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Phase-shifting algorithms for a finite number of harmonics: first-order analysis by solving linear systems Alejandro Téllez-Quiñones,* Daniel Malacara-Doblado, and Jorge García-Márquez Centro de Investigaciones en Óptica, A.C., Loma del Bosque 115, Col. Lomas del Campestre, C.P. 37150 León, Guanajuato, Mexico *Corresponding author:
[email protected] Received October 13, 2011; accepted November 30, 2011; posted December 6, 2011 (Doc. ID 156419); published March 12, 2012 From generalized phase-shifting equations, we propose a simple linear system analysis for algorithms with equally and nonequally spaced phase shifts. The presence of a finite number of harmonic components in the fringes of the intensity patterns is taken into account to obtain algorithms insensitive to these harmonics. The insensitivity to detuning for the fundamental frequency is also considered as part of the description of this study. Linear systems are employed to recover the desired insensitivity properties that can compensate linear phase shift errors. The analysis of the wrapped phase equation is carried out in the Fourier frequency domain. © 2012 Optical Society of America OCIS codes: 120.0120, 120.3180.
1. INTRODUCTION The topic of generalized phase-shifting algorithms (GPSAs), with nonequally spaced phase shifts, is something that has been analyzed in many scientific papers [1–12]. The approach given by Freischlad and Koliopoulos [13] is a practical method to analyze the phase-shifting algorithms (PSAs) when they are equally spaced, that is, with equally spaced phase shifts. The analysis of the PSAs described in [13] is based on the graphical behavior of amplitudes and phases of the frequency sampling functions (FSFs) associated to the wrapped phase equation. The graphical properties of these FSFs determine the insensitivity characteristics to error shifts and harmonics for an efficient design of every PSA. In a recent work [14], the authors explored the possibility of analyzing the GPSAs employing the Freischlad and Koliopoulos approach, and then a simple minimization procedure to optimize the phase-shifting interferometry (PSI) algorithms was proposed. In this paper, the authors will describe a first-order analysis for the equally and some nonequally spaced PSAs, using the previous mathematical expression defined in [12,14]. According to this, the tangent of the phase can be obtained not only from equally spaced shifts, but also with nonequally spaced shifts. As will be described in Section 2, the presence of harmonic components in the intensity patterns can be due to lightdetector nonlinearities. These nonlinearities are presented when the pixel array of sensors in the CCD camera is not well aligned. Although this problem is not significant for new highresolution devices, it is interesting from the theoretical point of view, especially when high-frequency fringes are present in the intensity patterns. The high-frequency fringes can be solved if the presence of harmonics is considered, because these harmonics can also be considered as positiondependent functions. The other error source that typically appears in the interferometric arrays for phase shifting is 1084-7529/12/040431-11$15.00/0
the miscalibration of the piezoelectric transducer (PZT). This error, in the majority of cases, is a linear error known as detuning. When this error is added to the phase shifts, it is interpreted as a slight deviation from the reference frequency in the intensity patterns. The detuning and other higher order error shifts have been well analyzed by many authors [15–19]. There are other error sources, for instance, capturing frame time and vibrations, but we are mainly interested in the first two mentioned sources. The Freischlad and Koliopoulos approach [13] considers that the values of the amplitudes of the FSFs should be zero in certain harmonic frequencies to have an algorithm insensitive to these harmonics. The insensitivity to harmonics is achieved when the orthogonality condition takes place on the phases of the FSFs, at least in the neighborhood of the fundamental frequency (local orthogonality). This condition is translated in an equal slope behavior for the FSF phases in the fundamental frequency. Moreover, if the local orthogonality condition is fulfilled and the amplitudes of the FSFs have zero slope and zero value in some harmonic frequencies, then the algorithm is said to be detuning insensitive to those harmonics. The insensitivity to harmonics has been analyzed in many papers, for example, [6–8,20,21]. In these works, the phase error analysis considers the intensity pattern as a series expansion in terms of cosines whose arguments have an infinite number of harmonic components. However, the PSI algorithms are obtained by considering only the fundamental harmonic, in other words, by considering a perfectly cosinusoidal intensity pattern equation. On the other hand, interesting mathematical expressions with finite number of harmonics are presented in [5,22,23]. Here we are interested in considering a finite number of harmonics and in proposing the equations to obtain the wrapped phase from a more general expression for the intensity patterns. © 2012 Optical Society of America
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Fig. 1.
Téllez-Quiñones et al.
Basic configuration in a Twyman–Green interferometer.
