Abstract

A computational study of line-focus generation was done using a self-written ray-tracing code and compared to experimental data. Two line-focusing geometries were compared, i.e., either exploiting the sagittal astigmatism of a tilted spherical mirror or using the spherical aberration of an off-axis- illuminated spherical mirror. Line focusing by means of astigmatism or spherical aberration showed identical results as expected for the equivalence of the two frames of reference. The variation of the incidence angle on the target affects the line-focus length, which affects the amplification length such that as long as the irradiance is above the amplification threshold, it is advantageous to have a longer line focus. The amplification threshold is physically dependent on operating parameters and plasma-column conditions and in the present study addresses four possible cases.

© 2011 Optical Society of America

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References

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    [CrossRef]
  2. I. N. Ross, J. Boon, R. Corbett, A. Damerell, P. Gottfeldt, C. Hooker, M. H. Key, G. Kiehn, C. Lewis, and O. Willi, “Design and performance of a new line focus geometry for x-ray laser experiments,” Appl. Opt. 26, 1584–1588 (1987).
    [CrossRef] [PubMed]
  3. R. Keenan, J. Dunn, V. N. Shlyaptsev, R. F. Smith, P. K. Patel, and D. F. Price, “Efficient pumping schemes for high average brightness collisional x-ray lasers,” Proc. SPIE 5197, 213–219 (2003).
    [CrossRef]
  4. W. Kruer, The Physics of Laser Plasma Interactions(Westview, 2003).
  5. M. Gruenig, C. Imesch, F. Staub, and J. E. Balmer, “Saturated x-ray lasing in Ni-like Sn at 11.9 nm using the grazing incidence scheme,” Opt. Commun. 282, 267–271 (2009).
    [CrossRef]
  6. D. Ursescu, “Grazing incidence pumped Zr x-ray laser for spectroscopy on Li-like ions,” Ph.D. Thesis (J. Gutenberg University, 2006).
  7. R. C. Elton, X-Ray Lasers (Academic, 1990).
  8. G. J. Pert, “Optimizing the performance of nickel-like collisionally pumped x-ray lasers,” Phys. Rev. A 73, 033809 (2006).
    [CrossRef]
  9. Y. Wang, M. A. Larotonda, B. M. Luther, D. Alessi, B. Berrill, V. N. Shlyaptsev, and J. J. Rocca, “Demonstration of high-repetition-rate tabletop soft-x-ray lasers with saturated output at wavelengths down to 13.9 nm and gain down to 10.9 nm,” Phys. Rev. A 72, 053807 (2005).
    [CrossRef]

2009

M. Gruenig, C. Imesch, F. Staub, and J. E. Balmer, “Saturated x-ray lasing in Ni-like Sn at 11.9 nm using the grazing incidence scheme,” Opt. Commun. 282, 267–271 (2009).
[CrossRef]

2006

G. J. Pert, “Optimizing the performance of nickel-like collisionally pumped x-ray lasers,” Phys. Rev. A 73, 033809 (2006).
[CrossRef]

2005

Y. Wang, M. A. Larotonda, B. M. Luther, D. Alessi, B. Berrill, V. N. Shlyaptsev, and J. J. Rocca, “Demonstration of high-repetition-rate tabletop soft-x-ray lasers with saturated output at wavelengths down to 13.9 nm and gain down to 10.9 nm,” Phys. Rev. A 72, 053807 (2005).
[CrossRef]

2003

R. Keenan, J. Dunn, V. N. Shlyaptsev, R. F. Smith, P. K. Patel, and D. F. Price, “Efficient pumping schemes for high average brightness collisional x-ray lasers,” Proc. SPIE 5197, 213–219 (2003).
[CrossRef]

1987

1985

I. N. Ross and E. M. Hodgson, “Some optical designs for the generation of high quality line foci,” J. Phys. E 18, 169–173(1985).
[CrossRef]

Alessi, D.

Y. Wang, M. A. Larotonda, B. M. Luther, D. Alessi, B. Berrill, V. N. Shlyaptsev, and J. J. Rocca, “Demonstration of high-repetition-rate tabletop soft-x-ray lasers with saturated output at wavelengths down to 13.9 nm and gain down to 10.9 nm,” Phys. Rev. A 72, 053807 (2005).
[CrossRef]

Balmer, J. E.

M. Gruenig, C. Imesch, F. Staub, and J. E. Balmer, “Saturated x-ray lasing in Ni-like Sn at 11.9 nm using the grazing incidence scheme,” Opt. Commun. 282, 267–271 (2009).
[CrossRef]

Berrill, B.

Y. Wang, M. A. Larotonda, B. M. Luther, D. Alessi, B. Berrill, V. N. Shlyaptsev, and J. J. Rocca, “Demonstration of high-repetition-rate tabletop soft-x-ray lasers with saturated output at wavelengths down to 13.9 nm and gain down to 10.9 nm,” Phys. Rev. A 72, 053807 (2005).
[CrossRef]

Boon, J.

Corbett, R.

Damerell, A.

