Abstract

Wereportthe experimental generation of a family of flattened Gaussian beams with bell-shaped, flattened, and annular intensity profiles in an electro-optically Q-switched Nd:YAG laser with a variable reflectivity mirror of a Gaussian reflectivity profile as an output coupler. The laser beams of different profiles were generated by modifying the resonator magnification. The propagation characteristics of the experimentally generated flat Gaussian beams were found to be in agreement with theory. To the best of our knowledge this is the first time such a family of flattened Gaussian beams is experimentally generated intracavity using a single variable reflectivity mirror.

© 2008 Optical Society of America

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References

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  1. D. L. Shealy and J. A. Hoffnagle, "Laser beam shaping profiles and propagation," Appl. Opt. 45, 5118-5131 (2006).
    [CrossRef] [PubMed]
  2. S. De Silvestri, P. Laporta, V. Magni, O. Svelto, and B. Majocchi, "Unstable laser resonators with super-Gaussian mirrors," Opt. Lett. 13, 201-203 (1988).
    [CrossRef] [PubMed]
  3. S. De Silvestri, V. Magni, O. Svelto, and G. Valentini, "Lasers with super-Gaussian mirrors," IEEE J. Quantum Electron. 26, 1500-1509 (1990).
    [CrossRef]
  4. H. Peng and S. W. Zhang, "Optimum design of Q-switched Nd:YAG lasers with super Gaussian mirrors," Proc. SPIE 2889, 436-440 (1996).
    [CrossRef]
  5. M. Morin and M. Poirier, "Graded reflectivity mirror unstable resonator design," Proc. SPIE 3267, 52-65 (1998).
    [CrossRef]
  6. F. Gori, "Flattened Gaussian beams," Opt. Commun. 107, 335-341 (1994).
    [CrossRef]
  7. C. Palma and V. Bagini, "Expansions of general beams in Gaussian beams," Opt. Commun. 116, 1-7 (1995).
    [CrossRef]
  8. S.-A. Amarande, "Beam propagation factor and the kurtosis parameter of flattened Gaussian beams," Opt. Commun. 129, 311-317 (1996).
  9. Y. Li, "New expressions for flat-topped light beams," Opt. Commun. 206, 225-234 (2002).
    [CrossRef]
  10. V. Bagini, R. Borghi, F. Gori, A. M. Pacilieo, and M. Santarsiero, "Propagation of axially symmetric flattened Gaussian beams," J. Opt. Soc. Am. A 13, 1385-1394 (1996).
    [CrossRef]
  11. J. A. Hoffnagle and C. M. Jefferson, "Design and performance of a refractive optical system that converts a Gaussian to a flattop beam," Appl. Opt. 39, 5488-5499 (2000).
    [CrossRef]
  12. J. A. Hoffnagle and C. M. Jefferson, "Beam shaping with a plano-aspheric lens pair," Opt. Eng. 42, 3090-3099 (2003).
    [CrossRef]
  13. M. Santarsiero and R. Borghi, "Correspondence between super-Gaussian and flattened Gaussian beams," J. Opt. Soc. Am. A 16, 188-190 (1999).
    [CrossRef]
  14. S.-A. Amarande, "Approximation of super-Gaussian beams by generalized flattened Gaussian beams," Proc. SPIE 3092, 345-348 (1997).
    [CrossRef]
  15. W. Koechner, "Solid-state laser engineering," 5th ed. (Springer-Verlag, 1999), p. 281.
  16. S. De Silvestri, P. Laporta, V. Magni, and O. Svelto, "Solid-state laser unstable resonators with tapered reflectivity mirrors: the super-Gaussian approach," IEEE J. Quantum Electron. 24, 1172-1177 (1988).
    [CrossRef]
  17. N. Hodgson and H. Weber, Optical Resonators: Fundamentals, Advanced Concepts and Applications (Springer-Verlag, 1997), p. 241.
  18. Y. Li, "Propagation and focusing of Gausssian beams generated by Gaussian mirror resonators," J. Opt. Soc. Am. A 19, 1832-1843 (2002).
    [CrossRef]

