High resolution heterodyne spectroscopy of the atmospheric methane NIR absorption Alexander Rodin,1,2,* Artem Klimchuk,1 Alexander Nadezhdinskiy,3 Dmitry Churbanov,1 and Maxim Spiridonov3 1
Moscow Institute of Physics and Technology, Institutsky dr. 9, 141700 Dolgoprudnyi, Russia 2 Space Research Institute (IKI), Profsoyuznaya 84/32, 117997 Moscow, Russia 3 Prokhorov General Physics Institute, Vavilova 38, 119991 Moscow, Russia *[email protected]
Abstract: The paper describes the concept of a compact, lightweight heterodyne NIR spectro-radiometer suitable for atmospheric sounding with solar occultations, and the first measurement of CO2 and CH4 absorption near 1.65 μm with spectral resolution λ/δλ~10 8. A highly stabilized DFB laser was used as local oscillator, while single model silica fiber Y-coupler served as a diplexer. Radiation mixed in the single mode fiber was detected by a balanced couple of InGaAs p-i-n diodes within the bandpass of ~3 MHz. Wavelength coverage of spectral measurement was provided by sweeping local oscillator frequency in the range of 1.1 cm1. With the exposure time of 10 min, the absorption spectrum of the atmosphere over Moscow has been recorded with S/N ~120, limited by shot noise. The inversion algorithm applied to this spectrum resulted in methane vertical profile with a maximum mixing ratio of 2148 ± 10 ppbv near the surface and column density 4.59 ± 0.02·1022 cm2. ©2014 Optical Society of America OCIS codes: (010.0280) Remote sensing and sensors; (010.1280) Atmospheric composition; (300.1030) Absorption; (300.6340) Spectroscopy, infrared; (040.2840) Heterodyne.
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1. Introduction High resolution spectroscopy is widely used in a variety of applications in space research, astrophysics, environmental science and technology as a powerful analytical tool allowing for accurate measurements of species abundance, isotopic ratios, velocity fields and other parameters of target objects. Spectroscopic methods are particularly efficient in characterization of rarified gases where Doppler broadening of the IR rotational lines dominates. Also in a number of applications related to climate studies, including spacecraft monitoring of greenhouse gases, CO2 and CH4, spectral resolution sufficient to distinguish individual rotational lines is required. Dedicated spacecraft missions GOSAT [1] and OCO [2] involve instruments characterized by resolving power exceeding 10 4, while ground stations operating in the TCCON network are usually equipped with Fourier transform spectrometers with ~0.01 cm1 resolution. High cost of acquisition and maintenance of such equipment remains a major factor limiting further expansion of the ground observing network and hence, overall efficiency of global greenhouse gas monitoring campaigns. Thus, a lightweight, compact and affordable instrument for spectral measurements with resolving power exceeding 106 in the near- and mid-IR spectral range is highly demanded in various research fields. High resolution spectroscopy (λ/δλ~107-108) allows for Doppler measurements of wind fields in the atmospheres of the Earth and other planets, implemented in the infrared spectral range in only a few instruments to date and resulted in seminal results on the dynamics of planetary atmospheres [3,4] and remote sensing of the Earth atmosphere [5,6]. High resolution laser spectroscopy has been proven to be a powerful method of in situ trace gas detection from various platforms, including aircraft and planetary rovers [7,8]. However, aircraft and spacecraft applications of passive heterodyne IR spectro-radiometry has not yet
#207910 - $15.00 USD Received 10 Mar 2014; revised 25 Apr 2014; accepted 25 Apr 2014; published 30 May 2014 (C) 2014 OSA 2 June 2014 | Vol. 22, No. 11 | DOI:10.1364/OE.22.013825 | OPTICS EXPRESS 13827
been progressed beyond laboratory demonstrators [9]. Most of infrared heterodyne instruments operate in the mid- and thermal IR range. Only few attempts of heterodyne spectro-radiometry have been made to date in the SWIR and NIR ranges [10,11], despite the availability of high precision lasers, detectors and fiber optics. In this paper we present the technique of heterodyne detection of solar radiation passed through the atmosphere in the range 1.1-2.1 μm with spectral resolution up to λ/δλ~10 8, first proposed in [12]. The core idea of the method is scanning local oscillator (LO) frequency and detection of the intermediate frequency (IF) in a narrow bandpass, with distributed feedback tunable diode used as LO and single mode optical fiber Y-coupler as beam combiner. Let D be detector responsivity, ES and ELO are components of the electric field associated with the radiation of the observed source and LO, respectively. Then the heterodyne signal may be expressed as the convolution of the source and LO radiation fields in spectral space: * ihet D ES ELO ES* ELO
0
0
D Re g LO ' d ' FS exp i 't iS t iLO t d ,
(1)
where gLO and FS are power spectral densities of the LO and signal, while φLO and φS are respective phases. Due to random phase variations in the thermal broadband radiation to be analyzed, within a narrow frequency interval Β the heterodyne component could be considered as white noise with the dispersion, proportional to the target spectral density FS:
ihet = 2 BiLO
dis , dω
(2)
where iLO is and the photocurrent value corresponding to the local oscillator, and dis dω is the photocurrent of the signal per unit frequency, providing its continual spectrum. Thus, providing B is narrow enough, heterodyne detection is reduced to the measurement of standard deviation of signal at the photomixer and subtraction of noise caused by other sources. According to the antenna theorem [13], there is a fundamental limitation on the aperture available in heterodyne technique, S 2 . Therefore the lack of signal level cannot be compensated by front-end optics, and a strong enough source is needed to provide acceptable measurement accuracy. Taking into account the quantum limit of detectable spectral density at 1.65 μm, hν ~1.43·1019 W/Hz, observations in the atmosphere in this spectral range could only be done observing direct sunlight. In this paper we present first results of high resolution observations of the atmospheric absorption in the of 1.65 μm CH 4 stretching overtone. Further development of the proposed technique may result in efficient remote sensing instruments for precise measurements of atmospheric composition, structure and dynamics. 2. The instrument A block diagram of the experimental setup is presented in Fig. 1. The setup includes the LO, an optical attenuator (OA), a bundle of single mode optical fiber, several single mode fiber couplers (FC), a reference channel including a low-pressure cell with methane and photodetector (D), detection block which consists of balanced photodetector (BD), amplitude detector (AD), ADC, and PC-based controller. Unlike other infrared heterodyne spectrometers built according to the classical scheme, it does not include an IF spectral analyzer.
