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Performance comparison of Fabry–Perot and Mach–Zehnder interferometers for molecular Doppler lidar based on fringe-imaging technique
Optik 125 (2014) 6291–6295
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Performance comparison of Fabry–Perot and Mach–Zehnder interferometers for molecular Doppler lidar based on fringe-imaging technique Linqiu Tan, Dengxin Hua ∗ , Li Wang, Tingyao He, Yufeng Wang, Meili Xing School of Mechanical and Precision Instrument Engineering, Xi’an University of Technology, Xi’an 710048, China
a r t i c l e
i n f o
Article history: Received 13 November 2013 Accepted 16 June 2014 Keywords: Doppler lidar Wind velocity measurement Fringe-imaging Mach–Zehnder interferometer Fringe-imaging Fabry–Perot interferometer Molecular backscattered signal
a b s t r a c t Fringe-imaging Fabry–Perot interferometer (FIFPI) and fringe-imaging Mach–Zehnder interferometer (FIMZI) used as frequency discriminator for incoherent molecular Doppler lidar were analyzed, respectively. For a pure molecular backscattered signal, performances (wind measurement sensitivity and signal-to-noise ratio) of both FIFPI and FIMZI systems were simulated based on the U.S. standard atmospheric model. Comparisons of two systems were made under the same emitting and receiving parameters with certain wind speed dynamic range. Simulated results show that, though relatively lower sensitivity to Doppler shift, the single-channel FIMZI system provides a factor of 1.3 times smaller error in the horizontal wind velocity than that of FIFPI at a range of 20 km. We expect that the FIMZI frequency discriminator would provide an effective technique to improve the measurement accuracy for incoherent molecular Doppler lidar. © 2014 Elsevier GmbH. All rights reserved.
1. Introduction In atmospheric wind field detection, Doppler lidar has been proved to be one of the most effective tools [1]. It can be realized by coherent and incoherent detection methods. Incoherent detection systems can further fall into two categories: (double) edge technique [1–4] and fringe imaging technique [5–10]. Both techniques have been analyzed and compared [11–13]. It demonstrates that, mathematically, the edge system is a special case of the fringe system where only one or two channels are considered [13]. For incoherent Doppler lidar based on fringeimaging technique, Fabry–Perot interferometers used as frequency discriminators were mainly proposed in current experimental arrangements [5–10]. However, the fringe-imaging Fabry–Perot interferometer (FIFPI) produces circular fringes, which do not match with conventional array detectors. An effective way to solve this problem was to use the circle-to-line interferometer optical system (CLIO), which converts the circular fringes into a linear pattern [8,14]. In recent years, Mach–Zehnder interferometer has also been proposed for backscattered light spectral analysis [15–19]. Compared with FIFPI, fringe-imaging Mach–Zehnder interferometer (FIMZI) provides equidistant linear parallel fringes instead of
∗ Corresponding author. Tel.: +86 029 8232 2441; fax: +86 029 8232 2441. E-mail address: [email protected] (D. Hua). http://dx.doi.org/10.1016/j.ijleo.2014.06.147 0030-4026/© 2014 Elsevier GmbH. All rights reserved.
circular rings. The record of the FIMZI fringe pattern is then noticeably easier than that of the FIFPI. Currently, the research on FIMZI for incoherent Doppler lidar is still in the theoretical stage [19], and it is necessary to make in-depth studies. In this work, the frequency discriminators based on FIFPI and FIMZI for incoherent Doppler lidar were presented, and their optical layouts were designed, respectively. Assuming a pure molecular signal with Gaussian spectral distribution, we simulated the performances of the FIFPI and FIMZI system, respectively, including transmission, sensitivity, detected photon counts of the backscattered signals, signal-to-noise ratio (SNR) and the error in the horizontal velocity component of the wind. Comparisons of the FIFPI system and FIMZI system were also analyzed. 2. Configuration of the incoherent Doppler lidar Fig. 1 shows a schematic diagram of an incoherent Doppler lidar system. An injection-seeded, single longitudinal mode, pulsed Nd:YAG laser at wavelength of 354.7 nm is selected as light source. The laser output is collimated by a beam expander to reduce the divergence and then split into two parts by beam splitter (BS). Most of light is transmitted into the atmosphere by M1, M2 and M3, the returned atmospheric backscattering light is collected by a Schmidt–Cassegrainian telescope and coupled into a fiber. The other part is also coupled into a fiber as the reference light. Both signals are passed through interferometer frequency discrimination
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L. Tan et al. / Optik 125 (2014) 6291–6295
transmission function of each path arriving on detector i (i = 1, 2) is [19] fMZi () =
1 · [1 + (−1)i cos(2nl)], 2
(2)
where l is the interferometer optical path difference. The interference pattern on each detector plane is therefore a set of sine fringes, and one complementary to the other. Assuming a pure molecular scattering with negligible broadenings due to Brillouin scattering, wind turbulence, and emitter line width, the power-normalized spectral distribution of the molecular signal is given by [19]
Fig. 1. Schematic diagram of an incoherent Doppler lidar.
