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Electromagnetic properties of inverted Π dielectric waveguides
Int. J. Electron. Commun. (AEÜ) 62 (2008) 349 – 355 www.elsevier.de/aeue
Electromagnetic properties of inverted dielectric waveguides José R. García∗ , Susana F. Fernández, Diego P. Ayuso, Miguel G. Granda Facultad de Ciencias, Applied Physics Department, University of Oviedo, C/ Calvo Sotelo, s/n. 33007 Oviedo, Spain Received 28 July 2006; accepted 9 May 2007
Abstract We present the inverted dielectric waveguide as an alternative for designing microwave devices as well as electromagnetic band gap structures. Inverted dielectric waveguides in cascade, and with high-index contrast, are enclosed by metallic rectangular waveguides. The generalized scattering matrix concept, together with the generalized telegraphist equations (GTE) formulism and the Modal-Matching Technique (MMT) were implemented for theoretical analysis. Numerical and experimental results confirm the possibilities of the new dielectric waveguide for designing single and low-cost microwave devices. 䉷 2007 Elsevier GmbH. All rights reserved. Keywords: Dielectric waveguides; Discontinuities; Microwave devices; Band-gap structures
1. Introduction At microwave frequencies, dielectric waveguides were extensively used for designing passive devices such as directional couplers, Y-junctions and transitions [1–6]. Single and multiple abrupt discontinuities in dielectric waveguides with rectangular cross-section were studied in depth [7–12]. For simplicity and design criteria, the image and slab dielectric waveguides were taken as canonical structure; however, microwave devices combining image dielectric waveguides, directed dielectric waveguides and metallic walls were successfully designed and implemented [1,2]. Besides, dielectric rods and dielectric posts were proposed for electromagnetic guiding and resonance proposals [13,14]. In the present paper, a new dielectric waveguide is introduced as a good candidate for microwave devices design. We denote it as inverted dielectric waveguide. Inverted dielectric waveguides, showing high contrast in the permittivity of the core, were connected in cascade and analyzed theoretically and experimentally. For this purpose, the
∗ Corresponding author. Tel.: +3485 103301; fax: +3485 103324.
E-mail address: [email protected] (J.R. García). 1434-8411/$ - see front matter 䉷 2007 Elsevier GmbH. All rights reserved. doi:10.1016/j.aeue.2007.05.005
inverted dielectric waveguides, with low- and high-core permittivity, were connected abruptly and alternatively inside metallic rectangular waveguides. The theoretical analysis was carried out by means of the Generalized Scattering Matrix Method (GSMM). In order to obtain the Generalized Scattering Matrix (GSM) of any cascaded set of abrupt discontinuities, the Generalized Telegraphist Equations (GTE) formulation [15] and the Modal-Matching Technique (MMT) [16–18] were applied. Different periodical configurations were tested theoretically and experimentally. Results for the reflection and transmission coefficients (moduli and phase) of the fundamental proper mode, as well as for higher-order modes, are given. The power conservation endorses the good performance of the theoretical analysis.
2. Theory Basically, the inverted dielectric waveguide consists of a square dielectric rod (permittivity 1 ), with a symmetrical and longitudinal channel. The channel may be air or any dielectric (permittivity 2 ), which can be removed and/or slided of a free way inside of the channel. Fig. 1 represents an inverted dielectric waveguide (a) and the cross-section
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Fig. 3. An abrupt discontinuity between two inverted dielectric waveguides, a and b, connected abruptly and enclosed by perfectly conducting walls.
connected in cascade and abruptly, as it is shown in Fig. 2. The complete mathematical algorithm is divided in two steps: (A) dielectric waveguides proper modes analysis (B) GSM evaluation: single and multiple discontinuities
2.1. Dielectric waveguides proper modes analysis
Fig. 1. Inverted dielectric waveguide (a) and cross-section of a shielded inverted dielectric waveguide (b). Geometrical and electrical parameters are shown.
Fig. 2. Multiple discontinuities set in inverted dielectric waveguides connected abruptly. Geometrical and electrical parameters are shown.
of an inverted dielectric waveguide inside a metallic rectangular waveguide (b). We are interested in solving the scattering electromagnetic problem inside metallic rectangular waveguides partially filled with n inverted dielectric waveguides
The method applies the MMT to analyze each single discontinuity. Fig. 3 is a side view of an abrupt discontinuity between two inverted dielectric waveguides enclosed by perfectly conducting walls. The application of the MMT requires the evaluation of the proper modes in the shielded dielectric waveguides. Slab and planar dielectric waveguides (1D) propagate TE and TM proper modes; however, rectangular and channel dielectric waveguides (2D) propagate y x hybrid modes, which can be approximated by Epq and Epq proper modes for media with diagonal dielectric tensors [19–25]. This is the case of the inverted dielectric waveguide. Assuming waveguides free from losses, small variations of the permittivity, and following the procedure shown in [26–29], the proper modes solutions in each shielded inverted dielectric waveguide are calculated. Surface, fast, evanescent and complex wave proper mode solutions can be obtained.
