Modified three-step iterative algorithm for phase-shifting interferometry

Research Article

Vol. 54, No. 18 / June 20 2015 / Applied Optics

5833

Modified three-step iterative algorithm for phase-shifting interferometry in the presence of vibration QIAN LIU,* YANG WANG, JIANGUO HE,

AND

FANG JI

Institute of Machinery Manufacturing Technology, China Academy of Engineering Physics, 64th Mianshan Road, Mianyang, Sichuan 621000, China *Corresponding author: [email protected] Received 25 March 2015; revised 20 May 2015; accepted 27 May 2015; posted 28 May 2015 (Doc. ID 236883); published 19 June 2015

To suppress the sensitivity of phase-shifting interferometry to mechanical vibration, a modified three-step iteration algorithm (MTIA) is proposed. The wavefront phase, x- and y-directional phase shifts are calculated in three individual steps in an iteration cycle. MTIA solves the tilt factors of phase shift through orthogonal decomposition and linear regression, avoiding the nonlinear optimization. The advantage of MTIA lies in its ability of calculating and compensating tilt-shifting error and fringe contrast variation caused by vibration. The simulations show that MTIA could provide more accurate results than an algorithm without contrast compensation. The superiority of MTIA is demonstrated experimentally. The experiment results show that MTIA could suppress the effect of vibration on surface measurement, which is of amplitude larger than the wavelength and of frequency over a large range. MTIA manifests good insensitivity to exposure time increase as well. © 2015 Optical Society of America OCIS codes: (120.3180) Interferometry; (120.2650) Fringe analysis; (050.5080) Phase shift; (120.6650) Surface measurements, figure. http://dx.doi.org/10.1364/AO.54.005833

1. INTRODUCTION Phase-shifting interferometry (PSI) has been regarded as the most accurate technique for wavefront measurement and has achieved widespread use in surface profilometry. PSI attains high resolution in depth and transverse space through collecting a set of interferograms in the temporal domain. When PSI is being carried out, the phase-shifting device, such as a piezoelectric transducer or wavelength-tuning laser, should make phase increment accurately be equal to preset value. Since phase shifting is implemented in a temporal domain, PSI is sensitive to time-dependent phenomena, such as mechanical vibration [1,2]. Vibration is probably the most significant error source for PSI subjected to environmental vibration. This sensitivity of PSI to vibration impairs the application of PSI to many practical cases, such as measurement of optics with very long focus length or in situ measurement. Much research has been carried out to develop methods insensitive to vibration for PSI. Transforming temporal phase shifting into the spatial domain [3–5] could measure wavefront instantaneously but suffers from its elaborate system. Considering the fact that the wavefront is invariable during the measuring period, researchers have developed many algorithms to extract wavefront phase from interferograms subjected to 1559-128X/15/185833-09$15/0$15.00 © 2015 Optical Society of America

vibration. These algorithms include detecting phase shifts from spatial-carrier interferograms [6–8], calculating wavefront phase by iteration [9–13] and some other methods [14,15]. These algorithms compensate the relative tilt between the test and reference wavefronts and have better performance than those algorithms that only consider the piston phase shifting. These algorithms assume the stationarity of the background and contrast of interference fringes, neglecting the contrast fluctuation of interferograms subjected to vibration. A CCD camera of an interferometer usually works in an integrating mode to collect enough light energy to form a high signalto-noise ratio (SNR) interferograms. Due to the averaging effect of a CCD camera, the interference fringe contrast varies with respect to phase-shifting speed, and the variation is unknown owing to the randomicity of environmental vibration. This is the reason why some fringes are blurred in an unsteady environment. The contrast variation is another significant error source of PSI [16]. Shuttering the camera is an effective way to suppress the contrast variation [13]; accordingly, a high-power laser source is required, otherwise, the SNR of the interferograms is significantly low. Hence, it is important to seek a mathematical method to solve the contrast variation problem for PSI in the presence of vibration. A spatial-carrier method

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Vol. 54, No. 18 / June 20 2015 / Applied Optics

