## Generalized formulations for aerial image based lens aberration metrology in lithographic tools with arbitrarily shaped illumination sources |

Optics Express, Vol. 18, Issue 19, pp. 20096-20104 (2010)

http://dx.doi.org/10.1364/OE.18.020096

Acrobat PDF (2588 KB)

### Abstract

In the current optical lithography processes for semiconductor manufacturing, differently shaped illumination sources have been widely used for the need of stringent critical dimension control. This paper proposes a technique for in situ measurement of lens aberrations with generalized formulations of odd and even aberration sensitivities suitable for arbitrarily shaped illumination sources. With a set of Zernike orders, these aberration sensitivities can be treated as a set of analytical kernels which succeed in constructing a sensitivity function space. The analytical kernels reveal the physical essence of partially coherent imaging systems by taking into account the interaction between the wavefront aberration and the illumination source, and take the advantage of realizing a linear and analytical relationship between the Zernike coefficients to be measured and the measurable physical signals. A variety of mainstream illumination sources with spatially variable intensity distributions were input into the PROLITH for the simulation work, which demonstrates and confirms that the generalized formulations are suitable for measuring lens aberrations up to a high order Zernike coefficient under different types of source distributions. The technique is simple to implement and will have potential applications in the in-line monitoring of imaging quality of current lithographic tools.

© 2010 OSA

## 1. Introduction

1. H. Nomura and T. Sato, “Techniques for measuring aberrations in lenses used in photolithography with printed patterns,” Appl. Opt. **38**(13), 2800–2807 (1999). [CrossRef]

4. F. Wang, X. Wang, and M. Ma, “Measurement technique for in situ characterizing aberrations of projection optics in lithographic tools,” Appl. Opt. **45**(24), 6086–6093 (2006). [PubMed]

5. B. W. Smith and R. Schlief, “Understanding lens aberration and influences to lithographic imaging,” Proc. SPIE **4000**, 294–306 (2000). [CrossRef]

7. F. Zernike, “Beugungstheorie des Schneidenverfahrens und seiner verbesserten form, der Phasenkontrastmethode,” Physica **1**(7-12), 689–704 (1934). [CrossRef]

8. H. van der Laan, M. Dierichs, H. van Greevenbroek, E. McCoo, F. Stoffels, R. Pongers, and R. Willekers, “Aerial image measurement methods for fast aberration setup and illumination pupil verification,” Proc. SPIE **4346**, 394–407 (2001). [CrossRef]

*σ*settings through a matrix of sensitivities, which is a function of the corresponding multiple NA/

*σ*settings. Wang et al recently reported a series of TAMIS based techniques to improve the measurement accuracy of coma and even aberrations by optimization of the test marks using phase-shifting gratings [9

9. Q. Yuan, X. Wang, Z. Qiu, F. Wang, M. Ma, and L. He, “Coma measurement of projection optics in lithographic tools based on relative image displacements at multiple illumination settings,” Opt. Express **15**(24), 15878–15885 (2007). [CrossRef] [PubMed]

12. Q. Yuan, X. Wang, Z. Qiu, F. Wang, and M. Ma, “Even aberration measurement of lithographic projection system based on optimized phase-shifting marks,” Microelectron. Eng. **86**(1), 78–82 (2009). [CrossRef]

13. J. K. Tyminski, T. Hagiwara, N. Kondo, and H. Irihama, “Aerial image sensor: in-situ scanner aberration monitor,” Proc. SPIE **6152**, 61523D (2006). [CrossRef]

14. T. Hagiwara, N. Kondo, I. Hiroshi, K. Suzuki, and N. Magome, “Development of aerial image based aberration measurement technique,” Proc. SPIE **5754**, 1659–1669 (2005). [CrossRef]

15. W. Liu, S. Liu, T. Zhou, and L. Wang, “Aerial image based technique for measurement of lens aberrations up to 37th Zernike coefficient in lithographic tools under partial coherent illumination,” Opt. Express **17**(21), 19278–19291 (2009). [CrossRef]

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

18. M. Mulder, A. Engelen, O. Noordman, R. Kazinczi, G. Streutker, B. van Drieenhuizen, S. Hsu, K. Gronlund, M. Degünther, D. Jürgens, J. Eisenmenger, M. Patra, and A. Major, “Performance of a programmable illuminator for generation of freeform sources on high NA immersion systems,” Proc. SPIE **7520**, 75200Y (2009). [CrossRef]

19. A. Engelen, M. Mulder, I. Bouchoms, S. Hansen, A. Bouma, A. Ngai, M. van Veen, and J. Zimmermann, “Imaging solutions for the 22nm node using 1.35NA,” Proc. SPIE **7274**, 72741Q (2009). [CrossRef]

