Two-wavelength interferometry combines measurement at two wavelengths and in order to increase the unambigous range (UR) for the measurement of an optical path difference. With the usual algorithm, the UR is equal to the synthetic wavelength , and the accuracy is a fraction of Λ. We propose here a new analytical algorithm based on arithmetic properties, allowing estimation of the absolute fringe order of interference in a noniterative way. This algorithm has nice properties compared with the usual algorithm: it is at least as accurate as the most accurate measurement at one wavelength, whereas the UR is extended to several times the synthetic wavelength. The analysis presented shows how the actual UR depends on the wavelengths and different sources of error. The simulations presented are confirmed by experimental results, showing that the new algorithm has enabled us to reach an UR of , much larger than the synthetic wavelength, which is only . Applications to metrology and fringe tracking are discussed.
© 2009 Optical Society of America
Absolute distance measurements are required in a wide range of applications, such as metrology , real-time fringe tracking for the cophasing of stellar interferometers [2, 3, 4], or segmented telescopes . When carried out with monochromatic light at the wavelength λ, the measurement of an optical path difference (OPD) suffers from a modulo λ ambiguity. To overcome this issue, techniques such as fringe counting  or phase unwrapping can be used to increase the unambiguous OPD range (UR), but these solutions rely on the spatial or temporal continuity of the phase. In the applications previously listed, a direct nonambiguous OPD measurement is made by using the differential information between several wavelengths .
First absolute distance measurements with several wavelengths were reported in 1898 when Benoît compared the excess fractions of the orders of interference at several wavelengths . Then, in order to test large aspheric mirrors, the method of two-wavelength interferometry (TWI) was proposed [8, 9], using two wavelengths and in order to generate a longer synthetic wavelength Λ defined as
Later, the introduction of numeric cameras and computers together with phase modulation–demodulation techniques considerably improved interferometric measurements [10, 11]. These methods have been applied to TWI, leading to phase-shifting TWI , multiple wavelength interferometry (MWI) , or tunable wavelength interferometry .
The TWI method was considerably improved when De Groot showed that the UR could be much larger than the synthetic wavelength by simply improving the data analysis . However, although de Groot’s principle for extending the synthetic wavelength has been shown with a clear physical background by Van Brug , the algorithms developed are not universal and cannot be applied to all wavelength couples. Later, de Groot’s methodology was used by Falaggis  who extended the UR in a multiple-wavelength interferometry context by using the generalized optimum wavelength selection . Löfdahl also developped a new algorithm for extending the synthetic wavelength, but this algorithm is iterative .
In this paper, we present a new TWI analytical algorithm that enables the measurement of the OPD over an UR that can be much larger than the synthetic wavelength Λ. After briefly recalling the basic TWI algorithm in Section 2, we develop a new algorithm in Section 3. Then, in Section 4, performance of this algorithm is investigated in terms of measurement error and maximum reachable UR, and we show its superior behavior compared with the basic TWI algorithm. Several applications, such as metrology or cophasing, are considered in Section 5. In Section 6, we present the experimental validation of the new algorithm.
2. BASIC ALGORITHM: TWI-1
An OPD measured at will be characterized by the fractional order of interference , defined by 
For any given number x, the nearest integer (rounded part) will be denoted , and the remaining part (wrapped part) will be denoted , so that the OPD can be written as
The use of the round and wrap functions instead of the more commmon floor and fractional part functions, respectively, is adopted in order to center the range of the estimated OPD around .
Because of the periodicity of the measured OPD when measured at the wavelength , it is not possible to know the rounded order of interference without additional information: what we are actually measuring is , or .
The use of OPD measurements at two wavelengths and allows the computation of the difference of the measured orders of interference. Because each is measured modulo 1, only the fractional part of is relevant, hence the basic TWI estimator (TWI-1),1).
Figure 1 shows the result of a simulation in which are plotted the measured orders of interference and at the wavelengths and , respectively, along with the wrapped difference of the orders of interference resulting in a Λ-periodic signal. The computation, with no additional error, is carried out with and (two common wavelengths for which stabilized sources are available), giving a synthetic wavelength [Eq. (1)]. Thus, even though the phase measurements are unambiguous over a dynamic range of at the most, the computation of enables us to increase the UR up to Λ.