#207910 - $15.00 USD Received 10 Mar 2014; revised 25 Apr 2014; accepted 25 Apr 2014; published 30 May 2014 (C) 2014 OSA 2 June 2014 | Vol. 22, No. 11 | DOI:10.1364/OE.22.013825 | OPTICS EXPRESS 13828
Fig. 1. A sketch of the experimental setup. LO – local oscillator, SMOF - single mode optical fiber; FC – fiber coupler; OA – optical attenuator; T – microtelescope; RC – reference gas cell; BD – balanced detector; AD- amplitude detector; ADC – analog-digital converter; PC – personal computer. Fiber connectors are shown as green boxes.
We used tunable distributed feedback laser from NTT-Electronics, operating at λ = 1.651 µm, as LO. The laser has built-in pigtailed fiber outlet to suppress unwanted feedback. Attenuator OA allows to control net power of the radiation registered by one of the shoulders of a balance detector BD in order to null out the LO constant signal. Broadband radiation from the Sun passed through the atmosphere is captured into a fiber by a spherical planoconvex lens with 1”, f = 50.0 mm, installed on tracking support, to provide full filling of the fiber field of view by sunlight. In order to minimize the shot noise, the spectrum of the solar radiation is limited by a passband entrance filter 12 nm wide, centered at 1.65 μm, so that the total power of the sunlight falling into a fiber is reduced from 3.5 μW without filter to about 200 nW. The Sun tracking system is connected to the instrument housed in the laboratory by a ~100 m single mode fiber. Angled physical connectors (APC) are used to avoid interference of reflected signals that may significantly affect measurement accuracy. Precisely controlled ramping of the diode laser pumping has been used to control LO frequency. Pumping current was modulated by a sequence of pulses having trapezoidal shape with a length of 5 ms and period of 6 ms, with 1 ms allocated to dead time. Since diode laser radiation wavelength depends on the pumping current, this mode results in sweeping the LO frequency by a quasi-linear periodic function. In order to provide feedback for more accurate frequency control, a portion of the LO radiation is passed through the reference cell filled by methane at ~10 Torr and detected by the photodetector D, as shown in Fig. 2 by red curve. Based on comparison of a signal from D and presumed spectral shape of methane line in the reference cell, a correction to the LO temperature is generated according to a special algorithm, which provides stabilization of the specified LO frequency variations with an accuracy of 600 kHz. This technique, now standard for diode laser spectroscopy, is described in more detail in [14,15]. In order to account for possible non-linearity of the LO frequency ramping, we performed also measurements of the LO radiation flux passed through the FabryPerot etalon. The distance between its adjacent maxima (blue curve in Fig. 2) corresponds to the resonator free dispersion range, equal to 0.0492 cm1. With the free dispersion range and the line position known a priori, it is straightforward to calibrate pumping current into LO frequency. In our case, the full frequency sweeping range was ~1.1 cm1. Radiation from the LO and from the telescope is delivered to a fiber Y-coupler and after coupling, to an InGaAs p-i-n diode used as a shoulder of the balanced mixer. The other shoulder was loaded by radiation passed from the LO and attenuated by the OA. The differential signal has been amplified in the preamplifier having feedback resistivity of 1.2 MΩ and bandpass from 500 kHz to 3 MHz. At the amplitude detector AD, a signal proportional to the mean amplitude of the signal acquired from BD is formed and then digitized by a 16-bit ADC with a sampling rate of 111 kHz. Assuming that the signal ξ(t) within BD bandpass obeys Gaussian statistics with standard deviation σ, an AD output
#207910 - $15.00 USD Received 10 Mar 2014; revised 25 Apr 2014; accepted 25 Apr 2014; published 30 May 2014 (C) 2014 OSA 2 June 2014 | Vol. 22, No. 11 | DOI:10.1364/OE.22.013825 | OPTICS EXPRESS 13829
is
2 , i.e. proportional to the heterodyne signal (2). AD signal has been averaged
over integration time up to tens of minutes. Its squared value gives information about the spectral density of the input radiation convolved with LO power spectrum, i.e. the desired spectrum of the target object. 1.6 1.4 Fabry-Perot 1.2
reference cell
Signal, V
1 0.8 0.6 0.4 0.2 0 20
30
40
50 60 pumping current, mA
70
80
90
Fig. 2. The signal at the detectors illuminated by LO radiation passed through the reference cell (red curve) and Fabry-Perot etalon (blue curve), versus LO pumping current. Peak absorption in the reference cell gives absolute frequency calibration; the distances between fringes in the Fabry-Perot etalon maxima provide information about frequency tuning within the whole spectral range..
In addition to heterodyne component (2), there are different noise sources in the system: photocurrent shot noise, LO relative intensity noise (RIN), and thermal noise of the detector. With the two photodetectors, each producing an independent shot noise, balanced detection increases minimal shot noise level by a factor of 2. The thermal noise is negligible due to high feedback resistivity and, accordingly, narrow detection bandpass. We found that the optimal photocurrent range where the overall noise is dominated with shot noise, is 50-700 μA. At this LO power the heterodyne signal is detected at the minimal level of two quantum limits, due to two shoulders of the balanced detector. Increasing detected LO power may lead to the loss of heterodyne detection quality due to RIN. It worth noting that the heterodyne signal reveals a similar dependence on the bandpass and the LO power as the shot noise, and further variations of these parameters would not improve the signal-to-noise (S/N) ratio. Due to similar reason, expected S/N ratio is the same throughout the whole LO frequency sweeping range, in spite of a substantial difference in detected power. 3. Observations Observations of the Sun were carried out on October 14th, 2013, between noon and 2 p.m., on the roof of Prokhorov Institute building in Moscow, with an elevation of the observing point being ~50 m above the ground. The detected signal random mean squared (RMS) intensity is shown in Fig. 3(a) versus LO pumping current.
#207910 - $15.00 USD Received 10 Mar 2014; revised 25 Apr 2014; accepted 25 Apr 2014; published 30 May 2014 (C) 2014 OSA 2 June 2014 | Vol. 22, No. 11 | DOI:10.1364/OE.22.013825 | OPTICS EXPRESS 13830
Fig. 3. (a) Shot noise (blue curve) and signal random mean-square deviation at the balanced detector versus LO pumping current during heterodyne observations of the Sun (blue curve). (b) Atmospheric transmission spectrum derived from data presented at Fig. 3(a). Narrow tip of the CO2 feature is caused by contribution of low-pressure stratospheric layers.
