Abstract

A rigorous analysis is performed on the reentrant nonplanar ring laser cavity constructed by a Herriott-type multipass cell. Since the cavity is highly nonplanar, the angle between the incident planes at each reflection becomes different from that of the image rotation angles. The beam rotation, astigmatism, and spherical aberration are considered to obtain a self-consistent solution of the Gaussian beam. It turns out that spherical aberration is an important issue for this nonplanar resonator. Without taking into account the spherical aberration, a stable resonator would be difficult to realize. Using a self-consistent Gaussian beam propagation method, the laser beam characteristics are solved analytically. The results are compared with that of the 2×2 ABCD method.

© 2005 Optical Society of America

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References

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  1. T. J. Kane, R. L. Byer, “Monolithic unidirectional single-mode Nd:YAG ring laser,” Opt. Lett. 10, 65–67 (1985).
    [CrossRef] [PubMed]
  2. J. L. Nightingale, J. K. Johnson, “Single frequency ring laser with two reflecting surfaces,” U.S. patent 5,052,815 (October 1, 1991).
  3. D. Herriott, H. Kogelnik, R. Kompfner, “Off-axis paths in spherical mirror interferometers,” Appl. Opt. 3, 523–526 (1964).
    [CrossRef]
  4. A. E. Siegman, Lasers (University Science, 1986), pp. 537–538.
  5. A. Sennaroglu, A. M. Kowalevicz, E. P. Ippen, J. G. Fujimoto, “Compact femtosecond lasers based on novel multipass cavities,” IEEE J. Quantum Electron. 40, 519–528 (2004).
    [CrossRef]
  6. H. Z. Cheng, P. L. Huang, S. L. Huang, F. J. Kao, “Reentrant two-mirror ring resonator for generation of a single-frequency green laser,” Opt. Lett. 25, 542–544 (2000).
    [CrossRef]
  7. P. L. Huang, C. R. Weng, H. Z. Cheng, S. L. Huang, “A passively Q-switched laser constructed by a two-mirror reentrant ring cavity,” Jpn. J. Appl. Phys. 40, Part 2, L508–L510 (2001).
    [CrossRef]
  8. S. L. Huang, Y. H. Chen, P. L. Huang, J. Y. Yi, H. Z. Cheng, “Multi-reentrant nonplanar ring laser cavity,” IEEE J. Quantum Electron. 38, 1301–1308 (2002).
    [CrossRef]
  9. A. Yariv, Optical Electronics in Modern Communication, 5th ed. (Oxford U. Press, 1997), p. 45.
  10. A. Sharma, A. Agrawal, “New method for nonparaxial beam propagation,” J. Opt. Soc. Am. A 21, 1082–1087 (2004).
    [CrossRef]
  11. J. A. Arnaud, “Nonorthogonal optical waveguides and resonators,” Bell Syst. Tech. J. 49, 2311–2347 (1970).
    [CrossRef]
  12. A. E. Siegman, “Laser beams and resonators: beyond the 1960s,” IEEE J. Sel. Top. Quantum Electron. 6, 1389–1399 (2000).
    [CrossRef]
  13. S. C. Sheng, “Optical-axis perturbation singularity in an out-of-plane ring resonator,” Opt. Lett. 19, 683–685 (1994).
    [CrossRef] [PubMed]
  14. A. A. Tovar, L. W. Casperson, “Generalized beam matrices: Gaussian beam propagation in misaligned complex optical systems,” J. Opt. Soc. Am. A 12, 1522–1533 (1995).
    [CrossRef]
  15. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1980), pp. 211–218.
  16. G. S. Monk, Light: Principles and Experiments (McGraw-Hill, 1937), pp. 424–426.
  17. J. A. Arnaud, H. Kogelnik, “Gaussian light beams with general astigmatism,” Appl. Opt. 8, 1687–1693 (1969).
    [CrossRef] [PubMed]

2004 (2)

A. Sennaroglu, A. M. Kowalevicz, E. P. Ippen, J. G. Fujimoto, “Compact femtosecond lasers based on novel multipass cavities,” IEEE J. Quantum Electron. 40, 519–528 (2004).
[CrossRef]

A. Sharma, A. Agrawal, “New method for nonparaxial beam propagation,” J. Opt. Soc. Am. A 21, 1082–1087 (2004).
[CrossRef]

2002 (1)

S. L. Huang, Y. H. Chen, P. L. Huang, J. Y. Yi, H. Z. Cheng, “Multi-reentrant nonplanar ring laser cavity,” IEEE J. Quantum Electron. 38, 1301–1308 (2002).
[CrossRef]

