Simulating the optical performance of a small-sized telescope with secondary optics for the Cherenkov Telescope Array

The Gamma-ray Cherenkov Telescope (GCT) is a small-sized telescope (SST) that represents one of three novel designs that are based on Schwarzschild–Couder optics and are proposed for use within the Cherenkov Telescope Array (CTA). The GAmma-ray Telescope Elements (GATE) program has led an effort to build a prototype of the GCT at the Paris Observatory in Meudon, France. The mechanical structure of the prototype, known as the SST-GATE prototype telescope, is now complete along with the successful installation of the camera. We present the results of extensive simulation work to determine the optical performance of the SST-GATE prototype telescope. Using the ROBAST software and assuming an ideal optical system, we ﬁnd the radius of the encircled point spread function ( θ 80 ) of the SST-GATE to be ∼ 1.3 arcmin ( ∼ 0.02 ° ) for an on-axis ( θ ﬁeld = 0 ◦ ) observation and ∼ 3.6 arcmin ( ∼ 0.06 ° ) for an observation at the edge of the ﬁeld of view ( θ ﬁeld = 4 . 4 ◦ ). In addition, this research highlights the shadowing that results from the stopping of light rays by various telescope components such as the support masts and trusses. It is shown that for on-axis observations the effective collection area decreases by approximately 1 m 2 as a result of shadowing components other than the secondary mirror. This is a similar loss ( ∼ 11%) to that seen with the current generation of conventional Davies–Cotton (DC) Cherenkov telescopes. An extensive random tolerance analysis was also performed and it was found that certain parameters, espe- cially the secondary mirror z-position and the tip and tilt rotations of the mirrors, are critical in order to contain θ 80 within the pixel limit radius for all ﬁeld angles. In addition, we have studied the impact upon the optical performance of introducing a hole in the center of the secondary mirror for use with pointing and alignment instruments. We ﬁnd that a small circular area (radius < 150 mm) at the cen- ter of the secondary mirror can be used for instrumentation without any signiﬁcant impact upon optical performance. Finally, we studied the impact of reducing the size of the primary mirror for the prototype telescope and found that this comes at the cost of poorer image quality and light collection eﬃciency for all ﬁeld angles, but at a signiﬁcant cost saving for a one-off prototype.


Introduction
The Cherenkov Telescope Array (CTA) [1] will be the world's first ground-based gamma-ray observatory open to the wider astrophysics community. To achieve an order of magnitude greater sensitivity compared to existing ground-based experiments, the such very high energies, gamma-rays interact with the atmosphere to trigger electromagnetic cascades ("air showers"), which emit a flash (duration of a few ns) of Cherenkov light in the direction close to the shower axis. This emission can be detected in the optical, near-infrared and ultra-violet range and is analyzed to reconstruct the arrival direction and energy of the primary gamma-rays.
CTA is currently investigating three SST designs [3] , each of which aims to combine a wide field of view (FOV) with a very good imaging resolution, all the while attempting to keep costs down in order to build ∼70 SSTs spread over a few km 2 . One of the designs employs a traditional 4 m Davies-Cotton optical system [4] [5] , while the other two employ Schwarzschild-Couder (SC) [6] [7] [8] configurations that comprise aspherical primary and secondary mirrors (see Fig. 1 ). In parallel, other research groups involved in CTA are pursuing an additional telescope type in the form of a 9 m SC telescope system [9] . The SC-based telescope systems proposed for CTA make use of a secondary mirror to reduce the plate scale and to allow for a wide FOV with a light-weight camera.
One of the SC-based SST designs proposed for CTA is called the Gamma-ray Cherenkov Telescope (GCT). The GAmma-ray Telescope Elements (GATE) program has led an effort to build a prototype of the GCT at the Paris Observatory in Meudon, France. The mechanical structure of the prototype is known as the SST-GATE prototype telescope. Thus the GCT is one of the SC configuration projects, which consists of telescope and camera subprojects: SST-GATE and Compact High Energy Camera (CHEC) [10] [11] . There are two prototype versions of CHEC: CHEC-M will use Multi-Anode Photo Multiplier Tubes (MAPMTs) and CHEC-S will use Silicon-based Photo Multipliers (SiPMs).
