Optimal polarization modulation for orthogonal two-axis Lloyd&#x2019;s mirror interference lithography

Xiuguo Chen; Zongwei Ren; Yuki Shimizu; Yuan-liu Chen; Wei Gao

doi:10.1364/OE.25.022237

1. Introduction

Diffraction gratings [1] play an important role in modern optical measuring systems, such as monochromators, spectrometers and optical encoders. Take the optical encoder as an example. As one of the most common displacement sensors for high precision applications, an optical encoder is generally composed of a special two-dimensional (2D) diffraction grating named the planar scale grating used as the measurement reference, an optical head for reading the scale graduations, and the related electronics for data acquisition and counting [2–4]. The displacement of the scale grating relative to the optical head induces phase shifts in the positive and negative first-order diffracted beams of the scale grating, which further lead to changes in the interferential signals. The displacement is finally obtained by analyzing the interferential signals. The measurement range of the encoder depends on the area of the 2D grating, which is typically larger than 10 mm × 10 mm [5]. The quality of the 2D grating, such as the X-, Y-directional pitch deviations and Z-directional out-of-flatness, directly influences the measurement uncertainty of the encoder. As such, it is required to fabricate the 2D scale gratings with uniform periods, and moreover, the associated diffraction efficiencies of both the negative and positive first-order diffracted beams need to be consistent for higher signal-to-noise ratios. Additionally, to achieve higher measurement resolution, the grating period is preferred to be as short as possible.

Over the past decades, many techniques have been developed for the fabrication of grating structures [6–10], of which the laser interference lithography such as the Lloyd’s mirror interference lithography is a promising technique for the fabrication of 2D gratings due to its simple optical configuration [10]. The conventional one-axis Lloyd’s mirror interferometer, which has a mirror aligned perpendicular to a substrate, is typically used to fabricate one-dimensional gratings. To fabricate 2D gratings, it is required to rotate the grating substrate 90° and then carry out a second exposure [11]. However, the 2D gratings fabricated in this two-step exposure will have different depths in the X- and Y-directions, because the grating structures generated in the first exposure will be influenced by the background light in the second exposure. As a result, the diffraction efficiencies of both the positive and negative first-order diffracted beams will not be consistent.

In order to solve the problem caused by the two-step exposure, an orthogonal two-axis Lloyd’s mirror interferometer has been developed [12–16]. Its advantage is that it can be used to fabricate large areas of orthogonal 2D grating structures with symmetric profiles at a single exposure. The orthogonal two-axis Lloyd’s mirror interferometer utilizes a corner-cube-like interferometer unit, which is essentially an extension of the one-axis Lloyd’s mirror interferometer from the two-beam interference to the multi-beam interference. However, compared with the one-axis Lloyd’s mirror interferometer, delicate polarization control is indispensable in the two-axis Lloyd’s mirror interferometer to achieve the required grating structures. Although it has been revealed that different combinations of the initial polarization states of incident beams to the corner-cube-like interferometer unit will lead to different interference fringes and then different 2D grating structures [14, 15], the optimization of initial polarization states of the incident beams by polarization modulation to achieve the preferred interference fringes has not been reported to the best of our knowledge.

To achieve the optimal polarization modulation, we first establish a three-dimensional (3D) polarization ray-tracing model to trace the evolution of polarization states of light through the corner-cube-like interferometer unit of the orthogonal two-axis Lloyd’s mirror interferometer. The 3D polarization ray-tracing model performs ray tracing in the global x-y-z coordinate system [17] rather than in the local p-s coordinate system as the conventional 2D Jones matrix formalism [18] does. The conventional 2D Jones matrix formalism involves tedious local coordinate transformations since the local coordinate of an output ray for one optical interface of the corner-cube-like interferometer is usually different from the local coordinate of the input ray for the next optical interface. In comparison, the 3D polarization ray-tracing model that represents polarization as a three-element electric filed vector in the global coordinate system can avoid the tedious local coordinate transformations, and thus provides an easier approach to trace the polarization evolution. With the established model, we then derive the optimal combination for the initial polarization directions of incident beams to achieve the preferred interference fringes. The comparison between the simulated and experimental interference fringes obtained under different combinations of initial polarization states of the incident beams has verified the feasibility of the established model as well as the achieved optimal polarization modulation.

2. The orthogonal two-axis Lloyd’s mirror interferometer

We have constructed an orthogonal two-axis Lloyd’s mirror interferometer [15]. As depicted in Fig. 1(a), the instrument mainly contains two units, namely the beam-shaping unit and the interferometer unit. The beam-shaping unit is composed of a spatial filter and a collimating lens with an effective focal length of 200 mm. The spatial filter consists of an objective lens with a numerical aperture of 0.65 and a pinhole with a diameter of 5 μm. The interferometer unit, as shown in Figs. 1(b) and 1(c), is composed of a pair of flat mirrors (X- and Y-mirrors) and a substrate. The two mirrors and the substrate are perpendicular to each other. A HeCd laser with the wavelength of λ = 441.6 nm was employed as the light source. Two kinds of mirrors will be employed in experiments. The mirror of the first kind (termed as Mirror 1 for convenience) is a glass substrate coated with an Al film layer that is thick enough to prevent light penetrating through it. The mirror of the second kind (termed as Mirror 2) is a glass substrate coated with, from the bottom to the top, an Al film layer (also thick enough) and a MgF₂ film layer. Ellipsometric measurements (M-2000 Ellipsometer, J. A. Woollam Co.) have been performed to accurately determine the thickness of each coated layer on the two kinds of mirrors. It was found that there was an oxide layer (Al₂O₃) on the top of the Al film layer for both Mirror 1 and Mirror 2. As for Mirror 1, the thickness of the Al₂O₃ layer was 4.6 nm. As for Mirror 2, the thicknesses of the Al₂O₃ layer and the MgF₂ layer were 11.2 nm and 217.1 nm, respectively. The complex refractive indices of Al, Al₂O₃, and MgF₂ at λ = 441.6 nm were 0.5952 – 5.3671j, 1.7810, and 1.3818, respectively.

