## Fourier-based analysis of moiré fringe patterns of superposed gratings in alignment of nanolithography

Optics Express, Vol. 16, Issue 11, pp. 7869-7880 (2008)

http://dx.doi.org/10.1364/OE.16.007869

Acrobat PDF (513 KB)

### Abstract

Considering the necessity of alignment in practical applications of photolithography, distribution of complex amplitude of moiré fringe patterns that are produced in superposition of two gratings is analyzed in the viewpoint of Fourier Optics and the relationship between fringes and properties of these two gratings is concluded by means of an analysis model. The rule of one-dimensional gratings (1D-gratings) is extended to other form of the gratings which have quasi-periodic repetitive structures. Especially, moiré fringes generated by the two superposed 1D-gratings (used in alignment of lithography) can be expressed by an arithmetical operation of two vectors which include enough information about these 1D-gratings. Numerical analyses regarding the moiré model and its application in the alignment process of lithography are carried out. Our computational analyses results show that the moiré fringes of the two extended gratings can be refined as a transformed fringe pattern of two standard 1D-gratings. Finally, the results also make it out that the fringes which have magnified periods versus that of two 1D-gratings are highly sensitive to relative shift of two gratings thus might be applicable in alignment of lithography or correlated fields.

© 2008 Optical Society of America

## 1. Introduction

5. G. Oster, M. Wasserman, and C. Zwerling, “Theoretical Interpretation of Moiré Patterns,” J. Opt. Soc. Am. A **54**, 169–175 (1964). [CrossRef]

6. Y. Nishijima and G. Oster, “Moiré Patterns: Their Application to Refractive Index and Refractive Index Gradient Measurements,” J. Opt. Soc. Am. A **54**, 1–5 (1964). [CrossRef]

7. O. Bryngdahl, “Moiré: Formation and interpretation,” J. Opt. Soc. Am. A **64**, 1287–1294 (1974). [CrossRef]

## 2. Theoretical models

### 2.1 Transmittance model of gratings and Fourier decomposition

*p*is the period of grating and

*d*is the width of grooves of grating. The transmittance coefficient can be expressed as an accumulation of a rectangle function

*rect*(

*x*). The gratings with other profiles can be defined in the same way. Hereinafter, we use the gratings with rectangular profile as an example to continue our analysis and gratings that can be expressed as the form of Eq. (1) are called standard 1D-gratings.

*x′*with a certain function

*T*(

*x*,

*y*) in Eq. (1), the 1D-gratings can be transformed into other curvilinear gratings. Here,

*T*(

*x*,

*y*)=

*C*denotes the layout of certain quasi-periodic structures as in Fig. 1. For each determined variable constant

*C*, this expression denotes a determined curve (like the particular nT grooves of 1D-gratings) among the transformed repetitive structures. As a result, a model of transmittance coefficient for the quasi-periodic repetitive structures can be obtained by the same way used in Eq. (1).

*T*(

*x*,

*y*) denotes the shape of them.

*E*(

*x*,

*y*) in the plane close to the back of gratings accords with the transmittance coefficient of gratings, [17] namely

*E*(

*x*,

*y*)=

*G*(

_{T}*x*,

*y*). Furthermore, the diffractive wave can also be regarded as an accumulation of different harmonics with discrete frequencies because of the periodicity of the gratings. When the planar wave with unit amplitude travels through the 1D-grating, the complex amplitude of the diffractive wave can therefore be decomposed as

*E*(

*f*)=∑

^{+∞}

_{n=-∞}

*a*(

_{n}δ*f*-

*nf*

_{0}) is the spectrum of the diffractive wave traveled through the 1D-grating,

*a*=

_{n}*df*

_{0}sin

*c*(

*ndf*

_{0}) is Fourier coefficient of rectangular function in Eq. (1) and

*f*=

_{0}*1*/

*p*. In the same way, the Fourier model of distribution of complex amplitude for those extended repetitive structures can be expressed as

### 2.2 Extraction model of moiré fringes distributions

#### 2.2.1 Moiré fringes distributions

*f*and

_{1}*f*. Then two gratings with the extended layout can be obtained by applying certain two transformations

_{2}*T*(

_{1}*x*,

*y*) and

*T*(

_{2}*x*,

*y*) to these two standard 1D-gratings respectively. As the discussion above, the complex amplitude of the diffractive wave behind two extended structures can be respectively expressed as

