2. Interferogram acquisition

The wave nature of light can be studied theoretically by a homogeneous partial differential equation of second order, which satisfies the superposition principle:

$$\nabla \mathcal{U}(\mathbf{x}, y, z, t) = \frac{1}{c^2} \frac{\partial \mathcal{U}(\mathbf{x}, y, z, t)}{\partial t}. \tag{1}$$

If two waves of the same frequency are superimposed on a point in space, they excite oscillations in the same direction:

$$\mathcal{U}I\_1(t) = A\_1 \cos(\omega t + \varphi\_1),\tag{2}$$

and

$$dL\_2(t) = A\_2 \cos(\omega t + \varphi\_2). \tag{3}$$

In the preceding equations and the subsequent ones in this section, we will drop the spatial dependence for displaying purposes. The amplitude of the resulting oscillation at that point is determined by the equation:

$$A^2 = A\_1^2 + A\_2^2 + 2A\_1A\_2\cos(\delta),\tag{4}$$

where

$$
\delta = \varphi\_1 - \varphi\_2.\tag{5}
$$

If the phase difference, δ, of the oscillations excited by waves remains constant with time, these waves are coherent. In the case of noncoherent waves, δ varies continuosly, taking any values with equal probability, so the average value of cos(δ) is zero. Therefore,

combines several regions with circular, straight and crossed fringes of varying density. Rapidly, it was recognized the need of automatic methods for fringe analysis. The first great advance arises with the development of the phase-shifting techniques. With those procedures, a set of interferograms is acquired with a phase shift among them. The phase shifts are usually introduced by a piezoelectric transducer moving the reference mirror in such way that the phase difference between two consecutive interferograms is a constant term. With phase-shifting techniques, it is possible to isolate the sine and cosine of the phase allowing the calculation of the wrapped phase distribution and consequently the continuous phase with an unwrapping algorithm. Another great success came with the method proposed by Takeda (also referred as the Fourier method) performing a band-pass filtering in the Fourier domain. The method of Takeda works only with interferograms that contain open fringes (patterns that consist in nearly straight fringes). In order to generate such interferograms, the reference beam (e.g., in a two arm interferometer) is tilted introducing a large carrier function to the phase. The Fourier transform of these interferograms is composed of three lobules, one at the center that corresponds to the background term and two lobules located symmetrically respect to the origin. One of this lobules and the one that is located at the origin are filtered out. The remaining spectrum is transformed back to the spatial domain from which the so-called wrapped phase can be calculated. A final step is to apply a phase unwrapping technique to recover the continuous phase. Interferometric measurements and fringe analysis techniques are a growing and fast-changing field of research. Through this

The wave nature of light can be studied theoretically by a homogeneous partial differential

c2

If two waves of the same frequency are superimposed on a point in space, they excite oscilla-

In the preceding equations and the subsequent ones in this section, we will drop the spatial dependence for displaying purposes. The amplitude of the resulting oscillation at that point is

<sup>2</sup> <sup>þ</sup> <sup>A</sup><sup>2</sup>

∂Uðx, y, z, tÞ

<sup>∂</sup><sup>t</sup> : (1)

U1ðtÞ ¼ A1cosðωt þ ϕ1Þ; (2)

U2ðtÞ ¼ A2cosðωt þ ϕ2Þ: (3)

<sup>2</sup> <sup>þ</sup> <sup>2</sup>A1A2cosðδÞ, (4)

chapter, we will review the most known procedures.

equation of second order, which satisfies the superposition principle:

<sup>A</sup><sup>2</sup> <sup>¼</sup> <sup>A</sup><sup>1</sup>

<sup>∇</sup>Uðx, <sup>y</sup>, <sup>z</sup>, <sup>t</sup>Þ ¼ <sup>1</sup>

2. Interferogram acquisition

2 Optical Interferometry

tions in the same direction:

determined by the equation:

and

where

$$A^2 = A\_1^2 + A\_2^2.\tag{6}$$

So, we can conclude that the intensity observed in the superposed point by noncoherent waves equals the sum of the intensities, which create each separately. However, if the difference δ = ϕ<sup>1</sup> − ϕ<sup>2</sup> is constant, the cos(δ) will also have a constant value over time, but own for each point in space, so that:

$$I = I\_1 + I\_2 + 2\sqrt{I\_1 I\_2} \cos(\delta) \tag{7}$$

where I<sup>1</sup> = A<sup>1</sup> <sup>2</sup> and I<sup>2</sup> = A<sup>2</sup> 2 . This superposition at a point in space results in a different sum of the intensities of the separate components intensity [1]. This phenomenon is known as interference. The light and dark areas that observed on screen placed in the region of interference are called interference fringes, and the fringes are intensity which changes from minimum to maximum, which together form a pattern commonly called interference pattern or interferogram, as can be seen in Figure 1.

The interference of two or more electromagnetic waves can be usually achieved in two ways: by division of the wave front and by division of the amplitude. A mechanism used for the division of the wave front is, for example, the Young's experiment or double slit that is showed in Figure 2.

A mechanism used for dividing the amplitude of the wave is, for example, the Michelson interferometer that is shown in Figure 3.

Both mechanisms produce a pattern of light and dark intensities in the plane of interference fringes. The image resulting from the interference is known as interferogram. When the interferogram is captured by a recording medium, that is, a photographic film or a CCD

Figure 1. Interference pattern or interferogram showing circular fringes.

Figure 2. Young's experiment, where a double slit produces two wave fronts that interfere on a screen.

Figure 3. Michelson interferometer.

camera, the process commonly involves some optical system, which introduces imperfections with respect to the ideal image. Such imperfections are known as aberrations. Aberrations can be classified as chromatic and monochromatic. Chromatic aberrations are present to illuminate the object with white light or polychromatic light, that is, light with different wavelengths. These aberrations are the only ones that can be predicted by the theory of the first order, which states that an optical system consisting of lenses has different focal lengths for different wavelengths. These variations are related to the change of refractive index with respect to wavelength causing that both the position and the image size are different for each wavelength. Monochromatic aberrations occur when the object is illuminated with monochromatic light (i.e., light of a single wavelength), and the reflected or transmitted light is registered by a recording medium. This type of aberration causes that the captured image of a punctual object is no longer a point, but a blurred point. Monochromatic aberrations can be calculated roughly in the third-order theory, using the first two terms of the expansion in power series of the sinθ and cosθ. Another alternative to calculate more accurately is to make the exact trigonometric trace rays through the system, where the deviation of the rays is calculated. These aberrations were studied in detail first by Ludwig von Seidel, hence often called Seidel aberrations, and are sphericity, coma, astigmatism, field curvature and distortion. These aberrations have a distinctive fringe distribution that appears frequently in the testing of a variety of objects with an interferometer. As the interferogram acquisition is only the capture of the interference phenomenon or interference pattern in a recording medium, it is convenient to study correcting aberrations problems. From these patterns, it can be known as the aberration coefficients using Zernike polynomials. Zernike polynomials have been successfully used in the recognition of patterns and image processing. Additionally, these have been used in astronomy to describe wavefront aberrations due to atmospheric turbulence and to describe wavefront aberrations in the human eye. This is because of the Seidel aberrations are related to the Zernike polynomials. Due to this relationship, polynomials are used to describe wavefront aberrations in order to calculate the aberration coefficients of a set of interferograms generated by an adaptive lens. These coefficients enable to describe the behavior of the aberrations present in the lens.
