EP 410: Stable Resonators and Gaussian Beams

Dinesh Kabra, Dept. of Physics, IITB

Contents Stable Curved Mirror CaviGes

Curved mirror caviGes ABCD matrices Cavity Stability Criteria

ProperGes of Gaussian Beams

PropagaGon of Gaussian Beams Gaussian beam properGes of two-mirror laser caviGes ProperGes of specific two-mirror laser caviGes Mode Volume

ProperGes of real laser beams

Curved Mirror Cavi,es As expected curved mirror would provide lower diffracGon losses in a laser cavity when compared to plane-parallel mirrors. There is a big choice R1, R2 and d Which will work?? Is there a Stability criterion?

Stable Curved Mirror Cavi,es For an arbitrary beam to evolve into a stable steady-state beam requires that the beam profile at the mirrors must duplicate itself aRer successive passes through the cavity. Stability criteria: beam remains concentrated within the cavity rather than diverging out of the cavity. Matrix method to analyze the propagaGon of beam. 2 X 2 matrices, or ABCD matrices, are convenient for analyzing the propagaGon of opGcal rays through systems of opGcal elements. These matrices acts on 2-element column vectors. The first element of the vector indicates the height of the ray (its distance from longitudinal axis of the cavity) and the 2nd element denotes the ray angle (again, with respect to the cavity axis). If each element in the system can be expresses by a matrix then the system as a whole can be represented as mulGplicaGon of these individual matrices.

ABCD Matrices

ABCD Matrices PropagaGon of a beam over a distance d This can be expressed in matrix form: You will note that we use small-angle approximaGons in this analysis. The difference in ray height is actually equal to dtan rather than d. As long as the cavity mirrors have diameters much less than the cavity length, this approximaGon is valid. ABCD matrix at boundary between materials with refracGve index n1 and n2:

ABCD Matrices Ray passing through a curved boundary: R = radius of curvature For R < 0 concave ; R > 0 convex boundary Beam reflect from a curved mirror of focal length f (R/2), the ABCD matrix is Also valid for thin lens with focal length f.

Right order of the matrices:

Cavity Stability Criteria We are now in posiGon to derive the stability criterion of a laser cavity. To begin, we consider a cavity composed of two mirrors of equal curvature R and focal length f = R/2, separated by a distance d along the axis.

PropagaGon of a ray over a distance of one pass through the cavity and then reflecGon from one of the mirror can be wriben as

In order to determine stability, we need to look at the values of r2 and 2 in comparison to the values of r1 and 1. If r2 > r1 and 2 > 1, then the beam will be on a diverging path and that would lead to instability aEer many passes. On the other hand if r2< r1 and 2 < 1, then we can conclude that the beam would tend toward stability, since it would always be abempGng to converge toward the axis.

Cavity Stability Criteria MathemaGcally, what this means is that we can ask whether soluGons exist for which If < 1, then we have the stability. Of course, this is simply an eigenvalue problem. For an arbitrary ABCD matrix, the problem reads: This equaGon will be saGsfied only if In our case, we have

Cavity Stability Criteria This equaGon has the form: Where x = and = 1-d/2f. The two soluGon of this equaGon are either both real or both imaginary, depending on the value of . The real soluGon occurs for | | > 1 and can be wriben as: The imaginary soluGon occur for | | < 1 and can be wriben as (In both cases, is a real number). Now lets check for N passes through the cavity. We dont need to mulGply the ABCD matric by itself N Gmes, since we can use the representaGon: In the case of | | > 1 , we have

Cavity Stability Criteria

This soluGon must diverge for large N, because one of e or e- must be greater than one. Therefore, when |1-d/2f| is greater than 1, the cavity is unstable. In the case of | | < 1 , we have This soluGon converges, since |eiN |=1. Therefore, when |1-d/2f|is less than 1, the cavity is stable. This can be put into more useful terms by wriGng out for | | < 1 : The value of will remain imaginary, leading to stability, only if

for spherical mirrors of radius R such that R = 2f

Cavity Stability Criteria

We assumed that the two mirrors had equal radius of curvatures, R. When the mirrors have unequal curvature, leading to two different focal lengths f1 and f2, the analysis proceeds in a similar fashion, except that the eigenvalue must be based on a full round trip of the cavity (rather than half-round trip). Which will result in Where s have been replaced by gs, dont get confuse with staGsGcal weights gs. Now, the Stability Criterion can be wriben as:

Cavity Stability Criteria

This condiGon can be expressed through a stability diagram. The shaded region denotes condiGons of stability. Its undesirable to design a cavity that lies on the stability boundary, as a slight misalignment in wrong direcGon will stop laser operaGon. Three parGcular points in the stability diagram refer to cavity designs shown, which are on verge of instability

Proper,es of Gaussian Beam In a gain medium located with a opGcal resonator, the TEM00 mode will have a Gaussian p r o fi l e a t t h e m i r r o r s . Furthermore, since the Fourier transform of Gaussian is Gaussian, it will have Gaussian profile at any longitudinal posiGon within the cavity, although width will change.

Beam waist is w0, it may occur outside or inside of cavity.

Proper,es of Gaussian Beam

This can also be wriben as zR is known as the Rayleigh range, depth of focus. The wavefront of a Gaussian beam has a curvature R at every point along the axis. At any posiGon z, this curvature is given by: (the curvature if infinite at the beam waist). The angular spread of a Gaussian beam in the region (z-z0) > zR is given as (see Figure in next slide)

Proper,es of Gaussian Beam