## Intuitive analysis of space-time focusing with double-ABCD calculation |

Optics Express, Vol. 20, Issue 13, pp. 14244-14259 (2012)

http://dx.doi.org/10.1364/OE.20.014244

Acrobat PDF (1121 KB)

### Abstract

We analyze the structure of space-time focusing of spatially-chirped pulses using a technique where each frequency component of the beam follows its own Gaussian beamlet that in turn travels as a ray through the system. The approach leads to analytic expressions for the axially-varying pulse duration, pulse-front tilt, and the longitudinal intensity profile. We find that an important contribution to the intensity localization obtained with spatial-chirp focusing arises from the evolution of the geometric phase of the beamlets.

© 2012 OSA

## 1. Introduction

1. K. Osvay, A. P. Kovacs, Z. Heiner, G. Kurdi, J. Klebniczki, and M. Csatari, “Angular dispersion and temporal change of femtosecond pulses from misaligned pulse compressors,” IEEE J. Sel. Top. Quantum Electron. **10**, 213–220 (2004). [CrossRef]

2. D. Oron, E. Tal, and Y. Silberberg, “Scanningless depth-resolved microscopy,” Opt Express **13**, 1468–1476 (2005). [CrossRef] [PubMed]

4. M. Durst, G. Zhu, and C. Xu, “Simultaneous spatial and temporal focusing in nonlinear microscopy,” Opt. Commun. **281**, 1796–1805 (2008). [CrossRef] [PubMed]

5. D. N. Vitek, D. E. Adams, A. Johnson, P. S. Tsai, S. Backus, C. G. Durfee, D. Kleinfeld, and J. A. Squier, “Temporally focused femtosecond laser pulses for low numerical aperture micromachining through optically transparent materials,” Opt. Express **18**, 18086–18094 (2010). [CrossRef] [PubMed]

6. D. N. Vitek, E. Block, Y. Bellouard, D. E. Adams, S. Backus, D. Kleinfeld, C. G. Durfee, and J. A. Squier, “Spatio-temporally focused femtosecond laser pulses for nonreciprocal writing in optically transparent materials,” Opt. Express **18**, 24673–24678 (2010). [CrossRef] [PubMed]

7. F. He, H. Xu, Y. Cheng, J. Ni, H. Xiong, Z. Xu, K. Sugioka, and K. Midorikawa, “Fabrication of microfluidic channels with a circular cross section using spatiotemporally focused femtosecond laser pulses,” Opt. Lett. **35**, 1106–1108 (2010). [CrossRef] [PubMed]

*because it improves the axial resolution for wide-field imaging*. In micromachining, we have shown that it strongly suppresses nonlinear propagation in a medium along the way to the focus, allowing machining on the back side of a transparent medium or on a surface immersed in water [5

5. D. N. Vitek, D. E. Adams, A. Johnson, P. S. Tsai, S. Backus, C. G. Durfee, D. Kleinfeld, and J. A. Squier, “Temporally focused femtosecond laser pulses for low numerical aperture micromachining through optically transparent materials,” Opt. Express **18**, 18086–18094 (2010). [CrossRef] [PubMed]

8. M. Coughlan, M. Plewicki, and R. Levis, “Parametric spatio-temporal control of focusing laser pulses,” Opt. Express **17**, 15808–15820 (2009). [CrossRef] [PubMed]

6. D. N. Vitek, E. Block, Y. Bellouard, D. E. Adams, S. Backus, D. Kleinfeld, C. G. Durfee, and J. A. Squier, “Spatio-temporally focused femtosecond laser pulses for nonreciprocal writing in optically transparent materials,” Opt. Express **18**, 24673–24678 (2010). [CrossRef] [PubMed]

9. P. G. Kazansky, W. Yang, E. Bricchi, J. Bovatsek, A. Arai, Y. Shimotsuma, K. Miura, and K. Hirao, ““Quill” writing with ultrashort light pulses in transparent materials,” Appl. Phys. Lett. **90**, 151120 (2007). [CrossRef]

10. W. Yang, P. G. Kazansky, Y. Shimotsuma, M. Sakakura, K. Miura, and K. Hirao, “Ultrashort-pulse laser calligraphy,” Appl. Phys. Lett. **93**, 171109 (2008). [CrossRef]

11. M. Durst, G. Zhu, and C. Xu, “Simultaneous spatial and temporal focusing for axial scanning,” Opt. Express **14**, 12243–12254 (2006). [CrossRef] [PubMed]

12. D. Zimmer, D. Ros, O. Guilbaud, J. Habib, S. Kazamias, B. Zielbauer, V. Bagnoud, B. Ecker, D. Hochhaus, and B. Aurand, “Short-wavelength soft-x-ray laser pumped in double-pulse single-beam non-normal incidence,” Phys. Rev. A **82**, 013803 (2010). [CrossRef]

13. O. E. Martinez, “Achromatic phase matching for second harmonic generation of femtosecond pulses,” IEEE J. Quantum Electron. **25**, 2464–2468 (1989). [CrossRef]

14. J. A. Fueloep, L. Palfalvi, M. C. Hoffmann, and J. Hebling, “Towards generation of mJ-level ultrashort THz pulses by optical rectification,” Opt. Express **19**, 15090–15097 (2011). [CrossRef]

