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Telescope design for 2 μm spacebased coherent wind lidar system
Optics Communications 315 (2014) 238–242
Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Telescope design for 2 μm spacebased coherent wind lidar system Xianying Ge, Siying Chen n, Yinchao Zhang, He Chen, Pan Guo, Taotao Mu, Jian Yang School of Optoelectronics, Beijing Institute of Technology, 5 South Zhongguancun Street, Haidian District, Beijing, China
art ic l e i nf o
a b s t r a c t
Article history: Received 13 February 2013 Received in revised form 24 October 2013 Accepted 12 November 2013 Available online 25 November 2013
The telescope is a key optical component which largely affects the signal intensity in coherent Doppler Lidar. In the paper the normalized signal to noise ratio(SNR) as a function of range z and diameter D of the telescope is investigated for pulsed coherent wind lidar, then the telescope aperture requirement for 300 km orbit is calculated and presented. After that the initial structure parameters for space-based (40 ) off-axis telescope are obtained with the PW method, furthermore, the initial structure has been optimized by the Zemax software. The telescope's RMS wavefront error is optimized to be less than 1/10λ (2 μm), and the tolerance requirement is discussed. The designed structure of the telescope is simple and compact, can be easily applied to other similar fields due to its high portability. & 2013 Elsevier B.V. All rights reserved.
Keywords: Telescope design Wavefront error Third-order aberration Tolerance
1. Introduction Atmospheric wind field is an important aspect in atmospheric physics, and the wind field data is very useful for military and civilian applications. For example, accurate detection of the wind field is helpful to the stability and accuracy of the launch and operation of the carriers running in the atmosphere [1,2], such as aircrafts, spacecrafts and missiles, conducive to the study of global climate change and improve the accuracy of weather forecast, beneficial to reduce the occurrence of various disasters by judging the typhoon trend, favourable to the research of atmospheric composition. As an active remote sensing tool, the lidar owns many good characteristics, for instance, high spatial and temporal resolutions, real-time detection, high mobility, facilitated control, and now it is one of the important means to measure atmospheric wind field. Currently, detecting the Doppler frequency shift is the main way for measuring atmospheric wind in the lidar method. There are two basic ways to weigh the value of Doppler shift. The first one is the direct detection method [3], which converts the change of frequency of the light to the change of power intensity. The second method is called the coherent (or heterodyne) detection, in which the beat-signal can be obtained by mixing the return signal and a local oscillator laser, moreover, the frequency of the beatsignal is the Doppler shift due to the moving particles apart from a fixed offset. Most of the lidar systems using coherent detection technology are established after the late 1980s, and they have
n
Corresponding author. Tel.: þ 86 10 68918398. E-mail address: [email protected] (S. Chen).
0030-4018/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.optcom.2013.11.020
been made significant progress in the past few decades, the greatest advance is the detection laser. The light source used in such systems were firstly 10.6 μm consecutive carbon dioxide laser [4], then developed to the pulse form which enlarged the detection distance [5], subsequently, the 1.06 μm solid pulsed laser [6] was chosen due to the reasons such as the volume and detection efficiency, and now the eye-safe 2 μm pulse laser attracts more attention [7,8]. At present both detection methods are under parallel development with their own characters. RMS wavefront deviation largely influences the lidar efficiency, it is reported that for a 3 dB loss of lidar efficiency, the RMS wavefront deviation must be between λ/5 and λ/10 [9], and for well designed systems the RMS wavefront deviation is less than λ/10 at 2 μm. To achieve such requirement, the telescopes for coherent lidar detection are gradually developed from the initial transmittance Galileo structure [10] to separated reception mode with double telescope [11] (bistatic system), and currently the reflecting telescopes are employed in most of the systems which serve both as transmitter and receiver. The volume of the system is continuously reduced, and the negative influence of heterodyne efficiency due to telescope's design and processing is also getting smaller. As the telescope acts as a beam expander in the transmitting path and a collector of weak signal in return optical path, its design directly affects the detection accuracy. Because the cost of large diameter transmission system is not high, at present the telescopes used in coherent wind measure systems are generally reflection-type. The high performance coherent systems requires that telescopes should be off-axis to avoid central obscuration, small wavefront error to improve the followed heterodyne detection efficiency, and without internal focus to avoid the air breakdown caused by high-power laser [12].
X. Ge et al. / Optics Communications 315 (2014) 238–242
In the paper we introduce the SNR calculation method and present the way to to obtain the best telescope aperture. First, we simplify SNR formula by ignoring the factors which are irrelevant with the telescope aperture; second, transverse coherence length is calculated according to Hufnagel–Valley model; finally, the optimum telescope aperture for spacebased coherent wind lidar is obtained from SNR formula under some fixed conditions. Meeting the aperture and design requirement the initial structure of the telescope is calculated with PW method, and the initial structure is optimized in Zemax software to achieve more reasonable parameters. Compared with the telescope reported in the literature [13], the diameter of the telescope in this paper is larger, more suitable for spaceborne coherent lidar needs.
