Theoretical Exposure Dose Modeling and Phase Modulation to Pattern a VLS Plane Grating with Variable-Period Scanning Beam Interference Lithography

Abstract

Variable-period scanning beam interference lithography (VP-SBIL) can be used to fabricate varied-line-spacing (VLS) plane gratings. The exposure phase modulation method to pattern a VLS grating with a desired groove density must be carefully devised. In this paper, a mathematical model of the total exposure dose for VLS plane grating fabrication is established. With model-based numerical calculations, the phase modulation effects of the parameters, including the fringe locked phase, fringe density, and step size, are analyzed. The parameter combinations for the phase modulation are compared and chosen, and the optimal coordinate for phase compensation is selected. The calculation results show that the theoretical errors of the groove density coefficients can be controlled within 1e-8. The mathematical model can represent the deposited exposure dose for patterning VLS gratings during the lithography process, and the chosen parameters and proposed phase modulation method are appropriate for patterning VLS gratings with VP-SBIL.

1. Introduction

A varied-line-spacing (VLS) plane grating is a straight groove periodic structure with a desired groove density function. VLS plane grating can self-focus and reduce aberration. The efficiency and resolving power of instruments can be improved. Compared with the concave grating, the substrate is easy to process. VLS plane gratings are widely used as key components in extreme ultraviolet spectrum analysis, monochromators, spectral imaging, and position sensors [1,2,3,4,5,6].

The most popular methods to pattern VLS plane gratings are diamond ruling, holographic imaging, and electron or laser beam lithography. Traditional direct-write methods such as diamond ruling and electron-beam or laser beam lithography can be extremely slow and expensive, and result in gratings with undesired phase errors [1,2,7]. Holographic gratings written by two-beam interference introduced a cost-effective path to more rapidly produce gratings with smoother phases. However, holographic methods require spherical or aspheric wavefront recording optics. The desired groove density frequently has difficulty satisfying geometrical constraints in spherical wavefront recording systems, and groove bending decreases the spectral resolution of the grating. Aspheric wavefront recording optics are complicated to design, manufacture, and align. The deviation and error from the recording optics introduce errors into the groove density distribution [8,9].

A variable-period scanning beam interference lithography (VP-SBIL) system is an effective tool to pattern high-fidelity general periodic structures in one or two dimensions within a short period. The VP-SBIL system uses small Gaussian beams with diameters of several micrometers to millimeters to form fringes in a small interference image. The image has a very high phase fidelity. The fringe period and orientation can be progressively changed in a desirable fashion. The fringe phase can be locked using a high-bandwidth fringe locking system. The grating substrate is scanned under the image using a high-performance stage. A uniform exposure dose is achieved by overlapping the subsequent scans. The patterning process is fast and cost-effective because the interference image contains several hundred or thousands of fringes, and expensive aspheric optics are not required in VP-SBIL [10,11,12].

Fabricating a VLS grating with VP-SBIL is not as straightforward as the direct-write method. The groove density of the grating represents the grating phase distribution, which is determined by the exposure phase modulation. To fabricate a straight groove VLS grating using VP-SBIL, three questions must be answered. First, how to express the total exposure dose for patterning a VLS grating with VP-SBIL? Second, which parameters are chosen for the exposure phase modulation? Third, how to cooperatively regulate these parameters to acquire the desired phase distribution? In the published articles about VP-SBIL, the optical topology and basic manufacturing principle for variable-period structures are illustrated. However, details about the phase modulation are not mentioned [10,11,12]. We attempt to provide possible and simple answers to these questions. This is a key issue for practical applications of VP-SBIL [13,14,15].

This paper establishes a model of the total exposure dose for a VLS plane grating patterned with VP-SBIL. The exposure phase distribution and exposure contrast in an intuitive manner are verified, and the phase modulation effects of the parameters are given. The parameter selection for the phase modulation and the optimal coordinate for phase compensation to pattern the VLS grating are proposed. The results for an actual VLS plane grating are given. Finally, the reasons for the differences in the phase modulation performance and phase compensation are discussed.

