Multiwavelength Frequency Modulated CW Ladar: The Effect of Refractive Index

Abstract

Frequency modulated continuous wave (FMCW) laser detection and ranging is a technique for absolute distance measurements with high performances in terms of resolution, non-ambiguity range, accuracy and fast detection. It is based on a simple experimental setup, thus resulting in cost restraint with potential wide spread, not only limited to research institutions. The technique has been widely studied and improved both in terms of experimental setup by absolute reference or active stabilization and in terms of data analysis. Very recently a multi-wavelength approach has been exploited, demonstrating high precision and non ambiguity range. The variability of refractive index along the path was not taken into account with consequent degradation of range accuracy. In this work we developed a simple model able to take into account refractive index effect in multi-wavelength FMCW measurement. We performed a numerical simulation in different atmospheric conditions of temperature, pressure, humidity and CO2 concentration showing a net improvement of range accuracy when refractive index modeling is used.

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Article
Multiwavelength Frequency Modulated CW Ladar:
The Effect of Refractive Index
Mariano Barbieri 1, Deborah Katia Pallotti 1,2 , Mario Siciliani de Cumis 1,2 and
Luigi Santamaria Amato 1,3,*
1ASI, Italian Space Agency, Space Geodesy Center “G. Colombo”, 75100 Matera, Italy;
mariano.barbieri@est.asi.it (M.B.); deborah.pallotti@asi.it (D.K.P.); mario.sicilianidecumis@asi.it (M.S.d.C.)
2CNR-INO, Istituto Nazionale di Ottica, Largo E. Fermi 6, 50125 Firenze, Italy
3CNR-INO, Istituto Nazionale di Ottica, Via Campi Flegrei 34, 80078 Pozzuoli, Italy
*Correspondence: luigi.santamaria@asi.it
Received: 5 August 2020; Accepted: 4 October 2020; Published: 8 October 2020


Abstract:
Frequency modulated continuous wave (FMCW) laser detection and ranging is a technique
for absolute distance measurements with high performances in terms of resolution, non-ambiguity
range, accuracy and fast detection. It is based on a simple experimental setup, thus resulting in
cost restraint with potential wide spread, not only limited to research institutions. The technique
has been widely studied and improved both in terms of experimental setup by absolute reference
or active stabilization and in terms of data analysis. Very recently a multi-wavelength approach
has been exploited, demonstrating high precision and non ambiguity range. The variability of
refractive index along the path was not taken into account with consequent degradation of range
accuracy. In this work we developed a simple model able to take into account refractive index
effect in multi-wavelength FMCW measurement. We performed a numerical simulation in different
atmospheric conditions of temperature, pressure, humidity and CO
2
concentration showing a net
improvement of range accuracy when refractive index modeling is used.
Keywords: laser ranging; length metrology; frequency modulated cw
1. Introduction
Precision measurement of non-cooperative target by laser ranging (LR) techniques had and
continues to have applications in several research fields
:
from satellites flying in defined formation
for large synthetic aperture telescope [
1
–
4
] used for fundamental physics test or extraterrestrial
planets search to earth based laser ranging. The earth based satellite laser ranging was successfully
demonstrated in 1964 and, since then, produced a huge amount of results in fundamental physics
experiments like: general relativity test (gravitomagnetic effect [
5
], Lense-Thirring [
6
,
7
] non
-
Newtonian
gravity [
8
]) or dark matter search [
9
,
10
]. However, it mostly finds application in fields like geodesy
and geodynamics, allowing , for example, the length-of-day determination or earth gravitational field
measurement [
11
] and contributing to the definition and the update of the terrestrial reference frame
(TRF). In addition, there are perspectives in space debris applications and multipurpose communication
experiments. The precise distance determination of near object has also several applications mostly
in industrial fields [
12
], particularly in automotive [
13
,
14
]. The main characteristics of laser ranging
measurements are: resolution of the detected length, non-ambiguity range and accuracy. A LR
system is based on a cw or pulsed laser source, collimation optics, a detector and a timing system.
A system based on pulsed source has large non-ambiguity range but low resolution whereas a cw
system shows a very small non-ambiguity range (of the order of the wavelength) but extraordinary
resolution. The accuracy, instead, does not depend on the particular laser source used. First of all, it is
Photonics 2020,7, 90; doi:10.3390/photonics7040090 www.mdpi.com/journal/photonics