However, it is noticeable that when , even though the wrapped difference of the measured orders of interference is zero, the measured orders of interference and are different from zero: this means that even though is Λ periodic, the states of interference at the measurement wavelengths and are not the same for and for . We propose to use this information in order to retrieve the actual OPD on a range still larger than Λ.
Another way to highlight this information is by illustrating the principle of OPD measurement with TWI thanks to a calliper rule (Fig. 2 ). The current OPD can be measured as the distance of the dashed line to the closest continuous marks (at or ): in this case, only the fractional parts are used, and the measurement is modulo . But this measurement does not take into account the relative position of the continuous marks at and . Figure 2 shows that the distance X between the two marks at and , defined as4), that we use in the algorithm described in the following.
3. NEW ALGORITHM: TWI-2
Since the OPD is measured at two different wavelengths, it is possible to write
Equation (7) relates two measured quantities and to two unknown integers: the rounded orders of interference and . Only one equation is usually not sufficient to estimate two unknowns without other constraints. However, one constraint is the integer values of the rounded orders of interference. Consequently, we propose to use arithmetic properties in order to retrieve the rounded order of interference , but it is also possible to retrieve the rounded order of interference , as will be shown in Subsection 4D.
Since the measurement wavelengths are never perfectly calibrated, the ratio of the wavelengths is known with an error :20].
Let us write the estimated ratio of the wavelengths as a quotient of two coprime natural numbers p and q plus an error that corresponds to a systematic error due to the approximation by a fraction of integers:
When measured, the wrapped orders of interference are corrupted by an error :12) can be further processed if its right-hand side can be converted into an integer without error, i.e., if12) modulo q to get rid of , it is possible to retrieve modulo q:
An important innovative step of our algorithm is to use Bézout’s identity in order to retrieve directly modulo q instead of modulo q. Bézout’s identity states, among other things, that if p and q are coprime, there exists an integer k such that
A dedicated algorithm, known as Euclid’s algorithm, allows one to find the integer k. Examples will be given in Section 4.
The dynamic range of Eq. (19) defines a new UR,17).
In this section, we developed a new algorithm that enables the measurement of the OPD over an UR that can be tuned by the choice of q. The only assumption made is related to all the encountered errors. The following section details how to deal with all the parameters.
4. PERFORMANCE OF TWI-2 ALGORITHM
This section addresses the trade-off between the different errors and the UR in order to optimize a system based on TWI-2. Then the algorithms TWI-1 and TWI-2 are compared.
4A. New Estimator Accuracy
The error of TWI-2, defined by Eq. (18), can have three origins. First, error on leads to a multiplicative error on . This is a classical issue for any interferometric measurement and will not be discussed further here. Second, since is an integer, the error on this term is null as soon as the condition in Eq. (14) is satisfied: this will be investigated in the next subsection. Third, the error on , directly resulting from the interferometric measurement at , is characterized by an error . Therefore, the new estimator has the same error as the monochromatic OPD estimator using the wavelength , but can maintain this capability over an UR that can be much larger than .
4B. Estimation of the Maximal Unambiguous Range
Equation (20) shows that the UR is given by the free parameter q. However, increasing q increases the total error ϵ as shown by Eq. (13), which must satisfy Eq. (14). Thus, in order to obtain a large UR, the sources of error , , and must be minimized.
This section deals with the maximal reachable UR for a given measurement error . The goal is to satisfy condition (14).
First, assuming good measurements, . Second, in the worst case, the errors of the fractional orders of interference measurements are added and are assumed to be smaller than an upper bound value of both measurements. Last, since the rounded order of interference has to be retrieved over the UR , we assume the worst case where and ( can only be retrieved modulo q). Then, using the triangle inequality, we have13) several terms, and we made only the approximation ; thus the inequality written in Eq. (21) is almost exclusively a strict inequality.
Then, assuming , Eq. (14) is satisfied with the sufficient condition
This inequality shows that a trade-off has to be made between p, q, , , and . Indeed, the higher and , the lower q, must be chosen, and vice versa. Thus, independently of the measurement error and the wavelength calibration, the UR is especially extended when their ratio can be approximated by a fraction with a high denominator q and a small error [Eq. (10)].
Based on empirical observations detailed in Appendix A, we propose to define the couples as4 in Appendix A.
