**2. Limitations of optical instruments**

This section briefly discusses some of the limitations of optical instruments. Many optical instruments use a microscope objective to magnify the features on the surface under test. There are two fundamental limitations of the optical instruments utilizing a microscope objective: the first is the numerical aperture (Na) of the object which is given by

$$\mathbf{N}\mathbf{a} = n\sin(\alpha)\tag{1}$$

where is the refractive index of the medium between the objective and the surface and is the acceptance angle of the aperture as shown in **Figure 1**. For optical instruments based on interference microscopy, a correction factor should be added to the interference pattern due to the impact of the Na. This correction can usually be estimated by well-known methods [1]. The second limitation is the optical resolution of the objective. The resolution determines the minimum distance between two lateral features on a surface that can be measured. The spatial resolution is approximately given by

$$\mathbf{s} = \mathcal{X} / 2\mathbf{\hat{n}a} \tag{2}$$

where is the wavelength of the incident radiation [2]. For a theoretically perfect optical system with a filled objective pupil, the optical resolution is given by the Rayleigh criterion, where the 1/2 in Eq. (2) is replaced by 0.61. If the objective is not optically perfect (i.e., aberration-free) or if a part of the beam is blocked (e.g., in a Mirau interference objective, or when a steep edge is measured), the value becomes higher (worse).

**Figure 1.** Numerical aperture of a microscope objective lens.

#### **2.1. Optical aberrations**

However, all two-beam interferometers suffer from the fact that they produce cos2

remarks.

82 Optical Interferometry

**2. Limitations of optical instruments**

spatial resolution is approximately given by

distributions. This fact makes two-beam interferometers unpopular to characterize strongly curved surfaces and steep edges because of the too high density of fringes which makes the feature too complex to measure. Multiple-beam interferometers are used to characterize these surfaces successfully thanks to the very sharp fringes. In this chapter, we present new frontiers in both two- and multiple-beam interferometers carried out by the author. As modern interferometers use a laser as the light source, spurious and speckle noises arise in the fringe pattern. Numerical techniques should be applied to the fringe pattern to suppress these spurious and speckle noises. In Section 2, limitations of optical instruments including optical aberrations and denoising and effect of noise on phase unwrapping are explained. In Section 3, fundamentals of interferometry with focus on two- and multiple-beam interferometers and their capabilities in testing film thickness, curvatures of strongly curved surfaces, and parallelism of a standard optical flat are described. It is worth mentioning that the in-line configuration of interferometry can feature finer sample spatial details compared with the offaxis configuration. However, using in-line configuration requires the time-sequent phaseshifting (PS) process to eliminate both zero-order and the twin image. Single-shot parallel phase-shifting technique is proposed for real-time measurement. In Section 4, single-shot parallel four-step phase-shifting Fizeau interferometer for three-dimensional (3-D) surface micro-topography measurement is explained. Section 5 gives concluding discussions and

This section briefly discusses some of the limitations of optical instruments. Many optical instruments use a microscope objective to magnify the features on the surface under test. There are two fundamental limitations of the optical instruments utilizing a microscope objective:

where is the refractive index of the medium between the objective and the surface and is the acceptance angle of the aperture as shown in **Figure 1**. For optical instruments based on interference microscopy, a correction factor should be added to the interference pattern due to the impact of the Na. This correction can usually be estimated by well-known methods [1]. The second limitation is the optical resolution of the objective. The resolution determines the minimum distance between two lateral features on a surface that can be measured. The

where is the wavelength of the incident radiation [2]. For a theoretically perfect optical system with a filled objective pupil, the optical resolution is given by the Rayleigh criterion, where the

(1)

/ 2Na (2)

Na sin( ) = *n* a

*s* = l

the first is the numerical aperture (Na) of the object which is given by

intensity

A system with aberrations has a wavefront phase surface that deviates from the ideal spherical wave. Aberrations are found in most practical imaging systems, and their effect reduces image quality. Aberrated systems tend to cause space-variant imaging, where the impulse response is not the same for each image point. **Figure 2** shows the representation of an ideal spherical and aberrated wavefronts. The difference between the ideal spherical wavefront and aberrated wavefront is a wavefront error *W*(*x,y*), where *x* and *y* are the coordinates in the pupil plane. It is worth noting that the source of aberrated wavefronts may come from the imperfections in the imaging optics.

**Figure 2.** (a) Spherical (sp) and aberrated (ab) wavefronts, (b) Seidel aberration coordinate definitions for the normalized exit pupil, and (c) Seidel aberration coordinate for the normalized image plane.

