.. include:: numbatdefs.txt .. _chap-josab-label: *********************** JOSA-B Tutorial Paper *********************** Introduction ------------------------------------------- Dr Christian Wolff and colleagues have used |NUMBAT| throughout their 2021 SBS tutorial paper `Brillouin scattering—theory and experiment: tutorial `_ published in J. Opt. Soc. Am. B. This set of examples works through their discussions of backward SBS, forward Brillouin scattering, and intermodal forward Brillouin scattering. As the calculations in this paper used |NUMBAT| with essentially the same code in these tutorials, we do not bother to include the original figures. Example 1 -- Backward SBS in a circular silica waveguide ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Figure 15 in the paper shows the fundamental optical field of a silica 1 micron waveguide, with the gain and other parameters shown in Table 2. The corresponding results generated with ``sim-josab-01.py`` are as follows: .. figure:: ./images/josab_tutorial/josab_01-fields/EM_E_field_01.png :width: 15cm Fundamental optical mode fields. .. figure:: ./images/josab_tutorial/josab_01-fields/AC_field_28.png :width: 15cm Elastic mode with largest SBS gain. .. figure:: ./images/josab_tutorial/josab_01-gain_spectra.png :width: 15cm Gain spectrum for the silica waveguide. .. raw:: latex \clearpage Example 2 -- Backward SBS in a rectangular silicon waveguide ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Figure 14 in the paper calculates the backwards SBS properties of a rectangular :math:`450x200` nm silicon waveguide. The corresponding results generated with ``sim-josab-02.py`` are as follows: .. figure:: ./images/josab_tutorial/josab_02a-gain_spectra.png :width: 15cm The fields and gain parameters are as follows: .. figure:: ./images/josab_tutorial/josab_02a-fields/EM_E_field_00.png :width: 15cm Fundamental optical mode fields. .. figure:: ./images/josab_tutorial/josab_02a-fields/AC_field_02.png Fundamental elastic mode fields for mode 2. .. figure:: ./images/josab_tutorial/josab_02a-fields/AC_field_06.png :width: 15cm Fundamental elastic mode fields for mode 6. We can reproduce Fig. 13 showing the elastic dispersion of this waveguide silicon waveguide using ``sim-josab-02b-acdisp.py``. .. figure:: ./images/josab_tutorial/josab_02b-disp-qnu.png :width: 15cm Acoustic dispersion diagram with modes categorised by symmetry as in Table 1 of "Formal selection rules for Brillouin scattering in integrated waveguides and structured fibers" by C. Wolff, M. J. Steel, and C. G. Poulton ``https://doi.org/10.1364/OE.22.032489`` .. raw:: latex \clearpage Example 3 -- Forward Brillouin scattering in a circular silica waveguide ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Figure 16 and Table 3 examine the same waveguides in the case of forward Brillouin scattering. These results can be generated with ``sim-josab-03.py`` and ``sim-josab-04.py``. Let's see the results for the silica cylinder first: .. figure:: ./images/josab_tutorial/josab_03-gain_spectra.png :width: 15cm Gain spectrum for forward SBS of the silica cylinder. .. figure:: ./images/josab_tutorial/josab_03-fields/EM_E_field_01.png :width: 15cm Fundamental optical mode field. .. figure:: ./images/josab_tutorial/josab_03-fields/AC_field_06.png :width: 15cm Elastic mode of maximum gain. .. figure:: ./images/josab_tutorial/josab_03-fields/AC_field_12.png :width: 15cm Elastic mode of second highest gain. .. raw:: latex \clearpage Example 4 -- Forward Brillouin scattering in a rectangular silicon waveguide ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ The corresponding results for the silicon waveguide can be generated with ``sim-josab-04.py``: .. figure:: ./images/josab_tutorial/josab_04-gain_spectra.png :width: 15cm Gain spectrum for forward SBS of the silicon waveguide. .. figure:: ./images/josab_tutorial/josab_04-fields/EM_E_field_00.png :width: 15cm Fundamental optical mode field. .. figure:: ./images/josab_tutorial/josab_04-fields/AC_field_06.png :width: 15cm Elastic mode of maximum gain. .. raw:: latex \clearpage Example 5 -- Intermodal Forward Brillouin scattering in a circular silica waveguide ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ For the problem of intermodal FBS, the paper considers coupling between the two lowest optical modes. The elastic mode of highest gain is actually a degenerate pair: .. figure:: ./images/josab_tutorial/josab_05-gain_spectra.png :width: 15cm Gain spectrum for intermodal forward SBS of the silica waveguide. .. figure:: ./images/josab_tutorial/josab_05-fields/EM_E_field_00.png :width: 15cm Fundamental optical mode field. .. figure:: ./images/josab_tutorial/josab_05-fields/EM_E_field_01.png :width: 15cm Second order optical mode field. .. figure:: ./images/josab_tutorial/josab_05-fields/AC_field_06.png :width: 15cm Elastic mode field of maximum gain. .. raw:: latex \clearpage Example 6 -- Intermodal Forward Brillouin scattering in a rectangular silicon waveguide ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Finally, the silicon waveguide generates extraordinarily high gain when operated in an intermodal configuration: .. figure:: ./images/josab_tutorial/josab_06-gain_spectra.png :width: 15cm Gain spectrum for intermodal forward SBS of the silicon waveguide. .. figure:: ./images/josab_tutorial/josab_06-fields/EM_E_field_00.png :width: 15cm Fundamental optical mode field. .. figure:: ./images/josab_tutorial/josab_06-fields/EM_E_field_02.png :width: 15cm Second order optical mode field. .. figure:: ./images/josab_tutorial/josab_06-fields/AC_field_02.png :width: 15cm Elastic mode field of maximum gain. .. raw:: latex \clearpage