Abstract

We introduce a simple and practical method to create ultrashort intense optical vortices for applications involving high-intensity lasers. Our method utilizes femtosecond laser pulses to laser etch grating lines into laser-quality gold mirrors. These grating lines holographically encode an optical vortex. We derive mathematical equations for each individual grating line to be etched, for any desired (integer) topological charge. We investigate the smoothness of the etched grooves. We show that they are smooth enough to produce optical vortices with an intensity that is only a few percent lower than in the ideal case. We demonstrate that the etched gratings can be used in a folded version of our 2f2f setup [Opt. Express 19, 7599 (2005)] to compensate angular dispersion. Finally, we show that the etched gratings withstand intensities of up to 1012W/cm2.

© 2007 Optical Society of America

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References

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  1. M. Protopapas, C. H. Keitel, and P. L. Knight, "Atomic physics with super-high intensity lasers," Rep. Prog. Phys. 60, 389-486 (1997).
    [CrossRef]
  2. S. L. Chin and P. Lambropoulos, eds., Multiphoton Ionization of Atoms (Academic, 1984).
  3. J. F. Nye and M. V. Berry, "Dislocations in wave trains," Proc. R. Soc. London Ser. A 336, 165-190 (1974).
    [CrossRef]
  4. H. Kogelnik and T. Li, "Laser beams and resonators," Appl. Opt. 5, 1550-1565 (1966).
    [CrossRef] [PubMed]
  5. L. Allen, S. M. Barnett, and M. J. Padgett, eds., Optical Angular Momentum (Institute of Physics, 2003).
    [CrossRef]
  6. A. E. Siegman, Lasers (University Science Books, 1986).
  7. M. Vasnetsov and K. Staliunas, eds., Optical Vortices (Nova Science, 1999).
  8. M. A. Bandres and J. C. Gutiérrez-Vega, "Ince-Gaussian beams," Opt. Lett. 29, 144-146 (2004).
    [CrossRef] [PubMed]
  9. M. A. Bandres and J. C. Gutiérrez-Vega, "Ince-Gaussian modes of the paraxial wave equation and stable resonators," J. Opt. Soc. Am. A 21, 873-880 (2004).
    [CrossRef]
  10. U. T. Schwarz, M. A. Bandres, and J. C. Gutiérrez-Vega, "Observation of Ince-Gaussian modes in stable resonators," Opt. Lett. 29, 1870-1872 (2004).
    [CrossRef] [PubMed]
  11. J. B. Bentley, J. A. Davis, M. A. Bandres, and J. C. Gutiérrez-Vega, "Generation of helical Ince-Gaussian beams with a liquid-crystal display," Opt. Lett. 31, 649-651 (2006).
    [CrossRef] [PubMed]
  12. K. Sueda, G. Miyaji, N. Miyanaga, and M. Nakatsuka, "Laguerre-Gaussian beam generation with a multilevel spiral phase plate for high intensity laser pulses," Opt. Express 15, 3548-3553 (2004).
    [CrossRef]
  13. K. J. Moh, X.-C. Yuan, D. Y. Tang, W. C. Cheong, and L. S. Zhang, "Generation of femtosecond optical vortices using a single refractive optical element," Appl. Phys. Lett. 88, 091103 (2006).
    [CrossRef]
  14. W. C. Cheong, W. M. Lee, X.-C. Yuan, and L. S. Zhang, "Direct electron-beam writing of continuous spiral phase plates in negative resist with high power efficiency for optical manipulation," Appl. Phys. Lett. 85, 5784-5786 (2004).
    [CrossRef]
  15. I. G. Mariyenko, J. Strohaber, and C. J. G. J. Uiterwaal, "Creation of optical vortices in femtosecond pulses," Opt. Express 19, 7599-7608 (2005).
    [CrossRef]
  16. J. Strohaber, C. Petersen, and C. J. G. J. Uiterwaal, "Efficient angular dispersion compensation in holographic generation of intense ultrashort paraxial beam modes," Opt. Lett. 32, 2387-2389 (2007).
    [CrossRef] [PubMed]
  17. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).
  18. K. Bezuhanov, A. Dreischuh, G. G. Paulus, M. G. Schätzel, and H. Walther, "Vortices in femtosecond laser fields," Opt. Lett. 29, 1942-1944 (2004).
    [CrossRef] [PubMed]
  19. J. F. James, A Student's Guide to Fourier Transforms (Cambridge U. Press, 1995).
  20. M. V. Berry, "Optical vortices evolving from helicoidal integer and fractional phase steps," J. Opt. A 6, 259-268 (2004).
    [CrossRef]
  21. J. Leach, E. Yao, and M. J. Padgett, "Observation of the vortex structure of a noninteger vortex beam," New J. Phys. 6, 71.1-71.8 (2004).
    [CrossRef]
  22. J. Leach and M. J. Padgett, "Observation of chromatic effects near a white-light vortex," New J. Phys. 5, 154.1-154.7 (2003).
    [CrossRef]
  23. J. Flamand, G. de Villèle, A. Cotel, B. Touzet, and S. Kane, "New MLD gratings adapted for tiling in petawatt-class lasers," J. Phys. IV 133, 601-605 (2006).
  24. M.-S. L. Lee, J.-C. Rodier, P. Lalanne, P. Legagneux, P. Gallais, C. Germain, and J. Rollin, "Blazed-binary diffractive gratings with antireflection coating for improved operation at 10.6 μm," Opt. Eng. 43, 2583-2588 (2004).
    [CrossRef]
  25. J. Leach, G. M. Gibson, M. J. Padgett, E. Esposito, G. McConnell, A. J. Wright, and J. M. Girkin, "Generation of achromatic Bessel beams using a compensated spatial light modulator," Opt. Express 14, 5581-5587 (2006).
    [CrossRef] [PubMed]

