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Astron. Astrophys. 340, 569-578 (1998)

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3. Interferometer specification

The most important issues of a multi-FPI spectrometer are the spacing ratios of the FPIs and the optical configuration, i.e. a collimated or a telecentric mount. The following two subsections describe the respective considerations and the solution adopted. The design goal was a dual FPI system, with the option to upgrade to a triple system. For TESOS we used two Queensgate ET-50 etalons with a clear aperture of 50 mm each. The etalons are coated for a wavelength range between 450 and 700 nm, with mean reflectivity of 94%. The surface quality is [FORMULA] before coating. Both etalons are driven by CS100 Controllers which stabilize the spacing and the parallelism via capacitance micrometers and piezo actuators. The servos operate in closed loop, eliminating non-linearity and hysteresis of the piezos.

3.1. Spacing ratios

The transmitted intensity for a single FPI as a function of the phase delay [FORMULA] between the plates is described by the Airy function

[EQUATION]

where R is the surface reflectivity, T the surface transmission and the coefficient F given by [FORMULA]. The phase difference [FORMULA] is given by

[EQUATION]

(In our case: µ=1, i.e. air between the plates). d is the plate separation, [FORMULA] the angle of incidence and [FORMULA] the wavelength. This leads to the typical transmission pattern with maxima of the order n, [FORMULA] for [FORMULA] and a free spectral range FSR, i.e. the spacing of two adjacent maxima, of FSR [FORMULA]. For large F the FWHM of a transmission peak [FORMULA] is given by

[EQUATION]

For the spectral resolution Res of a single FPI one obtains

[EQUATION]

with the finesse Fi, defined as the ratio FSR/[FORMULA] and commonly used to characterize the resolving power of etalons. Eqs. (3) and (4) are valid only for ideal etalon plates and pointlike sources. For details on the modification of the instrumental function and transmission line-width see e.g. Vaughan 1989. Taking into account possible transmission peak broadening mechanism like imperfect reflectivity and plate shapes, a value of 30 - 40 for the finesse is reached.

A combination of two (or more) etalons with different plate separations is used to enlarge the FSR. The resulting Airy function for a combined system is obtained by multiplying the individual functions. Near the common (or main) transmission peak at [FORMULA], the transmission function can be approximated by a Lorentz function [FORMULA] of the phase deviation [FORMULA] from the common peak:

[EQUATION]

[FORMULA] is the prefilter transmittance and [FORMULA] a phase difference describing the detuning of the prefilter transmission peak with respect to the common FPI peak at [FORMULA]; [FORMULA] to [FORMULA] denote the plate separations, [FORMULA] and [FORMULA] the spacing ratios of the etalon combination. (For details see Darvann & Owner-Petersen 1994). The spectral resolution for spacing ratios around 1 (Vernier ratio) is then given by

[EQUATION]

The resulting resolution is somewhat better than for larger spacing ratios, where the resolution is determined by the largest plate separation. However, for Vernier ratios one has to take more care of side lobes (ghosts) within the passband (see below).

For the largest plate separation we chose a value of 1.3 mm leading to a FSR of 0.1 nm (@500 nm). Together with a finesse of 40, we obtain a spectral resolution of at least 200.000. We calculated the Vernier spacing ratios of a dual and a triple system using the method described by Darvann and Owner-Petersen (1994). The criteria to optimize the spacing ratios are:

  • Max Ghost is the transmission amplitude of the strongest off band peak (ghost) of the combined FPI/prefilter system.

  • Stray light signal-to-noise ratio SNR is the ratio of integrated light from the main transmission peak of the system to the integrated light from all side lobes within the whole spectral range

Both parameters were combined to the single performance characterization value SNR/Max Ghost. Figs. 1 and 2 show examples for the calculation of a triple and a double FPI-system using a 1 nm FWHM prefilter for the triple system and a 0.3 nm FWHM prefilter for the double system. The complexity of the pattern decreases for increasing wavelength. A good solution at the blue end of our wavelength range therefore holds for all larger wavelengths. In the lower panel of Fig. 1 the brightest areas indicate good choices for the spacing ratios. To decide for the best spacing ratio within the variety of good combinations, we added three criteria:

  • Maximize the ratio SNR/Max Ghost for a slightly decentered prefilter passband,

  • Find a combination that works both for a double system and a possible upgrade of this combination to a triple system,

  • Optimize the spectral resolution for a double FPI system.

[FIGURE] Fig. 1. Max Ghost, SNR and SNR/Max Ghost for a triple FPI system with a 1 nm FHWM interference prefilter. Large values are bright, small values dark. The plate separation of the first FPI is [FORMULA]=1.3 mm. [FORMULA] is the spacing ratio [FORMULA] for the second, [FORMULA] the spacing ratio [FORMULA] for the third interferometer. The computation is made for a wavelength of 400 nm.

