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Spectrally modulated second harmonic emission from laser plasma filaments
Optics Communications North-Holland
106 ( 1994) 52-58
OPTICS
COMMUNICATIONS
Spectrally modulated second harmonic emission from laser plasma filaments D. Giulietti l, V. Biancalana, M. Borghesi, P. Chessa, A. Giulietti and E. Schifano 2 Istituto di Fisica Atomica e Molecolare, via de1 Giardino 7, 56127 Piss, Italy Received 6 October
1993
Spectra of forward emitted second harmonic light from laser interaction with filamentary plasmas have been experimentally studied. Rather regular modulations in the frequency domain have been observed into overall red-shifted spectra. The observed spectral features are consistent with self-phase-modulation of the intense laser light in growing filaments. A model accounts for this effect.
1. Introduction
The second harmonic (SH) of the laser frequency can be generated during interaction with inhomogeneous media [ 1 ] including laser produced plasmas. Detectable SH generation has been proved to occur also from interaction with plasmas well below the critical density, specially in condition of enhanced plasma density gradients and laser intensity gradients due to filamentation instability [ 2,3]. There is still some controversy on how accurate SH can be as a diagnostic of that instability [ 41. Even though imaging of the interaction region in the light of emitted SH is a very promising method, time resolved spectroscopy provides important information on the physics involved. In fact, time resolved spectra of SH, emitted at 90 degrees to the laser beam, evidenced the occurrence of sum frequency between the laser light and the Brillouin back-scattered light into the plasma [ 51, as previously suggested [ 21. In this letter we present time resolved spectra of SH emitted forward in the direction of the laser beam. The plasma parameters and interaction condition were the same as for ref. [ 5 1, while the spectral dispersion was higher in this latter experiment. The spectra presented here show unexpected features which seems
1 Dipartimento *
52
di Fisica, Universita di Pisa, Italy. Laboratoire pour 1’Utilisation des Laser Intenses, Ecole Polytechnique, Palaiseau, France.
to be explainable only in terms of self-phase modulation (SPM) of the laser light [ 1 ] crossing the plasma in condition of strong filamentation. SPM can affect the spectral profile of the laser light [ 6,7] and consequently the one of the second harmonic. In previous works some anomalies in the spectra of light backscattered by stimulated Brillouin scattering [8] and by reflection at critical density [9] were attributed to the same effect. In the following we present some typical spectra and discuss their features in terms of SPM, introducing also a simple model accounting for the SPM effect in a Nament and allowing a numerical evaluation of the spectra.
2. The experiment The experimental set-up is substantially the same as we described in a recent paper [ 10 1. A Nd:YAG laser (1 = 1.064pm) delivering up to 3 J in 3 ns pulses was used. The laser operated in a single-transverse but multi-longitudinal mode. Consequently the laser pulse was modulated in time by spikes of measured mean duration of about 50 ps, as expected from mode beating in the oscillator cavity. The laser beam was focused by an f /8 optics onto a thin foil plastic target up to a nominal intensity of 5 x 1OL3W/cm*. Foil thickness, laying in the range 0.5 to 1.0 ,um, was chosen in order to allow the plasma to become well underdense during the laser pulse. In
0030-4018/94/$07.00
@ 1994-Elsevier
Science B.V. All rights reserved n”.X- r\n*r, rn10 ,nr\nnn,, Y
Volume 106, number 1,2,3
OPTICS COMMUNICATIONS
