EFPI signal processing method providing picometer-level resolution in
cavity length measurement
Nikolai Ushakov, Leonid Liokumovich, Andrey Medvedev
Department of Radiophysics, St. Petersburg State Polytechnical University
ABSTRACT
In the current article an advanced method of EFPI baseline measurement by use of spectral function approximation is
proposed. The method provides an increase in EFPI baseline measurement precision and computational acceleration. The
method bases on two foundations provided by analysis of low-finesse Fabry-Perot interferometer model: reduction of
search domain and taking into account the most informative spectral intervals, providing a greater impact on the residual
of measured and theoretical EFPI spectral functions. Proposed signal processing method resulted in the EFPI baseline
measurement resolution less than 50 pm for the cavity length values between 20 and 700 µm.
Keywords: Extrinsic Fabry-Perot Interferometer, signal processing, picometer-level resolution, fibre optic sensors,
tunable laser, spectral measurements, phase measurements
1. Introduction
Fibre Extrinsic Fabry-Perot Interferometer (FEFPI, or EFPI) is a modification of an optical Fabry-Perot
interferometer, formed by two surfaces, reflecting the light. These surfaces can be represented by two oppositely faced
fibre ends (where both transmitted and reflected light can be analyzed), or a fibre end and a mirror (with a possibility for
only reflected signal analysis). Both these situations are illustrated in figure 1. Generally, in FEFPI an intensity of
reflected light is registered, demonstrating an oscillating relation to interferometer baseline L due to interferometric
effects1. Such short-base interferometer is attractive for different physical quantity sensors. Tube-type packaging of an
interferometer is applicable for obtaining constructions sensitive to temperature, pressure, strain, vibrations, etc2.
Figure 1. Structure of fibre EFPI with two fibres and with fibre and external mirror.
Generally, relation of reflected light intensity and cavity length I(L) is described by Airy function, however, while
the reflectivity of the mirrors is low and no collimation optics is used, multiple reflections can be neglected and EFPI can
be considered as a two-pass low-finesse interferometer. In this case the relation I(L) is expressed in a much simpler
way3,4:
I ( L, λ) = I 0 ( L) + Am ( L) ⋅ I ' ( L, λ)
 4πnL

I ' ( L, λ) = cos
+ ϕ( L, λ) ,
 λ

(1)
where n is the refractive index of the media between the mirrors, λ is the light wavelength and additional phase term ϕ(L,
λ) is due to phase shifts induced by the mirrors and the wavefront evolution during propagation inside the interferometer.
The constant term I0 and the amplitude of the interferometric oscillations Im besides dependence on the cavity length L,
are also related to mirrors reflectivity, their angular displacement, and in some cases (which aren’t considered in the
current paper) can depend on λ. Considering Gaussian distribution of mode propagating in the feeding fibre one can
apply the Gaussian beam theory to light propagating inside the interferometer. Hence the additional phase term in (1) can
be expressed in the following form5
 2 Lλ
ϕ( L, λ) = − arctan
 πnw 2
0


+π,


(2)
where w0 is an effective width of the Gaussian mode, summand π is stipulated by the reflection from the second
dielectric mirror.
In different tasks, such as reflective surface vibration measurements, the interferometer baseline can be
represented as L(t) = Lin + δL(t), where Lin is an insignificant initial value. In such a manner, δL(t) is the only measurand,
generally found by introducing an additional modulation (for example, harmonically varying L6) and applying one of the
standard demodulation methods, widely used in the fiber-optic interferometry7.
However, thanks to its short baseline, EFPI can be efficiently used in tasks where the measurand is either absolute
cavity length L, or a change of absolute cavity length with regard to some reference value L0, which relation to the
desired physical quantity can once be obtained by a simple calibration procedure.
There are known two general approaches for interferometer base L measurement. The first one is based on the
white light interferometry principle and requires a broadband light source and special scanning recovery
interferometer2,3. The second approach consists of measuring spectral transfer (or reflective) EFPI function S(λ). Taking
into account current progress in optical sensor interrogators, usually based on the tunable lasers, the second approach,
providing ability for EFPI spectral function registration, seems to be more favorable.
The simplest method estimating cavity length L from S(λ) function is based on its oscillating quasi-periodic
character and utilizes simple relation between L and number N of S(λ) quasi-periods3
L=
N ⋅ λ max λ min
,
2n ⋅ (λ max − λ min )
(3)
where λmin and λmax are boundary wavelengths of spectra measurements. In some cases other expressions of a similar
sense are used8.
A more advanced method, presented in works4,9 provides the value of L by approximating measured spectral
function S(λ) with expression (1). Taking into account the form of expression (1), it’s more convenient to represent S(λ)
as a sum of constant S0 and oscillating S’(λ) parts: S(λ)=S0+S’(λ). Then approximation procedure can be applied to S’(λ)
term with approximating function A0·cos(4πnL/λ+φ(L, λ)), A0 is equal to S’(λ) amplitude. Then the L value minimizing
the experimental and analytical EFPI spectral functions residual norm R(L) is found according to the least squares
method, R(L) is given by
 4πnL

