Optical design of the laser launch telescope via physical optics theorem for Laser Guide Star Facility
OO PTICAL DESIGN OF THE LASER LAUNCH TELESCOPE VIAPHYSICAL OPTICS THEOREM FOR L ASER G UIDE S TAR F ACILITY
A P
REPRINT
Yan Mo, ZhengBo Zhu, Zichao Fan, 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 22, 2021 A BSTRACT
The Laser Guide Star Facility (LGSF), as the most important part of the adaptive optics system ofthe large ground-based telescope, is aimed to generate multiple laser guide stars at the sodium layer.Laser Launch Telescope is employed to implement this requirement by projecting the Gauss beam tothe sodium layer with a small beam size in LGSF system. As the diffraction and interference effectsof laser’s long-distance transmission, the conventional optical design based on the geometrical opticsmechanism cannot achieve the expected laser propagation. In this paper, we propose a method todesign optical system for laser launch telescope based on the physical optics theorem to generate anacceptable light spot at the sodium layer in the atmosphere. Besides, a tolerance analysis methodbased on physical optics propagation is also demonstrated to be necessitated to optimize the system’sinstrumentation performance. The numerical results show that the optical design considering physicaloptics propagation is highly rewarding and even necessitated in many occasions, especially for laserbeam propagation systems. K eywords The Laser Guide Star Facility · Laser lunch telescope · Optical design · Physical optics propagation
The Laser Guide Star Facility (LGSF) is one of the most important parts of large ground-based telescopes to improvethe capability in high-resolution imaging of faint stars. Specifically, it is used to generate artificial laser guide starsfor adaptive optics (AO) systems to compensate for the perturbation caused by the atmosphere. The laser guide startechnique was firstly put forward by Happer et al in 1982 [1] and then the experiment was implemented by Primmerman et al [2]. In 2001, the laser guide star AO system with laser guide star was firstly installed on Keck I [3] and later onKeck II [4]. With the continuous progress of astronomical optical technology, the laser guide star system projectingseveral asterisms was assembled on Gemini telescope in 2011 [5], which demonstrated the well performance and highreliability of the LGSF. From then on, the LGSF has been widely used by many other famous observatories includingVLT [6], Subaru [7], and Thirty Meter Telescope (TMT) [8], etc.In order to project and expend the laser beam into sodium layer, a crucial component called Laser Launch Telescope(LLT)is employed in LGSF system. As for the optical requirements, the most important goal of LLT is to eliminate aberrationas much as possible to generate an acceptable light spot among the sodium layer at a predefined altitude (120km) andmaintain a high ratio of encircled energy. Specifically, the RMS wavefront error (WFE) of the LLT design should belimited to a reasonable value. A high encircled energy ratio means a high energy utilization efficiency, which contributesto produce bright artificial laser guide stars. ∗ 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
EBRUARY
22, 2021 lens1lens2 m m
Figure 1: Optical layout of LLTAs the long-distance of propagation of laser beam in free space, it is necessary to take the diffraction and interferenceeffects into consideration in the optical design process of LLT. In other words, the optical design of LLT should beimplemented in the physical optics theorem. However, it is almost impossible to conduct the optical design for LLTdirectly based on the physical optics theory simply relying on commercial optical design software such as ZEMAX. Asa rule of thumb, an optimal design process is to design the initial optical structure with the assumption of geometricaloptics approximation , and then optimize it based on the physical optics theorem.In this paper, we propose a method to design optical system for LLT based on the physical optics theorem. In thismethod, we firstly analyze the physical propagation model of laser beam, then design the initial optical system of LLTbased on the geometrical optics assumption. After that, the optical performance of the initial LLT is evaluated basedon physical optics. The RMS wavefront error (WFE) and the encircled energy ratio are selected as the criteria forevaluation of optical performance. Next, the optical system is updated with the physical optics theorem to achieve thepredefined optical requirements. Finally, we provide a tolerance analysis method based on Gauss beam propagation topredict the expected