Abstract

The equation for partially coherent high-numerical-aperture scalar imaging was originally derived by Cole et al. [Jpn. J. Appl. Phys. 31, 4110 (1992)]. Here I present an alternative derivation, based on the plane-wave spectral representation of propagation, which can, at least in some respects, be viewed as more straightforward.

© 2001 Optical Society of America

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References

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  1. See, for example, C. Mack, “Optical lithography modeling,” in Microlithography: Science and Technology, J. R. Sheats, B. W. Smith, eds. (Marcel Dekker, New York, 1998), Chap. 2, pp. 109–171.
  2. 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]
  3. M. Gu, Advanced Optical Imaging Theory (McGraw-Hill, New York, 1999), Chap. 6.
  4. D. G. Flagello, A. E. Rosenbluth, “Lithographic tolerances based on vector diffraction theory,” J. Vac. Sci. Technol. B 10, 2997–3003 (1992).
    [CrossRef]
  5. R. L. Gordon, D. G. Flagello, M. McCallum, “Deducing aerial image behavior from AIMS data,” in Optical Microlithography XIII, Proc. SPIE4000, C. J. Progler, ed., 734–743 (2000).
  6. H. Weyl, “Ausbreitung elektromagnetischer Wellen uber einem ebenen Leiter,” Ann. Phys. (Leipzig) 60, 481–500 (1919).
    [CrossRef]
  7. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1996), Chap. 3.
  8. P. C. Clemmow, The Plane Wave Spectrum Representation of Electromagnetic Fields (Pergamon, New York, 1966).
  9. J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975), Chap. 7.
  10. B. J. Thompson, “Image formation with partially coherent light,” Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1969), Vol. 7, pp. 171–230.

1992 (2)

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]

D. G. Flagello, A. E. Rosenbluth, “Lithographic tolerances based on vector diffraction theory,” J. Vac. Sci. Technol. B 10, 2997–3003 (1992).
[CrossRef]

1919 (1)

H. Weyl, “Ausbreitung elektromagnetischer Wellen uber einem ebenen Leiter,” Ann. Phys. (Leipzig) 60, 481–500 (1919).
[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]

Clemmow, P. C.

P. C. Clemmow, The Plane Wave Spectrum Representation of Electromagnetic Fields (Pergamon, New York, 1966).

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]

Flagello, D. G.

D. G. Flagello, A. E. Rosenbluth, “Lithographic tolerances based on vector diffraction theory,” J. Vac. Sci. Technol. B 10, 2997–3003 (1992).
[CrossRef]

R. L. Gordon, D. G. Flagello, M. McCallum, “Deducing aerial image behavior from AIMS data,” in Optical Microlithography XIII, Proc. SPIE4000, C. J. Progler, ed., 734–743 (2000).

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1996), Chap. 3.

Gordon, R. L.

R. L. Gordon, D. G. Flagello, M. McCallum, “Deducing aerial image behavior from AIMS data,” in Optical Microlithography XIII, Proc. SPIE4000, C. J. Progler, ed., 734–743 (2000).

Gu, M.

M. Gu, Advanced Optical Imaging Theory (McGraw-Hill, New York, 1999), Chap. 6.

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]

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975), Chap. 7.

Mack, C.

See, for example, C. Mack, “Optical lithography modeling,” in Microlithography: Science and Technology, J. R. Sheats, B. W. Smith, eds. (Marcel Dekker, New York, 1998), Chap. 2, pp. 109–171.

McCallum, M.

R. L. Gordon, D. G. Flagello, M. McCallum, “Deducing aerial image behavior from AIMS data,” in Optical Microlithography XIII, Proc. SPIE4000, C. J. Progler, ed., 734–743 (2000).

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]

Rosenbluth, A. E.

D. G. Flagello, A. E. Rosenbluth, “Lithographic tolerances based on vector diffraction theory,” J. Vac. Sci. Technol. B 10, 2997–3003 (1992).
[CrossRef]

Thompson, B. J.

B. J. Thompson, “Image formation with partially coherent light,” Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1969), Vol. 7, pp. 171–230.

Weyl, H.

H. Weyl, “Ausbreitung elektromagnetischer Wellen uber einem ebenen Leiter,” Ann. Phys. (Leipzig) 60, 481–500 (1919).
[CrossRef]

Ann. Phys. (Leipzig) (1)

H. Weyl, “Ausbreitung elektromagnetischer Wellen uber einem ebenen Leiter,” Ann. Phys. (Leipzig) 60, 481–500 (1919).
[CrossRef]

J. Vac. Sci. Technol. B (1)

D. G. Flagello, A. E. Rosenbluth, “Lithographic tolerances based on vector diffraction theory,” J. Vac. Sci. Technol. B 10, 2997–3003 (1992).
[CrossRef]

Jpn. J. Appl. Phys. (1)

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]

Other (7)

M. Gu, Advanced Optical Imaging Theory (McGraw-Hill, New York, 1999), Chap. 6.

R. L. Gordon, D. G. Flagello, M. McCallum, “Deducing aerial image behavior from AIMS data,” in Optical Microlithography XIII, Proc. SPIE4000, C. J. Progler, ed., 734–743 (2000).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1996), Chap. 3.

P. C. Clemmow, The Plane Wave Spectrum Representation of Electromagnetic Fields (Pergamon, New York, 1966).

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975), Chap. 7.