One of CTA's key motivations for considering the use of dualmirror SSTs is in the potential to reduce cost, which typically for a Davies-Cotton configuration is dominated by the camera construction, while taking advantage of new high-performance detector technologies. This is achieved through an optimized optical design combining a large FOV with very small aberration effects and a very compact plate scale. Compared to DC telescopes, calculating the tolerances (and hence optical performance) for an SC-optics telescope can be difficult due to the secondary mirror, aspherical surfaces and the very small spot size. This paper highlights the results of detailed simulations conducted to assess the optical performance of the SST-GATE prototype telescope, and for the first time, attempts to show comprehensive tolerance analysis of an SC Cherenkov telescope. Section 2 of this paper will provide a brief outline of the SST-GATE telescope and optical system. Section 3 will highlight simulations of the ideal SST-GATE optical performance. This will include discussion of the point spread function ( θ 80 ), effective collection area and telescope shadowing. Section 4 will show the results of a comprehensive tolerance analysis conducted for the SST-GATE telescope. Section 5 will highlight the results of a study undertaken to determine the feasibility of using a small area at the center of the secondary mirror for calibration equipment. Finally, Section 6 illustrates the impact on the optical performance expected as a result of trimming the primary mirror petals of the SST-GATE prototype telescope.

The SST-GATE telescope and optical system
The mechanical structure of the SST-GATE prototype telescope [12] [13] has been built at the Meudon site of the Paris Observatory in France [14] . Installation of 2 (out of 6) primary mirror petals (polished aluminum), the secondary mirror (polished aluminum), along with the CHEC-M camera is complete and the prototype telescope was inaugurated on the 1 st December 2015. Table 1 highlights the specifications of the telescope, which includes two mirror surfaces that can be characterized by 16 th order polynomials.
These aspherical mirror surfaces allow for an optimal θ 80 over the whole 9 degree field of view. The primary mirror is comprised of 6 individual petals while the secondary is a formed monolithic mirror. The original SST-GATE optical design concepts and tolerance analyses were conducted by Jürgen Schmoll and colleagues at Durham University. Since 2011 the project has been led by the SST-GATE group and in particular by teams at the Observatoire de Paris.
The introduction of a secondary mirror leads to ∼25% of light rays being obscured. The shadowing that results from the secondary means that a hole of 65 cm radius at the center of the primary mirror can be used for a laser-based alignment system. Each primary mirror segment is attached to the telescope's mechanical structure by three actuators allowing the position of each petal to be adjusted in order to achieve the best possible optical alignment. The secondary mirror will be monolithic and will also be attached to the mechanical structure using actuators to facilitate its alignment.
The SST-GATE ideal focal plane is ∼36 cm in diameter and has a 1 m radius of curvature. It is located between the primary and secondary mirror. The ideal focal plane is used for all calculations in this work; however the actual focal plane will be comprised of either MAPMTs or SiPMs. These will be positioned to match the focal plane curvature, but the detector front-end is likely to have a certain fraction of dead space ( ∼3 mm) between each module. In the case of the MAPMT modules this space will be smaller than a single pixel of 6 mm × 6 mm. The optical spot generated on the focal plane for an on-axis observation is expected to be of 4 mm diameter in terms of 80 percent encircled energy. The approximation of the curved focal plane by tiles (e.g. using MAPMT modules) introduces insignificant image spot shape changes, thus the results shown herein (produced using an ideal focal plane) should not be significantly different. The telescope design has been optimized to keep the maximum incidence angle of photons hitting the curved focal plane below 60 °with respect to the normal of the plane. At 60 °the visible pixel area for an incoming beam is halved as it scales with the cosine of the angle of incidence. In addition, there is the chance that detector surfaces become less efficient at larger angles due to Fresnel losses. Restricting the marginal angles to 60 °ensures that these effects are kept under some control to reduce losses at the detector plane. In fact the final model does not exceed the constraint significantly, with the largest marginal beam incidence at the field edge being 61 °.