[17] to detect the phase shifts and contrast fluctuation factors have been proposed. However, interferograms with closed fringes are ruled out of this method, since a spatial carrier is required. A contrast compensation iteration algorithm (CCIA) [18] has been developed to calculate and compensate contrast fluctuation, but the tilt shifting is not considered in CCIA. In this paper, we proposed an iterative algorithm to extract wavefront phase from interferograms subjected to vibration. The algorithm is a modification of our previous algorithm: a three-step iterative algorithm (TIA) [12]. TIA could calculate wavefront phase from tilt-shift interferograms by solving linear least-squares equations but could not handle interferograms with contrast fluctuation. In modified TIA (MTIA), CCIA is introduced to calculate and compensate the contrast fluctuations. We first describe MTIA and then verify the algorithm with simulations and experiments. At the end, the error of MTIA is discussed.

In this step, the time-dependent parameters are known. The theoretical intensity of interferogram is I tj
x; y  αj A
x; y  βj b
x; y cos Δj
x; y  βj c
x; y sin Δj
x; y;

where b
x; y  B
x; ycos ϕ
x; y, c
x; y  −B
x;y sin ϕ
x; y. Here, αj , βj , and Δj
x; y are known. Omitting the deriving process, we directly give the least-squares equation:

I tj
x; y  αj A
x; y  βj B
x; y cosφ
x; y  Δj
x; y; (1) where A
x; y and B
x; y are the background and modulation amplitude, respectively, and ϕ
x; y is the wavefront phase we seek. αj and βj are fluctuation factors of background and modulation, respectively. The fringe contrast, V j  βj B∕
αj A, is directly proportional to βj ∕αj . The superscript t denotes theoretical value. Δj
x; y is the phase shift at the coordinate
x; y in the jth interferogram. We assume that the measured surface is a rigid body, i.e., the wavefront is not distorted in the measurement. In the spatial domain, Δj
x; y is determined by a plane, named as the phase-shifting plane, which could be decomposed into phase shifts in the x and y directions. Hence, we have 8 < Δj
x; y  Δxj
x  Δyj
y Δxj
x  kxj x  δxj ;
2 : Δyj
y  kyj y  δyj where kxj and kyj are the x and y directional tilt factors of the jth phase-shifting plane, and δxj and δyj are translational values. In Eq. (1), except interference intensity I tj , all the other parameters are unknowns. A, B, and ϕ are space-dependent unknowns, while αj , βj , kxj , kyj , δxj , and δyj are time-dependent unknowns. To calculate the phase shift, the nonlinearity involved by tilt shifting is the key problem to solve. Taylor series expansion [9], subregion division [10], and nonlinear leastsquares fittings [11,13] have been applied to solve the problem. In TIA, we have decoupled the phase shifts in orthogonal directions and determined the tilt factors with linear regressions [12]. Here, two additional unknowns, contrast fluctuation factors, are involved in the problem, and the MTIA is described as follows. Each cycle of MTIA could be divided into three steps. Step 1. Calculate wavefront phase with phase shifts and contrast fluctuation factors known

U  M−1 V;

(4a)

U  A; b; cT ;

(4b)

where

P P 3 αj βj cos Δj αj βj sin Δj j 7 6P j P P j2 7 6 α β cos Δ 2 2 β cos Δ β sin Δ cos Δ 7; 6 j j j j j j j j M6 j 7 j j P 2 P 2 2 5 4P αj βj sin Δj βj sin Δj cos Δj βj sin Δj P

2

α2j

j

2. ALGORITHM DESCRIPTIONS For two-beam interference, considering the tilt-shift and contrast fluctuation, the intensity of the jth interferogram could be theoretically expressed as

(3)

j

j

(4c) and V

P j

αj I j

P βj I j cos Δj j

T P βj I j sin Δj : j

(4d)

For simplicity, the coordinate
x; y of U, V, M, A, b, c, I j , and Δj are omitted in Eqs. (4). The superscript T denotes the transpose of the matrix. From vector U, the wavefront phase could be calculated as φ
x; y  tan−1 −c
x; y∕b
x; y:

(5)

In this step, the contrast fluctuations are compensated when calculating the wavefront phase. Step 2. Calculate x-directional phase shift with wavefront phase and y-directional phase shift known As shown in Fig. 1, for all the pixels in the xth column, the x-directional phase shift, Δxj
x  kxj x  δxj , are uniform and are independent of y. With wavefront phase ϕ
x; y and y-directional phase shift Δyj
y known, the x-directional phase shift Δxj
x could be calculated with CCIA [18]. Interference aperture

Active pixels

Frame Jth index 2nd 1st

yth

Row index

x

y 1st xth Column index Fig. 1. Schematic of tilt shift decomposition in orthogonal (x and y) directions.

