Abstract

We propose a method to narrow the gap between the rigorous methods for the propagation of partially coherent light, which require excessive computational capacity, and the numerical methods used in practical engineering applications, where it is not clear how to handle spatial and temporal coherence in a statistically correct manner. As is the case for the latter methods, the numerical method described can deal with fields with a large spatial and temporal extent, which is necessary in practical applications such as laser fusion or optical lithography. However, the method also takes a few steps toward a more rigorous, yet efficient, representation of the optical field, which depends on detailed specified coherence properties of the radiation. The described method uses a set of independent monochromatic fields at different oscillation frequencies. The frequencies are chosen such that the statistical properties of the integrated intensity closely resemble those from a full-time trace treatment. Finally, we demonstrate the capabilities and limitations of the method with a few numerical examples of the propagation of a large field with a specified spatial and temporal coherence.

© 2006 Optical Society of America

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  1. H. Martinsson, T. Sandstrom, A. Bleeker, and J. D. Hintersteiner, 'Current status of optical maskless lithography,' J. Microlithogr. Microfabr. Microsyst. 4, 1-15 (2005).
    [CrossRef]
  2. Y. Granik, 'Illuminator optimization methods in microlithography,' in Proc. SPIE 5524, 217-229 (2004).
    [CrossRef]
  3. P. Michaloski, 'Requirements and designs of illuminators for microlithography,' in Proc. SPIE 5525, 1-10 (2004).
    [CrossRef]
  4. J. W. Goodman, Statistical Optics (Wiley, 2000).
  5. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).
  6. M. J. Bastiaans, 'Application of the Wigner distribution function to partially coherent light,' in Proc. SPIE 3729, 114-128 (1999).
    [CrossRef]
  7. H. Gross, 'Numerical propagation of partially coherent laser beams,' Opt. Laser Technol. 29, 257-260 (1997).
    [CrossRef]
  8. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).
  9. Q. Lin and Y. Cai, 'Fractional Fourier transform for partially coherent Gaussian Schell-model beams,' Opt. Lett. 27, 1672-1674 (2002).
    [CrossRef]
  10. A. Thaning, A. T. Friberg, and Z. Jaroszewicz, 'Synthesis of diffractive axicons for partially coherent light based on asymptotic wave theory,' Opt. Lett. 26, 1648-1650 (2001).
    [CrossRef]
  11. S. Y. Popov and A. T. Friberg, 'Design of diffractive axicons for partially coherent light,' Opt. Lett. 23, 1639-1641 (1998).
    [CrossRef]
  12. E. Wolf, 'New theory of partial coherence in the space-frequency domain. Part I: Spectra and cross-spectra of steady-state sources,' J. Opt. Soc. Am. 72, 343-351 (1982).
    [CrossRef]
  13. E. Wolf, 'New theory of partial coherence in the space-frequency domain. Part II: Steady-state fields and higher-order correlations,' J. Opt. Soc. Am. A 3, 76-85 (1986).
    [CrossRef]
  14. Y. Lin and J. Buck, 'Numerical modeling of the excimer beam,' in Proc. SPIE 3677, 700-710 (1999).
    [CrossRef]
  15. B. Eppich, G. Mann, and H. Weber, 'Spatial coherence: comparison of interferometric and non-interferometric measurements,' in Proc. SPIE 4969, 137-148 (2003).
    [CrossRef]
  16. B. L. Anderson and P. L. Fuhr, 'Twin-fiber interferometric method for measuring spatial coherence,' Opt. Eng. 32, 926-932 (1993).
    [CrossRef]
  17. P. De Santis, F. Gori, G. Guattari, and C. Palma, 'Synthesis of partially coherent fields,' J. Opt. Soc. Am. A 3, 1258-1262 (1986).
    [CrossRef]
  18. A. T. Friberg, E. Tervonen, and J. Turunen, 'Interpretation and experimental demonstration of twisted Gaussian Schell-model beams,' J. Opt. Soc. Am. A 11, 1818-1825 (1994).
    [CrossRef]
  19. B. J. Thompson and E. Wolf, 'Two-beam interference with partially coherent light,' J. Opt. Soc. Am. 47, 895-902 (1957).
    [CrossRef]
  20. X. Deng, B. Bihari, J. Gan, F. Zhao, and R. T. Chen, 'Fast algorithm for chirp transforms with zooming-in ability and its applications,' J. Opt. Soc. Am. A 17, 762-771 (2000).
    [CrossRef]

2005 (1)

