1. Introduction

In the past decade, femtosecond laser three-dimensional (3D) micromachining in glass materials has emerged as an important branch of femtosecond laser materials processing, which has enabled construction of not only integrated photonic devices (e. g., waveguide writing, polarization sensitive optics, quantum circuits, etc.) but also integrated microfluidic and optofluidic systems (e. g., lab on a chip devices, micro-total analysis systems, etc.) [1–9]. In the meantime, this new way of interaction of femtosecond laser with matter, i.e., irradiation of intense ultrafast pulses inside transparent materials such as glass and crystals, has led to many intriguing phenomena, such as refractive index modification [10], formation of nanovoids and periodic nanogratings [11–13], element redistribution [14], nanocrystallization [15], and very recently, the nonreciprocal writing [16–19]. Although significant effort has been made for understanding the underlying mechanisms behind the above-mentioned phenomena, the complete physical pictures are still lacking. Nevertheless, despite the incomplete understandings on these discoveries, some of the effects have already found important applications in integrated optics and microfluidics. Moreover, for high precision 3D micro and nanomachining, control of these novel phenomena at micrometer and even nanometer scales requires precise tailoring of light field within a focal spot, including the spatial (in all three dimensions) and/or temporal profiles, polarization direction, pulse front tilting, etc [20–23]. In particular, the simultaneous spatial and temporal beam shaping technique (also named “simultaneous spatial and temporal focusing (SSTF)”, “space-time focusing” or “temporal focusing (TF)”), which was originally developed for bio-imaging applications [24, 25], has been recently used in femtosecond laser micromachining of transparent materials, aiming at improving axial fabrication resolution [26, 27], eliminating nonlinear self-focusing [28, 29], and increasing the fabrication efficiency [30].

Recently, Vitek et al. reported that the spatiotemporally focused laser beam naturally creates a focal spot with a tilted pulse front, which can induce nonreciprocal writing in glass [31]. On the other hand, a complete characterization of the spatiotemporally focused femtosecond laser beam at its focus has not been done. In this work, we attempt to characterize the spatiotemporally focused femtosecond laser beam based on two-photon fluorescence excitation (TPFE) process. In such a manner, instead of examination of the fluence distribution in the focal volume, we focus more on the peak intensity distribution, which is more relevant to the nonlinear absorption processes during the irradiation of femtosecond laser in glass. We then perform a theoretical analysis to reveal the origin of such tilted peak intensity distribution. Our results contribute further to the understanding of interaction of spatiotemporally focused laser pulses with transparent materials, such as the nonreciprocal writing effect observed in glass [16–19]. More importantly, we show that the spatial profile of the focal spot of a spatiotemporally focused beam can be controlled by tuning the temporal chirp of the incident beam, providing a new approach for tailoring the focal spot for micromachining applications.

2. Experimental setup

The femtosecond laser system (Legend-Elite, Coherent Inc.) used in this experiment consists of a Ti:sapphire laser oscillator and amplifier, and a grating-based stretcher and compressor. In the normal operation mode (non-temporal focus, non-TF), transform-limited 800 nm, 2.5 mJ, 40 fs, p-polarized pulses with a spectral bandwidth of ~30 nm at a 1-kHz repetition rate are delivered after the compressor, which is a double pass through two parallel σ = 1200 grooves/mm gratings with an incident angle of 29.5°. For our temporal focusing configuration (TF-mode) [26–28], the double-pass compressor was bypassed (using flipping mirrors FM1, FM2) and the laser beam was directed through the single-pass grating compressor, consisting of two σ = 1500 grooves/mm gratings (Coherent Inc.), blazed for the incident angle of 53°(see Fig. 1). The maximum diffraction efficiency (p-polarization) of a single grating was estimated to be ~81%, so with only two grating reflections, the overall efficiency of the single-pass compressor is greater, resulting in a maximum laser power measured at ~3.4 W in a beam diameter of ~12.3 mm (1/e²).

 

Fig. 1 Schematic of the experiment setup for imaging the TPFE signal excited by the spatiotemporally focused pulses. FM1~2: flipping mirrors, Ap.: aperture, VF1~2: variable neutral density filter, L1~3: lenses, G1~2: gratings. (Inset shows the view angle of the camera).

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The angles of incidence for the single-pass compressor were set to i ~53° so that the ratio of third-order to second-order phase was the same as for the double-pass compressor. This was to minimize residual third-order phase on the compressed pulse. The distance d between the grating pair was adjusted to be ~730 mm, so as to compensate for the temporal dispersion of the beam induced by the transmitting optics such as the grating stretcher in the amplifier, ND filters, lenses, and so on. This separation is twice the distance required for a double-pass compressor. The gratings in the single-pass compressor have dimensions 50 mm × 110 mm, so as to ensure that all the frequency components can be collected. Both the separation and incident angle of the grating in the single-pass compressor were optimized to obtain the strongest ionization of air at the focus of the lens.