3. G. Zhu, J. van Howe, M. Durst, W. Zipfel, and C. Xu, “Simultaneous spatial and temporal focusing of femtosecond pulses,” Opt. Express **13**, 2153–2159 (2005). [CrossRef] [PubMed]

15. A. G. Kostenbauder, “Ray-pulse matrices: a rational treatment for dispersive optical systems,” IEEE J. Quantum Electron. **26**, 1148–1157 (1990). [CrossRef]

16. V. Chauhan, J. Cohen, and R. Trebino, “Simple dispersion law for arbitrary sequences of dispersive optics,” Appl. Opt. **49**, 6840–6844 (2010). [CrossRef] [PubMed]

*ω*–component can be considered to propagate independently. The aim of the double-ABCD technique is not so much to calculate the dispersion of pulse stretching/compression systems, for which there are other techniques [16

16. V. Chauhan, J. Cohen, and R. Trebino, “Simple dispersion law for arbitrary sequences of dispersive optics,” Appl. Opt. **49**, 6840–6844 (2010). [CrossRef] [PubMed]

18. C. G. Durfee, J. Squier, and S. Kane, “A modular approach to the analytic calculation of spectral phase for grisms and other refractive/diffractive structures,” Opt Express **16**, 18,004–18,016 (2008). [CrossRef]

*ω*) space, since in linear propagation the spatial and frequency components can be considered as separable functions. As part of this analysis, we derive a general approach to starting with a field that has a known Fresnel propagation on-axis and modifying it to include the effects of tilting its propagation direction at an angle to the optical axis. The resulting expression is then used to analyze the pulse in the spatial and spectral domains. Finally, in Section 3 we investigate the structure of the pulse in the spatial/temporal domains, where we can obtain insight into the origins of the axial intensity localization and the scaling with the degree of spatial chirp.

### 1.1. Optical systems for simultaneous spatial and temporal focusing

5. D. N. Vitek, D. E. Adams, A. Johnson, P. S. Tsai, S. Backus, C. G. Durfee, D. Kleinfeld, and J. A. Squier, “Temporally focused femtosecond laser pulses for low numerical aperture micromachining through optically transparent materials,” Opt. Express **18**, 18086–18094 (2010). [CrossRef] [PubMed]

6. D. N. Vitek, E. Block, Y. Bellouard, D. E. Adams, S. Backus, D. Kleinfeld, C. G. Durfee, and J. A. Squier, “Spatio-temporally focused femtosecond laser pulses for nonreciprocal writing in optically transparent materials,” Opt. Express **18**, 24673–24678 (2010). [CrossRef] [PubMed]

1. K. Osvay, A. P. Kovacs, Z. Heiner, G. Kurdi, J. Klebniczki, and M. Csatari, “Angular dispersion and temporal change of femtosecond pulses from misaligned pulse compressors,” IEEE J. Sel. Top. Quantum Electron. **10**, 213–220 (2004). [CrossRef]

4. M. Durst, G. Zhu, and C. Xu, “Simultaneous spatial and temporal focusing in nonlinear microscopy,” Opt. Commun. **281**, 1796–1805 (2008). [CrossRef] [PubMed]

*L*

_{1}) and a second lens (

*L*

_{2}) is placed near the front focal plane of

*L*

_{1}, so that the beam waist is focused tightly [Fig. 1(c), with

*z*

_{12}=

*f*

_{1}]. A similar configuration, with

*z*

_{12}=

*f*

_{1}+

*f*

_{2}produces an image of the grating at the focal plane of

*L*

_{2}[19

19. D. Oron and Y. Silberberg, “Harmonic generation with temporally focused ultrashort pulses,” J. Opt. Soc. Am. B **22**, 2660–2663 (2005). [CrossRef]

### 1.2. Direct Fresnel spatio-temporal beam propagation

*ω*

_{0}(and vacuum wavenumber

*k*

_{0}=

*ω*

_{0}/

*c*) propagating in the

*z*direction, with the beam waist (1/

*e*

^{2}radius

*w*) at the entrance of a lens of focal length

_{in}*f*. Each frequency component is laterally shifted at the lens entrance by the distance

*α*(

*ω*−

*ω*

_{0}), where

*α*is a parameter that describes the spatial chirp rate: The first term of the right-hand side represents the complex input spectrum, containing any input spectral phase

*ϕ*.

_{in}*z*> 0) by taking the spatial Fourier transform of Eq. (1) multiplied by the lens phase factor exp [−

*ik*

_{0}(

*x*

^{2}+

*y*

^{2})/2

*f*] to obtain the angular spectrum (with spatial frequency

*f*). The wavenumber is defined as

_{x}*k*

_{0}=

*ωn*(

*ω*)/

*c*. To propagate the field, the angular spectrum is multiplied by the Fresnel propagation phase. An inverse transform back to position space yields the spatio-spectral field For simplicity, we suppress the

*y*-dependence of the field, since that component propagates independently as a Gaussian beam. Since the propagation phase is a function of

*ω*through

*k*

_{0}, each frequency component propagates independently.