2. Telescope aperture optimization for spacebased coherent wind lidar In the lidar systems, the signal to noise ratio (SNR) is an important indicator to judge the lidar performance, the higher SNR in the detection range, the more accuracy for wind speed and direction after data inversion. In the relevant literatures [14] the principle that how to calculate the SNR in coherent wind lidar has been presented, it pointed out that SNR is related to the factors such as telescope aperture, heterodyne efficiency, transverse coherence length of the received field. But in these papers they didn't present the general sense of telescope aperture. In the following part, telescope aperture for space based lidar is optimized based on the signal to noise ratio formula. For shot-noise-limited coherent wind detection, the estimated value for the SNR is given by the following equation [15]: 2 3 !2 1 E0 ηβλK 2 π D2 4 z 2 π D2 D 25 SNR ¼ 1þ 1 þ ð1Þ f 2S0 8hBz2 4λ z where E0 is the laser pulse energy [J], η indicates the lidar system's optical/detection efficiency (electrons/photon), D means the telescope diameter [m], β(z) denotes the backscatter coefficient [m 1 sr 1], λ represents the laser wavelength [m], K is the oneway atmospheric transmission to range z [m 1], B means the system narrow bandwidth (1/τ) [Hz], h is the Planck's constant, h¼ 6.626 10 34, and S0 indicates the transverse coherence length of the received field in meters. From Eq. (1) it is obviously that SNR is related to many system parameters, in these parameters E0, η, λ, B are constant for a fixed system; β(z), K are related with the distance z. In order to analyze
239
the relationship between SNR and the telescope aperture, we can extract the items related to the telescope aperture, then Eq. (1) can be simplified: 2 3 !2 1 D2 4 z 2 π D2 D 25 SNR p 2 1 þ 1 þ ð2Þ f 2S0 z 4λ z
2.1. Detection range The signal to noise ratio is different for different detection range and telescope aperture. Coherent wind lidar systems can be divided into groundbased, airborne and spacebased systems according to the different detection range. Relatively speaking, spacebased systems have the biggest detection range and the greatest practical value, but by now there is still no such system running in the space due to the fund and other reasons [16]. In the article, we mainly analyzed the telescope aperture for spacebased systems, the satellite orbit is from 200 km to 600 km, detection range is 0–30 km from the earth surface, and finally we present one telescope design example for the 300 km orbit. 2.2. Transverse coherence length Transverse coherence length is closely related to the refractive index structure constant, in the atmosphere transverse coherence length can be expressed as [17]: " 5 # 35 Z R z′ 3 2 S0 ðzÞ ¼ Hk C n 2 ðz′Þ 1 dz′ ð3Þ z 0 where H ¼ 2:914383, k¼ 2π/λ is the wave number and z is the detection range. C n 2 is the value of the refractive index structure parameter of the atmosphere. We use Hufnagel–Valley model of the refractive index structure parameter to calculate the transverse coherence length, taking use of strong turbulence condition [18,19]: 2 21 h C 2n ðhÞ ¼ 0:00594 ð10 5 hÞ10 exp 27 1000 h h 16 þ 2:7 10 þ 1:7 10 14 exp exp 1500 100 ð4Þ where h is detection height from the ground, the range can be taken from 0 km to 30 km, in space application the height can be
Fig. 1. The normalized SNR intensity against distance z and the telescope aperture D.
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converted to the detection altitude, taken as 300 km as example. 2 21 300 z C 2n ðzÞ ¼ 0:00594 ½10 5 ð300 zÞ10 exp 27 1000 300 z 16 þ2:7 10 exp 1500 300 z ð5Þ þ1:7 10 14 exp 100 Taking Eq. (5) into Eq. (3) and substituting S0 in Eq. (2) with expression (3), then we can simulate and obtain the threedimensional figure which demonstrates the relationships between normalized SNR value and distance z, the telescope aperture D, shown as Fig. 1. It can be seen from Fig. 1 that the normalized SNR value increases with the telescope diameter first and then decreases at a certain orbit, the telescope diameter corresponding to the maximum SNR value increases with distance; simultaneously, the maximum SNR value also decreases with distance, which indicates that SNR would be gradually reduced as the space platform farther from earth, and at this time the laser energy need to be enhanced to compensate SNR degradation. Assuming spatial platform is 300 km from the earth surface, and the telescope diameter value corresponding to the maximum SNR value is about 0.83 m. But the design, process, and the final assembly of the telescope are difficult for the off-axis telescope systems. In the actual application the SNR value can be reduced to about 0.7 times, then the telescope diameter is about 0.5 m. In the following part we would introduce how to design and optimize a 0.5 m diameter off-axis telescope.
