Abstract

We have investigated the coupling efficiency and cavity loss associated with a ring cavity that has a hole in one of the focusing mirrors. The aperture provides a means through which intracavity high-harmonic generation can be coupled from the cavity. By studying different cavity geometries and input modes we have found that the integration of phase-plates on the focusing mirrors provides the best performance in terms of input coupling efficiency, cavity loss, and output-coupling of the generated high harmonic light.

© 2006 Optical Society of America

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References

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  1. M.J. Thorpe, K.D. Moll, R.J. Jones, B. Safdi, and J. Ye, "Broadband cavity ringdown spectroscopy for sensitive and rapid molecular detection," Science 311, 1595 (2006).
    [CrossRef] [PubMed]
  2. M.J. Thorpe, R.J. Jones, K.D. Moll, J. Ye, and R. Lalezari, "Precise measurements of optical cavity dispersion and mirror coating properties via femtosecond combs," Opt. Express 13, 882 (2005).
    [CrossRef] [PubMed]
  3. R.J. Jones and J. Ye, "High-repetition-rate coherent femtosecond pulse amplification with an external passive optical cavity," Opt. Lett. 29, 2812 (2004).
    [CrossRef] [PubMed]
  4. V.P. Yanovsky and F.W. Wise, "Frequency doubling of 100-fs pulses with 50% efficiency by use of a resonant enhancement cavity," Opt. Lett. 19, 1952 (1994).
    [CrossRef] [PubMed]
  5. F.O. Ilday and F.X. Kartner, "Cavity-enhanced optical parametric chirped-pulse amplification," Opt. Lett. 31, 637 (2006).
    [CrossRef] [PubMed]
  6. R.J. Jones, K.D. Moll, M.J. Thorpe, and J. Ye, "Phase-coherent frequency combs in the EUV via high-harmonic generation inside a femtosecond enhancement cavity," Phys. Rev. Lett. 94, 193201 (2005).
    [CrossRef] [PubMed]
  7. C. Gohle,  et al., "A frequency comb in the extreme ultraviolet," Nature 436, 234 (2005).
    [CrossRef] [PubMed]
  8. K.D. Moll, R.J. Jones and J. Ye, "Nonlinear dynamics inside femtosecond enhancement cavities," Opt. Express 13, 1672 (2005).
    [CrossRef] [PubMed]
  9. A. Siegman, Lasers, (University Science Books, Sausalito, California, 1986).
  10. G.B. Arfken and H.J. Weber, Mathematical methods for physicists, 4th ed., (Academic, San Diego, California, 1995).
  11. J. Peatross, J.L. Chaloupka, and D.D. Meyerhofer, "High-order harmonic-generation with an annular laser-beam," Opt. Lett. 19, 942 (1994).
    [CrossRef] [PubMed]
  12. E.W. Weisstein,"Generalized Hypergeometric Function." (MathWorld - A Wolfram Web Resource, 2006), http://mathworld.wolfram.com/GeneralizedHypergeometricFunction.html
  13. S.V. Fomichev, P. Breger, B. Carre, P. Agostini, and D.F. Zaretsky, "Non-collinear high-harmonic generation," Laser Phys. 12, 383 (2002).
  14. S.V. Fomichev, P. Breger, and P. Agostini, "Far-field distribution of third-harmonic generation by two crossed beams," Appl. Phys. B 76, 621 (2003).
    [CrossRef]

2006 (2)

M.J. Thorpe, K.D. Moll, R.J. Jones, B. Safdi, and J. Ye, "Broadband cavity ringdown spectroscopy for sensitive and rapid molecular detection," Science 311, 1595 (2006).
[CrossRef] [PubMed]

F.O. Ilday and F.X. Kartner, "Cavity-enhanced optical parametric chirped-pulse amplification," Opt. Lett. 31, 637 (2006).
[CrossRef] [PubMed]

2005 (4)

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

C. Gohle,  et al., "A frequency comb in the extreme ultraviolet," Nature 436, 234 (2005).
[CrossRef] [PubMed]

M.J. Thorpe, R.J. Jones, K.D. Moll, J. Ye, and R. Lalezari, "Precise measurements of optical cavity dispersion and mirror coating properties via femtosecond combs," Opt. Express 13, 882 (2005).
[CrossRef] [PubMed]

K.D. Moll, R.J. Jones and J. Ye, "Nonlinear dynamics inside femtosecond enhancement cavities," Opt. Express 13, 1672 (2005).
[CrossRef] [PubMed]

2004 (1)

2003 (1)

S.V. Fomichev, P. Breger, and P. Agostini, "Far-field distribution of third-harmonic generation by two crossed beams," Appl. Phys. B 76, 621 (2003).
[CrossRef]

2002 (1)

S.V. Fomichev, P. Breger, B. Carre, P. Agostini, and D.F. Zaretsky, "Non-collinear high-harmonic generation," Laser Phys. 12, 383 (2002).

