TE Mode

11 Key excerpts on "TE Mode"

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  Transmission Lines

  Equivalent Circuits, Electromagnetic Theory, and Photons

  • Richard Collier(Author)
  • 2013(Publication Date)
  • Cambridge University Press

    (Publisher)

  This means that of all the modes, the TE 10 is the one mode with the lowest cut-off frequency. In Figure 5.4(a) are shown the fields in a cross-section through the waveguide. The lines show the direction and not the amplitude of the fields. So, for instance, in the TE 10 mode both the electric and the magnetic fields vary sinusoidally with the horizontal direction and with a peak in the centre. Except for this mode, all the other modes can be made up of the shapes seen in the figures repeated in both directions. The cut-off frequencies of the various modes can be represented on a diagram as a ratio of the lowest cut-off frequency, as shown in Figure 5.4(b). This ratio is given by f cTE mn f cTE 10 ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m 2 þ 4n 2 p : ð5:46Þ For the next example, just the TE 10 mode will be described as this is the most commonly used mode and the higher order modes usually are not propagating. If they are produced by some irregularities in the waveguide, then they will rapidly decay and not propagate as they will be below their cut-off frequencies. These modes can store both electric and magnetic energy. Depending on which is larger, they can be represented in an equivalent circuit as either a capacitance or an inductance. However, even though they decay away they can interact if two irregularities or discontinuities are close to each other. In this case, the equivalent circuit model can give misleading results. 163 5.4 TE and TM waves within metallic rectangular boundaries TM 11 TM 21 TM 22 TE 10 (a) TE 11 TE 21 Figure 5.4(a) The fields for the first few modes in rectangular waveguide. The solid lines are the electric fields and the dotted lines the magnetic fields. These figures have been reproduced by permission of the Institution of Engineering and Technology from the Waveguide Handbook by N. Marcuvitz, published by the IET in 1985 as a reprint of the original publication by the McGraw-Hill Book Company in 1951.

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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)

  Since the direction of propagation is z, the solutions of the first group of equations are called the TE, transverse electric, modes. The solutions of the second group of equations are the TM, transverse magnetic, modes. In other words, all planar waveguide modes can be divided into TE and TM types. Since ε is only a function of x, the z variation of the fields must be the same in all layers. This is the consequence of the continuity of E y or H y for all z. For TE Modes, the transverse electric E y in Eq. (5.1) can now be written as a product of a function in y and a function in z, i.e. E y ðx; zÞ ¼ E y ðxÞE y ðzÞ. When all these considerations are taken into account, we obtain: ∂ 2 ∂ x 2 þ  ω 2 με ðxÞ  β 2   ! E y ðxÞ ¼ 0 ð5:2aÞ ∂ 2 ∂z 2 þ β 2  ! E y ðzÞ ¼ 0 ð5:2bÞ Similar equations exist for TM modes. Mathematically, Eq. (5.2) and its equivalent for TM modes are second-order differ- ential equations. All the TE Modes form a complete set of TE eigenfunctions, meaning that any arbitrary electric field polarized in the y direction with ∂=∂y ¼ 0 can be represented as a summation of TE Modes. Similarly, all the TM modes form a complete set of TM eigenfunctions, meaning at any arbitrary electric field polarized in the x direction with ∂=∂y ¼ 0 can be represented as a summation of TM modes. Any radiation field with arbitrary polarization needs to be decomposed first into TE and TM compo- nents, and then analyzed. 5.3 TE Modes of planar waveguides The planar TE Modes (i.e. modes with ∂=∂y ¼ 0) in the planar waveguides are eigen- solutions of the equation ∂ 2 ∂ x 2 þ ∂ 2 ∂ z 2 þ ω 2 με ðxÞ 2 4 3 5 E y ðxÞE y ðzÞ ¼ 0 ε ðxÞ ¼ n c 2 ε o x ≥ t ¼ n 2 2 ε o t > x > 0 ð5:3Þ ¼ n s 2 ε o 0 ≥ x H x ¼  j ωμ ∂ E y ∂ z ; H z ¼ j ωμ ∂ E y ∂ x 5.3 TE Modes of planar waveguides 113 Here, ε o is the free space electric permittivity. The boundary conditions are the continuity of the tangential electric and magnetic fields, E y and H z , at x = 0 and at x = t.

