Abstract

An analysis of the stability and astigmatic compensation of five- and six- or seven-mirror cavities for mode-locked dye lasers and simple relations for the folding angle to get a maximum stability region are given in this paper. Analytical relations referring to equivalent resonators are deduced. We draw attention to the lack of opportunity to use long cavity approximation to obtain stability diagrams and made some considerations on beam waist sizes.

© 1989 Optical Society of America

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References

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  1. H. W. Kogelnik, E. P. Ippen, A. Dienes, C. V. Shank, “Astigmatically Compensated Cavities for CW Dye Lasers,” IEEE J. Quantum Electron. QE-8, 373–379 (1972).
    [CrossRef]
  2. K. K. Li, A. Dienes, J. R. Whinnery, “Stability and Astigmatic Compensation Analysis of Five-Mirror Cavity for Mode-Locked Dye Lasers,” Appl. Opt. 20, 407–411 (1981).
    [CrossRef] [PubMed]
  3. K. K. Li, “Stability and Astigmatic Analysis of a Six-Mirror Ring Cavity for Mode-Locked Dye Lasers,” Appl. Opt. 21, 967–970 (1982).
    [CrossRef] [PubMed]
  4. H. Kogelnik, “Imaging of Optical Modes-Resonators with Internal Lenses,” Bell Syst. Tech. J. 44, 455–494 (1965).
  5. G. Herziger, H. Weber, “Equivalent Optical Resonators,” Appl. Opt. 23, 1450–1452 (1984).
    [CrossRef] [PubMed]

1984 (1)

1982 (1)

1981 (1)

1972 (1)

H. W. Kogelnik, E. P. Ippen, A. Dienes, C. V. Shank, “Astigmatically Compensated Cavities for CW Dye Lasers,” IEEE J. Quantum Electron. QE-8, 373–379 (1972).
[CrossRef]

1965 (1)

H. Kogelnik, “Imaging of Optical Modes-Resonators with Internal Lenses,” Bell Syst. Tech. J. 44, 455–494 (1965).

Dienes, A.

K. K. Li, A. Dienes, J. R. Whinnery, “Stability and Astigmatic Compensation Analysis of Five-Mirror Cavity for Mode-Locked Dye Lasers,” Appl. Opt. 20, 407–411 (1981).
[CrossRef] [PubMed]

H. W. Kogelnik, E. P. Ippen, A. Dienes, C. V. Shank, “Astigmatically Compensated Cavities for CW Dye Lasers,” IEEE J. Quantum Electron. QE-8, 373–379 (1972).
[CrossRef]

Herziger, G.

Ippen, E. P.

H. W. Kogelnik, E. P. Ippen, A. Dienes, C. V. Shank, “Astigmatically Compensated Cavities for CW Dye Lasers,” IEEE J. Quantum Electron. QE-8, 373–379 (1972).
[CrossRef]

Kogelnik, H.

H. Kogelnik, “Imaging of Optical Modes-Resonators with Internal Lenses,” Bell Syst. Tech. J. 44, 455–494 (1965).

Kogelnik, H. W.

H. W. Kogelnik, E. P. Ippen, A. Dienes, C. V. Shank, “Astigmatically Compensated Cavities for CW Dye Lasers,” IEEE J. Quantum Electron. QE-8, 373–379 (1972).
[CrossRef]

Li, K. K.

Shank, C. V.

H. W. Kogelnik, E. P. Ippen, A. Dienes, C. V. Shank, “Astigmatically Compensated Cavities for CW Dye Lasers,” IEEE J. Quantum Electron. QE-8, 373–379 (1972).
[CrossRef]

Weber, H.

Whinnery, J. R.

Appl. Opt. (3)

Bell Syst. Tech. J. (1)

H. Kogelnik, “Imaging of Optical Modes-Resonators with Internal Lenses,” Bell Syst. Tech. J. 44, 455–494 (1965).

IEEE J. Quantum Electron. (1)

H. W. Kogelnik, E. P. Ippen, A. Dienes, C. V. Shank, “Astigmatically Compensated Cavities for CW Dye Lasers,” IEEE J. Quantum Electron. QE-8, 373–379 (1972).
[CrossRef]

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Figures (8)

Fig. 1
Fig. 1

Cavity configurations for mode-locked dye lasers with (a) five, (b) six, and (c) seven mirrors. The short folded arms denoted by I and II contain the cell or jet streams at Brewster angle.

