TM Mode

11 Key excerpts on "TM Mode"

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  Analysis Methods for RF, Microwave, and Millimeter-Wave Planar Transmission Line Structures

  • Cam Nguyen(Author)
  • 2003(Publication Date)
  • Wiley-Interscience

    (Publisher)

  That is, the longitudinal components E z = 0 and H z 6D 0. Transverse Magnetic (TM) Wave or Mode: This wave has only the magnetic field in the transverse plane. That is, H z = 0 and E z 6D 0. Hybrid Wave or Mode: This wave is characterized as having both E z 6D 0 and H z 6D 0 and, therefore, is a combination of both TE and TM waves. Transverse Electromagnetic (TEM) Wave or Mode ‡ : The electric and magnetic fields of this wave have only transverse components. Both E z and H z are equal to zero. Note that, in general, different modes have different cutoff frequencies or cutoff wave numbers and hence different propagation constants. A rectangular waveguide is a special waveguide that has the same cutoff frequency for the corresponding TE mn and TM mn modes. When the modes have the same cutoff frequency, they are classified as degenerate modes. Otherwise, they are said to be nondegenerate modes. 2.11.2 Wave Solutions Assuming y h ( x,y,z) and y e ( x,y,z) have only longitudinal z components such as y h D a z h x,ye š h z 2.48a y e D a z e x,ye š e z 2.48b we can prove easily that the corresponding longitudinal electric and magnetic fields are equal to zero. This implies that the magnetic, h ( x,y) , and electric, e ( x,y) , scalar potentials may be used to determine the fields for TE and TM Modes, respectively. This result will be used to derive fields for the four principal classes of modes discussed earlier. Using the principle of superposition, we can express the fields in any wave- guide as a summation of those of TE and TM Modes. These fields are given, making use of Eqs. (2.42) and (2.44), as E D E TM C E TE D k 2 y e C W W Ð y e jω W ð y h 2.49a ‡ For TEM mode to exist exactly on a transmission line, all conductors must be perfect. WAVE TYPES AND SOLUTIONS 25 H D H TM C H TE D jωεW ð y e C k 2 y h C W W Ð y h 2.49b where the subscripts TE and TM indicate the TE and TM Modes, respectively.

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  Electromagnetic Field Theory Fundamentals

  • Bhag Singh Guru, Hüseyin R. Hiziroglu(Authors)
  • 2009(Publication Date)
  • Cambridge University Press

    (Publisher)

  In the following sections, instead of continuing with general solu-tions, we will tailor our solutions for the two types of modes we have mentioned earlier: the transverse magnetic (TM) mode and the trans-verse electric (TE) mode. 10.3 Transverse magnetic (TM) mode ................................. In the TM Mode, the magnetic field has components only in the trans-verse ( xy ) plane. In other words, the longitudinal ( z -direction) com-ponent of the magnetic field ˜ H z is zero, and ˜ E x , ˜ E y , and ˜ E z can all exist. The general solution of ˜ E z as given in (10.11a) must satisfy the bound-ary conditions at each of the four perfectly conducting walls of the waveguide. Note that ˜ E z represents a tangential field at each boundary. The tangential components of the E field must be continuous, and the E field inside a perfectly conducting wall is zero ; therefore, ˜ E z must be 509 10.3 Transverse magnetic (TM) mode zero at each boundary. Thus, the four boundary conditions are At x = 0 ˜ E z (0 , y , z ) = 0 (10.12a) At y = 0 ˜ E z ( x , 0 , z ) = 0 (10.12b) At x = a ˜ E z ( a , y , z ) = 0 (10.12c) At y = b ˜ E z ( x , b , z ) = 0 (10.12d) Applying the boundary conditions at x = 0 and y = 0, equation (10.11a), we obtain ˆ C 2 = 0 and ˆ C 4 = 0. Therefore, ˜ E z ( x , y , z ) = ˆ C 1 ˆ C 3 sin( Mx ) sin( Ny ) e − ˆ γ z or ˜ E z ( x , y , z ) = ˆ E zm sin( Mx ) sin( Ny ) e − ˆ γ z (10.13) where ˆ E zm = ˆ C 1 ˆ C 3 = E zm / φ , and E zm is the amplitude of ˆ E zm . When we apply the boundary conditions at x = a , we obtain sin( Ma ) = 0 or M = m π a for m = 0 , 1 , 2 , . . . (10.14) Finally, the application of boundary conditions at y = b yields sin( Nb ) = 0 or N = n π b for n = 0 , 1 , 2 , .

