The Stability Condition
Since for the optical resonator, we have assumed that radiation within the laser cavity propagates back and forth between plane-parallel mirrors in the form of a well-collimated beam. Because of diffraction effects, a perfectly collimated beam cannot be maintained with mirrors of finite extent and a fraction of the internal cavity radiation spills out around the edges of the mirrors
In general, the problem of excessive diffraction losses is overcome by curving the mirrors slightly inward toward the cavity; that is, by making them slightly spherical and concave inward. The fig is illustrated a pair of spherical mirrors each positioned with its center of curvature on a common line, the optic axis of the resonator
The mirrors are separated by a distance L, r1 is the radius of curvature of mirror M1, r2 that of mirror M2. By convention, the radius of curvature of a mirror is taken to be positive if the center of curvature of the mirror lies in the direction of the laser cavity; otherwise r is taken to be negative
In the figure, both radii are positive by this convention. The ray-tracing technique requires that we consider a ray of light within the cavity, initially very close to the optic axis and inclined at a very small angle with respect to it (such a ray is designated a paraxial ray), and that we observe its behavior as it is reflected bade and forth between the mirrors
If after a large number of reflections the ray is observed to diverge from the reso¬nator—the distance of the ray from the optic axis is found to grow as the number of reflections becomes large—we conclude that the particular resonator configuration being analyzed is characterized by high losses
If the ray is found to remain close to the optic axis, we conclude that the resonator has a low-loss configuration. By performing such an analysis for an arbitrary mirror separation L and arbitrary mirror curvatures r1 and r2, one can show that a single condition, called the stability condition, is satisfied by a low-loss resonator configuration
This condition is generally expressed in terms of two dimensionless quantities, the g-parameters of the resonator, defined by the equations
With these definitions, the
stability condition has the simple form
As long as this relationship is satisfied by the resonator, a paraxial ray continues to remain close to the optic axis, even after many reflections, and the mirror configuration is termed stable. If, on the other-hand,
the resonator is described as being
unstable, and upon multiple reflections a ray initially paraxial in the cavity diverges from the axis
In the cases where the product g1g2 equals zero or unity, the laser is on the boundary between stability and instability and is termed
marginally stable (شبه مستقر)
Some important configurations and indicated their stability (الأشكال)
The most common laser resonator is a nearly planar stable configuration or an equivalent configuration with one long radius-of-curvature mirror and one planar mirror. Such a configuration is relatively easy to align and is also efficient in the amplification process
Even though a particular resonator configuration is classified as unstable, it may still be quite useful. Indeed, unstable optical resonators have several very attractive features; (1) they can be highly efficient with respect to utilization of the active medium, even with very-short resonators
Note that the portion of the active medium near the optic axis contributes much to the ampli¬fication of the beam. In contrast, the entire volume of the active medium in an unstable laser resonator can contribute strongly to the amplification process, (2) their suitability for adjustable output coupling. Simply by changing the spacing between the resonator mirrors, one can adjust the output coupling over a wide range of values
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