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Phase-shifting interferometry

1. Introduction

Optical techniques have been employed to study heat and mass transfer in fluids. These techniques can be

broadly divided into three main classes. The first deals with the usage of dyes and bubbles to directly

visualize the motion of the fluid while the second deals with analyzing the frequency of radiation which

undergo changes, such as the Doppler shift, after getting scattered from particles. The third class includes

techniques that work with a monochromatic coherent light source such as a laser, and rely on the changes

in refractive index of the medium caused by changes in concentration, temperature or density variation.

Examples of these techniques are interferometry, shadowgraph and schlieren [1]. Here, change in

refractive index with the material density is made use of. Interferometry is appropriate when the change in

refractive index is very small while schlieren and shadowgraph allow large changes in refractive index.

In conventional interferometry the light beam from a laser is divided into two parts with the help of a

beam splitter. The first, known as the object beam, travels through the test section while, the other, known

as the reference beam travels through the compensation chamber. Interference between the reference and

the object beams takes place and interference fringes are formed. The interference pattern is recorded with

the help of a camera. One can extract information related to phase difference from the images. The

analysis of conventional interferograms can be tedious. It involves detection of coordinates of fringe

centers and the data is analyzed from this point onwards. The determination of the location of fringe

centers is simpler when the fringes are widely spaced but the number of data points is then, quite small.

The number of data points can be increased using wedge fringes wherein, a fixed number of fringes are

created by tilting the optical elements of the interferometer. This approach is useful when boundary

temperatures are clearly specified. The data analysis algorithm for post-processing requires

phase/refractive index information to be available on a rectangular grid. The fringe data has to be

subsequently interpolated to obtain the nodal values [2, 3]. Interpolation of data introduces additional

errors in image analysis.

Phase shifting interferometry (PSI) overcomes many of the shortcomings of conventional interferometry.

In PSI, interferograms are recorded by continuously changing the phase of the reference wavefront with

respect to the object wavefront. The wavefront is encoded as the variation of intensity patterns. Analysis

of PSI images is much simpler and does not require location of fringe centers. With PSI, the sensitivity of

measurement is greatly increased and small changes in concentration, temperature, and density can be

readily determined.

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1.1 Background

The initial development of phase shifting interferometry started in the 1960s (Carré 1966; Crate 1969)

while early applications were reported by Bruning et al. (1974), Wyant (1975), Hardy et al. (1977), and

Moore et al. (1973). Its application in applications related to fluid mechanics and heat transfer is of recent

origin.

In phase shifting interferometry, the relative phase of the reference beam is varied in a continuous manner

compared to the phase of the object beam. The interferograms are recorded with help of a CCD camera

connected to computer. By measuring the intensity variation for various phase shifts we can determine the

phase of the object wavefront relative to that of reference wavefront at the selected point.

Let the reference (r) and object (o) wavefronts be represented by

and

In equations (1) & (2), r is the phase of reference wave, (t) is the relative phase shift introduced in the

reference wave with respect to the object wave, o is the phase encountered by the object wave, Ar and Ao

are amplitudes of reference and object wavefronts.

The phase-shifted interferogram recorded at the camera/detector is formed by the interference of these

two wavefronts. The intensity at the detector can be represented as

In equation (4), is the phase difference between the object beam and the reference

beam which we now denote as . The above equation shows that intensity at each point varies with

time according to the time varying phase introduced in the reference beam. The above equation can

be written as

Here, I1 is the average intensity and I2 is the intensity modulation.

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2. Phase shifting interferometry implemented on a Mach-Zehnder configuration

Phase shifting interferometry can be implemented within a Mach-Zehnder interferometer in two ways: by

the use of a piezoelectric transducer or with the help of polarizers.

Method 1: Piezoelectric transducers

Piezoelectric transducer (PZT) converts electrical pulses into mechanical motion while the reversed

conversion is also possible. Such transducers work on the principle of the piezoelectric effect. The active

element is a piezoelectric crystal (or a polar material) with electrodes attached to its ends. When an

electric field is applied across the material, its molecules align in the direction of the electric field and

cause a change in dimension of the crystal.

