Characterization and control of peak intensity distribution at the focus of a spatiotemporally focused femtosecond laser beam Fei He,1 Bin Zeng,1 Wei Chu,1 Jielei Ni,1 Koji Sugioka,2 Ya Cheng,1,* and Charles G. Durfee3,4 1
State Key Laboratory of High Field Laser Physics, Shanghai Institute of Optics and Fine Mechanics, Chinese Academy of Sciences, P.O. Box 800-211, Shanghai 201800, China 2 RIKEN Center for Advanced Photonics, Hirosawa 2-1, Wako, Saitama 351-0198, Japan 3 Department of Physics, Colorado School of Mines, 1523 Illinois Street, Golden, CO 80401, USA 4 [email protected] * [email protected]
Abstract: We report on experimental examination of two-photon fluorescence excitation (TPFE) at the focus of a spatially chirped femtosecond laser beam, which reveals an unexpected tilted peak intensity distribution in the focal spot. Our theoretical calculation shows that the tilting of the peak intensity distribution originates from the fact that along the optical axis of objective lens, the spatiotemporally focused pulse reaches its shortest duration exactly at the focal plane. However, when moving away from the optical axis along the direction of spatial chirp of the incident pulse, the pulse reaches its shortest duration either before or after the focal plane, depending on whether the pulse duration is measured above or below the optical axis as well as the sign of the spatial chirp. The tilting of the peak intensity distribution in the focal spot of the spatiotemporally focused femtosecond laser beam can play important roles in applications such as femtosecond laser micromachining and bio-imaging. ©2014 Optical Society of America OCIS codes: (140.3390) Laser materials processing; (220.4000) Microstructure fabrication; (320.5540) Pulse shaping.
References and links 1.
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#206625 - $15.00 USD Received 17 Feb 2014; revised 28 Mar 2014; accepted 28 Mar 2014; published 15 Apr 2014 (C) 2014 OSA 21 April 2014 | Vol. 22, No. 8 | DOI:10.1364/OE.22.009734 | OPTICS EXPRESS 9734
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#206625 - $15.00 USD Received 17 Feb 2014; revised 28 Mar 2014; accepted 28 Mar 2014; published 15 Apr 2014 (C) 2014 OSA 21 April 2014 | Vol. 22, No. 8 | DOI:10.1364/OE.22.009734 | OPTICS EXPRESS 9735
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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, #206625 - $15.00 USD Received 17 Feb 2014; revised 28 Mar 2014; accepted 28 Mar 2014; published 15 Apr 2014 (C) 2014 OSA 21 April 2014 | Vol. 22, No. 8 | DOI:10.1364/OE.22.009734 | OPTICS EXPRESS 9736
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).
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 ( f1 = 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).
#206625 - $15.00 USD Received 17 Feb 2014; revised 28 Mar 2014; accepted 28 Mar 2014; published 15 Apr 2014 (C) 2014 OSA 21 April 2014 | Vol. 22, No. 8 | DOI:10.1364/OE.22.009734 | OPTICS EXPRESS 9737
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.
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 2win = 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
#206625 - $15.00 USD Received 17 Feb 2014; revised 28 Mar 2014; accepted 28 Mar 2014; published 15 Apr 2014 (C) 2014 OSA 21 April 2014 | Vol. 22, No. 8 | DOI:10.1364/OE.22.009734 | OPTICS EXPRESS 9738
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).