Photonics 2020,7, 90 2 of 10
related to experimental setup calibration (e.g., linearity of frequency sweep, electrical cable length
calibration or accuracy of timing system). Moreover, different environmental factors may influence the
accuracy: one of the main contributor is the refractive index. A great advance was the introduction
of multi-wavelength interferometry [
15
,
16
] that combines measurements at different wavelengths,
generating a longer ‘synthetic wavelength’ and, therefore, a larger ambiguity range while maintaining
same resolution. Unfortunately, extending the ambiguity range beyond a millimetre is very hard.
In 2009 Coddington et al. [
17
] used dual coherent frequency combs approach for ranging experiments,
obtaining at same time large non-ambiguity range, exceptional resolution and fast refresh rate. On the
downside, the setup used is very complex and expensive and so available only in state-of-the-art
laboratories. For this reason a very popular method that keeps costs low but good performance is
the frequency modulated continuous wave ranging (FMCW) [
18
–
20
]. In this technique a cw laser is
chirped at a constant rate K(Hz/s) and split in two parts: reference and probe beam. The probe beam
is collimated and sent to object whose distance must be measured, and the reflection is mixed with
reference beam on a photodetector. The electric signal is a beat note at frequency
f=
2
Lk/c
where
L
is the optical path, and
c
the speed of light. The performance of FMCW depends on the coherence
length of the used laser (related to spectral full width half maximum) and on total laser frequency
excursion [
21
] that, in turn, depends on chirp rate
K
. The accuracy is related to linearity of chirp
rate and to the equation used to model refractive index. The non-linearity of the chirp rate can be
compensated in an active way using a real time detection system and actuators [
22
,
23
]. Another way is
a post-processing correction by using a fine calibration up to 15 ppb accuracy level by using a frequency
comb [24–26] or ppm level with a simple and cheaper reference against a molecular spectrum [27].
Recently multi-wavelength approach has been applied to FMCW [
28
,
29
] demonstrating high
precision and non ambiguity range. In these experimental implementations the variability of refractive
index is not considered. Consequently, even if the range uncertainty is low, the overall accuracy of
range measurement is large since it does not consider the refractive index that masks the absolute
range value introducing a systematic uncertainty. In this paper we develop a model to take into
account the air refractive index in a FMCW multi-wavelength approach. The multiple observables
introduced in multi-wavelength approach allow to measure not only the range but also the average
atmospheric parameters (like pressure, temperature, humidity and CO
2
concentration) along the
path. Such parameters allow to calculate a sort of effective refractive index along the path enabling a
more accurate estimation of real absolute distance. In other words, in single wavelength FMCW the
atmospheric parameters are measured at laser station and used to calculate the refractive index.
In this way the refractive index along the path is approximated with the one at laser station.
The model developed in this paper allow to measure the effective refractive index along the path in
multiwavelength FMCW. Consequently, the refractive index along the path is approximated with the
effective refractive index (and not with index at laser station) with consequent improved accuracy.
Finally a comparison between two possible atmospheric conditions and a comparison between possible
multiple wavelengths has been performed.
2. Model
A general equation for object range (
R
) detection via beat frequency measurement of a chirped
signal (linearly-chirped with constant K) is









R1+n(x,λj)=c
2K·fj,j=1, . . . , m
x= ( t[◦C],p[Pa],h[%],xc[ppm])
λj[µm] = j-th laser wavelength, fj=j-th beat frequency observed
(1)
(2)
(3)
j
means that we are applying the same equation for
R
but using different values of laser wavelength
λj
hence measuring a different possibly specific beat frequency fj.

