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Temporal phase evaluation by fourieranalysis of fringe patterns with spatialcarrierJ. A. Quiroga a & J. A. Gómez-Pedrero aa Optics Department, Universidad Complutense de Madrid, Facultadde Ciencias Fisicas, Ciudad Universitaria s/n, 28040, Madrid E-mail:Published online: 03 Jul 2009.

To cite this article: J. A. Quiroga & J. A. Gómez-Pedrero (2001): Temporal phase evaluation byfourier analysis of fringe patterns with spatial carrier, Journal of Modern Optics, 48:14, 2129-2139

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JOURNAL, OF MODERN OP'I'ICS, 2001, VOL. 48, NO. 14, 2129-2139

Temporal phase evaluation by Fourier analysis of fringe patterns with spatial carrier

J . A. QUIROGA and J. A. GOMEZ-PEDRERO

Optics Department, Universidad Complutense de Madrid, Facultad de Ciencias Fisicas, Ciudad Universitaria s/n, 28040 Madrid; e-mail: aq@fis.ucm .es

(Received I9 March 2001; revision received 9 July 2001 )

Abstract. A new method is presented for the temporal evaluation of fringe patterns with spatial carrier. T h e proposed technique involves recording the irradiance fluctuations obtained when a linear variation of the set-up sensitivity is introduced. In this condition, the use of a spatial carrier introduces a linear temporal carrier frequency. In this way, Fourier analysis can be performed to obtain the phase and, finally, the quantity to be measured. T h e optimum conditions for the sensitivity variation have been studied in order to minimize the errors associated with the Fourier analysis. T h e technique has been applied to measure the distribution of ray deflections on the surface of two ophthalmic lenses using a deflectometric set-up.

1. Introduction In optical metrology, there are a number of techniques designed to measure a

physical quantity using fringe patterns. In general, we can consider the modulating phase of a fringe pattern as the product of a sensitivity and the physical quantity to be measured. Recently, several methods for the temporal phase evaluation of fringe patterns have been proposed. The objective of these techniques is to obtain the phase map point-wise, from the temporal irradiance variation, allowing in this way the evaluation of discontinuous phase fields, which are difficult to analyse with spatial phase evaluation techniques.

Basically, there are three approaches to solving this problem. In the first one, proposed by Huntley [ l , 21, several phase-shifted irradiance images are generated in each step as the sensitivity of the set-up is changed. From these images, the whole variation of the phase in terms of the sensitivity is computed for every point. Finally, the physical magnitude to be measured is obtained as the slope of the linear relationship between phase and sensitivity.

In the second approach [3,4], temporal irradiance fluctuations are generated by variation of the modulating phase. In this case, a Fourier analysis is performed to obtain the phase fluctuation, which is then unwrapped to obtain the total phase variation at a given point. The physical quantity to be measured is then obtained from this phase variation in different ways depending on the particular set-up. This technique has been applied to measure the shape or the deformation of a diffusing object with speckle interferometry. In this case, the change in the modulating phase was obtained by an object rotation, thus, introducing a linear

Journal of Modern Optics ISSN 0950-0340 print/ lSSN 1362-3044 online Ci 2001 Taylor & Francis Ltd http://www.tandf.co.uk/journals

DOI: io.1080~095003401i00~669t3

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2130 J . A. Quiroga and J . A. Gdmez-Pedrero

out-of-plane displacement for every object point. This displacement produces an irradiance fluctuation from which, the shape of the object is extracted.

The third solution [S], has also been applied to measure the shape of a diffusing object by speckle interferometry. In the proposed set-up, the wavelength of the light source (in this case a laser diode) is changed. In this way, variation in the modulating phase is achieved allowing temporal analysis by Fourier transform. Due to the unavoidable non-linearities introduced in the wavelength shift, it is necessary to perform an average of the derivative of the temporal signal. I t is also necessary to use a reference object in order to measure the surface topography.