hMolecular () = system and recorded by detectors. Then the wind speed can be retrieved based on Doppler frequency shift and the wind direction can be determined by beam scanning method [5,8]. Configurations of frequency discrimination systems with FIFPI and FIMZI are presented in Fig. 2. In FIFPI technique, the lidar returned signals are collimated and passed through a FIFPI, and form classic Fabry–Perot rings in the focal plane of the focusing lens (L2). By a 45◦ half angle reflecting cone as the primary optical element, the normal circular FIFPI fringe pattern can be transformed into a linear pattern and detected by a linear detector. The basic optical configuration is illustrated schematically in Fig. 2(a), where the cone is shown to be situated with its vertex on the optical axis in the focal plane of the focusing lens (L2) of the Fabry–Perot optical system [14]. The schematic of FIMZI is also presented in Fig. 2(b), where the output of fiber is collimated by lens L. A plane wave is divided into two plane waves by the beam splitter BS1. The two beams propagate through each arm of the interferometer, undergoing different optical paths, and are recombined by the beam splitter BS2. The mirrors and beam splitters can be adjusted to obtain a small angle ˛ between the two output waves and to form linear fringes [19]. The outputs are incident on a cylindrical lens and detected by a linear detector in each channel. It can be seen that the record of the FIMZI fringe pattern by linear detector is much easier than that of FIFPI and the FIMZI can easily be divided into two channels, resulting in doubled wind measurement sensitivity. In addition, the FIMZI can be designed with a compensated field to accept large extended optical sources without significant performance reduction [18,19], whereas the FIFPI can withstand only reduced extended sources [20]. This property can be obtained by inserting a field-compensation plate (FCP) of refractive index n and thickness t = nl0 /(n2 − 1) in the longer arm of the FIMZI (Fig. 2(b)) [19]. However, the classical FIMZI that involves a bulky arrangement of mirrors and beam splitters is deemed less stable and compact than FIFPI. 3. Mathematical descriptions
Tp 1 + (4F 2 /2 ) · sin2
2nd cos () + ϕ
=
20 c
2kT 1/2 m
(1)
where Tp accounts for the peak-transmission of Fabry–Perot interferometer, is the wave number of light, n is the refractive index of the optical propagation medium, d is the separation of the reflecting surfaces, is the angle of refraction at the first surface of interferometer, ϕ is the phase change of the internal reflection, and F is the fineness of interferometer. The schematic of a classical fringe-imaging FIMZI is presented in Fig. 2(b). For simplicity, we consider that the two beam splitters have the same transmission and reflection factors of 0.5, then the
,
(3)
(4)
,
where k is the Boltzmann constant, T is the temperature of the scattering medium, m is the average mass of atmospheric molecules, c is the velocity of light, and 0 is the wave number of emitted light. In Doppler lidar system, the Fabry–Perot interferometer transmission for a normalized molecular-scattering spectral distribution can be written as [1]
∞
TFP () =
hMolecular ( − )fFP ( )d .
(5)
−∞
Subsequently, the Mach–Zehnder interferometer transmission can be obtained by substituting fMZi for fFP . For a lidar transmitter operating at a wave number 0 with a pulse energy E0 , the number of molecular photons detected in channel j from an element of the atmosphere at range R with thickness R can be derived as [1] NM,j (R) =
S E0 T ˇM × R × exp[−2 hc0 nc R2 j j
R
(r)dr],
(6)
0
where j is detector channel number, S is the area of the lidar receiver, h is the Planck constant, nc is the number of detector channels, Tj and j are the transmission and quantum efficiency of the jth channel, respectively, ˇM and ˇA are the molecular and aerosol backscatter coefficient of the atmosphere per unit solid angle per unit length, respectively, and (r) is the extinction coefficient of the atmosphere as a result of the combined aerosol and molecular scattering and absorption effects. The aerosol photons NA,j detected in channel j can similarly be obtained by substituting the backscatter coefficient ˇA for ˇM . Generally, the measurement sensitivity is defined as the fractional change in the normalized measured signal for a unit velocity. The sensitivity to molecular backscatter can be given as [13]
nc
Based on multi-beam interference theory, the transmission function of FIFPI can be expressed as [19] fFP () =
with
1 ( − 0 )2 √ exp − ()2
j=1
2 =
NM,j ((1/NM,j )(∂NM,j /∂V ))
nc
j=1
NM,j
2
,
(7)
with V=
DOP · c , 20
(8)
where DOP is the Doppler frequency shift and V is the component of the wind velocity along the line-of-sight (LOS) of the laser. The error in the LOS wind velocity for a molecular system can be written as [13]
⎡ ⎤
2 −1/2 nc 1 ∂NM,j ⎦ ε=⎣ SNRj 2 × , j=1
NM,j
∂V
(9)
L. Tan et al. / Optik 125 (2014) 6291–6295
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Fig. 2. Configurations of frequency discrimination systems with FIFPI interferometer (a) and with FIMZI interferometer (b). L: lens; CLIO: circle-to-line interferometer optical system; M: mirror; BS: beam splitter; CYL: cylindrical lens; FCP: field-compensation plate.
