5. The Fourier method

observation plane, where a CCD camera sensor is placed to register an image of the resulting

In Figure 7, a set of four phase-shifted interferograms with phase shifts of π/2 is shown. A first interferogram with θ = 0 is seen in Figure 7a, and subsequent interferograms with θ = π/2, θ = π and θ = 3π/2 are observed in Figure 7b–d, respectively. The wrapped phase is showed in Figure 7e, and the unwrapped phase related with the shape of the optical element being tested is shown in Figure 7f. A three-dimensional representation of the unwrapped phase seen in

In applied phase-shifting interferometry, there are concerns about the presence of errors that may affect the accurate phase extraction from phase-shifted interferograms. A typical systematical source of error introduced PZT arises when there is miscalibration of the phaseshifting actuator, causing detuning in the phase extraction process. The inclusion of phaseshifting algorithms with more than three or four interferograms can be implemented in order

Figure 7. Four-step phase-shifted interferograms from (a) to (d), the wrapped phase (e) and the unwrapped phase (f).

interferogram that is then stored in a PC for subsequent processing.

Figure 7f is presented in Figure 8.

10 Optical Interferometry

to reduce this systematic error [5].

The method of Takeda or the Fourier method was developed in 1982 [6]. Unlike phase-shifting methods, see Ref. [7], a single interferogram with open fringes can be analyzed. This is useful when the object under study changes dynamically or environmental disturbances (vibrations or air turbulence) do not allow the use of phase-shifting methods unless special, and often, very expensive hardware is used to acquire several images simultaneously. However, in general, the accuracy and the dynamical range of the phase that can be measured are reduced. The Fourier method makes use of the fast Fourier transform technique to separate, in the frequency domain, the background and phase terms of the interferogram. Employing complex notation, an interferogram can be written as follows:

$$I(\mathbf{x}, y) = a(\mathbf{x}, y) + \frac{1}{2}b(\mathbf{x}, y)e^{i[\phi(\mathbf{x}, y) + \phi(\mathbf{x}, y)]} + \frac{1}{2}b(\mathbf{x}, y)e^{-i[\phi(\mathbf{x}, y) + \phi(\mathbf{x}, y)]},\tag{20}$$

where a(x, y) and b(x, y) are the background intensity and the modulation term, respectively. The phase term is denoted by φ(x, y) and finally, the symbol ϕ(x, y) denotes the lineal carrier function or tilt that is introduced usually by tilting the reference mirror in a two arm interferometer. The Fourier transform of the above expression can be written as:

$$\mathcal{F}\{I(\mathbf{x},y)\} = \hat{I}(\boldsymbol{\mu},\boldsymbol{\upsilon}) = \delta(\boldsymbol{\mu},\boldsymbol{\upsilon}) + E(\boldsymbol{\mu}+\boldsymbol{a},\boldsymbol{\upsilon}+\boldsymbol{\beta}) + E^\*(\boldsymbol{\mu}-\boldsymbol{a},\boldsymbol{\upsilon}-\boldsymbol{\beta}),\tag{21}$$

where (u, v) is the coordinates in the frequency domain, δ(u, v) is a delta function and E(u + α, v + β) and E\*(u-α, v-β) are complex conjugate functions that correspond to the transforms of the second and third terms of Eq. (20), respectively. The introduction of the linear carrier function, ϕ(x, y) = αx + βy, shifts the terms E(u, v) and E\*(u, v) in opposite directions in the frequency spectrum as can be seen noting that:

$$E(u,v) = \mathcal{F}\left\{\frac{1}{2}b(x,y)e^{i\phi(x,y)}\right\},\tag{22}$$

and

$$E(\mu+\alpha, \upsilon+\beta) = \mathcal{F}\left\{\frac{1}{2}b(\mathbf{x}, y)e^{i\phi(\mathbf{x}, y)}e^{i\wp(\mathbf{x}, y)}\right\},\tag{23}$$

similarly

$$E^\*(\mu, \upsilon) = \mathcal{F}\left\{\frac{1}{2}b(\mathbf{x}, y)e^{-i\phi(\mathbf{x}, y)}\right\},\tag{24}$$

and

$$\mathcal{E}^\*(\mu-\alpha, \upsilon-\beta) = \mathcal{F}\left\{\frac{1}{2}b(\mathbf{x}, y)e^{-i\phi(\mathbf{x}, y)}e^{-i\psi(\mathbf{x}, y)}\right\}.\tag{25}$$

Figure 9. Interferogram with open fringes (a) and its Fourier spectrum (logarithm of the module) (b).

