Optical design of a multi-object fiber-fed spectrograph system for Southern Spectroscopic Survey Telescope
OO PTICAL DESIGN OF A MULTI - OBJECT FIBER - FEDSPECTROGRAPH SYSTEM FOR S OUTHERN S PECTROSCOPIC S URVEY T ELESCOPE
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YueFan Shan, ZhengBo Zhu, Hao Tan, Donglin Ma ∗ School of Optical and Electronic Information and Wuhan National Laboratory of Opto-electronicsHuazhong University of Science and TechnologyWuhan 430074, China [email protected]
February 19, 2021 A BSTRACT
Southern Spectroscopic Survey Telescope (SSST) is a wide-field spectroscopic survey telescope thatChina plans to build in Chile in the next few years. As an instrument for astronomical spectroscopicsurvey, the multi-object and fiber-fed spectrograph (MOFFS) is one of the most important scientificinstruments for SSST. In this paper, we present a recommended optical design for the MOFFSsystem based on the Volume Phase Holographic Gratings (VPHG). The whole design philosophy andprocedure, including the analytic method to determine the initial structure, optimization proceduresof the VPHG and the camera groups, are demonstrated in detail. The numerical results of the finalobtained spectrograph show a superior imaging quality and a relatively high transmittance for thewhole working waveband and the field of view. The design method proposed in this paper can providea reference for the design of MOFFS accommodated in spectroscopic survey telescopes. K eywords Southern Spectroscopic Survey Telescope · spectrograph · Volume Phase Holographic Gratings · Spectroscopic Survey
Image and spectroscopy are two of the most important observational information in astronomical research. For theformer one, a lot of telescopes have been designed for wide-area deep image sky survey observation, such as the HubbleSpace Telescope (HST) [1], Large Synoptic Survey Telescope(LSST) [2], Chinese Space Station Telescope (CSST,under construction) [3] and James Webb Space Telescope (JWST, to be launched) [4]. As an indispensable complement,the southern multi-target spectral survey telescope will be the most important and widely used ground-based equipmentfor the next generation of wide-area image sky survey projects [5]. For the spectroscopy survey, the huge success ofTwo-Degree Field Galaxy Redshift Survey (2dFGRS) [6] and Sloan Digital Sky Survey (SDSS) [7] have proved thatthe large-field multi-object spectral survey telescope can maintain competitiveness and efficient scientific output fordecades. However, the multi-object spectroscopic survey telescope with an aperture larger than 6 meters and field ofview greater than 3 square degrees is a major gap of the parameter space for the current telescopes in service. Therefore,China has planed to build Southern Spectroscopic Survey Telescope (SSST) with an aperture of 6.5 meters and a fieldof view greater than ◦ × ◦ in Chile. SSST aims to perform an efficient spectral verification for wide-area image skysurvey and reveal the physics behind astronomical events.Two kinds of recommended optical designs for SSST have been proposed [8, 9], which are designed for the telescopebody. But the works about the spectrograph, which is also an important scientific instrument for SSST, have not beenreported. ∗ Shenzhen Huazhong University of Science and Technology, Shenzhen 518057, China a r X i v : . [ a s t r o - ph . I M ] F e b PREPRINT - F
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19, 2021Up to now, the most powerful implemented design of such kind of instruments is the Dark Energy SpectroscopicInstrument (DESI), which was designed for the 3.8-meter Mayall telescope [10]. It has a field of view of . ◦ and canrecord the spectrum of 5000 different targets at the same time. The working waveband of DESI ranges from 360nmto 980nm. Each spectrograph of DESI consists of three channels and each channel is equipped with a 4k×4k CCD toobtain the high-resolution spectral information from 500 different objects. However, the statement of the design processwas not detailed in [10].In this paper, we aim to design a multi-object and fiber-fed spectrograph (MOFFS) system with a working wavebandranging from 360nm to 1100nm to cover both near-infrared and visible regions. To make the MOFFS work in awider spectral range while maintaining a high resolution of spectral imaging, we divide the observing light into foursub-wavebands via 3 dichroic filters. For each sub-waveband, the specific optical channel is elaborately designedrespectively, and each channel is equipped with a 4k×4k CCD with a pixel size of 15 µ m. The numerical result showsthat the designed optical system has a superior optical performance for spectroscopy measurement. Figure 1 (a) illustrates the working mechanism of SSST, where light rays coming from the outer space are capturedby the telescope firstly, and then these rays enter into one ends of fibers which are located at the focal plane of thetelescope. The positions of these fiber’s ends on the focal plane can be mechanically adjusted. The other ends of thefibers are mounted linearly forming a so-called fiber slit. Finally, the light rays emitted from the fiber slit are transmittedto the CCD via the spectroscopic imaging system.
