Wide dynamic range homodyne interferometry method and its application for piezoactuator displacement measurements José Henrique Galeti,1 Paula Lalucci Berton,1 Cláudio Kitano,1 Ricardo Tokio Higuti,1,* Ronny Calixto Carbonari,2 and Emílio Carlos Nelli Silva3 1Universidade Estadual Paulista—UNESP, Department of Electrical Engineering, Avenida Brasil 56, 15385-000 Ilha Solteira, SP, Brazil 2Federal University of ABC, Centro de Engenharia, Modelagem e Ciências Sociais Aplicadas, Rua Santa Adélia, 166, 09210-170 Santo André, SP, Brazil 3Escola Politécnica da Universidade de São Paulo—EPUSP, Department of Mechatronics and Mechanical Systems Engineering, Av. Prof. Mello Moraes, 2231, 05508-030 São Paulo, Brazil *Corresponding author: tokio@dee.feis.unesp.br Received 9 July 2013; revised 3 September 2013; accepted 3 September 2013; posted 4 September 2013 (Doc. ID 193094); published 26 September 2013 Multiactuated piezoelectric flextensional actuators (MAPFAs) is a fast-growing technology in develop- ment, with a wide range of applications in precision mechanics and nanotechnology. In turn, optical inter- ferometry is an adequate technique to measure nano/micro-displacements and to characterize these MAPFAs. In this work, an efficient method for homodyne phase detection, based on a well-known Bessel functions recurrence relation, is developed, providing practical applications with a high dynamic range. Fading and electronic noise are taken into account in the analysis. An important advantage of themethod is that, for each measurement, only a limited number of frequencies in the magnitude spectrum of the photodetected signal are used, without the need to know the phase spectrum. The dynamic range for phase demodulation is from 0.2 to 100π rad (or 10 nm to 16 μm for displacement, using 632.8 nm wave- length). The upper range can be easily expanded by adapting the electronic system to the signal char- acteristics. By using this interferometric technique, a new XY nanopositioner MAPFA prototype is tested in terms of linearity, displacement frequency response, and coupling rate. © 2013 Optical Society of America OCIS codes: (120.3180) Interferometry; (120.0120) Instrumentation, measurement, and metrology; (120.5050) Phase measurement; (120.7280) Vibration analysis. http://dx.doi.org/10.1364/AO.52.006919 1. Introduction The main parameters that characterize actuator per- formance are displacement range, force, bandwidth, size, weight, and power consumption. In general, most actuators exhibit good performance in some of these characteristics but are inadequate in others. Examples of widely used actuators are the piezo- electric, magnetostrictive, microelectromechanical system (MEMS)-based electrostatic, MEMS-based electromagnetic, MEMS-based thermal, etc. [1]. Piezoelectric actuators are ubiquitous tools in a wide variety of applications, particularly those involving micro/nanomanipulation. These actuators have ex- cellent bandwidth and can generate mechanical forces in a compact design with small power con- sumption. However, they provide relatively small 1559-128X/13/286919-12$15.00/0 © 2013 Optical Society of America 1 October 2013 / Vol. 52, No. 28 / APPLIED OPTICS 6919 http://dx.doi.org/10.1364/AO.52.006919 displacement ranges [1] and, consequently, the raw piezoceramic displacements should be amplified in order to operate in multiscales. Conventional manipulator systems, based on pins and guides, present backlash and stiction of the revolute joints, as well as geometric and dimensional errors, preventing their application in micro/ nanomanipulation [2]. Therefore, different mecha- nisms must be developed to overcome this problem. One of the best solutions is to use flexure-based mechanisms, where conventional kinematic parts are replaced by flexure hinges. Novel multiactuated piezoelectric flextensional actuator (MAPFA) fami- lies have been developed using the topology optimi- zation method, which combines an optimization algorithm and the finite element method (FEM) in order to find the optimum topology of the mechanical parts of the system [3,4]. The actuators discussed in this paper are examples of MAPFAs, consisting essentially of multiflexible structures actuated by two or more piezoceramics that should generate different output displacements and forces at differ- ent points of the specific domain and at different directions. These types of mechanisms have a number of ad- vantages related to backlash, zero friction, and neg- ligible hysteresis [5]. Such piezoelectric positioners can be used in atomic force microscopes for position- ing the sample, in scanning tunneling microscopes for positioning the probe [6], lens and mirrors posi- tioners for laser interferometers [7], in cellular manipulation [8], in microsurgery [9], and others. However, despite being essential inMAPFA design (as XY nanopositioners, for example), compliant mechanisms produce an undesired effect, which is the cross talk between X and Y displacements [3,4]. That is, the displacement in a direction results in a spurious displacement in the perpendicular direction. This means that special care should be taken to solve this problem when designing ultrapre- cision nanopositioners. Consequently, measurements of displacement of such structures are of great impor- tance to verify analytical and FEM models, to access the material and device properties, to detect defects, and to evaluate performance. These data can also be used to provide an additional optimization of the device design. In this article, one MAPFA device, developed by using the topology optimization method [3,4], is experimentally analyzed to evaluate some of its char- acteristics such as movement output and coupling displacement. An elementary homodyne Michelson interferometer is used, where one of the beams im- pinges on the actuator vibrating surface. A major advantage of optical interferometry is its capability to measure physical quantities with a very high sensitivity. However, interferometry is also invaria- bly corrupted by large quasi-static phase random drifts due to external environmental disturbances. This phenomenon is well known as signal fad- ing [10,11]. A large number of phase-demodulation techniques have been described in the literature to detect the in- duced phase shift under fading: active homodyne [12], heterodyne [13], phase