Abstract

Recently, using parageometrical optics concepts, a hybrid, diffractive-refractive, lens triplet has been suggested to significantly improve the spatiotemporal resolution of light spots in multifocal processing with femtosecond laser pulses. Here, we carry out a rigorous wave-optics analysis, including the spatiotemporal nature of the wave equation, to elucidate both the spatial extent of the diffractive spots and the temporal duration of the pulse at the output plane. Specifically, we show nearly transform-limited behavior of diffraction maxima. Moreover, the temporal broadening of the pulse is related to the group velocity dispersion, which can be pre-compensated for in practical applications. Finally, some numerical simulations of the spatiotemporal wave field at the output plane in a realistic case are provided.

© 2007 Optical Society of America

1. Introduction

The capacity to concentrate energy in a reduced spatiotemporal volume is one of the most appealing features of ultrashort laser pulses. In this way, femtosecond lasers constitute an attractive tool for high-quality microstructuring and, even,nanostructuring. Tightly focused femtosecond pulses attain the energy required for multiphoton absorption, optical field ionization, and the corresponding Coulomb explosion. For a single refractive lens, large pulse bandwidth prevents from diffraction-limited focusing, both in the spatial and in the temporal domain [1]. Several methods have been reported to deal with this matter. In particular, temporal stretching of the focused pulse can be avoided by the use of focusing mirrors. Achromatic objectives [2, 3], annular apertures [4], and the combination of diffractive and refractive lenses [5-8] have also been employed to reduce temporal stretching to a great extent. The effect of wave front aberrations over the spatiotemporal light distribution in the focus has also been reported, with special emphasis deserved to spherocromatism [9-11]. Recently, realistic lenses were described by a combination of ray-tracing and wave optics methods [12].

Serial processing of large-scale structures is a time consuming process, which is performed by focusing femtosecond laser radiation onto the workpiece and next either scanning the beam or moving the sample. As alternative, parallel processing increases the scanning efficiency by using two-dimensional patterns of excitation spots that need only be dithered within the periodic bounds of the array [13]. Conventional implementations of multifocal processing utilize either microprisms arrays, microlenses arrays, or low-spot-density diffractive optical elements (DOEs) that are designed to generate an arbitrary pattern of light spots in the focal plane of the system [14-17]. High-speed parallel fabrication of microstructures has been performed by use of a spatial light modulator [18, 19]. DOE beam-splitters offer compactness,stability, efficiency, and uniformity [20]. Nevertheless, in the sub-100 fs range, the generation of well-defined focal intensity distributions based on DOEs is problematic because of the large spectral bandwidth of the pulsed light. The effect of the angular dispersion results in the spatiotemporal spread of the optical energy at the workpiece. Spatial broadening generates a distorted beam profile that is unsuitable for precise processing. Simultaneously, temporal stretching originates heat diffusion in the ablation process.

Several efforts have been conducted in the past few years to compensate for the angular dispersion in DOE-based beam-splitting of femtosecond light beams, with the temporal stretching remaining uncorrected. In the pioneering paper by Amako et al. [21], the femtosecond beam is splitted by means of a diffraction grating and the lateral walk-off between the different spectral components of each diffraction maximum is corrected by focusing the pulsed light with a diffractive lens. At this point it is worth mentioning that the focal length of a diffractive lens is proportional to the wave number of the incoming radiation, which is the key point to attain dispersion compensation. The pulse is temporally strechted at the focal plane, which leads to a considerable reduction of the laser peak power impiging over the workpiece. A Dammann grating and subsequent m-time density grating, arranged in a conventional double grating mounting, reduce the angular dispersion associated with the mth-order beam generated by the Damman element [22]. Here, pulse broadening due to diffraction is not as large as that of the grating pair used in other techniques because the low period of the Damman grating, 10 lines/mm. Finally, dispersion compensation in the spatial domain has allowed precise parallel microprocessing with a 40 nm bandwidth femtosecond laser pulse by using a 4 phase level DOE generating 22 focal points [23].

Recently, we have improved the limits of spatial and temporal resolution of light spots achievable in DOE-based multifocal processing [24]. We demonstrated that at least two diffractive lenses are needed to compensate for both the spatial and the temporal distortion associated with angular dispersion. In this case, a low-frequency diffraction grating splits the pulsed laser beam into several diffracted beams. These beams are gathered by a hybrid, diffractive-refractive, lens triplet and focused to an array of transform-limited, both spatial and temporal, light spots at its back focal plane. A resolution improvement of up to an order of magnitude has been demonstrated for a 20 fs pulse duration. In contrast with the heuristic treatment of Ref. [21], where the transmission of the femtosecond pulse through the lens triplet is analyzed in the framework of geometrical optics, here diffraction effects are taken into account. Note the geometrical optical theory only provides an estimate for the evaluation of the spatiotemporal resolution of light spots. The purpose of this paper is to carry out a wave optics analysis in detail, including the spatiotemporal nature of the wave equation, to elucidate the limits of both spatial focusing and temporal broadening of the pulse at the output plane. The paper contents of the paper is organized as follows. In Section 2 we derive the basic equation for the light transport of an ultrashort pulse through a misaligned optical setup within the framework of the Fresnel-Kirchhoff diffraction formula. The analysis is carried out in one step by means of generalized ray matrices. In Section 3, this general formulation is evaluated under a second order analysis. In Section 4, the above results are specifically applied to the conventional beam-splitting problem where the multibeams diffractive by a low-frequency diffraction grating are gathered with an achromatic lens doublet. Corresponding results achieved with the hybrid, diffractive-refractive, lens triplet are discussed in Section 5.Here, we emphasize the residual spatial and temporal stretching. Finally, in Section 6, we provide results of computer simulations to illustrate the dispersion compensation capabilities of our proposal.

2. Basic theory.

In this section we develop, for future interest, an analytical model to calculate the spatiotemporal dependence of the electrical field of an ultrashort pulse at the output plane of a misaligned optical setup, within the framework of the Fresnel-Kirchhoff diffraction formula. Following [25], the optical device is described by means of a 3x3 ray transfer matrix ABCDEF. To simplify the mathematical analysis and without loss of generality, we consider only one transverse coordinate. The extension to the two dimensional case is straightforward.

