Abstract

We report on a procedure to focalize few-cycle laser pulses in dispersive media with controlled waveform. Stationarity of the carrier-envelope phase for extended depth of focus is attained by shaping the spatial dispersion of the ultrashort beam. An adjustable group velocity is locally tuned in order to match a prescribed phase velocity at focus. A hybrid diffractive-refractive lens system is proposed to drive the wavefield to an immersion microscope objective under convenient broadband modulation. Numerical simulations demonstrate robustness over positioning of this dispersive beam expander.

©2008 Optical Society of America

1. Introduction

The so-called Gouy phase is the subject of continuous investigation [1, 2, 3] since in 1890 Louis G. Gouy published a celebrated paper [4] on the longitudinal phase delay of spherical beams. Its relevance is primarily purely academic; for instance, the superluminal phase velocity [5] found in the focal region of paraxial Gaussian beams may be understood in terms of this phase anomaly. Direct observation of Gouy phases has attracted the interest more recently in the framework of single-cycle focused beams [6, 7, 8, 9, 10]. In this context, on the contrary, it results of great importance for a wide variety of phenomena and applications involving high field physics and extreme nonlinear optics. The key point to highlight here is that Gouy phase dispersion is determinant in the waveform of broadband optical pulses [5], which is commonly parametrized by the carrier-envelope phase (CEP). Sensitivity to CEP is observed for instance in photoionization of atoms [11] and in photoelectron acceleration at metal surfaces mediated by surface plasmon polaritons [12, 13].

In general, focused pulses manifest strong phase dispersion along the focal region, so that enhanced spatial resolution is achieved in CEP-dependent phenomena. In some circumstances, however, increased depth of focus may be of convenience so that a stationary CEP should be required near the focus. It is remarkable that Gouy waves modes [14] show some control over on-axis phases demonstrating undistorted pulse focalization even in dispersive media. Practical realizations are driven by angular dispersion engineering of ultrashort laser beams. In this concern we have recently introduced the concept of dispersive imaging [15] as a tool for controlling the dispersive nature of broadband wavefields. Achromatic [16] and apochromatic [17] corrections of the angular spectrum of diffracted wavefields may be achieved with the use of highly-dispersive lenses such as kinoform-type zone plates. Incidentally, this is not a new idea; compensation of longitudinal chromatic aberrations [18, 19] and also diffraction-induced chromatic mismatching of Fraunhofer patterns [20, 21] have been proposed elsewhere.

In this paper we exploit dispersive imaging assisted by zone plates to gain control over the waveforms of focused laser beams as approaches to Gouy waves modes. For simplicity, the theoretical analysis is addressed to pulse Gaussian beams. We perform a lens system design using the ABCD matrix formalism providing fundamental attributes of a dispersive beam expander capable of adjusting conveniently the spatial dispersion of the collimated input beam. Finally, numerical simulations show CEP stationarity of few-cycle focused pulses in dispersive media along the optical axis near the focus.

2. Background

Let us consider a focused pulse. If we neglect pulse envelope distortion in the focal volume, a reasonable statement based on the short-path propagation in the region of interest, the on-axis temporal evolution of the (real) field can be conveniently given as

E(t,z)=E0(t')cos(ω0t'ϕ0),

where t′=t-φ̇0 denotes a local time, ϕ 0=φ 0-ω 0 φ̇0 is the carrier-envelope phase being φ(z,ω) the complex argument of the spectrum and ω 0 the carrier frequency. In the following, a dot over a parameter stands for a derivative with respect to ω, and a subscript 0 denotes its evaluation at the specific frequency ω=ω 0; therefore, the carrier phase φ 0 symbolizes φ(z,ω 0) and the envelope delay φ̇0 represents ωφ computed at ω=ω 0. Rigorously speaking, Eq. (1) is a sufficiently general expression and its validity is restricted to pulse envelopes E 0 described by symmetric even functions.

Stationarity of the CEP, i.e. conservation of φ 0 along with the z coordinate, entails a spatiotemporal evolution of the field E expressed in terms of an spatially-invariant waveform (irrespective of its amplitude). In free space, a rigorous stationary CEP has been reported solely for X-waves [22]. However, finite-energy focal pulses demonstrate a variation of ϕ 0 upon z due to material dispersion and, importantly, the presence of the Gouy phase. At most, a stationary CEP may be achieved in a small region around a point of interest (the focal point) reaching the stationarity condition zϕ 0=0. We conveniently introduce the local wavenumber kz(z,ω)=zφ; we point out that on-axis pulse propagation evolves at a local phase velocity νp(z)=ω 0/kz 0 and group velocity vg(z)=k ̇-1 z0. Therefore

zϕ0=ω0(1vp1vg),

and CEP stationarity is attained when the phase velocity matches the group velocity, vp=vg, (both denoted as v). In this case, and assuming the focal point is located at z=0, the spatial-temporal dynamics of E in its vicinity may be given in terms of the variable t -z/v. Indeed, this result is more general as derived here.

3. Stationarity of CEP in focus

For illustration, we consider a focused Gaussian beam of waist (and focus) at z=0 propagating through a nonabsorbing dielectric medium of refractive index n p=ck/ω. The on-axiswavefield of time-harmonic dependence exp(-iωt) results

ε(ω,z)=zRexp(ikz)iqε(ω,0),

where q(ω,z)=z-izR is the complex radius of curvature, zR=ks 2/2 is the Rayleigh range (s is the size of the beam waist), and ε (ω,0) is the focal spectrum. The argument of the complex field ε (omitting irrelevant terms) is φ=kz-φG, being φG=arctan(z/zR) the Gouy phase. Finally, the Gaussian local wavenumber is

kz=kzRz2+zR2,

which is lower than the wavenumber k associated with a plane wave.

 

Fig. 1. Phase velocity (dashed line) and group velocity (solid lines) represented in (a) for pulsed Gaussian beams of different 0( 0=34) propagating in fused silica at ω 0=3.14 fs-1. In (b) we also show the CEP evolution along the optical axis.

