## Femtosecond spatial pulse shaping at the focal plane |

Optics Express, Vol. 21, Issue 21, pp. 25010-25025 (2013)

http://dx.doi.org/10.1364/OE.21.025010

Acrobat PDF (2233 KB)

### Abstract

Spatial shaping of ultrashort laser beams at the focal plane is theoretically analyzed. The description of the pulse is performed by its expansion in terms of Laguerre-Gaussian orthonormal modes. This procedure gives both a comprehensive interpretation of the propagation dynamics and the required signal to encode onto a spatial light modulator for spatial shaping, without using iterative algorithms. As an example, pulses with top-hat and annular spatial profiles are designed and their dynamics analyzed. The interference of top-hat pulses is also investigated finding potential applications in high precision pump-probe experiments (without using delay lines) and for the creation of subwavelength ablation patterns. In addition, a novel class of ultrashort pulses possessing non-stationary orbital angular momentum is also proposed. These exotic pulses provide additional degrees of freedom that open up new perspectives in fields such as laser-matter interaction and micro-machining.

© 2013 OSA

## 1. Introduction

1. D. E. Leaird and A. M. Weiner, “Femtosecond direct space-to-time pulse shaping,” IEEE J. Quantum Electron. **37**, 494–504 (2001). [CrossRef]

2. V. Loriot, O. Mendoza-Yero, G. Mínguez-Vega, L. Bañares, and R. de Nalda, “Experimental demonstration of the quasy-direct space-to-time pulse shaping principle,” IEEE Photon. Technol. Lett. **24**, 273–275 (2012). [CrossRef]

3. N. Sanner, N. Huot, E. Audouard, C. Larat, and J. -P. Huignard, “Direct ultrafast laser micro-structuring of materials using programmable beam shaping, ” Opt. Laser Eng. **45**, 737–741 (2007). [CrossRef]

4. N. Sanner, N. Huot, E. Audouard, C. Larat, J.-P. Huignard, and B. Loiseaux, “Programmable focal spot shaping of amplified femtosecond laser pulses,” Opt. Lett. **30**, 1479–1482 (2005). [CrossRef] [PubMed]

4. N. Sanner, N. Huot, E. Audouard, C. Larat, J.-P. Huignard, and B. Loiseaux, “Programmable focal spot shaping of amplified femtosecond laser pulses,” Opt. Lett. **30**, 1479–1482 (2005). [CrossRef] [PubMed]

6. S. Akturk, X. Gu, P. Bowlan, and R. Trebino, “Spacio-temporal couplings in ultrashort laser pulses,” J. Opt. **12**, 093001 (2010). [CrossRef]

7. V. Arrizón, U. Ruiz, R. Carrada, and L. A. González, “Pixelated phase computer holograms for the accurate encoding of scalar complex fields,” J. Opt. Soc. Am. A **24**, 3500–3507 (2007). [CrossRef]

8. J. A. Rodrigo, T. Alieva, A. Cámara, Ó. Martínez-Matos, P. Cheben, and M. L. Calvo, “Characterization of holographically generated beams via phase retrieval based on Wigner distribution projections,” Opt. Express **19**, 6064–6077 (2011). [CrossRef] [PubMed]

## 2. Optical setup

12. Y. Toda, K. Nagaoka, K. Shimatake, and R. Morita, “Generation and spatiotemporal evolution of optical vortices in femtosecond laser pulses,” Electr. Eng. JPN **167**, 39–46 (2009). [CrossRef]

13. 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]

14. 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]

15. K. Bezuhanov, A. Dreischuh, G. G. Paulus, M. G. Schätzel, H. Walther, D. Neshev, W. Królikowski, and Y. Kivshar, “Spatial phase dislocations in femtosecond laser pulses,” J. Opt. Soc. Am. B **23**, 26–35 (2006). [CrossRef]

16. Ó. Martínez-Matos, J. A. Rodrigo, M. P. Hernández-Garay, J. G. Izquierdo, R. Weigand, M. L. Calvo, P. Cheben, P. Vaveliuk, and L. Bañares, “Generation of femtosecond paraxial beams with arbitrary spatial distributions,” Opt. Lett. **35**, 652–654 (2010). [CrossRef]