Research Article

Vol. 54, No. 18 / June 20 2015 / Applied Optics

Similar to CCIA, we force the background and modulation amplitude in the xth column to be uniform, i.e., A
x; y  B
x; y  1. We also assume the contrast fluctuation factors α and β are functions of x. Hence, the theoretical intensity of the xth column in the jth interferogram is I j0
x; y  αj
x  bj0
x cos φj0
x; y  c j0
x sin φj0
x; y; (6) bj0
x

Uj0
x  Mj0−1
xVj0
x;

(7a)

where Uj0
x   αj
x bj0
x

c j0
x T ;

(7b)

P P 3 cos φj0 sin φj0 N
x y 7 6P P P y 0 6 cos φ 0 2 φ0 07 cos sin φ cos φ 0 7; 6 j j j j Mj
x  6 y 7 y y P P 5 4P sin φj0 sin φj0 cos φj0 sin2 φj0 2

y

y

y

(7c)

Vj0
x 

P Ij y

P I j cos φj0 y

P y

I j sin φj0

T

:

(7d)

In Eqs. (7) the coordinate
x; y of I j and ϕj0 are omitted. N
x in matrix Mj0
x represents the total number of active pixels in the xth column. The summation sign represents the sum over the active pixels in the yth row. The x-directional phase shift in the xth column of the jth interferogram is Δxj
x  tan−1 −c j0
x∕bj0
x:

(8)

Uj0
x

The first element in is the background fluctuation factor αj
x. The amplitude fluctuation factor of the xth column in the jth interferogram is qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (9) βj
x  bj02
x  c j02
x: After calculating the x-directional phase shift Δxj for all the columns in the jth interferogram, unwrapping Δxj and making linear regression, we could obtain tilt factors k xj and translational value δxj . Step 3. Calculate y-directional phase shift with wavefront phase and x-directional phase shift known This step is similar to Step 2. The wavefront phase ϕ
x; y and x-directional phase shifts Δxj
x  kxj x  δxj are known. For the pixels in the yth row of the jth interferogram, we make A
x; y  B
x; y  1 and make denotations ϕj0 0
x; y  ϕ
x; y  Δxj
x, bj0 0
y  βj
y cos Δyj
y, and c j0 0
y  −βj
y sin Δyj
y. The least-squares equation for the yth row of the jth interferogram is Uj0 0
y  Mj0 0−1
yVj0 0
y; where

bj0 0
y

c j0 0
y T ;

(10b)

P P 3 cos φj0 0 sin φj0 0 N
y x 7 6P P P x 00 00 6 cos2 φj0 0 sin φj cos φj0 0 7 Mj0 0
y  6 x cos φj 7; x P 5 4P P x 00 00 00 2 00 sin φj sin φj cos φj sin φj 2

x

x

x

c j0
x

where  βj
x cos Δxj
x,  −βj
x sin Δxj
x, and ϕj0
x; y  ϕ
x; y  Δyj
y. Here, αj
x, bj0
x, and c j0
x are unknown and ϕj0
x; y is known. The least-squares equation for calculating x-directional phase shift is

and

Uj0 0
y   αj
y

5835

(10a)

(10c) and Vj0 0
y

P Ij  x

P I j cos φj0 0 x

P x

I j sin φj0 0

T

:

(10d)