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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)

  For the sake of mathematical simplicity, a particular traveling-wave mode for the waveguide with a rectangular cross section will be studied first. The same general technique employed for the coaxial cable, that of postulating a solution and testing it by substituting in Maxwell's equations, will be used initially. Afterward a more general mathematical approach will be tried. Figure 13-1 shows the geometry of the rectangular waveguide.

  FIGURE 13-1 Dimensions and coordinates for rectangular waveguide.

     

  13-1 TE

  m 0

  Propagating Modes

  The simplest and also the most commonly used mode in a rectangular waveguide has these properties: (1) the electric field is confined to planes perpendicular to the direction of propagation, whereas the magnetic field has both transverse and longitudinal components, and (2) the electric field is directed from one of the guide surfaces straight across to the opposite surface. Actually an infinite number of modes possess these properties in common; they are referred to as TE

  m 0

  modes by which the following is meant: (1) TE designates transverse electric (property 1 above), (2) the second subscript, 0, indicates an absence of variation in field configuration as a function of the shorter transverse direction (y direction in Fig. 13-1 ), and (3) the first subscript, m, which is an integer, designates the particular mode of this class. This third item will be described specifically in Sec. 13-la(2), after the TE

  m 0

  field patterns have been derived.

  a. Electric and Magnetic Fields

  For the ordinarily used approximation of the electric and magnetic fields, the waveguide walls may be assumed to be of infinite conductivity, in which case a traveling wave should propagate without attenuation.

  (1) Traveling-Wave Solution

  In view of the success experienced in finding traveling-wave solutions in unbounded space and in a coaxial cable, such a solution will be postulated here. The phase function β will be regarded as an unknown and may well differ from that for the TEM mode. In accordance with the properties just listed for the TE

  m 0

  mode, the E field will be assumed to consist solely of a y -directed component,

  Ey

  . However, because the component of E tangential to a perfectly conducting surface must vanish (Sec. 11-2b) and, as shown in Fig. 13-1 , such surfaces exist at x = 0 and at

  x —a, Ey

  will be assumed to be a function of x. The requirement that the divergence of D should vanish (Eq. 10-19 , with p = 0) will be met if the E thus chosen is not a function of y

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  Electromagnetic Field Theory Fundamentals

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

    (Publisher)

  Thus, the TE 00 mode cannot exist in a waveguide. However, the field components for the TE 01 and TE 10 modes are as follows. a) TE 01 mode: ˜ H x = 0 , ˜ E y = 0 , ˜ E z = 0 , (10.62a) ˜ H y = j β 01 b π ˆ H zm sin π b y e − j β 01 z (10.62b) ˜ H z = ˆ H zm cos π b y e − j β 01 z (10.62c) ˜ E x = j ωµ b π ˆ H zm sin π b y e − j β 01 z (10.62d) b) TE 10 mode: ˜ H y = 0 , ˜ E x = 0 , ˜ E z = 0 , (10.63a) ˜ H x = j β 10 a π ˆ H zm sin π a x e − j β 10 z (10.63b) ˜ H z = ˆ H zm cos π a x e − j β 10 z (10.63c) ˜ E y = − j ωµ a π ˆ H zm sin π a x e − j β 10 z (10.63d) The presence of ˜ E x and ˜ H y field components for the TE 01 mode indicates that the power flow can occur in the z direction ; thus, this mode can exist in the waveguide. Similarly, the field components ˜ E y and ˜ H x for the TE 10 mode result in the power flow in the z direction, suggesting that this is a viable mode. Thus, for the TE mn mode, either m or n can be zero, but not both. If a > b , then TE 10 is the lowest-order mode and TE 01 is the next lowest. The field patterns and the excitation schemes of TE 10 are illustrated in Figures 10.9 and 10.10. 521 10.4 Transverse electric (TE) mode Figure 10.9 Field patterns of TE 10 mode As in a TM wave, two distinct operating frequency regions exist for a TE Mode separated by the cutoff frequency f cmn = u p 2 m a 2 + n b 2 (10.64) Let us now investigate the various characteristics of TE Modes for the two distinct operating frequency regions. 10.4.1 Operation below cutoff frequency When the operating frequency is less than the cutoff frequency of a TE mn wave, we can write the propagation constant from equation (10.22) as ˆ γ mn = α mn = 1 u p ω 2 cmn − ω 2 (10.65) 522 10 Waveguides and cavity resonators Figure 10.10 Excitation schemes of TE 10 mode which indicates a complete attenuation of the wave inside the waveguide.

