Design of a monolithic Michelson interferometer
for fringe imaging in a near-field, UV,
direct-detection Doppler wind lidar
JONAS HERBST* AND PATRICK VRANCKEN
Institut für Physik der Atmosphäre (IPA), German Aerospace Center (DLR), Oberpfaffenhofen, Münchener Str. 20, 82234 Weßling, Germany
*Corresponding author: jonas.herbst@dlr.de
Received 18 April 2016; revised 19 July 2016; accepted 29 July 2016; posted 1 August 2016 (Doc. ID 263498); published 25 August 2016
The low-biased, fast, airborne, short-range, and range-resolved determination of atmospheric wind speeds plays a
key role in wake vortex and turbulence mitigation strategies and would improve flight safety, comfort, and
economy. In this work, a concept for an airborne, UV, direct-detection Doppler wind lidar receiver is presented.
A monolithic, tilted, field-widened, fringe-imaging Michelson interferometer (FWFIMI) combines the advantages
of low angular sensitivity, high thermo-mechanical stability, indep endence of the specific atmospheric conditions,
and potential for fast data evaluation. Desig n and integration of the FWFIMI into a lidar receiver concept are
described. Simulations help to evaluate the receiver design and prospect sufficient performance under different
atmospheric conditions.
© 2016 Optical Society of America
OCIS codes: (280.3640) Lidar; (120.3180) Interferometry; (280.3340) Laser Doppler velocimetry; (280.7060) Turbulence.
http://dx.doi.org/10.1364/AO.55.006910
1. INTRODUCTION
Wake vortices, gusts, and turbulence in clear air impose a major
risk in commercial air transport because onboard weather radars
cannot detect turbulence in clear air [1]. Recent radars provide
some turbulence detection functionality in the presence of
clouds and precipitation [2]. If encountered by another aircraft,
turbulence can cause unexpected rolling moments or abrupt
changes of altitude, which may result in damage to the plane
or injuries to the passengers [3,4].
In particular, wake vortices are a well-studied phenomenon
since their discovery in the early 20th centur y. Two rotating,
long-life vortices are produced at the wing tips of any aircraft.
Today’s traditional risk reducer is a standard minimum distance
of travel between any two planes, which is chosen according to
the aircrafts’ weights, such that the vortices have safely decayed
or subsided before the encounter. One way to meet the ongoing
trend of increasing passenger numbers is to increase the timing
frequency between any two planes without a reduction of
safety. Plate lines are a concept to increase the decay rate of
wake vortices that works only in ground proximity [5].
Further possibilities to reach this goal include a direct reaction
to the forces of the wake vortex on the aircraft by new flight
controller routines, examined by Looye et al. [6], or the remote
sensing of the disturbances caused by wake vortices and
turbulences.
Remote sensing is the only means for measuring wind
vectors in the near field (50–300 m), ahead of the aircraft.
This would allow for a fast reaction of the flight controller
(autopilot) [7,8]. Ehlers et al. deem a full-scan update rate
of the wind field measurement of 5–10 Hz appropriate for
their concept of wake impact alleviation control to work rea-
sonably well at a range of 60 m [8,9]. The control conce pt
includes a wake identification algorithm, which allows one
to reconstruct the wake vortex disturbance, and alleviates
its impact by specific control commands to compensate for
the determined disturbance. After a computation time of
200 ms for the first identification of the wake vortex, the con-
trol system continuously (typical sampling time in the order of
20 ms [9]) countervails the disturbances on the basis of the
determined wake vortex model. Note that the disturbance
reconstruction step allows the anticipation of the disturbances
at locations where no measurement was made (or not yet),
leading to rather complex relationships between the sensor
measurements (location, orientation, and quality) and the dis-
turbance rejection capability [8]. Actuator time delays, which
are assumed to be 100 ms, are compensated by predicting the
wake vortex impact on the aircraft for a moment 100 ms in
the future [9]. This control concept can be applied in very
similar ways for the mitigation of wake vortices, gusts, and
turbulences [10].
However, currently, no reliable sensing system for onboard
measurements of wind speeds exists, which could guarantee
that the safety standards of the International Civil Aviation
Organization are met in such a future scenario.