2. LIGHT-DETECTOR NONLINEARITIES As described in [24], the light detector may have an electronic output with a nonlinear relationship with the signal, even though the detectors are normally adjusted to work in its most linear region. Considering an interferometric phase-shifting arrangement such as the Twyman–Green interferometer shown in Fig. 1, if s is the intensity pattern detected by the sensor of a CCD camera, and st is the true intensity pattern obtained from the arrangement, we can write s st ε0 ε1 st ε2 s2t ;
(1)
where st a b cos ϕ − α, a is an average intensity, b is an amplitude function, ϕ is the phase difference between two light beams that interfere, and α is an arbitrary phase shift introduced by the PZT of the phase-shifting arrangement. The values ε0 , ε1 , and ε2 are the error coefficients added to the true intensity st . Equation (1) is an idealized form of the measured intensity s when the array of sensors in the CCD is not well aligned. A hypothetical example of this can be shown in Fig. 2. In this figure we have two small regions of two one-dimensional arrays of sensors; one of them is well aligned and the other is nonaligned. In the well-aligned array, the center positions of the sensors were multiplied by a normalizing scale factor. These positions are x 0, 1, 2, 3. In the nonaligned array, the center positions of the sensors are obtained by changing the values x of the aligned array with the transformation Hx −0.4 0.75x 0.15x2 . We can consider the func-
tion st x cosx, as an example of true one-dimensional intensity, which is achieved when ax 0, bx 1, ϕx x, and α 0. Then, the nonlinear transformation Hx can be considered as the Taylor expansion of another function ^ Hx; that is, ^ 0 H ^ 0 x0 x − x0 1∕2H ^ 0 x0 x − x0 2 : (2) Hx Hx ^ This function can be Hx arccosε0 1 ε1 cosx ε2 cos2 x; with the point x0 π∕2 and the values ε0 0.41022, ε1 0.11372 and ε2 −0.44268. Because the right-hand term of Eq. (2) is the second-order Taylor expan^ it implies that Hx ≈ Hx. ^ sion of the function H, However, the measured intensity must be sx st Hx cosHx, and from this we have ^ sx ≈ cosHx cosx ε0 ε1 cosx ε2 cos2 x st x ε0 ε1 st x ε2 s2 x;
(3)
which is the idealized expression given by Eq. (1). In a real general case, the approximation in Eq. (3) is considered as an equality because working with the model in Eq. (1) is more convenient than analyzing the composition sx st Hx, which is in terms of a theoretically unknown function H. On the other hand, it is possible to rewrite Eq. (1) as s a0 a1 cos ϕ − α a2 cos 2ϕ − α;
(4)
where a0 fa ε0 ε1 a ε2 a2 b2 ∕2g, a1 b ε1 b 2ε2 ab, and a2 1∕2ε2 b2 . In the same manner, if we consider a third-order nonlinearity in the detected intensity; that is, s st ε0 ε1 st ε2 s2t ε3 s3t ;
(5)
then we have s a0 a1 cos ϕ − α a2 cos 2ϕ − α a3 cos 3ϕ − α; Fig. 2. Two one-dimensional arrays of sensors. The positions x of the aligned array are normalized, and the positions of the nonaligned array are obtained from the transformation Hx −0.4 0.75x 0.15x2 .
(6)
where a0 fa ε0 ε1 a ε2 a2 b2 ∕2 ε3 a3 3ab2 ∕ 2g, a1 b ε1 b 2ε2 ab ε3 3a2 b 3∕4b3 , a2 b2 ∕2 ε2 3ε3 a, and a3 1∕4ε3 b3 . From the mathematical point
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of view, in these two cases, the presence of nonlinearities can be considered as the presence of harmonic components in the intensity pattern s. So in a general case, if the errors ε0 , ε1 and the nonlinearities εm of order m ≥ 2 are assumed independent of the phase shift α, we can consider a more general intensity pattern equation M X sk am cos mϕ − αk ; k 1; …; K; (7) m0
where sk is the kth intensity pattern obtained in the kth arbitrary phase shift αk and K ≥ 2M 1 is the minimum number required of intensity patterns captured, as we will see later. Equation (7) can be obtained from two properties that can be inferred from the Chebyshev polynomials [25]. Because the coefficient of the power p of the Chebyshev polynomial T p (with degree p) is a nonzero value, by simple induction, it is possible to prove that every polynomial with degree p, evaluated in cos x, can be rewritten as a linear combination of the first p 1 Chebyshev polynomials T 0 x 1; T 1 x x; T 2 x 2x2 − 1; T 3 x 4x3 − 3x; …; T p x 2xT p−1 x− T p−2 x, all of them evaluated in cos x. Because T p cos x cos px, Eq. (7) is inferred st;k a P by considering m b cos ϕ − αk and sk st;k M m0 εm st;k . The functions am with m ≠ 0 are amplitude functions for the mth harmonic component, and M is the maximum number of harmonic defined for the intensity pattern sk . The a0 function can be considered as the bias term of sk . All the functions am are assumed position dependent, such as the phase ϕ to recover, and the shifts αk are not necessarily equally spaced, as in the case of the generalized phase-shifting interferometry (GPSI) [1–12].