Dunn, J.

R. Keenan, J. Dunn, V. N. Shlyaptsev, R. F. Smith, P. K. Patel, and D. F. Price, “Efficient pumping schemes for high average brightness collisional x-ray lasers,” Proc. SPIE 5197, 213–219 (2003).
[CrossRef]

Elton, R. C.

R. C. Elton, X-Ray Lasers (Academic, 1990).

Gottfeldt, P.

Gruenig, M.

M. Gruenig, C. Imesch, F. Staub, and J. E. Balmer, “Saturated x-ray lasing in Ni-like Sn at 11.9 nm using the grazing incidence scheme,” Opt. Commun. 282, 267–271 (2009).
[CrossRef]

Hodgson, E. M.

I. N. Ross and E. M. Hodgson, “Some optical designs for the generation of high quality line foci,” J. Phys. E 18, 169–173(1985).
[CrossRef]

Hooker, C.

Imesch, C.

M. Gruenig, C. Imesch, F. Staub, and J. E. Balmer, “Saturated x-ray lasing in Ni-like Sn at 11.9 nm using the grazing incidence scheme,” Opt. Commun. 282, 267–271 (2009).
[CrossRef]

Keenan, R.

R. Keenan, J. Dunn, V. N. Shlyaptsev, R. F. Smith, P. K. Patel, and D. F. Price, “Efficient pumping schemes for high average brightness collisional x-ray lasers,” Proc. SPIE 5197, 213–219 (2003).
[CrossRef]

Key, M. H.

Kiehn, G.

Kruer, W.

W. Kruer, The Physics of Laser Plasma Interactions(Westview, 2003).

Larotonda, M. A.

Y. Wang, M. A. Larotonda, B. M. Luther, D. Alessi, B. Berrill, V. N. Shlyaptsev, and J. J. Rocca, “Demonstration of high-repetition-rate tabletop soft-x-ray lasers with saturated output at wavelengths down to 13.9 nm and gain down to 10.9 nm,” Phys. Rev. A 72, 053807 (2005).
[CrossRef]

Lewis, C.

Luther, B. M.

Y. Wang, M. A. Larotonda, B. M. Luther, D. Alessi, B. Berrill, V. N. Shlyaptsev, and J. J. Rocca, “Demonstration of high-repetition-rate tabletop soft-x-ray lasers with saturated output at wavelengths down to 13.9 nm and gain down to 10.9 nm,” Phys. Rev. A 72, 053807 (2005).
[CrossRef]

Patel, P. K.

R. Keenan, J. Dunn, V. N. Shlyaptsev, R. F. Smith, P. K. Patel, and D. F. Price, “Efficient pumping schemes for high average brightness collisional x-ray lasers,” Proc. SPIE 5197, 213–219 (2003).
[CrossRef]

Pert, G. J.

G. J. Pert, “Optimizing the performance of nickel-like collisionally pumped x-ray lasers,” Phys. Rev. A 73, 033809 (2006).
[CrossRef]

Price, D. F.

R. Keenan, J. Dunn, V. N. Shlyaptsev, R. F. Smith, P. K. Patel, and D. F. Price, “Efficient pumping schemes for high average brightness collisional x-ray lasers,” Proc. SPIE 5197, 213–219 (2003).
[CrossRef]

Rocca, J. J.

Y. Wang, M. A. Larotonda, B. M. Luther, D. Alessi, B. Berrill, V. N. Shlyaptsev, and J. J. Rocca, “Demonstration of high-repetition-rate tabletop soft-x-ray lasers with saturated output at wavelengths down to 13.9 nm and gain down to 10.9 nm,” Phys. Rev. A 72, 053807 (2005).
[CrossRef]

Ross, I. N.

Shlyaptsev, V. N.

Y. Wang, M. A. Larotonda, B. M. Luther, D. Alessi, B. Berrill, V. N. Shlyaptsev, and J. J. Rocca, “Demonstration of high-repetition-rate tabletop soft-x-ray lasers with saturated output at wavelengths down to 13.9 nm and gain down to 10.9 nm,” Phys. Rev. A 72, 053807 (2005).
[CrossRef]

R. Keenan, J. Dunn, V. N. Shlyaptsev, R. F. Smith, P. K. Patel, and D. F. Price, “Efficient pumping schemes for high average brightness collisional x-ray lasers,” Proc. SPIE 5197, 213–219 (2003).
[CrossRef]

Smith, R. F.

R. Keenan, J. Dunn, V. N. Shlyaptsev, R. F. Smith, P. K. Patel, and D. F. Price, “Efficient pumping schemes for high average brightness collisional x-ray lasers,” Proc. SPIE 5197, 213–219 (2003).
[CrossRef]

Staub, F.

M. Gruenig, C. Imesch, F. Staub, and J. E. Balmer, “Saturated x-ray lasing in Ni-like Sn at 11.9 nm using the grazing incidence scheme,” Opt. Commun. 282, 267–271 (2009).
[CrossRef]

Ursescu, D.

