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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 24 — Nov. 22, 2010
  • pp: 25143–25150
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Dynamic space shaping of intense ultrashort laser light with blazed-type gratings

I. Marienko, V. Denisenko, V. Slusar, and M. Soskin  »View Author Affiliations


Optics Express, Vol. 18, Issue 24, pp. 25143-25150 (2010)
http://dx.doi.org/10.1364/OE.18.025143


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Abstract

A beam shaper for dynamic transversal shaping of broadband laser pulses that utilizes a blazed ruled grating and a blazed-type grating simulated on Spatial Light Modulator was demonstrated. The introduced shaper scheme is an extension of 2f-2f scheme [Mariyenko, et al., Opt. Express 13, 7599 (2005)] where the two thin holograms with matched grating constants performed light shaping. The new scheme utilizes the diffraction gratings with different grating constants. Dispersion-free light shaping is achieved by means of the intermediate transversal light beam magnification. The magnification balances the mismatch in the grating constants resulting in total residual angular dispersion compensation. In turn, the magnified beam covers a greater area on the modulator matrix thus reducing the incident light power density by a value equal to square of the magnification factor. It translates to the safe-operation threshold extension of the modulator allowing shaping pulses that are powerful enough to be used in the applications. With a proper components selection, the throughput efficiency of the shaper can be well above 40%. A proper shaper operation was demonstrated with the 140-fs Ti:Sapphire oscillator. Theoretical calculations support the conclusions.

© 2010 OSA

1. Introduction

Optical Vortices (OVs) [1

1. M. Vasnetsov, and K. Staliunas, eds., “Optical Vortices,” in Horizons in World Physics, (Nova Science Publishers, Commack, New York, 1999) Vol. 228.

] brought the new meaning and new goals to space shaping of laser beams. Orbital Angular Momentum (OAM) associated with the Vortices happened to be well-defined [2

2. L. Allen, S. M. Barnett, and M. J. Padgett, Optical Angular Momentum, (Taylor & Francis, 2003).

] and used for microparticles manipulations [3

3. D. G. Grier, “A revolution in optical manipulation,” Nature 424(6950), 810–816 (2003). [CrossRef] [PubMed]

] as well as for demonstrating entanglement of OAM states of photons [4

4. A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412(6844), 313–316 (2001). [CrossRef] [PubMed]

] and the quantized rotation of atoms [5

5. M. F. Andersen, C. Ryu, P. Cladé, V. Natarajan, A. Vaziri, K. Helmerson, and W. D. Phillips, “Quantized rotation of atoms from photons with orbital angular momentum,” Phys. Rev. Lett. 97(17), 170406 (2006). [CrossRef] [PubMed]

]. These experiments utilized the multiple previously developed technologies (described and referenced in [1

1. M. Vasnetsov, and K. Staliunas, eds., “Optical Vortices,” in Horizons in World Physics, (Nova Science Publishers, Commack, New York, 1999) Vol. 228.

]) to generate and handle OVs in monochromatic and narrow-band laser light where powers of laser light are relatively low. The multiple studies of light filamentation in air and other media proposed using custom shaped femtosecond pulses of very high peak power, particularly in the form of OVs [6

6. L. Bergé, S. Skupin, R. Nuter, J. Kasparian, and J.-P. Wolf, “Ultrashort filaments of light in weakly ionized, optically transparent media,” Rep. Prog. Phys. 70(10), 1633–1713 (2007). [CrossRef]

]. Should local light intensity of a propagating laser pulse become so high to induce Kerr modulation and approach an optical damage threshold of a medium, the pulse propagation scenario acquires the dependence on a space shape of the propagating pulse. To ensure long distance propagation in transparent media the preliminary space shaping of the light was done that to include the use of: arrays of microlenses [7

7. K. Cook, R. McGeorge, A. K. Kar, M. R. Taghizadeh, and R. A. Lamb, “Coherent array of white-light continuum filaments produced by diffractive microlenses,” Appl. Phys. Lett. 86(2), 021105 (2005). [CrossRef]