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Research Artic le
1559-128X/16/256910-20 Journal © 2016 Optical Society of America
Amongst the remote sensing devices, lidar has the advantage
of speed and high local precision. Doppler wind lidars (DWL)
measure frequency changes caused by the Doppler effect of
molecules and aerosols moving with the ambient wind in
order to derive wind speed components along the line-of-sight
(LoS) of the laser beam. A distinction can be made between
coherent and incoherent (direct) DWLs.
Coherent DWLs often use IR laser light, which provides
sensitivity to backscatter from micron-sized aerosols [11,12].
Furthermore, IR laser technology is reliable and often asserted
eye-safe depending on the laser parameters, e.g., at low pulse
energies and low exposure durations [13]. The Doppler-shifted
signal scattered mainly from aerosols is superimposed coher-
ently with a frequency-shifted reference (local oscillator). The
Doppler shift is determined from the beat frequency by a fast
Fourier transform from the power density spectrum. Systems
with CO
2
-lasers, Tm:LuAG-lasers, and with Er-doped fiber
lasers have successfully been applied to measure wind speed s
at the ground level and in the boundary layer [14–16].
Turbulence detection in the troposphere at an altitude of
12 km with a cohere nt DWL has been demonstrated up to
9 km ahead of an aircraft with a range bin of 150 m [17]. At
these high altitudes, the concentration of aerosols is low and
the coherence of the received signal is reduced, which decreases
the signal-to-noise ratio (SNR) and the maximum range of
detection. As a consequence of the Fourier limit and the re-
quired coherence (small spectral bandwidth) properties of the
laser, the minimal pulse length and the spatial resolution are
limited [11], e.g., 170 ns and 30 m and a minimum detection
range of 150 m for the Halo Photonics 1.5-μm Streamline
pulsed coherent Doppler lidar [18].
Coherent DWLs have not yet been demonstrated to mea-
sure wind speeds reliably at high cruise flight altitudes (12 km)
or in clear air and with short range bins (15–30 m) in the near
field (50–300 m in front of an aircraft). Accurate range-resolved
LoS wind speed measurements with standard deviations of
approx. 1ms
−1
with comparable lidar geometry parameters
would be required for reliable feed-forward control, as pointed
out in a lidar parameter study by Ehlers et al. [8].
Although coherent DWLs might still be an option, only
direct-detection DWLs are considered from now on, because
they can rely on pure molecular scattering, pure scattering from
aerosols or a combination , and because the range gate length
can be made smaller than 30 m. Direct-detection DWLs may
consist of filters, which transmit only a certain spectral band-
width. From the amount of light that is transmitted through
the filters, the Doppler frequency shift can be determined.
Iodine-vapor DWLs using abs orption bands of iodine as
filters are limited to a laser wavelength of 532 nm. In airborn
lidar, UV wavelengths (λ) are preferred, because UV systems
may be designed eye-safe beyond a certain, acceptable distance
(see, e.g., [19]) and because of the high efficiency of Rayleigh
scattering in the UV, which is roughly proportional to λ
−4
(see Section 2.A).
The double-edge technique (DE) is based on two Fabry–
Perot interferometers with different optical path lengths
that determine the frequency of maximum transmission.
The Doppler shift is determined by the ratio of transmission
through these filters [20]. The transmission through the filters
strongly depends on the shape of the light scattering spectra,
which is why this method requires the knowledge of the
altitude (backscatter ratio, temperature, pressure). In practice,
the required separation of the Rayleigh and Mie channels, e.g.,
used in ALADIN [21], can only be circumvented by the use
of multiple filters or the equivalent fringe-imaging technique
(FI). These FI techniques have the main advantage that mea-
surements can be performed without knowledge of the shape
of the backscattered signal spectrum.
The principle of fringe imaging relies on the imaging of the
interference pattern of an interferometer on a position-sensitive
detector. The frequency shift between an unshifted reference
and a Doppler-shifted signal can be determined from the dis-
location of the interference pattern. A distinction can be made
between multi-wave and two-wave interference. The most
common multi-wave interferometer is the Fabry–Perot inter-
ferometer (FPI). When properly illuminated with divergent
light, the produced interference fringes are rings. As the light
is Doppler shifted, the radii of the rings change. This principle
was applied in the AWIATOR (“Aircraft Wing with Advanced
Technology Operation”) project [22,23]. However, the evalu-
ation of the interference patterns is delicate, time consuming,
and prone to errors due to possible dislocations of the ring
centers on a two-dimensional CCD array [24]. Furthermore,
typical two-dimensional detectors, such as CCD, are too slow
for range-resolved detection.