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  Principles of Lasers and Optics

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

    (Publisher)

  In Sections 3.2 to 3.4 , we will solve rigorously Maxwell’s equations and obtain all the modes for the asymmetric planar waveguide shown in Fig. 3.1 . We will identify the solutions thus obtained with the three types of waves discussed above. 3.1.1 TE and TM modes in planar waveguides The variation of the refractive index in the transverse direction is independent of z in Fig. 3.1 . From discussions of electromagnetic theory in classical electri-cal engineering textbooks, we know that modes for structures that have constant 76 Guided wave modes and their propagation transverse cross-section in the direction of propagation can often be divided into TE (transverse electric) type and TM (transverse magnetic) type. TE means no electric field component in the direction of propagation. TM means no magnetic field component in the direction of propagation. For the planar waveguide case, if we substitute ∂/∂ y = 0 into the ∇× e and ∇× h Maxwell equations, we obtain two separate groups of equations: ∂ E y ∂ z = µ∂ H x /∂ t , ∂ E y ∂ x = − µ∂ H z /∂ t , ∂ H z ∂ x − ∂ H x ∂ z = − ε∂ E y /∂ t ;                (3.1a) and ∂ H y ∂ z = − ε∂ E x /∂ t , ∂ H y ∂ x = ε∂ E z /∂ t , ∂ E z ∂ x − ∂ E x ∂ z = µ∂ H y /∂ t .                (3.1b) Clearly, E y , H x and H z are related only to each other, and H y , E x and E z are related only to each other. Since the direction of propagation is z , the solutions of the first group of equations ( 3.1a ) are the TE Modes. The solutions of the second group of equations ( 3.1b ) are the TM modes. Thus we have shown the separation of all planar waveguide modes into TE and TM types. Since ε is only a function of x , the z variation of the solution must be the same in all layers. This is a consequence of the requirement for the continuity of E y or H y for all z .

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  Introduction to Electromagnetic Waves with Maxwell's Equations

  • Ozgur Ergul(Author)
  • 2021(Publication Date)
  • Wiley

    (Publisher)

