Abstract

We introduce the so-called moment expansion of a defocused image as a tool for analyzing and improving the depth of focus in optical imaging. It is shown that a number of previously noted defocus phenomena can be readily derived or explained in terms of moment expansion. Some potential applications of the moment expansion to phase-shifting mask and pupil filter design for optical lithography are also briefly noted.

© 1995 Optical Society of America

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References

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  1. A. Papoulis, Signal Analysis (McGraw-Hill, New York, 1977).
  2. M. D. Levenson, N. S. Viswanathan, R. A. Simpson, IEEE J. Electron. Devices ED-29, 1828 (1982).
    [CrossRef]
  3. H. Watanabe, Y. C. Pati, R. F. Pease, Jpn. J. Appl. Phys. 33, 6790 (1994).
    [CrossRef]
  4. R. M. von Bünau, “Depth of focus enhancement in optical lithography,” Ph.D. dissertation (Stanford University, Stanford, Calif., 1995).
  5. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, (1968)
  6. T. Wilson, C. Sheppard, Theory and Practice of Scanning Optical Microscopy (Academic, New York, 1984).
  7. Y. Liu, A. K. Pfau, A. Zakhor, Proc. Soc. Photo-Opt. Instrum. Eng. 1674, 14 (1992).
  8. Y. C. Pati, T. Kailath, J. Opt. Soc. Am. A 11, 2438 (1994).
    [CrossRef]
  9. Y. C. Pati, Y. Wang, J. Liang, T. Kailath, Proc. Soc. Photo-Opt. Instrum. Eng. 2197, 314 (1994).

1994 (3)

H. Watanabe, Y. C. Pati, R. F. Pease, Jpn. J. Appl. Phys. 33, 6790 (1994).
[CrossRef]

Y. C. Pati, Y. Wang, J. Liang, T. Kailath, Proc. Soc. Photo-Opt. Instrum. Eng. 2197, 314 (1994).

Y. C. Pati, T. Kailath, J. Opt. Soc. Am. A 11, 2438 (1994).
[CrossRef]

1992 (1)

Y. Liu, A. K. Pfau, A. Zakhor, Proc. Soc. Photo-Opt. Instrum. Eng. 1674, 14 (1992).

1982 (1)

M. D. Levenson, N. S. Viswanathan, R. A. Simpson, IEEE J. Electron. Devices ED-29, 1828 (1982).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, (1968)

Kailath, T.

Y. C. Pati, Y. Wang, J. Liang, T. Kailath, Proc. Soc. Photo-Opt. Instrum. Eng. 2197, 314 (1994).

Y. C. Pati, T. Kailath, J. Opt. Soc. Am. A 11, 2438 (1994).
[CrossRef]

Levenson, M. D.

M. D. Levenson, N. S. Viswanathan, R. A. Simpson, IEEE J. Electron. Devices ED-29, 1828 (1982).
[CrossRef]

Liang, J.

Y. C. Pati, Y. Wang, J. Liang, T. Kailath, Proc. Soc. Photo-Opt. Instrum. Eng. 2197, 314 (1994).

Liu, Y.

Y. Liu, A. K. Pfau, A. Zakhor, Proc. Soc. Photo-Opt. Instrum. Eng. 1674, 14 (1992).

Papoulis, A.

A. Papoulis, Signal Analysis (McGraw-Hill, New York, 1977).

Pati, Y. C.

Y. C. Pati, T. Kailath, J. Opt. Soc. Am. A 11, 2438 (1994).
[CrossRef]

Y. C. Pati, Y. Wang, J. Liang, T. Kailath, Proc. Soc. Photo-Opt. Instrum. Eng. 2197, 314 (1994).

H. Watanabe, Y. C. Pati, R. F. Pease, Jpn. J. Appl. Phys. 33, 6790 (1994).
[CrossRef]

Pease, R. F.

H. Watanabe, Y. C. Pati, R. F. Pease, Jpn. J. Appl. Phys. 33, 6790 (1994).
[CrossRef]

Pfau, A. K.

Y. Liu, A. K. Pfau, A. Zakhor, Proc. Soc. Photo-Opt. Instrum. Eng. 1674, 14 (1992).

Sheppard, C.

T. Wilson, C. Sheppard, Theory and Practice of Scanning Optical Microscopy (Academic, New York, 1984).

Simpson, R. A.

M. D. Levenson, N. S. Viswanathan, R. A. Simpson, IEEE J. Electron. Devices ED-29, 1828 (1982).
[CrossRef]

Viswanathan, N. S.

M. D. Levenson, N. S. Viswanathan, R. A. Simpson, IEEE J. Electron. Devices ED-29, 1828 (1982).
[CrossRef]

von Bünau, R. M.

R. M. von Bünau, “Depth of focus enhancement in optical lithography,” Ph.D. dissertation (Stanford University, Stanford, Calif., 1995).

