Abstract

We introduce Bessel-Gauss beam enhancement cavities that may circumvent the major obstacles to more efficient cavity-enhanced high-field physics such as high-harmonic generation. The basic properties of Bessel-Gauss beams are reviewed and their transformation properties through simple optical systems (consisting of spherical and conical elements) are presented. A general Bessel-Gauss cavity design strategy is outlined, and a particular geometry, the confocal Bessel-Gauss cavity, is analyzed in detail. We numerically simulate the confocal Bessel-Gauss cavity and present an example cavity with 300 MHz repetition rate supporting an effective waist of 33 μm at the focus and an intensity ratio from the focus to the cavity mirror surfaces of 1.5 × 104.

© 2012 OSA

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  1. R. J. Jones, K. D. Moll, M. J. Thorpe, and J. Ye, “Phase-coherent frequency combs in the vacuum ultraviolet via high-harmonic generation inside a femtosecond enhancement cavity,” Phys. Rev. Lett.94(19), 193201 (2005).
    [CrossRef] [PubMed]
  2. C. Gohle, T. Udem, M. Herrmann, J. Rauschenberger, R. Holzwarth, H. A. Schuessler, F. Krausz, and T. W. Hänsch, “A frequency comb in the extreme ultraviolet,” Nature436(7048), 234–237 (2005).
    [CrossRef] [PubMed]
  3. J. Lee, D. R. Carlson, and R. J. Jones, “Optimizing intracavity high harmonic generation for XUV fs frequency combs,” Opt. Express19(23), 23315–23326 (2011).
    [CrossRef] [PubMed]
  4. A. Cingöz, D. C. Yost, T. K. Allison, A. Ruehl, M. E. Fermann, I. Hartl, and J. Ye, “Direct frequency comb spectroscopy in the extreme ultraviolet,” Nature482(7383), 68–71 (2012).
    [CrossRef] [PubMed]
  5. D. C. Yost, T. R. Schibli, and J. Ye, “Efficient output coupling of intracavity high-harmonic generation,” Opt. Lett.33(10), 1099–1101 (2008).
    [CrossRef] [PubMed]
  6. S. Holzberger, I. Pupeza, D. Esser, J. Weitenberg, H. Carstens, T. Eidam, P. Russbüldt, J. Limpert, T. Udem, A. Tünnermann, T. Hänsch, F. Krausz, and E. Fill, “Sub-25 nm high-harmonic generation with a 78-MHz repetition rate enhancement cavity,” QELS 2012, Postdeadline Paper QTh5B.7.
  7. I. Pupeza, T. Eidam, J. Rauschenberger, B. Bernhardt, A. Ozawa, E. Fill, A. Apolonski, T. Udem, J. Limpert, Z. A. Alahmed, A. M. Azzeer, A. Tünnermann, T. W. Hänsch, and F. Krausz, “Power scaling of a high-repetition-rate enhancement cavity,” Opt. Lett.35(12), 2052–2054 (2010).
    [CrossRef] [PubMed]
  8. K. D. Moll, R. J. Jones, and J. Ye, “Output coupling methods for cavity-based high-harmonic generation,” Opt. Express14(18), 8189–8197 (2006).
    [CrossRef] [PubMed]
  9. areJ. Weitenberg, P. Rußbüldt, T. Eidam, and I. Pupeza, “Transverse mode tailoring in a quasi-imaging high-finesse femtosecond enhancement cavity,” Opt. Express19(10), 9551–9561 (2011).
    [CrossRef] [PubMed]
  10. F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun.64(6), 491–495 (1987).
    [CrossRef]
  11. V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, and G. Schirripa Spagnolo, “Generalized Bessel-Gauss beams,” J. Mod. Opt.43, 1155–1166 (1996).
  12. C. J. R. Sheppard and T. Wilson, “Gaussian-beam theory of lenses with annular aperture,” IEE J Microwaves, Opt. Acoust.2(4), 105–112 (1978).
    [CrossRef]
  13. A. A. Al-Rashed and B. E. A. Saleh, “Decentered Gaussian beams,” Appl. Opt.34(30), 6819–6825 (1995).
    [CrossRef] [PubMed]
  14. J. D. Jackson, Classical Electrodynamics (John Wiley & Sons, 1999), Chap. 3.
  15. M. Santarsiero, “Propagation of generalized Bessel-Gauss beams through ABCD optical systems,” Opt. Commun.132(1-2), 1–7 (1996).
    [CrossRef]
  16. J. Rogel-Salazar, G. H. C. New, and S. Chávez-Cerda, “Bessel-Gauss beam optical resonator,” Opt. Commun.190(1-6), 117–122 (2001).
    [CrossRef]
  17. A. N. Khilo, E. G. Katranji, and A. A. Ryzhevich, “Axicon-based Bessel resonator: analytical description and experiment,” J. Opt. Soc. Am. A18(8), 1986–1992 (2001).
    [CrossRef] [PubMed]
  18. J. C. Gutiérrez-Vega, R. Rodríguez-Masegosa, and S. Chávez-Cerda, “Bessel-Gauss resonator with spherical output mirror: geometrical- and wave-optics analysis,” J. Opt. Soc. Am. A20(11), 2113–2122 (2003).
    [CrossRef] [PubMed]
  19. P. Pääkkönen and J. Turunen, “Resonators with Bessel-Gauss modes,” Opt. Commun.156(4-6), 359–366 (1998).
    [CrossRef]
  20. G. Abram, “High intensity femtosecond enhancement cavities,” M. Eng Thesis, MIT (2009).
  21. H. A. Haus, Waves and Fields in Optoelectronics (CBLS, 2004), Chap. 3.
  22. L. Yu, M. Huang, M. Chen, W. Chen, W. Huang, and Z. Zhu, “Quasi-discrete Hankel transform,” Opt. Lett.23(6), 409–411 (1998).
    [CrossRef] [PubMed]
  23. M. Guizar-Sicairos and J. C. Gutiérrez-Vega, “Computation of quasi-discrete Hankel transforms of integer order for propagating optical wave fields,” J. Opt. Soc. Am. A21(1), 53–58 (2004).
    [CrossRef] [PubMed]