fact previous measurements on 3012 harmonic [ 111 showed that 1.0 pm plastic foil irradiated at 10” W/cm’ produces a plasma whose maximum electron density at the peak of the pulse is about a quarter of the critical density, in rough agreement with a 1D selfsimilar model [ 121 for exploding foils. The electron temperature was estimated from the shift of the side emitted second harmonic [ 5 ] to be of the order of 500 eV, also in agreement with 30 / 2 harmonic measurements [ 111. SH radiation was collected in a forward direction within an angle off 8 degrees. The plasma region was imaged on the entrance slit of a 1 m spectrometer. Magnification was kept low in order to avoid the spectrometer slit making spatial resolution. The resulting entrance angle gave illumination of the whole diffraction grating. In this way we obtained a good spectral resolution, while the time resolution was limited by the time integration due to the grating extent. Time resolution of our spectra was about 200 ps. The spectral resolution was mainly determined by the entrance slit aperture and was about 0.4 A. An absolute spectral calibration was performed by a 5320 A reference wavelength, obtained by doubling the laser frequency with a KDP crystal. In fig. 1 three time-resolved spectra of SH light emitted forward are shown. The spiky structure of the laser pulse, in the time domain, enhanced by the nonlinear process of second harmonic emission, is apparent. Because of the limit evaluated above, the detection system did not time-resolve into a single spike, whose apparent duration is the instrumental one. Figure 2 shows the densitometric traces from negative films versus wavelength for three spikes identified in fig. 1 marked 2a, 2b and 2c, respectively. All the spectra were found to be red-shifted a few angstroms, like those shown in fig. 1. A small number of spikes showed a slightly blue-shifted tail in their spectrum. The spectral broadening was different spike-to-spike up to a width of some 5 A. This width is definitely much larger than the measured laser line width of 0.7 A at the fundamental and 0.4 A at the second harmonic obtained with the KDP crystal. A very interesting feature is that, for a few laser shots, some spikes produced SH light affected by rather regular and deep spectral modulations. A typical modulation scale was of the order of 1 A with spike-to-spike variations. The envelope of the spectral intensity pro-
1 March 1994
tile has its maximum at different positions spike-tospike, as evidenced by densitometric traces of fig. 2.
3. Model and computation The observed spectral features of the second harmonic, namely red shift, broadening, modulations and position of spectral intensity maxima, can be explained in terms of SPM of the laser light interacting with growing filaments. In fact, in the same experiment, a clear transition of the laser spot to a cluster of few pm sized filaments was observed [ 10 1. SPM becomes sensible when an intense e.m. wave induces very fast changes in the refractive index of the medium where the wave itself propagates [6,7,13151. In this experiment such changes are possible in the small structures generated by filamentation instability, where the high laser field depletes the plasma density in a very short time. As a consequence, the instantaneous laser frequency is modified by SPM, consequently affecting the second harmonic frequency. In principle the same effect could affect also the SH light, but it can be easily seen that the SPM of this latter produces smaller changes in its spectrum. In the following a model is described which, with some crude approximations, shows how SPM works on the laser light in a given filament. A few numerical results from the model are given. Some input data for the model were taken from previous measurements in the same conditions [ 5,10,11]. Some other parameters (e.g. filament length and time history of the filament density) were not well known and could vary spike-to-spike. For these input parameters we took arbitrary but quite realistic values. As a matter of fact let
(1)
be the phase of a plane wave propagating along the z-axis in a medium of refractive index p dependent on z and t, with wave-vector ko. +,, (t) is the phase term depending on the refractive index, which is responsible for the observed effects. The instantaneous angular frequency is 53
Volume 106, number
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OPTICS COMMUNICATIONS
1 March 1994
Fig. 1. Three time-resolved spectra of forward emitted second harmonic. The spectra are generally red shifted. Rather regular modulations are evident in the frequency domain for some spikes. Densitometric traces of the spikes marked as 2a, 2b and 2c follow in fig. 2. The position of the laser pulse in time is roughly indicated.
54
Volume 106, number 1,2,3
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OPTICS COMMUNICATIONS
Fig. 1. Continued.
