A novel design approach for dispersion-compensating chirped mirrors with greater-than-octave bandwidth is proposed. The commonly encountered problem of dispersion ripple is overcome by impedance matching via Brewster incidence in respect to the top-layer coating material. This approach totally suppresses undesired reflections off the interface to the ambient medium without any need for complicated matching sections. It is shown that Brewster-angled chirped mirrors can deliver ultrabroadband dispersion compensation over a much wider bandwidth than conventional double-chirped mirrors and without the mechanical complexity of back-deposition approaches. Due to their relatively simple structure, the sensitivity of the dispersion of the Brewster-angled designs towards growth errors is greatly reduced. Therefore, this new generation of chirped mirrors appears ideal for compression of continuum pulses with a potential of pulse durations in the single-cycle regime.
©2003 Optical Society of America
In the last few years, ultrafast pulse generation schemes have progressed well into the sub-5-fs regime. Pulses of only two optical cycles or even slightly less have been demonstrated [1, 2, 3]. This breakthrough has been made possible by sophisticated dispersion compensation schemes, which allow for wider bandwidths than pure bulk-optical approaches based on prisms or gratings. These advanced dispersion compensation schemes mostly involved chirped mirrors. Apart from chirped mirrors nearly octave-spanning dispersion compensation has also been achieved with liquid-crystal phase modulators in a zero dispersion delay line [4, 5]. Raman sideband generation appears to be another pulse compression method that may be pushed towards the optical octave and close to single-cycle pulse duration [6, 7]. However, even with demonstrated pulse durations of about 4 fs, dispersion compensation beyond one optical octave has neither been achieved with chirped mirrors nor any other method. Grating-based concepts encounter serious problems when orders start to overlap, i.e. at one optical octave. Design and manufacturing of chirped mirrors already becomes problematic when the bandwidth exceeds 0.6 optical octaves. Similar to chromatic correction in imaging systems, dispersion compensation becomes a challenging task when the bandwidth approaches the optical octave. In this Letter, we will explore a novel route towards pulse compression with single-cycle pulse duration based on a new generation of chirped mirrors.
2. Generations of chirped mirrors
The fundamental barrier to further extension of useful bandwidth of a chirped mirror can be tracked down to parasitic Gires-Tournois interferometer (GTI, ) effects, mainly arising at the interface between the chirped mirror stack and the ambient medium. Together with a reflection inside the mirror stack, the parasitic top reflection forms a variable-path GTI, which causes a modulation of the group delay dispersion vs. wavelength. This dispersion ripple overlays the desired dispersion characteristics of a chirped mirror and creates satellite pulses, which merge into a broad temporal pedestal upon multiple bounces off chirped mirrors . Generally, a tradeoff between bandwidth and suppression of dispersion ripple can be seen and limits the usefulness of chirped mirrors to much less than an optical octave.
After the initial demonstration of chirped mirrors , the ambient interface had been sub-sequently recognized as the main cause for dispersion ripple. The most straightforward way to suppress this effect is deposition of an antireflection coating on top of the high-reflecting mirror stack. It has been controversially discussed, whether to match from air to an average index of refraction of the layer pair  or to better match from air to one of the layer materials. The latter method has lead to the double-chirped mirror (DCM [12, 13]) design approach. Here a duty-cycle modulation between high and low index materials is used to additionally match the impedance adiabatically inside the mirror stack. These AR based strategies work very well up to approximately 300 nm bandwidth in the Ti:sapphire wavelength range, but are ultimately limited by the bandwidth of broadband AR coatings. For the typically used materials SiO2 and TiO2, e.g., a 300-nm bandwidth (0.6 optical octaves) results in a residual reflectivity of an AR coating of approximately 10-4 . Such coatings are very challenging in terms of growth error control. Provided total absence of growth errors, only 10-3 residual reflectivity can be reached for the full octave, regardless of the number of layers used in the AR section. A 10-3 suppression of GTI satellites, however, is insufficient in terms of dispersion ripple . Therefore, alternative approaches for impedance matching between ambient medium and mirror stack are required for octave-spanning dispersion compensation with chirped mirrors.
