There are growing demands for ultrashort lasers (with duration down to few-optical-cycle) in wavelength regimes that are not available by direct laser action^[1,2]. Nonlinear frequency conversions, e.g.,  second-harmonic generation (SHG), difference-frequency generation (DFG), optical parametric generation/amplification (OPG/OPA), have been widely adopted to fill the wavelength gaps^[3-5]. Efficient frequency conversion necessitates phase matching between interacting waves. In SHG, for example, if the phase-matching condition is not fulfilled, the second-harmonic waves generated at different positions in a crystal are out of phase—hence, destructively interfere^[6]. Angular phase matching in birefringent crystals has been widely used, where polarization and propagation directions (θ) of interacting waves are arranged to achieve Δk=0 at a specific wavelength^[7]. Broadband SHG, however, necessitates not only the first-order phase matching (i.e., Δk=0), but also the second-order matching (i.e., ∂Δk/∂ω=0, known as group-velocity matching) and even higher-order matching. The spectrally noncritical phase-matching solution, i.e., the solution for simultaneous first- and second-order phase matching, exists only at a specific wavelength for a certain crystal^[8], such as 1.5 µm for BBO^[9], 1.3 µm for LBO^[10], 1.55 µm for MgO-doped PPLN^[11], and 1 µm for partially deuterated KDP^[12]. To frequency-double ultrashort lasers at other wavelengths, thin nonlinear crystals of the order of hundreds of micrometers or less^[13] are used to reduce the impact of phase mismatch, at the cost of low conversion efficiency (∼20%) and limited pulse energy.

To remove the requirement of thin crystal, the phase mismatch should be compensated for completely in its nature of spatiotemporal coupling. The phase mismatch between the interacting waves in birefringent crystals, in essence, exhibits an interdependence of both the laser frequency (ω) and the wave-vector direction (θ). Hence, the phase mismatch is spatiotemporally coupled, which should be written as Δk(ω,θ). In this sense, the ability to manipulate the ultrashort pulses’ spatiotemporal couplings should be used as a new degree of freedom to fit in the spatiotemporally coupled phase-matching condition Δk(ω,θ) and achieve ideal broadband frequency conversion. Achromatic phase matching (APM)^[14–16] was proposed, accordingly, wherein the broadband fundamental is angularly dispersed such that each frequency component is injected into the nonlinear crystal along its individual phase-matching angle. However, the desired angular dispersion (one kind of spatiotemporal couplings) to compensate for Δk(ω,θ) is typically in the range of 10−4rad/nm. To introduce such an angular dispersion into the fundamental pulse, diffraction gratings having a groove density as low as ∼100 lines/mm were typically used^[17], at the cost of low diffraction efficiency (<80%). To reduce the energy loss in the low-density diffraction gratings, Baum et al. proposed an achromatic SHG scheme wherein the angular dispersion was introduced by a pair of prisms in combination with a focusing lens^[14]. However, the prism pair can hardly be used for the SHG of super-powerful ultrashort lasers in a large aperture. Besides, even in the scheme of APM, thin nonlinear crystals are still necessary; otherwise the angularly dispersed ultrashort pulse will quickly spread out in space and time.

In this paper, we present a different strategy to achieve spatiotemporally-coupled phase matching, which no longer necessitates low-density diffraction gratings or thin nonlinear crystals. To break this limit of the APM scheme where the phase-matching condition exhibits angular dispersion due to the nature of birefringence, we use a quasi-phase matching (QPM) crystal in a fan-out grating structure, wherein the phase-matching condition exhibits the spatiotemporal coupling characteristic of spatial chirp Δk(ω,x), rather than angular dispersion Δk(ω,θ). As schematically illustrated in Fig. 1(a), QPM crystal in a fan-out design is characterized by a linear variation of poling period along the transverse direction^[18]. Accordingly, the QPM wavelength varies along the transverse direction. By transforming an ultrashort fundamental pulse into a spatially chirped beam wherein the broadband spectrum is distributed “slice by slice” onto the QPM crystal with an appropriate poling period individually [Fig. 1(b)], ultrabroadband SHG can be achieved with high efficiency. Particularly, within each transverse slice of the fan-out QPM crystal, the fundamental wave is quasi-narrowband, which fundamentally cancels high-order phase mismatch, so that thick nonlinear crystals can be used.