2. Model of the Total Exposure Dose for VLS Grating Patterning

The desired groove density function of the VLS plane grating without higher-order aberration compensation can be expressed as follows

$$g\left(x\right)=n_{g0}+n_{g1}\left(x-W_g/2\right)$$

(1)

The desired phase variation of the VLS grating can be expressed as

$$\begin{array}{ll}{\Phi }_g\left(x\right)& =2\pi {\displaystyle \underset{0}{\overset{x}{\int }}g(x)}dx\\ & =2\pi \left[\left(n_{g1}/2\right)x^2+\left(n_{g0}-n_{g1}{W}_g/2\right)x\right]\end{array}$$

(2)

where the x coordinate is along the vector direction of the grating. It is assumed that x = 0 is at the margin of the grating substrate. W_g is the grating width along the x-axis; n_g0 is the groove density in the center of the grating with units of L/mm; n_g1 is the slope with units of L/mm²; and n_g0 and n_g1 are determined according to the aberration correction principle of the VLS grating and the structure of the spectrometer or the device, respectively.

In the VP-SBIL system, the step direction is along the x-axis, and the scanning direction is along the y-axis, which is parallel to the fringe direction. In this paper, the VLS grating was patterned with substrate parallel scanning. At the end of a scan, the stage stepped over and reversed direction for a new scan. The exposure phase was modulated during the step process, and the phase relative to the substrate remained unchanged during scanning. Other written modes such as Doppler scanning are not described here [16,17].

Let us assume a perfect stage scan, fringe period regulation, and fringe phase locking. The initial scan is at x = X₀ = 0. The fringe density in the initial scan is f₀. The dose delivered to the resist can be mathematically written as

$$D_0(x)=B(x)+A(x)\cos \left(2\pi x\cdot f_0+{\delta }_0\right)$$

(3)

where B(x) is the background dose. δ₀ is the locked phase introduced by the fringe locking unit in the initial scan. A(x) is the normalized Gaussian dose amplitude in the x-direction and is expressed as

$$A(x)=\exp (-2x^2/{\omega }^2)$$

(4)

where ω is the 1/e² radius of the Gaussian intensity envelope [16,17].

A uniform exposure dose is achieved by overlapping subsequent scans. After the first step with a step size S₁, x = X₁ = S₁ and its serial number k = 1. The dose deposited by the second scan (k = 1) is expressed as

$$D_1(x)=B(x-X_1)+A(x-X_1)\cos \left[2\pi \cdot f_1\cdot (x-X_1)+{\delta }_1\right]$$

(5)

where f₁ is the fringe density in the second scan, and δ₁ is the phase introduced by the fringe locking unit.

Similarly, the exposure dose deposited by the (k + 1)-th scan after k steps (k ≥ 1) is equal to the initial dose, but it has shifted in position due to the discrete stepping of the stage, which can be expressed with Equation (6). This is illustrated in Figure 1 (the fringe density is exaggerated for clarity).

$$D_k(x)=B(x-X_k)+A(x-X_k)\cos \left[2\pi f_k\cdot \left(x-X_k\right)+{\delta }_k\right]$$

(6)

where S_k is the k-th step size. The (k + 1)-th scan is at x = X_k = ${\displaystyle \sum _{i=1}^k{S}_i}$ after k steps. f_k and δ_k are the fringe density and locked phase introduced by the fringe locking unit in the (k + 1)-th scan, respectively.

f_k (k ≥ 1) can be written as $f_k=f_{k-1}+{\Delta }_k=f_0+{\displaystyle \sum _{i=1}^k{\Delta }_i}$, where Δ_k is the density increment between the (k + 1)-th scan and the k-th scan. Equation (6) can be rewritten as

$$D_k(x)=B(x-X_k)+A(x-X_k)\cos \left(2\pi f_{0}x+{\phi }_k\right)$$

(7)

where φ_k is the phase increment between the (k + 1)-th scan and the initial scan, which can be expressed as

$${\phi }_k\left(x\right)=2\pi \left(x{\displaystyle \sum _{i=1}^k{\Delta }_i-f_0}{X}_k-X_k{\displaystyle \sum _{i=1}^k{\Delta }_i}\right)+{\delta }_k$$

