Abstract

A resonator with an internal lens can be characterized by its equivalent g-parameters which give the regions of stability if the lens has variable focal length. However, other properties are not given by this equivalence and have to be investigated by means of the resonator matrix method. Beam diameter, divergence angle, number of transversal modes, positions of beam waists, stability, and the correlations of these parameters with position and focal length of the internal lens are discussed. Regions of low sensitivity to variations of the focal length are evaluated and compared with experimental results. For some special resonator configurations numerical results are presented.

© 1981 Optical Society of America

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References

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  1. W. Koechner, Solid State Laser Engineering (Springer, New York, 1976).
  2. J. P. Lörtscher, J. Steffen, G. Herziger, Opt. Quantum Electron. 7, 505 (1975).
    [CrossRef]
  3. J. Steffen, J. P. Lörtscher, G. Herziger, IEEE J. Quantum Electron. QE-8, 239 (1972).
    [CrossRef]
  4. H. Kogelnik, Bell Syst. Tech. J. 44, 455 (1965).
  5. N. Kurauchi, W. Kahn, Appl. Opt. 5, 1023 (1966).
    [CrossRef] [PubMed]
  6. H. Kogelnik, T. Li, Proc. IEEE 54, 1312 (1966).
    [CrossRef]
  7. A. Yariv, Quantum Electronics (Wiley, New York, 1975).
  8. R. Iffländer, H. P. Kortz, H. Weber, Opt. Commun. 29, 223 (1979).
    [CrossRef]
  9. W. Kleen, R. Müller, Laser (Springer, New York, 1969).
    [CrossRef]
  10. A. E. Siegman, An Introduction to Lasers and Masers (McGraw-Hill, New York, 1971).

1979 (1)

R. Iffländer, H. P. Kortz, H. Weber, Opt. Commun. 29, 223 (1979).
[CrossRef]

1975 (1)

J. P. Lörtscher, J. Steffen, G. Herziger, Opt. Quantum Electron. 7, 505 (1975).
[CrossRef]

1972 (1)

J. Steffen, J. P. Lörtscher, G. Herziger, IEEE J. Quantum Electron. QE-8, 239 (1972).
[CrossRef]

1966 (2)

1965 (1)

H. Kogelnik, Bell Syst. Tech. J. 44, 455 (1965).

Herziger, G.

J. P. Lörtscher, J. Steffen, G. Herziger, Opt. Quantum Electron. 7, 505 (1975).
[CrossRef]

J. Steffen, J. P. Lörtscher, G. Herziger, IEEE J. Quantum Electron. QE-8, 239 (1972).
[CrossRef]

Iffländer, R.

R. Iffländer, H. P. Kortz, H. Weber, Opt. Commun. 29, 223 (1979).
[CrossRef]

Kahn, W.

Kleen, W.

W. Kleen, R. Müller, Laser (Springer, New York, 1969).
[CrossRef]

Koechner, W.

W. Koechner, Solid State Laser Engineering (Springer, New York, 1976).

Kogelnik, H.

H. Kogelnik, T. Li, Proc. IEEE 54, 1312 (1966).
[CrossRef]

H. Kogelnik, Bell Syst. Tech. J. 44, 455 (1965).

Kortz, H. P.

R. Iffländer, H. P. Kortz, H. Weber, Opt. Commun. 29, 223 (1979).
[CrossRef]

Kurauchi, N.

Li, T.

H. Kogelnik, T. Li, Proc. IEEE 54, 1312 (1966).
[CrossRef]

Lörtscher, J. P.

J. P. Lörtscher, J. Steffen, G. Herziger, Opt. Quantum Electron. 7, 505 (1975).
[CrossRef]

J. Steffen, J. P. Lörtscher, G. Herziger, IEEE J. Quantum Electron. QE-8, 239 (1972).
[CrossRef]

Müller, R.

W. Kleen, R. Müller, Laser (Springer, New York, 1969).
[CrossRef]

Siegman, A. E.

A. E. Siegman, An Introduction to Lasers and Masers (McGraw-Hill, New York, 1971).

Steffen, J.

