Abstract

The far-field intensity distribution (FFID) of a beam generated by a phase-unifying mirror resonator was investigated based on scalar diffraction theory. Attention was paid to the parameters, such as obscuration ratio and reflectivity of the phase-unifying mirror, that determine the FFID. All analyses were limited to the TEM00 fundamental mode.

© 2005 Optical Society of America

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References

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  1. K. Yasui, M. Tanaka, S. Yagi, “Unstable resonators with phase-unifying coupler for high-power lasers,” Appl. Phys. Lett. 52, 530–531 (1988).
    [CrossRef]
  2. M. Piché, D. Catin, “Enhancement of modal feedback in unstable resonators using mirrors with a phase step,” Opt. Lett. 16, 1135–1137 (1991).
    [CrossRef] [PubMed]
  3. C. Giuri, M. R. Perrone, D. Flori, A. Piegari, S. Scaglione, “Phase shift of stepwise reflectivity profile mirrors,” Appl. Opt. 36, 2495–2498 (1997).
    [CrossRef] [PubMed]
  4. F. De Tomasi, P. Aghamkar, M. R. Perrone, “Phase-unifying mirrors for high-power XeF excimer lasers,” Appl. Phys. Lett. 82, 1809–1811 (2003).
    [CrossRef]
  5. K. Yasui, M. Tanaka, S. Yagi, “An unstable resonator with a phase-unifying output coupler to extract a large uniphase beam of a filled-in circular pattern,” J. Appl. Phys. 65, 17–21 (1989).
    [CrossRef]
  6. Y. Takenaka, M. Kuzumoto, K. Yasui, “High power and focusing cw CO2laser using an unstable resonator with a phase-unifying output coupler,” IEEE J. Quantum Electron. 27, 2482–2486 (1991).
    [CrossRef]
  7. G. Emiliani, A. Piegari, S. De Silvestri, P. Laporta, V. Magni, “Optical coatings with variable reflectance for laser mirrors,” Appl. Opt. 28, 2832–2837 (1989).
    [CrossRef] [PubMed]
  8. Y. Li, “Propagation and focusing of Gaussian beams generated by Gaussian mirror resonators,” J. Opt. Soc. Am. A 19, 1832–1843 (2002).
    [CrossRef]
  9. A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), pp. 913–917.
  10. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996).
  11. Y. Chen, Principles of Lasers (Zhejiang U. Press, HangZhou, China, 1996).

2003 (1)

F. De Tomasi, P. Aghamkar, M. R. Perrone, “Phase-unifying mirrors for high-power XeF excimer lasers,” Appl. Phys. Lett. 82, 1809–1811 (2003).
[CrossRef]

2002 (1)

1997 (1)

1991 (2)

M. Piché, D. Catin, “Enhancement of modal feedback in unstable resonators using mirrors with a phase step,” Opt. Lett. 16, 1135–1137 (1991).
[CrossRef] [PubMed]

Y. Takenaka, M. Kuzumoto, K. Yasui, “High power and focusing cw CO2laser using an unstable resonator with a phase-unifying output coupler,” IEEE J. Quantum Electron. 27, 2482–2486 (1991).
[CrossRef]

1989 (2)

K. Yasui, M. Tanaka, S. Yagi, “An unstable resonator with a phase-unifying output coupler to extract a large uniphase beam of a filled-in circular pattern,” J. Appl. Phys. 65, 17–21 (1989).
[CrossRef]

G. Emiliani, A. Piegari, S. De Silvestri, P. Laporta, V. Magni, “Optical coatings with variable reflectance for laser mirrors,” Appl. Opt. 28, 2832–2837 (1989).
[CrossRef] [PubMed]

1988 (1)

K. Yasui, M. Tanaka, S. Yagi, “Unstable resonators with phase-unifying coupler for high-power lasers,” Appl. Phys. Lett. 52, 530–531 (1988).
[CrossRef]

Aghamkar, P.

F. De Tomasi, P. Aghamkar, M. R. Perrone, “Phase-unifying mirrors for high-power XeF excimer lasers,” Appl. Phys. Lett. 82, 1809–1811 (2003).
[CrossRef]

Catin, D.

Chen, Y.