(8)

From Equation (8), the phase increment between the (k + 1)-th scan and k-th scan can be expressed as

$${\phi }_{kd}\left(x\right)={\phi }_k\left(x\right)-{\phi }_{k-1}\left(x\right)={\phi }_{kdx}\left(x\right)+{\phi }_{kdc}+\left({\delta }_k-{\delta }_{k-1}\right)$$

(9)

where φ_kdx(x) is the term varying with x, φ_kdx(x) = 2πxΔ_k. φ_kdc is the term independent of x. φ_kdc = −2π (X_k−1Δ_k + S_kf_k).

The resist on the substrate is deposited after N steps and N + 1 scans. The total exposure dose in the resist is the sum of all individual doses and can be expressed as

$$\begin{array}{ll}{D}_T(x)& =D_0(x)+D_1(x)+\cdots D_N(x)\\ & =D_{TB}(x)+A_T(x)\cos {\Psi }_T(x)\end{array}$$

(10)

where D_TB(x) is the total background dose, A_T(x) is the total dose amplitude, and Ψ_T(x) is the exposure phase distribution. They can be expressed with Equations (11) and (12).

$$D_{TB}\left(x\right)=B\left(x\right)+B\left(x-X_1\right)+\cdots B\left(x-X_k\right)$$

(11)

$$\begin{array}{ll}{A}_T(x)\cos {\Psi }_T(x)& =A\left(x\right)\cos \left(2\pi f_{0}x+{\phi }_0\right)+A\left(x-X_1\right)\cos \left(2\pi f_{0}x+{\phi }_1\right)\\ & +A\left(x-X_N\right)\cos \left(2\pi f_{0}x+{\phi }_N\right)\end{array}$$

(12)

From Equation (12), A_T(x) and Ψ_T(x) can be expressed as

$$\{\begin{array}{l}{A}_T\left(x\right)=\sqrt{E{\left(x\right)}^2+F{\left(x\right)}^2}\\ {\Psi }_T\left(x\right)=2\pi x{f}_0+\Psi \left(x\right)\\ \Psi \left(x\right)=\arctan \left[F\left(x\right)/E\left(x\right)\right]\\ E\left(x\right)=A\left(x\right)\cos {\phi }_0+A\left(x-X_1\right)\cos {\phi }_1+\cdots A\left(x-X_N\right)\cos {\phi }_N\\ F\left(x\right)=A\left(x\right)\sin {\phi }_0+A\left(x-X_1\right)\sin {\phi }_1+\cdots A\left(x-X_N\right)\sin {\phi }_N\end{array}$$

(13)

Let us assume that the resist on the substrate is uniform and linear to the exposure dose. Ψ_T(x) determines the theoretical phase distribution of the fabricated grating. The phase error between the fabricated grating and the desired grating can be expressed as

$${\Phi }_e(x)={\Psi }_T(x)-{\Phi }_g(x)$$

(14)

The patterned grating with the desired g(x) can be acquired under the condition of Φ_e(x) = 0, which is the goal of the phase modulation. The total exposure contrast can be expressed as

$$\gamma \left(x\right)=A_T\left(x\right)/D_{TB}\left(x\right)$$

(15)

Equations (10)–(13) are the mathematical exposure dose model for the VLS grating exposure with VP-SBIL in the parallel scanning style. If f_k is constant in all scans, the dose model coincides with that in equal-period SBIL (EP-SBIL) [18,19].

