现代光学计量中干涉条纹分析原理与应用详解

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"Fringe Pattern Analysis for Optical Metrology" 是一本深入探讨现代光学计量中光栅模式分析基本理论原理的专业书籍。该书旨在为实验者提供实用的方法，详细介绍如何从单个或多个光栅模式中恢复调制相位，涵盖了光学干涉、阴影莫尔、光栅投影、光弹性、莫尔干涉、莫尔衍射测量、全息干涉、剪切干涉、数字全息、斑点干涉以及角膜地形图等各种光学计量技术中的应用。作者Manuel Servin, J. Antonio Quiroga, 和 J. Moisés Padilla将读者带入一个广泛而细致的领域，从经典的干涉测量技术到新兴的数字成像和高级技术，如斑点成像计量学的进步。书中包含大约577幅插图和21张表格，以便于理解复杂的理论概念和实验操作步骤。这本书不仅涵盖了基础的数学处理，如傅立叶变换在红外光谱测量中的应用，还讨论了诸如光学商店测试和光学系统设计等实践应用。例如，"Fourier Transform Infrared Spectrometry" 由 Griffiths 和 DeHaseth合著，介绍了红外光谱技术的傅立叶变换方法，而Malacara的 "Optical Shop Testing" 则聚焦于实际光学元件的测试标准。此外，"Handbook of Optical Systems"系列由Singer, Totzeck, 和 Gross等人编撰，分别关注物理成像形成原理（Volume 2）和技术光学基础（Volume 1），这些全面的手册提供了关于光学系统设计与分析的深入指导。通过阅读"Fringe Pattern Analysis for Optical Metrology"，读者不仅能掌握光栅模式分析的核心原理，还能了解这些技术在现代精密测量中的实际应用，这对于光学工程师、计量科学家以及从事相关领域的研究人员来说，是一本极具价值的参考资源。

Digital Linear Systems

1.1

Introduction to Digital Phase Demodulation in Optical Metrology

In this chapter, we review the theory behind digital signals and their temporal

processing using linear time-invariant (LTI) systems. The analysis of digital LTI

systems is based on their impulse response h (t), their Z-transform H(z),their

frequency transfer function (FTF) H(𝜔), their harmonic response, and their stability

criteria. We then brieﬂy discuss the equivalence between phase-shifting algorithms

(PSAs) and quadrature linear ﬁlters tuned at the temporal phase-sampling rate

of 𝜔

radians per sample. Also, we analyze the aliasing phenomena produced by

high-order harmonic distortion of the continuous interferogram being sampled.

In this chapter, we also discuss regularized low-pass ﬁltering and its application

to fringe-pattern denoising. Convolution spatial ﬁlters (such as the 3 × 3 averaging

ﬁlter) mix up valid fringe data inside the interferogram boundaries with outside

background where no fringe data is deﬁned. This linear mixing of fringes and

background distorts the modulating phase near the interferogram boundaries. In

contrast, regularized linear ﬁlters optimally decouple the fringe data inside the

interferogram from the outside background.

Finally, we discuss the theory behind stochastic processes to analyze the response

of LTI systems to stochastic input signals X(t). We deﬁne and analyze their

probability density function (PDF), or f

(x), their ensemble average E{X},and

their stationary autocorrelation function R

(𝜏).Wethencontinuebydeﬁningthe

power spectral density (PSD) S

(𝜔) for X(t). This result is then used to show that

the input PSD, S

(𝜔) of X(t),changestoH(𝜔)

(𝜔) when processed by an LTI

system whose FTF is given by H(𝜔).

1.1.1

Fringe Pattern Demodulation as an Ill-Posed Inverse Problem

A fringe pattern is deﬁned as a sinusoidal signal where a continuous map, analogous

of the physical quantity being measured, is phase-modulated by an interferometer,

Moire system, and so on. An ideal stationary fringe pattern is usually modeled by

I(x, y)=a(x , y)+b(x, y) cos[𝜑(x, y)], (1.1)

Fringe Pattern Analysis for Optical Metrology: Theory, Algorithms, and Applications, First Edition.

Manuel Servin, J. Antonio Quiroga, and J. Mois

es Padilla.

 2014 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2014 by Wiley-VCH Verlag GmbH & Co. KGaA.

2 1 Digital Linear Systems

where {x, y}∈ℝ

; a(x, y) and b(x, y) are the background and local contrast func-

tions, respectively; and 𝜑(x, y) is the searched phase function.

In physics and mathematics, an inverse problem is a general framework that

is used to convert the observed measurements into information about a physical

object or system under study [1]. Clearly, Eq. (1.1) represents an inverse problem,

where the fringe pattern I(x, y) is our measurement and the searched information

is given by the phase 𝜑(x, y). An inverse problem is said to be well posed if

the mathematical model of a given physical phenomenon fulﬁlls the following

conditions:

•

A solution exists,

•

The solution is unique, and

•

The solution depends continuously on the data.