The output laser beam was truncated with an iris to have a diameter of ~10 mm so ensure beam circularity. The energy of the beam was varied with neutral density (ND) filters VF1~2. In order to create a greater ratio between the spatially chirped beam size and the input beam size, thereby leading to a stronger spatiotemporal coupling effect [32], a Galileo type telescope system consists of two convex lenses L1 ($f_1$ = 600 mm) and L2 ($f_2$ = 60 mm) was employed to reduce the beam diameter to 1 mm. In the context of the analysis by Durfee et al. [32], the beam aspect ratio for the spatial chirp (the ratio of the input beam diameter to the spatially chirped diameter) in our experiment is β = 30, which will be used later.

To visualize the spatiotemporally focused spot, a lens (L3) with a focal length of $f$ = 500 mm was employed to focus the spatially chirped beam into a 0.12 mg/mL sodium fluorescein diluted with water. The lens aperture was chosen to be 100 mm in diameter, to ensure that all the spectral components could be collected. The total volume of the fluorescein was ~1200 mL, which was contained in a 240 mm × 160 mm × 160 mm cuvette. In our experiment, the average laser power was adjusted to be ~50 mW (measured after VF2), which is below the threshold intensity to induce multiple filamentation in the fluorescein but sufficiently high to induce two-photon excited fluorescence. The distance between the back surface of L3 and the front surface of the cuvette was ~630 mm. The TPFE image could be directly captured by the digital camera from both top view and side view (Inset in Fig. 1).

3. Experimental results

Figures 2(a) and 2(c) show digital camera captured images of TPFE using the spatiotemporal focusing scheme with the parameters provided in the experimental section, observed from the XZ and YZ planes, respectively (the coordinate orientation is indicated in Fig. 1). It can be seen that the fluorescence intensity distribution in the XZ plane is tilted while in the YZ plane it appears symmetric. Such tilted intensity distribution in the XZ plane, which we will call an intensity plane tilt (IPT), has never been reported before. The characteristics of the IPT are fundamentally different from the well-known pulse front tilt (PFT), as will be described below in Sect. 4.

 

Fig. 2 TPFE images (a)-(d) with temporal focusing and (e)-(f) without temporal focusing (lens f = 500mm). (a) and (c) are TPFE images captured in the XZ and YZ planes, respectively, and (b), (d), (f) are corresponding numerical simulation TPFE signals. The color bar is shown at the bottom. Notice that the horizontal and vertical scale bars in (e) and (f) are different.

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For comparison, we also performed a similar TPFE experiment by focusing the femtosecond laser beam in non-TF mode, using the double-pass compressor instead of the single-pass compressor (see Fig. 1). The diameter of the incident beam in both arrangements was chosen to be 2w_in = 1 mm. The average power for non-TF and TF mode were the same at ~33 mW (measured after VF2). And all the other parameters were the same as those in Fig. 1. In particular, the exposure time of the camera for both of the two cases was set to be 1/4 s. The TPFE image obtained with conventional focusing is shown in Fig. 2(e), which demonstrates a much longer focal depth than that obtained with the TF mode. All results in Fig. 2 are consistent with the observations of highly localized focal spots formed in both thick glass sample and remote air enabled by use of temporal focusing with a low numerical aperture lens [28, 29].

To further examine the characteristics of this tilted focal plane, we varied the focal length of L3 in the TF mode, while the other experimental conditions were kept the same. Figures 3(a), 3(c), and 3(e) show the TPFE images with temporal focusing using lenses with different focal lengths of f = 1000 mm, 500 mm, and 250 mm, respectively. It can be seen that for all the focal lengths, the phenomenon of IPT always exists and the angle of tilt increases with decreasing focal length. As we will show later, this trend is consistent with our theoretical analysis.

 

Fig. 3 TPFE images with temporal focusing using lenses with focal lengths of (a) 1000 mm, (c) 500 mm and (e) 250 mm, respectively. (b), (d), (f) are corresponding numerical simulation TPFE signals. The color bar is shown at the bottom. Notice that the scale bar in (e) and (f) is smaller than that in (a) - (d).