*ω*space can be performed analytically. Under the assumption of limited spectral bandwidth, Durst

*et al*performed analytic inverse Fourier transform to the time domain [4

4. M. Durst, G. Zhu, and C. Xu, “Simultaneous spatial and temporal focusing in nonlinear microscopy,” Opt. Commun. **281**, 1796–1805 (2008). [CrossRef] [PubMed]

## 2. Frequency-space analysis and Double ABCD propagation

### 2.1. Structure of the spatially-chirped input beam

*e*

^{2}radii of this beam is the spatial chirp beam aspect ratio:

*β*, which, assuming there is no input chirp, stretches the input pulse duration by this factor. The shift in the peak of the local spectrum seen in Eq. (5) is (

_{BA}*x*/

*w*)

_{in}### 2.2. Plane wave analysis: origin of pulse-front tilt in focus

*et al.*[8

8. M. Coughlan, M. Plewicki, and R. Levis, “Parametric spatio-temporal control of focusing laser pulses,” Opt. Express **17**, 15808–15820 (2009). [CrossRef] [PubMed]

21. D. Oron and Y. Silberberg, “Spatiotemporal coherent control using shaped, temporally focused pulses,” Opt. Express **13**, 9903–9908 (2005). [CrossRef] [PubMed]

22. M. Coughlan, M. Plewicki, and R. Levis, “Spatio-temporal and-spectral coupling of shaped laser pulses in a focusing geometry,” Opt. Express **18**, 23973–23986 (2010). [CrossRef] [PubMed]

*ϕ*= 0), an angular spectral sweep that is linear in frequency leads to a phase function Here we follow the convention of Eq. (1) where beamlets with

_{in}*ω*>

*ω*

_{0}are displaced to positions

*x*> 0, leading to

*θ*< 0. The pulse front tilt (PFT) is defined as the spatial variation of the temporal peak of the pulse, which for a pulse without odd orders of spectral phase, is equal to the group delay evaluated at the central frequency. Since all frequency components are overlapped at the focal point we can calculate the frequency derivative of the spatially-dependent spectral phase to get the group delay

*ϕ*

_{1}(

*x*,

*ω*), then evaluate the result at

*ω*

_{0}:

*x*> 0; in the focus it is this side that leads in the PFT. Clearly, the tilt of the pulse front originates directly from the dependence of the beamlet angle with

*ω*. Departure from linearity in the angular chip can introduce curvature in the pulse front. Such a departure is expected in practical situations, since the angular dispersion of a diffraction grating is to first order linear in wavelength, not frequency. To estimate the magnitude of the PFT, we eliminate the focal length in Eq. (7) by using

*w*

_{in}= 2

*cf*/(

*ω*

_{0}

*w*

_{0}), where

*w*

_{0}is the 1/

*e*

^{2}radius in intensity of the focused spot size. We can also make use of the definition of the dimensionless spatial chirp rate

*β*[Eq. (3)] to substitute for

*α*, and replace the bandwidth by the transform-limited pulse duration

*τ*

_{0}= 2/Δ

*ω*. Simplifying, it is straightforward to show that the temporal shift of the pulse front is simply Note that the PFT depends only on the spatial chirp rate at the lens entrance, independent of the focusing conditions. Evaluating the second derivative of Eq. (6) to obtain the spatial dependence of the group delay dispersion (chirp),

*ϕ*

_{2}(

*x*), yields The spatial dependence of

*ϕ*

_{2}shows that in addition to the PFT, the pulse develops a spectral chirp that increases away from the optical axis. The second expression in Eq. (9) is an estimate of the broadening at

*x*=

*w*

_{0}. Since the broadening roughly corresponds to

*β*times the duration of an optical cycle, this chirping at the sides of the focus will be significant only if the pulse is extremely short. However, if the frequency crossing plane is arranged to be far from the beamlet waist position, this term could be much more important since the beam size at that position could be much larger than the beamlet waist size.

### 2.3. Tilt transformation of a forward propagated field

*ω*) that is propagating along the optical axis of the system (the

*z*-axis). We can use the Fresnel integral to find the field at any position downstream. Consider next the same beamlet tilted at

*z*= 0 at an angle

*θ*to the

_{x}*z*-axis by applying a linear phase ramp. This tilt can come from a prism, a grating, or from propagation off-center through a lens. The linear phase ramp in position space can be written as exp[

*ik*], where

_{x}x*k*= (

_{x}*ω*/

*c*)sin

*θ*≡ 2

_{x}*πf*

_{x0}. We next calculate the tilted field in terms of the on-axis Fresnel-propagated field.

*Ẽ*(

*f*−

_{x}*f*

_{x}_{0}, 0). To propagate the field, we multiply this by the non-paraxial propagator: The square root may be expanded in two ways: around the new beam direction (

*f*=

_{x}*f*

_{x}_{0}) and around the original

*z*-axis (

*f*= 0). The first expansion is more general, in that the beam direction change can be large, but the spread of the angular spectrum is small. The second expansion assumes that all angles are small: this corresponds to a direct Fresnel transform of the shifted field.