In this work, a new approach to the problem of temporal evaluation of fringe patterns is proposed. The method is focused on the case of fringe patterns with spatial carrier. Basically, this technique involves recording the irradiance fluctua- tions obtained when a linear variation of the set-up sensitivity is introduced. In this way the introduction of a spatial carrier leads to a point-wise linear temporal carrier. Thus, a temporal Fourier analysis can be carried out to obtain the quantity to be measured. The optimum conditions for the temporal carrier and the sensitivity variation in order to minimize the errors associated with the Fourier analysis have been determined.

This technique has been applied to the particular case of a fringe pattern with a linear spatial carrier, but we must point out that there are no restrictions on the spatial dependence of the spatial carrier. As an example, the method could also be applied to fringe patterns with circular spatial carriers.

This work is organized as follows: first, we give the theoretical foundations of the method and describe how optimum conditions are obtained. Then, we present the experimental results obtained using a deflectometric set-up and, finally, the conclusions are given.

2. Theoretical analysis

carrier, Let us consider the irradiance distribution of a fringe pattern with spatial

i , = a, + br * cos (A + $r), ( 1 )

a, being the background irradiance, b, the fringe modulation, $, the carrier phase and q5r the modulating phase. In equation (1) the subindex r indicates the spatial dependence of these magnitudes. We will suppose that the following relationships can be established

4, = S . hr, (2 a)

with S a scaling factor relating the phase 4, and the quantity to be measured h,. This factor is known as sensitivity, and its value depends on the nature of the magnitude h, and the type of experimental set-up (Moiri., interferometry, etc.) employed. On the other hand, as will be shown later, K, will act as a temporal carrier when sensitivity change with time. In these conditions, we have that equation (1) transforms into

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Temporal phase evaluation by Fourier analysis 2131

It should be noted that equation (3) is a general expression that describes the fringe pattern distributions obtained in a number of different experimental set- ups. If we introduce a temporal linear variation in the interferogram sensitivity, we will find, for each image point, a temporal fluctuation in the irradiance signal with a temporal frequency R, = h, + 6,. In these conditions, it is possible to employ the Fourier transform demodulation procedure described in [3], if R, is positive for every image point.

If S varies with time, we have a temporal irradiance distribution given by

ir(S) = a, + b, . cos (@,(S)) = a, + 6,. cos ( S . Or), (4)

in this way, it is possible to get the value of (a,(S) by means of the following expression

where W[ ] stands for the wrapping operator and ir(S) is the analytic signal obtained from the Hilbert transform [6] of the irradiance &(S) (this is why R, should be a positive number for every image point). The phase obtained by ( 5 ) is wrapped, so it is necessary to perform a 1D unwrapping in order to obtain the continuous phase. Once the phase a, is obtained from equation (5 ) and un- wrapped, it is possible to obtain the modulating frequency R, by means of a least squares fitting from the relationship @, = S . R,. Finally, the quantity h,, that we want to measure, is obtained from the following expression

where the analytic expression of K,, which is directly related with the carrier phase $,., is known. From equation (6), the role of 6, as a temporal carrier when S varies with time becomes clear. If this temporal sensitivity variation (as in our case) is linear, K, becomes a linear temporal carrier.

We now focus on the study of fringe patterns with linear spatial carrier. For simplicity we consider 6, = a . x and, consequently, R, = h, + a . x. In these conditions, if x varies between xo and xo +Ax, which are the limits of the field of view, and (Y is a positive quantity, the modulating frequency 0, remains positive, if the following condition holds

min (h,) XrJ >-, a (7)

where it is assumed that h, is a bounded function. Thus, if we have an estimate of the variation interval for h,, it is always possible to fix the coordinate origin (or, what is the same, the value of the offset XO) to ensure, according with equation (7), that R, will be positive for every image point.

