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K. Houairi and F. Cassaing Vol. 26, No. 12 /December 2009 /J. Opt. Soc. Am. A 2503

Two-wavelength interferometry: extended rangeand accurate optical path difference

analytical estimator

Kamel Houairi1,2,3,* and Frédéric Cassaing1,3

1ONERA, Optics Department, BP 72, 92322 Châtillon Cedex, France2CNES, 18 Avenue Édouard Belin, 31401 Toulouse Cedex 09, France

3PHASE, the high angular resolution partnership between ONERA, Observatoire de Paris,CNRS and University Denis Diderot, Paris 7, France

*Corresponding author: kamel.houairi@onera.fr

Received July 24, 2009; accepted September 24, 2009;posted September 30, 2009 (Doc. ID 114761); published November 4, 2009

Two-wavelength interferometry combines measurement at two wavelengths �1 and �2 in order to increase theunambigous range (UR) for the measurement of an optical path difference. With the usual algorithm, the URis equal to the synthetic wavelength �=�1�2 / ��1−�2�, and the accuracy is a fraction of �. We propose here anew analytical algorithm based on arithmetic properties, allowing estimation of the absolute fringe order ofinterference in a noniterative way. This algorithm has nice properties compared with the usual algorithm: it isat least as accurate as the most accurate measurement at one wavelength, whereas the UR is extended toseveral times the synthetic wavelength. The analysis presented shows how the actual UR depends on thewavelengths 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 17.3 �m, much larger than the syntheticwavelength, which is only �=2.2 �m. Applications to metrology and fringe tracking are discussed. © 2009Optical Society of America

OCIS codes: 120.3180, 120.2650, 120.3940, 110.5100.

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. INTRODUCTIONbsolute distance measurements are required in a wideange of applications, such as metrology [1], real-timeringe tracking for the cophasing of stellar interferom-ters [2–4], or segmented telescopes [5]. When carried outith monochromatic light at the wavelength �, the mea-

urement of an optical path difference (OPD) suffers frommodulo � ambiguity. To overcome this issue, techniques

uch as fringe counting [6] or phase unwrapping can besed to increase the unambiguous OPD range (UR), buthese solutions rely on the spatial or temporal continuityf the phase. In the applications previously listed, a directonambiguous OPD measurement is made by using theifferential information between several wavelengths �i.First absolute distance measurements with several

avelengths were reported in 1898 when Benoît com-ared the excess fractions of the orders of interference ateveral wavelengths [7]. Then, in order to test large as-heric mirrors, the method of two-wavelength interferom-try (TWI) was proposed [8,9], using two wavelengths �1nd �2 in order to generate a longer synthetic wavelengthdefined as

� =�1�2

�2 − �1with �2 � �1. �1�

Later, the introduction of numeric cameras and com-uters together with phase modulation–demodulation

1084-7529/09/122503-9/$15.00 © 2

echniques considerably improved interferometric mea-urements [10,11]. These methods have been applied toWI, leading to phase-shifting TWI [12], multiple wave-

ength interferometry (MWI) [13], or tunable wavelengthnterferometry [14].

The TWI method was considerably improved when Deroot showed that the UR could be much larger than the

ynthetic wavelength by simply improving the datanalysis [15]. However, although de Groot’s principle forxtending the synthetic wavelength has been shown withclear physical background by Van Brug [16], the algo-

ithms developed are not universal and cannot be appliedo all wavelength couples. Later, de Groot’s methodologyas used by Falaggis [17] who extended the UR in aultiple-wavelength interferometry context by using the

eneralized optimum wavelength selection [18]. Löfdahllso developped a new algorithm for extending the syn-hetic wavelength, but this algorithm is iterative [5].

In this paper, we present a new TWI analytical algo-ithm that enables the measurement of the OPD over anR that can be much larger than the synthetic wave-

ength �. After briefly recalling the basic TWI algorithmn Section 2, we develop a new algorithm in Section 3.hen, in Section 4, performance of this algorithm is inves-

igated in terms of measurement error and maximumeachable UR, and we show its superior behavior com-ared with the basic TWI algorithm. Several applications,uch as metrology or cophasing, are considered in Section

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2504 J. Opt. Soc. Am. A/Vol. 26, No. 12 /December 2009 K. Houairi and F. Cassaing

. In Section 6, we present the experimental validation ofhe new algorithm.