3. Telescope designed for space-based coherent wind lidar At present there is not any true sense of the global atmospheric wind lidar running in space, but some plans have been carried out. Taking use of the direct detection method ADM-Aeolus satellite is planned to launch by the end of 2013 [20]. Using the coherent detection mean researchers in Japan had proposed a plan called “Science plan on the wind measurements by the ISS/JEM-borne coherent Doppler lidar” [16], scientists in America had ever developed the SPARCLE equipment for the global wind measurement [21], but both projects eventually were suspended for technical and financial reasons. A new plan which intends to use hybrid technology combing coherent and direct method is “NexGen NPOESS” program, pre-system verification is ongoing. As analyzed in part 2, the telescope diameter is about 0.5 m for space coherent wind systems on orbit 300 km; apart from the field angle simulated from Siegman's antenna theorem [22], the field of view should also cover the lag angle of return beam, in some
systems the impact of the scan mode also need to be taken into account [13], here we just use the the detecting mode without scanning, then the field of view is small,about 0.055 mrad (half field of view). For large aperture of the off-axis telescope the twomirror system is unable to meet the requirements because the tolerances requirement is too high. After analyzation we choose the structure which is comprised by two mirrors and an aspherical lens, as shown in Fig. 2. 3.1. PW method introduction There are five monochromatic aberrations, spherical aberration, coma, astigmatism, field curvature and distortion, their respective third-order aberration coefficients are expressed by SI, SII, SIII, SIV, SV, and presented in PW form as follows [23]: 8 k k > 4 > SI ¼ ∑ hi P i þ ∑ hi K i > > > > i¼1 i¼1 > > > > k k k > 3 > > > SII ¼ ∑ hpi P i þ ∑ hpi hi K i þ J ∑ W i > > i¼1 i¼1 i¼1 > > > > < k h 2 k k h k 1 u pi pi 2 2 SIII ¼ ∑ P i þ ∑ hpi hi K i þ 2J ∑ W i þ J2 ∑ Δ i > i ¼ 1 hi i¼1 i ¼ 1 hi i ¼ 1hi ni > > > > k > n ′ ni > > SIV ¼ ∑ J 2 i ′ > > > ni ni r i > i ¼ 1 > > > 3 2 > ′ > > S ¼ k hpi P þ þ k h 3 h K þ 3J k hpi W þ J 2 k hpi ð 3 Δui þ ni ni Þ J 2 k 1 Δ 1 > ∑ 2 i ∑ pi i i ∑ 2 i ∑ ∑ 2 > : V ni ′ ni r i ni 2 i ¼ 1 hi i¼1 i ¼ 1 hi i ¼ 1 hi hi ni i ¼ 1hi
ð6Þ where P i ¼ ðΔui =ðΔð1=ni ÞÞÞ Δðui =ni Þ, W i ¼ ðΔui =ðΔð1=ni ÞÞÞ Δðui = ni Þ, K i ¼ ððni ′ ni Þe2i =r 3i Þ. hi, hpi respectively represent the heights of the axial and oblique rays on the ith surface, ui, ui ′ represent slope angles corresponding with the ith surface, ei denotes aspherical coefficient, ni, ni ′ represent the refractive index before and after ith surface, ri denotes the curvature radius. 2
3.2. Telescope design process and results In the subsequent section an off-axis telescope with a 500 mm diameter, 0.055 mrad field angle and 40 afocal magnification is presented. The structure of the telescope is shown in Fig. 2, here the lens is assumed to be a thin lens, the refractive index is n, similarly, parameters related to the contour are defined as follows:
α1 ¼
l2 h2 l′1 h1
ð7Þ
α2 ¼
l3 h3 l′2 h2
ð8Þ
β1 ¼
l′2 u2 ¼ l2 u′2
ð9Þ
β2 ¼
l′3 u3 ¼ l3 u′3
ð10Þ
In order to simplify the formula, the second surface of the thin lens is assumed to be plane. Combined with (7)–(10)the following expressions can be obtained from Gaussian optical theory: r2 ¼
α1 β 1 r 1 1 þ β1
r3 ¼
ðn 1Þα2 α1 β 1 r 1 2
d1 ¼ ð1 α1 Þ
Fig. 2. Telescope system diagram for space-based coherent wind lidar.
ð11Þ
d2 ¼
r1 2
ð13Þ
α1 β1 ð1 α2 Þ 2
ð12Þ
r1
ð14Þ
X. Ge et al. / Optics Communications 315 (2014) 238–242
241
Fig. 3. Off-axis wavefront error changes before and after optimization. Table 1 The initial structure parameters of the space-based telescope. Surface no.
Curvature radius/mm
The distance/mm
Conicoid coefficient
Off-axis value/mm
Refractive index
1 2 3 4
1800 421.6697 162.7788 Infinity
731.3110 731.3110 5 30
1.04286 2.71809 0.30306 —
500 100 12.5 12.5
— — 2.4462 —
Table 2 The optimized parameters of the space-based telescope. Surface no.
