Cutoff frequency

Radio communications

In radio communication, skywave communication is a technique in which radio waves are transmitted at an angle into the sky and reflected back to Earth by layers of charged particles in the ionosphere. In this context, the term cutoff frequency refers to the maximum usable frequency, the frequency above which a radio wave fails to reflect off the ionosphere at the incidence angle required for transmission between two specified points by reflection from the layer.

Waveguides

The cutoff frequency of an 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. The cutoff frequency is found with the 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 $${\displaystyle {\omega }_c=c\sqrt{{\left(\frac{m\pi }{a}\right)}^2+{\left(\frac{n\pi }{b}\right)}^2},}$$ where ${\displaystyle m,n\ge 0}$ are the mode numbers for the rectangle's sides of length ${\displaystyle a}$ and ${\displaystyle b}$ respectively. For TE modes, ${\displaystyle m,n\ge 0}$ (but ${\displaystyle m=n=0}$ is not allowed), while for TM modes ${\displaystyle m,n\ge 1}$.

The cutoff frequency of the TM₀₁ mode (next higher from dominant mode TE₁₁) in a waveguide of circular cross-section (the transverse-magnetic mode with no angular dependence and lowest radial dependence) is given by $${\displaystyle {\omega }_c=c\frac{{\chi }_{01}}{r}=c\frac{2.4048}{r},}$$ where ${\displaystyle r}$ is the radius of the waveguide, and ${\displaystyle {\chi }_{01}}$ is the first root of ${\displaystyle J_0(r)}$, the Bessel function of the first kind of order 1.

The dominant mode TE₁₁ cutoff frequency is given by $${\displaystyle {\omega }_c=c\frac{{\chi }_{11}}{r}=c\frac{1.8412}{r}}$$

However, the dominant mode cutoff frequency can be reduced by the introduction of baffle inside the circular cross-section waveguide. For a single-mode optical fiber, the cutoff wavelength is the wavelength at which the normalized frequency is approximately equal to 2.405.

Mathematical analysis

The starting point is the wave equation (which is derived from the Maxwell equations), $${\displaystyle \left({\mathrm{\nabla }}^2-\frac{1}{c^2}\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{t}^2}\right)\psi (\mathbf{r},t)=0,}$$ which becomes a Helmholtz equation by considering only functions of the form $${\displaystyle \psi (x,y,z,t)=\psi (x,y,z)e^{i\omega t}.}$$ Substituting and evaluating the time derivative gives $${\displaystyle \left({\mathrm{\nabla }}^2+\frac{{\omega }^2}{c^2}\right)\psi (x,y,z)=0.}$$ The function ${\displaystyle \psi }$ 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 $${\displaystyle \psi (x,y,z,t)=\psi (x,y)e^{i\left(\omega t-k_{z}z\right)},}$$ where ${\displaystyle k_z}$ is the longitudinal wavenumber, resulting in $${\displaystyle \left({\mathrm{\nabla }}_T^2-k_z^2+\frac{{\omega }^2}{c^2}\right)\psi (x,y,z)=0,}$$ 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 $${\displaystyle \psi (x,y,z,t)={\psi }_0{e}^{i\left(\omega t-k_{z}z-k_{x}x-k_{y}y\right)}.}$$ Thus for the rectangular guide the Laplacian is evaluated, and we arrive at $${\displaystyle \frac{{\omega }^2}{c^2}=k_x^2+k_y^2+k_z^2}$$ The transverse wavenumbers can be specified from the standing wave boundary conditions for a rectangular geometry cross-section with dimensions a and b: $${\displaystyle k_x=\frac{n\pi }{a},}$$ $${\displaystyle k_y=\frac{m\pi }{b},}$$ where n and m are the two integers representing a specific eigenmode. Performing the final substitution, we obtain $${\displaystyle \frac{{\omega }^2}{c^2}={\left(\frac{n\pi }{a}\right)}^2+{\left(\frac{m\pi }{b}\right)}^2+k_z^2,}$$ which is the dispersion relation in the rectangular waveguide. The cutoff frequency ${\displaystyle {\omega }_c}$ is the critical frequency between propagation and attenuation, which corresponds to the frequency at which the longitudinal wavenumber ${\displaystyle k_z}$ is zero. It is given by $${\displaystyle {\omega }_c=c\sqrt{{\left(\frac{n\pi }{a}\right)}^2+{\left(\frac{m\pi }{b}\right)}^2}}$$ 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.