Appl. Phys. B 80, 285–289 (2005)
Applied Physics B
DOI: 10.1007/s00340-004-1721-1
Lasers and Optics
c. vozzi m. nisoliu g. sansone s. stagira s. de silvestri
Optimal spectral broadening in hollow-fiber compressor systems National Laboratory for Ultrafast and Ultraintense Optical Science - INFM, Dipartimento di Fisica, Politecnico, Piazza L. da Vinci 32, Milano, Italy
Received: 10 November 2004 Published online: 21 December 2004 • © Springer-Verlag 2004
Supercontinuum generation in gas-filled hollow fibers is investigated using numerical simulation of the nonlinear propagation of light pulses in hollow waveguides. The use of the cascading hollow-fiber configuration allows one to significantly enhance the achievable spectral broadening, particularly in the high energy regime. General design criteria for a singleand a double-fiber configuration are presented, which allow the generation of high-energy supercontinua.
ABSTRACT
PACS 42.65.Re; 42.65.-k;
1
42.65.Jx
Introduction
Hollow fiber compression of femtosecond light pulses is a well-established compression technique [1] that has led to the generation of high-energy light pulses shorter than 5 fs [2–5]. Such pulses have opened the way to new applications in the field of extreme nonlinear optics, a light– matter interaction regime where the electric field of a light pulse, rather than the intensity profile, is relevant [6]. In particular, the use of few-optical-cycle pulses has proven to be essential for the generation of single attosecond pulses [7], using the process of high-order harmonic generation, as predicted by Christov el al. [8]. The use of the hollow-fiber technique by different research groups has stimulated theoretical investigation of various nonlinear processes, which influence the propagation of high-energy light pulses in gas-filled waveguides. In particular the onset of self-focusing in hollow fibers has been considered by Tempea and Brabec [9] and by Fibich and Gaeta [10]. The role of ionization in gas-filled fibers has been the subject of active investigations [11, 12]. Crossphase modulation in hollow fibers has been theoretically analyzed and experimentally employed for broadband ultraviolet light generation [13, 14] (see [15] for other references). The key element of this method is the use of guided-wave phase-matched optical parametric generation in gases, that preserves high conversion efficiency and good output beam quality while maintaining a short pulse width. In this paper we discuss general criteria for the design of a hollow-fiber setup, which allow one to choose the fiber u Fax: +39-02-2399-6126, E-mail:
[email protected]
characteristics (i.e., gas type and pressure, fiber length and radius), in order to maximize the spectral broadening of light pulses, with a given energy and duration. Spectral broadening induced by self-phase-modulation (SPM) in gas-filled hollow fibers can lead to the generation of high-energy supercontinua, covering more than two octaves, thus offering the possibility to generate pulses with a transform-limited duration below 2 fs. This requires the development of ultrabroadband dispersive delay lines. Moreover, compression of selected portions of the supercontinuum gives the possibility to generate sub-ten-fs pulses, tunable from the ultraviolet to the nearinfrared [16]. Using numerical simulations of the nonlinear propagation of a light pulse in a gas-filled fiber, we have determined the limits, in terms of spectral broadening, of a single and a double stage hollow-fiber compressor. We show that a simple analytical model, which describes the evolution of a Gaussian pulse in a Kerr medium, allows one to calculate the spectral broadening factor in very good agreement with the results of the numerical simulations. The paper is organized as follows: in Sect. 2 we briefly discuss the nonlinear equation, which will be used in the numerical simulation of the propagation of light pulses in hollow fibers. A simple analytical model, which describes the spectral broadening of Gaussian pulses in a Kerr medium, will be mentioned. In Sect. 3, general design criteria for the optimization of spectral broadening will be discussed, in the case of a single and a double fiber configuration. Finally Sect. 4 contains the conclusions. 2