_{x}*z*-axis is not necessarily small. In this case, we change variables to

*f*′

*=*

_{x}*f*−

_{x}*f*

_{x}_{0}. Then, noting that the projection of the

*k*-vector on the

*z*-axis is defined through

*k*out of the square root and expand for 2

_{z}*πf*′

*/*

_{x}*k*≪ 1: The tan

_{z}*θ*term in the exponential results from the ratio 2

_{x}*πf*

_{x}_{0}/

*k*=

_{z}*k*/

_{x}*k*. To transform back to position space, we use the shift theorem to represent the result in terms of a transform with respect to

_{z}*f*′

*, with the result Comparing to Eq. (2), we can see that the exponential quadratic in*

_{x}*f*′

*is the Fresnel propagator, and the term linear in*

_{x}*f*′

*will result in a shift in*

_{x}*x*. From this we conclude that we calculate the Fresnel propagation of the field

*without*the phase ramp, then make the two substitutions so that the propagation phase in front of the expression is exp[

*i*(

*k*+

_{x}x*k*)] instead of exp [

_{z}z*ik*

_{0}

*z*]. This derivation is a result that can be applied to general diffractive propagation, though in this paper we will restrict our attention to the propagation of angled Gaussian beams. Note that the substitution of

*k*for

_{z}*k*

_{0}in this derivation applies globally throughout the expression for the Fresnel-propagated field without the phase ramp. In the context of Gaussian beams, this modification will change the Rayleigh range of the beam in the expressions for the

*z*-dependence of the beam size, radius of curvature and the Gouy phase.

*πf*/

_{x}*k*

_{0}≪ 1. Note that this is implicitly assumed when the direct Fresnel transform described in Sect. 1.2 is performed. In this case, we expand the square root in Eq. (10) as is customary for Fresnel propagation. The propagated field takes the form where

*k*′

*=*

_{z}*k*

_{0}(1 − sin

^{2}

*θ*/2). The tilt transformation is simpler in this case, since there is no global change to

_{x}*k*

_{0}and only the

*x*coordinate is shifted: This is the form of the tilt transformation that will be used above in calculating the spatio-temporal propagation of spatially-chirped beams.

### 2.4. Angled Gaussian beam propagation

*ω*. For a coordinate system centered on the beam waist, the Gaussian beam can be written in term of the amplitude and phase

*E*(

*x*,

*y*,

*z*,

*ω*) =

*A*(

*x*,

*y*,

*z*,

*ω*)exp[

*iϕ*(

*x*,

*y*,

*z*,

*ω*)]. Using the sign convention that a forward-propagating plane wave is written as exp[

*i*(

*k*

_{0}

*z*−

*ωt*)], the well-known expression for the field is given by where the beam radius (

*w*), radius of curvature (

*R*) and the Gouy phase (

*η*) are given by Note that

*w*,

*R*and

*η*are implicitly functions of

*k*

_{0}(and

*ω*) through the Rayleigh range

*θ*is small so that we can use the transform described in Eqs. (14) and (15). To perform the tilt transformation on the Gaussian beamlet, the amplitude function undergoes the shift in the

_{x}*x*variable,

*x*→

*x*−

*z*sin

*θ*. The phase structure is important for the analysis of the pulse shape throughout the focus. For the paraxial case, This gives the complex field for a single-frequency Gaussian beam propagating at a specific angle to the optical axis. In an optical system that includes angular dispersion, we can obtain the beamlet angles from raytracing.

_{x}### 2.5. Combining raytracing with angled Gaussian beam propagation to obtain 3-D field

*θ*=

_{x}*α*(

*ω*−

*ω*

_{0})/

*f*. We then can expand spectral phase around

*ω*

_{0}to find the position-dependent group delay and chirp. Note that even though this paraxial treatment of the spatial chirp does not result in any angular dependence of the Rayleigh range, the Rayleigh range itself depends on frequency. If the focal spot size is considered to be frequency-independent,

*z*∝

_{R}*ω*. However, if the beam size at the lens entrance is independent of frequency, then the focused spot radius is inversely proportional to frequency, and

*z*∝ 1/

_{R}*ω*. To focus on the primary effects of the spatial chirp, we hold

*z*constant, treating the frequency-dependence of the Rayleigh range as a higher-order effect that is appreciable only for extremely wide bandwidth pulses. The framework presented below can be extended in a straightforward way to the more general cases.

_{R}*z*= 0, the PFT reduces to what we found earlier in Eq. (7). To better understand the

*z*–dependence of the PFT, we can simplify the Gaussian beam radius of curvature

*R*(

*z*) using Eq. (17). We can also make use of the simplifications leading to Eq. (8) to obtain the more intuitive form: The PFT is approximately zero far from the focus and develops within the beamlet confocal parameter. The magnitude of the PFT is directly proportional to the dimensionless chirp rate

*β*.

*x*-dependent chirp is the extension of the result that we found for the focal plane, Eq. (9). This term, which is small for low bandwidth pulses, decreases away from the focal plane just as the PFT does [see Eq. (21)]. The

*z*-dependent term in the first parenthesis is a new term that is important for the intensity localization of the spatio-temporal focus. This term is plotted in Fig. 2 for a value of

*β*= 10. Even though the pulse is ideally perfectly compressed at the focal plane, it develops chirp at either side of the focus. This tends to increase the duration of the pulse away from the focal plane.

*ϕ*

_{3}(

*x*,

*z*), and we find that it follows the same

*x*- and

*z*-dependence as

*ϕ*

_{2}(

*x*,

*z*), but with a leading factor of 1/

*ω*

_{0}. Therefore, if the expansion terms are assembled into a Taylor series, the contribution of the third-order to the net phase is smaller than that from the second-order by the factor (

*ω*−

*ω*

_{0})/

*ω*

_{0}. The geometric third-order phase is important only for large-bandwidth pulses; it will generally add to the increase of the pulse duration away from the focal plane.