In Eqs. (10) the coordinate
x; y of I j and ϕj0 0 are omitted. N
y in matrix Mj0 0
y represents the total number of active pixels in the yth row. Similarly, from vector U 0 0
y, we could get the y-directional phase shift Δyj
y and contrast fluctuation factors αj
y and βj
y. And then the y-directional tilt factors kyj and translational value δyj could be determined after unwrapping and linear regression. In Steps 2 and 3, we obtain αj and βj for each column and each row. Consequently, the averages of fluctuation factors are delivered to Step 1. When measurement encounters severe vibration, the contrast may deviate within an interferogram depending on the coordinate, named as intraframe contrast nonuniformity. Consequently, the obtained factors of row or column contrast of the same interferogram are different. From the deviations of β
x∕α
x or β
y∕α
y, the interferograms with nonuniform contrast are filtered out finally when the wavefront phase are calculated. The initial value of the iteration could be prejudgments of either phase shifts or wavefront phase. In an environment of slight vibration, the initial value of phase shifts could be the preset values. In an environment of strong vibration (large amplitude or high frequency), the actual values of phase shift may greatly deviate from preset values, in which case the iteration may not converge. For this case, a better choice of initial value is the estimation of wavefront phase. The wavefront could be estimated from spatial-carrier phase-shifting (SCPS) [19] or Fourier transform (FT) [20] methods, which could extract the wavefront phase from the single-shot interferogram. The instantaneous measurement with SCPS or FT method is resistant to vibration but suffers from obvious errors. Although there is obvious error in the wavefront phase estimated with SCPS or FT methods, the estimation is accurate enough as the initial value for MTIA. The convergence criterion of the iteration is the changes of calculated phase shift value between two adjacent cycles [12]. A threshold could be given before the iteration. For clarity, the flow of MTIA is exhibited in Fig. 2. Three steps in the MTIA cycle could follow different sequences, depending on the type of input for iteration. If the input is phase shift values, the sequence of cycle could be 1–2–3 or 1–3–2. If the input is wavefront phase estimation, the sequence could be 2–3–1 or 3–2–1. In MTIA, the decoupling of tilt shifting is realized by orthogonal decomposition and linear regression, and the

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Vol. 54, No. 18 / June 20 2015 / Applied Optics

Initial value (Wavefront phase or phase shifts)

ϕ(x,y) Average

kyj and δyj

αj, βj Δxj(x)

Step 1 CCIA

Step 2

Fig. 3. Computer-generated interferograms in simulations.

Step 3

Δyj(y) αj, βj

kxj and δxj

Convergence value

Fig. 2. Flow of MTIA.

contrast fluctuation calculation and compensation are realized with introduction of uniform fluctuation factors. The significant difference between MTIA and TIA is that, in MTIA, the fluctuation factors of background and modulation are determined in Steps 2 and 3 and compensated in Step 1. The determination and compensation of fluctuation factors add little computation burden to the algorithm due to their simplicity. MTIA combines the advantages of TIA and CCIA and could solve the wavefront phase from interferograms with tilt-shifting and contrast fluctuations.

phase error (profile of the 32nd row). The big error of the wavefront phase estimation with the SCPS method is suppressed and eliminated after several cycle iterations. After three cycles, the error reduces to a small amount and becomes almost stationary. For comparison, Fig. 5 shows the maximum residual errors of phase shift translational value with respect to iteration cycle number. The maximum residual error is defined as the maximum absolute value among the deviations of calculated phase shift from the exact value in one cycle. With MTIA, the error reaches 10−2 rad after only five cycles, while TIA has a larger calculation error due to its disability of compensating contrast fluctuation. Figure 6 represents the error map of a wavefront extracted with TIA and MTIA. The PV value of the error map with TIA is 3.63 × 10−2 wave, and the value with MTIA is only 7.8 × 10−3 wave. The simulation shows the improved performance of MTIA compared with TIA and verifies the capability of MTIA to handle the interferograms with contrast fluctuation. 4. EXPERIMENTS

3. COMPUTER SIMULATIONS In this section, the algorithm is first demonstrated with computer simulations. In the simulation, the wavefront phase is a spherical surface ϕ
x; y  2π
x 2  y 2 . The background A
x; y  140 exp−0.2
x 2  y 2  and the modulation amplitude B
x; y  110 exp−0.1
x 2  y 2 , where −1 ≤ x; y ≤ 1. Tilt factors of phase shifts and contrast fluctuation factors were generated randomly by computer. The distribution ranges of tilt factors and fluctuation factors are selected according to the observed data in experiments. The translational increase of phase shift is 1. With white noise of variance 4 introduced, eight interferograms were generated (see Fig. 3), of which the resolution is 64 × 64. The input of the iteration is the wavefront phase estimated from a spatial-carrier interferogram with the SCPS method. Phase dropout at the border of the wavefront could be supplemented with simple extrapolation, such as through the linear or spline method. The calculated result with MTIA after a 10-cycle iteration is exhibited in Table 1. The contrast fluctuation factors are normalized to the first interferogram. The maximum variance of the tilt factors is 0.2637 rad∕pixel, which indicates a 2.7 fringe variance for 64 × 64 resolution interferograms. The calculated result shows the feasibility of MTIA when coping with interferograms with tilt-shifting and contrast fluctuation. Figure 4 exhibits the evolution of the wavefront

