Lasers that produce femtosecond pulses require dispersion-controlled mirrors. Several researchers have reported on thin-film design methods for these mirrors that are similar to TMDs that produce a stopband. These mirror designs have increased bandwidth over quarter-wave stacks and phase control of the reflected group delay, or GD [also group-delay dispersion (GDD) or wavelength-dependent GDD]. For example, SzipÃ¶cs et al. reported the use of a monotonically varying quarter-wave stack (starting design) that is optimized to produce a required GDD as a function of wavelength. This monotonically varying stack is referred to as a chirped coating and is similar to progressive series stacks (see Sec. 2.3.2). Tempea et al. reported the use of a layer-thickness modulation method for chirped mirror coatings. This method generates a starting design with improved GDD by chirping the period of the modulated layers while simultaneously increasing the modulation amplitude. In this case, less optimization is required to adjust or smooth the GD of the starting design to target values. Matuschek et al. reported on double-chirped mirrors where more sophisticated analytical design methods produce starting designs that are very close to the desired reflectance and GD. Here, double-chirped starting designs have several features: independently chirped, low and high refractive-index layers; specific groups of layers chirped for impedance matching; and an impedance-matched, broadband antireflection (AR) coating placed on top of the mirror coating to further suppress oscillations in the GD. This starting design requires minimal optimization to improve its performance.
Southwell reported on the design of rugate filters using wavelet design methods to increase the bandwidth of the stopband. A wavelet is defined as a fully apodized sine wave refractive-index profile (see Fig. 1.6). This method increases the bandwidth of the resulting stopband by using several overlapping wavelets. These wavelet designs are free of stopband harmonics and reflection ripples in the passbands. Note that consideration of GD is not part of this design method.
In all of the above design methods, the objective is to accomplish one or both of the following: (1) to widen the bandwidth and increase the reflectance of the first-order stopband; and (2) to achieve the desired GD (and GDD) versus wavelength. This chapter starts by examining chirped TMDs from a more general perspective, like Chapters 2 and 3, but for only the spectral properties of multiple stopbands. Next, the GD of a chirped-TMD design is briefly discussed. This chapter concludes by considering a half-modulation TMD function and its effect on stopband positions and GD performance. The limitations of both methods are also addressed.
4.1 Chirped TMD Modulation
First, the spectral performance of chirped TMDs is evaluated for multiple stopbands and for the bandwidth of the produced stopbands.