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RESULTS OF THE OSIRIS COMMISSIONING

• LATEST NEWS
• OSIRIS Users Manual available
• OSIRIS cryostat repair status
• Final Call for ESO-GTC proposals
• New grisms for OSIRIS
• More about Zeropoint
• OSIRIS_MasterFlats_BroadBand_Imaging
• Field Of View (FOV) of OSIRIS
• OSIRIS CCDs Gap New
• Osiris CCDs
• Detector Quatum Eficiency
• Detector Setup
• Tunable Filters Optical Center New
• OSIRIS Grisms
• Sloan Filters of OSIRIS
• OSIRIS sort-order Spectral filters
• OSIRIS Interference filters
• Tunable Filter Widths New
• Fringing New
• Zeropoints
• Flexure New
• Calibration arcs

Latest News

15^th Jul 2011
Subject: OSIRIS Users Manual available

It is with pleasure that we can now announce the long-awaited users manual for OSIRIS. The first version of the OSIRIS Users Manual can be found here.
Although the manual will in further versions be improved and extended with information about the new observing modes that are still to be commissioning, we expect that in its current incarnation it already represents a useful document.

Comments and suggestions are most welcome.

1^st June 2011
Subject: OSIRIS cryostat repair status

The cryostat of OSIRIS has had a difficult youth, with serious problems affecting the day-to-day operation of the instrument and requiring continuous baby-sitting to ensure that temperature and pressure remained within operational bounds. This was an untenable situation and hence about a year ago a project was initiated to plan an intervention to repair the cryostat. With the successful completion of this work we can now put this bad episode behind us and look forward to a more mature exploitation of the OSIRIS instrument.

Early users of OSIRIS may remember the problems with high dark current and occasional loss of vacuum. Following several months of testing and modeling to identify the problem it was concluded that high thermal conductivity between the CCD and the outside world was likely the main culprit. A possible vacuum leak was also suspected. The planning and testing was complicated by the fact that throughout the process leading up to the intervention the engineers had to carry out their work while the cryostat was operational, limiting the scope and duration of any test that could be done.

In May the final intervention was carried out, dismantling the guts of the cryostat, including the CCD itself, the cold link to the liquid Nitrogen container, and, a very critical part, separating the lens that acts as a cryostat window from the CCD assembly. The mechanical tolerances being very small for the excellent image quality of the OSIRIS instrument meant that much effort had to be expended on modeling and on building a special tool to ensure that the re-assembly would reproduce the original image quality.

The thermal insulation between the CCD package and the exterior was much improved by replacing materials at the contact points with the external parts of the cryostat. In the process, the CCD surface was gently cleaned, removing some of the particles that had accumulated during the early life of the cryostat.

The work went according to plan and the objectives were fully achieved. The CCD temperature is now stable and the cryostat holds its vacuum over long periods of time. The CCD is currently being characterized again and is back in service.

We are grateful to our colleague engineers from the Isaac Newton Group of Telescopes for their assistance in the project. This work was made possible thanks also to the outstanding work carried out by Kevin Dee from Engineering & Project Solutions Ltd.

Custom-built allignment tool that was used to register the exact location of the CCD inside the cryostat, and to reproduce its positioning after the intervention to a tolerance of a few microns.

Picture of the bare CCD package during the intervention

9th Mar 2011
Subject: Final Call for ESO-GTC proposals

A final call has been issued by ESO for time under the ESO-GTC agreement. Details on how to apply may be found at the following URL: http://www.eso.org/sci/observing/phase1/lbn.html

15th Dic 2010
Subject: New grisms for OSIRIS

Five new grisms for OSIRIS have been taken into operation. These are grisms R2500U,V,R and I and R2000B. These grisms provide important and much-demanded medium-resolution capability for the instrument. Initially the grisms suffered from a quite strong ghost image of the spectrograph slit, but this has been corrected by adapting the baffling of the optics. This has been a delicate and precise task in order to avoid any vignetting that could be detrimental to the throughput. The ghosting has been nearly fully removed; what remains is a very small fraction of the total flux of the object that normally will have a negligible impact on the quality of the spectra. The efficiency of the grisms is shown to be good (R2500U and R2500I are awaiting on-sky testing). Technical details will be posted and updated on the OSIRIS instrument web page as they are obtained. As an example, below these lines you see a image and the extracted spectrum taken with the R2500 grism of a planetary nebula.