## 3. The structure of space-time focused beams

### 3.1. Calculation of the spatio-temporal field

*ω*–dependence of the beamlet angle

*θ*depend in a non-trivial way on

_{x}*ω*. The direct Fourier transform cannot be calculated analytically, but we can make use of the analysis above to expand the spectral phase to second-order in the frequency difference. Provided we ignore the frequency-dependence of the Rayleigh range, the amplitude functions are Gaussian functions, and analytic transform is possible. In order to do so, it is important to rearrange the expression for the spectral amplitude [Eq. (18)] so that the local bandwidth and center frequency is clear. We assume an input Gaussian spectrum with 1/

*e*amplitude half-width Δ

*ω*. By expanding the frequency-dependent angular terms in Eq. (18) and simplification, the amplitude function can be expressed in the simple form: The auxiliary

*z*-dependent functions defined in this expression are best represented in terms of the dimensionless variable

*ζ*=

*z*/

*z*. The

_{R}*x*-dependent beam radius is defined through and the local center frequency and the local bandwidth are Note that the local bandwidth is independent of the transverse coordinate. Although the local pulse duration is determined both by the local bandwidth and the degree of chirp, it is instructive to calculate the local bandwidth-limited pulse duration: At large distance from the focal plane,

*ζ*≫ 1, we find the pulse duration is longer than the transform-limited pulse duration by the factor

*β*. Thus

_{BA}*β*is the pulse duration contrast that we obtain from spatial-chirp focusing.

_{BA}*ω*is shifted away from

_{L}*ω*

_{0}when

*x*≠ 0. After calculating the inverse Fourier transform, this shift in center frequency does not affect the local pulse duration (since Δ

*ω*is independent of

_{L}*x*). However, it does result in a group delay offset. This offset adds to the group delay

*ϕ*

_{1}that was calculated from the expansion of the spectral phase around

*ω*

_{0}. This shift affects the detailed structure of the PFT off-axis and away from the frequency crossing plane.

*x*= 0,

*ω*=

_{L}*ω*

_{0}, we can use the second-order phase from Eq. (22), perform the inverse Fourier transform. If there is no input spectral chirp, the local pulse duration reduces to With an input chirp of

*ϕ*

_{2in}, the local on-axis pulse duration has a considerably more complicated form: Note that the input chirp can compensate the geometric phase over a narrow range (see Fig. 2). Within that range, the argument of the square root in Eq. (30) goes to unity, and the pulse duration goes to the bandwidth-limited value at the

*z*-position where the chirp cancellation takes place.

### 3.2. Contributions to the axial localization of the temporal intensity

*E*/[

_{in}*πw*

^{2}(

*z*)]. The other curves shown account for successively more of the localization contributions. The second widest curve shows the effect of focusing at a higher numerical aperture in the spatially-chirped direction. The beam fluence, ∝

*E*/[

_{in}*πw*(

_{x}*z*)

*w*(

*z*)], is lower away from the focal plane relative the the single beamlet. In the simplest view of space-time focusing, the increase in the pulse duration away from the focus results from the decrease of the local spectral width. The third widest curve accounts the decrease in the local bandwidth away from the focus [Eq. (27)], but neglects the geometric chirp. This axial intensity profile is further reduced by the geometric spectral chirp that is present within the confocal parameter of the focus [Eq. (29)] (center curve).

*I*

_{0}is the peak intensity at the focus. In the limit of no spatial chirp,

*β*→ 1, and the intensity follows the Lorentzian profile of a conventionally-focused Gaussian beam.

_{BA}*ϕ*

_{2}

*) can combine with the geometric chirp to shift the plane at which the pulse is compressed. This feature of space-time focusing has been used to scan the focal plane along the*

_{in}*z*-axis [2

2. D. Oron, E. Tal, and Y. Silberberg, “Scanningless depth-resolved microscopy,” Opt Express **13**, 1468–1476 (2005). [CrossRef] [PubMed]

11. M. Durst, G. Zhu, and C. Xu, “Simultaneous spatial and temporal focusing for axial scanning,” Opt. Express **14**, 12243–12254 (2006). [CrossRef] [PubMed]

*ϕ*

_{2}= 0 can be moved throughout the confocal parameter (see Fig. 2), the peak intensity and the localization suffer because the plane of zero chirp is moved to a position where the different frequency beamlets are not fully overlapped. (see Fig. 5(a)). The pulse duration is longer even though there is no spectral chirp because it is limited by the local bandwidth [Eq. (27)]. The increase in the beam area away from the wavelength crossing plane also decreases the peak intensity. The axial tuning is illustrated in Fig. 5(a).

23. A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy–Bessel wave packets as versatile linear light bullets,” Nat. Photon. **4**, 103–106 (2010). [CrossRef]

## 4. Discussion: scaling analysis of space-time focused beams

*ζ*: The value of the depth of focus for the spatially-chirped beam, relative to the depth of focus of the non-spatially-chirped focused beamlet is plotted as a dotted line in Fig. 6. A factor of 10 decrease in

_{DOF}*ζ*is obtained at a beam aspect ratio of

_{DOF}*β*≈ 4.1. It is important to observe that the beam aspect ratio is the sole parameter that controls the decrease of the depth of focus over the Gaussian beam limit. Therefore, the same localization can be obtained with ps-duration pulses as with fs pulses, provided the optical system produces sufficient spatial chirp for the desired value of

_{BA}*β*. Coherence is required, however: a broadband ns Q-switched pulse would not be increased by spreading the spectrum out spatially. The geometric spectral chirp effects described above would not lengthen the pulse duration away from the focus because such lengthening requires spectral phase coherence.