Fig. 1 (a) Schematic and (b) photograph of the constructed orthogonal two-axis Lloyd’s mirror interferometer; (c) photograph of the interferometer unit [15].

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The interferometer unit was mounted on a rotary stage having its rotational axis parallel with the A-axis as indicated in Fig. 1(a) so that the angle between the direction of the incident light and the substrate surface of the interferometer unit θ could be adjusted. Meanwhile, the interferometer unit was held stationary about the C-axis with the azimuthal angle ϕ = 45°. The definitions of θ and ϕ will be presented in Section 3. In addition, two half-wave plates (HWPs), as indicated in Fig. 1(b), were mounted in the ray path to modulate the initial polarization states of beams incident directly upon the substrate, the X- and Y-mirrors separately. Due to the limited space, only the polarization states of two incident beams can be modulated in each experiment. In experiments, a glass substrate with a size of 30 mm × 24 mm coated with layers of an adhesion promoting agent and a positive photoresist with thicknesses of 100 nm and 350 nm, respectively, was mounted on the substrate of the interferometer unit. A TEM₀₀ linearly polarized light (s-polarization) emitted from the laser source was made to successively pass through the spatial filter, then was collimated by the collimating lens, and finally was made incident to the interferometer unit. The glass substrate was exposed with the 2D interference fringes generated by the orthogonal two-axis Lloyd’s mirror interferometer, and was then developed by a NaOH solution with a volume concentration of 0.5%.

3. Theory

3.1 3D polarization ray-tracing model

The electric field of a plane wave is described by $e (r, t) = E \exp [j (ω t - k \hat{k} \cdot r)]$ , where $ω$ is the angular frequency, $\hat{k}$ is a unit vector along the propagation direction, $k = 2 π n / λ$ is the wavenumber in a medium with a refractive index of n (n = 1 for air), λ is the wavelength, r = [x, y, z]^T is the position vector, and the vector E = [E_x, E_y, E_z]^T contains the complex amplitudes and also defines the polarization state. In fact, the vector E here can be regarded as the 3 × 1 Jones vector [19], which is the extension of the conventional 2 × 1 Jones vector [18]. Since the discussion in this work focuses on monochromatic waves, we can thereby rewrite the electric field as $e (r) = E \exp (- j k \hat{k} \cdot r)$ by omitting the term $\exp (j ω t)$ for simplicity. Consider the evolution of polarization state of a ray through an optical system with N interfaces, as depicted in Fig. 2. For the whole optical system, there is a global coordinate system ${\hat{x}, \hat{y}, \hat{z}}$ , where $\hat{x}$ , $\hat{y}$ and $\hat{z}$ are unit vectors along the X-, Y- and Z-directions, respectively. To describe the polarization state, it is common to establish a local coordinate system ${\hat{p}, \hat{s}, \hat{k}}$ for each ray. Here, $\hat{p}$ is a unit vector in the plane of incidence and perpendicular to the propagation direction $\hat{k}$ , $\hat{s}$ is a also unit vector but perpendicular to the plane of incidence, and ${\hat{p}, \hat{s}, \hat{k}}$ constitutes a right-handed coordinate system.

Fig. 2 Schematic of the propagation of polarized light in a sequence of optical interfaces.

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As shown in Fig. 2, for any optical interface i (i = 1, 2, …, N), the incident propagation vector ${\hat{k}}_{i - 1}$ and the reflected (or refracted) propagation vector ${\hat{k}}_{i}$ may be different due to the light-matter interaction. The amplitude vectors E_i_–1 and E_i before and after the interaction are related by a 3 × 3 polarization ray-tracing matrix P_i [17]

E_{i} = P_{i} \cdot E_{i - 1} .

The polarization ray-tracing matrix P_i, which characterizes the change in the complex amplitudes induced by the interaction with the interface, depends on both the polarization effect of the interface and the incident and existing propagation vectors,

{\hat{k}}_{i - 1}

and

{\hat{k}}_{i}

. Specifically, P_i is represented by

P_{i} = T_{o u t, i} \cdot J_{i} \cdot T_{i n, i}^{- 1},

T_{i n, i} = [\begin{matrix} {\hat{p}}_{i} & {\hat{s}}_{i} & {\hat{k}}_{i - 1} \end{matrix}],

T_{o u t, i} = [\begin{matrix} {\hat{p}}^{'}_{i} & {\hat{s}}^{'}_{i} & {\hat{k}}_{i} \end{matrix}],

J_{i} = [\begin{matrix} r_{p p, i} & r_{p s, i} & 0 \\ r_{s p, i} & r_{s s, i} & 0 \\ 0 & 0 & 1 \end{matrix}],

where

T_{i n, i}^{- 1} = T_{i n, i}^{T}

projects E_i_–1 defined in the global coordinate system

{\hat{x}, \hat{y}, \hat{z}}

into the local coordinate system

{{\hat{p}}_{i}, {\hat{s}}_{i}, {\hat{k}}_{i - 1}}

,

T_{o u t, i}

projects the amplitude vector defined in the local coordinate system

{{\hat{p}}^{'}_{i}, {\hat{s}}^{'}_{i}, {\hat{k}}_{i}}

after the ith interface back into the global coordinate system

{\hat{x}, \hat{y}, \hat{z}}

, and

J_{i}

is the extended 3 × 3 Jones matrix that characterizes the polarization effect of the ith interface. The complex components

r_{p p, i}

,

r_{p s, i}

,

r_{s p, i}

and

r_{s s, i}

in

J_{i}

are the amplitude reflection (or transmission) coefficients, which can be calculated in the local coordinate system