Photonics 2020,7, 90 3 of 10
Following the formula proposed by Ciddor [
30
] for the excess component in air
n(x
,
λ)
compared
to the refractive index of the vacuum, the former depends on
λj
(from 0.3 to 1.69
µ
m in [
30
], formula
validity extended in mid infrared [
31
]),
t
the temperature in Celsius (
−
40 to
+
100
◦C
),
p
the pressure
in Pascal (80 to 120 kPa),
h
the fractional humidity in the
[
0, 1
]
interval,
xc
the CO
2
concentration (from
0 to 2000 ppm). Please see https://emtoolbox.nist.gov/Wavelength/Documentation.asp for detailed
explanation of the Ciddor formula used.
3. First order Taylor Approximation
Using first order Taylor approximation around a known refractive index value
n(x(0)
,
λj)
,
and using the notation of sum over repeated indexes:





















Rcj0+cji ·yi=c
2K·fj
cji =
∂n(x,λj)
∂xix(0),i=1, . . . , l
yi=xi−x(0)
i,xii-th element of x
cj0=1+n(x(0),λj)
(4)
(5)
(6)
(7)
In case of the Ciddor formula
l=
4, as we can see in Equation (2). Please notice that if we know
cij
, i.e.
x(0)
and the derivatives of
n
, and if we have
l+
1
=m
, we can invert the equations isolating
known and unknown quantities:



























P=R,R·y1, . . . , R·yl
C=


c10 c11 . . . c1l
.
.
..
.
.....
.
.
cm0cm1. . . cml



F=c
2·Kf1, . . . , fm
C·PT=FT=⇒PT=C−1·FT,C−1≡(a)ij
(8)
(9)
(10)
(11)
In the last equation we have inverted the linear system generated by the Taylor expansion of
n
(
C−1
is in fact the inverse of matrix
C
). In Equation (11) we have redefined the index convention for ease
of notation in the following treatment. The elements of
C−1
are redefined as
aij
, where now
j
sums
up with the beat frequencies in
F
while
i
identifies the components of the parameters vector
P
where
the range Rand the atmospheric parameters xare stored. Such convention will be kept from now on
unless otherwise stated.
From the above we can find Rand xvalues in terms of only known parameters:
Pi=aij ·Fj=⇒(P1=R=a1j·Fj=a1j·c
2·Kfj,i=1
xi−1=aij ·Fj/R+x(0)
i−1=aij ·c
2·Kfj/R+x(0)
i−1,i=2, . . . , m(12)
So the variance for Rand xiwill be (covariances excluded):









σ2
R=c
22
·h(a1j/K)2σ2
fj+ (a1jfj/K2)2σ2
Ki
σ2
xi=c
22
·"(aij /(K·R))2σ2
fj+ (aij fj/(K2·R))2σ2
K+aij fj
K·R22
σ2
R#
(13)
(14)

Photonics 2020,7, 90 4 of 10
4. Effect of Refractive Index Variability
In Equation (1) we assumed the refractive index has a constant value along the laser trajectory,
depending only on the constant values of the atmospheric parameters
x
. In this section we start
relaxing this assumption and provide an estimation of the error due to the refractive index variability.
We can think of the refractive index variation along the light path as an accumulation of
d fj
in
Equation (1) along the way. Namely







