1. INTRODUCTION

The generation of extremely short and ultrabroadband optical waveforms, which are custom-tailored within a single cycle of light, opens up unprecedented opportunities for the emerging field of waveform nonlinear optics, which is of primary importance, e.g., for the generation of intense isolated attosecond extreme-UV pulses [1,2], launching valence-electron wavepacket dynamics in atoms and molecules [3–5], relativistic laser–plasma interactions and laser-driven electron acceleration [6–9], and control of sub-cycle electron transport in solids [10]. However, the feasibility of studying nonlinear interactions of matter with intense sub-cycle waveforms critically depends on the availability of high-energy multi-octave-spanning carrier-envelope-phase-controlled optical pulses. In addition, the realization of the energy and bandwidth scalability of ultrashort optical transients provides a new enabling technology for the demonstration of bright coherent tabletop high-harmonic sources, especially in the water window and keV x-ray region [11,12]. Therefore, the generation and precise dispersion control of ever broader optical bandwidth is in high demand, in order to tailor the shortest light bursts in the time domain.

Recent progress in broadband waveform generation has produced coherent optical spectra with $>1$-octave bandwidth by sub-cycle waveform synthesis [13] and supercontinuum generation [3–5]. However, it is difficult to obtain sub-optical-cycle pulses with high pulse energy [i.e., at the millijoule (mJ) level], limited by the trade-off between the required high nonlinearity for the generation of ultrabroad bandwidth and the output beam quality in terms of the beam stability and beam profile distortion. The dispersion management of intense multi-octave spectra is extremely delicate and challenging, especially requiring ultrabroadband precision dispersive optics with high damage threshold. Active compression systems based on spatial light modulators with $>1$-octave bandwidths have been demonstrated [14,15]. However, they are hampered by the bandwidth and diffraction efficiency of the gratings used in $4\text{-}f$ systems. Therefore, they have not directly been applied to multi-octave-spanning high-intensity sources. Multilayer dielectric coatings, such as the ones used for chirped mirrors [16,17] and complementary mirror pairs [18–20], although not adaptable, have the advantage of potentially supporting multi-octave bandwidth with high reflectivity. Therefore, multilayer mirror designs are widely employed as robust solutions. 1.5-octave-wide chirped-mirror pairs [19] for pulse compression have been developed, and optical synthesizers based on hollow-core-fiber compressors with up to four channels have been realized with separate chirped-mirror designs [3–5]. The pulses covering different spectral ranges are individually recompressed before pulse recombination. While hollow-core-fiber compressors are limited in energy handling, a pulse synthesizer based on parametric amplifiers can further scale the energy to the multi-mJ range [13,21–24]. Then the maximum pulse energy reachable is ultimately limited by the peak power of the combined intense ultrashort pulse, which induces detrimental nonlinearities (i.e., $B$-integral) in the following optical beam path and particularly in the beam combiner optics and the window of vacuum chambers housing the experiment. In addition, the dispersion of the beam combiner around the edge of the high-reflection band is difficult to control. As a result, synthesized electric-field transients without spectral gaps are hard to achieve with dichroic mirrors (DMs): spectral gaps cannot be avoided in between the channels [3–5]. Therefore, to efficiently synthesize ultrabroad optical waveforms, precise dispersion matching between the transmission and reflection ports of the beam combiner is required. Broadband DMs with exquisite dispersion control are necessary but are not available yet, to the best of our knowledge. In this work, novel mirror designs for $>2$-octave-spanning waveform synthesizers delivering multi-mJ pulse energy [21–23] are introduced. We design, fabricate, and characterize the required laser optics [25], chirped dichroic mirrors (CDMs) for efficient splitting and coherent combining of pulses, and ultrabroadband double-chirped mirror (DCM) pairs for final compression. To avoid $B$-integral problems, we recombine the still slightly chirped pulses first on the CDMs and then use an ultrabroadband DCM pair as a final compressor unit. Figure 1 shows the use of these optics in an actual $>2$-octave-wide three-channel parametric synthesizer, which was recently demonstrated [21–23].