In the modification of the M-Z interferometer being discussed, the PZT is installed on the reference

mirror as shown in Figure 1. By discretely varying the voltage applied on the PZT the phase of the

reference beam can be changed. In the Mach-Zehnder configuration, the angle of incidence of the light

beam on the reference mirror is not normal and the phase varies according to the cosine of the angle of

incidence, Figure 2. A displacement x of the mirror leads to a change in path length equal to [2].

He-Ne Laser

BS1

BS2 Mirror2

Mirror1

PZT

Test section

Compensating section

Object beam

Reference beam

Camera

Data analysis

Figure 1: Phase shifting interferometry implemented within a Mach-Zehnder interferometer

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This technique has been employed by Duan and Shu [4] to study fluid convection during crystal growth.

Duan et al. [5] used this technique to study mass transfer and growth rate of protein crystals.

Since fringe formation results from path differences of the order of the wavelength of light, PSI is

affected by rapid and imprecise movement of the piezoelectric transducer. Even a small amount of

backlash and hysteresis can cause nonlinear phase shift and errors related to calibration. Within limits,

these errors can be minimized by using multi-step algorithms. PSI needs at least three interferograms for

fringe analysis; up to 20 step algorithms have been developed in the literature [6].

When a piezoelectric transducer is used, the phase-shifted interferograms are recorded sequentially using

a single camera. Most commercial cameras run at video rates and record 30 frames per second, yielding

an exposure time of 30ms. This time interval can be unacceptably long for measurements using PSI. This

approach is, therefore, not useful in cases where the process being studied involves rapid transients.

Method 2: Simultaneous phase shifting interferometry

To overcome the deficiency of time elapsed for each measurement, the method of simultaneous phase

shift measurement has been proposed. It involves the use of polarizers and CCD cameras as shown in

Figure 3. It is also known as the spatial phase shifting technique and is an application of the real time

phase shifting interferometer. It requires much less time compared to temporal phase shifting and captures

all the interferograms at practically the same instant. This method was originally proposed by Symthe and

Moore [7]. These authors employed beam splitters and polarizers and three or four CCD cameras.

Koliopolous [8] used as many cameras as the number of interferograms.

This method was used by Onuma et al. [9] to study concentration fields during a crystal growth process.

The laser beam, linearly polarized, is divided into reference and object beams with the help of a beam

splitter. The two beams are then orthogonally polarized. The object beam passes through the test cell and

combines with the reference beam at the second beam splitter. The beams do not interfere initially being

x

Path difference

=2xcos

Figure 2: Path difference created by the displacement of mirror 2

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orthogonally polarized. Subsequently, the light beams pass through a quarter-wave plate so that they are

circularly polarized and interfere to give an interference pattern. In [9], the authors used 3 CCD cameras

to record 3 phase-shifted interferograms. The use of many cameras makes the system complex and the

alignment of optics difficult as the cameras have to be aligned pixel by pixel to get the best results.

Fringe formation in the simultaneous phase-shifting interferometer can be better understood in the

following manner [10]. The left and right circularly polarized light after the quarter-wave plate can be

represented as

In equations (6) & (7), and are the reference and object beams respectively. The angle is

the phase difference between the object beam and the reference. A polarizer each is placed in the path of

the two beams. Let be the unit vector of the polarizer and let it be at an angle of with respect to .

Hence

He-Ne Laser

BS1

Polarizer1

Polarizer2

Test section

Compensation cell

BS2

Quarter- wave plate

Mirror 1

Mirror 2

cameras

Beam splitters

Figure 3: Simultaneous phase-shifting interferometer involving the use of polarizers and a quarter-wave plate

Polarizers

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Equation (9) simplifies to

The interferograms are formed due to an intensity variation given as

Equation (11) is similar to that obtained with a piezoelectric transducer. On varying the angle between the

polarizer and the vector , a collection of phase shifted interferograms can be obtained.