{\hat{p}, \hat{s}, \hat{k}}

according to the Fresnel equations [20]. In particular,

r_{p s, i} = r_{s p, i} = 0

for an isotropic interface. The relation between the local coordinate systems

{{\hat{p}}_{i}, {\hat{s}}_{i}, {\hat{k}}_{i - 1}}

before and

{{\hat{p}}^{'}_{i}, {\hat{s}}^{'}_{i}, {\hat{k}}_{i}}

after the ith interface is given by

{\hat{s}}^{'}_{i} = {\hat{s}}_{i} = \frac{{\hat{k}}_{i - 1} \times {\hat{k}}_{i}}{| {\hat{k}}_{i - 1} \times {\hat{k}}_{i} |},

{\hat{p}}^{'}_{i} = {\hat{s}}^{'}_{i} \times {\hat{k}}_{i},

{\hat{p}}_{i} = {\hat{s}}_{i} \times {\hat{k}}_{i - 1} .

A ray interacting with a sequence of optical interfaces can be represented by

E_{N} = P_{T o t a l} \cdot E_{1},

where

P_{T o t a l} = P_{N} \cdot P_{N - 1} \dots P_{i} \dots P_{2} \cdot P_{1}

denotes the net polarization ray-tracing matrix for the entire ray path. In addition, for any two amplitude vectors

E_{l}

and

E_{m}

, if they satisfy

E_{l}^{†} E_{m} = E_{m}^{†} E_{l} = 0,

the polarization states of the two beams associated with

E_{l}

and

E_{m}

are said to be orthogonal polarization state, where the superscript “†” stands for the Hermitian conjugate. The ellipses of polarization that correspond to a pair of orthogonal polarization states have equal and opposite handedness and their major axes are mutually orthogonal, such as the left- and right-circular polarization states. Note that here

E_{l}

and

E_{m}

should be defined in the same (global or local) coordinate system.

Figure 3 shows the optical configuration for the interferometer unit of the orthogonal two-axis Lloyd’s mirror interferometer. A collimated laser light incident to the interferometer unit will be divided into five beams, namely, 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), and the beams projected onto the substrate after being reflected by both the X- and Y-mirrors (beam 4 and beam 5). The total electric field incident upon the substrate will be

e (r) = \sum_{i = 1}^{5} E_{i} \exp (- j k {\hat{k}}_{i} \cdot r) .

Note that the amplitude vectors

E_{i}

and the propagation vectors

{\hat{k}}_{i}

of the five beams are all defined in the global coordinate system

{\hat{x}, \hat{y}, \hat{z}}

. Without loss of generality, we assume that the initial polarization states of beams 1~5 before their interaction with the interferometer unit are denoted as E₀_i (i = 1, 2, …, 5). Moreover, for convenience, E₀_i are all defined in the local coordinate system

{{\hat{p}}_{1}, {\hat{s}}_{1}, {\hat{k}}_{1}}

associated with beam 1, namely, E₀_i = [E_p_,0_i, E_s_,0_i, 0]^T. For example, E₀_i = [1, 0, 0]^T represents the common p-polarization, while E₀_i = [0, 1, 0]^T represents the common s-polarization. In the following part, we will give

E_{i}

and

{\hat{k}}_{i}

of the five beams, respectively.

Fig. 3 (a) Optical configuration for the interferometer unit of the orthogonal two-axis Lloyd’s mirror interferometer and (b) definition of the global coordinate system ${\hat{x}, \hat{y}, \hat{z}}$ , θ and ϕ in this paper, where θ is the angle between the direction of the incident beam and the XY-plane (substrate surface), and ϕ is the azimuthal angle between the X-axis and the plane of incidence associated with the beam (beam 1) incident upon the substrate [15].

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Beam 1: According to Fig. 3, the propagation vector ${\hat{k}}_{1}$ of beam 1 is

{\hat{k}}_{1} = [\begin{array}{l} - \cos θ \cos ϕ \\ - \cos θ \sin ϕ \\ - \sin θ \end{array}] .

Since beam 1 is directly projected onto the substrate, the only consideration when obtaining

E_{1}

is how to transform the initial polarization state

E_{01}

defined in

{{\hat{p}}_{1}, {\hat{s}}_{1}, {\hat{k}}_{1}}

to

{\hat{x}, \hat{y}, \hat{z}}

. According to Fig. 3, we have

E_{1} = T_{o u t, 1} \cdot E_{01},

T_{o u t, 1} = [\begin{matrix} {\hat{p}}_{1} & {\hat{s}}_{1} & {\hat{k}}_{1} \end{matrix}] = [\begin{matrix} - \sin θ \cos ϕ & \sin ϕ & - \cos θ \cos ϕ \\ - \sin θ \sin ϕ & - \cos ϕ & - \cos θ \sin ϕ \\ \cos θ & 0 & - \sin θ \end{matrix}] .

Beam 2: Beam 2 corresponds to the beam that is projected onto the substrate after being reflected by the X-mirror, whose propagation vector can be calculated by

{\hat{k}}_{2} = \frac{2 (- {\hat{n}}_{X} \cdot {\hat{k}}_{1}) \cdot {\hat{n}}_{X} + {\hat{k}}_{1}}{| 2 (- {\hat{n}}_{X} \cdot {\hat{k}}_{1}) \cdot {\hat{n}}_{X} + {\hat{k}}_{1} |},

where

{\hat{n}}_{X}

is the unit normal vector of the X-mirror. Ideally, the two mirrors and the substrate will be perpendicular to each other and

{\hat{n}}_{X} = \hat{x}

. However, in a practical optical configuration, it may be difficult to align the two mirrors with respect to the substrate without angular misalignment errors, as discussed in our previous work [15]. In this case,