], a deformable mirror to control the beam divergence angle [8

8. Z. Jin, J. Zhang, M. H. Xu, X. Lu, Y. T. Li, Z. H. Wang, Z. Y. Wei, X. H. Yuan, and W. Yu, “Control of filamentation induced by femtosecond laser pulses propagating in air,” Opt. Express 13(25), 10424–10430 (2005). [CrossRef] [PubMed]

], periodic meshes to improve the multi-filamentation control [9

9. V. P. Kandidov, N. Ak¨ozbek, M. Scalora, O. G. Kosareva, A. V. Nyakk, Q. Luo, S. A. Hosseini, and S. L. Chin, “Towards a control of multiple filamentation by spatial regularization of a high-power femtosecond laser pulse,” Appl. Phys. B 80(2), 267–275 (2005). [CrossRef]

], and others. Also, it was shown theoretically [10

10. A. Vinçotte and L. Bergé, “Femtosecond optical vortices in air,” Phys. Rev. Lett. 95(19), 193901 (2005). [CrossRef] [PubMed]

], that OV-shaping of femtosecond pulses increases revolutionally robustness of the beam upon propagation in air allowing much longer propagation. The other application to potentially benefit from custom space shaping of ultrashort and intense pulses is free electron laser in the part where a shaped laser pulse initiates the electron beam in radio-frequency photoinjector. A uniform ellipsoidal beam [11

11. M. Reiser, Theory and Design of Charged Particle Beams, (Wiley, NY, 2005)

], being a highly desired shape, happens to be technically challenging to create [12

12. C. Limborg-Deprey and P. Bolton, “Optimum electron distributions for space charge dominated beams in photoinjectors,” Nucl. Instrum. Methods. Phys. Res, Sect. A 557, 106 (2006). [CrossRef]

].

The already developed techniques for shaping of ultra-short narrow-band laser pulses are not suitable for spectrally broad light because chromaticity is not taken into account. The proposed grating-based 4f scheme [13

13. K. Bezuhanov, A. Dreischuh, G. G. Paulus, M. G. Schätzel, and H. Walther, “Vortices in femtosecond laser fields,” Opt. Lett. 29(16), 1942–1944 (2004). [CrossRef] [PubMed]

] and 2f-2f scheme [14

14. I. G. Mariyenko, J. Strohaber, and C. J. G. J. Uiterwaal, “Creation of optical vortices in femtosecond pulses,” Opt. Express 13(19), 7599–7608 (2005). [CrossRef] [PubMed]

] handle broadband spectra properly allowing custom pulse shaping. They, however, provide relatively low throughput efficiency because of the low diffraction efficiency of the used gratings. The other shaper design that utilizes two counter-set prisms and computer-generated grating [15

15. A. Schwarz and W. Rudolph, “Dispersion-compensating beam shaper for femtosecond optical vortex beams,” Opt. Lett. 33(24), 2970–2972 (2008). [CrossRef] [PubMed]

] has throughput as high as 22% and the bandwidth exceeding 100 nm. Its shaping capability is limited by the uncompensated diffraction spread of the beam upon propagation from the input prism to the diffraction grating. The listed techniques use diffraction gratings (recorded on dichromate gelatin or simulated with LC Spatial Light Modulator (SLM)) that allow custom patterning to encode gratings with the desired pulse shapes. At the same time, the material of such gratings imposes the low-damage-threshold limits on average and peak power of the light pulse being shaped. Some threshold data for dichromate gelatin can be found in [15

15. A. Schwarz and W. Rudolph, “Dispersion-compensating beam shaper for femtosecond optical vortex beams,” Opt. Lett. 33(24), 2970–2972 (2008). [CrossRef] [PubMed]

]. As to SLM, the peak power damage threshold for ultrashort laser pulses is not determined yet [16

16. Communications with HOLOEYE Photonics AG.

], but believed to be much lower than the threshold of metal-coated blazed gratings. In contrast, standard metal coated blazed gratins have high damage threshold, but do not provide with an option of custom pattern encoding. So, a space pulse shaper that combines the capability of custom shaping with high power throughput and high damage threshold will be useful for shaping of ultrashort light pulses that are both broadband and intense.