2007 (1)

2006 (4)

J. Flamand, G. de Villèle, A. Cotel, B. Touzet, and S. Kane, "New MLD gratings adapted for tiling in petawatt-class lasers," J. Phys. IV 133, 601-605 (2006).

J. Leach, G. M. Gibson, M. J. Padgett, E. Esposito, G. McConnell, A. J. Wright, and J. M. Girkin, "Generation of achromatic Bessel beams using a compensated spatial light modulator," Opt. Express 14, 5581-5587 (2006).
[CrossRef] [PubMed]

J. B. Bentley, J. A. Davis, M. A. Bandres, and J. C. Gutiérrez-Vega, "Generation of helical Ince-Gaussian beams with a liquid-crystal display," Opt. Lett. 31, 649-651 (2006).
[CrossRef] [PubMed]

K. J. Moh, X.-C. Yuan, D. Y. Tang, W. C. Cheong, and L. S. Zhang, "Generation of femtosecond optical vortices using a single refractive optical element," Appl. Phys. Lett. 88, 091103 (2006).
[CrossRef]

2005 (1)

I. G. Mariyenko, J. Strohaber, and C. J. G. J. Uiterwaal, "Creation of optical vortices in femtosecond pulses," Opt. Express 19, 7599-7608 (2005).
[CrossRef]

2004 (9)

M.-S. L. Lee, J.-C. Rodier, P. Lalanne, P. Legagneux, P. Gallais, C. Germain, and J. Rollin, "Blazed-binary diffractive gratings with antireflection coating for improved operation at 10.6 μm," Opt. Eng. 43, 2583-2588 (2004).
[CrossRef]

K. Bezuhanov, A. Dreischuh, G. G. Paulus, M. G. Schätzel, and H. Walther, "Vortices in femtosecond laser fields," Opt. Lett. 29, 1942-1944 (2004).
[CrossRef] [PubMed]

M. V. Berry, "Optical vortices evolving from helicoidal integer and fractional phase steps," J. Opt. A 6, 259-268 (2004).
[CrossRef]

J. Leach, E. Yao, and M. J. Padgett, "Observation of the vortex structure of a noninteger vortex beam," New J. Phys. 6, 71.1-71.8 (2004).
[CrossRef]

W. C. Cheong, W. M. Lee, X.-C. Yuan, and L. S. Zhang, "Direct electron-beam writing of continuous spiral phase plates in negative resist with high power efficiency for optical manipulation," Appl. Phys. Lett. 85, 5784-5786 (2004).
[CrossRef]