The separation of the terms in the Fourier spectrum due to the introduction of a linear carrier can be observed in Figure 9. An open-fringe interferogram and its Fourier spectrum can be seen in Figure 9a and b. It can be noted that the three terms of Eq. (21) are clearly separated. The central peak corresponds to the delta function δ(u, v) while the adjacent lobules correspond to E(u + α, v + β) and E\*(u - α, v - β). The interferogram was constructed as follows:

$$a(x, y) = 120e^{-(x^2 + y^2)} \tag{26}$$

$$b(x, y) = 124e^{-(x^2 + y^2)},\tag{27}$$

$$\phi(\mathbf{x}, y) = 2\pi \left\{ 4e^{-4\left[\left(\mathbf{x} - 0.2\right)^2 + \left(y - 0.3\right)^2\right]} \mathbf{-} \mathbf{S} e^{-6\left[\left(\mathbf{x} + 0.2\right)^2 + \left(y + 0.3\right)^2\right]} \right\},\tag{28}$$

and

function or tilt that is introduced usually by tilting the reference mirror in a two arm interfer-

where (u, v) is the coordinates in the frequency domain, δ(u, v) is a delta function and E(u + α, v + β) and E\*(u-α, v-β) are complex conjugate functions that correspond to the transforms of the second and third terms of Eq. (20), respectively. The introduction of the linear carrier function, ϕ(x, y) = αx + βy, shifts the terms E(u, v) and E\*(u, v) in opposite directions in the frequency

> 2 bðx, yÞe

> > 2 bðx, yÞe

2 bðx, yÞe

2 bðx, yÞe

iφðx, yÞ 

> iφðx,yÞ e iϕðx,yÞ

−iφðx,yÞ 

> −iφðx, yÞ e −iϕðx,yÞ

ðu−α, v−βÞ, (21)

, (22)

, (23)

, (24)

: (25)

ometer. The Fourier transform of the above expression can be written as:

spectrum as can be seen noting that:

and

12 Optical Interferometry

similarly

and

<sup>F</sup>fIðx, <sup>y</sup>Þg ¼ <sup>~</sup>Iðu, <sup>v</sup>Þ ¼ <sup>δ</sup>ðu, <sup>v</sup>Þ þ <sup>E</sup>ð<sup>u</sup> <sup>þ</sup> <sup>α</sup>, <sup>v</sup> <sup>þ</sup> <sup>β</sup>Þ þ <sup>E</sup>�

<sup>E</sup>ðu, <sup>v</sup>Þ ¼ <sup>F</sup> <sup>1</sup>

<sup>ð</sup>u, <sup>v</sup>Þ ¼ <sup>F</sup> <sup>1</sup>

<sup>ð</sup>u−α, <sup>v</sup>−βÞ ¼ <sup>F</sup> <sup>1</sup>

Figure 9. Interferogram with open fringes (a) and its Fourier spectrum (logarithm of the module) (b).

<sup>E</sup>ð<sup>u</sup> <sup>þ</sup> <sup>α</sup>, <sup>v</sup> <sup>þ</sup> <sup>β</sup>Þ ¼ <sup>F</sup> <sup>1</sup>

E�

E�

$$
\varphi(\mathbf{x}, y) = 2\pi (16\mathbf{x} + 20y) \tag{29}
$$

where x and y vary from 1 to −1 along the vertical and horizontal directions and the interferogram was multiplied by a circular function of radius one.

In order to recover the phase, we need to isolate one of the lateral lobules of the Fourier domain. To this end, we employ a band-pass filter. The filtered spectrum is then transformed back to the spatial domain to obtain If(x, y) and the wrapped phase is found with the arctangent function of the ratio of the imaginary and real parts of If(x, y) as shown in Figure 10. Figure 10a shows the band-pass filter H(u, v), the filtered spectrum can be observed in Figure 10b and finally the wrapped phase is shown in Figure 10c.