2.2. GSM evaluation: single and multiple discontinuities Regarding Fig. 3, we denote as waveguide a and waveguide b the shielded inverted dielectric waveguides located at z < 0 and z > 0, respectively, of the abrupt discontinuity, at z = 0. Once the proper modes of each inverted dielectric waveguide were calculated, the application of the MMT
J.R. García et al. / Int. J. Electron. Commun. (AEÜ) 62 (2008) 349 – 355
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Fig. 4. General representation of a cascaded set of N abrupt discontinuities in shielded dielectric waveguides.
procedure at each discontinuity provides its single GSM [26–29]. This process is applied to each discontinuity so that, finally, we have the same number of single GSMs as discontinuities. In our case, the GSM of a single discontinuity symbolizes a matrix of electromagnetic ports (proper modes) corresponding to physical ports; besides, we are interested in multiple discontinuities in cascade. Fig. 4 is a general representation of a cascaded set of N abrupt discontinuities in arbitrary dielectric waveguides enclosed by perfectly conducting walls. We denote the shielded dielectric waveguides using the labels: a, b, c, . . . , n, n + 1. The capitals: M, N, P , Q, . . . , R, T , U represent the number of proper modes to be taken in the successive waveguides, respectively. So, n + 1 different waveguides give N cascaded abrupt discontinuities, labelled as: I, II, III, . . . , N − 1, N . The discontinuities are separated by the dielectric waveguide lengths: l1 , l2 , l3 , . . . , ln−1 . The successive single GSMs are denoted as [S I ], [S II ], [S III ], . . . , [S N−1 ], [S N ]. To determine the total GSM, [S T ], we join the N GSMs, two by two and correlatively [27,30,31]. Finally, we obtain the total GSM, [S T ], corresponding to the N discontinuities. It has the form T T (M × U )] [S11 (M × M)] [S12 T [S ] = , (1) T (U × M)] T (U × U )] [S21 [S22 where M and U represent, respectively, the number of proper modes in the empty metallic rectangular waveguides located before and after the complete shielded dielectric structure.
3. Results In order to demonstrate the efficiency of the method for analyzing inverted dielectric waveguides, as well as the possible application of this new dielectric waveguide for microwave devices design purpose, we have obtained experimental results which are compared with the theoretical ones. As we are interested in the electromagnetic performance of the shielded dielectric structures, the input and output
Fig. 5. Top and cross section views of three periodical configurations built in inverted dielectric waveguides connected abruptly. In all cases: 1 = 2.1 and 2 = 10. Dimensions in mm.
physical ports (i = 1, 2; j = 1, 2) are always empty metallic rectangular waveguides. Consequently, our scattering results of interest refer to the fundamental and higher-order modes solutions supported by the empty rectangular metallic waveguides enclosing the dielectric waveguides. Evidently, the number of proper modes in our empty metallic waveguides depends on the cross-section dimensions and frequency range under analysis. The inverted dielectric waveguides were built in teflon (r1 = 1 = 2.1), and fixed to the wider side of the rectangular metallic waveguide with double adhesive tape, symmetrically, as shown in Fig. 1(b). The dielectric block in the channel was alumina (r2 =2 =10). The cross dimensions of the rectangular metallic waveguide were 22.86×10.16 mm2 , typically used in the X-band frequencies. The experimental results were taken with the HP-8510 network analyzer. In all cases, results of the reflection and transmission coefficients, both moduli and phase, for the fundamental proper mode were measured and compared with the theoretical ones. For this reason, although the complete generalized scattering matrix was calculated, only the first column of the submaT , S T ] and [S T , S T ] is relevant because we are trices [S11 21 12 22 interested, specifically, in the fundamental proper mode. We will denote by R and T the reflection, S11 , and transmission, S21 , coefficients for the fundamental proper mode. The results presentation starts for the dielectric configurations shown in Fig. 5: three inverted dielectric waveguides in cascade with air–alumina–air in the channel (a), one inverted dielectric waveguide with alumina filling the
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Fig. 7. Theoretical and experimental results for structure 5(b). Reflection, R, and transmission, T, coefficients, versus frequency, for the fundamental proper mode: moduli. Basic modes number: 15. Proper modes number in all waveguides: 10.
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(c) Fig. 6. Theoretical and experimental results for structure 5(a). Reflection, R, and transmission, T, coefficients, versus frequency, for the fundamental proper mode: moduli (a), power conservation (b) and phase (c). Basic modes number: 15. Proper modes number in all waveguides: 10.
channel (b) and the same length as in case (a) and 15 inverted dielectric waveguides in cascade with alumina–air–alumina, . . . as dielectric in the channel (c).
Higher order modes power (%)
Phase (degrees)
R, experimental R, theory T, experimental T, theory
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R1 , 1st mode R2 , 2nd mode T1 , 1st mode T2 , 2nd mode Total power
20 0 18.0 18.5 19.0 19.5 20.0 20.5 21.0 21.5 22.0 F (GHz)
(b) Fig. 8. Theoretical and experimental results for structure 5(c). Reflection, R, and transmission, T, coefficients, versus frequency, for the fundamental proper mode: moduli (a) and higher-order modes power conservation (b). Basic modes number: 30. Proper modes number in all waveguides: 20.
