Abstract

Optical proximity correction (OPC) and phase shifting masks (PSM) are resolution enhancement techniques (RET) used extensively in the semiconductor industry to improve the resolution and pattern fidelity of optical lithography. In this paper, we develop generalized gradient-based RET optimization methods to solve for the inverse lithography problem, where the search space is not constrained to a finite phase tessellation but where arbitrary search trajectories in the complex space are allowed. Subsequent mask quantization leads to efficient design of PSMs having an arbitrary number of discrete phases. In order to influence the solution patterns to have more desirable manufacturability properties, a wavelet regularization framework is introduced offering more localized flexibility than total-variation regularization methods traditionally employed in inverse problems. The proposed algorithms provide highly effective four-phase PSMs capable of generating mask patterns with arbitrary Manhattan geometries. Furthermore, a double-exposure optimization method for general inverse lithography is developed where each exposure uses an optimized two-phase mask.

©2007 Optical Society of America

1. Introduction

Due to resolution limits of optical lithographic systems, the electronics and photonics industry have relied on 2D and 3D resolution enhancement techniques (RET) to compensate and minimize mask distortions as they are projected onto semiconductor wafers [1, 4, 5]. Resolution in optical lithography obeys the Rayleigh criterion resolution (R)=kλNA, where λ is the wavelength, NA is the numerical aperture taking on values around 0.9 for most lithography systems used today, and k is the process constant which can be minimized through RET methods [6, 7, 8, 9]. RET methods manipulate the local amplitude and phase features of the optical wave-front to pre-compensate for imaging distortions. Optical and process correction (OPC) methods modify the mask amplitude by the addition of sub-resolution features to the mask pattern [5]. Phase shifting masks (PSMs), commonly attributed to Levenson [10], induce phase shifts in the transmitted field which has a favorable constructive or destructive interference effect. Thus, a suitable modulation of both the phase and the intensity of the incident light can be used to effectively compensate for some of the resolution-limiting phenomena in optical diffraction. The challenge is then to develop a general systematic method to design pre-compensated masks also referred to as inverse lithography technology (ILT).

Several approaches to inverse lithography have been proposed in the literature. These range from heuristic and empirically-based design rules to computationally expensive optimization-based inverse algorithms. Sherif, et al. derived an iterative approach to generate binary masks [11]. Liu and Zakhor developed a binary and phase shifting mask design strategy based on the branch and bound algorithm and simulated annealing [12]. Pati-Kailath exploited a class of approximations for partially coherent imaging systems to develop sub-optimal projections onto convex sets for PSM designs [13]. In addition, Erdmann proposed automatic optimization of the mask and illumination parameters with a genetic algorithm [14]. Pang, et al. gave an overview of ILT and provided some simulations to demonstrate the benefit of ILT [15]. Granik described and compared solutions of inverse mask problems [16]. All of the methods mentioned above, however, are not based on gradient type optimization and thus the searching process for a suitable solution is either computationally expensive or not efficient. Recently, Poonawala and Milanfar introduced a novel optimization framework for inverse lithography based on a pixel-based, continuous function formulation, well suited for gradient-based search [?, 17]. Based on a steepest descent search, this approach exploits the rich theory of regularized iterative optimization [18].

Letting M(x,y) be the input mask to an optical lithography system T{∙}, approximated as a low-pass spatial filter followed by a soft threshold operation, the output pattern is denoted as Z(x,y) = T{M(x,y)}. Given a N × N desired output pattern Z *(x,y), the goal of PSM design is to find the optimized M(x,y) called M̂(x,y) such that the distance D = d(Z(x,y), Z *(x,y)) = d(T{M(x,y)}, Z *(x,y)) is minimized, where d(∙,∙)is the mean square error criterion. The PSM inverse lithography optimization problem can thus be formulated as the search of M̂(x,y) over the N × N complex space C N×N such that

M̂(x,y)=argminM(x,y)CN×Nd(T{M(x,y)},Z*(x,y)).

With the goal of attaining two-phase-level optimized masks, the system in Ref. [17] further constrained M(x,y) in Eq. (1) such that M(x,y) = cos(θ(x,y)), thus searching over the range of θ and thus constraining the value of M in the interval M(x,y) ∈ (-1,1) and in essence representing masks limited to two phases. While this approach is very effective in some cases, the end result is that the search generally fails to generate adequate PSM for mask patterns having arbitrary Manhattan geometries [19]. According to the “Four-Phase Theorem” described in Ref. [13], given an arbitrary pattern with a Manhattan geometry, a phase-shifting mask used to synthesize the image pattern must use a minimum of four distinct phase levels.

The main goal of this paper is thus to overcome this limitation of the Poonawala, et al. algorithm in Ref. [17], so as to obtain a generalized synthesis algorithm capable of generating arbitrary mask patterns. This is accomplished as follows: First, the iterative optimization framework is re-formulated where the search trajectory is unconstrained in the complex plane. The optimization problem is thus broadly formulated as in Eq. (1). As expected, the resultant mask patterns obtained by Eq. (1) have arbitrary complex pixel values, and consequently a post-processing step is used to quantize the patterns into the desired four-phase-level, shifting-mask patterns.

A second contribution of the paper is the introduction of a new regularization framework. Inverse lithography is an ill-posed problem where numerous input patterns can lead to the same binary output pattern. Regularization in ILT seeks to bias the solution space to sample solutions that have lower manufacturabilty complexity. We introduce an effective detail-reduction approach referred to as “wavelet penalty” regularization.

Finally, in the event that four-phase masks are difficult to fabricate, but the goal is to still synthesize masks with arbitrary geometry, a third contribution of the paper is the development of a double-exposure PSM optimization method, where at each stage the PSM masks are constrained to have two phases. The two-stage exposure method can lead to high fidelity output pattern reproduction at the expense of a more complex exposure process. Poonawala and Milan-far, independently from our work, have simultaneously developed a different double-exposure method [20].

2. PSM design and problem formulation

The methods developed in this paper extend and generalize the model and approach proposed in Ref. [17], illustrated in Fig. 1. The complex-valued mask is the input of the system, where the phase pattern of the propagating light is modified by a phase shifting material overcoating the mask. Light propagating through the mask pattern is affected by diffraction and mutual interference—a phenomena described by the Hopkins Diffraction Model [21, 22]. The light that transmitted through the mask reaches a light-sensitive photoresist, which is subsequently developed through the use of solvents. The thickness of the remaining photoresist after development is proportional to the exposure dose exceeding a given threshold intensity. In a positive photoresist process, almost all the photoresist material remains in the low-exposure area on the wafer and is removed in the high-exposure area. Between these two extremes is the transition region. For mathematical simplicity, it is assumed that when the light field exceeds a threshold, the exposed area becomes a high-exposure area, otherwise, a low-exposure area. Thus, a hard threshold operation can adequately represent the exposure effect described above and the output pattern of the optical system is binary. This is an approximate model for the field transfering through the mask. In Fig. 1, |∙| is the element-by-element absolute operation. In this paper, only coherent imaging is considered, and the aerial image formation process is approximated by a convolution between the mask pattern and a Gaussian low pass filter h. In the pixel-based algorithm, pixelsize=resolution(R)=kλNA. The standard deviation of the Gaussian low-pass filter h is σ=R×NAλ=k. The output of the convolution and the absolute operation model is the electrical field amplification distribution of the aerial image. Further, since the derivative of the sigmoid function exists, it is used to approximate the hard threshold function. The hard threshold function is a shifted unit step function U(x-tr), which is approximated by the sigmoid function

sig(x)=11+exp[a(xtr)],

where tr is the process threshold, and a dictates the steepness of the sigmoid function.

 figure: Fig. 1.

Fig. 1. Approximated forward process model. (Ref. [17], Fig. 1).

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The mask is represented by an M N×N matrix with a N 2 × 1 equivalent raster scanned vector representation m̲N2 × 1 which is denoted as m̲ for short notation. Following the definitions above, the following notation is used:

  1. A convolution matrix H is a N 2 × N 2 matrix denoting an equivalent two-dimensional low-pass filter h.
  2. The desired N × N binary output pattern is denoted as Z *. It is the desired light distribution sought on the wafer. Its vector representation is denoted as ẕ*.
  3. The output of the sigmoid function is the N × N real-valued image denoted as Z = sig{|H{M}|}. The equivalent vector is denoted as ẕ ∈ ℜN2 × 1.
  4. The hard threshold version of Z is the binary output pattern denoted as Zb. Its equivalent vector is denoted as ẕb ∈ ℜ N2 × 1, with all entries constrained to 0 or 1.
  5. The electrical field of the input mask is denoted as the complex-valued N × N matrix M and it is equivalently represented as m̲ ∈ C N2 × 1, where C N2 × 1 is the set of N 2 × 1 complex vector. All entries in M and m̲ can take on unrestricted complex values.
  6. The pole-constrained mask Mp is the quantization of M. The pixel magnitudes of Mp are quantized to 0 or 1. The pixel phases are quantized to several discrete phase levels. Its equivalent vector is denoted as m̲pC N2 × 1
  7. The optimized N × N complex-valued mask denoted as M̂ minimizes the distance between Z and Z *, ie,

    M̂=argminMd(sig{H{M}},Z*).