The engineerable period of a QPM grating structure provides a new degree of freedom to optimize phase matching. Figure 1(b) presents the basic layout for ultrabroadband SHG based on a periodically poled lithium niobate (PPLN) in a fan-out grating structure. A 4f optical setup is adopted to spatially disperse the incident 1ω wave. The frequency components of 1ω wave are angularly dispersed by G, and then focused by M1 to a diffraction-limited line beam at the Fourier plane (i.e., the back focal plane of M1). The fan-out PPLN is placed at the Fourier plane. It is worth noting that the 4f system in our scheme is quite different from those used in pulse shaping or those in frequency-domain optical parametric amplification (FOPA)^[19], wherein the signal wavelength stays the same. In our scheme, however, the aiming output (2ω) has a different wavelength from the input (1ω). Therefore, it necessitates a precise arrangement of the 4f system parameters to deliver a collimated 2ω output without or with least spatiotemporal coupling distortions.

We first deduce the spatiotemporal evolution of the 1ω and 2ω fields under the assumption of prefect phase matching. A spatiotemporal Gaussian-shaped 1ω pulse with a beam width of D and a Fourier-transform-limited duration of τ0 is assumed, whose complex amplitude in the spatial-spectral domain can be written as a1ωin(x,ω)=a0exp(−x22D2)exp[−ω22(Δω)2],(1)where Δω(=1/τ0) denotes the frequency width. We only consider one transverse coordinate, x, because the spatial dependence of the electric field on the other transverse spatial coordinate y is uninvolved in any kind of spatiotemporal couplings, so that it has less impact onto the spatiotemporal phase matching and thus conversion bandwidth of SHG, and its impact on the conversion efficiency can be estimated separately as a(x,y,ω)=a(x,ω)ay(y). All the mentioned widths refer to the half-width at 1/e intensity. After being diffracted by G1, the 1ω laser becomes an angularly dispersed beam as given by a1ωG1(x,ω)=a0exp(−x22D2)exp[−ω22(Δω)2]exp(ik1β1ωx),(2)where β1 is the angular dispersion provided by G1. In the Fourier plane, the 1ω laser thus evolves into a spatially chirped beam as given by a1ωFP(x,ω)=a1exp(−ik12fx2)exp(−ik1β12ω2f2)⁢exp[−ω22(Δω)2]exp(−ik1β1ωx)⁢exp[−(x+β1fω)22σ2],(3)where k1 is the wavenumber, f is the focal length of M1, and σ=f/(k1D) is the focal spot size for each monochromatic component. As indicated by the term exp[−(x+β1fω)2/2σ2], different frequency components (ω) are focused onto distinct transverse positions (x=−β1fω). The spatial separation of the frequency components can be described by the spatial-chirp coefficient defined as ζ1=dxdω=−β1f.(4)

