Abstract

In optical lithography, light diffracted from photo mask structures has been customarily assumed to be constant with the angle of incidence of the light illuminating the photo mask. As numerical aperture increases to unity and beyond, to cope with the continuous demand for shrinking integrated circuits, device dimensions, and densities, this approximation is no longer valid. In this paper we use the physical theory of diffraction to study, understand, and model the variation of light diffracted from photo mask structures of the order of the wavelength, with the angle of light incidence. We present a semianalytical model that is fast, accurate, and compatible with existing professional software in this domain. The accuracy of the model is studied using the finite-difference time-domain technique and is shown to be below 5% at the image plane, within angles of incidence between ±20°.

© 2010 Optical Society of America

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References

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  5. T. Tawfik and E. Tejnil, “Characterizing OPC model accuracy versus lens induced polarization effects in hyper NA immersion lithography,” Proc. SPIE 7122, 712245 (2008).
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  6. A. Erdmann, G. Citarella, P. Evanschitzky, H. Schermer, V. Philipsen, and P. De Bisschop, “Validity of Hopkins approximation in simulations of hyper NA (NA>1) line-space structures for an attenuated PSM mask,” Proc. SPIE 6154, 61540G (2006).
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2009 (1)

T. Tawfik, A. Morshed, and D. Khalil, “Modeling mask scattered field at oblique incidence,” Proc. SPIE 7274, 727433 (2009).
[CrossRef]

2008 (3)

K. Adam and M. Lam, “Hybrid Hopkins-Abbe method for modeling oblique angle mask effects in OPC,” Proc. SPIE 6924, 69241E (2008).
[CrossRef]

T. Tawfik and E. Tejnil, “Characterizing OPC model accuracy versus lens induced polarization effects in hyper NA immersion lithography,” Proc. SPIE 7122, 712245 (2008).
[CrossRef]

J. T. Azpiroz, G. W. Burr, A. E. Rosenbluth, and M. Hibbs, “Massively-parallel FDTD simulations to address mask electromagnetic effects in hyper-NA immersion lithography,” Proc. SPIE 6924, 69240Y (2008).
[CrossRef]

2007 (3)

M. Saied, F. Foussadier, J. Belledent, Y. Trouiller, I. Schanen, E. Yesilada, C. Gardin, J. C. Urbani, F. Sundermann, F. Robert, C. Couderc, F. Vautrin, L. LeCam, G. Kerrien, J. Planchot, C. Martinelli, B. Wilkinson, Y. Rody, A. Borjon, N. Morgana, J.-L. Di-Maria, and V. Farys, “3D mask modeling with oblique incidence and mask corner rounding effects for the 32nm node,” Proc. SPIE 6730, 673050(2007).
[CrossRef]

P. Liu, Y. Cao, L. Chen, G. Chen, M. Feng, J. Jiang, H.-Y. Liu, S. Suh, S.-W. Lee, and S. Lee, “Fast and accurate 3D mask model for full-chip OPC and verification,” Proc. SPIE 6520, 65200R (2007).
[CrossRef]

K. Lai, A. E. Rosenbluth, G. Han, J. Tirapu-Azpiroz, J. Meiring, A. Goehnermeier, B. Kneer, M. Totzeck, L. de Winter, W. de Boeij, and M. van de Kerkhof, “Modeling polarization for hyper-NA lithography tools and masks,” Proc. SPIE 6520, 65200D(2007).
[CrossRef]

2006 (1)

A. Erdmann, G. Citarella, P. Evanschitzky, H. Schermer, V. Philipsen, and P. De Bisschop, “Validity of Hopkins approximation in simulations of hyper NA (NA>1) line-space structures for an attenuated PSM mask,” Proc. SPIE 6154, 61540G (2006).
[CrossRef]

2005 (1)

R. Socha, X. Shi, and D. LeHoty, “Simultaneous source mask optimization (SMO),” Proc. SPIE 5853, 180–193 (2005).
[CrossRef]

2004 (1)

B. Streefkerk, J. Baselmans, W. Gehoel-van Ansem, J. Mulkens, C. Hoogendam, and M. Hoogendorp, “Extending optical lithography with immersion,” Proc. SPIE 5377, 285–305(2004).
[CrossRef]

1975 (1)

C. L. Andrews and D. Margolis, “Elemental derivation of the Babinet principle in electromagnetic form,” Am. J. Phys . 43, 672–676 (1975).
[CrossRef]

Adam, K.

K. Adam and M. Lam, “Hybrid Hopkins-Abbe method for modeling oblique angle mask effects in OPC,” Proc. SPIE 6924, 69241E (2008).
[CrossRef]

K. Adam, “Domain decomposition methods for the electromagnetic simulation of scattering from three-dimensional structures with applications in lithography,” Ph.D. dissertation (University of California, Berkeley, 2001).

Andrews, C. L.

C. L. Andrews and D. Margolis, “Elemental derivation of the Babinet principle in electromagnetic form,” Am. J. Phys . 43, 672–676 (1975).
[CrossRef]

Azpiroz, J. T.

J. T. Azpiroz, G. W. Burr, A. E. Rosenbluth, and M. Hibbs, “Massively-parallel FDTD simulations to address mask electromagnetic effects in hyper-NA immersion lithography,” Proc. SPIE 6924, 69240Y (2008).
[CrossRef]

J. T. Azpiroz, “Analysis and modeling of photomask near-fields sub-wavelength deep ultraviolet lithography with optical proximity corrections,” Ph.D. dissertation (University of California, Los Angeles, 2004).

Baselmans, J.

B. Streefkerk, J. Baselmans, W. Gehoel-van Ansem, J. Mulkens, C. Hoogendam, and M. Hoogendorp, “Extending optical lithography with immersion,” Proc. SPIE 5377, 285–305(2004).
[CrossRef]

Belledent, J.