_{BA}*M*

^{2}is often used to characterize aberrated beams [24]. As the multimode content increases, the value of

*M*

^{2}increases, and for a given spot size, the effective Rayleigh range is decreased by the

*M*

^{2}factor. Therefore the effective depth of focus can be reduced by creating a beam with a wide distribution of transverse spatial modes. However, such a beam would require each mode must be correctly phased with all the others at the focus. Conventional ways to produce multimode beams, with random phase plates or by coupling the beam into a multimode fiber, would leave each mode with a different phase, effectively reducing the coherence of the beam.

*ζ*. It is instructive to compare the depth of focus that can be attained by filling the aperture of the optic with a larger beam. If one increases the beam size entering a lens of a fixed focal length by a factor

_{DOF}*n*, the spot size decreases by 1/

*n*and the confocal parameter increases by

*n*

^{2}. Since the spatially-chirped beam requires a larger aperture than the beamlet by the factor

*β*, we can plot as a reference the curve

_{BA}**18**, 18086–18094 (2010). [CrossRef] [PubMed]

*ϕ*= ∫

_{NL}*k*

_{0}

*n*

_{2}

*I*(

*z*)

*dz*through the medium with nonlinear refractive index,

*n*

_{2}(e.g. see [25]). To evaluate how the spatial chirp affects this nonlinear interaction, we can calculate

*ϕ*by integrating over the complete unperturbed axial intensity profile for the cases with and without spatial chirp. In Figs. 6(a) and 6(b), the ratio of these two integrals, the relative B-integral is shown as a solid curve. To reduce the B-integral by a factor of 10, for example, we can use

_{NL}*β*≈ 5.5. It is well known that for a Gaussian beam the threshold for self-focusing depends on the peak

_{BA}*power*, not the peak

*intensity*: in the tight-focusing limit, where the nonlinear medium extends beyond the Rayleigh range to either side of the focal plane, a smaller focal spot leads to higher intensity but also a shorter interaction length (2

*z*). With space-time focusing, it is possible to decrease the interaction length (the depth of focus, Eq. (32)), which leads to an increased threshold for self-focusing. For the same reason, space-time focusing can reduce the effects of ionization

_{R}*defocusing*.

*et al.*[7

7. F. He, H. Xu, Y. Cheng, J. Ni, H. Xiong, Z. Xu, K. Sugioka, and K. Midorikawa, “Fabrication of microfluidic channels with a circular cross section using spatiotemporally focused femtosecond laser pulses,” Opt. Lett. **35**, 1106–1108 (2010). [CrossRef] [PubMed]

26. F. He, Y. Cheng, J. Lin, J. Ni, Z. Xu, K. Sugioka, and K. Midorikawa, “Independent control of aspect ratios in the axial and lateral cross sections of a focal spot for three-dimensional femtosecond laser micromachining,” New J. Phys. **13**, 083014 (2011). [CrossRef]

*ρ*as the confocal parameter divided by the FWHM of the focal spot: where

_{G}*n*is the refractive index in the medium. It is difficult to obtain

*ρ*= 1 with conventional focusing. Even if the beam is focused at F/1,

_{G}*ρ*= 3.4. Figure 7 shows the focal volume aspect ratio for the spatial chirp focusing case. The beam aspect ratio for which

_{G}*ρ*= 1 is at the value of

_{ST}*β*where the curves cross the dotted line. Creating a spherical focal volume requires larger spatial chirp as the spot size is increased. For

_{BA}*w*

_{0}= 10

*μm*,

*ρ*= 1 at

_{ST}*β*≈ 11 (solid line); for

_{BA}*w*

_{0}= 20

*μm*a beam aspect ratio of

*β*≈ 15 is required (dashed line).

_{BA}## 5. Summary

19. D. Oron and Y. Silberberg, “Harmonic generation with temporally focused ultrashort pulses,” J. Opt. Soc. Am. B **22**, 2660–2663 (2005). [CrossRef]

27. M. Durst, A. Straub, and C. Xu, “Enhanced axial confinement of sum-frequency generation in a temporal focusing setup,” Opt. Lett. **34**, 1786–1788 (2009). [CrossRef] [PubMed]

## Acknowledgments

## References and links

1. | K. Osvay, A. P. Kovacs, Z. Heiner, G. Kurdi, J. Klebniczki, and M. Csatari, “Angular dispersion and temporal change of femtosecond pulses from misaligned pulse compressors,” IEEE J. Sel. Top. Quantum Electron. |

2. | D. Oron, E. Tal, and Y. Silberberg, “Scanningless depth-resolved microscopy,” Opt Express |

3. | G. Zhu, J. van Howe, M. Durst, W. Zipfel, and C. Xu, “Simultaneous spatial and temporal focusing of femtosecond pulses,” Opt. Express |

4. | M. Durst, G. Zhu, and C. Xu, “Simultaneous spatial and temporal focusing in nonlinear microscopy,” Opt. Commun. |

5. | D. N. Vitek, D. E. Adams, A. Johnson, P. S. Tsai, S. Backus, C. G. Durfee, D. Kleinfeld, and J. A. Squier, “Temporally focused femtosecond laser pulses for low numerical aperture micromachining through optically transparent materials,” Opt. Express |