Maruyama et al. [11] and Guo et al. [12] replaced the three different cameras and beam splitters by a 3-

CCD sensor (color) camera. This camera was able to record three interferograms jointly at the same

instant of time. Overall, this method uses polarization of the test and the reference beams and needs a

complex optical arrangement and synchronization of the cameras.

Miller [13] constructed a pixelated phase mask dynamic interferometer and described a procedure by

which one camera at a time was used to record four phase-shifted interferograms. A pixelated phase mask

was a micro-polarizer wire array for imaging polarimetry and is described by Nordien [14]. The micro-

polarizer array was installed just ahead of the detector. It could be directly used for circularly polarized

light but for linearly polarized light, a quarter-wave plate was used in combination with the phase mask.

The phase mask used had alternate out-of-phase signals along the column, Figure 4. Each 2×2 polarizer

matrix is shown in the figure and is known as a ‘super pixel’. Each pixel has unique phase shift and

distinct transfer function. The pixels with a given transfer function were combined into a single

interferogram to generate a continuous phase map. The data thus obtained was four phase shifted

interferograms, which were analyzed with the help of the 4-bucket algorithm [13].

0 90

180 270

Quarter-wave plate

Phase mask

Detector

Figure 4: A super pixel of a microplarizer array

Figure 5: Optical layout for linearly polarized light

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Min [15] employed Michelson-like architecture and polarization unit that generates two interferograms

with phase shifts at a single camera unit. It has an advantage over other grating techniques in the

sense that the distance between the interferometric fringes is adjustable. It makes use of non-polarizing

beam splitter which splits the beam into two. These two beams are reflected by mirrors M1 and M2

aligned at distinct angles with the optic axis. The reflected rays are circularly polarized by means of a

quarter-wave plate. The two beams pass through an array of two polarizers with their axis inclined at an

angle of π/4 (radians). Then two π/2 phase shifted interferograms are recorded by means of a single

camera. The two interferograms thus obtained are processed by two-step phase shifting algorithm. This

technique can also be applied on a Mach-Zehnder configuration [15].

α

β

Collimating lens

Quarter-wave plate

Polarizer array

Detector

M1

M2

Figure 7: Parallel two step phase shifting interferometry [15]

Polarizing interferometer

Aperture

Collimating lens Detector

Phase mask

Figure 6: Pixelated phase mask dynamic interferometer, [13]

+

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3. Integrating bucket technique

In the phase shifting methods discussed above, the ways in which data is collected depends on the optical

elements used after the second beam splitter. The phase can be varied in a continuous as well as discrete

manner. It is preferable to vary phase in a continuous fashion because discrete steps may introduce

vibrations in the detector. The continuous phase shifting method, also known as the integrating bucket

technique was developed by Wyant (1975). It allows the phase to vary linearly with time. The integration

period is the frame time of the detector or the time required to collect the data, whichever is less [2].

If we take Δ as the integration period, the intensity of the signal is given as

On integration

Equation (13) simplifies to

Owing to continuously varying phase, the above equation shows that the contrast of the signal decreases

as a sinc function but this effect is negligible in most of the algorithms used.

4. Data analysis algorithms

Quite a few data analysis algorithms have been developed to calculate the phase map from

interferograms. A few of them are described below.

4.1 Three-step algorithm

The intensity equation has three unknowns (I1, I2, and θ) and so at least three phase equations are needed

to get the phase map [2]. Let the three phase shifts be –α, 0, α. The corresponding equations for these

phase shifts can be written as

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We can rewrite equations (15) to (17) using trigonometric identities as

(20)

The value of can be obtained from equations (18) to (20) as

When = 90º we get

4.2 Four-step algorithm

In this algorithm four phase steps are given as input and four interferograms are recorded [2]. We take

phase shift α as 0, π/2, π, and 3π/2. The corresponding intensity equations for these phase shifts can be

written as

Using trigonometric identities, equations (23) to (26) simplify to

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These equations can be solved for phase at every point on the interferogram. Although

only three interferograms are needed for the calculation, the fourth one is included for

computational ease.