We now present an optimization procedure in order to minimize the error associated with the Fourier method. Suppose that, for each point of the image, the irradiance signal has been measured for a discrete set of sensitivity values, {S(j)},=,,,,.v, N being the number of samples. Consider also a linear variation in the sensitivity, in such a way that

S( j ) = S( 1 ) + [ . ( j - l), j = 1 . . . N , (8)

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2132 J. A. Quiroga and J . A. Go’mez-Pedrero

< being a constant factor. From these values of the set-up sensitivity, we can find the following expression for the phase values

@ , . ( j ) = S ( 1 ) . R r + f . R , . ( j - 1 ) , j = 1 ... N . (9)

Thus, the phase difference between two consecutive images is

@,( j + 1) - a,( j) = E * 0, = E . (h, + (Y . x), j = 1 . . . N , (10)

and this equation implies a constant temporal sampling of the phase for each location r. We call q, the phase variation normalized by n-. According to this definition,

As is stated by the sampling theorem it is necessary that qr E [0,1] (when qr = 1 we are sampling at the Nyquist frequency). From equation (1 l ) , the maximum and minimum possible values for qr (qmin and q,,,) are given by

Thus, by controlling the experimental conditions (given by the parameters XO, Ax and I) and having a priori information about h,, it is possible to establish the limits of the variation interval for 4,. Conversely, if we know the variation interval of q,, it is possible to obtain the optimum experimental conditions.

I t is well known that, in the Fourier transform technique, the signal must be sampled at a frequency as close as possible to the Nyquist frequency (that is, qmin,qmax l ) , and with the maximum possible number of cycles (given by 0.5 . q, . N for each image location r).

As we have a linear sensitivity variation, we will found a monochromatic irradiance signal for each point-see equation (4). In this case, the discrete Fourier transform algorithm will introduce a distortion in the signal spectrum if the extension of the irradiance signal does not correspond to an integer number of cycles. This distortion could be especially noticeable if a small number of samples (low values of N ) is used without any kind of signal pre-processing. In our case, N is the number of acquired frames, which we want to keep within 30 to 50 due to the image size and computer storage limitations. For instance, if we want to evaluate the phase in an experiment with a temporal series of 30 images of 512 x 512 pixels we would need about 8Mb using 1 bytelpixel, which is an affordable quantity for the capabilities of standard computers. Moreover, the processing time is propor- tional to the number of acquired frames.

In order to avoid this spectrum distortion, a pre-processing procedure has been implemented. First the signal is interpolated to 10 times the original sampling frequency. Then, the interpolated signal is cropped in such a way that the remaining signal extension corresponds to an integer number of cycles. Finally, this interpolated and cropped signal is used to obtain the phase using Fourier analysis. T o show this behaviour, a numerical simulation has been performed. Figure l(a) shows a simulated interpolated signal, and superposed the cropped

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Temporal phase evaluation by Fourier analysis

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Figure 1. ( a ) Plot of a typical irradiance signal, the solid curve represents the original signal while the solid-point curve represents the cropped signal, note that the extension of the cropped signal represents an integer number of periods. ( h ) and (c) plots of the F F T modulus of the original and cropped signal, respectively. Note the distortion presented by the lobes of the FFT corresponding to the uncropped signal disappear for the Fourier transform of the cropped one.

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2134 J . A. Quiroga and J . A. Gdmez-Pedrero

- uncropped 11 GI

-1 5 5 10 15 20 25 30

Sensitivity

Figure 2. Scheme of the experimental set-up employed to measure ray deflection fields of ophthalmic lenses.

version. As can be seen in Figures l(b) and l(c), where the Fourier spectrum of these signals has been plotted, the distortion of the Fourier spectrum presented by the interpolated signal (see figure l (b)) is removed in the corresponding spectrum of the cropped signal shown in figure l(c).

Figure 2 shows how the signal processing described in the preceding paragraph reduces the errors associated with Fourier analysis. Specifically, the difference between the recovered phase and theoretical one for the signals shown in figure l(a) is plotted. As can be seen, the error is reduced in the case of the processed signal, specially at the borders.

