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 42View Image 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 n and m. We can also see how the size of the intensity distribution increases with the mode index, while the peak intensity decreases.
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Figure 42: This plot shows the intensity distribution of Hermite–Gauss modes unm. 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 44View Image shows the amplitude and phase distribution of several higher-order Hermite–Gauss modes. Some further features of Hermite–Gauss modes:

Note that these are special features of Gaussian beams and not generally true for arbitrary beam shapes. Figure 43View Image, 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.

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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.
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Figure 44: These plots show the amplitude distribution and wave front (phase distribution) of Hermite–Gaussian modes unm (labeled as HGnm in the plot). All plots refer to a beam with λ = 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.

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