Abstract

We introduce an approximation to the Hopkins integral for partially coherent optical systems. The average coherence approximation yields results close to the Hopkins integral for a wide range of coherent transfer functions and illumination functions and is far less computationally demanding than the full Hopkins integral.

© 1996 Optical Society of America

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References

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  1. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, Calif., 1968), p. 106.
  2. H. H. Hopkins, “The concept of partial coherence in optics,” Proc. R. Soc. London Ser. A 208, 263–277 (1951).
    [CrossRef]
  3. Y. Liu, A. K. Pfau, A. Zakhor, “Systematic design of phase-shifting masks with extended depth of focus and/or shifted focus plane,” IEEE Trans. Semicond. Manuf. 6, 1–21 (1993).
    [CrossRef]
  4. D. C. Cole, E. Barouch, U. Hollerbach, S. A. Orszag, “Derivation and simulation of higher numerical aperture scalar aerial images,” Jpn. J. Appl. Phys. 31, 4110–4119 (1992).
    [CrossRef]
  5. Y. C. Pati, T. Kailath, “Phase-shifting masks for microlithography: automated design and mask requirements,” J. Opt. Soc. Am. A 11, 2438–2452 (1994).
    [CrossRef]
  6. J. Perina, Coherence of Light, 2nd ed. (Reidel, Dordrecht, The Netherlands, 1985), Chap. 4.
  7. L. Mandel, “Concept of cross-spectral purity in coherence theory,” J. Opt. Soc. Am. 51, 1342–1350 (1961).
    [CrossRef]
  8. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1965), p. 526.
  9. W. H. Carter, E. Wolf, “Coherence and radiometry with quasihomogeneous planar sources,” J. Opt. Soc. Am. 67, 785–796 (1977).
    [CrossRef]

1994

1993

Y. Liu, A. K. Pfau, A. Zakhor, “Systematic design of phase-shifting masks with extended depth of focus and/or shifted focus plane,” IEEE Trans. Semicond. Manuf. 6, 1–21 (1993).
[CrossRef]

1992

D. C. Cole, E. Barouch, U. Hollerbach, S. A. Orszag, “Derivation and simulation of higher numerical aperture scalar aerial images,” Jpn. J. Appl. Phys. 31, 4110–4119 (1992).
[CrossRef]

1977

1961

1951

H. H. Hopkins, “The concept of partial coherence in optics,” Proc. R. Soc. London Ser. A 208, 263–277 (1951).
[CrossRef]

Barouch, E.

D. C. Cole, E. Barouch, U. Hollerbach, S. A. Orszag, “Derivation and simulation of higher numerical aperture scalar aerial images,” Jpn. J. Appl. Phys. 31, 4110–4119 (1992).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1965), p. 526.

Carter, W. H.

Cole, D. C.

D. C. Cole, E. Barouch, U. Hollerbach, S. A. Orszag, “Derivation and simulation of higher numerical aperture scalar aerial images,” Jpn. J. Appl. Phys. 31, 4110–4119 (1992).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, Calif., 1968), p. 106.

Hollerbach, U.

D. C. Cole, E. Barouch, U. Hollerbach, S. A. Orszag, “Derivation and simulation of higher numerical aperture scalar aerial images,” Jpn. J. Appl. Phys. 31, 4110–4119 (1992).
[CrossRef]

Hopkins, H. H.

H. H. Hopkins, “The concept of partial coherence in optics,” Proc. R. Soc. London Ser. A 208, 263–277 (1951).
[CrossRef]

Kailath, T.

Liu, Y.

Y. Liu, A. K. Pfau, A. Zakhor, “Systematic design of phase-shifting masks with extended depth of focus and/or shifted focus plane,” IEEE Trans. Semicond. Manuf. 6, 1–21 (1993).
[CrossRef]

Mandel, L.

Orszag, S. A.

D. C. Cole, E. Barouch, U. Hollerbach, S. A. Orszag, “Derivation and simulation of higher numerical aperture scalar aerial images,” Jpn. J. Appl. Phys. 31, 4110–4119 (1992).
[CrossRef]

Pati, Y. C.