We introduce the universal space shaper – 2f-2f shaper with magnification – for broadband ultrashort laser pulses that is capable of forming custom light shapes dynamically in the transversal plane of the propagating pulse. Due to scalable magnification, the shaper is suitable for processing pulsed laser radiation that features both high average and peak intensity. The shaper consists of SLM encoded with custom phase blazed-type grating, a highly efficient blazed metal grating, and a focusing lens. The use of SLM enable allows both dynamic and custom light shaping according to the properly encoded hologram grating. The custom shaping includes also focusing the output laser radiation to create specific focal shapes. The use of two highly efficient blazed gratings provides high throughput efficiency of the shaper. The lens makes the gratings planes conjugate and introduces the transversal magnification. Due to the magnification, the energy fluence incident onto SLM spreads over a greater area (by the square of the magnification factor) thus increasing an effective safe operation threshold of the SLM. The enlarged transversal area of the beam covers more SLM pixels enabling a higher density of the light field sampling. The temporal characteristics of the pulse that has been shaped remain unchanged from an original one.

2. Model

The desired space shape of the light beam can be encoded in phase/amplitude pattern of a thin holographic grating that is conveniently simulated on SLM matrix. When such hologram is illuminated with readout monochromatic plane wave the light diffracted into the first diffraction order contains the encoded shape. In contrast to the case of monochromatic light, the readout broadband light brings up the problem of angular dispersion. In order to apply this holographic technique to broadband light dispersion compensation is required to account for the dispersion caused by the SLM holographic grating. It is accomplished with another grating that produces matching angular dispersion of an opposite sign. This is realized with the optical scheme composed of the two gratings as shown on Fig. 1
Fig. 1 shows the optical scheme of 2f-2f shaper with magnification. The red line illustrates the space transformations of a broadband laser beam.
. The first input grating G1 disperses the oncoming laser light within each diffraction order. The second grating – output grating G2 – encoded in SLM shapes the dispersed light and introduces its own angular dispersion in each diffraction order. The lens L1 placed in between of the gratings flips the sign of angular dispersion introduced by the first grating.

Finally, after diffracting onto the two gratings and passing the lens each diffraction order is dispersed by angle in general case. The value of the final dispersion for each order depends on its original path and the transversal magnification. In our analysis of the beam transformations through the optical setup, we use the Kirchhoff-Fresnel integral. It is applied here to Gaussian beam that is usually generated by lasers:
Einexp{x2+y2w02}
(1)
Where E in is electric field amplitude incident onto the grating G1, x is the transversal coordinate (y is dropped further because of no relevant modulation in this direction), w 0 is the waist size of the beam. The wave amplitude E 1 diffracted into m 1 diffraction order is described by the formula (2):
E1exp{x2w02+i​  n1K1x}
(2)
where K 1 is grating constant. Propagating further, this wave passes the thin lens L1 placed at the distance z 1 from the input grating. Right after the lens its amplitude becomes
E2exp{ik2(z1LR)(xin1K1kLR)2n12K122kLRikx22f}
(3)
Where k is wavenumber, L R = kw 0 2/2 is Rayleigh range, f is focal length of the lens L1. The last term in the formula (3) corresponds to phase modulation caused by the lens. When the wave propagates the distance z 2 and reaches the output grating, its amplitude takes the following form:

E3exp{x2w02(z1z2)2+ik2fz1z2x2iz1z2n1K1x}
(4)

The lens is in the position to image the face of the input grating onto the output grating. In this case, the ratio of the distances z 1 and z 2 defines the transversal beam magnification:z1z2=M. Finally, the magnified beam diffracts onto the output grating with grating constant K 2. The amplitude of the wave diffracted into n 2 diffraction order takes the form
Eoutexp{x2(Mw0)2+ik2Mfx2in1K1Mx+in2K2x}
(5)
The first term inside the brackets defines Gaussian profile magnified transversally M times. The second term is the quadratic phase factor imparted to the wave by the lens and free space propagation. The last two terms have diffraction origin and determine angular dispersion. To suppress the angular dispersion the combined value of those terms is to be zero. From this we have the equation that is the condition for the dispersion-free beam:
n1K1M+n2K2=0
(6)
This equation relates the parameters of the optical components of the beam shaper scheme shown on Fig. 1. If, for example, n 1 = n 2 = 1 and M = −1 (negative magnification sign corresponds to a flipped sign of the angular dispersion) then K 2 = K 1 and the shaper becomes 2f–2f scheme as shown in [14

14. I. G. Mariyenko, J. Strohaber, and C. J. G. J. Uiterwaal, “Creation of optical vortices in femtosecond pulses,” Opt. Express 13(19), 7599–7608 (2005). [CrossRef] [PubMed]

].

