1. Introduction

Optical low-coherence reflectometry (OLCR), also called OCDR (Optical Coherence Domain Reflectometry), is a powerful continuous-wave (CW) white-light interferometric technique used primarily for testing fiber optical components and integrated optical waveguides [1–6]. An OLCR setup consists of a broadband source, e.g. a superluminescent diode (SLD), and a Mach-Zehnder-type interferometer. When unbalanced, the optical powers of signal and reference arm add up at the receiver and no coherent interference signal is detected. On the contrary, an interferometric signal is obtained if the optical path difference is smaller than the coherence length (which is inversely proportional to the optical bandwidth). During the measurement the optical length in the reference arm usually is changed so that different parts of the device under test enter - one after the other - the coherence range causing an interference signal. Therefore, the range of path change determines the measurement range (commonly of the order of centimeters and limited by a mechanically moved reference mirror) while the optical bandwidth and the coherence length of the source, respectively, determine the spatial resolution (which can be as low as a few micrometers). Within the coherence length the detection can possibly be quantum limited (down to about one signal photon per temporal resolution element).

An alternative CW reflectometric measurement technique, called optical frequency-domain reflectometry (OFDR), is based on a narrowband source which is swept in its optical frequency during the measurement (see e.g. [7]). It is the optical counterpart of frequency-modulated continuous-wave (FMCW) radar. Since the swept frequency range can be as large as 10 THz very high spatial resolutions can be obtained. Basically, however, FMCW-Lidar is not suited for velocity measurements because different locations lead to different detector frequencies as different target velocities do likewise via the Doppler effect. That problem can be solved by sequential up- and downshifts of the optical frequency for moving hard targets. That is not an option, however, for distributed moving targets such as aerosols in the atmosphere [8]. Very recently attempts have been made in that direction with a particular combination of CW and pulsed Lidar using frequency-stepped lasers realized by amplified and frequency shifting fiber optical recirculating delay lines [9, 10].

Furthermore, there are continuous wave (CW) Lidar systems systems – widely used for wind sensing – the spatial resolution of which is based on focusing the emitted light to a particular location [11–13]. They rely on the fact that the close-to-focus range primarily contributes to the received backscattered signal. A typical spatial resolution obtained in that way is Δz = 10 m at a measurement distance z = 100 m and it scales about quadratically with z.

On the other hand many reflectometric optical measurement methods in fiber optics and Lidar use pulsed sources. In such a case the pulse width determines the spatial resolution and it usually is of the order of meters.

In this paper an OLCR-Lidar system is proposed [14] and investigated which translates systems with micrometer resolution to systems with meter resolution. As an application we study remote wind speed measurements. Our approach cures the main disadvantages of CW-Lidars: poor and poorly controllable spatial resolution and vulnerability by strong out-of-focus scatterers. Moreover, it is of extraordinary importance that the location to be resolved can be shifted numerically after the measurement. That means that a single measurement already contains the information about all locations in sharp contrast to ordinary OLCR and ordinary CW Lidar. As compared to pulsed systems our measurement technique is not limited by short observation times and the related problem of frequency resolution versus spatial resolution. To the best of our knowledge similar work has not been done before.

2. Proposed OLCR Lidar system

A schematic view of the proposed setup is shown in Fig. 1. It consists of an interferometer with a single-mode fiber delay line in the reference path which serves as a local oscillator (LO). The signal path is partly fiber-optical comprising an acousto-optical frequency shifter (acousto-optic modulator, AOM; frequency shift f_AOM) and an optical circulator, both preferably in fiber pigtailed versions. The other part of the signal path is in free space with the incident laser light propagating into the atmosphere and the backscattered light from aerosols returning to the circulator. Signal and reference light are subsequently combined by means of a 3dB directional coupler with photodetectors at either output. The two electrical signals are subtracted from each other in order to get so-called balanced detection with the DC-signal nulled and the interference signal doubled in order to cancel the relative intensity noise (RIN) of the source. Therefore, RIN is not further considered in this paper. For the time being the center of the coherence zone in the atmosphere is determined by the choice of the delay in the reference path. The states of polarization of signal and reference light are supposed to be matched by proper – preferably fiber optical – polarization adjustment. The local oscillator power is assumed to be large enough so that the thermal detector noise is negligible as compared to the LO shot noise.

 

Fig. 1 Low coherence CW-Lidar setup

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A vital part of the measurement setup is a CW broadband source with Gaussian optical power density spectrum S_L and adjustable spectral width Δν_L of the order of 10 to 100 MHz:

To begin with let us assume that backscattered light solely comes from a small length element δz in the atmosphere located at a distance z with an optical power δP being returned into the transmitting fiber. Moreover, let us assume - for the time being - that the scattering particles are fixed. The corresponding delay time with respect to the reference arm is t_d = 2z/c − n_refl_ref/c, where l_ref and n_ref denote the length and effective refractive index of the delay fiber, respectively. In that case the currents of the photodetectors (a) and (b) are

 

Fig. 2 Division of measurement range into zones of length Δz_c

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In an actual measurement, the narrowband part of the electrical power density spectrum has to be identified. According to Eq. (8) it is governed by the scattered power from the matched zone so that its width Δz_c can be considered as the spatial resolution of our ”coherence gated” CW Lidar system. Hence, the spatial resolution is constant in a given measurement. Moreover, with our synthetic laser source presented in section 3 the coherence length Δz_c can be electronically varied at will in a wide range. In an ordinary CW Lidar, on the contrary, spatial discrimination relies on focusing the transmitted laser beam to a particular location (see e.g. [18–20] and the appendix of this paper). As a result high spatial resolution can hardly be achieved with ordinary CW Lidar, the resolution obtained decreases with distance and the resolution is only moderately well determined since the spatial sensitivity variation is approximately given by a Lorentzian (rather than a Gaussian) function.

