r exp ( (r/w o ) 2 ).

16 174 Menchem nd Mond Similrly, the remining elements of the boundry conditions t ζ = + re obtined. The mtrix system of Eqs. (18) (18d) is solved to obtin the coefficients (A 1, B 1, etc). x -z 2w Phse fronts z= { +z Propgtion lines Figure 6. Propgting Gussin bem. According to the Gussin bems [18] the prmeter w is the minimum spot-size t the plne z = (see Fig. 6), nd the electric field t the plne z = is given by E = E exp[ (r/w o ) 2 ]. The modes excited t ζ = in the wveguide by the conventionl CO 2 lser IR rdition (λ =1.6 µm) re closer to the TEM polriztion of the lser rdition. The TEM mode is the fundmentl nd most importnt mode. This mens tht cross-section of the bem hs Gussin intensity distribution. The reltion between the electric nd mgnetic fields [18] is given by E/H = µ /ɛ η, where η is the intrinsic wve impednce. Suppose tht the electric field is prllel to the y- xis. Thus the components of E y nd H x re written by the fields E y = E exp[ (r/w o ) 2 ] nd H x = (E /η ) exp[ (r/w o ) 2 ]. After Gussin bem psses through lens nd before it enters to the wveguide, the wist cross-sectionl dimeter (2w ) cn then be pproximtely clculted for prllel incident bem by mens of w = λ/(πθ) (fλ)/(πw). This pproximtion is justified if the prmeter w is much lrger thn the wvelength λ. The prmeter of the wist cross-sectionl dimeter (2w ) is tken into ccount in our method, insted of the focl length of the lens (f). The initil fields t ζ = + re formulted by using the Fresnel coefficients of the trnsmitted fields [19] s follows E r + (r) =T E (r)(e e (r/wo)2 sin θ), E θ + (r) =T E (r)(e e (r/wo)2 cos θ), H r + (r) = T H (r)((e /η )e (r/wo)2 cos θ), H θ + (r) =T H (r)((e /η )e (r/wo)2 sin θ), (21) (21b) (21c) (21d) where E + ζ = H + ζ =,T E (r) =2/[(n(r)+1], T H (r) =2n(r)/[(n(r)+1],

17 Progress In Electromgnetics Reserch, PIER 61, nd n(r) =[ɛ r (r)] 1/2. The index of refrction is denoted by n(r). The trnsverse components of the fields re finlly expressed in form of trnsfer mtrix functions for n rbitrry vlue of δ p s follows: E r (r, θ, ζ) =E r + (r)e jkhζζ jωµ R h ζ cos 2 θcos 2 (δ p ) C m S1 (ζ)j 1 (ψ) m jωµ R h ζ sin θ cos θcos 2 (δ p ) D m S1 (ζ)j 1 (ψ) m + jωµ h 2 ζ sin θ CS1 m (ζ)j 1 (ψ) r m jωµ h 2 ζ cos θ DS1 m (ζ)j 1 (ψ) r m + 1 R sin θ cos θcos2 (δ p ) m A m S2(ζ)J 1 (ξ) + 1 R sin2 θcos 2 (δ p ) m B m S2 (ζ)j 1 (ξ) +h ζ cos θ m A m S2(ζ) dj 1 dr (ξ)+h ζ sin θ m B m S2 (ζ) dj 1 dr (ξ), (22) where h ζ =1+(r/R) sin θcos 2 (δ p ),Ris the rdius of the cylinder, δ p is the step s ngle, ψ =[P 1m (r/)] nd ξ =[P 1m (r/)]. The coefficients re given in the bove eqution, for instnce { } A m S1(ζ) =L 1 A1m (s) s 2 + k 2 (r)h 2, (23) ζ { } A m S2(ζ) =L 1 sa1m (s) s 2 + k 2 (r)h 2, (23b) ζ where m =1,...N, 3 N 5. (23c) Similrly, the other trnsverse components of the output fields re obtined (see Appendix C). The first fifty roots (zeros) of the equtions J 1 (x) = nd dj 1 (x)/dx = my be found in tbles [2, 21]. The inverse Lplce trnsform is performed in this study by direct numericl integrtion in the Lplce trnsform domin by the

