1. Introduction

The present paper concerns the technique of temporal shaping of ultrashort pulses by acting on the spectral phase of the pulse. In one of the most common approaches, a desired angular frequency dependent phaseshift φ(ω) is imprinted on the spectrum of a pulse by inserting a pixellated spatial light phase modulator (SLM) across the pulsed beam which is spectrally dispersed and recollimated by diffraction gratings in a zero dispersion unit [1]. This technique is very versatile and can generate a wide variety of complex shapes, the temporal profile being obtained as the inverse Fourier transform of the modulated spectral field amplitude [1]. Application in time resolved spectroscopy and controlled femtosecond chemistry were demonstrated [2]. One particularly simple temporal function of high practical interest is that of splitting one incident pulse into two temporally neighboring pulses. These for instance are waveforms of interest for two step machining [3]. Such function can be performed by an SLM in several ways, one example being the introduction of a phase jump in the spectral phase [1, 4]. We are showing here that a monolithic waveguide grating structure can be used for imprinting a wavelength dependent phase variation on an incident beam. Such element is a resonant grating composed of a slab waveguide of corrugated surface which exhibits the effect of 100% resonant reflection first demonstrated and analyzed in the mid-eighties by Sychugov et al. [5]. Usually, this highly wavelength, angularly and polarization selective effect is used for its sharp high reflection peak. The sudden phase change associated with the reflection peak is generally discarded. However, Usievich et al. [6] showed that the phase variation is of utmost importance when such resonant mirror is associated with another element into a complex optical system. This is precisely the case in the present work where the spectral phase only, not the modulus of the reflection, is the matter of concern. The effect of 0^th order reflection from a corrugated surface on the temporal shape of an incident ultra-short pulse was investigated by Ichikawa et al. [7] who showed that a metal corrugation induces strong amplitude and phase dispersion effects on the spectral reflection which can lead to an alteration of the reflected temporal pulse shape even though no electromagnetic resonance is excited. They reported in particular a pulse compression effect of 20%.

The present paper will first establish the relationship between the type of spectral phase provided by a lossless resonant grating and the expectable temporal shape. Then the example of a functional demonstrator will be presented.

2. The spectral phase of reflection on a mirror-based resonant grating

It was shown in a previous publication [8] that, for a plane wave of spatial frequency k exciting a mode of a grating waveguide, the reflection coefficient r in the zero^th order can be expressed in the neighborhood of the synchronism condition as the sum of two quantities: a constant term r0 assimilated to a Fresnel reflection coefficient and a polar function related with the excitation of the mode into the slab waveguide. The spectral response of a resonant grating can thus be written as a function of the angular frequency ω :

where ω₀ is the angular frequency for synchronous mode excitation, Δω the spectral resonance width and a_ω is a complex constant depending essentially on the coupling of the incidence wave to the mode and the diffraction efficiency of this mode into the incidence medium.

The aim in the present temporal shaping operation of ultrashort pulses is to have the whole spectrum reflected in the zero^th order with the phase shift placed approximately in the middle so as to get a pure phase modulation. This condition can simply be achieved by placing a zero order grating waveguide on top of a background mirror to avoid losses due to diffraction orders and transmission. The mirror will be considered as lossless in what follows.

It was also demonstrated in [8] that the graphical representation of the reflection coefficient r from a resonant grating in the complex plane is a circle reaching theoretically 100% at resonant reflection and centered at point C located at :

In the case of a resonant grating based on a lossless mirror as sketched in Fig. 1, the reflection coefficient modulus is always 1. Thus the reflectivity circle representing r(ω) is the unit circle with its center at the origin of the complex plane (C=0). In this particular case the constant a_ω and the resonance spectral width Δω are closely linked. Expressing a_ω in equation (2) set to zero gives :

 

Fig. 1. Sketch of the resonant mirror principle. The mode is excited using the −1^st order diffracted into the dielectric layer.

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Substituting a_ω given by (3) in the polar function (1), factorizing by r₀, and finally simplifying the result, yields :

This complex ratio can be written as a pure phase :

where * represents the convolution product. It is worth noting that the modulus of the complex function r(ω) is always 1. Thus, considering the argument of r(ω) expressed in (5), the spectral phase φ(ω) induced by reflection on a mirror-based resonant grating is essentially an arctangent function of 2π amplitude, centered at the synchronism angular frequency ω₀ and scaled by the resonance spectral width Δω :

where φ₀ is the constant phase of the off-resonance reflection coefficient.