Assuming that a lucky couple has been found, Eq. (22) leads to the sufficient condition to correctly estimate the rounded order of interference :
This quadratic equation in q enables one to find the maximal number that verifies it:
Equation (25) can be simplified for each error regime:
Equation (27) also shows the diffference between the calibration and the measurement error regimes: to increase the UR by a factor of 2, a gain of 2 is expected for in the measurement error regime, but a gain of 4 is expected for in the calibration error regime.
4C. Comparison with TWI-1
Therefore, whereas the UR is imposed by the two wavelengths with TWI-1, TWI-2 allows one to tune the UR extension, which can be very high, as will be discussed in Section 5.
Thus, four major benefits of TWI-2 are clearly evident. First, as soon as is larger than 1, the UR of the new algorithm is larger than the UR of TWI-1. Second, according to Subsection 4A, the accuracy of TWI-2 is directly proportional to the measurement wavelength , and thus can be much better than the accuracy of TWI-1, which is proportional to the synthetic wavelength. Third, when , the synthetic wavelength is smaller than the wavelength ; thus TWI-1 is rather useless, whereas TWI-2 can still be used. Fourth, the measurement wavelengths cannot be chosen too close with TWI-1 because the measurement error of the differential phase is amplified by Λ, which is inversely proportional to [cf. Eq. (1)]. When the measurement wavelengths are close, in order to extend the UR with TWI-1, large values of p and q are required for TWI-2 to obtain a rational approximation of the ratio with and thus to increase the UR compared with the use of the synthetic wavelength [Eq. (28)]. Besides, according to Eq. (13), when p and q are large, very small values of the errors are required in order to use the algorithm presented in Section 3. Thus, the extension of the UR with TWI-2 is easier when the measurement wavelengths are not close.
4D. Other Estimators
In Section 3, starting from Eq. (15), the algorithm developed estimates the OPD with Eq. (18) based on the rounded order of interference from Eq. (19). But it is also possible to estimate the rounded order of interference from Eq. (12):
According to Eq. (30), the UR of the estimator is , and since , we have . Moreover, Eq. (29) shows that the estimator has the same error as the monochromatic OPD estimator using the wavelength . Therefore, if , then is better than , since .
However, if the instrument is limited by repeatability errors rather than bias errors, then the existence of two different estimators allows the derivation of a new OPD estimator, obtained by averaging and , with the same UR but with an error reduced by a factor of about 2.
5. EXAMPLES OF APPLICATION
We show here two typical applications of TWI-2, with different operating conditions.
We consider here the case of distance measurement over a large range. Two bright and stabilized sources are assumed in order to minimize the errors and .
This section deals with the specification of the OPD measurement error when the OPD UR is imposed. The requirement of the OPD UR leads directly to a lower bound limit of the integer q [Eq. (20)] and thus to several couples . For any given couple , it is possible to retrieve an upper bound to the measurement error; we call this limit the error margin . Assuming a large denominator q, from Eq. (22), we have
Table 1 shows with a numerical example different possible rational approximations of the ratio as expressed in Eq. (10). Of course, all the possible couples are not illustrated. The chosen measurement wavelengths are the yellow-orange and the red transitions of Helium–neon, and we assume that the wavelengths are known with an accuracy better than , which leads to .
Moreover, even though the specification on the measurement error globally decreases when q increases, i.e., when the UR increases, the example in Table 1 shows that with the couple , it is not possible to extend the UR with the algorithm developed in Section 3. Thus, the choice of the rational approximation of the ratio is fundamental.
Table 1 shows that if the order of interference measurement error can be lower than , then it is possible to increase up to a factor of 16 the UR compared with the synthetic wavelength and thus to reach an UR of .
5B. Fringe Tracking in Multiple-Aperture Optics
We consider here the case of a multiple-aperture instrument observing a broadband object. A central issue is the cophasing of the subaperture array on the central fringe (at 0 OPD). In this goal, the light from the object at the output of the interferometric instrument is split into several spectral channels from which the phase can be tracked and the central fringe identified. The allocation of the spectral bands is a trade-off among the object spectrum, the perturbation amplitude, and the signal-to-noise ratio, which can be much smaller than in Subsection 5A.
Such an analysis has been performed for Pegase , a Darwin /TPF  pathfinder. For these missions, based on the coherent combination of beams reflected by formation-flying spacecraft, the performance of the cophasing system is critical. The strategy selected for Pegase is to use the near-IR part of the spectrum where the star is the brightest, with only two spectral channels, to minimize detection noise, but with maximum width, in order to maximize the flux in each channel . This leads, for Pegase, to and , leading to two polychromatic interferograms centered at and , with similar coherence lengths (at half-maximum) . Therefore, since the synthetic wavelength is , fringes outside the coherence length can be seen, but the OPD can not be computed without ambiguity.