17. J. Atencia, M.-V. Collados, M. Quintanilla, J. Marín-Sáez, and I. J. Sola, “Holographic optical element to generate achromatic vortices,” Opt. Express **21**, 21057–21062 (2013). [CrossRef]

18. J. Strohaber, T. D. Scarborough, and C. J. G. J. Uiterwaal, “Ultrashort intense-field optical vortices produced with laser-etched mirrors,” Appl. Opt. **46**, 8583–8590 (2007). [CrossRef] [PubMed]

22. 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). [CrossRef] [PubMed]

*π*radians distributed in 256 (8-bit) gray levels, allowing a proper sampling for the secondary maximum analyzed in the examples of this work. Spatial shaping of ultrashort pulses at the focus of a convergent lens using SLMs is feasible to be realized experimentally with current state-of-the-art technology.

## 3. Relationship between signal and pulse via the LG modes

*E*

_{0}, corresponding to each frequency component

*ω*at the SLM plane (

*z*= 0), can be written in cylindrical coordinates (

*r*

_{0},

*ϕ*

_{0}) as where

*S*(

*ω*) is the spectral amplitude of the input pulse and

*R′*(

*r*

_{0},

*ϕ*

_{0}) is the encoded complex signal in the dynamically controllable hologram. It has been considered an incident pulse with Gaussian spatial distribution whose beam waist size,

*β*, is identical for all spectral components being the waist located at the SLM plane. Under these assumptions we define

*R*(

*r*

_{0},

*ϕ*

_{0}), for the whole spectrum. A proper choice of

*R*(

*r*

_{0},

*ϕ*

_{0}) modulates spatially the pulse at the focal plane achieving the required spatial distribution.

*f*is placed at the distance

*s*from the SLM being the field evaluated at the focal plane

*z*=

*s*+

*f*. The field propagation is described by the paraxial theory considering nondispersive aberration-less lenses, such as dichroic lenses or spherical mirrors. Under such conditions, the field at the focus expressed in cylindrical coordinates (

*r*,

*ϕ*), is where is defined by the arrangement geometry [23] and

*FT*is the spatial Fourier Transform operator that relates the field at the SLM with the field at the focal plane.

*ℰ*, as a function of the temporal and spatial coordinates is obtained by integrating (3) on the whole frequency range as Numerical integration of (5) is a cumbersome problem. An alternative method to find the diffraction of optical pulses by hard apertures were proposed recently in [11

11. R. J. Mahon and J. A. Murphy, “Diffraction of an optical pulse as an expansion in ultrashort orthogonal Gaussian beam modes,” J. Op. Soc. Am. A **30**, 215–226 (2013). [CrossRef]

*LG*being the Laguerre-Gaussian beams of an orthonormal base explicitly giving by [24

_{n,m}24. V. Lakshminarayanan, M. L. Calvo, and T. Alieva, *Mathematical Optics: Classical, Quantum and Computational Methods* (CRC Press, 2012). [CrossRef]

*A*whose explicit expression is The

_{n,m}*LG*modes of the expansion, with radial

_{n,m}*n*, and azimuthal

*m*indexes have a doughnut-shaped beam profile with a OAM of order

*m*(topological charge

*m*). In (7),

*w*

_{0}is the beam waist size of the expansion elements and

25. E. Abramochkin, E. Razueva, and V. Volostnikov, “General astigmatic transform of Hermite-Laguerre-Gaussian beams,” J. Opt. Soc. Am. A **27**, 2506–2513 (2010). [CrossRef]

*w*= 2

*fc/w*

_{0}

*ω*, one can finally obtain the expression for the pulselets: In summary, the propagation of the pulse to the focus is just found by calculating the coefficients

*A*that relate the signal at the SLM (6) to the wavefield at the focus (8). It could be said that the spatio-temporal features of the pulse are fully described by the pulselets

_{n,m}*ℒ𝒢*, allowing a physically intuitive representation of the field.