If 〈ΔP(m_c)〉 denotes the expected scattered optical power in the transmitting fiber from zone #m_c the expected peak values of the narrowband and broadband spectral parts are

The powers P_coh and P_tot are random variables because they consist of contributions of many scatterers the electric fields of which are superimposed with random phases. Therefore they behave like the intensity I in a speckle pattern [21]. For a coherent superposition the standard deviation σ_I of the intensity is known to be equal to the mean intensity 〈I〉 (unity contrast). Since the electric power spectral density $S_{\text{narrow}}^{\left(p\right)}$ of the narrow spectral part is proportional to the optical power P_coh = ΔP(m_c = 0) it follows the same statistics:

Thus, different - but fixed - spatial arrangements of the scattering particles cause different optical powers, different detector currents and different electric power spectral densities as quantified in Eq. (12) – Eq. (15). Actually, when the particles move, they will get in and out of the illuminated volume, in particular by transverse wind components, and their relative positions will change with time. That phenomenon yields a temporal fluctuation of the detector current amplitude. As a consequence the narrowband part of the spectrum will be broadened from Δf_rec to some value Δf_sp (speckle induced spectral broadening) which is of the order or below 1 MHz in most cases being of interest here. We approximate the resulting spectral shape by a Gaussian function and replace Δf_rec by Δf_sp in Eq. (12). In the applications outlined below we will always choose Δf ≫ Δf_sp so that the speckle broadening is irrelevant for the broad spectral part S_broad(f). Hence, the expectation of the electric power density spectrum is

 

Fig. 3 Simulated normalized electrical power density spectrum for power ratios 〈P_tot〉 / 〈P_coh〉 = [1, 10, 100]

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Let us call the vicinity of the spectral peak the ”upper level” and the adjacent spectral range the ”lower level” and let us define the signal-to-noise ratio SNR according to

In the case where the speckle noise is negligible and the shot noise (SN) due to the local oscillator power P_LO prevails the SNR is given by

3. Synthetic broadband laser source with adjustable power density spectrum

The optical power density spectrum is an important attribute of continuous wave (CW) lasers in general and laser diodes in particular. In single longitudinal mode operation the finite spectral linewidth is caused by the presence of incoherent spontaneous emission inside the active medium. Depending on the laser under considerations and the operation conditions the linewidth may vary from the sub-kHz to the multi-MHz range. The shape of the spectrum is known to be Lorentzian. The electric field E(t) of the emitted light has an almost constant amplitude. Its phase Θ(t), however, fluctuates randomly due to the spontaneous transitions which continually add power to the oscillation field thereby causing the spectral broadening observed. The phase noise can be modeled by using the Fokker-Planck equation.

Some applications such as ordinary interferometry, conventional Doppler Lidar or coherent optical communications require narrowband sources, e.g. with linewidths below 10 kHz. As already summed up in the introductory remark of Sec. 1 broadband sources on the other hand with bandwidths in the THz range are used in white-light interferometry, optical low-coherence reflectometry (OLCR) and low-coherence tomography.

In Sec. 2 it has been shown that optical low-coherence reflectometry can also be the basis of CW Lidar systems with resolution in the range of meters. Such Lidar systems are particularly beneficial for remote wind velocity measurement (see Sec. 5). One of their key elements is a source with adjustable spectral width in the range of tens of MHz and a Gaussian rather than a Lorentzian spectrum since the respective temporal coherence function γ₁₁ falls off much more rapidly.

The optical source we propose and use afterwards consists of a laser diode and an electro-optical phase modulator as schematically shown in Fig. 4. The laser diode employed is supposed to oscillate in a single transverse and single longitudinal mode and it is considered to be sufficiently monochromatic for our purposes, say having a linewidth of 100 kHz or below. Such laser diodes are readily available, in particular in the 1500 nm range of optical communication. The output light is transmitted through an optical phase modulator which also is an off-the-shelf telecom component, using e.g. LiNbO₃ as electro-optic material. The modulator is assumed to be driven by a noise-like electrical signal. It is evident, hence, that the electric field of the optical output is given by

 

Fig. 4 Setup for source with adjustable spectral width

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Due to lack of respective answers we applied the following heuristic and iterative approach: We start with a random phase function ψ(f_opt) and calculate a first approximation to Θ(t) by applying Eq. (25)

