1. Introduction

Since the advent of the powerful Ti-Sapphire femtosecond oscillators with chirped-pulse amplifiers more than a decade ago, the interaction of ultra-short, high power laser pulses with transparent glass materials has attracted much attention because of its potential application in micromachining [1–3] due to its deterministic character as pointed out by Du [4] and studied by Joglekar et al[2]. This process has also been used in fabricating waveguides [5–11], gratings [12,13], directional couplers [14,15], and optical storage [16] in these materials.

By using μJ pulses from systems of Ti-Sapphire femtosecond oscillators and chirped-pulse amplifiers, a number of researchers [16, 17–19] have modified the structure of transparent dielectrics through optical breakdown or filamentation depending on peak intensity and power of the impinging beams. The mechanisms of material modification in fused silica by these two processes are seemingly different. Plasma generation in the optical breakdown process generates permanent changes in refractive index or damage such as voids. In contrast, the filamentation process generates weaker plasma, has free electron density that is below the critical density [18], and creates reversible index change and good quality waveguides [20]. The repetition rate of the chirped-pulse amplifier is in the range of kHz, and optical breakdown generated by this amplifier is believed to be generated by independent, single laser pulses. However, optical breakdown can also be generated by the cumulative action of a number of pulses from a high repetition rate femtosecond laser. This suggests that the free electron gas generated by the sub-threshold pulses does not completely relax down to the valence band; some of it may be trapped by defect states which are below the band gap energy [21]. These trapped electrons could be more easily excited into the conduction band than the ones in the valence band. Another possible cause is that sub-threshold pulses create absorptive centers which increase the material absorption. After the optical breakdown starts, the high repetition rate (25 MHz) femtosecond laser pulses can sustain a continuous optical breakdown in BK7 glass.

As an approach for obtaining high pulse energy from standard Ti:Sapphire laser oscillators without using an amplifier, we extended the cavity length of the standard Ti:Sapphire laser to increase pulse energy, which was a concept pioneered by Cho et al. [22]. However, we followed a simpler design of Libertun et al [23] as shown in Fig. 1. We could not generate filamentation with the tens-of-nJ pulse energy from the output of the extended-cavity Ti:Sapphire oscillator. However, we tightly focused these laser beams down into the glass material and successfully induced optical breakdown – creating a permanent index change in glass. This process enabled us to fabricate waveguides in glass, using below-threshold power of self-focusing.

The 26 MHz extended-cavity femtosecond laser has been used to write waveguides in zinc-borosilicate glass (0211) and alkali-zinc-silicate glass (IOG10) by an optical breakdown process [11]. However, this process generates void-like structures in fused silica [11], and it was also observed by our group. A possible cause for this behavior is because the recombination in fused silica is about 120 fs [11], where the plasma extinguishes between consecutive incident laser pulses.

In our work, we investigated in detail the temporal and spatial properties of the optical breakdown process in BK7 glass using an extended cavity femtosecond Ti-sapphire laser. We will describe how we extended our laser and the experimental set-up in part 2, experimental results in part 3 along with discussions, a computer model for the time evolution optical breakdown in part 4, and a conclusion in part 5.

2. Extended-cavity femtosecond laser and Experimental set-up

2.1 Extended-cavity femtosecond laser

Conventional Ti-Sapphire femtosecond lasers operate at 80 MHz with output pulse energies of a few nJ and pulse widths commonly in the range of 50 to 100 fs. With this pulse energy, we could not generate optical breakdown in BK7 glass even when we used high NA objective lens mainly because nearly half of the energy of the laser beam is lost due to Fresnel loss through focusing optics of the objective lens and other optics in the set-up.

We used a pair of 2-m radius-of-curvature concave mirrors in a 4 focal length 1:1 folded telescope arrangement to extend the output coupler arm of the laser [23]. We added 4 m onto the ~ 2 m of the original cavity length. Since we lowered the repetition rate of our laser, we increased the pulse energy traveling inside the laser cavity; as a result, we increased the self phase modulation. In order to mode-lock our laser stably and avoid the multiple pulsing instability, we increased the negative group velocity dispersion by increasing the distance between the two fused silica intra-cavity prisms by 80 mm on the original distance of 655 mm, and to decreased the self phase modulation, we used a 25%-transmission output coupler instead of the original 12% one. We pumped our laser at 9 watts of multi-line argon, producing 44 nJ pulse energy at 25 MHz. We were able to keep our laser single pulsing for several hours by locking it in a slightly positive group velocity dispersion regime. The pulse width was about 100 fs after the external prism compressor.