20. Y. Granik and K. Adam, “Analytical approximations of the source intensity distributions,” Proc. SPIE **5992**, 599255 (2005). [CrossRef]

## 2. Theory

*J*(

**r**

*) to accurately characterize an arbitrarily shaped source with spatially variable intensity distributions, where*

_{c}**r**

*is the normalized pupil vector that can be expressed in normalized polar coordinates as (*

_{c}*r*,

_{c}*θ*) or in normalized Cartesian coordinates as (

_{c}*f*,

_{c}*g*). According to Hopkins theory of partially coherent imaging [21

_{c}21. H. Hopkins, “On the diffraction theory of optical images,” Proc. R. Soc. A **217**(1130), 408–432 (1953). [CrossRef]

**r**

*is a vector in the pupil plane, which can be expressed in normalized polar coordinates as (*

_{m}*r*,

_{m}*θ*) corresponding to the pitch and orientation of the binary grating;

_{m}*TCC*is transmission cross coefficient given by

*H*(

**r**,

*h*) is the pupil function given by

*k*= 2π/λ is the wave number; λ is the wavelength of the monochromatic light source;

*W*(

**r**,

*h*) is the total aberrated wavefront given by

*W*(

_{odd}**r**), the lens even aberration

*W*(

_{even}**r**), and the defocus aberration that is induced by another even-type aberration

*h*from the ideal focal plane;

*NA*is the image-side numerical aperture of the projection lens.

*φ*[

**r**

*,*

_{m}*J*(

**r**

*),*

_{c}*h*] and amplitude

*φ*[

**r**

*,*

_{m}*J*(

**r**

*)] of the + 1st-order spectrum at the ideal focal plane corresponding to*

_{c}*h*= 0, and an axial shift value

*h*=

*D*[

**r**

*,*

_{m}*J*(

**r**

*)] satisfying the condition*

_{c}*W*(

**r**,

*h*) in the lithographic tools, we have approximated formulas

*φ*[

**r**

*,*

_{m}*J*(

**r**

*)] and*

_{c}*D*[

**r**

*,*

_{m}*J*(

**r**

*)] as [15*

_{c}15. W. Liu, S. Liu, T. Zhou, and L. Wang, “Aerial image based technique for measurement of lens aberrations up to 37th Zernike coefficient in lithographic tools under partial coherent illumination,” Opt. Express **17**(21), 19278–19291 (2009). [CrossRef]

*S*is the integral region which is determined by the intersection of two different shift pupils and represented in normalized Cartesian coordinates (

*f*,

_{c}*g*) as the shaded area shown in Fig. 1 .

_{c}

*R**(*

_{n}**r**

*) indicates the*

_{m}*n*th Zernike polynomial,

*n_odd*and

*n_even*indicate the Zernike indexes for odd aberration and even aberration respectively, Eqs. (2) and (3) can be written as:

*F*[

_{n_odd}**r**

*,*

_{m}*J*(

**r**

*)] and*

_{c}*G*[

_{n_even}**r**

*,*

_{m}*J*(

**r**

*)] as two sensitivities for odd aberration and even aberration respectively, and they have the formulations of:*

_{c}15. W. Liu, S. Liu, T. Zhou, and L. Wang, “Aerial image based technique for measurement of lens aberrations up to 37th Zernike coefficient in lithographic tools under partial coherent illumination,” Opt. Express **17**(21), 19278–19291 (2009). [CrossRef]

*σ*to represent a conventional source with a circular top-hat distribution, Eqs. (8) and (9) will be simplified to have the identical formulations shown as Eqs. (20) and (21) in Ref. 15

**17**(21), 19278–19291 (2009). [CrossRef]

**r**

*or polar coordinates as (*

_{m}*r*,

_{m}*θ*).

_{m}_{2}to Z

_{37}under the smooth conventional illumination shown in Fig. 5 . As expected, the analytical kernels have the same symmetric properties along the angle

*θ*at a fixed

_{m}*r*as the corresponding Zernike polynomials do. Therefore, similar to the Zernike polynomials, the analytical kernels can be categorized into 1

_{m}*θ*kernels (

_{m}*n_odd*= 2, 3, 7, 8, 14, 15, 23, 24, 34, 35), 3

*θ*kernels (

_{m}*n_odd*= 10, 11, 19, 20, 30, 31), and 5

*θ*kernels (

_{m}*n_odd*= 26, 27) for odd aberrations, together with 0

*θ*kernels (

_{m}*n_even*= 4, 9, 16, 25, 36, 37), 2

*θ*kernels (

_{m}*n_even*= 5, 6, 12, 13, 21, 22, 32, 33), and 4

*θ*kernels (

_{m}*n_even*= 17, 18, 28, 29) for even aberrations.