  It should be emphasized that propagating modes propagate with different velocities: i.e. lower-order modes propagate slower than higher-order modes. On the other hand, since most electromagnetic energy is car-ried by lower-order modes, the mode with the smallest mode numbers is called the dominant (fundamental) mode. In a rectangular waveguide, the cutoff frequencies for TE 01 and TE 10 modes can be written f 01 c = 1 2 π √ με π b = 1 2 b √ με (7.307) 478 Introduction to electromagnetic waves with Maxwell’s equations y = 0.5 m y x x z 200 MHz 100 MHz Cutoff z = 0.2 m – 0.5 0.5 – 0.5 0.5 – 0.5 0.5 V/m Figure 7.76 The real part of the z component of the electric field intensity for the TM 11 mode in a 2 × 1 m rectangular waveguide (a vacuum inside) at three different frequencies: i.e. at 200 MHz (propagating), at approximately 168 MHz (cutoff), and at 100 MHz (evanescent). The values are shown with respect to position on the y = 0 . 5 m plane (with respect to z ∈ [0 , 6] m and x ∈ [0 , 2] m) and on the z = 0 . 2 m plane (with respect to x ∈ [0 , 2] m and y ∈ [0 , 1] m). and f 10 c = 1 2 π √ με π a = 1 2 a √ με , (7.308) respectively. Therefore, depending on the values of a and b , either TE 01 or TE 10 is the dominant mode of a rectangular waveguide. 70 70 We can write field expressions for TE 10 as H TE x (¯ r ) = γB 10 a π sin πx a exp( − γ 10 z ) E TE y (¯ r ) = − jωμB 10 a π sin πx a exp( − γ 10 z ) H TE z (¯ r ) = B 10 cos πx a exp( − γ 10 z ) E TE x (¯ r ) = 0 , H TE y (¯ r ) = 0 , ¯ E TE z = 0 . Similarly, for TE 01 , we have E TE x (¯ r ) = jωμB 01 b π sin nπy b exp( − γ 01 z ) H TE y (¯ r ) = γ 01 B 01 b π sin nπy b exp( − γ 01 z ) H TE z (¯ r ) = B 01 cos πy b exp( − γ 01 z ) H TE x (¯ r ) = 0 , E TE y (¯ r ) = 0 , ¯ E TE z = 0 . Similar to the wave impedance in a homogeneous space, one can define a wave impedance inside a waveguide.

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  Microwave Engineering

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

    (Publisher)

  At a given operating frequency f only those modes having f > f c will propagate; modes with f < f c will lead to an imaginary β (or real α), meaning that all field compo- nents will decay exponentially away from the source of excitation. Such modes are referred to as cutoff modes, or evanescent modes. If more than one mode is propagating, the wave- guide is said to be overmoded. From (3.22) the wave impedance that relates the transverse electric and magnetic fields is Z TE = E x H y = −E y H x = k η β , (3.86) where η = √ µ/ is the intrinsic impedance of the material filling the waveguide. Note that Z TE is real whenβ is real (a propagating mode) but is imaginary when β is imaginary (a cutoff mode). The guide wavelength is defined as the distance between two equal-phase planes along the waveguide and is equal to λ g = 2π β > 2π k = λ, (3.87) which is thus greater than λ, the wavelength of a plane wave in the medium filling the guide. The phase velocity is v p = ω β > ω k = 1/ √ µ, (3.88) which is greater than 1/ √ µ , the speed of light (plane wave) in the medium. In the vast majority of waveguide applications the operating frequency and guide dimensions are chosen so that only the dominant TE 10 mode will propagate. Because of the practical importance of the TE 10 mode, we will list the field components and derive the attenuation due to conductor loss for this case. Specializing (3.81) and (3.82) to the m = 1, n = 0 case gives the following results for the TE 10 mode fields: H z = A 10 cos π x a e − jβ z , (3.89a) E y = − j ωµa π A 10 sin π x a e − jβ z , (3.89b) H x = jβ a π A 10 sin π x a e − jβ z , (3.89c) E x = E z = H y = 0. (3.89d) 114 Chapter 3: Transmission Lines and Waveguides The cutoff wave number and propagation constant for the TE 10 mode are, respectively, k c = π/a, (3.90) β =  k 2 − (π/a) 2 .

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  Microwave Engineering

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

    (Publisher)