Perina, J.

J. Perina, Coherence of Light, 2nd ed. (Reidel, Dordrecht, The Netherlands, 1985), Chap. 4.

Pfau, A. K.

Y. Liu, A. K. Pfau, A. Zakhor, “Systematic design of phase-shifting masks with extended depth of focus and/or shifted focus plane,” IEEE Trans. Semicond. Manuf. 6, 1–21 (1993).
[CrossRef]

Wolf, E.

Zakhor, A.

Y. Liu, A. K. Pfau, A. Zakhor, “Systematic design of phase-shifting masks with extended depth of focus and/or shifted focus plane,” IEEE Trans. Semicond. Manuf. 6, 1–21 (1993).
[CrossRef]

IEEE Trans. Semicond. Manuf.

Y. Liu, A. K. Pfau, A. Zakhor, “Systematic design of phase-shifting masks with extended depth of focus and/or shifted focus plane,” IEEE Trans. Semicond. Manuf. 6, 1–21 (1993).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Jpn. J. Appl. Phys.

D. C. Cole, E. Barouch, U. Hollerbach, S. A. Orszag, “Derivation and simulation of higher numerical aperture scalar aerial images,” Jpn. J. Appl. Phys. 31, 4110–4119 (1992).
[CrossRef]

Proc. R. Soc. London Ser. A

H. H. Hopkins, “The concept of partial coherence in optics,” Proc. R. Soc. London Ser. A 208, 263–277 (1951).
[CrossRef]

Other

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, Calif., 1968), p. 106.

J. Perina, Coherence of Light, 2nd ed. (Reidel, Dordrecht, The Netherlands, 1985), Chap. 4.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1965), p. 526.

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Figures (5)

Fig. 1
Fig. 1

Schematic diagram of the general optical system described by the Hopkins equation.

Fig. 2
Fig. 2

Schematic representation of the system configuration in the average coherence approximation (x can be an n-dimensional vector, in general).

Fig. 3
Fig. 3

One-dimensional slit in (a), after imaging through a finite aperture (b), yields the patterns in (c) when either the Hopkins integral (solid curve) or the ACA (dashed curve) is used. Here J(x, x′) is a sinc function half as wide as the image.

Fig. 4
Fig. 4

Pattern in (a) after imaging through a finite aperture becomes (b) under the average coherence approximation and (c) by use of the full Hopkins equation. The aperture size is half of the image bandwidth, and coherence diameter is half of the image size; average error/pixel, 0.95%.

Fig. 5
Fig. 5

ACA error versus normalized coherence diameter for a one-dimensional finite aperture undergoing Fresnel diffraction.

Equations (23)