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Particularly, the phase velocity of the Gaussian beam in terms of the normalized axial coordinate ζ=z/z R0 is

vp=c[np010(1+ζ2)]1,

where z R0 is the Rayleigh range at ω 0 and the Gaussian length =kzR/np. Note that 0=k 2 0 s 2 0/2n p0 also gives the area (conveniently normalized) of the beam waist at ω 0. On the other hand, the group velocity yields

vg=c[ng0+0(1-ζ2)0(1+ζ2)2]1,

where ng=ck̇ is the group index, and ℱ=ω żR/zR. In order to understand its significance, let us conceive a Gaussian beam exhibiting an invariant ℱ within a given spectral band around ω 0; this case would consider a dispersive Rayleigh range zR=z R0 (ω/ω 0), a model employed elsewhere [10]. Therefore, ℱ parametrizes the longitudinal dispersion of the Gaussian beam; this can be deduced also from the relationship ℱ=ω ̇/-1. More importantly, velocity matching vp=vg at focus is found if

0=(1+0Δn0),

being Δn=ng-np the difference of group index and refractive index in the dispersive medium. Equivalently Eq. (7) reads ̇0=- 2 0Δn 0/ω 0. We conclude that CEP stationarity requires a given spatial dispersion of the wavefield to balance dispersion of the dielectric material Δn 0. In particular, Δn=0 in vacuum and therefore ℱ=-1, independently of the beam length.

In Fig. 1(a) we compare graphically the phase velocity and the group velocity of pulse Gaussian beams of 0=34 with several ratios ℱ0 propagating in fused silica. At the mean frequency ω 0=3.14 fs-1 we have n p0=1.458 and n g0=1.478. As expected, isodiffracting Gaussian beams evidencing an invariant Rayleigh range (=0) exhibit a constant group velocity along the optical axis. Of particular importance is the case 0=-1.685 obtained from Eq. (7), since phase and group velocities matches at the focal point. In this case the CEP, written with zero initial phase (ϕ0=0 at ζ=0) as

ϕ0(ζ)=arctan(ζ)0Δn0ζ0ζζ2+1,

proves stationarity features at the origin, where ∂ζϕ0=0, as shown in Fig. 1(b). Moreover, Eq. (7) leads the on-axis CEP evolution to ultraflattened curves in the focal region since, in fact, ζ=0 is a saddle point where 2 ζϕ0=0. Consequently, a quasi-matching of phase and group velocities is also demonstrated in the neighborhood of the focus. At the boundaries of the focal volume, |ζ|=1, a significant mismatch of velocities is clear, which is quantitatively equivalent for different values of ℱ0. However, the CEP accumulates monotonically (in our numerical examples) such differences, in a certain way, displaying the mismatch history of the phase and group velocities and, as a consequence, the CEP provides its lowest (absolute) value for ℱ0=-1.685 in Fig. 1(b).

4. Optical system

Established that focalization of ultrashort laser beams is commonly produced with a CEP running loosely within the focal depth, here we analyze a procedure inducing controlled spatial dispersion to achieve a stationary CEP. We assume the (collimated) laser Gaussian beam, of input Rayleigh range z Rin=ωs 2 in/2c, propagates in vacuum and impinges over an objective lens in order to produce the required focused field of z R embedded in a medium of refractive index np. For convenience we assume a nondispersive infinity-corrected microscope objective of focal length f, and also beam truncation is ignored. Under the Debye approximation, the width of the Gaussian beam at the back focal plane

s=2fksin,

provided the Rayleigh range of the focused beam z Rf [23]. In this model, the focused laser beam is free of low-Fresnel-number focal shifts and longitudinal chromatic aberrations. As a consequence, the Gaussian lengths of the input (ℒin) and focused () beams satisfies in=ω 2 f 2/c 2 np, whereas longitudinal dispersion of the fields obeys in +=-Δn/np. Rather than concerning the focused pulse, we cast Eq. (7) showing explicitly the CEP constraint for the input Gaussian beam instead, yielding

in0=1+Δn0np0(ω02f2c2in01).

As a consequence, dispersive tailoring of the laser beam reaching in0 of Eq. (10) leads to CEP stationarity near the focus of the converging field. If focusing is performed in vacuum (Δn 0=0), in=1 characterizing input pulsed fields of a Gaussian width that is independent upon the frequency, on other hand a well-established assumption in numerous studies.[24]

When focusing is carried out in dispersive bulk media, apparently, Eq. (10) is not generally satisfied for the input laser beam. In principle, modification of f and in0 with different microscope objectives and beam expanders may result of practical convenience; however, these laser-beam tunes are produced at the cost of a field resizing at focus. This is evident if we express Eq. (10) as in0=1+Δn 0 ( 0-n-1 p0). In a majority of applications this is undesirable. We might conserve typical lengths of inputs and focused beams at ω 0 if, alternatively, spatial dispersion of the laser pulse is altered modifying in0. In this case we switch the group velocity of the pulse at focus in order to match the given phase velocity. Next we propose an optical arrangement specifically designed to convert a given (collimated) pulsed Gaussian beam of Gaussian length ̃in0 and dispersion ̃in0, violating Eq. (10), into an ultrashort collimated laser beam of the same length in0=̃in0 but in0 satisfying Eq. (10), as a previous step to the focusing action performed by the microscope objective.

 

Fig. 2. Schematic diagram of the focusing setup: L 1 and L 2 are components of the diffractive doublet; L 3 is the refractive lens and MO is the microscope objective.

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First we may give general features of the required system using the ABCD matrix formalism. A perfect replica (image) of the Gaussian beam is generated if we impose B=0 and C=0, simultaneously. This afocal system provides a lateral magnification A=D -1 of the image and, therefore, z Rin=A 2 z̃Rin is inferred. From ̃in0=̃in0 we derive that A 2 0=1; that is, beam expansion is unitary at ω 0. Also we obtain that

in0=˜in0+2ω0A˙0A0.

As a consequence, dispersion of the matrix element A is required in order to change the spatial dispersion properties of the Gaussian beam, switching from ̃in0 to in0. This is of relevance also concerning the temporal response of the ABCD system; the spectrum of the field at the output plane is altered being multiplied by the spectral modifier A -1. Under strong corrections at this preprocessing stage, the spectrum of the collimated laser beam may be greatly distorted [25]. Ultimately, a spectral shift of the mean (carrier) frequency may be induced, which urges to recalculate Eq. (10); this effect is neglected here.

We propose the spatio-spectral processing of the Gaussian beam to be performed by the thin-lens triplet depicted in Fig. 2. This is inspired in the setup shown in [15]. Firstly we introduce a doublet composed of kinoform-type zone plates (L 1 and L 2 in the plot) of dispersive focal lengths f 1=f 10 ω/ω 0 and f 2=f 20 ω/ω 0, respectively, followed by a nondispersive refractive lens (L 3) of focal length f 3=f 30. Phase-only diffractive lenses may have nearly-100% efficiency at ω 0; neglecting losses due for instance to material absorption and reflections, the kinoform-lens efficiency may be estimated as [26]

η(ω)=sin⁡c2(ωω01),

where sinc(x)=sin(πx)/(πx). We point out that Eq. (12) ignores material chromatic dispersion in the lens to simplify our discussion; this assumption may be given in practice using for instance a diffractive mirror [27]. Therefore, the strength of the response η decreases severely at increasing spectral shifts due to the appearance of undesirable diffraction orders. This is a relevant effect for subcycle pulses, however being negligible in ultrashort beams of sufficiently narrow power spectrum as considered below. Interestingly, diffractive optical elements with high-efficiency responses, significantly flatter than η from Eq. (12) and spanning the visible wavelength range, have been reported in [28] using twisted nematic liquid crystals.