. BASIC ALGORITHM: TWI-1n OPD measured at �i will be characterized by the frac-

ional order of interference mi, defined by [19]

mi = �/�i. �2�

For any given number x, the nearest integer (roundedart) will be denoted x, and the remaining part (wrappedart) will be denoted x, so that the OPD can be written as

� = �mi + mi��i with �mi � Z

mi � �� − 0.5,0.5��. �3�

The use of the round and wrap functions instead of theore commmon floor and fractional part functions, re-

pectively, is adopted in order to center the range of thestimated OPD around �=0.

Because of the �i periodicity of the measured OPD �hen measured at the wavelength �i, it is not possible tonow the rounded order of interference mi without addi-ional information: what we are actually measuring is6 i�i, or m6 i.

The use of OPD measurements at two wavelengths �1nd �2 allows the computation of the difference m of theeasured orders of interference. Because each mi is mea-

ured modulo 1, only the fractional part of m is relevant,ence the basic TWI estimator (TWI-1),

�� = m� with m = wrap �m6 1 − m6 2�. �4�

he UR of this estimator is the synthetic wavelength �efined in Eq. (1).Figure 1 shows the result of a simulation in which are

ig. 1. Orders of interference measured at �1 and �2 are plottedf interference is plotted as a solid line. In this simulation, there

lotted the measured orders of interference m6 1 and m6 2 athe wavelengths �1 and �2, respectively, along with therapped difference of the orders of interference m result-

ng in a �-periodic signal. The computation, with no ad-itional error, is carried out with �1=1.31 �m and �21.55 �m (two common wavelengths for which stabilizedources are available), giving a synthetic wavelength �8.46 �m [Eq. (1)]. Thus, even though the phase mea-urements are unambiguous over a dynamic range of �2t the most, the computation of m enables us to increasehe UR up to �.

However, it is noticeable that when �= ±�, even thoughhe wrapped difference m of the measured orders of inter-erence is zero, the measured orders of interference m1nd m2 are different from zero: this means that evenhough m is � periodic, the states of interference at theeasurement wavelengths �1 and �2 are not the same for

=0 and for �=�. We propose to use this information inrder to retrieve the actual OPD on a range still largerhan �.

Another way to highlight this information is by illus-rating the principle of OPD measurement with TWIhanks to a calliper rule (Fig. 2). The current OPD can beeasured as the distance of the dashed line to the closest

ontinuous marks (at �1 or �2): in this case, only the frac-ional parts mi �i= �1,2�� are used, and the measurements modulo �i. But this measurement does not take into ac-ount the relative position of the continuous marks at �1nd �2. Figure 2 shows that the distance X between thewo marks at �1 and �2, defined as

X��� = m2�2 − m1�1, �5�

hanges each time a tick (related to the wavelength �1 or2) is passed along the OPD axis, enabling a measure-ent of the absolute OPD. It is this information, which is

ted and dashed lines, respectively. The difference m of the ordersadditional measurement error.

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K. Houairi and F. Cassaing Vol. 26, No. 12 /December 2009 /J. Opt. Soc. Am. A 2505

ot used by TWI-1 recalled in Eq. (4), that we use in thelgorithm described in the following.

. NEW ALGORITHM: TWI-2ince the OPD is measured at two different wavelengths,

t is possible to write

OPD = �1�m1 + m1� = �2�m2 + m2�. �6�

onsequently, we have

r�m1 + r�m1 = m2 + m2, �7�

here r� is the ratio of the measurement wavelengths �1nd �2:

r� = �1/�2. �8�

Equation (7) relates two measured quantities m1 and˙ 2 to two unknown integers: the rounded orders of inter-erence m1 and m2. Only one equation is usually not suf-cient to estimate two unknowns without other con-traints. However, one constraint is the integer values ofhe rounded orders of interference. Consequently, we pro-ose to use arithmetic properties in order to retrieve theounded order of interference m1, but it is also possible toetrieve the rounded order of interference m2, as will behown in Subsection 4.D.

Since the measurement wavelengths are never per-ectly calibrated, the ratio of the wavelengths r� is knownith an error �r:

r� = r� + �r. �9�

he error �r can be typically equal to about 10−5, but itan be reduced to 10−9 at least [20].