optical performance of LLT with instrumentation errors. For the finally obtained optical system,the encircled energy ratio within a diameter of 233mm at 120km exceeds . considering the tolerance allocation.And the largest RMS WFE is less than 0.016 λ with the working temperature ranging from -5 ◦ C to 20 ◦ C. The LLT is a laser beam expander essentially. To avoid the extremely tight optical and mechanical tolerances, we choosethe Galilei telescope as the initial structure of LLT. Additional advantages of this choice are the compact configurationand the avoidance of the internal focal point, which is beneficial to the mechanical fabrication. Only a single workingwavelength (589nm) is considered, and a Galilei telescope configuration with two singlets is employed. The opticallayout of LLT is shown in Fig. 1.For the general applications, the working temperature ranges from -5 ◦ C to 20 ◦ C is considered. To reduce the difficultyof mechanical alignment, the distance between Lens 1 and Lens 2 is set as 850mm in consideration of the generaldemands. Besides, the distance between Lens 1 and Lens 2 is expected to be adjustable within a range of ±0.25 mm tocompensate the performance degradation due to the manufacturing and assembly errors as well as the environmentaldisturbance. We choose the field of view (FOV) as 0.06 ◦ to match the general AO systems. As mentioned above, themain goal of LLT is to produce a small light spot at the sodium layer while maintaining high energy efficiency asmuch as possible, we select the RMS WFE and the enclosed energy ratio as the criteria for the evaluation of opticalperformance. Based on the science requirement of general AO systems, the radius of the light spot is usually limited to233mm at 120km in altitude. The design specifications are expressly presented in Table 1.Table 1: Design Requirements for LLT
Parameters ValueWavelength 589nmPupil position Lens 1RMS WFE 0.037 λ FOV 0.06 ◦ Working temperature -5 ◦ C ∼ ◦ CDistance between Lens 1 and Lens 2 850 ± PREPRINT - F
EBRUARY
22, 2021
FOV / ° E F W S M R / λ -5 ℃ ℃ ℃ Figure 2: RMS WFE Vs FOV for different temperature
The spherical aberration is the main error source that contributes the enclosed energy loss in the sodium layer. Based onSeidel sums [9], all of the primary aberrations coefficients of the given optical system can be numerically calculated.For the spherical aberration: S = 14 φ y ( AX − BXM + CM + D ) (1) W = S (2)where φ is the refractive power, M represents the position or conjugate parameter, X denotes the bending parameterand A , B , C , D are constants related to the refractive index. The position or conjugate parameter M is given by: M = u (cid:48) + uu (cid:48) − u = 1 + m − m (3)where u and u (cid:48) are the paraxial marginal ray angle before and after the lens respectively, m stands for the magnification.The bending parameter X is determined by: X = c + c c − c (4)where c and c are curvatures of a lens. From Eq. (1), it is obvious that the spherical aberration S depends on thesquare of the bending parameter X . Therefore a suitable choice of bending allocation for two lenses is necessitated tominimize spherical aberration. During the optimization progress, the location of the beam waist is constrained. Theoptimized optical parameters are listed in Table 2.To evaluate the optical performance of the obtained optical system at different temperatures, the thermal analysisis implemented. Three different working temperatures including -5 ◦ C, 9 ◦ C and 20 ◦ C are considered. The distancebetween Lens 1 and Lens 2 is selected as the compensator. As illustrated in Fig. 2, the initial design shows a good opticalperformance under different working conditions and meets the requirement of RMS WFE over full FOV. However, thecompensation distance reaches 0.5134mm, which may not satisfy the mechanical constraint.Table 2:
Optical parameter for LLT initial design
Element Material Curvature Thickness ConicRadius(mm) (mm)Lens1-S1 SILICA 397.239 70.022 -0.411Lens1-S2 1641.119 920.000Lens2-S1 SILICA -68.700 15.236 -0.972Lens2-S2 1120.1453
PREPRINT - F
EBRUARY
22, 2021 lens1lens2 m m k m Figure 3: Gauss Beam propagation of LLT