  (3.127) The first TE Mode to propagate is the mode with the smallest p ′ nm , which from Table 3.3 is seen to be the TE 11 mode. This mode is therefore the dominant circular waveguide mode and the one most frequently used. Because m ≥ 1, there is no TE 10 mode, but there is a TE 01 mode. The transverse field components are, from (3.110) and (3.121), E  = −jn k 2 c  (A cos n − B sin n)J n (k c )e −jz , (3.128a) E  = j k c (A sin n + B cos n)J ′ n (k c )e −jz , (3.128b) H  = −j k c (A sin n + B cos n)J ′ n (k c )e −jz , (3.128c) H  = −j n k 2 c  (A cos n − B sin n)J n (k c )e −jz . (3.128d) TABLE 3.3 | Values of p ′ nm for TE Modes of a Circular Waveguide n p ′ n1 p ′ n2 p ′ n3 0 3.832 7.016 10.174 1 1.841 5.331 8.536 2 3.054 6.706 9.970 3.4 Circular Waveguide 115 The wave impedance is Z TE = E  H  = −E  H  = k  . (3.129) In the above solutions there are two remaining arbitrary amplitude constants, A and B. These constants control the amplitude of the sin n and cos n terms, which are independent. That is, because of the az- imuthal symmetry of the circular waveguide, both the sin n and cos n terms represent valid solutions, and both may be present in a specific problem. The actual amplitudes of these terms will depend on the excitation of the waveguide. From a different viewpoint, the coordinate system can be rotated about the z-axis to obtain an h z with either A = 0 or B = 0. Now consider the dominant TE 11 mode with an excitation such that B = 0. The fields can be written as H z = A sin J 1 (k c )e −jz , (3.130a) E  = −j k 2 c  A cos J 1 (k c )e −jz , (3.130b) E  = j k c A sin J ′ 1 (k c )e −jz , (3.130c) H  = −j k c A sin J ′ 1 (k c )e −jz , (3.130d) H  = −j k 2 c  A cos J 1 (k c )e −jz , (3.130e) E z = 0.

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  An Introduction to the Theory of Microwave Circuits

  • K. Kurokawa(Author)
  • 1969(Publication Date)
  • Academic Press

    (Publisher)

  If some of the y„'s are equal to zero, the coefficients of the corresponding terms must be replaced by either (3.123) or (3.124) depending on the particular group concerned. In the previous section, it was shown that electromagnetic fields in a wave-guide could be divided into three groups, TEM, TE, and TM modes, assuming the exponential field variation with z. However, because of this assumption, it was not clear whether or not all possible electromagnetic fields in a waveguide could be expressed as a linear combination of those modes. In this section, the exponential field variation was not assumed (indeed, nonexponential field variation appeared in the particular case of y n = 0), and we have demonstrated that every possible electromagnetic field in a waveguide is given by the above expressions and that no other functional forms need be considered. This is a far stronger assertion than we could have made before; however, without this guarantee, the discussions in Sections 3.7 and 3.8 would have little meaning. Suppose the transverse components of both E and H are given on a refer-ence plane perpendicular to the axis of the waveguide. Let the position of this reference plane be at z = 0. Then, from (3.125) and (3.126), we have E,(0) = ( + B n ) , H f (0) = k x E in Z 0 -l (A n - B n ) which give simultaneous equations for A n and B n , j E,(0).E r „ dS = A n + B n , j H,(0).k x E,„ dS = Z^{A n - B„) 5.5. General Theory of Waveguides 137 These equations determine A n and B n for each n and, hence, the electro-magnetic field in the waveguide uniquely. In other words, the transverse components of E and H at a reference plane are sufficient to specify the electromagnetic field in the waveguide. This is true even when some of the y„'s are equal to zero.

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  Fundamentals of Electromagnetics with Engineering Applications

  • Stuart M. Wentworth(Author)
  • 2017(Publication Date)
  • Wiley

    (Publisher)

