**4. Relative phase reconstruction with Fourier transform method (FTM)**

#### **4.1 Relative and absolute phase reconstruction**

Before addressing the analysis methods, it is helpful to note how the measured phase is relative to a known reference. Unlike radio frequency methods, the phase reference is generally unknown in optical interferometry. Thus, the term phase reconstruction implicitly means *relative* phase reconstruction. The phase difference developed in Section 3.1 is the subject of the phase reconstruction methods and referred to as the phase function and phase profile in this section.

#### **4.2 1D fringe analysis**

When the interference pattern has a uniform behavior along one dimension such as *κ<sup>y</sup>* ¼ 0 then each fringe line can be visually analyzed to measure the amount of line shift. In the case of a uniformly tilted line, i.e. *κ<sup>y</sup>* 6¼ 0 and *κ<sup>z</sup>* 6¼ 0 then the image can be rotated such that *κ<sup>y</sup>* ¼ 0. Also, the principle component analysis can be performed on the fringe line to measure the shift, but firstly, requires a method to extract the fringe line. The approach in [8] treated the intensity image as a surface and extracted the contours for each fringe line in the image. After averaging, normalization and fitting of the lines to a Gaussian curve for each wavelength, the line-integrated densities were recovered and inverted with the Abel transform to calculate the electron and atom densities. The 1D approach is unlikely to work well for closed loop fringe lines that occur in surface profilometry and collisional plasma interferometry.

#### **4.3 2D fringe analysis**

The 2D fringe analysis is applicable to temporal and spatial analysis, 1D and 2D interference patterns, and straight lines and open and closed curves. Approaches to reconstruct *ϕ*ð Þ *y*, *z* include principal component analysis [27, 28], phase shifting [29], wavelet analysis [30] and Fourier analysis [10, 15]. The latter has become popular in profilometry, 3D shape reconstruction and more recently measurement of nano-scale and femtosecond observables. In much of the literature the original work of [10] has become known as the Fourier Transform Method (FTM).

FTM is well-known for 2D phase reconstruction in optical metrology of surface flatness [31], surface height, strain [32] and defect [33] profilometry, surface motion [34], and 3D shape measurement. However, the spatial-carrier fringe behavior has also been used to observe extreme physical phenomenon [35] such as magnetic fields, electron waves and ultra-violet lithography, as well as beam propagation [36], plasma property measurement [37], refractive index studies of polymeric substrates [38], nano-scale surface metrology [39], corneal topography [40] and biological tissue characterization [41].

The essence of the approach is to analyze the 2D Fourier spectrum of the fringe pattern and to extract the phase function using a digital homodyne receiver process. In treating the interferogram as a 2D signal with spatial dimensions and spatial frequencies *κ<sup>y</sup>* and *κz*, the perturbation is a modulating signal to be recovered as in a communication system. FTM first demodulates the image by the spatial frequency and then filters the baseband content. The solution is succinctly and elegantly described as a two-step algorithm [6, 12, 13, 35]. The listing in Algorithm 1 includes two additional steps to estimate the spatial frequencies and to unwrap the phase function.

**Algorithm 1.** Fourier Transform Method [12].

**Given** input image *I y*ð Þ¼ , *<sup>z</sup> a y*ð Þþ , *<sup>z</sup> b y*ð Þ , *<sup>z</sup>* cos *<sup>κ</sup>yy* <sup>þ</sup> *<sup>κ</sup>zz* <sup>þ</sup> *<sup>ϕ</sup>*ð Þ *<sup>y</sup>*, *<sup>z</sup>* � �. Measure spatial frequencies *κ<sup>y</sup>* and *κz*.