3. Phase Modulation Effects of the Parameters and Parameter Selection

3.1. General Parameters and Numerical Calculation

As shown in Equations (8) and (13), Ψ_T(x) is mainly determined by the locked phase δ_k, fringe density f_k, and step size S_k from all of the scans. Fortunately, all of these parameters can be adjusted in a VP-SBIL system. However, Ψ_T(x) cannot be expressed with a simple mathematical form. It is difficult to directly solve the equation Φ_e(x) = 0. The numerical calculation based on the exposure model is an effective means of analysis.

For simplicity, assume that the contrast in a single scan is 1, which implies that A (x) = B (x) in Equation (3). A grating in a surface physics beamline monochromator is taken as an example. The parameters of the grating are W_g = 30 mm, n_g0 = 1200 L/mm, and n_g1 = −0.7783 L/mm² [20]. To avoid fringe smearing in VP-SBIL, a spot diameter in the range of 10–100 microns is optimal [11]. We chose ω = 100 μm for the numerical calculation and analysis. To achieve a uniform exposure dose, the adjacent scans overlap. As described in previous research on EP-SBIL, the step size is smaller, the uniformity and contrast decrease caused by phase error are better, but the exposure is slower. If the fringe period is constant in all scans, a step size of 0.9 times the 1/e² radius (StepRatio = S_k/ω = 0.9) produces a better dose uniformity than 1% [18,19]. VLS grating fabrication requires fringe phase modulation in each step process, which implies that phase inconsistency in the adjacent scans is inevitable. The step size should decrease with increasing the coefficient n_g1 to satisfy the exposure contrast demand. As StepRatio is not a crucial parameter for phase modulation, we arbitrarily chose StepRatio = 0.8 for the small n_g1 value in this paper.

With the above general parameters, the numerical calculation was developed with Matlab. To minimize the numerical artifact that may arise from an abrupt data cutoff, the total step-scan width W_t is larger than W_g (W_t = W_g + 1 mm), and a sufficiently wide x range [−5 mm, W_t + 5 mm] is chosen for calculation. To guarantee calculation precision, the calculation step is taken to be 1/100 times ω. Based on the exposure model, the exposure doses are calculated and accumulated step by step in the same way as those in a real exposure process, and Ф_e(x) and γ(x) are calculated according to Equations (14) and (15). Because of the lack of sufficient overlap at the start and end scan segments, the data from x = 0 to x = W_g are the focus. To observe the effect of δ_k, f_k, and S_k on Ψ_T(x), three phase modulation manners by varying these parameters in an intuitive way with other parameters remaining unchanged are demonstrated. These approaches appeared to be feasible and needed to be verified, and the modulation effects of the parameters were analyzed. According to the numerical calculation results, the parameters for phase modulation were chosen.

3.2. Phase Modulation Effects of the Parameters

3.2.1. Phase Modulation by Varying Only δ_k

Set the fringe density to a constant value of f_k = f₀ = n_g0 − n_g1 W_g/2. It is equal to the desired density at x = 0. The step size is an integral multiple of the fringe period as that in a traditional EP-SBIL, which implies that S_k = N_steps/f₀ and X_k = kN_steps/f₀. N_steps influences the dose uniformity and exposure contrast. For general VLS gratings with a small coefficient n_g1, the grating density is mainly determined by n_g0. Therefore, N_steps is chosen as a constant expressed as

$$N_{steps}=round\left(\omega \cdot StepRatio\cdot n_{g0}\right)$$

(16)

Under these conditions, Equation (8) changes to a simple form φ_k(x) = δ_k. If δ_k is used for phase modulation, all fringes in the image have identical phase variations, which implies that δ_k cannot be expressed as a function of the x coordinate.