Curvature radius/mm
The distance/mm
Conicoid coefficient
Off-axis value/mm
Refractive index
1 2 3 4
1800 422.3369 163.7008 Infinity
730.5195 745.2456 5 30
1.0642 2.9064 0.2563 —
500 100 12.5 12.5
— — 2.4462 —
According to the previous statement, the magnification is 40 , so α1 α2 ¼ 0:025; considering the volume and compactness it can be assumed that d1 ¼ d2 ; with a relatively high refractive index and transmittance at a wavelength of 2 μm ZnSe glass can be selected, the refractive index would be calculated as 2.4462 from Sellmeier equation. Using these conditions in Eq. (6) the following results could be obtained: 8 2 > < e1 ¼ 1:04286 e2 2 ¼ 2:71809 > : 2 e3 ¼ 0:30306
ð15Þ
It is assumed that the curvature radius of the first surface is 1800 mm, the ZPL code can be programmed in accordance with the relationship between r 2 ; r 3 ; d1 ; d2 and r 1 with the expressions (11)–(14),the system parameters input Zemax are listed in Table 1. The Seidel aberration coefficients of spherical aberration, coma, astigmatism, field curvature in Zemax are all zero, verifying the calculation results are correct. Next taking the wavefront as optimal condition, the three aspherical coefficients, thicknesses and curvature
Fig. 4. The optimized optical path diagram of the telescope.
radiuses are made variable to optimize, and the equilibrium between primary aberration and senior aberration is reached quickly. The three aspheric coefficients are optimized to 1.0642, 2.9064 and 0.2563, little change compared with the value of the initial
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Table 3 The tolerance data as the RMS wavefront error less than 1/10λ (2 μm). No.
ΔR/mm
Δe2
Δd/mm
Dx/mm
Dy/mm
Tx/(′)
Ty/(′)
1 2 3 4 3–4
70.02 70.04 70.1 — —
70.0002 70.0015 70.05 — —
7 0.01 7 0.15 7 0.5 — —
7 0.005 7 0.008 7 0.015 7 0.2 7 0.03
7 0.005 7 0.008 7 0.015 7 0.2 7 0.03
7 0.09 7 0.19 73 73 7 1.8
7 0.09 7 0.19 73 73 7 1.8
Note: ΔR represents the tolerances of curvature radius;Δe2 represents conicoid coefficient tolerances;Δd is the thickness tolerance; Dx,Dy represents the eccentric in the x, y direction; Tx,Ty represents the tilt in the x, y direction;3–4 indicates the collimating lens.
structure, the parameters after optimization are listed in Table 2. Offaxis wavefront error changes are shown in Fig. 3. The RMS wavefront error of 0 field changes from 0.2109waves to 0.0068waves, 1 field decreases from 0.2110waves to 0.0068waves, optimized optical path shows in Fig. 4. Tolerance analysis is carried out for ensuring that the RMS wavefront error of the entire system can be smaller than the 1/10λ (2 μm), the final tolerances data is listed in Table 3. So far telescope design has been completed for spacebased coherent wind lidar systems. Here the aspheric coefficients, curvature radius, thicknesses are made variables to be optimized, under other conditions refractive index also could be changed for smaller volume or other structural forms. 4. Conclusion The telescope is a key optical component in coherent wind lidar system, its quality largely influences heterodyne detection efficiency. In the paper we introduce the way to calculate the aperture for spacebased coherent wind lidar system, and present the aperture needed for 300 km orbit. The requirements of coherent detection for telescope such as off-axis, little wavefront error, no internal focus, high-magnification in the space-based detection make telescope's design and process difficult. From these requirements, the initial structure of the space-based telescope system is calculated with third-order aberration theory, then system parameters are optimized on the basis of the initial structures with the
help of Zemax software, finally tolerance analysis results for spacebased telescope are given in the off-axis state, providing the data support for subsequent processing. The deduced aberration formula can be used not only in coherent wind lidar, but also suitable for other similar afocal systems.
Acknowledgement This work was supported by the National Natural Science Foundation of China (NSFC) projects no. 61178072.
