Influences of misalignment errors of optical components in an orthogonal two-axis Lloyd's mirror interferometer

Yuki Shimizu; Ryo Aihara; Zongwei Ren; Yuan-Liu Chen; So Ito; Wei Gao

doi:10.1364/OE.24.027521

1. Introduction

A diffraction grating is one of the key components [1] in modern optical measuring instruments such as optical encoders [2]. In an encoder system, a laser beam is made incident to a scale grating so that displacement of a measurement target, a side surface of which the scale is attached in most of the cases, can be detected by monitoring interference signals generated by superimposing diffracted light rays from the scale [2]. Measurement will be carried out by using the grating structures as scale graduations. With the employment of a signal interpolation technique, a measurement resolution of the state-of-the-art linear encoder system can reach to several-ten pm, while the measurement range, which is determined by the length of grating, can reach to several-ten meters [3]. Meanwhile, multi-axis encoders employing two-dimensional (2D) grating structures as a scale have also been developed for simultaneous measurement of the multi degree-of-freedom displacement [4–6]. In order to realize a high measurement resolution and accuracy of the multi-axis encoders, the 2D gratings to be used as their scales are required to have following features. Firstly, it is necessary to fabricate 2D grating structures having uniform grating periods. Secondly, diffraction efficiencies of both the negative and positive diffraction beams need to be consistent for higher signal to noise ratios of the interference signals. Finally, the grating period is preferred to be as short as possible to achieve further higher measurement resolution.

Among several fabrication process for the grating structures [7–11], laser interference lithography (LIL) is a promising method for fabrication of the 2D grating structures [11]. In LIL, a laser beam is split into two beams by division of amplitude or division of wavefront method. The two coherent beams will be superimposed and will generate interference fringes, which can be utilized for a pattern exposure. Although many optical configurations have been developed so far, the Lloyd's mirror interferometer is a good candidate for fabrication of the grating structures due to its simple optical configuration [11]. The conventional one-axis Lloyd's mirror interferometer, which has a mirror aligned perpendicularly with respect to a substrate, can fabricate one-dimensional grating structures. In addition, it can fabricate 2D grating structures with second exposure process after rotating the grating substrate 90° [12]. However, the grating structures generated in the first exposure will be influenced by the background light in the second exposure, and the depths of the grating structures will be different in the X- and Y- directions. As a result, diffraction efficiencies of both the negative and positive diffraction beams will not be consistent; this is a fatal problem for the encoder system since it directly affects the signal-to-noise ratio of the interference signals.

In order to solve the problem caused by the two-step exposure, several optical configurations, which can generate 2D grating structures at a single exposure, have been developed [13–16]. Among them, an orthogonal two-axis Lloyd's interferometer [16] can fabricate orthogonal 2D grating structures having a symmetric profile, which is required for consistent diffraction efficiencies in both the positive and negative directions. However, we have found that visible stripes, which affect the uniformity of the diffraction efficiency of the fabricated grating, always appear on the fabricated grating structures. Although modulation of the interference patterns as a consequence of the misalignment of the angle between two interferometer mirrors is predicted in computer simulation [16], a detailed mechanism of how the visible stripes appear on the fabricated grating structures has not been clear to the best of the knowledge of the authors.

In responding to the background described above, in this paper, theoretical analysis and experiments are carried out to investigate the mechanism of how the visible stripes appear on the grating structures. Influences of the misalignment errors of optical components in the orthogonal two-axis interferometer are theoretically investigated. Computer simulation is also carried out to simulate the visible stripes to be generated under the influences of misalignment errors of the optical components. Furthermore, a modified optical configuration for the orthogonal two-axis Lloyd's mirror interferometer is proposed, and its feasibility is verified in experiments.