One approach to increase the useful bandwidth of chirped mirrors to one optical octave has been termed back-side coated chirped-mirror (BASIC, ) or tilted-front-interface mirror (TFI, ). The idea behind both approaches is to use an ambient medium with the same index of refraction as one of the layer materials. This design approach requires the deposition of the chirped coating on the rear surface of the substrate, or optically contacting a thin substrate on top of the layer stack. Interference between the chirped mirror structure and the front surface is avoided by wedging the cover substrate or using different curvatures of the two surfaces. For the compensation of material dispersion, one can only afford very thin cover substrates of a few tens or hundreds of microns thickness, which makes the mechanical construction of such mirrors cumbersome. In the following, a method is proposed that allows for dispersion control over more than one optical octave, without the need for complex optical assemblies, multiple coating runs, or any kind of matching section. In addition, the novel approach is fairly robust towards layer deposition errors, an aspect that has often been neglected in previous attempts to overcome the octave barrier in chirped mirror design.
3. Brewster-angled chirped mirrors
As discussed in the introduction, impedance matching problems at the interface to the ambient medium are considered the main obstacle in reaching a bandwidth of more than one optical octave with little dispersion oscillations. Previous attempts to suppress these effects have always modified the interface to the ambient medium, which provokes other detrimental effects like bandwidth reduction or the mechanical complexity of back-deposition approaches. The most straightforward way of suppressing undesired reflections, namely Brewster-angle orientation, has not been exploited yet for the design of chirped mirrors. The question may arise whether the Brewster orientation should be calculated for the low-index or high index material, or for some average index. As the dispersion ripple arises from the interface between air and the top layer material, a minimum dispersion ripple is only observed in the vicinity of the Brewster angle calculated for the top layer material. This strongly discourages the average index approach  and requires double-chirping  in the coating for impedance matching from one of the layer materials to the symmetric layer pair as a second step. Still there is the choice of selecting the high-index or low-index material for the top layer.
Figure 1 shows the Brewster angle vs. wavelength, calculated from measured data of sputtered coating materials . This calculation clearly reveals, that for the low-index material the Brewster angle only varies by 0.2–0.4° for an octave. The high-index material, however, shows much stronger Brewster-angle dispersion and is therefore much less suited as the cover material. A schematic drawing of a Brewster-angled chirped mirror and one representative optical path in this structure are depicted in Fig. 2.
As Brewster-angle orientation removes the prevalent source of multiple path interference, we can use a simple approach for designing an initial layer sequence from a given group delay dispersion -GDD(ω) to be compensated. Our approach is a simplified and adapted version of the WKB approach introduced in . We ignore the interference of multiples paths for one wavelength, i.e. assume that a well-defined classical turning point exists for each and every wavelength in the coating. Then the dispersion properties of the coating can be simply mapped by a frequency-dependent path length l(ω). The group delay dispersion resulting from such a path l(ω) is then written as
where c is the speed of light and the factor 2 accounts for passage back and forth through the mirror structure. From the known dependence l(ω), one can track the optical path through the mirror stack ( in Fig. 2) to find the layer pair (t i-1, ti) with matching Bragg wavelength
Here ti designates the physical thickness of the ith layer of the stack with index of refraction ni. Typical coatings consist of two alternating materials of high and low index of refraction, n hi and n lo, respectively. ϑi is the internal beam angle in layer i and related to the incident angle ϑ in by Snell’s law.
Equation 2 relates optical path lengths in the angled stack to physical thicknesses ti of the individual layers. For symmetric Bragg layers t i-1 n i-1cosϑ i-1=tinicosϑi=λB/4. Impedance matching is achieved by variation of the duty cycle of the layer pairs . This method is called double-chirping. For the angled stack in Fig. 2, we define the double-chirp coefficient
For κ=-1, e.g., a low-index λ B/2 layer is deposited and the high-index layer has zero thickness. κ=0 refers to symmetric λ B/4 layers. Finally, κ=1 corresponds to the situation of a 100% high-index duty cycle. Typically, a slow ramp from κ=-1 to 0 is used in the topmost 10 to 20 layers of a chirped mirror coating.
