# Longitudinal-section mode

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'''Longitudinal-section modes''' are a set of a particular kind of [electromagnetic transmission modes](/source/Mode_(electromagnetism)) found in some types of [transmission line](/source/transmission_line).  They are a subset of '''hybrid electromagnetic modes''' (HEM modes).<ref>Zhang & Li, p. 188</ref>  HEM modes are those modes that have both an [electric field](/source/electric_field) and a [magnetic field](/source/magnetic_field) component longitudinally in the direction of travel of the propagating wave.  Longitudinal-section modes, additionally, have a component of either magnetic or electric field that is zero in one transverse direction.  In longitudinal-section electric (LSE) modes this field component is electric.  In longitudinal-section magnetic (LSM) modes the zero field component is magnetic.  Hybrid modes are to be compared to [transverse mode](/source/transverse_mode)s which have, at most, only one component of either electric or magnetic field in the longitudinal direction. 

== Derivation and notation ==
There is an analogy between the way [transverse mode](/source/transverse_mode)s (TE and TM modes) are arrived at and the definition of longitudinal section modes (LSE and LSM modes).  When determining whether a structure can support a particular TE mode, one sets the electric field in the {{mvar|z}} direction (the longitudinal direction of the line) to zero and then solves [Maxwell's equations](/source/Maxwell's_equations) for the [boundary condition](/source/boundary_condition)s set by the physical structure of the line.  One can just as easily set the electric field in the {{mvar|x}} direction to zero and ask what modes that gives rise to.  Such modes are designated LSE<sup>{x}</sup> modes.  Similarly there can be LSE<sup>{y}</sup> modes and, analogously for the magnetic field, LSM<sup>{x}</sup> and LSM<sup>{y}</sup> modes.  When dealing with longitudinal-section modes, the TE and TM modes are sometimes written as LSE<sup>{z}</sup> and LSM<sup>{z}</sup> respectively to produce a consistent set of notations and to reflect the analogous way in which they are defined.<ref name=ZhangLi294/>

Both LSE and LSM modes are a [linear superposition](/source/superposition_principle) of the corresponding TE and TM modes (that is, the modes with the same suffix numbers).  Thus, in general, the LSE and LSM modes have a longitudinal component of both electric and magnetic field.  Likewise the LSM modes are found by setting one of the transverse components of magnetic field to zero with analogous results.<ref name=ZhangLi294>Zhang & Li, pp. 294–299</ref>

== Occurrence ==
LSE and LSM modes can occur in some types of [planar transmission line](/source/planar_transmission_line) with non-homogeneous transmission media.  There are some structures that are unable to support a pure TE or TM mode and consequently the transmission mode must necessarily be hybrid.<ref>Zhang & Li, pp. 332</ref>

== References ==
{{reflist|23em}}

== Bibliography ==
* Zhang, Kequian; Li, Dejie, ''Electromagnetic Theory for Microwaves and Optoelectronics'', Springer, 2013 {{ISBN|3-662-03553-7}}.

Category:Wave mechanics
Category:Electromagnetic radiation
Category:Microwave transmission

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