The procedure outlined above has been carried out by a simple MATLAB code. Usually, we used a time discretization of δt = 1 ns and N = 2¹⁴ = 16384 data points. The optical bandwidths used were in the range of 10 MHz < Δν_L < 100 MHz corresponding to coherence lengths Δz_c = c · 2/(πΔν_L) between 2 m and 20 m. Examples of the power density spectra obtained with Δν_L = 10 MHz and 100 MHz are shown in Fig. 5(a) and 5(b) along with the target Gaussian spectra which - apart from weak spectral fluctuations - are very closely reached. Cutouts of the phase Θ(t) in a time interval of 1μs (i.e. 1000 data points) are shown at the bottom. In an actual application the complete set of discrete data Θ_n, n = 1 ... N, has to be stored one single time in a digital memory and used to drive the phase modulator. For repeated measurements it can be used again or other predetermined representations may be used. The mean number of phase jumps per time interval increases with increasing spectral bandwidth Δν_L. In the example of Fig. 5(b) there are about 400 phase jumps out of 16000 phase values, i.e. the jump rate is around 2.5 percent. If the phase modulator has a maximum frequency above 1 GHz (actually 40 GHz versions are readily available) the rate of phase jumps is not significant. Otherwise, the actual phase function will be slightly corrupted when compared to the numerical one.

 

Fig. 5 Normalized optical power density spectrum for Δν_L = 10 MHz (a) and Δν_L = 100 MHz (b), green curve: Gaussian target spectrum, blue curve: iteration result; bottom: cutout of phase Θ(t)

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As mentioned above the temporal degree of coherence γ₁₁(τ) of the output radiation field is determined by the inverse Fourier transform of the power density spectrum S_L(ν). A Gaussian spectrum according to Eq. (1) yields an absolute value of γ₁₁(τ) as given above in Eq. (3). In the remainder of this paper we will use a single numerical data set as an example – both for simulations (section 6) and for experimentally driving the phase modulator (section 7) - with Δν_L = 38.2 MHz and Δz_c = 5 m, respectively. The corresponding optical power density spectrum and the associated temporal degree of coherence are shown in Fig. 6(a) and 6(b). The latter is presented in a logarithmic scale 20 log₁₀ |γ₁₁| because the actual numerical curve (shown in blue) and the ideal gaussian one (shown in green) are indistinguishable on a linear scale. According to Fig. 6(b) the deviation of |γ₁₁(τ)| from the target function Eq. (3) is of the order of 10⁻³ outside the coherence peak.

 

Fig. 6 Result from iteration and Gaussian target function for Δν_L = 38.2 MHz: a) Normalized optical power density spectrum; b) temporal degree of coherence |γ₁₁|

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In a low-coherence reflectometer as well as in a low coherence Lidar the absolute value |γ₁₁| is of utmost importance because the measured signal should be a measure of the power originating from the coherence zone |τ| < τ_c, only. Actually, contributions in the measured electrical power density spectrum originating from reflections outside the coherence range are suppressed by |γ₁₁|². From Fig. 6(b) it is obvious that an excellent suppression is obtained when using the synthetic phase noise source presented above - much better than the Lorentzian-type spatial sensitivity obtained by beam focusing in ordinary CW-Lidars.

4. Post-measurement numerical range scanning

The signal to be detected stems from length elements close to the point where signal and reference path are of equal optical length (which we called above the matched zone). That location therefore is given by the length of the reference fiber and can be changed by changing the length of the reference fiber l_ref. In this section we will show that the matched zone can also be shifted afterwards by a suitable numerical evaluation of the measured data.

According to Sec. 2, Eq. (4) and Eq. (5) the detector signal and its spectrum caused by scattered light from a length element δz strongly depends on the phase difference ϕ(t;t_d) = Θ(t + t_d) − Θ(t). The matched zone is the vicinity of the location z = n_ref ·l_ref/2 where the delay time t_d = 0. In order to shift the matched zone to another location the analytical signal corresponding to the real detector signal - which is readily available by suppressing the spectral part at negative frequencies and consists of terms exp {j[ω_AOMt + Θ(t + t_d) − Θ(t)]}, cp. Eq. (4) - can be multiplied by a function

That procedure is made possible because the phase data Θ(t) are known (see Sec. 3) in sharp contrast to the situation in optical low-coherence reflectometry with natural laser sources. Thus, a single measurement can be repeatedly evaluated in order to get the response of different locations. That is by far more convenient and less time consuming than physically changing the setup and repeating the measurement and also means a tremendous advantage of our measurement scheme compared to ordinary CW-Lidars. Furthermore the numerical range-scanning feature of our system implies that the delay fiber in the reference arm can even be omitted.

5. Low-coherence Doppler wind Lidar system

Lidar systems are known to be useful for a large variety of applications [22]. One of them is the remote measurement of wind velocities which is based on the Doppler frequency shift caused by moving scattering particles. The Doppler effect directly translates the line-of-sight velocity v_LOS (to be determined) to the optical frequency shift

In this paper we propose a CW Doppler wind Lidar system where the spatial resolution primarily originates from the coherence properties of the source. The scheme has been presented already in Fig. 1 and discussed in Sec. 2. The only change to be made is to replace the fixed frequency shift f_AOM caused by the AOM by f_AOM + f_D. The detector signal is digitized and the electrical power spectral density is calculated by FFT. As shown in Sec. 2 this spectrum has a narrow peak which nearly exclusively stems from the matched zone (m_c = 0) of width Δz_c. Contributions of other locations (with other velocities, in general) are strongly suppressed according to a Gaussian function exp[−2(t_d/τ_c)²].