• Intuitively varying δ_k according to Ф_g(x)

Intuitively, changing the δ_k value with the desired Ф_g phase in the (k + 1)-th scan appears to be feasible, which can be written as

$${\delta }_k={\Phi }_g\left(X_k\right)=2\pi \left[\left(n_{g1}/2\right)X_k{}^2+\left(n_{g0}-n_{g1}{W}_g/2\right)X_k\right]$$

(17)

Ф_e(x) is shown in Figure 2, and it is basically piecewise linear with a transition region between the two adjacent linear segments. The exposure contrast is shown in Figure 3. The troughs of the exposure are in contrast with an amplitude close to zero. We performed a differential operation to Ф_e(x). The differences in Ф_e(x) are also shown in Figure 3. The x coordinates of the cusps in Ф_e(x) coincide with those of the lowest contrasts. The frequency components of the exposure contrast are extracted using FFT (Fast Fourier Transform). Two main frequency components are approximately located at F_p1 = |0.5n_g1|/(2π) and F_p2 = 1/S_k. The F_p1 component is influenced by the linear variation of the grating density. The F_p2 component is caused by the phase inconsistency in adjacent scans, which also exists in Ф_e(x), as shown in Figure 3. Ф_e(x) and γ(x) cannot satisfy the demands of grating fabrication.

2.

Phase modulation effect by only varying δ_k according to X_k

To modulate the phase by only varying δ_k, δ_k is different at different X_k. As shown in Equation (2), the desired phase variation is a quadratic polynomial. More generally, δ_k can be written in an analogous form expressed as

$${\delta }_k=2\pi \left(a_{\delta 2}{X}_k{}^2+a_{\delta 1}{X}_k+a_{\delta 0}\right)$$

(18)

Through numerical calculation and theoretical analysis, the modulation effects of δ_k are as follows.

(1) a_δ2 = a_δ1 = 0, and φ_k = δ_c = 2πa_δ0. The phase increments are constant for all scans. From Equation (13), Ψ_T(x) can be expressed as

$${\Psi }_T\left(x\right)=2\pi f_{0}x+{\delta }_c$$

(19)

The patterned grating is an equal-period grating with density f₀. The constant phase shift introduced by δ_c does not affect the grating characteristics. From Equation (15), γ(x) = 1.

(2) a_δ2 = 0, φ_k = δ_k = 2π (a_δ1X_k + a_δ0). Ψ_T(x) is a linear function with the F_p2 frequency component. The slope is influenced by a_δ1, and the constant term and γ(x) are determined by a_δ1 and a_δ0. If the absolute value of a_δ1 is small, the slope of Ψ_T(x) is approximately 2π (f₀ + a_δ1), which implies that a_δ1 can be used to change the period of the patterned grating. This conclusion is consistent with the phenomenon found in an EP-SBIL in Ref. [21]. γ(x) is an average exposure contrast plus an F_p2 frequency component. With a constant step size and small a_δ1, a_δ1 is smaller, the average exposure contrast is higher, and the amplitude of the F_p2 component is smaller. With the general parameters for numerical calculation, the average γ(x) is better than 0.9 when |a_δ1| < 1.4. To maintain the exposure contrast, the adjustment range of the grating density is limited to ±1.4 L/mm.

(3) a_δ2 ≠ 0, a_δ1 ≠ 0 and a_δ0 ≠ 0. Ψ_T(x) and Ф_e(x) are basically piecewise quadratic polynomials. δ_k expressed with Equation (17) is a special case of this situation. Ф_e(x) in Figure 2 is the piecewise quadratic with a quadratic coefficient of zero. The results are similar to those in Figure 2 and Figure 3. Ф_e(x) increases with x, and γ(x) fluctuates between 0 and 1. The cusps of Ф_e(x) and the lowest γ(x) are located at the same x coordinates, and their spatial frequency is |a_δ2|/(2π). There is also the F_p2 frequency component caused by the phase inconsistency in the adjacent scans.

In other words, the desired VLS grating cannot be obtained by only varying δ_k in the form in Equation (18). δ_k can compensate for a certain phase error or change the density of an equal-period grating in a small range.

3.2.2. Phase Modulation by Varying f_k and δ_k

Instinctively, the VLS grating can be patterned by directly varying the fringe period according to g(x). In the (k + 1)-th scan, f_k is set to the desired density g(X_k) of this scan, and Δ_k = f_k − f_k−1 = n_g1S_k. S_k is not used to modulate the phase in this manner. We chose a constant value of S_c = StepRatio·ω, and X_k = kS_c.

