Single Axial Mode And Spectral Hole Burning

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Reflection at a surface takes place under the condition that the field amplitude is zero at the reflecting surface. As a result, the axial modes i of wavelength λi inside a laser cavity can be defined by their number ni of sine-wave half cycles that fit exactly into the laser cavity. The optical gain curve of the active medium of a crystalline solid-state laser has a wavelength range of typically 10-100 nm, so that many (~104) axial modes inside a few-cm long cavity fall within the optical gain curve and may therefore start lasing.

Can someone explain why nevertheless only one axial mode starts lasing in a homogeneously broadened gain medium at pump threshold?

Can someone explain the phenomenen of spectral hole burning?

Another question is, is there someone who can help me with a few other questions that i send a pm with a pdf file?
 
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...so that many (~104) axial modes inside a few-cm long cavity fall within the optical gain curve and may therefore start lasing.
yes and that is, what typically happens in solid state lasers unless you use a mode-selective element.
why nevertheless only one axial mode starts lasing in a homogeneously broadened gain medium at pump threshold?
The gain curve has a maximum. The mode closest to that maximum will start lasing first. That is the threshold.
spectral hole burning
You may want to read http://www.physics.montana.edu/faculty/rebane/Research/Tutorials/Hole_burning/HoleBurning_00.htm first and then ask more specifically.
 
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