Design of the folded resonators for thin-disk lasers 79
80 Laser experiments and discussion Design of single folded multi-and single-transverse-mode resonators A calculation of the stability parameter of the roundtrip matrix−1<((A+D)/2)<1and of the Gaussian beam radius on the disk is shown for a resonator folded via the thin disk in Figure 5.28.
In the symmetric case of L1-disk = L2-disk and two equal OC mirrors (in (a) and (b)), the beam radius on the disk decreases for L1-disk = L2-disk → fM1. Therefore, this resonator is suitable for multi-transverse-mode operation. The asymmetric resonator, (calculated in (c) and (d)) can be more beneficial for fundamental mode operation, when the areas with a comparable beam radius are bigger. Single-transverse-mode operation can be achieved for wdisk ≈ 480µm for a pump spot of 1.2 mm. This yields to, e.g., L1-disk = 21 cm and L2-disk=71 cm. For almost equal resonator arm lengths, areas close toL1-disk≈L2-disk≈24 cm at the edge of the stability zone can be used in case of a symmetric resonator design.
0.0 0.2 0.4 0.6 0.8 1.0 0.0
0.2 0.4 0.6 0.8 1.0
0.0 0.2 0.4 0.6 0.8 1.0 0.0
0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
0.0 0.2 0.4 0.6 0.8 1.0
0.0 0.2 0.4 0.6 0.8 1.0 0.0
0.2 0.4 0.6 0.8 1.0
L 1-disk(m)
L2-disk(m)
L 1-disk(m)
L2-disk(m)
wdisk(µm)
(a) (b)
L 1-disk(m)
L2-disk(m)
L 1-disk(m)
L2-disk(m)
wdisk(µm)
RM1=RM2=0.5 m Rdisk= 5 m RM1=RM2=0.5 m
Rdisk= 5 m
RM2= 1 m RM1= 4 m Rdisk= 5 m
RM2= 1 m RM1= 4 m Rdisk= 5 m
(c) (d)
100 300 500 700 900
(A+D)/2 (A+D)/2
- 0.5 0 0.5 - 0.5 0 0.5
100 300 500 700 900
Figure 5.28: (a)/(c) Plot of the stability parameter and (b)/(d) Gaussian beam radius on the thin disk for a symmetric resonator in (b) and for an unsymmetric resonator in (d).
Design of the folded resonators for thin-disk lasers 81 Resonator design of single-transverse-mode resonators with a small internal focus An internal focus can be achieved when the resonator is folded via a mirror with the ROC ROCM1. For a small ROC this mirror can focus the beam in a distance of approx.ROCM1/2. This focus can be imaged back onto this point by a 2f-2f image by a mirror with anROCM2 in a distance from the mirrorM1of approx.ROCM1/2+ROCM2. The mode size on the mirror M2can be controlled withROCM2. A largerROCM2requires a longer distance from the focus and because of the constant Rayleigh length the beam radius on M2 increases.
Diffraction losses can become relevant for beam radii comparable to the mirror dimensions or for a low AOI onM1. In this case the beam between the two mirrors can overlap with the beam coming from the disk to the first mirror. In addition, already for an AOI on M1 of approx. 5°, the tangential and sagittal beam radii on the mirrorM1 can differ by 10 %.
Caused by the astigmatism, which is also relevant for the pump spot, the probability of an oscillation of higher order modes can be increased. A free space of approx. 10 cm between the two mirrors suggests the use ofROCM2=10 cm. The smallest ROC which was available at the time of the experiments was ROCM1 = 5 cm. Thus, this corresponds to a length betweenM1and M2 of approx. 12.5 cm. With the available mirror mounts, this leads to an AOI on the mirror M2 larger than 3°. A beam, which is central pointed on the mirror M2 necessitates a minimum AOI of approx. 6° for 1-inch optics. To find the working point, also the required Gaussian beam radius on the thin disk of 480 µm has to be considered. This suggests the minimization of a figure of merit (FOM) with the target beam radius on the disk wdisk-want and the required waist radiusw0-want according to
F OM= v u t
wdisk wdisk-want
2
−1 +
w0 w0-want
2
−1
. (5.1)
0.0 0.1 0.2 0.3 0.4
101
Position in resonator (m)
Fundamentalmoderadius (µm)
400 600
Frequency (MHz)
Intensity (a.u)
(a) (b)
M1(f) M2(2f-2f) Disk
L1-disk= 33 cm L
1-2= 12.52 cm
ROCdisk= 5 m ROCM2= 5 cm ROCM2= 10 cm
wdisk= 503 µm wM1= 533 µm wM1= 2114 µm
w0 = 15.8 µm
FSR = 329 MHz
40kHz
102 102 103
Figure 5.29: (a) Caustic of the resonator with a strong internal focus. (b) Frequency spectrum of the resonator. The linewidth has to be scaled in dependence on the resonator losses.