2001 (1)

P. L. Huang, C. R. Weng, H. Z. Cheng, S. L. Huang, “A passively Q-switched laser constructed by a two-mirror reentrant ring cavity,” Jpn. J. Appl. Phys. 40, Part 2, L508–L510 (2001).
[CrossRef]

2000 (2)

1995 (1)

1994 (1)

1985 (1)

1970 (1)

J. A. Arnaud, “Nonorthogonal optical waveguides and resonators,” Bell Syst. Tech. J. 49, 2311–2347 (1970).
[CrossRef]

1969 (1)

1964 (1)

Agrawal, A.

Arnaud, J. A.

J. A. Arnaud, “Nonorthogonal optical waveguides and resonators,” Bell Syst. Tech. J. 49, 2311–2347 (1970).
[CrossRef]

J. A. Arnaud, H. Kogelnik, “Gaussian light beams with general astigmatism,” Appl. Opt. 8, 1687–1693 (1969).
[CrossRef] [PubMed]

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1980), pp. 211–218.

Byer, R. L.

Casperson, L. W.

Chen, Y. H.

S. L. Huang, Y. H. Chen, P. L. Huang, J. Y. Yi, H. Z. Cheng, “Multi-reentrant nonplanar ring laser cavity,” IEEE J. Quantum Electron. 38, 1301–1308 (2002).
[CrossRef]

Cheng, H. Z.

S. L. Huang, Y. H. Chen, P. L. Huang, J. Y. Yi, H. Z. Cheng, “Multi-reentrant nonplanar ring laser cavity,” IEEE J. Quantum Electron. 38, 1301–1308 (2002).
[CrossRef]

P. L. Huang, C. R. Weng, H. Z. Cheng, S. L. Huang, “A passively Q-switched laser constructed by a two-mirror reentrant ring cavity,” Jpn. J. Appl. Phys. 40, Part 2, L508–L510 (2001).
[CrossRef]

H. Z. Cheng, P. L. Huang, S. L. Huang, F. J. Kao, “Reentrant two-mirror ring resonator for generation of a single-frequency green laser,” Opt. Lett. 25, 542–544 (2000).
[CrossRef]

Fujimoto, J. G.

A. Sennaroglu, A. M. Kowalevicz, E. P. Ippen, J. G. Fujimoto, “Compact femtosecond lasers based on novel multipass cavities,” IEEE J. Quantum Electron. 40, 519–528 (2004).
[CrossRef]

Herriott, D.

Huang, P. L.

S. L. Huang, Y. H. Chen, P. L. Huang, J. Y. Yi, H. Z. Cheng, “Multi-reentrant nonplanar ring laser cavity,” IEEE J. Quantum Electron. 38, 1301–1308 (2002).
[CrossRef]

P. L. Huang, C. R. Weng, H. Z. Cheng, S. L. Huang, “A passively Q-switched laser constructed by a two-mirror reentrant ring cavity,” Jpn. J. Appl. Phys. 40, Part 2, L508–L510 (2001).
[CrossRef]

H. Z. Cheng, P. L. Huang, S. L. Huang, F. J. Kao, “Reentrant two-mirror ring resonator for generation of a single-frequency green laser,” Opt. Lett. 25, 542–544 (2000).
[CrossRef]

Huang, S. L.

S. L. Huang, Y. H. Chen, P. L. Huang, J. Y. Yi, H. Z. Cheng, “Multi-reentrant nonplanar ring laser cavity,” IEEE J. Quantum Electron. 38, 1301–1308 (2002).
[CrossRef]

P. L. Huang, C. R. Weng, H. Z. Cheng, S. L. Huang, “A passively Q-switched laser constructed by a two-mirror reentrant ring cavity,” Jpn. J. Appl. Phys. 40, Part 2, L508–L510 (2001).
[CrossRef]

H. Z. Cheng, P. L. Huang, S. L. Huang, F. J. Kao, “Reentrant two-mirror ring resonator for generation of a single-frequency green laser,” Opt. Lett. 25, 542–544 (2000).
[CrossRef]

Ippen, E. P.

A. Sennaroglu, A. M. Kowalevicz, E. P. Ippen, J. G. Fujimoto, “Compact femtosecond lasers based on novel multipass cavities,” IEEE J. Quantum Electron. 40, 519–528 (2004).
[CrossRef]

Johnson, J. K.