*k*,

_{1}*k*are the integers except 0, and

_{2}*f*is the standardized frequency. Thus twofold sum of Eq. (5) is decomposed into many partial sums. Compared with Eq. (4), Eq. (6) can be regarded as the form obtained by applying a compound transformation of

*T*(

*x*,

*y*) to

*E*(

*x′*), which can be considered as the complex amplitude distribution of the diffractive wave behind certain 1D-gratings (the profile of which may not be rectangular any longer) with standard frequency

*f*. Here, we have

*k*,

_{1}*k*), partial sum series in Eq. (7) will converge to a periodic-distributed pattern which is similar to the layout of standard 1D-gratings, and this pattern would become another patterns denoted by Eq. (6) while transformed by

_{2}*T*(

*x*,

*y*) in Eq. (8). We call the pattern denoted by Eq. (6) (

*k*,

_{1}*k*) moiré fringes and the pattern denoted by Eq. (7) standard 1D moiré fringes. This process can be concluded as that moiré fringes generated by superposition of two geometrically transformed 1D-gratings are equivalent to the patterns obtained by application of a compound transformation to a certain 1D-distributed moiré fringes. We call it “

_{2}*conclusion 1*” hereinafter. The relationship between the 1D-distributed moiré fringes and these two 1D-gratings can be described by Eq. (7), whereas the relationship between the compound transformation and two transformations of the original 1D-gratings can be expressed by Eq. (8).

#### 2.2.2 Analysis of 1D-distributed moiré fringes

*θ*is slope of the rotated 1D-grating with frequency of

_{1}*f*(seen in Fig. 2(b)) and

_{1}*θ*is the slope of the other 1D-grating with frequency of

_{2}*f*

_{2}. And the complex amplitude of (

*k*,

_{1}*k*) moiré fringes behind these two superposed gratings can be obtained according to Eq. (6) and standardized according to Eq. (4) as

_{2}*T*=(

_{e}*x*,

*y*)=

*x*sin

*θ*-

_{e}*y*cos

*θ*, where

_{e}*θ*,

_{e}*f*are slope and frequency of moiré fringes respectively.

_{e}*f*and

_{1}*f*are individually rotated by

_{2}*T*(

_{1}*x*,

*y*) and

*T*(

_{2}*x*,

*y*) in Eq. (9) and Eq. (10), the (

*k*,

_{1}*k*) moiré fringes generated by two rotated gratings can be regarded as a set of standard 1D-fringes with frequency of

_{2}*f*rotated by

_{e}*T*(

_{e}*x*,

*y*) in Eq. (11). Furthermore, the rotated fringes can be expressed by a vector which is the sum of two vectors on be half of two original 1D-gratings. We called it “

*conclusion 2*” hereinafter.

*F⃗*

_{1}=

*k*

_{1}

*f*

_{1}exp(

*iθ*

_{1}) to denote the 1D-grating with frequency

*f*and slope

_{1}*,*

**θ**_{1}*F⃗*

_{2}=

*k*

_{2}

*f*

_{2}exp(

*iθ*

_{2}) to denote the other 1D-grating with frequency

*f*and slope

_{2}*respectively. As a result, the vector*

**θ**_{2}*F⃗*=

_{e}*f*exp(

_{e}*iθ*) on be half of their moiré fringes can be obtained by sum of

_{e}*F⃗*

_{1}and

*F⃗*

_{2}, namely

*F⃗*=

_{e}*F⃗*

_{1}+

*F⃗*

_{2}. Thus

#### 2.2.3 Analysis of special moiré fringes distributions

*different frequencies*by

*different transformations*are in accordance with the rule of

*conclusion 1*. Whereas, according to Eq. (6) no resolvable fringes are generated by two gratings transformed from two 1D-gratings with the

*same frequency*by

*the same transformation*because the (1,-1) moiré fringes disappeared. Especially, Eq. (12) indicates that the two superposed 1D-gratings with the same frequency and the same slope generated no resolvable fringes because the (1,-1) moiré fringes disappeared too. The moiré fringes of the two superposed 1D-gratings with different frequencies and different slopes are in accordance with the rule of

*conclusion 2*. Therefore, we had four cases here:

*different frequencies vs. different transformations, the same frequency vs. the same transformation, the same frequency vs. different transformations*, and

*different frequencies vs. the same transformation*. The other two cases are discussed below in detail.