R( L) = S ' (λ) − A0 cos
+ ϕ( L, λ)  .
 λ

(4)
Using such method, a sub-nanometer resolution of cavity length measurement was achieved4. In paper9 a
combination of the fringe analysis and the least-squares approximation is proposed, resulting in the cavity length
measurement resolution less than 0.1 nm.
Despite the attractiveness of such method, its practical implementation causes some problems, requiring further
improvements. The problem of the measurement accuracy enhancement always remains actual, especially the challenge
of abrupt measurement errors at a half of the central wavelength, clarified below. Another problem is the computational
efficiency of the cavity length estimation. The realization of R(L) global minimum search by direct enumeration with
extremely high accuracy < nm requires an enormous amount of calculations.
2. Approximation method features and problem statement
The task of measuring L ranged from tens to thousands micrometers with sub-nanometer accuracy, much less than
the optical wavelength, requires discussion of the physical meaning of the actual EFPI baseline. Performing such
measurement, one desires the outcome to be as close to the real cavity length as possible. However, in a context of such a
high accuracy, it’s not clear what the measurand is. Taking into account such issues as abruptness of the surfaces forming
the Fabry-Perot cavity, generally exceeding tens nanometers, angular mismatch between these surfaces; the resulting
measured cavity length has a sense of some effective optical path difference. Other causes of accuracy restriction are the
issues of the practical spectra measurement devices such as additional phase increment inside the interferometer, errors
of wavelength stamps in the measured spectra, photodetector and electronic noises and so on, aren’t strictly taken into
account in current consideration. However, the problems of defining the baseline absolute value with extremely high
accuracy don’t eliminate the importance of increasing the resolution of such measurements.
The main aim of the current work is fibre EFPI baseline measurement resolution enhancement. Therefore as the
desired measurand we assume the equivalent baseline Leq of an ideal low-finesse interferometer with spectral function
given by (1). We assume the limits of the cavity length measurement resolution to be determined by intrinsic fluctuations
and distortions produced by the optical spectrum analyzer and propose signal processing approach for eliminating their
influence.
Since the modern measurement devices utilize digital information registration and processing, function S(λ) is
represented as an array of point pairs {Si, λi}, the same takes place for oscillating part of spectral function S’(λ). As a
result, the target function for original spectra approximation can be expressed as
R( L) =
∑ (S ' i − A0 I ' ( L, λ i ) )2
,
(5)
i
where the sum is calculated over all i numbers.
Also it’s worth mentioning some characteristic properties of functions S’(λ) and R(L) and peculiarities of resultant
value Lr deviations from the real value L0. In ideal case S(λ) is described by (1) with some fixed baseline value L=L0. In
such a way the oscillating part S'(λ) = A0·cos[4πnL0/λ)] is quasi-periodic with quasi-period generally depending on two
quantities: λ0 = (λmax + λmin)/2 and n·L0, therefore resulting in the estimate (3).
However, slight deviations of the interferometer baseline provide negligible changes of S’(λ) oscillation period.
Hence, EFPI cavity length measurement according to (3) provides poor accuracy (about several µm). At the same time,
considering the relation of S’(λ) itself with L0, illustrated in figure 2 (a), one can see that small changes of the cavity
length brings a noticeable “shift” of S’(λ) relation.
1
1
0
0
−1
1500
1520
1540
1560
1580
1600
−1
1500
1520
1540
1560
1580
1600
Figure 2. Examples of theoretical Ι’(λ, L) dependences according to (1). а) – L=40 µm (solid line), L=40,1 µm (dash), L=40,2 µm
(dot); b) – L=40 µm, L=40+1.55/2 µm and L=40 – 1.55/2 µm.
On the other hand, changing L0 at value λ0/2n causes a shift of S'(λ) at one quasi-period, resulting in nearly the
same shape of dependence. As illustrated in figure 2 (b), S'(λ)|L ≈ S'(λ)|L±λ0/2. This circumstance causes oscillations of
R(L) function, illustrated in figure 3, (a).
a)
b)
S’ , L), a.u.
S’ , L), a.u.
1
1
0
0
−1
1500
1520
1540
1560
1580
1600
, nm
R(L), a.u.
−1
1500
1520
1540
1580
1600
, nm
R(L), a.u.
L0
L0
1
1
0.5
0.5
0
98
1560
99
100
101
102
L, m.
0
98
99
100
101
102
L, m.
Figure 3. Examples of functions S'(λ, L0) and corresponding R(L) for L0=100 µm. a) – S'(λ, L0) calculated directly according to (1); b)
– S'(λ, L0) calculated as a sum of (1) and additive noises.
The global minimum corresponds to the perfect agreement of approximating function argument L and real EFPI
baseline L0, R(L0) = 0 in an ideal case. Besides the global minimum, parasitic lateral minimums appear, advocated at
integer numbers of λ0/2n. These minimums correspond to the neighbor longitudinal modes of the considered Fabry-Perot
cavity. Since the value R(L) in the lowest lateral minimum δ is nonzero, the global minimum can be distinguished from
the laterals. However, due to the inevitable presence of noises and distortions in the practical spectra S'(λ), the R(L)
function properties are considerably changed, see figure 3, (b). First of all, in this case R(L0) is no longer equal to zero,
the valley containing the global minimum gets broader, resulting in a less precise localization of the desired Lr. In
addition, the difference between the global and the lowest lateral minimums decreases and even can become negative. In
that case an abrupt error at λ0/2n happens (λ/2-failure).
Taking into account preceding considerations, the task of approximation algorithm improvement consists of
finding such preprocessing and modified target function R’(L), that will on the one hand provide a lower value of the
global minimum located in a sharper valley, and on the other hand maximize the value δ' = R'(L0± λ0/2n) – R'(L0),
therefore reducing the probability of abrupt λ/2-failures.
Considering the computational difficulty of the approximation algorithm it’s likely for the modified target
function to decrease the amount of required calculations. Also an improvement of the approximation algorithm
convergence to the desired quantity would be an effort.
3. PROPOSED FITTING METHOD
As one can see in figure 3, function R(L) has very complex structure with a great number of local extreme points.
However, a detailed analysis of function I’(λ, L) shows that these extremes are located equidistantly with shift λ0/2.
Hence, finding one (maybe local) extreme point and then consequently checking neighbor extremums will result in a
significant speedup of the global minimum search.
3.1. Reduction of the processed data amount
The first feature we propose in order to enhance the accuracy of the approximation method is to perform a
reduction of experimental data, by taking into account only those parts of S’(λ), providing a greater impact on the
residual of the measured and the theoretical spectral functions of the Fabry-Perot interferometer. In such a manner, the
expression (5) is replaced by
R ' ( L) =
∑ (S ' i − A0 I ' ( L, λ i ))2
,
(6)
i∈I sel
Isel is a reduced set of indexes determining the analyzed data. Forming rules for Isel are introduced below. Proposed
additional features are based on oscillation character of residual between measured and analytical spectra
D(λ, L0, δL) = S'(λ, L0) – I'(λ, L0+δL),
(7)
L0 is actual effective cavity length which value we are interested in.
These arrangements suppress the impact of the spectral function noises on the cavity length registration, resulting
in a great increase of the fibre EFPI measurements stability. Also as a consequence of the data size reduction, an amount
of calculations is reduced by several times. Below two general situations with different properties, exhibiting some
features, important for the enhancement of the approximation approach, are discussed.
At first let us consider the situation of relatively small difference δL (from picometers to nanometers). Then
function D(λ, L0, δL) can be expressed from (7) in the following term
D(λ, L0 , δL) ≅
4πn ⋅ δL
 4πnL0

