OSIRIS - Tunable Filter Basic Principles

In its simplest form, a Fabry­Perot filter (FPF) consists of two plane parallel transparent plates which are coated with films of high reflectivity and low absorption. The coated surfaces are separated by a small distance (typically m m to mm) to form a cavity which is resonant at specific wavelengths. Light entering the cavity undergoes multiple reflections (Figure 1) with the amplitude and phase of the resultant beams depending on the wavelength. At the resonant wavelengths, the resultant reflected beam interferes destructively with the light reflected from the first plate­cavity boundary and all the incident energy, in the absence of absorption, is transmitted. At other wavelengths, the FPF reflects almost all of the incident energy.

Performance of an ideal FPF

The general equation for the intensity transmission coefficient of an ideal FPF (perfectly flat plates used in a parallel beam) as a function of wavelength is:

where T is the transmission coefficient of each coating (plate-cavity boundary), R is the reflection coefficient , d is the plate separation, m is the refractive index of the medium in the cavity (usually air, m =1) and q is the angle of incident light. Thus, the FPF transmits a narrow spectral band at a series of wavelengths given by

where m is an integer known as the order of interference. The peak transmission of each passband is

where A is the absorption and scattering coefficient of the coatings (A = 1 - T - R); and the minimum transmission, halfway between the resonant wavelengths is

Therefore, the contrast between the maximum and minimum transmission intensities is

For a FPF contrast greater than 100, the reflection coefficient R of the coatings needs to be greater than or about 0.82.

The wavelenght spacing between passbands, known as the inter-order spacing or free spectral range, is about

which is obtained from Equation 1.2 by setting consecutive integral values of m. Each passband has a bandwidth (dl ), full width at half­peak transmission, given by

derived from Equation 1.1. The ratio of inter-order spacing to bandwidth is called the finesse

For an ideal FPF, it is given by

Thus, we can see that the resolving power of a FPF is equal to the product of the order and the finesse

Limitations

It is apparent from the above equations that to obtain a higher resolution for a given order or to obtain a wider inter­order spacing for a given resolution, the finesse needs to be increased. For a finesse greater than 100, a reflection coefficient R of greater than or about 0.97 is necessary (Equation 1.9). However, so far we have considered the ideal situation where the plates are flat and parallel, and the incoming light is parallel. In particular, Equations 1.1, 1.3-1.5, 1.7 and 1.9 refer to this situation using the subscript r to distinguish the results from a real filter. In practice, plate defects and the angular size of the beam limit the maximum finesse obtainable.

The effective finesse (N) is approximately given by

where N_r is the reflective finesse from Equation 1.9, N_d is the defect finesse (due to plate defects) and N_a is the aperture finesse (due to the solid angle of the beam).

The defect finesse

where d d is a length scale related to deviations from flat parallel plates. The exact details depend on the type of deviations (Atherton et al. [RD1]). A FPF manufactured with N_d~ 80 and a reflection coefficient of 0.97 (N_r~ 100) performs with a finesse of about 60.

The aperture finesse

where W is the solid angle of the cone of rays passing through the FPF. This equation is related to the l dependence on q in Equation 1.2. In terms of astronomical imaging, the effect of aperture finesse is negligible for most objects in the field of view of a telescope. For example, an object which is one degree across (in the collimated beam) imaged with m =50 has N_a ~ 500 according to Equation 1.13. A more relevant analysis to consider the change in central wavelength of the filter as the ray angle q is varied in Equation 1.2. For example, a change in ray angle from 1º to 3º produces a change of 0.1% in the central wavelength of the filter at any given order. Therefore, at high resolving powers (~ 1000), a FPF may not be truly monochromatic across a desired field of view. 