Solution of the differential equation (1) requires two boundary conditions, which are given in the form l(ω 1)=0 and l(ω 2)=l max and define the bandwidth of the coating Δω=|ω 2-ω 1|. The maximum available path length l max is typically dictated by manufacturing constraints. For the simplest case of a constant GDD, a solution lc is readily obtained
Once differential equation (1) is solved, one can directly compute the Bragg wavelength as a function of optical path length
where ω(l) is the inverse function of the solution l(ω) of Eq. (1). For simplicity, let us ignore double chirping of the coating in the calculation of the initial layer sequence. Then the ti’s are readily determined from the chirp law Eq. (5) using the recurrence
The index i numbers physical layer thicknesses ti starting from the interface to air, i.e. opposite to typical numbering conventions of coating manufacturers. In this paper, we have always surmised that the index of the top layer i=1 is low, as this simplifies the design procedure and also yields a wider bandwidth. Odd indices i therefore identify low-index layers, even indices high-index materials.
Equations (1–6) are a straightforward extension of the procedure outlined in  and can be applied to arbitrary incidence angles. Apart from the different chirp law required to design a chirped mirror at non-normal incidence the reflectivity is also modified. Assuming p-polarization, the Fresnel reflectivity r at an interface between materials with index n lo and n hi is now written as
A reduction of r requires an increase of the number of layers if the same reflectivity and bandwidth of the coating are to be achieved. With layer material data from , one calculates a change from r 0=0.23 at normal incidence to r Brewster=0.17 at Brewster’s angle, i.e. a 35% decrease of reflectivity per interface in the stack. On the other hand, however, Brewster-angled mirrors do not require AR matching sections, which in some broadband mirror designs require more than 15 layers . In typical broadband DCM coatings, the chirped mirror stack (without AR coating) consisted of 50 layers. Assuming that the decrease of reflectivity has to be compensated by a similar increase of the number of layers, one arrives at an estimate of about 70 layers for the Brewster-angled coating. The decrease of the Fresnel reflectivity is therefore widely compensated by the absence of matching sections. However, when Brewster-angled chirped mirrors are used to cover a wider bandwidth, then one also has to increase the number of layers. In the following we use 120 layers to cover 1.2 octaves rather than 70 layers for 0.7 octaves. These numbers may serve as a rough estimation for the numbers of layers to be expected for octave-spanning mirror coatings. An initial design computed with the procedure outlined in this section is shown in Fig. 3.
4. A coating design example for Brewster-angled chirped mirrors
Figure 3 shows a simple layer structure suitable for compensation of 40 fs2 dispersion per bounce (see Fig. 4). One bounce therefore compensates the second-order dispersion of a 1-mm path length in an optical glass. The layer thickness sequence follows a simple relationship with an added double-chirp section in the first 10 layer pairs designed according to . Doublechirping was found essential for the Brewster approach to work. The stack has a total thickness of 10 microns and consists of 120 layers. In the following, an incident angle of ϑB=56.2° was assumed (compare Fig. 1). The resulting spectral reflectivity, group delay (GD) and group delay dispersion (GDD) characteristics at Brewster incidence are depicted in Fig. 4. Even without computer-optimization, the coating already exhibits a high reflectivity over more than one octave together with a reasonably smooth GDD. The GDD ripple amounts to about 50 fs2 (rms) for the octave from 400–800 nm. The GD exhibits similar oscillations with an rms value of 2.5 fs. Tuning the incident angle away from Brewster’s angle dramatically increases these values. To further reduce the dispersion ripple at Brewster’s angle, an attempt was made to improve coating properties by computer optimization. The optimized coating and its properties are also shown in Figs. 3 and 4. By comparing with the unoptimized design, one can see that the coating now extends all the way up to 1000 nm, both with smooth dispersion and reasonably high reflectivity. Some bandwidth at 400–430 nm is lost during the optimization process. Still, the resulting coating exhibits an average reflectivity of 99.5%, a GD ripple of 0.5 fs (rms), and a GDD ripple of only 8 fs2 in an 1.2 octave bandwidth. This coating therefore covers the entire bandwidth from 430–1000 nm, i.e. nearly double the bandwidth of the broadest DCMs demonstrated to date.