_{n,m}## 4. Pulselets features

*S*(

*ω*) =

*δ*(

*ω*−

*ω*

_{0}). According to (12) the wavelet (the monochromatic pulselet)

*w*

_{1}= 2

*fc/w*

_{0}

*ω*

_{0}is the beam waist size at the focal plane for the angular frequency

*ω*

_{0}, and is a position-dependent delay time.

26. R. L. Phillips and L. C. Andrews, “Spot size and divergence of Laguerre Gaussian beams of any order,” Appl. Opt. **22**, 643–644 (1983). [CrossRef] [PubMed]

*ω*

_{0}/2

*fc*)(1 −

*s/f*)

*r*

^{2}giving by (14) that slightly shifts the beam focus towards higher distances. In this case the focal plane,

*z*=

*s*+

*f*, and the beam focus are not coincident. The factor (−

*i*)

^{2n+|m|+1}in (13) appears to be the accumulated Gouy phase shift [27

27. M. F. Erden and H. M. Ozaktas, “Accumulated Gouy phase shift in Gaussian beam propagation through first-order optical systems,” J. Opt. Soc. Am. A **14**, 2190–2194 (1997). [CrossRef]

*ℒ𝒢*. We describe the propagation by a conventional temporal Gaussian signal: where

_{n,m}*ω*

_{0}is the central angular frequency. The analytical expression for the pulselets is deduced expanding the modified Laguerre polynomial in a power series in (12), performing the integral and taking into account the Rodrigues formula [28], giving

*H*(.) is the Hermite polynomials of index

_{l}*l*and

*ρ*

_{0}=

*fcτ*

_{0}/

*w*

_{0}is a constant that normalizes the transverse distance

*r/ρ*

_{0}. Equation (16) reduces to (13) when

*τ*

_{0}→ ∞.

*t*

_{p}/τ_{1}in (16). The exponential factor

*s*+

*f*)/

*c*, which is removed for simplicity, and by the spherical arrival-time (1/2

*fc*)(1 −

*s/f*)

*r*

^{2}. In the case

*s*=

*f*, the pulse peak follows on a plane arrival time. The spatial distribution of the pulselets are formed by the finite combination of the Hermite polynomials

*H*(

_{l}*t*

_{p}/τ_{1}). In addition, the OAM of the pulselets is defined by exp(

*imϕ*). As it has the same order

*m*than the

*LG*mode, the OAM is preserved in the propagation of each pulselet to the focus.

_{n,m}*ℒ𝒢*

_{5,3}. This is a representative case because exhibits an appreciable OAM (order

*m*= 3) and a complex intensity profile composed by six concentric rings (

*n*= 5), see Fig. 2(a). The parameters used in the simulations are feasible to be experimentally realized,

*f*= 15 cm,

*τ*

_{0}= 15 fs,

*λ*

_{0}= 800 nm and

*w*

_{0}= 150

*μ*m. For these values it is obtained

*ρ*

_{0}= 4.5 mm and a transverse extension characterized by the effective radius,

*β*

_{3}∼ 670

*μ*m for the

*β*

_{3}are

*τ*

_{1}= 15.16 fs and

*ω*

_{1}= 2.3010

^{15}Hz, very close to

*τ*

_{0}= 15.00 fs and

*ω*

_{0}= 2.3610

^{15}Hz, respectively. Thus, it is expected that the pulse duration and the angular frequency at positions where the intensity is non-negligible slightly differs from the original values of the incident pulse.

*s*=

*f*is depicted, while Figs. 2(c)–2(d) display the intensity evolution of

*ℒ𝒢*

_{5,3}, computed from (16), for the cases

*s*=

*f*and

*s*= 5

*f*, respectively. These are representative cases because the way in which the pulse arrives to the focal plane varies depending on the value

*s*, as it will be shown below. In these figures, the axial extent is displayed in relation to

*β*

_{3}, while the temporal axis is in terms of the time duration

*τ*

_{0}of the input signal. The dashed lines delimits the temporal HWHM centered on the pulse peak intensity. As illustrated, plots along the spatial coordinate in Figs. 2(b)–2(c) are very similar when evaluated at

*t*= 0. This is because the rings of the pulselet arrive at the focal plane simultaneously, no matter the transverse position. The case

*s*= 5

*f*does not meet this condition and the pulse rings follow a spherical arrival-time [see Fig. 2(d)]. The pulse delay from

*r*=

*β*

_{3}to the center,

*r*= 0, is

*t*∼ 20 fs, higher than

_{p}*τ*

_{0}. Therefore, the shape of the instantaneous intensity at the focal plane changes very quickly in time as it is highlighted by the dashed line on Fig. 2(d).