6. | D. N. Vitek, E. Block, Y. Bellouard, D. E. Adams, S. Backus, D. Kleinfeld, C. G. Durfee, and J. A. Squier, “Spatio-temporally focused femtosecond laser pulses for nonreciprocal writing in optically transparent materials,” Opt. Express |

7. | F. He, H. Xu, Y. Cheng, J. Ni, H. Xiong, Z. Xu, K. Sugioka, and K. Midorikawa, “Fabrication of microfluidic channels with a circular cross section using spatiotemporally focused femtosecond laser pulses,” Opt. Lett. |

8. | M. Coughlan, M. Plewicki, and R. Levis, “Parametric spatio-temporal control of focusing laser pulses,” Opt. Express |

9. | P. G. Kazansky, W. Yang, E. Bricchi, J. Bovatsek, A. Arai, Y. Shimotsuma, K. Miura, and K. Hirao, ““Quill” writing with ultrashort light pulses in transparent materials,” Appl. Phys. Lett. |

10. | W. Yang, P. G. Kazansky, Y. Shimotsuma, M. Sakakura, K. Miura, and K. Hirao, “Ultrashort-pulse laser calligraphy,” Appl. Phys. Lett. |

11. | M. Durst, G. Zhu, and C. Xu, “Simultaneous spatial and temporal focusing for axial scanning,” Opt. Express |

12. | D. Zimmer, D. Ros, O. Guilbaud, J. Habib, S. Kazamias, B. Zielbauer, V. Bagnoud, B. Ecker, D. Hochhaus, and B. Aurand, “Short-wavelength soft-x-ray laser pumped in double-pulse single-beam non-normal incidence,” Phys. Rev. A |

13. | O. E. Martinez, “Achromatic phase matching for second harmonic generation of femtosecond pulses,” IEEE J. Quantum Electron. |

14. | J. A. Fueloep, L. Palfalvi, M. C. Hoffmann, and J. Hebling, “Towards generation of mJ-level ultrashort THz pulses by optical rectification,” Opt. Express |

15. | A. G. Kostenbauder, “Ray-pulse matrices: a rational treatment for dispersive optical systems,” IEEE J. Quantum Electron. |

16. | V. Chauhan, J. Cohen, and R. Trebino, “Simple dispersion law for arbitrary sequences of dispersive optics,” Appl. Opt. |

17. | F. Druon, M. Hanna, G. Lucas-Leclin, Y. Zaouter, D. Papadopoulos, and P. Georges, “Simple and general method to calculate the dispersion properties of complex and aberrated stretchers-compressors,” J. Opt. Soc. Am. B |

18. | C. G. Durfee, J. Squier, and S. Kane, “A modular approach to the analytic calculation of spectral phase for grisms and other refractive/diffractive structures,” Opt Express |

19. | D. Oron and Y. Silberberg, “Harmonic generation with temporally focused ultrashort pulses,” J. Opt. Soc. Am. B |

20. | J. Goodman, |

21. | D. Oron and Y. Silberberg, “Spatiotemporal coherent control using shaped, temporally focused pulses,” Opt. Express |

22. | M. Coughlan, M. Plewicki, and R. Levis, “Spatio-temporal and-spectral coupling of shaped laser pulses in a focusing geometry,” Opt. Express |

23. | A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy–Bessel wave packets as versatile linear light bullets,” Nat. Photon. |

24. | A. E. Siegman, |

25. | R. W. Boyd, |

26. | F. He, Y. Cheng, J. Lin, J. Ni, Z. Xu, K. Sugioka, and K. Midorikawa, “Independent control of aspect ratios in the axial and lateral cross sections of a focal spot for three-dimensional femtosecond laser micromachining,” New J. Phys. |

27. | M. Durst, A. Straub, and C. Xu, “Enhanced axial confinement of sum-frequency generation in a temporal focusing setup,” Opt. Lett. |

**OCIS Codes**

(220.2560) Optical design and fabrication : Propagating methods

(320.1590) Ultrafast optics : Chirping

(320.7160) Ultrafast optics : Ultrafast technology

**ToC Category:**

Ultrafast Optics

**History**

Original Manuscript: April 26, 2012

Manuscript Accepted: May 28, 2012

Published: June 12, 2012

**Citation**

Charles G. Durfee, Michael Greco, Erica Block, Dawn Vitek, and Jeff A. Squier, "Intuitive analysis of space-time focusing with double-ABCD calculation," Opt. Express **20**, 14244-14259 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-13-14244