The band-pass filter has the following form:

$$H(\mu, \upsilon) = e^{-3000 \left[ \left( \mu - 0.2 \right)^2 + \left( \upsilon - 0.2 \right)^2 \right]^2},\tag{30}$$

Figure 10. Band-pass filtering process of the Fourier spectrum seen in Figure 9. Pass-band filter (a), filtered spectrum (b) and recovered wrapped phase (c).

where u and v vary from 1 to −1 in vertical and horizontal directions, respectively. The filtered spectrum becomes,

$$
\tilde{I}\_f(\mu, \upsilon) = \tilde{I}(\mu, \upsilon) H(\mu, \upsilon). \tag{31}
$$

Transforming back to spatial domain we obtain:

$$I\_f(\mathbf{x}, y) = \mathcal{F}^{-1}\left\{\tilde{I}\_f(\mu, v)\right\}.\tag{32}$$

The wrapped phase is found by:

$$\psi\_w(\mathbf{x}, y) = \operatorname{atan2}\left\{ \operatorname{imag}[I\_f(\mathbf{x}, y)], \operatorname{real}[I\_f(\mathbf{x}, y)] \right\} \tag{33}$$

where the atan2() function accepts two arguments corresponding to the sine and cosine and returns the results modulo 2π. This wrapped phase, however, is not the desired one because it contains the introduced tilt that is not part of the object information. The wrapped desired phase is found with:

$$\phi\_w(\mathbf{x}, y) = \operatorname{atan2}\left\{ \sin[\psi\_w(\mathbf{x}, y) - \varphi(\mathbf{x}, y)], \cos[\psi\_w(\mathbf{x}, y) - \varphi(\mathbf{x}, y)] \right\}.\tag{34}$$

The final step is to apply an unwrapping method to obtain the continuous phase related with the object under study. This last procedures are shown in Figure 11a and b where the wrapped phase ϕw(x, y) and the unwrapped reconstructed phase ϕr(x, y) can be appreciated, respectively.

The Fourier method is not the unique procedure for phase retrieval from one interferogram with open fringes. Besides the Fourier approach there are other procedures in the spatial domain including phase locked loop [8] and spatial carrier phase-shifting methods [9, 10].

Figure 11. Phase reconstruction results. Wrapped phase (a) and unwrapped continuous phase (b).

## 6. Phase unwrapping

where u and v vary from 1 to −1 in vertical and horizontal directions, respectively. The filtered

n <sup>~</sup>Ifðu, <sup>v</sup><sup>Þ</sup> o

where the atan2() function accepts two arguments corresponding to the sine and cosine and returns the results modulo 2π. This wrapped phase, however, is not the desired one because it contains the introduced tilt that is not part of the object information. The wrapped desired

The final step is to apply an unwrapping method to obtain the continuous phase related with the object under study. This last procedures are shown in Figure 11a and b where the wrapped phase ϕw(x, y) and the unwrapped reconstructed phase ϕr(x, y) can be appreciated, respectively. The Fourier method is not the unique procedure for phase retrieval from one interferogram with open fringes. Besides the Fourier approach there are other procedures in the spatial domain including phase locked loop [8] and spatial carrier phase-shifting methods [9, 10].

imag½Ifðx, yÞ�,real½Ifðx, yÞ�

sin½ψwðx, yÞ−ϕðx, yÞ�, cos½ψwðx, yÞ−ϕðx, yÞ�

Ifðx, <sup>y</sup>Þ ¼ <sup>F</sup><sup>−</sup><sup>1</sup>

n

ψwðx, yÞ ¼ atan2

n

Figure 11. Phase reconstruction results. Wrapped phase (a) and unwrapped continuous phase (b).