*s*=

*f*evaluated at

*t*= −

*τ*

_{0}, 0,

*τ*

_{0}, while in Fig. 3(b) it is displayed the case

*s*= 5

*f*evaluated at the times

*t*= −2.5

*τ*

_{0}, −

*τ*

_{0}, 0,

*τ*

_{0}. The snapshot

*t*= −2.5

*τ*

_{0}in Fig. 3(a) has been not represented because of the intensity at this time is negligible. For comparative purposes the monochromatic case with

*s*=

*f*is also depicted. As illustrated, the intensity distribution in Fig. 3(a) for

*t*= 0 is very close to the intensity of

*t*= −

*τ*

_{0}and

*t*=

*τ*

_{0}, in accordance with Fig. 2. Besides, the phase at

*t*= 0 is almost identical to the one of the monochromatic case, while in previous and latter times it acquires a spherical phase, convergent for

*t*= −

*τ*

_{0}and divergent for

*t*=

*τ*

_{0}. The pulselet evolves first converging to the focal plane until the phase become plane, beginning to diverge in a posterior time. This is the optimal experimental configuration: the instantaneous intensity of the pulselet is as close as possible to its monochromatic counterpart. Pulselets with

*s*= 5

*f*behave quitte different. Its peak intensity has a spherical arrival-time. Hence, the larger diameter ring appears first and, as the pulse evolves in time, the most close to the center appears at latter times. Such a behavior is interpreted as following: the pulse get into focus in a plane different to the focal plane. Thus, there is a spherical phase at all times analyzed, as the first line of Fig. 3(b) points out.

*w*

_{0}= 150

*μ*m, was used. As pointed out in (6), the pulselets features characterize the space-time dependence of the wavefield at the focus through the expansion method. It is then convenient to analyze the properties of the pulselets when the waist size dimension of the LG expansion is relatively large,

*w*

_{0}= 3 mm. In this case, a small transverse extension characterized by the effective radius,

*β*

_{3}∼ 34

*μ*m, has been obtained. The spatio-temporal evolution of the pulselet is carried out by

*t*

_{p}/τ_{1}, that is independent on

*w*

_{0}when evaluated at

*β*

_{3}. Thereby, the instantaneous intensity of

*ℒ𝒢*

_{5,3}for

*s*=

*f*is also represented by Fig. 2(a) and Fig. 3(a), both scaled by the new value

*β*

_{3}. When

*s*= 5

*f*, the delayed spherical arrival-time evaluated at

*β*

_{3}is ∼ 10

^{−2}fs, which is negligible compared to

*τ*

_{0}. The pulse peak follows on an approximate plane arrival-time so that the pulselet features are close to the case

*s*=

*f*. This conclusion also applies to longer pulses, shorter focal distances and larger beam waist size at the SLM.

## 5. Spatial pulse shaping at the focus

*f*(

*r*,

*ϕ*). To this,

*f*(

*r*,

*ϕ*) must be expanded in terms of a LG basis particularized for the central frequency

*ω*

_{0}: finding the new expansion coefficients

*B*. The relationship between

_{n,m}*B*and

_{n,m}*A*can be derived by comparing (18) with the wavefield at the focus for the central frequency,

_{n,m}*ℰ*, evaluated in

_{Mon}*t*= 0: obtaining that and

*w*

_{0}= 2

*fc/w*

_{1}

*ω*

_{0}. The ultrashort spatial pulse at the focus is then computed by (8) and the signal to be codified in the dynamically controllable hologram is found by (6), in both cases using the coefficients

*A*giving by (20). This approach avoids a hard calculation arose from the iterative algorithms used to solve the inverse problem. Notice that these algorithms developed for monochromatic light, for example [5], are limited to pulses > 130 fs and they only compute the instensity distribution at the focus. By the procedure described above, these limitations are overcomed since the electric field of the ultrashort pulse for any pulse duration is obtained directly.