3. Experiment

The broadband shaper was assembled according to the optical scheme shown on the Fig. 1. To test the shaper, we used Ti:Sapphire oscillator that generates transform-limited 140-fs pulses at 86-MHz repetition rate. The corresponding spectral width of the pulse is 6.4 nm centered at 810 nm. The laser has an option to operate in CW mode at the same wavelength. The original output laser beam in both operating modes is divergent and astigmatic with the measured horizontal-to-vertical ratio of 3:2. The misaligned telescope was added to the scheme to compensate for the beams divergence; the beam astigmatism was not corrected. The images were captured with CCD camera.

The optical scheme parameters were configured in accordance with the Eq. (6). The input grating G1 is ruled, blazed, and gold coated for the best performance in NIR. It has the grating constant of 60 lines/mm and the diffraction efficiency of 62% into the first diffraction order as measured in Littrow configuration. A focusing lens with 30-cm focal length is taken as the lens L1. The SLM – the basic HoloEye SLM LC-2002 – has 1.3 Mpixels over the area of 26.6 x 20 mm – serves as the output diffractive element when encoded with a diffraction grating. The maximum phase modulation for passing light with wavelength of 800 nm amounts π [16

16. Communications with HOLOEYE Photonics AG.

]. We encoded the SLM with the grating with grating constant of 15.6 lines/mm. According to the Eq. (6), to compensate for the dispersion induced by the input grating in the first diffraction order, the magnification is to be M = 3.8. This determines the position of the lens L1 in between of the gratings: 37.8 cm from the input grating and 145.2 cm from the output grating. In such configuration, the residual dispersion of the output wave is to be zero for some diffraction orders; particularly, for the first diffraction order.

To demonstrate the principle of femtosecond pulse shaping, we encoded the SLM with OV of the topological charge m = 3. The phase grating pattern was calculated according to the two-wave interference approach [17

17. V. Y. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Screw Dislocations in Light Wavefronts,” J. Mod. Opt. 39(5), 985–990 (1992). [CrossRef]

, 18

18. N. R. Heckenberg, R. McDuff, C. P. Smith, and A. G. White, “Generation of optical phase singularities by computer-generated holograms,” Opt. Lett. 17(3), 221–223 (1992). [CrossRef] [PubMed]

], that results in phase variation proportional to 12[1+sin(mϕKx)], where φ is azimutal phase. The phase variation amplitude was equal to π. The actual phase variation did not follow sin exactly because of nonlinear dependence between the computer-generated phase values and the appropriate ones displayed on the SLM. The discrepancy was not large, so, we did not correct for that. The reconstructed pattern is shown on the Fig. 2a
Fig. 2 Reconstructed intensity “fingerprints” of OV of charge m = 3 in broadband laser field, (a), and CW laser field, (c); the angular dispersion is compensated for broadband field. (b) The cross-cut of the intensity distribution for both beams along the red lines: three zero points identify the individual single-charged vortex cores.
where the beam contains three single-charge vortices instead of the original OV with charge m = 3. This is to be compared to the pattern on the Fig. 2c where the reconstruction is performed on the OV in monochromatic light when the laser operates in CW mode. As one can see, the patterns are almost identical, besides the diffraction fringes contrast that is lower on Fig. 2a due to broadband smoothening. In both cases the vortex cores of the individual vortices are clearly dark. The OV cores positions and the beam intensity profiles match well as shown in the cross-cut intensity profile, Fig. 2b, taken along the red lines, Figs. 2a,c. It verifies that the holographic reconstruction is correct and the angular dispersion is compensated across the beam.