18 176 Menchem nd Mond residue method, s follows f(ζ) =L 1 [ f(s)] = 1 σ+j f(s)e sζ ds = Res[e sζ f(s); Sn ]. 2πj σ j n (24) By using the inverse Lplce trnsform (24) we cn compute the output trnsverse components in the rel plne nd the output power density t ech point t ζ = (Rφ c )/ cos(δ p ). The integrtion pth in the right side of the Lplce trnsform domin includes ll the singulrities ccording to Eq. (24). All the points S n re the poles of f(s) nd Res[e sζ f(s); Sn ] represent the residue of the function in specific pole. According to the residue method, two dominnt poles for the helicl wveguide re given by ( s = ±j k(r)h ζ = ±j k(r) 1+ r ) R sin θcos2 (δ p ). Finlly, knowing ll the trnsverse components, the ζ component of the verge-power density Poynting vector is given by S v = 1 { } 2 Re E r H θ E θ H r, (25) where the sterisk indictes the complex conjugte. The totl verge-power trnsmitted long the guide in the ζ direction cn now be obtined by the integrl of Eq. (25) over the wveguide cross section. Thus, the output power trnsmission is given by T = 1 2π { } Re E r H θ E θ H r rdrdθ. (26) 2 Lossy medium cse In liner lossy medium, the solution is obtined by replcing the permitivity ɛ by ɛ c = ɛ j(σ/ω) in the preceding mthemticl expressions, where ɛ c is the complex dielectric constnt nd σ is the conductivity of the medium. The coefficients re obtined directly from the lgebric equtions (18) (18d) nd re expressed s functions in the Lplce trnsform domin. To stisfy the metllic boundry conditions of circulr cross-section we find the new roots P (new) 1m nd P (new) 1m of the equtions J 1 (z) = nd dj 1 (z)/dz =, respectively, where z is complex. Thus, from the requirement tht the coefficients vnish, the new roots P (new) 1m nd P (new) 1m re clculted by developing

19 Progress In Electromgnetics Reserch, PIER 61, into the Tylor series, in the first order t 1/σ. The new roots in the cse of lossy medium re complex. The complex Bessel functions re computed by using NAG subroutine [22]. The explntion is given in detil in Ref. [16]. It should be noticed tht theoreticl method [23] hs been developed lso for curved wveguide with rectngulr cross section. Severl exmples computed on Unix system re presented in the next sections, in order to demonstrte the results of this proposed method for helicl wveguide ( spce curved wveguide for n rbitrry vlue of the step s ngle of the helix) in the cse of hollow wveguide. We suppose tht the trnsmitted fields of the initil fields (TEM mode in excittion) re formulted by using the Fresnel coefficients (21) (21d). 4. NUMERICAL RESULTS The next exmples represent the cse of the hollow wveguide with metllic lyer (Ag) coted by thin dielectric lyer (AgI). For silver hving conductivity of (ohm m) 1 nd the skin depth t 1.6 µm is m. Three test-cses re demonstrted for smll vlues of the step s ngle (δ p ). In these cses, the condition (2) becomes δ p 2( + δ m )/(2πR). Note tht for smll vlues of the step s ngle, the helicl wveguide becomes pproximtely toroidl wveguide (see Fig. 5), where the rdius of the curvture of the helix (ρ) cn then be pproximtely by the rdius of the cylinder (R). The first test-cse is demonstrted for the stright wveguide (R ). The results of the output trnsverse components of the fields nd the output power density ( S v ) (e.g., Fig. 7()) show the sme behvior of the solutions s shown in the results of Ref. [16] for the TEM mode in excittion. The result of the output power density (Fig. 7()) is compred lso to the result of published experimentl dt [24] (see lso in Fig. 7(b)). This comprison shows good greement ( Gussin shpe) s expected, except for the secondry smll propgtion mode. In this exmple, the length of the stright wveguide is 1 m, the dimeter (2) of the wveguide is 2 mm, the thickness of the dielectric lyer [d (AgI) ]is.75 µm, nd the minimum spot-size (w ) is.3 mm. The refrctive indices of the ir, dielectric lyer (AgI) nd metllic lyer (Ag) re n () =1,n (AgI) =2.2, nd n (Ag) =13.5 j75.3, respectively. The vlue of the refrctive index of the mteril t wvelength of λ =1.6 µm is tken from the tble compiled by Miygi et l. in Ref. [5]. The second test-cse is demonstrted in Fig. 8() for the toroidl dielectric wveguide. Fig. 8(b) shows the experimentl result tht