*φ*[

**r**

*,*

_{m}*J*(

**r**

*)] and*

_{c}*D*[

**r**

*,*

_{m}*J*(

**r**

*)], which leads to be not only accurate but also efficient for aberration in-line monitoring under arbitrary illumination sources. If we concentrate on a set of Zernike coefficient from Z*

_{c}_{2}to Z

*(*

_{N}*N*is the highest concerned Zernike order), the corresponding analytical kernels from Z

_{2}to Z

*are able to together construct a generalized sensitivity function space. As shown in Fig. 4 , after obtaining the physical signals from through-focus images by Fourier transform (FT), it is easy to map the*

_{N}*φ*[

**r**

*,*

_{m}*J*(

**r**

*)] and*

_{c}*D*[

**r**

*,*

_{m}*J*(

**r**

*)] into the sensitivity function space where the weights of corresponding kernel components indicate the Zernike coefficients to be measured, which is the essential physical meaning of linear Eqs. (6) and (7).*

_{c}**Φ**=

**FZ**

_{odd}and

**D**=

**GZ**

_{even}, where

**Φ**and

**D**are vectors respectively containing the phase shifts and the axial shifts of the + 1st-order intensity spectrums at the setting of 36 binary gratings;

**Z**

_{odd}and

**Z**

_{even}are unknown vectors to be measured, and respectively containing the Zernike coefficients for odd aberration and even aberration;

**F**and

**G**are two matrixes of sensitivities for measuring odd aberration and even aberration respectively. Since there are more equations than unknowns, both two sets of equations become over-determined and Zernike coefficients form Z

_{2}up to Z

_{37}can be solved by the least-square method.

## 3. Simulation

*σ*= 0.31), annular (

*σ*

_{i}/

*σ*

_{o}= 0.4/0.7), dipole (

*σ*

_{i}/

*σ*

_{o}/degree = 0.4/0.7/45°), and quadrupole (

*σ*

_{i}/

*σ*

_{o}/degree = 0.4/0.7/45°) shapes, which were respectively constructed by convolving the corresponding top-hat distribution sources shown in the lower row in Fig. 5 with a Gaussian kernel diffusion length of 0.1 [20

20. Y. Granik and K. Adam, “Analytical approximations of the source intensity distributions,” Proc. SPIE **5992**, 599255 (2005). [CrossRef]

_{2}up to Z

_{37}were used as inputs for the simulation. The comparison between the input and measured aberrated wavefronts is illustrated in Fig. 7. It is clear that the measured wavefront values are in good agreement with the input ones with absolute measurement errors less than 3.5mλ. Figure 8 shows the measurement errors of individual Zernike coefficients from Z

_{2}up to Z

_{37}for the input aberrated wavefront 1. The upper chart represents a comparison of the input Zernike coefficients and the measured values, and the lower chart represents the absolute errors of Zernike coefficients. The measured values of the Zernike coefficients are noted to coincide quite closely with the input values. From the simulation result, the absolute errors of all Zernike coefficients are less than 0.62mλ. From this observation and lots of other simulation results for different arbitrarily shaped sources, it is found that the proposed technique yields a superior quality of wavefront estimate with an accuracy on the order of mλs and accuracy of Zernike coefficients on the order of 0.1mλs.

**F**and

**G**, which are determined directly by the effective source function

*J*(

**r**

*) together with the set of binary gratings*

_{c}**r**

*(i.e., pupil sampling scheme). It is promising to enhance the overall performance by optimizing either of the effective source function, the pupil sampling scheme, or both of them. It is also interesting to note that if we obtain the aerial image intensities with only several fixed binary gratings but at multiple illumination source and NA settings, our technique will be quite similar to the commonly used TAMIS technique [8*

_{m}8. H. van der Laan, M. Dierichs, H. van Greevenbroek, E. McCoo, F. Stoffels, R. Pongers, and R. Willekers, “Aerial image measurement methods for fast aberration setup and illumination pupil verification,” Proc. SPIE **4346**, 394–407 (2001). [CrossRef]

## 4. Conclusion

## Acknowledgements

## References and links

1. | H. Nomura and T. Sato, “Techniques for measuring aberrations in lenses used in photolithography with printed patterns,” Appl. Opt. |