To study light propagation, we first consider one spectral component of frequency ω within the spectrum of the pulsed beam with carrier frequency ωo. A Fourier backtransformation of this solution then gives the desired spatiotemporal dependence of the field. Figure 1 shows the geometry to which we will refer in what follows. A monochromatic wave with amplitude Uin (x;ω) is assumed to be incident at the input plane of a misaligned optical device consisting of a low-frequency diffraction grating, with spatial period p, cascaded with a rotationally symmetric,but otherwise arbitrary, ABCD focusing setup. The grating splits the pulsed radiation into several diffracted beams. These beams are gathered by the collecting optics and focused to an array of light spots. For the n-order diffracted beam, the light amplitude at the output plane can be evaluated through the equation

 

Fig. 1. Schematic diagram of a Fourier beam splitter. The system, form the input grating to the output plane, is fullydescribed through the ABCDEF ray transfer matrix.

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Uoutxω=ω2πcBiexp[iωLc]exp[iDBω2cx2]×
×Uin(x′;ω)exp[iABω2cx′2]exp[iωcBx′(xE)]dx.

Here, the symbol L stands for the on-axis optical path length between the input and the output planes and c is the speed of the light. Only the case of a low-frequency diffraction grating is considered so that the paraxial approximations remains valid, but the full frequency-dependence of the grating equation is retained. In general, the matrix coefficients A, B, C, and D are wavelength-dependent. Throughout this paper, we will refer to the value of any wavelength-dependent parameter evaluated at the carrier frequency by employing the subscript o, for example, Ao = A(ωo). For the case of the optical device sketched in Fig. 1, the spatial-shift coefficient E is given, for the n-order diffraction maxima, as

E=nB2πcωp.

The output instantaneous irradiance distribution Iout (x;t) is obtained as the modulus square of the inverse (temporal) Fourier transform of Eq. (1), i.e.,

Ioutxt=ʃUoutxωexp[iω˜t]dω˜2,

with ω~ =ωωo. Generally, for complex optical systems, Eqs. (1) and (3) do not have an analytical solution and they must be solved numerically [12].

We further proceed by assuming a transform-limited Gaussian-shaped, both spatial and temporal, input pulse. In mathematical terms, the input electric field is given by Uin(x;t)=exp⌊−x 2 / 4σ 2 x⌋exp⌊−t 2 / 4σ 2 t⌋, where σx and σt denote the root-mean-square (rms) width of the spatial and temporal irradiance profiles, respectively. Recall that for a Gaussian pulse the full width at 1/e 2 of the intensity profile is four times the rms width. For this waveform, the input field in the spectral domain is (Uin(x;ω)= exp⌊−x 2 / 4σ 2 x⌋exp⌊−σ 2 t ω 2⌋. After introduction of the above waveform into Eq. (1), we obtain

Uoutxω=ω2πcBiexp[iωLc]exp[σt2ω˜2]×
exp[iωB2c(Dx2σx2A(xE)24σ′2)]exp[(xE)24σ′2],

with σ2 = σ 2 x A 2+(c 2 B 2 / 4ω 2 σ 2 x). Note that the last exponential term is a spatial Gaussian function peaked at x=E and with spatial extent σ. Generally, for a fixed diffraction order of the grating, each temporal frequency of the pulsed provides a different value of E, as indicated in Eq. (2), causing an angular dispersion.

3. Second order analysis.

We begin by considering the factor ω/B in front of the exponential functions. It has been shown that, for a singlet lens, we can approximate ω/Bωo/Bo with a negligible error for pulse durations longer than 15 fs in the focal region [10]. Of course, the error is even smaller for wavelength corrected focusing systems.

Equation (4) is further simplified by the use of the Taylor series expansion up to second order around ωo of the argument inside the phase exponentials. Namely,

ωLcαo+α1ω˜+α22ω˜2and
ωB2c(Dx2σx2A(xE)24σ′2)βo(x)+β1(x)ω˜+β2(x)2ω˜2,

where, of course,

αi=iωi(ωLc)ω=ωoandβi(x)=iωi(ωB2c{Dx2σx2A(xE)24σ′2})ω=ωo.

Whenever the bandwidth of the pulse is only a small fraction of the carrier frequency, a linear approximation for the spatial-shift term E can be done

E2πcnpωoBo[1+ω˜(1BoBωω=ωo1ωo)].

Equation (7) can be written in a compact way as E = Eo + E 1 ω~. We also note that wavelength-dependence of coefficient E is the leading mechanism responsible for chromatic distortion of the output field. Thus, we assume σ (ω) ≡ σ (ωo). Finally, we introduce Eqs. (5) and (7) into Eq. (4) and we find

Uoutxω=ωo2πcBoiexp[i(αo+βo(x))]exp[i(α1+β1(x))ω˜]exp[i(α2+β2(x))ω˜22]×
exp[(σt2+E124σ′2(ωo))ω˜2]exp[(xEo)24σ′2(ωo)]exp[2(xEo)E1ω˜4σ′2(ωo)].

Relation (8) is the basic one for the following discussion. It describes, in the spectral domain and up to second order,the transformation of a short light pulse by the system of Fig. 1. The phase term that contains αo and βo(x) does not contribute to the field irradiance and will be omitted for further calculations. The exponential term linear with ω~ provides the group delay, GD. Specifically, coefficient α 1 produces a non-disturbing shift in the arrival time of the pulse and the presence of β 1(x) leads to the x-dependent temporal pulse distortion. The group-velocity dispersion, GVD, caused by the lenses material and diffraction, responsible for pulse stretching is included in α 2 and β 2(x), respectively. Furthermore, the spectral narrowing of the spectrum of the pulse caused by the angular dispersion is controlled by the quantity E 1/2σ (ωo). Of course, in the temporal domain, it results in the stretching of the pulse. Equation (8) also accounts for a spatial Gaussian function peaked at Eo and scaled the factor σ(ω). Finally, the last exponential term is related to the coupling between spatial and temporal coordinates.

4. Conventional grating-based spot array generator.

A schematic representation of the setup is shown in Fig. 2. The paraxial focal length of the achromatic doublet L1 is f(ω), with ∂f(ω)/∂ωw=wo = 0, and the output plane is located at a distance fo from the lens, that corresponds to the focal plane for ωo. The doublet has a wavelength-dependent on-axis optical path length, L = n 1 (ω)d 1 + n 2 (ω)d 2, where n 1,2 refers to the refractive index and d 1,2 to the center thickness of each material. The thickness of the glass substrate of the diffractive grating has been neglected in this analysis, as it is usually smaller compared to L. The ABCD matrix corresponding to light propagation between the grating and the output plane is

ABCD=1fof(ω)fo(2fof(ω))1f(ω)1fof(ω).