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4. Numerical analysis

Theoretically, simulation of the two-photon excitation of dye with spatiotemporally focused femtosecond laser beam can be achieved using Fresnel diffraction theory [24, 26, 27]. The normalized light field of a spatially dispersed pulse E₁ at the entrance aperture of lens thus can be expressed as [27]

For our conditions (f = 500 mm, beamlet width 2w₀ = 1mm, spatially chirped width β_BA⋅2w₀ = 30mm), the effective F-numbers are sufficiently large that the paraxial approximation condition can be well satisfied, which makes it reasonable to use the Fresnel diffraction theory in the paraxial approximation for the simulation [33]. The generalized Fresnel diffraction integral acting on the input spectrum is

The field of the spatio-temporally focused beam written in the time domain is derived by calculating the inverse Fourier transform of E₂. The excited two-photon fluorescence signal based on the TF scheme can be obtained as

For convenience, we set axial position of the geometrical focus of each focusing mode to be z = 0, as shown in the later plotting. Most variables used in the simulation are tabulated in Table 1.

Table 1. List of variables used for simulation

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The TPFE intensity distributions, in both the XZ and YZ planes, can then be obtained with Eq. (3). On the other hand, TPFE intensity distribution obtained using the conventional focusing system (non-TF) can be calculated by substituting α = 0 into Eq. (1). The aspect ratios of the experimentally detected fluorescence signal are in agreement with the simulation results, as shown in Figs. 2 and 3. It should be noted that in our previous work in Ref [27], the slowly varying envelope approximation (i.e., the wave number k(ω) was approximately considered to be the center wave number k₀) was employed to obtain an analytical expression of the focal intensity distribution, hence the IPT effect was masked in the calculation.

5. Origin of the tilted intensity distribution

The pulse front tilt (PFT), the dependence of the arrival time of the pulse on the transverse coordinate, is controlled by the combination of angular or lateral spatial chirp with spectral chirp [32, 34]. In the present case of the spatiotemporal focusing, the temporal chirp is pre-compensated and ϕ₂ = 0, hence the PFT is introduced by the angular dispersion alone. In our experiment the pulse is spatially chirped in the transverse direction before the lens, and the lens converts this transverse spatial chirp to angular chirp.

Following the analysis in Ref [32], we can treat a particular frequency component as an individual beamlet. The phase function, under the paraxial approximation can be written as

 

Fig. 4 Illustration of the relative orientations of the spatial chirp (top), pulse front tilt (lower left), and intensity plane tilt with the IPT angle indicated (lower right). The PFT is represented as a snapshot so that the leading edge of the pulse is at larger z = ct.

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For positions close to z = 0, the denominator of the second parenthesis in the right side of Eq. (6) can be neglected. Working with this approximation again, the pulse duration of this spatiotemporally focused beam can be analytically expressed as [28]:

Intuitively, the slope of IPT is proportional to the angular chirp rate and inversely proportional to the focal length of L3, which is also evidenced in Fig. 3. In our experiment, the IPT angle is calculated to be ~26° by using Eq. (8). Obviously, for conventional focusing (α = 0), it should result in θ_t = 0 since no IPT effect will occur. This is exactly what we have observed in our experiment.

From Eq. (7), it is clear that the pulse duration will be at a minimum along a plane that is tilted with respect to the z axis. This intensity plane tilt (IPT) is distinct from what is called the pulse front tilt (PFT), which describes the arrival time of the pulse across the beam for a fixed value of z.

For any amount of angular spatial chirp, the IPT will be present, but whether the effect will be noticeable is not obvious from the expression. Since spatiotemporal focusing confines the axial intensity profile compared to the Rayleigh range of the beamlet, we can compare the axial shift on the x axis at the beam radius to the axial width (the depth of focus). In the limit of large spatial chirp rate, we can obtain an approximate expression for z_DOF, the axial full-width at half-maximum (FWHM) from the analytic expression for the axial intensity profile (Eq. (31) in Ref [32].). z_DOF can be expressed in the simple form (see Appendix 1)

Using Eq. (10) and the parameters in Table 1, the depth of focus (z_DOF) is calculated to be ~1.3 mm, which is also in good agreement with both the numerical simulation and experiment results shown in Figs. 1(a) and 1(b), respectively.

Calculating the ratio in the limit β² >>1 it is found that

In the present experiment, β ≈30 and Δω/ω₀ ≈0.03, so at x = w₀, δz/z_DOF is approximately 1.03, consistent with the modeling shown in Fig. 2. When the beam down collimation telescope was removed we had β ≈3.2 and δz/z_DOF ≈0.14 and the effect was not noticeable. For comparison, in the experiment on nonreciprocal writing by Vitek et al. [31], β ≈15 and the fractional bandwidth was closer to the present experiment (the measured pulse duration of 74 fs was limited in that experiment by third-order dispersion introduced by the single-pass grating compressor). Thus in that experiment, we estimate that δz/z_DOF ≈0.65.