 

Fig. 1. The schematic of an extended-cavity Ti-Sapphire oscillator

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2.2 Experimental set-up

In order to detect and study the stability of optical breakdown in BK7 glass, we recorded the broadband light emitted from the optical breakdown in the pump beam focusing region, the transmitted and reflected pump beam, and the transmitted probe beam, which propagated collinearly with the pump beam. The experimental set-up for this experiment is shown in Fig. 2.

 

Fig. 2. Experimental set-up

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The output laser pulses were variably attenuated through the combination of a half-wave plate and a polarizer. The laser pulses went through an optical switch, which consists of a half-wave Q-switch (KD*P, Inrad-PC1086); its switching time was about 35 ns. Before the laser pulses went into the sample, they went through an optical isolator that consisted of a polarizer and a quarter wave plate. The function of the optical isolator was to keep the laser from unlocking-- caused by the back-reflection from the optical breakdown in glasses. The radius at 1/e² of the pump pulses was 1.2 mm before these pump pulses were focused into a BK7 glass sample by a 100X, 1.3 NA, oil immersion objective lens (Zeiss) with a focal length of 1.65 mm. The light emitted by the optical breakdown region included the broadband fluorescence from the plasmas. The back-reflected pump pulses were partially collected by the objective lens. Photodiode 2 collected the back-reflected laser pulses from the sample reflected by the polarizer 2 and was followed by the transmission through an 800 nm band pass filter. The photomultiplier (PMT 1) collected the fluorescence from the generated plasmas that transmitted through the 700 nm short pass filter. The spectrometer (Spectra Pro 2300i, Acton Research) monitored the fluorescence spectrum of the plasmas. A small portion of the argon laser beam was used as a probe to monitor the optical breakdown. It passed through the sample and a 500 nm band pass filter and was collected by PMT 2. Photodiode 1 collected the transmitted pump light. The signals from PMT 1, PMT 2, photodiode 1, and photodiode 2 were transferred to a four-channel 500 MHz oscilloscope (Tektronix, TDS5054B) for data collection.

3. Experimental results and discussion

3.1 The build-up time to optical breakdown

The beginning of optical breakdown was established by the following observations that occurred at the same time [24].

By varying the incident pulse energy, we were able to start optical breakdown right after the first pulse or after a number of pulses. We set the optical switch opening time for 2 μsec and measured the power of the broadband light emitted from the plasma, the transmitted pump pulses centered at 800 nm (Ti:Sapphire laser), and the back-reflected pump pulses on every opening of the optical switch. The laser beam interacted with a new area of the sample on every opening of the optical switch. In this measurement, the transmitted probe signal centered at 500 nm (Argon ion laser) collected by PMT 2 was perturbed by electrical noise generated by the optical switch. Consequently, we will show the transmitted probe signal in the averaging mode only.

 

Fig. 3. Powers of (a) the broadband light emitted by the plasma, (b) the transmitted pump pulses, (c) the back-reflected pump pulses, and (d) the transmitted probe beam in the first 1.7 μsec after the optical switch was open. The pump pulse energy was 14 nJ.