*A. The same frequency vs. different transformations*. When two superposed gratings are transformed from the same standard 1D-grating with frequency of

*f*by different transformations

*T*

_{1}and

*T*

_{2}, the amplitude distribution of the (1,-1) moiré fringes can be deduced from Eq. (6) as

*T*

_{1}(

*x*,

*y*)-

*T*

_{2}(

*x*,

*y*). Especially, if two superposed gratings are rotated 1D-gratings, their descriptive vectors is

*F⃗*

_{1}=

*f*

_{1}exp(

*iθ*

_{1}) and

*F⃗*

_{2}

*f*

_{2}exp(

*iθ*

_{2}). Correspondingly, the vector

*F⃗*which on behalf of the 1D-distributed moiré pattern can be expressed as the

_{e}*difference*of vectors

*F⃗*

_{1}and

*F⃗*

_{2}according to the

*conclusion 2*, namely

*F⃗*=

_{e}*F⃗*

_{1}-

*F⃗*

_{2}.

*B. The same transformation vs. different frequencies*. According to Eq. (6), the complex amplitude distribution of (1,-1) moiré fringes produced by two superposed gratings that are transformed from two standard 1D-gratings with different frequencies

*f*and

_{1}*f*by the same transformation

_{2}*T*can be expressed as

*f*is slightly bigger than

_{1}*f*, namely

_{2}*f*>

_{1}*f*. Obviously the pattern denoted by Eq. (14) can be regarded as a set of standard 1D-distributed moiré fringes with frequency of

_{2}*f*-

_{1}*f*transformed by the same function

_{2}*T*. Now the period of the low-frequency fringes is magnified with respect to that of two original gratings, namely

*p*=

*1*/|

*f*-

_{1}*f*|=

_{2}*p*/(

_{1}p_{2}*p*-

_{2}*p*).

_{1}*T*

_{1}=

*x*+Δ

*x*and

*T*

_{2}=

*x*, i.e. the 1D-gratings with frequency

*f*

_{1}move left by Δ

*x*with respect to the other 1D-grating. The moiré fringes distribution before and after movement can be respectively expressed as

*of the grating with frequency*

**Δ**x*f*leads to a left shift of

_{1}**Δ**

*xf*

_{1}/(

*f*-

_{1}*f*)=

_{2}*/(*

**Δ**xp_{2}*p*-

_{2}*p*) of the fringes. Vice versa, a left shift

_{1}*of the grating with frequency*

**Δ**x*f*results in a left shift of the fringes by Δ

_{2}*xp*

_{1}/(

*p*

_{2}-

*p*

_{1}). Therefore, the phenomenon that tiny displacement of a grating leads to large displacement of corresponding moiré fringes can be applied in some fields such as the alignment of nanolithography.

## 3. Results analyses and discussions

*T*(

_{1}*x*,

*y*)-0.9

*T*(

_{2}*x*,

*y*) to the standard 1D-distributed moiré fringes denoted by Eq. (7) are shown in Fig. 3(f). These two results indicate that the (1,-1) moiré fringes extracted according to the rule of

*conclusion 1*are the same as the fringes directly observed from superposition of two corresponding gratings.

*slightly different frequencies*by the

*same transformation*. It is apparent that moiré fringes obtained in the two different approaches are the same, and the period of the fringes is magnified with respect to that of two original 1D-gratings. Take the 1D-grating with the period

*p*

_{1}=1/0.9 in Fig. 3(b) for example, the period of its moiré fringes in Fig. 4(a) is magnified by

*p*/|

_{1}*p*-

_{1}*p*

_{2}**|≈10**times. As the discussions in section 2.2.3 B, this result can be interpreted that the frequency of (1,-1) moiré fringes generated by the two superposed gratings with

*slightly different frequencies*equals to the difference of two original frequencies. Likewise, the transverse displacement of two gratings led to a magnified displacement of corresponding moiré fringes.

*f*=2 and slope of

_{1}*=30°. Figure 5(b) shows the other standard 1D-gratings with frequency*

**θ**_{1}*f*=1 and slope

_{2}*=90°. According to the vector’s arithmetic operation rule drawn in*

**θ**_{2}*conclusion 2*, (1, -1) moiré fringes of these two 1D-gratings can be expressed by the vector

*F⃗*

_{1}-

*F⃗*

_{2}=3

^{1/2}which means that the frequency and slope of their moiré fringes is

*f*=3

_{e}^{1/2}and

*θ*=0°. The extracted (1,-1) moiré fringes shown in Fig. 5(c) prove in accordance with this result.