{\hat{n}}_{X} \neq \hat{x}

. We thereby use

{\hat{n}}_{X}

rather than

\hat{x}

to take the angular misalignment errors into account if necessary. Specifically, we denote

δ_{X Y}

and

δ_{X Z}

as the angular misalignment errors of the X-mirror with respect to the Y- and Z-axes, respectively, and

δ_{X Y}

and

δ_{X Z}

are positive when they are rotated in a counter-clockwise direction. After rotation, the normal vector of the X-mirror becomes

\begin{array}{l} {\hat{n}}_{X} = R_{Y} (δ_{X Y}) R_{Z} (δ_{X Z}) \hat{x} \\ = [\begin{matrix} \cos δ_{X Y} & 0 & - \sin δ_{X Y} \\ 0 & 1 & 0 \\ \sin δ_{X Y} & 0 & \cos δ_{X Y} \end{matrix}] [\begin{matrix} \cos δ_{X Z} & \sin δ_{X Z} & 0 \\ - \sin δ_{X Z} & \cos δ_{X Z} & 0 \\ 0 & 0 & 1 \end{matrix}] [\begin{array}{l} 1 \\ 0 \\ 0 \end{array}], \end{array}

or

\begin{array}{l} {\hat{n}}_{X} = R_{Z} (δ_{X Z}) R_{Y} (δ_{X Y}) \hat{x} \\ = [\begin{matrix} \cos δ_{X Z} & \sin δ_{X Z} & 0 \\ - \sin δ_{X Z} & \cos δ_{X Z} & 0 \\ 0 & 0 & 1 \end{matrix}] [\begin{matrix} \cos δ_{X Y} & 0 & - \sin δ_{X Y} \\ 0 & 1 & 0 \\ \sin δ_{X Y} & 0 & \cos δ_{X Y} \end{matrix}] [\begin{array}{l} 1 \\ 0 \\ 0 \end{array}], \end{array}

where

R_{Y} (δ_{X Y})

and

R_{Z} (δ_{X Z})

are rotation matrices from the rotation group SO(3). According to the 3D polarization ray-tracing calculus given in Eqs. (1)-(4), the amplitude vector

E_{2}

of beam 2 can be calculated by

E_{2} = P_{2} \cdot T_{o u t, 1} \cdot E_{02} = T_{o u t, 2} \cdot J_{X} \cdot T_{i n, 2}^{- 1} \cdot T_{o u t, 1} \cdot E_{02},

T_{i n, 2} = [\begin{matrix} {\hat{p}}_{2} & {\hat{s}}_{2} & {\hat{k}}_{1} \end{matrix}],

T_{o u t, 2} = [\begin{matrix} {\hat{p}}^{'}_{2} & {\hat{s}}^{'}_{2} & {\hat{k}}_{2} \end{matrix}],

where the local coordinate systems

{{\hat{p}}_{2}, {\hat{s}}_{2}, {\hat{k}}_{1}}

before and

{{\hat{p}}^{'}_{2}, {\hat{s}}^{'}_{2}, {\hat{k}}_{2}}

after the X-mirror can be constructed according to Eq. (3);

J_{X}

is the Jones matrix of the X-mirror and is determined by the coating layers on the mirror. When calculating

J_{X}

, the incidence angle for the light incident upon the X-mirror is required, which is given by

θ_{1 X} = π - \arccos ({\hat{n}}_{X} \cdot {\hat{k}}_{1})

.

Beam 3: Beam 3 corresponds to the beam that is projected onto the substrate after being reflected by the Y-mirror, whose propagation vector can be calculated by

{\hat{k}}_{3} = \frac{2 (- {\hat{n}}_{Y} \cdot {\hat{k}}_{1}) \cdot {\hat{n}}_{Y} + {\hat{k}}_{1}}{| 2 (- {\hat{n}}_{Y} \cdot {\hat{k}}_{1}) \cdot {\hat{n}}_{Y} + {\hat{k}}_{1} |},

where

{\hat{n}}_{Y}

is the unit normal vector of the Y-mirror. The angular misalignment errors of the Y-mirror with respect to the X- and Z-axes can be taken into account in a similar manner to Eq. (10). The amplitude vector

E_{3}

of beam 3 can be similarly calculated by

E_{3} = P_{3} \cdot T_{o u t, 1} \cdot E_{03} = T_{o u t, 3} \cdot J_{Y} \cdot T_{i n, 3}^{- 1} \cdot T_{o u t, 1} \cdot E_{03},

T_{i n, 3} = [\begin{matrix} {\hat{p}}_{3} & {\hat{s}}_{3} & {\hat{k}}_{1} \end{matrix}],

T_{o u t, 3} = [\begin{matrix} {\hat{p}}^{'}_{3} & {\hat{s}}^{'}_{3} & {\hat{k}}_{3} \end{matrix}],

where

J_{Y}

is the Jones matrix of the Y-mirror, and the incidence angle for the light incident upon the Y-mirror is

θ_{1 Y} = π - \arccos ({\hat{n}}_{Y} \cdot {\hat{k}}_{1})

.

Beam 4: Beam 4 corresponds to the beam that is projected onto the substrate after being reflected by the X- and Y-mirrors successively, whose propagation vector is calculated by

{\hat{k}}_{4} = \frac{2 (- {\hat{n}}_{Y} \cdot {\hat{k}}_{2}) \cdot {\hat{n}}_{Y} + {\hat{k}}_{2}}{| 2 (- {\hat{n}}_{Y} \cdot {\hat{k}}_{2}) \cdot {\hat{n}}_{Y} + {\hat{k}}_{2} |} .