H. Martinsson, T. Sandstrom, A. Bleeker, and J. D. Hintersteiner, 'Current status of optical maskless lithography,' J. Microlithogr. Microfabr. Microsyst. 4, 1-15 (2005).
[CrossRef]

2004 (2)

Y. Granik, 'Illuminator optimization methods in microlithography,' in Proc. SPIE 5524, 217-229 (2004).
[CrossRef]

P. Michaloski, 'Requirements and designs of illuminators for microlithography,' in Proc. SPIE 5525, 1-10 (2004).
[CrossRef]

2003 (1)

B. Eppich, G. Mann, and H. Weber, 'Spatial coherence: comparison of interferometric and non-interferometric measurements,' in Proc. SPIE 4969, 137-148 (2003).
[CrossRef]

2002 (1)

2001 (1)

2000 (1)

1999 (2)

Y. Lin and J. Buck, 'Numerical modeling of the excimer beam,' in Proc. SPIE 3677, 700-710 (1999).
[CrossRef]

M. J. Bastiaans, 'Application of the Wigner distribution function to partially coherent light,' in Proc. SPIE 3729, 114-128 (1999).
[CrossRef]

1998 (1)

1997 (1)

H. Gross, 'Numerical propagation of partially coherent laser beams,' Opt. Laser Technol. 29, 257-260 (1997).
[CrossRef]

1994 (1)

1993 (1)

B. L. Anderson and P. L. Fuhr, 'Twin-fiber interferometric method for measuring spatial coherence,' Opt. Eng. 32, 926-932 (1993).
[CrossRef]

1986 (2)

1982 (1)

1957 (1)

Anderson, B. L.

B. L. Anderson and P. L. Fuhr, 'Twin-fiber interferometric method for measuring spatial coherence,' Opt. Eng. 32, 926-932 (1993).
[CrossRef]

Bastiaans, M. J.

M. J. Bastiaans, 'Application of the Wigner distribution function to partially coherent light,' in Proc. SPIE 3729, 114-128 (1999).
[CrossRef]

Bihari, B.

Bleeker, A.

H. Martinsson, T. Sandstrom, A. Bleeker, and J. D. Hintersteiner, 'Current status of optical maskless lithography,' J. Microlithogr. Microfabr. Microsyst. 4, 1-15 (2005).
[CrossRef]

Buck, J.

Y. Lin and J. Buck, 'Numerical modeling of the excimer beam,' in Proc. SPIE 3677, 700-710 (1999).
[CrossRef]

Cai, Y.

Chen, R. T.

De Santis, P.

Deng, X.

Eppich, B.

B. Eppich, G. Mann, and H. Weber, 'Spatial coherence: comparison of interferometric and non-interferometric measurements,' in Proc. SPIE 4969, 137-148 (2003).
[CrossRef]

Friberg, A. T.

Fuhr, P. L.

B. L. Anderson and P. L. Fuhr, 'Twin-fiber interferometric method for measuring spatial coherence,' Opt. Eng. 32, 926-932 (1993).
[CrossRef]

Gan, J.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).

J. W. Goodman, Statistical Optics (Wiley, 2000).

Gori, F.

Granik, Y.

Y. Granik, 'Illuminator optimization methods in microlithography,' in Proc. SPIE 5524, 217-229 (2004).
[CrossRef]

Gross, H.

H. Gross, 'Numerical propagation of partially coherent laser beams,' Opt. Laser Technol. 29, 257-260 (1997).
[CrossRef]

Guattari, G.

Hintersteiner, J. D.

H. Martinsson, T. Sandstrom, A. Bleeker, and J. D. Hintersteiner, 'Current status of optical maskless lithography,' J. Microlithogr. Microfabr. Microsyst. 4, 1-15 (2005).
[CrossRef]

Jaroszewicz, Z.

Lin, Q.

Lin, Y.

Y. Lin and J. Buck, 'Numerical modeling of the excimer beam,' in Proc. SPIE 3677, 700-710 (1999).
[CrossRef]

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).

Mann, G.

B. Eppich, G. Mann, and H. Weber, 'Spatial coherence: comparison of interferometric and non-interferometric measurements,' in Proc. SPIE 4969, 137-148 (2003).
[CrossRef]

Martinsson, H.

H. Martinsson, T. Sandstrom, A. Bleeker, and J. D. Hintersteiner, 'Current status of optical maskless lithography,' J. Microlithogr. Microfabr. Microsyst. 4, 1-15 (2005).
[CrossRef]

Michaloski, P.

P. Michaloski, 'Requirements and designs of illuminators for microlithography,' in Proc. SPIE 5525, 1-10 (2004).
[CrossRef]

Palma, C.

Popov, S. Y.