J.R. García et al. / Int. J. Electron. Commun. (AEÜ) 62 (2008) 349 – 355
Fig. 9. Top and side views of five inverted dielectric waveguides in cascade. Cross-section dimensions are shown (mm).
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Fig. 11. Top view and cross-section of a dielectric periodical configuration, in inverted dielectric waveguides, analyzed at microwave frequencies: 23 inverted dielectric waveguides in cascade (2 = 1 (air) in odd-order waveguides and 2 = 10 in even-order waveguides).
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Fig. 10. Theoretical results for the periodical structure in Fig. 9 when li (i =1, 2, 3, 4, 5)=30 mm. Reflected, R, and transmitted, T, coefficients, versus frequency, for the fundamental proper mode.
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In Fig. 6(a)–(c), theoretical results are compared with the experimental ones for the reflection, R, and transmission, T, coefficients of the fundamental proper mode for the structure in Fig. 5(a). Both moduli and phase are shown, as well as the power conservation characteristics. Fig. 7 compares the theoretical and experimental results for the reflection, R, and transmission, T, coefficients of the fundamental proper mode for the structure in Fig. 5(b). The results for the 15 inverted dielectric waveguides connected in cascade (Fig. 5(c)) are shown in Fig. 8(a)–(b). In order to explore in depth the possibilities of the inverted dielectric waveguide for microwave design proposes, additional numerical results were obtained. We have analyzed a periodical structure consisting of five inverted dielectric waveguides connected in cascade, as shown in Fig. 9. Fig. 10 shows the reflected, R, and transmitted, T, coefficients of the fundamental proper mode, for the periodical structure in Fig. 9. According to the previous results, EBG structures can be obtained at microwave frequencies by using shielded inverted dielectric waveguides, appropriately designed and connected in cascade. In our case, several different periodical configurations were designed and tested to
Fig. 12. Theoretical results for the periodical structure in Fig. 11. Reflected, R, and transmitted, T, coefficients, versus frequency, for the fundamental proper mode.
demonstrate this behavior. As an example, Fig. 11 shows a periodic structure and the cross section dimensions. In this case, the physical parameters were as follows: 23 inverted dielectric waveguides; conducting walls cross section dimensions (mm); a = 22.86, b = 10.16; basic modes number: 20; proper modes number: M = N = P = · · · = 20; 2 = 2.1 (teflon), 2 = 10 (alumina) in odd-order inverted dielectric waveguides and 2 = 1 (air) in even-order inverted dielectric waveguides. Fig. 12 presents the moduli of the reflection, R, and transmission, T, coefficients for the fundamental proper mode. An EBG, from 9.125 to 10 GHz, was obtained. The authors continue exploring the possibilities of the shielded inverted dielectric waveguide for designing filters and electromagnetic band gaps devices at microwave frequencies. Recent results confirm very interesting performances and will be published in the future.
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4. Conclusions The procedure described in this paper provides good accuracy in finding the proper modes in arbitrary dielectric waveguides bounded by perfectly conducting walls with rectangular cross section. Surface, fast, evanescent and complex proper modes can be evaluated. By using these wave solutions, the method combines the MMT technique with the GSM concept, resulting in a successful and powerful tool to evaluate the electromagnetic scattering caused by abrupt discontinuities in cascade between dielectric waveguides. The inverted dielectric waveguide was introduced as a good candidate for microwave devices design. Different periodical configurations in inverted dielectric waveguides, showing high contrast in the permittivity of the core, were connected in cascade inside a metallic rectangular waveguide, and analyzed theoretically and experimentally. In all cases, an excellent agreement between theory and experiment was noticed. Additional numerical results have shown the possibility of designing low-cost microwave passive devices, by using rectangular metallic waveguides enclosing inverted dielectric waveguides in cascade.
Acknowledgments This work was supported by the Projects from MEC (TIC2002-02300), (TEC2005-05541) and from FICY IB05151C1. Special thanks to Prof. A. Mediavilla, from University of Cantabria, for taking measurements.
References [1] Rodríguez J, Prieto A. Wide-band directional couplers in dielectric waveguide. IEEE Trans Microwave Theory Tech 1987;35:681–7. [2] Solano MA, Prieto A, Rodríguez J. Reduction of losses in dielectric waveguide Y-junctions with wide aperture angles. Int J Electron 1987;63:741–9. [3] Strube J, Arndt F. Rigorous hybrid-mode analysis of the transition from rectangular waveguide to shielded dielectric image guide. IEEE Trans Microwave Theory Tech 1985;33:391–401. [4] Malherbe JAG, Cloete JH, Lösch JE. A transition from rectangular to nonradiating dielectric waveguide. IEEE Trans Microwave Theory Tech 1985;33:539–43. [5] Christ A, Hartnagel HL. Three-dimensional finite difference method for the analysis of microwave device embedding. IEEE Trans Microwave Theory Tech 1987;35:688–96. [6] Katzier H. Streuverhalten elektromagnetischer eellen bei sprunghaften übergängen geschirmter dielektrischer leitungen. AEÜ 1984;38:290–6. [7] Brooke GH, Kharadly MMZ. Step discontinuities on dielectric waveguides. Electron Lett 1976;12:473–5. [8] Brooke GH, Kharadly MMZ. Scattering by abrupt discontinuities on planar dielectric waveguides. IEEE Trans Microwave Theory Tech 1982;30:760–70.