(2) Let us now consider the laser radiation passing through a filament of length L. Electron density inside the filament will decrease in time and correspondingly p will increase. We simply assume p to be uniform into the filament. In this case the difference from Q and laser frequency 09 results to be
In fig. 3 the thin curve represents the relative difference S 8 / 00 versus time for neodymium laser light as determined by using the electron density given by the thick curve to evaluate eq. (3). The red shift is due to the increase of refractive index. The electron density is assumed to vary from an initial value of nc/4 down to n,/40 in a time of 50 ps. The lower density is arbitrary, but the result is rather insensitive to that value, provided it is much smaller than the critical density. 50 ps is the typical duration of one spike in the laser emission and it is comparable with the transit time rr = R/us of a sound wave across a filament of radius R. In our experimental conditions v, x 2 x 10’ cm/s. The value of R was taken from the measurements of
the filament cross-section obtained in the same experimental conditions [IO]. In these conditions we expect that the density will decrease in the filament during the whole spike, thus giving a net red shift. The shift shown in fig. 3 refers to a 50 pm length filament. Both spectral shift and broadening of laser light can be roughly estimated from the range of values covered by 6Q. This estimation substantially agrees with the mean values of shift and broadening observed in experimental spectra of the second harmonic, either for spikes showing spectral modulations or not. The Fourier transform of the electric field provides a more rigorous calculation of shift, broadening, and other spectral parameters as the number of spectral intensity peaks and location of maximum of the envelope of the spectral intensity. To this purpose let us consider the Fourier transform of the laser electric field after propagation in the filament +oO E(w)
I&(t)
=
exp(i@)
exp(iwt)
dt
J --03 +oO = I --oo
&(t)
where f(t)
exp[if(t)l
dt,
= (o - oo)t + dr(t)
(4) and &(t)
is the 55
Volume 106, number 1,2,3 1000
OPTICS COMMUNICATIONS
‘r
1 March 1994
0.0
a
*0 tL
-3.5
x $=
-7.0
I
0
-I
20
40 time (psec)
6 SH shift (A)
80
Fig. 3. Time evolution of plasma density is represented by the thick curve (left axis reports density values relative to the critical one, for &J = 1.064pm). The thin curve describes the time behaviour of the relative shift of the instantaneous frequency of laser light.
b 2 3
60
800
electric field amplitude of the impinging laser radiation. From eq. (3) and fig. 3 we observe that each instant t.4 corresponds to another instant tB in which the radiation has the same pulsation D but different phase, as defined by eq. ( 1) Consequently, assuming a much slower time scale for E,J (t ) than for a, the Fourier transform can be approximated to [ 161
.
6
i
4
5
6
SH sh:ft (A)3
J?(W) = [2ki/f”(tA)]1’2Eo(t)
C
+
[2ni/f”(tB)]“‘EO(t)
=
VA
exp[if(tA)] exp[if(fB)l
exp[if(fA)l
+
UB
exp[if(tB)l,
where f ” (tA ) is the second timederivative Then the spectral density results @%)I2 +
=
luA12
+
(5)
of f (t ).
luB12
21UAll~Blcos[f(tB)
-
f(tA)l,
(6)
where the last term is the interference one and is responsible for the spectral modulation. The condition for a maximum of order m in the spectrum is
-22
-1
0
1 SH sift
4
5
6
(A:
Fig. 2. Densitometric traces of spikes marked as 2a, 2b and 2c in fig. 1. Five and three modulations are respectively evident in the first and in the other two red-shifted spectra. Intensity maxima are located at different spectral positions.
f(tB)
-
f(tA)
=
As far as tA and tB converge to thn (see fig. 3 1, m = 0. For tA and tg far apart from ZM,the number M of observable maxima is obtained M%
[b(tB)
-&(tA)hAX
2n 56
(7)
d!n.
=
&
2n ’
(8)
Volume 106, number 1,2,3
1 March 1994
OPTICS COMMUNICATIONS
that in terms of the parameters used in the model above reduces to M F;:(koL.l2~) M(t),
(9)
Ap (t ) being the maximum variation of the refractive index during the filament formation. With the same choice of input values as in fig. 3 we obtain a number of maxima of the order of 5, which is equal or close to the observed number of maxima in the SH spectra of many spikes.