Taking into account Eqs. (2) and (4), we find

E=2cnπfopωo(1ω˜ωo)andσ′(ωo)=12cfoσxωo.

The achromatic requirement for the doublet leads to β1 (x) = 0. Following the paper by Kempe et al. [3], we take into account that 2n(ω)/∂ω 2w=wo = C (2) ∂n(ω)/∂ωw=wo/ωo where C (2) results from the Sellmeier equation and it is typically of the order of unity [3]. In this way 2 f(ω)/∂ω 2w=wo = 0 and β 2(x) = 0.

We note that the narrowing of the spectrum of the pulse at the diffraction maxima Eo is the most important cause for temporal broadening. The GVD of a conventional lens, provided by α 2, is a second order effect that causes a change in the phase of the different spectral components. For the optical device in Fig. 2, GVD effects are at least two orders of magnitude smaller than angular dispersion effects. Consequently the output spatiotemporal intensity is provided by

Ioutx′t=exp[(xEo)22σ′x2]exp[(tα1)22σ′t2],

where

σ′t2=σt2(1+4n2π2σx2p2ωo2σt2)andσ′x2=σ2(ωo)(1+4n2π2σx2p2ωo2σt2).

Equation (11) demonstrates that the output irradiance is simply the product of two expanded Gaussian functions. In this way, for the nth-order diffraction maximum, we obtain a relative stretching in the temporal domain, σ2 t /σ 2 t, which is just the same expression as the relative broadening for the spatial domain σ2 x/σ 2 (ωo). Note that the spatial extent increases with the diffraction order n. So diffraction spots are becoming more and more elliptically distorted. Some illustrative examples of the lack of spot resolution for multiple spots will be provided in Section 6.

 

Fig. 2. Sketch of the conventional grating based multifocal device.

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5. DOE-based beam splitter.

We focus our attention on the system shown in Fig. 3. It is constituted by an achromatic doublet L1 and two kinoform diffractive lenses DL1 and DL2 with image focal length Z = Zoω/ωo and Z′ = Zo ω/ωo, respectively. The input grating, located at a distance z from L, is illuminated by the Gaussian-shaped input pulse. Axial distances fo, d and d′ denote arbitrary but fixed spacings between the different elements of the system. Following Ref. [24], a set of angular dispersion-compensated focal spots corresponding to diffraction maxima are achieved at the output plane when matrix coefficients satisfy A(ωo)=0, (B/ω)/∂ω|ωo = 0 and ∂A/∂ω|ωo =0. The above conditions are fulfilled when d 2 = −Zo Zo and d′ = −d 2/(d + 2Zo) [24].

 

Fig. 3. Representation of the dispersion-compensation hybrid device for spot array generation.

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To proceed with the wave optics analysis, we compute the overall wave matrix ABCD corresponding to light propagation between the grating plane and the output plane. It results from the product of wave matrices associated either with free-space propagation or to a passage through a lens multiplied in the reverse of the order in which the operations are encountered, as

ABCD=1d0110ωoZ′oω11d0110ωoZoω1
1fod01101f(ω)11z01.

It is straightforward matter to show that

E=Eo=2πncZofopωo(d+2Zo),andσ′(ωo)=cfoZo2ωoσx(d+2Zo).

Thus, E1=0 and, from Eq. (8), the output spatiotemporal intensity is given by

Ioutxt=exp[(xEo)22σ′2(ωo)]exp[(tα1)22σ′t2]

where σ2 t = σ 2 t + (α 2 + β 2(x)/2σt)2. To reach the above equation, several facts must be indicated. On the one hand, we consider light intensity only at the vicinity of the different diffraction maxima xEo. Thus, β 1(x) ≅ β 1(Eo), that produces a non-disturbing temporal delay at the focal spot, and β 2(x) ≅ β 2(Eo). Furthermore, from Eq. (5), we deduce that ∣β 1(x)∣ ≪ α 1. Note that the GVD introduced by diffraction, β 2(x), shows an anomalous dispersion that is larger for off-axis spots. Equation (15) indicates that the spatial width of the n-order diffraction maximum is essentially the one achieved for CW illumination for the frequency ωo. Dispersion compensation drastically reduces the lateral walk-off between the different spectral components at the light spots. It results in an improved available bandwidth and, thus, in a negligible temporal broadening of the energy. In fact, the temporal width of the output pulse is limited, in a first order approximation, by the total GVD. These statements are completely general and not only restricted to Gaussian beams, so our proposed system is a suitable tool for high precision multispot processing with minimal thermal damage.

6. Comparative numerical simulations.

We perform computer simulations to verify the goodness of our theoretical approach. In the numerical recreation, a 100 lines/inch diffraction grating is illuminated by a Ti:Sapphire laser producing femtosecond Gaussian pulses centered at λo = 2πc/ωo =800 nm with an input beam size of σx=5mm. Both in Figs. 2 and 3, the achromatic doublet is a biconvex lens of fo=80 mm made of BK7 (crown) and SF5 (flint) with center thickness of dBK7= 5 mm and dSF5= 2.5 mm. The consideration of the dispersion formula for these materials leads to α1=39.5 ps and α2=470 fs2. In Fig. 3, the diffractive lenses have an image focal length of Zo = −Zo =70 mm and the axial distances are d′=d=70 mm and z=80 mm.

The behaviour of both multifocal generators in the Fresnel approximation is simulated by means of Eqs. (3) and (4). We emphasize no approximation is made in order to compute the instantaneous intensity. In Fig. 4(a) we plot the relative spatial broadening, σx/σ(ωo), against σt. The fifth-order diffraction maxima is considered. With a different scaling, we also show in the graph the same ratio for the setup of Fig. 2. It is apparently a noteworthy improvement of the spatial resolution. Analog results are obtained for the relative temporal broadening and shown in Fig. 4(b). In both plots, for the conventional grating-based spot array generator, there is a perfect matching between the numerical simulations and the results that would be obtained through approximated Eq. (12). In the case of the DOE-based beam splitter, the approximated Eq. (15) is no valid for pulses shorter of 50 fs, as can be seen from Fig. 4(a). For this short-pulse duration, there is not a simple analytical solution.

 

Fig. 4. Relative stretching versus the input pulse width for the fifth-order diffraction maximum (n=5) after focusing with an achromatic doublet (dashed line) and the system proposed in Fig. 3 (continuous line) for: (a) the spatial domain and (b) the temporal domain.

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We also calculate the integrated intensity,

I(t)=Ioutxtdt.