Let us write the estimated ratio of the wavelengths r�

s a quotient of two coprime natural numbers p and qp ,q�N\ �0,1�� plus an error �f ��f�R� that correspondso a systematic error due to the approximation by a frac-ion of integers:

r� =p

q+ �f with ��f� �

1

2q. �10�

When measured, the wrapped orders of interference m6 ire corrupted by an error � :

λ 1

λ 2

m2

λ2

λ1

m1

0

X(λ2)

ig. 2. Analogy of OPD measurements with TWI and a calliperan be measured (modulo �i) with respect to the closest �i tick. Buelated to the absolute OPD value.

i

m6 i = mi + �i. �11�

he error �i takes into consideration the measurement er-or and the nonlinearity along with systematic errors,uch as any offset, which can arise from longitudinalhromatism, for instance.

Taking into account Eqs. (9)–(11), Eq. (7) can be writtens

q�m6 2 − r�m6 1� = pm1 − qm2 + �, �12�

here the total error is

� = q�m1 − �1���f − �r� + q�2 − p�1 − qm6 1�r. �13�

quation (12) can be further processed if its right-handide can be converted into an integer without error, i.e., if

��� � 1/2. �14�

ndeed, if this condition is satisfied, and by taking the in-eger part of Eq. (12) modulo q to get rid of m2, it is pos-ible to retrieve p�m1 modulo q:

q�m6 2 − r�m6 1� pm1 �mod q�, �15�

here mod is the modulo operator.An important innovative step of our algorithm is to use

ézout’s identity in order to retrieve directly m1 modulo qnstead of p�m1 modulo q. Bézout’s identity states,mong other things, that if p and q are coprime, there ex-sts an integer k such that

k � p 1 �mod q�. �16�

A dedicated algorithm, known as Euclid’s algorithm, al-ows one to find the integer k. Examples will be given inection 4.From Eqs. (15) and (16), it can be written that

k � q�m6 2 − r�m6 1� m1 �mod q�. �17�

owever, the modulo operator is not centered around �0; consequently, we would prefer the final OPD estima-

or TWI-2:

� = �1�m1 + m6 1�, �18�

here the rounded order of interference m1 is computedith

λ 2

λ 1m1

OPD

m2

X(δ )

he current OPD is marked with a bold dashed line, and its valueelative position of the closest �1 and �2 ticks gives an information

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2506 J. Opt. Soc. Am. A/Vol. 26, No. 12 /December 2009 K. Houairi and F. Cassaing

m1 = q � wrapk

q� q�m6 2 − r�m6 1�� . �19�

Equation (19) returns the rounded order of interference¯ 1, assuming the error condition written in Eq. (14).

The dynamic range of Eq. (19) defines a new UR,

�� = q�1, �20�

mposed by the bounds of the wrap function �±1/2� or al-ernatively by the modulo operator in Eq. (17).

In this section, we developed a new algorithm that en-bles the measurement of the OPD over an UR that cane tuned by the choice of q. The only assumption made iselated to all the encountered errors. The following sec-ion details how to deal with all the parameters.

. PERFORMANCE OF TWI-2 ALGORITHMhis section addresses the trade-off between the differentrrors and the UR in order to optimize a system based onWI-2. Then the algorithms TWI-1 and TWI-2 are com-ared.

. New Estimator Accuracyhe error of TWI-2, defined by Eq. (18), can have threerigins. First, error on �1 leads to a multiplicative errorn �. This is a classical issue for any interferometric mea-urement and will not be discussed further here. Second,ince m1 is an integer, the error on this term is null asoon as the condition in Eq. (14) is satisfied: this will benvestigated in the next subsection. Third, the error on6 1, directly resulting from the interferometric measure-

ent at �1, is characterized by an error �1. Therefore, theew estimator has the same error as the monochromaticPD estimator using the wavelength �1, but can main-

ain this capability over an UR that can be much largerhan �1.

. Estimation of the Maximal Unambiguous Rangequation (20) shows that the UR is given by the free pa-ameter q. However, increasing q increases the total erroras shown by Eq. (13), which must satisfy Eq. (14). Thus,

n order to obtain a large UR, the sources of error ��m�, ��r�,nd ��f� must be minimized.This section deals with the maximal reachable UR for a

iven measurement error �m. The goal is to satisfy condi-ion (14).

First, assuming good measurements, �11�m1. Sec-nd, in the worst case, the errors �i of the fractional or-ers of interference measurements mi are added and aressumed to be smaller than an upper bound value �m ofoth measurements. Last, since the rounded order of in-erference has to be retrieved over the UR �� =q�1, we as-ume the worst case where m1=0.5 and m1=q /2 (m1 cannly be retrieved modulo q). Then, using the triangle in-quality, we have

2��� q2��f� + q2��r� + 2�p + q���m� + q��r�. �21�

o obtain this inequality, we bounded from Eq. (13) sev-ral terms, and we made only the approximation � m ;

1 1

hus the inequality written in Eq. (21) is almost exclu-ively a strict inequality.