    Its equivalent vector is denoted as m̲^C N2 × 1.

  8. The pole-constrained optimized mask p is the quantization of . Its equivalent vector is denoted as m̲^ pC N2 × 1.

Given the gray level pattern ẕ = sig{|H{m̲}|}, the ith entry in this vector can be represented as

i=11+exp[aj=1N2hijj+atr],i=1,...N2,

where hij is the i, jth entry of the filter. In the optimization process, m̲^ is searched to minimize the L 2 norm of the difference between ẕ and ẕ*. Therefore, m̲^ = argminm̲^{F(m̲)}, where the cost function F(∙) is defined as:

F(m¯)=z¯*z¯22=i=1N2(zi*zi)2=i=1N2(zi*11+exp[aj=1N2hijm¯j+atr])2.

3. Generalized PSM design algorithm

As stated before, two-phase-level PSMs are not adequate to generate arbitrary patterns with Manhattan geometries [13] and phase conflicts are likely to arise. According to the Four – Phase Theorem [13], the number of discrete phase levels used in PSM design should be at least four, in order to avoid conflicts and ambiguities in the assignment of phases. The inverse model-based algorithms in Ref. [17] is not suited for optimization of masks having more than two-phase levels (0 and π). This drawback motivates us to develop a generalized algorithm, admitting an arbitrary number of discrete phase levels, which overcomes this limitation.

Let ṟ and θ̱ be the magnitude and phase components of the complex-valued mask

m¯k=r¯kejθ¯k,k=1,...,N2,

where j=1,θ¯k(,) and k ∈ [0,1]. The bound-constrained optimization is then reduced to an unconstrained optimization problem using following parametric transformation,

r¯k=1+cosϕ¯k2,k=1,...,N2,

where ϕ¯k ∈ (-∞,∞). Substituting k, in Eq. (6) with Eq. (7), we have,

m¯k=1+cosϕ¯k2ejθ¯k,k=1,...,N2.

Defining the vector θ̲ = [θ 1,…,θ N2]T and ϕ̲ = [ϕ 1,…,ϕ N2]T, the optimization problem is formulated as:

(θ̂¯,ϕ̂¯)=argmin(θ̲,ϕ̲){F(θ̲,ϕ̲)},

where the cost function is:

F(θ̲,ϕ̲)=z¯*z¯22=i=1N2(zi*zi)2
=i=1N2(zi*11+exp[a(k=1N2hik1+cosϕ̲k2ejθ̱k)2+atr])2.

The steepest-descent method is used to optimize the above problem. The gradients ∇F(θ̲,ϕ̲)θ̲ and ∇F(θ̲,ϕ̲)ϕ̲ can be calculated as follows [19]:

Fθ̲ϕ̲θ̲=d¯θ̲=2a×+cosϕ̲2sinθ̲{HT[(z¯*z¯)z¯(z¯)H(m¯R)T(m¯)]}
2a×+cosϕ̲2cosθ¯{HT[(z¯*z¯)z̲⊙(z¯)H(m¯I)T(m¯)]},
Fθ̲ϕ̲ϕ̲=d¯ϕ̲=a×sinϕ̲cosθ̲{HT[(z¯*z¯)(z¯)H(m¯R)T(m¯)]}
+a×sinϕ̲sinθ̲{HT[(z¯*z¯)z̲⊙(z¯)H(m¯I)T(m¯)]},

where ∇F(θ̲) ∈ R N2 × 1, ⊙ is the element-by-element multiplication operator, and T(m¯)=[H(m¯R)2+H(m¯I)2]12. R and I are the real part and the imaginary part of m̲. 1̱ = [1,…, 1]TR N2 × 1 Assuming θ̲k and ϕ̲k are the kth iteration results, then at the k + 1th iteration:

θ̲k+1=θ̲ksθ̲d¯θ̲k,
ϕ̲k+1=ϕ̲ksϕ̲d¯ϕ̲k,

where sθ̲ and sϕ̲ are the step-sizes. The iterative optimization above, in general, leads to complex-valued solutions that are not constrained to a discrete number of magnitude and phase levels. Therefore a post-processing step is needed to obtain the pole-constrained optimized mask, m̲^ p. Since m̲^ is complex-valued, a two-step quantization process is followed. First the magnitudes are quantized by a global threshold tm as |m̲^ pi| = U(|m̲^ i|-tm), where i = 1,…, N 2. The phases are subsequently quantized to the nearest prescribed discrete phase level. We define the pattern error E as the distance between the desired output pattern Z * and the actual binary output pattern Zb:

E=i=1N2z¯i*z¯bi=i=1N2z¯i*ΓHm¯pi.

When the pattern error is reduced to a tolerable level, the steepest descent iteration is stopped.

In order to demonstrate the effect of the number of phases used in PSM design, consider the desired pattern shown in Fig. 2. The top horizontal block cross-connects two vertical parallel blocks. The phases assigned to the two parallel adjacent blocks cannot be the same, so as to exploit the PSM principle. Since the horizontal top block connects the parallel blocks, and since it would be desirable that there are no gaps introduced in the image, the horizontal block must be assigned an intermediate phase value that is distinct from the phases of the two parallel blocks. Therefore, two-phase levels are not sufficient to attain this goal. The conflict can be eliminated with the use of four phases. In Fig. 2, the patterns in the top row and the middle row illustrate the input patterns. The output patterns are shown in the bottom row. The three images in the bottom row show the output patterns of (left) the desired pattern (ẕ*), (center) the complex-valued optimized mask (m̲^), and (right) the pole-constrained optimized mask (m̲^ p). The aerial image formation process is approximated by a 11 × 11 Gaussian low pass filter with k = 14, in the sigmoid function, we assign parameters a = 80 and tr = 0.5. The global threshold is tm = 0.5. The shape of the image used to initialize the iterative algorithm is the same as the desired binary output pattern Z *. For ϕ_k, we assign the phase of π5 corresponding to the areas having a magnitude of 1 and a phase of 4π5 for the areas with magnitude of 0. The phase assignment must be done a-priori and phases in neighboring blocks are assigned alternately. Because of the numerous sine and cosine functions in Eq. (11) and Eq. (12), we intentionally avoid assigning θ_k and ϕ_k the values of 0, π2, π or 3π2. Otherwise, Eq. (11) or Eq. (12) may reduce to a zero update, terminating the iteration. For a four-phase-level mask design, empirical observations show that an efficient assignment of phase values to θ_k is to select phases in the set of π4,3π4,5π4 and 7π4. Further, regions around a block should be assigned a phase value that is π2 different from that of the block and on the same side of the imaginary axis. This approach has proved efficient in our extensive simulation analysis. As a comparison to the four-phase PSM design, the experiment using just two phases is illustrated in Fig. 3. Note that a gap appears on the top connection of the synthesized mask, as expected.

 figure: Fig. 2.

Fig. 2. Top row (input masks), left to right: desired pattern, magnitude of the optimized complex-valued mask and optimized pole-constrained mask obtained using a threshold tm. The middle row indicates the phases of the optimized complex-valued mask. The bottom row indicates the corresponding binary output patterns. The parameters used in the simulation are a = 80, tr = 0.5, tm = 0.5, 11 × 11 Gaussian low pass filter with k = 14, sϕ_ = 2 and sθ_ = 0.01. Green, red and blue represent 0, 1 and -1 respectively in the top and bottom rows; Dark blue, light blue, yellow and red represent π4,3π4,5π4 and 7π4 respectively in the middle row.

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 figure: Fig. 3.

Fig. 3. Left to right: desired pattern, pole-constraied optimized mask and binary output pattern. sϕ_ = 2, sθ_ = 0.01. Red and blue represent 0 and π phases respectively.

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4. Regularization

As illustrated in Fig. 2, the optimized mask patterns contain numerous details, which may bring difficulty to mask fabrication. Most of the details consist of singular transmission pixels. One approach to reduce image detail is through regularization during the optimization process [18, 17]. Regularization is formulated as follows:

m̂¯=argminm̂¯{F(m¯)+γR(m¯)},

where F(m̲) is the data-fidelity term and R(m̲) is the regularization term which is used to reduce the solution space and constrain the optimized results. γ is the user-defined parameter to reveal the weight of the regularization. In the following, we will discuss pole penalty and the wavelet penalty.

4.1. Pole penalty

The pole penalty is used to constrain the magnitude and phase of each pixel value in the optimized mask to several prescribed discrete levels, so the pole penalty is divided into the magnitude pole penalty and the phase pole penalty. Poonawala and Milanfar used the pole penalty approach favoring a two-phase-level solution. In order to extend the pole-penalty method to the algorithm developed in Section 3, it is generalized to admit an arbitrary number of phase levels.

The magnitude pole-penalty term is:

Rpole(m¯)=i=1N2i2=T.