20 178 Menchem nd Mond S v [W/m 2 ] x [mm] y [mm] 1. () (b) Figure 7. The output power density ( =1mm, d (AgI) =.75 µm, λ = 1.6 µm, w =.3 mm, n () = 1, n (AgI) = 2.2, n (Ag) = 13.5 j75.34, nd the length of the stright wveguide is 1 m), for R : () theoreticl result; (b) experimentl result. ws received in the lbortory of Croitoru t Tel-Aviv University. This experimentl result ws obtined from the mesurements of the trnsmitted CO 2 lser rdition (λ =1.6 µm) propgtion through hollow tube covered on the bore wll with silver nd silver-iodide lyers (Fig. 4), where the initil dimeter (ID) is 1 mm (nmely, smll bore size). The output modl profile is gretly ffected by the bending, nd the theoreticl nd experimentl results (Figs. 8() 8(b)) show tht in ddition to the min propgtion mode, severl other secondry modes nd symmetric output shpe pper. The mplitude of the output

21 Progress In Electromgnetics Reserch, PIER 61, S v [W/m 2 ] x [mm] y [mm].6 () (b) Figure 8. Solution of the output power density ( =.5 mm, d (AgI) =.75 µm, λ = 1.6 µm, w =.2 mm, n () = 1, n (AgI) = 2.2, n (Ag) =13.5 j75.3, R =.7m, φ = π/2, nd ζ = 1 m): () theoreticl result; (b) experimentl result. power density ( S v ) is smll for bending rdii (R), nd the shpe is fr from Gussin shpe. This result grees with the experimentl results, but not for ll propgtion modes. The experimentl result (Fig. 8(b)) is influenced by the bending nd dditionl prmeters (e.g., the roughnes of the internl wll of the wveguide) which re not tken into ccount theoreticlly. In this exmple, =.5 mm, R =.7m, φ = π/2, nd ζ = 1 m. The thickness of the dielectric lyer [d (AgI) ]is.75 µm (Fig. 4), nd the minimum spot size (w ) is.2 mm. The vlues of the refrctive indices of the ir, dielectric lyer (AgI) nd

22 18 Menchem nd Mond T (norm. units) Mode-model Experimentl dt /R (1/m) Figure 9. The theoreticl mode-model s result nd the experimentl result [11] where the hollow metllic wveguide (Ag) is covered inside the wlls with AgI film. The output power trnsmission s function of 1/R for δ p =, where ζ =.55 m, where =1.2mm, d (AgI) =.75 µm, w =.1mm, λ =1.6 µm, n () =1, n (AgI) =2.2, nd n (Ag) =1. metllic lyer (Ag) re n () =1,n (AgI) =2.2, nd n (Ag) =13.5 j75.3, respectively. In both theoreticl nd experimentl results (Figs. 8() 8(b)) the shpes of the output power density for the curved wveguide re not symmetric. The third test-cse is demonstrted in Fig. 9 for the toroidl dielectric wveguide. The theoreticl mode-model s result nd the experimentl result [11] re demonstrted in normlized units where the length of the curved wveguide (ζ) is.55 m, the dimeter (2) of the wveguide is 2.4 mm, nd the minimum spot size (w ) is.1 mm. This comprison (Fig. 9) between the theoreticl mode-model nd the experimentl dt [11] indictes good greement. For ll the exmples, our theoreticl mode-model tkes into ccount only the dielectric losses nd the bending losses, in conjunction with the problem of the propgtion through curved wveguide. The experimentl result [11] tkes into ccount dditionl prmeters (e.g., the roughnes of the internl wll of the wveguide) which re not tken into ccount theoreticlly. In spite of the differences, the comprison shows good greement. For smll vlues of the bending (1/R) inthe cse of smll step s ngle, the output power trnsmission is lrge nd decreses with incresing the bending. Figure 1demonstrtes the influence of the step s ngle (δ p ) nd the rdius of the cylinder (R) on the rdius of curvture of the helix (ρ) long the ζ-xis of the helix. Three results re demonstrted for three vlues of δ p (δ p =,.4,.8). For n rbitrry vlue of rdius of cylinder (R), the rdius of curvture of the helix (ρ) is lrge for lrge vlues of the step s ngle nd decreses with decresing the vlue

