Freezing the carrier-envelope phase of few-cycle light pulses about a focus

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1 Freezing the carrier-envelope phase of few-cycle light pulses about a focus Miguel A. Porras 1 and Péter Dombi 2 1 Departamento de Física Aplicada, Universidad Politécnica de Madrid, Rios Rosas 21, ES Madrid, Spain 2 Research Institute for Solid-State Physics and Optics, Konkoly-Thege M. út 29-33, H-1121 Budapest, Hungary, Abstract: We study the effects of dispersive media and structures on the carrier-envelope phase (CEP) shift of focused few-cycles pulses. For phase-sensitive interactions with matter requiring focusing in vacuum, the variation of the CEP through the focal region can be significantly slow down by inserting a dispersive slab of adequate thickness between the focusing system and the focus. The focal CEP shift can also be slow down in experiments requiring focusing in a dispersive medium by a suitable choice of the dispersive propagation distance up to the focus Optical Society of America OCIS codes: ( ) Femtosecond phenomena; ( ) Phase; ( ) Ultrafast nonlinear optics. References and links 1. A. Apolonski, P. Dombi, G.G. Paulus, M. Kakehata, R. Holzwarth, Th. Udem, Ch. Lemell, K. Torizuka, J. Burgdörfer, T.W. Hänsch, and F. Krausz, Observation of light-phase-sensitive photoemission from a metal, Phys. Rev. Lett. 92, (2004). 2. F. Lindner, G.G. Paulus, H.Walther, A. Baltuska, E. Goulielmakis, M. Lezius, and F. Krausz, Gouy phase shift for few-cycle laser pulses, Phys. Rev. Lett. 92, (2004). 3. T. Tritschler, K. D. Hof, M. W. Klein, and M. Wegener, Variation of the carrier-envelope phase of few-cycle laser pulses owing to the Gouy phase: a solid-state-based measurement, Opt. Lett. 30, (2005). 4. P. Dombi, A. Apolonski, Ch. Lemell, G. G. Paulus, M. Kakehata, R. Holzwarth, Th. Udem, K. Torizuka, J. Burgdörfer, T. W. Hänsch and F. Krausz, Direct measurement and analysis of the carrier-envelope phase in light pulses approaching the single-cycle regime, New J. Phys. 6, 39 (2004). 5. F. Krausz and Misha Yu. Ivanov, Attosecond physics, Rev. Mod. Phys. 81, (2009). 6. A. Baltuska, Th. Udem, M. Uiberacker, M. Hentschel, E. Goulielmakis, Ch. Gohle, R. Holzwarth, V. S. Yakovlev, A. Scrinzi, T. W. Hansch, and F. Krausz, Attosecond control of electronic processes by intense light fields, Nature 421, 611 (2003). 7. T.M. Fortier, P. A. Roos, D. J. Jones, S.T. Cundiff, R. D. R. Bhat, and J. E. Sipe, Carrier-envelope phasecontrolled quantum interference of injected photocurrents in semiconductors, Phys. Rev. Lett. 92, (2004). 8. P. A. Roos, Q. Qraishi, S. T. Cundiff, R. D. R. Bhat, and J. E. Sipe, Characterization of quantum interference control of injected currents in LT-GaAs for carrier-envelope phase measurements, Opt. Express 11, (2003). 9. O. D. Mücke, T. Tritschler, M. Wegener, F. X. Kaertner, U. Morgner, G. Khitrova, and H. M. Gibbs, Carrier wave Rabi flopping: role of the carrier-envelope phase, Opt. Lett. 29, (2004). 10. S. E. Irvine, P. Dombi, G. Farkas, and A. Y. Elezzabi, Influence of the carrier-envelope phase of few-cycle pulses on ponderomotive surface-plasmon electron acceleration, Phys. Rev. Lett. 97, (2006). 11. P. Dombi and P. Rácz, Ultrafast monoenergetic electron source by optical waveform control of surface plasmons, Opt. Express 16, (2008). 12. A. Cavalieri, Attosecond spectroscopy in condensed matter, Nature 449, (2007). (C) 2009 OSA 26 October 2009 / Vol. 17, No. 22 / OPTICS EXPRESS 19424