where SNRj is the signal-to-noise ratio for channel j. For a molecular Doppler lidar system, the SNR in channel j is [13] SNRj =
NA,j + NM,j (NA,j + NM,j + NB,j )1/2
,
(10)
where NB,j is background counts (including dark counts, solar photons, detector noise, etc.) in channel j. Note that for a molecular system, any aerosol signals can improve the SNRj because the aerosol spectral signature is narrower than the molecular [13]. The SNRj , as given by Eq. (10), is for a single channel. For a fringeimaging system, the composite signal-to-noise ratio is given as [1]
⎡ ⎤1/2 nc 1 1 ⎦ . =⎣
SNR
j=1
SNRj 2
(11)
For a measurement at a zenith angle ω, the error in the horizontal component of the wind velocity is [1] εH =
ε . sin (ω)
(12)
4. Wind measurement errors for FIFPI and FIMZI 4.1. Simulation conditions We assume that the lidar systems based on FIFPI and FIMZI have similar instrument constants (e.g., lasers, telescopes, detectors, etc.) except for frequency discrimination systems. The FIMZI possesses two channels while only one for FIFPI. The comparison here, was made between the single-channel FIMZI (e.g., channel 2) and the FIFPI. For fringe-imaging systems, the wind speed dynamic range is set by the interferometer free spectral range (FSR) and is typically larger than that of edge systems [13]. To further simplify the analysis and equalize the comparison, we assumed that the wind speed dynamic ranges for FIFPI and FIMZI were equal. For the FIMZI system, it has been proved that a single fringe spacing on the detector width is optimal and the least number of channels per fringe spacing is 10 [19]. Another instrumental parameter that can be optimized is the interferometer initial optical path difference l0 , which can be given as √ −1 l0 = ( 2) .
Fig. 3. Interferometer transmission for a molecular backscattered signal at a temperature of 250 K as a function of the detector channel. Solid line: a full free spectral range imaged onto detector 2 for FIMZI; Dashed line: 0.6 free spectral ranges imaged onto the detector for FIFPI.
a 32-channel detector and 0.6 orders displayed onto the detector [13]. Simulated parameters of lidar system are given in Table 1. 4.2. Interferometer transmission Considering that two beam splitters of FIMZI have transmissions and reflections equal to 0.5, the transmission TMZ2 and TFP of FIMZI and FIFPI for a molecular backscattered signal at a temperature of 250 K are shown in Fig. 3, respectively. In the simulation, the phase change of the internal reflection in the Fabry–Perot interferometer plates was ignored. From Fig. 3, it can be seen that the peak transmission approaches 0.8 for FIMZI, while only 0.31 for FIFPI. 4.3. Wind measurement sensitivity Based on Eq. (7), the wind measurement sensitivity to molecular backscatter for the FIFPI system and the single-channel FIMZI system are shown in Fig. 4, respectively. It can be seen that, the sensitivity of the single-channel FIMZI molecular system is
(13)
This difference l0 corresponds to 3.16 cm for a lidar operation wavelength 0 = 354.7 nm and a scattering medium temperature T = 250 K [19]. Since the molecular spectrum is wide, it is practical to display less than a full order onto the detector for FIFPI system while still maintaining a large LOS dynamic range (1681 m/s) which is equal to that of FIMZI for the measurement. The FIFPI spacing was determined to be 0.95 cm with a reflectivity of 71.5%, assuming
Fig. 4. Sensitivity to molecular backscattered signal for the single-channel FIMZI system and the FIFPI system.
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L. Tan et al. / Optik 125 (2014) 6291–6295
Table 1 Simulated parameters for molecular Doppler lidar. Shared parameters
Values
Interferometer parameters
FIFPI
Wavelength Pulse energy Laser 1/e width Received area Zenith Detector channels Detector efficiency
354.7 nm 300 mJ 60 MHz 0.166 m2 45◦ 32 40%
Fineness (F) Peak transmission (Tp ) Free spectral range (FSR) Optical path difference (l) Air refractive index (n) Number of orders imaged LOS dynamic range
9.32 – 0.66 1 52.6 m−1 (15.8 GHz) 31.6 m−1 (9.48 GHz) 1.9 cm ( = 0) 3.16 cm (␣ = 0) 1 1 0.6 1 1681 m/s 1681 m/s
Fig. 5. Simulations of the detected photon counts for the signal backscattered from the atmosphere for the single-channel FIMZI system and the FIFPI system.