We used a fast Fourier transform code to obtain numerically the spectra at the fundamental’ frequency. The choice of parameters was the same as above. We considered a spike having alternatively two slightly different positions (I ) and (II) in time with respect to the density history. The time profile of the two spikes is shown in fig. 4 by thin curves and the plasma density by a thick curve. In both cases modulated spectra were obtained (fig. 4 (I) and (II) ) as we observed experimentally. Of course the assumed decreasing density can only produce red shift from the original laser frequency. The blue tail occasionally observed in the experimental spectra must be due to light interacting marginally with the filament boundary, where the electron density may have a transient increase. Notice that the maximum of the spectral intensity is located at the red side of the spectrum in case (I) and at the opposite side in case (II). The simulation thus suggests that different positions of the intensity maxima observed in the spectra of the second harmonic (see fig. 1) correspond to different delays between the intensity peak of the spike and the time of the minimum density in the filament. Variations in this delay can be due to either different plasma conditions or to spikes of different intensity. It is relevant that, in our experimental spectra, modulations are clearly visible only in a limited number of spikes for some laser shots. This is due to the fact that usually more than one filament is produced by a single spike: in the 100 ps framed images [ lo] we observed up to some 5-6 filaments per spike. Although they were local&d in a small region in the centre of the laser spot and seemed to have comparable intensity, even small differences in the initial density could produce different modulations, then washing out the spectrum. Only in a few cases filaments have almost the same evolution and sum “coherently” in a deeply modulated spectrum.
0
h
4
20
40 time (ps)
60
80
12
c
.$ 8.0 2 2 4.0
-2
0
2 %&A)
6
8
10
Fig. 4. A laser spike is represented with two different time positions (thin lines) with respect to the density time profile (thick line) . The two positions are labelled as ‘I’ and ‘IP. Below the results of FFT calculations are then reported showing modulated spectra of laser light whose maxima are either away from the laser wavelength (case T) or close to it (case ‘IT’).
57
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1,2,3
OPTICS COMMUNICATIONS
4. Conclusion In conclusion
the spectra of the second harmonic light emitted forward show considerable broadening and red-shift, if compared to SH obtained by an ordinary frequency-doubling crystal. Furthermore some spikes of SH emission show clear spectral modulation, a novel evidence in this kind of spectra. We show that these features are consistent with self-phasemodulation of the laser light in filaments growing into the plasma. Simple calculations already support such a conclusion, and computation by FFT code, based on simplified spike profiles and density evolution in the filaments, gives spectra closely looking like the experimental ones. The number of relative maxima and the position of the absolute maximum in the spectrum of a single spike have been shown to be related to the parameters of the filamentary structures and to the timing of their evolution with respect to the laser light peaks. As a consequence this kind of spectroscopy is rather promising in order to characterise filamentary regions of laser-interacting plasma.
Acknowledgements The authors are grateful to F. Bianconi, I. Deha and L.A. Gizzi for relevant scientific contributions and valuable suggestions. The work of the IFAM technical team was determinant to the success of our measure ents The research programme on laser-plasma f” *. interaction is fully supported by Cons&ho Nazionale delle Ricerche, Italy.
58
1 March 1994
References [ 1] Y.R. Shen, The principles of nonlinear optics (Wiley, New York, 1984) Ch. 1, 17, 28 and references therein.