M. Saied, F. Foussadier, J. Belledent, Y. Trouiller, I. Schanen, E. Yesilada, C. Gardin, J. C. Urbani, F. Sundermann, F. Robert, C. Couderc, F. Vautrin, L. LeCam, G. Kerrien, J. Planchot, C. Martinelli, B. Wilkinson, Y. Rody, A. Borjon, N. Morgana, J.-L. Di-Maria, and V. Farys, “3D mask modeling with oblique incidence and mask corner rounding effects for the 32nm node,” Proc. SPIE 6730, 673050(2007).
[CrossRef]

Borjon, A.

M. Saied, F. Foussadier, J. Belledent, Y. Trouiller, I. Schanen, E. Yesilada, C. Gardin, J. C. Urbani, F. Sundermann, F. Robert, C. Couderc, F. Vautrin, L. LeCam, G. Kerrien, J. Planchot, C. Martinelli, B. Wilkinson, Y. Rody, A. Borjon, N. Morgana, J.-L. Di-Maria, and V. Farys, “3D mask modeling with oblique incidence and mask corner rounding effects for the 32nm node,” Proc. SPIE 6730, 673050(2007).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed.(Pergamon, 1987).

Burr, G. W.

J. T. Azpiroz, G. W. Burr, A. E. Rosenbluth, and M. Hibbs, “Massively-parallel FDTD simulations to address mask electromagnetic effects in hyper-NA immersion lithography,” Proc. SPIE 6924, 69240Y (2008).
[CrossRef]

Cao, Y.

P. Liu, Y. Cao, L. Chen, G. Chen, M. Feng, J. Jiang, H.-Y. Liu, S. Suh, S.-W. Lee, and S. Lee, “Fast and accurate 3D mask model for full-chip OPC and verification,” Proc. SPIE 6520, 65200R (2007).
[CrossRef]

Chen, G.

P. Liu, Y. Cao, L. Chen, G. Chen, M. Feng, J. Jiang, H.-Y. Liu, S. Suh, S.-W. Lee, and S. Lee, “Fast and accurate 3D mask model for full-chip OPC and verification,” Proc. SPIE 6520, 65200R (2007).
[CrossRef]

Chen, L.

P. Liu, Y. Cao, L. Chen, G. Chen, M. Feng, J. Jiang, H.-Y. Liu, S. Suh, S.-W. Lee, and S. Lee, “Fast and accurate 3D mask model for full-chip OPC and verification,” Proc. SPIE 6520, 65200R (2007).
[CrossRef]

Citarella, G.

A. Erdmann, G. Citarella, P. Evanschitzky, H. Schermer, V. Philipsen, and P. De Bisschop, “Validity of Hopkins approximation in simulations of hyper NA (NA>1) line-space structures for an attenuated PSM mask,” Proc. SPIE 6154, 61540G (2006).
[CrossRef]

Couderc, C.

M. Saied, F. Foussadier, J. Belledent, Y. Trouiller, I. Schanen, E. Yesilada, C. Gardin, J. C. Urbani, F. Sundermann, F. Robert, C. Couderc, F. Vautrin, L. LeCam, G. Kerrien, J. Planchot, C. Martinelli, B. Wilkinson, Y. Rody, A. Borjon, N. Morgana, J.-L. Di-Maria, and V. Farys, “3D mask modeling with oblique incidence and mask corner rounding effects for the 32nm node,” Proc. SPIE 6730, 673050(2007).
[CrossRef]

De Bisschop, P.

A. Erdmann, G. Citarella, P. Evanschitzky, H. Schermer, V. Philipsen, and P. De Bisschop, “Validity of Hopkins approximation in simulations of hyper NA (NA>1) line-space structures for an attenuated PSM mask,” Proc. SPIE 6154, 61540G (2006).
[CrossRef]

de Boeij, W.

K. Lai, A. E. Rosenbluth, G. Han, J. Tirapu-Azpiroz, J. Meiring, A. Goehnermeier, B. Kneer, M. Totzeck, L. de Winter, W. de Boeij, and M. van de Kerkhof, “Modeling polarization for hyper-NA lithography tools and masks,” Proc. SPIE 6520, 65200D(2007).
[CrossRef]

de Winter, L.

K. Lai, A. E. Rosenbluth, G. Han, J. Tirapu-Azpiroz, J. Meiring, A. Goehnermeier, B. Kneer, M. Totzeck, L. de Winter, W. de Boeij, and M. van de Kerkhof, “Modeling polarization for hyper-NA lithography tools and masks,” Proc. SPIE 6520, 65200D(2007).
[CrossRef]

Di-Maria, J.-L.

M. Saied, F. Foussadier, J. Belledent, Y. Trouiller, I. Schanen, E. Yesilada, C. Gardin, J. C. Urbani, F. Sundermann, F. Robert, C. Couderc, F. Vautrin, L. LeCam, G. Kerrien, J. Planchot, C. Martinelli, B. Wilkinson, Y. Rody, A. Borjon, N. Morgana, J.-L. Di-Maria, and V. Farys, “3D mask modeling with oblique incidence and mask corner rounding effects for the 32nm node,” Proc. SPIE 6730, 673050(2007).
[CrossRef]

Erdmann, A.

A. Erdmann, G. Citarella, P. Evanschitzky, H. Schermer, V. Philipsen, and P. De Bisschop, “Validity of Hopkins approximation in simulations of hyper NA (NA>1) line-space structures for an attenuated PSM mask,” Proc. SPIE 6154, 61540G (2006).
[CrossRef]

Evanschitzky, P.

A. Erdmann, G. Citarella, P. Evanschitzky, H. Schermer, V. Philipsen, and P. De Bisschop, “Validity of Hopkins approximation in simulations of hyper NA (NA>1) line-space structures for an attenuated PSM mask,” Proc. SPIE 6154, 61540G (2006).
[CrossRef]

Farys, V.