The mechanical structure of the SST-GATE telescope [15] is made of steel. The main mechanical components include a tower with fork mount and the azimuth and elevation drives, which are based on electric torque motors. The rest of the mechanical system is based on a mast and truss structure, again made from steel, and using a Serrurier [16] configuration which mounts the secondary support structure and camera support structure on the telescope. The preliminary SST-GATE design (now referred to as the 2014 design ) had three camera support trusses, fixing the camera to the support of the secondary mirror, and three pairs of masts that connect the secondary mirror to the main frame. scope 2014 design evolved as changes were made to reinforce the structure. The prototype telescope constructed at the Observatoire de Paris follows this updated design (now referred to as the final design ) which includes a fourth pair of secondary support masts (see Fig. 3 ). Furthermore all the secondary support masts have a smaller diameter compared to those of the 2014 design . In addition, the number of camera support trusses has been reduced to 2 and a pair of thin camera support bars (12 mm diameter) has been introduced.
Finally, the telescope uses an adaptable counter-weight to balance the loads and the overall telescope mass is expected to be approximately 8 tonnes, making the SST-GATE telescope a relatively light-weight Cherenkov telescope.

Simulations of the ideal optical performance
The detailed studies of the SST-GATE optical performance discussed here were carried out principally with the ROOT [17] based ROBAST [18] ray tracing software. The commercially available Zemax [19] ray tracing software was also used for some studies and cross-checks were also made with the sim_telarray software [20] , which is also used for the published performance simulations of CTA. Details of the ray-tracing software used are limited to ROBAST , but for the results herein we indicate (if appropriate) which package was used.
A three dimensional (3D) computer model of the core SST-GATE optical components was constructed using the ROOT geometry library and additional ROBAST classes. Fig. 3 illustrates the computational model of the final design SST-GATE telescope used for the ROBAST ray tracing simulations in Sections 3, 5 and 6 . Fig. 2 shows the ROBAST computational model of the 2014 design used for the tolerance analysis conducted in Section 4 .
Non-sequential ray tracing was performed in ROBAST using two methods: parallel rays and rays randomly distributed within a cone from a point at a given distance. The ray tracing of parallel rays was done using a square grid configuration with a single side length of 2.2 × R primary where R primary is the radius of the primary mirror. In total ∼1.6 × 10 5 rays were distributed equally over this square grid structure and the parallel rays were traced from a height of ∼8 m above the telescope. For the second method the rays were traced from a point situated at 10 km above the telescope, a similar height to that where Cherenkov light is produced in air showers induced by astrophysical gamma-rays. The cone used to confine the paths of the randomly generated rays had a ground radius of 1.1 × R primary . In total ∼1.5 × 10 5 rays were randomly distributed within this cone. Both methods lead to consistent results for sources close to infinity. For this work the random cone ray-tracing method is used for all the quoted values unless otherwise specified.
For the purposes of determining the optical performance parameters an ideal focal plane is used. Photons focused at the ideal focal plane, shown in Fig. 4 , are then used to calculate the various performance parameters.

Point spread function and imaging performance
Within CTA, the imaging performance of a telescope is characterized with θ 80 and this is defined as the radius that contains 80} of the photons which form the optical point spread function. For this work, θ 80 was calculated by firstly determining the spot center of mass (COM) which is defined as the arithmetic mean of the photon distributions along both the focal plane x-axis and y-axis. Beginning with an arbitrarily-sized circle centered on the spot image COM, the fraction of photons contained within the circle was calculated. The circle radius was increased in steps of 0.001 cm until the fraction of photons contained within the circle was equal to or greater than the required 80%. This encircled method was conducted for each of the field angles tested, as shown by the red circles in Fig. 4 . At a field angle θ field = 4 . 5 • the light spot hits the physical limit of the focal plane, within the SST-GATE computational model constructed for this work. This analysis is limited to a field angle of θ field = 4 . 4 • which approximately corresponds to the center-point of a pixel if it were placed flush to the edge of the 9 degree field of view.