[ 21 J.A. Stamper, R.H. Lehmberg, A. Schmitt, M.J. Herbst, F.C. Youn& J.H. Gardner and S.P. Obenschain, Phys. Fluids 28 (1985) 2563 [3] D. Batani, F. Bianconi, A. Giulietti, D. Giulietti and L. Nocera, Optics Comm. 70 (1989) 38. [4] P.E. Young, H.A. Baldis, T.W. Johnston, W.L. Kruer and K.G. Estabrook, Phys. Rev. Lett. 63 (1989) 2812. [5] A. Giulietti, D. Giulietti, D. Batani, V. Biancalana, L. Gizzi, L. Nocera and E. Schifano, Phys. Rev. Lett. 63 (1989) 524. [6] C. Yamanaka, T. Yamanaka, J. Mizui and N. Yamaguchi, Phys. Rev. A 11 (1975) 2138. [ 71 C. Labaune, E. Fabre, A. Michard and F. Briand, Phys. Rev. A 32 (1984) 577 [8] T. Afshar-Rad, S. Coe, A. Giulietti, D. Giulietti and 0. Willi, Europh. L&t. 15 (1991) 745. [9] J.A. Tarvin and R.J. Schroeder, Phys. Rev. Lett. 47 (1981) 341. [lo] V. Biancalana, M. Borghesi, P. Chessa, I. Deha, A. Giulietti, D. Giulietti, L.A. Gizzi and 0. Willi, Europhys. Lett. 22 (1993) 175. [ 111 D. Giulietti, V. Biancalana, D. Batani, A. Giulietti, L. Gizzi, L. Nocera and E. Schifano, 11Nuovo Cimento 13 D (1991) 845. [ 121 R.A. London and M.D. Rosen, Phys. Fluids 29 ( 1986) 3813. [13] F. Shimizu, Phys. Rev. Lett. 19 (1967) 1097. [ 141 T.K. Gustafson, J.P. Taran, H.A. Haus, J.R. Lifsitz and P.L. Kelley, Phys. Rev. 177 (1969) 306. [ 151 J.L. Bother, J.C. Griesemann, M. Louis-Jacquet and M. Decroisette, Optics Comm. 16 (1976) 262. [ 161 J.D. Jackson, Classical electrodynamics, 2nd Ed. (Wiley, New York, 1975).
106 ( 1994) 52-58
OPTICS
COMMUNICATIONS
Spectrally modulated second harmonic emission from laser plasma filaments D. Giulietti l, V. Biancalana, M. Borghesi, P. Chessa, A. Giulietti and E. Schifano 2 Istituto di Fisica Atomica e Molecolare, via de1 Giardino 7, 56127 Piss, Italy Received 6 October
1993
Spectra of forward emitted second harmonic light from laser interaction with filamentary plasmas have been experimentally studied. Rather regular modulations in the frequency domain have been observed into overall red-shifted spectra. The observed spectral features are consistent with self-phase-modulation of the intense laser light in growing filaments. A model accounts for this effect.
1. Introduction
The second harmonic (SH) of the laser frequency can be generated during interaction with inhomogeneous media [ 1 ] including laser produced plasmas. Detectable SH generation has been proved to occur also from interaction with plasmas well below the critical density, specially in condition of enhanced plasma density gradients and laser intensity gradients due to filamentation instability [ 2,3]. There is still some controversy on how accurate SH can be as a diagnostic of that instability [ 41. Even though imaging of the interaction region in the light of emitted SH is a very promising method, time resolved spectroscopy provides important information on the physics involved. In fact, time resolved spectra of SH, emitted at 90 degrees to the laser beam, evidenced the occurrence of sum frequency between the laser light and the Brillouin back-scattered light into the plasma [ 51, as previously suggested [ 21. In this letter we present time resolved spectra of SH emitted forward in the direction of the laser beam. The plasma parameters and interaction condition were the same as for ref. [ 5 1, while the spectral dispersion was higher in this latter experiment. The spectra presented here show unexpected features which seems
1 Dipartimento *
52
di Fisica, Universita di Pisa, Italy. Laboratoire pour 1’Utilisation des Laser Intenses, Ecole Polytechnique, Palaiseau, France.
to be explainable only in terms of self-phase modulation (SPM) of the laser light [ 1 ] crossing the plasma in condition of strong filamentation. SPM can affect the spectral profile of the laser light [ 6,7] and consequently the one of the second harmonic. In previous works some anomalies in the spectra of light backscattered by stimulated Brillouin scattering [8] and by reflection at critical density [9] were attributed to the same effect. In the following we present some typical spectra and discuss their features in terms of SPM, introducing also a simple model accounting for the SPM effect in a Nament and allowing a numerical evaluation of the spectra.
