n-point spatial phase-measurement techniques for non...

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Oprio and Lasers in Engineering 24 (1996) 365-379 Copvrlght 0 1996 Elsevier Science LimIted

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N-point Spatial Phase-measurement Techniques for Non-destructive Testing

Katherine Creath & Joanna Schmit

Optical Sciences Center, University of Arizona, Tucson, AZ 85721, USA

(Received 19 April 1995; revised version received 23 August 1995; accepted 23 August 1995)

ABSTRACT

Spatial phase-measurement interferometry techniques commonly used in non-destructive testing are affected by a number of fundamental error sources. This paper focuses on the major limitations for phase calculations using standard N-point algorithms. These limitations include: the wrong carrier frequency, unequally spaced fringes, detector non- linearities and variations in the dc fringe intensity. The character and magnitude of these errors are quantified by using computer simulations. A displacement measurement of a bent plate using different algorithms on the same data shows that practical limitations are unequally-spaced fringes and variations in dc intensity and fringe visibility. These errors limit the resolution of this type of measurement to about a tenth of a wave r.m.s.

1 INTRODUCTION

Many different techniques for quantitative phase measurement from fringe patterns have been developed.‘-3 A wide range of applications for these techniques in optical testing and metrology makes them important for non-destructive testing. These techniques can be classified into two groups: temporal and spatial. Temporal phase-measurement (TPM) techniques measure the phase of a single point in an interferogram as the phase difference between the test and reference beams is changed in a controlled way. The most popular type of these are N-frame techniques which measure a sequence of interferograms with known relative phase shifts. Spatial phase-measurement (SPM) techniques extract the phase information from a single interferogram which has a

365

366 Katherine Creath, Joanna Schmit

large number of tilt fringes acting as a carrier frequency. There are two types of SPM techniques. One of these processes the fringe data in the spatial domain using the same equations as the N-frame TPM techniques. The other processes data in the Fourier domain using different algorithms and will not be discussed in this paper.

The advantage of the SPM techniques is that only one image is necessary instead of at least three images separated in time as in the TPM. This advantage enables the analysis of dynamic events or measurements in adverse conditions. Thus, these techniques are most commonly used in situations where a test object is not going to stay in a static state long enough to obtain multiple frames of fringe data. Another advantage is that SPM techniques do not require a special device to shift the phase. One disadvantage is that the requirements on the detector are more stringent than in TPM techniques. This is because the detector array must resolve a large number of fringes and the detector sensitivity should be uniform over the whole array. Because the single interferogram in SPM techniques must contain high frequency fringes which are obtained by adding tilt between the beams, unwanted aberrations may be introduced to the measured wavefront which will increase systematic errors. Often the magnitude of the phase error is not constant over the whole field, as we show in this paper.

The major errors encountered in SPM techniques are the wrong carrier frequency (miscalibrated tilt or phase shift error), unequally spaced fringes, non-linearities in the detection system, and variations in the dc fringe intensity. Because there are so many different phase- measurements techniques, only the most common algorithms have been studied in this work.

2 SPATIAL PHASE-MEASUREMENT TECHNIQUES

N-point techniques use a number of data points with different known phase shifts using N adjacent pixels in an array. The phase shift between pixels is generated by adding tilt fringes across the interferogram. For the class of algorithms which use 90” phase shifts between adjacent pixels, each fringe must cover four pixels. Likewise, for algorithms which assume a 120” phase shift, one fringe must cover three pixels. This is getting close to the Nyquist sampling limit.

The intensity (irradiance) recorded by a detector for a single interferogram can be written as

I(4 Y) =&4x, L’){I + Y(-% Y) cos [COG, y) + (Y(x, y)]}, (I)

N-poinf spatial phase-measurement techniques 367

where 1(x, y) is the measured intensity at a single pixel x, y, 1,(x, y) is the unknown average intensity, y(x, y) is the unknown fringe visibility, cp(x, y) is the unknown phase of the measured wavefront, and a(x, y) is the known phase shift between the test and reference wavefronts. Three unknowns (I”, y, cp) imply at least three data points are needed to find the phase distribution. There are a number of different equations which can be used for these techniques.