The amplitude vector

E_{4}

of beam 4 is calculated by

E_{4} = P_{4} \cdot P_{2} \cdot T_{o u t, 1} \cdot E_{04} = T_{o u t, 4} \cdot J_{Y} \cdot T_{i n, 4}^{- 1} \cdot P_{2} \cdot T_{o u t, 1} \cdot E_{04},

T_{i n, 4} = [\begin{matrix} {\hat{p}}_{4} & {\hat{s}}_{4} & {\hat{k}}_{2} \end{matrix}],

T_{o u t, 4} = [\begin{matrix} {\hat{p}}^{'}_{4} & {\hat{s}}^{'}_{4} & {\hat{k}}_{4} \end{matrix}] .

The incidence angle for the light reflected from the X-mirror incident upon the Y-mirror is

θ_{2 Y} = π - \arccos ({\hat{n}}_{Y} \cdot {\hat{k}}_{2})

. Note that the Jones matrices

J_{Y}

in Eqs. (13) and (15) will be different if

θ_{1 Y} \neq θ_{2 Y}

.

Beam 5: Beam 5 corresponds to the beam that is projected onto the substrate after being reflected by the Y- and X-mirrors successively, whose propagation vector is calculated by

{\hat{k}}_{5} = \frac{2 (- {\hat{n}}_{X} \cdot {\hat{k}}_{3}) \cdot {\hat{n}}_{X} + {\hat{k}}_{3}}{| 2 (- {\hat{n}}_{X} \cdot {\hat{k}}_{3}) \cdot {\hat{n}}_{X} + {\hat{k}}_{3} |} .

The amplitude vector

E_{5}

of beam 5 is calculated by

E_{5} = P_{5} \cdot P_{3} \cdot T_{o u t, 1} \cdot E_{05} = T_{o u t, 5} \cdot J_{X} \cdot T_{i n, 5}^{- 1} \cdot P_{3} \cdot T_{o u t, 1} \cdot E_{05},

T_{i n, 5} = [\begin{matrix} {\hat{p}}_{5} & {\hat{s}}_{5} & {\hat{k}}_{3} \end{matrix}],

T_{o u t, 5} = [\begin{matrix} {\hat{p}}^{'}_{5} & {\hat{s}}^{'}_{5} & {\hat{k}}_{5} \end{matrix}] .

The incidence angle for the light reflected from the Y-mirror incident upon the X-mirror is

θ_{3 X} = π - \arccos ({\hat{n}}_{X} \cdot {\hat{k}}_{3})

. Note that the Jones matrices

J_{X}

in Eqs. (11) and (17) will also be different if

θ_{1 X} \neq θ_{3 X}

.

According to Eqs. (6)-(17), the 2D fringe patterns generated by the interference among the five beams can be calculated by

I (r) = {[e (r)]}^{†} e (r) = \sum_{l = 1}^{5} {| E_{l} |}^{2} + 2 \sum_{l = 2}^{5} \sum_{m < l} Re {E_{m}^{†} E_{l} \exp [j k ({\hat{k}}_{m} - {\hat{k}}_{l}) \cdot r]},

where Re{⋅} stands for the real part of a complex value. Note that when calculating the 2D interference fringes, it is necessary to distinguish the interference (illumination) regions of the five beams on the XY-plane. For beams 1, 2 and 3, the interference regions cover the whole XY-plane, while the interference regions for beams 4 and 5 only cover half of the XY-plane. Specifically, the interference region for beam 4 satisfies x ≥ y and the interference region for beam 5 satisfies x ≤ y, which suggests that beam 4 and beam 5 will never interfere with each other. According to Eq. (18), the periods of the fringe patterns g_lm (l < m ≤5) generated by the interference between two of those beams will be

g_{l m} = \frac{2 π}{k | {\hat{k}}_{l} - {\hat{k}}_{m} |} .

According to Eq. (19), we know that the angles θ and ϕ as defined in Fig. 3 determine the final periods of the fringe patterns.

Interference can occur between two of the five beams, and will generate equally spaced line interference fringes having a period g_lm calculated by Eq. (19). The fringes are aligned along the direction perpendicular to the vector ${\hat{k}}_{m} - {\hat{k}}_{l}$ . Table 1 summarizes the line interference fringes generated by two of those beams using the established model. In the calculation, Mirror 1 was employed as the X- and Y-mirrors. The initial polarization states of beams 1, 2 and 3 were all s-polarization. The angle between the direction of the incident light and the XY-plane was set to be θ = 71.805° (ϕ = 45°), so that the periods of the interference fringes generated by beams 1 and 2, beams 1 and 3, beams 2 and 4, beams 2 and 5, beams 3 and 4, and beams 3 and 5 are 1 μm, while those generated by beams 2 and 3, beams 1 and 4, and beams 1 and 5 are $1 / \sqrt{2}$ μm. The projection of the two beams associated with each interference fringe on the XY-plane is also presented. As can be seen in Table 1, the same line interference fringe can be generated by several combinations of the beams.

Table 1. Interference fringe patterns generated by each pair of beams

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3.2 Optimal polarization modulation

As revealed in our previous work [15], the interference fringes generated by beams 1, 2 and 3 are enough for the fabrication of 2D grating structures, and moreover, different combinations of the initial polarization states of these three beams (E₀₁, E₀₂ and E₀₃) will lead to different interference fringes. We thereby focus on the interference fringes generated by these three beams, and try to find the optimal combination of the initial polarizations states of beams 1, 2 and 3 to achieve the preferred interference fringes. Ideally, we would like to acquire square (or rhombic) interference fringes, so that the diffraction efficiencies along the X- and Y-directions would be consistent. According to the line interference fringes presented in Table 1, square interference fringes would be achieved, if we could prevent the interference between beams 2 and 3, and meanwhile the interference fringe generated by beams 1 and 2 along the X-direction is identical to that generated by beams 1 and 3 along the Y-direction.

The non-interference between beams 2 and 3 indicates that the polarization states of beams 2 and 3 after their interaction with the X- and Y-mirrors are orthogonal, namely,

E_{2}^{†} E_{3} = E_{3}^{†} E_{2} = 0.