30th Sept 2010
Subject: More about Zeropoint

Daily monitorizing of the aforementioned OSIRIS broadband zeropoints yields to a decrease of 0.5-0.7 mag due to presence of thick cirrus (spectroscopic nights), that shows the excellent sensitivity of the instrument even in bad observing conditions. Unusual presence of dust in the atmosphere above the observatory also produces a slightly decrease of this sensibility, but typically no larger than 0.2-0.3 mag respect to the standard zeropoint values.

27th Sept 2010
Subject: OSIRIS MasterFlats BroadBand Imaging

1. The flat fielding homogeneity in each of the OSIRIS Sloan filters is better than 2.5% over the full unvignetted FOV of the instrument, except in Sloan u', where fluctuations up to 5-6% respect to the mean value (caused by the filter coating) are found as measures from many twilight flat fields.
2. Day to day fluctuations in the flatfields are less than 0.05% , and less than 0.1% week to week. Hence, Sky Flat fields obtained with OSIRIS are well usable up to within a week before or after the observations.
3. Comparisons with SuperFlats derived from GTC scientific observations during Bright time (Sky background values around 15,000 - 20,000 ADUs on average) show no variations with respect to the standard Sky Flats up to such a low level as 0.01%, hence they can be considered practically identical for scientific purposes.
   These percentage variations are measured globally, while of course locally, due to dust particles that can come and go, the variations may be larger. Moreover, differences between the night-sky and the twilight spectrum may result in suble flat fielding differences.
4. By combining more than 200 Sky Flats in each filter, obtained during the GTC scientific operations during July, August and September 2010, we have constructed a series of MasterFlats frames that can be retrieved here. Flat fields were all obtained by using the GTC automatic sequence for SkyFlats, that allows to get a series of flats with exposure times always larger than 1 s (to minimize possible photometric effects due to OSIRIS shutter) and a maximum exposure time of about 20 s (where the detection of stars is notable), with an average of 35,000-40,000 ADUs in each individual image. Rejection parameters were chosen accordingly to use 150 flatfields in each filter, except in Sloan u', where only 75 were used. MasterFlats are available separately for each CCD of OSIRIS (as they have a slightly different gain and bias level).
5. Comparisons between Sky Flats and Dome Flats taken with OSIRIS show that these latter are not as good as Sky Flats for the photometry, due to inhomogeneities in the GTC Dome illumination. Differences up to 10-15% are found in CCD2, although they are as small as 2% in CCD1. Therefore Dome Flats are only recommended for obtaining reliable OSIRIS photometry in CCD1 and as last choice in CDD2. In any case, GTC policy of taking FlatFields in a regular basis produces that a series of Sky Flatfields will be always available within a week of any scientific observation, hence Dome Flats will be unecessary.

Master Flats

• MasterFlat u'
• MasterFlat g'
• MasterFlat r'
• MasterFlat i'
• MasterFlat z'

17th Mar 2010
Subject: OSIRIS CCD Dark Current status

Recently a new cryostat for the OSIRIS detectors was taken into use with the aim to address a number of problems, the most critical of which for regular science observations was the very high dark current resulting from an excessive temperature of the CCD that was not correctly reported by the CCD thermometry system. A redesign of the thermal coupling between the liquid Nitrogen container and the CCD has resulted in a notable improvement of the dark current, which is now at acceptable levels of about 6 ADU per hour for a 2 x 2 binned pixel. Concerns remain present about the vacuum hold time and the temperature stability of the CCD, but in spite of that for astronomical purposes the detector is operational.

21st Feb 2010
Subject: New OSIRIS SNR Calculator

From Monday 15 February the latest version of the OSIRIS SNR calculator is available. This version (v1.2) includes true dark current measurements and filter efficiencies, grism (not all of them) and the red TF.

This update has been made possible through the efforts of the OSIRIS team's observers and especially that of Antonio Cabrera Lavers (GTC). We also thank the comments from many users. So I keep encouraging to send comments, suggestions, mistakes etc...

The New OSIRIS SNR Calculator is at:

http://www.ifca.unican.es/users/gserrano/OsirisETC/

(Although it will change its location shortly).