_{n,m}*s*=

*f*. This condition simplifies (20) to

*A*∞ (−

_{n,m}*i*)

^{−2n−|m|−1}

*B*. For comparative purposes, the same coefficients

_{n,m}*A*will be used to find the pulses for

_{n,m}*s*= 5

*f*. The parameters in the simulations are the same than in Section 4.

### 5.1. Spatial top-hat and annular ultrafast beam profiles

3. N. Sanner, N. Huot, E. Audouard, C. Larat, and J. -P. Huignard, “Direct ultrafast laser micro-structuring of materials using programmable beam shaping, ” Opt. Laser Eng. **45**, 737–741 (2007). [CrossRef]

29. E. Constant, A. Dubrouil, O. Hort, S. Petit, D. Descamps, and E. Mével, “Spatial shaping of intense femtosecond beams for the generation of high-energy attosecond pulses,” J. Phys. B: At. Mol. Opt. Phys. **45**, 074018 (2012). [CrossRef]

*a*is calculated at the focal plane considering

*s*=

*f*. This profile has no OAM so that the terms of the expansion with

*m*≠ 0 are identically zero. The coefficients

*B*

_{n}_{,0}are shown in the inset of Fig. 4(a) evaluated for

*a*= 1.4 mm and a finite numbers of terms giving by

*M*= 30. Following [11

11. R. J. Mahon and J. A. Murphy, “Diffraction of an optical pulse as an expansion in ultrashort orthogonal Gaussian beam modes,” J. Op. Soc. Am. A **30**, 215–226 (2013). [CrossRef]

*w*

_{1}= 255

*μ*m, corresponding to the value

*w*

_{0}= 150

*μ*m at the SLM. The finite series is obtained by changing the ∞ in the summation operator in (6) and (8) by the finite number

*M*. In Fig. 4(a), the ideal top-hat (thick gray line) is compared to the fitting (blue line), while Fig. 4(b) compares the signal (thick gray line) at the SLM, obtained by Fourier Transforming the top-hat function [

*R*(

*r*

_{0},

*ϕ*

_{0}) ∝

*J*

_{1}(

*ω*

_{0}

*ar/cf*)

*/*(

*ω*

_{0}

*ar/cf*)], to the LG expansion (blue line) calculated with

*w*

_{0}= 150

*μ*m and the coefficients

*A*giving by (20). Notice that the signal in Fig. 4(b) should be encoded in computer generated holograms with the carrier frequency (not shown on the figure) following the procedure [7

_{n,m}7. V. Arrizón, U. Ruiz, R. Carrada, and L. A. González, “Pixelated phase computer holograms for the accurate encoding of scalar complex fields,” J. Opt. Soc. Am. A **24**, 3500–3507 (2007). [CrossRef]

8. J. A. Rodrigo, T. Alieva, A. Cámara, Ó. Martínez-Matos, P. Cheben, and M. L. Calvo, “Characterization of holographically generated beams via phase retrieval based on Wigner distribution projections,” Opt. Express **19**, 6064–6077 (2011). [CrossRef] [PubMed]

*ε*, is [30

_{M}30. E. Cagniot, M. Fromager, and K. Ait-Ameur, “Modeling the propagation of apertured high-order Laguerre-Gaussian beams by a user-friendly version of the mode expansion method,” J. Opt. Soc. Am. A **27**, 484–491 (2010). [CrossRef]

*U*‖

^{2}is the integration of the square modulus of the original function over the whole transverse plane. The value obtained is

*ε*

_{30}= 0.0096 indicating that the fitting deviates from the original function less than 1%. We obtain a good fitting by

*M*= 30, a reasonable low value to the method be numerically highly efficient.