In order to estimate the orthogonality between any two beams l and m, we introduce the degree of orthogonality (DoO) defined as

γ_{l m} = | E_{l}^{†} E_{m} | .

According to Eq. (21), we know that

γ_{l m} = γ_{m l}

and

0 \leq γ_{l m} \leq 1

. Note that here we assume that the initial polarization states E₀₁, E₀₂ and E₀₃ are all normalized. According to Eq. (18), we can derive the interference fringe I_lm between any two beams l and m as

I_{l m} = {| E_{l} |}^{2} + {| E_{m} |}^{2} + 2 γ_{l m} \cos [k ({\hat{k}}_{m} - {\hat{k}}_{l}) \cdot r + φ_{l m}],

where

φ_{l m} = \arctan (Im (E_{m}^{†} E_{l}) / Re (E_{m}^{†} E_{l}))

, and Im(⋅) and Re(⋅) denote the imaginary and real parts of a complex number, respectively. According to Eq. (22), we further obtain the contrast of the interference fringe I_lm as

C_{l m} = \frac{\max (I_{l m}) - m i n (I_{l m})}{\max (I_{l m}) + m i n (I_{l m})} = \frac{2 γ_{l m}}{{| E_{l} |}^{2} + {| E_{m} |}^{2}},

where max(I_lm) and min(I_lm) denote the maximal and minimal values of I_lm, respectively. Obviously,

0 \leq C_{l m} \leq 1

. To make the interference fringe generated by beams 1 and 2 along the X-direction identical to that generated by beams 1 and 3 along the Y-direction, we have

C_{12} = C_{13} .

Note that here we omit the initial phases

φ_{12}

and

φ_{13}

in the interference fringes I₁₂ and I₁₃ generated by beams 1 and 2 and beams 1 and 3, respectively.

For simplicity, we assume that the initial polarization states of beams 1, 2 and 3 are all linear polarization, and are represented by E₀_i = [cosα_i, sinα_i, 0]^T, where α_i (i = 1, 2, 3) are the orientation angles of linear polarization and α_i ∈ [–π/2, π/2]. As mentioned in Section 2, due to the limited space, only the polarization states of two of the above three beams can be modulated by the HWPs in Fig. 1(b) in each measurement. In the following part, we assume the polarization state of beam 1 is s-polarization, namely α₁ = π/2 and E₀₁ = [0, 1, 0]^T, which is also identical to the polarization state of the light source, while the polarization states of beams 2 and 3 are modulated by the HWPs. It is found that, when α₁ = π/2 (or α₁ = 0),

α_{2} = - α_{3}

is a solution to Eq. (24), provided that the polarization effects of X- and Y-mirrors are identical (namely, X- and Y-mirrors have identical Jones matrices). In addition, the substitution of Eqs. (11)a) and (13a) into Eq. (20) leads to

a_{11} \cos α_{2} \cos α_{3} + a_{12} \cos α_{2} \sin α_{3} + a_{21} \sin α_{2} \cos α_{3} + a_{22} \sin α_{2} \sin α_{3} = 0,

where a_ij are elements of matrix A given by

A = [\begin{matrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{matrix}] = T_{o u t, 1}^{T} \cdot P_{2}^{†} \cdot P_{3} \cdot T_{o u t, 1} .

According to Eqs. (25) and (26), we have

α_{2} = - α_{3} = arc \tan \frac{a_{21} - a_{12} \pm \sqrt{{(a_{12} - a_{21})}^{2} + 4 a_{11} a_{22}}}{2 a_{22}} .

Note that Eq. (28) is only valid when

{(a_{12} - a_{21})}^{2} + 4 a_{11} a_{22} \geq 0

. Otherwise, it indicates that Eq. (26) has no solution. In this case, the optimal combination of α₂ and α₃ can be achieved by

({\hat{α}}_{2}, {\hat{α}}_{3}) = \underset{α_{2}, α_{3} \in Θ}{\arg \min} γ_{23},

where Θ = [–π/2, π/2]. In addition, as revealed in Eq. (28), if Eq. (28) is valid, there always exist two combinations of α₂ and α₃ that could completely prevent the interference between beams 2 and 3. According to Eq. (27), we know that the matrix A exclusively depends on the polarization effects of the X- and Y-mirrors as well as the angles θ and ϕ, which suggests the optimal combination of initial polarization states of beams 2 and 3 can be determined for any given X- and Y-mirrors and the angle θ (ϕ = 45°).

4. Results and discussions

In this section, we will first investigate the interference fringes generated by beams 1, 2 and 3 and make a comparison with the experimental results under different combinations of initial polarization states of the three beams (E₀₁, E₀₂ and E₀₃) to verify the established polarization ray-tracing model. To do this, we have also made a modification to the interferometer unit of the orthogonal two-axis Lloyd’s mirror interferometer. As indicated in Fig. 1(c), physical filters were placed on the surfaces of both X- and Y-mirrors to eliminate the influence of beams 4 and 5. In the experiments, the initial polarization state of beam 1 was fixed at s-polarization, while the initial polarization states of beams 2 and 3 were modulated by rotating the fast axes of the two HWPs in the developed optical setup. In addition, the angle between the direction of the incident light and the XY-plane was set to be θ = 71.805°, so that the periods of the 2D interference fringes in both the X- and Y-directions would be 1 μm.