1994 (2)

Agostini, P.

S.V. Fomichev, P. Breger, and P. Agostini, "Far-field distribution of third-harmonic generation by two crossed beams," Appl. Phys. B 76, 621 (2003).
[CrossRef]

S.V. Fomichev, P. Breger, B. Carre, P. Agostini, and D.F. Zaretsky, "Non-collinear high-harmonic generation," Laser Phys. 12, 383 (2002).

Breger, P.

S.V. Fomichev, P. Breger, and P. Agostini, "Far-field distribution of third-harmonic generation by two crossed beams," Appl. Phys. B 76, 621 (2003).
[CrossRef]

S.V. Fomichev, P. Breger, B. Carre, P. Agostini, and D.F. Zaretsky, "Non-collinear high-harmonic generation," Laser Phys. 12, 383 (2002).

Carre, B.

S.V. Fomichev, P. Breger, B. Carre, P. Agostini, and D.F. Zaretsky, "Non-collinear high-harmonic generation," Laser Phys. 12, 383 (2002).

Chaloupka, J.L.

Fomichev, S.V.

S.V. Fomichev, P. Breger, and P. Agostini, "Far-field distribution of third-harmonic generation by two crossed beams," Appl. Phys. B 76, 621 (2003).
[CrossRef]

S.V. Fomichev, P. Breger, B. Carre, P. Agostini, and D.F. Zaretsky, "Non-collinear high-harmonic generation," Laser Phys. 12, 383 (2002).

Gohle, C.

C. Gohle,  et al., "A frequency comb in the extreme ultraviolet," Nature 436, 234 (2005).
[CrossRef] [PubMed]

Ilday, F.O.

Jones, R.J.

Kartner, F.X.

Lalezari, R.

Meyerhofer, D.D.

Moll, K.D.

M.J. Thorpe, K.D. Moll, R.J. Jones, B. Safdi, and J. Ye, "Broadband cavity ringdown spectroscopy for sensitive and rapid molecular detection," Science 311, 1595 (2006).
[CrossRef] [PubMed]

M.J. Thorpe, R.J. Jones, K.D. Moll, J. Ye, and R. Lalezari, "Precise measurements of optical cavity dispersion and mirror coating properties via femtosecond combs," Opt. Express 13, 882 (2005).
[CrossRef] [PubMed]

K.D. Moll, R.J. Jones and J. Ye, "Nonlinear dynamics inside femtosecond enhancement cavities," Opt. Express 13, 1672 (2005).
[CrossRef] [PubMed]

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

Peatross, J.

Safdi, B.

M.J. Thorpe, K.D. Moll, R.J. Jones, B. Safdi, and J. Ye, "Broadband cavity ringdown spectroscopy for sensitive and rapid molecular detection," Science 311, 1595 (2006).
[CrossRef] [PubMed]

Thorpe, M.J.

M.J. Thorpe, K.D. Moll, R.J. Jones, B. Safdi, and J. Ye, "Broadband cavity ringdown spectroscopy for sensitive and rapid molecular detection," Science 311, 1595 (2006).
[CrossRef] [PubMed]

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

M.J. Thorpe, R.J. Jones, K.D. Moll, J. Ye, and R. Lalezari, "Precise measurements of optical cavity dispersion and mirror coating properties via femtosecond combs," Opt. Express 13, 882 (2005).
[CrossRef] [PubMed]

Wise, F.W.

Yanovsky, V.P.

Ye, J.

Zaretsky, D.F.

S.V. Fomichev, P. Breger, B. Carre, P. Agostini, and D.F. Zaretsky, "Non-collinear high-harmonic generation," Laser Phys. 12, 383 (2002).

Appl. Phys. B (1)

S.V. Fomichev, P. Breger, and P. Agostini, "Far-field distribution of third-harmonic generation by two crossed beams," Appl. Phys. B 76, 621 (2003).
[CrossRef]

Laser Phys. (1)

S.V. Fomichev, P. Breger, B. Carre, P. Agostini, and D.F. Zaretsky, "Non-collinear high-harmonic generation," Laser Phys. 12, 383 (2002).