Wavefront optical path length (OPD) is commonly described by a polynomial series. The Seidel series is used by optical designers because the terms have straightforward mathematical relationships to factors such as lens type and position in the image plane. Another series, Zernike polynomials, is used in optical testing and applications where the aberrations do not have a simple dependency on the system parameters. Both formulations assume a circular pupil. Seidel polynomials are often used to describe monochromatic aberrations for rotationally symmetric optical systems, such as most lenses and mirrors. A common form that is applied in conventional imaging systems is described by [3]

$$W(\hat{\mathbf{i}}\_0; \boldsymbol{\rho}, \boldsymbol{\theta}) = \sum\_{j, m, n} W\_{k \, l \, m} \hat{\mathbf{i}}\_0^k \, \boldsymbol{\rho}^l \cos^m \boldsymbol{\theta}; \, k = 2j + m, \quad l = 2n + m,\tag{3}$$

where is a normalized radial distance in the exit pupil and is the angle in the exit pupil as shown in **Figure 2(b)**. For computational reasons, the angle is defined here relative to the *x*axis in a counter-clockwise direction. However, note that this angle is often defined relative to the *y-*axis in traditional aberration treatments. The normalized exit pupil has a radius of 1 where the physical coordinates (*x, y*) are divided by the exit pupil radius to get normalized coordinates (, ). <sup>0</sup> is the normalized image height, defined along the axis in the imaging plane as indicated in **Figure 2(c)**. The normalized image height is the physical height of a given point in the image divided by the maximum image radius being considered. Since the Seidel polynomials assume a rotationally symmetric system, the pupil and image plane coordinate systems are simply rotated to find the wavefront OPD function for an image point that is off the axis. The indices *j, m, n*, and so forth, in Eq. (3), are a numbering and power scheme. are the wavefront aberration coefficients, and the five primary Seidel aberrations correspond to + = 4. These primary aberrations are known as spherical aberration, coma, astigmatism, field curvature, and distortion. The coefficients have units of distance (μm), although they are usually discussed relative to the optical wavelength (i.e., so many "waves").

For simulation purposes, it is convenient to convert from polar to Cartesian coordinates. Referring to **Figure 2(b)**,

$$
\rho = \sqrt{\hat{\mathbf{x}}^2 + \hat{\mathbf{y}}^2} \quad \text{and} \quad \rho \cos \theta = \hat{\mathbf{x}}, \tag{4}
$$

and the primary aberrations are then written as

$$\begin{split} W(\hat{l\_0}; \hat{\mathbf{x}}, \hat{\mathbf{y}}) &= W\_d(\hat{\mathbf{x}}^2 + \hat{\mathbf{y}}^2) + W\_{\text{o40}}(\hat{\mathbf{x}}^2 + \hat{\mathbf{y}}^2)^2 + W\_{13}\hat{l}\_0(\hat{\mathbf{x}}^2 + \hat{\mathbf{y}}^2)\hat{\mathbf{x}} \\ &+ W\_{222}\hat{l}\_{\text{o}}^2 \hat{\mathbf{x}}^2 + W\_{220}\hat{l}\_{\text{o}}^2(\hat{\mathbf{x}}^2 + \hat{\mathbf{y}}^2) + W\_{31}\hat{l}\_{\text{o}}^3 \hat{\mathbf{x}}. \end{split} \tag{5}$$

The first term in this series is not one of the five primary aberrations, but is a defocus term. It is the wavefront OPD that is "created" in moving the image plane along the optical axis from the paraxial focus position. The second, third, fourth, and fifth Seidel aberration terms in Eq. (5) are spherical, coma, astigmatism, field curvature, and distortion, respectively. Simulating the effects of these aberrations by plotting some wavefront OPD surfaces is shown in **Figure 3**. **Figure 3** illustrates that spherical aberration (*W*040) and field curvature (*W*220) are wavefront curvature-like terms that are spherically symmetric with respect to the pupil coordinates. Coma (*W*131) and astigmatism (*W*222) are not spherically symmetric and depend on the image point position.

Zernike polynomials, is used in optical testing and applications where the aberrations do not have a simple dependency on the system parameters. Both formulations assume a circular pupil. Seidel polynomials are often used to describe monochromatic aberrations for rotationally symmetric optical systems, such as most lenses and mirrors. A common form that is

