Abstract

The magnitudes and locations of the beam waists in both the sagittal and the tangential planes have been found by means of the ABCD matrix method for a triangular resonator. Equilateral and isosceles resonators are discussed, and curves are given from which resonators with astigmatism-free beams can be designed. A frequency-doubled triangular Nd ring laser has been constructed after this design, and it is demonstrated that this laser emits a single longitudinal mode with a circular TEM00 Gaussian beam.

© 2000 Optical Society of America

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References

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  1. W. W. Rigrod, “The optical ring resonator,” Bell Syst. Tech. J. 44, 907–916 (1965).
    [CrossRef]
  2. 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]
  3. N. Jamasbi, J. C. Diels, L. Sarger, “Study of a linear femtosecond laser in passive and hybrid operation,” J. Mod. Opt. 35, 1891–1906 (1988).
    [CrossRef]
  4. R. W. Dunn, “Design of a triangular active ring laser 13 m on a side,” Appl. Opt. 37, 6405–6409 (1998).
    [CrossRef]
  5. T. Yoshino, T. Sumimoto, “Polarization properties of a triangular ring laser having a discharge tube with Brewster angle windows,” Appl. Opt. 9, 1831–1833 (1970).
    [CrossRef] [PubMed]
  6. J. J. Liang, S. T. Lau, M. H. Leary, J. M. Ballantyne, “Unidirectional operation of waveguide diode ring lasers,” Appl. Phys. Lett. 70, 1192–1194 (1997).
    [CrossRef]
  7. A. Yariv, Optical Electronics, 4th ed. (Saunders, Philadelphia, Pa., 1991).
  8. F. A. Jenkins, H. E. White, Fundamentals of Optics (McGraw-Hill, Tokyo, 1976).

1998 (1)

1997 (1)

J. J. Liang, S. T. Lau, M. H. Leary, J. M. Ballantyne, “Unidirectional operation of waveguide diode ring lasers,” Appl. Phys. Lett. 70, 1192–1194 (1997).
[CrossRef]

1988 (1)

N. Jamasbi, J. C. Diels, L. Sarger, “Study of a linear femtosecond laser in passive and hybrid operation,” J. Mod. Opt. 35, 1891–1906 (1988).
[CrossRef]

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]

1970 (1)

1965 (1)

W. W. Rigrod, “The optical ring resonator,” Bell Syst. Tech. J. 44, 907–916 (1965).
[CrossRef]

Ballantyne, J. M.

J. J. Liang, S. T. Lau, M. H. Leary, J. M. Ballantyne, “Unidirectional operation of waveguide diode ring lasers,” Appl. Phys. Lett. 70, 1192–1194 (1997).
[CrossRef]

Diels, J. C.

N. Jamasbi, J. C. Diels, L. Sarger, “Study of a linear femtosecond laser in passive and hybrid operation,” J. Mod. Opt. 35, 1891–1906 (1988).
[CrossRef]

Dienes, A.

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]

Dunn, R. W.

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]

Jamasbi, N.

N. Jamasbi, J. C. Diels, L. Sarger, “Study of a linear femtosecond laser in passive and hybrid operation,” J. Mod. Opt. 35, 1891–1906 (1988).
[CrossRef]

Jenkins, F. A.

F. A. Jenkins, H. E. White, Fundamentals of Optics (McGraw-Hill, Tokyo, 1976).

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]

Lau, S. T.

J. J. Liang, S. T. Lau, M. H. Leary, J. M. Ballantyne, “Unidirectional operation of waveguide diode ring lasers,” Appl. Phys. Lett. 70, 1192–1194 (1997).
[CrossRef]

Leary, M. H.

J. J. Liang, S. T. Lau, M. H. Leary, J. M. Ballantyne, “Unidirectional operation of waveguide diode ring lasers,” Appl. Phys. Lett. 70, 1192–1194 (1997).
[CrossRef]

Liang, J. J.