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Figures 3(a)–3(c) show the powers of the broadband light emitted by the plasma, transmitted pulses, and back reflected pump pulses in the first 1.7 μsec after the optical switch opened. The incident pulse energy in this run was 14 nJ, and the data was collected in a single opening of the optical switch. The optical breakdown started on the first pulse, indicating that the pulse energy of 14 nJ was at or above the single pulse threshold for the 100X objective lens used in this experiment. Figure 3(a) shows the signal from the PMT 1 which detected the plasma emission right after the first pulse. The transmitted pump pulses were drastically reduced after the first two pulses, and it decreased to the noise level after about 1 μs or 25 pulses (Fig. 3(b))-- indicating the growing in size of the electron critical-density region. We also saw the effect of every pulse of the first 14 pump pulses on the plasma emission. Figure 3(a) also showed the fast rise and fast decay of the plasma emission signal between pulses, which indicated the recombination of free electrons and positive ions [25]. The energy released from these combinations heated up the ionic background, which gave the slow rise in the broadband light emitted by the plasma. The continuity of the broadband light emitted from the plasma and the behavior of the transmitted pulses with respect to time (Fig. 3(b)) show the continuous growing of the critical-density plasma region. Figure 3(c) shows the back reflected pump pulses. The first pump pulse was partially reflected. The pump pulses were reflected, perhaps due to the change of the refractive index caused by the electron-critical density region. The back reflected pump pulses increased to a maximum at the fifth pulse, which showed that this region was growing in size, and then the reflected pump pulses decreased at a lower pace compared to the transmitted pump pulses. The reason for the decrease of the reflected pulses was perhaps because the focusing region changed phase (solid to melt) or damaged. The back-reflected pump pulses reached the noise level at about 1.5 μs after the plasma started. Thus, after the optical breakdown existed for about 1.5 μs, the plasma absorbed approximately 90% of the pump pulses. The plasma strongly absorbed the energy from the pump pulses that heated up the electron gas, and the hot electron gas interacted with the ionic background through two processes:

Figure 3(d) showed the power of the transmitted probe beam in the averaged mode. In this mode, we averaged out the electronic noise introduced by the optical switch. This data was taken in 20 openings of the optical switch; the laser beam interacted with the same area of the sample in the 20 openings of the switch. This figure shows that at the onset of the optical breakdown, the dense plasma in the focal region started to reduce the transmission of the probe beam. At about 1.4 μs after the plasma was created, the transmission of the probe beam was reduced to nearly zero, and the plasma completely blocked the probe beam. The fluorescence emitted by the plasma still increased after the probe beam was completely blocked, indicating several possibilities:

Figure 3(a) shows that at the pump pulse energy of 14 nJ, the optical breakdown started right at the first pulse. By decreasing the pump pulse energy, the optical breakdown started on subsequent pulses. Figure 4 shows the traces of the power of the broadband light emitted by the plasma, the transmitted pump pulses, and the back-reflected pump pulses when the incident pulse energy was 8 nJ. The data was collected in the first 2 μs after the optical switch opened and in a single opening of the optical switch. The laser beam interacted with a new area of the sample in each run. As shown in Figs. 4(a) and 4(b), the optical breakdown was started by the cumulative actions of the first 8 pulses after the optical switch opened. After the optical breakdown started for about 1.4 μs, the powers of the transmitted pump pulses and the back-reflected pump pulses decreased to the noise level as indicated in Figs. 4(b) and 4(c). This shows that the plasma in the optical breakdown region strongly absorbed the energy of the pump pulses.

 

Fig. 4. Powers of (a) the broadband light emitted by the plasma, (b) the transmitted pump pulses, and (c) the back reflected pump pulses in the first 2 μsec after the optical switch was opened. The incident pulse energy was 8 nJ.

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Data similar to Figs. 3 and 4 was collected for a range of different pulse energies. We measured the build-up time to optical breakdown, which was the time between the first complete transmitted pump pulse and the onset of the broadband light emitted from the plasma created by the optical breakdown. Figure 5 shows the build-up time to optical breakdown as a function of the pump pulse energy. At higher pulse energy, it took less time to optical breakdown, and the build-up time had smaller uncertainty. This information showed that the optical breakdown process generated by the output of an extended cavity femtosecond laser was more deterministic as the pump pulse energy was higher, perhaps due to the uncertainty in the build-up time caused by the fluctuation in laser pulse energy. For pulse energies of 14 nJ or higher, our results showed that the optical breakdown in BK7 glass started right at the first pump pulse. For pulse energy lower than 14 nJ, we found that the optical breakdown was generated by the accumulated action of many pulses. The number of pulses required to induce optical breakdown showed a clear dependency on the pulse energy. For pulse energies close to the threshold, it could take many pulses to start optical breakdown; perhaps, sub-threshold laser pulses prior to optical breakdown modified the BK7 glass at the focal region and made it more absorptive. This behavior also indicated that the sample in the focal region did not return to the original state after it was interacted with a laser pulse.