23 Progress In Electromgnetics Reserch, PIER 61, ρ (m) δ p =. δ p =.4 δ p = /R (1/m) Figure 1. The rdius of curvture of the helix (ρ) long the ζ-xis of the helix s function of 1/R, where R is the rdius of the cylinder. Three results re demonstrted for three vlues of δ p (δ p =,.4,.8). 1 T (norm. units) δ p =. δ p =.4 δ p =.7.2 δ p =.8 δ p =.9 δ p = /R (1/m) Figure 11. The results of the output power trnsmission of the helicl wveguide s function of 1/R, where R is the rdius of the cylinder. Six results re demonstrted for six vlues of δ p (δ p =,.4,.7,.8,.9, 1), where ζ = 4m, = 1 mm, w =.6 mm, n d =2.2, nd n (Ag) =13.5 j75.3. of δ p. On the other hnd, for n rbitrry vlue of the step s ngle, the rdius of curvture of the helix (ρ) is lrge for lrge vlues of the rdius of the cylinder (R), nd decreses with decresing the vlue of the rdius of the cylinder. The min contribution of this pper is demonstrted in Fig. 11, in order to understnd the influence of the step s ngle (δ p ) nd the rdius of the cylinder (R) on the output power trnsmission, defined in Eq. (26). Six results re demonstrted for six vlues of δ p (δ p =,.4,.7,.8,.9, 1.), where ζ =4m, = 1 mm, w =.6 mm,

24 182 Menchem nd Mond S v [W/m 2 ] S v [W/m 2 ] x [mm] y [mm] x [mm] y [mm] 1 () (b) S v [W/m 2 ] S v [W/m 2 ] x [mm] y [mm] x [mm] y [mm] 1 (c) (d) Figure 12. The results of the output power density s functions of the step s ngle (δ p ) nd the rdius of the cylinder (R), where ζ =1m, = 1 mm, w =.3 mm, n d =2.2, nd n (Ag) =13.5 j75.3: (). δ p =.4, nd R =.7 m; (b). δ p =.8, nd R =.7 m; (c). δ p =.4, nd R = 1 m; (d). δ p =.8, nd R =1m. n d =2.2 nd n (Ag) =13.5 j75.3. For n rbitrry vlue of R, the output power trnsmission is lrge for lrge vlues of δ p nd decreses with decresing the vlue of δ p. On the other hnd, for n rbitrry vlue of δ p, the output power trnsmission is lrge for lrge vlues of R nd decreses with decresing the vlue of R. Note tht for smll vlues of the step s ngle, the rdius of curvture of the helix (ρ) cn be pproximted by the rdius of the cylinder (R). In this cse, the output power trnsmission is lrge for smll vlues of the bending (1/R), nd decreses with incresing the bending. Thus, this model cn be useful tool to find the prmeters (δ p nd R) which will give us the improved results (output power trnsmission) of hollow wveguide in the cses of spce curved wveguides. Figures 12() (d) show the results of the output power density s functions of the step s ngle (e.g., δ p =.4,.8) nd the rdius of the cylinder (e.g., R =.7 m, 1 m). For these results ζ = 1 m, where = 1 mm, w =.3mm, n d =2.2, nd n (Ag) =13.5 j75.3. For δ p =.4, the mplitude of the output power density is smll

