A recent discussion led me to understand how this topic can be confusing for a great many people. Hopefully the illustration below and the discussion that follows will help. There's a lot of misinformation and misconceptions out there. First, as the old saying goes, a picture (or three) is worth 1000 words:
Click for larger.
At the top panel A we see the absorption line spectrum of the Sun. Buried inside the H alpha absorption line of the photosphere, we can see there is the much fainter emission line produced by the chromosphere. For solar H alpha filters, we ideally want only to see the chromosphere emission line and none of the light intruding from the photosphere coming from outside the absorption line - aka parasitic continuum from the photosphere.
In panel B, we can see that the absorption line of the photosphere is only about 1.3 Angstroms wide. Beyond this +/- 0.65 A of the H alpha emission line, parasitic continuum begins to appear. Also shown in panel B is the transmission profiles of a single 0.7 A etalon filter, and the transmission profile of two such filters in series - aka "double stacking." Note that each filter basically has a Lorentzian distribution transmission profile, and the "tails" or "wings" near the bottom are much wider than the Full Width Half Maximum (the width of the profile at 50% transmission peak) - aka "bandpass" - of 0.7 and 0.5 A, which both reside well within the absorption line. The most important effect of double stacking therefore is not the reduction of the FWHM (it's actually irrelevant - if you could have a filter with a square wave transmission profile 1 A wide it would be perfect!); it's the reduction or suppression of the transmission profile wings or tails, which reduces the amount of parasitic continuum passing through the filter system.
In panel C, we can see the effects of this for H alpha observation and imaging – with both exposures “normalized” for the brightness of the prominences. On the Left, the single 0.7 FWHM filter has a lot of continuum from the photosphere passing through, and this "noise" decreases the contrast of the chromosphere's disc detail. Note sunspot detail is quite visible. Photosphere continuum is also evident as the "double limb" where the outline of the photosphere lies 2000 km below the general edge of the chromosphere, which is dimmer. On the Right, we see the effect of double stacking, which pushes the filter profile more towards a Gaussian distribution profile and suppresses the amount of parasitic continuum from the photosphere leaking through (note the decreased visibility of the sunspot detail and disappearance of the "double limb" of the photosphere), and greatly improves disc contrast, while having virtually no effect on prominence visibility.
However, an overall decrease in peak filter transmission does make double stacked images relatively dimmer, and why increased aperture and ERF/blocking filter modifications can be employed to increase overall filter system transmission and improve image brightness. Some additional issues arise with double stacking, including reflections between the two etalons that need to be addressed – usually via slight tilting one of the etalons; and with internal etalons, the additional collimator optics can lead to more scattered light and brighter backgrounds. A circular polarizer can be employed to remove this, but also decreases image brightness even further. For double stacking, I generally prefer two front mounted etalons, followed by a front and internal, and lastly two internal filters. Rear mounted filters are monolithic systems incorporating many filter elements and polarizers, and therefore are rather difficult to double stack – but this has been done, and is easiest with using a front etalon (without the need of the blocking filter).