Again, we consider the fifth diffraction order and pulse durations: (a) σt= 50 fs and (b) σt= 21.24 fs (which corresponds to a FWHM of 50 fs). For comparison, the output gaussian beam obtained with a monochromatic laser emitting in 800 nm is also drawn. For the 50 fs pulse duration, the ouput beam obtained with our proposal and with the CW is the same, so lines overlap. For shorter pulse duration a slight difference between both cases can be observed. Be aware of the clear improvement in the spatial spot size elongation in comparison with the conventional setup.

 

Fig. 5. Integrated intensity profiles of the 5th diffraction order of a 100 lines/inch diffractive grating after focusing with an achromatic doublet (dashed line), the system of Fig. 3 (continuous line) and a monochromatic wave of 800 nm (dash/dot line) for an input pulse duration of: (a) σt= 50 fs and (b) σt= 21.24 fs.

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Finally, Fig. 6 shows an animation of the output spatiotemporal intensity distribution for the 5th diffraction maxima when the input pulse width varies from σt= 200 fs to 50 fs. The output profile, calculated numerically by using Eqs. (3) and (4), is shown in the right part of the figure for the achromatic doublet and in the left part for the DOE-based system. The size and duration of the output pulse is clearly reduced for short pulses when the system show in Fig. 3 is used. Consequently, due to the spatiotemporal concentration of power, our proposed setup is a good instrument for parallel microprocessing.

 

Fig. 6. (752 KB) This video shows the spatiotemporal profiles of the 5th diffraction order of a 100 lines/inch diffractive grating for an input pulse width that varies from σt = 200 to 50 fs. The right part of the figure is obtained after focusing with an achromatic lens double and the left part by focusing with the DOE-based system. Note that the time origin is chosen arbitrarily [Media 1]

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7. Conclusions

We have demonstrated a simple, chromatic dispersion compensation method for multifocal generation with femtosecond pulses that uses a combination of a fan-out grating and a hybrid diffractive-refractive device. The system provides an order of magnitude correction with respect to the spatiotemporal elongation obtained by a refractive doublet. For a transform-limited Gaussian pulses we modelled the spatiotemporal intensity of each diffraction order and we show that, for pulses shorter than 50 fs, we get dispersion-free pulse behavior, i.e. the transversally spread made by diffraction is corrected and the pulse duration is conserved. In summary, we propose an optical setup to greatly improve, with no scanning, the spatiotemporal resolution in parallel processing.

Acknowledgments

This research was funded by the Dirección General de Investigación Científica y Técnica, Spain, under the project FIS2004-02404 and FEDER. We also acknowledge financial support from the Generalitat Valenciana (grant GV05/110).

References and links

1. Z. Bor, “Distorsion of femtosecond laser pulses in lenses and lens systems,” J. Mod. Opt. 35,1907–1918 (1988). [CrossRef]  

2. Z. Bor, “Distorsion of femtosecond laser pulses in lenses,” Opt. Lett. 14,119–121 (1989). [CrossRef]   [PubMed]  

3. M. Kempe, U. Stamm, B. Wilhelmi, and W. Rudolph, “Spatial and temporal transformation of femtosecond laser pulses by lenses and lens systems,” J. Opt. Soc. Am. B 9,1158–1165 (1992). [CrossRef]  

4. M. Kempe, U. Stamm, and B. Wilhelmi, “Spatial and temporal transformation of femtosecond laser pulses by lenses with annular aperture,” Opt. Commun. 89,119–125 (1992). [CrossRef]  

5. 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]  

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

7. R. Piestun and D. A. B. Miller, “Spatiotemporal control of ultrashort optical pulses by refractive-diffractive-dispersive structured optical elements,” Opt. Lett. 26,1373–1375 (2001). [CrossRef]  

8. U. Fuchs, U. D. Zeitner, and A. Tünnermann,“Hybrid optics for focusing ultrashort laser pulses,” Opt. Lett. 31,1516–1518 (2006). [CrossRef]   [PubMed]  

9. M. Kempe and W. Rudolph, “Impact of chromatic and spherical aberration on the focusing of ultrashort light pulses and lenses,” Opt. Lett. 18,137–139 (1993). [CrossRef]   [PubMed]  

10. M. Kempe and W. Rudolph, “Femtosecond pulses in the focal region of lenses,” Phys. Rev. A 48,4721–4729 (1993). [CrossRef]   [PubMed]  

11. G. O. Mattei and M. A. Gil, “Spherical aberration in spatial and temporal transforming lenses of femtosecond laser pulses,” Appl. Opt. 38,1058–1064 (1999). [CrossRef]  

12. U. Fuchs, U. D. Zeitner, and A. Tünnermann, “Ultra-short pulse propagation in complex optical systems,” Opt. Express 13,3852–3861 (2005). [CrossRef]   [PubMed]  

13. S. Nolte, “Micromachining” in Ultrafast Optics: Technology and Application, M. E. Fermann, A. Galvanauskas, and G. Sucha, eds., (Marcel Dekker, New York,2003 ).

14. C. Momma, S. Nolte, G. Kamlage, G. von Alvensleben, and A. Tünnermann, “Beam delivery of femtosecond laser radiation by diffractive optical elements,” Appl. Phys. A 67,517–520 (1998). [CrossRef]  

15. Y. Nakata, T. Okada, and M. Maeda, “Fabrication of dot matrix, comb, and nanowire structures using laser ablation by interfered femtosecond laser beams,” Appl. Phys. Lett. 81,4239–4241 (2002). [CrossRef]  

16. L. Sacconi, E. Froner, R. Antolini, M. R. Taghizadeh, A. Choudhury, and F. S. Pavone, “Multiphoton multifocal microscopy exploiting a diffractive optical element,” Opt. Lett. 28,1918–1920 (2003). [CrossRef]   [PubMed]  

17. S. Matsuo, S. Juodkazis, and H. Misawa, “Femtosecond laser microfabrication of periodic structures using a microlens array,” Appl. Phys. A 80,683–685 (2005). [CrossRef]  

18. Y. Hayasaki, T. Sugimoto, A. Takita, and N. Nishida, “Variable holographic femtosecond laser processing by use of spatial light modulator,” Appl. Phys. Lett. 87,031101 (2005). [CrossRef]  

19. S. Hasegawa, Y. Hayasaki, and N. Nishida, “Holographic femtosecond laser processing with multiplexed phase Fresnel lenses,” Opt. Lett. 31,1705–1707 (2006). [CrossRef]   [PubMed]  