0.170% ms−1 , which is a constant and independent of the wind speed. However, the sensitivity of the FIFPI molecular system is about 0.226% ms−1 , which decreases lightly with the LOS wind speed increasing due to an incomplete free spectral range displayed onto the detector. Apparently, the single-channel FIMZI system has a lower sensitivity, which illustrates that the transmission spectrum of FIMZI is less sharp than that of FIFPI. However, if we take the two complementary channels into account, the sensitivity of FIMZI will approach 0.340% ms−1 and will be higher than that of FIFPI. 4.4. SNR for the molecular system According to 1976 U.S. standard atmospheric model [21], the detected backscattered photon counts with a single shot and a vertical resolution of 300 m are shown in Fig. 5. The Doppler shift was considered zero. It is clear that the single-channel FIMZI detected photon counts whether for aerosol or for molecular components are much more than that of FIFPI because of a higher transmission (see also Fig. 3). Without considering the background noise, the SNR in a 20-shot average for a molecular system is shown in Fig. 6. It
FIMZI
Fig. 7. Simulated errors in the horizontal velocity component of the wind as a function of the LOS range. The solid line and dashed line corresponds to the singlechannel FIMZI system and the FIFPI system, respectively.
can be seen that the SNR for the single-channel FIMZI is higher than that of FIFPI, and the highest LOS ranges are about 15 km for FIMZI and 10.6 km for FIFPI at the SNR of 10. 4.5. Wind measurement error The error in the horizontal wind velocity for a measurement is given by Eqs. (9) and (12). It can be found from the sensitivity and the SNR of the molecular system, which were discussed above. The simulated error in the horizontal velocity component of the wind is shown in Fig. 7. In the simulation, the horizontal wind velocity was considered 20 m/s. As the LOS range increases from 0 to 20 km, the error for the FIFPI system increases more rapidly than that of the single-channel FIMZI. For a 300 m vertical resolution and a 20-shot average, errors of smaller than 5.6 and 4.3 m/s were obtained from the ground to a range of 20 km for the FIFPI and the single-channel FIMZI system, respectively. We can see that the single-channel FIMZI system provides a factor of 1.3 times smaller measurement error than that of FIFPI at the range of 20 km. 5. Conclusions
Fig. 6. SNR for the molecular system as a function of the LOS range. The solid line and dashed line corresponds to the single-channel FIMZI system and the FIFPI system, respectively.
Molecular Doppler lidar systems based on fringe-imaging technique were described in this work. In particular, frequency discrimination systems with FIFPI and FIMZI were analyzed. By assuming a pure molecular signal with Gaussian spectral distribution, we simulated the performances of FIFPI and FIMZI systems, respectively. Obtained results show that the single-channel FIMZI provides higher peak transmission and SNR, and a factor of 1.3 times smaller horizontal wind error than that of FIFPI, though with a relatively lower sensitivity. In addition, the FIMZI can be employed for extended sources using field-compensation, which, however cannot be achieved in FIFPI system. These improvements validate the feasibility of the FIMZI system used as frequency discrimination in Doppler lidar. We expect that the supplied simulation would provide numerical foundation for its further use.
L. Tan et al. / Optik 125 (2014) 6291–6295
Acknowledgements This work was supported by the Young Scientists Fund of the National Natural Science Foundation of China granted No. 61205189 and the Specialized Research Fund for the Doctoral Program of Higher Education of China granted No. 20106118120009. References [1] C.L. Korb, B.M. Gentry, C.Y. Weng, Edge technique: theory and application to the lidar measurement of atmospheric wind, Appl. Opt. 31 (1992) 4202–4213. [2] J.A. McKay, Modeling of direct detection Doppler wind lidar. I. The edge technique, Appl. Opt. 37 (1998) 6480–6486. [3] C. Flesia, C.L. Korb, Theory of the double-edge molecular technique for Doppler lidar wind measurement, Appl. Opt. 38 (1999) 432–440. [4] C.Y. She, J. Yue, Z.A. Yan, J.W. Hair, J.J. Guo, S.H. Wu, Z.S. Liu, Direct-detection Doppler wind measurements with a Cabannes–Mie lidar: A. Comparison between iodine vapor filter and Fabry–Perot interferometer methods, Appl. Opt. 46 (2007) 4434–4443. [5] V.J. Abreu, J.E. Barnes, P.B. Hays, Observations of winds with an incoherent lidar detector, Appl. Opt. 31 (1992) 4509–4514. [6] M.J. McGill, W.R. Skinner, T.D. Irgang, Analysis techniques for the recovery of winds and backscatter coefficients from a multiple-channel incoherent Doppler Lidar, Appl. Opt. 36 (1997) 1253–1268. [7] J.A. McKay, Modeling of direct detection Doppler wind lidar. II. The fringe imaging technique, Appl. Opt. 37 (1998) 6487–6493. [8] T.D. Irgang, P.B. Hays, W.R. Skinner, Two-channel direct-detection Doppler lidar employing a charge-coupled device as a detector, Appl. Opt. 41 (2002) 1145–1155.