Then, assuming q�1, Eq. (14) is satisfied with the suf-cient condition

q2��f� + q2��r� + 2�p + q���m� 1. �22�

This inequality shows that a trade-off has to be madeetween p, q, �f, �r, and �m. Indeed, the higher �m and �r,he lower q, �f must be chosen, and vice versa. Thus, in-ependently of the measurement error and the wave-ength calibration, the UR is especially extended whenheir ratio r� can be approximated by a fraction with aigh denominator q and a small error �f [Eq. (10)].Based on empirical observations detailed in Appendix

, we propose to define the couples �p ,q� as

q��f� � 1/2, the worst couples;

q��f� � 1/10, the good couples;

q2��f� � 1/3, the lucky couples. �23�

ince the couple �p ,q� is a free parameter and the maxi-al UR is a priori expected, we consider lucky couples in

he rest of this section. Lucky couples can be found by aystematic search, performed once for all during the sys-em design, as illustrated by Fig. 4 in Appendix A.

Assuming that a lucky couple has been found, Eq. (22)eads to the sufficient condition to correctly estimate theounded order of interference m1:

q2��r� + 2q�1 + r����m� 2/3. �24�

This quadratic equation in q enables one to find theaximal number qmax that verifies it:

qmax ��1 + r����m�

��r�� 1 + s2 − 1�, �25�

ith

s = 2

3

��r�

�1 + r����m�. �26�

his parameter marks the limit between the calibrationnd the measurement errors: the experiment is limited byalibration error when s�1 and by measurement errorhen s�1. Finally, the calibration and measurement er-

ors are well balanced when s�1.Equation (25) can be simplified for each error regime:

qmax ��1

3�1 + r����m�if s 1

0.82

��r�if s � 1

1

��r�0.58 −

0.24

s � if s � 1� . �27�

Equation (27) gives a rough estimation of the maximaleachable integer qmax considering lucky couples, and con-equently Eq. (27) gives a rough estimation of the maxi-al UR reachable with Eq. (20).

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K. Houairi and F. Cassaing Vol. 26, No. 12 /December 2009 /J. Opt. Soc. Am. A 2507

Equation (27) also shows the diffference between thealibration and the measurement error regimes: to in-rease the UR by a factor of 2, a gain of 2 is expected for�m� in the measurement error regime, but a gain of 4 isxpected for ��r� in the calibration error regime.

. Comparison with TWI-1t is straightforward to show that the increase of the URf TWI-2, given by Eq. (20), with respect to the UR ofWI-1 can be expressed from Eqs. (1), (8), and (10) as

��

�=

q�1

�r� − 1�−1�2� q − p. �28�

Therefore, whereas the UR is imposed by the two wave-engths �i with TWI-1, TWI-2 allows one to tune the URxtension, which can be very high, as will be discussed inection 5.Thus, four major benefits of TWI-2 are clearly evident.

irst, as soon as q−p is larger than 1, the UR of the newlgorithm is larger than the UR of TWI-1. Second, accord-ng to Subsection 4.A, the accuracy of TWI-2 is directlyroportional to the measurement wavelength �1, and thusan be much better than the accuracy of TWI-1, which isroportional to the synthetic wavelength. Third, when2�2�1, the synthetic wavelength is smaller than theavelength �2; thus TWI-1 is rather useless, whereasWI-2 can still be used. Fourth, the measurement wave-

engths cannot be chosen too close with TWI-1 becausehe measurement error of the differential phase is ampli-ed by �, which is inversely proportional to �2−�1 [cf. Eq.

1)]. When the measurement wavelengths are close, in or-er to extend the UR with TWI-1, large values of p and qre required for TWI-2 to obtain a rational approximationf the ratio r� with q−p�1 and thus to increase the URompared with the use of the synthetic wavelength [Eq.28)]. Besides, according to Eq. (13), when p and q arearge, very small values of the errors are required in ordero use the algorithm presented in Section 3. Thus, the ex-ension of the UR with TWI-2 is easier when the measure-ent wavelengths are not close.