 

Fig. 1. Schematic of a $>2$-octave-wide three-channel parametric synthesizer [21–23]. CEP, carrier-envelope phase; WL, white light; CDM, chirped dichroic mirror; OPA, optical parametric amplifier; DCM, double-chirped mirror; BOC, balanced optical cross correlator.

Download Full Size | PDF

2. DUAL-ADIABATIC-MATCHING STRUCTURE

In principle, chirped mirrors are designed as dispersive optical interference coatings with low-/high-index dielectric layer pairs to achieve dispersion management [16–20,26]. In chirped mirrors, the Bragg wavelength is chirped so that different wavelengths penetrate different depths into the mirror until Bragg reflection, giving rise to a wavelength-dependent group delay (GD). Figure 2(a) shows the simple-chirped (SC) structure, where the Bragg wavelength is monotonically increased with each quarter-wave layer pair to provide negative dispersion, as well as a broader high-reflection bandwidth than the Fresnel reflection bandwidth of a fixed layer pair [26,27]. However, pronounced GD ripples are usually observed, as shown in Fig. 2(d). These ripples are due to the impedance mismatch between free space and the multilayer grating, which acts as a Gires–Tournois interferometer. Since the period of the oscillations in the spectral domain determines the position of a satellite in the time domain upon reflection, the suppression of the adverse GD ripples by achieving impedance matching to avoid internal resonances is of great importance especially in few-cycle pulse generation. The double-chirped (DC) structure [16], as shown in Fig. 2(b), has been proposed to achieve impedance matching by adiabatically tapering the impedance in the front layer pairs, which is shown to be equivalent to an adiabatic chirp in the thickness of the high-index layer in addition to the chirp of the center wavelength of the Bragg mirror. However, a multilayer mirror design covering more than 2 octaves in bandwidth, which requires multi-octave impedance matching, has not been achieved so far. Here, we introduce a dual-adiabatic-matching (DAM) structure [25], as shown in Fig. 2(c), that generates also a double chirp in the back section of the mirror approaching the substrate, adiabatically tapering the impedance again to provide high transmission for long wavelengths. The front and back chirped high-index layers perform dual adiabatic impedance matching, providing (1) high reflectivity and smooth GD over the high-reflectivity range of the mirror and (2) high transmission with sidelobe suppression outside the high-reflectivity range, respectively.

 

Fig. 2. (a)–(c) Structures and (d) reflectivity/GD of different chirped mirror structures: (a) a design of the SC structure monotonically increases the Bragg wavelength with quarter-wave layer pairs to provide negative dispersion; (b) the DC design tapering the impedance in the front section based on the SC layers features reduced GD ripples in the high-reflectivity range; (c) the proposed DAM structure further introduces another impedance-matching section in the back layers, resulting in high transmittance for longer wavelengths; low, low-index material (e.g., ${\mathrm{SiO}}_2$); high, high-index material (e.g., ${\mathrm{TiO}}_2$).

Download Full Size | PDF

Because of the smooth transmittance with smooth GD behavior as shown in Fig. 2(d), the DAM structure can be used as a $>2$-octave-spanning dispersive CDM. With the impedance-matching features shown in Fig. 2(d), we implement the DAM structure to design a CDM. To realize the idea experimentally, the analytical DAM structure is employed as an initial design, which provides a high-reflection window for the spectral range of 0.45–1 μm and transmission in the range of 1.1–2.5 μm for $p$-polarized laser beams with an incidence angle of 45°. The numerically optimized result using a fast algorithm [28] is shown in Fig. 3(a). With an antireflection (AR) coating in the initial few layers to provide impedance matching from air to the low-index coating material, the following structure preserves the DAM structure. The GD in reflection is designed to compensate a 0.52 mm optical path in fused silica over the spectral range from 0.45–1.3 μm, which is even broader than the high-reflectivity range of 0.45–1.1 μm, as shown in Fig. 3(b).

 

Fig. 3. (a) Structure and (b) the designed/measured reflectivity/GD of the CDM, as well as the corresponding GD design goals. The imperfect transmittance around 0.8 μm and the 5% reflection above 1.1 μm are intentionally created to supply a BOC for waveform synthesis; low, low-index material, ${\mathrm{SiO}}_2$; high, high-index material, ${\mathrm{TiO}}_2$; GD_R, designed/measured GD in reflection; GD_T, the transmitted designed/measured GD; FS, fused silica.