2 13. D. Faccio, A. Lotti, M. Kolesik, J.V. Moloney, S. Tzortzakis, A. Couairon, and P. Di Trapani, Spontaneous emergence of pulses with constant carrier-envelope phase in femtosecond filamentation, Opt. Express 16, (2008). 14. C. J. Zapata-Rodríguez, and M. A. Porras, Controlling the carrier-envelope phase of few-cycle focused laser beams with a dispersive beam expander, Opt. Express 16, (2008). 15. M. A. Porras, Characterization of the electric field of focused pulsed Gaussian beams for phase-sensitive interactions with matter, Opt. Lett. 34, (2009). 16. A. E. Siegman, Lasers, University Science books, Mill Valley, California (1986). 17. M. A. Porras, Diffraction effects in few-cycle optical pulses, Phys. Rev. E 65, (2002). 18. S. Feng, and H. G. Winful, Spatiotemporal structure of isodiffracting ultrashort electromagnetic pulses, Phys. Rev. E 61, (2000). 1. Introduction It is well-known that the change of the phase of a few-cycle pulse through a focus can be a major obstacle for the observation of the dependence of fundamental physical light-matter interaction phenomena on the phase of an ultrashort laser pulse [1, 2, 3, 4]. This effect can play a role both in light-atom and in light-solid interactions. The most prominent example is high harmonic generation (HHG), which is the routine technique used for attosecond pulse generation [5]. It was demonstrated that the high harmonic spectrum generated in a jet of noble gas atoms is highly sensitive to the phase of the ultrashort, infrared driver pulse [6], or in more technical terms, to the so-called carrier-envelope phase (CEP), or phase of the carrier oscillations at the instant of maximum amplitude. The result is the generation of different HHGbased attosecond pulses for different values of the CEP [6], both in terms of their energy and their pulse shape. It is therefore important to keep the CEP constant in the interaction region where the harmonics are generated. Various light-solid interactions were also found to depend on the CEP [1, 3, 4, 7, 8, 9, 10, 11]. In this case the interaction geometry can be even more complex since in some experiments the target surface is placed at close to grazing incidence with respect to the laser beam (to maximize the normal component of the field vector) [1, 4, 12], or focusing is carried out through a dispersive medium to enable surface plasmon coupling [10, 11]. It is therefore necessary to examine the spatial variation of the CEP in all of these cases i) in order to gain a deeper knowledge of the phase distribution in the interaction region and ii) to establish whether it is possible to achieve a spatially constant CEP in the vicinity of focus where the intensity is highest based on a proper choice of focusing conditions. In the experiments where the material target is placed in vacuum, the CEP shift originates from Gouy s phase solely. When the pulse is focused into a dispersive medium, the CEP variation can be much more pronounced, blurring the phase-sensitivity of the interaction [10, 11]. Two methods have been suggested to minimize this problem. One is the use of conical waves with equal phase and group velocities [13]. For the Gaussian-like beams emitted by phasestabilized lasers the Gouy s phase shift of π of each Gaussian beam Fourier component is unavoidable, but the imprinting of a suitable variation of the spot size with frequency prior to focusing can result in a flattened variation of the CEP in the focal region. [3, 14, 15] This technique requires however complex refractive and diffractive optics, which is hardly appropriate for few-cycle pulses. In this paper we pursue another approach based on the slight variation of the position of the focus with frequency induced by dispersive media placed either in the focusing path or in the focus itself. A small and controlled amount of dispersion in the position of the focus is seen to result in negligible changes in the pulse envelope upon passage through the focus for few-cycle pulses, and in turn can result in a significant reduction of the CEP shift. First, we find general expressions for the CEP shift experienced by few-cycle pulsed Gaussian beams with arbitrarily shaped temporal envelope in the focal volume under the joint effects of focusing and dispersive propagation. We then find that the CEP can be locally frozen along a focus placed in vacuum (C) 2009 OSA 26 October 2009 / Vol. 17, No. 22 / OPTICS EXPRESS 19425

