page_title: Transverse mode
section_title: Laser modes
hierarchical_section_title: Transverse mode / Laser modes
caption_attribution_description: English: Hermite-Gaussian transverse mode patterns. Bigger and better version created with the help of the Python script below Русский: Моды оптического резонатора с квадратными зеркалами #!/usr/bin/env python import sys import Image from math import * from scipy.special.orthogonal import hermite size = 320 I_0 = 1. w = size/4. m = int(sys.argv[1]) n = int(sys.argv[2]) def linear_to_sRGB(l): # Formula from http://www.w3.org/Graphics/Color/sRGB if l &lt;= 0.00304: l = 12.92*l else: l = 1.055*pow(l,1.0/2.4) - 0.055 return 255.0*l # First, let's make a floating-point image of the raw intensities. raw = Image.new('F', (size,size)) high = 0 for x in range(1, size, 2): for y in range(1, size, 2): I = I_0 * (hermite(m)(sqrt(2)*x/w)*exp(-x**2/w**2))**2 \ * (hermite(n)(sqrt(2)*y/w)*exp(-y**2/w**2))**2 if I &gt; high: high = I raw.putpixel((size/2+(x-1)/2,size/2+(y-1)/2), I) raw.putpixel((size/2+(x-1)/2,size/2-(y+1)/2), I) raw.putpixel((size/2-(x+1)/2,size/2+(y-1)/2), I) raw.putpixel((size/2-(x+1)/2,size/2-(y+1)/2), I) print 'row ' + str((x+1)/2) + ' of ' + str(size/2) + ' complete' # Now, let's normalize them and export them as sRGB. cooked = Image.new('L', (size,size)) for x in range(size): for y in range(size): l = raw.getpixel((x,y))/high cooked.putpixel((x,y), linear_to_sRGB(l)) print 'row ' + str(x+1) + ' of ' + str(size) + ' complete' cooked.save('plot-' + str(m) + '-' + str(n) + '.png')
context_page_description: A transverse mode of electromagnetic radiation is a particular electromagnetic field pattern of the radiation in the plane perpendicular to the radiation's propagation direction. 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's optical 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.
context_section_description: In a laser with cylindrical symmetry, the transverse mode patterns are described by a combination of a Gaussian beam profile with a Laguerre polynomial. The modes are denoted TEMₚₗ where p and l are integers labeling the radial and angular mode orders, respectively. The intensity at a point (r,φ) (in polar coordinates) from the centre of the mode is given by:
  where ρ = 2r²/w², Lˡ
  ₚ is the associated Laguerre polynomial of order p and index l, and 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. It is the 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 TEM₀ᵢ* mode, the so-called doughnut mode, is a special case consisting of a superposition of two TEM₀ᵢ modes (i = 1, 2, 3), rotated 360°/4i with respect to one another.
  The overall size of the mode is determined by the Gaussian beam radius w, and this may increase or decrease with the propagation of the beam, however the modes preserve their general shape during propagation. Higher order modes are relatively larger compared to the TEM₀₀ mode, and thus the fundamental Gaussian mode of a laser may be selected by placing an appropriately sized aperture in the laser cavity.
  In many lasers, the symmetry of the optical resonator is restricted by polarizing elements such as Brewster's angle windows. In these lasers, transverse modes with rectangular symmetry are formed. These modes are designated TEMₘₙ with m and n being the horizontal and vertical orders of the pattern. The electric field pattern at a point (x,y,z) for a beam propagating along the z-axis is given by
  where , , , and  are the waist, spot size, radius of curvature, and Gouy phase shift as given for a Gaussian beam;  is a normalization constant; and  is the kᵗʰ physicist's Hermite polynomial. The corresponding intensity pattern is
  The TEM₀₀ mode corresponds to exactly the same fundamental mode as in the cylindrical geometry. Modes with increasing m and n show lobes appearing in the horizontal and vertical directions, with in general (m + 1)(n + 1) lobes present in the pattern. As before, higher-order modes have a larger spatial extent than the 00 mode.
  The phase of each lobe of a TEMₘₙ is offset by π radians with respect to its horizontal or vertical neighbours. This is equivalent to the polarization of each lobe being flipped in direction.
  The overall intensity profile of a laser's output may be made up from the superposition of any of the allowed transverse modes of the laser's cavity, though often it is desirable to operate only on the fundamental mode.
caption: Rectangular transverse mode patterns TEM(mn)
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