2006 (1)

2003 (1)

J. A. Hoffnagle and C. M. Jefferson, "Beam shaping with a plano-aspheric lens pair," Opt. Eng. 42, 3090-3099 (2003).
[CrossRef]

2002 (2)

2000 (1)

1999 (1)

1998 (1)

M. Morin and M. Poirier, "Graded reflectivity mirror unstable resonator design," Proc. SPIE 3267, 52-65 (1998).
[CrossRef]

1997 (1)

S.-A. Amarande, "Approximation of super-Gaussian beams by generalized flattened Gaussian beams," Proc. SPIE 3092, 345-348 (1997).
[CrossRef]

1996 (3)

S.-A. Amarande, "Beam propagation factor and the kurtosis parameter of flattened Gaussian beams," Opt. Commun. 129, 311-317 (1996).

H. Peng and S. W. Zhang, "Optimum design of Q-switched Nd:YAG lasers with super Gaussian mirrors," Proc. SPIE 2889, 436-440 (1996).
[CrossRef]

V. Bagini, R. Borghi, F. Gori, A. M. Pacilieo, and M. Santarsiero, "Propagation of axially symmetric flattened Gaussian beams," J. Opt. Soc. Am. A 13, 1385-1394 (1996).
[CrossRef]

1995 (1)

C. Palma and V. Bagini, "Expansions of general beams in Gaussian beams," Opt. Commun. 116, 1-7 (1995).
[CrossRef]

1994 (1)

F. Gori, "Flattened Gaussian beams," Opt. Commun. 107, 335-341 (1994).
[CrossRef]

1990 (1)

S. De Silvestri, V. Magni, O. Svelto, and G. Valentini, "Lasers with super-Gaussian mirrors," IEEE J. Quantum Electron. 26, 1500-1509 (1990).
[CrossRef]

1988 (2)

S. De Silvestri, P. Laporta, V. Magni, and O. Svelto, "Solid-state laser unstable resonators with tapered reflectivity mirrors: the super-Gaussian approach," IEEE J. Quantum Electron. 24, 1172-1177 (1988).
[CrossRef]

S. De Silvestri, P. Laporta, V. Magni, O. Svelto, and B. Majocchi, "Unstable laser resonators with super-Gaussian mirrors," Opt. Lett. 13, 201-203 (1988).
[CrossRef] [PubMed]

Appl. Opt. (2)

IEEE J. Quantum Electron. (2)

S. De Silvestri, V. Magni, O. Svelto, and G. Valentini, "Lasers with super-Gaussian mirrors," IEEE J. Quantum Electron. 26, 1500-1509 (1990).
[CrossRef]

S. De Silvestri, P. Laporta, V. Magni, and O. Svelto, "Solid-state laser unstable resonators with tapered reflectivity mirrors: the super-Gaussian approach," IEEE J. Quantum Electron. 24, 1172-1177 (1988).
[CrossRef]

J. Opt. Soc. Am. A (3)

Opt. Commun. (4)

F. Gori, "Flattened Gaussian beams," Opt. Commun. 107, 335-341 (1994).
[CrossRef]

C. Palma and V. Bagini, "Expansions of general beams in Gaussian beams," Opt. Commun. 116, 1-7 (1995).
[CrossRef]

S.-A. Amarande, "Beam propagation factor and the kurtosis parameter of flattened Gaussian beams," Opt. Commun. 129, 311-317 (1996).