According to Eq. (3), the local bandwidth of the spatially chirped 1ω wave can be deduced as Δω1FP=σ|ζ1|=−1k1β1D=−Δθ1ωβ1,(5)which is an important parameter that determines the SHG performance at each transverse position. Equation (5) indicates that the local bandwidth is determined only by the divergence angle of the incident 1ω beam Δθ1ω=1/(k1D) and the angular dispersion β1 provided by the diffraction grating G1 and is unrelated to the total bandwidth of the incident 1ω beam. In practice, such a local bandwidth can be reduced by expanding the incident beam width D and/or by adopting a higher groove-density grating. The spatiotemporal amplitude of the fundamental wave can be deduced via Fourier transformation^[20], A1ωFP(x,t)=A1exp[−x22(ζ1Δω)2]exp[−t22(k1β1D)2]exp(−ixtζ1),(6)where the term exp(−ixt/ζ1) manifests the spatiotemporal coupling in the form of wavefront rotation. Equation (6) indicates that the fundamental wave in the Fourier plane does not exhibit pulse-front tilt (i.e., a transverse-position-dependent arrival time), and the temporal duration is uniformly ΔT=−k1β1D across the beam^[20]. For a monochromatic beam incident into G1, this duration ΔT=−k1β1D corresponds to the overall delay resulting from the different arrival times (at the Fourier plane) of different spatial slices of the incident beam. For an ultrashort Fourier-transform-limited beam, this duration ΔT=−k1β1D is exactly the Fourier-transform-limited duration determined by the local bandwidth [Eq. (5)] after spatial dispersion, as ΔT·Δω1FP=1.

Under the assumption of perfect phase matching, the spatial-spectral amplitude of the 2ω laser produced in the Fourier plane can be directly written as a2ωFP(x,ω)=a2exp(−ik22fx2)exp[−ik2β22(2ω)2f2]⁢exp[−ω24(Δω)2]exp[−ik2β2(2ω)x]⁢exp{−[x+β2f(2ω)]2σ2},(7)where k2(=2k1) is the wavenumber of the 2ω beam, and β2=β1/2 is the angular dispersion coefficient of the 2ω beam. M2, with the same focal length as M1, is then used to perform an inverse Fourier transform in the spatial domain and restore the spatially chirped 2ω beam into an angularly dispersed beam without spatial chirp at the back focal plane of M2, where the complex amplitude of the 2ω field can be deduced as a2ωG2(x,ω)=a3exp(−x2D2)exp[−ω24(Δω)2]exp(−i2k2β2ω).(8)

A comparison of Eq. (8) with Eq. (2) indicates that a collimated and distortion-free SHG output can be obtained if we adopt a diffraction grating G2 with the half-angular dispersion coefficient of G1 (i.e., β2=β1/2). In practice, we can use a diffraction grating that has a double groove density of G1 for G2. After removing the angular dispersion by G2, the 2ω field translates into an ultrashort pulse without spatiotemporal couplings as a2ωout(x,ω)=a3exp(−x2D2)exp[−ω24(Δω)2].(9)

We next present a design of a 4f system for the SHG of a broadband 1ω laser centered at 810 nm with a full width at half-maximum (FWHM) bandwidth of 50 nm (corresponding to Δω=144THz in half-width at 1/e intensity). G1 adopts a diffraction grating with a groove density of 1200 lines/mm, which provides an angular dispersion coefficient of β1=−0.478mrad/THz for the 1ω laser [Fig. 2(a)]. And the second diffraction grating G2, with a groove density of 2400 lines/mm, provides an angular dispersion of β2=−0.239mrad/THz for the 2ω laser centered at 405 nm. The two concave mirrors M1 and M2 have the same focal length of f=100mm. The incident 1ω pulse is assumed to be a Fourier-transform-limited one in Gaussian shape, with τ0=12.1fs in duration and D=0.5mm in beam width. At the back focal plane of the concave mirror M1 (i.e., the Fourier plane), the frequency components are dispersed in space sufficiently. In consequence, the local bandwidth is as narrow as 0.22 nm, corresponding to a pulse duration of ∼3.1ps [Fig. 2(b)]. The 1ω laser is featured by a spatial-chirp coefficient of dx/dω=4.78×10−2mm/THz. Assuming a perfect phase matching, a spatially chirped 2ω beam is produced, characterized by a spatial-chirp coefficient of dx/dω=2.39×10−2mm/THz [Fig. 2(c)] and a pulse duration of ∼2.2ps [Fig. 2(d)].