The unequal fringe density in the adjacent scans introduces the phase inconsistency. As described in Section 3.2.1, δ_k cannot compensate for the phase error varying with x. However, with δ_k in Equation (20), the fringe phase difference between the (k + 1)-th scan and the k-th scan at a certain x_comk coordinate can be compensated.

$${\delta }_k={\delta }_{k-1}-{\phi }_{kdx}\left(x_{comk}\right)-{\phi }_{kdc}$$

(20)

With f₀ = n_g0 − n_g1W_g/2, Ф_e(x) with different δ_k are shown in Figure 4. Ф_e(x) is generally a piecewise quadratic polynomial with δ_k = 0 or δ_k = δ_k−1 − φ_kdc. The x coordinates of the cusps in Ф_e(x) coincide with those of the lowest contrasts. Ф_e(x) with x_comk = (X_k + X_k−1)/2 is close to 0, and Ф_e(x) with x_comk = X_k and x_comk = X_k−1 have a cumulative characteristic. For small VLS gratings, the cumulative effects can be neglected. However, with increasing the grating size, the cumulative Φ_e(x) may have a great impact. The F_p2 frequency component in Ф_e(x) can be acquired by removing the linear polynomial from Ф_e(x), as shown in Figure 5. The F_p2 frequency components indicate that the phase inconsistences in the adjacent scans are not fully compensated. The peak-to-valley amplitude of the phase fluctuation is close to 2 × 10⁻⁴ rad, which is smaller than 1/30,000 groove period. The grating groove position error caused by the phase fluctuation is smaller than 0.028 nm for the grating example. Compared with the phase errors in nanometer scale caused by the substrate and resist roughness, development, and metrology frame, this small phase fluctuation can usually be neglected for general VLS gratings.

We performed a quadratic polynomial fitting to Ψ_T(x). The fitting polynomial can be expressed as

$$\begin{array}{ll}{\Psi }_{Tp}\left(x\right)& =2\pi \left(a_{\Psi 2}{x}^2+a_{\Psi 1}x+a_{\Psi 0}\right)\\ & =2\pi {\displaystyle \underset{0}{\overset{x}{\int }}{g}_p(x)}dx+{\Psi }_{Tp0}\end{array}$$

(21)

where g_p(x) = n_gp0 + n_gp1(x − W_g/2) can be treated as the theoretical groove density of the patterned grating. Ψ_Tp0 = 2πa_Ψ0 is the constant component of Ψ_Tp(x), which has little effect on the characteristics of the grating. The groove density coefficients with different phase compensations are shown in

Table 1. g_p(x) under the condition of f_k = g(X_k) and phase compensation with δ_k.

xcomk	gp(x) Coefficients		Relative Errors of gp(x) Coefficients
	ngp0, L/mm	ngp1, L/mm2	ε(ng0) = |ngp0 − ng0|/ng0	ε(ng1) = |ngp1 − ng1|/ng1
Xk	1200.0311	−0.7783	2.5943 × 10−5	<1 × 10−8
(Xk + Xk−1)/2	1200.0000	−0.7783	<1 × 10−8	<1 × 10−8
Xk−1	1199.9689	−0.7783	−2.5943 × 10−5	<1 × 10−8

. The theoretical groove densities g_p(x) are close to the desired value, especially g_p(x) with x_comk = (X_k + X_k−1)/2.

The exposure contrasts γ(x) are shown in Figure 6. The exposure contrasts are better than 0.99, and γ(x) with x_comk = (X_k + X_k−1)/2 is better than the others. The resist has low-frequency filtering characteristics, so the influence of the F_p2 fluctuation can be neglected [22,23]. The contrast can satisfy the demands of the grating exposure.

