High complexity femtosecond pulse duplicator

This paper presents a theoretical and numerical study of a 0-π fan-out phase grating placed in the Fourier plane of a spatio-spectral pulse shaper followed by a spherical focusing lens. It is shown that this device acts as a high complexity femtosecond pulse duplicator designed for two source interferometry. At the focus of the lens, the electric field displays two spatially separated intense spots in which relative delay can be continuously tuned over 4 orders of magnitude, typically from a few attoseconds to a few tens of femtoseconds. Because the two pulses do not spatially overlap, their intensity remains unchanged when the relative delay is smaller than the pulse duration. Detailed simulations of the shaped electric field are presented. © 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement With the development of extreme ultra-violet (XUV) sources of coherent emission, many experimental techniques developed in the optical range have been transferred to the short wavelength domain. This includes techniques making use of two XUV pulses with adjustable delay such as Ramsey spectroscopy [1,2], heterodyne spectroscopy [3,4] and Fourier transform spectroscopy [5–10], interferometry [11] and pump/probe experiments [12–14]. Experimental schemes providing two XUV pulses of adjustable delays are also central in ultrafast spectroscopy, may it be attosecond-pump/attosecond-probe [8] or attosecond interferometry experiments [15]. In the absence of beam splitters in the XUV spectral range, several experimental arrangements were developed [16–23], some of which consist in splitting a mid-infrared laser pulse in two replicas used to generate attosecond pulses from two spatially separated sources through High order Harmonic Generation (HHG) in gas [24,25] . In this case, the delay between the two XUV pulses is directly controlled by the delay between the two replicas of the HHG driver field. Still, controlling the delay between two XUV pulses with attosecond precision remains challenging and requires high mechanical stability with current state of the art interferometers. In the present work, we introduce a new common-path optical interferometer based on a 0-π Fan-Out Phase Grating placed in the Fourier plane of a spatio-spectral pulse shaper. This interferometer is designed to create two spatially separated intense near-infrared spots at focus of a spherical lens of focal length f1 placed downstream of the pulse shaper. These two spots can be used to generate two attosecond pulses through HHG in the gas phase. The relative delay between the two pulses can be tuned continuously with attosecond precision over a large delay range. In contrast with previous work based on one-dimensional pulse-shapers [26,27], this setup generates two pulses which intensity remains unchanged whether the pulses are synchronized or delayed and therefore allows to drive non-linear processes with the same intensity for delays shorter or longer than the pulse duration. Hereafter we describe this new experimental arrangement and report on a theoretical and numerical study of its capabilities. Recently a 0-π phase step was used to convert a TEM0,0 (Transverse Electromagnetic Mode) femtosecond pulse into a TEM0,1 [28,29] resulting at focus of a lens in two spatially separated phase-locked bright spots. These two intense lobs can be used to drive HHG in a gas jet, thus providing a simple and robust two-arms XUV interferometer. In this scheme, the phase difference between the two arms is set to π and cannot be adjusted which is an advantage in terms of stability but limits the flexibility of the experimental arrangement. More recently, a 0-π Square-Wave Phase Grating [30] was introduced to circumvent this limitation. In this case, the two bright spots observed at focus of a lens correspond to two replicas of the incoming laser pulse (with small #398627 https://doi.org/10.1364/OE.398627 Journal © 2020 Received 26 May 2020; accepted 11 Jun 2020; published 13 Jul 2020 Research Article Vol. 28, No. 15 / 20 July 2020 / Optics Express 22248 distortions), populating the 1st and -1st orders of diffraction of the phase grating. The phase difference between the two replicas can be continuously tuned between 0 and 2π by shifting the position of the grating in its own plane. The phase difference between the two orders of diffraction is defined by the ratio between the vertical offset of the grating and the groove size. Here again the phase difference between the two beams is very robust since it is decided by a single optical element used in transmission. The limitations of this setup are that only the relative phase between the two beams of the interferometer can be controlled and the scanning range is limited to 2π. Here, the proposed scheme is based on a 0-π Fan-Out Phase Grating (FOPG) placed in the Fourier plane of a spatio-spectral pulse shaper (see Fig. 1). The delay between the 1st and -1st orders of diffraction of the FOPG can be continuously tuned over a broad range with high precision. This experimental arrangement offers the same stability as the 0-π Square-Wave Phase Grating since the delay between the two pulse replicas only depends on the vertical position of the FOPG. In Eq. (1), we introduce P(y,ω, y0), an analytical expression representing the FOPG: ∀(y, y0,ω) ∈ R2 × R∗,+,

distortions), populating the 1 st and -1 st orders of diffraction of the phase grating. The phase difference between the two replicas can be continuously tuned between 0 and 2π by shifting the position of the grating in its own plane. The phase difference between the two orders of diffraction is defined by the ratio between the vertical offset of the grating and the groove size. Here again the phase difference between the two beams is very robust since it is decided by a single optical element used in transmission. The limitations of this setup are that only the relative phase between the two beams of the interferometer can be controlled and the scanning range is limited to 2π. Here, the proposed scheme is based on a 0-π Fan-Out Phase Grating (FOPG) placed in the Fourier plane of a spatio-spectral pulse shaper (see Fig. 1). The delay between the 1 st and -1 st orders of diffraction of the FOPG can be continuously tuned over a broad range with high precision. This experimental arrangement offers the same stability as the 0-π Square-Wave Phase Grating since the delay between the two pulse replicas only depends on the vertical position of the FOPG.