d fj=2K
c·1+n(x,λj)dr
fj(R) = ZRd fj
(15)
(16)
the longer the path, the bigger will be the beat frequency observed. To test the performance of
the proposed approach, we evaluate the integral (16) using simplified models for the variation of
atmospheric parameters
x
between the target and the laboratory where the measure is performed.
In the next section we consider the case of
x
fluctuating around an average value; this simplifies the
analytical treatment and the exact solution, allowing for direct estimation over the entire validity range
of Ciddor formula. Then we generalize to the case of a linear gradient of
x
between lab and target,
solving numerically integral (16) also for very extreme differences between lab and target in order to
test the performance of the proposed approach in limit cases of input variability. We also sketch up the
analytical treatment in the general case when
x
has an unknown three-dimensional spatial distribution
to be chosen.
4.1. Refractive Index Fluctuation around an Average Value
In general the atmospheric variables
x
have a three-dimensional distribution in space. For instance,
increasing altitude of the target means the light trajectory passes through a gradient of temperature;
same usually happens with pressure and the other parameters
x
. In this section we will assume that
the variability of
n
is dependent on fluctuations of the
x
spatial distribution around an average value
with a certain distribution
g
. This allows an easier analytical treatment of (16) producing a simple
formula to test the performance of the proposed approach that can be evaluated over the entire
x
validity range of Ciddor formula. Such an assumption can be mathematically stated as follows:
ZRn(r)dr →ZRdr Zg(n)ndn ≡R·E[n](17)
Hence from (16)
fj(R) = 2K
cZR1+n(x(r),λj)dr
=2K
cR+ZRn(x(r),λj)dr
=2K
cR+ZRdr Zgj(n)ndn
=2K
cR1+E[nj]
(18)
(19)
(20)
(21)
This last equation gives an estimation of the beat frequency when
R
is known. We can substitute
it in Equation (12)-dropping the first index for simplicity of notation so that now a1jis aj
R∗=ajfj·c
2K=aj·R1+E[nj]
σrel
R∗=R∗−R
R=∑
jaj+aj·E[nj]−1
(22)
(23)

Photonics 2020,7, 90 5 of 10
σrel
R∗
can be seen as an estimate of the relative error due to the usage of Equation (12) for range
estimation
R∗
. Notice that even when
E[nj]
goes to zero, such error does not necessary go to zero
as well. E[nj]could be estimated as n(E[x],λj); the ajare estimated from the derivatives ∂xinapplied
at x(0)=E[x].
Similar results can be found for the xquantities substituting Equation (21) into Equation (12):
x∗
i−1=aij ·c
2·Kfj/R+x(0)
i−1=aij ·1+E[nj]+E[xi−1]
x(0)
i−1≡E[xi−1]
σrel
x∗=x∗
i−1−E[xi−1]
E[xi−1]=1
E[xi−1]∑
jaij +ai j ·E[nj]
(24)
(25)
(26)
Now we evaluate
σrel
R∗
varying the
xi
values across the range reported in Equation (2) and by
calculating E[nj]. We see that σrel
R∗is never higher than 10−11:
max
x(σrel
R∗)<1.5 ·10−12 (27)
In addition to the above uncertainties, the National Institute of Standards and Technology (NIST)
estimates the error using the empirical Ciddor formula, that is unrelated to the uncertainties of the
input atmospheric parameters. This takes the form of an adjustment to the formula accounting for
various effects (please see https://emtoolbox.nist.gov/Wavelength/Documentation.asp#AppendixAV
for detailed explanation and formula). By direct evaluation we estimate the error to be
∼
10
−8
, while
assuming higher values only at the very extreme values of the formula’s validity.
4.2. General Case
In the general case, Equation (17) is not valid anymore and we need to consider the spatial
distribution of n:ZRn(r)dr →ZTrajectory≡L
n(x(l))dl (28)
and we can rewrite the equations above this way



































fj(R) = 2K
cZL1+n(x(l),λj)dl
=2K
cR1+ZL
n(x(l),λj)
Rdl
=2K
cR1+Nj(L,R)
σrel
R∗=∑
jaj+aj·Nj(L,R)−1
where Nj(L,R)≡ZL
n(x(l),λj)
Rdl
(29)
(30)
(31)
(32)
(33)
4.3. Linear Variation of Atmospheric Parameters
In order to have a numeric estimation of the error
σrel
R∗
in case of atmospheric parameters
x
that
change values between the observer (the laboratory where the laser is positioned) and the target,
we have chosen a specific value of the range
R
, a specific linear variation of
x
with light trajectory
between the lab and the target, and performed numerically the integral in Equation (16). This provides
an estimation of the real observed beat frequency fj.