It is instructive to compare the IPT to the PFT. The PFT, i.e., the tilt in x-t in the focal plane arises from the calculation of the variation of the group delay with position. At z = 0, we can write ϕ₁(x) = αω₀x/(cf) = βτ₀x/w₀ whereas in Ref [32]. we have used τ₀Δω = 2, and w₀ = 2cf/(ω₀w_in) to simplify the expression. The temporal shift of the pulse at the spot radius is βτ₀. For positions x > 0 in the focal region, the pulse arrives later (larger group delay) and the z position of maximum intensity is farther from the lens (z > 0). Figure 4 illustrates the relative orientations of the tilt in the peak intensity plane in the XZ plane.

6. Control of IPT

Since the IPT gives rise to an asymmetric focal spot shape, it is generally undesirable in many applications, such as ultrafast laser material processing, nonlinear fluorescence microscopy, two-photon optogenetic applications [35–37], and so on. As we have shown in Sect. 5, the IPT results from the intrinsically varying chirp of the local light field in the focal plane, thus one may expect that manipulation of IPT could be achieved by tuning the temporal chirp of the output laser pulses from the amplifier. Unfortunately, as we will show below, this idea is not completely correct. Nevertheless, our results also show that by taking into account the fanning out of the beamlets over the depth of focus, the IPT can be reduced with sufficiently large initial chirp of the output laser pulses from the amplifier.

To vary the chirp of output laser beam, we simply varied the distance between the grating pair G1-G2. The amount of the second-order phase was given as [38]

Table 2. List of variables used for the analytical calculation

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By increasing the temporal chirp of the incident pulses in our experiment, it was noticed that the TPFE signals became weak, so the exposure time of the digital camera was increased to 1 s, which is longer than that used in former cases. Figure 5 shows TPFE images using TF with different input temporal chirp values. It was observed that the temporally focused spot shifted its position away from the geometrical focal point as the SOP increasing, which has been proven earlier in the two-photon bio-imaging experiment with temporally focused beam [39, 40]. When the initial temporal chirp of the incident laser beam becomes sufficiently large i.e., ϕ_2in > 4000 fs², the IPT angle starts to decrease with the increasing SOP. This trend is also evidenced by the corresponding simulation results, as shown in Fig. 5.

 

Fig. 5 TPFE images using temporal focusing with input SOP values of (a) ϕ_2in = 0, (c) ϕ_2in = 500 fs², (e) ϕ_2in = 1000 fs², (g) ϕ_2in = 2000 fs², (i) ϕ_2in = 4000 fs², and (k) ϕ_2in = 6000 fs², respectively. (b), (d), (f), (g), (j) and (l) are corresponding numerical simulation TPFE signals. The color bar is shown at the bottom.

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The experimental observations in Fig. 5 can be understood following the analysis in Sect. 4. With the SOP, the second order spectral phase can be expressed as:

Table 3. Estimation of z_d with respect to ϕ_2in

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One can find that for relatively small values of SOP (i.e., ϕ_2in < 2000 fs²), the simple expression Eq. (14) reproduces both the experimental and the numerical simulation results. Compared with Eq. (4) in Ref [39], our calculation describes more precisely the relation between the axial shift of the temporally focused spot and the chirp of the incident laser beam. When ϕ_2in > 2000 fs², the temporal focus will shift farther away from the geometric focal point of the lens along z-axis and located out of the depth of focus (1.3 mm), thus the paraxial approximation fails and Eq. (14) becomes invalid. For ϕ_2in = 0, the temporal focused spot and the geometrical focus overlap.

The change of the IPT angle with the varying SOP can be obtained by calculating the slope of the zero-SOP curve ϕ₂(x, z) = 0 (see Appendix 2). Since in such circumstance, we have

The IPT angle as a function of the SOP is obtained as

Again, Eq. (16) is only valid when z_d << z_R. Under this condition, for small input chirps, the change in the focal plane tilting angle will be negligible using Eq. (16), which is evidenced in Figs. 5(a)–5(f). To increase the value of z_d, the incident laser pulses should be strongly temporally chirped before they reach the external grating pair, whereas for very large chirp values the approximation involved in derivation of Eqs. (14)–(16) would be inappropriate. In such a case, our experimental results and calculations based on the paraxial Fresnel diffraction model show that IPT can be reduced with sufficiently large initial chirp of the output laser pulses, which may benefit some applications such as femtosecond laser micro processing and two-photon fluorescence bio-imaging.