<sup>~</sup>Ifðu, <sup>v</sup>Þ ¼ <sup>~</sup>Iðu, <sup>v</sup>ÞHðu, <sup>v</sup>Þ: (31)

o

: (32)

o

: (34)

(33)

spectrum becomes,

14 Optical Interferometry

phase is found with:

Transforming back to spatial domain we obtain:

φwðx, yÞ ¼ atan2

The wrapped phase is found by:

Phase unwrapping is a common step to finally find a continuous phase for several fringe analysis techniques such as phase shifting, Fourier, the phase synchronous and others methods that use the arctangent function of the ratio of the sine and cosine of the phase to obtain a wrapped phase. In its simplest form, phase unwrapping consists in adding or subtracting 2π terms to the pixel being unwrapped if a difference greater that π is found with a previous pixel already unwrapped [11]. The phase unwrapping problem in one dimension can be observed in Figure 12. The wrapped phase found with the arctangent function is seen in Figure 12a and the unwrapped continuous phase is showed in Figure 12b.

The described procedure works well only for wrapped phases with no inconsistencies and low noise levels, however, delivers wrong results when dealing with noisy wrapped phases or those obtained from interferograms with broken or unconnected fringes. A more consistent approach is achieved with the least squares phase unwrapping method [12]. The least square technique integrates the discretized laplacian of the phase. To this end, the laplacian of the phase is calculated as follows:

$$L\_{i,j} = \phi\_{i+1,j}^x - \phi\_{i,j}^x + \phi\_{i,j+1}^y - \phi\_{i,j}^y,\tag{35}$$

where

$$\phi\_{i,j}^{x} = \operatorname{atan2}\left[\sin\left(\phi\_{i,j}^{w} - \phi\_{i-1,j}^{w}\right), \cos(\phi\_{i,j}^{w} - \phi\_{i-1,j}^{w})\right] p\_{i,j} p\_{i-1,j},\tag{36}$$

and

$$\phi\_{i,j}^y = \operatorname{atan2}\left[\sin\left(\phi\_{i,j}^w - \phi\_{i,j-1}^w\right), \cos(\phi\_{i,j}^w - \phi\_{i,j-1}^w)\right] p\_{i,j} p\_{i,j-1}.\tag{37}$$

In the last equations, we have used pixel subscript notation in order to limit the extension of the equations. The pupil function pi, j is defined as equal to one where we have valid data and zero otherwise. One may note that if pi, j = pi+1, j = pi-1, j = pi, j+1 = pi, j-1 = 1, then

$$L\_{i,j} = -4\phi\_{i,j} + \phi\_{i+1,j} + \phi\_{i-1,j} + \phi\_{i,j+1} + \phi\_{i,j-1}.\tag{38}$$

Solving for the phase in the above equation, we obtain that the unwrapping problem under the least square approach consist in the resolution of a linear system of equations, as:

Figure 12. Wrapped (a) and unwrapped phase (b).

An iterative technique that solves the above system of linear equations is the overrelaxation method in which the following equation is iterated until convergence:

$$
\phi\_{i,j}^{k+1} = \phi\_{i,j}^k - \frac{(d\phi\_{i,j}^k - \phi\_{i-1,j}^{k+1} - \phi\_{i+1,j}^k - \phi\_{i,j-1}^{k+1} - \phi\_{i,j+1}^k + L\_{i,j})w}{d},\tag{40}
$$

where

$$d = p\_{i-1,j} + p\_{i+1,j} + p\_{i,j-1} + p\_{i,j+1}.\tag{41}$$

In the last equations, k is the iteration number and w is a parameter of the overrelaxation method that must be set between 1 and 2. Results of the least square technique are showed in Figure 13. A two-dimensional wrapped phase is seen Figure 13a, the phases differences are shown in Figure 13b and c, respectively. The laplacian of the phase is showed in Figure 13d,

Figure 13. Least square results to unwrap the phase. Wrapped phase (a), phase differences in the x (b) and y (c) directions, laplacian of the phase (d), unwrapped phase (e) and rewrapped phase (f).

the reconstructed phase in a two-dimensional view is observed in Figure 13e and finally, for comparison purposes the reconstructed rewrapped phase is also showed in Figure 13f. One may observe that the original wrapped phase and the reconstructed wrapped phase are slightly different, this is because the least square method recovers the phase with an arbitrary constant term, however this term is usually neglected since doesn't carry any information of the object being measured.