The spectral phase induced by the reflection on a mirror-based resonant grating is now fully defined according to (6).

3. The temporal response

The output profile in the temporal domain e_o(t) of a pulse of initial spectrum E_i(ω), after experiencing a spectral modulation H(ω), is given by the Fourier transform of the product E_i(ω) H(ω) [1]. In the case of a mirror-based waveguide grating, the spectral modulation is a pure phase function given by (5). The translation factor δ(ω-ω₀) can be neglected if the angular frequency ω₀ exciting the mode is close to the central angular frequency of the pulse spectrum : it will only result in a linear temporal phase in the output pulse. The constant phase term φ₀ does not affect the temporal profile either. The transfer function H(ω) of the mirror-based resonant grating can be simply considered as :

The temporal response h(t) of this linear filter is obtained by calculating the inverse Fourier transform using basic Fourier transform functions and properties :

where δ is the Dirac distribution and δ′ and δ″ respectively its first and second derivatives. Convoluting a function with the derivative of the Dirac distribution consists in taking the derivative of the function. Setting f(t)e=^-Δω|t|, the successive derivatives are :

where sign represents the sign distribut ion. Consequently, the temporal transfer function h(t) of the resonant mirror is equal to :

where U is the Heaviside distribution. The electric field e_o(t) of the pulse after reflection on the mirror-based waveguide grating can then be expressed as a convolution product of the temporal transfer function with the electric field e_i(t) of the incident Gaussian pulse:

Using the definition of the convolution product, we can write :

Operating the change of variable $X=\frac{\tau }{p}+\frac{p·\Delta \omega }{2}$ leads to :

where erf is the error function. After simplification and considering that the Gaussian parameter p is linked to the incident Gaussian pulse duration Δt at half maximum by $p=\frac{\Delta t}{\sqrt{2\ln 2}}$, the electric field e_o(t) of a Gaussian pulse after reflection on a mirror-based waveguide grating of resonance width Δω is :

In expression (13), the linear temporal phase induced by the translation of the pulse spectrum at the angular frequency ω₀ is neglected. The temporal expression of the electric field of the ultrashort pulse experiencing the phase modulation is composed of two main terms. The first one is the product of a decreasing exponential and an error function between brackets starting from zero : an asymmetric post-pulse is created with a slowly decreasing amplitude. The second term is the input pulse which leads to a pre-pulse close to a Gaussian. It appears clearly from (13) that the effect of the 2π arctangent phaseshift located at the middle of the pulse spectrum is to give rise to two subpulses. It is worth noting that the electric field of the shaped pulse expressed in (13) is practically real. The temporal phase of the pulse remains essentially constant with a π step located at the saddle point: the pulse can be considered as chirpless. Figure 2 illustrates the temporal shape of the reflected pulse assuming a Gaussian input pulse.

 

Fig. 2. Intensity of a 130 fs Gaussian input pulse after reflection onto a mirror-based resonant grating for different spectral widths.

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The first calculations are made for a 130 fs full width at half maximum (FWHM) pulse. There are two free parameters in the spectral phase given by the resonant reflection : the first one is the location of mode excitation ω₀ which must be close to the central angular frequency of the input pulse, the second one is the resonance spectral width Δω of the arctangent function. This last parameter can be chosen rather freely to define the balance between the two subpulses and the temporal spacing between them. It should however neither be too small since the phase would be seen as nearly constant across the pulse spectrum, nor too large since a sudden 2π phase jump would not be seen, unlike in [4], where a π phase step does lead to pulse splitting. As a result of a number of scans of (13) with different spectral widths, the product ΔωΔt considering a 130 fs incident pulse centered at a 800 nm wavelength should be comprised between 0.12 rad and 1.26 rad to ensure that the amplitude of the smallest subpulse is larger than 10% of that of the largest subpulse. Concerning the effect of an off-centered phase modulation within the spectrum, it results in a decrease of the pulse contrast. The more the phaseshift is off-centered, the higher is the minimum of intensity and the more the output temporal profile tends toward the initial Gaussian pulse.