D. Ursescu, “Grazing incidence pumped Zr x-ray laser for spectroscopy on Li-like ions,” Ph.D. Thesis (J. Gutenberg University, 2006).

Wang, Y.

Y. Wang, M. A. Larotonda, B. M. Luther, D. Alessi, B. Berrill, V. N. Shlyaptsev, and J. J. Rocca, “Demonstration of high-repetition-rate tabletop soft-x-ray lasers with saturated output at wavelengths down to 13.9 nm and gain down to 10.9 nm,” Phys. Rev. A 72, 053807 (2005).
[CrossRef]

Willi, O.

Appl. Opt.

J. Phys. E

I. N. Ross and E. M. Hodgson, “Some optical designs for the generation of high quality line foci,” J. Phys. E 18, 169–173(1985).
[CrossRef]

Opt. Commun.

M. Gruenig, C. Imesch, F. Staub, and J. E. Balmer, “Saturated x-ray lasing in Ni-like Sn at 11.9 nm using the grazing incidence scheme,” Opt. Commun. 282, 267–271 (2009).
[CrossRef]

Phys. Rev. A

G. J. Pert, “Optimizing the performance of nickel-like collisionally pumped x-ray lasers,” Phys. Rev. A 73, 033809 (2006).
[CrossRef]

Y. Wang, M. A. Larotonda, B. M. Luther, D. Alessi, B. Berrill, V. N. Shlyaptsev, and J. J. Rocca, “Demonstration of high-repetition-rate tabletop soft-x-ray lasers with saturated output at wavelengths down to 13.9 nm and gain down to 10.9 nm,” Phys. Rev. A 72, 053807 (2005).
[CrossRef]

Proc. SPIE

R. Keenan, J. Dunn, V. N. Shlyaptsev, R. F. Smith, P. K. Patel, and D. F. Price, “Efficient pumping schemes for high average brightness collisional x-ray lasers,” Proc. SPIE 5197, 213–219 (2003).
[CrossRef]

Other

W. Kruer, The Physics of Laser Plasma Interactions(Westview, 2003).

D. Ursescu, “Grazing incidence pumped Zr x-ray laser for spectroscopy on Li-like ions,” Ph.D. Thesis (J. Gutenberg University, 2006).

R. C. Elton, X-Ray Lasers (Academic, 1990).

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Figures (8)

Fig. 1
Fig. 1

Geometry of the line-focus generation with a spherical reflector oriented for three grazing incidence angles on the target. In the sketch the target position is kept fixed, and the reflector is moved correspondingly. The radius of curvature (ROC) is taken as in the Bern x-ray laser facility (“BeAGLE” system). The sketch visualizes the equivalence between the two discussed line-focusing geometries; i.e., thick traces indicate tilted reflectors and thin traces indicate an off-axis large-aperture illuminated reflector.

Fig. 2
Fig. 2

Schematic of the focusing reflectors, here for the case of a 45 ° grazing angle. (a) Astigmatic line focus by tilted mirror ( M tilt ) with tangential focus ( F T , length parallel to the page) and sagittal focus ( F S , length normal to the page) at the target position; (b) off-axis illumination of the nontilted mirror ( M off ) to exploit its spherical aberration for a line focus ( F SA ) at the target position.

Fig. 3
Fig. 3

Calculated line-focus profiles produced by means of (a) sagittal focus with tilt-induced astigmatism and (b)  off-axis spherical aberration, as a function of incidence angle (top), driver beam spatial profile (middle), and driver beam diameter (bottom).

Fig. 4
Fig. 4

Line-focus length (a) as a function of incidence angle at 80 mm diameter and (b) as a function of beam diameter for 45 ° incidence, determined with various computational methods discussed in the text.

Fig. 5
Fig. 5

Width- and peak-normalized line-focus irradiance profiles produced by means of (a) sagittal focus with tilting-induced astigmatism and (b) off-axis spherical aberration as a function of incidence angle (top), beam profile (middle), and beam diameter (bottom).

Fig. 6
Fig. 6

(a) Normalized line-focus irradiance distributions as a function of incidence angle from Fig. 3a. One can define four important cases for an amplification threshold, as explained in the text (cases are labeled as A, B, C, D). (b) Depending on the four cases, the amplification length will have an optimum (high threshold) or not (low threshold), as a function of incidence angle.

Fig. 7
Fig. 7

Spot diagrams obtained by ray tracing with an additional target tilt of 2 ° to obtain the indicated incidence angles.

Fig. 8
Fig. 8

Comparison of experimental and computational results. The experimental curve was obtained with an incidence angle of 45 ° , a beam diameter of 60 mm (FWHM), and a super-Gaussian profile of order N = 6 .

Tables (1)

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Table 1 Parameter Settings

Equations (4)

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f T = ( R / 2 ) cos θ
f S = ( R / 2 ) / cos θ ,
I ( r ) = I o Exp [ 2 ( r / r o ) 2 N ] ,
L = f 1 ( sin ϑ + d 4 f ) 2 f 1 ( sin ϑ d 4 f ) 2 .

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