In Eq. (1), we introduce P(y, ω, y 0 ), an analytical expression representing the FOPG: with 1 d(ω) = ω 2πc 1 and where d(ω) is the groove size of the FOPG for the optical frequency ω, y 0 is an offset along the y axis referenced to the optical axis, and c 1 is a critical parameter of the FOPG linking the spatial frequency 2π d(ω) with the optical frequency ω. c 1 is homogeneous to a velocity and determines the distanceỹ 1 = f 1 c c 1 (where c is the speed of light) between the first order of diffraction of the FOPG and the optical axis in the focal plane of f 1 .
The phase function φ(y, ω, y 0 ), defined such as P(y, ω, y 0 ) = e iφ(y,ω,y 0 ) , takes only two values: 0 and π. In practice, the FOPG can be a phase plate etched in fused silica or BK-7 glass [31] with different depth depending on the value of the index of refraction of the glass at ω so that the phase depth is always almost π. The continuous diffractive optical element shaping technique offers etching precision down to 10 nm [31], yielding an error of the order of 1% on the phase depth in the near-infra red region. If the phase depth slightly differs from π (say by 10% as a reference) the main effect is to populate the 0 th order of diffraction of the FOPG, the intensity of which is low enough to avoid driving non-linear effects in this order of diffraction [30]. This however does not affect the relative phase between the different orders of diffraction of the FOPG which is fully determined by y 0 . P(y, ω, y 0 ) is a d(ω)-periodic function that can be expanded in its spatial frequency components through a discrete Fourier sum: where ∀n ∈ Z, ∀ω ∈ R * ,+ , a n (y 0 , ω) = sinc nπ 2 e −iφ n (y 0 ,ω) with φ n (y 0 , ω) = nφ 1 (y 0 , ω) = 2πn y 0 d(ω) = nωt 1 (y 0 ) and t 1 (y 0 ) = y 0 c 1 .
We assume the Fresnel or Fraunhofer approximations to be valid and ∀ω ∈ R * ,+ , d(ω) σ, where σ is the characteristic dimension of the spatial extent of the input beam. In these  [32] geometry followed by a spherical lens of focal length f 1 . Adapted from [33]. The two gratings have the same characteristics. The two lenses are cylindrical (with the same focal length f) so that the spectral components are focused in the Fourier plane but the beam remains unaffected along the other dimension and keeps its size along the vertical dimension y throughout the pulse shaper. The 0-π Fan-Out Phase Grating (FOPG) placed in the Fourier plane of the pulse shaper is represented in the inlet below and to the right of the optical scheme. Its vertical position can be adjusted for example using a vertical translation stage and induce an offset y 0 defined with respect to the optical axis. In the inlet, the x-axis corresponds to spectral shaping while the y-axis corresponds to spatial shaping. The 4f-line's gratings are assumed to show linear dispersion in the range of interest so that x is directly proportional to ω [34]. For a given spectral component, the phase of the electric field is shaped along y with a 0-π square-wave phase grating. The groove size and the depth of the grating are wavelength dependent yielding a phase depth of π for all wavelengths. The color code is used to specify the physical depth of the grating for a spectrum centered at 800 nm and covering from 775 nm to 828 nm. The index of refraction of Corning 7980 computed with Sellmeier coefficients were used. A spatio-temporal representation of the shaped electric field at focus of f 1 when the two replicas are synchronized is presented in the inlet below and to the left of the optical scheme.