References [1] J.K. Esser, J.S. Lindoerfer, J. Behav. Decis. Making 2 (1989) 167. [2] T. Sakimura, Y. Watanabe, T. Ando, et al., 1.5-μm high average power laser amplifier using an Er,Yb:glass planar waveguide for coherent doppler LIDAR. SPIE 8526 (2012) 1. [3] Z.S. Liu, S.H. Wu, B.Y. Liu, Opt. Laser Technol. 39 (3) (2007) 541. [4] R.M. Huffaker, Appl. Opt. 9 (1970) 1026. [5] M.J. Post, R.E. Cupp, Appl. Opt. 29 (1990) 4145. [6] J.G. Hawley, R. Targ, R. Bruner, et al., Performance of a l μm, 1-joule Coherent Launch Site Atmospheric Wind Sounder. SPIE 1694 (1992) 110. [7] P. Laporta, M. Marano, L. Pallaro, et al., Opt. Lasers Eng. 37 (2002) 447. [8] C.T. Wu, Y.L. Ju, Q. Wang, Z.G. Wang, et al., Opt. Commun. 284 (2011) 9948. [9] T. Fujii, T. Fukuchi, Laser Remote Sensing, Taylor & Francis Group (2005) 615. [10] A.A. Woodfield, J.M. Vaughan, Airspeed and Wind Shear Measurements with an Airborne CO2 CW Laser. AGARDograph 272 (1983) 7.1. [11] M. Harris, G. Constant, C. Ward, Appl. Opt. 41 (2001) 1501. [12] F. Chen, A. Anees, A. Farzin, Design and Analysis of a Spaceborne Lidar Telescope. SPIE 2540 (1995) 68. [13] A. Anees, A. Farzin, F. Chen, et al., Design and Fabrication of a Compact Lidar Telescope, SPIE 2832 (1996) 34. [14] R.M. Huffaker, Proc. IEEE 84 (1996) 181. [15] S.W. Henderson, P.J.M. Suni, C.P. Hale, et al., IEEE Trans. Geosci. Remote Sens. 31 (1993) 4. [16] S. Ishii, T. Iwasaki, M. Sato, et al., Future doppler lidar wind measurement from space in Japan. SPIE 8529 (2012) 1. [17] K.P. Chan, D.K. Killinger, Appl. Opt. 31 (1992) 4915. [18] R.J. Sasiela, J. Opt. Soc. Am. A 9 (1992) 1398. [19] X.R. Wang, L. Guo, Y.J. Liu, et al., Opt. Commun. 306 (2013) 42. [20] D.G.H. Tan, E. Andersson, M. Fisher, et al., Q. J. R. Meteorol. Soc. 133 (2007) 381. [21] G.D. Emmitt, SPARCLE: An Approved Shuttle Mission to Demonstrate Tropospheric Wind Sensing Using a Coherent2micron Doppler Lidar. SPIE 3439 (1998) 31. [22] A.E. Siegman, Appl. Opt. 5 (1966) 1588. [23] J.H. Pan, The Design, Manufacture and Test of the Aspherical Optical Surfaces, Suzhou: Suzhou University Press (2004) 12.
Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Telescope design for 2 μm spacebased coherent wind lidar system Xianying Ge, Siying Chen n, Yinchao Zhang, He Chen, Pan Guo, Taotao Mu, Jian Yang School of Optoelectronics, Beijing Institute of Technology, 5 South Zhongguancun Street, Haidian District, Beijing, China
art ic l e i nf o
a b s t r a c t
Article history: Received 13 February 2013 Received in revised form 24 October 2013 Accepted 12 November 2013 Available online 25 November 2013
The telescope is a key optical component which largely affects the signal intensity in coherent Doppler Lidar. In the paper the normalized signal to noise ratio(SNR) as a function of range z and diameter D of the telescope is investigated for pulsed coherent wind lidar, then the telescope aperture requirement for 300 km orbit is calculated and presented. After that the initial structure parameters for space-based (40 ) off-axis telescope are obtained with the PW method, furthermore, the initial structure has been optimized by the Zemax software. The telescope's RMS wavefront error is optimized to be less than 1/10λ (2 μm), and the tolerance requirement is discussed. The designed structure of the telescope is simple and compact, can be easily applied to other similar fields due to its high portability. & 2013 Elsevier B.V. All rights reserved.
Keywords: Telescope design Wavefront error Third-order aberration Tolerance
1. Introduction Atmospheric wind field is an important aspect in atmospheric physics, and the wind field data is very useful for military and civilian applications. For example, accurate detection of the wind field is helpful to the stability and accuracy of the launch and operation of the carriers running in the atmosphere [1,2], such as aircrafts, spacecrafts and missiles, conducive to the study of global climate change and improve the accuracy of weather forecast, beneficial to reduce the occurrence of various disasters by judging the typhoon trend, favourable to the research of atmospheric composition. As an active remote sensing tool, the lidar owns many good characteristics, for instance, high spatial and temporal resolutions, real-time detection, high mobility, facilitated control, and now it is one of the important means to measure atmospheric wind field. Currently, detecting the Doppler frequency shift is the main way for measuring atmospheric wind in the lidar method. There are two basic ways to weigh the value of Doppler shift. The first one is the direct detection method [3], which converts the change of frequency of the light to the change of power intensity. The second method is called the coherent (or heterodyne) detection, in which the beat-signal can be obtained by mixing the return signal and a local oscillator laser, moreover, the frequency of the beatsignal is the Doppler shift due to the moving particles apart from a fixed offset. Most of the lidar systems using coherent detection technology are established after the late 1980s, and they have
n
Corresponding author. Tel.: þ 86 10 68918398. E-mail address: [email protected] (S. Chen).
0030-4018/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.optcom.2013.11.020
been made significant progress in the past few decades, the greatest advance is the detection laser. The light source used in such systems were firstly 10.6 μm consecutive carbon dioxide laser [4], then developed to the pulse form which enlarged the detection distance [5], subsequently, the 1.06 μm solid pulsed laser [6] was chosen due to the reasons such as the volume and detection efficiency, and now the eye-safe 2 μm pulse laser attracts more attention [7,8]. At present both detection methods are under parallel development with their own characters. RMS wavefront deviation largely influences the lidar efficiency, it is reported that for a 3 dB loss of lidar efficiency, the RMS wavefront deviation must be between λ/5 and λ/10 [9], and for well designed systems the RMS wavefront deviation is less than λ/10 at 2 μm. To achieve such requirement, the telescopes for coherent lidar detection are gradually developed from the initial transmittance Galileo structure [10] to separated reception mode with double telescope [11] (bistatic system), and currently the reflecting telescopes are employed in most of the systems which serve both as transmitter and receiver. The volume of the system is continuously reduced, and the negative influence of heterodyne efficiency due to telescope's design and processing is also getting smaller. As the telescope acts as a beam expander in the transmitting path and a collector of weak signal in return optical path, its design directly affects the detection accuracy. Because the cost of large diameter transmission system is not high, at present the telescopes used in coherent wind measure systems are generally reflection-type. The high performance coherent systems requires that telescopes should be off-axis to avoid central obscuration, small wavefront error to improve the followed heterodyne detection efficiency, and without internal focus to avoid the air breakdown caused by high-power laser [12].