On analyzing Eq. (1.1), one can see that the phase function 𝜑(x, y) cannot be

directly estimated since it is screened by two other unknown functions, namely

a(x, y) and b(x, y). Additionally, 𝜑(x , y) can only be determined modulo 2π because

the sinusoidal fringe pattern I(x, y) depends periodically on the phase (2π phase

ambiguity); and its sign cannot be extracted from a single measurement without

a priori knowledge (sign ambiguity) because of the even character of the cosine

function [cos(𝜑)=cos

(−

𝜑)]. Finally, in all practical cases, some noise n(x, y) is

introduced in an additive and/or multiplicative manner, and the fringe pattern may

suffer from a number of distortions, degrading its quality and further screening

the phase information [2, 3].

It must be noted that, even if careful experimental setups could prevent

the screening of 𝜑(x, y) due to the unknown signals a(x, y), b(x, y),and

n(x, y), one would still have to deal with the sign ambiguity and the 2π

phase ambiguity. Because of these ambiguities, the solution for this inverse

problem is not unique; this is illustrated in Figure 1.1, where several

phases (from an inﬁnite number of possibilities) produce exactly the same

sinusoidal signal.

In short, the phase demodulation of a fringe pattern, as the one modeled in

Eq. (1.1), can be viewed as an ill-posed inverse problem where some sort of

regularization process is required in order to obtain a proper phase estimation.

However, despite its intrinsic difﬁculties, it is rather easy to visualize a possible

solution for this inverse problem. First, let us rewrite Eq. (1.1) by means of the

complex representation of the cosine function

I(x, y )=a(x, y)+

b(x, y){exp[i𝜑(x, y)] + exp[−i𝜑(x, y)]}. (1.2)

Now, if somehow one is able to isolate one of the analytic signals in Eq. (1.2), say,

(1∕2)b(x, y) exp[i𝜑(x, y)],wehave

tan 𝜑(x, y)=

Im{(1∕2)b(x, y) exp[i𝜑(x, y)]}

Re{(1∕2)b(x, y) exp[i𝜑(x, y)]}

, (1.3)

where b(x, y) ≠ 0. Computing the arc-tangent of the above formula, one obtains a

wrapped estimation of the phase under study, that is, 𝜑(x, y)mod2π.Thus,the

4 1 Digital Linear Systems

where we deﬁne (locally) this nabla operator as

𝛻 =



∂

∂x

∂

∂y

∂

∂t



. (1.6)

For instance, for a stationary phase (which shows no explicit time dependence)

given by in 𝜑(x, y), and a spatial carrier c(x, y), the following condition must be

fulﬁlled:





∂c

∂x





∂c

∂y







∂𝜑

∂x





∂𝜑

∂y



. (1.7)

The spatial and/or temporal carriers are of extreme importance in modern inter-

ferometry: ﬁrst of all, their presence allows us to solve the sign ambiguity since in

general cos(𝜑 + c) ≠ cos(−𝜑 + c).Theyalsoallowustoisolatetheanalyticsignal

(1∕2)b(x, y) exp[i𝜑(x, y)] which practically solves the phase demodulation process

(the proof of this last point will be postponed until we review some basic concepts

of Fourier analysis). Some typical examples of the carrier functions are as follows:

•

linear temporal carrier [5, 6]

(t)=𝜔

t; (1.8)

•

tilted (spatial) carrier [7, 8]

(x, y)=u

x + v

y; (1.9)

•

conic carrier [9]

(𝜌)=𝜔

𝜌; 𝜌(x, y)=



+ y

; (1.10)

•

2 × 2 pixelated carrier [10–12]

exp[ic

(x, y)] = exp



i𝜔





∗∗

∞



m=0

∞



n=0

𝛿(x − 2m, y − 2n), (1.11)

where 𝜔

=π∕2, and ∗∗ is the two-dimensional convolution operation;

•

3 × 3 pixelated carrier [13]

exp[ic

(x, y)] = exp







i𝜔







123

894

765













∗∗

∞



m=0

∞



n=0

𝛿(x − 3m, y − 3n), (1.12)

where 𝜔

= 2π∕9.

Since digital interferometry is a research area under continuous development, it

is impossible to list all useful spatiotemporal carriers; again, these are just some

commonly used examples. For illustrative purposes, in Figures 1.2–1.5 we show

how these carriers modify the fringe pattern.

The temporal linear carrier approach (shown in Figure 1.2) allows us to demod-

ulate closed-fringe interferograms [5, 6]. However, this method is not useful (in

principle) to study fast-varying phenomena since it requires a(x, y), b(x, y),and

𝜑(x, y) to remain stationary during the phase-step acquisition.

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