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] 2 2 (ω − ω0 )2 1 2 [ x − α (ω − ω0 ) ] + y + − − E1 ( x, y , ω ) = A0exp − i exp φ ω ω ( ) , (1) 2 in 0 Δω 2 2 win2
where A0 is the constant field amplitude, ω0 the carrier frequency, Δω the bandwidth (1/e2 half width) of the pulses, win the initial beam waist (1/e2), respectively. φ2in is the initial secondorder phase (SOP), α(ω-ω0) is the linear shift of each spectral component at the entrance aperture of the lens, with detailed derivation of α in the Appendix in Ref [27]. For our conditions (f = 500 mm, beamlet width 2w0 = 1mm, spatially chirped width βBA⋅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 E2 ( x, y,z,ω ) =
exp(ikL) ik E ( ξ ,η ,ω )exp A (ξ 2 + η 2 ) − 2 ( xξ + yη ) + D( x 2 + y 2 ) dξ dη , (2) iλ B 1 2B
#206625 - $15.00 USD Received 17 Feb 2014; revised 28 Mar 2014; accepted 28 Mar 2014; published 15 Apr 2014 (C) 2014 OSA 21 April 2014 | Vol. 22, No. 8 | DOI:10.1364/OE.22.009734 | OPTICS EXPRESS 9739
where L is the overall path along the optical axis, A, B, C, and D are elements of the ABCD matrix used for describing the optical elements, and are defined as
A B 1 nz 1 l 1 C D = 0 1 0 1 −1 / f
0 , 1
(3)
with n the refractive index of the water, l the length of the air gap between L3 and the cuvette. Thus the overall optical path is L = l + nz. 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 ∞
∞
−∞
−∞
2 I 2 P ( x, y , z ) ∝ I TF ( x, y, z, t ) dt =
−1 {E2 ( x, y , z, ω )} dt , 4
(4)
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 Laser system
Grating pair
Focal lens
λ0: Central wavelength in vacuum c: The speed of light in vacuum Δλ: Spectral bandwidth (FWHM) Δω: Spectral bandwidth (half width at 1/e2)
800 nm 3 × 108 m/s 30 nm
win: Input beam radius (1/e2) φin: Input spectral phase σ: Groove density i: Incident angle γ:The −1st order diffraction angle of the central beamlet d: distance between the parallel grating pair α: Angular chirp rate
0.5 mm 0 1500 line/mm 53° arc sin (sin i –σλ0)
f: Focal length in air
2 ln2 / 2π cΔλ /λ02 (5.2 × 1013 rad/s)
730 mm dσλ0cosi/(ω0cos3γ) (2.9 × 10−16 m.s/rad) 500 mm (f = 250 mm, 1000 mm in Fig. 3 for comparison)
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 precompensated 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
#206625 - $15.00 USD Received 17 Feb 2014; revised 28 Mar 2014; accepted 28 Mar 2014; published 15 Apr 2014 (C) 2014 OSA 21 April 2014 | Vol. 22, No. 8 | DOI:10.1364/OE.22.009734 | OPTICS EXPRESS 9740
1 2
φ ( x, y , z, ω ) = n(ω )k0 x sin θ x + n(ω )k0 z 1 − sin 2 θ x − η ( z ) + n(ω )k0
( x − z sin θ x ) 2 + y 2 , (5) 2 R( z )
where n is the refractive index of the medium, k0 is wave number of the pulses in the vacuum, R(z) is the radius of the curvature of the beam’s wavefront and η(z) is the Gouy phase shift. We define the frequency dependence of the beamlet angle as sinθx ≈θx = α(ω-ω0)/f . For future reference the spatial chirp rate is related to the angular chirp rate α through β = αΔω/win = 2α/(τ0win). When the spectral phase is expanded to second order, and by neglecting the material dispersion of air and water, we can obtain the spatial dependence of the pulse chirp: x τ 0 β z τ 02 β 2 n − 2 ω + 4 1 / z R2 w z z R 0 0
φ2 ( x, z ) =
,
(6)
where τ0 = 2/Δω is the transform limited pulse duration. This spectral phase originates from the evolution of the wavefront curvature of each of the beamlets. It is evident in this expression that there is a “line” in the XZ plane where this geometric phase is zero. Solving for where φ2 = 0 we can get z ( x) =
2 x Δω 2nf z = x, β w0 ω0 R αω0
(7)
where the focal spot size is w0 = λ0f/(πwin),and the beamlet Rayleigh range is zR = πnw02/λ0. Thus the value of the IPT angle θt, defined as the angle between the tilted intensity plane and the positive z-axis (indicated in Fig. 4), can be evaluated as: tan θ t =
αω0
. (8) 2nf 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]: 2
k2 x x + k2 z z 1 α2 2 ≈ τ 0 1 + 4 2 2 ( x − z tan θ t ) , 2 c f τ τ 0 0
τ ( x , z ) =τ 0 1 +
(9)
where k2x = ∂φ2/∂z and k2z = ∂φ2/∂z are group velocity dispersion (GVD) coefficients in the x and z directions, respectively. Clearly, the pulse duration in this situation is dependent on both the propagation distance z and the lateral distance x away from the geometrical focus, which is quite different from that in Ref [24]. the pulse duration only depend on the propagation distance z. 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 θt = 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
#206625 - $15.00 USD Received 17 Feb 2014; revised 28 Mar 2014; accepted 28 Mar 2014; published 15 Apr 2014 (C) 2014 OSA 21 April 2014 | Vol. 22, No. 8 | DOI:10.1364/OE.22.009734 | OPTICS EXPRESS 9741
of large spatial chirp rate, we can obtain an approximate expression for zDOF, the axial fullwidth at half-maximum (FWHM) from the analytic expression for the axial intensity profile (Eq. (31) in Ref [32].). zDOF can be expressed in the simple form (see Appendix 1) 2 3 zR . (10) 1+ β 2 Using Eq. (10) and the parameters in Table 1, the depth of focus (zDOF) 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 zDOF ≈
z ( x) 1 x Δω 1 αΔω 2 ≈ β = x, zDOF 3 w0 ω0 2 3c f
(11)
where the spatial chirp rate β and the angular chirp rate α are employed in the two forms, respectively. From this expression we can see that at x = w0, the IPT will cause a shift along z direction (δz) that can be comparable to zDOF for spatial chirp rate sufficiently larger than βΔω/ω0. This relative IPT is actually independent of the focusing conditions (f and w0), and solely dependent on the spatial chirp rate and the fractional bandwidth.