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[9] M.C. Hirschberger, G. Ehret, Simulation and high-precision wavelength determination of noisy 2D Fabry–Perot interferometric rings for direct-detection Doppler lidar and laser spectroscopy, Appl. Phys. B 103 (2011) 207–222. [10] Y.Y. Huang, J.J. Makela, G.R. Swenson, Simulations of imaging Fabry–Perot interferometers for measuring upper-atmospheric temperatures and winds, Appl. Opt. 51 (2012) 3787–3800. [11] J.A. McKay, The edge filter and fringe imaging for laser Doppler wind speed measurement, Proc. SPIE 3065 (1997) 420–427. [12] J.A. McKay, D. Rees, Space-based Doppler wind lidar: modeling of edge detection and fringe imaging Doppler analyzers, Adv. Space Res. 26 (2000) 883–891. [13] M.J. McGill, J.D. Spinhirne, Comparison of two direct-detection Doppler lidar techniques, Opt. Eng. 37 (1998) 2675–2686. [14] P.B. Hays, Circle to line interferometer optical system, Appl. Opt. 29 (1990) 1482–1489. [15] Z.Y. Liu, T.K. Kobayashi, Differential discrimination technique for incoherent Doppler lidar to measure atmospheric wind and backscatter ratio, Opt. Rev. 3 (1996) 47–52. [16] D. Bruneau, Mach–Zehnder interferometer as a spectral analyzer for molecular Doppler wind lidar, Appl. Opt. 40 (2001) 391–399. [17] D. Bruneau, J. Pelon, Simultaneous measurements of particle backscattering and extinction coefficients and wind velocity by lidar with a Mach–Zehnder interferometer: principle of operation and performance assessment, Appl. Opt. 42 (2003) 1101–1114. [18] D. Bruneau, A. Garnier, A. Hertzog, J. Porteneuve, Wind–velocity lidar measurements by use of a Mach–Zehnder interferometer, comparison with a Fabry–Perot interferometer, Appl. Opt. 43 (2004) 173–181. [19] D. Bruneau, Fringe-imaging Mach–Zehnder interferometer as a spectral analyzer for molecular Doppler wind lidar, Appl. Opt. 41 (2002) 503–510. [20] Y.F. Wang, D.X. Hua, L. Wang, Q. Yan, Z.R. Zhou, S.C. Li, Analysis of measurement error of wind lidar caused by propagation properties of Gaussian beam, Optik 123 (2012) 1293–1296. [21] US Standard Atmosphere, US Government Printing Office, Washington, DC, 1976.
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Performance comparison of Fabry–Perot and Mach–Zehnder interferometers for molecular Doppler lidar based on fringe-imaging technique Linqiu Tan, Dengxin Hua ∗ , Li Wang, Tingyao He, Yufeng Wang, Meili Xing School of Mechanical and Precision Instrument Engineering, Xi’an University of Technology, Xi’an 710048, China
a r t i c l e
i n f o
Article history: Received 13 November 2013 Accepted 16 June 2014 Keywords: Doppler lidar Wind velocity measurement Fringe-imaging Mach–Zehnder interferometer Fringe-imaging Fabry–Perot interferometer Molecular backscattered signal
a b s t r a c t Fringe-imaging Fabry–Perot interferometer (FIFPI) and fringe-imaging Mach–Zehnder interferometer (FIMZI) used as frequency discriminator for incoherent molecular Doppler lidar were analyzed, respectively. For a pure molecular backscattered signal, performances (wind measurement sensitivity and signal-to-noise ratio) of both FIFPI and FIMZI systems were simulated based on the U.S. standard atmospheric model. Comparisons of two systems were made under the same emitting and receiving parameters with certain wind speed dynamic range. Simulated results show that, though relatively lower sensitivity to Doppler shift, the single-channel FIMZI system provides a factor of 1.3 times smaller error in the horizontal wind velocity than that of FIFPI at a range of 20 km. We expect that the FIMZI frequency discriminator would provide an effective technique to improve the measurement accuracy for incoherent molecular Doppler lidar. © 2014 Elsevier GmbH. All rights reserved.
1. Introduction In atmospheric wind field detection, Doppler lidar has been proved to be one of the most effective tools [1]. It can be realized by coherent and incoherent detection methods. Incoherent detection systems can further fall into two categories: (double) edge technique [1–4] and fringe imaging technique [5–10]. Both techniques have been analyzed and compared [11–13]. It demonstrates that, mathematically, the edge system is a special case of the fringe system where only one or two channels are considered [13]. For incoherent Doppler lidar based on fringeimaging technique, Fabry–Perot interferometers used as frequency discriminators were mainly proposed in current experimental arrangements [5–10]. However, the fringe-imaging Fabry–Perot interferometer (FIFPI) produces circular fringes, which do not match with conventional array detectors. An effective way to solve this problem was to use the circle-to-line interferometer optical system (CLIO), which converts the circular fringes into a linear pattern [8,14]. In recent years, Mach–Zehnder interferometer has also been proposed for backscattered light spectral analysis [15–19]. Compared with FIFPI, fringe-imaging Mach–Zehnder interferometer (FIMZI) provides equidistant linear parallel fringes instead of
∗ Corresponding author. Tel.: +86 029 8232 2441; fax: +86 029 8232 2441. E-mail address: [email protected] (D. Hua). http://dx.doi.org/10.1016/j.ijleo.2014.06.147 0030-4026/© 2014 Elsevier GmbH. All rights reserved.