J. P. Lörtscher, J. Steffen, G. Herziger, Opt. Quantum Electron. 7, 505 (1975).
[CrossRef]

J. Steffen, J. P. Lörtscher, G. Herziger, IEEE J. Quantum Electron. QE-8, 239 (1972).
[CrossRef]

Weber, H.

R. Iffländer, H. P. Kortz, H. Weber, Opt. Commun. 29, 223 (1979).
[CrossRef]

Yariv, A.

A. Yariv, Quantum Electronics (Wiley, New York, 1975).

Appl. Opt. (1)

Bell Syst. Tech. J. (1)

H. Kogelnik, Bell Syst. Tech. J. 44, 455 (1965).

IEEE J. Quantum Electron. (1)

J. Steffen, J. P. Lörtscher, G. Herziger, IEEE J. Quantum Electron. QE-8, 239 (1972).
[CrossRef]

Opt. Commun. (1)

R. Iffländer, H. P. Kortz, H. Weber, Opt. Commun. 29, 223 (1979).
[CrossRef]

Opt. Quantum Electron. (1)

J. P. Lörtscher, J. Steffen, G. Herziger, Opt. Quantum Electron. 7, 505 (1975).
[CrossRef]

Proc. IEEE (1)

H. Kogelnik, T. Li, Proc. IEEE 54, 1312 (1966).
[CrossRef]

Other (4)

A. Yariv, Quantum Electronics (Wiley, New York, 1975).

W. Kleen, R. Müller, Laser (Springer, New York, 1969).
[CrossRef]

A. E. Siegman, An Introduction to Lasers and Masers (McGraw-Hill, New York, 1971).

W. Koechner, Solid State Laser Engineering (Springer, New York, 1976).

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Figures (18)

Fig. 1
Fig. 1

Spherical resonator with the lenslike active medium.

Fig. 2
Fig. 2

Stability diagram with the lines of increasing refractive power D: g1,g2, position of the resonator if D = 0.

Fig. 3
Fig. 3

Beam envelopes inside the spherical resonator near the regions of instability.

Fig. 4
Fig. 4

Fundamental mode divergence Θ01, Θ02 vs refractive power D (qualitatively): (a) spherical resonator; (b) plane–plane resonator.

Fig. 5
Fig. 5

Beam envelopes inside the plane–plane resonator near the regions of instability.

Fig. 6
Fig. 6

Multimode divergence Θm2 vs refractive power D for a plane–plane resonator. Lens position, d2 = 0.5 m; radius of aperture, R = 0.005 m; parameter, d1/m; Θ2,l = upper limit of divergence according to Eq. (27).

Fig. 7
Fig. 7

Normalized relation between the multimode divergence Θmi vs refractive power D. Parameter is the lens position δj = dj/L. R = radius of aperture, L = d1 + d2. Inserted are experimental data (Sec. IV). Θi,l = upper limit of divergence.

Fig. 8
Fig. 8

Beam envelope inside the plane–spherical resonator near the regions of instability.

Fig. 9
Fig. 9

Fundamental mode divergence Θ01, Θ02 vs refractive power D (qualitatively) for the plane–spherical resonator (S1 = plane mirror): (a) DII < DIII; (b) DII > DIII: —, Θ01; - - - -, Θ02; – · – · –, Θ01, Θ02 for DII = DIII.

Fig. 10
Fig. 10

Fundamental mode divergence Θ01 (flat mirror output) vs refractive power D for the plane–spherical resonator. R = radius of aperture = 0.003 m; radius of curvature of mirror S2: ρ2 = 2 m: (a) parameter d1/m = lens position, d2 = 0.5 m; (b) parameter d2/m, d1 = 0.5 m.

Fig. 11
Fig. 11

Multimode divergence of the plane–spherical resonator: d1 = 0.06 m; d2 = parameter; R = 0.003 m; - - - -, Θm1; —, Θm2. Number in brackets indicate d2 for Θm1.

Fig. 12
Fig. 12

Oscilloscope traces showing the refractive power: (a) warm-up and cooling phase; (b) several pulses in the stationary state; (c) upper trace: increasing refractive power during the pulse; lower trace: intensity of the flashlamp pulse.

Fig. 13
Fig. 13

Experimental setup for measuring the refractive power of a laser rod. PIN-diode monitors the intensity variations of an expanded He–Ne probe laser. I0 is the measured intensity for D = 0.