M. Saied, F. Foussadier, J. Belledent, Y. Trouiller, I. Schanen, E. Yesilada, C. Gardin, J. C. Urbani, F. Sundermann, F. Robert, C. Couderc, F. Vautrin, L. LeCam, G. Kerrien, J. Planchot, C. Martinelli, B. Wilkinson, Y. Rody, A. Borjon, N. Morgana, J.-L. Di-Maria, and V. Farys, “3D mask modeling with oblique incidence and mask corner rounding effects for the 32nm node,” Proc. SPIE 6730, 673050(2007).
[CrossRef]

Feng, M.

P. Liu, Y. Cao, L. Chen, G. Chen, M. Feng, J. Jiang, H.-Y. Liu, S. Suh, S.-W. Lee, and S. Lee, “Fast and accurate 3D mask model for full-chip OPC and verification,” Proc. SPIE 6520, 65200R (2007).
[CrossRef]

Foussadier, F.

M. Saied, F. Foussadier, J. Belledent, Y. Trouiller, I. Schanen, E. Yesilada, C. Gardin, J. C. Urbani, F. Sundermann, F. Robert, C. Couderc, F. Vautrin, L. LeCam, G. Kerrien, J. Planchot, C. Martinelli, B. Wilkinson, Y. Rody, A. Borjon, N. Morgana, J.-L. Di-Maria, and V. Farys, “3D mask modeling with oblique incidence and mask corner rounding effects for the 32nm node,” Proc. SPIE 6730, 673050(2007).
[CrossRef]

Gardin, C.

M. Saied, F. Foussadier, J. Belledent, Y. Trouiller, I. Schanen, E. Yesilada, C. Gardin, J. C. Urbani, F. Sundermann, F. Robert, C. Couderc, F. Vautrin, L. LeCam, G. Kerrien, J. Planchot, C. Martinelli, B. Wilkinson, Y. Rody, A. Borjon, N. Morgana, J.-L. Di-Maria, and V. Farys, “3D mask modeling with oblique incidence and mask corner rounding effects for the 32nm node,” Proc. SPIE 6730, 673050(2007).
[CrossRef]

Gehoel-van Ansem, W.

B. Streefkerk, J. Baselmans, W. Gehoel-van Ansem, J. Mulkens, C. Hoogendam, and M. Hoogendorp, “Extending optical lithography with immersion,” Proc. SPIE 5377, 285–305(2004).
[CrossRef]

Goehnermeier, A.

K. Lai, A. E. Rosenbluth, G. Han, J. Tirapu-Azpiroz, J. Meiring, A. Goehnermeier, B. Kneer, M. Totzeck, L. de Winter, W. de Boeij, and M. van de Kerkhof, “Modeling polarization for hyper-NA lithography tools and masks,” Proc. SPIE 6520, 65200D(2007).
[CrossRef]

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, 7th ed. (Academic, 2007).

Han, G.

K. Lai, A. E. Rosenbluth, G. Han, J. Tirapu-Azpiroz, J. Meiring, A. Goehnermeier, B. Kneer, M. Totzeck, L. de Winter, W. de Boeij, and M. van de Kerkhof, “Modeling polarization for hyper-NA lithography tools and masks,” Proc. SPIE 6520, 65200D(2007).
[CrossRef]

Hibbs, M.

J. T. Azpiroz, G. W. Burr, A. E. Rosenbluth, and M. Hibbs, “Massively-parallel FDTD simulations to address mask electromagnetic effects in hyper-NA immersion lithography,” Proc. SPIE 6924, 69240Y (2008).
[CrossRef]

Hoogendam, C.

B. Streefkerk, J. Baselmans, W. Gehoel-van Ansem, J. Mulkens, C. Hoogendam, and M. Hoogendorp, “Extending optical lithography with immersion,” Proc. SPIE 5377, 285–305(2004).
[CrossRef]

Hoogendorp, M.

B. Streefkerk, J. Baselmans, W. Gehoel-van Ansem, J. Mulkens, C. Hoogendam, and M. Hoogendorp, “Extending optical lithography with immersion,” Proc. SPIE 5377, 285–305(2004).
[CrossRef]

Jiang, J.

P. Liu, Y. Cao, L. Chen, G. Chen, M. Feng, J. Jiang, H.-Y. Liu, S. Suh, S.-W. Lee, and S. Lee, “Fast and accurate 3D mask model for full-chip OPC and verification,” Proc. SPIE 6520, 65200R (2007).
[CrossRef]

Kerrien, G.

M. Saied, F. Foussadier, J. Belledent, Y. Trouiller, I. Schanen, E. Yesilada, C. Gardin, J. C. Urbani, F. Sundermann, F. Robert, C. Couderc, F. Vautrin, L. LeCam, G. Kerrien, J. Planchot, C. Martinelli, B. Wilkinson, Y. Rody, A. Borjon, N. Morgana, J.-L. Di-Maria, and V. Farys, “3D mask modeling with oblique incidence and mask corner rounding effects for the 32nm node,” Proc. SPIE 6730, 673050(2007).
[CrossRef]

Khalil, D.

T. Tawfik, A. Morshed, and D. Khalil, “Modeling mask scattered field at oblique incidence,” Proc. SPIE 7274, 727433 (2009).
[CrossRef]

Kneer, B.

K. Lai, A. E. Rosenbluth, G. Han, J. Tirapu-Azpiroz, J. Meiring, A. Goehnermeier, B. Kneer, M. Totzeck, L. de Winter, W. de Boeij, and M. van de Kerkhof, “Modeling polarization for hyper-NA lithography tools and masks,” Proc. SPIE 6520, 65200D(2007).
[CrossRef]

Lai, K.