Photonics 2020,7, 90 6 of 10
Then we used Equation (12) to measure the range as per the multi-frequency method above
exposed, and compared this prediction
R∗
with the known value
R
. To this aim
,
we assigned to
x(0)
the xvalues as measured in the lab.
Similarly we used
fj
also to estimate the range in the single-frequency classical case
R∗∗
,
using Equation (1), and compared it with the known
R
. For this case the refractive index used
is calculated from the xin the lab conditions as well.
The results of the relative error are reported in Table 1. In the table we also report the example
wavelengths of a Nd:YAG laser and its harmonics generated in nonlinear crystals. They produce beat
frequencies of
fbeat =
13.5483, 13.5484, 13.5485, 13.5487, 13.5489 GHz for
R=
10,101 m, with a chirp
rate
K
= 201 MHz/
µ
s, the same order of magnitude as in [
27
]. Similar relative errors are found for
R=
100,301 m. Table 1shows two sets of range measurements obtained from two linear variations
of atmospheric parameters between lab and target. The two sets are chosen in order to have a
large excursion in the lab-vs-target atmospheric value (case 1) and a more mild condition (case 2).
The atmospheric parameters are shown in Table 2.
For
λ=
1.0645
µ
m and more extreme variation (case 1), the refractive index between the lab and
the target varies from 1.00020745 to 1.000342853.
Ps=
101,325 Pa is the standard atmospheric pressure.
For completeness, we calculated the expected error using multiple wavelength generated by
electro-optic modulator (see Appendix B). In this case the vicinity of the generated wavelengths
limits the performance of the multi-wavelength approach. Furthermore also the error due to the
numerical inversion of matrix
C
needs to be addressed. In fact matrix
C
is close to a singular matrix,
due to the terms with the derivatives of
n
being very small compared with the terms with
n
. Hence the
system is solved numerically, with a certain error on the solution. As described in Appendix A, this
error turns out to be small compared with the errors above, and can be neglected.
Table 1.
Relative errors obtained from one (
(R∗∗ −R)/R
) and multi-wavelength (
(R∗−R)/R
)
approach in two different atmospheric conditions (Case 1 and Case 2 as reported in Table 2). The first
column of the table reports the number of wavelengths used by the new method, while the second
column reports their values.
Case 1 Case 2
#fbeat λ(µm) R(m) (R∗−R)/R(R∗∗ −R)/R(R∗−R)/R(R∗ ∗ −R)/R
1 1.0645 10,101 6.4 ×10−51.8 ×10−5
2 1.0645, 0.53225 10,101 −2.5 ×10−5−1.9 ×10−6
3 1.0645, 0.53225, 0.35483 10,101 −5.0 ×10−14 −3.1 ×10−13
4 1.0645, 0.53225, 0.35483, 0.266125 10,101 −5.6 ×10−14 6.6 ×10−14
5 1.0645, 0.53225, 0.35483, 0.266125, 0.2129 10,101 −1.0 ×10−14 −3.2 ×10−14
Table 2. Linear variation of atmospheric parameters x.
Case 1 Case 2
Laboratory Target Laboratory Target
Temperature (◦C ) 30 −20 20 10
Pressure (kPa) Ps−20 Ps+10 Ps−5Ps+5
Humidity (fraction) 1 0 0.8 0.5
CO2(ppm) 350 550 350 550
5. Conclusions
In the last two decades, laser ranging techniques have been improved, due to fast development
of frequency metrology tools [
32
–
34
] and performing laser sources (continuous wave [
35
–
39
] and
pulsed [
40
,
41
]). Such enhancements affect particularly performances and reliability for end users.