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### References

- K. Osvay, A. P. Kovacs, Z. Heiner, G. Kurdi, J. Klebniczki, and M. Csatari, “Angular dispersion and temporal change of femtosecond pulses from misaligned pulse compressors,” IEEE J. Sel. Top. Quantum Electron.10, 213–220 (2004). [CrossRef]
- D. Oron, E. Tal, and Y. Silberberg, “Scanningless depth-resolved microscopy,” Opt Express13, 1468–1476 (2005). [CrossRef] [PubMed]
- G. Zhu, J. van Howe, M. Durst, W. Zipfel, and C. Xu, “Simultaneous spatial and temporal focusing of femtosecond pulses,” Opt. Express13, 2153–2159 (2005). [CrossRef] [PubMed]
- M. Durst, G. Zhu, and C. Xu, “Simultaneous spatial and temporal focusing in nonlinear microscopy,” Opt. Commun.281, 1796–1805 (2008). [CrossRef] [PubMed]
- D. N. Vitek, D. E. Adams, A. Johnson, P. S. Tsai, S. Backus, C. G. Durfee, D. Kleinfeld, and J. A. Squier, “Temporally focused femtosecond laser pulses for low numerical aperture micromachining through optically transparent materials,” Opt. Express18, 18086–18094 (2010). [CrossRef] [PubMed]
- D. N. Vitek, E. Block, Y. Bellouard, D. E. Adams, S. Backus, D. Kleinfeld, C. G. Durfee, and J. A. Squier, “Spatio-temporally focused femtosecond laser pulses for nonreciprocal writing in optically transparent materials,” Opt. Express18, 24673–24678 (2010). [CrossRef] [PubMed]
- F. He, H. Xu, Y. Cheng, J. Ni, H. Xiong, Z. Xu, K. Sugioka, and K. Midorikawa, “Fabrication of microfluidic channels with a circular cross section using spatiotemporally focused femtosecond laser pulses,” Opt. Lett.35, 1106–1108 (2010). [CrossRef] [PubMed]
- M. Coughlan, M. Plewicki, and R. Levis, “Parametric spatio-temporal control of focusing laser pulses,” Opt. Express17, 15808–15820 (2009). [CrossRef] [PubMed]
- P. G. Kazansky, W. Yang, E. Bricchi, J. Bovatsek, A. Arai, Y. Shimotsuma, K. Miura, and K. Hirao, ““Quill” writing with ultrashort light pulses in transparent materials,” Appl. Phys. Lett.90, 151120 (2007). [CrossRef]
- W. Yang, P. G. Kazansky, Y. Shimotsuma, M. Sakakura, K. Miura, and K. Hirao, “Ultrashort-pulse laser calligraphy,” Appl. Phys. Lett.93, 171109 (2008). [CrossRef]
- M. Durst, G. Zhu, and C. Xu, “Simultaneous spatial and temporal focusing for axial scanning,” Opt. Express14, 12243–12254 (2006). [CrossRef] [PubMed]
- D. Zimmer, D. Ros, O. Guilbaud, J. Habib, S. Kazamias, B. Zielbauer, V. Bagnoud, B. Ecker, D. Hochhaus, and B. Aurand, “Short-wavelength soft-x-ray laser pumped in double-pulse single-beam non-normal incidence,” Phys. Rev. A82, 013803 (2010). [CrossRef]
- O. E. Martinez, “Achromatic phase matching for second harmonic generation of femtosecond pulses,” IEEE J. Quantum Electron.25, 2464–2468 (1989). [CrossRef]
- J. A. Fueloep, L. Palfalvi, M. C. Hoffmann, and J. Hebling, “Towards generation of mJ-level ultrashort THz pulses by optical rectification,” Opt. Express19, 15090–15097 (2011). [CrossRef]
- A. G. Kostenbauder, “Ray-pulse matrices: a rational treatment for dispersive optical systems,” IEEE J. Quantum Electron.26, 1148–1157 (1990). [CrossRef]
- V. Chauhan, J. Cohen, and R. Trebino, “Simple dispersion law for arbitrary sequences of dispersive optics,” Appl. Opt.49, 6840–6844 (2010). [CrossRef] [PubMed]
- F. Druon, M. Hanna, G. Lucas-Leclin, Y. Zaouter, D. Papadopoulos, and P. Georges, “Simple and general method to calculate the dispersion properties of complex and aberrated stretchers-compressors,” J. Opt. Soc. Am. B25, 754–762 (2008). [CrossRef]
- C. G. Durfee, J. Squier, and S. Kane, “A modular approach to the analytic calculation of spectral phase for grisms and other refractive/diffractive structures,” Opt Express16, 18,004–18,016 (2008). [CrossRef]
- D. Oron and Y. Silberberg, “Harmonic generation with temporally focused ultrashort pulses,” J. Opt. Soc. Am. B22, 2660–2663 (2005). [CrossRef]
- J. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts and Company Publishers, 2004).
- D. Oron and Y. Silberberg, “Spatiotemporal coherent control using shaped, temporally focused pulses,” Opt. Express13, 9903–9908 (2005). [CrossRef] [PubMed]
- M. Coughlan, M. Plewicki, and R. Levis, “Spatio-temporal and-spectral coupling of shaped laser pulses in a focusing geometry,” Opt. Express18, 23973–23986 (2010). [CrossRef] [PubMed]
- A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy–Bessel wave packets as versatile linear light bullets,” Nat. Photon.4, 103–106 (2010). [CrossRef]
- A. E. Siegman, Lasers, 1st ed. (University Science Books, 1986).
- R. W. Boyd, Nonlinear Optics, 3rd ed. (Academic Press, 2008).
- F. He, Y. Cheng, J. Lin, J. Ni, Z. Xu, K. Sugioka, and K. Midorikawa, “Independent control of aspect ratios in the axial and lateral cross sections of a focal spot for three-dimensional femtosecond laser micromachining,” New J. Phys.13, 083014 (2011). [CrossRef]
- M. Durst, A. Straub, and C. Xu, “Enhanced axial confinement of sum-frequency generation in a temporal focusing setup,” Opt. Lett.34, 1786–1788 (2009). [CrossRef] [PubMed]

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