[9] Koshiba M, Ooshi K, Miki T, Suzuki M. Finite-element analysis of the discontinuities in a dielectric slab waveguide bounded by parallel plates. Electron Lett 1982;18:33–4. [10] Koshiba M, Suzuki M. Boundary-element analysis of dielectric slab waveguide discontinuities. Appl Opt 1986;25:828–9. [11] Weisshaar A, Tripathi VK. Modal analysis of step discontinuities in graded-index dielectric slab waveguides. J Lightwave Technol 1992;10:593–602. [12] Schmidt R, Russer P. Modelling of cascade coplanar waveguide discontinuities by the mode-matching approach. IEEE Trans Microwave Theory Tech 1995;43:2910–7. [13] Chung-I GH, Hesham AA. Multiple dielectric post in a rectangular waveguide. IEEE Trans Microwave Theory Tech 1986;34:883–9. [14] Leviatan Y, Sheaffer GS. Analysis of inductive dielectric posts in rectangular waveguide. IEEE Trans Microwave Theory Tech 1987;35:48–59. [15] Rodríguez J, Crespo RD, Fernández S, Palacios SL, Virgós JM. Comments on: verification of generalized telegraphists equations applied to dielectric waveguide problems. Appl Opt 1984;33:356–7. [16] Wexler A. Solution of waveguide discontinuities by modal analysis. IEEE Trans Microwave Theory Tech 1967;15: 508–17. [17] James GL. On the problem of applying mode-matching techniques in analyzing conical waveguide discontinuities. IEEE Trans Microwave Theory Tech 1983;31:718–23. [18] Rodríguez J, Hidalgo O, Fernández S, Ibañez I. Investigación de nuevos efectos electromagnéticos causados por estructuras periódicas en guías dieléctricas: evaluación experimental y predicciones teóricas. In: Optoel03 Conference, Madrid, Spain, 2003. p. 26–9. [19] Marcatili EAJ. Dielectric rectangular waveguide and directional couplers for integrated optics. Bell Syst Tech J 1969;48:2071–102. [20] Knox RM, Toulios PP. Rectangular dielectric image lines for millimeter integrated circuits. In: Wescon Conference. Los Angeles, 1970. [21] Knox RM, Toulios PP. Integrated circuits for the millimeter through optical frequency range. Symposium on Submillimeter Waves, 1970, p. 497–516. [22] Steinberg RA, Giallorenzi TG. Modal fields of anisotropic channel waveguides. J Opt Soc Am 1977;67:523–33. [23] Solbach K, Wolff I. The electromagnetics fields and phase constants of dielectric image lines. IEEE Trans Microwave Theory Tech 1978;26:266–74. [24] Suhara T, Handa Y, Nishihara J, Koyama J. Analysis of optical channel waveguides an directional couplers with graded-index profile. J Opt Soc Am 1979;6:807–15. [25] Strake E, Bava GP, Montrosset I. Guided modes of Ti:LiNbO3 channel waveguides: a novel quasi-analytical technique in comparison with the scalar finite-element method. J Lightwave Technol 1988;6:1126–35. [26] Rodríguez J, Crespo R, Palacios SL, Virgós JM. Phase constants and scattering properties of circular dielectric waveguides. Int J Electron 1992;73(2):385–93. [27] Rodríguez J, Crespo RD, Fernández S, Pandavenes J, Olivares J, Carrasco S, Ibañez I, Virgós JM. Radiation losses on discontinuities in integrated optical waveguides. Opt Eng 1999;38(11):1896–906.