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E 0 ( x , y ) 2 = E ( x , y ) E * ( x ˜ , y ˜ ) J ( x , y ; x ˜ , y ˜ ) × K ( x , y ; x , y ) × K * ( x ˜ , y ˜ ; x , y ) d x d y d x ˜ d y ˜ ,
E 0 ( x ) 2 = E ( x ) E * ( x ˜ ) J ( x , x ˜ ) K ( x , x ) × K * ( x ˜ , x ) d x ˜ d x ˜ .
f ( x , x ) = K ( x , x ^ ) 2 μ ( x , x ^ ) d x ^ K ( x , x ˜ ) 2 d x ˜ , μ ( x , x ^ ) = J ( x , x ^ ) [ J ( x , x ) J ( x ^ , x ^ ) ] 1 / 2 ,
E 0 ( x ) 2 = E ( x ) * K ( x , x ) 2 ,             E ( x ) coherent , E 0 ( x ) 2 = E ( x ) 2 * K ( x , x ) 2 ,             E ( x ) incoherent ,
K ( x , x ) 2 = K c ( x , x ) 2 + K i ( x , x ) 2 = [ f ( x , x ) ] 1 / 2 K ( x , x ) 2 + [ 1 - f ( x , x ) ] 1 / 2 K ( x , x ) 2 ,
E 0 ( x ) 2 = E ( x ) * K c ( x , x ) 2 + E ( x ) 2 * K i ( x , x ) 2 = E ( x ) * [ f ( x , x ) ] 1 / 2 K ( x , x ) 2 + E ( x ) 2 * [ 1 - f ( x , x ) ] 1 / 2 K ( x , x ) 2
E 0 ( x ) 2 = | E ( x ) [ f ( x , x ) ] 1 / 2 K ( x , x ) d x | 2 + E ( x ) 2 [ 1 - f ( x , x ) ] K ( x , x ) 2 d x .
f ( x , y , x , y ) = K ( x , y , x ^ , y ^ ) 2 μ ( x , y , x ^ , y ^ ) d x ^ d y ^ K ( x , y , x ˜ , y ˜ ) 2 d x ˜ d y ˜ ,
E 0 ( x , y ) 2 = | E ( x , y ) [ f ( x , y , x , y ) ] 1 / 2 × K ( x , y , x , y ) d x d y | 2 + E ( x , y ) 2 [ 1 - f ( x , y , x , y ) ] K ( x , y , x , y ) 2 d x d y .
K ( x , y , x , y ) = K ( x - x , y - y ) , J ( x , y , x ^ , y ^ ) = J ( x - x ^ , y - y ^ ) ,
K ( x , x ) = sinc ( x - x d ) = d π ( x - x ) sin [ π ( x - x ) d ] ,
K ( x , x ) 2 = sinc 2 ( x - x d ) ,
J ( x , x ˜ ) = sinc ( x - x ˜ a ) = a π ( x - x ˜ ) sin [ π ( x - x ˜ ) a ] ,
f ( x , x ) = 1 sinc 2 ( x ˜ - x d ) d x ˜ sinc 2 ( x ˜ - x d ) × sinc ( x ˜ - x a ) d x ˜ .
f ( x , x ) = 1 sinc 2 ( x ^ d ) d x ^ sinc 2 ( x ^ d ) sinc ( x ^ - x ¯ a ) d x ^ ,
f ( x , x ) sinc 2 ( x - x d ) .
J ( x , x ˜ , x ) = [ f ( x , x ) f ( x ˜ , x ) ] 1 / 2 + { [ 1 - f ( x , x ) ] × [ 1 - f ( x ˜ , x ) ] } 1 / 2 δ ( x - x ˜ ) .
e ( x ) = E ( x ) E * ( x ˜ ) K ( x , x ) K * ( x ˜ x ) J ( x , x ˜ ) d x d x ˜ - E ( x ) E * ( x ˜ ) K ( x , x ) K * ( x ˜ , x ) × J ( x , x ˜ , x ) d x d x ˜ = E ( x ) E * ( x ˜ ) K ( x , x ) K * ( x ˜ , x ) [ J ( x , x ˜ ) - J ( x , x ˜ , x ) ] d x d x ˜ .
f ( x , x ) = δ ( x ^ - x ) J ( x , x ^ ) d x ^ = J ( x , x ) e ( x ) = E ( x ) E * ( x ˜ ) K ( x , x ) × K * ( x ˜ , x ) J ( x , x ˜ ) d x d x ˜ - { | E ( x ) [ J ( x , x ) ] 1 / 2 K ( x , x ) d x | 2 + E ( x ) 2 1 - J ( x , x ) K ( x , x ) 2 d x } = E ( x ) 2 [ 1 - J ( x , x ) - 1 + J ( x , x ) ] = 0.
E 0 ( x ) 2 = E ( x ) * K ( x ) 2 ,
J ( x , x ) = δ ( x - x ) f ( x , x ) = K ( x , x ) 2 J ( x , x ) K ( x , x ) 2 d x = 0 ,
E 0 ( x ) 2 = E ( x ) 2 * K ( x ) 2 ,
2 ( 2 n 2 log n ) + n 2 + 2 n 2 log n + 2 [ 2 ( 2 n 2 log n ) + n 2 + 2 n 2 log n ] = 3 n 2 ( 5 log n + 1 ) .

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