6. Simulation results

In Sec. 2 an OLCR Lidar method has been proposed and analyzed. Regardless and independent of the formulae derived there we present results of numerical simulations in this section which mimic the actual scattering and detection process. The simulation starts with an optical field with constant amplitude and synthetic noisy phase (cp. Sec. 3) so that the optical power density spectrum is a Gaussian with a predetermined spectral width. The time domain data are sampled with a time increment of 1 ns and 2¹⁴ = 16384 data points corresponding to 16μs are taken for a single measurement. The responses of individual length elements - many of them in each coherence zone #m_c with random optical phases which, moreover, change randomly from measurement to measurement - are superimposed yielding a speckle-like total signal. In each coherence zone #m_c the line-of-sight velocity and, hence, the Doppler frequency shift is assumed to be constant. The power weighting of different coherence zones is actually given by the transmitted beam (see appendix) and the scattering properties of the atmosphere. For demonstration reasons we choose truncated weighting profiles of the form

In the Fig. 7–10 the received power level is assumed to be large enough so that the speckle noise is dominant rather than the LO shot noise. In Fig. 7 the scattered light is assumed to originate from M_c = 11 coherence zones with uniform power weighting w̱ = [1 1 1 1 1 1 1 1 1 1 1] so that μ = 11. It shows the narrow spectral peak from the central zone residing on the broad socket originating from the remaining zones. The central part is magnified in the inset of Fig. 7. The peak to socket ratio is about 0.9/0.2 and hence close to the expected value of 4.46 calculated from Eq. (12) and Eq. (13). The SNR as defined by Eq. (17) is about 57, close to the expected value. The center of gravity of the narrow peak was determined to be 81.24 MHz which is very close to the true position of 81.27 MHz. The simulation results shown in Fig. 8 hold for the same conditions, except for the fact that the power weighting w(m_c = −1) of the coherence zone directly adjacent to the central zone is increased to 10. In accordance with Eq. (13) and Eq. (20) that results in an increased socket (and a slightly decreased SNR). There is a weak indication of a second peak at 77.5 MHz which corresponds to the adopted Doppler shift of −2.5 MHz of the zone m_c = −1. In Fig. 9 the weighting of the three zones m_c = −4 through m_c = −2 is increased to 10 so that μ = 38. No indication is noticeable of additional Doppler peaks. Thus, Fig. 8 and 9 prove the excellent OLCR suppression of Doppler peaks away from the central coherence zone so that strong scatterers like clouds are not particularly detrimental. In Fig. 10 the SNR for uniform weighting is given as a function of the number M_av of averages. It is found by repeating the averaging process several times and yields a result which slightly outnumbers the values of Sec. 2, Eq. (20) which predicts SNR = 57 after 100 averages. It has to be taken into account, however, that the SNR after averaging is a random variable, too, so that after a single averaging process - of e.g. 100 averages - the SNR can be smaller or larger than the expected value.

 

Fig. 7 Normalized power spectral density for uniformly weighted zones w(m_c) ≡ 1; inset: central peak

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Fig. 8 Normalized power spectral density for increased scattered power from outside the matched zone: w̱ = [1 1 1 1 10 1 1 1 1 1 1]

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Fig. 9 Normalized power spectral density for increased scattered power from outside the matched zone: w̱ = [1 1 10 10 10 1 1 1 1 1 1]

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Fig. 10 SNR for uniform weighting w(m_c) ≡ 1 as function of averages M_av

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In Fig. 11 the power weighting in the eleven zones is assumed to be uniform again. However, the expected received scattered power from the matched zone is assumed to be 〈P_coh〉 = 250 fW only (i.e. 〈p(z)〉 = 50 fW/m) so that the shot noise of the local oscillator is dominant. Nevertheless, the narrow peak is clearly visible. The expected SNR after 100 averages (found by repeating the averaging process 100 times) was determined to be 16. That is in splendid accordance with Eq. (21) which yields SNR_SN = 17 in this case (assuming a responsivity of ℛ = 1A/W in Eq. (21) as well as in the simulation). Moreover, the mean value of the center frequency is found to be 81.30 MHz with a standard deviation of 0.15 MHz which is very close to the frequency of 81.27 MHz assigned to the matched zone even in this noisy case. In Fig. 12 the expected received power 〈P_coh〉 is increased to 1 pW which is about the value of the critical power between shot noise and speckle noise dominance.

 

Fig. 11 Normalized power spectral density for uniformly weighted zones w(m_c) ≡ 1 with 〈P_coh〉 = 250 fW

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Fig. 12 Normalized power spectral density for uniformly weighted zones w(m_c) ≡ 1 with 〈P_coh〉 = 1 pW

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In Fig. 13(a) and 13(b) the signal data of Fig. 7 are re-evaluated by applying the procedure presented in Sec. 4 in order reveal the scattering from other locations. In Fig. 13(a) the matched zone is shifted by +4Δz_c = +20 m, in Fig. 13(b) by −3Δz_c = −15 m, i.e. to the coherence zones m_c = 4 and m_c = −3, respectively. The narrow Doppler peaks at 90 MHz and 72.5 MHz are clearly visible.