2. The experiment The experimental set-up is substantially the same as we described in a recent paper [ 10 1. A Nd:YAG laser (1 = 1.064pm) delivering up to 3 J in 3 ns pulses was used. The laser operated in a single-transverse but multi-longitudinal mode. Consequently the laser pulse was modulated in time by spikes of measured mean duration of about 50 ps, as expected from mode beating in the oscillator cavity. The laser beam was focused by an f /8 optics onto a thin foil plastic target up to a nominal intensity of 5 x 1OL3W/cm*. Foil thickness, laying in the range 0.5 to 1.0 ,um, was chosen in order to allow the plasma to become well underdense during the laser pulse. In
0030-4018/94/$07.00
@ 1994-Elsevier
Science B.V. All rights reserved n”.X- r\n*r, rn10 ,nr\nnn,, Y
Volume 106, number 1,2,3
OPTICS COMMUNICATIONS
fact previous measurements on 3012 harmonic [ 111 showed that 1.0 pm plastic foil irradiated at 10” W/cm’ produces a plasma whose maximum electron density at the peak of the pulse is about a quarter of the critical density, in rough agreement with a 1D selfsimilar model [ 121 for exploding foils. The electron temperature was estimated from the shift of the side emitted second harmonic [ 5 ] to be of the order of 500 eV, also in agreement with 30 / 2 harmonic measurements [ 111. SH radiation was collected in a forward direction within an angle off 8 degrees. The plasma region was imaged on the entrance slit of a 1 m spectrometer. Magnification was kept low in order to avoid the spectrometer slit making spatial resolution. The resulting entrance angle gave illumination of the whole diffraction grating. In this way we obtained a good spectral resolution, while the time resolution was limited by the time integration due to the grating extent. Time resolution of our spectra was about 200 ps. The spectral resolution was mainly determined by the entrance slit aperture and was about 0.4 A. An absolute spectral calibration was performed by a 5320 A reference wavelength, obtained by doubling the laser frequency with a KDP crystal. In fig. 1 three time-resolved spectra of SH light emitted forward are shown. The spiky structure of the laser pulse, in the time domain, enhanced by the nonlinear process of second harmonic emission, is apparent. Because of the limit evaluated above, the detection system did not time-resolve into a single spike, whose apparent duration is the instrumental one. Figure 2 shows the densitometric traces from negative films versus wavelength for three spikes identified in fig. 1 marked 2a, 2b and 2c, respectively. All the spectra were found to be red-shifted a few angstroms, like those shown in fig. 1. A small number of spikes showed a slightly blue-shifted tail in their spectrum. The spectral broadening was different spike-to-spike up to a width of some 5 A. This width is definitely much larger than the measured laser line width of 0.7 A at the fundamental and 0.4 A at the second harmonic obtained with the KDP crystal. A very interesting feature is that, for a few laser shots, some spikes produced SH light affected by rather regular and deep spectral modulations. A typical modulation scale was of the order of 1 A with spike-to-spike variations. The envelope of the spectral intensity pro-
1 March 1994
tile has its maximum at different positions spike-tospike, as evidenced by densitometric traces of fig. 2.
3. Model and computation The observed spectral features of the second harmonic, namely red shift, broadening, modulations and position of spectral intensity maxima, can be explained in terms of SPM of the laser light interacting with growing filaments. In fact, in the same experiment, a clear transition of the laser spot to a cluster of few pm sized filaments was observed [ 10 1. SPM becomes sensible when an intense e.m. wave induces very fast changes in the refractive index of the medium where the wave itself propagates [6,7,13151. In this experiment such changes are possible in the small structures generated by filamentation instability, where the high laser field depletes the plasma density in a very short time. As a consequence, the instantaneous laser frequency is modified by SPM, consequently affecting the second harmonic frequency. In principle the same effect could affect also the SH light, but it can be easily seen that the SPM of this latter produces smaller changes in its spectrum. In the following a model is described which, with some crude approximations, shows how SPM works on the laser light in a given filament. A few numerical results from the model are given. Some input data for the model were taken from previous measurements in the same conditions [ 5,10,11]. Some other parameters (e.g. filament length and time history of the filament density) were not well known and could vary spike-to-spike. For these input parameters we took arbitrary but quite realistic values. As a matter of fact let
(1)
be the phase of a plane wave propagating along the z-axis in a medium of refractive index p dependent on z and t, with wave-vector ko. +,, (t) is the phase term depending on the refractive index, which is responsible for the observed effects. The instantaneous angular frequency is 53
Volume 106, number
L&3
OPTICS COMMUNICATIONS
1 March 1994
Fig. 1. Three time-resolved spectra of forward emitted second harmonic. The spectra are generally red shifted. Rather regular modulations are evident in the frequency domain for some spikes. Densitometric traces of the spikes marked as 2a, 2b and 2c follow in fig. 2. The position of the laser pulse in time is roughly indicated.