Fiber SlitheadFiber Slit Collimator MirrorFold MirrorCamera
VPHG
Fibers
Fiber slitCCD
Image
Telescope Focal plane (a) (b)
Figure 1: (a) Schematic diagram of SSST; (b) optical layout of one channel of MOFFS.Figure 1(b) shows the optical layout of one channel of the MOFFS system, which is composed of a fiber slit, a collimatormirror, a folder mirror, a dichroic filter (not drawn), a volume phase holographic grating (VPHG), and a camera unit.The light emitted from the fiber slit is firstly collimated by the collimator mirror. Then the light beam is divided into 4channels corresponding to 4 sub-wavebands with the help of 3 dichroic filters. After that, the light is dispersed by theVPHG forming the mixed filed of views, and each field of view corresponds to a specific wavelength. Finally, the lightbeam propagates to CCD via the camera unit. Based on the relevant research experience and the specific application ofSSST, we summarize the design technical specifications for the spectrometer optical system in Table 1.Table 1:
Design specification
Wavelength range 360nm - 1100nmSpectral resolution <0.06nm/pixelTransmittance >86% (360nm - 390nm)>90% (390nm - 1100nm)RMS radius 15 µ m2 PREPRINT - F
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Focal ratio degradation (FRD) is the most important factor to determine the throughput loss of the Fiber-fed spectro-graphs [11, 12]. Because of the FRD effect, the divergence angle of the beam emitted from the fiber will be larger thanthat of the beam entering into the fiber. The light rays beyond the acceptance solid angle of the spectrometer systemlead to the FRD loss. So, the magnitude of the FRD effect is greatly determined by the focal ratio of the incident light[13, 14]. According to Lawrence W. Ramsey’s study [14], F/ ∼ F/ is the best choice of f-ratio to minimize theFRD effect. As a rule of thumb, we set the f-ratio as F/ . in our design. The fiber slit is the object plane of MOFFS. The parameters of fiber slit are selected according to DESI [10]. The slitlength is 120.9mm and contains 500 fibers which are equally spaced, as shown in Fig. 2(a). d s l ... N Collimator Fiber SlitVPHG(a) (b) d= μ m l = mm N= Figure 2: (a) Configuration of the fiber slit; (b) Schematic diagram of collimator system.
