Waveguide Introduction & Analysis Setup

Transcription

1 4347 Applied letromagnetis Topi 5a Waveguide Introdution & Analsis Setup Leture 5a These notes ma ontain oprighted material obtained under fair use rules. Distribution of these materials is stritl prohibited Slide 1 Leture Outline What is a waveguide? Governing equations for waveguide analsis. Setup for TM, T, and TM modes Setup for analing hbrid modes Setup for analing slab waveguides Note, this leture overs just setting up Mawell s equations for solution. This leture does not attempt to obtain solutions. Leture 5a Slide 1

2 What is a Waveguide? Leture 5a Slide 3 What is a Waveguide? A waveguide is a struture that onfines the propagation of waves to a single path. The are pipes for eletromagneti waves. Leture 5a Slide 4

3 Waveguide Modes The field inside a waveguide must obe Mawell s equations. This limits what field onfigurations are possible into a disrete set. ah solution is alled a mode. ah mode looks different and behaves differentl inside the waveguide. Leture 5a Slide 5 Slab Vs. Channel Waveguides Slab waveguides onfine energ in onl one transverse diretion. Channel waveguides onfine energ in both transverse diretions. Confinement Confinement Leture 5a Slide 6 3

4 Map of Waveguides (LI Media) Transmission Lines Contains two or more ondutors. No low frequen utoff. omogeneous as TM mode. as T and TM modes. Inhomogeneous Supports onl quasi (TM, T, & TM) modes. Waveguides Confines and transports waves. Supports higher order modes. Metal Shell Pipes Pipes nlosed b metal. Does not support TM mode. as a low frequen utoff. as one or less ondutors. Usuall what is implied b the label waveguide. Dieletri Pipes Composed of a ore and a ladding. Smmetri waveguides have no low frequen utoff. omogeneous Channel Waveguides Single nded oaial stripline mirostrip oplanar Supports T and TM modes retangular irular dual ridge Confinement along two aes. T & TM modes onl supported in irularl smmetri guides. Inhomogeneous optial Fiber photoni rstal rib Differential buried parallel plate oplanar strips Supports T and TM modes onl if one ais is uniform. Otherwise supports quasi TM and quasi T modes. Slab Waveguides Confinement onl along one ais. Supports T and TM modes. Interfaes an support surfae waves. Leture 5a shielded pair slotline no uniform ais (no T or TM) uniform ais (has T and TM) dieletri Slab large area parallel plate interfae Slide 7 Notes on Transmission Lines Contains two or more ondutors No low frequen utoff. Works down to DC. Supports TM, T, and TM modes when the dieletri is homogeneous Supports higher order modes, not just TM. Serve more as a iruit element than a wave devie Ver ompat for low frequen signals. Tend to be loss at ver high frequenies (> 10 G) due to skin effet and dieletri loss. Leture 5a Slide 8 4

5 Notes on Metal Pipe Waveguides Contains on a single ondutor as a low frequen utoff below whih there is no propagation of waves. Supports T and TM waves onl if dieletri is homogeneous. Field onfined to inside of the waveguide. Less loss for ver high frequen waves. Prohibitivel large sie at low frequenies. Leture 5a Slide 9 Notes on Dieletri Waveguides Does not ontain an metals Smmetri dieletri waveguides have not lowfrequen utoff. Smmetri waveguides (e.g. slabs & irularl smmetri) support T and TM modes. Most have a low frequen utoff below whih no waves an propagate. brid modes still tend to be strongl linearl polaried. Optial fibers are dieletri waveguides. Field etends outside of the ore. Leture 5a Slide 10 5

6 Channel Waveguides for Integrated Optis Stripe waveguide Diffused waveguide Buried strip waveguide Buried rib waveguide Rib waveguide Strip loaded waveguide Leture 5a Slide 11 Channel Waveguides for Radio Frequenies Isolated Wire Twisted Pair Transmission Line Retangular Waveguide Shielded Pair Transmission Line Coaial Cable Leture 5a Slide 1 6

7 Channel Waveguides for Printed Ciruits Transmission lines are metalli strutures that guide eletromagneti waves from DC to ver high frequenies. Mirostrip Parallel Plate Transmission Line Stripline Slot Line Coplanar Line Leture 5a Slide 13 Strutures Supporting Surfae Waves Surfae Plasmon Polariton (SPP) Dakonov Surfae Wave Bloh Surfae Wave Leture 5a Slide 14 7

8 Notes on Waveguides Waveguides support an infinite number of disrete modes The modes have a onstant amplitude profile that just aumulates phase as it propages. Modes have utoff frequenies, below whih the are not supported and dea ver quikl. Leture 5a Slide 15 Governing quations for Waveguide Analsis Leture 5a Slide 16 8