Sandstrom, T.

H. Martinsson, T. Sandstrom, A. Bleeker, and J. D. Hintersteiner, 'Current status of optical maskless lithography,' J. Microlithogr. Microfabr. Microsyst. 4, 1-15 (2005).
[CrossRef]

Tervonen, E.

Thaning, A.

Thompson, B. J.

Turunen, J.

Weber, H.

B. Eppich, G. Mann, and H. Weber, 'Spatial coherence: comparison of interferometric and non-interferometric measurements,' in Proc. SPIE 4969, 137-148 (2003).
[CrossRef]

Wolf, E.

Zhao, F.

J. Microlithogr. Microfabr. Microsyst. (1)

H. Martinsson, T. Sandstrom, A. Bleeker, and J. D. Hintersteiner, 'Current status of optical maskless lithography,' J. Microlithogr. Microfabr. Microsyst. 4, 1-15 (2005).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (4)

Opt. Eng. (1)

B. L. Anderson and P. L. Fuhr, 'Twin-fiber interferometric method for measuring spatial coherence,' Opt. Eng. 32, 926-932 (1993).
[CrossRef]

Opt. Laser Technol. (1)

H. Gross, 'Numerical propagation of partially coherent laser beams,' Opt. Laser Technol. 29, 257-260 (1997).
[CrossRef]

Opt. Lett. (3)

Proc. SPIE (5)

Y. Granik, 'Illuminator optimization methods in microlithography,' in Proc. SPIE 5524, 217-229 (2004).
[CrossRef]

P. Michaloski, 'Requirements and designs of illuminators for microlithography,' in Proc. SPIE 5525, 1-10 (2004).
[CrossRef]

M. J. Bastiaans, 'Application of the Wigner distribution function to partially coherent light,' in Proc. SPIE 3729, 114-128 (1999).
[CrossRef]

Y. Lin and J. Buck, 'Numerical modeling of the excimer beam,' in Proc. SPIE 3677, 700-710 (1999).
[CrossRef]

B. Eppich, G. Mann, and H. Weber, 'Spatial coherence: comparison of interferometric and non-interferometric measurements,' in Proc. SPIE 4969, 137-148 (2003).
[CrossRef]

Other (3)

J. W. Goodman, Statistical Optics (Wiley, 2000).

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).

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Figures (6)

Fig. 1
Fig. 1

Probability distribution P ( W ) for the integrated intensity W of a pulse with a Lorentzian line shape and a Gaussian envelope. The simulations were performed with the frequency-domain method, the full-time trace method, and a Γ distribution with M degrees of freedom. The stochastic fields are normalized such that W t = M for each value of M.

Fig. 2
Fig. 2

Double-pinhole setup for the numerical simulation examples.

Fig. 3
Fig. 3

(a) Intensity distribution of a single monochromatic field in plane A. (b) The intensity distribution of a portion of a single monochromatic field in plane B. (c) Intensity distribution of an interference pattern for a single monochromatic field in plane C.

Fig. 4
Fig. 4

Fringe formation in plane C with the curves showing the integrated intensity cross section. The integrated intensity is calculated for a large number of fields ( 10 3 ) ; the hole separation in plane B is (a) 2.0, (b) 4.0, (c) 5.7 mm .

Fig. 5
Fig. 5

Probability distribution of V P , the visibility of the fringes in plane C. (a)–(c) Case for Gaussian pulses with M t = 2.5 , 6.0 , and 50, respectively. Roughly, these pulses may be interpreted as increasingly longer pulses with a duration M t τ c . Each graph shows the distribution for three different temporal delays of the light in the lower light path, τ = 0 τ c , 0.66 τ c and 1.8 τ c , respectively. Also plotted are the results of a full-time trace treatment.

Fig. 6
Fig. 6

Three planes involved in the two-step propagation from plane P 1 to plane P 2 in a homogeneous medium.

Equations (37)