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.
In the present experiment, β ≈30 and Δω/ω0 ≈0.03, so at x = w0, δz/zDOF 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/zDOF ≈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/zDOF ≈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/(ω0win) 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
#206625 - $15.00 USD Received 17 Feb 2014; revised 28 Mar 2014; accepted 28 Mar 2014; published 15 Apr 2014 (C) 2014 OSA 21 April 2014 | Vol. 22, No. 8 | DOI:10.1364/OE.22.009734 | OPTICS EXPRESS 9742
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. Table 2. List of variables used for the analytical calculation Variable name
φ φ1 φ2 φ2in
θx θt θT β βBA zR zd zDOF δz τ0 w0 ω0
Description Phase of the TF beam Group delay of the TF beam Second-order phase (group delay dispersion) of the TF beam Input second-order phase Frequency-dependent angle of the beamlets IPT angle without input second-order phase IPT angle with input second order phase Spatial chirp rate Spatial chirp rate beam aspect ratio [32] Rayleigh range of the beamlet The distance of the temporal focus shift away from the geometrical focus Depth of the focus of the TF beam (FWHM) Axial intensity shift of focus of the TF Beam Transform-limited pulse duration (FWHM) Focal spot size of the beamlet Central frequency of the spectrum
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]
φ2 = −
−3/2 λ 3d σ 2 1 − (λσ − sin i ) 2 , 2 πc
(12)
which is proportional to the distance between G1 and G2. For our current configuration, φ2in ≈5.3Δd × 106 fs2, where Δd is the distance of the grating away from its chirp-free position in our setup. Again all the other experimental conditions were the same as those in Sect. 2. The corresponding theoretical analysis involving SOP can be developed by varying the parameter φ2in in Eq. (1). For convenience, most variables used in this analysis are listed in Table 2.
#206625 - $15.00 USD Received 17 Feb 2014; revised 28 Mar 2014; accepted 28 Mar 2014; published 15 Apr 2014 (C) 2014 OSA 21 April 2014 | Vol. 22, No. 8 | DOI:10.1364/OE.22.009734 | OPTICS EXPRESS 9743
Fig. 5. TPFE images using temporal focusing with input SOP values of (a) φ2in = 0, (c) φ2in = 500 fs2, (e) φ2in = 1000 fs2, (g) φ2in = 2000 fs2, (i) φ2in = 4000 fs2, and (k) φ2in = 6000 fs2, respectively. (b), (d), (f), (g), (j) and (l) are corresponding numerical simulation TPFE signals. The color bar is shown at the bottom.
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
#206625 - $15.00 USD Received 17 Feb 2014; revised 28 Mar 2014; accepted 28 Mar 2014; published 15 Apr 2014 (C) 2014 OSA 21 April 2014 | Vol. 22, No. 8 | DOI:10.1364/OE.22.009734 | OPTICS EXPRESS 9744
i.e., φ2in > 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. 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: x τ 0 β z τ 02 β 2 n − 2 4 1 / z R2 w z z ω + R 0 0
φ2 ( x, z ) =
+ φ2in ,
(13)
where z = 0 is set to be the axial position of the geometric focal plane. The input second-order phase φ2in can balance the geometric phase at an axial position zd away from the geometric focal plane. Working with Eq. (13) we can solve for on-axis positions with zero second-order phase: φ2(x = 0, z = zd) = 0. There are two solutions for zd; the position closer to z = 0 results in higher intensity because there is a larger bandwidth and smaller focal spot size. Neglecting the denominator term in second parentheses in Eq. (13), and making use of βτ0 = 2α/win we find: zd ≈ φ2in
win2 Δω 2 = φ z zR , R 2 in nα 2 nβ 2
(14)
where in the second form we make use of the time-bandwidth relation τ0Δω = 2 . In the present investigation, we can use Eq. (14) to evaluate the results in Fig. 5. For comparison, we also estimate the values of zd with respect to φ2in from the experimental and numerical simulation in Fig. 5, respectively, as listed in Table 3. The experimental values of zd can be obtained by directly measuring the distance shift of the peak position of the TPFE signals away from this peak position with zero-SOP input (Fig. 5(a)). Table 3. Estimation of zd with respect to φ2in zd (mm)
φ2in (fs2) Experimental Numerical Analytical
500 0.5 0.5 0.5
1000 1.0 1.0 1.0
2000 1.9 2.0 2.0
4000 3.8 3.9 4.1
6000 5.2 5.4 6.1
One can find that for relatively small values of SOP (i.e., φ2in < 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 φ2in > 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 φ2in = 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 1 1 ⋅ . tan θ t 1 − 2 zd2 / z R2 The IPT angle as a function of the SOP is obtained as z ' ( x ) |x = 0 ≈
tan θT = (1 − 2 zd2 / zR2 ) tan θt .