For each pixel value, the corresponding penalty is the quadratic function r(i) = |mi 2, i = 1,…,N 2. According to Eq. (17), the gradient of Rpole(m̲) is ∇Rpole(m̲) = 2|m|̲, thus, the cost function in Eq. (5) is adjusted as J(m̲) = F(m̲)+γpoleRpole(m̲). In the phase penalty, the phases are constrained to the closest phase levels. If we use two-phase levels, the regularization term is:

Rpole(θ̲)=i=1N2[sin(2θ̲iπ2)+1]2=[sin(2θ̲π2)+]T[sin(2θ̲π2)+].

For each pixel value, the corresponding penalty depicted by solid line in Fig. 4 is r(θ̲i)=[sin(2θ̲iπ2)+1]2i=1,...,N2. According to Eq. (18), the gradient of Rpole(θ̲) is:

Rpole(θ̲)=4[sin(2θ̲π2)+]Tcos(2θ̲π2).

If four-phase levels are considered, the regularization term is then obtained as:

Rpole(θ̲)=i=1N2[sin(4θ̲i3π2)+1]2=[sin(4θ̲i3π2)+]T[sin(4θ̲i3π2)+].

For each pixel value, the corresponding penalty depicted by the dash-dot line in Fig. 4 is:

r(θ̲i)=[sin(4θ̲i3π2)+1]2,i=1,...,N2

and the gradient of Rpole(θ̲) is:

Rpole(θ̲)=8[sin(4θ̲3π2)+]Tcos(4θ̲3π2).

Using the pole penalty described in Eq. (17) and Eq. (20), the experiment shown in Fig. 2 can be repeated. The result is illustrated in Fig. 5. Comparison between Fig. 2 and Fig. 5 shows that the pole penalty leads to fewer transmission pixels and thus fewer details in the attained mask pattern. However, some details still remain in the mask pattern. In order to remove those details, an additional penalty can be considered. The wavelet penalty regularization method is introduced next for this goal.

 figure: Fig. 4.

Fig. 4. Phase penalty for two-phase levels and four-phase levels

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 figure: Fig. 5.

Fig. 5. Left to right: desired pattern, pole-constrained optimized mask magnitude, pole-constrained optimized mask phases (Dark blue is π4, light blue is 3π4, yellow is 5π4 and red is 7π4 and binary output pattern. sϕ_ = 2, sθ_ = 0.01, γpole,ϕ_ = 0.001 and γpole,θ_ = 0.0001.

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4.2. Wavelet penalty

4.2.1. Wavelet penalty

Since typical mask patterns encountered in circuitry are piecewise smooth images, the Haar wavelet is used as the building block. Consider a N × N (assume N is even) image M N×N, where mij represents the (i, j) matrix element. The 1-depth Haar wavelet transform of the image above, ignoring the scale parameters, leads to the level-one approximation coefficient block and three detail coefficient blocks, each block of size N2×N2. Specifically, the approximation coefficient block is AN2×N2, where

aij=m(2(i1)+1)(2(j1)+1)+m(2(i1)+1)(2(j1)+2)+m(2(i1)+2)(2(j1)+1)+m(2(i1)+2)(2(j1)+2),

for i,j=1,...,N2. The horizontal, vertical and diagonal detail coefficient blocks are respectively,

hij=m(2(i1)+1)(2(j1)+1)m(2(i1)+1)(2(j1)+2)+m(2(i1)+2)(2(j1)+1)m(2(i1)+2)(2(j1)+2),
vij=m(2(i1)+1)(2(j1)+1)+m(2(i1)+1)(2(j1)+2)m(2(i1)+2)(2(j1)+1)m(2(i1)+2)(2(j1)+2),
dij=m(2(i1)+1)(2(j1)+1)m(2(i1)+1)(2(j1)+2)m(2(i1)+2)(2(j1)+1)+m(2(i1)+2)(2(j1)+2),

for i,j=1,...,N2. The approximation coefficient block represents the low-frequency component of the image and the other three detail coefficient blocks represent the high-frequency components or the details of the image. Further, using Eq. (24) to Eq. (26), the total energy in the detail components is:

Edetail=i=1N2j=1N2(hijhij*+vijvij*+dijdij*).

In order to remove details in the mask, the energy of the detail components should be reduced during the optimization process. Although Edetail contains many terms, there are just three terms relative to a specific mask element mij. This property is convenient for calculating the energy differential of the detail components with respect to each pixel value mij. We refer to this property as the “localization property”. The partial derivatives of Edetail with respect to ϕ̲ and θ̲ are as follows [19]:

Edetailϕ̲(2(i1)+p)(2(j1)+q)=sinϕ̲(2(i1)+p)(2(j1)+q)×Re[ejθ̲(2(i1)+p)(2(j1)+q)×(3m¯(2(i1)+p)(2(j1)+q)
m¯(2(i1)+p1)(2j1)+q)(2(i1)+p)(2(j1)+q1)(2(i1)+p1)(2(j1)+q1))],
Edetailθ̲(2(i1)+p)(2(j1)+q)=(1+cosϕ̲(2(i1)+p)(2(j1)+q))×Re[(j)ejθ̲(2(i1)+p)(2(j1)+q)
×(3m¯(2(i1)+p)(2(j1)+q)(2(i1)+p1)(2(j1)+q)m¯(2(i1)+p)(2(j1)+q1)
m¯(2(i1)+p1)(2(j1)+q1))],

where i,j=1,...,N2; p,q = 1 or 2; p 1 = (p + 1) mod 2 and q 1 = (q + 1) mod 2. From Eq. (28) and Eq. (29), the gradient of Edetail can be calculated and the cost function can be adjusted as:

J(m¯)=F(m¯)+γpoleRpole(m¯)+γwaveletEdetail(m¯).

The experiments of Fig. 5 are then repeated using the wavelet penalty. The results are illustrated in Fig. 6, where it can be seen that the wavelet penalty removes many small peaks at the bottom of the two parallel bars in Fig. 5. These results illustrate the efficiency of the wavelet penalty. Regularization invariably will have a tradeoff reducing the pattern details while often increasing the pattern errors. Since the wavelet penalty removes small assisting blocks, the distortions of the output images are increased.

 figure: Fig. 6.

Fig. 6. Left to right: desired pattern, pole-constrained optimized mask magnitude, phase and binary output pattern. sϕ_ = 2, sθ_ = 0.01, γpole,ϕ_ = 0.01, γpole,θ_ = 0.001. Wavelet regularization uses γwavelet,ϕ̲ = 0.2 and γwavelet,θ_ = 0.001.

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 figure: Fig. 7.

Fig. 7. Left to right: desired pattern, pole-constrained optimized mask magnitude, phase and binary output pattern. sϕ_ = 2, sθ_ = 0.01, γpole,ϕ_ = 0.01, γpole,θ_ = 0.001, γtv,ϕ_ = 0.3 and γtv,θ_ = 0.001.

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Another well-known penalty to remove details is total variation penalty. Generalized to the complex domain, the gradients of the total variation penalty are given as:

RTV(ϕ̲)=[QxTsgn(QxTf¯)+QyTsgn(QyTf¯)]sinϕ¯Re[(m¯z¯*)ejθ̲]12f¯,
RTV(θ̲)=[QxTsgn(QxTf¯)+QyTsgn(QyTf¯)](+cosϕ̲)Re[(m¯z¯*)ejθ̲(j)]12f¯,

where f̲ is the activation pattern and j = |j - ẕ* j| for j = 1,…, N 2. Using the total variation penalty in our approach, the results are presented in Fig. 7. Comparing the results in Figs. 5–7, the following observations can be made. Firstly, the Haar wavelet is suitable for the piecewise smooth image. Secondly, given a set error range in the attained mask patterns in all simulations tested, the Haar wavelet penalty removes more details than the total variation penalty. Finally, the Haar wavelet penalty makes the shape of the blocks constructing the mask more regular and closer to the Haar basis functions. The Haar wavelet penalty thus leads to a set of advantages that can be attributed to the rectangular shape of the Haar basis waveforms.

4.2.2. Localized wavelet penalty

In the experiment above, the wavelet penalty is applied to the entire mask pattern without any local discrimination. Thus, equal penalty was assigned to details in all regions. In practice, details may be intolerable in some special mask regions, while some details may be permissible in other regions. Because of the “localization property” of the Haar wavelet penalty, we can assign regional weights to the penalty term. Thus, a localized wavelet penalty is effective for achieving local discrimination. To this end, the cost function is adjusted as:

J(m¯i)=F(m¯i)+γpoleRpole(m¯i)+ω(i)γwaveletEdetail(m¯i),

where ω(i), i = 1,…N 2 are the weight coefficients and may be changed in different spatial regions. The experimental result using localized wavelet penalty is presented in Fig. 8. The experiment shown in Fig. 8 has the same parameters as the one shown in Fig. 6, except for the regional weights. We placed a higher cost, ω(i) = 1.6, to the gap area between the vertical bars, and we assigned lower cost, ω(i) = 0.7, to other regions. Comparing the results in Fig. 6 and Fig. 8, it is observed that the localized wavelet penalty removes more details in the gap, but tolerates slightly more details in other regions.

 figure: Fig. 8.