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[28] Rodríguez J, Hidalgo O, Fernández S, Ibañez I, Domenech G. Photonic devices in waveguide periodical structures: generalized analysis of multiple discontinuities. Opt Eng 2002;41(8):1947–56. [29] Rodríguez J, Fernández S, García M, Pozo DF. Photonic effects caused by multiple discontinuities in optical waveguides. IEE Proc Optoelectron 2005;152(5):263–8. [30] Vanblaricum FG, Mittra R. A modified residue-calculus technique for solving a class of boundary value problempart II: waveguide phased arrays modulated surfaces and diffraction grating. IEEE Trans Microwave Theory Tech 1969;17(6):310–9. [31] Rodríguez J. Contribution to the analysis and optimization of passive devices in dielectric waveguides. PhD dissertation, University of Santander, Spain, 1987. José Rodríguez García is a Professor at the University of Oviedo (Spain). He received the Teaching degree from the University of Oviedo, and the BSc Extraordinary PhD degrees in physics from the University of Santander (Spain). From 1982 to 1988 he worked in electromagnetic field analysis on dielectric waveguides in the Electronics Department of the University of Santander. As member of the COST-216 European Project, he has worked at the ETH (Zurich) as well as for the CTNE and NESTLE Companies. Since 1988 he has been a Professor at the University of Oviedo and since 1993 Supervisor and Coordinator of the Secondary School. He is a member of the Electromagnetism Academy (USA) and member of the SPIE society. He was selected as Man of the Year 1997 by the ABI and he was included in: Who’s Who in Electromagnetics, Who’s Who in the World and Who’s Who in Contemporary Achievements. He conducts research in the following areas: electromagnetic field theory, modelling, characterization and evaluation of integrated optical waveguides and devices.
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Diego Francisco Pozo Ayuso received the degree in Physics from the University of Oviedo, Spain. Since 2003 he is working in optical waveguide fabrication and characterization in the Physics Department of the University of Oviedo. In 2005 he received a grant to doctoral courses from this University. Moreover, this year he received the pedagogic aptitude for teachers of Secondary Education, obtained a master in management of the environment from Asturias Business School and worked in an environmental control company. He is currently working as researcher for a coordinated project between the inmunoelectroanalisis group and the integrated optic group of the University of Oviedo. His research interests are fabrication and characterization of optical waveguides, capillary electrophoresis microchips (MCE) and biosensors microdevices based on Mach–Zehnder interferometer.
Miguel García Granda received his degree in Physics in 2003 from the University of Oviedo (Spain). During 2003 he joined the Hanh-Meitner Institut in Berlin (Germany) where he worked in chalcopyrite solar cells fabrication techniques and structural characterization. In 2004 he received a doctoral grant from the Spanish Ministry of Education and Science. Currently, he is pursuing his PhD which is directed towards the study of optical waveguides design, fabrication and characterization, as well as Mach–Zehnder optical modulators, within a cooperation framework between the Universities of Oviedo (Spain) and Paderborn (Germany).
Electromagnetic properties of inverted dielectric waveguides José R. García∗ , Susana F. Fernández, Diego P. Ayuso, Miguel G. Granda Facultad de Ciencias, Applied Physics Department, University of Oviedo, C/ Calvo Sotelo, s/n. 33007 Oviedo, Spain Received 28 July 2006; accepted 9 May 2007
Abstract We present the inverted dielectric waveguide as an alternative for designing microwave devices as well as electromagnetic band gap structures. Inverted dielectric waveguides in cascade, and with high-index contrast, are enclosed by metallic rectangular waveguides. The generalized scattering matrix concept, together with the generalized telegraphist equations (GTE) formulism and the Modal-Matching Technique (MMT) were implemented for theoretical analysis. Numerical and experimental results confirm the possibilities of the new dielectric waveguide for designing single and low-cost microwave devices. 䉷 2007 Elsevier GmbH. All rights reserved. Keywords: Dielectric waveguides; Discontinuities; Microwave devices; Band-gap structures
1. Introduction At microwave frequencies, dielectric waveguides were extensively used for designing passive devices such as directional couplers, Y-junctions and transitions [1–6]. Single and multiple abrupt discontinuities in dielectric waveguides with rectangular cross-section were studied in depth [7–12]. For simplicity and design criteria, the image and slab dielectric waveguides were taken as canonical structure; however, microwave devices combining image dielectric waveguides, directed dielectric waveguides and metallic walls were successfully designed and implemented [1,2]. Besides, dielectric rods and dielectric posts were proposed for electromagnetic guiding and resonance proposals [13,14]. In the present paper, a new dielectric waveguide is introduced as a good candidate for microwave devices design. We denote it as inverted dielectric waveguide. Inverted dielectric waveguides, showing high contrast in the permittivity of the core, were connected in cascade and analyzed theoretically and experimentally. For this purpose, the
∗ Corresponding author. Tel.: +3485 103301; fax: +3485 103324.
E-mail address: [email protected] (J.R. García). 1434-8411/$ - see front matter 䉷 2007 Elsevier GmbH. All rights reserved. doi:10.1016/j.aeue.2007.05.005
inverted dielectric waveguides, with low- and high-core permittivity, were connected abruptly and alternatively inside metallic rectangular waveguides. The theoretical analysis was carried out by means of the Generalized Scattering Matrix Method (GSMM). In order to obtain the Generalized Scattering Matrix (GSM) of any cascaded set of abrupt discontinuities, the Generalized Telegraphist Equations (GTE) formulation [15] and the Modal-Matching Technique (MMT) [16–18] were applied. Different periodical configurations were tested theoretically and experimentally. Results for the reflection and transmission coefficients (moduli and phase) of the fundamental proper mode, as well as for higher-order modes, are given. The power conservation endorses the good performance of the theoretical analysis.