The blue curve in Fig. 3(a) shows the measured signal in case of only LO radiation falls to the detector and corresponds to double shot noise level with minor additive component coming from noise sources other than the shot noise. The factor of 2 appears due to balanced detection scheme implying two photodetectors. The red curve in Fig. 3(a) shows measured signal in the case if LO coupled with solar radiation is detected. The departure of the red curve from shot noise level is the heterodyne signal shown in Fig. 3(b) by red curve. The apparent irregularity of this curve is a result of spectral absorption features along the atmospheric path. After subtraction of the “zero noise level”, i.e. noise caused by the detector and electronics measured with LO being off, pure shot noise has been scaled by a factor 0.08 equal to the S/N ratio of a single measurement, resulting in the assumed baseline presented in Fig. 3(b) by blue curve. Frequency calibration procedure described above and continuum estimation based on the scaling of the shot noise level [blue curve in Fig. 3(b)] gives the atmospheric absorption spectrum in terms of relative transmittance, as shown in Fig. 4. With the exposure time equal to 10 min, S/N~120 is reached. As the shot noise is not the only source of noise in the system, albeit the dominating one, a special procedure may be needed to calibrate zero transmission level. We had to correct the absorption spectrum by a constant value ~0.026 in order to compensate the suspected drift of zero level in the instrument during the long exposure. 1
CH4
transmission
0.8 0.6 0.4 0.2 0 6056.4
CO2Solar
6056.6
CH4
6056.8 6057 6057.2 wavenumber, cm-1
6057.4
Fig. 4. Atmospheric transmission spectrum derived from data presented at Fig. 3(a). Narrow tip of the CO2 feature is caused by contribution of low-pressure stratospheric layers.
#207910 - $15.00 USD Received 10 Mar 2014; revised 25 Apr 2014; accepted 25 Apr 2014; published 30 May 2014 (C) 2014 OSA 2 June 2014 | Vol. 22, No. 11 | DOI:10.1364/OE.22.013825 | OPTICS EXPRESS 13831
The spectrum shown in Fig. 4 reveals CO2 line at 6056 cm1 with a completely resolved profile, and a quartet of overlapping CH4 lines centered at 6057.1 cm1. Other features of the observed spectra include a weak CH4 absorption line on the longwave wing of the main quartet and a Fraunhofer line in the solar spectrum at 6056.65 cm1. The latter line was compared with the reference solar spectrum from [16] in order to calibrate zero transmission level as mentioned above. The analysis of CO2 absorption will be presented in more detail elsewhere. In this paper we focus on the stronger 6057.1 cm1 methane feature measured at 2 p.m., when the peak absorption is unsaturated. An attempt to retrieve available information from the resolved profile of mutually overlapping CH 4 line group is presented below. 4. Methane profile retrievals The retrieval algorithm includes forward model, aimed to reproduce the observed spectrum, and the inverse problem solution. Due to small angular size of the Sun, the forward model is based on simplified radiative transfer equation with zero scattering coefficient. Meteorological data (temperature profile and surface pressure) have been borrowed from the ERA interim database [17] with adopted time and location of the observations. The atmospheric radiative transfer model includes 100 plane-parallel layers and extends from the ground to ztop = 40 km. Neglecting the atmosphere above this altitude may result in the transmission error less than 0.001, which does not exceed the uncertainty of the experiment. Solar spectrum obtained by Eureca and Atlas satellites [16] and spectral information from HITRAN2012 database [18] are adopted. The corrected heterodyne data along with synthetic absorption spectra calculated with different forward model parameters are presented in Fig. 5(a), 5(b). The inverse problem of methane mixing ratio profile CH 4 z retrieval can be expressed using the equation ztop
K , z z dz , h
CH 4
0 , 1 ,
(3)
0
where is the optical depth calculated from the measured transmittance. K h , z is the absorption kernel, i.e. the product of molecular absorption crossection of methane for IR radiation frequency ν within the spectral range of the instrument 0 , 1 and air number density at specified altitude z. The parameters h and δ refer to uncertainty of the absorption kernel K and the measured opacity τ respectively. Evidently, in general Eq. (3) represents an ill-posed problem, and for the solution to be practical, some regularization method must be applied. Thus, we retrieved methane vertical profile using Tikhonov regularization by smoothing functional, which takes into account a priori information about first guess profile 0 CH z . Lacking the information about target values at particular altitudes, e.g. in the upper 4 part of the retrieved profile, the regularization procedure always selects a priori values unless the measurements contradict this assumption. In our case, mean methane profile for the Northern hemisphere obtained by the Atmospheric Infrared Sounder onboard AQUA satellite [19] was adopted as first guess. The Tikhonov smoothing functional gives the compromise between the residual minimization and solution smoothness, which is controlled by the regularization parameter :
K v , z z dz v dv v
M
1
z
top
h
v
0
0
z z z
z
top
2
0
0
dz
top
z z
2
0
0
dz
(4)
#207910 - $15.00 USD Received 10 Mar 2014; revised 25 Apr 2014; accepted 25 Apr 2014; published 30 May 2014 (C) 2014 OSA 2 June 2014 | Vol. 22, No. 11 | DOI:10.1364/OE.22.013825 | OPTICS EXPRESS 13832
As to achieve the best precision of the solution parameter α must be determined according to the S/N level δ, operator uncertainty level h. For retrieval of α, so called generalized smoothing functional method [20,21] is applied. As in the forward model the atmosphere is split into N = 100 layers, with the height of one layer hz zmax N , a complaint step in wavenumber h 1 0 N is adopted. Discretizing and differentiating the functional (4) results in a solvable linear system of equations:
h Kˆ Kˆ h E Lˆ Lˆ ˆ h Kˆ ˆ h ˆ
T
T
0 z
T
0 z
0
LˆT ˆ '0 .
(5)
Here the matrix Kˆ corresponds the kernel function K ( , z ) , vectors ˆ and ˆ correspond to functions z and respectively, matrix Lˆ corresponds to the first derivative operator. Soluton to Eq. (5) is equivalent to an optimal solution to the inverse problem (3), i.e. the desired vertical profile of absorber abundance.
Fig. 5. Methane profile retrievals procedure: (a) Measured transmittance spectrum in the vicinity of CH4 6057.1 cm1 feature (black curve) versus synthetic spectra corresponding to first guess methane profile (blue) and best fit (red). (b) The same, but in terms of optical depth along the line of sight. Error bars correspond to statistical errors of opacity measurements caused by shot noise (c) Retrieved methane vertical profile (red curve) and first guess (blue curve). Confidence intervals are marked by red dots. (d) Variations of the retrieved profile resulting from data perturbation within standard deviation (black) and from perturbation of the implied thermal profile by 5K (red).