2. Principle of the orthogonal two-axis Lloyd's mirror interferometer

An optical configuration for the orthogonal two-axis Lloyd's mirror interferometer is based on the traditional one-axis Lloyd’s mirror interferometer [11], which can generate one-dimensional grating structures. Figures 1(a) and 1(b) show schematics of the traditional one-axis Lloyd's mirror interferometer and the orthogonal two-axis Lloyd's mirror interferometer, respectively. In the optical configuration for the orthogonal two-axis Lloyd's mirror interferometer, another mirror referred to as the Y-mirror in this paper is added to that for the traditional one-axis Lloyd’s mirror interferometer in such a way that the normal of Y-mirror is perpendicular to both the normals of the substrate and the X-mirror. By using these two mirrors, a collimated laser light made incident to the interferometer can be divided into four beams; the beam directly projected onto the substrate (beam 1); the beam projected onto the substrate after being reflected by the X-mirror (beam 2); the beam projected onto the substrate after being reflected by the Y-mirror (beam 3); the beams projected onto the substrate after being reflected by both the X- and Y-mirrors (beam 4 and 4'). The 2D fringe patterns generated by the interference among the four beams can be calculated by superimposing electric fields of the beams. An electric field of a laser beam can be expressed as follows [17]:

{\vec{E}}_{i} = E_{i} \cdot {\vec{e}}_{i} \exp (j {\vec{k}}_{i} \cdot \vec{r} + γ_{i}) (i =1, 2, 3, 4)

where

\vec{r}

is the position vector, E_i is the real electric field amplitude,

{\vec{e}}_{i}

is a unit vector in the polarization direction of the laser beam, γ_i is the initial phase, and

{\vec{k}}_{i}

is the wave vector. Denoting the angle between the direction of incident beam and the substrate surface as θ_i, and the azimuthal angle of the laser light with respect to the X-axis as φ as shown in Fig. 1(c), the wave vector of each beam in the orthogonal two-axis Lloyd's mirror interferometer can be expressed as follows [16]:

{\vec{k}}_{1} = \frac{2 π}{λ} (\begin{array}{l} - \cos θ_{1} \cos ϕ \\ - \cos θ_{1} \sin ϕ \\ - \sin θ_{1} \end{array}), {\vec{k}}_{2} = \frac{2 π}{λ} (\begin{array}{l} \cos θ_{2} \cos ϕ \\ - \cos θ_{2} \sin ϕ \\ - \sin θ_{2} \end{array}), {\vec{k}}_{3} = \frac{2 π}{λ} (\begin{array}{l} - \cos θ_{3} \cos ϕ \\ \cos θ_{3} \sin ϕ \\ - \sin θ_{3} \end{array}), {\vec{k}}_{4} = \frac{2 π}{λ} (\begin{array}{l} \cos θ_{4} \cos ϕ \\ \cos θ_{4} \sin ϕ \\ - \sin θ_{4} \end{array})

where λ is a wavelength of the laser light. The intensity distribution of the interference fringe

I (\vec{r})

can therefore be calculated as follows:

I (\vec{r}) = \sum_{ℓ = 1}^{4} E_{ℓ}^{2} + 2 \sum_{m = 2}^{4} \sum_{n < m} E_{m} E_{n} {\vec{e}}_{m} \cdot {\vec{e}}_{n} \cos {({\vec{k}}_{n} - {\vec{k}}_{m}) \cdot \vec{r} + φ_{n} - φ_{m}}

According to Eq. (3), the periods of the interference fringes g_nm to be generated by the interference between the two of those beams can be expressed as follows:

g_{n m} = \frac{2 π}{| {\vec{k}}_{m} - {\vec{k}}_{n} |} (n < m \leq 4)

As a consequence of the superimposition of the interference fringes, the 2D fringe patterns can be generated simultaneously. Assume φ = 45° and the mirrors and substrate are orthogonal with each other, which means θ₁ = θ₂ = θ₃ = θ₄, the X- and Y-directional fringe periods g_x and g_y, respectively, can be expressed as follows:

Fig. 1 Optical configurations for the Lloyd's mirror interferometer: (a) Traditional one-axis Lloyd's mirror interferometer; (b) Orthogonal two-axis Lloyd's mirror interferometer; (c) Definition of θ and φ in this paper.