2012 (1)

A. Cingöz, D. C. Yost, T. K. Allison, A. Ruehl, M. E. Fermann, I. Hartl, and J. Ye, “Direct frequency comb spectroscopy in the extreme ultraviolet,” Nature482(7383), 68–71 (2012).
[CrossRef] [PubMed]

2011 (2)

2010 (1)

2008 (1)

2006 (1)

2005 (2)

R. J. Jones, K. D. Moll, M. J. Thorpe, and J. Ye, “Phase-coherent frequency combs in the vacuum ultraviolet via high-harmonic generation inside a femtosecond enhancement cavity,” Phys. Rev. Lett.94(19), 193201 (2005).
[CrossRef] [PubMed]

C. Gohle, T. Udem, M. Herrmann, J. Rauschenberger, R. Holzwarth, H. A. Schuessler, F. Krausz, and T. W. Hänsch, “A frequency comb in the extreme ultraviolet,” Nature436(7048), 234–237 (2005).
[CrossRef] [PubMed]

2004 (1)

2003 (1)

2001 (2)

A. N. Khilo, E. G. Katranji, and A. A. Ryzhevich, “Axicon-based Bessel resonator: analytical description and experiment,” J. Opt. Soc. Am. A18(8), 1986–1992 (2001).
[CrossRef] [PubMed]

J. Rogel-Salazar, G. H. C. New, and S. Chávez-Cerda, “Bessel-Gauss beam optical resonator,” Opt. Commun.190(1-6), 117–122 (2001).
[CrossRef]

1998 (2)

P. Pääkkönen and J. Turunen, “Resonators with Bessel-Gauss modes,” Opt. Commun.156(4-6), 359–366 (1998).
[CrossRef]

L. Yu, M. Huang, M. Chen, W. Chen, W. Huang, and Z. Zhu, “Quasi-discrete Hankel transform,” Opt. Lett.23(6), 409–411 (1998).
[CrossRef] [PubMed]

1996 (2)

M. Santarsiero, “Propagation of generalized Bessel-Gauss beams through ABCD optical systems,” Opt. Commun.132(1-2), 1–7 (1996).
[CrossRef]

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, and G. Schirripa Spagnolo, “Generalized Bessel-Gauss beams,” J. Mod. Opt.43, 1155–1166 (1996).

1995 (1)

1987 (1)

F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun.64(6), 491–495 (1987).
[CrossRef]

1978 (1)

C. J. R. Sheppard and T. Wilson, “Gaussian-beam theory of lenses with annular aperture,” IEE J Microwaves, Opt. Acoust.2(4), 105–112 (1978).
[CrossRef]

Alahmed, Z. A.

Allison, T. K.