The observed transmission spectrum is shown in the Fig. 5(a) by black curve. Corresponding optical depth along the line of sight is presented in the panel (b) within narrower spectral range in order to emphasize its behavior in the vicinity of the methane feature tip. The synthetic transmission [Fig. 5(a)] and optical depth [Fig. 5(b)] corresponding
#207910 - $15.00 USD Received 10 Mar 2014; revised 25 Apr 2014; accepted 25 Apr 2014; published 30 May 2014 (C) 2014 OSA 2 June 2014 | Vol. 22, No. 11 | DOI:10.1364/OE.22.013825 | OPTICS EXPRESS 13833
to first guess methane vertical profile and the best fit to data, are presented by blue and red curves, respectively. In spite of strong pressure broadening in the troposphere, not only the shape of the methane feature allows to estimate overall column abundance of the absorber, but also to distinguish variations in its vertical distribution. In particular, profile with more abundant stratospheric methane would result in more distinct double-tip character of the CH4 feature due to increasing contribution of less broadened absorption lines, as shown in Fig. 5(b). The retrieved methane profile [Fig. 5(c)] demonstrates higher mixing ratio in the first scale height compared to the assumed model profile, well expected in the megalopolis center [22–24]. In the upper part of the profile, where the data reveal weak sensitivity to methane abundance, retrieved profile approaches the first guess values due to regularization procedure described above. However, the reference to first guess profile is not the only source of uncertainty in the retrieved profile. To estimate its sensitivity to the uncertainty in both measured and synthetic transmittance spectra we repeated the retrieval procedure with random perturbations in data within one sigma level (about 0.01 in terms of transmittance) and in the assumed thermal profile within 5 K. 100 test retrievals have been made with each type of perturbations. The resulting variability in the retrieved profile is shown in Fig. 5(d) in terms of one-sigma RMS variations. Retrieved profile variations corresponding to perturbed data are shown by magenta curves, and those corresponding to perturbed temperature profile by black curves. In both cases of perturbations the retrievals sensitivity is limited by 10 ppb, with the exception of the lower part of the profile where the tendency to lower values is revealed. Thus the methane abundance variations may be evaluated with relative accuracy much better than 1%, which fits the requirements of greenhouse gas monitoring. 5. Conclusions The experiment described in this paper has proven feasibility of heterodyne detection in the near infrared range using commercial tunable diode laser as local oscillator and single mode fiber optics for beam combining. By means of heterodyne technique, a completely resolved methane absorption feature in the atmosphere has been measured in the solar occultation mode with RMS uncertainty equal to ~0.008 at net exposure time of 10 min. A minimal detectable signal of 1021 W/Hz is expected, with the accuracy being limited by double shot noise level. Achieved spectral resolution is determined by LO line width and stability and constitutes about 2.5 MHz, which corresponds to resolving power of λ/δλ~10 8. Due to higher spectral resolution, lower sensitivity to atmospheric temperatures, humidity and vibrations, compared to heterodyne measurements in the thermal IR spectral range, the technique described in this paper provides accuracy comparable with much more complicated high resolution measurements now used in TCCON stations. Higher spectral resolution, longer integration time and broader spectral coverage achieved due to LO stabilization by means of an external reference gas cell provides certain advantages compared to recently published examples of NIR heterodyne spectro-radiometry [10,11], in particular, the capability of absorber vertical profiling. Relative simplicity of the proposed scheme opens a perspective to employ this scheme for high resolution spectroscopy in various applications. In particular, it may allow solar occultation observations of CO2, CО, CH4, H2S, C2H4 and other gases from spacecraft, airborne or ground-based platforms. Acknowledgments This work has been supported by the Ministry of Education and Science of Russian Federation, grant #11.G34.31.0074. We thank Anna Fedorova and Oleg Korablev for helpful discussions and comments, and anonymous reviewers whose remarks have allowed us to improve the quality of the manuscript.
#207910 - $15.00 USD Received 10 Mar 2014; revised 25 Apr 2014; accepted 25 Apr 2014; published 30 May 2014 (C) 2014 OSA 2 June 2014 | Vol. 22, No. 11 | DOI:10.1364/OE.22.013825 | OPTICS EXPRESS 13834
Moscow Institute of Physics and Technology, Institutsky dr. 9, 141700 Dolgoprudnyi, Russia 2 Space Research Institute (IKI), Profsoyuznaya 84/32, 117997 Moscow, Russia 3 Prokhorov General Physics Institute, Vavilova 38, 119991 Moscow, Russia *[email protected]
Abstract: The paper describes the concept of a compact, lightweight heterodyne NIR spectro-radiometer suitable for atmospheric sounding with solar occultations, and the first measurement of CO2 and CH4 absorption near 1.65 μm with spectral resolution λ/δλ~10 8. A highly stabilized DFB laser was used as local oscillator, while single model silica fiber Y-coupler served as a diplexer. Radiation mixed in the single mode fiber was detected by a balanced couple of InGaAs p-i-n diodes within the bandpass of ~3 MHz. Wavelength coverage of spectral measurement was provided by sweeping local oscillator frequency in the range of 1.1 cm1. With the exposure time of 10 min, the absorption spectrum of the atmosphere over Moscow has been recorded with S/N ~120, limited by shot noise. The inversion algorithm applied to this spectrum resulted in methane vertical profile with a maximum mixing ratio of 2148 ± 10 ppbv near the surface and column density 4.59 ± 0.02·1022 cm2. ©2014 Optical Society of America OCIS codes: (010.0280) Remote sensing and sensors; (010.1280) Atmospheric composition; (300.1030) Absorption; (300.6340) Spectroscopy, infrared; (040.2840) Heterodyne.