In this section, the MTIA is first validated by experiments and then the resistance to vibration is investigated. First, two experiments were carried out to test the performance of MTIA applying to PSI in practical vibration. The experiments were conducted on a Fizeau-type interferometer, of which the aperture is Φ100 mm and the laser wavelength is 632.8 nm. The phase shift increase is π∕4, and 13 interferograms are collected in a measurement. In the fast mode, the CCD resolution is 250 × 250, the exposure time is 0.8 ms, and the frame rate is 65 Hz. The interferometer is set on a vibration-isolating platform. The measuring object is a flat surface of Φ50 mm. Figure 7 shows the exact surface, of which the PV value is 0.165 wave and the RMS value is 4.38 × 10−2 wave. In the experiment, a piezoelectric transducer was employed to generate vibration via pushing the mirror holder along the direction of the interference cavity. The vibration is a sinusoidal wave of 1 μm amplitude. And the frequencies of vibration are 2 and 10 Hz, respectively, in two measurements. Before the measurements, a spatial-carrier interferogram was captured under vibration. The wavefront phase was extracted with the SCPS method from the spatial-carrier interferogram, of which the PV value of the surface is 0.215 wave, which is much larger than the exact value. With the estimated wavefront phase as the initial value, the measured surfaces of two measurements were calculated after six cycle iterations and are shown in Fig. 8.

Research Article

5

−26.40 −8.74 7.00 0.956 1.075 23.85 19.59 4 0.900 1.113 −16.59 −10.01 3 1.008 0.895

4 3

−6.61 −10.05 2 0.999 0.999 3.08 2.67 1 0.952 1.157

2 Frame

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Initial

Error of profile (rad)

0 -0.5 0.5

Cycle 1

0 -0.5 0.2

Cycle 2

0 -0.2 0.02 0 -0.02

Cycle 3

Cycle 4

0.02 0 -0.02 10

20

30 Pixel

40

50

60

Fig. 4. Evolution of wavefront profile error.

101 Max residual error (rad)

7

−9.64 −15.60 5.99 1.047 1.016 15.88 13.46 4.99 0.998 1.147

6 5

23.85 19.59 4.00 0.902 1.127

3

−6.61 −10.04 1.998 1.001 1.011

2

3.10 2.69 0.990 0.955 1.171

8

−26.37 −8.75 7 0.951 1.065

7

−9.65 −15.63 6 1.043 1.002

6

15.87 13.45 5 0.992 1.136

4

0.5

TIA MTIA

100

10-1

10-2 1

k x
×10−2 rad∕pixel k y
×10−2 rad∕pixel δ
rad α β

Calculated Values Real Values Table 1. Calculated Results with MTIA

−16.61 −10.00 3.00 1.011 0.900

8

Vol. 54, No. 18 / June 20 2015 / Applied Optics

2

3

4 5 6 7 Iteration cycles

8

9

10

Fig. 5. Maximum residual error versus iteration cycle.

The PV and RMS values of the surface in the first measurement are 0.168 wave and 4.32 × 10−2 wave, respectively. The values in the second experiment are 0.169 wave and 4.38 × 10−2 wave, respectively. The distributions and values of the surfaces measured in the experiment meet the exact surface well, although some slight ripples are present. The residual ripples may result from calculation errors and intraframe contrast nonuniformity and could be simply removed with phase averaging. As a comparison, the surfaces reconstructed with TIA are exhibited in Fig. 9, of which the ripples are obvious, since the contrast fluctuations are not compensated in TIA, and the ripples could not be removed with phase averaging. The comparison shows the superiority of MTIA over TIA. MTIA, meanwhile, provides the phase shifts and contrast of the interferograms collected in a vibration environment. The calculated phase shifts and contrast fluctuation factors in the

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Vol. 54, No. 18 / June 20 2015 / Applied Optics

x 10-2

x 10-3

1.5

10 20

1

10

0.5

20

0

30

3 2 1

30 0

-0.5 40

40

-1

-1 50

50

-2

60

-3

-1.5 60

10

20

30

40

50

-2 (wave)

60

10

20

(a)

30

40

50

60

(wave)

(b)

Fig. 6. Wavefront error map with (a) TIA and (b) MTIA.