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Γ ( r 1 , t 1 , r 2 , t 2 ) = E ( r 1 , t 1 ) E * ( r 2 , t 2 ) ,
Γ A ( ρ 1 , ρ 2 ) = A ( ρ 1 ) A * ( ρ 2 ) .
A ( ρ ) = K ( ρ , ρ ) N ( ρ ) d ρ ,
Γ N ( ρ 1 , ρ 2 ) = δ ( ρ 1 ρ 2 ) ,
K ( ρ , ρ ) = K s ( r , r ) K t ( t , t ) .
K ( ρ , ρ ) = K ( ρ ρ ) = K s ( r r ) K t ( t t ) ,
A ( ρ ) = K ( ρ ρ ) N ( ρ ) d ρ
A ( ρ ) = A e ( ρ ) K ( ρ ρ ) N ( ρ ) d ρ .
A ( r ) = A e ( r ) K s ( r r ) N ( r ) d r .
A ( t ) = A e ( t ) K ( t t ) N ( t ) d t A e ( t ) U ( t ) ,
W t pulse A ( t ) A * ( t ) d t = I e ( t ) I U ( t ) d t ,
W t = I e ( t ) I U ( t ) d t = I U ( t ) I e ( t ) d t ,
W t 2 = [ I e ( t ) I U ( t ) d t ] 2 = I e ( t ) I e ( t ) I U ( t ) I U ( t ) d t d t .
I U ( t ) I U ( t ) = I U ( t ) I U ( t ) + U ( t ) U * ( t ) 2 ,
W t 2 = [ I U ( t ) I e ( t ) d t ] 2 + Γ U ( τ ) 2 [ I e ( t ) I e ( t + τ ) d t ] d τ ,
σ W t 2 W t 2 W t 2 = Γ U ( τ ) 2 ( I e I e ) ( τ ) d τ ,
M t W t 2 σ W t 2 = I U ( t ) 2 [ I e ( t ) d t ] 2 Γ U ( τ ) 2 ( I e I e ) ( τ ) d τ .
{ U ν 1 = ν min ( x , y ) , U ν 2 = ν 1 + Δ ν ( x , y ) , , U ν N f ( x , y ) } .
U ν m 2 = S ( ν m ) ,
W f = m = 1 N f U ν m 2 .
W f = m = 1 N f U ν m 2 = m = 1 N f S ( ν m ) 1 Δ ν ν min ν max S ( ν ) d ν .
σ W f 2 = m = 1 N f σ U ν m 2 2 .
σ W f 2 = m = 1 N f S ( ν m ) 2 1 Δ ν ν min ν max S ( ν ) 2 d ν .
M f W f 2 σ W f 2 = [ m = 1 N f S ( ν m ) ] 2 m = 1 N f S ( ν m ) 2 [ ν min ν max S ( ν ) d ν ] 2 Δ ν ν min ν max S ( ν ) 2 d ν .
Δ ν [ ν min ν max S ( ν ) d ν ] 2 M t ν min ν max S ( ν ) 2 d ν ,
N f = ν max ν min Δ ν .
M t = [ π 2 4 ( ln 2 ) 2 ( c 0 Δ T FWHM Δ λ FWHM λ 0 2 ) 2 + 1 ] 1 2 .
W max = m = 1 N f U ν m , 1 + U ν m , 2 e 2 π i ν m τ e i α 2 = m = 1 N f U ν m , 1 2 + m = 1 N f U ν m , 2 2 + 2 R [ e i α m = 1 N f U ν m , 1 U ν m , 2 * e 2 π i ν m τ ] = m = 1 N f U ν m , 1 2 + m = 1 N f U ν m , 2 2 + 2 m = 1 N f U ν m , 1 U ν m , 2 * e 2 π i ν m τ ,
V P = W max W min W max + W min = 2 m = 1 N f U ν m , 1 U ν m , 2 * e 2 π i ν m τ m = 1 N f U ν m , 1 2 + m = 1 N f U ν m , 2 2 .
τ c Γ ( τ ) Γ ( 0 ) 2 d τ = λ 0 2 c 0 Δ λ FWHM 2 ln 2 π
U d ( u , v ) = 1 i λ L 1 e i k ( L 1 + u 2 + v 2 2 L 1 ) U 1 ( x , y ) e i k x 2 + y 2 2 L 1 e i k x u + y v L 1 d x d y ,
U d ( u , v ) = 1 i λ L 2 e i k ( L 2 + u 2 + v 2 2 L 2 ) U 2 ( ξ , η ) e i k ξ 2 + η 2 2 L 2 e i k ξ u + η v L 2 d ξ d η ,
U 2 ( ξ , η ) = L 2 L 1 e i k ( L 2 L 1 + ξ 2 + η 2 2 L 2 ) F 1 [ e i k u 2 + v 2 2 ( 1 L 1 1 L 2 ) F [ U 1 ( x , y ) e i k x 2 + y 2 2 L 1 ] ] ,
Δ d , 1 = K λ N Δ 1 L 1 ,
Δ d , 2 = K λ N Δ 2 L 2 ,
Δ 2 = L 2 L 1 Δ 1 .
U 2 ( ξ , η ) = e i k Δ L F 1 [ e i k x 2 + k y 2 2 k Δ L F [ U 1 ( x , y ) ] ] ,

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