⋅ sin
+ ϕ( L0 ) ,
λ
 λ

(8)
as illustrated in figure 4, (a). In this case the greatest values of D(λ, L0, δL) function (shown as bold lines in figure 4, (a))
correspond to the slopes of the original function S'(λ), emphasized with bold lines in figure 4, (b).
Under these conditions reduction of the processed data is done by “cutting” both spectra S'(λ) and I'(λ, L) at levels
a1 and –a1 and substituting into (6) data range Isel1, given by
i∈ Isel1 if |S'i|<a1·A0.
(9)
Typical a1 values are about 0.5 ÷ 0.7, which provide good robustness and a reduction of calculations amount in
about 2-3 times. In such a manner, approximation procedure is carried out using modified target function R1(L), obtained
from (6) by performing summation over the index set Isel1 given by (9). In figure 4 the examples of I'(λ, L) and D(λ, L0,
δL) are shown for the case of the following values: spectral range λ=[1510; 1590] nm, L0=150 µm, δL=1 nm and a1=0.5.
Figure 4. Examples of D(λ, L0, δL) (a) and S’(λ, L) (b), L0=150 µm and δL=1 nm.
In figure 4, (b) the bold lines emphasize spectra parts selected for the analysis according to rule (9), in figure 4,
(a) bold lines accent that data parts selected according to (9) provide a greater part (more than 80%) of the residual
D(λ, L0, δL) norm. The amount of processed data is reduced in about 3 times.
The second situation we would like to consider is δL nearly equal to λ0/2n. Considering the difference D(λ,
L0, λ0/2n), one can show that its amplitude differs with respect to wavelength, instead of residuals for spectra
corresponding to slightly differing cavity lengths. Again, omitting trivial mathematical developments, for the case of
δL≈λ0/2n the expression (7) can be expressed in the following form
λ −λ