quadrature [14], and synthetic heterodyne [15], among many others [10,11]. There is general agreement that heterodyne techniques, with picometer resolution, are useful when the displacement exceeds an interference fringe, while homodyne techniques are often em- ployed when the displacement does not exceed the fringe spacing [16]. However, homodyne interferom- eters still have interesting characteristics, such as ease of use, low cost, and small periodic error level, which are much more pronounced in heterodyne interferometers [17]. Moreover, with the availability of inexpensive processing power, some computational signal-processing complexity is often preferable to recover the optical path length from the measured photocurrent, instead of additional optics and elec- tronics that are required by other techniques. Be- cause of this, a new passive homodyne-detection method, based on the analysis of the photodetected signal spectrum, is presented. Essentially the method is used to recover the value of themodulation index (x), which is then related to the desired dis- placement amplitude (ΔL). Well-established methods based on spectrum analysis, such as the J1…J4 [18] and J1…J6 [19], for example, have the advantages of being simple, di- rect, self-consistent, and immune to fading. However, they have reduced dynamic ranges, between 0.2 and 3.8 rad and between 0.035 and 6.1 rad, respectively. A method described by Pernick in 1973 [20] was among the first proposed to detect the value of the modulation index x from the spectral components of the photodetected signal. This method was not spe- cially named, as happened with the J1 max, J1null, J1∕J2, J1∕J3 [21], J1…J4 [18], and J1…J6 [19] meth- ods, for example. In this text, it will be referred to as the Pernick method, in honor of its creator. According to the Pernick method, the photodetec- tor output at three harmonics, selected according to a chosen n value (1; 2; 3;…), are used to calculate the dynamic phase-modulation index. The Pernick method is elegant and mathematically consistent, but, curiously, to the authors’ knowledge, no practical implementation of the method has been published in the literature, as happened, for example, with the J1…J4 method, published 16 years later. Furthermore, in principle, the Pernick method should allow the exact calculation of x using indis- tinctly n � 2 or n � 3, for example, and also should not depend on the current value of x, which might take any value between zero and infinity. That is, any values of x could be measured independently of the selected value of n. However, as noted by the authors (and as will be shown in the next sections), there may be severe limitations when taking into account the highly restrictive effects of electronic noise and signal fading, which were not considered by Pernick. 6920 APPLIED OPTICS / Vol. 52, No. 28 / 1 October 2013 It is not the sole objective of this work just to revisit the method originally proposed by Pernick and test it in experiments. The main contribution of this paper is the proposal of an efficient algorithm with a large dynamic range and computational ro- bustness for displacement measurement. In view of the severe problems intrinsic to the Pernick method, which will be discussed below, an algorithm to properly select the integer value of n is proposed, according to the modulation index value (x) to be measured. Therefore, by taking advantage of a well-known Bessel function recurrence relation and simple com- putational resources, a method for automatic mea- surements of dynamic phase, subject to effects of electronic noise and environmental drifts, is pre- sented. Modulation indexes x ranging from 0.2 rad up to 100π rad can be measured using the proposed method. The method is direct (no transcendental equations), is self-consistent (no self-calibration needed), does not present phase ambiguity problems (due to oscillatory behavior of Bessel functions), is immune to fading, and has a large dynamic range, wide enough for practical measurements of MAPFA displacements. Another advantage of the proposed approach is that only the harmonic magnitudes are necessary, regardless of their phases. Consequently, it does not need algebraic sign correction algorithms, as the modified J1…J4 method [22]. This paper is divided into six sections, including this introduction. In Section 2, the MAPFA used in this work is presented. In Section 3, the original Pernick method is examined, pointing out its limita- tions when electronic noise and fading are consid- ered. The modified Pernick method is proposed in Section 4, and some computational simulations are presented to confirm its efficiency. Experiments about the MAPFA linearity, frequency response, and coupling rate are presented in Section 5. Conclusions are presented in Section 6. 2. Multiactuated Piezoelectric Flextensional Actuators The design of a MAPFA is relatively complex, as it involves several piezoceramics, with a large number of actuating displacements and mainly due to the displacement cross coupling, which is quite critical. The cross coupling is undesirable and imposes a limitation in the device efficiency and control. A comprehensive discussion on the design of these mechanisms and compliant nanopositioners by using the topology optimization method can be found in [3,4,23]. The XY nanopositioner shown in Fig. 1 is a MAPFA with two piezoceramics (PZT-5A, American Piezoceramics) glued with epoxy resin to an alumi- num structure and has been designed to have maxi- mum displacements (Δx and Δy) at its right corner. The displacement in X (Y) direction, for example, is generated by exciting the upper (lower) piezocer- amic, while the displacement in Y (X) direction is the cross-coupled one. The aluminum structure is fabricated by wire electron discharge machining technique, and the two piezoceramics have dimen- sions 20 mm × 6 mm × 1 mm in directions 1, 2, and 3, respectively. The electrodes are deposited on plane 1–2, and the piezoceramic is polarized in direction 3. Reflective tape (3M, Scotchlite 7610, formed by micro glass beads with 60 μm diameter) was adhered at the measurement points of interest in order to reflect a significant portion of the optical beam of the sensor arm of the interferometer. A quantitative measure of the cross talk between X and Y displacements in an XY nanopositioner can be given in terms of percentage by Sxy � Δys∕Δx [23], where Δx is the desired displacement in the X direc- tion (the generated displacement), and Δys is the spurious coupled displacement in the Y direction. Consequently, it is important to decrease Sxy in the XY nanopositioner design step [3,4]. It should be emphasized that the XY nanoposi- tioners designed and manufactured by the authors should operate in a static or quasi-static regime in practical applications, below the first resonance frequencies [3,4,23]. As the first resonance frequency is not a parameter that is taken into account in the present MAPFAs computational design, interferom- etry plays an important role in the measurement of frequency response, as well as linearity, bandwidth, and cross coupling. 