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A Tellez Quinones PSAFAFNOHFOABSLS 2012

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A Tellez Quinones PSAFAFNOHFOABSLS 2012
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  Phase-shifting algorithms for a finite number of harmonics: first-order analysis bysolving 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: alejandroteq@cio.mx  Received October 13, 2011; accepted November 30, 2011;posted December 6, 2011 (Doc. ID 156419); published March 12, 2012Fromgeneralizedphase-shiftingequations,weproposeasimplelinearsystemanalysisforalgorithmswithequallyand nonequally spaced phase shifts. The presence of a finite numberof harmonic components in the fringes of theintensity patterns is taken into account to obtain algorithms insensitive to these harmonics. The insensitivity todetuning for the fundamental frequency is also considered as part of the description of this study. Linear systemsare employed to recover the desired insensitivity properties that can compensate linear phase shift errors. TheanalysisofthewrappedphaseequationiscarriedoutintheFourierfrequencydomain. ©2012OpticalSocietyof 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 hasbeen analyzed in many scientific papers [1 – 12]. The approachgiven by Freischlad and Koliopoulos [13] is a practical methodto analyze the phase-shifting algorithms (PSAs) when they areequally spaced, that is, with equally spaced phase shifts. Theanalysis of the PSAs describedin [13] is based onthe graphicalbehavior of amplitudes and phases of the frequency samplingfunctions (FSFs) associated to the wrapped phase equation.The graphical properties of these FSFs determine the insen-sitivity characteristics to error shifts and harmonics for an ef-ficient design of every PSA. In a recent work [14], the authorsexplored the possibility of analyzing the GPSAs employing theFreischlad and Koliopoulos approach, and then a simple mini-mization procedure to optimize the phase-shifting interfero-metry (PSI) algorithms was proposed. In this paper, theauthors will describe a first-order analysis for the equallyand some nonequally spaced PSAs, using the previous math-ematical expression defined in [12,14]. According to this, the tangent of the phase can be obtained not only from equallyspaced shifts, but also with nonequally spaced shifts. As will be described in Section 2, the presence of harmoniccomponents in the intensity patterns can be due to light-detector nonlinearities. These nonlinearities are presentedwhen the pixel array of sensors in the CCD camera is not wellaligned. Although this problem is not significant for new high-resolution devices, it is interesting from the theoretical pointof view, especially when high-frequency fringes are presentin the intensity patterns. The high-frequency fringes canbe solved if the presence of harmonics is considered, be-cause these harmonics can also be considered as position-dependent functions. The other error source that typicallyappears in the interferometric arrays for phase shifting isthe miscalibration of the piezoelectric transducer (PZT). Thiserror, in the majority of cases, is a linear error known asdetuning. When this error is added to the phase shifts, it isinterpreted as a slight deviation from the reference frequencyin 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 frametime and vibrations, but we are mainly interested in the firsttwo mentioned sources.The Freischlad and