[FIGURE] Fig. 2. SNR/Max Ghost for a tandem FPI system with a 0.3 nm FHWM interference prefilter. Large values are bright, small values dark. The plate separation of the first FPI is [FORMULA]=1.3 mm. [FORMULA] is the spacing ratio [FORMULA] for the second etalon. The line plot corresponds to a wavelength of 400 nm. The [FORMULA] chosen for TESOS is indicated by a dotted line.

The first criterion is important, because any decentering of the prefilter causes a small change in the SNR/MaxGhost pattern. The FWHM of the prefilters for a double system is only 0.3 nm and even slight variations of the ambient temperature cause a passband shift. Fig. 2 is the equivalent to Fig. 1 for a double system. The value chosen for the double system is indicated in the Figure. The spacing ratios for a future upgrade of TESOS to a triple system are summarized in Table 1.


[TABLE]

Table 1. FPI spacing ratios of TESOS. FWHM(PF) is the required width of the interference prefilters.


3.2. Optical configuration

There are essentially two optical mounting possibilities which can be adopted for the design of an FPI spectrometer:

[FORMULA] Collimated (classical) mounting
The FPIs are mounted near the image of the entrance pupil of the telescope (Fig. 3). Every image point corresponds to rays with a specific angle with respect to the optical axis propagating through the FPIs. The effective plate separation is a function of this angle, which results in a wavelength gradient across the field of view (FOV). With respect to the center of the FOV, the wavelength of the transmission maximum near the edges will be blue-shifted.

[FIGURE] Fig. 3. Mounting concepts for FPI interferometers. In the collimated mounting, the telescope aperture (PS) is imaged into the interferometer(s), L2 is the reimaging lens. At the FPI location the beam is collimated. In the telecentric configuration, lenses L1 and L2 project the solar image (FS) into the interferometer. Lenses L3 and L4 image the solar image onto the detector (FP2).

Moreover, all rays distributed across the pupil image (and the interferometer plates) with equal inclination will form one image point. Therefore non-uniformity of the plate surfaces or parallelism errors will result in a broadening of the transmitted spectral profile. So the spectral resolution is not only given by the reflection of the interferometer coatings but also by the flatness and parallelism of the plates.

[FORMULA] Telecentric mounting
In the telecentric configuration the pupil image is collimated and the FPIs are located near the image plane. Additional optical components reimage the sun to the camera. At the location of the etalon(s) the beams which form the image have the same cone angle at each point across the field of view. The maximum ray angle within this cone is a function of the f-number of the telecentric optics (L1 & L2 in Fig. 3, which determines the FOV together with a given clear diameter of the FPI. To attain the calculated spectral resolution of the system, the angle of this ray cone has to be minimized.

Since every position on the interferometer belongs to exactly one point within the solar image, there are no systematic wavelength shifts over the FOV due to angle variations. The spectral resolution is given by the plate reflectivity and the f-number of the optical system (Fig. 4).

[FIGURE] Fig. 4. Resolution of an FPI in telecentric configuration as a function of the f-ratio. The dashed curve indicates the usable FOV for a 50 mm FPI. A and B mark the two fields of view used for TESOS.

To avoid image contamination due to dust on the interferometer plates or due to inhomogeneities within the coatings, the FPIs are shifted slightly away from the focal plane. Light from one image point now covers a small area (1 mm diameter) on the interferometer plates causing a small broadening of the transmitted profile due to small scale variations of the FPI-spacing (micro roughness). Large-scale variations of the plate shape or parallelism errors lead to small shifts of the transmission profile over the FOV. But these variations are usually much smaller than the wavelength dependence of the collimated mounting. The telecentric mounting allows (and requires) to trade spectral resolution against FOV. TESOS was built with two fields of view with different spectral resolution(Fig. 4):

  • A: FOV: 50 arcsec; spectral resolution 320.000,

  • B: FOV: 100 arcsec; spectral resolution 160.000.

The resolution numbers above refer to a wavelength of 500 nm. The reimaging optics following the interferometer set, provides a simple means to blocking ghost images by an aperture stop. In Table 2 we summarize the advantages and disadvantages of the two mounting concepts. Although collimated mounting offers a higher spectral resolution, we preferred the telecentric solution for TESOS. Collimated mounting suffers from the risk of deteriorating the image quality (Darvann & Owner-Peterson 1994).


[TABLE]

Table 2. Comparison between collimated and telecentric interferometer mount.


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© European Southern Observatory (ESO) 1998

Online publication: November 9, 1998
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