Having made the analysis of the pulse splitting mechanism, it is now possible to set the structure parameters leading to the desired temporal characteristics. The first step is to adjust the opto-geometrical parameters of the structure to obtain an angular frequency for mode excitation ω₀ corresponding approximately to the central angular frequency of the input pulse spectrum. For fixed incidence angle and grating period this amounts to an adjustment of the thickness of the slab waveguide. The second step implies to adjusting the spectral width Δω so that the splitting of the pulse occurs with the desired characteristics as shown in Fig. 2. This is achieved by gradually increasing the grating depth and decreasing the homogenous waveguide layer thickness so that the equivalent thickness and effective index of the structure remain essentially constant. In doing so, the resonance spectral width increases monotonically with the grating depth while the angular frequency for mode excitation ω₀ remains essentially unchanged. This procedure permits to obtain the desired Δω, therefore the desired split temporal profile.

4. Temporal pulse splitting monolith

As stated in section 2, a reflectivity close to 100% is needed for the whole pulse spectrum. Adding a mirror as a substrate to a waveguide grating permits to also reflect the non-resonant spectral components. A further condition is to prevent the non-resonant spectral components from being diffracted by the coupling grating into propagating orders. This can be ensured by achieving the mode coupling by the −1^st order of the grating since then all diffraction orders are cut off within the pulse bandwidth. This implies a small grating period. To minimize metal losses, a sufficient distance between waveguide and mirror is required to electromagnetically isolate the modal field from the mirror. A low index buffer layer will thus be inserted between the waveguide slab and the mirror.

 

Fig. 3. Fabricated resonant mirror. (a) Sketch of the layered structure with corrugation in the last HfO₂ layer. (b) Corresponding spectral phase of the TM reflection under 46° incidence.

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The structure which was fabricated is sketched in Fig. 3(a). It consists in an alternation of high index HfO₂ (n_h=2.06) and low index SiO₂ (n_l=1.46) layers deposited by ion beam sputtering. The mirror is composed of a gold film on which a 6-layer dielectric mirror is deposited to decrease the metal reflection losses over the pulse spectrum. A thick low index buffer layer is inserted between the grating waveguide and the mirror. This is particularly important for achieving a phase only modulator. Note that the last high index λ/4 layer is slightly thicker; this is to avoid the excitation of a spurious mode in the multilayer mirror which would generate an absorption peak in the spectrum near resonance. Finally a 350 nm pitch binary grating is etched at 77 nm depth in the last high index layer. The expected spectral phase of the fabricated structure is given in Fig. 3(b).

Illuminating this structure with a TM-polarized 800 nm input pulse under a 46° incidence angle is expected to lead to pulse splitting. The resonance spectral width Δω of the structure is calculated numerically using a commercial software developed by N. Lyndin [9] based on the true-mode method [10]. Δω is found to be around 3.7 rad/ps. The temporal intensity obtained after reflection of a Δt=85 fs Gaussian pulse corresponding to the experimental situation on the resonant mirror was calculated numerically by considering the square of the modulus of the electric field expressed in (13). A first pulse of similar length as the incident one and containing approximately 30% of the output energy should be followed 150 fs later by a twice longer pulse containing approximately 70% of the energy. The maximal intensities of the two pulses are approximately the same. The total losses, mostly induced by the metallic mirror of the structure, are measured to be around 15%. The splitting effect was demonstrated experimentally using a 85 fs pulse from a Coherent Vitess 800 oscillator at a repetition rate of 80 MHz and an energy of 3nJ/pulse. To demonstrate the splitting of the 85 fs Gaussian pulse after reflection on the resonant mirror, the temporal profile was characterized using a second order intensity cross-correlation method. The background free second order intensity cross-correlation signal is depicted in Fig. 4(a) as a function of the relative delay, indicating the temporal form of the shaped pulse. All the pulses are normalized in energy.

Experimental cross-correlation results (crosses) in Fig. 4(a) demonstrate the occurrence of pulse splitting. The temporal delay between the two subpulses is about twice the incident pulse duration at half maximum (160 fs). For the grey dotted curve, the output pulse is taken to have a theoretical electric field as calculated in equation (13). The cross-correlation signal calculation of this shaped pulse is numerically performed considering a Gaussian incident pulse then represented in Fig. 4(a). Experimental results (cross-scatter graph) do confirm the theoretical result developed in the previous sections (grey curve), although the two normalized cross-correlation curves exhibit a slight discrepancy. This slight difference can be mainly attributed to energetic losses induced by an amplitude modulation in the pulse spectrum. Actually, the fabricated structure was not energetically optimized since a lossy chromium layer was deposited to prevent a possible delamination of the dielectric multilayer on the low adhesion gold surface. The total losses were measured to be around 15 % of the whole incident pulse energy. These losses are mainly due to metal absorption of the modal field and show as an absorption peak in the spectrum centered at the angular frequency for synchronous mode excitation ω₀. In order to model the spectral response of a mirror-based waveguide grating with losses, let us consider this absorption peak inducing an amplitude modulation in addition to the theoretical phase modulation (6). The resulting amplitude modulation is here modeled by a Lorentzian function with a spectral width Δω equivalent to the one of the mirror-based waveguide grating and a minimum R_min adjusted so that 15% losses of the incident pulse energy occur. This induces a slight amplitude modulation of the Gaussian pulse spectrum as shown in Fig. 4(b). Thus the spectral response r(ω) becomes :

