1. Introduction

Optical Vortices (OVs) [1] 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] and used for microparticles manipulations [3] as well as for demonstrating entanglement of OAM states of photons [4] and the quantized rotation of atoms [5]. These experiments utilized the multiple previously developed technologies (described and referenced in [1]) 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]. 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], a deformable mirror to control the beam divergence angle [8], periodic meshes to improve the multi-filamentation control [9], and others. Also, it was shown theoretically [10], 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], being a highly desired shape, happens to be technically challenging to create [12].

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] and 2f-2f scheme [14] 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] 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]. As to SLM, the peak power damage threshold for ultrashort laser pulses is not determined yet [16], 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 . 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.

 

Fig. 1 shows the optical scheme of 2f-2f shaper with magnification. The red line illustrates the space transformations of a broadband laser beam.

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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:

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:$-\frac{z_1}{z_2}=M$. 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

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]. 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, 18], that results in phase variation proportional to $\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\left[1+\sin (m\varphi -Kx)\right]$, 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 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.

 

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.

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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], 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], 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.

Another figure to demonstrate the shaping capability is “focal spot”. We simulated a focusing lens on the SLM by encoding it with a phase grating where phase varies proportionally to $\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\left[1+\sin (a[x^2+y^2]-bx)\right]$. The coefficients a and b define a focal length of the holographic lens and a spatial period of the encoded grating respectively. Its spatial period happened to be twice as large as in previous case. According to Eq. (6), the second diffraction order of the output grating G₂ is to be taken to compensate for the dispersion. In the experiment, we obtained the following transversal intensity profiles of the focused beams: Fig. 3a and 3c represent the pulsed and CW beams focal spots respectively. Figure 3b shows their x cross-cut intensity profiles along red lines. The profiles match closely thus verifying the dispersion-free shaping. A discrepancy is attributed to the small size difference between the intensity profiles of the original CW and pulsed laser modes.

 

Fig. 3 Intensity patterns of the Ti:S output laser beam focused by the simulated lens into a spot. Pattern (a) represents the focusing of pulsed laser output, (c) shows focused CW laser beam. The cross-cut of the intensity distribution for both beams along red lines is shown on the graph (b).

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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 magnification happened to be related to angular dispersion. This allows pairing commercial metal-coated blazed gratings of relatively high grooves density (sub-hundred to hundreds grooves/mm) with SLM grating that has lines density of about 10 lp/mm. This opens up a number of possibilities that were not available previously with 2f-2f shaper. First, it is an increased/scaled safe operation threshold of the shaper. It becomes possible because the magnified laser beam spreads over a greater surface of an SLM thus reducing the light power density the SLM matrix is to withstand by the factor of the magnification squared. For example, the safe 2f-2f SLM shaper operation at the repetitive fluence of 10¹⁰ W/cm² for 50-fs Ti:S pulses [21] can be extended by 100 times when 10X magnification is added to the scheme. At the same time, the magnification increases sampling density for a light pattern to be diffracted by the factor of magnification squared. It not only makes the shaping finer but also allows simulation of saw-tooth type phase gratings with diffraction efficiency close to 100%. For example, if NIR-optimized SLM with transmission efficiency of 80% and 90% diffraction efficiency is used (the same as blazed input grating) then total throughput efficiency would become 43% – the value that makes the technique experimentally feasible for the applications. The selection of right components for the shaper along with a proper magnification would increase the throughput efficiency of the shaper to even higher value. And, finally, dynamic shaping capability can be provided at any rate up to 60 Hz – typical maximum LC matrix refresh rate for SLM devices.

Also, we would like to point out that the 4f scheme [13] can be upgraded with magnification as well providing the same relation between the magnification factor and angular dispersion; the magnification value depends on a targeted application. Such scheme was experimentally realized in [22] where the SLM served as an input dispersive element and a low-dispersion prism served as an output dispersion compensator. The use glass prizm sets the damage threshold of the shaper to even higher value as compared to a metal blazed grating. The scheme featured internal lateral demagnification of the dispersed light and was used for while-light (from halogen lamp) vortex formation and targeted the experimental investigation of the chromatic effects in the vicinity of the vortex core. Then, this 4f optical arrangement with demagnification was successfully used for creation of white-light (from a spatially coherent supercontinuum source), achromatic Bessel and Laguerre-Gaussian beams [23]. Later, those exotic white-light beams were used for microparticles rotation [24].

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.