Abstract

A laser light field oscillating in a multitude of transverse modes is analyzed to estimate the degree of coherence of the field. The phasor amplitude of the laser light is described using the expression for a confocal resonator field derived by Boyd and Gordon [ Bell Sys. Tech. J. 40, 489 ( 1961)]. The fringe visibilities around the optical axis are calculated by using a one-dimensional model with two sampling apertures. Light containing only even modes has a visibility of 1. The odd modes have a great influence on the degradation of visibilities.

© 1989 Optical Society of America

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References

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  1. V. Pol, J. H. Bennewitz, G. C. Escher, M. Feldman, V. A. Firtion, T. E. Jewell, B. E. Welcomb, J. T. Clements, “Excimer laser-based lithography: a deep ultraviolet wafer stepper,” in Optical Microlithography V, H. L. Stover, ed., Proc. Soc. Photo-Opt. Instrum. Eng.633, 6–16 (1986).
    [CrossRef]
  2. M. Sasago, M. Endo, K. Ogawa, T. Ishihara, “Half-micron photolithography using a KrF excimer laser stepper,” in 1986 International Electron Device Meeting Technical Digest (Institute of Electrical and Electronics Engineers, New York, 1986), pp. 316–319.
    [CrossRef]
  3. B. J. Thompson, E. Wolf, “Two-beam interference with partially coherent light,”J. Opt. Soc. Am. 47, 895–902 (1957).
    [CrossRef]
  4. B. J. Thompson, “Illustration of the phase change in two-beam interference with partially coherent light,”J. Opt. Soc. Am. 48, 95–97 (1958).
    [CrossRef]
  5. B. J. Thompson, R. Sudol, “Finite-aperture effects in the measurement of the degree of coherence,” J. Opt. Soc. Am. A 1, 598–604 (1984).
    [CrossRef]
  6. A. S. Marathay, D. B. Pollock, “Young’s interference fringes with finite-sized sampling apertures,” J. Opt. Soc. Am. A 1, 1057–1059 (1984).
    [CrossRef]
  7. F. T. S. Yu, Y. W. Zhang, “Fringe visibility of dual-aperture sampling with partially coherent illumination,” Appl. Opt. 25, 3191–3196 (1986).
    [CrossRef] [PubMed]
  8. G. D. Boyd, J. P. Gordon, “Confocal multimode resonator for millimeter through optical wavelength masers,” Bell Sys. Tech. J. 40, 489–508 (1961).
  9. J. W. Goodman, Statistical Optics (Wiley, New York, 1985).
  10. T. E. Jewell, J. H. Bennewitz, G. C. Escher, V. Pol, “Effect of laser characteristics on the performance of a deep UV projection system,” in Lasers in Microlithography, D. J. Ehrlich, J. Y. Tsao, J. S. Batchelder, eds., Proc. Soc. Photo-Opt. Instrum. Eng.774, 124–132 (1987).
    [CrossRef]

1986 (1)

1984 (2)

1961 (1)

G. D. Boyd, J. P. Gordon, “Confocal multimode resonator for millimeter through optical wavelength masers,” Bell Sys. Tech. J. 40, 489–508 (1961).

1958 (1)

1957 (1)

Bennewitz, J. H.

T. E. Jewell, J. H. Bennewitz, G. C. Escher, V. Pol, “Effect of laser characteristics on the performance of a deep UV projection system,” in Lasers in Microlithography, D. J. Ehrlich, J. Y. Tsao, J. S. Batchelder, eds., Proc. Soc. Photo-Opt. Instrum. Eng.774, 124–132 (1987).
[CrossRef]

V. Pol, J. H. Bennewitz, G. C. Escher, M. Feldman, V. A. Firtion, T. E. Jewell, B. E. Welcomb, J. T. Clements, “Excimer laser-based lithography: a deep ultraviolet wafer stepper,” in Optical Microlithography V, H. L. Stover, ed., Proc. Soc. Photo-Opt. Instrum. Eng.633, 6–16 (1986).
[CrossRef]

Boyd, G. D.

G. D. Boyd, J. P. Gordon, “Confocal multimode resonator for millimeter through optical wavelength masers,” Bell Sys. Tech. J. 40, 489–508 (1961).

Clements, J. T.

V. Pol, J. H. Bennewitz, G. C. Escher, M. Feldman, V. A. Firtion, T. E. Jewell, B. E. Welcomb, J. T. Clements, “Excimer laser-based lithography: a deep ultraviolet wafer stepper,” in Optical Microlithography V, H. L. Stover, ed., Proc. Soc. Photo-Opt. Instrum. Eng.633, 6–16 (1986).
[CrossRef]

Endo, M.

M. Sasago, M. Endo, K. Ogawa, T. Ishihara, “Half-micron photolithography using a KrF excimer laser stepper,” in 1986 International Electron Device Meeting Technical Digest (Institute of Electrical and Electronics Engineers, New York, 1986), pp. 316–319.
[CrossRef]

Escher, G. C.