82 Laser experiments and discussion dp(mm) ROCdisk(m) ROCM1(cm) ROCM2 (cm) L1-disk (cm) L1-2(cm)
1.2 5 5 10 33 12.52
1.4 5 7 15 54 18.5
1.8 5 8 13 55 17
2.7 5 11.2 11 88 16.54
3.6 5 15 10 75 17.38
5.4 5 20 20 125 29.75
10 5 30 15 200 29.29
Table 5.7: Suggested resonator designs for different pump spot diameters. The target beam radius on the disk was0.4·dpump and the target waist 20 µm.
The optimization of Equation 5.1 was performed for additional pump spot diameters larger than 1.2 mm to explore the general feasibility for larger pump powers. Inclusion of the ROC of the two OC as parameters into the optimization turned out to be necessary. In a second step, the optimized ROCs were set to common available ROCs and the optimization was performed again. Two examples are shown in Figure 5.32. The desired length of L1-2 would have to be adjusted for a pump spot diameter of 3.6 mm to a precision better than 100 µm. In case of a pump spot diameter of 10 mm the necessary precision is increased to approx. 10 µm. The results of the resonator design and optimization can be obtained from Table 5.7. The alignment sensitivity of this resonator can be gathered from Figure 5.30 and Figure 5.31. In Figure 5.30 (a) the plot of the stability parameter is shown in dependency on L1−d iskand L1−2. It can be seen, that the stable range of L1−2 decreases for an increasing L1−d isk. In Figure 5.30 (b) region plots of different requirements (inequalities) are depicted.
The overlap of the three regions defines the target area of the resonator. The working point of Figure 5.29 is depicted by the horizontal and the vertical lines. A better insight into the
0.2 0.4 0.6 0.8 1.0 0.124
0.126 0.128 0.130 0.132 0.134
(a) (A+D)/2
L 1-2(m)
L1-disk(m)
(b)
L 1-2(m)
L1-disk(m)
wM2< 2.5 mm w0< 16 µm, > 10 µm wdisk< 500 µm, > 450µm
0.2 0.4 0.6 0.8 1.0 0.125
0.127 0.128
0.126 - 1
- 0.5 0 0.5 1
Figure 5.30: (a) Stability parameter for the resonator, shown in Figure 5.29.(b) Region plot for different inequalities of the resonator’s properties.
Design of the folded resonators for thin-disk lasers 83
15 20
5 10 15 20
0.4 1.2 2
Waist M2 M1 Disk
15 20
1 2 3
12.50 12.55 12.60 12.65 12.70
Beam radius on element (mm) Beam radius on element (mm)
(a) (b)
Resonator waist radius (µm)
L1-2(cm) ROCdisk(m)
L1-disk = 33 cm, ROCdisk = 5 m L1-2 = 12.52 cm, L1-disk = 33 cm
Resonator waist radius (µm)
Waist M2 M1 Disk
Figure 5.31: Variation of the beam and waist radius in dependence on the (a) ROC of the disk and (b) the length L1-2between the two focusing mirrors.
stability can be gained from Figure 5.31. In (a) the ROC of the disk is varied and in (b) the distance between the mirror M1 and the mirror M2 is. The remaining parameters are kept constant. An increase of the ROC yields to a lower waist radius, while the radius on the disk is nearly constant. Therefore, this resonator should not exhibit a large sensitivity in respect to changes of the ROC. In contrast, a strong dependence on the length can be seen.
A resonator design for fundamental mode operation and a 20 µm waist can be found for common pump spot diameters. At the same time one has to consider the astigmatism and possible losses by diffraction.
L 1-2(cm)
L1-disk(m)
L 1-2(cm)
L1-disk(m)
(a) (b)
1)ROCM1 = 13.9 cm, ROCM2= 8.1 cm ROCDisk = 5 m , dp= 3.6 mm
2) ROCM1 = 15 cm, ROCM2= 10cm w0< 20 µm, > 15 µm
wdisk< 1500 µm, > 1300 µm
w0< 20 µm, > 15 µm wdisk< 4500 µm, > 4300µm ROCDisk = 5 m , dp= 10 mm
1)ROCM1 = 30.2 cm, ROCM2= 9.5 cm 2) ROCM1 = 30 cm, ROCM2= 15cm Requires AOI < 1 °
Requires AOI < 1 °
1.99 2.00 2.01
29.28 29.29 29.30
0.3 0.5 0.7 0.9 17.35
17.36 17.37 17.38 17.39 17.40
Figure 5.32: Stability zones for resonators with tight focus and a large pump spot of (a) 3.6 mm and (b) 10 mm. Further parameters of the resonators can be found in Table 5.7.
84 Laser experiments and discussion