25 Progress In Electromgnetics Reserch, PIER 61, S v [W/m 2 ] (d) (c) (b) () y [mm] Figure 13. The output mplitude, the width of the Gussin shpe, nd the shift of the centrl pek in the sme cross section of Figs. 12() (d) where x = nd y =[.5mm, +.5 mm]. Four exmples re demonstrted for: () δ p =.4, nd R =.7m; (b) δ p =.8, nd R =.7m; (c) δ p =.4, nd R = 1 m; (d) δ p =.8, nd R = 1 m. The output modl profile is gretly ffected for smll vlues of δ p nd R, s shown in cse (). (e.g., ( S v =.5W/m 2 ) s the rdius of the cylinder is smll (e.g., R =.7m), nd the output shpe (Fig. 12()) is fr from Gussin shpe (Fig. 12(d)). By incresing only the step s ngle from δ p =.4 to δ p =.8 where R =.7 m, the mplitude of the output power density is greter nd lso the output shpe is chnged (Fig. 12(b)). By incresing only the rdius of the cylinder from.7 m to 1 m where δ p =.4, the result of the output power density ( S v ) shows Gussin shpe, nd the mplitude of the output power density is chnged from.5 W/m 2 (Fig. 12()) to.8 W/m 2 (Fig. 12(c)). Now, by incresing the step s ngle from δ p =.4 toδ p =.8 nd lso by incresing the rdius of the cylinder from.7 m to 1 m, the result becomes Gussin shpe (Fig. 12(d)), where the mplitude is chnged from.5 W/m 2 (Fig. 12()) to.9 W/m 2. Fig. 12() shows tht in ddition to the min propgtion mode, severl other secondry modes pper, where δ p =.4 nd R =.7m. Figures 13() (d) show the output mplitude, the width of the Gussin shpe, nd the shift of the centrl pek in the sme cross section of Figs. 12() (d), where x = nd y =[.5mm, +.5 mm]. These results represent the output power density s functions of the step s ngle (δ p ) nd the rdius of the cylinder (R), for the sme prmeters of Figs. 12() (d). By incresing only the step s ngle from δ p =.4 to δ p =.8 where R =.7m, the mplitude nd the

26 184 Menchem nd Mond width of the Gussin shpe re greter (Fig. 13(b)). In ddition, the output shpe is improved from symmetric shpe (Fig. 13()) to the symmetric shpe (Fig. 13(b)). The pek of the output shpe (Fig. 13(b)) is closer to y =, s regrding to the result s shown in Fig. 13(). By incresing only the rdius of the cylinder of the helix from.7 m to 1 m where δ p =.4, the symmetric shpe ppers (Fig. 13(c)). In this cse the mplitude nd the width of the Gussin shpe re greter thn the mplitude nd the width of the Gussin shpe s shown in Fig. 13(). Now, by incresing the step s ngle from δ p =.4 toδ p =.8 nd lso by incresing the rdius of the cylinder of the helix from.7 m to 1 m, the mplitude nd the width of the Gussin shpe (Fig. 13(d)) re greter thn the mplitude nd the width of the Gussin shpe in the previous cses. The symmetric shpe (Gussin shpe) is shown in Fig. 13(d) where δ p =.8nd R =1m. From the bove results we cn see tht the output power trnsmission, the mplitude of the output power density ( S v ), nd the output Gussin shpe re improved by incresing the step s ngle or the rdius of the cylinder of the helix, in the cses of spce curved wveguides. The output modl profile is gretly ffected by smll prmeters of R nd δ p, (e.g., R =.7m, nd δ p =.4). For smll vlues of the step s ngle, the helicl wveguide becomes toroidl wveguide, where the rdius of the curvture of the helix (ρ) cn then be pproximtely by the rdius of the cylinder (R). For smll vlues of R (e.g., R =.7m), the output shpes of the fields nd the output power density ( S v ) re fr from Gussin shpe, s shown in Fig. 12() nd Fig. 13(), for instnce. The mplitude of the output power density is smll (e.g., ( S v ) =.4 W/m 2 ) s the rdius of the cylinder is smll (e.g., R =.7m), nd the output shpe is fr from Gussin shpe, s shown in Fig. 13(), for instnce. The symmetric output shpe ppers in this cse, the width of the output Gussin shpe is smller with regrd to the width of the output Gussin shpe of the symmetric cse (Fig. 13(d)). This mode model enbles us to understnd the influence of the step s ngle (δ p ) nd the rdius of the cylinder (R) on the output results (output power trnsmission, etc.). The output power trnsmission is improved by incresing the step s ngle or by incresing the rdius of the cylinder of the helix. The best results re obtined by incresing the vlue of δ p nd lso by incresing the vlue of R. Thus, this model cn be useful tool to find the prmeters (δ p nd R) which will give us the improved results (output power trnsmission, output power density, etc.) of hollow wveguide in the cses of spce curved wveguides, nd for ppliction in the medicl nd industril fields.

