1. Introduction

The self-focusing of ultra-short laser pulses in condensed media can give rise to one of the most incredible effects in nonlinear optics: the supercontinuum (SC). In this phenomenon the spectral content of the pulse is dramatically broadened and can span from the ultraviolet to the infrared, manifesting itself as an intense white disk surrounded by a concentric rainbow pattern. Ever since its discovery in 1970 [1] it has been the subject of intense research striving to better understand the various processes involved in its generation. It is now widely understood that self-phase modulation (SPM) is the main cause of spectral broadening although other effects such as stimulated Raman scattering and four wave mixing contribute to spectral features [2]. More recently theoretical investigations have shown that self-steepening of the laser pulse can explain the asymmetry characteristic of the SC spectrum [3]. Self-focusing also arises as a result of the change in transverse refractive index, this effect increases the local intensity which in turn assists the intensity dependent SPM process. The threshold power for these effects has been found to coincide and is typically several megawatts for a range of transparent condensed media [4]. At pulse powers well above the threshold (~100MW) the beam is observed to fragment and form multiple self-focused white-light filaments. These filaments form as a result of aberrations on the beam’s spatial profile therefore the spatial distribution is inherently random, the temporal distribution however exhibits a cone-shaped distribution for a Gaussian input beam. This cone-shaped pattern is due to the center of the pulse having a higher local intensity than the periphery and since self-focusing is an intensity dependent effect this means that the filaments originating from the center of the beam will form at shorter distances [5]. The propagation distance at which filamentation occurs can be controlled by adding negative chirp to the pulse. The pulse is then temporally recompressed by the group velocity dispersion of the medium. This means that the range at which the filament forms may be controlled by the rate of chirp [6].

Early applications of such a broadband white-light source include its use as a probe in time-resolved pump probe spectroscopy experiments [7]. However subsequent advances in laser technology now mean that femtosecond pulses with gigawatt powers are readily achievable allowing the generation of white-light continuum filaments in media with lower n₂, this led to the first generation of a continuum source in gases [8]. More recently the development of transportable terrawatt level systems [6] means that self-focusing may now be excited in the atmosphere giving rise to the generation of WLC at long range in narrow filaments that do not diffract over many kilometers. This development has opened a new front in the field of remote sensing and light detection and ranging (LIDAR) experiments. The SC source offers the ability to excite several absorption bands simultaneously when probing various gases in the atmosphere, furthermore the large IR component can now allow the detection of certain aerosols and rare biological compounds [9].

Motivated by the potential of such applications, recent research has aimed to control the filamentation process and their distribution. Méchain et al. achieved a degree of control by introducing field gradients and phase changes on the input beam in air [10], Schroeder et al performed similar investigations for controlling filaments in water [11]. Other methods for filament control have utilized the effects of beam astigmatism and ellipticity [12, 13]. In this paper Fresnel diffraction from a hard-edge aperture is used to control filament formation, thus a focusing lens is not necessary. The experiment demonstrates that filaments are formed at distances from the aperture that coincide with local intensity maxima in the near field diffraction pattern.

2. Experiment

A Ti:Sapphire laser was combined with a regenerative amplifier system to provide 440µJ pulses centred around 800nm. The repetition rate was set to 1kHz for the duration of the experiment and the pulse length of 135fs (FWHM) was continuously monitored using a single shot autocorrelator (SSA). The energy could be varied by a neutral density (ND) wheel. A schematic of the experimental configuration is depicted in Fig. 1.

 

Fig. 1. Experimental set-up for inducing filament formation.

Download Full Size | PDF

A thin circular aperture of radius a=0.825mm was placed in the beam. The e^-2 radius of the incident beam was ω₀=3.1mm and the transmission through the aperture was 13% in agreement with the theoretical value for a Gaussian beam. The pulse then propagated through free space and was incident on a 50mm long block of BK270 glass. The z position of the glass block could be varied by using the translation stage and since the Fresnel diffraction pattern varies with z, this gave control over the spatial profile incident on the glass block.

A lens and camera were also mounted on the translation stage, allowing transverse images of the beam profile to be imaged onto a CCD camera. The lens and camera could also be positioned orthogonal to the glass to obtain images of the generated filaments. The position of the filaments inside the glass was measured by eye using a ruler.

3. Results and discussion

In order to measure the profile of the Fresnel diffraction pattern the glass and lens were removed, the camera was then translated from the plane of the aperture at z=0 to z=1m and a series of images were captured - beyond 1m the beam enters the Fraunhofer regime. In the Fresnel regime the axial part of the beam oscillates between maxima and minima, this may be explained by recognising that Fresnel diffraction due to the plate modifies the wavefront to produce intensity variations across the beam aperture. The field amplitude is given by the Huygens-Fresnel integral [14] shown in Eq. (1)

where a is the radius of the aperture, ρ is distance between a point source on the wavefront in the plane of the aperture and the point of observation, dS₀ is an elemental area in the plane of the aperture and θ is the angle between the direction of ρ and the normal to dS₀. This may be solved to obtain an analytical solution for the axial intensity dependence on z as shown in Eq. (2).

This equation is plotted along with the experimental values in Fig. 2. The axial intensity oscillates with decreasing frequency as z increases and the intensity decays monotonically to zero in the far-field after the final maximum located at z=a²/mλ (m=odd integer) which corresponds to (m=1) z=85cm, for our aperture and operating wavelength. Note that axial minima are located at values of z corresponding to even m.

 

Fig. 2. Theoretical and measured axial intensity of the Fresnel diffraction profile. Inset: Images of transverse profile at various maxima and minima.