We point out that the matrix elements are necessarily dispersive in virtue of the ω-dependent character of f 1 and f 2. The coupling distance of the diffractive doublet is d 1, and the distance from L 2 to L 3 is d 2. Furthermore, we assume that L 1 is placed at the waist plane of the input Gaussian beam of Rayleigh range z̃Rin. After propagating through the triplet, the output field of dispersion-compensated z Rin=-Im{qin} is examined at a distance

d3=[d1d2(d1+d2)f2]f3d1(d2f2f3)+f2(f3d2)

from L 3 where the waist image is found (B=0). For completeness, let us remind the well-known matrix equation for Gaussian beams

qin=Aq˜in+BCq˜in+D,

being q̃in=-iz̃Rin the complex radius of the input laser beam at the plane of L 1.

Distance d 3 depends upon ω so that we have control over the waist location of the imaged Gaussian beam for a single frequency. This is of concern when placing the microscope objective; thus let us impose d 3 at ω=ω 0 to be the distance from L 3 to the objective lens (see Fig. 2). This longitudinal chromatic dispersion of the waist plane leads to a dispersive spherical wave-front in the entrance plane of the objective. Ultimately this fact yields a longitudinal chromatic aberration of the broadband focused wavefield that may be neglected upon evaluation of the Gaussian Fresnel number NG=Re{qin}/zRin [23] of the beam impinging onto the objective lens; specifically |NG|≪1 should be satisfied.

Collaterally, a parametric solution satisfying, simultaneously, B=0 and C=0 cannot be found. A sufficient condition may be established by imposing C 0=0 and Ċ0=0 so that C vanishes in the vicinity of the carrier frequency. Such a tradeoff is given when

f3=d2+(d1f10)2d1,
f20=(d1f10)2f10.

This approach suggests that afocality of the the imaging setup is not rigorous but stationary around ω=ω 0.

Finally d 2=f 10-d 1 yields an unitary magnification A 0=1; alternatively we may consider d 2=3 f 10-d 1-2f 2 10/d 1 giving A 0=-1. Additionally, if

f10=d1(1+2Δin0)

(in both cases), where the mismatching Δℱ in0= in0-̃in0, then Ȧ0=A 0 Δℱ in0/2ω 0 as requested in (11). We conclude that the focal length of each lens composing the hybrid diffractive-refractive triplet is determined by the parameter Δ in0 and the positive axial distance d 1. This particular procedure cannot be applied if -4≤Δ in0<0 demanding a negative value of the axial distance d 2. In these cases we might consider a different arrangement: a triplet with L 1 being a nondispersive thin lens and L 3 a zone plate instead. Following the analysis given above, it can be proved that a satisfactory dispersive processing is also provided using positive values of d 1 and d 2 in the interval Δ in0<0.

 

Fig. 3. Evolution of the wavefield and envelope at different on-axis points ζ of the focal volume. Negative values of ζ are drawn in dashed lines and positive values in solid lines. Subfigures (a)–(b) correspond to isodiffracting Gaussian beams focused onto bulk fused silica. Time domain is given in terms of the local time t′. In (c)–(d) we employ a CEP-stationarity procedure.

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5. Numerical verification

Let us illustrate the validity of our approach with a numerical simulation. We consider an isodiffracting Gaussian beam focused by a microscope objective lens of f=4 mm, which propagates in fused silica near the focus. From a practical point of view, oil-immersion objectives are suitable to get rid of longitudinal chromatic aberration and focal shifts. Input widths s in=0.8 mm ( in0=3.5 107) allow that aperturing may be neglected in immersion objectives of NA>0.75. Numerical computations are performed with the Fresnel-Kirchhoff diffraction formula. At focus we employ a bandlimited signal ε of normalized (amplitude) spectrum 1-2Ω24 for |Ω|<1, where Ω=(ω-ω 0)/Δ0. The mean frequency is ω 0=3.14 fs-1 and the width of the spectral window is 0.8ω0, i.e. Δ0=0.4ω 0, providing a 4.8 fs (FWHM) transform-limited optical pulse. Figures 3(a) and 3(b) show on-axis waveforms near the focal point. The field envelopes are unaltered in spite of material dispersion; however, the carrier shifts inside the envelope in response of the velocity mismatching, v p=0.700c and vg=0.677c, evaluated at focus.

Insertion of the dispersive beam expander of Fig. 2 provides focal waveforms shown in Figs. 3(c) and 3(d). The contribution due to unwanted higher orders of a given kinoform lens is neglected; considering an input signal s like that observed at focus, ε, the integrated (mean) efficiency for the output,

η̅=11s2ηdΩ11s2dΩ,

theoretically reaches a value η̄=0.954, revealing that higher-order foci carry only 4.6% of the input intensity. Correspondingly, the spectral amplitude shaping on the transmitted beam may be disregarded; the spectral modifier √η altering the input signal s through a kinoform zone plate would induce a small pulse stretching (from 4.8 fs) up to 4.9 fs FWHM. We observe that CEP is significantly stabilized around the focus, where 0=34 and 0=-1.685. In order to minimize the Fresnel number, we have selected d 1=200 mm (for A 0=1) giving |NG| < 0.4 in our spectral window. Eqs. (15)(17) yield values for the focal distances of the triplet, f 10=439 mm, f 20=-130 mm, and f 3=525 mm; also d 2=f 10-d 1=239 mm. Importantly, evaluation of Eq. (13) provides a negative axial distance at ω 0; this incongruency is overcome placing the microscope objective immediately behind L 3 (d 3=0), as considered in the numerical simulations. This procedure is robust since such an adjustment still maintains the value of vg switched up to 0.700c; however, a minor asymmetric pulse broadening is observable. We point out that a relay system might be employed in cases where pulse distortions are dramatic. Placed between the dispersive beam expander and the microscope objective, the relay system may translate the appropriately-corrected virtual pattern of the broadband Gaussian pulse to the entrance plane of the focusing element.