The theoretical electric field of the output pulse obtained after a reflection onto such a structure is numerically calculated as the inverse Fourier transform of the product of the spectral modulation r(ω) with the incident theoretical pulse spectrum [1,11]. The cross-correlation operation in the presence of losses in the mirror-based waveguide grating is then numerically performed as explained previously for the theoretical results. Experimental results (cross-scatter graph) and numerical results considering 15% losses of the incident pulse energy in the structure (black curve) are in a remarkably good agreement. This confirms that the absorption peak induced by the metallic mirror is the cause of the discrepancy between theoretical and experimental results. Nevertheless, the opto-geometrical parameters of the structure can be optimized to avoid losses due to modal field absorption by the metallic substrate.

 

Fig. 4. (a) Cross-correlation of the temporal profile obtained after reflection of a 85 fs incident Gaussian pulse onto the resonant mirror (crosses : experimental results, grey dotted line: theoretic results and black line: numerical calculation of the cross-correlation signal considering 15% energy losses). (b) Theoretical spectral intensity of the Gaussian incident pulse before (dashed line) and after reflection (solid line). Losses are considered.

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Having demonstrated that a single monolithic resonant diffractive element can perform femtosecond pulse splitting, we investigate below how the temporal distance between subpulses can be extended and possibly made variable.

5. Adjustable temporal pulse shaping using a mirror-based waveguide grating

The time delay between the subpulses achieved by the described monolithic element is very short. Moreover, the spectral modulation is fixed as the opto-geometrical parameters cannot vary. This element can be inserted in a laser system like a simple mirror but its functional flexibility is very limited. One way to develop the flexibility of the analyzed monolithic pulse splitter and to expand its application potential is to resort to a cascading scheme whereby the beam is reinjected inside the structure using a second standard or resonant mirror as sketched in Fig. 5. Each injection induces a 2π arctangent phaseshift in the pulse spectrum. The total phase modulation is the sum of all these phaseshifts and the phase modulation range increases : the pulse energy is spread over a larger time scale. The second asset of this set-up is the possibility to adjust the phase modulation which makes the mirrored resonant grating more flexible.

 

Fig. 5. Basic set-up for adjustable temporal pulse shaping using cascaded phaseshifts.

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There are three degrees of freedom : the number N of reflections of the pulse on the mirror-based waveguide grating, the incidence angle θ_i, and the tilt angle between the two mirrors γ. The incidence angle θ_n of each injection depends on the incidence angle θ_i and the injection angle γ. If γ is non-zero, the incidence angle θ_n changes from one incidence to the next : each resonance is characterized by a different angular frequency for mode excitation ω_n and a slightly different spectral width Δω_n. The total spectral phase modulation is :

The number of reflections permits to choose the number of phaseshifts in the spectral domain. The total spectral phase modulation is limited by the size and the losses of the structure. However, the device remains small and essentially monolithic. Besides, it was demonstrated that the metal losses can be minimized since the fabricated sample involves a lossy Cr adhesion layer. The incidence angle θ_i permits to spectrally translate the whole phase modulation. Finally, the tilt angle γ permits to adjust the constant spectral interval between the different phaseshifts. These three degrees of freedom are sufficient to achieve various types of temporal profiles ranging from a single asymmetric pulse to multi-pulses among which an interesting sequence is determined by pulse doubling with adjustable temporal separation and subpulse energy ratio. This case is numerically studied hereafter.