Download Full Size | PDF

To combine spectra with a spectral overlap, the simultaneous dispersion management in both reflection and transmission, especially around the edge of the high-reflection band, is very important. The GD oscillations in reflection of a DM in this range are usually rather large due to the strong resonant interferences originating from internal Fabry–Pérot effects. The backside impedance-matching section in the DAM structure reduces these typically observed GD oscillations. Since the reflection from the back impedance-matching region is suppressed, the etalon resonance is reduced between the quarter-wave Bragg stacks and the back impedance-matching section. On the other hand, the GD in transmission is subject to the Kramers–Kronig relation [29]:

3. DISPERSION-MATCHED SCHEME FOR COHERENT BEAM COMBINING

To match the beam dispersion between the reflection port and the transmission port, the dispersion for the two combined beam paths in the spectrally overlapping region should be the same after the CDM, which is similar to a previous scheme based on a dispersion-matched neutral beam splitter [32]. A proposed scheme with a CDM and a dielectric plate in port 1 is shown in Fig. 4(a). Let us denote the GDD of the coating between the air and the substrate interfaces with reflection R1, transmission T1, reflection R2, and transmission T2 by ${\mathrm{GDD}}_{\mathrm{R}1}$, ${\mathrm{GDD}}_{\mathrm{T}1}$, ${\mathrm{GDD}}_{\mathrm{R}2}$, and ${\mathrm{GDD}}_{\mathrm{T}2}$, respectively. Also, the GDD for a single pass through the substrate of the CDM and the dielectric plate are denoted as ${\mathrm{GDD}}_{\mathrm{S}}$ and ${\mathrm{GDD}}_{\mathrm{P}}$, respectively. In the design, we will match the GDD of the substrate with the GDD of the plate and ${\mathrm{GDD}}_{\mathrm{R}1}$ (i.e., ${\mathrm{GDD}}_{\mathrm{S}}={\mathrm{GDD}}_{\mathrm{R}1}+{\mathrm{GDD}}_{\mathrm{P}}$). The GDD for each optical path is then given by

 

Fig. 4. (a) Schematic diagram of a dispersion-matched system based on the CDM. (b) The reflection (purple), transmission (red), and the combined spectra (green, blue, and orange dashed) in port 4 with different delays between port 1 and port 2 in (a), as well as the incoherently combined spectrum (black). Constructively/destructively interfered spectrum over the whole transition range (1.0–1.1 μm) can be obtained as the green/blue curve. (Inset) The optimized beam combining efficiency is $>90\%$, even including the $\sim 8\%$ total interface reflection losses of the matching plate in port 1.

Download Full Size | PDF

For a lossless coating of the CDM, the following relationships are generally valid [27,32]: ${\mathrm{GDD}}_{\mathrm{T}1}={\mathrm{GDD}}_{\mathrm{T}2}$ and ${\mathrm{GDD}}_{\mathrm{R}1}+{\mathrm{GDD}}_{\mathrm{R}2}={\mathrm{GDD}}_{\mathrm{T}1}+{\mathrm{GDD}}_{\mathrm{T}2}$. Assuming the transmittance of the CDM in the transition region is smooth enough, the corresponding GDD is negligible (i.e., ${\mathrm{GDD}}_{\mathrm{T}1}\approx 0$). With these additional conditions for a lossless and slowly varying transmission coating in the transition region, the GDD for each beam path to port 4 becomes identical:

Since the dispersion is matched, the spectrum in the spectrally overlapping region in port 4 can constructively be combined with matched spectral phase over the overlapping region, which further boosts the combination efficiency. The purple/red curves in Fig. 4(b) show the reflected/transmitted spectra measured using a white-light source from port 1 and port 2, respectively. The combined spectra are also shown as the green, blue, and orange dashed curves in Fig. 4(b), as well as the incoherently combined spectrum as a reference: interference fringes (orange dashed curve) are observed due to the optical delay between the beams from the two input ports. By fine-tuning the optical delay, constructively/destructively interfered spectra over the spectrally overlapping region can be obtained as the green/blue curves, respectively. The beam combining efficiency with the green spectrum is $>90\%$ over the transition range, even including the $\sim 8\%$ total interface reflection losses of the silica plate in port 1.