3 by insertion of one or several dispersive slabs of suitable thicknesses in the focusing path, i.e., between the focusing system and the focus. In experiments requiring focusing into a dispersive medium, the CEP can be also frozen by an adequate choice of the length of dispersive medium up to the focal volume. We point out that our results apply under focusing conditions such that the pulse temporal envelope remains nearly unchanged within the focal volume (as is usually needed in application experiments), though the envelope may, of course, experience strong dispersive reshaping on propagation from the focusing system up to the focal volume. In the configurations for CEP freezing in a focus in vacuum or in a medium, the broadening and distorting action of the slab or medium, can be compensated with standard dispersion management techniques (e.g., by second- and third-order dispersion pre-compensation in the input pulse in front of the focusing element), being then possible to obtain nearly-transform limited, few-cycle pulses with nearly invariant envelope and CEP in the focal region. 2. Pulsed Gaussian beam focusing in presence of dispersive media The focusing system (henceforth the lens) is illuminated by a collimated pulsed Gaussian beam, whose monochromatic constituents are the plane Gaussian beams ( ) iωr 2 E in (ω,r)=p(ω)exp, (1) 2c ˆq in where ˆq in = iωs 2 in /2c = il, s in is the (generally ω-dependent) spot size, and L = ωs 2 in /2c the Rayleigh range. The function p(ω) is a broadband function about an optical frequency ω 0 (usually defined as the mean value of p(ω) 2 ), r is a radial coordinate in the transversal plane, and c the speed of light in vacuum. The ABCD ray matrix for focusing and propagation through a sequence of media is ( ) ( ) A B 1 B/ f B =, (2) C D 1/ f 1 where f is the focal length, and B = d j /n j (ω)+z/n(ω) for media of thicknesses d j and indexes n j (ω). Throughout this paper the coordinate z measures the distance from the entrance plane of the last medium, whose refraction index is n(ω), and where the focal region is assumed to be located. The simplest situation is the focusing into a dispersive medium filling the space beyond the lens, in which case B = z/n(ω), and z is the distance from the lens. Two focusing configurations are of particular interest for us. Figure 1(a) illustrates the focusing in vacuum through a dispersive slab of thickness δ and refraction index n s (ω) placed a distance d from the lens (d + δ/n s (ω) < f ), in which case B = d + δ/n s (ω)+z, and z is the distance from the slab. Figure 1(b) shows the focusing inside a dispersive medium of refraction index n(ω) separated a distance d < f from the lens, for which B = d +z/n(ω) and z is the distance from the entrance plane of the medium. In presence of dispersive media, the location of the geometrical focus z f depends on frequency and is determined by the condition B = f, yielding [ z f = f d ] j n(ω) (3) n j (ω) [e. g., z f = f d δ/n s (ω) and z f =(f d)n(ω) and in the respective cases of Figs. 1(a) and (b)]. It is convenient to introduce the ω-dependent parameter b = B f = d j n j (ω) + z n(ω) f = z z f n(ω) Z n(ω), (4) (C) 2009 OSA 26 October 2009 / Vol. 17, No. 22 / OPTICS EXPRESS 19426

4 where Z z z f determines the position of the plane z of interest from the geometrical focus at each frequency. = B I M > @ Fig. 1. Focusing (a) in vacuum through a dispersive slab, and (b) in a dispersive medium. According to Gaussian beam transformation formulas, the monochromatic Gaussian beam components at a distance z are given by [16] E(ω,r,z)= p(ω) A + B/ ˆq in exp ( iωr 2 2c ˆq ) exp (i ω ) c S where S = n j (ω)d j + n(ω)z is the optical path along the optical axis, and where ˆq =(A ˆq in + B)/(C ˆq in + D). For the ABCD matrix in Eq. (2), one obtains and ˆq = b + E(ω,r,z)=p(ω) ˆq( f ) exp ˆq (5) f 1 + il/ f, (6) ( iωr 2 2c ˆq ) exp (i ω ) c S, (7) where ˆq( f ) stands for ˆq evaluated at b = f. In practical settings focusing is tight enough so that the Rayleigh ranges L of all Gaussian beam constituents in front of the lens are much larger than f, in which case the focal shift is negligible, i.e., the waist of each Gaussian beam is not shifted from