J. L. Nightingale, J. K. Johnson, “Single frequency ring laser with two reflecting surfaces,” U.S. patent 5,052,815 (October 1, 1991).

Kane, T. J.

Kao, F. J.

Kogelnik, H.

Kompfner, R.

Kowalevicz, A. M.

A. Sennaroglu, A. M. Kowalevicz, E. P. Ippen, J. G. Fujimoto, “Compact femtosecond lasers based on novel multipass cavities,” IEEE J. Quantum Electron. 40, 519–528 (2004).
[CrossRef]

Monk, G. S.

G. S. Monk, Light: Principles and Experiments (McGraw-Hill, 1937), pp. 424–426.

Nightingale, J. L.

J. L. Nightingale, J. K. Johnson, “Single frequency ring laser with two reflecting surfaces,” U.S. patent 5,052,815 (October 1, 1991).

Sennaroglu, A.

A. Sennaroglu, A. M. Kowalevicz, E. P. Ippen, J. G. Fujimoto, “Compact femtosecond lasers based on novel multipass cavities,” IEEE J. Quantum Electron. 40, 519–528 (2004).
[CrossRef]

Sharma, A.

Sheng, S. C.

Siegman, A. E.

A. E. Siegman, “Laser beams and resonators: beyond the 1960s,” IEEE J. Sel. Top. Quantum Electron. 6, 1389–1399 (2000).
[CrossRef]

A. E. Siegman, Lasers (University Science, 1986), pp. 537–538.

Tovar, A. A.

Weng, C. R.

P. L. Huang, C. R. Weng, H. Z. Cheng, S. L. Huang, “A passively Q-switched laser constructed by a two-mirror reentrant ring cavity,” Jpn. J. Appl. Phys. 40, Part 2, L508–L510 (2001).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1980), pp. 211–218.

Yariv, A.

A. Yariv, Optical Electronics in Modern Communication, 5th ed. (Oxford U. Press, 1997), p. 45.

Yi, J. Y.

S. L. Huang, Y. H. Chen, P. L. Huang, J. Y. Yi, H. Z. Cheng, “Multi-reentrant nonplanar ring laser cavity,” IEEE J. Quantum Electron. 38, 1301–1308 (2002).
[CrossRef]

Appl. Opt. (2)

Bell Syst. Tech. J. (1)

J. A. Arnaud, “Nonorthogonal optical waveguides and resonators,” Bell Syst. Tech. J. 49, 2311–2347 (1970).
[CrossRef]

IEEE J. Quantum Electron. (2)

A. Sennaroglu, A. M. Kowalevicz, E. P. Ippen, J. G. Fujimoto, “Compact femtosecond lasers based on novel multipass cavities,” IEEE J. Quantum Electron. 40, 519–528 (2004).
[CrossRef]

S. L. Huang, Y. H. Chen, P. L. Huang, J. Y. Yi, H. Z. Cheng, “Multi-reentrant nonplanar ring laser cavity,” IEEE J. Quantum Electron. 38, 1301–1308 (2002).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (1)

A. E. Siegman, “Laser beams and resonators: beyond the 1960s,” IEEE J. Sel. Top. Quantum Electron. 6, 1389–1399 (2000).
[CrossRef]

J. Opt. Soc. Am. A (2)

Jpn. J. Appl. Phys. (1)

P. L. Huang, C. R. Weng, H. Z. Cheng, S. L. Huang, “A passively Q-switched laser constructed by a two-mirror reentrant ring cavity,” Jpn. J. Appl. Phys. 40, Part 2, L508–L510 (2001).
[CrossRef]

Opt. Lett. (3)

Other (5)

A. Yariv, Optical Electronics in Modern Communication, 5th ed. (Oxford U. Press, 1997), p. 45.

J. L. Nightingale, J. K. Johnson, “Single frequency ring laser with two reflecting surfaces,” U.S. patent 5,052,815 (October 1, 1991).

A. E. Siegman, Lasers (University Science, 1986), pp. 537–538.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1980), pp. 211–218.

G. S. Monk, Light: Principles and Experiments (McGraw-Hill, 1937), pp. 424–426.

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Figures (7)

Fig. 1
Fig. 1

(a) Schematic drawing of the multiple reentrant ring laser setup, (b) the rays in the cavity of a nonplanar figure-8 ring laser. I/C and O/C are the input and output couplers, M 1 and M 2 are the spherical centers of the output and input couplers, R is the radius of curvature of the couplers, Ω is the incident angle of the beam on the couplers, L is the cavity length, d is the distance between the laser spot on the couplers and the optical axis, and 2 l is the distance between laser spots on the two couplers in the projection of the x z or y z planes.