Wang, Y.

Y. C. Pati, Y. Wang, J. Liang, T. Kailath, Proc. Soc. Photo-Opt. Instrum. Eng. 2197, 314 (1994).

Watanabe, H.

H. Watanabe, Y. C. Pati, R. F. Pease, Jpn. J. Appl. Phys. 33, 6790 (1994).
[CrossRef]

Wilson, T.

T. Wilson, C. Sheppard, Theory and Practice of Scanning Optical Microscopy (Academic, New York, 1984).

Zakhor, A.

Y. Liu, A. K. Pfau, A. Zakhor, Proc. Soc. Photo-Opt. Instrum. Eng. 1674, 14 (1992).

IEEE J. Electron. Devices (1)

M. D. Levenson, N. S. Viswanathan, R. A. Simpson, IEEE J. Electron. Devices ED-29, 1828 (1982).
[CrossRef]

J. Opt. Soc. Am. A (1)

Jpn. J. Appl. Phys. (1)

H. Watanabe, Y. C. Pati, R. F. Pease, Jpn. J. Appl. Phys. 33, 6790 (1994).
[CrossRef]

Proc. Soc. Photo-Opt. Instrum. Eng. (2)

Y. C. Pati, Y. Wang, J. Liang, T. Kailath, Proc. Soc. Photo-Opt. Instrum. Eng. 2197, 314 (1994).

Y. Liu, A. K. Pfau, A. Zakhor, Proc. Soc. Photo-Opt. Instrum. Eng. 1674, 14 (1992).

Other (4)

A. Papoulis, Signal Analysis (McGraw-Hill, New York, 1977).

R. M. von Bünau, “Depth of focus enhancement in optical lithography,” Ph.D. dissertation (Stanford University, Stanford, Calif., 1995).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, (1968)

T. Wilson, C. Sheppard, Theory and Practice of Scanning Optical Microscopy (Academic, New York, 1984).

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Figures (1)

Fig. 1
Fig. 1

Plot of the point-spread function Jinc2(r) of a confocal microscope and its curvature.

Equations (20)

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g z ( x , y ) f ( x , y ) * h z ( x , y ) , h z ( x , y ) = 1 i z exp [ - π ( x 2 + y 2 ) / i z ] ,
f ( x , y ) F F ( u , v ) ,
f ( x , y ) = F ( u , v ) exp [ i 2 π ( u x + v y ) ] d u d v .
H z ( ρ ) = F [ h z ( x , y ) ] = exp ( i π ρ 2 ) ,
G z ( u , v ) = H z ( ρ ) F ( u , v ) .
exp ( i π z ρ 2 ) = 1 + i π z 1 ! ρ 2 + + ( i π z ) n n ! ( ρ ˙ 2 ) n + .
G z ( u , v ) = F ( u , v ) + i π z 1 ! ρ 2 F ( u , v ) + ( i π z ) n n ! ( ρ 2 ) n F ( u , v ) + .
( 2 ) n f ( x , y ) F ( - 4 π 2 ρ 2 ) n F ( u , v ) ,
g z ( x , y ) = n = 0 1 n ! ( i π z - 4 π 2 ) n ( 2 ) n f ( x , y ) .
m 2 m 2 n x 2 m y 2 n h z ( x , y ) d x d y = ( m + n ) ! m ! n ! ( i π z - 4 π 2 ) m + n .
| | f | | 2             f ( x , y ) , f ( x , y ) ,
η ( f , g )             f ( x , y ) / | | f | | , g ( x , y ) / | | g | | ,
η ¯ = - ρ 2 F ( u , v ) 2 d u d v / | | F | | | | ρ 2 F | |
2 f ( x , y ) = - 4 π 2 [ ρ 1 2 f 1 ( x , y ) + ρ 2 2 f 2 ( x , y ) ] .
g z ( x , y ) = n = 0 ( i π z ρ 1 2 ) n n ! f 1 ( x , y ) + n = 0 ( i π z ρ 2 2 ) n n ! f 2 ( x , y ) = exp ( i π z ρ 1 2 ) f 1 ( x , y ) + exp ( i π z ρ 2 2 ) f 2 ( x , y ) .
ɛ 2 ( f )             | | f ( x , y ) - g z ( x , y ) | | 2
ɛ 2 ( f ) = 2 [ 1 - f ( x , y ) ,     g z ( x , y ) ] 2 [ 1 - f ( x , y ) , g z ( x , y ) ] .
g z ( x , y ) f ( x , y ) + i π z - 4 π 2 2 f ( x , y )
η ( f , g z ) 2 1 - 1 2 ( π z 4 π 2 ) 2 | | 2 f | | 2 ( 1 - η ¯ 2 ) .
ɛ 2 ( f ) ( π z 4 π 2 ) 2 [ | | 2 f | | 2 ( 1 - η ¯ 2 ) ]

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