B. J. Thompson, “Image formation with partially coherent light,” Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1969), Vol. 7, pp. 171–230.

See, for example, C. Mack, “Optical lithography modeling,” in Microlithography: Science and Technology, J. R. Sheats, B. W. Smith, eds. (Marcel Dekker, New York, 1998), Chap. 2, pp. 109–171.

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Equations (43)

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2+nx2k2ϕx=0.
p2=n2k2,
pz=±γβ, with γβ=n2k2-β21/2=n2k2-px2-py21/2.
ϕx=ϕρ, z= d2βAβexpiβ·ρ+iγβz.
ϕ˜β, z= d2ρ2π2exp-iβ·ρϕρ, z,
Aβ=ϕ˜β, 0= d2ρ2π2exp-iβ·ρϕρ, 0,
ϕ˜β, z=Aβexpiγβz=ϕ˜β, 0expiγβz,
ϕρ, z= d2ρ  d2β2π2expiβ·ρ-ρ+iγβzKρ-ρ, z ϕρ, 0= d2ρKρ-ρ, zϕρ, 0.
 d2β2π2expiSβ12πidet2S/βiβjsp1/2×expiSsp,
Sβiβsp=0  Spxpxsp,pysp=0,  Spypxsp,pysp=0,
2Spx22Spxpy2Spxpy2Spy2
0=pxpxx+pyy+γpx, pyzpxsp, pysp=x-z pxspγpxsp, pysp, 0=pypxx+pyy+γpx, pyzpxsp,pysp=y-z pxspγpxsp, pysp.
βxsp=pxsp=k xr,  βysp=pxsp=k yr,
Ssp=kr,
 dpxdpy2π2expiSpx, py12πi k cosθexpikrr=1λicosθexpikrr,
γ=k-β22k+.
KFresnelρ-ρ, z=expikz2π2  d2β expiβ·ρ-ρ-izβ2/2k=expikziλzexpik2zρ-ρ2.
0=Vd3xϕ*2+n2k2ϕ-ϕ2+n*2k2ϕ*=Vd3xϕ*2ϕ-ϕ*2ϕ+k2n2-n*2ϕ*ϕ.
0=Vd3x·ϕ*ϕ-ϕϕ*+k2  d3xn2-n*2ϕ*ϕ.
0=Vd3x·ϕ*ϕ-ϕϕ*,
0=Sd2sn·ϕ*ϕ-ϕϕ*.
Qx=ϕx*ϕx-ϕxϕx*.
 d2xϕ*zϕ-ϕzϕ*z1= d2xϕ*zϕ-ϕzϕ*z2.
 d2βγβ|ϕ˜β, z1|2= d2βγβ|ϕ˜β, z2|2.
ρI=xIyI=MxxMxyMyxMyyxOyO=M·ρO.
ϕIρI, zI= d2ρOPρI, ρO, zI, zOϕOρO, zO.
ϕ˜Iβ, zI=0=|detM|ϕ˜OMT·β, zO=0,
ϕ˜Iβ, zI=0=ϕ˜β, z1, ϕ˜OMT·β, zO=0=ϕ˜β=MT·β, z2.
γβ1/2ϕ˜Iβ, zI=0=|detM|γMT·β1/2×ϕ˜OMT·β, zO=0.
 d2βγβ|ϕ˜Iβ, zI=0|2= d2β|detM|γMT·β|×ϕ˜OMT·β, zO=0|2= d2βγβ|ϕ˜Oβ, zO=0|2,
ϕ˜Iβ, zI=|detM|1/2γMT·βγβ1/2 expiγβzI-iγMT·βzOϕ˜OMT·β, zO=|detM|1/21-MT·β/k21-β/k21/4×expiγβzI-iγMT·βzO×ϕ˜OMT·β, zO.
ϕ˜Iβ, zI=θNA-|β|/k|detM|1/21-MT·β/k21-β/k21/4×expi2πwβ+iγβzI-iγMT·βzOϕ˜OMT·β, zO.
ϕ˜OMT·β, zO=0 for |β|/k>NA.
PρI, ρO, zI, zO= d2β2π2 θNA-|β|/k×|detM|1/21-MT·β/k21-β/k21/4×expiβ·ρI-M·ρO+i2πwβ+iγβzI-iγMT·βzO.
ϕ˜Oβ, zO= d2βTβ, β, zO, zSϕ˜Sβ, zS.
ϕSρ= d2suρ, sAs
A*sAs=Isδ2s-s=Isδsx-sxδsy-sy.
ϕS*ρϕSρ= d2sd2su*ρ, suρ, s×A*sAs= d2sd2su*ρ, suρ, sIs×δ2s-s= d2su*ρ, suρ, sIs
ϕSρ= d2s expis·ρAKs,
ϕS*ρϕSρ= d2sIKsexpis·ρ-ρ,
ϕSρ= d2sfρ-sACs,
ϕS*ρϕSρ= d2sICsf*ρ-sfρ-s,
IρI, zI=|ϕIρI, zI|2=ϕI*ρI, zIϕIρI, zI= d2ρO  d2ρOP*ρI, ρO, zI, zO×PρI, ρO, zI, zOϕO*ρO, zOϕOρO, zO= d2ρO  d2ρOP*ρI, ρO, zI, zO×PρI, ρO, zI, zOT*ρOTρO×ϕS*ρO, zOϕSρO, zO,

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