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  Principles of Optics for Engineers

  Diffraction and Modal Analysis

  • William S. C. Chang(Author)
  • 2015(Publication Date)
  • Cambridge University Press

    (Publisher)

  When there are scattering centers or absorption losses in the substrate or cladding, the evanescent decay is slower, and the attenuation rate larger. For this reason, higher-order guided-wave modes often have larger attenuation rate. Thus TM Modes may have higher scattering loss. The distinction between TE and TM Modes is very important in applications. TE modes are excited by input radiation that has an electric field polarized in the y direction. TM Modes are excited by input radiation that has an electric field polarized in the x direction. Most waveguide structures support both types of modes. The perfor- mance of the devices depends which modes have been excited. In ideal straight waveguides, the electric fields of TE and TM Modes do not interact with each other except at discontinuities or defects in the waveguide. At each defect or discontinuity, we need the sum of all TE and TM Modes to satisfy the boundary condition. Thus TM as well as TE modes may be excited by the incident mode (or modes) at any defect or discontinuity. The scattered modes constitute the scattering loss. 5.5 Guided waves in planar waveguides There are various applications, such as the Star coupler, acousto-optical scanner and RF spectral analyzer [4], that use planar waveguides. It is important to learn about the properties of generalized planar guided waves, how they focus or collimate and how to excite them in various configurations. 5.5.1 The orthogonality of modes The orthogonality condition is important to analyze how modes function and interact in a waveguide. 126 Optical waveguides and fibers

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  Principles of Lasers and Optics

  • William S. C. Chang(Author)
  • 2005(Publication Date)
  • Cambridge University Press

    (Publisher)

  Similarly to TE guided wave modes, TM planar guided wave modes inside the film with index n 1 can also be described by a plane wave that has a magnetic field polarized in the y direction. It is totally internally reflected back and forth from the two boundaries, in a propagation direction in the xz plane making an angle θ n with 88 Guided wave modes and their propagation 3.27 3.26 3.25 3.24 3.23 3.22 3.21 3.20 3.19 3.18 0 2 p 4 p 6 p 8 p kt b n / k n = 0 n = 1 n = 2 n = 3 n = 4 Figure 3.4. Propagation wave number of TM n modes in epitaxially grown GaAs waveguides. Solid line, n = 0.10; , n = 0.08; ◦ , n = 0.06; × , n = 0.04; • , n = 0.02. The copyright figure is taken from ref. [ 1 ] with permission from Elsevier. respect to the x axis. The n th TM guided wave modes can be found by requiring the round trip phase shift to be 2 n π . The TM planar guided wave modes are orthogonal to each other and to the TE modes. When TM Modes are normalized, 1 2 Re   +∞ −∞ H yn E ∗ xm dx   = β n 2 ω +∞ −∞ H n H ∗ m 1 ε ( x ) dx = δ nm (3.16) and B 2 n = 4 ωε 0 β n      n 2 1 n 2 3 p n      p 2 n + h 2 n h 2 n + n 2 1 n 2 3 2 p 2 n      + n 2 1 n 2 2 q n      q 2 n + h 2 n h 2 n + n 2 1 n 2 3 q 2 n      + t      − 1 . (3.17) 3.3 TM planar waveguide modes 89 3.3.5 TM planar substrate modes For the substrate TM Modes, the y component of the magnetic field is H (s) ( x , z ; β ) = B (s) sin( ht + φ ) e − p ( x − t ) e − j β z , x ≥ t , H (s) ( x , z ; β ) = B (s) sin( hx + φ ) e − j β z , t > x > 0 , H (s) ( x , z ; β ) = D (s) e − j ρ x + D (s) ∗ e + j ρ x e − j β z , 0 ≥ x ,      (3.18) D (s) = B (s) 2 sin φ + j n 2 2 h cos φ n 2 1 ρ , tan[( h / k ) kt + φ ] = − n 2 3 h n 2 1 p . D and B are obtained from the orthogonalization and normalization conditions, β 2 ω +∞ −∞ H (s) ( β ) H (s) ∗ ( β ) /ε ( x ) dx = δ ( ρ − ρ ) , (3.19) D (s) D (s) ∗ = ωε 0 n 2 2 βπ .

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  Transmission Lines and Wave Propagation

  • Philip C. Magnusson, Andreas Weisshaar, Vijai K. Tripathi, Gerald C. Alexander(Authors)
  • 2017(Publication Date)
  • CRC Press

    (Publisher)

  10 mode.