3. Pernick Method Laser metrology is the technique of choice for high- accuracy displacement measurements. Two-beam interferometers are often used in practical metrology systems because they can ensure displacement measurement traceability [24]. Among the different available configurations, the Michelson interferom- eter is adequate for many practical applications [25]. By considering a sinusoidal input phase shift described by Δϕ�t� � x sin�ωst� ϕs�, where the fre- quency ωs (rad∕s) is known, x (rad) is the unknown modulation index, and ϕs (rad) is a random initial phase, the photodetected voltage at the output of a homodyne Michelson interferometer can be ex- panded in Fourier series, resulting in [18] Fig. 1. XY nanopositioner prototype. 1 October 2013 / Vol. 52, No. 28 / APPLIED OPTICS 6921 v�t� � A � 1 2 � F � Q 2 J0�x� −P X∞ i�1 J2i−1�x� sin��2i − 1��ωst� ϕs�� �Q X∞ i�1 J2i�x� cos�2i�ωst� ϕs�� �� ; (1) where Ji�x� are Bessel functions of the first kind, and order i � 1; 2; 3;…, Q � cos ϕ0, and P � sin ϕ0 de- pend on the random phase ϕ0. The proportionality constant A depends on the optical source intensity, the photodetector responsivity, and the signal condi- tioning circuit gain. Factor F�0 ≤ F ≤ 1� is related to the interference fringe visibility and ϕ0 is a static phase (in principle). Temperature fluctuations, air turbulence, and mechanical vibrations at the inter- ferometer site can cause large variations in the phase ϕ0. For these reasons, in this work, ϕ0 will be denoted by ϕ0�t�, which corresponds to a random and slow phase component. The odd and even harmonics of v�t� have amplitudes V2i−1�x� � AFPJ2i−1�x� (2a) and V2i�x� � AFQJ2i�x�; (2b) respectively. Usually the information corresponding to the DC term J0�x� is not used because it suffers large influence from electronic and ambient noise [26]. Pernick established the following expression to obtain the modulation index x from the harmonics amplitudes [20]: x2 � 4n�n� 1��n� 2�Jn�1�x� �n� 2�Jn−1�x� � 2�n� 1�Jn�1�x� � nJn�3�x� ; (3) which can be easily obtained from the well-known re- currence relation Jn�1�x� � 2nJn�x�∕x − Jn−1�x�. The substitution of Eqs. (2a) and (2b) in (3) is the essence of the method, which proposes to estimate the modulation index from the harmonic components of v�t�: �x0�2 � 4n�n� 1��n� 2�Vn�1�x� �n� 2�Vn−1�x� � 2�n� 1�Vn�1�x� � nVn�3�x� ; (4) where x0 is the estimated (or measured), and x is the expected values of the modulation indexes. In Eq. (4), the AFP or AFQ factors are canceled, which means that in this technique there is no dependence on random phase ϕ0�t�, fringe visibility, photodetector responsivity, or optical source oscillations. Conse- quently, in principle, the method is direct, self- consistent, and immune to fading. By choosing an odd value of n in Eq. (4), it is necessary to use three even harmonics. For example, for n � 1: V0, V2, and V4; for n � 3: V2, V4, and V6, and so on. By choosing an even value of n, three odd harmonics are used (for n � 2: V1, V3, and V5; for n � 4: V3, V5, and V7, and so on). According to Pernick, the user should look for the best value of n according to the experimental cir- cumstances and suggests that the value n � 1 should be avoided, once it would be difficult to distinguish V0�x� from the DC component �A∕2� in v�t� given in Eq. (1). As discussed below, it is not possible to use a single value of n in Eq. (4) to estimate the value of x in a wide range. Furthermore, the method has a resolu- tion limit, which is the minimum finite modulation index x that can be measured, which depends on the random noise. In order to consider the effect of electronic noise on the photodetected signal, a deter- ministic formulation proposed in [27] will be fol- lowed, which is valid for noise voltages that vary with frequency f according to 1∕f m, where m is a positive integer that depends on the noise type. Using this approach, the noise-free voltage compo- nents in Eqs. (2a) and (2b) are substituted by V2i−1�x� � AF�PJ2i−1�x� �K∕�2i − 1�m� (5a) and V2i�x� � AF�QJ2i�x� �K∕�2i�m�; (5b) where K is a noise factor defined by K � ΔV1∕AF, and ΔV1 is the noise voltage measured at the funda- mental frequency. For the particular value of m � 1 used in [27], the noise is generated by semiconductor junctions in optical and electronic components (laser, photodetector, amplifier, acquisition system, etc.). This kind of noise will be employed in this paper only for comparison with other authors, once the experi- mental measurements show, in the frequency bands explored in this work, that white noise (m � 0) predominates. For example, by substituting Eqs. (5a) and (5b) in (4), using n � 2, m � 1, ϕ0�t� � π∕4 rad, and K � 0.0011, which is the same as those used in [19] [this value of K is conservative because in this work the authors have measured values of the order of 0.0004 for white noise (m � 0)], the curve of x0 as a function of x is presented in Fig. 2(a). The case for n � 3 is shown in Fig. 2(b). The relative percent error, Δxr � 100jx0 − xj∕x, is also shown in the figures. Moduli and phases of Vi�x� were taken into account in the calculation of the results shown in these figures. Ideally, Eq. (4) should lead to the relation x0�x� � x in the absence of noise. However, when x ≪ 1 rad, Vn�x� have very small magnitudes for n > 1, which can be below the electronic noise level in the system. Consequently, there is a lower limit in resolution. Otherwise, even without considering the effect of noise, for x � 6.38 rad the numerator and denomina- tor of Eq. (3) become zero, originating a singularity of 6922 