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#207910 - $15.00 USD Received 10 Mar 2014; revised 25 Apr 2014; accepted 25 Apr 2014; published 30 May 2014 (C) 2014 OSA 2 June 2014 | Vol. 22, No. 11 | DOI:10.1364/OE.22.013825 | OPTICS EXPRESS 13826
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for continuous measurements of the concentration of methane, carbon dioxide, water and its isotopes,” Appl. Phys. B 109(3), 505–510 (2012). G. Thuillier, M. Hersé, D. Labs, T. Foujols, W. Peetermans, D. Gillotay, P. C. Simon, and H. Mandel, “The Solar Spectral Irradiance from 200 to 2400 nm as measured by the SOLSPEC Spectrometer from the Atlas and Eureca Missions,” Sol. Phys. 214(1), 1–22 (2003). http://apps.ecmwf.int/datasets/data/interim_ full_ daily. L. S. Rothman, I. E. Gordon, Y. Babikov, A. Barbe, D. C. Benner, P. F. Bernath, M. Birk, L. Bizzocchi, V. Boudon, L. R. Brown, A. Campargue, K. Chance, E. A. Cohen, L. H. Coudert, V. M. Devi, B. J. Drouin, A. Fayt, J.-M. Flaud, R. R. Gamache, J. J. Harrison, J.-M. Hartmann, C. Hill, J. T. Hodges, D. Jacquemart, A. Jolly, J. Lamouroux, R. J. LeRoy, G. Li, D. A. Long, O. M. Lyulin, C. J. Mackie, S. T. Massie, S. Mikhailenko, H. S. P. Müller, O. V. Naumenko, A. V. Nikitin, J. Orphal, V. Perevalov, A. Perrin, E. R. Polovtseva, C. Richard, M. A. H. Smith, E. Starikova, K. Sung, S. Tashkun, J. Tennyson, and G. C. Toon, “The HITRAN2012 molecular spectroscopic database,” J. Quant. Spectrosc. Radiat. Transf. 130, 4–50 (2013). X. Xiong, C. Barnet, E. Maddy, J. Wei, X. Liu, and T. S. Pagano, “Seven years’ observation of mid-upper tropospheric methane from Atmospheric Infrared Sounder,” Remote Sens. 2(11), 2509–2530 (2010), doi:10.3390/rs2112509. M. Buchwitz, M. Reuter, H. Bovensmann, D. Pillai, J. Heymann, O. Schneising, V. Rozanov, T. Krings, J. P. Burrows, H. Boesch, C. Gerbig, Y. Meijer, and A. Löscher, “Carbon Monitoring Satellite (CarbonSat): assessment of scattering related atmospheric CO2 and CH4 retrieval errors and first results on implications for inferring city CO2 emissions,” Atmos. Meas. Tech. Discuss. 6, 4769–4850 (2013). E. Choi, K. Choi, and S.-M. Yi, “Non-methane hydrocarbons in the atmosphere of a Metropolitan City and a background site in South Korea: Sources and health risk potentials,” Atmos. Environ. 45(40), 7563–7573 (2011). G. Gorchakov, E. Semoutnikova, A. Karpov, and E. Lezina, “Air Pollution in Moscow Megacity”, Advanced Topics in Environmental Health and Air Pollution Case Studies,” Moldoveanu, A. (Ed.), (2011). Available from: http://www.intechopen.com/books/advanced-topics-in-environmental-health-and-air-pollution-case-studies/airpollution-in-moscow-megacity T. R. Tsai, R. A. Rose, D. Weidmann, and G. Wysocki, “Atmospheric vertical profiles of O3, N2O, CH4, CCl2F2, and H2O retrieved from external-cavity quantum-cascade laser heterodyne radiometer measurements,” Appl. Opt. 51(36), 8779–8792 (2012). M. C. Geibel, J. Messerschmidt, C. Gerbig, T. Blumenstock, H. Chen, F. Hase, O. Kolle, J. V. Lavrič, J. Notholt, M. Palm, M. Rettinger, M. Schmidt, R. Sussmann, T. Warneke, and D. G. Feist, “Calibration of columnaveraged CH4 over European TCCON FTS sites with airborne in-situ measurements,” Atmos. Chem. Phys. 12(18), 8763–8775 (2012).
1. Introduction High resolution spectroscopy is widely used in a variety of applications in space research, astrophysics, environmental science and technology as a powerful analytical tool allowing for accurate measurements of species abundance, isotopic ratios, velocity fields and other parameters of target objects. Spectroscopic methods are particularly efficient in characterization of rarified gases where Doppler broadening of the IR rotational lines dominates. Also in a number of applications related to climate studies, including spacecraft monitoring of greenhouse gases, CO2 and CH4, spectral resolution sufficient to distinguish individual rotational lines is required. Dedicated spacecraft missions GOSAT [1] and OCO [2] involve instruments characterized by resolving power exceeding 10 4, while ground stations operating in the TCCON network are usually equipped with Fourier transform spectrometers with ~0.01 cm1 resolution. High cost of acquisition and maintenance of such equipment remains a major factor limiting further expansion of the ground observing network and hence, overall efficiency of global greenhouse gas monitoring campaigns. Thus, a lightweight, compact and affordable instrument for spectral measurements with resolving power exceeding 106 in the near- and mid-IR spectral range is highly demanded in various research fields. High resolution spectroscopy (λ/δλ~107-108) allows for Doppler measurements of wind fields in the atmospheres of the Earth and other planets, implemented in the infrared spectral range in only a few instruments to date and resulted in seminal results on the dynamics of planetary atmospheres [3,4] and remote sensing of the Earth atmosphere [5,6]. High resolution laser spectroscopy has been proven to be a powerful method of in situ trace gas detection from various platforms, including aircraft and planetary rovers [7,8]. However, aircraft and spacecraft applications of passive heterodyne IR spectro-radiometry has not yet
#207910 - $15.00 USD Received 10 Mar 2014; revised 25 Apr 2014; accepted 25 Apr 2014; published 30 May 2014 (C) 2014 OSA 2 June 2014 | Vol. 22, No. 11 | DOI:10.1364/OE.22.013825 | OPTICS EXPRESS 13827
been progressed beyond laboratory demonstrators [9]. Most of infrared heterodyne instruments operate in the mid- and thermal IR range. Only few attempts of heterodyne spectro-radiometry have been made to date in the SWIR and NIR ranges [10,11], despite the availability of high precision lasers, detectors and fiber optics. In this paper we present the technique of heterodyne detection of solar radiation passed through the atmosphere in the range 1.1-2.1 μm with spectral resolution up to λ/δλ~10 8, first proposed in [12]. The core idea of the method is scanning local oscillator (LO) frequency and detection of the intermediate frequency (IF) in a narrow bandpass, with distributed feedback tunable diode used as LO and single mode optical fiber Y-coupler as beam combiner. Let D be detector responsivity, ES and ELO are components of the electric field associated with the radiation of the observed source and LO, respectively. Then the heterodyne signal may be expressed as the convolution of the source and LO radiation fields in spectral space: * ihet D ES ELO ES* ELO
0
0
D Re g LO ' d ' FS exp i 't iS t iLO t d ,
(1)
where gLO and FS are power spectral densities of the LO and signal, while φLO and φS are respective phases. Due to random phase variations in the thermal broadband radiation to be analyzed, within a narrow frequency interval Β the heterodyne component could be considered as white noise with the dispersion, proportional to the target spectral density FS:
ihet = 2 BiLO
dis , dω
(2)
where iLO is and the photocurrent value corresponding to the local oscillator, and dis dω is the photocurrent of the signal per unit frequency, providing its continual spectrum. Thus, providing B is narrow enough, heterodyne detection is reduced to the measurement of standard deviation of signal at the photomixer and subtraction of noise caused by other sources. According to the antenna theorem [13], there is a fundamental limitation on the aperture available in heterodyne technique, S 2 . Therefore the lack of signal level cannot be compensated by front-end optics, and a strong enough source is needed to provide acceptable measurement accuracy. Taking into account the quantum limit of detectable spectral density at 1.65 μm, hν ~1.43·1019 W/Hz, observations in the atmosphere in this spectral range could only be done observing direct sunlight. In this paper we present first results of high resolution observations of the atmospheric absorption in the of 1.65 μm CH 4 stretching overtone. Further development of the proposed technique may result in efficient remote sensing instruments for precise measurements of atmospheric composition, structure and dynamics. 2. The instrument A block diagram of the experimental setup is presented in Fig. 1. The setup includes the LO, an optical attenuator (OA), a bundle of single mode optical fiber, several single mode fiber couplers (FC), a reference channel including a low-pressure cell with methane and photodetector (D), detection block which consists of balanced photodetector (BD), amplitude detector (AD), ADC, and PC-based controller. Unlike other infrared heterodyne spectrometers built according to the classical scheme, it does not include an IF spectral analyzer.
#207910 - $15.00 USD Received 10 Mar 2014; revised 25 Apr 2014; accepted 25 Apr 2014; published 30 May 2014 (C) 2014 OSA 2 June 2014 | Vol. 22, No. 11 | DOI:10.1364/OE.22.013825 | OPTICS EXPRESS 13828
Fig. 1. A sketch of the experimental setup. LO – local oscillator, SMOF - single mode optical fiber; FC – fiber coupler; OA – optical attenuator; T – microtelescope; RC – reference gas cell; BD – balanced detector; AD- amplitude detector; ADC – analog-digital converter; PC – personal computer. Fiber connectors are shown as green boxes.
We used tunable distributed feedback laser from NTT-Electronics, operating at λ = 1.651 µm, as LO. The laser has built-in pigtailed fiber outlet to suppress unwanted feedback. Attenuator OA allows to control net power of the radiation registered by one of the shoulders of a balance detector BD in order to null out the LO constant signal. Broadband radiation from the Sun passed through the atmosphere is captured into a fiber by a spherical planoconvex lens with 1”, f = 50.0 mm, installed on tracking support, to provide full filling of the fiber field of view by sunlight. In order to minimize the shot noise, the spectrum of the solar radiation is limited by a passband entrance filter 12 nm wide, centered at 1.65 μm, so that the total power of the sunlight falling into a fiber is reduced from 3.5 μW without filter to about 200 nW. The Sun tracking system is connected to the instrument housed in the laboratory by a ~100 m single mode fiber. Angled physical connectors (APC) are used to avoid interference of reflected signals that may significantly affect measurement accuracy. Precisely controlled ramping of the diode laser pumping has been used to control LO frequency. Pumping current was modulated by a sequence of pulses having trapezoidal shape with a length of 5 ms and period of 6 ms, with 1 ms allocated to dead time. Since diode laser radiation wavelength depends on the pumping current, this mode results in sweeping the LO frequency by a quasi-linear periodic function. In order to provide feedback for more accurate frequency control, a portion of the LO radiation is passed through the reference cell filled by methane at ~10 Torr and detected by the photodetector D, as shown in Fig. 2 by red curve. Based on comparison of a signal from D and presumed spectral shape of methane line in the reference cell, a correction to the LO temperature is generated according to a special algorithm, which provides stabilization of the specified LO frequency variations with an accuracy of 600 kHz. This technique, now standard for diode laser spectroscopy, is described in more detail in [14,15]. In order to account for possible non-linearity of the LO frequency ramping, we performed also measurements of the LO radiation flux passed through the FabryPerot etalon. The distance between its adjacent maxima (blue curve in Fig. 2) corresponds to the resonator free dispersion range, equal to 0.0492 cm1. With the free dispersion range and the line position known a priori, it is straightforward to calibrate pumping current into LO frequency. In our case, the full frequency sweeping range was ~1.1 cm1. Radiation from the LO and from the telescope is delivered to a fiber Y-coupler and after coupling, to an InGaAs p-i-n diode used as a shoulder of the balanced mixer. The other shoulder was loaded by radiation passed from the LO and attenuated by the OA. The differential signal has been amplified in the preamplifier having feedback resistivity of 1.2 MΩ and bandpass from 500 kHz to 3 MHz. At the amplitude detector AD, a signal proportional to the mean amplitude of the signal acquired from BD is formed and then digitized by a 16-bit ADC with a sampling rate of 111 kHz. Assuming that the signal ξ(t) within BD bandpass obeys Gaussian statistics with standard deviation σ, an AD output
#207910 - $15.00 USD Received 10 Mar 2014; revised 25 Apr 2014; accepted 25 Apr 2014; published 30 May 2014 (C) 2014 OSA 2 June 2014 | Vol. 22, No. 11 | DOI:10.1364/OE.22.013825 | OPTICS EXPRESS 13829
is
2 , i.e. proportional to the heterodyne signal (2). AD signal has been averaged
over integration time up to tens of minutes. Its squared value gives information about the spectral density of the input radiation convolved with LO power spectrum, i.e. the desired spectrum of the target object. 1.6 1.4 Fabry-Perot 1.2
reference cell
Signal, V
1 0.8 0.6 0.4 0.2 0 20
30
40
50 60 pumping current, mA
70
80
90
Fig. 2. The signal at the detectors illuminated by LO radiation passed through the reference cell (red curve) and Fabry-Perot etalon (blue curve), versus LO pumping current. Peak absorption in the reference cell gives absolute frequency calibration; the distances between fringes in the Fabry-Perot etalon maxima provide information about frequency tuning within the whole spectral range..