In addition to the encircled method described above, we also derived θ 80 using an ensquared method. The process was similar to the encircled method except that 80% of the photons need to be contained within a square, the half −length of which defined the θ 80 value. Starting at the spot image COM an arbitrary square was defined. The square half −length was increased in steps of 0.001 cm until the fraction of photons contained within the square was equal to or greater than the required 80%. From an error budget point of view, determining θ 80 using the ensquared method is preferable as the imaging pixels are also square. From a performance budget point of view, the tolerance analysis results highlighted in Section 4 provides the system engineers with the technical margins for which the optical performance remains within specifications. For example, a θ 80 = 2 mm for an ideal op- tical performance restricts any increase in the spot radius due to misalignments of the mirrors or camera or other effects to 1 mm, above which the spot ceases to be fully contained inside a pixel.
The SST-GATE θ 80 value derived using ROBAST was compared against θ 80 values calculated using different software packages. As previously mentioned, ray-tracing analysis for the SST-GATE was also conducted using the commercially available ray-tracing software Zemax . In addition, we also used the Monte Carlo simulation software used in CTA sim_telarray . The evolution of θ 80 as a function of field angle for these independently derived SST-GATE θ 80 values is shown in Fig. 5 . All of the θ 80 values were calculated for a point source at a distance of 10 km using either the encircled

Effective collection area and shadowing analysis
The effective collection area for the SST-GATE telescope was calculated using A effective = N focused N simulated × A simulation where N is the number of photons and A the area. Therefore, this definition does not include photon detection efficiency or mirror reflectance. The simulation area depends on the method of ray-tracing used. For the parallel rays method the ray-traced photons were distributed over a square grid area of single side length 2.2 × R primary where R primary is the radius of the primary mirror, thus A simulation = (2 . 2 × R primary ) 2 . For the random cone method the simulation area is defined by the cone thus A simulation = π (1 . 1 × R primary ) 2 cos θ field .
Using the definitions outlined above, Fig. 6 illustrates the derived effective collection area for a single SST-GATE telescope. For an on-axis observation, adding the structural components to the computational model results in a decrease in the effective collection area (for a single rotation angle) from ∼9.2 m 2 to ∼8.1 m 2 . Thus with the key shadowing components included in the computational model, the SST-GATE effective collection area is expected to be A effective ≈ 8.1 m 2 for on-axis observations decreasing to A effective ≈ 7.3 m 2 toward the FOV edge. Varying the source distance changes the effective collection area by less than 2%. . Shadowing from the secondary mirror, the secondary mirror support masts, the camera support trusses and the camera support bars is clearly visible in both panels. Shadowing from the camera housing envelope is only visible when observing at large field angles ( θ field > 2 °). The black dot denotes the center-point of the secondary mirror and the solid gray line shows the secondary mirror circumference.
One major advantage of using ROBAST to conduct the SST-GATE optical performance simulations, is that it is possible to keep a record of all the light rays that are stopped by single or multiple telescope components included in the 3D computational model. Using the parallel ray-tracing method we determined both the fraction of light rays stopped by a specific telescope component as well as the fraction of light rays lost at the telescope focal plane due to shadowing caused by these components. Fig. 7 illustrates a sample (5%) of the traced light rays being stopped by the SST-GATE camera support trusses and the camera support bars for an off-axis The SST-GATE shadowing is dominated by the secondary mirror that is half the diameter of the primary mirror. This causes a light loss of ∼25%. As already mentioned additional shadowing arises from the support mechanics (e.g. masts and trusses) as well as the camera. The camera casts a shadow onto the secondary mirror for rays coming from the primary mirror. For most field angles this area coincides with the secondary mirror shadow on the primary mirror, and is only visible on the secondary mirror at large field angles. Fig. 8 illustrates the shadowing effects seen on the For each of the structural components included in the computational model it is possible to determine the total number of rays stopped as a function of field angle. Moreover, it is possible to determine the effective collection area change as a function of field angle that results from each of these shadowing components.