2. | H. Nomura, K. Tawarayama, and T. Kohno, “Aberration measurement from specific photolithographic images: a different approach,” Appl. Opt. |

3. | J. Sung, M. Pitchumani, and E. G. Johnson, “Aberration measurement of photolithographic lenses by use of hybrid diffractive photomasks,” Appl. Opt. |

4. | F. Wang, X. Wang, and M. Ma, “Measurement technique for in situ characterizing aberrations of projection optics in lithographic tools,” Appl. Opt. |

5. | B. W. Smith and R. Schlief, “Understanding lens aberration and influences to lithographic imaging,” Proc. SPIE |

6. | L. Zavyalova, A. Bourov, and B. W. Smith, “Automated aberration extraction using phase wheel targets,” Proc. SPIE |

7. | F. Zernike, “Beugungstheorie des Schneidenverfahrens und seiner verbesserten form, der Phasenkontrastmethode,” Physica |

8. | H. van der Laan, M. Dierichs, H. van Greevenbroek, E. McCoo, F. Stoffels, R. Pongers, and R. Willekers, “Aerial image measurement methods for fast aberration setup and illumination pupil verification,” Proc. SPIE |

9. | Q. Yuan, X. Wang, Z. Qiu, F. Wang, M. Ma, and L. He, “Coma measurement of projection optics in lithographic tools based on relative image displacements at multiple illumination settings,” Opt. Express |

10. | Z. Qiu, X. Wang, Q. Bi, Q. Yuan, B. Peng, and L. Duan, “Translational-symmetry alternating phase shifting mask grating mark used in a linear measurement model of lithographic projection lens aberrations,” Appl. Opt. |

11. | Z. Qiu, X. Wang, Q. Yuan, and F. Wang, “Coma measurement by use of an alternating phase-shifting mask mark with a specific phase width,” Appl. Opt. |

12. | Q. Yuan, X. Wang, Z. Qiu, F. Wang, and M. Ma, “Even aberration measurement of lithographic projection system based on optimized phase-shifting marks,” Microelectron. Eng. |

13. | J. K. Tyminski, T. Hagiwara, N. Kondo, and H. Irihama, “Aerial image sensor: in-situ scanner aberration monitor,” Proc. SPIE |

14. | T. Hagiwara, N. Kondo, I. Hiroshi, K. Suzuki, and N. Magome, “Development of aerial image based aberration measurement technique,” Proc. SPIE |

15. | W. Liu, S. Liu, T. Zhou, and L. Wang, “Aerial image based technique for measurement of lens aberrations up to 37th Zernike coefficient in lithographic tools under partial coherent illumination,” Opt. Express |

16. | A. K. Wong, Resolution Enhancement Techniques, (SPIE Press, 2001). |

17. | F. Schellenberg, “Resolution enhancement technology: The past, the present, and extensions for the future,” Proc. SPIE |

18. | M. Mulder, A. Engelen, O. Noordman, R. Kazinczi, G. Streutker, B. van Drieenhuizen, S. Hsu, K. Gronlund, M. Degünther, D. Jürgens, J. Eisenmenger, M. Patra, and A. Major, “Performance of a programmable illuminator for generation of freeform sources on high NA immersion systems,” Proc. SPIE |

19. | A. Engelen, M. Mulder, I. Bouchoms, S. Hansen, A. Bouma, A. Ngai, M. van Veen, and J. Zimmermann, “Imaging solutions for the 22nm node using 1.35NA,” Proc. SPIE |

20. | Y. Granik and K. Adam, “Analytical approximations of the source intensity distributions,” Proc. SPIE |

21. | H. Hopkins, “On the diffraction theory of optical images,” Proc. R. Soc. A |

**OCIS Codes**

(110.4980) Imaging systems : Partial coherence in imaging

(110.5220) Imaging systems : Photolithography

(120.0120) Instrumentation, measurement, and metrology : Instrumentation, measurement, and metrology

(220.1010) Optical design and fabrication : Aberrations (global)

**ToC Category:**

Imaging Systems

**History**

Original Manuscript: August 2, 2010

Manuscript Accepted: August 28, 2010

Published: September 3, 2010

**Citation**

Wei Liu, Shiyuan Liu, Tielin Shi, and Zirong Tang, "Generalized formulations for aerial image based lens aberration metrology in lithographic tools with arbitrarily shaped illumination sources," Opt. Express **18**, 20096-20104 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-19-20096