K. Lai, A. E. Rosenbluth, G. Han, J. Tirapu-Azpiroz, J. Meiring, A. Goehnermeier, B. Kneer, M. Totzeck, L. de Winter, W. de Boeij, and M. van de Kerkhof, “Modeling polarization for hyper-NA lithography tools and masks,” Proc. SPIE 6520, 65200D(2007).
[CrossRef]

Lam, M.

K. Adam and M. Lam, “Hybrid Hopkins-Abbe method for modeling oblique angle mask effects in OPC,” Proc. SPIE 6924, 69241E (2008).
[CrossRef]

LeCam, L.

M. Saied, F. Foussadier, J. Belledent, Y. Trouiller, I. Schanen, E. Yesilada, C. Gardin, J. C. Urbani, F. Sundermann, F. Robert, C. Couderc, F. Vautrin, L. LeCam, G. Kerrien, J. Planchot, C. Martinelli, B. Wilkinson, Y. Rody, A. Borjon, N. Morgana, J.-L. Di-Maria, and V. Farys, “3D mask modeling with oblique incidence and mask corner rounding effects for the 32nm node,” Proc. SPIE 6730, 673050(2007).
[CrossRef]

Lee, S.

P. Liu, Y. Cao, L. Chen, G. Chen, M. Feng, J. Jiang, H.-Y. Liu, S. Suh, S.-W. Lee, and S. Lee, “Fast and accurate 3D mask model for full-chip OPC and verification,” Proc. SPIE 6520, 65200R (2007).
[CrossRef]

Lee, S.-W.

P. Liu, Y. Cao, L. Chen, G. Chen, M. Feng, J. Jiang, H.-Y. Liu, S. Suh, S.-W. Lee, and S. Lee, “Fast and accurate 3D mask model for full-chip OPC and verification,” Proc. SPIE 6520, 65200R (2007).
[CrossRef]

LeHoty, D.

R. Socha, X. Shi, and D. LeHoty, “Simultaneous source mask optimization (SMO),” Proc. SPIE 5853, 180–193 (2005).
[CrossRef]

Liu, H.-Y.

P. Liu, Y. Cao, L. Chen, G. Chen, M. Feng, J. Jiang, H.-Y. Liu, S. Suh, S.-W. Lee, and S. Lee, “Fast and accurate 3D mask model for full-chip OPC and verification,” Proc. SPIE 6520, 65200R (2007).
[CrossRef]

Liu, P.

P. Liu, Y. Cao, L. Chen, G. Chen, M. Feng, J. Jiang, H.-Y. Liu, S. Suh, S.-W. Lee, and S. Lee, “Fast and accurate 3D mask model for full-chip OPC and verification,” Proc. SPIE 6520, 65200R (2007).
[CrossRef]

Mack, C.

C. Mack, Fundamental Principles of Optical Lithography(Wiley, 2007).
[CrossRef]

Margolis, D.

C. L. Andrews and D. Margolis, “Elemental derivation of the Babinet principle in electromagnetic form,” Am. J. Phys . 43, 672–676 (1975).
[CrossRef]

Martinelli, C.

M. Saied, F. Foussadier, J. Belledent, Y. Trouiller, I. Schanen, E. Yesilada, C. Gardin, J. C. Urbani, F. Sundermann, F. Robert, C. Couderc, F. Vautrin, L. LeCam, G. Kerrien, J. Planchot, C. Martinelli, B. Wilkinson, Y. Rody, A. Borjon, N. Morgana, J.-L. Di-Maria, and V. Farys, “3D mask modeling with oblique incidence and mask corner rounding effects for the 32nm node,” Proc. SPIE 6730, 673050(2007).
[CrossRef]

Meiring, J.

K. Lai, A. E. Rosenbluth, G. Han, J. Tirapu-Azpiroz, J. Meiring, A. Goehnermeier, B. Kneer, M. Totzeck, L. de Winter, W. de Boeij, and M. van de Kerkhof, “Modeling polarization for hyper-NA lithography tools and masks,” Proc. SPIE 6520, 65200D(2007).
[CrossRef]

Morgana, N.

M. Saied, F. Foussadier, J. Belledent, Y. Trouiller, I. Schanen, E. Yesilada, C. Gardin, J. C. Urbani, F. Sundermann, F. Robert, C. Couderc, F. Vautrin, L. LeCam, G. Kerrien, J. Planchot, C. Martinelli, B. Wilkinson, Y. Rody, A. Borjon, N. Morgana, J.-L. Di-Maria, and V. Farys, “3D mask modeling with oblique incidence and mask corner rounding effects for the 32nm node,” Proc. SPIE 6730, 673050(2007).
[CrossRef]

Morshed, A.

T. Tawfik, A. Morshed, and D. Khalil, “Modeling mask scattered field at oblique incidence,” Proc. SPIE 7274, 727433 (2009).
[CrossRef]

Mulkens, J.

B. Streefkerk, J. Baselmans, W. Gehoel-van Ansem, J. Mulkens, C. Hoogendam, and M. Hoogendorp, “Extending optical lithography with immersion,” Proc. SPIE 5377, 285–305(2004).
[CrossRef]

Philipsen, V.

A. Erdmann, G. Citarella, P. Evanschitzky, H. Schermer, V. Philipsen, and P. De Bisschop, “Validity of Hopkins approximation in simulations of hyper NA (NA>1) line-space structures for an attenuated PSM mask,” Proc. SPIE 6154, 61540G (2006).
[CrossRef]

Pistor, T. V.

T. V. Pistor, “Electromagnetic simulation and modeling with applications in lithography,” Ph.D. dissertation (University of California, Berkeley, 2001).

Planchot, J.

M. Saied, F. Foussadier, J. Belledent, Y. Trouiller, I. Schanen, E. Yesilada, C. Gardin, J. C. Urbani, F. Sundermann, F. Robert, C. Couderc, F. Vautrin, L. LeCam, G. Kerrien, J. Planchot, C. Martinelli, B. Wilkinson, Y. Rody, A. Borjon, N. Morgana, J.-L. Di-Maria, and V. Farys, “3D mask modeling with oblique incidence and mask corner rounding effects for the 32nm node,” Proc. SPIE 6730, 673050(2007).
[CrossRef]

Robert, F.

