Question

Explain the origin of Transverse Electromagnetic Modes (TEM modes) in a commercial laser. Use words and diagrams as necessary.

Answer 1

A transverse mode of a beam of electromagnetic radiation is a particular electromagnetic field pattern of radiation measured in a plane perpendicular (i.e., transverse) to the propagation direction of the beam. Transverse modes occur in radio waves and microwaves confined to a waveguide, and also in light waves in an optical fiber and in a laser'soptical resonator.

Transverse modes occur because of boundary conditions imposed on the wave by the waveguide. For example, a radio wave in a hollow metal waveguide must have zero tangential electric field amplitude at the walls of the waveguide, so the transverse pattern of the electric field of waves is restricted to those that fit between the walls. For this reason, the modes supported by a waveguide are quantized. The allowed modes can be found by solving Maxwell's equations for the boundary conditions of a given waveguide.

However in any sort of waveguide where boundary conditions are imposed by a physical structure, a wave of a particular frequency can be described in terms of a transversemode (or superposition of such modes). These modes generally follow different propagation constants. When two or more modes have an identical propagation constant along the waveguide, then there is more than one modal decomposition possible in order to describe a wave with that propagation constant (for instance, a non-central Gaussian laser mode can be equivalently described as a superposition of Hermite-Gaussian modes or Laguerre-Gaussian modes which are described below).

Modes in waveguides can be further classified as follows:

• Transverse electromagnetic (TEM) modes: neither electric nor magnetic field in the direction of propagation.
• Transverse electric (TE) modes: no electric field in the direction of propagation. These are sometimes called H modes because there is only a magnetic field along the direction of propagation (H is the conventional symbol for magnetic field).
• Transverse magnetic (TM) modes: no magnetic field in the direction of propagation. These are sometimes called E modes because there is only an electric field along the direction of propagation.
• Hybrid modes: non-zero electric and magnetic fields in the direction of propagation.

Hollow metallic waveguides filled with a homogeneous, isotropic material (usually air) support TE and TM modes but not the TEM mode. In coaxial cable energy is normally transported in the fundamental TEM mode. The TEM mode is also usually assumed for most other electrical conductor line formats as well. This is mostly an accurate assumption, but a major exception is microstrip which has a significant longitudinal component to the propagated wave due to the inhomogeneity at the boundary of the dielectric substrate below the conductor and the air above it. In an optical fiber or other dielectric waveguide, modes are generally of the hybrid type.

In rectangular waveguides, rectangular mode numbers are designated by two suffix numbers attached to the mode type, such as TE_mn or TM_mn, where m is the number of half-wave patterns across the width of the waveguide and n is the number of half-wave patterns across the height of the waveguide. In circular waveguides, circular modes exist and here m is the number of full-wave patterns along the circumference and n is the number of half-wave patterns along the diameter.

Laser modes:

In a laser with cylindrical symmetry, the transverse mode patterns are described by a combination of a Gaussian beam profile with aLaguerre polynomial. The modes are denoted TEM_pl where p and l are integers labeling the radial and angular mode orders, respectively. The intensity at a point r,? (in polar coordinates) from the centre of the mode is given by:

where ? = 2r²/w², and L_p^l is the associated Laguerre polynomial of order p and index l. w is the spot size of the mode corresponding to the Gaussian beam radius.

With p=l=0, the TEM₀₀ mode is the lowest order, or fundamental transverse mode of the laser resonator and has the same form as a Gaussian beam. The pattern has a single lobe, and has a constant phase across the mode. Modes with increasing p show concentric rings of intensity, and modes with increasing l show angularly distributed lobes. In general there are 2l(p+1) spots in the mode pattern (except for l=0). The TEM_0i* mode, the so-called doughnut mode, is a special case consisting of a superposition of two TEM_0i modes (i=1,2,3), rotated 360