Non-linear propagation in gas-filled hollow fiber
Equations that describe the propagation of a light pulse through a gas-filled hollow fiber are the same as for propagation in standard optical fibers. Assuming propagation in z direction, the electric field of the mode in the hollow fiber can be written as [17] E(r, ω) = F(x, y)A(z, ω) exp[iβ(ω)z], where F(x, y) is the transverse mode distribution, A(z, ω) is the mode amplitude and β(ω) is the propagation constant. In this way the propagation equation splits in two different equations for mode amplitude and mode transverse distribution. The equation for F(x, y) can be solved with first order perturbation theory. The fundamental mode sustained by single mode hollow fiber is the hybrid mode E H11 characterized by a radial intensity profile I0 (r) = I0 J02 (2.405r/a) where r is the radial coordinate, I0 is the pulse peak intensity, J0 (r) is
286
Applied Physics B – Lasers and Optics
the zero order Bessel function and a is the radius of the fiber core [18]. The equation which describes time propagation of the mode amplitude, A, through the fiber can be derived from the one in the frequency domain with an inverse Fourier transform. We will use the retarded frame T = t − z/vg , where vg is the group velocity of the pulse. The equation describing the evolution through √ the fiber of the normalized amplitude U(z, T) = A(z, T)/ P0 , where P0 is the pulse peak power, is the following: ∂U α i ∂2U 1 ∂3U + U + β2 2 − β3 3 ∂z 2 2 ∂T 6 ∂T i ∂ 2 2 = iγP0 |U| U + (|U| U) . ω0 ∂T
2.405 2π
2
λ20 ν2 + 1 √ 2a3 ν2 − 1
(1)
ω0 = 2πc/λ0 is the central laser frequency. Finally, the nonlinear parameter γ is defined by γ = n 2 ω0 /(cAeff ), where: c is the speed of light in vacuum; n 2 is the nonlinear index coefficient; Aeff 0.48 πa2 is the effective mode area. We assume an input normalized field amplitude of the form
(4)
The first term in the squared bracket of (1) is related to the SPM process, the second one is responsible for selfsteepening. The integration of the propagation equation (1) is performed with the split-step Fourier method [17] in which time derivatives are evaluated in the frequency domain with a fast-Fourier-transform algorithm. A particularly important parameter for the applications is the broadening factor, defined as the ratio between the bandwidths of the pulse at the output and at the input of the fiber: F = ∆ω/(∆ω)0 . Using the results of the numerical simulations, it is possible to calculate the rms bandwidth by calculating the variance (∆ω)2 = ω2 − ω2 2 2 ω |g(ω)|2 dω ω|g(ω)|2 dω = − |g(ω)|2 dω |g(ω)|2 dω
(6)
where ϕm is the maximum phase shift, given by ϕm = γP0 L eff and L eff = [1 − exp(−αL)]/α. Using (6) and (5) it is possible to derive a simple expression for the broadening factor [19] (7)
In the following section we will discuss design criteria for the hollow-fiber compression of light pulses with energy in the millijoule range. We will compare the results of the computer simulations with the analytical model.
(2)
λ0 is the light wavelength in the gas and ν is the ratio between the refractive indices of the external (fused silica) and internal (gas) media. Moreover n d β βn = (n = 2, 3) (3) dωn ω=ω0
T2 U(0, T) = exp − 2 . 2T0
α T2 2 2 exp − 2 exp iϕm e−T /T0 A(L, T) = exp − 2 2T0
4 2 1/2 F = 1 + √ ϕm . 3 3
In (1) α/2 is the field attenuation constant [18]: α = 2
In this case, the field amplitude at the output of a gas-filled fiber with length L is given by
(5)
where g(ω) = F [U(T)], and F indicates the √ Fourier transform operator. Using (5) we have (∆ω)0 = ( 2T0 )−1 . As it will be shown in the next section, it is instructive to compare the results, in terms of spectral broadening, of the complete numerical simulations, with a simple analytical model, which does not include dispersion and self-steepening.