The combination of a fan-out PPLN and a spatially chirped fundamental provides the possibility of ultrabroadband SHG. Two key factors need to be considered deliberately. The first one is the match between the local bandwidth of the 1ω beam and the acceptance bandwidth of the QPM crystal at each transverse position. The second one is the match between the spatial-chirp coefficient of the fundamental and the period gradient of the fan-out grating.

QPM is characterized by a periodic reversal of the sign of the nonlinear coefficient to offset the accumulated phase mismatch by the material dispersion. For a PPLN in fan-out design, the varying poling period Λ(x) leads to a space-dependent phase mismatch as given by ΔkQ(x,λ)=k2(x,λ/2)−2k1(x,λ)−2πΛ(x).(10)

With such a phase mismatch, the process of spatially chirped QPM can be numerically simulated by solving the nonlinear coupled-wave equations under the slowly varying amplitude approximation, ∂A2ω∂z+∑n=13(−i)n−1n!k(n)∂A2ω∂t=−iω2deffn2cπA1ω2eiΔkQ(x)z,(11a)∂A1ω∂z+∑n=13(−i)n−1n!k(n)∂A1ω∂t=−i2ω1(x)deffn1cπA1ω*A2ωe−iΔkQ(x)z,(11b)where A1ω and A2ω refer to the spatiotemporal complex amplitude of the spatially chirped 1ω and 2ω laser, respectively. ω1(x) and ω2(x) denote their respective space-dependent frequency. n1 and n2 are their respective refractive indices. deff is the effective nonlinear coefficient, and c is the speed of light in vacuum. The first term on the left-hand side of Eq. (11) describes the propagation of the 1ω (2ω) pulse envelope along the z direction. The second term describes the dispersion of the 1ω (2ω) pulse inside the QPM crystal, up to the third order. The term on the right-hand side describes the coupling between the 1ω and 2ω components. In the following simulations, a 5% MgO-doped PPLN in type-0 SHG (eee) configuration is assumed, so that deff=d33=28.4pm/V in Eq. (11)^[21].

First, to cover a spectral range from 760nm(λmin) to 860nm(λmax), the poling period range of the fan-out PPLN can be calculated as Λmin=2.33μm and Λmax=3.32μm, according to the relation, Λmax=2πk2(x,λmax2)−2k1(x,λmax),(12a)Λmin=2πk2(x,λmin2)−2k1(x,λmin).(12b)

Second, to cover the local bandwidth (0.22 nm in our simulated case) of the spatially chirped 1ω laser at each position x, the length of the fan-out PPLN should be properly designed to provide an acceptance bandwidth that is wider than the local bandwidth of 1ω laser [Eq. (5)]. Under the assumptions of dispersion-free and pump nondepletion (∂A1ω/∂z=0), Eq. (11) has an approximate analytical solution as given by I2ω(x,L)=32L2deff2n12n2λ22cε0I1ω2sinc2[ΔkQ(0,λ)L2].(13)L is the crystal length, and ε0 is the vacuum permittivity. From Eq. (13), the local acceptance bandwidth Δλlocal can thus be deduced as |ΔkQ(λ0±Δλlocal)L|=2π.(14)

Taking the central frequency component, for example, the QPM grating period can be calculated as Λ0=2.81μm. A PPLN length of L=1mm leads to a local acceptance bandwidth of Δλlocal=0.26nm [Fig. 3(a)], which is larger than the local bandwidth (0.22 nm) of the spatially chirped 1ω beam.