_{e}*conclusion 1*. And each couple of these quasi-periodic structures is transformed from the same standard 1D-grating by different transformations

*T*and

_{1}*T*. Significantly, each set of fringes can be considered as that a set of standard 1D-distributed moiré fringes are transformed by the difference of two corresponding transformations

_{2}*T*and

_{1}*T*. This result also indicates that the superposition of the curvilinear structures may also generate 1D-distributed moiré fringes and the layout of gratings can be synthesized according to the distribution of the corresponding fringes.

_{2}## 4. Applicability in alignment of nanolithography

*f*=1 and

_{1}*f*=0.9 in the alignment of photolithography. Figure 7(a) shows moiré fringes of two aligned marks and fringes in Figs. 7(b)–7(d) correspond to two superposed marks that are misaligned by 1, 2, and 4 grating periods, respectively. We can also make it out from Fig. 7 that the number of misaligned grating periods is half the number of curves that pass through the common point in the pattern.

_{2}*x*leads to a magnified reverse shift Δ

*L*of the two sets of moiré fringes on the pattern, that is

*m*of period of moiré fringes only when it satisfies the condition Δ

*x*=[2

*p*

_{1}

*p*

_{2}/(

*p*

_{1}+

*p*

_{2})]·

*m*/2, where

*p*=2

_{average}*p*

_{1}

*p*

_{2}/(

*p*

_{1}+

*p*

_{2}) is the weighted average of periods of two gratings,

*m*=0, 1, 2….

*p*/4,

_{average}*p*/8 and

_{average}*p*/16 respectively. Whereas the pattern in Fig. 8(f) corresponds to two superposed marks that are aligned or misaligned by

_{average}*p*/2.

_{average}m## 5. Summary

## Acknowledgment

## References and links

1. | L. Raleigh, “On the manufacture and theory of diffraction gratings,” Philos. Mag. |

2. | J. Guid, |

3. | J. Guid, |

4. | I. G. O Kafri, |

5. | G. Oster, M. Wasserman, and C. Zwerling, “Theoretical Interpretation of Moiré Patterns,” J. Opt. Soc. Am. A |

6. | Y. Nishijima and G. Oster, “Moiré Patterns: Their Application to Refractive Index and Refractive Index Gradient Measurements,” J. Opt. Soc. Am. A |

7. | O. Bryngdahl, “Moiré: Formation and interpretation,” J. Opt. Soc. Am. A |

8. | O. Bryngdahl, “Moiré and higher grating harmonics,” J. Opt. Soc. Am. A |

9. | I. Amidror and R. D. Hersch, “Fourier-based analysis of phase shifts in the superposition of periodic layers and their moire effects,” J. Opt. Soc. Am. A |

10. | I. Amidror and R. D. Hersch, “Fourier-based analysis and synthesis of moirés in the superposition of geometrically transformed periodic structures,” J. Opt. Soc. Am. A |

11. | M. C. King and D. H. Berry, “Photolithographic mask alignment using moire techniques,” J. Vac. Sci. Technol. B |

12. | K. Hara, Y. Uchida, T. Nomura, S. Kimura, D. Sugimoto, A. Yoshida, H. Miyake, T. Iida, and S. Hattori, “An alignment technique using diffracted moire signals,” J. Vac. Sci. Technol. B |

13. | A. Moel, E. E. Moon, R. D. Frankel, and H. I. Smith, “Novel on-axis interferometric alignment method withsub-10 nm precision,” J. Vac. Sci. Technol. B |

14. | J. H. Lee, C. H. Kim, Y.-S. Kim, K. M. Ho, K. Constant, W. Leung, and C. H. Oh, “Diffracted moire fringes as analysis and alignment tools for multilayer fabrication in soft lithography,” Appl. Phys. Lett. |

15. | N. H. Li, W. Wu, and S. Y. Chou, “Sub-20-nm alignment in nanoimprint lithography using Moire fringe,” Nano Lett. |

16. | M. Muhlberger, I. Bergmair, W. Schwinger, M. Gmainer, R. Schoftner, T. Glinsner, C. Hasenfuss, K. Hingerl, M. Vogler, H. Schmidt, and E. B. Kley, “A Moire method for high accuracy alignment in nanoimprint lithography,” Microelectron. Eng. |