APPLIED OPTICS / Vol. 52, No. 28 / 1 October 2013 the type 0∕0 at this point. This would generate only a single point discontinuity in the curve x0�x�, and its consequences would not be significant. However, as can be seen in Fig. 2(a), this discontinuity also de- pends on noise, and the error increases to intolerable levels in a finite width around x � 6.38 rad. Other 0∕0 singularities exist for n � 2 at 9.76, 13.01, 16.22 rad, and so on. Consequently, the Pernick method for n � 2 has limited dynamic range, be- tween 0.1790 and 5.9476 rad, approximately, accord- ing to the discussion in Section 4. There is a similar behavior for n � 3, which has a dynamic range between 0.4274 and 7.3328 rad. An additional problem related to the Pernick method refers to the algebraic signs of Bessel func- tions, which should be corrected when the magnitude (or modulus) of the spectral components are used in Eqs. (2)–(4), without considering the phase. This was also a problem encountered by Jin et al. [22] for the J1…J4 method, which operated only up to 3.83 rad. At this value of x, J1�x� becomes negative for the first time, and by using only the magnitude information, the dynamic range of this method is limited, between 0.2 (due to noise) and 3.83 rad, approximately [19]. To solve this problem and indefinitely extend the dy- namic range of the J1…J4 method, Jin et al. proposed themodified J1…J4 method [22], in which the knowl- edge of the initial phase ϕs is necessary, in order to correct the spectral components algebraic signs in Eqs. (2a) and (2b) for higher values of x. However, the arbitrary phase ϕs depends on the initial instant of acquisition and sampling of the interferometric signal. In order to take these matters into account in the analysis, a complicated algorithm to correct the algebraic signs was proposed, which can expand the dynamic range up to approximately 5.1 rad only. This is because the authors of the modified J1…J4 method did not take into account the effect of elec- tronic noise. Actually, around x � 5.1 rad, for the J1…J4 method (modified or no), there is a 0∕0 singu- larity [19] that, also considering electronic noise, re- sults in a finite region where the error increases to intolerable values. Besides these two problems in the Pernick method, there is another one, if, at the measurement instant, the random phase assumes a value that is an integer multiple of π∕2 rad. That is, if ϕ0�t� � pπ∕2 rad, p � 0;�1;�2;�3;…, then P � 0 or Q � 0 in Eq. (2), resulting in 0∕0 singularities in Eq. (4) for n even and odd, respectively. This random value of ϕ0�t� should rarely occur and would have no more consequences than generating one or other point out of the linear- ity curve x0�x�. Furthermore, this specific measure- ment could be repeated later and the problematic data discarded. However, the experiments could take longer because the rate of change of ϕ0�t� is slow, and the authors of this work have been observing that these singularity points occur often in open-loop ho- modyne interferometers and in normal laboratory conditions. In order to complete the discussion about the per- formance of the Pernick method, the relative percent error as a function of modulation index x and phase ϕ0 for n � 2, m � 1, K � 0.0011, 0 ≤ x ≤ 15 rad, and 0 ≤ ϕ0 ≤ 2π rad are shown in Fig. 3. This tridimen- sional graph is called a bathtubs graph, for obvious reasons. The value of x0 was obtained from themoduli and phases of the harmonics using Eq. (4). The error values were truncated to 10% in order to make the interpretation of the results easier. The percentage error shown in Fig. 2(a) corresponds to the particular case of this graph when ϕ0�t� � π∕4 rad. The error can exceed 100% in several regions: close to x � 0, close to values of x that cause 0∕0 singularities, and close to the instants when ϕ0�t� � pπ∕2 rad (in this case, for 0, π and 2π rad). The occurrence of many bathtubs in this error graph means there are defi- ciencies in the method. Similar graphs would be obtained for other values of n in Eq. (4). By observing Fig. 2 again, one can conclude that the Pernick method, when applied with a single value of n, results in a limited dynamic range, as happens with the J1…J, modified J1…J4, and Fig. 2. Relation between x0 and x for the Pernick method with (a) n � 2 and (b) n � 3. The percent relative error Δx is also represented. Fig. 3. Bathtubs graph for the Pernick method using n � 2. 1 October 2013 / Vol. 52, No. 28 / APPLIED OPTICS 6923 J1…J methods, for example [19]. However, unlike the other methods, the Pernick method can be adapted to be used for any integer value of n, each one with its own dynamic range. For a determined value of n, there are large errors in the vicinities of x � 0 and near the first 0∕0 singularity, but in between these values the error is extremely small. Consequently, each value of n corresponds to a par- ticular dynamic range. These dynamic ranges are complementary, which give rise to the proposed method, with large global dynamic range. 4. n-Commuted Pernick Method In this work, a generalization of the Pernick method is proposed, which is the denominated n-Commuted Pernick method (n-CPM). The objective is to estab- lish a standard and computationally simple pro- cedure to estimate values of x as high as 100π rad (or higher), by selecting an adequate value of n in Eq. (4), according to the information contained in the photodetected signal v�t� itself. The challenge is to do it when there is fading, which is manifested by the low-frequency random phase ϕ0�t�, and work- ing only with the magnitude of Vn�x�, without consid- ering the phase. In principle, the selection of an adequate value of n can be accomplished by more than one way, but the one that is proposed, because of its simplicity and efficiency, is: Proposition: Given a value of x and considering ϕ0�t� a random signal, then, if k is the order of the higher magnitude harmonic jVkj of the photode- tected signal, a good choice for n in Eq. (4) will be n � k� 1, which means that the harmonics of inter- est will be of order n − 1 � k, n� 1 � k� 2 and n� 3 � k� 4. This will be valid always that k ≥ 3. There are two ambiguity cases for k � 1 and k � 2. When k � 1 and jV5∕V1j < 0.6, then n � 2; other- wise, if jV5∕V1j ≥ 0.6, then n � 4. When k � 2 and jV6∕V2j < 0.6, then n � 3; otherwise, if jV6∕V2j ≥ 0.6, then