wavefront sensor_wavefront sensor supplier-Yucheng Light Sense

Lateral Shearing Interferometry Testing Technology

4-Wave LateralShearing Interferometry [1] is a high-precision optical measurement technique that uses two identical wavefronts, with only a tiny lateral offset between them, to interfere in their overlapping region. The most notable feature of a 4-Wave LateralShearing Interferometry system, compared to traditional Twyman-Green or Fizeau interferometers, is that it doesn't have a reference optical path, and therefore, it doesn't need a reference mirror.

The advantages of 4-Wave LateralShearing Interferometry include:

Simplified system structure and significantly reduced cost, especially notable in measuring complex surface forms.

Elimination of system errors introduced by the reference mirror's surface form.

Easy achievement of a completely common optical path, which eliminates interference from environmental factors during the measurement process.

Applications of Lateral Shearing Interferometry

Based on these characteristics, 4-Wave LateralShearing Interferometry finds extensive applications not only in traditional optical inspection fields, such as (aspheric) surface form measurement [2] and angular displacement measurement [3], but it's also a primary method for implementing Fourier transform spectrometers in spectral imaging or polarimetric spectral imaging. Furthermore, it has wide-ranging applications in emerging fields like quantitative phase microscopy (DPM), optical coherence tomography (OCT), and optical diffraction tomography (ODT) [4].

Classification of Lateral Shearing Interferometry

4-Wave LateralShearing Interferometry can be categorized into diffraction-based and non-diffraction-based methods, which differ significantly in their design principles and optical path structures.

Diffraction-based methods typically use gratings to generate multiple diffracted wavefronts that have different propagation directions but are identical in shape to the measured light wave. Current research in this area primarily focuses on multiwave lateral shearing interferometry using compound gratings, where a single interferogram is formed by the superposition of multiple sets of sheared wavefronts.

Non-diffraction-based methods generally use parallel plates, prisms, polarizing elements, or some special optical path structures to produce lateral shear. This approach typically generates only one set of sheared wavefronts in a single direction, meaning that phase deviations perpendicular to that direction cannot be reflected in the interferogram. Therefore, two interference systems (with mutually orthogonal shear directions) are often required.

The biggest advantage of diffraction-based methods is their ability to achieve a set of orthogonal lateral shears, thus avoiding the need for two optical paths. However, they are limited by manufacturing processes; the actual diffracted light waves from gratings inevitably have system errors compared to theoretical designs (including wavefront shape errors, diffraction direction deviations, and stray diffraction orders). To minimize these system errors, extremely high precision is required for grating fabrication, leading to high costs. This chapter primarily focuses on non-diffraction-based dual-wavefront lateral shearing interferometry, while Chapter 4 will introduce diffraction-based multiwave lateral shearing interferometry.

Basic Concepts of 4-Wave LateralShearing Interferometry

Theoretical Basis of 4-Wave LateralShearing Interferometry

4-Wave LateralShearing Interferometry differs from other interferometric systems in that it doesn't have a 'reference wave' and a 'test wave.' Instead, the two light waves participating in the interference are both 'copies' of the original light wave carrying information about the object being measured. That is, they have identical wavefront shapes but are laterally shifted relative to each other. From a hardware perspective, the key to shearing interferometry is generating these two wavefronts. The structure that produces these two wavefronts is called the shearing device, and the lateral offset between the two wavefronts is called the shear amount.

A 4-Wave LateralShearing Interferometry system has a simpler structure than traditional interferometric systems; generally, it consists of only one optical path. Excluding the light source, beam shaping device, and the subsequent imaging and acquisition devices, its main structure can be represented as follows: the wavefront under test, $W_{0}$ , passes through the shearing device to produce two sheared wavefronts ( $W_{1}$ and $W_{2}$ ) with a certain lateral offset. These two wavefronts then interfere in their overlapping region, as shown in Figure 3-1.

Figure 3-1: Schematic Diagram of a 4-Wave LateralShearing Interferometry System Principle. (a) Test wavefront with a plane wave as the template; (b) Test wavefront with a spherical wave as the template.

There are two common forms of test wavefronts: as shown in Figure 3-1(a), a wavefront with a plane wave as the template, and as shown in Figure 3-1(b), a wavefront with a spherical wave as the template. The former uses a plane wave as the reference standard, and the test information reflects the deviation of the test wavefront relative to the standard plane wave, typically used for optical component surface inspection. The latter uses a spherical wave as the reference standard, and the test information reflects the deviation between the test wavefront and the reference spherical wave, generally used for inspecting the exit wavefront of optical systems (often with a converging light wave incident on the shearing device).

