A formalism in terms of Hertz potentials is presented describing sum-frequency generation in a uniaxial non-linear crystal. A scheme is proposed consisting in aligning the sideways propagation of extraordinary waves in combination with phase-matching. Simplified paraxial equations describing this situation are obtained. Particular attention is paid to the generation of second harmonics.A formalism in terms of Hertz potentials is presented describing sum-frequency generation in a uniaxial non-linear crystal. A scheme is proposed consisting in aligning the sideways propagation of extraordinary waves in combination with phase-matching. Simplified paraxial equations describing this situation are obtained. Particular attention is paid to the generation of second harmonics.