Photonics 2020,7, 90 7 of 10
Multi-wavelength approach is widely used in interferometry and ranging fields with pulsed laser and
recently has been applied to FMCW. In this context is fundamental a careful study of the refractive index
variation due to atmospheric parameters (i.e., pressure, temperature, humidity and CO
2
concentration).
In our approach we focused on the accuracy of length estimation and we performed a numerical
simulation in order to compare with the single wavelength approach. The result is an improvement
on the accuracy limited only by accuracy of Ciddor formula. In addition, a careful modeling of the
atmospheric influence on a continuous wave laser beam (amplitude or frequency modulated or even
not modulated) enables new applications in the communication, meteorological and metrological fields.
Author Contributions:
Conceptualization, L.S.A.; simulation and model development, M.B.; validation and
discussion of the results, M.S.d.C. and D.K.P. All authors have read and agreed to the published version of
the manuscript.
Funding:
This research has been supported by: the Italian Space Agency through projects MOST and WhiTech
and fruitful interaction with projects “OT4CLIMA” (D.D. 2261 del 6.9.2018, PON R&I 2014-2020 e FSC) from
Ministero della Pubblica Istruzione and project QOMBS from EU commission.
Conflicts of Interest: The authors declare no conflict of interest.
Appendix A. Errors Due to Numerical Evaluation
The system (11) is more precisely solved to find
P
by means of numerical methods than by
calculating the inverse
C−1
, due to the high degree of singularity of matrix
C
. There are different
techniques to do so. The technique, among those we tried, that finds the solution
P∗
of the system (11)
producing the smallest error on the vector
F
is the LU decomposition method. If
e
is the error on
F
produced by the decomposition we can write:
C·P∗T=FT+e(A1)
To perform LU decomposition we used the numpy linalg.solve function that relies on the fortran
LAPACK library-one of the standards for high precision and performance scientific computing [
42
,
43
],
that returns the error
e∼
10
−12
m for every component. We can use
e
to propagate the relative error
eRon the range R. In fact we can manipulate the first equation of system (11) as follows
R(c10 +
l
∑
i=1
c1iyi) = F1=⇒dR
dF1
= (c10 +
l
∑
i=1
c1iyi)−1
eR=1
R
dR
dF1
·e1=(c10 +∑l
i=1c1iyi)−1
R·e1∼10−16
(A2)
(A3)
since R∼104m and (c10 +∑l
i=1c1iyi)−1∼1.
The method proposed in this paper has been implemented as a python script, with the usage of
standard routines of the SciPy ecosystem [
43
]. No proprietary and specifically dedicated software was
needed, neither any particular setting. The numerical evaluation of integral in Equation (16) has been
interpolated over 1000 points implementing a classical trapezoidal algorithm. No significant change is
observed for higher number of points. Scripts and materials can be made available upon request to
the authors.
Appendix B. Multiple Frequencies Generation by Means of Electro-Optic Modulator
The use of an electro-optic modulator is an alternative approach with respect to a non linear
crystal for the generation of multiple frequencies from one laser frequency. These frequencies are
symmetrically spaced by a fixed gap around a central frequency. A gap value of 40 GHz is now
available in commercial devices . This produces very close values of the wavelength and increases
the relative error
σrel
R∗
. In Figure A1 we report how the relative error decreases with the increase of the
frequency gap for a central wavelength of 1.55
µ
m. We can see that only for high not experimentally

Photonics 2020,7, 90 8 of 10
feasible gaps the error is sufficiently reduced. A slightly smaller, but quite similar, error is seen using a
wavelength of 1.0645 µm.
Figure A1. σrel
R∗
relative error (absolute value) vs. frequency gap via electro-optic modulator (central
wavelength 1.55
µ
m). Different colors correspond to different number of frequencies used by the
proposed approach to range estimation. It can be seen that for 3 and 4 (red and green colors) frequencies
the method reaches good levels of sensitivity only for unfeasible high gap values, while for 2 frequencies
only (blue color) the relative error never significantly declines in the interval considered, looking like a
constant value in comparison to the other cases.
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