If desired, the constant term in the retrieved phase may be corrected easily in the following form:

$$\phi\_c = \phi\_r \text{--attan2}[\sin\left(\phi\_r \neg \phi\_w\right), \cos(\phi\_r \neg \phi\_w)].\tag{42}$$

The wrapped phase seen in Figure 13a was constructed as follows:

$$\phi\_w = \operatorname{atan2}[\sin(\phi), \cos(\phi)],\tag{43}$$

where

An iterative technique that solves the above system of linear equations is the overrelaxation

In the last equations, k is the iteration number and w is a parameter of the overrelaxation method that must be set between 1 and 2. Results of the least square technique are showed in Figure 13. A two-dimensional wrapped phase is seen Figure 13a, the phases differences are shown in Figure 13b and c, respectively. The laplacian of the phase is showed in Figure 13d,

Figure 13. Least square results to unwrap the phase. Wrapped phase (a), phase differences in the x (b) and y (c) directions,

laplacian of the phase (d), unwrapped phase (e) and rewrapped phase (f).

− φ<sup>k</sup>þ<sup>1</sup> <sup>i</sup>,j−1<sup>−</sup> <sup>φ</sup><sup>k</sup>

<sup>i</sup>,jþ<sup>1</sup> <sup>þ</sup> Li,jÞ<sup>w</sup>

<sup>d</sup> <sup>¼</sup> pi<sup>−</sup>1,<sup>j</sup> <sup>þ</sup> piþ1,<sup>j</sup> <sup>þ</sup> pi,j−<sup>1</sup> <sup>þ</sup> pi,jþ<sup>1</sup>: (41)

<sup>d</sup> , (40)

method in which the following equation is iterated until convergence:

φ<sup>k</sup>þ<sup>1</sup> <sup>i</sup>,<sup>j</sup> <sup>¼</sup> <sup>φ</sup><sup>k</sup> i,j − <sup>ð</sup>dφ<sup>k</sup> i,j − φ<sup>k</sup>þ<sup>1</sup> i−1,j − φ<sup>k</sup> iþ1,j

where

16 Optical Interferometry

$$\phi\_{i,j} = 2\pi \left\{ \begin{array}{c} 16\left[ \left( \mathbf{x}\_i \right)^2 + \left( \mathbf{y}\_j \right)^2 \right]^2 - 18\left[ \left( \mathbf{x}\_i \right)^2 + \left( \mathbf{y}\_j \right)^2 \right] + \\ 5y\_j(\mathbf{x}\_i)^2 - 4\mathbf{x}\_i(y\_j)^2 + 3y\_j\mathbf{x}\_i + 2\mathbf{x}\_i - 2y\_j + 0.56 \end{array} \right\}. \tag{44}$$

In the above equations xi and yj are range variables that vary from −1 to 1, the wrapped phase was multiplied by an annular pupil function with an exterior radius of 198 pixels, while the interior radius was 60 pixels for an image size of 400 + 400. The convergence of the reconstructed phase seen in Figure 2e was reached after 700 iterations; the overrelaxation parameter w was set to 1.99, which is usual for large images. The reconstructed phase with the constant term corrected and the phase error, ε = φ = φc, can be appreciated in Figure 14a and b. The maximum error was about 0.00005 radians.

Finally, results on a noisy wrapped phase are presented. Random noise with uniform distribution in the range of −π/4 to π/4 radians were added to the phase to obtain the wrapped phase

Figure 14. Unwrapped phase and phase error. Three-dimensional view of the reconstructed and corrected phase (a) and phase error (b).

Figure 15. Results on noisy measurements. Wrapped phase with noise (a), retrieved phase (b) and wrapped retrieved phase (c).

seen in Figure 15a, the unwrapped phase is shown in Figure 15b, and the rewrapped reconstructed phase is observed in Figure 15c. As can be noticed the least square method is a very reliable technique that works with any pupil configuration and stands noisy measurements.