Fig. 8. Left to right: desired pattern, pole-constrained optimized mask magnitude, phase and binary output pattern. sϕ_ = 2, sθ_ = 0.01, γpole,ϕ_ = 0.001, γpole,θ_ = 0.0001, γwavelet,ϕ_ = 0.2 and γwavelet,θ_ = 0.001. The gap between the vertical bars has regional weight of 1.6. The other regions have regional weight of 0.7.

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It should be noted that for some special patterns, four-phase levels are not necessary to avoid the phase assignment conflict. For instance, the parallel-bar pattern in Fig. 9 can be attained by a two-phase mask. As shown in Fig. 9, the generalized PSM algorithm is also capable of designing two-phase masks. Using the localized wavelet penalty, the attained mask design is shown in Fig. 10.

 figure: Fig. 9.

Fig. 9. Left to right: desired pattern, pole-constrained optimized mask and binary output pattern. sϕ_ = 2, sθ_ = 0.01, γpole,ϕ_ = 0.001, γpole,θ_ = 0.0001, γwavelet,ϕ_ = 0.03 and γwavelet,θ_ = 0.001.

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5. Double exposure optimization method

As an alternative method to the four-phase PSM, a double-exposure optimization method can be developed to avoid the phase conflict. In this method, the photoresist layer is exposed twice, each with a two-phase mask. Assume that the two masks are M 1 and M 2. Using the parametric transformations:

m¯1k=r¯1kejθ̲1k=1+cosϕ̲1k2ejθ̲1k,k=1,...,N2,
m¯2k=r¯2kejθ̲2k=1+cosϕ̲2k2ejθ̲2k,k=1,...,N2,

The corresponding output images are:

z¯1k=11+exp[ak=1N2hik1+cosϕ̲1k2ejθ̲1k+atr],k=1,...,N2,
z¯2k=11+exp[ak=1N2hik1+cosϕ̲2k2ejθ̲2k+atr],k=1,...,N2,

The superposition of ẕ1 and ẕ2 is the final output pattern ẕ,ẕk = U(ẕ1k + ẕ2k - 1), k=1,…,N 2, where U(∙) is a unit step function. Since the step function’s derivative will introduce a Dirac impulse term that is inconvenient for further analysis, a simple approximation is given by the hyperbolic tangent function z¯k=U(z¯1k+z¯2k1)12[tanh(z¯1k+z¯2k1)+1],k=1,...,N2. The cost function is then calculated as:

F=F(θ̲1,ϕ̲1,θ̲2,ϕ̲2)=z¯*z¯22=i=1N2{zi*12[tanh(z¯1i+z¯2i1)+1]}2.
 figure: Fig. 10.

Fig. 10. Left to right: desired pattern, pole-constrained optimized mask and binary output pattern. sϕ_ = 2, sθ_ = 0.01, γpole,ϕ_ = 0.001, γpole,θ_ = 0.0001, γwavelet,ϕ_ = 0.03 and γwavelet,θ_ = 0.001. The regions for the first and the fourth bars have regional weight of 0.8. The regions for the second and the third bars have regional weight of 1.1. The other regions have regional weight of 0.7.

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Therefore, the gradients ∇F θ̲1, ∇F ϕ̲1, ∇F θ̲2 and ∇F ϕ̲2 can be calculated as:

Fθ̲p=a×+cosϕ̲p2sinθ̲p{HT[(z¯*z¯)sech2(z¯1+z¯2)z¯p
(p)H(m¯pR)T(m¯,p)]}
a×+cosϕ̲p2cosθ̲p{HT[(z¯*z¯)sech2(z¯1+z¯2)z¯p
(z¯)pH(m¯pI)T(m¯,p)]},
Fϕ̲p=a2×sinϕ̲pcosθ̲p{HT[(z¯*z¯)sech2(z¯1+z¯2)z¯p
(p)H(m¯pR)T(m¯,p)]}
+a2×sinϕ̲psinθ̲p{HT[(z¯*z¯)sech2(z¯1+z¯2)z¯p
(z¯)pH(m¯pI)T(m¯,p)]},

where p=1or2andT(m¯,p)=[H(m¯pR)2+H(m¯pI)2]12. Both pole penalty and wavelet penalty can be applied to the double-exposure optimization method. The experiment shown in Fig. 2 is repeated in Fig. 11, where a double-exposure method is used. It is obvious that the pattern error is effectively reduced. The double-exposure optimization method is indeed effective, but requires more complicated processing and longer fabrication time. As mentioned in the introduction, independently from our work, Poonawala and Milanfar have developed a double-exposure inverse lithography method. Although similar in concept, these two approaches are different. Their method optimizes the inverse lithography process prior to a threshold operator, whereas the double-exposure method presented in this paper encompasses the threshold operator that models the photoresist development stage.

 figure: Fig. 11.

Fig. 11. Left to right: mask for first exposure, mask for second exposure and the output pattern. s ϕ̲1 = 4, s θ̲1 = 0.01, s ϕ̲2 = 4, s θ̲2 = 0.01, γ pole,ϕ_1 = 0.015, γ pole,θ_1 = 0.001, γ wavelet,ϕ_1 = 0.5, γ wavelet,θ_1 = 0.003, γ pole,ϕ_2 = 0.015, γ pole,θ_2 = 0.001, γ wavelet,ϕ_2 = 0.5 and γ wavelet,θ_2 = 0.003.

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6. Conclusion

This paper studies and models the optical system for generalized inverse lithography. A pixel-based mask representation is used and the MSE between the output pattern and the desired pattern is used to measure the pattern error. The cost function approach used enables the algorithm to search for a solution in the entire complex plane, and as such avoiding possible phase conflicts. In order to control the amount of details, the pole penalty terms are used to curb the transmission regions on the mask pattern and constrain the phases to several prescribed discrete levels. Another contribution of this paper is the introduction of the wavelet penalty used to reduce the mask complexity. Because of the “localization property”, regional weighting can be applied to different areas on the mask pattern. Finally, a double-exposure optimization method is introduced. It is capable to avoid the phase conflict and results in much less pattern error as well.

References and links

1. S. V. G. J. Schneider, J. Murakowski, and D. W. Prather, “Combination lithography for photoniccrystal circuits,” of Vacuum Science and Technology B 22(1), 146–151 (2004). [CrossRef]  

2. J. M. M. J. M. P. Yao, G. J. Schneider, and D. W. Prather, “Micro/nano lithography realized by chemical printing,” in Proceedings of SPIE - The International Society for Optical Engineering, vol. 6151I of Emerging Lithographic Technologies X, p. 61511N (2006).

3. D. W. P. E. D. W. P. Yao, G. J. Schneider, and D. J. O’Brien, “Fabrication of three-dimensional photonic crystals with multilayer photolithography,” Optics Express 13, 2370–2376 (2005). [CrossRef]   [PubMed]  

4. B. L. M. J. M. D. W. P. E. D. W. P. Yao, G. J. Schneider, and D. J. O’Brien, “Multilayer three-dimensional photolithography with traditional planar method,” Applied Physics Letters 85, 2920–2922 (2004). [CrossRef]  

5. A. K. Wong, Resolution enhancement techniques, vol. 1 (SPIE Press, 2001). [CrossRef]  

6. S. A. Campbell, The science and engineering of microelectronic fabrication, 2nd ed. (Publishing House of Electronics Industry, 2003).

7. F. Schellenberg, “Resolution enhancement technology: The past, the present, and extensions for the future, Optical Microlithography,” Proc. SPIE 5377, 1–20 (2004). [CrossRef]  

8. F. Schellenberg, Resolution enhancement techniques in optical lithography (SPIE Press, 2004).

9. A. W. M. L. W. L. L. Liebmann, S. Mansfield, and T. Dunham, “TCAD development for lithography resolution enhancement,” IBM Journal of Research and Development pp. 651–665 (2001). [CrossRef]  

10. N. S. V. M. D. Levenson and R. A. Simpson, “Improving resolution in photolithography with a phase-shifting mask,” IEEE Trans. Electron Devices ED-29, 1828–1836 (1982). [CrossRef]  

11. B. S. S. Sherif and R. Leone, “Binary image synthesis using mixed integer programming,” IEEE Transactions on Image Processing 4(9), 1252–1257 (1995). [CrossRef]   [PubMed]  

12. Y. Liu and A. Zakhor, “Binary and phase shifting mask design for optical lithography,” IEEE Transactions on Semiconductor Manufacturing 5(2) (1992). [CrossRef]  

13. Y. C. Pati and T. Kailath, “Phase-shifting masks for microlithography: Automated design and mask requirements,” Optical Society of America 11 (1994).

14. T. F. B. T. A. Erdmann, R. Farkas, and G. Kokai, “Towards automatic mask and source optimization for optical lithography,” Optical Microlithography, Proc. SPIE 5377, 646–657 (2004).

15. Y. L. L. Pang and D. Abrams, “Inverse lithography technology (ILT): What is the impact to the photomask industry?” Proc. SPIE (2006). [CrossRef]  

16. Y. Granik, “Illuminator optimization methods in microlithography,” Optical Microlithography Proc. SPIE 5754, 217–229 (2005).