Y. Chen, Principles of Lasers (Zhejiang U. Press, HangZhou, China, 1996).

De Silvestri, S.

De Tomasi, F.

F. De Tomasi, P. Aghamkar, M. R. Perrone, “Phase-unifying mirrors for high-power XeF excimer lasers,” Appl. Phys. Lett. 82, 1809–1811 (2003).
[CrossRef]

Emiliani, G.

Flori, D.

Giuri, C.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996).

Kuzumoto, M.

Y. Takenaka, M. Kuzumoto, K. Yasui, “High power and focusing cw CO2laser using an unstable resonator with a phase-unifying output coupler,” IEEE J. Quantum Electron. 27, 2482–2486 (1991).
[CrossRef]

Laporta, P.

Li, Y.

Magni, V.

Perrone, M. R.

F. De Tomasi, P. Aghamkar, M. R. Perrone, “Phase-unifying mirrors for high-power XeF excimer lasers,” Appl. Phys. Lett. 82, 1809–1811 (2003).
[CrossRef]

C. Giuri, M. R. Perrone, D. Flori, A. Piegari, S. Scaglione, “Phase shift of stepwise reflectivity profile mirrors,” Appl. Opt. 36, 2495–2498 (1997).
[CrossRef] [PubMed]

Piché, M.

Piegari, A.

Scaglione, S.

Siegman, A. E.

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), pp. 913–917.

Takenaka, Y.

Y. Takenaka, M. Kuzumoto, K. Yasui, “High power and focusing cw CO2laser using an unstable resonator with a phase-unifying output coupler,” IEEE J. Quantum Electron. 27, 2482–2486 (1991).
[CrossRef]

Tanaka, M.

K. Yasui, M. Tanaka, S. Yagi, “An unstable resonator with a phase-unifying output coupler to extract a large uniphase beam of a filled-in circular pattern,” J. Appl. Phys. 65, 17–21 (1989).
[CrossRef]

K. Yasui, M. Tanaka, S. Yagi, “Unstable resonators with phase-unifying coupler for high-power lasers,” Appl. Phys. Lett. 52, 530–531 (1988).
[CrossRef]

Yagi, S.

K. Yasui, M. Tanaka, S. Yagi, “An unstable resonator with a phase-unifying output coupler to extract a large uniphase beam of a filled-in circular pattern,” J. Appl. Phys. 65, 17–21 (1989).
[CrossRef]

K. Yasui, M. Tanaka, S. Yagi, “Unstable resonators with phase-unifying coupler for high-power lasers,” Appl. Phys. Lett. 52, 530–531 (1988).
[CrossRef]

Yasui, K.

Y. Takenaka, M. Kuzumoto, K. Yasui, “High power and focusing cw CO2laser using an unstable resonator with a phase-unifying output coupler,” IEEE J. Quantum Electron. 27, 2482–2486 (1991).
[CrossRef]

K. Yasui, M. Tanaka, S. Yagi, “An unstable resonator with a phase-unifying output coupler to extract a large uniphase beam of a filled-in circular pattern,” J. Appl. Phys. 65, 17–21 (1989).
[CrossRef]

K. Yasui, M. Tanaka, S. Yagi, “Unstable resonators with phase-unifying coupler for high-power lasers,” Appl. Phys. Lett. 52, 530–531 (1988).
[CrossRef]

Appl. Opt. (2)

Appl. Phys. Lett. (2)

F. De Tomasi, P. Aghamkar, M. R. Perrone, “Phase-unifying mirrors for high-power XeF excimer lasers,” Appl. Phys. Lett. 82, 1809–1811 (2003).
[CrossRef]

K. Yasui, M. Tanaka, S. Yagi, “Unstable resonators with phase-unifying coupler for high-power lasers,” Appl. Phys. Lett. 52, 530–531 (1988).
[CrossRef]

IEEE J. Quantum Electron. (1)

Y. Takenaka, M. Kuzumoto, K. Yasui, “High power and focusing cw CO2laser using an unstable resonator with a phase-unifying output coupler,” IEEE J. Quantum Electron. 27, 2482–2486 (1991).
[CrossRef]

J. Appl. Phys. (1)

K. Yasui, M. Tanaka, S. Yagi, “An unstable resonator with a phase-unifying output coupler to extract a large uniphase beam of a filled-in circular pattern,” J. Appl. Phys. 65, 17–21 (1989).
[CrossRef]

J. Opt. Soc. Am. A (1)

Opt. Lett. (1)

Other (3)

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), pp. 913–917.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996).