 4πnL

(10)
D (λ, L0 , λ 0 2 n) ≅ 2π ⋅  0
+ 1 ⋅ sin 
+ ϕ .
λ
λ


0


In such a manner, finding quadrature component for I’(λ) (for example, by taking a Hilbert transform of the initial
function S’(λ)) and multiplying it at linearly decreasing function with opposite maximal and minimal values, one will
obtain a function which is in a very good correspondence with the residual D(λ, L0, λ0/2n). After that, since we are
interested in analyzing only the data providing a greater influence on the D(λ, L0, δL) norm, a collection of indexes Isel2
determining the reduced data set can be formed with expression below
2λ − λ i
2λ 0 − λ min
(11)
i ∈ I sel 2 if Η ' i ⋅ 0
> a 2 ⋅ A0
,
λ0
λ0
where Η'i denotes the discrete representation of the S’(λ) Hilbert transform and typical a2 values are about 0.3÷0.5. In
this case, the approximation procedure is carried out using another modified target function R2(L), derived from (6) by
performing summation over the indexes Isel2 determined by (11). In figure 5 the examples of I’(λ, L) and D(λ, L0, δL) are
shown for spectral range λ=[1510; 1590] nm, L0=150 µm, δL=0.775 µm and a2=0.5.
Figure 5. Examples of D(λ, L0, δL) (a) and S’(λ, L) (b), L=150 µm and δL=0.775 µm.
Again, as in figure 4, in figure 5, (b) bold lines emphasize spectra parts selected for analysis according to rule
(11), in figure 5, (a) bold lines illustrate the principle of expression (11) and accent that data points selected according to
(11) provide a great part (more than 80%) of the spectra residual norm. The amount of processed data is reduced in about
5 times.
3.2. R(L) global minimum search algorithm
Proposed signal processing algorithm can be divided into four steps. On the first step preliminary estimate of the
cavity length is provided. Three following steps are developed in order to reduce the search domain to several narrow
intervals, therefore extremely increasing the performance of the proposed method. Initial step of the search algorithm is
decreased at each stage, providing an increasing accuracy of R(L) extreme point localization. Also data reduction
operations, discussed in section 3.1 are applied to the spectral functions in order to improve the accuracy of the cavity
length measurement and reduce the amount of calculations.
Proposed algorithm is described below in a more detail:
1. Preliminary value of the cavity length LE is found according to the number of the measured spectra oscillation
periods using expression (3);
2. Preliminary value LE is specified with a higher accuracy by the use of the binary search algorithm10 (shortly,
BSA). The summation in (6) is performed over the spectra points Isel1 defined by (9). Initial value is set to LE, initial step
to λ0/8. Search is terminated after m0 BSA stages are performed and step becomes less than λ0/2m0+3. In such a manner,
one reaches the extreme point L1 located in the interval [LE – λ0/2; LE + λ0/2], with precision around λ0/2m0+3. Optimal
value for m0 is about 7–8, providing the cavity length estimation accuracy about 1 nm.
3. The second step finds the coordinate of one of the R(L) minimums (maybe local). However, taking into account
I'(λ, L) features, one can conclude a set of a possible coordinates of the global minimum. Therefore points
L2k = L1+k· λ0/2, |k| ≤ kmax – integer, are taken as initial values for the third stage, kmax value depends on the spectra
oscillation periods number Nos. In this case the approximation is performed with the use of the residual function R2(L),
obtained from (6) by summation over indexes Isel2 given by (11). The coordinates of the extreme points are adjusted
using BSA, converging to resulting points L3k. For each of them the residual norm is equal to Rk=R2(L3k). In such a
manner, the minimal value of the residual norm Rk0 corresponds to the value L3k0 best corresponding to the desired cavity
length value. Again, about 7 stages of the binary search algorithm are executed for each L2k point, providing the
uncertainty of L3k0 about 10 pm.
4. On the last step value L3k0 is adjusted using the BSA, approximating the data set reduced according to (9)
(although values δL are about 1-10 picometers, which is much smaller than on the second stage, function D(λ, L0, δL) is
still defined by (8) and is pretty similar to the one shown in figure 4, a). About 5 stages of the binary search algorithm are
executed providing the uncertainty of the resulting value Lr less than 1 pm.