2. Theory Basically, the inverted dielectric waveguide consists of a square dielectric rod (permittivity 1 ), with a symmetrical and longitudinal channel. The channel may be air or any dielectric (permittivity 2 ), which can be removed and/or slided of a free way inside of the channel. Fig. 1 represents an inverted dielectric waveguide (a) and the cross-section
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Fig. 3. An abrupt discontinuity between two inverted dielectric waveguides, a and b, connected abruptly and enclosed by perfectly conducting walls.
connected in cascade and abruptly, as it is shown in Fig. 2. The complete mathematical algorithm is divided in two steps: (A) dielectric waveguides proper modes analysis (B) GSM evaluation: single and multiple discontinuities
2.1. Dielectric waveguides proper modes analysis
Fig. 1. Inverted dielectric waveguide (a) and cross-section of a shielded inverted dielectric waveguide (b). Geometrical and electrical parameters are shown.
Fig. 2. Multiple discontinuities set in inverted dielectric waveguides connected abruptly. Geometrical and electrical parameters are shown.
of an inverted dielectric waveguide inside a metallic rectangular waveguide (b). We are interested in solving the scattering electromagnetic problem inside metallic rectangular waveguides partially filled with n inverted dielectric waveguides
The method applies the MMT to analyze each single discontinuity. Fig. 3 is a side view of an abrupt discontinuity between two inverted dielectric waveguides enclosed by perfectly conducting walls. The application of the MMT requires the evaluation of the proper modes in the shielded dielectric waveguides. Slab and planar dielectric waveguides (1D) propagate TE and TM proper modes; however, rectangular and channel dielectric waveguides (2D) propagate y x hybrid modes, which can be approximated by Epq and Epq proper modes for media with diagonal dielectric tensors [19–25]. This is the case of the inverted dielectric waveguide. Assuming waveguides free from losses, small variations of the permittivity, and following the procedure shown in [26–29], the proper modes solutions in each shielded inverted dielectric waveguide are calculated. Surface, fast, evanescent and complex wave proper mode solutions can be obtained.
2.2. GSM evaluation: single and multiple discontinuities Regarding Fig. 3, we denote as waveguide a and waveguide b the shielded inverted dielectric waveguides located at z < 0 and z > 0, respectively, of the abrupt discontinuity, at z = 0. Once the proper modes of each inverted dielectric waveguide were calculated, the application of the MMT
J.R. García et al. / Int. J. Electron. Commun. (AEÜ) 62 (2008) 349 – 355
351
Fig. 4. General representation of a cascaded set of N abrupt discontinuities in shielded dielectric waveguides.
procedure at each discontinuity provides its single GSM [26–29]. This process is applied to each discontinuity so that, finally, we have the same number of single GSMs as discontinuities. In our case, the GSM of a single discontinuity symbolizes a matrix of electromagnetic ports (proper modes) corresponding to physical ports; besides, we are interested in multiple discontinuities in cascade. Fig. 4 is a general representation of a cascaded set of N abrupt discontinuities in arbitrary dielectric waveguides enclosed by perfectly conducting walls. We denote the shielded dielectric waveguides using the labels: a, b, c, . . . , n, n + 1. The capitals: M, N, P , Q, . . . , R, T , U represent the number of proper modes to be taken in the successive waveguides, respectively. So, n + 1 different waveguides give N cascaded abrupt discontinuities, labelled as: I, II, III, . . . , N − 1, N . The discontinuities are separated by the dielectric waveguide lengths: l1 , l2 , l3 , . . . , ln−1 . The successive single GSMs are denoted as [S I ], [S II ], [S III ], . . . , [S N−1 ], [S N ]. To determine the total GSM, [S T ], we join the N GSMs, two by two and correlatively [27,30,31]. Finally, we obtain the total GSM, [S T ], corresponding to the N discontinuities. It has the form T T (M × U )] [S11 (M × M)] [S12 T [S ] = , (1) T (U × M)] T (U × U )] [S21 [S22 where M and U represent, respectively, the number of proper modes in the empty metallic rectangular waveguides located before and after the complete shielded dielectric structure.
3. Results In order to demonstrate the efficiency of the method for analyzing inverted dielectric waveguides, as well as the possible application of this new dielectric waveguide for microwave devices design purpose, we have obtained experimental results which are compared with the theoretical ones. As we are interested in the electromagnetic performance of the shielded dielectric structures, the input and output
Fig. 5. Top and cross section views of three periodical configurations built in inverted dielectric waveguides connected abruptly. In all cases: 1 = 2.1 and 2 = 10. Dimensions in mm.