*M*= 30 possessing smaller radius,

*a*= 70

*μ*m. In this case, the best fitting beam waist size is

*w*

_{1}= 12.7

*μ*m, corresponding to

*w*

_{0}= 3 mm at the SLM. As discussed in Section 4, the behavior of the pulselets with

*w*

_{0}= 3 mm for

*s*=

*f*and

*s*= 5

*f*are very similar. The same behavior is then expected for a top-hat ultrafast beam profile. This statement can be demonstrated by a direct substitution of the new parameters in (8). Hence, both cases,

*s*=

*f*and

*s*= 5

*f*, are represented by Fig. 4(c) but normalized by the new value

*a*= 70

*μ*m.

*M*= 30. This function is composed by a top-hat with radius

*a*= 350

*μ*m and a single-annular beam of width

*a*. The best fitting beam waist size at the focal plane is the same than in Fig. 4,

*w*

_{1}= 255

*μ*m. The results by performing an equivalent analysis to that performed in Fig. 4, are shown in Fig. 5. A good fitting (

*ε*

_{30}= 0.033) is obtaining by just using 31 terms, even for hard functions with sharp edges.

### 5.2. Interference of spatial top-hat ultrafast beam profiles

*r*

_{0}= 0. A displacement of the signal by a distance ±

*b*from

*r*

_{0}= 0 along a transverse axis of the SLM, says

*x*

_{0}, generates a exp (±

*iωbx/fc*) phase at the focal plane, in accordance with the shift property of the Spatial Fourier Transform [23]. This phase term modifies the position-dependent delay time, The pulse at the focal plane is computed by (8), with the coefficients

*A*being calculated according to Section 5.1, and the pulselets giving by (16) with

_{n,m}*a*= 1.4 mm, is shown in Fig. 6(a), while Fig. 6(b) depicts the interference of two top-hats having a size of

*a*= 70

*μ*m. In these plots

*s*=

*f*and

*b*= ±1.4 mm.

*x*= 0, is simultaneously reached. The parameter

*b*defines the angle of incidence of each individual pulse as well as the interference pattern in the center of the figure. The lapse of time between the pulses varies linearly along

*x*in a well-defined fashion. The greatest lapse of time, shown in Fig. 6(a), is Δ

*t*= 5

*τ*

_{0}, although it can be easily changed just by varying the value

*b*in the computer generated hologram. According to (22), Δ

*t*= 2

*ba/fc.*In addition, both pulses fully overlap in the focal plane (but in different times) showing exactly the same intensity along

*x*. These features can be useful in high precision spatial pump-probe experiments [32

32. H. J. Wörner, J. B. Bertrand, D. V. Kartashov, P. B. Corkum, and D. M. Villeneuve, “Following a chemical reaction using high-harmonic interferometry,” Nature **466**, 604–607 (2010). [CrossRef] [PubMed]

*b*and the relative phase of the signal at the SLM. Furthermore, signals involving the generation of a large number of pulses can be used to create interference patterns with hard profiles, such as

*x*and

*ϕ*,

*n*and

*m*, are the interference period, relative phase, and integers, respectively. Proper choice of

*n*and

*m*can perform subwavelength ablation patterns of constant depth, controlled by the signal in an approach with no moving parts.

### 5.3. Pulses with nonstationary orbital angular momentum

*s*>

*f*the intensity will have a spherical arrival-time, so that first pulselet arriving to the focus will be that possessing the higher waist and, thus, in latter times, those having smaller waists. The OAM evolves in time from the outside to the center achieving a global nonstationary orbital angular momentum.

*LG*

_{0,5}and

*LG*

_{0,−5}modes with

*w*

_{0}= 50 and 400

*μ*m, respectively, in the approach

*s*= 5

*f*. The ultrashort pulse is composed by two pulselets arriving at the focal plane in different times [see Fig. 7(a)]. A detailed analysis of the transverse intensity is displayed in Figs. 7(b)–7(c) for the snapshots

*t*= −4

*τ*

_{0}and

*t*= 0, respectively. As it can be seen, the topological charge drastically jumps from

*m*= 5 to

*m*= −5 as the pulse evolves in time.