Gap­scanning etalons

In order to manufacture a tunable FPF, which can change the central wavelength for a given order, it is necessary to be able to adjust either the refractive index of the cavitym , the plate separation d or the angle q (as can clearly be seen from Equation 1.2). In a gap­scanning etalon, the plate separation can be controlled to extremely high accuracy.  In recent years, these etalons have undergone considerable improvements. It is now possible to move the plates between any two discrete spacings at very high frequencies (200 Hz or better) with no hysteresis effects while maintaining l /2000 parallelism (measured at 633 nm). The etalon spacing is maintained by three piezo­electric transducers as discussed below.

Piezo­electric transducers

Piezo­electric materials undergo dimensional changes in an applied electric field. Conversely, they develop an electric field when strained mechanically. Under an applied electric field, a piezoelectric crystal deforms along all its axes. It expands in some directions and contracts in others. The dimensional change (expansion or contraction) of a piezo­electric material is a smooth function of the applied electric field. The material is sufficiently stiff that piezo­electric transducers (PZTs) can respond on sub­microsecond time scales (Atherton [RD2]). The resolution is limited only by the precision with which the electric field can be controlled. For this reason, PZTs are commonly used for rapid switching and sensing, as indeed they are in the Queensgate etalons. However, all piezo­electric materials exhibit hysteresis, particularly in the relationship between the voltage applied and the amount of expansion. Thus, a servo­control system is required to tune the spacing between two plates to high accuracy.

FCapacitance micrometry

In a seminal paper, Jones & Richards [RD3] show that capacitance micrometry can be used to detect motions on scales as small as 10^-15m. Using this basic method, Queensgate Instruments have developed a capacitance bridge system to monitor parallelism and spacing of a Fabry­Perot etalon (Hicks et al. [RD4]). Information from the capacitance bridge is used to drive PZTs in a closed loop control system to maintain the parallelism and spacing. In Figure 3, we show the basic structure of a gap­scanning etalon. There are two x­channel and two y­channel capacitors, and a fifth ?reference? capacitor which monitors the spacing with respect to a fixed reference capacitor in the circuit. The two etalon plates can be kept parallel with an accuracy of l /200 for many weeks at a time.

Coatings

Reflective coatings are now laid down with ionic bombardment which allows for much higher integrity and uniformity in the coating response. After coatings are evaporated onto the plates, the flatness is typically l /140. Thus the defect finesse is about 70 (at 633 nm).

Reflectance phase of coatings

The analysis of Fabry­Perot filters in previous section does not take into account the wavelength­ dependent phase change inherent in reflections between the optical coatings on the inner­plate surfaces. Such coatings reflect the design wavelength (819 nm for the RTTF) with zero phase change, but incur a lead and lag elsewhere.

Atherton[RD2] re­writes the transmission function for a FPF (Eqn. 1.1) as

to take account of the phase change on reflection (el ). One effect of this is to shift the wavelength passbands (orders) at a given gap spacing, i.e., the equation for constructive interference (Eqn. 1.2) becomes

where n is an integer.

Apart from altering the positions of the passbands, the bandwidths (Eqn. 1.7) are altered if there is a dispersion effect (i.e., e l varies with wavelength). In this case, the full width at half­peak transmission is given by

To first­order (i.e., de l =dl is constant), the inter­order spacing is also altered by the same fractional amount and therefore the finesse remains approximately the same (Atherton [RD2]). The important effect, in terms of tunable­filter performance, is the change in resolving power. The resolving power can be written as

For the TTF, de l /dl is positive and, therefore, the resolving power is higher than predicted by non­reflectance­phase theory (Eqn. 1.10). Effectively, a FPF with a gap spacing of d acts like a FPF (of constant ffl - ) with a larger gap spacing:

The last term can be regarded as an effective gap­spacing offset. Note that this equation applies to the effective gap spacing in terms of resolving power, the wavelength peaks of the orders are governed by Equation 1.15.

 For the Blue TTF, the effective offset is not expected to be as high because of the l dependence in Equation 1.18. The reason for this lower dependence is that the Blue TTF operates in higher orders than the Red TTF, with the same gap spacing. This theory has not been rigorously tested.