M. Saied, F. Foussadier, J. Belledent, Y. Trouiller, I. Schanen, E. Yesilada, C. Gardin, J. C. Urbani, F. Sundermann, F. Robert, C. Couderc, F. Vautrin, L. LeCam, G. Kerrien, J. Planchot, C. Martinelli, B. Wilkinson, Y. Rody, A. Borjon, N. Morgana, J.-L. Di-Maria, and V. Farys, “3D mask modeling with oblique incidence and mask corner rounding effects for the 32nm node,” Proc. SPIE 6730, 673050(2007).
[CrossRef]

Rody, Y.

M. Saied, F. Foussadier, J. Belledent, Y. Trouiller, I. Schanen, E. Yesilada, C. Gardin, J. C. Urbani, F. Sundermann, F. Robert, C. Couderc, F. Vautrin, L. LeCam, G. Kerrien, J. Planchot, C. Martinelli, B. Wilkinson, Y. Rody, A. Borjon, N. Morgana, J.-L. Di-Maria, and V. Farys, “3D mask modeling with oblique incidence and mask corner rounding effects for the 32nm node,” Proc. SPIE 6730, 673050(2007).
[CrossRef]

Rosenbluth, A. E.

J. T. Azpiroz, G. W. Burr, A. E. Rosenbluth, and M. Hibbs, “Massively-parallel FDTD simulations to address mask electromagnetic effects in hyper-NA immersion lithography,” Proc. SPIE 6924, 69240Y (2008).
[CrossRef]

K. Lai, A. E. Rosenbluth, G. Han, J. Tirapu-Azpiroz, J. Meiring, A. Goehnermeier, B. Kneer, M. Totzeck, L. de Winter, W. de Boeij, and M. van de Kerkhof, “Modeling polarization for hyper-NA lithography tools and masks,” Proc. SPIE 6520, 65200D(2007).
[CrossRef]

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, 7th ed. (Academic, 2007).

Saied, M.

M. Saied, F. Foussadier, J. Belledent, Y. Trouiller, I. Schanen, E. Yesilada, C. Gardin, J. C. Urbani, F. Sundermann, F. Robert, C. Couderc, F. Vautrin, L. LeCam, G. Kerrien, J. Planchot, C. Martinelli, B. Wilkinson, Y. Rody, A. Borjon, N. Morgana, J.-L. Di-Maria, and V. Farys, “3D mask modeling with oblique incidence and mask corner rounding effects for the 32nm node,” Proc. SPIE 6730, 673050(2007).
[CrossRef]

Schanen, I.

M. Saied, F. Foussadier, J. Belledent, Y. Trouiller, I. Schanen, E. Yesilada, C. Gardin, J. C. Urbani, F. Sundermann, F. Robert, C. Couderc, F. Vautrin, L. LeCam, G. Kerrien, J. Planchot, C. Martinelli, B. Wilkinson, Y. Rody, A. Borjon, N. Morgana, J.-L. Di-Maria, and V. Farys, “3D mask modeling with oblique incidence and mask corner rounding effects for the 32nm node,” Proc. SPIE 6730, 673050(2007).
[CrossRef]

Schermer, H.

A. Erdmann, G. Citarella, P. Evanschitzky, H. Schermer, V. Philipsen, and P. De Bisschop, “Validity of Hopkins approximation in simulations of hyper NA (NA>1) line-space structures for an attenuated PSM mask,” Proc. SPIE 6154, 61540G (2006).
[CrossRef]

Shi, X.

R. Socha, X. Shi, and D. LeHoty, “Simultaneous source mask optimization (SMO),” Proc. SPIE 5853, 180–193 (2005).
[CrossRef]

Socha, R.

R. Socha, X. Shi, and D. LeHoty, “Simultaneous source mask optimization (SMO),” Proc. SPIE 5853, 180–193 (2005).
[CrossRef]

Streefkerk, B.

B. Streefkerk, J. Baselmans, W. Gehoel-van Ansem, J. Mulkens, C. Hoogendam, and M. Hoogendorp, “Extending optical lithography with immersion,” Proc. SPIE 5377, 285–305(2004).
[CrossRef]

Suh, S.

P. Liu, Y. Cao, L. Chen, G. Chen, M. Feng, J. Jiang, H.-Y. Liu, S. Suh, S.-W. Lee, and S. Lee, “Fast and accurate 3D mask model for full-chip OPC and verification,” Proc. SPIE 6520, 65200R (2007).
[CrossRef]

Sundermann, F.

M. Saied, F. Foussadier, J. Belledent, Y. Trouiller, I. Schanen, E. Yesilada, C. Gardin, J. C. Urbani, F. Sundermann, F. Robert, C. Couderc, F. Vautrin, L. LeCam, G. Kerrien, J. Planchot, C. Martinelli, B. Wilkinson, Y. Rody, A. Borjon, N. Morgana, J.-L. Di-Maria, and V. Farys, “3D mask modeling with oblique incidence and mask corner rounding effects for the 32nm node,” Proc. SPIE 6730, 673050(2007).
[CrossRef]

Tawfik, T.

T. Tawfik, A. Morshed, and D. Khalil, “Modeling mask scattered field at oblique incidence,” Proc. SPIE 7274, 727433 (2009).
[CrossRef]

T. Tawfik and E. Tejnil, “Characterizing OPC model accuracy versus lens induced polarization effects in hyper NA immersion lithography,” Proc. SPIE 7122, 712245 (2008).
[CrossRef]

Tawfik, T. M.

T. M. Tawfik, “Optical lithography enhancement techniques for microelectronics,” M.S. thesis (Ain Shams University, Cairo, Egypt, 2009).