circular rings. The record of the FIMZI fringe pattern is then noticeably easier than that of the FIFPI. Currently, the research on FIMZI for incoherent Doppler lidar is still in the theoretical stage [19], and it is necessary to make in-depth studies. In this work, the frequency discriminators based on FIFPI and FIMZI for incoherent Doppler lidar were presented, and their optical layouts were designed, respectively. Assuming a pure molecular signal with Gaussian spectral distribution, we simulated the performances of the FIFPI and FIMZI system, respectively, including transmission, sensitivity, detected photon counts of the backscattered signals, signal-to-noise ratio (SNR) and the error in the horizontal velocity component of the wind. Comparisons of the FIFPI system and FIMZI system were also analyzed. 2. Configuration of the incoherent Doppler lidar Fig. 1 shows a schematic diagram of an incoherent Doppler lidar system. An injection-seeded, single longitudinal mode, pulsed Nd:YAG laser at wavelength of 354.7 nm is selected as light source. The laser output is collimated by a beam expander to reduce the divergence and then split into two parts by beam splitter (BS). Most of light is transmitted into the atmosphere by M1, M2 and M3, the returned atmospheric backscattering light is collected by a Schmidt–Cassegrainian telescope and coupled into a fiber. The other part is also coupled into a fiber as the reference light. Both signals are passed through interferometer frequency discrimination
6292
L. Tan et al. / Optik 125 (2014) 6291–6295
transmission function of each path arriving on detector i (i = 1, 2) is [19] fMZi () =
1 · [1 + (−1)i cos(2nl)], 2
(2)
where l is the interferometer optical path difference. The interference pattern on each detector plane is therefore a set of sine fringes, and one complementary to the other. Assuming a pure molecular scattering with negligible broadenings due to Brillouin scattering, wind turbulence, and emitter line width, the power-normalized spectral distribution of the molecular signal is given by [19]
Fig. 1. Schematic diagram of an incoherent Doppler lidar.
hMolecular () = system and recorded by detectors. Then the wind speed can be retrieved based on Doppler frequency shift and the wind direction can be determined by beam scanning method [5,8]. Configurations of frequency discrimination systems with FIFPI and FIMZI are presented in Fig. 2. In FIFPI technique, the lidar returned signals are collimated and passed through a FIFPI, and form classic Fabry–Perot rings in the focal plane of the focusing lens (L2). By a 45◦ half angle reflecting cone as the primary optical element, the normal circular FIFPI fringe pattern can be transformed into a linear pattern and detected by a linear detector. The basic optical configuration is illustrated schematically in Fig. 2(a), where the cone is shown to be situated with its vertex on the optical axis in the focal plane of the focusing lens (L2) of the Fabry–Perot optical system [14]. The schematic of FIMZI is also presented in Fig. 2(b), where the output of fiber is collimated by lens L. A plane wave is divided into two plane waves by the beam splitter BS1. The two beams propagate through each arm of the interferometer, undergoing different optical paths, and are recombined by the beam splitter BS2. The mirrors and beam splitters can be adjusted to obtain a small angle ˛ between the two output waves and to form linear fringes [19]. The outputs are incident on a cylindrical lens and detected by a linear detector in each channel. It can be seen that the record of the FIMZI fringe pattern by linear detector is much easier than that of FIFPI and the FIMZI can easily be divided into two channels, resulting in doubled wind measurement sensitivity. In addition, the FIMZI can be designed with a compensated field to accept large extended optical sources without significant performance reduction [18,19], whereas the FIFPI can withstand only reduced extended sources [20]. This property can be obtained by inserting a field-compensation plate (FCP) of refractive index n and thickness t = nl0 /(n2 − 1) in the longer arm of the FIMZI (Fig. 2(b)) [19]. However, the classical FIMZI that involves a bulky arrangement of mirrors and beam splitters is deemed less stable and compact than FIFPI. 3. Mathematical descriptions
Tp 1 + (4F 2 /2 ) · sin2
2nd cos () + ϕ
=
20 c
2kT 1/2 m
(1)
where Tp accounts for the peak-transmission of Fabry–Perot interferometer, is the wave number of light, n is the refractive index of the optical propagation medium, d is the separation of the reflecting surfaces, is the angle of refraction at the first surface of interferometer, ϕ is the phase change of the internal reflection, and F is the fineness of interferometer. The schematic of a classical fringe-imaging FIMZI is presented in Fig. 2(b). For simplicity, we consider that the two beam splitters have the same transmission and reflection factors of 0.5, then the
,
(3)
(4)
,
where k is the Boltzmann constant, T is the temperature of the scattering medium, m is the average mass of atmospheric molecules, c is the velocity of light, and 0 is the wave number of emitted light. In Doppler lidar system, the Fabry–Perot interferometer transmission for a normalized molecular-scattering spectral distribution can be written as [1]
∞
TFP () =
hMolecular ( − )fFP ( )d .