The reconstructed intensity patterns do not resemble “dougnut” intensity distribution of the original OV with topological charge m = 3 for both broadband and monochromatic light. In general, the obtained pattern is a result of the vortex break down during the reconstruction transformations into three single-charged vortices; it is natural process. OVs of high orders are non-generic [19

19. J. F. Nye, Natural Focusing and Fine Structure of Light, (Institute of Physics, 1999).

], i.e. unstable to small field perturbations what, in turn, originate from the optical aberrations. In general, this explains the vortex break down. Particularly, the given patters contain the isolated astigmatic vortices that are also lined up. As we mentioned before, the original output laser beam is astigmatic; it experiences two focuses after passing a spherical focusing lens. These beam transformations are similar to the transformations of the transversal structure of the higher order OV beam when it passes an astigmatic lens. The gained astigmatism breaks down the high-order vortex into a set of individual vortices of a single charge that are astigmatic and lined up [20

20. V. G. Denisenko, A. Minovich, A. S. Desyatnikov, W. Krolikowski, M. S. Soskin, and Y. S. Kivshar, “Mapping phases of singular scalar light fields,” Opt. Lett. 33(1), 89–91 (2008). [CrossRef]

], as in our case. This explains the obtained OV beam structure (Fig. 2c) for monochromatic case. This transformation process of the beam transversal profile has no dependence on light wavelength what explains the same OV beam structure (Fig. 2a) for the broadband beam.

The measured throughput efficiency happened to be 1.2%. Such low value is accounted for two reasons. First, we used the SLM optimized for visible spectra, not for NIR light. This introduces significant Fresnel losses at multiple optical surfaces of the LC screen. Second, and the most significant reason is low diffraction efficiency of the simulated gratings because of low space resolution of the LC matrix: we could use a 3-pixel line per grating period.

4. Discussion and summary

The considered femtosecond-pulses shaper based on 2f-2f scheme with magnification demonstrated a correct space shaping as tested with 140-fs pulses. Both created shapes – OV with topological charge m = 3 and the focal spot – were reproduced correctly for both pulsed and CW laser modes. The angular spectra dispersion, being proper to the proposed scheme, is created inside the shaper by its input grating and completely compensated by its output grating. The output grating also shapes a laser beam transversally imparting it with the desired shape.

The proper magnification imparts the presented shaper with critically important capabilities to handle ultrashort and intense light (typically above 1012 W/cm2) and, therefore introduces the experimentally simple technique of custom and dynamic light shaping to ultrashort and ultraintense pulses shaping to be used in the described applications.

Acknowledgements

The experimental part of the work was performed at the Femtosecond Laser Complex, National Academy of Sciences of Ukraine. We thank P. Kornelyuk for his help during the experiments. VS, VD, and MS were supported by the National Academie of Sciences of Ukraine (projects nos. 2.16.1.4 and 2.16.1.7) and the International Science and Technology Center (project no. STCU 4687).

References and links

1.

M. Vasnetsov, and K. Staliunas, eds., “Optical Vortices,” in Horizons in World Physics, (Nova Science Publishers, Commack, New York, 1999) Vol. 228.

2.

L. Allen, S. M. Barnett, and M. J. Padgett, Optical Angular Momentum, (Taylor & Francis, 2003).

3.

D. G. Grier, “A revolution in optical manipulation,” Nature 424(6950), 810–816 (2003). [CrossRef] [PubMed]

4.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412(6844), 313–316 (2001). [CrossRef] [PubMed]

5.

M. F. Andersen, C. Ryu, P. Cladé, V. Natarajan, A. Vaziri, K. Helmerson, and W. D. Phillips, “Quantized rotation of atoms from photons with orbital angular momentum,” Phys. Rev. Lett. 97(17), 170406 (2006). [CrossRef] [PubMed]

6.

L. Bergé, S. Skupin, R. Nuter, J. Kasparian, and J.-P. Wolf, “Ultrashort filaments of light in weakly ionized, optically transparent media,” Rep. Prog. Phys. 70(10), 1633–1713 (2007). [CrossRef]

7.

K. Cook, R. McGeorge, A. K. Kar, M. R. Taghizadeh, and R. A. Lamb, “Coherent array of white-light continuum filaments produced by diffractive microlenses,” Appl. Phys. Lett. 86(2), 021105 (2005). [CrossRef]

8.