physical ports (i = 1, 2; j = 1, 2) are always empty metallic rectangular waveguides. Consequently, our scattering results of interest refer to the fundamental and higher-order modes solutions supported by the empty rectangular metallic waveguides enclosing the dielectric waveguides. Evidently, the number of proper modes in our empty metallic waveguides depends on the cross-section dimensions and frequency range under analysis. The inverted dielectric waveguides were built in teflon (r1 = 1 = 2.1), and fixed to the wider side of the rectangular metallic waveguide with double adhesive tape, symmetrically, as shown in Fig. 1(b). The dielectric block in the channel was alumina (r2 =2 =10). The cross dimensions of the rectangular metallic waveguide were 22.86×10.16 mm2 , typically used in the X-band frequencies. The experimental results were taken with the HP-8510 network analyzer. In all cases, results of the reflection and transmission coefficients, both moduli and phase, for the fundamental proper mode were measured and compared with the theoretical ones. For this reason, although the complete generalized scattering matrix was calculated, only the first column of the submaT , S T ] and [S T , S T ] is relevant because we are trices [S11 21 12 22 interested, specifically, in the fundamental proper mode. We will denote by R and T the reflection, S11 , and transmission, S21 , coefficients for the fundamental proper mode. The results presentation starts for the dielectric configurations shown in Fig. 5: three inverted dielectric waveguides in cascade with air–alumina–air in the channel (a), one inverted dielectric waveguide with alumina filling the
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0
0 -5
-5
R, T (dB)
R, T (dB)
-10 -15 -20 -25
R, experiment R, theory T, experiment T, theory
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-10 -15 -20 -25 8.0
-40 8.0
8.5
9.0
(a)
9.0
9.5
100
10.0 10.5 11.0 11.5 12.0 F (GHz)
Fig. 7. Theoretical and experimental results for structure 5(b). Reflection, R, and transmission, T, coefficients, versus frequency, for the fundamental proper mode: moduli. Basic modes number: 15. Proper modes number in all waveguides: 10.
80 60
R (%) T (%) R+T (%)
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20
-10
0 8.0
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R, T (dB)
Reflected,R, and transmitted, T, power (%)
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(b) 350 300
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R, experimental
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-35 -40 8.0
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250
(a)
200 100
150 100 50 0 8.0
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9.5 10.0 10.5 11.0 11.5 12.0 F (GHz)
(c) Fig. 6. Theoretical and experimental results for structure 5(a). Reflection, R, and transmission, T, coefficients, versus frequency, for the fundamental proper mode: moduli (a), power conservation (b) and phase (c). Basic modes number: 15. Proper modes number in all waveguides: 10.
channel (b) and the same length as in case (a) and 15 inverted dielectric waveguides in cascade with alumina–air–alumina, . . . as dielectric in the channel (c).
Higher order modes power (%)
Phase (degrees)
R, experimental R, theory T, experimental T, theory
80 60 40
R1 , 1st mode R2 , 2nd mode T1 , 1st mode T2 , 2nd mode Total power
20 0 18.0 18.5 19.0 19.5 20.0 20.5 21.0 21.5 22.0 F (GHz)
(b) Fig. 8. Theoretical and experimental results for structure 5(c). Reflection, R, and transmission, T, coefficients, versus frequency, for the fundamental proper mode: moduli (a) and higher-order modes power conservation (b). Basic modes number: 30. Proper modes number in all waveguides: 20.
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Fig. 9. Top and side views of five inverted dielectric waveguides in cascade. Cross-section dimensions are shown (mm).
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Fig. 11. Top view and cross-section of a dielectric periodical configuration, in inverted dielectric waveguides, analyzed at microwave frequencies: 23 inverted dielectric waveguides in cascade (2 = 1 (air) in odd-order waveguides and 2 = 10 in even-order waveguides).
0
0
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-15
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R T
EBG
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R, T (dB)
R, T (dB)
-5
-30 -40 -50
8.0
8.5
9.0
9.5 10.0 10.5 11.0 11.5 12.0 F (GHz)
Fig. 10. Theoretical results for the periodical structure in Fig. 9 when li (i =1, 2, 3, 4, 5)=30 mm. Reflected, R, and transmitted, T, coefficients, versus frequency, for the fundamental proper mode.
-60
R T
-70
EBG
-80 8.0
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In Fig. 6(a)–(c), theoretical results are compared with the experimental ones for the reflection, R, and transmission, T, coefficients of the fundamental proper mode for the structure in Fig. 5(a). Both moduli and phase are shown, as well as the power conservation characteristics. Fig. 7 compares the theoretical and experimental results for the reflection, R, and transmission, T, coefficients of the fundamental proper mode for the structure in Fig. 5(b). The results for the 15 inverted dielectric waveguides connected in cascade (Fig. 5(c)) are shown in Fig. 8(a)–(b). In order to explore in depth the possibilities of the inverted dielectric waveguide for microwave design proposes, additional numerical results were obtained. We have analyzed a periodical structure consisting of five inverted dielectric waveguides connected in cascade, as shown in Fig. 9. Fig. 10 shows the reflected, R, and transmitted, T, coefficients of the fundamental proper mode, for the periodical structure in Fig. 9. According to the previous results, EBG structures can be obtained at microwave frequencies by using shielded inverted dielectric waveguides, appropriately designed and connected in cascade. In our case, several different periodical configurations were designed and tested to
Fig. 12. Theoretical results for the periodical structure in Fig. 11. Reflected, R, and transmitted, T, coefficients, versus frequency, for the fundamental proper mode.