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### References

- H. Nomura and T. Sato, “Techniques for measuring aberrations in lenses used in photolithography with printed patterns,” Appl. Opt. 38(13), 2800–2807 (1999). [CrossRef]
- H. Nomura, K. Tawarayama, and T. Kohno, “Aberration measurement from specific photolithographic images: a different approach,” Appl. Opt. 39(7), 1136–1147 (2000). [CrossRef]
- J. Sung, M. Pitchumani, and E. G. Johnson, “Aberration measurement of photolithographic lenses by use of hybrid diffractive photomasks,” Appl. Opt. 42(11), 1987–1995 (2003). [CrossRef] [PubMed]
- F. Wang, X. Wang, and M. Ma, “Measurement technique for in situ characterizing aberrations of projection optics in lithographic tools,” Appl. Opt. 45(24), 6086–6093 (2006). [PubMed]
- B. W. Smith and R. Schlief, “Understanding lens aberration and influences to lithographic imaging,” Proc. SPIE 4000, 294–306 (2000). [CrossRef]
- L. Zavyalova, A. Bourov, and B. W. Smith, “Automated aberration extraction using phase wheel targets,” Proc. SPIE 5754, 1728–1737 (2005). [CrossRef]
- F. Zernike, “Beugungstheorie des Schneidenverfahrens und seiner verbesserten form, der Phasenkontrastmethode,” Physica 1(7-12), 689–704 (1934). [CrossRef]
- H. van der Laan, M. Dierichs, H. van Greevenbroek, E. McCoo, F. Stoffels, R. Pongers, and R. Willekers, “Aerial image measurement methods for fast aberration setup and illumination pupil verification,” Proc. SPIE 4346, 394–407 (2001). [CrossRef]
- Q. Yuan, X. Wang, Z. Qiu, F. Wang, M. Ma, and L. He, “Coma measurement of projection optics in lithographic tools based on relative image displacements at multiple illumination settings,” Opt. Express 15(24), 15878–15885 (2007). [CrossRef] [PubMed]
- Z. Qiu, X. Wang, Q. Bi, Q. Yuan, B. Peng, and L. Duan, “Translational-symmetry alternating phase shifting mask grating mark used in a linear measurement model of lithographic projection lens aberrations,” Appl. Opt. 48(19), 3654–3663 (2009). [CrossRef] [PubMed]
- Z. Qiu, X. Wang, Q. Yuan, and F. Wang, “Coma measurement by use of an alternating phase-shifting mask mark with a specific phase width,” Appl. Opt. 48(2), 261–269 (2009). [CrossRef] [PubMed]
- Q. Yuan, X. Wang, Z. Qiu, F. Wang, and M. Ma, “Even aberration measurement of lithographic projection system based on optimized phase-shifting marks,” Microelectron. Eng. 86(1), 78–82 (2009). [CrossRef]
- J. K. Tyminski, T. Hagiwara, N. Kondo, and H. Irihama, “Aerial image sensor: in-situ scanner aberration monitor,” Proc. SPIE 6152, 61523D (2006). [CrossRef]
- T. Hagiwara, N. Kondo, I. Hiroshi, K. Suzuki, and N. Magome, “Development of aerial image based aberration measurement technique,” Proc. SPIE 5754, 1659–1669 (2005). [CrossRef]
- W. Liu, S. Liu, T. Zhou, and L. Wang, “Aerial image based technique for measurement of lens aberrations up to 37th Zernike coefficient in lithographic tools under partial coherent illumination,” Opt. Express 17(21), 19278–19291 (2009). [CrossRef]
- A. K. Wong, Resolution Enhancement Techniques, (SPIE Press, 2001).
- F. Schellenberg, “Resolution enhancement technology: The past, the present, and extensions for the future,” Proc. SPIE 5377, 1–20 (2004). [CrossRef]
- M. Mulder, A. Engelen, O. Noordman, R. Kazinczi, G. Streutker, B. van Drieenhuizen, S. Hsu, K. Gronlund, M. Degünther, D. Jürgens, J. Eisenmenger, M. Patra, and A. Major, “Performance of a programmable illuminator for generation of freeform sources on high NA immersion systems,” Proc. SPIE 7520, 75200Y (2009). [CrossRef]
- A. Engelen, M. Mulder, I. Bouchoms, S. Hansen, A. Bouma, A. Ngai, M. van Veen, and J. Zimmermann, “Imaging solutions for the 22nm node using 1.35NA,” Proc. SPIE 7274, 72741Q (2009). [CrossRef]
- Y. Granik and K. Adam, “Analytical approximations of the source intensity distributions,” Proc. SPIE 5992, 599255 (2005). [CrossRef]
- H. Hopkins, “On the diffraction theory of optical images,” Proc. R. Soc. A 217(1130), 408–432 (1953). [CrossRef]

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