Note that, it is impossible to directly project the beam waist to the sodium layer by the above obtained optical systemaccording to the relationship between beam waist location and waist size, which is given by w = w (cid:115) (1 + z f ) (5)where w is the semi-diameter of the waist, z represents the propagation distance, and f demotes the rayleigh range. Theoutput beam /e diameter is set as 240mm to avoid beam clipping [10] referring to the existing designs [11, 12, 13].Based on the simulation result, the optimum beam waist locates at 34km in altitude with a waist radius of 100mm. Asthe long-distance propagation of laser beam in free space, the diffraction and interference effects can not be ignored.To precisely evaluate the optical performance of this specific optical system, the physical propagation model of lasershould be analyzed. The Gauss beam propagation model based on the physical simulation of the above obtained LLTsystem is shown in Fig. 3.As the brightness and the beam quality is highly required by the AO System, the encircled energy ratio inside a specificradius is chosen as one of the most important assessment criteria of optical performance. The amplitude of an idealcollimated Gauss beam can be represented by: A ( r ) = a exp( − r w ) . (6)And the corresponding irradiance is calculated as: I ( r ) = I exp( − r w ) (7)where r denotes the light spot radius and w represents the specific value of r when the irradiance equals I /e . Asexpressed in Eq. (7), the beam brightness and quality is relevant to w , which is given by Eq. (5). Therefore, the ratio ofthe encircled energy of an ideal Gauss beam can be calculated as: E ( r = a ) = (cid:82) a (cid:82) π I ( r )2 πrdrdθ (cid:82) ∞ (cid:82) π I ( r )2 πrdrdθ (8)The normalized irradiance distribution at 120km is shown in Fig. 4. Only . of energy is encircled inside thecircular domain with a radius of 233mm, which cannot satisfy the science requirement as listed in Table 1. This resultindicates that the optical design optimization procedure under geometrical optics evaluation criterion is ineffective forthe long-distance propagation of laser beam. A precise optical design is necessary, for instance, the optical design basedon the physical optics theorem is needed. In Section 3.A, the location of the beam waist is constrained in the optimization process while minimizing the RMSWFE, however, the optical performance can not meet the requirements as analyzed in Section 3.B. In this Section,the merit function is replaced by the physical optics evaluation criterion. Meanwhile, the geometrical ray tracingperformance should be satisfied.As mentioned above, a shorter compensation distance is preferred considering the instrumentation, and we make atrade-off discussion of material choice to minimize the compensation distance. Three different kinds of typical optical4
PREPRINT - F
EBRUARY
22, 2021 -400 -320 -240 -160 -80 0 80 160 240 320 400
Radius/mm R e l a t i v e I rr a d i a n ce -233 233 Figure 4: The encircled energy ratio of the initial design -5 -4 dn / d T Temperature/
SILICAF2BK7 ℃ Figure 5: Temperature coefficient of the absolute refractive index of three different optical glassesglass including SILICA, F2, and BK7 are taken into consideration. The temperature coefficient of the absolute refractiveindex for a specific material can be expressed: dndT = n − n ( D + 2 D ∆ T + 3 D ∆ T + E + 2 E ∆ Tλ − λ T K (9)where n represents the refractive index relative to vacuum; ∆ T is the temperature difference; λ stands for the wavelength; D , D , D , E , E , and λ T K are constants depending on the glass type. The temperature coefficient of the absoluterefractive index of three different types of optical glass is shown in Fig. 5. It is obvious that F2 and BK7 has a relativelylower temperature coefficient of refractive index compared with SILICA, and we choose F2 and BK7 as the newmaterials for LLT and conduct the optimization with the physical optics evaluation criterion on the initial design. Thedistance between Lens 1 and Lens 2 is selected as a compensator. The final positions of the optical components afterthe optimization process are presented in Table 3. The optical performance is also evaluated by RMS WFE and theencircled energy ratio with different working temperatures. The relationship between RMS WFE and FOV is shown inFig. 6. It shows that the optimized design shows good optical performance in terms of RMS WFE and satisfies thepredefined specifications well. The largest RMS WFE is less than 0.016 λ in the temperature range of -5 ◦ C to 20 ◦ C.The compensation distance between Lens 1 and Lens 2 is reduced to 0.033mm, which is far less than that of the initialdesign.Similarly, we evaluate the encircled energy ratio within a diameter of 233mm at 120km in altitude, as shown in Fig. 7,wherein 96.10 % energy efficiency is achieved. All of these results demonstrate the optical design based on the physicaloptics theorem is necessitated for the laser’s long-distance propagation after the initial design is constructed based ongeometrical ray tracing. 5 PREPRINT - F
EBRUARY
22, 2021Table 3:
Optical parameter for LLT initial design