3.2.3. Phase Modulation by Varying f_k, S_k and δ_k

The fringe density f_k = g(X_k), and the step size S_k varies according to the fringe density. Here, we set S_k equal to an integer N_steps multiplied by a variable period value. N_steps is chosen to be the same constant expressed with Equation (16). The variable period value can be chosen as the fringe period in the adjacent last scan. S_k can be expressed as

$$S_k=N_{steps}/f_{k-1}$$

(22)

The phase difference can also be compensated with δ_k in Equation (20). With f₀ = n_g0 − n_g1W_g/2 and the different phase compensation with δ_k, Φ_e(x) and γ(x) are shown in Figure 7 and Figure 8, respectively. Except for the condition of δ_k = δ_k−1 − φ_kdc, Φ_e(x) is small, and the average contrasts are close to 1. The phase error from adjacent scans still causes F_p2 fluctuations in Φ_e(x) and γ(x).

As shown in Figure 7, Φ_e(x) with δ_k = 0 is coincident with Φ_e(x) with δ_k = δ_k−1 − φ_kdx(X_k) − φ_kdc, which implies that the step size expressed with Equation (22) can compensate for the phase difference at X_k. We assume that the phase difference at x_comk must be compensated, where x_comk is located between X_k−1 and X_k. x_comk can be expressed with X_k−1 and X_k as

$$x_{comk}=X_{k-1}+r\left(X_k-X_{k-1}\right)$$

(23)

where r = (x_comk − X_k−1)/(X_k − X_k−1) is the ratio. If δ_k = 0 for all scans, from Equations (9) and (23), the phase difference at x_comk in the k-th scan and (k + 1)-th scan can be expressed as

$$\begin{array}{ll}{\phi }_{kd}\left(x_{comk}\right)& ={\phi }_{kdx}\left(x_{comk}\right)+{\phi }_{kdc}\\ & =-2\pi S_k\left[\left(r{f}_{k-1}+(1-r)f_k\right)\right]\end{array}$$

(24)

From Equation (24), φ_kd(x_comk) can be compensated with S_k, expressed as

$$S_k=\frac{N_{steps}}{r{f}_{k-1}+(1-r)f_k}$$

(25)

If S_k is other values, such as an actual step size with a stage motion error, δ_k can be used for phase compensation, which implies that S_k and δ_k have similar phase modulation effects.

In this manner, the groove density coefficients with different phase compensations are shown in

Table 2. g_p(x) under the condition of f_k = g(X_k) and phase compensation with S_k.

xcomk	gp(x) Coefficients		Relative Errors of gp(x) Coefficients
	ngp0, L/mm	ngp1, L/mm2	ε(ng0)	ε(ng1)
Xk	1200.0311	−0.7783	2.5943 × 10−5	−2.5946 × 10−5
(Xk + Xk−1)/2	1200.0000	−0.7783	<1 × 10−8	<1 × 10−8
Xk−1	1199.9689	−0.7783	−2.5943 × 10−5	2.5946 × 10−5

.

3.3. Parameter Selection for Phase Modulation

We demonstrate three intuitive phase modulation manners and analyze the phase modulation effects of the parameters, including the locked phase δ_k, fringe density f_k, and step size S_k. The results show that the desired phase distribution can be acquired by varying the parameter combinations as follows: (a) f_k and δ_k; (b) f_k and S_k; (c) f_k, S_k, and δ_k. These parameters can be selected for phase modulation.

In an actual SBIL system, it is difficult to modulate the exposure phase by only varying S_k because of the stage positioning error. The stage positioning error can be corrected with δ_k according to the phase compensation relationship. Under this condition, an actual stage can be considered an ideal stage without a positioning error, similar to that in an EP-SBIL system. Therefore, parameter combinations (a) and (c) are applicable to practical applications. Comparing the groove density errors in Table 1 and Table 2, the parameter combination (a) has a better performance, and x_comk = (X_k + X_k−1)/2 is the optimal coordinate for phase compensation.