The Taurus Tunable Filter (TTF)

The TTF, manufactured by Queensgate Instruments Ltd., has the appearance of a conventional Fabry­Perot etalon in that it comprises two highly polished glass plates. Unlike conventional Queensgate etalons, the TTF also incorporates very large piezo­electric stacks (which determine the plate separation) and high performance coatings over half the optical wavelength range. The plate separation can be varied between about 2m m and 12 m

The TTF is used in the collimated beam of the TAURUS­2 focal reducer available at both the 3.9­m Anglo­Australian (AAT) and 4.2­m William Herschel (WHT) telescopes. It has largely removed the need for buying arbitrary narrow and intermediate interference filters, as one can tune the bandpass and its centroid by selecting the plate spacing. The spacing of the plates is controlled to extremely high accuracy with capacitance micrometry. This approach to tunable imaging has existed for about 20 years, although TTF is the first of its kind in terms of both wavelength and bandpass accessibility.

The highly polished plates are coated for optimal performance over 370-960 nm using two ?arms? (separate etalons). The coating reflectivity determines the shape and degree of order separation of the instrumental profile. This is fully specified by the coating finesse, N, which has a quadratic dependence on the coating reflectivity. The TTF was coated to a finesse specification of N = 40 which means that the separation between periodic profiles is forty times the width of the instrumental profile. At such high values, the profile is Lorentzian to a good approximation. For a given wavelength, changes in plate spacing, d, correspond to different orders of interference, m. This in turn, dictates the resolving power (mN) according to the finesse.

Charge shuffling

Central to almost all modes of TTF use is charge shuffling. Charge shuffling is movement of charge along the CCD between multiple exposures of the same frame, before the image is read out. An aperture mask ensures only a section of the CCD frame is exposed at a time. For each exposure, the tunable filter is systematically moved to different gap spacings in a process called frequency­switching. This way, a region of sky can be captured at several different wavelengths on the one image. Alternatively, the TTF can be kept at fixed frequency and charge shuffling performed to produce time­series exposures.

The TTF plates can be switched anywhere over the physical range 2 to 12 m m at rates in excess of 100 Hz, although in most applications, these rates rarely exceed 0.1 Hz. If a shutter is used, this limits the switching rate to about 1 Hz. Charge on a large format CCD can be moved over the full area at rates approaching 10 Hz: it is only when the charge is read out through the amplifiers that this rate is greatly slowed down. The TTF exploits the ability of certain large format CCDs to move charge up and down many times before significant signal degradation occurs. In this way, it is possible to form discrete images taken at different frequencies where each area of the detector may have been shuffled into view many times to average out temporal effects

Order sorters

A Fabry­Perot Filter clearly gives a periodic series of narrow passbands. To use a FPF with a single passband, it is necessary to suppress the transmission from all the other bands that are potentially detectable. This is done by using conventional filters, called order sorters because they are used to select the required FPF order.

For the TTF which has a finesse of around 40, at low resolution (l /d l = 300), conventional broadband UBV RIz filters suffice. At high resolution (l /d l = 1000), eight intermediate band filters are used to sub­divide the wavelength range of each arm.

Resolutions and wavelength ranges

Ideally, we would like one TF which has a range of resolving powers from at least 300 to 1000 at all wavelengths from 365 nm to 1000 nm. However, high reflectivity on the coated surfaces is hard to achieve across such a large optical range. The full wavelength range can instead be covered by two tunable filters, a ?red arm? (ROTF) and a ?blue arm? (BOTF) of the OSIRIS­TF system.