In this context, the extension of the central-fringe acquisition range outside the short synthetic wavelength is welcome and has motivated the algorithm presented in this paper.
6. EXPERIMENTAL VALIDATION
In order to demonstrate the feasibility of future formation-flying missions, a laboratory demonstrator called Persee is under integration at Observatoire de Paris-Meudon . Persee uses a similar spectral allocation than Pegase, but the broadband star has been replaced by two laser sources: one laser diode at and one superluminescent light emitting diode (SLED) at with a full width at half-maximum. The goal of the polychromatic SLED is to locate the central fringe thanks to its coherence length . With this configuration, the coherence lengths for each measurement are much larger than the synthetic wavelength .
This setup is thus an intermediate case between the two presented in Section 5. Because of the rather large spacing between the two wavelengths, TWI-1 is not relevant, as Λ is nearly equal to . The goal is thus to extend the UR up to typically , i.e., to have .
As explained in Section 4, the calibration of the ratio as well as the calibration of the measurement wavelengths is necessary. The calibration of Persee’s cophasing system, based on the knowledge of the behavior of the delay lines, led to an estimation of the measurement wavelengths:
Moreover, under typical conditions, the detector and photon noises along with the chromatism of our experiment are such that the maximum measurement error of the order of interference is
While the denominator and the fractionnal approximation error is not too large, Table 2 gives different possible rational approximations of the ratio and the fractional error . According to Eq. (21) we show an upper bound value of the total error ϵ for each rational approximation.
We recall that the total error ϵ must be lower than to be sure that the algorithm developed in Section 3 can be used to extend the OPD UR [cf. Eq. (14)]. The largest integer q for which this condition is satisfied is . Thus, we made the following rational approximation for the ratio :
Figure 3 shows that the algorithm developed in this paper enabled us to reach an UR of , whereas the UR or TWI-1 is only . Thus, only thanks to extended data processing and an accurate choice of free parameters, the UR was multiplied by a factor of 8.
This paper has shown that with a very simple signal processing based on arithmetic properties, the conventional TWI-1 algorithm can be considerably improved. We showed that the novel algorithm TWI-2 can reach an unambiguous range (UR) much larger than synthetic wavelength and is at least as accurate as the most accurate measurement at one wavelength. In addition, the TWI-2 algorithm is well adapted when the measurement wavelengths are not too close, and it also analytical and thus can be implemented for real-time absolute order of interference measurement.
The extension of the UR is especially increased by choosing the measurement wavelengths so that their ratio can be approximated with very good accuracy by a rational fraction. We show that the calibration of the ratio of the wavelengths is of the highest importance in order to maximize the optical path difference (OPD) UR. We also showed that the choice of the rational approximation of the wavelength ratio directly affects the OPD UR.
The TWI-2 algorithm has been experimentally validated with two relatively separated spectral bands ( and ) and it has enabled us to reach an UR of , much larger than the synthetic, which is only .
APPENDIX A. RATIONAL APPROXIMATION ANALYSIS
The ratio of the measurement wavelengths has to be expressed as a rational function in order to estimate the rounded order of interference with the algorithm developed in Section 3 [Eq. (19)]. When the wavelengths are such that is not fractional, a large number of couples can be considered. And some couples are better than others. In general, increasing the approximation accuracy requires larger q values. For example, π can be approximated by or , with respective errors of about and . This shows that by multiplying q by a factor of about 16, the accuracy has been increased by a factor of about 5000. A better approximation is , with an accuracy of , but this requires a q 300 times larger, for an increase in accuracy of only . As shown by Eq. (22), the relevant figure of merit is . We will call a “lucky couple” a fraction with an exceptionnaly small with respect to the value of q, so that can be bound.
Figure 4 shows the rational approximation error for . This plot shows that a few lucky couples (and their replicates) can be found. It also shows that the figure of merit is lower than for these couples. This value has been selected by computing similar graphs for various values of , so that a few lucky couples are always present.
The authors appreciate the financial support of CNES and ONERA. This work also received the support of PHASE, the high-angular-resolution partnership between ONERA, Observatoire de Paris, CNRS and Université Paris Diderot.
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