Abstract

We derive an asymptotically accurate formula for the beam waist of ultrashort, tightly focused fundamental linearly polarized, radially polarized, and azimuthally polarized modes in free space. We compute the exact beam waist via numerical cubature to ascertain the accuracy with which our formula approximates the exact beam waist over a broad range of parameters of practical interest. Based on this, we describe a method of choosing parameters in the model given the beam waist and pulse duration of a laser pulse.

© 2014 Optical Society of America

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  1. G. Rodríguez-Morales and S. Chávez-Cerda, Opt. Lett. 29, 430 (2004).
    [CrossRef]
  2. Y. I. Salamin and C. H. Keitel, Phys. Rev. Lett. 88, 095005 (2002).
    [CrossRef]
  3. V. Marceau, A. April, and M. Piché, Opt. Lett. 37, 2442 (2012).
    [CrossRef]
  4. Y. Jiang, T. Narushima, and H. Okamoto, Nat. Phys. 6, 1005 (2010).
    [CrossRef]
  5. H. Schwoerer, B. Liesfeld, H.-P. Schlenvoigt, K.-U. Amthor, and R. Sauerbrey, Phys. Rev. Lett. 96, 014802 (2006).
    [CrossRef]
  6. N. Dudovich, D. Oron, and Y. Silberberg, Nature 418, 512 (2002).
    [CrossRef]
  7. V. Marceau, C. Varin, and M. Piché, Opt. Lett. 38, 821 (2013).
    [CrossRef]
  8. C. F. R. Caron and R. M. Potvliege, J. Mod. Opt. 46, 1881 (1999).
    [CrossRef]
  9. Q. Lin, J. Zheng, and W. Becker, Phys. Rev. Lett. 97, 253902 (2006).
    [CrossRef]
  10. A. April, in Coherence and Ultrashort Pulse Laser Emission, F. J. Duarte, ed. (InTech, 2010), pp. 355–382.
  11. C. Varin, M. Piché, and M. A. Porras, J. Opt. Soc. Am. A 23, 2027 (2006).
    [CrossRef]
  12. G. A. Deschamps, Electron. Lett. 7, 684 (1971).
    [CrossRef]
  13. M. Couture and P.-A. Bélanger, Phys. Rev. A 24, 355 (1981).
    [CrossRef]
  14. M. Lax, W. H. Louisell, and W. B. McKnight, Phys. Rev. A 11, 1365 (1975).
    [CrossRef]
  15. Y. I. Salamin, Opt. Lett. 34, 683 (2009).
    [CrossRef]
  16. C. J. R. Sheppard and S. Saghafi, Phys. Rev. A 57, 2971 (1998).
    [CrossRef]
  17. Z. Ulanowski and I. K. Ludlow, Opt. Lett. 25, 1792 (2000).
    [CrossRef]
  18. P. Saari, Opt. Express 8, 590 (2001).
    [CrossRef]
  19. S. Feng and H. G. Winful, Phys. Rev. E 61, 862 (2000).
    [CrossRef]

2013 (1)

2012 (1)

2010 (1)

Y. Jiang, T. Narushima, and H. Okamoto, Nat. Phys. 6, 1005 (2010).
[CrossRef]

2009 (1)

2006 (3)

Q. Lin, J. Zheng, and W. Becker, Phys. Rev. Lett. 97, 253902 (2006).
[CrossRef]

C. Varin, M. Piché, and M. A. Porras, J. Opt. Soc. Am. A 23, 2027 (2006).
[CrossRef]

H. Schwoerer, B. Liesfeld, H.-P. Schlenvoigt, K.-U. Amthor, and R. Sauerbrey, Phys. Rev. Lett. 96, 014802 (2006).
[CrossRef]

2004 (1)

2002 (2)

Y. I. Salamin and C. H. Keitel, Phys. Rev. Lett. 88, 095005 (2002).
[CrossRef]

N. Dudovich, D. Oron, and Y. Silberberg, Nature 418, 512 (2002).
[CrossRef]

2001 (1)

2000 (2)

S. Feng and H. G. Winful, Phys. Rev. E 61, 862 (2000).
[CrossRef]

Z. Ulanowski and I. K. Ludlow, Opt. Lett. 25, 1792 (2000).
[CrossRef]

1999 (1)

C. F. R. Caron and R. M. Potvliege, J. Mod. Opt. 46, 1881 (1999).
[CrossRef]

1998 (1)

C. J. R. Sheppard and S. Saghafi, Phys. Rev. A 57, 2971 (1998).
[CrossRef]

1981 (1)

M. Couture and P.-A. Bélanger, Phys. Rev. A 24, 355 (1981).
[CrossRef]

1975 (1)

M. Lax, W. H. Louisell, and W. B. McKnight, Phys. Rev. A 11, 1365 (1975).
[CrossRef]

1971 (1)

G. A. Deschamps, Electron. Lett. 7, 684 (1971).
[CrossRef]

Amthor, K.-U.