Other interferometers are designed with an inclination of
one of their mirrors to produce a pattern of linear interference
fringes. The shift of the linear fringe can be determined with
fast, linear detectors.
A multi-wave type is the Fizeau interferometer. The com-
plex fringe shape of a deformed Airy can be accounted for
by the use of proper system parameters [25 ]. The fringe shape
depends strongly on the field angle of the incident light, but
the deformation and contrast loss impede us from being able
to use it with extended sources, as in the presently described
application (see Section 3.A).
For the purpose of collecting range-resolved backscattered
light in close range (50–300 m) in front of the aircraft with
an equivalent region of total overlap, a telescope with a large
field of view (FOV) (about 4 mrad) is required. This large
FOV produces, independent of the setup design (free beam
or fiber coupled), an important angular distribution after col-
limation. The interferometer needs to be field widened in order
to accept a broad range of incident angles without a loss of
contrast.
The necessary field widening can be realized with two-wave
interferometers. Liu and Kobayashi proposed to use a Mach–
Zehnder interferometer in a direct-detection DWL, using a
two-channel differential discrimination method (DMZ),
similar to the DE technique using the FPI [26]. Bruneau
considered a four-channel-based version (QMZ) [27] and an
equivalent field-widened fringe-imaging Mach–Zehnder inter-
ferometer with inclined mirrors (FIMZ) [28], both optimized
for Rayleigh scattering. Bruneau and Pelon showed that the
concept can be used to measure wind speeds [29]. An applica-
tion of this principle is Ball Aerospace and Technologies Corp.’s
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Vol. 55, No. 25 / Se ptember 1 2016 / Applied Optics 6911
Optical Autocovariance Wind Lidar [30], which was proposed
to measure wind speeds from the international space station
[31]. Recently, a modified DMZ using three wavelengths was
proposed for an Fe Doppler lidar for wind measurements from
ground to thermosphere [32].
Multiple filters can be created with a Michelson interferom-
eter as well. Cezard et al. considered a dual fringe-imaging
Michelson interferometer (FIMI) with inclined mirrors for
the measurement of the wind speed and other air parameters
(temperature, scattering ratio, density) [33].
The monolithic Michelson interferometer design we present
in this work is based on the same fringe-imaging principle.
In contrast to the cited test setup, our field-widened, fringe-
imaging Michelson interferometer (FWFIMI) design provides,
however, the thermo-mechanical stability and design features
necessary for fast, range-resolved, and airborne measurements
of wind speeds in the near field in front of an aircraft. The
advantages of a monolithic FIMI are detailed more closely
below.
There is a long tradition of the design and application of
monolithic, “field-widened Michelson interferometers” (FWMI)
or “wide-angle Michelson interferometers” (WAMI). The basic
idea of FWMI was developed in 1941 [34]. FWMIs can be built
temperature compensated as solid model, where the two inter-
ferometer arms consist of different glasses at a certain length
ratio, or as air-spaced models, where the air arm mirror is fixed
by spacers of the same or different materials as the glass arm
and may be positioned with a piezo feedback control. They have
been realized with cube [35] or hexagonal beam splitters [36]
and as a Doppler asymmetric spatial heterodyne version [37],
just to name a few.
For the presently considered application of measuring
LoS wind speeds that are range resolved in the near field, we
consider only two-wave interferometers to be adequate. In par-
ticular we consider a FIMI for the following reasons: first, a
FIMI with slanted mirrors produces linear fringes, which
can be imaged on fast, linear detectors for range-resolved
detection. Secondly, airborne interferometers call for stability
with respect to vibrations and temperature. An FIMI is more
easily built in a monolithic way than an FIMZ. Third, a mono-
lithic FIMI can be constructed to be both field widened
(FWFIMI) and temperature compensated. Finally, our design
of a monolithic FWFIMI can be arranged to be tilted to the
incident light, enabling a two-channel operation, in which case
the FIMI can reach the theoretical performance of the FIMZ
(see Section 2.B).