20. S. Sinzinger and J. Jahns, Microoptics (Wiley-VCH, Weinheim,2003). [CrossRef]  

21. J. Amako, K. Nagasaka, and N. Kazuhiro, “Chromatic-distorsion compensation in splitting and focusing of femtosecond pulses by use of a pair of diffractive optical elements,” Opt. Lett. 27,969–971 (2002). [CrossRef]  

22. Y. Kuroiwa, N. Takeshima, Y. Narita, S. Tanaka, and K. Hirao, “Arbitrary micropatterning method in femtosecond laser microprocessing using diffractive optical elements,” Opt. Express 12,1908–1915 (2004). [CrossRef]   [PubMed]  

23. G. Li, C. Zhou, and E. Dai, “Splitting of femtosecond laser pulses by using a Damman grating and compensation gratings,” J. Opt. Soc. Am. A 4,767–772 (2005). [CrossRef]  

24. G. Mínguez-Vega, J. Lancis, J. Caraquitena, V. Torres-Company, and P. Andrés, “High spatiotemporal resolution in multifocal processing with femtosecond laser pulses,” Opt. Lett. 31,2631–2633 (2006). [CrossRef]   [PubMed]  

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References

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  1. Z. Bor, "Distorsion of femtosecond laser pulses in lenses and lens systems," J. Mod. Opt. 35, 1907-1918 (1988).
    [CrossRef]
  2. Z. Bor, "Distorsion of femtosecond laser pulses in lenses," Opt. Lett. 14, 119-121 (1989).
    [CrossRef] [PubMed]
  3. M. Kempe, U. Stamm, B. Wilhelmi, and W. Rudolph, "Spatial and temporal transformation of femtosecond laser pulses by lenses and lens systems," J. Opt. Soc. Am. B 9, 1158-1165 (1992).
    [CrossRef]
  4. M. Kempe, U. Stamm, and B. Wilhelmi, "Spatial and temporal transformation of femtosecond laser pulses by lenses with annular aperture," Opt. Commun. 89, 119-125 (1992).
    [CrossRef]
  5. 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]
  6. E. Ibragimov, "Focusing of ultrashort laser pulses by the combination of diffractive and refractive elements," Appl. Opt. 34, 7280-7285 (1995).
    [CrossRef] [PubMed]
  7. R. Piestun and D. A. B. Miller, "Spatiotemporal control of ultrashort optical pulses by refractive-diffractive-dispersive structured optical elements," Opt. Lett. 26, 1373-1375 (2001).
    [CrossRef]
  8. U. Fuchs, U. D. Zeitner, and A. Tünnermann,"Hybrid optics for focusing ultrashort laser pulses," Opt. Lett. 31, 1516-1518 (2006).
    [CrossRef] [PubMed]
  9. M. Kempe and W. Rudolph, "Impact of chromatic and spherical aberration on the focusing of ultrashort light pulses and lenses," Opt. Lett. 18, 137-139 (1993).
    [CrossRef] [PubMed]
  10. M. Kempe and W. Rudolph, "Femtosecond pulses in the focal region of lenses," Phys. Rev. A 48, 4721-4729 (1993).
    [CrossRef] [PubMed]
  11. G. O. Mattei and M. A. Gil, "Spherical aberration in spatial and temporal transforming lenses of femtosecond laser pulses," Appl. Opt. 38, 1058-1064 (1999).
    [CrossRef]
  12. U. Fuchs, U. D. Zeitner, and A. Tünnermann, "Ultra-short pulse propagation in complex optical systems," Opt. Express 13, 3852-3861 (2005).
    [CrossRef] [PubMed]
  13. S. Nolte, "Micromachining" in Ultrafast Optics: Technology and Application, M. E. Fermann, A. Galva-nauskas and G. Sucha, eds., (Marcel Dekker, New York, 2003).
  14. C. Momma, S. Nolte, G. Kamlage, G. von Alvensleben, and A. Tünnermann, "Beam delivery of femtosecond laser radiation by diffractive optical elements," Appl. Phys. A 67, 517-520 (1998).
    [CrossRef]
  15. Y. Nakata, T. Okada, and M. Maeda, "Fabrication of dot matrix, comb, and nanowire structures using laser ablation by interfered femtosecond laser beams," Appl. Phys. Lett. 81, 4239-4241 (2002).
    [CrossRef]
  16. L. Sacconi, E. Froner, R. Antolini, M. R. Taghizadeh, A. Choudhury, and F. S. Pavone, "Multiphoton multifocal microscopy exploiting a diffractive optical element," Opt. Lett. 28, 1918-1920 (2003).
    [CrossRef] [PubMed]
  17. S. Matsuo, S. Juodkazis, and H. Misawa, "Femtosecond laser microfabrication of periodic structures using a microlens array," Appl. Phys. A 80, 683-685 (2005).
    [CrossRef]
  18. Y. Hayasaki, T. Sugimoto, A. Takita, and N. Nishida, "Variable holographic femtosecond laser processing by use of spatial light modulator," Appl. Phys. Lett. 87, 031101 (2005).
    [CrossRef]
  19. S. Hasegawa, Y. Hayasaki, and N. Nishida, "Holographic femtosecond laser processing with multiplexed phase Fresnel lenses," Opt. Lett. 31, 1705-1707 (2006).
    [CrossRef] [PubMed]
  20. S. Sinzinger and J. Jahns, Microoptics (Wiley-VCH, Weinheim, 2003).
    [CrossRef]
  21. J. Amako, K. Nagasaka, and N. Kazuhiro, "Chromatic-distorsion compensation in splitting and focusing of femtosecond pulses by use of a pair of diffractive optical elements," Opt. Lett. 27, 969-971 (2002).
    [CrossRef]
  22. Y. Kuroiwa, N. Takeshima, Y. Narita, S. Tanaka, and K. Hirao, "Arbitrary micropatterning method in femtosecond laser microprocessing using diffractive optical elements," Opt. Express 12, 1908-1915 (2004).
    [CrossRef] [PubMed]
  23. G. Li, C. Zhou, and E. Dai, "Splitting of femtosecond laser pulses by using a Damman grating and compensation gratings," J. Opt. Soc. Am. A 4, 767-772 (2005).
    [CrossRef]
  24. G. Mínguez-Vega, J. Lancis, J. Caraquitena, V. Torres-Company, and P. Andrés, "High spatiotemporal resolution in multifocal processing with femtosecond laser pulses," Opt. Lett. 31, 2631-2633 (2006).
    [CrossRef] [PubMed]
  25. O. E. Martínez, "Matrix formalism for pulse compressors," IEEE J. Quantum Electron. 24, 2530-2536, (1988).
    [CrossRef]