. Other Estimatorsn Section 3, starting from Eq. (15), the algorithm devel-ped estimates the OPD with Eq. (18) based on theounded order of interference m1 from Eq. (19). But it islso possible to estimate the rounded order of interference

¯ 2 from Eq. (12):

Table 1. Order of Interference Measurement E

p q k ��f�

43 45 22 1.2�10−4

65 68 45 4.5�10−4

193 202 157 1.3�10−5

343 359 157 7.7�10−7

aFor the yellow-orange ��1=604.613 nm� and the red ��2=632.816 nm� transitioatio of the measurement wavelengths is well calibrated: �� � =1.6�10−6.

r

�� = �2�m2 + m6 2�, �29�

here the rounded order of interference m2 is computedith

m2 = p � wrap l

p� q�r�m6 1 − m6 2�� , �30�

here the integer l is such that

l � q 1 �mod p�. �31�

According to Eq. (30), the UR of the �� estimator is

�� =p�2, and since q�1�p�2, we have �

�� ���. Moreover,

q. (29) shows that the estimator �� has the same error2�2 as the monochromatic OPD estimator using theavelength �2. Therefore, if ��2����1�, then � is better than

ˆ�, since �1��2.However, if the instrument is limited by repeatability

rrors rather than bias errors, then the existence of twoifferent estimators allows the derivation of a new OPDstimator, obtained by averaging � and ��, with the sameR �� but with an error reduced by a factor of about 2.

. EXAMPLES OF APPLICATIONe show here two typical applications of TWI-2, with dif-

erent operating conditions.

. Metrologye consider here the case of distance measurement over a

arge range. Two bright and stabilized sources are as-umed in order to minimize the errors �m and �r.

This section deals with the specification of the OPDeasurement error when the OPD UR is imposed. The re-

uirement of the OPD UR leads directly to a lower boundimit of the integer q [Eq. (20)] and thus to several couplesp ,q�. For any given couple �p ,q�, it is possible to retrieven upper bound to the measurement error; we call thisimit the error margin �margin. Assuming a large denomi-ator q, from Eq. (22), we have

�margin =1 − q2���f� + ��r��

2�p + q�. �32�

Table 1 shows with a numerical example different pos-ible rational approximations of the ratio r� as expressedn Eq. (10). Of course, all the possible couples are notllustrated. The chosen measurement wavelengths arehe yellow-orange ��1=604.613 nm� and the red ��2632.816 nm� transitions of Helium–neon, and we as-

nd Error Margin for Different Couples „p ,q…a

f� ��margin� �� �� /�

5 4.2�10−3 27.2 �m 2No margin Bad couple �p ,q�

3 5.1�10−4 122.1 �m 94.9�10−4 217.1 �m 16

e helium–neon, leading to a synthetic wavelength �=13.6 �m. We assume that the

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2508 J. Opt. Soc. Am. A/Vol. 26, No. 12 /December 2009 K. Houairi and F. Cassaing

ume that the wavelengths are known with an accuracyetter than 0.5 pm, which leads to ��r�=1.6�10−6.Table 1 confirms the analysis of appendix A: the frac-

ional error does not necessarily decrease when q in-reases. Indeed, the lucky couple �p ,q�= �343,359� isuch better than the couple �p ,q�= �65,68� for instance.Moreover, even though the specification on the mea-

urement error ��margin� globally decreases when q in-reases, i.e., when the UR increases, the example in Tableshows that with the couple �p ,q�= �65,68�, it is not pos-

ible to extend the UR with the algorithm developed inection 3. Thus, the choice of the rational approximationf the ratio r� is fundamental.

Table 1 shows that if the order of interference measure-ent error can be lower than 4.9�10−4, then it is possible

o increase up to a factor of 16 the UR compared with theynthetic wavelength �=13.6 �m and thus to reach anR of 217 �m.

. Fringe Tracking in Multiple-Aperture Opticse consider here the case of a multiple-aperture instru-ent observing a broadband object. A central issue is the

ophasing of the subaperture array on the central fringeat 0 OPD). In this goal, the light from the object at theutput of the interferometric instrument is split into sev-ral spectral channels from which the phase can beracked and the central fringe identified. The allocation ofhe spectral bands is a trade-off among the object spec-rum, the perturbation amplitude, and the signal-to-noiseatio, which can be much smaller than in Subsection 5.A.