Ignacio González Serrano (OSIRIS Team)

1st Nov 2009
Subject: OSIRIS status update

The previous information note about OSIRIS touched on the problem of the high dark current of the CCD. This remains the current most serious problem that we are facing, as it affects a large fraction of the scheduled science programmes. The OSIRIS team, in close collaboration with GTC is working vigorously to resolve this problem. Because of the delicate nature and the amount of work it entails there is not yet a detailed time table of when we can expect this to be a problem of the past.

The multi-object observing mode of OSIRIS is suffering delays in its implementation. At this point in time it seems unlikely that science observations can commence during semester 2009B. For some of the approved science projects switching to long-slit spectroscopy mode may be an option, but for faint objects requiring long exposure times this will probably not be a viable option due to the high dark current of the CCD. Hence the highest priority now is placed on reducing the dark current.

The four higher resolution grisms VPH2500 are not yet available. They are currently undergoing integration and testing by the OSIRIS team of the IAC.

CCD binning has been fully implemented and 2x2 binning is now used as the standard observing mode. Windowed readout is undergoing final tests at the telescope. More exotic CCD readout modes such as charge shuffling and frame transfer modes will be commissioned at some later stage.

The red-optimized tunable filter is now in regular science use. Probably due to flexure in the filter the wavelength stability is not perfect, which implies the need to avoid certain instrument rotator positions and the need for repeated wavelength calibrations. This increases the observing overheads (see note of June 19th, 2009). Our hope is that through long-term characterization of the flexure profile this effect can be modelled, thus reducing the need for calibrations.

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October 5th, 2009.
Subject: OSIRIS CCD dark current problem

It recently transpired that the dark current of the OSIRIS CCD is much higher than what would normally be considered acceptable for astronomical scientific use. This is due to the higher than optimal operating temperature of the CCD inside its temporary cryostat. This cryostat has a limited hold-time for liquid Nitrogen and has its temperature stabilization deactivated. This implies that the dark current is relatively high, and variable in time. In particular during the warmer summer months this has been the case. We recently learned that the effects are more severe than was anticipated and it seems to have been getting worse recently.

This problem is under active investigation and in the mean time we take regular dark exposures to monitor the effects. At the moment typical dark currents that are measured are of the order of 2 electrons per pixel per second.

The extra noise level is significant and will affect the quality of the data. Until this problem is resolved we will take this into account when planning the observations. This shortcoming will be corrected in the coming months as the detector package will the tranferred to the final cryostat that will be able to hold a stable and sufficiently low temperature. The new cryostat has already been procured and is currently undergoing operational verification tests at the Instituto de Astrofísica de Canarias.

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July 30th, 2009.
Subject: OSIRIS Standard binning in spectroscopy mode

CCD binning is currently undergoing tests at the telescope. Earlier problems with binning have been resolved, and if the trials go well binning will be supported from the start of semester 2009B onwards (Sep 1st 2009). Following consultation with some users and the instrument specialists, it has been decided to define standard binning of the OSIRIS detector in spectroscopy mode as 2 by 2 pixels, identical to the standard binning in the other science modes of the instrument.

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June 19th, 2009.
Subject: Status of the tuneable filter

Over the last few months much work has been invested in understanding and characterizing the problems that were encountered in the wavelength stability of the red-optimized tuneable filter. The most important problem is a significant drift in wavelength as a function of the instrument rotator position. Drifts as large as 1.3nm are seen.

This rotator position dependence in itself appears to be stable and is now sufficiently well characterized in order to allow us to start using the tuneable filter. The tuneable filter can be used in the whole range, except for -30 < alpha < 30 of the instrument rotator position in order to minimize the wavelength drift. At the start of each observation the filter wavelength position will have to be calibrated. For long observations this may have to be done more than once. This implies an increase in observing overheads.

It should also be noted that the practical use of the tuneable filter is more restrictive than was originally anticipated. The minimum width achievable is 1.2nm. There is also a maximum width, depending on the wavelength range, as follows:

1. 2.0nm for lambda < 800nm
2. 1.5nm for 800nm < lambda < 850nm
3. 1.2nm for lambda > 850nm

The range of operation of the tuneable filter is from 651nm to 934.5nm.