Tejnil, E.

T. Tawfik and E. Tejnil, “Characterizing OPC model accuracy versus lens induced polarization effects in hyper NA immersion lithography,” Proc. SPIE 7122, 712245 (2008).
[CrossRef]

Tirapu-Azpiroz, J.

K. Lai, A. E. Rosenbluth, G. Han, J. Tirapu-Azpiroz, J. Meiring, A. Goehnermeier, B. Kneer, M. Totzeck, L. de Winter, W. de Boeij, and M. van de Kerkhof, “Modeling polarization for hyper-NA lithography tools and masks,” Proc. SPIE 6520, 65200D(2007).
[CrossRef]

Totzeck, M.

K. Lai, A. E. Rosenbluth, G. Han, J. Tirapu-Azpiroz, J. Meiring, A. Goehnermeier, B. Kneer, M. Totzeck, L. de Winter, W. de Boeij, and M. van de Kerkhof, “Modeling polarization for hyper-NA lithography tools and masks,” Proc. SPIE 6520, 65200D(2007).
[CrossRef]

Trouiller, Y.

M. Saied, F. Foussadier, J. Belledent, Y. Trouiller, I. Schanen, E. Yesilada, C. Gardin, J. C. Urbani, F. Sundermann, F. Robert, C. Couderc, F. Vautrin, L. LeCam, G. Kerrien, J. Planchot, C. Martinelli, B. Wilkinson, Y. Rody, A. Borjon, N. Morgana, J.-L. Di-Maria, and V. Farys, “3D mask modeling with oblique incidence and mask corner rounding effects for the 32nm node,” Proc. SPIE 6730, 673050(2007).
[CrossRef]

Ufimtsev, P. Y. A.

P. Y. A. Ufimtsev, Fundamentals of Physical Theory of Diffraction (Wiley, 2007).
[CrossRef]

Urbani, J. C.

M. Saied, F. Foussadier, J. Belledent, Y. Trouiller, I. Schanen, E. Yesilada, C. Gardin, J. C. Urbani, F. Sundermann, F. Robert, C. Couderc, F. Vautrin, L. LeCam, G. Kerrien, J. Planchot, C. Martinelli, B. Wilkinson, Y. Rody, A. Borjon, N. Morgana, J.-L. Di-Maria, and V. Farys, “3D mask modeling with oblique incidence and mask corner rounding effects for the 32nm node,” Proc. SPIE 6730, 673050(2007).
[CrossRef]

van de Kerkhof, M.

K. Lai, A. E. Rosenbluth, G. Han, J. Tirapu-Azpiroz, J. Meiring, A. Goehnermeier, B. Kneer, M. Totzeck, L. de Winter, W. de Boeij, and M. van de Kerkhof, “Modeling polarization for hyper-NA lithography tools and masks,” Proc. SPIE 6520, 65200D(2007).
[CrossRef]

Vautrin, F.

M. Saied, F. Foussadier, J. Belledent, Y. Trouiller, I. Schanen, E. Yesilada, C. Gardin, J. C. Urbani, F. Sundermann, F. Robert, C. Couderc, F. Vautrin, L. LeCam, G. Kerrien, J. Planchot, C. Martinelli, B. Wilkinson, Y. Rody, A. Borjon, N. Morgana, J.-L. Di-Maria, and V. Farys, “3D mask modeling with oblique incidence and mask corner rounding effects for the 32nm node,” Proc. SPIE 6730, 673050(2007).
[CrossRef]

Wilkinson, B.

M. Saied, F. Foussadier, J. Belledent, Y. Trouiller, I. Schanen, E. Yesilada, C. Gardin, J. C. Urbani, F. Sundermann, F. Robert, C. Couderc, F. Vautrin, L. LeCam, G. Kerrien, J. Planchot, C. Martinelli, B. Wilkinson, Y. Rody, A. Borjon, N. Morgana, J.-L. Di-Maria, and V. Farys, “3D mask modeling with oblique incidence and mask corner rounding effects for the 32nm node,” Proc. SPIE 6730, 673050(2007).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th ed.(Pergamon, 1987).

Wong, A.

A. Wong, Resolution Enhancement Techniques in Optical Lithography (SPIE Press, 2001).
[CrossRef]

Yesilada, E.

M. Saied, F. Foussadier, J. Belledent, Y. Trouiller, I. Schanen, E. Yesilada, C. Gardin, J. C. Urbani, F. Sundermann, F. Robert, C. Couderc, F. Vautrin, L. LeCam, G. Kerrien, J. Planchot, C. Martinelli, B. Wilkinson, Y. Rody, A. Borjon, N. Morgana, J.-L. Di-Maria, and V. Farys, “3D mask modeling with oblique incidence and mask corner rounding effects for the 32nm node,” Proc. SPIE 6730, 673050(2007).
[CrossRef]

Am. J. Phys (1)

C. L. Andrews and D. Margolis, “Elemental derivation of the Babinet principle in electromagnetic form,” Am. J. Phys . 43, 672–676 (1975).
[CrossRef]

Proc. SPIE (10)

P. Liu, Y. Cao, L. Chen, G. Chen, M. Feng, J. Jiang, H.-Y. Liu, S. Suh, S.-W. Lee, and S. Lee, “Fast and accurate 3D mask model for full-chip OPC and verification,” Proc. SPIE 6520, 65200R (2007).
[CrossRef]

B. Streefkerk, J. Baselmans, W. Gehoel-van Ansem, J. Mulkens, C. Hoogendam, and M. Hoogendorp, “Extending optical lithography with immersion,” Proc. SPIE 5377, 285–305(2004).
[CrossRef]

K. Lai, A. E. Rosenbluth, G. Han, J. Tirapu-Azpiroz, J. Meiring, A. Goehnermeier, B. Kneer, M. Totzeck, L. de Winter, W. de Boeij, and M. van de Kerkhof, “Modeling polarization for hyper-NA lithography tools and masks,” Proc. SPIE 6520, 65200D(2007).
[CrossRef]