its geometrical focus z f. This constitutes the Debye approximation to Eqs. (6) and (7), which is described by ˆq b if 2 /L = b il R, (8) where L R f 2 /L is the ω-dependent Rayleigh distance, or half-depth of focus of the focused Gaussian beams in vacuum, and by E(ω,r,z) p(ω) f ˆq exp ( iωr 2 2c ˆq ) exp (i ω ) c S. (9) Since 1/ ˆq 1/ ˆR + i2c/ωs 2, the reduced radius of curvature ˆR = n(ω)/r of the wave fronts and the spot size at the plane z are given by where s 2 f = 2cL R/ω is the spot size at the focus z f. ˆR = b + L 2 R/b, s 2 = s 2 f (1 + b 2 /L 2 R), (10) (C) 2009 OSA 26 October 2009 / Vol. 17, No. 22 / OPTICS EXPRESS 19427

5 3. The carrier-envelope phase shift in the focal region From Eq. (9), the pulse temporal form at a position (r,z) is obtained by the inverse temporal Fourier transform E(t,r,z)= 1 2π dωe(ω,r,z)exp( iωt)= 1 dωa(ω,r,z)exp{ i[ωt ϕ(ω,r,z)]}, 2π (11) where we have introduced the amplitude a(ω,r,z) > 0 and the phase ϕ(ω,r,z) of the Gaussian beam constituents, which from Eqs. (8) and (9) are given by a(ω,r,z)= f ) 1 ( L R 1 +(b/l R ) 2 exp r2 s 2, (12) and ϕ(ω,r,z)= π ( ) b 2 tan 1 + ωr2 L R 2c ˆR + ω S. (13) c The CEP of E(t,r,z) is defined as the phase Φ of the optical oscillations at the instant of time of maximum pulse amplitude. To obtain it, we extract from E(t,r,z) the oscillations at the carrier frequency ω 0 by writing E(t,r,z) =A(t,r,z)exp{ i[ω 0 t ϕ 0 (r,z)]}, where ϕ 0 (r,z) ϕ(ω 0,r,z) is the phase at the carrier frequency, and where the envelope is given by A(t,r,z)= 1 dω p(ω)a(ω,r,z)exp{i[ϕ(ω,r,z) ϕ 0 (r,z)]}exp{ i(ω ω 0 )t}. (14) 2π Writing the phase as the power series ϕ(ω,r,z)=ϕ 0 (r,z)+ϕ 0 (ω ω 0)+ϕ 0 (ω ω 0) 2 /2+..., where prime signs stand for differentiation with respect to ω and subindexes 0 for evaluation at ω 0, the envelope is also given by A(τ,r,z)= 1 { [ ]} 1 dω p(ω)a(ω,r,z)exp i 2π 2 ϕ 0 (r,z)(ω ω 0 ) 2 + exp{ i(ω ω 0 )τ}, (15) where we have introduced the local time τ = t ϕ 0 (r,z). If at the lens plane the pulse amplitude A peaks, e.g., at t = 0, and A does not experience any change during propagation, the time of arrival of the peak at a point (r,z) would be t = ϕ 0 (r,z), that is, τ = 0. However, this is generally not true owing to significant envelope reshaping in the dispersive media beyond the lens and some (generally weak) envelope reshaping induced during the focusing process. [17] Mathematically, envelope reshaping is described in Eq. (15) by the second- and higherorder derivatives of ϕ(ω,r,z) with respect to ω, and by the dependence of a(ω,r,z) with ω, as described in Ref. [17]. At the focus of the pulse, defined here as the geometrical focus (r = 0,z = z f,0 ) for the monochromatic constituent at the carrier frequency ω 0, the amplitude A will generally reach a maximum value at a local time τ p generally different from zero. If the argument of the envelope A at this time is φ, the envelope can be conveniently written as A(τ,0,z f,0 )=Ã(τ,0,z f,0 )exp(iφ), where Ã(τ,0,z f,0 ) is real at τ p, and the pulse form can be written as E(t,0,z f,0 )=Ã(τ,0,z f,0 )exp{ i[ω 0 t ϕ 0 (0,z f,0 ) φ]}. The CEP at the focus is the phase of the carrier oscillations (the argument of the last exponential) at τ p, i.e., Φ(0,z f,0 )= ω 0 τ p ω 0 ϕ 0 (0,z f,0)+ϕ 0 (0,z f,0 )+φ. Though the envelope experiences usually strong reshaping from the lens up to the focal region in presence of dispersive media, experiments usually require negligible envelope reshaping within the focal region, and these are designed to fulfill this requirement. Negligible envelope reshaping in the focal region is usually achieved by choosing the focal length f and the spot size (C) 2009 OSA 26 October 2009 / Vol. 17, No. 22 / OPTICS EXPRESS 19428