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June 19th, 2009.
Subject: Status of R2500 grisms

Of the four R2500 VPH grisms the R2500R and R2500I have now been delivered to the OSIRIS instrument team at the Instituto de Astrofísica de Canarias. If all goes well with testing and mounting of these two grisms the expectation is to have them available in the instrument for science use towards the end of August. Unfortunately this implies that there will be very little time to carry out science observations with these grisms before the end of semester 09A. A final date for the delivery of the R2500U and R2500V grisms remains uncertain.

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April 16th, 2009.
Subject: OSIRIS CCD binning mode

Due to existing limitations of the CCD controller software observations in binning mode with the OSIRIS CCD have been suspended until further notice.

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April 10th, 2009.
Subject: Delay in the delivering of the OSIRIS R2500 grism

Although the OSIRIS team have done their very best to press for timely delivery of these grisms, the company producing the optics is suffering major delays. Time scales for delivery keep slipping, and therefore we cannot guarantee that the R2500 grism set will be available soon. However, we are pleased to note that the R2000B VPH grism is now available in OSIRIS. PIs may want to consider using the R2000B grism instead of the R2500 grism. PIs of proposals already approved wishing to change their Observing Blocks accordingly would need to contact your support astronomer to re-open the Phase-2 tool for your proposal.

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March 16th, 2009.
Subject: Limitations of the OSIRIS red tunable filter

The PI of the OSIRIS instrument inform us that limitations in the use of the tunable filter have come to light during the commissioning process.

Specifically:

1. The maximum FWHM achievable is of around 2.0 nm
2. Beyond 850nm only FWHM around 1.2 nm are available

Due to these limitations you may wish to alter the Observing Blocks for your project. If you have already submitted the Phase-2 form please contact your support astronomer and we will re-instate your access to the database.

Due to problems encountered during the commissioning of the tunable filter the scientific programmes that require the tunable filter cannot starts until the mid-May 2009 at the earliest.

On this page you will find results from the OSIRIS commissioning process that may assist you in planning your observations. This page will be regularly updated as we will learn more about the instrument and commissioning new modes.

Note that at the present time the Tunable Filter is not yet operational for technical problems ocurred during the integration process.

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Field Of View (FOV) of OSIRIS

The field of view of OSIRIS is projected onto two CCDs that are separated by a small gap. The OSIRIS field center is off-axis relative to the telescope field center. (Observatory staff will take care of the correct pointing of the instrument, so there is no need to incorporate this offset when preparing your observations). Moreover, part of the field is strongly vignetted by mechanisms in the OSIRIS instrument, which causes the peculiar dark patterns in the accompanying picture.

Below we show an image with the default pointing positions for all the OSIRIS observing modes. Note that the pointing for the the longslit spectroscopy have been chosen based on columns free of bad pixels.

OSIRIS' Field Of View. We can see the gap between the two CCDs. On the left the vigneting produced by components inside the instrument is shown.

The OSIRIS vignetting is important in the first 500 pixels of CCD1 and the first 500 pixels of the bottom of the image in both CCDs (though this is less striking than that produced by the collimator and filter wheels in CCD1). Therefore one needs to be careful when doing photometry of stars near that area. This vignetting defines a maximum unvignetted field of 7.8x 7.8arcmin (The figure says it is 7.8 'x 8.5', but does not consider the vignetting in the lower part of both CCDs.)

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OSIRIS CCDs Gap

The distance between the CCDs (gap) is 9.2" (approximately 72 pixels), rather than 8" as it was previously thought. This distance is measured in the central area of the CCDs, because it changes slightly from top to bottom (the CCD2 is inclined by about 0.06 degrees from the vertical direction). The CCD2 also is shifted about 0.5 "(= 4 pix) down respect to the CCD1.

The image below shows a mosaic made with these two conditions (It shows the trace of an asteroid).

Accurate information for the astrometric coefficients of OSIRIS will be incorporated soon.

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OSIRIS CCDs

The two OSIRIS science CCDs are of type E2V 44-82, back illuminated devices with 2048 x 4096 pixels each. The control system offers a wide range of readout modes and gain settings, but for the time being the standard modes that are being used are:

• Slow readout at 100kHz, with gain of approximately 1.18e-/ADU. Full-frame readout takes about 84 seconds.
• Fast readout at 200kHz with gain of approximately 0.95 e-/ADU. Full-frame readout takes about 42 seconds.

In these modes the detector linearity is guaranteed up until the full 16 bits signal. Exact linearity curves will be posted here soon.