In addition to the grating example in Section 3.1, other VLS gratings, including another grating in a monochromator (n_g0 = 400 L/mm, n_g1 = −0.2594 L/mm and W_g = 80 mm) [20] and a grating in a diagnostic spectrometer (n_g0 = 600 L/mm, n_g1 = −0.7869 L/mm and W_g = 30 mm) [24], are designed for further verification. The results are consistent with the above results. Compared with the existing fabrication methods of VLS gratings, the relative errors of the groove density can satisfy the technical requirements [1,2,3,8,20,24], and the exposure contrast can satisfy the demands for VLS grating fabrication.

4. Discussion

Although S_k and δ_k have similar phase compensation effects, the results show that parameter combination (a) performs better, and x_comk = (X_k + X_k−1)/2 is the optimal coordinate for phase compensation. The reason for the phenomenon will be discussed.

First, let us focus on the performance differences with different parameter combinations. In parameter combination (b) and (c), S_k varies with f_k−1, as expressed in Equation (22). For the VLS grating example in this paper, S_k increases with k. With an increasing S_k, the phase inconsistency from adjacent scans has a greater impact on the phase error and exposure contrast. Therefore, the overall performances with parameter combination (b) and (c) decrease with increasing S_k, and a constant S_k = S_c maintains a consistent performance with combination (a).

Second, the selection of x_comk and the cause of the cumulative characteristic in Φ_e(x) with x_comk ≠ (X_k + X_k−1)/2 are discussed. As shown in Equation (9), after the phase difference at x_comk is compensated with δ_k or S_k, the residual phase in φ_kdx(x) is expressed as

$$\begin{array}{ll}{\phi }_{kd\_res}\left(x\right)& ={\phi }_{kdx}\left(x\right)-{\phi }_{kdx}\left(x_{comk}\right)\\ & =2\pi {\Delta }_k\left(x-x_{comk}\right)\end{array}$$

(26)

As shown in Figure 1, the overall phase compensation performance in the x range between X_k−1 and X_k is mainly determined by the k-th scan and (k + 1)-th scan. The overall phase compensation error can be estimated with the integral of φ_{kd_res}(x) in the range [X_k−1, X_k] as

$$\begin{array}{ll}{I}_k\left(x_{comk}\right)& ={\displaystyle \underset{X_{k-1}}{\overset{X_k}{\int }}{\phi }_{kd\_res}\left(x\right)}dx\\ & =2\pi {\Delta }_k{S}_k\left[\left(X_k+X_{k-1}\right)/2-x_{comk}\right]\end{array}$$

(27)

Although Equation (27) is not a strict mathematical deduction based on the exposure model, it provides a tool for the analysis. The absolute value of I_k is smaller, and the phase compensation performance is better. If x_comk = (X_k + X_k−1)/2, the best phase compensation performance can be obtained with I_k = 0, and it is not affected by S_k. Therefore, (X_k + X_k−1)/2 is a good selection for x_comk. If x_comk ≠ (X_k + X_k−1)/2, the phase compensation performance degrades with |I_k(x_comk)| > 0. As shown in Figure 4 and Figure 7, I_k(X_k) and I_k(X_k−1) have the same absolute value with opposite signs, which is consistent with the cumulative Φ_e(x) with different variation directions.

5. Conclusions

In this paper, a mathematical model of the total exposure dose for patterning a VLS grating with VP-SBIL is established, and the phase modulation method of the exposure is proposed. According to the model, the phase modulations in three intuitive manners are calculated, and the phase modulation effects of the parameters, including the locked phase, fringe density, and step size, are analyzed. The phase errors with phase compensation at different coordinates are compared and discussed, and the optimal coordinate for phase compensation is presented. The results show that three parameter combinations can be selected for phase modulation, and the phase modulation performance with the locked phase and the fringe density is better than those with other parameter combinations. The average value of the coordinates in two adjacent scans is the optimal coordinate for phase compensation. The theoretical error between the patterned grating and the desired grating can be controlled within 1e-8, and the requirements of the grating phase distribution and exposure contrast can be satisfied. The proposed exposure dose model and exposure phase modulation method are suitable for VLS planar grating fabrication with VP-SBIL.