m n	2	3	4	4’
1	$g_{12} = λ / (\sqrt{2} \cos θ)$	$g_{13} = λ / (\sqrt{2} \cos θ)$	$g_{14} = λ / (2 \cos θ)$	$g_{14'} = λ / (2 \cos θ)$
2		$g_{23} = λ / (2 \cos θ)$	$g_{24} = λ / (\sqrt{2} \cos θ)$	$g_{24'} = λ / (\sqrt{2} \cos θ)$
3			$g_{34} = λ / (\sqrt{2} \cos θ)$	$g_{34'} = λ / (\sqrt{2} \cos θ)$

	$g_{12} = λ / (\sqrt{2} A_{12} \sqrt{1 + \cos θ \cos (θ + 2 Δ φ_{X Y}) \sin (2 Δ φ_{X Z}) - \sin θ \sin (θ + 2 Δ φ_{X Y})})$ $A_{12} = \cos ((\arctan (\tan θ \cos (ϕ - Δ φ_{X Z})) + \arctan (- \tan (θ + 2 Δ φ_{X Y}) \cos (ϕ - Δ φ_{X Z}))) / 2)$
	$g_{13} = λ / (\sqrt{2} A_{13} \sqrt{1 + \cos θ \cos (θ + 2 Δ φ_{Y X}) \sin (2 Δ φ_{Y Z}) - \sin θ \sin (θ + 2 Δ φ_{Y X})})$ $A_{13} = \cos ((\arctan (\tan θ \cos (ϕ - Δ φ_{Y Z})) + \arctan (- \tan (θ + 2 Δ φ_{Y X}) \cos (ϕ - Δ φ_{Y Z}))) / 2)$
	$g_{14} = λ / (\sqrt{2} A_{14} \sqrt{1 + \cos θ \cos (θ + 2 Δ φ_{X Y} + 2 Δ φ_{Y X}) \cos (2 Δ φ_{X Z} + 2 Δ φ_{Y Z}) - \sin θ \sin (θ + 2 Δ φ_{X Y} + 2 Δ φ_{Y X})})$ $A_{14} = \cos ((\arctan (\tan θ \cos (Δ φ_{X Z} + Δ φ_{Y Z})) + \arctan (- \tan (θ + 2 Δ φ_{X Y} + 2 Δ φ_{Y X}) \cos (Δ φ_{X Z} + Δ φ_{Y Z}))) / 2)$
	$g_{14'} = λ / (\sqrt{2} A_{14'} \sqrt{1 + \cos θ \cos (θ + 2 Δ φ_{X Y} + 2 Δ φ_{Y X}) \cos (2 Δ φ_{X Z} + 2 Δ φ_{Y Z}) - \sin θ \sin (θ + 2 Δ φ_{X Y} + 2 Δ φ_{Y X})})$ $A_{14'} = \cos ((\arctan (\tan θ \cos (Δ φ_{X Z} + Δ φ_{Y Z})) + \arctan (- \tan (θ + Δ φ_{X Y} + 2 Δ φ_{Y X}) \cos (Δ φ_{X Z} + Δ φ_{Y Z}))) / 2)$
	$g_{23} = λ / (\sqrt{2} A_{23} \sqrt{1 + \cos (θ + 2 Δ φ_{X Y}) \cos (θ + 2 Δ φ_{Y X}) \cos (2 Δ φ_{X Z} + 2 Δ φ_{Y Z}) - \sin (θ + 2 Δ φ_{X Y}) \sin (θ + 2 Δ φ_{Y X})})$ $A_{23} = \cos ((\arctan (\tan (θ + 2 Δ φ_{X Y}) \cos (θ - Δ φ_{Y Z})) + \arctan (- \tan (θ + 2 Δ φ_{Y X}) \cos (θ - Δ φ_{Y Z}))) / 2)$
	$g_{24} = λ / (\sqrt{2} A_{24} \sqrt{1 + \cos (θ + Δ φ_{X Y}) \cos (θ + 2 Δ φ_{X Y} + 2 Δ φ_{Y X}) \sin (4 Δ φ_{X Z} + 2 Δ φ_{Y X}) - \sin (θ + 2 Δ φ_{X Y}) \sin (θ + 2 Δ φ_{X Y} + 2 Δ φ_{Y X})})$ $A_{24} = \cos ((\arctan (\tan (θ + 2 Δ φ_{X Y}) \cos (ϕ + 2 Δ φ_{X Z} + Δ φ_{Y Z})) + \arctan (- \tan (θ + 2 Δ φ_{X Y} + 2 Δ φ_{Y X}) \cos (ϕ + 2 Δ φ_{X Z} + Δ φ_{Y Z}))) / 2)$