The range of resolving powers is determined by the gap scanning range (variation of the plate separation (d), the reflectivity (R) the defect finesse (N_d) and there is also some effect from tilting the TF (we shall assume normal incidence [q = 0] in this section 3.1).< /p>

The gap scanning range determines the orders that are observable for each wavelength; for an air cavity (m = 1), the order

It would be desirable to have large dynamic range in to enable a large range in resolving power. The range in d is set by the technology of the TF to be about 2-12 m m. This gives observable orders from 4 to 24 at 1000 nm and from 11 to 65 at 365 nm.

The resolving power is equal to mN where N is the effective finesse which is determined by the defect finesse and the reflective finesse (ignoring aperture finesse; see Section 1.2). The defect finesse will be proportional to the wavelength (Equation 1.12), so we can write it as

where a _d is the ?spectral defect finesse? with units of 1/wavelength. The spectral defect finesse can be greater than 100 m m^-1 in Queensgate etalons.

The overall finesse is then given by

Defects reduce the efficiency as well as the overall finesse of a FPF. The efficiency can be defined as

where the first term is due to defects (Bland & Tully [RD5]) and the second term is due to absorption and scattering (Equation 1.3; the coefficient is the efficiency loss for a single reflection). The defect finesse needs to be more than about twice the reflective finesse for efficiency greater than 0.9, in the absence of significant loss due to absorption and scattering.

In order to determine the coating reflectivity, we shall set four requirements for the OSIRIS TF:

mN < 300 for the lowest order with d ³d_min

mN ³ 1000 for the highest order with d £ d_max

e ¤ 0.8 (Equation 3.4);

N ¤ 25 (equivalent to C ¤ 250 determined for an ideal FPF using Equation 1.5).

Requirements 1 & 2 relate to resolving powers (300 to 1000), while 3 & 4 relate to FPF performance (the efficiency and, the contrast between the maximum and minimum transmission intensities). Requirement 4 is also related to the ratio between inter­order spacing and resolution (Equation 1.7). Figure 6 shows the region of reflectivities that are set by these requirements for a TF with gap scanning from 2m m to 12m m, a spectral defect finesse of 110m m^-1 and an absorption and scattering coefficient of 0.002. Below 410 nm, Requirement 3 or 4 must be relaxed ora_d must be higher. A higher a_d pushes the boundary for Requirement 3 up and slightly lowers the boundary for Requirement 4.

Assuming constant reflectivity for the arms of the TF. This means that we want R ~ 0.91 for the blue arm and R~ 0.94 for the red arm. Figure 5 shows an example of the resolving powers versus wavelength for such a TF. The blue arm goes from 365 nm to 650 nm, while the red arm goes from 600 nm to 1000 nm.

In fact, the reflectivity is not the same across the bandwidth of a ?constant reflectivity coating?. The RMS­variation of reflectivity with wavelength increases with bandwidth (thus the need for two arms) and it increases with decreasing mean reflectivity. Therefore, it may be better to have higher mean reflectivities than 0.91 and 0.94 in order to reduce the RMS­variation across the wavelength range of each arm. This variation of reflectivity needs further investigation before deciding on the specifications of the coatings (mean reflectivity and wavelength range). As can be seen from Figure 6, it could be desirable to have an increasing reflectivity with wavelength. Coatings which produce this effect could also be investigated.

Etalon specifications

The manufacturer of the TF is Queensgate Instruments Ltd. Two etalons and two `CS100 controllers' are needed for OSIRIS. ET70 and ET100 etalons (sizes in mm) were recommended as possible choices by Chris Pietraszewski . The ET85 was not recommended because of problems obtaining plate flatness. In a recent email correspondence, Chris Pietraszewski wrote "All the ET85 etalons that we have built in the last 5 years have been troublesome and could end up being marginal for the TF work. We do not know why this should be but I would think carefully before attempting low­gap flat etalons in this size."

Table 1 show the main optical specification for the OSIRIS etalons.