3
Hollow-fiber pulse compression: design criteria
3.1
Single hollow fiber
The broadening factor F , for a given input pulse energy and duration, can be increased in various ways: (i) increasing the fiber length; (ii) increasing the gas pressure, in order to increase the nonlinearity strength; (iii) decreasing the mode area. The fiber length is limited by propagation losses of the fundamental mode, by distortion of the temporal pulse shape and by practical reasons: it is not trivial to achieve long fibers with uniform inner surface and constant radius; fibers longer than about one meter are difficult to finely align. In the following we will assume that the maximum fiber length is L = 1 m. In order to have a weak coupling from the fundamental transverse mode of the fiber to higher-order modes [9], the maximum pulse peak power is limited by the critical power for self focusing, Pcr = λ20 /(2κ2 p) [20], where κ2 is the ratio between the nonlinear index coefficient and the gas pressure p. In the following we will use the following condition: P0 /Pcr < 0.3, which imposes an upper limit to the gas pressure: pmax = 0.15
λ20 κ2 P0
(8)
Using (8), the phase shift ϕm can be written in the following way: ϕm = 0.3πλ0
L eff f Aeff
(9)
where f = p/ pmax. In order to fix the fiber radius, ionization effects have to be considered, which impose an upper limit to the peak intensity of the input pulse, and therefore, a lower limit, amin , to the fiber radius. For a proper operation of the hollow-fiber compression we require that the variation of the refractive index induced by the Kerr effect, ∆n = κ2 pI , is much larger than the change of the refractive index induced by gas ionization, ∆n p = ω2p /2ω20 , where ωp = e2 e /(m e ε0 ) is the plasma frequency, e and m e are the electron charge and mass, e is the free-electron density in the gas, which has been
VOZZI et al.
Optimal spectral broadening in hollow-fiber compressor systems
calculated using the Ammosov–Delone–Krainov (ADK) theory [21] as follows t e = 0 1 − exp − w(t ) dt . (10) −∞
In (10) 0 is the neutral atomic density and w(t) is the ADK ionization rate, ∗ 4Ωp 2n −1 4Ωp w(t) = Ωp |Cn∗ |2 exp − (11) ωt 3ωt where Ip Ωp = , h
e|E(t)| ωt = , 2m e Ip
∗
n =Z
Iph Ip
1/2 ,
∗
|Cn∗ |2 =
22n n ∗ Γ(n ∗ + 1)Γ(n ∗ )
(12)
where: Z is the net resulting charge of the ion; Ip and Iph are the ionization potential of the atomic species under consideration and of the hydrogen atom, respectively; Γ(x) is the Euler gamma function. Figure 1 shows the calculated minimum fiber radius as a function of the input pulse energy, calculated for various pulse durations, assuming ∆n ≥ 103∆n p and a fiber filled with helium. From numerical calculations it turns out that the dependence of the minimum radius, amin , on the energy, E0 , and duration, T0 , of the input pulse, can be well fitted by the following simple expression: β
amin = AT0−α E0
287
a 1-m-long hollow fiber, with radius a = amin , obtained from (13), and a central light wavelength λ0 = 780 nm. In order to have the minimum effective area, it would be convenient to use helium, but, if the peak power of the input pulses is low, the maximum gas pressure, given by (8), is correspondingly high. For example, considering 150-fs pulses with energy E0 = 1 mJ, the maximum pressure in helium is pmax = 35.4 bar. Therefore, in the case of low-energy pulses, it is useful to choose different noble gases, characterized by larger κ2 values (e.g., in the case of argon κ2,Ar 34 κ2,He ). In this way the maximum pressure decreases; the drawback is that the minimum radius imposed by ionization increases, thus leading to a smaller broadening factor. Figure 2 shows by dots the broadening factor, calculated using the numerical simulations, for different values of the input pulse duration, ranging from 25 fs up to 150 fs, for two values of the pulse energy. In the case of 1-mJ pulses, the hollow fiber is filled with argon, in the case of 5-mJ pulses the fiber is filled with helium. The maximum broadening factor, Fmax , obtained at the maximum pressure ( f = 1), decreases upon increasing the pulse energy. Moreover, at a given energy, Fmax increases with pulse duration. In Fig. 2 we have also reported, by dashed lines, the broadening factor calculated using the analytical formula (7). The agreement between the simple analytical model and the simulations is remarkable. The expression (7) slightly underestimates the real broadening factor at high pressure, mainly because it does not include the self-steepening effect, which becomes increasingly important upon increasing the gas pressure.