If we want to carry out this pre-processing procedure with harmonic signals sampled near the Nyquist limit, the interpolation will not work well due to the lack of support. This means that, if we want to interpolate the irradiance signal properly we must have 4 to 10 points per cycle. Thus, a compromise must be found between the requirements of the Fourier transform method (high number of cycles) and the interpolation procedure (high number of points per cycle) for a fixed number of frames, N . For instance, when N = 50, we found that appropriate values for the variation limits of qr are &,in = 0.25 and (Imax = 0.5. This leads to irradiance signals with 6 to 12 cycles. For higher values of N , (Imax should be decreased (improving the interpolation). On the other hand, if N is reduced, then &in should be increased (improving the sampling).

In accordance with the preceding discussion, it was found necessary to keep the values of qr between the optimum limits [&in, (Imax] in order to minimize the errors associated with the Fourier transform method. This implies that, according to equations ( l l a ) and (l lb), the values of the magnitudes [ and xg must be fixed. These optimum values are given by the following equations

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Temporal phase evaluation by Fourier analysis 2135

Note that it is necessary to check that the optimum value 20 given by equation (12b) satisfies the condition stated by equation (7) to ensure the correct phase extraction from the Fourier analysis. Otherwise, the interval [&,in, qmax] must be changed until this condition holds.

In short, the proposed method for a temporal evaluation of fringe patterns can be described as follows: starting with an irradiance distribution given by equation (4), we take a discrete set of irradiance images for different values of the set-up sensitivity, with the sensitivity change given by equation (8). From these images discrete set of phase values {@r(j)}j=,,..N is obtained, using the Fourier analysis method described above-see equation (5). In this way, we have for every image point a set of pairs { S ( j ) , @ r ( j ) } j = , , , , N . The next step is to obtain the value of the temporal modulating frequency, R, by performing a linear regression with the pairs { S ( j ) , Qr( j )} j=, , , .N. Finally, the magnitude h, is obtained from R, by means of equation (6).

In the particular case of fringe patterns with linear carrier, equations (12a) and (12b) permit calculation of the optimum values for two parameters of the experi- ment: the offset xg and the rate of sensitivity change <. With these values, errors associated with the Fourier analysis procedure are reduced, thus allowing a reduced number of samples.

3. Application to deflectometry, experimental results The method has been applied to a deflectometric set-up for the analysis of

ophthalmic lenses. Deflectometry is a well-known technique for the study of phase objects [7, 81. The particular set-up described in [7] (see figure 3 ) has been used. The use of computer monitors-LCD panels for the generation of test patterns has been reported by Asundi et al. [9] and Teipen et al. [lo] in the fields of Moire testing and MTF measurements. Recently, for the particular case of the set-up shown in figure 3 , the use of a computer monitor instead of a printed grating has been proposed in [ l 13. Computer-generated gratings have the ability to be changed without moving parts, which enables the selection of the correct pitch for a particular problem. In our experiment instead of a printed grating we have used the monitor of a lap-top computer in order to generate the sinusoidal gratings.

Figure 3 . Difference between the recovered phase and the theoretical one for the two signals (cropped and uncropped) represented in figure l ( a ) . Note how the error is reduced for the cropped signals specially in the borders.

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2136 J . A. Quiroga and J . A. Gomez-Pedrero

The deflection maps of two ophthalmic lenses, a monofocal lens and a bifocal lens, have been measured in order to show the ability of the proposed method to analyse discontinuous phase objects.

In these conditions, if we call {6 , (x ,y ) , 6 y ( x , y ) } the horizontal and vertical components of the ray deflection introduced by the phase object at a given point P, we can write the following expression for the irradiance distribution at the CCD image plane

where p is the fringe pitch. In equation ( 1 3 ) , it is supposed that the fringes presented on the computer screen are oriented in the vertical direction. It is clear that we have in this case, a fringe pattern with linear carrier. Comparing with equations ( 3 ) and subsequent expressions, we can make the following identifica- tions: h, = 6 , (x ,y ) , S = 2 7 r . d / p and a = d - ' . In order to apply the Fourier evaluation technique, it is necessary to change the sensitivity of the set-up. This can be done by changing the fringe pitch p in the following way

which leads to a linear sensitivity variation

( 1 5 ) 2 ~ . d 2 ~ . d

S ( j ) = - - ( j - 1 ) j = 1 . . . N .