V. Pol, J. H. Bennewitz, G. C. Escher, M. Feldman, V. A. Firtion, T. E. Jewell, B. E. Welcomb, J. T. Clements, “Excimer laser-based lithography: a deep ultraviolet wafer stepper,” in Optical Microlithography V, H. L. Stover, ed., Proc. Soc. Photo-Opt. Instrum. Eng.633, 6–16 (1986).
[CrossRef]

T. E. Jewell, J. H. Bennewitz, G. C. Escher, V. Pol, “Effect of laser characteristics on the performance of a deep UV projection system,” in Lasers in Microlithography, D. J. Ehrlich, J. Y. Tsao, J. S. Batchelder, eds., Proc. Soc. Photo-Opt. Instrum. Eng.774, 124–132 (1987).
[CrossRef]

Feldman, M.

V. Pol, J. H. Bennewitz, G. C. Escher, M. Feldman, V. A. Firtion, T. E. Jewell, B. E. Welcomb, J. T. Clements, “Excimer laser-based lithography: a deep ultraviolet wafer stepper,” in Optical Microlithography V, H. L. Stover, ed., Proc. Soc. Photo-Opt. Instrum. Eng.633, 6–16 (1986).
[CrossRef]

Firtion, V. A.

V. Pol, J. H. Bennewitz, G. C. Escher, M. Feldman, V. A. Firtion, T. E. Jewell, B. E. Welcomb, J. T. Clements, “Excimer laser-based lithography: a deep ultraviolet wafer stepper,” in Optical Microlithography V, H. L. Stover, ed., Proc. Soc. Photo-Opt. Instrum. Eng.633, 6–16 (1986).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, New York, 1985).

Gordon, J. P.

G. D. Boyd, J. P. Gordon, “Confocal multimode resonator for millimeter through optical wavelength masers,” Bell Sys. Tech. J. 40, 489–508 (1961).

Ishihara, T.

M. Sasago, M. Endo, K. Ogawa, T. Ishihara, “Half-micron photolithography using a KrF excimer laser stepper,” in 1986 International Electron Device Meeting Technical Digest (Institute of Electrical and Electronics Engineers, New York, 1986), pp. 316–319.
[CrossRef]

Jewell, T. E.

V. Pol, J. H. Bennewitz, G. C. Escher, M. Feldman, V. A. Firtion, T. E. Jewell, B. E. Welcomb, J. T. Clements, “Excimer laser-based lithography: a deep ultraviolet wafer stepper,” in Optical Microlithography V, H. L. Stover, ed., Proc. Soc. Photo-Opt. Instrum. Eng.633, 6–16 (1986).
[CrossRef]

T. E. Jewell, J. H. Bennewitz, G. C. Escher, V. Pol, “Effect of laser characteristics on the performance of a deep UV projection system,” in Lasers in Microlithography, D. J. Ehrlich, J. Y. Tsao, J. S. Batchelder, eds., Proc. Soc. Photo-Opt. Instrum. Eng.774, 124–132 (1987).
[CrossRef]

Marathay, A. S.

Ogawa, K.

M. Sasago, M. Endo, K. Ogawa, T. Ishihara, “Half-micron photolithography using a KrF excimer laser stepper,” in 1986 International Electron Device Meeting Technical Digest (Institute of Electrical and Electronics Engineers, New York, 1986), pp. 316–319.
[CrossRef]

Pol, V.

V. Pol, J. H. Bennewitz, G. C. Escher, M. Feldman, V. A. Firtion, T. E. Jewell, B. E. Welcomb, J. T. Clements, “Excimer laser-based lithography: a deep ultraviolet wafer stepper,” in Optical Microlithography V, H. L. Stover, ed., Proc. Soc. Photo-Opt. Instrum. Eng.633, 6–16 (1986).
[CrossRef]

T. E. Jewell, J. H. Bennewitz, G. C. Escher, V. Pol, “Effect of laser characteristics on the performance of a deep UV projection system,” in Lasers in Microlithography, D. J. Ehrlich, J. Y. Tsao, J. S. Batchelder, eds., Proc. Soc. Photo-Opt. Instrum. Eng.774, 124–132 (1987).
[CrossRef]

Pollock, D. B.

Sasago, M.

M. Sasago, M. Endo, K. Ogawa, T. Ishihara, “Half-micron photolithography using a KrF excimer laser stepper,” in 1986 International Electron Device Meeting Technical Digest (Institute of Electrical and Electronics Engineers, New York, 1986), pp. 316–319.
[CrossRef]

Sudol, R.

Thompson, B. J.

Welcomb, B. E.

V. Pol, J. H. Bennewitz, G. C. Escher, M. Feldman, V. A. Firtion, T. E. Jewell, B. E. Welcomb, J. T. Clements, “Excimer laser-based lithography: a deep ultraviolet wafer stepper,” in Optical Microlithography V, H. L. Stover, ed., Proc. Soc. Photo-Opt. Instrum. Eng.633, 6–16 (1986).
[CrossRef]

Wolf, E.

Yu, F. T. S.