On solving equations (27) to (30), we get

Dividing equation (31) by (32)

Equation (33) can be rearranged to yield

This is used to calculate phase at every point. The optical path difference at every point can be

obtained from the relation

4.3 Carré’s algorithm

In this algorithm the phase shift is considered to be an unknown. The phase difference between two

successive phase shifts is considered to be 2α and so the phase steps of -3α, -α, α and 3α are introduced.

The intensity equations for the interferograms are given by

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In this approach, we have four unknowns and four equations, as compared to three unknowns in the

previous formulations. This is because the reference phase is also an unknown quantity. Equations (36) to

(39) can be expanded using trigonometric identities as follows:

For calculating the reference and object phases we determine

From equations (44) & (45) we get

The phase difference in successive phase shifts is assumed to be 2α. Using the equations given above, the

object phase can be calculated as

The phase calculated above depends on the point intensities. This algorithm requires that the phase

difference between successive phase shifts remains constant. Phase unwrapping cannot be directly applied

here as the numerator and denominator contain the square root term while in other algorithms there is no

such term involved.

4.4 Hariharan’s algorithm

Hariharan recorded five interferograms to determine the phase map [16]. The phase shifts were equal to

2α, -α, 0, α, 2α. Using these phase shifts, the intensity equation of the interferograms can be written as:

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Using trigonometric identities these equations can be solved to yield

The above equation can be solved for a known value of α to get the phase map.

5. Errors in PSI

5.1 Phase shift errors arise due to nonlinear movement of the PZT or vibrations. In PSI, all

measurements depend on known phase shifts. The intensity recorded by the detector is a sinusoidal

function of the phase shift introduced in the reference arm of the detector. At times, there can be

increments or decrements in the phase shift and the resultant phase does not remain a subdivision of 2π.

The data points then do not accurately represent the phase map. If there is a small error in the reference

phase, there is a corresponding error in the measured phase [19]. In order to minimize this error a number

of algorithms have been developed. Carré’s algorithm stated earlier calculates phase shift at each step and

hence is insensitive to such kind of errors. The algorithms by Schwider [17] and Hariharan [16] are also

insensitive to this error.

5.2 Vibration is a major source of error in phase shifting interferometry. Precautions must be taken in this

respect. Vibration isolation mount and vibration damping are quite essential in this context.

5.3 Quantization error comes into picture due to digitization of the output from the detector. Digitization

is carried out within the camera using 8 bits (256 levels) or 12 bits (4096 levels) to quantize the output.

Brophy has derived the quantization errors for most of the algorithms reported above [17]. It has been

shown that the standard deviation (σN) due to q quantization levels in an N-step algorithm is given by

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5.4 Stray reflections also cause errors in the phase map. When using lasers as a light source, extra fringes

are formed at the detector due to stray reflections. The light so reflected adds into the object beam in

amplitude and phase giving rise to errors in phase. The best way to reduce such reflections is to use a

laser whose coherence length is not very large.

6. Phase unwrapping

The phase data available from the interferograms has 2π discontinuity. Phase unwrapping is carried on the

data points to remove this 2π discontinuity and the phase thus available is a continuous representation.

Unwrapped phase is calculated by adding or subtracting 2π to the obtained phase. It is considered that the

phase difference between the adjacent pixels should be less than π and if the phase difference is greater

than π then 2π is added or subtracted to the phase of one of the pixel. This process is repeated for all the

data points on the phase map [18].

Phase unwrapping is carried out in one as well as two dimensions. One dimensional phase unwrapping is

relatively easier to implement compared to the two dimensional.

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7. Some Experimental Images

Figure 8: An example of observations by the phase shift interferometer. (a) Original interferogram, (b) two-dimensional phase distribution picture with 4π phase connection, (c) 8π phase connection (d) and three-dimensional concentration profile are shown. [9]

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Figure 9: The interferograms of four steps phase shift (a) to (d), phase calculation (e), phase unwrapping (f), the distribution of refractive index gradient (g) and the contour map (h).[4]

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References

1) S. Verma, Y. M. Joshi and K. Muralidhar, “Optical Interferometers: Principles and applications in transport phenomena”, to appear in Interferometry - Principles and Applications, Edited by Mark E. Russo, Nova Publishers, USA, [60 pages] ISBN: 978-1-61209-347-5 (2011).

