Double Higgs Production at the Linear Colliders and the Probing of the Higgs Self-Coupling

We study double Higgs production in the $e^+ e^-$ and $\gamma \gamma$ modes of the linear collider. It is also shown how one can probe the scalar potential in these reactions. We discuss the effective longitudinal $W$ approximation in $\gamma \gamma$ processes and the $W_L W_L$ luminosities in the two modes of a high-energy linear collider. A generalised non-linear gauge-fixing condition, which is particularly useful for tree-level calculations of electroweak processes for the laser induced collider, is presented. Its connection with the background-field approach to gauge fixing is given.