  In usual applications of rectangular waveguides, transmission in the TE10 mode alone is sought, and the guide dimensions are chosen with respect to frequency so as to ensure this. Hence transverse-magnetic modes will exist only in evanescent form in the vicinity of guide discontinuities.

  FIGURE 13-4 TMn propagating mode in rectangular waveguide.

  (2) TE Modes

  For general analysis of TE modes one substitutes Eqs. 13-74 and 13-75 in Eq. 13-73 , and that in turn in Eq. 13-63 , so that the E-field components parallel to the guide surfaces may be found:

  The boundary condition that the tangential components of E vanish along the guide surfaces fixes most of these constants:

  From Eq. 13-76 ,

  This is the same as the equation for the phase function of TM Modes, Eq. 13-79 ; Eq. 13-82 for cutoff frequencies of TM Modes is also applicable here for TE modes.

  Equations 13-79 and 13-84 may be restated in shortened form:

  The results just found may be combined with Eqs. 13-70 ,13-83,13-73,13-62, and 13-63 to yield expressions for the field components.

  Let

  It should be noted that either m or n (but not both) may be set equal to zero for TE modes and still yield nontrivial results, but that both m and n must be nonzero for TM Modes. Setting n equal to zero reduces the TE solution just given to the form developed in Eqs. 13-8 and 13-9 . TE0n-mode patterns resemble the TEw0 ones, except that they are turned 90° in space (in terms of Fig. 13-1 ) such that the E-field variation takes place in the shorter transverse direction.

  Except for a square waveguide, the TE10 mode has a lower cutoff frequency than the TE01 mode; in any case it has a lower cutoff frequency than any other TE mode, and, as noted in Sec. 13-4d(2), than any TM Mode. Hence it is properly called the dominant mode.

  e. Waveguide Characteristic Impedances

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  Plasma Dynamics for Aerospace Engineering

  • Joseph J. S. Shang, Sergey T. Surzhikov(Authors)
  • 2018(Publication Date)
  • Cambridge University Press

    (Publisher)

  On the other hand, the transverse electric (TE) waves for the electric field intensity E vanish. The transverse waves are designated as TE m,n or TM m,n to be the transverse electric and magnetic waves of order mn, and the lowest cut-off frequency will occur for m and n equal to unity or zero. In fact, the lowest TE wave frequency in a rectangular guide is the TE 1,0 wave, which has the lowest cut-off frequency and is called the dominant wave. The higher-order waves (larger values of m and n) are generated at higher frequencies in order to be propagated within a waveguide of given physical dimensions. Figure 2.1 presents the linearly polarized electromagnetic waves in free space and propagating through a waveguide in transverse electric mode TE 11 . The electric field strength E and magnetic field strength H are traveling in phase but perpendicular to each other. To demonstrate this peculiar wave structure in a waveguide, the computational simulated magnetic field is projected on the top sur- face of a rectangular cross-section waveguide and the electric field is displayed on the bottom and side surfaces of the waveguide. The numerical results are projected onto surfaces perpendicular to each other, so the distinct polarized structures can be clearly discerned. The polarized and mutually perpendicular electric and magnetic components of a traveling wave are the unique feature of the transverse electromagnetic waves. The analytic solution for a lossless dielectric medium in a rectangular wave- guide is well known, and the electromagnetic wave can propagate unimpeded (Stratton 1953). As has been discussed previously, there is a low frequency limit for which a linear polarized transverse magnetic and electric wave can be trans- mitted through a waveguide with a given cross-section dimension. An electromag- netic wave also rapidly attenuates in a conducting medium in the collision process.

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  Microwave Engineering

  • David M. Pozar(Author)
  • 2012(Publication Date)
  • Wiley

    (Publisher)