Koliopoulos approach [13] considersthat the values of the amplitudes of the FSFs should be zeroin certain harmonic frequencies to have an algorithm insensi-tive to these harmonics. The insensitivity to harmonics isachieved when the orthogonality condition takes place onthe phases of the FSFs, at least in the neighborhood of thefundamental frequency (local orthogonality). This conditionis translated in an equal slope behavior for the FSF phasesin the fundamental frequency. Moreover, if the local orthogon-ality condition is fulfilled and the amplitudes of the FSFs havezero slope and zero value in some harmonic frequencies, thenthe algorithm is said to be detuning insensitive to those har-monics. The insensitivity to harmonics has been analyzed inmany papers, for example, [6 – 8,20,21]. In these works, the  phase error analysis considers the intensity pattern as a seriesexpansion in terms of cosines whose arguments have aninfinite number of harmonic components. However, the PSIalgorithms are obtained by considering only the fundamentalharmonic, in other words, by considering a perfectly co-sinusoidal 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 proposingthe equations to obtain the wrapped phase from a moregeneral expression for the intensity patterns. Téllez-Quiñones  et al.  Vol. 29, No. 4 / April 2012 / J. Opt. Soc. Am. A 4311084-7529/12/040431-11$15.00/0 © 2012 Optical Society of America   2. LIGHT-DETECTOR NONLINEARITIES  As described in [24], the light detector may have an electronicoutput with a nonlinear relationship with the signal, eventhough the detectors are normally adjusted to work in its mostlinear region. Considering an interferometric phase-shiftingarrangement such as the Twyman – Green interferometer shown in Fig. 1, if   s  is the intensity pattern detected by thesensor of a CCD camera, and  s t  is the true intensity patternobtained from the arrangement, we can write s    s t    ε 0    ε 1 s t    ε 2 s 2 t ;  (1)where  s t    a   b  cos  ϕ  −  α   ,  a  is an average intensity,  b  is anamplitude function,  ϕ  is the phase difference between twolight beams that interfere, and  α   is an arbitrary phase shift in-troduced by the PZT of the phase-shifting arrangement. The values  ε 0 ,  ε 1 , and  ε 2  are the error coefficients added to the trueintensity  s t .Equation(1) is an idealized formof 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 thisfigure we have two small regions of two one-dimensional ar-rays of sensors; one of them is well aligned and the other isnonaligned. 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 changingthe values  x   of the aligned array with the transformation  H    x     − 0 . 4    0 . 75   x     0 . 15   x  2 . We can consider the func-tion  s t   x     cos   x   , as an example of true one-dimensional in-tensity, which is achieved when  a   x     0 ,  b   x     1 ,  ϕ   x     x  ,and  α     0 . Then, the nonlinear transformation  H    x    can beconsidered as the Taylor expansion of another function ^  H    x   ; that is,  H    x     ^  H    x  0    ^  H  0   x  0   x   −  x  0    1 ∕ 2  ^  H  0   x  0   x   −  x  0  2 :  (2)This function can be  ^  H    x     arccos  ε 0    1    ε 1  cos   x    ε 2  cos 2   x   ;  with the point  x  0    π  ∕ 2  and the values ε 0    0 . 41022 ,  ε 1    0 . 11372  and  ε 2    − 0 . 