(13)
where α 0.45, β 0.51 and A is a constant, which depends on the gas. The numerical values of the constant A for various noble gases (in S.I. units) are the followings: AHe 2.62 × 10−9 msα J−β , ANe 1.14 AHe , AAr 1.79 AHe , AKr 2.08 AHe . Using the complete numerical simulations, we have then calculated the maximum broadening factors, as a function of the gas pressure, for different values of the input pulse energy, and pulse duration. In the following we will assume
Broadening factor F as a function of the pressure ratio, f = p/ pmax , in a one-m-long hollow fiber with radius a = amin , for various input pulse durations. (a) Input pulse energy 1 mJ, fiber filled with argon; (b) input pulse energy 5 mJ, fiber filled with helium. The dots are obtained by the complete numerical simulations; dashed lines are obtained by (7)
FIGURE 2
Minimum fiber radius, amin , as a function of the input pulse energy, calculated for various pulse durations, assuming ∆n ≥ 103 ∆n p in a helium-filled hollow fiber. τ0 is the full-width at half maximum pulse duration, τ0 = 1.665 T0 FIGURE 1
288
Applied Physics B – Lasers and Optics
FIGURE 3 Maximum broadening factor as a function of the fiber radius, in a one-m-long hollow fiber filled with gas at pressure p = pmax , calculated using (7)
Taking advantage of the quantitative validity of (7), we have then calculated the maximum broadening factor ( f = 1 in (9)) as a function of the fiber radius. The result is shown in Fig. 3, where we have considered a 1-m-long hollow fiber. We point out that such result, which does not take into account the limitations imposed by gas ionization, does not depend on the gas inside the fiber, the pulse energy and duration. Using a 1-m-long fiber, the maximum broadening factor ( Fmax 72) could be obtained assuming a radius a 52 µm. In the case of a smaller radius, the losses, given by (2), severely limit the fiber effective length, L eff . If we consider the limitation imposed by gas ionization, it is possible to conclude that, in order to maximize the spectral broadening at a given pulse energy, E0 , and duration, T0 , the following procedure has to be followed: (i) the minimum radius imposed by ionization is calculated using (13); (ii) if amin < 52 µm, the correct radius is a = 52 µm, and the achievable broadening factor is F 72; (iii) if amin > 52 µm, the fiber radius is a = amin and the corresponding maximum broadening factor can be obtained (as a first approximation) from Fig. 3. In the last case, Fmax decreases upon increasing the fiber radius. This explains why Fmax increases with pulse duration, as shown in Fig. 2. This is due to the fact that the minimum radius imposed by the gas ionization decreases upon increasing the pulse duration, thus leading to an increase of the maximum broadening factor, as shown in Fig. 3. Moreover, Fmax decreases upon increasing the pulse energy, because the minimum radius increases with energy. We note that, upon increasing the pulse energy, the spectral broadening achievable with a single hollow-fiber, could be not enough to obtain, after compression, sub-5-fs pulses. In the case of a 5-mJ pulse (see Fig. 2b), the maximum spectral broadenings correspond to a transform-limited pulse duration of about 6 fs, for an input pulse duration larger than ∼ 50 fs. In the following section we will see that the use of two hollow fibers, separated by a compression stage, allows one to overcome this limitation and to obtain spectral broadening corresponding to transform-limited pulses in the sub-3-fs domain. 3.2
Cascading hollow fibers
To simulate the performances of the cascading configuration, we performed the propagation of the pulse in three
steps: (i) pulse propagates along the first fiber; (ii) an ideal compressor, with a 80%-transmission, is assumed, so that the pulse at the input of the second fiber is transform limited; (iii) the compressed pulse propagates through the second fiber. Since the aim of the cascading configuration is to extend the spectral broadening factor at high input pulse energies, in the following we will consider a 5-mJ pulse at the input of the first fiber, whose parameters are chosen as described in the previous section: helium; gas pressure p1 = f 1 p1,max (0 ≤ f 1 ≤ 1), where p1,max is given by (8); fiber length L 1 = 1 m; fiber radius a1 = a1,min , fixed by the ionization. The pulse energy at the input of the second fiber takes into account the transmission losses of the first one and an additional loss factor introduced by the compressor stage between the two fibers. The characteristics of the second stage are the following: helium at pressure p2 = f 2 p2,max (0 ≤ f2 ≤ 1); fiber length L 2 = 1 m; fiber radius a2 = a2,min , fixed by the ionization. Using the numerical simulations, we have calculated the broadening factor F = (∆ω)2 /(∆ω)0 , where (∆ω)2 is the bandwidth of the pulse at the output of the second fiber, obtained using (5). The