Abstract

We derive an expression for the lower bound of the phase-space beam quality factor M2 of an ultrashort pulse. We show that the condition M21 does not, in general, hold for such pulsed beams. Rather, the minimum value of M2 depends on the pulse spectrum. We also show that M2 attains minimum only for pulsed beams whose spot size varies with frequency as ω12, and that the radial intensity profile of the best-quality pulsed beam can be non-Gaussian.

© 2008 Optical Society of America

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References

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  1. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).
  2. An extensive list of references on beam quality (up to 1998) can be found at http://www.stanford.edu/~siegman/beam_quality_ref_list.html.
  3. A. E. Siegman, Proc. SPIE 1224, 2 (1990).
    [Crossref]
  4. A. Caprara and G. C. Reali, Opt. Lett. 17, 414 (1992).
    [Crossref] [PubMed]
  5. J. J. Chang, Appl. Opt. 33, 2255 (1993).
    [Crossref]
  6. F. Encinas-Sanz, J. Serna, C. Martinez, R. Martinez-Herrero, and P. M. Mejias, IEEE J. Quantum Electron. 34, 1835 (1998).
    [Crossref]
  7. A. Mafi and J. V. Moloney, J. Lightwave Technol. 23, 2267 (2005).
    [Crossref]
  8. J-C. Diels and W. Rudolph, Ultrashort Laser Pulse Phenomena, 2nd ed. (Elsevier, 2006).
  9. T. Brabec and F. Kraunz, Rev. Mod. Phys. 72, 545 (2000).
    [Crossref]
  10. I. P. Christov, Opt. Commun. 53, 364 (1985).
    [Crossref]
  11. A. E. Kaplan, J. Opt. Soc. Am. B 15, 951 (1998).
    [Crossref]
  12. G. P. Agrawal, Opt. Commun. 157, 52 (1998).
    [Crossref]
  13. G. P. Agrawal, Opt. Commun. 167, 15 (1999).
    [Crossref]
  14. M. A. Porrras, Phys. Rev. A 60, 5069 (1999).
    [Crossref]
  15. D. Q. Lu, W. Hu, Y. Zheng, and Z. Yang, Opt. Commun. 228, 217 (2003).
    [Crossref]
  16. Q. H. Zou and B. D. Lu, Optik (Jena) 117, 259 (2006).
  17. S. A. Ponomarenko, G. P. Agrawal, and E. Wolf, Opt. Lett. 29, 394 (2004).
    [Crossref] [PubMed]
  18. S. A. Ponomarenko and E. Wolf, Opt. Lett. 25, 663 (2000).
    [Crossref]

2006 (1)

Q. H. Zou and B. D. Lu, Optik (Jena) 117, 259 (2006).

2005 (1)

2004 (1)

2003 (1)

D. Q. Lu, W. Hu, Y. Zheng, and Z. Yang, Opt. Commun. 228, 217 (2003).
[Crossref]

2000 (2)

S. A. Ponomarenko and E. Wolf, Opt. Lett. 25, 663 (2000).
[Crossref]

T. Brabec and F. Kraunz, Rev. Mod. Phys. 72, 545 (2000).
[Crossref]

1999 (2)

G. P. Agrawal, Opt. Commun. 167, 15 (1999).
[Crossref]

M. A. Porrras, Phys. Rev. A 60, 5069 (1999).
[Crossref]

1998 (3)

A. E. Kaplan, J. Opt. Soc. Am. B 15, 951 (1998).
[Crossref]

G. P. Agrawal, Opt. Commun. 157, 52 (1998).
[Crossref]

F. Encinas-Sanz, J. Serna, C. Martinez, R. Martinez-Herrero, and P. M. Mejias, IEEE J. Quantum Electron. 34, 1835 (1998).
[Crossref]

1993 (1)

1992 (1)

1990 (1)

A. E. Siegman, Proc. SPIE 1224, 2 (1990).
[Crossref]

1985 (1)

I. P. Christov, Opt. Commun. 53, 364 (1985).
[Crossref]

Appl. Opt. (1)

IEEE J. Quantum Electron. (1)