  (7.70) (7.71) As with the TM case, we see that if both m and n are zero then all the fields disappear. But we do have fields if only one of m or n is zero. For instance, for the TE 10 mode, since n = 0 there will be no E xs or H ys . The instantaneous expressions for the rest of the fields for the TE 10 mode are as follows: (7.72) (7.73) (7.74) Normalized values of these fields in a cross section of guide are shown in Figure 7.13. H x y z t a H x a t z x o , , , sin sin ( ) = − ⎛ ⎝ ⎞ ⎠ − ( ) β β β π π ω β u 2 2 E x y z t a H x a t z y o , , , sin sin ( ) = − ⎛ ⎝ ⎞ ⎠ − ( ) ωμ β β π π ω β u 2 2 H x y z t H x a t z z o , , , cos cos ( ) = ⎛ ⎝ ⎞ ⎠ − ( ) π ω β H j n b H m x a n y b e y o j z s u = − ⎛ ⎝ ⎞ ⎠ ⎛ ⎝ ⎞ ⎠ − β β β π π π β 2 2 cos sin E j n b H m x a n y b e x o j z s u = − ⎛ ⎝ ⎞ ⎠ ⎛ ⎝ ⎞ ⎠ − ωμ β β π π π β 2 2 cos sin 360  Chapter 7. Waveguide Figure 7.13 TE 10 field plots are constant in the y direction. 0 –10 –5 10 5 20*Hz, (at y = b/2) –10 –20 –15 0 –5 10*Ey, (at y = b/2) 20*Hx, (at x = a/2) x 20 10 0 30 40 x 20 10 0 30 40 0 10 20 15 5 x 20 10 0 30 40  7.3 DIELECTRIC WAVEGUIDE Metallic rectangular waveguide finds practical application at microwave frequencies typi- cally between 1 and 40 GHz. However, as we have seen, the dimensions of rectangular waveguide operating in the TE 10 mode must be on the same order as the wavelength, and at higher frequencies it becomes impractical to fabricate such small guide. For instance, at an optical frequency of 300 THz, the wavelength is only 1 μm. Making a metallic waveguide at such dimensions would be a serious challenge even for MEMS technology. It is unlikely that such waveguide would be desirable even if it could be built, because ohmic losses in the diminishing conductive skin of the walls would be excessive. Dielectric waveguide overcomes these problems. Before turning our attention to fiber optic waveguide, we will first consider the simpler case of a rectangular slab of waveguide as shown in Figure 7.1c.

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  Electromagnetic Waves

  Propagation in Complex Matter

  • Ahmed Kishk(Author)
  • 2011(Publication Date)
  • IntechOpen

    (Publisher)

  We stress again the use of the coefficient of potential is essential for the formulation of the three-line system. Hence, the ordinary TEM mode propagation of the EM wave is achieved only when the symmetrization is introduced for the electric circuit in the circumstance. We shall calculate the electromagnetic power of the three-conductor transmission-line system. P ( x ) = 1 4 ( V ∗ 1 I 1 + V ∗ 2 I 2 + V ∗ 3 I 3 + V 1 I ∗ 1 + V 2 I ∗ 2 + V 3 I ∗ 3 ) (76) = 1 4 ( V ∗ n I n + V ∗ c I c + V ∗ a I a + V n I ∗ n + V c I ∗ c + V a I ∗ a ) It is interesting to calculate the change of the power with distance so that we can pick up only the terms which change with distance. We take the time dependence of all the modes as exp ( − j ω t ) . We can work out the change rate dP ( x ) dx on the basis of Eqs. (65) and (69) in exactly the same way as the case of the one-conductor transmission line. We write only the final result. dP ( x ) dx = − 1 2 ( R n | I n ( x ) | 2 + R c | I c ( x ) | 2 + R a | I a ( x ) | 2 (77) + R nc ( I n ( x ) I ∗ c ( x ) + I ∗ n ( x ) I c ( x )) + R na ( I n ( x ) I ∗ a ( x ) + I ∗ n ( x ) I a ( x )) + R ca ( I c ( x ) I ∗ a ( x ) + I ∗ c ( x ) I a ( x ))) − 1 2 M ω c ( I I ∗ t ( l , x ) I a ( x ) + I I , x t ( l ) I ∗ a ( x )) − j 1 2 Mc parenleftbigg Q I ∗ t ( l , x ) dI a ( x ) dx − Q I t ( l . x ) dI ∗ a ( x ) dx parenrightbigg This expression agrees with the one of the line-antenna (62), when the change rate is expressed with the total current by using the relation I a ( x ) = 1 2 I t ( x ) . It is interesting to point out that the change of the electric power is made by the resistance terms and the antenna mode terms. 250 Electromagnetic Waves Propagation in Complex Matter Electrodynamics of Multiconductor Transmission-line Theory with Antenna Mode 19 There are two effects in the antenna mode terms. One is a term associated with emission and the other is a term associated with absorption.

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