 

Fig. 5. Time to optical breakdown as a function of pulse energy for BK7 glass. Each data point is an average of ten data acquisitions.

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3.2 The instability of the optical breakdown

Figures 3 and 4 showed the beginning of the optical breakdown at various pulse energies. We studied the instability of the optical breakdown by keeping the energy of the pump pulses low and letting the optical switch open for roughly 200 μs. Figure 6 shows powers of the broadband light emitted by the plasma, the transmitted pump pulses, and back reflected pump pulses with the optical switch open for about 170 μs. The incident pulse energy was 6.7 nJ, and the data was collected in a single opening of the optical switch. It took about 4 μs for the optical breakdown to start. The power of the broadband light emitted by the plasma increased for about 15 μs but then decreased, and the transmission of the pump pulses increased. At the beginning of the optical breakdown, a dense plasma was formed in the focal region, and it strongly absorbed the energy of the pump pulses. Because of the thermal diffusion process, this dense plasma region expanded and could become larger than the focal region. The density of the free electron gas of this dense plasma region increased, which lead to the decrease of the refractive index of this region [26], then it defocused the pump pulses [17, 27]. As the dense plasma region became larger than the focal region, the intensity of the pump pulses in the focal region became lower than before because of the absorption, defocusing, and shielding generated by the free electron gas in the plasma region. The focal region also lost energy due to the thermal diffusion process. As a result of these effects, the optical breakdown might not have been sustained in the focal region. Thus, the pump pulses in the focal region could not sustain the optical breakdown, and it gradually decayed and extinguished. However, the optical breakdown came back right away with a drastic reduction in the transmission of the pump pulses (Fig. 6(b)) and the appearance of the back-reflected pump pulses (Fig. 6(c)). The optical breakdown went through this cycle three times in the 170 μs opening of the optical switch. As the optical breakdown extinguished and started for the third time, only a small amount of transmitted and back reflected pulses appeared, indicating that during the time that the optical breakdown extinguished and came back, the plasma in the focal region and its surrounding still survived and absorbed energy of the pump pulses.

 

Fig. 6. Powers of (a) the broadband light emitted by plasma, (b) the transmitted pump pulses, and (c) the reflected pump pulses, where the optical switch was open for about 170 μs. The incident pulse energy was 6.7 nJ.

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We also collected the data above for 15-nJ pump pulses (Fig. 7), which was above the single-shot threshold (14 nJ). We saw a small instability in the optical breakdown at 50 μs after its start. The time of this instability was about the same as the first instability when the pulse energy was low (Fig. 6(a)). After the instability, the optical breakdown became stable as shown in Fig. 7. In this case, after the pump beam was blocked, the fluorescence of the plasma decayed for about 15 μsec before we started to see a change in the transmission of the probe beam. This showed that it took 15 μsec for the focal region to cool, and it allowed the probe beam to be transmitted through. This implied that the size of the plasma region was much larger than the focal region.

 

Fig. 7. Powers of (a) the broadband light emitted by plasma, (b) the transmitted pump pulses, and (c) the transmitted probe beam. The incident pulse energy was 15 nJ.

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At the beginning of the optical breakdown process, the dense plasma region occupied only a small portion of the focal region, and this dense plasma strongly absorbed the energy of the pump pulses. As a result, the dense plasma not only grew in volume, it also heated up the ionic background in the focal region and its surroundings. Figure 8(a) shows an image of a modified region when the pulse energy was 8 nJ. With the optical switch open for 1.8 μs, the build-up time to optical breakdown was about 280 ns, and the optical breakdown lasted for about 1.5 μs. This image was taken by positive phase contrast microscopy. The image consisted of a dark ring surrounding a very light circular region. In images produced by positive phase contrast microscopy, the darker color represented higher refractive index; thus, the refractive index of the dark annular region was higher than that of the central region and its surroundings. The diameter of the modified region was about 4.5 μm and should be larger than the size of the focal region. During the optical breakdown, the focal region and its surrounding heated up and melted [3], having its volume expand and push the material outward. Consequently, the area surrounding the melt region had higher density than the average density of the sample. When the pump beam was blocked, the surroundings of the melt region cooled first, and then the central melt region cooled. As a result, the surrounding region still had a higher density and the central region had a lower density. This process introduced stress in the modified region, and possibly it became birefringent [28]. Thus, the refractive index of the modified region changed non-uniformly because of the thermal process introduced by the optical breakdown.