A. Cingöz, D. C. Yost, T. K. Allison, A. Ruehl, M. E. Fermann, I. Hartl, and J. Ye, “Direct frequency comb spectroscopy in the extreme ultraviolet,” Nature482(7383), 68–71 (2012).
[CrossRef] [PubMed]

Al-Rashed, A. A.

Apolonski, A.

Azzeer, A. M.

Bagini, V.

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, and G. Schirripa Spagnolo, “Generalized Bessel-Gauss beams,” J. Mod. Opt.43, 1155–1166 (1996).

Bernhardt, B.

Carlson, D. R.

Chávez-Cerda, S.

Chen, M.

Chen, W.

Cingöz, A.

A. Cingöz, D. C. Yost, T. K. Allison, A. Ruehl, M. E. Fermann, I. Hartl, and J. Ye, “Direct frequency comb spectroscopy in the extreme ultraviolet,” Nature482(7383), 68–71 (2012).
[CrossRef] [PubMed]

Eidam, T.

Fermann, M. E.

A. Cingöz, D. C. Yost, T. K. Allison, A. Ruehl, M. E. Fermann, I. Hartl, and J. Ye, “Direct frequency comb spectroscopy in the extreme ultraviolet,” Nature482(7383), 68–71 (2012).
[CrossRef] [PubMed]

Fill, E.

Frezza, F.

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, and G. Schirripa Spagnolo, “Generalized Bessel-Gauss beams,” J. Mod. Opt.43, 1155–1166 (1996).

Gohle, C.

C. Gohle, T. Udem, M. Herrmann, J. Rauschenberger, R. Holzwarth, H. A. Schuessler, F. Krausz, and T. W. Hänsch, “A frequency comb in the extreme ultraviolet,” Nature436(7048), 234–237 (2005).
[CrossRef] [PubMed]

Gori, F.

F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun.64(6), 491–495 (1987).
[CrossRef]

Guattari, G.

F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun.64(6), 491–495 (1987).
[CrossRef]

Guizar-Sicairos, M.

Gutiérrez-Vega, J. C.

Hänsch, T. W.

Hartl, I.

A. Cingöz, D. C. Yost, T. K. Allison, A. Ruehl, M. E. Fermann, I. Hartl, and J. Ye, “Direct frequency comb spectroscopy in the extreme ultraviolet,” Nature482(7383), 68–71 (2012).
[CrossRef] [PubMed]

Herrmann, M.

C. Gohle, T. Udem, M. Herrmann, J. Rauschenberger, R. Holzwarth, H. A. Schuessler, F. Krausz, and T. W. Hänsch, “A frequency comb in the extreme ultraviolet,” Nature436(7048), 234–237 (2005).
[CrossRef] [PubMed]

Holzwarth, R.

C. Gohle, T. Udem, M. Herrmann, J. Rauschenberger, R. Holzwarth, H. A. Schuessler, F. Krausz, and T. W. Hänsch, “A frequency comb in the extreme ultraviolet,” Nature436(7048), 234–237 (2005).
[CrossRef] [PubMed]

Huang, M.

Huang, W.

Jones, R. J.

Katranji, E. G.

Khilo, A. N.

Krausz, F.

Lee, J.

Limpert, J.

Moll, K. D.

K. D. Moll, R. J. Jones, and J. Ye, “Output coupling methods for cavity-based high-harmonic generation,” Opt. Express14(18), 8189–8197 (2006).
[CrossRef] [PubMed]

R. J. Jones, K. D. Moll, M. J. Thorpe, and J. Ye, “Phase-coherent frequency combs in the vacuum ultraviolet via high-harmonic generation inside a femtosecond enhancement cavity,” Phys. Rev. Lett.94(19), 193201 (2005).
[CrossRef] [PubMed]

New, G. H. C.

J. Rogel-Salazar, G. H. C. New, and S. Chávez-Cerda, “Bessel-Gauss beam optical resonator,” Opt. Commun.190(1-6), 117–122 (2001).
[CrossRef]

Ozawa, A.

Pääkkönen, P.

P. Pääkkönen and J. Turunen, “Resonators with Bessel-Gauss modes,” Opt. Commun.156(4-6), 359–366 (1998).
[CrossRef]

Padovani, C.

F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun.64(6), 491–495 (1987).
[CrossRef]

Pupeza, I.

Rauschenberger, J.

Rodríguez-Masegosa, R.

Rogel-Salazar, J.

J. Rogel-Salazar, G. H. C. New, and S. Chávez-Cerda, “Bessel-Gauss beam optical resonator,” Opt. Commun.190(1-6), 117–122 (2001).
[CrossRef]

Ruehl, A.