44268 . Because theright-hand term of Eq. (2) is the second-order Taylor expan-sion of the function  ^  H  , it implies that  H    x    ≈  ^  H    x   . However,the measured intensity must be  s   x     s t   H    x     cos   H    x   ,and from this we have s   x    ≈  cos  ^  H    x     cos   x     ε 0    ε 1  cos   x     ε 2  cos 2   x    s t   x     ε 0    ε 1 s t   x     ε 2 s 2   x   ;  (3)which is the idealized expression given by Eq. (1). In a realgeneral case, the approximation in Eq. (3) is considered asan equality because working with the model in Eq. (1) is moreconvenient than analyzing the composition  s   x     s t   H    x   ,which is in terms of a theoretically unknown function  H  .On the other hand, it is possible to rewrite Eq. (1) as s    a 0    a 1  cos  ϕ  −  α     a 2  cos  2  ϕ  −  α   ;  (4)where  a 0   f a    ε 0    ε 1 a    ε 2  a 2   b 2 ∕ 2 g ,  a 1    b    ε 1 b  2 ε 2 ab  , and  a 2    1 ∕ 2  ε 2 b 2 . In the same manner, if we consid-er a third-order nonlinearity in the detected intensity; that is, s    s t    ε 0    ε 1 s t    ε 2 s 2 t    ε 3 s 3 t ;  (5)then we have s    a 0    a 1  cos  ϕ  −  α     a 2  cos  2  ϕ  −  α   a 3  cos  3  ϕ  −  α   ;  (6) where  a 0   f a    ε 0    ε 1 a   ε 2  a 2   b 2 ∕ 2    ε 3  a 3   3 ab 2 ∕ 2 g ,  a 1    b    ε 1 b   2 ε 2 ab   ε 3  3 a 2 b   3 ∕ 4  b 3  ,  a 2    b 2 ∕ 2  ε 2    3 ε 3 a  , and  a 3    1 ∕ 4  ε 3 b 3 . From the mathematical point Fig. 1. Basic configuration in a Twyman – Green interferometer.Fig. 2. Two one-dimensional arrays of sensors. The positions  x   of the aligned array are normalized, and the positions of the non-aligned array are obtained from the transformation  H    x     − 0 . 4  0 . 75   x    0 . 15   x  2 .432 J. Opt. Soc. Am. A / Vol. 29, No. 4 / April 2012 Téllez-Quiñones  et al.  of view, in these two cases, the presence of nonlinearities canbe considered as the presence of harmonic components in theintensity pattern  s . So in a general case, if the errors  ε 0 ,  ε 1  andthe nonlinearities  ε  m  of order   m  ≥  2  are assumed independentof the phase shift  α  , we can consider a more general intensity pattern equation s  k   X  M  m  0 a  m  cos   m  ϕ  −  α   k  ; k    1 ; … ;K;  (7)where  s  k  is the  k th intensity pattern obtained in the  k th arbi-trary phase shift  α   k  and  K   ≥  2  M     1  is the minimum number required of intensity patterns captured, as we will see later.Equation (7) can be obtained from two properties that canbe inferred from the Chebyshev polynomials [25]. Becausethe coefficient of the power   p  of the Chebyshev polynomial T   p  (with degree  p ) is a nonzero value, by simple induction, itis possible to prove that every polynomial with degree  p , eval-uated 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     2  x  2 −  1 ; T  3   x     4  x  3 −  3  x;  … ; T   p   x     2  xT   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 by considering  s t;k    a  b  cos  ϕ  −  α   k   and  s  k    s t;k   P  M  m  0  ε  m s  mt;k . The functions  a  m with  m  ≠  0  are amplitude functions for the  m th harmonic com- ponent, and  M   is the maximum number of harmonic definedfor the intensity pattern  s  k . The  a 0  function can be consideredas the bias term of   s  k . All the functions  a  m  are assumed posi-tion dependent, such as the phase  ϕ  to recover, and the shifts α   k  are not necessarily equally spaced, as in the case of thegeneralized phase-shifting interferometry (GPSI) [1 – 12]. 