Read noise is better than 5 electrons in both readout modes, but much higher noise levels have been measured intermittently on the telescope. This problem is under investigation.

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Detector Quatum Eficiency

Here we show a graph of the detector quantum efficiency.

To view a larger version of this chart click here.

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Detector Setup

The following “standards” modes have been defined for OSIRIS. They depend on the observing mode selected: imaging, spectroscopy or acquisition. The acquisition mode is generally used for test images, for instance to check quickly the number of counts in flat images, through slit images… This mode has a significant high noise pattern so it is not suitable for scientific cases. Next table shows the parameters that define each standard mode:

	Imaging	Spectroscopy	Acquisition
Readout configuration	CCD1+CCD2_A	CCD1+CCD2_A	CCD1+CCD2_A
Readout velocity	200 kHz	100kHz	500kHz
Ident. Gain	9.5	4.75	4.75
Gain (e-/ADU)	0.95	1.18	1.46
Binning (X x Y)	2 x 2	2 x 2	2 x 2
Readout time	15.3 sec	~40 sec	7.8 sec
Actual readout noise	~4.5 e-	~3.5 e-	~8 e-
Expected readout noise	< 5 e-	< 5 e-	< 5 e-

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Tunable Filters Optical Center

The TF optical center is located in the pixel (2118, 1966) from the pixel (1,1) of CCD1, including the 50 pixels of overscan. Thus, lies within the gap of the CCDs, and 20 pixels away from the right edge of the CCD1. The wavelength observed with the TF relative to this point changes following the law:

λ= λ₀ * [1-0.0007952 * r (arcmin) ²]

The images below show a picture of prominent sky lines taken at 7325 A , which clearly shows the center of the system. The last image indicates the 4arcmin radius that assures the osbervations will not have any contamination of other interference orders in the filter.

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OSIRIS Grisms

Next table presents the main characteristics of the OSIRIS grisms. These results have been measured during the instrument commissioning.

ID	Name	λc(A)	Δλc(A)	D (A/pix)	R	Efficiency(*)
R300B	MBR_G1	4560	3600 -10000	2.48(45600A)	389(4560A)	Image
R300R	MBR_G2	6865	4800 10000	3.87(6865A)	376(6865A)	Image
R500B	BR_G1	4830	3440 - 7600	1.77(4830A)	580(4830A)	Image
R500R	BR_G2	7310	4800 - 10000	2.44(7319A)	634(7319A)	Image
R1000B	IR_G2.1	5510	3630 - 7500	1.06(5510A)	1100(5510A)	Image
R1000R	IR_G3.2	7510	5100 - 10000	1.31(7510A)	1212(7510A)	Image

^(*) The efficiency curves shown here are supplied by the manufacturer and simulated using the program GSolver. Shortly we will show efficiency curves measurements obtained at the telescope.

The spectroscopic order sorter is needed for the observations with the red grisms (R300R, R500R, and R1000R), to avoid the second order contamination in the spectra.

Here we show the graphs of the resolution (R) versus wavelength (λ(A)) for the three red grisms.

R300

R500

R1000

R2000B

R2500R

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Sloan Filters of OSIRIS

Transmision measurements of the OSIRIS Sloan filters. These filters are optimized for the visible range (250nm-1000nm).

The working conditions in these filters are:

• They are placed in the beam collimator
• They are inclined 10º with respect to the optical axis of the instrument, therefore their central wavelength [λ_c(10º)] is shifted with respect to the nominal central wavelength [λ_c(0º)]. Its bandwidth [Δλ] changes slightly, but the transmission curve shape is hardly altered.
• The light beams from the field cover a range between -2º y 22º

The measures shown here were obtained at the IAC optical laboratories.

Filter	λc(0º) [Δλ(0º)] nm	λc(10º) nm	Δλ(10º) nm	Transmision (10º)	Transmision (0º)
u'(*)	350.0 [60.0]	--	--	--	--
g'	481.5 [153]	479.5	149	81.81	Image
r'	641.0 [176]	638.5	171	94.79	Image
i'	770.5 [151]	768.0	144	88.57	Image
z'	969.5 [261]	969.5	261	96.41	Image

^(*)Data from the filter u ' standard definition.

As can be seen, the inclination of the filters with respect to the optical axis does not imply any significant difference, except the displacement of a few nm at shorter lengths, with respect to its nominal wavelength.