ˆ ˆ (; ,) <sup>=</sup> å cos ; 2 , 2 , =+ =+ *kl m*

 q

where is a normalized radial distance in the exit pupil and is the angle in the exit pupil as shown in **Figure 2(b)**. For computational reasons, the angle is defined here relative to the *x*axis in a counter-clockwise direction. However, note that this angle is often defined relative to the *y-*axis in traditional aberration treatments. The normalized exit pupil has a radius of 1 where the physical coordinates (*x, y*) are divided by the exit pupil radius to get normalized coordinates (, ). <sup>0</sup> is the normalized image height, defined along the axis in the imaging plane as indicated in **Figure 2(c)**. The normalized image height is the physical height of a given point in the image divided by the maximum image radius being considered. Since the Seidel polynomials assume a rotationally symmetric system, the pupil and image plane coordinate systems are simply rotated to find the wavefront OPD function for an image point that is off the axis. The indices *j, m, n*, and so forth, in Eq. (3), are a numbering and power scheme. are the wavefront aberration coefficients, and the five primary Seidel aberrations correspond to + = 4. These primary aberrations are known as spherical aberration, coma, astigmatism, field curvature, and distortion. The coefficients have units of distance (μm), although they are usually discussed relative to the optical wavelength (i.e., so many "waves").

For simulation purposes, it is convenient to convert from polar to Cartesian coordinates.

0 0 0

The first term in this series is not one of the five primary aberrations, but is a defocus term. It is the wavefront OPD that is "created" in moving the image plane along the optical axis from the paraxial focus position. The second, third, fourth, and fifth Seidel aberration terms in Eq. (5) are spherical, coma, astigmatism, field curvature, and distortion, respectively. Simulating the effects of these aberrations by plotting some wavefront OPD surfaces is shown

22 2 2 2 3 222 220 311

ˆˆ ˆ ˆ ˆˆ ˆ () . = ++ + + +

ˆ ˆ (;,) ( ) ( ) ( ) ˆˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ

2 2 2 22 2 2

*W ix W i x y W ix* (5)

2 2

0 040 131 0

+ + ++ *Wi xy W x y W x y W i x y x <sup>d</sup>*

 =+ = *x y* ˆ ˆand cos , r q

r

and the primary aberrations are then written as

(3)

*x*ˆ (4)

*Wi W i k j m l n m*

r

applied in conventional imaging systems is described by [3]

0 0 , ,

r q

84 Optical Interferometry

Referring to **Figure 2(b)**,

*jmn*

*klm*

**Figure 3.** Example wavefront OPD surface and contour plot using Seidel 5. These represent phase surfaces that are applied in the exit pupil.

Because of the great coherence of the laser light, the fringe pattern may be easily obtained. This advantage of laser light makes most of the modern interferometers use a laser as the light source. In fact, this advantage can also be a serious disadvantage, as spurious and speckle noises arise. Special precautions must be taken into account to suppress these spurious and speckle noises. Some of them are practical such as inserting many stops in the optical system and the others are numerical such as applying windowed Fourier transform (WFT) [4] and flat fielding with apodization techniques [5].

### **2.2. Denoising and effect of noise on phase unwrapping**

When an optically rough surface is illuminated by an expanded laser beam, the formed image is a speckle pattern (bright and dark spots). Noise can have catastrophic effects on the phaseunwrapping process. Application of simple filtering techniques in classical image processing to suppress speckle noise tends to do more harm than good, because they blur the image indiscriminately. Alternative techniques must be applied to obtain a clean interference pattern. Once a clean image is obtained, unwrapping process is applied easily for reconstruction. Let us see how the noise affects on phase unwrapping, suppose that we have a discrete signal whose amplitude exceeds the range [−π,π] as shown in **Figure 4(a)**. We can wrap the signal () by calculating the sinusoidal and the cosinusoidal values of (). The four quadrant arctangent function (atan2) of sin(*x*) and cos(*x*) is then calculated using the following equation:

**Figure 4.** (a) Continuous phase, (b) wrapped phase, and (c) the phase unwrapped signal.

$$a \tan 2(u, \nu) = \begin{cases} \tan^{-1}(u/\nu) & \text{1st.} \quad quadrant \\ \tan^{-1}(u/\nu) + \pi & \text{2nd.} \quad quadrant \\ \tan^{-1}(u/\nu) - \pi & \text{3rd.} \quad quadrant \\ \tan^{-1}(u/\nu) & \text{4th.} \quad quadrant \end{cases} \tag{6}$$

where *u* and *v* are real numbers. We can express the wrapping process mathematically as () = () . The 2π jumps that are present in the wrapped phase signal that is shown in **Figure 4(b)** must be removed in order to return the phase signal () to a continuous form and hence make the phase usable in any analysis or further processing. This process is called phase unwrapping and has the effect of returning a wrapped phase signal to a continuous phase signal that is free from 2π jumps. We can express the unwrapping process mathematically as () = ()+2, where () is the unwrapped phase signal and *k* is an integer.

