7.7 Properties of higher-order Hermite–Gauss modes
Some of the properties of Hermite–Gauss modes can easily be described using cross sections of the field
intensity or field amplitude. Figure 42 shows such cross sections, i.e., the intensity in the x-y plane, for a
number of higher-order modes. This shows a x-y symmetry for mode indices and . We can also see
how the size of the intensity distribution increases with the mode index, while the peak intensity
Figure 42: This plot shows the intensity distribution of Hermite–Gauss modes . One can see
that the intensity distribution becomes wider for larger mode indices and the peak intensity decreases.
The mode index defines the number of dark stripes in the respective direction.
Similarly, Figure 44 shows the amplitude and phase distribution of several higher-order
Hermite–Gauss modes. Some further features of Hermite–Gauss modes:
- The size of the intensity profile of any sum of Hermite–Gauss modes depends on z while its
shape remains constant over propagation along the optical axis.
- The phase distribution of Hermite–Gauss modes shows the curvature (or radius of curvature)
of the beam. The curvature depends on z but is equal for all higher-order modes.
Note that these are special features of Gaussian beams and not generally true for arbitrary beam shapes.
Figure 43, for example, shows the amplitude and phase distribution of a triangular beam at the point where
it is (mathematically) created and after a 10 m propagation. Neither the shape is preserved nor does it
show a spherical phase distribution.
Figure 43: These top plots show a triangular beam shape and phase distribution and the bottom
plots the diffraction pattern of this beam after a propagation of z = 5 m. It can be seen that the
shape of the triangular beam is not conserved and that the phase front is not spherical.
Figure 44: These plots show the amplitude distribution and wave front (phase distribution) of
Hermite–Gaussian modes (labeled as HGnm in the plot). All plots refer to a beam with
= 1 µm, w = 1 mm and distance to waist z = 1 m. The mode index (in one direction) defines
the number of zero crossings (along that axis) in the amplitude distribution. One can also see that
the phase distribution is the same spherical distribution, regardless of the mode indices.