F. Encinas-Sanz, J. Serna, C. Martinez, R. Martinez-Herrero, and P. M. Mejias, IEEE J. Quantum Electron. 34, 1835 (1998).
[Crossref]

J. Lightwave Technol. (1)

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

Opt. Commun. (4)

G. P. Agrawal, Opt. Commun. 157, 52 (1998).
[Crossref]

G. P. Agrawal, Opt. Commun. 167, 15 (1999).
[Crossref]

I. P. Christov, Opt. Commun. 53, 364 (1985).
[Crossref]

D. Q. Lu, W. Hu, Y. Zheng, and Z. Yang, Opt. Commun. 228, 217 (2003).
[Crossref]

Opt. Lett. (3)

Optik (Jena) (1)

Q. H. Zou and B. D. Lu, Optik (Jena) 117, 259 (2006).

Phys. Rev. A (1)

M. A. Porrras, Phys. Rev. A 60, 5069 (1999).
[Crossref]

Proc. SPIE (1)

A. E. Siegman, Proc. SPIE 1224, 2 (1990).
[Crossref]

Rev. Mod. Phys. (1)

T. Brabec and F. Kraunz, Rev. Mod. Phys. 72, 545 (2000).
[Crossref]

Other (3)

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).

An extensive list of references on beam quality (up to 1998) can be found at http://www.stanford.edu/~siegman/beam_quality_ref_list.html.

J-C. Diels and W. Rudolph, Ultrashort Laser Pulse Phenomena, 2nd ed. (Elsevier, 2006).

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Figures (3)

Fig. 1
Fig. 1

Spectral form factor as a function of δ ω 0 for pulses whose spectrum is centered at frequency ω 0 .

Fig. 2
Fig. 2

Pulse spectra as functions of ω ω * for q = 1 , and form factors as functions of q for four values of ω 0 ω * .

Fig. 3
Fig. 3

Radial intensity profile for two values of ω 0 ω * . The dashed curve shows a Gaussian profile of width σ 0 for comparison.

Equations (19)

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ρ 2 = 0 d ω d 2 ρ ρ 2 U ( ρ , ω ) 2 0 d ω d 2 ρ U ( ρ , ω ) 2 ,
S ( ω ) d 2 ρ U ( ρ , ω ) 2 .
s 2 = 0 d ω d 2 s s 2 , J ( k s , ω ) 0 d ω d 2 s J ( k s , ω ) ,
J ( k s , ω ) = ( 2 π k ) 2 cos 2 θ U ̃ ( k s , ω ) 2 .
s 2 = 0 d ω d 2 ( k s ) s 2 U ̃ ( k s , ω ) 2 0 d ω d 2 ( k s ) U ̃ ( k s , ω ) 2 .
Φ ( α ) = 1 W 0 d ω d 2 ρ ( f * f ) 0 ,
f ρ U ( ρ , ω ) + α ( ω ) U ( ρ , ω ) ,
W = 0 d ω d 2 ρ U 2 = 0 d ω d 2 ( k s ) U ̃ 2 .
d 2 ρ U * U = d 2 ( k s ) k 2 s 2 U ̃ 2 ,
ρ 2 2 α 0 F s ( ω 0 ) + α 0 2 k 0 2 s 2 0 ,
F s ( ω 0 ) 0 d ω ( ω 0 ω ) S ( ω ) 0 d ω S ( ω ) ,
M 2 k 0 s 2 ρ 2 ,
M 2 F s ( ω 0 ) .
S ( ω ) { ( ω ω * ) q e a ω if ω ω * 0 if ω ω * ,
ρ U ( ρ , ω ) + α ( ω ) U ( ρ , ω ) = 0 .
U ( ρ , ω ) = A ( ω ) exp [ ρ 2 2 σ 2 ( ω ) ] ,
σ ( ω ) = σ 0 ( ω 0 ω ) 1 2 .
I ( ρ ) 0 d ω U ( ρ , ω ) 2 .
I ( ρ ) = 1 4 π 0 d ω S ( ω ) σ 2 ( ω ) exp [ ρ 2 2 σ 2 ( ω ) ] .

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