 

Fig. 8. The positive phase contrast image of the modified region. The incident pulse energy was 8nJ; the optical breakdown lasted for 1.5 μs (a) and 847 μs (b).

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We also used positive phase contrast microscopy to observe and measure the size of the modified region by optical breakdown of the same pulse energy (8 nJ) at four different durations of optical breakdown: 44 μs, 73 μs, 277 μs, and 847 μs. The corresponding diameters of the modified regions were 9.8 μm, 11.1 μm, 18.3 μm, and 18.3 μm, respectively. The diameters of the modified regions were the same for the last two time durations of optical breakdown, and they were much bigger than the focal region with the modified region reaching its maximum size at this pulse energy (8 nJ). This was evidence that the material inside this region was modified by the thermal effect. Figure 8(b) shows the positive phase contrast image of the modified region when the optical breakdown lasted for 847 μs with 8 nJ pulse energy. From this image, we saw the structure of the modified region more clearly than that of Fig. 8(a). It consisted of three regions: a very light color circular center region and two darker annular regions with the outermost region being darkest. Thus, the refractive index of the outermost region was highest, and the circular central region was lowest. We also saw a sharp change in the refractive index between regions. The structures of the modified regions were the same of all of these cases; we did not see any modulation in the size of the modified region correlating to the oscillation of the optical breakdown at this pulse energy. Thus, the modification in the sample created by the thermal effect was irreversible. Figure 9 shows the diameters of the modified regions as a function of the optical breakdown time.

 

Fig. 9. Diameters of the modified regions at five different time durations of the optical breakdown. The pulse energy was 8 nJ. It showed that the modified region reached its maximum size of 18.3 μm after the optical breakdown lasted for 277 μs.

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We also observed and measured the modified regions by optical breakdown at the pulse energy of 15 nJ, which was above the single pulse damage threshold (14 nJ) at four different time durations of optical breakdown: 1.4 μs, 11 μs, 43 μs, and 105μs The diameters of the modified regions were 5.5 μm, 11.7 μm, 14.6 μm, and 19.8 μm, respectively. The structure of the modified regions for the breakdown times of 1.4 μs, 11 μs, and 43 μs was similar to that of the 8 nJ cases. However, when the optical switch opened for 105 μs, there was a darker area at the center of the modified region – seen clearer in the image of the modified region with the optical switch opened for 15 seconds, which is shown in Fig. 10. The diameter of the modified region was 62.5 μm, and it stayed the same as we exposed the sample to the pump pulses for longer dwell times. We could see a dark area and a ring in the center of the modified region. The dimension of the modified region was a lot bigger than the focusing region; this fact indicated that the origin of the material modification was due to thermal diffusion [11].

 

Fig. 10. The positive phase contrast image of the modified region, where the pump pulse energy was 15 nJ and the optical switch was opened for 15 seconds.

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3.3 The spectrum of the broadband light emitted by the optical breakdown process

Figure 11 shows the emission spectrum from optical breakdown in BK7 glass. We set the pump laser beam for this work to be linearly polarized because it was easier for us to study the relation between the polarization of the fluorescence emitted by the optical breakdown and the pump laser beam. The peak at 800 nm was due to the scattering of incident laser pulses from the optical breakdown region and was linearly polarized. However, the remainder of the spectrum spanned a broad wavelength range. This broadband light was shown in Fig. 3(a). It consisted of two components: the fast rise and fast decay after every pump pulse and a CW background which is the dominant contribution to the broadband light. We also found that the broadband light was unpolarized and omni-directional. The CW and omni-directional emission confirmed that super-continuum generation did not contribute to the spectrum, since the power of the pump pulses was about 150 kW – well below the critical power for self focusing in BK7 glass, which was in the order of megawatts. Emission spectra at different pulse energies were qualitatively the same shape aside from an overall intensity change. This spectrum looked like that of a quasi-black body, having its peak wavelength at about 700 nm. We used Wien’s displacement law to find that the black body temperature was about 4000 ⁰K and this temperature should be attributed to the temperature of the ionic background. This information indicated that the modified regions shown in Figs. 8 and 10 went through a thermal process.

