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We study the gauge invariance of laser-matter interaction. The velocity gaugewhere the vector potential is expanded to the $n$-th order with respect to thespatial coordinate, and the length gauge where the electric and magnetic fieldsare expanded to the $n$-th and $(n-1)$-th orders, respectively, are mutuallygauge-transformed, describing the physically equivalent situation. The latterincludes up to the electric $2^{n+1}$-pole and magnetic $2^n$-pole interactionsas well as two extra terms. The finding serves to develop consistentnonperturbative simulation methods beyond the electric dipole approximation.

Gauge Invariance beyond the Electric Dipole Approximation