demonstrate this behavior. As an example, Fig. 11 shows a periodic structure and the cross section dimensions. In this case, the physical parameters were as follows: 23 inverted dielectric waveguides; conducting walls cross section dimensions (mm); a = 22.86, b = 10.16; basic modes number: 20; proper modes number: M = N = P = · · · = 20; 2 = 2.1 (teflon), 2 = 10 (alumina) in odd-order inverted dielectric waveguides and 2 = 1 (air) in even-order inverted dielectric waveguides. Fig. 12 presents the moduli of the reflection, R, and transmission, T, coefficients for the fundamental proper mode. An EBG, from 9.125 to 10 GHz, was obtained. The authors continue exploring the possibilities of the shielded inverted dielectric waveguide for designing filters and electromagnetic band gaps devices at microwave frequencies. Recent results confirm very interesting performances and will be published in the future.
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4. Conclusions The procedure described in this paper provides good accuracy in finding the proper modes in arbitrary dielectric waveguides bounded by perfectly conducting walls with rectangular cross section. Surface, fast, evanescent and complex proper modes can be evaluated. By using these wave solutions, the method combines the MMT technique with the GSM concept, resulting in a successful and powerful tool to evaluate the electromagnetic scattering caused by abrupt discontinuities in cascade between dielectric waveguides. The inverted dielectric waveguide was introduced as a good candidate for microwave devices design. Different periodical configurations in inverted dielectric waveguides, showing high contrast in the permittivity of the core, were connected in cascade inside a metallic rectangular waveguide, and analyzed theoretically and experimentally. In all cases, an excellent agreement between theory and experiment was noticed. Additional numerical results have shown the possibility of designing low-cost microwave passive devices, by using rectangular metallic waveguides enclosing inverted dielectric waveguides in cascade.
Acknowledgments This work was supported by the Projects from MEC (TIC2002-02300), (TEC2005-05541) and from FICY IB05151C1. Special thanks to Prof. A. Mediavilla, from University of Cantabria, for taking measurements.
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[28] Rodríguez J, Hidalgo O, Fernández S, Ibañez I, Domenech G. Photonic devices in waveguide periodical structures: generalized analysis of multiple discontinuities. Opt Eng 2002;41(8):1947–56. [29] Rodríguez J, Fernández S, García M, Pozo DF. Photonic effects caused by multiple discontinuities in optical waveguides. IEE Proc Optoelectron 2005;152(5):263–8. [30] Vanblaricum FG, Mittra R. A modified residue-calculus technique for solving a class of boundary value problempart II: waveguide phased arrays modulated surfaces and diffraction grating. IEEE Trans Microwave Theory Tech 1969;17(6):310–9. [31] Rodríguez J. Contribution to the analysis and optimization of passive devices in dielectric waveguides. PhD dissertation, University of Santander, Spain, 1987. José Rodríguez García is a Professor at the University of Oviedo (Spain). He received the Teaching degree from the University of Oviedo, and the BSc Extraordinary PhD degrees in physics from the University of Santander (Spain). From 1982 to 1988 he worked in electromagnetic field analysis on dielectric waveguides in the Electronics Department of the University of Santander. As member of the COST-216 European Project, he has worked at the ETH (Zurich) as well as for the CTNE and NESTLE Companies. Since 1988 he has been a Professor at the University of Oviedo and since 1993 Supervisor and Coordinator of the Secondary School. He is a member of the Electromagnetism Academy (USA) and member of the SPIE society. He was selected as Man of the Year 1997 by the ABI and he was included in: Who’s Who in Electromagnetics, Who’s Who in the World and Who’s Who in Contemporary Achievements. He conducts research in the following areas: electromagnetic field theory, modelling, characterization and evaluation of integrated optical waveguides and devices.
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Diego Francisco Pozo Ayuso received the degree in Physics from the University of Oviedo, Spain. Since 2003 he is working in optical waveguide fabrication and characterization in the Physics Department of the University of Oviedo. In 2005 he received a grant to doctoral courses from this University. Moreover, this year he received the pedagogic aptitude for teachers of Secondary Education, obtained a master in management of the environment from Asturias Business School and worked in an environmental control company. He is currently working as researcher for a coordinated project between the inmunoelectroanalisis group and the integrated optic group of the University of Oviedo. His research interests are fabrication and characterization of optical waveguides, capillary electrophoresis microchips (MCE) and biosensors microdevices based on Mach–Zehnder interferometer.
Miguel García Granda received his degree in Physics in 2003 from the University of Oviedo (Spain). During 2003 he joined the Hanh-Meitner Institut in Berlin (Germany) where he worked in chalcopyrite solar cells fabrication techniques and structural characterization. In 2004 he received a doctoral grant from the Spanish Ministry of Education and Science. Currently, he is pursuing his PhD which is directed towards the study of optical waveguides design, fabrication and characterization, as well as Mach–Zehnder optical modulators, within a cooperation framework between the Universities of Oviedo (Spain) and Paderborn (Germany).