Interferogram Analysis For Optical Testing

Zacarias Malacara, Manuel Servín

• Publisher: CRC Press
• Year: 2018
• ISBN:
  9781351836913
• Edition: 2
• Topic: Ciencias físicas
• Subtopic: Física

In this day of digitalization, you can work within the technology of optics without having to fully understand the science behind it. However, for those who wish to master the science, rather than merely be its servant, it's essential to learn the nuances, such as those involved with studying fringe patterns produced by optical testing interferometers.When Interferogram Analysis for Optical Testing originally came to print, it filled the need for an authoritative reference on this aspect of fringe analysis. That it was also exceptionally current and highly accessible made its arrival even more relevant. Of course, any book on something as cutting edge as interferogram analysis, no matter how insightful, isn't going to stay relevant forever. The second edition of Interferogram Analysis for Optical Testing is designed to meet the needs of all those involved or wanting to become involved in this area of advanced optical engineering. For those new to the science, it provides the necessary fundamentals, including basic computational methods for studying fringe patterns. For those with deeper experience, it fills in the gaps and adds the information necessary to complete and update one's education. Written by the most experienced researchers in optical testing, this text discusses classical and innovative fringe analysis, principles of Fourier theory, digital image filtering, phase detection algorithms, and aspheric wavelength testing. It also explains how to assess wavefront deformation by calculating slope and local average curvature.

1

Review and Comparison of the Main Interferometric Systems

1.1 TWO-WAVE INTERFEROMETERS AND CONFIGURATIONS USED IN OPTICAL TESTING

Two-wave interferometers produce an interferogram by superimposing two wavefronts, one of which is typically a flat reference wavefront and the other a distorted wavefront whose shape is to be measured. The literature (e.g., Malacara, 1992; Creath, 1987) provides many descriptions of interferometers; here, we will just describe some of the more important aspects.

An interferometer can measure small wavefront deformations with a high accuracy, of the order of a fraction of the wavelength. The accuracy in a given interferometer depends on many factors, such as the optical quality of the components, the measuring methods, the light source properties, and disturbing external factors, such as atmospheric turbulence and mechanical vibrations. It has been shown by Kafri (1989), however, that the accuracy of any interferometer is limited. He proved that, if everything else is perfect, a short coherence length and a long sampling time can improve the accuracy. Unfortunately, a short coherence length and long measuring time combined make the instrument more sensitive to mechanical vibrations. In conclusion, the uncertainty principle imposes a fundamental limit to the accuracy that depends on several parameters but is of the order of 1/1000 of the wavelength of the light.

Figure 1.1 Two interfering wavefronts.

To study the main principles of interferometers, let us consider a two-wave interferogram with a flat wavefront that has a positive tilt about the y-axis and a wavefront under analysis, for which the deformations with respect to a flat wavefront without tilt are given by W(x, y). This tilt is said to be positive when the wavefront is as shown in Figure 1.1. The complex amplitude in the observation plane, where the two wavefronts interfere, is the sum of the complex amplitudes of the two waves as follows:

$E_1(x,y)=A_1(x,y)\exp ikW(x,y)+A_2(x,y)\exp i(kx\sin \text{θ})$

(1.1)

where A₁ is the amplitude of the light beam at the wavefront under analysis, A₂ is the amplitude of the light beam with the reference wavefront, and k = 2π/λ. Hence, the irradiance is:

$E_1(x,y)\cdot E_1^{*}(x,y)=A_1^2(x,y)+A_2^2(x,y)+2A_1(x,y)A_2(x,y)\cos k[x\sin \text{θ}-W(x,y)]$

(1.2)

Figure 1.2 Irradiance as a function of phase difference between the two waves along the light path.

where the symbol * denotes the complex conjugate of the electric field. Here, we have introduced optional tilt θ about the y-axis between the two wavefronts. The irradiance function, I(x, y), may then be written as:

$I(x,y)=I_1(x,y)+I_2(x,y)+2I_1(x,y)I_2(x,y)\cos k[x\sin \text{θ}-W(x,y)]$

(1.3)

where I₁(x, y) and I₂(x, y) are the irradiances of the two beams, and the phase difference between them is given by φ = k(xsinθ – W(x, y)). This function is shown graphically in Figure 1.2.

For convenience, Equation 1.3 is frequently written as:

$I(x,y)=a(x,y)+b(x,y)\cos k[x\sin \text{θ}-W(x,y)]$

(1.4)

Assuming that the variations in the values of a(x, y) and b(x, y) inside the interferogram aperture are smoother than the variations of the cosine term, the maximum irradiance in the vicinity of the point (x, y) in this interferogram is given by:

$I_{\text{max}}(x,y)={(A_1(x,y)+A_2(x,y))}^2=I_1(x,y)+I_2(x,y)+2\sqrt{I_1(x,y)I_2(x,y)}$

(1.5)

and the minimum irradiance in the same vicinity is given by:

$I_{\text{min}}(x,y)={(A_1(x,y)-A_2(x,y))}^2=I_1(x,y)+I_2(x,y)-2\sqrt{I_1(x,y)I_2(x,y)}$

(1.6)

The fringe visibility, v(x, y), is defined by:

$v(x,y)=\frac{I_{\text{max}}(x,y)-I_{\text{min}}(x,y)}{I_{\text{max}}(x,y)-I_{\text{min}}(x,y)}$

(1.7)

Hence, we may find:

$v(x,y)=\frac{I_{\text{max}}(x,y)-I_{\text{min}}(x,y)}{I_{\text{1}}(x,y)-I_2(x,y)}=\frac{b(x,y)}{a(x,y)}$

(1.8)

Using the fringe visibility,...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Table of Contents
• Chapter 1 Review and Comparison of the Main Interferometric Systems
• Chapter 2 Fourier Theory Review
• Chapter 3 Digital Image Processing
• Chapter 4 Fringe Contouring and Polynomial Fitting
• Chapter 5 Periodic Signal Phase Detection and Algorithm Analysis
• Chapter 6 Phase-Detection Algorithms
• Chapter 7 Phase-Shifting Interferometry
• Chapter 8 Spatial Linear and Circular Carrier Analysis
• Chapter 9 Interferogram Analysis with Moiré Methods
• Chapter 10 Interferogram Analysis without a Carrier
• Chapter 11 Phase Unwrapping
• Chapter 12 Wavefront Curvature Sensing
• Index