A. Cingöz, D. C. Yost, T. K. Allison, A. Ruehl, M. E. Fermann, I. Hartl, and J. Ye, “Direct frequency comb spectroscopy in the extreme ultraviolet,” Nature482(7383), 68–71 (2012).
[CrossRef] [PubMed]

Rußbüldt, P.

Ryzhevich, A. A.

Saleh, B. E. A.

Santarsiero, M.

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, and G. Schirripa Spagnolo, “Generalized Bessel-Gauss beams,” J. Mod. Opt.43, 1155–1166 (1996).

M. Santarsiero, “Propagation of generalized Bessel-Gauss beams through ABCD optical systems,” Opt. Commun.132(1-2), 1–7 (1996).
[CrossRef]

Schettini, G.

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, and G. Schirripa Spagnolo, “Generalized Bessel-Gauss beams,” J. Mod. Opt.43, 1155–1166 (1996).

Schibli, T. R.

Schirripa Spagnolo, G.

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, and G. Schirripa Spagnolo, “Generalized Bessel-Gauss beams,” J. Mod. Opt.43, 1155–1166 (1996).

Schuessler, H. A.

C. Gohle, T. Udem, M. Herrmann, J. Rauschenberger, R. Holzwarth, H. A. Schuessler, F. Krausz, and T. W. Hänsch, “A frequency comb in the extreme ultraviolet,” Nature436(7048), 234–237 (2005).
[CrossRef] [PubMed]

Sheppard, C. J. R.

C. J. R. Sheppard and T. Wilson, “Gaussian-beam theory of lenses with annular aperture,” IEE J Microwaves, Opt. Acoust.2(4), 105–112 (1978).
[CrossRef]

Thorpe, M. J.

R. J. Jones, K. D. Moll, M. J. Thorpe, and J. Ye, “Phase-coherent frequency combs in the vacuum ultraviolet via high-harmonic generation inside a femtosecond enhancement cavity,” Phys. Rev. Lett.94(19), 193201 (2005).
[CrossRef] [PubMed]

Tünnermann, A.

Turunen, J.

P. Pääkkönen and J. Turunen, “Resonators with Bessel-Gauss modes,” Opt. Commun.156(4-6), 359–366 (1998).
[CrossRef]

Udem, T.

Weitenberg, J.

Wilson, T.

C. J. R. Sheppard and T. Wilson, “Gaussian-beam theory of lenses with annular aperture,” IEE J Microwaves, Opt. Acoust.2(4), 105–112 (1978).
[CrossRef]

Ye, J.

A. Cingöz, D. C. Yost, T. K. Allison, A. Ruehl, M. E. Fermann, I. Hartl, and J. Ye, “Direct frequency comb spectroscopy in the extreme ultraviolet,” Nature482(7383), 68–71 (2012).
[CrossRef] [PubMed]

D. C. Yost, T. R. Schibli, and J. Ye, “Efficient output coupling of intracavity high-harmonic generation,” Opt. Lett.33(10), 1099–1101 (2008).
[CrossRef] [PubMed]

K. D. Moll, R. J. Jones, and J. Ye, “Output coupling methods for cavity-based high-harmonic generation,” Opt. Express14(18), 8189–8197 (2006).
[CrossRef] [PubMed]

R. J. Jones, K. D. Moll, M. J. Thorpe, and J. Ye, “Phase-coherent frequency combs in the vacuum ultraviolet via high-harmonic generation inside a femtosecond enhancement cavity,” Phys. Rev. Lett.94(19), 193201 (2005).
[CrossRef] [PubMed]

Yost, D. C.

A. Cingöz, D. C. Yost, T. K. Allison, A. Ruehl, M. E. Fermann, I. Hartl, and J. Ye, “Direct frequency comb spectroscopy in the extreme ultraviolet,” Nature482(7383), 68–71 (2012).
[CrossRef] [PubMed]

D. C. Yost, T. R. Schibli, and J. Ye, “Efficient output coupling of intracavity high-harmonic generation,” Opt. Lett.33(10), 1099–1101 (2008).
[CrossRef] [PubMed]

Yu, L.

Zhu, Z.

Appl. Opt. (1)

IEE J Microwaves, Opt. Acoust. (1)

C. J. R. Sheppard and T. Wilson, “Gaussian-beam theory of lenses with annular aperture,” IEE J Microwaves, Opt. Acoust.2(4), 105–112 (1978).
[CrossRef]

J. Mod. Opt. (1)

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, and G. Schirripa Spagnolo, “Generalized Bessel-Gauss beams,” J. Mod. Opt.43, 1155–1166 (1996).