Download Full Size | PDF

The glass sample was then placed on the translation stage and moved from z=0 to z=1m for various energies. It was found that at certain positions of glass input surface a filament was formed inside the glass. In earlier work [15] it was shown that filament formation coincided with axial maxima and the non-linear Schrödinger equation was solved to show how the peak intensity caused by diffraction induced self-focusing and filamentation. At certain energies and positions a second filament occurred. Figures 3 and 4 show the filament positions inside the glass block as a function of z for various pulse energies, the 50mm point represents the output plane of the glass. A plot of the measured axial intensity is also shown.

 

Fig. 3. Filament position inside glass block (right Y axis, red and green lines) recorded for various post-aperture pulse energies. The measured axial beam intensity from Fig. 2. is also plotted (left Y axis, blue line).

Download Full Size | PDF

 

Fig. 4. Filament position inside glass block for maximum post-aperture energy of 62µJ (right Y axis, red and green lines) recorded for various post-aperture pulse energies. The measured axial beam intensity from Fig. 2. is also plotted (left Y axis, blue line).

Download Full Size | PDF

At an energy of 30µJ a filament is observed when the glass is positioned at 25cm. The filament forms just before the output side of the glass and moves closer into the glass as the translation stage is moved away from the aperture. As the block is translated the filament moves to 43mm from the input surface before rapidly moving back towards the output plane and disappearing. This is shown in Fig. 3(a) and it also occurs when the axial intensity undergoes its penultimate maximum. At 40µJ, a second filament is formed but a single filament occurs when z is less than 17.5cm. At 50µJ a filament is formed at the last axial maximum. At the maximum post-aperture energy of 62µJ some weak filaments are formed when the glass is located at z<17.5cm. The second filament forms behind and, as far as could be judged by eye, in line with the first. For example, Fig. 4 for z=23.5cm the first filament occurs 20mm from the front surface of the block and the second filament occurs 28mm from the front surface. From earlier work [5, 16] and direct observation, the filament length is 2–3mm and only one filament can fit within the depth of focus (which we define as the FWHM in z of an axial maximum) for m=5, 7 etc. It is suggested that the ionization induced in the filament causes self-diffraction [17] and the pulse diverges into neutral medium by which time the pulse has propagated out of the region of maximum intensity. However, the depth of focus for m=3 is significantly longer, ~10mm (Fig. 2: theoretical curve) thus, as the pulse diffracts into the neutral medium, the axial intensity is still high enough to induce a second self-focus and a second filament. It is known from earlier work [18] that the induced focal length depends on b², where b is the e^-1 intensity radius of the beam or, in this case, the radius of the axial peak. The semi-empirical equation that describes this relationship is [18]

Where P_crit is the critical threshold power and k=2πn₀/λ is the wave number. Therefore, as the pulse propagates towards an axial maximum the peak intensity rises due to Fresnel diffraction and self-focusing is induced. Evidence for self-focusing is clearly indicated by the fact that the trajectory of the focus (green curves in Fig. 2) occurs slightly to the left (smaller z) than the position of the axial maximum. Interestingly, no second filament is observed for m=1 where the depth of focus is >80mm. At this point the pulse is entering the far field and the axial maximum is much wider and comparable in size to the aperture diameter. The central peak is therefore wider and the intensity gradient lower. The induced focal length is therefore expected to be much longer- indeed the second filament forms toward the back of the block confirming the longer induced focal length. A second filament might be expected to form at a greater distance beyond the length of the block used.

These filaments form in spite of the reduction in beam energy caused by the aperture, due to the refractive index dependence on beam intensity and not power. Immediately after the iris this intensity is distributed approximately equally across the beam in a top-hat profile. The radial intensity profile at this point shows a large intensity gradient formed by this axial peak. This produces a large refractive index gradient across this part of the beam. This allows the phase front at the outer edges of this peak to advance faster than the centre, leading to self-focusing and the formation of a filament.

When the circular iris is removed the two filaments disappear and no WLC is generated even though the pulse power is greater by a factor of ~7.7. This can be explained by considering the change in spatial beam profile. For an aperture diameter less than that of the beam the Fresnel diffraction pattern for z corresponding to axial maxima has a central peak surrounded by bright rings. This is shown in the Fig. 2 insets. The pictures show the spatial profile for z=29.0cm (m=1) consisting of a central spot and a distinct dark ring surrounded by a broad bright ring (this outer ring appears to have structure but this is an artefact of the camera). For z=17.5cm (m=3) the central spot is much smaller and is surrounded by a broad dark ring and fainter outer ring. When the aperture is removed the beam has its familiar Gaussian profile with e^-2 radius 3.1mm. Thus, insertion of the aperture causes the central peak to narrow and increases the radial intensity gradient. In this case it will take a shorter propagation length of glass for the beam to collapse to a filament since the beam is now narrower and the induced focal length shorter [18].

4. Conclusion

It may be concluded that experiments show that changing the spatial profile of the incident beam can induce the generation of WLC filaments. By inserting a circular aperture into the path of the beam a Fresnel diffraction pattern is generated causing the central portion of the beam to oscillate between maxima and minima. The change in beam profile has a direct effect on the intensity dependent refractive index, allowing the central peak of the diffraction pattern to act like a strongly focussing lens compared with the weaker focusing that would occur from the Gaussian profile incident on the aperture. It is shown that translating a sample of B270 Crown glass along the z axis results in the formation of WLC filaments at certain locations. Furthermore, an increase in the width of the circular aperture, while the glass was at the maxima, led to the disappearance of the filaments. This experiment demonstrates that the onset of filamentation can be controlled by varying the beam aperture whereas other methods involve complex negative chirping techniques. The creation of axial filaments is of importance in remote sensing applications where the pointing stability must be maintained.