H. Schwoerer, B. Liesfeld, H.-P. Schlenvoigt, K.-U. Amthor, and R. Sauerbrey, Phys. Rev. Lett. 96, 014802 (2006).
[CrossRef]

April, A.

V. Marceau, A. April, and M. Piché, Opt. Lett. 37, 2442 (2012).
[CrossRef]

A. April, in Coherence and Ultrashort Pulse Laser Emission, F. J. Duarte, ed. (InTech, 2010), pp. 355–382.

Becker, W.

Q. Lin, J. Zheng, and W. Becker, Phys. Rev. Lett. 97, 253902 (2006).
[CrossRef]

Bélanger, P.-A.

M. Couture and P.-A. Bélanger, Phys. Rev. A 24, 355 (1981).
[CrossRef]

Caron, C. F. R.

C. F. R. Caron and R. M. Potvliege, J. Mod. Opt. 46, 1881 (1999).
[CrossRef]

Chávez-Cerda, S.

Couture, M.

M. Couture and P.-A. Bélanger, Phys. Rev. A 24, 355 (1981).
[CrossRef]

Deschamps, G. A.

G. A. Deschamps, Electron. Lett. 7, 684 (1971).
[CrossRef]

Dudovich, N.

N. Dudovich, D. Oron, and Y. Silberberg, Nature 418, 512 (2002).
[CrossRef]

Feng, S.

S. Feng and H. G. Winful, Phys. Rev. E 61, 862 (2000).
[CrossRef]

Jiang, Y.

Y. Jiang, T. Narushima, and H. Okamoto, Nat. Phys. 6, 1005 (2010).
[CrossRef]

Keitel, C. H.

Y. I. Salamin and C. H. Keitel, Phys. Rev. Lett. 88, 095005 (2002).
[CrossRef]

Lax, M.

M. Lax, W. H. Louisell, and W. B. McKnight, Phys. Rev. A 11, 1365 (1975).
[CrossRef]

Liesfeld, B.

H. Schwoerer, B. Liesfeld, H.-P. Schlenvoigt, K.-U. Amthor, and R. Sauerbrey, Phys. Rev. Lett. 96, 014802 (2006).
[CrossRef]

Lin, Q.

Q. Lin, J. Zheng, and W. Becker, Phys. Rev. Lett. 97, 253902 (2006).
[CrossRef]

Louisell, W. H.

M. Lax, W. H. Louisell, and W. B. McKnight, Phys. Rev. A 11, 1365 (1975).
[CrossRef]

Ludlow, I. K.

Marceau, V.

McKnight, W. B.

M. Lax, W. H. Louisell, and W. B. McKnight, Phys. Rev. A 11, 1365 (1975).
[CrossRef]

Narushima, T.

Y. Jiang, T. Narushima, and H. Okamoto, Nat. Phys. 6, 1005 (2010).
[CrossRef]

Okamoto, H.

Y. Jiang, T. Narushima, and H. Okamoto, Nat. Phys. 6, 1005 (2010).
[CrossRef]

Oron, D.

N. Dudovich, D. Oron, and Y. Silberberg, Nature 418, 512 (2002).
[CrossRef]

Piché, M.

Porras, M. A.

Potvliege, R. M.

C. F. R. Caron and R. M. Potvliege, J. Mod. Opt. 46, 1881 (1999).
[CrossRef]

Rodríguez-Morales, G.

Saari, P.

Saghafi, S.

C. J. R. Sheppard and S. Saghafi, Phys. Rev. A 57, 2971 (1998).
[CrossRef]

Salamin, Y. I.

Y. I. Salamin, Opt. Lett. 34, 683 (2009).
[CrossRef]

Y. I. Salamin and C. H. Keitel, Phys. Rev. Lett. 88, 095005 (2002).
[CrossRef]

Sauerbrey, R.

H. Schwoerer, B. Liesfeld, H.-P. Schlenvoigt, K.-U. Amthor, and R. Sauerbrey, Phys. Rev. Lett. 96, 014802 (2006).
[CrossRef]

Schlenvoigt, H.-P.

H. Schwoerer, B. Liesfeld, H.-P. Schlenvoigt, K.-U. Amthor, and R. Sauerbrey, Phys. Rev. Lett. 96, 014802 (2006).
[CrossRef]

Schwoerer, H.

H. Schwoerer, B. Liesfeld, H.-P. Schlenvoigt, K.-U. Amthor, and R. Sauerbrey, Phys. Rev. Lett. 96, 014802 (2006).
[CrossRef]

Sheppard, C. J. R.

C. J. R. Sheppard and S. Saghafi, Phys. Rev. A 57, 2971 (1998).
[CrossRef]

Silberberg, Y.

N. Dudovich, D. Oron, and Y. Silberberg, Nature 418, 512 (2002).
[CrossRef]

Ulanowski, Z.