0.08 0.06

50

0.04 100

0.02 0

150 -0.02 -0.04

200

-0.06 50

100

150

(wave)

200

Fig. 7. Exact value of surface in the experiments.

experiment are shown in Figs. 10 and 11, respectively. The tilt caused by vibration is smaller under low-frequency vibration than that under high-frequency vibration. The maximum tilt factor is 0.048 rad∕pixel, which indicates a two-wave

wavefront tilt over the aperture. Due to the displacement caused by mechanical vibration, the translational values of phase shifts greatly deviate from preset values. The large deviation may result in failure if the preset values of phase shift are selected as the input of iteration. The contrast is greatly influenced by vibration, especially high-frequency vibration. If the vibration frequency f v and frame rate f r satisfy the relation f v  0.5nf r ,
n  1; 2; …, the vibration influences little on translational value, but the tilt-shifting is still present, and the practical environment contains a wide variety of vibration. Hence, the vibration-resistant algorithm is necessary to extract wavefront phase for PSI in the presence of vibration. Increases of the phase-shifting speed and frame rate help to heighten the resistance of MTIA to vibration. Then, an experiment was carried out to statistically investigate the measurement errors of MTIA. First, the measurement error of MTIA under various harmonic vibrations was studied. In the experiment, the amplitude of vibration was 1 μm and the exposure time of CCD was 0.8 ms. The RMS error of a figure as a function of vibration frequency is shown in Fig. 12. The frequency is normalized to frame rate. The figure

0.08 0.06

50

0.08 50

0.06

0.04 100

0.02

0.04 100

0.02

0 150

-0.02 -0.04

200

0

150

-0.02 -0.04

200

-0.06 50

100

150

(a)

200

(wave)

-0.06 50

100

150

200

(wave)

(b)

Fig. 8. Extracted wavefront with MTIA from experiment under vibration of 1 μm amplitude. (a) 2 Hz frequency. (b) 30 Hz frequency.

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0.1

0.08 0.06

50

5839

0.08 50 0.06

0.04 100

0.02 0

150

0.04

100

0.02 0

150

-0.02 -0.04

200

-0.02 -0.04

200

-0.06

-0.06 50

100

150

200

50

(wave)

100

(a)

150

200

(wave)

(b)

Fig. 9. Extracted wavefront with TIA from experiment under vibration of 1 μm amplitude. (a) 2 Hz frequency. (b) 30 Hz frequency.

x 10-2 x y

2Hz

14

x y

30Hz

Translational value (rad)

Tilt factors (rad/pixel)

1 0 -1 -2 -3 -4

Preset value 2Hz 30Hz

12 10 8 6 4 2 0 -2

-5

0

2

4

6 8 10 Frame index

12

14

-4 0

2

4

6 8 10 Frame index

(a) Fig. 10.

14

(b)

Calculated phase shifts in the experiment in the presence of vibration.

of the Φ50 mm flat was measured with six phase averaging. The result indicates that the error of MTIA is almost constant below the frequency of 0.8 times frame rate and rises slowly when the frequency increases. The error of TIA is close to that of MTIA in the low-frequency region and rises quickly in the mid- and high-frequency region. Under high-frequency

vibration, MTIA could still retrieve wavefront phase, although the error become larger, while TIA fails due to very large retrieved phase error. Then, the measurement error versus shuttering speed was studied. The measurements under vibration of 0.4 μm amplitude and different frequencies were implemented

8

1.05

x 10-3 TIA MTIA

7 1

RMS error (wave)

Normalized contrast

12

0.95

0.9

0.85 0

2Hz 30Hz 2

4

6 8 10 Frame index

6 5 4 3 2 1

12

14

Fig. 11. Calculated contrast fluctuation factors in the experiment in the presence of vibration.