Zhang, Y. W.

Appl. Opt. (1)

Bell Sys. Tech. J. (1)

G. D. Boyd, J. P. Gordon, “Confocal multimode resonator for millimeter through optical wavelength masers,” Bell Sys. Tech. J. 40, 489–508 (1961).

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (2)

Other (4)

J. W. Goodman, Statistical Optics (Wiley, New York, 1985).

T. E. Jewell, J. H. Bennewitz, G. C. Escher, V. Pol, “Effect of laser characteristics on the performance of a deep UV projection system,” in Lasers in Microlithography, D. J. Ehrlich, J. Y. Tsao, J. S. Batchelder, eds., Proc. Soc. Photo-Opt. Instrum. Eng.774, 124–132 (1987).
[CrossRef]

V. Pol, J. H. Bennewitz, G. C. Escher, M. Feldman, V. A. Firtion, T. E. Jewell, B. E. Welcomb, J. T. Clements, “Excimer laser-based lithography: a deep ultraviolet wafer stepper,” in Optical Microlithography V, H. L. Stover, ed., Proc. Soc. Photo-Opt. Instrum. Eng.633, 6–16 (1986).
[CrossRef]

M. Sasago, M. Endo, K. Ogawa, T. Ishihara, “Half-micron photolithography using a KrF excimer laser stepper,” in 1986 International Electron Device Meeting Technical Digest (Institute of Electrical and Electronics Engineers, New York, 1986), pp. 316–319.
[CrossRef]

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Figures (7)

Fig. 1
Fig. 1

Configuration of the one-dimensional two-aperture sampling system to be analyzed.

Fig. 2
Fig. 2

Relative amplitude and phase of phasor amplitude Ua(s; λ) for (a) N = 300 and (b) N = 450, in which Ua(s; λ) is obtained as the summation from 0th to (N − 1)th modes. The phasor amplitude is described by using the approximate expression for a confocal resonator field.8

Fig. 3
Fig. 3

Relation between the spectral bandwidth Δλ and the fringe visibility.

Fig. 4
Fig. 4

Effect of the number of transverse modes N on (a) the fringe visibility and (b) the relative phase of the fringe for the phasor amplitude Ua(s; λ).

Fig. 5
Fig. 5

Effect of the number of transverse modes N on (a) the fringe visibility and (b) the relative phase of the fringe for the phasor amplitude Ua(s; λ), in which the visibility is calculated every 30 modes.

Fig. 6
Fig. 6

Effects of the number of transverse modes on (a) the fringe visibility and (b) the relative phase of the fringe for the phasor amplitude Ub(s; λ). Ne is the number of even modes, and No is the number of odd modes. The phasor ampitude Ub(s; λ) is obtained by adding No odd modes to the phasor amplitude Ue(s; λ) composed of Ne even modes.

Fig. 7
Fig. 7

Dependence of the visibility on the deviation δ of the phasor amplitude center from the optical axis. The phasor amplitude consists of only even modes.

Equations (15)

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A n ( s ; λ ) = Γ ( n / 2 + 1 ) Γ ( n + 1 ) H n ( s σ ) exp ( - s 2 2 σ 2 ) × exp { - i [ ( k σ ) 2 + s 2 2 σ 2 - ( n + 1 ) π 2 ] } ,
σ 2 = b k ,
U ( s ; λ ) = n A n ( s ; λ ) .
J s ( s 1 , s 2 ; λ ) = U ( s 1 ; λ ) U * ( s 2 ; λ ) .
J u ( u 1 , u 2 ; λ ) = 1 4 ( λ f ) 2 - a a J s ( s 1 , s 2 ; λ ) × exp { i k 4 f [ ( u 1 - s 1 ) 2 - ( u 2 - s 2 ) 2 ] } d s 1 d s 2 .
J u ( u 1 , u 2 ; λ ) = J u ( u 1 , u 2 ; λ ) t ( u 1 ; λ ) t * ( u 2 ; λ ) ,
t ( u ; λ ) = { exp ( - i k 2 f u 2 ) for h - r u h + r and - h - r u - h + r 0 otherwise .
J x ( x 1 , x 2 ; λ ) = 1 4 ( λ f 2 ) - J u ( u 1 , u 2 ; λ ) × exp { i k 4 f [ ( x 1 - u 1 ) 2 - ( x 2 - u 2 ) 2 ] } d u 1 d u 2 .
I ( x ; λ ) = J x ( x , x ; λ ) .
I ( x ) = λ 0 - Δ λ / 2 λ 0 + Δ λ / 2 I ( x ; λ ) d λ ,
V = I max - I min I max + I min ,
V = 2 ( I a I b ) 1 / 2 I a + I b μ a b ,
U a ( s ; λ ) = n = 1 N A n - 1 ( s ; λ ) .
U b ( s ; λ ) = U e ( s ; λ ) + n = 1 N o A 2 n - 1 ( s ; λ ) ,
U e ( s ; λ ) = n = 1 N e A 2 ( n - 1 ) ( s ; λ ) ,

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