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OSIRIS sort-order Spectral filters

This is a sort-order filter and it is optimized to be used along with the diffraction grims.
The working conditions in this filter are the same as those described for the OSIRIS Sloan filters.

The measurements obtained in the laboratory for this filter are as follows:

Filter	λc(0º) [Δλ(0º)] nm	λc(10º) nm	Δλ(10º) nm	Transmision (10º)	Transmision (0º)
GR	546.0 [108]	546.0	108	94.67	Image

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OSIRIS Interference filters

Results obtained by measuring the efficiency in the visible range (540nm - 1000nm) for the OSIRIS red interference filters. These filters are optimized for the spectral red bands (from 650 nm), which are used as sort-order filters by the OSIRIS red tunable filters.
The working conditions in this filter is already specified for the OSIRIS Sloan filters.

The measures obtained in the laboratory for these filters are:

ID Filter	λc(0º) [Δλ(0º)] nm	λc(10º) nm	Δλ(10º) nm	Transmision (10º)	Transmision (0º)
643Bp28	643.0 [28]	640.0	28	91.79	Image
648Bp28	648.5 [27]	646.0	28	91.01	Image
657Bp35	657.0 [36]	652.5	37	91.55	Image
666Bp36	667.5 [35]	664.0	36	90.47	Image
680Bp43	679.0 [40]	675.0	40	88.83	Image
694Bp44	695.0 [44]	691.5	45	92.87	Image
709Bp45	708.0 [42]	704.5	43	95.30	Image
723Bp45	723.0 [40]	718.0	42	91.49	Image
738Bp46	737.5 [43]	732.5	45	94.12	Image
754Bp50	754.0 [46]	749.0	48	92.44	Image
770Bp50	770.0 [48]	766.0	48	95.15	Image
785Bp48	786.5 [47]	783.0	48	95.41	Image
802Bp51	800.0 [50]	796.0	52	95.18	Image
819Bp52	818.5 [55]	814.5	57	95.18	Image
838Bp58	838.0 [56]	834.0	56	95.54	Image
858Bp58	859.0 [58]	854.0	56	95.49	Image
878Bp59	879.0 [60]	874.5	59	95.50	Image
893Bp50	893.5 [51]	888.5	51	94.07	Image
902Bp40	904.5 [39]	900.0	38	94.71	Image
919Bp41	919.0 [36]	915.0	36	95.61	Image
924Bp34	924.5 [33]	920.0	32	94.28	Image
927Bp34	929.0 [32]	922.5	33	91.97	Image
932Bp34	932.5 [31]	928.0	32	93.22	Image
936Bp35	937.5 [31]	933.0	32	94.88	Image
940Bp35	942.0 [34]	936.0	34	92.27	Image

As can be seen, the inclination of the filters with respect to the optical axis does not imply any significant difference, except the displacement of a few nm at shorter lengths, with respect to its nominal wavelength.

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Tunable Filter Widths

In addition to the information of the maximum tunable widths with the TFs as a function of wavelength (see table above); the plot below shows the variation of the available widths as a function of wavelength. The minimum width is 1.2nm for all the wavelength to avoid contamination due to other orders in a circular FOV of 4arcmin radius (this value is also the maximum usable width from 8500 A).

To avoid further confusion it must be emphasized that the operating range of the TF is 6510 A - 9340 A.

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Fringing

The measured value of fringing in the OSIRIS CCD is <1% for λ < 9000 A and 5% for λ > 9500 A, so it is relevant only in the range z '. The plot below shows an example of fringing vs wavelength.

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Zeropoints

During semester 2009A, we have measured OSIRIS zeropoints for the Sloan filters using standard stars. The following values are (for airmass= 1):

Filter	Zeropoint	Extinction	ETC(OSIRIS)
u'	25.75 ± 0.09	0.531 ± 0.063	25.54
g'	28.74 ± 0.09	0.216 ± 0.023	28.63
r'	29.16 ± 0.06	0.111 ± 0.043	28.74
i'	28.75 ± 0.05	0.085 ± 0.027	28.69
z'	28.34 ± 0.08	0.163 ± 0.025	27.90

These values are consistent, and in some cases even slightly better, that those used in the OSIRIS ETC. The estimated photometric transformations (using an arbitrary zeropoint of 25 mag) are:

• u – u₀ = -