Fig. 2
Fig. 2

Equivalent empty resonator for the five-mirror cavity. ( M 1 , M 2 ) is its equivalent two-mirror resonator.

Fig. 3
Fig. 3

Stability regions of a five-mirror cavity obtained either by analytical or matrix formalism for sagittal and tangential planes. Intersection is the stable region. Parameters used are: l = 60 cm, l3 = 60 cm, R1 = 5 cm, R2 = 10 cm, R4 = 10 cm, t1 = t2 = 0, θ = 6.57°.

Fig. 4
Fig. 4

Stability diagram of a five-mirror cavity. Intersection is the stable region. Parameters used are the same as in Fig. 3, but the thicknesses t1 and t2 of the dye streams are t1 = 0.075 cm, t2 = 0.3 cm which satisfy Eq. (6).

Fig. 5
Fig. 5

Equivalent resonator for the six- or seven-mirror ring cavity.

Fig. 6
Fig. 6

Stability region of a six- or seven-mirror ring cavity obtained either by analytical or detailed matrix formalism for sagittal (vertical lines) and tangential planes (horizontal lines). Intersection is the stable region. Parameters used are: l1 = 60 cm, l2 = 180 cm, R1 = 10 cm, R2 = 10 cm, θ = 6.5°, t1 = t2 = 0.

Fig. 7
Fig. 7

The square ratio of the waist sizes ω2 and ω1 of the beam in the folded arms II and I, respectively, of a five-mirror cavity calculated for pairs (δ1,δ2) arranged on curves 1, 2, and 3 taken in the middle of the major stability regions a, c, and e, respectively. Solid and broken lines indicate the ratio values greater and less than unity, respectively. Parameters used are the same as in Fig. 3.

Fig. 8
Fig. 8

The square ratio of the waist sizes ω2 and ω1 to the beam in the short folded arms of a six- or seven-mirror ring cavity calculated for pairs (δ1,δ2) arranged on curves 1 and 2 taken in the middle of the major stability regions a, b, c, and d, respectively. Solid and broken lines indicate the ratio values greater and less than unity, respectively. Parameters used are the same as in Fig. 6.

Equations (17)

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f t = ( R cos θ ) / 2 ,
f s = R / ( 2 cos θ ) ,
d 1 = R 1 + f 2 + δ 1 ,
d 2 = f 3 + f 4 + δ 2
d 1 = 2 f 1 + δ 1 ,
d 2 = 2 f 2 + δ 2
θ = A cos [ ( t 1 N + t 1 2 N 2 + R 2 2 ) / R 2 ] = A cos { [ t 2 N + t 2 2 N 2 + ( R 3 + R 4 ) 2 ] / ( R 3 + R 4 ) }
θ = A cos [ ( t 1 N + t 1 2 N 2 + R 1 2 ) / R 1 ] = A cos [ ( t 2 N + t 2 2 N 2 + R 2 2 ) / R 2 ]
t 1 / R 2 = t 2 / ( R 3 + R 4 )
t 1 / R 1 = t 2 / R 2
R 1 = R 1 f 2 2 / δ 1 ( R 1 + δ 1 ) ,
R 2 = f 3 2 f 4 2 / δ 2 [ δ 2 ( l 3 f 4 ) f 4 2 ] ,
I = l f 2 f 3 f 2 2 / ( R 1 + δ 1 ) f 3 2 ( l 3 f 4 ) / [ δ 2 ( l 3 f 4 ) f 4 2 ] .
M 0 = M 1 M 2 M 8 = ( A 0 B 0 C 0 D 0 ) ,
| ( A 0 + D 0 ) / 2 | 1 .
A 0 = 1 + δ 1 / f 1 + δ 2 / f 2 + δ 1 ( 2 f 2 l 1 l 2 ) / f 1 2 l 2 δ 2 / f 2 2 + δ 1 δ 2 [ 1 / f 1 2 + 1 / ( f 1 f 2 ) ( l 1 + l 2 ) / ( f 1 2 f 2 ) l 2 / ( f 1 f 2 2 ) + l 1 l 2 / ( f 1 2 f 2 2 ) ] ,
D 0 = 1 + δ 1 / f 1 + δ 2 / f 2 + 2 f 1 δ 2 / f 2 2 l 1 δ 2 / ( f 1 f 2 ) + δ 1 δ 2 [ 1 / ( f 1 f 2 ) + 1 / f 2 2 l 1 / ( f 1 2 f 2 ) ] .

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