### 7. Phase recovery from lateral shearing interferograms

Lateral shearing interferometry is a very important field in experimental optical measurements, in which, the test beam interferes with a laterally displaced version of itself instead of a reference beam. The resulting fringe patterns are thus related with the object wavefront derivative in a given direction. This is very useful when the object information of interest is related with the derivative as in strain analysis or when the dynamical range of the object wave front is too high that cannot be measured with direct interferometry. Let us consider a laterally shear interferogram with a beam displacement in the x direction a quantity Δx, we obtain:

$$I^{\mathbf{x}}(\mathbf{x},y) = B(\mathbf{x},y) + \mathbb{C}(\mathbf{x},y)\cos[\psi^{\mathbf{x}}(\mathbf{x},y)],\tag{45}$$

where

$$
\psi^x(\mathbf{x}, y) = \phi(\mathbf{x}, y) \neg \phi(\mathbf{x} - \Delta \mathbf{x}, y). \tag{46}
$$

In Eq. (45) B(x, y) is the background intensity and C(x, y) is the modulation term. The objective is to retrieve the undisplaced phase ϕ(x, y). To this end, we need, at least, another laterally shear interferogram with a beam displacement in the y direction another quantity Δy, obtaining:

$$I^y(\mathbf{x}, y) = B(\mathbf{x}, y) + \mathbb{C}(\mathbf{x}, y) \cos[\psi^y(\mathbf{x}, y)],\tag{47}$$

where

$$
\psi^y(\mathbf{x}, y) = \phi(\mathbf{x}, y) \neg \phi(\mathbf{x}, y - \Delta y). \tag{48}
$$

Let us considered that we have retrieved the phase differences ψ<sup>x</sup> (x, y) and ψ<sup>y</sup> (x, y) by means of a phase-shifting technique as is depicted in Figure 16 and Figure 17 for the x and y directions, respectively. A set of four phase-shifted interferograms can be seen in Figure 16a. The wrapped phase differences ψwx(x, y) and the unwrapped phase ψ<sup>x</sup> (x, y) are observed in Figure 16e and f, respectively. The modulation term and a sheared pupil function px (x, y) are shown in Figure 16g and h, respectively. The sheared pupil px (x, y) is found after normalization from zero to one and thresholding the normalized modulation term. A second set of phase-shifted interferograms are seen Figure 17a and d. The wrapped phase differences ψwy(x, y) and the unwrapped phase ψ<sup>y</sup> (x, y) are observed in Figure 17e and f, respectively. The modulation term and a sheared pupil function py (x, y) are shown in Figure 4g and h, respectively. In a similar way to the first set of phase-shifted interferograms, the pupil py (x, y) is found after normalization from zero to one and thresholding the normalized modulation term. The sheared pupils px (x, y) and py (x, y) are useful to find the undisplaced pupil p(x, y), where the original wave front ϕ(x, y) have valid data, since px (x, y) = p(x, y)p(x − Δx, y) and py (x, y) = p(x, y)p((x, y) − Δy)

Once the phase differences θ<sup>x</sup> (x, y) and θ<sup>y</sup> (x, y) are known, we can use the next procedure to find the searched phase ϕ(x, y) observing that:

seen in Figure 15a, the unwrapped phase is shown in Figure 15b, and the rewrapped reconstructed phase is observed in Figure 15c. As can be noticed the least square method is a very reliable technique that works with any pupil configuration and stands noisy measurements.

Figure 15. Results on noisy measurements. Wrapped phase with noise (a), retrieved phase (b) and wrapped retrieved

Lateral shearing interferometry is a very important field in experimental optical measurements, in which, the test beam interferes with a laterally displaced version of itself instead of a reference beam. The resulting fringe patterns are thus related with the object wavefront derivative in a given direction. This is very useful when the object information of interest is related with the derivative as in strain analysis or when the dynamical range of the object wave front is too high that cannot be measured with direct interferometry. Let us consider a laterally shear interferogram with a beam displacement in the x direction a quantity Δx, we obtain:

<sup>ð</sup>x, <sup>y</sup>Þ ¼ <sup>B</sup>ðx, <sup>y</sup>Þ þ <sup>C</sup>ðx, <sup>y</sup>Þcos½ψ<sup>x</sup>

In Eq. (45) B(x, y) is the background intensity and C(x, y) is the modulation term. The objective is to retrieve the undisplaced phase ϕ(x, y). To this end, we need, at least, another laterally shear interferogram with a beam displacement in the y direction another quantity Δy,

<sup>ð</sup>x, <sup>y</sup>Þ ¼ <sup>B</sup>ðx, <sup>y</sup>Þ þ <sup>C</sup>ðx, <sup>y</sup>Þcos½ψ<sup>y</sup>

ðx, yÞ�, (45)

ðx, yÞ�, (47)

ðx, yÞ ¼ φðx, yÞ−φðx−Δx, yÞ: (46)

7. Phase recovery from lateral shearing interferograms

I x

I y ψx

where

phase (c).