Since a minimum of three points are necessary to solve eqn (1) for phase, the phase measurement can be done using the following equation:

cp = tan ’ [

fi (4 - 12) 1 21, - I* - I3 ’

(2)

where I,, Z2 and Z3 are intensities recorded at either a single detector point in three different interferograms or at all three adjacent points in single interferogram with phase shifts of (Y = O”, 120” and 240 (0,2x/3, 4x/3). This technique will be referred to as the three-point (120”) technique. (Note that the phase shift will only be used in reference to this technique because all of the other algorithms assume 90” phase shifts.)

Since relative phase shifts of 90” (z/2) between data points are the most common, phase can also be determined using three points with phase shifts of 1y = 0”, 90” and 180” (0, 7r/2, z). The phase would then be calculated using

I, - I? cp = tan’ I_I .

[ 1 I 2

This technique will be referred to as the three-point technique. The very common four-point technique uses four intensity values

with 7r/2 relative phase shifts between points. It is written in the form

(4)

A technique which is comparably insensitive to many systematic errors based on the work of Schwider et al4 was introduced by Haritharan et al.’ and is widely used in present-day commercial interferometers. This five-point technique employs five data points which have relative shifts of 90”. This simple algorithm is written as

cp = tan’ {

2[& - 141 . 21, - I, - z, I

The last N-point technique used for this work was proposed by

368 Katherine Creath. Joanna Schmit

Car& and assumes a constant phase shift from point to point which is not necessarily equal to 90”. The phase is retrieved by applying the equation

cp = tan ’ 1 {[(I1 - 4) + (12 - 4)1[3G - !A) - (4 - 14)lY

CL + 4) - (4 + 14) 1. (6)

As with most phase-measurement techniques, these N-point techniques determine the phase modulo 2~. Phase ambiguities must then be removed using a phase unwrapping technique. In addition, the carrier frequency tilt fringes will result in a large tilt of the measured surface. The tilt due to the carrier fringes then needs to be subtracted in order to obtain the test surface.

3 EFFECTS OF MAJOR FUNDAMENTAL ERRORS

Errors always affect phase measurement calculations. In N-point techniques the signal which we want to extract contributes significantly to the error, These errors can be analytically determined and have been shown by Larkin & Oreb to depend upon the harmonics of the signal.’ However, for the purposes of our work, we prefer to simulate the various errors we encounter so that we can see their interaction. This also helps determine digitization errors and errors due to the finiteness of the data set.

3.1 Miscalibrated tilt (phase shift) error

Ideally, in N-point techniques, the phase shift (phase difference) between adjacent pixels should equal (Y (either 120” or 90” for the agorithms discussed in this paper). This implies that each fringe should cover three or four pixels depending upon the algorithm. A change in wavefront tilt will cause a change in the phase shift between pixels. This is illustrated in Fig. 1 for 90” phase shifts between pixels. Introducing a

CALIBRATED

l/4 h TILT MISCALIBRATION

(12.5%)

FFg. 1. Illustration of tilt miscalibration.

N-poim spatial phase-measurement rechniques 369

different fringe spacing is equivalent to miscalibration of the phase shifter in the N-frame TPM techniques. The difference between the tilt needed for (Y phase shifts and the actual tilt (the miscalibration error) will be noticeable in the retrieved phase. This residual tilt can be subtracted from the phase data. However, high frequency errors due to the miscalibrated tilt will be very noticeable.

Two error terms are observed which depend on the amount of miscalibrated tilt. They are a high frequency error (at twice the carrier fringe frequency when the miscalibration is small) and an envelope error function modulating at four times the phase error frequency (four ripples for each fringe of miscalibration in the carrier frequency). Figure 2 shows the phase error for one wave of additional tilt across the 256 points which have 85.3 waves of tilt (120” phase shifts) or 64 waves of tilt (900 phase shifts) in the carrier frequency. All tilt has been removed from the retrieved phase by fitting the data to a line in a least-squares sense. Because of the sampling frequency, the three-point (120”) algorithm shows a slower high frequency oscillation and the

3-Point (120”)

3-Point

4-Point

0 90 180 270 360

Phase of Miscalibrated Tilt (Degrees)

(a)

3-Point (120”)

3-Point

0 90 180

Phase of Miscalibrated Tilt (Degrees)

Fig. 2. (a) One wave tilt miscalibration over 256 data points. (b) Expansion of (a) showing detail of high frequency error.