It is noteworthy that the above analysis is restricted to the second order phase, and the model is typically valid for ϕ_2in < 2000 fs². However, it is well known that a single-pass compressor/stretcher with incident angle not perfectly optimized may give rise to strong additional third-order phase (TOP), which may also introduce unexpected distortions and shifts in multi-photon process [41]. Ref [32] points out that excessive TOP will make the on-axis intensity not sensitive to the second order phase. Nevertheless in the current configuration, the grating angles of the single pass compressor were optimized for minimum TOP, so the influence of the residual TOP is weak. Moreover, while the change in the compressor separation changes SOP and TOP together, we find this does not noticeably change the IPT results. One can find an estimation of the spatiotemporal distortion of the pulses introduced by the TOP in Appendix 3.

7. Conclusions

To conclude, we have experimentally demonstrated visualization of TPFE at the focus of a spatiotemporally focused femtosecond laser beam. Our observation, which is based on a nonlinear optical process, reveals a novel IPT effect in the focal spot, resulting from the intrinsically varying chirp of the local light field in the focal plane. Due to this unusual effect, the shortest pulse duration of the spatiotemporally focused laser beam does not always occur in the focal plane. We also show that the IPT can hardly be affected by adding small amount of temporal chirp into the incident femtosecond laser pulses. However, with sufficiently large initial temporal chirps in the incident femtosecond pulses, the IPT can be reduced mainly because of the fanning out of the beamlets over the depth of focus.

Appendix

A.1 Derivation the expression for the depth of focus (z_DOF) for Eq. (10)

The axial intensity profile from Eq. (31) in Ref [32]. can be rewritten as:

(17)$$I\left(z\right)=I_0\sqrt{\frac{1}{(1+z^2/z_R^2)(1+{\beta }_{BA}^4{z}^2/z_R^2)}},$$

where${\beta }_{BA}=\sqrt{1+{\beta }^2}$. Solving for the half-maximum distance of z_H = z_DOF/2 in Eq. (17) yields

(18)$$(1+z_H^2/z_R^2)(1+{\beta }_{BA}^4{z}_H^2/z_R^2)=4.$$

In our experiment, β_BA ≈ 30, making the anzatz that z_H << z_R, we neglect the term in the first parentheses and obtain

(19)$$z_{DOF}\approx \frac{2\sqrt{3}}{{\beta }_{BA}^2}{z}_R=\frac{2\sqrt{3}}{1+{\beta }^2}{z}_R.$$

Checking our assumption, the approximation made above is valid when β_ΒΑ >> 3^1/4.

A.2 Derivation the expression of the tilt angle of the temporal focus (θ_T) for Eq. (16)

Instituting βτ₀ = 2α/w_in into ϕ₂(x, z) = 0 for Eq. (13), we can obtain

(20)$$\frac{2\alpha }{{\omega }_0{w}_{in}{w}_0}x\text{-}\frac{{\alpha }^2}{z_R{w}_{in}^2}z\text{+}\frac{{\varphi }_{2\text{in}}}{n{z}_R^2}{z}^2+\frac{{\varphi }_{2\text{in}}}{n}=0.$$

Taking the first-order derivation of z with respect to x for Eq. (A.4), we have

(21)$$z\text{'}{|}_x=\frac{2w_{in}{z}_R}{\alpha {\omega }_0{w}_0}\cdot \frac{1}{1-2{\varphi }_{2in}\cdot \frac{z}{n{z}_R}\cdot \frac{w_{in}^2}{{\alpha }^2}}=\frac{1}{\tan {\theta }_t}\cdot \frac{1}{1-2z_{d}z/z_R^2}.$$

Considering the derivation at x = 0 and from Eq. (14), we have z(x)|_x=0 ≈ z_d. Thus, the first order derivation of z(x) at x = 0 can be written as

(22)$$z\text{'}\left(x\right){|}_{x=0}\approx \frac{1}{\tan {\theta }_t}\cdot \frac{1}{1-2z_d^2/z_R^2}.$$

A.3 Evaluation the spatio-temporal distortion induced by the third order phase (TOP)

For the single-pass grating compressor, the third order phase and the second order phase has a function of [38]

(23)$${\varphi }_{3in}=-\frac{3\lambda }{2\pi c}\cdot {\varphi }_{2in}\cdot \frac{1+\lambda \sigma \sin i-{\sin }^{2}i}{1-{(\lambda \sigma -\sin i)}^2}.$$

In the current experimental configuration (incident angle i =53°, groove density σ = 1500/mm), we have ϕ_3in ≈ −2ϕ_2in fs³, where ϕ_2in is measured in fs². When taking into consideration the residual TOP caused by the single-pass grating, we can rewrite Eq. (1) as