2006 (3)

2005 (4)

G. Li, C. Zhou, and E. Dai, "Splitting of femtosecond laser pulses by using a Damman grating and compensation gratings," J. Opt. Soc. Am. A 4, 767-772 (2005).
[CrossRef]

S. Matsuo, S. Juodkazis, and H. Misawa, "Femtosecond laser microfabrication of periodic structures using a microlens array," Appl. Phys. A 80, 683-685 (2005).
[CrossRef]

Y. Hayasaki, T. Sugimoto, A. Takita, and N. Nishida, "Variable holographic femtosecond laser processing by use of spatial light modulator," Appl. Phys. Lett. 87, 031101 (2005).
[CrossRef]

U. Fuchs, U. D. Zeitner, and A. Tünnermann, "Ultra-short pulse propagation in complex optical systems," Opt. Express 13, 3852-3861 (2005).
[CrossRef] [PubMed]

2004 (1)

2003 (1)

2002 (2)

Y. Nakata, T. Okada, and M. Maeda, "Fabrication of dot matrix, comb, and nanowire structures using laser ablation by interfered femtosecond laser beams," Appl. Phys. Lett. 81, 4239-4241 (2002).
[CrossRef]

J. Amako, K. Nagasaka, and N. Kazuhiro, "Chromatic-distorsion compensation in splitting and focusing of femtosecond pulses by use of a pair of diffractive optical elements," Opt. Lett. 27, 969-971 (2002).
[CrossRef]

2001 (1)

1999 (1)

1998 (1)

C. Momma, S. Nolte, G. Kamlage, G. von Alvensleben, and A. Tünnermann, "Beam delivery of femtosecond laser radiation by diffractive optical elements," Appl. Phys. A 67, 517-520 (1998).
[CrossRef]

1995 (1)

1993 (2)

1992 (3)

1989 (1)

1988 (2)

Z. Bor, "Distorsion of femtosecond laser pulses in lenses and lens systems," J. Mod. Opt. 35, 1907-1918 (1988).
[CrossRef]

O. E. Martínez, "Matrix formalism for pulse compressors," IEEE J. Quantum Electron. 24, 2530-2536, (1988).
[CrossRef]

Amako, J.

Andrés, P.

Antolini, R.

Bor, Z.

Z. Bor, "Distorsion of femtosecond laser pulses in lenses," Opt. Lett. 14, 119-121 (1989).
[CrossRef] [PubMed]

Z. Bor, "Distorsion of femtosecond laser pulses in lenses and lens systems," J. Mod. Opt. 35, 1907-1918 (1988).
[CrossRef]

Caraquitena, J.

Choudhury, A.

Dai, E.

G. Li, C. Zhou, and E. Dai, "Splitting of femtosecond laser pulses by using a Damman grating and compensation gratings," J. Opt. Soc. Am. A 4, 767-772 (2005).
[CrossRef]

Froner, E.

Fuchs, U.

Gil, M. A.

Hasegawa, S.

Hayasaki, Y.

S. Hasegawa, Y. Hayasaki, and N. Nishida, "Holographic femtosecond laser processing with multiplexed phase Fresnel lenses," Opt. Lett. 31, 1705-1707 (2006).
[CrossRef] [PubMed]

Y. Hayasaki, T. Sugimoto, A. Takita, and N. Nishida, "Variable holographic femtosecond laser processing by use of spatial light modulator," Appl. Phys. Lett. 87, 031101 (2005).
[CrossRef]

Hirao, K.

Ibragimov, E.

Juodkazis, S.

S. Matsuo, S. Juodkazis, and H. Misawa, "Femtosecond laser microfabrication of periodic structures using a microlens array," Appl. Phys. A 80, 683-685 (2005).
[CrossRef]

Kamlage, G.

C. Momma, S. Nolte, G. Kamlage, G. von Alvensleben, and A. Tünnermann, "Beam delivery of femtosecond laser radiation by diffractive optical elements," Appl. Phys. A 67, 517-520 (1998).
[CrossRef]

Kazuhiro, N.

Kempe, M.

M. Kempe and W. Rudolph, "Impact of chromatic and spherical aberration on the focusing of ultrashort light pulses and lenses," Opt. Lett. 18, 137-139 (1993).
[CrossRef] [PubMed]

M. Kempe and W. Rudolph, "Femtosecond pulses in the focal region of lenses," Phys. Rev. A 48, 4721-4729 (1993).
[CrossRef] [PubMed]

M. Kempe, U. Stamm, B. Wilhelmi, and W. Rudolph, "Spatial and temporal transformation of femtosecond laser pulses by lenses and lens systems," J. Opt. Soc. Am. B 9, 1158-1165 (1992).
[CrossRef]

M. Kempe, U. Stamm, and B. Wilhelmi, "Spatial and temporal transformation of femtosecond laser pulses by lenses with annular aperture," Opt. Commun. 89, 119-125 (1992).
[CrossRef]

Kuroiwa, Y.

Lancis, J.

Li, G.

G. Li, C. Zhou, and E. Dai, "Splitting of femtosecond laser pulses by using a Damman grating and compensation gratings," J. Opt. Soc. Am. A 4, 767-772 (2005).
[CrossRef]

Maeda, M.

Y. Nakata, T. Okada, and M. Maeda, "Fabrication of dot matrix, comb, and nanowire structures using laser ablation by interfered femtosecond laser beams," Appl. Phys. Lett. 81, 4239-4241 (2002).
[CrossRef]

Martínez, O. E.

O. E. Martínez, "Matrix formalism for pulse compressors," IEEE J. Quantum Electron. 24, 2530-2536, (1988).
[CrossRef]

Matsuo, S.

S. Matsuo, S. Juodkazis, and H. Misawa, "Femtosecond laser microfabrication of periodic structures using a microlens array," Appl. Phys. A 80, 683-685 (2005).
[CrossRef]

Mattei, G. O.

Miller, D. A. B.

Mínguez-Vega, G.

Misawa, H.

S. Matsuo, S. Juodkazis, and H. Misawa, "Femtosecond laser microfabrication of periodic structures using a microlens array," Appl. Phys. A 80, 683-685 (2005).
[CrossRef]

Momma, C.

C. Momma, S. Nolte, G. Kamlage, G. von Alvensleben, and A. Tünnermann, "Beam delivery of femtosecond laser radiation by diffractive optical elements," Appl. Phys. A 67, 517-520 (1998).
[CrossRef]

Nagasaka, K.