(5)
−∞
Subsequently, the Mach–Zehnder interferometer transmission can be obtained by substituting fMZi for fFP . For a lidar transmitter operating at a wave number 0 with a pulse energy E0 , the number of molecular photons detected in channel j from an element of the atmosphere at range R with thickness R can be derived as [1] NM,j (R) =
S E0 T ˇM × R × exp[−2 hc0 nc R2 j j
R
(r)dr],
(6)
0
where j is detector channel number, S is the area of the lidar receiver, h is the Planck constant, nc is the number of detector channels, Tj and j are the transmission and quantum efficiency of the jth channel, respectively, ˇM and ˇA are the molecular and aerosol backscatter coefficient of the atmosphere per unit solid angle per unit length, respectively, and (r) is the extinction coefficient of the atmosphere as a result of the combined aerosol and molecular scattering and absorption effects. The aerosol photons NA,j detected in channel j can similarly be obtained by substituting the backscatter coefficient ˇA for ˇM . Generally, the measurement sensitivity is defined as the fractional change in the normalized measured signal for a unit velocity. The sensitivity to molecular backscatter can be given as [13]
nc
Based on multi-beam interference theory, the transmission function of FIFPI can be expressed as [19] fFP () =
with
1 ( − 0 )2 √ exp − ()2
j=1
2 =
NM,j ((1/NM,j )(∂NM,j /∂V ))
nc
j=1
NM,j
2
,
(7)
with V=
DOP · c , 20
(8)
where DOP is the Doppler frequency shift and V is the component of the wind velocity along the line-of-sight (LOS) of the laser. The error in the LOS wind velocity for a molecular system can be written as [13]
⎡ ⎤
2 −1/2 nc 1 ∂NM,j ⎦ ε=⎣ SNRj 2 × , j=1
NM,j
∂V
(9)
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Fig. 2. Configurations of frequency discrimination systems with FIFPI interferometer (a) and with FIMZI interferometer (b). L: lens; CLIO: circle-to-line interferometer optical system; M: mirror; BS: beam splitter; CYL: cylindrical lens; FCP: field-compensation plate.
where SNRj is the signal-to-noise ratio for channel j. For a molecular Doppler lidar system, the SNR in channel j is [13] SNRj =
NA,j + NM,j (NA,j + NM,j + NB,j )1/2
,
(10)
where NB,j is background counts (including dark counts, solar photons, detector noise, etc.) in channel j. Note that for a molecular system, any aerosol signals can improve the SNRj because the aerosol spectral signature is narrower than the molecular [13]. The SNRj , as given by Eq. (10), is for a single channel. For a fringeimaging system, the composite signal-to-noise ratio is given as [1]
⎡ ⎤1/2 nc 1 1 ⎦ . =⎣
SNR
j=1
SNRj 2
(11)
For a measurement at a zenith angle ω, the error in the horizontal component of the wind velocity is [1] εH =
ε . sin (ω)
(12)
4. Wind measurement errors for FIFPI and FIMZI 4.1. Simulation conditions We assume that the lidar systems based on FIFPI and FIMZI have similar instrument constants (e.g., lasers, telescopes, detectors, etc.) except for frequency discrimination systems. The FIMZI possesses two channels while only one for FIFPI. The comparison here, was made between the single-channel FIMZI (e.g., channel 2) and the FIFPI. For fringe-imaging systems, the wind speed dynamic range is set by the interferometer free spectral range (FSR) and is typically larger than that of edge systems [13]. To further simplify the analysis and equalize the comparison, we assumed that the wind speed dynamic ranges for FIFPI and FIMZI were equal. For the FIMZI system, it has been proved that a single fringe spacing on the detector width is optimal and the least number of channels per fringe spacing is 10 [19]. Another instrumental parameter that can be optimized is the interferometer initial optical path difference l0 , which can be given as √ −1 l0 = ( 2) .
Fig. 3. Interferometer transmission for a molecular backscattered signal at a temperature of 250 K as a function of the detector channel. Solid line: a full free spectral range imaged onto detector 2 for FIMZI; Dashed line: 0.6 free spectral ranges imaged onto the detector for FIFPI.
