A linear laser cavity illustrated in the following, comprises one concave mirror of radius of curvature, R and a partially reflecting flat mirror which acts as an output-coupler (O/C). The mirrors are separated by a distance, L and the cavity is in air for which you can assume the refractive index, n = 1. Output Flat mirror (O/C) Cavity length, L Laser field Mirror, radius of curvature, R The cavity shown here includes a representative stable laser beam profile. i) Sketch and label an equivalent linear wave-guide (unfolded representation) that would reproduce the same field distribution (q-parameters), for which the mirror of this laser cavity is represented as a lens (f = R/2). Add to this a representation of the laser (spatial) beam profile and wave-fronts. (2 marks) ii) From your equivalent wave-guide and using matrices from lectures or workshops, determine the system matrix, MRT for one complete round trip of a ray, r. Start your ray propagation from the flat mirror i.e., consider planes 1 and 2 to be equivalent to that of the flat mirror. (2 marks) iii) Derive the relationship for q at the O/C in terms of L and R, that is necessary for stable laser operation and deduce the range of cavity length, L that supports stable laser operation. (2 marks) iv) Starting from your expression for q in part (iii), determine the waist size of the laser at the output coupler for R = 60.0 cm, L = 50.0 cm and 20 = 600 nm. (2 marks) v) Determine the radius of curvature of the wave-front at the curved mirror. Explain your reasoning and any arguments made. (2 marks)