X. Ge et al. / Optics Communications 315 (2014) 238–242
In the paper we introduce the SNR calculation method and present the way to to obtain the best telescope aperture. First, we simplify SNR formula by ignoring the factors which are irrelevant with the telescope aperture; second, transverse coherence length is calculated according to Hufnagel–Valley model; finally, the optimum telescope aperture for spacebased coherent wind lidar is obtained from SNR formula under some fixed conditions. Meeting the aperture and design requirement the initial structure of the telescope is calculated with PW method, and the initial structure is optimized in Zemax software to achieve more reasonable parameters. Compared with the telescope reported in the literature [13], the diameter of the telescope in this paper is larger, more suitable for spaceborne coherent lidar needs.
2. Telescope aperture optimization for spacebased coherent wind lidar In the lidar systems, the signal to noise ratio (SNR) is an important indicator to judge the lidar performance, the higher SNR in the detection range, the more accuracy for wind speed and direction after data inversion. In the relevant literatures [14] the principle that how to calculate the SNR in coherent wind lidar has been presented, it pointed out that SNR is related to the factors such as telescope aperture, heterodyne efficiency, transverse coherence length of the received field. But in these papers they didn't present the general sense of telescope aperture. In the following part, telescope aperture for space based lidar is optimized based on the signal to noise ratio formula. For shot-noise-limited coherent wind detection, the estimated value for the SNR is given by the following equation [15]: 2 3 !2 1 E0 ηβλK 2 π D2 4 z 2 π D2 D 25 SNR ¼ 1þ 1 þ ð1Þ f 2S0 8hBz2 4λ z where E0 is the laser pulse energy [J], η indicates the lidar system's optical/detection efficiency (electrons/photon), D means the telescope diameter [m], β(z) denotes the backscatter coefficient [m 1 sr 1], λ represents the laser wavelength [m], K is the oneway atmospheric transmission to range z [m 1], B means the system narrow bandwidth (1/τ) [Hz], h is the Planck's constant, h¼ 6.626 10 34, and S0 indicates the transverse coherence length of the received field in meters. From Eq. (1) it is obviously that SNR is related to many system parameters, in these parameters E0, η, λ, B are constant for a fixed system; β(z), K are related with the distance z. In order to analyze
239
the relationship between SNR and the telescope aperture, we can extract the items related to the telescope aperture, then Eq. (1) can be simplified: 2 3 !2 1 D2 4 z 2 π D2 D 25 SNR p 2 1 þ 1 þ ð2Þ f 2S0 z 4λ z
2.1. Detection range The signal to noise ratio is different for different detection range and telescope aperture. Coherent wind lidar systems can be divided into groundbased, airborne and spacebased systems according to the different detection range. Relatively speaking, spacebased systems have the biggest detection range and the greatest practical value, but by now there is still no such system running in the space due to the fund and other reasons [16]. In the article, we mainly analyzed the telescope aperture for spacebased systems, the satellite orbit is from 200 km to 600 km, detection range is 0–30 km from the earth surface, and finally we present one telescope design example for the 300 km orbit. 2.2. Transverse coherence length Transverse coherence length is closely related to the refractive index structure constant, in the atmosphere transverse coherence length can be expressed as [17]: " 5 # 35 Z R z′ 3 2 S0 ðzÞ ¼ Hk C n 2 ðz′Þ 1 dz′ ð3Þ z 0 where H ¼ 2:914383, k¼ 2π/λ is the wave number and z is the detection range. C n 2 is the value of the refractive index structure parameter of the atmosphere. We use Hufnagel–Valley model of the refractive index structure parameter to calculate the transverse coherence length, taking use of strong turbulence condition [18,19]: 2 21 h C 2n ðhÞ ¼ 0:00594 ð10 5 hÞ10 exp 27 1000 h h 16 þ 2:7 10 þ 1:7 10 14 exp exp 1500 100 ð4Þ where h is detection height from the ground, the range can be taken from 0 km to 30 km, in space application the height can be
Fig. 1. The normalized SNR intensity against distance z and the telescope aperture D.