The proposed measurement concept is detailed more closely
in the following, considering certain assumptions: we consider
different lidar transmitters (lasers) used in WALES/DELICAT
(“WAter vapor Lidar Experiment in Space” project/
“Demonstration of LIdar based Clear Air Turbulence” project)
[38,39], AWIATOR [23], and MULTIPLY (ESA project). We
assume a moderately sized, airborne-compatible telescope of
about 15 cm diameter in a monostatic configura tion. Two
receiver concepts are presented: free beam and fiber coupled.
In a free-beam arrangement, the image of the fringes on the
detector is range dependent and changes with the misalignment
of the laser beam with respect to the telescope (see Section 3.A).
Even with a field-widened interferometer, a fiber-coupled
design may be preferred in the context of bias reduction [40]
in comparison to a free-beam setup (see Section 3.A). In a fiber-
coupled setup, the backscattered collimated light is coupled
into a large-core multimode fiber. The scrambling properties
of the fiber produce a constant far field of the out-coupl ed light
and a constant illumination of the interferometer, independent
of the position and the angular orientation of the light focused
on the multimod e fiber core during coupling (see Section 4).
The fiber-coupled concept is enhanced by the application of a
two-lens optical scrambler to increase the far-field scrambling
gain [41] and by mechanical vibrations for speckle reduction.
Because of the finite extension of the fiber core, the recolli-
mated light is expected to come by a centered range- and
misalignment-independent angular distribution. The FWFIMI
has a net inclination angle fixed such that one linear fringe
is imaged on a linear detector, which allows range-resolved
measurements in the near field in front of the aircraft.
To begin with, we summarize in Section 2 the fundamentals
of light scattering and of the FIMI as a direct-detection DWL,
and we compare the theoretical performances of different
direct-detection DWLs for the measurement of wind speeds.
In Section 3, the lidar geometry requirements and all aspects
of our design of a temperature-compensated FWFIMI are de-
scribed in detail. In Section 4, the integration of the FWFIMI
in a lidar receiver prototype for the measurement of wind
speeds in the near field is detailed. In Section 5, we evaluate
the proposed receiver concept in terms of expected perfor-
mance, using simulations, while taking into acco unt the detec-
tor and speckle noise during the data evaluation.
2. FUNDAMENTALS
A. Atmospheric Backscattering Spectrum and
Single-Scattering Lidar Equation
The atmospheric backscattering spectrum has contributions
from light scattering by molecules (“Rayleigh–Brillouin”
scattering, rotational and vibrational Raman scattering) [42]
and from light scattering by aerosols/hydrometeors. The
Rayleigh–Mie-Laser spectrum (RMLS ) is introduced here to
model the spectral contributions from Rayleigh–Brillouin
scattering by molecules, from scattering by spherical particles
[43] and from the lineshape of the laser as Gaussian-shaped
lines [24].
The quasi-elastic molecular (“Rayleigh–Brillouin”) scatter-
ing spectrum (the so-called Cabannes line composed of the
Landau–Placzek line and the Brillouin doublet) is the result
of coherent scattering, which dominates molecular scattering,
and therefore, the scattered light is mostly polarized [44]. For
a DWL only, this central part is considered. The shape of the
Cabannes line depends on the density of the scatterers.
Accordingly, different regimes (hydrodynamic: e.g., gas-liquid
mixtures [45], kinetic: atmosphere, and Knudsen: thin gases)
can be discriminated. If the mean free path between the ther-
mally moving molecules is large, a Gaussian lineshape
(Knudsen) can be assumed. As the pressure increases and the
temperature decreases, density fluctuations moving at acoustic
speeds deform the lineshape (kinetic regime), until at the
6912 Vol. 55, No. 25 / September 1 2016 / Applied Optics
Research Article
hydrodynamic limit, two acoustic side bands (Brillouin lines)
appear.
The scattering properties of aerosols in the atmosphere, such
as the lidar ratio and particle depolarization rato, are highly
dependent on their type and shape, and there is large variability
[46]. Here, the simplifying assumption of spherical particles is
made (the Mie theory is valid only for spherical particles [43]).
This allows us to describe the backscattering from aerosols as
purely elastic and without depolarization.