a 32-channel detector and 0.6 orders displayed onto the detector [13]. Simulated parameters of lidar system are given in Table 1. 4.2. Interferometer transmission Considering that two beam splitters of FIMZI have transmissions and reflections equal to 0.5, the transmission TMZ2 and TFP of FIMZI and FIFPI for a molecular backscattered signal at a temperature of 250 K are shown in Fig. 3, respectively. In the simulation, the phase change of the internal reflection in the Fabry–Perot interferometer plates was ignored. From Fig. 3, it can be seen that the peak transmission approaches 0.8 for FIMZI, while only 0.31 for FIFPI. 4.3. Wind measurement sensitivity Based on Eq. (7), the wind measurement sensitivity to molecular backscatter for the FIFPI system and the single-channel FIMZI system are shown in Fig. 4, respectively. It can be seen that, the sensitivity of the single-channel FIMZI molecular system is
(13)
This difference l0 corresponds to 3.16 cm for a lidar operation wavelength 0 = 354.7 nm and a scattering medium temperature T = 250 K [19]. Since the molecular spectrum is wide, it is practical to display less than a full order onto the detector for FIFPI system while still maintaining a large LOS dynamic range (1681 m/s) which is equal to that of FIMZI for the measurement. The FIFPI spacing was determined to be 0.95 cm with a reflectivity of 71.5%, assuming
Fig. 4. Sensitivity to molecular backscattered signal for the single-channel FIMZI system and the FIFPI system.
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Table 1 Simulated parameters for molecular Doppler lidar. Shared parameters
Values
Interferometer parameters
FIFPI
Wavelength Pulse energy Laser 1/e width Received area Zenith Detector channels Detector efficiency
354.7 nm 300 mJ 60 MHz 0.166 m2 45◦ 32 40%
Fineness (F) Peak transmission (Tp ) Free spectral range (FSR) Optical path difference (l) Air refractive index (n) Number of orders imaged LOS dynamic range
9.32 – 0.66 1 52.6 m−1 (15.8 GHz) 31.6 m−1 (9.48 GHz) 1.9 cm ( = 0) 3.16 cm (␣ = 0) 1 1 0.6 1 1681 m/s 1681 m/s
Fig. 5. Simulations of the detected photon counts for the signal backscattered from the atmosphere for the single-channel FIMZI system and the FIFPI system.
0.170% ms−1 , which is a constant and independent of the wind speed. However, the sensitivity of the FIFPI molecular system is about 0.226% ms−1 , which decreases lightly with the LOS wind speed increasing due to an incomplete free spectral range displayed onto the detector. Apparently, the single-channel FIMZI system has a lower sensitivity, which illustrates that the transmission spectrum of FIMZI is less sharp than that of FIFPI. However, if we take the two complementary channels into account, the sensitivity of FIMZI will approach 0.340% ms−1 and will be higher than that of FIFPI. 4.4. SNR for the molecular system According to 1976 U.S. standard atmospheric model [21], the detected backscattered photon counts with a single shot and a vertical resolution of 300 m are shown in Fig. 5. The Doppler shift was considered zero. It is clear that the single-channel FIMZI detected photon counts whether for aerosol or for molecular components are much more than that of FIFPI because of a higher transmission (see also Fig. 3). Without considering the background noise, the SNR in a 20-shot average for a molecular system is shown in Fig. 6. It
FIMZI
Fig. 7. Simulated errors in the horizontal velocity component of the wind as a function of the LOS range. The solid line and dashed line corresponds to the singlechannel FIMZI system and the FIFPI system, respectively.
can be seen that the SNR for the single-channel FIMZI is higher than that of FIFPI, and the highest LOS ranges are about 15 km for FIMZI and 10.6 km for FIFPI at the SNR of 10. 4.5. Wind measurement error The error in the horizontal wind velocity for a measurement is given by Eqs. (9) and (12). It can be found from the sensitivity and the SNR of the molecular system, which were discussed above. The simulated error in the horizontal velocity component of the wind is shown in Fig. 7. In the simulation, the horizontal wind velocity was considered 20 m/s. As the LOS range increases from 0 to 20 km, the error for the FIFPI system increases more rapidly than that of the single-channel FIMZI. For a 300 m vertical resolution and a 20-shot average, errors of smaller than 5.6 and 4.3 m/s were obtained from the ground to a range of 20 km for the FIFPI and the single-channel FIMZI system, respectively. We can see that the single-channel FIMZI system provides a factor of 1.3 times smaller measurement error than that of FIFPI at the range of 20 km. 5. Conclusions
Fig. 6. SNR for the molecular system as a function of the LOS range. The solid line and dashed line corresponds to the single-channel FIMZI system and the FIFPI system, respectively.
Molecular Doppler lidar systems based on fringe-imaging technique were described in this work. In particular, frequency discrimination systems with FIFPI and FIMZI were analyzed. By assuming a pure molecular signal with Gaussian spectral distribution, we simulated the performances of FIFPI and FIMZI systems, respectively. Obtained results show that the single-channel FIMZI provides higher peak transmission and SNR, and a factor of 1.3 times smaller horizontal wind error than that of FIFPI, though with a relatively lower sensitivity. In addition, the FIMZI can be employed for extended sources using field-compensation, which, however cannot be achieved in FIFPI system. These improvements validate the feasibility of the FIMZI system used as frequency discrimination in Doppler lidar. We expect that the supplied simulation would provide numerical foundation for its further use.
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