results are shown as dots in Fig. 4, for two durations of the pulses at the input of the first fiber: the broadening factor continuously increases upon increasing the gas pressure in the second fiber. In the simulations shown in Fig. 4 we have assumed the maximum gas pressure in the first fiber ( f1 = 1). Figure 5 shows the calculated spectra of the pulses at the output of the two fibers assuming f 1 = f 2 = 1, for an input pulse duration τ0 = 100 fs. It is evident that the use of the cascading configuration allows one to significantly improve the spectral broadening. Since the pulse spectrum at the input of the second fiber is not Gaussian, the simple analytical formula (6) and (7) cannot be directly used. Nevertheless, it is instructive to compare the results of the simulations, in terms of broadening factor, with the analytical model, assuming that the pulse at the input of the second fiber is Gaussian (in time and frequency). In this case the broadening factor, F2 = (∆ω)2 /(∆ω)1 , of the second fiber can be calculated using (7) and (9). The total broadening factor is F = F1 F2 , where F1 is the broadening factor of the first fiber. The broadening factor calculated using the an-
FIGURE 4 Spectral broadening of cascading hollow fibers as a function of the pressure ratio, f 2 = p2 / p2,max , in the second fiber, for two durations of 5-mJ input pulses. The dots are the results of the numerical simulations; dashed curves are calculated using the analytical model, as described in the text. Parameters used in the calculations: helium in both fibers; L 1 = L 2 = 1 m; a1 = a1,min ; p1 = p1,max ; a2 = a2,min
VOZZI et al.
Optimal spectral broadening in hollow-fiber compressor systems
289
increases with p1 . Therefore, in order to obtain the maximum broadening we have to use the maximum pressure in both fibers. 4
FIGURE 5 Calculated spectra of the pulses at the output of the first (dashed line) and of the second (solid line) fiber, assuming a 100-fs, 5-mJ input pulse. Parameters used in the simulations: (a) first fiber: helium at pressure p1 = p1,max = 4.7 bar; a1 = a1,min = 146 µm; (b) second fiber: helium at pressure p2 = p2,max = 0.41 bar; a2 = a2,min = 430 µm; pulse energy at the input of the second fiber 3.68 mJ
alytical formula is shown by the dashed curves in Fig. 4. The agreement between the model and the simulations is remarkable. Therefore, for a first design of the hollow-fiber cascading setup, it is possible to use the results of the model. We have then calculated the total broadening factor as a function of the pressure of the gas in the first fiber, in order to find the condition for the maximum spectral broadening. Figure 6 shows the results of the simulations (dots) and of the analytical model (dashed curve), in very good agreement with the simulations. The fiber parameters are the followings: helium in both fibers; L 1 = L 2 = 1 m; a1 = a1,min , and a2 = a2,min , determined by the ionization; p2 = p2,max. Spectral broadening continuously increases upon increasing f 1 . We point out that, upon increasing the pressure p1 , the pulse duration at the input of the second fiber decreases, and the minimum radius of the second fiber increases. Therefore, the broadening factor between the first and the second fiber, F2 , continuously decreases. Nevertheless, the total broadening factor F = F1 F2
FIGURE 6 Spectral broadening of cascading hollow fibers as a function of the pressure ratio, f 1 = p1 / p1,max , in the first fiber, for two durations of 5-mJ input pulses. The dots are the results of the numerical simulations; dashed curves are calculated using the analytical model. Parameters used in the calculations: helium in both fibers; L 1 = L 2 = 1 m; a1 = a1,min ; a2 = a2,min ; p2 = p2,max
Conclusions
Using numerical simulations of the nonlinear propagation of a light pulse in a gas-filled fiber, we have determined the limits, in terms of spectral broadening, of a single and a double stage hollow-fiber compressor. The use of the cascading hollow-fiber configuration allows one to significantly enhance the achievable spectral broadening, particularly in the high energy regime. We have shown that a simple analytical model, which describes the evolution of a Gaussian pulse in a Kerr medium, allows one to calculate the spectral broadening factor in very good agreement with the results of the numerical simulations. General criteria for the design of an ultrabroadband hollow-fiber setup have been discussed. The possible applications of such supercontinua are the generation of single-cycle light pulses, or the generation of sub-10-fs pulses tunable from the ultraviolet to the near-infrared. ACKNOWLEDGEMENTS This work was partially supported by the European Community’s Human Potential Programme under project MRTN-CT-2003-505138 (XTRA), and by INFM under the project ‘Clusters as nano-environments for laser-induced extreme states of matter and chemical reactions’.