Element Material Curvature Thickness ConicRadius(mm) (mm)Lens1-S1 BK7 392.831 70.000 -0.378Lens1-S2 1356.645 850.000Lens2-S1 F2 -82.124 13.000 -0.824Lens2-S2 6546.516 E F W S M R / λ FOV / ° -5 ℃ ℃ ℃ Figure 6: RMS WFE vs Field of View under different temperatures
Due to the special working environment of LLT, it is necessary to analyze its optical performance in different situations.In this research, we choose RMS WFE as the performance metric to allocate the tolerance. Assigning wavefront errorsto each optical element of the LGSF system will be used to guide the fabrication of each optical component and thedesign of mechanical devices. The final RMS WFE budget for LLT should be limited to 0.037 λ . A reasonable toleranceallocation of optical parameters is given after the sensitivity being analyzed. The estimated RMS WFE shows that theallocation can confirm to the performance deviation requirement well. As we discussed in Section 3, optical simulationbased on the physical optics model is much more convinced than geometrical ray tracing. Hence, during the tolerancingprocess, we take the encircled energy ratio within a diameter of 233mm at 120km as the merit function in Monte Carlotolerance analysis to estimate the expected physical optics propagation performance of LLT based on the allocation thatwe made above. -400 -320 -240 -160 -80 0 80 160 240 320 400 Radius/mm R e v i t a l e I ec n a i d a rr -233 233 Figure 7: The encircled energy ratio of the optimized design.6
PREPRINT - F
EBRUARY
22, 2021
Set of assigned Tolerance
Evaluate Physical Optics
PerformanceLLT DesignEncircled Energy
Acceptable ? Encircled Energy Acceptable ? Adjust Compensator
Further Performance Simulation with Assigned Tolerance
Yes YesNo No
Figure 8: Tolerance analysis flow chart.Firstly, the tolerance parameters need to be assigned. Each of the design parameters is perturbed within the range oftolerance allocation following a modified Gaussian normal distribution, which is given by: p ( t ) = 1 √ πσ exp( − t σ ) . (10)After the tolerance being assigned, a perturbed LLT module is created to evaluate the physical optics performance. Ifthe enclosed energy of the perturbed module is unacceptable, a changeable parameter will be selected as a compensator.Hence, we perform the optimization of the distance between Lens 1 and Lens 2 of the perturbed module to compensatefor the energy loss caused by the allocated tolerance parameter via the physical optics propagation simulation. Duringthe optimization process, the compensation distance is restricted to 0.5mm. If the physical optics performance of thecompensated module is acceptable, the enclosed energy ratio will be output as the result of each tolerance iteration. Ifnot, the tolerance parameters are considered tight, and the tolerance iteration will be performed again until the enclosedenergy ratio is acceptable. The tolerance process is shown in Fig. 8.This process has been implemented in each iteration to guarantee a proper compensation distance and toleranceallocation. After 400 times of Monte Carlo analysis, a resealable tolerance allocation has been obtained as listed inTable 4 considering the achievable fabrication ability, and parts of the Monte Carlo results is shown in Fig. 9. It isobvious that the optimized system is not sensitive to fabrication when tolerance allocation is being considered. Theenclosed energy ratio is larger than . for all working conditions, which demonstrates that the provided tolerancemethod and allocation are convinced. Table 4: Design Requirements for LLT
Parameters (mm) Tolerance (mm)Lens1 surface1 Radius 0.1Lens1 surface2 Radius 0.4Lens1 thickness 0.2Lens2 surface1 Radius 0.1Lens2 surface2 Radius 0.4Lens2 thickness 0.1
To summarize, we have provided a recommended optical design method for laser launch telescope based on the physicaloptics theorem which is aimed to generate multiple laser guide stars at sodium layer. The optical design starts with theinitial optical system design based on the geometrical optics assumption, and then we optimize the optical system viathe physical optics theorem. Besides, the tolerance analysis is also provided to evaluate the feasibility of instrumentationbased on the physics optics propagation. The simulation results show that the proposed optical design method based on7
PREPRINT - F
EBRUARY
22, 2021 C u m u l a t i v e F re qu e n c y Enclosed Energy
Figure 9: Monte Carlo tolerance result.precise physical optics propagation is highly rewarding and even necessitated for the laser beam propagation systems.We believe that our work might provide a good guidance for researchers to design similar laser propagation systems inthe future.