(24)$$E_1\left(x,y,\omega \right)=A_0\text{exp}\left[-\frac{{\left(\omega -{\omega }_0\right)}^2}{\Delta {\omega }^2}+i\frac{1}{2}{\varphi }_{2\text{in}}{\left(\omega -{\omega }_0\right)}^2+i\frac{1}{6}{\varphi }_{3\text{in}}{\left(\omega -{\omega }_0\right)}^3\right]\text{exp}\left\{-\frac{{\left[x-\alpha (\omega -{\omega }_0)\right]}^2+y^2}{w_{in}^2}\right\}.$$

By performing the similar calculations as those in Sect. 4, we can obtain the spatial and temporal profile (I_TF(x, y, z, t)) of the pulse and the intensity distribution of the TPFE signals (I_2P(x, y, z)) at the focus.

Fig. 6 shows the temporal profile of the pulses at the temporal focus with different input second order phase and corresponding residual third order phase induced by the single-pass grating pair in our experimental configuration. For comparison, pulse profiles without input spectral phase and those with only input SOP are also plotted, respectively. It can be seen from Figs. 6(a)–6(c) that for current bandwidth, the residual TOP does not affect too much to the temporal profile of the pulse. However, for 10-fs pulses, even small amount of TOP will deform the pulses by inducing side wings to the temporal profile, as shown in Fig. 7.

 

Fig. 6 Comparison of the temporal profile of the pulse at the temporal focus with (a) τ₀ = 40 fs, ϕ_2in = 2000 fs², ϕ_3in = −4000 fs³, (b) τ₀ = 40 fs, ϕ_2in = 4000 fs², ϕ_3in = −8000 fs³, (c) τ₀ = 40 fs, ϕ_2in = 6000 fs², ϕ_3in = −12000 fs³ and (d) τ₀ = 10 fs, ϕ_2in = 2000 fs², ϕ_3in = −4000 fs³, respectively. The red curves are pulse profiles without input SOP and TOP, and the green curves are pulse profiles with only input SOP, while the blue curves are those with both input SOP and TOP.

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Fig. 7 Numerical simulation of the TPFE signals using temporal focusing with (a) ϕ_2in = 0 fs², ϕ_3in = 0 fs³, (b) ϕ_2in = 2000 fs², ϕ_3in = 0 fs³, (c)ϕ_2in = 2000 fs², ϕ_3in = −4000 fs³, (d) ϕ_2in = 4000 fs², ϕ_3in = 0 fs³, (e) ϕ_2in = 4000 fs², ϕ_3in = −8000 fs³, (f) ϕ_2in = 6000 fs², ϕ_3in = 0 fs³, and (g) ϕ_2in = 6000 fs², ϕ_3in = −12000 fs³ respectively. The color bar is shown at the bottom.

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The influence of the input TOP to the spatial profile of the intensity distribution of a TF beam is shown in Fig. 7. It is found that the profile shows no evident discrepancy for ϕ_2in < 2000 fs² and |ϕ_3in| < 4000 fs³. With input SOP and TOP increasing, the depth of the focus increases and the IPT angle decreases. In particular, the residual TOP of the grating pair intensifies this trend.

The above analysis shows that in our typical configuration, the spatial and temporal distortion caused by the input TOP is negligible for ϕ_2in < 2000 fs².

Acknowledgments

The work is supported by NSFC (Nos. 61327902, 61275205, 11104294, 61108015), Shanghai Natural Science Foundation (No. 11ZR1441600), and the Program of Shanghai Subject Chief Scientist (11XD1405500). C. D. acknowledges support from the Air Force Office of Scientific Research under programs FA9550-10-1-0394, FA9550-10-0561 and FA9550-12-1-0482. C. D. would also like to thank Jeff Squier at CSM for useful discussions.

References and links

1. K. M. Davis, K. Miura, N. Sugimoto, and K. Hirao, “Writing waveguides in glass with a femtosecond laser,” Opt. Lett. 21(21), 1729–1731 (1996). [CrossRef]   [PubMed]  

2. A. Marcinkevičius, S. Juodkazis, M. Watanabe, M. Miwa, S. Matsuo, H. Misawa, and J. Nishii, “Femtosecond laser-assisted three-dimensional microfabrication in silica,” Opt. Lett. 26(5), 277–279 (2001). [CrossRef]   [PubMed]  

3. L. Sansoni, F. Sciarrino, G. Vallone, P. Mataloni, A. Crespi, R. Ramponi, and R. Osellame, “Polarization entangled state measurement on a chip,” Phys. Rev. Lett. 105(20), 200503 (2010). [CrossRef]   [PubMed]  