In addition to heterodyne component (2), there are different noise sources in the system: photocurrent shot noise, LO relative intensity noise (RIN), and thermal noise of the detector. With the two photodetectors, each producing an independent shot noise, balanced detection increases minimal shot noise level by a factor of 2. The thermal noise is negligible due to high feedback resistivity and, accordingly, narrow detection bandpass. We found that the optimal photocurrent range where the overall noise is dominated with shot noise, is 50-700 μA. At this LO power the heterodyne signal is detected at the minimal level of two quantum limits, due to two shoulders of the balanced detector. Increasing detected LO power may lead to the loss of heterodyne detection quality due to RIN. It worth noting that the heterodyne signal reveals a similar dependence on the bandpass and the LO power as the shot noise, and further variations of these parameters would not improve the signal-to-noise (S/N) ratio. Due to similar reason, expected S/N ratio is the same throughout the whole LO frequency sweeping range, in spite of a substantial difference in detected power. 3. Observations Observations of the Sun were carried out on October 14th, 2013, between noon and 2 p.m., on the roof of Prokhorov Institute building in Moscow, with an elevation of the observing point being ~50 m above the ground. The detected signal random mean squared (RMS) intensity is shown in Fig. 3(a) versus LO pumping current.
#207910 - $15.00 USD Received 10 Mar 2014; revised 25 Apr 2014; accepted 25 Apr 2014; published 30 May 2014 (C) 2014 OSA 2 June 2014 | Vol. 22, No. 11 | DOI:10.1364/OE.22.013825 | OPTICS EXPRESS 13830
Fig. 3. (a) Shot noise (blue curve) and signal random mean-square deviation at the balanced detector versus LO pumping current during heterodyne observations of the Sun (blue curve). (b) Atmospheric transmission spectrum derived from data presented at Fig. 3(a). Narrow tip of the CO2 feature is caused by contribution of low-pressure stratospheric layers.
The blue curve in Fig. 3(a) shows the measured signal in case of only LO radiation falls to the detector and corresponds to double shot noise level with minor additive component coming from noise sources other than the shot noise. The factor of 2 appears due to balanced detection scheme implying two photodetectors. The red curve in Fig. 3(a) shows measured signal in the case if LO coupled with solar radiation is detected. The departure of the red curve from shot noise level is the heterodyne signal shown in Fig. 3(b) by red curve. The apparent irregularity of this curve is a result of spectral absorption features along the atmospheric path. After subtraction of the “zero noise level”, i.e. noise caused by the detector and electronics measured with LO being off, pure shot noise has been scaled by a factor 0.08 equal to the S/N ratio of a single measurement, resulting in the assumed baseline presented in Fig. 3(b) by blue curve. Frequency calibration procedure described above and continuum estimation based on the scaling of the shot noise level [blue curve in Fig. 3(b)] gives the atmospheric absorption spectrum in terms of relative transmittance, as shown in Fig. 4. With the exposure time equal to 10 min, S/N~120 is reached. As the shot noise is not the only source of noise in the system, albeit the dominating one, a special procedure may be needed to calibrate zero transmission level. We had to correct the absorption spectrum by a constant value ~0.026 in order to compensate the suspected drift of zero level in the instrument during the long exposure. 1
CH4
transmission
0.8 0.6 0.4 0.2 0 6056.4
CO2Solar
6056.6
CH4
6056.8 6057 6057.2 wavenumber, cm-1
6057.4
Fig. 4. Atmospheric transmission spectrum derived from data presented at Fig. 3(a). Narrow tip of the CO2 feature is caused by contribution of low-pressure stratospheric layers.
#207910 - $15.00 USD Received 10 Mar 2014; revised 25 Apr 2014; accepted 25 Apr 2014; published 30 May 2014 (C) 2014 OSA 2 June 2014 | Vol. 22, No. 11 | DOI:10.1364/OE.22.013825 | OPTICS EXPRESS 13831
The spectrum shown in Fig. 4 reveals CO2 line at 6056 cm1 with a completely resolved profile, and a quartet of overlapping CH4 lines centered at 6057.1 cm1. Other features of the observed spectra include a weak CH4 absorption line on the longwave wing of the main quartet and a Fraunhofer line in the solar spectrum at 6056.65 cm1. The latter line was compared with the reference solar spectrum from [16] in order to calibrate zero transmission level as mentioned above. The analysis of CO2 absorption will be presented in more detail elsewhere. In this paper we focus on the stronger 6057.1 cm1 methane feature measured at 2 p.m., when the peak absorption is unsaturated. An attempt to retrieve available information from the resolved profile of mutually overlapping CH 4 line group is presented below. 4. Methane profile retrievals The retrieval algorithm includes forward model, aimed to reproduce the observed spectrum, and the inverse problem solution. Due to small angular size of the Sun, the forward model is based on simplified radiative transfer equation with zero scattering coefficient. Meteorological data (temperature profile and surface pressure) have been borrowed from the ERA interim database [17] with adopted time and location of the observations. The atmospheric radiative transfer model includes 100 plane-parallel layers and extends from the ground to ztop = 40 km. Neglecting the atmosphere above this altitude may result in the transmission error less than 0.001, which does not exceed the uncertainty of the experiment. Solar spectrum obtained by Eureca and Atlas satellites [16] and spectral information from HITRAN2012 database [18] are adopted. The corrected heterodyne data along with synthetic absorption spectra calculated with different forward model parameters are presented in Fig. 5(a), 5(b). The inverse problem of methane mixing ratio profile CH 4 z retrieval can be expressed using the equation ztop
K , z z dz , h
CH 4
0 , 1 ,
(3)
0
where is the optical depth calculated from the measured transmittance. K h , z is the absorption kernel, i.e. the product of molecular absorption crossection of methane for IR radiation frequency ν within the spectral range of the instrument 0 , 1 and air number density at specified altitude z. The parameters h and δ refer to uncertainty of the absorption kernel K and the measured opacity τ respectively. Evidently, in general Eq. (3) represents an ill-posed problem, and for the solution to be practical, some regularization method must be applied. Thus, we retrieved methane vertical profile using Tikhonov regularization by smoothing functional, which takes into account a priori information about first guess profile 0 CH z . Lacking the information about target values at particular altitudes, e.g. in the upper 4 part of the retrieved profile, the regularization procedure always selects a priori values unless the measurements contradict this assumption. In our case, mean methane profile for the Northern hemisphere obtained by the Atmospheric Infrared Sounder onboard AQUA satellite [19] was adopted as first guess. The Tikhonov smoothing functional gives the compromise between the residual minimization and solution smoothness, which is controlled by the regularization parameter :
K v , z z dz v dv v
M
1
z
top
h
v
0
0
z z z
z
top
2
0
0
dz
top
z z
2
0
0
dz
(4)
#207910 - $15.00 USD Received 10 Mar 2014; revised 25 Apr 2014; accepted 25 Apr 2014; published 30 May 2014 (C) 2014 OSA 2 June 2014 | Vol. 22, No. 11 | DOI:10.1364/OE.22.013825 | OPTICS EXPRESS 13832
As to achieve the best precision of the solution parameter α must be determined according to the S/N level δ, operator uncertainty level h. For retrieval of α, so called generalized smoothing functional method [20,21] is applied. As in the forward model the atmosphere is split into N = 100 layers, with the height of one layer hz zmax N , a complaint step in wavenumber h 1 0 N is adopted. Discretizing and differentiating the functional (4) results in a solvable linear system of equations:
h Kˆ Kˆ h E Lˆ Lˆ ˆ h Kˆ ˆ h ˆ
T
T
0 z
T
0 z
0
LˆT ˆ '0 .