n � 5. This is a sufficient condition to choose an adequate value of n, but it is not the only one. For example, another possibility is to choose n � k − 1 (with other thresholds for the ambiguity cases when k � 1 or 2), but in this case it is necessary to correct the algebraic sign of the Bessel functions. Additionally, some har- monic components may have very low magnitudes within the range of x associated to n, resulting in problems due to noise. In this work, the dynamic range is defined as in [19,27]; it is lower-limited by the minimum detect- able phase shift (MDPS) and upper-limited when the error exceeds 0.05 rad. The MDPS was defined in order to estimate the resolution of the method in the presence of noise and is the value of x when the absolute error satisfies Δx � jx0 − xj � x in the presence of noise. Following this procedure, the dynamic ranges for each value of n (up to 10) in Eq. (4) are obtained and presented in Table 1. Modu- lus and phase of Vn�x� are taken into account to obtain the dynamic ranges. The different ranges have superpositions that are suitable for establish- ing a robust method in the presence of noise and fad- ing in practical cases. Although this analysis has been made only for a few values of n, it can be easily extended to values as large as n � 100, for example. Before explaining the proposition, it is instructive to revise the behavior of the Bessel functions Ji�x� and the harmonics Vi�x� in Eq. (2) in the absence of noise. The Bessel functions Ji�x� are decaying os- cillatory functions of x and the first maximum of J1�x�, which occurs at x � 1.8 rad, is higher than the first maximum of J2�x� at x � 3.2 rad, which is higher than the first maximum of J3�x� at x � 4.2 rad, and so on. The same happens for the second maxima and so on. However, in Eq. (4), the photode- tected voltage magnitudes Vn are proportional to these Bessel functions multiplied by the random functions P � sin ϕ0�t� (odd harmonics) or Q � cos ϕ0�t� (even harmonics). Then, if at measurement instant t0 the phase is ϕ0�t0� � π∕8 rad, for example, then the first maximum of jV1�x�j is 0.2225AF, which is now smaller than the first maximum of jV2�x�j, which is 0.4482AF. Consequently, the maximum harmonic magnitude jVkj for a determined value of x depends on the values of Jn�x� and ϕ0�t� at the measurement instant. That is, for a given value of x the higher harmonic of v�t� in Eq. (1) can be ran- domly odd or even, depending on the current value of x and ϕ0�t�. Furthermore, as the same factor cos ϕ0�t� multiplies all the even harmonics and sin ϕ0�t� multiplies all the odd harmonics, for a determined value of ϕ0�t�, all even harmonics in the spectrum of v�t� increase (decrease) while all odd harmonics decrease (increase) simultaneously and in the same proportion. As a consequence, it is convenient to analyze the behavior of the envelope of maxima as a function of x for several values of ϕ0�t�. This analysis will be done for several ranges of ϕ0�t� and, for clarity, only for 0 < x < 18 rad in the cases studied here, but can be easily extended for higher values of x. A. Range of ϕ0�t� between π∕2� 0.3156π rad and between 3π∕2� 0.3156π rad As the proposition uses the information relative to the harmonics with maximum magnitude, it is Table 1. Dynamic Range as a Function of na n Lower Limit [rad] Upper Limit [rad] 2 0.1790 5.9476 3 0.4274 7.3328 4 0.7618 8.5824 5 1.1616 9.7818 6 1.6113 10.9534 7 2.0997 12.1064 8 2.6187 13.2461 9 3.1622 14.3752 10 3.7524 15.4956 aMagnitude and phase of the harmonic Vi�x� are taken into account. 6924 APPLIED OPTICS / Vol. 52, No. 28 / 1 October 2013 convenient to know the interval of x where each har- monic has maximum magnitude. This is shown as the envelope of maxima in Fig. 4: the magnitudes of maxima of Eq. (2), normalized by AF, for �0 < x < 12� rad and three values of ϕ0�t�: π∕2, 0.3634π, and 0.3156π rad. For the values of ϕ0�t� considered in this subsec- tion, the following observations can be made: as P � sin ϕ0�t� ∼ 1 (in modulus) and Q � cos ϕ0�t� ∼ 0, the even harmonics are always smaller than the odd har- monics; the length of the range of x corresponding to each harmonic k which has maximum envelope, re- mains constant and is independent of ϕ0�t�; as the value of x increases from zero, there is a region where jV1�x�j is maximum for 0 ≤ x ≤ 3.0558 rad; then jV3�x�j is maximum from 3.0558 to 5.1363 rad; then jV1�x�j is maximum again, from 5.1363 to 5.6164; then jV5�x�j for 5.6164 ≤ x ≤ 7.5019 rad. The authors have observed that for higher values of x after this point, the sequence of maxima are related to func- tions jV7�x�j, jV9�x�j, jV11�x�j, etc. This was verified up to x � 100π rad, but is possibly valid until infinity. Consequently, there is only one ambiguity case, re- lated to function jV1�x�j (for k � 1), which is maxi- mum at two disjoint intervals. This ambiguity case is provided in the proposition and will be further discussed in Subsection 4.D. In order to analyze other intervals for ϕ0�t�, a func- tion that indicates which harmonic envelope of order k is larger for each pair of �x;ϕ0� was generated for 0 ≤ x ≤ 10 rad (without loss of generality) and 0 ≤ ϕ0�t� ≤ π∕2 rad, as shown in Fig. 5. Actually, this is obtained from the top view of a three-dimensional surface of jVkj versus x and ϕ0�t�, and the values of k where jVkj are maximum and are shown in gray col- ors. Due to the symmetry of the problem, only the first quadrant is considered. The analyzed interval in this subsection corre- sponds to the portion between ϕ0�t� � 0.3156π and π∕2 in Fig. 5. It is now clearer from the figure that in this interval of ϕ0�t�, the envelope of maxima are related to odd-order harmonics only, and the cor- responding ranges of x in which each harmonic is larger do not change with ϕ0�t� for each value of k. Now it is useful to compare the intervals where these functions are maximum with the intervals re- lated to the dynamic ranges for each value of n (or k). By using the information from Fig. 5 and Table 1, Fig. 6(a) is obtained, where the regions in white represent the x intervals corresponding to the maxi- mum magnitude harmonic of order k (odd), in the range of ϕ0�t� considered in this subsection. The regions in gray (which contain the white strips) re- present the dynamic ranges for each value of n, as in Table 1 [considering modulus and phase of Fig. 4. Envelope of maxima jVk�x�j∕AF and three values of ϕ0�t�: 0.3156π, 0.3634π, and π∕2 rad. The numbers above the peaks indicate the order of the harmonic with a larger magnitude in that interval. Fig. 5. Function that indicates which harmonic envelope of order k is larger for each pair of �x;ϕ0�. Fig. 6. Information relative to dynamic range (gray strips), envelope of maxima (white strips), and large error (black strips) for each value of k (odd). (a) ϕ0�t� between π∕2� 0.3156π rad and between 3π∕2� 0.3156π rad. (b) ϕ0�t� � 0.2641π rad. Ambiguity case k � 1 is shown in yellow strip. 