In our simulations in Section 5, we consider an approxima-
tion of the (kinetic) S6 model [47]. It describes the kinetic re-
gime by the sum of three Gaussian functions (henceforth called
the G3 model) [48]. Here, for a simple analytical description of
the theoretical performance of the FIMI, the Knudsen model
(neglecting the Brillouin doublet) is presented in the following
to produce the RMLS. The neglected Brillouin contribution
does not affect the spectrum’s central frequency and has
therefore no effect on the performance of the wind speed mea-
surements [33]. Furthermore, there is no effect on the contrast
factor G in the vicinity of the optimal FSR determined in
Section 5.B.
The RMLS consists of the weighted sum of the Gaussian
molecular scattering peak and the Gaussian aerosol scattering
peak, convolved with the Gaussian laser lineshape,
I
RMLS
ν
1
R
b
1
ffiffiffiffiffi
2π
p
σ
G
exp
−
ν− ν
c
ffiffiffi
2
p
σ
G
2
1−
1
R
b
1
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2πσ
2
L
σ
2
w
p
exp
−
ν− ν
c
ffiffiffi
2
p
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
σ
2
L
σ
2
w
p
2
:
(1)
Here, R
b
is the particle scattering ratio given by R
b
1 β
Mie
∕β
Ray
with the Mie and Rayleigh backscattering
coefficients [m
−1
sr
−1
] β
Mie
and β
Ray
. Values of β
Ray
for differ-
ent altitudes H can be obtained by β
Ray
pH∕
R
air
T Hm
air
× 550∕λ
L
nm
4
× 5.45 × 10
−32
after Collis
and Russell [49], where pH and T H are altitude-depen-
dent pressures and absolute temperatures obtained from an
atmospheric model. Here, R
air
287.058 J∕kgK is the gas
constant of air, and m
air
4.811 × 10
−26
kg is the mass of an
air molecule. A more complete model provided by Bucholtz
[50] includes the dispersion of the refractive index of air,
the anisotropy of air molecules, and the dispersion of the
depolarization factor of air. The differential scattering cross
section and backscattering coefficients calculated with the
simplified model stated above are 4.5% smaller than those cal-
culated by Bucholtz. This approximation has, however, a neg-
ligible influence on the absolute values of the signal-to-noise
ratios calculated in Section 5 (deviation: ≈ 2.3%). The values
of β
Mie
are scaled from values determined by Vaughan at
10.6 μm[51], using β
Mie
are scaled from values determined by
Vaughan at 10.6 μm[51], using β
Mie
β
Mie
10.6 μm×
10.6∕λ
L
μm×−0.104×lnβ
Mie
10.6 μm− 0.62. The total
backscattering coefficient is β β
Ray
β
Mie
. The Doppler
shift [Hz] is defined as Δν ν
c
− ν
L
2ν
L
u
r
∕c, where ν
L
is the frequency of the laser, ν
c
is the Doppler-shifted central
frequency, and u
r
is the LoS wind speed. σ
L
Δν
L
∕
ffiffiffiffiffiffiffiffiffiffiffiffi
8ln2
p
is the standard deviation [Hz] of the Gaussian laser line
shape, where Δν
L
is the laser linewidth (FWHM) [Hz].
σ
G
σ
2
Ray
σ
2
L
1∕2
is the standard deviation in Hz of the
Rayleigh–Laser spectrum, whereby σ
Ray
2∕λ
L
k
B
TN
A
∕
m
air
1∕2
is the standard deviation in Hz of the Rayleigh spec-
trum, where k
B
is the Boltzmann constant, N
A
is the Avogadro
constant, and T is the air temperature in the scattering volume.
λ
L
is the wavelength of the laser. σ
w
ffiffiffiffiffiffiffiffi
4∕3
p
∕ν
L
u
r:m:s
is the
broadening due to the r.m.s. wind speed u
r:m:s
at flight level.
Typical, conservative values of u
r:m:s
and σ
w
at H
30–40; 000 ft are 1.7 m/s [52] and 5.5 MHz for moderate
turbulence, which is about 3% as broad as the WALES trans-
mitter lineshape.