18 Optical Interferometry

obtaining:

where

Figure 16. Recovery of the phase differences in the x direction. Set of four sheared interferograms acquired under a phaseshifting technique (a) to (d), wrapped phase differences (e), unwrapped phase differences (f), modulation (g) and recovered sheared pupil (h).

Figure 17. Recovery of the phase differences in the y direction. Set of four sheared interferograms acquired under a phase-shifting technique (a) to (d), wrapped phase differences (e), unwrapped phase differences (f), modulation (g) and recovered sheared pupil (h).

$$
\psi\_{i,j}^{\mathbf{x}} p\_{i-a,j} - \psi\_{i+a,j}^{\mathbf{x}} p\_{i+a,j} = \phi\_{i,j} (p\_{i-a,j} - p\_{i+a,j}) - \phi\_{i-a,j} p\_{i-a,j} - \phi\_{i+a,j} p\_{i+a,j}, \tag{49}
$$

and

$$
\psi\_{i,j}^y p\_{i,j-b} - \psi\_{i,j+b}^y p\_{i,j+b} = \phi\_{i,j} (p\_{i,j-b} - p\_{i,j+b}) - \phi\_{i,j-b} p\_{i,j-b} - \phi\_{i,j+b} p\_{i,j+b}.\tag{50}
$$

In the above equations we have changed the (x, y) dependence by pixel subscript notation. As described before, pi, j is the undisplaced pupil and the displacement quantities Δx and Δy are rounded to the nearest integer in pixel dimensions obtaining a and b, respectively. Adding Eq. (49) and Eq. (50) and solving for the phase, we obtain:

$$
\phi\_{i,j} = \frac{F + G}{H},
\tag{51}
$$

where

$$H = p\_{i+a,j} + p\_{i-a,j} + p\_{i,j+b} + p\_{i,j-b},\tag{52}$$

$$F = \psi\_{i,j}^x p\_{i-a,j} - \psi\_{i+a,j}^x p\_{i+a,j} + \psi\_{i,j}^y p\_{i,j-b} - \psi\_{i,j+b}^y p\_{i,j+b},\tag{53}$$

and

Digital Processing Techniques for Fringe Analysis http://dx.doi.org/10.5772/66474 21

$$G = \phi\_{i-a,j} p\_{i-a,j} + \phi\_{i+a,j} p\_{i+a,j} + \phi\_{i,j-b} p\_{i,j-b} + \phi\_{i,j+b} p\_{i,j+b}.\tag{54}$$

Eq. (51) represents a linear system of equations; however, it is ill posed because there are more unknowns than equations due to the effects of the sheared pupils. Nevertheless, a regularization term may be aggregated to overcome this problem [13–15]. The regularization term is in the form of discrete Laplacians of the phase among adjacent pixels. The following equation that incorporates the regularization term is iterated until convergence:

$$\phi\_{i,j}^{k+1} = \phi\_{i,j}^k - \frac{H\phi\_{i,j}^k - (F+G) + a(L\_{i+1,j}^x - 2L\_{i,j}^x + L\_{i-1,j}^x + L\_{i,j+1}^y - 2L\_{i,j}^y + L\_{i,j-1}^y)}{H},\tag{55}$$

where

$$L\_{i,j}^x = (\phi\_{i+1,j} - 2\phi\_{i,j} + \phi\_{i-1,j})p\_{i+1,j}p\_{i-1,j},\tag{56}$$

and

ψx i,j pi<sup>−</sup>a,<sup>j</sup> −ψ<sup>x</sup> iþa,j

recovered sheared pupil (h).