Table 1 OSIRIS Tunable Filter optical specification

Etalon Material	fused silica
Etalon clear aperture	100 mm of diameter
Gap scanning range	between 2 m m and 12 m m
Inner plate reflectivity for TF blue	0.91 ± 0.005 between 365nm to 670 nm
Inner plate reflectivity for TF red	0.94 ± 0.005 between 620 nm to 1000 nm
Outer plate AR coatings	less than 0.01
Plate flatness	l /200 at 633nm
Coating flatness	l /140 at 633nm

Etalon finesse

The finesse is the parameter that defines the performance of a real etalon. In this section will be analysed the finesse for OSIRIS TFs for obtaining the etalon characteristics in the following sections.

Reflection finesse.

The reflection finesse is defined by the equation 1.9.

For the blue OSIRIS tunable filter (BOTF), which has a reflection, specified of 91%, the reflection finesse will be 33.3. For the red OSIRIS tunable filter (ROTF), which has a reflection, specified of 94%, the reflection finesse will be 50.8

Defect finesse.

The defect finesse is defined by the equation 1.12. The deviations from flat parallel plates for OSIRIS tunable filters are limited by the coating flatness specification. In this way the defect finesse at l =633 nm will be 70 for both TFs, but this property have a wavelength dependence as well the coating flatness specification. This dependence is showed in equation 3.2 where the spectral defect finesse for the OSIRIS tunable filter is a = N_d / l = 110 nm^-1. With the spectral defect finesse is possible to know the defect finesse for whole spectral range. Figure 7 and Figure 8 show the defect finesse for OSIRIS TFs.

Figure 7 Defect finesse for BOTF.

Figure 8 Defect finesse for ROTF

Effective finesse.

The effective finesse is defined by the equation 1.11. As mentioned in section 1.2 the aperture finesse is negligible for most astronomical objects and observational modes. Then the effective finesse have a contribution from the reflective finesse and from the defect finesse. Figure 9 and Figure 10 show the effective finesse for OSIRIS TFs.

Figure 9 Effective finesse for BOTF

Figure 10 Efective finesse for ROTF

Resolution.

Knowing the effective finesse, the resolution power can be obtained using equation 1.10. Figure 11 shows the range of resolution power for the Blue and Red OSIRIS tunable filters in they defined spectral ranges. Note the spectral range overlapping in order to have the Ha region available for both TF with a comparable performance.
 

Figure 11. Resolving power.

The resolving power is the ratio of the central wavelength to the passwidth (equation 1.10) and this ratio depends with the order. It is useful to know what is the spectral range and the passband range available in an order. Figure 12 and Figure 13 show theses parameters for both OSIRIS TFs. In these figures, the lines represent the position of the wavelength of maximum transmission for each of the orders (number in red), the Y axis shows the plate separation and numbers in the corners of the lines are the bandwidth for extreme wavelengths for each of orders. Tables for different wavelength with detailed information are included in Annex B.

Figure 12 Wavelength of peak transmission for orders in the BOTF. Numbers in red are the selecting order, and numbers at line extremes are the full width in nm at that point.

Figure 13 Wavelength of peak transmission for orders in ROTF. Numbers in red are the selecting order, and numbers at line extremes are the full width in nm at that point.

Efficiency

The TF efficiency can be obtained using equation 3.4. Assuming a constant reflectivity, the efficiency has dependence with the wavelength through the finesse but not wirh the observing mode. Both OSIRIS TFs have an efficiency around 80%. Figure 14 and Figure 15 show the efficiency for OSIRIS tunable filters

Figure 14 Efficiency for BOTF.

Figure 15 Efficiency for ROTF

Contrast

 

Equation 1.4 defines the contrast for an ideal tunable filter. For a real tunable filter the equation for the contrast becomes

From this expression it is possible obtain Figure 16 and Figure 17 for the OSIRIS TFs contrast. < p>

Figure 16 Contrast for BOTF

Figure 17 Contrast for ROTF

Tuning errors. Stability.