In this design, a single spherical mirror is selected as the collimator as shown in Fig. 2(b). The light rays emanatingfrom both ends of the fiber slit are collimated to be a parallel light beam by the collimator, the angle between these twobeams can be approximately calculated as tan θ lr , (1)where θ is the angle between the two beams, l is the length of the fiber slit equaling 120.9mm, and r is the radius ofcurvature of the collimator mirror. It is clear that the size of the VPHG is determined by the diameter of the collimatedparallel beam. A larger collimated beam can definitely guarantee a better optical performance. While large collimatedbeam means high cost of VPHG. So, a trade-off between the optical performance and the cost of VPHG has to bemade. As illustrated in Fig. 3, a larger collimator makes a smaller average RMS radius, i.e. better optical performance.When the diameter of the collimator increases to 170mm, the descent rate of the average RMS radius tended to be anuneconomical level. So, we choose the diameter of collimator as 170mm in this design.The radius of curvature of the collimator mirror can be obtained as follows F / · D = r (2)where F/ is the focal ratio of the beam emitted from the fiber slit, D is the diameter of the collimator mirror. Asmentioned above, F/ is 3.6 and D is 170mm. So, we can get r =1224mm and θ ≈ . ◦ . As a result, the distancefrom the fiber slit to the collimator mirror can be calculated as D · F/ mm.3 PREPRINT - F
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130 140 150 160 170 180 190 200 s u i d a r S M R e g a r e v A E R f o a r e m a C D ( μ m ) Diameter of collimated parallel beam(mm) D i a m e t e r o f V P HG ( mm ) RMS
VPHG
Figure 3: Average RMS radius of RED Camera Vs diameter of collimated beam.
For the waveband ranginig from λ to λ , the average spectral resolution can be approximately expressed as ∆ λ = ( λ − λ ) N , (3)where N is the number of pixels along the x or y direction of CCD. Considering the requirement of spectral resolution,we divide the whole waveband into four channels as listed in Table 2. The layout of the whole optical system isillustrated in Fig. 4, and the corresponding parameters of the 3 dichroics are listed in Table 3.Table 2: Waveband for each channel.
Channel Wavelength rangeBLUE 360nm-580nmRED 560nm-740nmNIR 720nm-900nmIR 877nm-1100nmTable 3:
Parameter requirements of dichroics.
Dichroic RED Pass NIR and IR Pass IR PassTransmission Band 580nm-740nm 740nm-1100nm 900nm-1100nmReflection Band 360nm -560nm 360nm -720nm 360nm -877nmCrossover Region 560nm -580nm 720nm -740nm 877nm -900nmCrossover Width 20nm 20nm 22.5nm4
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Collimator
MirrorBLUE CameraRED
Camera
IR Camera
NIR CameraIR Pass
Dichroic
RED Pass
Dichroic NIR and IR Pass
Dichroic
Figure 4: Design layout of the whole system of multi-object fiber-fed spectrographs.
Figure 5(a) detailedly shows the optical layout of one of the four channels of the spectrometer. VPHG is the kerneloptical element of the whole system dispersing the incident light into different angles for the different wavelength oflight.Supposing that the diffraction angles β and β are for the light with a shorter wavelength λ and the longer wavelength λ , respectively. Their relation can be specified as (cid:26) d (sin β + sin i ) = k · λ d (sin β + sin i ) = k · λ , (4)where d is the grating period, i donates the incident angle and k represents the diffraction order. The divergence angle (cid:52) β of the dispersed beam can be calculated as ∆ β = β − β = arcsin( k · λ d − sin i ) − arcsin( k · λ d − sin i ) . (5)In this design, only +1 diffraction order is considered, which means that k = 1 in Eq. (5). Since the initial structureof the camera group lens of the spectrometer is rotationally symmetrical about the optical axis, the dispersion angle5 PREPRINT - F
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19, 2021can be approximated to the field of view of the camera group, which means (cid:52) β ≈ θ . The line density of VPHG foreach channel is determined based on the conventional choice [10, 15], the incident angle and dispersion angle arealso calculated as listed in Table 4. These parameters are the basis to determine the initial optical structure of thespectrometer. β β d Δ i β Figure 5: (a) Design layout of one channel of MOFFS; (b) Schematic diagram of grating’s dispersion effect.Table 4:
Initial parameters of the VPHG.