Varin, C.

Winful, H. G.

S. Feng and H. G. Winful, Phys. Rev. E 61, 862 (2000).
[CrossRef]

Zheng, J.

Q. Lin, J. Zheng, and W. Becker, Phys. Rev. Lett. 97, 253902 (2006).
[CrossRef]

Electron. Lett. (1)

G. A. Deschamps, Electron. Lett. 7, 684 (1971).
[CrossRef]

J. Mod. Opt. (1)

C. F. R. Caron and R. M. Potvliege, J. Mod. Opt. 46, 1881 (1999).
[CrossRef]

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

Nat. Phys. (1)

Y. Jiang, T. Narushima, and H. Okamoto, Nat. Phys. 6, 1005 (2010).
[CrossRef]

Nature (1)

N. Dudovich, D. Oron, and Y. Silberberg, Nature 418, 512 (2002).
[CrossRef]

Opt. Express (1)

Opt. Lett. (5)

Phys. Rev. A (3)

M. Couture and P.-A. Bélanger, Phys. Rev. A 24, 355 (1981).
[CrossRef]

M. Lax, W. H. Louisell, and W. B. McKnight, Phys. Rev. A 11, 1365 (1975).
[CrossRef]

C. J. R. Sheppard and S. Saghafi, Phys. Rev. A 57, 2971 (1998).
[CrossRef]

Phys. Rev. E (1)

S. Feng and H. G. Winful, Phys. Rev. E 61, 862 (2000).
[CrossRef]

Phys. Rev. Lett. (3)

Y. I. Salamin and C. H. Keitel, Phys. Rev. Lett. 88, 095005 (2002).
[CrossRef]

H. Schwoerer, B. Liesfeld, H.-P. Schlenvoigt, K.-U. Amthor, and R. Sauerbrey, Phys. Rev. Lett. 96, 014802 (2006).
[CrossRef]

Q. Lin, J. Zheng, and W. Becker, Phys. Rev. Lett. 97, 253902 (2006).
[CrossRef]

Other (1)

A. April, in Coherence and Ultrashort Pulse Laser Emission, F. J. Duarte, ed. (InTech, 2010), pp. 355–382.

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Figures (3)

Fig. 1.
Fig. 1.

Color maps of the Poynting vector’s z-component for a single-cycle LP01 pulse of parameters k0a=500 and s=1 at various propagation times. (a), (b), and (c), respectively, show the pulse before, during, and after it arrives at the focal plane. The arrows in (b) indicate the waist diameter [corresponding radius defined in Eq. (5)]. Note that the second irradiance moment is evaluated on the focal plane at the pulse peak and carrier peak. The waist radius for this particular case is about 2.9λ.

Fig. 2.
Fig. 2.

Color maps of numerically obtained (a) exact pulse duration τ and (b) exact beam waist wexact of the LP01 mode as a function of a and s. (c) and (d) contain the same information as (a) and (b), respectively, except for the TM01 (or TE01) mode. The dotted lines are contour lines, which show relative invariance of pulse duration with a, especially at large a, in (a) and (b).

Fig. 3.
Fig. 3.

Relative error ε of the beam waist formulas in (a) Eq. (1) and (b) Eq. (4) for the LP01 mode; (c) and (d) present the information in (a) and (b), respectively, in a different form; (e) and (f) correspond to (c) and (d) for the TM01 (or TE01) mode. The color maps corresponding to (a) and (b) for the TM01 (or TE01) mode are visually very similar and thus omitted.

Equations (13)

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wpulse=2ak0ss+1+ς02/2,
Ψ˜=sin(kR)Rexp(ka)[ωsexp(sωω0)θ(ω)],
Ψ=1R(f+s1fs1),
wCW=2k01+(k0a)21.
wexact=ς0r2SzdxdySzdxdy,
Er=Re{AjraR},η0Hφ=Re{BrR},
rdSzdxdy=Crd+1drRe{rjaR2[3k0R3g13j(s+1)sR2g2+(s+1)(s+2)k0s2Rg3]}Re{rR[j(s+1)sR2g2+(s+1)(s+2)k0s2Rg3+]},
rdSzdxdy=Cd(R2)(R2+a2)1+d/2(R2+b2)6+O(s),
k0wexact=k023(b2a2)+O(s3/2)=4k0as3[1+O(s+sk0a)]=k0wpulse+O(s3/2+s3/2k0a),
rdSzdxdy=C0Δdρ(2a2ρ)d/2ρ(1+hρ)2s6[1+O(1k0a)],
k0wexact=2k0ass+3/2+O(1k0a)=k0wpulse+O(1k0a),
Ex=Re{Ax2R2A++BjaR}Ey=η0Hx=Re{AxyR2}η0Hy=Re{Ay2R2A++BjaR}.
τ=t2SzdxdydtSzdxdydt,

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