Fig. 14
Fig. 14

Multimode divergence of different resonators with Nd:glass rod type Schott LG 706, 6 × 100 mm; d1 = 0.033 m; ρ1 = plane; ρ2 = plane (□), 5 m (○), and 2 m (△): (a) d2 = 0.15 m; (b) d2 = 0.30 m; (c) d2 = 0.45 m; (d) d2 = 0.60 m; —, theory.

Fig. 15
Fig. 15

Multimode divergence of different resonators with Nd:YAG rod, 6 × 90 mm; d1 = 0.025 m; ρ1 = plane; ρ2 = plane (□), 5 m (△), 3 m (◇), 2 m (○): (a) d2 = 0.165 m; (b) d2 = 0.64 m; —, theory.

Fig. 16
Fig. 16

Comparison of experimental values with theoretical predictions for D = 0, L = d1 + d2.

Fig. 17
Fig. 17

Multimode divergence of a directly coated plane–plane Nd:YAG rod, 6 × 90 mm vs pumping power P: ○, experimental; —, theory.

Fig. 18
Fig. 18

Multimode divergence of four different directly coated plane–spherical Nd:glass rods (rods 1 … 4 from Table III): △, □, ○, ◇: experiment; —, theory.

Tables (3)

Tables Icon

Table I Instabilities and Critical Values of the Refractive Power or g*-Parameters, Respectively, for the Spherical Resonator

Tables Icon

Table II Asymptotic Values of the Beam Parameters at the Instabilities for the Spherical (sp), the Plane–Plane (pl), and the Plane–Spherical (pl/sph) Resonator(ρ1 = ∞)

Tables Icon

Table III Characteristic Values of P0,Pc, for Different Laser Rods a

Equations (55)