6 s in on the lens such that the depth of focus is much smaller than the second- and higher-order dispersion lengths for the few-cycle pulse, in the case of focusing in a dispersive medium. Envelope reshaping is usually assumed to be negligible in case of focusing in vacuum, as sustained also by Ref. [17]. Under the assumption of negligible focal envelope reshaping, the envelope A(r, z) at any point (r,z) in the focal volume will be characterized by the same peak local time τ p and argument φ. The CEP at (r,z) will then be given, as above, by Φ(r,z)= ω 0 τ p ω 0 ϕ 0 (r,z)+ϕ 0(r,z)+φ, and the CEP shift from the pulse focus by ΔΦ(r,z)=[ ω 0 ϕ 0(r,z)+ϕ 0 (r,z)] [ ω 0 ϕ 0(0,z f,0 )+ϕ 0 (0,z f,0 )]. (16) For the Gaussian beam phase in Eq. (13), long but straightforward calculations yield ω 0 ϕ 0(r,z)+ϕ 0 (r,z) = ω 0 c S 0ω 0 π ( ) 2 b0 tan 1 L R,0 ( ) L ) R,0 b 0 /L R,0 ω 0 (1 L R,0 1 +(b 0 /L R,0 ) 2 2r2 s ω 0b [ ) ] 0 1 (1 L R,0 1 +(b 0 /L R,0 ) 2 2r2 + r2, (17) where, as above, subindexes 0 stand for evaluation of the ω-dependent quantities S, b, L R and s at ω 0. Using the relations n(ω)=ck(ω)/ω, where k(ω) is the propagation constant in the last medium, b = Z/n(ω), b = z f Zn (ω)/n(ω) 2, and introducing the length L 1 =[k (ω)ω k(ω)] 1, we obtain from Eqs. (16) and (17) our final result for the CEP shift from the focus: ΔΦ(r,z) = Z ( ) 0 tan 1 Z0 L 1,0 n 0 L R,0 [ ( ) ( ) ] 1 2 ( ) Z0 Z0 + ( ) 2 g 0 + γ 0 1 2r2 r 2 Z0 n 0 L R,0 n 0 L R,0 s 2 + γ 0 0 s 2. (18) n 0 L R,0 In this equation Z 0 = z z f,0 is an axial coordinate with origin at the pulse focus, or geometrical focus at the carrier frequency ω 0. The length L 1,0 =[k 0 ω 0 k 0 ] 1 characterizes the axial distance at which the carrier oscillations and envelope are significantly shifted (one radian shift) for plane pulse propagation in the last medium (L 1,0 = in vacuum), n 0 L R,0 is the carrier halffocal depth, s 0 is the carrier Gaussian spot size at any position Z 0 from the pulse focus as given, from Eq. (10), by s 2 0 = s2 f,0 [1 +(Z 0/n 0 L R,0 ) 2 ], where s 2 f,0 = 2cL R,0/ω 0 is the carrier spot size at the pulse focus. The first term in Eq. (18) is the CEP shift due to material dispersion for plane pulses, and the second term is Gouy s phase shift upon passage through a focus. These intrinsic CEP shifts are modified by the remainder terms in Eq. (18) that depend on the specific focusing geometry. The term with g 0 = (L R,0 /L R,0)ω 0 has been described previously for on-axis [3] and off-axis points [15], and accounts for the effect of the dependence of the focal spot size with ω about ω 0. Since L R = f 2 /L, the parameter g 0 is also given by g 0 = L 0 ω 0, (19) L 0 and hence is determined by the variation of the spot size with frequency in the input pulse, e.g., g 0 = 1 for constant s in (s f inversely proportional to ω), g 0 = 0 for s in ω 1/2 (s f ω 1/2 too, s 2 0 s 2 0 (C) 2009 OSA 26 October 2009 / Vol. 17, No. 22 / OPTICS EXPRESS 19429

7 or isodiffracting pulse [18]), and g 0 = 1 for s in ω 1 (s f constant). The new parameter γ 0 = z f,0 n 0 L R,0 ω 0 (20) in Eq. (18) is non-negligible if the variation of the focus position with frequency induced by dispersion is comparable with the pulse focal depth n 0 L R,0. The terms with γ 0 account for the effect of this dispersion in the focus position on the CEP shift, as described in detail below. Though Eq. (18) gives the CEP shift at any point in the focal volume, we limit our discussion to the CEP shift along on-axis points in the focal volume, where the intensity is maximum and phase-sensitive interactions with matter are greatly enhanced. A detailed study of the off-axis CEP shift in vacuum has been performed in Ref. [15], and could be similarly performed here. Two application examples of Eq. (18) are studied in the following sections. 