J. J. Liang, S. T. Lau, M. H. Leary, J. M. Ballantyne, “Unidirectional operation of waveguide diode ring lasers,” Appl. Phys. Lett. 70, 1192–1194 (1997).
[CrossRef]

Rigrod, W. W.

W. W. Rigrod, “The optical ring resonator,” Bell Syst. Tech. J. 44, 907–916 (1965).
[CrossRef]

Sarger, L.

N. Jamasbi, J. C. Diels, L. Sarger, “Study of a linear femtosecond laser in passive and hybrid operation,” J. Mod. Opt. 35, 1891–1906 (1988).
[CrossRef]

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]

Sumimoto, T.

White, H. E.

F. A. Jenkins, H. E. White, Fundamentals of Optics (McGraw-Hill, Tokyo, 1976).

Yariv, A.

A. Yariv, Optical Electronics, 4th ed. (Saunders, Philadelphia, Pa., 1991).

Yoshino, T.

Appl. Opt. (2)

Appl. Phys. Lett. (1)

J. J. Liang, S. T. Lau, M. H. Leary, J. M. Ballantyne, “Unidirectional operation of waveguide diode ring lasers,” Appl. Phys. Lett. 70, 1192–1194 (1997).
[CrossRef]

Bell Syst. Tech. J. (1)

W. W. Rigrod, “The optical ring resonator,” Bell Syst. Tech. J. 44, 907–916 (1965).
[CrossRef]

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]

J. Mod. Opt. (1)

N. Jamasbi, J. C. Diels, L. Sarger, “Study of a linear femtosecond laser in passive and hybrid operation,” J. Mod. Opt. 35, 1891–1906 (1988).
[CrossRef]

Other (2)

A. Yariv, Optical Electronics, 4th ed. (Saunders, Philadelphia, Pa., 1991).

F. A. Jenkins, H. E. White, Fundamentals of Optics (McGraw-Hill, Tokyo, 1976).

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Figures (10)

Fig. 1
Fig. 1

(a) General triangular resonator with three different mirror radii, R 1, R 2, and R 3, side lengths, a, b, and c, and angles, α1, α2, and α3. (b) One period of the equivalent lens waveguide for light running anticlockwise in the resonator. When the ABCD matrix elements are computed, the round trip is started and ended at the black spots just in front of f 1.

Fig. 2
Fig. 2

(a) Equilateral triangular resonator with identical mirrors. (b) Equivalent lens waveguide, which is identical to that of a simple two-mirror symmetrical resonator.

Fig. 3
Fig. 3

Beam waist ratio w oS /w oT between waists in the sagittal (S) and the tangential (T) planes versus the ratio a/ R between side length and mirror radius for an equilateral resonator with one plane mirror and two identical spherical mirrors. Solid curves, the waist ratio for the beam between the identical mirrors; dashed curves, the waist ratio for the beam waist located at the plane mirror.

Fig. 4
Fig. 4

Beam waist ratio w oS /w oT between waists in the sagittal (S) and the tangential (T) planes versus the ratio a/ R between side length a and mirror radius R for an isosceles resonator (90°–45°–45° triangle) with one spherical mirror with radius R 1 and two identical spherical mirrors with radii R, where R 1/R = 1.75. Solid curves, the waist ratio for the beam between the identical mirrors; dashed curves, the waist ratio for the two beams located between mirrors R 1 and R.

Fig. 5
Fig. 5

Difference between beam waist locations z S in the sagittal plane and z T in the tangential plane divided by mirror radius R versus a/ R for the resonator in Fig. 4.

Fig. 6
Fig. 6

Beam waist ratio w oS /w oT between waists in the sagittal (S) and the tangential (T) planes versus the ratio a/ R between side length a and mirror radius R for an isosceles resonator (90°–45°–45° triangle) with one plane mirror and two identical spherical mirrors. Solid curves, the waist ratio for the beam between the identical mirrors; dashed curves, the waist ratio for the beam waist located at the plane mirror.