Miscalibrated Tilt Error (%)

Fig. 3. Peak-to-valley tilt miscalibration

3-Point (120”)

3-Point

4-Point

5Point Carrh

error.

envelope function looks more complicated. The errors are the same as for the 90” techniques, but they are aliased, and thus downshifted in frequency from the 90” techniques. The amplitudes of the errors for a given phase shift miscalibration are the same as in the N-frame TPM techniques; however, the character of the error is much different due to the fast phase changes from pixel to pixel.” The CarrC technique has no error because it automatically compensates for this type of error.

Peak-to-valley (PV) phase errors are presented in Fig. 3 for phase shift errors between adjacent pixels in the range -20% to +20%. The PV error functions are the same as for the N-frame TPM techniques with the Carr6 technique (PV = 0), and five-frame techniques are the least sensitive to a miscalibrated phase shift. The most sensitive to miscalibration error is the three-point (120”) technique. This is mostly due to the sampling interval and how close this frequency is to the Nyquist limit. When 90” phase shifts are used, the sampling frequency increases and the error decreases for the N-point techniques. What this study shows is that whenever the carrier frequency in the interferogram is not correct, there will be sinusoidal errors, of the type shown in Fig. 2, present in the calculated phase. This also has implications for the measurement limits of this technique in terms of surface deviations.

3.2 Unequally spaced fringes

Unequally spaced fringes are similar to having a miscalibrated tilt, except that the error is not constant across the interferogram. The amount of error in any one calculated point depends upon the phase

N-point spatial phase-measurement techniques

(4

3 3-Point

$

(120”)

3-Point

E 0. 4-Point W

8 5-Point m z Carrb

0 128 256

Pixel Number

371

3-Point

0 50 100

Pixel Number

Fig. 4. (a) Error due to five waves of added quadratic phase. (b) Close-up of (a) for the two three-point techniques.

differences between the N pixels surrounding the point where the phase is calculated. To illustrate this effect, five fringes worth of quadratic phase were added across the array for all of the algorithms. For the 90 phase shift algorithms, the phase shift starts out at 90” at pixel number 1 and gradually increases to a value of about 104” when five fringes of quadratic phase are added. The resulting phase error (Fig. 4a) shows the four cycles per fringe that the linear error showed. The difference is that the spacing varies because the fringe spacing varies. As the phase shift between pixels gets larger, the error increases linearly with the phase shift error. There is also a high frequency oscillation at twice the carrier fringe frequency as with the miscalibrated tilt. [This is aliased to a frequency of 1.5 times the carrier fringe frequency for the three-point (120”) technique.] A magnified view of the phase error in the four-point technique for this example is shown in Fig. 4b. The PV errors for any point in an interferogram can be estimated from Fig. 3 by determining the local amount of miscalibrated tilt. The phase error due to the unequally spaced fringes is not constant over the whole interferogram and depends strongly on the local slopes of the measured wavefront.


I’ = I (Linear)

I’ = I + 6 I’ (2nd Order)

Detector Nonlinearity

6 = -25%

0.0 0.5 1.0 Incident Intensity

Fig. 5. Actual versus recorded intensity values for a second-order detection non- linearity.

3.3 Detection non-linearities

A detector can have a non-linear response to the incident intensity. A partial solution to this problem is to adjust the gain, enabling the camera to work in the most linear portion of gain curve. The most common non-linearity in a detector is of second order and can be written as I’ = I + SZ*, where I’ is detected intensity, I is the incident intensity, and S is a normalized error in the detected intensity. Figure 5 shows recorded intensity versus the actual intensity and the effect on the fringe intensity.

Detection non-linearity errors also produce high frequency errors. Figure 6 shows the error functions for the three-point technique with 10% second-order detection non-linearity. For SPM techniques, the detection non-linearity error has a high frequency signature which is related to the carrier fringe frequency. The amplitude of the error depends upon the initial phase of first sampling point. For example,

0.01 3 F

itial Phase

g 0”

k 0.00 45” W

t 22.5”

iz -0.01

0 1 2

Number of Fringes

Fig. 6. Ten per cent second-order detector non-linearity for three-point technique using sampling points with different initial phases.

N-point spatial phase-measurement techniques 373

-0.01 0 180 360

Phase (Degrees)

Fig. 7. Relationship between initial phase and maximum phase error for both three-point techniques.

Fig. 7 illustrates the magnitude of the phase error due to a detection non-linearity as a function of initial phase for 10% detection non- linearity using the three-point and three-point (120”) techniques. No error will be seen for the three-point technique (solid line) when the initial phase is 45” and there is no tilt miscalibration or unequally spaced fringes. When the initial phase is o”, the error will be a maximum. When the samples are unequally spaced, the detector non-linearity error will be more prevalent.