Y. Li, "New expressions for flat-topped light beams," Opt. Commun. 206, 225-234 (2002).
[CrossRef]

Opt. Eng. (1)

J. A. Hoffnagle and C. M. Jefferson, "Beam shaping with a plano-aspheric lens pair," Opt. Eng. 42, 3090-3099 (2003).
[CrossRef]

Opt. Lett. (1)

Proc. SPIE (3)

S.-A. Amarande, "Approximation of super-Gaussian beams by generalized flattened Gaussian beams," Proc. SPIE 3092, 345-348 (1997).
[CrossRef]

H. Peng and S. W. Zhang, "Optimum design of Q-switched Nd:YAG lasers with super Gaussian mirrors," Proc. SPIE 2889, 436-440 (1996).
[CrossRef]

M. Morin and M. Poirier, "Graded reflectivity mirror unstable resonator design," Proc. SPIE 3267, 52-65 (1998).
[CrossRef]

Other (2)

N. Hodgson and H. Weber, Optical Resonators: Fundamentals, Advanced Concepts and Applications (Springer-Verlag, 1997), p. 241.

W. Koechner, "Solid-state laser engineering," 5th ed. (Springer-Verlag, 1999), p. 281.

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Figures (7)

Fig. 1
Fig. 1

Theoretically transmitted profile from Gaussian mirror for various resonator magnifications.

Fig. 2
Fig. 2

Schematics of resonator and beam profiling setup.

Fig. 3
Fig. 3

(Color online) (a) Output spatial profile of a resonator with a Gaussian mirror, magnification M = 1.5 . (b) Theoretical and experimental fit of the output spatial profile for resonator M = 1.5 .

Fig. 4
Fig. 4

(Color online) (a) Output spatial profile of a resonator with a Gaussian mirror, magnification M = 1.8 . (b) Theoretical and experimental fit of the output spatial profile for resonator M = 1.8 .

Fig. 5
Fig. 5

(Color online) (a) Output spatial profile of a resonator with a Gaussian mirror, magnification M = 2 . (b) Theoretical and experimental fit of the output spatial profile for resonator M = 2 .

Fig. 6
Fig. 6

(Color online) (a) Output spatial profile of a resonator with a Gaussian mirror, magnification M = 2.6 . (b) Theoretical and experimental fit of the output spatial profile for resonator M = 2.6 .

Fig. 7
Fig. 7

Theoretical and recorded propagation profiles of FGB for resonator M = 2 .

Tables (1)

Tables Icon

Table 1 Summary of Resonator Magnifications, Rear Mirrors, and Length; Calculated R 0 Values; Spot Size Predicted and Measured Values; Theoretical Fit of FGB Order; and Output Energy of Different Resonator Configuration Experiments

Equations (19)

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I out ( r ) = { 1 R 0 exp [ 2 ( r ω m ) n ] } I 0 exp [ 2 ( r ω i ) n ] ,
M = g + g 2 1 ,
g = 2 g 1 g 2 1 ,
g 1 = ( 1 L R 1 ) ,
g 2 = ( 1 L R 2 ) ,
ω i = ω m ( M n 1 ) 1 / n .
I out ( r ) = { 1 R 0 exp [ 2 β ( r ω i ) 2 ] } I 0 exp [ 2 ( r ω i ) 2 ] ,
I out ( r ) = { 1 R 0 exp [ 2 ( M 2 1 ) ( r ω i ) 2 ] } I 0 × exp [ 2 ( r ω i ) 2 ] .
U N ( r , z ) = A v 0 v N ( z ) exp { i [ k z φ N ( z ) ] } × exp [ ( i k 2 R N ( z ) 1 v N 2 ( z ) ) r 2 ] × n = 0 N C n L n ( 2 r 2 v N 2 ( z ) ) exp [ 2 i n φ N ( z ) ] ,
R ( Z ) = z + ( z R 2 z ) ,
v ( Z ) = v 0 1 + ( z z R ) 2 ,
φ ( Z ) = tan 1 ( z z R ) ,
C n = ( 1 ) n m = n N ( m n ) 1 2 m ,
Z R = π v 0 2 λ ,
I ( r , z ) = U N ( r , z ) [ U N ( r , z ) ] * .
v 0 = w 0 N .
Γ ( N + 1 , N + 1 ) 2 Γ ( N + 1 ) 2
π w 0 2 / λ .
T = 1 R 0 M 2 .

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