Fig. 2
Fig. 2

Equivalent views of beam propagation in a round trip: (a) paraxial and nonorthogonal approach, (b) nonparaxial and orthogonal approach.

Fig. 3
Fig. 3

Stability of the empty ring laser cavity of exact solution. (a) Two-dimensional mapping on B 2 4 A C . The dark region is the stable area and gets darker if the value of B 2 4 A C is more negative. (b) A comparison of the stable area for the linear cavity (all the area to the right of the L = 2 R line) and ring (the crossed area) with different cavity parameters.

Fig. 4
Fig. 4

Comparison of (a) spot size and (b) radius of curvature for the exact and approximated (solid curve) solutions. The triangle and rhombus marks are in the y , z and x , z planes, respectively. The cross marks are the average of ω x and ω y .

Fig. 5
Fig. 5

Comparison of (a) mode size and (b) radius of curvature at z = 0 of the empty laser cavity with exact and approximated solutions.

Fig. 6
Fig. 6

Stabilities of the ring laser cavity by (a) exact ( B 2 4 A C ) and (b) approximated ( A B C D method) solutions with various effective thicknesses of the gain media, which is defined by [ 1 ( 1 n ) ] ( t R ) . Here, n and t are the index and physical thickness of the gain medium, respectively.

Fig. 7
Fig. 7

Minimum side shift required for various thicknesses of the gain media.

Tables (1)

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Table 1 Coordinates of the Four Incident Planes

Equations (39)

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d 0 = d 2 ,
cos Ω = R R 2 + d 2 ,
l = R L 2 = 1 2 R 2 d 2 ,
a = R 2 + d 2 ,
R z c ( Θ ) = [ cos Θ sin Θ 0 sin Θ cos Θ 0 0 0 1 ] .
x ̂ i = R z c ( 90° ) x ̂ i , y ̂ i = R z c ( 90° ) y ̂ i , z ̂ i = R z c ( 90° ) z ̂ i .
R T = R cos Ω in the tangential plane ,
R S = R cos Ω in the sagittal plane ,
R eff = R [ 2 1 1 ( d R ) 2 ] .
R T = R eff cos Ω in the tangential plane ,
R S = R eff cos Ω in the sagittal plane .
Ψ ( x , y , z ) = A 0 exp [ j k 2 ( x 2 q x + y 2 q y ) ] ,
Φ 0 = k 2 ( x 2 q x 0 + y 2 q y 0 ) .
Φ 1 = k 2 ( x 2 q x 1 + y 2 q y 1 ) ,
Φ 1 m = k 2 ( x 2 q x 1 m + y 2 q y 1 m ) ,
1 q x 1 m = 1 q x 1 2 cos Ω R eff , 1 q y 1 m = 1 q y 1 2 R eff cos Ω .
O R = [ cos Θ sin Θ sin Θ cos Θ ] .
Φ 1 m r = k 2 ( x 2 q y 1 m + y 2 q x 1 m ) .
Φ 2 = k 2 ( x 2 q x 2 + y 2 q y 2 ) ,
Φ 2 m = k 2 ( x 2 q x 2 m + y 2 q y 2 m ) ,
Φ 2 m r = k 2 ( x 2 q y 2 m + y 2 q x 2 m ) ,
q x 2 = q y 1 m + a ,
q y 2 = q x 1 m + a ,
1 q x 2 m = 1 q x 2 2 cos Ω R eff ,
1 q y 2 m = 1 q y 2 2 R eff cos Ω .
q x 0 = q y 2 m + a 2 ,
q y 0 = q x 2 m + a 2 .
A q x 0 2 + B q x 0 + C = 0 ,
A q y 0 2 B q y 0 + C = 0 ,
A = 4 R 2 R eff 2 R eff cos 2 Ω ,
B = 2 R eff a sin 2 Ω ,
C = a 2 ( 3 2 R eff R ) + 3 2 R eff R 2 2 R eff 2 R .
A = 2 R [ 2 ( 2 R 2 + d 2 ) ( 2 R 2 d 2 R ) ( R 2 + d 2 ) R 2 d 2 ]
q x 0 = B + i B 2 4 A C 2 A ,
q y 0 = B + i B 2 4 A C 2 A
1 q x 0 = 1 R x i λ π ω x 2 ,
1 q y 0 = 1 R y i λ π ω y 2 .
2 ω x = 2 ω y = 2 λ R π ,
R x = R y ,

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