2) H. Schreiber and J. H. Bruning, “Phase shifting interferometry,” in Optical Shop Testing, 3rd ed., D. Malacara, ed. (Wiley, 2007), pp. 547–666.

3) D. Malacara, M. Servín, and Z. Malacara, Interferogram Analysis for Optical Testing, 2nd edition, CRC Press, Boca Raton (2005).

4) L. Duan and J.Z. Shu, “The convection during NaClO3 crystal growth observed by the phase shift interferometer”, Journal of Crystal Growth, Elsevier, 223(1-2), pp. 181-188 (2000).

5) L. Duan, Q. Kang, W.R. Hu, G.P. Li, D.C. Wang, “The mass transfer process and the growth of protein crystals”, Biophysical Chemistry, Elsevier, 97(2-3), pp. 189-201(2002).

6) J. Hayes and J. Millerd, “Dynamic Interferometry: Getting Rid of the Jitters”, 4D Technology Corporation.

7) Smythe and Moore, “Instantaneous phase measuring interferometry”, Opt. Eng., 23, pp. 361–365 (1984)

8) C. L. Koliopoulos, “Simultaneous phase shift interferometer”, Proc. SPIE, 1531, pp. 119–133 (1991).

9) K. Onuma, K. Tsukamoto and S. Nakadate, “Application of real time phase shift interferometer to the measurement of concentration field”, Journal of Crystal Growth, 129 (3-4), pp. 706-718 (1992).

10) E. M. Frins, “Polarization-shifting method for step interferometry”, Pure Appl. Opt., 7, pp. 53-60 (1998).

11) S. Maruyama, T. Shibata and K. Tsukamoto, “Measurement of diffusion fields of solutions using real-time phase-shift interferometer and rapid heat-transfer control system”, Experimental Thermal and Fluid Science, Elsevier, 19 (1), pp. 34-48 (1988).

12) Z. Guo, S. Maruyama and A. Komiya, “Rapid yet accurate measurement of mass diffusion coefficients by phase shifting interferometer”, J. Phys. D: Appl. Phys. 32 (9), pp. 995 (1999).

13) J. E. Millerd, N. J. Brock, J. B. Hayes, M. B. North-Morris, M. Novak, and J. C. Wyant, “Pixelated Phase-Mask Dynamic Interferometer,” Proc. SPIE, 5531, pp. 304-314 (2004).

14) Nordin, “Micorpolarizer array for infrared imaging polarimetry”, Journal of Optical Society of America A: Optics, Image Science and Vision, 16(5), pp.1168-1174 (1999).

15) J. Min, B. Yao, P. Gao, R. Guo, J. Zheng, and T. Ye, “Parallel phase-shifting interferometry based on Michelson-like architecture”, Applied Optics, 49,pp. 6612-6616 (2010).

16) P. Hariharan, B. F. Oreb, and T. Eiju, “Digital Phase-Shifting Interferometry: A Simple Error-Compensating Phase Calculation Algorithm”, Applied Optics, 26 (13), pp. 2504-2506 (1987).

17) C. P. Brophy, “Effect of Intensity Error Correlation on the Computed Phase of Phase-shifting Interferometry”, Journal of Optical Society of America A, 7 (4), pp. 537-541 (1990).

18) J. Schwider , “Phase Shifting Interferometry: Reference Phase Error Reduction”, Applied Optics, 28 (18), pp. 3889-3892 (1989).

19) J. C. Wyant, “Use of an ac Heterodyne Lateral Shear Interferometer with Real-Time Wavefront Correction Systems”, Applied Optics, 14 (11), pp. 2622-2626 (1975).

20) K. Onuma, K. Tsukamoto and S. Nakadate, “In situ study of surface phenomena by real time phase shift interferometry”, Journal of crystal growth, Elsevier, 137 (3-4), pp. 610-622 (1994).

Prepared by Veena Singh

MT (Laser Technology Program) IIT Kanpur