The two wavefronts involved in shearing are generated by the shearing device and should theoretically have identical intensity and phase distributions, with only a certain lateral shift. Therefore, interference will occur in their overlapping region. It's important to note that the phase distribution corresponding to the interferogram in 4-Wave LateralShearing Interferometry is the difference between the two sheared wavefronts (referred to as the differential wavefront) rather than the test wavefront itself. Thus, unlike traditional interferometric systems, a 4-Wave LateralShearing Interferometry interferogram does not directly reflect information about the object being measured. Furthermore, since interference only occurs within the overlapping range of the two wavefronts, the size and contour of the effective region of a 4-Wave LateralShearing Interferometry interferogram also differ from those of traditional interferometers. Figure 3-2 compares the system structures and interferograms of 4-Wave LateralShearing Interferometry and traditional interferometry (Michelson type).

Figure 3-2: Two types of interferometry systems and their corresponding interferograms.

(a) 4-Wave LateralShearing Interferometry system ( $W_{0}$ is the wavefront under test, $W_{1}$ and $W_{2}$ are the sheared wavefronts); (b) Traditional interferometry system ( $W_{0}$ is the wavefront under test, $W_{1}$ is the reference wavefront); (c) 4-Wave LateralShearing Interferometry interferogram; (d) Traditional interferogram (where (c) and (d) correspond to the same wavefront under test).

Mathematical Model of 4-Wave LateralShearing Interferometry

Shear Amount and Shear Ratio

Before quantitatively describing some mathematical relationships in 4-Wave LateralShearing Interferometry, it's crucial to define two important concepts involved in this field: shear amount and shear ratio.

First, let's define the shear amount. For a system where the wavefront under test is an approximately plane wave, the shear amount refers to the lateral offset between the two wavefronts participating in the shearing interference. It's important to note that 'lateral' here can refer to any direction within the observation plane perpendicular to the system's optical axis.

As shown in Figure 3-3(a), the wavefront under test, $W_{0}$ , with a width $D$ , is represented by a dashed line. The two wavefronts generated by its lateral offset (referred to as sheared wavefronts) are represented by solid lines. If the offset direction forms an angle $θ$ with the x-axis, the shear amount $S$ is the magnitude of this offset. For convenience in subsequent calculations, the shear direction is typically chosen to align with one of the coordinate axes of the established coordinate system, for instance, the x-axis, as shown in Figure 3-3(b). It's common for the same wavefront under test to produce different interference patterns when subjected to different shear directions.

Figure 3-3: Schematic diagram for defining shear amount. (a) Shearing along an arbitrary direction; (b) Shearing along a coordinate axis (x-axis direction).

In practical testing, the size of the beam aperture can change. To more reasonably describe the relative size of the overlapping region of the two wavefronts on the observation plane, the concept of shear ratio is defined. Here, the definition of shear ratio

β

is given as the ratio of the shear amount

S

to the aperture

D

It's important to clarify that equation (3-1) is merely the definition of the shear ratio. In practical calculations, the shear ratio is generally determined by its mathematical relationship with the system's optical and structural parameters. These mathematical relationships must be established before constructing a 4-Wave LateralShearing Interferometry system.The shear ratio is a critical parameter in 4-Wave LateralShearing Interferometry measurements, closely related to performance indicators like system sensitivity and dynamic range. Sensitivity ( $σ$ ), defined as the ratio of the differential wavefront to the wavefront under test, affects the interferogram contrast. Its mathematical relationship with $β$ is not linear, but when $β$ is very small, $σ$ can be considered approximately proportional to $β$ . An excessively low $σ$ leads to a decrease in interference fringe contrast, thus $σ$ generally determines the lower limit of $β$ .The dynamic range (DNR), typically considered the maximum measurable wavefront slope, is primarily limited by the Shannon sampling theorem of the detector (theoretically, the detector needs to sample at least 2 pixels per fringe; in practice, 4-8 pixels are needed). Its relationship with $β$ is more complex. One discernible pattern is that when the magnitudes of both the wavefront under test and the differential wavefront change monotonically and $β$ is small, DNR is inversely proportional to $β$ , which determines the upper limit of $β$ . The relationships among $σ$ , DNR, and $β$ can be graphically simulated using a set of function curves.

Figure 3-4 illustrates a one-dimensional cosine function simulating the wavefront under test. The two solid curves represent the mutually shifted sheared wavefronts, while the dashed curve represents the differential wavefront. The horizontal axis represents the coordinate, and the vertical axis represents the phase magnitude (i.e., the function value). As the shear ratio

β

(reflected in the degree of mutual offset between the two solid curves) continuously increases from 0.05 to 0.20, from Figure 3-4(a) to (d), the peak-to-valley (PV) value of the dashed curve representing the differential wavefront continuously increases, and the curve's slope continuously increases (using the coordinate origin as the reference point). Based on the previous analysis of

σ

and DNR, during this process, the sensitivity

σ

gradually increases, while the dynamic range DNR gradually decreases.