4. Freezing the carrier-envelope phase shift in a focus in vacuum If the focus is placed in vacuum (n 0 = 1, L 1,0 = ), Eq. (18) at r = 0 yields ΔΦ(z)= tan 1 Z/L R (Z/L R )+g 1 +(Z/L R ) 2 + γ (Z/L R ) 2 1 +(Z/L R ) 2, (21) where we have omitted all subscripts 0 for conciseness. Figures 2(a), (b) and (c) (dashed curves), show the CEP shift ΔΦ in the case that the space between the lens and the focus is empty (γ = 0) for several values of g on the lens. As is well-known, the CEP shift differs from Gouy s phase except for g = 0. [15] In the particular case with g = 1, the CEP is locally frozen about the pulse focus Z = 0. [3, 15] There is no information in the literature, however, concerning the exact value of g for collimated few-cycle pulses before focusing, though measurements of the CEP shift indicate that typically g does not exceed unity. [2, 3] Methods proposed for changing g of few-cycle pulses to its optimum value for CEP stationarity are based on the knowledge of its initial value, and involve complex refractive and diffractive optics, which will induce significant energy losses and irreversible pulse deterioration in practice. Fig. 2. On-axis CEP shift ΔΦ(z) as given by Eq. (21) for the indicated values of g and γ. The dashed curves correspond to γ = 0 (no dispersion-induced focal shift). We point out here that, irrespective of the value of g, a slight variation of the focus position with ω (γ 0) induced by insertion of dispersive media in the focusing path can significantly slow down the CEP shift of nearly transform-limited, few-cycle pulses in the focal region, and does not involve appreciable envelope reshaping in this region. The solid curves in Figs. 2(a), (b) and (c) show the CEP evolution for several values of γ. If g 1, the CEP shift features a horizontal flex point in the second half of the focal region for γ =(1 g 2 ) 1/2 [γ = 1,0.87 and 0 in Figs. 2(a), (b) and (c), respectively], and a slight oscillatory behavior with a minimum and a maximum for slightly higher values of γ. (C) 2009 OSA 26 October 2009 / Vol. 17, No. 22 / OPTICS EXPRESS 19430

8 The simplest arrangement to achieve the above effect is placing a single dielectric slab in the focusing path. If as in Fig. 1(a), d is the distance from the lens to the slab, δ is the slab thickness and n s (ω) n(ω) (to lighten the notation) its refraction index, the position of the focus is z f = f d δ/n, and its derivative is z f = δn /n 2. Eq. (20) then yields γ = δ/nl R kl 1, where we have used the relation n ω/n = 1/kL 1, and where n, k and L 1 refer to the dispersive properties of the slab at ω 0. Given a half-depth of focus L R = f 2 /L and a slab material, its thickness for nearly flat CEP in the second half of the focus is of the order of δ nl R kl 1. (22) The actual thickness may range from zero up to this value or slightly larger, depending on the unknown value of g and whether we wish an horizontal flex point or slight oscillatory behavior, and can be controlled in practice with two dielectric wedges. Fig. 3. (a) Pulse forms at the focus z f (black solid curve) and at z f +2L R (red dashed curve), and (b) axial CEP shifts from Eq. (21) (solid curve) and numerically calculated (open circles) for focusing in vacuum without insertion of slab. (c) and (d), and (e) and (f) show the same physical quantities as (a) and (b) but with the insertion of a slab in the focusing path. In (c) and (d) second- and third-order dispersion introduced by the slab were compensated in the input pulse; in (e) and (f) only second-order dispersion was compensated. Details of the focusing geometry, slab material and input pulse are given in the text. In Fig. 3 we test these predictions in a simple case. For a carrier wavelength of 800 nm (ω 0 = fs 1 ), focal length f = 100 mm, and input carrier spot size s = 10 mm, the halfdepth of focus is L R = mm. Taking the intermediate value g = 0.5, we evaluated from Eqs. (7) and (11) (i.e., without the Debye approximation) the pulse form along the axis about the focus and after passage through a dielectric slab made of fused silica placed at d = 90 mm from the lens. From Sellmeier relation, the slab thickness must be δ = nl R kl 1 = 3.87 mm in order to produce the quasi-stationary behavior of the CEP of Fig. 2(b) with γ = 1. For reference, Figs. 3(a) and (b) illustrate the case that no slab is inserted in the focusing path. For an input two-cycle pulse of Gaussian spectrum p(ω)=exp[δt 2 (ω ω 0 ) 2 /4] (Δt = fs, or intensity FWHM Δt 2ln2= fs), the pulse forms at the focus [black curve in Fig. 3(a)] and at 2L R beyond it (red curve) present shifted CEPs by about 1 rad. In Fig. 3(b) the CEP shifts along z, extracted directly from the pulse forms (open circles) and predicted by Eq. (21) with g = 0.5 and γ = 0 (solid curve) are compared. Equation (21) is accurate because focusing does not appreciably distorts the pulse envelope in the focal region. (C) 2009 OSA 26 October 2009 / Vol. 17, No. 22 / OPTICS EXPRESS 19431