Nakata, Y.

Y. Nakata, T. Okada, and M. Maeda, "Fabrication of dot matrix, comb, and nanowire structures using laser ablation by interfered femtosecond laser beams," Appl. Phys. Lett. 81, 4239-4241 (2002).
[CrossRef]

Narita, Y.

Nishida, N.

S. Hasegawa, Y. Hayasaki, and N. Nishida, "Holographic femtosecond laser processing with multiplexed phase Fresnel lenses," Opt. Lett. 31, 1705-1707 (2006).
[CrossRef] [PubMed]

Y. Hayasaki, T. Sugimoto, A. Takita, and N. Nishida, "Variable holographic femtosecond laser processing by use of spatial light modulator," Appl. Phys. Lett. 87, 031101 (2005).
[CrossRef]

Nolte, S.

C. Momma, S. Nolte, G. Kamlage, G. von Alvensleben, and A. Tünnermann, "Beam delivery of femtosecond laser radiation by diffractive optical elements," Appl. Phys. A 67, 517-520 (1998).
[CrossRef]

Okada, T.

Y. Nakata, T. Okada, and M. Maeda, "Fabrication of dot matrix, comb, and nanowire structures using laser ablation by interfered femtosecond laser beams," Appl. Phys. Lett. 81, 4239-4241 (2002).
[CrossRef]

Pavone, F. S.

Piestun, R.

Rudolph, W.

Sacconi, L.

Sharp, T. E.

Stamm, U.

M. Kempe, U. Stamm, B. Wilhelmi, and W. Rudolph, "Spatial and temporal transformation of femtosecond laser pulses by lenses and lens systems," J. Opt. Soc. Am. B 9, 1158-1165 (1992).
[CrossRef]

M. Kempe, U. Stamm, and B. Wilhelmi, "Spatial and temporal transformation of femtosecond laser pulses by lenses with annular aperture," Opt. Commun. 89, 119-125 (1992).
[CrossRef]

Sugimoto, T.

Y. Hayasaki, T. Sugimoto, A. Takita, and N. Nishida, "Variable holographic femtosecond laser processing by use of spatial light modulator," Appl. Phys. Lett. 87, 031101 (2005).
[CrossRef]

Taghizadeh, M. R.

Takeshima, N.

Takita, A.

Y. Hayasaki, T. Sugimoto, A. Takita, and N. Nishida, "Variable holographic femtosecond laser processing by use of spatial light modulator," Appl. Phys. Lett. 87, 031101 (2005).
[CrossRef]

Tanaka, S.

Torres-Company, V.

Tünnermann, A.

von Alvensleben, G.

C. Momma, S. Nolte, G. Kamlage, G. von Alvensleben, and A. Tünnermann, "Beam delivery of femtosecond laser radiation by diffractive optical elements," Appl. Phys. A 67, 517-520 (1998).
[CrossRef]

Wilhelmi, B.

M. Kempe, U. Stamm, and B. Wilhelmi, "Spatial and temporal transformation of femtosecond laser pulses by lenses with annular aperture," Opt. Commun. 89, 119-125 (1992).
[CrossRef]

M. Kempe, U. Stamm, B. Wilhelmi, and W. Rudolph, "Spatial and temporal transformation of femtosecond laser pulses by lenses and lens systems," J. Opt. Soc. Am. B 9, 1158-1165 (1992).
[CrossRef]

Wisoff, P. J.

Zeitner, U. D.

Zhou, C.

G. Li, C. Zhou, and E. Dai, "Splitting of femtosecond laser pulses by using a Damman grating and compensation gratings," J. Opt. Soc. Am. A 4, 767-772 (2005).
[CrossRef]

Appl. Opt. (3)

Appl. Phys. A (2)

C. Momma, S. Nolte, G. Kamlage, G. von Alvensleben, and A. Tünnermann, "Beam delivery of femtosecond laser radiation by diffractive optical elements," Appl. Phys. A 67, 517-520 (1998).
[CrossRef]

S. Matsuo, S. Juodkazis, and H. Misawa, "Femtosecond laser microfabrication of periodic structures using a microlens array," Appl. Phys. A 80, 683-685 (2005).
[CrossRef]

Appl. Phys. Lett. (2)

Y. Hayasaki, T. Sugimoto, A. Takita, and N. Nishida, "Variable holographic femtosecond laser processing by use of spatial light modulator," Appl. Phys. Lett. 87, 031101 (2005).
[CrossRef]

Y. Nakata, T. Okada, and M. Maeda, "Fabrication of dot matrix, comb, and nanowire structures using laser ablation by interfered femtosecond laser beams," Appl. Phys. Lett. 81, 4239-4241 (2002).
[CrossRef]

IEEE J. Quantum Electron. (1)

O. E. Martínez, "Matrix formalism for pulse compressors," IEEE J. Quantum Electron. 24, 2530-2536, (1988).
[CrossRef]

J. Mod. Opt. (1)

Z. Bor, "Distorsion of femtosecond laser pulses in lenses and lens systems," J. Mod. Opt. 35, 1907-1918 (1988).
[CrossRef]

J. Opt. Soc. Am. A (1)

G. Li, C. Zhou, and E. Dai, "Splitting of femtosecond laser pulses by using a Damman grating and compensation gratings," J. Opt. Soc. Am. A 4, 767-772 (2005).
[CrossRef]

J. Opt. Soc. Am. B (1)

Opt. Commun. (1)

M. Kempe, U. Stamm, and B. Wilhelmi, "Spatial and temporal transformation of femtosecond laser pulses by lenses with annular aperture," Opt. Commun. 89, 119-125 (1992).
[CrossRef]

Opt. Express (2)

Opt. Lett. (8)

J. Amako, K. Nagasaka, and N. Kazuhiro, "Chromatic-distorsion compensation in splitting and focusing of femtosecond pulses by use of a pair of diffractive optical elements," Opt. Lett. 27, 969-971 (2002).
[CrossRef]

G. Mínguez-Vega, J. Lancis, J. Caraquitena, V. Torres-Company, and P. Andrés, "High spatiotemporal resolution in multifocal processing with femtosecond laser pulses," Opt. Lett. 31, 2631-2633 (2006).
[CrossRef] [PubMed]

L. Sacconi, E. Froner, R. Antolini, M. R. Taghizadeh, A. Choudhury, and F. S. Pavone, "Multiphoton multifocal microscopy exploiting a diffractive optical element," Opt. Lett. 28, 1918-1920 (2003).
[CrossRef] [PubMed]