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converted to the detection altitude, taken as 300 km as example. 2 21 300 z C 2n ðzÞ ¼ 0:00594 ½10 5 ð300 zÞ10 exp 27 1000 300 z 16 þ2:7 10 exp 1500 300 z ð5Þ þ1:7 10 14 exp 100 Taking Eq. (5) into Eq. (3) and substituting S0 in Eq. (2) with expression (3), then we can simulate and obtain the threedimensional figure which demonstrates the relationships between normalized SNR value and distance z, the telescope aperture D, shown as Fig. 1. It can be seen from Fig. 1 that the normalized SNR value increases with the telescope diameter first and then decreases at a certain orbit, the telescope diameter corresponding to the maximum SNR value increases with distance; simultaneously, the maximum SNR value also decreases with distance, which indicates that SNR would be gradually reduced as the space platform farther from earth, and at this time the laser energy need to be enhanced to compensate SNR degradation. Assuming spatial platform is 300 km from the earth surface, and the telescope diameter value corresponding to the maximum SNR value is about 0.83 m. But the design, process, and the final assembly of the telescope are difficult for the off-axis telescope systems. In the actual application the SNR value can be reduced to about 0.7 times, then the telescope diameter is about 0.5 m. In the following part we would introduce how to design and optimize a 0.5 m diameter off-axis telescope.
3. Telescope designed for space-based coherent wind lidar At present there is not any true sense of the global atmospheric wind lidar running in space, but some plans have been carried out. Taking use of the direct detection method ADM-Aeolus satellite is planned to launch by the end of 2013 [20]. Using the coherent detection mean researchers in Japan had proposed a plan called “Science plan on the wind measurements by the ISS/JEM-borne coherent Doppler lidar” [16], scientists in America had ever developed the SPARCLE equipment for the global wind measurement [21], but both projects eventually were suspended for technical and financial reasons. A new plan which intends to use hybrid technology combing coherent and direct method is “NexGen NPOESS” program, pre-system verification is ongoing. As analyzed in part 2, the telescope diameter is about 0.5 m for space coherent wind systems on orbit 300 km; apart from the field angle simulated from Siegman's antenna theorem [22], the field of view should also cover the lag angle of return beam, in some
systems the impact of the scan mode also need to be taken into account [13], here we just use the the detecting mode without scanning, then the field of view is small,about 0.055 mrad (half field of view). For large aperture of the off-axis telescope the twomirror system is unable to meet the requirements because the tolerances requirement is too high. After analyzation we choose the structure which is comprised by two mirrors and an aspherical lens, as shown in Fig. 2. 3.1. PW method introduction There are five monochromatic aberrations, spherical aberration, coma, astigmatism, field curvature and distortion, their respective third-order aberration coefficients are expressed by SI, SII, SIII, SIV, SV, and presented in PW form as follows [23]: 8 k k > 4 > SI ¼ ∑ hi P i þ ∑ hi K i > > > > i¼1 i¼1 > > > > k k k > 3 > > > SII ¼ ∑ hpi P i þ ∑ hpi hi K i þ J ∑ W i > > i¼1 i¼1 i¼1 > > > > < k h 2 k k h k 1 u pi pi 2 2 SIII ¼ ∑ P i þ ∑ hpi hi K i þ 2J ∑ W i þ J2 ∑ Δ i > i ¼ 1 hi i¼1 i ¼ 1 hi i ¼ 1hi ni > > > > k > n ′ ni > > SIV ¼ ∑ J 2 i ′ > > > ni ni r i > i ¼ 1 > > > 3 2 > ′ > > S ¼ k hpi P þ þ k h 3 h K þ 3J k hpi W þ J 2 k hpi ð 3 Δui þ ni ni Þ J 2 k 1 Δ 1 > ∑ 2 i ∑ pi i i ∑ 2 i ∑ ∑ 2 > : V ni ′ ni r i ni 2 i ¼ 1 hi i¼1 i ¼ 1 hi i ¼ 1 hi hi ni i ¼ 1hi
ð6Þ where P i ¼ ðΔui =ðΔð1=ni ÞÞÞ Δðui =ni Þ, W i ¼ ðΔui =ðΔð1=ni ÞÞÞ Δðui = ni Þ, K i ¼ ððni ′ ni Þe2i =r 3i Þ. hi, hpi respectively represent the heights of the axial and oblique rays on the ith surface, ui, ui ′ represent slope angles corresponding with the ith surface, ei denotes aspherical coefficient, ni, ni ′ represent the refractive index before and after ith surface, ri denotes the curvature radius. 2
3.2. Telescope design process and results In the subsequent section an off-axis telescope with a 500 mm diameter, 0.055 mrad field angle and 40 afocal magnification is presented. The structure of the telescope is shown in Fig. 2, here the lens is assumed to be a thin lens, the refractive index is n, similarly, parameters related to the contour are defined as follows:
α1 ¼
l2 h2 l′1 h1
ð7Þ
α2 ¼
l3 h3 l′2 h2
ð8Þ
β1 ¼
l′2 u2 ¼ l2 u′2
ð9Þ
β2 ¼
l′3 u3 ¼ l3 u′3
ð10Þ
In order to simplify the formula, the second surface of the thin lens is assumed to be plane. Combined with (7)–(10)the following expressions can be obtained from Gaussian optical theory: r2 ¼
α1 β 1 r 1 1 þ β1
r3 ¼
ðn 1Þα2 α1 β 1 r 1 2
d1 ¼ ð1 α1 Þ
Fig. 2. Telescope system diagram for space-based coherent wind lidar.