(15)
(16)
Again, Eq. (16) is only valid when zd 31/4.
A.2 Derivation the expression of the tilt angle of the temporal focus (θT) for Eq. (16) Instituting βτ0 = 2α/win into φ2(x, z) = 0 for Eq. (13), we can obtain
φ φ 2α α2 xz + 2in2 z 2 + 2in = 0. nz R n ω0 win w0 z R win2
(20)
#206625 - $15.00 USD Received 17 Feb 2014; revised 28 Mar 2014; accepted 28 Mar 2014; published 15 Apr 2014 (C) 2014 OSA 21 April 2014 | Vol. 22, No. 8 | DOI:10.1364/OE.22.009734 | OPTICS EXPRESS 9746
Taking the first-order derivation of z with respect to x for Eq. (A.4), we have z ' |x =
2 win zR
αω0 w0
⋅
1 z win2 ⋅ 1 − 2φ2in ⋅ nz R α 2
=
1 1 ⋅ . tan θ t 1 − 2 zd z / z R2
(21)
Considering the derivation at x = 0 and from Eq. (14), we have z(x)|x=0 ≈ zd. Thus, the first order derivation of z(x) at x = 0 can be written as z ' ( x ) |x = 0 ≈
1 1 . ⋅ tan θ t 1 − 2 zd2 / z R2
(22)
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] 3λ 1 + λσ sin i − sin 2 i ⋅ φ2in ⋅ . (23) 2π c 1 − (λσ − sin i ) 2 In the current experimental configuration (incident angle i =53°, groove density σ = 1500/mm), we have φ3in ≈ −2φ2in fs3, where φ2in is measured in fs2. When taking into consideration the residual TOP caused by the single-pass grating, we can rewrite Eq. (1) as
φ3in = −
2 2 (ω − ω0 )2 1 1 2 3 [ x − α (ω − ω0 ) ] + y E1 ( x, y , ω ) = A0exp − + i φ2in (ω − ω0 ) + i φ3in (ω − ω0 ) exp − . 2 2 2 6 win Δω (24) By performing the similar calculations as those in Sect. 4, we can obtain the spatial and temporal profile (ITF(x, y, z, t)) of the pulse and the intensity distribution of the TPFE signals (I2P(x, y, z)) at the focus.
Fig. 6. Comparison of the temporal profile of the pulse at the temporal focus with (a) τ0 = 40 fs,
φ2in = 2000 fs2, φ3in = −4000 fs3, (b) τ0 = 40 fs, φ2in = 4000 fs2, φ3in = −8000 fs3, (c) τ0 = 40 fs, φ2in = 6000 fs2, φ3in = −12000 fs3 and (d) τ0 = 10 fs, φ2in = 2000 fs2, φ3in = −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.
#206625 - $15.00 USD Received 17 Feb 2014; revised 28 Mar 2014; accepted 28 Mar 2014; published 15 Apr 2014 (C) 2014 OSA 21 April 2014 | Vol. 22, No. 8 | DOI:10.1364/OE.22.009734 | OPTICS EXPRESS 9747
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. 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 φ2in < 2000 fs2 and |φ3in| < 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.
Fig. 7. Numerical simulation of the TPFE signals using temporal focusing with (a) φ2in = 0 fs2, φ3in = 0 fs3, (b) φ2in = 2000 fs2, φ3in = 0 fs3, (c)φ2in = 2000 fs2, φ3in = −4000 fs3, (d) φ2in = 4000 fs2, φ3in = 0 fs3, (e) φ2in = 4000 fs2, φ3in = −8000 fs3, (f) φ2in = 6000 fs2, φ3in = 0 fs3, and (g) φ2in = 6000 fs2, φ3in = −12000 fs3 respectively. The color bar is shown at the bottom.
The above analysis shows that in our typical configuration, the spatial and temporal distortion caused by the input TOP is negligible for φ2in < 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 FA9550-12-10482. C. D. would also like to thank Jeff Squier at CSM for useful discussions.
