Coupling into a single-mode fiber with a (a) matched electric field, (b) first order null and (c) second order null. First row: electric field projected onto the fiber. Second row: intensity on the fiber face. Third row: coupling efficiency for off-center fibers. The matched mode couples well into the fiber, even for small off-center positions. The first-order null has no throughput: its odd electric field ensures a zero overlap integral in Eq. (1). Off-center positions however still transmit because the odd structure is lost. The second-order employs an even electric field where the contribution of the central peak is canceled by the two sidebands. This creates a much broader null when decentering the fiber.