 

Fig. 13 Simulation of post-hoc shifting the matched zone; a) shift by +4Δz_c = +20 m; b) shift by −3Δz_c = −15 m

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7. Experimental results

Experimental work with the low-coherence Lidar system presented above has been done in the laboratory so far. The actual setup closely follows the schematic view of Fig. 1 with the source being a combination of a laser diode and a phase modulator according to Fig. 4. For the time being some of the components used are laboratory instruments. For future field measurements they can easily be replaced by OEM modules so that our approach will share the properties of small footprint, light weight, low cost and simple data processing with other CW Lidars. The results shown below focus on the crucial specific features of our approach rather than on those - such as heterodyne and quantum limited detection - which it has in common with established Lidar systems:

7.1. Synthetic broadband laser source

The synthetic laser source is realized as described in Sec. 3. It consists of a narrowband laser diode with a wavelength λ = 1550 nm and a phase modulator. The latter is driven by data as obtained from the iterative procedure also described in Sec. 3. In all the measurements discussed below a single data set with 2¹⁴ values is used which generates a Gaussian optical power density spectrum with a width of Δν_L = 38.2 MHz corresponding to a coherence length of Δν_L = 5 m. The same data set has also been used in the simulations of Sec. 6.

To experimentally determine the bandwidth of our artificially broadened source we make use of the delayed-self-heterodyne-method [15, 16], which is widely used for the measurement of narrow optical bandwidths. Its setup consists of a strongly imbalanced fiber optical Mach-Zehnder-interferometer, where the length difference of the arms of the interferometer has to exceed the coherence length of the optical source under test, in other words the delay time t_d has to exceed the coherence time τ_c. The expected coherence length in our case is 5 m. To meet the above requirement we use a length difference of 500 m of optical fiber, resembling 725 m of air or a time delay of t_d = 2.4μs. At the interferometer output two incoherent signals interfere and lead to the so called ”incoherent interference” with an electrical power spectral density (PSD) the width of which is expected to be $\sqrt{2}$ times broader than the optical power spectrum of the source (for Gaussian PSD); see Sec. 2.

The experimental result, normalized by the total electrical power P_el, is shown in Fig. 14. As expected it has a rugged appearance. However, a Gaussian fit which is also shown in Fig. 14 yields a 1/e-width of Δf_el = 55.4 MHz, corresponding to an optical bandwidth of $\mathrm{\Delta }{\nu }_{\text{L}}=55.4\text{MHz}/\sqrt{2}=39.2\text{MHz}$ which is very close to the expected value of Δν_L = 38.2 MHz. The smooth expected Gaussian spectrum would be obtained by averaging many spectra either with different delays t_d or by using different numerical data sets Θ_n for the phase modulation which are readily available.

 

Fig. 14 Normalized measured electrical power spectral density of the synthetic broad-band source with delayed-self-heterodyne-method, Δf_el = 55.4 MHz, $\mathrm{\Delta }{\nu }_{\text{L}}=\mathrm{\Delta }{f}_{\text{el}}/\sqrt{2}=39.2\text{MHz}$, N = 16384 data points, δf = 1/(N * δt), M_av = 1, 16384 data points

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7.2. Experimental demonstration of low coherence Lidar principle

To test the low coherence Lidar in the lab, the setup is set to a measurement distance of z = 10 m, i.e. the laser beam is focused to that distance and the matched zone is positioned there by choosing an appropriate length of the fiber optical reference arm. The measurement resolution is set to Δz = 5 m through the choice of the laser bandwidth of Δν_L = 38.2 MHz. These test conditions remain the same for all measurements.

Four measurements using a retroreflecting film as at target were carried out and the respective results are shown in Fig. 15 and Fig. 16. The target is placed once in the center of zone m_c = 0 at z = 10 m and once in the center of zone m_c = −1 at z = 5 m. At each position we are taking measurements once on a static target and once on a moving target. The moving target is simulated by fixing the retroreflecting film to a rotating cylinder, the static target is the same, but with the rotation stopped. From the narrow peaks in Fig. 15(a) and 16(a) we can determine the speed of movement from the Doppler shift Eq. (30). From Fig. 15(a) we find for the static target in zone m_c = 0 a narrow peak at f = f_AOM = 80 MHz corresponding to a Doppler shift of f_D = 0 MHz as expected with no movement. In Fig. 16(a) the PSD of the moving target is shown and clearly a shift of the narrow peak is visible, which can be determined to be f_D = −2.54 MHz, denoting a movement speed of v_LOS = −2.0 m/s with λ = 1550 nm, i.e. away from the observer.