Third, we move to calculate the poling period gradient of the 1 mm-long fan-out PPLN. To match the frequency gradient of the spatially chirped 1ω beam [Eq. (4)] along the x axis, ω1(x)=ω0−xβ1f,(15)the poling period gradient of the fan-out PPLN can be obtained as Λopt(x)=2πk2−2k1=πc(n2−n1)ω0×11−xβ1fω0.(16)

Obviously, Λopt(x) is not a strictly linear function of the transverse position x. The global beam width of the spatially chirped 1ω beam is β1fΔω, where Δω is the fundamental bandwidth. Since Δω is typically much smaller than the central frequency ω0, Eq. (16) can be approximated by Taylor expansion to the first order, Λ(x)≈πc(n20−n10)ω0×(1+xβ1fω0),(17)where n10 and n20 stand for the refractive indices at the central frequencies of the fundamental and second harmonic, respectively. The red line in Fig. 3(b) plots the result of such a linear approximation, which shows a very good agreement with the solution of Eq. (16) as depicted by the black line.

Another important parameter for a fan-out PPLN is the tilt angle of domain walls. As shown in Fig. 1(a), in a fan-out pattern, domain walls tilt away from the x axis progressively toward the end of the crystal. The tilt angle of the outmost domain wall is determined as θtilt=arctan{Nv[Λmax−Λmin]2(xmax−xmin)},(18)wherein xmin and xmax denote the transverse positions of the minimum (λmin) and maximum (λmax) wavelengths, respectively. Nν is the number of domains. To avoid the risk of adjacent domains running together in fabrication, the tilt angles of domain walls are typically required to be no larger than 2°^[18]. In our simulated case, there have been xmin=−7.31mm and Λmin=2.33μm for λmin=760nm at one edge, and xmax=6.47mm and Λmax=3.32μm for λmax=860nm at the other edge; the tilt angle is calculated to be 0.93°, well within the acceptable level.

In spatially chirped QPM, the ultrabroadband 1ω input gets frequency-doubled slice by slice at different transverse positions. The 2ω output can be regarded as the superposition of different spectral slices. The spatial overlap of the adjacent spectral slices with minor frequency shift will lead to amplitude modulations in both the spatial and spectral profiles of the spatially chirped 2ω beam at the output of the fan-out PPLN. For comparison, we numerically simulate the process of SHG in a multi-period PPLN, as shown in Fig. 4(a), which can be regarded as the prototype of a fan-out PPLN. The simulations are conducted for a fundamental without spatiotemporal couplings of 5 nm in bandwidth centered at 810 nm and 1 mm-long PPLN with three grating periods Λ1=2.80μm, Λ2=2.81μm, and Λ3=2.82μm. Figure 4(b) shows the second-harmonic spectra produced by each PPLN section, which clearly show that the spectra produced by the adjacent PPLN of different periods overlap with each other. Figure 4(c) depicts the 2ω spectrum produced by the multigrating PPLN with these three poling periods. It shows that the spatial overlap of the three individual spectra produced by each grating results in spectral amplitude modulation.

We next move to characterize the performance of spatially chirped QPM in the saturation regime, where Eq. (13) is invalid. We numerically simulate the SHG process based on Eq. (11), wherein the fundamental depletion as well as media dispersion is all included. For operation around saturation the 1ω pulse incident onto G1 is assumed to have an initial intensity of 10GW/cm2. After spatial dispersion, the spatially chirped fundamental in the Fourier plane has an intensity of 70MW/cm2. Under such an intensity, the SHG conversion efficiency reaches the maximum (77%) with a crystal length of 1 mm, as shown in Fig. 5(a). To assess the SHG efficiency of the whole system, the optical loss induced by the diffraction gratings and concave mirrors must be taken into account. Given the ultrabroadband characteristic of the fundamental and second-harmonic waves, gold-coated gratings and concave mirrors should be used, which have a typical reflectivity of ∼95% for each piece, so that the overall SHG efficiency is about 77%×(95%)4≈62%.