1 October 2013 / Vol. 52, No. 28 / APPLIED OPTICS 6925 Vn�x�]. The regions in black correspond to large er- rors, due to noise. As can be seen, the white strips are accommodated loosely inside the low-error gray strips and far away from the large-error black strips. A detailed study of the Bessel functions leads to the conclusion that, in the white strips intervals in Fig. 6(a), the functions Jn−1�x�, Jn�1�x�, and Jn�3�x� used in Eq. (3) have the same algebraic sign. This means that it is not necessary to use any sign correc- tion algorithm, as occurs with the modified J1…J4 method. This is valid for any value of x larger than the MDPS, which was verified in this work up to x � 100π rad. Then the choice of n � k� 1 is ad- equate and will result in low-estimation errors for x. This will not be the case only in the lower limit of the dynamic range of x for k � 1 (or n � 2), which is adjacent to a black strip, resulting in a limited res- olution given by the MDPS. The yellow strip for n � 4 in Fig. 6(a), which is adjacent to the white strip for k � 3, corresponds to the ambiguity case (see Subsection 4.D). As only the odd harmonics are used in Eq. (4), in this ϕ0�t� interval, there is only the term P � sin ϕ0�t�; there is no term Q � cos ϕ0�t� and consequently no problems when ϕ0�t� � pπ∕2 rad, p � �1;�3;…. Obviously, the values of ϕ0�t� for p � 0;�2;�4… do not cause any problems because they are out of the range considered in this subsection. B. Range of ϕ0�t� between 0� 0.1974π rad and between π � 0.1974π rad In this range, which can be identified (first quadrant) in Fig. 5, the even harmonics are higher than the odd ones because the modulus of factor Q � cos ϕ0�t� is close to one, except in the interval x ≤ 2.205 rad, where the first harmonic (k � 1) may be larger. For a fixed value of ϕ0�t� and increasing values of x, the harmonic with order k � 1 is larger, then k � 2, then k � 4, then again k � 2 (another ambiguity case, for 6.3816 < x < 6.6417 rad, which will be treated later), then k � 6, and so on. Furthermore, except for the ranges for k � 1 and k � 2 (the first one), all the other ranges have fixed width in x, independently of ϕ0�t�. Therefore, by increasing ϕ0�t�, the values of x when jV1j is maximum stretch within the range 0 ≤ xV1 ≤ 2.205 rad. Moreover, the values of x where jV2j is maximum for the first time span within the �xV1 ; 4.2011�rad range. When ϕ0�t� � 0 rad, the V1 component vanishes [as P � sin ϕ0�t� becomes null], existing as only maximum jVkj for even values of k. As represented in Fig. 7(a), the ranges of x where the envelope of a determined harmonic k is larger are represented by white strips. In the particular case for k � 2 (n � 3) the maximum possible range (up to x � 4.2011 rad) was used, and the case k � 1 will be discussed below. As before, the dynamic ranges ob- tained from Table 1 are represented by gray strips, and the regions with intolerable errors are repre- sented by black strips. As can be seen, each white strip accommodates loosely in its corresponding dy- namic range, and then there will be a negligible error in the estimation of x if that value of k (or n) is used. When the first harmonic is higher, a new white strip for k � 1 in Fig. 6(a) could be considered, for x between the MDPS (x � 0.1790 rad) and 2.205 rad, and thus it is a subset of the first one. This range could also be accommodated in the dynamic range for n � 2 (k � 1) in Table 1, so it was not plot- ted in Fig. 7(a). In the white strip ranges, the Bessel functions Jn−1�x�, Jn�1�x�, and Jn�3�x�, used in Eq. (3), have the same algebraic sign, and no sign correction algo- rithm is necessary. Consequently, in this range of ϕ0�t�, only the harmonic with k � 1 or with even or- ders can be maximum. By first considering the case of even k, only the factor Q � cos ϕ0�t� is used in Eq. (4), and, as the factor P � sin ϕ0�t� is not used, there will be no singularity problems when ϕ0�t� � pπ rad, p � �1;�3;…. When k � 1, the sit- uation fits perfectly to the case of Subsection 4.A, being valid all considerations made there, regarding singularities and change of algebraic sign. Conse- quently, the n � k� 1 choice is again adequate. The ambiguity case when k � 2 [yellow strip adja- cent to the white strip for k � 4 in Fig. 7(a)] will be discussed in Subsection 4.D. C. Remaining Values of ϕ0�t� In this subsection, the remaining intervals of ϕ0�t� will be analyzed, and it will be shown that the case considered here is even less restrictive than the ones described in Subsections 4.A and 4.B. As observed in Fig. 5, for ϕ0�t� � π∕2 rad, for exam- ple, the odd harmonics are maxima. When the value of the modulation index is increased, the first harmonic jV1j becomes maximum for the first time for x between 0 and 3.0558 rad, and then a second time for x between 5.1363 and 5.6164 rad. Consider- ing now ϕ0�t� � 0.2641π rad in Fig. 5, the region where jV1j becomes maximum for the second time Fig. 7. Information relative to dynamic range (gray strips), envelope of maxima (white strips), and large error (black strips) for each value of k (even). (a) ϕ0�t� between 0� 0.1974π rad and between π � 0.1974π rad. (b) ϕ0�t� � 0.2641π rad. Ambiguity case k � 2 is shown in yellow strip. 