27 Progress In Electromgnetics Reserch, PIER 61, CONCLUSIONS The min objective ws to generlize the method [16] from the curved dielectric wveguide (pproximtely plne curve) with circulr cross-section to helicl wveguide ( spce curved wveguide for n rbitrry vlue of the step s ngle of the helix) with circulr crosssection. Another objective ws to demonstrte the bility of the model to solve prcticl problems with inhomogeneous cross-section in the cse of hollow wveguides. The generlized mode model with the two bove objectives provides us numericl tool for the clcultion of the output fields, output power density, nd output power trnsmission for n rbitrry vlue of the step s ngle of the helix (δ p ), in the cse of hollow wveguide. Three test-cses were demonstrted for smll vlues of the step s ngle. The first test-cse ws demonstrted for the stright wveguide (R ). The results of the output trnsverse components of the fields nd the output power density (e.g., Fig. 7()) show the sme behvior of the solutions s shown in the results of Ref. [16] for the TEM mode in excittion. The result of the output power density (Fig. 7()) ws compred lso to the result of the previous published experimentl dt [24] (see lso in Fig. 7(b)). The comprison shows good greement ( Gussin shpe) s expected, except for the secondry smll propgtion mode. The second nd third test-cses were demonstrted in the cse of the toroidl dielectric wveguide. The second test-cse ws demonstrted (Fig. 8()) where the initil dimeter (ID) is 1 mm (nmely, smll bore size). The output modl profile is gretly ffected by the bending. The theoreticl nd experimentl results (Figs. 8() (b)) show tht in ddition to the min propgtion mode, severl other secondry modes nd symmetric shpe pper. The mplitude of the output power density is smll s the bending rdius (R) is smll, nd the shpe is fr from Gussin shpe. In the third test-cse, the result of the output power trnsmission (Fig. 9) ws compred to the experimentl dt [11]. Our theoreticl model tkes into ccount only the dielectric losses nd the bending losses, in conjunction with the problem of the propgtion through curved wveguide. The experimentl result (Fig. 8(b)) nd the experimentl result [11] tke into ccount dditionl prmeters (e.g., the roughnes of the internl wll of the wveguide) which re not tken into ccount theoreticlly. In spite of the differences, the comprisons show good greements. For smll vlues of the bending (1/R) in the cse of smll step s ngle, the output power trnsmission is lrge nd decreses with incresing the bending.

28 186 Menchem nd Mond The results of the rdius of curvture of the helix (ρ) s function of 1/R nd the results of the output power trnsmission of the helicl wveguide s function of 1/R re shown in Fig. 1nd in Fig. 11, respectively, where R is the rdius of the cylinder. For n rbitrry vlue of R, the rdius of curvture of the helix (ρ) nd the output power trnsmission re lrge for lrge vlues of δ p nd decrese with decresing the vlue of δ p. On the other hnd, for n rbitrry vlue of δ p, the rdius of curvture of the helix (ρ) nd the output power trnsmission re lrge for lrge vlues of R nd decrese with decresing the vlue of R. The results of the output power density s functions of the step s ngle (δ p ) nd the rdius of the cylinder (R) re shown in Figs. 12() (d). The output mplitude, the width of the Gussin shpe, nd the shift of the centrl pek re shown in Figs. 13() (d), in the sme cross section of Figs. 12() (d). In ddition to the min propgtion mode, severl other secondry modes pper in Figs. 12() nd 13(), where δ p =.4 nd R =.7 m, nd the output modl profile is gretly ffected in this cse. By incresing only the step s ngle or the rdius of the cylinder of the helix, the mplitude of the output power density nd the width of the Gussin shpe re greter nd the output shpe is chnged from symmetric shpe to the symmetric shpe. The best results in these exmples re shown in Figs. 12(d) nd 13(d), by incresing the vlue of δ p nd lso by incresing the vlue of R. This mode model enbles us to understnd the influence of the step s ngle (δ p ) nd the rdius of the cylinder (R) on the output results (output power trnsmission, etc.). The output power trnsmission is improved by incresing the step s ngle or by incresing the rdius of the cylinder of the helix. The best results re obtined by incresing the vlue of δ p nd lso by incresing the vlue of R. Thus, this model cn be useful tool to find the prmeters (δ p nd R) which will give the improved results (output power trnsmission, output power density, etc.) of hollow wveguide in the cses of spce curved wveguides, nd for ppliction in the medicl nd industril fields. APPENDIX A. By using the Serret-Frenet reltions for sptil curve, we cn find the curvture (κ) nd the torsion (τ) for ech sptil curve tht is chrcterized by θ = const nd r = const for ech pir (r, θ) in the rnge. This is chieved by using the helicl trnsformtion introduced in equtions (3) (3c). The loction vector for the helicl trnsformtion of the