3. GPSAs FOR INTENSITY PATTERNS WITH A FINITE NUMBER OF HARMONICS Taking Eq. (7) as the model to consider, the difference between consecutive intensity patterns is Δsk1 sk1 − sk k
M X
for all p 1; 2; …; M and k 1; 2; …; K − 1. Another 2M sets of unknown scalars, each one with K − 1 elements, can be considered. They are fB1k g; fB2k g; …; fBM fA1k g; k g, 2 M fAk g; …; fAk g with the properties K−1 X
Bpk λpk Δ cos mαk1 0; k
k1 K −1 X
K−1 X
Apk μpk Δ
M X
μpk Δsk1 k
sin mϕλpk Δ
M X
Z
Ω
Ω
1
m p;
~ p ds B
Ω
A ds
∼p
R
(11)
∼p
Ω
B αds∕dαdα
Ω
A αds∕dαdα
R
~ p α sin mαdα 0; B
~ p α cos mαdα 0; A
Ω
(9)
Ω
~ p α cos mαdα B
Z
m1
sin mϕμpk Δ sin mαk1 k ;
Z
Ω
am cos mϕμpk Δ cos mαk1 k
∀ m ≠ p;
∼p
:
(13)
~ p α, A ~ p α, and sα PM am The functions in Eq. (13), B m0 cos mϕ − ∼α are considered dependent of α. Two of them, p ~ p α and A α, are termed as weight factor functions, and B P because ds M m1 mam sin mϕ − αdα, they require to satisfy
Z
sin mαk1 k ;
0
~ p λp Bp and A ~ p μp Ap . Here, we will call Eq. (12) where B k k k k k k the Riemann–Stieltjes expression of the PSI algorithm, because the sums in the numerator and denominator of the quotient are Riemann–Stieltjes sums [26]. Considering the Riemann–Stieltjes integrals, if we have an interval Ω, and a partition of this interval given by α1 < α2 < … < αK , which cannot necessarily be regular (with equally spaced shifts), but with ‖Δα‖ max1≤k≤K−1 fΔαk1 k g sufficiently small (K sufficiently large), then Eq. (12) can be written as tan pϕ R
m1
∀ m 1; 2; …; M;
for all p 1; 2; …; M. For each p, Eqs. (10) and (11) determine two linear systems with 2M equations and K − 1 unknowns, then it is required to have at least K 2M 1 intensity patterns to solve these systems. By solving these two linear systems for each p, we can find the scalar solutions fBpk g and fApk g, and then we have a determined wrapped phase equation given by PK−1 ~ p k1 Bk Δsk tan pϕ Pk1 ; (12) k1 K−1 ~ p k1 Ak Δsk
(8)
am cos mϕλpk Δ cos mαk1 k
k1
m1
λpk Δsk1 k
cos mαk1 k
(10)
m p;
Apk μpk Δ sin mαk1 0; k
k1
R
for k 1; …; K − 1. In Eq. (8), we have Δ cos mαk1 k cos mαk1 − cos mαk and Δ sin mαk1 sin mα − k1 k sin mαk . From now on, we will employ the abbreviated notation Δhk1 Δhαk1 hαk1 − hαk hk1 − hk , to k k refer any consecutive difference where h is considered as a dependent function of the phase shift α. Now, let us define 1 2 M 2M sets of scalars fλ1k g; fλ2k g; …; fλM k g and fμk g; fμk g; …; fμk g, each one with K − 1 elements. These sets have arbitrary numbers different from zero and we will call these numbers the sensitivity factors. So, we have
1
k1 K−1 X
∀ m 1; 2; …; M; 0 ∀ m ≠ p;
Bpk λpk Δ sin mαk1 k
am cos mϕΔ cos mαk1 k
sin mϕΔ sin mαk1 k ;
433
~ p α sin mαdα A
∀ m 1; …; M; γp;
mp
0;
m≠p
;
(14)
∀ m 1; …; M; −γp;
mp
0;
m≠p
;
(15)
γp being a nonzero value that can be assumed dependent of p. So, for K large, Eq. (12) approximates Eq. (13) by taking
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~ p α and A ~p A ~ p α , where α is a number con~p B B k k k k k tained in the interval αk ; αk1 . As an example of functions that satisfy Eqs. (14) andR (15), we can consider Ω −π; π, ~ p α γp cospα∕ cos2 pαdα; B Ω R ~ p α −γp sinpα∕ sin2 pαdα. and A Ω
αj −αK1−j ;
j 1; …; τ;
ατ1 0;
then 2τ−j2 −Δ cos •α2τ−j1 ; Δ cos •αj1 j
Δ sin •α2τ−j2 Δ sin •αj1 j 2τ−j1 ;
4. ORTHOGONALITY FOR ALL FREQUENCIES WITH ZERO SLOPE
(16)
In a similar way to the description in [14], it is easily concluded that the FSFs that represent the numerator and denominator of the algorithm defined by Eq. (16) are two frequency N f and G D f , respectively. They dependent functions G satisfy (17)
−∞
where Sf is the Fourier transform [27] of st m0 am N f and G D f are complex cos mϕ − ωt. The functions G P ~ conjugate of the Fourier transforms of g t K−1 N k1 Bk δt − PK−1 ~ tk1 − δt − tk and gD t k1 Ak δt − tk1 − δt − tk , respectively, where δ is the Dirac delta function. Here the phase shifts are defined by αk ωtk 2πf r tk , where tk can be considered as discrete-time and f r is a reference frequency. The FSFs are
D f G
X
~ k Δ cos •αk1 i A k
X
~ k Δ sin •αk1 ; B k ~ k Δ sin •αk1 ; A k
(18)
where the sums run from k 1; …; K − 1, • f ∕f r and i is the imaginary unity. An interesting particular case of orthogonality for the FSFs at all frequencies is achieved when the two phases of the functions in Eq. (18) are linear and separated by an odd integer factor of π∕2 with zero slope, that is for instance when X N f i ~ k Δ sin •αk1 AmN f expiπ∕2; G B k X k1 ~ Ak Δ cos •αk AmD f expi · 0; GD f
(19)
X
So, assuming
∀ f:
j1
j1
τ X
∼
∼
Bj − B2τ−j1 Δ cos •αj1 j ;
j1
X∼ Ak Δ sin •αk1 k …
τ X
∼
∼
Aj A2τ−j1 Δ sin •αj1 j :
(23)
j1
Thus, from Eq. (23), Eq. (20) is satisfied if ∼
∼
∼
Bj B2τ−j1 BK −j ;
∼
Aj −AK−j ;
j 1; …; τ:
(24)
N f i G
τ X
∼
2Bj Δ sin f α∕f r j1 j ;
j1
D f G
τ X
∼
2Aj Δ cos f α∕f r j1 j :
(25)
j1
Similar results are obtained for an algorithm with an even number of steps, that is K 2τ. When K is even, we can assume αj −α2τ1−j −αK1−j ;
j 1; …; τ;
(26)
and to have FSFs, perfectly orthogonal for all frequencies, it is required that ∼
∼
∼
∼
Bj B2τ−j BK−j ; j 1; …; τ − 1;
∼
Aj −AK−j ; ∼
Bτ 0;
∼
Aτ 0.