Figure 3-4: Schematic diagram illustrating the change in sensitivity and dynamic range with the shear ratio.

(a) Shear ratio of 0.05; (b) Shear ratio of 0.10; (c) Shear ratio of 0.15; (d) Shear ratio of 0.20.

Mathematical Representation of 4-Wave LateralShearing Interferometry

Let

W (x, y)

represent the wavefront under test, where

(x, y)

are the normalized aperture coordinates. If we set the shear direction along the x-axis and the shear ratio as

β

, then the two mutually displaced sheared wavefronts,

W_{1}

and

W_{2}

, generated by the shearing device in a 4-Wave LateralShearing Interferometry system can be expressed as:

If the wavefront under test is assumed to be a distorted plane wave, then the complex amplitude of the interference field

E

in the overlapping region in a 4-Wave LateralShearing Interferometry system can be expressed as:

In the equation, $A$ represents the amplitude of the wavefront under test. The corresponding interference pattern is:

From equation (3-4), it's evident that the difference between the two sheared wavefronts (defined as the differential wavefront, denoted as $Δ W$ ) determines the fringe distribution on the interference plane in 4-Wave LateralShearing Interferometry. It can be expressed as:

In the equation, $n$ represents the fringe order. When the shear ratio $β$ is very small, equation (3-5) can be rewritten in differential form for a 4-Wave LateralShearing Interferometry system:

Equation (3-6) quantifies the relationship between the shear ratio $β$ and the system's sensitivity $σ$ and dynamic range (DNR) in 4-Wave LateralShearing Interferometry. First, based on our previous definition, $σ$ can be expressed as:

When $β$ is very small, substituting equation (3-6) into equation (3-7) shows that $σ$ is proportional to $β$ in a 4-Wave LateralShearing Interferometry system:

For dynamic range (DNR), the discussion becomes a bit more complex. First, for 4-Wave LateralShearing Interferometry, the Shannon sampling theorem applies to the slope of the differential wavefront. Taking the commonly required 8-pixel sampling per wavelength fringe as an example, the slope of the differential wavefront needs to satisfy the following condition:

In the equation, $N$ is a constant coefficient introduced during the normalization of pixel coordinates. Therefore, the right side of the inequality is a constant, denoted as $ρ_{0}$ . When the conditions of equation (3-6) are met, equation (3-9) can be rewritten as:

In the equation, the maximum slope of the wavefront gradient, $\nabla W$ , is inversely proportional to $β$ . If both $W$ and $Δ W$ are monotonic functions at this point, then the maximum value of the wavefront gradient itself is also inversely proportional to $β$ . This means the dynamic range (DNR) of the 4-Wave LateralShearing Interferometry system can be expressed as:

Wavefront Aberration and Interference Pattern

Actual wavefronts under test are typically approximate plane or spherical waves with wavefront aberrations (which can be understood as aberrations in geometrical optics). To evaluate the quality of these wavefronts, it's necessary to quantitatively describe the wavefront aberrations.

Zernike Polynomials [5], also known as Zernike equations (ZF), are a commonly used method. Their basis is orthogonal on the unit circle and corresponds to geometric aberrations, making them highly suitable for image quality evaluation. Using Zernike Polynomials, the wavefront under test can be represented as:

In the equation, Zernike Polynomials are the basis, $a_{j}$ are the polynomial coefficients, and $N$ is the number of polynomial terms. Details regarding Zernike Polynomials have been thoroughly introduced in Section 1.4 of this book and will not be repeated here. For 4-Wave LateralShearing Interferometry, however, the description of the differential wavefront is more crucial. Based on equation (3-12), the Zernike polynomial representation of the differential wavefront, known as Difference Zernike Polynomials, can be derived:

In the equation, the basis of the Difference Zernike Polynomials is the difference between the bases of the two sheared wavefronts' Zernike Polynomials. The shear ratio is a crucial parameter in Difference Zernike Polynomials, as it dictates the shape of the resulting interference pattern in 4-Wave LateralShearing Interferometry. Table 3-1 presents the individual interference patterns corresponding to the first 16 terms of the polynomial when the shear ratio is 0.2.

The above article is excerpted from

Book Title: Novel Common-Path Interferometers

Authors: Yang Yongying, Ling Tong

Publisher: Zhejiang University Press

4-Wave LateralShearing Interferometry: Basic Concepts

Basic Concepts of 4-Wave LateralShearing Interferometry

Theoretical Basis of 4-Wave LateralShearing Interferometry

Get details

4-Wave LateralShearing Interferometry: Basic Concepts

Basic Concepts of 4-Wave LateralShearing Interferometry

Theoretical Basis of 4-Wave LateralShearing Interferometry

分享至微信分享

Get details

分享至微信