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

Nature (2)

C. Gohle, T. Udem, M. Herrmann, J. Rauschenberger, R. Holzwarth, H. A. Schuessler, F. Krausz, and T. W. Hänsch, “A frequency comb in the extreme ultraviolet,” Nature436(7048), 234–237 (2005).
[CrossRef] [PubMed]

A. Cingöz, D. C. Yost, T. K. Allison, A. Ruehl, M. E. Fermann, I. Hartl, and J. Ye, “Direct frequency comb spectroscopy in the extreme ultraviolet,” Nature482(7383), 68–71 (2012).
[CrossRef] [PubMed]

Opt. Commun. (4)

F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun.64(6), 491–495 (1987).
[CrossRef]

M. Santarsiero, “Propagation of generalized Bessel-Gauss beams through ABCD optical systems,” Opt. Commun.132(1-2), 1–7 (1996).
[CrossRef]

J. Rogel-Salazar, G. H. C. New, and S. Chávez-Cerda, “Bessel-Gauss beam optical resonator,” Opt. Commun.190(1-6), 117–122 (2001).
[CrossRef]

P. Pääkkönen and J. Turunen, “Resonators with Bessel-Gauss modes,” Opt. Commun.156(4-6), 359–366 (1998).
[CrossRef]

Opt. Express (3)

Opt. Lett. (3)

Phys. Rev. Lett. (1)

R. J. Jones, K. D. Moll, M. J. Thorpe, and J. Ye, “Phase-coherent frequency combs in the vacuum ultraviolet via high-harmonic generation inside a femtosecond enhancement cavity,” Phys. Rev. Lett.94(19), 193201 (2005).
[CrossRef] [PubMed]

Other (4)

S. Holzberger, I. Pupeza, D. Esser, J. Weitenberg, H. Carstens, T. Eidam, P. Russbüldt, J. Limpert, T. Udem, A. Tünnermann, T. Hänsch, F. Krausz, and E. Fill, “Sub-25 nm high-harmonic generation with a 78-MHz repetition rate enhancement cavity,” QELS 2012, Postdeadline Paper QTh5B.7.

G. Abram, “High intensity femtosecond enhancement cavities,” M. Eng Thesis, MIT (2009).

H. A. Haus, Waves and Fields in Optoelectronics (CBLS, 2004), Chap. 3.

J. D. Jackson, Classical Electrodynamics (John Wiley & Sons, 1999), Chap. 3.

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Figures (8)

Fig. 1
Fig. 1

Decentered Gaussians and Bessel-Gauss beam construction. (a) A single decentered Gaussian component for constructing a Bessel-Gauss beam (b) A decentered component for constructing a generalized Bessel-Gauss beam.

Fig. 2
Fig. 2

Bessel-Gauss beam types. (a) - (c) Illustrations of the r-z plane cross-section of gBG, BG, and mBG beams respectively. (d) - (f) Plots of the amplitude in the r-z plane for gBG (λ = 1 μm, w0 = 200 μm, φ = 0.21°, r0 = 0.25 mm), BG (λ = 1 μm, w0 = 200 μm, φ = 0.29°). and mBG (λ = 1 μm, w0 = 200 μm, r0 = 1 mm) beams respectively.

Fig. 3
Fig. 3

BG beam and intensity gain. (a) Plot of amplitude cross-section in the z = 0 plane of a BG beam with λ = 1 μm, w0 = 200 μm, and semi-aperture angle φ = 0.29°. Cross-section of the focus in the y-direction is on the right with 2wB labeled. (b) Plot of approximate (orange dashed) and exact (solid green) intensity gain of BG beams with λ = 1 μm, w0 = 30 μm, and semi-aperture angles φ of 1°, 2°, 3°, and 4° at distance z. The intensity gain of a Gaussian beam with λ = 1 μm and w0 = 30 μm (blue curve) is also included. (c) Plot of amplitude cross-section in the z = 20 cm plane of the BG beam from plot (a). Cross-section in the y-direction is included on the right with w and rc labeled.