17. A. Poonawala and P. Milanfar, “OPC and PSM design using inverse lithography: A non-linear optimization approach,” in Proceedings of the SPIE, vol. 6154, pp. 1159–1172 (San Jose, CA, 2006).

18. C. Vogel, Computational methods for inverse problems (SIAM Press, 2002). [CrossRef]  

19. X. Ma and G. R. Arce, “Generalized inverse lithography methods for phase-shifting mask design,” in Proceedings of SPIE, vol. 65200U (2007).

20. A. Poonawala and P. Milanfar, “Double Exposure Mask Synthesis using Inverse Lithography,” submitted to Journal of Microlithography, Microfabrication, and Microsystems.

21. M. Born and E. Wolfe, Principles of optics (Cambridge University Press, 1999).

22. N. Cobb and A. Zakhor, “Fast sparse aerial image calculation for OPC,” BACUS Symposium on Photomask Technology, Proc. SPIE 2440, 313–327 (1995).

References

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  1. S. V. G. J. Schneider, J. Murakowski, and D. W. Prather, “Combination lithography for photoniccrystal circuits,” of Vacuum Science and Technology B 22(1), 146–151 (2004).
    [Crossref]
  2. J. M. M. J. M. P. Yao, G. J. Schneider, and D. W. Prather, “Micro/nano lithography realized by chemical printing,” in Proceedings of SPIE - The International Society for Optical Engineering, vol. 6151I of Emerging Lithographic Technologies X, p. 61511N (2006).
  3. D. W. P. E. D. W. P. Yao, G. J. Schneider, and D. J. O’Brien, “Fabrication of three-dimensional photonic crystals with multilayer photolithography,” Optics Express 13, 2370–2376 (2005).
    [Crossref] [PubMed]
  4. B. L. M. J. M. D. W. P. E. D. W. P. Yao, G. J. Schneider, and D. J. O’Brien, “Multilayer three-dimensional photolithography with traditional planar method,” Applied Physics Letters 85, 2920–2922 (2004).
    [Crossref]
  5. A. K. Wong, Resolution enhancement techniques, vol. 1 (SPIE Press, 2001).
    [Crossref]
  6. S. A. Campbell, The science and engineering of microelectronic fabrication, 2nd ed. (Publishing House of Electronics Industry, 2003).
  7. F. Schellenberg, “Resolution enhancement technology: The past, the present, and extensions for the future, Optical Microlithography,” Proc. SPIE 5377, 1–20 (2004).
    [Crossref]
  8. F. Schellenberg, Resolution enhancement techniques in optical lithography (SPIE Press, 2004).
  9. A. W. M. L. W. L. L. Liebmann, S. Mansfield, and T. Dunham, “TCAD development for lithography resolution enhancement,” IBM Journal of Research and Development pp. 651–665 (2001).
    [Crossref]
  10. N. S. V. M. D. Levenson and R. A. Simpson, “Improving resolution in photolithography with a phase-shifting mask,” IEEE Trans. Electron Devices ED-29, 1828–1836 (1982).
    [Crossref]
  11. B. S. S. Sherif and R. Leone, “Binary image synthesis using mixed integer programming,” IEEE Transactions on Image Processing 4(9), 1252–1257 (1995).
    [Crossref] [PubMed]
  12. Y. Liu and A. Zakhor, “Binary and phase shifting mask design for optical lithography,” IEEE Transactions on Semiconductor Manufacturing 5(2) (1992).
    [Crossref]
  13. Y. C. Pati and T. Kailath, “Phase-shifting masks for microlithography: Automated design and mask requirements,” Optical Society of America 11 (1994).
  14. T. F. B. T. A. Erdmann, R. Farkas, and G. Kokai, “Towards automatic mask and source optimization for optical lithography,” Optical Microlithography, Proc. SPIE 5377, 646–657 (2004).
  15. Y. L. L. Pang and D. Abrams, “Inverse lithography technology (ILT): What is the impact to the photomask industry?” Proc. SPIE (2006).
    [Crossref]
  16. Y. Granik, “Illuminator optimization methods in microlithography,” Optical Microlithography Proc. SPIE 5754, 217–229 (2005).
  17. A. Poonawala and P. Milanfar, “OPC and PSM design using inverse lithography: A non-linear optimization approach,” in Proceedings of the SPIE, vol. 6154, pp. 1159–1172 (San Jose, CA, 2006).
  18. C. Vogel, Computational methods for inverse problems (SIAM Press, 2002).
    [Crossref]
  19. X. Ma and G. R. Arce, “Generalized inverse lithography methods for phase-shifting mask design,” in Proceedings of SPIE, vol. 65200U (2007).
  20. A. Poonawala and P. Milanfar, “Double Exposure Mask Synthesis using Inverse Lithography,” submitted to Journal of Microlithography, Microfabrication, and Microsystems.
  21. M. Born and E. Wolfe, Principles of optics (Cambridge University Press, 1999).
  22. N. Cobb and A. Zakhor, “Fast sparse aerial image calculation for OPC,” BACUS Symposium on Photomask Technology, Proc. SPIE 2440, 313–327 (1995).

2005 (2)

D. W. P. E. D. W. P. Yao, G. J. Schneider, and D. J. O’Brien, “Fabrication of three-dimensional photonic crystals with multilayer photolithography,” Optics Express 13, 2370–2376 (2005).
[Crossref] [PubMed]

Y. Granik, “Illuminator optimization methods in microlithography,” Optical Microlithography Proc. SPIE 5754, 217–229 (2005).

2004 (4)

T. F. B. T. A. Erdmann, R. Farkas, and G. Kokai, “Towards automatic mask and source optimization for optical lithography,” Optical Microlithography, Proc. SPIE 5377, 646–657 (2004).

B. L. M. J. M. D. W. P. E. D. W. P. Yao, G. J. Schneider, and D. J. O’Brien, “Multilayer three-dimensional photolithography with traditional planar method,” Applied Physics Letters 85, 2920–2922 (2004).
[Crossref]

F. Schellenberg, “Resolution enhancement technology: The past, the present, and extensions for the future, Optical Microlithography,” Proc. SPIE 5377, 1–20 (2004).
[Crossref]

S. V. G. J. Schneider, J. Murakowski, and D. W. Prather, “Combination lithography for photoniccrystal circuits,” of Vacuum Science and Technology B 22(1), 146–151 (2004).
[Crossref]

2001 (1)

A. W. M. L. W. L. L. Liebmann, S. Mansfield, and T. Dunham, “TCAD development for lithography resolution enhancement,” IBM Journal of Research and Development pp. 651–665 (2001).
[Crossref]

1995 (2)

B. S. S. Sherif and R. Leone, “Binary image synthesis using mixed integer programming,” IEEE Transactions on Image Processing 4(9), 1252–1257 (1995).
[Crossref] [PubMed]

N. Cobb and A. Zakhor, “Fast sparse aerial image calculation for OPC,” BACUS Symposium on Photomask Technology, Proc. SPIE 2440, 313–327 (1995).

1994 (1)

Y. C. Pati and T. Kailath, “Phase-shifting masks for microlithography: Automated design and mask requirements,” Optical Society of America 11 (1994).

1992 (1)

Y. Liu and A. Zakhor, “Binary and phase shifting mask design for optical lithography,” IEEE Transactions on Semiconductor Manufacturing 5(2) (1992).
[Crossref]

1982 (1)

N. S. V. M. D. Levenson and R. A. Simpson, “Improving resolution in photolithography with a phase-shifting mask,” IEEE Trans. Electron Devices ED-29, 1828–1836 (1982).
[Crossref]

Abrams, D.

Y. L. L. Pang and D. Abrams, “Inverse lithography technology (ILT): What is the impact to the photomask industry?” Proc. SPIE (2006).
[Crossref]

Arce, G. R.

X. Ma and G. R. Arce, “Generalized inverse lithography methods for phase-shifting mask design,” in Proceedings of SPIE, vol. 65200U (2007).

Born, M.

M. Born and E. Wolfe, Principles of optics (Cambridge University Press, 1999).

Campbell, S. A.

S. A. Campbell, The science and engineering of microelectronic fabrication, 2nd ed. (Publishing House of Electronics Industry, 2003).

Cobb, N.

N. Cobb and A. Zakhor, “Fast sparse aerial image calculation for OPC,” BACUS Symposium on Photomask Technology, Proc. SPIE 2440, 313–327 (1995).

Dunham, T.

A. W. M. L. W. L. L. Liebmann, S. Mansfield, and T. Dunham, “TCAD development for lithography resolution enhancement,” IBM Journal of Research and Development pp. 651–665 (2001).
[Crossref]

Erdmann, T. F. B. T. A.

T. F. B. T. A. Erdmann, R. Farkas, and G. Kokai, “Towards automatic mask and source optimization for optical lithography,” Optical Microlithography, Proc. SPIE 5377, 646–657 (2004).

Farkas, R.