Z. Jin, J. Zhang, M. H. Xu, X. Lu, Y. T. Li, Z. H. Wang, Z. Y. Wei, X. H. Yuan, and W. Yu, “Control of filamentation induced by femtosecond laser pulses propagating in air,” Opt. Express 13(25), 10424–10430 (2005). [CrossRef] [PubMed]

9.

V. P. Kandidov, N. Ak¨ozbek, M. Scalora, O. G. Kosareva, A. V. Nyakk, Q. Luo, S. A. Hosseini, and S. L. Chin, “Towards a control of multiple filamentation by spatial regularization of a high-power femtosecond laser pulse,” Appl. Phys. B 80(2), 267–275 (2005). [CrossRef]

10.

A. Vinçotte and L. Bergé, “Femtosecond optical vortices in air,” Phys. Rev. Lett. 95(19), 193901 (2005). [CrossRef] [PubMed]

11.

M. Reiser, Theory and Design of Charged Particle Beams, (Wiley, NY, 2005)

12.

C. Limborg-Deprey and P. Bolton, “Optimum electron distributions for space charge dominated beams in photoinjectors,” Nucl. Instrum. Methods. Phys. Res, Sect. A 557, 106 (2006). [CrossRef]

13.

K. Bezuhanov, A. Dreischuh, G. G. Paulus, M. G. Schätzel, and H. Walther, “Vortices in femtosecond laser fields,” Opt. Lett. 29(16), 1942–1944 (2004). [CrossRef] [PubMed]

14.

I. G. Mariyenko, J. Strohaber, and C. J. G. J. Uiterwaal, “Creation of optical vortices in femtosecond pulses,” Opt. Express 13(19), 7599–7608 (2005). [CrossRef] [PubMed]

15.

A. Schwarz and W. Rudolph, “Dispersion-compensating beam shaper for femtosecond optical vortex beams,” Opt. Lett. 33(24), 2970–2972 (2008). [CrossRef] [PubMed]

16.

Communications with HOLOEYE Photonics AG.

17.

V. Y. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Screw Dislocations in Light Wavefronts,” J. Mod. Opt. 39(5), 985–990 (1992). [CrossRef]

18.

N. R. Heckenberg, R. McDuff, C. P. Smith, and A. G. White, “Generation of optical phase singularities by computer-generated holograms,” Opt. Lett. 17(3), 221–223 (1992). [CrossRef] [PubMed]

19.

J. F. Nye, Natural Focusing and Fine Structure of Light, (Institute of Physics, 1999).

20.

V. G. Denisenko, A. Minovich, A. S. Desyatnikov, W. Krolikowski, M. S. Soskin, and Y. S. Kivshar, “Mapping phases of singular scalar light fields,” Opt. Lett. 33(1), 89–91 (2008). [CrossRef]

21.

J. Strohaber, I. Mariyenko, and C. J. G. J. Uiterwaal, “Generation and diagnosis of focused ultrashort pulses of complex light,” Proc. SPIE Vol. 6483 (2007).

22.

J. Leach and M. J. Padgett, “Observation of chromatic effects near a white-light vortex,” N. J. Phys. 5, 1–7 (2003). [CrossRef]

23.

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(12), 5581–5587 (2006). [CrossRef] [PubMed]

24.

A. J. Wright, J. M. Girkin, G. M. Gibson, J. Leach, and M. J. Padgett, “Transfer of orbital angular momentum from a super-continuum, white-light beam,” Opt. Express 16(13), 9495–9500 (2008). [CrossRef] [PubMed]

OCIS Codes
(050.1950) Diffraction and gratings : Diffraction gratings
(050.2770) Diffraction and gratings : Gratings
(320.1590) Ultrafast optics : Chirping
(320.5540) Ultrafast optics : Pulse shaping
(050.1965) Diffraction and gratings : Diffractive lenses
(050.4865) Diffraction and gratings : Optical vortices

ToC Category:
Ultrafast Optics

History
Original Manuscript: August 16, 2010
Revised Manuscript: September 25, 2010
Manuscript Accepted: September 28, 2010
Published: November 17, 2010