ð11Þ
d2 ¼
r1 2
ð13Þ
α1 β1 ð1 α2 Þ 2
ð12Þ
r1
ð14Þ
X. Ge et al. / Optics Communications 315 (2014) 238–242
241
Fig. 3. Off-axis wavefront error changes before and after optimization. Table 1 The initial structure parameters of the space-based telescope. Surface no.
Curvature radius/mm
The distance/mm
Conicoid coefficient
Off-axis value/mm
Refractive index
1 2 3 4
1800 421.6697 162.7788 Infinity
731.3110 731.3110 5 30
1.04286 2.71809 0.30306 —
500 100 12.5 12.5
— — 2.4462 —
Table 2 The optimized parameters of the space-based telescope. Surface no.
Curvature radius/mm
The distance/mm
Conicoid coefficient
Off-axis value/mm
Refractive index
1 2 3 4
1800 422.3369 163.7008 Infinity
730.5195 745.2456 5 30
1.0642 2.9064 0.2563 —
500 100 12.5 12.5
— — 2.4462 —
According to the previous statement, the magnification is 40 , so α1 α2 ¼ 0:025; considering the volume and compactness it can be assumed that d1 ¼ d2 ; with a relatively high refractive index and transmittance at a wavelength of 2 μm ZnSe glass can be selected, the refractive index would be calculated as 2.4462 from Sellmeier equation. Using these conditions in Eq. (6) the following results could be obtained: 8 2 > < e1 ¼ 1:04286 e2 2 ¼ 2:71809 > : 2 e3 ¼ 0:30306
ð15Þ
It is assumed that the curvature radius of the first surface is 1800 mm, the ZPL code can be programmed in accordance with the relationship between r 2 ; r 3 ; d1 ; d2 and r 1 with the expressions (11)–(14),the system parameters input Zemax are listed in Table 1. The Seidel aberration coefficients of spherical aberration, coma, astigmatism, field curvature in Zemax are all zero, verifying the calculation results are correct. Next taking the wavefront as optimal condition, the three aspherical coefficients, thicknesses and curvature
Fig. 4. The optimized optical path diagram of the telescope.
radiuses are made variable to optimize, and the equilibrium between primary aberration and senior aberration is reached quickly. The three aspheric coefficients are optimized to 1.0642, 2.9064 and 0.2563, little change compared with the value of the initial
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Table 3 The tolerance data as the RMS wavefront error less than 1/10λ (2 μm). No.
ΔR/mm
Δe2
Δd/mm
Dx/mm
Dy/mm
Tx/(′)
Ty/(′)
1 2 3 4 3–4
70.02 70.04 70.1 — —
70.0002 70.0015 70.05 — —
7 0.01 7 0.15 7 0.5 — —
7 0.005 7 0.008 7 0.015 7 0.2 7 0.03
7 0.005 7 0.008 7 0.015 7 0.2 7 0.03
7 0.09 7 0.19 73 73 7 1.8
7 0.09 7 0.19 73 73 7 1.8
Note: ΔR represents the tolerances of curvature radius;Δe2 represents conicoid coefficient tolerances;Δd is the thickness tolerance; Dx,Dy represents the eccentric in the x, y direction; Tx,Ty represents the tilt in the x, y direction;3–4 indicates the collimating lens.
structure, the parameters after optimization are listed in Table 2. Offaxis wavefront error changes are shown in Fig. 3. The RMS wavefront error of 0 field changes from 0.2109waves to 0.0068waves, 1 field decreases from 0.2110waves to 0.0068waves, optimized optical path shows in Fig. 4. Tolerance analysis is carried out for ensuring that the RMS wavefront error of the entire system can be smaller than the 1/10λ (2 μm), the final tolerances data is listed in Table 3. So far telescope design has been completed for spacebased coherent wind lidar systems. Here the aspheric coefficients, curvature radius, thicknesses are made variables to be optimized, under other conditions refractive index also could be changed for smaller volume or other structural forms. 4. Conclusion The telescope is a key optical component in coherent wind lidar system, its quality largely influences heterodyne detection efficiency. In the paper we introduce the way to calculate the aperture for spacebased coherent wind lidar system, and present the aperture needed for 300 km orbit. The requirements of coherent detection for telescope such as off-axis, little wavefront error, no internal focus, high-magnification in the space-based detection make telescope's design and process difficult. From these requirements, the initial structure of the space-based telescope system is calculated with third-order aberration theory, then system parameters are optimized on the basis of the initial structures with the
help of Zemax software, finally tolerance analysis results for spacebased telescope are given in the off-axis state, providing the data support for subsequent processing. The deduced aberration formula can be used not only in coherent wind lidar, but also suitable for other similar afocal systems.
Acknowledgement This work was supported by the National Natural Science Foundation of China (NSFC) projects no. 61178072.
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