Such an analysis has been performed for Pegase [21], aarwin [22]/TPF [23] pathfinder. For these missions,ased on the coherent combination of beams reflected byormation-flying spacecraft, the performance of theophasing system is critical. The strategy selected for Pe-ase is to use the near-IR part of the spectrum where thetar is the brightest, with only two spectral channels, toinimize detection noise, but with maximum width, in or-

er to maximize the flux in each channel [24]. This leads,or Pegase, to 0.8–1.05 �m and 1.05–1.5 �m, leading towo polychromatic interferograms centered at �10.92 �m and �2�1.22 �m, with similar coherence

engths (at half-maximum) Lc�3.5 �m. Therefore, sincehe synthetic wavelength is ��3.4 �m, fringes outsidehe coherence length can be seen, but the OPD can not beomputed without ambiguity.

In this context, the extension of the central-fringe ac-uisition range outside the short synthetic wavelength iselcome and has motivated the algorithm presented in

his paper.

. EXPERIMENTAL VALIDATIONn order to demonstrate the feasibility of futureormation-flying missions, a laboratory demonstratoralled Persee is under integration at Observatoire dearis-Meudon [25]. Persee uses a similar spectral alloca-

ion than Pegase, but the broadband star has been re-laced by two laser sources: one laser diode at �1830 nm and one superluminescent light emitting diode

SLED) at �2�1320 nm with a 60 nm full width at half-aximum. The goal of the polychromatic SLED is to lo-

ate the central fringe thanks to its coherence lengthc2�30 �m. With this configuration, the coherence

engths for each measurement are much larger than theynthetic wavelength ��2.2 �m.

This setup is thus an intermediate case between thewo presented in Section 5. Because of the rather largepacing between the two wavelengths, TWI-1 is not rel-vant, as � is nearly equal to �2. The goal is thus to ex-end the UR up to typically Lc2, i.e., to have q�36.

As explained in Section 4, the calibration of the ratio r�

s well as the calibration of the measurement wave-engths is necessary. The calibration of Persee’s cophasingystem, based on the knowledge of the behavior of the de-ay lines, led to an estimation of the measurement wave-engths:

�1 = 824 nm ± 3 nm,

�2 = 1332 nm ± 3 nm, �33�

nd an independent estimation of their ratio r�:

r� = 0.6180 with ��r� = 1 � 10−4. �34�

hese error values may seem large for the use of high-esolution metrology but are absolutely proper for the usef cophasing systems.

Moreover, under typical conditions, the detector andhoton noises along with the chromatism of our experi-ent are such that the maximum measurement error of

he order of interference is

�m = 0.005. �35�

Let us assume that a lucky couple can be found. Then,he error parameter expressed in Eq. (26) is s=1.001.hus, Eq. (27) can be used to estimate the maximal reach-ble integer qmax; we obtain qmax=34.While the denominator q�34 and the fractionnal ap-

roximation error �f is not too large, Table 2 gives differ-nt possible rational approximations of the ratio r�

0.6180 and the fractional error �f. According to Eq. (21)e show an upper bound value max����� of the total error �

or each rational approximation.We recall that the total error � must be lower than 1/2

o be sure that the algorithm developed in Section 3 cane used to extend the OPD UR [cf. Eq. (14)]. The largest

Table 2. Fractional Error and Bound Value of theTotal Error for Different Couples „p ,q…a

p q ��f� max��� � �

3 5 1.8�10−2 0.275 8 7�10−3 0.298 13 2.6�10−3 0.3411 18 6.9�10−3 1.313 21 1�10−3 0.4214 23 9.3�10−3 2.618 29 2.7�10−3 1.421 34 3.5�10−4 0.54

ar =0.6180 �m, � =1�10−4 and � =0.005.

� r m

iTt

toq

p=Tcb

7TcTsalmalcf

ct

broWt

d�rw

AATervsctcpabb�romas

r

Frr

K. Houairi and F. Cassaing Vol. 26, No. 12 /December 2009 /J. Opt. Soc. Am. A 2509

nteger q for which this condition is satisfied is q=21.hus, we made the following rational approximation forhe ratio r�:

�1

�2�

13

21. �36�

For this fractional approximation, we found that the in-eger k=13 verifies Eq. (16). Figure 3 shows the estimatedrder of interference m6 1 [Eq. (19)] with the values p=13,=21, and k=13.Figure 3 shows that the algorithm developed in this pa-

er enabled us to reach an UR of �� =17.3 �m �UR21�1�, whereas the UR or TWI-1 is only ��2.2 �m.hus, only thanks to extended data processing and an ac-urate choice of free parameters, the UR was multipliedy a factor of 8.