 

Fig. 11. Broadband emission spectrum from optical breakdown in BK7 glass generated by 9.4 nJ pulses.

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4. Optical absorption calculations

As the 25-MHz train of femto-second laser pulses interacted with the glass target, our optical measurements showed several characteristic features. On a few-microsecond time scale (Figs. 3 and 4), the transmission of the pump and probe beams started high and decreased, plasma emission increased, and reflection of the pump beam at first increased and then decreased. On a longer time scale (Figs. 6), the optical signals displayed slow variations suggestive of relaxation oscillations. In order to investigate these characteristic features, we performed a series of calculations of the interaction of an electromagnetic (EM) wave with plasma. These calculations addressed the transmission, absorption, and reflection of the EM wave from a prescribed electron-density profile. The calculations were one-dimensional and did not treat the evolution of the plasma self consistently. Consequently, the calculations were in no regard comprehensive. Nevertheless, the results were enlightening and demonstrated a plausible scenario for the plasma dynamics.

We modeled the plasma electrons as being a classical gas of free electrons in a background of immobile neutralizing ions and immobile neutral atoms. We assumed that the electron-ion plasma was quasi-neutral and that the ionization fraction was small. The spatially varying electron density corresponds to a spatially varying plasma frequency ${\omega }_p=\sqrt{n_e}{e}^2/m_e\epsilon$, where n_e is the electron density, m_e is the electron mass, and ε is the permittivity of the medium. (The plasma frequency is the natural oscillation frequency of a perturbed electron caused by the restoring electrostatic field.) An EM wave of radian frequency ω in plasma will drive oscillations in the electrons. Collisions between the electrons and background atoms at frequency v will damp the electron oscillations and also damp the EM wave. In the absence of collisions (v = 0), the wave will propagate if ω > ω_p and will be evanescent (“cutoff”) if ω Ã < _p. With collisions, that sharp boundary at ω = ω_p becomes blurred.

Following standard electromagnetic theory and using complex phasor notation, one may write a one-dimensional equation for the transverse component of electric field of a TEM wave propagating in the ẑ direction as

This equation can be written in terms of dimensionless quantities, i.e. the dimensionless electron density ω ² _p/ω ², the dimensionless collision frequency v/ω, and the dimensionless characteristic scale length ωz/c = 2πz/λ. The term in brackets in Eq. (1) is the propagation constant for the wave.

We solved Eq. (1) numerically for a range of conditions intended to represent the general features of our optical-breakdown experiment. Within the constraints of our model, we find that the electron density must reach approximately the “critical” electron density (where ω/(ω _p =1) in order for significant reflection of the incident wave to occur. At the experimental laser wavelength (800 nm), ω = 2.4 × 10¹⁵/s. The critical electron density for this wavelength is 2.6 ×10²¹/cm³. If it required 4.5 eV of energy to generate one free electron, then a single 10 nJ laser pulse could generate 1.4 × 10¹⁰ electrons. This many electrons would fill a cube 2.1 wavelengths on a side to the critical density. As a result of this order-of-magnitude estimate, as well as the numerical aperture of the final focusing lens, the extent of the plasma region was expected to be typically a small multiple of the laser wavelength (0.8 μm).

The electron-neutral collision frequency v was taken to be approximately 4 × 10¹⁴/s based on the work of Fischetti and DiMaria [28], and, consequently, we used v/ω ≅ 0.2 for all our calculations. This value, and other numerical values in our calculation, may not be precise. The reader should pay most attention to the qualitative features of the results.

 

Fig. 12. Schematic representation of an electromagnetic wave interacts with plasma. The solid curve is the normalized, time-averaged electric field magnitude. The dashed curve is the normalized electron density, which is assumed parabolic in shape, two wavelengths in extent at the base. The standing-wave oscillations in field at the left indicate the presence of forward-going and reflected EM waves.