T. F. B. T. A. Erdmann, R. Farkas, and G. Kokai, “Towards automatic mask and source optimization for optical lithography,” Optical Microlithography, Proc. SPIE 5377, 646–657 (2004).

Granik, Y.

Y. Granik, “Illuminator optimization methods in microlithography,” Optical Microlithography Proc. SPIE 5754, 217–229 (2005).

Kailath, T.

Y. C. Pati and T. Kailath, “Phase-shifting masks for microlithography: Automated design and mask requirements,” Optical Society of America 11 (1994).

Kokai, G.

T. F. B. T. A. Erdmann, R. Farkas, and G. Kokai, “Towards automatic mask and source optimization for optical lithography,” Optical Microlithography, Proc. SPIE 5377, 646–657 (2004).

Leone, R.

B. S. S. Sherif and R. Leone, “Binary image synthesis using mixed integer programming,” IEEE Transactions on Image Processing 4(9), 1252–1257 (1995).
[Crossref] [PubMed]

Levenson, N. S. V. M. D.

N. S. V. M. D. Levenson and R. A. Simpson, “Improving resolution in photolithography with a phase-shifting mask,” IEEE Trans. Electron Devices ED-29, 1828–1836 (1982).
[Crossref]

Liebmann, A. W. M. L. W. L. L.

A. W. M. L. W. L. L. Liebmann, S. Mansfield, and T. Dunham, “TCAD development for lithography resolution enhancement,” IBM Journal of Research and Development pp. 651–665 (2001).
[Crossref]

Liu, Y.

Y. Liu and A. Zakhor, “Binary and phase shifting mask design for optical lithography,” IEEE Transactions on Semiconductor Manufacturing 5(2) (1992).
[Crossref]

Ma, X.

X. Ma and G. R. Arce, “Generalized inverse lithography methods for phase-shifting mask design,” in Proceedings of SPIE, vol. 65200U (2007).

Mansfield, S.

A. W. M. L. W. L. L. Liebmann, S. Mansfield, and T. Dunham, “TCAD development for lithography resolution enhancement,” IBM Journal of Research and Development pp. 651–665 (2001).
[Crossref]

Milanfar, P.

A. Poonawala and P. Milanfar, “OPC and PSM design using inverse lithography: A non-linear optimization approach,” in Proceedings of the SPIE, vol. 6154, pp. 1159–1172 (San Jose, CA, 2006).

A. Poonawala and P. Milanfar, “Double Exposure Mask Synthesis using Inverse Lithography,” submitted to Journal of Microlithography, Microfabrication, and Microsystems.

Murakowski, J.

S. V. G. J. Schneider, J. Murakowski, and D. W. Prather, “Combination lithography for photoniccrystal circuits,” of Vacuum Science and Technology B 22(1), 146–151 (2004).
[Crossref]

O’Brien, D. J.

D. W. P. E. D. W. P. Yao, G. J. Schneider, and D. J. O’Brien, “Fabrication of three-dimensional photonic crystals with multilayer photolithography,” Optics Express 13, 2370–2376 (2005).
[Crossref] [PubMed]

B. L. M. J. M. D. W. P. E. D. W. P. Yao, G. J. Schneider, and D. J. O’Brien, “Multilayer three-dimensional photolithography with traditional planar method,” Applied Physics Letters 85, 2920–2922 (2004).
[Crossref]

Pang, Y. L. L.

Y. L. L. Pang and D. Abrams, “Inverse lithography technology (ILT): What is the impact to the photomask industry?” Proc. SPIE (2006).
[Crossref]

Pati, Y. C.

Y. C. Pati and T. Kailath, “Phase-shifting masks for microlithography: Automated design and mask requirements,” Optical Society of America 11 (1994).

Poonawala, A.

A. Poonawala and P. Milanfar, “OPC and PSM design using inverse lithography: A non-linear optimization approach,” in Proceedings of the SPIE, vol. 6154, pp. 1159–1172 (San Jose, CA, 2006).

A. Poonawala and P. Milanfar, “Double Exposure Mask Synthesis using Inverse Lithography,” submitted to Journal of Microlithography, Microfabrication, and Microsystems.

Prather, D. W.

S. V. G. J. Schneider, J. Murakowski, and D. W. Prather, “Combination lithography for photoniccrystal circuits,” of Vacuum Science and Technology B 22(1), 146–151 (2004).
[Crossref]

J. M. M. J. M. P. Yao, G. J. Schneider, and D. W. Prather, “Micro/nano lithography realized by chemical printing,” in Proceedings of SPIE - The International Society for Optical Engineering, vol. 6151I of Emerging Lithographic Technologies X, p. 61511N (2006).

Schellenberg, F.

F. Schellenberg, “Resolution enhancement technology: The past, the present, and extensions for the future, Optical Microlithography,” Proc. SPIE 5377, 1–20 (2004).
[Crossref]

F. Schellenberg, Resolution enhancement techniques in optical lithography (SPIE Press, 2004).

Schneider, G. J.

D. W. P. E. D. W. P. Yao, G. J. Schneider, and D. J. O’Brien, “Fabrication of three-dimensional photonic crystals with multilayer photolithography,” Optics Express 13, 2370–2376 (2005).
[Crossref] [PubMed]

B. L. M. J. M. D. W. P. E. D. W. P. Yao, G. J. Schneider, and D. J. O’Brien, “Multilayer three-dimensional photolithography with traditional planar method,” Applied Physics Letters 85, 2920–2922 (2004).
[Crossref]

J. M. M. J. M. P. Yao, G. J. Schneider, and D. W. Prather, “Micro/nano lithography realized by chemical printing,” in Proceedings of SPIE - The International Society for Optical Engineering, vol. 6151I of Emerging Lithographic Technologies X, p. 61511N (2006).

Schneider, S. V. G. J.

S. V. G. J. Schneider, J. Murakowski, and D. W. Prather, “Combination lithography for photoniccrystal circuits,” of Vacuum Science and Technology B 22(1), 146–151 (2004).
[Crossref]

Sherif, B. S. S.

B. S. S. Sherif and R. Leone, “Binary image synthesis using mixed integer programming,” IEEE Transactions on Image Processing 4(9), 1252–1257 (1995).
[Crossref] [PubMed]

Simpson, R. A.

N. S. V. M. D. Levenson and R. A. Simpson, “Improving resolution in photolithography with a phase-shifting mask,” IEEE Trans. Electron Devices ED-29, 1828–1836 (1982).
[Crossref]

Vogel, C.

C. Vogel, Computational methods for inverse problems (SIAM Press, 2002).
[Crossref]

Wolfe, E.

M. Born and E. Wolfe, Principles of optics (Cambridge University Press, 1999).

Wong, A. K.

A. K. Wong, Resolution enhancement techniques, vol. 1 (SPIE Press, 2001).
[Crossref]

Yao, B. L. M. J. M. D. W. P. E. D. W. P.

B. L. M. J. M. D. W. P. E. D. W. P. Yao, G. J. Schneider, and D. J. O’Brien, “Multilayer three-dimensional photolithography with traditional planar method,” Applied Physics Letters 85, 2920–2922 (2004).
[Crossref]

Yao, D. W. P. E. D. W. P.

D. W. P. E. D. W. P. Yao, G. J. Schneider, and D. J. O’Brien, “Fabrication of three-dimensional photonic crystals with multilayer photolithography,” Optics Express 13, 2370–2376 (2005).
[Crossref] [PubMed]

Yao, J. M. M. J. M. P.

J. M. M. J. M. P. Yao, G. J. Schneider, and D. W. Prather, “Micro/nano lithography realized by chemical printing,” in Proceedings of SPIE - The International Society for Optical Engineering, vol. 6151I of Emerging Lithographic Technologies X, p. 61511N (2006).

Zakhor, A.

N. Cobb and A. Zakhor, “Fast sparse aerial image calculation for OPC,” BACUS Symposium on Photomask Technology, Proc. SPIE 2440, 313–327 (1995).

Y. Liu and A. Zakhor, “Binary and phase shifting mask design for optical lithography,” IEEE Transactions on Semiconductor Manufacturing 5(2) (1992).
[Crossref]

Applied Physics Letters (1)

B. L. M. J. M. D. W. P. E. D. W. P. Yao, G. J. Schneider, and D. J. O’Brien, “Multilayer three-dimensional photolithography with traditional planar method,” Applied Physics Letters 85, 2920–2922 (2004).
[Crossref]

BACUS Symposium on Photomask Technology, Proc. SPIE (1)

N. Cobb and A. Zakhor, “Fast sparse aerial image calculation for OPC,” BACUS Symposium on Photomask Technology, Proc. SPIE 2440, 313–327 (1995).