Comparing this expression with equation (8), we can identify the parameters S( 1 ) = 27r * d/pmin and < = 27r. d . L( pmin)-' - ( pmaX)-'] / ( N - l ) , respectively.

T o measure the deflection maps it is necessary to obtain the optimum values for the offset xo and rate of sensitivity change < as it was stated in previously. We want to take 50 images, which implies that qr must vary within the interval [ 0 . 2 , 0 . 5 ] . On the other hand, we have fixed the values of the distance d and minimum pitch value pmin. As the lens power is approximately known, the maximum and minimum values of the deflection 6*, for the two lenses can be estimated. Finally, the optimum values for xo and < have been calculated using equations ( 1 2 a ) and (12b) . Table 1 presents the values of the experimental parameters employed to measure the deflection map of each lens.

Figures 4(a) and 4(b) show the horizontal component of the deflection (6,) obtained for each lens. For the bifocal lens, it can be appreciated how the temporal evaluation process is able to obtain the discontinuous deflection due to the bifocal segment. T o test results, comparisons have been made with the values

Table 1 . Experimental parameters used in the measurement of the ray deflection fields for the spherical and bifocal lenses. The values of xg (the offset) and [ have been obtained using the optimization procedure described in the text.

d max(6,) min(6,) xo Prnin Pmax

(mm) N (rad) (rad) (cm) t (mm) (mm)

Spherical 280 50 0.05 -0.05 221.3 1.129 27.8 213.3 Bifocal 430 50 0.1 -0.1 212.8 1.782 26.9 206

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Temporal phase evaluation by Fourier analysis 2137

Figure 4. ( a ) Distribution of the horizontal component of the measured ray deflection 6x along the surface of the spherical lens. ( b ) Distribution of 6, along the surface of the bifocal lens, note the discontinuous character of this latter distribution according with the nature of the lens employed.

obtained using an automatic focimeter (Humphrey L650). Figure 5(a) plots the deflection profiles obtained with temporal evaluation and focimeter, along line A-B of figure 4(a ) . The profiles along line C-D of figure 4(b) are shown in figure 5 ( b ) . Good agreement is found between the experimental values obtained by temporal demodulation and those given by the focimeter. The same experi- mental accuracy with was achieved with the proposed method and with the focimeter.

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Figure 5. Comparison of the experimental values (continuous line) of 6 , with those given by an automatic focimeter (square plot) for (a) the spherical and (b) the bifocal lens. The plots show good agreement between the two methods.

4. Conclusions A new approach to the problem of temporal evaluation of fringe patterns is

proposed. The method is focused to study the case of fringe patterns with spatial carrier. The introduction of a spatial carrier leads to a point-wise temporal carrier when the set-up's sensitivity varies with time. This permits phase evaluation by the Fourier transform technique.

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Temporal phase evaluation by Fourier analysis 2139

The optimum conditions that should be accomplished by the temporal carrier as well as the sensitivity variation in order to minimize errors associated with Fourier analysis have been determined

Finally, the technique has been applied for a deflectometric study of ophthal- mic lenses. In this application the potential of the technique to measure continuous as well as discontinuous phase distributions is demonstrated.

An important feature of the method is that no restriction exists in the spatial dependence of the carrier. For example, we could also apply our method to fringe patterns with circular spatial carriers.

Acknowledgments We wish to thank the kind and helpful comments of D r Agustin Gonzalez-

Cano. We also wish to thank the financial support given to this work by the Comisi6n Interministerial de Ciencia y Tecnologia, proyecto TAP98-0701 and European Union, proyect INDUCE, BRPR-CT97-0805.

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[9] ASUNDI, A. K., 1993, O p t . Eng., 32, 107.

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