Nature (1)

C. Gohle,  et al., "A frequency comb in the extreme ultraviolet," Nature 436, 234 (2005).
[CrossRef] [PubMed]

Opt. Express (2)

Opt. Lett. (4)

Phys. Rev. Lett. (1)

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

Science (1)

M.J. Thorpe, K.D. Moll, R.J. Jones, B. Safdi, and J. Ye, "Broadband cavity ringdown spectroscopy for sensitive and rapid molecular detection," Science 311, 1595 (2006).
[CrossRef] [PubMed]

Other (3)

A. Siegman, Lasers, (University Science Books, Sausalito, California, 1986).

G.B. Arfken and H.J. Weber, Mathematical methods for physicists, 4th ed., (Academic, San Diego, California, 1995).

E.W. Weisstein,"Generalized Hypergeometric Function." (MathWorld - A Wolfram Web Resource, 2006), http://mathworld.wolfram.com/GeneralizedHypergeometricFunction.html

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Figures (7)

Fig. 1.
Fig. 1.

Schematic of the ring cavity under investigation. A hole of radius a is drilled in one of the curved mirrors to allow the high-harmonic light to escape from the cavity. The curved mirrors have radius of curvature R 1 and R 2 and the distances of separation between the mirrors are denoted as di .

Fig. 2.
Fig. 2.

(a) Comparison of expected cavity loss for a gaussian (ν = 0) and donut-mode (ν = 1) beam coupled to a cavity with a hole drilled in one of the mirrors. The losses were also approximated by integrating the power of the unperturbed beam over the hole area which are shown as solid and dashed lines for the Gaussian and donut mode, respectively. (b) Percentage of light that can be coupled to cavity mode using an ideal gaussian or donut-mode.

Fig. 3.
Fig. 3.

(a) Comparison of loss for the donut mode at varying levels of cavity stability. (b) Fortunately, even after renormalizing the hole size by the mode diameter, for larger hole sizes the cavity loss is less when the cavity is operated near the edge of stability where the intracavity focus is tighter (△ and ☐).

Fig. 4.
Fig. 4.

Ninth-harmonic intensity profiles at the apertured mirror for gaussian, donut (ν =1), and TEM01 input beams. The high-harmonic donut has a ν = 9 character. The white circle represent the 225-μm diameter aperture.

Fig. 5.
Fig. 5.

(a) Intracavity loss near the inner edge of stability for a cavity that has phase masks integrated on the curved mirrors. A hybrid mode which has ν = 0 and ν = 1 character in different parts of the cavity is supported. For comparison, the round-trip loss without the integrated masks is also computed for an input gaussian and donut mode. (b) The hybrid mode can also be excited with similar coupling efficiency as the donut mode.

Fig. 6.
Fig. 6.

Output coupling efficiency of the HHG for a cavity (a) without and (b) with phase masks integrated onto the concave mirrors. Note the vertical scale difference between (a) and (b)

Fig. 7.
Fig. 7.

Output coupling method using noncollinear geometry. The length of the optical cavity is twice the laser-cavity length such that two pulses inside the cavity can simultaneously focus into a gas sample. The noncollinearly generated HHG is output coupled through a gap between the two focusing mirrors.

Equations (8)

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u ν ( r , ϕ ) = p = 0 c p 2 2 π p ! ( p + ν ) ! L p ν ( 2 r 2 w 2 ) w ( 2 r w ) ν e ik r 2 R e r 2 w 2 e i ν ϕ e i ψ p , ν
A D 2 D 1 c = β c
A p , m = p ! ( p + ν ) ! m ! ( m + ν ) ! S p , m ν , ν
S p , m ν , η = 2 a 2 w 0 2 L p ν ( y ) L m ν ( y ) y η e y d y
S p , m ν , η = L p ν ( y 0 ) L m ν ( y 0 ) ( y 0 ) η e y 0 i = 0 p 1 S i , m ν , η j = 0 m 1 S p , j ν , η + η S p , m ν , η 1 .
N q , p = 0 L q 1 ( y ) L p 0 ( y ) y e y q + 1 d y = Γ ( p 1 2 ) Γ ( q + 1 2 ) 3 F 2 ( p , 3 2 , q ; q + 1 2 , p + 3 2 ; 1 ) 4 π q + 1 Γ ( p + 1 ) Γ ( q + 1 )
N q , p = ( 4 q + 8 p q 7 8 p 2 + 16 p ) 2 p ( 1 + 2 q 2 p ) N q , p 1 ( p 1 ) ( 5 + 2 q 2 p ) p ( 1 + 2 q 2 p ) N q , p 2
β c = A N D 2 N T D 1 c

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