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M R ( E i ) = | A i B i C i D i | .
1 q i = ( D i - A i ) - i ^ [ 4 - ( A i + B i ) 2 ] 1 / 2 2 B i = 1 ρ i - i ^ λ π · w i 2 ,
1 q i , ext = n m ( D i - A i ) - i ^ n [ 4 - ( A i + D i ) 2 ] 1 / 2 2 B i ,
w i 2 = λ 0 π · n · 2 B i [ 4 - ( A i + D i ) 2 ] 1 / 2 ,
1 q i ( z ) = 1 q i , ext + z = 1 ρ i , ext ( z ) - i ^ · λ 0 w i , ext 2 ( z ) .
Θ o i , ext 2 = λ 0 π n m 2 ( D i - A i ) 2 + n 2 [ 4 - ( A i + D i ) 2 ] 2 B i n · [ 4 - ( A i + D i ) 2 ] 1 / 2 ,
Θ o i , ext = w i 2 B i · { n m 2 ( D i - A i ) 2 + n 2 [ 4 - ( A i + D i ) 2 ] 1 / 2 } .
Θ ω m i = ( m + 1 / 2 ) 1 / 2 · w i ,
w Θ m i = ( m + 1 / 2 ) 1 / 2 · Θ o i ,
m = R 2 / w o L 2 - 1 / 2 ,
Θ m i , ext = Θ o i , ext ( R / w o L )             R > w o L ,
Θ m i , ext = R 2 B i · w i w o L { n m 2 ( D i - A i ) 2 + n 2 [ 4 - ( A i + D i ) 2 ] } 1 / 2 .
M th = | cos b l n r sin b b - n r l · b · sin b cos b | , b 2 = P / P c ,
D = n r l · b 2 ( 1 - 1 6 b 2 + ) , h 1 = h 2 = l 2 n r ( 1 + 1 12 b 2 + ) ,
g i = 1 - ( d 1 + d 2 ) / ρ i ,
g i * = g i - D d j ( 1 - d i / ρ i ) .
g i * = 1 - L * / ρ i * ;
L * = d 1 + d 2 - D d 1 d 2 ;
ρ i * = ρ i L * / ( D d j ρ i + L * ) .
w i 2 = λ 0 L * π · [ g j * g i * ( 1 - g 1 * g 2 * ) ] 1 / 2 ,
A i = 2 g 1 * g 2 * - 1 + 2 L * · g j * / ρ i ,
B i = 2 g j * L * ,
D i = 2 g 1 * g 2 * - 1 - 2 L * · g j * / ρ i ,
C i = A i B i / D i ,
1 q i = 1 ρ i - i ^ [ g 1 * g 2 * ( 1 - g 1 * g 2 * ) ] 1 / 2 g j * · L * .
Θ o i , ext 2 = λ 0 π L * n · n m 2 g j * ( L * / ρ i ) 2 + n 2 g i * ( 1 - g 1 * g 2 * ) [ g 1 * g 2 * ( 1 - g 1 * g 2 * ) ] 1 / 2 .
w o i 2 = λ 0 L * π n [ g 1 * g 2 * ( 1 - g 1 * g 2 * ) ] 1 / 2 g j * ( L * / ρ i ) 2 + g i * ( 1 - g 1 * g 2 * ) ,
L o i = L * g j * ( L * / ρ i ) g j * ( L * / ρ i ) 2 + g i * ( 1 - g 1 * g 2 * ) .
Θ o i = λ o / π w o i .
w o L 2 = w o i 2 [ 1 + ( d i - L o i ) 2 / z R i 2 ]
z R i = n π w o i 2 / λ 0 .
( Θ m i , ext L * R ) 2 = [ n m 2 g j * ( L * / ρ i ) 2 + n 2 g i * ( 1 - g 1 * g 2 * ) ] · [ g j * ( L * / ρ i ) 2 + g i * ( 1 - g 1 * g 2 * ) ] g 1 * g 2 * ( 1 - g 1 * g 2 * ) + [ d i L * · { g j * ( L * / ρ i ) 2 + g i * ( 1 - g 1 * g 2 * ) } - g j * ( L * / ρ i ) ] 2 .
Θ i , l , s 2 = R 2 ± { R 4 - [ 2 λ 0 ( d i - L o i ) / π ] 2 } 1 / 2 2 ( d i - L o i ) 2 .
Θ i , l = R / ( d i - L o i ) ,
Θ i , s = λ 0 / π · R .
g 1 * · g 2 * = 1             g 1 * · g 2 * = 0.
Θ m i L R = [ D L ( 1 - δ j D L ) 1 - 2 ( 1 - δ j ) δ j D L ] 1 / 2
D min max · L = 1 2 δ 1 δ 2 · [ 1 ± ( 2 δ j - 1 ) 1 / 2 ]
( Θ m i , min max · L R ) 2 = δ j ± ( 2 δ i - 1 ) 1 / 2 2 δ i · δ 1 · δ 2 .
Θ i , s · L R = λ 0 · L π · R 2 = 1 π · F ,
Θ i , l · L R = 1 δ i .
d 1 / d 2 < 1 - d 2 / ρ 2 .
( Θ m 1 · L R ) 2 = [ ( 1 - δ 1 δ 2 · D L ) 2 · g 2 * g 1 * ( 1 - g 1 * g 2 * ) + δ 1 2 ] - 1 ,
δ 1 + δ 2 = 1 ,
g 1 * = 1 - δ 2 · D L ,
g 2 * = 1 - δ 1 · D L - γ 2 ( 1 - δ 1 δ 2 D L ) ,
δ i = d i / L             γ 2 = L / ρ 2             L = d 1 + d 2 .
M th , int = | cos b l b · sin b - b l sin b cos b | .
A 1 = D 1 = cos 2 b - l ρ 2 sin 2 b b ,
B 1 = l ( sin 2 b b l ρ 2 · 1 - cos 2 b b 2 ) ,
C 1 = A 1 D 1 / B 1 .
( Θ m 1 · l R n r ) 2 = 1 - ( cos 2 b - l ρ 2 sin 2 b b ) 2 ( sin 2 b b - l ρ 2 1 - cos 2 b b 2 ) 2 .
Θ m 1 · l R · n r = ( D · l / n r ) 1 / 2 .
( Θ m 1 · l R · n r ) 2 l / ρ 2 1 - l / ρ 2 { 1 + D ρ 2 / n r 1 - l / ρ 2 · [ 1 - 2 l 3 ρ 2 ( 1 - l / ρ 2 ) ] } .
D = D 0 ( P / P 0 ) α .

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