The structure presented in section 4 was modeled using the true-mode method [10] to determine the angular frequency ω_n and the resonance width Δω_n as a function of the incidence angle. Fixing the incidence angle θ_i, the tilt angle γ and the number of reflections, and using (15), it is possible to determine the phase modulation induced by the cascading setup, then the temporal shape of the pulse at the output of the monolithic shaper. Results of temporal intensity and phase of double pulses simulated with the structure of Fig. 3(a) are depicted in Fig. 6. The two subpulses obtained are temporally larger than the incident pulse but they remain clearly distinguishable.

 

Fig. 6. Examples of intensity, phase and instantaneous frequency for double pulses with larger delay obtained after 8 reflections of a 130 fs-Gaussian pulse: (a) tilt angle γ=0.2 (b) tilt angle γ=0.5.

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In Fig. 6 we have represented as well the instantaneous angular frequency of the shaped pulse. In contrast with the double pulse obtained with a single reflection, the double pulse obtained in the cascading scheme is complexly chirped, with high angular frequencies in the first pulse and low angular frequencies in the second one. As an example, the corresponding spectrum of the pulse depicted in Fig. 6(a) is represented in amplitude and phase in Fig. 7. The spectral phase corresponds to the phase modulation induced by the cascaded resonant mirror after 8 reflections.

 

Fig. 7. Example of cascaded phase modulation (solid line) induced in the spectrum of a 130 fs- Gaussian pulse (dashed line). This modulation results in the shaped pulse of Fig. 6(a).

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Fig. 8. (a) Theoretical evolution of the temporal shape of a 130 fs-Gaussian pulse upon a slight variation of the incidence angle. (b) Theoretical evolution of the delay between the two subpulses as a function of the tilt angle considering 8 reflections on the structure depicted in Fig. 3(a).

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The influence of the tilt angle on the delay between these two subpulses is represented in Fig. 8(a) : the delay increases as the tilt angle decreases. For 8 reflections on the structure of section 4, the two subpulses can be delayed up to 1.2 ps without significant temporal profile modification. For smaller values of the tilt angle, i.e. larger delays, the number of reflections has to be increased to smooth the pulse shape. As a further illustration of the flexibility of the cascading scheme, Fig. 8(b) shows the evolution of the temporal profile in the case of a variation of the incidence angle. This parameter is easy to adjust and permits to play with the relative energy contained into the two subpulses. Fig. 6 and 8 show that a mirror-based waveguide grating used in a cascading setup does have the potential of a flexible element to shape double pulses with adjustable relative energies and delay within a few picoseconds.

7. Conclusion

The effect of a resonant grating mirror on the temporal shape of an ultrashort pulse was demonstrated. This element is shown theoretically to induce a pure phase modulation acting on the spectrum and, thus, on the temporal profile of an ultrashort pulse. The specific spectral phase modulation leads to pulse splitting with a short delay between subpulses. The shape is fixed by the physics of the resonance: the only adjustable parameter is the ratio of the two subpulse amplitudes by adjusting the spectral resonance width of the resonant mirror. The latter can be regulated by acting on the opto-geometrical parameters of the structure. The resonant mirror is a fixed phase-only modulator which can be inserted in a laser system like a simple mirror at a determined incident angle. A concept is proposed to extend the temporal distance between subpulses and to possibly make it variable. The solution is to re-inject the reflected beam into the resonant mirror to get cascaded phaseshifts. Three geometrical degrees of freedom permit to generate a wide variety of temporal profiles. The resulting doubling of the subpulse delay was demonstrated experimentally.

The question remains on the practicality of this device, considering the high flexibility achieved with present programmable light modulators to generate complex temporal pulse forms. We believe that, for example, practical applications can be envisaged in the field of laser micromachining. In many cases, laser material processing demands require high throughput and precision. Benefits can be found if the light modulation can take advantage on the transient changes of the excited materials [12]. These changes in the optical properties can be induced by electronic excitation, e.g. considering free carrier response on the dielectric function in band-gap materials or temperature influence on electronic collision rates. Additionally, optical transformations can be derived following structural transitions, either electronically-induced or mediated by a thermal path. In the latter case, the relevant time scale is given by the electron-phonon coupling strength. Considering various ways of transiently changing the laser energy coupling efficiency and its effect in material processing, the relevant times scale can be estimated to be up to several ps and multi pulse sequences with variable delay and relative amplitudes may become relevant. The presented device has the advantage to achieve double pulse sequence based on a simple reflection effect. However, as a drawback, a certain elongation of the pulse form remains. We are presently testing the device in micromachining applications.