IBM Journal of Research and Development (1)

A. W. M. L. W. L. L. Liebmann, S. Mansfield, and T. Dunham, “TCAD development for lithography resolution enhancement,” IBM Journal of Research and Development pp. 651–665 (2001).
[Crossref]

IEEE Trans. Electron Devices (1)

N. S. V. M. D. Levenson and R. A. Simpson, “Improving resolution in photolithography with a phase-shifting mask,” IEEE Trans. Electron Devices ED-29, 1828–1836 (1982).
[Crossref]

IEEE Transactions on Image Processing (1)

B. S. S. Sherif and R. Leone, “Binary image synthesis using mixed integer programming,” IEEE Transactions on Image Processing 4(9), 1252–1257 (1995).
[Crossref] [PubMed]

IEEE Transactions on Semiconductor Manufacturing (1)

Y. Liu and A. Zakhor, “Binary and phase shifting mask design for optical lithography,” IEEE Transactions on Semiconductor Manufacturing 5(2) (1992).
[Crossref]

Optical Microlithography Proc. SPIE (1)

Y. Granik, “Illuminator optimization methods in microlithography,” Optical Microlithography Proc. SPIE 5754, 217–229 (2005).

Optical Microlithography, Proc. SPIE (1)

T. F. B. T. A. Erdmann, R. Farkas, and G. Kokai, “Towards automatic mask and source optimization for optical lithography,” Optical Microlithography, Proc. SPIE 5377, 646–657 (2004).

Optical Society of America (1)

Y. C. Pati and T. Kailath, “Phase-shifting masks for microlithography: Automated design and mask requirements,” Optical Society of America 11 (1994).

Optics Express (1)

D. W. P. E. D. W. P. Yao, G. J. Schneider, and D. J. O’Brien, “Fabrication of three-dimensional photonic crystals with multilayer photolithography,” Optics Express 13, 2370–2376 (2005).
[Crossref] [PubMed]

Proc. SPIE (1)

F. Schellenberg, “Resolution enhancement technology: The past, the present, and extensions for the future, Optical Microlithography,” Proc. SPIE 5377, 1–20 (2004).
[Crossref]

Vacuum Science and Technology B (1)

S. V. G. J. Schneider, J. Murakowski, and D. W. Prather, “Combination lithography for photoniccrystal circuits,” of Vacuum Science and Technology B 22(1), 146–151 (2004).
[Crossref]

Other (10)

J. M. M. J. M. P. Yao, G. J. Schneider, and D. W. Prather, “Micro/nano lithography realized by chemical printing,” in Proceedings of SPIE - The International Society for Optical Engineering, vol. 6151I of Emerging Lithographic Technologies X, p. 61511N (2006).

A. K. Wong, Resolution enhancement techniques, vol. 1 (SPIE Press, 2001).
[Crossref]

S. A. Campbell, The science and engineering of microelectronic fabrication, 2nd ed. (Publishing House of Electronics Industry, 2003).

F. Schellenberg, Resolution enhancement techniques in optical lithography (SPIE Press, 2004).

Y. L. L. Pang and D. Abrams, “Inverse lithography technology (ILT): What is the impact to the photomask industry?” Proc. SPIE (2006).
[Crossref]

A. Poonawala and P. Milanfar, “OPC and PSM design using inverse lithography: A non-linear optimization approach,” in Proceedings of the SPIE, vol. 6154, pp. 1159–1172 (San Jose, CA, 2006).

C. Vogel, Computational methods for inverse problems (SIAM Press, 2002).
[Crossref]

X. Ma and G. R. Arce, “Generalized inverse lithography methods for phase-shifting mask design,” in Proceedings of SPIE, vol. 65200U (2007).

A. Poonawala and P. Milanfar, “Double Exposure Mask Synthesis using Inverse Lithography,” submitted to Journal of Microlithography, Microfabrication, and Microsystems.

M. Born and E. Wolfe, Principles of optics (Cambridge University Press, 1999).

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Figures (11)

Fig. 1.
Fig. 1. Approximated forward process model. (Ref. [17], Fig. 1).
Fig. 2.
Fig. 2. Top row (input masks), left to right: desired pattern, magnitude of the optimized complex-valued mask and optimized pole-constrained mask obtained using a threshold tm . The middle row indicates the phases of the optimized complex-valued mask. The bottom row indicates the corresponding binary output patterns. The parameters used in the simulation are a = 80, tr = 0.5, tm = 0.5, 11 × 11 Gaussian low pass filter with k = 14, s ϕ _ = 2 and s θ _ = 0.01. Green, red and blue represent 0, 1 and -1 respectively in the top and bottom rows; Dark blue, light blue, yellow and red represent π 4 , 3 π 4 , 5 π 4 and 7 π 4 respectively in the middle row.
Fig. 3.
Fig. 3. Left to right: desired pattern, pole-constraied optimized mask and binary output pattern. s ϕ _ = 2, s θ _ = 0.01. Red and blue represent 0 and π phases respectively.
Fig. 4.
Fig. 4. Phase penalty for two-phase levels and four-phase levels
Fig. 5.
Fig. 5. Left to right: desired pattern, pole-constrained optimized mask magnitude, pole-constrained optimized mask phases (Dark blue is π 4 , light blue is 3 π 4 , yellow is 5 π 4 and red is 7 π 4 and binary output pattern. s ϕ _ = 2, s θ _ = 0.01, γpole, ϕ _ = 0.001 and γpole, θ _ = 0.0001.
Fig. 6.
Fig. 6. Left to right: desired pattern, pole-constrained optimized mask magnitude, phase and binary output pattern. s ϕ _ = 2, s θ _ = 0.01, γpole, ϕ _ = 0.01, γpole, θ _ = 0.001. Wavelet regularization uses γwavelet,ϕ̲ = 0.2 and γwavelet, θ _ = 0.001.
Fig. 7.
Fig. 7. Left to right: desired pattern, pole-constrained optimized mask magnitude, phase and binary output pattern. s ϕ _ = 2, s θ _ = 0.01, γpole, ϕ _ = 0.01, γpole, θ _ = 0.001, γtv, ϕ _ = 0.3 and γtv, θ _ = 0.001.
Fig. 8.
Fig. 8. Left to right: desired pattern, pole-constrained optimized mask magnitude, phase and binary output pattern. s ϕ _ = 2, s θ _ = 0.01, γpole, ϕ _ = 0.001, γpole, θ _ = 0.0001, γwavelet, ϕ _ = 0.2 and γwavelet, θ _ = 0.001. The gap between the vertical bars has regional weight of 1.6. The other regions have regional weight of 0.7.
Fig. 9.
Fig. 9. Left to right: desired pattern, pole-constrained optimized mask and binary output pattern. s ϕ _ = 2, s θ _ = 0.01, γpole, ϕ _ = 0.001, γpole, θ _ = 0.0001, γwavelet, ϕ _ = 0.03 and γwavelet, θ _ = 0.001.
Fig. 10.
Fig. 10. Left to right: desired pattern, pole-constrained optimized mask and binary output pattern. s ϕ _ = 2, s θ _ = 0.01, γpole, ϕ _ = 0.001, γpole, θ _ = 0.0001, γwavelet, ϕ _ = 0.03 and γwavelet, θ _ = 0.001. The regions for the first and the fourth bars have regional weight of 0.8. The regions for the second and the third bars have regional weight of 1.1. The other regions have regional weight of 0.7.
Fig. 11.
Fig. 11. Left to right: mask for first exposure, mask for second exposure and the output pattern. s ϕ̲1 = 4, s θ̲1 = 0.01, s ϕ̲2 = 4, s θ̲2 = 0.01, γ pole, ϕ _ 1 = 0.015, γ pole, θ _ 1 = 0.001, γ wavelet, ϕ _ 1 = 0.5, γ wavelet, θ _ 1 = 0.003, γ pole, ϕ _ 2 = 0.015, γ pole, θ _ 2 = 0.001, γ wavelet, ϕ _ 2 = 0.5 and γ wavelet, θ _ 2 = 0.003.

Equations (52)