The wavelength tuning error of OSIRIS TFs shall be less than 10% of TFs bandwith. The tuning error is directly related with error in the etalon plates separation. Following equation 3.1

then

Where d d is the maximum error allowed for the TF plate separation. As N is related to the wavelength, the allowing error will be a constant for both etalons, it can be seen in Figure 18.

The plate separation maximum error is due a three error contribution:

Tuning resolution

Temperature stability

Electronic stability

Figure 18 Plate separation acceptable error for OSIRIS tunable filters.

Gap scanning

The plate spacing has an upper limit of about 12 m m set by the long­range piezo stacks. The lower limit is nominally zero, except moving the plates too close to each other could damage the coated surfaces due to dust particles. Therefore, a lower limit of about 2 m m is set in practice. The tuning resolution is 0.49 nm.
 

Electronic and environmental stability

For the CS100 controller, the electronic drift induced in the cavity spacing of an etalon is 0 ± 50 pmK^-1. The cavity displacement from electronic noise is 10 pm Hz^-1/2 RMS.

The best reported stability of an etalon system, over a period of weeks, was 1 part in 10⁸ with the etalon held in a sealed cell to within ± 0.01 K and the CS100 held to within ± 0.1 K of a fixed temperature. The etalon was operating at 643.8 nm with a cavity spacing of 5.000 mm (1 part in 10⁸ of this spacing is 50 pm).

The observational temperature range for OSIRIS are between -2ºC and 19ºC. A total range of 21ºC can be done in an observational run, in this case a drift of 1.0 nm in the cavity spacing could be induced. Following GTC requirements, the temperature change in a hour is 1.9ºC, in this contitions a drift of 0.01 nm in the cavity spacing could be induced.

Change in central wavelength

The primary goal of a tunable filter is to provide a monochromatic field over as large a detector area as possible. With the present TTF, however, the field of view is not strictly monochromatic. The effect is most acute at high orders of interference. Figure 19 a) shows how wavelength gradients (or phase effects) are evident from a ring pattern of atmospheric OH emission lines across the TTF field. In this particular case we see rings at different wavelength appearing within the one order. The circular pattern is not centrally aligned due to tilting of the plates (~10 to 15^o) to deflect ghost images from the beam.

Wavelengths are longest at the centre and get bluer the further one moves off-axis. For instruments such as TTF, the wavelength as a function of off-axis angle q (see Eq. 1.2). This change in wavelength, compared to the central wavelength when q = 0º,can be written as

where q _sky is the angular distance on the sky away from the central axis of the etalon. For OSIRIS, the wavelength changes are

Table 2 Wavelength changes in OSIRIS

 

fcoll	1.20 m
d l /l (q sky = 1 arcmin)	-0.08%
d l /l (q sky = 2 arcmin)	-0.34%
d l /l (q sky = 3 arcmin)	-0.76%
d l /l (q sky = 4 arcmin)	-1.36%

Thus, we can see that an FPF is not truly monochromatic across an eight-arcminute-diameter field of view even if the etalon is not tilted.

Figure 19 Images showing the removal of the atmospheric OH emission lines from a raw TTF image before (a) and after (b) cleaning. Only half the full TTF field is shown in each case.

For a filed view with |d l/l | less than a certain tolerance, the angular diameter is proportional to f_cclland the solid angle is proportional to f²_coll

Following the definition given by Bland-Hawthorn & Jones [RD6] the monochromatic field will be the size of the Jacquinot spot, the central region of the ring pattern. In this region the wavelength changes less than the etalon bandwidth which verify:

Where N is the effective finesse of the etalon.

Then, the monochromatic field is a region which subtent a angle j that can be written as

For a particular etalon, the size of the Jacquinot spot depends on order m alone. Eqn. (4.7) shows how the spot covers increasingly larger areas on the detector as the filter is used at lower orders of interference. The absolute wavelength change across the detector remains the same, independently of order. However, its effect relative to the bandpass diminishes as m decreases.