17. | L. Q. T, |

**OCIS Codes**

(050.2770) Diffraction and gratings : Gratings

(070.0070) Fourier optics and signal processing : Fourier optics and signal processing

(120.4120) Instrumentation, measurement, and metrology : Moire' techniques

(220.1140) Optical design and fabrication : Alignment

**ToC Category:**

Fourier optics and signal processing

**History**

Original Manuscript: February 1, 2008

Revised Manuscript: April 18, 2008

Manuscript Accepted: May 3, 2008

Published: May 16, 2008

**Citation**

Shaolin Zhou, Yongqi Fu, Xiaoping Tang, Song Hu, Wangfu Chen, and Yong Yang, "Fourier-based analysis of moiré fringe patterns of superposed gratings in alignment of nanolithography," Opt. Express **16**, 7869-7880 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-11-7869

Sort: Year | Journal | Reset

### References

- L. Raleigh, "On the manufacture and theory of diffraction gratings," Philos. Mag. 4, 81-93 (1874).
- J. Guid, The Interference Systems of Crossed Diffraction Gratings (Oxford University Press, London, 1956).
- J. Guid, Diffraction Gratings as Measuring Scales, (Oxford University Press, London,1960).
- I. G. O. Kafri, The Physics of Moire Metrology, (Wiley, New York,1989).
- G. Oster, M. Wasserman, and C. Zwerling, "Theoretical Interpretation of Moiré Patterns," J. Opt. Soc. Am. A 54, 169-175 (1964). [CrossRef]
- Y. Nishijima and G. Oster, "Moiré Patterns: Their Application to Refractive Index and Refractive Index Gradient Measurements," J. Opt. Soc. Am. A 54, 1-5 (1964). [CrossRef]
- O. Bryngdahl, "Moiré: Formation and interpretation," J. Opt. Soc. Am. A 64, 1287-1294 (1974). [CrossRef]
- O. Bryngdahl, "Moiré and higher grating harmonics," J. Opt. Soc. Am. A 65, 685-694 (1975). [CrossRef]
- I. Amidror and R. D. Hersch, "Fourier-based analysis of phase shifts in the superposition of periodic layers and their moire effects," J. Opt. Soc. Am. A 13, 974-987 (1996). [CrossRef]
- I. Amidror and R. D. Hersch, "Fourier-based analysis and synthesis of moirés in the superposition of geometrically transformed periodic structures," J. Opt. Soc. Am. A 15, 1100-1113 (1998). [CrossRef]
- M. C. King and D. H. Berry, "Photolithographic mask alignment using moire techniques," J. Vac. Sci. Technol. B 11, 2455-2458 (1972).
- K. Hara, Y. Uchida, T. Nomura, S. Kimura, D. Sugimoto, A. Yoshida, H. Miyake, T. Iida, and S. Hattori, "An alignment technique using diffracted moire signals," J. Vac. Sci. Technol. B 7, 1977-1979 (1989). [CrossRef]
- A. Moel, E. E. Moon, R. D. Frankel, and H. I. Smith, "Novel on-axis interferometric alignment method with sub-10 nm precision," J. Vac. Sci. Technol. B 11, 2191-2194 (1993). [CrossRef]
- J. H. Lee, C. H. Kim, Y.-S. Kim, K. M. Ho, K. Constant, W. Leung, and C. H. Oh, "Diffracted moire fringes as analysis and alignment tools for multilayer fabrication in soft lithography," Appl. Phys. Lett. 86, 204101-204101 -- 204101-204103 (2005).[DOI: 10. 1063/1.1927268]
- N. H. Li, W. Wu, and S. Y. Chou, "Sub-20-nm alignment in nanoimprint lithography using Moire fringe," Nano Lett. 6, 2626-2629 (2006). [CrossRef] [PubMed]
- M. Muhlberger, I. Bergmair, W. Schwinger, M. Gmainer, R. Schoftner, T. Glinsner, C. Hasenfuss, K. Hingerl, M. Vogler, H. Schmidt, and E. B. Kley, "A Moire method for high accuracy alignment in nanoimprint lithography," Microelectron. Eng. 84, 925-927 (2007). [CrossRef]
- L. Q. T, Physical Optics, (China Machine Press, Beijing, 1986).

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.