The interfered fringes in the spectrally overlapping region are extremely sensitive to the delay between the pulses from the two input ports, which benefit the precise active stabilization of the relative timing.

4. $\mathsf{>}\mathsf{2}$-OCTAVE-WIDE HIGH-REFLECTION DCM PAIRS

The DAM structure lends itself to the design of an ultrabroadband DCM pair as the final compression unit, which is key enabling technology to realize $>2$-octave multi-mJ sub-cycle waveforms without running into $B$-integral problems that destroy the pulse quality. The proposed structure particularly reflects the light within its Bragg wavelength and provides smooth transmittance for the longer wavelengths, which is a proper impedance-matching section to reduce the Gires–Tournois effect in the chirped mirror design. As a result, cascading the DAM structure in the front layers, as an ultrabroadband impedance-matching section, makes it possible to achieve $>2$-octave bandwidth DCMs, as shown in Fig. 5. The ultrabroadband DCM pair is designed and optimized for compensating 1.44 mm optical path in fused silica in the spectral range from 0.49 to 1.05 μm, the high-reflectivity range of the CDM, and 0.32 mm optical path in ZnSe in the range of 1.05–2.3 μm, the transmission range of the CDM. For $>1$-octave-wide chirped mirrors, mirrors providing negative chirp are advantageous, because short-wavelength light is reflected by the top layers, whereas long-wavelength light penetrates deeper and mostly sees an average index of the thinner top layers. To demonstrate the broadband impedance matching of the cascaded-DAM-like structures, Figs. 5(c) and 5(d) show the reflectivity of the structures from the ambient air to the specific layers pointed at by the arrows in Figs. 5(a) and 5(b), respectively. As the number of front layers increases, starting from the air, the high-reflectivity band expands to longer wavelengths, which is determined by the Bragg wavelength of the layer pairs. The arrows in Figs. 5(a) and 5(b) point to the end layer of each DAM-like structure, providing broadband AR coating (impedance matching) with $<5\%$ reflection to the design wavelength of 2.3 μm. The designed/measured reflectivity and GD of the pair are shown in Fig. 6. The average reflectivity of the ultrabroadband DCM pair is $>90\%$ and the calculated peak-to-peak values of the averaged residual GD ripples are controlled to $<5\mathrm{fs}$ over $>2$-octave bandwidth. In Fig. 6, the dispersion and reflectivity measurements using a home-built white-light interferometer and a photospectrometer, respectively, show excellent agreement with the design targets.

 

Fig. 5. (a) and (b) Structure of the mirror pair of the optimized DAM DCM pair. (c) and (d) AR behavior for the longer wavelengths in the cascaded DAM sections indicated by the arrows in the mirror structure in (a) and (b), respectively; low, low-index material, ${\mathrm{SiO}}_2$; high, high-index material, ${\mathrm{TiO}}_2$.

Download Full Size | PDF

 

Fig. 6. Reflectivity and GD of the ultrabroadband DCM pair: the calculation shown as the blue/red dashed lines and the solid lines show the measurement results. The dispersion of the DCM pair compensates a 1.44-mm optical path in fused silica and a 0.32-mm optical path in ZnSe for ranges of 0.49–1.05 μm and 1.05–2.3 μm, respectively.