Parallel rays for 10 field angles θ field (0 °, 4.4 °) and 8 rotation angles around the z-axis φ(0 °, 360 °) were traced and analyzed in order to determine the shadowing that results from the major SST-GATE components: the secondary support masts, the camera support trusses, the camera housing envelope and the camera lid envelope. The secondary mirror is a fundamental design characteristic of the SST-GATE telescope and is thus not considered to be a shadowing component that adversely affects the performance parameters like effective collection area. Fig. 9 shows the expected loss in effective collection area as a function of field angle that results when the shadowing components (i.e. masts, trusses and camera envelope) are added to the computational model.  the trusses. This is evident by comparing Fig. 9 with [22] ( Fig. 3 therein) which shows the shadowing contribution of the 2014 design .

Depth of field
An aspect of the telescope design we evaluated was the influence of the distance of an air shower with respect to the telescope. The SST-GATE optics were designed for an object distance of 10 km, but this is just an average value and while the distance decreases with increasing altitude of the observatory, it increases with zenith distance of the observation. To evaluate any change of the image quality we tested three separate models with an object distance of 5 km, 10 km and 10 0 0 0 km (representing infinity) respectively. Fig. 10 shows that the derived θ 80 values change by < 1 arcmin for all field angles and between the different object distances analyzed. Thus, for the observation of Cherenkov air showers, there is no significant change in the image quality along the depth of field.

Plate scale distortion
In the final optical design, a certain residual aberration still remains due to a small variation of the plate scale with the field angle. A Zemax analysis of this effect shows that the distortion is of the barrel-type, meaning the image gets ∼2.5%} larger at the field edge. Fig. 11 illustrates this effect which has to be taken into account to enable a precise trajectory estimate for the gamma-ray showers observed.

Photon time spread
The CTA design specifications for the SST stipulates that the telescope must focus light with an optical time spread of < 1.5 ns. Using ROBAST , Fig. 12 illustrates the average time interval (as a function of field angle) between the arrival on the focal plane of the first and last ray-traced photon. The error bars show the standard deviation of the time-spread photons. As expected the photon time spread increases for large field angles as the photons travel Fig. 11. Shown here is the barrel-type distortion seen for the SST-GATE optical design. This is seen when comparing the end-points of the traced rays on the focal plane for the non-distorted rays (black filled circles) and the distorted rays (yellow filled circles). For the purposes of illustration we have exaggerated the barrel-type distortion by making the effect 10 times stronger. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 12.
Shown here is the average time interval (as a function of field angle) between the arrival on the focal plane of the first and last ray-traced photon. The error bars show the standard deviation of the photon arrival times for each field angle. This shows that the SST-GATE optical design meets the CTA time-spread requirements (which are standard deviation < 1.5 ns). a longer path length through the telescope optics. These findings confirm that the SST-GATE optical design successfully meets the CTA time spread specifications.

Tolerance analysis
The preliminary SST-GATE design work was conducted with Zemax and included an analysis of the design's tolerances. In a second step we derived results with ROBAST for a single parameter tolerance analysis as well as a (more realistic) random tolerance analysis, in which all free parameters are allowed to vary at the same time. In addition, we also tested two parameters not studied with Zemax : the margin between the primary mirror petals and the mirror surface micro-roughness. The purpose of the tolerance analysis is to help guide a strategy for SST-GATE alignment procedures. The tolerance analysis was conducted prior to the SST-GATE telescope design update and the ray-tracing was done using the 2014 design model. As already mentioned in Fig. 13. Shown here is a sketch highlighting the position and rotation axes of a single primary mirror petal used for the tolerance analysis. The same translation and rotation definitions were used for the monolithic secondary mirror tolerance tests.

Table 2
The SST-GATE single parameter tolerances derived using Zemax (top section) and the new parameters tested using ROBAST (bottom section).