. CONCLUSIONhis paper has shown that with a very simple signal pro-essing based on arithmetic properties, the conventionalWI-1 algorithm can be considerably improved. Wehowed that the novel algorithm TWI-2 can reach an un-mbiguous range (UR) much larger than synthetic wave-ength and is at least as accurate as the most accurate

easurement at one wavelength. In addition, the TWI-2lgorithm is well adapted when the measurement wave-engths are not too close, and it also analytical and thusan be implemented for real-time absolute order of inter-erence measurement.

The extension of the UR is especially increased byhoosing the measurement wavelengths ��1 ,�2� so thatheir ratio can be approximated with very good accuracy

ig. 3. Experimental results obtained with Persee’s cophasingesent the measured orders of interference m1 and m2, plotted asounded order of interference m1.

y a rational fraction. We show that the calibration of theatio of the wavelengths is of the highest importance inrder to maximize the optical path difference (OPD) UR.e also showed that the choice of the rational approxima-

ion of the wavelength ratio directly affects the OPD UR.The TWI-2 algorithm has been experimentally vali-

ated with two relatively separated spectral bands (�1824 nm and �2�1332 nm) and it has enabled us to

each an UR of 17.3 �m, much larger than the synthetic,hich is only �=2.2 �m.

PPENDIX A. RATIONAL APPROXIMATIONNALYSIS

he ratio of the measurement wavelengths r� has to bexpressed as a rational function in order to estimate theounded order of interference m1 with the algorithm de-eloped in Section 3 [Eq. (19)]. When the wavelengths areuch that r� is not fractional, a large number of �p ,q�ouples can be considered. And some couples are betterhan others. In general, increasing the approximation ac-uracy requires larger q values. For example, can be ap-roximated by 22/7 or 355/113, with respective errors ofbout 10−3 and 2.10−7. This shows that by multiplying qy a factor of about 16, the accuracy has been increasedy a factor of about 5000. A better approximation is 104,348/33,215, with an accuracy of 3�10−10, but this

equires a q 300 times larger, for an increase in accuracyf only 103. As shown by Eq. (22), the relevant figure oferit is q2��f�. We will call a “lucky couple” a fraction with

n exceptionnaly small ��f� with respect to the value of q,o that q2��f� can be bound.

Figure 4 shows the rational approximation error ��f� for= . This plot shows that a few lucky couples (and their

when an OPD range of ±11 �m is applied. Sawtooth lines rep-nd dashed lines, respectively. The steps represent the estimated

systemsolid a

�

rmhvp

ATaPOP

R

1

1

1

1

1

1

1

1

1

1

2

2

2

2

Fc uples).

2510 J. Opt. Soc. Am. A/Vol. 26, No. 12 /December 2009 K. Houairi and F. Cassaing

eplicates) can be found. It also shows that the figure oferit q2��f� is lower than 1/3 for these couples. This value

as been selected by computing similar graphs for variousalues of r�, so that a few lucky couples are alwaysresent.

CKNOWLEDGMENTShe authors appreciate the financial support of CNESnd ONERA. This work also received the support ofHASE, the high-angular-resolution partnership betweenNERA, Observatoire de Paris, CNRS and Universitéaris Diderot.

EFERENCES1. Y. Gursel, “Picometer-accuracy, laser-metrology gauge for

Keck interferometer differential-phase subsystem,” Proc.SPIE 4838, 995–1010 (2003).

2. M. Shao, M. M. Colavita, B. E. Hines, D. H. Staelin, and D.J. Hutter, “The Mark III stellar interferometer,” Astron.Astrophys. 193, 357–371 (1988).

3. P. R. Lawson, “Group-delay tracking in optical stellarinterferometry with the fast Fourier transform,” J. Opt.Soc. Am. A 12, 366–374 (1995).

4. F. Delplancke, F. Derie, F. Paresce, A. Glindemann, F. Lévy,S. Lévêque, and S. Ménardi, “PRIMA for the VLTI—Science,” Astrophys. Space Sci. 286, 99–104 (2003).

5. M. G. Löfdahl and H. Eriksson, “Algorithm for resolving 2 ambiguities in interferometric measurements by use ofmultiple wavelengths,” Opt. Eng. 40, 984–990 (2001).

6. P. Juncar, H. Elandaloussi, M. E. Himbert, J. Pinard, andA. Razet, “A new optical wavelength ratio measurementapparatus: the fringe counting sigmameter,” IEEE Trans.Instrum. Meas. 46, 690–695 (1997).