References [1] W Happer, GJ MacDonald, CE Max, and FJ Dyson. Atmospheric-turbulence compensation by resonant opticalbackscattering from the sodium layer in the upper atmosphere.
JOSA A , 11(1):263–276, 1994.[2] Charles A Primmerman, Daniel V Murphy, Daniel A Page, Byron G Zollars, and Herbert T Barclay. Compensationof atmospheric optical distortion using a synthetic beacon.
Nature , 353(6340):141–143, 1991.[3] Jason CY Chin, Peter Wizinowich, Randy Campbell, Liz Chock, Andrew Cooper, Ean James, Jim Lyke, JoeMastromarino, Olivier Martin, Drew Medeiros, et al. Keck i laser guide star adaptive optics system. In
AdaptiveOptics Systems III , volume 8447, page 84474F. International Society for Optics and Photonics, 2012.[4] Jason CY Chin, Peter Wizinowich, Ed Wetherell, Scott Lilley, Sylvain Cetre, Sam Ragland, Drew Medeiros,Kevin Tsubota, Greg Doppmann, Angel Otarola, et al. Keck ii laser guide star ao system and performance withthe toptica/mpbc laser. In
Adaptive Optics Systems V , volume 9909, page 99090S. International Society for Opticsand Photonics, 2016.[5] Céline d’Orgeville, Sarah Diggs, Vincent Fesquet, Benoit Neichel, William Rambold, François Rigaut, AndrewSerio, Claudio Araya, Gustavo Arriagada, Rodrigo Balladares, et al. Gemini south multi-conjugate adaptive optics(gems) laser guide star facility on-sky performance results. In
Adaptive Optics Systems III , volume 8447, page84471Q. International Society for Optics and Photonics, 2012.[6] W Hackenberg, D Bonaccini Calia, B Buzzoni, M Comin, C Dupuy, F Gago, IM Guidolin, R Guzman, R Hol-zloehner, L Kern, et al. Assembly and test results of the aof laser guide star units at eso. In
Adaptive OpticsSystems IV , volume 9148, page 91483O. International Society for Optics and Photonics, 2014.[7] Yosuke Minowa, Yutaka Hayano, Hiroshi Terada, Tae-Soo Pyo, Shin Oya, Masayuki Hattori, Mai Shirahata,Hideki Takami, Olivier Guyon, Vincent Garrel, et al. Subaru laser guide adaptive optics system: performance andscience operation. In
Adaptive Optics Systems III , volume 8447, page 84471F. International Society for Opticsand Photonics, 2012.[8] M Li, CC Jiang, K Wei, et al. Design of the tmt laser guide star facility.
Opto-Electronic Engineering , 45(3):170735,2018.[9] H Gross, H Zügge, M Peschka, and F Blechinger. Handbook of optical systems, aberration theory and correctionof optical systems (2015).
Chap , 29:1–70.[10] Ronald Holzlöhner, Domenico Bonaccini Calia, and Wolfgang Hackenberg. Physical optics modeling andoptimization of laser guide star propagation. In
Adaptive Optics Systems , volume 7015, page 701521. InternationalSociety for Optics and Photonics, 2008. 8
PREPRINT - F
EBRUARY
22, 2021[11] Domenico Bonaccini, Eric Allaert, Constanza Araujo, Enzo Brunetto, Bernard Buzzoni, Mauro Comin, Martin JCullum, Richard I Davies, Canio Dichirico, Philippe Dierickx, et al. The vlt laser guide star facility. In
AdaptiveOptical System Technologies II , volume 4839, pages 381–392. International Society for Optics and Photonics,2003.[12] Céline d’Orgeville, Brian Bauman, Jim Catone, Brent Ellerbroek, and Richard Buchroeder. Gemini north andsouth laser guide star systems requirements and preliminary designs. In
International Symposium on OpticalScience and Technology , 2002.[13] Corinne Boyer, Brent Ellerbroek, Luc Gilles, and Lianqi Wang. The tmt laser guide star facility. In1st AO4ELTconference-Adaptive Optics for Extremely Large Telescopes