where r is the inverse mean life for the transition E2 to E 2 Transverse modes of a laser 2.1 Motivation Intuitively, we expect the irradiance pattern of a laser beam to be circular if the beam is shined on a material, we expect to observe a circular hole burned on the material. However, we will see that it is possible to obtain other transverse modes, for which the resulting irradiance patterns are not circular. We will first study the origin of these effects. We will then see how they can be removed, to ensure a highly focused output beam 2.2 Analogy to Fraunhofer diffraction Imagine a simple setup in which there are two concave mirrors with focal length f, which implies that their radii of curvature are R=2f. Place these mirrors confocally, with the concave sides of the mirrors facing each other nagine a beam beginning at the point z=0, reflecting off each mirror, and returning to the point z=0. The total length of this path is 4f Using some hand waving, we can say that this setup is very similar to the previous setup that we considered when studying Fraunhofer diffraction The path length of the apparatus in that case was also 4f, where f was the focal length of the thin field lenses placed between the aperture, transform da We saw in the case of Fraunhofer diffraction that the optical disturbance at the image plane is the fourier transform of that at the transform plane n this case. the concave mirrors function as the thin lenses. and so the transform and image planes are both located at the center point of the two confocal concave lenses After many reflections of the beam, we expect by symmetry that the ptical disturbance at z=0 should be the same before and after reflection off one of the mirrors. Combined with the previous result, this implies that the optical disturbance must be a function that is equal to its Fourier transformwhere Γ is the inverse mean life for the transition E2 to E1. 2 Transverse modes of a laser 2.1 Motivation Intuitively, we expect the irradiance pattern of a laser beam to be circular; if the beam is shined on a material, we expect to observe a circular hole burned on the material. However, we will see that it is possible to obtain other transverse modes, for which the resulting irradiance patterns are not circular. We will first study the origin of these effects. We will then see how they can be removed, to ensure a highly focused output beam. 2.2 Analogy to Fraunhofer diffraction Imagine a simple setup in which there are two concave mirrors with focal length f, which implies that their radii of curvature are R = 2f. Place these mirrors confocally, with the concave sides of the mirrors facing each other. Imagine a beam beginning at the point z = 0, reflecting off each mirror, and returning to the point z = 0. The total length of this path is 4f. Using some hand waving, we can say that this setup is very similar to the previous setup that we considered when studying Fraunhofer diffraction. The path length of the apparatus in that case was also 4f, where f was the focal length of the thin field lenses placed between the aperture, transform, and image planes. We saw in the case of Fraunhofer diffraction that the optical disturbance at the image plane is the fourier transform of that at the transform plane. In this case, the concave mirrors function as the thin lenses, and so the transform and image planes are both located at the center point of the two confocal concave lenses. After many reflections of the beam, we expect by symmetry that the optical disturbance at z = 0 should be the same before and after reflection off one of the mirrors. Combined with the previous result, this implies that the optical disturbance must be a function that is equal to its Fourier transform. 4