6. Conclusions

In summary, we investigate the problem of focusing a few-cycle optical field with stationarity of the signal within the depth of focus. Following theoretical concerns, we establish basic requirements for an optical setup to drive pulsed beams with adjustable spatial dispersion. Ultimately group velocity is tuned at the focal point in order to match the prescribed phase velocity. We also present a dispersive beam expander consisting of a hybrid diffractive-refractive triplet capable of preparing the spatiotemporal response of ultrashort Gaussian pulses to maintain a stationary CEP along the optical axis when focused within dispersive media. Robustness of the optical arrangement is demonstrated upon coupling with immersion microscope objectives.

Acknowledgments

This research was funded by Ministerio de Ciencia e Innovación (grant HU2007-0020) and Generalitat Valenciana (grants GV/2007/043 and GVPRE/2008/005). Carlos J. Zapata-Rodríguez also acknowledges financial support from Ministerio de Ciencia e Innovación (Sub-programa de estancias de movilidad de profesores investigadores en centros extranjeros).

References and links

1. A. Rubinowicz, “On the anomalous propagation of phase in the focus,” J. Opt. Soc. Am. 54, 931–936 (1938).

2. R. W. Boyd, “Intuitive explanation of the phase anomaly of focused light beams,” J. Opt. Soc. Am. 70, 877–880 (1980). [CrossRef]  

3. S. Feng and H. G. Winful, “Physical origin of the Gouy phase shift,” Opt. Lett. 26, 485–487 (2001). [CrossRef]  

4. L. G. Gouy, “Sur une propriété nouvelle des ondes lumineuses,” Compt. Rendue Acad. Sci. (Paris) 110, 1251–1253 (1890).

5. M. A. Porras, “Diffraction effects in few-cycle optical pulses,” Phys. Rev. E 65, 026,606 (2002). [CrossRef]  

6. S. Feng, H. G. Winful, and R. W. Hellwarth, “Gouy shift and temporal reshaping of focused single-cycle electro-magnetic pulses,” Opt. Lett. 23, 385–387 (1998). [CrossRef]  

7. Z. L. Horváth and Z. Bor, “Reshaping of femtosecond pulses by the Gouy phase shift,” Phys. Rev. E 60, 2337–2346 (1999). [CrossRef]  

8. A. B. Ruffin, J. V. Rudd, J. F. Whitaker, S. Feng, and H. G. Winful, “Direct observation of the Gouy phase shift with single-cycle Terahertz pulses,” Phys. Rev. Lett. 83, 3410–3413 (1999). [CrossRef]  

9. F. Lindner, G. G. Paulus, H. Walther, A. Baltuška, E. Goulielmakis, M. Lezius, and F. Krausz, “Gouy phase shift for few-cycle laser pulses,” Phys. Rev. Lett. 92, 113,001 (2004). [CrossRef]  

10. 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, 753–755 (2005). [CrossRef]   [PubMed]  

11. E. Goulielmakis, M. Uiberacker, R. Kienberger, A. Baltuska, V. Yakovlev, A. Scrinzi, T. Westerwalbesloh, U. Kleineberg, U. Heinzmann, M. Drescher, and F. Krausz, “Direct measurement of light waves,” Science 305, 1267–1269 (2004). [CrossRef]   [PubMed]  

12. S. E. Irvine, P. Dombi, G. Farkas, and A. Elezzabi, “Influence of the carrier-envelope phase of few-cycle pulses on ponderomotive surface-plasmon electron acceleration,” Phys. Rev. Lett. 97, 146,801 (2006). [CrossRef]  

13. P. Dombi and P. Rácz, “Ultrafast monoenergetic electron source by optical waveform control of surface plasmons,” Opt. Express 16, 2887–2893 (2008). [CrossRef]   [PubMed]  

14. M. A. Porras, C. J. Zapata-Rodríguez, and I. Gonzalo, “Gouy wave modes: undistorted pulse focalization in a dispersive medium,” Opt. Lett. 32, 3287–3289 (2007). [CrossRef]   [PubMed]  

15. C. J. Zapata-Rodríguez and M. T. Caballero, “Ultrafast beam shaping with high-numerical-aperture microscope objectives,” Opt. Express 15, 15,308–15,313 (2007). [CrossRef]  

16. C. J. Zapata-Rodríguez and M. T. Caballero, “Isotropic compensation of diffraction-driven angular dispersion,” Opt. Lett. 32, 2472–2474 (2007). [CrossRef]   [PubMed]  

17. C. J. Zapata-Rodríguez, M. T. Caballero, and J. J. Miret, “Angular spectrum of diffracted wave fields with apochromatic correction,” Opt. Lett. 33, 1753–1755 (2008). [CrossRef]   [PubMed]  

18. T. E. Sharp and P. J. Wisoff, “Analysis of lens and zone plate combinations for achromatic focusing of ultrashort laser pulses,” Appl. Opt. 31, 2765–2769 (1992). [CrossRef]   [PubMed]  

19. E. Ibragimov, “Focusing of ultrashort laser pulses by the combination of diffractive and refractive elements,” Appl. Opt. 34, 7280–7285 (1995). [CrossRef]   [PubMed]  

20. G. M. Morris, “Diffraction theory for an achromatic Fourier transformation,” Appl. Opt. 20, 2017–2025 (1981). [CrossRef]   [PubMed]  

21. G. Mínguez-Vega, E. Tajahuerce, M. Fernández-Alonso, V. Climent, J. Lancis, J. Caraquitena, and P. Andrés, “Dispersion-compensated beam-splitting of femtosecond light pulses: Wave optics analysis,” Opt. Express 15, 278–288 (2007). [CrossRef]   [PubMed]  

22. M. A. Porras, G. Valiulis, and P. D. Trapani, “Unified description of Bessel X waves with cone dispersion and tilted pulses,” Phys. Rev. E 68, 016,613 (2003). [CrossRef]  

23. C. J. Zapata-Rodríguez, “Debye representation of dispersive focused waves,” J. Opt. Soc. Am. A 24, 675–686 (2007). [CrossRef]  

24. A. E. Siegman, Lasers (University Science Books, Mill Valley, 1986).

25. C. J. Zapata-Rodríguez, “Focal waveforms with tunable carrier frequency using dispersive aperturing,” Opt. Commun. 281, 4840–4843 (2008). [CrossRef]  

26. V. Moreno, J. F. Román, and J. R. Salgueiro, “High efficiency diffractive lenses: Deduction of kinoform profile,” Am. J. Phys. 65, 556–562 (1997). [CrossRef]  