(5)
Here the matrix Kˆ corresponds the kernel function K ( , z ) , vectors ˆ and ˆ correspond to functions z and respectively, matrix Lˆ corresponds to the first derivative operator. Soluton to Eq. (5) is equivalent to an optimal solution to the inverse problem (3), i.e. the desired vertical profile of absorber abundance.
Fig. 5. Methane profile retrievals procedure: (a) Measured transmittance spectrum in the vicinity of CH4 6057.1 cm1 feature (black curve) versus synthetic spectra corresponding to first guess methane profile (blue) and best fit (red). (b) The same, but in terms of optical depth along the line of sight. Error bars correspond to statistical errors of opacity measurements caused by shot noise (c) Retrieved methane vertical profile (red curve) and first guess (blue curve). Confidence intervals are marked by red dots. (d) Variations of the retrieved profile resulting from data perturbation within standard deviation (black) and from perturbation of the implied thermal profile by 5K (red).
The observed transmission spectrum is shown in the Fig. 5(a) by black curve. Corresponding optical depth along the line of sight is presented in the panel (b) within narrower spectral range in order to emphasize its behavior in the vicinity of the methane feature tip. The synthetic transmission [Fig. 5(a)] and optical depth [Fig. 5(b)] corresponding
#207910 - $15.00 USD Received 10 Mar 2014; revised 25 Apr 2014; accepted 25 Apr 2014; published 30 May 2014 (C) 2014 OSA 2 June 2014 | Vol. 22, No. 11 | DOI:10.1364/OE.22.013825 | OPTICS EXPRESS 13833
to first guess methane vertical profile and the best fit to data, are presented by blue and red curves, respectively. In spite of strong pressure broadening in the troposphere, not only the shape of the methane feature allows to estimate overall column abundance of the absorber, but also to distinguish variations in its vertical distribution. In particular, profile with more abundant stratospheric methane would result in more distinct double-tip character of the CH4 feature due to increasing contribution of less broadened absorption lines, as shown in Fig. 5(b). The retrieved methane profile [Fig. 5(c)] demonstrates higher mixing ratio in the first scale height compared to the assumed model profile, well expected in the megalopolis center [22–24]. In the upper part of the profile, where the data reveal weak sensitivity to methane abundance, retrieved profile approaches the first guess values due to regularization procedure described above. However, the reference to first guess profile is not the only source of uncertainty in the retrieved profile. To estimate its sensitivity to the uncertainty in both measured and synthetic transmittance spectra we repeated the retrieval procedure with random perturbations in data within one sigma level (about 0.01 in terms of transmittance) and in the assumed thermal profile within 5 K. 100 test retrievals have been made with each type of perturbations. The resulting variability in the retrieved profile is shown in Fig. 5(d) in terms of one-sigma RMS variations. Retrieved profile variations corresponding to perturbed data are shown by magenta curves, and those corresponding to perturbed temperature profile by black curves. In both cases of perturbations the retrievals sensitivity is limited by 10 ppb, with the exception of the lower part of the profile where the tendency to lower values is revealed. Thus the methane abundance variations may be evaluated with relative accuracy much better than 1%, which fits the requirements of greenhouse gas monitoring. 5. Conclusions The experiment described in this paper has proven feasibility of heterodyne detection in the near infrared range using commercial tunable diode laser as local oscillator and single mode fiber optics for beam combining. By means of heterodyne technique, a completely resolved methane absorption feature in the atmosphere has been measured in the solar occultation mode with RMS uncertainty equal to ~0.008 at net exposure time of 10 min. A minimal detectable signal of 1021 W/Hz is expected, with the accuracy being limited by double shot noise level. Achieved spectral resolution is determined by LO line width and stability and constitutes about 2.5 MHz, which corresponds to resolving power of λ/δλ~10 8. Due to higher spectral resolution, lower sensitivity to atmospheric temperatures, humidity and vibrations, compared to heterodyne measurements in the thermal IR spectral range, the technique described in this paper provides accuracy comparable with much more complicated high resolution measurements now used in TCCON stations. Higher spectral resolution, longer integration time and broader spectral coverage achieved due to LO stabilization by means of an external reference gas cell provides certain advantages compared to recently published examples of NIR heterodyne spectro-radiometry [10,11], in particular, the capability of absorber vertical profiling. Relative simplicity of the proposed scheme opens a perspective to employ this scheme for high resolution spectroscopy in various applications. In particular, it may allow solar occultation observations of CO2, CО, CH4, H2S, C2H4 and other gases from spacecraft, airborne or ground-based platforms. Acknowledgments This work has been supported by the Ministry of Education and Science of Russian Federation, grant #11.G34.31.0074. We thank Anna Fedorova and Oleg Korablev for helpful discussions and comments, and anonymous reviewers whose remarks have allowed us to improve the quality of the manuscript.
#207910 - $15.00 USD Received 10 Mar 2014; revised 25 Apr 2014; accepted 25 Apr 2014; published 30 May 2014 (C) 2014 OSA 2 June 2014 | Vol. 22, No. 11 | DOI:10.1364/OE.22.013825 | OPTICS EXPRESS 13834