## 6. Discussion and conclusion

*x*. On the one hand, these results could be applied in high precision pump-probe experiments without delay lines. On the other hand, the interference of small size top-hat pulses could create subwavelength ablation patterns in an approach with no moving parts.

## Acknowledgments

*Ministerio de Economía y Competividad*under projects TEC 2011-23629, CTQ2008-02578/BQU, CTQ2012-37404-C02-01 and Consolider SAUUL CSD2007-00013 and from

*Conselho Nacional de Desenvolvimento Científico e Tecnológico*(CNPq), Brazil, under project 477260/2010-1, are acknowledged. P.V. acknowledges a PQ fellowship of CNPq. The publication of this work was supported by

*Servicio Nacional de Aprendizagem Industrial*(SENAI) - DR/Bahia, Brazil.

## References

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2. | V. Loriot, O. Mendoza-Yero, G. Mínguez-Vega, L. Bañares, and R. de Nalda, “Experimental demonstration of the quasy-direct space-to-time pulse shaping principle,” IEEE Photon. Technol. Lett. |

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12. | Y. Toda, K. Nagaoka, K. Shimatake, and R. Morita, “Generation and spatiotemporal evolution of optical vortices in femtosecond laser pulses,” Electr. Eng. JPN |

13. | K. Bezuhanov, A. Dreischuh, G. G. Paulus, M. G. Schätzel, and H. Walther, “Vortices in femtosecond laser fields,” Opt. Lett. |

14. | I. G. Mariyenko, J. Strohaber, and C. J. G. J. Uiterwaal, “Creation of optical vortices in femtosecond pulses,” Opt. Express |

15. | K. Bezuhanov, A. Dreischuh, G. G. Paulus, M. G. Schätzel, H. Walther, D. Neshev, W. Królikowski, and Y. Kivshar, “Spatial phase dislocations in femtosecond laser pulses,” J. Opt. Soc. Am. B |

16. | Ó. Martínez-Matos, J. A. Rodrigo, M. P. Hernández-Garay, J. G. Izquierdo, R. Weigand, M. L. Calvo, P. Cheben, P. Vaveliuk, and L. Bañares, “Generation of femtosecond paraxial beams with arbitrary spatial distributions,” Opt. Lett. |

17. | J. Atencia, M.-V. Collados, M. Quintanilla, J. Marín-Sáez, and I. J. Sola, “Holographic optical element to generate achromatic vortices,” Opt. Express |

18. | J. Strohaber, T. D. Scarborough, and C. J. G. J. Uiterwaal, “Ultrashort intense-field optical vortices produced with laser-etched mirrors,” Appl. Opt. |

19. | A. Schwarz and W. Rudolph, “Dispersion-compensating beam shaper for femtosecond optical vortex beams,” Opt. Lett. |

20. | 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. |

21. | K. Bezuhanov, A. Dreischuh, G. G. Paulus, M. G. Schätzel, H. Walther, D. Neshev, W. Królikowski, and Y. Kivshar, “Spatial phase dislocations in femtosecond laser pulses,” J. Opt. Soc. Am. B |

22. | I. Marienko, V. Denisenko, V. Slusar, and M. Soskin, “Dynamic space shaping of intense ultrashort laser light with blazed-type gratings,” Opt. Express |

23. | J. W. Goodman, |

24. | V. Lakshminarayanan, M. L. Calvo, and T. Alieva, |

25. | E. Abramochkin, E. Razueva, and V. Volostnikov, “General astigmatic transform of Hermite-Laguerre-Gaussian beams,” J. Opt. Soc. Am. A |

26. | R. L. Phillips and L. C. Andrews, “Spot size and divergence of Laguerre Gaussian beams of any order,” Appl. Opt. |

27. | M. F. Erden and H. M. Ozaktas, “Accumulated Gouy phase shift in Gaussian beam propagation through first-order optical systems,” J. Opt. Soc. Am. A |

28. | A. Jefferey and H. H. Day, |

29. | E. Constant, A. Dubrouil, O. Hort, S. Petit, D. Descamps, and E. Mével, “Spatial shaping of intense femtosecond beams for the generation of high-energy attosecond pulses,” J. Phys. B: At. Mol. Opt. Phys. |