7. M. R. Benoît, “Application des phénomènes d’interférence àdes déterminations métrologiques,” J. Phys. (France) 3,57–68 (1898).

8. J. C. Wyant, “Testing aspherics using two-wavelengthholography,” Appl. Opt. 10, 2113–2118 (1971).

ig. 4. Illustration of the rational approximation error �f of bouple according to Eq. (23) (the worst, the good, or the lucky co

9. C. Polhemus, “Two-wavelength interferometry,” Appl. Opt.12, 2071–2074 (1973).

0. K. Creath, “Phase measurement interferometrytechniques,” in Progress in Optics, E. Wolf, ed. (North-Holland, 1988), vol. XXVI, chap. V, pp. 349–393.

1. J. Schwider, “Advanced evaluation techniques ininterferometry,” in Progress in Optics, E. Wolf, ed. (North-Holland, 1990), vol. XXVIII, chap. IV, pp. 371–359.

2. Y.-Y. Cheng and J. C. Wyant, “Two-wavelength phaseshifting interferometry,” Appl. Opt. 23, 4539–4543 (1984).

3. R. Dändliker, Y. Salvadé, and E. Zimmermann, “Distancemeasurement by multiple-wavelength interferometry,” J.Opt. 29, 105–114 (1998).

4. A. Wada, M. Kato, and Y. Ishii, “Large step-heightmeasurements using multiple-wavelength holographicinterferometry with tunable laser diodes,” J. Opt. Soc. Am.A 25, 3013–3020 (2008).

5. P. J. de Groot, “Extending the unambiguous range of two-color interferometers,” Appl. Opt. 33, 5948–5953 (1994).

6. H. van Brug and R. G. Klaver, “On the effective wavelengthin two-wavelength interferometry,” Pure Appl. Opt. 7,1465–1471 (1998).

7. K. Falaggis, D. P. Towers, and C. E. Towers, “A hybridtechnique for ultra-high dynamic range interferometry,”Proc. SPIE 7063, 70630X (2008).

8. C. E. Towers, D. P. Towers, and J. D. C. Jones, “Generalizedfrequency selection in multifrequencyinterferometry,” Opt.Lett. 29, 1348–1350 (2004).

9. M. Born and E. Wolf, Principles of Optics (Pergamon,1986).

0. G. Bönsch, “Wavelength ratio of stabilized laser radiationat 3.39 �m and 0.633 �m,” Appl. Opt. 22, 3414–3420(1983).

1. J. M. Le Duigou, M. Ollivier, A. Léger, F. Cassaing, B.Sorrente, B. Fleury, G. Rousset, O. Absil, D. Mourard, Y.Rabbia, L. Escarrat, F. Malbet, D. Rouan, R. Clédassou, M.Delpech, P. Duchon, B. Meyssignac, P.-Y. Guidotti, and N.Gorius, “Pegase: a space-based nulling interferometer,”Proc. SPIE 6265, 62651M (2006).

2. A. Léger, J. M. Mariotti, B. Mennesson, M. Ollivier, J. L.Puget, D. Rouan, and J. Schneider, “The DARWIN project,”Astrophys. Space Sci. 241, 135–146 (1996).

3. C. Beichman, P. Lawson, O. Lay, A. Ahmed, S. Unwin, and

ction p /q for q� �2,3�105�. It also shows the different types of

y a fra

2

2

K. Houairi and F. Cassaing Vol. 26, No. 12 /December 2009 /J. Opt. Soc. Am. A 2511

K. Johnston, “Status of the terrestrial planet finderinterferometer (TPF-I),” Proc. SPIE 6268, 62680S (2006).

4. K. Houairi, F. Cassaing, J.-M. Le Duigou, B. Sorrente, S.Jacquinod, and J.-P. Amans, “PERSEE, the dynamicnulling demonstrator: recent progress on the cophasing

system,” Proc. SPIE 7013, 70131W (2008).

5. F. Cassaing, J.-M. Le Duigou, J.-P. Amans, M. Barillot, T.Buey, F. Henault, K. Houairi, S. Jacquinod, P. Laporte, A.Marcotto, L. Pirson, J.-M. Reess, B. Sorrente, G. Rousset,V. Coudé du Foresto, and M. Ollivier, “Persee: a nullingdemonstrator with real-time correction of external

disturbances,” Proc. SPIE 7013, 70131Z (2008).