conditions, we can apply results of the theory of optimal beam splitting using phase gratings [35,36]. In particular, the energy in each order of diffraction is determined by the modulus of the corresponding Fourier expansion coefficients of the periodic function representing the grating, |p n |, which does not depend on ω for a 0-π FOPG. The relative phase between the different diffraction orders is given by the phase of the Fourier coefficients. For example, the phase difference between the 1 st and -1 st orders of diffraction is given by φ 1 (y 0 , ω) − φ −1 (y 0 , ω) = 2ωt 1 (y 0 ). For a given offset of the FOPG y 0 , this phase difference is linearly proportional to ω which corresponds to a group delay equal to 2 × t 1 (y 0 ) between the two orders of diffraction. This group delay can be linearly adjusted by changing y 0 following the relation t 1 (y 0 ) = y 0 c 1 . The relation between the delay t 1 (y 0 ) and the vertical offset y 0 is at the core of this work. For a realistic situation where λ = 800 nm and d(800 nm) = 2.667 mm and considering that c 1 = d(λ) λ c where λ is the wavelength corresponding to ω, one finds c 1 = 3 × 10 3 c. This means that the proposed experimental arrangement is equivalent to a Michelson or Mach-Zender interferometer for which the speed of light would be c 1 2 ∼ 1, 500 × c. In such case, the time it takes for light to cover the same length difference between the two arms of the interferometer is 1,500 times shorter at c 1 compared to c. In other terms, the 0-π FOPG placed in the Fourier plane of a spatio-spectral pulse shaper is 1,500 times more sensitive than a Michelson or Mach-Zender interferometer. Another illustration of the precision of the proposed experimental arrangement is given by 1 c 1 ∼ 1.1 ps/m (to be compared to 1 c ∼ 3.3 ns/m). Moving the FOPG in its plane by 1 µm induces a delay of ∼ 2.2 as to be compared with the delay of ∼ 3.3 fs induced by a length difference of 1 µm between the two arms of a traditional interferometer. This extreme precision is common to the proposed scheme and to previous works based on 1D pulse shapers [26,27] with the novelty here that the delay between the two pulses can be smaller than the pulse duration without affecting the intensity of the pulses.
We now take a closer look at how E(y, ω), the electric field of a laser pulse defined in the Fourier plane of the zero-delay line just upstream of the FOPG, gets shaped while propagating through the proposed arrangement. From Eq. (2), we derive S(y, ω, y 0 ) which describes the electric field right after the FOPG in the Fourier plane of the spatio-spectral pulse shaper: ∀(y, y 0 , ω) ∈ R 2 × R * ,+ , S(y, ω, y 0 ) = E(y, ω) × P(y, ω, y 0 ) Equation (3) shows that S(y, ω, y 0 ) is a coherent sum of replicas of E(y, ω). Each replica features a characteristic linear spectral phase e −iωt 2n+1 (y 0 ) and wave front tilt dispersion e 2πi(2n+1) y yω [37]. The combined action of the cylindrical lens and grating downstream the FOPG can be described by a frequency-time Fourier transform. Using Eq. (3) with 2π(2n + 1) y d(ω) = y y 0 ωt 2n+1 (y 0 ), we getS(y, t, y 0 ) which describes the electric field at the exit of the spatio-spectral pulse shaper:S Equation (4) shows that, at the exit of the zero dispersion line, the shaped electric field is composed of replicas ofĒ (y, t), the field at the entrance of the zero dispersion line. On the optical axis (for y = 0), the time delay between two consecutive replicas is 2 × t 1 (y 0 ). This delay can be traced back to the linear spectral phase emphasized in Eq. (3). All replicas feature pulse front tilt [37], that is a linear delay in the arrival time of the maximum intensity across y. The pulse front tilt is independent of y 0 and characteristic of the order of diffraction of the FOPG. It can be associated with the wave-front tilt dispersion described in Eq. (3). These properties are illustrated in Fig. 2.   