Table 2 presents the comparison between the simulated and experimental interference fringes generated by beams 1, 2 and 3 under different combinations of initial polarization states of these three beams. The experimental fringes shown in the 3rd row of Table 2 were measured by an atomic force microscope (AFM). We use the triplet of orientation angles (α₁, α₂, α₃) to represent the combination of initial polarization directions of beams 1, 2 and 3. For the special case of α_i = 0° or 90° (i = 1, 2, 3), we do not use the specific angle values but “p” or “s” to represent the corresponding p- or s-polarization. As shown in Table 2, different combinations of initial polarization states of beams 1, 2 and 3 indeed lead to different interference fringes. Notice that, since the photoresist used in the experiments was a positive photoresist, the regions in the simulated interference fringes with high intensities will be removed in the subsequent development process. In other words, the bright regions in the experimental interference fringes (3rd row of Table 2) correspond to the regions with low intensities in the simulated interference fringes (2nd row of Table 2). In addition, for both the simulation and experimental results shown in Table 2, Mirror 1 was employed as the X- and Y-mirrors. It is worth pointing out that Mirror 2 showed similar results to Mirror 1. Therefore, we did not present the corresponding results for Mirror 2. As can be observed from Table 2, after considering the possible errors in the interferometer and the fabrication processes, the simulated interference fringes exhibit reasonable agreement with the experimental results under different combinations of initial polarization states of the three beams.

Table 2. Comparison between the simulated and experimental interference fringes generated by beams 1, 2 and 3 under different combinations of initial polarization states of the three beams

View Table | View all tables in this article

Since different combination of initial polarization states of beams 1, 2 and 3 will lead to different interference fringes, as presented in Table 2, we then try to find the optimal combination of initial polarization states of these three beams. Figures 4(a) and 4(b) present the mapping of the DoO of beams 2 and 3 (namely γ₂₃) for Mirror 1 and Mirror 2, respectively, under different combinations of initial orientation angles α₂ and α₃. As can be observed, Mirror 1 and Mirror 2 exhibit a similar DoO mapping, which therefore leads to similar results with Mirror 2 to those shown in Table 2 with Mirror 1 as mentioned above. Moreover, both Mirror 1 and Mirror 2 show smaller DoO near the region of α₂ = –α₃. Figure 4(c) further presents the variation of γ₂₃ with respect to α₃ for Mirror 1 and Mirror 2 by taking α₂ = –α₃. As can be observed, Mirror 1 and Mirror 2 have a similar γ₂₃ curve but show a small offset due to the different coating layers. It can be observed from Fig. 4(c) that γ₂₃ > 0.04 for both Mirror 1 and Mirror 2, which suggests that actually there does not exist a combination of α₂ and α₃ that could completely prevent the interference between beams 2 and 3 with either Mirror 1 or Mirror 2. This amazing conclusion can be verified according to Eq. (28), since it was found that ${(a_{12} - a_{21})}^{2} + 4 a_{11} a_{22} < 0$ for both Mirror 1 and Mirror 2. According to Fig. 4(c), we derive that γ₂₃ shows the smallest value of 0.048 at α₃ = 33.5° for Mirror 1 and of 0.041 at α₃ = 62.2° for Mirror 2, respectively. Nevertheless, as shown Fig. 4(c), there is essentially not a large difference between γ₂₃ at the optimal value of α₃ and those at other angles. Moreover, the values of γ₂₃ are less than 0.1 at most of the angles of α₃ for both Mirror 1 and Mirror 2, which might not be discerned in actual experimental interference fringes due to errors. Take Mirror 1 as an example. When α₃ = 0 (corresponding to p-polarization) and α₃ = 45°, the associated DoO between beams 2 and 3 are γ₂₃ = 0.07 and γ₂₃ = 0.05, respectively. Therefore, as shown in Table 2, the interference fringes corresponding to the combinations of initial polarization states of (s, p, p) and (s, –45°, + 45°) show similar and good patterns. For the combination of (s, + 45°, –45°), the corresponding interference fringe is a little worse than those obtained at (s, p, p) and (s, –45°, + 45°), as confirmed by the results in Table 2, because of the higher DoO of γ₂₃ = 0.12 at α₃ = –45°, as revealed in Fig. 4(c). For the combinations of (s, + 45°, + 45°) and (s, s, + 30°), the corresponding interference fringes are even worse due to the much higher DoO values, as revealed in Fig. 4(a). In addition, the DoO mapping in Figs. 4(a) and 4(b) also shows a slow change in DoO near the region of α₂ = –α₃, which suggests a good tolerance towards the misalignment errors in α₂ and α₃. A slight misalignment error in the optimal combination of initial polarization states of beams 1, 2 and 3 will not induce a noticeable change in the interference fringes.

Fig. 4 The mapping of the DoO of beams 2 and 3 (γ₂₃) for (a) Mirror 1 and (b) Mirror 2 under different combinations of initial orientation angles α₂ and α₃; (c) The variation of γ₂₃ with respect to the initial orientation angle α₃ of beam 3 for Mirror 1 and Mirror 2, where α₂ = –α₃.

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We then performed experiments at the above achieved optimal combination of initial polarization states of beams 1, 2 and 3. Figures 5(a) and 5(b) present the simulated interference fringes under the optimal combination of (s, –33.5°, + 33.5°) for Mirror 1 and of (s, –62.2°, + 62.2°) for Mirror 2, respectively. As can be observed, Mirror 1 and Mirror 2 show similar interference fringes. Figures 5(c) and 5(d) present the corresponding AFM images of the experimental interference fringes under the optimal combinations of initial polarization states for Mirror 1 and Mirror 2, respectively. A reasonable agreement between the simulated and experimental interference fringes can be observed from Fig. 5. Figure 6 presents the representation of polarization states of beams 1, 2 and 3 (E₁, E₂ and E₃) after the interaction with the Lloyd’s mirror interferometer under the corresponding optimal combination of their initial polarization states. Due to the diattenuation induced by the mirrors, the polarization states of beams 2 and 3 will be restricted within a unit sphere. Since beam 1 is directly incident upon the substrate, the polarization state of E₁ shown in Figs. 6(a) and 6(b) is still s-polarization but is represented in the global coordinate system. As for beams 2 and 3, they are first incident upon the X- and Y-mirrors, respectively, and the reflected beams are then projected onto the substrate. After interaction with the mirrors, the polarization states of beams 2 and 3 are no longer linear polarization but become elliptical polarization due to the polarization effects (retardance and diattenuation) of the mirrors. Moreover, the polarization states of beams 2 and 3 are nearly orthogonal (note that the γ₂₃ only approaches to but is not equal to zero for both Mirror 1 and Mirror 2). We should also note the difference in the polarization states of beams 2 and 3 after their interaction with Mirror 1 and Mirror 2 from Figs. 6(a) and 6(b). Take beam 2 as an example. The polarization state of beam 2 after its interaction with Mirror 1 becomes right-handed elliptical polarization while becomes left-handed elliptical polarization after its interaction with Mirror 2.

