Example of a canonical optical resonator in (a) diagrammatic form and (b) rendered using the ray tracing software POV-Ray (Ref. 7). Each mirror is a segment of a sphere; the respective radii of the spheres are and , giving the mirrors focal lengths and . The mirrors are separated by a distance in the direction.
Views inside different symmetric resonators. The sequence of images was calculated for a fixed resonator length , but different radii of curvature as calculated from Eq. (1) for values between (top left) and (bottom right). The mirror diameter , as it is for all the figures. (As is increased the mirror appears to become smaller due to the edge of the mirror moving further away from the camera as its shape becomes more convex.) Each image shows the view from near the center of the resonator toward the first mirror. The camera was positioned on the resonator axis a distance in front of the second mirror. Reference 12 contains a POV-Ray script to create these views and a movie of the view changing continuously as is increased from to .
Details of the transition from stability to instability for two resonators: the same resonator as in Fig. 2 (left), which has slightly golden mirrors, and the same resonator, but with perfectly reflecting and therefore colorless mirrors (right). In both image sequences, is increased from 0.99 to 1.02. The other parameters of the resonators are the same as in Fig. 2. A movie of the transition (in reverse) for colorless mirrors is available (Ref. 12).
Imaging of an object placed inside geometrically stable resonators. The object is a semitransparent letter P in (a) a transverse plane near the center of the resonator. (b) A plane-plane resonator shows the familiar hall-of-mirrors effect, creating an infinite series of equidistant images. Such a resonator is exactly at the edge of stability . Truly stable resonators, like the symmetric resonator with shown in (c) create a more complex series of images. In both cases the resonator length , the letter P is positioned at , and the camera is positioned at .
Location of the images of the letter P in Fig. 4(c). The diagram at the top indicates the positions and relative magnification of the images due to the first 40 round trips in the resonator (some images are outside the displayed range and are not marked). Each image is marked by an arrow, whose horizontal position indicates the image’s position and whose length indicates the image’s relative magnification. Numbers above or below an arrow indicate the number of round trips after which the image is formed [always starting with light traveling to the right; image positions due to light traveling to the left are not shown, and no such images are visible in Fig. 4(c)]. The object is in the plane marked P. The four picture frames at the bottom show the view from the camera position (, marked by an eye; the camera is looking toward the right mirror). In Fig. 4 the depth of field is infinite; it is reduced here so that individual planes come into focus. In POV-Ray, a finite depth of field is achieved by setting the parameter aperture to values greater than 0 (here we use 0.7). The different frames focus on the object plane and the planes that contain the first , second , and third images. A movie of the focus varying smoothly over the range and a POV-Ray script to create individual views is available (Ref. 12). The size at which the image is seen depends not only on its relative magnification, but also on its distance from the camera.
Creation of fractal eigenmodes inside a confocal resonator, an example of a geometrically unstable resonator. The confocal plane is also a self-conjugate plane, here with magnification . The semitransparent obstacle in the confocal plane (a) simulates the intensity pattern of the diffraction pattern from a slit aperture inside the resonator. (b) The view from within the resonator, and in particular the brightness cross section along the center [top curve in (c)], shares the basic fractal character and many details with the intensity cross section through the fractal eigenmode of a resonator with the same magnification and a slit aperture, calculated using a full wave-optical simulation [bottom curve in (c) and Fig. (7b) in Ref. 11]. In (a) the number of blocks is greatly reduced for clarity; (b) and (c) were calculated with approximately 200 blocks and, to avoid complicating the resulting view, with the light source moved to a position directly above the resonator axis and with the floor removed.
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