Fig. 2. Spatio-temporal representation of the electric field shaped by a 0-π FOPG placed in the Fourier plane of a spatio-spectral pulse shaper a) at the exit of the pulse shaper, b) and c) at focus of a lens with focal length f 1 = 50 cm placed downstream of the zero-delay line. The envelop (respectively the field) is represented in a) and c) (resp. b)). The envelop and field atỹ +1 andỹ −1 are shown in d). The pulse is defined in the (y, ω) space. The spectrum is defined as a Gaussian of width 35 nm centered at 800 nm yielding a pulse duration of 10 fs. The groove size of the FOPG at 800 nm is 2.667 mm so that t 1 (y 0 ) (as) ∼ y 0 (µm). y 0 is set to 50 mm. For a positive value of y 0 , the delay induced in the +1 st (respectively −1 st ) order of diffraction of the grating is negative (resp. positive). The pulses shown in a), b) and c) can be linked and correspond to different orders of diffraction of the FOPG. The pulse front tilt [37] observed in a) is positive (respectively negative) for the pulses associated with the positive (resp. negative) orders of diffraction resulting in the focusing of the orders above (resp. below) the optical axis (ỹ = 0). The tilt observed in a) increases with the order of diffraction of the FOPG, resulting in the focusing away from the optical axis illustrated in c). The simulations are based on discrete Fourier transform of the field defined in the Fourier plane of the pulse shaper. The values of y 0 and the pulse width have been chosen to well separate the replicas in time and clarify the illustration.
We now take a look atS(ỹ, t, y 0 ), the shaped electric field at focus of a f 1 focusing lens placed downstream the zero-delay line. In the Fraunhofer approximation, the electric field at focus of a lens is well described by a spatial Fourier transform of the incoming field, as described in Eq. (6) in whichỹ is the vertical spatial coordinate in the focal plane of the lens. Equation (2) is used with 2π(2n + 1) y d(ω) = ky f 1ỹ 2n+1 , whereỹ 2n+1 = (2n + 1) ×ỹ 1 and k = ω c , to establish Eq. (5) where two Fourier transformations are applied to E(y, ω). Since the two dimensions are independent, the two Fourier transformations are commutative and we apply first the Fourier transformation along y before applying the one along ω to ease the demonstration.
Equation (7) shows thatS(ỹ, t, y 0 ), the shaped electric field at focus of the f 1 lens, is made of replicas ofẼ(ỹ, t), the incoming field focused by the same lens, with no distortion. The replicas are distributed along the vertical dimension and separated by a distance of 2 ×ỹ 1 . The spatial offset of each replica is a direct consequence of the pulse front tilt described in Eq. (4).ỹ 1 is independent of ω, meaning that all spectral components of one replica focus in the same position. This is a direct consequence of the linear relation between the groove size of the FOPG and the wavelength: the FOPG is designed to avoid spectral dispersion at focus of a lens and is primarily motivated by wave-vector shaping [38,39]. The group delay between two replicas distant by 2 ×ỹ 1 from each other is 2 × t 1 (y 0 ). This delay is linearly proportional to y 0 , can be made positive, negative or null and can take uncommonly small values over large ranges. The intensity of the different replicas is directly determined by |p n | 2 . The two most intense replicas are the two closest to the optical axis and correspond to the 1 st and −1 st orders of diffraction with |p +1 | 2 = |p −1 | 2 ∼ 0.4. The overall efficiency of an anti-reflection coated phase plate based FOPG placed in the Fourier plane of a 2D pulse shaper can therefore be better than 70% considering the use of state-of-the-art gratings [40]. Also, the damage threshold of fused silica phase plates is well above the requirements for shaping femtosecond pulses used for HHG in gas [28].
In conclusion, we have introduced and analysed an experimental setup allowing to duplicate femtosecond pulses and spatially separate them at focus of a lens using wave-vector shaping. The delay between the two most intense replicas can be set to extremely small values of the order of a few attoseconds over a range of typically tens of femtoseconds whilst the intensity in the replicas remain unchanged, thus the title of the present paper. Technically, the main difficulty is to choose experimental parameters to prevent the different orders of diffraction to leak into each other inducing undesired intensity modulations [5,30] which should be relatively easy [5]. This setup could be used to generate several 100 eV photons through HHG with short mid-infrared pulses [41] and realize Fourier transform spectroscopy at unprecedented small wavelengths and attosecond pump -attosecond probe experiments with unprecedented time resolution.

Disclosures
The author declares no conflicts of interest.