Citation
I. Marienko, V. Denisenko, V Slusar, and M. Soskin, "Dynamic space shaping of intense ultrashort laser light with blazed-type gratings," Opt. Express 18, 25143-25150 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-24-25143


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References

  1. M. Vasnetsov, and K. Staliunas, eds., “Optical Vortices,” in Horizons in World Physics, (Nova Science Publishers, Commack, New York, 1999) Vol. 228.
  2. L. Allen, S. M. Barnett, and M. J. Padgett, Optical Angular Momentum, (Taylor & Francis, 2003).
  3. D. G. Grier, “A revolution in optical manipulation,” Nature 424(6950), 810–816 (2003). [CrossRef] [PubMed]
  4. A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412(6844), 313–316 (2001). [CrossRef] [PubMed]
  5. M. F. Andersen, C. Ryu, P. Cladé, V. Natarajan, A. Vaziri, K. Helmerson, and W. D. Phillips, “Quantized rotation of atoms from photons with orbital angular momentum,” Phys. Rev. Lett. 97(17), 170406 (2006). [CrossRef] [PubMed]
  6. L. Bergé, S. Skupin, R. Nuter, J. Kasparian, and J.-P. Wolf, “Ultrashort filaments of light in weakly ionized, optically transparent media,” Rep. Prog. Phys. 70(10), 1633–1713 (2007). [CrossRef]
  7. K. Cook, R. McGeorge, A. K. Kar, M. R. Taghizadeh, and R. A. Lamb, “Coherent array of white-light continuum filaments produced by diffractive microlenses,” Appl. Phys. Lett. 86(2), 021105 (2005). [CrossRef]
  8. Z. Jin, J. Zhang, M. H. Xu, X. Lu, Y. T. Li, Z. H. Wang, Z. Y. Wei, X. H. Yuan, and W. Yu, “Control of filamentation induced by femtosecond laser pulses propagating in air,” Opt. Express 13(25), 10424–10430 (2005). [CrossRef] [PubMed]
  9. V. P. Kandidov, N. Ak¨ozbek, M. Scalora, O. G. Kosareva, A. V. Nyakk, Q. Luo, S. A. Hosseini, and S. L. Chin, “Towards a control of multiple filamentation by spatial regularization of a high-power femtosecond laser pulse,” Appl. Phys. B 80(2), 267–275 (2005). [CrossRef]
  10. A. Vinçotte and L. Bergé, “Femtosecond optical vortices in air,” Phys. Rev. Lett. 95(19), 193901 (2005). [CrossRef] [PubMed]
  11. M. Reiser, Theory and Design of Charged Particle Beams, (Wiley, NY, 2005)
  12. C. Limborg-Deprey and P. Bolton, “Optimum electron distributions for space charge dominated beams in photoinjectors,” Nucl. Instrum. Methods. Phys. Res, Sect. A 557, 106 (2006). [CrossRef]
  13. K. Bezuhanov, A. Dreischuh, G. G. Paulus, M. G. Schätzel, and H. Walther, “Vortices in femtosecond laser fields,” Opt. Lett. 29(16), 1942–1944 (2004). [CrossRef] [PubMed]
  14. I. G. Mariyenko, J. Strohaber, and C. J. G. J. Uiterwaal, “Creation of optical vortices in femtosecond pulses,” Opt. Express 13(19), 7599–7608 (2005). [CrossRef] [PubMed]
  15. A. Schwarz and W. Rudolph, “Dispersion-compensating beam shaper for femtosecond optical vortex beams,” Opt. Lett. 33(24), 2970–2972 (2008). [CrossRef] [PubMed]
  16. Communications with HOLOEYE Photonics AG.
  17. V. Y. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Screw Dislocations in Light Wavefronts,” J. Mod. Opt. 39(5), 985–990 (1992). [CrossRef]
  18. N. R. Heckenberg, R. McDuff, C. P. Smith, and A. G. White, “Generation of optical phase singularities by computer-generated holograms,” Opt. Lett. 17(3), 221–223 (1992). [CrossRef] [PubMed]
  19. J. F. Nye, Natural Focusing and Fine Structure of Light, (Institute of Physics, 1999).
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