Channel BLUE RED NIR IRWavelength range 360nm-580nm 560nm-740nm 720nm-900nm 877nm-1100nmLinear density 900 lines/mm 1050 lines/mm 1000 lines/mm 850 lines/mmIncident angle 25.05° 24.11° 24.2° 34.67°Grating spectral dispersion 11.3632° 11.2825° 11.2818° 11.2819°
The size of the photosensitive region of 4K×4K CCD equals 61.44mm×61.44mm. Taking the alignment tolerance andedge effects of CCD into consideration [10], only the central region of 59mm×59mm is considered. The focal length ofthe camera group can be obtained as f = a (cid:18) θ (cid:19) − , (6)where a is the margin length of the working area of CCD. With the information given above, the value of f is calculatedas 298.66mm. As shown in Fig. 6, the fiber slit is positioned at the optical axis of the spherical collimator mirror. To avoid thevignetting effect caused by the fiber slit as well as the fiber bundle, a flat fold mirror is located physically around thefiber slit. The reflected beam is divided into 4 channels by the 3 dichroics. Since most glasses have a strong spectralabsorption around 360nm, the blue channel is designed as a reflective unit before the camera group lens.
The initial optical structure of the camera group optical system is determined by the focal length and the field angle thathave been provided in Eq. (1) and Eq. (6). The optical performance requirements for the camera groups are listed inTable 5.Firstly, we choose the Petzval lens system as the initial structure as shown in Fig. 7(a), which consists of two positivelenses [16, 17]. The power of the whole optical system is equally distributed in two lenses. And, a field flattener isusually added near the image plane to improve the optical performance as shown in Fig. 7(b) [18]. To further simplifythe design, our design is promoted to have a triplet lens, a single positive lens, and a field flattener as shown in Fig. 8(a).6
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Fold MirrorFiber Slit
BLUE Grating
RED Grating
IR Grating
NIR GratingDichroics
Figure 6: Design layout of optical arrangement.Table 5:
Parameter requirements of the camera group lens.
Channel BLUE RED NIR IRWavelength range 360nm-580nm 560nm-740nm 720nm-900nm 877nm-1100nmSpectral resolution < . / pixel < . / pixel < . / pixel < . / pixel Transmittance > − > > > > − RMS radius <13 µ m <13 µ m <13 µ m <13 µ mFocal length 298.66mm 298.66mm 298.66mm 298.66mmTo correct the field curvature introduced by the collimator mirror, we use a “biconical” surface as the front surface ofthe field flattener as shown in Fig. 8(b). The sag height of the biconical surface is expressed as z = c x x + c y y (cid:112) − (1 + k x ) c x x − (1 + k y ) c y y , (7)where c x and c y are curvatures of the surface in x and y directions, and k x and k y are the conic coefficients of thesurface in x and y directions. Considering the cost, the spectral absorption characteristics, the refractive index, and theAbbe number, the selected glass combinations for each channel are listed in Table 6. Since most glasses have strongabsorption for the blue spectrum, the transmittance plays a major role in the glass choice in the BLUE channel. Themain reason for choosing fused silica as the material of the last lens is that it is not radioactive [10]. After the initialstructure of the camera group, the VPHG and the collimator mirrors are independently designed, and then we take a7 PREPRINT - F
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Figure 7: Evolution of the initial structure model for the camera group. (a) (b)
Figure 8: Evolution of the initial structure model for the camera group: (a) Camera lens system; (b) Biconical surfaceof the field flattener.further optimization of the whole optical system to improve the optical performance. In the optimization process, weuse the diameter of the collimated beam and the size of the spectral imaging area as the constraints. We optimize the linedensity and the incident angle of the grating while constraining the dispersion angle. The optimization is implementedin ZEMAX [19].
The final obtained camera group lens for each channel are shown in Fig. 9, and the final grating parameters are listed inTable 7. The final optimization parameters of the camera group lens in each channel obtained by the optical designsoftware ZEMAX are shown in Table 8 to 11.
The transmittance is one of the most important metrics of optical performance for the spectrograph system. Sincethe system has no vignetting effect, we mainly consider two loss factors for the system, i.e. the absorption of glassesTable 6:
Glass material of camera group lens.