(27)
∼
where AmN and AmD represent amplitude functions. Let us verify the necessary conditions to obtain Eq. (19). In the case of an algorithm with an odd number of steps, this is K 2τ 1, it is required that ~ k Δ cos •αk1 0; B k X ~ k Δ sin •αk1 0; A k
τ ∼ τ ∼ X X Bj Δ cos •αj1 B2τ−j1 Δ cos •α2τ−j2 j 2τ−j1
From Eqs. (21) and (24), the FSFs are reduced to
PM
X
∼
R∞ N f df Sf G tan ϕ R−∞ ∞ D f df ; Sf G
~ k Δ cos •αk1 i B k
X∼ Bk Δ cos •αk1 k
PK−1 ~ Bk Δsk1 k : tan ϕ Pk1 k1 K−1 A ~ k1 k Δsk
X
j 1; …; τ: (22)
From Eq. (22) we have
Although with Eq. (12) it is possible to recover the phase ϕ multiplied by its corresponding harmonic factor p, that is pϕ, the main interest in this manuscript is when p 1. So, ~ k and A ~ k to we are going to analyze Eq. (12) referring by B ~ p and A ~ p for p 1. For this case, Eq. (12) is the scalars B k k
N f G
(21)
(20)
The value of Bτ in Eq. (27) can be any number, because Δ cos •ατ1 0; however, without loss of generality it τ can be assumed to be equal to zero.
5. INSENSITIVITY TO DETUNING AND HARMONICS IN THE PRESENCE OF DETUNING The algorithm in Eq. (12), when p 1, is insensitive to the harmonics m 2; …; M, but in many cases it is not insensitive to those harmonics in the presence of detuning. The sensitivp1 ity factors λp1 k , μk , can be changed until the insensitivity to detuning in the fundamental frequency (m 1) is satisfied, but the insensitivity to the harmonics (m > 1) in the presence
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Pτ
of detuning cannot necessarily be achieved at the same time. However, based on the results in the previous section, we can consider certain equations to design an algorithm insensitive to detuning and harmonics in the presence of detuning. Let us consider the case of K 2τ 1, and Eqs. (21) and (24). In order to work with linear systems, the degree of insensitivity analyzed here will be of the first order. From Eq. (25) we have
AmN f
τ X
AmD f
τ X
2Aj Δ
cos f α∕f r j1 j ;
(28)
j1
dAmN ∕df
τ X
∼
2Bj 1∕f r Δα
τ X
∼
2Aj −1∕f r Δα sin f α∕f r j1 j :
(29)
One of the main conditions to be satisfied by any PSA should be the equal amplitude property in f r , that is Pτ ∼ AmN f r AmD f r 1, which is rewritten as j1 Bj Pτ ∼ j1 j1 2Δ sin αj j1 Aj 2Δ cos αj 1, as well as
τ ∼ τ ∼ X X Bj 2Δ sin αj1 Aj 0 j j1
τ ∼ τ ∼ X X Bj 0 Aj 2Δ cos αj1 1; j 1:
The terms
Pτ
j1
j1
∼
Pτ
j1
Aj 0 and
Bj 0 in Eq. (30) are equal to ∼
zero because the products Aj 0 and Bj 0 for j 1; 2; …; τ are equal to zero. These terms were added to the expressions in Eq. (30) to denote two linear equations with 2τ unknowns ∼
∼
Aj and Bj . To have detuning insensitivity, it is required that the amplitudes have a tangential behavior in f r ; that is, dAmN f r ∕df dAmD f r ∕df . This implies
τ ∼ τ ∼ X X Bj 2∕f r Δα cos αj1 Aj 2∕f r Δα sin αj1 j j 0. j1
Aj 0 0;
∼
32
∼
j1
Bj 2∕f r Δα cos 2αj1 j
j1
Bj 2∕f r Δα cos 3αj1 j
∼
Pτ
∼
j1
Aj 0 0;
j1
Aj 0 0;
Pτ
∼
∼ Pτ ∼ Bj 2∕f r Δα cos Mαj1 j1 Aj 0 0; j Pτ ∼ Pτ ∼ j1 j1 Bj 0 j1 Aj −2∕f r Δα sin 2αj 0; .. . Pτ ∼ Pτ ∼ j1 j1 Bj 0 j1 Aj −2∕f r Δα sin Mαj 0.