29 Progress In Electromgnetics Reserch, PIER 61, coordintes (3) (3c) is given by ( ) r = (R + r sin θ) cos(φ c )+rsin(δ p ) cos θ sin(φ c ) î ( ) + (R + r sin θ) sin(φ c ) r sin(δ p ) cos θ cos(φ c ) ĵ ( ) + r cos θ cos(δ p )+Rφ c tn(δ p ) ˆk, (A1) where φ c =(ζ/r) cos(δ p ), R is the rdius of the cylinder, nd (r, θ) re the prmeters of the cross-section. The tngent vector is given by T =(dr/dζ) =(dr/dφ c )/(dζ/dφ c ). The norml vector is given by N =(1/κ)(dT /dζ), nd the binorml vector is given by B = T N. The rte of the chnge of the tngent vector relted to the prmeter ζ mesurses the curvture, nd is given by dt /dζ = (dt /dφ c )/(dζ/dφ c ). The curvture of the helix is constnt for constnt vlues of the rdius of the cylinder (R), the step s ngle (δ p ) nd the prmeters (r, θ) of the cross-section. The curvture is given by the first Serret-Frenet eqution of curve r(ζ) in the spce ccording to dt /dζ = κn. Thus, the curvture is κ = dt dζ = 1+C t R(1 + tn 2 (δ p )+C t ), (A2) where C t = r2 R 2 sin2 θ +2 r r2 sin θ + R R 2 sin2 (δ p )cos 2 θ, nd the rdius of curvture is given by ρ =1/κ. The rte of the chnge of the binorml vector relted to the prmeter ζ mesurses the torsion, nd is given by db/dζ = (db/dφ c )/(dζ/dφ c ). The torsion of the helix is constnt for constnt vlues of the rdius of the cylinder (R), the step s ngle (δ p ) nd the prmeters (r, θ) of the cross-section. The torsion is given by the second Serret-Frenet eqution of curve r(ζ) in the spce ccording to db/dζ = τn. Thus, the torsion is τ = db dζ = tn δ p R(1 + tn 2 (δ p )+C t ), (A3) where C t is given bove, nd the rdius of torsion is given by σ =1/τ.

30 188 Menchem nd Mond APPENDIX B. The elements of the mtrices (, etc.) re given by: ( ) r r = J 1 (P 1m )J 1 P 1m rdrδ 1n, ( ) r r 1 = g(r)j 1 (P 1m )J 1 P 1m rdrδ 1n, ( ) r r 2 = J 1 (P 1m )J 1 P 1m r 3 drδ 1n, ( ) r r 3 = g(r)j 1 (P 1m )J 1 P 1m r 3 drδ 1n, ( ) 4 = g 2 r r (r)j 1 (P 1m )J 1 P 1m r 3 drδ 1n, ( ) 5 = k 2 r r g(r)j 1 (P 1m )J 1 P 1m rdrδ 1n, ( ) r r 6 = J 1 (P 1m )J 1 P 1m r 5 drδ 1n, ( ) r r 7 = g(r)j 1 (P 1m )J 1 P 1m r 5 drδ 1n, ( ) ( ( ) P1m r r 8 = g r J 1 P 1m )J 1 P 1m rdr, ( ) r r 9 = g r J 1 (P 1m )J 1 P 1m r 2 drδ 1n, ( ( ) P1m r r 1 = J 1 )J 1 P 1m r 2 drδ 1n, ( ( ) P1m r r 11 = g r J 1 )J 1 P 1m r 3 drδ 1n, ( ) 12 = g 2 r r (r)j 1 (P 1m )J 1 P 1m r 5 drδ 1n, ( p 1m 13 = g r J r ( ) r 1 )J 1 P 1m drδ 1n,