6926 APPLIED OPTICS / Vol. 52, No. 28 / 1 October 2013 disappears because it is surpassed by jV4j. It is also evident that, unlike the previous cases examined in Subsections 4.A and 4.B, both even and odd harmon- ics can now assume maximum values. An interesting fact to be noticed is that, within this remaining range of ϕ0�t�, if the largest harmonic is odd (for a given x), the extent of its strip is now less than or equal to that of Fig. 6(a). In Fig. 6(b), the white strips correspond to the new range of maximum for odd k and the par- ticular case where ϕ0�t� � 0.2641π rad. Now each new white strip is a subset of the corresponding strip that was shown in Fig. 6(a), and therefore all their properties are the same: it accommodates itself loosely inside each dynamic range associated with the index n (established in Table 1), presents no prob- lemswith singularities when ϕ0�t� � π∕2 or π rad, and requires no algorithms for algebraic sign corrections. In summary, inside this complementary range of ϕ0�t�, if k is associated with a maximum magnitude, the choice n � k� 1 is again adequate. Moreover, this case is less restrictive than the one analyzed in Subsection 4.A because each white strip in Fig. 6(b) accommodates inside the corresponding white strip in Fig. 6(a). The same can be concluded when the maximum corresponds to harmonic jVkj for even k. If ϕ0�t� � 0 rad in Fig. 5, the even harmonics are maxima, and jV2j becomes maximum for the second time in the range of x between 6.3816 and 6.6417 rad. Using again the case where ϕ0�t� � 0.2641π rad as an example, Fig. 5 reveals that the region where the second harmonic (k � 2) becomes maximum for the second time disappears, being exceeded by com- ponent jV5j. Furthermore, if the highest harmonic for a given x is even, the extent of its strip is also less than or equal to those in Fig. 7(a), as shown in Fig. 7(b) for ϕ0�t� � 0.2641π rad. Again, as each new white strip in Fig. 7(b) is a subset of the corre- sponding strips that were shown in Fig. 7(a), they re- tain all their properties (is accommodated loosely inside each dynamic range, presents no problems with singularities when ϕ0�t� � π∕2 or π rad, and re- quires no algorithms for algebraic sign corrections). D. Ambiguity Cases for k � 1 and k � 2 The ambiguity cases occur when, as x increases, the harmonic with k � 1 or k � 2 becomes maximum for the second time. This constitutes a problem just be- cause it is more adequate to work only with harmonic magnitudes, without considering the phases. For k � 1 and referring to the interval of ϕ0�t� analyzed in Subsection 4.A, this range comprises the interval 5.1363 < x < 5.6164 rad. A detailed analysis of the magnitudes given by Eq. (2a) leads to the conclusion that if the harmonic with maximum envelope has order k � 1, and if the ratio jV5∕V1j < 0.6, for the first time, then x will be between 0 and 3.5057 rad, which comprises the range where jV1�x�j is maxi- mum for the first time. Consequently, the choice n � k� 1 can still be applied, and the value n � 2 should be used in Eq. (4). On the other hand, if jV5∕V1j ≥ 0.6, for the first time, then x will be between 3.5057 and 8.1493 rad, which comprises the range where jVkj, for k � 1, is maximum for the second time. Again, the value n � 2 could be used, and a second white strip could be accommo- dated within the gray strip (region of small errors) for n � 2 in Fig. 6(a). However, it would be necessary to apply a sign correction algorithm for V1 because it becomes negative in this gray strip region for n � 2. On the other hand, if this second strip (now in yellow) is accommodated within the gray strip for n � 4, there is no need to sign correction. Consequently, the value n � 4 should be used in Eq. (4). The ambiguity case for k � 2 is treated in a similar way: if, in the range of ϕ0�t� considered in Subsection 4.B, the harmonic with maximum magni- tude has order k � 2, and if jV6∕V2j < 0.6 for the first time, then x will be between 0 and 4.6615 rad. Then the practical rule n � k� 1 is followed and n � 3 is used in Eq. (4). Otherwise, if jV6∕V2j ≥ 0.6 for the first time, corresponding to 4.6615 < x < 9.3745 rad (which comprises the range where jV2j is maximum for the second time), then the choice n � 5 is sug- gested in Eq. (4), related to the yellow strip in Fig. 7(a), where it is not necessary to use any algo- rithm of algebraic sign correction. In both ambiguity cases, there will be no problems with singularities of the type ϕ0�t� � pπ∕2 rad, p � �1;�2;… According to Fig. 5, for the range of ϕ0�t� discussed in Subsection 4.C, the regions for k � 1 and k � 2 for the second time become subsets of those shown in Figs. 6(a) and 7(a) (and hence are treated the same way) or disappear (consequently, there is no need to concern). E. Simulations Using the n-CPM By considering 0 ≤ x ≤ 15 rad, 0 ≤ ϕ0�t� ≤ 2π rad, m � 1, and K � 0.0011, the relative percent error Δxr � 100jx0 − xj∕x is numerically calculated in Matlab using the n-CPM. The result is shown in Fig. 8, where the error was truncated to 10% for x close to zero. As can be observed, the bathtubs present in the original method disappear, showing the efficiency of the proposed method. Fig. 8. Bathtub graph for the n-CPM. 1 October 2013 / Vol. 52, No. 28 / APPLIED OPTICS 6927 In the following, Eq. (4) was simulated by consid- ering −46 dBV additive white Gaussian noise and a value of ϕ0�t� varying randomly between 0 and π∕2 rad. This kind of noise (for m � 0) fits more to the real case found in practical experiments in the present work. The result is shown in Fig. 9, in terms of the estimated value of the modulation index x0 as a function of x (in circles). Some values of n used at each point are represented in the figure, illustrating how the algorithm based on the proposition selects the best value of n for the estimation of x. The straight line corresponds to the curve x0�x� � x. The value of ϕ0�t� used in the simulation is also plot- ted (triangles, right vertical axis). When ϕ0�t� ∼ 0, the even harmonics have larger magnitudes, and an odd value of n is selected. When ϕ0�t� ∼ π∕2 rad, the odd harmonics have larger magnitudes, and an even value of n is selected. The error between x0 and x is very small (less than 0.07%). 