9 Figures 3 (c)-(f) illustrate the effect of introducing the silica slab of thickness δ = 3.87 mm. In order to obtain a nearly transform-limited pulse in the focal region, the input broadband Gaussian spectrum is pre-chirped to compensate for the chirp introduced by the slab, so that the pulse strongly compresses down to a nearly transform-limited two-cycle pulse during the propagation in the slab. Specifically, in Figs. 3(c) and (d) the chirp compensates for both secondand third-order slab dispersion, and in Figs. 3(e) and (f) for second-order dispersion only (in case that a fully transform-limited pulse is not required). Independently of the actual pulse forms reached in the focal region, these are seen to experience negligible CEP shifts in the second-half of the focal region and beyond. In Figs. 3(d) and (f), the CEP shifts predicted from Eq. (21) with g = 0.5 and γ = 1 (solid curves) are seen to fit well to the actual CEP shifts (open circles). Small discrepancies arise from slight envelope reshaping along the focal region. In fact, no appreciable envelope reshaping during propagation through the focus is observable at the scale of Figs. 3(c) and (f). We stress that CEP stationarity results from the dispersion in the focal position induced by the slab placed in the focusing path, pre-chirping being only a way to obtain a few-cycle pulse in the focal region. In fact, the same CEP stationarity effect is observed if the input pulse is not chirped, but the pulse in the focal region is in this case broadened by the slab dispersion. Also, placing the slab before the focusing element (as done for other purposes in many experiments [2]) has no effect on the CEP evolution along the focus. Note also that stationary CEP over focal depths L R longer than in the preceding example would require thicker slabs, which can make compensation of the slab dispersion difficult. In these situations, multiple slabs can be used with appropriate thicknesses δ i and refraction indexes n j such that the condition of CEP stationarity in the focal depth L R, which reads now δ i /(n i k i L 1,i ) L R, is satisfied, at the same time that dispersion in the multilayered structure is compensated up to a certain order. This analysis is beyond the scope of this paper. 5. Slowing down the carrier-envelope phase shift in a focus in a dispersive medium It is also possible to slow down the CEP in experiments involving focusing in a dispersive medium, as in Fig. 1(b). In this case, Eq. (18) yields ΔΦ(z)= Z/L 1 tan 1 Z/nL R (Z/nL R )+g 1 +(Z/nL R ) 2 + γ (Z/nL R ) 2 1 +(Z/nL R ) 2 (23) along the optical axis, where n and L 1 refer here to the medium where the pulse is focused, and all subindexes 0 meaning evaluation at the carrier frequency are omitted. As previously reported [14], the CEP is stationary about Z = 0 for input pulse with g = 1 + nl R /L 1. This requires artificially high g if L 1 is of the order or smaller than nl R, i.e., if material dispersion significantly shifts the CEP in the focal region. Instead, fixing suitably the distance from the entrance of the medium up to the focus can produce a similar effect. For any g < 1 + nl R /L 1, the CEP in Eq. (23) presents a horizontal flex point in the second half of the focus for γ [(1 g+nl R /L 1 )(1+g+2nL R /L 1 )] 1/2, or slightly larger. This is illustrated in Fig. 4 for several values of the input g [from (a) to (c)] and several dispersion strengths along the focus, or ratios nl R /L 1 (blue, red and black curves from stronger to weaker dispersion strengths). In the arrangement of Fig. 1(b), z f =(f d)n, z f =(f d)n, and Eq. (20) yields γ = z f /kl 1 nl R. Thus, for given depth of focus nl R and dispersive medium, there exists an optimum thickness z f of the medium up to the focus for stationary CEP, which can be controlled, e. g., by moving the lens or, in the experiments of Ref. [11], by lateral translation of the prism. Figure 5 illustrates the CEP slowing down in the focusing with f = 50 mm of a pulse of carrier frequency ω 0 = fs 1 in fused silica. Taking s in = 10 mm independent of ω, the half-depth of focus nl R equals to L 1, and g = 1. If the entrance plane of the medium is placed (C) 2009 OSA 26 October 2009 / Vol. 17, No. 22 / OPTICS EXPRESS 19432