Figure 20 Monochromatic field for the BOTF at wavelength =372.7nm, working in order 11 and a tilt of 0 degrees.

Figure 21 Monochromatic field for the BOTF at wavelength =372.7nm, working in order 11 and a tilt of 5 degrees.

Figure 22. Monochromatic field for the BOTF at wavelength =372.7nm, working in order 11 and a tilt of 10.5 degrees.

A tilt of the etalon produces a translation of the monochromatic field along the detector. Figure 20, Figure 21 and Figure 22 show the monochromatic field for a particular etalon with different tilt angles. The etalon is tuned at the same wavelength and the same order. The monochromatic field drift on the detector it is evident in theses figures.

Vignetting limit and tilting angles

In order to avoid ghosts the TF could be allowed to tilt between 0º and about 20º. The minimum requirement for the TF is only about 10º so that most ghost images are reflected out of the field. Usually the TF will only be tilted to angles of a few degrees.

An accuracy of about 1º for tilting of the TF and the order­sorter filters is sufficient

Vignetting study

To calculate the minimum diameter of a FPF that is needed to transmit light without vignetting, we need to consider the tilt angle, the beam diameter and the angular size of the field in the collimated beam. Figure 23 shows a tilted FPF, the critical rays are those coming from the edge of the field of view in the opposite direction to the tilt. Various angles and lengths are defined in the figure: g is the tilt angle; f is the half­angle of the collimated beam,

where q _v is the angular diameter of the field on the sky; b is the beam diameter and; D is the aperture diameter and L is the length of the etalon. For the critical light rays coming from the edge of the field of view, q₀ is the incidence angle on the etalon, q₀ = f + g , and q₁ is the refracted angle in the glass, n₁sinq₁ = n₀ sinq₀. If we assume that the vignetting limit is reached at both ends of the etalon (as in the figure), then

where

Therefore, to avoid vignetting

with 
 
 

To determine the minimum diameter for various collimators, we consider an etalon which is 60 mm long and consider the beam diameter to be the long pupil dimension. Table 3 shows the minimum aperture diameter required to avoid vignetting of five­ and eight­ arcminute­diameter fields of view at various tilt angles. To avoid ghost images arising from internal reflections within the system (Bland & Tully [RD6]), it is desirable to allow tilt angles upto at least the half­angle of the collimated beam (f )The bottom row in Table 3 refers to this condition.

If the short pupil dimension is used for the beam diameter, the required aperture diameter is reduced by about eight percent. This would be a reasonable limit to take because the vignetting would only be marginal at the edge of the field. On the other hand, if the pupil is not strictly in the middle of the etalon (it could be positioned at the coated surface which is off­centre), or the etalon is longer than 60 mm, then the required aperture diameter is increased.

The clear­aperture diameter is guaranteed to be 100 mm by Queensgate. However, the surfaces (outside and inner­plate) are coated over the glass­aperture diameter of 105 mm in order to allow for edge effects. Should we take D in Equation 4.11 to be 100 or 105 mm? If D is taken as 105 mm then there will be edge effects for some light coming from the edge of the FOV. These edge effects will occur with the AR coatings on the outside surfaces but will not occur on the inner­plate surfaces. Therefore, there will be some minor loss of light because the anti reflection is not performing ideally in the edge region of the aperture. The resulting loss will be far less than true vignetting of the collimated beam.

Calibration

The calibration of the TF is made using arc lamps. The TF is gap scanned, through at least one free­spectral range, and the charge is shuffled between each exposure. In this way, the spectrum of a lamp is obtained and the gap scanning is calibrated.

The lamp requirement for tunable filters is stringent. These are traditionally located near the A&G box out of the direct view of the instrument path. A diffuser disc is placed across the light path so as to redirect the calibration source towards the instrument in such a way as to simulate crudely the telescope f/ratio. The diffuser disc diameter is matched carefully to the size of the convergent beam at the position of the disc.