33
Equations (30)–(33) generate a linear ∼ ∼system of 4M − 1 equations with 2τ K − 1 unknowns (Bj , Aj , j 1; …; τ). To solve this system, it is necessary at least that K − 1 4M − 1, but this implies K 4M, which is impossible. However, to have insensitivity to harmonics in the presence of detuning, in the majority of these harmonics, m 2; …; M − 1, we can avoid the last equality in Eq. (33). Thus, with 4M − 1 − 1 4M − 2 equations, the system can be solved when K 4M − 1 and the corresponding matrix of the system has the inverse. The same 4M − 2 linear equations can be obtained for ∼
(30)
∼
j1 ∼
j1
Pτ
j1
j1
j1
Aj 0 0;
Finally, the insensitivity to harmonics in the presence of detuning is achieved when dAmN mf r ∕df dAmD mf r ∕df 0, m 2; …; M. Then it is required that
.. . Pτ
j1
dAmD ∕df
∼
∼
j1
∼ Pτ ∼ Bj 2Δ sin Mαj1 j1 Aj 0 0; j Pτ ∼ Pτ ∼ j1 j1 Bj 0 j1 Aj 2Δ cos 2αj 0; .. . Pτ ∼ Pτ ∼ j1 j1 Bj 0 j1 Aj 2Δ cos Mαj 0.
Pτ
cos f α∕f r j1 j ;
Bj 2Δ sin 3αj1 j
Pτ
j1
Pτ
whose derivatives with respect to f are
j1
.. . Pτ
∼
∼
Bj 2Δ sin 2αj1 j
Pτ
2Bj Δ sin f α∕f r j1 j ;
j1
∼
j1
435
j1
(31)
The insensitivity to harmonics without the presence of detuning is obtained when AmN mf r AmD mf r 0, m 2; …; M. So, it is sufficient to set
∼
the case K 2τ, where the unknowns would be Bj , Aj , j 1; …τ − 1, having at least K 4M intensity patterns to solve the system. Independently of the values αk that were chosen and regardless the fact that K was odd or even, the existence of the inverse matrix of the system could be questionable. However, the cases when the matrix is singular (noninvertible) are rarely found.
6. DETUNING INSENSITIVE PSAs: DETUNING INSENSITIVITY TO CONSECUTIVE HARMONICS Our main interest is the design of PSAs with equally spaced phase shifts because they are employed in the classical experimental arrangements of PSI. However, the use of Eqs. (30)– (33) is interesting in the GPSAs to appreciate the scope of these equations. Thus, before starting the examples with equally spaced shifts, some GPSAs will be shown. As it was described above, Eqs. (21) or (26) should be assumed to reduce the mathematical expressions for the FSFs, and then our examples of GPSAs will not be too generalized. For the PSAs with equally spaced shifts, this limitation is not an obstacle for practical purposes, because the phase ϕ is always calculated with a piston term. Then, this piston term is interpreted as a translation parameter that can be added to the phase shifts.
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This means that a PSA with shifts α α1 ; α2 ; α3 ; α4 0; 1; 2; 3; 4 (in radians), is equivalent to a second PSA with α −1.5; −0.5; 0.5; 1.5. Instead of ϕ, the recovered phase would be ϕ − 1.5 with this second PSA. Let us propose some examples of GPSAs by considering arbitrary phase shifts that satisfy Eq. (21) or Eq. (26). Our goal is to calculate the weight factors in the Eq. (16). The calculation of these weight factors has been described in [12,14]; however, in this manuscript we will employ the system given by Eqs. (30)–(33) to obtain algorithms with the properties of insensitivity previously analyzed. If it is required to have an algorithm insensitive to the second harmonic with an odd number of steps, then K should have the form 4M − 1 42 − 1 7. Since K 2τ 1 7, then τ 3, so we can suppose for instance that the phase shifts are given by the vector α −4.5; −3.5; −2; 0; 2; 3.5; 4.5. From this assumption, we can obtain the weight factors
is detuning insensitive and also insensitive to the second harmonic. Now let us propose an algorithm insensitive to the second harmonic with K 2τ 4M 8 steps. If the phase shifts α −4.5; −3.7; −3.5; −3; 3; 3.5; 3.7; 4.5 are considered, then we have τ 4 and ∼
B −1.3023; 7.9869; −2.7384; ∼
A −1.9358; 8.5394; −0.3891;
(36)
from Eqs. (30)–(33) for K even. The vectors in Eq. (36) have τ − 1 3 coefficients to define the algorithm. The other coefficients are given by Eq. (27), then the algorithm is tan ϕ −1.3023Δs21 7.9869Δs32 − 2.7384Δs43 0 · Δs54 − 2.7384Δs65 7.9869Δs76 − 1.3023Δs87 ∕−1.9358Δs21
∼
∼
∼
∼
8.5394Δs32 − 0.3891Δs43 0 · Δs54 0.3891Δs65
∼
∼
∼
∼
− 8.5394Δs76 1.9358Δs87 :
B B1 ; B2 ; B3 0.5429; 2.2799; 4.0835; A A1 ; A2 ; A3 1.0589; 1.0589; 0.3040;
(34)
(37)
The amplitudes that represent the algorithm given in Eq. (37) are shown in Fig. 3(b). Another example with K 2τ 1 11 is given by α −10.5; −9.4; −8.4; −7.4; −3.5; 0; 3.5; …; 10.5. An algorithm with these shifts can be insensitive to the harmonics m 2, 3, because K 4M − 1 11 implies M 3. The first τ 5 weight factors of this algorithm are
by solving the system given by Eqs. (30)–(33) (without consid∼ ering the last equality in Eq. (33). In this case the vectors B and ∼ A in Eq. (34) have τ 3 coefficients that correspond to the 2τ 6 unknowns of the system. So, the algorithm is tan ϕ 0.5429Δs21 2.2799Δs32 4.0835Δs43 4.0835Δs54 2.2799Δs65 0.5429Δs76 ∕1.0589Δs21 1.0589Δs32
∼
0.3040Δs43 − 0.3040Δs54 − 1.0589Δs65 − 1.0589Δs76 ;
B 0.2697; 0.7963; 1.3278; 1.6425; 1.6792; ∼
(35)
A 0.8486; 2.9111; 2.7907; 2.4387; 0.1169;
from Eqs. (16), (24) and (34) for τ 3. Figure 3(a) shows the amplitudes of the FSFs of Eq. (35). If the concepts of insensitivity to detuning and harmonics are interpreted for the GPSAs such as the authors described them in [14], then this algorithm
(38)
for the numerator and denominator of Eq. (16), respectively. The corresponding amplitudes of this GPSA are shown in Fig. 3(c).