An interesting case arises for the three-point (120”) technique. This technique is insensitive to second-order detection non-linearity when the tilt is properly calibrated, but for a different reason. The three- frame (120”) technique is, in fact, sensitive to second harmonics even for a correct phase shift.‘,’ The character of the phase error for one fringe (360“ tilt in phase) in the interferogram and a 10% detection non-linearity is depicted by the dashed line in Fig. 7. Three identical ripples in the phase error (every 120”) are noticed for this one intensity fringe. This indicates that the error has a constant value for each of the three samples. The constant phase offset depends upon the initial phase and is of no consequence because the measurement is relative and the mean is subtracted out. This indicates the relative insensitivity of this technique to detection non-linearities. However, when the fringes are unequally spaced, this error will not stay hidden.

The PV phase errors for a range of second-order detection non- linearities are presented in Fig. 8 for different initial phase values. There is no noticeable error for the three-point (1209, four-point and five-point techniques when the correct carrier-fringe frequency is used;“’ however, when tilt miscalibration is present, the detector non-linearity will generate an error in addition to the tilt miscalibration error. Since it is difficult to control the initial phase for the fringe sampling with real

374 Katherine Creath. Joanna Schmit

3 0.050

g

‘0 k 0.025

8 E

h 0.000

Initial Phase

+ 22.5”

-20 -10 0 10 20

2nd-Order Nonlinear Detector Error (%)

Fig. 8. Peak-to-valley second-order non-linear detector error for three-point technique using sampling points with different initial phases.

data, it is possible and likely that the maximum error will be obtained. This maximum PV error versus the amount of second-order non-linear detection error with no tilt miscalibration is plotted in Fig. 9. The maximum error amplitude for the N-point techniques is equal to the equivalent N-frame technique.

3.4 DC intensity variation across fringes

It is nearly impossible to have a uniform intensity distribution across an entire image plane. All of the commonly used phase measurement algorithms assume that the dc intensity is the same for all of the samples used to calculate the phase. Because the N-point techniques

7 0.050

g b wt 0.025

f a’

2 0.000

3-Point (120”)

4-Point

5-Point

/

-20 -10 0 IO 20

end-Order Nonlinear Detector Error (%)

Fig. 9. Maximum peak-to-valley second-order non-linear detector error with no tilt miscalibration.

N-point spatial phase-measurement techniques 37s

I? 3-Point (120”)

$ 3-Point

g 0. 4-Point w

% 5-Point ([I if Carrb

0 128 256

Pixel Number

Fig. 10. Phase error due to Gaussian dc intensity variation.

used adjacent pixels to determine phase, phase errors can occur if the dc intensity varies across the image plane. To illustrate this limitation, we have simulated a change in dc intensity across the array as a Gaussian. This simulates the Gaussian falloff in intensity for most laser beams. To exaggerate the error, the value of the dc intensity at the edge of the array is set to fall off to 1% of the center.

The phase error due to the dc intensity variation is shown in Fig. 10. This error becomes linearly larger from the center to the edge of the array and contains a high frequency error at the fringe frequency. All of the techniques are sensitive to this error. At the edges of the array, the error has a maximum PV value of about 0.03 waves for all of the techniques. The three-point (120”) and Carre techniques are marginally better than the other techniques: however, they have a small residual added tilt. In places on the interferogram where the dc intensity varies little, the error is small; and where the intensity varies most the error is largest, as expected. This error can be kept below 0.01 PV waves by keeping the dc intensity within a factor of 2 at the edges relative to the center.

4 OUT-OF-PLANE DISPLACEMENT MEASUREMENT EXAMPLE

Up to this point, this work has concentrated on computer simulations of

errors to get an idea of the measurement limitations. These algorithms have been implemented with a number of different interferometers. As an example of how these techniques work on real data, a grating interferometer was used in conjunction with a reference beam to


Fig. 11. Displacements fringes due to out-of-plane distortion.

determine the out-of-plane displacement of a flat plate which has been bent slightly.” A binary representation of the measured fringe pattern (digitized to 8 bits) is shown in Fig. 11. Notice that these fringes are not exactly straight and that their widths vary, which indicates variations in dc fringe intensity and fringe visibility across the image. Figures 12 and

80

I

20 I I I I

40 60 60

pixels

Fig. 12. Displacement map using the four-point technique (PV = 4.0 waves).