5. Experimental Results A He–Ne laser (Newport, 15 mW, λ � 632.8 nm) is used as an optical source in an open-loop homodyne Michelson interferometer. The laser is divided by a 50∕50 beam splitter in the sensor and reference arms of the interferometer. The sensor arm impinges on a slender mirror (reflective tape), which is glued to the XY nanopositioner shown in Fig. 1, while the refer- ence arm impinges on a fixed mirror. Precision micro- metric stages for translation, rotation, and tilt are used to interferometer alignment, so that the rays re- flected from the mirrors return to the beam splitter and are superimposed at the active area of a photo- diode (ThorLabs, PDA 55 Amplified Si Photodetector, 10 MHz bandwidth, 400–1100 nm range). In the ex- periments, the interferometric signal v�t� is digitized by an oscilloscope (Tektronix TDS 2022), and each acquisition window has 2500 samples. The data is transferred to a computer and processed in Matlab. The signals are Hamming-windowed and zero- padded before spectral computation. The sampling rate is adjusted, so that there are at least 10 signal periods per acquisition window in order to result in a good spectral resolution. On the other hand, the ac- quisition window should not be too long because, if ϕ0�t� has a significant drift in this interval, v�t� is not periodic, and the series expansion in Eq. (1) cannot be applied. The XY nanopositioner was driven by a sinusoidal voltage by using a function generator (Agilent 22330A) and a power amplifier (A. A. Lab Systems, A-301HS Piezo Amplifier), reaching voltages up to 350 V (peak-to-peak) for frequencies smaller than 1 kHz. Laboratory measurements showed that white noise was predominant, with −61.5 dBV, approxi- mately. The modulation index x was then calculated using the n-CPM, and the maximum output displace- ment is obtained from ΔL � xλ∕4π, where λ is the laser wavelength. The XY nanopositioner is an MAPFA with high mechanical flexibility for generated and coupled movements. Due to the flexibility, large dynamic displacements can be generated in the structure, even using low-input voltages. The generated move- ment displacements can be observed in Fig. 10 for sinusoidal input voltages from 0 to 160 V (peak) at the frequencies of 370 and 930 Hz. Each point in the curve (and in the other results) corresponds to the mean obtained from 10measurements. There is a lin- ear response starting from 0.2 rad, and the MAPFA linear length-to-voltage sensitivities resulted in 13.4 and 20.4 nm∕V (peak), at the frequencies of 370 and 930 Hz, respectively. The cross-movement linearity response is shown in Fig. 11 for voltages between 0 and 52 V (peak) at the frequencies of 290 and 910 Hz. The coupled movement also shows a linear response, with linear length-to-voltage sensitivities of 2.95 and 11.1 nm∕V Fig. 9. Simulation of the n-CPM showing the selection of the best values of n for some points. Fig. 10. XY nanopositioner generated movement linearity curves at the frequencies of 370 and 930 Hz. 6928 APPLIED OPTICS / Vol. 52, No. 28 / 1 October 2013 (peak) at the frequencies of 290 and 910 Hz, respec- tively. The coupled movements measured in this prototype have significant amplitudes that are larger than desired, which could demand a redesign of the device. The frequency response shows the MAPFA mechanical resonance frequencies as well as the quasi-static range of operation. The MAPFA gener- ated and coupled movements frequency response, in terms of the displacement-to-applied voltage ratio, is shown in Fig. 12, for frequencies between 90 and 1270 Hz. Each point in the frequency response curve is the slope obtained from a linearity curve. There are resonances around 230, 390, 990, 1190, and 1230 Hz. As observed in Fig. 12, at these frequencies the coupled displacement have similar magnitudes as the generated movements, which is a nonintuitive result. From the data of generated and coupled move- ments (Fig. 12), the coupling rateSxy can bemeasured as a function of frequency. The result is shown in Fig. 13. As a general result, this XY nanopositioner should operate below 200 Hz approximately, far away from mechanical resonances. Furthermore, in this frequency band the coupled displacements are around 25%, which is larger than desired. The differ- ence between the computational simulations and prototype may be related to manufacturing issues such as the bonding of the piezoceramics to the metal structure [23]. A thick bonding layermay compromise the displacement transmission from the piezoceramic to the metallic structure. This will require an im- provement of the manufacturing process. 6. Conclusions In this article, some classical methods based on homodyne interferometry and the photodetected sig- nal spectrum were mentioned, but that have prob- lems related to resolution, reduced dynamic range, and singularities close to the condition ϕ0�t� � pπ∕2 rad, p � 1; 2; 3;…, where ϕ0�t� is a random sig- nal. In order to overcome most of these limitations, this work proposed to use the recurrence relation in Eq. (3) and create a new phase-demodulation tech- nique by adding to it an algorithm used to choose an adequate value of n in Eq. (4). The proposed n-CPM is homodyne, passive, direct, self-consistent, without phase ambiguity problems, immune to fading, does not demand algebraic signal correction and has a large dynamic range. As limitations, the n-CPM has relatively poor res- olution, measuring modulation indexes larger than 0.2 rad (or 10 nm for the laser used), operates only with low-distortion sinusoidal signals, and does not allow the characterization of nonlinear devices. Actually, in a nonlinear regime, there is no direct relationship between the applied voltage and output displacement waveforms and neither with the Fig. 11. XY nanopositioner coupled movement linearity curves at the frequencies of 290 and 910 Hz. Fig. 12. Generated and coupled XY nanopositioner displacement frequency response, in terms of modulation index- and displacement-to-applied voltage ratio. Fig. 13. Coupling rate Sxy as a function of frequency. 1 October 2013 / Vol. 52, No. 28 / APPLIED OPTICS 6929 waveform of the interferometer-induced optical phase shift. Consequently, the periodic signal re- quirements for the Fourier series expansion in Eq. (1) is no longer satisfied. However, this character- istic is intrinsic to all conventional methods based on the signal spectrum analysis discussed in the text. Simulations considering 1∕f voltage additive noise revealed that the n-CPM presents a maximum per- cent error of only 1% (3%) in the 0.6 rad (0.36 rad) to 100π rad modulation index range. In this dynamic range, the method eliminates all imperfections in the bathtubs graphs of the original Pernick method, which occur due to 0∕0 and ϕ0�t� � pπ∕2 rad singularities. Large displacements can be measured with the n-CPM, as those generated by the XY nanopositioner shown in Fig. 1, whose flexibility is very high. 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