	$g_{24'} = λ / (\sqrt{2} A_{24'} \sqrt{1 - \cos (θ + 2 Δ φ_{X Y}) \cos (θ + 2 Δ φ_{X Y} + 2 Δ φ_{Y X}) \sin (2 Δ φ_{Y Z}) - \sin (θ + 2 Δ φ_{X Y}) \sin (θ + 2 Δ φ_{X Y} + 2 Δ φ_{Y X})})$ $A_{24'} = \cos ((\arctan (\tan (θ + 2 Δ φ_{X Y}) \cos (ϕ - Δ φ_{Y Z})) + \arctan (- \tan (θ + 2 Δ φ_{X Y} + 2 Δ φ_{Y X}) \cos (ϕ - Δ φ_{Y Z}))) / 2)$
	$g_{34} = λ / (\sqrt{2} A_{34} \sqrt{1 - \cos (θ + 2 Δ φ_{Y X}) \cos (θ + 2 Δ φ_{X Y} + 2 Δ φ_{Y X}) \sin (2 Δ φ_{X Z}) - \sin (θ + 2 Δ φ_{Y X}) \sin (θ + 2 Δ φ_{X Y} + 2 Δ φ_{Y X})})$ $A_{34} = \cos ((\arctan (- \tan (θ + 2 Δ φ_{Y X}) \cos (ϕ - Δ φ_{X Z})) + \arctan (\tan (θ + 2 Δ φ_{X Y} + 2 Δ φ_{Y X}) \cos (ϕ - Δ φ_{X Z}))) / 2)$
	$g_{34'} = λ / (\sqrt{2} A_{34'} \sqrt{1 + \cos (θ + 2 Δ φ_{Y X}) \cos (θ + 2 Δ φ_{X Y} + 2 Δ φ_{Y X}) \sin (2 Δ φ_{X Z} + 4 Δ φ_{Y Z}) - \sin (θ + 2 Δ φ_{Y X}) \sin (θ + 2 Δ φ_{X Y} + 2 Δ φ_{Y X})})$ $A_{34'} = \cos ((\arctan (- \tan (θ + 2 Δ φ_{Y X}) \cos (ϕ + Δ φ_{X Z} + 2 Δ φ_{Y Z})) + \arctan (\tan (θ + Δ φ_{X Y} + 2 Δ φ_{Y X}) \cos (ϕ + Δ φ_{X Z} + 2 Δ φ_{Y Z}))) / 2)$

Item	number	unit
Light wavelength	441.6	nm
Angle between the direction of incident light and the substrate surface (θ)	71.805	degree
Mesh size on the substrate surface	0.1 (X) 0.1 (Y)	μm
Data points (Simulated area)	10000 (X) × 10000 (Y) (1 × 1)	points (mm²)

Abstract

1. Introduction

2. Principle of the orthogonal two-axis Lloyd's mirror interferometer

3. Influence of the misalignment errors of the optical setup

3.1 An overview of the theoretical analysis

3.2 Angular misalignment errors of the mirrors

3.3 Deviations of the direction of laser light

4. Computer simulation and experiments

4.1 Influences of the misalignment errors

4.2 Elimination of the visible stripes due to the misalignment errors

5. Conclusions

Funding

References and links

Cited By

Figures (16)

Tables (4)

Equations (7)

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