#206625 - $15.00 USD Received 17 Feb 2014; revised 28 Mar 2014; accepted 28 Mar 2014; published 15 Apr 2014 (C) 2014 OSA 21 April 2014 | Vol. 22, No. 8 | DOI:10.1364/OE.22.009734 | OPTICS EXPRESS 9748
State Key Laboratory of High Field Laser Physics, Shanghai Institute of Optics and Fine Mechanics, Chinese Academy of Sciences, P.O. Box 800-211, Shanghai 201800, China 2 RIKEN Center for Advanced Photonics, Hirosawa 2-1, Wako, Saitama 351-0198, Japan 3 Department of Physics, Colorado School of Mines, 1523 Illinois Street, Golden, CO 80401, USA 4 [email protected] * [email protected]
Abstract: We report on experimental examination of two-photon fluorescence excitation (TPFE) at the focus of a spatially chirped femtosecond laser beam, which reveals an unexpected tilted peak intensity distribution in the focal spot. Our theoretical calculation shows that the tilting of the peak intensity distribution originates from the fact that along the optical axis of objective lens, the spatiotemporally focused pulse reaches its shortest duration exactly at the focal plane. However, when moving away from the optical axis along the direction of spatial chirp of the incident pulse, the pulse reaches its shortest duration either before or after the focal plane, depending on whether the pulse duration is measured above or below the optical axis as well as the sign of the spatial chirp. The tilting of the peak intensity distribution in the focal spot of the spatiotemporally focused femtosecond laser beam can play important roles in applications such as femtosecond laser micromachining and bio-imaging. ©2014 Optical Society of America OCIS codes: (140.3390) Laser materials processing; (220.4000) Microstructure fabrication; (320.5540) Pulse shaping.
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#206625 - $15.00 USD Received 17 Feb 2014; revised 28 Mar 2014; accepted 28 Mar 2014; published 15 Apr 2014 (C) 2014 OSA 21 April 2014 | Vol. 22, No. 8 | DOI:10.1364/OE.22.009734 | OPTICS EXPRESS 9734
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#206625 - $15.00 USD Received 17 Feb 2014; revised 28 Mar 2014; accepted 28 Mar 2014; published 15 Apr 2014 (C) 2014 OSA 21 April 2014 | Vol. 22, No. 8 | DOI:10.1364/OE.22.009734 | OPTICS EXPRESS 9735
40. M. E. Durst, G. Zhu, and C. Xu, “Simultaneous spatial and temporal focusing in nonlinear microscopy,” Opt. Commun. 281(7), 1796–1805 (2008). 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).
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, #206625 - $15.00 USD Received 17 Feb 2014; revised 28 Mar 2014; accepted 28 Mar 2014; published 15 Apr 2014 (C) 2014 OSA 21 April 2014 | Vol. 22, No. 8 | DOI:10.1364/OE.22.009734 | OPTICS EXPRESS 9736
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).
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 ( f1 = 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).
#206625 - $15.00 USD Received 17 Feb 2014; revised 28 Mar 2014; accepted 28 Mar 2014; published 15 Apr 2014 (C) 2014 OSA 21 April 2014 | Vol. 22, No. 8 | DOI:10.1364/OE.22.009734 | OPTICS EXPRESS 9737
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.
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 2win = 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
#206625 - $15.00 USD Received 17 Feb 2014; revised 28 Mar 2014; accepted 28 Mar 2014; published 15 Apr 2014 (C) 2014 OSA 21 April 2014 | Vol. 22, No. 8 | DOI:10.1364/OE.22.009734 | OPTICS EXPRESS 9738
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).
