# Adaptive Generation and Diagnostics of Linear Few-Cycle Light Bullets

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Linear Light Bullets

#### 2.1. Medium-Free Generation of Linear Light Bullets

#### 2.2. Ambiguity of the Poynting Vector Maps of Nondiffracting Beams

_{2}tend to become nearly identical again so that a wavefront-division based detection technique can be well applied to characterize the beams. This is of relevance to the experiments we will report on.

**Figure 1.**Ambiguity of local wavefronts of nondiffracting beams in comparison to a convergent beam (simplified ray representation): (

**a**) Gaussian-type beam with unambiguous Poynting vectors (red arrows) as detected by a Shack-Hartmann wavefront sensor at a certain plane WS

_{1}(gray); (

**b**) Bessel beam with an ambiguity in the superposition zone WS

_{2}(gray) and a ring-shaped unambiguous angular distribution at a distance WS

_{3}behind the superposition zone (gray). Linear light bullets in absence of a medium are created by constructive interference in superposition zones like shown in (

**b**).

**Figure 2.**Intensity profiles of (

**a**) a Bessel and (

**b**) a needle beam schematically represented for an arbitrary plane perpendicular to the propagation axis (color-coded, red = maximum intensity, green = minimum intensity). Needle beams correspond to Bessel beams which are truncated exactly at the first zero (self-apodized truncation).

#### 2.3. Nonlinear and Linear Light Bullets: Brief Remark on the Terminology

#### 2.4. Pulsed Needle Beams and Highly Localized Wavepackets

**Figure 3.**Self-apodized truncation condition for the aperture-less Bessel-like needle pulses (schematically, after ref. [44]). Curve on the separated blue plane: radial Bessel intensity profile, red area: central lobe, white area: not generated outer parts (Λ = diameter of central lobe, z = propagation axis, D = axicon diameter, θ

_{max}= maximum allowed half conical angle, Δz

_{min}= minimum extension of the needle-shaped nondiffracting zone, J

_{0}= zero order first kind Bessel function).

_{max}is determined by the finite diameter D of the initial transversal field under self-apodized conditions [44]. For a wavelength λ, it amounts roughly to

_{max}= D

^{2}/4λ

_{NB}of the pulsed needle beam can be represented (after some transformation) by the following Equations (2)–(7):

_{0}(r,k

_{r}) represents a wavelength-dependent Bessel distribution function in radial direction, Φ(r,t) is a complex phase term, τ

_{a}a radial-dependent time separation between two interfering pulses and τ

_{0}an input pulse duration (standard deviation). In good approximation, one can replace the integral in Equation (3) by the expression

## 3. Experimental Section

**Figure 4.**Experimental setup for adaptive shaping and detection of paraxial linear few-cycle light bullets (schematically). For collinear autocorrelation, a balanced interferometer generates two identical replica of the pulse. The variable time delay (Δτ) is induced by the length variation (Δz) of the interferometer arm with a piezo actuator. The SLM shapes the light bullets in finite but extended zones of stable nondiffracting propagation. The combination with second order nonlinear conversion and 2D detection enables to extract spatially resolved pulse duration via image autocorrelation, as well as wavefront curvature, vortex characterization, and highly resolved time delay mapping.

## 4. Results and Discussion

**Figure 5.**Flexible generation of few-cycle light bullets with the characteristics of spatially oscillation-free, temporally nondiffracting highly localized wavepackets (HLWs): (

**a**) gray value distributions corresponding to the phase maps of axicons; (

**b**) measured time-integrated intensity maps of a set of simultaneously formed light bullets of different spatial structures in a transversal plane (disks, lines, rings and stadiums). Both the new types of shaping phase elements and the HLWs represent generalizations of the conventional approaches of axicons and nondiffracting beams, respectively. Arrays of programmable HLWs enable to realize time-wavefront sensors with spatially encoded spots and flexible array geometries.

**Figure 6.**Nondiffracting intensity propagation of a stadium-shaped light bullet (time integrated, field of view 2.7 × 2.7 mm

^{2}, detected after SHG at 400 nm, initial pulse duration 6.5 fs).

**Figure 7.**Temporal autocorrelation function of a stadium-shaped few-cycle light bullet at a distance of 80 mm;

**left**: cut through the major (long) axis;

**right**: cut through the minor (short) axis.

**Figure 8.**Temporal autocorrelation function of a stadium-shaped light bullet measured at three different axial distances (80 mm, 100 mm and 120 mm). For a sech

^{2}-pulse shape, one can estimate a nearly constant pulse duration of about 6.6 fs for the propagating light bullet.

^{2}-pulse shape, the corresponding final pulse duration is 6.6 fs at an input pulse duration of 6.5 fs. Thus, the results indicate a very stable propagation behavior also in temporal domain within the fluctuations (the estimated error bar was about 0.1 fs). In another experiment, the spectral phase of a circular light bullet (pulsed needle beam, depth of the nondiffracting zone 1 m) was characterized with few-cycle spatially integrated spectral phase interferometry for direct electric field reconstruction (FC-SPIDER, Figure 9). The retrieved final pulse duration (τ

_{M}= 6.4 fs) was found to be within the error bar as well.

**Figure 9.**Phase transfer in a circular nondiffracting light bullet (pulsed needle beam, length of the nondiffracting zone about 1 m), measured with an FC-SPIDER (APE); black curve: temporal phase, dashed red curve: pulse calculated for the Fourier transform limited case, blue curve: pulse retrieved from measuring data. τ

_{FL}and τ

_{M }are the corresponding pulse durations, respectively. The distance of the entrance window of the SPIDER from the SLM was about 20 cm (the effective distance was higher because of the additional internal path in the SPIDER system). The diameter of the light bullet at the entrance of the SPIDER was 1 mm.

## 5. Conclusions

## Acknowledgments

## Conflict of Interest

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Bock, M.; Grunwald, R. Adaptive Generation and Diagnostics of Linear Few-Cycle Light Bullets. *Appl. Sci.* **2013**, *3*, 139-152.
https://doi.org/10.3390/app3010139

**AMA Style**

Bock M, Grunwald R. Adaptive Generation and Diagnostics of Linear Few-Cycle Light Bullets. *Applied Sciences*. 2013; 3(1):139-152.
https://doi.org/10.3390/app3010139

**Chicago/Turabian Style**

Bock, Martin, and Ruediger Grunwald. 2013. "Adaptive Generation and Diagnostics of Linear Few-Cycle Light Bullets" *Applied Sciences* 3, no. 1: 139-152.
https://doi.org/10.3390/app3010139