Channel BLUE RED NIR IRFront group H-K9LGT H-QK3L H-QK3L H-QK3LQF8 H-F2 H-F2 H-F2H-QK3L H-QK3L H-QK3L H-QK3LRear group H-K9LGT H-K9LGT H-K9LGT H-K9LGTField flatter F_SILICA F_SILICA F_SILICA F_SILICA PREPRINT - F
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19, 2021Table 7:
Optimized parameters of the VPH gratings.
Channel BLUE RED NIR IRWavelength range 360nm-580nm 560nm-740nm 720nm-900nm 877nm-1100nmLinear density 901 lines/mm 1016 lines/mm 979 lines/mm 792 lines/mmIncident angle 15.11° 21.432° 24.452° 24.036°Grating spectral dispersion 11.5318° 10.9880° 10.9359° 10.9199°Table 8:
Optimization parameters of camera group lens in BLUE channel.
Element Material Curvature Thickness ConicRadius(mm) (mm)
Triplet − S1 H-K9LGT 287.430 25.163 -0.954
Triplet − S2 QF8 -1957.749 5.643 0
Triplet − S3 H-QK3L 156.250 54.996 0
Triplet − S4 — 2588.211 188.565 0 Rear − S1 H-K9LGT 223.149 90.000 -0.603
Rear − S2 a — -443.488 136.757 -2.641 Flat − S1 c F_SILICA -191.595 5.000 1
Flat − S2 c — 212.592 5.000 0 CCD d — infinite — 0 a Aspheric surface coefficients: α = 3 . − , α = − . − . b Biconical surface:
X Curvature Radius = − . , X Conic = − . c Biconical surface:
X Curvature Radius = infinite , X Conic = 0 . d Slit image: slit angle with X axis = 1 . ◦ . Table 9:
Optimization parameters of camera group lens in RED channel.
Element Material Curvature Thickness ConicRadius(mm) (mm)
Triplet − S1 H-QK3L 239.989 48.368 -0.843
Triplet − S2 H-F2 -370.194 11.359 0
Triplet − S3 H-QK3L 250.850 42.000 0
Triplet − S4 — -787.564 184.632 0 Rear − S1 H-K9LGT 204.668 55.000 -0.192
Rear − S2 — -832.669 110.718 0 Flat − S1 F_SILICA -134.887 16.000 -2.786
Flat − S2 a CCD b infinite — 0 a Biconical surface:
X Curvature Radius = 967 . , X Conic = 0 . b Slit image: slit angle with X axis = − . ◦ . PREPRINT - F
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19, 2021 (a) (b) (c) (d)
Figure 9: 2D layout of the optimized camera group lens in each channel:(a) BLUE channel; (b) RED channel; (c) NIRchannel; (d) IR channel.Table 10:
Optimization parameters of camera group lens in NIR channel.
Element Material Curvature Thickness ConicRadius(mm) (mm)
Triplet − S1 H-QK3L 260.508 46.493 -1.076
Triplet − S2 H-F2 -273.735 15.990 0
Triplet − S3 H-QK3L 180.575 47.999 0
Triplet − S4 — -659.152 151.243 0 Rear − S1 H-K9LGT 287.219 38.352 -0.886
Rear − S2 — -413.081 161.364 0 Flat − S1 F_SILICA -138.478 16.001 -2.586
Flat − S2 a CCD b infinite — 0 a Biconical surface:
X Curvature Radius = 603 . , X Conic = 0 . b Slit image: slit angle with X axis = − . ◦ . and the reflection loss on the optical surfaces. Figure 10 shows the transmittance curves of the four channels. Theanti-reflection coatings on the lenses are assumed to have a transmission of 99% in all channels. The result shows thatthe transmittance meets the requirements given in Table 5.As shown in Fig. 11, the RMS radius is less than 12.2 µ m for the RED, NIR, and IR channels, and is less than 13 µ mfor the BLUE channel. The 2D layout of the focal plane is shown in Fig. 12. We can find that the distortions of thespectral images are well controlled(the final optimization parameters of each channel obtained by the optical designsoftware ZEMAX are provided in the appendix). And these results meet the predefined requirements. To make the spectrometer be friendly to fabrication and alignment, the tolerance budgets are eased in ZEMAX referringto Table 12. A thousand trials of Monte Carlo tolerancing have been performed, and Table 13 lists the five cases whichhave the biggest negative impact on RMS radius of each channel. Analysis results show that the RMS radius of eachchannel with 90% confidence is less than 15 µ m for our proposed error budgets.10 PREPRINT - F
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19, 2021Table 11:
Optimization parameters of camera group lens in IR channel.