  As with the TE modes, m ≥ 1, so there is no TM 10 mode. From (3.110), the transverse fields can be derived as E ρ = − jβ k c ( A sin nφ + B cos nφ) J  n (k c ρ)e − jβ z , (3.141a) E φ = − jβ n k 2 c ρ ( A cos nφ − B sin nφ) J n (k c ρ)e − jβ z , (3.141b) H ρ = j ω n k 2 c ρ ( A cos nφ − B sin nφ) J n (k c ρ)e − jβ z , (3.141c) H φ = − j ω k c ( A sin nφ + B cos nφ) J  n (k c ρ)e − jβ z . (3.141d) The wave impedance is Z TM = E ρ H φ = −E φ H ρ = ηβ k . (3.142) Calculation of the attenuation for TM Modes is left as a problem. Figure 3.12 shows the attenuation due to conductor loss versus frequency for various modes of a circular wave- guide. Observe that the attenuation of the TE 01 mode decreases to a very small value with 3.4 Circular Waveguide 127 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 1 3 5 7 9 11 13 15 TE 11 cutoff TM 01 cutoff TE 01 cutoff Frequency (GHz)  c (dB/m) TM 01 TE 01 TE 11 FIGURE 3.12 Attenuation of various modes in a circular copper waveguide with a = 2.54 cm. f c TE 11 TE 21 TM 01 TM 11 TM 21 TM 02 TE 01 TE 31 TE 41 TE 12 2 3 1 0 f c(TE 11 ) FIGURE 3.13 Cutoff frequencies of the first few TE and TM Modes of a circular waveguide relative to the cutoff frequency of the dominant TE 11 mode. increasing frequency. This property makes the TE 01 mode of interest for low-loss trans- mission over long distances. Unfortunately, this mode is not the dominant mode of the circular waveguide, so in practice power can be lost from the TE 01 mode to lower order propagating modes. Figure 3.13 shows the relative cutoff frequencies of the TE and TM Modes, and Table 3.5 summarizes results for wave propagation in circular waveguide. Field lines for some of the lowest order TE and TM Modes are shown in Figure 3.14. EXAMPLE 3.2 CHARACTERISTICS OF A CIRCULAR WAVEGUIDE Find the cutoff frequencies of the first two propagating modes of a Teflon-filled circular waveguide with a = 0.5 cm.

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  Microwave Engineering

  • David M. Pozar(Author)
  • 2021(Publication Date)
  • Wiley

    (Publisher)

  and is real for propagating modes and imaginary for cutoff modes. The cutoff frequencies for the TM mn modes are also the same as those of the TE mn modes, as given in (3.84). The guide wavelength and phase velocity for TM Modes are also the same as those for TE modes. Observe that the field expressions for  E and  H in (3.101) are identically zero if either m or n is zero. Thus there is no TM 00 , TM 01 , or TM 10 mode, and the lowest order TM Mode to propagate (lowest f c ) is the TM 11 mode, having a cutoff frequency of f c 11 = 1 2 √  √ (  a ) 2 + (  b ) 2 , (3.103) which is seen to be larger than f c 10 , the cutoff frequency of the TE 10 mode. The wave impedance relating the transverse electric and magnetic fields for TM Modes is, from (3.26), Z TM = E x H y = −E y H x =  k . (3.104) Attenuation due to dielectric loss is computed in the same way as for TE modes, with the same result. The calculation of attenuation due to conductor loss is left as a problem. Figure 3.8 shows attenuation versus frequency for some TE and TM Modes in a rectangular waveguide. Table 3.2 summarizes results for TE and TM wave propagation in rectangular waveguides, and Figure 3.9 shows the field lines for several of the lowest order TE and TM Modes. EXAMPLE 3.2 CHARACTERISTICS OF A RECTANGULAR WAVEGUIDE Consider a length of Teflon-filled, copper K-band rectangular waveguide having dimensions a = 1.07 cm and b = 0.43 cm. Find the cutoff frequencies of the first five propagating modes. If the operating frequency is 15 GHz, find the attenuation due to dielectric and conductor losses. Solution From Appendix H, for Teflon,  r = 2.08 and tan  = 0.0004. From (3.84) the cutoff frequencies are given by f c mn = c 2 √  r √ ( m a ) 2 + ( n b ) 2 .

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  Nonlinear Surface Electromagnetic Phenomena

  • H.-E. Ponath, G.I. Stegeman(Authors)
  • 2012(Publication Date)
  • North Holland

    (Publisher)

  These include electronic distortion, the molecular Kerr effect and électrostriction (thermal). All of these are accounted for in the theory given below. Early work (Alanakyan 1967) that used a perturbation theory to investigate nonlinear TM waves in materials of lower symmetry has not yet been extended to the strongly nonlinear regime. On a more general level, a study of nonlinear T M wave propagation is important because it has long been realised that the polarisation of an electromagnetic wave plays a non-trivial role in the self-focussing of light beams (Pohl 1970). For example, stationary nonlinear T M waves in bulk nonlinear media have been analysed in some detail and it has been shown that solitons can be formed with a solitary TM waveguide channel that is wider than the corresponding channel for TE waves. In addition to this kind of polarisation dependence, many systems exist that, in the linear (zero power) limit, only support TM surface-guided waves. These guides have always attracted attention and involve waveguide structures in which there exist one or more components that have negative dielectric functions such as metal films or substrates. These, as hinted at above, also possess a strong frequency dependent dielectric function and behave rather differently in different parts of the electromagnetic spectrum. 3.2.1. The field equations Suppose that the T M nonlinear electromagnetic waves, propagate, in the xz-plane along the x-axis in an isotropic medium with wave number k x and angular 162 A.D. Boardman et al. frequency ω, and have the form B= [0, B y (z), 0]e i ( f c --w i ) , (3.116) where the field components E x (z) and E z (z) are related to B y (z) through the equation dz -ik x E z = ioeB y . (3.117) For notational convenience the dependence of E x , E z and B y upon ζ is suppressed in eq. (3.117) and in all subsequent expressions. A further improve-ment in the presentation results from a conversion to dimensionless coordinates.