27. J. M. Bendickson, E. N. Glytsis, and T. K. Gaylord, “Metallic surface-relief on-axis and off-axis focusing diffractive cylindrical mirrors,” J. Opt. Soc. Am. A 16, 113–130 (1999). [CrossRef]  

28. C. Oh and M. J. Escuti, “Achromatic diffraction from polarization gratings with high efficiency,” Opt. Lett. 33, 2287–2289 (2008). [CrossRef]   [PubMed]  

References

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  1. A. Rubinowicz, “On the anomalous propagation of phase in the focus,” J. Opt. Soc. Am. 54, 931–936 (1938).
  2. R. W. Boyd, “Intuitive explanation of the phase anomaly of focused light beams,” J. Opt. Soc. Am. 70, 877–880 (1980).
    [Crossref]
  3. S. Feng and H. G. Winful, “Physical origin of the Gouy phase shift,” Opt. Lett. 26, 485–487 (2001).
    [Crossref]
  4. L. G. Gouy, “Sur une propriété nouvelle des ondes lumineuses,” Compt. Rendue Acad. Sci. (Paris) 110, 1251–1253 (1890).
  5. M. A. Porras, “Diffraction effects in few-cycle optical pulses,” Phys. Rev. E 65, 026,606 (2002).
    [Crossref]
  6. S. Feng, H. G. Winful, and R. W. Hellwarth, “Gouy shift and temporal reshaping of focused single-cycle electro-magnetic pulses,” Opt. Lett. 23, 385–387 (1998).
    [Crossref]
  7. Z. L. Horváth and Z. Bor, “Reshaping of femtosecond pulses by the Gouy phase shift,” Phys. Rev. E 60, 2337–2346 (1999).
    [Crossref]
  8. A. B. Ruffin, J. V. Rudd, J. F. Whitaker, S. Feng, and H. G. Winful, “Direct observation of the Gouy phase shift with single-cycle Terahertz pulses,” Phys. Rev. Lett.  83, 3410–3413 (1999).
    [Crossref]
  9. F. Lindner, G. G. Paulus, H. Walther, A. Baltuška, E. Goulielmakis, M. Lezius, and F. Krausz, “Gouy phase shift for few-cycle laser pulses,” Phys. Rev. Lett. 92, 113,001 (2004).
    [Crossref]
  10. 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, 753–755 (2005).
    [Crossref] [PubMed]
  11. E. Goulielmakis, M. Uiberacker, R. Kienberger, A. Baltuska, V. Yakovlev, A. Scrinzi, T. Westerwalbesloh, U. Kleineberg, U. Heinzmann, M. Drescher, and F. Krausz, “Direct measurement of light waves,” Science 305, 1267–1269 (2004).
    [Crossref] [PubMed]
  12. S. E. Irvine, P. Dombi, G. Farkas, and A. Elezzabi, “Influence of the carrier-envelope phase of few-cycle pulses on ponderomotive surface-plasmon electron acceleration,” Phys. Rev. Lett. 97, 146,801 (2006).
    [Crossref]
  13. P. Dombi and P. Rácz, “Ultrafast monoenergetic electron source by optical waveform control of surface plasmons,” Opt. Express 16, 2887–2893 (2008).
    [Crossref] [PubMed]
  14. M. A. Porras, C. J. Zapata-Rodríguez, and I. Gonzalo, “Gouy wave modes: undistorted pulse focalization in a dispersive medium,” Opt. Lett. 32, 3287–3289 (2007).
    [Crossref] [PubMed]
  15. C. J. Zapata-Rodríguez and M. T. Caballero, “Ultrafast beam shaping with high-numerical-aperture microscope objectives,” Opt. Express 15, 15,308–15,313 (2007).
    [Crossref]
  16. C. J. Zapata-Rodríguez and M. T. Caballero, “Isotropic compensation of diffraction-driven angular dispersion,” Opt. Lett. 32, 2472–2474 (2007).
    [Crossref] [PubMed]
  17. C. J. Zapata-Rodríguez, M. T. Caballero, and J. J. Miret, “Angular spectrum of diffracted wave fields with apochromatic correction,” Opt. Lett. 33, 1753–1755 (2008).
    [Crossref] [PubMed]
  18. T. E. Sharp and P. J. Wisoff, “Analysis of lens and zone plate combinations for achromatic focusing of ultrashort laser pulses,” Appl. Opt. 31, 2765–2769 (1992).
    [Crossref] [PubMed]
  19. E. Ibragimov, “Focusing of ultrashort laser pulses by the combination of diffractive and refractive elements,” Appl. Opt. 34, 7280–7285 (1995).
    [Crossref] [PubMed]
  20. G. M. Morris, “Diffraction theory for an achromatic Fourier transformation,” Appl. Opt. 20, 2017–2025 (1981).
    [Crossref] [PubMed]
  21. G. Mínguez-Vega, E. Tajahuerce, M. Fernández-Alonso, V. Climent, J. Lancis, J. Caraquitena, and P. Andrés, “Dispersion-compensated beam-splitting of femtosecond light pulses: Wave optics analysis,” Opt. Express 15, 278–288 (2007).
    [Crossref] [PubMed]
  22. M. A. Porras, G. Valiulis, and P. D. Trapani, “Unified description of Bessel X waves with cone dispersion and tilted pulses,” Phys. Rev. E 68, 016,613 (2003).
    [Crossref]
  23. C. J. Zapata-Rodríguez, “Debye representation of dispersive focused waves,” J. Opt. Soc. Am. A 24, 675–686 (2007).
    [Crossref]
  24. A. E. Siegman, Lasers (University Science Books, Mill Valley, 1986).
  25. C. J. Zapata-Rodríguez, “Focal waveforms with tunable carrier frequency using dispersive aperturing,” Opt. Commun. 281, 4840–4843 (2008).
    [Crossref]
  26. V. Moreno, J. F. Román, and J. R. Salgueiro, “High efficiency diffractive lenses: Deduction of kinoform profile,” Am. J. Phys. 65, 556–562 (1997).
    [Crossref]
  27. J. M. Bendickson, E. N. Glytsis, and T. K. Gaylord, “Metallic surface-relief on-axis and off-axis focusing diffractive cylindrical mirrors,” J. Opt. Soc. Am. A 16, 113–130 (1999).
    [Crossref]
  28. C. Oh and M. J. Escuti, “Achromatic diffraction from polarization gratings with high efficiency,” Opt. Lett. 33, 2287–2289 (2008).
    [Crossref] [PubMed]

2008 (4)

2007 (5)

2006 (1)