Test
Mirror Axis Range −0 . 14 • · · · + 0 . 14 • Position Primary X (translation) −5 mm · · · + 5 mm Y (translation) −3 mm · · · + 3 mm Z (translation) −5 mm · · · + 5 mm Secondary X (translation) −5 mm · · · + 5 mm Y (translation) −2 mm · · · + 2 mm Z (translation) −1 mm · · · + 1 mm Roughness Primary  Fig. 13 shows a sketch of a single primary mirror petal including the position and rotation axes used for this tolerance analysis work. The same translation and rotation definitions were used for the monolithic secondary mirror tolerance tests. Table 2 highlights the Zemax derived tolerances for the SST-GATE optical model. Using ROBAST , these parameters and ranges were then tested in the single parameter tolerance analysis in order to keep the θ 80 value within the pixel limit radius i.e. for 80% of the spot image photons to be contained within a single pixel.
Only the parameters which appear to have the greatest influence on the stability of θ 80 will be discussed. These include the translation of the secondary mirror (and thus of the focal plane) in the z-direction, the tipping and tilting of the primary mirror petals and the mirror surface roughness. The translation of the focal plane in the z-direction appears to significantly affect the resulting θ 80 value. Fig. 14 shows that even  Table 2 for changes in primary petal z-positions, which is expected as only 1 / 6 th of the mirror contributes to the deterioration of the encircled energy. However, θ 80 is sensitive to changes in the z-position of the primary (i.e. all petals) and secondary mirrors. When altering the secondary mirror z-position and keeping all other values in their ideal position this affects two separation distances; the separation distance between primary mirror and the secondary mirror and the separation distance between the secondary mirror and the focal plane. The dominant effect is the change in distance between the secondary mirror z-position and the focal plane. In this respect the mechanical design is based on a priority link between the secondary mirror and the focal plane, and the allowed tolerance for the secondary mirror z-position relative to the focal plane is smaller to a maximum deviation of −1 mm ≤ z ≤ 1 mm as reflected in Fig. 14 .
In the preliminary tolerance analysis conducted with Zemax , the SST-GATE spot image was found to be sensitive to rotational movement about 2 axes of the primary mirror petals; sagittal rotation or tipping of the petals toward and away from the optical axis and tangential rotation or tilting of the petals on their sides (both shown in Fig. 13 ).
The sensitivity of θ 80 to tilting and tipping of the primary mirror petals is shown in Fig. 15 . This illustrates that for θ 80 to stay within the pixel limit radius for all field angles, tilting of the petals (all in the same way) should be kept within the range −0 . 01 • ≤ Likewise Fig. 16 also illustrates that in order for the θ 80 value to stay within the pixel limit radius for all field angles, tipping of the primary mirror petals (all in the same way) should be kept within  Using the ROBAST library, it is possible to perform a tolerance analysis of the SST-GATE mirror quality in a very simple approach that does not treat explicitly the actual mirror surface structure and the wavelength dependence of the diffusion it causes. Typically, for Cherenkov telescopes the mirror surface deviation from the ideal shape ("form deviation"), which is much larger than the  Table 1 ). incident wavelengths, is the dominant effect in determining the optics performance. Such "form deviations" can have an impact on the PSF, but the study of their impact is beyond the scope of this work. In addition, one has to account for the irregularities of the mirror surface at size scales close to the wavelength of the incident light. This "surface roughness" leads to the scattering of a fraction of the incident photons. For simplicity, we approximate the form deviation and the scattering by a single Gaussian. Within ROBAST this reflection angle distribution (which we will now simply refer to as roughness) is simulated as a two-dimensional Gaussian such that when the roughness is equivalent to 1 • , the projected onedimensional angular distribution around the ideal reflection angle will be a Gaussian of 1 • width ( σ roughness = 1 • ). Fig. 17 shows that by increasing the mirror roughness this results in a smearing of the spot image at the focal plane. The left panel shows the spot image for an on-axis ( θ field = 0 • ) observation with a mirror roughness of 0.001 °set for both the primary and secondary mirrors . The right panel shows the same spot for an on-axis observation, but with the mirror roughness parameter (for both the primary and secondary mirrors) increased by an order of magnitude to 0.01 °. Fig. 18 illustrates the sensitivity of θ 80 to different roughness values for both the primary and secondary mirrors. Due to the longer path length for each ray, θ 80 is more sensitive to changes in the roughness of the primary mirror compared to that of the secondary mirror. However, it is assumed that the quality of both mirrors will be identical and thus the single parameter tolerance analysis shown here was conducted for both mirrors simultaneously. It appears that if all telescope elements are perfectly aligned, then an SST-GATE mirror roughness resulting in a diffusion angle width of 0.013 °produces a θ 80 value within the pixel limit radius for all field angles except the largest analyzed. However, it should be noted that even at this roughness value θ 80 at a field angle of 1.5 °is roughly a factor of 2.4 worse than a mirror which produces an order of magnitude smaller diffusion angle.