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  Wave Propagation

  Theories and Applications

  • Yi Zheng(Author)
  • 2013(Publication Date)
  • IntechOpen

    (Publisher)

  The main limitation of these methods is the high computation time when electrical-large structures are considered. With the increasing performance of computers, research in electromagnetic modeling through numerical methods has been continuously advancing. New techniques have merged and the capability of the existing methods has been improved. A modal approach based in the Transmission Line Matrix 262 Wave Propagation Theories and Applications Electromagnetic Wave Propagation Modeling for Finding Antenna Specifications and Positions in Tunnels of Arbitrary Cross-Section 3 (TLM) method, is considered to analyze the wave propagation in tunnels. A reduced TLM formulation, is employed to find the mode field distribution of the uniform guiding structure. Adequate boundary conditions are additionally used to to limit the computation volume. The modeling approach can be applied for a tunnel with arbitrary cross section, including lossy walls. 2.1. Brief overview The modal approach is a rigorous technique, allowing us to have an appropriate physical insight of the propagation in tunnel environments. It allows us to express the fields in terms of modes and generalize the applicability of our algorithm for tunnels with arbitrary cross section. This theory is based on functional analysis. The total fields in a guiding structure, traveling in the ± z direction, can be expanded by the sum of a set of linearly-independent functions Ψ . Their mutual orthogonality is not necessarily required. The space spanned by these functions is given by: vector E guided ( vector r ) = ∑ N n = 1 w n ( vector r ) Ψ vector E ( vector r ) = ∑ N n = 1 w n ( vector r ) ˆ E n ( x , y ) e − ( α n ± j β n ) z vector H guided ( vector r ) = ∑ N n = 1 w n ( vector r ) Ψ vector H ( vector r ) = ∑ N n = 1 w n ( vector r ) ˆ H n ( x , y ) e − ( α n ± j β n ) z , (1) where vector E guided and vector H guided are the total guided-electric and magnetic fields, respectively, N is the total number of modes.

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  Advanced Electromagnetic Waves

  • Saad Osman Bashir(Author)
  • 2015(Publication Date)
  • IntechOpen

    (Publisher)

  In the case of a good conductor, an approximation by a real number for Γ is presented in [28]. However, Γ is in general complex and would alter the shape of the excitation pulses, which cannot be accounted for in the TLM method [7]. 3.2. Application to confined environments modeled by arbitrary cross-section oversized waveguides: 2.5D TLM approach The analysis of electrically large structures like tunnels involves a high computation time by using full-wave methods like TLM. However, some astutenesses can be considered to efficiently model them, which translates in finding an approach to simplify the analysis of oversized waveguides. From the geometrical point of view, tunnels can have any arbitrary shape, and this constraint has to be considered. From the electromagnetic point of view, waves propagating in tunnels can be classified in modes. This problem is well known in electromagnetic theory. As we shall see, a full-wave model based on a modal decomposition of the waves for the analysis of the radio propagation in tunnels at high frequencies is presented. Besides, the concept is used to further reduce the complexity of this electromagnetic problem. 3.2.1. The TLM modal approach Modal approaches in frequency domain are based on the expansion of fields in terms of modes. In general, total fields can be represented as the sum of modes as shown in (5): Advanced Electromagnetic Waves 244 F ( x , y , z ) = N ∑ i = 1 A i ( x , z ) e -( α i + j β i ) y , (5) where N is the total number of modes, A i ( x , z ) the amplitude of the i -th mode at the coordinate ( x , z ) in the plane perpendicular to the propagation direction y , α i its attenuation, and β i its phase constant. The mode amplitude A i ( x , z ) gives the electric or magnetic field distribution of the mode. The phase constant β represents the change in phase along the path traveled by the wave. Lastly, the attenuation constant is a measure of losses in the structure.

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