54
Volume 106, number 1,2,3
1 March 1994
OPTICS COMMUNICATIONS
Fig. 1. Continued.
(2) Let us now consider the laser radiation passing through a filament of length L. Electron density inside the filament will decrease in time and correspondingly p will increase. We simply assume p to be uniform into the filament. In this case the difference from Q and laser frequency 09 results to be
In fig. 3 the thin curve represents the relative difference S 8 / 00 versus time for neodymium laser light as determined by using the electron density given by the thick curve to evaluate eq. (3). The red shift is due to the increase of refractive index. The electron density is assumed to vary from an initial value of nc/4 down to n,/40 in a time of 50 ps. The lower density is arbitrary, but the result is rather insensitive to that value, provided it is much smaller than the critical density. 50 ps is the typical duration of one spike in the laser emission and it is comparable with the transit time rr = R/us of a sound wave across a filament of radius R. In our experimental conditions v, x 2 x 10’ cm/s. The value of R was taken from the measurements of
the filament cross-section obtained in the same experimental conditions [IO]. In these conditions we expect that the density will decrease in the filament during the whole spike, thus giving a net red shift. The shift shown in fig. 3 refers to a 50 pm length filament. Both spectral shift and broadening of laser light can be roughly estimated from the range of values covered by 6Q. This estimation substantially agrees with the mean values of shift and broadening observed in experimental spectra of the second harmonic, either for spikes showing spectral modulations or not. The Fourier transform of the electric field provides a more rigorous calculation of shift, broadening, and other spectral parameters as the number of spectral intensity peaks and location of maximum of the envelope of the spectral intensity. To this purpose let us consider the Fourier transform of the laser electric field after propagation in the filament +oO E(w)
I&(t)
=
exp(i@)
exp(iwt)
dt
J --03 +oO = I --oo
&(t)
where f(t)
exp[if(t)l
dt,
= (o - oo)t + dr(t)
(4) and &(t)
is the 55
Volume 106, number 1,2,3 1000
OPTICS COMMUNICATIONS
‘r
1 March 1994
0.0
a
*0 tL
-3.5
x $=
-7.0
I
0
-I
20
40 time (psec)
6 SH shift (A)
80
Fig. 3. Time evolution of plasma density is represented by the thick curve (left axis reports density values relative to the critical one, for &J = 1.064pm). The thin curve describes the time behaviour of the relative shift of the instantaneous frequency of laser light.
b 2 3
60
800
electric field amplitude of the impinging laser radiation. From eq. (3) and fig. 3 we observe that each instant t.4 corresponds to another instant tB in which the radiation has the same pulsation D but different phase, as defined by eq. ( 1) Consequently, assuming a much slower time scale for E,J (t ) than for a, the Fourier transform can be approximated to [ 161
.
6
i
4
5
6
SH sh:ft (A)3
J?(W) = [2ki/f”(tA)]1’2Eo(t)
C
+
[2ni/f”(tB)]“‘EO(t)
=
VA
exp[if(tA)] exp[if(fB)l
exp[if(fA)l
+
UB
exp[if(tB)l,
where f ” (tA ) is the second timederivative Then the spectral density results @%)I2 +
=
luA12
+
(5)
of f (t ).
luB12
21UAll~Blcos[f(tB)
-
f(tA)l,
(6)
where the last term is the interference one and is responsible for the spectral modulation. The condition for a maximum of order m in the spectrum is
-22
-1
0
1 SH sift
4
5
6
(A:
Fig. 2. Densitometric traces of spikes marked as 2a, 2b and 2c in fig. 1. Five and three modulations are respectively evident in the first and in the other two red-shifted spectra. Intensity maxima are located at different spectral positions.
f(tB)
-
f(tA)
=
As far as tA and tB converge to thn (see fig. 3 1, m = 0. For tA and tg far apart from ZM,the number M of observable maxima is obtained M%
[b(tB)
-&(tA)hAX
2n 56
(7)
d!n.