T. Tawfik and E. Tejnil, “Characterizing OPC model accuracy versus lens induced polarization effects in hyper NA immersion lithography,” Proc. SPIE 7122, 712245 (2008).
[CrossRef]

A. Erdmann, G. Citarella, P. Evanschitzky, H. Schermer, V. Philipsen, and P. De Bisschop, “Validity of Hopkins approximation in simulations of hyper NA (NA>1) line-space structures for an attenuated PSM mask,” Proc. SPIE 6154, 61540G (2006).
[CrossRef]

M. Saied, F. Foussadier, J. Belledent, Y. Trouiller, I. Schanen, E. Yesilada, C. Gardin, J. C. Urbani, F. Sundermann, F. Robert, C. Couderc, F. Vautrin, L. LeCam, G. Kerrien, J. Planchot, C. Martinelli, B. Wilkinson, Y. Rody, A. Borjon, N. Morgana, J.-L. Di-Maria, and V. Farys, “3D mask modeling with oblique incidence and mask corner rounding effects for the 32nm node,” Proc. SPIE 6730, 673050(2007).
[CrossRef]

J. T. Azpiroz, G. W. Burr, A. E. Rosenbluth, and M. Hibbs, “Massively-parallel FDTD simulations to address mask electromagnetic effects in hyper-NA immersion lithography,” Proc. SPIE 6924, 69240Y (2008).
[CrossRef]

K. Adam and M. Lam, “Hybrid Hopkins-Abbe method for modeling oblique angle mask effects in OPC,” Proc. SPIE 6924, 69241E (2008).
[CrossRef]

R. Socha, X. Shi, and D. LeHoty, “Simultaneous source mask optimization (SMO),” Proc. SPIE 5853, 180–193 (2005).
[CrossRef]

T. Tawfik, A. Morshed, and D. Khalil, “Modeling mask scattered field at oblique incidence,” Proc. SPIE 7274, 727433 (2009).
[CrossRef]

Other (11)

T. M. Tawfik, “Optical lithography enhancement techniques for microelectronics,” M.S. thesis (Ain Shams University, Cairo, Egypt, 2009).

P. Y. A. Ufimtsev, Fundamentals of Physical Theory of Diffraction (Wiley, 2007).
[CrossRef]

www.panoramictech.com.

M. Born and E. Wolf, Principles of Optics, 7th ed.(Pergamon, 1987).

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, 7th ed. (Academic, 2007).

C. Mack, Fundamental Principles of Optical Lithography(Wiley, 2007).
[CrossRef]

T. V. Pistor, “Electromagnetic simulation and modeling with applications in lithography,” Ph.D. dissertation (University of California, Berkeley, 2001).

K. Adam, “Domain decomposition methods for the electromagnetic simulation of scattering from three-dimensional structures with applications in lithography,” Ph.D. dissertation (University of California, Berkeley, 2001).

J. T. Azpiroz, “Analysis and modeling of photomask near-fields sub-wavelength deep ultraviolet lithography with optical proximity corrections,” Ph.D. dissertation (University of California, Los Angeles, 2004).

http://www.itrs.net/Links/2007ITRS/2007_Chapters/2007_Lithography.pdf.

A. Wong, Resolution Enhancement Techniques in Optical Lithography (SPIE Press, 2001).
[CrossRef]

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Figures (10)

Fig. 1
Fig. 1

Configuration of case study.

Fig. 2
Fig. 2

Babinet’s principle applied to a rectangular aperture. Fields incident onto and diffracted from (a) a rectangular aperture and (b) a complementary screen are equivalent by means of Babinet’s principle.

Fig. 3
Fig. 3

Illustration of the incidence angles to diffracting edges.

Fig. 4
Fig. 4

Error introduced by considering G φ constant over the integration limits.

Fig. 5
Fig. 5

As aspect ratio increases, polarization coupling significantly diminishes.

Fig. 6
Fig. 6

1D ISO line CD error at different off-axis point source illuminations at (a) the image plane, (b) + 50 nm defocus, and (c) 50 nm defocus.

Fig. 7
Fig. 7

Field intensity at (a) the image plane, (b) + 50 nm defocus, and (c) 50 nm defocus.

Fig. 8
Fig. 8

32 nm CD through pitch test patterns at (a) the image plane, (b) + 50 nm defocus, and (c) 50 nm defocus.

Fig. 9
Fig. 9

45 nm CD through pitch test patterns at (a) the image plane, (b) + 50 nm defocus, and (c) 50 nm defocus.

Fig. 10
Fig. 10

65 nm CD through pitch test patterns at (a) the image plane, (b) + 50 nm defocus, and (c) 50 nm defocus.

Equations (36)