20 Optical Interferometry

ψy i,j pi,j−b−ψ<sup>y</sup>

Eq. (49) and Eq. (50) and solving for the phase, we obtain:

<sup>F</sup> <sup>¼</sup> <sup>ψ</sup><sup>x</sup> i,j pi<sup>−</sup>a,<sup>j</sup> −ψ<sup>x</sup> iþa,j

and

where

and

piþa,<sup>j</sup> <sup>¼</sup> <sup>φ</sup>i,<sup>j</sup>

<sup>i</sup>,jþ<sup>b</sup>pi,jþ<sup>b</sup> <sup>¼</sup> <sup>φ</sup>i,<sup>j</sup>

ðpi<sup>−</sup>a,<sup>j</sup>

Figure 17. Recovery of the phase differences in the y direction. Set of four sheared interferograms acquired under a phase-shifting technique (a) to (d), wrapped phase differences (e), unwrapped phase differences (f), modulation (g) and

In the above equations we have changed the (x, y) dependence by pixel subscript notation. As described before, pi, j is the undisplaced pupil and the displacement quantities Δx and Δy are rounded to the nearest integer in pixel dimensions obtaining a and b, respectively. Adding

<sup>φ</sup>i,<sup>j</sup> <sup>¼</sup> <sup>F</sup> <sup>þ</sup> <sup>G</sup>

piþa,<sup>j</sup> <sup>þ</sup> <sup>ψ</sup><sup>y</sup>

i,j

pi,j−b−ψ<sup>y</sup>

<sup>−</sup>piþa,<sup>j</sup>

Þ−φi−a,<sup>j</sup>

pi<sup>−</sup>a,<sup>j</sup>

<sup>H</sup> <sup>¼</sup> piþa,<sup>j</sup> <sup>þ</sup> pi<sup>−</sup>a,<sup>j</sup> <sup>þ</sup> pi,jþ<sup>b</sup> <sup>þ</sup> pi,j−b, (52)

<sup>−</sup>φ<sup>i</sup>þa,<sup>j</sup>

<sup>ð</sup>pi,j−b−pi,jþ<sup>b</sup>Þ−φi,j−bpi,j−b−φi,jþ<sup>b</sup>pi,jþ<sup>b</sup>: (50)

<sup>H</sup> ; (51)

<sup>i</sup>,jþ<sup>b</sup>pi,jþ<sup>b</sup>, (53)

piþa,<sup>j</sup>

, (49)

$$L\_{i,j}^y = (\phi\_{i,j+1} - 2\phi\_{i,j} + \phi\_{i,j-1})p\_{i,j+1}p\_{i,j-1}.\tag{57}$$

Here, α is a parameter that controls the effects of the regularization term. The phase reconstruction is seen in Figure 18. A two-dimensional view of the retrieved phase is observed in Figure 18a, the same phase but in a three-dimensional perspective is shown in Figure 18b and the phase error (the actual phase minus the reconstructed one) can be appreciated in Figure 18c. This reconstruction was achieved after 800 iterations with a parameter α = 0.1 obtaining a maximum error of about 0.0004 radians.

The actual phase was constructed as follows:

$$\phi\_{i,j} = 2\pi \left\{ \begin{aligned} 20\left[\left(\mathbf{x}\_i\right)^2 + \left(\mathbf{y}\_i\right)^2\right]^2 - 24\left[\left(\mathbf{x}\_i\right)^2 + \left(\mathbf{y}\_i\right)^2\right] \neg 6\left(\mathbf{x}\_i\right)^2 y\_j \\ + 8\mathbf{x}\_i \left(\mathbf{y}\_j\right)^2 + 5\mathbf{x}\_i y\_j + 4\mathbf{x}\_i \neg 5y\_j + 0.45 \end{aligned} \right\}, \tag{58}$$

Figure 18. Phase reconstruction from lateral sheared interferograms. Reconstructed phase (a), three-dimensional view of the reconstructed phase (b) and the phase error (c).

where xi and yj are range variables that vary from −1 to 1 in both vertical and horizontal directions. The pupil function is a circular one with a radius of 170 pixels. The shear distances were set to Δx = a = 12 pixels and Δy = b = 12 pixels.