Y. Chen, Principles of Lasers (Zhejiang U. Press, HangZhou, China, 1996).

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Figures (5)

Fig. 1
Fig. 1

Schematic of a Gaussian beam diffracted by a PU mirror.

Fig. 2
Fig. 2

Far-field intensity distribution and encircled power distributions versus a far-field angle with R0 = 0.5.

Fig. 3
Fig. 3

Far-field encircled power distributions versus far-field angle for a variety of obscuration ratios with R0 = 0.5.

Fig. 4
Fig. 4

Far-field encircled power distributions versus far-field angle at a variety of reflectivities with ∊ = 0.7.

Fig. 5
Fig. 5

Normalized intensity in the diffraction limit versus center reflectivity of a PU mirror for a variety of obscuration ratios ∊.

Equations (14)

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U ( x 1 , y 1 ) = U ( x , y ) t ( x , y ) × exp [ - i k z ( x x 1 + y y 1 ) ] d x d y ,
U ( x , y , z ) = c exp [ - x 2 + y 2 ω 2 ( z ) ] exp { - i [ - k ( x 2 + y 2 ) 2 R ( z ) ] } ,
c = ω 0 ω ( z ) exp { - i [ k z - arctan ( λ z π ω 0 2 ) ] } ,
ω = ω 0 1 + ( z / z r ) 2 1 / 2 ,
R ( z ) = z [ 1 + ( z r / z ) 2 ] ,
z r = π ω 0 2 / λ .
t ( x , y ) = circ ( r / R ) - R 0 circ ( r / ɛ R )             ( 0 < ɛ < 1 ) ,
U ( ρ 1 , ϕ 1 ) = c 0 R 0 2 π exp ( - ρ 2 ω 2 ) [ circ ( ρ R ) - R 0 circ ( ρ ɛ R ) ] exp [ - i k ρ 2 2 R ( z ) ] × exp [ - i k z ρ ρ 1 cos ( ϕ - ϕ 1 ) ] ρ d ρ d ϕ = 2 π c 0 R exp ( - ρ 2 ω 2 ) [ circ ( ρ R ) - R 0 × circ ( ρ ɛ R ) ] exp [ - i k ρ 2 2 R ( z ) ] J 0 ( k z ρ ρ 1 ) ρ d ρ ,
U ( ρ 1 , ϕ 1 ) = 2 π c 0 R exp ( - ρ 2 ω 2 ) × exp [ - i k ρ 2 2 R ( z ) ] J 0 ( k z ρ ρ 1 ) ρ d ρ - 2 π c R 0 0 ɛ R exp ( - ρ 2 ω 2 ) × exp [ - i k ρ 2 2 R ( z ) ] J 0 ( k z ρ ρ 1 ) ρ d ρ .
1 u 2 = 1 ω 2 + i k 2 R ( z ) , v = k ρ z k θ .
U ( θ ) = 2 π c 0 R exp ( - ρ 2 u 2 ) J 0 ( v ρ ) ρ d ρ - 2 π c R 0 0 ɛ R exp ( - ρ 2 u 2 ) J 0 ( v ρ ) ρ d ρ .
U ( θ ) = π R 2 c exp ( - R 2 u 2 ) [ n = 1 2 n J n ( k R θ ) ( k R θ ) n ( R 2 u 2 ) n - 1 ] - π ɛ R 2 c R 0 exp [ - ( ɛ R ) 2 u 2 ] × [ n = 1 2 n J n ( k ɛ R θ ) ( k ɛ R θ ) n ( ɛ 2 R 2 u 2 ) n - 1 ] .
U ( P 0 ) = π R 2 c exp ( - R 2 u 2 ) [ n = 1 1 n ! ( R 2 u 2 ) n - 1 ] - π ɛ R 2 c R 0 exp [ - ( ɛ R ) 2 u 2 ] [ n = 1 1 n ! ( ɛ 2 R 2 u 2 ) n - 1 ] .
I ( p ) = U ( p ) U * ( p ) ,

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