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E inc = y i E o exp { j k ( x i cos ( γ o ) + z i sin ( γ o ) ) } ,
H inc = 1 η o k × E inc ,
E D = E FR D + E PO D ,
E P O C = j η o k D y D y D x D x ( z i × 2 H inc C ) e j k R 4 π R d x i d y i = j k D y D y D x D x E inc C e j k R 2 π R d x i d y i = j η o k D y D y D x D x H inc e j k R 2 π R d x i d y i ,
E x C x i = η o H x x i = ( z i × E y y i ) , E y y i = z i × E x C x i ,
E y y i = j η o k ( z i × x i ) D y D y D x D x H x inc e j k R 2 π R d x i d y i = y j k D y D y D x D x E y inc e j k R 2 π R d x i d y i ,
E y = y i j k E o { D y D y D x D x exp { j k [ x i cos ( γ o ) + R ] } 2 π R d x i d y i } .
E FR = C ( E inc tang F θ + η o H inc tang G θ ) θ + ( η o H inc tang G ϕ ) ϕ e j k R 2 π R d l ,
E FR C = 2 η o D y D y H y C { G θ 1 θ + G φ 1 φ } G ( R 1 ) d y i + 2 D x D x E x C F θ 2 G ( R 2 ) d x i θ 2 η o D y D y H y C { G θ 3 θ + G φ 3 φ } G ( R 3 ) d y i 2 D x D x E x C F θ 4 G ( R 4 ) d x i θ ,
E FR C = 2 η o G φ 1 D y D y H y C G ( R 1 ) d y i φ + 2 F θ D x D x E x C G ( R 2 ) d x i θ 2 η o G φ 3 D y D y H y C G ( R 3 ) d y i φ 2 F θ D x D x E x C G ( R 4 ) d x i θ .
E FR = y ^ 2 E o sin ( γ o ) { G φ 1 e j k D x cos ( γ o ) D y D y G ( R 1 ) d y i + G φ 3 e j k D x cos ( γ o ) D y D y G ( R 3 ) d y i + F θ D x D x e j k x i cos ( γ o ) G ( R 2 ) d x i + F θ D x D x e j k x i cos ( γ o ) G ( R 4 ) d x i } .
G ( R 2 ) = e j k R 2 4 π R 2 e j k ( R o + 2 x x i 2 y D y 2 R o ) 4 π R o , G ( R 4 ) = e j k R 4 4 π R 4 e j k ( R o + 2 x x i + 2 y D y 2 R o ) 4 π R o ,
H 0 ( 1 ) ( k d ) = 1 j π e j k d 2 + ξ 2 d 2 + ξ 2 d ξ ,
E FR = y ^ 2 E o sin ( γ o ) { j π G φ 1 e j k D x cos ( γ o ) H o ( 1 ) ( k z 2 + ( x + D x ) 2 ) + j π G φ 3 e j k D x cos ( γ o ) H o ( 1 ) ( k z 2 + ( x D x ) 2 ) + F θ D x e j k R o π R o cos ( k y D y R o ) sinc ( k D x ( x R o + cos ( γ o ) ) } ,
θ inc max = sin 1 ( NA R ) ,
F θ = ( A θ 3 B θ 2 C θ + D ) ( 2 cos 2 ( γ o ) sin ( γ o ) ) ; at oblique incidence = 1 4 θ 2 + 1 2 ; at normal incidence ,
A = 1 sin ( γ o ) cos ( γ o ) cos ( γ o ) 2 , B = 1 sin ( γ o ) , C = 2 ( 1 sin ( γ o ) ) cos ( γ o ) cos ( γ o ) , D = ( 2 ( 1 sin ( γ o ) ) ) / ( 2 cos 2 ( γ o ) sin ( γ o ) ) ,
G φ oblique G φ + Δ ( γ o ) ; φ inc = γ o > π / 2 G φ Δ ( γ o ) ; φ inc = γ o < π / 2 ,
E FR = y ^ 2 E o sin ( γ o ) { M G j π G φ H o ( 1 ) F θ D x e j k R o π R o cos ( k y D y R o ) sinc ( kD x ( x R o + cos ( γ o ) ) ) ,
M G = cos ( ψ ) ( 1 j Δ ( γ o ) tan ( ψ ) G φ ) ,
I image ( x , y ) = i 1 2 | E image , i ( x , y ) | 2 ,
E image , i = E PO + M G sin ( γ o ) E FR ,
I image = ( J ( f , g ) P ( κ , ζ ) P * ( κ , ζ ) d f d g ) O ( f , g ) O ( f , g ) e 2 π j [ ( f f ) x + ( g g ) y ] d f d g d f d g ,
Δ E E PO = E FR E PO = Δ d ( 2 D x + 2 D y ) ( D x D y ) = boundary layer area total area ,
Δ d = 1 M G sin ( γ o ) .
F θ ( θ , φ ) = [ U ( σ , π 2 ) + U ( σ , 3 π 2 ) ] sin ( θ ) ,
cos ( σ ) = cos 2 ( γ o ) + cos ( γ o ) cos ( θ ) sin 2 ( γ o ) ,
U ( σ , π 2 ) = A ( σ ) 4 sin 2 ( γ o ) 1 cos ( γ o ) cos ( θ ) + cos 2 ( γ o ) ,
U ( σ , 3 π 2 ) = B ( σ ) 4 sin 2 ( γ o ) ,
A ( σ ) = cot ( σ + π / 2 4 ) cot ( σ π / 2 4 ) = ( 2 ( cos 2 ( σ 1 4 ) cos 2 ( π 8 ) ) ) 1 ,
B ( σ ) = tan ( σ + π / 2 4 ) tan ( σ π / 2 4 ) = ( 2 ( cos 2 ( σ 1 4 ) cos 2 ( π 8 ) ) ) 1 ,
cos 2 ( σ 1 4 ) = 1 2 { 1 + 1 + cos ( σ 1 ) 2 } .
F θ = sin ( θ ) 1 + cos ( σ 1 ) ( sin 2 ( γ o ) ) ( cos ( σ 1 ) ) sin ( θ ) cos ( γ o ) cos ( θ ) + cos 2 ( γ o ) .
F θ = cos ( θ o ) ( 1 + cos ( γ o ) cos ( θ ) sin 2 ( γ o ) 1 ) cos ( γ o ) ( sin ( θ o ) + cos ( γ o ) ) .
F θ = ( A θ 3 B θ 2 C θ + D ) ( 2 cos 2 ( γ o ) sin ( γ o ) ) ,
A = 1 sin ( γ o ) cos ( γ o ) cos ( γ o ) 2 , B = 1 sin ( γ o ) , C = 2 ( 1 sin ( γ o ) ) cos ( γ o ) cos ( γ o ) , D = ( 2 ( 1 sin ( γ o ) ) ) / ( 2 cos 2 ( γ o ) sin ( γ o ) ) .

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