K. Sueda, G. Miyaji, N. Miyanaga, and M. Nakatsuka, "Laguerre-Gaussian beam generation with a multilevel spiral phase plate for high intensity laser pulses," Opt. Express 15, 3548-3553 (2004).
[CrossRef]

M. A. Bandres and J. C. Gutiérrez-Vega, "Ince-Gaussian beams," Opt. Lett. 29, 144-146 (2004).
[CrossRef] [PubMed]

M. A. Bandres and J. C. Gutiérrez-Vega, "Ince-Gaussian modes of the paraxial wave equation and stable resonators," J. Opt. Soc. Am. A 21, 873-880 (2004).
[CrossRef]

U. T. Schwarz, M. A. Bandres, and J. C. Gutiérrez-Vega, "Observation of Ince-Gaussian modes in stable resonators," Opt. Lett. 29, 1870-1872 (2004).
[CrossRef] [PubMed]

2003 (1)

J. Leach and M. J. Padgett, "Observation of chromatic effects near a white-light vortex," New J. Phys. 5, 154.1-154.7 (2003).
[CrossRef]

1997 (1)

M. Protopapas, C. H. Keitel, and P. L. Knight, "Atomic physics with super-high intensity lasers," Rep. Prog. Phys. 60, 389-486 (1997).
[CrossRef]

1974 (1)

J. F. Nye and M. V. Berry, "Dislocations in wave trains," Proc. R. Soc. London Ser. A 336, 165-190 (1974).
[CrossRef]

1966 (1)

Appl. Opt. (1)

Appl. Phys. Lett. (2)

K. J. Moh, X.-C. Yuan, D. Y. Tang, W. C. Cheong, and L. S. Zhang, "Generation of femtosecond optical vortices using a single refractive optical element," Appl. Phys. Lett. 88, 091103 (2006).
[CrossRef]

W. C. Cheong, W. M. Lee, X.-C. Yuan, and L. S. Zhang, "Direct electron-beam writing of continuous spiral phase plates in negative resist with high power efficiency for optical manipulation," Appl. Phys. Lett. 85, 5784-5786 (2004).
[CrossRef]

J. Opt. A (1)

M. V. Berry, "Optical vortices evolving from helicoidal integer and fractional phase steps," J. Opt. A 6, 259-268 (2004).
[CrossRef]

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

J. Phys. IV (1)

J. Flamand, G. de Villèle, A. Cotel, B. Touzet, and S. Kane, "New MLD gratings adapted for tiling in petawatt-class lasers," J. Phys. IV 133, 601-605 (2006).

New J. Phys. (2)

J. Leach, E. Yao, and M. J. Padgett, "Observation of the vortex structure of a noninteger vortex beam," New J. Phys. 6, 71.1-71.8 (2004).
[CrossRef]

J. Leach and M. J. Padgett, "Observation of chromatic effects near a white-light vortex," New J. Phys. 5, 154.1-154.7 (2003).
[CrossRef]

Opt. Eng. (1)

M.-S. L. Lee, J.-C. Rodier, P. Lalanne, P. Legagneux, P. Gallais, C. Germain, and J. Rollin, "Blazed-binary diffractive gratings with antireflection coating for improved operation at 10.6 μm," Opt. Eng. 43, 2583-2588 (2004).
[CrossRef]

Opt. Express (3)

J. Leach, G. M. Gibson, M. J. Padgett, E. Esposito, G. McConnell, A. J. Wright, and J. M. Girkin, "Generation of achromatic Bessel beams using a compensated spatial light modulator," Opt. Express 14, 5581-5587 (2006).
[CrossRef] [PubMed]

K. Sueda, G. Miyaji, N. Miyanaga, and M. Nakatsuka, "Laguerre-Gaussian beam generation with a multilevel spiral phase plate for high intensity laser pulses," Opt. Express 15, 3548-3553 (2004).
[CrossRef]

I. G. Mariyenko, J. Strohaber, and C. J. G. J. Uiterwaal, "Creation of optical vortices in femtosecond pulses," Opt. Express 19, 7599-7608 (2005).
[CrossRef]