9 Steps for Waveguide Analsis 1. Draw the waveguide. Assume a form of the solution. Outer regions must dea eponentiall or be equal to ero. 3. Substitute solution into Mawell s equations. 4. Simplif equations based on the geometr of the waveguide. 5. Manipulate equations into a differential equation to solve. This is alled the governing equation. 6. Solve the governing equation in eah homogeneous region of the waveguide. 7. Connet the solutions in eah region using boundar onditions. 8. Calulate the overall field solution. 9. Use the field solution to alulate the waveguide parameters suh as, Z 0, and the profile of the fields. Leture 5a Slide 17 Various Wave quations 1. Mawell s Curl quations j j 3. Wave quation in LI Media 1 k k0 k wave number. Wave quation in General Media j j j j j 1 Leture 5a Slide Wave quation Deouples k 0 k 0 k 0 We an solve these equations independentl. 9

10 pand Mawell s quations We must anale waveguides using Mawell s equations. j j The two url equations epand into a set of si oupled partial differential equations. j j j j j j We have si field omponents to solve for:,,,,, and. Yikes!! Leture 5a Slide 19 General Form of Solution for Waveguides A mode in a waveguide has the following general mathematial form.,,, j 0 e omple amplitude, mode shape aumulation of phase in diretion j e phase onstant This means we an solve the problem b just analing the ross setion in the - plane. This redues to a twodimensional problem. 0, 3D Leture 5a Slide 0 D 10

11 Animation of a Waveguide Mode Leture 5a Slide 1 Assume the Form of the Solution For a waveguide uniform in the diretion, the solution will have the form j j,, e,,, e, 0 0 If we substitute this solution into our si equations, we get j j j j 0, j j j j j j Things are a little more simple, but we still have si field omponents to solve for. Leture 5a Slide 11

12 Setup for TM, T, and TM Modes Leture 5a Slide 3 istene Conditions for TM, T, and TM Modes TM modes onl eist in transmission lines with two or more ondutors embedded in a homogeneous fill. Supports TM No TM T and TM modes onl eist in waveguides with a homogeneous fill or in waveguides with a uniform ais like slabs and irularl smmetri guides. Supports T and TM No T or TM Leture 5a Slide 4 1

13 Goal of Following Derivation j j j j 0, j j j j j j j k j k j 0, k j 0, k?? We now onl need to find and. k 0 k 0 Leture 5a Slide 5 Redue the Number of Terms to Solve (1 of ) 0, j j0, q. 1a j j0, q. 1d 0, j j0, q. 1b 0, j0, q. 1 Step 1 Solve q. (1e) for 0,. j j 1 j j0, q. 1e j 0, q. 1 f Step Substitute this epression to q. (1a) to eliminate 0,. 1 j j j j Step 3 Reall that k = and solve this new epression for 0,. j 0, k Leture 5a Slide 6 13

14 Redue the Number of Terms to Solve ( of ) Step 4 Derive three more similar equations. Solve q. (1d) for 0,, substitute that epression into q. (1b) and solve for 0,. j 0, k Solve q. (1b) for 0,, substitute that epression into q. (1d) and solve for 0,. j 0, k Solve q. (1a) for 0,, substitute that epression into q. (1e) and solve for 0,. j 0, k Leture 5a Slide 7 Redued Set of quations Step 5 Define the utoff wave number k as k k We now have all of the transverse field omponents epressed in terms of the longitudinal omponents. j 0, k j 0, k j k j k Now all that we have to do is determine 0, and 0,. The remaining field omponents an be alulated from just these two terms. Leture 5a Slide 8 14

15 ow Do We Find 0, and 0,? Reall that in LI media, our wave equation simplified to 1 1 k 0 k 0 k 0 k 0 k 0 k 0 k 0 k 0 Substituting our solution into the bottom equations above gives k k 0 0 Leture 5a Slide 9 Solution Categories If 0, = 0, = 0, we obtain a transverse eletromagneti (TM) solution beause all of the field omponents are transverse to the diretion of propagation. Analsis redues to an eletrostatis problem. If 0, = 0 and 0, 0, we obtain a transverse eletri (T) solution beause the eletri field has no longitudinal omponent. If 0, 0 and 0, = 0, we obtain a transverse magneti (TM) solution beause the magneti field has no longitudinal omponent. If 0, 0 and 0, 0, we obtain a hbrid solution. In LI media, 0, and 0, are solved independentl. This means solutions an be obtained in an ombination. This is the origin of TM, T, TM, and hbrid modes. k 0 0 k0, 0, 0, Solution 0 0 TM 0 T 0 TM brid Leture 5a Slide 30 15

16 TM Analsis (1 of 3) For TM waves, we have 0, = 0, = 0. Under this ondition, Mawell s equations redue to 0, j j q. 1a j 0, j0, q. 1b j q. 1 0, j j q. 1d j 0, j 0, q. 1e j q. 1 f j j q. a j j q. b 0 q. j j q. d j j q. e 0 q. f Leture 5a Slide 31 TM Analsis ( of 3) After dropping 0, and 0,, we get j j q. a j j q. b 0 q. j j q. d j j q. e 0 q. f Solve q. (d) for 0,. 0, Substitute 0, into q. (b). j j k We see that for TM k Previousl, we defined k k. If = k, then k = 0 impling that there is no utoff frequen for the TM mode. In summar, for the TM mode we have k and k 0 no utoff Leture 5a Slide 3 16