Much the best way to achieve proper diffusion is with an integrating sphere around the calibration source placed directly in the beam, and a condensing lens which focuses the light along the optical path. This has the added advantage that low power lamps produce a much greater flux than in the traditional off­axis arrangement.
 

Tunable filter considerations
 

It is important that the lamp be properly diffused over the tunable filter entrance aperture, and that the lamp provide 3 - 5 emission lines for the intermediate-band order sorters. For the broadband (e.g., u?g?r?i?z?) order sorters, we require a similar number of lines although 2 - 3 bright lines tends to suffice. Typically, a CuAr lamp can be used in the r?i?z? region, a Neon lamp in the g?r? region and a Hg lamp in the u?g? region.

By virtue of the smaller wavelength, the blue tunable filter is used at smaller plate spacings to achieve moderate resolving powers. If the light is not properly diffused, the calibration spectrum shows the hallmarks of phase reflectance (Jones & Bland­Hawthorn[RD3]), i.e., spurious phase shifts. This arises from the strong angular dependence of the coating reflectivity at small plate spacings.
 

TF order sorters

Using an order­sorter filter and the TF etalon, single­order observations can be made at any wavelength, within the range of the filter, as long as the inter­order spacing is larger than the bandwith of the filter. Also, observations can be made with smaller inter-order spacings if the order is close enough to the central wavelength of the filter such that adjacent orders are outside the range of the filter.

The inter­order spacing is given by Equation 1.6. Figure 24 shows the range of these spacings for the two­arm TF (R = 0.91 and R = 0.94), while Figure 25 shows the transmission profile at two different plate separations, 2 m m and 12 m m. The spacings can vary from 6 nm (around order 65 at 365 nm [d = 12 m m]) to 200 nm (between order 5 at 800 nm and order 4 at 1000 nm [d = 2m m]). For resolving powers between 300 and 1000, the spacings can vary from about 10 nm to 150 nm.

Standard UBV RI filters or u, g, r, i, z filters (the SDSS set) suffice as order sorters for some low resolution observations (l /d l £ 300). In Figure 24, the approximate bandwidths of UBV RI filters and sixteen selected intermediate­band filters (with l /d l ~ 18 for the red arm and ~ 28 for the blue arm) are shown. Wavelength gaps of about 2% have been left between the intermediate­band filters because they can be tuned blueward by tilting. A tilt angle of 11.5º corresponds to a change in wavelength of ­2% for an interference filter. Figure 26 shows the maximum observable resolving power versus wavelength for the set of sixteen filters, allowing for tilt angles of upto 12º.

For the red arm, there is a series of seven filters with bandwidths between 32 nm and 56 nm; and for the blue arm, there is a series of nine filters with bandwidths between 13 nm and 22 nm. More are required for the blue arm because of the lower finesse. These filters are adequate for resolving powers of upto at least 800. At resolving powers of 1000, wavelengths near the edges of the order­sorter filters cannot be observed. To allow for resolving powers of 1000 at all wavelengths, either, more filters need to be used, or, the finesse needs to be higher. The problem with increasing the finesse is that the efficiency will decrease and low resolving powers (below 300) may become impossible.

The final selection of order­sorter filters will depend on the coating reflectivities and wavelength ranges of the TF arms, which are still to be confirmed. Additionally, there is also the consideration of sky emission lines. It could be desirable to chose the order­sorter filters so that they fall inbetween the wavelengths of strong sky lines. Other points to consider are the range of tilting and the probable transmission profiles for each filter.

Barr Associates is a manufacturer of filters. The total cost of TF order­sorter filters may be close to US$100 000. For an extra US$200 dollars per filter, special AR coatings for the wavelength range of the filter could be applied to both sides, giving reflectivity of less than 0.2%. This is important for reducing ghost images.

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Last update August 5, 2005, by Héctor Castañeda