S. Hasegawa, Y. Hayasaki, and N. Nishida, "Holographic femtosecond laser processing with multiplexed phase Fresnel lenses," Opt. Lett. 31, 1705-1707 (2006).
[CrossRef] [PubMed]

Z. Bor, "Distorsion of femtosecond laser pulses in lenses," Opt. Lett. 14, 119-121 (1989).
[CrossRef] [PubMed]

R. Piestun and D. A. B. Miller, "Spatiotemporal control of ultrashort optical pulses by refractive-diffractive-dispersive structured optical elements," Opt. Lett. 26, 1373-1375 (2001).
[CrossRef]

U. Fuchs, U. D. Zeitner, and A. Tünnermann,"Hybrid optics for focusing ultrashort laser pulses," Opt. Lett. 31, 1516-1518 (2006).
[CrossRef] [PubMed]

M. Kempe and W. Rudolph, "Impact of chromatic and spherical aberration on the focusing of ultrashort light pulses and lenses," Opt. Lett. 18, 137-139 (1993).
[CrossRef] [PubMed]

Phys. Rev. A (1)

M. Kempe and W. Rudolph, "Femtosecond pulses in the focal region of lenses," Phys. Rev. A 48, 4721-4729 (1993).
[CrossRef] [PubMed]

Other (2)

S. Sinzinger and J. Jahns, Microoptics (Wiley-VCH, Weinheim, 2003).
[CrossRef]

S. Nolte, "Micromachining" in Ultrafast Optics: Technology and Application, M. E. Fermann, A. Galva-nauskas and G. Sucha, eds., (Marcel Dekker, New York, 2003).

Supplementary Material (1)

» Media 1: AVI (752 KB)     

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Figures (6)

Fig. 1.
Fig. 1.

Schematic diagram of a Fourier beam splitter. The system, form the input grating to the output plane, is fullydescribed through the ABCDEF ray transfer matrix.

Fig. 2.
Fig. 2.

Sketch of the conventional grating based multifocal device.

Fig. 3.
Fig. 3.

Representation of the dispersion-compensation hybrid device for spot array generation.

Fig. 4.
Fig. 4.

Relative stretching versus the input pulse width for the fifth-order diffraction maximum (n=5) after focusing with an achromatic doublet (dashed line) and the system proposed in Fig. 3 (continuous line) for: (a) the spatial domain and (b) the temporal domain.

Fig. 5.
Fig. 5.

Integrated intensity profiles of the 5th diffraction order of a 100 lines/inch diffractive grating after focusing with an achromatic doublet (dashed line), the system of Fig. 3 (continuous line) and a monochromatic wave of 800 nm (dash/dot line) for an input pulse duration of: (a) σt = 50 fs and (b) σt = 21.24 fs.

Fig. 6.
Fig. 6.

(752 KB) This video shows the spatiotemporal profiles of the 5th diffraction order of a 100 lines/inch diffractive grating for an input pulse width that varies from σt = 200 to 50 fs. The right part of the figure is obtained after focusing with an achromatic lens double and the left part by focusing with the DOE-based system. Note that the time origin is chosen arbitrarily [Media 1]

Equations (21)

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

U out x ω = ω 2 πcBi exp [ i ωL c ] exp [ i D B ω 2 c x 2 ] ×
× U in ( x′ ; ω ) exp [ i A B ω 2 c x′ 2 ] exp [ i ω cB x′ ( x E ) ] dx .
E = nB 2 πc ωp .
I out x t = ʃ U out x ω exp [ i ω ˜ t ] d ω ˜ 2 ,
U out x ω = ω 2 πcBi exp [ i ωL c ] exp [ σ t 2 ω ˜ 2 ] ×
exp [ i ω B 2 c ( D x 2 σ x 2 A ( x E ) 2 4 σ′ 2 ) ] exp [ ( x E ) 2 4 σ′ 2 ] ,
ωL c α o + α 1 ω ˜ + α 2 2 ω ˜ 2 and
ω B 2 c ( D x 2 σ x 2 A ( x E ) 2 4 σ′ 2 ) β o ( x ) + β 1 ( x ) ω ˜ + β 2 ( x ) 2 ω ˜ 2 ,
α i = i ω i ( ωL c ) ω = ω o and β i ( x ) = i ω i ( ω B 2 c { D x 2 σ x 2 A ( x E ) 2 4 σ′ 2 } ) ω = ω o .
E 2 πcn p ω o B o [ 1 + ω ˜ ( 1 B o B ω ω = ω o 1 ω o ) ] .
U out x ω = ω o 2 πc B o i exp [ i ( α o + β o ( x ) ) ] exp [ i ( α 1 + β 1 ( x ) ) ω ˜ ] exp [ i ( α 2 + β 2 ( x ) ) ω ˜ 2 2 ] ×
exp [ ( σ t 2 + E 1 2 4 σ′ 2 ( ω o ) ) ω ˜ 2 ] exp [ ( x E o ) 2 4 σ′ 2 ( ω o ) ] exp [ 2 ( x E o ) E 1 ω ˜ 4 σ′ 2 ( ω o ) ] .
A B C D = 1 f o f ( ω ) f o ( 2 f o f ( ω ) ) 1 f ( ω ) 1 f o f ( ω ) .
E = 2 cn π f o p ω o ( 1 ω ˜ ω o ) and σ′ ( ω o ) = 1 2 c f o σ x ω o .
I out x′ t = exp [ ( x E o ) 2 2 σ′ x 2 ] exp [ ( t α 1 ) 2 2 σ′ t 2 ] ,
σ′ t 2 = σ t 2 ( 1 + 4 n 2 π 2 σ x 2 p 2 ω o 2 σ t 2 ) and σ′ x 2 = σ 2 ( ω o ) ( 1 + 4 n 2 π 2 σ x 2 p 2 ω o 2 σ t 2 ) .
A B C D = 1 d 0 1 1 0 ω o Z′ o ω 1 1 d 0 1 1 0 ω o Z o ω 1
1 f o d 0 1 1 0 1 f ( ω ) 1 1 z 0 1 .
E = E o = 2 πnc Z o f o p ω o ( d + 2 Z o ) , and σ′ ( ω o ) = c f o Z o 2 ω o σ x ( d + 2 Z o ) .
I out x t = exp [ ( x E o ) 2 2 σ′ 2 ( ω o ) ] exp [ ( t α 1 ) 2 2 σ′ t 2 ]
I ( t ) = I out x t dt .

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