S. E. Irvine, P. Dombi, G. Farkas, and A. Elezzabi, “Influence of the carrier-envelope phase of few-cycle pulses on ponderomotive surface-plasmon electron acceleration,” Phys. Rev. Lett. 97, 146,801 (2006).
[Crossref]

2005 (1)

2004 (2)

E. Goulielmakis, M. Uiberacker, R. Kienberger, A. Baltuska, V. Yakovlev, A. Scrinzi, T. Westerwalbesloh, U. Kleineberg, U. Heinzmann, M. Drescher, and F. Krausz, “Direct measurement of light waves,” Science 305, 1267–1269 (2004).
[Crossref] [PubMed]

F. Lindner, G. G. Paulus, H. Walther, A. Baltuška, E. Goulielmakis, M. Lezius, and F. Krausz, “Gouy phase shift for few-cycle laser pulses,” Phys. Rev. Lett. 92, 113,001 (2004).
[Crossref]

2003 (1)

M. A. Porras, G. Valiulis, and P. D. Trapani, “Unified description of Bessel X waves with cone dispersion and tilted pulses,” Phys. Rev. E 68, 016,613 (2003).
[Crossref]

2002 (1)

M. A. Porras, “Diffraction effects in few-cycle optical pulses,” Phys. Rev. E 65, 026,606 (2002).
[Crossref]

2001 (1)

1999 (3)

Z. L. Horváth and Z. Bor, “Reshaping of femtosecond pulses by the Gouy phase shift,” Phys. Rev. E 60, 2337–2346 (1999).
[Crossref]

A. B. Ruffin, J. V. Rudd, J. F. Whitaker, S. Feng, and H. G. Winful, “Direct observation of the Gouy phase shift with single-cycle Terahertz pulses,” Phys. Rev. Lett.  83, 3410–3413 (1999).
[Crossref]

J. M. Bendickson, E. N. Glytsis, and T. K. Gaylord, “Metallic surface-relief on-axis and off-axis focusing diffractive cylindrical mirrors,” J. Opt. Soc. Am. A 16, 113–130 (1999).
[Crossref]

1998 (1)

1997 (1)

V. Moreno, J. F. Román, and J. R. Salgueiro, “High efficiency diffractive lenses: Deduction of kinoform profile,” Am. J. Phys. 65, 556–562 (1997).
[Crossref]

1995 (1)

1992 (1)

1986 (1)

A. E. Siegman, Lasers (University Science Books, Mill Valley, 1986).

1981 (1)

1980 (1)

1938 (1)

1890 (1)

L. G. Gouy, “Sur une propriété nouvelle des ondes lumineuses,” Compt. Rendue Acad. Sci. (Paris) 110, 1251–1253 (1890).

Andrés, P.

Baltuska, A.

E. Goulielmakis, M. Uiberacker, R. Kienberger, A. Baltuska, V. Yakovlev, A. Scrinzi, T. Westerwalbesloh, U. Kleineberg, U. Heinzmann, M. Drescher, and F. Krausz, “Direct measurement of light waves,” Science 305, 1267–1269 (2004).
[Crossref] [PubMed]

Baltuška, A.

F. Lindner, G. G. Paulus, H. Walther, A. Baltuška, E. Goulielmakis, M. Lezius, and F. Krausz, “Gouy phase shift for few-cycle laser pulses,” Phys. Rev. Lett. 92, 113,001 (2004).
[Crossref]

Bendickson, J. M.

Bor, Z.

Z. L. Horváth and Z. Bor, “Reshaping of femtosecond pulses by the Gouy phase shift,” Phys. Rev. E 60, 2337–2346 (1999).
[Crossref]

Boyd, R. W.

Caballero, M. T.

Caraquitena, J.

Climent, V.

Dombi, P.

P. Dombi and P. Rácz, “Ultrafast monoenergetic electron source by optical waveform control of surface plasmons,” Opt. Express 16, 2887–2893 (2008).
[Crossref] [PubMed]

S. E. Irvine, P. Dombi, G. Farkas, and A. Elezzabi, “Influence of the carrier-envelope phase of few-cycle pulses on ponderomotive surface-plasmon electron acceleration,” Phys. Rev. Lett. 97, 146,801 (2006).
[Crossref]

Drescher, M.

E. Goulielmakis, M. Uiberacker, R. Kienberger, A. Baltuska, V. Yakovlev, A. Scrinzi, T. Westerwalbesloh, U. Kleineberg, U. Heinzmann, M. Drescher, and F. Krausz, “Direct measurement of light waves,” Science 305, 1267–1269 (2004).
[Crossref] [PubMed]

Elezzabi, A.

S. E. Irvine, P. Dombi, G. Farkas, and A. Elezzabi, “Influence of the carrier-envelope phase of few-cycle pulses on ponderomotive surface-plasmon electron acceleration,” Phys. Rev. Lett. 97, 146,801 (2006).
[Crossref]

Escuti, M. J.

Farkas, G.

S. E. Irvine, P. Dombi, G. Farkas, and A. Elezzabi, “Influence of the carrier-envelope phase of few-cycle pulses on ponderomotive surface-plasmon electron acceleration,” Phys. Rev. Lett. 97, 146,801 (2006).
[Crossref]

Feng, S.

Fernández-Alonso, M.

Gaylord, T. K.

Glytsis, E. N.

Gonzalo, I.

Goulielmakis, E.

F. Lindner, G. G. Paulus, H. Walther, A. Baltuška, E. Goulielmakis, M. Lezius, and F. Krausz, “Gouy phase shift for few-cycle laser pulses,” Phys. Rev. Lett. 92, 113,001 (2004).
[Crossref]

E. Goulielmakis, M. Uiberacker, R. Kienberger, A. Baltuska, V. Yakovlev, A. Scrinzi, T. Westerwalbesloh, U. Kleineberg, U. Heinzmann, M. Drescher, and F. Krausz, “Direct measurement of light waves,” Science 305, 1267–1269 (2004).
[Crossref] [PubMed]

Gouy, L. G.

L. G. Gouy, “Sur une propriété nouvelle des ondes lumineuses,” Compt. Rendue Acad. Sci. (Paris) 110, 1251–1253 (1890).

Heinzmann, U.

E. Goulielmakis, M. Uiberacker, R. Kienberger, A. Baltuska, V. Yakovlev, A. Scrinzi, T. Westerwalbesloh, U. Kleineberg, U. Heinzmann, M. Drescher, and F. Krausz, “Direct measurement of light waves,” Science 305, 1267–1269 (2004).
[Crossref] [PubMed]

Hellwarth, R. W.