REFERENCES 1 M. Nisoli, S. De Silvestri, O. Svelto: Appl. Phys. Lett. 68, 2793 (1996) 2 M. Nisoli, S. De Silvestri, O. Svelto, R. Szipöcs, K. Ferencz, C. Spielmann, S. Sartania, F. Krausz: Opt. Lett. 22, 522 (1997) 3 B. Schenkel, J. Biegert, U. Keller, C. Vozzi, M. Nisoli, G. Sansone, S. Stagira, S. De Silvestri, O. Svelto: Opt. Lett. 28, 1987 (2003) 4 K. Yamane, Z. Zhang, K. Oka, R. Morita, M. Yamashita, A. Suguro: Opt. Lett. 28, 2258 (2003) 5 G. Sansone, G. Steinmeyer, C. Vozzi, S. Stagira, M. Nisoli, S. De Silvestri, K. Starke, D. Ristau, B. Schenkel, J. Biegert, A. Gosteva, U. Keller: Appl. Phys. B 78, 551 (2004) 6 T. Brabec, F. Krausz: Rev. Mod. Phys. 72, 545 (2000) 7 M. Hentschel, R. Kienberger, C. Spielmann, G.A. Reider, N. Milosevic, T. Brabec, P. Corkum, U. Heinzmann, M. Drescher, F. Krausz: Nature (London) 414, 509 (2001) 8 I.P. Christov, M.M. Murnane, H. Kapteyn: Phys. Rev. Lett. 78, 1251 (1997) 9 G. Tempea, T. Brabec: Opt. Lett. 23, 762 (1998) 10 G. Fibich, A.L. Gaeta: Opt. Lett. 25, 335 (2000) 11 G. Tempea, T. Brabec: Opt. Lett. 23, 1286 (1998) 12 M. Nurhuda, A. Suda, K. Midorikawa, M. Hatayama, K. Nagasaka: J. Opt. Soc. Am. B. 20, 2002 (2003) 13 C.G. Durfee III, S. Backus, H.C. Kapteyn, M.M. Murnane: Opt. Lett. 24, 697 (1999) 14 L. Misoguti, S. Backus, C.G. Durfee III, R. Bartels, M.M. Murnane, H.C. Kapteyn: Phys. Rev. Lett. 87, 13 601 (2001) 15 A.M. Zheltikov: Physics-Uspekhi 45, 687 (2002) 16 M. Nisoli, G. Sansone, S. Stagira, C. Vozzi, S. De Silvestri, O. Svelto: Appl. Phys. B 75, 601 (2002) 17 G.P. Agrawal: Nonlinear Fiber Optics, 2nd edn. (Academic, San Diego, CA 1995) 18 E.A.J. Marcatili, R.A. Schmeltzer: Bell Syst. Tech. J. 43, 1783 (1964) 19 S.C. Pinault, M.J. Potasek: J. Opt. Soc. Am. B 2, 1318 (1985) 20 N. Milosevic, G. Tempea, T. Brabec: Opt. Lett. 25, 672 (2000) 21 M. Ammosov, N. Delone, V. Krainov: Sov. Phys. JETP 64, 1191 (1986)