Fig. 4
Fig. 4

gBG beam transformations. (a) Example 1 geometry: an mBG beam reflecting from a curved mirror. (b) r-z plane cross-section of numerically simulated amplitude for example 1 (note z-axis corresponds to reflecting geometry). (c) r-direction cross-sections of field's spatial amplitude and phase at the end of propagation (numerically simulated (blue) and analytical (red-dashed)). (d) Example 2 geometry: an mBG beam reflecting from a reflecting axicon. (e) and (f) are as (b) and (c) but for example 2. (g) Example 3 geometry: an mBG beam reflecting from a toroidal optic. (h) and (i) are as (b) and (c) but for example 3.

Fig. 5
Fig. 5

Conventional Gaussian cavities and gBG cavities. (a) Conventional Gaussian cavity (b) Revolving the conventional Gaussian cavity about its central axis (dashed), we obtain a cavity supporting gBG modes.

Fig. 6
Fig. 6

Single-mode selection in the confocal BG cavity. (a) Cavity mirror with patterned annular (donut-shaped) region of high-reflectivity. (b) Cross-section of patterned cavity mirror with incident beams.

Fig. 7
Fig. 7

Patterned-mirror confocal BG cavity simulation. (a) r-z plane cross-section of fundamental BG mode amplitude. (b) Normalized mode intensity at mirror surface plotted against r (as labeled in (a)). (c) Mode intensity at focus plotted against r (same normalization as (b) and labeled in (a)).

Fig. 8
Fig. 8

Patterned mirror confocal cavity scaling. (a) Intensity gain, Ig, scaling with repetition rate (i.e. cavity length and mirror radius of curvature). (b) Effective waist, weff, scaling with repetition rate (i.e. cavity length and mirror radius of curvature). For all cavities in these plots Δr = 3.1wmin.

Equations (19)

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u(r,θ,z=0)= a 0 exp( r 2 / w 0 2 )exp( iβrcos( θγ ) )
u(r,θ,z)= a 0 q 0 q(z) exp( ik 2q(z) ( r 2 + r c 2 (z)2r r c (z)cos(θγ) ) )exp( iβrcos(θγ) )
q(z)= q 0 +z ; r c (z)=( β/k )z
U BG (r,z)= 0 2π u(r,θ,z)dγ = A 0 q 0 q(z) exp( ik 2q(z) ( r 2 + r c 2 (z) ) ) J 0 ( βr k r c (z) q(z) r )
U gBG (r,z)= A 0 q 0 q(z) exp( ik 2q(z) ( r 2 + r c 2 (z) ) ) J 0 ( βr k r c (z) q(z) r )
q(z)= q 0 +z ; r c (z)= r 0 +( β/k )z
I P foc = 2P π ( w eff foc ) 2 , w eff foc = w 0 e β 2 w 0 2 /4 I 0 ( β 2 w 0 2 /4 ) w 0 w B 3
I P FF (z)= 2P π ( w eff FF (z)) 2 , w eff FF (z) 2 2π w(z) r c (z)
Bessel-Gauss Beam: I g BG (z)C ( φ φ G ) 2 ( z z 0 ) 2 , C= 12 2π 2.4 12.5 Gaussian Beam: I g G (z) ( z z 0 ) 2
ϕ gBG (r)= ik 2R(L) r 2 +iarg( J 0 ( βr k r c (L) q(L) r ) )
ϕ gBG (r) ik 2R(L) r 2 +( ( L z 0 )( r 0 z 0 ) β k ) ik 1+ ( L/ z 0 ) 2 r
Conical ( ϕ con =iαkr) : q 0 '=q(L) ; r 0 '= r c (L) ; β =βkα
Spherical ( ϕ sph = ik r 2 / 2f ) : q 0 '= q(L) q(L) / f+1 ; r 0 '= r c (L) ; β'=β k r c (L) f
arg( J 0 ( βr k r c (L) q(L) r ) )=arg( J 0 ( u+iv ) )
u= βr 1+ ( L/ z 0 ) 2 kr 1+ ( L/ z 0 ) 2 ( L z 0 )( r 0 z 0 )
v= βr 1+ ( L/ z 0 ) 2 ( L z 0 ) kr 1+ ( L/ z 0 ) 2 ( r 0 z 0 )
arg( J 0 ( u+iv ) )arg( cos(u+ivπ/4 ) ) = tan 1 ( tan(uπ/4 )tanhv ) u+π/4 =( ( L z 0 )( r 0 z 0 ) β k ) k 1+ ( L/ z 0 ) 2 r+ π 4
BG-like Beam: v= β 2 w 0 2 4 ( 2 ( L/ z 0 ) 2 1+ ( L/ z 0 ) 2 )
mBG-like Beam: v= r 0 2 w 0 2 ( 2 1+ ( L/ z 0 ) 2 )

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