4. W. Xiong, Y. S. Zhou, X. N. He, Y. Gao, M. Mahjouri-Samani, L. Jiang, T. Baldacchini, and Y. F. Lu, “Simultaneous additive and subtractive three-dimensional nanofabrication using integrated two-photon polymerization and multiphoton ablation,” Light Sci. Appl. 1(4), e6 (2012). [CrossRef]  

5. Y. Cheng, K. Sugioka, and K. Midorikawa, “Microfluidic laser embedded in glass by three-dimensional femtosecond laser microprocessing,” Opt. Lett. 29(17), 2007–2009 (2004). [CrossRef]   [PubMed]  

6. F. Chen and J. R. Vazquez de Aldana, “Optical waveguides in crystalline dielectric materials produced by femtosecond-laser micromachining,” Laser Photon. Rev. 8(2), 251–275 (2014).

7. F. He, Y. Cheng, L. Qiao, C. Wang, Z. Xu, K. Sugioka, K. Midorikawa, and J. Wu, “Two-photon fluorescence excitation with a microlens fabricated on the fused silica chip by femtosecond laser micromachining,” Appl. Phys. Lett. 96(4), 041108 (2010). [CrossRef]  

8. K. Sugioka and Y. Cheng, “Femtosecond laser processing for optofluidic fabrication,” Lab Chip 12(19), 3576–3589 (2012). [CrossRef]   [PubMed]  

9. Y. Liao, J. Song, E. Li, Y. Luo, Y. Shen, D. Chen, Y. Cheng, Z. Xu, K. Sugioka, and K. Midorikawa, “Rapid prototyping of three-dimensional microfluidic mixers in glass by femtosecond laser direct writing,” Lab Chip 12(4), 746–749 (2012). [CrossRef]   [PubMed]  

10. R. R. Gattass and E. Mazur, “Femtosecond laser micromachining in transparent materials,” Nat. Photonics 2(4), 219–225 (2008). [CrossRef]  

11. Y. Shimotsuma, P. G. Kazansky, J. Qiu, and K. Hirao, “Self-organized nanogratings in glass irradiated by ultrashort light pulses,” Phys. Rev. Lett. 91(24), 247405 (2003). [CrossRef]   [PubMed]  

12. V. R. Bhardwaj, E. Simova, P. P. Rajeev, C. Hnatovsky, R. S. Taylor, D. M. Rayner, and P. B. Corkum, “Optically produced arrays of planar nanostructures inside fused silica,” Phys. Rev. Lett. 96(5), 057404 (2006). [CrossRef]   [PubMed]  

13. S. Kanehira, J. Si, J. Qiu, K. Fujita, and K. Hirao, “Periodic nanovoid structures via femtosecond laser irradiation,” Nano Lett. 5(8), 1591–1595 (2005). [CrossRef]   [PubMed]  

14. Y. Liu, M. Shimizu, B. Zhu, Y. Dai, B. Qian, J. Qiu, Y. Shimotsuma, K. Miura, and K. Hirao, “Micromodification of element distribution in glass using femtosecond laser irradiation,” Opt. Lett. 34(2), 136–138 (2009). [CrossRef]   [PubMed]  

15. S. K. Sundaram and E. Mazur, “Inducing and probing non-thermal transitions in semiconductors using femtosecond laser pulses,” Nat. Mater. 1(4), 217–224 (2002). [CrossRef]   [PubMed]  

16. 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(15), 151120 (2007). [CrossRef]  

17. W. Yang, P. G. Kazansky, and Y. P. Svirko, “Non-reciprocal ultrafast laser writing,” Nat. Photonics 2(2), 99–104 (2008). [CrossRef]  

18. W. Yang, P. G. Kazansky, Y. Shimotsuma, M. Sakakura, K. Miura, and K. Hirao, “Ultrashort-pulse laser calligraphy,” Appl. Phys. Lett. 93(17), 171109 (2008). [CrossRef]  

19. P. S. Salter and M. J. Booth, “Dynamic control of directional asymmetry observed in ultrafast laser direct writing,” Appl. Phys. Lett. 101(14), 141109 (2012). [CrossRef]  

20. R. Stoian, M. Boyle, A. Thoss, A. Rosenfeld, G. Korn, I. V. Hertel, and E. E. B. Campbell, “Laser ablation of dielectrics with temporally shaped femtosecond pulses,” Appl. Phys. Lett. 80(3), 353–355 (2002). [CrossRef]  