=
&
2n ’
(8)
Volume 106, number 1,2,3
1 March 1994
OPTICS COMMUNICATIONS
that in terms of the parameters used in the model above reduces to M F;:(koL.l2~) M(t),
(9)
Ap (t ) being the maximum variation of the refractive index during the filament formation. With the same choice of input values as in fig. 3 we obtain a number of maxima of the order of 5, which is equal or close to the observed number of maxima in the SH spectra of many spikes.
We used a fast Fourier transform code to obtain numerically the spectra at the fundamental’ frequency. The choice of parameters was the same as above. We considered a spike having alternatively two slightly different positions (I ) and (II) in time with respect to the density history. The time profile of the two spikes is shown in fig. 4 by thin curves and the plasma density by a thick curve. In both cases modulated spectra were obtained (fig. 4 (I) and (II) ) as we observed experimentally. Of course the assumed decreasing density can only produce red shift from the original laser frequency. The blue tail occasionally observed in the experimental spectra must be due to light interacting marginally with the filament boundary, where the electron density may have a transient increase. Notice that the maximum of the spectral intensity is located at the red side of the spectrum in case (I) and at the opposite side in case (II). The simulation thus suggests that different positions of the intensity maxima observed in the spectra of the second harmonic (see fig. 1) correspond to different delays between the intensity peak of the spike and the time of the minimum density in the filament. Variations in this delay can be due to either different plasma conditions or to spikes of different intensity. It is relevant that, in our experimental spectra, modulations are clearly visible only in a limited number of spikes for some laser shots. This is due to the fact that usually more than one filament is produced by a single spike: in the 100 ps framed images [ lo] we observed up to some 5-6 filaments per spike. Although they were local&d in a small region in the centre of the laser spot and seemed to have comparable intensity, even small differences in the initial density could produce different modulations, then washing out the spectrum. Only in a few cases filaments have almost the same evolution and sum “coherently” in a deeply modulated spectrum.
0
h
4
20
40 time (ps)
60
80
12
c
.$ 8.0 2 2 4.0
-2
0
2 %&A)
6
8
10
Fig. 4. A laser spike is represented with two different time positions (thin lines) with respect to the density time profile (thick line) . The two positions are labelled as ‘I’ and ‘IP. Below the results of FFT calculations are then reported showing modulated spectra of laser light whose maxima are either away from the laser wavelength (case T) or close to it (case ‘IT’).
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Volume 106, number
1,2,3
OPTICS COMMUNICATIONS
4. Conclusion In conclusion
the spectra of the second harmonic light emitted forward show considerable broadening and red-shift, if compared to SH obtained by an ordinary frequency-doubling crystal. Furthermore some spikes of SH emission show clear spectral modulation, a novel evidence in this kind of spectra. We show that these features are consistent with self-phasemodulation of the laser light in filaments growing into the plasma. Simple calculations already support such a conclusion, and computation by FFT code, based on simplified spike profiles and density evolution in the filaments, gives spectra closely looking like the experimental ones. The number of relative maxima and the position of the absolute maximum in the spectrum of a single spike have been shown to be related to the parameters of the filamentary structures and to the timing of their evolution with respect to the laser light peaks. As a consequence this kind of spectroscopy is rather promising in order to characterise filamentary regions of laser-interacting plasma.
Acknowledgements The authors are grateful to F. Bianconi, I. Deha and L.A. Gizzi for relevant scientific contributions and valuable suggestions. The work of the IFAM technical team was determinant to the success of our measure ents The research programme on laser-plasma f” *. interaction is fully supported by Cons&ho Nazionale delle Ricerche, Italy.
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1 March 1994
References [ 1] Y.R. Shen, The principles of nonlinear optics (Wiley, New York, 1984) Ch. 1, 17, 28 and references therein.
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