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] 2 2 (ω − ω0 )2 1 2 [ x − α (ω − ω0 ) ] + y + − − E1 ( x, y , ω ) = A0exp − i exp φ ω ω ( ) , (1) 2 in 0 Δω 2 2 win2
where A0 is the constant field amplitude, ω0 the carrier frequency, Δω the bandwidth (1/e2 half width) of the pulses, win the initial beam waist (1/e2), respectively. φ2in is the initial secondorder phase (SOP), α(ω-ω0) is the linear shift of each spectral component at the entrance aperture of the lens, with detailed derivation of α in the Appendix in Ref [27]. For our conditions (f = 500 mm, beamlet width 2w0 = 1mm, spatially chirped width βBA⋅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 E2 ( x, y,z,ω ) =
exp(ikL) ik E ( ξ ,η ,ω )exp A (ξ 2 + η 2 ) − 2 ( xξ + yη ) + D( x 2 + y 2 ) dξ dη , (2) iλ B 1 2B
#206625 - $15.00 USD Received 17 Feb 2014; revised 28 Mar 2014; accepted 28 Mar 2014; published 15 Apr 2014 (C) 2014 OSA 21 April 2014 | Vol. 22, No. 8 | DOI:10.1364/OE.22.009734 | OPTICS EXPRESS 9739
where L is the overall path along the optical axis, A, B, C, and D are elements of the ABCD matrix used for describing the optical elements, and are defined as
A B 1 nz 1 l 1 C D = 0 1 0 1 −1 / f
0 , 1
(3)
with n the refractive index of the water, l the length of the air gap between L3 and the cuvette. Thus the overall optical path is L = l + nz. 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 ∞
∞
−∞
−∞
2 I 2 P ( x, y , z ) ∝ I TF ( x, y, z, t ) dt =
−1 {E2 ( x, y , z, ω )} dt , 4
(4)
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 Laser system
Grating pair
Focal lens
λ0: Central wavelength in vacuum c: The speed of light in vacuum Δλ: Spectral bandwidth (FWHM) Δω: Spectral bandwidth (half width at 1/e2)
800 nm 3 × 108 m/s 30 nm
win: Input beam radius (1/e2) φin: Input spectral phase σ: Groove density i: Incident angle γ:The −1st order diffraction angle of the central beamlet d: distance between the parallel grating pair α: Angular chirp rate
0.5 mm 0 1500 line/mm 53° arc sin (sin i –σλ0)
f: Focal length in air
2 ln2 / 2π cΔλ /λ02 (5.2 × 1013 rad/s)
730 mm dσλ0cosi/(ω0cos3γ) (2.9 × 10−16 m.s/rad) 500 mm (f = 250 mm, 1000 mm in Fig. 3 for comparison)
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 precompensated 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
#206625 - $15.00 USD Received 17 Feb 2014; revised 28 Mar 2014; accepted 28 Mar 2014; published 15 Apr 2014 (C) 2014 OSA 21 April 2014 | Vol. 22, No. 8 | DOI:10.1364/OE.22.009734 | OPTICS EXPRESS 9740
1 2
φ ( x, y , z, ω ) = n(ω )k0 x sin θ x + n(ω )k0 z 1 − sin 2 θ x − η ( z ) + n(ω )k0
( x − z sin θ x ) 2 + y 2 , (5) 2 R( z )
where n is the refractive index of the medium, k0 is wave number of the pulses in the vacuum, R(z) is the radius of the curvature of the beam’s wavefront and η(z) is the Gouy phase shift. We define the frequency dependence of the beamlet angle as sinθx ≈θx = α(ω-ω0)/f . For future reference the spatial chirp rate is related to the angular chirp rate α through β = αΔω/win = 2α/(τ0win). When the spectral phase is expanded to second order, and by neglecting the material dispersion of air and water, we can obtain the spatial dependence of the pulse chirp: x τ 0 β z τ 02 β 2 n − 2 ω + 4 1 / z R2 w z z R 0 0
φ2 ( x, z ) =
,
(6)
where τ0 = 2/Δω is the transform limited pulse duration. This spectral phase originates from the evolution of the wavefront curvature of each of the beamlets. It is evident in this expression that there is a “line” in the XZ plane where this geometric phase is zero. Solving for where φ2 = 0 we can get z ( x) =
2 x Δω 2nf z = x, β w0 ω0 R αω0
(7)
where the focal spot size is w0 = λ0f/(πwin),and the beamlet Rayleigh range is zR = πnw02/λ0. Thus the value of the IPT angle θt, defined as the angle between the tilted intensity plane and the positive z-axis (indicated in Fig. 4), can be evaluated as: tan θ t =
αω0
. (8) 2nf 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]: 2
k2 x x + k2 z z 1 α2 2 ≈ τ 0 1 + 4 2 2 ( x − z tan θ t ) , 2 c f τ τ 0 0
τ ( x , z ) =τ 0 1 +
(9)
where k2x = ∂φ2/∂z and k2z = ∂φ2/∂z are group velocity dispersion (GVD) coefficients in the x and z directions, respectively. Clearly, the pulse duration in this situation is dependent on both the propagation distance z and the lateral distance x away from the geometrical focus, which is quite different from that in Ref [24]. the pulse duration only depend on the propagation distance z. 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 θt = 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
#206625 - $15.00 USD Received 17 Feb 2014; revised 28 Mar 2014; accepted 28 Mar 2014; published 15 Apr 2014 (C) 2014 OSA 21 April 2014 | Vol. 22, No. 8 | DOI:10.1364/OE.22.009734 | OPTICS EXPRESS 9741
of large spatial chirp rate, we can obtain an approximate expression for zDOF, the axial fullwidth at half-maximum (FWHM) from the analytic expression for the axial intensity profile (Eq. (31) in Ref [32].). zDOF can be expressed in the simple form (see Appendix 1) 2 3 zR . (10) 1+ β 2 Using Eq. (10) and the parameters in Table 1, the depth of focus (zDOF) 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 zDOF ≈
z ( x) 1 x Δω 1 αΔω 2 ≈ β = x, zDOF 3 w0 ω0 2 3c f
(11)
where the spatial chirp rate β and the angular chirp rate α are employed in the two forms, respectively. From this expression we can see that at x = w0, the IPT will cause a shift along z direction (δz) that can be comparable to zDOF for spatial chirp rate sufficiently larger than βΔω/ω0. This relative IPT is actually independent of the focusing conditions (f and w0), and solely dependent on the spatial chirp rate and the fractional bandwidth.