Amplitude
10 5 0 −5 −10
0
1
2
3
4
5
6
7
8
9
10
6
7
8
9
10
6
7
8
9
10
Normalized frequency
(a)
Amplitude
40 20 0 −20 −40
0
1
2
3
4
5
Normalized frequency
(b) Amplitude
10 5 0 −5 −10
0
1
2
3
4
5
Normalized frequency
(c) Fig. 3. (Color online) Amplitudes of the FSFs for some GPSAs. (a) Algorithm of Eq. (35). (b) Algorithm of Eq. (37). (c) Algorithm with weight factors of Eq. (38).
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detuning, and insensitivity to the harmonic M, in some cases the system gives solutions with additional insensitivity properties for some harmonics m > M. This behavior is observed in the last two examples given by Eqs. (39)–(40). These additional properties depend on the size of the step chosen and the symmetries of the functions sine and cosine that define the coefficients of the linear system. To get insensitivity to the harmonics m 2, 3, algorithms with K 11 or K 12 steps are required. For example α π∕3−5; −4; …; −1; 0; 1; …; 4; 5, whose weight factors are given by
Now let us verify some PSAs with equally spaced shifts. A seven-sample algorithm with shifts α π∕2−3; −2; −1; 0; 1; 2; 3 can be considered. Then the first τ 3 coefficients for the numerator and denominator of this algorithm are ∼
B −0.1518; −0.2054; 0.1429; ∼
A −0.0268; 0.2232; 0.2500:
(39)
This algorithm is insensitive to detuning and insensitive to the second harmonic in the presence of detuning. Its amplitudes are shown in Fig. 4(a), where it can be seen that the algorithm is also insensitive to the fourth and eight harmonics, and insensitive in the presence of detuning to the sixth and tenth harmonics. Another algorithm with eight steps α 2π∕5−3.5; −2.5; −1.5; −0.5; 0.5; 1.5; 2.5; 3.5 can be obtained by solving Eqs. (30)–(33). Its first τ − 1 3 weight factors are
∼
B −0.0573; −0.1812; −0.2180; −0.0532; 0.1781; ∼
A −0.0437; −0.0041; 0.1625; 0.2896; 0.1667:
∼ ∼
(40)
This algorithm is insensitive to detuning and insensitive to the second, third, fifth, seventh, eighth, and tenth harmonics. Its amplitudes are shown in Fig. 4(b). Our algorithms are orthogonal for all frequencies, and then they have a peak-to-valley (P-V) phase error given by 1 PVEf 1 − AmN f ∕AmD f ; 2
(42)
The amplitudes of the algorithm implied by Eq. (42) are shown in Fig. 5(a). In this figure it is shown that the algorithm is insensitive to detuning, insensitive in the presence of detuning to the harmonics 2, 3, 4, 8, 9, 10, and insensitive to the harmonic 6. On the other hand, with an even number of steps, for instance α 5π∕12−5.5; −4.5; …; −0.5; 0.5; …; 4.5; 5.5, we have
B −0.2172; −0.3984; −0.2351; A −0.1317; 0.0683; 0.3236:
437
∼
B −0.0384; −0.1785; −0.3622; −0.3687; −0.1482; ∼
A −0.0099; 0.0003; 0.1297; 0.3426; 0.3453:
(43)
This algorithm is insensitive to detuning, and insensitive in the presence of detuning to the harmonics 2 and 3, as it can be seen in Fig. 5(b). The P-V errors of the algorithms in Eqs. (42)–(43) are shown in Fig. 5(c) for the same reference f r 1. For M 4 we can consider 4M − 1 15 shifts given by α π∕3−7; −6; …; −1; 0; 1; …; 6; 7, and solving Eqs. (30)–(33) we have
(41)
for frequencies close to the reference f r [24]. The P-V errors for the algorithms in Eqs. (39) and (40) are shown in Fig. 4(c) for f r 1. Although the system of Eqs. (30)–(33) [without the last equality in Eq. (33)] allows obtaining algorithms with insensitivity to the first (M − 1) harmonics in the presence of
Amplitude
2 1 0 −1 −2 0
1
2
3
4
5
6
7
8
9
10
6
7
8
9
10
Normalized frequency
(a)
Amplitude
2 1 0 −1 −2 0
1
2
3
4
5
Normalized frequency P−V phase error [rad]
(b) 0.1 case (a) case (b)
0.08 0.06 0.04 0.02 0 0.8
0.85
0.9
0.95
1
1.05
1.1
1.15
1.2
Normalized frequency
(c)
Fig. 4. (Color online) (a) Amplitudes for the algorithm with Eq. (39). (b) Amplitudes for the algorithm with Eq. (40). (c) P-V errors for the algorithms with Eqs. (39)–(40).