Hof, K. D.

Horváth, Z. L.

Z. L. Horváth and Z. Bor, “Reshaping of femtosecond pulses by the Gouy phase shift,” Phys. Rev. E 60, 2337–2346 (1999).
[Crossref]

Ibragimov, E.

Irvine, S. E.

S. E. Irvine, P. Dombi, G. Farkas, and A. Elezzabi, “Influence of the carrier-envelope phase of few-cycle pulses on ponderomotive surface-plasmon electron acceleration,” Phys. Rev. Lett. 97, 146,801 (2006).
[Crossref]

Kienberger, R.

E. Goulielmakis, M. Uiberacker, R. Kienberger, A. Baltuska, V. Yakovlev, A. Scrinzi, T. Westerwalbesloh, U. Kleineberg, U. Heinzmann, M. Drescher, and F. Krausz, “Direct measurement of light waves,” Science 305, 1267–1269 (2004).
[Crossref] [PubMed]

Klein, M. W.

Kleineberg, U.

E. Goulielmakis, M. Uiberacker, R. Kienberger, A. Baltuska, V. Yakovlev, A. Scrinzi, T. Westerwalbesloh, U. Kleineberg, U. Heinzmann, M. Drescher, and F. Krausz, “Direct measurement of light waves,” Science 305, 1267–1269 (2004).
[Crossref] [PubMed]

Krausz, F.

F. Lindner, G. G. Paulus, H. Walther, A. Baltuška, E. Goulielmakis, M. Lezius, and F. Krausz, “Gouy phase shift for few-cycle laser pulses,” Phys. Rev. Lett. 92, 113,001 (2004).
[Crossref]

E. Goulielmakis, M. Uiberacker, R. Kienberger, A. Baltuska, V. Yakovlev, A. Scrinzi, T. Westerwalbesloh, U. Kleineberg, U. Heinzmann, M. Drescher, and F. Krausz, “Direct measurement of light waves,” Science 305, 1267–1269 (2004).
[Crossref] [PubMed]

Lancis, J.

Lezius, M.

F. Lindner, G. G. Paulus, H. Walther, A. Baltuška, E. Goulielmakis, M. Lezius, and F. Krausz, “Gouy phase shift for few-cycle laser pulses,” Phys. Rev. Lett. 92, 113,001 (2004).
[Crossref]

Lindner, F.

F. Lindner, G. G. Paulus, H. Walther, A. Baltuška, E. Goulielmakis, M. Lezius, and F. Krausz, “Gouy phase shift for few-cycle laser pulses,” Phys. Rev. Lett. 92, 113,001 (2004).
[Crossref]

Mínguez-Vega, G.

Miret, J. J.

Moreno, V.

V. Moreno, J. F. Román, and J. R. Salgueiro, “High efficiency diffractive lenses: Deduction of kinoform profile,” Am. J. Phys. 65, 556–562 (1997).
[Crossref]

Morris, G. M.

Oh, C.

Paulus, G. G.

F. Lindner, G. G. Paulus, H. Walther, A. Baltuška, E. Goulielmakis, M. Lezius, and F. Krausz, “Gouy phase shift for few-cycle laser pulses,” Phys. Rev. Lett. 92, 113,001 (2004).
[Crossref]

Porras, M. A.

M. A. Porras, C. J. Zapata-Rodríguez, and I. Gonzalo, “Gouy wave modes: undistorted pulse focalization in a dispersive medium,” Opt. Lett. 32, 3287–3289 (2007).
[Crossref] [PubMed]

M. A. Porras, G. Valiulis, and P. D. Trapani, “Unified description of Bessel X waves with cone dispersion and tilted pulses,” Phys. Rev. E 68, 016,613 (2003).
[Crossref]

M. A. Porras, “Diffraction effects in few-cycle optical pulses,” Phys. Rev. E 65, 026,606 (2002).
[Crossref]

Rácz, P.

Román, J. F.

V. Moreno, J. F. Román, and J. R. Salgueiro, “High efficiency diffractive lenses: Deduction of kinoform profile,” Am. J. Phys. 65, 556–562 (1997).
[Crossref]

Rubinowicz, A.

Rudd, J. V.

A. B. Ruffin, J. V. Rudd, J. F. Whitaker, S. Feng, and H. G. Winful, “Direct observation of the Gouy phase shift with single-cycle Terahertz pulses,” Phys. Rev. Lett.  83, 3410–3413 (1999).
[Crossref]

Ruffin, A. B.

A. B. Ruffin, J. V. Rudd, J. F. Whitaker, S. Feng, and H. G. Winful, “Direct observation of the Gouy phase shift with single-cycle Terahertz pulses,” Phys. Rev. Lett.  83, 3410–3413 (1999).
[Crossref]

Salgueiro, J. R.

V. Moreno, J. F. Román, and J. R. Salgueiro, “High efficiency diffractive lenses: Deduction of kinoform profile,” Am. J. Phys. 65, 556–562 (1997).
[Crossref]

Scrinzi, A.

E. Goulielmakis, M. Uiberacker, R. Kienberger, A. Baltuska, V. Yakovlev, A. Scrinzi, T. Westerwalbesloh, U. Kleineberg, U. Heinzmann, M. Drescher, and F. Krausz, “Direct measurement of light waves,” Science 305, 1267–1269 (2004).
[Crossref] [PubMed]

Sharp, T. E.

Siegman, A. E.

A. E. Siegman, Lasers (University Science Books, Mill Valley, 1986).

Tajahuerce, E.

Trapani, P. D.

M. A. Porras, G. Valiulis, and P. D. Trapani, “Unified description of Bessel X waves with cone dispersion and tilted pulses,” Phys. Rev. E 68, 016,613 (2003).
[Crossref]

Tritschler, T.

Uiberacker, M.

E. Goulielmakis, M. Uiberacker, R. Kienberger, A. Baltuska, V. Yakovlev, A. Scrinzi, T. Westerwalbesloh, U. Kleineberg, U. Heinzmann, M. Drescher, and F. Krausz, “Direct measurement of light waves,” Science 305, 1267–1269 (2004).
[Crossref] [PubMed]

Valiulis, G.

M. A. Porras, G. Valiulis, and P. D. Trapani, “Unified description of Bessel X waves with cone dispersion and tilted pulses,” Phys. Rev. E 68, 016,613 (2003).
[Crossref]

Walther, H.

F. Lindner, G. G. Paulus, H. Walther, A. Baltuška, E. Goulielmakis, M. Lezius, and F. Krausz, “Gouy phase shift for few-cycle laser pulses,” Phys. Rev. Lett. 92, 113,001 (2004).
[Crossref]

Wegener, M.