A limitation of the single parameter tolerance test is that one has to assume all other parameters are fixed at their optimal value. In practice, this may not be the case and θ 80 may be very sensitive to a combination of misalignments. Thus a random tolerance test was also conducted for the SST-GATE telescope using the findings of the single tolerance analysis to help define the parameter ranges. From these parameter ranges a value was selected randomly for all the telescope's degrees of freedom being analyzed. Assuming that the real misalignments of the telescope The value of θ 80 shown here is derived assuming both the primary and secondary mirrors are manufactured with the same surface roughness. It appears that a surface roughness better than 0.013 °results in θ 80 staying within the pixel limit radius for all field angles except the largest.
components are normally distributed about their ideal positions, the parameter values were randomly selected using a Gaussian distribution. Table 3 highlights the new parameter ranges using input from the single parameter tolerance analysis (see Table 2 ). Two random tolerance tests were conducted: the first, called a "broad sigma" test, required that 99.73% (3 σ ) of the randomly selected parameter values fall within the tolerance range defined from the single parameter analysis, and the second, called a "narrow sigma" test, required 99.73% of the randomly selected parameters to fall within a manually adjusted, smaller range (cf. last column of Table 3 ), resulting in a θ 80 distribution closer to the ideal one. Table 3 also highlights the mean ( μ) and 3 σ values used to create the normal distributions for the random selection of the parameters in both the broad and narrow tests. The roughness value was fixed for both mirror surfaces as this is not expected to change in the short term. For both tests the average θ 80 values fall within the pixel limiting radius, but the standard deviation of the broad test results exceeds the pixel limit at the largest field angles (above 3.5 °).

Secondary hole analysis
An additional optical performance test of the general SST-GATE design was an analysis of the effects that result from introducing a hole or dead space at the center of the secondary mirror. The reason for doing this is to determine whether it is possible to fit additional instrumentation into this space that could be used for telescope pointing and alignment calibration.
An opaque circular mask, which will be called a central hole for simplicity, was added to the center of the secondary mirror in ROBAST . Ray-tracing using the random cone method was per- ) from the ideal θ 80 value i.e. no central hole (solid black line, filled circles). It is clear that, for the radii we tested, θ 80 is not significantly sensitive to the addition of a hole in the center of the secondary mirror.
Using the method described in Section 3.2 , the effective collection area as a function of field angle was also calculated for secondary mirrors with different central hole radii as shown in   radius 150 mm is feasible, as even with a hole of this size, the effective collection area residual remains above ∼94% for all field angles.

Primary mirror petal analysis
The SST-GATE prototype telescope [12] constructed at the Paris Observatory has primary mirror petals that are smaller than those of the GCT proposed for the CTA [23] . This is due to the fact that the construction of a single prototype telescope required a compromise solution for the primary mirror petals due to size constraints of existing mirror molds. If the GCT design is chosen for mass production, the cost for a larger, dedicated mirror mold will be acceptable.