17 TM Analsis (3 of 3) In LI media, reall that the wave equation was k 0 But for the TM mode, k = 0. k 0 0 The wave equation redues to Laplae s equation from eletrostatis. Leture 5a Slide 33 Alternate Derivation of TM Analsis The TM mode in a transmission line has no utoff frequen (k = 0). This means that it an be analed as 0 and the problem redues to an eletrostatis problem. Derivation Mawell s equations for eletrostatis 0 q. 3a D0 q. 3b D q. 3 V q. 3d For isotropi dieletris V 0 Substitute q. (3) into q. (3b). 0 q. 4 Substitute q. (3d) into q. (4). V 0 For homogeneous dieletris V 0 Leture 5a Slide 34 17

18 T Analsis in LI Media We are free to set 0, = 0 and 0, 0. This means we do not have to solve for 0,. We onl have to obtain a solution for 0,. k 0 k 0 An added benefit of this solution approah is that 0, is tangential to all boundaries in a waveguide. For T analsis, the other field omponents are alulated just from 0,. j j k k j j 0, k k From this, we an alulate the impedane. j 0, k k ZT We must still determine b j solving the wave equation. k Leture 5a Slide 35 TM Analsis in LI Media We are free to set 0, 0 and 0, = 0. This means we do not have to solve for 0,. We onl have to obtain a solution for 0,. k 0 k 0 An added benefit of this solution approah is that 0, is tangential to all boundaries in a waveguide. For TM analsis, the other field omponents are alulated just from 0,. j j k k j j k k From this, we an alulate the impedane. j 0, k ZTM j k k Leture 5a Slide 36 18

19 Setup for Analing brid Modes Leture 5a Slide 37 brid Mode Analsis (1 of 3) To setup a solution to Mawell s equations, we bak up to Mawell s equations in linear and isotropi media (i.e. an be inhomogeneous). 0, j j j j 0, j q. 1a q. 1b q. 1 Solve q. (1) for 0, and solve q. () for 0,. j j j j j q. a q. b q. 1 0, j q. 3a 1 j q. 3b Substitute q. (3a) into qs. (a) and (b), & substitute q. (3b) into qs. (1a) and (1b) q. 4a q. 4b 1 1 0, 1 1 q. 5a q. 5b Leture 5a Slide 38 19

20 brid Mode Analsis ( of 3) We an write our four remaining equations more ompatl as q , q. 7 Full Wave Analsis Solve q. (7) for the magneti field omponents , = q Solve q. (8) into q. (6) to arrive at the final wave equation to be solved , 0 Yikes!! This is tpiall solved numeriall on a omputer. Leture 5a Slide 39 brid Mode Analsis (3 of 3) Quasi LP Analsis Reogniing that the hbrid modes tend to be strongl linearl polaried, we an make a simplifing approimation that the ross oupling between 0, and 0, is weak and an be negleted. Under this ondition, our governing equation separates into two independent equations, one for eah LP mode , Quasi LP Analsis These final two equations simplif even more for homogeneous dieletris and for slab waveguides. The are still tpiall solved numeriall on a omputer. Leture 5a Slide 40 0

21 Setup for Analing Slab Waveguides Leture 5a Slide 41 Geometr and Solution e,, Amplitude Profile 0 j Wave osillations phase onstant Leture 5a Slide 4 1

22 LI Slab Waveguide Analsis (1 of 3) Given this geometr 0 0, j j q. 1a 0, j j0, q. 1b 0, j q. 1 0, j j q. 1d j j0, q. 1e j q. 1 f Leture 5a Slide 43 LI Slab Waveguide Analsis ( of 3) We see that Mawell s equations have deoupled into two sets of equations. j j q. a 0, 0, j0, j0, q. b 0, j0, q. j j q. d 0, j j0, q. e j 0, q. f T Mode (i.e. 0, = 0) TM Mode (i.e. 0, = 0) j j0, q. 3a j j q. 3b j q. 3 0, j j0, q. 3d j j q. 3e j q. 3 f Leture 5a Slide 44

23 LI Slab Waveguide Analsis (3 of 3) We an derive wave equations b substituting the nd and 3 rd equations into the 1 st. T Mode 1 k 0 q. 4a 0, q. 4b 1 j q. 4 TM Mode 1 k 0 q. 5a 0, q. 5b 1 0, j q. 5 Leture 5a Slide 45 Tpial Modes in a Slab Waveguide ffetive refrative indies T Modes n ore =.0 n lad = 1.5 ffetive refrative indies TM Modes n ore =.0 n lad = 1.5 Leture 5a Slide 46 3

24 Remarks About Slab Waveguide Analsis Waves are onfined in onl one transverse diretion. Waves are free to spread out in the uniform transverse diretion Propagation within the slab an be restrited to a single diretion without loss of generalit. Mawell s equations rigorousl deouple into two distint modes. No approimations are neessar Leture 5a Slide 47 4