N-point spatial phase-measurement techniques

80

I I I / / , _i

0 20 40 60 60

pixels

Fig. 13. Displacement map using the five-point technique (PV = 3.8 waves).

13 show displacement maps determined using the four-point and five-point algorithms. In both, there are noticeable oscillations due to one or more of the fundamental errors outlined in this paper. The predominant ones are unequally spaced fringes and variations in dc intensity and fringe visibility across the data set. The five-point algorithm has much less error than the four-point algorithm. The difference between these two calculations is shown in Fig. 14. An error at the fringe frequency dominates. It has a PV of 1.4 waves and an r.m.s. of 0.1 waves. This example illustrates the limitations of this technique. However, there are times when a resolution of 0.1 wave

0 1 1 I / / I I I

0 20 40 60 80

pixels

Fig. 14. Difference between Figs 13 and 14 (PV = 1.4 waves; r.m.s. = 0.1 waves).


r.m.s. is perfectly acceptable when the object you are measuring will not stay in a static state long enough to use temporal techniques.

5 CONCLUSIONS

A study of limitations in spatial phase-measurement techniques shows that there are sizable errors which need to be considered. Choosing a fringe-pattern analysis technique requires weighing the effects of different errors to determine the best technique. Each application needs to be considered separately. Tilt (phase shift) miscalibration or unequally spaced fringes does not result in significant error when the Carre or five-point techniques are used. However, it causes large errors in both three- and four-point techniques. When there are detector non-linearities present, the three (120”)-, four- or five-point techniques are preferred. For variations in dc fringe intensity all techniques have the same sensitivity with the exceptions of the three-point (120”) and Carre techniques, which are a bit better. Errors due to this effect can be minimized by keeping the intensity at the edges of the interferogram greater than 50% that at the center. The test of any of these algorithms comes by measuring real data. The example shown in this paper clearly shows the limitations. However, for most applications in non- destructive testing, objects do not stay stationary and these techniques can be very useful.

REFERENCES

1. Robinson, D. W. & Reid, G. T. (eds), Interferogrum Analysis, Ch. 5, pp. 141-193, Ch. 4, pp. 94-140, Ch. 6, pp. 194-229. Institute of Physics, Bristol, 1993.

2. See, for example, Creath, K., Phase-shifting holographic interferometry. In Holographic Interferometry, Principles and Methods (edited by P. K. Rastogi), Ch. 5, pp. 109-150. Springer, Berlin, 1994.

3. Schwider, J., Advanced evaluation techniques in interferometry. In Progress in Optics (edited by E. Wolf), Vol. XXIX, pp. 271-359. Elsevier, Amsterdam.

4. Schwider, J., Burow, R., Elssner, K.-E., Grzanna, J., Spolaczyk, R. & Merkel, K., Digital wave-front measuring interferometry: some systematic error sources. Appl. Opt., 22 (1983) 3421-3432.

5. Hariharan, P., Oreg, B. F. & Eiju, R., Digital phase-shifting interferometry: a simple error compensating phase calculation algorithm. Appl. Opt., 26 (1987) 2504-2505.

6. Carre, P., Installation et utilization du comparateur photoelectrique et

K-point spatial phase-measurement techniques 379

interferentiel du Bureau International des Poids et Mesures. Metrologia, 2 (1966) 13-23.

7. Larkin, K. G. & Oreb, B. F., Design and assessment of symmetrical phase-shifting algorithms. J. Opt. Sot. Am., A9 (1992) 1740-1748.

8. Schmit, J., Creath, K. & Kujawinska, M., Spatial and temporal phase- measurement techniques: a comparison of major error sources in one- dimension. Proc. SPIE, 1755 (1992) 202-211.

9. Hibino, K., Oreb, B. F., Farrant, D. I. & Larkin, K. G., Phase shifting for nonsinusoidal waveforms with phase-shift error. 1. Opt. Sot. Am., Al2 (1995) 761-768.

10. Schmit, J. & Creath, K., Extended averaging technique for derivation of error-compensating algorithms in phase-shfting interferometry. Appl. Opt., 34 (1995) 3610-3619.

11. Schmit, J. & Patorski, K., A novel approach to high-sensitivity grating interferometry. Opt. Lasers Engng, to be published.

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