 

Fig. 15 Lidar Measuremtent with static target; (a) Normalized power density spectrum with target at z = 10 m, inside matched zone m_c = 0, inset: closeup of central peak of; (b) Normalized power density spectrum with target at z = 5 m inside zone m_c = −1

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Fig. 16 Lidar Measuremtent with moving target; (a) Normalized power density spectrum with target at z = 10 m, inside matched zone m_c = 0, inset: closeup of central peak of; (b) Normalized power density spectrum with target at z = 5 m inside zone m_c = −1

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Placing the target at a distance z = 5 m, inside zone m_c = −1 leads to the spectra shown in Fig. 15(b) and Fig. 16(b) for the static target case and the moving target case, respectively. Here we can see the broad and low amplitude spectra with a bandwidth of about 39 MHz, identifying these measurements clearly as signals from outside the matched zone. Thus, these examples demonstrate the range resolving and speed measurement abilities of our low coherence Lidar.

7.3. Post measurement shift of matched zone

As already detailed in Sec. 4 the exact knowledge of the phase Θ(t) provides the possibility of moving the matched zone numerically after the measurement to any desired location. To demonstrate this feature we set up an experiment with two targets in the beam path, one moving target at a distance of z = 10 m centered at zone #m_c = 0 and one static target at a distance of z = 5 m centered at zone #m_c = −1. The static target is formed by laterally moving a reflecting film inside the beam, thus reflecting a fraction of about 25% of the light. The rest is passed on to the moving target. Fig. 17(a) shows the originally measured PSD with a peak at 77 MHz, revealing a target at a speed of v_LOS = −2.3 m/s inside the resolution range. A reevaluation after multiplication of the measurement data with

 

Fig. 17 (a) Evaluation of data for moving target at z = 10 m ; (b) Reevaluation of data: numerical shift of matched zone to location of static target at z = 5 m

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8. Summary and conclusions

A continuous-wave Lidar system has been presented, primarily meant for remote sensing of wind velocities. It is based on principles of optical low-coherence reflectometry (OLCR), using, however, a laser source with artificially broadened optical power density spectrum, e.g. with Gaussian shape. On that basis it is featuring a number of specific characteristics, in particular:

All of these three characteristics are in sharp contrast to ordinary CW wind sensing systems. When compared to pulsed Lidar systems it stands out by its simple setup and the absence of the problem of velocity versus spatial resolution.

Our Lidar concept may also be useful for other applications such as range-resolved depolarization diagnosis and range-resolved determination of gas concentrations by means of differential absorption Lidar (DIAL), provided scattering particles are available.

Appendix: Transmitted laser beam and received backscattered power

The problem of transmitting light into the atmosphere and receiving backscattered light in a monostatic Lidar system has been addressed previously (see e.g. [18–20]). It is crucial in the case of an ordinary CW-Lidar because the spatial resolution in that case is based on focusing the laser beam. It also plays an important role in a low-coherence Lidar as proposed in Sec. 2 because it determines - together with the scattering properties of the atmosphere - the signal level and the level of the broadband background of the measured electrical power density spectrum. In this appendix we revisit the problem in more detail in fiber optical terms and we derive formulae for the launching efficiency and the received power per length element, respectively, in closed form.

As depicted in Fig. 1 the interferometer-to-atmosphere terminal consists of a single-mode fiber endface and a lens with focal length f. It is shown again in Fig. 18 along with the introduction of some parameters used in the analysis to follow. The task of the terminal is to form a waist of the transmitted beam at a desired distance z₁ and to collect the backscattered light originating from particles in the atmosphere. Since the near-field distribution of a standard single-mode fiber can be well approximated by a Gaussian function [23]

(33)$$E_{\text{f}}\left(x_{\text{f}},y_{\text{f}}\right)={\widehat{E}}_{\text{f}}\cdot \text{exp}\left(-\frac{x_{\text{f}}^2+x_{\text{f}}^2}{w_0^2}\right)$$

with w₀ denoting the spot radius, the well-known theory of Gaussian beams [24] can be applied in the direction of forward propagation. The location z₁ of the right hand side waist is related to the location z₀ = f + δz₀ of the fiber endface by

(34)$$z_1=f+\frac{f^2\left(z_0-f\right)}{{\left(z_0-f\right)}^2+{\left(\frac{\pi w_0^2}{\lambda }\right)}^2}$$

and the spot radius of the remote waist is given by

(35)$$\frac{1}{w_1^2}=\frac{1}{w_0^2}{\left(1-\frac{z_0}{f}\right)}^2+\frac{1}{f^2}{\left(\frac{\pi w_0}{\lambda }\right)}^2$$

It is worth mentioning that - according to Eq. (34) - no remote waist exists beyond z_1max = f + f²/(2z_R0) where z_R0 denotes the fiber’s Rayleigh distance $z_{\text{R}0}=\pi w_0^2/\lambda$, i.e. for instance beyond z_1max = 600 m in the case f = 0.25 m, w₀ = 5 μm, λ = 1500 nm. For z₁ < z_1max/2 the remote spot radius can be very well approximated by w₁ = w₀ · z₁/ f (there is a second solution with large w₁ of the order of λf/(πw₀); that solution is not useful in general because it leads to very low launching efficiencies of the backscattered light). The spot radius variation with z′ (distance from the remote waist) is given by

(36)$$w\left(z^{\prime }\right)=w_1\sqrt{1+{\left(\frac{z^{\prime }}{z_{R1}}\right)}^2}$$

with $z_{R1}=\pi w_1^2/\lambda$ denoting the remote Rayleigh range.

