Editing Cutoff frequency (section)

==Waveguides==
The cutoff frequency of an [[waveguide (electromagnetism)|electromagnetic waveguide]] is the lowest frequency for which a mode will propagate in it. In [[fiber optics]], it is more common to consider the '''cutoff wavelength''', the maximum [[wavelength]] that will propagate in an [[optical fiber]] or [[Waveguide (optics)|waveguide]]. The cutoff frequency is found with the [[Characteristic equation (calculus)|characteristic equation]] of the [[Helmholtz equation]] for electromagnetic waves, which is derived from the [[electromagnetic wave equation]] by setting the longitudinal [[wave number]] equal to zero and solving for the frequency.  Thus, any exciting frequency lower than the cutoff frequency will attenuate, rather than propagate.  The following derivation assumes lossless walls.  The value of c, the [[speed of light]], should be taken to be the [[group velocity]] of light in whatever material fills the waveguide.

For a rectangular waveguide, the cutoff frequency is
<math display="block"> \omega_{c} = c \sqrt{\left(\frac{m \pi}{a}\right)^2 + \left(\frac{n \pi}{b}\right) ^2}, </math>
where <math>m,n \ge 0</math> are the mode numbers for the rectangle's sides of length <math>a</math> and <math>b</math> respectively. For TE modes, <math> m,n \ge 0</math> (but <math> m = n = 0</math> is not allowed), while for TM modes <math> m,n \ge 1 </math>.

The cutoff frequency of the TM<sub>01</sub> mode (next higher from dominant mode TE<sub>11</sub>) in a waveguide of circular cross-section (the transverse-magnetic mode with no angular dependence and lowest radial dependence) is given by 
<math display="block"> \omega_{c} = c \frac{\chi_{01}}{r} = c \frac{2.4048}{r},</math>
where <math>r</math> is the radius of the waveguide, and  <math>\chi_{01}</math> is the first root of <math>J_{0}(r)</math>, the [[Bessel function]] of the first kind of order 1.

The dominant mode TE<sub>11</sub> cutoff frequency is given by<ref>{{Cite book |last=Hunter |first=I. C. |title=Theory and design of microwave filters |date=2001 |publisher=Institution of Electrical Engineers |others=Institution of Electrical Engineers |isbn=978-0-86341-253-0 |location=London |url=https://www.worldcat.org/oclc/505848355 |oclc=505848355 |pages=214}}</ref>
<math display="block"> \omega_{c} = c \frac{\chi_{11}}{r} = c \frac{1.8412}{r}</math>

However, the dominant mode cutoff frequency can be reduced by the introduction of baffle inside the circular cross-section waveguide.<ref>{{Cite journal |last1=Modi |first1=Anuj Y. |last2=Balanis |first2=Constantine A. |date=2016-03-01 |title=PEC-PMC Baffle Inside Circular Cross Section Waveguide for Reduction of Cut-Off Frequency |url=https://ieeexplore.ieee.org/document/7422717 |journal=IEEE Microwave and Wireless Components Letters |volume=26 |issue=3 |pages=171–173 |doi=10.1109/LMWC.2016.2524529 |s2cid=9594124 |issn=1531-1309|url-access=subscription }}</ref> For a [[single-mode optical fiber]], the cutoff wavelength is the wavelength at which the [[normalized frequency (fiber optics)|normalized frequency]] is approximately equal to 2.405.

===Mathematical analysis===
The starting point is the wave equation (which is derived from the [[Maxwell equations]]),
<math display="block">
  \left(\nabla^2-\frac{1}{c^2}\frac{\partial^2}{\partial{t}^2}\right)\psi(\mathbf{r},t)=0,
</math>
which becomes a [[Helmholtz equation]] by considering only functions of the form 
<math display="block">
  \psi(x,y,z,t) = \psi(x,y,z)e^{i \omega t}.
</math>
Substituting and evaluating the time derivative gives
<math display="block">
  \left(\nabla^2 + \frac{\omega^2}{c^2}\right) \psi(x,y,z) = 0.
</math>
The function <math> \psi </math> here refers to whichever field (the electric field or the magnetic field) has no vector component in the longitudinal direction - the "transverse" field.  It is a property of all the eigenmodes of the electromagnetic waveguide that at least one of the two fields is transverse. The ''z'' axis is defined to be along the axis of the waveguide.

The "longitudinal" derivative in the [[Laplacian]] can further be reduced by considering only functions of the form 
<math display="block">
  \psi(x,y,z,t) = \psi(x,y)e^{i \left(\omega t - k_{z} z \right)},
</math>
where <math>k_z</math> is the longitudinal [[wavenumber]], resulting in
<math display="block">
  \left(\nabla_{T}^2 - k_{z}^2 + \frac{\omega^2}{c^2}\right) \psi(x,y,z) = 0,
</math>
where subscript T indicates a 2-dimensional transverse Laplacian.  The final step depends on the geometry of the waveguide.  The easiest geometry to solve is the rectangular waveguide.  In that case, the remainder of the Laplacian can be evaluated to its characteristic equation by considering solutions of the form 
<math display="block">
  \psi(x,y,z,t) = \psi_{0}e^{i \left(\omega t - k_{z} z - k_{x} x - k_{y} y\right)}.
</math>
Thus for the rectangular guide the Laplacian is evaluated, and we arrive at
<math display="block">
  \frac{\omega^2}{c^2} = k_x^2 + k_y^2 + k_z^2
</math>
The transverse wavenumbers can be specified from the standing wave boundary conditions for a rectangular geometry cross-section with dimensions {{mvar|a}} and {{mvar|b}}:
<math display="block"> k_{x} = \frac{n \pi}{a},</math>
<math display="block"> k_{y} = \frac{m \pi}{b},</math>
where {{mvar|n}} and {{mvar|m}} are the two integers representing a specific eigenmode. Performing the final substitution, we obtain
<math display="block"> \frac{\omega^2}{c^2} = \left(\frac{n \pi}{a}\right)^2 + \left(\frac{m \pi}{b}\right)^2 + k_{z}^2,</math>
which is the [[dispersion relation]] in the rectangular waveguide. The cutoff frequency <math>\omega_{c}</math> is the critical frequency between propagation and attenuation, which corresponds to the frequency at which the longitudinal wavenumber <math>k_{z}</math> is zero. It is given by
<math display="block"> \omega_{c} = c \sqrt{\left(\frac{n \pi}{a}\right)^2 + \left(\frac{m \pi}{b}\right)^2}</math>
The wave equations are also valid below the cutoff frequency, where the longitudinal wave number is imaginary. In this case, the field decays exponentially along the waveguide axis and the wave is thus [[Evanescent wave|evanescent]].