Tasks 1. A 'ring' laser cavity as illustrated in the following diagram comprises one concave mirror (CM) of radius of curvature R, tilted by an angle to each arm of a cavity in the horizontal plane i.e., so as to fold the cavity with an net angle of 20 between the arms of the cavity. It also comprises a high reflecting flat mirror (M) and a partially reflecting flat mirror which acts as an output-coupler (OC). The mirrors are arranged to make a triangular cavity path in air. The optical path length for one complete circuit of the cavity (a round- trip, RT) is d. Caustics for a stable laser field are also illustrated. M 30 OC Laser field 20 CM (a) Give a labelled sketch for the unfolded equivalent representation of this cavity as a wave-guide in each of: (i) the horizontal plane and; (ii) the vertical plane. For each sketch, sketch caustics for a laser field supported by your wave-guide illustrating a main difference that you would expect between the two cases and for clarity, label this difference. Clearly label your two sketches with the parameters given in the description above, including the focal length of your equivalent-lenses in terms of R and D. Note that for an unfolded wave-guide, a curved mirror is represented by a lens, and it helps to consider symmetries when sketching the guided laser field. [2] Background U₁(x) = exp(-inx²/qiλo) eq. 1 1/9: = 1/ - ίλο/πω eq. 2 q₁ = inσ²/λo + d₁ eq. 2' where the reduced radius of curvature, R = R/n and λo is the wavelength for the field where in vacuum. For a laser cavity which supports a stable (reproducing) light-field, the q-parameter for that light-field taken at any chosen plane satisfies: q Aq+B Cq+D eq. 3 where A, B, C and D are elements of the round-trip system matrix, MRT, that would transform a ray, r, as defined in this module through one complete round trip of the cavity starting from the chosen plane. MRT is not a unit matrix and so only a specific value of q will satisfy this equation. This defines the fundamental (Gaussian) mode of the laser cavity. For a lens or curved mirror tilted by relative to a horizontal primary ray, the effective focal length is modified to: fhor=fcos fvert=f/cose eq. 4 eq. 4' where for and fvert are the modified focal lengths for rays contained in the horizontal and vertical planes respectively and f = R/2 for a curved mirror.

Tasks 1. A 'ring' laser cavity as illustrated in the following diagram comprises one concave mirror (CM) of radius of curvature R, tilted by an angle to each arm of a cavity in the horizontal plane i.e., so as to fold the cavity with an net angle of 20 between the arms of the cavity. It also comprises a high reflecting flat mirror (M) and a partially reflecting flat mirror which acts as an output-coupler (OC). The mirrors are arranged to make a triangular cavity path in air. The optical path length for one complete circuit of the cavity (a round- trip, RT) is d. Caustics for a stable laser field are also illustrated. M 30 OC Laser field 20 CM (a) Give a labelled sketch for the unfolded equivalent representation of this cavity as a wave-guide in each of: (i) the horizontal plane and; (ii) the vertical plane. For each sketch, sketch caustics for a laser field supported by your wave-guide illustrating a main difference that you would expect between the two cases and for clarity, label this difference. Clearly label your two sketches with the parameters given in the description above, including the focal length of your equivalent-lenses in terms of R and D. Note that for an unfolded wave-guide, a curved mirror is represented by a lens, and it helps to consider symmetries when sketching the guided laser field. [2] Background U₁(x) = exp(-inx²/qiλo) eq. 1 1/9: = 1/ - ίλο/πω eq. 2 q₁ = inσ²/λo + d₁ eq. 2' where the reduced radius of curvature, R = R/n and λo is the wavelength for the field where in vacuum. For a laser cavity which supports a stable (reproducing) light-field, the q-parameter for that light-field taken at any chosen plane satisfies: q Aq+B Cq+D eq. 3 where A, B, C and D are elements of the round-trip system matrix, MRT, that would transform a ray, r, as defined in this module through one complete round trip of the cavity starting from the chosen plane. MRT is not a unit matrix and so only a specific value of q will satisfy this equation. This defines the fundamental (Gaussian) mode of the laser cavity. For a lens or curved mirror tilted by relative to a horizontal primary ray, the effective focal length is modified to: fhor=fcos fvert=f/cose eq. 4 eq. 4' where for and fvert are the modified focal lengths for rays contained in the horizontal and vertical planes respectively and f = R/2 for a curved mirror.