3. GPSAs FOR INTENSITY PATTERNS WITHA FINITE NUMBER OF HARMONICS Taking Eq. (7) as the model to consider, the differencebetween consecutive intensity patterns is Δ s  k  1  k    s  k  1  −  s  k  X  M  m  1 a  m  cos   m ϕ  Δ  cos   m α    k  1  k   sin   m ϕ  Δ  sin   m α    k  1  k   ;  (8)for   k    1 ; … ;K   −  1 . In Eq. (8), we have  Δ  cos   m α    k  1  k   cos   m α   k  1   −  cos   m α   k   and  Δ  sin   m α    k  1  k    sin   m α   k  1  − sin   m α   k  . From now on, we will employ the abbreviatednotation  Δ h  k  1  k    Δ h  α    k  1  k    h  α   k  1   − h  α   k    h  k  1  − h  k , torefer any consecutive difference where  h  is considered as a dependent function of the phase shift  α  . Now, let us define 2  M   sets of scalars  f  λ 1  k g ; f  λ 2  k g ; … ; f  λ  M  k  g  and  f  μ 1  k g ; f  μ 2  k g ; … ; f  μ  M  k  g ,each one with  K   −  1  elements. These sets have arbitrarynumbers different from zero and we will call these numbersthe sensitivity factors. So, we have  λ  p k Δ s  k  1  k   X  M  m  1 a  m  cos   m ϕ   λ  p k Δ  cos   m α    k  1  k    sin   m ϕ   λ  p k Δ  sin   m α    k  1  k   ;  μ  p k Δ s  k  1  k   X  M  m  1 a  m  cos   m ϕ   μ  p k Δ  cos   m α    k  1  k    sin   m ϕ   μ  p k Δ  sin   m α    k  1  k   ;  (9) for all  p    1 ; 2 ; … ;M   and  k    1 ; 2 ; … ;K   −  1 . Another   2  M  sets of unknown scalars, each one with  K   −  1  elements, canbe considered. They are  f  B 1  k g ; f  B 2  k g ; … ; f  B  M  k  g ,  f  A 1  k g ; f  A 2  k g ; … ; f  A  M  k  g  with the properties X  K  − 1  k  1  B  p k  λ  p k Δ  cos   m α    k  1  k    0 ;  ∀  m    1 ; 2 ; … ;M; X  K  − 1  k  1  B  p k  λ  p k Δ  sin   m α    k  1  k    0  ∀  m  ≠  p; 1  m    p; (10) X  K  − 1  k  1  A  p k  μ  p k Δ  sin   m α    k  1  k    0 ;  ∀  m    1 ; 2 ; … ;M; X  K  − 1  k  1  A  p k  μ  p k Δ  cos   m α    k  1  k    0  ∀  m  ≠  p; 1  m    p; (11) for all  p    1 ; 2 ; … ;M  . For each  p , Eqs. (10) and (11) determine two linear systems with  2  M   equations and  K   −  1  unknowns,then it is required to have at least  K     2  M     1  intensity pat-terns to solve these systems. By solving these two linear sys-tems for each  p , we can find the scalar solutions  f  B  p k g  and f  A  p k g , and then we have a determined wrapped phase equationgiven bytan   p ϕ   P  K  − 1  k  1 ~  B  p k Δ s  k  1  k P  K  − 1  k  1 ~  A  p k Δ s  k  1  k ;  (12)where  ~  B  p k    λ  p k  B  p k  and  ~  A  p k    μ  p k  A  p k . Here, we will call Eq. (12)the Riemann – Stieltjes expression of the PSI algorithm,because the sums in the numerator and denominator of thequotient are Riemann – Stieltjes sums [26]. Considering theRiemann – Stieltjes integrals, if we have an interval  Ω , and a  partition of this interval given by  α  1  <  α  2  <  …  <  α   K  , whichcannot necessarily be regular (with equally spaced shifts),but with  ‖ Δα  ‖    max 1 ≤  k ≤  K  − 1 f Δα   k  1  k  g  sufficiently small (  K   suf-ficiently large), then Eq. (12) can be written astan   p ϕ   R  Ω ~  B  p d s R  Ω  A ∼  p d s  R  Ω  B ∼  p  α   ds ∕ d α   d α  R  Ω  A ∼  p  α   ds ∕ d α   d α  :  (13)The functions in Eq. (13),  ~  B  p  α   ,  ~  A  p  α   , and  s  α    P  M  m  0  a  m cos   m  ϕ  −  α    are considered dependent of   α  . Two of them, ~  B  p  α    and  A ∼  p  α   , are termed as weight factor functions, andbecause  ds   P  M  m  1  ma  m  sin   m  ϕ  −  α   d α  , they require tosatisfy Z  Ω ~  B  p  α   sin   m α   d α     0 ;  ∀  m    1 ; … ;M; Z  Ω ~  B  p  α   cos   m α   d α     γ    p  ; m    p 0 ; m  ≠  p;  (14) Z  Ω ~  A  p  α   cos   m α   d α     0 ;  ∀  m    1 ; … ;M; Z  Ω ~  A  p  α   sin   m α   d α     − γ    p  ; m    p 0 ; m  ≠  p;  (15) γ    p   being a nonzero value that can be assumed dependent of   p . So, for   K   large, Eq. (12) approximates Eq. (13) by taking Téllez-Quiñones  et al.  