The main purpose of this paper is to present a method to design tunable quadrature filters in phase shifting interferometry. From a general tunable two-frame algorithm introduced, a set of individual filters corresponding to each quadrature conditions of the filter is obtained. Then, through a convolution algorithm of this set of filters the desired symmetric quadrature filter is recovered. Finally, the method is applied to obtain several tunable filters, like four and five-frame algorithms.
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Several kinds of methods to design quadrature filters are found in the literature and references therein [1–6]. However, most of them give an algorithm designed for a specific phase step [1–8]. In a previous work, a theory to design tunable filters was presented . Then, in this paper another method to obtain tunable filters is introduced. Like in the previous work , the quadrature conditions of the filter and errors like the phase shift detuning error and bias modulation are represented as geometrical conditions to be satisfied by the filter. Thus, from these conditions and the desired errors to minimize, a set of two-frame filters for each condition is obtained. Hence, trough the convolution of this set of individual filters we recover the desired filter as an algebraic problem. Finally, the method is applied to obtain novel results for tunable four and five-frame algorithms.
It should be noticed that is the column vector of frames, and N and D are the desired numerator and denominator row vectors; then, for a symmetrical filter the corresponding temporal impulse response is given by [6–9]Eqs. (1) and (2) are recovered easily. In the previous work it was proved that the Fourier transform of h(t) is the real function [7,8]. For an α step, any quadrature filter satisfies the two conditions ;. That is, the filter cuts off both ω = 0 and ω = α frequencies. Therefore, the condition for a filter tuned onto the right side and insensitive to the mth order phase shift detuning error is [1–7]
In other words, m gives the order of insensitivity to the phase shift error, and is the mth derivate of with regard to ω. In the same way, the condition to be satisfied by a filter which is insensitive to the mth order bias variation error is ,
2. Convolution algorithm
Assuming that the Fourier transform of a filter can be factorized in two functions such as , where and are the Fourier transforms of and which are an n and m order filters respectively, the individual estimated phases and are given by
Hence, the desired filter is obtained from the expression , where ∗ denotes the temporal discrete convolution, and becomes,
In other words, as mentioned before, a new (n + m-1) frame filter from two individual filters is obtained. Likewise, the design of a tunable quadrature filter is seen as an algebraic problem without the use of Fourier formalisms. The convolution properties allow this case to be extended for three or more filters.
3. Design of tunable filters
The design of a quadrature filter order M implies that only M-1 parameters are free, two of which are the quadrature conditions and the other M-3 are used to compensate some errors.
3.1 The tunable two-frame algorithm
The general form of a symmetric two-frame algorithm is ,
Tuning this filter to cut off the frequency , the condition is satisfied. Hence, the solution of Eq. (10) is given by the ratio . And from Eq. (9) the corresponding tunable two-frame filter and the ratio N/D are respectively,
Furthermore, this two-frame filter has the following Fourier transform
Therefore, from Eq. (3) the Fourier transform and the corresponding algorithm for a filter insensitive to the 0, 1, 2 … and m orders phase shift detuning error are respectivelyEq. (13), for α = 0, the other quadrature condition is recovered and its Fourier transform becomesEq. (11) the corresponding two-frame algorithm that cuts off the frequency isEq. (15) for α = 0, the corresponding algorithm is
In the same way, other possible conditions working over the frequency are,Eqs. (12) and (20) the respective algorithm becomes
3.2 The tunable three-frame algorithm
The Fourier transform that meets the two quadrature conditions H(0) = 0 and H(α) = 0 is
This function is the product of the two filters that cuts off individually both frequencies ω = 0 and ω = α. Then, in terms of the corresponding two-frame filters, and through the convolution algorithm Eq. (8), the desired N/D becomesEq. (7) the estimated phase is expressed as [1–5],
3.3 The tunable four-frame algorithm
Like Eq. (1), the general form for the estimated phase of a four-frame algorithm is
From Eq. (13), the Fourier transform that cuts off the frequencies , and is
That is, two conditions correspond to the necessary quadrature conditions, and is the particular condition that cuts off the frequency β. Then, by applying the corresponding two-frame filters, the result is
From Eq. (8) the last two terms become,Eqs. (7) and (8) the estimated phase is expressed as
This is the general four-frame filter that cuts off both frequencies, the α step and an arbitrary frequency β. Thus, by using the geometrical condition , a new four-frame algorithm is obtained. Therefore, as reported in  we named it “tunable four-frame algorithm in X”,5]. On the other hand, Eq. (30) for gives a filter that cuts off twice the same frequency α, then the derivative at frequency α is zero. That is to say, that the filter becomes insensitive to the linear phase shift detuning error, and gives the reported four-frame algorithm class B . Finally, Eq. (30) for recovers the reported four-frame algorithm class C , which is a four-frame algorithm insensitive to the linear bias modulation error.
3.4 The tunable five-frame algorithm
For the α step and the two arbitrary frequencies β and γ, the Fourier transform of the filter is,
In case 2, and becomes,
In case 3, and gives
In case 4, and results
In case 5, the filter meets the conditions and gives
This is a tunable five-frame algorithm which is simultaneously insensitive to linear bias modulation and linear phase shift detuning errors. It comprises the results reported in . To conclude, in case 6 the tunable filter insensitive to second order phase shift detuning error is,
Finally, from [7,8], the detuning error of the function gives for the angle Δ. Therefore, for the detuning error is . Thus, cases 3 and 5 have q = 2, while case 6 has q = 3, and the others cases have the value q = 1.
A new method to design quadrature filters is presented. The design problem is reduced to an algebraic problem that through the convolution of a set of two-frame filters gives the desired phase without the use of Fourier’s formalisms. Each individual condition of the filter corresponds with a specific two-frame filter. Therefore, several new tunable four and five-frame symmetrical algorithms are reported. This method is easily evaluated numerically.
This work was partially supported by CONACyT México through scholarship granted #175434. The authors also acknowledge the help granted by MDD. J. J. Lozano in the revision of this paper.
References and links
3. D. W. Phillion, “General methods for generating phase-shifting interferometry algorithms,” Appl. Opt. 36(31), 8098–8115 (1997). [CrossRef]
5. H. Schreiber, J. H. Brunning, and J. E. Greivenkamp, “Phase shifting interferometry,” in Optical Shop Testing, D. Malacara ed., (John Wiley & Sons, 2007).
6. M. Afifi, K. Nassim, and S. Rachafi, “Five-frame phase-shifting algorithm insensitive to diode laser power variation,” Opt. Commun. 197(1-3), 37–42 (2001). [CrossRef]