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Figure 12 describes schematically the computational geometry and the EM wave behavior. The electron plasma density (dashed line) was assumed to have a parabolic shape with variable height and width. In Fig. 12, the width shown is 2λ at the base. The choice of shape was somewhat arbitrary; the only important point is that the density varied smoothly in space and did not have discontinuous changes. Any other smoothly varying plasma shape would produce qualitatively similar results. The EM wave (solid line) was incident from the left. Some of the wave reflected from the plasma due to the spatially dependent index of refraction caused by the electrons. That reflection caused the standing waves in intensity on the left. Inside the plasma, the wave was partially absorbed by collisional damping. The field that was not absorbed was transmitted to the right. We identify the computed reflection and transmission of the EM wave with the experimental measurements of the reflection of the pump beam and the transmission of both the pump and probe beams. We identify the absorption of the EM wave with the plasma emission measured in the experiment.

 

Fig. 13. Computed laser power transmission (top), reflection (middle) and absorption (bottom) for a range of plasma sizes and densities. For all cases the plasma was parabolic in shape with v/ω = 0.2. The dots in the center graph show a possible trajectory for the growth of the plasma electron density and plasma size with repeated pulsing of the laser. The dashed line shows a possible trajectory for the relaxation of the plasma once it grows too big for the laser power to support.

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Figure 13 shows the computed power transmission, reflection, and absorption of the EM wave for a range of electron densities and plasma thicknesses. These results were obtained by numerical integration of Eq. (1). The series of dots in the middle graph in Fig. 13 represents a possible trajectory in plasma density and thickness for a sequence of laser pulses. The nearby arrow indicates the progression of the plasma with successive pulses. The first pulse, represented by the lowest point, indicates the generation of a small, low-density “hot spot” of plasma. With successive pulses, the plasma density increases towards, and possibly somewhat exceeding, the critical density. Simultaneously, the size of the plasma grows. The indicated trajectory corresponds to the experimental results: it shows increasing absorption of the pump beam (increasing plasma emission), decreasing transmission of the pump and probe beams, and increasing and subsequent decreasing reflection of the pump beam.

As successive laser pulses added energy to the plasma, one might expect that, eventually, a steady state would have been reached. Experimentally, a steady state was not observed at low pump energy. We speculate that the lack of steady state and apparent relaxation oscillations (Fig. 6) may have been caused by the spatial growth of the plasma. As the plasma grew larger, the spatial gradients became smaller such that the laser energy was absorbed over a larger region. At some point, the specific deposition of laser energy was insufficient to balance electron recombination losses and the plasma relaxed over a large volume. Alternatively, the plasma may have grown in the direction of the incident laser beam into a region where the focus was larger, with a similar result. In Fig. 13, a possible trajectory for plasma relaxation is indicated by the dashed line. The associated arrow indicates plasma relaxation first in density, then in size. This trajectory follows a path of increasing transmission, reduced absorption, but minimal reflection, as shown in Fig. 6.

As stated above, these calculations are merely suggestive of the plasma dynamics because they are one-dimensional and do not treat the plasma self-consistently. In spite of these limitations, the overall behavior of our simple wave-propagation analysis is generally consistent with the experimental results.

5. Conclusion

We used the output of an extended-cavity femtosecond laser to investigate the optical breakdown started by single laser pulse and by the accumulated action of many laser pulses. From these two cases, by inspecting the recordings of the broadband light emitted by the plasma, the transmitted and reflected pump beams and also from the transmitted probe beam, we learned the evolution of the critical region. We showed the broadband light emitted from the optical breakdown consists of two components: The fast rise and fast decay after each pump pulse and a CW background, we speculated the fast component is due to the recombination of free electrons and positive ions, and a CW component from a quasi-black body of the ionic background generated by the excess energy released from the recombination. We also studied the spectrum of the broadband light emitted from the optical breakdown, it was of a quasi black body, and we learned the bulk damage morphology in the femtosecond regime and the instability of the optical breakdown. We presented a model of electromagnetic wave propagation in plasma that, while not conclusive, suggests behavior consistent with the major experimental observations.