Vol. 29, No. 4 / April 2012 / J. Opt. Soc. Am. A 433  ~  B  p k    ~  B  p  α    k   and  ~  A  p k    ~  A  p  α    k  , where  α    k  is a number con-tained in the interval  α   k ; α   k  1  .As an example offunctionsthatsatisfy Eqs. (14) and (15), we can consider   Ω    − π  ; π   , ~  B  p  α     γ    p  cos   p α   ∕ R  Ω  cos 2   p α   d α  ; and  ~  A  p  α     − γ    p  sin   p α   ∕ R  Ω  sin 2   p α   d α  . 4. ORTHOGONALITY FOR ALLFREQUENCIES WITH ZERO SLOPE  Although with Eq. (12) it is possible to recover the phase  ϕ multiplied by its corresponding harmonic factor   p , that is  p ϕ , the main interest in this manuscript is when  p    1 . So,we are going to analyze Eq. (12) referring by  ~  B  k  and  ~  A  k  tothe scalars  ~  B  p k  and  ~  A  p k  for   p    1 . For this case, Eq. (12) istan  ϕ   P  K  − 1  k  1 ~  B  k Δ s  k  1  k P  K  − 1  k  1 ~  A  k Δ s  k  1  k :  (16)In asimilar way to the descriptionin[14], it is easily concludedthat the FSFs that represent the numerator and denomi-nator of the algorithm defined by Eq. (16) are two frequency-dependent functions   G  N    f    and   G  D   f   , respectively. Theysatisfytan  ϕ   R  ∞−∞  S    f    G  N    f   d  f  R  ∞−∞  S    f    G  D   f   d  f ;  (17)where  S    f    is the Fourier transform [27] of   s  t   P  M  m  0  a  m cos   m  ϕ  −  ω t  . The functions   G  N    f    and   G  D   f    are complexconjugate of the Fourier transforms of   g  N   t   P  K  − 1  k  1 ~  B  k  δ   t  − t  k  1   −  δ   t  − t  k   and  g  D  t   P  K  − 1  k  1 ~  A  k  δ   t −  t  k  1   −  δ   t −  t  k  , re-spectively, where  δ   is the Dirac delta function. Here the phaseshifts are defined by  α   k    ω t  k    2 π   f   r   t  k , where t  k  can be con-sidered as discrete-time and  f   r   is a reference frequency.The FSFs are  G  N    f    X ~  B  k Δ  cos  • α    k  1  k    i X ~  B  k Δ  sin  • α    k  1  k  ;  G  D   f    X ~  A  k Δ  cos  • α    k  1  k    i X ~  A  k Δ  sin  • α    k  1  k  ;  (18) where the sums run from  k    1 ; … ;K   −  1 ,  •    f  ∕  f   r   and i is theimaginary unity. An interesting particular case of orthogonality for the FSFsat all frequencies is achieved when the two phases of the func-tions in Eq. (18) are linear and separated by an odd integer factor of   π  ∕ 2  with zero slope, that is for instance when  G  N    f     i X ~  B  k Δ  sin  • α    k  1  k    Am  N    f   exp  i π  ∕ 2  ;  G  D   f    X ~  A  k Δ  cos  • α    k  1  k    Am  D   f   exp  i  ·  0  ;  (19) where  Am  N   and  Am  D  represent amplitude functions.Let us verify the necessary conditions to obtain Eq. (19). Inthe case of an algorithm with an odd number of steps, this is  K     2 τ     1 , it is required that X ~  B  k Δ  cos  • α    k  1  k    0 ; X ~  A  k Δ  sin  • α    k  1  k    0 ;  ∀  f:  (20) So, assuming α   j     − α   K   1 −  j  ; j     1 ; … ; τ  ;  α  τ   1    0 ;  (21)then Δ  cos  • α    j   1  j     − Δ  cos  • α   2 τ  −  j   22 τ  −  j   1 ; Δ  sin   • α    j   1  j     Δ  sin  • α   2 τ  −  j   22 τ  −  j   1 ; j     1 ; … ; τ  :  (22) From Eq. (22) we have X  B ∼  k Δ  cos  • α    k  1  k  X τ   j   1  B ∼  j  Δ  cos  • α    j   1  j    X τ   