### 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
decreases.
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.