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∼
B 0.0955; 0.1178; −0.0010; −0.1117; −0.1457; 0.0323; 0.1066; ∼
A 0.0227; −0.0884; −0.1276; −0.1060; 0.0607; 0.2058; 0.0556:
(44)
From Fig. 6(a), the algorithm given by Eq. (44) is insensitive to detuning, insensitive in the presence of detuning to the harmonics 2, 3, 4, 8, 9, 10, and insensitive to the sixth harmonic. In the case of an algorithm with 4M 16 shifts and α 10π∕31−7.5; −6.5; …; −0.5; 0.5; …; 6.5; 7.5, we have
4M 20 shifts given by α π∕4−9.5; −8.5; …; −0.5; 0.5; …; 8.5; 9.5, then the solution of Eqs. (30)–(33) is ∼
B 0.0713; 0.1727; 0.0952; −0.0056; −0.0869; −0.1540; − 0.1254; 0.0243; 0.0832; ∼
A 0.0393; −0.0247; −0.1146; −0.1208; −0.1185; − 0.0289; 0.1125; 0.2058; 0.0770:
(47)
∼
B −0.0080; −0.0512; −0.1580; −0.3002; −0.3707; −0.2815; − 0.0962; ∼
A −0.0046; −0.0194; −0.0221; 0.0443; 0.1954; 0.3215; 0.2598:
(45)
The amplitudes of this algorithm are shown in Fig. 6(b). The algorithm is insensitive to detuning and insensitive in the presence of detuning to the harmonics 2, 3, 4. For M 5 we can consider K 4M − 1 19 steps given by the shifts α 2π∕9−9; −8; …; −1; 0; 1; …; 8; 9, which implies ∼
This algorithm is insensitive to detuning, insensitive in the presence of detuning to the harmonics 2, 3, 4, 5, 6, 10 and insensitive to the eighth harmonic. Figure 7(b) shows the amplitudes of this algorithm, and Fig. 7(c) shows the corresponding P-V errors. Finally, in the case of M 6, an algorithm with K 4M − 1 23 steps given by α π∕6−11; −10; …; −1; 0; 1; …; 10; 11 can be considered. Its solution is ∼
B −0.0082; −0.0369; −0.0878; −0.1498; −0.2019; −0.2204; − 0.1891; −0.1068; 0.0090; 0.1246; 0.1997; ∼
B 0.0111; 0.0109; −0.0387; −0.1228; −0.1888; −0.1788;
A −0.0167; −0.0386; −0.0479; −0.0281; 0.0274; 0.1107;
− 0.0703; 0.0989; 0.2415;
0.1996; 0.2632; 0.2724; 0.2110; 0.0833:
∼
(48)
A −0.0670; −0.1039; −0.1402; −0.1274; −0.0442; 0.0814; 0 .1798; 0.1774; 0.0441:
(46)
In this case, Eq. (46) has amplitudes such as those shown in Fig. 7(a). It is insensitive to detuning and insensitive in the presence of detuning to the harmonics 2, 3, 4, 5, 6, 7, with insensitivity to the ninth harmonic. If an algorithm has K
Like the previous algorithm, this one is insensitive to detuning. However, it is insensitive in the presence of detuning to the harmonics 2, 3, 4, 5, 6, 7, 8, 9, 10, as can be shown in Fig. 8(a). If K 4M 24, and the shifts are given by α π∕4−11.5; −10.5; …; −0.5; 0.5; …; 10.5; 11.5, when the system proposed is solved, we have
Amplitude
2 1 0 −1 −2
0
1
2
3
4
5
6
7
8
9
10
6
7
8
9
10
Normalized frequency
(a) Amplitude
2 1 0 −1 −2
0
1
2
3
4
5
Normalized frequency P−V phase error [rad]
(b) 0.06 case (a) case (b)
0.04 0.02 0 0.8
0.85
0.9
0.95
1
1.05
1.1
1.15
1.2
Normalized frequency
(c)
Fig. 5. (Color online) (a) Amplitudes for the algorithm with Eq. (42). (b) Amplitudes for the algorithm with Eq. (43). (c) P-V errors for the algorithms with Eqs. (42)–(43).
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439
Amplitude
2 1 0 −1 −2 0
1
2
3
4
5
6
7
8
9
10
6
7
8
9
10
Normalized frequency
(a) Amplitude
2 1 0 −1 −2 0
1
2
3
4
5
Normalized frequency P−V phase error [rad]
(b) 0.08 case (a) case (b)
0.06 0.04 0.02 0 0.8
0.85
0.9
0.95
1
1.05
1.1
1.15
1.2
Normalized frequency
(c)
Fig. 6. (Color online) (a) Amplitudes for the algorithm with Eq. (44). (b) Amplitudes for the algorithm with Eq. (45). (c) P-V errors for the algorithms with Eqs. (44)–(45).