Opt. Lett. (5)

Proc. R. Soc. London (1)

J. F. Nye and M. V. Berry, "Dislocations in wave trains," Proc. R. Soc. London Ser. A 336, 165-190 (1974).
[CrossRef]

Rep. Prog. Phys. (1)

M. Protopapas, C. H. Keitel, and P. L. Knight, "Atomic physics with super-high intensity lasers," Rep. Prog. Phys. 60, 389-486 (1997).
[CrossRef]

Other (6)

S. L. Chin and P. Lambropoulos, eds., Multiphoton Ionization of Atoms (Academic, 1984).

L. Allen, S. M. Barnett, and M. J. Padgett, eds., Optical Angular Momentum (Institute of Physics, 2003).
[CrossRef]

A. E. Siegman, Lasers (University Science Books, 1986).

M. Vasnetsov and K. Staliunas, eds., Optical Vortices (Nova Science, 1999).

J. F. James, A Student's Guide to Fourier Transforms (Cambridge U. Press, 1995).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).

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Figures (12)

Fig. 1
Fig. 1

Computer-generated binary grating patterns based on Eq. (2), for Φ 0 = 1 2 π . Each pattern consists of 1000 × 1000 pixels, where pixels for which T = 0 are rendered black and T = 1 white. The pattern on the left has no encoded vortex ( m = 0 ) . The pattern on the right has a single fringe bifurcation ( m = 1 ) ; when used as a hologram it gives rise to vortices with topological charge ± 1 in the ± first diffraction order.

Fig. 2
Fig. 2

Binary (line) grating holograms constructed using Eqs. (4), (6), and (8). The holograms are for a topological charge m equal to a, 0; b, 1; c, 2; d, 3; e, 4; and f, 5 all for Φ 0 = 0 . Labeled on each hologram are the grating line numbers n for each line.

Fig. 3
Fig. 3

(Color online) Schematic of the setup used for laser etching the mirrors. The laser beam is shown entering the setup from the top right. A mirror mounted at 45° with respect to the incoming beam sends the beam through an achromatic microscope objective with a focal length of 30.8   mm . The objective was mounted to a manual vertically positioned translation stage having a micrometer resolution. The beam is focused onto a mirror that is mounted on a motorized XY translation stage controlled by stepper motors. The resulting resolution in the translation is 5 μ m in both the X and the Y directions.

Fig. 4
Fig. 4

Diagram displaying the method used to etch the gratings as a single continuous line. Arrows show the direction of motion of the focused laser beam relative to the mirror. For clarity, the center lines in this m = 3 grating are shown separated. Each of these center lines was etched down to the center of the grating and back up the exact path.

Fig. 5
Fig. 5

(Color online) Gray-scale microscopic images of the LG mirrors with 20 × magnification for gratings with topological charges (top row, left to right) 1, 2, and 7. The gray portions represent the reflective gold coating, while the black curves are the regions where the gold has been removed by the focused femtosecond radiation. The bottom row shows the same images, overlaid with the skeleton lines (red online, gray in print) calculated from Eq. (3).

Fig. 6
Fig. 6

(Color online) When the laser focus is moved too fast we obtain a trail of separate pits instead of the desired continuous groove, as this microscopic image shows. Here, pits burned by individual laser pulses no longer spatially overlap. The width of the pits is approximately 20 μ m .

Fig. 7
Fig. 7

Images at various magnifications of an m = 1 LG mirror produced by laser etching a laser-quality gold mirror (gray is the gold plating and black is the groove). The grating constant is 100 μ m . The images give an impression of the quality in which our setup can currently laser etch grating lines.

Fig. 8
Fig. 8

(Color online) Analysis of groove smoothness: Close-up images of LG mirrors etched with an average laser power of a, 100 and c, 60   mW . The red lines (gray in print) are the groove edges as determined with a MATLAB edge-detection routine. The groove widths were measured at regular intervals, and histograms of the groove width distribution are shown: panel b is the histogram for the 100   mW groove in a, and panel d for the 60   mW groove in c. Black curves are Gaussian fits, with a full width at half-maximum of b, 4.3 and d, 2.4 μ m . We ascribe the improved smoothness of c compared to a to the lowering of the laser power used in the etching process.