A study was conducted in order to determine how smaller primary mirror petals would impact the θ 80 value and the effective collection area. To simulate the impact of trimmed petals this study implements a very simple approach. Triangular masks are overlaid above the proposed petals as if the petals themselves were trimmed. Fig. 23 illustrates the SST-GATE final design computational model with the masks overlaid.
Using the random cone ray tracing method, Fig. 24 shows the spot images generated on the ideal focal plane for different field angles. Trimming or masking the primary mirror petals results in spot images with a flower type shape. Essentially the image quality deteriorates and this effect is exaggerated as the field angle increases.
The cyan-colored circles in Fig. 24 show the θ 80 values fitted to the spot images for each of the field angles analyzed. Despite the image deterioration seen, the effect on θ 80 is negligible. Using these spot images, Fig. 25 Fig. 26 illustrates the effective collection area calculated for the SST-GATE prototype with trimmed primary mirror petals. As Trimming or masking the primary mirror petals results in the deterioration of the spot image. This effect becomes more noticeable as the field angle increases. As before (see Fig. 4 ) the center of the camera is to the left for angles > 0 °. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) the mask decreases the mirror surface area this has a direct impact on the effective collection area of the telescope and hence the effective collection area of the SST-GATE prototype (solid red line with open diamonds) is smaller relative to the effective collection area calculated for the ideal SST-GATE telescope proposed (solid black line with open circles). The effective collection area for the prototype is ∼15% smaller for all field angles analyzed. Fig. 26. The top panel shows the effective collecting area as a function of field angle for a single SST-GATE prototype telescope (solid red line with open diamonds) using a mask to simulate the trimmed primary mirror petals. The effective collection area is smaller relative to the effective collection area calculated for ideal SST-GATE telescope proposed (solid black line with open circles). As shown in the bottom panel, trimming the primary petals results in an effective collection area that is approximately ∼ 15% smaller for all field angles. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Conclusions
In conclusion this work highlights some significant findings to be used in the continued development of the SST-GATE prototype telescope, and hence GCT proposed for the CTA. For all field angles the ideal encircled θ 80 is smaller than the size of the camera pixels under consideration, leading to an optimal resolution of the air shower image on the camera. It is also shown that the very good θ 80 results presented in this paper which were produced using the ROBAST software are in excellent agreement with independent analysis conducted using different ray-tracing software, sim_telarray and Zemax . Furthermore, we have studied for the first time the expected loss in the effective collection area for the SST-GATE telescope due to the telescope's shadowing components such as the secondary mirror support masts, the camera support trusses and the camera housing envelope including the camera lid. For an on-axis observation we see the shadowing results in the effective collection area decreasing by approximately 1 m 2 . Reducing the shadowing from the additional components (which is at a similar level to that seen with current DC telescopes for on-axis observations) is unlikely to significantly improve the effective collection area performance relative to the cost of making any such improvements.
In addition, the results of a first extensive tolerance analysis highlighted in this paper show the need to carefully control certain parameters which greatly influence the stability of θ 80 , especially the secondary mirror z-position, and the tip and tilt rotations of both the primary and secondary mirrors. For example the knowledge of the separation distance between the secondary mirror and the focal plane is required to sub-millimeter precision and thus there is a need for innovative instrumentation that will enable this to be easily, cost effectively and precisely monitored.
This paper also highlights that it is feasible to use a small circular area at the center of the secondary mirror, which is foreseen to be equipped with instrumentation for calibration and pointing accuracy measurement.
Concerning the prototype telescope, we have investigated the effect of reducing the size of the primary mirror petals. Such a reduction helps to significantly reduce the costs associated with constructing a single SST-GATE prototype while still meeting the CTA specifications. However this comes at the cost of poorer image quality (particularly for large offsets) and light collection efficiency for all field angles.
Finally, all of the SST-GATE optical performance simulation results shown in this paper have helped to optimize the technical trade-off required with the construction of the SST-GATE prototype [24] . A number of additional alignment and pointing studies for the SST-GATE telescope are still in progress and the intention is to publish these at a later date.