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.
In the present experiment, β ≈30 and Δω/ω0 ≈0.03, so at x = w0, δz/zDOF 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/zDOF ≈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/zDOF ≈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/(ω0win) 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
#206625 - $15.00 USD Received 17 Feb 2014; revised 28 Mar 2014; accepted 28 Mar 2014; published 15 Apr 2014 (C) 2014 OSA 21 April 2014 | Vol. 22, No. 8 | DOI:10.1364/OE.22.009734 | OPTICS EXPRESS 9742
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. Table 2. List of variables used for the analytical calculation Variable name
φ φ1 φ2 φ2in
θx θt θT β βBA zR zd zDOF δz τ0 w0 ω0
Description Phase of the TF beam Group delay of the TF beam Second-order phase (group delay dispersion) of the TF beam Input second-order phase Frequency-dependent angle of the beamlets IPT angle without input second-order phase IPT angle with input second order phase Spatial chirp rate Spatial chirp rate beam aspect ratio [32] Rayleigh range of the beamlet The distance of the temporal focus shift away from the geometrical focus Depth of the focus of the TF beam (FWHM) Axial intensity shift of focus of the TF Beam Transform-limited pulse duration (FWHM) Focal spot size of the beamlet Central frequency of the spectrum
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]
φ2 = −
−3/2 λ 3d σ 2 1 − (λσ − sin i ) 2 , 2 πc
(12)
which is proportional to the distance between G1 and G2. For our current configuration, φ2in ≈5.3Δd × 106 fs2, where Δd is the distance of the grating away from its chirp-free position in our setup. Again all the other experimental conditions were the same as those in Sect. 2. The corresponding theoretical analysis involving SOP can be developed by varying the parameter φ2in in Eq. (1). For convenience, most variables used in this analysis are listed in Table 2.
#206625 - $15.00 USD Received 17 Feb 2014; revised 28 Mar 2014; accepted 28 Mar 2014; published 15 Apr 2014 (C) 2014 OSA 21 April 2014 | Vol. 22, No. 8 | DOI:10.1364/OE.22.009734 | OPTICS EXPRESS 9743
Fig. 5. TPFE images using temporal focusing with input SOP values of (a) φ2in = 0, (c) φ2in = 500 fs2, (e) φ2in = 1000 fs2, (g) φ2in = 2000 fs2, (i) φ2in = 4000 fs2, and (k) φ2in = 6000 fs2, respectively. (b), (d), (f), (g), (j) and (l) are corresponding numerical simulation TPFE signals. The color bar is shown at the bottom.
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
#206625 - $15.00 USD Received 17 Feb 2014; revised 28 Mar 2014; accepted 28 Mar 2014; published 15 Apr 2014 (C) 2014 OSA 21 April 2014 | Vol. 22, No. 8 | DOI:10.1364/OE.22.009734 | OPTICS EXPRESS 9744
i.e., φ2in > 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. 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: x τ 0 β z τ 02 β 2 n − 2 4 1 / z R2 w z z ω + R 0 0
φ2 ( x, z ) =
+ φ2in ,
(13)
where z = 0 is set to be the axial position of the geometric focal plane. The input second-order phase φ2in can balance the geometric phase at an axial position zd away from the geometric focal plane. Working with Eq. (13) we can solve for on-axis positions with zero second-order phase: φ2(x = 0, z = zd) = 0. There are two solutions for zd; the position closer to z = 0 results in higher intensity because there is a larger bandwidth and smaller focal spot size. Neglecting the denominator term in second parentheses in Eq. (13), and making use of βτ0 = 2α/win we find: zd ≈ φ2in
win2 Δω 2 = φ z zR , R 2 in nα 2 nβ 2
(14)
where in the second form we make use of the time-bandwidth relation τ0Δω = 2 . In the present investigation, we can use Eq. (14) to evaluate the results in Fig. 5. For comparison, we also estimate the values of zd with respect to φ2in from the experimental and numerical simulation in Fig. 5, respectively, as listed in Table 3. The experimental values of zd can be obtained by directly measuring the distance shift of the peak position of the TPFE signals away from this peak position with zero-SOP input (Fig. 5(a)). Table 3. Estimation of zd with respect to φ2in zd (mm)
φ2in (fs2) Experimental Numerical Analytical
500 0.5 0.5 0.5
1000 1.0 1.0 1.0
2000 1.9 2.0 2.0
4000 3.8 3.9 4.1
6000 5.2 5.4 6.1
One can find that for relatively small values of SOP (i.e., φ2in < 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 φ2in > 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 φ2in = 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 1 1 ⋅ . tan θ t 1 − 2 zd2 / z R2 The IPT angle as a function of the SOP is obtained as z ' ( x ) |x = 0 ≈
tan θT = (1 − 2 zd2 / zR2 ) tan θt .