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M ̂ ( x , y ) = arg min M ( x , y ) C N × N d ( T { M ( x , y ) } , Z * ( x , y ) ) .
sig ( x ) = 1 1 + exp [ a ( x t r ) ] ,
M ̂ = arg min M d ( sig { H { M } } , Z * ) .
i = 1 1 + exp [ a j = 1 N 2 h ij j + a t r ] , i = 1 , . . . N 2 ,
F ( m ¯ ) = z ¯ * z ¯ 2 2 = i = 1 N 2 ( z i * z i ) 2 = i = 1 N 2 ( z i * 1 1 + exp [ a j = 1 N 2 h ij m ¯ j + a t r ] ) 2 .
m ¯ k = r ¯ k e j θ ¯ k , k = 1 , . . . , N 2 ,
r ¯ k = 1 + cos ϕ ¯ k 2 , k = 1 , . . . , N 2 ,
m ¯ k = 1 + cos ϕ ¯ k 2 e j θ ¯ k , k = 1 , . . . , N 2 .
( θ ̂ ¯ , ϕ ̂ ¯ ) = arg min ( θ̲ , ϕ̲ ) { F ( θ̲ , ϕ̲ ) } ,
F ( θ̲ , ϕ̲ ) = z ¯ * z ¯ 2 2 = i = 1 N 2 ( z i * z i ) 2
= i = 1 N 2 ( z i * 1 1 + exp [ a ( k = 1 N 2 h i k 1 + cos ϕ̲ k 2 e j θ̱ k ) 2 + a t r ] ) 2 .
F θ̲ ϕ̲ θ̲ = d ¯ θ̲ = 2 a × + cos ϕ̲ 2 sin θ̲ { H T [ ( z ¯ * z ¯ ) z ¯ ( z ¯ ) H ( m ¯ R ) T ( m ¯ ) ] }
2 a × + cos ϕ̲ 2 cos θ ¯ { H T [ ( z ¯ * z ¯ ) z̲⊙ ( z ¯ ) H ( m ¯ I ) T ( m ¯ ) ] } ,
F θ̲ ϕ̲ ϕ̲ = d ¯ ϕ̲ = a × sin ϕ̲ cos θ̲ { H T [ ( z ¯ * z ¯ ) ( z ¯ ) H ( m ¯ R ) T ( m ¯ ) ] }
+ a × sin ϕ̲ sin θ̲ { H T [ ( z ¯ * z ¯ ) z̲⊙ ( z ¯ ) H ( m ¯ I ) T ( m ¯ ) ] } ,
θ̲ k + 1 = θ̲ k s θ̲ d ¯ θ̲ k ,
ϕ̲ k + 1 = ϕ̲ k s ϕ̲ d ¯ ϕ̲ k ,
E = i = 1 N 2 z ¯ i * z ¯ b i = i = 1 N 2 z ¯ i * Γ H m ¯ p i .
m ̂ ¯ = arg min m ̂ ¯ { F ( m ¯ ) + γ R ( m ¯ ) } ,
R pole ( m ¯ ) = i = 1 N 2 i 2 = T .
R pole ( θ̲ ) = i = 1 N 2 [ sin ( 2 θ̲ i π 2 ) + 1 ] 2 = [ sin ( 2 θ̲ π 2 ) + ] T [ sin ( 2 θ̲ π 2 ) + ] .
R pole ( θ̲ ) = 4 [ sin ( 2 θ̲ π 2 ) + ] T cos ( 2 θ̲ π 2 ) .
R pole ( θ̲ ) = i = 1 N 2 [ sin ( 4 θ̲ i 3 π 2 ) + 1 ] 2 = [ sin ( 4 θ̲ i 3 π 2 ) + ] T [ sin ( 4 θ̲ i 3 π 2 ) + ] .
r ( θ̲ i ) = [ sin ( 4 θ̲ i 3 π 2 ) + 1 ] 2 , i = 1 , . . . , N 2
R pole ( θ̲ ) = 8 [ sin ( 4 θ̲ 3 π 2 ) + ] T cos ( 4 θ̲ 3 π 2 ) .
a i j = m ( 2 ( i 1 ) + 1 ) ( 2 ( j 1 ) + 1 ) + m ( 2 ( i 1 ) + 1 ) ( 2 ( j 1 ) + 2 ) + m ( 2 ( i 1 ) + 2 ) ( 2 ( j 1 ) + 1 ) + m ( 2 ( i 1 ) + 2 ) ( 2 ( j 1 ) + 2 ) ,
h i j = m ( 2 ( i 1 ) + 1 ) ( 2 ( j 1 ) + 1 ) m ( 2 ( i 1 ) + 1 ) ( 2 ( j 1 ) + 2 ) + m ( 2 ( i 1 ) + 2 ) ( 2 ( j 1 ) + 1 ) m ( 2 ( i 1 ) + 2 ) ( 2 ( j 1 ) + 2 ) ,
v i j = m ( 2 ( i 1 ) + 1 ) ( 2 ( j 1 ) + 1 ) + m ( 2 ( i 1 ) + 1 ) ( 2 ( j 1 ) + 2 ) m ( 2 ( i 1 ) + 2 ) ( 2 ( j 1 ) + 1 ) m ( 2 ( i 1 ) + 2 ) ( 2 ( j 1 ) + 2 ) ,
d i j = m ( 2 ( i 1 ) + 1 ) ( 2 ( j 1 ) + 1 ) m ( 2 ( i 1 ) + 1 ) ( 2 ( j 1 ) + 2 ) m ( 2 ( i 1 ) + 2 ) ( 2 ( j 1 ) + 1 ) + m ( 2 ( i 1 ) + 2 ) ( 2 ( j 1 ) + 2 ) ,
E detail = i = 1 N 2 j = 1 N 2 ( h i j h i j * + v i j v i j * + d i j d i j * ) .
E detail ϕ̲ ( 2 ( i 1 ) + p ) ( 2 ( j 1 ) + q ) = sin ϕ̲ ( 2 ( i 1 ) + p ) ( 2 ( j 1 ) + q ) × Re [ e j θ̲ ( 2 ( i 1 ) + p ) ( 2 ( j 1 ) + q ) × ( 3 m ¯ ( 2 ( i 1 ) + p ) ( 2 ( j 1 ) + q )
m ¯ ( 2 ( i 1 ) + p 1 ) ( 2 j 1 ) + q ) ( 2 ( i 1 ) + p ) ( 2 ( j 1 ) + q 1 ) ( 2 ( i 1 ) + p 1 ) ( 2 ( j 1 ) + q 1 ) ) ] ,
E detail θ̲ ( 2 ( i 1 ) + p ) ( 2 ( j 1 ) + q ) = ( 1 + cos ϕ̲ ( 2 ( i 1 ) + p ) ( 2 ( j 1 ) + q ) ) × R e [ ( j ) e j θ̲ ( 2 ( i 1 ) + p ) ( 2 ( j 1 ) + q )
× ( 3 m ¯ ( 2 ( i 1 ) + p ) ( 2 ( j 1 ) + q ) ( 2 ( i 1 ) + p 1 ) ( 2 ( j 1 ) + q ) m ¯ ( 2 ( i 1 ) + p ) ( 2 ( j 1 ) + q 1 )
m ¯ ( 2 ( i 1 ) + p 1 ) ( 2 ( j 1 ) + q 1 ) ) ] ,
J ( m ¯ ) = F ( m ¯ ) + γ pole R pole ( m ¯ ) + γ wavelet E detail ( m ¯ ) .
R T V ( ϕ̲ ) = [ Q x T sgn ( Q x T f ¯ ) + Q y T sgn ( Q y T f ¯ ) ] sin ϕ ¯ R e [ ( m ¯ z ¯ * ) e j θ̲ ] 1 2 f ¯ ,
R T V ( θ̲ ) = [ Q x T sgn ( Q x T f ¯ ) + Q y T sgn ( Q y T f ¯ ) ] ( + cos ϕ̲ ) R e [ ( m ¯ z ¯ * ) e j θ̲ ( j ) ] 1 2 f ¯ ,
J ( m ¯ i ) = F ( m ¯ i ) + γ pole R pole ( m ¯ i ) + ω ( i ) γ wavelet E detail ( m ¯ i ) ,
m ¯ 1 k = r ¯ 1 k e j θ̲ 1 k = 1 + cos ϕ̲ 1 k 2 e j θ̲ 1 k , k = 1 , . . . , N 2 ,
m ¯ 2 k = r ¯ 2 k e j θ̲ 2 k = 1 + cos ϕ̲ 2 k 2 e j θ̲ 2 k , k = 1 , . . . , N 2 ,
z ¯ 1 k = 1 1 + exp [ a k = 1 N 2 h i k 1 + cos ϕ̲ 1 k 2 e j θ̲ 1 k + a t r ] , k = 1 , . . . , N 2 ,
z ¯ 2 k = 1 1 + exp [ a k = 1 N 2 h i k 1 + cos ϕ̲ 2 k 2 e j θ̲ 2 k + a t r ] , k = 1 , . . . , N 2 ,
F = F ( θ̲ 1 , ϕ̲ 1 , θ̲ 2 , ϕ̲ 2 ) = z ¯ * z ¯ 2 2 = i = 1 N 2 { z i * 1 2 [ tanh ( z ¯ 1 i + z ¯ 2 i 1 ) + 1 ] } 2 .
F θ̲ p = a × + cos ϕ̲ p 2 sin θ̲ p { H T [ ( z ¯ * z ¯ ) sech 2 ( z ¯ 1 + z ¯ 2 ) z ¯ p
( p ) H ( m ¯ p R ) T ( m ¯ , p ) ] }
a × + cos ϕ̲ p 2 cos θ̲ p { H T [ ( z ¯ * z ¯ ) sech 2 ( z ¯ 1 + z ¯ 2 ) z ¯ p
( z ¯ ) p H ( m ¯ p I ) T ( m ¯ , p ) ] } ,
F ϕ̲ p = a 2 × sin ϕ̲ p cos θ̲ p { H T [ ( z ¯ * z ¯ ) sech 2 ( z ¯ 1 + z ¯ 2 ) z ¯ p
( p ) H ( m ¯ p R ) T ( m ¯ , p ) ] }
+ a 2 × sin ϕ̲ p sin θ̲ p { H T [ ( z ¯ * z ¯ ) sech 2 ( z ¯ 1 + z ¯ 2 ) z ¯ p
( z ¯ ) p H ( m ¯ p I ) T ( m ¯ , p ) ] } ,

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