Fig. 5 The simulated interference fringes under the optimal combination of initial polarization states of (s, –33.5°, 33.5°) for (a) Mirror 1 and of (s, –62.2°, 62.2°) for (b) Mirror 2. The corresponding AFM images of the experimental interference fringes obtained under the optimal combination of initial polarization states for (c) Mirror 1 and (d) Mirror 2.

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Fig. 6 Representation of polarization states of beams 1, 2 and 3 (E₁, E₂ and E₃) after their interaction with the Lloyd’s mirror interferometer under the associated optimal combination of initial polarization states in the global coordinate system for (a) Mirror 1 and (b) Mirror 2, where the sphere has a radius of 1.

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As mentioned above, there does not exist a combination of α₂ and α₃ that could completely prevent the interference between beams 2 and 3 with either Mirror 1 or Mirror 2 used in our interferometer. We then further investigated whether or not it was possible to completely remove the interference between beams 2 and 3. We performed simulations for a mirror similar to Mirror 1 but with a thickness of 2 nm for the top oxide (Al₂O₃) layer. Recall that the thickness of the oxide layer for Mirror 1 was 4.6 nm. Figure 7(a) presents the variation of γ₂₃ with respect to the initial orientation angle α₃ of beam 3. As shown in Fig. 7(a), there are two optimal angles (α₃ = –14.7° and α₃ = 28.6°) that could lead to γ₂₃ = 0. These two optimal angles can also be directly obtained according to Eq. (28). A perfect square interference fringe can be observed from Fig. 7(b), which was simulated under one optimal combination of initial polarization states of (s, 14.7°, –14.7°). The simulation result under another optimal combination of initial polarization states of (s, –28.6°, 28.6°) was not presented due to the same interference fringe to Fig. 7(b). This is really amazing, since the mirror used in the simulation is nearly identical to Mirror 1 except an only 2.6 nm difference in the thickness of the top oxide layer. This suggests that the polarization effects of the mirrors used in the two-axis Lloyd’s mirror interferometer, which was usually ignored in the previous literature [12–16], should be taken into account in order to achieve accurate prediction of the interference fringes and the optimal combination of initial polarization states.

Fig. 7 (a) The variation of γ₂₃ with respect to α₃ for a virtual mirror similar to Mirror 1 but with a thickness of 2 nm for the oxide (Al₂O₃) layer (α₂ = –α₃), (b) The simulated interference fringe for the mirror under the optimal combination of initial polarization states of (s, 14.7°, –14.7°).

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5. Conclusion

The optimization of initial polarization states of the incident beams (beams 1, 2 and 3) has been investigated to achieve the preferred (square or rhombic) interference fringes for an orthogonal two-axis Lloyd’s mirror interferometer. To this end, we first established a novel 3D polarization ray-tracing model to trace the evolution of polarization states of light through the orthogonal two-axis Lloyd’s mirror interferometer. The established 3D polarization ray-tracing model also took into account of the polarization effects of the mirrors used in the interferometer, which was usually ignored in the previous literature. With the established model, we have derived the optimal conditions for the polarization states of beams 1, 2 and 3 to achieve the preferred interference fringes, namely, the polarization states of beams 2 and 3 are orthogonal to prevent their interference and the contrast of the interference fringe generated by beams 1 and 2 is equal to that of the interference fringe generated by beams 1 and 3. With the above optimal conditions, we have also derived the optimal combination of the initial polarization directions of beams 1, 2 and 3.

Experiments have been performed using a home-made orthogonal two-axis Lloyd’s mirror interferometer. Two kinds of mirrors (Mirror 1 and Mirror 2) were employed in the experiments. Thicknesses of the coating layers for the two kinds of mirrors have been accurately determined by ellipsometric measurements. The comparison between the simulated and experimental interference fringes obtained under different combinations of initial polarization states of beams 1, 2 and 3 has verified the feasibility of the established model as well as the achieved optimal polarization modulation. We has also found that, for either Mirror 1 or Mirror 2, there actually did not exist an optimal combination of the initial polarization directions of beams 2 and 3 that could completely prevent their interference. However, for a virtual mirror that was nearly identical to Mirror 1 except an only 2.6 nm difference in the thickness of the top natural oxide layer, we found that there were two optimal combinations of the initial polarization directions of beams 2 and 3 that could completely prevent their interference. This suggested that the polarization effects of the mirrors used in the two-axis Lloyd’s mirror interferometer should not be ignored to achieve accurate prediction of the interference fringes and the optimal combination of initial polarization directions of incident beams.

It should also be noted that the achieved optimal combinations of the initial polarization directions of beams 1, 2 and 3 shown in Fig. 4 are specific. However, the theory presented in this work, especially the 3D polarization ray-tracing model, is expected to be applicable for not only the orthogonal two-axis Lloyd’s mirror interference lithography, but also other multi-beam interference lithography techniques where polarization control plays an important role. Future work will probe the optimal polarization modulation for more general 2D grating structures in sub-wavelength scale.

Funding

Japan Society for the Promotion of Science (JSPS).

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