j   1  B ∼ 2 τ  −  j   1 Δ  cos  • α   2 τ  −  j   22 τ  −  j   1  X τ   j   1   B ∼  j   −  B ∼ 2 τ  −  j   1  Δ  cos  • α    j   1  j   ; X  A ∼  k Δ  sin  • α    k  1  k   …   X τ   j   1   A ∼  j     A ∼ 2 τ  −  j   1  Δ  sin  • α    j   1  j   :  (23) Thus, from Eq. (23), Eq. (20) is satisfied if   B ∼  j     B ∼ 2 τ  −  j   1    B ∼  K  −  j  ; A ∼  j     −  A ∼  K  −  j  ; j     1 ; … ; τ  :  (24)From Eqs. (21) and (24), the FSFs are reduced to  G  N    f     i X τ   j   1 2  B ∼  j  Δ  sin   f  α  ∕  f   r    j   1  j   ;  G  D   f    X τ   j   1 2  A ∼  j  Δ  cos   f  α  ∕  f   r    j   1  j   :  (25) Similar results are obtained for an algorithm with an evennumber of steps, that is  K     2 τ  . When  K   is even, we can as-sume α   j     − α  2 τ   1 −  j     − α   K   1  −  j  ; j     1 ; … ; τ  ;  (26)and to have FSFs, perfectly orthogonal for all frequencies, it isrequired that  B ∼  j     B ∼ 2 τ  −  j     B ∼  K  −  j  ; A ∼  j     −  A ∼  K  −  j  ; j     1 ; … ; τ   −  1 ; B ∼ τ     0 ; A ∼ τ     0 .  (27) The value of   B ∼ τ   in Eq. (27) can be any number, because Δ  cos  • α   τ   1 τ     0 ; however, without loss of generality itcan be assumed to be equal to zero. 5. INSENSITIVITY TO DETUNING ANDHARMONICS IN THE PRESENCE OFDETUNING The algorithm in Eq. (12), when  p    1 , is insensitive to theharmonics  m    2 ; … ;M  , but in many cases it is not insensitiveto those harmonics in the presence of detuning. The sensitiv-ity factors  λ  p  1  k  ,  μ  p  1  k  , can be changed until the insensitivity todetuning in the fundamental frequency (  m    1 ) is satisfied,but the insensitivity to the harmonics (  m >  1 ) in the presence 434 J. Opt. Soc. Am. A / Vol. 29, No. 4 / April 2012 Téllez-Quiñones  et al.  of detuning cannot necessarily be achieved at the same time.However, based on the results in the previous section, we canconsider certain equations to design an algorithm insensitiveto detuning and harmonics in the presence of detuning. Let usconsider the case of   K     2 τ     1 , and Eqs. (21) and (24). In order to work with linear systems, the degree of insensitivityanalyzed here will be of the first order. From Eq. (25) we have  Am  N    f    X τ   j   1 2  B ∼  j  Δ  sin   f  α  ∕  f   r    j   1  j   ; Am  D   f    X τ   j   1 2  A ∼  j  Δ  cos   f  α  ∕  f   r    j   1  j   ;  (28) whose derivatives with respect to  f   are  dAm  N  ∕ df    X τ   j   1 2  B ∼  j   1 ∕  f   r   Δα   cos   f  α  ∕  f   r    j   1  j   ;  dAm  D ∕ df    X τ   j   1 2  A ∼  j   − 1 ∕  f   r   Δα   sin   f  α  ∕  f   r    j   1  j   :  (29) One of the main conditions to be satisfied by any PSA should be the equal amplitude property in  f   r  , that is  Am  N    f   r     Am  D   f   r     1 , which is rewritten as  P τ   j   1  B ∼  j   2 Δ  sin  α    j   1  j     P τ   j   1  A ∼  j   2 Δ  cos  α    j   1  j      1 , as well as X τ   j   1  B ∼  j   2 Δ  sin   α    j   1  j     X τ   j   1  A ∼  j   0   1 ; X τ   j   1  B ∼  j   0   X τ   j   1  A ∼  j   2 Δ  cos   α    j   1  j      1 :  (30)The terms  P τ   j   1  A ∼  j   0   and  P τ   j   1  B ∼  j   0   in Eq. (30) are equal tozero because the products  A ∼  j   0   and  B ∼  j   0   for   j     1 ; 2 ; … ; τ  are equal to zero. These terms were added to the expressionsin Eq. (30) to denote two linear equations with  2 τ   unknowns  A ∼  j   and  B ∼  j  .To have detuning insensitivity, it is required that theamplitudes have a tangential behavior in  f   r  ; that is,  dAm  N    f   r   ∕ df     dAm  D   f   r   ∕ df   . This implies X τ   j   1  B ∼  j   2 ∕  f   r   Δα   cos  α    j   1  j     X τ   j   1  A ∼  j   2 ∕  f   r   Δα   sin  α    j   1  j      0 .