 

Fig. 18 Gaussian beam of the Lidar setup

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In order to analyze the reception of scattered light let us consider first a single point-like scatterer at the location (x,y,z) which isotropically scatters a power of $\delta P_{\text{sc}}^{\left(1\right)}$. The power launched back into the transmitting fiber can be specified as

(37)$$\delta P^{\left(1\right)}=\eta \cdot \frac{A_{\text{lens}}}{4\pi z^2}\cdot \delta P_{\text{sc}}^{\left(1\right)}$$

where z = z₁ + z′. It is the product of the power incident onto the lens area A_lens and a launching efficieny η(x,y,z). The latter can be determined from the overlap integral of the field distribution $E_{\text{sc}}^{\left(1\right)}$ formed in the plane of the fiber endface and the fiber modal field E_f to be excited:

(38)$$\eta =\frac{{\left|\iint E_{\text{sc}}^{\left(1\right)}\left(x_{\text{f}},y_{\text{f}}\right)\cdot E_{\text{f}}^{*}\left(x_{\text{f}},y_{\text{f}}\right)\text{d}{x}_{\text{f}}\text{d}{y}_{\text{f}}\right|}^2}{\iint {\left|E_{\text{sc}}^{\left(1\right)}\left(x_{\text{f}},y_{\text{f}}\right)\right|}^2\text{d}{x}_{\text{f}}\text{d}{y}_{\text{f}}\cdot \iint {\left|E_{\text{f}}\left(x_{\text{f}},y_{\text{f}}\right)\right|}^2\text{d}{x}_{\text{f}}\text{d}{y}_{\text{f}}}$$

The incident scattered wave is a spherical wave, truncated and focused by the lens. Therefore, when assuming that z ≫ f scalar wave optics states that E_sc is given by an Airy function J₁(ζ)/ζ, $\zeta =2\pi r_{\text{lens}}\sqrt{x_{\text{f}}^2+y_{\text{f}}^2}/\left(\lambda f\right)$. That field can be well approximated by a Gaussian field

(39)$$E_{\text{sc}}^{\left(1\right)}\approx {\widehat{E}}_{\text{sc}}^{\left(1\right)}\cdot \text{exp}\left(-\frac{x_{\text{f}}^2+y_{\text{f}}^2}{w_0^2}\right)$$

provided the lens radius is chosen appropriately:

(40)$$r_{\text{lens}}=1.29\frac{\lambda f}{\pi w_0}$$

That condition [20] just means that the lens should neither be too small nor to large but chosen in a way that it is well illuminated by the transmitted beam with a relative intensity at the lens boundary of exp(−2 ·(1.29)²) ≈ 0.04. That’s what we will assume in the following and it leads to an analytical expression of η. The calculation is fairly straightforward but long-winded because the following factors which are highlighted in Fig. 19 have to be taken into account:

• if the point-like scatterer is off-axis its image will be offset by δx_f, δy_f with respect to the fiber axis
• in this case, moreover, the wavefronts of $E_{\text{sc}}^{\left(1\right)}$ incident on the fiber endface are tilted, i.e. the phase is not constant in the fiber plane
• depending on the longitudinal location z of the scatterer under consideration the image is at different locations f +δz_f (lens law)
• since in general the location f +δz_f of the image does not coincide with the location of the fiber endface f + δz₀ the incident wavefronts in the plane of the fiber endface will have a curvature

 

Fig. 19 Factors considered for calculation of η(x,y,z)

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All this will reduce the launching efficiency η calculated from Eq. (38). With some minor approximations the final result is

(41)$$\eta \left(x,y,z\right)={\eta }_{\text{long}}\left(z\right)\cdot \text{exp}\left[-\frac{{\left(\delta x_{\text{f}}\right)}^2+{\left(\delta y_{\text{f}}\right)}^2}{w_{\eta }^2}\right]$$

where

$$\begin{array}{l}{\eta }_{\text{long}}\left(z\right)=4/\left\{{\left[\frac{{\tilde{w}}_0\left(z\right)}{w_0}+\frac{w_0}{{\tilde{w}}_0\left(z\right)}\right]}^2+{\left[\frac{k{w}_0{\tilde{w}}_0\left(z\right)}{2R_{\text{curv}}}\right]}^2\right\}\\ {\tilde{w}}_0^2\left(z\right)=w_0^2\left[1+{\left(\frac{\delta z_f-\delta z_0}{z_{R0}}\right)}^2\right]\\ R_{\text{curv}}=\left(\delta z_f-\delta z_0\right)\left[1+\frac{z_{R0}^2}{{\left(\delta z_f-\delta z_0\right)}^2}\right]\\ \frac{1}{w_{\eta }^2}=\frac{2}{{\tilde{w}}_0^2}+\text{Re}\left\{\left[\frac{k^2}{2}{\left(\frac{1}{f}-\frac{1}{R_{\text{curv}}}+j\frac{2}{k{\tilde{w}}_0^2}\right)}^2\right]/\left(\frac{1}{w_0^2}+\frac{1}{{\tilde{w}}_0^2}+j\frac{k}{2R_{\text{curv}}}\right)\right\}\\ k=2\pi /\lambda ;\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\delta x_{\text{f}}\approx x\cdot \frac{f}{z};\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\delta y_{\text{f}}\approx x\cdot \frac{f}{z}\end{array}$$

It is interesting to note that η(x = 0, y = 0, z′ = 0) = η_long(z′ = 0) ≈ 1.