10 Fig. 4. CEP shift from Eq. (23) for the indicated values of g and nl R /L 1. For the dashed curves γ = 0, and for the solid curves γ takes the value resulting in a horizontal flex point. at d = mm from the lens, then the pulse focuses at z f =(f d)n = mm inside the medium, which yields the optimum value γ = 2 for CEP freezing. For a pulse of Gaussian spectrum p(ω)=exp[δt 2 (ω ω 0 ) 2 /4] (Δt = fs), Fig. 5(a) evidences that the foci of the extreme frequencies ω 0 2/Δt in the spectrum (red and blue curves) are sizeably shifted from the focus at ω 0, remaining nevertheless within the pulse depth of focus (2nL R at ω 0 ). Figure 5(b) shows that the CEP shifts (open circles) evaluated numerically from Eqs. (7) and (11) (including a chirp in p(ω) to compensate for second- and third-order dispersion in the propagation up to the pulse focus), reproduce the slowing down of the CEP predicted by Eq. (23) (solid curve), while the CEP shift due only to material dispersion and Gouy s phase shift (dashed curve), i.e., in absence of dispersion-induced focal shift, would be about three times larger in the secondhalf of the focus. The pulse remains nearly transform-limited upon passage through the focus. The small distortion in the amplitude and phase of the pulse envelope, observed in Fig. 5(c) as pulse narrowing and blue shift of the oscillations, cause the deviations of the CEP shift from the prediction of theory [open circles and solid curve in Fig. 5(b)]. Fig. 5. For focusing of a two-cycle pulse in fused silica: (a) Caustics surfaces s(z) about the focus for the carrier-frequency (black curve) and sideband red and blue frequencies (red and blue curves). (b) CEP shifts numerically calculated (open circles), from Eq. (23) (solid curve), and from Eq. (23) neglecting the effect of dispersion-induced focal shift (dashed curve). (c) Pulse forms at the focus z f (solid black curve) and at z f + nl R (dashed red curve). See the text for the focusing geometry, input pulse and medium thickness. 6. Conclusions Equation (18) for the phase shift experienced by few-cycle pulses in a focal volume, including the effects of dispersive media placed in the focal volume or before it, is expected to be useful in the design and interpretation of experiments involving phase-sensitive interactions of focused few-cycle pulses with matter. As relevant examples, we have shown that the CEP shift in the focal volume due to Gouy s phase and/or material dispersion can be significantly reduced, not only by introducing dispersion in the focal spot size, as previously reported, but also, and more (C) 2009 OSA 26 October 2009 / Vol. 17, No. 22 / OPTICS EXPRESS 19433

11 feasibly, by means of a dispersion in the focus position. For experiments requiring focusing in vacuum or in dispersive media, the appropriate amount of dispersion in the focus position for optimum CEP freezing is easily controlled with the thickness of the dispersive media placed in the focusing path or in the focal volume. Acknowledgements M. A. P. acknowledges financial support from projects Acción Integrada HU and Acción Integrada HH of the Ministerio de Ciencia e Innovación of Spain. P. D. is a grantee of the János Bolyai Research Scholarship of the Hungarian Academy of Sciences and acknowledges support from the Hungarian Scientific Research Fund (OTKA project F60256) (C) 2009 OSA 26 October 2009 / Vol. 17, No. 22 / OPTICS EXPRESS 19434

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