(31)The insensitivity to harmonics without the presence of detuning is obtained when  Am  N    mf   r     Am  D   mf   r     0 ,  m    2 ; … ;M  . So, it is sufficient to set P τ   j   1  B ∼  j   2 Δ  sin  2 α    j   1  j     P τ   j   1  A ∼  j   0    0 ; P τ   j   1  B ∼  j   2 Δ  sin  3 α    j   1  j     P τ   j   1  A ∼  j   0    0 ; ... P τ   j   1  B ∼  j   2 Δ  sin   M  α    j   1  j     P τ   j   1  A ∼  j   0    0 ; P τ   j   1  B ∼  j   0   P τ   j   1  A ∼  j   2 Δ  cos  2 α    j   1  j      0 ; ... P τ   j   1  B ∼  j   0   P τ   j   1  A ∼  j   2 Δ  cos   M  α    j   1  j      0 .  32  Finally, the insensitivity to harmonics in the presenceof detuning is achieved when   dAm  N    mf   r   ∕ df    dAm  D   mf   r   ∕ df     0 ,  m    2 ; … ;M  . Then it is required that P τ   j   1  B ∼  j   2 ∕  f   r   Δα   cos  2 α    j   1  j     P τ   j   1  A ∼  j   0    0 ; P τ   j   1  B ∼  j   2 ∕  f   r   Δα   cos  3 α    j   1  j     P τ   j   1  A ∼  j   0    0 ; ... P τ   j   1  B ∼  j   2 ∕  f   r   Δα   cos   M  α    j   1  j     P τ   j   1  A ∼  j   0    0 ; P τ   j   1  B ∼  j   0   P τ   j   1  A ∼  j   − 2 ∕  f   r   Δα   sin  2 α    j   1  j      0 ; ... P τ   j   1  B ∼  j   0   P τ   j   1  A ∼  j   − 2 ∕  f   r   Δα   sin   M  α    j   1  j      0 .  33  Equations (30) – (33) generate a linear system of   4  M   −  1  equa-tions with  2 τ     K   −  1  unknowns (  B ∼  j  ,  A ∼  j  ,  j     1 ; … ; τ  ). To solvethis system, it is necessary at least that  K   −  1    4  M   −  1 , butthis implies  K     4  M  , which is impossible. However, to haveinsensitivity to harmonics in the presence of detuning, in themajority of these harmonics,  m    2 ; … ;M   −  1 , we can avoidthe last equality in Eq. (33). Thus, with   4  M   −  1   −  1   4  M   −  2  equations, the system can be solved when  K     4  M   − 1  and the corresponding matrix of the system has the inverse.The same  4  M   −  2  linear equations can be obtained for the case  K     2 τ  , where the unknowns would be  B ∼  j  ,  A ∼  j  ,  j     1 ; … τ   −  1 , having at least  K     4  M   intensity patterns tosolve the system. Independently of the values  α   k  that werechosen and regardless the fact that  K   was odd or even, theexistence of the inverse matrix of the system could be ques-tionable. However, the cases when the matrix is singular (noninvertible) are rarely found. 6. DETUNING INSENSITIVE PSAs:DETUNING INSENSITIVITY TOCONSECUTIVE HARMONICS Our main interest is the design of PSAs with equally spaced phaseshifts because theyare employed in the classical experi-mental 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 withequally spaced shifts, some GPSAs will be shown. As it wasdescribed above, Eqs. (21) or (26) should be assumed to re- duce the mathematical expressions for the FSFs, and then our examples of GPSAs will not be too generalized. For the PSAswith equally spaced shifts, this limitation is not an obstacle for  practical purposes, because the phase  ϕ  is always calculatedwith a piston term. Then, this piston term is interpreted as a translation parameter that can be added to the phase shifts. Téllez-Quiñones  et al.  Vol. 29, No. 4 / April 2012 / J. Opt. Soc. Am. A 435
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