The amount of backscattered light received by the lidar is
calculated with the single-scattering lidar equation [53]ina
monochromatic approximation. The amount of time-resolved
EM wave power detected, imagined here as number of photons
per range gate n
p
(no assumptions on the nature of light) [54]is
given by
n
p
ν
L
;RE
L
ΔR
hν
L
A
R
2
ξrη
R
η
T
β exp
−2
Z
R
0
αdr
: (2)
Here, R is the distance [m] of the light scattering volume in
front of the telescope. ΔR is the length [m] of the range gate.
E
L
is the transmitted energy of one laser pulse with pulse
duration τ
p
. h 6.62610
−34
Js is Planck’s constant. A is
the receiver telescope area [m
2
]. ξr is the range-dependent
overlap function. η
R
and η
T
are the receiver and transmitter
loss factors. α α
Ray
s
α
Ray
a
α
Mie
is the overall atmos-
pheric extinction coefficient [1/m], where α
Ray
s
8π∕3β
Ray
is the molecular extinction [1/m], α
Ray
a
is the molecular absorp-
tion [1/m], and α
Mie
is the extinction and absorption by
aerosols. For monodispersed spherical particles, the aerosol
extinction is α
Mie
k
0
β
Mie
[55]. Here, a constant extinction-
to-backscatter ratio of k
0
50 sr is assumed.
B. Theoretical Performance of a Fringe-Imaging
Michelson Interferometer
In this section, the princ iple of direct-detection DWLs
based on the FIMI is summarized, and the FIMI’s optimized
theoretical performance is compared with other direct-
detection DWL methods.
The monochromatic transmission function (TF) of a
Michelson interferometer with inclined mirrors is cosine shaped
and varies in space along the x-axis. It can be written as
Ix;y;νFI
0
1 V cosϕ: (3)
Here, the linear interference fringes are aligned perpendicular to
the x-axis and parallel to the y-axis. V is the instrumental inter-
ference contrast; its contributions are described in Section 3.E
and F. ϕ 2πν∕cOPD
0
− 2θx is the fringe phase. OPD
0
is the fixed optical path length difference between the arms.
Assuming dispersion-free media in the interferometer arms,
OPD
0
is equal to c∕FSR, where FSR is the free spectral range.
The FSR is the width of one fringe period in [Hz]. θ is the angle
of inclination in the x-direction rotated along the y-direction.
θ creates a linear variation of OPD
0
within the illuminated area
of width d
w
. It determines the amount of periods N
p
of the TF.
To image exactly N
p
fringe periods, θ equals N
p
λ
L
∕2d
w
.The
prefactor F 0.5 accounts for the reflection losses of a
Research Article
Vol. 55, No. 25 / Se ptember 1 2016 / Applied Optics 6913
Michelson interferometer. The instrument function is the con-
volution of the laser lineshape with the TF. The received instru-
ment function (I
F
) is the convolution of the RMLS [Eq. (1)]
with the TF [Eq. (3)]:
I
F
x;y;νFI
0
1 W T;α cosϕ Δϕ: (4)
The resulting interference pattern is shifted in phase by
Δϕ 4π∕FSRλ
L
u
r
and has a reduced global fringe contrast
W T;αV × GFSR,where
GFSRexp
−2
πΔν
L
FSR
2
×
1
R
b
exp−2πσ
Ray
∕FSR
2
1 −
1
R
b
: (5)
The LoS wind speed u
r
is determined by measuring the phase
shift Δϕ between a reference instrument function and a Doppler
frequency-shifted received instrument function, which are
both imaged sequentially on a position-sensitive detector.
In the above description, the FIMI is a multichannel spectral
analyzer. A general way to find the optimal FSR setting of the
FIMI for the measurement of wind speeds is to introduce a
penalty factor κ
VLOS
, comparing the interferometer with an
ideal spectral analyzer (ISA). An ISA performs the perfect spec-
tral analysis because it is composed of an infinite number of
sampling channels, which sample the spectrum with Dirac-type
transmission functions. In an ISA, there is no loss of informa-
tion, energy, or spectral content. An interferometer (like the
FIMI) mixes the photons spatially and spectrally and therefore
underperforms compared to the ISA.
Bruneau and Cezard et al. derived expressions for the
optimal fixed optical path difference of the fringe-imaging
Mach–Zehnder interferometer (OPD
FIMZ
≈ 3cmat 250 K)
[28] and the fringe-imaging Michelson interferometer
(OPD
FIMI
2.8 cm at R
b
1, T 273 K)[33], respectively.
