TABLE I
M = 1 eigenvalues. The magnitudes and phases of the eigenvalues for the six lowest loss modes with N = 10, M = 1 are calculated and compared with numerical results and predictions from Weinstein. The numerical results are taken from a graph in Ref. 2. |λ|,θ = present results; |λI|,θI = numerical integration1,2; |λw|,θw = Weinstein predictions.14
ηl | |λ| | |λI| | |λw| | θ | θI | θw |
---|
00 | 0.996 | 0.996 | 0.996 | 2.34 | 2.3 | 2.33 |
01 | 0.989 | 0.989 | 0.989 | 5.95 | 6.0 | 5.93 |
02 | 0.980 | | 0.981 | 10.67 | | 10.64 |
10 | 0.978 | | 0.977 | 12.33 | | 12.30 |
03 | 0.969 | | 0.970 | 16.48 | | 16.43 |
11 | 0.964 | | 0.963 | 19.91 | | 19.86 |
TABLE II
Magnitudes of eigenvalues for cavities with curved mirrors. The magnitudes of the full eigenvalue γ of the six lowest loss modes for the N = 10 resonator with various marror curvatures are given. The curvature is given in terms of g − 1 = −L/R. The stable region corresponds to g − 1 = 0.
g − 1 | −0.010 | −0.008 | −0.006 | −0.004 | −0.002 | 0 | 0.002 | 0.004 | 0.006 | 0.008 | 0.010 |
---|
| | | | | | | | | | | |
---|
ηl | | | | | | | | | | | |
---|
00 | 0.9999 | 0.9998 | 0.9996 | 0.9990 | 0.9978 | 0.9956 | 0.9913 | 0.9840 | 0.9744 | 0.9632 | 0.9508 |
01 | 0.9991 | 0.9984 | 0.9973 | 0.9955 | 0.9927 | 0.9889 | 0.9834 | 0.9762 | 0.9672 | 0.9574 | 0.9451 |
02 | 0.9966 | 0.9948 | 0.9923 | 0.9889 | 0.9846 | 0.9798 | 0.9736 | 0.9660 | 0.9573 | 0.9431 | 0.9245 |
10 | 0.9949 | 0.9925 | 0.9894 | 0.9856 | 0.9812 | 0.9777 | 0.9742 | 0.9718 | 0.9705 | 0.9635 | 0.9476 |
03 | 0.9913 | 0.9883 | 0.9845 | 0.9798 | 0.9744 | 0.9691 | 0.9626 | 0.9551 | 0.9441 | 0.9419 | 0.8504 |
11 | 0.9857 | 0.9816 | 0.9770 | 0.9722 | 0.9675 | 0.9643 | 0.9607 | 0.9572 | 0.9541 | 0.9486 | 0.9139 |
TABLE III
Phases of eigenvalues for cavities with curved mirrors. The phases in degrees of the eigenvalue γ for the modes in Table II are given.
g − 1 | −0.010 | −0.008 | −0.006 | −0.004 | −0.002 | 0 | 0.002 | 0.004 | 0.006 | 0.008 | 0.010 |
---|
| | | | | | | | | | | |
---|
ηl | | | | | | | | | | | |
---|
00 | 8.12 | 7.27 | 6.31 | 5.21 | 3.92 | 2.34 | 0.36 | −2.11 | −5.12 | −8.62 | −12.55 |
01 | 16.26 | 14.59 | 12.76 | 10.73 | 8.47 | 5.95 | 3.09 | −0.10 | −3.59 | −7.39 | −11.48 |
02 | 24.52 | 22.12 | 19.55 | 16.79 | 13.83 | 10.67 | 7.20 | 3.52 | −0.02 | −3.98 | −7.49 |
10 | 24.64 | 22.34 | 19.92 | 17.41 | 14.85 | 12.33 | 9.80 | 7.51 | 5.43 | 3.55 | 5.31 |
03 | 33.03 | 30.04 | 26.89 | 23.57 | 20.09 | 16.48 | 12.57 | 8.53 | 4.42 | 3.63 | 5.31 |
11 | 33.59 | 30.88 | 28.12 | 25.35 | 22.59 | 19.91 | 17.14 | 14.50 | 11.93 | 9.18 | 7.49 |
TABLE IV
s parameter for marginally stable cavities. The magnitude of the complex parameter |s| is given for each of the modes in Table II. The geometric-optics result is |s| = 2n + l, and is approximated by the stable cavities. Near M = 1, |s| becomes large as the core term approaches a Bessel function.
g − 1 | −0.010 | −0.008 | −0.006 | −0.004 | −0.002 | 0 | 0.002 | 0.004 | 0.006 | 0.008 | 0.010 |
---|
| | | | | | | | | | | |
---|
ηl | | | | | | | | | | | |
---|
00 | 0.002 | 0.003 | 0.007 | 0.020 | 0.087 | ∞ | 0.867 | 0.918 | 1.117 | 1.383 | 1.679 |
01 | 1.005 | 1.012 | 1.033 | 1.095 | 1.342 | ∞ | 1.124 | 0.731 | 0.901 | 1.213 | 1.540 |
02 | 2.024 | 2.051 | 2.115 | 2.279 | 2.827 | ∞ | 2.069 | 0.922 | 0.601 | 0.768 | 1.026 |
10 | 2.039 | 2.081 | 2.174 | 2.402 | 3.113 | ∞ | 2.768 | 1.615 | 1.129 | 0.859 | 0.698 |
03 | 3.074 | 3.144 | 3.285 | 3.605 | 4.562 | ∞ | 3.491 | 1.734 | 0.850 | 0.727 | 0.672 |
11 | 3.143 | 3.261 | 3.484 | 3.957 | 5.260 | ∞ | 4.744 | 2.875 | 1.985 | 1.393 | 0.994 |
TABLE V
Magnitudes of eigenvalues for M = 1.02 cavities. The magnitudes of the full eigenvalue γ for six low order modes are given for various Fresnel numbers.
N | 10 | 20 | 30 | 40 | 50 |
---|
| | | | | |
---|
ηl | | | | | |
---|
00 | 0.9953 | 0.9979 | 0.9984 | 0.9984 | 0.9983 |
01 | 0.9885 | 0.9952 | 0.9968 | 0.9973 | 0.9975 |
02 | 0.9793 | 0.9917 | 0.9948 | 0.9959 | 0.9964 |
10 | 0.9774 | 0.9910 | 0.9946 | 0.9962 | 0.9971 |
03 | 0.9686 | 0.9875 | 0.9924 | 0.9943 | 0.9953 |
11 | 0.9641 | 0.9858 | 0.9916 | 0.9941 | 0.9954 |
TABLE VI
Phases of eigenvalues for M = 1.02 cavities. The phase shifts for the modes in Table V are given.
N | 10 | 20 | 30 | 40 | 50 |
---|
| | | | | |
---|
ηl | | | | | |
---|
00 | 2.16 | 0.86 | 0.28 | −0.15 | −0.54 |
01 | 5.68 | 2.56 | 1.30 | 0.50 | −0.12 |
02 | 10.33 | 4.89 | 2.79 | 1.55 | 0.65 |
10 | 12.06 | 5.91 | 3.63 | 2.38 | 1.57 |
03 | 16.08 | 7.81 | 4.71 | 2.94 | 1.71 |
11 | 19.62 | 9.80 | 6.24 | 4.31 | 3.06 |