21. Y. Cheng, K. Sugioka, K. Midorikawa, M. Masuda, K. Toyoda, M. Kawachi, and K. Shihoyama, “Control of the cross-sectional shape of a hollow microchannel embedded in photostructurable glass by use of a femtosecond laser,” Opt. Lett. 28(1), 55–57 (2003). [CrossRef]   [PubMed]  

22. C. Mauclair, A. Mermillod-Blondin, N. Huot, E. Audouard, and R. Stoian, “Ultrafast laser writing of homogeneous longitudinal waveguides in glasses using dynamic wavefront correction,” Opt. Express 16(8), 5481–5492 (2008). [CrossRef]   [PubMed]  

23. Y. Shimotsuma, M. Sakakura, P. G. Kazansky, M. Beresna, J. Qiu, K. Miura, and K. Hirao, “Ultrafast manipulation of self-assembled form birefringence in glass,” Adv. Mater. 22(36), 4039–4043 (2010). [CrossRef]   [PubMed]  

24. G. Zhu, J. van Howe, M. E. Durst, W. Zipfel, and C. Xu, “Simultaneous spatial and temporal focusing of femtosecond pulses,” Opt. Express 13(6), 2153–2159 (2005). [CrossRef]   [PubMed]  

25. D. Oron, E. Tal, and Y. Silberberg, “Scanningless depth-resolved microscopy,” Opt. Express 13(5), 1468–1476 (2005). [CrossRef]   [PubMed]  

26. 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(7), 1106–1108 (2010). [CrossRef]   [PubMed]  

27. 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(8), 083014 (2011). [CrossRef]  

28. B. Zeng, W. Chu, H. Gao, W. Liu, G. Li, H. Zhang, J. Yao, J. Ni, S. L. Chin, Y. Cheng, and Z. Xu, “Enhancement of peak intensity in a filament core with spatiotemporally focused femtosecond laser pulses,” Phys. Rev. A 84(6), 063819 (2011). [CrossRef]  

29. 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(17), 18086–18094 (2010). [CrossRef]   [PubMed]  

30. D. Kim and P. T. C. So, “High-throughput three-dimensional lithographic microfabrication,” Opt. Lett. 35(10), 1602–1604 (2010). [CrossRef]   [PubMed]  

31. 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(24), 24673–24678 (2010). [CrossRef]   [PubMed]  

32. C. G. Durfee, M. Greco, E. Block, D. Vitek, and J. A. Squier, “Intuitive analysis of space-time focusing with double-ABCD calculation,” Opt. Express 20(13), 14244–14259 (2012). [CrossRef]   [PubMed]  

33. J. W. Goodman, Introduction to Fourier Optics (Roberts and Company, 2005).

34. S. Akturk, X. Gu, E. Zeek, and R. Trebino, “Pulse-front tilt caused by spatial and temporal chirp,” Opt. Express 12(19), 4399–4410 (2004). [CrossRef]   [PubMed]  

35. K. Itoh, W. Watanabe, S. Nolte, and C. Schaffer, “Ultrafast processes for bulk modification of transparent materials,” MRS Bull. 31, 620–625 (2006).

36. E. Papagiakoumou, F. Anselmi, A. Bègue, V. de Sars, J. Glückstad, E. Y. Isacoff, and V. Emiliani, “Scanless two-photon excitation of channelrhodopsin-2,” Nat. Methods 7(10), 848–854 (2010). [CrossRef]   [PubMed]  

37. B. K. Andrasfalvy, B. V. Zemelman, J. Tang, and A. Vaziri, “Two-photon single-cell optogenetic control of neuronal activity by sculpted light,” Proc. Natl. Acad. Sci. U.S.A. 107(26), 11981–11986 (2010). [CrossRef]   [PubMed]  

38. S. Backus, C. G. Durfee, M. M. Murnane, and H. C. Kapteyn, “High power ultrafast lasers,” Rev. Sci. Instrum. 69(3), 1207–1223 (1998). [CrossRef]  

39. M. E. Durst, G. Zhu, and C. Xu, “Simultaneous spatial and temporal focusing for axial scanning,” Opt. Express 14(25), 12243–12254 (2006). [CrossRef]   [PubMed]  

40. M. E. Durst, G. Zhu, and C. Xu, “Simultaneous spatial and temporal focusing in nonlinear microscopy,” Opt. Commun. 281(7), 1796–1805 (2008). [CrossRef]   [PubMed]  

41. B. Chatel, J. Degert, and B. Girard, “Role of quadratic and cubic spectral phases in ladder climbing with ultrashort pulses,” Phys. Rev. A 70(5), 053414 (2004). [CrossRef]