Demodulate *I* to baseband and low-pass filter using the Fourier Transform as

$$\mathcal{L}(\mathcal{y}, \boldsymbol{z}) = h(\mathcal{y}, \boldsymbol{z}) \* \left[ I(\mathcal{y}, \boldsymbol{z}) \boldsymbol{e}^{i\kappa\_{\mathcal{Y}} \mathcal{Y}} \boldsymbol{e}^{i\kappa\_{\mathcal{X}} \boldsymbol{z}} \right], \tag{13}$$

Recover the wrapped phase *ϕ* as

$$\phi(\boldsymbol{y},\boldsymbol{z}) = \arctan\left(\frac{\mathcal{F}m\{\boldsymbol{c}(\boldsymbol{y},\boldsymbol{z})\}}{\mathcal{R}e\{\boldsymbol{c}(\boldsymbol{y},\boldsymbol{z})\}}\right).\tag{14}$$

Unwrap phase: *<sup>ϕ</sup>*<sup>~</sup> <sup>¼</sup> *unwrap*ð Þ *<sup>ϕ</sup>* . **return** unwrapped phase *<sup>ϕ</sup>*~ð Þ *<sup>y</sup>*, *<sup>z</sup>* .

In (13), *h* is an ideal 2D rectangular window function with size *Qy* � *Qz*, and *Qy* ¼ 2*π=κ<sup>y</sup>* � � and *Qz* <sup>¼</sup> d e <sup>2</sup>*π=κ<sup>z</sup>* where d e� denotes the ceiling function. The last step to unwrap the phase may be accomplished with a variety of techniques. To demonstrate the FTM approach, an interference image is simulated as *I y*ð Þ¼ , *z a y*ð Þþ , *z b y*ð Þ , *<sup>z</sup>* cos *<sup>κ</sup>yy* <sup>þ</sup> *<sup>κ</sup>zz* <sup>þ</sup> *<sup>ϕ</sup>*ð Þ *<sup>y</sup>*, *<sup>z</sup>* � � with spatial frequencies *<sup>κ</sup><sup>y</sup>* <sup>¼</sup> *<sup>π</sup>=*5 rad � <sup>m</sup>�<sup>1</sup> and *<sup>κ</sup><sup>z</sup>* <sup>¼</sup> *<sup>π</sup>* rad � <sup>m</sup>�1. The phase function *<sup>ϕ</sup>* is defined as

$$\phi(y, z) = \pi \exp\left[ -\mathbf{0}.\mathbf{1}(z/\mathbf{1}\mathbf{0} + y - \mathbf{0}.\mathbf{1})^2 \right],\tag{15}$$

and the background and contrast functions are specified as

$$a(\mathbf{y}, \mathbf{z}) = \mathbf{0}.\mathbf{6} + \eta \delta(\mathbf{y}, \mathbf{z}),\tag{16}$$

$$b(\mathbf{y}, z) = \mathbf{1}.\tag{17}$$

In (16), *<sup>γ</sup>* � <sup>N</sup> 0, *<sup>σ</sup>*<sup>2</sup> ð Þ is a random variable from the standard normal distribution with zero mean and variance *σ*2. The variance is set to produce phase noise with 10-dB signal-to-noise ratio.

The simulated phase and interferogram are shown in **Figure 6a** and **b**. The phase profile causes fringe lines similar to what are observed with a single exploded wire. However, the radiometric effects are ignored in the example. The results of the different stages of Algorithm 1 are also shown in **Figure 6c–f**. The spectrum of *g* has three components: a direct-current (DC)-like term, the spatial carrier, and the carrier's conjugate.

#### **Figure 6.**

*Example of FTM using simulated noisy 532-nm interferogram. (a) phase, (b) image, (c) signal spectrum, (d), filter spectrum, (e) filtered signal spectrum, (f) recovered phase.*

The DC component centered at *κ<sup>y</sup>* ¼ *κ<sup>z</sup>* ¼ 0 is mainly due to the background *a* but often includes residual energy from the carrier and its conjugate. The region of interest is centered at the spatial carrier frequency *κy*, *κ<sup>z</sup>* and has a spectral shape determined by *<sup>c</sup>* <sup>¼</sup> *be<sup>i</sup><sup>ϕ</sup>*. Therefore, the phase recovery is sensitive to the design of the filter *h*. **Figure 6f** shows the result when the ideal rectangular impulse response is used per [12] and has a noticeable ripple and effect of Gibbs phenomenon. The root mean square error between *ϕ* and the reconstructed and unwrapped phase *ϕ*~ is 0*:*67 radians.