The penalty factor κ
VLOS
(by Cezard et al. for pure Rayleigh
scattering) compares the Cramer–Rao bounds (CRBs) of
the FIMI and the ISA. The CRBs are the respective lowest-
achievable standard deviations of an unbiased estimator.
Cezard et al. used a maximum-likelihood estimator approach
(which asymptotically reaches the CRB) for inversion and
obtained the CRBs of the wind speed as diagonal elements of
the inverse Fischer matrices of the FIMI and the ISA. The
underlying assumptions are that the signal is shot-noise limited,
obeys a Poisson statistic, and that the different channels are
statistically independent. κ
VLOS
can be written as a function
of the FSR,
κ
VLOS
ε
FIMI
ε
ISA
d
c
FSR
ffiffiffi
2
p
c
1 −
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 − V
2
exp
−8
c
d
c
FSR
2
s
−1∕2
: (6)
Here, ε
FIMI
and ε
ISA
are the CRBs for the FIMI and the ISA.
d
c
2c∕πσ
Ray
ffiffiffiffi
T
p
is the coherence length of the Rayleigh
signal. If the FSR is too large, the fringe phase sensitivity
S 4π∕FSRλ
L
in rad/(m/s) with respect to the Doppler
shift is small. If the FSR is too small, the fringe constrast is
too small for an efficient determination of the fringe phase.
The FSR is optimized for the “worst” condition, where no aero-
sols contribute to backscattering (R
b
1). Here, the contrast
factor G is equal to 66%. In Fig. 1, κ
VLOS
is plotted as a func-
tion of the FSR at 273 K for R
b
1. The plot includes contrast
factors GFSR for R
b
1 (green) and R
b
2 (magenta) and
the phase sensitivity SFSR (black).
Cezard et al. showed that when R
b
increases, the global con-
trast increases, thus producing lower penalty factors, and that a
decrease of the temperature by 40 K decreases κ
VLOS
by 10%
[33]. The best measurement performance is thus expected at
low temperatures and high scattering ratios. The optimal
FSR value of 10.7 GHz ≈ 6.8 × σ
Ray
(at R
b
1, T 273 K)
is found at the minimum: κ
VLOS
4.4 (dotted line).
Table 1 lists the penalty factors of other DWL techniques
obtained in similar ways for comparison.
The two filter-based techniques, the double-edge Fabry–
Perot (DFP) and the DMZ, have good theoretical performance,
but they are sensitive to the Rayleigh–Mie backscattering scat-
tering ratio and require inversion of the lidar signal to correct
this [27]. The fringe-imaging Fabry–Perot technique (FIFP) is
complicated by the evaluation of circular fringe patterns or
the complexi ties of circle-to-line converters [59]. The fringe-
imaging Fizeau interferometer (FIFI) provides linear fringes.
However, the fringe shape is very sensitive to the incident an-
gular distribution (see Section 3.A) and does require collimated
light, which complicates range-dependent measurements in the
near field (50–300 m). This is not the case with field-widened
Michelson and Mach–Zehnder interferometers. The QMZ and
FIMZ techniques have very low penalty factors and do not re-
quire knowledge of the scattering ratio. It can be seen that the
penalty factor for the FIMI is about two times the penalty
factor for the FIMZ. This is because half of the light is back
reflected [factor F, Eq. (3)].
A possible option to decrease the FIMI penalty factor by a
factor of two is to tilt the FIMI at a small angle with respect to
Table 1. Penalty Factors for Wind Speed Measurement
Technique κ
VLOS
Technique κ
VLOS
DFP 2.4 [56] DMZ 1.65 [27]
FIFP 3.1 [57], 2–4[58] QMZ 2.3 [27]
FIMI 4.4 [33] FIMZ 2.3 [28]
Fig. 1. Penalty factor of wind speed measurement κ
VLOS
(blue),
contrast factor G(FSR) (green: R
b
1, magenta: R
b
2), and phase
sensitivity S (black) as a function of FSR for 273 K, R
b
1 at a wave-
length of 355 nm.
6914 Vol. 55, No. 25 / September 1 2016 / Applied Optics
Research Article