The final conclusion about the uncertainty of the proposed EFPI baseline measurement algorithm was drawn
without taking into account the presence of noisy components in the measured spectra. Practical values of the
measurement resolution will be presented below.
4. EXPERIMENTAL DEMONSTRATION
Proposed signal processing method was implemented experimentally. Spectra measurements were performed
using the optical sensor interrogator NI PXIe 4844 (analysis range 1510÷1590 nm, spectral resolution 4 pm), operating at
a sample rate about 1 Hz. Signal processing was implemented in LabVIEW running on the PXI controller in a real-time
mode. Examined interferometer was formed by the two fibre ends packaged in the PC connectors fixed in a hollow
ceramic tube as illustrated in figure 6, (b). The air gap L0 between the fibre ends was varied from tens µm up to 4 mm.
Figure 6. Experimental setup.
The left fibre (figure 6, b) was rigidly screwed to the tube, while the right one was fixed by the friction in the
tube. Reflectivity of the mirrors were about 4% for the left one (Fresnel reflections at the glass-air boundary) and about
10% for the right one, which was formed by the dielectric evaporation on the fibre end. The assembled interferometer
was placed into a thermally isolated box in order to suppress the influence of the environment for intrinsic measurement
resolution estimation. Experimental spectral functions for different cavity lengths are shown in figure 7.
Figure 7. Examples of experimental EFPI spectra for different cavity lengths: a – ~56 µm, b – ~287 µm, c – ~803 µm, d – ~3901 µm.
For better clearness an enlarged fragment of the largest baseline spectra is shown.
In figure 8 plots illustrating the resolution of the static cavity length measurement during more than 20 min are
shown. Results for two baseline values (about 100.5 and 530.8 µm) are presented. Standard deviations of the illustrated
dependencies are 47 pm for ~100 µm cavity and 27 pm for ~530 µm cavity.
Figure 8. Experimental results for cavity length measurement resolution, a – <Lr> ≈ 100.5 µm, b – <Lr>≈530.8 µm. Dash lines show
measurement standard deviations.
In another experiment the examined interferometer was first slowly heated and then cooled (inside the box)
during about 100 minutes. An FBG temperature sensor was placed in the same box for the temperature calibration. In
figure 9, (a) a comparison of the FBG sensor indications (calculated by standard software coming with the interrogator)
and the measured EFPI baseline is shown. In order to estimate the resolution achievable in the dynamic cavity length
measurements we have subtracted the trend from the data using the wavelet filtering of the experimental curve. The
residual characterizes the possible resolution of the dynamic measurements and is illustrated in figure 9, (b). Standard
deviation of the residual is 16 pm for the whole measurement and drops to 10 pm at 1-minute temporal intervals.
Figure 9. Comparison of FBG temperature sensor data with EFPI cavity length measurements (a). Residual between experimental data
and estimated “low-frequency” trend (b), dash lines show measurement standard deviations.
The difference between EFPI and FBG data is likely to be due to the different thermal capacity of the fibre grating
and the ceramic tube forming the interferometer. Discretization step of FBG data is defined by spectral resolution of the
optical sensor interrogator.
5. Conclusion
A novel signal processing for the EFPI cavity length measurement providing high resolution is developed.
Measurement procedure is based on approximating the measured EFPI spectral function with the analytical expression.
Proposed improvements of the approximation procedure are based on the analysis of the fibre EFPI mathematical model.
As a result the robustness and the noise-immunity of the EFPI baseline measurement is sufficiently increased. The
proposed signal processing method was implemented in LabVIEW and tested experimentally. An ability of measuring
EFPI baselines in range from 10 to 4000 µm with sub-nanometer resolution is shown. For cavity lengths from 20 to 700
µm the standard deviation of the measured quantity on temporal intervals about 10 minutes varied from 15 to 50 pm,
decreasing down to 10 pm on temporal intervals about 1 minute.
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