Next, we analyze the realistic case with many scatterers being illuminated by the transmitted Gaussian beam. Since the phases of the scattered waves are random the scattered powers of the individual scatterers as well as the backscattered powers launched in the fiber add up on average. The total scattered power of a layer of thickness δz therefore can be expressed as

(42)$$\delta P_{\text{sc}}\left(z\right)=\delta z\cdot \iint I\left(x,y,z\right)\rho \left(x,y,z\right)\text{d}x\text{d}y$$

where ρ(x,y,z) is the product of scattering cross section σ_sc(x, y) and the number of scatterers per volume element, n_sc(x, y,z), while

(43)$$I\left(x,y,z\right)=\frac{2}{\pi }\frac{P_0\text{exp}\left[-a\left(z\right)\right]}{w^2\left(z\right)}\text{exp}\left[-2\frac{x^2+y^2}{w^2\left(z\right)}\right]$$

denotes the intensity of the transmitted Gaussian beam. For simplicity, the scattering density ρ(x,y,z) can be approximated by ρ̄(z) where the bar means averaging over x and y so that δP_sc(z) = ρ̄ (z) · P₀ · exp(−a(z)) · δz with P₀ being the transmitted laser power and $a\left(z\right)={\int }_0^z\alpha \left(\zeta \right)\text{d}\zeta$ the attenuation up to the location z. The scattering density ρ̄(z) is equal to the attenuation constant α_sc caused by particle scattering multiplied by the value of the normalized scattering diagram in backward direction which is unity in the isotropic case. The corresponding power launched into the transmitting fiber - named δP in sec. 2, Eq. (4) ff - can be expressed as

(44)$$\delta P=\delta z\cdot \iint I\left(x,y,z\right)\overline{\rho }\left(z\right)\eta \left(x,y,z\right)\text{d}x\text{d}y\cdot \frac{A_{\text{lens}}}{4\pi z^2}\cdot \text{exp}\left[-a\left(z\right)\right]$$

Defining the laterally averaged launching efficiency η̄(z) by

(45)$$\delta P\left(z\right)=\overline{\eta }\left(z\right)\cdot \frac{A_{\text{lens}}}{4\pi z^2}\delta P_{\text{sc}}\left(z\right)\cdot \text{exp}\left[-a\left(z\right)\right]$$

yields

(46)$$\begin{array}{ll}\overline{\eta }\left(z\right)\hfill & =\frac{\iint \text{exp}\left[-2\frac{x^2+y^2}{w^2\left(z\right)}\right]\eta \left(x,y,z\right)\text{d}x\text{d}y}{\iint \text{exp}\left[-2\frac{x^2+y^2}{w^2\left(z\right)}\right]\text{d}x\text{d}y}\hfill \\ \hfill & ={\eta }_{\text{long}}\left(z\right)/\left[1+\frac{1}{2}{\left(\frac{f}{z}\right)}^2\frac{w^2}{w_{\eta }^2}\right]\hfill \end{array}$$

Thus, the received power per length element δz depends on z according to

(47)$$p\left(z\right)=\frac{\delta P\left(z\right)}{\delta z}=\overline{\eta }\left(z\right)\frac{A_{\text{lens}}}{4\pi z^2}\overline{\rho }\left(z\right)\cdot P_0\text{exp}\left[-2a\left(z\right)\right]$$

The integral of p(z) yields the total received scattered power. An example is shown in Fig. 20 for a homogeneously scattering atmosphere and negligible attenuation (a(z) ≈ 0) along with the related launching efficiency η̄(z) for a measurement distance (i.e the distance to the focus) of z₁ = 100 m. From Fig. 20 it is evident that the close-to-focus range contributes most to the received power. The ratio of the integral of p(z) and its maximum value can be taken as a measure of the spatial resolution obtained by focusing

(48)$$\mathrm{\Delta }{z}_{\text{focus}}=\frac{\int p\left(z\right)\text{d}z}{p_{\text{max}}}$$

In the example of Fig. 20 we have Δz_focus = 35 m. Increasing the distance to the focus to z₁ = 200 m while keeping the other parameters constant raises this value to Δz_focus = 128 m. These values are a measure of spatial resolution in conventional CW Lidars without using low coherence resolution. A further severe deterioration - specified by ρ̄(z) - will occur in conventional CW Lidars if out-of-focus parts of the atmosphere show enhanced scattering caused e.g. by a cloud.

 

Fig. 20 Received power per length element p(z) and launching efficiency η̄(z)

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On the contrary, a CW-Lidar with coherence resolution as presented in Sec. 2 can offer a better and constant spatial resolution (e.g. Δz_c = 5 m; cp. Sec. 2). Moreover, it is only marginally affected by out-of-focus scatterers.

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