Download Full Size | PDF

Even broader design bandwidth of DCM pairs is possible but, depending on the applications, limited by the required reflectivity and dispersion control. For example, it is difficult to design chirped mirrors with $>99.5\%$ reflection over $>2$-octave-wide bandwidth used as intracavity mirrors for precise dispersion compensation in a broadband laser oscillator. The laser-induced damage threshold of the lower bandgap dielectric coating material, ${\mathrm{TiO}}_2$ in our case, would hamper the energy scalability of the ultrabroadband source based on the DCMs. The threshold fluence of ion-beam sputtering ${\mathrm{TiO}}_2$ films is measured to be $>0.1\mathrm{J}/{\mathrm{cm}}^2$ using a 25 fs Ti:sapphire amplifier [33]. Although the repetition rate [34], pulse duration, and center wavelength of the optical source, as well as the coating structures of the DCMs, would affect the breakdown fluence, sub-cycle optical waveform synthesizers supporting up to several tens of mJ of pulse energy with a $>1{\mathrm{cm}}^2$ beam size should be possible based on the multilayer chirped mirrors.

We evaluate the expected pulse distortions introduced by the demonstrated ultrabroadband DCM pair on the waveform synthesizer output using the experimental combined second-stage optical parametric amplifier (OPA) output spectra of Ref. [22]. As the scheme shows in Fig. 1, the spectrum below 1 μm is reflected by the CDM, and the spectrum above 1.1 μm is transmitted. The combined pulse, with 1.88 fs FWHM transform-limited pulse duration, is still chirped when passing through the combiner substrate and the vacuum-chamber window to decrease the peak intensity and thus reduce the $B$-integral, and the ultrabroadband DCM pair can compensate the dispersion of the combined waveform afterward. The pulse distortions were examined by considering the measured reflectivity and residual GD errors of the mirrors (i.e., starting from the flat-phase waveform and accounting for the measured GD deviation of the mirrors from the design goal, the GD of 1.44 mm optical path in fused silica or of 0.32 mm optical path in ZnSe), as shown in Fig. 7. Figure 7(b) shows the synthesized waveform after final compression by the ultrabroadband DCM pair according to Figs. 5 and 6. The measured residual phase error of the DCM pair, is $<0.1\pi \mathrm{rad}$ (i.e., $<\lambda /20$), as the red dotted curve in Fig. 7(a) shows, enabling pulse compression very close to its transform limit. Pulse compression over $>2$-octave-bandwidth can thus be achieved for the first time, to the best of our knowledge.

 

Fig. 7. Evaluated pulse characteristics in (a) frequency domain and (b) time domain after the ultrabroadband DCM pair. The black curve in (a) is the measured combined OPA spectra, as well as the transform-limited electric-field waveform (TL) in (b). The red curves show the pulse after the mirror pair, as well as the residual phase shown as dotted line in (a). The duration of the pulses depicted in the inset as the black and red lines are 1.88 and 1.93 fs, respectively.

Download Full Size | PDF

5. CONCLUSIONS

We proposed a DAM structure that enables $>2$-octave-wide dispersion control. The proposed structure is similar to refractive-index apodization in fiber Bragg gratings [35,36]: the DAM structure adiabatically matches the impedance of both the Bragg stacks and low-index material, providing not only high reflectivity with smooth GD in reflection at the Bragg wavelength, but also a broadband AR coating for the longer wavelengths with smooth GD in transmission. Dispersive CDMs based on the DAM structure can thus be designed and fabricated to achieve both beam combination and dispersion compensation with $>2$-octave bandwidth. With the demonstrated CDM and the dispersion-matched scheme, the combined spectrum from different input ports can be optimized. In principle, near-100%-constructive-interfered beam combining over the whole spectrally overlapping region can be achieved; here, $>90\%$ of beam combining efficiency was demonstrated experimentally, taking Fresnel losses of interfaces into account. In chirped-mirror designs, a broadband AR coating as an impedance-matching section is critical to reduce the amplitude of GD ripples induced by the Gires–Tournois effect. As a result, cascades of DAM structures in the front section of a chirped-mirror design for an ultrabroadband DCM pair are demonstrated, resulting in the most broadband chirped mirror designed so far. The designed CDM and DCM pairs are key enabling technology for a $>2$-octave multi-mJ parametric synthesizer supporting 1.9 fs FWHM waveforms without running into $B$-integral problems that destroy the pulse quality. No temporal pulse broadening is found due to the use of CDM and DCM pairs. The results indicate that the proposed designs will pave the way for the development of intense ultrabroadband parametric synthesizers and in general dispersion control of multi-octave-bandwidth coherent optical sources [37].