Resonato r

4.4.1 Estimation of Mode-Sizes; Stability Ranges

The full layout of the resonator for intracavity frequency-doubling is shown in Figure 4.7. A four-mirror bow-tie arrangement was used to keep the angles of incidence on the two curved mirrors to a minimum, thereby limiting the astigmatism in the output beam produced by these mirrors. In practice, the mimimum angle achievable was approximately 30° because of the size of the oven housing the LBO crystal, which had a diameter of 30 mm. A large oven was required to give thermal stability for the temperature-tuned LBO crystal.

TGG Faraday Zero-order

rotator x/2 plate Diode-bar

Plane, HR @ 1pm Plane, HR @ 1pm or output coupler @ 1pm ,1pm output Nd:YAG slab Diode-bar ROC = -100 mm HR @ 1pm LBO crystal in heated oven 0.5pm output ROC = -100 mm HR @ 1pm, HT @ 0.5pm

Figure 4.7. Single-mode intracavity-doubled Nd.YAG slab ring-laser.

Estimation of the mode size at a given point for various cavity configurations was made by finding the relevant ABCD matrix for one round-trip and requiring that the complex beam parameter q reproduce itself after one trip round the ring. In other words we seek the solution of the equation

where the complex beam parameter q is given by

1 ^ 1 /A

q{z) R{z) 7tnw^{z)

Stable cavities are characterised by real values of the wavefront radius of curvature R(z)

and the 1/e electric field radius w(z). For a round-trip matrix ABCD these quantities are given by

2B and w = ■■

71

C { A - D f

B

2B^

The results of such an analysis for an empty four-mirror bow-tie ring with a plane mirror separation of 250 mm and an angle of 30° are shown in Figure 4.8. The cavity is stable for curved mirror separations between 100 and 115 mm. Even at the large separation end (towards 115 mm) where the saggital mode radius increases rapidly, the mode radius is less than 600 pm so that the 1.3 mm aperture of the slab amplifier is quite adequate. However, the astigmatism introduced by the large internal angle of 30° is quite clear. Figure 4.9 shows the behaviour when the Nd: YAG slab is introduced into the cavity midway between the plane mirrors. The distance from the plane mirrors to the slab faces was taken to be 120 mm. The stability range remains the same as before although there is a significant increase in the mode radii. Figures 4.10 and 4.11 model the mode radius in the slab and the waist radius at the focus of the ring when a 25 mm LBO crystal is placed midway between the two curved mirrors. The range of curved mirror separations for which the resonator is stable is now 9 8 - 1 0 9 mm. The tangential mode radius in the slab is now quite sensitive to the curved mirror separation. The waist at the focus of the ring is minimised by setting the cui ved mirror separation close to the stability limit of » 109 mm. At this point the astigmatism is quite severe, the saggital waist radius being around 60 pm while the tangential waist radius is about 30 pm.

An important question to be answered was whether the Boyd and Kleinman focusing condition could be achieved in the LBO crystal while maintaining a stable cavity configuration. For a 25 mm length of LBO, optimum focusing is achieved for a confocal parameter of 4.4 mm. The corresponding waist radius needed is given by

(4.5) V Tvn

which in this case is 31 pm. The conclusion was therefore that by operating the laser with the curved mirrors as far apart as possible optimum focusing could be achieved in the tangential plane, although the « 60 pm saggital waist meant that optimum nonlinear coupling could not be achieved. The fact that optimum nonlinear coupling coefficient could not be attained was unfortunate given that the modest non-linear coefficient of LBO and the modest gain over linear loss for a ring laser together meant that non-linear under-coupling was likely even if the Boyd and Kleinman conditions could be reached. However the use of a ring laser enabled single-frequency operation to be reliably achieved at all puming levels, and the type I non-critical phase-matching in LBO meant

that the horizontally polarised laser mode was not disturbed by birefringence effects in the nonlinear crystal.

501)

"3 r»(io

101)

10X15 107,5 111.25

Cui'>ed m irro r se p aratio n (m m ) 1(1!)

1500

IIS 103.75 107.5 111.25 C urved m irro r separation (m m ) inn

Figure 4.8. Mode sizes in the empty ring. Figure 4.9. Modes sizes with slab in the ring.

100(1

no

104

C u rv ed m irro r se p a ra tio n (m m )

Figure 4.10. Mode sizes in the slab for doubled laser.

S 50 •S -I"

')« I (10 102

C u rv ed m irro r sep a ra tio n (m m )101 I or,

Figure 4.11. Waist sizes in LBO crystal.

4.4.2 Unidirectional Operation

Single-longitudinal mode operation was achieved by causing only one of the two possible directions around the ring to lase. The travelling wave in the ring has the effect of eliminating spatial hole burning so that only the mode with the highest gain operates. Unidirectional operation is achieved in a ring laser by introducing a differential loss between the two directions so that the lower-loss direction is preferentially amplified and establishes itself, excluding the other direction. Unidirectional lasing was achieved in this case by inserting a Faraday rotator and a half­ wave plate into the cavity, so that for one direction there was no loss on entering the Nd:YAG slab at the Brewster angle, while for the other direction, rotation of the (linear) polarisation out of the plane of the ring caused significant loss to be introduced. The principle is illustrated in Figure 4.12. The method relies on the non-reciprocal rotation

B' A' C

D'

Figure 4.12. Action o f an optical diode.

produced by the Faraday-rotator, i.e. the sense of rotation depends on the direction of travel for a fixed direction of the magnetic field through the TGG material. At point A in Figure 4.12, the laser is polarised in the plane of the ring. After passing through the Faraday rotator, the plane of polarisation is rotated, say in right-handed sense, by a certain angle which is typically a few degrees. The half-wave plate C is rotated so that one of its axes bisects the angle that the polarisation direction at B makes with the horizontal. Thus at D the polarisation is restored to the horizontal plane and no loss is experienced on entering the Nd:YAG slab at the Brewster angle. For the opposite direction (starting at A') the half-wave plate is encountered first, giving an effective rotation which is the same as that produced in going from A to B. The polariation at C is therefore the same as that at B. However on passing through the Faraday rotator the rotation sense is now left-handed so that the plane of polarisation is rotated further away from the horizontal plane rather than being restored. Consequently this direction will experience Fresnel loss on entering the NdiYAG slab.

4.5 Laser Perform ance