The phase-unwrapped signal is shown in **Figure 4(c)**. The wrapped phase signal that is shown in **Figure 4(a)** is a very simple signal to unwrap. This is because () is a simulated signal that does not contain any noise, but if the signal is noisy, a fake-phase wrap may be produced by noise in the signal. The existence of a fake wrap will affect the unwrapping of the sample. Let us use a computer simulation to illustrate this. Suppose that we have the discrete signal () and then we add white noise to this signal as noise() = () + white noise. The noise variance is set to a higher value of 0.8. The original noisy signal is shown in **Figure 5(a)**. Wrapping and unwrapping phase signals are shown in **Figure 5(b)** and **(c)**, respectively. As shown in **Figure 5**, the higher noise level has seriously affected the phase-unwrapping process and the phase unwrapping of the signal became a challenging task. This is due to the existence of a fake wrap in the signal.

Once a clean image is obtained, unwrapping process is applied easily for reconstruction. Let us see how the noise affects on phase unwrapping, suppose that we have a discrete signal whose amplitude exceeds the range [−π,π] as shown in **Figure 4(a)**. We can wrap the signal () by calculating the sinusoidal and the cosinusoidal values of (). The four quadrant arctangent function (atan2) of sin(*x*) and cos(*x*) is then calculated using the following equation:

**Figure 4.** (a) Continuous phase, (b) wrapped phase, and (c) the phase unwrapped signal.

ì ï

ï î

of a fake wrap in the signal.

86 Optical Interferometry


tan ( / ) 2 . tan 2( , ) tan ( / ) 3 .

<sup>ï</sup> <sup>+</sup> <sup>=</sup> <sup>í</sup> ï -

tan ( / ) 1 .

p

p

*u v nd quadrant a uv u v rd quadrant*

*u v st quadrant*

*u v th quadrant*

(6)

tan ( / ) 4 .

where *u* and *v* are real numbers. We can express the wrapping process mathematically as () = () . The 2π jumps that are present in the wrapped phase signal that is shown in **Figure 4(b)** must be removed in order to return the phase signal () to a continuous form and hence make the phase usable in any analysis or further processing. This process is called phase unwrapping and has the effect of returning a wrapped phase signal to a continuous phase signal that is free from 2π jumps. We can express the unwrapping process mathematically as () = ()+2, where () is the unwrapped phase signal and *k* is an integer.

The phase-unwrapped signal is shown in **Figure 4(c)**. The wrapped phase signal that is shown in **Figure 4(a)** is a very simple signal to unwrap. This is because () is a simulated signal that does not contain any noise, but if the signal is noisy, a fake-phase wrap may be produced by noise in the signal. The existence of a fake wrap will affect the unwrapping of the sample. Let us use a computer simulation to illustrate this. Suppose that we have the discrete signal () and then we add white noise to this signal as noise() = () + white noise. The noise variance is set to a higher value of 0.8. The original noisy signal is shown in **Figure 5(a)**. Wrapping and unwrapping phase signals are shown in **Figure 5(b)** and **(c)**, respectively. As shown in **Figure 5**, the higher noise level has seriously affected the phase-unwrapping process and the phase unwrapping of the signal became a challenging task. This is due to the existence

**Figure 5.** (a) Original noisy signal, (b) the phase wrapped signal, and (c) the phase unwrapped signal. Here, the noise variance has been increased to a value of 0.8.

Some techniques such as windowed Fourier transform and flat fielding with apodization may be used to suppress the noise and solve the problem of fake wrap. **Figure 6(a)** shows a simulated noisy closed fringe, and a clean image is shown in **Figure 7(b)** obtained using WFT technique. Profiles at the middle of **Figure 6(a)** and **(b)** are shown in **Figure 6(c)**. **Figure 7(a)** shows an experimental inline interferogram of a nano-pattern taken by Mach-Zehnder interferometer and a clean image is shown in **Figure 7(b)** obtained using flat fielding with apodization technique. Profiles at the middle of **Figure 7(a)** and **(b)** are shown in **Figure 7(c)** up and down, respectively.

**Figure 6.** A simulated noisy closed fringes (a), a clean image of **Figure 7(a)** obtained using WFT technique (b), and profiles at the middle of **Figure 7(a)** and **(b) (c)**.

**Figure 7.** An empirical inline interferogram of a nano-pattern object (a), a clean image of **Figure 7(a)** obtained using flat fielding with apodization technique (b), and profiles at the middle of **Figure 7(a)** and **(b)**, up and down, respectively, (c).

Several basic interferometric configurations are used in optical-testing procedures, but almost all of them are two-beam interferometers. In Section 3, we review fundamentals of interferometry with focus on two- and multiple-beam interferometers and its capability in featuring the topography of surfaces.