Westerwalbesloh, T.

E. Goulielmakis, M. Uiberacker, R. Kienberger, A. Baltuska, V. Yakovlev, A. Scrinzi, T. Westerwalbesloh, U. Kleineberg, U. Heinzmann, M. Drescher, and F. Krausz, “Direct measurement of light waves,” Science 305, 1267–1269 (2004).
[Crossref] [PubMed]

Whitaker, J. F.

A. B. Ruffin, J. V. Rudd, J. F. Whitaker, S. Feng, and H. G. Winful, “Direct observation of the Gouy phase shift with single-cycle Terahertz pulses,” Phys. Rev. Lett.  83, 3410–3413 (1999).
[Crossref]

Winful, H. G.

Wisoff, P. J.

Yakovlev, V.

E. Goulielmakis, M. Uiberacker, R. Kienberger, A. Baltuska, V. Yakovlev, A. Scrinzi, T. Westerwalbesloh, U. Kleineberg, U. Heinzmann, M. Drescher, and F. Krausz, “Direct measurement of light waves,” Science 305, 1267–1269 (2004).
[Crossref] [PubMed]

Zapata-Rodríguez, C. J.

Am. J. Phys. (1)

V. Moreno, J. F. Román, and J. R. Salgueiro, “High efficiency diffractive lenses: Deduction of kinoform profile,” Am. J. Phys. 65, 556–562 (1997).
[Crossref]

Appl. Opt. (3)

Compt. Rendue Acad. Sci. (Paris) (1)

L. G. Gouy, “Sur une propriété nouvelle des ondes lumineuses,” Compt. Rendue Acad. Sci. (Paris) 110, 1251–1253 (1890).

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (2)

Opt. Commun. (1)

C. J. Zapata-Rodríguez, “Focal waveforms with tunable carrier frequency using dispersive aperturing,” Opt. Commun. 281, 4840–4843 (2008).
[Crossref]

Opt. Express (3)

Opt. Lett. (7)

Phys. Rev. E (3)

Z. L. Horváth and Z. Bor, “Reshaping of femtosecond pulses by the Gouy phase shift,” Phys. Rev. E 60, 2337–2346 (1999).
[Crossref]

M. A. Porras, “Diffraction effects in few-cycle optical pulses,” Phys. Rev. E 65, 026,606 (2002).
[Crossref]

M. A. Porras, G. Valiulis, and P. D. Trapani, “Unified description of Bessel X waves with cone dispersion and tilted pulses,” Phys. Rev. E 68, 016,613 (2003).
[Crossref]

Phys. Rev. Lett. (3)

A. B. Ruffin, J. V. Rudd, J. F. Whitaker, S. Feng, and H. G. Winful, “Direct observation of the Gouy phase shift with single-cycle Terahertz pulses,” Phys. Rev. Lett.  83, 3410–3413 (1999).
[Crossref]

F. Lindner, G. G. Paulus, H. Walther, A. Baltuška, E. Goulielmakis, M. Lezius, and F. Krausz, “Gouy phase shift for few-cycle laser pulses,” Phys. Rev. Lett. 92, 113,001 (2004).
[Crossref]

S. E. Irvine, P. Dombi, G. Farkas, and A. Elezzabi, “Influence of the carrier-envelope phase of few-cycle pulses on ponderomotive surface-plasmon electron acceleration,” Phys. Rev. Lett. 97, 146,801 (2006).
[Crossref]

Science (1)

E. Goulielmakis, M. Uiberacker, R. Kienberger, A. Baltuska, V. Yakovlev, A. Scrinzi, T. Westerwalbesloh, U. Kleineberg, U. Heinzmann, M. Drescher, and F. Krausz, “Direct measurement of light waves,” Science 305, 1267–1269 (2004).
[Crossref] [PubMed]

Other (1)

A. E. Siegman, Lasers (University Science Books, Mill Valley, 1986).

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Figures (3)

Fig. 1.
Fig. 1. Phase velocity (dashed line) and group velocity (solid lines) represented in (a) for pulsed Gaussian beams of different 0( 0=34) propagating in fused silica at ω 0=3.14 fs-1. In (b) we also show the CEP evolution along the optical axis.
Fig. 2.
Fig. 2. Schematic diagram of the focusing setup: L 1 and L 2 are components of the diffractive doublet; L 3 is the refractive lens and MO is the microscope objective.
Fig. 3.
Fig. 3. Evolution of the wavefield and envelope at different on-axis points ζ of the focal volume. Negative values of ζ are drawn in dashed lines and positive values in solid lines. Subfigures (a)–(b) correspond to isodiffracting Gaussian beams focused onto bulk fused silica. Time domain is given in terms of the local time t′. In (c)–(d) we employ a CEP-stationarity procedure.

Equations (18)

Equations on this page are rendered with MathJax. Learn more.

E ( t , z ) = E 0 ( t ' ) cos ( ω 0 t ' ϕ 0 ) ,
z ϕ 0 = ω 0 ( 1 v p 1 v g ) ,
ε ( ω , z ) = z R exp ( i k z ) i q ε ( ω , 0 ) ,
k z = k z R z 2 + z R 2 ,
v p = c [ n p 0 1 0 ( 1 + ζ 2 ) ] 1 ,
v g = c [ n g 0 + 0 ( 1 - ζ 2 ) 0 ( 1 + ζ 2 ) 2 ] 1 ,
0 = ( 1 + 0 Δ n 0 ) ,
ϕ 0 ( ζ ) = arctan ( ζ ) 0 Δ n 0 ζ 0 ζ ζ 2 + 1 ,
s = 2 f k s i n ,
in 0 = 1 + Δ n 0 n p 0 ( ω 0 2 f 2 c 2 in 0 1 ) .
in 0 = ˜ in 0 + 2 ω 0 A ˙ 0 A 0 .
η ( ω ) = sin⁡ c 2 ( ω ω 0 1 ) ,
d 3 = [ d 1 d 2 ( d 1 + d 2 ) f 2 ] f 3 d 1 ( d 2 f 2 f 3 ) + f 2 ( f 3 d 2 )
q in = A q ˜ in + B C q ˜ in + D ,
f 3 = d 2 + ( d 1 f 10 ) 2 d 1 ,
f 20 = ( d 1 f 10 ) 2 f 10 .
f 10 = d 1 ( 1 + 2 Δ in 0 )
η ̅ = 1 1 s 2 η d Ω 1 1 s 2 d Ω ,

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