Scale-invariant nonlinear optics in gases

Nonlinear optical methods have become ubiquitous in many scientific areas, from fundamental studies of timeresolved electron dynamics to microscopy and spectroscopy applications. They are, however, often limited to a certain range of parameters such as pulse energy and average power. Restrictions arise from, for example, the required field intensity as well as from parasitic nonlinear effects and saturation mechanisms. Here, we identify a fundamental principle of nonlinear light-matter interaction in gases and show that paraxial nonlinear wave equations are scaleinvariant if spatial dimensions, gas density, and laser pulse energy are scaled appropriately. As an example, we apply this principle to high-order harmonic generation and provide a general method for increasing peak and average power of attosecond sources. In addition, we experimentally demonstrate the implications for the compression of short laser pulses. Our scaling principle extends well beyond those examples and includes many nonlinear processes with applications in different areas of science.