Figures 5(b)–5(d) present the spectral, temporal, and spatial profiles of the 2ω pulse right behind the fan-out PPLN, while Figs. 5(e)–5(g) illustrate the 2ω pulse profiles at the output of G2. A spatially chirped 2ω pulse has been produced [Fig. 5(b)], which exhibits obvious amplitude modulations in both the spectral [Fig. 5(b)] and spatial profiles [Fig. 5(d)]. Such amplitude modulations are induced by the local spectral clipping due to the local acceptance band and the spatial overlap of the clipped spectra produced by adjacent x positions, as has been illustrated in Fig. 4. The modulation depth in the spectrum is as low as 0.05%. The 2ω pulse covers a spectral range from 370 to 440 nm [Fig. 5(b)], indicating that broadband SHG of the fundamental from 760 to 860 nm is realized. In time domain, deformation is observed in the 2ω pulse [Fig. 5(c)], which is caused by the temporal walk-off between the 1ω and 2ω pulses in the PPLN. At the output of G2, the 2ω pulse resorts to an ultrashort one with negligible spatiotemporal couplings [inset in Fig. 5(e)], and the pulse duration is 10.2 fs [Fig. 5(f)], which is about seven optical cycles. In the spatial profile, sidelobes appear in the output 2ω beam, as shown in the inset of Fig. 5(g), which originates from the spatial amplitude modulation imprinted onto the 2ω beam in the Fourier plane due to spatial-spectral coupling. Nevertheless, more than 96% energy is concentrated in the main beam spot.

Finally, we move to analyze the upper limit of the conversion bandwidth of our proposed scheme. Since Λopt(x) as given by Eq. (16) is not a strictly linear function of the transverse position x, the compatibility of the fan-out PPLN and the spatially chirped 1ω wave is bound to deteriorate with the increase of the fundamental bandwidth. Figure 6(a) depicts the deviation of the local poling period Λ(x) from the optimum poling period Λopt(x), and the frequency-dependent phase mismatch induced by this deviation is calculated and shown by the red line in Fig. 6(b). Due to the large phase mismatch experienced by the edge wavelength components, especially at the end of the short wavelength, the maximum wavelength range of the second-harmonic wave is 347–500 nm, as plotted by the black line in Fig. 6(b). Such a spectral range supports the production of a second-harmonic pulse centered at 405 nm with a Fourier-transform-limited duration of ∼4.2fs (full width at 1/e2 intensity), as illustrated in Fig. 6(c).

In conclusion, we theoretically demonstrate the scheme of spatially chirped QPM as well as its application in the production of a 10 fs (∼7 optical cycles) 2ω pulse centered at 405 nm via broadband SHG with a conversion efficiency as high as 77%. The conversion efficiency and bandwidth rely on the match between the frequency gradient of the spatially chirped beam and the poling period gradient of the fan-out QPM grating, as well as the match between the local bandwidth of the spatially chirped beam and the acceptance bandwidth of QPM. For the spatially chirped fundamental, its frequency gradient and local bandwidth are primarily determined by the angular dispersion ability of the first diffraction gratings used in the zero-dispersion 4f optical system and the diffraction divergence angle of the incident fundamental. The design principles as well as the quantitative formula for the 4f optical system and the space-dependent poling period of the fan-out QPM grating are presented. The spatiotemporal distortions of the second-harmonic wave induced by the space-dependent spectral reshaping in the Fourier plane are also investigated, which shows that the ultrabroadband second-harmonic wave produced by a fan-out QPM grating is accompanied by spectral amplitude modulation with a modulation depth as tiny as 0.05%. Due to the spatial-spectral coupling in the Fourier plane, such spectral modulation only induces the production of weak sidelobes in the second-harmonic beam output from the 4f optical system, while it does not degrade the temporal compression of the second-harmonic pulse. Further research shows that the theoretical limit of the conversion bandwidth for our proposed scheme is approximately 150 nm in the second-harmonic wave, which supports the production of a 4.2 fs, 2ω pulse centered at 405 nm. Such an ultrabroadband high-efficiency SHG scheme has important application for the generation of few-optical-cycle ultrashort lasers in the visible and ultraviolet spectral regions.