0

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Normalized vibration frequency

Fig. 12. RMS error of MTIA under harmonic vibration compared to TIA. The vibration frequency is normalized to frame rate.

x 10-2 MTIA 10Hz 20Hz 30Hz

1

TIA 10Hz 20Hz 30Hz

0.5

0

2

3

4

5

6

Exposure time (ms) Fig. 13. RMS error of MTIA and TIA versus exposure time under different vibrations.

x 10-3 4

x 10-5 Tilt factor Translational value

2

1

0

0

1 2 3 4 5 6 7 Phase shift distribution range s (rad)

Error of translational value (rad)

Error of tilt factors (rad/pixel)

2

0 8

Fig. 14. Calculation error of phase shift with respect to phase shift distributions.

with changing shuttering speed. The RMS errors versus exposure time are shown in Fig. 13. The error of MTIA keeps almost stationary, although the exposure time increases, while the error of TIA keeps stationary at the low-frequency region only and rises quickly at the high-frequency region when the exposure time increase. The experiment result indicates the resistance of MTIA to vibration over a large frequency range. The insensitivity of MTIA to exposure time shows its advantages of compensating the contrast fluctuation that TIA does not have. Despite the insensitivity to exposure time, MTIA may fail if the exposure time is larger than two times the oscillation period of vibration, since the fringes are blurred seriously due to the averaging effect of a camera. 5. ERROR ANALYSIS Calculation error of MTIA mainly comes from the solution of least-squares equations, i.e., Eqs. (4), (7), and (10). According to the Cramer principle, the solution error of linear equation will be suppressed if the determinants of matrixes M, M 0 0 ,

Tilt factor Translational value

1.5

3

1

2

0.5

1

0 1

x 10-2 4

x 10-3

2

0

1

2 3 Fringe number N

4

5

Error of translational value (rad)

1.5

RMS error (wave)

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Vol. 54, No. 18 / June 20 2015 / Applied Optics

Error of tilt factors (rad/pixel)

5840

0

Fig. 15. Calculation error of phase shift with respect to fringe number of interferograms.

and M 0 0 are very large. The determinant of M depends on the phase shifts and contrast fluctuation factors. Since the values of contrast fluctuation factors range in a small region near one, their influence on the determinant is weaker. Hence, the phase shifts have a dominant influence on the determinant of M. The determinants of M 0 and M 0 0 depend on wavefront phase and phase shifts at the same time. Computer simulations were carried out to statistically study the influence of diversity of phase shift and wavefront phase on the calculation error. In the simulations, the calculation error is represented by the error of calculated phase shift. The calculation error versus phase shift distribution range is shown in Fig. 14. The error decreases when the distribution range increases, and the error approaches a constant if the phase shifts are distributed in a range of at least π. The calculation error versus fringe number in an interferogram is shown in Fig. 15. For an interference pattern, a fringe means phase diversity of 2π. Interferograms containing 1.5 fringes generate small errors for MTIA. Therefore, to suppress calculation errors, the diversity of phase shift and wavefront phase should be ensured, i.e., the phase should not concentrate at a small region. 6. CONCLUSIONS To apply PSI to measurement in the presence of vibration, we have proposed a modified three-step iteration algorithm (MTIA), which could extract wavefront phase from interferograms with tilt-shifting and contrast variance. MTIA combines the best features of two existing methods: tilt-shifting decoupling and contrast compensation. The calculation efficiency is high, since only linear least-squares fittings are involved in the algorithm, and MTIA is compatible with an almost arbitrary wavefront. The simulation results show that MTIA could achieve higher accuracy than algorithms without contrast compensation. The experiments under practical vibration demonstrate the capability to suppress the wavefront phase error caused by vibration. MTIA could suppress the error caused by vibration over a large frequency space and is insensitive to a camera’s exposure time. The analysis indicates that, to ensure high accuracy, the phase shift and wavefront should keep phase diversity. Since no special hardware is required for the

Research Article interferometer, MTIA provides a high-accuracy and low-cost solution for the measurement with PSI in the presence of vibration.

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