∼
B −0.0361; −0.0034; 0.0611; 0.1669; 0.0977; 0.0081; −0.1131; − 0.1597; −0.0746; 0.0024; 0.0794; ∼
A 0.0495; 0.0873; 0.0852; 0.0241; −0.0687; − 0.1305; −0.1045; 0.0288; 0.0770; 0.1089; 0.0770:
(49)
This algorithm is also detuning insensitive, and insensitive in the presence of detuning only to the harmonics 2, 3, 4, 5, 6, 8, 10. Figure 8(b) shows the amplitudes of this algorithm, and Fig. 8(c) shows its P-V error in comparison with the algorithm of Eq. (48).
Amplitude
2 1 0 −1 −2
0
1
2
3
4
5
6
7
8
9
10
6
7
8
9
10
Normalized frequency
(a) Amplitude
2 1 0 −1 −2
0
1
2
3
4
5
Normalized frequency P−V phase error [rad]
(b) 0.1 case (a) case (b)
0.08 0.06 0.04 0.02 0 0.8
0.85
0.9
0.95
1
1.05
1.1
1.15
1.2
Normalized frequency
(c) Fig. 7. (Color online) (a) Amplitudes for the algorithm with Eq. (46). (b) Amplitudes for the algorithm with Eq. (47). (c) P-V errors for the algorithms with Eqs. (46)–(47).
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Amplitude
1.5 1 0.5 0 −0.5
0
1
2
3
4
5 6 Normalized frequency
7
8
9
10
7
8
9
10
(a) Amplitude
2 1 0 −1 −2
0
1
2
3
4
5 6 Normalized frequency
P−V phase error [rad]
(b) 0.04 case (a) case (b)
0.03 0.02 0.01 0 0.8
0.85
0.9
0.95
1 Normalized frequency
1.05
1.1
1.15
1.2
(c)
Fig. 8. (Color online) (a) Amplitudes for the algorithm with Eq. (48). (b) Amplitudes for the algorithm with Eq. (49). (c) P-V errors for the algorithms with Eqs. (48)–(49).
7. DISCUSSION AND CONCLUSIONS Here we have proposed the use of linear systems to solve the problem of the generation of PSAs with detuning insensitivity and insensitivity to a finite number of consecutive harmonics. The insensitivity to harmonics is achieved even in the presence of detuning in the fundamental frequency. This approach is an easy way to define PSAs that can compensate both possible linear errors added to the equally spaced shifts and the presence of a certain number of harmonics in the intensity patterns when they are assumed position dependent. As was described, if the equation system conformed by Eqs. (10)–(11) is solved, then an algorithm for the tangent of the phase in Eq. (12) can be obtained. This algorithm is insensitive to the harmonics m 2; 3; …; M for p 1. Moreover, with the same system, the tangent of the phase multiplied by the harmonic p ≥ 2 can also be obtained, opening the possibility to confirm the presence of harmonics in the intensity patterns. In this manuscript, we did not give examples of algorithms constructed by the solution of the system with Eqs. (10)–(11). These equations produce insensitivity to harmonics without the presence of detuning, which means amplitudes of the FSFs with nonzero slope in the corresponding harmonic frequencies. To get zero slope in these harmonics, we simplified the expressions of the FSFs as Eq. (25), working with algorithms orthogonal at all frequencies. When the FSFs are not reduced, such as in Eq. (25), their complex N j and jG D j can have corners with zero value in the modules jG harmonic frequencies 1 < m ≤ M. Then it follows that the corresponding differentiable amplitudes AmN and AmD are strictly monotone functions (locally in those harmonic frequencies). Thus these amplitudes cannot have zero slope in those frequencies. However, when the reduction in Eq. (25) is achieved, the complex modules of the FSFs, and consequently their amplitudes, can have zero value and zero slope in the harmonic frequencies from Eqs. (32)–(33).
In a similar manner, the detuning-insensitive condition is not necessarily achieved from the system given by Eqs. (10)– (11). Thus, by considering Eq. (31), the insensitivity to detuning is easily implemented. The second system conformed by Eqs. (30)–(33) behaves better to compensate the detuning and harmonics. With this last system, we have a first-order insensitivity degree of detuning and harmonics. However, from the simulations, it can be seen that some examples would have, at least in the reference frequency (f r 1), a second- or thirdorder insensitivity degree. Then those algorithms would be also insensitive to some nonlinear errors inherent to the phase shifts. To confirm this, it would be enough with the calculation of the second- (or third-) order derivatives of the amplitudes. If the second derivatives coincide in f r , then the algorithm is insensitive to the quadratic phase shift errors, if the second and third derivatives coincide in f r , then the algorithm is insensitive to cubic phase shift errors, etc. For algorithms with a higher order of insensitivity, the desired conditions in the FSFs are nonlinear equations. In such a case, to solve the problem a nonlinear system of equations could be implemented and optimization procedures [28–30] could be employed. A possible solution could be found by the minimization of a quadratic cost functional that reflects the desired conditions of insensitivity, as we proposed in a previous work [14]. However, the design of PSAs that can compensate higher order errors in the shifts is a different topic that has been extensively treated by many authors [5–8,16–19,20]. The approaches in these references can be applied to the simplification of many equations that are presented in an extensive higher order analysis.
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