Fig. 7
Fig. 7

Triangular ring laser design for a diode-laser- (DL-) pumped Nd:YVO4 laser. The isosceles resonator with angles 2u = 45° is used, and the laser becomes single frequency because of inclusion of a Faraday rotator with a TGG crystal. Furthermore, the laser radiation is frequency doubled by the KTP crystal, which is placed in the beam waist whose astigmatism has been compensated for (see Fig. 6).

Fig. 8
Fig. 8

(a) Beam profiles for the green light emitted from the ring laser shown in Fig. 7. Both vertical and horizontal profiles are shown, indicating that the beam is a TEM00 Gaussian beam. (b) Fabry–Perot scan showing that only one longitudinal mode is emitted. The free spectral range of the interferometer is 2 GHz.

Fig. 9
Fig. 9

(a) Equilateral resonator with one plane mirror and two identical spherical mirrors. (b) Equivalent lens waveguide.

Fig. 10
Fig. 10

(a) Isosceles resonator with one spherical mirror with radius R 1 and two identical mirrors with radii R. (b) Equivalent lens waveguide.

Equations (29)

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A=1-a+cf2-af3+acf2f3, B=a+b+c-ab+bcf2-ab+acf3+abcf2f3, C=-1f1-1f2-1f3+a+cf1f2+af1f3+cf2f3-acf1f2f3, D=1-af1-bf1-cf1-bf2-b+cf3+ab+bcf1f2+ab+acf1f3+bcf2f3-abcf1f2f3.
fiT=Ri cosαi22
fiS=Ri2 cosαi2
-2<A+D<2
zb=A-D2C,  wob2=λ2π|C|n4-A+D21/2.
zc=f2+b-zb-f2λf2πwobw2, woc=λf2πw,
w2=wob21+λb-zb-f2πwob22,
c2=a2+b2-2ab cos α1, sin α2=a/γ, sin α3=b/γ, γ=c/sin α1,
wo2=λ2πna4f-a,
wiR2=λRi2πn,
woS2w1R2=aR2.309-aR1/2, woT2w1R2=aR1.752-aR1/2.
A=D=1-3af+a2f2, B=3a-4a2f+a3f2, C=a-2ff2.
wo12=λπn|a-2f|aa-2fa-f3f-a1/2,
wo22=λ2a3f-a2π2wo12.
wo1S2w1R2=21.154-aRaRaR-1.154aR-1×1.731-aR1/2,
wo1T2w1R2=20.866-aRaRaR-0.866aR-1×1.30-aR1/2,
wo2S2w1R2=2w1Rwo1S2aR1.731-aR,
wo2T2w1R2=2w1Rwo1T2aR1.30-aR
A=1-2a+cf+acf2, B=a-ff2ac-cf-2af, C=-1f1-2f+2a+cff1+cf2-acf2f1 D=1-2a+cf-2a+cf1+2aa+cff1+acf2-a2cf2f1.
wo14w1R4=X-F2XF-XY+YFF+2F1-X2XF-XY+YF+2YF1-4FF1XY-2XF-YF+F2-YF1+2FF12,
X=aR, Y=cR=2X sinα12, F=fR, F1=f1R.
F1T=cosα122, F1S=12 cosα12, FT=Γ2, FS=12Γ,
Γ=cosα22=1+sinα1221/2.
z1=R2X-FXY-2XF-YF2XF-XY+YF-F2+YF1-2FF1.
wo24w1R4=2XF-XY+YF2XF-XY+YF+2YF1-4FF1X-FF+2F1-X.
A=D=1-2a+cf+acf2, B=a-ff2ac-cf-2af, C=-2f+cf2.
0<1-af1-c2f<1.
wo14w1R4=4 X-F2XF-XY+YFY-2F.
wo24w1R4=2XF-XY+YFY-2FX-F.

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