(15)
(16)
Again, Eq. (16) is only valid when zd 31/4.
A.2 Derivation the expression of the tilt angle of the temporal focus (θT) for Eq. (16) Instituting βτ0 = 2α/win into φ2(x, z) = 0 for Eq. (13), we can obtain
φ φ 2α α2 xz + 2in2 z 2 + 2in = 0. nz R n ω0 win w0 z R win2
(20)
#206625 - $15.00 USD Received 17 Feb 2014; revised 28 Mar 2014; accepted 28 Mar 2014; published 15 Apr 2014 (C) 2014 OSA 21 April 2014 | Vol. 22, No. 8 | DOI:10.1364/OE.22.009734 | OPTICS EXPRESS 9746
Taking the first-order derivation of z with respect to x for Eq. (A.4), we have z ' |x =
2 win zR
αω0 w0
⋅
1 z win2 ⋅ 1 − 2φ2in ⋅ nz R α 2
=
1 1 ⋅ . tan θ t 1 − 2 zd z / z R2
(21)
Considering the derivation at x = 0 and from Eq. (14), we have z(x)|x=0 ≈ zd. Thus, the first order derivation of z(x) at x = 0 can be written as z ' ( x ) |x = 0 ≈
1 1 . ⋅ tan θ t 1 − 2 zd2 / z R2
(22)
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] 3λ 1 + λσ sin i − sin 2 i ⋅ φ2in ⋅ . (23) 2π c 1 − (λσ − sin i ) 2 In the current experimental configuration (incident angle i =53°, groove density σ = 1500/mm), we have φ3in ≈ −2φ2in fs3, where φ2in is measured in fs2. When taking into consideration the residual TOP caused by the single-pass grating, we can rewrite Eq. (1) as
φ3in = −
2 2 (ω − ω0 )2 1 1 2 3 [ x − α (ω − ω0 ) ] + y E1 ( x, y , ω ) = A0exp − + i φ2in (ω − ω0 ) + i φ3in (ω − ω0 ) exp − . 2 2 2 6 win Δω (24) By performing the similar calculations as those in Sect. 4, we can obtain the spatial and temporal profile (ITF(x, y, z, t)) of the pulse and the intensity distribution of the TPFE signals (I2P(x, y, z)) at the focus.
Fig. 6. Comparison of the temporal profile of the pulse at the temporal focus with (a) τ0 = 40 fs,
φ2in = 2000 fs2, φ3in = −4000 fs3, (b) τ0 = 40 fs, φ2in = 4000 fs2, φ3in = −8000 fs3, (c) τ0 = 40 fs, φ2in = 6000 fs2, φ3in = −12000 fs3 and (d) τ0 = 10 fs, φ2in = 2000 fs2, φ3in = −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.
#206625 - $15.00 USD Received 17 Feb 2014; revised 28 Mar 2014; accepted 28 Mar 2014; published 15 Apr 2014 (C) 2014 OSA 21 April 2014 | Vol. 22, No. 8 | DOI:10.1364/OE.22.009734 | OPTICS EXPRESS 9747
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. 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 φ2in < 2000 fs2 and |φ3in| < 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.
Fig. 7. Numerical simulation of the TPFE signals using temporal focusing with (a) φ2in = 0 fs2, φ3in = 0 fs3, (b) φ2in = 2000 fs2, φ3in = 0 fs3, (c)φ2in = 2000 fs2, φ3in = −4000 fs3, (d) φ2in = 4000 fs2, φ3in = 0 fs3, (e) φ2in = 4000 fs2, φ3in = −8000 fs3, (f) φ2in = 6000 fs2, φ3in = 0 fs3, and (g) φ2in = 6000 fs2, φ3in = −12000 fs3 respectively. The color bar is shown at the bottom.
The above analysis shows that in our typical configuration, the spatial and temporal distortion caused by the input TOP is negligible for φ2in < 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 FA9550-12-10482. C. D. would also like to thank Jeff Squier at CSM for useful discussions.
#206625 - $15.00 USD Received 17 Feb 2014; revised 28 Mar 2014; accepted 28 Mar 2014; published 15 Apr 2014 (C) 2014 OSA 21 April 2014 | Vol. 22, No. 8 | DOI:10.1364/OE.22.009734 | OPTICS EXPRESS 9748