30. | E. Cagniot, M. Fromager, and K. Ait-Ameur, “Modeling the propagation of apertured high-order Laguerre-Gaussian beams by a user-friendly version of the mode expansion method,” J. Opt. Soc. Am. A |

31. | J. T. Foley and E. Wolf, “Anomalous behavior of spectra near phase singularities,” J. Op. Soc. of Am. A |

32. | H. J. Wörner, J. B. Bertrand, D. V. Kartashov, P. B. Corkum, and D. M. Villeneuve, “Following a chemical reaction using high-harmonic interferometry,” Nature |

33. | R. W. Ziolkowski and J. B. Judkins, “Propagation characteristics of ultrawide-bandwidth pulsed Gaussian beams,” J. Opt. Soc. Am. A |

34. | S. Feng and H. G. Winful, “Higher-order transverse modes of isodiffracting pulses,” Phys. Rev. E |

35. | J. C. Shane, M. Mazilu, W. M. Lee, and K. Dholakia, “Effect of pulse temporal shape on optical trapping and impulse transfer using ultrashort pulsed laser,” Opt. Express |

36. | M. Duocastella and C. B. Arnold, “Bessel and annular beams for materials processing,” Laser Photonics Rev. |

37. | C. Hnatovsky, V. G. Shvedov, W. Krolikowski, and A. V. Rode, “Materials processing with tightly focused femtosecond vortex laser pulse,” Opt. Lett. |

38. | J. Hamazaki, R. Morita, K. Chujo, Y. Kobayashi, S. Tanda, and T. Omatsu, “Optical-vortex laser ablation,” Opt. Express |

39. | K. Toyoda, F. Takahashi, S. Takizawa, Y. Tokizane, K. Miyamoto, R. Morita, and T. Omatsu, “Transfer of light helicity to nanostructures,” Phys. Rev. Lett. |

40. | L. T. Vuong, T. D. Grow, A. Ishaaya, A. L. Gaeta, G. W. ’ t Hooft, E. R. Eliel, and G. Fibich, “Collapse of optical vortices,” Phys. Rev. Lett. |

41. | I. J. Sola, V. Collados, L. Plaja, C. Méndez, J. San Román, C. Ruiz, I. Arias, A. Villamarín, J. Atencia, M. Quintanilla, and L. Roso, “High power vortex generation with volume phase holograms and non-linear experiments in gases,” Appl. Phys. B |

42. | A. Vincote and L. Bergé, “Femtosecond optical vortices in air,” Phys. Rev. Lett. |

43. | P. Hansinger, A. Dreischuh, and G. G. Paulus, “Vortices in ultrashort laser pulses,” Appl. Phys. B |

44. | M. K. Bhuyan, F. Courvoisier, P.-A. Lacourt, M. Jacquot, L. Furfaro, M. J. Withford, and J. M. Dudley, “High aspect ratio taper-free microchannel fabrication using femtosecond Bessel beams,” Opt. Express |

45. | A. Picón, J. Mompart, J. R. Vázquez de Aldana, L. Plaja, G. F. Calvo, and L. Roso, “Photoionization with orbital angular momentum beams,” Opt. Express |

46. | C. T. Schmiegelow and F. Schmidt-Kaler, “Light with orbital angular momentum interacting with trapped ions,” Eur. Phys. J. D |

**OCIS Codes**

(050.0050) Diffraction and gratings : Diffraction and gratings

(320.5540) Ultrafast optics : Pulse shaping

(090.1995) Holography : Digital holography

(070.7145) Fourier optics and signal processing : Ultrafast processing

(070.6120) Fourier optics and signal processing : Spatial light modulators

**ToC Category:**

Ultrafast Optics

**History**

Original Manuscript: July 22, 2013

Revised Manuscript: September 25, 2013

Manuscript Accepted: September 25, 2013

Published: October 11, 2013

**Citation**

Ó. Martínez-Matos, P. Vaveliuk, J. G. Izquierdo, and V. Loriot, "Femtosecond spatial pulse shaping at the focal plane," Opt. Express **21**, 25010-25025 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-21-25010

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