Most remarkably, the basic structure of the layer sequence remains conserved during the optimization process. Close to the interface to air, the main effect in the optimization process seems to be a modification of the double-chirp section. The simple linear chirp in the duty-cycle is replaced by a more complex function with a strong oscillatory component in the first 40 layers close to the interface to air. This corrugated double-chirping serves to further suppress the dispersion oscillations. Additionally, changes at the bottom of the layer structure greatly increase the reflectivity of the coating beyond 800 nm.
The behavior of the GDD for varying angle of incidence (ϑ in = 50°–60°) is shown in the movie sequence Fig. 5. The calculated rms spread of the dispersion oscillations is also shown close to the left axis as an error bar. It can be clearly seen that even in this relatively small angular interval the reduction of dispersion oscillations is already dramatic. In addition, Fig. 6 contains a summary of a simulation over an even wider range of input angles. At normal incidence, the GDD ripple amounts to more than 1000 fs2. This value is reduced to 8 fs2 at Brewster incidence. At angles larger than Brewster’s angle, dispersion ripple starts to increase again.
Figure 7 further illustrates the concept of Brewster-angled chirped mirrors. Based on the simulations already discussed, we illustrate the effect of the dispersion ripple on shaping of an ultrashort optical pulse as a function of incident angle. For this purpose, we assume that an unchirped Gaussian-shaped pulse with a 140-THz bandwidth encountered a dispersion of +400 fs2, i.e. the equivalent of 1 cm of a light optical glass. Ideal recompression of such a pulse should yield the original pulse duration of about 3 fs. The mirror structure compensates for this material dispersion in 10 bounces. Dispersion oscillations cause imperfections of the recompression, which mainly manifest themselves as a broad pedestal structure but also widen the pulse at its half maximum. This is most dramatically seen far away from Brewster incidence, where the pulse extends over nearly a picosecond and exhibits many uncoordinated satellite pulses. The peak intensity is reduced to about 10% of the optimum value. Towards higher angles of incidence, the continuum structure appears more and more compressed until, at about 45° incidence, two GTI satellite pulses show up, located at ±60 fs delay from the center of gravity of the pulse. At 50° incidence, secondary GTI satellites become visible. In the vicinity of Brewster’s angle all satellite pulses are strongly suppressed and nearly the entire pulse energy is confined into a single pulse. Directly at Brewster’s angle, satellites are suppressed to below 10-4 of the main pulse. Finally, going to higher incident angles than Brewster’s angle, the isolated pulse rapidly decays into the temporal continuum structure, traversing the scenes already described in reverse order.
The characteristics of the pulse in Fig. 7 are also summarized in Fig. 6. From the simulations, one can conclude that a severe degradation of pulse quality sets in at about 5 degrees deviation from the nominal Brewster’s angle. At below 52° and above 58° incidence, the pulse is compressed to 5 fs rather than the optimum 3.5 fs at Brewster incidence. This is a surprisingly large window for the operation of Brewster-angled mirrors, which makes it clear that slight errors in the angular alignment on the order of one degree can easily be tolerated in this design approach.
5. Robustness of the method
Dispersion control with chirped mirrors generally requires very precise control of layer deposition accuracies. From previously reported examples, one can conclude that the demand on deposition accuracies is growing dramatically in the vicinity of an optical octave bandwidth [3, 13]. For conventional chirped mirrors, a strong increase of the sensitivity to growth errors has been found for the top layers of a coating, i.e. those layers that form the matching section. As the mirror stack has already reached a very high reflectivity during deposition of the last few layers, growth errors in these layers only cause tiny changes of the coating’s transmission. The increased sensitivity of the top layers is often accompanied by the presence of very thin layers. These thin layers are very difficult to monitor via the coating transmission. The simultaneous occurrence of poor monitoring conditions and an enhanced sensitivity of the top layers is the dilemma in manufacturing chirped mirrors.