Fig. 9
Fig. 9

(Color online) Schematic of Michelson interferometer used to observe intensity and interference patterns of optical vortices produced by laser-etched LG mirrors (BS is a beam splitter). A removable diverging lens, L, was used to create the spherical reference beam. The vortex beam is drawn as a solid red line (solid gray lines in print) and the reference beam is drawn as dotted blue lines (dotted gray lines in print). By removing the diverging lens a plane reference wave can be interfered with the vortex beam. We investigated the diffraction orders + 1 , 0, and 1 (yellow boxes).

Fig. 10
Fig. 10

(Color online) Intensity profiles and interference patterns for optical vortices with charges 1, 2, and 7 (left, center, and right columns). First row (a)–(c): far-field images of optical vortices taken in the focus of a 1 m lens. Second row (d)–(f): interferograms of the optical vortices with a plane reference wave; these images mimic the LG-grating pattern. Third row (g)–(i): images of the optical vortices interfering with a spherical reference beam, creating a spiral intensity pattern. The vortex charge is confirmed by counting the number of intertwined spiral arms in these interferograms.

Fig. 11
Fig. 11

(Color online) Schematic of the folded 2 f 2 f setup (angles, object dimensions and relative object positions not to scale for clarity). Ultrashort laser pulses enter from the top right, and then propagate along the following path: bottom half of the LG mirror, containing the line grating of which a part is shown in Fig. 2a; S, order-selecting aperture, blocking all diffraction orders but + 1 ; L1, plano-convex lens with focal length f = 100   cm ; M, folding mirror; top half of the LG mirror, containing a grating with the vortex fingerprint of which an example appears in Fig. 2b; L2 is a collimating lens; CCD is a CCD camera. The dimensions and relative positions of the optical elements are as follows: the etched part of the LG mirror is square, 2   cm × 2   cm ; distance between the LG mirror and L1 is 100 cm; distance between L1 and M is 150 μ m ; distance between the LG mirror and L2 is 25 cm; CCD is located 120 cm behind L2. Diffraction orders ( 1 , 0 , + 1 ) are indicated as black numbers on a yellow background. A few colored rays are shown to remind the reader that there is spatial chirp, with gray rays indicating that spatial chirp is absent. Note that we show just three colors—in reality, the frequency spectrum is, of course, continuous.

Fig. 12
Fig. 12

(Color online) Intensity profiles and interference patterns for optical vortices with topological charges 1, 2, and 7 (left, center, and right columns) after compensation of angular chirp using the folded 2f–2f setup. First row (a)–(c): far-field images of the + 1 order (compensated) optical vortices. Second row (d)–(f): far-field images of the 1 order (uncompensated) optical vortices. Third row (g)–(i): images of the compensated optical vortices in the focus, interfering with a spherical reference wave. This wave was created by placing a pinhole in the zero-order beam. As in Fig. 10, the vortex charge is confirmed by counting the number of intertwined spiral arms in the interferograms.

Equations (12)

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I ( x , y ) = 1 2 { 1 + cos [ m   arctan ( y x ) 2 π K x + Φ 0 ] } .
R ( x , y ) = { 0 cos ( Φ ) 0 1 cos ( Φ ) > 0 .
m   arctan ( y x ) = 2 π K x + arccos ( c ) [ mod   2 π ]
= 2 π ( n + K x ) .
y ( x ) = x   tan [ 2 π m ( n + K x ) ] .
x > 0 1 2 π < θ < 1 2 π ,
x < 0 1 2 π < θ < 3 2 π .
x > 0 : 1 K ( m 4 + n ) x 1 K ( m 4 n ) , n < m 4 ;
x < 0 : 1 K ( m 4 n ) x 1 K ( 3 m 4 n ) , n > m 4 .
θ = { 1 2 π , y > 0 undefined, y = 0 1 2 π , y < 0 .
n = { m 4 , y > 0 undefined, y = 0 m 4 , y < 0 .
1 f eff = 1 f first   pass + 1 f second   pass = 2 f f eff = 1 2 f = 50   cm ,

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