Element Material Curvature Thickness ConicRadius(mm) (mm)
Triplet − S1 H-QK3L 249.801 48.005 -0.979
Triplet − S2 H-F2 -267.096 12.000 0
Triplet − S3 H-QK3L 175.583 47.006 0
Triplet − S4 — -727.455 137.670 0 Rear − S1 H-K9LGT 299.867 46.438 -1.304
Rear − S2 — -383.637 161.330 0 Flat − S1 F_SILICA -135.802 16.000 -2.914
Flat − S2 a CCD b infinite — 0 a Biconical surface:
X Curvature Radius = − . , X Conic = 0 . b Slit image: slit angle with X axis = 0 . ◦ .
300 400 500 600 700 800 900 1000 11000.860.870.880.890.9
BLUE
RED
NIR IR Wavelength(nm) T r a n s m i ss i on Figure 10: The throughput of the camera group lenses. The blue, red, green and purple curves are for the BLUE, RED,NIR and IR channels respectively.
To summarize, we propose a MOFFS system for the wide-field spectroscopic survey telescopes, especially for China’sSSST. The proposed design is demonstrated to have superior optical performance with good image quality, high opticalthroughput, and low cost of glass materials. A brief tolerance analysis of the proposed design indicates a moderate errorbudget for optical alignment and optical surface quality, which demonstrates its good manufacturability. And we believethat the design ideas, including the calculation of initial structures, optimization procedures of the VPHG parameters aswell as the camera groups, can provide a good reference and guidance to researchers for the future instrumentation ofsuch kind of astronomical spectrograph systems.
Funding.
National Natural Science Foundation of China (61805088); Science, Technology, and Innovation Commissionof Shenzhen Municipality (JCYJ20190809100811375); Key Research and Development Program of Hubei Province(2020BAB121); Fundamental Research Funds for the Central Universities (2019kfyXKJC040); Innovation Fund ofWNLO.
Disclosures.
The authors declare no conflicts of interest.
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19, 2021Table 12:
Tolerances setting in ZEMAX of the calculation.
Tolerances BLUE RED NIR IRSurface tolerancesRadius ±0.05mm ±0.05mm ±0.05mm ±0.05mmThickness ±0.05mm ±0.05mm ±0.05mm ±0.05mmDecenter in X ±0.05mm ±0.05mm ±0.05mm ±0.05mmDecenter in Y ±0.05mm ±0.05mm ±0.05mm ±0.05mmTilt in X ±0.01° ±0.05° ±0.03° ±0.05°Tilt in Y ±0.01° ±0.05° ±0.03° ±0.05°Element tolerancesDecenter in X ±0.05mm ±0.05mm ±0.05mm ±0.05mmDecenter in Y ±0.05mm ±0.05mm ±0.05mm ±0.05mmTilt in X ±0.01° ±0.05° ±0.03° ±0.05°Tilt in Y ±0.01° ±0.05° ±0.03° ±0.05°Index tolerancesIndex ±0.0003 ±0.0005 ±0.0004 ±0.0005Abbe 1% 1% 1% 1%
Table 13:
Tolerances setting in ZEMAX of the calculation.
Offenders
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