In the following, we want to convince ourselves of the utility of the Brewster-angle design approach. For this purpose, we check the sensitivity of the coating parameters to random deposition errors. For a realistic estimation of the feasibility of such a coating, we account for difficulties in optically monitoring individual layers. For deposition of each and every layer of the optimized coating sequence in Fig. 3, the average transmission change per nm deviation from the nominal layer thickness is calculated. Averaging is carried out for the wavelength range from 300–1100 nm, i.e. the range covered by silicon photo detectors. These values are used as weights for computing otherwise randomly distributed growth errors. This means that the difficult-to-monitor top layers are varied more strongly in the simulations than the bottom layers of the coating.
Statistical deviations of the coating GDD from the designed value are shown in Fig. 8 for a wide spread of growth errors (0.01–1 nm). The picture is based on about 10000 calculations of the GDD of the optimized coating in Fig. 3. The coating GDD is relatively immune towards growth errors in the wavelength region below 500 nm. Generally, the sensitivity of the GDD increases with growing wavelength. Average growth errors of a few 0.01 nm would be required to exploit the full potential of the approach in terms of suppression of dispersion oscillations, but this is well out of reach even with the most advanced manufacturing capabilities. Average growth errors on the order of 0.1 nm or 0.2 nm appear to be at the verge of current state-of-the-art manufacturing capabilities [2, 15]. This deposition accuracy relates to the light blue shade in Fig. 8. The overall GDD of the mirror would still be well in the negative dispersion region, an indication for the practicality of this approach, which would even allow for slightly increased deposition errors.
Compared to back deposition approaches, the full dispersion of the coating can be used to compensate for material dispersion, as no cover slide has to be compensated. Rotating the mirror from Brewster’s angle to normal incidence, the spectral characteristics of the mirror are shifted to longer wavelengths as the path length scales with the cosine of the internal angle, compare Eq. (2). At normal incidence the high-reflectivity regions of the coating appear shifted towards longer wavelengths. This opens an additional window for monitoring coatings that extend well into the blue spectral range. Monitoring Brewster coatings at normal incidence is another decisive advantage of the new coating design approach.
A new approach to chirped mirror design is introduced. Use of chirped mirrors at Brewster incidence can strongly reduce dispersion ripple and avoid impedance mismatch problems at the interface to the ambient medium. We show by example that it is possible to compensate for group delay dispersion over a bandwidth of about 1.2 octaves whilst maintaining a high reflectivity. This is significantly more bandwidth than any previous mirror design technique allowed so far. It also exceeds the bandwidth of any other dispersion compensation technology, including prism or grating sequences and 4f -shapers, which are limited to one octave by overlapping grating orders.
The Brewster-angle approach may not be the universal solution to intracavity dispersion compensation, but it is excellently suited for extra-cavity compression of white-light continua. This compression is currently the most demanding application in terms of bandwidth and probably the only one that currently demands far greater than octave dispersion compensation. The layer sequences in the Brewster-angle design approach are structurally relatively simple, which results in a robustness of the approach towards growth errors. No complicated matching sections are required. Given current growth monitoring capabilities, reliable manufacturing of such mirrors appears promising.
Slightly modified designs can also be useful for the design of wideband dielectric mirrors at 45° incidence. Compared to normal incidence, the demands on an AR matching section would be strongly reduced for 45° incidence and p-polarization. A hybrid DCM-Brewster design may then serve to provide a broadband reflectivity with very smooth dispersion. Such mirror structures would be useful to extend the bandwidth of high power mirror designs, as they are used in Ti:sapphire amplifier chains. Quite generally, going to non-normal angles of incidence appears to be a promising new route and opens up new application areas of chirped mirrors.
The original coating design computer code, on which the simulations are based, was developed by N. Matuschek at ETH Zürich. The author further acknowledges G. Arisholm, U. Griebner, and N. Matuschek for carefully proof-reading the manuscript and their suggestions for some clarifications.
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